Statistical Physics of a Linear Polymer Chain

M. V. Vol'kenshtein and O. B. Ptitsyn

1. Introduction

Linear polymers constitute one of the important objects of study in modern physics and chemistry. Linear polymers include a number of materials of very great importance in technology: rubber, plastics, fibrous substances. The use of these materials is based chiefly on their specific mechanical and dielectric properties.

The mechanical properties of a number of polymers are especially striking; in this respect polymers have no analogue among solid and liquid bodies of another structure. Thus, natural and synthetic rubbers are characterized by high elasticity—the ability to undergo very large reversible elastic deformations with very low elastic moduli, tens of thousands of times smaller than the moduli of such, for example, elastic materials as steel. The mechanical and other properties of high polymers depend to a great extent on temperature and on the conditions of application. The time factor is of particular importance here. One and the same material behaves in essentially different ways depending on whether it is subjected to rapid or slow deformations. In short, high-polymer substances are characterized by special properties that distinguish them from other gases, liquids, and solids. Thanks to the many years of efforts of a number of chemists, the chemical structure of a large number of high polymers is now known. Here an especially significant role belongs to Soviet science in the person of Academician S. V. Lebedev, who carried out the world’s first synthesis of artificial rubber.

Linear polymers are long chain molecules in which one and the same monomer unit is repeated many times.

Let us give examples of the structure of some of the most important polymers:

\[ \left[-\mathrm{CH_2}-\mathrm{CH_2}-\right]_n \qquad \text{Polyethylene} \]

\[ \left[-\mathrm{CH_2}-\mathrm{C}(\mathrm{CH_3})_2-\right]_n \qquad \text{Polyisobutylene} \]

\[ \left[-\mathrm{CH_2}-\mathrm{CH}(\mathrm{C_6H_5})-\right]_n \qquad \text{Polystyrene} \]

\[ \left[-\mathrm{CH_2}-\mathrm{CH}(\mathrm{OH})-\right]_n \qquad \text{Polyvinyl alcohol} \]

\[ \left[-\mathrm{CH_2}-\mathrm{CH}=\mathrm{CH}-\mathrm{CH_2}-\right]_n \qquad \text{Polybutadiene} \]

\[ \left[-\mathrm{CH_2}-\mathrm{C}(\mathrm{CH_3})=\mathrm{CH}-\mathrm{CH_2}-\right]_n \qquad \text{Polyisoprene (natural rubber)} \]

The number of repeating identical units—the degree of polymerization—can reach tens and hundreds of thousands. Further examples the reader will find in the specialized literature ^1, ^2. The overwhelming majority of known polymers are organic compounds, howe—

…such as, for example, a substance like plastic sulfur

\[ \backslash \mathrm{S}/\mathrm{S}\backslash \mathrm{S}/\mathrm{S}\backslash \mathrm{S}/ \]

also constitutes a high polymer. We call substances built of long chain molecules without transverse bonds linear polymers. Alongside linear polymers there are known two-dimensional and three-dimensional spatial polymers, in which the individual chains are joined by transverse bonds. It is obvious that graphite may be assigned to two-dimensional polymers, and to three-dimensional ones, on the one hand, diamond, and on the other, vulcanized rubber.

The task of the physical theory of linear polymers consists in the quantitative explanation of their special properties and, ultimately, in establishing the connection between these properties and the chemical structure of the polymer. It is obvious that the ideal theory would be one that made it possible to predict the physical properties of a polymeric material on the basis of its chemical structure. At present we are still far from such a state of polymer physics. However, in this field of knowledge there already exist major achievements, some of which we shall discuss in the present article.

The chemical properties of high polymers are distinctive, but in them the characteristics of the low-molecular units from which they are built are preserved to a considerable extent. Thus, for example, the chemical reactions of polyethylene (see above) are similar to the chemical reactions of any low-molecular paraffin, since separate groups \(\mathrm{CH}_2\), which are part of both substances, are responsible for these reactions. At the same time, the physical properties of a polymer are characterized by a qualitative peculiarity, a substantial difference from the properties of low-molecular substances. In this sense a high-molecular compound is similar to a new aggregate state of a low-molecular substance. It is immediately evident that these peculiarities of the physical properties are determined by the fact that a polymer molecule is a long chain molecule consisting of a large number of identical units. Here we encounter a very vivid example of the transition of quantity into quality.

The physical theory of polymers must, obviously, proceed from consideration of the joint behavior of a very large number of identical objects—the units of the chain. Thus both an individual polymer molecule and their aggregate in a specimen of polymeric material are characteristic objects of statistical physics. However, this proposition is not yet sufficiently justified by the presence of a large number of identical elements. In order for such an aggregate to be the subject of statistical theory, it is necessary that these elements—individual units of the chain or groups of consecutively arranged units—possess their own degrees of freedom.

and, consequently, the system as a whole—a large number of internal degrees of freedom. As we shall see, such a situation is indeed realized in the case of a long polymer molecule. It was precisely on the basis of statistical physics that an explanation could be given for the highly elastic properties of rubber. The study of the thermal phenomena that accompany the deformation of rubber shows that the internal energy of the specimen does not change when it is stretched; only its entropy changes. The equilibrium state of the specimen in the absence of external forces applied to it corresponds to the maximum entropy; any deformation is accompanied by a decrease in entropy. In other words, upon deformation rubber passes from a more probable state to a less probable state. Therefore the deformation is reversible. In this sense the elasticity of rubber is similar to the elasticity of an ideal gas. This analogy, however, should not be assigned especially great significance—the rubber-like state has very little in common with the gaseous state and much more in common with the liquid and solid states.

Modern polymer physics constitutes a broad field of knowledge that touches upon the most diverse branches of physical science. A relatively simple matter is the theoretical consideration of the properties of isolated polymer molecules, which we encounter in dilute solutions. The configuration of a chain molecule, subject to continuous fluctuations, can be studied by means of statistical physics. The results of theoretical analysis can be compared with the results of experimental investigation of the structure of isolated polymer molecules in solutions by methods of light scattering, dynamic double refraction, diffusion, etc., and also with the results of studies of the thermodynamics of polymer solutions. Despite the great variety of the corresponding statistical problems, the study of isolated chains is relatively the simplest. Incomparably more complicated is the theory of an aggregate of polymer molecules in bulk. At the same time, it is precisely this theory that is essential for practice, which deals not with isolated chain molecules but with macroscopic samples. Here we encounter a double statistics—each polymer chain is an ensemble of the separate elements of which it is constructed, and at the same time it is part of a macroscopic ensemble of intertwined and connected chains constituting a specimen of polymeric material.

Along with the study of the thermodynamically equilibrium states of polymers, the investigation of the time course of stresses and deformations is very important. We have already pointed out the special significance of the time factor in polymer physics. Thus not only statistical thermodynamics, but also the statistical kinetics of chain molecules is a topical field of physics.

Statistical physics of linear polymers is built on the basis of the achievements of the statistical theory of real gases and liquids, and the theory of diffusion processes.

This article is the first part of a review in which the statistical theory of an isolated linear chain molecule will be considered. The second part of this review will be devoted to the statistical theory of the mechanical properties of a macroscopic specimen, and, finally, the third part to certain kinetic problems. Modern statistical physics of polymers is far from exhausted by the problems considered in the present review. For data relating to questions not touched upon here, see the available monographic literature 3—9.

2. DISTRIBUTION FUNCTION OF THE DISTANCE BETWEEN THE ENDS OF A CHAIN

In this part we shall consider the properties of an isolated polymer molecule. Such molecules are encountered in highly dilute solutions of polymers in solvents whose molecules interact only weakly with the polymer molecule. Thus, in our consideration we abstract from intermolecular interaction. In addition, we restrict ourselves to the consideration of chains of unbranched linear polymers.

The study of branched polymers is of considerable scientific and practical interest; however, theoretical ideas concerning the influence of branching on the dimensions and properties of polymer molecules require separate detailed consideration, going beyond the scope of the present article.

A polymer molecule consists of a very large number of links and always contains a large number of single valence bonds in the chain. Thus, in polyethylene all the bonds are single, while in polyisoprene three quarters of the bonds are single (see above, p. 502). It is known that around single bonds internal rotation is possible, hindered to a certain degree owing to the interaction of atoms not bonded to one another by valence forces ^10. The presence of these internal degrees of freedom determines the flexibility of the polymer chain—its ability to assume a large number of different, but approximately energetically equivalent, configurations.

Let us consider a chain consisting of an aggregate of links possessing a certain freedom of rotation relative to one another. It is obvious that the degree of correlation between the directions of these links decreases rapidly with increasing distance between them measured along the chain (Fig. 1). The valence angle between two neighboring links is fixed; the third link can occupy a whole series of positions on the cone shown in Fig. 1, and after another three or four links have passed, we already completely lose any correlation

with the direction of the first link. Consequently, if atoms of links sufficiently far apart from one another are connected by straight lines, then the directions of these straight lines will be practically independent (Fig. 2). This makes it possible to replace the consideration of a real polymer chain by an investigation of the properties of a simpler model consisting of such freely jointed statistical elements[^11].

Fig. 1. Chain of a linear polymer.

It is obvious that the division of a real chain into statistical elements is to some extent arbitrary, although the smallest number of links in them is determined by the concrete properties of the molecule—the magnitude of the valence angle and the degree of hindrance to internal rotation. Arbitrarily assigned lengths and the number of statistical elements do not, however, enter into the final results of the calculations (see Section 4).

A freely jointed chain consisting of \(Z\) elements (\(Z \gg 1\)) of length \(b\), as a result of fluctuations, assumes a large number of equally probable configurations. Each of these configurations can be characterized by the value of some geometric parameter referring to the chain as a whole. It is rational to choose as such a parameter

Fig. 2. Division of a polymer chain into statistical elements.

the distance from the beginning to the end of the chain—the “length” of the chain \(h\). Different values of \(h\) correspond to different numbers of equally probable configurations. Therefore the probability of different values of \(h\) is different. Let us find the statistical distribution function of the values of \(h\)[^11],[^12], neglecting the circumstance that two statistical elements cannot be located in one and the same region of space.

Following Kuhn\(^{11}\), we shall begin by considering the one-dimensional case. We shall project each statistical element onto some axis \(z\). As a consequence of the equal probability of all orientations of the statistical element,*) the mean square length of its projection is equal to (Fig. 3)

\[ \overline{b_z^2}=\frac{b^2}{4\pi}\int_0^\pi\int_0^{2\pi}\cos^2\vartheta\sin\vartheta\,d\vartheta\,d\varphi=\frac{b^2}{3} \tag{2.1} \]

and, consequently, in moving along the chain by one element, on the average we shall take along the \(z\)-axis a step of length

\[ \frac{b}{\sqrt{3}} \]

in the positive or negative direction. It is obvious that, on the average, the numbers of positive and negative steps are equal and, for this reason, the mean value of the projection of the “length” of the chain on the \(z\)-axis is zero. Let us calculate the probability that, after making \(Z\) steps along the chain, we shall advance by \(a\) steps in the positive direction along the \(z\)-axis. For this it is necessary that the number of positive steps be equal to

\[ Z_+=\frac{Z}{2}+\frac{a}{2}. \]

At the same time the number of negative steps is

\[ Z_-=\frac{Z}{2}-\frac{a}{2}, \]

since

\[ Z_+ + Z_- = Z. \]

Fig. 3. Projection of a statistical element on the \(z\)-axis.

The probability that the number of positive steps is \(Z_+\), and that of negative steps \(Z_-\), is equal to

\[ W_{Z_+Z_-}=\frac{Z!}{Z_+!Z_-!}\left(\frac{1}{2}\right)^Z . \tag{2.2} \]

Substituting into (2.2) the values of \(Z_+\) and \(Z_-\) and making use of Stirling’s formula, we find (for large \(Z\)):

\[ W_a = W_{\frac{Z}{2}+\frac{a}{2},\,\frac{Z}{2}-\frac{a}{2}} = \mathrm{const}\, e^{-\frac{a^2}{2Z}} . \tag{2.3} \]

If \(a\) excess positive steps have been made, then the projection of the “length” of the chain is equal to

\[ h_z=\frac{ba}{\sqrt{3}}. \]

*) See on this point below, Section 3.

