7 Physical Properties of Polymers and their Solutions By J. M. G. COWIE Department of Chemistry University of Stirling Stirling FK9 4LA 1 Physical Chemistry of Polymer Systems Introduction.-It is now four years since the last Annual Report on the physical properties of polymer systems appeared. As that concentrated predominantly on work appearing up to and including 1969 this Report will highlight the main trends since then covering the period 1970-73. No attempt is made to include every reference on the topics covered which will be concerned only with synthetic organic polymers. Polymer Solutions -’Iheory.-‘It is well known that statistical mechanics provides a tool for the description of the relationship between the macroscopic behaviour of substances and/or molecular properties.Clearly the same principles apply to polymer science as to the study of small molecules.’ These words open Hiromi Yamakawa’s book’ on modern polymer solution theory which is undoubtedly the most important treatise on polymer science to appear during the period covered by this Report. Much of its strength lies in the fact that the author is a leading authority in this field and has contributed extensively to the development of the subject. The treatment of polymers in solution is based on two major concepts (i) the recbgnition that a polymer chain can be approximated by a random-flight model and (ii) that in very dilute solutions the polymer molecules are sufficiently well separated to allow the analogy of a ‘gas’ of polymers in a liquid continuum to be drawn.The main theme in the book is a description of a number of theories based on these premises which are now generally classified under the heading ‘the two-parameter theory’. This states that the various properties of dilute polymer solutions may be described in terms of only two fundamental quantities (r2)o the unperturbed mean-square end-to-end distance of the polymer chain and 2 the excluded-volume parameter. In order to demonstrate how these may be formulated Yamakawa begins by dealing with the various distributions and averages used to describe a single flexible linear polymer chain as a function of the number of chain segments n and the mean bond length 1. In the simplest case H. Yamakawa ‘Modern Theory of Polymer Solutions’ Harper and Row New York 1971.173 J. M. G. Cowie the statistical properties of the random-flight chain can be analysed by the Markoff method to yield (r2)r = n12 where (r2)f is the mean-square end-to-end distance of a chain with unrestricted rotation about main-chain bonds. This is an unrealistic description and the restrictions imposed by short-range effects are then .outlined leading to the replacement of 1 by ‘u’,defined as the unperturbed effective bond length and (r2), = nu2 now refers to an unperturbed flexible chain with only short-range interactions operative. In addition this section encompasses the problems of ring branched star and stiff-chain conformations. Since publication further work has appeared on the shape of random-flight chains2v3 and the radius of gyration of ring polymer^.^ The unperturbed chain only exists when the polymer solution is pseudo-ideal -at the theta temperature for the system-and it is much more likely that the polymer will be dissolved in a good solvent which expands the coil.This means that the problem of long-range interactions must be met and this is introduced in the next chapter with a consideration of these long-range effects as embodied in the parameter p. Here p describes an effective volume from which all chain segments are excluded except one. The excluded-volume effect causes chain expansion and hence deviation from the simple Markoff model. To allow for this the dimensions of the expanded chain (r2) can be expressed in the form (r2> = (r’>,a The parameter aR is the linear expansion factor which Flory demonstrated to be a function of n but the exact relation depends on the theoretical approximations.The excluded-volume problem can be dealt with on two levels. The first re- cognizes that perturbation theory can be used directly to derive an expression for the coil dimensions in terms of a power expansion in p which can be applied to polymers dissolved in poor solvents where p is vanishingly small. For polymers dissolved in good solvents the situation is more complex as the excluded-volume effects are large and this requires the introduction of approximations. Much of this part of the book is devoted to the various treatments while concentrating predominantly on the two-parameter concept.Both approaches use the same relation for the excluded-volume parameter Z Z = (+na2)fflnL As fin2 can be deduced from thermodynamic measurements the following chapter is concerned with the thermodynamic behaviour of dilute solutions. This is followed by two extensive chapters on light scattering and the frictional behaviour of polymers before returning to an analysis of experimental data using the two-parameter approach. The latter is ultimately a test of whether or * K. SolE and W. H. Stockmayer J. Chem. Phys. 1971,54 2756. K. Sole J. Chem. Phys. 1971 55 335. K. SolE Macromolecules 1972 5. 705. Physical Properties of Polymers and their Solutions not the various solution properties such as the intrinsic viscosity [q] the second virial coefficient A, and the expansion factor can be reduced to a common dependence on 2 for data derived in different solvents and at different tempera- tures.Here disagreements begin to appear which will be mentioned later but in general the two-parameter theory appears well founded. The use of extrapola-tion methddp to .obtain unperturbed dimensions from data obtained in good solvents is now widely used but the precise form of the relation such as the widely employed Stockmayer-Fixman equation (SF) [q]/M* = KO + C@oBM* depends on the coefficients in the perturbation series a; = 1 + CIZ -... at small 2,where a, is the expansion coefficient derived from [q] measurements. Experimental results show that the original SF equation derived for a series with C = 1.55 led to curved plots at high values of 2.Empirical modifications gave two equations [q]/M*= KO +0.346QOBM* for 0 < a < 1.6 and [q]/M* = 1.05KO+ 0.287@0BM* for 0 < a; < 2.5 which suggests that C should lie in the range 1.05 < C < 1.55.Theoretical justification for these relations has been presented by Yamakawa and Tanaka,' who have obtained a value of C = 1.06 from a rigorous evaluation of the Zimm- Hearst theory. Yamakawa's book contains a wealth of detailed information but for those who wish a more distilled version of the salient features some of the original papers or short reviews may be of use ;in particular those on the excluded-volume effect6 and viral coefficients' are helpful.Subsequent contributions to a theoretical understanding of dilute-solution behaviour have been made by Bloomfield and McKenzie,' who estimated the excluded-volume effect while treatments ofstiff chains without excluded volumeg and the frictional coefficients of coils'o have been published. Improved distribution functions characterizing one- two- and threedimen- sional radii of gyration (S2)* have been presented by Forsman et UI.,"~'~ who formulated the problem of the dependence of (S2) on n in terms of a series of H. Yamakawa and G. Tanaka J. Chem. Phys. 1971,55 3188. 'H. Yamakawa Pure Appl. Chem. 1972,31 179. ' E. Casassa Pure Appl. Chem. 1972 31 151. V. A. Bloomfield and D. S. McKenzie J. Chem. Phys. 1970,52,628. H. Yamakawa and M. Fujii Macromolecules 1973 6 407.lo A. Horta Makromol. Chem. 1972 154 63. " R. Hoffman and W. C. Forsman J. Polymer Sci. Part A-2 Polymer Phys. 1972 10 607. S. K. Gupta and W. C. Forsman Macromolecules 1972 5 779. 176 J. M. G.Cowie convolution integrals. Solution of these leads to results in good agreement with the asymptotic values obtained by Fixman and the final expression indicates a fifth-power dependence. A similar fifth-power dependence of a on n was obtained by Fujita and Norisuye' and by Alexandro~icz~~*' using Monte Carlo techniques. Light-scattering theory has also received some attention. Horta16 has used the boson formulation of Fixman to examine the effect of the excluded volume on the particle-scattering factor P(0) of linear chains; however a lack of ap- propriately good experimental data did not allow an adequate test of the result to be carried out.A companion paper" examined the magnitude of the error involved when a Gaussian approximation is used to calculate P(0)and shows that this could become significant if measurements are made using incident light with short wavelengths or if large coils are being studied. The mean-squared optical anisotropy (7') of a polymer has also come under consideration; Patterson and Flory" have derived a method of calculating (y') from measurements of the depolarized Rayleigh scattering from dilute solutions. The method has been tested for n-alkanes" and oligomers of poly(ethy1ene oxide)' dissolved in optically isotropic liquids. Assignment of appropriate values for the bond optical anisotropies in conjunction with rotational isomeric state theory gave (y2) in good agreement with the experimental results.An extensive series of papers by Huggins outlines a new approach to the interpretation of the thermodynamics of polymer The model adopted is one in which the liquid is composed of chemically uniform segments with average contact surfaces and energies. These remain constant at any given temperature but the relative number of contact areas for each contact type can vary to minimize the Gibbs free energy of the system. Ditonic systems (containing only two types of segment 1 and 2) have been treated and the thermodynamic quantities are derived as a function of these parameters. Thus the enthalpic contribution to the interaction parameter arises from the equilibrium between the various types of contact between the segment surfaces ;the relevant parameters l3 H.Fujita and T. Norisuye J. Chem. Phys. 1970,52 11 15. l4 Z. Alexandrowicz and Y. Accad Macromolecules 1973 6 251. Z. Alexandrowicz Macromolecules 1973 6 255. l6 A. Horta European Polymer J. 1970 6 1253. A. Horta Macromolecules 1970 3 371. G. D. Patterson and P. J. Flory J.C.S. Faraday 11 1972,68 1098. G. D. Patterson and P. J. Flory J.C.S. Faraday ZZ 1972 68 11 11. 2o M. L. Huggins J. Phys. Chem. 1970 74 371. 21 M. L. Huggins Polymer 1971 12 389. 22 M. L. Huggins J. Polymer Sci. Par! C,Polymer Symposia 1971,33 55. 23 M. L. Huggins J. Phys. Chem. 1971,75 1255. 24 M.L. Huggins Macromolecules 1971 4 274. 25 M. L. Huggins J. Paint Technol. 1972 44 55. 26 M. L. Huggins Pure Appl. Chem. 1972 31 245. 27 M. L. Huggins Polymer J. 1973,4 502. M. L. Huggins. Polymer J. 1973 4 511. Physical Properties of Polymers and their Solutions 177 are E~, the energy difference between (1-1) or (2-2) contacts and (1-2) contacts and r,, the ratio of the segment 2 contacting surfaces to that of segment 1. Both must be derived from experimental heats of mixing. The entropic contribution comes from the segment orientational randomness and the non-random distri- bution of intersegment contacts and involves a parameter k,! which must be determined empirically at present. The theory has been applied with some success to selected non-polar systems but it remains to be adequately tested only when more data are available which may enable more precise interpretations of the several new parameters to be made.Polymer Solutious -Experimental.-A number of studies on the hydrodynamic behaviour of homopolymers in dilute solution have been published. These fall into two main categories those concerned with the verification and elaboration of dilute-solution theory and those reporting measurements of solution para- meters such as Mark-Houwink equations unperturbed dimensions characteristic ratios and steric factors (a). The latter group are more numerou~~~-~~ and 29 G. Moraglio G. Gianotti F. Zoppi and U. Bonicelli European Polymer J. 1971 7 303. 30 C. Booth and R. Orme Polymer 1970 11 626.31 R. Jerome and V. Desreux European Polymer J. 1970 6 171. 32 K. S. V. Srinivasan and M. Santappa Polymer 1973 14 5. 33 H. L. Wagner and C. A. J. Hoeve J. Polymer Sci. Polymer Phys. Edn. 1973 11 1 189. 34 A. Cervenka Makromol. Chem. 1973 170 239. 35 V. N. Tsvetkov 1. N. Shtennikova E. I. Rjumtsev and Yu. P. Getmanchuk EurzqGan Polymer J. 197I 7 767. 36 A. H. Fawcett and K. J. Ivin Polymer 1972 13 439. 37 N. Hadjichristidis M. Devaleriola and V. Desreux European Polymer J. 1972 8 1193. 38 M. Tricot and V. Desreux Makromol. Chem. 1971 149 185. 39 J. Stejskal M. J. Benes P. Kratochvil and J. Peska J. Polymer Sci. Polymer Phys. Edn. 1973 11 1803. 40 R. H. Marchessault K. Okamura and C. J. Su Macromolecules 1970 3 735.41 J. Cornibert R. H. Marchessault H. Benoit and G. Weill Macromolecules 1970 3 741. 42 (a) T. Matsumoto N. Nishioka and H. Fujita J. Polymer Sci. Part A-2 Polymer Phys. 1972 10 23; (b)A. Kotera T. Saito Y.Shimoda and N. Onda Reports Progr. Polymer Phys. Japan 1971 14 35. 43 G. Gianotti U. Bonicelli and D. Borghi Mukromol. Chem. 1973 166 235. 44 A. R. Shultz A. L. Bridgman E. M. Hadsell and C. R. McCullough J. Polymer Sci. Part A-2 Polymer Phys. 1972 10 273. 45 G. Moraglio G. Gianotti and U. Bonicelli European Polymer J. 1973 9 623. 46 A. Stokes European Polymer J. 1972 6 719. 47 G. Allen and J. McAinsh European Polymer J. 1970 6 1635. 48 K. Matsumura Polymer J. 1970 1 322. 49 Y. Izumi K. Shinbo N. Kato A. Chiba and Y.Miyake Polymer J. 1973 4 183. 50 Y. Izumi and Y. Miyake Polymer J. 1973 4 205. 51 K. Matsuo and W. H. Stockmayer J. Polymer Sci.,Polymer Phys. Edn. 1973 11 43. 52 M. Pizzoli G. Stea G. Ceccorulli and G. B. Gechele European Polymer J. 1970 6 1219. 53 G. Ceccorulli M. Pizzoli and G. Stea Makromol. Chem. 1971 142 153. 54 G. Ceccorulli and M. Pizzoli Chimica e Industria 1972 54 420. 55 K. Sakato and M. Kurata Polymer J. 1970 1 260. 56 G. Sitaramaiah and D. Jacobs Polymer 1970 11 165. 57 L. A. Utracki Polymer J. 1972 3 551. 58 G. Sitaramaiah and D. Jacobs Makromol. Chem. 1973 164 237. Table 1 Mark-Houwink relations for homopolymers in dilute solution [q] = KM' with concentration units g cm-I 4 00 w* cm g Polymer Solvent Temp./K mol-* V 0 Ref.Polybutene Phenetole 8 = 337.5 11.3 (isotactic) Anisole 6 = 362 11.1 -29 Diphenyl ether 8 = 421 10.3 0.50 Polybutene Phenetole 8 = 334 10.5 (atactic) Anisole 8 = 356 10.8 -29 Diphenyl ether 8 = 414 10.4 0.50 Poly(but-1-ene oxide) Benzene 298 1.59 Hexane 298 1.43 n-Butanol 298 1.96 730 1.71 30 Isopropyl alcohol 8 = 303 11.10 0.50 Poly(t-butyl acrylate) Butanone 298 0.32 Acetone 298 0.47 Methanol 298 1.60 640 2.30 31 Pentane 298 2.20 0.57 Hexane 8 = 297.2 4.90 0.50 Poly(ethy1 acrylate) Acetone 308 4.15 Butanone 308 2.03 673 2.15 32 n-pro pano 8 = 312 7.89 Polyethylene 1-Chloronaphthalene 403 5.55 1,2,4-TrichIorobenzene 403 3.92 106.3 -33 Chlorobenzene 408 4.48 0.7 1 -34 Poly(buty1 isocyanate) Carbon tetrachloride 298 3.16 1.20 -35 % Poly(cyc1ohexene sulphone) Dioxan 298 0.57 -1.24 36 a Benzene 298 1.33 is Poly(cyc1opentene sulphone) Dioxan 298 0.53 0.76 -1.17 36 9 Poly(cyclohexy1 methacrylate) Benzene 298 0.35 0.77 Cyclohexane 298 0.89 9 Dioxan 298 1.20 590 2.50 37 2.Butanone 298 1.23 0.65 n-Butanol 8 = 296 4.46 0.50 Poly(pheny1 methacrylate) Benzene 298 0.57 "a 9- Dioxan 298 0.54 Butanone 298 0.95 670 2.80 Acetone 298 1.49 "a Poly(deca hydro-8-naphthyl -600 2.90 37 4 met hacry late) Poly(B-naphthyl met hacryla te) -3. 660 3.10 37 Poly(4-chlorophenyl metha- Dioxan 298 0.61 h crylate) Benzene 298 0.92 605 2.61-38 % Carbon tetrachloride 298 2.00 0.58 2.75 2 Poly-[ 1-(2-hydroxyethyl)-O.5M-KCI 298 0.40 0.70 -3.10 39 L" trimethylammonium benzene- 3 sulphonate methacrylate] 5 Poly-[ 142-hydroxyethy1)-O.5M-KCl 298 0.26 0.7 1 2.74 39 8 pyridinium sulphonate methacry late] h f Poly(8-h ydroxybutyrate) Chloroform 303 0.77 40 2,2,2-Trifluoroethanol 303 2.5 1 41 3 Polyisobutene Heptane 298 1.63 i? Cyclohexane 298 1.19 -42a L.l Isoamyl isovalerate 8 = 295.1 11.40 0.50 Di-isobutyl ketone 8 = 335.1 -717 1.74 42b trans-Pol yperitanamer Toluene 303 5.2 1 0.69 C yclohexane 303 5.69 99 1 1.26 43 Isoamyl acetate e= 311 23.40 Poly-(2,6-diphenyl-I ,4-Chloroform 298 3.21 phenylene oxide) Benzene 323 1.97 0.64 44 Polypropene Isoamyl acetate e = 307 16.85 (at ac t ic) Isobutyl acetate 0 = 331 15.85 Biphenyl 8 = 402 12.80 0.50 45 Diphenyl ether e = 419 12.56 Pol ypropene Biphenyl 8 = 398 14.1 5 (isotactic) Diphenyl ether e = 416 13.0 0.50 45 L 00 0 Table1 (Contimred) 102K/ 10" cm3 g-9 (r;lM)f Polymer Solvent Temp./K mol-+ V /cm 0 Ref Poly(propene sulphide) Benzene 304 0.50 0.78 -46 Polysulphone A Dimethylformamide 298 0.33 0.64 753 '1 47 Poly(o-chlorostyrene) Butanone e = 298 4.60 0.50 -2.15 48 Pol y(p-chlorostyrene) ---2.29 49.50 Poly(p-fluorast yrene) Carbon tetrachloride e = 298 8.28 Benzene 298 4.05 -51 Chloroform 298 1.61 Butanone 298 1.11 0.73 Pol y(m-fluorostyrene) Carbon tetrachloride 298 6.56 Benzene 298 1.53 -51 Chloroform 298 1.28 0.70 Butanone 298 1.38 Pol y@-methoxyst yrene) Toluene 298 1.05 Toluene 296.4 0.92 Chlorocyclohexane 296.4 1.77 n-Amy1 acetate 296.4 5.50 0.52 636 2.40 52-Methyl isobutyl ketone 8 = 296.4 6.40 0.50 54 t-But ylbenzene e = 325.2 7.40 0.50 Isoamyl acetate 8 = 348.0 6.90 0.50 -Dichlorodecane 8 = 365.6 Poly(a-met hy lstyrene) Benzene 303 1.03 0.72 650 Poly(N-vin ylcarbazole) Benzene 298 3.05 0.58 Cyclohexanone 298 2.00 i:::] Tetrahydrofuran 298 1.44 0.67 619 Chloroform 298 1.36 Tetrachloroethylene 298 1.29 0.68 Poly(viny1 chloride) Mesityl oxide 298 Di-2-ethylhexyl 298 phthalate Diethyl phthalate 1' 577 2.84 58 298 22.0 0.46 0"::; 0.50 Poly-( 3-vinylpyridine) o-Dichlorobenzene 298 0.012 Tetrahydro fur an 298 0.032 Chloroform e = 298 0.051 Bisphenol A polycarbonate -920 1.35 88 182 J.M. G. Cowie pertinent data are summarized in Table 1. These contain a number of 0-solvents for the various polymers which have enabled direct measurement of the un- perturbed dimensions to be made but these have also been estimated in a large number of cases by extrapolation procedures such as the Stockmayer-Fixman method mentioned earlier. Of particular interest are the large values for o obtained for the methacrylate series37 with very bulky ester side-groups poly(N- ~inylcarbazole),~~ and poly-( 3-~inylpyridine).