GENERAL DISCUSSIONProf. R. L. Scott (University of California, Los Angeles) said : In the past some ofus have regarded concepts like the " regular solution ", the van der Waals mixture,and the Flory-Huggins equation for chain-molecules as arising from " random " modelsin the sense that these correspond closely to molecular distributions produced entirelyby the steep repulsive forces unperturbed by the weaker intermolecular attractions.However, I do not want to get involved in a semantic quibble; strictly speaking" random mixing " cannot be so interpreted and must be restricted to the definitiongiven in Rowlinson's paper.Nonetheless, it is worthwhile to consider the basic assumptions which lead to theScatchard-Hildebrand equation. Scatchard assumed " (1) that the mutual energyof two molecules is a function only of their relative position and orientations .. . anddepends not at all on the nature of the other molecules between or around them or onthe temperature; and (2) that the distribution function for any pair of molecules isalso independent of the temperature and the nature of the other molecules present."When this derivation is carried out explicitly in terms of pair distribution functions 2 9 3"Correct " 41 Irk 912rrrI .rrFIG. 1.-Radial distribution functions (schematic) for a dilute solution of large spheres in smallspheres (oZ2 = 30, The predictions of the random mixing and regular solution approximationsare compared with more nearly correct functions.G. Scatchard, Chem.Rev., 1931, 8, 321 ; Trans. Faraday SOC., 1937,33, 160.J. H. Hildebrand and S. E. Wood, J. Chem. Phys., 1933,1,817.J. H. Hildebrand and R. L. Scott, Regular Solutions (Prentice Hall, Englewood Cliffs, N.J.,1962), chap. 7.7GENERAL DISCUSSION 77it is clear that this is not a " random mixture ". Specifically, in the random mixturethe pair distribution function gas(r) is the same function for all a/? (e.g., 11, 12, and 22)in a particular mixture, but may vary with the composition (e.g., with mole fractionx,). Conversely the assumptions of Scatchard and Hildebrand require that gas beindependent of composition, although g, 1, g12, and g22 can be different ; one usuallyassumes that they scale with the collision diametersFig. 1 illustrates these differences schematically for a mixture of a dilute solute (2)of large spheres in a solvent (1) of small spheres.The various gas are sketched for the" random mixture " and the " regular solution " and compared with the " correct "function. At infinite dilution, the three gll are identical. The g12 and g22 for therandom mixture, the same as g,,, are patently absurd. The scaled g,, and g22 forthe regular solution are probably nearly right in the region of thc first maximum, butquite unrealistic at greater distances. The fact that the regular solution gas makessome allowance for differences in size may perhaps account in part for the somewhatbetter GE values in Rowfinson's table 1.The comparisons in Rowlinson's table 1 might be extended to include the excessentropy and enthalpy separately.For the van der Waals (one-fluid and two-fluid)and Flory-Huggins approximations the free energy coefficient A arises entirely fromentropy contributions, while that for the regular solution (more accurately thesolubility parameter) approximation is entirely an energy contribution. Singer'sMonte Carlo calculations suggest that the excess Gibbs free energy GE is the smalldifference between larger RE and TsE terms.such that ga&) = f(r/aag).Prof. G. Rehage (University of Ckausthal) said : Concerning the applicability of theHuggins-Flory-theory to non-polar, binary liquid mixtures with phase separation,the critical point of phase separation is theoretically given byV,, and VO2 are the molar volumes of the pure liquid components, #2,k is the volumefraction of the component 2 at the critical point.We assume that the component 2is an associated substance with unknown molecular weight. From the measuredvalues of $2,k and Vo, one can calculate Voz, when the theory is applicable. Onecan calculate from the known volume of the monomer the degree of association ofthe component 2. For the phase separation behaviour of sulphur in the solventschlorobenzene, mustard oil, benzylchloride, aniline ; p,p'-dichlorodiethylsulphide(mustard gas), benzene and toluene we found from the known atomic volume of thesulphur from the number of sulphur atoms in the molecule the value of 7.8+ 1.7.The critical points were in the temperature range 117-180°C. From cryoscopicand ebullioscopic determinations of the molecular weight in the range from -75to +277"C the molecular size of sulphur is S8 in all solvents.Thus, the Huggins-Flory-theory gives good accordance with the directly measured