Reflections on Theories of Polymer Solutions

J. J. HERMANS, Chemstrand Research Center, Inc., Durham, North Carolina

Introduction

Historically, polymer solutions are classified as colloidal systems, and although the very definition of colloid has undergone considerable changes since the word was coined by Graham in 1861, this classification is still legitimate. Graham characterized a colloid particle by its low rate of diffusion. Nowadays, a particle is called colloidal simply on the basis of its size, a classification that is similar to the distinction between infrared, visible, ultraviolet, and other electromagnetic waves. It is sufficient if one of the three particle dimensions is in the colloidal range; the other two may be of the order of ordinary molecular dimensions. On the basis of both this new definition and Graham’s original one, most polymers in solution are designated as colloidal.

The intimate relation of polymer science to colloid science has been stressed because i t serves as an illustration of the great role which polymer science has played in the clarification of ideas. The name colloid is derived from the Greek kolla, meaning glue, and for a long time the systems studied by colloid chemists were chemically ill-defined and complicated systems of which little was known. These systems showed properties that were poorly understood. In fact, it was believed by many that these properties violated some of the well-established thermodynamic laws of solutions. This situation, however, has changed drastically. It is true that a science which is bold enough to include among its objects of study such complicated and incompletely known systems as gums, proteins, polysaccharides, and complexes of proteins and polysaccharides cannot be expected to give the correct interpretation of all solution properties ob- served, but our general understanding of the basic phenomena is in- comparably better today than in the early days of colloid science, and there can be no doubt that this development originated largely in the physical chemistry of polymers.

It is now generally recognized that the peculiar thermodynamic proper- ties of polymer solutions, such as the large deviations from Van’t Hoff’s law for osmotic pressure and from Raoult’s law for vapor pressure are attributable to the large size of the solute molecules. In particular, the entropy of mixing a polymer with a low molecular weight solvent is quite different from that found in mixtures of compounds of low molecular weight.

51

52 J. J. HERMANS

The reason for this was understood, at least qualitatively, at an early date. Reference is made, for example, to the work of Meyer' and that of Staverman and Van Santen.2 Also, a statistical mechanical treatment of dumbbell-shaped molecules in a solvent consisting of spherical molecules was given by Fowler and Rushbrooke3 as early as 1937. However, a quantitative treatment for polymer molecules of large chain length was not given until 1942 when Flory and, independently, Huggins developed their lattice theories. R/liller14 who used a different method, derived essentially the same result. Later developments have been made, and although some of them avoid the use of the lattice model, the basic results are similar.

The present thermodynamic theories of polymer solutions may be sum- marized as follows. The convenient and appropriate concentration vari- able is volume fraction rather than mole fraction. The heat of mixing, AH, as a function of volume fraction is not essentially different from that in solutions of low molecular weight compounds, but the entropy of mixing, AS, is highly assymmetrical. As a result, in those cases in which the heat, of mixing is positive, even moderate values of AH can overcompensate A S a t low concentrations of polymer. This explains why demixing is such a general phenomenon in polymer solutions. In particular, two different polymers in the same solvent are seldom compatible.

With regard to the osmotic pressure, Van't Hoff's law is obeyed at the limit of zero polymer concentration; for uncharged polymer molecules, an expansion of osmotic pressure T in terms of polymer concentration (weight per unit volume) gives

T = RT[c/M + A?c' + . . .] where M is the molecular weight of the solute and Az, in analogy with the theory of non-ideal gases, is called the second virial coefficient. Its numerical value, in regard to order of magnitude, is not essentially different from that in solutions of low molecular weight compounds, and the dif- ference between these compounds and polymer solutions is therefore mainly a matter of comparison between A d and c / M . For a polymer, the molecular weight M is large, and it is not so much the deviation A&' itself, but rather its value relative to the term c /M which is the predominant factor in determining the peculiar thermodynamic properties.

Although this brief summary of the results obtained is essentially valid, it ignores all the details and refinements that have engaged polymer chemists over the last 20 years. In this chapter no attempt is made to go into the details, but some of the problems will be briefly indicated. Before doing so, however, mention is made of two other areas that have seen a vigorous revival as a result of polymeric studies: the hydrodynamic properties of colloidal solutions and the light scattering by colloidal particles.

