摘要:

Solutions of n-Alkanes:Volume-related Properties and the Principle of CongruenceBY A. J. B. CRUICKSHANK AND C. P. HICKSSchool of Chemistry, University of Bristol, Bristol BS8 1TSReceived 2nd January, 1970The relationship of the simple segment theory to the Orwoll-Flory’ theory is examined ; thissuggests a restatement of the principle of congruence as five postulates. These are tested in turnagainst the molar volume, thermal expansivity and isothermal compressibility measurements ofDfaz Peiia and Benitez de Soto on the system n-hexane+n-hexadecane, using the methods of Patter-son and Bardin. The molar volume data are analyzed also by a method based on the principle ofcorresponding states, which includes direct comparison of each solution with a reference n-alkane.The Patterson-Bardin analyses accord with the analyses of the molar volumes, with reservationsbecause of an inconsistency in the thermal expansivity results.The solutions appear to follow thesame reduced equation of state as do the n-alkanes, but their equivalent substances depart signifi-cantly from the locus of reducing parameters defined by the n-alkanes. The implications of thisdeparture are discussed.Theories applicable to binary solutions of n-alkanes try to define, for everysolution, an equivalent substance whose configurational thermodynamic propertiesdiffer from those of the solution only by the so-called combinatorial term, in thefree energy and entropy. The simple segment theoriesy1. and that of Orwoll andFloryy3* both correlate the free energies of mixing of binary solutions of n-alkanes,each using two adjustable parameters.The Orwoll-Flory theory correlates also theenthalpies and volumes of mixing (with one parameter only), whereas the simplesegment theory requires an additional assumption in order to fit the enthalpies ofmixing. Comparison of the two theories raises the question whether the equivalentsubstance is always a n-alkane in the sense that its characteristic properties all lie onthe smooth curve against carbon number defined by the pure n-alkanes. Thisquestion can be answered satisfactorily only in terms of the reduced equation of state,Q(j, T) = 0. The measurements of Diaz Pe5a and Benitez de Soto of themolar volume and its temperature and pressure derivatives at p = 0, over the fullcomposition range of the system n-hexane + n-hexadecane, are of adequate precisionfor this purpose.This theory l*SIMPLE SEGMENT THEORYsupposes that at low T and p = 0, the configurational intrinsicenergy U depends solely on the relative numbers of two types of segment (A-type,middle; B-type, end), irrespective of how these are distributed between the two mole-cular species, the two types of segment being specified so as to have the samevalue of the effective coordination number z.The free energy of mixing, GM, isexpressed as the sum of interactional and combinatorial terms,where r = xlrl +xZrz, rl: being the number of segments per molecule of species i,and xi its mol fraction, b1 is the proportion of B-type segments in species i, w, is anenergy parameter dependent upon the segment specification (the second parameter),106GM = NWab(b1- b2)”4142 3- G Y (1A .J . 3. CRUICKSHANK AND C. P. HICKS 107both being evaluated by fitting eqn (1) to experimental results, and = xlrl/(xlrl +x2r2), the segment fraction of species i in the solution.The only non-empirical justification for eqn (1) is a corresponding states treat-mentY6 following Prig~gine,~ of A-type and B-type segments, which assumes thathypothetical liquids of all A-type and all B-type segments, respectively, would obeythe same reduced equation of state, though with T* not necessarily bearing the samerelation to the configurational intrinsic energy. This treatment givesGM = [ H a a l f N Z E a * a l [ ~ N z ( 2 & ~ b - E,*,-c:b)l(bl - b2)2r#l 4 2 -k' equation of state ' terms + GE, (2)where the gij are the energy parameters of the two-segment potentials.If the productof the two square brackets in eqn (2)is identified with Nwub in eqn (I), the twoequationsdiffer only in the " equation of state " terms, whose coefficients are also configurationalproperties of the hypothetical all-A-type liquid. The fitting of eqn (2) to infinite-dilution GY values is not sensitive to the form of the equation of state terms, but theirnecessary existence precludes physical interpretation of the best-fit value of w,.Although the corresponding states treatment assumes the solution equivalentsubstance to be the hypothetical n-alkane defined bywhere nes may be non-integral, there is no way of testing whether the best-fit value ofwub correlates also the configurational properties of the pure n-alkanes.However,the best-fit value of WUb for the infinite-dilution GY values in eighteen binary n-alkanesystems near 300 K (which decreases with increasing temperature, and changes onthe addition of a third type of segment 8, is larger than that indicated by fitting then-alkane critical temperatures to the segment model? To correlate heats of mixingwe must suppose that the first term in eqn (1) gives, not HM, but U', cf. McGlashanMorcom and William~on.~nes = xlnl +X2n2, (3)THEORY OF ORWOLL AND FLORYThis theory 3* uses an explicit three-parameter equation of state (requiringT:, P'z and p: to characterize each n-alkane) to formulate the " equation of state "terms, cf.eqn (2); this equation of state takes U(config.) to be proportional toreciprocal volume. The leading term in the derived expression for AM is formallysimilar to that in eqn (2) except that E$ is replaced by qll/C (modified by segmentsurface area ratios consequent upon setting v* the same for both types of segment).The " equation of state " terms are formulated in the T:, V: and p t found from thepure n-alkanes, and expressed as functions of the carbon number n, cf. eqn (17)-(20)of ref. (3). The solution equivalent substance is assumed in the derivation to bethe hypothetical n-alkane defined by eqn (3), and in principle its 2-2, Ves and p,*, aredeterminate. In practice, the parameter analogous to the Wab of eqn (11,is treated as adjustable, and optimized with respect to the HM for thirteen systems,mostly at x2 = 0.5.This optimum value of I' cannot be identified with that for thepure n-alkanes, implying a departure from the principle of congruence. A secondadjustable parameter rs is required to fit the GY data.Patterson and Bardin lo have made corresponding states correlations of molarvolume V, thermal expansivity ap and isothermal compressibility PT for the n-alkanesat p = 0 ; they find p z to be nearly independent of n, implying that r is small. Theincrease in p*, with increasing n found by Orwoll and Flory is attributed lo t108 PRINCIPLE OF CONGRUENCEimposing the Tonks-Eyring equation of state. Allowing for differences in the formwla-tion of the " end effect ", it is concluded that r = 1.4 for binary n-alkane solutionsis " unreasonably large ',lo, and incompatible with the VM meas~rements.~.It isnot clear whether this is entirely due to the equaton of state or partly imposedby the HM values.The foregoing is inconclusive, as is speculation about w& in the simple segmenttheory ; but together they suggest that while the departure from congruence compatiblewith the VM is very small, that required to fit the HM may be much larger. We notethat using for solutions a value of the segment interchange parameter different fromthat which would correlate the n-alkane T: andpz need not alter the form of TZ =T,*,(n,,), etc. ; it is equivalent to adding a term like x1x2A in eqn (3).PRINCIPLE OF CONGRUENCEWe restate this principle * as follows : (i) the n-alkanes follow the same reducedequation of state, @(r $) = 0, so that to each n-alkane may be assigned reducingparameters T:, V: andp; ; (ii) the n-alkanes fall in sequence of carbon number II on asmooth curve in T*, V*, p* space ; (iii) the equivalent substance for a binary n-alkanesolution follows the same reduced equation of state as do the n-alkanes; (iv) theequivalent substance is characterized by the point corresponding to its effectivecarbon number, nes on the n-alkane T*, V*, p* curve ;(v) the effective carbon numberis that given by eqn (3).Assumptions (iv) and (v) seem not to be substantiated by the applications 4* loof the segment theories.They have not been tested directly in terms of the reducedequation of state. Patterson and Bardin lo have tested (i) and (ii) in this way; theyconclude that both are valid, and also that p* changes by less than 1 % over therange of n-alkanes studied.METHOD OF ANALYSISIn the following, we take assumptions (i) and (ii) as valid.1° Then assumption(iii) implies that (TaJ3 is a single-valued function of T for each 8, so that plots of(Tap),,o against log T, for n-alkane solutions, or n-alkanes, i and j , should super-impose on displacement along the log T axis by log (Tf/T'). Likewise, plots of(p/lT)? or (P&)~=, against log p should be superimposable, with relative displacementlog (pr/pT). An equivalent procedure lo is to plot against log (PT)p=O.Finally, plots of (Ta,>-&o against log V should be superimposable, with relativedisplacement log (G/Vj*).The curves are closer to straight lines if is usedin the first plot and (Tap)p=o in the third.Assumption (iv) implies, in addition to (iii), that for each binary n-alkane solution(es), compared to a reference n-alkane (r) the three ratios pZJp?, Tz/TF and VzlV?,should locate the same value for nes. Experimental uncertainty precludes this testof (iv) being conclusive (see below), so further tests are required.Provided that (iii) is valid, corresponding states theory requireswherehe, = V,/V:.The values of V , , and V, at each temperature define a line in thef,h plane, and thelines for different temperatures will show a consensus, definingf,, and he,, and thenceTz and Vz.If (iv) is also valid, then the consensus point, As, he,, will lie on thef,h line interpolated between the reference n-alkane and its immediate homologuesA. J . B . CRUICKSHANK AND C. P. HICKS 109i.e., the line established by Patterson and Bardin.l0 Considering only solutionswhose nes according to eqn (3) would be integers, and using as reference substance ineach case the corresponding n-alkane, minimizes the effect of deviation from eqn (4).If the pointf,,, he, coincides with the point, 1 .O, 1 .O, then both (iv) and (v) are validated.If (iii) and (iv) are valid, every solution will obeyVedT) = UT), P = 0, (5)for some value of n, = nes, irrespective of whether or not (v) is obeyed. If (v) isinvalid, the solution compositions for which xlnl+x2n2 is an integer will not haveintegral n,,, so that locating the nr to satisfy eqn (5) may involve interpolating Vrbetween the integer values of n,.This procedure was used by Fernhndez-Garcia,Stoeckli and Boissonas.14 It is better to locate, for each reference n-alkane, thex2 for which eqn (5) is satisfied; here the interpolation uncertainty is only that onP! rather than that on V,, and Y"(x) may be checked by direct measurement. Wethus locate, at each experimental temperature, the values of x2 for which n,, is integral.The difference An,is then plotted against n,,. This plot is not necessarily identical to that of An againstx2 but is likely to be indistinguishable from it. If (iv), as well as (v), is invalid, Anwill be temperature-dependent.An = n,,-xlnl-x2n2 (6)SYSTEM n-HEXANE + n-HEXADECANEIn addition to the smoothed results of Diaz Peiia and Benitez de S O ~ O , ~ we usethe VM data listed in table 1 which gives the coefficients a, b and c inwhere x2 is the rnol fraction of n-hexadecane.were taken from Orwoll and Floryfor which we prefer the data of Egl0ff.l'.VM = x&a + b(x, - x,) + c(x, - x#], (7)Molar volumes of the n-alkanes(table 5), except for n-heptane and n-octane,TABLE 1 .-vM (1.m0l-l) IN THE SYSTEM II-HEXANE+ n-HEXADECANETK -a -b -C ref.coefficients in eqn (6), x lo3293 2.06 0.225 0 14298 2.30 0.99 0 15298 2.1284 1.2404 1.0320 16308 2.4651 1.1948 1 S658 16-b = 0.9038f0.01369 t+0.000003 t 2- a = 1.3712+0.02129 t+0.000346 t 2- C = 0.4226+0.00591 t+0.000025 t 2* t = T-273293-333*The (Tap);d0 values calculated from the (a VM/8T)p= recorded by Diaz Pefia andBenitez de Soto,' using their values for (a V6/aT)p and (a V, 6/aT)p, give curves againstlogloT whose gradients at logloT = 2.5 change smoothly from - 11.9 at x2 = 0to - 10.3 at x2 = 1, so that none of the curves will superimpose.These values of(aV6/aT), and (aVl ,/aT>, differ systematically from other results.3* l7 Assumingthat systematic errors might be cancelled in (a V'/aT), calculated according towe have recalculated (a VJaT), from the recordedFlory(avy/aT)p = (avx/aT),-xl(a v6/aT)p-x2(av1 6 / a T ) p , (8)(a V'/aT),, using the Orwoll-(table 5 ) values for (aVG/aT)p and (av16/i?T)p.The recalculated (Tap)<J110 PRINCIPLE OF CONGRUENCElog,o(T/K)FIG. 1 .--(Tap);: ,, [recalculated from the (a W / a T ) p of ref. (5), using (8 v6/dT)p, (a V16/d r), ,v6 and V16 from ref. (3), table 51, against logloT; x2 is mol fraction n-hexadecane. The broken lines,reading downward, are n-tetradecane, n-undecane and n-octane .loglo(PT/N-' m2)FIG. 2.-(u) (Tcc,),=o as first calculated from ref. (5) against -loglopr, also from ref. (5); readingdownward, x2 = 0, 0.1,0.2,0.3,0.5,0.8,1.0. (b) (Tor,),= o recalculated as in fig. 1 against -loglofirfrom ref. (5); reading downward, x2 = 0, 0.3, 0.5, 0.8, 1.0A. J . B . CRUICKSHANK AND C. P . HICKS 111are plotted in fig. 1, together with those for n-octane, n-undecane and n-tetradecanefrom the same s o ~ r c e .~ The separation in log,,T between any pair of solutioncurves, includiiig the pure components, changes by less than 2 "/o over the range of(Tct,,)$o. The curves for n-octane and n-undecane show gradients which areconsistent neither with each other nor with those for n-hexane, n-tetradecane andn-hexadecane, but their divergences from the curves for x2 = 0.2 and x2 = 0.5,respectively, are smaller than from the first-calculated (Tap)$ o.The first-calculated (Tap),=o values are plotted against loglo& in fig. 2a, and therecalculated values of fig. 1 are similarly plotted in fig. 2b. The latter supports theconclusion lo that p* is nearly constant for the n-alkanes, and shows the same to holdfor these solutions.values are the more reliable.They are plotted against log,,V in fig.3, together with the curves for n-heptane andn-undecane (from ref. (3)). All are parallel straight lines, and the n-heptane andn-undecane lines are indistinguishable from the x2 = 0.1 and x2 = 0.5 lines, respec-tively, in agreement with previous work.18Fig. 1 and 2 suggest that the recalculatedlO(TRp)p= 0FIG. 3.-10g101/ against TO(^)^=^ recalculated as in fig. 1 ; reading downward, x2 = 1.0, 0.9, 0.8 . . ., 0.2, 0.1, 0. The broken lines are, n-undecane and n-heptane, from ref. (3), table 5.Fig. 1-3 indicate (subject to the doubt due to the discrepancies in the ap measure-iiients3* 5 9 17) that the n-hexane + n-hexadecane solutions probably obey the samereduced equation of state as do the n-alkanes, and that they have p z and V z closeto those for the " congruent '' n-alkane indicated by eqn (3), T: being probablyfurther off.It is not firmly established whether or not the solution equivalentsubstance is a n-alkane in the sense of (iv), nor whether, if so, the congruence pre-scription (v) for nes is exact.Fig. 4 shows thef,h plots for x2 = 0.5 and x2 = 0.8 ; these are representative ofall the plotsfor x2 = 0.4 to x2 = 0.9. The plot for x2 = 0.1 shows no clearconsensus ;those for 0.2 and 0.3 show consensus points displaced from 1.0, 1.0 in the directionopposite to fig. 4. The consensus points are listed in table 2.The existence of consensus points implies that these solutions do follow the sam112 PRINCIPLE OF CONGRUENCEreduced equation of state as do the n-alkanes; but comparison of the consensuspoints with thef,h line defined for the n-alkanes by Patterson and Bardin lo (seefig.4) shows that the solutions do not obey assumption (iv) of congruence. TheFIG. 4.-(a) f against h according to eqn (4), for x2 = 0 . 5 at various temperatures ; heavy brokenline, f against h for the n-alkanes in the region of n = 11, interpolated from eqn (2.4) and (5.1) ofref. (10). (b), the same for x2 = 0.8 ; heavy broken line, fagainst h for the n-alkanes lo in the regionof IZ = 14.magnitude of cfes- 1) is in every case larger than that of (hes- 1), where the reversewould be true if (iv) were valid. Also, both ( f e s - 1) and (lies- 1) apparently changesign as x2 decreases from 0.4 to 0.3.To check for inconsistency in the data, we haveapplied the test devised for (v). Values of An calculated according to eqn (6) for aselection of VM results are plotted against the carbon number of the reference n-alkaneTABLE 2.XOMPARISON OF EQUIVALENT SUBSTANCE WITH REFERENCE II-ALKANEX 20.10.20.30.40.50.60.70.80.9*r789101112131415103 (fa- 1) 103 (ha- 1)0.5 fl.5 0.3 &0.8-2.3 f l . 0 - 0.5 f0.5- 7.0 f l . O - 1.8 f0.50.5 f0.5 0.7 f0.23.0f1.0 1.2 f0.47.5 f1.0 2.5 k0.4no good V, values available,5.4 f0.6 1.6 &0.21.0 f 1 .o 0.5 f0.4in fig. 5. The smoothed results of table 1 were used, but no points are plotted fromoutside the range of each author’s experimental measurements. The curve on fig.5is that for An = 0.024The points for different temperatures tend to run in sequence, but the separationis large in comparison with the experimental uncertainty, and the order i s reversedbetween x2 ~ 0 . 3 and x2 > 0.4. The points in fact reflect the intersections (on fig. 4)cf. eqn (l), corresponding to (iv) being validA. J . B . CRUICKSHANK AND C . P . HICKS 113of the solutionJh lines (one for each temperature) with the n-alkanef,h line. Thedeparture from congruence indicated by fig. 5 alone (ignoring the temperature-dependence of An) is a dramatic underestimate, due to the partial compensation ofthe &- 1) and (lies-- 1) factors, as shown by the mean slope of the solutionJh lineson fig. 4.N 3046 7 8 9 10 I 1 12 13 14 I5 16nesFIG.5.-Calculated values of An = n,--xlnl -x2n2 to satisfy eqn (5) against n, ; Vdata from table 1 .Error bar shows change in An for change of 1. mol-l in VM or V,. Symbols, reading downwardat nes = 12, represent 293, 298, 303, 308, 313, 323, 333 K, respectively.DISCUSSIONPresupposing that eqn (3) is valid for he,, Patterson and Bardin lo concluded thatthe n-hexane + n-hexadecane solutions probably do not follow the n-alkane reducedequation of state. By separating the principle of congruence for solutions into (iii),(iv) and (v), we show that while these solutions probably do follow the n-alkanereduced equation of state, their equivalent substances are not generally n-alkanes inthe sense of (iv) differing especially in their T&. The change in sign of (&- 1) withincreasing x2, which is apparent in the analysis of the VM data, cf.table 2, is apparentalso in the (Ta,) data, in fig. 1, and, less obviously, in fig. 3.Since n-hexane+n-hexadecane is, according to the GM and HM measurements, atypical binary n-alkane system, we must assume that its equation of state behaviouris also typical. The data of table 2 show that detailed analysis of the VM measure-ments on this system contradict the basic assumptions of the segment model.Consequently, we cannot expect this model, even with modification of assumption(v), to correlate simultaneously the VM, HM and GM results for binary n-alkane systems.On the other hand, either of the segment models discussed above gives a moderatelyprecise correlation for any one of these functions of mixing114 PRINCIPLE OF CONGRUENCEWe gratefully record the influence of Dr.D. Patterson on this work, and thankC. P. H. thanks the S.R.C him for communicating ref. (10) to us before publication.for the award of a studentship.A. J. B. Cruickshank, B. W. Gainey and C. L. Young, fians. Faraday Soc., 1968, 6 4 , 3 3 7 .Faradizy Soc., 1969, 65,2356.R. A. Orwoll and P. J. Flory, J. Amer. Chem. Soc., 1967, 89,6814.R. A. Orwoll and P. J. Flory, J. Amer. Chem. SOC., 1967, 89,6822.1965, 61,1163.C. P. Hicks, unpublished.2A. J. B. Cruickshank, B. W. Gainey, C. P. Hicks, T. M. Letcher and C. L. Young, Tram,' M. Dim Peiia and M. Benitez de Soto, Anales Real Soc. Espan". Fis. Quinz. (Madrid), Ser. B.,' I. Prigogine, Molecular Theory of Solutions (North Holland, 1957), chap. 17. * A. J. B. Cruickshank, C. P. Hicks and R. W. Moody, XXII I.U.P.A.C. Congr. (Sydney, 1969).M . L. McGlashan, K. W. Morcom and A. G. Williamson, Trans. Faraday Soc., 1961,57,601.lo D. Patterson and J. M. Bardin, Trans. Faraday Soc., in press (privately communicated).J. N. Brarnsted and J. Koefoed, Kgl. Danske Videnskab. Selskab., Mat.-Fys. Me&., 1946,22,1.l2 Th. Holleman and J. Hijmans, Mol. Phys., 1961,4,91.I3 J. S. Rowlinson, Liquids andLiquid Mixtures, 2nd ed. (Butterworth, 1969), chap. 8.l4 J. G. FernAndez-Garcia, F. Stoeckli and Ch. G. Boissonas, Helu. chim. Acta, 1966, 49, 1983.E. L. Heric and J. G. Brewer, J. Chem. Eng. Data, 1969, 14, 55.l6 J. D. Gbmez-Ibhfiez and Chia-Tsun Liu, J. Phys. Chem., 1963,67,1388.G. Egloff, Physical Constants of Hydrocarbons, vol. V, (Reinhold, N.Y., 1953).A. Desymter and J. H. van der Waals, Rec. Trau. Chim., 1958, 77,53

ISSN:0366-9033    

DOI:10.1039/DF9704900106

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Thermodynamic Properties of some Polymer Solutions atElevated TemperaturesBY A. H. LIDDELL AND F. L. SWINTONDepartment of Pure and Applied Chemistry, University of Strathclyde,Glasgow, C. 1. ScotlandReceived 23rd January, 1970The enthalpies of mixing at infinite dilution have been measured for the four systems polyiso-butylene+ benzene, fn-pentane, +n-hexane and 4n-octane from room temperature up to thevicinity of the lower critical solution temperatures. The results are analyzed in terms of the statisticaltheories of polymer solutions due to Prigogine and to Flory. Both theories give a reasonable qualita-tive description of the experimental results.The discovery by Freeman and Rowlinson,' that most polymer solutions undergophase separation at a lower critical solution temperature (LCST) at high temperaturesin the vicinity of the gas-liquid critical temperature of the low molecular weightsolvent, invalidated the majority of statistical theories of polymer solutions in useat that time.The existence of a LCST implies a negative (exothermal) enthalpy ofmixing and a negative entropy of mixing, whereas theories of the Flory-Huggins orMiller-Guggenheim type predict positive (endothermal) enthalpies of mixing and largepositive entropies of mixing. These earlier theories correctly predicted the occurrenceof an upper critical solution temperature (UCST) found experimentally for manypolymer + solvent systems at low reduced temperatures. Since the discovery of theLCST phenomenon, Patterson et ~ 2 .~ 9 have shown that this effect is predicted by thecorresponding states treatment of polymer solutions developed by Prigogine althoughquantitative agreement between theory and experiment is generally poor. RecentlyFlory et al. have carried out a similar analysis of experimental LCST 5* using astatistical treatment of polymer solutions also based on a corresponding states ap-proach that is similar in inany ways to the Prigogine theory.The estimation of HM, the enthalpy of mixing of a binary liquid system, and of thevariation of HM with temperature, CP, is a stringent test of any theory of fluid mix-tures. The present paper describes the experimental measurement of HM at lowconcentrations of polymer for four polymer + solvent systems from room temperatureup to the vicinity of the LCST.The results are analyzed in terms of the Prigogineand Flory theories.EXPERIMENTALThe LCST were measured by the conventional method in which the samples are sealedin heavy-walled Pyrex tubes and are heated slowly. The tubes were 150 mm long and 1.6 mint. diam. The furnace and experimental procedure have been described previously.8The enthalpies of mixing were measured using a commercial Calvet microcalorimeterwhich can operate from room temperature to around 480 K. The specially designed stain-less-steel mixing cell, in which the components were allowed to mix in the absence of a vapouror an air space, has been described previou~ly.~ All measurements made above the normalboiling point of a particular solvent were carried out under a constant pressure of 16 bar.11116 THERMODYNAMIC PROPERTIES OF POLYMER SOLUTIONSi .* a * a )The pressure coefficient of the enthalpy of mixing, ((&Y'/&)~ = P- T(i3 YM/dT)p, couldbe neglected because of the relatively low experimental precision.The hydrocarbon solvents used were obtained from Newton Maine Limited and were of aguaranteed purity in excess of 99 %. Two samples of Vistanex polyisobutylene were re-ceived from the Esso Chemical Company. Sample A had a viscosity average molecularweight of 7 . 2 ~ lo4 and a density of 0.917 f0.002 g ~ r n - ~ at 298.15 K which compares wellwith the density quoted by Eichinger and Flory lo for material of similar molecular weight.Sample A was a viscous liquid and was used for all the enthalpy of mixing measurements.Sample B was a rubbery solid and had a viscosity average molecular weight of 2 .8 ~ lo6.RESULTSThe experimental phase diagrams for the four binary systems polyisobutylene(sample A)+benzene, +n-octane, +n-hexane and +n-pentane are shown in fig. 1.The LCST occur at a polymer weight fraction of 0.025-0.030 in all cases and the LCSTare listed in table 1. Also listed in table 1 are the LCST of the high-molecular-weightpolymer (sample B) in the same solvents. These values agree well with the figuresquoted by Patterson et aZ.ll* l2 for polymer of infinite molecular weight.TABLE LO LOWER CRITICAL SOLUTION TEMPERATURES OFPIB + HYDROCARBON SYSTEMSthis work, ref. (11) solvent this work,sample A sample Bbenzene 540.5 534 -n-pentane 373.5 349 344n-hexane 428.5 407 402n-octane 506 489 47A .H. LIDDELL AND F. L. SWINTON 117The experimental enthalpies of mixing are listed in table 2. The measurementswere performed at high dilution and the values quoted are expressed in J mol-l ofrepeat unit of polymer (56 g). Each value listed is the average of five separate deter-minations and the uncertainty is + 3 % at room temperature increasing to +6 % atthe highest temperatures. Absolute temperatures were measured using a singlejunction thermocouple and were only recorded to 1 K.HM at infinite dilution, HE, has been measured for the polyisobutene + benzenesystem by several other workers around room temperature and the present results arecompared with literature values in table 3.The values of Delmas et aL2 and ofTABLE 2.-ENTHALPIES OF MEUNG AT INFINITE DILUTION, ffz, FOR PIB f HYDROCARBON MIXTURESPIB + benzeneTIK H z / J mol-1300323343375394423437453+ 1076 + 10003- 879 + 699 + 515 + 197- 26- 264PIB + n-pentaneTlK H ~ I J rnol--l303 - 159333 - 193352 - 253365 - 306PIB + n-hexaneT / K Ht/Jmol-303324348373393408423433-100- 130- 163- 205- 297- 373- 502- 557PlB + n-octaneT / K H z / ~ m o i - 1303 - 17324 - 42348 - 50373 - 59393 - 75423 -2200Waters et aZ.l are in excellent agreement with the present results, the value of Bianchi l4is in moderate agreement, while the figure quoted by Tager and Podlesnjak l5 is incomplete disagreement with all the other values. The higher molecular weight of thepolymer used by Tager and Podelsnjak would tend to give a more positive H," andtheir results are further suspect because they state that H," did not vary within experi-mental error throughout the temperature range 303-343 K.It seems probable thatTABLE 3.-Hz FOR PIB+BENZENE, COMPARISON WITH OTHER RESULTSTJK Mu of PIB H: /J mol- 1 ref.297.7 3x lo4 +lo88 2303.2 4.5 x 104 3-900 14303 2 . 6 ~ lo6 + 450 15298 6.3 x 104 +lo97 13300 7 . 2 ~ 104 3-1 076 present resultscomplete dissolution of the polymer was not obtained. The results given in table 2were all obtained at temperatures below the LCST except the result at 433 K forPIB + n-hexane. This temperature is 4;5 K above the measured LCST for this systemand the reason that a spurious point did not result is that the LCST was determined a118 THERMODYNAMIC PROPERTIES OF POLYMER SOLUTIONSa pressure equal to the saturation vapour pressure of the solvent, 9 bar at that temp-erature, whereas the calorimetric measurements were made under an applied pressureof 16 bar.An increase in applied pressure raises an LCST.The experimental results for all four systems are characterized by large negativevalues of both C? and (dC,M/dT) as can be seen from fig. 2. Negative values forCpM are a necessary condition for systems that exhibit a UCST lying below a LCST andare found in most binary systems when the molar volumes of the two componentsdiffer appreciably, e.g.n-hexane + n-hexadecane,l benzene + diphenylmethane. l7200 350 400 450TIKFIG. 2.-The experimental enthdpies of mixing at infinite dilution. 0, PIB+benzene; 0, PIB+n-pentane ; CD , PLB + n-hexane ; a, PIB + n-octane.DISCUSSIONhave shown that the corresponding states, average potentialtheory of polymer solutions due originally to Prigogine predicts the existence of aLCST in polymer solutions and can correlate the LCST and H," at constant temp-erature for a single polymer with a homologous series of solvents successfully. Theexpressions for the thermodynamic functions of mixing are found in terms of certainconfigurational properties of a reference substance, z, the co-ordination number of thepolymer + solvent lattice and the three interaction parameters 8, p and 6.Patterson et aL2.6 = (&22-&11)/ElI ; (la)P = (~22-r11)/r11 ; (Wand 8 = [elz-(&11+&22)/211&11, (14where & l l and rll are the coordinates in energy and length of the minimum in the(intermolecular potential, distance) function for a 1-1 interaction etc.When theBerthelot relation is applied to eqn (lc), and the further simplifying assumptionsare made that terms in 6, p and 6 higher than those of the second power are neglectedand that z+w, the expression for H," becomesH z = (h - TCJ(- a2/4 - 9p2) -fiT2(dCP/dT)(d2 + 9/r2 - 6S/r2). (2A. H. LIDDELL AND F. L. SWINTON 119h, Cp and (dCJdT) are configurational properties of the reference substance, takento be the polymer in this case, and r, is the number of segments composing the solvent.The first of the two terms on the r.h.s.of eqn (2) is numerically positive and is almostindependent of temperature, while, assuming (dCJdT) to be positive and independentof temperature, the last term in eqn (2) is negative and proportional to T2. H,"should, to a first approximation, vary linearly as T2. This is found not to be the caseexperimentally. The experimental values of HZ for the PIB + n-hexane system areplotted as a function of T2 in fig. 3 and deviate considerably from Iinearity. Eqn (2)does, however, correctly predict that both CpM and (dC,M/dT) are negative. Anapproximate test of eqn (2) can be made for the PIB+n-hexane system by assumingthe following reasonable values for the various parameters, h = -40 kJ mol-I,Cp = 60 J mol-1 K-l, 6 = 0.1, p = 0 and r2 = 3.Few experimental values ofdCJdT are available so this was used as an adjustable parameter to obtain a fitbetween theory and experiment at 300 K, resulting in a reasonable value of dC,/dT =+6.7 mJ mol-l K-2. HfoM was then estimated at higher temperatures and the theor-etical straight line shown in fig. 3 was obtained, in fair agreement with the experi-mental curve.0-60010 13 16 192721104 KZFIG. 3.-The experimental results for PJB-1-n-hexane.-0-, expt ; - - - , Prigogine theory ; . . . . . ., Flory theory.The other recent theory of polymer solutions that predicts the existence of a LCSTis that due to Flory.' Patterson, Bhattacharyya and Picker have shown that theFlory and the Prigogine theories are related in many ways, the main difference beingin the choice of intermolecular potential.Flory has derived an expression for H:,(3)u*, T*, and p * are the so-called characteristic parameters of the two components andd etc. are reduced quantities related to the characteristic parameters, ij = v]u* etc. alis the coefficient of thermal expansion of the solvent and N2 is the weight of the polymerrepeat unit. s1 and s2 are the number of surface sites per unit core volume and theratio sz/sl can be estimated from the molecular dimensions of the component mole-cules. X12 is an interchange energy similar in many ways to the Prigogine 6 and isusually treated as an adjustable parameter. Numerical values of the various para-H," = N2(~2*/~1>(PZ*f(v"1/~2)- 1 -a1m -TT/T;)l+(l +aJ)(s2ls,)X,21120 THERMODYNAMIC PROPERTIES OF POLYMER SOLUTIONSmeters in eqn (3) can be determined from a, the coefficient of thermal expansion and y,the thermal pressure coefficient of the components, and have been calculated prev-iously for PIB and n-hexane by F10ry.~. lo These values are given in table 4 togetherwith experimental values of a1 at various temperatures.TABLE 4.-FLORY PARAMETERS FOR PIB AND n-HEXANE AT 298 Ku*/cm3g-1 p*/J cm-3 T*/KPIB 0.949 3 447.7 7 580n-hexane 1.155 4 423.4 4 437T/K 300 350 400CCJ~O-3 K-I 1.390 1.69 2.20s Z / S ~ = 0.55The characteristic parameters p*, v* and T* should be independent of temperatureif the theoretical equation of state is a good representation of the experimental dataand X,, should likewise be temperature independent.A value of XI, was found byusing the experimental value of H," at 300 K in eqn (3), X,, = 9.75 J ~ r n - ~ . Thisvalue of X,, was then used to calculate H," at higher temperatures, part of the result-ant curve being shown in fig. 3. The theory fails completely to represent the experi-mental data adequately although, as for the Prigogine theory, both CpM and dC,M/dTare correctly predicted to be negative. It is possible to force a fit between theory andexperiment at all temperatures by treating XI as a temperature-dependent parameter.A value of Xlz = 51.2 J ~ m - ~ is required at 400 K and so large a variation in thisparameter cannot be justified.Both the Prigogine and the Flory theories fail badly in the quantitative predictionof the variation of Hg with temperature for the PIB + n-hexane system.The theoriesfail to a similar degree when they are applied to the three other polymer+solventmixtures. A fully satisfactory theory of the equilibrium properties of polymersolutions has yet to be developed.We thank the S.R.C. for the award of a research studentship to A.H.L.P. 1. Freeman and J. S. Rowlinson, Polymer, 1960, 1,20.G. Delmas, D. Patterson and T. Somcynsky, J. Polymer Sci., 1962, 57, 79.D. Patterson, S. N. Bhattacharyya and P. Picker, Trans. Faraday Soc., 1968, 64, 648.I. Prigogine, The Molecular Theory of Solutions (North-Holland, Amsterdam, 1957), chap. 17.P. J. Flory, J. L. Ellenson and B. E. Eichinger, Macromol., 1968,1, 279.B. E. Eichinger and P. J. Flory, Tram. Paraahy Soc., 1968, 64,2035. ' P. J. Flory, R. A. Orwoll and A. Vrij, J. Amer. Chem. Soc., 1964,86, 3515.R. J. Powell, F. L. Swinton and C. L. Young, J. Chem. Thermodynamics, 1970, 2, in press.A. H. Liddell and F. L. Swinton, J. Sci. Instr., 1970, in press.lo B. E. Eichinger and P. J. Flory, Macrornol., 1968, 1,285.l1 D. Patterson, G. Delmas and T. Somcynsky, Polymer, 1967, 8, 503.l2 J. M. Bardin and D. Patterson, Polymer, 1969, 10, 247.l3 C. Watters, H. Daoust and M. Rinfret, Can. J. Chem., 1960, 38, 1087.l4 V. Bianchi, E. Pedemonte and C. Rossi, Makromol. Chem., 1966, 92, 114.l6 J. A. Friend, J. A. Larkin, A. Maroudas and M, L. McGlashan, Nature, 1963, 198, 683.l7 M. La1 and F. L. Swinton, Trans. Farahy SOC., 1967, 63, 1596.A. Tager and A. Podlesnjak, presented at IUPAC meeting (Toronto, 1968)