values. Incidentally,all investigated systems with sulphur had an upper critical p0int.lDr. K. N. Marsh (University of Exeter) said: Some recent calculations of theexcess functions of simple binary liquid mixtures lead to the conclusion that it is theprescription for extending the equations of state to the mixtures which primarilydetermines the excess functions and not the equation of state itself. The mostsuccessful prescriptions to date are those due to van der Waals as discussed byRowlinson. Young has discussed the adequacy of the v.d.W.prescription for theG. Rehage, unpublished results.K. N. Marsh, M. L. McGlashan and C. Warr, Trans. Furday SOC., submitted78 GENERAL DISCUSSIONa term in explaining the critical phenomena for mixtures of octamethylcyclotetra-siloxane (a large globular molecule) with small globular molecules. The van derWads prescription for the b term has been found to be less adequate for these mixtures.The equation of state for a hard sphere mixture given by Lebowitz and used in thetheory by Snider and Herrington can be considered as another prescription forthe b term. This prescription gives excess volumes for the above mixtures an order ofmagnitude better than those calculated using the v.d.W. b prescription. Calculationsalso show that for these systems, the hard sphere mixture equation gives a muchbetter representation of the composition dependence of the excess entropy comparedwith the v.d.W.prescription for the b term.Dr. M. Rigby (Queen Elizabeth College, London) said : Rowlinson has mentionedthe difficulty of assessing the effects of molecular shape. It is relevant to consider theequation of state of non-spherical hard molecules since it seems probable that thesewill provide a basis for describing the geometrical structure of real non-sphericalmolecules, as hard spheres do for spherical molecules. The scaled particle theoryhas been applied to systems of convex hard molecules of generalized shape andleads to an equation of state in which the molecular shape may be characterized by asingle parameter similar to that introduced by I ~ i h a r a .~It was not previously possible to investigate the accuracy of the scaled particletheory for such systems since no exact data were available either for the equation ofstate or for the higher virial coefficients. Recent calculations of the third andfourth virial coefficients for prolate spherocylinders have shown that the theoryappears to be applicable to non-spherical hard molecules with similar accuracy tothat found for hard spheres, and it may thus provide a useful basis for the study ofnon-spherical molecules. Preliminary calculations with an equation of state of thegeneralized van der Waals type using the scaled particle equation for the hard corecontributions show that the volumetric properties are not very sensitive to the valueof the shape parameter.Prof.C. Domb (King’s College, University of London) said : In his paper, Edwardsmakes the statement : “ My belief is that until the continuum models of polymershave been fully understood one will not obtain mastery over the problems of realpolymers. . . ”. I should like to comment on what information can be derived aboutthe continuum model from lattice models. One always has a slightly guilty feelingwhen working with a lattice model since the lattice is clearly an artificial construction.Nevertheless, if one allows walks to second- and higher-order neighbours of the lattice,it is possible to simulate any magnitude of excluded volume and to obtain a coherentseries of lattice approximations to any given continuum model.Some possibleexamples of this are illustrated in fig. 1.In practice, this has not been pursued very far but nevertheless calculations havebeen undertaken for a variety of different lattices. The particular lattice structurehad no physical significance, and hence any properties which depend on latticestructure cannot have any relevance for the continuum model. However, a numberof properties have emerged which are apparently independent of lattice structure,J. L. Lebowitz, Phys. Rev. A , 1964, 133: 895.N. S. Snider and T. M. Herrington, J. Chem. Phys., 1967, 47,2248.R. M. Gibbons, Mol. Phys., 1969, 17, 81.A. Tsihara and T. Hayashida, J. Phys. SOC. Japan, 1951,6,40.M. Rigby, J. Chenr. PJys., in pressGENERAL DISCUSSION 79and it is reasonable to expect that these same properties are valid for a continuummodel.for which parallelproperties have emerged as being independent of lattice structure, and we also havethe advantage of exact analytic solutions in two dimensions which confirm thisindependence.In table 1, I have drawn up a list of quantities which seem to be latticeAnalogy with the Ising model is helpful in this connection+a, allowed steps; 0, forbidden pointsFIG. 