~~ At the other extreme low values of o have been reported for substituted polys~lphones~~ and polysulphone A,47 the latter being approximately unity. The values suggest that these are very flexible polymers ;Allen47 interprets the result as indicating the rigidity of the backbone segment containing the phenylene groups with free rotation about the ether oxygen whereas I~in~~ points out that the low values of o could indicate very similar energy levels for the trans and &gauche rotations about C-S bonds.Steric parameters have also been reported for poly(p-brom~styrene)~~ and polyacrylonitrile.60 Also reported are dilute- solution studies on poly(dimethylsiloxane),6' poly(methylphenylsiloxane),62 poly(fluoroalkoxy-phosphazenes),63and poly(trimethylsiloxanetitanoxanes),64 whose properties can be rationalized if the repeat unit is assumed to contain a cube-shaped Ti,,O, structure. A helix-coil transition was detected in poly@- hydroxybutyrate) in mixed solvent media:' and a relatively rigid rod-like conformation with a fold helical structure has been proposed4' for the polymer in 2,2,2-trifluoroethanol.The conformation of oligomeric poly(propene glycols) has been examined in both polar and non-polar solvents. In water65 and other high-dielectric media66 a tightly coiled conformation is in evidence whose structure is thought to be like an impermeable disc composed of three tightly folded rods. This is believed to be the best model for chains with M -= 1000. Meyerhoff et ~ studied molecular 1.~~9~~ weights up to 4OOO in acetone and benzene and found that gaussian coil behaviour was attained above M = 2000 in these solvents. In acetone A2was negative at low M then rose to a maximum at M = 400 followed by a decrease; this be-haviour has been observed in other polymers.Small-angle X-ray scattering was used to measure coil sizes but these did not agree with the classical theories. Modification of the expression for the frictional coefficient to [f]= 16.4(SZ)f 59 P. Karayannidis and A. Dondos Makromol. Chem. 1971 147 135. 6o K. Kamide and T. Terakawa Makromol. Chem. 1972 155 25. * J. Brezezinski Z. Czlonkowska-Kohutnicka B. Czarnecka and A. Kornas-Calka European Polymer J. 1973 9 733. 62 K. A. Andrianov S. A. Pavlova I. I. Tverdokhlekova N. V. Pertsova and A. Temni-kovskii Vysokomol. Soedinenii (A) 1972 14 18 16. b' G. L. Hagnauer and N. S. Schneider J. Polymer Sci. Parr A-2 Polymer Phys. 1972 10 699. 64 D. E. G. Jones and J. W. Lorimer Polymer 1972 13 265. 65 L.Sandell and D. A. I. Goring Macromolecules 1970 3 50. 66 L. Sandell and D. A. I. Goring Macromolecules 1970 3 54. 67 G. Meyerhoff Makromol. Chem. 1971 145 189. 68 G. Meyerhoff U. Moritz R. G. Kirste and W. Heitz European Polymer J. 1971 7 933. Physical Properties of Polymers and their Solutions 183 gave a better description of the experimental data where the proportionality constant is now larger than that theoretically predicted. One particularly important quantity which can provide significant information to aid elucidation of chain conformation is the temperature coefficient of the mean-square unperturbed dimension expressed as (d In (r2)o/dT). Although important this coefficient is also difficult to measure accurately and agreement between workers is often poor.It can be determined in a number of ways by measuring (r2), in different O-solvents by measuring [q] as a function of temperature and using an extrapolation technique or from thermoelastic measurements. A number of authors have reported such measure-mentS,29,45,50.69-71 which are listed in Table 2. Table 2 Temperature coeficients of unperturbed chain dimensions Method and Temp. lo3dln(r2),/dT Polymer range/K /K-Ref Pol ybutene Several &solvents -0.80 29 (isotactic) 333423 Pol ybutene 293413 -0.40 29 (atactic) Poly(ethy1ene Thermoelastic -0.14 to -0.28 71 oxide) d[ql/d T -0.85 to -1.13 Polypropene Two 8-solven’s -3.0 45 (isotactic) 398-416 Pol y propene Four O-solvents -1.8 45 (atactic) 3074 1 9 Poly(dimethy1-S-F plots from [q] in +0.8 69 siloxane) bromobenzene 353-393 Poly(met hylphenyl- [q]in isopropylcyclohexane -3.0 69 siloxane) 298-348 Poly(b-chloro-Eleven 8-solvents +0.48 50 styrene) 258.3-348.7 Poly(p-met hoxy- Four O-solvents 0 52 styrene) 296.4-365.6 Poly(tetrahydr0-Five &mixtures of -2.0 70 furan) chlorobenzene-octane KO 283-353 Light-scattering measurement + 11.0 70 of (S)& Natural rubber Thermoelastic +0.38 to +0.54 403,404 (cross-linked) The discrepancies obtained when different measuring techniques are used are highlighted in a comparison between the negative values reported for poly- b~tene~~ in a selection of &solvents and the positive values reported by Mark and Flory ”from stress-temperature measurements using amorphous networks.69 G. G. Kartasheva E. G. Erenburg and I. Ya. Poddubnyi Vysokomol. Soedinenii (B) 1972 14 665. lo J. M. Evans and M. B. Hugh European Polymer J. 1970 6 1161. ’’ F. de Candia V. Vittoria U. Bianchi and E. Patrone Macromolecules 1972 5 493. ’* J. E. Mark and P. J. Flory J. Amer. Chem. SOC.,1965. 87 1423. 184 J. M. G.Cowie Even data from the same laboratory may vary as with poly(ethy1ene oxide),’l where thermoelastic and viscosity measurements led to negative coefficients which differed both from each other and quite markedly from the +0.23 x K-I reported by Mark and Fl~ry,~~ again from thermoelastic effects. Considerable criticism has been levelled at the use of the thermoelastic method for measurement of (d In (r2)o/dT) but many workers support the technique and de Candia’s71 results and criticisms have been refuted.74 The situation is still rather confused; good agreement between thermoelasti~~~ and d[q]/dT measurements6’ for poly(dimethylsi1oxane) have been obtained although the values are higher than in earlier work by Ciferri76 and Bianchi et Of course one should not ignore the possibilities of specific solvent effects.de Candia’s7’ results were solvent-dependent but it was suggested that the reasons for the discrepancy between the methods could be due to intermolecular contributions tof (the energy term in the force-temperature measurement) which should have a strictly intramolecular character. Similar intermolecular contributions could come from the creation of supermolecular structures in the swollen network.Alternatively specific solvent effects could appear in measurements of [q]e when a variety of 8-solvents are used. This effect has been carefully examined by Kotera et al.,78 who measured [q]e for polystyrene in a range of 8-solvents which were classified into four homologous groups cycloparaffins esters alcohols and chloroparafhs. The resulting scatter was considerable and gave a range of (dln (r2)o/dT) values between (-0.3 and -2.0) x K-’ . Care in the selection of solvents is also emphasized by Evans and who used mixed solvents to obtain consolute liquids. Their value for the temperature coefficient as obtained from viscosity measurements was in good agreement with the theoretically expected value but direct measurement of the unperturbed dimensions by light scattering gave a vastly different value.In theory the latter method is most direct and should lead to an accurate evaluation of (d In (r2)o/dT) but in this system there was a significant preferential adsorption effect which apparently led to considerable coil expansion over that of the unperturbed state. These authors79 have also presented a method of analysing the variations in In [q] which allows evaluation of both 8 temperature and (d In (r2)o/dT). The dependence of (r2)if on solvent has been discussed by Dondos and Benoit,” who have found that polar solvents in particular can alter dimensions. This has been reported also for poly(viny1 chloride).81 For ternary systems they have obtained a distinct relationship between KO and the excess free energy of mixing AGE for the binary liquid mixture.In general KO (and hence the coil 73 J. E. Mark and P. J. Flory J. Amer. Chem. Soc. 1965 87 1415. 74 J. E. Mark and P. J. Flory Macromolecules 1973 6 300. 75 J. E. Mark and P. J. Flory J. Amer. Chem. Soc. 1964 86 138. 76 A. Ciferri Trans. Faraday SOC. 1961 57 853. 77 U. Bianchi E. Patrone and M. Dalpiaz Makromol. Chem. 1965,84 230. ’I3 A. Kotera. T. Saito N. Yamaguchi K. Yoshizaki Y. Yanagisawa and H. Tsuchiya Reports Progr. Polymer Phys. Japan 1971 14 31. 79 J. M. Evans M. B. Hugh and R. F. T. Stepto Mukromol. Chem. 1971 146 91. A. Dondos and H. Benoit Macromolecules 1971 4 279. L. A. Utracki Polymer J. 1972 3 551.. Physical Properties of Polymers and their Solutions 185 dimensions) is larger than expected when AGE is positive and smaller when AGE is negative. Again reports tend to vary ;mixed 8-solvents of increasing polarity toluene-octane (1 1.4) CC1,-MeOH (5.3 I) benzene-MeOH (4.7 l) CHC1,-Pr‘OH (1.39 l) and dichloroethylene-acetone (1.4 1) had no reported effect on the unperturbed dimensions of polychloroprene.82 The effect of thermo- dynamic interactions on the coil size of poly(p-ch10rostyrene)~~ showed little variation when the polymer was dissolved in either an exothermic or endothermic solvent. A further complication can arise in certain instances when the solvent may aid a conformational change in the polymer as the temperature is increased.Thus conformational changes have been observed to occur in poly(p-bromosty- and poly(methy1 metha~rylate)~~’ rene),830,bpoly-(2-~inylpyridine),~~’ when dissolved in benzene dioxan acetone or THF. The conformational transition could be suppressed by addition of a solvating polar solvent which apparently maintained the chain rigidity as the temperature was raised. This suggests that the changes involved only short-range interactions perhaps indicating the presence of some helical sequences which would be stabilized by solvents such as chloroform or DMF. A related study on the behaviour of poly(methy1 metha- ~rylate)~~ in mixed solvent media showed that a conformational change could be induced by altering the solvent composition. Unperturbed dimensions have also been reported for p~lyacrylates,~’ and poly(isobuty1 methacrylate),86 which was found to be less extended than the related n-butyl ester.The temperature dependence of the unperturbed dimen- sions of a series of polyesters poly(tetramethy1ene adipate) poly(octamethy1ene sebacate) and poly(&-caprolactone) showed that an increase in the methylene content of the chain actually led to an increase in the characteristic parameter.87 A second group of papers concerned primarily with testing dilute-solution theory have examined the hydrodynamic behaviour of narrow-distribution polymer samples in great detail. One of the open questions to be answered is what is the asymptotic behaviour of the expansion factor a? Here there is some disagreement.The various research groups interested in these problems find that their data agree at low a but diverge at large a when measurements are made in good solvents. Two groups of opinion exist and the major points of difference are (i) the value of the interpenetration function Y = A2M2/4n*N,(S2)*; (ii) whether or not a is uniquely dependent on 2. Group I believe that Y reaches an asymptotic value between 0.25 and 0.3 and that a depends only on 2,whereas group I1 82 A. V. Gevorkian and L. Kh. Simonyan Uch. Zap. Erevan. Univ. Estestu Nauk. 1971 No. 2 47. 83 (a)P. Karayannidis and A. Dondos Makromol. Chem. 1971 147 135; (6) A. Dondos P. Rempp and H. Benoit ibid. 1973 171 135. 84 A. Dondos Makromol Chem. 1972 162 113. ” S. A. Pavlova G.I. Timofeeva and V. M. Men’shov J. Polymer Sci.,Part C Polymer Symposia 1972 39 I 13. sf M. M. Zafar R. Mahmood and S. Wadooi Makromol. Chem. 1972 160 313. ” M. R. Knecht and H. G. Elias Makromol. Chem. 1972 157 1. W. R. Moore and M. A. Uddin European Polymer J. 1970 6 121. 186 J. M. G.Cowie suggest that the value is lower Y < 0.2 and that data cannot be reduced to a common dependence of a on 2. A number of papers support the group I view with studies using poly(p-bromostyrene),8g~go p~ly(p-methylstyrene),~'poly-~tyrene,~~,~~ and polyisobutene,42 where Y was always greater than 0.25 for large a. Lower values of Y were obtained by Kato et a1.94395using monodisperse poly(a-methylstyrene) but these results are regarded by Yamakawa as peculiar to Kato's samples.This criticism is hardly justified and more constructive reasons can be given centred on the different methods of data analysis which can be employed. The method of calculating coil sizes differs; for example group I uses square-root plots from the light-scattering measurements whereas group I1 select (S2) to gain best agreement between the experimental and theoretical particle scattering functions. The calculation of 2is also open to variation. The original work by Berryg6 on polystyrene only gave partial support to the two- parameter theory but recalculation of his data improved the agreement sig- nificantly. The recalculation involved the method in which Z was derived. This parameter depends on the binary cluster integral P and this cannot be estimated directly from experiment.Berry assumed a linear relation of the form B = BOU -6/77 where Po is a temperature-independent constant. Calculation of Po from data at or near the 6 temperature allows fi and hence 2 to be obtained over a limited temperature range. This linear relationship can hardly be valid over a wide range however if one accepts the existence of a lower critical solution temperature (LCST) in the system which would necessitate a parabolic form for the P-T relation. This was considered by Ei~hinger,'~ who obtained a first-order correction from the value of the LCST and corrected the over-estimated values of 2 reported by Berry. This brought the data more into line with the Yamakawa group. A recent paper by Nakatag8 outlines an alterna- tive method for calculating P (expressed in the form B = P/m2,where m is the molecular weight of a segment) which is founded on a semi-empirical equation of Stockmayer.Here Bo is obtained from the intercept of a plot of (1 -6/T)/A against M2(1 -6/T)/(S2)*,but again a linear form of the B-T relation is assumed. One additional point of interest is that Nakata has derived an ap- proximate closed form for the Yamakawa-Tanaka' perturbation expansion which is a5 -= 1.9142 89 Y.Noguchi A. Aoki G. Tanaka and H. Yamakawa J. Chem. Phys. 1970,52 2651. 90 K. Takashima G. Tanaka and H. Yamakawa Polymer J. 1971 2,245. 91 G. Tanaka S. Imai and H. Yamakawa J. Chem. Phys. 1970,52,2639. 92 A. Yamamoto M. Fujii G. Tanaka and H. Yamakawa Polymer J.1971,2,799. 93 M. Nakata Makromol. Chem. 1971 149 99. 94 T. Kato K. Miyaso I. Noda T. Fujimoto and M. Nagasawa Macromolecules 1970 3 777. 95 1. Noda K. Mizutani T. Kato T. Fujimoto and M. Nagasawa Macromolecules 1970 3 787. 9h G. C. Berry J. Chem. Phys. 1966,44,4550; 1967,46 1338. 97 B. E. Eichinger J. Chem. Phys. 1970 53 561. M. Nakata Makromol. Chem. 1973 167 273. Physical Properties of Polymers and their Solutions This is again a fifth-power dependence of 01 on 2,and at present the consensus of opinion favours this form. Additional experimental support for the Yamakawa- Tanaka relations comes from Bohdanecky and Petrus," who examined the excluded-volume effect in [q] from measurements near the &temperature for a number of systems.The equation derived to estimate C1was (d In [t,~]/dT),-(d In K$dT) = C,(4nbN,)-'((S2)o/M)-' x (dA,/dT),Mf A value of C z 1.06 was obtained in good agreement with Yamakawa and Tanaka. Although the evidence tends to support the validity of the two-parameter approach and the Yamakawa school of thought experimental limitations may keep this an open question for some time. The values quoted by Kato et al.94 for coil sizes do differ slightly from those reported earlier,"' and improved mea- surement of (S') could help in data analysis. This could be achieved by carrying out light-scattering measurements at much lower angles and a specially designed instrument for this purpose has been described."' Methods have also been proposed for more accurate treatment of light-scattering data102"~b and for measuring coil sizes down to about 10 .,.Io3 The application of heterogeneity corrections,' 04*'O5 as applied to extrapolation techniques used to calculate the unperturbed dimensions has been described based on a Zimm-Schulz distribu- tion function.Such improvements may still not be enough as an extensive study of butyl rubber in seven failed to produce conclusive evidence concerning the asymptotic behaviour of Y in spite of a wealth of good data. Indeed one cannot help wondering whether solvent effects play a bigger part than anticipated in such studies and to what extent the reported conformation change in p~ly(p-bromostyrene)*~" affects the study reported by Takashima et aLgO Star-and Combbranched Polymers.-The success of the two-parameter theory in providing a framework within which a unified treatment of dilute-solution para- meters is possible might tend to lead to a complacent attitude.One might well consider that the existing points of difference are relatively minor but this is rapidly dispelled when branched-chain structures are considered. It soon becomes obvious that the existing theory is limited to linear polymers and cannot account for the dilute-solution behaviour of non-linear structures. 9q M. Bohdanecky and V. Petrus European Polymer J. 1972,8 893. loo J. M. G. Cowie S. Bywater and D. J. Worsfold Polymer 1967 8 105. I0' H. Utiyama and Y. Tsunashima Appl. Optics 1970 9 1330. Io2 (a) H. Fujita Polymer J.1970 1 537; (b) W. Miller and R. F. T. Stepto European Polymer J. 1971 7 65. Io3 I. N. Serdyuk and S. K. Grenader J. Polymer Sci. Part B Polymer Letters 1972 10 241. Io4 W. Sutter and A. Kuppel Makromol. Chem. 1971 149 271. lo' R. E. Bareiss Makromol. Chem. 1973 170 251. '06 J.-G. Zilliox Makromol. Chem. 1972 156 121. J. M. G.Cowie For branched and star-shaped polymers differences immediately emerge (i) the &temperature is lower than for the linear polymer (ii) coil expansion is decreased by branching and (iii) the radius of gyration of branched and star structures is considerably larger than calculated from random-flight statistics. Systematic studies are complicated by the increased number of variables; in addition to differences in backbone chain length both the length and frequency of the branches can be altered and will affect the solution behaviour.Several excellent papers have been published by the Strasbourg grouplo6-' lo in which star-and comb-branched polystyrenes were examined by light scattering viscosity and diffusion in good and poor solvents. Zilliox'06 observed a decrease in &temperature which was a function of the branch length. Candau and Rempp'" found that the geometric dimensions of the polymer increased with the frequency and length of the branches and that the coil expansion depended on the number but not the length of the branches. The lowering of the 8-temperature' ' was studied for branched polystyrene and branched polyisoprene where it was found that the &temperatures measured by two techniques namely A and precipitation were always lower than for the linear homologues and decreased with increase in branching frequency.As the segment density is always much higher in the branched polymers the idea of multiple contacts has been used to modify the classical dilute-solution theory. A general equation for the calculation of the unperturbed dimensions of comb polymers has been developed by Tung' ' ' and the Flory-Orofino theory has been extendedlog to include three terms in the series expansion of the segment interaction coefficients which results in a further term in the expansion describing the thermodynamic functions. This new term depends on the third power of the segment concentration and is finite when A = 0.Consequently 8-conditions defined in terms of A = 0 have no physical meaning in these systems and in addition Candau et suggest that branched molecules are also non-gaussian under the conventional &condition defined by a linear expansion factor of unity. Similar conclusions have been reached by other workers.' 