ISSN:0366-9033    

DOI:10.1039/DF9704900115

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Dilute and Concentrated Solutions of a Heterocyclic PolymerBY G. C. BERRYMellon Institute, Carnegie-Mellon University, Pittsburgh, Pa.Received 12th January, 1970Properties of the heterocyclic polymer prepared by the reaction of naphthalene-lY4,5,8-tetra-carboxylic acid with 3,3'-diaminobenzidene are discussed. Light scattering, viscometric and freezingpoint depression measurements on dilute solutions lead to the conclusion that in strong acids BBB isprotonated and takes on a flexible coil conformation in dilute solutions. Viscometric data onconcentrated solutions likewise follow the behaviour expected for coiled chains, but exhibit thebehaviour expected for entangled chains at unusually low concentrations, and a segmental mobilitythat decreases rapidly with increasing polymer concentration.These data, together with X-raydiffraction studies on the undiluted polymer, are discussed in terms of short range interactions leadingto stacked aggregates in very concentrated solutions and the bulk polymer.The need for polymers stable at temperatures in excess of 400°C has led to thedevelopment of polymers with heterocyclic rings in the backbone of the chain.These materials have typically, although not universally, exhibited (i) poor solubilityand (ii) a high softening temperature, often with no softening prior to decomposition.This report concerns studies on BBB, the polymer of this class prepared by the reactionof naphthalene-1,4,5,8-tetracarboxylic acid with 3,3'-diaminobenzidine to give achain with the (idealized) structure :(both the " trans " and " cis " isomers of the repeat unit are possible).The poly-merization has been discussed by Van Deusen.' Some aspects of the dilute solutionproperties have been presented previously. The dilute solution studies describedherein, which include freezing point depression, light scattering and viscosity measure-ments, are interpreted to show that the BBB chain behaves as a flexible coil in dilutesolutions. The viscometric studies on concentrated solutions show that there isappreciable intermolecular interaction giving rise to a viscosity for concentratedsolutions (i) that depends markedly on polymer concentration and (ii) that exhibits amolecular weight dependence characteristic of entangled chains at unusually lowpolymer concentrations.The implication of these findings on the properties of theundiluted polymer is explored, and it is concluded that in the undiluted state there mustbe short range order even though the polymer is not crystalline in the usual sense.Evidence of this order from X-ray diffraction data is described.12122 SOLUTIONS OF A HETEROCYCLIC POLYMEREXPERIMENTALMATERIALSVat Orange 7 (Colour Index 71105) and Vat Red 15 dyestuff (Colour Index 71100) weresupplied by the National Aniline Division of Allied Chemical Corporation and by AmericanHoechst, respectively. Both compounds were precipitated three times from dilute solutionsin 96 % sulphuric acid, washed exhaustively and dried. Infra-red spectra showed no traceof any sulphate group, and mass spectroscopy revealed both products to be pure compoundswith molecular weight 412.The BBB polymer 51165 was supplied by Dr.R. L. Van Deusen, Wright-Patterson AirForce Base, Ohio. The polymer was prepared by polymerization in polyphosphoric acid at185°C with a concentration of 0.13 mol/I. of each reactant. The unfractionated polymerhas an intrinsic viscosity of 1.7 dl/g in 96 % H2S04 and a weight average molecular weightof 6.5 x lo4. Infra-red spectroscopy shows that imide groups representing incompletecyclization are absent, or present at exceedingly low concentration.2 Dissolution of thepolymer in aqueous 5 M KOH causes an irreversible reduction in [q] of 15 % after 1 h ; nofurther reduction in [q], achieved by additional aging in the alkaline solution.Analysis ofthe molecular weight distribution of the original and the base-treated polymer by analyticalexclusion chromatography shows that the entire distribution of species is shifted towardlower chain lengths by the base treatment. The reduction in chain length is believed to bethe result of chain scission at a few, randomly placed weak links, probably caused by incom-pletely cyclized structures. The magnitude of the observed reduction in [q] corresponds to1 scission per 7 initial chains for polymer 51 165.Polymer 4-22 was prepared at Mellon Institute by Mr. V. Ammons by polymerization inpolyphosphoric acid at 180°C with a concentration of 0.03 mol/1. of each reactant. Theunfractionated polymer has an intrinsic viscosity of 0.51 dl/g in 96 % H2SO4 and [q] is un-affected by dissolution of the polymer in aqueous 5 M KOH.The preparation of series I fractions by preparative scale exclusion chromatography hasbeen described.2 Series I1 fractions were similarly prepared, except that the primaryfractions were not refractionated in this case.Series I11 fractions were prepared by successiveprecipitation from a strong alkali solution. The 51165 polymer was soaked overnight in 7 Mammonium hydroxide and then dissolved by the addition of 1 M potassium hydroxide.After a cloud point was developed by the addition of further KOH, the precipitate wasallowed to settle and was removed. Successive fractions were recovered in like fashion.The recovered polymer was washed with water, dissolved in sulphuric acid and reprecipitatedin water, washed exhaustively with water and dried.If the alkaline-treated polymer is recovered and washed with anhydrous alcohol, a brownsolid distinctly different from authentic BBB is obtained.This material has no absorptionat 1705 cm-l characteristic of the carbonyl group found in BBB. Thorough extraction watercauses the brown solid to revert BBB; i.e., the infra-red spectra of the solid and the elec-tronic spectra of sulphuric acid solutions of the water washed material are identical to thespectra of authentic BBB.Practical grade methane sulphonic acid was distilled under vacuum and stored undernitrogen prior to use.FREEZING POINT DEPRESSIONThe melting temperature was determined on solutions in 100 % sdphuric acid in the cryo-stat shown schematically in fig.1. The 100 % sulphuric acid was prepared by titration ofreagent grades 96 % sulphuric acid and fuming sulphuric acid to give a freezing temperatureof 10.364"C, compared to the empirically determined freezing temperature of 10.371 for the100 % sulphuric acid.5 The freezing temperature of the solvent did not change sijpificantlyover a 10 month storage period under argon. Special precautions were taken during thepreparation and transfer of solutions to prevent contamination with moisture. A slow flowof argon was maintained over the solution in the cryostat at all times. As a check on tech-niques, the freezing point depression of potassium sulphate was measured and found toagree with reported v a l ~ e s .~ The freezing temperature of 100 % sulphuric acid held in thG. C. BERRY 123cryostat for 20 days did not change significantly. Melting temperatures were determinedby following the temperature of a slowly melting slurry of the partially frozen solution asmelting proceeded following procedures given by Rossini and coworkers. 5* ti Complete detailswill be given elsewhere.’ARGON /OUTLETFIG. 1.-Schematic drawing of the freezing temperature apparatus used with solutions in sulphuricacid.INTRINSIC VISCOSITY AND LIGHT SCATTERINGViscosity of solutions prepared to have a relative viscosity qrel in the range 1.1 <rrel < 1.8were determined in suspended level Ubbelohde viscometers. The usual plots according tothe relations :qsplc = [qI+k”ql2c+ .. .In q,& = [ql- (+- k’)[q12c+ . . .,with ysp = ~ 7 ~ 1 - 1 , were used to determine [q] from data on four concentrations. In allcases, k’ was in the range +<k‘<* usually observed. Additional data on solutions ofpolymer 51165 were obtained on the rotating cup viscometer * operated at a shear rate of0.7 s-l.Methods of sample preparation for light scattering, the scattering apparatus, and thetreatment of the data have been described.2 Additional corrections necessary for fluores-cence emission at the scattering wavelength of 6328 A are discussed in the following. Briefly,small light scattering cells allowing in situ centrifugation of the solutions were used with aphotometer constructed at Mellon Institute.A HeNe laser (6328A) was used as the lightsource to minimize absorption of light by the polymer solutions.VISCOMETRY ON CONCENTRATED SOLUTIONSSimple capillary viscometers of the type described by Fox et aL9 were used These werespecially constructed to prevent contamination of the sample by moisture. The capillar124tube and sample container were held in a glass tube with gas inlet and outlet ports. Theglass tube was placed in a constant temperature water bath in which water evaporation washeld to a minimum with a layer of oil on the surface. Viscosities were calculated in triplicatefrom flow times between four successive marks on the capillary as the liquid was forced upinto the capillary by a pressure differential supplied with dry nitrogen gas.Capillaries withbores ranging from 0.5 to 2 mm were used to cover the viscosity range studied. In no casewas it possible to achieve a substantial variation in the shear rate in order to be certain that aNewtonian viscosity was being measured. The shear rates at the wall are around 10 s-l orless, however, so appreciable departures from a viscosity independent of shear rate would beunexpected.SOLUTIONS OF A HETEROCYCLIC POLYMERRESULTSDILUTE SOLUTION STUDIESFREEZING POINT DEPRESSIONThe freezing point depression of two model compounds of the BBB repeat unithas been determined in 100 % sulphuric acid. The compounds Vat Orange 7 andVat Red 15 serve as the trans and cis isomers of the repeat units of BBB, respectively(with protons replacing the interrepeat unit bond).The data, given in table 1, wereanalyzed according to the method of Gillespie4 with the assumption that the inter-action of the solute S with the acid is simple protonation :S+xH,S04+SH~"+xHS0,.Thus, the degree of protonation x was calculated from the freezing point depression #(relative to the hypothetical undissociated acid) asO(l + 0.0020) md -- l+x = 6.l2m m 'where m is the molality of the solute and md that of the dissociation products ofsulphuric acid. The latter can be deduced from m and x by a rapidly convergenttrial and error procedure lo provided the basic assumption concerning the nature ofthe interaction between the solute and acid is correct. The values of x for VatOrange 7 scatter about a value near 2.An integral value of x should be expectedunless a weakly basic protonation site is involved giving rise to partial protonation.A value distinctly less that 2 is indicated for Vat Red 15. Definitive interpretation ofthese data will require some independent experiment, such as a conductivity determin-ation in order to assess the underlying premise concerning the nature of the solute-acidintera~tion.~ At present, it appears that the model compounds, and thus presumablyBBB as well, are protonated in strong acids. Protonation probably involves the-N= pyridene nitrogen, Previous studies showed that these dyestuffs have apKb value similar to that usually found for the pyridene nitrogen in imidazole com-pounds.11 It may be that the proximity of these groups in the Vat Red isomer is thereason that x is less than 2 for this isomer. Similar proximity effects have been notedfor dicarboxylic acids.INTRINSIC VISCOSITY STUDIESIt was not possible to carry out all dilute solution studies in sulphuric acid sincethe higher molecular weight fractions of BBB were incompletely dissolved, except inacids approaching 100 % H2S04.Work in this reagent was avoided since BBB isreadily sulphonated in acids slightly richer in H2S04.2 By contrast, all of the fractionsobtained in this study were soluble in methane sulphonic acid, (MSA), and the poly-mer could be recovered unchanged by simple precipitation with methanolG . C . BERRY 125It is assumed that BBB is protonated in methane sulphonic acid.Comparison ofthe electronic absorption spectra of solutions of BBB in MSA and in sulphuric acidshow them to be nearly identical, supporting our assumption. Accordingly, BBB ispresumed to be a polyelectrolyte in methane sulphonic acid. Nonetheless, excessivechain expansion is not expected (in either MSA or sulphuric acid) owing (i) to the highTABLE FREEZING POINT DEPRESSION OF MODEL COMPOUNDS IN100 % SULPHURIC ACIDsolute solute degree ofmolality e(K) protonation xVat Orange 7 0.003 7 0.299 2.22Vat Orange 7 0.007 8 0.346 2.13Vat Orange 7 0.019 3 0.537 2.24Vat Red 15 0.004 0 0.278 1.25Vat Red 15 0.006 5 0.305 1 S Odielectric constant of the solvent which reduces the effective interaction distance ofthe charged groups on the polyion, and (ii) to the presence of simple electrolytes inaddition to the counterions of the polyion.The additional simple electrolyte origin-ates from the dissociation products expected in MSA (or sulphuric acid). This doesnot imply that expansion effects are absent, but only that they should be moderate inmagnitude. Data, given in table 2, for the intrinsic viscosity of an unfractionatedBBB in sulphuric acid as a function of water content, and in sulphuric acid containingpotassium sulphate, support this contention. Thus, [y] decreases as the concentrationof ions produced by self-dissociation of sulphuric acid is increased with increasingwater content, or as the concentration of the simple electrolyte is increased by directaddition of potassium sulphate.Addition of potassium sulphate can bring aboutprecipitation of BBB if enough of the salt is added to produce a supersaturated solu-tion. It is noteworthy that [q] is about the same for solution in 100 % sulphuric acidand methane sulphonic acid.TABLE 2.-EFFECT OF WATER AND ADDEDELECTROLYTE ON THE INTRINSIC VISCOSITY OFBBB IN SULPHURIC ACID104 % sulphuric acid 2.7899.7 % sulphuric acid 2.0897.4 % sulphuric acid 1.9497 % sulphuric acid 1.42solvent [??I (w3)sat. with potassiumsulphateThe dependence of [q] on shear rate for an uiifractionated BBB sample was ex-amined in solution in 97 % sulphuric acid by comparison of [y] determined in 8couette viscometer with [q] determined in a capillary viscometer. These instru-ments were operated at shear rates of 0.7 and 900 s-I, respectively (the latter figurerepresenting the maximum shear rate at the capillary wall).Values of [y] agreedwithin 4 %, indicating a negligible dependence of [q] on shear rate for a sample with[q] = 1.74 dl/g.Measurements on [q] as a function of temperature with the unfractionated 51 165and series 11-5 fraction in methane sulphonic acid gave the small, negative temperaturecoefficient d In [q]/dT = -2.5 x (“C-l)126 SOLUTIONS OF A HETEROCYCLIC POLYMERAs for solutions in sulphuric acid, values of [q] in MSA depend on the watercontent of the MSA. For example, the addition of 3 % water to freshly distilledMSA caused a reduction of 9 % in [q] for a sample with [q] = 3.0 dl/g in the freshlydistilled solvent.Values of [q] listed in table 3 have been normalized to remove thissolvent effect. These values of [q] are plotted in fig. 2 against the weight average-0 - -It1 I I I I 1 1 1 I I I I I ITABLE DATA ON SOLUTIONS OF BBB IN METHANE SULPHONIC ACIDsample [TI (dug) 1 O-SMW ~O'*(S*)LS (cmz) logloq for w-2 = 0.025T = 35°C (q in poise)51165 2.66 0.654-22 0.74 0.1 11-3 5.48 1.59 40.41-5 4.09 0.87 10.01-1 I 2.87 0.67 13.0r-18 1.96 0.42 -11-111-211-311-411-511-611-711-95.65 1.26 28.04.16 0.70 17.03.403.042.22 0.38 6.02.071.881.68 0.30 -- -- I- -- -111-1 3.90 1.35 37.2111-1 3.90 1.26 29.7111-2 3.37 0.85 30.9111-3 2.99 0.70 25.5111-6 1.83 0.30 -111-7 1.25 - -0.53--1.8231.4551.1671.1460.7940.73 61.141--0.8710.3610.01 G.C. BERRY 127molecular weight M, determined by light scattering studies. The correlation of [pl]with M, is much better within a given fractionation series than it is among all of thedata taken together.LIGHT SCATTERING STUDIESLight scattering studies on solution of BBB in methane sulphonic acid are hamp-cred by absorption and fluorescence, even at the wavelength 6328A of the Ne-Helaser source used, although this wavelength is on the tail of the visible absorptionband.2 Previous data given by us 2* l3 are in error owing to the neglect of the fluo-rescence encountered at this wavelength. Fluorescence emission was too feeble to bedetected with an Aminco-Bowman spectrofluomroeter, but was comparable to thescattered light in intensity in some cases.The excess Rayleigh ratio R was calculatedaswhere U”(8) is the observed intensity at angle 8 with vertically polarized incident lightbut no analyzer, U&(8) is the observed fluorescence emission at angle 8, U;(O) is theobserved solvent scatter at angle 0, k is an instrument calibration, andwhere OD,’: is the optical density in a 10 rnm cell and I is the pathlength of the lightscattering cell (expressed in mm). The fluorescence emission was determined as thedifference between the observed intensity with no filter before the photomultiplier andthat with a 6328 A band-pass filter between the scattering cell and the photomultiplier.The product V,VI = U,Vl(8) sin 8 was averaged over 8 to determine Us@) on theassumption that UFy, is independent of 8.It was found empirically that U& could becorrelated with the solute concentration, or equivalently OD::, by the relationwhereInsertion of eqn (3) and (4) into (2) yields the result,where k, = 0.230 3 I, and q = OD::. Differentiation with respect to q gives someidea of the uncertainty in R associated with an uncertainty in q :~ ( 0 ) = k sin 8{p[uv(8) - u;,(8)] - u;;(8)), (2)p = exp (0.2303 Z OD::) (3)PFUhloD:,O = KF1 (4)( 5 )(6)pF = exp (0.2303 I OD::/2) = p’,R(8) = k{sin i3Uv(0) exp (k,q)-KFlq exp (klq/2)-sin 8u@)>,3111 R - k,-v exp(-klq/2)[l+klq/2] ---i?q 1-vexp(-k,q/2)--pexp(-k,q)’ (7)wherep = Uv(0)/U;(O) and v = &/sin 8 Uv(0). For a typical BBB solution is MSA,kl = 3, v = 3 and p = 0.10.Use of these values and an assumed uncertainty 0.02for OD: gives values of AR/R that increase from - 0.4 to 8 % as q increases from 0.2to 1.0. These are regarded as realistic estimates of the uncertainties in c/R over theconcentration range studied.A second possible difficulty in the interpretation of the light scattering data onsolution of BBB in MSA is the effect of the complex refractive index m. If we repre-sent m as n - in’, then the refractive index increment that should be used to analyze thedata for a weakly absorbing system such as this is I rn-no I/c instead of (n-n,)/c,where no is the refractive index of the s01vent.l~ However, n’ is related to the extinc-tion coefficient E for absorption byexp ( --EcZ) = exp (-4nn’Z/d) (8128 SOLUTIONS OF A HETEROCYCLIC POLYMERso that(9)Since E = 2.77 x lo3 cm3/g for 3, = 6328 A, the difference between I m -no I/c and(n-no)/c can be neglected.Values of M, and the mean square radius of gyration (s2) were computed fromc/R(c,B) in the usual way :e= owhere R(c,B) is the value of R at concentration c and angle 8.The subscript LSindicates that (s2) is an averaged value over all species if the sample is heterodisperse.Values of M, and (s2),, are listed in table 3 and log (s2)Ls is plotted against log Mwin fig. 3. Values of the second virial coefficient A2 were always positive, but are notreported owing to the large absorption and fluorescence corrections which stronglyaffect the value calculated for A2 from the data.10-4 iwW4 6 8 10 201' I i I I I ' 1 14 5log MwFIG.3.-Mean square radius of gyration as a function of niolecular weight for BBB in methanesulphonic acid; symbols as in fig. 1.CON CENTRATED SOLUTIONSDEPENDENCE OF VISCOSITY O N TEMPERATUREThe ratio of the viscosity y(T) at temperature T to the viscosity y(35) at 35°C isshown in fig. 4 for representative data on samples varying in molecular weight fromlo4 to 6.5 x lo4. These data show that the apparent activation energy EApp calculateG . C. BERRY 1 29t("C)60 40 30 2 0 10 0I I IFIG. 4 . T h e viscosity at temperature T 2divided by that at 35°C as a function of 1/T 3for solutions of BBB in methane sulphonic sacid : 0, unfractionated 51 165 ; 0, un- Afractionated 4-22 ; 0, series 11-S ; 0, series &-,11-7 ; 6, series 111-1 ; 0, series IIZ-8.From %top to bottom, the curves are for solutionswith weight fractions 0.135, 0.085 to 0.087, $0.035 to 0.039, and 0.010. w 03I -r-r I 1m 1.3X8 1.2-2- -(0) -k II I--FIG. 5.-The Vogel parameters CI andTo as function of the weight fractionwz of BBB in methane sulphonicacid.100 -50 - --(b)-O' oh2 ode od6 0.68 kt0 0.12 Ot4 &!S 0-18w2130 SOLUTIONS OF A HETEROCYCLIC POLYMERas Rd In y/d(T-l) depends strongly on the weight fraction w2 of polymer and thatEApp is not independent of T, even over the relatively narrow temperature intervalstudied. These data have been fitted to the Vogel relation to determine the para-meters cc and To as functions of w2 :This correlation is not unique since the temperature span is not wide, but the analysishas been made on the assumption that a is a slowly varying linear function of w2, asis usually the case.3 This constraint allows both a and To to be uniquely determinedas a function of w2 with the results shown in fig. 5.The values of To and a are inde-pendent of the mofecular weight over the range of variables studied. The strongdependence of To on w2 provides one part of the dependence of q on w2.In y = A + [l/a(T- To)]. (1 2)DEPENDENCE OF THE VISCOSITY ON MOLECULAR WEIGHT CONCENTRATIONThe usual relation used to correlate 11 with M,,, and the volume fraction 42 is givenbyY = FC, (13)F = ( ~ , 1 6 ) X ( X / ~ , ) " 9 (14)whereandHere r: is the friction factor per hydrodynamic unit with molecular weight m,, ( s ~ } ~ isthe unperturbed radius of gyration of the chain, v2 the specific volume of the polymer,and X, a constant found to be about 400 x 10'' for many flexible chain polymer^.^The friction factor is given bywhere a and To are the constants discussed above and to is a parameter assunied to beindependent of T, M and w2, but perhaps specific for each polymer.We shall setv , ~ ~ / v , equal to w2 over the concentration range employed in these studies, where u1is the specific volume of the solvent.Although eqn (13)-(15) have been found to correlate data on a variety of polymers,they have never before been applied to a polymer with as large a repeat unit as that ofBBB.In the following, we associate a hydrodynamic unit with the entire repeatunit of BBB since this unit is inflexible. This group then corresponds to the '' chainatom " discussed in ref. (3). We neglect the temperature dependence of X in thefollowing analysis, based on the weak dependence of [q], and hence (s2>, on T.Since 5 and F both depend on w2, the concentration dependence of q comes from twosources, and it is necessary to separate these in order to interpret the data in a meaning-ful way. For example, a plot of log y against log w2 without this separation at 35°Cfor an unfractionated BBB polymer with M = 6.4 x lo4 is shown in fig. 6. A steeplyrising function of q against w2 is observed. An attempt to correlate rj with w;, wheren is some positive number, would fail since n would have to depend on w2, approachinga value of 10 as w2 increased to 0.17. Fig.7 shows log ((w2)/5(0.025) against log w2,with values of [(w,) the friction factor at concentration w2 computed at 35°C using Toand a from fig. 5. These data are used to compute log [q((0.025)/5(w2)], whichdirectly reflects log F, shown in fig. 6. The structural factor F is proportional toM,";4 at least for log w2Mw> 3.0. Since ( s ~ ) ~ is proportional to M,v, this is just thebehaviour expected for a flexible chain polymer.a = 2.4 if X L X , ; a = 0 if X<X,,x = ~ S 2 > o ( ~ 2 1 ~ s V 2 * >lnc = ln~,+[l/a(T-T0)], (1 5G . C . BERRY 131-2 - I 0log w2FIG. 6.-Viscoinetric data at 35°C for a solution of BBB in methane sulphonic acid.Solid points :log q against log w2. Open points : log r-log[~(wz)/r(0.025)] against log w2 (cf. fig. 5).1 I I I-2 -Ilog wzFIG. 7.-The friction factor <(w2) at 35°C for a solution with weight fraction w2 of BBB in methanesulphonic acid divided by that for a solution with w2 = 0.025 as a function of wz132 SOLUTIONS OF A HETEROCYCLIC POLYMERFig. 8 shows log [y[(0.025)/[(w2)] at 35°C against log tv,M, for all of the samplesinvestigated. Values of M, used are smoothed values taken from the individualcorrelations between [q] and M, shown in fig. 2 for each of the three fractionationseries. Values of log y for those solutions with w2 = 0.025 are given in table 3, anddata not listed on solutions with concentrations over the interval 0.01 < w2 <0.14 arealso included in fig.8. Portions of the data with yccw,M, and y ~ c ( w , M , ) ~ * ~ can beclearly discerned. The unexpected appearance of two separate sets of data with11 CC(W,M,)~-~ will be discussed below.log w2Mw log WZ[?lFIG. 8.-Visconietric data at 35°C for solutions of BBB in methane sulphonic acid as a function ofmolecular weight and weight fraction of polymer. Log ?-log [C(wz)/C(0.025)] is shown against (a)log wzMw, and (b) log w2[q]. Points are 9, series I1 fractions ; <), series 111 fractions ; @, uufrac-tionated 51165 ; and 9, unfractionated 4-22.DISCUSSIQNThe data on [y] as a function of M, do not give a single relation between [q] andM,,, for all three fractionation series. We can offer no definitive explanation for thisbehaviour, but some differences in sample preparation that may be responsible can beconsidered.Series I fractions were obtained by a double fractionation on a prepara-tive scale porous silica Such a separation is made according to molecularsize rather than molecular weight. Series I1 fractions were obtained by a singlefractionation on a porous silica column, and so should be similar in character to seriesI fractions, but more heterogeneous. Series 111 fractions were obtained by successiveprecipitation from a strong alkali solution, (cf. experimental section). In this method,the concentration of potassium hydroxide is increased in successive incrementscausing precipitation of samples with successively lower molecular weights.Analysisof the fractions on an analytical scale porous silica column showed that this single-stage fractionation did not produce fractions as narrowly dispersed in molecular sizeas the separation on the porous silica column. Fraction 111-1 was nearly as polyG. C. BERRY 133dispersed as the starting material, with succeeding fractions having a polydispersitysomewhat greater than that observed for series I1 fractions.The difference between separation techniques based on chain dimensions or onchain length may account for the discrepancies observed in the correlation of [y] withM. Although the molecular weights are high for condensation polymers, owingpartly to the large value of the mol weight per repeat unit, the average number ofbonds N about which rotational isomerization can occur is modest compared tonormal values for aliphatic chains of comparable molecular weight.Thus, N is only250 for M about lo5 for BBB, whereas a polystyrene chain of this molecular weightwould have nearly 2,000 bonds about which rotation can occur. Three of the sixpossible isomers of the BBB repeat unit would lead to chain conformers with a largeradius of gyration, and could be expected to augment the value of ( s ~ ) ~ / M for a chainif present.2 Estimate of the dimension for a chain with free rotation about thesingle bonds interconnecting the repeat units is not possible without some knowledgeof the distribution of the six isomers of the repeat unit present. It is suggested,however, that the effects of isomer content are large enough, and the chain lengthsshort enough that a fractionation based on (s2) can produce samples with a largervalue of (s2>/M at a given molecular weight than a fractionation based on chainlength through solubility.This fractionation effect may be the reason that the highest series I11 fraction has anappreciable lower value of [q] than the series I and I1 fractions (note that [q] usuallyreflects (s2)).The data on (s2)Ls against Mw do not support this conjecture directlyin that (s2),,/Mw appears to be independent of the fractionation method. However,(S2>LS, given byis proportional to M, if (s2),/M, is a constant. Thus, polydispersity corrections willeffectively lower the value of (s2),,/Mw for fractions I11 relative to the values forfractions I and 11, yielding the expected relationship between values of (s2)/M forsamples with the same M.It is not suggested that any of the specimens deviate markedly from a coiledconformation in dilute solution.The data given in fig. 3 show that ( s ~ } , ~ increases alittle less rapidly than M,. Certainly, (s2),, is not proportional to M: as would berequired for a rod-like conformer. The large value of (s2),,/Mw of -2.5 x(cm2) indicated by the data in fig. 3 is compatible with the long length of the rigidrepeat unit and the large bond angles of some of the repeat unit isomers2Use of the Fox-Flory relation :for non-draining coils, together with the experimental values of <s2)Ls and M, yieldscalculated values of [q] that are too large by an order of magnitude.Thus, forM,,, = lo5, <s2),, is about 2.5 x cm2, so that [r] is predicted to be 49 dl/g,compared to an observed value of 4 dl/g. A polydispersity correction would tend toreduce this discrepancy, but only by a factor of 2 at most.Alternatively, for a completely free-draining chain, the relation derived by Debyegiveswhere 5" is the friction factor per hydrodynamic unit of molecular weight m,, qo is thesolvent viscosity and N is the Avogadro number. Estimates of 5" with experimenta134 SOLUTIONS OF A HETEROCYCLIC POLYMERvalues of [q], ( s ~ } ~ and qo, with m, = 410 corresponding to one complete repeat unitof BBB, yields the value (I" = 7.35 x dyne s cm-l. Estimate of the Stokesdiameter 0, with (I" and the Stokes relation leads to an implausibly small value of0.7A for 9,.Application of a molecular weight heterogeneity correction to (s2)Lswill increase [" (or D,) approximately by the ratio MJM,, so D, would still be improb-ably small even with this correction. Small values of 0, are not without precedent l 5in studies of the transport properties of ions in hydrogen bonding solvents, and areoften attributed to a " solvent structure " breaking role of the solute. Whatever theexplanation of the small value of D,, it appears that the values observed for [q], M,,and are more nearly consistent with the freely-draining model for viscous flowthan with the more usually adopted non-draining model ; and this is compatible withthe large values of (s2)/M observed for BBB.The analysis of the temperature, molecular weight and concentration dependenceof the viscosity of concentrated solutions given above suggests that the usual relationsgiven by eqn (13) can describe the data.The datum for sample 111-1 lies significantlybelow the line described by the data on series I1 fractions in fig. 8a. The discrepancyis such that about the same reduction in (s') is implied by the correlations of both [y]and q with M,. The data for sample 111-1 and the unfractionated sample 51165,which are about equally polydisperse, follow the same correlation, reinforcing theassertion that the distribution of species is the important factor in the observedbehaviour.In an effort to account for the dependence of (s2) on the structural factor F, thevalues of yc-1 are shown as a function of [q] in fig.8b. It is assumed that [y] isproportional to (s2), as would be true for a freely draining chain, so that in terms of[q], the viscosity q is given byInspection of fig. 8b reveals that the data are more nearly reduced to a singlefunction than when plotted against w2Mw, with a change in a occurring for (w2[y]),equal to 0.035 5. The data on the unfractionated sample 51 165 tend to fall below theline for the series I1 fractions, and the viscosity of samples of the unfractionated 4-22polymer with w2 = 0.139 falls well below the line with slope 3.4, but on the extensionof the line with slope 1.0. Conversion of [q] to M shows that the change of slopefrom 1.0 to 3.4 occurs for (M4'), = 230, or for a value of X equal to 12x 1O-l'compared to the more usual value of 400x 10-17.3 Usually, (M&)= is independentof 42,3 so that this result would normally predict entanglement for BBB of any chainlength in the bulk (&- = 1).The very low value of (Mq5JC is here interpreted to mean that the slippage ofentanglement junctions is very low.According to the entanglement model of Bueche,the parameter X, (or M,) should be universally proportional to a number f ( s ) thatincreases rapidly with decreasing slippage at entanglement junctions as expressed bythe parameter s that varies between the limits of zero for a couple resulting in noconstraint on the molecular motion and unity for a permanent c ~ u p l e .~ In terms ofthis model, s is typically 3 for many polymers, whereas the low value of X, observedfor BBB solutions would require s to be about 3.The large value of s, or equivalently, the small value of X,, suggests an interchaininteraction at the entanglement couple that is less easily relieved than is usual. Thedeviation of the data on sample 4-22 with w2 = 0.139, and the tendency of the data onthe unfractionated 51 165 sample to lie below the correlation for the fractions suggestthat the presence of a substantial percentage of very low molecular weight species, sucG. C. BERRY 135as should be found in an unfractionated condensation polymer, can effect a decreasein s, or an increase in X,, unlike the behaviour usually 0b~erved.l~The steeply increasing values of5 with increasing w2 shown in fig.7 indicate thatthe segmental mobility diminishes rapidly with increasing w2. The properties ofsolutions with w2 in excess of 0.20 will receive further comment below, but it is alreadyevident in fig. 7 that segmental mobility would be limited at such concentrations(independent of the molecular weight). A rapid increase of 5 with increasing con-centration is usually attributed to the proximity of the experimental temperatures tothe glass temperature T, of the undiluted p~lyrner,~ which itself increases with in-creasing 42. Thus, the increase of [ with increasing w2 is correlated with an increasingvalue of To such that Tg-To is about constant. This interpretation would assign avery high value of Tg to the polymer, since that of the MSA solvent must be low (of theorder 100 K).We believe that this is not the best interpretation of the data in thiscase, however, but rather that the segmental mobility decreases markedly with in-creasing w2 owing to an increased number of tenacious intermolecular interactions,leading ultimately to a network polymer formed by secondary bonds for solutionsmore concentrated than those discussed thus far. The low value of the parameter Xcis consistent with this suggestion.The presence of tenacious intermolecular interactions can also be deduced fromthe magnitude of 5 and the properties of solutions with w2 in excess of 0.20. Thecorrelation shown in fig. 8 together with eqn (13)-(15) can be used to compute thevalue of the segmental friction factor for concentrated solutions of BBB with the resultthat for T = 298 K and w2 = 0.025, 5 = 1.35 x dyn s cm-'. The value of locan be computed with the values of a and To in fig.5 with the result C0 = 2.04 x 10-13dyn s cm-l, which is not atypical of values found for other polymer^.^ The rapidincrease of f with w2 brings 5 to a value of 2.35 x dyn s cm-l for w2 = 0.20.Even though this (extrapolated) value of [ is much smaller than the value of 5 of theorder lo2 dyn s crn-l, or greater, typically observed for polymers at the glass temp-erature, attempts to prepare homogeneous solutions of BBB with w2 of 0.20 or greaterby direct mixing of solvent and polymer have not been successful. The preparationsremain heterogeneous even after weeks of slow agitation at 70°C.More concentratedsolutions have been prepared by evaporation of MSA from solutions prepared origin-ally with w2 ca. 0.10. Viscoelastic measurements have not been made on thesesolutions, but they are not simply more viscous fluids than solutions with lower w2 ;they have more the properties of a network polymer than of a fluid.Complete evaporation of MSA yields a film that has a golden hue. Small angleX-ray scattering on such a film reveals a scattering intensity i such that log i decreasescontinuously with increasing sin20/2. The slope of this plot would correspond toscattering species of the order lOOA in dimension, and larger. Wide angle X-rayscattering studies reveal that the diffraction band at a Bragg spacing of 3.5-3.6A ispreferentially oriented parallel to the plane of the film, but otherwise unoriented,whereas a spacing at 7 A is oriented perpendicular to the plane of the film, but other-wise unoriented.These same bands are seen without orientation in BBB samplesprepared by coagulation from acid solution.2 Diffraction studies on drawn fibres l 3kindly supplied by Dr. A. J. Rosenthal, Celanese Research Company, Summit, NewJersey, show the 3.5-3.6 A and 7 A bands strongly oriented perpendicular to the fibreaxis.These observations suggest that there is an increasing amount of supramolecularstructure in solutions of BBB with increasing polymer concentration. The 3.5-3.6 ABragg spacing has been identified with the interplanar separation of the (nearly)planar BBB repeat units in a direction normal to the molecular plane.2 It is suggeste136 SOLUTIONS OF A HETEROCYCLIC POLYMERthat regions of ordered chains packed with their repeat units so aligned with the mol-ecular pIanes parallel are formed as the solution is concentrated by the loss of MSAsolvent. These regions, which may be crudely slab-shaped, then align with each otherwith the surface of the slab parallel to that of the film as MSA is completely removed,giving rise to the mutually perpendicular orientation of the 3.5 and 7Essentially, the same type of supramolecular structure can be expected in BBBsamples obtained by coagulation from acid solutions.The small angle X-rayscattering studies on such a polymer show the same type of scattering curve as thatfound for the evaporated film, although less intense.The formation of films ofcoagulated BBB by rolling,2 although the polymer will not flow under moderate shearstress, may involve the orientation of such supramolecular units. The unusuallytenacious entanglement couple observed for BBB solutions may also be a rnanifesta-tion of the tendency of the repeat units to align with each other. Studies of possibleyield stresses and other rheological properties of these solutions are now under way toassess this possibility.The model of BBB discussed suggests that the intractibility of BBB should not beviewed as the result of an unusually high glass temperature (in excess of 6OO0C), butrather as the result of extensive crosslinking by secondary bonds. It seems probablethat similar considerations should apply to other heterocyclic polymers. In some ofthese, hydrogen bonding can be expected as well as the interplanar stacking proposedhere for BBB.spacing.The assistance of S. M. Liwak and J. S . Burke in carrying out some of the experi-mental work is acknowledged. This study has been supported in part by the AirForce Materials Laboratory, Wright-Patterson Air Force Base, under Contract no.F33615-69-C-2168.R. L. Van Deusen, J. Polymer Sci. B, 1966, 4, 21 1.G. C. Berry and S.-P. Yen, in Symp. Polymerization and Polycondensation Processes, ed. byR. F. Gould, Adv. Chem. Series (Amer. Chem. Soc., 1970), no. 91, p. 734.G. C. Berry and T. G Fox, Adv. Polymer Sci., 1967, 5, 261.R. J. Gillespie and E. A. Robinson, in Non-Aqueous Solvent Systems, ed. T. C. Waddington(Academic Press, New York, 1965), chap. 4.W. J. Taylor and F. D. Rossini, J. Res. Nat. Bur. Stand., 1944, 32, 197.’ A. R. Glascow, N. C. Krowskop and F. D. Rossini, Anal. Chern., 1950,22, 1521.’ G. C. Berry and P. R. Eisaman, to be published.* G. C. Berry, J. Chem. Phys., 1967, 46, 1338.T. G Fox, S. Gratch and S. Loshaek, in Rheology, ed. F. R. Eirich (Academic Press, NewYork, 1965), vol. 1.K. Hofmann, Imidazole and its Derivatives (Interscience Publishers, New York, 1953).lo S. J. Bass, R. J. Gillespie and E. A. Robinson, J. Chem. Soc., 1960, 714.l 2 E. A. Robinson and S. A. A. Quadri, Can. J. Chem., 1967, 45,2391.l 3 G. C. Berry and T. G Fox, J . Macromol. Sci.-Chem. A , 1969, 3, 1125.l4 A. C. van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, New York, 1957),l 5 R. L. Kay and D. F. Evans, J. Phys. Chem., 1966,70,2325.l7 V. R. Allen and T. G Fox, J. Chem. Phys., 1964, 41, 337.Is W. H. Gloor, Polymer Preprints, 1968, 9, 1174.p. 85 ff.F. Bueche, Physical Chemistry ofPolymers (Interscience Publishers, New York, 1962), chap. 3