1.-Lattice models which approach more closely to a continuum.TABLE 1 .-SELF-AVOIDING WALK-LATTICE INDEPENDENT PARAMETERSking analogueno. of walks Cn ~ / i ’ ’ t l U (susceptibility)no. of polygons un -pn+ (specific heat)mean sq. length { R i ) WnY (range of coherence)probability distribution Pn(R) mexp - (R/C,)~ (spin pair correlation function)correlation between steps (ujui) -(I@, t)/n”-Y (critical equation of state)(s = iln, t = (n-j)/n)(and other moments of t,9(s, t ) )radius of gyration <Sn”)/<Rn2>independent and have listed the Ising analogues.The exponents a, p, and y havebeen known for some time (particularly y which was first introduced by Florymany years ago). However, the parameter 6 which represents the deviation from aGaussian distribution, is not so well known ; it has the value 4 in two dimensions and512 in three dimensions. More recently it has been suggested that the completeC. Domb, A&. Chem. Phys., 1969, 15,229.P. J. Flory, J. Chem. Phys., 1949, 17, 303.C . Domb and F.T. Hioe, J . Chem. Phys., 1969,51,1915, 192080 GENERAL DISCUSSIONcorrelation function between steps is also lattice independent and is parallel to thecritical equation of state for the Ising model. It is the moments of this function whichare related to ratios like ( S z } / ( R : } , and which are thus also lattice independent.Dr. J. L. Martin (King’s College, University of London) said: It has often beenconsidered that the behaviour of quite short self-avoiding walks on a regular latticewill provide a reasonable picture of a polymer chain in dilute solution, provided thereis no strong mutual attraction between remote parts of the chain. If this is true, thencorrelations between different parts of such walks are of interest.The simplest typeof correlation is that between the two extremities of a walk. To study this, the end-point distribution is required : cn(P) is the total number of self-avoiding walks of nsteps starting at the origin and finishing at the point P. Distributions for the commonlattices have been obtained by a combination of recurrence relations and directenumeration ; and the computation has been pushed almost to the limit of computercapacity. The work has been done in collaboration with a student, M. G. Watts.The accepted wisdom is that the mean-square end-to-end length of a walk of nsteps on a chosen latticeu, = (&@)r?)/(Cc,(~)) - nY . const.P Pas n--+ 00, where y depends only on the dimensionality of the lattice. We have made amuch more restrictive assumption (essentially that u, is a coefficient in a generatingfunction with a “ simple ” singularity) :un/un-l = 1 + y/(n - a) + O( 1 /n3).Such an assumption leads to(nn + 1 - ~ n > ( u n - un - 1)u,2-un+1un-1 *Yn == y+O(l/n2),whence we obtain a sequence of estimates for y.Some values of y- for the triangular and FCC lattices aretriangular FCCn Yn9 1.487910 1.4860I 1 1.487512 1.4874n Yn6 1.20577 1.20418 1.20269 1.2013These sequences suggest the estimates :with considerable confidence, since the yn are not varying rapidly.y(triangu1ar) - 1.487 ; y(FCC) somewhat less than 1.20,Prof. W.H. Stockmayer (Dartmouth College, U.S.A.) said: Monte Carlo methodswill be very useful in confronting the complicated dynamical problems posed bychain entanglements.Suitable formulations of chain dynamics for this purpose maybe of several kinds. For example, chains on simple lattices can diffuse by a successionof randomly selected local jumps, as in the studies of Verdier 1*2 and MonnerieP. H. Verdier and W. H. Stockmayer, J . Chem. Phys., 1962, 36,227.P. H. Verdier, J. Chem. Phys., 1966, 45, 21 18 and 2122GENERAL DISCUSSION 81and Geny. g 2 Alternatively, an appropriate Langevin equation for the familiarbead-and-spring model invites Monte Carlo treatment of the random Brownianforces. The latter method has already been applied to an entanglement problem(unwinding of a chain wrapped around a stick, as an idealization of DNA unwinding)by Simon and Zimm.4Prof.P. J. FIory (Stanford University) said: With reference to Domb's andMartin's calculations on self-avoiding random walks, I would point out that forpolymer chains convergence to an expression of the form (r2)ccnY, in which y = 6/5,is attained only for very large values of the number of bonds n. Treatment of theeffect of excluded volume in the smoothed density approximation clearly shows thisto be so. According to numerous experiments on polymer solutions, even at chainlengths n = 104-105 the empirical value of the exponent y falls appreciably below1.20. Certainly, a bond of a real chain should not be equated to one step of a randomwalk, but the facior relating them should be of the order of 10 and not 103-104.That the limiting