12-' l4 Both melt and solution viscosities of star and branched polystyrenes have been examined.' ''-' Four- and six-branched star polystyrenes had lower limiting lo' F. Candau P. Rempp and H. Benoit Compt. rend. 1971 273 C 1733. lo' F. Candau and P. Rempp European Polymer J. 1972 8 757. Io9 F. Candau P. Rempp and H. Benoit Macromolecules 1972 5 627. ll0 F. Candau C. S. Strazielle and H. Benoit Mukromol. Chem. 1973 170 165.L. H. Tung J. Polymer Sci. Polymer Phys. Edn. 1973 11 1247. IL2 G. C. Berry J. Polymer Sci.,Part A-2 Polymer Phys. 1971 9 687. IL3 J.-C. Meunier and R. G. van Leemput Mukromol. Chem. 1971 147 191. lL4 J. Pannell Polymer 1971 12 558. J. E. L. Roovers and S. Bywater Macromolecules 1972 5 384. IL6 M. Kurata M. Abe M. Iwana and M. Matsushima Polymer J. 1972 3 729. IL7 L. A. Utracki and J. E. L. Roovers Macromolecules 1973 6 366. 11' I. Noda T. Horikawa T. Kato T. Fujimoto and M. Nagasawa Macromolecules 1970 3 795. l9 K. Kamada and H. Sato Polymer J. 1971 2,489. lZo K. Kamada and H. Sato Polymer J. 1971 2 593. IZ1 J. Pannell Polymer 1972 13 2. Iz2 T. W. Bates European Polymer J. 1972 8 19. Iz3 T. Fujimoto H. Narukawa and M. Nagasawa Macromolecules 1970 3 37.Physical Properties of Polymers and their Solutions 189 viscosity numbers [& than the corresponding linear polymers [?,?I1 and the ratio [&/[v]l was larger in poor solvents than in good.115 The corresponding ratio for the sedimentation constants was higher than expected for star polymers' l5 but close to the theoretical prediction for randomly branched structures. l6 Similar results were obtained for randomly branched poly(methy1 metha- crylate),' 19,1 2o where the Zimm-Kilb theory' 24 for the viscosity of branched polymers could be applied near the &temperature although results were less conclusive in good solvents. This can be contrasted with data for comb- branched polystyrene which were in better agreement with the Thurmond-Zimm theory.'25 Melt viscosities were also found to be dependent on branch frequency and structure.'21-'25 These studies on branched-chain structures suggest that solution theory may require a complete rethinking to broaden the coverage and encompass both linear and branched chains.Quasielastic Laser-light Scattering.-The frictional properties of polymers in solution have received somewhat less attention. The concentration dependence of the sedimentation constant in the region of the &temperature is found to be in accord with the Pyun-Fixman theory for p~lystyrene'~~.'~' and poly(a- methylstyrene).'28 Sedimentation measurements have also been used to detect long-chain branching in poly(viny1 chloride).' 29 Conventional diffusion measure- ments have been rep~rted'~~-'~~ which were concerned mainly with the con- centration dependence but renewed interest in diffusion measurements is being stimulated by the development of a relatively new technique quasi-elastic laser- light scattering.The principles of light scattering are well established. When a polymer solution is irradiated by light from a monochromatic source the concentration and density fluctuations in the system arising from Brownian motion cause light to be scattered. As the scattering is from polymers which are undergoing continual random motion a Doppler shift will be observed. Consequently the light scattered will have a frequency distribution differing from that of the incident beam. This spectrum of frequencies consists of a Rayleigh line centrally located about the incident frequency and two symmetrical Brillouin wings one on either side.Iz4 B. H. Zimm and R. W. Kilb J. Polymer Sci.,1959 37 19. IZs C. D. Thurmond and B. H. Zimm J. Polymer Sci. 1952 8,477. 12' T. Tsuja and H. Fujita Polymer J. 1973 4 409. V. Petrus I. Danihel and M. Bohdanecky European Polymer J. 1971 7 143. A. Kotera T. Saito and T. Hamada Polymer J. 1972 3 421. W. I. Bengough and G. F. Grant European Polymer J. 1971 7 203. C. J. Vadovic and C. P. Colver J. Polymer Sci. Polymer Phys. Edn. 1973 11 389. A. Kotera T. Saito H. Matsuda and K. Takemura Reports Progr. Polymer Phys. Japan 1971 14 39. 13' A. Kotera N. Yamaguchi K. Takemura and N. Takahashi Reports Progr. Polymer Phys. Japan 1972 15,63.133 A. Kotera and H. Matsuda Reports Progr. Polymer Phys. Japan 1972 15 67. 134 B. Porsch and M. Kubin European Polymer J. 1973 9 1013. 190 J. M. G.Cowie The width of the frequency distribution of the Rayleigh peak can be related to the causal effects of polymer chain motion. According to theory135 the shape of the Rayleigh line should have a continuous Lorentzian distribution emanating from the translational diffusion of the molecules. The frequency distribution can be expressed as 0 const.(p//lr) Z(0) = (0-Oo)2 + p' where 16n2fi2DTsin2 (8/2) B= 1 and liis the solvent refractive index DTthe translational diffusion constant 8 the scattering angle and lothe wavelength of the incident beam of incident angular frequency coo.Thus if the broadening of the Rayleigh line can be measured information on the diffusion constant and possibly intramolecular chain motions can be obtained. This is technically rather difficult as the Doppler shift is usually very small for polymer solutions but by using a laser light source with narrow band widths and improved techniques in optical heterodyne and homodyne (self beat) spectroscopy meaningful results can be obtained. The methods have been extensively reviewed by C~U,'~~ Peticolas,' 37 and recently by Jamieson and Maret,'38 among others. A considerable amount of work has been published on biological macromolecules but synthetics have also been studied. Reed and Frederi~k'~'analysed the spectra for low- and high-molecular-weight polystyrene in cyclohexane and found increasing deviations from a Lorentzian distribution with increasing chain length.It was suggested that these deviations were due to the presence of contributions from other relaxation processes such as rotary diffusion and intramolecular motion but while the actual interpretation was inconclusive it did show that the calculation of DTmay be complicated by secondary effects. A further study of the problem using polystyrene in butan-2- showed that highly accurate measurements are necessary to detect the extent of the deviations even at high molecular weight. Latex spheresI4' or dilute ludox solutions'42 have been used for calibration purposes and theoretical expressions for the spectrum expected for polydisperse latex spheres have been derived.143a9b The molecular weight and concentration 13' R. Pecora J. Chem. Phys. 1968,49 1032. 136 B. Chu Ann. Rev. Phys. Chem. 1970 145. 13' W. L. Peticolas Adv. Polymer Sci. 1972 9 285. 138 A. M. Jamieson and A. R. Maret Chem. SOC. Rev. 1973,2 325. 139 T. F. Reed and J. E. Frederick Macromolecules 1971 4 72. 140 0. Kramer and J. E. Frederick Macromolecules 1972 5 69. S. P. Lee W. T. Scharnuter and B. Chu J. Polymer Sci.,Part A-2 Polymer Phys. 1972 10 2453. 14' D. B. Sellen Polymer 1970 11 374. 143 (a) D. S. Thompson J. Chem. Phys. 1971 54 1411; (b) D. S. Thompson J. Phys. Chem. 1971,75 789. Physical Properties of Polymers and their Solutions 191 dependence of DT have been reported for polystyrene'u with an accuracy of f3 % while the importance of removing dust from solutions'45 has also been emphasized.King and his c~-workers'~~-~~~ ha ve examined the concentration dependence of DT for polystyrene in good and poor solvents expressed in the form DT(c)= Di(1 + k,~+ k;c2 + .. .) and compared their results with the theories of Pyun-Fixman Yamakawa and Imai. In &solvents none of the theories describe ki which was always negative for cyclohexane solutions but there was a trend towards the Pyun-Fixman theory at low molecular weights. In butan-2-one k was negative at low M tending towards zero at about M = 3 x lo' and again agreement between theory and experiment was disappointing. Different diffusional modes,' 'O which are dependent on the ionic strength have been detected in solutions of partially hydrolysed polyacrylamide.Polydispersity effects have also been calculated for coils' 51*1'2 and rods,'53 together with the distribution functions for both homodyning and heterodyning techniques. It has been shown'" that D' values calculated from heterodyning spectra are close to 2-average values but that the spectral shape is relatively insensitive to the sample molecular-weight distribution. Form factors have been derived for both flexible and stiff linear chains.lS4 The Rayleigh spectra can be used to calculate M,,but the value was found to depend on the distribution function chosen for the calculation.' 55 Alternatively one can measure the Brillouin scattering and compare the ratio J of the intensity of the Rayleigh peak with the sum of the intensities of the side peaks.This is described by Carpenter et ~1.l'~ and the form of the equation is BKc (-1 ~ = + 2A2c + 3A,c2 + . (J -J,) M where Jo is the value of J for pure solvent c is the polymer concentration and the constants are B and K. Results were in good agreement with other methods. In general the technique of laser-light scattering has many advantages. Direct measurement of D' at a given concentration does not involve extensive extrapola- tion and only requires small amounts of material. Measurements are rapid and 144 N. C. Ford F. E. Karasz and J. E. M. Owen Discuss. Faraday SOC.,1970 No. 49 p. 228. 145 0. Kramer and J. E. Frederick Macromolecules 1971,4 613. 14' T.A. King and W. I. Lee J. Phys. (E) 1972 5 1091. 14' T. A. King A. Knox W. 1. Lee and J. D. G. McAdam Polymer 1973 14 151. 148 T. A. King A. Knox and J. D. G. McAdam Polymer 1973 14,293. 14' T. A. King A. Knox and J. D. G. McAdam Chem. Phys. Letters 1973,19 351. A. Jamieson and C. T. Presley Macromolecules 1973. 6 358. "' J. E. Frederick T. F. Reed and 0.Kramer Macromolecules 1971 4 242. W.-N. Huang E. Vrancken and J. E. Frederick Macromolecules 1973 6 58. H. Maeda and N. Saito Polymer J. 1973,4 309. S. Fujime and M. Maruyama Macromolecules 1973 6 237. 155 T. F. Reed Macromolecules 1972 5 771. G. A. Miller F. I. San Filippo and D. K. Carpenter Macromolecules 1970 3 125. 192 J. M. G. Cowie independent of the magnitude of DT,thereby enabling slowly diffusing species to be studied with reasonable accuracy.Although there are still problems as- sociated with data analysis the method has other applications yet to be explored and areas of study such as aggregation effect^,'^^-'^^ the kinetics of rapid equilibria and macromolecular reactions and various other relaxation pheno- mena merit immediate investigation. Polymer Solutions -Thermodynamics.-The majority of workers exploring the field of polymer solution thermodynamics are concerned with measurements made in dilute solution. A large number have been published by Maron and his co-workers'60 in which the systematic analysis of a variety of systems using the Maron theory,16' has been carried out. These are predominantly heat of dilution measurements but other measurements mainly for polystyrene' 62 or ethylene have been reported.The interdependence of the enthalpic and entropic contributions to the second virial coefficient was examined by Lechner and S~hulz,'~~ who showed that the data should be converted to reduced quantities if a realistic comparison was to be made between polymers with widely differing chain flexibility and geometries. It was found that the reduced values of A2 and A2,swere predominantly functions of the enthalpy contribution. Wolf' compared the temperature and pressure dependence of these quantities with the predictions of the Prigogine correspond- ing states theory,' 66 but only qualitative agreement was obtained. At the other end of the concentration scale gas-liquid chromatography (g.1.c.) has been applied to the measurement of thermodynamic data at very high polymer concentration^.'^^-'^^ The polymer under study was used as the V.A. Bloomfield and J. A. Benbasat Macromolecules 1971 4 609. I 58 J. A. Benbasat and V. A. Bloomfield Macromolecules 1973 6 593. I 59 S. Fujime Macromolecules 1973 6 36 1 160 (a) C. A. Daniels S. H. Maron and P. J. Livesey J. Macromof. Sci. 1970 B4,47; (b)S. H. Maron and F. E. Filisko ibid. 1972 B6,79; (c)S. H. Maron and F. E. Filisko ibid. p. 41 3 ;(6)S. H. Maron and F. E. Filisko ibid. p. 57; (e)S. H. Maron and M.4. Lee ibid. 1973 B7 29; U,S. H. Maron and M.4. Lee ibid. p. 47; (g) S. H. Maron and M.-S. Lee ibid. p. 61 ;(h)C. A. Daniels and S. H. Maron ibid. 1972 B6,1. 16' S.H. Maron J. Polymer Sci. 1959 38 329. 162 (a)G. Lewis and A. F. Johnson Polymer 1970,11 336; (b)S. Morimoto Bull. Chem. SOC.Japan 1971 44 879; (c) K. Tamura S. Murakami and R. Fujishiro Polymer 1973 14 237. Ib3 A. Kagemoto Y. Itoi Y. Baba and R. Fujishiro Makromof. Chem. 1971 150 255. 164 (a) M. D. Lechner and G. V. Schulz European Pofymer J. 1973 9 723; (6) M. D. Lechner and G. V. Schulz Makromol. Chem. 1973 172 161. 16' B. A. Wolf J. Polymer Sci. Part A-2 Polymer Phys. 1972 10 847. Ib6 I. Prigogine (with the collaboration of A. Bellemans and V. Mathot) 'The Molecular Theory of Solutions' North Holland Amsterdam and Interscience New York 1957. 16' (a) D. Patterson Y. B. Tewari H. P. Schreiber and J. E. Guillet Macromolecules 1971,4 356; (b)W.R. Summers Y. B. Tewari and H. P. Schreiber ibid. 1972 5 12; (c) Y. B. Tewari and H. P. Schreiber ibid. p. 329; (6)H. P. Schreiber Y. B. Tewari and D. Patterson J. Polymer Sci. Polymer Phys. Edn. 1973 11 15. 16* W. E. Hammers and C. L. de Ligny Rec. Trav. chim. 1971,90 912. (a)N. F. Brockmeier R. W. McCoy and J. A. Meyer Macromolecules 1972 5 130; (b)N. F. Brockmeier R.W. McCoy and J. A. Meyer ibid. p. 464. Physical Properties of Polymers and their Solutions stationary phase and the solvents were introduced into the carrier gas. Retention times can then be related to the interaction parameter xfor very concentrated solutions. The method described by Schreiber et ~1.'~~ yields activity coefficients and vapour-polymer equilibrium ratios at infinite dilution both rapidly and accurately.A variation'69 of the method allows xto be estimated as a function of solution composition in the range 0-10 % solvent in polymer. This is achieved by using higher concentrations of solvent (up to 60molX) in the carrier gas. The method has been used successfully to estimate xfor-systems composed of hydrocarbon solvents and rubber '67c p~lyethylene,'~~~*'~~~ polyisobutene,'68 polystyrene,' 69aand poly(dimethylsiloxane).'67b This technique is a useful addition to thermodynamic studies but perhaps the most significant contribution in the past ten years to the interpretation of the thermodynamics of polymer solutions is the development of the Flory-Prigogine corresponding states theory where the Flory theory'70 is a particular case of the more general Prigogine approach.'66 One particularly attractive feature of the theory is that it is a three-parameter treatment in which the particular importance of the free volumes of the components of the solution is emphasized. The new factor which accounts for the disparity in size between a polymer chain and a solvent molecule i.e. 'the free volume dissimilarity' has a significant influence on the thermodynamic properties of the solution. Also introduced by Flory is the concept of segment surface areas or sites which is similar to the ideas of Huggins mentioned earlier. The various terms in the theory can be estimated from a knowledge of structural factors and the equation of state parameters but for the exchange interaction parameter X,,a value must be assigned arbi- trarily.The theory has been tested17' using experimental data derived from calori- metry osmotic pressure and vapour sorption measurements. Enthalpies of mixing have been determined for polyisobutene in hydrocarbon solvents,' 72 polylactones,' 73 poly(ethy1ene oxide) and poly(propene oxide),' 74 and polybut- 1-ene.'75 In all cases the theory failed to predict the enthalpy change in a quanti- tative manner. Adjustment of X12could improve this but the value was often unrealistic.' 72 Large negative values of X12were obtained for both poly(ethy1ene oxide) and poly(propene oxide) which were attributed to specific interactions between polymer and solvent possibly of a charge-transfer nature. The observa- tion that X,,passed through a minimum with increasing solvent (n-alkane)17' chain length was attributed to loss of order in the higher alkanes.Poor agreement 170 (a)P. J. Flory. J. Amer. Chem. SOC.,1965 87 1833; (b)P. J. Flory J. L. Ellenson and B. E. Eichinger. Macromolecules 1968 1 279; (c) P.J. Flory Discuss. Furuduy SOC. 1970 No.49 p. 7. 17' (a)P. J. Flory and H. Hocker Trans. Furaday Soc. 1971 67 2251 ;(b)P. J. Flory and H. Hocker ibid. p. 2258; (c)H. Hocker H. Shih and P. J. Flory ibid. p. 2270. 17* A. H. Liddell and F. L. Swinton Discuss. Faraduy Suc.. 1970 No.49 p. 115. 17' G. Manzini and V. Crescenzi Polymer 1973 14 343. '74 (a)C. Booth and C. J. Devoy Polymer 1971 12 309; (6)C. Booth and C. J. Devoy ibid. p. 320. 17' G. Delmas and P. Tancrede European Polymer J.1973 9 199. J. M. G. Cowie was also found for entropy of dilution and the excess-volume parameter'76 in poly(dimethylsi1oxane) solutions. As the equation of state parameters assume an important role in these calculations these have received attention.' 