Thus,

\[ W(h_z)=\mathrm{const}\, e^{-\frac{3h_z^2}{2Zb^2}}. \tag{2.3a} \]

The constant is found from the normalization condition:

\[ \int_{-\infty}^{+\infty} W_{h_z}\,dh_z=1. \]

We have:

\[ W_{h_z}\,dh_z=\left(\frac{3}{2\pi Zb^2}\right)^{1/2} e^{-\frac{3h_z^2}{2Zb^2}}\,dh_z . \tag{2.4} \]

Similar expressions can also be obtained for the two other independent coordinate directions. The final expression for the distribution function in the three-dimensional case has the form

\[ W_{h_x,h_y,h_z}\,dh_x\,dh_y\,dh_z = W_{h_x}W_{h_y}W_{h_z}\,dh_x\,dh_y\,dh_z = \]

\[ = \left(\frac{3}{2\pi Zb^2}\right)^{3/2} e^{-\frac{3}{2Zb^2}(h_x^2+h_y^2+h_z^2)} \,dh_x\,dh_y\,dh_z . \tag{2.5} \]

We have obtained a Gaussian distribution function for the values \(h_x\), \(h_y\), \(h_z\), characteristic of the distribution of any random variables. It is easy to see that the simple problem considered is entirely analogous to the stochastic problem of the mean path traversed by a freely diffusing particle\({}^{13}\). The agreement will be complete if we put the number of elements \(Z\) in correspondence with the time factor in the diffusion equation, and the length of the statistical element \(b\) with the quantity \(\sqrt{6D}\), where \(D\) is the diffusion coefficient of the particle.

From (2.5) follows the distribution function of the absolute values of the “lengths” of the chain:

\[ W(h)\,h^2\,dh= \left(\frac{3}{2\pi Zb^2}\right)^{3/2} 4\pi e^{-\frac{3h^2}{2Zb^2}}h^2\,dh . \tag{2.6} \]

Hence the most probable value of \(h^2\) is

\[ h_0^2=\frac{2}{3}Zb^2 . \tag{2.6a} \]

It is easy to see that

\[ \bar h=\int_0^\infty h W_h\,dh = \sqrt{\frac{8}{3\pi}}\sqrt{Zb^2} \tag{2.7} \]

and

\[ \bar h^2=Zb^2 . \tag{2.8} \]

Thus, the average “length” of the chain is of the order of magnitude \(b\sqrt{Z}\), whereas its maximum length is \(bZ\), i.e., as a result of fluctuations the chain is strongly coiled (we recall that \(Z \gg 1\)).

We arrive at the conclusion that, for a sufficiently large degree of polymerization, a polymer molecule in solution is not a rigid rod (as Staudinger originally assumed), but a statistically entangled coil, whose average linear dimensions are \(\sqrt{Z}\) times smaller than its maximum length. Each extended configuration of the chain is, in itself, no less probable than its coiled configuration. However, whereas the state of maximum extension \((h=bZ)\) can be realized in only one way, strongly coiled states, for which \(h \ll bZ\), can be realized in an enormous number of ways. Precisely because such states are the most probable, the chain spends most of its time in them. Experimental data obtained in studies of polymer molecules in solution confirm this conclusion of the theory (see Sections 4, 7).

Formulas (2.4)—(2.8) are valid only for sufficiently long chains (large \(Z\)), since in deriving them we made use of Stirling’s formula. Moreover, since we applied Stirling’s formula not only to the quantities \(Z!\) and \(\left(\dfrac{Z}{2}+\dfrac{\alpha}{2}\right)!\), but also to \(\left(\dfrac{Z}{2}-\dfrac{\alpha}{2}\right)!\), the Gaussian distribution of chain “lengths” obtained by us is valid if not only \(Z \gg 1\), but also \(Z \gg \alpha\). The latter condition means that the distance between the ends of the chain is considerably smaller than its maximum length, for which \(Z=\alpha\).

Thus, the Gaussian distribution is applicable only to sufficiently long and sufficiently strongly coiled chains.

In fact, formula (2.6), for example, gives a finite (although very small) probability that \(h > Zb\), which, of course, is meaningless.

A more exact distribution function was obtained by V. Kuhn and Grün\({}^{14}\). Kuhn and Grün calculated the probability of various distributions of the orientations of statistical elements in a chain with a given distance between its ends. This probability is equal to

\[ W=Z!\prod_k \frac{\left(\dfrac{\sin\vartheta_k\cdot \Delta\vartheta_k}{2}\right)^{n_k}}{n_k!}, \tag{2.9} \]

where \(n_k\) is the number of statistical elements that form, with the vector connecting the ends of the chain, angles from \(\vartheta_k\) to \(\vartheta_k+\Delta\vartheta_k\).

The numbers \(n_k\) must satisfy the following two obvious conditions:

\[ \sum_k n_k = Z \tag{2.10} \]

and

\[ \sum_k n_k \cos \vartheta_k \cdot b = h . \tag{2.11} \]

The most probable values \(n_k\) must make expression (2.9) a maximum subject to the additional conditions (2.10) and (2.11).

Using the usual method for solving problems of constrained extrema, and replacing the condition that expression (2.9) be a maximum by the condition that its logarithm be a maximum, we have

\[ \frac{\partial \ln W}{\partial n_k}+\alpha+\beta \cos \vartheta_k=0, \tag{2.12} \]

where \(\alpha\) and \(\beta\) are Lagrange multipliers.

From (2.9), applying Stirling’s formula, we have

\[ \frac{\partial \ln W}{\partial n_k} = \ln\left(\frac{\sin \vartheta_k \cdot \Delta \vartheta_k}{2}\right) -\ln n_k . \tag{2.13} \]

Substituting (2.13) into (2.12), we find that the most probable values \(n_k\) are equal to

\[ n_k=\frac{1}{2} e^\alpha \cdot e^{\beta \cos \vartheta_k}\cdot \sin \vartheta_k \cdot \Delta \vartheta_k . \tag{2.14} \]

The quantities \(\alpha\) and \(\beta\) are determined from conditions (2.10) and (2.11), by substituting (2.14) into them and passing from summation to integration:

\[ \begin{aligned} \alpha &= \ln\left(\frac{Z\beta}{\operatorname{sh}\beta}\right),\\ \beta &= L^{-1}\left(\frac{h}{Zb}\right), \end{aligned} \tag{2.15} \]

where \(L^{-1}\) is the function inverse to the Langevin function:

\[ L(x)=\operatorname{cth} x-\frac{1}{x}. \]

Thus,

\[ dn_{\vartheta} = \frac{Z\beta}{\operatorname{sh}\beta} e^{\beta \cos \vartheta} \cdot \frac{1}{2}\sin \vartheta\, d\vartheta . \tag{2.14a} \]

Returning to the problem of interest to us, that of finding a more exact form of the function \(W(h)\), we note that the probability of a given distance between the ends \(h\) must obviously be proportional to the probability of the most probable distribution \(n_k\), corres-

sponding to this distance. From (2.12) we have:

\[ \ln W=-an-\beta \overline{\cos \vartheta}. \tag{2.16} \]

With the aid of (2.14a) it is easy to show that

\[ \overline{\cos \vartheta}=L(\beta)=\frac{h}{Zb}. \tag{2.17} \]

Substituting (2.15) and (2.17) into (2.16), we obtain:

\[ \ln W(h)=\mathrm{const}-Z\left[\ln \frac{\beta}{\operatorname{sh}\beta}+\frac{h}{Zb}\,\beta\right]. \tag{2.18} \]

Differentiating this expression with respect to \(h\), it is easy to verify that

\[ \frac{d\ln W(h)}{dh}=-\frac{\beta}{b}. \]

Thus:

\[ \ln W(h)=-\frac{1}{b}\int_{0}^{h}\beta\,dh. \tag{2.18a} \]

From (2.18) and (2.18a), respectively, we have\({}^{15}\):

\[ \left. \begin{aligned} W(h)h^{2}\,dh &=B\left(\frac{\operatorname{sh}\beta}{\beta}\right)^{Z} e^{-\frac{\beta h}{Z}}h^{2}\,dh,\\[6pt] W(h)h^{2}\,dh &=B e^{-\frac{1}{b}\int_{0}^{h}\beta\,dh}h^{2}\,dh. \end{aligned} \right\} \tag{2.19} \]

Making use of the expansion of \(\beta\) in a series

\[ \beta=L^{-1}\left(\frac{h}{Zb}\right) =3\frac{h}{Zb}+\frac{9}{5}\left(\frac{h}{Zb}\right)^{3} +\frac{297}{175}\left(\frac{h}{Zb}\right)^{5}+\cdots, \tag{2.20} \]

from (2.19) we obtain:

\[ W(h)h^{2}\,dh =Be^{-\frac{3}{2}\frac{h^{2}}{Zb^{2}} -Z\left[\frac{9}{20}\left(\frac{h}{Zb}\right)^{4} +\frac{99}{350}\left(\frac{h}{Zb}\right)^{6}+\cdots\right]} h^{2}\,dh. \tag{2.21} \]

It follows from this that formulas (2.19) pass into (2.6) if \(h\ll Zb\).

For \(h=Zb\), \(\beta\) tends to infinity, and \(W(h)\), according to formulas (2.19), tends to zero. For \(h>Zb\), \(\beta\), and consequently \(W(h)\), lose their meaning. Thus, formulas (2.19) correctly describe the behavior of the chain not only for very small, but also for very large \(h\).

Finally, Treloar\({}^{16,8}\) obtained an exact distribution function for the “length” of a chain consisting of freely jointed elements.

Its formula is obtained without any approximations and therefore is applicable to chains with any number of links, beginning with \(Z=1\), and for any \(h\). It has the form:

\[ W(h)h^2\,dh = \]

\[ = \frac{h}{2b^2}\frac{Z^{Z-2}}{(Z-2)!} \sum_{s=0}^{k}(-1)^s \frac{Z!}{s!(Z-s)!} \left(m-\frac{s}{Z}\right)^{Z-2}dh, \tag{2.22} \]

where

\[ m=\frac{1}{2}\left(1-\frac{h}{Zb}\right), \tag{2.22a} \]

and the upper limit of summation \(k\) is determined from the condition

\[ k \leq mZ \leq k+1. \tag{2.22b} \]

In Fig. 4, graphs taken from work\({}^{8}\) are reproduced for the dependence

\[ \ln \frac{W(h)}{4\pi h^2} \]

on

\[ \left(\frac{h}{Zb}\right)^2 \]

for the cases \(Z=6,\ 25\), and \(100\). Curves \(a\) correspond—

Fig. 4. Distribution function for the distance between the ends of a chain with \(Z=6,\ 25\), and \(100\). Curves \(a\) — Gaussian approximation (2.6), curves \(b\) — Langevin approximation (2.19), curves \(в\) — exact formula (2.22).

—correspond to formula (2.6) (Gaussian approximation), curves \(b\) to formula (2.19) (Langevin approximation), and curves \(в\) to the exact formula (2.22). It is clearly seen from the figure that formula (2.6) is valid only for \(h \ll Zb\). As for formula (2.19), even for chains consisting of several tens of statistical elements it gives a sufficiently good approximation over the entire range of values of \(h\).