form of the foregoing relationship can be ascertained with fewerthan 20 steps is diacult to reconcile with other evidence, both from experiment andfrom theory.Dr.R. F. T. Stepto (University of Manchester) said : In agreement with Flory's remarks,one thinks that since the advent and acceptance of the rotational-isomeric state modelfor real polymer chains, the problem of excluded volume would be approached moremeaningfully by using this model than by using lattice models. Using the rotational-isomeric state model and Lennard-Jones expressions for the energies of interactionbetween segments we have obtained preliminary results which indicate that the valueof the exponent y depends to a large extent on the actual Lennard-Jones parametersused.Dr. A. J. Hyde (University of StrathcZyde) said: In view of the different approachesused by Edwards and Domb, the one considering very long chains with equilibriumpopulations of knots and entanglements, and the other considering the enumerationof very short chains on lattices the shorter members of which are not long enough toshow excluded volume effects let alone knots, I would ask whether the agreementbetween the exponents in ((R2), n) relationships obtained by the different methods isa confirmation of the essential truth underlying both approaches, or is to someextent fortuitous.Prof. S.F. Edwards (University of Manchester) said: In reply to Hyde, in anequilibrium distribution the equilibrium number of entanglements will automaticallybe present, so that in a calculation of an (R2, n) relationship, topology can be com-pletely ignored.However, I would make this comment on Domb and Martin's work :I tried to estimate some while ago the molecular weight of a chain showing theasymptotic (R2, n) relationship. It came out very large, agreeing with Flory's paperon this subject. However, this is when one uses realistic forces and effective steplengths. Domb, in using a lattice, is using " forces " which are hard to translateinto chemical constants. The criterion for the use of R2Kn, or R 2 ~ n 6 / 5 (or what-ever it is), is the largeness or smallness of (Z/Z0)3'5 (l/n)'/'" where Z is the effective1 L. Monnerie and F. Geny, J. Chim. Phys., 1969, 66, 1691.F. Geny and L. Monnerie, J. Chim. Phys., 1969,66, 1708.R.Zwanzig, Ado. Chem. Phys., 1969,15,325.E. M. Simon and B. H. Zirnm, J. Statistical Phys., 1969, 1'4182 GENERAL DISCUSSIONstep length and Z, a length characteristic of the excluded volume. How precisely oneassigns these in a lattice is not clear, and I suspect that a lattice has an extremelystrong excluded volume effect relative to any known chemical constants.Prof. C. Domb (King’s College, Uiziversity of London) said: The essence of myargument is the analogy with the Ising model for which exact analytic solutions existin two dimensions. The self-avoiding walk configurations represent one contributionto the king model solution’but it is the dominant contribution, and if all other contri-butions are ignored, it gives a good approximation to the known results.It seemsreasonable to assume therefore that rates of convergence are the same in the twoproblems, and this is borne out by the smooth and steady behaviour of numericaldata (such as shown by Martin) and by the agreement between exact enumerationsand Monte Carlo results. I certainly agree that there are problems for which con-vergence is very slow (e.g., when strong attractive forces are present). But forpurely repulsive forces I think that convergence is quite rapid and this indicates thatknots do not contribute significantly to the (R2, n) relationship. However, chainsof length 15-20 on a tetrahedral lattice are quite long enough for the pure excluded-volume effect to be felt.Dr. R. UlJman (Ford Motor Co., Michigan) said: In eqn.2 of Edwards’ paper,the differential equation for the velocity and position of a polymer chain of restrictedcurvature is given. The second moment of this equation is the Kratky-Porod resultfor semi-flexible chains.l Does he have the solution for this differential equation?If a polymer chain is in a chain-folded arrangement in the solid, it will have lessthan the equilibrium number of knots which he would predict. Does he have anyidea of the time it would take to form these knots once the polymer is dissolved?Would this be long enough to permit an experiment in which the change in radius ofgyration with time could be determined?Prof. S. F. Edwards (University of Marzchester) said: In reply to Ullman, thedifferential equation quoted is the simplest which allows for curvature, i.e., the simplestmethod of introducing a persistence length a.It can be solved because it is a versionof Hermite’s equation, which has an inhomogeneous solution given