ISSN:0366-9033    

DOI:10.1039/DF9704900121

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Small Angle Light Scattering by Ordered Polymer SolutionsBY V. G. BARANOVU. S.S. R., LeningradInstitute of Macromolecular Compounds of the Academy of SciencesReceived 2nd January, 1970The niethod of sniall angle scattering of polarized light permits one to investigate fast processesof ordering of polymer solutions in the course of crystallization, and in particular, to study the kineticsof spherulite growth. The registration velocity of the scattering patterns was not lower than 2photos s-l, the exposure did not exceed 3 s, and the range of the size registered was from a fractionof a micron to several microns. This method was used for studying the temperature dependenceof the spherulite growth rate during isothermal crystallization of polypropylene solutions in decalinin the absence of the " kinetic memory " effect.The growth rate followed fairly closely the generalthermodynamic relationships. Crystallization of the polypropylene solution in decalin carried outwith stretching of the jet leads to the growth of spherulites flattened with respect to the stretchingdirection. When the take-up velocity grows, the flattening of the spherulites increases and theirradius decreases. A general formal explanation is given of the phenomenon of the flattening of thespherulites in the course of their growth in a mechanical field and of the peculiarities of the super-molecular order formation during molecular orientation.Investigation of ordering of polymer solutions during their isothermal or non-isothermal crystallization is of interest for at least two reasons.First, one mayobtain definite information on the structure and properties of the solutions themselvesand, when the concentration dependencies are studied, it provides information onthe melts or amorphous polymers. The latter problem is of fundamental importanceand is far from being solved. Secondly, morphological investigations of singlepolymer crystals obtained from solutions enable us to understand more profoundlythe structure and properties of the bulk polymer. Probably? it is no coincidencethat the scientific study of the structure of bulk crystallized polymers began with thediscovery and analysis of lamellar monocrystals of polymers grown from very diluteso1utions.l By increasing the polymer concentration it is possible to pass throughthe entire spectrum of the polymer morphology from single crystals to spherulite.Moreover, the solutions permit one to vary over a wide range the ordering of theprecrystalline phase by means of agitation, heating and change in concentration whilethe melt, in principle, probably cannot be greatly disordered since the energy ofdisordering should be comparable with the energy of breaking of a chemical bond.The usual methods of investigating the polymer morophology (X-ray diffractioninvestigations, electron and optical microscopy) are not always applicable to solutions,especially for kinetic and dynamic experiments involving fast processes.Con-sequently, our laboratory has developed a modified technique of small angle scatteringof polarized light specially applicable for kinetic studies.2EXPERIMENTALIn contrast t o the usual methods of polarized light scattering 3* we have used a broad(10-20 mm) incident light beam.This enabled us to attain a great increase in the scatteringvolume without changing the celI thickness (0.5-1 .O mm) and a corresponding decrease in13138 SMALL ANGLE LIGHT SCATTERINGthe exposure. This modification is described in detail in ref. (2), ( 5 ) and here we note onlythat it enabled us to register the scattering patterns at a velocity of not less than 2 photoss-I while the exposure did not exceed 3 s.The kinetics of the spherulite growth in solutions of polypropylene in decalin were studiedas follows. The solution in a sealed rectangular cell was placed in an air thermostat at atemperature T,>T;.After maintaining it at Tl for a certain time it was placed in an oilthermostat at a temperature T2tT;. The windows in the second thermostat permittedone to begin the registration of scattering patterns immediately. The precision of thermo-statting in the first thermostat was k2"C and in the second one it was fO.l"C.Besides carrying out experiments on the effect of Tl and T2 upon the kinetics of thespherulite growth, we have investigated the processes of ordering in solutions of polypropylenein decalin under non-isothermal conditions with simultaneous cooling and stretching of thesolution jet. The scheme of these experiments is given in fig. 1. In those cases when itwas impossible to observe the scattering patterns directly in the jet, the scattering patternof the " as-spun " filament was registered according to the method of V01kov.~-v2 i13 2FIG.1.-Registration scheme of light scattering by stretched jet. 1, hot solution; 2, take-upapparatus ; 3, laser with polarizer ; 4, analyzer and registering apparatus.We used the isotactic polypropylene manufactured in Czechoslovakia, Brno, with adegree of isotacticity greater than 97 % and a viscosity-average mol. wt cu. 100 000; thedecalin was of C.P. grade.RESULTSThe following change in scattering pattern occurs during isothermal crystallizationfrom solution : the distance between discrete reflexes is reduced and their intensityincreases (fig.2). The position of the maximum of the scattering intensity 0, permitsone to find the spherulite radius R from the equationwhere A' is the wavelength of incident light in the medium. Thus, it is possible tofind the change in R with time (fig. 3) and to determine the rate G of the spherulitegrowth at the given temperature T,.R = 2A'(7~ sin en,)-1, (1V. G . BARANOV 139t 6)FIG. 3.-Change in the spherulite radius with time. TI = 155"C, T2 = 25"C, C = 10 %.For a preceding study in our lab~ratory,~ we have reported some results on thedependence of G on T I , T2 and some other parameters. The most interesting resultwas the so-called effect of " kinetic memory ". When the solution was heated toTI = const, which only slightly exceeded T,, and the melting-crystallization cyclesat T2 = const were repeated, the spherulite growth rate increases appreciably withthe growth in the number of cycles.Melting of a sample at TI which greatly exceedT, does not produce this effect. Since we were interested in the dependence of Gon T2, the results in fig. 4 were obtained under conditions that excluded the effectof kinetic memory.When the crystallization in solution occurs during stretching, the small anglescattering pattern of polarized light has the form shown in fig. 5. Jt is possible toobtain from this figure both the mean radius of the spherulite from eqn (I) and thedeviation of its shape from the spherical one from the equationit, = tan !in#,43nr( bE,"2Z JX uI000000120 40 60T2 "CFIG.4.-G as a function of Tz , TI = 15S0C, C = 10 %140 SMALL ANGLE LIGHT SCATTERINGwhere A1 is the ratio of the maximum size of the spherulite to its minimum size andpm is the azimuthal direction of maximum scattering in the plane of the pattern. Fig.5 shows that the largest spherulite size is normal to the direction of the jet axis.Fig. 6 gives R and R1 as a function of the take-up velocity V2. (We could not measuredirectly the stresses occurring during stretching but according to some data thestresses are proportional to V2.) In all the cases under investigation, R decreasesand A1 increases with increasing V2.86nW 244201,YI I I 1 I? 4 A 8 !r3FIG. 6.-R and hl as functions of the take-up velocity.v 2 x 10 (rn s-1)DISCUSSIONIt is usually assumed lo that the secondary nucleation on the surface of the growingspherulite is the elementary act of the spherulite growth and the molecular chains ofthe new crystals are as a rule l1 tangential (or almost tangential) to the spheruliteradius. The spherulite growth rate is determined by the following general equation l2G = N exp (- E,,/RT- AF*/RT), (3)where N is either constant or inversely proportional to T ; ED is the energy of chaintransfer to the crystallizing surface and AF* is the free energy of the formation of anucleus having a critical size and R is the gas constant.Usually ED is identifiedwith the activation energy of viscous flow and the relationships of ED with tempera-ture may be expressed by the equation of Williams-Landell-Ferry.13 In this workwe assume that the contribution of the first term of the exponent is negligible. Whenthe nucleation of a cylindrical nuclei is three-dirnensional,l2AF* = 8no,pi/(A fu)2, (44AF* = 4a,ou/Afu, (4b)and when it is two-dimensional,where 0, and a; are the free energies of the end and the lateral surfaces, respectively,and AL is the free energy of fusion per monomer unit ; it can be written asAh = AH,-TAS,,.( 5 V. G. BARANOV 141Usually it is assumed l 2 that AH, and ASu are equal to their equilibrium values,hence(6)According to the above approximations, during crystallization of the unperturbedstate, log G should be linearly dependent on Tm/T(AT), or on T i / T ( A T ) 2 accordingto the type of nucleation.12 As a rule the inadequate precision of the experimentaldata l4 does not permit one to distinguish between these two cases; however, thedependence of log G on T,/T(AT) and T,2/(AQ2T is nearly always linear.14 Fig.7shows that for crystallization from solution both dependences are also practicallyA h = AHu(Tm-T)/Tm = AH,AT/T,.linear.- 6- 8(TmITAT)X lo2i 2 3 4I I I I[T2/T(AT)2] x 10FIG.7.-Log G as a function of T:/T(AT)” and TmITAT for data in fig. 4.As already mentioned, the elementary act of the spherulite growth is the foldingof the macromolecular chain. When the macromolecular chain is oriented, itsfolding and crystallization in the direction of orientation are favourable for the freeenergy of the formation of a nucleus of critical size.In contrast, its folding andcrystallization in the direction perpendicular to the orientation require additionalenergy AFII. Hence, the growth rates in these two directions will be different andmay be formally written asGI = Nexp [-ED/RT-(AF*-AFl)/RT], (74GI] = N exp [ -ED/RT-(AF* + AFII)/RT]. (74In A1 = (AF, +AFII)/RTConsequently, according to the scheme in fig. 8, In A1 under the condition that thegrowth in linear (Gl/GllzRl/Rll = &) is given byAt present, the physical meaning of the values AFl and AF,, is not sufficiently clear,but most probably they are associated with the changes in the entropy of the solutionwhen a mechanical field is present, and both these values grow with increase in theelongation of the jet.Thus, the growth rate of the spherulite or of its relic formincreases infinitely in the direction normal to the direction of jet stretching, while inthe direction of the stretching it drops to values negligible in comparison with the(8142 SMALL ANGLE LIGHT SCATTERINGfirst rate. In fact, crystalline aggregations obtained by crystallization of the solutionin the gap between two rotating coaxial cylinders have the shape of flat lamellaethreaded on a single filament.15 The largest size of the lamellae is normal to thedirection of the force stretching the macromolecules. These structures were calledthe shish-kebab structures.16. The above considerations may provide the mostlikely explanation of the appearance of these structures which should be consideredas degenerate forms of the spherulite morphology under the condition GI, / < GI.tpFIG.S.-Schenie of the spherulite flattening mechanism.One other process of crystallization in the polymer systems undergoing stretchingis possible. This is the orientational crystallization in the course of which the mole-cules become completely or partially extended in the stretching direction and arecrystallized without folding. Experimentally this morphology was not observed forsolutions ; nevertheless there are some indications of its possible existence. l’.0P (N m-2)FIG. 9.-Relationship of morphology and stress during crystallization and stretchingV. G . BARANOV 143Thus, the general picture of the effect of stretching on the morphology and crystal-lization kinetics of the polymer systems may be shown in the scheme in fig.9. Inthis figure the ordinate gives the parameter connected with the overall crystallizationkinetics and the abscissa gives the stresses developing during stretching. The firstzone corresponds to the growthof flattened spherulites, the second one to the structuresof the shislz-kebab type, and the third one to purely orientational crystallization.In addition to a change in the overall rate of crystallization, onemayexpect changesin the geometry and in the type of nucleation. At present, this suggestion is confirmedonly by the experiments of Kim and Mande1kern.l' They have found that a changein the elongation of natural rubber undergoing crystallization changes the Avramiconstant from ca.4 to 1 ; i.e., the nucleation becomes unidimensional and not three-dimensional.The author is indebted to Prof. S . Ya. Frenkel for his valuable suggestions anddiscussions and to Mr. A. V. Kenarov for the preparation and testing of samples ofthe polypropylene solutions.' P. H. Geil, Polymer Single Crystals (Wiley & Sons, 1963), chap. 2.A. V. Kenarov, V. G. Baranov and S. Ya. Frenkel, Vysokomolekul. Soedin. A, 1969, 11, 1725.M. B. Rhodes and R. S. Stein, J. Polymer. Sci., 1965,45,531.S . Ya. Frenkel, V. G. Baranov and T. I. Volkov, J. Polymer Sci. C, 1967, 16,1655.V. G. Baranov, A. V. Kenarov and T. I. Volkov, J. Polymer Sci., in press.T . I. Volkov, Vysokomolekul. Soedin. A, 1967, 9,2734. ' T. I. Volkov, G. S. Farshyan, V. G. Baranov and S. Ya. Frenkel, Vysokomolekul. Soedin. A,1969, 11,108.K. A. Gasparyan, Ya. Holoubek and V. G. Baranov, Vysokomolekul. Soedin. A, 1968,10,86.A. Ziabicki, Man-Made Fibre, ed. H. F. Mark, S. M. Atlas and E. Cemia (Intersc. Publ.,Wiley & Sons, 1965), vol. 1, pp. 13-94.l o W. B. Barnes, W. G. Luetzel and F. P. Price, J. Phys. Chem., 1962, 65,1742.A. Keller, J. Polymer Sci., 1955, 17,291.L. Mandelkern, N. L. Jain and H. Kim, J. Polyiner Sci., 1968, A2-6,165.l3 J. H. Magill, J. Polymer Sci., 1967, A2-5,89.l4 L. Mandelkern, Crystallization of Polyiners (McGraw-Hill, 1964).l6 R. B. Williamson and W. F. Busse, J. Appl. Phys., 1967, 38,4187.A. J. Pennings and A. M. Kiel, Kolloid-Z., 1965, 205, 160.S. Ya. Frenkel, V. G. Baranov, N. G. Belnikevitch and Yu. N. Panov, Vysokomdekul. Soedin.,1964,6,1917.H. Kim and L. Mandelkern, J. Polymer Sci., 1968, A2-6, 181

ISSN:0366-9033    

DOI:10.1039/DF9704900137

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Liquid-Liquid Equilibria in Multicomponent Polymer S ys ternsBY R. KONINGSVELDCentraal Laboratorium, DSM, Geleen, The Netherlands.Received 19th January, 1970The effect of polydispersity on liquid-liquid phase relationships in equilibrium is reviewed andfound to be similar for all the kinds of systems examined. As a rule, it is not allowed to identify acloud-point curve with a binodal (coexistence curve), nor the maximum in a cloud-point curve with acritical point. In any kind of system (solvent-polymer ; polymer 1-polymer 2 ; solvent-non-solvent-polymer ; solvent-polymer 1-polymer 2) measurement of the consolute state opens the way towardsaccurate determination of interaction parameters. Use of the critical opalescence for the latterpurpose is subject to complications.Liquid-liquid phase relationships in systems containing macromolecules are influ-enced by the polydispersity existing in virtually all polymers.It is the object of thepresent paper to review this effect in various types of systems. We shall first considerquasi-binary mixtures which consist of a single low-molecular-weight solvent and apolydisperse polymer. By checking the effect of an increase of the single-solventchain length up to values representative of (monodisperse) polymers we considerpolymer compatibility. Quasi-ternary mixtures represent a type of systems frequentlyencountered in polymer literature. They may either consist of a solvent, a non-solventand a polydisperse polymer, or contain a single solvent and two polymers differingin chemical structure.More complicated systems are conceivable, but we shallrestrict ourselves here to those mentioned above. It will appear that phase relation-ships in all types of systems are affected in a similar way by the polydispersity of themacromolecular constituents.When trying to derive values of molecular parameters from measured phaserelationships, the polydispersity of the samples must not be ignored since otherwiseerrors amounting to orders of magnitude may be introduced. Such quantitativetreatments are based on a free enthalpy (Gibbs free energy) of mixing function arisingfrom a statistical-mechanical treatment. In this paper we shall use the expressionderived by Flory 1* and Huggins 3-5 who employed the lattice model for the liquidstate.Flory,’ and Patterson.8 The original Flory-Huggins expression, however, has theadvantage of being much simpler than the functions arising from the later treatments.In addition, it provides a correct qualitative description of thermodynamic propertiesof macromolecular systems and can easily be made quantitative along empiricalTompa 9* l1 and Scott l2 formulated the Flory-Huggins expression for ternaryMore refined theories have been advanced; we mention those of Hugginslines.2, 5 .9, 10systems. Accounting for polydispersity,AGIRT = 4os; In 4o +x4irn; In 4i + c ~ + + ~ s i In $ j +where AG denotes the free enthalpy of mixing per mol of lattice sites ; R and T have144i j901404+90240$+g124J/, (1R. KONINGSVELD 1 45the usual meaning; #o, 4 and $ are the volume fractions of solvent and the wholepolymers 1 and 2, respectively ; 4, and $J those of polymer species i (in polymer 1) andj (in polymer 2); gol, gO2 and g12 represent free-enthalpy correction parametersaccounting for the solvent-polymer 1, solvent-polymer 2 and polymer 1 -polymer 2interactions ; so, m, and sj stand for the chain lengths of solvent-, species i- and speciesj-molecules, expressed as the number of occupied lattice sites.We shall deal with two-liquid-phase equilibrium states only.At equilibrium,phase-relationships can be calculated by equating the chemical potential in the twophases for every component. The numerical calculation procedures have been orwill be p~blished.l~-~' Application of eqn (1) requires that assumptions be made forthe molecular-weight distributions of the polymers.Use has been made of the fami-liar logarithmic normal and exponential functions, here defined as weight distributions(2)(3)where W = w(M)dM = total mass of polymer, normalized to 1 g. B2 = 2 In 6 ;b = MJM, ; il = (2- b)/(b- 1) ; z = (A+ l)/Mn. M,,M, = the number- andweight-average molecular weights. Distribution functions deviating in shape fromthose given by eqn (2) and (3) can be constructed by adding two or more exponentialor logarithmic normal functions. l4The difference in composition between two phases may decrease if the conditionsare changed. Eventually, when the two phases have become identical, the consoluteor critical state is reached.Such changes of the conditions can be brought about bychanging T at constant pressure or vice versa, or by varying the overall compositionof the system at constant p and T. Critical points are located on the spinodal, whichis the locus of points on the AG(41,42 . . ., 4h1,t,h2 . . .) surface (at constant p and T )where the curvature of the latter is zero in one direction. According to Gibbs l8 thespinodal is defined by (constant p and T )w(M) :log normal: W ( M ) = w~M,~B-'~-+M exp [ - B ~ In2 ( b 1 * 5 ~ ~ n - 1 ) J ,exponential : w(M) = w?+2[r(L+ 2)I-l MA+l exp [-(A+ 1) M/M,I, s,where xi and x, stand for the independent composition variables, which are so chosenas to be represented by the volume fractions qbi and $j of all polymer species.For the critical point,18J,, = 0in which Jcr is the determinant formed from Jsp by replacing the elements in anyhorizontal line bya J S P l a 4 1 , a J s P l a 4 2 - * a J s p l a $ l , aJSPla$z * * 'The conditions (4) and (5) have been worked out for eqn (1) in which the g may bearbitrary functions of the overall composition of the mixture.lg The resulting equa-tions are highly involved and instead of writing them out we shall refer to some specialcases in the next sections.Strictly speaking, using eqn (1) we exclude any type of macromolecules other thanlinear homopolymers.However, in a general sense, the following considerations maybe expected to have qualitative validity for other molecular structures as far as theeffect of polydispersity in chain length is concerned.2146 LIQUID-LIQUID EQUILIBRIAI .QUASI-BINARY SYSTEMSIf we introduce so = 1 and + j = 0 (allj), eqn (1) describes mixtures of a singlelow-molecular-weight solvent and a polydisperse polymer. We first consider a poly-mer containing two components differing in chain length. Tompa 9 p l1 has compre-hensively dealt with such ternary mixtures ; his phase diagrams are schematicallysummarized in fig. 1, which shows the extension of a miscibility gap in the temperature-composition space. Consideration of such ternary systems has the advantage thatthe phase diagram is still visualizable, and yet brings out the essential differencesbetween truly binary and quasi-binary mixtures of the present type whatever thenumber of macromolecular components.T4PIFIG.1.-Miscibility gap in a ternary system containing two macromolecular homologues P1 and P2and a single small-molecule solvent S. The chain lengths of P1 and P2 (mi and mz) are differentm2 > m AA2C5B, quasi-binary section (cloud-point curve) ; A2, precipitation threshold ; CC5C',locus of critical points ; DC5E, quasi-binary section of the spinodal surface (- - -).In the binary lateral face TSP2 we have the usual situation with the critical point Cat the maximum of the coexistence curve (binodal). Spinodal and binodal have acommon tangent at C. The binodal can be measured by changing the temperatureof a homogeneous system of known concentration and noting the temperature ofincipient phase separation (cloud point).If such a measurement is carried out on anon-binary polymer solution. e.g., on mixture X dissolved in S, one obtains the cloud-point czirve AA2C5B. The orientation of the tie lines (A2B2 ; AK ; Q'Q" ; LB)discloses that such a cloud-point curve should not be identified with a coexistencecurve. Quite generally, it can be shown 21 that the critical point (C,) in this kind ofquasi-binary system is shifted towards a concentration value higher than that of thR. KONINGSVELD 147maximum A2. Tompa 11* 22 proposed to denote this maximum by the termprecipita-tion threshold. The quasi-binary section DC5E of the spinodal surface has a commontangent with the cloud-point curve at the critical point.When plotting phase relationships in a quasi-binary system in a planar diagramthe graph paper in fact coincides with TSX.The coexisting-phase compositions (i.e.,Qr and Q”) lie outside TSX, but can be projected on this plane. On doing so fordifferent temperatures, one obtains coexistence curves whose location depends on theoverall polymer concentration.l** 23 Fig. 1 reveals a narrowing of the projectedcoexistence region upon an increase of the overall concentration. The phases B2 andK are on the verge of appearing and coexist with cloud-points at sub-critical concen-trations. Above the critical concentration we have L coexisting in the same way withB. Projecting these points onto TSX, we obtain the shadow curve, which is thecoexistence curve relating to the cloud-point curve. Shadow and cloud-point curvesintersect in the critical point, which also lies on the coexistence curve for an overallconcentration equal to the critical.For all other concentrations the coexistencecurves consist of two detached branches, representing the dilute and concentratedphases. If the whole-polymer volume fraction 4 is smaller than its critical value (bc,the dilute-phase branch ends in the cloud point at # and the concentrated-phasebranch in the shadow concentration. The reverse situation exists if +> 4=.These observations are quite general and independent of the number of compo-nents. Two-dimensional phase diagrams can be calculated for arbitrary molecular-weight distributions; fig. 2 gives some examples. Evidently, the shape of thediagram is governed by the polymer composition.An increase of a(= MJM,) isaccompanied by a distortion of the diagram (Mz = the z-average molecular weight ;The preceding theoretical considerations have found experimental verification,e.g., in the extensive work reported by Rehage et al. on the system polystyrene+cycl~hexane.~~~ 25 Fig. 3 is based upon their data.b = M,/M,,).05205148 LIQUID-LIQUID EQUILIBRIA0.53r( G055I005 0.1 0 0.1 54(4FIG. 2.Two-dimensional phase diagrams for quasi-binary systems. Cloud-point curves : - ;shadow curves, - - - ; coexistence curves, - - - ; (overall volume fractions 4 are indicated) ; spin-odals, -; critical points, 0 ; so = 1 ; gol independent of +; m/Mm (a) Whole polymerdistribution : sum of two exponentialfunctions, Mw = 1 3 1 .7 ~ lo3 ; b(= Mw/Mn) = 5 ; a(= Mz/Mw)= 2 ; 4c = 0.039 308 ; (b) whole polymer distribution : sum of two exponential functions, Mw =131.7 x lo3 ; b = 5 ; a = 7 ; cjC = 0.071 104 ; (c) whole polymer distributions : Mw = 131.7 x lo3 ;1, monodisperse (b = a = 1) ; 2, logarithmic normal (b = a = 2) ; 3, logarithmic normal (b = a =10)R . KONINGSVELD 149I.FIG. 3.-Experimental two-dimensional phase diagram for a polystyrene-cyclohexane system.Experimental data of Rehage et aLZ49 25 Cloud-point curve, -; shadow curve, - - - ; co-existence curves, - for 2 (0) ; 6( a), 10( A), 15 (V) and 20 (a) % whole polymer concentration(wt- %) ; critical point, determined with the phasevolume-ratio method, @. Characteristics ofpolystyrene sample, Mn = 2.1 x lo5 ; Mw = 3.46 x lo5 ; Mz = 5.5 x lo5 ; b = 1.6 ; a = 1.6.The spinodals and critical points in fig.2 were calculated with the equationsobtained by applying eqn (4) and (5) to eqn (l), subject to the conditions mentionedin the first paragraph of this section. The resulting equations are :I4CRITICAL POINT :6 ( a Y o l / a 4 ) c - 3(1 - 2 4 c ) ( a 2 g 0 1 1 a 4 2 ) c - 4 0 , c ~ c ( a 3 ~ o l l a 4 3 ~ c =so1(l-~,)-2--mz(4cm,)-2. (7)If gol is independent of 4, and so = 1, eqn (6) and (7) reduce to the well-knownexpressions previously derived by Stockmayer.26Eqn (6) reveals that distributions differing on any point except on weight-average,will share the same spinodal in a quasi-binary plot. This holds for the examples infigure 2 which show different cloud-point curves, touching the same spinodal atdifferent critical points (if M, differs, see eqn (7)).1411.COMPATIBILITY OF POLYMERSAn increase of so, the chain length of the solvent, broadens the miscibility gap andmoves it towards lower values of gol. At high values of so, a small positive value ofthe interaction parameter gol suffices to bring about incompatibility. Fig. 4 illus-trates this feature. In practice, the long-chain solvent, in fact, a polymer, also has amolecular-weight distribution. Cloud-point curves in mixtures of polymers, as faras they are measurable, may be expected to have widely differing shapes accordin150 LIQUID-LIQUID EQUILIBRIAas the distributions of the two constituents vary. That this must be so is indicated byfig. 5, which shows some spinodals calculated with the appropriate function.19(SPMOD AL)%I2 - 2(1--2+)%Z/a4 - o$a2g12/a42 = ($sw)-l + ( o m V Y 9 (8)0.60.060.1 R.KONINGSVELD 151J -- I I I T I0.2 0.4 0.6 0-84(4FIG. 4.-Calculated cloud-point (-) and shadow (- - -) curves for various values of the solventchain Iength SO. SpinodaIs, - ; criticaI points, 0 ; gol independent of 4. Molecular weightdistribution of constituent 2 (concentration 4) : sum of two exponential functions, Mw = 2 x lo5 ;b = 2 ; u = 2; m/M = lo-*. (a) SO = 1 ; (b) SO = 10 ; (c) SO = 100; (d) SO = 1000.The locations of the critical points were calculated with(CONSOLUTE STATE),where s, and s, are the weight- and z-average chain lengths of the ‘‘ solvent ” polymerand the interaction parameter is denoted by g12.The locations of spinodals and critical points indicate that the miscibility gap will,as a rule, be shifted towards the axis of the component with the shorter chains.If th152 LIQUID-LIQUID EQUILIBRIAopposite behaviour should be found, this must be attributed either to a large differ-ence between the a values of the two polymers or, perhaps more likely, to a concentra-tion dependence of g1 2 . Depending on the value of ag, Jaq5, the unstable region maybe found on either side of the diagram. From Fig. 6 substantializes this statement.00.10.2 40.30.400.10.20.30-4FIG. 5.-Spinodals calculated for various sets of sw/mw values, indicated in the figure. Drawncurves : g12 independent of 4; critical point for different uo/u ratios( 0); right-hand branch: 0.5;0.2 ; 0.1 ; 0.05 ; 0.02 ; 0.01 ; 0.001 (from left to right ; as far as they are indicated) ; left-hand branch,2 ; 5 ; 10 ; 20 ; 50 ; 100 ; 1000 (from right to left).The critical points 0 at the maxima refer toao/a = 1 ; uo and a stand for sz/sw and mz/mw. The spinodals - - - and - - - relate to a concentra-tion-dependent g12 (ag12/a+ = -0.1 and +0.1, respectively).this point of view one is inclined to attribute the unexpected locations of some cloud-point curves reported by Allen, Gee and Nicholson 27 to a strong concentration de-pendence of the interaction parameter in their system. They studied mixtures ofpolyisobutenes and silicones of relatively low molecular weight and found miscibilitygaps shifted towards the pure silicone axis.The silicones, however, had the longeraverage chain lengths of the two constituents.The maxima of the spinodals in fig. 5 are noteworthy. In truly binary mixturesthey coincide with the critical points. In the present quasi-binary ones the consolutestate may evidently also occur at the maximum of the spinodal, if the two polymershave special a values. If 9 1 2 does not depend on 4, these values must be equal. Ifg12 depends on 4 the two a values must either be equal to one (binary mixture) or berelated in a definite way given by the value of ag,,/a#. In this respect (critical pointat maximum of spinodal) a multicomponent system may behave as if it were binary.It remains to be ascertained whether this feature also exists in the other aspects inwhich polydispersity is revealedR.KONINGSVELD 1530005G0.1 00-150 0 40.0 6' 0 0 80.10g, = - 0.1I 1 1 1 1 10-2 0.4 0.64(40.4 06 0.8d(b)FIG. 6.-Calculated cloud-point (-) and shadow (----) curves and spinodals (-) for twoindicated values of agI2/8+( = g1 ; go is the concentration-independent term in gI2( d)). Long-chain solvent, so = 10 ; second constituent, polydisperse polymer (distribution : sum of two expo-nential functions, Mw = lo4, b = 5, a = 7). Critical points, 0.111. QUASI-TERNARY SYSTEMSQuasi-ternary systems come in two classes, viz., mixtures of a solvent, a non-solventand a polydisperse polymer, and mixtures of a solvent (good or poor) and two differentpolymers (mixtures of three polymers will be left out of consideration).Ternar154 LIQUID-LIQUID EQUILIBRIAdiagrams with monodisperse polymer components were calculated for either type ofsystem by Tompa 9 9 l1 and by Scott.12 We now discuss the effect of polydispersityon their diagrams. Calculations on this point have also been reported by Baysaland Stockmayer.28SOLVENT-NON-SOLVENT-POLYMERAddition of a second polymer component to a truly ternary mixture brings out theessential features in which polydispersity reveals itself. They are quite similar toquasi-binary systems. Since there are at least four components we must restrictourselves to an isothermal section; this is shown schematically in fig. 7.The twolateral faces NS-S-P1 and NS-S-P2 of the tetrahedron represent Tompa’s and Scott’sternary diagrams. The coexisting phases are located on a binodal surface within thetetrahedron and the tie lines will generally be so oriented as to fall outside a quasi-ternary section like ABC.SNSFIG. 7.-Miscibility gap in a quaternary system containing two macromolecular homologues P1 andPz, a solvent S and a non-solvent NS. The chain lengths of PI and P2 differ (mz >ml). ABC,cloud-point curve of polymer mixture X ; - 0 - - 0 -, critical line ; , coexisting-phase composi-tions relating to system A. Tie line, - - - -.Cloud-point curves in quasi-ternary systems can be measured by adding non-solvent to solutions of the polymer (say X) in solvent S and noting the amount yadded before phase separation sets in.Obviously, a plot of y against the wholepolymer concentration is identical to the quasi-ternary intersection NS-S-X of thebinodal surface. As a rule, the critical point will not coincide with the precipitationthreshold B at which y = yth. The coexisting-phase compositions can be projectedonto NS-S-X; this yields coexistence curves the location of which depends on thR . KONINGSVELD 155whole polymer concentration in a similar way to quasi-binary systems. Someexamples calculated by means of eqn (l), are collected in fig. 8 and 9. The extensionand location of the miscibility gap are determined primarily by the values of theinteraction parameters. The polydispersity of the polymer, however, reveals itself innon-negligible details, in the same way as it does in quasi-binary systems.0.1 AdFIG.8.-Coexistence curves in a quasi-ternary phase diagram calculated on the basis of eqn (1) for anexponential molecular weight distribution X (M, = 131.7 x lo3 ; b = 2 ; a = 1.5). Values of theinteraction parameters are indicated (gol~goz/glz). Coexistence 4 = 0.01, unless stated otherwise.curves, -, --- ; tie lines, -, - - -.SOLVENT-POLYMER ~ - P O L Y M E R 2Phase relations in solutions of two different polymers in a mutual solvent (good orpoor) may be very complicated. A quasi-ternary diagram (fig. 10) contains all thefeatures noted in $I and 11. In addition, however, the threefold interactions maygive rise to extra phenomena, not observed in the quasi-binary lateral faces.Werefrain from a complete description and only make some general qualitative remarks.A fuller treatment will be given in a forthcoming paper.19The two cloud-point curves SAIB1 and SA2B2 are intersections of the cloud-pointsurface within the prism with the lateral quasi-binary faces TSX1 and TSX2. Onthese faces, all features described in $1 are present. Hence, there must exist spinodaland shadow surfaces within the prism, while the coexisting phases can be projectedonto coexistence surfaces. Similar observations may be made on the third lateralface (4 11). Further, critical solution phenomena of the upper and lower consolutetypes may occur on either face. Consequently, phase behaviour of this type of quasi-ternary systems may vary considerably and be very complex156 LIQUID-LIQUID EQUILIBRIAFIG.9.-Calculated cloud-point (-),shadow (- - -) and coexistence (- - -)curves in a quasi-ternary system, go1 =0 ; gO2 = 0; g12 = 1. Polymer distri-butions: Mw = 1 3 1 . 7 ~ lo3; XI : expo-nential, b = 2, a = 1.5; X 2 : mono-disperse ; X3, logarithmic normal ;b = a = 10. Spinodals, - ; criticalpoints, 0. Some tie lines of the trulyternary systeni (A',) in fig. 9b areindicated (- - - - -).NS 0.05 0. I x4(bFIG. 1 1 .-Cloud-point surfaces in a quasi-ternary system composed of diphenylether, linear poly-ethylene (Mn = 7.9 x lo3 ; Mw = 7.6 x lo4) and crystallizable polypropylene ( M , = 5.2 x lo4 ;M~ = 6 . 4 ~ 105).To fucepage 157.R . KONINGSVELD 157T 1sFIG.10.-Quasi-ternary pliasc diagram for a solvent S and two polydisperse polymers XI and Xz.SAIBl and SA2B2, quasi-binary cloud-point curves ; 0 - - 0 -, critical line ; - , spinodals;- - -, shadow curves ; Al and Az, quasi-binary precipitation thresholds.Fig. 1 1 shows an experimental example referring to the system diphenylether-linearpolyethylene-crystallizable polypropylene. There are two miscibility gaps and these --_- _._-__ -I-.-- c,,,+t-- rpt, C-A - A ,. - - i - ~ : - . - i - . I - - - . --.--A- L-c:,, -r +t- --:---A C;UlIlt: VGly GlUbG 1 U ~ ~ l l I E ; l . lllt: IllSL, ill it 1t:lilLlVE;ly IUW C;UIlC;GlIl1illlUl~ Ul LUG IllliLGUpolymer, is the most obvious and arises from the cloud-point curves in the two quasi-binary systems solvent-polymer.Within the prism, however, this cloud-pointsurface, rises to a maximum (quasi-ternary precipitation threshold). Hence, attemperatures above the quasi-binary thresholds, quasi-ternary mixtures may still bepartially miscible. This effect may be due either to the molecular-weight distributionsof the polymers or, as pointed out by T ~ m p a , ~ to special values of the interactionparameters.The possible occurrence of such closed miscibility gaps jeopardizes the mutualsolvent method as a means for establishing polymer compatibility. Judged from thepresent example, phase separation of a solution of two polymers in a common solventdoes not necessarily involve that the polymers themselves are incompatible.The second miscibility gap in fig.11 has been found to be closed at higher temp-eratures (- 280°C). Whether it is also closed at higher concentrations of the mixedpolymer remains to be established. Its absence in the two quasi-binary solvent-polymer systems indicates that its origin must be sought in the polymer-polymerinteraction, whether or not in conjunction with the solvent.I V . DETERMINATION OF INTERACTION PARAMETERSThe examples presented so far indicate that measurable differences in the locationof miscibility gaps may accompany relatively small variations in the interactio158 LIQUID-LIQUID EQUILIBRIAparameters. Hence, measurement of phase relationships should conversely giveaccess to values of these parameters in a sensitive way. The experimental techniquesare simple ; with quasi-binary mixtures containing a low-molecular-weight solventthey have been applied freq~ently.~*~* O* 14* 9 * 2 0 p 2 3 * 29 Polymer mixtures presentproblems in that the viscosity of the systems soon becomes too high upon an increaseof the average molecular weights.Measurements on low-molecular-weight sampleshave been reported.27* 30 Quasi-ternary systems are generally well manageable, buttheir description involves considerable experimental effort. 9 2 9 9In all these studies the polydispersity of the samples must be taken into account ifthe conclusions are to have quantitative ~ignificance.~~ Accurate determination ofmolecular weight distributions, if possible at all, is a tedious Fortunately, thecritical point is a well-defined state in the phase diagram, being determined as it is bytwo average molecular weights only.Therefore, it provides the obvious key to theproblem.--I$ n-e.a Polystyrene -Cycl ohex drle(I:'- xic,'be0.2 00.180 16+ 0.10IN"10-8.I0.0 5 0.1 0 0.154FIG. 12.-Concentration dependence of the interaction parameter for two systems. The magnitudeof the estimated error in the ordinate is indicated. The abscissa in fig. 12a gives x*, the segment molfraction (Grundmolenbruch).As shown elsewhere, the consolute state in multicomponent systems can be ac-curately traced by determining the phase-volume ratio as a function of temperatureand overall polymer concentration. O* l4 These measurements, combined withcloud-point data, reveal the critical state in terms of 4c and T,, the critical volumefraction and temperature.Such data have, so far, been obtained only on quasi-binarysystems of which two examples are presented here.14* 34Having established $c and T, for a number of samples differing in M, and M,, wR . KONINGSVELD 159use eqn (7) to check the concentration dependence of the interaction parameter.Assuming the concentration dependence of Sol to have the form 9* 35SO1 = 90 +91+ f92#2 $- - - (10)the right-hand side of eqn (7), which contains measured quantities, will be : zero, ifgol is independent of + ; constant, if gol is linear in 46 ; linear in #, if gol is quadraticin 4, etc. In this approach it has been assumed that the temperature dependence ofgol is limited to go.Arguments supporting this assumption can be advanced lo* l4 ;however, a temperature dependence of the other gt coefficients can also be taken intoaccount, if necessary.Fig. 12 shows two examples of such an analysis, one (polystyrenefcyclo-hexane 34) pointing to a linear dependence of the right-hand side of eqn (7) on 4, theother (polyethylene + diphenylether) to a zero value within the experimental error.Eqn (6) can be used for calculating go and determining its temperature function from aplot of $qi0 against 1/Tc. This is illustrated in fig. 13 ; both plots show the usual lineardependence.I -. --LI.-- ---A- lo’ 3.30 3.35 3.40 3.45 350(2.50 d&-I .--- . L .A L---2.35 1.40 2.4 5 2.50 I ’ I 03-0103pFIG. 13.-Temperature dependence of the interaction parameter from critical miscibility data. Theblack dot denotes the 8 temperature taken from literature.The gO1(Q3,T) function thus obtained for polystyrene +cyclohexane is in goodagreement with those found by other methods (light-scattering, osmotic and vapourpressure measurements).In the description and quantitative prediction of phas160 LIQUID-LIQUID EQUILIBRIAbehaviour it is superior to the For polyethylene +diphenylether the g(T)function derived from critical miscibility data could so far only be checked againstChiang's value for the 9 temperature (161.4"C 37), which compares favourably withthe extrapolated value of 161.8"C. However, phase relationships could be predictedwith great accuracy.38Debye proposed the use of the critical opalescence, i.e., for determining the criticalcon~entration.~~ The angular dissymmetry of the light scattered at angles of 45 and135", for instance, should show a maximum at $c at the critical temperature. Thismethod was used with polymer solutions as well as with small-molecule mixtures byDebye et aL40 and by Borchard and Rehage 44 showed, however,that the maximum dissymmetry occurs at a concentration lower than the critical.The correctness of their finding can be verified by means of Debye et aZ. own data.Ifwe assume that the M, and M, values of their polystyrene samples were equal we cancalculate the critical whole-polymer volume fractions in cyclohexane their samplesshould have had, and compare these values with those at which they found maximumdissymmetry.We have used eqn (7) and the g(Q) function derived from criticalmiscibility data 34 for the purpose. The 4c values thus calculated are too smallbecause any deviation of M, from M, tends to shift 4c to higher values. Table 1 liststhe two sets of data and gives strong support to Borchard and Rehage's discovery.41-43TABLE 1 .-SYSTEM POLYSTYRENE+ CYCLOHEXANECritical whole-polymer volume fractions from critical opal-escence and from eqn (7)of ref. (40) dissymmetry * dc ( e w 7)A 0.068 > 0.0923 0.056 > 0.079C 0.050 > 0.073D 0.046 >0.061E 0.043 > 0.060F 0.028 > 0.043G 0.020 > 0.032data of Debye, Coll and W~errnann.~~sample notation maximumThese remarks do not suggest that the critical-opalescence method would beinvalid.It should, however, be adapted to the systems on hand ; these are not binaryand not strictly regular as those in the model employed by D e b ~ e . ~ ~ These deviationsmight be responsible for the discrepancy.Finally, a small value of M,/M, in a sample still does not necessarily justifyidentification of threshold and critical concentrations. First, the critical concentra-tion is determined by M, and M, and a small value of M,/M, may well go with a muchlarger value of MZ/Mw.l4* 3 2 9 45 In fact, Rehage et aZ.32* 45 reported critical andthreshold concentrations differing by a factor 2.4 although the M,,,/M, value of theirpolystyrene sample was 1.07. The M&, ratio, however, was 1.4. Secondly, evenif Mw/Mn andM,/Mw are small, the difference between threshold and critical concentra-tions is often enhanced by a strong dependence of the interaction parameter onconcentration.The author is much indebted to Messrs R.J. Aarssen, P. H. Hermans and L. A.Kleintjens for their valuable assistaiice in the preparation of the manuscriptR. KONINGSVELD 161P. J. Flory, J. Chem. Phys., 1942,10, 51 ; 1944,12,425.P. J. Flory, PrincQles of Polymer Chemistry (Cornell University Press, 1953).M. L. Huggins, Ann. N. Y. Acad., 1942,43, 1.M. L. Huggins, J. Amer. Chem. SOC., 1942,64, 1712.M. L. Huggins, Physical Chemistry of High Polymers (John Wiley, Inc., New York, 1958).M. L. Huggins, J. Amer. Chem. SOC., 1964,86,3535.P. J. Flory, J. Amer. Chem.SOC., 1965,87, 1833.H. Tompa, Polymer Solutions (Butterworth, London, 1956).* D. Patterson, Rubber Rev., 1967,40, 1.lo R. Koningsveld, Ado. Colloid Znterface Sci., 1968, 2, 151.l1 H. Tompa, Trans. Farauizy SOC., 1949,45, 1142.l2 R. L. Scott, J. Chem. Phys., 1949,17,268,279.l3 L. H. Tug, J. Polymer Sci., 1962, 61,449.l4 R. Koningsveld and A. J. Staverman, J. Polymer Sci. A , 2,1968,6, 305, 325, 349, 367, 383.K. Kamide, T. Ogawa and M. Matsumoto, Chem. High Polymers (Japan), 1968,25,788.l6 K. solc, Coll. Czech. Chem. Comm., in press.R. Koningsveld, R. J. Aarssen, L. A. Kleintjens and C. J. H. Schouteten, to be published.J. W. Gibbs, Collected Works (Dover Publications Reprint, New York, 1961), vol. I, p. 132.l9 R. Koningsveld, H. A. G. Chermin and M. Gordon, Proc. Roy. Soc. A , in press.2o R. Koningsveld, G. A. M. Diepen and H. A. G. Chermin, Rec. Trav. Chim., 1966, 85, 504.'l M. Gordon, H. A. G. Chermin and R. Koningsveld, Macromol., 1969, 2, 207.22 H. Tompa, Trms. Faraday SOC., 1950,46,970.23 R. Koningsveld and A. J. Staverman, Kolloid 2.2. Polymere, 1967,218, 114.24 G. Rehage, D. Moller and 0. Ernst, Makromol. Chem., 1965,88,232.25 G. Rehage and D. Moller, J. Polymer Sci. C, 1967, 16, 1787.26 W. H. Stockmayer, J. Chem. Phys., 1949, 17, 588.27 G. Allen, G. Gee and J. P. Nicholson, Polymer, 1961, 2, 8.28 B. M. Baysal and W. H. Stockmayer, J. Polymer Sci. A2, 1967,5, 315.29 A. R. Shultz and P. J. Flory, J. Amer. Chem. Soc., 1952,74,4760; 1953,75, 3888, 5631.30 D. McIntyre, N. Rounds and E. Campos-Lopez, Polymer Preprints A.C.S., 1969, 10, 531.31 G. Allen, G. Gee and J. P. Nicholson, Polymer, 1960, 1, 56.32 G. Rehage and R. Koningsveld, J. Polymer Sci. B, 1968, 6,421.33 see, e.g., R. Koningsveld, Adu. Polymer Sci., 1970,7, 1.34 R. Koningsveld, L. A. Kleintjens and A. R. Shultz, to be published.35 H. Tompa, Compt. rend., 2e Re'union de Chim. Phys. (Paris, 1952).36 G. Rehage, Kunstofe-Plastics, 1963, 53, 605.37 R. Chiang, J. Phys. Chem., 1965, 69, 1645.38 R. Koningsveld and A. J. Staverman, Kolloid 2.2. Polymere, 1966,210, 151 ; 1967,220, 31.39 P. Debye, J. Chem. Phys., 1959,31,680.40 See P. Debye, H. Coll and D. Woermann, J. Chem. Phys., 1960,33, 1746.41 D. McIntyre, A. M. Wims and J. H. O'Mara, Polymer Preprints A.C.S., 1965, 6, 1037.42 A. A. Tager and V. M. Andreeva, J. Polymer Sci. C, 1967,16,1145.43 V. E. Eskin and A. E. Nesterov, J. Polymer Sci. C, 1967,16, 1619.44 W. Borckhard and G. Rehage, 6th IUPAC Microsymposium (September 1969, Prague).45 G. Rehage and W. Wefers, J. PoZymer Sci. A2, 1968,6,1683