Systematic light-scattering studies have been made since the classical work of Rayleigh and of Tyndall in the 19th century, but when the method

POLYMER SOLUTIONS 53

was re-introduced by Debye in the United States and, independently, by Putzeys in Belgium, for the purpose of studying macromolecules, a vigorous activity in this field developed, and many interesting new contributions were made. Similarly, the viscosity of suspensions had been a study ob- ject for many years, but Staudinger's introduction of the viscosity number as a tool to determine the molecular weight of long-chain molecules was the beginning of a thorough reexamination of the problem. Staudinger's relation [ q ] = KM between intrinsic viscosity and molecular weight has long since been replaced by that of Mark:

[ q ] = KM"

and a considerable amount of theoretical work has been devoted to the subject. The fact remains that this revival, like several others, originated with research on polymers.

Thus, when we look back on the last 20 or 30 years, it is easy to see what made polymers such a fascinating field of study. There exists no essential qualitative difference between macromolecules and micromolecules. All differences that do exist are quantitative, not qualitative. This quantita- tive difference is a result of the molecular size, and it is often of such a nature as to throw some new and unexpected light on old and well-estab- lished laws. There is almost always some new aspect that has to be de- veloped when an old method is applied to macromolecules, and conversely this has had an invigorating and fertilizing effect on physical chemistry in general.

Some Basic Problems in the Theory of Polymer Solutions What makes the study of polymer solutions fascinating on the one hand,

and somewhat frustrating on the other hand, is the great complexity of the theoretical problems involved. It may be worthwhile to substantiate this statement in some detail for the particular case of the hydrodynamic properties.

Even when the restriction is made to solutions that are sufficiently dilute Go negIect all interaction between the polymer molecules (a state of affairs that is never realized experimentally) , there is still considerable room for guesswork and controversy.

If polymer molecules in solution could be treated as rigid spheres, ellipsoids, or rods, straightforward methods and comparatively simple theoretical results could be used for sedimentation rate and intrinsic viscosity determinations. It is true that these results have been derived for particles dispersed in a continuum rather than in a solvent that itself has a molecular structure, but this situation is not new, and a wealth of experimental data exists to prove that this is not too serious a matter. The difficuIties begin when one considers the more realistic model of the random coil, because the calculation of the hydrodynamic interaction between the monomer units requires knowledge of the distances between these units, and this immediately raises the problem of the excluded volume

54 J. J. HERMANS

effect which is a complicated many body problem for which a rigorous solution is not known. Moreover, the mathematical treatment of the hydrodyr~amic interaction is riot rigorous either: use is made of averaging procedures6s6 which may introduce serious errors. It, is riot surl)rising that the simplest approach to this complex problem is the one that has found the greatest following. In this approach it is assumed that the hydro- dynamic interaction between the monomer units is so pronounced that it is legitimate to treat the entire structure, essentially, as an impenetrable particle (equivalent particle concept). This circumvents the intricacies of the hydrodynamic problem and leaves merely the problem of the molecu- lar dimensions. Even when this is done, the question remains whether the equivalent particle may be considered as a sphere (Flory’) or perhaps, rather, an ellipsoid (Sadrons). Also, for all but very low velocity gradients, it is important to know whether the equivalent particle is deformable (Cerfg) and, if so, whether the response of the particle shape to external forces is essentially instantaneous or requires time.

Notwithstanding these uncertainties, the development of the equivalent particle concept by Flory has been extraordinarily successful in describing a great body of data. One of its greatest merits lies in the fact that it relates the hydrodynamic properties to the thermodynamic ones. The reason for this is that not only the hydrodynamic behavior but also the second virial coefficient is governed by the molecular dimensions.

However, this statement is so simple that it may easily be quite mislead- ing. The earlier statement, that in the theories based on the equivalent particle concept one obviates the problem of the hydrodynamic interactions and is left with “merely the problem of the molecular dimensions” was not meant to imply that this reduces the problem to a simple one. As indicated already, no rigorous theory of the excluded volume effect has thus far been developed. The simplest approach to this problem7 consists in assuming that the polymer molecule may be treated as a droplet of a polymer solu- tion and applying macroscopic thermodynamics to this system. More rigorous treatments on the basis of perturbation theory do not go beyond the first few terms of a series expansion1°~11*12 in terms of the square root of the molecular weight, and this series converges only slowly.