Thus, the question of distribution functions for a chain consisting of freely jointed statistical elements has now been considered in the literature with sufficient completeness. However, in real polymer molecules, as is known, there is no free jointing. The orientation of each \(k\)-th link of the chain is characterized, in the coordinate system connected with the preceding \((k-1)\)-st link, by two angles—the fixed valence angle \(\alpha\) and the angle of rotation \(\varphi\), the different values of which have different probabilities in accordance with the hindering potential \(U(\varphi)\). If the molecule consists of identical links, i.e. if \(\alpha\) and \(U(\varphi)\) do not depend on the number \(k\), then the distribution of orientations of each link relative to the preceding link is the same for all links. However, relative to the coordinate system connected with the polymer molecule as a whole, each link will have its own distribution of orientations, depending on the distribution of orientations of the preceding link. Here we encounter a typical domain of application of A. A. Markov’s method of chains\(^{17,18,13}\), which makes it possible to find the distribution function for a certain resultant quantity if the distribution functions for its components are known.

As applied to our problem, A. A. Markov’s method gives\(^{13}\):

\[ W(\mathbf h)=\frac{1}{8\pi^3}\int_{-\infty}^{+\infty} e^{-i(\boldsymbol{\rho},\mathbf h)} A_N(\boldsymbol{\rho})\,d\boldsymbol{\rho}, \tag{2.23} \]

where

\[ A_N(\boldsymbol{\rho})=\prod_{k=1}^{N}\int_{-\infty}^{+\infty}\tau_k(\mathbf l_k)e^{i(\boldsymbol{\rho},\mathbf l_k)}\,d\mathbf l_k. \tag{2.23a} \]

Here \(\tau_k(\mathbf l_k)\) is the distribution of orientations of the link \(\mathbf l_k\) in the coordinate system connected with the polymer chain as a whole, and \(N\) is the number of links in the chain.

If the directions \(\mathbf l_k\) are distributed randomly, i.e. all functions \(\tau_k\) are spherically symmetric (a freely jointed chain), then from (2.23) and (2.23a) it is not difficult to obtain\(^{13}\):

\[ W(\mathbf h)=\frac{1}{2\pi h}\int_{0}^{\infty}\sin \rho h \left\{ \prod_{k=1}^{N}\frac{\sin \rho l_k}{\rho l_k} \right\}\rho\,d\rho, \tag{2.24} \]

where

\[ h=|\mathbf h|,\qquad \rho=|\boldsymbol{\rho}|,\qquad l_k=|\mathbf l_k| \quad (\text{cf.}^{19}). \]

If, in addition, all \(l_k\) are equal to one another and \(N\gg 1\), then (2.24) goes over into (2.5) (with \(Z\) replaced by \(N\), and \(b\) by \(l\), since in this case we regard each link of the chain as a statistical element).

If all the functions $\tau_k(\mathbf l_k)$ were devoid of spherical symmetry, but identical to one another, then in the chain there would be a preferred direction of orientation of the vectors $\mathbf l_k$, the same for all links. Then the function $W(\mathbf h)$ obtained from (2.23) and (2.23a) would correspond to a random distribution about a certain systematic displacement. If the problem stated above, leading to the function (2.5), is analogous to the problem of a freely diffusing particle, then we are now dealing with an analogue of the diffusion of a particle in an external field. In this case

\[ \overline{h^2}=c_1N+c_2N^2 \quad (c_1,c_2=\mathrm{const}). \tag{2.25} \]

However, in the real case of chains with fixed valence angles and hindered internal rotation that interests us, the functions $\tau_k(\mathbf l_k)$, although devoid of spherical symmetry, are nevertheless different from one another. Therefore, although each link has certain preferred directions of orientation with respect to its neighbors, as a whole the chain, for sufficiently large $N$, will be devoid of any preferred direction, and the distribution $W(\mathbf h)$, in contrast to the case of identical $\tau_k(\mathbf l_k)$, must retain a random character. Undoubtedly, obtaining the function $W(\mathbf h)$ for different and not spherically symmetric $\tau_k(\mathbf l_k)$ would be of considerable fundamental interest; however, this problem is extremely difficult. Separate attempts undertaken in this direction by a number of authors (see, for example, $^{20}$) were not successful. However, the simple considerations already given above show that if the correlation between the orientations of the links decreases sufficiently rapidly as their separation from one another along the chain increases, then in the limiting case $N\to\infty$ the function $W(\mathbf h)$ will be Gaussian $^{21,22}$.

Of some interest from this point of view are also the results of Treloar $^{23,8}$, who considered the distribution function of the “length” of chains composed of small, freely jointed segments several links long. In calculating the distribution function of the “length” of each segment, the presence of fixed valence angles was taken into account, while rotation about single bonds was assumed to be free. Calculations carried out for such a model (for polyethylene and polyisoprene) gave good agreement with the results obtained for freely jointed chains with a specially chosen number of statistical elements. This shows that the general form of the function $W(\mathbf h)$ does not depend on whether we divide the chain into statistical elements of fixed or variable length. It is clear that in fact the length of the statistical element varies, and by the quantity $b$ one must understand its mean value.

The foregoing shows that we may, with a certain justification, use the results obtained for the model of a freely joi...

jointed chain, i.e., by Gaussian and Langevin functions, when describing the properties of real polymer chains. It is necessary, however, to make one more essential reservation.

All the preceding arguments were of a purely geometrical character, whereas real polymer molecules are not geometrical but physical entities. The links of these molecules occupy a definite volume and interact with one another in a definite way, attracting one another at large distances and repelling one another at small distances. A particularly important role must be played by the repulsion of links from one another at small distances, connected with the fact that two links cannot simultaneously be in one and the same volume element. This effect, to which Section 6 is devoted, compels us to exclude from consideration all configurations in which such “self-intersection” of the chain occurs. It is clear that taking into account the corrections indicated above, i.e., passing from the geometrical model to a real polymer chain, may change our results concerning the function \(W(\mathbf{h})\) in a very substantial way.

3. SHAPE AND DIMENSIONS OF FREELY JOINTED CHAINS

In order to obtain more detailed information on the geometry of a freely jointed fluctuating chain (its mean transverse dimensions, radius of inertia, distances between various pairs of atoms, etc.), it is useful to solve the following auxiliary problem. Let us consider three atoms of the chain with numbers \(i<k<l\) and determine the probability that the \(k\)-th atom is in a given volume element, if the positions of the \(i\)-th and \(l\)-th atoms are fixed. The probability that atom \(k\) lies at the point \((x,y,z)\), if atom \(l\) lies at the point \((x',y',z')\) and atom \(i\) lies at the point \((x'',y'',z'')\), is equal to the probability that a chain of \(k-i\) links, beginning at the point \((x'',y'',z'')\), ends at the point \((x,y,z)\):

\[ W_1=\left(\frac{3}{2\pi (k-i)b^2}\right)^{3/2} \exp\left\{-\frac{3}{2(k-i)b^2}\left[(x-x'')^2+(y-y'')^2+(z-z'')^2\right]\right\} \]

(cf. (2.5)), multiplied by the probability that a chain of \(l-k\) links, beginning at the point \((x',y',z')\), ends at the same point \((x,y,z)\):

\[ W_2=\left(\frac{3}{2\pi (l-k)b^2}\right)^{3/2} \exp\left\{-\frac{3}{2(l-k)b^2}\left[(x-x')^2+(y-y')^2+(z-z')^2\right]\right\}. \]

Multiplying \(W_1\) and \(W_2\) and carrying out the normalization, we obtain:

\[ \begin{aligned} &W_{kli}(x,y,z;\,x',y',z';\,x'',y'',z'')= \left(\frac{3(l-i)}{2\pi(k-i)(l-k)b^2}\right)^{3/2}\times \\ &\quad \times \exp\left\{ -\frac{3(l-i)}{2(k-i)(l-k)b^2} \left[ \left(x-\frac{x'(k-i)+x''(l-k)}{l-i}\right)^2+\right.\right.\\ &\quad \left.\left. +\left(y-\frac{y'(k-i)+y''(l-k)}{l-i}\right)^2+ \left(z-\frac{z'(k-i)+z''(l-k)}{l-i}\right)^2 \right]\right\} \end{aligned} \tag{3.1} \]

\[ (i<k<l). \]

We shall use this expression to calculate the probability of a given position of the \(k\)-th atom of a chain with fixed ends. In this case \(i=0,\ l=Z,\ x''=y''=z''=0\) (we assume that the chain begins at the origin of coordinates), \(x'=y'=0,\ z'=h\) (we direct the \(z\)-axis along the “length” of the chain). We have:

\[ W_k(x_k,y_k,z_k;\,h)\,dx_k\,dy_k\,dz_k = \left(\frac{3Z}{2\pi k(Z-k)b^2}\right)^{3/2}\times \]

\[ \times \exp\left\{ -\frac{3Z}{2b^2k(Z-k)} \left[ x_k^2+y_k^2+\left(z_k-\frac{kh}{Z}\right)^2 \right]\right\}. \tag{3.2} \]

The mean values of the coordinates of the \(k\)-th atom in this case are equal to

\[ \overline{x}_k=\overline{y}_k=0;\qquad \overline{z}_k=\frac{kh}{Z}, \tag{3.3} \]

\[ \left. \begin{aligned} \overline{x_k^2}=\overline{y_k^2}&=\frac{1}{3}b^2\frac{Z-k}{Z}\,k,\\ \overline{z_k^2}&=h^2\left(\frac{k}{Z}\right)^2+\frac{1}{3}b^2\frac{Z-k}{Z}\,k. \end{aligned} \right\} \tag{3.4} \]

Hence

\[ \overline{h_k^2} = \overline{x_k^2}+\overline{y_k^2}+\overline{z_k^2} = h^2\left(\frac{k}{Z}\right)^2 + b^2\frac{Z-k}{Z}\,k . \tag{3.5} \]

The last expression was obtained by Kacalsky, Konzle, and Kuhn\(^{24}\). It is easy to see that if \(h^2\) is equal to its mean value \(Zb^2\), then \(\overline{h_k^2}=kb^2\). Let us find the value of \(k\) corresponding to the greatest value of \(\overline{h_k^2}\). We obtain that

\[ k_{(\mathrm{max})}=\frac{b^2Z^2}{2(b^2Z-h^2)}, \tag{3.6} \]

where \(k\leq Z\). If \(h^2>\frac{1}{2}b^2Z\), and, in particular, if \(h^2\) is equal to its mean value, i.e. \(Zb^2\), then expression (3.6) loses

Statistical Physics of a Linear Polymer Chain

sense ($k$ is either negative or greater than $Z$). This means that

\[ (\overline{h_k^2})_{\text{max}}=\overline{h_z^2}=h^2. \tag{3.7} \]

If \(h^2 \ll \frac{1}{2} b^2 Z\), then \(k_{(\text{max})}<Z\), and, if \(h^2 \to 0\),

\[ k_{(\text{max})}\to \frac{Z}{2}. \]

Thus, on the average, the distance between the beginning and the end of the chain is the greatest in comparison with the distance between any other pair of its atoms, and therefore it really has the meaning of the chain length. However, \(h\) loses this meaning for configurations in which it is appreciably smaller than its mean value.

To characterize the mean transverse dimensions of a fluctuating chain, let us define the mean square distance of the \(k\)-th atom from the \(z\)-axis—the straight line connecting the beginning and the end of the chain. According to (3.4), we have

\[ \overline{r_k^2}=\overline{x_k^2}+\overline{y_k^2} =\frac{2}{3} b^2 \frac{Z-k}{Z}\,k . \tag{3.8} \]

The distribution function for \(r_k\) is obtained from (3.2) by passing to cylindrical coordinates and integrating over \(z_k\) and over the angle:

\[ W(r_k) r_k\,dr_k = \frac{3Z}{k(Z-k)b^2} e^{-\frac{3Z r_k^2}{2k(Z-k)b^2}} r_k\,dr_k . \tag{3.9} \]

\(\overline{r_k^2}\) attains its maximum value, equal to

\[ \overline{r^2}=\frac{1}{6} Z b^2 \tag{3.10} \]

at \(k=\frac{Z}{2}\). The distribution function of the distance of the middle atom of the chain from the \(z\)-axis, according to (3.9) (where in this case \(k=\frac{Z}{2}\)), has the form

\[ W(r)r\,dr = \frac{12}{Zb^2} e^{-\frac{6r^2}{Zb^2}} r\,dr, \tag{3.11} \]

\(\overline{r^2}\) is the mean square “width” of the coiled chain. Let us note, incidentally, that in the work of Sadron\(^{25}\) the following formula is given for \(r^2\):

\[ r^2=h_{Z/2}^2-\frac{1}{4}h_z^2 \]

(in our notation). It is easy to see that this formula is incorrect for the mean values, for the mean-square values, and for the most probable values of the quantities entering into it.