ISSN:0366-9033    

DOI:10.1039/DF9704900144

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

GENERAL DISCUSSIONProf. A. Silberberg (Weizmmn Institute of Science, Israel) said: In the usualmethod of deriving the Flory-Huggins expression,l a long-chain, linear, highlyflexible structure for the solute molecule is assumed. This, however, is not arequirement and a more general derivation can be given., One assumes mixing with-out volume change and divides the solute molecule, arbitrarily, into r subunits equalin size to the solvent molecule. This subdivision is completely conceptual and weassume that upon mixing with solvent each one out of the set of r sub-units, whileremembering its topological connection to the others, can occupy any position in thespace available for the mixture, i.e., can become arbitrarily far separated from theothers. It follows that n1 solvent molecules of volume V,, and n2 solute molecules ofvolume rV, (subdivided as above) can be distributed over a space of total volume,V = nlVl+m2Vl (1)in (n, +m2)! distinguishable ways if all the solvent molecules and all the solutemolecules and their sub-units are regarded as distinguishable.Dropping the distinction among solvent molecules and among solute moleculebut maintaining distinguishability within each set of sub-units composing a solutemolecule, i.e., retaining the topological, if not the physical, connectivity of eachsolute molecule, reduces the number of distinguishable arrangement to(2) (n, + m,) ! In, ! n, !To correct this expression for the dismemberment of the solute molecules we mayargue as follows.In (2) it is assumed that each solute subunit has the entire volumeVat its disposal irrespective of the real structural restrictions.This can be true onlyfor one, say the first, sub-unit in the solute molecule, the other (r - 1) sub-units arerestricted in some way to a parametric volume V* associated with the first sub-unitand not to the total volume V. Hence we have overestimated the number of con-figurations in (2) by a factor (V*/V>nz("-l). With this correction we arrive at thefollowing expression for the microcanonical partition function of the mixture :(3)where u), and u)= are mixture-independent factors associated with each solvent andsolute molecule respectively. We now use (3) to write the microcanonical partitionfunction for the unmixed species 1 and 2 respectively :R(n,, 11,) = o ~ ~ o ~ [ ( ~ ~ + rn,) !Inl !n, !](V*/V)nz(r- '1,which together with (3) give for the free energy of mixing (no internal energy change isassumed)AG = kTIn [a(n,)~(nl>/~(n,, n2)l= kT In [nl ! (m2)!/(nl +rn2)!][(nl +rn2)/(rn,)]"2(r- l )= (n, + rn2)kT[x1 In u1 + x2 In 24, (5)'see, e.g., C.Tanford, Physical Chemistry of Macromolecules (John Wiley, New York, 1961),p. 204.A. Silberberg, J. Chem. Phys., 1968, 48,2835, appendix.1 6GENERAL DISCUSSION 163where x1 and x2 are the mole fractions and u1 and u2 are the volume fraction of thecomponents.The Flory-Huggins mixing equation (5) was thus derived without detailed specifica-tion of how the solute molecules are built up. The derivation depends only on theassumption that a parameter V* exists that (V*/ V)n2(r- l) is an adequate correctionfactor for connectivity and most important that the parameter V* does not change ingoing from the unmixed state of pure 2 to the mixture.Mr.C. P . Hicks (Uniuersity of Bristol) (communicated): It has sometimes beensuggested that partial molar volumes should be used in the calculation of volumefractions in the place of the molar volumes of the pure components. While it isimpossible to say that it is correct to use pure data, it is certainly wrong to use partialmolar volumes instead. The relationship V, = xlvl +x2V2 must not be taken tosuggest that is the effective volume of component i in the mixture, i.e., the partof the total volume occupied by, or to be ascribed to,l component i.In fact, thepartial molar volume need not bear any particular relationship to the effective volume ;the former merely shows how the total volume will change for vanishingly smallincrements of the component. This is not a fine distinction : consider the extremesize difference case mentioned in Rowlinson’s paper where small molecules (1) arefitted into the spaces between large molecules (2). Here, until a certain criticalconcentration of 1 is reached, Vl = 0, but this is clearly not the volume which themolecules of 1 occupy. After this point, vl tends to the pure molar volume of 1,reaching it at once if the size ratio is large enough.This naive argument, based on a well-ordered structure, is difficult to extend to afluid, but there is experimental evidence to show that occasionally Vl<O, as, e.g.,near the liquid-vapour critical point in methane + propane,2 methane + i-~entane,~methane + 2-methylpentane and ethane + n-~entane,~ all of which are “ simple ”systems.The use of partial molar volumes to assess effective volumes will sometimeslead to nonsensical values, and even when they are not so I suggest that it would bebetter to use the pure molar volumes as a not necessarily accurate estimate of theeffective volume of each component.Dr. H . Tompa (Union Carbide European Research Associates, Brussels) said :Flory has referred to an old suggestion of mine to consider the effects of a possibleconcentration dependence of the interaction parameter, though this suggestion hasalso been made by others.This encourages me to bring up another old suggestionof mine, also not original since it has also been made before by others, in particularby Munster : all theories of polymer solutions assume that the force field around themolecules or polymer segments is spherical, in other words, that there are noorientation effects. However, clearly the interaction energy, e.g., of two molecules ofbenzene depends on whether the molecules are “ face to face ” or ‘‘ face to edge ” or“ edge to edge ’’ or in any other mutual orientation, and the same applies, e.g., tothe phenyl group of polystyrene. The mixing of two components will interfere withthe state of orientation and will lead in particular to an additional term in the entropyof mixing.5 I would suggest that the discrepancies with experiment which remaineven in the greatly refined theory which Flory has just presented might well be due tosuch orientation effects.J.S. Rowlinson, Liquids and Liquid Mixtures, 2nd ed. (Butterworth, London, 1969), p. 106.* P. L. Chueh and J. M. Prauznitz, Ameu. Znst. Chem. Eng. J., 1967, 13, 1099.A. J. Davenport, J. S. Rowlinson and G. Saville, Trans. Faraduy SOC., 1966,62, 322.H. H. Reamer, V. Berry and B. H. Sage, J . Chem. Eng. Data, 1961, 6,185.H. Tompa, J. Chem. Phys., 1953, 21, 250164 GENERAL DISCUSSIONDr. R. F. T. Stepto (University of Mmchester) said: In agreement with Flory'sremarks, one believes that since the advent and acceptance of the rotational-isomericstate model for real polymer chains, the problem of excluded volume could beapproached more meaningfully by using this model than by using lattice models.Using the rotational-isomeric state model and Lennard-Jones expressions for theenergies of interaction between segments we have obtained preliminary results whichindicate that the value of the exponent y depends to a large extent on the actualLennard-Jones parameters used.Prof. R.L. Scott (University of California, Los Angeles) said: Flory has reporteddifficulties in fitting the thermodynamic properties of solutions of polydimethyl-siloxanes to his equation of state theory with a single parameter XI2. Similardifficulties were encountered by S. P. Koh working with Prof. G. N. Malcolm inNew Zealand.In their unpublished work on polydimethylsiloxane in CCl, andCHC13, they found it necessary to assume different values of X12 to fit GE, RE, and rE. I suggest that part of the difficulty may arise from the fact that polydimethyl-siloxane is a very bulky molecule with a chain cross-section much larger than thatof a solvent molecule. In such a case the usual Flory-Huggins configurational termis appreciably in error and contributes too large a term to the excess entropy.Adjusting X12, which is essentially an energy parameter, is insufficient and a secondparameter for the local configurational entropy (as distinct from equation of stateeffects) may be necessary.Prof. G. N. Malcolm (Massey University, New Zealand) (communicated): Theresults obtained by Dr.S. P. Koh for solutions of polydimethylsiloxane (PDMS)(with number average relative molecular mass of 1655+30) in CCl, and CHC13 aresummarized in the following table. The mixing functions are expressed as the totalchange in the property on mixing divided by the total volume of the unmixedcomponents.Scott has described our experience in attempting to fit these results to the equationof state theory of Flory.MAXIMUM VALUES OF MIXING FUNCTIONS AT 278.68 K AND 1 atm.system AmHmax/J cm-3 AmGmax/J cm-3 103AmV/cm3 cm- 3*0.10 f 0.25 f 008PDMS (A)+ CCI, (B) 0.06 at $A = 0.35PDMS (A)+ CHC13 (B) 0.14 at ( b ~ = 0.55- 5.89 at $A = 0.70- 5.50 at +A = 0.75- 2.13 at 4~ = 0.60- 3.83 at @A = 0.60Prof. P. J. Flory (Stanford University) said : I agree with Scott that a large disparitybetween the diameter of the polymer chain and that of the solvent should affect theentropy of mixing. The diameter of the PDMS chain is comparatively large, and insolutions of PDMS with CHC13 or CCl, this difference may act to diminish the entropyof mixing.But in the systems PDMS + cyclohexane and PDMS + chlorobenzenethis difference is not great. Moreover, the excess volume as well as the entropydepart from our theory. The former would not be altered by modification of thecombinatorial entropy. Hence, we must look elsewhere for the explanation of thediscrepancy between theory and experiment in this case.Dr. M. L. Huggins (Arcadia Institute for Scientijic Research l) said: For comparisonwith other theories, I outline briefly my new theory of the thermodynamic properties135 Northridge Lane, Woodside, Calif.94062GENERAL DISCUSSION 165of solutions.' It is based on concepts I have previously discussed with regard topolymer solutions, but is simplified by omission of some factors that are usuallyunimportant.Like Flory in his new theory, outlined in the Spiers Memorial Lecture, I deal withsegment surfaces, but the surface area that enters into the equations is the averagearea that makes contact with other surfaces. The energies in the equations are theaverage contact energies per unit area of contact, for each type of contact. For asystem containing two types of segments there are three types of contact. I assumethe relative contact areas for these three types to be governed by an equilibriumconstant, equal to unity for randomness of contacts.With these assumptions, equations are deduced for the total energy of the systemand derived quantities as functions of the variables mentioned.For excess enthalpiesof mixing the contact energies enter only as the energy change for replacement ofcontacts between like segments by contacts between unlike segments. Similarly, theequations for excess volumes of mixing involve the volume change when contactsbetween like segments are replaced by contacts between unlike segments. UnlikeFlory, I do not assume the energy and volume change parameters to be the same;each is deduced from experimental data.An important feature of my theory is the transferability of the magnitudes of theparameters from one system to another containing the same segment types.Thistheory gives excellent agreement with experiment, when tested by good data on avariety of systems.Prof. J. D. Ghmez-Ibhiiez ( WesZeyan University, Conn.) said : Relative to solutionsin which n-alkanes are one of the components, it may be of interest to report somerecent measurements on binary mixtures of cyclohexane with n-alkanes carried outin our laboratory which together with others, available in the literature, may permitmaking some generalizations about the effect of chain length on the excess thermo-dynamic functions. We have measured the enthalpies of mixing of cyclohexanewith n-nonane and with n-dodecane. Similar measurements have been made byLundberg with n-heptane and with n-hexadecane.In all four cases the enthalpiesof mixing are positive and increase in magnitude with increasing length of the alkane,being in fact a linear function of n, the number of carbons in the chain. For a givenalkane the magnitude of HE decreases with increasing temperature, this effect beingmore pronounced as the length of the alkane increases.Our recent measurements of the excess Gibbs free energy of mixing of cyclo-hexane with n-eicosane, together with our previous ones with n-hexadecane andwith n-dodecane show a similar linear dependence on chain length, and the threebinary systems exhibit also a relation of congruence. In all three cases GE is negative,and increases in absolute value with increasing length of the alkane.For a givenalkane, GE becomes more negative with increasing temperature. The excess volumesof these mixtures, on the other hand, are positive and linear function (around themaximum) not of n, but of l / n . 5It may be of interest to report also on some preliminary results obtained for the sys-tems formed by n-hexadecane with 2,2,4-trimethylpentane and with transdecaline. Forthe first system, GE is small but positive, also decreasing with increasing temperature,M. L. Huggins, J. Phys. Chem., 1970, 74, 371.G. W. Lundberg, J. Chem. Eng. Data, 1964,9, 193.J. D. G6mez-IbBfiez and J. J. C. Shieh, J. Phys. Chem., 1965,69, 1660.J. D. G6mez-IbAiiez, J. J. C. Shieh and E. M. Thorsteinson, J . Phys. Chem., 1966, 70, 1998.J.D. G6mez-IbAfiez and C. T. Liu, J . Phys. Chem., 1961, 65,2148166 GENERAL DISCUSSIONand HE has a value of about 200 J for the equimolar mixture at 25”. For thesecond system GE is about zero at 25” and HE is about 50 J at x = 0.5. We believethat these type of measurements may yield valuable empirical information which mayhelp to elucidate the effect size and shape, previously referred to in this Discussion,may have on the excess thermodynamic functions.For the mixtures formed by cyclohexane with n-alkanes we have calculated, fromequations of state, and also following the procedure of Abe and Flory,, theparameter X I , of Flory’s theory. We find that the value for X , , thus calculatedpredicts rather well the experimental values for GE, but the predictions for the excessenthalpies, entropies and volumes of mixing are much less satisfactory.Dr. C.L. Young (University of Leeds) said : Fig. 2 of Scott and van Konynenburgis in surprisingly good agreement with the experimental data for hydrocarbons butthe boundaries between type 11, I11 and IV behaviour are strongly dependent on thecombining rule for a12. The van der Waals combining rule for a,, appears to befairly accurate for mixtures of widely differing size molecules. It may be comparedwith the geometric mean and Hudson and McCoubrey combining rules for theinteraction parameter E , ~ :a,2/o:za = E12(vdW) = ~:/,’~~~~[o:/l”o:/2”/~~,]~, van der Waals;q2(G.M.) = ~:/1~~;/22, geometric mean;E~ ,(H.M.) = E: i2&:(2, [o: /:o~/z”lo, ,16j”, Hudson and McCoubrey,TABLE COMPARISON OF CALCULATED AND EXPERIMENTAL VALUES OF <system no.van der WaalsOMCTS (a)+ cyclopentane 1 0.937OMCTS+ cyclohexane 2 0.953OMCTS+ cycloheptane 3 0.964OMCTS+ cyclo-octane 4 0.974OMCTS + 2,3-dimethylbutane 5 0.965OMCTS+ benzene 6 0.938OMCTS+ carbontetrachloride 7 0.943ethane+ n-heptane 8 0.955propane+ n-octane 9 0.968n-pentane + n-tridecane 10 0.966n-hexane+ nitrogen 11 0.921n-hexme+ methane 12 0.932n-pentane + methane 13 0.947argon + methane 14 0.978(a) OMCTS = octamethylcyclotetrasiloxane.(b) assuming the ionization potentials are the same.Hudson andMcCoubrey (h)0.878 (b)0.907 (6)0.929 (b)0.948 (6)0.930 (b)0.879 (b)0.889 (6)0.9100.9370.9340.8330.8620.8910.952experimental0.935 (c)0.961 (c)0.979 (c)0.981 (c)0.968 (c)0.940 (c)0.953 (c)0.945 (d)0.972 (d)0.983 (c)0.87 (e)0.96 (f 10.97 (f)0.981 (9)(c) unpublished results obtained from critical temperature data.(d) obtained from critical temperature data taken from sources given in C.P. Hicks and C. L.( e ) obtained from virial coefficients, C. L. Young, Ph.D. Thesis, (University of Bristol, 1967).cf) obtained from virial coefficients data of E. M. Dantzler, C. M. Knobler and M. L. Windsor,(9) obtained from free energy of mixing data, T. W. Leland, J. S. Rowlinson, G. A. Sather and(h) G. M. Hudson and J. C. McCoubrey, Trans. Furahy SOC., 1960,56,761.Young, Trans. Furaduy SOC., in press.J. Phys. Chem., 1968,72,676.1.D. Watson, Trans. Farahy SOC., 1969,65,2034.R. A. Orwoll and P. J. Flory, J. Amer. Chem. SOC., 1967,89,6814.A. Abe and P. J. Flory, J. Amer. Chem. Suc., 1965,87,1838.A. J. B. Cruickshank, M. L. Windsor and C. L. Young, Trans. Faruday Soc., 1966,62,2341GENERAL DISCUSSION 167where fl is a function of the ionization potential and is nearly unity for mixtures ofhydrocarbons.For equal size molecules the three combining rules are equivalent. Unfortunatelythere are few data available to test these rules for molecules of differing sizes. Assum-ing T'OCE and YCcco3 and the Lorentz rule,l o12 = (oll +az2)/2, values of 5 defined0 60 50 40 30 2L0 96 0 9 5 0 . 9 4 0 99 0 98 0 9 7k WW)FIG. 1 .-Comparison of experimental and calculated interaction parameters.The numbers referto the systems given in table 1. - - - , geometric mean; - - , Hudson and McCoubrey; Il = 12.by = E~JE~~(G.M.) have been determined from gas-liquid critical data mixedsecond virial coefficients and free energy of mixing data. These are compared withthe combining rules in table 1. The values of 5 derived from gas-liquid criticaltemperatures have been calculated using a refined version of the method given byRowlinson.2* Values of A t given byA5 = [El2 -&12(VdWI/&12(G.M.)are plotted against <(VdW) in fig. 1. The data for second virial coefficients have alarger uncertainty and have been omitted. In general, for chemically similar mole-cules, the van der Waals rule is superior to the two other more widely used combiningrules.Dr.R. Ullman (Ford Motor Co., Michigan) said: Scott stated that end groups onnormal alkanes must be treated separately from chain segments near the centre ofthe molecule in order to obtain consistent experimental results for thermodynamicl A. J. B. Cruickshank, M. L. Windsor and C. L. Young, Trans. Faraday Soc., 1966,62,2341.J. S . Rowlinson, Liquids and Liquid Mixtures (Buttenvorths, London 2nd ed. 1969).C. P. Hicks and C. L. Young, to be published168 GENERAL DISCUSSIONexcess functions. In this connection, n.m.r. spectroscopy studies of normal alkanesin aromatic solvents by K. J. Liu demonstrate this difference in a direct way. Thesingle main methylene peak in the proton magnetic resonance spectrum of CI6Hs4in a-chloronaphthalene changes to a doublet for CI8H3* and higher paraffins.lThis has been explained as a difference between end segments and central segments ofthe molecule which first becomes apparent when the paraffin chain becomes longenough so that folded conformations of the molecule similar to those in polyethylenelamellae attain some stability.2Mr.A. S. Teja and Prof. J. S . Rowlinson (Imperial College, London) said : Naturalgas consists principally of methane, with some nitrogen and smaller amounts ofhydrocarbons up to about decane. In developing methods of predicting the thermo-dynamic properties of such mixtures we have made calculations similar to those ofScott and van Konynenburg except that we have substituted the true equation of stateof methane for the van der Waals equation.We have retained van der Waals'srules for obtaining a characteristic energy and volume to represent the mixture, i.e.,eqn. (19), together with (13) and (28), of their paper. The higher alkanes do notreduced equation of state as methane and we allow for this bycalled shape factors, whose magnitude is determined for eachacentric factor o. ,conform to the sameusing what Leland4substance by Pitzer's502 0 3 3 0 0 400FIG. l.-Calculated gas-liquid critical curves for binary mixtures of three hydrocarbons withmethane. The points are experimental.The calculated gas-liquid critical lines are shown in fig. 1, in which this treatmentreproduces well the observed maxima in p" for mixtures with ethane, propane andn-butane. Fig.2 shows that the calculated values of the critical pressure for mixtureswith n-pentane are changed significantly in mixtures rich in the heavier componentif the shape factors are omitted, i.e., if it is assumed that n-pentane obeys the sameK. J. Liu, J. Polymer Sci. A2, 1967, 5, 1209; 1968, 6,947.K. J. Liu and R. Ullman, J. PaZymer Sci. A2, 1968, 6,451.A. J. Vennix and R. Kobayashi, Amer. Inst. Chem. Eng. J., 1969,15, 926.J. W. Leach, P. S. Chappelear and T. W. Leland, Amer. Inst. Chem. Eng. J., 1968, 14,568.A. J. Davenport and J. S. Rowlinson, Truns. Furaday SOC., 1963,59,78GENERAL DISCUSSION 169reduced equation of state as methane. The difference is much less in mixtures richin methane, which is the region of interest for the calculation of the properties ofnatural gas, and the region in which separation of a second liquid phase might occur.We find that octane is the first n-alkane for which we predict the separation of asecond liquid phase at, approximately 193-195 K.In practice, n-hexane is the firstto show this behavi0ur.l The discrepancy is, in part, due to the use of the geometric-mean rule for the calculation of the cross-energy e12, since a lower value of thisparameter favours separation, and, in part, due to the assumption that it is possible tocalculate the properties of hydrocarbons with from six to eight carbon atoms fromthose of methane by what is essentially a first-order perturbation treatment. Never-theless the agreement with experiment is reasonable for a calculation which has noadjustable parameters.The maximum pc found in mixtures with n-octane is 2005 0 t - I I I I I I I I I I I I I l k 12 0 0 3 0 0 403T/KFIG. 2.-Gas-liquid critical curves for the system methane+n-pentane, calculated with andwithout correction for the non-spherical shape of pentane. The points are experimental.bar (experiment, 270 bar). Patterson and Delmas calculated critical curves bystarting from the opposite assumption, viz., that solutions of n-octane etc. in methanecan be regarded as polymer solutions and described by the theory developed byPrigogine and his colleagues. This led to a calculated maximum p" of about 500 barin mixtures with n-octane.It appears therefore that for solutions in methane of paraffin hydrocarbons up toabout C5 it is adequate to describe the system in terms of molecules each of a single" segment ", but of different size, shape and energy.From C6 to Clo this approachbecomes quantitatively less acceptable, but is still as sound as describing the system asa polymer solution.Prof. D. Patterson (McGilZ University) (communicated): Teja and Rowlinsonmention our calculated value of 500 bar for the maximump, along the critical locusfor the methane + octane system, which compares poorly with the experimental valueof 270 bar. This figure comes from a calculation not only of the critical locus, butalso of critical constants of the pure components, i.e., pc of the components are notfixed by experiment as in the work of Teja and Rowlinson.The predicted values ofT, for the pure components were very good, but thep,, and the pressures along theA. J. Davenport and J. S. Rowlinson, Trans. Faraday SOC., 1963, 59, 78.D. Patterson and G. Delmas, Trans. Faraday Suc., 1969, 65, 7081 70 GENERAL DISCUSSIONcritical locus, were all too high by a factor of about two. If thep, of the componentsare fixed by experiment, as in curve (e) of fig. 3 of our paper,l for the methane+hexane case, one has a prediction of the critical locus which may be compared withthat of Teja and Rowlinson. For methane with hexane, octane and decane themaximum p , values are, respectively, about 180 (200), 260 (270) and 370 (360) bar,experimental values being in brackets. Our main purpose was not accuracy but togive apriori predictions of a variety of phenomena.The shortest n-alkane to showlimited miscibility with methane was predicted to be C, ; with ethane it was Czo andwith propane CS2. Experimentally, the alkanes are c g y C19 and with propane abranched C3, is still miscible. The main interest is that the same calculation couldpredict the phase behaviour of polyethylene in ethane, propane, butane and pentaneand the general phase behaviour of binary systems of liquids of widely differentvolatility.Mr. C. P. Hicks (University of Bristol) said: Like Teja and Rowlinson, I havebeen working on improved methods of predicting the critical points of binary mixturesby solution of the complete stability relationships.2 Like Rowlinson I have foundthatintroducing experimentally determined equation of state properties greatly improvesquantitative agreement with experimental measurements on gas-liquid critical points.I have used corresponding states together with the Taylor series about the criticalpoint of a pure substance derived by Davis and Rice to define thep,V,Tpropertiesof the mixture equivalent substance (es), and so have been restricted to the regionwhere both I T/Tzs- 1 I and I V/V,"s - 1 I are less than 0.3.The gas-liquid criticalpoints of most mixtures which have been studied experimentally fall within theselimits, and a comparison with experimental results for a wide range of mixturesof hydrocarbons, fluorocarbons, silanes and siloxanes has been carried out.Of the recipes for the equivalent substance reducing parameters, the so-called vander Waals prescription has been, in general, the most successful, although for binaryn-alkane mixtures a prescription based on the principle of congruence has been equallysuccessful.Unlike the solution to the stability relations used by Scott and van Konynenburgwhich is specific to the van der Waals equation, the programme I have written willaccept any equation of state, and has recently been extended to locate all the criticalpoints in a given region of the (V/Vzs, T/T,"J plane by stepping along the locus ofzeroes of either (d2G/8~2)T,p or (d3G/dx3)T,p.Fortunately, the few comparableresults which have been calculated using the van der Waals equation agree with thework of van Konynenburg.6Prof.D. Patterson (McGiZZ University, Montreal) said: Predictions made with thetheory of Scott et al. based on the van der Waals equation of state must be quali-tatively similar to those found with the corresponding states theory.' In that theory,a simple model of the liquid such as the van der Waals will not predict either anupper critical solution temperature (UCST) or HE>O for a chain-molecule mixtureif the chains axe of completely identical segments. A good measure of the strengthD. Patterson and G. Delmas, Trans. Faruhy Soc., 1969,65,708.A. J. B. Cruickshank and C. P. Hicks, 22nd I. U.P.A.C. Congr. (Sydney, 1969).B. W. Davis and 0. K. Rice, J. Chem. Phys., 1967,47,5043.C. L. Young, private communication.T. W. Leland, J.W. Rowlinson and G. A. Sather, Trans. Furaduy SOC., 1968,64, 1447.P. H. van Konynenburg, Ph.D. Diss. (U.C.L.A., 1968).D. Patterson and G. Delmas, Tram. Farad'y SOC., 1969,65,708GENERAL DISCUSSION 171of the intermolecular force fields around a segment is the reduction parameter P* =U*/V* which corresponds to the van der Waals equation of state quantity a/b2.Using the principle of corresponding states, Bardin and I have shown that P* isconstant for the normal alkanes from butane to polymethylene, so the n-alkanes areessentially chains of identical segments. However when an approximate equation ofstate such as the van der Waals or the Flory-Orwoll-Vrij is used, P* is found to varywith n. This is seen in the paper of Scott et al. where alb20c ((n+0.88)l(n+ 1.32)j2.It is because aZ1/bl1 # ajz/b2, that HE is given positive by eqn.