77 Interest has also beer) focused on the concentration dependence of the interaction para- meter x1.17' Okazawa and Kanek~'~'' observed that the temperature depen- dence was in agreement with the predictions of the Flory-Prigogine theory and used the data to predict LCST for polyisobutene-n-alkane solutions but the quantitative agreement was rather poor. In general the agreement between theory and experiment is bad for polymer solutions; in fact it is poor even for smaller molecules'79*'80 and aspects of the failure of the corresponding states theory have been discussed.Table 3 LCST for quasi-binary solutions at M = co LCS T/K Polymer Solvent (M = 00) $1 Ref Polyethylene n-Pentane -353 n-Hexane 406.3 181a n-Heptane 446.9 -1.2 2;i n-Octane 483.0 -1.1 Pol ypropene Diethyl ether 383 -0.24 (at ac t ic) n-Pentane 397 -0.30 181b n-Hexane 441 -0.33 1 n-Heptane 48 3 -0.43 Polystyrene Cyclohexane 486 -1.19 Methylcyclohexane 484 -0.94 1 Toluene 550 -1.92 Benzene 523 -1.79 Butan-2-one 422 -0.5921 Cyclopentane 427 -0.858 Butan-2-one 417.9 Benzene" 514 Et h y1benzene" 568 Poly(p-chlorostyrene) Isopropyl acetate 348.7 -0.360 t-Butyl acetate 338.4 -0.323 n-Pentyl alcohol 323.1 -0.489 1 n-Propyl alcohol 300.8 -0.696 Poly(a-methylstyrene) Cyclohexane 456 Methylcyclohexane 434 181h Butyl chloride 412 Measured at finite molecular weight.' P. J. Flory and H. Shih Macromolecules 1972 5 761. "' (a) H. Shih and P. J. Flory Macromolecules 1972 5 758; (b)H. Hocker G. J. Blake and P. J. Flory Trans. Faraday SOC.,1971 67,2251 ; (c) Y. Tsujita T. Nose and T. Hata Polymer J. 1972 3 581 ;(d)D. A. Isacescu I. V. Ionescu M. Leca C. Roncca D. Mihaita and E. Rizescu European Polymer J. 1971 7,913. (a) H. Vink European Polymer J. 1971 7,141 1; (6)A. Muramoto Polymer J. 1970 1 450; (c) T. Okazawa and M. Kaneko ibid. 1971 2 747; (6)J. Pouchly and D. Patterson Macromolecules 1973 6,465. D.W.Dreifus and D. Patterson. Trans. Faraday Soc..1971 67 631. IB0W. E. Hammers C. L. de Ligny and L. H. Vaas J. Polymer Sci. Polymer Phys. Edn. 1973 11 499. Physical Properties of Polymers and their Solutions A major drawback is the observation that the characteristic parameters derived for the components vary with temperature instead of remaining constant. This can be overcome to some extent by a judicious choice of an appropriate reference temperature. It should be stressed however that the tnost encouraging feature is that qualitative agreement is obtained and many of the inadequacies of the classical theory such as the occurrence of volume changes on mixing and the existence of an LCST are accounted for if somewhat crudely. Although still in a rudimentary state the qualitative agreement achieved by the theory should encourage refinements to the proposed model which may lead to improvement.In this respect the investigation of lower critical solution phenomena could prove helpful. Phase Equilibria and the LCST.-The existence of ad LCST is now accepted as being a general phenomenon of polymer solutions and can be explained in a satisfactory qualitative manner by the three-parameter Flory-Prigogine theory. Some attempts have been made to obtain a quantitative prediction of the LCST. Flory extrapolated equation-of-state data for polystyrene to temperatures in the vicinity of the LCST and calculated OLCST = 474 K for cyclohexane solutions.171c This is in reasonable agreement with that quoted in Table 3 where the results from a number of sources are summarized.Good agreement was also obtained for polyethylene solutionslgl" when the Flory equation was used but this was not so impressive when the Patterson-Prigogine equation was employed mainly owing to the neglect of the interactior) parameter. The molecular weight dependence of LCST has been expressed as by Patterson and Delmas,lg2 where r is the ratio of the molar volumes of polymer and solvent c1 is determined by the number of external degrees of freedom of the solvent molecule 7 is a parameter reflecting the free-volume change vz is a term characterizing molecular differences between solvent and polymer seg-ments and vl is the reduced volume of the solvent. The spread between the upper and lower critical solution temperatures and their variation with molecular weight have been successfully described for a number of systems qualitati~ely.'~''~~ To obtain absolute agreement it is usually necessary to employ a temperature-shift factor but recent work suggests '*I (a)F.Hamada K. Fujisawa and A. Nakajima. Pol-vmer J. 1973 4 316; (b)J. M. G. Cowie and I. J. McEwen J. Polymer Sci. Polymer Phys. Edn. 1974 It 441 ;(c)S.Saeki N. Kuwahara S. Konno and M. Kaneko Macromolecules 1973 6,246; (d)S. Sacki N. Kuwahara S. Konno and M. Kaneko ibid. p. 589; (e) Y. Baba Y. Fujita and A. Kagemoto Makromol. Chem. 1973,164,349; (f)V. M. Andreeva A. A. Anikeeva S. A. Vshivkov and A. A. Tager Vysokomol. Soedinenii (B) 1970,12,789; (g)Y. Izumi and Y. Miyake Polymer J. 1972 3 647; (h) J. M. G. Cowic and I.J. McEwen un- published results; (i) A. V. Gevorkyan L. Kh. Simonyan A. A. Gevorkyan and G. A. Chukhadzhyan Vysokomol. Soedinenii (E) 1972 14 745. I* ' D. Patterson and G. Delmas J. Polymer Sci. Part C Polymer Symposia 1970 30 1. J. M. G. Cowie that selection of equation-of-state data at the appropriate temperature will reduce the discrepancy.' 83 Interesting phase diagrams have been measured for polymers dissolved in poor solvents.' 84 The high molecular weight fractions were insoluble but those of lower molecular weight were found to be soluble and 'hour glass'-shaped cloud-point curves were obtained for polystyrene in acetone and to a lesser extent in diethyl ether. The main factor contributing to this behaviour is the free volume difference reflected in the very high expansion coefficients of the solvents.This affects the value of the interaction parameter x to the extent that increasing the chain length of the polymer forces x above the critical value at finite M and the two critical solution temperatures coalesce. These systems have no unique theta temperature. Similar behaviour was observed in ternary systems by Wolf et UL''~ for a solvent-non solvent-polymer system and by Cowie and McEwen'86 for a co-solvent system. At least two types of LCST can be distinguished. Some systems exhibit phase separation at temperatures below the boiling point of the solvent. Examples of these are given in Table 3 for poly(p-chlorostyrene) and are normally attributed to specific solvent-polymer interactions.The more commonly observed LCST occur above the solvent boiling point and must be studied in closed systems where the pressure now exceeds atmospheric. Consequently the influence of pressure on polymer solutions is of importance in this context. Pressure can have a signifi- cant effect on the cloud-point curves as demonstrated for polystyrene-acetone ;''' application of 20 bar opened up the hour-glass phase diagram to produce separate UCST and LCST. However as the pressures generated during an LCST measure- ment are normally much lower the LCST is usually only a degree or two higher than the equivalent value at atmospheric pressure; for example (dT/dP) = 0.44 K bar-' for polyisobutene-2-methylbutanesolutions.'87u Some important work has appeared on the pressure dependence of chain dimensions and the second virial coefficient measured by light scattering.' 88 In general A2 increased with applied pressure but the influence on the chain dimensions is less regular ; t'he effects are related to the variation produced in x or Z by the pressure.Phase equilibria in the upper critical region continue to be subjected to careful examination. A number of attempts have been made to modify the Flory- Huggins equations by incorporating the concentration dependence of the interaction parameter x. While some authors use a truncated power series Koningsveld and Kleintjens' have derived a closed expression which proved Ia3 J. M. G. Cowie and I. J. McEwen Macromolecules in the press. (a)K. S. Siow G. Delmas and D.Patterson Macromolecules 1972,5,29; (b)J. M. G. Cowie A. Maconnachie and R. J. Ranson Macromolecules 1971 4 57. Ia5 B. A. Wolf J. W. Breitenbach and H. Senftl J. Polymer Sci. Part C Polymer Sympo- sia 1970 345. ls6 J. M. G. Cowie and I. J. McEwen J. C. S. Faraday I 1974,70 171. (a) L. Zeman J. Biros G. Delmas and D. Patterson J. Phys. Chem. 1972,76 p. 1206; (b)L. Zeman and D. Patterson ibid. p. 1214. lSa (a) G. V. Schulz and M. Lechner J. Polymer Sci. Part A-2 Polymer Phys. 1970 8 1885; (b) G. V. Schulz and M. Lechner European Polymer J. 1970 6 945; (c) D. Gaeckle and D. Patterson Macromolecules 1972 5 136; (d)C. J. McDonald and S. Claesson Chem. Scripta 1973 4 155. ls9 R. Koningsveld and L. A. Kleintjens Macromolecules 1971 4 637. Physical Properties of Polymers and their Solutions 197 successful for linear polymers.A more general form covering copolymers and branched chains was subsequently developed by Kennedy et ~l.,'~~ who pre- sented closed-form expressions for the chemical potential spinodal and con- solute states. Afurther extension of the treatment to include the possible tempera- ture dependence of x was made by Emmerik and smolder^,'^' who have also carried out a comprehensive examination of the phase equilibria in binary and ternary solutions containing poly-(2,6-dimethyl-l,4-phenyleneoxide). Calcula- tions by SO~C,'~~ using a modified Flory-Huggins approach illustrate the vast differences in shape that the cloud-point curve can adopt for different molecular weight distributions.Bimodal cloud-point curves for narrow-distribution polymers in two-polymer-ne-solvent systems have also been found. 193 The thermodynamic constraints on the O-temperature have been discussed by Kennedy'94 and a more rigorous definition 'the O-temperature is that tempera- ture at which the kcond virial coefficient is zero in a specified limit M - a,for any polymer-solvent system at a given pressure' is proposed. It is suggested that on this basis the conventional extrapolation procedures used to estimate T = 8 using phase separation data are in error. This is illustrated for poly- styrene-cyclohexane for which a O-temperature about 3 K lower than the pres- ently accepted value is estimated. A lower &temperature for the same system was also reported by Kuwahara et ~1.,'~'but this was attributed to the use of ultra-dry solvent.Spinodal and consolute state relations have also been treated for quasi- ternary systems on the basis of the lattice the~ry.'~~",'~~ While the theoretical treatments lead to expressions for both the cloud-point curves and spinodal conditions it is normally the cloud-point curves which are measured experimentally. The suggestion that the spinodal decomposition in a polymer solution was impossible to measure'97 has been refuted by recent work. Light-scattering theory shows that the scattering intensity will increase to very high values in the vicinity of the spinodal. The procedure developed by Scholte' 98 to measure spinodal curves depends on plotting the scattering intensity at I9O J.W. Kennedy M. Gordon and R. Koningsveld J. Polymer Sci. Part C Polymer Symposia 1972 39 43. 19' (a)P.T. van Emmerik and C. A. Smolders J. Polymer Sci. Part C Polymer Symposia 1972 39 73; (b) P. T. van Emmerik and C. A. Smolders ibid. p. 31 1; (c) P. T. van Emmerik and C. A. Smolders European Polymer J. 1973,9 157; (4P. T. van Emmerik and C. A. Smolders ibid. p. 293; (e) P. T. van Emmerik C. A. Smolders and W. Geymayer ibid. p. 309. 19' K. Sole Macromolecules 1971 3 665. 193 (a)R. Koningsveld H. A. G. Chermin and M. Gordon Proc. Roy. Soc. 1970 A139 331; (6) D. G. Welygan and C. M. Burns J. Polymer Sci. Polymer Letters Edn. 1973 11 339. 194 (a)J. W. Kennedy J. Polymer Sci. Part C,Polymer Symposia 1972,39,71; (b)H.A. G. Chermin and J. W. Kennedy Macromolecules 1972 5 655. 195 N. Kuwahara M. Nakata and M. Kaneko Polymer 1973 14,415. 196 (a) R. Koningsveld Chem. Zvesri 1972 26 263; (6) H. A. G. Chermin Ph.D. thesis University of Essex 197 1. 19' R. Koningsveld Adv. Colloid Interface Sci. 1968 2 151. 19' (a)Th. G. Scholte J. Polymer Sci. Part A-2 Polymer Phys. 1971,9 1553; (b) Th. G. Scholte J. Polymer Sci. Part C Polymer Symposia 1972 39 28 1. 198 J. M. G. Cowie zero angle for a given concentration as a function of reciprocal temperature. This is repeated for a number of concentrations to build up the curve. Laser homodyne spectroscopy has been used in a similar fashion but the authors express some doubt as to the accuracy of their extrapolation method.'99 Both procedures rely on measurements made above the cloud-point curve and so extrapolations to spinodals are often of necessity rather long.A much more promising technique has been developed by Gordon et dZoO The principle of the method called pulse induced critical scattering (PICS) is to effect very rapid critical scattering measurements in both the stable and metastable regions. Use of a laser beam and small samples are features of the experimental design and an important aspect of the method is that measurements of scattering intensities which need only be relative are complete before actual phase separation can occur in the metastable region (less than 5 s). Penetration into the metastable region between the cloud-point curve and the spinodal is achieved using rapid thermal pulses from a base temperature located above the cloud-point curve.In between pulses the solution is returned to the base temperature. Although an extrapolation procedure is still required (recipro- cal scattering intensity against temperature) the penetration into the metastable region is sufficiently deep to make this very short and quite accurate. As the shape of the spinodal is strongly dependent on the weight and Z-average mole- cular weights the development of the PICS technique could provide a useful tool for polymer characterization. Few publications on the method exist at present but a recent one2' gives a comprehensive description of the application of PICS to the study of thermo- dynamic and kinetic effects in polymer solutions.Also confirmed was the dependence of the spinodal loci on M and to some extent on M,. The application of light scattering to the measurement of thermodynamic functions,202 critical opalescence,203 and an estimation of particle size in and the mechanism of spinodal decomposition204 have been reported. An interesting method of measuring the Flory parameter X,,was reported by Booth and Pickles,20s who examined the phase separation in mixtures of low molecular weight poly(ethy1ene oxide) and poly(propene oxide) liquids. Large values of X,,were found once again. Differential thermal analysis (d.t.a.) has also been used to study phase equilibria LCST,Z06 and the solution process of polyolefin crystals.207 19' N. Kuwahara D. V.Fenby M. Tamsky and B. Chu J. Chem. Phys. 1971,55 1140. M. Gordon J. M. G. Cowie B. Ready and J. Goldsbrough U.K.P.61071/1970; 275311971;275411971 ;275511971. 201 K. W. Derham J. Goldsbrough and M. Gordon paper presented at 12th IUPAC Microsymposium Prague August 1973. 'O' Th. G. Scholte European Polymer J. 1970 6 1063. '03 (a)W. Borchard Ber. Bunsengesellschaftphys. Chem. 1972,76 224; (6)B. Chu ibid. p. 202; (c)A. Vrij and M. W. J. van den Esker J.C.S. Faraday II 1972 68 513. *04 (a)J. J. van Aartsen European Polymer J. 1970 6 919; (b) J. J.van Aartsen ibid. p. 1105. 'OS C. Booth and C. J. Pickles J. Polymer Sci.,Polymer Phys. Edn. 1973 11 595. '06 (a)A. Kagemoto Y.Baba and R. Fujishiro Makromol. Chem. 1972,154 105; (6)K. Tamura Y. Baba K. Nakatsukasa and R.Fujishiro Polymer J. 1972 3 28. lo' H. P. Schreiber J. Appl. Polymer Sci. 1972 16 539. Physical Properties of Polymers and their Solutions Refractive Index and Cohesive Energy Density.-Light-scattering studies require accurate measurement of the refractive index increment (dfi/dc) but standard instrumentation is often limited by the range of refractive indices which can be covered. This problem has been recognized by Kratochvil and Babka,208 who have modified the cell in the Brice-Phoenix instrument to allow highly refractive liquids to be handled. The basic cell design remains unaltered but by replacing the inclined plane face with glass having fi = 1.62 total reflection is avoided up to a value of fi = 1.72 for the measured liquid.Cell modification is also described for a Jena interfer~meter.~" Lorimer2" has developed a relation based on the theory of Onsager and Bottchen for the refractive index which is apparently more accurate than the Lorenz-Lorentz equation. The influence of molecular weight on (dfi/dc) has also been examined2' ' and results indicate that (dfi/dc) will depart from its asymptotic limit in the range M = 5 x lo4 to lo5 for polystyrene. A somewhat lower limit of M = lo4 for polystyrene in benzene was obtained by Hert and Strazielle2' lC and these authors also examined (dfi/dc) in mixed solvent systems taking into consideration the additional variables of temperature and mixture composition. The possibility that (dfi/dc) can vary with chain length is often overlooked and must be considered in light-scattering measurements involving short chains.A relation between (dfi/dc) and the density increment has been used to calculate the partial specific volume of a polymer in solution,212 and a method of calculating the refractive index of a polymer solution from a knowledge of the density and the specific refraction of the components has been outlined.213 Cohesive energy density (CED) measurements have also been reported and the novel use of gas chromatography for this purpose has been described. The CED of p~lyethylene~'~ was estimated by determining the retention time of solvents passing over a polyethylene-coated column and correlating this with the activity coefficient at infinite dilution. The assignment of a CED value to random copoly- mers has been discussed by Schneier.21 An attempt to extend the Hildebrand concept of solubility parameter [= (CED)*] to the prediction of polymer-solvent miscibility by Chen2' involves a two-dimensional approach.In a purely qualitative sense the method appears to be as effective as the more complex three-dimensional approach of Hansen. A more rigorous approach to solubility-parameter theory2" illustrates the apparent compatibility of this concept with the newer Flory theory of the liquid state. It has been shown that correspondence with the predictions of the Flory '08 P. Kratochvil and J. Babka J. Appf. Polymer Sci. 1972 16 1053. '09 I. Baltog C. Ghita and L. Ghita European Polymer J. 1970 6 1299. 'lo (a)J. W. Lorimer Polymer 1972 13.46; (b)J.W. Lorimer ibid. p. 274. 'I1 (a)J. W. Lorimer and D. E. G. Jones Polymer 1972 13 52; (b)D. Margerison D. R. Bain and B. Kiely ibid. 1973,14 133; (c) M. Hert and C. Strazielle European Polymer J. 1973 9 543. "'Th. G. Scholte J. Polymer Sci. Part A-2 Polymer Phys. 1972 10 519. 'I3 H.Looyenga J. Polymer Sci. Polymer Phys. Edn. 1973 11 1331. 'I4 S. K. Ghosh Makromol. Chem. 1971 143 181. '15 B. Schneier J. Appl. Polymer Sci. 1972 16 1515. 'I6 (a) &-A. Chen J. Appl. Polymer Sci. 1971 15 1247; (b) S.-A. Chen ibid. 1972 16 1603. 