摘要:

Dependence of the Gel Point on Molecular Structure and Reaction Conditions BY R. F. T. STEPTO Department of Polymer and Fibre Science, The University of Manchester Institute of Science and Technology, Manchester, M60 1 QD Received 2nd January, 1974 The interpretation of gelation in irreversible polymerisations is discussed in terms of a combination of the Frisch and Kilb theories. According to this a branching parameter, A, is determined from the equation A = (acCf- 1)- 1)/((1- ac)a&- l)), and is interpreted using the relationship A = ($rub2)* ~$(l,~)/iVc~~~. In these expressions ccc is the critical value of the branching coefficient, f is the branch-pomt functionality, u is the number of bonds in the smallest ring which can form, b is the effective bond length of these bonds, d(l, t) is a constant, and text is a measure of the average concen- tration of a reaction system.The interpretation is applied to data from polyurethane and polyester forming systems in which f = 3. Two expressions for c,,t are tried-the average initial and average gel-point concentration of reactive groups, (cao+ cb0)/2 and (cacf ct,,J/2, respectively. The polyester results in particular are satisfactorily interpreted in this way, and over-a11 the results indicate a preference for the use of ( ~ a c + q,,-)/2. With this expression the correct limiting behaviour of X at small c;; is found, and the values of b derived from the preceding equations are in better accord with those expected from solution properties. Although the classical theory of gelation due to Flory and to Stockmayer provides the basic explanation of the occurrence of gelation in non-linear polymerisa- tions, it is well-known that the gel point often occurs at extents of reaction larger than those predicted.Such excess reaction can result from the unequal reactivities of like functional groups, and from the formation of ring structures through intramolecular reaction. The former cause is particularly prevalent in reaction mixtures used in commercial resin preparation. However, even when equal reactivity is assured intramolecular reaction occurs to an appreciable extent, even in the absence of solvent, as shown by the data of Stanford and Stepto.2 General formulations of polymerisation statistics to include both unequal reactivity and intramolecular reaction have been described by Gordon and co- workers.3-5 Here, two cases can be distinguished, the equilibrium and non-equili- brium. The non-equilibrium case is the more difficult to treat but is perhaps the more relevant as in many polymerisations based on reversible reactions chemical equilibrium is not allowed to establish itself, and in addition many polymerisations employ irreversible reactions. Descriptions of the non-equilibrium process are essentially in terms of sets of rate equations which describe the interconversion of molecular In addition, the equations can be defined so as to allow computation of the gel point without detailed calculations of the concentrations of all the molecular species. The sets of equations so far developed for irreversible non-linear polymerisations are accurate only at small extents of r e a ~ t i o n , ~ ~ and because of the algebraic and computational effort required in their extension it is pertinent to enquire how far gel points in such polymerisations can be predicted and explained in terms of simple 6970 GEL POINTS analytical expressions.The factors which influence intramolecular reaction are dilution of the reaction system, the distribution of displacement lengths between reactive groups on the same molecule, and in a non-linear polymerisation the numbers of pairs of groups on a given molecule which can react together. Ideally, an expres- sion describing the dependence of the gel point on intramolecular reaction should relate to all three factors. If, in addition, unequal reactivity is to be accounted for, the work of Gordon and Scantlebury indicates that it is doubtful whether such an expression can be found.Case has developed equations for the gel point in the absence of intramolecular reaction, which account for what may be termed the intrinsic unequal reactivities rather than the induced unequal reactivities (or sub- stitution effects 4, of like functional groups. Several approximate theories of gelation which relate the gel point to the amount of intramolecular reaction have been published. Of these, the theories of Frisch l 1 and Kilb l3 lead to expressions which may be simply applied to gelation data from the polymerisations offfunctional reactants (RA,), and from the polymer- isations of f-functional and difunctional reactants (RA2 + RB,). However, in order to examine critically the performance of these expressions it is necessary to have data from systems in which like groups are known to have equal reactivity, and in which the previously mentioned factors affecting intramolecular reaction have been varied systematically.Such data on polyurethane and polyester forming systems 6* 9* 14-16 exist, and the aim of this paper is to use this data and the Frisch and Kilb theories to interpret the dependence of the gel point on molecular structure (chain lengths, and chain structures of reactants), and on reaction conditions (dilution, and ratios of reactants). Initially, the Frisch and Kilb theories are discussed and compared. KILB’S THEORY OF GELATION For RA2 +RB, and RA, polymerisations, the classical theory of gelation gives aJf-1) = 1 (1) at the gel point.a is the branching coefficient, it being the probability of passing along one sequence of bonds from a randomly chosen branch unit to another branch unit, given that the chosen unit has already one of its groups reacted, i.e., that the chosen unit is already part of a chain. Given one group reacted, there are f- 1 possible routes to the next unit so that aCf- 1) is the probability that the next unit is reached. When this probability is equal to unity unlimited molecules are formed, and the gel point is reached. As indicated in eqn (1) this occurs when a = a,. Given equal reactivities, a = PaPb for RA2 + RB, polymerisations, and for RA, polymerisations a = pa, with pa and Pb the (fractional) extents of reaction of A- and B-groups, respectively.Henceforth, the symbol a, will be used merely to represent PaPb or pa at the gel point, rather than the critical value of the branching coefficient. Kilb’s theory l3 shows that at gelation, with small amounts of intramolecular reaction, a,(f- 1)(1 -A’) = 1, where A‘ is an intramolecular branching parameter. A‘ 4 1, and is the probability that a sequence of bonds from a chosen branch unit in a chain leads not to a new branch unit but back to one already in the chain. Eqn (2) was derived for RA2 + RB, polymerisations. However, it holds true for RA, reactions as there is still a fixed number (zero) of reactant units between branch points. (A more complicated expression l 3 is required for RA2+RB2 +RB, reactions, as here the number of units between branch points is of infinite range).R.F. T. STEPTO 71 In detail, 3,’ is the total probability of intramolecular reaction, and may be written as the sum of the probabilities over all sizes of rings, i.e., A’ = C r, . (3) n= 1 In this sum, rl is the probability that a ring forms with the preceding branch point (the chosen branch point), r2 the probability that one forms with the branch point preceding that, and so on. Remembering that the molecules are highly branched, this expression will underestimate the amount of intramolecular reaction as it assumes only one possible ring structure of each size. In other words, it assumes a linear sequence of branch units each with one free functionality, which in RA,+RB, systems can be an A- or a B-group.Kilb gives a molecular interpretation to A‘ by an approximate application of Jacobson and Stockmayer’s expression for the equilibrium ring and chain con- centrations in linear polymers. An expression for A’ similar to Kilb’s may be derived by more straightforward arguments, and these show clearly the approximations involved. Consider a terminal group on a growing chain (in the RA2 + RB, case this could be A- or B-), and assume that the rate of reaction intramolecularly or intermolecularly is proportional to the concentration of groups belonging to the chain (tint) or external to the chain (text), respectively. This assumption requires that the intrinsic rate of reaction is insensitive to whether intra- or intermolecular reaction is occurring.It neglects orientation effects which may be prevalent in intramolecular reacti0n.l The probability of intramolecular reaction (A’) is then (4) clnt is defined by the chain statistics of the growing molecule. Taking the position of the terminal group as reference, the concentration at this position of reactive groups separated by vn bonds may be written as Cint Cint + Cext Cext N - Cint - . A’ = ~ 3 1 - (moles functional groups per unit volume). (A) N v is the number of bonds in the smallest ring which can form, and summing over all ring sizes (n) 4(1,+) = X:=l 1%-3, as defined by Truesdell.18 The value of 4(1, 4) is 2.612. Eqn (5) assumes Gaussian chain statistics. This is inexact not only because the chain is not linear, but also because the simultaneous concentrations due to different points of the same chain (i.e., with n = 1,2,.. .) are assumed independent. In addition, v must be sufficiently large for Gaussian statistics to apply as the first term in the sum +(l, +) constitutes 38 % of its value. The final expression for A‘ is Apart from the fact that molecular complexity will change with extent of reaction and will affect the value of b, cine is to a first approximation independent of extent of72 GEL POINTS reaction. In contrast, text is the instantaneous concentration of external groups, so that I’ will increase as the reaction proceeds. The use of a single value of A’ implies that some average value of text is employed, and at present there is no inde- pendent way of determining this.For RA2 + RB, polymerisations Kilb proposed cext proportional to c,,, the initial concentration of A-groups. However, the data of Peters and Stepto l4 on polyurethane forming systems show clearly that the amount of intramolecular reaction is not a function of the initial concentration of difunctional reactant alone. In general, results on poly~ethane,~* l4 and polyester l5 forming systems, which contain data with r(=c,,/cb0) not equal to 1, show that text is best taken as proportional to Cao + cbo, and not to c,, or cbo alone. In the present paper the average initial concentration, (cao+ cbo)/2, will be used as one estimate of text. This retains the symmetry of the Kilb theory with respect to A- and B-groups, and the interpretation of text as the concentration of external A-groups around a B-group, or vice-versa.An alternative expression is the average concentration at gel, (c,, + cbc)/2. The use of these two expressions implies that one is seeking to characterise the total intramolecular reaction up to the gel point in terms of the ring-chain competition occurring at single points in the reaction, namely at the start of the reaction and at the gel point. The corresponding expressions for RA, polymerisations are c,, and cat. FRISCH’S THEORY OF GELATION Frisch l1 accounts more completely than Kilb for the molecular complexity of the growing chains, and hence for the numerous opportunities for intramolecular reaction, but still employs a single intramolecular branching parameter, A. Frisch’s expression, which is valid for small I, is a,(f- 1)(1-(1 -ac)A+O((l -a,)2)) = 1.(7) Rearrangement, and the neglect of terms O((1 -a,)2) to give a, in terms of ;t yields (for R # 0) = --f(-&+(j-l)(I-I)’ (1-2) 2;t 1-2 4a )’ At I = 0, a, = df- 1)-l. Expansion of the binomial in eqn (8) with retention of the first two terms yields for small 1. The identity of eqn (9) and Kilb’s expression, eqn (2), enables the same molecular interpretation to be placed on I as on A’, at least for small A. The difference between the two theories is the functional dependencies predicted for a, upon 1 and A’. These are shown in fig. 1 for the case f = 3. The expansion of the binomial in eqn (8) is valid for 0 < A < (f+ 1 - 2 J f ) / ( f - 1) and for A > df+ 1 +2df)/cf- l), that is, for f = 3 , O < A < 0.268 and R > 3.732.However, fig. 1 shows that the Kilb approxi- mation will be inaccurate at values of a, sensibly distinct from 2. In contrast, the combination of the Frisch and Kilb theories, taking the functional dependence of a, upon ;1 from the former, and the molecular interpretation of ;t (or A’) from the latter, gives an expression which may more reasonably be applied to gelation data.g* Note that in eqn (7), 1 is not restricted to the range 0 < 3, < 1, but that in the limit a, equal to I, A is infinite. (In eqn (8), the range of values of a, is covered by taking the positive root for R < 1 and the negative root for I > 1.) This behaviour can ac(f- 1)(1-A) = 1, (9)R. F. T. STEPTO 73 arise because of neglect of higher terms in eqn (7).It means that although values of ;1 > 1 may be given by experimental data, any molecular interpretation of A must be characteristic of the behaviour near 3, = 0. FIG. 1.-Relationship between ct;' and the ring-forming parameters A and A' of Frisch and Kilb theories for f = 3. COMPARISON WITH EXPERIMENTAL DATA The experimental data to be considered are from polyurethane and polyester forming reactions of the type RA,+RB,. In all cases the reactions were with trio1 prepolymers having oxypropylene chains. The polyurethane forming reac- tions 6* 9* 14* l6 used hexamethylene diisocyanate (HDI) in benzene at 70", in nitro- benzene at 105", 80" and 50", and decamethylene diisocyanate (DDI) in nitrobenzene at 80". The polyester forming reactions used adipoyl and sebacoyl chlorides in diglyme (diethylene glycol dimethyl ether) at 60".No density corrections were made to the data and concentrations are expressed in units of mol kg-l. 1 . KILB'S THEORY AND cert According to eqn (2) and (6), a,' should be a linear function of A', with A' pro- portional to the dilution of a reaction mixture (c&t). With regard to the two alternatives for text, namely, (cao + Cba)/2 and (cac + cbc)/2, the presentation of most of the data with text proportional to ca0 + cbo has been given before 9* 14* l5 and will not be repeated here. However, fig. 2 shows a;' as a function of (cat + Cbc)-' for some of the polyurethane l4 and polyester l5 forming systems. In general, with respect to both average initial and average gel dilutions, linear behaviour is not observed, and in view of the subsequent interpretation according to Frisch's expression this indicates that the data lie outside the range of a,' to which Kilb's theory may be74 GEL POINTS applied (see fig.1). (It may be noted that the use of (C,,+C~~)-~ as abscissa gave apparent linear behaviour for the polyurethane data.14 In the light of the more recent results 6* 9* 15* l6 this appears to be because the data covered only a small range of values of I I I I I I 4 2 3 5 6 0 (cat+ %)-'/kg mol-I FIG. 2.-Polyurethanes l4 and polyester^.'^ a, against dilution at gel. X , + - polyurethanes (HDI+ OXTMP*)/nitrobenzene/l05" ; v = 34 ; (a) Cao = 1.77 mol kg-' ; (b) Cao = 1.18 mol kg-' ; (c) Cao = 0.89 mol kg-'. 0, O-polyesters/diglyrne/6O0. 1 .-adipoyl chloride + LHT240* ; v = 37 2.-sebacoyl chloride + LHT240 ; v = 41 3.-adipoyl chloride+ LHT112* ; v = 66 4.-sebacoyl chloride+ LHT112 ; v = 70 5.-adipoyl chloride + LG56* ; v = 132 6.-sebacoyl chloride+ LG56 ; v = 136 *OFTMP--oxypropylated trimethylol propane *LHT240, LHTl12-oxypropylated 1,2,6 hexane trio1 *LG56-oxypropylated glycerol In fig.2, the curves for the polyester systems show that the excess reaction at gelation is a decreasing function of v, as required by the expression for tint, eqn (5). This also occurred with (cao + cb0)-l as abscissa. The polyurethane results come from three series of experiments, each at an approximately constant initial isocyanate concentration (cao). Each of the series defines a separate curve, and the same behaviour was found with (ca,+ cb0)-l as abscissa.The results show that at a given text the number of rings formed increases with initial isocyanate concentration. This is consistent with other ring-data on this reaction in both linear l9 and non- linear polymerisations,2 and is probably due to peculiarities of the reaction.lg However, it may be that the average initial or gel concentration is too simple an expression to use for c,,,. This can be resolved if more systematic data, with respect to ratios of components, were obtained for other systems. For the majority of the polyester reactions, r s 1, and no systematic variation in the ratio of reactive groups was undertaken.R. F. T . STEPTO 75 2 . THE FRISCH-KILB THEORY AND EVALUATION OF b As stated previously, the combined theories of Frisch and Kilb provide a more useful basis for the interpretation of gelation data.mental values of ct, via eqn (7), i.e., 2 is evaluated from and is interpreted according to eqn (6), with 1. cc c&t. j0/* I. 6 4 ./’ 0.2 0.4 0.6 0.8 1.0 the experi- (10) - 1.2 (cao 4- cbo)-’ /kg m01-l FIG. 3.-Polyurethanes and polyesters. h against initial dilution. Legend as for fig. 2. Plots of 2 against ( C , , + C ~ ~ ) - ~ and (caC+cbc)-I are presented in fig. 3 and 4 for the same systems as in fig. 2. The polyester results follow essentially the behaviour predicted. Some curvature is present when ( C , , + C ~ ~ ) - ~ is used as abscissa, but this may be expected in view of the previously discussed approximations in the derivations of eqn (6) and (7).For the polyurethane systems (caC+cbc)-l would appear to be the better approximation to c&:, as direct proportionality between 2 and (c,,+ Cbo)-’ is not observed. (The lines a, b, c in fig. 3 for the polyurethane results are merely straight lines through the sets of points. Deviations from linearity must occur at smaller values of A in order that the results extrapolate to the origin). Fig. 5 and 6 show further results 2 s 9* l6 for polyurethane forming systems. These generally follow Frisch-Kilb behaviour at the smaller values of 2, although the scatter of points about the curves drawn is larger than with the previous results. This again may be due to the additional dependence of ring formation on isocyanate concentration. Most of the reactions refer to r E 1, and the effect of isocyanate concentration was not systematically investigated. Again more deviations from linearity occur when (c,,+cbC)-l is used as abscissa. The plots in fig.3 to 6 indicate that, provided ;1 is not too large, the results are in accord with the functional dependence of A upon a, predicted by eqn (lo), with text76 2.0 1.5 2.c I 4 1.; h 0.E 0.4 - - GEL POINTS b AC I 2 3 4 5 (Cat + &)-'/kg mol-' FIG. 4.-Polyurethanes and polyesters. A against dilution at gel. Legend as for fig. 2. X x I/ / / / 0 0.5 1.0 1.5 2.0 2.5 (cao + cb~)-l /kg mol-' FIG. 5.-Poly~rethanes.~* 9* l6 h against initial dilution. 1 .-HDI + LHT240*/nitrobemene/8Oo ; v = 36 2.-DDI+ LHT240/nitrobenzene/80° ; v = 40 3.--HDI+ LHTl12*/nitrobenzene/8O0 ; v = 66 4.-HDI + LG56*/nitrobenzene/8O0 ; v = 112 5.-HDI + LHT240/nitrobenzene/50° ; v = 36 6.-HDI + LG56/benzene/7O0 ; v = 1 15 * see legend to fig.2.R. F. T. STEPTO 77 taken to be proportional to the initial or the gel point concentration of reactive groups. In addition, from the polyurethane results at 105", there is some indication that use of the gel point concentration is more generally valid. (cat+ &-'/kg mol-' FIG. 6.-Polyurethanes. h against dilution at gel. Legend as for fig. 5. TABLE VA VALUES OF b ACCORDING TO FRISCH-KILB THEORY system V (0 b l m (ii) blnm polyesters (diglyme/60°) adipoyl+ LHT240 37 0.235 0.403 sebacoyl+ LHT240 41 0.240 0.408 adipoyl + LHTll2 66 0.207 0.343 sebacoyl+ LHTll2 70 0.227 0.368 adipoyl+ LG56 132 0.202 0.320 sebacoyl+ LG56 136 0.208 0.324 polyurethanes (nitrobenzene/8O0) HDI + LHT240 36 0.22 0.370 DDI+ LHT240 40 0.22 0.370 HDI+ LHTll2 66 0.20 0.310 HDI+ LG56 112 0.16 0.260 polyurethanes (nitrobenzene/50") HDI+ LHT240 36 0.20 0.350 polyurethanes (benzene/70") HDI+ LG56 115 0.16 0.260 polyurethanes l4 (nitrobenzene/l05") HDIf OPTMP 34 - 0.30-0.3578 GEL POINTS It was found previously 9* that the slopes of lines obtained with (cao+ cb0)-l as abscissa were not in proportion to v-*.This can be due to variations in b with chain structure. However, if cext is equated with (c,,+cbo)/2, and with (ca,+cbc)/2, b can be evaluated from eqn (6) and the slopes of the lines, or the slopes of the curves at small 2, and a quantitive assessment of this equation made. Values of b so derived are shown in table 1.Both sets show the same internal consistency with respect to the polyester results and the polyurethane results at 80". Here, for a given difunctional reactant an increase in v means a larger proportion of oxypropylene units as compared to methylene units in a ring, and results in a decrease in the value of b. This is in accord with the known more flexible nature of the polyoxypropylene as compared to the polymethylene chain. The values of b for these chains are approximately 20* 21 0.34 and 0.40 nm, respectively. In addition, amongst the polyesters, the change from adipoyl to sebacoyl chloride produces an increase in b in accord with an increased proportion of methylene units in the smallest ring. The range of values of b from the polyurethane forming reactions at 105" are the maximum and minimum values given by the three lines in fig.4. No values were deducible from fig. 3 as Frisch-Kilb behaviour was not observed with (cao + cbO)-l as abscissa. In absolute terms the values of b derived with (cac + cbc)/2 equal to c,,, are in the better agreement with the values of b expected from solution properties. This indicates that a majority of the rings are formed near the gel point in accord with the rapid increase in the numbers of functional groups per molecule at this point in the reaction. CONCLUSIONS Particularly for the polyester forming systems, the Frisch expression, eqn (lo), provides a satisfactory explanation of the observed gel points, given that 2 is inter- preted according to eqn (6). The unknown quantities in A, namely, b and text, cannot be determined separately from gelation data.The structure of the theory is such that an average value of cext is required. Apart from the polyurethane results at 105" the data indicate that either the average initial or the average gel-point concentration of functional groups can be used for small values of A. Support for the latter expression is found in that the values of b so derived are in better agreement with values from solution properties.20* 21 The simple description used here, eqn (10) and (6), may be contrasted with the more rigorous description of irreversible non-linear polymerisations in terms of sets of rate equation^.^^ In the latter, the quantity text varies automatically as the polymerisation proceeds, and the ring-forming parameter is essentially tint.Thus, the uncertainty attached to text does not exist. The author wishes to acknowledge helpful discussions with Dr. J. L. Stanford during the preparation of this paper. P. J. Flory, Priticiples of Polymer Chemistry (Cornell Univ. Press, Ithaca, 1953), chap. IX. J. L. Stanford and R. F. T. Stepto, IUPAC International Symposium of Macromolecules, Univ. of Aberdeen, 1973, abstract E32. M. Gordon and W. B. Temple, Makronrol. Chem., 1972,160,263. M. Gordon and G. R. Scantlebury, J. Chem. SOC. B, 1967, 1. W. B. Temple, Makromol. Chern., 1972, 160, 277. J. L. Stanford, Ph.0. Thesis (Univ. of Manchester, 1972). H. Jacobson and W. H. Stockmayer, J. Clrem. Phys., 1950, 18, 1600. L. C. Case, J. Polytner Sci., 1957, 26, 333. W. Hopkins, R. H. Peters and R. F. T. Stepto, Polymer, 1974, 15, 315.R. F. T. STEPTO 79 lo F. E. Harris, J. Chem. Phys., 1955, 23, 1518. l 1 H. L. Frisch, paper presented to 128th Meeting Amer. Chem. SOC., Polymer Division, Minne- apolis, 1955. C. A. J. Hoeve, J. Polymer Sci., 1956, 21, 11. l3 R. W. Kilb, J. Phys. Chem., 1958,62,969. l4 R. H. Peters and R. F. T. Stepto, in The Chemistry of PoZymerisation Processes (Monograph No. 20, SOC. Chem. Ind., London, 1965), p. 157. R. S. Smith and R. F. T. Stepto, Makromol. Chem., 1974, 175, 2365. l6 W. Hopkins, Ph.D. Thesis (Univ. of Manchester, 1967). l7 P. J. Flory and J. A. Semlyen, J. Amer. Chem. Soc., 1966,88,3209. l9 R. F. T. Stepto and D. R. Waywell, Makromol. Chem., 1972,152,263. 2o M . Kurata, M. Iwama and K. Kamada, in Polymer Handbook, ed. J . Brandrup and E. H. 21 P. J. Flory, Statistical Mechanics of Chain Molecules (Interscience Publishers, London, 1969) C. A. Truesdell, Ann. Math., 1945,46, 144. Immergut (Interscience Publishers, London, 1966), sect. IV-1. p. 40.

ISSN:0301-7249    

DOI:10.1039/DC9745700069

出版商:RSC    

年代:1974

数据来源: RSC

摘要:

GENERAL DISCUSSION Prof. A. Silberberg (Israel) said: It is in principle impossible for a true homo- polymer to form a gel. If the groups which characterize the polymer can interact to give rise to an energetically favourable, i.e., attractive situation, the homopolymer could not be soluble in the solvent medium. For a homopolymer to be soluble the net interaction between polymer segments must be repulsive. If, therefore, gels are formed it is safe to assume that the polymer possesses some copolymer character. For example, polymethacrylic acid, although chemically uniform, is a stereo copolymer and we may assume that the energetic interactions between two polymer segments in this and similar cases can move from repulsive to attractive, depending upon the nature of the stereo regular environment of the two interacting segments.While the attractive configurations may perhaps be only slightly more rare than the repulsive one, leaving the polymer with an overall repulsive character, a cross-link of some perma- nence can arise in these cases if a whole sequence of attractively interacting segments, along one section of one chain, interacts with another such sequence on another chain and builds up a cooperative unit of sufficient strength and sufficiently long chemical relaxation time. Dr. 0. Smidsr#d (Trondheim) (communicated): I should like to comment on a statement made by Silberberg : " It is in principal impossible for a true homopolymer to form a gel ". This is of course true for equilibrium gels, but in general it may not be so for non-equilibrium gels where there may be a kinetic hindrance to the formation in a given situation, of the thermodynamically favoured precipitate.All experience with fractions of agarose, rc-carrageenan and alginate suggest that the modulus gets higher the closer you get to a true homopolymer in composition ((AB),-polymers are then regarded as homopolymers). Dr. G. S. Park (UWIST, Cardzfl) said: In agreement with the remarks of Prof. Silberberg we attributed gel formation in poly(viny1 chloride) to its stereocopolymeric character.2 It is not necessary for the linking points to be crystalline for a gel to be formed and interesting examples of non crystallized linked gels are given by ABA block-copolymers in media that are solvents for B units but not for the A units.The correlation between X-ray crystallinity and the possibility of gel formation in our copolymers, however, has led us to conclude that the linking points in these gels are crystallites. Prof. F. Franks (Unilever) said: Implicit in the use of eqn (4) of Park's paper is the assumption that the copolymer gel can be equated to crystalline polyvinylidene chloride and presumably the enthalpy of gel melting is equated to the latent heat of fusion of the crystalline homopolymer. It is found that in polysaccharide gels, crystalline remnants can exist well above the observed gel melting temperature and the authors mention that this is also a possibility with their copolymer. If no estimate can be obtained of the fraction of crystallites which persist in the sol phase, then what is the quantitative significance of AHv, or indeed AH,? A.Silberberg and P. F. Mijnlieff, J. Polymer Sci., A2 1970, 8, 1089. M. A. Harrison, P. H. Morgan and G. S. Park, European Polymer J., 1972, 8, 1361. 80GENERAL DISCUSSION a1 Dr. G. S . Park (UWIST, Card@) said: As Franks remarks, the persistence of crystallites at temperatures above T, creates a problem. A similar problem is found when other methods are used to detect the ultimate melting point in copolymers. Flory has pointed out that eqn (4) is applicable to the disappearance of those crystal- lites that have been made from the coming together of very long sequences of repeating crystallisable units. Since the probability of occurrence of these long units is very small, the concentration of crystallites at the ultimate melting point of the polymer is minute and so X-ray methods, specific volume and heat content measurements tend to underestimate the true crystalline melting point.The underestimation that comes from equating T, to the ultimate melting point may be somewhat greater than is found in the other method. An attempt to deal with this situation has been made by Taka- hashi et aL2 They have equated the gel melting point with the point at which each polymer chain goes through at least two crystalline regions. For multifunctional junction points this considerably overestimates the critical number of crystallites required for network formation. Nevertheless, it is interesting to look at the relation- ship that they obtain for the gel melting point.This can be expressed in the form 1 1 -ln(l-x)--ln(l-$)--ln N + - r V V 5 V V where the degree of polymerization is N , 0 is the excess surface free energy per mole associated with the terminal unit at the end of the crystallite (the surface free energy per unit), 5 is the crystallite length and the other symbols have the same meaning as in our paper. For small values of x we can now rewrite eqn (1) in the following way Here, A4 is the mean molecular weight, per monomer unit of the polymer chain, v is the specific volume of the polymer, and the other symbols have the same meaning as in our paper. Eqn (2) can be compared with the various relationships used in our paper and when 0 is small it can be reduced to eqn (5) of our paper and hence gives the values of AHv as predicted by that equation. Comparison with eqn (2) of our paper gives which gives an energy having the same meaning as AHx.The ratio of alnC/a(l/T,) to VvAHv is 5 even if 0 is not small so the crystallite lengths given in table 2 are also those that would be predicted from the relationship of Takahashi et al. Prof. C. A. Smolders (Enschede) said: The dependence of gel melting points on polymer concentration at constant copolymer composition could, in principle, also be treated on the basis of a concentration dependence of x. In a comparable T,(c) study of the system poly(2,6 dimethyl-l,4 phenylene oxide) P. J. Flory, Trans. Faraday Soc., 1955, 51, 848. A. Takahashi, T. Nakamura and I. Kagawa, Polymer J., 1972,3,207 ; A. Takahashi, Polymer J., 1973, 4, 379.P. T. van Emmerik and C. A. Smolders, European Polymer J., 1973, 9,293.82 GENERAL DISCUSSION in toluene, where indeed the concentration dependence of x is noticeable, a theory of melting point depression could be worked out quantitatively and checked by experi- ments. Dr. G. S. Park (UWIST, Card@) said: The correct prediction of the effect of toluene on the melting point of 2,6-dimethyl-1,4 phenylene oxide from x values obtained from cloud point determinations is an interesting demonstration of the validity of eqn (4) of our paper for the melting of homopolymers. In our studies of poly(viny1 chloride)-dioxan gels we made an attempt to obtain the concentration dependence of the gel melting point by assuming a constant x in eqn (4). This gave the wrong sign to By substituting the experimental values of T, and 4 in eqn (4) we were able to calculate the x values at each pair of T' and 4 values but x values obtained by other methods were not available for comparison.A similar situation would hold for the copolymer gels. Prof. C. A. Smolders (Enschede) said: Gel melting points of thermoreversible gels can be nicely determined by differential scanning calorimetry. When using this technique one should keep the system at a temperature below 7" for a sufficiently long time for the gelling process to proceed. In a specific system ~tudied,~ viz. poly(2,6-dimethyl-l,4phenylene oxide) in toluene, this meant that we had to keep the sample at the low temperature for at least one hour, before the endothermic peak on reheating showed up, that enabled the determination of T,.Dr. D. S . Reid (Unilever) said: Application of the van't Hoff isochore assumes that junctions are formed by a two-statey all-or-none process, and also that the junc- tions are monodisperse (i.e., the crystallites are all of the same length ). If these conditions are not met, the enthalpies estimated by this type of approach can be seriously in error. An example which shows just how wrong van't Hoff enthalpies can be is contained in the paper by Reid et al. where, due to the polydisperse nature of the species associating to form junctions, the vant' Hoff enthalpy corresponds to a junction size which is only about 30 % of M, of a junction. In view of this, I wonder why the authors have not carried out more extensive calorimetric studies in order to measure directly the enthalpy changes which occur during the melting of the gel, and correlate these with the composition of the polymer, and the nature of the solvent.Dr. G. S . Park (UWIST, Card$) said: I was surprised that Reid et al. found such large differences between the van't Hoff and the calorimetric enthalpies. It would be interesting if differences of this size occur in our AH values but it does not appear possible to obtain calorimetric values in our system. In the study of helix-coil transmissions made by Reid et al. it is assumed that the whole of the polymer under- goes the transition and so it is easy to calculate the enthalpy change per polymer mole- cule or per polymer unit from the total heat change that takes place in the system.The crystalline-amorphous transition that produces gel melting in our copolymer gels or for instance in poly(viny1 chloride) gels involves only a portion (and probably a very minor portion) of the polymer. Neither the number of network junction points nor the number of monomer units involved are known and so measurements of total heat changes would not allow the required values of the enthalpy per junction point to be calculated. P. T. van Emmerick and C. A. Smolders, European Polymer J., 1973, 9, 157. M. A. Harrison, P. H. Morgan and G. S. Park, European Polymer J., 1972, 8, 1461. P. T. van Emmerik and C. A. Smolders, European Polymer J., 1973, 9, 293.GENERAL DISCUSSION 83 Dr. I. Tar (Lorand Eiitv6s University, Hungary) said : The Harrison-Morgan-Park method (1971) was applied to our model, gelatin+water in 1972.Gelatin does not belong to the same group of gels as Park’s vinylidene chloride copolymers according to the system introduced by Flory in his Introduction. In spite of this, it could be of interest to show some of our results here, because we worked with exactly the same method and looked at a similar problem, the effect of solvent properties on the gel melting point and the heat of gelation, as did Park and his coworkers. We achieved melting points (T,) for isoelectric gelatin gels (pH = 5.33) of various concentrations using the H-M-P microcapillary viscometer and worked out results for AHusing the method of Eldridge and Ferry (1954). It was our aim to show if well known phenomena, such as the strong salting-out effect of Me” chlorides, or the structuring effect due to nonionic surfactants manifest themselves in reasonable AH changes.This being the case, it is our opinion that AH data are useful to detect changes brought about by co-solved substances in the gelling system. heat of gelation AHlkcal mol-1 character and concentration of the co-solved substance (g/dl) hydrophile ionic NaCl BaCI2 Na2S04 5.0 - 68.4 5.0 -61.4 5.0 - 62.9 nonionic PEG M = 550 2.0 - 67.1 PEG M = 2000 1 .o - 60.4 amphiphle ionic NaDS c 1 2H2 5 S04Na 0.056 - 63.4 a nonionic BEROL 043 C18H37E01 oOH 1.0 - 68.6 b M = 711,04 2.0 - 74.5 BEROL 08 c 1 8 H3 7E05 0 OH 0.5 - 74.5 M = 2473, 2 1 .o - 75.6 Rousselot photogelatin in dist. water pH = 5 , 33 - 67.45 2.4 a M.Kustos, Thesis, 1974 (Budapest) ; b Z s . Fodor, Thesis, 1972 (Budapest). Dr. G. S. Park (UWLW, Cardif) said: It is gratifying that the values Dr. Tar obtains for AH correlate with other known effects such as structuring and salting out of the gelatin system. It would be interesting to investigate parallel changes in rigidity as we are now doing in poly (vinyl chloride)-plasticizer systems. Dr. M. Pyrlik and Prof. G. Rehage (Clausthal) (partly communicated) : We wish to comment on the method of determining gel melting points used by Park et al., and, as a consequence, on the evaluation of the results. The method is able to determine a certain state of fluidity of the system, in other words, one measures an iso-viscous state of the melting gel. We believe that it is hardly possible to get the melting point of the largest crystallites (which clearly melt at I.Tar, Zs. Fodor and E. Wolfram, Magy. Ke’m. Foly’oirat, 1973, 79, 532.84 GENERAL DISCUSSION the highest gel temperatures of the melting range) by measuring a viscosity property of the system. This can be proved by the fact that melting of the crystallites, which have to be regarded as the junction points of the network, is not indicated solely by the complete dissociation of the network. Even at the temperature of the commencement of flow, larger aggregates-parts of the network structure-are present in the melt indicating that only a certain proportion of all the crystalline junctions are already molten. We have performed dynamic mechanical measurements on thermally reversible gels with stereospecific PMMA and have found that even at very low rates of temperature rise (3”/h) the temperature dependence of the loss modulus G” (and, consequently, of the viscosity) is by no means a step function but-over a temperature of some 40°C-a gliding Also, far below the commencement of flow, which is measured in the apparatus of Park et al., the slope of the (G”, temperature) curve is considerable, according to our results.0,2 0,4 0,s 0,8 Y z FIG. 1.-Phase diagram of the system gelatine+water. Ts = melting temperature; y2 = weight fraction of gelation. Method : DSC. The disappearance of the temperature dependence of the storage modulus G’ is a better criterion, because “ normal ” macrornolecular solutions, in most cases, do not show a strong temperature dependence of the storage modulus.Also in a melting gel small parts of the dissociating network network structure contribute to Gr ; only after complete melting of the largest crystallites does the storage modulus reach a minimum value. This is the case usually above the temperature, at which the gel is molten. Determinations of gel melting points by measurement of viscous properties can, therefore, not yield a thermodynamic equilibrium value for the melting point of the crystallites. Evaluation of these melting temperatures according to an existing theory, for example the Flory theory of melting point depression or the copolymer theory, is not really possible. Evaluation of melting points using the Ferry-Eldridge relation- M. Pyrlik, W. Borchard, G.Rehage and E. P. Uerpmann, Angew. Makromol. Chem., 1974,36, 133. M. Pyrlik and G. Rehage, Rheol. Acta, in press. M. Pyrlik and G. Rehage, Kolloid-2. 2. Polymere, in press.GENERAL DISCUSSION 85 ship seems to be possible, if the slopes of the concentration dependence of the measured and the equilibrium melting pionts are the same. However, it must be stated, that the Ferry-Eldridge relationship can describe the concentration dependence of the melting points of the crystallites only over a certain concentration range. Most of the solvent-polymer systems have a phase diagram showing a eutectic. Consequently, the gel melting points of gels with crystalline junction points must have similar concentration dependences showing a point of inflection. The figure shows the melting points of an aqueous gelatin gel against the concen- tration, measured using D.S.C.techniques.l The curve is identical with the right hand side of a common eutectic phase diagram. It is clear that the Ferry-Eldridge equation is only applicable to the very low concentration part of the curve, which shows a distinct increase of gel melting points with concentration. Application of the relationship to the main part of the curve (with low concentration dependence) yields results for the crystallization enthalpy AHcr, which are apparently too high. Dr. G. S . Park (UWIST, Cardifl) (communicated) : Rather than giving a measure of an isoviscous state, we believe that our technique gives a good approximation to the temperature at which transition from completely recovei able elastic deformation to a state of non-recoverable viscous flow occurs.This is the sol/gel transition temperature and is independent of the actual value of the viscosity. For this reason the shape of the (G”, temperature) plot does not appear to have relevance for our measurements. Nevertheless, I would agree that unmelted crystallites are still present at temperatures above our gel melting points and the suggestion of Pyrlik and Rehage that the disappearance of temperature dependence in the storage modulus, G’, could give a better measure of the disappearance of the last network traces in gelling systems is useful. Our measurements determine the point at which the number of crystallites reaches such a critical value that the continuous network disappears and so the kind of treatment referred to in our reply to Franks should be applicable.Prof. M. Gordon (University of Essex) : Edwards’s paper presents a fundamental and comprehensive physical theory of polymer dynamics. As in his equilibrium theories, Edwards uses a framework of continuum mathematics. Not only is the polymer chain smoothed to a continuous line, but an infinite number of integrations (Wiener integrals) may be carried out on suitable functions. Now the elementary graph theory, inherent in the usual discrete descriptions of chains and networks (consisting of point-atoms and line-bonds) ought to be contained in Edwards’s more sophisticated formalism, presumably by virtue of an isomorphism of the underlying operator algebras of the continuum and graph-theoretical operators, which is largely unexplored. I have two specific questions : (a) Edwards’ formulation covers the whole range from liquid solutions, through the critical entanglement point, to covalently cross- linked gels, and should, therefore, deal with the transition regions. Near the gel point, long-range effects embodied in the “ extinction probability ” u (see paper by Burchard et al., this Discussion) dominate the properties. For instance, in reversible gelation, the gel point is an Ehrenfest transition of order five.Part of this order comes from the fact that the modulus is a second derivative of the free energy, but three further (temperature) differentiations of the modulus are required before we encounter a discontinuity at the gel point.This is due to the fact that the modulus (or concentra- tion of active network chains) is proportional to (1 - v ) ~ , and (1 - u) vanishes at the gel K. Bergmann, Diplomarbeit (TH Clausthal, 1974).86 GENERAL DISCUSSION point (cf. Burchard et al., this Discussion, eqn (39), using eqn (37) and (38)). Thus the high order of the transition derives from the unprecedented effect of long-range cor- relations embodied in the extinction probability, and this is well supported by experiments on widely different amorphous gelling systems. Edwards’s formulation appears on the surface to be entirely a local one (a primary chain wriggling in a tube). How does a segment of the primary chain know that it is connected by covalent paths to the surface, and how is the transition covered in the continuum formulation? (b) Gel formation is not dependent on pre-existing primary chains.In an f- functional polycondensate, active network chains can be traced after the gel point, but are only brought about by the linking process itself. The discrete (“ graph-like- state ”) formalism covers all gelling systems with the same equations, when suitable parameters and functions are substituted (see DuSek, this Discussion, eqn (5)-(7)). Thus thef-functional condensates also have a fifth-order gel transition, as can be derived from Dobson and Gordon’s eqn (12) and (13).3 Can Prof. Edwards generalise his formulation to free it from the restriction to pre-existing primary chains? Prof. S . F. Edwards (Science Research Council) said: The use of functional integration methods is merely a convenient notation.The content is entirely con- ventional. Thus the discrete nature of atoms and bonds can be built in if one wishes. This puts in a lot of detail however which is unnecessary to describe long range effects, and since it is these long range effects which differentiate polymer physics from other areas, a formalism which permits the discarding of unwanted detail seems worth pursuing. This comment covers Gordon’s point (b) : there is no difficulty in writing the statistical mechanics of any system in these terms, it is indeed standard practice in all of normal solid or liquid state physics in the second quantization formalism (which need have nothing to do with quantum theory). As regards (a), since the formalism is merely a representation of the usual set up, when one puts in the same assumption, one gets out the same conclusions.I must confess some surprise at Gordon’s confidence that the transition is exactly of order five, since I would have thought an exact answer would imply a solution of well known but hitherto insoluble problems such as the percolation problem. However the ferroelectric transition is soluble whilst the ferromagnetic is not, so perhaps that is right. I am not an expert in this matter, and my contribution is confined to study on either side of the transition except in as much as I and Grant have studied the dynamics of diffusion through the transition (4.v.). However it is clear how the transition would arise: the radius of the confining pipe tends to infinity at the transition.Dr. B. Warburton (London University) said: Some gelling processes in practice take place in a surface.4* I would like to ask Edwards if his theory is sufficiently flexible in its generality to be applicable to two dimensional behaviour of chains. At first sight it would appear that equations of the type (1.1) and (1 -2) of his paper have an equivalent form in two dimensions but when complex topological considerations arise, such as those discussed at the end of the present paper and in the author’s M. Gordon, Trud, 1969, Meshdinarodnoi Conf. Kautsh. i. Resine Chemia, Moscow, 1971. see e.g., S. Strella and A A Bibeau, J. Macrornol. Sci., 1966, 1,417. G. R. Dobson and M. Gordon, J. Chem. Phys., 1965,43,705 ; Rubb.Chem. and Technol., 1966, 39, 1472. K. Wibberley, A study of some surface properties of solutions of salts of arabic acid, Ph. D. Thesis (University of London, 1963). E. Shotton, K. Wibberley and A. Vazin, Proc. 4th Znt. Congr. Surface Act., 1967, 11, 1211.GENERAL DISCUSSION 87 it would seem that any theory for the surface would need previous publications,l* to start ab initio. E.g., the invariant expression, I = II(dAxdB)-V(A-B)-l (eqn (2.22) reference (3)) for the condition of encirclement of one curve by another would be meaningless in two dimensions. We have been studying the formation of gel films at aqueous/air and aqueous/oil interfaces in connection with a programme of research into the action of polysacch- aride emulsifying agent^.^'^ It would appear that good emulsifyers, judged by the mechanical stability of the oil-in-water emulsion formed, form a viscoelastic solid or gel film at the interface but in order to allow the emulsion to have complete mobility in its continuous phase no gelation must occur in the bulk solution. Prof.S. F. Edwards (Science Research Council) said : The surface gelation, like all surface phenomena, cannot be a truly two dimensional phenomenon, but must correspond to a state of affairs which decays away quickly as one moves away from the surface. A combination of the theory of surface tension effects with the normal gelation should be possible, but I have not attempted it. The invariant still has a meaning under these conditions. In a mathematically exact two dimensional problem it becomes the angle swept out by a surface chain about a point (say the end of a chain at right angles to the surface).Entropic problems can be solved exactly in that limit, but I suspect that is not really helpful for the three dimensional problems, consequently the thickness of the surface is essential. Prof. A. Silberberg (Israel) said: Many years ago it occurred to me that the single effect on which flow in condensed polymer systems depends is a concertina-like contraction and expansion of the macromolecular chain along sections of its length. Brownian motion of the segments in this direction should be less inhibited and thus relative sliding of neighbouring parallel chain sections is enabled. Developing this model one can immediately derive the 3.5 power dependence of steady state viscosity on polymer molecular weight.These results were never published since I discovered that Eyring had years earlier already written a short paper in which similar ideas were outlined in their essentials. Prof. P. J. F'lory (Stanford) said : A note of caution should be recorded on the use of the freely-jointed model for real chains of limited length. Detailed calculations have been carried out by Drs. Yoon, Conrad and Chang in our laboratory on poly- methylene, poly(dimethylsi1oxane) and polypeptide chains using a three-dimensional expansion in Hermite tensor polynomials. These calculations show the density distributions for chain vector r for fewer than about 50 bonds to be highly anisotropic. In keeping with Monte Carlo calculations of Semlyen and coworkers and of Fixman and Alben on the distribution of scalar r, the probability density exhibits a minimum S.F. Edwards, The statistical mechanics of rubbers in Polymer Networks, Structural and Mech- anical Properties, ed. A. 3. Chompff and S. Newman (Plenum Press, New York, London 1971), p. 83. S, F. Edwards, Disc. Faraday SOC., 1970, 49, 43. E. Shotton and R. F. White in Emulsion Rheology, ed. P. Sherman (Pergamon Press, London, B. Warburton, The properties and kinetics of formation of some hydrocolloid surface films, Ph. D. Thesis (University of London, 1972). M. S h e d and B. Warburton, Polymer, 1974, 15,253. 1963), p. 59.88 GENERAL DISCUSSION near r = 0 where the usual formulation in terms of a Gaussian predicts a maximum. Thus, the departures from the Gaussian distribution are severe, this being in addition to inadequacies of the widely used relation < r 2 ) = n1b2 where n1 is the number of hypothetical bonds and b is the length of one of them. Calculations of ring closure probabilities that (i) rely on this approximation, (ii) involve assumption of Gaussian density at r = 0, and (iii) take no account of the possible effects of angular correlations as noted by Stepto, may be subject to considerable error for short chains.On the other hand, (i) and (ii) affect such calculations oppositely, and may therefore be partially compensatory. The approximate correlation of b values in table 1 of Stepto’s paper with those derived from measurements of the unperturbed dimensions of longer chains may perhaps be explained in this way.If so, the agreement must be regarded as fortuitous. Secondly, I should like to suggest that revision of Kilb’s theory to take account of the array of all isomeric species containing x branches may lead to a lower, rather than a larger A’. Thus, in the simple case of a single kind of reacting group A, only the linear isomer A* -AA-AA-A, etc. 1 A I A I A commencing with the chosen group A* and having exactly one unreacted A group at each “ generation ” is considered. It we take the average over all of the ramified “ x-mers ” that can be constructed about an arbitrarily selected root A*, then the expected number of unreacted A groups at the first generation obviously will be < 1 ; in the limit x+ co it is 3. At the second generation, it turns out to be 1 .O in the same limit.Since the ring closures of short length make the largest contribution, it does not seem to follow that elaboration of Kilb’s theory to take account of the full array of isomeric structures would necessarily raise 2‘. For higher generations, the expectancy must become > 1. Dr. G. J. Howard (UMIST) said: An experimental technique which may be used to examine the extent of intramolecular reaction in reactions of the type discussed by Stepto is to locate the critical condition at which gelation will just fail to occur : this may be carried by adjustment of the dilution of the system or by alteration of the ratio of the reacting functional groups. Some 10 years ago, Bates measured the time to gel in the reaction between a low molecular weight styrene/allyl alcohol copolymer and some aliphatic di-acid chlorides.The figure shows a plot of the reciprocal gel times against the acyl chloride/hydroxyl ratio at constant polyol concentration. [Curves 1, 2, 3 are for one crosslinker at three (increasing) concentrations of the styrene/allyl alcohol copolymer. Curve 4 is the same concentration of polyol as 2 but with a shorter crosslinking reactant (adipoyl chloride rather than suberyl chlor- ide)]. As may be seen, it is relatively easy to locate the limiting reactant ratios. These are the conditions whereby the minor component is just consumed at the critical con- dition for gelation. The intercepts on the axis represent the critical branching coefficient, or its reciprocal. The main point I wish to make is that it is possible to learn something of the intramolecular wastage without the necessity of determining extents of reaction by functional group analysis of the reacting systems.The rather ill-defined polyfunctional component in these reactions perhaps makes these results unsuitable for a critical assessment of the theoretical approaches dis- R. F. Bates, Ph.D. Thesis (Manchester, 1965) ; R. F. Bates and G. J. Howard, J. Polymer Sci., C, 1967, 16, 921.GENERAL DISCUSSION 89 cussed by Stepto. However, it should be noted that, within experimental error the intercepts are reciprocally related. This implies that the term Cext is satisfactorily expressed by the initial polyol concentration alone in this system ; CA + C,, whether taken as initial or final concentrations, varies by a factor of about 2, typically, from one intercept to the other.Within the concentration range studied, the Kilb parameter II’ of Stepto’s eqn (2) is linearly related to the reciprocal concentration of the polyol, but the plot does not go to the origin-indeed it lies rather distant from the origin. The parameter ;1 from Stepto’s eqn (10) is, however, directly related to the reciprocal polyol concentration, i.e., by lines similar to those in Stepto’s fig. 5 but with (CA); as the axis. Dr. R. F. T. Stepto (UMIST) said : Howard’s reanalysis of the results of Bates and Howard appears to give support to the use of the Frisch expression in the analysis of gel-point data. The precise definition of cext to be used in the expression for 3, is still unresolved. The gelation data discussed in the paper (particularly those of Peters and Stepto) show that a, depends on the concentrations of both reactants, and c,, + cbo and c,, + cbc were chosen for text as the simplest expressions displaying such a dependence.The apparent insensitivity of ;1 to c,, (initial concentration of difunctional component) shown by the data of Bates and Howard may be due to the high functionality of the polyol employed by them. In other words, text could depend not only on the con- centrations but also on the functionalities of the reactants. Certainly, more compli- cated expressions than c,, + cbo and c,, + cbc are required to bring the data of Peters and Stepto onto single curves. The method used by Bates, which obviated the necessity of determining extents of reaction at gelation, does have the disadvantage that several gel times have to be determined for each a, evaluated.Dr. S. B. Ross-Murphy (University of London) said: I would like to ask Stepto how a, the gel point conversion, was measured in the cyclisation experiments mentioned in his paper? Further, with reference to fig. 2 of this paper, (which shows plotted against dilution) to what accuracy could a, be measured? One would expect that the parameter 2 and the “ effective bond length ” b of table 1 would be very sensitive to small errors in a,.90 GENERAL DISCUSSION Dr. R. F. T. Stepto (UMIST) said: The method of determining a, was through the extrapolation of (conversion, time) curves to the gel time.The gel time was measured as that time at which the reaction mixture climbed the stirrer in a sealed reaction flask. A test of the reliability of this method in absolute terms has been reported by Bates and Howard,' who showed it corresponded to the maximum time of zero gel fraction. The accuracy to which a;' was typically determined may be seen from the results given by Peters and Stepto.2 In general, a;' was determined to better than & 1 %, and duplicate experiments, not reported in the literature, confirm this. The effects of the uncertainty in a, on the values derived for A and b are negligible. This applies particularly to b which is directly proportional to A-+. Dr. J. A. Semlyen (University of York) said : Cyclization processes frequently take place in gelation reactions and have been mentioned already by Flory in his Introduction and in several papers presented at this Discussion. In Stepto's paper, reference was made to the Jacobson and Stockmayer expression for estimating intramolecular reactions in his systems.In particular, chains were assumed to obey the Gaussian expression for the probability of intramolecular cyclization. He emphasised that v (the number of bonds in the smallest ring that can form) must be sufficiently large for Gaussian statistics to apply as the first term in the sum @(l, 3) (of eqn (5)) constitutes 38 % of its value. Values of v for his polyesters and polyurethanes range from 34 to 136. Very briefly, I would like to present evi- dence from a single aliphatic polyester system which suggests that his values of v may indeed be sufficiently large.Recently, Dr. Jones in our laboratory has measured the molar cyclization equi- librium constants for rings in poly(decamethy1ene adipate) and compared these values with those calculated by the Jacobson and Stockmayer theory, which were made by assuming that chains adopt random-coil conformations and obey Gaussian statistics. The discrepancy between experiment and theory is small for the cyclic dimer ((O~(CH2)lo-O~CO-(CH2),CO)2) with 36 skeletal bonds and it is negligible for the larger cyclics, with 54, 72 and 90 skeletal bonds respectively. Therefore, on the basis of these results, Stepto's use of eqn ( 5 ) appears to be valid for the systems he has studied. Dr. I(. D&ek (Czechoslovak Academy of Sciences) said: It should be pointed out that the present theories of cyclization in crosslinking reactions are based on the conformational statistics of linear sequences of bonds connecting the reacting func- tionalities and neglect correlations due to already existing cycles.Therefore, the theories are asymptotically exact at zero degree of cyclization. The cycles already present influence the probability for the reacting functionalities to come into close contact : if the chain stiffness remained constant the already existing cycles would increase the cyclization probability. This follows also from the application of the random flight statistics to simple cyclic structures carrying reacting functionalities. However, the cycles may increase also the chain stiffness which has an opposite effect on cyclization.According to the results of our statistical treatment of the crosslinking polymerization of diallyl phthalate, the neglect of correlations due to existing cycles seems to underestimate cyclization, especially in the early stages of polymerization. When the degree of cyclization is increased by carrying out the polymerization in the presence of diluents, the fraction of crosslinks wasted in cycles does not grow and R. F. Bates and G. J. Howard, J. Polymer Sci. C, 1967, 16, 921. R. H. Peters and R. F. T. Stepto, S.C.I. Monograph, No. 20, 1966, p. 164.GENERAL DISCUSSION 91 unsaturation does not fall so fast as expected and the stiffening (and possibly the excluded volume) effect seems to become more important. Dr.R. F. T. Stepto (UMZST) (communicated) : Flory’s note of caution concerning the use of Gaussian statistics for short chains finds support not only from the results of the calculations he mentions, but also from the earlier experimental work of Semlyen and co-workers on ring-chain equilibria, involving oligomers of substituted siloxanes and other monomers. However, the results just quoted by Semlyen on ring concentrations in poly(decaniethy1ene adipate) indicate that, for linear chains which are similar in flexibility and numbers of bonds to the chains forming the smallest rings in our systems, the assumption of Gaussian statistics near Y = 0 is justified, at least numerically. The use of Gaussian statistics for gelling systems can also be criticised on the grounds that one is concerned with the mutual concentrations of pairs of chain ends, not of linear chains, but of branched chains.In this case, given one chain end, the configurational behaviour of the other ends is interdependent as they are connected by sequences of skeletal bonds. This is mentioned in the paper following eqn (5) and is emphasised by DuSek’s remarks. However, the approach has been taken that the errors inherent in the Frisch-Kilb approach are more serious than those resulting from the use of these statistics. The point raised by Flory that Kilb’s counting may overestimate, rather than underestimate A’, is an interesting one. Extension of his calculations to higher generations is required before any definite conclusions can be reached. The difficulty of enumerating the isomeric species and the possible sites for intramolecular reaction of course increases with the generation number, but such enumerations could lead to alternative expressions for the gelation condition. The resulting expressions, however, would still refer only to ring formation in infinite species at a particular point in the reaction. The calculations given show clearly that the probabilities r, of eqn (3) should contain weighting factors which are a function of n. In general, such factors will depend also on$ Indeed, a similar calculation for f= 4 shows that at gelation, and in the limit of zero rings, the expected number of unreacted A groups at the first generation is > 1. Gaussian statistics are only a first approximation. Prof. M. Gordon (University of Essex) said: The paper by Burchard and co-workers illustrates how well correlations in structural statistics can be dealt with by the graph- theoretical formalism which lies at the root of the cascade method. Effects due to local correlations are transmitted over long distances here; their fig. 1 shows how a difference in reactivity of two neighbouring functionalities, a local feature, has conse- quences a long distance from this locality. The necessary information is carried from generation to generation up the “ family tree ”, by an elegant implementation of the facility for labelling of the auxiliary variables of the generating functions. In this way, exact calculation of many measurable statistical parameters is achieved despite the correlation trouble. One aspect makes the advance in theory specially important : chemical reactions affecting a gel molecule are always subject to correlations of indefinitely long range, owing to packing restrictions among other reasons, and further development of mathematical tools to cope with this situation is urgent. see M. Gordon and T. C . Parker, Proc. Roy. SOC. Edin. A , 1970/71, 69, 181.