(17) and that aUCST is predicted at low temperatures. For the particular case of a normal alkanemixture this prediction seems to be an artefact caused by an imprecise equation ofstate.Turning to the origin of the lower critical solution temperature (LCST), accordingto corresponding states theory the necessary positive excess free energy arises from thevolume changes of the components during the mixing process which give an overallnegative value of VE at high temperature. However, these volume changes havetheir origin in a large difference in free volume between the components, equivalentto a large difference between the reduction parameters T* or the vapour-liquid criticaltemperatures T,.Correspondingly, in the present theory based on the van der Waalsequation of state, I believe that the condition for an LCST must be that values of T,or a/b are very different for the two components. (This condition is not in contradic-tion with an n-alkane system having a/b2 equal for the components.) In the van derWaals theory, the condition of equal T, is <+[ = 0, and in fig. 2 this gives a linecutting diagonally across the diagram parallel to the boundary where type I1 phasebehaviour becomes type IV, but at a considerable distance from it. Since type IVbehaviour involves an LCST one has qualitative agreement between predictions of thetwo theories. However, the line of equal T, also passes through the region of type111-HA behaviour, and this is surprising since this corresponds to the occurrence of aUCST and a LCST which have coalesced.Returning to the corresponding states theory, for the ethane + alcohol mixturesstudied by Kuenen, the difference in T* or T, is due to the wide difference in strengthsof intermolecular forces of the components.For n-alkane mixtures, however, thedifference in T* and T, is due, not to a difference in intermolecular forces, but to adifference in chain length. . . or more properly to a difference in the number ofexternal degrees of freedom per chain carbon (structural factor of Prigogine). Thesetwo effects are distinguishable in the corresponding states treatment, but not in thatbased on the van der Waals equation where the constant a depends on the molecularsize and also on the intermolecular forces.In this respect, I believe that the statementin Rowlinson's paper that the LCST in mixtures of n-alkanes is primarily an energyeffect could be misleading. The energy he refers to is that of the whole molecule,and thus both molecular size and intermolecular forces affect this quantity.Prof. R. L. Scott (University of California, Los Angeles) said: We are not pro-moting the van der Waals equation as an accurate equation of state or as even thebest available two parameter one. It is merely one of the simplest, and one in whichthe basic physical features, an energy of attraction and a repulsive core, appear in astraightforward way. At least over certain ranges of temperature and density, otherforms for f ( n ) in a generalized van der Waals equation (Rowlinson's eqn (7)) arebetter ; one such is the Flory form f ( n ) = [l -(cn)+]-l.The fact that various theoretical models and approximate equations predictsubstantially the same properties for mixtures of n-alkanes is, I think, strong evidenceD. Patterson and J.M. Bardin, Trans. Faraday Suc., 1970, 66, 321I72 GENERAL DISCUSSIONthat these experimentally observed properties arise from " physical " features commonto all these models-heats of mixing which are positive but quite small at low temper-tures, and substantial differences in the gas-liquid critical temperatures of the purecomponents ; the latter produces very great differences in the relative degrees ofexpansion over the close-packed volume, negative excess volumes, and ultimatelyincomplete miscibility at high temperatures.1.c0.E0.650.40.20.0t:FIG.1.-Types of phase equilibria for mixtures obeying the relation a12 = (ai The solidlines are the boundaries calculated using an ideal configurational term (see fig. 2 of paper); thedashed lines arise when the Flory configurational term is used.For the appropriate values of a and b, the van der Wads equation with an idealconfigurational entropy of mixing (as in our eqn (5)) yields an excess Gibbs freeenergy GE which is invariably positive for n-alkane mixtures at ordinary temperatures.The experimental values of GE for such mixtures at 25°C are all weakly negative.However, if one substitutes the Flory configurational term for the ideal term (addingthe term RT[xl In (bll/b,,,)+x2 In (bzz/b,)] to the right-hand side of our eqn.(5)), thevan der Waals equation gives negative values for GE over an intermediate range oftemperature.Since submitting our paper we have made more calculations on phase equilibriain mixtures obeying the geometric mean for a12 using a free energy modified with theFlory configurational expression. Fig. 1 shows (as dashed lines) the resultingboundaries for type 11, 111, and IV behaviour. The differences are only quantitativeexcept as one approaches the limit of very large differences in size (chain length), i.e.,near ( = 1. Here there is a real difference in kind, for the IV-I11 boundary nowterminates about 5 = 0.12, instead of at = 0.00 for the ideal configurational term ;this places polymethylene solutions (e.g., 6 + 00) in the correct region.OtherwiseGENERAL DISCUSSION 173the new boundaries improve the agreement for the ethane systems at the cost ofpoorer agreement for the methane systems.One important qualitative question remains : is a difference in cohesive energydensity (a/b2) between end and middle groups (e.g., between the CH3 and the CH2)physically real or necessary? We have introduced it into our treatment, as Floryhas in his. Patterson does not and gets similar results. Perhaps there is no experi-mental way of settling this, in which case the question is not operationally significant,but it still merits discussion. Some suggestive (but admittedly inconclusive) pointsare :(a) The excess enthalpies RE of n-alkane mixtures are invariably positive at lowtemperatures. The transition to the solid state precludes an experimental test, butif RE were to remain positive down to 0 K, an upper critical solution point would beinevitable.All equations of state of the van der Waals type require differences incohesive energy density to produce positive RE ; however, they oversimplify the heatcapacities of the liquids and cannot yield the negativevalues of dCJdTwhich Pattersonoffers as an alternative explanation.(b) The boiling points and enthalpies of vaporization of branched hydrocarbons areinvariably lower than those of the corresponding n-alkanes, a phenomenon easy toexplain in terms of cohesive energy density differences between CH3, CH2, CH, etc.(c) Other homologous series appear different from the n-alkanes in some of theseproperties.Thus fluorocarbons behave (in terms of our model) as if CF, and CF2were much more nearly equivalent (in cohesive energy density a/b2) than the corres-ponding CH3 and CH2.Prof. P. J. Flory (Stanford University) (communicated) : Statements in the paper byPatterson et al. and its predecessors convey the erroneous impression that ow theoryrests on commitment to a (3, GO) intermolecular pair potential. This potential, which ofcourse is quite unrealistic, is introduced by Patterson and co-workers as an artifice forovercoming aberrations inherent in the cell model, and thus to reconcile this modelwith a van der Waals intermolecular energy.Familiar arguments leading to anenergy of this form for a liquid make no stipulation concerning the intermolecularpotential, apart from the requirement that it be of short range. We have adhered tothis line of reasoning and certainly do not subscribe to a (3, GO) pair potential.Prof. D. Patterson (McGill University, Montreal) (communicated): We have forbrevity coupled the terms “ (3 - co) potential ” and “ Flory theory ”. This could indeedlead to the erroneous impression mentioned, but we hope our point of view is clearlyexpressed in our paper. If m and n are varied in eqn (27) of our paper a spectrum ofintermolecular energy-volume dependences are obtained, at least in a formal way.One of them corresponding to rn = 3, n+m is the van der Waals Y-l relationship,which may be obtained on fundamental grounds, e.g., as described in Rowlinson’spaper. These energy-volume dependences are usually coupled with, to within aconstant factor, the Eyring-Hirschfelder cell partition function, as in work by Prigogineand by Flory and their collaborators.Flory has strongly criticized the cell modeland has recalled that the “ cell ” partition function can be supported by other argu-ments. However, the resulting equations of state all show similar discrepancies whencompared with n-alkane (and other) equation-of-state data. In view of this, transcen-dent importance should not be attached apriori to any one of these models. (Indeed,a hole model gives the best results of those tested.) There seems to be an advantageR.Simha and T. Somcynsky, Macromolecules, 1969, 2, 3421 74 GENERAL DISCUSSIONin following Prigogine and formulating the corresponding states theory of chain-molecule liquids and mixtures in a general way. Then, particular models may beused to obtain the properties of the " reference liquid " and predictions can be madeand tested.Mr. A. J. B. Cruickshank (Univers.Sity of Bristol) said: Current corresponding-states theories cannot fit GE and H E simultaneously without additional assumptions,even at p-0. Some authors suspect the '' combining rules " defining fe, and he,rather than the obvious but probably unimportant shortcomings of the commonlyused equations of state. Others suggest departures from the principle of correspond-ing states.I suggest that this situation calls for semi-empirical investigation, ratherthan abstract model building. If the pure components and their solutions obeyG ~ P , T) = fesGT(PIkes, T i f e s ) + RT In he,, (1)where subscript r denotes the reference species and subscript as either a pure substanceor a solution, and ke, may or may not be equal tofes/hes, then,l for a binary mixture, x,of 1 and 2 at p+O,GE = ( H ~ - 3 T C ~ , r ) f E - ~ T C ~ , , ( 1 / f ) E + ... + RT[ln" h + x x , In (4Jxi)], (2)(3)(4)iH E = (HT- TC;,,) f E + ...,vE = (vr- Y,Tap,r)hE+ V,Tap,,(h/f)E + - - - 9SE = fes(x>-xJi-x2f2, etc.whereThese equations present only two problems : to specify f,,, he,, k,, ; and to evaluateHT, C;I*,r, etc.Since V, = V,(T) is experimentally accessible the generalized form of eqn (4)given in our paper allows us to discover empirical " combining rules " instead ofdeducting them from artificial models.This procedure will expose any significantdeparture from the principle of corresponding states ; otherwise it will specifyf,,, he,,etc. It is not restricted to the particular case of solutions of homologues. There isno certainty thatf,,, he, should have the same values in all of eqn (2)-(4) ; the solutionmight not obey the same reduced equation of state as the components over the fullrange of density ; but this seems an unlikely complication.The role of the assumed equation of state in current theories is always to defineHF, C:,,, etc., or, equivalently, to give absolute values for the reducing parametersp:, VT, TF.Different equations of state indeed give similar values for (HT-+TC;,,),but are the differences between the Cp*,r important in relation to the misfit betweenHE and GE? The empirical alternative is to use experimental GE = GE(x) andHE = HE(x), together with the empirical combining rules, to evaluate H,*, C,*,r, etc.The complete equation of state is not required, but the truncated series forms ofeqn. (2) and (3) must be used and higher-order terms can be included only whenHE(x) is known over an adequate range of temperature.As to the n-alkane binary systems, there is no advantage in further discussingwhether the segment interchange parameter r is large or small, since it depends onthe configurational properties of the hypothetical " all-middle " n-alkane (dictated bythe imposed equation of state) as well as on the shape of T: = T:(n). Moreover,J.S . Rowlinson, Liquids and Liquid Mixtures, 2nd ed., (Butterworth, 1%9), pp. 312, 313.R. A. Orwoll and P. J . Flory, J . Amer. Chem. Soc., 1967,89,6822GENERAL DISCUSSION 1.75table 2 of our paper suggests that the departure from strict congruence indicated forfes = fes(x) in the system n-hexane + n-hexadecane cannot be described simply byadjusting the value of I? appropriate to the pure n-alkanes. One way of gaininginsight into this problem might be to attack first the theoretical interpretation of theobserved variation with n of the fn, h,, (relative to a selected member) for varioushomologous series.Mr.C. P. Hicks (University of Bristol) said : It seems that our paper did not givesufficient emphasis to one implication of the principle of congruence. Most theories ofsolutions when applied to homologous series assume that the non-combinatorial partof the equation of state of a mixture of members of the series is identical to that of oneof the members. Usually the congruence formula(1) - n = xlnl +x2n2is used to identify the particular member. If this is done it follows that the excessvolume of the mixture can just as validly be calculated byVM = v--(xlv,+x,v,) (2)as by the full theory, for cases in which i? is integer. Eqn (2) has been used by previousworkers 1* only when they are concerned with testing congruence.On this basis allsuch theories which reproduce the pure substance volumes well, and assume eqn (l),must in principle predict the same excess volume for the mixture. Differencesbetween theories must be associated with inadequate correlation of the pure homologuemolar volumes.TABLE 1 .-vM FOR EQUIMOLAR n-HEXANE+ n-HEXADECANE- V M I C , ~ mol-1expt. ref. (4) ref. (3) eqn (2)TIK303 0.58 0.52 0.65 0.65323 0.82 0.73 0.89 0.88The theory of Orwoll and Flory admits of departure from eqn (l), but predictsVM for the equimolar mixture of n-hexane and n-hexadecane which agree closelywith those predicted by eqn (2) (see our table 1). The divergence of the valuespredicted by Patterson and Bardir~,~ assuming eqn (l), may be due to their reducingvolume correlation ignoring a small curvature in V* = V*(n).A 0.1 % departurefrom linearity would suffice.Dr. M. L. Huggins (Arcadia Institute for ScientiJic Research ’) said: I wouldreport briefly on the application of my new solution theory to polymer solutions.Treating a solvent molecule and a mer of the polymer each as a single segment andassuming random mixing of the intersegment contacts, the theoretical equations arevery simple :Here vA, and ks are volume, energy and entropy parameters and r, is the ratio ofJ. D. G6mez-IbBiiez and Chia-Tsun Liu, J . Phys. Chem., 1963, 67, 1388.J. Hijmans and Th. Holleman, Ado. Chem. Phys., 1969, 16, 223.R. A. Orwoll and P. J. Flory, J . Amer. Chem. SOC., 1967, 89, 6822.D.Patterson and J. M. Bardin, Trans. Faraday Soc., 1970, 66, 321.135 Northridge Lane, Woodside, Calif. 94062.M. L. Huggins, J. Phys. Chem., 1970,74, 371176 GENERAL DISCUSSIONthe contacting surface areas of the two types of segments. These parameters arededuced from data on this or other systems containing the same segment types.f2 is the fraction of the contacting segment surface area that is polymer surface. Itis related to mol fractions and to the volume fraction of polymer by the equationsn is the average number of mers per polymer molecule and ru is the ratio of the averagevolume of a mer to that of a solvent molecule.These equations have been applied to the available experimental data for solutionsof natural rubber and benzene at 25°C.Considering the scattering of the data points,the agreement is good.Prof. G. Rehage (University of Clausthal) said : With regard to the paper byLiddell and Swinton, the exact conditions for the existence of a critical point ofphase separation in a binary mixture areAG is the molar free enthalpy of the mixture and x the independent mol fraction.The subscript k indicates the critical point. For an upper critical point,(a2AG/dx2), = 0 ; (d3AG/dx3), = 0. (1)(d2AH/aX2), <o ; (a2As/aX2)k < 0. ( 2 4For a lower critical point,(a2AH/dx2)k > 0 ; (d2AS/dx2), > 0.AB is the molar enthalpy of mixing and AS the molar entropy of mixing. Usuallyin the neighbourhood of a critical point the following rule is valid: the value andcurvature of the AH(x)- and the AS(x)-functions have the opposite sign.'^From this we have instead of the rigorous inequalities (2) the simpler statements :AHk>O; ASk>O (UCST)AH,<O; A&<O (LCST)Usually criterion (3) is confirmed experimentally, but it is possible that the functionsA n and AS have a --shaped course with eventually positive and negative values.Then the eqn (3a) and (3b) may be totally or partly invalid.Dr.Henryk Eisenberg (The Weizmann Institute of Science, Israel) said: I wouldreport that phase separation, sometimes of the liquid-liquid equilibrium type, has beenobserved in a number of highly charged multicomponent polyelectrolyte systems.The phase diagrams can be conveniently shifted by a change in the concentration ofthe simple electrolyte, and specific salt effects are also often observed.Alkali saltsof polyvinylsulphonic acid in aqueous solution, separate into two liquid phases athigh concentrations of added monovalent electrolytes. The phenomenon is highlyspecific with respect to the monovalent electrolytes investigated. In the alkalihalide series the order of specificity is NaCl< KCl > RbCl and KI > KBr > KCl ;no phase separation occurs under similar conditions with HCl, LiCl, CsCl andNH4Cl. Reversal of the phase separation is achieved by further increase in theconcentration of the salts which at intermediate concentration lead to the liquid-G. Rehage, 2. Nutuvforsch, 1955, lOa, 301.R. Haase and G. Rehage, 2. Elektrochem., 1955,59,994.H. Eisenberg and G.Ram Mohan, J. Phys. Chem., 1959,63,671GENERAL DISCUSSION 171liquid equilibrium. Significantly higher monovalent salt concentrations are requiredfor the salting out of salts of polystyrenesulphonic acid.l The barium salt of poly-styrenesulphonic acid is unusual in showing both upper and lower consolute tempera-tures in solution. Polyriboadenylic acid (Poly A) in aqueous solution shows phaseseparation in NaCl solutions more concentrated than 1 M.2 Under such conditionspoly A is soluble at low and high temperatures, but only partially miscible in a rangecentred on 3540°C. The extent of the range varies with molecular weight, andpolymer and salt concentration. Upper and lower Flory (theta) temperatures weredetermined for a number of systems.In this way it was possible to determine (bylight scattering) the radius of gyration at a number of temperatures, unperturbed bysolvent effects, thus separating long-range solvent-dependent influences on molecularconformation from the influences of short-range interactions due to purine basestacking.Prof. W. Byers Brown (Manchester) said: I would like to comment on the inter-molecular forces between the polymer molecules in ternary solutions containing amonomer solvent and two polymers, of the type discussed by Koningsveld. I am notconcerned with the polydisperse character of the polymers, but simply with whethertwo polymers are compatible or incompatible in a given solvent. The first carefulstudy of systems of this type, by Allen, Gee and Nicholson in the paper (ref. (31)quoted by Koningsveld) revealed several unusual features : (a) the Flory-Huggins xparameter derived from the miscibility measurements was much larger than thatderived from the heat of mixing; (b) the Flory-Huggins equation predicted noimmiscibility for average molecular weights M less than about lo4, but in fact speciallyprepared polymers with around 102-103 were still immiscible ; (c) the cloud-point,indicating the onset of immiscibility, occurred at very low concentrations correspond-ing roughly to close packing of the freely-coiled polymer chains.The last pointsuggests that long-range forces between the different polymer molecules in solutionmay be responsible for incompatibility. In the Faraday Society Discussion onintermolecular forces in 1965, McLachlan presented an interesting paper applyingthe Lifschitz dielectric theory to the effect of the medium on the long-range Londondispersion forces.Two molecules 1 and 2 at a large distance R apart will alwaysattract each other in vacuu with an interaction energy which may be writtenwithAE = -C/R6 (1)a3C = "s al(iu)a2(iu)du,n owhere a,(u) and a2(m) are the dynamic polarizabilities for frequency m, and i = ,/T.McLachlan showed that in the presence of a solvent (0) which could be treated as acontinuous dielectric medium, the van der Waals coefficient (2) becomes(3)= zJa ba,(iu)Aa,(iu)du, n &iu)where Aak(o) is the excess polarizability of a solute (k) molecule over that of thesolvent, and E ~ ( o ) is the permittivity (dielectric constant) of the solvent, both forfrequency o.The important point is that Aal and Aa, may be positive or negative,W. R. Carroll and H. Eisenberg, J. PoZymer Sci., A-2, 1960, 4, 599.H. Eisenberg and G. Felsenfeld, J. Mol. Biol., 1967, 30, 17.A. D. McLachlan, Disc. Faraday SOC., 1965, 40,239178 GENERAL DISCUSSIONso that C may be negative, corresponding to long-range repulsion between the mole-cules.The excess polarizability Actk of a solute molecule in solution may be found experi-mentally from the variation in the permittivity of the solution, E, with concentrationNk (molecules per unit volume) byIf E varies linearly with concentration, as is usually the case, thenAuk = (&k-&o)V;/47t, (5)where E ~ ( o ) is the permittivity of the solute, and ui is its molecular volume.Sincecc(iu) decreases monotonically with u, in most cases the sign of the integral (3) for Cwill be determined by the signs of Aal and Act2 for zero frequencies. In thesecircumstances it follows from (3) and (5) that the sign of C will be the same as that ofAE1AE2 = 6 1 - EO)(E2 -&oh (6)where the E are static permittivities, ~~(0). If go lies outside the range cl to E ~ , then(6) is positive, and the molecules have long-range attraction, as usual. However,if E~ falls in between E~ and E ~ , then (6) is negative, and the dispersion forces arerepulsive at long range.The above theory is probably of little interest for ternary solutions of smallmolecules, because the assumption of a dielectric continuum is unrealistic. However,it seems appropriate for the interaction of two large polymer molecules in the presenceof a solvent of small molecules.In this case the total interaction energy could beobtained by applying eqn. (1) and (3) to the interaction of segments of the twopolymers, summing over the polymer chains, and then averaging over their distribu-tions. It would, of course, be more in keeping with the spirit of the Lifshitz theory toapply it directly to the polymer situation. However, in either version of the theory,the sign of the long-range dispersion forces will be determined by an expressionsimilar to (3), and in the simplest case by the sign of (6).TABLE 1POLYMERSpolybutadiene (1)polystyrene (2)SOLVENTS40)2.202.32sign of experimentallyAqAez compatibleXXtoluene 2.30 -carbon tetrachloride 2.24 -cyclohexane 2.01 + ?It therefore seems worth-while to see if the data on compatibility in ternary solu-tions correlate with the sign of the dielectric product (6).A preliminary examinationfor the polymers polybutadiene (PB) and polystyrene (PS) is given in the table,2 butA. D. McLachlan, Disc. Faraday SOC., 1965, 40,239.I am grateful to Dr. D. J. Buckley of Esso Research Laboratories, Linden, N.J., for drawingmy attention to the phenomenon, and to him and Dr. E. 0. Forster for the data in thetableGENERAL DISCUSSION 179is inconclusive in the absence of a solvent in which they are compatible. It is well-known that most polymers are incompatible in most solvents.According to the ideasuggested here, PB and PS should be compatible in cyclohexane, provided they aresoluble. It may well be, and indeed it is likely, that a negative value of the van derWaals coefficient C will only mean that immiscibility is delayed, and can still occur athigher concentrations, when many more short-range interactions between the polymerchains are possible. I should like to ask Koningsveld if he has any experimentalresults which could confirm or contradict the above theory.Prof. R. L. Scott (University of California, Los Angeles) (communicated): ByersBrown’s provocative comments do not seem directly relevant to the compatibility oftwo polymers in a given solvent. Phase separation depends primarily upon aninterchange energy which is proportional to (Cll + C22-2C12), where Cap is theconstant in Byers Brown’s eqn (1) for the interaction between two molecules a and p.If the sign of Cap is given by eqn (6) and if we may further assume approximate pro-portionality, thenThe permittivity c0 of the solvent has disappeared from the equation, suggesting thatthe detailed nature of the solvent is largely immaterial, in agreement with earliertreatments of compatibi1ity.l.The data in Byers Brown’s table 1 are not incon-sistent with this conclusion since the behaviour in cyclohexane is unknown.Dr. R. Koningsveld (Geleen) said: Byers Brown’s suggestion is certainly worthinvestigating. Macromolecular systems have considerable technological importanceand any method allowing predictions to be made about their possible states would bewelcome.One would like to be able to estimate values of the interaction parametersg (see eqn (1) of our paper) without elaborate measurements being required. How-ever, also in view of earlier attempts in this direction, e.g., by Sta~erman,~ Smalland B~rrell,~ there are no grounds for great optimism. Miscibility of macromolecularsystems is governed by a subtle balance between the entropy and enthalpy of mixing.Considerable changes in phase relations may proceed with very small variations ofthe g. This is particularly true if two polymers are involved. Fig. 6 of our papershows that a concentration dependence of g may then give rise to drastic changes inphase behaviour.The end of Byers Brown’s comment also suggests this. It seemsto be questionable whether the proposed procedure will result in the high precisionneeded. At least for the system diphenylether + polyethylene +polypropylene, thesign of AelAc2 does not seem to be sufficient. Estimated values of the E in thetemperature range of interest are 1.8 for each of the two polymers and about 2.6 forthe solvent. Yet it is not only that the system is partially miscible, it also showstwo miscibility gaps. One of these is absent if the polypropylene is replaced by theatactic variety (ref. (19), our paper), which, however, does not affect the E values.It may be that a more refined treatment will yieId better results, but it remains anopen question whether it would reveal such details.Nevertheless, no effort shouldbe dropped before its working value has been established. It cannot be stressedenough that the whole search is for subtle effects so that it is inadmissible to ignorethe marked influence of polydispersity and concentration dependence of g. In fact,.W. H. Stockmayer, A.C.S. Meeting (Atlantic City, N.J., April, 1949).R. L. Scott, J. Chem. Phys., 1949, 17, 279.A. J. Staverman, Physica, 1937,4, 1141 ; Verfironiek, 1962,35,284.P. A. Small, J. Appl. Chem., 1953, 3, 71.H. Burrell, Oficial Digest, 1955, 726; 1957, 1069, 1159180 GENERAL DISCUSSIONone, or both, of these features may well be the cause of the discrepancies (a), (6) and(c) that Byers Brown cites from the work of Allen, Gee and Nicholson (ref. (31) ofour paper).Prof. W. H. Stockmayer (Dartmouth College, U.S.A.) said: One might fear thatnon-classical behaviour in the critical region (of the type well known for non-polymericsystems) might vitiate some of the quantitative conclusions based on critical misci-bility measurements. Apparently there have been no theoretical considerations ofthis question for polymer solutions. In practice, the fear above expressed fortu-nately appears groundless, to judge by the excellent agreement which Koningsveldobtains with other methods of determining interaction parameters.Prof. J. S. Rowlinson (Imperial College, London) said: I agree with Stockmayerthat the non-classical singularities that have been found at the critical points of binarymixtures of non-polymeric substances will surely be present also in multicomponentpolymeric solutions of the kind discussed by Koningsveld. Nevertheless the classicalconditions for a critical point, eqn (4) and (5) of Koningsveld’s paper, remain asnecessary conditions which must be satisfied even if there are additional (and weaker)non-classical singularities.Dr. R. Koningsveld (Geleen) said: One comment of a general nature might beadded. In polymer solution work one frequently comes across the terms lower andupper critical solution temperature, which, in many cases, are used to denote the1559I501452 4 6 8 1 0polyethylene concentration ( % by weight)FIG. 1.-Cloud-point curve for an n-hexane+polyethylene system (Marlex 6009 ; Mn = 8 x lo3 ;Mw = 1.8 x 10’ ; Mz = 1 . 2 ~ lo6). The dashed curve indicates the estimated location of the quasi-binary spinodal. The critical point (black dot) was determined by the phase-volume ratio method.threshold temperatures. It is not mere hair-splitting to object against this error.The figure presented herewith, referring to a system composed of n-hexane and alinear polyethylene, shows that the threshold and critical temperatures may diffeGENERAL DISCUSSION 181by about 10°C and the threshold and critical concentrations by a factor 6.l FollowingTompa,2 one would prefer that future references be made to upper and lower criticalsolution (or miscibility) phenomena, which are associated with lower and upperprecipitation thresholds (LPT, UPT). In quantitative theoretical treatments, inparticular, thresholds and critical states should not be mixed up, in other words, thepolydispersity of the polymer should always be taken into account.M. Gordon, H. A. G. Chermand and R. Koningsveld, Macromolecules, 1969, 2, 207.H. Tompa, Trans. Faraday Soc., 1950,46,970