by Mehler. If(~+$$+pw”n’ G(x, x’, t , t ’ ) = 6 ( ~ - ~ ‘ ) 6 ( t - t ’ ) , ) then0G == [ sinh ~ ( t - t’)]‘exp (-+(x’+x’~) coth ~ ( t - t ’ ) + x x ’ cosech o(t-l‘)f.This solution is readily extended to any differential equation of purely quadraticstructure.The calculation of the time to reach equilibrium from some totally remote state isextremely difficult even in conventional statistical mechanics (e.g., for a liquid dropsuddenly finding itself in a vacuum and becoming a low density vapour). However,a first step is to calculate the fluctuation time of particular configurations withinequilibrium and I think that problem is soluble and hope to get a solution to it.Prof.S. F. Edwards (University of Munchester) said: To include variationsA simplified in curvature of a polymer one must include torsion into the calculation.Rec. trav. chim., 1949, 68, 1106GENERAL DISCUSSION 83version of torsional energy is just I r"'(s) I 2. If one has a Boltzmann weight factorfor curvature and torsion one has a weight(where kT is included in the constants a, b) for each configuration. The probabilitythat a chain has r(s) = r, r'(s) = v, r"(s) = q, is then given by-+v-+q-+-v2+--q2+ a a 8 3 3a2 ~- 8:)p(r,v,q; s) = 0. { as ar av 21 21 a 2 b 2 . 6 a qThis is a soluble equation (see reply to Ullman).[r"' . (r' x r'')]2/[r"2]2.This still leads to a differential equation, but an awkward one :The exact form for torsion is more complicated, the energy being proportional to= 0.The kind of polymers discussed by North which are stiff but have occasional sharpbends can be described by differential equations likea a a 3 2 w-+v-+q-+----u + ar av 21where W(q) is a function with a principal minimum at the origin, but subsidiaryminima at the preferred kink directions. For the solution to such equations resultsin the Kramers-type expression, see my paper.Prof. C. Domb (King's College, University of London) said: I would suggest cautioni n regard to using direct enumeration and Monte Carlo methods in the presence ofattractive forces.The enumerations which Martin presented converge rapidly andsmoothly and the area of doubt is confined to whether the exponent should be 1.20 or1.195. In either case we can find a fairly accurate representation of the asymptoticbehaviour of the mean square length of a chain. However, when there are attractiveforces present, the rate of convergence is much slower, and it is more difficult to drawreliable conclusions about asymptotic behaviour.Both Fisher and Hileyl in early enumerations, and Mazur and McCracken2 intheir Monte Carlo work have suggested that when attractive forces are present, theexponents depend on the force of attraction; for example, y becomes a function ofJ/kT where J represents the force of attraction. Recent work by one of my formerst~dents,~ P.G. Watson, has cast doubt on this conclusion, and suggests that, althoughit may be valid as a representation of the mean-square end-to-end distance for acertain range of n, it is not valid asymptotically.Watson has drawn the following conclusions from some exact analytical work.(a) The exponents are unchanged in going from a normal self-avoiding walk to a self-avoiding walk with no near neighbour contacts. This suggests that exponents areunchanged by repulsive forces, a conclusion in agreement with Mazur and McCrackenbut at variance with Fisher and Hiley. (b) For a particular lattice in two dimensions,the extended Kagome lattice (fig. l), Watson has shown that an exact solution can' M. E. Fisher and B. J. Hiley, J. Chem. Phys., 1961,34,1253.J.Mazur and F. L. McCrackin, 1968, J. Chem. Phys., 1968,49,648.1'. G. Watson, J. Plrys. Chcm, 1969, L28, and further results privately communicated84 GENERAL DISCUSSIONbe obtained with attractive forces. If the generating function of closed polygons fora walk with no near neighbours isZU2&*" (1)then the generating function for walks with an interaction isL=u2,xn(1 + wx2)" [w = exp (J/kT)].From this and analogous formulae it is possible to show rigorously that this type ofattraction does not change the exponents.FIG. 1 .