The interaction between the segments of different polymer molecules, and thus the second virial coefficient, must be derived on a similar basis,1a*14*1s*16 and this again is not a trivial matter. The entire problem has been summarized in a recent article” and will not be further discussed here. Some remarks will be made, however, on the hydrodynamic aspects which enter the picture when the equivalent particle concept is abandoned.

Even though the hydrodynamics of the equivalent particle have been exceedingly helpful in relating the intrinsic viscosity to the second virial coefficient, thereby clarifying to a large extent the influence of the solvent and of the temperature, it cannot be said that this is the last word on the subject. Whether the hydrodynamic interaction of the monomer units is sufficiently large to apply the model of the equivalent particle depends

POLYMER SOLUTIONS 55

on the average density of the monomer units in the random coil and on its overall dimensions. There are strong indication~1~~1~ that cellulose derivatives, and deoxyribonucleic acid in solution, are partially free drained. And, quite generally, some free-draining is to be expected in all polymer molecules at the low molecular weight end.

The theories that have been developed to calculate the hydrodynamic interaction of the monomer units take the point of view that each unit can be replaced by a force acting on the liquid. This force is proportional to the velocity of the monomer unit relative to the adjacent liquid; the proportionality constant [ is called frictional coefficient. As a result of this force there is liquid flow, at some distance from the monomer unit considered, which affects the other units. The first to apply this idea was Burgers.20 Application to polymer molecules was made later.5v6s21

Unfortunately, the set of equations that results from this concept cannot be solved. The problem can be solved only for certain average quantities, and the errors involved in the averaging procedure have, to the author's knowledge, never been estimated.

In addition to the constant [, the theory contains a parameter which determines the overall dimensions of the molecule, but this is a parameter that can in principle be derived from the angular dependence of light scattering, provided the molecular dimensions are not too small compared to the wavelength of the light used. Thus, by combining light-scattering data with the intrinsic viscosity or the sedimentation rate, the frictional coefficient [ can be calculated. It has often been considered a serious shortcoming of the theory that almost always turns out to be much smaller than would be expected on the basis of Stokes' law. One expects [ to be approximately equal to 6 ~ 3 ~ r 0 where 70 is the viscosity of the solvent and ro is the radius of the monomer unit. In reality, the coefficient p frequently turns out to be more than 10 times smaller. However, Marrinan and H e r n a n ~ ' ~ . ~ ~ have shown that this discrepancy is not always real. The theoretical equations can be made independent of Gaussian statistics. If, at the same time, the equations are extended so as to be applicable to a homologous polymer mixture, the [-values derived from the experimental data are quite reasonable. l8

The procedure suggested by Marrinan and Hermans is not the only one that has been proposed to extend the theory to non-Gaussian chains and to correct for polydispersity. A much more detailed analysis of the Kirk- wood-Riseman theory for non-Gaussian chains was developed by Kurata,2a and the comparison with experimental data has been extensive.'', 22 Ref- erence is made also to the work of HearstlZ3 and of Ullman.24

Peterlin's calculation^^^ in connection with this problem make use of the very simple elastic dumbbell model, which replaces the actual polymer molecule by its two end points and accounts for the chain connectivity by an elastic force between these end points. This model mas used extensively by KuhnZ6; as shown by Peterlin it leads, in many cases, to correct results.

Notwithstanding the approximations made, there is no doubt that these

56 J. J. HERMANS

more recent theories constitute a substantial improvement over earlier work. At the same time it should be kept in mind that they are based on the model of the random coil and that this is not necessarily the only model to be considered. The suggestion has often been made that many polymer molecules in solution assume a helical form, or that a t least parts of the molecule are helical. This would be true, in particular, for molecules with a high degree of stereoregularity. We will consider this question in the next section.