Fixing the “length” \(h\) and the “width” \(r\), let us determine the “thickness” of the coiled chain \(\rho\). To this end we draw the \(y\)-axis perpendicular to the \(z\)-axis through the atom with number \(Z/2\). Using expression (3.1), in which now (for \(k<Z/2\)) \(i=0\), \(l=Z/2\), \(x''=y''=z''=0\), \(x'=0\), \(y'=r\), \(z'\) arbitrary, and integrating over \(y, z\), and \(z'\) from \(-\infty\) to \(+\infty\), we find:

\[ W_k(x_k)=\left(\frac{3Z}{2\pi k(z-2k)b^2}\right)^{1/2} e^{-\frac{3Zx_k^2}{2k(Z-2k)b^2}}. \tag{3.12} \]

\[ \left(k<\frac{Z}{2}\right). \]

In order to obtain the distribution function for the distance of the \(k\)-th atom from the plane \((yz)\), \(\rho_k=|x_k|\), this expression must be multiplied by two, since to each pair of values \(\pm x_k\) there corresponds one value of \(\rho_k\):

\[ W_k(\rho_k)\,d\rho_k= \left(\frac{6Z}{\pi k(Z-2k)b^2}\right)^{1/2} e^{-\frac{3Z\rho_k^2}{2k(Z-2k)b^2}}\,d\rho_k \tag{3.12a} \]

\[ \left(k<\frac{Z}{2}\right). \]

Hence

\[ \overline{\rho_k^2}=\frac{1}{3}b^2\frac{Z-2k}{Z}\,k \tag{3.13} \]

\[ \left(k<\frac{Z}{2}\right). \]

\(\overline{\rho_k^2}\) acquires its maximum value, equal to

\[ \overline{\rho^2}=\frac{1}{24}Zb^2 \tag{3.14} \]

at \(k=Z/4\).

The distribution function for \(\rho\), according to (3.12) (where \(k=Z/4\)), has the form

\[ W(\rho)\,d\rho= \left(\frac{48}{\pi Zb^2}\right)^{1/2} e^{-\frac{12\rho^2}{Zb^2}}\,d\rho. \tag{3.15} \]

Analogous formulas can also be obtained for \(k=3Z/4\).

The greater part of the results presented above were obtained in a somewhat different way by Kuhn\(^{11}\) and Huggins\(^{26}\).

Thus, the quantities \(\sqrt{\overline{h^2}}\), \(\sqrt{\overline{r^2}}\), \(\sqrt{\overline{\rho^2}}\), defined by us as the “length,” “width,” and “thickness” of a chain molecule, rela—

are related as \(1 : \dfrac{1}{\sqrt{6}} : \dfrac{1}{2\sqrt{6}}\). In this case the molecule is not a triaxial ellipsoid, but has the shape of a “bean.” In fact, if in a molecule with fixed ends we fix the direction (the \(y\)-axis) in which the atom with number \(\dfrac{Z}{2}\) lies, then it is clear that the density of elements in this direction will be greater than in the direction perpendicular to it. A similar argument is valid for the atom with number \(\dfrac{Z}{4}\), which fixes the \(x\)-axis.

The meaning of the results obtained is as follows. If some chain begins at the origin of coordinates, then over a sufficiently long time, owing to fluctuations, its configurations corresponding to the given value \(h\) fill a sphere of radius \(h\). Consequently, the time-averaged configuration of the chain in space is a sphere. If we now consider only those configurations for which the chain, beginning at the origin, ends at a given point on the surface of this sphere, then such configurations fill an ellipsoid of revolution, whose major axis is equal to \(h\), while the minor axis, for a given value of \(h\), may have different values \(r\), corresponding to the distribution function (3.10). In this case the atom with number \(\dfrac{Z}{2}\) is situated on the circle of the largest transverse section of the ellipsoid. If, finally, we restrict ourselves to configurations corresponding to a definite position of the atom with number \(\dfrac{Z}{2}\) on this circle, then they fill a figure having the shape of a “bean.” Thus the “bean” is the configuration of the chain in a coordinate system rigidly connected with it (a molecularly immobile coordinate system). The foregoing is explained by Fig. 5. It is obvious that such an asymmetric shape of the molecule does not contradict the spherical symmetry of the function (2.6) and the cylindrical symmetry of the function (3.10). The matter reduces to a different choice of the statistical ensemble.

Fig. 5. Configuration of a polymer chain.

Let us note that the results set forth above indicate the independence of the dimensions of chains in three dimensions (“length,” “width”

and “thickness”). Thus, \(h\), \(r\), and \(\rho\) may be regarded as independently varying variables. It is obvious, however, that this conclusion cannot be correct for strongly extended chains, since it is clear that as \(h \to Zb\), \(r\) and \(\rho\) must tend to zero (all the calculations given above were made in the Gaussian approximation, i.e., for the case \(h \ll Zb\)).

In fact, all our calculations were based on the function (2.5), which was obtained by multiplying three functions of the type (2.4). But this multiplication is legitimate only so long as we may assume that the mean-square projections of the length of the statistical element \(b\) on three mutually perpendicular directions are equal to one another and are equal to \(\frac{b^2}{3}\) (see (2.1)). For a more detailed consideration of this question we must calculate the mean value \(\overline{b_x^2}\) for a chain with fixed \(h\). Using (2.14a), we readily obtain[^27]:

\[ \overline{b_x^2}=\frac{h\beta}{Z\beta} \]

or, taking (2.20) into account:

\[ \overline{b_x^2}=\frac{1}{3}b^2\left[1-\frac{3}{5}\left(\frac{h}{Zb}\right)^2+\cdots\right]. \]

It follows that for \(h \ll Zb\), \(\overline{b_x^2}=\frac{1}{3}b^2\), and consequently the dimensions of the chain in three mutually perpendicular directions do not depend on one another.

It should be pointed out, however, that this independence remains valid only insofar as we do not take into account the finite volume occupied by the chain links, nor the interaction of links separated from one another along the chain by large distances (cf. the end of Section 1 and Section 6).

In a number of cases, the most convenient (and directly experimentally determinable—see Section 7) geometrical characteristic of a chain is not the “length” \(h\), but the mean square radius of inertia \(\overline{R^2}\). To calculate this quantity, let us define the position of the \(k\)-th atom of the chain relative to the origin of coordinates by the vector \(\mathbf{h}_{0k}\) \((x_{0k}; y_{0k}; z_{0k})\), and the position of the center of gravity of the chain by the vector \(\mathbf{h}_{0c}\) \((x_{0c}; y_{0c}; z_{0c})\). Then the mean square radius of inertia of a chain consisting of identical atoms is represented by the expression

\[ \overline{R^2}=\frac{1}{Z}\sum_{k=0}^{Z}\overline{s_k^2}, \tag{3.16} \]

where

\[ \mathbf{s}_k=\mathbf{h}_{0k}-\mathbf{h}_{0c}. \]

Obviously,

\[ \mathbf{h}_{0c}=\frac{1}{Z}\sum_{i=0}^{Z}\mathbf{h}_{0i}. \]

We have:

\[ \overline{s_k^2}=\overline{h_{0k}^2}-2\overline{(\mathbf{h}_{0k},\mathbf{h}_{0c})}+\overline{h_{0c}^2}= \]

\[ =\overline{h_{0k}^2}-\frac{2}{Z}\sum_{i=0}^{Z}\overline{(\mathbf{h}_{0k},\mathbf{h}_{0i})} +\frac{1}{Z^2}\sum_{i,j=0}^{Z}\overline{(\mathbf{h}_{0i},\mathbf{h}_{0j})}. \]

\(\overline{h_{0k}^2}\) is computed without any difficulty by means of the function (2.6),

\[ \overline{h_{0k}^2}=kb^2. \]

To compute the scalar products it is necessary to know the probability of simultaneous occurrence of the \(i\)-th and \(k\)-th atoms in volume elements characterized by the vectors \(\mathbf{h}_{0i}\) and \(\mathbf{h}_{0k}\). This probability is equal to the product of a function of type (2.6) and a function of type (3.1). Thus, it is not difficult to obtain

\[ \overline{(\mathbf{h}_{0k},\mathbf{h}_{0c})}=kb^2-\frac{k^2b^2}{2Z} \]

and

\[ \overline{h_{0c}^2}=\frac{1}{3}Zb^2, \]

whence

\[ \overline{s_k^2}=Zb^2\left\{\frac{1}{3}-\frac{k}{Z}\left(1-\frac{k}{Z}\right)\right\} \tag{3.17} \]

and

\[ \overline{R^2}=\frac{1}{6}Zb^2. \tag{3.18} \]

Formulas (3.15) and (3.16) were published without derivation by Debye \(^{28}\). A derivation of formula (3.18), somewhat different from ours, was proposed by Zimm and Stockmayer \(^{29}\) (see also \(^{30}\) and \(^{120}\)). Let us establish the connection between the mean square radius of inertia of the chain and its “length,” “width,” and “thickness.” For this it is necessary to carry out calculations similar to those just described, but instead of the function (2.6) to use the expression for the probability of a definite position of the atom under consideration in the chain with fixed values \(h, r\), \(\rho_{Z/4}, \rho_{3Z/4}\).

This expression is easily obtained from (3.1). The result has the following form:

\[ \overline{R_{h,r,\rho_{Z/4},\rho_{3Z/4}}^{\,2}} =\frac{35}{576}Zb^2+\frac{1}{12}h^2+\frac{1}{12}r^2+ \]

\[ +\frac{5}{48}\left(\rho_{Z/4}^{2}+\rho_{3Z/4}^{2}\right) -\frac{3}{16}\rho_{Z/4}\rho_{3Z/4}. \tag{3.19} \]

The presence in this formula of a term that does not depend on \(h\), \(r\), \(\rho\) shows that, for small values of these parameters, they cease to characterize the dimensions of the chain.

If in the chain only the value of \(h\) is fixed, while \(r\) and \(\rho\) are equal to their mean values, i.e.

\[ \overline{r^2}=\frac{1}{6}Zb^2,\qquad \overline{\rho^2_{Z/4}}=\overline{\rho^2_{3Z/4}}=\frac{1}{24}Zb^2,\qquad \overline{\rho_{Z/4}}=\overline{\rho_{3Z/4}}=0, \]

then from (3.19) we obtain:

\[ \overline{R_h^2}=\frac{1}{12}Zb^2+\frac{1}{12}h^2. \tag{3.20} \]

We shall need this formula later (see § 6).

To obtain more detailed information on how well the quantities \(\overline{h^2}\), \(\overline{r^2}\), and \(\overline{\rho^2}\) characterize the mean dimensions of the chain, it is of interest to compare these maximum mean dimensions of a molecule with its mean maximum dimensions.

Fig. 6. External dimensions of a polymer chain.