ISSN:0301-7249    

DOI:10.1039/DC9745700080

出版商:RSC    

年代:1974

数据来源: RSC

摘要:

Rheological Monitoring of the Formation of Polyvinyl Alcohol-Congo Red Gels BY H. BELTMAN AND J. LYKLEMA" Laboratory for Physical and Colloid Chemistry of the Agricultural University, De Dreijen 6, Wageningen, Netherlands Received 10th December, 1973 The formation of polyvinyl alcohol (PVA)-Congo Red (CR) gels has been followed rheologically. These gels are almost purely elastic, their storage modulus G is entropically determined. The time dependency of G has been used to analyze the kinetics of the gelling process. The network build-up proceeds in two stages, the first of which being the formation of a superstructure in the PVA molecule. The main function of CR is the promotion of this superstructure formation. Polyvinyl alcohol (PVA) gels readily in the presence of a number of low molecular weight substances, among which Congo Red (CR) is one of the most powerful.' PVA can even form gels in its own right : concentrated solutions of well-hydrolyzed PVA can develop detectable elasticity upon standing. From a molecular point of view, the PVA-molecule is relatively simple.Hence the study of the PVA-CR system may help to further our insight into the complex phenomena of gelling. In the investigation to be described, an attempt is made to obtain information on the kinetics of gelling. The build-up of a three-dimensional network must be reflected in the time dependence of the storage modulus G'(t). This quantity, in con- junction with the (in this case less important) loss modulus G"(t) has been measured in a recent apparatus, the so-called rheometer.Although the interpretation still contains several uncertainties, some trends can be definitely recognized. EXPERIMENTAL CHEMICALS The PVA samples are commercial products ex Kurashiki, Japan. They have been pre- pared from polyvinyl acetate (PVAc) by hydrolysis. Two samples have been studied, with degrees of hydrolysis of 98.5 and 88 % respectively. They are abbreviated as PVA-98.5 and PVA-88. For both samples the DP is about 1750. The width of the m.w. distribution is not known. The majority of the experiments to be described has been done with the 98.5 sample. Congo Red was obtained from Merck (BRD) and used without further treatment. Water was demineralized over a Dowex ion exchanger till K < 2 x 10-6 cm-'. PREPARATION OF GELS Except where otherwise stated, gels have been prepared as follows.To a PVA solution (obtained from a 10 % stock solution by dilution), CR was added with stirring and refluxing at 85°C. The mixture was kept for 10min at this temperature to ensure complete dis- solution of all components and the disruption of any preformed structure in the polymer. It was then brought into the measuring cylinder of the rheometer and cooled rapidly to the measuring temperature (25°C except where otherwise stated). Some gelling during the cooling stage could not be prevented, but it was always minor in comparison with the same upon keeping the mixture at 25°C during which the process was rheologically investigated. 92H. BELTMAN A N D J . LYKLEMA 93 THE RHEOMETER This is an instrument developed by Duiser and Den Otter, T.N.O., Delft, Netherlands.Its mode of construction and operation have been described elsewhere.2* Essentially, it consists of two coaxial cylinders between which the sample is brought. The inner cylinder, which is suspended through a torsion wire is brought into an oscillatory motion whereby the amplitude and frequency o = 2nv of the driving system can be varied between 0 and 0.25 rad and 2x and 3 x lo2 s-l respectively. Due to the hiscoelastic properties of the sample there is in general a phase shift and amplitude difference between the applied motion and the motion of the inner cylinder. These two quantities can be measured, using a specially constructed mirror system and from them the two moduli G’ and G” can be calcu- lated.The system is thermostatted. The reproducibility of the measurement depends mainly on the mode of preparation and cooling. The overall repeatability is within 15 %. The accuracy of the rheometer is about 1 %. RESULTS AND DISCUSSION GENERAL BEHAVIOUR OF G ‘ ( o , t ) A N D G”(C0, t ) The main purpose of our investigation is to probe the gelling process rheologically, so that a number of G’(t) and G”(t) curves have been measured. It soon appeared that PVA-CR gels, also during the formation stage, are almost entirely elastic. G’ is independent of o over the whole frequency range, whereas G” is low, less than 2 % of G’. For this reason, most attention is paid to G‘(t). However, with other gelling agents, like borax, there is considerable dissipation in this frequency range and G”(w) can become as high as G’(o).Fig. 1 illustrates this for a typical case. I: I- .-.-.- .-.-.-.- *-.-*-*-* -,*- *- G to L I - - ~ - L .___ 1 .- -I- to j 0-. r I 1‘: wlrad s-‘ FIG. 1 .-The storage and loss moduli of a PVA (4 %)-CR (2.4 %) gel, aged for 200 h ; T = 25 “C. Fig. 2-4 show various influences on the rate of gelling, as monitored by G(t). Because of the invariance of G’ with o (fig. I), the choice of o is a matter of convenience. In the experiment, represented by fig. 2, the CR concentration cCR has been chosen to be in excess. In that case a strong dependence on the PVA-concentration c, is observed. The ordinate axis intercepts reflect the gel build-up during the cooling prior to the systematic measurement. It is seen that on a weight basis the gelling rate is much less sensitive to cCR than to c,.Fig. 4 illustrates the temperature dependency. Unlike most chemical processes the rate of gelling decreases with temperature, the temperature coefficient is considerable. At 35”C, G’(t) has an inflection point, suggesting the occurrence of an induction period. Fig. 3 is the counterpart of fig. 2.94 FORMATION OF PVA-CR GELS 2500 2000 N I 1500 E c b IOOG 5oc I . -~ I I._-.-..- 1 3000 6000 9000 t Is FIG. 2.-The influence of the PVA concentration (cp in %) on the rate of gelation; CCR = 5 %; T = 25°C. CCR = 3 % N I E 0 3 000 6000 9000 t Is FIG. 3.-The influence of the CR concentration (CCR in %) on the rate of gelatioil ; cp = 5 % ; T = 25°C.H. BELTMAN AND J. LYKLEMA 95 APPLICABILITY AND APPLICATION OF THE THEORY OF RUBBER ELASTICITY For a purely elastic rubber-like gel the shear force z, as a function of the shearing strain y can be written as 4b gkTv 2s = 7’.In this equation, v is the number of chain sections in a volume Vcontributing to the elasticity. In a fully-developed gel with many cross-links per polymer v = 2n, where n is the number of cross-links in V. The dimensionless factor g accounts inter alia for permanent entanglements, contributing to the gel strength; in first 250 200 150 N I E z 2 u 100 50 11s FIG. 4.-The influence of the temperature on the rate of gelation. cp = 4 %; CCR = 1.6 %. approximation their number is proportional to n and hence to v, at least if v = 2n amlies. Whether or not non-Dermanent entanglements contribute to z.de.pends on o. There are arguments to presume that they play no role in our case. In the first place, G’ is independent of o (fig. l), indicating that they are either non-existent or, if they do exist, they do not relax at all or are already fully relaxed at the highest cc) studied. Viscosity studies indicate also that in the cp range studied intermolecular entangle- ments do not yet O C C U ~ . ~ Eqn (1) applies if the elasticity is purely entropic, the length of the chain sections must be large enough to enable the calculation of their entropy as a function of extension by statistical means. The PVA-CR gels studied display linear elasticity : G’ is independent of amplitude up to 0.20 rad (y = 0.96), whereas most experiments have been carried out at about96 FORMATION OF PVA-CR GELS 0.10 rad (7 = 0.48).The elastic modulus G, in our case identical to G', follows from G = zJy. Hence G = gkTvJV. (2) In a purely entropic gel, G' must increase proportionally with T, whereas in an energetic gel, G' decreases with T. Measurement of G'(T) therefore provides a power- ful procedure of discrimination. In our system, such measurements are possible but not without complications, because in all thermo-reversible gels n changes also with 2'. This problem can be overcome by exploiting the relatively slow relaxation of n(T) as compared to the almost instantaneous adaptation of the chain section en- tropies to temperature alterations. Experimentally, the procedure consists of rapidly changing Tfollowed by rapid measurements of G', both with increasing and decreasing T.Two typical results are represented in fig. 5. Although there is some spread in the data, which is mainly due to incorporating two directions of measurement, it is evident 37 35 33 m E : : s B - I L 12 1 \ E 10 8 I I I I 290 295 300 305 T/K FIG. 5.-The influence of Ton the storage modulus using rapid temperature changes. PVA-CR gel, cp = 4 %. (1) CCR = 3.2 %; (2) CCR = 1.6 %. that GIRT is independent of T. Energetic elasticity is clearly excluded. In passing we note that doubling cCR leads to roughly a threefold increase of G', indicating that v increases more than proportionally with c,. Our finding that PVA-CR gels display entropic elasticity corroborates work by Hirai.' In conclusion, it appears that eqn (2) applies to our system, and hence the time derivative of G' may be used as a direct measure for the rate av/at at which new chain sections become part of the network.€3. BELTMAN AND J .LYKLEMA 97 SECONDARY STRUCTURES I N THE POLYMER Before analyzing the kinetics of the gelling process, attention must be paid to the possible formation of a certain extent of ordering in the PVA molecule itself prior to gelation. There is much circumstantial evidence for this. Concentrated PVA solutions can form gels upon standing provided the acetate content is smalL8* The pronounced inhibiting effect of a small fraction of acetate groups in the chain is difficult to explain if random H-bridges between PVA chains are responsible for the net- work formation. However, if gelling is preceded by some kind of structure formation in the PVA chain, this becomes conceivable since it is known that due to the co- operative action of steric hindrance the formation of superstructures is greatly reduced.A cold concentrated PVA-98 solution is coloured blue by Iz, as is amylose, which suggests inclusion in akind of helix. As for PVAgels, the colour is thermoreversible.1° Upon cooling the colour develops at a rate comparable with the rate of gelling in the absence of gelling agents. Gelling agents like Congo Red typically improve this colour reaction. Substances like NaCNS that inhibit the colour formation suppress also the gelation, if added to the PVA prior to the addition of the gelling agent. All of this has led us to the conclusion that it is necessary to account for structure formation in the PVA chain prior to the network formation by CR.A similar explanation can be offered for an observation made by Ferry,l1 working with thermo-reversible gelatin gels. At given composition and pH, the modulus of the gelatine gel was found to decrease with T. At a given T, the same value of G’ was attained, independent of whether the final situation was approached from below (is., T rising) or from above (T decreasing). However, the rate of the process was much faster if measured with rising T. This can probably be explained if some super- structure is required for gelation. If approaching from a high T, this superstructure needs firstly to be built up, whereas upon approach from below it is already there. Finally, our own observation that the rate of the gelling decreases with increasing temperature can hardly be explained otherwise than that the structure formation is less at higher T.KINETICS OF GELLING According to the theory of gelation,12 incipient gelling with random cross-linking starts if the number of cross-links n is about half the number of original chains. It is the gradual build-up of the network after this that is reflected in the G’(t) curves of fig. 2-4. Using eqn (2), these curves can be rearranged into v ( t ) curves, the slopes of which are a measure for the rate of gelling. We consider this rate at t -+ 0, obtainable by extrapolation, and investigate the influences of cp and cCR. By subsequently plotting the initial rate logarithmically against cp in excess of CR and conversely the overall initial reaction rate could well be represented by This result is insensitive to the value assigned to g, provided g is a constant.How- ever, if absolute values of v are wanted a definite value must firstly be assigned to g. The uncertainty in the first exponent is about 5 %, that in the second about 15 %. Fifth-order reactions are unusual phenomena although also from Ferry’s measure- ments l3 on the gelation of gelatin a fourth power in cp could be inferred. Never- theless, eqn (3) probably does not represent the detailed mechanism adequately. It has already been ascertained that the process proceeds probably in two consecutive steps. The rate of the first step, the formation of a superstructure, is considerably reduced by increase of T.If T is high enough this step becomes the limiting factor. 57-D98 FORMATION OF PVA-CR GELS Note the induction period at 35°C in fig. 4. In agreement with this, we also found lengthy induction periods with gelling agents like resorcinol and sucrose, all sub- stances that are weaker gel-formers with PVA and, with I,, weaker blue colour pro- motors than CR. If this picture is correct, the relationship G'(t) at 25°C (fig. 2 and 3) must be en- tirely determined by the cross-linking process, because at t = 0 the first step is virtually completed. It is this step that should lead to the c: factor in the overall rate eqn (3). It has already been argued that n is not large compared to the number of molecules. Hence there must be considerable loss of cross-links.The extent of loss decreases with increasing c,, as can, e.g., be judged from the fact that for a complete gel G' increases with cf. As a consequence of this, PVA becomes progressively more effec- tive in cross-linking with increasing concentration. This can qualitatively explain the high power of c, in (3). Quantitatively, the following equations may be written : (1) the structure formation of PVA, induced by CR (with PVAst denoting structurized PVA) k i PVA+CR -+ PVAst ; (4) (2) the cross-linking process, which is almost certainly bimolecular, k2 2PVAst -+ fce + (1 -f),... Here ce stands for effective cross-link and cs for a spoiled one. f i s the fraction of effective links (0 < f < 1). Following F l ~ r y , ~ " after incipient gelatioil the formation of each new cross-link leads to two contributing segments, i.e., Then 50 25 c,, = 2v.dc,,/dt = k2 fc;VAst = 2(dv/dt). 0 2 L 6 8 t 3 / m i d FIG. 6.-The increase of the number of participating chain sections as a function of time. gel, CCR = 5 % ; cp indicated ; T L- 25°C. PVA-CRH . BELTMAN A N D J . LYKLEMA 99 If, at the onset of the development of the network, reaction (4) is completed, the 4th power in (3) would requiref = const. c;. In later stages of the process,fincreases, partly due to the formation of permanent entanglements, accounted for by the factor g in (1) and (2). The rate of the process is then predominantly diffusion controlled, as is illustrated in fig. 6. v(t3) is a straight line through the origin. The slope increases more than proportionally with c, because of the relatively increasing effectivity of PVA with increasing concentration.Our model does not call for an explicit role of CR in the cross-linking process. Although such a role is not excluded, the main function of CR is to induce (or stabilize) the formation of structurized PVA. Several additional arguments can be invoked to support this. First, there is the close analogy (with respect to time, tem- perature and concentration dependency) with the gelation of gelatin,' which proceeds also without gelling agent. Secondly, if Congo Red is chemically reduced by hydro- sulphite it is an ineffective gelling agent for PVA. However, if the Congo Red is reduced in an existing PVA-Congo Red gel, the gel structure remains.l This pheno- menon also suggests that Congo Red does not play an essential role in the cross-link itself, but only in processes leading towards cross-linking.The quantitative effect of acetate groups provides still another argument. If intermolecular CR bridges are formed between isolated random chains, 12 % of Ac-groups would reduce the number of cross-links to about 88 %. However, if CR molecules give some kind of intra- molecular bonding, leading to the formation of, say, a helical structure with repeating turns requiring a sequence o f p OH groups in a series, the inhibition of 12 % of Ac groups would rather be of the order of (0.88)Px 100%. Actually, the gel strength is reduced by about 90 %, which fits better in the second than in the first picture. The strong influence of 12 % acetate groups is also demonstrated in concentrated PVA solutions.At low temperatures, PVA-98 solutions gel gradually but PVA-88 solutions do not gel. The difference between PVA-CR and PVA-borax gels also illustrates our point. The enthalpy of a cross-link in a PVA-CR gel has been esti- mated at 175 kJ by Arakawa,14 that of a cross-link in a PVA-borax gel at 22 kJ by Schultz and Myers.15 These results suggest about eight H-bridges in the former against one in the latter, indicative of more structuring with CR. This structure formation was detectable by X-ray analysis in PVA-CR gels but not in PVA-borax gels. * 9 ' The difference is also reflected in the rheological properties of the two gels : in PVA-borax gels, G' is a decreasing function of w (i.e., at the w observed by us we are already beyond the rubber plateau) whereas G"(cu) passes through a maximum and is much larger than in PVA-CR gels.Phenomenologically, the PVA-borax system behaves as a highly viscous gel, in which self-healing takes place after dis- rupture. The average life time z of a cross-link is about 4 s at room temperature. On the other hand, PVA-CR gels behave as an elastic material ; they are not self-healing and z > 2 h (the upper limit studied by us). All these facts corroborate our theory, which requires structure formation in the PVA as an essential first step towards gelling. The authors thank Dr. M. van den Tempe1 and Dr. A. J. M. Segeren for useful criticism. C. Dittmar and W. J. Priest, J. Polymer Sci., 1955, 18, 275. J. A. Duiser, Thesis (Leiden, 1965). J. L. den Otter, Thesis (Leiden, 1967). (a) P. J. Flory, Chem. Rev., 1944, 35, 51 ; (6) P. J. Flory, PrincQdes of Polymer Chemistry (Cornell Univ. Press, 1953), chap. 11. H. Beltman, Meded. LandbouwhogeschooI, to be published (1975).100 FORMATION OF PVA-CR GELS F. Bueche, Physical Properties of PoZymers (Interscience Publ., 1962). M. M. Zwick and C. van Bochove, Textile Res. J., 1964,34,417. Kurashiki Poval (Kurashiki Rayon Co., Ltd., Osaka, Japan). lo M. M. Zwick, J. Appl. Polymer Sci., 1965, 9, 2393. J. D. Ferry, J. Amer. Chem. Soc., 1948, 70, 2244. l2 P. J. Flory, J. Amer. Chem. Soc., 1941, 63,3083, 3086; J. Phys. Chem., 1942.46, 132; W H. Stockmayer, J. Chem. Phys., 1943, 11,45; 1944, 12, 125. l 3 J. D. Ferry, Ado. Protein Chem., 1948, 4, 2. l4 K. Arakawa, Birll. Inst. Chem. Soc. Japan, 1959, 32, 1248. l5 R. K. Schultz and R. R. Myers, Macromolecules, 1969, 2, 281. l6 N. Okada, and I. Sakurada, Birll. Inst. Chem. Res. Kyoto Univ., 1951, 26, 94 ; Chem. Abstr., ’ N. Hirai, Bull. Znst. Chern. Res., Kyoto Univ., 1955, 33, 21; Chem Abst. 50, 651i. 47, 1423h.

ISSN:0301-7249    

DOI:10.1039/DC9745700092

出版商:RSC    

年代:1974

数据来源: RSC

摘要:

Parameters of the Swelling Equation and Network Structure BY K. DUSEK Institute of Macromolecular Chemistry, Czechoslovak Academy of Sciences, 162 06 Prague 6, Czechoslovakia Received 30th November, 1973 The functionality of a crosslink which appears in the swelling equation differs from the chemical functionality and represents the number average of elastically-active chains issuing from an elastically-active cross-link, independently of the type of the crosslinking process. Relationships have been derived for the polycondensation of an $functional monomer, for alternating copoly- condensation of two monomers, and for the crosslinking of primary chains which allow the determina- tion of the effective functionalityf, either from the composition of the gel phase or its calculation by means of crosslinking statistics based on the theory of cascade processes (Gordon-Good).In close proximity of the gel point,f, is always three and its increase is the faster the more positive the substitu- tion effect in the monomer unit. Up to comparatively low values of the gel fraction (10-20 %), fe in most cases does not depart too much from three and increases steeply to the final value only at a virtually negligible sol fraction. The change in the free enthalpy AG for mixing of a solvent with polymer systems arising by crosslinking reactions may be written 1. as a sum of two contributions, AG = AGmi,+AGe1, (1) where AGmjx relates to the mixing of a solvent with a crosslinked polymer having un- stretched chains, while AGel is a contribution due to a change in the dimensions of the elastically-active network chains (EANC).The term AGmi, has been derived on the basis of a lattice model and the quasi-chemical equilibrium approach, when primary structural units by which the network is formed (monomer units or primary chains) bearing specific contact points (surfaces) are placed on a lattice. Among these contact points there are some (crosslinkable) which are capable of quantitative coupling (at the expense of a entropy change), either with contact points of the same kind, or with similar contact points of a different kind. This simulates the actual process of network formation by various types of reactions. The other contact points, the number of which also varies as the reaction proceeds, interact physically with each other and with the contact points of the solvent.The magnitude of the equilibrium sorption of the solvent is determined by the equality of the chemical potentials of the solvent in both phases. We have demon- strated l * that the contribution of mixing to the chemical potential of the solvent may be obtained in a closed form as a function of surface and volume frac- tions of the solvent and polymer, of the fractions of the contact surfaces, and of the interaction free energies for the formation of the respective contacts. If one assumes for AGel the Gaussian behaviour of EANC (Wall-Flory's approach 3, and carries out a partial expansion of (AP&~ into a power series of the volume fraction of the polymer, &, one obtains a relationship for the total change in the chemical potential of the solvent Aps, co APS/RT = In (1-4p)+4: C Q i ~ ~ + [ ~ - ( 1 / ~ ) ( ~ ~ / ~ ~ ) ( ~ ~ / 2 - 1 ) ] ~ ~ + i = O ve(VsIQ[Ch$(+,")' - 4 g I 7 1011 02 PARAMETERS OF THE SWELLING EQUATION where f is the number average number of segments per primary unit (monomer unit, primary chain), G, vp and 6, respectively, are the molar volume of the solvent and the average molar volume of the polymer segment in the system and in the gel; +g is the volume fraction of the gel component in the system, and 4: is its value at net- work formation (assuming that the network chains are in the relaxed state during network formation), NJ2 is the number of bonds formed per primary unit, and 17, is the number of EANC per segment.The sum Qi& has the meaning of a concectra- tion-dependent interaction parameter x ; the coefficients Q, comprise the effect of thermodynamic interactions and are dependent on the size of the primary units and on the number of the bonds formed.They may be expressed analytically if the respective interaction energies and the magnitudes of the contact surfaces are knowii. For the case of crosslinking of simple primary chains bearing only one type of non- crosslinkable contact points it has been demonstrated that the parameter 3~ increases with increasing & and the degree of crosslinking, if the interaction free energy is positive, but that it may decrease with concentration for a negative value. The situation observed for polycondensation networks is more complex, since here one has to consider more types of non-crosslinkable contact points and major changes during the reaction (e.g., formation of an ester bond due to the reaction between the carboxyl and the hydroxyl group).Eqn (2) may be used directly for measuring sorption equilibria between a cross- linked system (containing both sol and gel) and, e.g., solvent vapours, as long as the crosslinked system does not separate into two condensed phases. N, is known from the course of the reaction, v,, e.g., from mechanical measurements, and +g may be determined from parallel extraction of soluble fractions. Thus for the determination of x the treatment of a closslinked system may be sometimes more useful than the treatment of the swelling equilibrium of an extracted gel. If we now pass to the sorption of solvent by an extracted gel (e.g., swelling in pure solvent), all quantities must be related to the gel phase (index g).A simple trans- formation of eqn (2) gives a relationship analogous to Flory’s swelling equation (cf. ref. (5)) : 00 i = O ~,-m- = in (1 -4g) + ~,+X42,+V,(~1~)~~~(4g0)8-~4~i, (3) where x is a variable corresponding to the sum in eqn (2), and B replaces 2/fin the original version (f is the chemical functionality of the crosslink), so that (4) B = 1 - (NJ2 - l ) / ~ , f ~ . It is evident that B varies with the extent of reaction (crosslinking) ; as a consequence, f cannot be constant either. Ng in eqn (4) has then the following meaning for various types of crosslinking reaction * * : No =fag Ng = nlga,,f, + n 2 p 2 , f 2 for homopolycondensation, or for alternating polycondensation of monomer 1 and 2, or Ng = v,Y, for the crosslinking of the primary chains In the above expressions, a, is the con- version of the functionalities in the gel, f is the chemical functionality of the cross- link, nip is the molar fraction of the monomer units i in the gel, and vp is the fraction of the crosslinked segments in the gel.K .DUSEK 103 In principle, N , may be determined experimentally or calculated by means of cross- linking statistics. In the following section we shall apply crosslinking statistics based on the theory of cascade processes 6-8 (cf. a.lso ref. (2), (4)) and derive relation- ships for the effective functionality of the crosslink-f, in the swclling equation. MEANING AND DERIVATION OF THE FUNCTIONALITY f e I N THE SWELLING EQUATION By analogy with Flory's swelling equation we write .f, s 2/B = 2veFg/(v&-Ng/2+ l), (8) and express the parameters on the right-hand side by using the crosslinking statistics with the first shell substitution effect (fsse), iizmely, using the probability generating function (pgf) for the ties T(8).The pgf for the number of offsprings of a primary unit in the root of probability trees for a reaction of units of one type is given by Fo(8) = C piei, (9) i where p i is the probability that i bonds lead from one unit in the root to the units in the first generation. The pgf for the number of units borne by the unit in the first and fol1owir.g generations is and the extinction probability u is defined by Pgf for ties (or active bonds, active reacted functionalities) i.e., those that are part of an infinite sequence of bonds may be written as For a reaction of several types of primary units, it is possible to perform generalization in the form of vectorial generating functions (cf.ref. (7)) which for alternating poly- condensation of two monomers of type 1 and 2 degenerates to if the coefficients p l i or p 2 j denote the probability that from a monomer unit of type 1 or 2, i o r j bonds lead to monomer units of type 2 or 1, respectively. These relation- ships may also involve cyclization, if the coefficients pr are related to units giving rise to i intermolecular bonds. In the appendix,f, is expressed in terms of the moments of T(8) and the proof is given that fe is the average number of EANC issuing from elastically-active cross- links (i.e., from those issuing at least three EANC); consequently,I 0 4 PARAMETERS OF THE SWELLING EQUATION if zt is the fraction of crosslinks with i EANC.(A10) and (A15) of the Appendix we obtain for homopolycondensation, f e = CWG(1)- T W ) - G ( O ) ) + n2(7X)-- G(0)- T’;(0))3/ for alternating polycondensation, and for crosslinking of the primary chains. The symbol T’(k) or T”(k) designates the first or second derivatives of T(8) with respect to 8 at 8 = k. By using relationships (9)-( 16) one may transform eqn (18)-(20) into (1 8a)-(20a) : By rearrangement of eqn (A4), fe = (T’( 1) - T’(0) - T”(O))/( 1 - T(0) - T’(0) - T”(0)/2) (18) [ 1 - n,(T,(O) + TXO) + T;(O)) - n,(T,(O) + W O ) + 7-;(0)/2)1 (19) fe = 4(1 +T’(l)-T(O)-2T’(0))/(2+T’(l)-2T(O)-3T’(O)) (20) fe = [( 1 - vI2fa( 1 - F;(v))lI[ 1 - Fo(u) --fa( 1 - fe = C[2(1 - U 1 ) ( 1 - % ) - 4 - 2)2)2F;1(v2)-n2(l -vd2Fi2(v1)11 + (+)(I - u)Fi(u))l; (184 [l - ~ l ~ o l ( ~ 2 ) - ~ 2 ~ 0 2 ( ~ 1 ~ + ~ ( ( ~ -v2)u1 +(I -u2)”l;;1(u2)/2+ (1 - U , > U , + (1 - vd21;.i2(u1)/2)1 ; fc = 4[1 -Fo(~)+y(l -~)(1-2~)]/[2-2~0(~)+~(1 -~)(1-3~)], (194 (204 c n l a l f l = n2a2f2> where y = v,,v is the crosslinking index.DISCUSSION The finding that the effective functionality dfe) in the swelling equation is a number- average number of EANC originating from an elastically-active crosslink in the gel is in accord with the conception of crosslinking entropy derived by F l ~ r y , ~ if applied to real systems. However, the quasi-chemical equilibrium approach involves surface fractions of the contact points ; consequently, crosslinking entropy contains also terms with higher powers of the volume fraction of the polymer which affect x, but not f e .In real systems the fractions and effective functionalities of elastically-active crosslinks are not obvious, but the analytis offered here demonstrates that fe may be calculated either from experimental data (eqn (5)-(8)), or from crosslinking statistics At the gel point,f, = 3 independently of the chemical functionality of crosslinks The increase infe with proceeding crosslinking after the gel point is illustrated in fig. 1 (a-c) and 2 by several examples of ring-free equilibrium-controlled homopoly- condensation and alternating polycondensation of two monomers involving both a linear positive (N > 1) or negative (N < 1) fsse (as well as random crosslinking of the primary chains with different distributions).(The definition of the respective pgf of the substitution parameter N, as well as further treatment, have been explained else- where 2-4* 7-9.) Thef, is plotted against reduced conversion where a, is the conversion at the gel point. The examples make it clear that : (a) the increase inf, from 3 to the final value is the steeper the more positive the substitution effect ; (b) for alternating copolycondensation and stoichiometrically non-equivalent ratio of the functional groups (fig. I(c)), f e will never attain a value corresponding to the chemical functionality of the crosslink, f; (c) f e does not depart greatly from 3 (eqn (1 8)-(20))* %d = (a - (21)K. DUSEK 105 within a comparatively broad range of the experimentally-available weight fraction of the gel, w,, if the substitution effect is not strongly positive, or the distribution very wide.For instance, for chemically tetrafunctional crosslinks, fe increases in most cases to attain 3.1-3.2 at wg = 0.85, which may be used for estimatingf, of sys- tems with a measurable sol content. The procedure described above may also be used in cases when cyclization becomes operative. Then, quantities in eqn (5)-(8) of the coefficients of pgf must be related to intermolecular connections. In crosslinking statistics, cyclization up to the gel 4.0 3.5 3.0 4.0 q 3.5 3.0 4.0 3.5 i 2 A I "'"0 0 .5 I .o 0 . 5 I %ed, w g FIG. l(a-c).-Dependence of the effective functionality fe in the swelling equation on the reduced conversion of functional groups Crred and on the weight fraction of the gel, wg, for networks obtained by homopolycondensation or by alternating ring-free copolycondensation. (a) homopolycondensa- tion of a tetrafunctional monomer with a linear fsse ; numbers near curves designate the value of the substitution parameter N ; (b) copolycondensation of a tetrafunctional (1) and bifunctional (2) monomer with a linear fsse, 1 iVl = 1.5, N2 = 1.5,2 Ni = 1, N2 = 1, 3 Nl = 0.5, N2 = 0.5 ; (c) co- polycondensation of a tetrafunctional and bifunctional monomer in a stoichiometrically non- equivalent ratio of the functional groups without fsse ; the initial ratio of the number of functional groups: 1 1 : 1, 2 1 : 1.1, 3 1 : 1.5, 4 1 : 2.106 PARAMETERS OF THE SWELLING EQUATION point may be treated satisfactorily lo ; the simplest assumption after the gel point is that all gel-gel reactions are intermoIecuIar and that the cyclization proceeds only within the s01.'~ This assumption was used for the treatment of the kinetically- controlled chain polymerization of a bisunsaturated monomer with a strong cycliza- tion.12 Fig. 3 shows that the dependences offe on the reduced conversion of the double bonds differ from each other, but the dependences on the gel fraction are practically identical.The experimental data that may be employed for the determination of& are scarce. It would be possible to use a procedure suggested by Mark et al.l3* l4 according to which the coefficient B (eqn (3)) may be determined from the equilibrium elastic moduli of samples having different network density and dilution at crosslinking, but i ---l----- Yhfc, "'g FIG.2.-Dependence of the effective functionality fe in the swelling equation on the relative cross- linking index y/yc and the weight fraction of the gel wg for networks obtained by ring-free random crosslinking of the primary chains. Number average degree of polymerization of the primary chains r, = 1 0 0 ; distribution type: 1, homodisperse chains; 2, most probable distribution; 3, Schulz distribution with the polydispersity index rw/rn = 5 . 4. w c, 0 . 5 0.5 ared, wg FIG. 3.-The effect of cyclization on the effective functionality f, during radical crosslinking polymer- ization of a bisunsaturated monomer in dependence on the reduced conversion of double bonds a,,d and on the weight fraction of the gel wg.Monodisperse primary chains with m = 50 ; 1, without cyclization ; 2, with cyclization using the spanning-tree approximation and Gaussian statistics for cyclization probability l 2 (distance between the double bonds 9 links, number of links in a statistic segment 2).K. DUBEK 107 swollen in equilibrium to the same degree. While using this method for loosely crosslinked diethyleneglycol methacrylate networks (the Mooney-Rivlin constant C2 which is practically zero) we found that in most casesf, varies between 3 and 4. The polydimethylsiloxane networks investigated by Johnson and Mark l 4 are not very suitable because of the high value of the constant C,.If one assumes v, to be pro- portional to the elasticity modulus (cf. ref. (14)), f, varies from 4 up to physically unwarranted high values. On the other hand, for ve cc C1 (Mooney-Rivlin),f, de- creases for strongly diluted networks below the physically reasonable value of 3. One must not forget, however, that the method is based on the assumption that the para- meter 31 is independent of cross-linking density, which may be a source of error. APPENDIX EXPRESSION OF THE EFFECTIVE FUNCTIONALITY .f, I N TERMS OF CROSSLINKING STATISTICS The effective functionality fe is defined by eqn (S), and the quantities N,, ve and F, for the individual cases (eqn ( 3 4 7 ) ) are expressed in terms of the moments of probability generating functions (9)-(16).The summing indices are i or j . (a) HOMOPOLYCONDENSATION OF AN f-FUNCTIONAL MONOMER (4 =fag, OdiGf) The number of reacted functionalities in the gel per monomer unit in the system is the difference between the number of all reacted functionalities and reacted functionalities in the sol. This difference is divided by the number of monomer units in the gel (cf. eqn (10)-(12)) : By using pgf (12) and the definition equation (11) for u, the numerator may be rearranged to become Only monomer units with more than two active reacted functionalities contribute to the number of EANC; each such bond contributes to the number of EANC by 3, so that We obtain from eqn (A2) and (A3) after substitution into eqn (8) and rearrangements, which is just the average number of EANC issuing from one elasticaly-active monomer unit, i.e., those which gives rise to three and more EANCs.(6) ALTERNATING COPOLYCONDENSATION OF MONOMER 1 A N D 2 (Ng = f1a1,n1,+f2~2gn20, 0 < i < fl, 0 d j < f2). The fraction of monomer units 1 in the gel is given by the relationship, because the gel contains units having at least one active reacted functionality. The number of reacted functionalities of type 1 per monomer unit in the gel is obtained similarly to the case of homopolycondensation :108 so that PARAMETERS OF THE SWELLING EQUATION and This again is the average number of EANC issuing from one elastically-active monomer unit. (c) RANDOM CROSSLINKING OF PRIMARY CHAINS (N, = VeFg, 0 < i < 00) In this case the pgf& and Timplicitly include the distribution of the degrees of polymer- ization of the primary chains (cf.ref. (8)). The number average degree of polymerization of the primary chains in the gel is 7;0 = wgrn/(C t i ) (A111 1 (wg is the weight fraction of the gel and r,, is the number average degree of polymerization of the primary chains), since the numerator is the number of segments in the gel and the de- nominator is the number of particles in the gel per primary chain in the system. The fraction of crosslinked segments in the gel is obtained by simiIar reasonings as for po 1 ycondensa t ion vg = (C i t i + tl)/rnwg* (A 12) VeFg = Ne/(C ti> 6413) 1 The product vei, is the number of EANC per primary chain in the gel, so that if N, is the number of EANC per primary chain in the system. The calculation of Ne com- pared to polycondensation is somewhat different, since the EANC are considered to be also segment sequences within the primary chain between active crosslinks.The procedure has been explained in ref. (8) or ref. (4) (N, was designated there by pe) : On substitution into eqn (8) and rearrangement we obtain fe = 4(C i t i + C ti)/(C i4+2 C t i ) . 2 2 2 2 It is possible to prove thatf, has again the same meaning as for polycondensation. We consider a primary chain having i active crosslinked segments (ACU) (from which a sequence of segments leads to infinity), of which i-2 are inner and 2 are end segments. If an inner ACU is connected with an end ACU, the crosslink is effectively trifunctional, similarly to the case when an end ACU is connected with an inner ACU. Connection between two inner ACU gives an effectively tetrafunctional crosslink. Their fractionK. DUSEK 109 and the fraction of effectively trifunctional crosslinks is It is evident that is identical with eqn (Al5). K. DuSek, J. Polymer Sci. (Polymer Symp.), 1972, 39, 83. K. DuSek, J. Polymer Sci. (Polymer Phys.), 1974, 12, 1089. F. T. Wall and P. J. Flory, J. Chem. Phys., 1951, 19, 1435. K. DuSek, J. Polymer Sci. (Polymer Symp.), 1973, 42, 701. P. J. Flory, J. Chem. Phys., 1950, 18, 108. M. Gordon, Proc. Roy. SOC. A, 1962,268,240. G. R. Dobson and M. Gordon, J. Chem. Phys., 1965, 43, 705. M. Gordon and G. R. Scantlebury, Trans. Faraday Soc., 1964, 60, 604. M. Gordon, T. C . Ward, and R. S . Whitney, Polymer Networks. Structure and Mechanical Properties, ed. A. J. Chompff and S. Newman (Plenum Press, New York, 1971), vol. 1. ’ M. Gordon and G. N. Malcolm, Proc. Roy. Soc. A, 1966,295,29. lo M. Gordon and G. R. Scantlebury, J. Polymer Sci. C, 1968, 16, 3933. l 2 K. DuSek and M. Ilavsky, J. Polymer Sci. (Polymer Symp.), in press. l3 J. E. Mark, J. Amer. Chem. Soc., 1970,92,7252. l4 R. M. Johnson and J. E. Mark, Macromolecules, 1972, 5, 41.