'I ' J. Biros L. Zeman and D. Patterson Macromolecules 197 1 4 30. J. M.G. Cowie theory can be obtained by treating free-volume effects in the solubility-parameter approach. This method of approaching the problem of polymer solution thermo- dynamics has also been used by Dayank2l8 Utracki219 has attempted to correlate the solubility parameter of the solvent with the binary cluster integral or at least the non-polar contribution to this parameter.89 Evidence for such a relationship was found and fell into two distinct groupings one for linear and another for non-linear solvent molecules.This highlights the need for a symmetry factor in the solubility-parameter theory which may have an important bearing on the thermodynamics of polymer solubility. Mixed Solvents .-When specific conditions are required for polymer solution study (e.g.pseudo-ideal solutions) they are often obtained most easily by means of mixed solvents. Mixtures can be used successfully if the attendant problems are recognized.These include selective adsorption of one component of the solution by the polymer and the effect this has on the polymer dimensions and other dilute-solution parameters. The problem of the dependence of (r2)if on the thermodynamic properties of the solvent mixture has been studied most notably by Dondos and Benoit.220 They have concluded that (r2)$ depends both on the solvent-solvent interactions and on the polarity of the polymer. In all cases KO which is derived from Stock- mayer-Fixman plots is taken as a measure of (r2)i and the excess free energy AGE of the mixture is regarded as the thermodynamic function of interest. One can also express AGE as the interaction parameter x12. Comparison of KO obtained for a single theta solvent with KY measured in a mixture shows that when AGEis positive then KY is larger than expected and smaller when AGEis negative.This is expressed in a linear relationship between {KY/[&KB(l) + 42K&2)]} and AGE where 4 is the volume fraction KY,KB(l),and KB(2)are the values obtained in the mixture and in liquid components (1) and (2) respectively. This linear relation holds good for a mixture of two good solvents but when solvent-precipitant mixtures were studied deviations were observed. The linear relation was reaffirmed when corrections were made for preferential adsorption effects in the solutions. Benoit and Dondos22’ have shown that if preferential adsorption is evident in a consolute mixture for which A2 = 0 then the slope of the Stockmayer-Fixman plot is not zero as expected but is proportional to the preferential adsorption parameter A*.This was not con- firmed for poly(p-methoxystyrene) in a number of where the slope was always found to be zero even when A* # 0. It has been suggested220b that medium-range effects are responsible for the variation of (r2)8 with x12 J. Dayantis J. Polymer Sci. Part C Polymer Symposia 1972 39 35. (a)L. A. Utracki J. Appl. Polymer Sci. 1972 16 1167; (6) L. A. Utracki Polymer J. 1972 3 551. (a)A. Dondos and H. Benoit European Polymer J. 1968,4,561; (b)A. Dondos and H. Benoit ibid. 1970 6 1439; (c) A. Dondos and H. Benoit Macromolecules 1971 4 279; (4A. Dondos Compt. rend. 1971 272 C 1419; (e) A. Dondos and H. Benoit Macromolecules 1973 6 242.’” A. Dondos and H. Benoit J. Polymer Sci. 1969,7 335. ’’’ A. Mattiussi E. Conti and G. B. Gechele European Polymer J. 1972.8. 429. Physical Properties of Polymers and their Solutions 201 but Pouchly and Patterson22 have proposed an alternative explanation. They have considered the concentration dependence of x represented as x = xo + ~‘4~ and have introduced this into the expression for the effective interaction parameter x of a mixture x = u1(x?3 +-d343) -I-u2(x:3 4-x:343) -u~u2[x12-2xT(1 -&)I where ui = 4i/(4i+ +j) and xT is a ternary interaction parameter included to compensate for deviations from the value of AGM calculated using the Flory- Huggins theory. In this context xT should always have the same sign as xI2 and be numerically smaller than (x12/2).The expression for x can be used to modify the Stockmayer-Fixman equation and to introduce a factor which predicts the variation of K with x12 as observed experimentally. Light scattering is the most commonly used means of studying preferential adsorption in mixed solvent but n.m.r. relaxation methods2’ and the ultracentrifuge226 have also been used to some effect. Fujishige and Elias2” reported that the atactic form of poly(methy1 methacry- late) preferentially adsorbed acetone to a much greater extent than the isotactic form in acetone-CHC13 mixtures but that little difference in A* could be detected when benzene replaced CHCl . In contrast no difference in A* could be found for the different stereo-regular forms by Kratochvil et ~1.,~’*who suggested that aggregation effects may have interfered with Fujishige’s measurements.Small differences in A* between the stereo-regular forms of poly(methy1 methacrylate) in other mixtures have been reported,225d but it appears that stereostructure is not a major influence on the magnitude of A*. Perhaps more important is the effect of chain length ; it was shown that A* increased with decreasing molecular weight.229 This is a result of the probability of intramolecular segment contacts upsetting the number of binding sites available for solvent on a polymer chain and examination of polydispersity substantiates this view. This has important consequences in the study of comb- and star-branched pol~rners,’~’ where the dependence of A* on molecular weight is more complicated than for 223 J.Pouchly and D. Patterson Macromolecules 1973. 6 465. 224 (a) B. Chaufer B. Sebille and C. Quiveron Compr. rend. 1972 274 C 764; (b) L. Moldovan and C. Strazielle Makromol. Chem. 1970 140 201 ; (c) B. Sedlatek P. Kratochvil and D. Strakova Coll. Czech. Chem. Comm. 1972 37 970; (6)Z. Bardet H. Maillols and J. Maillols J. Chim. phys. 1973 70 615. 225 (a)S. Brownstein S. Bywater and J. M. G. Cowie Trans. Faraday SOC.,1969,65,2480; (b) H. Lutje. Makromol. Chem. 1971 142 81 ; (c) H. Lutje J. Polymer Sci. Part C Polymer Symposia; 1972 39 325; (d)K. Sat0 and A. Nishioka Polymer J. 1972 3 245. z26 (a)J. M. G. Cowie R. Dey and J. T. McCrindle Polymer J. 1971.2 88; (b)A. Rosen-thal Macromolecules 1972 5 310.227 (a) S. Fujishige and H.-G. Elias Makromol. Chem. 1972 155 127; (6)S. Fujishige and H.-G. Elias ibid.,p. 137. 228 P. Kratochvil D. Strakova and D. K. Carpenter Makromol. Chem. 1972 162 275. 229 A. Dondos and H. Benoit Makromol. Chem. 1970 133 119. 230 (a) M. Hert C. Strazielle and H. Benoit Makromol. Chem. 1973 172 169; (6)E. F. Casassa Polymer J. 1972 3 517. 23’ M. Hert C. Strazielle and H. Benoit Makromol. Chem. 1973 172 185. 202 J. M. G. Cowie linear polymers. The increase in preferential adsorption at shorter chain lengths in linear polymers was manifest in higher coil expansion near the O-p~int.~~~ Some insight into the mechanism of co-solvency is obtained from preferential adsorption It is found that at the co-solvent composition which is most compatible with the polymer A* is very close to zero.As the composition is changed there is preferential adsorption of the liquid in the mixture which has been depleted by this change. In effect the preferential adsorption can be seen as a mechanism which attempts to regain the composition of the best mixture by attracting the deficient solvent component to the vicinity of the polymer coil when the polymer is dissolved in a mixture whose composition is not the optimum. The synergistic effects of co-solvents have been reported for polyethylene,234 poly(methy1 metha~rylate),~~~ and poly-(2-hydroxymethacry-Some progress in formulating a satisfactory theoretical description of selective adsorption has been made.237 Excluded-volume effects in mixed solvents have received further attention.The advantages of mixed solvent systems in the study of copolymer solutions were examined by Kratoch~il.~~~ Segment heterogeneity can cause problems in copolymer solutions and optical masking of one component can allow the study of the other by light scattering. This can be achieved most easily by using solvent mixtures to obtain the correct refractive index General selection rules are proposed to ease the problem of solvent selection. Selective adsorption in copolymer solutions can have a marked effect on refractive index increment values and A* is found to be largest for alternating copolymers where the number of heterogeneous contacts is a maximum. Deviations due to selective adsorption effects decreased through random to block copolymers where the required corrections were small.Acoustic and Electro-optical Studies.-The application of acoustical methods to the study of polymer solutions has recently been reviewed by Pethri~k~~~" and covers most of the relevant work to date. The acoustic technique is potentially useful for studying molecular motions and relaxation processes for both polar and non-polar polymers whereas the more widely employed dielectric measurements 232 A. Dondos K. Viras and F. Aroni European Polymer J. 1973,9 1051. 233 (a)J. M. G.Cowieand J. T. McCrindle European Polymer J. 1972,8 1185; (b)J. M. G. Cowie and J. T. McCrindle ibid. p. 1325. 234 N. Das and S. R. Palit J. Polymer Sci.Polymer Phys. Edn. 1973 11 1025. 235 P. C. Deb and S. R. Palit Makromof. Chem. 1973 166,227. 236 K. Ddek and B. Sedlafek European Pol.vmer J. 1971,7 1275. 237 (a)M. Yamamoto J. L. White and D. McLean Pol,vmer 1971,12 290; (6)J. Pouchly A. Zivny and K. Sole Coll. Czech. Chem. Comm. 1972 37 988; (c) A. Zivny and J. Pouchly J. Polymer Sci. Part A-2 Polymer Phys. 1972 10 1467 1481. 238 M. Yamamoto and J. L. White Macromolecules 1973 5 58. 239 (a) Z. Tuzar P. Kratochvil and D. Strakova European Polymer J. 1970 6 11 13; (b) P. Kratochvil and Z. Tuzar Chem. Ztyesti. 1971 25 190; (c) J. PodeSva and P. Kratochvil European Polymer J. 1972 8 1179; (6)P. Kratochvil B. SedlaEek D. Strakova and Z. Tuzar Makromol. Chem. 1973 166 265. 240 (a)R. A. Pethrick J. Macromol.Sci. 1973 C9,91; (6) A. M. North Chem. SOC. Rev. 1972 1 49. Physical Properties of Polymers and their Solutions 203 are restricted to polar materials. Recent advances in this field have been reviewed by North.240b Some interesting work on rigid siloxane polymers,241 including ladder struc- tures and stiff-chain polyis~cyanates~~~ has appeared. The poly( butyl isocyanate) chain has been studied most often and exhibits a high degree of order in solution. The chain conformation was observed to change from a rigid rod-like shape at low molecular weight to a flexible coil at high molecular weight.242e Tsvetkov concluded that the molecule has a planar cis conformation. Other confirmed the transition from rod to coil and agree that the onset of flexibility occurs above M = 5 x lo4.There is also a tendency to favour the idea of a helical form at low M,242cSbut the data are inconclusive. Other poly(alky1 isocyanates) exhibit similar propertie~,~~~~*~J but in contrast the aromatic derivatives were found to behave like statistical coils with flexibilities comparable with other synthetic polymers. 242e Mobile liquid-crystalline structures were detected in solutions of poly(pheny1- methacrylic esters) of cetyl- and nonyl-oxybenzoic acids which emanate from the extremely long ester ~ide-chains.~~~ Polymer Compatibility.-Mixtures or blends of two or more polymers are rarely compatible and will tend to form heterogeneous solids. The extent of the incom- patibility affects the properties of the blend and can be assessed in several ways.A critical discussion of the methods based on the viscosities of solutions244 comprising liquid (l),polymer (2) polymer (3) has been presented by Vasile and S~hneider.~~’ These authors have suggested that two types of compatibility can be distinguished (i) a true compatibility and (ii) a pseudocompatibility arising from specific interactions between functional groups. Systems in the second category are believed to have a very slow rate of phase separation and con- sequently appear to be compatible.246 Light scattering is also a useful tool to study these ternary systems.247 Cantow and his co-workers have used a technique whereby polymer (2) is dissolved in an isorefractive mixture of solvent (1) and polymer (3).This optical masking of the 241 (a) V. N. Tsvetkov K. A. Andrianov G. I. Okhrimenko and M. G. Vitovskaya European Polymer J. 1971,7 1215; (6)V. N. Tsvetkov Makromol. Chem. 1972 160 1; (c) V. N. Tsvetkov K. A. Andrianov N. N. Makarova M. G. Vitovskaya E. 1. Rjumtsev and I. N. Shtennikova European Polymer J. 1973 9 27. 242 (a)H. Plummer and B. R. Jennings European Polymer J. 1970,6 171 ;(b)L. J. Fetters and H. Yu Macromolecules 1971,4,385; (c)B. R. Jennings and B. L. Brown European Polymer J. 197 1,7 805 ;(6)T. C. Troxell and H. A. Scheraga Macromolecules 197 1 4 528; (e)V. N. Tsvetkov I. N. Shtennikova E. I. Rjumtsev and Yu. P. Getmanchuk European Polymer J. 1971 7 767; u> L. J. Fetters J. Polymer Sci. Part B Polymer Letters 1972 10 577. 243 V.N. Tsvetkov E. I. Rjumtsev I. N. Shtennikova E. V. Korneeva B. A. Krentsel and Yu. B. Amerik European Polymer J. 1973 9 481. 244 (a) D. Feldman and M. Rusu European Polymer J. 1971 7 215; (b)B. Bohner D. Berek and S. Florian ibid. 1970 6 471 ;(c) D. Feldman and M. Rusu ibid. p. 627. C. Vasile and I. A. Schneider Makromol. Chem. 1971,141 127. 246 C. Vasile and I. A. Schneider European Polymer J. 1973 9 1063. 247 (a) R. Kuhn H.-J. Cantow and S. B. Liang Angew. Makromol. Chem. 1971 18 93; (b)R. Kuhn S. B. Liang and H.-J. Cantow ibid. p. 101 ;(c) R. Kuhn V. Bugdahl and H.-J. Cantow ibid. p. 109; (4R. Kuhn V. Bugdahl and H.-J. Cantow ibid. p. 109. 204 J. M.G. Cowie second polymer component means that the behaviour of the first polymer can be studied separately and the interactions compared in the presence and absence of polymer (3).This then provides a measure of the compatibility of the two polymers in that solvent. Solution calorimetry248 and differential scanning calorimetry249 have been applied ;the latter will show whether or not two glass transitions are present in the mixture and this together with mechanical measurements provides an indication of the heterogeneity in a mixture. To some extent the compatibility of polymers cast in a film can be controlled by the solvent used to prepare the mixture,250 and this is also true for copolymers. The problem has been approached on the basis of the interaction parameters x and while it is generally accepted that two polymers in solution are incompatible when xZ3is unfavourable Zeman and Patterson251 have shown that this is only true at high polymer concentration or when x12and x1 are of equal magnitude.In dilute solution then the controlling factor is the difference between x12 and XI 3. Random Copolymer Solutions.-Incompabitility effects are not confined to blends or mixtures of polymers ;copolymers will also exhibit properties arising from adverse interactions between comonomers. In random copolymers this is manifest in deviations from property additivity rules based on the composition which are caused by the presence of heterogeneous contacts between the two or more types of monomer unit in the system. Even when the comonomers are relatively compatible as in poly(styrene-co-p-rnetho~ystyrene),~~~ the Mark- Houwink exponents for the copolymers are lower and the unperturbed dimen- sions are always larger than those of the parent homopolymers.Some enhancement of solubility was obtained in methylcyclohexane solutions of p~ly(styrene-co-a-methylstyrene).~ 53 The &temperatures plotted as a func- tion of copolymer composition were always below the line joining the &values for the homopolymers. This is in agreement with the solution behaviour of statistical and alternating copolymers of styrene and methyl metha~rylate.'~~ In cyclohexanol which is a &solvent for both homopolymers there is a distinct lowering of the O-temperature with the alternating copolymer being a more effective depressant than the random structure.This trend was borne out by the fact that the block copolymers had much the same O-temperature as the homopolymers although this could have been a fortuitous elimination of inter- and intra-molecular interactions. Calculation of the characteristic parameter C showed that for block copolymers C could be approximated 248 P. Novakov Ch. Konstantinov and P. Mitanov J. Appl. Polymer Sci.,1972 16 1827. 249 G. A. Zakrzewski Polymer 1973 14 347. 250 M. Bank J. Leffingwell and C. Thies Macromolecules 1971 4 43. 251 I. Zeman and D. Patterson Macromolecules 1973 5 513. 252 (a) M. Pizzoli and G. Ceccorulli European Polymer J. 1972 8 769; (b) M. Piuoli G. Ceccorulli and G. Stea Makromol. Chem. 1973 164 273. 253 D. J. Goldwasser and D. J. Williams Macromolecules 1973 6 353.2s* T. Kotaka T. Tanaka H. Ohnuma Y. Murakami and H. Inagaki Polymer J. 1970 1 245. Physical Properties of Polymers and their Solutions 205 from the compositional average of the homopolymers but for random and alternating copolymers C was larger than predicted by the compositional rule. This discrepancy was greatest for the alternating structures and so the extent of the deviations could be related to the proportion of heterogeneous diads present in the chain. Long-range effects embodied in x,were more difficult to establish. No effect of sequence length on the sedimentation behaviour of block and random copolymers of butadiene and a-methylstyrene could be detected,’ 55 but this is probably not the most sensitive diagnostic technique.Other workers256 have examined the internal plasticizing effect of copolymerization as reflected in the glass transition temperature and mechanical response. Dilute-s~lution~~~ properties and solubility parameters’ * have been reported. BIock Copolymer Solutiolrs.-Recent improvements in synthetic techniques have led to the preparation of well-characterized block copolymers. This in turn has stimulated interest in their dilute-solution behaviour and morphology. Several workers have attempted to reduce the difficulties and rather tedious effort involved when trying to establish the true M of a copolymer by light scattering. Normally M has to be determined from measurements in at least three solvents which have as widely differing refractive indices as possible.259 Kratochvil et a1.260have outlined the conditions which should achieve the greatest accuracy in light scattering for a copolymer with comonomers A and B ; the three solvents should meet the requirements (i) first solvent :vA and vg both high and of the same sign ; (ii) second solvent lvAI large vg = 0; (iii) third solvent vA = 0 lv9l large; where v = (dfi/dc).The task of solvent selection can be made easier if mixed solvents are used and v is measured after equilibrium dialysis to eliminate possible errors arising from preferential adsorption effects. When this is done the classical relationship for copolymers derived by Bushuk and Benoit261 can be used. Urwin and Girolamo262 have established a computer program to handle the combinations of data required for the calculation of M,,but they found that high-precision experimental work was required for accurate analysis.Light-scattering data 25’ K. F. Elgert and E. Seiler Makromol. Chem. 1972 151,83. 