-Extended Kagome lattice.However, this corresponds only to a short-range attraction since the lattice hasthe peculiarity that a self-avoiding walk could not fold up along it. Watson hassubsequently extended his work to a Kagome lattice in which long-range attractionsare also taken into account.By using a star-triangle transformation, he is able toshow that the exponent of one lattice with a sufficiently weak attractive force isidentical to that of another lattice with a repulsive force.In fact, Watson's approach suggests an alternative method of dealing with attrac-tive forces on a general lattice, e.g., the triangle or face-centred-cubic. We start witha no-contact walk and insert contacts one at a time, taking account of their mutualattraction and repulsion. This is illustrated in fig. 2. A single contact can be intro-duced along every step of the walk. However, if we introduce two contacts, threedifferent cases of interaction arise, the first two corresponding to repulsion (orexclusion) and the third to attraction.Similarly, we can introduce three contacts, etc.If the generating function for no contacts is given byFdx) = Xcnd', (3)F(x) = ~ c n o ~ @ n ( x , w), (4)then the generating function with contacts is given bywhere the cc contact interaction 'I function is given bGENERAL DISCUSSION 85Any one familiar with Ising model expansions will recognize this type of pattern andmaking the usual assumption about asymptotic behaviour, we find thatwhere@n(& w> - [&, Wll",#(x, w) = 1 + 2 x ~ + x ~ [ ( - a ~ ~ - a ~ ~ ) w ~ + a ~ ~ w ~ ] + ... .(6)(7)When we pass to a folded chain corresponding to strong interaction, we know fromthe work of W. J. C . Orr that there is a finite entropy and no long-range order.Using our knowledge of the Ising model, this again tentatively suggests that there is nosharp phase transition.FIG.2 . 4 ~ 2 ) No-contact walk in which contacts can be inserted on each step (shown dotted).(b) Possible interactions between a pair of contacts.Any conclusions drawn from the analysis of the last paragraph must be tentative.But it suggests the possibility that the asymptotic exponent remains unchanged evenin the presence of attractive forces. However, since the asymptotic behaviour isdetermined by all roots of the equationit is likely that the expression becomes complicated for large w. However, thedominant asymptotic term for very large n is of the formThis opens up the possibility that for a single chain there is no single 8 temperatureat which the expansion factor is unity for all n.For a particular n there is such a8 temperature but this temperature is dependent on n.1lPo = XO(X, w) (8)(R: (w)} A 0 [a(w)]-W (9)Prof. T. Shimanouchi (University of Tokyo, Japan) said : The Raman spectra ofn-paraffin in the crystalline state show a progression of bands in the low-frequencyW. J. C. Orr, Trans. Faraday SOC., 1947, 43, 1286 GENERAL DISCUSSIONregion. They are assigned to the accordion-like vibration and its harmonics.'*The appearance of those longitudinal acoustic vibrations shows that the moleculestake the extended zigzag conformation in the crytalline state. This is the case evenfor C94H190, for which the progression is well displayed.The frequencies of the progression of bands are expressed byv = (m/21,n)J(E/P), (1)where m, lo, n, E and p are the order of the harmonics, the length of the CH2 unit,the number of carbon atoms, Young's modulus of the n-paraffin molecule and thedensity of the molecular rod, respectively. The frequency v is expressed as a functionof m/n.Schaufele and the author measured and assigned these acoustic vibrationbands for eight normal paraffins from C18H38 to Cg4HIg0. The obtained frequenciesare all on one line given by eqn (1) for small values of m/n and E = 3.58 x 10l2dyn/cm2. This line is in agreement with the v 5 curve (the CCC deformation vibration)of the frequency against phase difference diagram of the polyethyleneFrom the v 5 and vg curves of this diagram we can also obtain information about theelastic constants for the in-plane bending, the out-of-plane bending and the twistingmodes of the n-paraffin molecular rod (fig. 1).6 00500.-I 400\ 'Eh u 5 3 0 0$L?z2001000-- FIG. 1.-Dispersion curve for acousticvibrations of polymethylene chain. v5,CCC deformation mode; vg, CC tor-sional mode ; part A, longitudinal mode ;part B, in-plane bending mode ; part C,twisting mode ; part D, out-of-planebending mode.-0 TIphase differenceA similar treatment can be used for the molecular rods of polyvinyl chloride,polypropylene, polyoxymethylene, etc. Among them, the properties of the a-helixof polyamino acids are of interest. Itoh and the author obtained the frequencyagainst phase difference curve of the polyalanine a-helix and gave the value ofYoung's modulus and the relationship between the frequencies of the accordion-likevibrations and the lengths of the a-heli~.~ This relationship may be used for esti-mating the lengths of a-helixes in natural globular proteins from the Raman spectra.The elastic constants for the bending and twisting modes of the helix can also beobtained from the dispersion curve.S. Mizushima and T. Shimanouchi, J. Amer. Chem. SOC., 1949,71,1320.R. F. Schaufele and T. Shimanouchi, J. Chem. Phys., 1967,47, 3605.M. Tasumi, T. Shimanouchi and T. Miyazawa, J. MoZ. Spectr., 1962,9,261; 1963,11,422.K. Itoh and T. Shimanouchi, Biopolymers, 1970,9, 383