ISSN:0301-7249    

DOI:10.1039/DC9745700101

出版商:RSC    

年代:1974

数据来源: RSC

摘要:

Interparticle Forces in Montmorillonite Gels BY I. C. CALLAGHAN? AND R. H. OTTEWILL" School of Chemistry, University of Bristol Received 3rd January, 1974 Aqueous dispersions of homoionic sodium montmorillonite have been studied under compressive and decompressive conditions in order to determine the internal pressure of the system as a function of the clay and electrolyte concentrations. Complementary measurements were also made of the electrokinetic properties of the montmorillonite particles. Compression of a sodium montmorillonite dispersion from ca. 2 % to ca. 65 % w/w gave a continuous curve of pressure against distance of plate separation up to pressures greater than 100 atmospheres.: On subsequent decompression, however, a different curve was obtained and a hysteresis effect was found to occur in the pressure against distance curves.It is considered that the initial curve represents the behaviour of a dis- ordered clay system, i.e., the initial gel, whereas the decompression curve represents the behaviour of a system in which the plates have become ordered into a parallel array. The elastic modulus of the latter system was determined. The results strongly suggest that the gel properties of montmorillonite dispersions are the consequence of long range electrostatic interactions. The gelation of dispersions of montmorillonitic clays is a well known phenomenon have been studied in some detail. and both the swelling properties of montmorillonites 2* and the rheological properties of montmorillonite gels Early observations on the phase separation of montmorillonite dispersions were made by Langm~ir.~ It was suggested by van Olphen that the edges of the plate-like particles carried a positive charge and he attributed gelation to the attraction between the positively charged edges of the plates and the negatively charged faces which enabled a " cubic cardhouse " of linked plates to be ~reated.~ Norrish ' suggested that the repulsive force caused by interacting double layers was responsible for the gel structure.In the present work, the internal pressure between the plates due to the electrical double layers has been measured directly as a function of the electrolyte concentration of the dispersion medium and the distance of separation between the plate surfaces (weight fraction of the clay) using a technique previously developed in these laboratories.8* The results obtained essentially support the contention of Norrish that the interplay of electrical double layer forces can be of considerable importance in gelling processes. EXPERIMENTAL The distilled water was doubly distiiled from an all-Pyrex apparatus. The sodium chloride used was BDH, A.R.materia1. The montmorillonite was a sample of bentonite (No. 26) from Clay Spur, Wyoming, as prepared for the American Petroleum Institute Clay Mineral Standards Project No. 49. Approximately 30 g of this material in the dry form was dispersed in 1 din3 of distilled water containing 1 cm3 of 30 % v/v hydrogen peroxide per 100 g of clay. The dispersion was warmed to about 333 K, held at this temperature for a few minutes and then left to cool.The dispersion was centrifuged at 4000 r.p.m. for 20 min in order to remove gritty impurities such as particles of silica. Conversion to sodium montmorillonite was achieved by re- dispersing the clay particles in sodium chloride solution (1 mol dm-3) and agitating vigorously t present address : Newcastle Technical Centre, Procter and Gamble Ltd., Newcastle on Tyne. 1 atmosphere E 1.0133 x lo2 kN m-2 E 1.0133 x lo6 dyn cm-*. 1101. C . CALLAGHAN AND R . H . OTTEWILL 111 for several days. The dispersion was then centrifuged at 38 0oOg for about 1 h and the sodium montmorillonite removed as a gel. The latter material was redispersed in fresh sodium chloride solution (1 mol dm-3) and allowed to stand for 5 days. The salt was then removed by dialysis against distilled water.The sodium montmorillonite was recovered as a gel with a slight yellow coloration by centrifugation at 38 OOOg. The latter material was used for the preparation of dispersions by mixing with a sodium chloride solution of the required concentration, and then dialysing against a large volume of solution of the same con- centration, with several changes. The dialysis was considered to have reached a satisfactory state when the conductivity of the dialysate and the sodium chloride solution were identical within experimental limits. The cation exchange capacity of the sodium montmorillonite, determined by a method based on the technique of Fraser and Russell,1o was found to be 83.7 mequivalents per 100 g of dry clay.MICROELECTROPHORESIS EXPERIMENTS Mobility determinations were made in an apparatus similar in design to that described by Alexander and Saggers l 1 but fitted with a slit ultramicroscope. The glass cell used for observations was of the form suggested by Mattson and was immersed in a water bath maintained at room temperature (ca. 293 Kj. The sodium montmorillonite samples were prepared by adding 25 cm3 of an 0.04 % w/v clay dispersion to 25 cm3 of a sodium chloride solution at twice the required salt concentra- tion. The dispersions were then dialysed against the appropriate salt concentration until the conductivities of the dialysate and the salt solution became equal. The mobility values quoted are the mean obtained by determining the velocities of thirty particles in each direction of motion, i.e., with opposite electrode polarities, in the cell.COMPRESSION MEASUREMENTS The equipment used for measuring the volume of a clay dispersion under an applied constant pressure has been described previously in some detail.* The clay particles were con- fined between a rubber membrane and a millipore filter arranged so that the electrolyte expelled by the application of pressure was maintained in equilibrium with that of the dis- persion. Under these conditions the total volume of the system was also maintained constant but the ratio of clay dispersion to equilibrium electrolyte changed. The use of a servo-mechanism on the hydraulic system enabled the applied pressure to be accurately maintained. The stainless steel pressure cell was immersed in a water bath maintained at a temperature of 298.2 K.RESULTS ELECTROPHORESIS STUDIES ON SODIUM MONTMORILLONlTE Fig. 1 gives the (mobility, pH) curves obtained with sodium montmorillonite particles at different concentrations of sodium chloride. All the results were obtained with montmorillonite samples which had been aged for several weeks in order to make them comparable with the time duration of the compression experiments. Fresh samples showed a slightly different behaviour l 3 in the region of pH 5 but this effect disappeared on ageing the samples and the results became identical with those shown. The mobility data were found to be reproducible after ageing provided that the systems were not subjected to a pH of less than 3.0 or greater than 10.6.The trend of the results shown in fig. 1 indicates, within experimental error, very little variation of mobility with pH. It was anticipated, following the arguments of van Olphen,6 that the change of the edge sites from positive to negative might have caused an increase in negative mobility with increase in pH as has been observed with kaolinite.21 The absence of this effect would suggest that the edge sites do not make any substantial112 MONTblORILLONITE GELS contribution to the charge on montmorillonites and that the electrokinetic properties are dominated by the charges on the faces. The mobility of the particles showed a slight decrease as the electrolyte concentration was increased from to mol - 4 I I I I I dm-3 and then increased as the salt concentration was increased to 10-1 rnol dm-3.The zeta-potentials of the particles were calculated using the Smoluchowski equation where [ = zeta-potential, E = relative static permittivity and y = viscosity. The results obtained at pH 6, the pH of the compression experiments, are listed in table 1. It must be noted, however, that eqn (1) may not give a true representation of the electrokinetic potential for plate-like particles since, as the plates rotate, the lines of viscous flow and electric field do not necessarily remain parallel to the particle surface. u = E5f411V (1) TABLE 1 .-ZETA-POTENTIALS OF SODIUM MONTMORILLONITE PARTICLES AT pH 6 electrolyte concentration I mol dm-3 mobility/pm m V-1 s- 1 zeta-potentiallmv 10-4 3.l(f0.2)x 44 10-3 2.8(+0.2)x 40 2.7(*0.2)x 38 lo-' 3.5(+0.2)x 50 COMPRESSION STUDIES O N SODIUM MONTMORILLONITE The data obtained for the compression of a sodium montmorillonite dispersion in lod4 mol dm-3 sodium chloride are shown in fig.2, in semi-logarithmic form, as pressure against the distance of separation between the surfaces of the plates (Ho). The value of the latter quantity was calculated from the expression Ho = 2V/mA where Y = volume of liquid in a dispersion containing m g of montmorillonite and A is the specific surface area ; the latter was taken as 800 m2 g-l. The expression assumes a parallel arrangement of the plates in the cell and under conditions where this has been established gives good agreement with the Ho values obtained by low- angle X-ray diffraction measurement.8r. c.CALLAGHAN AND R. H . OTTEWILL 113 The results obtained on the initial compression of the sample to ca. 108 atmos- pheres are shown as a dashed curve in fig. 2. Subsequent decompression gave a different curve, but on decreasing the applied pressure to a small value and then compressing again the data fell on the decompression curve; this is shown by the full line in fig. 2. Subsequent recompressions gave the same curve and, as can be seen from the figure, good agreement was obtained between the new data obtained in the present work and those obtained in the previous study.3 The w/w concentra- tions of clay are also shown on fig. 2. The range of concentrations covered was from about 2 % to 65 %. I I I I 1 I I 1 2 4 6 8 I0 12 14 16 18 2 0 2 2 distance Ho /nm FIG.2.-Equilibrium pressure against the distance between the plate surfaces for a sodium mont- morillonite dispersion in 0, fist compression ; 8, first decompression ; A, second compression ; Open symbols, present work ; full symbols, data from previous work.* The results obtained in other salt concentrations were very similar in form. The data obtained in mol dm-3 are given in fig. 3 and those obtained in lo-' mol dm-3 in fig. 4. The area formed by the loop between the initial compression curve and the final curve is smaller for the experiments carried out at the lower salt concentrations. Within the experimental conditions used loop closure was not ob- tained at the lower pressure end of the cycle although it was nearly achieved at the highest salt concentration. The reason for this was that it was not possible to expand the clay volume contained in the cell sufficiently for the sample to return to the con- dition of being a dilute dispersion.Despite the difficulties of obtaining loop closure, an estimate has been made of the areas contained within the loops. The results are plotted as a function of salt concentration in fig. 5. The loop areas show a smooth change with increase of salt concentration, and, in particular, a smooth decrease over the range mol dm-3 sodium chloride solution. and to lo-' mol dm-3.114 8 0 - 7 0 - a 6 0 - 8 2 50- E 3 40- a 30- 2 0 - a U WJ 10- MONTMORlLLONlTE GELS distance Ho /nm FIG. 3.-Equilibrium pressure against the distance between the plate surfaces for sodium mont- morillonite in (a) mol dm-3 sodium chloride (a) (6) mol dm-3 sodium chloride solution and (6) solution. --CI-~ initial compression values ; -A-, final pressure values.distance Ho/nm FIG. 4.-Equilibrium pressure against the distance between the plate surfaces for a sodium mont- morillonite dispersion in lo-' mol dm-3 sodium chloride solution. -@-, first compression ; -@-y first decompression ; - A--, second compression.I . C. CALLAGHAN AND R . H. OTTEWILL 115 log (electrolyte conc./mol dm-3) FIG. 5.-Loop area in arbitrary units against log (electrolyte concentration) for sodium montmorillonite dispersions in sodium chloride solution of various concentration. DISCUSSION The form of the final curve of pressure against distance of surface separation, as for example the full line in fig.2, can be interpreted in terms of the electrostatic repulsion between clay plates in a parallel array. On the basis of the theory of Derjaguin and Landau l 5 and Verwey and Overbeek l6 the total pressure between two parallel plates in an electrolyte can be written as, where PR is the force of electrostatic repulsion per square centimetre of plate and is given by,5* l6 PR = 2nkT(cosh(u) - 1) (3) with n = number of ions per unit volume and u = vetjHo/z/kT; tjHo/2 is the electro- static potential midway between the plates and u the valency of the electrolyte ions. PA is the pressure created by the van der Waals attraction and, as shown previously,* this term is small for the very thin, 1 nm thick, plates of montmorillonite and can be neglected at Ho values greater than 3 nm. It is, moreover, still small for H4 values between 2 and 3 nm.A detailed correlation between the experimental results and the theory of electrical double layer interaction has been given e1~ewhere.l~ It has been demonstrated using similar arguments to those of Chen and Levine that good agreement can be ob- tained between theory and experiment provided that due allowance is made for the transfer of electrolyte between the clay dispersion and the external electrolyte volume, i.e., the system is treated as one of constant total volume. This agreement supports the contention that parallel alignment occurs after the system has been subjected to high pressures. Confirmation of the alignment was also obtained by X-ray measure- ments which indicated that the distribution of Ho values was rather narrow.The theory of electrostatic repulsion thus appears adequate to explain the results obtained down to Ho values of the order of 2 nm. The existence of strong hydration forces seems unlikely at distances greater than 2 nm but their effect at shorter distances is still not clear. The electrophoretic data obtained do not provide any evidence for the presence of an oxide type l4 edge charge on montrnorillonite to the extent that it exists on kaolinite. Moreover, the edge area of a montmorillonite plate is very small compared with the face area and any charge on the edges would probably be swamped by the116 MONTMORILLONITE GELS negative charge on the extensive face areas. Thus if the isopotential lines are con- sidered around a montmorillonite particle these will run parallel to the faces with effects at the corners causing some possible extension over the edges.On a schematic basis, therefore, the approach of least repulsion between two particles would be an edge-to-edge one but an edge-to-face approach is also possible with one plate perpendicular to the other. Some direct evidence for the latter type of association has been found by Schweitzer and Jennings using electro-optical measurements of aged montmorillonite sols. In a very dilute dispersion (< I %) it seems unlikely that either structure would be maintained owing to the extensive Brownian motion, when the particles would exist as small transient associated units or single particles. With increase in particle number concentration, however, allowing for the fact that as each particle rotates a substantial volume is swept out, double layer overlap can occur at distances of the order of 100 nm or greater. The double layer forces thus restrict the motion of the particles and a disordered structure is formed.With thicker plates of higher density materials, e.g., tungstic oxide, gravitational forces play a much more significant role than they do with montmoril- lonite and the plates rotate and settle into parallel arrays so that the downward gravitational component is balanced by an upward electrostatic repulsion. These constitute the so-called Schiller layers. With montmorillonite, although the plates do not orientate under gravity, they do so under the influence of a compressive force.However, it should be mentioned that Langmuir found evidence for the separation of montmorillonite gels into an anisotropic phase and an isotropic phase of lower clay concentration. The interpretation of the dashed curves in fig. 2, 3 and 4 is, therefore, that they represent the gradual rearrangement of the thermally disordered gel structure into an ordered parallel array. This could be regarded as a laminar liquid-crystalline state in which the colloidal particles are individually stable but ordered with respect to each other. In the disordered state the system might be regarded as loosely floc- culated but a more detailed interpretation would depend on whether edge-face contacts occur in the manner suggested by van Olphen' or whether, as suggested by the present work, the structure is a much looser one not necessarily involving physical contact between the particles.Increasing the electrolyte concentration, with its consequent compression of the electrical double layer (smaller 1 /K), means that the isopotential lines move closer to the surface. This diminishes the electrostatic repulsive forces and, as can be seen from fig. 3 and 4, the particles are pushed much closer together before the same repulsive force is experienced as at lower electrolyte concentrations. The work done against the repulsive forces in bringing two parallel plates, of area 1 cm2, to a separation distance of Ho from an infinite distance is given by, w = -2 P d(H0/2) = - J y P dH,. (4) The area APCB in fig. 4 represents this term.The total work done in destroying the disordered gel structure and opposing the repulsive forces is AOCB and hence the work done in ordering the structure is AOCP. For the results obtained in 10-1 mol drnV3, where loop closure was almost effected, the work term estimated from the area AOCP was found to be 2.7 mN m-1 (2.7 erg cm-*). This is essentially a co- hesive energy term and as such suggests a rather weak cohesion between the plates in the disordered structure. It is notable that the loop area in fig. 4 is smaller than that at the lower salt concentrations and an interpretation of this effect is that, with the decrease in magnitude of the double layer forces, the plates can rotate more easilyI . C. CALLAGHAN AND R. H. OTTEWILL 117 into the parallel positions under the influence of a compressive force.This supports the hypothesis that electrical double layer forces are responsible for gel formation in sodium montmorillonite systems and as they become of shorter range gelation is less favoured. One further piece of information which can be derived from the data obtained is the elastic modulus of the gel. The elastic or bulk modulus, E, of a system can be defined as, E = - VdP/dV ( 5 ) and thus if the applied compressive force P is considered to act in a direction perpen- dicular to a parallel array of plates we obtain per cm2 of plate, E = -Ho dP/dHo. (6) Thus from eqn (6) an estimate of the elastic modulus can be made as a function of the clay concentration. The results obtained from the final compression curve for sodium montmorillonite in mol dm-3 are shown in fig.6 as a plot of log E against log c, I 1.2 1.4 1.6 1.8 2.0 log (4 % w/w) FIG. 6.-Log E against log c for a sodium montmorillonite dispersion in mol d r 3 sodium chloride solution. Data from final compression curve. where c = the concentration of montmorillonite expressed as a w/w percentage. A good linear curve is obtained obeying the equation, log E = 2.30 + 3.20 log C. (7) The values of E at low clay concentrations are of the order of lo3 N m-2 (lo4 dyn cm-2), corresponding to a jelly-like material, and at high clay concentrations the values are of the order of lo7 N m-2 (lo8 dyn cm-2) corresponding to a glass-like material.20 The results at the lower clay concentrations which overlap with those obtained by van O l ~ h e n , ~ using a dynamic method, are in reasonably good agreement118 MONTMORILLONITE GELS with his values.He found a relationship of the form E = k(c- c,J3 where k was a constant and c, a constant obtained from the Bingham yield stress. These results demonstrate the elastic (reversible) behaviour of an ordered mont- morillonite system and show that in principle the elastic modulus could be calculated theoretically by the combination of eqn (3) and (6). We wish to thank the Science Research Council and Messrs English China Clays Ltd. for the award of a CAPS studentship during the tenure of which this work was carried out. We also wish to thank Dr. B. Jepson for a number of helpful discussions. H. van Olphen, An Introduction to CIuy Colloid Chemistry (Interscience, London, 1963), p. 93 et seq. K. Norrish, Disc. Furuduy SOC., 1954, 18, 120. K. Norrish and J. A. Rausell-Colom, Proc. 10th Nut. Conf. CZuys and CIuy Min. (Pergamon Press, New York, 1961), p. 123. H. van Olphen, Proc. 4th Nut. Conf. CIuys and CZuy Min. (Nat. Res. Council Nat. Acad. Sci. (US.), Washington, D.C., 1956), p. 204. I. Langmuir, J. Chem. Phys., 1938,6, 873. H. van Olphen, Disc. Furaday SOC., 1951, 11, 82. H. van Olphen, J. CoIZoid Sci., 1962, 17, 660. L. Barclay and R. H. Ottewill, Spec. Disc. Faruday SOC., 1970, 1, 138. L. Barclay, A. Harrington and R. H. Ottewill, KoIIoidZ., 1972, 250, 655. l o A. D. Fraser and J. R. Russell, CIuy Minerals, 1969, 8, 229. l1 A. E. Alexander and L. Saggers, J. Sci. Instr., 1948, 25, 374. l2 S. Mattson, J. Phys. Chem., 1928, 32, 1532; 1933, 27, 223. l3 I. C. Callaghan and R. H. Ottewill, in press. l4 A. W. Flegmann, J. W. Goodwin and R. H. Ottewill, Proc. Brit. Ceramic SOC., 1969, 31. l6 E. J. W. Verwey and J. Th. G. Overbeek, Theory of StnbiIity of Lyophobic CoIIoids (Elsevier, l7 C. S. Chen and S. Levine, J. C. S. Furuduy ZI, 1972,68,1497. l8 J. Schweitzer and B. R. Jennings, J. Colloid Znterf. Sci., 1971, 37, 443. l9 P. Bergmann, P. Low-Beer and H. Zocher, 2. phys. Chem. A, 1938, 181, 301. 2o R. Houwink, EZusticity, PZusticity and Structure of Mutter (Dover Publications, Inc., New York, 21 A. W. Flegmann, Ph.D. Thesis (Cambridge, 1967). B. V. Derjaguin and L. Landau, Actu physkochim., 1941, 14, 633. Amsterdam, 1948). 1958).