256 (a)E. F. Jordan G. R. Riser B. Artymyshyn S. Smith and A. N. Wrigley J. Polymer Sci. Polymer Phys. Edn. 1973,11 1475; (b)E. F. Jordan B. Artymyshyn G. R. Riser J. Nidock and A. N. Wrigley J. Appl. Polymer Sci. 1973 17 1545; (c) E. F. Jordan G. R. Riser B. Artymyshyn and A. N. Wrigley ibid. p. 1569. 257 (a)A. Dondos European Polymer J. 1971,7,405; (b) M. Morimoto and Y. Okamoto J. Appl. Polymer Sci. 1972 16 2795; (c) K. S. V. Srinivasan and M. Santappa J. Polymer Sci. Polymer Phys. Edn. 1973 11 331. 258 B. Schneier J.Polymer Sci. Part B Polymer Letters 1972 10 245. 259 T. Kotaka T. Tanaka and H. Inagaki Polymer J. 1972,3 327. 260 P. Kratochvil B. Sedlaeek D. Strakova and Z. Tuzar Makromol. Chem. 1971 148 271. 261 W. Bushuk and H. Benoit Canad. J. Chem. 1958,36 1616. 262 J. R. Urwin and M. Girolamo Makromol. Chem. 1971 142 161. 206 J. M. G. Cowie can be used to estimate the chemical heterogeneity of copolymers,263 although the sensitivity of the method may well be over-rated. A nomogram for this purpose has been constructed264 and tested for model systems,265 but in practice it works most effectively for high molecular weight samples with low hetero- genei ty. Many authors use osmotic pressure266 or sedimentation equilibrium methods259 to determine the molecular weight.AB and ABA poly(styrene-b- isoprene) copolymers have been used frequently in solution studies because of the availability of good samples. Girolamo and Urwin266 established O-condi- tions for AB poly(styrene-b-isoprene) using cloud-point and A2 = 0 criteria. In contrast to random and block poly(styrene-co-methyl metha~rylate),~~~ the &temperature in several solvents passed through a maximum as the copolymer composition was varied except in cyclohexane solutions where 8 increased monotonically. In this solvent there was evidence of intramolecular phase separation a common feature of many copolymer solutions. The maximum &temperature is believed to occur when the structure is as random as possible with the maximum number of hetero-contacts in the solution.When the un- perturbed dimensions were measured267 they were found to be a linear function of copolymer composition and the characteristic ratio could be calculated from where x is the mole fraction. The general principle that the properties of these copolymers are close to the weighted averages of the homopolymers was con- firmed by Prud’homme et u1.268uExceptions to this rule are found for copolymers dissolved in selective solvents in which intramolecular phase separation occurs but even in such cases it may depend on the temperature of measurement. When [q] was measured as a function of temperature for cyclohexane solutions of AB poly(styrene-b-isoprene) a discontinuity was detected at a temperature called the transition temperature (q) A significant lowering of by Urwin.268b*c the unperturbed dimension was found on decreasing the temperature from above to below Tpand this was large enough to be interpreted as a conformational transition.Below Tp,the polystyrene block collapsed in the poor solvent cyclo- hexane but the isoprene block was selectively expanded. Above Tp the poly- styrene block expanded and created a somewhat poorer environment for the polyisoprene block ;this tended to produce a much more even distribution of segments in solution with an interpenetration of the blocks. Consequently the number of hetero-contacts increased above Tp and this random distribution means that C^,”can again be calculated from the composition rule given above. 263 J.Lamprecht C. Strazielle J. Dayantis and H. Benoit Makromol. Chem. 1971 148 285. 264 J. VorliCek and P. Kratochvil J. Polymer Sci. Polymer Phys. Edn. 1973 11 855. 265 J. VorliCek and P. Kratochvil J. Polymer Sci. Polymer Phys. Edn. 1973 11 1251. 266 M. Girolamo and J. R. Urwin European Polymer J. 1972,8 299. 267 J. R. Urwin and M. Girolamo Makromol. Chem. 1972 160 183. 268 (a) J. Prud’homme J. E. L. Roovers and S. Bywater European Polymer J. 1972 8 901; (6) M. Girolamo and J. R. Urwin ibid. 1971 7 693; (c) J. R. Urwin and M. Girolamo ibid. p. 785. Physical Properties of Polymers and their Solutions 207 This behaviour has been observed for the same block copolymers in other selective solvents i.e. poor for one block but good for the other such as decalin and methylcy~lohexane.~~~ Agreement is not unanimous as Plante et ~21.~~’ found no conformation transition between 283 and 333 K for AB or ABA blocks in toluene dioxan isobutyl methyl ketone or cyclohexane.Sharp breaks in [ql-temperature curves were detected by D~ndos~~l in solutions of ABA poly- (methyl methacrylate-b-styrene). Again at low temperatures there was a segregated conformation with few heterocontacts but as the temperature rose the conformation became increasingly gaussian as the number of hetero-contacts increased. This conformational change was substantiated by light scatter- ing.271,272 are of the opinion that the two-parameter Girolamo and Ur~in~~~ theory is valid for copolymers in good solvents where substantial interpenetration of the blocks is evident.In selective solvents this is probably true above Tp but below this temperature the segregation of blocks makes this a doubtful assumption and such concepts as a &state become open to question. In selective solvents block segregation can lead to micelle foimation. This effect has been studied by light scattering274 in solutions of ABA poly(styrene-b- butadiene). Addition of ethanol to a dioxan solution of this block copolymer selectively precipitated the butadiene block but the solvated polystyrene blocks were able to hold the copolymer in solution. In doing so micelles were formed of average size 52 nm. The general condition for micellization is then use of a solvent which is good for one block but a precipitant for the other and this can be achieved most easily through the use of mixed solvents.The solubilization of homopolymer in a micelle and the difficulties this can present when attempting to separate homopolymer from copolymer in a mixture have been recognized.275 Micellar formation has been detected in AB poly(styrene-b-dimethylsiloxane) solutions,276 but in ABA the preferred conformation was one of randomly interpenetrating coils in toluene butan-2-one or cyclohexane. The solution properties of graft copolymers are also strongly dependent on the solvent and Price and have shown that micelles form in poly(styrene-g- isoprene) with the graft chains holding the main chain in solution. The incompatibility of the component blocks in a copolymer can depend on a number of variables such as the molecular weight the nature of the solvent 269 (a)J.R. Urwin and M. Girolamo Makromol. Chem. 1971 150 179; (b)J. R. Urwin and M. Girolamo Austral. J. Chem. 1971 24 729. 270 J. P. Plante N. Ho-Duc and J. Prud’homme European Polymer J. 1973 9 77. 27’ (a)A. Dondos Makromol. Chem. 1971 147 123; (b) A. Dondos P. Rempp and H. Benoit Polymer 1972 13 97. 272 T. Tanaka T. Kotaka and H. Inagaki Polymer J. 1972 3 338. 273 M. Girolamo and J. R. Urwin European Polymer J. 1972 8 I1 59. 2’4 Z. Tuzar and P. Kratochvil Makromol. Chem. 1972 160 301. 275 (a)Z. Tuzar and P. Kratochvil Makromol. Chem. 1973 170 177; (6) A. Skoulious P. Helffer Y. Gallot and J. Selb ibid. 1971 148 305. 276 (a)J. C. Saam D. J. Gordon and S.Lindsey Macromolecules 1970 3 1; (b) M. J. Owen and T. C. Kendrick ibid. p. 455. ’17 W. G. Davies and D. P. Jones Ind.and Eng. Chem. (Product. Res. and Development) 1971 10 168. 278 C. Price and D. Woods Polymer 1973 14 82. 208 J. M. G. Cowie or the chain architecture.279 Differences are obvious between the behaviour of di- and tri-block polymers;259 generally speaking AB blocks will dissolve in selective solvents for either block but in ABA blocks the solubility is governed by the A subchains rather than the central block. Optical anomalies in light-scattering measurements were highlighted by Prud’homme and Bywater.280 Normal Zimm plots were obtained when the refractive index of the solvent used was significantly different from either block but distorted plots arising from intermolecular interferences appeared when the solvent refractive index approached that of either block.When conditions for microphase separation in block copolymers are calculated as a function of the interaction parameter xfor a copolymer of fixed composition and molecular weight they predict that phase separation becomes increasingly unlikely as the number of blocks in the chain increases.28’ This isin agreement with the experimental observations. Block Copolymer Morphology.-The microphase separation and solvent effects observed in solutions of block copolymers bear some relation to the ultimate morphology of the block copolymer in the bulk state. It is generally accepted that a ‘domain’ structure is formed with each type of block either aggregating into limited regions (domains) or forming a matrix in which domains of the other block are embedded.The fine structure has been studied using X-ray (including small angle) scattering phase-contrast microscopy and electron microscopy. In the latter method staining with OsO has proved useful for copolymers containing dienes282-284 and poly(ethy1ene-g-vinyl acetate).285 Several etching techniques have been proposed for multiphase blends,286 ABS and high-impact polystyrene,287 a freeze-etching method for latex particles,288 and a freeze- etching-replication process for samples of both block and graft copolymers in preparation for electron micro~copy.~ 89 Several structural forms have been identified and described variously as (i) hexagonal (ii) reverse hexagonal (iii) linear or lamellar and (iv) irreg~lar.~~’.~~’ The type of structure obtained during film casting290” is found to be dependent 279 H.Ohnuma T. Kotaka and H. Inagaki Polymer J. 1970 1 716. 280 J. Prud’homme and S. Bywater Macromolecules 1971,4 543. 281 S. Krause Macromolecules 1970 3 84. *” P. R. Lewis and C. Price Polymer 1972 13 20. 283 C. Price A. G. Watson and M. T. Chow Polymer 1972 13 333. 284 T. Uchida. T. Soen T. Inoue and H. Kawai J. Polymer Sci. Part A-2 Polymer Phys. 1972 10 101. ”’ Y. Jyo C. Nozaki and M. Mutsuo Macromolecules 1971,4 517. 286 G. C. Eastmond and E. G. Smith Polymer 1973 14 509. 287 C. B. Bucknall I. C. Drinkwater and W. E. Keast Polymer 1972 13 115.288 R. Reed and J. R. Barlow Polymer 1972 13 226. 289 C. Price and D. Woods European Polymer J. 1973 9,827. 290 (a) A. Douy and B. Gallot Makromol. Chem. 1973 165 297; (6) M. Gervais A. Douy and B. Gallot Mol. Crystals Liquid Crystals 1971 13 289; (c) A. Douy and B. Gallot Makromol. Chem. 1972 156 81. 291 H. Kawai T. Soen T. Inoue T. Ono,and T. Uchida Mem. Fac. Eng. Kyoto Uniu. 1971 33. 383. Physical Properties of Polymers and their Solutions 209 on the solvent the temperature copolymer composition and the rate of sample preparation. Lewis and Price282-292 found that for films of ABA poly(styrene-b-diene) the slower rate of casting resulted in greater domain ordering. Uchida et al.284 obtained five types of domain structure by changing the fractional composition of two components in a solvent mixture but this aspect of selective solvent control has not been studied very fully.The effect of chain structure on morphology was examined by Price et al.283 using AB poly(styrene-b-isoprene) but although an ordered arrangement of quite regular domains was found for single chains and both three- and four- branched star conformations the detailed morphology was apparently unaffected by the chain geometry. As the rate of evaporation can play a significant part in altering the final morphology of a sample Gallot and Sadr~n~~~ have attempted to ‘freeze’ the structure present in a solution by photopolymerizing methyl methacrylate added to the solution to form a restraining network. In this way they were able to observe a gradual change from spherical to cylindrical to lamellar structures as a function of increasing copolymer concentration in the initial solution.Much of the work on the effect of external factors on the domain structure of copolymer samples has been comprehensively covered by Kawai et who provide a good synopsis of work in this area up to 1971. More recently a review paper by Sadron and Gal10t~~~ has appeared which focuses attention on the relation between the concentration of copolymers in selective solvents and the mesomor- phic structures formed. A study of solvent localization in these mesomorphic phases has also been made.295 A quite specific and very regular three-dimensional orthorhombic lattice structure composed of spherical polystyrene aggregates in a polybutadiene matrix has been proposed296 for samples of ABA poly(styrene-b-butadiene) and low-angle X-ray scattering data have been interpreted as fitting this face- centred cubic structure.The model has been criticized297 mainly on the grounds that the difficulties encountered in analysing the results cast serious doubt on this postulate :indeed the reported X-ray data may not even match the suggested A similar view has been expressed by Kim,299 who concedes that although a superlattice-like structure may exist to some degree,300 the existence of a highly ordered structure remains unproven. 292 P. R. Lewis and C. Price Polymer 1971 12 258. 293 B. Gallot and C. Sadron Macromolecules 1971,4 514. 294 C.Sadron and B. Gallot Makromol. Chem. 1973 164 301. 295 A. Douy and B. Gallot Compr. rend. 1973,276 C 391. 296 (a) D. McIntyre and E. Campos-Lopez Macromolecules 1970,3 322; (6)E. Campos-Lopez D. McIntyre and L. J. Fetters ibid. 1973,6 415. 297 A. Skoulious Macromolecules 1971 4 268. 298 W. R. Krigbaum S. Yazgan and W. R. Tolbert J. Polymer Sci. Polymer Phys. Edn. 1973 11 51 I. 299 H. Kim Macromolecules 1972 5 594. 300 G. Kampf H. Kromer and M. Hoffmann J. Macromol. Sci. 1972 B6,167. 210 J. M.G.Cowie A statistical thermodynamic treatment of phase separation301 showed that polystyrene domain formation was favoured by surface-energy terms in ABA poly(styrene-b-butadiene). The domain sizes were found to be inversely pro- portional to temperature and thermal treatment could lead to a reorganization of the internal structure increasing its regularity.302 The thermodynamic factors controlling stable domain formation showed that results could be sensitive to differences in the cohesive energy den~ity.~'~.~'~ A treatment of phase separa- tion in a two-block polymer as a mutual excluded-volume effect suggested that only partial separation would occur unless the solvent was poor enough to act as a selective precipitant for one The influence of the Benard effect on the anisotropy of films cast from solution has been highlighted by Krigbaum et The Benard effect describes thecellular ordering of convection currents during solvent evaporation which leads to macroscopic hexagonal cell structures in the film and ordering in the polymer phase.Quite marked changes in morphology can be obtained if impurities such as homo polymer^^^^ or di-blocks in tri-block preparations3" are present. Small quantities of impurities can be tolerated and often create a more regular distribu- tion of domains but solubilization of larger quantities enlarges the domains so much that the phase boundaries become diffuse. This can have a distinct influence on the mechanical properties.308 1onomers.-The ionomers constitute an interesting group of ionic copolymers. They are now generally defined as random copolymers prepared from vinyl and acid monomers which are capable of forming intermolecular ionic bonds. The morphology is again believed to be a domain structure with ionic groups clustering together to form polar domains in a matrix of amorphous polymer.There are differences of opinion as to the size of the ionic aggregate and two models have been used to account for the properties (i) a homogeneous model and (ii) a cluster In the homogeneous model the ionic groups are believed to be distributed as dimers throughout an amorphous phase with no aggregation larger than 2nm. Evidence to support this picture has been 30* U. Bianchi E. Pedemonte and A. Tuturro Polymer 1970 11,268. 302 E. Pedemonte A. Tuturro U. Bianchi and P. Devetta Polymer 1973 14 145. 303 D. F. Leary and M. C. Williams J. Polymer Sci. Polymer Phys. Edn. 1973 11 345. 304 M. Gervais G. Jouan and B. Gallot Compt. rend. 1972 275 C 1243. 305 J.Pouchly A. Zivny and A. Sikora J. Polymer Sci. Part A-2 Polymer Phys. 1972 10 151. 306 T. Inoue T. Soen T. Hashimoto and H. Kawai Macromolecules 1970,3 87. 307 L. J. Fetters B. H. Meyer and D. McIntyre J. Appl. Polymer Sci. 1972 16 2079. (a) K. E. Cunningham M. Auerbach and W. J. Floyd J. Appl. Polymer Sci. 1972 16. 163; (b) R. E. Cunningham and M. L. Wise ibid. p. 107. 309 E. P. Otacka J. Macromol. Sci.,1971 C5,275. Physical Properties of Polymers and their Solutions 211 obtained by Otacka and Kwei3" and by Roe.311 The results of Marx et and others314 lend credence to the cluster model. Eisenberg and Navrati13' have tended to adopt an intermediate description. They found that a breakdown in linear viscoelastic theory occurred when the mole % of salt in styrene-methacrylic acid copolymers exceeded 6%.This was interpreted and substantiated on theoretical as indicating the presence of dimers at low ionic concentrations which increased in size to give trimers tetramers and so on as the amount of acid monomer in the ionomer increased. Eventually microphase separation and clusters could occur at high ionic concentrations. A new model proposed by Marx et ~1.,~fits these observations. The acid aggregates are pictured as being homogeneously dispersed throughout the amorphous phase and each aggregate contains two or more acid groups depend- ing on the copolymer composition and the amount of water present. The aggre- gates are never as large as suggested in the cluster model and so this is much closer to the homogeneous model in concept.The revitalized 'fringe-micelle' structure has also been suggested as a suitable model structure.318 Viscoelastic Properties of Solutiom.