ISSN:0366-9033    

DOI:10.1039/DF9704900162

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

3. Dynamic Properties of SolutionsModels for Chain Molecule Dynamics in Dilute Solution*BY W. H. STOCKMAYER, W. GOBUSH, Y. CHIKAHISA AND D. K. CARPENTERDept. of Chemistry, Dartmouth College, Hanover, New Hampshire 03755, USAReceived 27th February, 1970The behaviour of the Zimm model for flexible chain dynamics is illustrated by the dielectricresponse for arbitrary field strength and by a calculation of the intrinsic viscosity from an appropriatetinie correlation function. The stochastic dynamics of a chain with constant bond Iengths and weakcorrelations between adjacent bond directions are developed and applied to the dielectric experiment.Introduction of rotational diffusion in parallel to local conformational changes appears to offer auseful approach to the problem of internal viscosity.The purpose of this paper is twofold : (i) to add a few remarks to the extensiveliterature dealing with the bead-and-spring models of Rouse, Zimm and others fordescribing the conformational dynamics of long chains in solution; and then (ii) toillustrate more fully some of the properties of alternate stochastic models, which mayturn out to be more convenient than the spring models for some applications andin any event offer some conceptual advantages.No attempt is made to discuss applica-tions of the models to all the possible experimental techniques, and the selection isthus somewhat personal. Only linear molecules obeying gaussian equilibrium chainstatistics are considered specifically.THE BEAD-SPRING MODELSThe reader is assumed to be familiar with the models of Rouse and Zimm.2In the former, hydrodynamic interactions between chain elements are ignored, whilein the latter these have been introduced in a pre-averaged representation introducedearlier by Kirkwood and Ri~eman.~ The premature averaging has the effect ofmaintaining linearity in the chain coordinates and thus preserving the normal-coordinate transformation which paves the way to useful practical results. It canapparently sometimes lead to non-physical results 4 9 when examined with care,but this seems not to cause trouble with any of the final formulae for transport co-efficients or relaxation times.An alternative to the pre-averaged Oseen interactionhas recently been offered by Rotne and PragerY6 but it is harder to apply.In matrix form the diffusion equation for the Zimm model may be written as(1)wheref(r,t) is the distribution function at time t for the coordinates {ro,rl, .. .,rN]of the N f l beads of the model. The symbol r denotes the column vector whoseelements are the bead coordinates. Further, vo is a column vector giving the fluidvelocity (when unperturbed by the hydrodynamic interactions between beads) atthe locus of each bead, 5 is the friction constant of a bead and b is the root-mean-square displacement length of a spring (c' submolecule ") between two successive* supported by a grant from the U.S. National Science Foundation182i3flat = -VT fvo + VT(kT/()H [Vf + f VUJkT+ 3b-2fAr]STOCKMAYER, GOBUSH, CHIKAHISA AND CARPENTER 183beads.The potential energy Ue describes the effect of external fields, and the N x Nsquare matrix A, with elementsexpresses the linear sequence of spring forces between the beads. (Here, as through-out this paper, we shall assume the chains to be long enough so that end effects maybe neglected.) Finally, the hydrodynamic-interaction matrix H has the structureA,, = 2 ; A,,jfr = - 1 ; zero otherwise, (2)H j k = j k + (l- jk)(1/6.7E.rfO)(rG1>= 6 j k + (I - 6 jk)/(6n3)3bqo I j - k I *. (3)This model exhibits no shear dependence of the intrinsic viscosity,2 and an explicitsolution to eqn. (1) can be written down for arbitrarily large shear rates. Corres-pondingly, the deformation of the molecule during flow is predicted to be withoutlimit, in strong contradiction to the experimental indications.' An analogous defectis found for an electric field.Let the electric charges eo, el, . . ., e N of the beadsform the elements of a column vector e, and place the solution in an electric fieldF, which may depend on time but not on position. The potential energy is thusU, = -eTr F. We use this in eqn. (1) and introduce the normal coordinates Rby the transformation r = QR, obtaining after some rearrangement,where E = QTe, y = F/kT, and the matrices A = Q-lHAQ and N = Q-lHQ-lTare diagonal. For the steady state in an alternating field of any magnitude, thesolution can be verified to be(C/kT)af/at = (a/aR)= [N(af/aR) - Nfgy- 3 b - 2 A . ] (4)f = exp C-(3/2b2)Dp(Rp- (Rp>)219 ( 5 )(R,) = (yepb2/3pp)(l + iwz;)-l : 7; = b2c/3kTAp.(6)Pwhere the ,up are the elements of the diagonal matrix M = QTAQ andAs the last equation shows, the chain is on the average deformed in direct proportionto the field, without limit, as is to be expected for gaussian springs. This is hardly apractical concern in the dielectric case as it is in viscoelasticity.Our other excercise with the Zimm model is to compute the intrinsic viscosityfrom an appropriate time correlation function. We start from the general resultfor the shear viscosity of a fluidq = (&)I:( Jxy(t)JxY(O)> exp (- iot)dt,in whichJxy = xmi2ii)i + Ey,F$"),i ithe sum being over all particles in the volume. V. The ith particle has mass m,,velocity components SI and pi, and is acted on by the x-component of force Fp).The time correlation function of Jxy is over an equilibrium ensemble, and the viscosityis for an oscillating shear of circular frequency o.To apply the above result to a dilute polymer solution, we recognize that onlythe dzJierence in J between the solution and the pure solvent is important, and alsothat at infinite dilution each polymer chain contributes independently. We thusfin184 MODELS FOR CHAIN MOLECULE DYNAMICSwhere N is Avogadro’s number, M is the polymer molecular weight and qo is thesolvent viscosity. Most important, AJm is defined as due entirely to the mutualinteractions among the elements of a single chain molecule.Eqn (8) defines J in terms of both coordinates and velocities of the particles.The diffusion eqn (I), on the other hand, treats only of a distribution in coordinatespace; and indeed this is the case with almost all present approaches to polymerchain dynamics. To apply the correlation function recipe directly, it would thereforebe necessary to consider the Brownian motion of the chain in the full phase space.gThe construction of Langevin equations for chains in coordinate space or phasespace is discussed by 2wanzig.l’ It seems possible, however, to avoid this step,and instead to pass from eqn (8) to a purely coordinate-space definition of Jxy by asuitable averaging over the momentum space, as in the standard 9 9 l 1 theory ofBrownian motion.We believe that when this is done the result (which we have notyet succeeded in establishing in general) will be the following :where the sum is over all elements (segments, beads, or whatever they are called)of a polymer chain, and the two force terms are respectively called “ diffusion ”and “ bond ” contributions.The former, which is directly related to the velocityterm in the original specification of Jxy, is justFYI’ = - kT(a In f /axi),while the second isFYI‘’ = - aUin,/axi,in which Uint is the intramolecular potential energy governing the bonding in thechain. A simple test of eqn (10)1(12) is afforded by the rigid dumb-bell, for whichthe correct intrinsic viscosity is obtained. When the Zimm model is used, one findsfrom eqn (l), for the field-free zero-flow case, that the “ diffusion ” term vanishes ;and the other term givesJxy = CyiFP’ = yT(3kT/b2)Ax = (3kT/b2)YTMX, (13)iwhere X and Y are components of the normal coordinates R.Therefore, invokingthe orthogonality of the normal coordinates, we find<Jxy(t>JXy(o)> = ( 3 w b 2 ) 2 z P ; < Yp(0 yp(o)>(xp(t)xp~o~>P= (kT)2z exp (- t/zp> (14)(15)Pin which the relaxation times are half those in the earlier dielectric example :zP = 7;/2 = b2[/6kThp.The correlation functions used in obtaining the above result can be reconstructedfrom eqn (1) or may be found in a paper of Verdier.12 Substitution of eqn (14)into (9) leads to the well-known formula[rl = (NkTIMr,)C~p(l+ ioZ,)- l, (16)Pwhich has usually been obtained by solving eqn (1) with vo specified for simple shearflow.A correlation-function treatment of undiluted polymer (for which the free-draining Rouse model is appropriate, as shown by Bueche 13) has been given by Pao.14Many refinements of the bead-spring models must unfortunately be omitted fromthe present discussion. We mention in passing particularly the consideration oSTOCKMAYER, GOBUSH, CHIKAHISA A N D CARPENTER 185stiff-chain models by Harris and Hearst l5 and by Saito and coworkers l6 ; theperturbation theory of Fixman l7 which permitted use of the more correct un-averagedform of the Oseen hydrodynamic interaction formula in a calculation of the vis-cosity ; and the boson representation of Fixman l9 which permitted introductionof a good approximation to the excluded volume effect between different parts ofthe chain.LOCAL-JUMP STOCHASTIC MODELThe close connection between the problem of random flights and the diffusionequation has long been known,g and it seems natural therefore to investigate modelsin which chain motion results from repeated local segmental rearrangements. Itappears that Verdier 2o first exploited this possibility (which had been pointed outearlier by G.W. King) in his Monte Carlo study of chains moving on simple cubiclattices. Since the effects of excluded volume were included in this and some ofVerdier's later work,12 a purely analytical approach was not convenient or easy.Orwoll and Stockmayer 21 produced analytical treatments of a freely-jointed non-lattice chain (without excluded volume) and also of a one-dimensional chain modelwith correlations of arbitrary strength between the directions of successive links.They worked entirely with average values of chain coordinates and did not derive adiffusion equation.Independently, Monnerie 22 treated chains on diamond lattices ;while Iwata and Kurata showed 23 that the Verdier lattice chain without excludedvolume leads, for the longer relaxation processes, directly to the Rouse diffusionequation. In the following paragraphs we give an account of a slightly modifiedversion of the Orwoll-Stockmayer model.Let the chain molecule consist of Nf 1 beads, joined by N bonds each of lengthb. The direction of the bond from bead i-1 to bead i is given by the unit vector al.If the spatial position vector of the zeroth bead is ro, the location and conformationof the chain is then specified by the set of N+ 1 vectors (ro,al,az,.. .,aN}.Forinterior beads (i # 0 or N ) the motion consists of a jump or " flip " whereby thevectors cI and af+l exchange values : if a prime is used to denote the bond vectorsafter such a flip, the process corresponds toThis simple process is essentially the same as that used by Verdier 20* l2 in his workon cublic lattice chains. The terminal beads require a different specification of theirmotion, but we shall be concerned here only with long chains and thus do not troublewith end effects. In what follows, we shall also avoid explicit consideration of thespatial location of the chain and deal only with its conformation.Let the probability that the chain at time t has the conformation (c1,a2,.. .,aN} =(a"] be designated by p(a",t). If w,dt denotes the probability that bead i willexecute a flip during the time interval dt from the above conformation to the n6wconformation (. . .,~:,af+~,. . .}, the time evolution obeys the master equationTo vary the chain conformation, beads are allowed to move one at a time.6; = ai+l and G ; + ~ = ai. (17)ap(aN,t)/ar = - p(aN,t)zwi + C W i P ( . . . ,q+ l,bi,. . . t).i iThe last term follows from (17) ; it is slightly inaccurate because we have ignored thespecial condition of the terminal beads.A choice of the flip probabilities wi must now be made. In our earlier work onthe freely-jointed chain 21 we employed the formwi = a(l --ai ali J186 MODELS FOR CHAIN MOLECULE DYNAMICSwith I a I < 1, which permitted weighting the mobility of a bead according to theinstantaneous bond angle.This feature however, adds no results of interest; forchains in three dimensions it simply increases the rates of all relaxation processes bya factor 1 +(a/3), and we shall not retain it further. Here we focus attention onchains which have a weak correlation in the directions of nearest-neighbour links.Putting the equilibrium value of this correlation as(19)we recall that the equilibrium mean-square length of the chain is, in general,(20)but for the kinetic analysis to follow we must keep /? small. We then specify thefollowing expression for the flip probabilities of interior beads :<a, GlS1)eq = B,<r2)e, = Nb2(1 +8)1(1 -P) ;Wi =a(I+3BG,-i *Gi+1+3BGi* Gi+2).(21)This apparently arbitrary choice appears to be the simptest one that is consistent withthe requirement of microscopic reversibility. This consistency is demonstrated inthe appendix.We now focus attention on the first moments of the non-equilibrium distribution.The average value of one of the bond vectors isTo evaluate the time-dependence of this quantity, we multiply the master equation(1 8) by G~ and integrate over all configuration space. Noting that the jth bond maybe re-oriented by flips of both beadj- 1 and beadj, we observe that this can be writtenasdqj/dt = <aj-lw(aj-l,a~)+ai+lW(Gj,bj+ l ) - ~ j w ( e j - I,aj)-ajw(aj,aj+ I)), (23)where the averages are over the non-equilibrium ensemble.Combining this equationwith the specified flip probability of (21), we finda- 'dqj/dt = - 2qj + qj - 1 + qj + 1 + 3B[ (Gj- I(Gj- 2 ~ j ) ) + ( ~ j - 1(Gj- 1 ~j + 1)) +( G j + l ( G j - l Gj+J)+(Gj+l(Gj. e j + 2 ) ) - ( ~ j ( ~ j - 2 Gj)>-(Gj(Gj Gj+d)-2<oj(@j-l ~ j + l ) ) I * (24)With neglect of all terms of Ow2), the triple correlations which appear in (24) arejust those of a freely-jointed chain, which are easily evaluated (see appendix) interms of linear averages. The final result is thenIn more compact form, this can be written aswhere q is a column vector whose elements are all the qr and B is a square N x Nmatrix with elementsa-ldql/dt = -2Qr+(1 +B)(e-l +%+l)-B(q,-2+e+2).(25)&dq/dt = -Bq, (26)B , = 2,BtJ = -(I + p ) for i = j + l ,= fl for i= j+2,= 0 otherwiseSTOCKMAYER, GOBUSH, CHIKAHISA A N D CARPENTERThe matrix B is diagonalized by the transformation Q-lBQ = A, whereand the eigenvalues of B areIn the above result, we have assumed N large and thus ignored any distinction betweenN and N + 1, and as always have suppressed terms of O(/?”).The normal coordinates 4Jt) related to average chain coordinates form a columnvector 5 defined by the orthogonal transformation187Q j p = (2/NIa sin ( j p d N ) ,An = 4(1- 3p) sin2 (pnl2N) + 168 sin4 (pn12N).(28)(29)The normal coordinates therefore relax according towithThe slower normal modes ( p <N) display the same relaxation spectrum as a Rousechain or a freely-jointed chain & starting out proportional to p 2 , but deviations setin for the faster modes.From (29) and (31) the ratio of longest to shortest relaxationtimes is given byThus, the over-all relaxation spectrum is broadened as the energetic preference forextended conformations (p positive) is increased.Although we have avoided direct consideration of the displacement of the chainas a whole, it is not difficult to relate the model parameters to the translationaldiffusion coefficient D. We may evaluate the mean square displacement per unittime of the centre of mass as a result of the flip process, taken over an equilibriumensemble. This iszl/N2zN = 4(1 +4/3)/n2. (32)in which the displacement of bead i at a flip isIf we use the flip probability of (21) and perform the averages with the aid of theequilibrium distribution of (Al-4), we obtainwhere the last equality illustrates the connection between the basic flip frequency aand the effective friction constant C of a bead in the model.We also note that forthe slow relaxations (p < N ) we may combine (31) with (20) and (34) to getwhich is precisely the Rouse-model result.With the exception of an occasional factor of 3, all the above results are the sameas those given earlier for a one-dimensional chain when these are taken only toterms of first order in /I. As yet, we have obtained no useful results for strongcorrelations in more than one dimension. In any case, this crude model showsexplicitly that local chain structure must explicitly influence the high-frequency partof the relaxation spectrum.Some further embellishments of the model may be of interest.Thus, it is easyAri = b(~$ - GJ = b ( ~ i + 1 - ~i).D = ab2(1 -p)IN = KTINC, (34)l/TI, = 3n2p2D/(r2),, (35188 MODELS FOR CHAIN MOLECULE DYNAMICSto treat heterogeneous chains with different flip rates within the repeat unit. Thegeneralization is precisely the same as for heterogeneous one-dimensional lattices.As in that case, the spectrum (of relaxation as contrasted to vibration) falls into twoor more branches. Conceivably such behaviour could be observed in some realheteratomic polymeric chains.The elementary flip process expressed by eqn (17) is not the only possible onethat might be considered for non-lattice chains.For example, the jump might bedefined to rotate the plane defined by vectors q and G ~ + ~ through some angle lessthan 180". In general, such jumps will not lead to simple results, but two specialcases may be worth noting, although the results at present are not complete : (a) ifthe jumps are equally frequent for all rotation angles between zero and 180", therelaxation spectrum can be obtained and is identical with the case worked out above ;and (b) if the jumps are restricted to very small angles, the master equation shouldpass over into an appropriate diffusion equation in the space of the bond vectors.(The original 180" process developed above is not ergodic : bond directions are onlyexchanged among the N members of the set but no new directions can develop.Physically this appears innocuous.)Introduction of the Oseen-Kirkwood-Riseman hydrodynamic interactionsbetween pairs of chain elements is possible in the scalar average approximationcorresponding to the explicit calculations of Kirkwood and Riseman and of Zimm.Since these interactions are of prime importance only for the slowest relaxationmodes, which are identical to those of the bead-spring model in the free-drainingcase, no new results or predictions of consequence are to be expected from such atreatment.The prediction of relaxation and response functions for specific applicationsof the foregoing stochastic model is straightforward for tensile and dielectric relaxation.In these cases the appropriate time correlation function can be computed from theresults given above in straightforward fashion ; the earlier paper gives illustrationsfor the one-dimensional chain.Alternatively, it is possible to return directly to themaster equation and employ suitably modified jump probabilities ; this is a procedureanalogous to considering the diffusion equation of the bead-spring model in thepresence of an external field. As an example, we take the dielectric case, whichinvolves a simple generalization of the treatment given by Debye 24 for a two-positionmodel of a single dipole. Let the electric moment fixed to bond i be a vector m , q ,and place the chain in an uniform electric field F, which may be time-dependent.The flip probabilities must now be taken aswhere wp refers to the field-free expression of (21) and y = F/kT.The above expres-sion is correct only to terms linear in the electric field. It satisfies the requirementsof microscopic reversibility. When the above equation for wi is used in the masterequation and the linear average of the bond coordinates is taken, the result in matrixform isHere rn is a column vector whose elements are the mf, and the square matrix A is thefamiliar Rouse matrix, equal to the B matrix for p = 0.Now the ensemble average dipole moment for the chain is given by { p) = rnT<a) =mTq, where mT is the transpose of m. In an alternating field with circular frequencycr) this iswi(Bi,n.i+ 1) = wP(ai,ai+ 1)(1 -miai Y -mi+ 1bi+ 1 Y), (36)a-ldqldt = -Bq+(l -P)Amy/3.(37){ p} = (y/3)CP2,[ 1 + 2 cos (np/N)] [ 1 + COT;] - , (38)STOCKMAYER, GOBUSH, CHIKAHISA AND CARPENTERin which the elements of the column vector Qm are symbolized by189A few details are shown in the appendix. Except for effect of the bond correlationsas explicitly given by the terms in p, this equation resembles that found 25* 26 for theRouse-Zimm chain, and it gives similar predictions for specific dipolar structures ;for example, if all bond dipoles point along the chain in the same direction, then thedominant relaxation mode is the longest ( p = 1); while if the dipoles alternate indirection it is the shortest ( p = N ) .Neither the above stochastic model nor the bead-spring model allows rotationaldiffusion of the macromolecule without attendant changes of conformation ; ineach case there is only one elementary dissipative process.Thus these models donot provide for the possibility that relative rates of rotational and conformationaldiffusion might vary greatly with chain structure. This limitation is of no consequencein the theory of the zero-frequency Newtonian viscosity of polymer solutions, but itbecomes important for the effects of frequency or shear rate on viscosity as well as onother properties. A second kinetic parameter, the " internal viscosity ", was intro-duced for this purpose by Kuhn and K ~ h n , ~ ' and the concept has been developedby Cerf 28 and later by Pete~-lin.~~ In the most successful form of the theory, thenormal modes of the pristine bead-spring model are retained, but each mode isprovided with an additional damping term inversely proportional to mode number,i.e., the internal viscosity has the greatest effect on the most localized motions.Although this device unquestionably is a step in the right direction, it is beset withconceptual difficulties which are well summarized in a review by Jane~chitz-Kriegl.~~An alternative approach by Budtov and Gotlib 31 includes the fact that in a stricttreatment one must introduce non-linear terms which mix the normal modes, but it islimited to small perturbations.A different approximate approach to the above problem is now offered.It isbased on the assumption that for long chains the rotational diffusion tensor onlyrarely departs widely from spherical symmetry.We can thus generalize the previously-described stochastic model by introducing rotational diffusion of the entire chain asa second independent fundamental dissipative process in parallel to the local confor-mational flips, and by assuming that this process corresponds to the rotationaldiffusion of a sphere with an invariant rotational diffusion constant 0,. Theconvenience of this approximation is that the normal coordinates for the conforma-tional rearrangements are preserved in the rotating frame; and its physical basis isconceptually simple.When the above notion is introduced into the model, eqn (26) must be amendedto&dq/dt = - (B+ 2D&. (40)11~; = 4,+2D,. (41)The normal coordinates are unchanged, but the relaxation times are now given byDetailed applications of this extended model will be worked out elsewhere (e.g.,frequency dependence of the intrinsic viscosity), but one illustration may suffice toshow its possible utility.As mentioned earlier, chains with bond dipoles alternatingin direction are predicted to show dielectric dispersion at a frequency correspondingto the shortest normal-mode relaxation time zN. According to (31) this is independentof chain length, but in the modified eqn (41) for sufficiently short or sufficientl190 MODELS FOR CHAIN MOLECULE DYNAMICSsluggish chains (a < D,) the relaxation process goes over into rotational diffusion,which depends significantly on chain length. Such situations are known experi-mentally.26* 32For example, therivalry between conformational and rotational diffusion must always be greatest forrelatively short chains, and if the equilibrium conformational statistics show strongdepartures from gaussian behaviour in the significant range of molecular weights themodel is surely untrustworthy.Finally, reference should be made to a discussion by Williams 33 of dielectricrelaxation in relatively simple molecules capable of inversion or internal rotation,with results resembling that of eqn (41).Some limitations of the extended model will be apparent.APPENDIX1.FLIP RATES FOR CHAIN WITH WEAK BOND CORRELATIONSIf correlatons between the directions of different chain links extend only to those betweennearest neighbours, the equilibrium configurational probability of the chain has the formN-1pe4(cN) = const.n u(zi); zi = ci . ci+l, ( A 0i = 1where U(zi) = exp [- E(zj)/kT] is the statistical weight or Boltzmann factor for a given bondpair. The simplest possible dependence of the correlation energy on zi is linear, and so ifthe correlations are weak we may putE(Zj) = -E’zj+. . ;U(ZJ = 1 + (E’/kT)zt+. . . . (A21in which case a positive value of E’ would signify that obtuse bond angles are favoured, asmight best permit the model to mimic real chains. It follows thatp = (zi) = SI:r,a(zi)d~i/S]:u(ri)dzi = (&)+ ..., (A31so that (A2) may be rewritten asU(Zi) = 1+3Pzj+. . STOCKMAYER, GOBUSH, CHIKAHISA A N D CARPENTER 19 12. TRIPLE CORRELATIONS I N THE FREELY-JOINTED CHAINWith unit Cartesian vectors ex, e,, and e, we have<ai(ai a j > ) = ex(xTxj + xiyiyj + xizizj) + ey(xix,x + y:yi + yjzizj> +ez(zixixj+ziyjyj+ z:zj).At equilibrium the bond directions in the freely jointed chain are not correlated, so that(x2xj) = ( x f ) ( x j ) = ( x j ) / 3 , etc.; and (xiyiy,) = ( X i u i ) ( y j ) = 0.If all three bonds in a triple correlation are different, as in (ai(aj ak)), the average vanishes.Thus (24) of the main text leads to (25).Hence(aj(ai GI)) = ( ~ j ) / 3 .(A913. DIELECTRIC RESPONSEFor an alternating field the steady-state solution of (37) takes the form(a-liu+B)q = ( ~ / 3 ) ( 1 -P)Am. (A 10)Now we introduce the normal coordinates by q = Q4 and multiply both sides of (A10) byQ-' to get(a-liw+A)t = (y/3)(1 -P)Q-'Arn= (y/3)(1 -P>A"P, ( A l l )( A 12)in which Q-'BQ = A, Q-lAQ = A', and Q-lm = P.Thusin which the property Q-I = Q = QT has been invoked. From the eigenvalues (29) wefind, to first order terms in p,(p) = mTq = PT4 = (y/3)(1-p)PTAo(a-'io+A)-'P(1 - j3)Ai/J., = 1 + 2p cos (np/N) + . . . , (A13)which Ieads at once to (38).( ~ ' ) ~ ~ y / 3 . From (381, we have to evaluateCP;[1+2/3 cos (np/N) =It is interesting to check on the requirement that at zero frequency we must get (p.) =PN N N c c mjmk( $) sin 2 sin *[ N 1 + 2p c o s z ] . (A14)p = l j = 1 k = lWith the aid of the identitiesandthe foregoing triple sum can be reduced toN NBut alsowhich agrees to first order in p192 MODELS FOR CHAIN MOLECULE DYNAMICSP.E. Rouse, J. Chem. Phys., 1953,21,1272.B. H . Zimm, J. Chem. Phys., 1956,24,269.J. G. Kirkwood and J. Riseman, J. Chem. Phys., 1948,16,565.R. E. DeWames, W. F. Hall and M. C. Shen, J. Chem. Phys., 1967,46,2782.R. Zwanzig, J. Kiefer and G. H. Weiss, Proc. Nat. Acad. Sci., 1968, 60,381.J. Rotne and S. Pragner, J. Chem. Phys., 1969,50,4831.F. R. Cottrell, E. W. Merrill and K. A. Smith, J. Polymer Sci., A-2, 1969, 7,1415. * R. Zwanzig, Ann. Rev. Phys. Chem., 1965,16,67.S. Chandrasekhar, Rev. Mod. Phys., 1943, 15,l.1969, 15,305.lo R. Zwanzig, in Stochastic Processes in Chemical Physics, K. E. Shuler, ed.; Adu. Chem. Phys.,l 1 H. D. Brinkman, Physica, 1956, 22,29.l2 P. H. Verdier, J. Chem. Phys., 1966,45,2118, 2122.l3 F. Bueche, J. Chem. Phys., 1952,20,1959.l4 Y. H. Pao, J. Macromol. Sci., B, 1967, 1,289.l6 N. Saito, K. Takahshi and Y. Yunoki, J. Phys. SOC. Japan, 1967, 22,219.l7 M. Fixman, J. Chem. Phys., 1965,42,3821.l9 M . Fixman, J. Chem. Phys., 1966,45,785, 793.2o P. H. Verdier and W. H . Stockmayer, J. Chem. Phys., 1962,36,227.21 R. A. Orwoll and W. H. Stockmayer, in Stochastic Processes in ChemicalPhysics, K. E. Shuler,22 K. Iwata and M. Kurata, J. Chem. Phys., 1969,50,4008.23 L. Monnerie, I. U.P.A.C. Symp. Macromol. Chem. (Toronto, 1968).24 P. Debye, Polar Molecules, 1929 (Dover Publication (reprint), New York), chap. V.25 W. H. Stockmayer and M. E. Baur, J. Amer. Chem. SOC., 1964,86,3485.26 W. H. Stockmayer, Pure. Appl. Chem., 1967, 15,539.27 W. Kuhn and H. Kuhn, Helu. chim. Acta, 1945,28,1533.28 R. Cerf, J. Chim. Phys., 1969,66,479.29 A. Peterlin, J. Polymer Sci., A-2, 1967, 5,179.30 H. Janeschitz-Kriegl, Adv. Polymer Sci., 1969, 6,170.31 V. P. Budtov and Yu. Ya. Gotlib, Yysokomolekul. Soedin., 1965,7,478.32 T. W. Bates, K. J. Ivin and G. Williams, Trans. Faraday Soc., 1967, 63,1694.33 G. Williams, Trans. Faraday Soc., 1968, 64,1219.R. A. Harris and J. E. Hearst, J. Chem. Phys., 1966, 44,2595.C. W. Pyun and M. Fixman, J. Chem. Phys., 1965,42,3838.ed.; Ado. Chem. Phys., 1969, 15,305