ISSN:0301-7249    

DOI:10.1039/DC9745700110

出版商:RSC    

年代:1974

数据来源: RSC

摘要:

GENERAL DISCUSSION Prof. M. Gordon (University of Essex) said : With reference to fig. 4 of the paper by Beltman and Lyklema, the gel point region is lost in the period of temperature equi- libration at 15” and 25”, but at 35°C gelation is slow enough to show the gel point, where G’ rises from zero, very close to the time origin. This rise from zero starts with zero slope (dG’/dt = 0) as predicted by the generalised statistical theory based on the Scanlan-Case definition of an active network chain. This generalisation covers the complete range of network formation from the gel-point onwards and supersedes previous approximations like chain-end corrections. An active network chain is a chain connecting two active junction points, i.e., two points from each of which ut least three independent paths can be traced through the network to the surface of the specimen.Three paths are required to fix the mean position of the junction in three- dimensional space and to prevent it relaxing after deformation of the sample. (Stat- istically, it is almost certain that any such independent path, once started, will lead to billions of connections to the surface, by virtue of the criticality of branching statistics). The slow initial build-up of modulus (dG’/dt = 0) is due to the inefficiency of cross-linking in producing active network chains, as just defined, just after gelation. This is also the reason for the high-order Ehrenfest transition in the reversible case (see my remarks to Edwards’s paper). As shown previously (ref. (19) of Burchard et al., this Discussion), the plot of modulus versus number of cross-links is always sigmoid.It is easily shown that a sigmoid shape is mandatory even if the number of crosslinks is replaced as abscissa by a time axis via the kinetics of cross-linking, provided the kinetics follows any positive reaction order : this is how I explain the bottom curve in fig. 5. The corollary that initially v increases more than proportionally to cCR squares with the remark by Beltman and Lyklema just below their very convincing fig 5. In view of the highly magnified temperature scale in fig. 5, could the slight upward drift of modulus with rising temperature displayed by the points indicate a reduction of modulus from energetic sources : i.e., the chains would appear to prefer a stretched configuration from the purely energetic standpoint ? Mr.H. Beltman and Prof. J. LykIema (Wageningen) said: In replying to Gordon’s interesting suggestion it appears expedient to distinguish two problems : (a) the actual gelling kinetics in the strict phenomenological sense, i.e., the growth of the number of contributing chain segments and cross-links as a function of time, and (b) the interpretation of the nature of the cross-links. Gordon’s remark concerns especially the first and amounts to a more generalized approach. If instead of more or less postulating the linearity G‘(v) as in our eqn (2) and attributing deviations to rate-influencing contributions of precursor-structures, as done by us, a more generalized G’(v) relation is used, the sigmoid shape can auto- matically be accounted for.We think that this is a worth-while suggestion deserving perhaps to be worked out further for our systems. However, we must not forget that there is at the same time much circumstantial evidence in favour of a definite precursor-structure effect. Part of the evidence is accumulated in our paper, but comparative work with other gelling agents (e.g., resorcinol) can also be mentioned. Such an effect belongs to category (b). There J. Scanlan, J. Polymer Sci., 1960, 43, 501 and L. C. Case, J. PoZymer Sci., 1960, 45, 397. 119120 GENERAL DISCUSSION is some danger that these chemically interesting features are overlooked if G'(t) is interpreted solely on the basis of the kinetics, suggested by Gordon. We tend to conclude that Gordon's suggestion is a valuable alternative, especially if it is so applied that the precursor-structure concept is not a priuri renounced.With respect to Gordon's remark concerning our fig. 5 we refer to our reply to Prim and Flory, who h.ave raised essentially the same point. Pro€ W. Prim (Syracuse University) said : Previous work on crosstinked swollen poly(vinylalcoho1) has shown that there is invariably an energy contribution to the elasticity as indeed one would expect in view of the conformational energetics.'. It seems therefore that your conclusion " energetic elasticity is excluded " should be somewhat modified. Prof. M. Gordon (University of Essex) said: A dynamic transition occurs in the simple system of molten linear polyethylene gelled by irradiation (fig.I). This figure, with data by Prof. Pechhold of Stuttgart, was previously published in Russian. The I I 50 100 dose/Mrad FIG. 1.-Measurements by W. Pechhold (Stuttgart) of the real part of the compression modulus of radiation cross-linked polyethylene as a function of dose in Mrad. x , Hz; n, lo-' Hz; A, 2.6 Hz ; A, 120 Hz; 0, 1.2 kHz; ., 12 kHz. The line Q is the theoretical asymptote, and it was found that at 400 Mrad measurements at all the frequencies shown lie on this line. The asymptotic line is drawn to point at the origin, because the pointy= 1 must lie very close to the origin. storage modulus yields the equilibrium modulus curve at very low frequency Hz). It has the typical shape I have repeatedly emphasised in this discussion. But at increasing frequency, the storage modulus G' (not G"!) goes through a maximum H.Abe and W. Prim, J. Polymer Sci. C, 1963, 2, 527. A. Nakajima and H. Yanagawa, J. Phys. Chem., 1963,67,654.GENERAL DISCUSSION 121 along the cross-link density axis. The maximum is seen to approach the gel point at high frequency ; the gel point is too close to the origin (zero radiation dose) to study this in detail. The mechanism behind the effect of frequency on G‘ is likely to involve the sol fraction : the sol fraction is mobile but its concentration falls rapidly towards zero after the gel point. Its molecular weight averages rapidly decline also. Since small sol molecules would require the application of high frequencies to contribute to the storage modulus, the whole shape of the diagram seems qualitatively explained in this way.Further work on this dynamic transition would be valuable. Prof. P. J. Flory (Stanford) said: It seems to me that the most significant con- clusion to be drawn from Beltman and Lyklema’s fig. 5 is that the temperature coeffi- cient of G’/Tis small. The chains in these networks are bound together by crystalline domains, and one might have expected the alteration of these domains with deforma- tion to enhance the elastic compliance, the more so the lower the temperature, with the results that G’/T should increase with temperature. The fact that both G’ and G” are nearly independent of the frequency argues against the appreciable occurrence of such processes at the small deformations here involved. On the other hand, as Prins points out, the elastic response of a poly(viny1 alcohol) network can scarcely be expected to be purely entropic. The conformational ener- getics of the PVA chains are such as to lead one to expect its energy to vary with elongation.Hence, for the rubber elastic deformation of a network of fixed structure, G’/T should change with temperature. The magnitude of this change must depend on the stereoregularity of the PVA. Mr. H. Beltman and Prof. J. Lyklema (Wageningen) said: We thank Prins and Flory for their remarks. With regard to the occurrence or not of an energetic com- ponent to the elasticity, we note in the first place that the very fact that energetic elasticity has been observed experimentally in other PVA-gels does not necessarily have a bearing on our results, since PVA-gels, prepared under different conditions may vary widely with respect to their viscoelastic properties. The gels Prins is referring to are concentrated (20-40 % PVA as compared with 2-6 % in our case) and have probably shorter chains between cross-links, so that the relative energetic contri- bution could be higher.Moreover, Prins et al. prepared their gels in an entirely different way, e.g., by spinning. Spinning has a strong ordering effect on PVA; even a spun network without a cross-linking agent can only be dissolved in boiling water whereas our gels all melt at about 50°C. It seems possible that the chains in spun PVA-gels have a greater energetic contribution to the elasticity than ours. The theoretical argument that the conformational energetics of the PVA-chain requires an energetic component of G’ remains, of course, valid.It would require a decrease of G’/T with increasing temperature. However, linear regression analysis of our data did not confirm this, even a slight increase was found. Not too much value must be attached to this, though, because the measuring stretch is short in view of the mentioned intrinsic difficult of “ melting ” of cross-links. Also Hirai has found a positive value for d(G’/T)/dT. In conclusion, there seems to be no reason to alter the view-point that the elasticity is mainly entropic, although we leave room for an (as yet unconfirmed) small energetic contribution. With respect to the presence of crystalline domains, see our reply to Franks, C .Bayer, Chem. Unserer Zeit, 1968, 2, 61. N. Hirai, Bull. Inst. Chem. Rex Kyoto Univ., 1955, 33, 21.I22 GENERAL DISCUSSION Dr. M. Sheriff (Ui?ii*crsitj? of London) said: It was established by Beltman and Lyklema that for the gel : PVA (4 %)-CR (2.4 %) aged for 200 h at 25”C, both G’(w) and G”(w) were independent of o over the range of to 102 rad s-l, and it was then assumed that both these parameters were frequency independent for all the gels investigated, and that the choice of co was a matter only of experimental convenience. This assumption is incorrect, and may lead to serious errors in the interpretation of results for the following reasons. The maximum time over which the gelling process was investigated was only 2 % of the ageing time for the 200 h gel, and any inferences as to the frequency behaviour of a gel made over such a wide time interval will be subject to considerable error.Furthermore, as the gel structure develops the viscoelastic response of the system will change from that of a viscoelastic liquid to that of a rubbery viscoelastic solid. It has been well established that, to a first approximation, for a viscoelastic liquid (corresponding to the terminal zone of the viscoelastic spectrum) G’ is proportional to frequency and G” is proportional to (frequency)2.L It follows, therefore, that in the initial stages of gelation the choice of frequency is not a matter of experimental convenience but one that can have a profound effect on the magnitude of G’ and G”. When the gel structure becomes substantial, e.g., at long times the viscoelastic res- ponse will correspond to that of a rubbery solid in the plateau zone, where G’ and G” should be independent of frequency. This behaviour was observed for the gel aged for 200 h.Another factor which might have an effect on the values of G’ and G” is whether or not the applied frequency interferes with the gelling process, as it has been observed that above certain frequencies there is serious interference in the development of a two- dimensional surface gel of Acacia senegal at the air-water interface2 Whether this phenomenon also occurs in the three-dimensional bulk gel has still to be investigated. Mr. H. Beltman and Prof. J. LykIema (Wageningen) said: Sherriff is correct in stating that there is some element of uncertainty involved in the extrapolation of the o-functionality, observed after completion of the gel, to the initial stage of gelation, where it is practically impossible to study the co-dependence.Although it is not feasible to justify our extrapolation rigorously, we can forward at least the following arguments in favour of it. (1) The independence of G and G” of o applies to any gel, even those with very few cross-links per unit volume, i.e., gels resemble other gels in an early state of gelati on. (2) The G’(co) and G”(02) dependences, referred to by Sheriff, reflect relaxation of cross-links during the measurement. No such relaxation is observed after completion of the gel. Assuming that the nature of the cross-links is the same in the various stages of the gelling process, no relaxation is to be expected in the initial stages.(3) For safety’s sake we choose o = 5.28 rad s-’, which is to the high side of the spectrum, thus maximally suppressing any relaxation. Sherriff’s second remark concerns the possible interference of the oscillatory deformation with the gelling process. There is no indication of that with our system : the deformations are far below the maximum admissible limit of linear viscoelasticity and intermittent interruption of the oscillations had no detectable effect on G’ nor on G”. We conclude that the measurement itself does not affect the cross-link density. Accepting in PVA-CR gels the absence of any influence of the measurement as correct, we nevertheless leave room for such an interference in other systems, notably J.D. Ferry, Viscoelastic Properties of Polymers (Wiley, New York. 2nd edn, 1970). M Sherriff and B Warburton, to be published.GENERAL DISCUSSION 123 in those exhibiting no linear viscoelasticity and in systems where the linear behaviour occurs only over a narrow range. Prof. F. Franks (Unilever) said : Beltman and Lyklema raise a number of inter- esting points, some of which are also touched upon in the papers by Harrison et al. and Eagland et al., in particular the requirement for a highly conformation-specific gel precursor. Whereas with gelatin this is almost certainly an intramolecular process involving collagen type triple helical segments, in the case of PVA this specific intra- molecular conformational arrangement appears to resemble closely the crystal struc- ture of anhydrous PVA.X-ray studies have shown that the diffraction pattern characteristic of the monoclinic lattice of crystalline PVA also develops in PVA solutions (in glycerol, ethylene glycol or water) on cooling. Thus no solvent swelling of the crystalline regions is observed.' The authors suggest that the cross-links arise from helical PVA regions but I do not know of any instance where carbon chains adopt such conformations. The results are more compatible with the formation of syndiotactic regions, such as occur in crystalline PVA. I do not think that too much importance can be attached to the difference in the G'(t) curves shown in fig. 4. Surely the type of behaviour observed for the 35" curve is the one to be expected, i.e., initially aG'/at > 0.Considering the moi- xular nature of the gel this is surprising and I wonder how much reliance can be placed on the truly horizontal nature of the plots, bearing in mind the small tempera- ture interval. The observed effects of NaCNS in suppressing gel formation makes me suspect the solvent involvement in the conformational rearrangement and/or gelling and this also points to a considerable enthalpic component. It would be interesting to ascertain whether the Hofmeister series can be applied to the effect of ions on the stability of the gel. Finally to study the function of Congo Red would it be possible to use low molec- ular weight oligomers of PVA? It might be expected that these would still exhibit the conformational transition to form microcrystallites without however giving rise to gels.This type of approach has been of use in studying the nature of the junction zones in carrageenan and analogues of ~t-gelatin.~ Fig. 5 is taken as evidence that the PVA gels are entropic. Mr. H. Beltman and Prof. J. Lyklema ( Wageningefz) said: Franks has made a number of points, the first of which concerns the possibility of a superstructure formation. Franks argues that this superstructure is not a kind of helix but a semi- crystalline region. At the outset we would bring to the fore that we did nowhere emphasize that the superstructure should be a helix; just for sake of argument the term " helical " has been invoked in the concluding section. Nevertheless, there are some arguments which favour a helix-like structure just as there are arguments against it.These have been mentioned in part in the paper but it might be advisable to mention some of them and some other ones again. We recall the blue colouring of (aged) PVA-solutions with 12, parallelling the blue colouring of amylose, and occurring with a rate that is comparable with the rate of (incipient) gelation. The promotion of this colouring by boric acid or by the G. Rehage, Kunststofle, 1963, 53, 605. K. A. Piez and M. R. Sherman, Biochem., 1970, 9, 4134, J. Horacek, J. Chern. Prumsyl., 1962, 12, 385. * D. S. Reid, T. A. Boyce, A. H. Clark and D. A. Rees, paper at this Discussion.124 GENERAL DISCUSSION introduction of F groups as well as its disappearance upon heating parallel these influences on gelation.All of this points too strongly towards the possibility of helix- like structures to be fortuitous, though it is not a definite proof. X-ray data, providing they are properly interpreted, support the concept of micro- crystalline regions. However, if these are due to syndioactic regions, as suggested by Franks, the vastly differing influences of differing gelling agents as Congo Red, resorcinol and borax are not easy to explain. The points raised by Franks with respect to our fig. 4 (behaviour of G’(t)) and 5 have been dealt with in our discussions with Gordon, Prins and Flory, to which we refer. We have not made a systematic study of the effect of electrolytes on the gelling. This could be an interesting topic for further investigation. Some work in this field has been reported by Dittmar and Priest (our paper, ref.(1)). Effects are observed only at high concentrations and only between different anions was it possible to discriminate ; not between (monovalent) cations. Moreover, NaCNS is effective in much Iower concentrations than those studied by Dittmar and Priest. This may point to a different mechanism, though not conclusively. For further information, NaCNS discolours the blue PVA + I, complex and prevents the autonomous gelling of a highly concentrated PVA-solution. Congo Red promotes the formation of the blue PVA + I2 complex, even in the presence of NaCNS. Also the rate of gelling of PVA/Congo Red is only slightly influenced by NaCNS. Our general conclusion is that a search for the physical interpretation of the blue colouring will presumably provide us with the clue for discrimination between the role of microcrystalline and helical structures.We have indeed found that low MW PVA’S are poor gel-formers. For example, a 3 % PVA-solution of M-25 000 did not give any detectable gelling though blue colouring still occurred with iodine. However, we do not think that this helps one to distinguish between microcrystalline regions or more helix-like structures. Franks’ last remark concerns the MW-effect. Dr. D. S. Reid (Unilever) said : What evidence have Beltman and Lyklema that the initial refluxing at 85°C produces complete dissolution of all components, and disruption of any preformed structures in the polymer? In previous work involving agarose, we found that complete dissolution was difficult.The only way to be sure of complete solution was to compare the results obtained for gelation temperature, etc., after differing pretreatments. Have the authors checked that their results are unchanged if they use a higher temperature for the refluxing stage. If the kinetics are altered, it would indicate that the initial treatment is insufficient to disperse all nuclea- tion sites. Mr. H. Beltman and Prof. J. LykIema ( Wageningen) said: Although the refluxing temperature was not varied, we did vary the refluxing time within certain limits and found the obtained gel properties unaffected by it. This applies to the stock solution of PVA as well as to the system after addition of Congo Red. The cloudiness disappears upon heating.For this reason only fresh PVA-solutions have been used. Upon standing the 10 % PVA solution becomes slightly cloudy. Dr. A. B. Fasina and Dr. R. F. T. Stepto (UMZST) said : The concept of an effective elastic functionality is an interesting one, but there are several points worthy of clarification. H. C. Haas and R. L. MacDonald, J. Polymer Sci. Al, 1972, 10, 1617.GENERAL DISCUSSION 125 fc is a parameter which can be evaluated experimentally (from swelling), or theoretically from crosslinking statistics. Because of network defects (free ends, closed loops, and entanglements), there is always the problem of the correspondence between chemical crosslinks and the number of EANC. However, in the statistical evaluation it is assumed that every monomer unit in the gel with more than two reacted functionalities makes a contribution.Will this assumption not result in differences between experimental and calculated values off, ? For stoichiometric polycondensations, fe apparently increases in value from 3 at the gel point tof(the chemical functionality) at complete reaction. However, it is not immediately obvious howf, can be used whenfitself is equal to 3. Isf, still a useful concept for such systems or should some other effective functionality be sought ? The concept of an effective chemical functionality (fc) has of course been used previously to characterise gelation in polycondensations. Here, f, is determined simply from the experimental value of a, and the gelation condition a, = l/(fc- 1). This concept is related only to the gel point, but it indicates that networks can form providedf, 2 2.This lower value of an effective functionality at gel is presumably because the sol fraction is included in the definition offc (but not in the definition offe) so thatf, as such cannot be used to describe the network. The approximate treatment of cyclisation used by DuSek, in which intra- molecular reaction is excluded in the gel, although justified by mathematical exped- iency, appears to be unrealistic. For sufficiently concentrated systems, and subject to the chain configurational properties of the reactants, it is known that cyclisation can be neglected in the sol fraction. However, intramolecular connections are an essential part of the gel structure and they will increase as the reaction proceeds from gelation to complete reaction.Dr. K. Dusek (Czechoslovakia) said : The effective functionality f, can be deter- mined from swelling provided x and v, are known. The present statistical evaluation fully covers only the free ends effect and in principle can be extended to include cyclization and trapped entanglements. The progress is dependent on the develop- ment of crosslinking statistics covering these effects. For ring formation it has been shown that the evaluation is still valid if one takes into account only intermolecular connections.2 When the chemical functionality is 3, thenf, is also 3 over the whole range on conversion. Fogiel’s fc is a corrected chemical functionality of crosslinks in the whole system, but fe is related only to the elastically effective crosslinks which exist in the gel; crosslinks in the sol as well as crosslinks in the gel that are not elastically effective are not counted.Cyclization in the gel has been discussed in detail in ref. (3). If one regarded the gel as a single giant macromolecule, all gel-gel reactions might be consideied as intramolecular, but it is not so with respect to the elastic properties of the gel. Essentially only loops formed within elastically active network chains between two crosslinks are to be counted as intramolecular and not contributing to the modulus, but even they gradually disappear when their segments become crosslinked. The fraction of elastically ineffective crosslinks engaged in elastically inactive loops is thus expected to be very small at high degrees of crosslinking. Fogiel, Macromolecules, 1969, 2, 581.K. DuSek, J. Polymer Sci. C, 1973, 42, 701. M. Gordon, T. C. Ward and R. S. Whitney, Pulpier Networks, ed. A. J. Chompff (Plenum, New York, 1971).126 GENERAL DISCUSSION Dr. D. J. Walsh (Manchester University) said: Could DuSek please enlarge on the physical meaning of an effective functionality of 3 in a system where the chemical functionality is 4. Also could he comment on what effect would have been observed in his calcula- tions if instead of using eqn (3), which is based on Flory’s expression for the free energy of deformation of a network, he had used one based on the expressions derived by James and Guth or any of the other alternative theories. Dr. I(. DuBek (Czechoslovakia) said : The concept of the effective functionality, fe, is related to the equilibrium elastic response of the network including swelling.In the network only such crosslinks that are a part of the gel and from which at least three independent paths issue to the surface of the sample are elastically effective (elastically effective crosslinks, EEC). Other crosslinks do not contribute to the number of elastically active network chains (EANC) although the number of their reacted functionalities may exceed the value of 2. In the vicinity of the gel point almost all of the few EEC are effectively three-functional because the chance of issuing additional paths to the surface (infinity) is extremely low. Further beyond the gel point this probability increases and so doesf,.Eventuallyf, approaches the chemical functionality for a network without free ends. The effective functionality of crosslinks in the network is thus a topological parameter, but its contribution to the free energy of deformation depends on the model. Following essentially the Flory-Wall approach one gets the result given, e.g., by eqn (8), but the procedure could be used also for different contributions due to stretching of EANC provided the additivity of the mixing and elastic terms is valid. Within the framework of the James-Guth theory, fe affects spatial fluctuations of crosslinks around their equilibrium positions and consequently may have an effect on the elastic modulus. Dr. B. Launay (Massy- France) said: Referring to the paper of Callaghan and Ottewill, I wonder whether the values of E calculated by eqn (6) and shown on fig.6, are really representative of a bulk modulus. As a matter of fact a bulk modulus is measured during the hydrostatic compression of a constant mass of a product; in your experience water is expelled from the gel. Do you think that E is nevertheless related to the bulk modulus of the gel? Prof. R. H. Ottewill (University of Bristol) said : Our experiments were carried out with a constant mass of montmorillonite in the compression cell and compression- decompression cycles demonstrated that the system was a reversible one. The displacements involved in the application of pressure were those in the “ lattice of montmorillonite plates ”, the number of units in the lattice remaining constant.The small displacement at each point corresponds to a particular volume fraction of the solid. I think it is therefore reasonable to assume that the figures obtained give a representation of the elasticity modulus of the montmorillonite gel-lattice. Dr. J. W. Goodwin and Mr. R. W. Smith (Bristol University) said: Electrostatic repulsive forces can result in gel formation in dispersions of spherical particles. This effect can be observed with small particle size polystyrene lattices during the dialysis procedure that is often used after emulsion polymerisation to remove electrolyte, surface active agents and reaction products. Polystyrene latices are dispersions of rigid spherical particles with a surface cliarge arising froin bound ionic groups. It has127 I I 1 1 I 4 0 5 0 6 0 7 0 interparticle separation Ho/nm FIG.1.-Shear modulus as a function of interparticle separation for polystyrene latex. 0, 1 x 1O-j mol dm-3 sodium chloride ; 0, 1 x mo1 dm-3 sodium chloride ; -, calculated values. I I I - 5.0 -4.0 -3.0 -2.0 log ionic strength FIG. 2.-Shear modulus as a function of ionic strength for polystyrene latex in 1 : 1 electrolyte at a volume fraction of 0.186 (Ho = 50 nm). 0, experimental values ; - , calculated values.128 GENERAL DISCUSSION been shown i * that monodisperse latices form ordered arrays and that the ordering process is the result of the net repulsive forces between the particles. The shear modulus of a polystyrene latex has been measured as a function of the volume frac- tion over a range of ionic strengths.The apparatus was built to the design of van O l ~ h e n . ~ It consisted of two parallel discs which were mounted on piezo-electric crystals. The separation of the discs could be varied fram 1 mm to 2 cm. A small amplitude shear wave was produced by feeding an electrical pulse to the lower crystal and movement of the upper disc was detected by the upper crystal. By displaying both signals on a twin-beam oscillo- scope and photographing the image, the time taken for the shear wave to traverse the distance between the discs was measured. For a Hookean material, the shear modu- lus, G, is related to the density of the suspension, p , and the propagation velocity of the shear wave, v, thus : G = v2p. A polystyrene latex was prepared by the emulsion polymerisation of styrene.The latex was extensively dialysed against distilled water, concentrated by evaporation and samples with different volume fractions were dialysed against sodium chloride solutions to give the chosen ionic strengths. Before each measurement, the suspen- sions were centrifuged to remove any aggregates and the polymer content of each suspension was determined by drying weighed samples to constant weight. The number average particle size was determined from electron microscopy to be 85.6 nm (lo3 particles were measured) and the electrokinetic potential (5) was found to be -50 mV. Some typical experimental data are shown in the figures. The surface to surface separation of the particles (Ho) was calculated from the volume fraction of the suspensions assuming hexagonal arrays.Fig. 1 shows that the shear modulus increases rapidly with decreasing interparticle separation over the range of volume fractions studied. It can be seen that an appreciable elastic modulus was observed at a sodium chloride concentration of mol dm-3 at a volume fraction as low as 0.18. The theoretical curves were calculated by assuming that only nearest neighbour interactions were important at the large separations used and that with the small displacements due to the simple shearing motion there was displacement between adjacent layers, but that displacement within a layer was negligible. The net restoring force per particle in each layer (li,) is : F, = Fi(cos Bi sin 4i) 1 where Fi is the net force between two particles along a line joining the centres, is the angle between that line and the x-z plane, while 4i is the angle between the x-axis and the projection of the line on the x-z plane.Now the shear modulus is related to the slope of the force-distance curve so for the regular array of particles : G = P. A. Hiltner and 1. M. Krieger, J. Phys. Chem., 1969,73,2386. L. Barclay, A. Harrington and R. H. Ottewill, KuZZuid Z. Z. PuZymere, 1972, 60, 966. R. W. Smith, MSc. Thesis (Bristol, 1973). H. van OIphen, C/ays and Clay Min., 1956, 4, 204. A. E. H . Love, Treatise on the Mathematical Theory of Elasticity (4th edn., London, 1934).GENERAL DISCUSSION 129 where a is the particle radius and Vtot is the total potential energy of interaction between adjacent particles calculated from the Derjaguin-Landau-Verwey-Overbeek theory of colloid stabi1ity.l The calculated curves of shear modulus as a function of inter- particle separation are of the same form as the experimental curves with the best agreement at sodium chloride concentration of mol dm-3. It is suggested that many electrostatically stabilised dispersions should have a measurable elastic modulus when the volume fraction is sufficiently high to produce large repulsive forces between particles due to the overlap of the electrical double layers. Prof. A. Silberberg (Israel) said: In a study of gel formation in well fractionated montmorillonite systems Dr. A. Posner and I (unpublished results) found that there was an optimum platelet size for gel formation which suggested to us that the inter- action in these systems was not simple and could involve edge to face interactions. Prof. R. H. Ottewill (University of Brisfol) said: It is not possible completely to rule out edge-face interactions and, indeed, our initial compression results suggest that these might well exist in some form in the original low concentration gel. Our contention is, however, that these are random contacts formed by thermal motion and that they are probably not a consequence of electrostatic interaction between the negatively charged faces and the positively charged edges. Although ion-exchange experiments indicate a small number of positively charged sites on the edges of the particles it must be remembered that the edge area is very small compared with the face area and that the surface charge density of the faces is very high. Our evidence is that in the sodium montmorillonite examined the edge electrostatic effects are screened by the long range electrostatic interactions of the faces. These arguments, however, could not be applied to kaolinites where edge-face interactions are very important. ' H. R. Kruyt, Cuffoid Science, Vol. 1 (Elsevier, Amsterdam, 1952). 57--E