Formulas (3.5), (3.8), and (3.18) show that \(\overline{h_k^2}\), \(\overline{r_k^2}\), and \(\overline{\rho_k^2}\) are maximal respectively for \(k=Z\), \(Z/2\), and \(Z/4\). Thus, \(\overline{h_Z^2}\), \(\overline{r_{Z/2}^2}\), and \(\overline{\rho_{Z/4}^2}\) represent the maximum mean dimensions of the chain in three mutually perpendicular directions. Otherwise one poses the problem of the mean maximum dimensions of the molecule; in this case one averages not the distances between any fixed points in the chain, but the distances between such points (different at every instant) which at a given instant are separated from one another by the greatest distance. It is clear that the mean maximum dimensions of the chain computed in this way are always greater than its maximum mean dimensions (see Fig. 6).

The problem just set forth has been considered by a number of authors\(^{31\text{–}34}\) on the basis of calculating the probability that the chain reaches a certain boundary but does not cross it. This probability is without difficulty

is determined by analogy with the problem of the diffusion of particles in a vessel with absorbing walls. Without dwelling on the details of the corresponding calculations, we shall give only their results. The length of the chain, i.e., the mean distance between its two points most remote from one another[^32]:

\[ \overline{H}=1.4\sqrt{\overline{Zb^2}}=1.4\sqrt{\overline{h^2}}=1.5\overline{h}. \tag{3.21} \]

Thus, the mean maximum dimensions of the chain are one and a half times greater than its maximum mean dimensions. The mean distance from the beginning of the chain to the segment most remote from it[^38]:

\[ \overline{L}=1.2\sqrt{\overline{Zb^2}}=1.2\sqrt{\overline{h^2}}=1.3\overline{h}\;*). \tag{3.22} \]

The mean maximum projection of the chain onto a given direction[^32]:

\[ \overline{X}=0.92\sqrt{\overline{Zb^2}}=0.92\sqrt{\overline{h^2}}=\overline{h}. \tag{3.23} \]

For comparison, let us calculate from (2.4):

\[ \overline{|x|}=0.46\sqrt{\overline{Zb^2}}=0.46\sqrt{\overline{h^2}}=0.5\overline{h}. \]

Thus,

\[ \overline{X}=2\,\overline{|x|}. \]

The mean maximum transverse dimensions of the coil[^32]

\[ \overline{Q}=0.7\sqrt{\overline{Zb^2}}=0.7\sqrt{\overline{h^2}}=0.8\overline{h}. \tag{3.24} \]

Let us note that

\[ \overline{Q}=\frac{1}{2}\overline{H}. \]

The mean distance from the line joining the ends of the molecule to the segment most remote from it:

\[ \overline{S}=0.62\sqrt{\overline{Zb^2}}=0.62\sqrt{\overline{h^2}}=0.67\overline{h}. \tag{3.25} \]

For comparison, let us calculate from (3.11):

\[ \overline{r}=\frac{1}{2}\sqrt{\frac{\pi}{6}}\sqrt{\overline{Zb^2}} =0.36\sqrt{\overline{Zb^2}} =0.36\sqrt{\overline{h^2}} =0.39\overline{h}. \]

Thus,

\[ \overline{Q}=2\overline{r} \quad \text{and} \quad \overline{S}=1.7\overline{r}. \]

These results show that, in characterizing the dimensions of a chain by means of the quantities \(\overline{h^2}\), \(\overline{r^2}\), and \(\overline{\rho^2}\), we obtain somewhat understated—

*) In work[^3] a somewhat different value is given:

\[ \overline{L}=1.14\sqrt{\overline{Zb^2}}. \]

… values, although the order of magnitude is conveyed correctly. However, as we shall see below (see Section 7), experiments with polymer solutions give us not the mean maximum dimensions of the chain \(\overline H\), but the quantities \(\overline{h^2}\) or \(\overline{R^2}\).

4. GEOMETRY OF THE CHAIN MOLECULE

The results obtained in the preceding sections referred to a model of a chain consisting of \(Z\) freely jointed elements of length \(b\). In order to apply these results to real polymers, it is necessary to relate them to molecular constants. As was already said above, the quantities \(Z\) and \(b\) are, in themselves, arbitrary to a certain degree; however, all final formulas pertaining to the Gaussian chain contain only their combination \(Zb^2\), which, as we shall see below, can be expressed through molecular constants characterizing the real chain: the number and lengths of the links, the valence angles, and the degree of hindrance of internal rotation. Thus the final formulas pertaining to the Gaussian chain do not depend on the manner in which the chain is divided into statistical elements, i.e., they contain no arbitrariness*).

According to equation (2.8)

\[ \overline{h^2}=Zb^2. \]

Therefore, in order to express \(Zb^2\) through molecular constants, it is necessary to calculate independently \(\overline{h^2}\) for a real polymer chain. This geometrical problem was solved for chains with free rotation by Eyring\({}^{35}\). Subsequently S. E. Bresler and Ya. I. Frenkel\({}^{36}\), and after them a number of foreign scientists\({}^{22,37-43}\), proposed formulas applicable to various cases of hindrance of internal rotation. In its general form this question was recently considered by the authors of the present article and by T. M. Birshtein\({}^{44-46}\)**).

Let us consider a real chain of a linear polymer consisting of \(N\) links of length \(l\). Assign to each link a vector \(\mathbf l_i\), where \(i=1,2,\ldots,N\). The distance between the ends of the chain is determined by the vector

\[ \mathbf h=\sum_{i=1}^{N}\mathbf l_i . \tag{4.1} \]

*) As we saw above, formulas pertaining to the Langevin approximation depend not only on \(Zb^2\), but also on \(Z\) and \(b\) separately. Usually \(Z\) and \(b\) are chosen so that both the mean quadratic and the maximum length of the freely jointed chain coincide with the corresponding dimensions of the real chain.

**) Analogous results were obtained independently of us by the Chinese scientist O-Chin Tang\({}^{120}\).

Therefore,

\[ \overline{h^2}=l^2\left\{N+2\sum_{i=2}^{N}\sum_{j=1}^{i-1}\overline{(i,j)}\right\}, \tag{4.2} \]

where \(\overline{(i,j)}\) is the mean cosine of the angle between the vectors \(\mathbf l_i\) and \(\mathbf l_j\). To compute \(\overline{(i,j)}\), let us associate with each vector \(\mathbf l_i,\mathbf l_j\) a right-handed coordinate system whose \(z\)-axis is directed along the vector (Fig. 7). Then, if \(A_k\) is the matrix of the cosines of the angles between the axes of the \((k-1)\)-th and \(k\)-th coordinate systems, then

\[ \overline{(i,j)}= \left( \prod_{k=j}^{i-1} A_k \right)_{33}. \tag{4.3} \]

Assuming that the rotations of the individual links are independent of one another*) and that all links are identical, we have:

\[ \overline{\prod_{k=j}^{i-1} A_k} = \prod_{k=j}^{i-1} \overline{A_k} = (\overline{A})^{\,i-j}. \tag{4.4} \]

Fig. 7. Coordinate systems in a polymer chain.

To raise the matrix to a power, we reduce it to diagonal form:

\[ B^{-1}\overline{A}B=\Lambda, \tag{4.5} \]

where

\[ \Lambda= \begin{pmatrix} \lambda_1 & 0 & 0\\ 0 & \lambda_2 & 0\\ 0 & 0 & \lambda_3 \end{pmatrix} \]

(\(\lambda_n\) are the eigenvalues of the matrix \(\overline{A}\)).

Obviously,

\[ \overline{(i,j)}=\{B\Lambda^{\,i-j}B^{-1}\}_{33}, \tag{4.6} \]

and

\[ \overline{h^2}=l^2\left[N+2\{BLB^{-1}\}_{33}\right], \tag{4.7} \]

*) See, in this connection, the end of Section 6.

where

\[ L= \begin{pmatrix} f(\lambda_1) & 0 & 0\\ 0 & f(\lambda_2) & 0\\ 0 & 0 & f(\lambda_3) \end{pmatrix} \]

and

\[ f(\lambda_n)=\lambda_n^2\left[ \frac{N-1}{\lambda_n(1-\lambda_n)} - \frac{1-\lambda_n^{N-1}}{(1-\lambda_n)^2} \right] \qquad (n=1,2,3). \]

For \(N\gg 1\), formula (4.7) takes the form\({}^{45}\)

\[ \overline{h^2}=Nl^2\left[1+2(\overline{A}G)_{33}\right], \tag{4.8} \]

where

\[ G=B \begin{pmatrix} \dfrac{1}{1-\lambda_1} & 0 & 0\\ 0 & \dfrac{1}{1-\lambda_2} & 0\\ 0 & 0 & \dfrac{1}{1-\lambda_3} \end{pmatrix} B^{-1}. \tag{4.8a} \]

If identical links form valence angles \(\pi-\alpha\) with one another, and the angle of rotation about the \(k\)-th bond is \(\varphi_k\) (Fig. 7), then the matrix \(A_k\) can be written in the form

\[ A_k= \begin{pmatrix} -\cos\varphi_k\cos\alpha & -\sin\varphi_k & \cos\varphi_k\sin\alpha\\ -\sin\varphi_k\cos\alpha & \cos\varphi_k & \sin\varphi_k\sin\alpha\\ \sin\alpha & 0 & \cos\alpha \end{pmatrix}. \tag{4.9} \]

On averaging we obtain

\[ \overline{A}= \begin{pmatrix} -\eta\cos\alpha & -\varepsilon & \eta\sin\alpha\\ -\varepsilon\cos\alpha & \eta & \varepsilon\sin\alpha\\ \sin\alpha & 0 & \cos\alpha \end{pmatrix}, \tag{4.10} \]

where \(\eta\) and \(\varepsilon\) are the mean values of the cosine and sine of the angle of internal rotation:

\[ \eta=\overline{\cos\varphi} = \frac{ \displaystyle \int_{-\pi}^{+\pi} e^{-U(\varphi)/kT}\cos\varphi\,d\varphi }{ \displaystyle \int_{-\pi}^{+\pi} e^{-U(\varphi)/kT}\,d\varphi }, \qquad \varepsilon=\overline{\sin\varphi}. \tag{4.11} \]

\(U(\varphi)\) is the potential energy of internal rotation.

From (4.8) follows the formula46 *)

\[ \overline{h^2}=Nl^2\cdot \frac{1+\cos\alpha}{1-\cos\alpha}\cdot \frac{1-\eta^2-\varepsilon^2}{(1-\eta)^2+\varepsilon^2}. \tag{4.12} \]

If the substituents on the chain are symmetrical, then the potential function \(U(\varphi)\) is even and \(\varepsilon=0\). This is the case for such polymers as, for example, polyethylene and polyisobutylene. In this case formula (4.12) assumes the well-known form41, 42, 22:

\[ \overline{h^2}=Nl^2\frac{1+\cos\alpha}{1-\cos\alpha}\frac{1+\eta}{1-\eta}. \tag{4.13} \]

Thus, we have expressed the mean-square length of the chain in terms of the molecular constants \(N\), \(l\), \(\alpha\), \(\eta\), and \(\varepsilon\).

If rotation is strongly hindered, then \(\eta\) is close to 1 and \(\varepsilon\) is close to 0, and formula (4.12) goes over into the formula of S. E. Bresler and Ya. I. Frenkel36

\[ \overline{h^2}=Nl^2\frac{1+\cos\alpha}{1-\cos\alpha}\cdot \frac{2}{1-\eta}. \tag{4.14} \]

This formula is applicable to very long and stiff chains.

If rotation is completely free, then \(\eta=\varepsilon=0\), and from (4.12) we obtain Eyring’s formula35

\[ \overline{h^2}=Nl^2\frac{1+\cos\alpha}{1-\cos\alpha}. \tag{4.15} \]

The regions of applicability of formulas (4.13)—(4.15) to an accuracy of 5% are shown in Fig. 8. From formulas (4.7) and (4.12) follow also all the other expressions appearing in the literature37, 38, 39. We shall not dwell on this here (see44).

Fig. 8. Regions of applicability of formulas (4.13)—(4.15) (to an accuracy of 5%).