ISSN:0366-9033    

DOI:10.1039/DF9704900182

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Low Frequency Molecular Modes in Liquid HydrocarbonsBY P. A. EGELSTAFF* AND D. H. C. HARRIS"Materials Physics Division, A.E.R.E., HarwellReceived 16th January, 1970The spectral distribution of slow neutrons scattered at low angles by straight chain hydrocarbonliquids has been measured. For pentane, measurements were made at six different points in PVTspace, and the data analyzed by comparison with various models. Properties studied include thediffusion coefficient, the activation energy for diffusion, the time for a diffusive step, the size of aproton's thermal cloud and the spectral density for low frequency ( ~ 1 0 ' ~ Hz) modes of motion.The diffusion data agree well with those obtained by other techniques but the magnitude of the diffus-ive step time is not well determined for these liquids.A pronounced peak in the low frequencyregion of the spectral density function has been discovered, and may be associated with short timelocal bonding between molecules.The thermal motions of atoms and molecules in liquids is a poorly understoodsubject. For many years the methods of study consisted of macroscopic measure-ments (e.g., of diffusion and viscosity coefficients) supplemented by the techniques ofnuclear magnetic resonance and light scattering which provide information in limitedfrequency ranges only. However, recently neutron scattering (e.g., Egelstaff hasbeen developed to the stage where it can provide dynamical information on a time anddistance scale suitable for the study of atomic motions.In particular, because of thehigh scattering cross section for protons and because their scattering is incoherent,hydrogeneous liquids have been a favourite subject of study. For example, Larssonet aL2 studied glycerol over a range of temperatures, and pentane in detail at lowtemperatures. Saunderson and Rainey have studied a number of alcohols at twotemperatures. In addition, considerable work has been devoted to the study of water(e.g., review by Larsson 4).Much of this work has been dominated by two features : first, the influence of thehydrogen bond, and secondly, experimental limitations, e.g., set by the relatively poorresolution and the non-constancy of the momentum transfer. Nevertheless, theneutron method can separate short and long time motions, that is separate diffusivefrom vibrational or librational modes.AlsoY29 sometimes there is an apparentdiffusion constant for molecules (if measured on a scale of s) which is greaterthan that measured by the macroscopic methods.In the present series of experiments several improvements over the previous workhave been achieved. First, through the use of a multi-chopper time-of-flight spectro-meter a higher resolution of both incident and scattered neutron beams has beenobtained. Secondly, through the use of incident energies large compared with theexpected energy transfer, it has been possible to work with the momentum transferalmost constant at each angle of scatter. Thirdly, in order to simplify the interpreta-tion non-hydrogen bonded liquids were chosen for study and measurements weremade using three different straight-chain hydrocarbons.Finally, the experiments onpentane covered both the liquid and dense gas regions (see fig. 1).* experimental work carried out while on attachment (1965/66) to Atomic Energy of CanadaLtd. Chalk River, Ontario.0 191947 5 'E2+a ---8 5 0 'aMOLECULAR MODES I N LIQUID HYDROCARBONSX213360)(I900 0.2 0.4 0.6 0.8density/(g ~ r n - ~ )FIG. 1.-State diagram showing the relative positions of the six runs on pentane. X indicates thepressure and density at each temperature ("C shown), and the dashed line marks the liquid-gas co-existence curve.The results show that in these systems the diffusion constant at room temperatureand for times near 10-l2 s is the same (within the experimental error) as that obtainedfrom macroscopic measurements.It has also provided information on the spectraldensity of the velocity correlation function for the low frequency modes of motion ofthe whole molecule. New methods of analyzing such data are discussed and used inthis paper.THEORETICAL BACKGROUNDThe scattering of neutrons by condensed systems has been treated by Van Hove,5and expressed in terms of space-time correlation functions. For hydrocarbons weneglect the coherent scattering by hydrogen (2 barns) and carbon (5 barns) and discussonly the incoherent scattering by hydrogen (80 barns) ; then differential scatteringcross section can be written asd20/dlRdw = b2(k/ko)Ss( Q,w), (1)where Q2 = k2+kg-2kko cos 8 ; o = h(ki$-k2)/2in ; ko and k are the wavevectors of the neutron before and after scattering, m is the neutron mass, b is theincoherent scattering length of hydrogen, 8 the scattering angle and SZ is a solid angle.The function S,( Q,w) is the Fourier transformation of the self-correlation functionG, of the hydrogen nuclei, i.e.,1 Pwhere G,(r,t) is the chance of finding a proton at r at time t, if it was at the origin at atime denoted by t = 0P .A ~ G E L S T A F F AND D. H . c . HARRIS 195We are concerned with contributions to G,(r,t) arising from translational vibra-tional and rotational motions, and if we assume that the rotational motions are notcoupled to the others we can express the intermediate scattering function I, asI,(Q,t) = JGs(r,t) exp (iQ.r)drzIR(Q,t) k ( Q , t ) , (3)where Ik(Q,t) includes all motions which are not rotations and thus includes all thetranslational (including vibrational) modes of motion.We now assume further thatthe Gaussian approximation can be used for Ig and that I, is given by the theory ofSears ' in which the molecules are treated as rigid rotators. This is justified by thefact that both the neutron energy transfers and the value of kT are negligible comparedto the C-H mode energy :where w(t) is a width function (e.g., Vineyard 6, related to the mean square displace-ment of the proton (from the origin) at the time t, j , is the Zth spherical Bessel functionand d is the radius of the rotational motion (assumed isotropic and that an averagevalue of d can be used).The rotational relaxation functions, Fl(t) have the followingproperties :Fo(t) = 1, for all t ; Fl(0) = 1, for all 1.For rotational diffusion,8 which is analogous to Fick's law for translational diffusion,where 0, is the rotational diffusion coefficient. This model is adequate for our pur-pose as we are concerned with the long-time behaviour (i.e., small values of Q and w).The scattering law, Ss(Q,w), is the time-Fourier-transform of eqn (4) and, since(see later) we measured neutron time-of-flight '1: our observed cross-section isFl(t) = exp -Z(Z+l)Drt, (5)d2a m d2a m o b 2I,(Q,t) exp (- iot)dt,where zo is the time-of-flight of the incident neutrons. We can generate the crosssections for various models of the proton's motion by using eqn (4) in eqn (6), and forthis purpose we must evaluate w(t).Again, we are only interested in the long timebehaviour of w(t).where u2 is a mean square displacement caused by short period modes, D is thetranslational diffusion constant and c is a step time in a multi-step diffusion process.Egelstaff and Schofield modified this expression to include collisions after the mannerof the Langevin equation. For convenience they introduced the following mathe-matical form,If we take the Z = 0 term of eqn (4) in eqn (6) we would expect the central part of theneutron spectrum to be given by the expression,In the limit t + 00, we have the form,w(t)+2D(t - c) + u2, (7)w(t) = 2D( Jt2 + c2 - C) + u2.(8)The use of either eqn (7) or (8) in this expression did not fit all the data (fig. 6), andtherefore we concluded that either higher-order rotational terms are needed in (4) orthat (8) should be modified to include the effect of low-frequency translational mode196 MOLECULAR MODES IN LIQUID HYDROCARBONSof motion. To simulate the latter effect we modified eqn (8) to the form (with a < c),w(t) = 2 0 Real( J(t + ia)' + c2 - Jc2 - a'> -I- u2 (10)where a is a constant to be determined. With this expression the Fourier integral ineqn (9) becomes2n LJ'wexp{-iot-- 'co Q 2 W ) } 2 dt=exp ( -- Q;2)J+m -co S+ (Q,o')S- ( Q , o - w')dw' (1 1 a)whereand Kl is the Bessel function of the second kind. Eqn (10) with a 21 c gave cross sec-tions significantly different to those obtained via eqn (8).Thus, in summary, the data analysis employs the above theory through the follow-ing steps : (i) an (intuitive) separation of low Q and u) data from the remainder inorder to separate out the long-time behaviour of the system ; (ii) the area of this partis given byif terms of order N are included in the separation (i). (iii) For sufficiently small Qonly the I = 0 term appears in (12) and only the limiting form (7) for w(t) is required-in this limit the width of the spectrum is proportional to the translational diffusionconstant.(iv) For (small) Q values which are larger than those used in (iii) a modelof the motion is required-we calculate the neutron cross section using eqn (6) and (4)with the functional forms given at eqn ( 5 ) and (10).EXPERIMENTALPROCEDUREThe measurements of neutron differential scattering cross-sections were carried out on theChalk River phased rotor spectrometer.1° Initially, data were taken on samples of pentane,heptane and dodecane (at room temperature) using incident neutrons of reciprocal velocity448 p/m.Subsequent measurements using pentane at elevated temperatures employedincident neutrons of 550 p / m reciprocal velocity. The neutron burst closely approximatedto a Gaussian distribution (fig. 5a) with a standard deviation at the specimen of 3.0 ps for thedata using 448 pslm and 3.7 p for 550 p / m neutrons. The standard deviation of the velo-city distribution in the incident beam was 1.2 p/m.The neutrons were scattered by the specinien into eleven LiF scintillation detectors ofarea 12 in.x 3 in. placed 3 m from the specimen position and scanning 4" to 24" in steps of 2".Each counter was 0.7" wide and its mean angle determined to f0.1". The spectrum of thescattered neutrons was obtained by feeding the output of each detector to a time sorter using270 time channels of 4 ,us (or 6 p) duration for each detector. The incident flux was mon-itored throughout the measurements using three fission chambers mounted in the directbeam. By time sorting the pulses from the monitors the reciprocal velocity was obtained andthe incident spectrum monitored continuously. For elastic scattering the full width at half-height of the observed spectrum was AT/Z = 1 %.The measurements on each sample lasted 5-10 days and several runs were added togetherafter careful comparison.To remove uncertainties due to background fluctuations, thesample was oscillated in and out of the beam at 15-min intervals. During the sample outperiod an identical sample container to the one containing the sample, was placed in thebeam. Thus, a simple subtraction of the counts obtained in the two cases (with slightadjustment for difference in incident flux observed on the monitor and for sample absorption)gave the scattered neutron intensity from the sampleP. A . EGELSTAFF AND D . H . C. HARRIS 197Detector efficiencies are required to determine the differential scattering cross section loabsolutely. These were measured on a scale relative to the monitor before and after theexperiments, by recording the scattering from a slab of vanadium, of known cross section.To increase flux and thus improve statistical accuracy the incident spectrum was broadenedfor the efficiency measurements. This was justifiable since variation of efficiency with energyis small for the detectors used.TABLE 1 .--CHEMICAL COMPOSITION OF SAMPLEShydrocarbon purity, mol % impuritiesn-pentane 99.98 iso-pentanen-hep tane 99.92 cis-1, 2-dimethylcyclopentane ;3-ethyl pentanen-dodecane 99.70 unknownMeasurements on pentane at room temperature only, were made using the rotatingcrystal spectrometer at Chalk River.For these measurements incident neutrons of 1000ps/m reciprocal velocity were scattered into detectors placed at nine different angles rangingfrom 14 to 32".Outputs from each detector were time sorted into 512 time channels of 8 psduration on a 3.2-m flight path. Results observed on this machine were compared to theresults obtained on the phased rotor spectrometer and found to be in close agreement. Forelastic scattering the full width at half-height of the observed spectrum was Azlz = 2 %.Sample holders for room temperature (27") use were a modification of the type describedby Stiffler and Carpenter.I2 They were made from aluminium alloy with windows of foil0.001 in. thickness. The sample thickness contained between the windows was 0.012 in.with a uniformity of fO.OOO 5 in. When mounted at 45" to the incident beam about 10 % ofthe incident neutrons were scattered from a sample area of 2 in.x 1 in. Table 1 gives thecomposition of each sample.Pentane specimens at elevated temperatures were contained in a sample holder developedby Ball and Cocking.13 Briefly, it consisted of an array of thin walled stainless steel tubes(0.032 in. diam.) heated by passing an electric current through the tubes themselves. Thespacing of the tubes is such that a minimum variation in sample thickness is achieved with thetube array inclined at 45". Temperature control to within *3" was achieved. Table 2 andfig. 1 shows the physical parameters used in this work. In the liquid region, the data fallnearly on to the isochores p = 0.6 and p = 0.4. In addition, a dense gas point at p-0.2was taken.TABLE 2.-m'vSICAL PARAMETERS FOR n-PENTANE SPECIMENSdensity diffusion coefficientgcm-3 10-5 cm*/s run no.temp., "C pressure atm1, 2 27 1 0.626 5.753 90 30 0.56 134 150 30 0.46 265 190 43 0.40 326 213 67 0.38 447 300 58 N O . 18 60PVT data were taken froin ref (14) and (1 5)RESULTSCORRECTIONSQ VARIATIONThe variables measured in a neutron scattering experiment are the wave vectortransfer Q and energy transfer hm. At the small angles and for the incident energiesused in these measurements Q was substantially constant for any one detector ove198 MOLECULAR MODES I N LIQUID HYDROCARBONSthe range of the observed neutron spectrum. Fig. 2 shows the small variation of Qwith co for each of the detector positions (the Q range for the detectors was 0.24 to1.47 Awl).The limits of the quasi-elastic and the inelastic scattering from pentane atroom temperature are shown.IFIG. 2.-An (qQ) diagram for the 11 detectors used in this experiment. The full lines representthe (a,Q) loci for the several angles (in degrees) ; the dotted lines represent a typicd extent of thequasi-elastic scattering and the dashed lines show the extent of the inelastic scattering.BACKGROUND SUBTRACTION AND DATA REDUCTIONThe background was monitored continuously during an experiment by inter-changing an empty holder for the specimen. The amount of scattering from thecanning material in the room temperature sample holders was about 5 % of the samplescattering, whereas for the high temperature samples the scattering from the stainlesssteel tubes was 20 % of that from the pentane.Subtraction of the sample in and outdata provides net counts against time channel. Fig. 3 shows the data from run 3on pentane (table 2). The full range of these spectra is 3 meV on a centre energy of18 meV, confirming that only the low frequency part of S(Q,cu) is being observed.The data were reduced to differential scattering cross section (in barns/ster.)against reciprocal velocity (in p/m) by means of the SCATTERN programme on theG-20 computer at Chalk River. The programme computes differential cross-sectionsfrom the formula :d 2 q NE, a, -- --dQd7 EARAT H(1- T)’ (13)where N = net counts per channel, E = efficiency of detector relative to monitor,AQ = solid angle subtended by detector, At = channel width in ps/m, a, = totalscattering cross section (barns) at incident energy, H = beam monitor count, E, =monitor efficiency, and T = sample transmission.This programme also evaluateFIG. 3.-Summary of the data from run 3 on pentane (table 2) : this is a typical set of results from 10counters and illustrates the decrease in amplitude and increase in width with increasing scatteringangle. The full lines are drawn " by eye " through the points.[To face page 198P . A . EGELSTAFF A N D D . 11. C. HARRIS 199the scattering cross section do/dQ for the quasi-elastic region or for the whole spec-trum, by summing over a suitable number of time channels.I ’_t!,t--/ _ - - __-- 0 -00 #-MeMULTIPLE S CAT T ER I N G CORRECT I 0 NThe theory applies to single scattering only and therefore it is necessary to apply acorrection for multiple scattering.In this section we consider the magnitude andspectral shape of the multiple scattered component in the observed data. As anexample, we discuss the dodecane sample at room temperature which scattered 17 %of the incident neutron beam. From the results of Vineyard l6 on isotropic scatteringthe multiple intensity should amount to 30 % of the primary scattering. Fig. 4shows the result of adding counts obtained from the lowest angle in groups of tentime channels, and it is seen that the spectrum consists of a large narrow elastic peak**2~-:0 , I , - . 0‘- - -0---*-*- c.’-superimposed on a broad low level background. This background includes themultiple scattered neutrons and the inelastic component.We expect the multiplescattering to be broadly independent of angle l6 while the single scattering variessystematically with angle. From plots of intensity against detector angle the amountof inelastic scattering at 4” was estimated to be about 0.16 of the area of the broadbackground (at high angles the inelastic spectrum dominates the data). Thus, about0.84 of this area is due to multiple scattering and fig. 4 shows its broad spectral shape.From this reasoning the multiple scattered component observed in fig. 4 was estimatedto be 26 5 % of the total intensity at that angle, which is in agreement with the pre-diction of 30 %. Because the spectral shape of the multiple scattering was so broadit is of little importance when evaluating half widths of the quasi-elastic peaks.How-ever, corrections for multiple scattering using Vineyard’s theory were made to thescattering cross-section data da/dQ.ANALYSIS OF DATASPECTRAL WIDTHSIn order to obtain a spectral width it is necessary to remove the resolution widthfroin the observed spectrum, and for this purpose we assumed that the observed datawere given by the convolution of a Lorentzian-shaped scattering cross section with aGaussian resolution function. Fig. 5a shows the pulse shape seen by the monitorscompared to a Gaussian curve. From such data (corrected for sample geometry) aLorentzian/Gaussian convolution was fitted to the observed spectra and the Lorent-zian half-width at half-height for the best fit was obtained.Fig. 5b and c show thefits obtained for pentane (run 3), which are typical for all runs200 MOLECULAR MODES IN LIQUID HYDROCARBONSIn order to compare the width data for different temperatures and differenthydrocarbons they were converted to a non-dimensional form for the width AB andmomentum transfer squared M2 :where AT is the width (in p/m) of the total intensity curve. Fig. 6a shows AB plottedagainst M2 for pentane, heptane and dodecane at room temperature. The values ofD for these samples were obtained from Douglas and McCall,’* Fishman l9 and VanGeet and Adamson.20 The data appear to lie above the simple diffusion lien,indicating that some process is being observed in addition to translational diffusion.a b-25 - 2 0 -15 -10 - 5 0 5 10 I S 2 0 25time channels from “elastic” channelFIG.5.-ExampIes of quasi-elastic peak analysis ; theoretical curves are shown by the full and dashedlines and experimental points are shown as full circles. (a), comparison of monitor 2 with Gaussianresolution function; (b) and (c), pentane at 90°C and scattering angles of 10 and 18” respectively-the fulI line is a convolution of the Gaussian of 5(u) with a Lorentzian curve adjusted to fit the peak ;the dashed line shows a similar calculation using the horizontal dashed line as a base line (to representthe level of inelastic scattering under the quasi-elastic peak), and the dotted line is a theoretical curvebased on eqn (10) using a/c = 0.98.P.A . EGELSTAFF AND D. H . C . HARRIS 20 1The diffusion constaiits for pentane were obtained in the manner described below.Fig. 6b shows AB against M2 for pentane. On this plot, simple diffusion theory givesa universal straight line and all the data approximate to it near the origin with adeviation becoming more pronounced as Q is raised, although the scatter on the pointspreclude a detailed analysis. The additional width (i.e., >DQ2) is removed as wemove from the liquid to the gas regions suggesting that it is associated with hinderedrotational or hindered translational modes in the denser liquid.Further fits to the data were made after an estimated inelastically scattered back-ground had been subtracted : this was estimated by averaging a few channels on eitherside of the elastic peak and taking this average as a uniform inelastic level. Thismodification lead to negligible changes in the evaluated half-width at the low angleswhere the amount of inelastic scattering is small, but had a significant effect at thehigher angles (fig.5c).aM 2bFIG. 6.-Plot of width AB of total intensity curve against momentum transfer squared M2, in reducedunits. (a), n-dodecane - - - ; n-heptane X-data points ; n-pentane . . . . ; (all at 27°C) ; full line,simple diffusion. (b) n-pentane under the conditions listed in table 2. 4, data points of run 1 ; -- run3; .... run4; ---- run 5 ; - - - - run 6 ; 0, data points of run 7 ; the full line issimple diffusion theory.The full width at half height AE (in eV) for the quasi-elastic scattering was plottedas a function of Q’.Points close to the origin lay on a straight line. The value ofthe width tends to simple diffusion (2r2 D Q2) for low values of Q and the self-diffusioncoefficient D can be calculated from the straight portion of the curve. The values ofD thus obtained for pentane are plotted against reciprocal temperature in fig. 7. Thevalues obtained below room temperature by Douglas and McCall l8 using an n.m.r.spin echo technique are also shown (this w r k was apparently carried out at 1 atm).At the temperature common to both techniques (room temperature) there is excellentagreement. Because of this agreement we believe that the neutron determination ofthe diffusion coefficient is correct for the other runs and we have used these values intable 2 and throughout this paper.Activation energies were determined for threeloci, the isochores p = 0.6 and 0.4 and the isobar at 1 atm (from Douglas andMcCall 18). Table 3 is a summary of these data, and shows that the constant volumeactivation energy is greater than that for constant pressure, but less than that (3-4kcal/mol) for a hydrogen bonded liquid202 MOLECULAR MODES IN LIQUID HYDROCARBONSThe diffusion model of Egelstaff and Schofield has been fitted to these data andvalues of the interaction time (or diffusive step time) appearing in their formula (eqn(8)) have been determined. Table 4 is a summary of these values for pentane. Thevalues obtained are less than 10-l2 s (i.e., less than for other liquids presumably3 90 0.4 ;:; 4 1505 1 906 213 0.5TABLE 3 .-SUMMARY OF ACTIVATION ENERGIEScondition activation energyP = 1 atm 1.6 kcal/molp -0.6 2.5 kcal/molp -0.4 2.4 kcal/mol132644’ f 2 0 % 32 *I5 %because the inelastic subtraction was not satisfactory.The trend of these data (whichmay be correct on a relative scale) are to increase with increasing diffusion coefficientin qualitative agreement with the theory of free diffusion and opposite to that ex-pected for jump diffusion.7 300 0.8 60t\I .5 3.9 2.5 3.0 3.5 4 0 4.5(1031~) ~ - 1FIG. 7.Diffusion coefficients plotted against 1/T. A, data of Douglas and McCall; -+- isochore p 4 . 6 ; . . $ . ., isochore p 4 . 4 .AREA ANALYSISThe scattering cross section dcr/da for dodecane at room temperature is plottedagainst Q2 in fig.8a. Two values of the cross section are obtained for each value of Q2P . A . EGELSTAFF AND D. H . C. HARRTS 203one by integrating the differential cross section over the quasi-elastic peak only and theother by including the inelastic scattering as well. The latter presents little difficultysince it is only necessary to choose the limits of the integration correctly to include allthe scattering. The slopes of the lines In (do/dQ) against Q2 for pentane are plotted0 - 2 I T0 100 7 0 0 3 0 0temp., "CFIG. %--(a) Area of spectra against Q2 for dodecane at room temperature - - x - -, area of totalspectra ; -0-, area of quasi-elastic spectra. (b) slope of total area plots for n-pentane ; the fullline is a theoretical curve.in fig.8b and it is seen that the slope increases with increasing temperature. Thetheoretical line includes two effects (i) the non-constancy of Q (fig. 2), and (ii) theDebye-Waller factor for the internal molecular modes at ho-40 meV.2* Thelatter effect is the larger. The curve of da/dSZ against Q2 at room temperature fordodecane and heptane had no detectable slope (fig. Sa), showing that the Debye-Waller factor for the internal modes is unity within experimental error.Evaluation of the quasi-elastic area is more difficult because the quasi-elasticTABLE S.-"THERMAL CLOUD " RADIIsampleheptanedodecanepent anepentanepentanepentanepentanepentanetemp. "C2727279015019021 3300value of (3u2)3.A2.11.42.42.13.43.53.54.204 MOLECULAR MODES I N LIQUID HYDROCARBONSpeak is not clearly distinguishable from the inelastic scattering : the data at the lowerQ values are the more reliable. Thus, we used only those data for which dc/dQ hadfallen by less than a factor of 2 from its value at Q = 0. The quasi-elastic intensityvaries with angle with a distribution described by eqn (12). If the rotational terms sumto unity, then the data should yield a value for the thermal cloud radius or (3u2)*.The value of u2 was calculated from the slopes of lines such as those shown in fig. 8a.Table 5 gives the values of (3u2)>3 for each case (the error is 15 %). The values areof reasonable magnitude and show a plausible trend.This lends some justificationfor the above approach to the rotational terms in (12).ANALYSIS I N TERMS OF LOW-LYING ENERGY LEVELSTwo experimental observations (the wings of the spectra of fig. 5 being broaderthan a Lorentzian and the widths being greater than simple diffusion, fig. 6) suggestthat, for part of our data, another physical process is being observed. It is likelythat this process involves molecular rotation or libration. For the free pentane mole-cule the value of (2kT/Z)* is several meV in our temperature range (where I is themoment of the molecule about a chosen axis). In the condensed state we may expectn 3 W;-.. ..,wFIG. 9.-Examples of the spectral density of the velocity correlation function for the model of eqn(10).The o scale is in units of kT/M*D and the z(o) scale is in the inverse of this unit. The dottedline corresponds to a/c = 0.98 while the full line corresponds to a/c = 0.96.a rotational or librational spectrum to extend to higher energies. The low frequencypart may be described either as the rotational diffusion process (eqn (5)) or energytransfers to quasi-librational states. In the latter case we use eqn (IO), and maydiscuss such states through the spectral density function ~ ( c D ) , i.e.,+co a2 w(t) exp (- iwt)+t.z(0) = 'j 4x atThe important parameter is the ratio a/c; and this model implies a generalization ofusual interaction time formula of the Langevin equation, i.e.P . A . EGELSTAFF AND D . H . C. HARRIS 205where M* is the mass of the particle treated by the Langevin equation.Since we aredialing with molecular rotations as well as translations and the most significant rotationwell be about the -C-C- axis, we have used M* = 12 proton masses in this paper.Thus c may be calculated in terms of a/c from eqn (16) if the D is taken from table 2.Fig. 9 shows two examples of the function z(o) of this model. The complete intensitycurve was fitted by variation of the single parameter a/c, it being required that a singlechoice of a/c fit each of the eleven angles. Examples of this are shown at fig. 5b and c(dotted line) in which the theoretical cross section has been folded with the experi-mental resolution function (fig. 5a). It is seen that the whole of these data can befitted by this model, so indicating that the spectral density exhibits a peak similar tothose shown in fig.9. Table 6 summarizes the values of the parameter a/c for pentane,and the corresponding value of c for M* = 12. These values should be contrastedwith those of table 4.TABLE 6.-PARAMETERS IN LIBRATIONAL MODEL FITTED TO PENTANE DATArun no.(table 2)1, 234567best value c(10-12 s)> 0.98 > 200.98 450.96 260.96 300.95 270.93 20of a / c ifM*/M= 12In addition the rotational diffusion model can be fitted to the data using eqn (4)and (5). Here, two parameters are required (d and 0,) for the description of therotations, and further parameters are required to describe the function w(t) in eqn (4).If, e.g., eqn (8) for w(t) is used, the constants c and D, might be varied keeping D and dfixed (d is some average molecular dimension).It is possible to fit the data in thisway with c and D, about 10-l2 s. The significance of this fit is similar to the abovemodel, since it confirms the existence of a wing to S,(Q,w) interpretable in terms ofhindered rotational motion.DISCUSSIONWe have studied the translational and rotational behaviour of some hydrocarbonsin the liquid state for a time scale of - 10-l2 s and a distance scale - 10 A. Thediffusion constant for translation determined by neutron scattering agrees with thatdetermined by n.m.r. spin-echo method, and also gives reasonable activation energies.At temperatures and pressures above the critical region the molecules are makingtranslational movements of - 5 A between collisions and the diffusion process maynearly be described by the Langevin equation.We attempted to work along twoisochores, but for technieal reasons were not able to follow them exactly. At lowertemperatures-especially at room temperature-we observed that the data could notbe fitted by a simple theory of translational motion. Instead, we employed a treat-ment based on the assumption that rotational modes add intensity to the wings of thequasi-elastic peak. In this way we could fit the data and show that low frequencyrotational or librational modes were being observed. The spectral density of thesemodes has the form shown in fig. 9 (which is determined by the a]c ratio only).Auseful way to discuss the peak in z(w), is to assume it represents short time localbonding of the molecules206 MOLECULAR MODES IN LIQUID HYDROCARBONSFig. 5 and 6 summarize the data from this point of view ; the peak widths are greaterthan given by simple diffusion theory, and the peak shape includes a wing which iswider than a Lorentzian curve. A Lorentzian plus a constant is a poor fit also asshown by the dashed lines in fig. 5.The theoretical discussion of these results is limited by uncertainty about the truemotion of the molecules. Conversely, the experimental data are not sufficientlyaccurate to fix more than one model parameter. It is therefore possible to fit thedata by a number of rotational models, each of which gives a slightly different shapeto Ss(Q,o), without being able to distinguish between them except on the basis of thephysical significance of the single parameter determined.The conventional separationinto a quasi-elastic peak and a background is not satisfactory for two reasons. First,because such a separation is not intuitively obvious-see fig. 2 and 5-and musttherefore be fairly arbitrary. Secondly, table 4 summarizes the diffusion data ob-tained this way, the interaction times seem unrealistic compared to those for otherliquids and the lengths I derived from the formula D = Z2/6c are small. Therefore,this approach, which works well for some liquids,l* does not appear to be satisfac-tory for hydrocarbons. In the other method (using eqn (10)) no such separation isneeded, and the parameter c is now more nearly related to the position of the peakin fig.9 than to the width of this curve. On this model the width is aboutM'kD/kT, and for realistic times ( w 10-l2 s) we need an M* larger than thevalue of 12M employed here. This, however, is reasonable since co-operativemovements are involved (i.e., several molecules may move) and the mass of a numberof carbon atoms might be included in Me'. The value of c would be modified fromthe values in table 6 by the same ratio. Thus, although the parameters of this modelappear arbitrary, the values are probably within reasonable physical limits.We conclude that the further exploitation of this method would reveal detailsof the molecular modes which are reflected in the shape of S'( Q,o).For this purposea higher statistical accuracy is required and the isochores should be followed withgreater precision. Both of these steps would seem to be technically feasible at thepresent time.We thank the Chalk River Nuclear Laboratories of Atomic Energy of Canada Ltd.for their hospitality during the course of this work and the scattering law group atC.R.N.L. for their assistance with the measurements and data reduction. Also, weare grateful to Mr. C. R. T. Heard of A.E.R.E. for writing computer programmesfor the several models and the numerical evaluation of the model cross sections.P. A. Egelstaff, An Introduction to the Liquid State (Academic Press, London, 1967).K. E. Larsson and U. Dahlborg, Physica, 1964,30,1561; K. E. Lamon et al., 1966, Phys. Reo.,151, 126 ; U. Dahlborg, et al., Neutron Inelastic Scattering, Vol. 1 (I.A.E.A., Vienna, 1969), 1,58.D. H. Saunderson and V. S. Rainey, Inelastic Scattering of Neutrons in Solids and Liquids(I.A.E.A., Vienna), 1963, 1, 413.K. E. Larsson, Thermal Neutron Scattering, ed. P. A. Egelstaff (Academic Press, London),1965)) chap. 8.L. Van Hove, Phys. Reu., 1954,95,249.G. H. Vineyard, Phys. Rev., 1958, 110, 999.V. F. Sears, Can. J. Phys., 1966,44, 1279, 1299; 45,237.W. H. Furry, Phys. Rev., 1957,107, 7.P. A. Egelstaff and P. Schofield, Nucl. Sci. Eng., 1962, 12,220.lo P. A. Egelstaff, S . J. Cocking and T. K. Alexander, AECL Report CRRP 1078,1960 ; rotors ofhigher resolution were used for this experiment.B. N. Brockhouse, Inelastic Scattering of Neutrons in Solids and Liquids (I.A.E.A., Vienna,1961)) p. 113P. A . EGELSTAFF A N D D. H . C. HARRIS 207'* C. D. Stiffler and J. M. Carpenter, University of Michigan report 03712-3-T, 1963.l3 A. Ball and S. J. Cocking, J. Sci. Instr., 1964, 41, 376.l4 A. L. Lee and R. T. Ellington, J. Chem. Eng. Data, 1965, 10, 101.l 5 J. Timmermans, Physico-Chemical Constants of Pure Organic Compounds (Elsevier, 1969,l6 G. H. Vineyard, Phys. Rev., 1964, 96, 93.l7 D. M. Thorson, unpublished.I. C. Douglas and D. W. McCall, J. Phys. Chenz., 1958, 62, 1102.l9 E. Fishman, J. Phys. Chern., 1955, 59,469.'O A. L. Van Geet and A. W. Adamson, J. Phys. Chem., 1964, 68,238.vol. 1 and 2.S. Mizushima and T. Sinianouti, J. Arner. Chern. Soc.. 1949, 71. 1320