Other departures from random coil statistics have been considered as well. In particular, when the angle between successive segments in the chain is small, the model of the “worm-like chain” becomes a good ap- proximation. The statistics of this model were considered by Kratky and P ~ r o d , ~ ~ Daniels,28 and Hermans and U l l m ~ i n . ~ ~ Sedimentation rates for these molecules were discussed by Hearst and Sto~kmayer .~~ The treat- ment of the intrinsic viscosity was given by Eisner and Ptitsyn31; applica- tions to cellulosic molecules appear to be promising.32

So much for the problems of molecular dimensions and hydrodynamic interaction, which govern the intrinsic viscosity a t the limit of zero shear rate. Additional problems arise when rate effects must be taken into ac- count, for example, as in an oscillatory shear. In this case, the contribution of the polymer molecules to the viscosity of the solution becomes frequency dependent, and there is a phase difference between the shear rate and the shear stress. Likewise, in a steady shear rate, when the velocity gradient is not negligibly small, the viscosity depends on the rate of shear.

The mathematical formulation of this kind of problem in terms of the statistical mechanics of rate processes was developed by Kirkwood. 33

For random coils it has led to interesting results when applied to an in- geneous model that was suggested by This model replaces the actual polymer chain (assumed to be Gaussian) by a series of beads, each of which is attached to its two neighbors by a Hookean spring. If the force constant of this spring is chosen properly, the chain has the correct statistical dimensions. At the same time, the Hookean nature of the springs insures that the equation of motion for each bead is linear. This has the great mathematical advantage that the set of equations for the entire assembly of beads can easily be reduced, by a linear transformation, to normal coordinates, which allows a simple solution. It is found35 that the intrinsic viscosity for this model is independent of the velocity gradient, irrespective of whether or not hydrodynamic interaction of the beads is taken into account.

It is indisputable that the model is rather artificial, and whether it is liked or disliked is very much a matter of taste. It is often said that creative scien’tists have much in common with artists, and science with art. The use of an elegant model may well be considered a case in point. A molecular model in a scientific theory has much the same relation- ship to reality a7 an impressionist painting has to the subject portrayed. The parallel may be drawn even further: There is an element of art in the

I’OLY MER SOLUTIONS 57

construction of a model on the part of the scientist, but art is required also on the part of the student who considers the model. In particular, the observer must have the ability, not only to see those aspects that the artist wants him to see, but aIso not to see what he is not supposed to see.

There is a story in the Tao, that treasure of Chinese wisdom, which tells us of a prince who was the proud owner of a stable of superb horses. These had been acquired with the help of a faithful servant who had an incredible knowledge and understanding of horses. There came a time, however, that the prince bought a horse on the advice of someone else, who had written to him from a remote province. And when the horse arrived, the prince was deeply disappointed because, he said, ‘(He wrote to me that he was sending a brown mare, and in reality the horse is a black stallion. If he cannot even see the difference between a brown mare and a black stallion. . . ”. But his servant did not agree. “Truly,” he said, “in him I have found my master. What is unimportant in the horse he does not even see.”

The fact that the intrinsic viscosity for the model considered turns out to be independent of the shear rate is attributed by Zimm35 to the linearity of the equations used. An alternative theory was developed by P e t e r l i ~ ~ ~ ~ on the basis of the elastic dumbbell model. It was concluded that the intrinsic viscosity must be expected first to decrease with increasing gradi- ent but later to increase again. The physical origin of this increase in [v] at higher shear rates lies in the fact that the coiled molecule becomes extended and, as a result, the chain segments become more exposed in- dividually to the action of the sheared liquid. In other words, the molecule becomes more free drained. Several factors were neglected in this theory, and an experimental verification3’ can be expected only in liquids of very high viscosity.

One important factor that is not taken into account in either Zimm’s or Peterlin’s theory is the possiblity that a chain molecule has what KuhnZ6 called “internal viscosity” or “shape resistance”: It does not respond instantaneously to external forces, but needs time to change its shape, not only because it is imbedded in a viscous liquid (external viscosity) but also because potential barriers in the polymer molecule oppose rapid changes in shape (internal viscosity).