The method developed by the authors of this paper for calculating \(\overline{h^2}\) is also applicable to chains consisting of different links. Thus, for chains consisting of two alternating types of bonds, which are characterized by different mean cosines of the angle of rotation \(\eta_1\) and \(\eta_2\) (\(\varepsilon_1=\varepsilon_2=0\)), we obtain:

\[ \overline{h^2}=Nl^2\frac{1+\cos\alpha}{1-\cos\alpha}\cdot \frac{1-\eta_1\eta_2}{(1-\eta_1)(1-\eta_2)}. \tag{4.16} \]

*) Formula (4.12) pertains to chains in which all \(R\)-groups are attached on one and the same side. Obviously, the dimensions of chains with asymmetrical substituents must depend substantially on the stereoisomerism in the chain. See on this below.

It is easy to see that when $\eta_1=\eta_2=\eta$, formula (4.16) reduces to the simpler expression (4.13).

Of greatest interest among problems of this type is the comparison of the lengths of chains of trans-polyisoprene (natural rubber) and cis-polyisoprene (gutta-percha). Such a comparison was carried out by Uoll$^{47}$ and Benua$^{38}$ for a model that does not take account of hindered rotation, and gave the following results:

\[ \sqrt{\overline{h^2}}=2.90\cdot\sqrt{N}\ \text{\AA}\qquad(\text{trans}), \]

\[ \sqrt{\overline{h^2}}=2.01\cdot\sqrt{N}\ \text{\AA}\qquad(\text{cis}). \]

For comparison, let us note that an aliphatic chain with free rotation (see (4.15)) has the length

\[ \sqrt{\overline{h^2}}=2.18\cdot\sqrt{N}\ \text{\AA}. \]

Thus, a rubber molecule proves to be approximately $44\%$ longer than a gutta-percha molecule. Recently Markovich$^{30}$, remaining within the framework of Uoll’s model, carried out a more detailed study of the dependence of the dimensions of molecules of the polybutadiene type on the percentage content in them of cis- and trans-configurations (and also on the ratio of linkages of the types 1,2 and 1,4). However, in view of the crudeness of the model used (neglect of hindered rotation), no particular quantitative significance should be attached to all these results.

It must be emphasized that formulas (4.12)—(4.15) are applicable only to long chains with $N\gg 1$. For small $N$ other expressions are obtained. For example, instead of (4.13), from (4.7) we obtain$^{44}$:

\[ \overline{h^2} = Nl^2 \frac{1+\cos\alpha}{1-\cos\alpha} \cdot \frac{1+\eta}{1-\eta} - 2l^2 \frac{\cos\alpha\cdot(1-\eta^2)+2\eta}{(1-\cos\alpha)^2(1-\eta)^2} +\Delta \tag{4.17} \]

(where the term $\Delta$ may in practice be neglected in almost all cases). Failure to understand this circumstance leads to errors. Thus, in a recent work by Ubbelohde and Woodward$^{48}$ an attempt was made to apply formula (4.13) to low-molecular paraffins up to n-octane $(N=7)$. According to (4.11), the quantity $\eta$ in formula (4.13) must decrease with increasing temperature. Consequently, the quantity $\overline{h^2}$ must also decrease. Having considered experimental data on the viscosity of paraffins in the vapor state, which do not show this decrease, Ubbelohde and Woodward arrive at the conclusion that existing ideas about the nature of the potential $U(\varphi)$ are incorrect (cf.$^{49}$). In reality, formula (4.13) is inapplicable to this case, and instead one must use formula (4.16). It is easy to show that, with a correct calculation in the temperature interval from $+80$ to $+200^\circ$C, to which

...to which the data under discussion belong, there should be no substantial dependence of \(\bar h^2\) on temperature. Therefore, these data in no way contradict the generally accepted ideas about the retarding potential \(U(\varphi)\).

The theory set forth shows that real polymer molecules are strongly coiled and have a globular shape. This, however, does not apply to chains with small values of \(N\). V. N. Tsvetkov and V. A. Marinin \(^{50}\) investigated the Kerr effect in solutions of alcohols \(C_nH_{2n+1}OH\) with \(n\) from 3 to 26 and established that these molecules have no noticeable coiling and practically constitute fairly rigid formations. The data on the vibrational spectra of \(n\)-paraffins \(^{51}\) say the same.

On the other hand, there is a multitude of diverse experimental evidence for the strong coiling of long polymer chains in solutions.

A convincing direct confirmation of such coiling is the work of V. N. Tsvetkov and E. V. Frisman \(^{52,6}\), who established that the birefringence of polystyrene in a magnetic field decreases with increasing degree of polymerization. This shows that the longer chains are more strongly coiled and therefore more isotropic than the shorter ones. Obviously, if there were no coiling, the birefringence would have to increase linearly with increasing degree of polymerization.

The study of the dynamo-optical effect in solutions of polyisobutylene (V. N. Tsvetkov and E. V. Frisman \(^{53,6}\)) shows that the intrinsic anisotropy of a polymer molecule in solution is of the same order of magnitude as the anisotropy of the monomer, as it should be for a statistically coiled chain.

The data on viscosity and diffusion of polymer solutions also speak of the strong coiling of the chain. Direct determinations of the dimensions of polymer molecules in solutions from the angular asymmetry of the intensity of scattered light \(^{54}\) and by the method recently proposed by V. N. Tsvetkov \(^{55}\) also give results confirming the conclusion about the strong coiling of the chain (see more on this in Section 7).

5. ROTATIONAL-ISOMERIC THEORY OF LINEAR POLYMERS

Determination of the potential function of internal rotation \(U(\varphi)\) in general is associated with considerable difficulties. In this paragraph we shall consider the principal facts relating to internal rotation about single bonds and its influence on the dimensions and flexibility of polymer chains.

As investigations of the thermodynamic properties of low-molecular substances containing single bonds have shown, internal rotation in molecules is always hindered to one degree or another \(^{56,57}\).

The cause of the hindrance is the repulsion of nonbonded atoms, which for the most part leads to a more stable trans configuration of the molecule. Thus, for example, in ethane the configuration shown in Fig. 9, a, is more stable than that shown in Fig. 9, b. The difference in energy of these configurations of ethane is 2750 cal/mole. The experimental values of the potential barriers to internal rotation, obtained from thermodynamic data, in the case of hydrocarbons are satisfactorily reproduced by the empirical relation^58

Fig. 9. Ethane molecule: a) trans configuration, b) cis configuration.

\[ U=\sum_{i,k}\frac{C_{ik}}{r_{ik}^{5}}, \tag{5.1} \]

where \(C_{ik}\) are constants, and \(r_{ik}\) is the distance between the nonbonded atoms \(i\) and \(k\). Recently an attempt was made to justify this dependence on the basis of considering the electrostatic interaction of quadrupoles—the quadrupole interaction^59. This attempt met with objections^60. Apparently, the forces hindering internal rotation do not reduce to any one definite type^61, ^62, ^63. Their character and the order of magnitude of their energy are probably the same as in the case of intermolecular interaction.

The approximate form of the function \(U(\varphi)\) for ethane may be written as

\[ U(\varphi)=\frac{1}{2}U_0(1-\cos 3\varphi), \tag{5.2} \]

Fig. 10. Potential of internal rotation in the ethane molecule.

where \(U_0\) is the value of the potential barrier (Fig. 10). In the case when there is only one stable configuration, the potential \(U(\varphi)\) has the form^36 (Fig. 11)

\[ U(\varphi)=\frac{1}{2}U_0(1-\cos \varphi). \tag{5.3} \]

If the molecule lacks axial symmetry, then internal rotation leads to the appearance of several energetically nonequivalent—

... configurations, the so-called rotational isomers.^10 Their properties and structure can be studied by the methods of molecular optics, spectroscopy, etc.^64,65

Potentials (5.2) and (5.3) are unsuitable for characterizing internal rotation in the presence of rotational isomerism (Fig. 12). In this case one may use, for example, a function of the following type:^49

\[ \begin{aligned} U(\varphi)={}& \frac{1}{2}U_0\{x(1-\cos\varphi)+{}\\ &+(1-x)(1-\cos3\varphi)\} \end{aligned} \tag{5.4} \]

(see also^42). There have been a number of attempts to apply these approximate expressions to the calculation of the quantity \(\eta=\cos\varphi\) in the case of polymer chains. Hindered rotation in such chains was first considered in the cited work of S. E. Bresler and Ya. I. Frenkel,^36 who limited themselves to considering torsional oscillations about the single equilibrium position—the potential (5.3). The calculation of \(\eta\) with the aid of the function \(U(\varphi)\) (5.3) can be carried out in closed form. From (4.11) and (5.3) it follows^38:

\[ \eta=-i\,\frac{J_1\!\left(i\frac{U_0}{2kT}\right)} {J_0\!\left(i\frac{U_0}{2kT}\right)}, \tag{5.5} \]

where \(J\) are Bessel functions. Calculation of \(\eta\) with the aid of (5.4) requires numerical integration. Further refinement of the form of \(U(\varphi)\) is difficult and should lead to very cumbersome calculations.

Fig. 11. Potential of internal rotation when there is only one stable configuration.

Fig. 12. Potential of internal rotation in the presence of rotational isomerism.

Internal rotation in polymer chains always leads to energetically nonequivalent configurations, since the highest symmetry of an element of a polymer chain is \(C_i\) or \(C_2\). Instead of considering a continuous potential function \(U(\varphi)\) of the type (5.4) or of some more complicated form, one may regard a linear polymer as a mixture of rotational isomers.^66,67

In the zeroth approximation we shall assume that each bond can have only certain equilibrium positions with discrete values \(\varphi_i\). For example, singling out one bond \(C—C\) in a polyethylene molecule, we shall write its formula in the form

\[ \mathrm{H_3C(CH_2)_nH_2C—CH_2(CH_2)_mCH_3}. \]

On rotation about this bond one obtains the rotational isomers \(t\), \(d\), and \(l\) (Fig. 13a), similar to the rotational isomers of \(n\)-butane (Fig. 13b). The trans isomer is the most stable.

Fig. 13a. Rotational isomerism in polyethylene.

It is obvious that the integration (4.11) in this case may be approximately replaced by the much simpler summation

\[ \eta = \frac{\displaystyle \sum_{i=1}^{n} g_i \cos \varphi_i} {\displaystyle \sum_{i=1}^{n} g_i} \quad ^*) \tag{5.6} \]

\(^*)\) In the following approximation, taking into account small torsional oscillations about the equilibrium position for each rotational isomer, we obtain, for potential barriers \(U_i\) much larger than \(kT\),

\[ \eta = \frac{\displaystyle \sum_{i=1}^{n} g_i \cos \varphi_i \left(1-\frac{kT}{U_i}\right)} {\displaystyle \sum_{i=1}^{n} g_i} \]

(see \(^{67}\)).

where \(g_i\) is the statistical weight of the \(i\)-th isomer,

\[ g_i=e^{-\frac{F_i}{kT}}, \tag{5.7} \]

\(F_i\) is the free energy of the isomer. In an approximation that does not take torsional vibrations into account, the difference in the free energies of the rotational isomers is equal to the difference in their internal energies: \(\Delta F_i \simeq \Delta U_i\). If

Fig. 136. Rotational isomerism in n-butane.

there were no rotational isomerism and all \(g_i\) were equal to one another, we would obtain

\[ \eta=\frac{1}{n}\sum_{i=1}^{n}\cos\varphi_i \tag{5.8} \]

and if the energy minima are situated at equal distances from one another, then

\[ \eta=\frac{1}{n}\sum_{i=1}^{n}\cos\frac{2\pi}{n}i=0, \tag{5.8a} \]

i.e., at arbitrarily high potential barriers \(\eta=0\), as in the case of completely free rotation.