ISSN:0366-9033    

DOI:10.1039/DF9704900193

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Fickian and Non-Fickian Diffusion in High Polymer SystemsBY G. REHAGE, 0. ERNST AND J. FUHRMANNPhysikalisch-Chemisches Institut der Technischen Universitat Clausthal, B. R.D.Received 12th June, 1970The normal and the anomalous diffusion in binary polymer solutions are considered. After asurvey of the theory of diffusion it is demonstrated which criteria must be fulfilled so that the Fickiandiffusion occurs. For transport between the solvent and a solution, or between two solutions,normal diffusion is found. Experimental results with polystyrene+solvent systems are given. Withgood solvents the diffusion coefficient increases strongly with increasing polymer concentration,passes through a maximum at medium concentrations and decreases by several decades at highpolymer concentrations.This concentration dependence arises because the diffusion coefficient isnot only a transport coefficient, but also contains a thermodynamic factor. With poor solvents,which show a phase separation of the polymer at low temperatures, the concentration dependenceof the diffusion coefficient is even more complicated. With increasing concentration of polymer thediffusion coefficient first decreases, passes through a minimum and then increases again. In thiscase there is also a maximum at medium concentrations and a strong decrease at high polymerconcentrations. The minimum for a binary system is located at the critical point of the system.Normally In D is a linear function of 1/T. Deviations from the linear course can be explainedby the temperature dependence of the thermodynamic factor.For anomalous diffusion the l/t-relations are no longer valid, since the diffusion coefficient dependsnot only on concentration, but also explicitly on time.The anomalous diffusion is due to the factthat superimposed on the normal diffusion is another process. If a solvent penetrates intoa glassy polymer, then relaxation processes are superimposed on the diffusion process. The polymerchanges from the glassy state into a state of internal thermodynamic equilibrium. Experimentallythe continuous range between the pure glassy polymer and the pure solvent can be divided into threeparts of an open system. In the first and the third part only diffusion occurs, while in the secondpart diffusion and structural relaxation are superimposed.Diffusion is mass transport caused by concentration gradients, or more exactly,by gradients of the chemical potential. For binary non-electrolyte systems, withwhich we are concerned in the following, we can write, if we consider diffusion in onedirection only,Wl dP2 J1 = -all--aa Z 12- aZ4% aP2a Z a2J 2 = -a21--a22--whereJ1,J2 = mass fluxes of the components 1 and 2,p1,p2 = chemical potentials of the components 1 and 2,afk = phenomenological coefficients,z = space coordinate.Since we consider only the diffusion in one direction, we have not to allow for thevector character of the flux.The Gibbs-Duhem-equation is20G . REHAGE, 0. ERNST AND J . FUHRMANN 209where p,, p 2 are the mass concentrations of the two components, e.g., in g ~ r n - ~ .From the eqn (1) and (2) it follows thata11 a12 a , = -- - 3 P l .cz ' P1 P2J1 = -alp,-dPu, a 2 2 a21a2 P2 P1J 2 = -a2p2-; a2 = ---.Further we can write, for liquid mixtures, which are only considered here,J1 = P1 (Vl-w),J 2 = P2 @ 2 - 4 ,(34where v l , and v2 are the mean particle velocities and w is the mean volume velocity.For w, we may writewhere q1 and 'p2 are the volume fractions of the two components.(4) it follows thatw = CPlVl +P2V2 (5)From eqn (3) anda1 = (01 - W M - a P l l a z )a2 = (02 - w ) f c - 3P2PZ)( 6 4(6b)Eqn (6a) and (6b) imply that the coefficients al and a, are the relative velocities of theparticle species 1 and 2, and are related to the driving forces, i.e., negative to thegradients of the chemical potentials.A mean particle velocity, related to the drivingforce, is defined as mobility. It follows thatwhere u1 and u2 are the mobilities of the two particle species related to the meanvolume velocity.The first Fickian law for diffusion isJ , = -D,ap,/aZ,J~ = - ~ ~ a p ~ / a z .Dl = ~ l ( P l a P l / a P l ) ,D2 = U2(P2aP2/aP2).By comparing the coefficients with eqn ( 7 ) one finds thatFurther for a binary system where D1 = D2,1 thenD = u l ( P l ~ P l / a P , ) = %(P2aP2/aP2). (10)Eqn (10) shows that the diffusion coefficient contains a hydrodynamic or transportfactor u1 or u2 and a thermodynamic factor p l a p l / a p l or pzap2/dp2. To obtainmobilities therefore, the diffusion coefficient and the concentration dependence of thechemical potentials must be known.Instead of eqn (10) one can writ210 FICKIAN AND NON-FICKIAN DIFFUSIONwhere fl and f 2 are the friction coefficients.For infinite dilution there are well-known relationships, e.g., the Einstein-Stokes eqn for spherical particles.2 By meansof the thermodynamics of irreversible processes one can derive the following equivalentrelation :where s1 and s2 are the sedimentation coefficients, i.e., the mean particle velocitiesdivided by the gravitational or centrifugal acceleration, M1 and M2 the molar masses, vl and v’. the partial specific volumes of the two components and p the density of thesolution. At infinite dilution one has the Svedberg eqn, which is of great importancefor measurements with the ultracentrifuge.From eqn (lo)-( 12), the followingrelations between the transport coefficients are obtained :u2 = llfi = s2/M2(1-F2p). (13b)The quantities s1 and s2 can be measured directly; so one can evaluate ul, u2 oi-fl,f 2 respectively from eqn (13). If u1 and u2 and the diffusion coefficients D are known,one can evaluate from eqn (10) the concentration dependence of the chemicalpotential.FICKIAN DIFFUSION(i) For normal diffusion the Fickian equations are valid. Eqn (8) defines thediffusion coefficient when it is independent of the concentration gradient. Fromthe local mass balance it follows, in absence of convection and chemical reactions,thatIn eqn (14) D may depend on the concentration p1 or p 2 respectively.But thediffusion coefficient is not allowed to depend explicitly on the time t. If D is indepen-dent of the concentration, the second Fickian eqn is valid, i.e.,For the evaluation of D one measures the concentration pi and its dependence ont and z and integrates the partial differential equation, with specific initial andboundary conditions, and obtains close expressions with constant values of D.But in polymer systems, the diffusion coefficient generally depends strongly on theconcentration, especially over a large concentration range. Boltzmann has shownhow one can then proceed for a system that is infinitely large and when D onlydepends on concentration and not explicitly on time34* 35* 38. So, by introduction of anew variable, 11 = z / Jt, one obtains an ordinary differential equation from the partialdifferential eqn (15), with the initial conditions expressed by q.The latter is onlypossible for q = i- 00. The initial conditions are, if the concentration of the solvent p1is considered,11 = +a, = 0 ; dplldv = 0, (1 6 4(1 6b) 11 = -a, P1 = Pol;where pol is the density of the pure solvent.dPIld11 = 0G . REHAGE, 0. ERNST AND J . FUHRMANN 21 1After substitution,-and after integration and resubstitution, the diffusion coefficient D at the concentra-tion p1 is given byOne must know the concentration-distance curve and must evaluate integrals andtangents, as can be seen from eqn (1 8). This graphical method, gives the concentra-tion dependence of the diffusion coefficient.We now have to find criteria for normaldiffusion, i.e., when the Ficks eqn are valid. At u] = 00, apl/dq vanishes, but Dhas a finite value in the whole concentration range; it therefore follows from eqn(17) thatP i =Po1qdp, = O . s P l = oThis is the Matano condition4.(ii) From the relation p1 = p l ( z / J t ) , it is evident that at a given concentrationp1 with diffusion proceeding the space coordinate will change such that Z-J Jt.This is the Jt-law, which is always observed for unperturbed diffusion. It is validalso, e.g., for solvent absorption by a foil in the early stage. For one-dimensionaldiffusion it follows for a constant surface q for the absorbed amount of solvent in ahalf-infinitely extended body thatfa, +oo0 0m1= 41 pldz = 41 Jtdu].(19)This is valid at any time.per surface unit from the formula,From this one obtains the amount of solvent absorbedThe inlegral is a constant, so thatFor swelling of a crosslinked polymer one can characterize the constant of propor-tionality in more detail.5(iii) If one evaluates the concentration-distance diagram at different times, oneshould obtain within the accuracy of the measurements the same curve for the plotIf these criteria are fulfilled, the diffusion is normal. As yet, volume change onmixing has not been considered ; in general it is very small (order of magnitude 1 %)and does not affect appreciably the results for the accuracy of measurements withliquid mixtures.33 The absence of convection is also presumed.This means inpractice that the mean volume velocity equals zer0.lP1 = Pl(Y).*EXPERIMENTALThe diffusion process for a pure solvent and a dilute solution or for two solutions ofdifferent concentrations, always follows a normal course. We performed such measurementson polystyrene (M, = 180 OOO) with a number of solvents in the temperature range 20-60°C.*The special form of these criteria, which must be fulfilled in vitreous solidified systems arepublished el~ewhere.21 2 FICKIAN AND NON-FICKIAN DIFFUSIONThe concentration range was kept as large as possible. The highest concentration, at whichmeasurements could be done yet, was about 40 % polymer. The mass concentration is thenabout 400 g I-'. The measurements were performed in an interferometer, the opticalarrangement being that of Jamin.6 Into the measuring and comparison cells of the diffusioncuvette two solutions differing in concentration about 2 % were placed.From the inter-ferograms one can evaluate at a given initial concentration the plot of concentration at acertain time.The analysis was not made according to the procedure of Matano, which is laboriousand has some disadvantages. Secondly,the Matano plane, which usually is the initial dividing plane, must be known exactly. Forthe two solutions in layers one usually does not get initially a sharp dividing line, so that thefixing of this plane is uncertain.We used a new procedure, based on the existence of the so-called " common points " ofstoke^.^-^ He found empirically that for all systems, for a diffusion coefficient which wasindependent of the concentration or a linear function of it, all concentration-distance curvespass through two points, equidistant from the Matano plane.In fig.1 (C- CII)/CI- CII) is plotted against q* = z / 2 2 / 5 . C is the mol concentrationof one of the components, z the space coordinate, t the time and 5 the diffusion coefficientfor C = (CI+ C11)/2. CI and CII are the initial concentrations of the two solutions. The curvesare valid for different values of &/&I, where DI and DII are the diffusion coefficients at theconcentrations CI and CJI. The coordinates of the common points are according to Stokes :First, drawing a tangent to a curve is not exact.q* = + 0.66 ; (C- CII)/(CI- CII) = 0.176 ;q* = -0.66 ; (C- CII)/(C~- C~I) = 0.824.The existence of the common points has been confirmed by Sauer and Freise for a linearconcentration dependence of the diffusion coefficient.Deviations from the linearity must beconsiderable to cause invalidity.lO- 2.0FIG. l.-(C-C'~~)/(C~--C~~) against r)* at constant D or for a linear concentration dependence of D.z = space coordinate; t , time; A, B, common points; D is the diffusion coefficient at C =(C1+C11)/2. @, DI/DII = 7.11 ; @, DI/DII = 1 ; @, DI/DII = 0.141.For fig. 2, starting from the dividing line, one evaluates the distances Z" and z', at which(C-CI~/(CI-CII) = 0.176 and 0.824 respectively. From the definition of q* and thecoordinates of the " common points " one obtains the mean diffusion coefficient 5 at thepoint (CI+ C11)/2 according to the relation :5 = ~ ' / 0 .6 6 ~ X 4twithz = ( I 2' I + I 2' I )/2.The method avoids the difficulties of fixing the Mataiio plane and the evaluation of thG. REHAGE, 0. ERNST AND J. FUHRMANN 213tangent. Because the method gives the average value, one replaces the real concentration-distance curve by a steplike curve. The agreement is naturally better the smaller thedifferences between the initial concentrations CI and CII.2' 0 2" + zFIG. 2.-Concentration-distance curve (schematically). A and B are the common points.RESULTS AND DISCUSSIONIn the above manner numerous diffusion measurements were 9 nWith good solvents the diffusion coefficient increases strongly with increasing ploymerconcentration and passes through a maximum (see also ref. (28)).In fig. 3 this is2E20nIv, 150 "EuI. 2Q 10X5CFIG. 3.-Concentration dependence of the diffusion coefficient for the system polystyrene+ethyl-benzene at different temperatures. xz = segment mol fraction of the polymer. x: = mz/(nl +nz) ;nl, n2 = mol numbers of solvent and polymer ; r = degree of polymerization214 FICKIAN AND NON-FICKIAN DIFFUSIONshown for the system PST + ethylbenzene. The initial large increase of the diffusion co-efficient is unexpected, because the viscosity also increases considerably with increasingpolymer concentration. Thus, diffusion and viscosity are not reciprocally related,as is usually assumed.The maximum value of the diffusion coefficient is locatedat concentrations at which the solutions are already highly viscous. Measurementswith polystyrene (PST) in a different interferometer, in which the whole concentrationrange from the pure solvent to the pure polymer could be investigated, showed thatD at polymer concentrations of 60-70 % strongly decreased.12 In the glassy rangethe diffusion coefficient decreases to values of about 10-l2 cm2 s-l, while it can reachvalues of order cm2 s-' at the maximum. Thus, there is a change of severaldecades in the same system. The strong decrease at large polymer concentrationsis related to the vitreous solidification. The maximum at medium concentrationsdenotes that the diffusion coefficient, as eqn (10) shows, is not a mere transportcoefficient, but also contains a thermodynamic factor.We evaluated the latter fromosmotic measurements on PST-solutions and separated it from the transportcoefficient, i.e., the mobility.6* l310080PI LY 8v) 60c(OI4rl 0X 3" LO200 0- I 0 *2 0.3x;FIG. 4.-Concentration dependence of the mobility u2 of the polymer component in the systempolystyrene + chlorobenzene.In fig. 4 is given the concentration dependence of the mobility u2 of the polymermolecules in the system PST + chlorobenzene. As expected, the mobility is smaller,the more concentrated the solution. In contrast, the thermodynamic factorp2(i?p2/i?p2) generally increases with increasing polymer concentration. Thecooperative effect of these two factors can lead to extreme values in the diffusion-concentration curve.The reciprocal value of the mobility is the molar frictionCoefficient. The friction coefficient f2 of the polymer component increases witG. REHAGE, 0. ERNST AND J . FUHRMANN 215increasing polynier concentration inore rapidly than linearly. The concentrationdependence of the friction coefficient is similar to that of the viscosity. In simplecases proportionality may exist, as is so for Stokes' law. Thus, there is a directrelation between the viscosity and the friction coefficient (or the reciprocal value ofthe mobility) but not between the viscosity and the diffusion coefficients. The latteris related to the viscosity in a complicated manner through the thermodynamic factor.The relation between the diffusion coefficient and the viscosity is simple if the thermo-dynamic factor equals RT (R = gas constant).From the diffusion coefficients, thethermodynamic factors and the densities we evaluated the sedimentation coefficientsby eqn (12). The concentration dependence of the sedimentation coefficient of thepolymer was similar to that of the mobility. The calculated values were in goodagreement with those measured directly. **3.0!AN 20EWk 40 O C30 "C28OCv25.20 -0.1 0.2~ ~~0 0: I 0.2xz*FIG. 5.-Phase separation curve (phase equilibrium curve) and concentration dependence of thediffusion coefficient for the system polystyrene+cyclohexane.For systems with poor solvents which are inclined to phase separation, the con-centration dependence of the diffusion coefficient is even more complicated.Inthis case one finds at temperatures immediately above the critical temperature adistinct minimum in the diffusion-concentration curve at smaller concentrations.The maximum again appears at medium concentrations. In a binary system thecondition ap2/dp2 = 0 must be valid at a critical phase separation point. Becausethe mobility u2 cannot become infinite, however, according to eqn (10) the diffusioncoefficient has to vanish at the critical point. Diffusion measurements at the criticalpoint are not feasible because of the critical opalescence. But somewhat above thecritical point the diffusion-concentration curve should show a minimum.This isindeed the case, as is shown in fig. 5 for the system PST+cyclohexane216 FICKIAN AND NON-FICKIAN DIFFUSIONThe minima are located near the critical concentration. In the hoinogeneousstate polymer solutions can be considered as binary systems despite the fact thatthere is a distribution function of the polymer component, consequently eqn (10)-(12)are applicable. However, the ‘‘ quasi-binary ” polymer systems have now to beconsidered as multicomponent systems. This problem has been investigated,following the work of Tompa and Stockmayer, especially by Koningsveld andStaverman and by We only state here that there exists not only one, but afamily of phase equilibrium curves. The curve in fig. 5 is the “ closed phase separa-tion curve ” of the system, the maximum of which equals the critical point.The temperature dependence of the diffusion coefficient is usually described bythe relationD = D , exp (-AD/RT), (21)where AD is the activation energy of the diffusion and D, the diffusion coefficient atan intinitely high temperature. But often the temperature dependence is morecomplicated.This is due to the fact that the diffusion coefficient is not a meretransport coefficient. Because the mobility represents solely a transport coefficient,however, one can assume that its temperature dependence can be expressed by therelationu2 = u2, exp (- A,/RT).A, is the activation energy of the mobility and u200 the mobility at an infinitely hightemperature. From eqn (10) and (22) one then obtains the relationship :D = U2a3 exp (-Au/RT)P2aP2/aP2, (23)orIn D = In u2, - (AJRT) + In (p,ap2/ap2).So the temperature dependence of the diffusion coefficient depends not only on theactivation energy A,, but also on the temperature dependence of the thermodynamicfactor.25 For ideal dilute solutions, ideal mixtures and athermal solutions one canwrite eqn (24) in the form :In D = const - (Au/RT) + In T (25)and in this temperature range In T virtually decreases linearly with 1/T.Therefore,a term is added to the activation energy A,, which comes solely from the temperaturedependence of InT. It is about 20-30 % of AD. The diffusion activation energyAD contains this term, as eqn (21) and (25) show. Therefore, this quantity is a“ virtual ” activation energy.According to swelling measurements,2 the systemPST+ethylbenzene is nearly athermal. One would expect therefore a linear plot ofIn D with l/Ty and this has been found e~perimentally.~~If mixing enthalpy and entropy depend strongly on the temperature, as withphase-separating systems, then the temperature dependence of the thermodynamicfactor has a great influence on the activation energy of the diffusion. AD is tempera-ture dependent and In D no more a linear function of 1/T. This has been measuredalso for the system PST + cyclohexane.Fig. 6 and 7 show that the temperature dependence of the diffusion coefficient isvery similar to that of the thermodynamic factor, consequently the temperaturedependence of the thermodynamic factor may be the reason for a temperaturedependence of the activation energy of the diffusionG .REHAGE, 0. ERNST AND J . FUHRMANN30 7r(I v1N E < 2.0-24XIsO-21 7% = 0.15Y2' 0.01I I IFIG. system polystyrene+polymer.1/Tx lo3FIG. 7.-Logarithm of the thermodynamic factor Cz (dpz/dC2) as a function of 1/T for the systempolystyrene + cyclohexane. Cz = mol concentration of PST218 FICKIAN AND NON-FICKIAN DIFFUSIONNON-FICKIAN DIFFUSIONIf the diffusion coefficient at constant temperature depends not only on theconcentration, but also explicitly on time, then the simple Jt-relations are no longervalid. The time dependence of e.g., solvent absorption by, or desorption from afoil, is then more c~mplicated.~* 3 2 9 3 6 * 37 For a detailed investigation of theprocesses,one has to measure the concentration-distance diagrams. We did this for PST+solvent systems, using good and poor solvents.A micro-interferometer wasdeveloped, which works in a similar way to the instrument of Robinson and Crankon the Fabry-Perot-principle. 2* 27-29 We investigated the diffusion of solvents intoglassy PST and measured the concentration-distance curves. Anomalies are causedby the fact that the normal diffusion processes have superimposed on it relaxationprocesses. The relaxation phenomena arise because the polymer after penetrationof the solvent tends to reach a state of internal thermodynamic equilibrium, if theconcentration, which limits the glassy state, is exceeded.29Z / ~ O - ~ cmFIG.8.-Subdivision of the continuous system of the mixed phase between the pure glassy polymer(xp = 0) and the pure solvent ( x r = 1). In the concentration interval xyG1 < x: Q X ~ E , the enlarge-ment of which is shown on the right, the diffusion has superimposed on it the relaxation from theglassy state into a state of internal thermodynamic equilibrium (range II). The neighbouring ranges(I and IU), in which only diffusion processes occur, interact with range 11. z = space coordinate.Fig. 8 shows the concentration profile of the system PST+ toluene at 30°C after50 min diffusion time as a function of the segment mol fraction x: of the solvent.It is useful to distinguish between three domains.30 In range I (glassy mixed phase)in the concentration interval 0 < xT < xTGl the relaxation processes can be neglectedcompared with the diffusion process, so that the diffusion is not influenced by relaxa-tions.In range I1 with xfG,<xf<xTE, both relaxation and diffusion processesoccur at the same time with comparable velocities. At xTGl the molecular motionsof polymer molecules begin, and is the freezing concentration. At this concen-tration all polymer molecules have reached the state of the internal thermodynamicequilibrium. Thus, within range I1 the transition occurs from the glassy state into astate of the internal thermodynamic equilibriumG. REHAGE, 0. ERNST AND J . FUHRMANN 219With glassy, oriented samples at xTGI, the initial change of birefringence occursby rearrangement of the polymer molecules and at xTE the orientation of the moleculesdisappears completely.In range I11 with xTE < xT < 1 the relaxation velocity is highcompared with the diffusion velocity, so that diffusion occurs during internal thermo-dynamic equilibrium. The diffiision is not disturbed by relaxalion. All the parts ofthe system are open, so the entire transport in the whole system is influenced by theprocesses within parts of the system and by the interaction of these parts through theboundaries of these systems. Depending on the quality of the solvents, single partsof the system can become autonomous, i.e., the disturbing influence of the neighbour-ing ranges is negligible. In this case processes occurring in the whole system can bedescribed quantita t i ~ e l y .~I0.3 0-5 0.7 0.9PPST/g/Cm3FIG. 9.-Concentration dependence of the diffusion coefficient D in the system polystyrene+ tolueneat 29.9 "C after different diffusion times: V , 4 min; 0, 16 min; +, 36 min; 0, 50 min. Thefull curve was measured by Rehage and Ernst during internal thermodynamic equilibrium. PPST =mass concentration of the polystyrene.This may be shown with two examples. In the system PST + toluene at 30°C, therange I11 is autonomous. The diffusion in this range is influenced only slightly by therelaxation processes in range 11. The relaxation in range 11 is therefore controlledby diffusion in range 111, i.e., the entire transport process is nearly normal despitethe occurrence of relaxation processes, and the Fickian diffusion laws are approxi-mately valid.In the system PST+toluene the entire transport process occurs, as ifPST were not in the glassy state. Toluene is a good solvent for PST.Fig. 9 shows that the diffusion-concentration curves evaluated at different timesare not greatly different from each other." A curve, obtained fiom measurements* In this case the evaluation was made according to Boltzman and Matano220 FICKIAN AND NON-FICKIAN DIFFUSIONbetween two solutions in a state of internal thermodynamic equilibrium, was notgreatly different from these curves.8*In the system PST+ cyclohexane the behaviour is quite different. Cyclohexaneis a poor solvent. In the temperature range below 30°C separation into two phasesoccurs. It follows from the diffusion measurements that at 7"C, range11 is autonomous,i.e., the diffusion in the range I11 is strongly influenced by the relaxation in range IT.In addition this demonstrated that the concentration profile (i.e., all variables ofstate) are independent of time within range I1 (stationary state).Therefore in thisrange the relaxation velocity is independent of time, i.e., in range I1 a constant amountof polymer relaxes per unit time and this passes through the boundary of range I1into range 111. Therefore the amount of polymer in range I11 increases linearly withthe time, as is shown in fig. 10. This is a typical anomalous behaviour, and it is nolonger possible to evaluate diffusion coefficients.2G100LiL60 I00tlhFIG. 10.-Amount of polymer, which passes per unit area through the boundary between range I1into range 111.The slope of the line gives the stationary relaxation velocity (2x g/cm2 h)for the structural relaxation in the system PST+cyclohexane at 6.9"C. The limits of integration23 and 24 are given by the distance of the boundaries of range TU.These two examples show that subdivision into the three system parts I, I1 andI11 is a useful procedure. In this way it is possible according to the system to separatethe prevailing diffusion from relaxation and the prevailing relaxation from diffusion,if both processes occur at the same time. In other words, depending on the sol-vent, the diffusion can be, despite any superimposed process, normal or anomalous.In general complicated and different anomalous diffusion occurs because of the inter-action of the parts of the open In general, the diffusion seems to be" anomalous " when other time-dependent processes, e.g., relaxation processes, aresuperimposed on the diffusion process.Naturally in this case the Fickian eqn are no longer valid.They have to beextended by means of the thermodynamics of the irreversible processes, if one knowsthe mechanism of s~perirnposing.~G . KEIIAGE, 0. ERNST AND J . FUHRMANN 221R. Haase, Thermodynamik der irreuersiblen Prozesse, (Steinkopff-Verlag, Darmstadt 1963),v. 299 ff.R. Haase, Thermodynamik der Mischphasen, (Springer-Verlag, Berlin-Gottingen-Heidelberg1956), p. 583.C. Boltzmann, Wied. Ann., 1894, 53,959.C.Matano, Japan. J. Phys., 1933, 8, 109.G. Rehage, Symposium uber Makromolekiile in Wiesbaden, 1959, I1 A 15.0. E r s t , Diplomarbeit (Aachen, 1959).0. Ernst, Thesis, (Aachen, 1962).G. Rehage and 0. Ernst, DECHEMA-Monogruphie Band 49, (Verlag Chemie, WeinheimlBergstr., 1964), p. 157.G. Rehage and 0. Ernst, Kolloid-2.2. Polymere, 1964, 21, 64.’ R. H. Stokes, Trans. Faraduy Soc., 1952,48, 887.lo F. Sauer and V. Freise, private communication.I’ J. Fuhrmann, Diplomarbeit, (Aachen, 1963).l3 H. J. Palmen, Thesis, (Aachen, 1965).l4 see ref. (1) p. 182 ; ref. (2), p. 308.l5 H. Tompa, C. R. 2e Reunion Sac. Chim. Phys., (Paris, 1952), p . 163.16H. W. Stockmayer, J. Chem. Phys., 1949,17, 588.I7 G. Rehage, D. Moller and 0. Ernst, Makromol. Chem., 1965, 88, 232.I 8 G. Rehage and D. Moller, J. Polymer Sci. C, 1967, 16, 1787.’O R. Koningsveld and A. J. Staverman, J , Polymer Sci. A-2, 1968, 6,305, 325,349,367, 383. ’’ R. Koningsveld and A. J . Staverman, Kolloid-Z. 2. Polymere, 1967, 218, 114; 220, 31.22 R. Koningsveld, Ado. Colloid Interface Sci., 1968, 2, 151.23 G. Rehage and R. Koningsveld, J. Polymer Sci. By 1968, 6,421.24 G. Rehage and W. Wefers, J. Polymer Sci. A-2, 1968, 6, 1683. ’’ G. Rehage 2. Naturforsch., 1964, 19a, 823.26 G. Rehage, Kolloid-2. Z. Polymere, 1964, 196, 97. ’’ C. Robinson, Proc. Roy. Soc. A, 1950, 204, 339.28 J. Crank and C. Robinson, Proc. Roy. SOC. A, 1951, 204, 549. ’’ J. Fuhrmann, Thesis, (Aachen, 1967).G. Rehage and J. Fuhrmann, 2. phys. Chem., N.F., 1967,56,232.31 J. Fuhrmann and G. Rehage, 2. phys. Chem., N.F., 1969, 67, 291.32 G. Rehage, Kunststoffe, 1963, 53, 605.33 F. Sauer and V. Freise, Ber. Bunsenges. phys. Chem., 1962, 66, 353.34 W. Jost, Diffusion in Solids, Liquids, Gases, (Academic Press, New York, 1960), p. 31.35 J. Crank, The Mathematics of Diflusion, (Clarendon Press, Oxford, 1957), p. 14836 G. S. Park, Diflusion in Polymers, ed. by J. Crank and G. S. Park, (Academic Press, London,37 H. Fujita, Fortschr. Hochpolymeren-Forschung, 1961, 3, 1.38 W. Jost, Diffusion, (Steinkopff-Verlag, Darmstadt, 1957), p. 45.R. Koningsveld, Thesis, (Leiden, 1967).New York, 1958), p. 141

ISSN:0366-9033    

DOI:10.1039/DF9704900208

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Light Scattering Spectra and Dynamic Properties of Macro-molecular Solutions*BY R. PECORADepartment of Chemistry, Stanford University, Stanford, California 94305Received 12th January, 1970A brief account of the applications of Iight scattering spectroscopy to the study of the dynamicproperties of dilute macromolecular solutions is presented. In experiments of this type, light isscattered by thermal fluctuations of the molecules in solution about equilibrium. When the circum-stances permit, light scattering may be used to study translational' diffusion, rotational diffusion,intramolecular structural relaxation processes, and chemical reaction rate constants of macromole-cules in dilute solution.Dynamic properties of macromolecules in solution have been studied by polymerphysicists for many years.Numerous studies have been made of intrinsic viscosity,and translational and rotational diffusion of solutions of rigid macromolecules andtheir counterparts for flexible molecules. These properties are especially valuable fortheir use in obtaining macromolecular structural information and for characterizingpolymer samples. In this paper, we describe the application of '' low frequency "(Rayleigh) light scattering spectroscopy to the study of these solution dynamic prop-erties. " High frequency " (Raman) light scattering spectroscopy has been used formany years to study the molecular vibrations of large molecules. We exclude Ramanscattering from our discussion. Raman scattered lines usually have frequencyshifts from the incident frequency greater than about 5 0 ~ m - ~ while Rayleigh scatteringfrequency shifts are usually much less than 50 cm-I.Rayleigh scattering has been used to study static properties of macromolecules insolution. Much information has, for instance, been obtained about molecularsizes and shapes, molecular weights, solution virial coefficients and solution poly-dispersity by this method.This information is obtained by studying the integrated(over frequency) scattered light as a function of scattering angle. However, when theRayleigh scattered light is frequency analyzed, information about the dynamicproperties mentioned above may be obtained in addition to the usual informationabout the static properties. We outline some of the essential physical ideas thatenter into the theory of this light scattering spectrum and quote some of the resultsapplicable to simple systems.Full derivations of the results either have been or willbe presented elsewhere. We also exclude any discussion of experimental techniquehere since this will be discussed by other authors.In most applications, it is expected that detailed interpretation of the spectrum interms of the familiar transport coefficients will be very difficult. This will almostcertainly be true for polydisperse polymer solutions. For solutions of this type, thespectrum itself might, along with properties measured by other techniques, be used tocharacterize the system.THEORYIn this section, we outline the scattering theory for " small " molecules in diluteBy " small " molecules we mean those in which the molceule is too* work supported by The National Science Foundation (U.S.A.)22R .PECORA 223small to exhibit intramolecular interference effects of the type described in the nextsection. The molecule must, however, be large enough to have a polarizability muchlarger than those of the solvent molecules. All solute molecules are consideredidentical and scattering from the solvent is assumed to be negligible.We consider a special scattering geometry. The incident light beam propagatesin the x-y plane with propagation vector ko and frequency coo and the scattered beampropagates in the x-direction with propagation vector and frequency 0,. Theincoming beam has its electric field polarized along with z-direction.The important parameters on which the scattering will depend are the propagationvector shiftand the frequency shiftFor the small frequency changes involved in the present considerationswhere A is the wavelength -of the incident light in the medium and 8, is the anglebetween ko and k,, the scattering angle.The component of scattered light intensity per unit volume with scattered electricvector oscillating along the z-direction is given byK E ko-k, (1)w = 00-0,.K = (4n/A) sin (8J2) (2)I V " ( W 4 = ! y ~ ~ ) a z z ( t ) a z z ( 0 ) - exp {ix [~ct>-r(o)l>> exp (- imdt, (3)and the component with electric vector oscillating along the y-direction isi-a2I H v ( w 4 = - {ayz(t)ayz(0) exp (ix .[r(t)-r(O)]}) exp (- iot)dt, (4) 2n -cowhere A is a constant independent of the fluctuations in the scattering medium anda,,(t) and ayz(t) are the zz and yz components, respectively, of the polarizability tensorof a solute molecule at time t measured in the laboratory-fixed system described above.The position of the centre of mass of a molecule at time t is given by r(t). The angularbrackets denote an ensemble average and p is the number of solute molecules per unitvolume of solution.The sample considered is macroscopically in equilibrium and the light wave ismonitoring the microscopic fluctuations around the macroscopic equilibrium state.The cli depend on time because of microscopic rotational motions of molecules andthe r(t) because of microscopic translational motion of molecules.From eqn (3) and(4) the light wave " sees " a function of position and orientation of a given moleculeat time zero and then again at time t. It then multiplies the function evaluated atthe two times together and averages over the equilibrium ensemble. The Fouriertransform of this " correlation function " taken at the frequency shift cu gives thelight scattering spectrum.The z-axisof a molecule-fixed coordinate system is along the molecular symmetry axis and alland al denote the components of the molecular polarizability tensor parallel andperpendicular to this axis. If this molecular symmetry axis has polar and azimuthalorientation angles 8, 4 with respect to the laboratory-fixed coordinate system, thenandwhere the Ylm(O,$) are spherical harmonics.We confine our calculation to molecules of cylindrical symmetry.. , Z ( 6 4 ) = (all +2a1)/3+(16445)* (all -al> Y2,0(0,4),~ Y Z ( 8 9 4 ) = i(2n/l5)+ (q - 0 r d Y2,1(8,4>+ Y2,--1(094)19(5)(6224 LIGHT SCATTERING SPECTRATo evaluate the spectrum, we substitute eqn ( 5 ) and (6) into (3) and (4) and eval-uate the averages involved.In order to do this we must know the mechanisms oftranslation and rotation in the liquid. We assume that translation occurs by transla-tional diffusion characterized by a translational diffusion coefficient D and thatrotation occurs by rotational diffusion characterized by a rotational diffusion coeffi-cient 0. We furthermore assume that these processes are independent of eachon the right hand sides of eqn.(3) and (4) may now be evaluatedother.2*to yieldThe expressionsandIf the moleculesthen consists solelyare optically isotropic, all = aL = a and IHu = 0.of a single Lorentzian line,The spectrumpAa2 rc2D~ u u ( w 4 = - n w ~ + ( K ~ D ) ~ ' (9)with half-width at half-maximumAm, = rc2D = D ( 4 ~ / 4 ~ sin (8,/2). (10)As the scattering angle approaches zero (forward scattering), the spectral half-widthgoes to zero as sin2(0,/2). Measurements of the Am+ and a plot of them against x2yields a straight line whose slope is the translational diffusion coefficient. Experi-ments of this type are now c~mrnon-place.~-~If the molecule is highly anisotropic (all # al) then this same result may be obtainedby measurement of both I,, and IHu and calculation of Iv,,-$IHv from the data.Meas-urement of IHv alone near the forward direction where 7c2D = 0, yields the rotationaldiffusion coefficientThis theory is highly simplified. More sophisticated treatments of the scatteringfrom solutions of optically isotropic molecules use a hydrodynamic approach fromthe outset.2* 9-10 The scattering is then related to the space-time correlation functionof the dielectric constant fluctuations (A&(r,t)A&(O,O)). This space-time correlationfunction is evaluated by expanding the A& in terms of the fluctuations of relevanthydrodynamic quantities (pressure, entropy, concentration, etc.) and then solving thehydrodynamic equations for the regression of these quantities.The scattered spec-trum is thus obtained in terms of transport coefficients. Our result eqn (9) may beobtained from the hydrodynamic approach by making several approximations, mostof which are applicable in the dilute solution limit.1°LARGE OPTICALLY ISOTROPIC MOLECULESMore complicated spectra than those of 92 arise if the molecule is very large.Since, in molecules of this type, segments on a given molecule may be highly corre-lated with segments on the same molecule a large space and time distance away, thesecorrelated relative motions of the segments must usually be taken into account incomputing the spectrumR. PECORA 225Eqn (3) must be modified by considering the basic scattering units to be thesegments (assumed identical) of a given Thenwhere ag)(t) is the zz component of the polarizability tensor of the ith segment in thelaboratory-fixed coordinate system ; r,(t) is the position of the centre of mass of theith segment ; p is number density of molecules in the solution, and n is the number ofsegments per molecule.We treat here only optically isotropic molecules and notefrom eqn (5) that a$ is a constant and may be taken outside the integral.Each ri(t) may be expressed as the sum of the position of the molecular centre ofmass in the laboratory system R(t) and the position of the ith segment relative to acoordinate system with origin at the molecular centre of mass bi(t),ri(t) = R(t) + b,(t).Eqn (1 1) may be written in the form- exp ( i ~ [b,(t)- b,(O)]) exp (- icot)dt (13)where a is the polarizability of the molecule. It has been assumed in eqn (13) thatthe molecular centre of mass motion is independent of the intramolecular motion.We now write bj(t) in terms of its zero time value bj(0) and a displacement lj(t),bj(t) = bj(0) + lj(t).The intramolecular motion will not affect the spectrum iffor all j and all times relevant to the experiment.If eqn (14) holdsK l j ( t ) < 1. (14)where P(8,) is the particle interference factor well-known to polymer physicists.lIf P(8,) = 1, eqn (9) is obtained for the spectrum.If the polymer is uniform, rigid and spherically symmetric, no additional frequencyshift due to the relative motion of the segments (rotational diffusion in this case) cantake place.There is no change in the configuration of segments upon molecularrotation that can be “ seen ” by the light wave.3The simplest case that shows the “ intramolecular ” motion effect is that of a longrigid, thin, uniform rod.3* l 1 If it is assumed that translation of the rod is by trans-lational diffusion, rotation by rotational diffusion and, furthermore, that the transla-tion and rotation are independent of each other, thenThe spectrum now consists of essentially two Lorentzians with relative strengths givenby the squares of sums containing spherical Bessel functions of order 0 and 2, res-pectively. These strengths may be numerically evaluated in terms of a dimensionless226 LIGHT SCATTERING SPECTRAparameter rCL equal to the product of K and the length of the rod L.The Lorentziancontaining the rotational diffusion coefficient is important only for ~ L s 5 , i.e., forlong rods observed at large K (small A, large scattering angle). This is true since therelevant fluctuation (rotation) must be large enough for the light wave to " see " thedifference between the segment distributions at various times (see eqn (14)).Calculations of this type have been performed for the Rouse-Zimm model offlexible-coils l2 and for the once-broken rod l3 with similar results. So far, only therod theory has been experimentally ~0nfirmed.l~ No calculations have as yet beenexplicitly performed on the effects of large molecular size on IHv(~,w).The results given above must be modified when dealing with polydisperse polymersolutions.In this case the solution may be considered to be a multicomponent sys-tem. Each component will have different dynamical constants and structure, givingrise to its own spectrum. Hence, to obtain the resultant spectrum, the equationsgiven above and in $2 must be averaged over the whole polymer distribution. Quan-tities such as the molecular weight (or size) dependence of the rotational and transla-tional diffusion coefficients and intramolecular relaxation times as well as the distri-bution function must be known. One might also, under favourable circumstances,expect to use the light scattering spectrum to determine the molecular weight distri-bution of the sample. The scattering spectrum might, in addition, be used to studyaggregation in biological materials.A calculation of the effects of polydispersity on light scattering from polydisperserods and gaussian coils has been performed using the Schulz two-parameter unimodalmolecular weight distribution function.CHEMICAL REACTIONSThe light scattering theory developed above may be extended to systems in whichthe solute molecules are in dynamic chemical equilibrium.We quote here some of themajor theoretical assumptions and results applicable to an especially simple case. 6-1The solute in a dilute solution is in dynamic chemical equilibrium between twostates A and B with in general different optical and dynamic propertieskfA+B,where kf is the rate constant for the transformation A+B and kb that for the reversetransformation B+A.It is assumed that (1) the molecules A and B are optically isotropic ; (2) moleculesA and B translate by translational diffusion with equal translational diffusion coeffi-cients denoted by D ; (3) the duration of the transformations A+B are so short thatthe molecules do not have time to diffuse during the transformation.kbWhen these conditions are fulfilled the spectrum is given by- DK2 ] (17) AM rc2D , ANT[- (Drc2+z-l) L u ( w 4 = - n m2 + ( K ~ D ) ~ n w 2 + ( D K ~ + z- ')' co2 + ( D K ~ ) ~whereand M and N depend on the equilibrium concentrations CA, CB and polarizabilitiesaA, aB of species A and B, respectively, and the total concentration C = CA+ CB ;M = C ~ a i + C&)N = kjcAcCi f k,C,ai - (k,CA f kbCB)aAa, (19R .PECORA 227When, in addition to equal diffusion coefficients, the two species have equalpolarizabilities, N = 0 and the reaction rate no longer contributes to the spectrum.When r1 % x2 D (fast reactions andlor observation at low scattering angle) the contri-bution of translational diffusion to the spectrum may be ignored and the reactionrelaxation time z obtained directly from the spectrum. If the reaction is " turned off "i.e., kf = k, = 0, then the spectrum is that of a mixture of two independent specieswith concentrations CA and CB.More general expressions for the spectrum of light scattered from fluids of chem-ically-reacting, optically isotropic molecules have been obtained from hydrodynamictheory by Blum and Salzburg.20 Only one experiment observing chemical reactionrelaxation times by light scattering has been published.2SUMMARY AND DISCUSSIONThe technique of light scattering spectroscopy is potentially a very useful tool inthe study of dynamic properties of macromolecules in solution.It has alreadybecome a standard method of measuring macromolecular translational diffusioncoefficients. It will probably become a useful method for the study of rotationaldiffusion, intramolecular relaxation processes for flexible polymers, chemical reactionrate constants, and solution polydispersity. It is a relatively easy experimentaltechnique once the basic methods are learned. It requires only small amounts ofsample and is non-destructive. One studies only the thermally induced fluctuationsaround the macroscopic equilibrium state.No external perturbing force is required.Several laboratories are now engaged in the experimental development of thisfield. Much theoretical work also remains to be done. The results given in thispaper were given only for limiting cases. For example, theories for concentratedmacromolecular solutions and for large, flexible, optically anisotropic molecules havenot yet been formulated. This field will be rapidly developed in the next few years.K. Stacey, Light Scatterirtg in Physical Chemistry (Butterworths Sci. Publ., London, 1956).R. Pecora, J. Chem. Phys., 1964, 40,1604.R. Pecora, J. Chem. Phys., 1968,49, 1036.H. Z. Cummins, N. Knable and Y. Yeh, Phys. Rev. Letters, 1964, 12, 150.S. B. Dubin, J. G. Lunacek and G. B. Benedek, Proc. Nut. Acad. Sci., 1967, 57, 1164.N. C. Ford, W. Lee and F. E. Karasz, J. Chem. Phys., 1969,50, 3098. ' M. J. French, J. C. Angus and A. G. Walton, Science, 1969, 163, 345.A. Wada, N. Suda, T. Tsuda and K. Soda, J. Chem. Phys., 1969,50,31.R. D. Mountain, Rev. Mod. Phys., 1966,38, 205.R. Pecora, J. Chem. Phys., 1968,48,4126.R. Pecora, MacromoZ., 1969, 2, 31.Abstract TDI.Y. Tagami and R. Pecora, J. Chem. Phys., 1969,51, 3293, 3298.B. J. Berne, J. M. Deutch, J. T. Hynes and H. L. Frisch, J. Chem. Phys., 1968, 49, 2864.B. J. Berne and P. Pecora, to be published.R. Pecora, Photochemistry of MacromoZecuZes ed. R. F. Reinisch (Plenum, New York), inpress.lo R. D. Mountain and J. M. Deutch, J. Chem. Phys., 1969, 50, 1103.'* R. Pecora, J. Chem. Phys., 1965,43, 1562; 1968, 49, 1032.l4 H. Z. Cumins, F. D. Carlson, T. J. Herbert and G. Woods, Biophys. J., 1968, 8, A95.I7 B. J. Berne and R. Pecora, J. Chem. Phys., 1969, 50,783 ; 1969,51,475.2o L. Blum and Z. M. Salzburg, J. Chein. Phys., 1968, 48, 2292; 1969, 50, 1654.21 Y. Yeh and R. N. Keeler, J. Chem. Phys., 1969, 51, 1120

ISSN:0366-9033    

DOI:10.1039/DF9704900222

出版商:RSC    

年代:1970

数据来源: RSC