New Trends

No attempt was made in the previous section to discuss all the problems encountered in solution theory. Very little attention was paid to thermo- dynamic aspects and none whatever to light scattering (recent reviews of light-scattering problems can be found in ref. 38 and 39). Several other subjects have not even been mentioned: adsorption of polymers at inter- f a c e ~ ~ ~ . ~ ~ the spreading of polymers in mono layer^,^^ properties of poly- electrolyte^,^^ and many others. Looking back on this vigorous develop- ment, it is clear that the field of polymers has been and will continue to be

58 J. J. IIERMANS

a marvelous playground, both delightful and instructive, for statistical mechanics, the theory of rate processes, hydrodynamics, and other disci- plines. Is it possible to say where one will go from here? The answer to such a question is bound to be speculative, but some predictions can be made with a fair amount of confidence.

In the first place there is still room for improvements in the theory. This is true for the molecular dimensions and the hydrodynamic interaction, but even more so for the effect of the shear rate and the birefringence of flow.

The overall dimensions that were discussed in the previous section are determined primarily by interactions between segments that are not close neighbors in the chain or, if interaction between such neighbors does play a role, their effect may to a large degree be masked by the effect of forces between nonneighboring segments. In connection with stereoregularity it is of importance to find phenomena which depend in a more unique manner on the configuration of neighbors in the chain. Such phenomena are: nuclear magnetic resonance,44 dipole infrared a b ~ o r p t i o n , ~ ~ depolarization of scattered light.47 Another technique is flow birefringence, because the anisotropy of a chain segment depends on the t a ~ t i c i t y . ~ ~

Interesting further developments may be expected from the concept of the helical shape. The work of Liquori et. al.49 on the energy of interaction between the various atoms in a polymer chain has already been extremely successful in predicting the helical form in the solid state. Evidence for a helical shape in solution has been derived from infrared'spe~tra~~ and optical rotation.60 Very recently, direct evidence has been claimed for syndio- tactic poly(methy1 methacrylate) in benzene by Kirste and Wunderlichsl on the basis of low angle x-ray scattering.

Benoit and ~o-workers~~ have found that the radius of gyration of poly- styrene molecules in several solvents and the second virial coe5cient in these solvents show an abrupt change in magnitude at a temperature which varies from about 55°C. to about 80"C., depending on the nature of the solvent. This transition was demonstrated also by means of the de- polarization of scattered light.63 It is tempting to assumeK2 that the phe- nomenon is due to a process which is similar in nature to the helix-coil transition in protein solutions.

It is interesting to note in connection with helical structures that Liquoris4 has recently found evidence for the belief that when methyl methacrylate is polymerized in a polar solvent in the presence of isotactic polymethyl- methacrylate, the polymer formed is syndiotactic, indicating a template- type growth of a synthetic polymer in dilute solution.

It is safe to say that stereoregularity will play an increasing role in solution studies in the near future. There are other areas that will be very fruitful, but it is not easy to predict to what extent progress will be made, and in what order. Three fields of study are mentioned explicitly: ultracentrifugation, solution properties of copolymers, and the properties of concentrated solutions.

POLYMER SOLIJTIONY 59

Xone of these is new. The use of the ultracentrifuge is well-established, but in recent years we have witnessed a renewed interest and a considerable improvement in techniques. Moreover, until comparatively recently, the initiative was mostly in the hands of biochemists, and there is a wide, rela- tively unexplored field still open for the study of synthetics. Special men- tion must be made of density gradient cen t r i f~ga t ion ,~~ in which use is made of a mixture of solvents to establish a density gradient in the cell. As a result of this gradient, the polymer collects in a band around the posi- tion where its buoyant density is equal to the density of the solvent mixture, and since the width of this band depends on the molecular weight, the method may be used to obtain information on molecular weight and molecu- lar weight d i s t r i b ~ t i o n . ~ ~ , ~ ~ RIoreover, the position of the band depends on the buoyant density of the polymer, and this in turn depends not only on chemical composition but also on molecular structure. It has been found, for example, that the partial specific volume of a polymer in a solvent may depend on the t a ~ t i c i t y . ~ ~ This explains why density centrifugation may give information about tacticity, 58 especially because the density difference is sometimes enhanced by preferential absorption of one of the solvents by the polymer.59