Thus, in a thermodynamically equilibrium state the quantities \(\eta\) and \(\varepsilon\), and consequently also the degree of coiling of the chain, are determined not by the heights of the potential barriers separating the rotational isomers, but by the differences in the energies of the latter. The kinetic behavior of the chain, i.e. its relaxation properties, must, of course, depend in an essential way on the barriers to internal rotation.

At a given temperature, a polymer chain may be characterized by a definite value of the equilibrium constant of the rotational isomers. Let the chain consist of \(N\) links. \(N_1\) links are in the position \(\varphi_1\), \(N_2\)—in the position \(\varphi_2\), etc.

Then

\[ N_i=\frac{N e^{-\frac{F(\varphi_i)}{kT}}}{\sum_{j=1}^{N} e^{-\frac{F(\varphi_i)}{kT}}}, \tag{5.9} \]

\[ \sum_i N_i=N. \]

For given values of \(N_i\), the rotational isomers can be distributed along the chain in a large number of different ways. This number is equal to

\[ \frac{N!}{\prod_i N_i!}. \tag{5.10} \]

All these distributions correspond to one and the same chain energy and therefore they are equiprobable. Each distribution corresponds to a definite value of the chain “length” \(h\). However, this does not mean that all values of \(h\), from 0 to the maximum, are equiprobable, since different values of \(h\) correspond to different numbers of distributions.

It is evident that, in principle, it is possible to derive a formula for the mean square length of the chain, of the type (4.13), by assigning from the very beginning statistical weights to the rotational isomers. This problem is solved by the method of Markov chains\(^{17,18}\) and was considered by one of us in Ref. \(^{67}\) for the plane case. The result of the calculation naturally proved to correspond to formula (4.13). In an analogous manner, evidently, the distribution function of chain lengths can also be derived.

We see that the degree of coiling of a polymer chain—its thermodynamic flexibility—turns out to be an explicit function of the statistical weights of the rotational isomers and, consequently, of the temperature. Indeed, the factor \(\frac{1+\eta}{1-\eta}\) in formula (4.13) can be represented in the form

\[ \frac{1+\eta}{1-\eta} = \frac{ \sum_{i=1}^{n} e^{-\frac{F(\varphi_i)}{kT}}(1+\cos\varphi_i) }{ \sum_{i=1}^{n} e^{-\frac{F(\varphi_i)}{kT}}(1-\cos\varphi_i) }. \tag{5.11} \]

Let us consider some of the simplest cases. For polymers of the polyethylene and polyisobutylene type we have: \(\varphi_1=0^\circ,\ \varphi_2=120^\circ,\ \varphi_3=240^\circ\).

Taking the free energy relative to the value \(F(\varphi_1)\), we have: \(F(\varphi_1)=0\), \(F(\varphi_2)=F(\varphi_3)=\Delta F \simeq \Delta U\). Consequently,

\[ \frac{1+\eta}{1-\eta} = \frac{2+e^{-\frac{\Delta U}{kT}}}{3e^{-\frac{\Delta U}{kT}}}. \tag{5.11a} \]

The smaller \(\Delta U\) is, the closer \(\eta\) is to zero and the greater the degree of coiling of the chain. From obvious stereochemical considerations

Fig. 13b. Rotational isomerism in polyisobutylene.

it is clear that a polyisobutylene chain must be more coiled (possess greater thermodynamic flexibility) than a polyethylene chain.

Indeed, the rotational isomers of polyisobutylene differ very little from one another (Fig. 13b), whereas in polyethylene they differ substantially (Fig. 13a). Fig. 14 shows an approximate course of the potential curves \(U(\varphi)\) for polyethylene and polyisobutylene.

Fig. 14. Potential of internal rotation in polyethylene (curve \(a\)) and polyisobutylene (curve \(b\)).

The degree of coiling of polymer chains with asymmetric substituents (polystyrene, polyvinyl chloride, polyvinyl alcohol, etc.), built according to the scheme \((-\mathrm{CH}_2-\mathrm{CHR}-)_n\), depends substantially on whether the \(R\)-groups are arranged on different sides or on the same side of the chain. In the first case

(\(dl\)-configuration, see Fig. 15, a) the chain dimensions must be larger than in the second case (\(dd\)-configuration, see Fig. 15, b).

Apparently, in the majority of polymers of the type under consideration the stereoisomers are distributed along the chain in a more or less random manner. However, for example, in polyvinyl chloride an ordered \(dl\)-arrangement of the substituents is realized \(^{1,71}\). It is possible that the stereoisomerism in polymer chains depends to some extent on the polymerization conditions, and that this is one of the reasons determining the different viscosity and different ability to crystallize of samples of one and the same polymer obtained at different temperatures.

a)

b)

Fig. 15. Stereoisomerism in polystyrene.

We see that simple stereochemical considerations make it possible to draw substantial conclusions about the properties of the chain. At the same time, the rotational-isomeric theory also provides an opportunity for more detailed quantitative calculations of \(\eta\) and \(\varepsilon\). Starting from the empirical formula (5.1) and the known experimental values of the potential barriers for ethane and propane molecules, one can determine the constants \(C_{ik}\), characterizing the mutual repulsion of the groups H and H, H and \(\mathrm{CH}_3\), \(\mathrm{CH}_3\) and \(\mathrm{CH}_3\). We have:

\[ C_{\mathrm{H},\mathrm{H}} = 4.4 \cdot 10^5 \frac{\mathrm{cal}\cdot \text{\AA}^5}{\mathrm{mol}}, \qquad C_{\mathrm{H},\mathrm{CH}_3} = 7.6 \cdot 10^5 \frac{\mathrm{cal}\cdot \text{\AA}^5}{\mathrm{mol}}, \]

\[ C_{\mathrm{CH}_3,\mathrm{CH}_3} = 13.1 \cdot 10^5 \frac{\mathrm{cal}\cdot \text{\AA}^5}{\mathrm{mol}}. \]

The potential barriers of isobutane and tetramethylmethane calculated with the aid of these values are in good agreement with experiment. Table 1 gives the results of calculations for several polymers68.

Table 1

Polymer	Bond about which rotation takes place	Rotational isomers, in degrees	$\Delta U$ (cal/mol)	$\eta$
Polyethylene	$\mathrm{H_2C — CH_2}$	0, 120, 240	$\Delta U_{120}=\Delta U_{240}=800$	0.5
Polyisobutylene	$\mathrm{H_2C — C(CH_3)_2}$	0, 120, 240	$\Delta U_{120}=\Delta U_{240}=180$	0.1
Polyisoprene (1,4) cis (natural rubber)	$\mathrm{CH=CH}$	180	$(\infty)$	$-1$
Polyisoprene (1,4) cis (natural rubber)	$\mathrm{CH — CH_2}$	90, 270	$\Delta U_{270}=0$	$0.3^{*}$
Polyisoprene (1,4) cis (natural rubber)	$\mathrm{C(CH_3)—CH_2}$	90, 270	$\Delta U_{270}=0$	$0.1^{*}$
Polyisoprene (1,4) cis (natural rubber)	$\mathrm{CH_2 — CH_2}$	0, 120, 240	$\Delta U_{120}=\Delta U_{240}=1000$	0.5

* In the case of rotation about a single bond adjacent to a double bond, because of the smoothness of the potential curve (see Fig. 16), formula (5.6) cannot be used. The values of $\eta$ given in the table were obtained by numerical integration using formula (4.11).

The rotational isomers whose energy is taken to be zero are set in bold type; $\varepsilon$ in these polymers is equal to zero.

The method of calculation employed makes it possible, along with the values of $\Delta U$, $\eta$, and $\varepsilon$, to determine also the general form of the curve $U(\varphi)$. For rotation about single bonds adjacent to double bonds, the form of the curves $U(\varphi)$ differs substantially from that shown in Figs. 12 and 14. Figure 16 shows the corresponding curve for rotation about the bond $\mathrm{=CH — CH_2—}$ in rubber, and the configurations corresponding to the minima and maxima of the curve. Rotation proves to be practically free over a fairly wide interval of angles $\varphi$. This is due to the stereochemistry of the double bond—the planar arrangement of all bonds adjacent to it. Such an arrangement leads to a smoothing of the potential curve, since the configurations favorable

for some pairs of interacting atoms prove to be unfavorable for others.

At the present time there is still no direct experimental evidence for the existence of rotational isomerism in polymers. At the present stage of investigation, the rotational-isomeric theory should be regarded as a method for considering the structures of polymer chains. On the basis of simple stereochemical considerations, this method makes it possible to relate the physical properties of polymer chains to their chemical structure. The rotational-isomeric theory is directly applicable

Fig. 16. Potential of internal rotation about a single bond adjacent (curve a) and nonadjacent (curve b) to a double bond.

to isolated polymer chains. The physical properties of polymers in bulk require, for their explanation, that account be taken, along with the behavior of individual chains, of intermolecular interaction. This interaction must have a very strong effect on the form of the potential curve of internal rotation—on the values of \(U_0\) and \(\Delta U\)—in the case of polar polymers. Indeed, the intermolecular interaction here is of the same order as the intramolecular mutual repulsion of unbonded atoms, and the quantities \(U_0\) and \(\Delta U\) should in this case be regarded not as characteristics of an individual molecule, but as characteristics of the medium as a whole. However, if the polymer contains no polar groups, its properties in bulk should be more directly connected with the behavior of the individual chains. Thus, the high-elastic properties of rubber are evidently connected with practically free rotation about bonds adjacent to double bonds, as established for isolated chains.

The rotational-isomeric theory concerns the thermodynamically equilibrium properties of chains. Let us show that the consequences of this

… theoretical considerations make it possible to interpret quantitatively certain properties of polymers in bulk—the melting of crystalline polymers.

Polymers crystallize either under large extensions or in the unstretched state. There exist polymers that crystallize in both ways, as well as polymers that cannot at all be obtained in crystalline form. A crystalline polymer differs substantially from a crystalline low-molecular substance. A polymer is never

Fig. 17. Diagram of the structure of a crystalline polymer.

crystallized throughout. There is no boundary separating the crystalline and amorphous phases: the same chain molecules pass through both crystalline and amorphous regions (Fig. 17). The crystalline regions may have different sizes depending on the crystallization conditions (cooling rate, temperature at which crystallization takes place, etc.). The sizes and degree of ordering of the crystallites, as well as the very fact of crystallization, can be established directly by X-ray and electron-diffraction methods, and also on the basis of the macroscopic physical properties of the specimen. Owing to the different sizes of the crystallites and the different degrees of their ordering, a polymer does not possess a strictly definite melting temperature of crystals, but is characterized by a more or less broad interval \(T_{\text{m}}\). It is characteristic that this interval corresponds to considerably lower temperatures than might have been expected. If we extrapolate the values of \(T_{\text{m}}\) for

low-molecular substances constructed from rigid molecules to large values of the molecular weights, we shall obtain much higher \(T_{\mathrm{m}}\) than those observed for high polymers. The values of \(T_{\mathrm{m}}\) of a polymer are practically independent of the degree of polymerization. The \(T_{\mathrm{m}}\) of crystalline polymers increases with increasing degree of stretching. Further information relating to this very interesting chapter of polymer physics the reader will find in the special literature\(^3, 7, 8, 69, 70, 71\).