ISSN:0301-7249    

DOI:10.1039/DC9745700119

出版商:RSC    

年代:1974

数据来源: RSC

摘要:

Dynamics of Water and Amphiphile Molecules in Lamellar Liquid Crystalline Phases BY J. B. HAYTER,* A. M. HECHT,~ J. W. WHITE Physical Chemistry Laboratory, South Parks Road, Oxford AND G. J. T. TIDDY Unilever Port Sunlight Research Laboratory, Wirral, Cheshire Received 25th January, 1974 Thin layers of water of controlled thickness may be prepared between the fluorocarbon sheets in the ammonium perfluoro-octanoate/water system at 23°C. Neutron scattering in this system is almost exclusively from the inter lamellar water due to the low scattering cross-section of fluorine and carbon. In a first series of experiments, the modification of water diffusion as a function of layer thickness has been studied and the anisotropy of this effect determined by using oriented samples. The results are compared with nuclear magnetic resonance spin-echo measurements, which measure the same phenomenon but on a timescale approximately lo6 times longer.The difference between these two types of measurements allows some tentative conclusions to be made about the presence of cracks and holes in the lamellae. In the second series of experiments the motion of probe molecules (for example octanol) has been studied in the fluorocarbon layer to determine diffusion kinetics in this region. It is well known that fatty acids and their salts, with or without another component such as a long chain alcohol, form a wide range of pbases in water, depending upon the composition and the temperature. For example, sodium caprylate and water with a little decanol are believed to have phases at room temperature which range from isotropic mixtures through hexagonal close packed cylinders to lamellar liquid crystalline aggregations. Such lamellar mesophases are of particular interest to us here since, by the addition of extra water, they may be swelled to oriented gels.Their structures have been determined, in part, by X-ray diffraction and consist of bilayers of the fatty acid amphiphile molecules separated by layers of water. Amongst the systems showing these properties, the ammonium perfluoro-octanoate- water system shows a particularly simple phase diagram at 23°C consisting of three well-defined regions : an isotropic phase, an intermediate lamellar lyotropic phase and possibly at high concentrations a solid plus lamellar mixed phase.2 At 23°C the range of water layer spacings embraced by the lamellar phase is from about 9 8, to 27 A.The swelling behaviour resembles that in clay+ water systems where only partial crystalline and osmotic swelling occurs. By adding up to 10% by weight of octanol to this system at 23"C, the range of water layer spacings between the lamellae can be increased to several hundred Bngstrom units.2 In this paper we report neutron scattering and nuclear magnetic resonance measurements of the transla- t Now at Cermo, University of Grenoble, Grenoble, France. 1 Now at the Institut Laue-Langevin, Grenoble: France. 130J . B . HAYTER, A. M . HECHT, J . w . WHITE AND G. J . T . TIDDY 131 tional diffusion of water molecules in the inter-lamellar region and at right angles to the lamellae, as well as some preliminary results on the rotational diffusion of the octanol molecules within the bilayers themselves.Neutron spectroscopy has great potential for understanding the relationships between the structure of the phases and the molecular dynamics in such partially crystalline materials. This is because thermal neutrons have wavelengths between 1 and 10A and energies of about one milli-electron volt (8 cm-1 in optical units). The first property means that, for normal scattering angles, diffraction effects from structures of periodicity 1 to 10008, may be readily observed in conjunction with inelastic excitations in the structures. The second feature follows from the low energy of the neutron, which makes detection of changes in its energy quite feasible either by time-of-flight analysis or by measuring the scattered neutron wavelength with a crystal analy~er.~ These features have already allowed the method to be deployed successfully for studies of the inter-atomic forces between molecules 4-6 molecular vibrations 7* * the dynamics of polymers 9 9 lo and liquids." In addition to these features, which define the range of applicability of neutron inelastic scattering spectroscopy, there are also some unique features of the method which establish a contrast between its results and those using electro-magnetic radi- ation.Chief of these, in the present context, is the controllable observation time and range of the neutron experiment. Additionally, the absence of optical selection rule allows all molecular motions, appropriately weighted by the atomic scattering cross sections,12 to be observed by the technique.At the momentum transfers used in the experiments reported here, the observation time scales vary from about lo-'' to s and the range of observation for diffusive motions of the molecular centres of mass, between 10 and 0.5A. As a result it is worthwhile to compare the neutron measurements of translational diffusion with those from nuclear magnetic resonance spin echo experiments.13 Here the time scale of observation is about s and the range of observation for diffusing molecules about lo4 A. One particularly valuable aspect of neutron scattering from molecules is the selec- tive weighting of the scattering cross section by atomic scattering factors within the system.Here this weighting depends upon the very strong scattering cross section of hydrogen compared to the scattering cross sections from fluorine? carbon, oxygen and deuterium nuclei. Thus in the simple two-component system the scattering from the lamellae is always < 1/10 and often < 1/20 of the scattering from the inter- lamellar water. By using D,O, instead of H20, to prepare the phases it is possible to re-emphasize the scattering from the lamellar region and, in particular when octanol is present in the amphiphilic region, its scattering can be selectively observed with good signal-to-noise ratios compared to the scattering from either the perfluoro- alkyl chains or the intervening water. Formally, for a monatomic system we may express the incoherently scattered neutron intensity from nucleus, v, for srngular frequency change 8co and for solid angle change d i 2 , as the differential scattering cross section, d2a/dRi30. This may in turn be connected with the dynamics of the system by the expression a20 k = -b:S(Q, 0 ) 8Rdo k , where S(Q, co) is the scattering law or the intensity frequency-momentum hyper- surface which characterizes the structure and dynamics of the ~arnple.'~ The energy, Aco, and momentum, AQ, transferred to or from the neutron in the scattering event are defined in terms of the neutron mass, rn, and its incident and scattered wave- vectors k,, k by132 LAMELLAR LIQUID CRYSTALLINE PHASES Often the dimensionless momentum and energy transfers a = h2 Q2/2mkT, p = h / k T are convenient to use instead of Q and a.The scattering law for a given nucleus may in turn be related by fourier transforms either to the eigenvalues and eigen- vectors of the time independent Schrodinger equation for the system, where this representation is appropriate, or to the space-time correlation functions, G(r, t), solutions of the full time-dependent Schrodinger equation. l4 For a single nucleus, v, of scattering length b,, the scattering law can be expressed as the space time fourier transform of a propagator, S(Q, o) = 1 exp i[Q r - ot]G(r,!t) dr dr. (3) For nuclei like hydrogen, the scattering is largely incoherent and the appropriate correlation function for eqn (3) is the auto correlation function Gs(rv, t ) for the vth atom. For a number of different atoms in the sample, all with different scattering lengths, b,, the total cross section for the system (considering only the incoherent scattering part) is the sum of the individual scattering laws S,(Q, o) weighted by the squared scattering lengths b$ for the constituents. a20 k m a 0 ko ,, = - b:(inc)S,(Q, w).(4) By choosing the incident momentum and the scattering angles over which inelastic scattering is observed, conditions suitable for observing this part of the whole scatter- ing from the sample can be set up in a neutron experiment. In principle, therefore, complete structural and dynamical information is available from the inelastic neutron scattering experiment. The limitations at present derive largely from the spectrometer resolutions in energy (frequency) transfer, and momen- tum space.These are restricted by the flux of neutrons available to the first mono- chromator and, with present medium flux reactors, the energy resolution of chopper spectrometers in the frequency range 0.5 cm-l to 500 cm-l (75 p eV to 75 meV) is between 0.05 and 15 cm-l. These are the limiting factors on our interpretation of the present experimental results, but the situation is changing rapidly with the intro- duction of novel instruments of greatly improved energy resolution such as the MARX (absolute resolution about 60 peV (0.48 cm-l) F.W.H.M.) which also has excellent momentum resolution, the back scattering spectrometer,l absolute resolution about 3.5 x cm-l) and the spin-echo spectro- meter l7 (absolute energy resolution better than cm-' i.e., about 100 MHz).eV (i.e., 2.5 x eV; EXPERIMENTAL PREPARATION AND PROPERTIES OF THE SAMPLES STUDIED Perfluoro-octanoic acid (Fluorochem Limited, 97 % min.) was neutralized using 0.880 ammonia and dried over potassium hydroxide under a vacuum. The pure product was recrystallized from a petroleum ether/butanol mixture and dried under vacuum for several days. The samples of ammonium perfluoro-octanoate+ water, APO-H20, were prepared from weighed-out quantities of the components using a variety of mixing techniques. This was done because repeated measurements indicated a variability in diffusion coefficient with different preparative methods. The method described previously,2 of mixing componentsJ . B. HAYTER, A. M. HECHT, J .w . WHITE AND G . J . T . TIDDY 133 by centrifugation through a constriction many times was first used. It was found to give results which were reproducible if good equilibration (over a month) was allowed. Alterna- tively, heating the weighed components in a sealed tube to 80°C, where the liquid crystal melted, followed by cooling and equilibration for one day was used. This gave no different results for water diffusion than dispersion of the mixture using sonication for five minutes at 298 K with equilibration for one day subsequently but longer sonication (over 30 min) gave samples with higher D values. The samples of the two component system ranging in water content from 60 % by weight (an isotropic phase) to 20 % by weight ( d ~ ~ o = 9 A) were prepared by 5 min sonica- tion.The mixture after cooling to room temperature was spread and sheared between thin aluminium foils which were then sealed to prevent escape or exchange of water vapour. These planar samples had good orientation of the perfluoro-alkyl chains (typically the co shaft rocking curve was better than 5" and sometimes better than 20") and they were stable for periods of many months at room temperature. The sample of ammonium perffuoro-octanoate-D20 was prepared by low power ultra- sonic mixing at a temperature of about 60°C for 3 hr and subsequent cooling. The samples containing octanol were prepared by the method used for the two component system. Neutron inelastic scattering measurements were made on the two phase samples using the 6H time of flight spectrometer on the DIDO reactor, A.E.R.E.Harwell and the water layer thickness determined for each sample by neutron diffraction on the same specimen before and after the inelastic experiment. To make these measurements, the long wave- length diffractometer installed at 7H2R on the PLUTO reactor, A.E.R.E. Harwell, was used.20 This instrument uses an incident white spectrum filtered by a guide tube and the sample is followed by a graphite monochromator. It thus has an effective, 4.70A neutron beam, which makes measurements on crystals whose c-axis spacing is 10 to 50 A quite convenient. Samples of the three component system were measured in the same way. Preliminary experiments on the anisotropy of water diffusion, were done using the MARX spectrometer AEK Riss, Denmark, both because of its high resolution and because it is readily adapted to make measurements with the vector momentum transfer, Q, along and perpendicular to the water layers.The self diffusion coefficient measurements were carried out using a Bruker-physic variable frequency pulsed n.m.r. spectrometer (B-KR 32%) operating at 16.0 and 35.0 MHz with the Bruker-physic field gradient unit accessory (B-KR-300z 18). The pulsed field gradient technique 18* l9 was used with the following instrumental conditions : n / 2 pulse length = 2-3 p s : field gradient pulse length - 1 ms : time between field gradient pulses - 34 ms. All D values were measured relative to the value for water at 23°C (2.384~ cm2 s-').19 The calibration of the instrument was set up at the beginning of each day and checked several times during the day to avoid errors due to temperature and other drift.Samples for n.m.r., with oriented lamellae were prepared by placing the non-oriented sample contained in a 0.1 x 1 . 0 ~ 3 cm3 silica U.V. cell in the n.m.r. spectrometer with the 0.1 cm side oriented along the magnetic field direction. The sample was melted by raising the temperature to 353 K and then cooled in the magnetic field. The resulting sample had lamellae oriented with the alkyl chains aligned along the magnetic field and perpendicular to the broad face of the cell. Perpendicular and parallel orientations of the sample with respect to the magnetic field were found by adjusting the sample position for minimum or maxi- mum height of the echo after a n12- t - ?I: pulse sequence in the presence of a field gradient.RESULTS NEUTRON DIFFRACTION MEASUREMENTS In the aluminium containers used for the neutron scattering studies, shearing produces alignment of the lamellae with the water layers parallel to the aluminium sheets. For both the H20 and D,O containing samples, lamellar crystallites were well aligned by this technique. Fig. 1 and 2 show the neutron diffraction patterns, for 8,28 scans using the 7H2R diffractometer on PLUTO reactor, A.E.R.E. Harwell. The incident wavelength was 4.70A and the resolution was about 0.24 degrees in 28134 LAMELLAR LIQUID CRYSTALLINE PHASES ammonium fiuoro-octanoate-H20, AP09; d = 37.8 A lyotropic liquid crystal at 296 K B=7.21' r( - j \ / \. \.-.-.- \ ./. . _._.-./.' I l 5 28- ,6 13 . . L+ . . . . . . . . . . - ZERO ANGLE = -0.33' 1000 -500 0 100 50 FIG. 1.-8, 28 neutron diffraction scan of a shear oriented sample of ammonium perfluoro- octanoate + HzO mesophase (7H2R diffractometer). at angles below 20 degrees. To obtain peak widths, this high resolution 7H2R machine was most useful. The curves show a pronounced change in diffraction pattern arising from the substitution of D20 for H20 in the system. The ratio of the first, second and third orders of diffraction for the H20 and D20 samples ammonium perfluoro-octanoate-D20 at 296 K lyotropic liquid crystal d = 37.15 + O x 8, r r J J :t%;P;:* r I 41% D20 59% APFO 1st ordtr 8.5.61' ze=7.22* . . i neutron scattering angle 28, degrees Fro. 2.-0,28 neutron diffraction scan of a shear oriented sample of ammonium perfluoro-octanoate- DzO mesophase (7H2R diffractometer).J .B . HAYTER, A . M. HECHT, J . w . WHITE AND G . J . T . TIDDY 135 were about 50 : 9 : 1 : and 60 : 2 : I : respectively. Present indications are that better sample orientation might be achieved by magnetic orientation so that higher orders of diffraction might be seen and a one dimensional fourier transform used to get the positions of the ammonium ions if they are located at the amphiphile/water interface. In addition to the low angle reflections from the layer structure, it was possible, by tilting the sample into the neutron beam, to observe off-axis reflections associated with the interchain spacing. For example, in the ammonium perfluoro-octanoate- H20 sample illustrated in fig.1, a broad reflection at 28 = 29.5" which we index as the [l , 11 reflection of the layer lattice was found with the Curran diffractometer. When so assigned, this reflection gives an interchain spacing of 5.2 A. This gives an area of about 20 A2 per molecule or 40 A2 per head group which is not an unreason- able value when compared with chain area measurements using surface chemical techniques. A similar reflection was found in the D,O samples although this was broader and indexed on an interchain spacing of about 4.8 Ak0.3. From these diffraction data we can establish the sizes and inter-lamellar geometry of the particles responsible for the liquid crystalline gel. In all cases, the d spacings for the layers were calculated from both the first and second order of diffraction maxima.The water layer spacings derived from them will be listed below, together with the water diffusion constants measured by inelastic scattering. From the breadths of the powder diffraction peaks, it is possible to infer values of the crystallite sizes in the basal plane and in the c direction of the liquid crystals.20 Typically, the breadths of (001) reflections measured with the 7H2R diffractometer (resolution A28 = 0.24" were about 0.7" full width at half maximum. This gives an L, value of about 550A. By contrast, the peaks associated with the hexagonal packing of the perfluoroalkyl chains were about one degree wide (using the Curran diffractometer), which gives a layer lattice extension, La of about 250 to 400A. These figures represent upper limits to the extent of the layers and have not been corrected fully for the resolution functions of the neutron diffractometersPlused. INELASTIC SCATTERING MEASUREMENTS AMMONIUM PERFLUORO-OCTANOATE f WATER TWO COMPONENT SYSTEM At 296 K, water layer thicknesses between 25 A and about 8 8, were conveniently prepared.For the thickest layer studied (about 22A) the percentage by weight of water is 45 % and the ratio of the water scattering to the scattering from the amphi- phile-salt is e 25/1. In the thinnest water layer samples, the ratio was about 10/1. In their general appearance the quasi-elastic and inelastic neutron spectra from the included water closely resemble those from water layers in clay minerals,8* 21 and by the, predominantly H20, scattering shown in fig.5, (upper spectra). In the inelastic region there is a well developed peak at energy transfer near 450cm-l associated with the water torsional motions. This is connected by a fairly featureless region of scattering to the quasi-elastic region. The quasi-elastic peak is noticeably broader than the machine resolution function and this broadening increases both with momen- tum transfer and with the thickness of the water layer at a given momentum transfer. A more detailed analysis of the spectra will be given elsewhere. If one assumes that Fick's law applies to the diffusive motions of the centres of mass of the water molecules in this system, the scattering law in eqn (1) is a Lorentzian in energy (fico) and given by136 LAMELLAR LIQUID CRYSTALLINE PHASES It can be seen that this Lorentzian has a width at half height which is proportional to the square of the momentum transfer, Q2 The diffusion coefficient, D, may therefore be obtained by finding the limiting slope at small Q, of a plot of the energy broadening against the momentum transfer squared.The experimentally measured quasi-elastic peak is a convolution of the Lorent zian with the instrumental Gaussian resolution function. The true Lorentzian widths were obtained from astrophysical 22 tables of the Voigt function and by a method 23 using the peak height and area. The criterion of successful deconvolution was agreement of the two AE values. This was corroboration of the assumed Lorentzian- Gaussian convolution. Table 1 summarizes the diffusion coefficients obtained from neutron scattering by this type of analysis.The plots of AE against Q2 were good straight lines even up to (momentum transfers)2 of about 3.0A-2. AE = 2DQ2. (6) TABLE 1 .-SUMMARY OF DIFFUSION MEASUREMENTS IN AMMONIUM PERFLUORO-OCTANOATES HzO SYSTEM d(water)/A isotropic 22 19.5 17 13 10 9 20 0.046 0.0515 0.059 0.077 0.10 0.1 1 0.05 LAMELLAE :omposition (% H2O) 60 45 40 36 30 22 20 41 D / 10s cm2 s-* 1.61 1.39 1.38 1.32 1.24 1.21 1.03 1.40 U scan, Qli layers V scan, QL layers FIG. 3.-Vector diagram summarizing the quasielastic scattering conditions on the incident and final neutron wavevectors ko, k and the crystal orientation for momentum transfer Q parallel (U scans) and perpendicular (V scans) to the water layers in APO-H20.ANISOTROPY OF THE WATER DIFFUSION I N ORIENTED SAMPLES An exploratory study to detect any anisotropy in the diffusion of the inter-layer water for oriented crystals of the two component APO-H20 system was carried out using the MARX spectrometer at AEK Risar. In such oriented systems the diffusion must be characterized by a tensor D. Since the momentum transfer, Q, is a vector it is possible to determine some of the elements of the tensor by orienting Q with respect to the crystalline directions of the lamellar sample. Q may be placed along or perpendicular to the layers and varied in magnitude by changing both the sampleJ . B . HAYTER, A . M . HECHT, J . w . WHITE AND G. J . T. TIDDY 137 orientation and the counter orientation simultaneously in the neutron scattering experiment. Fig.3 shows the vector diagrams which illustrate the relationship between Q, and the incident and outgoing momenta k,, k, of the neutron involved in the scattering event. got 200 ‘1 Q2/A-” Q2/,&-’ Q2/A-2 FIG. 4.-Quasi-elastic analysis of the neutron scattering from the tensor diffusion of water in thin, 10 A, and thick, 22 A, layers. Fig. 4 show the diffusion plots for Q perpendicular and parallel to the layer directions and two different water layer thicknesses at 296 K. The results are tenta- tive because of poor counting statistics, but are strongly indicative of a rather smaller anisotropy than might have been expected even in the thinnest layer studied (9&. It was these results which led us to make careful measurements, with nuclear magnetic resonance spin echo, of the anisotropy of water diffusion in these systems.In this the effect of the different ranges of observation for neutron and n.m.r. experiments was also tested. SPIN ECHO MEASUREMENTS OF DIFFUSION IN TWO COMPONENT APO-HZO POLYCRYSTALLINE SAMPLES The pulsed field gradient method for measuring self-diffusion coefficients consists in measuring the echo height (h,) following a n/2 - t - 71 radio frequency pulse sequence with the application of equal field gradient (f.g.) pulses between the n/2 and ?I pulses and between the n pulse and the resulting echo. This is followed by a measurement of the echo height (h,) and the self-diffusion coefficient is given by equation138 LAMELLAR LIQUID CRYSTALLINE PHASES where y is the gyromagnetic ratio of the nucleus whose diffusion coefficient is measured, 6 is the duration of the field gradient pulses At is the time interval between field gradient pulses and G, is the amplitude of the field gradient pulses.D is obtained from a graph of In (h,/h,) against L as a2, At or Gt are varied. In practice, the absolute determination of G, is tedious and inaccurate and so a standard such as water is used and the measurements are made relative to it. In the unoriented APO-H20 samples, very considerable difficulty in measuring reliable D values was encountered. The measurements were less reproducible than measure- ments on other samples and so a thorough study of the dependence of the diffusion constant measured on sample preparation and on aging was made.In parallel, studies of the dependence of the observed diffusion constant on the pulse sequence and other experimental parameters were made. TABLE 2.-DEPENDENCE ON WATER CONTENT OF Dr, THE DIFFUSION COEFFICIENT RELATIVE TO WATER AT 296 K, FOR UNORIENTED APO+HzO SAMPLES instrument conditions : S = 1.05 ms, At = 34.5 ms, t = 20 ms) % water 35 32.5 30 27.5 25 22.5 20 temp./K 296.1 296.3 296.6 296.8 297.1 295.4 295.6 D r = DIDH~o 0.46 0.45 0.46, 0.44 0.46 0.31 0.30 0.28 One possible systematic error that would arise in measuring diffusion of water in restricted environments has been discussed by Tanner and S t e j ~ k a l . ~ ~ They report that restricted diffusion leads to non-linear relationships between log (h,/h,) and L. No such dependence was seen in our samples and D values found by varying Gt at different, constant, At values were the same.Another possible source of complica- tion arises because exchange between the ammonium ion and water protons takes place in these samples. The exchange time is probably quite close to the n.m.r. time scale of observation. This results in a more rapid than normal attenuation of the echo amplitude after a n/2-t-n pulse sequence, the effect being larger as t is increased. For low t values (t < 1 ms) the echo observed is derived from both non- exchanged water and ammonium ion protons, while at longer t values ammonium protons undergo the exchange process more frequently than water protons, because of their lower concentration and the echo becomes almost completely due to non- exchanged water protons (t > 10 ms).To check this effect, D measurements were made using pulse field gradient technique in the presence of a sequence of 180" pulses, as described by Packer and co-worker~.~~ The experimental error for these measurements is at least +lo %, and they indicate that D values measured at low values of t (1 to 5 ms) are not reduced by more than 10 % of the value at longer t. To investigate the dependence of D on composition in the polycrystalline samples, it was decided to measure the D values of the samples prepared and stored for one month under the same conditions, (at 301 K) and using the same instrument settings The results are listed in Table 2 and show that the measured self-diffusion coefficients are apparently independent of concentration down to 27.5 % water where a sudden reduction occurs.J .B . HAYTER, A . M . HECHT, J . w. WHITE AND G . J . T . TIDDY 139 SPIN ECHO MEASUREMENTS ON ORIENTED SAMPLES Because of indications from the neutron measurements that the anisotropy of the water diffusion tensor was fairly small and because of the difficulties encountered with polycrystalline samples, some measurements were made on the magnetically oriented samples. The orientation was checked by observing the light transmitted through the short axis of the sample using crossed polarizer and analyzer on a micro- scope. Only those samples in which little or no transmitted light was observed were used. Values of D (parallel) and D (perpendicular) relative to standard water were measured and are shown in table 3.The measurements were difficult to make because of low sensitivity but do appear to indicate that D (parallel) and D (per- TABLE 3.--0, VALUES FOR ORIENTED SAMPLES CONTAINING 35 % WATER (296f2 K) glass slides 0.56 0.40 (orientation by shearing) u.-v. cells (i) 0.8 0.36 (magnetically oriented) (ii) 0.7 0.3 (magnetically oriented) (iii) 0.52 0.26 (magnetically oriented) (iv) - 0.28 (magnetically oriented) Du DI neutron scattering 0.55 ammonium perfluoro-octanoate, octanol, water lamellar mesophases at 296 K 8.0 4.0 2.0 . . neutron time of flightlps m-1 FIG. 5.-Neutron time of flight spectra at scattering angles 8 = 54" and 90" to the incident beam direction for the ammonium perfluoro-octanoate+ water + 10 % octanol system at 296 K. Upper spectra APO + HnO + octanol.Lower spectra APO + D20 + octanol.140 LAMELLAR LIQUID CRYSTALLINE PHASES pendicular) are the same order of magnitude. The variability in the results indicates the very considerable difficulty associated with such n.m.r. measurements but the general trend of D (parallel) and D (perpendicular) to be scattered on either side of the neutron value is apparent. NEUTRON SCATTERING MEASUREMENTS ON THE THREE COMPONENT SYSTEM AMMONIUM PERFLUORO-OCTANOATE-WATER-10 % OCTANOL At room temperature, swelling of the two component APO-H20 system is limited to water layer thicknesses of about 27 A. To obtain larger water layer thicknesses, and so to bridge the gap between these model gels and biologically interesting gel structures, it is necessary to work with the system containing a few per cent of octanol.The phase diagram of this three component system has not been mapped in detail but well defined regions of lamellar, lyotropic order exist and we have worked within these regions. The system also offers for the first time a chance of determining a degree of mobility of chains in the amphiphilic region by observing the neutron scattering from the included octanol molecules. These molecules effectively act as a probe and we can, in principle, observe both their translational and rotational diffusion to obtain the linear and rotational friction constants within the layer. The system also lends itself well to the isotopic or atomic substitution method ammonium perfluoro-octanoate, octanol, water lamellar mesophases at 296 K 2.0 3:*1 9=54O .. QZ= 1.24A.' "O* 1.0 t 8-90' . . . - . - Qz= 3.0 A ' I energy transfer/meV FIG. 6.-Quasielastic neutron scattering laws, S(a, 8) calculated from the data of fig. 5 contrasting the dependence of the spectra on scattering angle and cross section weighting.J . B . HAYTER, A . M. HECHT, J . w. WHITE AND G. J. T. TIDDY 141 whereby we can extract, selectively, information from the neutron scattering spectrum using eqn (4). Fig. 5 contrasts the time of flight neutron scattering spectra taken from the H,O three component system and from the equivalent system made up with D20. In the inelastic region of the spectra, and this is especially noticeable at the higher angle of scattering (0 = 90') the H20 system shows the strong inelastic peak associ- ated with water torsions at a time of flight of about 300 ps m-' (ca.450 cm-I). This is absent in the D20 samples where the scattering from water is rather less than the scattering from octanol. The remainder of the inelastic spectrum is rather featureless. This is in fact quite useful for the subsequent quasi-elastic analysis because it makes corrections for inelastic scattering less difficult. In the quasi-elastic region, near neutron time of flight 1300 p s m-l, the strong peak is broader at higher angles of scattering for both samples. In the case of the sample prepared with light water the scattering is almost entirely from the H20 quasi-elastic scattering analysis ammonium perfluoro-octanoate, octanol, water mesophase, at 296 K 1 1 I - ! --- -4.5 nt u 0 0.84 t t- I t 9 3 .a .-.0 0 0 0 0 0 2 0 . 4 4 b I 1 0 . (momentum transfer)' (Q'/A-") 0, HzO ; a, D20 ; +, vanadium FIG. 7.-Analysis of the quasi-elastic peak intensity and width dependence on squared momentum transfer Q '.142 LAMELLAR LiQUID CRYSTALLINE PHASES (scattering from water compared to the scattering from the rest of the system is about 6 : 1). In the deuterated sample the scattering from octanol compared to the scattering from all other sources is about 2 : 1. When the data are reduced to the form of the scattering law, S(cr, p) it can be seen that the quasi-elastic scattering has an almost symmetrical and almost Lorentzian shape around zero energy transfer. The scattering laws derived from the data of fig. 5 are shown in fig.6. The differential broadening as a function of scattering angle and between the system where water is observed as opposed to octanol, can be readily seen from this figure. TABLE 4.-sUMMARY OF DIFFUSION MEASUREMENTS IN THE THREE COMPONENT APO+ WATER+ 10 % OCTANOL SYSTEMS composition ~/105cm* s-1 d-IlA-1 70 %ItH20 1.64 61 0.01 6 50 %tH20 1.51 27.4 0.036 20 % H20 1.01 8 0.125 70 % D20 1.23 61 0.016 50 %D20 1.02 27.4 0.036 20 % D2O 0.99 8 0.125 A quasi-elastic analysis showing the variation of peak heights and peak widths for the deuterated and undeuterated sample of the same water layer thickness with squared momentum transfer is shown in fig. 7. Also plotted in fig. 7 is the energy width of the scattering from vanadium, a perfectly incoherent scatterer, which defines the resolution function of the spectrometer. It can be seen from the upper part of fig.7 that there is a weak Bragg reflection at a momentum transfers of about 1.2 A-l which does interfere with the quasi-elastic analysis at, and somewhat above, this momentum transfer. Its effects may be seen in the (energy broadening, momentum transfer squared) plots in the lower half of the figure as a dip in the plot for the deuter- ated sample. This occurs because the quasi-elastic scattering is rather narrower in the region of Bragg reflections than it would be if only incoherent scattering were contributing. The figures show data for intermediate water layer thicknesses. Bragg effects are less pronounced in the H20 sample and for samples with greater water content. Even for the thinnest layers, work at low momentum transfers makes it possible to obtain an " effective diffusion coefficient " for scattering from inter-layer water and from octanol depending on whether the H,O or D20 samples are being considered.The results are summarized in table 4. A general conclusion is that in the H 2 0 system the diffusion constants change in line with the expectations from water in the two component APO-H20 systems mentioned above. In the deuterated samples a much smaller variation in the " effective diffusion constant " is found and, indeed, almost no change in diffusion constant between the 50 % D20 and 20 D20 sample can be detected. DISCUSSION CENTRE OF MASS DIFFUSION The inter-lamellar water in gels formed by clay minerals, sodium montmorillonite, lithium montmorillonite and lithium vermiculite has been studied by neutron inelastic scattering 2 1 and it has been shown that the diffusion coefficient is logarithmically related to the inverse of the water Iayer thickness for water layers between about 5 AJ .B. HAYTER, A . M. HECHT, J . w. WHITE AND G. J . T. TIDDY 143 and 60A. The data from tables 1 and 4 now suggest that a similar relationship holds for water diffusion in the ammonium perfluoro-octanoate gel systems. In fig. 8 we plot the logarithm of the diffusion coefficient determined by neutron scattering measurement as a function of the inverse water layer thickness determined by diffraction from the same samples. It is at once obvious that the results here and in table 1 suggest only a weak dependence of diffusion coefficient on water layer thickness in these samples.The measured diffusion coefficients may be, if anything, slightly low for points at high momentum transfers in the (AE, Q2) plots, (previous section) have been given quite high weighting in the analysis because the dependences are best represented by almost straight lines. Good agreement between the Area and Table methods was found throughout and so the results are, at the least, self- consistent. H20 diffusion in ammonium perfluoro-octanoate gels at 296 K 4 Oe2- .4 G v) c 3 z 0.1 2 M CI 0 1 0 \ +?\ - -+y+ I I \ FIG. &-Water diffusion in ammonium perfluoro-octanoate gels as a function of inverse water layer thickness. Two differences between the behaviour shown in fig. 8 and that in the clays, can be noticed.First, if a straight line is drawn through the points its slope is approxi- mately 2 A while that found for the clay+ water systems was about 5 A. Secondly, whereas the line drawn through the points for the clay mineral experiments extra- polated within experimental error to the bulk diffusion coefficient of water at the measuring temperature, this is certainly not so for the ammonium perfluoro-octanoate system. The limiting diffusion coefficient is about 1.75 x cm2 s-l. It should be noticed, though, that the points from the three component system fall on the same line as those from the two component system. Again this supports the argument of self-consistency amongst the data. There it was seen that the logarithmic behaviour of D with inverse layer thickness could be explained if there was a term, in the enthalpy of activation for diffusion between the charged sheets, linear in the reciprocal of the water layer thickness.A simplified thermodynamic model can be constructed using the Kelvin equation and, on this basis, the aqueous layer in APO is much less sensitive to the mean inter-lamellar field (calculated assuming complete dissociation of APFO) than was the water in lithium montmorillonite for example. Fig. 8 can be discussed in an analogous manner to the clay minerals.21144 LAMELLAR LIQUID CRYSTALLINE PHASES The first signs of this difference are already apparent at the microscopic level at high momentum transfers. The quasi-elastic broadening at Q’ = 3 A-’ for approxi- mately 8 A thick layers of water in APFO/H20 and lithium montmorillonite is about 0.4 meV and 0.15 meV respectively and the diffusion plots for the clay are far from linear. At these momentum transfers the scattering from rotational diffusion is more strongly weighted than at lower Q and we may infer that the rotational cor- relation times are rather longer in the clay minerals than in APO-H20 for the same water layer thickness.The details of these phenomena will be explored further as higher energy and momentum resolutions become available. Because the spin relaxation time of the included water is short, due to chemical exchange, it is clear that nuclear magnetic resonance measurements on this system present unusual difficulty. The caesium perfluoro-octanoate-water system avoids this effect and experiments with it have so far been in good qualitative agreement with the general conclusions above, showing accord between the diffusion coefficients measured by neutron scattering and those determined by nuclear magnetic resonance measurements.The rather low anisotropy found by nuclear magnetic resonance measurements supports the exploratory measurements done with neutron scattering on the two component system. Because the observation range of the nuclear magnetic resonance studies is up to six orders of magnitude greater than that for neutron scattering, it appears that microscopic and bulk transport perpendicular to the predominant orientation direction of the layers is relatively fast. This may indicate that the bi- layers are quite porous to the transport of water or that there is a very large amount of inter-crystalline water in the sample.We think that the latter possibility is not very great, particularly in the oriented samples where relatively large crystallite sizes are present as indicated by the diffraction determined values of La and L, and from electron micrographs of frozen s a i n p l e ~ . ~ ~ And so we are left with the conclusion that water diffusion is relatively fast through our compacted crystalline bi-layers. ROTATIONAL DIFFUSION In the APO +water + 10 % octanol system, the most important result is the rather weak dependence of the quasi-elastic broadening on Q2 for the systems where octanol scattering predominates (table 4). At the energy resolution used, this behaviour suggests that rotational diffusion of the probe amphiphile is a much more important contributor to the dynamics on the s time scale than is the diffusion of the molecular centre of mass.Inspection of the scattering law diagrams of fig. 6 supports this view. The curves for APO+H,O+ 10 % octanol are good approxi- mations to a Voigt profile, but the curves for APO + D20 + 10 % octanol suggest a narrow peak superimposed on a broader low intensity feature. This is most notice- able in S(a, /I) for Q2 = 1.24 A-’ and less. Near AE = -0.65 meV there is a break in the smooth rise from - 1.5 meV. The effect requires resolution improved by about a factor of 3-5 times for clarification. Once again, the high crystallinity of the bi-layers in these compounds suggest attractive future experiments such as deter- mination of anisotropy of the centre of mass motion in two dimensions and the diffusion mechanism from the scalar momentum transfer dependence.CONCLUSIONS It is possible to study the microscopic aspects of water diffusion in model bi-layers with good signal to background by neutron inelastic scattering. At momentum transfers around 1 A-l the diffusion can be analyzed by the classical diffusion equationJ . B . HAYTER, A . M. HECHT, J . w. WHITE AND G . J . T. TIDDY 145 and the diffusion constants found from neutron scattering are in good agreement with those from nuclear magnetic resonance spin echo studies. The diffusion of amphiphile molecules can also be studied selectively by using the atomic substitution method,26 although with spectrometer resolutions of about 0.5 cm-' , the quasi-elastic scattering is strongly dominated by rotational diffusion of the amphiphile about its long axis with a rotational correlation time of about lo-' ' s.We thank Dr. P. A. Reynolds and Dr. K. Carneiro of AEK Rim, Denmark for their help with the MARX measurements. K. Fontell, L. Mandell, H. Lehtinen and P. Edwall, Acta Polytechnica Scand., Ch. 74, 1968. G. J. T. Tiddy, J.C.S. Faraday I, 1972, 68, 608. See for example, G. C. Stirling, Ch. 2 in Chemical Applications of Thermal Neutron Scattering, Ed. B. T. M. Willis (Oxford University Press, 1973). W. Hanke and H. Bilz, Neutron Inelastic Scattering, 1972 Proc. Grenoble Symposium I.A.E.A. Vienna 1972, p. 3. G. S. Pawley, ref. (4), p. 175. J. K. Kjems, P. A. Reynolds and J. W. White, J. Chem. Phys., 1974, 60. ' J. W. White in Moleculer Spectroscopy 1971, Ed. P. Hepple (Inst. of Petroleum, 1971), p. 231. * J. W. White, Neutron Inelastic Scattering 1972. Proc. Grenoble Symposium, I.A.E.A. Vienna 1972, p. 315. G. Allen, ref. (S), p. 261. l o J. W. White in Polymer Science, Ed. A. D. Jenkins (North Holland, Amsterdam, 1972), Ch. 27. K. E. Larsson, Neutron Inelastic Scattering, Vol. I Proc. Copenhagen Symposium, I.A.E.A., Vienna 1968, p. 397. "J. W. White in Chemical Application of Thermal Neutron Scattering. Ed. B. T. M. Willis (Oxford Univ. Press, 1973), Ch. 3. l 3 H. G. Hertz, Proc. 24 Reunion de la Societe de chemie physique Orsay, July 1973 (D. Reidel Co., 1974). l 4 W. M. Lomer and G. G. Low in Thermal Neutron Scattering. Ed. P. A. Egclstaff (Academic Press, N.Y., 1965), Ch. 1. * 5 J. K. Kjems, P. A. Reynolds and J. W. White, J. Chem. Phys., 1972, 56, 2928. A. Heidermann and B. Alefeld, Neutron Inelastic Scattering 1972, Proc. Grenobli: Symposium, I.A.E.A. Vienna 1972, p. 851. F. Mezei, 2. Phys., 1972, 255, 146. l 8 G. J. T. Tiddy, Symp. Faraday Soc., 1971, 5, 150. l9 K. J. Packer, C. Rees and J. D. Tomlinson, Mol. Phys., 1970, 18,421. 2o B. E. Warren, Phys. Rev., 1941,59, 693. 22 J. Tudor Davies and J. M. Vaughan, Astrophys. J., 1963, 137, 1302. 23 H. D. Middendorf, J. Nucl. Sci. Instrum., 1974. 24 J. E. Tanner and E. 0. Stejskal, J. Chem. Phys., 1968,49, 1768. 2 5 D. A. B. Backnall, J. S. Clunie and J. F. Goodman, Liquid Crystals, 1969, 2, 1. 26 B. K. Aldred, R. L. Eden and J. W. White, Disc. Faraday SOC., 1967,43, 169. S. Olejnik and J. W. White, Nature, Phys Sci., 1972, 236, 15.