Density gradient centrifugation is particularly helpful in the study of compositional d i s t r i b u t i ~ n , ~ ~ * ~ ~ where other methods are scarce. It is likely that there will be a difference in buoyant density also between a block co- polymer and a random copolymer even when they have the same chemical composition. This is of practical interest in relation to recent work on the solution properties of copolymers. Preparative polymer chemistry has reached the stage where the structure of a molecule can be controlled be- tween wide limits: random copolymers versus block idem, branching of a given type, and so on. The solution properties of block and graft co- polymers are often extremely interesting. As indicated in the introduction, chemically different polymers are usually not compatible; a separation into two phases takes place. Something similar, on the intramolecular scale, occurs when a block or graft copolymer is dissolved: The segments of one type tend to avoid the segments of the other type. In a liquid that is a good solvent for one but a bad solvent for the other, the molecular structure becomes comparable to that found in micellar soap solutions; the insoluble segments tend to form a micelle that is protected from precipitation by solvated segments of the soluble type. This phenomenon can become quite pronounced when the polymer molecules form aggregates. A review of several solution properties can be found in reference 61, which also discusses some phenomena in a mixture of solvents. It is clear that the combinations of polymer segments of different kinds, of the number and length of the blocks, and of solvents and solvent mixtures that, may be studied are prac- tically unlimited. The t h e ~ r y ~ * , ~ ~ for these sy5tCms is still in its iiifniicy.

Finally, there is n large area of research in polymer solutions that thus far has not been explored to the extent that it deserves. The theories of intrinsic viscosity, intrinsic sedimentation rate, etc., are theories for the

60 J. J. HERMANS

isolated polymer molecule. They play an important role in the study of molecular weight and molecular weight distribution and in the study of structural aspects, but with all due respect for these theories, sophisticated though they may be, the concentrations of almost all polymer solutions of technical importance are entirely outside the range of validity of the calcula- tion. It is true that in many cases attempts have been made to calculate also the second-order term, i.e., the effect of interaction between pairs of solute molecules, but the treatment is often only approximate and, niore- over, it still does not reach the concentration range of technical interest. Isolated attempts have been made, however, to reach a better understand- ing of concentrated polymer solutions, and three of them are mentioned:

A comparatively simple device64 is to adopt the treatment derived for the isolated molecule but to assume that this molecule is moving in a liquid with a viscosity that is not that of the solvent (vo), but is some higher, effective, viscosity v*. Whereas at the limit of infinite dilution the viscos- ity of the solution in terms of polymer concentration c is equal to

7 = 70 + 770[91C,

one sets

?I = 70 + ?I*[vlc

when the concentration is finite. Similar equations are written for the flow birefringence, and the results are surprisingly good.64

A special model was used by Lodge65 who assumed the existence of a network with temporary junctions and applied the theory of rubber elastic- ity to this network.

The approach made by Simha and co-workers66 consists of a search for reduced variables that represent the viscosity of a polymer solution by a universal function, i.e., independent of molecular weight or temperature or both. It was found that for good solvents this is achieved to a good approximation if the specific viscosity is expressed in units of [ q ] c and the concentration in units co. In other words, a plot of vsp/ [q ]c versus C/Q is approximately independent of molecular weight. Here c is the polymer concentration, [ q ] the intrinsic viscosity, and Q the concentration at which the polymer coils, visualized as spheres, begin to overlap. Using the Flory-Fox equation for the intrinsic viscosity of the equivalent particle, and assuming hexagonal close packing for the estimate of cot it is possible to express co in terms of [v], namely: co = l.OS[v]-l. Although this works reasonably well in good solvents, considerable discrepancies are found in poor solvents. Simha observed, however, that there appears to exist a relation between the shift factor co and the critical concentration c, at which phase separation occurs, and that this relation applies also in poor solvents.

Theoretically the most sophisticated approach to concentrated solutions was made recently by Fixman and Peterson,67 who derived a concentration- dependent intermolecular potential from a fluctuation theory and used this

POLYMEIZ SOLUTIONS 61

to obtain the radial distribution function and the thermodynamic proper- ties. Further developments will show whether this kind of approach will be useful also in connection with the hydrodynamic properties.

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