-The viscosity of a dilute polymer solution is commonly used to estimate the molecular weight of a polymer. Such solutions are also viscoelastic. This is not quite so obvious but arises because the flow of solvent restricts the number of possible conformations a chain can adopt thereby reducing the entropy and raising the free energy which is elastic energy. To explain the phenomenon a 'bead and spring' model was adopted first by Rouse then by Zimm and later by Tschoegl and used to relate the molecular parameters such as solvent viscosity and relaxation time to the frequency dependence of the storage modulus (G') and the loss modulus (G").The theories contain just one adjustable hydrodynamic interaction parameter h* but are valid only at infinite dilution. This presents experimental difficulties as the elasticity of the polymer chain tends to be swamped by the solvent viscosity and very sensitive measure- ments are required to detect this contribution. Progress towards this end has been made gradually over a number of years and with the development of the 310 E. P. Otacka and T. K. Kwei Macromolecules 1968 1,401. 311 R. J. Roe Amer. Chem. Soc. Div. Polymer Chem. Polymer Preprints 1971 12 730. 312 (a) P. J. Phillips J. Polymer Sci. Part B Polymer Letters 1972 10 443; (b) P. J. Phillips F. A. Emerson and W. J. McKnight Macromolecules 1970 3 767.313 C. L. Marx J. A. Koutsky and S. L. Cooper J. Polymer Sci. Part B Polymer Letters 1971,9 167. 314 (a)F. L. Binsbergen and G. F. Kroon Macromolecules 1973,6 145; (6)T. Kajiyama T. Oda R. S. Stein and W. J. McKnight ibid. 1971 4 198; (c) T. Kajiyama R. S. Stein and W. J. McKnight J. Appl. Phys. 1970 41 4361. (a) A. Eisenberg and M. Navratil J. Polymer Sci. Part B Polymer Letters 1972 10 537; (b)A. Eisenberg and M. Navratil Macromolecules 1973 6 604. 'I6 A. Eisenberg Macromolecules 1970 3 147. 317 C. L. Marx D. F. Caulfield and S. L. Cooper Macromolecules 1973,6 344. 318 R. G. L. Johnson B. W. Delf and W. J. McKnight J. Polymer Sci. Polymer Phys. Edn. 1973 11 571. 212 J. M. G. Cowie Schrag-Birnboim multiple lumped res~nator,~'~*~~' measurements over a wide range of frequency and viscosity are now possible which can be extrapolated to infinite dilution with accuracy.Experimental data in 0-solvents for p~lystyrene,~~' poly(a-p~lybutadiene,~~~ methyl~tyrene),~~~ are all in excellent agreement and poly(dimethyl~iloxane)~~~ with the Zimm theory if a value of h* = 0.25 is used and the frequency of measure- ment is not too high. In good solvents the agreement is poorer and h* must be altered to a lower value.323 In some cases the Tschoegl theory is best if a non- gaussian factor is introduced. The results suggest that at least in the low- frequency range the viscoelastic behaviour is relatively insensitive to chemical structure. This alters markedly with the introduction of branching which affects the longer relaxation times.Osaki and S~hrag~~' have evaluated the Zimm-Kilb theory for star polymers using exact eigenvalues for star-branched chains of equal-length branches. The shapes of the theoretical curves are now a function of the number of branches but agreement with experiment though good requires use of a wider range of h* values possibly because of the non-gaussian character of branched structures. For star-shaped polystyrene with nine arms the agreement was good if h* = 0.4 was used for &solvents and 0.25 for a-chloronaphthalene solutions.326 Similarly polybutadiene stars were close to the Zimm-Kilb predictions for h* = 0.1.322In comb-branched p~lystyrene~~' solutions both the low-frequency [G'] and reduced steady-state compliance J:R were found to be relatively insensitive to branching.Work on these branched structures makes it quite clear that both the hydrodynamic interaction and the segment concentration are important factors controlling the viscoelastic behaviour of polymer solutions. The Zimm theory is invalid for high frequencies but as high-viscosity solvents create similar conditions deviations in viscous liquids have been examined for poly~tyrene.~~~.~~~ The behaviour is close to the predictions of the Peterlin theory which is a bead and spring model modified by the introduction of an 'internal viscosity' factor.330 This measures the opposition of the polymer to the rate of change of its shape in solution but more data are required before the physical significance of the parameter is properly understood.Two approaches to the study of viscoelasticity in polymer solutions are possible;one can either develop better experimental methods for infinite dilution ' J. L. Schrag and R. M. Johnson Rec. Sci. Instr. 1971 42 224. 320 J. L. Schrag and J. D. Ferry Faraday Symposia 1972 No. 6 p. 182. 32 R. M. Johnson J. L. Schrag and J. D. Ferry Polymer J. 1970 1 742. 322 K. Osaki Y. Mitsuda R. M. Johnson J. L. Schrag and J. D. Ferry Macromolecules 1972 5 17. 323 K. Osaki J. L. Schrag and J. D. Ferry Macromolecules 1973 5 144. 324 T. C. Warren J. L. Schrag and J. D. Ferry Macromolecules 1973 6,467. 325 (a)K. Osaki and J. L. Schrag J. Polymer Sci. Polymer Phys. Edn. 1973 11 549; (6) K.Osaki Macromolecules 1973 5 141. 326 Y. Mitsuda K. Osaki J. L. Schrag and J. D. Ferry Polymer J. 1973 4 24. 327 Y. Mitsuda J. L. Schrag and J. D. Ferry Polymer J. 1973 4 668. 328 D. J. Massa J. L. Schrag and J. D. Ferry Macromolecules 1971 4 210. 329 K. Osaki and J. L. Schrag Polymer J. 1971 2 541. 330 A Peterlin J. Polymer Sci. Part B Polymer Letters 1972 10 101. Physical Properties of Polymers and their Solutions 213 measurements or formulate a better theory to incorporate concentrated solutions. The latter approach has been attempted by Everage and Gordon331 using continuum mechanics to relate the mechanical behaviour to molecular weight molecular weight distributions and the solution interaction parameter. The theory has produced a constitutive equation which remains to be tested.On moving from dilute to concentrated solutions and ultimately to the bulk polymer the importance of entanglements and molecular weight distribution increases. Results in concentrated solutions that aggregates form whose behaviour is closer to the Rouse theory and only when these break down to discrete molecules in dilute solution is there agreement with the Zimm theory. Melt viscosities qo of narrow-distribution polystyrene showed only a gradual change in slope when plotted as a function of chain length which was inter- preted on the basis of only partial entanglement.332 Earlier work on melt viscosities exhibited a sharp change in slope and the value of the critical entangle- ment molecular weight M was estimated from the break point.A re-examination of these data suggests that a continuous function of the form qo = KIMw+ K2Mw34 is a much better repre~entation.~~~ The steady-state compliance J is also a useful quantity which can be used to gain some insight to the molecular mechanism operatihg during the slow de- formation of the polymer. The dependence of J on A4 has been examined above and below M,. Above M, J was independent of M,334while below J was proportional to M in agreement with the Rouse theory. For four- and six-branched polystyrene the effective entanglement molecular weight Me above which J," became independent of M was given by Me = M,/$2 where $ is the volume fraction of the polymer.335 The concentration dependence of J has been found to be fairly complex.336 The importance and influence of branching in viscoelastic measurements has been stressed337 and J," has been found to be much higher for comb338 and four-branch polystyrene339 than for the linear chains.331 (a) R. J. Gordon and A. E. Everage J. Appl. Polymer Sci. 1971 15 1903; (b) A. E. Everage and R. J. Gordon ibid. 1972 16 1967. 332 R. I. Wolkowicz and W. C. Forsman Macromolecules 1971 4 184. 333 M. M. Cross Polymer 1970 11 238. 334 (a) N. Nemoto M. Moriwaki H. Odani and M. Kurata Macromolecules 1971 4 21 5; (6)W. C. Uy and W. W. Graessley ibid. p. 458; (c)S. Onogi T. Masuda and K. Kitagawa ibid. 1970 3 109; (d)M.Fujiyama and H. Awaya J. Appl. Polymer Sci. 1972 16 275; (e) H. Odani N. Nemoto S. Kitamura M. Kurata and M.Tamura Polymer J. 1970 1 356; cr)T. Masuda K. Kitagawa and S. Onogi ibid. p. 418; (g) N. Nemoto H. Odani and M. Kurata Macromolecules 1972 5 531. 335 L. A. Utracki and J. E. L. Roovers Macromolecules 1973 6 373. 336 (a) N. Nernoto T. Ogawa H. Odani and M. Kurala Macromolecules 1972 5 641 ; (6)Y. Einaga K. Osaki M. Kurata and M. Tarnura ibid. p. 635; (c) Y. Einaga K. Osaki M. Kurata T. Sugie and M. Tamura ibid. 1973 6 598; (d)M. Sakai T. Fujimoto and M. Nagasawa. ibid. 1972 5 786. 33' T. Masuda Y. Nakagawa Y. Ohta and S. Onogi Polymer J. 1972 3 92. 338 (a)T. Fujimoto H. Kajiura M. Hirose and M. Nagasawa Polymer J. 1972 3 181; (b)T. Fujirnoto H. Narukawa and M. Nagasawa Macromolecules 1970 3 57. 339 T. Masuda T. Ohta. and S. Onogi Macromolecules 1971 4 763.214 J. M. G.Cowie A new and potentially fruitful area of viscoelastic study has been opened with the availability of sharp-distribution polymer samples.340 Improved correlation between mechanical behaviour and molecular-weight distribution is now possible and by creating blends of these fractions the sensitive relations between composi- tion and properties can be demonstrated.341 The effect of chain-length dis- tribution has been most noticeable in the rubbery region342 and binary poly- styrene blends show a two-step rubbery plateau suggesting that two types of entanglement coupling may The effects of entanglement on relaxation times have been considered in a modified Rouse theory for polydisperse polymers.344 Order in Amorphors Polymers.-A growing body of evidence exists which supports the idea that limited order exists in the glassy and rubbery phases of amorphous polymers and that the random-coil conformation is not uniformly .~~~ adopted throughout the sample.Geiszler et ~1 detected a transition at ap- proximately 15-20 K above the glass-transition temperature 3,in butadiene- acrylonitrile copolymers which suggests that some degree of order was present in the rubbery region. As more energy was required to destroy these regions completely than was necessary to overcome the glass transition the existence of a mesomorphic phase above 5was postulated. This could be a result of ordering and association among nitrile groups in the polymer. The phenomenon has been confirmed in other polymers polyisobutene films were found to contain small amounts of highly ordered and a modular morphology has been detected in amorphous p~lystyrene,~~~~~~' poly(ethy1ene tere~hthalate),~~~ poly~arbonate,~~~~~~' polyethylene and Heueu rubber.352 Additional evidence of ordering in polyisobutene was obtained in a study of the glass transition using positron annihilation as a probe.353 Measurement of the distribution of positron lifetimes indicated that about 40-50 % of the sample was in an ordered state even up to 80-90 K above <.340 D. J. Plazek and V. M. O'Rourke J. Polymer Sci. Part A-2 Polymer Phys. 1971 9 209. 341 (a) See papers in 'Proceedings of 5th International Congress on Rheology,' ed. S. Onogi 1970 Vol. 3; (b) K.Murakami K. Ono K. Shiina T. Ueno and M. Matsuo Polymer J. 1971 2 698. 342 W. F. Knoff I. L. Hopkins and A. V. Tobolsky Macromolecules 1971 4 750. 343 T. Masuda K. Kitagawa T. Inoue and S. Onogi,Macromolecules 1970 3 116. 344 E. Menefee J. Appl. Polymer Sci. 1972 16 2215. 345 W. A. Geiszler J. A. Koutsky and A. T. Dibenedetto J. Appl. Polymer Sci. 1970 14 89. 346 S. Krishnamurthy and D. Mclntyre J. Polymer Sci. Part A-2 Polymer Phys. 1972 10 647. 347 A. Siegmann and P. H. Geil J. Macromol. Sci. 1970 E4,239. 348 G. S. Y. Yeh J. Macromol. Sci. 1972 B6,451. 349 P. J. Harget and A. Siegmann J. Appl. Phys. 1972,43,4357. 3s0 W. Lin and E. J. Kramer J. Appl. Phys. 1973,44 4288. 351 A. Siegmann and P. H. Geil J. Macromol. Sci. 1970 B4,557. 352 G. S.Markova Y. K. Ovchinnikov and E.B. Bokhyan Reprint 111-42 IUPAC Helsinki 1972. 353 J. R. Stevens and R. M. Rowe J. Appl. Phys. 1973,44,4328. Physical Properties of Polymers and their Solutions 215 To account for these observations a two-phase structural model has been proposed for the amorphous state.354 The model comprises ordered domains of 24nm diameter with distinct boundaries embedded in a truly random intergrain region. The excess free volume is believed to be located in the inter- domain region. A similar type of domain model for linear amorphous polymers introduces the concept of permanent and transitory interactions between loops and strands at the surfaces of quite large domains.355 Both types of interactions influence the mechanical properties but only the transitory interactions affect the flow be- haviour.Data supporting the concept of lateral ordering in amorphous polymers have been reviewed by Yeh356 and this together with the more recent work casts serious doubt on the idea of a structureless amorphous state. Ordered structures may arise from (i) lateral ordering (ii) helical sequences or (iii) departure from random-coil statistics in the melt and indeed Boyer is of the opinion3” that an irregularly folded chain is a more likely conformation in the amorphous state than a random coil. 2 Molecular Motions Glass Transitions.-A theory of the glassy state based on a hole concept for simple liquids has been offered by Nose.358 The glass is represented as a quasi- equilibrium state with holes frozen into it and these distinguish it from the liquid or crystalline state.Calculation of the pressure dependence of shows that (dT$dP) < A/l/Aa,where Aa and A/l are the thermal expansion and compressi- bility changes on moving from a glass to a rubber. The isofree volume theory predicts an equality for this relation and the inequality suggests that either the entropy or the enthalpy are the controlling factors and so the approach to an understanding of 3 should be isoenergetic. Support for the isoenergetic theory comes from Ichihara et af.,359who have studied the compression of glassy polymers. These authors conclude that if pressure is applied to all polymer glasses they will eventually reach a state of compaction in which all possess the same enthalpy.They have confirmed the predictions for (d T‘dP). Consequently they favour the isoenergetic description of 3,for this reason and also because no compaction of a glass should occur at if the fractional free volume is constant. This would be contrary to experimental 354 G. S. Y. Yeh J. Macromol. Sci. 1972 B6,465. 355 (a) S. M. Aharoni J. Appl. Polymer Sci. 1972 16 3275; (6) S. M. Aharoni ibid. 1973 17 1507. 356 G. Yeh Crit. Rev. Macromol. Sci. 1972 1 173. 357 R. F. Boyer presented at the Seventh Swinburne Award Address London 1972. 358 (a)T. Nose Polymer J. 1971 2 124; (b)T. Nose ibid. p. 427; (c)T. Nose ibid. p. 437; (6)T. Nose ibid. p. 445; (e)T. Nose ibid. 1972 3 1 ; v) T. Nose ibid. p. 196. 359 (a) S. Ichihara A.Komatsu Y. Tsujita T. Nose and T. Hata Polymer J. 1971 2 530; (b)S. Ichihara A. Komatsu and T. Hata ibid.,p. 644; (c)S. Ichihara A. Komatsu and T. Hata ibid. p. 650. 216 J. M. G. Cowie observation. The theory bears some similarity to the Miller360 criterion for T which is expressed in terms of the conformational entropy. Quach and Si~nha~~' have adopted the view that although experiment confirms the inequality (dT$dP) < (Ap/Aa),the experimental definitions of the various quantities may be in error. New definitions were presented within the framework of a modified cell theory for liquids and polymers,362 which revert once more to agreement with the free-volume theory. Re-examination of these definitions has led G~ldstein~~~ to the opinion that the original ones are appropriate and the evidence in support of an isoenergetic theory is correct.The debate continues. and The chain-length dependence of .T for p~lypropene~~~poly(viny1 was found to bear some relation to the predictions of the Gibbs- Dimarzio theory but this was not true for poly(dirnethylsilo~ane),~~~ for which was only slightly dependent on M and even then only for quite short chains. When ionic polymers were examined the relationship was more ~omple~~~~,~~~ and of the general form 7 = 4q/4fB where A and B are constants characteristic of the polymer qis the cation charge and a is the internuclear distance between cation and anion at closest approach. A review paper by Ei~enberg~~~ contains a more detailed description of these systems.The effect of tacticity on T has been examined for p~lypropene,~~~ poly(t-butylethylene oxide),370 poly(methy1 metha~rylate),~~ and poly(ethy1 a-chloro- a~rylate).~~~ The expected difference in was found for the two a-substituted polymers with the syndiotactic form having the higher <. The reverse was true for poly(t-butylethylene oxide) where <for the isotactic form was largest. A large part of the research effort in this area has been centred on the effect of crystallinity on q. Jordan et ~1.~'~ have examined the influence of side-chain crystallinity on the T of selected copolymers of alkyl acrylates incorporating either n-octadecyl acrylate or vinyl stearate as comonomer. The expected increase in was observed when the concentration of the comonomer was 360 A.A. Miller Macromolecules 1970 3 674. 361 (a)A. Quach and R. Simha J. Appl. Phys. 1971,42,4592; (b)A. Quach and R. Simha J. Phys. Chem. 1972,76,416. 362 T. Somcynsky and R. Simha J. Appl. Phys. 1971,42,4545. 363 M. Goldstein J. Phys. Chem. 1973 77 667. 364 J. M. G. Cowie European Polymer J. 1973,9 1041. 365 G. Pezzin F. Zilio-Grandi and P. Sanmartin European Polymer J. 1970 6 1053. 366 J. M. G. Cowie and I. J. McEwen Polymer 1973 14 423. 367 A. Eisenberg H. Matsuura and T. Yokoyama Polymer J. 1971 2 117. 368 A. Eisenberg and K. Takahashi J. Non-Cryst. Solids,1970 3 279. 369 A. Eisenberg Macromolecules 1971,4 125. 370 N. Doddi W. C. Forsman and C. C. Price Macromolecules 1971,4 648. 37' S. Bywater and P.M. Toporowski Polymer 1972 13 94. 372 B. Wesslen R. W. Lenz W. J. McKnight and F. E. Karasz Macromolecules 1971,4 24. 