ISSN:0301-7249    

DOI:10.1039/DC9745700130

出版商:RSC    

年代:1974

数据来源: RSC

摘要:

Laser Light Scattering by Polymer Gels BY K. L. WUN, G. T. FEKE AND (the late) W. PRINS Dept. of Chemistry, Syracuse University, Syracuse, New York 13210 Received 6th December, 1973 The frequency-integrated, absolute Rayleigh-Debye scattering measured as a function of scattering angle down to lo, contains information about the supramolecular structural order exhibited by gels. For covalently crosslinked, swollen polymer networks this provides a measure for the degree of spatial non-randomness in crosslinking. The non-randomness index for a series of poly(2-hydroxy- ethyl methacrylate) (PHEMA) networks is found to vary systeniatically by a factor of over a hundred depending on the history of the network formation. Intensity-fluctuation spectroscopy at any given scattering angle provides a probe for the local viscoelasticity of a gel without applying an external driving force.Swollen PHEMA networks are found to exhibit a spectrum of long s) relaxation times. In dilute aqueous agarose systems both above and below the sol-+gel transition a similar spectrum of relaxation times is found. During the transition, the autocorrelation function exhibits oscillatory behaviour for several hours. This is attributed to the mass flow taking place during the microphase separation. Upon inducing the sol-tgel transition in agarose solutions by quenching to various temperatures below the phase boundary, the frequency-integrated Rayleigh-Debye scattering is found to vary systematically a thousand fold. A Bragg maximum appears, caused by the regularly spaced spontaneous concentration fluctuations which occur in a nucleation-free, spinodal phase decomposition.Polymer gels are defined as disperse systems of at least one polymer in at least one diluent and exhibiting the properties of an elastic solid. Polymer gels can be formed by copolymerizing bifunctional and multifunctional monomers in the presence of a diluent. A polymer network formed in the absence of diluent, can subsequently be swollen so as to yield a gel. Gels of a similar nature are obtained if already existing polymer chains are covalently linked by a crosslinking agent. A different type of polymer gel is formed by the physical aggregation of existing polymer chains into a three-dimensional structure, usually with the chains in somewhat ordered conforma- tions. The nature of the diluent may be such that this second type of gel structure superposes itself on a covalently linked structure.The intensity of the light scattered by the polymer chains in gels measured as a function of the scattering vector K (IKj = (47r/4 sin 812) provides information about the supramolecular structural order exhibited by the gels. If a laser beam of high monochromaticity is used as the light source, one can at each K-value also perform intensity fluctuation spectroscopy and thus obtain the frequency content of the Rayleigh scattering. This provides information about the dynamics of the scatter- ing entities. In the following, we apply these two light scattering techniques 10 covalently crosslinked swollen polymer networks and to aggregating polymer solu- tions which form thermoreversible gels.QUASI-ELASTIC LIGHT SCATTERING THEORY AND DATA ACQUISITION The light scattering of most gels is predominantly caused by fluctuations in the Only a minor amount of scattering arises froin fluctuations in We, therefore, consider the gels to be describable by an isotropic mean polarizability. local anisotropy. 146K . L. WUN, G. T. FEKE AND W. PRINS 147 pokdrizability, which exhibits fluctuations, q(r, t ) , in space and time. The Rayleigh ratio for polarized incident light in the Rayleigh-Debye approximation is then given by 1 6n4 y(r, z) exp (iK r) exp ( iooz) exp ( - iwz) dr dz (1) -03 where I(K, w) is the scattered intensity at a scattering vector K and frequency w, Ro is the distance from the scattering volume V to the detector, I , the incident intensity and 2, the wavelength of the incident light in uacuo.The space-time correlation function y(r, z) is defined by Integrating over the scattering volume one has with y(r, z) exp (iK.r) dr. Integrating over the frequency one has 16n4 1 +a3 2: 2n -03 -* = -(q2)- J+a T(K, z) exp ( i w o ~ ) exp (- im) dz da, = ( 16n4p:>(?2>r(K, 0). (3) According to the Wiener-Khinchine theorem the spectral power density is equal to the Fourier transform of the time correlation function of the scattered field <E(K, t)E*(M, t +z)). Upon normalization we thus have J T(K, z) exp (iwoz) exp ( - iwz) dz R(K7 4 - -al +03 (E(K, z)E*(K, t + z ) ) -m<@K, f)E*(K, f)> exp (- iwz) dz. (4) Data acquisition is conveniently carried out by counting the anode pulses generated by the photons arriving at the cathode of a square-law detector (photomultiplier).In our instrumentation, a digital autocorrelator and mini computer are employed,2 covering a time domain of 100 ns-1 s or a frequency domain of 10 M Hz to 1 Hz.148 LIGHT SCATTERING BY GELS The pulse counting correlation function is related to the field correlation function as follows : (n(K, t)n(K, t + z)) = (n(K, t))2 + (n(K, t))”r2((n, z) + 6(z)(n(K, t ) ) . Here n(K, t ) denotes the number of pulses accumulated at K and time t. normalization and rejection of the z = 0 channel (shot noise), one has Upon Eqn (5) shows that experimentally the square of the normalized field correlation function (eqn (4)) is obtained.The same instrumentation is used to determine the frequency-integrated Rayleigh ratios given in eqn (3). The number of pulses per lops channel are collected at a scattering vector K and in the K = 0 position for about 10-30 s, stored and averaged in the minicomputer. The Rayleigh ratio then follows from its definition (eqn (1)) : where Q is the solid angle viewed by the detector, S the cross-section of the beam, d the thickness of the sample and V = Sd. Neutral density filters are used to reduce the (n(0, t ) ) count. Corrections for turbidity, reflection and refraction are applied as usual.4 COVALENTLY CROSS-LINKED SWOLLEN POLYMER NETWORKS For a randomly-crosslinked swollen polymer network the frequency-integrated Rayleigh scattering (in excess of that caused by the diluent) derives from the thermally- induced fluctuations in the polymer segment density.These fluctuations can be evaluated if an expression for the free energy of the gel is available. In a good diluent the equilibrium degree of swelling (and thus the average segment density) is deter- mined by the sum of a (negative) free energy of dilution, which drives the swelling and a (positive) network free energy, which limits the swelling. Employing the Flory- Huggins expression for the former and ideal rubber elasticity for the latter, the Rayleigh ratio for polarized incident light is In this result n is the refractive index of the gel, 4 the volume fraction of polymer, zll the molecular volume of the diluent, x the Flory-Huggins interaction parameter, v the number density of network chains,fthe functionality of a crosslink, qo the refer- ence degree of swelling (which is simply related to the degree of dilution at which crosslinking takes place) and ( R b ) is the mean square radius of gyration of all seg- ments belonging to a given crosslink. Values for the network parameters can be obtained by measuring the degree of swelling and the elasticity modulus of the gels.5 In reality, the crosslinking is often non-random, so that the gel will possess spatial variations in the local degree of swelling, which willenhance the light scattering.One can now quite generally describe the scattering in terms of eqn (3) : sin K r 11: Kr 16n4 R(K) = -(q2)r(K, 0) = - ( q 2 ) y(r)-4nr2 dr. 2;: A2K. L . WUN, G . T. FEKE AND W. PRINS 149 In principle, y(r) is extractable from eqn (8) by Fourier inversion but if y(r) is assumed to be a sum of two or more Gaussians : y(r) = C xi exp (-r2/a;), C xi = 1.(9) Eqn (8) can be integrated to yield (10) 16n4 II;: R(K) = -(<r12>[x Xia?n3 exp (- K2a:/4)]. The typical example shown in fig. 1, demonstrates that a plot of log R(K) versus sin2 8/2 down to 8 = 1" can be resolved into two straight lines corresponding to two Gaussians in y(r). A reliable extrapolation to K = 0 can thus be made and the para- meters al, a2, X and <q2) determined. Upon comparing the experimental (non- random, nr) Rayleigh ratio [(R(0)Inr with that theoretically obtained for a randomly (r) crosslinked gel (eqn (7)), a quantitative non-randomness index (NRI) can be assigned 0.01 L ' I I I I I I I I I I 0 1 2 3 4 5 6 7 0 9 10 I 1 12 103 x sin2 el2 FIG.1 .-The logarithm of the Rayleigh ratio of a covalently crosslinked poly(2-hydroxyethylmetha- crylate) gel (sample 805) fully swollen in ethylene glycol, plotted against sin2 8/2. The dashed lines indicate that a resolution into two Gaussians is possible, so that al, a2, Xand <q2> can be obtained. for poly(2-hydroxyethyl methacrylate) and ethylene- glycoldimethacrylate in the presence of varying amounts of ethylene glycol. The gels were measured in equilibrium swelling and were visually clear. Decreasing dilution of the monomer mixture decreases the NRI more than a hundred-fold. Such a decrease is to be expected because less microgel formation takes place at lower dilution.6 Similarly, increased crosslinking leads to a lower NRI because of a more uniform crosslinking.The NRI has an important bearing on the mechanical behaviour of polymer networks because it reflects the existence of regions of widely different crosslinking. Excessive non-random crosslinking will place an excessive Table 1 shows some results150 LIGHT SCATTERING BY GELS 1.0- n 0 0.5 n d W z 0 strain on a small portion of the polymer chains in a mechanically loaded network, so that poor mechanical performance may result. a a a - a - a a . - * a - *.. a * . a a a m * a m o m - * a m I I 1 I I I a * * : a a - ,-arr. - a , TABLE 1 .-EXPERIMENTAL AND THEORETICAL RAYLEIGH RATIOS AND THE NON-RANDOMNESS CRYLATE) NETWORKS, SWOLLEN IN ETHYLENE GLYCOL. THE FIRST TWO DIGITS IN THE SAMPLE THIRD DIGIT INDICATES THE LEVEL OF CROSSLINKING (IN 10-5 MOL ETHYLENE GLYCOL DI- INDEX (NRI) FOR A SERIES OF COVALENTLY CROSSLINKED POLY(2-HYDROXY ETHYL METHA- DESIGNATION INDICATE THE VOLUME PERCENT MONOMER DURING POLYMERIZATION AND THE ETHYLENE GLYCOL DIMETHACRYLATE PER ML MIXTURE).sample 01 x lO4/cm 4 x lO4/crn X <+> [R(0)lnr/cm-l [R(0)lr/cm-l NRI 205 1.97 5.91 0.767 6 . 4 2 ~ lo-" 1.88 1 . 1 9 ~ 10-4 1 . 5 8 ~ 104 405 1.49 2.86 0.504 4 . 6 4 ~ 10-l' 3.30 24.2 x 10-4 1.37 x 103 703 1.59 4.12 0.517 1 . 8 1 ~ 10-l' 3.50 1 4 . 9 ~ 10-4 2.05 x 103 805 1.01 2.78 0.617 6 . 0 1 ~ 10-l' 2.89 228 x 1.26 x lo2 705 1.15 4.54 0.694 3.71 x 10-lo 3.17 21.1 x 1 . 5 0 ~ lo3 710 1.01 3.11 0.717 1 . 1 8 ~ 5.94 480x 1 . 2 3 ~ lo2 The dynamics of a gel structure, i.e., its viscoelasticity as driven by thermal motion, is reflected in the time correlation function of the scattered field (eqn (4)), the square of which can be determined by intensity fluctuation spectroscopy (eqn (5)).One of us has reported on this before and Tanaka, Hocker and Benedek * have recently developed an approximate model for the dynamical behaviour of a randomly- crosslinked swollen polymer network. Their model predicts a single exponential decay : In this result, E is Young's inodulus and y the frictional constant of the gel. The latter is defined by AP/L = 72, if AP/L is a pressure gradient across the gel of length L as a result of which diluent is set in motion with a velocity i. These authors find that the autocorrelation functions of crosslinked poly(acry1amid) hydrogels are indeed single exponential decays within experimental accuracy. The Ely values which they obtain from the light scattering experiment agree well with the macroscopically determined E and y.ray r)/r(K, 0) = exp (-m) = exp [ - (E/y)K2r]. (12)K . L. WUN, G . T . FEKE AND W. PRINS 151 It thus appears that the viscoelasticity of a gel can be measured by means of light scattering without an externally imposed deformation. Since, moreover, the scattered light is collected from a very small volume element (typically cm3), it is the ZocaZ viscoelasticity which is probed. Spatial variations in network dynamics can therefore be investigated simply by moving different portions of the gel into focus. We have investigated several PHEMA gels in ethylene glycol over an angular range of 35" < 6 < 80".We find a to be proportional to K2 in accordance with eqn (12) if we force the data to fit a single exponential decay. However, visual inspection of any of our autocorrelation results (fig. 2) makes it clear that a single exponential fit does not adequately describe the data. A further data analysis was therefore undertaken as follows. Although the precision in each time channel can be set to 1 % simply by continued sampling until this precision is reached, the reproducibility of the data is much poorer. It was observed that thermal and swelling equiIibrium of a newly handled gel in the light scattering cell is only established after long times (typically 24 h). In the early stages an oscillating behaviour of the autocorrelation is sometimes observed, presum- ably caused by convection currents.If the cell is left undisturbed final equilibrium data were fourid to be reproducible to about 8 %. A polynomial fit programme to the natural logarithm of the data points was therefore used with as many terms as required to reach the 8 % reproducibility of the experimental data. Most of our data required at least a third-order fit. The higher-order terms reflect the existence of more than a single exponential. The coefficients of the polynomial fit are simply related to the moments of the function F@, a) which is defined by -)=I r(Ky F(K, a) exp (-az) da. Eqn (1 3) replaces eqn (1 2). If the fitting polynomial at any given K-value is represented by (ao(K) + a,(K)z + a2(K)z2 + .. .) then, according to the method of c~mulants,~ the moments are = [u2F@, a) da = 24K)+&K). The reliability of higher moments than a2@) is not high.g The magnitude of the variance V(K) = I ~ - = ) ~ I + / = ) = l 2 a 2 ~ ) l + / a 1 ~ ) is therefore taken as a measure of the deviation of the correlation function from a single exponential decay. For a single decay V = 0. In our case, V ranges from 0.64 to 2.8, depending on K and on the type of gel. Jt is thus clear that all our gels exhibit more than one relaxation process in their local viscoelasticity. In view of our earlier finding that the PHEMA gels are highly non-random (see table l), it is possible that within the small scattering volume there are in our case indeed several gel relaxation processes.However, even for a randomly cross-linked polymer network one would expect a spectrum of relaxation times because the approximate model underlying eqn (12) is inadequate. It ought to be replaced by the more realistic Rouse-type model of Gaussian network chains. This model generates a viscoelastic spectrum (eqn (13)) with many long relaxation times, caused by the coupling of the polymer chains into a network.1° However, no theo- retical derivation of F a , a) has been undertaken as yet.152 LIGHT SCATTERING BY GELS REVERSIBLE AGGREGATION OF POLYMER CHAINS AND THE SOL-GEL TRANSITION In dilute solutions of polymers the time correlation function of the scattered field is determined mainly by the translational diffusion constant D of the polymer chains, although at sufficiently high molecular weights the amplitude r2(K, 0) of the larger internal modes of motion (a, = 7;') may become sufficiently large to give an addi- tional measurable decay constant at large K-values l1 : We have investigated the aggregation processes in dilute aqueous solutions of agarose (an alternating copolymer of (1 +4) linked 3,6 anhydro-a-L-galactose and (1 4 3 ) linked P-D-galactose) in the concentration range 0.1 to 1.0 wt.%. At SUE- ciently high temperatures the pulse counting correlation function may be represented by a single exponential decay with a decay rate determined by the average diffusion constant, D, of the various molecular aggregates of agarose present in the solution. Extreme care has to be taken to eliminate externally induced thermal convection currents.This was accomplished by a temperature controlled jacket with narrow openings for the incident and scattered beams. As the temperature is decreased, B also decreases indicating that more aggregation has occurred. At temperatures slightly above the sol-gel transition temperature, typically 40-50°C, the pulse counting correlation function can no longer be represented by a single exponential decay. Instead, (fig. 3) we obtain what appears to be a sum of 1.0 n 0 0 0 5 10 15 20 25 30 35 40 45 50 5X1O2 71s FIG. 3.-The pulse counting autocorrelation function, P ( K , T/rZ(K, 0) (eqn (9, one dot for each of the 50 channels of 2 ms) for a 0.3 wt. % aqueous agarose solution at 56°C and a scattering angle of 55". Clearly, the data deviate from the best fit single exponential decay (solid curve). decays.The additional decays may reflect internal modes of motion of the large aggregates, or microgel particles. We cannot rule out the possibility, however, that the additional decays result from an increasing polydispersity of the microgel particles. When agarose solutions are cooled to temperatures below the sol-gel transition temperature, we observe the onset of a damped oscillatory pulse counting correlation function (fig. 4). We believe that this phenomenon is most likely associated with the mass flow that accompanies the gelation process. The oscillatory correlation function persists for several hours but eventually disappears as the gelation process is completed.K . L. WUN, G .T . FEKE AND W. PRINS 153 The time evolution of previously published optical rotation and frequency integrated light scattering data l 2 also provides evidence for the long duration of the gelation process. $ 5 h k " O ~ - 0 O- 0 10 20 30 40 50 60 70 80 90 100 3.33 x lo2 71s FIG. 4.-The pulse counting autocorrelation function (eqn (5), one dot for each of the 100 channels of 3 ms) for a 1.0 wt. %aqueous agarose system below the sol+gel transition temperature at a scattering angle of 30". The measurement was obtained 3 h after the initiation of the gelation process. Once the agarose gels have equilibrated, we have been able to measure correlation functions in only the most dilute (-0.12 wt. %) samples. We find a sum of expo- nential decays, similar to that shown in fig.2 for a typical swollen PHEMA network, reflecting a spectrum of viscoelastic relaxation times. In more concentrated samples we found that the signal to noise ratio was too low to obtain measurable results. In the light of our present results it appears that the earlier data quoted as ref. (7), indicating damped oscillatory correlation functions in microphase separated gels of agarose and poly(viny1 alcohol), must be re-interpreted. It was conjectured that the oscillatory correlation functions were due to thermally excited low frequency vibra- tions present in the equilibrium gel. Since we have now found that the equilibrated gels do not exhibit oscillatory correlation functions, it seems that the measurements reported in ref. (7) were obtained from samples in which gelation was not yet completed.In the case of PVA, the incompleteness of the gelation process is connected with the slow crystallization of PVA in the concentrated polymer phase. In the course of a closer examination of the sol + gel transition, we have found the frequency-integrated light scattering to be very strongly dependent on the thermal history. Fig. 5 shows that the frequency-integrated Rayleigh ratios of 1 % agarose gels may differ by as much as a factor of 1000, depending on the quenching tempera- ture. The sol --+ gel phase transition temperature l2 at this concentration is about 4142°C and rapid supercooling was achieved by switching the cell from one circula- ting thermostat to another in about 5 s. Fig. 5 also shows a maximum in the Rayleigh ratio which shifts to larger scattering angles when the temperature jump is increased.Such a maximum can be either due to the formation of very monodisperse microgel particles which become suspended in the gel matrix in a random array, or to an inter-particle Bragg spacing of regularly placed microgel particles in the gel matrix.12 No maximum is observed when a very dilute, and therefore non-gelling, agarose solution is cooled.12 The scattered in- tensity and its angular dependence indicates that in this case microgel particles of1 54 LIGHT SCATTERING BY GELS larger dimensions than in the hot solution are also formed. Since no maximum is observed, the microgel particles are not very monodisperse. It seems therefore more likely that the scattering maximum observed in the gels is due to a dominant inter- particle Bragg spacing.Such a regular spacing might result from a non-nucleated, spinodal decomposition mechanism along the lines suggested by Cahn's the0ry.l I oc IC r( E 2 n m W & I 0. I - - - 0 32.5 C 29 "C I 2 3 4 s 6 7 8 9 1 0 103xsin2 8/2 FIG. 5.-The logarithm of the frequency integrated Rayleigh ratio (eqn (3)) plotted against sin2 8/2 for 1 % aqueous agarose gels, obtained by cooling the solutions from 80°C to the indicated tempera- tures. Note the appearance of a maximum and its shifting to wider scattering angles with increasing supercooling below the phase boundary of the sol -+ gel transition at 42°C. In order to test whether the gel formation is nucleation controlled ur results from the spontaneous concentration waves of Cahn's theory, the time-evolution of the intensity of scattered light after a temperature jump into the thermodynamically unstable region of the system has to be measured at various K ~a1ues.l~ Such work is in progress but has not yet been completed.Financial support of NSF grant No. GP-33755 is gratefully acknowledged. R. D. Mountain and J. M. Deutch, J. Chem. Phys., 1969,50, 1103. R. W. Wynaendts van Resandt, Master's Thesis (Department of Electrical Engineering, Syracuse University, 1973). E. Jakeman, J. Phys. A. 1970,3,201. R. S. Stein and J. J. Keane, J. PoZymer Sci., 1955, 17, 21. K. L. Wun and W. Prins, J. PoZymer Sci. (Polymer Physics Edition) submitted for publication. K. DuSek, ColZ. Czech. Chem. Commun., 1968, 33, 1100; Brit. Polymer J., 1970, 2, 257. W. Prins, L. Rimai, A. J. Chompff, Macromolecules, 1972, 5, 104. T. Tanaka, L. 0. Hocker and G. B. Benedek, J. Chem. Phys., 1973, 59, 5151. D. E. Koppel, J. Chem. Phys., 1972,57,4814. lo see, e.g., J. D. Ferry, Visco-elastic Properties of PoZymers (J. Wiley and Sons, New York, 1970), l 1 see, e.g., R. Pecora, Ann. Rev. Biophys. Bioengin., 1972, 1, 257. l2 E. Pines and W. Prins, Macromolecules, 1973, 6, 888. p. 437 ff.K . L . WUN, G . T. FEKE AND W . FRINS 155 l 3 J. W. Cahn, J. Chem. Pltys., 1965, 42,93 ; see also J. Goldsbrough, Sci. Prog., 1972, 60, 281. 14J. J. Van Aartsen and C. Smolders, European Polymer J., 130, 6, 1105. Note added in proofi We have shown (G. T. Feke and W. Prins, “ Spinodal Phase Separation in a Macro- molecular Sol+Gel Transition,” Macromolecules, 1974,7, that the Bragg maxima shown in fig. 5 are indeed due to the formation of a supramolecular structure governed by phase separation proceeding via spinodal decomposition. We have explained the shifting and the broadening of the maxima with lower quenching temperatures in terms of the spinodal decomposition mechanism. We have also shown that the absence of a maximum in the upper curve in fig. 5 is the result of phase separation proceeding via nucleation and growth since, in this case, the quenching temperature is higher than the spinodal temperature.