373 (a)E. F. Jordan D. F. Feldeisen and A. N. Wrigley J. Polymer Sci. Part A-I Polymer Chem. 1971 9 1835; (6) E. F. Jordan B. Artymyshyn and A. N. Wrigley ibid. p. 3349; (c)E. F. Jordan ibid. p. 3357. Physical Properties of Polymers and their Solutions 217 sufficient to allow significant crystallization of its side chain to occur. Multiple transitions are commonly observed in polymer samples but when there is ap- preciable crystallinity the interpretation of the nature of each transition can become confused. Nowhere is this more true than in the data reported for polyethylene; approximately 50 papers have appeared since 1953 and as yet there is no unanimous acceptance of one value for T.The most frequently reported centre on 250 195 and 150 K and the lowest one is usually termed the y-relaxation. especially the earlier reports favour the higher value as T for semi-crystalline polyethylene and the middle value for amorphous samples. Recently Stehling and Mandelke~n~~~ and Beatty and Kara~z,~~~ in particular have come to the conclusion that the y-transition is the q for amorphous material. This is based on observed heat-capacity and expansion- coefficient changes in this temperature region of sufficiently high magnitude to represent a glass transition. The reader is recommended to recent reviews by B~yer,~~~ who has gathered and attempted to weigh the extensive data.It is soon obvious that an unambi- guous interpretation of multiple transitions is extremely difficult and Boyer has applied certain criteria which are believed to characterize the glass transition. These are (i) AaT = 0.113 (ii)a < = 0.164 and (iii) AC,T = 105J g- '. Sharma et aL3" have gathered relevant data for a wide range of polymers and have shown that criterion (i) is invalid as the factor is not constant but varies with q. The criticism is partially offset38' by using integrals in the expression to allow for the temperature dependence and free volume. B~yer~~~~ has concentrated mainly on C data to make the point that he favours T = 195 & 10 K for amorphous polyethylene. The implication of this proposal is that a sub-glass molecular motion must then be associated with a heat-capacity change and theoretically it could be detected using differential scanning calorimetry.No one has reported this as yet probably because sufficiently sensitive instrumentation has only recently become available. Methods of predicting q for polymers and copolymers have been proposed. We~land~~' has presented a method based on group increments which are assigned and totalled for a given chain. JohnstonjE3 has developed an additive 374 K. H. Illers Kolloid-Z. 1972 250 426. 375 G. T. Davis and R. K. Eby J. Appl. Phys. 1973,44,4274. 3'6 S. H. S. Chang Amer. Chem. SOC. Div. Polymer Chem. Polymer Reprints 1972 13 322. 377 F. C. Stehling and L. Mandelkern Macromolecules 1970 3 242. 378 C.L. Beatty and F. E. Karasz Bull. Amer. Phys. Soc. 1971 16 1391. 379 (a)R. F. Boyer Thermal Analysis 1971,3 3; (6) R. F. Boyer presented at the Seventh Swinburne Award Address London 1972; (c) R. F. Boyer J. Macromol. Sci. 1973 B7 487; (d)R. F. Boyer Macromolecules 1973 6 288. 380 S. C. Sharma L. Mandelkern and F. C. Stehling J. Polymer Sci. Part B Polymer Letters 1972 10 345. 38' (a) R. Simha and C. E. Weil J. Macromol. Sci. 1970 B4,215; (6) R. Simha ibid. 1971 5 331. 382 H. G. Weyland P. J. Hoftyzer and D. W. Van Krevelen Polymer 1970 11 79. 383 (a) N. W. Johnston J. Macromol. Sci. 1973 A7 531; (b) N. W. Johnston Macro-molecules 1973 6 453. 218 J. M. G. Cowie relationship for copolymers using the values for the homopolymers and the sequence distribution of the comonomers in the chain.Neutron Scattering.-Access to cold neutron sources (i% 0.5 nm) has provided an exciting new probe for the study of motion and chain dimensions in polymer systems. Low-angle scattering from deuteriated poly(methy1 methacrylate) glasses384 and polystyrene in deuteriated polystyrene matrix,385 in solution,386 and in the have yielded measurements of the radius of gyration in good agreement with those estimated by conventional procedures. Low-frequency motions in polymers have been investigated using quasi-elastic neutron scattering and an excellent review by Allen and Higgin~~~~ covers much of the work including polymer systems up to 1973. It can be expected that this technique will contribute valuable information to the study of long-range conformational motion and side-group movement in polymeric chains.Dynamic Mechanical Measurements.-Thermomechanical spectra yield infor- mation on molecular motions in a polymer sample throughout the whole range of mechanical states. Dynamic methods such as the torsional pendulum generate these spectra but may require substantial specimen sizes. A semi-micro method developed by Gillham called torsional braid analysis (TBA) overcomes this by using a composite specimen comprising an inert-glass braid impregnated with polymer. Additional advantages are that non-self-supporting polymers are easily handled and in situ reactions can be studied during a measure- ment. The method however only gives relative values of the mechanical prop- erties.Gillham389 has reviewed the technique and its applications up to 1972. Since then some interesting work has been reported on an unusual family of poly(carbaborane4oxane) high-temperature elastomers.390 Two different types of carbaborane cages were incorporated in the chain and compositional changes could be used to alter the melting and glass-transition temperatures. TBA is readily suited to the measurement of the thermomechanical response of high-temperature and relatively intractable polymers such as polynorborna-2,5- diene,391poly(phenylq~inoxaline),~~~ and has been used to and polyimide~,~~~ 384 R. G. Kirste W. A. Kruse and J. Schelten Makromol. Chem. 1972 162 299. ’13’ D. G. H. Ballard G. D. Wignall and J. Schelten European Polymer J.1973 9,965. 386 (a)H. Benoit D. Decker J. S. Higgins C. Picot J. P. Cotton B. Farnoux G. Jannink and R. Ober Nature Phys. Sci. 1973,245 13; (6)J. P.Cotton B. Farnoux G. Jannink and C. Strazielle J. Polymer Sci. Polymer Symposia 1973 42 981. ”’ (a)J. P. Cotton B. Farnoux G. Jennink J. Mons and C. Picot Compf. rend. 1972 275 C 175; (6)J. P. Cotton B. Farnoux G. Jennink C. Picot and G. C. Summerfield J. Polymer Sci. Polymer Symposia 1973 42 807. 388 G. Allen and J. S. Higgins Reports Progr. Phys. 1973 36 1073. 389 J. K. Gillham Crif. Rev. Macromol. Sci.,1972 1 83. 390 (a) M. B. Roller and J. K. Gillham J. Appl. Polymer Sci. 1972 16 3095; (6) M. B. Roller and J. K. Gillham ibid. p. 3105; (c) M. B. Roller and J. K. Gillham ibid. 1973 17 2141.391 M. B. Roller J. K. Gillham and J. P. Kennedy J. Appl. Polymer Sci. 1973 17 2223. 392 (a) J. M. Augl and H. J. Booth J. Polymer Sci.,Polymer Chem. Edn. 1973 11 2179; (b)J. M. Augl and H. J. Booth ibid. p. 2195. 393 J. K. Gillham J. Appl. Polymer Sci. 1972 16 2595. Physical Properties of Polymers and their Solutions 219 monitor thermal cross-linking and chain-cyclization reactions394 of polymers on the braid. Two damping peaks detected during the curing of thermosetting resins have been identified as representing the gel point and the glass transition in the system ;395 thus TBA is capable of following the property changes in a cross- linking resin and shows that this depends on the curing temperature. For temperatures above only gelation is observed ; for temperatures below only vitrification occurs ;and at intermediate temperatures both are detected.TBA has proved useful for the thermomechanical study of poly(dimethy1-siloxane) liquids366 and for investigating the damping at the melting transition of poly(ethy1ene oxide).396 A large number of papers have been devoted to the systematic and detailed investigation of the mechanical response of various polymer systems but as they are of specific interest rather than general they will not be catalogued here. Of these a few of special interest describe the polarization of polymer films to form electrets.397 A strong d.c. electric field is applied at high temperature and then the sample is cogled under an applied electric field. The depolarization current is then measured as a function of temperature and sudden changes occur which can be correlated with molecular motion in the glass or crystalline phase.The method could prove more sensitive to low-frequency motions than the established viscoelastic techniques. Elastomers.4ne point of contention remaining unresolved in the molecular theory of rubber elasticity centres on the form of the expression for the free energy of network deformation. The incorporation of a logarithmic form and the value of B in the equation AG,,JRT = (u/2)(lf + i;+ A -3) -Bu In Axl.,,lz are open to argument. As it is difficult to prepare networks with a precisely known structure to allow calculation of u the number of elastically effective cross-links one must use methods which eliminate u from the calculations.This can be achieved by comparing networks swollen in different solvents and cross- linked in the swollen state and in this way Froelich et have confirmed the need for the logarithmic term. They also estimate B = 2/f wheref is the function- ality of the cross-link. A general conclusion arrived at by a number of workers is that networks prepared in solution have different topologies from those cross- linked in the bulk state and that the former behave ideally obeying gaussian theory.399 This could be due to a lower proportion of physical entanglements 394 J. K. Gillham and K. C. Glazier J. Appl. Polymer Sci. 1972 16 2153. 39s P. G. Babayevsky and J. K. Gillham J. Appl. Polymer Sci. 1973 17 2067. 396 B.Hartmann Polymer 1972 13,460. 397 (a)T. Takamatsu and E. Fukada Polymer J. 1970 1 101 ;(6) J. van Turnhout ibid. 1971 2 173; (c) E. Fukada and T. Sakurai ibid. p. 656; (d)E. Sacher J. Macromol. Sci.,1972 B6,365; (e)E. Sacher ibid. p. 377; U,H. Sasabe and S. Saito Polymer J. 1972 3 624. 398 D. Froelich D. Crawford T. Rozek and W. Prins Macromolecules 1972 5 100. 399 (a)R. M. Johnson and J. E. Mark Macromolecules 1972,541 ;(b)C. Price G. Allen F. de Candia M. C. Kirkham and A. Subramanian Polymer 1970 11 486; (c) W. Brostow Macromolecules 1971,4 742; (4F. de Candia ibid. 1972,5 103 220 J. M. G. Cowie which contribute to the non-ideality of the network being formed in the solution cross-linking process. Alternatively the non-ideal behaviour may originate from ordering in the amorphous state which is reduced when cross-linking of the swollen polymer takes place.Some evidence of the presence of mesomorphic phases in cross-linked polymers has been obtained by Prin~.~" The Flory-Huggins theoretical approach to network swelling has been developed by Treloar for cylinders subjected to torsion and axial e~tension.~" The method has been successfully applied to the common p~lydienes.~~~ Thermo-elastic measurements of rubber in torsion403 gave a value of d In (r2)o/dT of +0.43 x K-' in good agreement with later results by Price et uL404 Contributions to theory have come from Goebel and T~bolsky,~'~ who have formulated a new equation for the volume dilation of a rubber during extension and Ei~hinger,~'~ who used graph theory to calculate distribution functions for perfect phantom networks.Some work on the properties of interpenetrating networks (IPN) has been published. Sperling et uL407 prepared IPN from the incompatible pair poly(ethy1 acrylate) and poly(methy1 methacrylate) but could only detect one broad glass transition rather than two. Bamford et aL408found two values when poly(viny1 trichloroacetate) was cross-linked with polystyrene or poly(methy1 methacrylate) and a system similar to an AB block copolymer with a domain structure was formed. A more careful definition of the types of IPN by S~erling,~" points out that a 'joined' structure is composed almost entirely of intramolecular cross- links and is easier to prepare. A 'sequential' IPN is one in which both polymers are in the network form and ideally all cross-links are intermolecular.Sequential IPN of poly(dimethylsi1oxane) and poly(methy1 methacrylate) were quite opaque had two values and behaved somewhat like a thermoplastic elastomer. Electron micrographs support the interpenetrating and reveal a cellular morphology of about 100nm and phase domains of lOnm in some IPN.41 Limited control over the morphology could be effected by altering the composition of the network components. 400 (a) E. Pines and W. Prins J. Polymer Sci. Part A-2 Polymer Phys. 1972 10 719; (6) M. Ilavsky and W. Prins Macromolecules 1970 3 425. 40 ' (a)L. G. R. Treloar Polymer 1972,13,195; (6)K. M. Loke M. Dickinson and L. G. R. Treloar ibid.p. 203. 402 A. N. Gent and T. H. Kuan J. Polymer Sci. Polymer Phys. Edn. 1973 11 1723. 403 P. H. Boyce and L. G. R. Treloar Polymer 1970 11 21. 404 C. Price K. A. Evans and F. de Candia Polymer 1973 14 338. 405 (a)A. V. Tobolsky and J. C. Goebel Macromolecules 1970 3 556; (6) J. C. Goebel and A. V. Tobolsky ibid. 1971 4 208. 406 (a)B. E. Eichinger Macromolecules 1972,5 496; (6) B. E. Eichinger ibid. p. 647. 407 L. H. Sperling D. W. Taylor M. L. Kirkpatrick H. F. George and D. R. Bardman J. Appl. Polymer Sci. 1970 14 73. 408 C. H. Bamford G. C. Eastmond and D. Whittle Polymer 1971 12 247. 409 L. H. Sperling and H. D. Sarge J. Appf. Polymer Sci. 1972 16 3041. 'I0 A. J. Curtis M. J. Covitch D. A. Thomas and L. H. Sperling Polymer Eng. Sci. 1972 12 101.(a) V. Huelck D. A. Thomas and L. H. Sperling Macromolecules 1972 5 340; (6) V. Huelck D. A. Thomas and L. H. Sperling ibid. p. 348. Physical Properties of Polymers and their Solutions 221 Crystallization.-The Avrami equation is still widely used in the analysis of crystallization kinetics but it has many limitations. The exponent n is not always an integer as predi~ted,~' 2414 although it has been used successfully for poly- (propene oxide)?" nylon 6,416nylon 8,417and nylon 12,418if one allows for the effects of secondary crystallization. The temperature of crystallization also altered n;414,416 for polyethylene Mandelkern4' found that the Avrami equation was adhered to at high temperature while at lower temperatures the onset of deviations from the theory was a function of both A4 and the crystallizing temperature.Price and Th~rnton~~' found that n was only slightly affected if the crystal-growth pattern was assumed to be rod-like rather than spherical. A more radical approach was made by Danusso et uI.,~~'who proposed a new two-range nucleation model. They argued that the Avrami theory leads to a two-parameter equation with an ambiguous character for n which is related to both growth and nucleation. A three-parameter equation was developed to overcome this defect but it appears to have excited little interest from workers in the field. More recently Gandica and Magi11422 developed a universal rela- tionship for polymer crystallization which leads to a corresponding states equation and can be used to provide a master curve for polymers at various temperatures crystal modes and tacticities.Booth and Hay4' 'believe that fractionation during isothermal crystallization may complicate the kinetic analysis but the contrary view is expressed by Kamide and Yamag~chi.~'~ Fractionation is sometimes claimed to be respon-sible for lamellar but again this is disputed by those who prefer to think this is due to annealing of the An attempt to resolve the problem of fractionation effects426 was unsuccessful because the data could not be inter-preted in a satisfactory way. La~ritzen~'~ has developed a kinetic theory of lamellar growth rate based on the assumption that isothermal crystallization proceeds through .the growth of lamellae which then provide a substrate for 'I2 C.Borri S. Briickner V. Crescenzi G. Della Fortuna A. Mariano and P. Scarazzato European Polymer J. 197 1 7 15 15. 'I3 A. Booth and J. N. Hay Polymer 1971 12 365. 'I4 S. Gogolewski and E. Turska J. Appl. Polymer Sci. 1972 16 1959. 'I5 (a)C. Booth D. V. Dodgson and I. H. Hillier Polymer 1970 11 11 ;(6)D. R. Beech and C. Booth ibid. 1972 13 355. 416 (a)E. Turska and S. Gogolewski Polymer 1971,12,616; (6)E. Turska and S. Gogolew-ski ihid. p. 629. 'I7 G. Ceccorulli and F. Manescalchi Makromol. Chem. 1973 168 303. F. Manescalchi R. Rossi and A. Mattiussi European Polymer J. 1973 9 601. 'I9 E. Ergoz J. G. Fatou and L. Mandelkern Macromolecules 1972 5 147. 420 F. P. Price and J. M. Thornton J. Appl.Phys. 1973,44 4312. ''I F. Danusso G. Tieghi and V. Felderev European Polymer J. 1970 6 1521. 422 A. Gandica and J. H. Magill Polymer 1972 13 595. 423 K. Kamide and K. Yamaguchi Makromol. Chem. 1972 16 219. '"T. Kawai M. Hosoi and K. Kamide Makromol. Chem. 1971 146 55. 425 A. Mehta and B. Wunderlich Makromol. Chem. 1972 153 327. 426 F. C. Stehling E. Ergoz and L. Mandelkern Macromolecules 1971 4 672. 427 (a)J. I. Lauritzen J. Appl. Phys. 1973,44,4353; (6)J. I. Lauritzen and J. D. Hoffman ibid. p. 4340. J. M. G. Cowie further growth; this remains to be tested. Some progress in the theoretical description of copolymers has been made.428 The experimentally determined melting temperature of a polymer T is normally lower than the thermodynamic melting point T; of the perfect crystal.The major limiting factor to the attainment of T is the thickness of lamellar crystals and the difference between T and Ti can be expressed in terms of the lamellar thickness [ and end interfacial energy ere. In the Flory-Vrij theory partially crystallized chains are taken into account and the equation is T = T:[1 -(2ae/AhC)]/[l -(RT In I/AhtC)] where Ah is the enthalpy of fusion t the number of times a chain folds into a crystal and I is a parameter to allow for different degrees of order in the crystal. Thus a low T can be associated with a high 0 and a low [. The main emphasis has been on the determination of oeand its behaviour with chain length. Booth and his co-~orkers~~’ have concentrated predominantly on poly(ethy1ene oxide).With short chains available the possibility arises of obtaining crystallites with different chain-folding morphologies. They found that low molecular weight polymer formed lamellar crystals with thicknesses which corresponded to extended chains once folded twice folded and higher. The crystal type was also a function of M the molecular weight distribution and the crystallizing temperature. An increase in oewith M was observed but credecreased when the extent of folding increased for any given chain length. fa to^^^' agrees essentially with this general trend. The effect of different end groups was also Substitution of the OH terminal group by C1 phenoxy- or acetoxy-groups raised T because of an increase in cre.This led to the conclusion that hydrogen- bonding in the hydroxy-terminated polymer stabilizes the interfacial layers.For BAB poly(ethy1ene oxide-b-propylene oxide) cre was found to increase markedly432 as the length of the propylene oxide block increased and T was depressed accordingly. The depression of the corresponding ABA blocks was not nearly so great. 428 E. Helfand and J. I. Lauritzen Macromolecules 1973 6 63 1. 429 (a)D. R. Beech C. Booth D. V. Dodgson R. R. Sharpe and J. R. S. Waring Polymer 1972,13,73;(b)D. R. Beech C. Booth I. H. Hillier and C. J. Pickles European Polymer J. 1972 8 799; (c) D. R. Beech C. Booth C. J. Pickles R. R. Sharpe and J. R. S. Waring Polymer 1972 13 246; (d)P. C. Ashman and C. Booth ibid. p. 459. 430 J. M. Barrales-Rienda and J.G. Fatou Polymer 1972 13 407. 431 (a)C. Booth J. M. Bruce and M. Buggy Polymer 1972,13,475;(b)P. C. Ashman and C. Booth ibid. 1973 14 300. 432 (a) C. Booth and C. J. Pickles J. Polymer Sci.,Polymer Phys. Edn. 1973 11 249; (b) C. Booth and D. V. Dodgson ibid. p. 265.