ISSN:0301-7249    

DOI:10.1039/DC9745700146

出版商:RSC    

年代:1974

数据来源: RSC

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GENERAL DISCUSSION Prof. F. Franks (Unilever) said: Fig. 8 of the paper by Hayter et al. shows that the extrapolated values of Dd -t do not correspond to the self diffusion coefficient of bulk water. This result is at odds with studies on other lamellar systems, notably clays and phospholipids.' Two possible explanations can be advanced for this unacceptable result : (1) If d is determined by X-ray diffraction then the measured length corresponds to the sum of the water layer and the amphiphile and it is assumed that the addition of water to the anhydrous amphiphile system does not affect the mean chain length or area/molecule. (2) Finer and Darke have shown that by plotting D against d-l linear regions are obtained which reflect the different diffusion states of the inter lamellar water.Fig. 4 indicates that the anisotropy of D increases as d decreases, as would be expected. However, if the range of observation of the motion is only < lOA then the observed low anisotropy is not surprising. On the question of lack of anisotropy as shown by the n.m.r. results (does not eqn (7) assume isotropic diffusion?) this, as well as the irreproducibility, might be due to bilayer defects and end effects. Over a period of 1 ms the distance diffused is > 1 pm. It is very difficult to produce lamellar systems of such dimension without serious defects which would allow water to diffuse out of the interlamellar channel and apparently diffuse through the bilayer, as suggested by the authors. Prof. A. Silberberg (Israel) said: Dr. Windrich and I have found that water penetrates fatty acid multilayers built up on glass slides by the Blodgett technique.These effects were studied and evaluated by their effects on the spectra of dye molecules incorporated into a selected one of the transferred monolayers as a function of the distance of this dye containing layer from the surface. Dr. D. S. Reid (Unilever) said: In scanning calorimetric studies of agarose gelation,2 we found that it was very difficult to obtain complete dispersion of the agarose. We finally settled on solution at 120°C in an autoclave, followed by filtra- tion. There, therefore, may still be a significant concentration of nuclei present after heating to 80°C. Whilst these may not be effective in a fast quenching experiment, I wonder how Prins' results (fig.5) would change if solutions were heated to >80°. Also, I wonder how relevant results from such quenching experiments are to the under- standing of agarose gels formed under slower cooling conditions. We have found, for example, that though agarose normally gels at around 40"C, a 1 % sample held for 24 h at 50°C will gel. Clearly, kinetic factors are very important in this gelation. A quenching experiment will not give us information on the nature of these kinetic barriers, and the final gel obtained may be very different from that obtained by slower cooling. Prof. W. Prim (Syracuse University) said: (1) The important point of 5 fig. in our paper is the occurrence of a Bragg spacing upon quenching to sufficiently low temper- * D. S. Reid and D.J. Tibbs, Thermal Analysis, Proc. 3rd I.C.T.A. (Davos, 1971), ed. H. G. E. G. Finer and A. Darke, Chem. Phys. Lipids, 1974, 12,l. Wiedemann (Birkhauser Verlag, Basel), vol. 3, p. 423. 156GENERAL DISCUSSION 157 atures. We have proved (see " note added in proof " above) that this spacing is caused by a nucleation-free spinodal phase decomposition mechanism. In so far as autoclaving seems to eliminate residual agarose aggregation, one might expect the regularity caused by the spinodal mechanism to be possibly enhanced upon quenching auto- claved agarose solutions to below the spinodal. (2) Slow gelling at a temperature just below the phase boundary leads to a gel structure without a Bragg spacing because now the phase decomposition takes place through a normal nucleation and growth mechanism.The residual aggregates in our samples heated at 80°C are prime candidates for the initiation of the gelation process, giving rise to a very inhomogeneous structure with large domains of polymer-rich phase. If the residual aggregation is removed by autoclaving, the gelation will have to take place through homogeneous, rather than heterogeneous nucleation and it is difficult to predict whether the final structure will be very inhomogeneous with large domains-and thus a lot of light scattering-or less inhomogeneous with many small domains-and thus less light scattering. Prof. P. J. Flory (Stanford) said: Is it conceivable that the long-range correlations could be due merely to fluctuations of cross-linking density that are entirely of random statistical origin ? Such an explanation would avoid the implication of nonuniformity in the chemical processes involved in network formation.Secondly, can Prins clarify the term " . . . radius of gyration of all segments belonging to a given crosslink "? The basis for assigning segments to individual cross-linkages is not obvious. Moreover, irrespective of how the segments should be apportioned to the various cross-linkages, in most networks these sets of segments do not occupy discrete regions of space; their domains must overlap copiously. It is essential in this connection to distinguish spatial neighbours from topological ones, the latter being first neighbours in sequence along a given primary molecule. The topological domain may envelop 10 to 100 spatial neighbours.Prof. W. Prins (Syracuse University) said: (1) The observed light scattering of swollen networks formed by a vinyl/divinyl crosslinking copolymerization is typic- ally a hundred times larger than the scattering arising from the thermal fluctuations in segment density. The latter are calculated by assuming that all chains between crosslinks have the same contour length. It is hard to imagine that a distribution of contour lengths of any reasonable width will cause the thermal fluctuations in segment density to increase by a factor of 100. Similarly, it is hard to see why a distribution of contour lengths centring around chains with say 100 monomer units, would give 5000 to lOOOOA correlation distances in the light scattering as observed experi- mentally. For these two reasons we believe to be dealing with spatially non-random networks. So far we have not carried out a calculation quantitatively to corroborate the above statements. (2) In our treatment the radius of gyration refers to the topological domain of the p , segments around each crosslink, but the local volume fraction, q5(ro) of polymer segments takes into account contributions at ro from the crosslink concentrations v, (ro + r) at any location ro + r by integration over r : 40-0) = Vl {P(r)vc(ro + r> dr where p ( r ) is the topological segment density belonging to a given crosslink and u, is the volume of a segment. The spatial overlap of topological domains is thus fully accounted for. W. Prins et a/., J. Polytner Sci. (Phys. Ed.), 1974, 12, 533.

ISSN:0301-7249    

DOI:10.1039/DC9745700156

出版商:RSC    

年代:1974

数据来源: RSC

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Gelation of Globular Proteins BY M. P. TOMBS Unilever Research Laboratory, Colworth House, Sharnbrook, Bedfordshire Received 22nd October 1973 Globular proteins form gels as a resuIt of aggregation to form strands followed by interaction of the strands to form the gel mesh. An approximate (pore size, concentration) relationship can be predicted from selected models of the aggregation process, which is consistent with that determined from electron micrographs of gels. A limited random aggregation may plausibly lead to gelation, and the mode of aggregation is the main quantitative factor determining concentration requirements. Globular proteins form gels in conditions which would be expected to, and are often known to, disrupt the native structure of the protein. Thus, gels typically form when solutions are heated, or in urea solutions or comparable disrupting agents.Events may be complex; for example, glycinin will gel at 20°C in urea+alcohol mixtures ; these gels melt at 5O-6O0C, and then set again at 70-80°C to give irreversible thermostable gels. No doubt both covalent and non-covalent interchain links are involved. (“ Chain ” will always be the peptide chain. “ Strands ’’ are the rnesh- forming structures in the gel, which may not be single peptide chains). As a rule, to obtain gels from globular proteins requires concentrations an order of magnitude higher than, for example, from gelatin or the gel forming carbohydrates. It is sot obvious how more or less spherical particles could reasonably form a structure such as a gel mesh. Electron micrographs of gels (e.g., fig.1) show a mesh structure which may be imagined as built up from strands ; and the dimensions of the strands and the pore sizes show that they must arise by aggregation of the individual particles. This paper reports an investigation of some possible aggregation processes and attempts to predict expected pore sizes from models ranging from highly-oriented to random aggregation. STRAND LENGTHS: SIMPLE CASES IN A REGULAR CUBIC ARRAY Consider a peptide chain of weight 30 000 mol-’, containing about 280 amino-acid residues. The length of each residue is 3.67A, ignoring the terminals and any proline residues. Thus, the fully extended length of the chain is about 99OA and. taking the specific volume as 0.75 cm3 g-l, 1 cm3 of pure protein contains 2.66 x 1019 particles and a total length I of chain of 2.64 x 1014 cm (2.6 x lo9 krn!).We imagine chains like these linked end to end, at some volume concentration c, and folded into a regular cubic array contained in a centimetre cube. Ifp, is the pore size and d the thickness of the strand then, approximately, and, if p , is large compared with d, this further simplifies to from which for this casep; = 1.13 x 10-14/c cm. (pr+q2 = 311; (1) pr” = 311, (2) Here I is the total length of strands 158T available to form the mesh. At the other extreme, a compact spherical molecule of 30 000 mol-l would have a diameter of about 42 A, and if these aggregate like a " string of beads ", then I = 11.2 x 10l2 cm ~ m - ~ and p: = 2.68 x 10-13/ccm.In an intermediate case, e.g., after thermal disruption, the effective diameter might be about 78 A, and p f = 1.44 x 10-13/c cm. Thus, the concentration requirement to produce a gel of specified pore size is strikingly different depending on the postulated arrangement. Table 1 shows the concentrations required for a regular cubic array for p = 300 A in these three cases, and the pore sizes expected for c = 0.01. RANDOM AGGREGATIONS The simple cases considered above required a highly oriented interaction of a molecule (in three different configurations) to form strands, and then a regular arrangement of the strands. However, in addition to variable configurations of the particle, we must also consider both a possibly random aggregation of the initial particle to form the strands, and a possibly random arrangement of the strands so obtained to form the gel mesh.Protein molecules do not show completely random aggregation because the surface of protein molecules is not uniform with respect to the probability that contact will lead to adhesion, though we do not know how non-random it is. In general, random aggregation of spherical particles might be expected to produce larger, more or less spherical, particles. Of interest is the distribution of shapes of such an assembly of random aggregatec and the effect of a small degree of non-randomness, because the total length of strand available to form the gel will depend on the shape-distribution curve of the aggregated particles. Highly oriented cases are relatively easy : the " string of beads " model described above is one such case, and the problem is to allow for the random arrangement of such oriented aggregates.Ogston has treated the problem of the distribution of spaces in a random array of fibres : he finds the mzan pore size p can be described approximately by (p)2 = 0.25/2. (3) The interpretation of pore size in this theory is the diameter of the largest sphere that could just be contained within the pore. Comparing this with eqn (2) for the regular array, and in a random array the mean pore size is smaller. The relative concentration requirements for different models found for the regular array remains unchanged in the random case and this must be generally true for the conversion of any regular array to the random case.Thus, eqn (3) (which is only approximate because it takes the gel strand thickness as negligible and this is not justified in many real gels) may be used to allow for the effects of random arrangement of strands, providing an estimate of I can be obtained. Table 1 includes calculations for the " string of beads '' strands in a random array. We have attempted to allow for random aggregation to form strands by finding the shape distribution curve, and the mean maximum dimension of the aggregates. Thus, the contributions made, on average, by each initial particle to the total possible strand length can be calculated, and I estimated for any initial number of particles. The model used was to start with a particle, and then add others randomly by using a matrix reference and random numbers, up to a specified number of particles n.To begin, we generated two-dimensional aggregates and found the maximum dimension 0 . 2 9 ~ ~ = p (4)160 GELATION OF GLOBULAR PROTEINS (particle centre to particle centre) of the aggregate. For a ten-particle aggregate, the maximum dimension distribution from 30 trials and a visualisation are shown in fig. 2. Half the particles would be more asymmetric than the one illustrated and one may imagine that gels such as those illustrated in fig. 1 could be built up from such aggre- gates (cf. ref. (3)). I 2 3 4 5 6 7 8 9 centre-to-centre distance FIG. 2.-Random two-dimensional aggregation. At the bottom, the distribution of maximum dimension of a ten-particle aggregate. The units are molecular diameters, measured centre to centre.The actual maximum dimension requires addition of one molecular diameter. At the top, a repre- sentation of an approximately average particle : half the distribution would be more asymmetric. A two-dimensional aggregate corresponds to a particular type of oriented inter- action : we were also able to obtain the maximum dimension for a three-dimensional case (see appendix for details). From the mean maximum dimension, the average contribution made by each initial particle to the available strand length was found, and then, by applying eqn (3), the (concentration, pore size) relationship for random aggregates in a random array. Some figures are given in table 1. They show, as expected, that the larger the aggregate the smaller the contribution from each particle.A weakness of these calculations is that we have made all aggregates the same size and the next step is to find the contribution when the particles are allowed to aggregate to a mean value of n, and also to introduce elements of non-randomness into the aggregation process. The model is not, therefore, particularly realistic ; its main interest is in revealing how a random aggregation might lead to gels of the type observed.A 0 C FIG. 1.-EIectron micrographs of embedded, sectioned gels. Positive stain with uranyl acetate. A, Arachin, c = 0.15 made by reducing the pH from 12 to 4. B, Arachin, c = 0.15 made by heating at 110°C at pH5 in 4 % (w/v) NaCI soIution. C. Bovine serum albumin, c = 0.02, made by [To facepage 160 dissolving protein in 8 M urea solution.TABLE 1 .-THE EFFECT OF CONFORMATION, MODE OF AGGREGATION AND RANDOM ARRANGEMENT OF STRANDS ON (PORE-SIZE, CONCENTRATION) RELATIONSHIPS mean maximum distance in aggregate (units of molecular diameters) - 3.713 5.289 6.695 7.274 8.065 averago contribution to strand length per particklA 990 78 42 24.94 46.3 19.8 13.2 10.76 8.68 7.6 36.7 24.5 20.0 16.1 14.1 c required for p = 3008( in a regular case : n = degres of aggregation dm F molecular diameter kmlcm3 array extended chain random coil compact sphere 2-d aggregate n = lodm = 42 A 2-d aggregate n = lOd, = 78 A 3-4 aggregate n = 1Odm = 42 A n = 20 n = 30 n = 4 Q n = 50 n = lOdm = 78A n = 20 It = 30 n = 4 0 n = 50 2 .6 4 ~ 1014 2.08 x 1013 11.2x 10l2 6 . 6 4 ~ loi2 12.3 x 10l2 5.28 x 10l2 3 .5 2 ~ 10'' 2.88 x 10l2 2 . 3 2 ~ 10l2 2.02x loi2 9.79x 10l2 6 . 5 4 ~ 10l2 5.34x 10l2 4 . 2 9 ~ 10l2 3 . 7 6 ~ 10l2 1.26 x 10-3 1.6 x 0.297 x lo-' 0.501 x 10-1 0 . 2 7 ~ 10-1 0.63 x lo-' 0.95 x lo-' 1.15 x 10-1 1.43 x 10-1 1.64x 10-1 0.340~ 10-1 0.509~ 10-l 0.623 x 10-l 0.776~ 10-1 0.885 x lo-' c required for p = 300 8( in a random array 1.04~ 10-4 1.33 x 10-3 0.247~ 0.417~ 0.225~ 0.524~ lo-' 0.786~ 0.961 x lo-' 1 . 1 9 ~ 1.37 x 0.282~ lo-' 0.423 x 0.518 x 0.645 x lo-' 0 . 7 4 ~ p obtained from c = 0.1 or 0.01, a random array A A 9.7 31 34.6 109 47.2 149 61 193 45 142 68.8 217 84 266 93 294 104 328 105 351 51 159 62 195 68 216 76 241 81 257 p for c = 0.01, for a regular array A 107 375 513 ' 665 489 2 750 5 917 v1 101 3 1131 1210 548 672 744 83 1 886I62 GELATION OF GLOBULAR PROTEINS The rcL.ults arc 5uniniariscd in table 1.The concentrations required to produce a p value of 300& and the pore sizes expected from 1 (v/v) and 10 % (v/v) solu- tions, were calculated for a random array by using eqn (3), or for a regiilar array by eqn (2). Surprisingly, the requirements for a "string of beads '' model and an M = 10 random aggregate are not very different. However, the absolute concentra- tion requirements for the random array are uniformly lower than general experience suggests : for the regular array, requiring an order of magnitude larger concentrations, these requirements are of the right order. Any case where the strands are not ran- doiii1y arranged should fall between these limits, subject to the approximations already noted.ELECTRON MICROGRAPHS OF GELS Over the past few years we have collected electron micrographs of a number of gcls, made by methacrylate embedding and sectioning. Quantitative data from such photographs are obviously suspect. The most readily available parameters are a mean pore size and a mean strand diameter but as fig. 1 shows, these are not casy to obtain. In particular, it is easy to miss small pores, while diiiiensional changes during embedding lead to errors in concentration. Both these effects make i t likely that thc estimated p values are liable to substantial errors. Some results are given in t3ble 2, where I has been calculated froin a suitable aggregation iiiodel a i d from an cstiniatc of dmade directly from the photograph.In view of the approximation used, the agrccnient hetween predicted and estimated value is as good as can be expected, though it is interesting that predicted values are almost always lower than those found. TABLE 2.-cOMPARISON OF MEASURED AND PREDICTED PORE SIZES ex~ullplc c glycinin hcatcd 100" 0.038 i n 4 (x NaCl 0.075 0.1 13 0.15 albumin 0.01 5 arachin 0.15 arachin f 0.15 niensured measured r? d I" P b I C ( P ) 545 100 4.Sx 10'' 229 7 . 6 7 ~ 10'' 180 333 188 2 . 7 ~ 10" 304 1 5 . 1 5 ~ 10" 128 211 134 8 . 0 3 ~ 10'' 176 2 2 . 8 3 ~ 10'' 105 320 100 1 3 . 9 ~ 1 0 ' ~ 115 3 . 0 3 ~ 1 0 ~ ~ 91 330 100 1.89x 10" 363 3 . 0 3 ~ 10'' 287 200 100 1 8 . 9 ~ 10'' 115 3 . 0 3 ~ 10" 91 1000 500 0 . 7 6 ~ 10" 570 - - calculated from c = hr(0.5c/)2 ; 0 from eyn (4) ; C from 3 - 4 ti = 50, CIM = 42 A case in table I in all examplcs ; d:f as shown in fig.1, C , Bt A. The main conc1usion, that the postulated mode of zggregation lias a mzjor quantitative effect on the (pore-size, concentration) relationship, follows from the nature of eqn (3), which appears to be consistent with the limited experimental data, at least to an order of magnitude. Also, it seems reasonable to explain the relatively high concentration required to obtain gels from globular protejns as dependent on such effects. It is perhaps surprising that random aggregations can lead to gel formation, but this follows from the shape distribution curves of limited random aggregates. It also scenis reasonable to suppose that as the aggregates increase in size, the eventual result will be IL population of less favourable shapes and fewer particles which would then form coagulates. Seen in this way, gelation is a particular kind of limited aggregation process.Albumin heated at different temper;itures shows m;irkcdly different modes of aggregation, leading to either gelation or coagulation.' In this case, the interactionM. P. TOMBS 163 alters from a highly oriented one to a less oriented one. Both larger random aggre- gates, and an increase in the randomness of aggregation can lead to a smaller effective strand length, and coagulation rather than gelation. More detailed models of aggregation processes could be constructed, and best pursued in conjunction with gel measurements on a well-characterised protein where some information on surface structure is available. In the relatively rare cases where highly oriented interactions, such as interchain disulphides, predominate it should be possible to test eqn (3) on the random arrangement of strands without the additional complications introduced by a random aggregation to form the strands.I am much indebted to Mr. J. M. Stubbs for electron micrographs and particularly to Mr. L. J. Aspinall for calculations on three-dimensional aggregates, and the appen- dix. M. P. Tombs in Proteins us Human Foods, ed. R. A. Laurie (Butterworths, London, 1970). A. G. Ogston, Trans. Faraday Soc., 1958,54, 1754. J. Fessler, Nature, 1956, 177, 439. APPENDIX COMPUTER SIMULATION OF THE RANDOM CLUMPING OF SPHERES The model used is as follows. An initial specified clump, of touching unit-diameter spheres, is considered to be stationary throughout the simulation.Other unit diameter spheres travelling in straight-line paths collide with the clump, and possibly stick to it. All such collisions are assumed equally likely to result in sticking. The path directions are assumed random, in the following sense. For an arbitrary spherical region, stationary with respect to the initial clump, each successive intersection of this region by the projected path of the centre of a travelling sphere is equally likely to be from any angle in 3-d space ; and for a particular angle the probability distribution of the point of intersection on any circle, which is a projection of the spherical region from this direction, is uniform. The probability of a collision resulting in sticking does not need to be known since this only affects the time-course of clump growth and not the distribution of clump shapes which is of interest.The probability that a new sphere sticks at a particular point on the current clump is proportional to Ez=, (probability that the nth collision is the next collision to result in sticking) x (probability that the nth collision is at this specified point), and hence is propor- tional to the probability that the nth collision is at the specified point which is independent of n. The algorithm adds spheres successively to the initially specified clump until the required size is reached. At each addition, the following steps are carried out. (These exploit the fact that in order for a sphere in the clump to be hit by a travelling sphere, the path of the latter’s centre must intersect in an “extended sphere”, with a diameter of two units concentric with the sphere in the clump.) (1) The equation of a “ containing sphere ” which contains all the ‘‘ extended spheres ” in the current clump is calculated.(2) A random direction in 3-d space is selected. (3) The equations of the circular projections from this direction of the ‘‘ containing sphere ” and the “ extended spheres ” are calculated. (4) A random point is selected in the circle which is the projection of the “ containing sphere.” (5) If this point does not fall in any of the circles which are the projections of the ‘‘ extended spheres ”, a return to step (2) is made. Otherwise, the point of first contact is calculated, and the new sphere is added to the clump. All probability-density functions are generated via transformation of pseudo-random uniform variates Ui in the range (0, 1). Successive values are produced by scaling down integers calculated from The first value is calculated by reference to the clock. Z(n+ 1) = (n)x 16 807 (Mod 2 31)164 GELATION OF GLOBULAR PROTEINS A direction in 3-d space can be defined by the two spherical polar co-ordinates 4,8. In order that the selected directions should have the required distribution values of 4 and 8 are chosen such that (b = 2nu1; 8 = cos-' (1-2u2). A point in a circle can be defined by the polar co-ordinates Y, 8. In order to produce a uniform distribution over the circle, values of r and 8 are chosen such that where R is the radius of the containing sphere. 8 = 2 m 3 ; r = R(u~)*,

ISSN:0301-7249    

DOI:10.1039/DC9745700158

出版商:RSC    

年代:1974

数据来源: RSC