The rotational-isomeric theory makes it possible to explain the differences in the temperatures and heats of fusion of different polymers\(^ {72}\). These differences are very large and have substantial practical significance. Thus, polyethylene and gutta-percha are largely crystalline at room temperature and accordingly lack high-elastic properties. Conversely, natural rubber crystallizes either at a sufficiently low temperature or upon stretching, while polyisobutylene in the unstretched state cannot be obtained in crystalline form at all. The substances mentioned are hydrocarbons, and the energy of intermolecular interaction in them should be of one and the same order. It follows from this that the difference in the melting temperatures of these substances is connected chiefly with the difference in the entropies of fusion. The melting of crystals amounts to the disappearance of long-range order. In the present case this means long-range order in the arrangement of the links of one and the same chain and in the relative arrangement of the links of different chains. We assume that in the case of polymer crystals the entropy of fusion is determined primarily by the change in the relative arrangement of the links belonging to one and the same chain. The relative arrangement of the links of different chains is determined mainly by the latent heat of fusion. Let us assume that in the amorphous state a polymer chain is a mixture of rotational isomers with different free energies \(F_i\). The statistical sum of such a mixture can be written in the form

\[ G = N! \prod_i \frac{g_i^{N_i}}{N_i!}, \tag{5.12} \]

where \(N\) is the total number of links of the chain, \(N_i\) is the number of links with energy \(F_i\), \(g_i = e^{-F_i/kT}\). The free energy of the mixture is equal to

\[ F = -kT \ln G \tag{5.13} \]

and, for large \(N\) and \(N_i\),

\[ F = NkT \sum_i \nu_i \ln \frac{\nu_i}{g_i}, \tag{5.14} \]

where \(\nu_i N = N_i\).

Entropy of the chain

\[ S=-\left(\frac{\partial F}{\partial T}\right)_{V,N_i} =-Nk\sum_i \nu_i \ln \frac{\nu_i}{g_i} +NkT\sum_i \nu_i \frac{d\ln g_i}{dT}. \tag{5.15} \]

In the state of thermodynamic equilibrium

\[ \nu_i^{(0)}=\frac{g_i}{\sum_k g_k}. \tag{5.16} \]

and

\[ \left. \begin{aligned} F^{(0)}&=-NkT\ln\left(\sum_k g_k\right),\\[4pt] S^{(0)}&=Nk\left\{\ln\left(\sum_k g_k\right) +T\frac{\partial}{\partial T}\ln\left(\sum_k g_k\right)\right\},\\[4pt] U^{(0)}&=F^{(0)}+TS^{(0)} =N\frac{\sum_k g_k u_k}{\sum_k g_k} =N\bar u, \end{aligned} \right\} \tag{5.17} \]

\(\bar u\) is the internal energy per link, averaged over all rotational isomers.

From X-ray and electron-diffraction studies of low-molecular and high-molecular hydrocarbons and their derivatives it is known that in the crystalline state they have a trans configuration\(^{71,73}\). In this state

\[ \left. \begin{aligned} F_{\mathrm{tr}}&=NF_1,\\ S_{\mathrm{tr}}&=NS_1,\\ U_{\mathrm{tr}}&=NU_1, \end{aligned} \right\} \tag{5.18} \]

where the index 1 refers to the single trans isomer in the present case. It follows from this that melting of a crystalline polymer is associated with the following changes in the thermodynamic functions:

\[ \left. \begin{aligned} \Delta F&=F^{(0)}-F_{\mathrm{tr}} =-NkT\ln\left(\sum_k x_k\right),\\[4pt] \Delta S&=Nk\left\{\ln\left(\sum_k x_k\right) +\frac{\overline{\Delta U}}{kT}\right\},\\[4pt] \Delta U&=N\cdot\overline{\Delta U}, \end{aligned} \right\} \tag{5.19} \]

where

\[ x_k=\frac{g_k}{g_1}=e^{-\frac{F_k-F_1}{kT}}=e^{-\frac{\Delta F_k}{kT}}\simeq e^{-\frac{\Delta U_k}{kT}}. \tag{5.20} \]

At the melting point the relation holds

\[ T_{\mathrm{m}}=\frac{\Delta Q}{\Delta S'}=\frac{\Delta U+p\Delta V+\Delta E}{\Delta S+\Delta \Sigma}, \tag{5.21} \]

where \(\Delta Q\) is the change in heat content (latent heat of melting), \(p\) is the pressure, \(\Delta V\) is the change in volume, \(\Delta E\) is the change in the energy of intermolecular interaction, \(\Delta S\) is the change in the internal entropy of the chains, and \(\Delta \Sigma\) is the change in intermolecular entropy. The term \(p\Delta V\) may be neglected. Moreover, we neglect the quantity \(\Delta \Sigma\) in comparison with \(\Delta S\), i.e., as was already indicated above, we assume that on melting the principal role is played by the change in intramolecular entropy. Indeed, extrapolating the dependence of \(T_{\mathrm{m}}\) on \(N\) for rigid low-molecular paraffins, we obtain for polymers considerably higher melting temperatures than those that actually occur. This is explained by the fact that for rigid molecules \(\Delta S=0\), whereas for polymers \(\Delta S\), on the contrary, is large. For rigid chains we would have

\[ T_{\mathrm{m}}^{*}=\frac{\Delta E}{\Delta \Sigma}, \]

and for flexible ones

\[ T_{\mathrm{m}}=\frac{\Delta E+\Delta U}{\Delta S+\Delta \Sigma}. \]

Since

\[ T_{\mathrm{m}}\ll T_{\mathrm{m}}^{*}, \]

then

\[ \Delta U\cdot\Delta\Sigma\ll\Delta E\cdot\Delta S, \]

i.e.,

\[ \Delta \Sigma \ll \frac{\Delta E}{\Delta U}\Delta S. \]

The quantities \(\Delta E\) and \(\Delta U\) are of the same order; therefore the indicated inequality means that

\[ \Delta\Sigma\ll\Delta S. \]

Indeed, since the number of internal degrees of freedom in a polymer chain is much greater than the number of external ones, it is natural that the main part of the entropy change upon melting is connected precisely with the internal degrees of freedom (cf. \(^{74}\)).

Taking into account the simplifications made, we have:

\[ kT_{\text{pl}}\cdot \ln\left(\sum_k x_k\right)=\Delta\varepsilon, \tag{5.22} \]

where

\[ N\Delta\varepsilon=\Delta E. \]

Table I gives the values of \(\Delta U\) calculated by us for several of the simplest polymers. Equation (5.22) and Table I make it possible, proceeding from the experimental value of the melting interval \(T_{\text{pl}}\), for example for rubber (from \(-45^\circ\text{C}\) to \(+15^\circ\text{C}^{70}\)), to calculate the corresponding interval of values of \(\Delta\varepsilon\): from 750 to 1000 calories per mole of isoprene units.

The heat of fusion of rubber

\[ \Delta Q=\Delta\varepsilon+\overline{\Delta U} \]

turns out to lie in the interval \(950\text{--}1250\ \text{cal/mol}\) of isoprene units. The experimental value, estimated by Parks\({}^{75}\) from data on the melting of partially crystallized rubber\({}^{76}\), is \(1200\ \text{cal/mol}\) of isoprene units. Assuming that for polyethylene and polyisobutylene \(\Delta\varepsilon\) has approximately the same value as for rubber, and using the data of Table I, we obtain the following values of the temperature and latent heat of fusion for these polymers (see Table II).

Table II

Polymer	\(T_{\text{pl}}^\circ\text{C}\), calculated	\(T_{\text{pl}}^\circ\text{C}\), experimental	\(\Delta Q\), cal/mol of bonds, calculated	\(\Delta Q\), cal/mol of bonds, experimental
Polyethylene	\(+50 \div +110\)	\(+60 \div +120^{70}\)	\(600 \div 700\)	\(800^{77}\)
Polyisobutylene	\(-90 \div -40\)	Does not crystallize \(T_{\text{pl}}<T_{\text{glass}}=-70^\circ\text{C}\)	\(400 \div 500\)	—
Natural rubber	—	\(-45 \div +15^{70}\)	\(400 \div 500\)	\(300^{76}\)

It is obvious that the assumption of the constancy of \(\Delta\varepsilon\) for all hydrocarbon polymers cannot be considered entirely legitimate, since the change in energy upon crystallization depends not only on the energy of intermolecular interaction, but also on how much

greatly changes upon crystallization the density of the specimen, i.e., the packing factor. Therefore, the greater crystallinity of polyethylene in comparison with polyisobutylene or natural rubber is explained not only by the lesser flexibility of its molecules, but also by better packing. The packing factor is also responsible for the differences between the crystallinity of natural rubber and gutta-percha.

Thus, we consider the mechanism of melting of a crystalline polymer to be directly related to the flexibility of the chain. Polymers melt at comparatively low temperatures as a consequence of a considerable increase in entropy upon transition to the amorphous state, owing to the appearance of a mixture of rotational isomers. The more flexible the chain is, i.e., the smaller the difference in the energies of the rotational isomers \(\Delta U\), the larger the number of coiled rotational isomers in the mixture, and the lower the melting temperature of the polymer. According to the rotational-isomeric theory, \(T_{\mathrm{m}}\) does not depend on the degree of polymerization \(N\), if the latter is sufficiently high.

6. STATISTICS OF REAL POLYMER MOLECULES

As we have already indicated above (see Section 2), the usual statistics of linear polymers essentially describes the configurations assumed by an infinitely thin thread. It is precisely with this that the circumstance is connected that the classical distribution function for the distance between the ends of a chain

\[ W(h)h^2\,dh= \left(\frac{3}{2\pi NA^2}\right)^{3/2} 4\pi e^{-\frac{3h^2}{2NA^2}}h^2\,dh \tag{6.1} \]

\[ \left( A^2=l^2\frac{1+\cos\alpha}{1-\cos\alpha}\, \frac{1+\eta}{1-\eta} \right) \]

is completely analogous to the distribution function for the distance traveled by a freely diffusing particle in time \(t=N\), provided that the diffusion coefficient is

\[ D=\frac{1}{6}A^2. \]

In reality, the various parts of this thread interact with one another. First of all, we are interested in the main part of this interaction, consisting in the fact that no pair of links in the chain can simultaneously be in one and the same volume element (the so-called volume effects). In analogy with the motion of a diffusing particle, this means that the particle must diffuse in such a way as never to enter the same place twice. Thus, we are dealing with the diffusion of a particle that does not intersect its own path. In this case, from the entire set of a priori equally probable configurations that a polymer chain can assume, a large number of configurations turn out to be forbidden.

Since in reality only configurations can occur in which not a single pair of links falls at one and the same point, the number of forbidden configurations turns out to be very large even at comparatively small densities of links inside the polymer coil. It is quite obvious that, since the density of links is greater in strongly coiled configurations than in extended ones, it is precisely among the former that the largest number of forbidden configurations will be found. This means that volume effects reduce the relative number of strongly coiled configurations and thereby increase the average dimensions of the chain. The theoretical calculation of this increase in the average dimensions constitutes the principal problem of the theory of volume effects.

Above we saw that the existence in the chain of fixed valence angles and hindered internal rotation likewise leads to an increase in the average dimensions of the chain. However, allowance for these factors does not change the general form of the distribution function \(W(h)\) (equation (6.1)), but is reflected only in the expression for \(A^2\): for a freely jointed chain \(A^2=l^2\), for a chain with fixed valence angles

\[ A^2 = l^2 \frac{1+\cos\alpha}{1-\cos\alpha}, \]

for a chain with fixed valence angles and hindered internal rotation:

\[ A^2 = l^2 \frac{1+\cos\alpha}{1-\cos\alpha}\cdot \frac{1+\eta}{1-\eta}. \]

It was natural first of all to try to take account also of volume effects while retaining a function of the type (6.1), changing only \(A^2\) in the corresponding way. It is clear, however, that attempts of this kind constitute only an extremely crude approximation, since the Gaussian function (6.1) is applicable only to the description of a set of random events whose probabilities do not depend on one another. Meanwhile, the meaning of volume effects consists precisely in the fact that the probability that some link will fall into the element of volume under consideration drops to zero if this element is already occupied by another atom. Therefore the application of Gaussian statistics of independent random events in this case is illegitimate. Nevertheless, the first attempts at a theoretical treatment of volume effects were made on the basis of retaining the Gaussian function.

The question of volume effects was first considered in the work cited above by W. Kuhn \(^{11}\). Proceeding from the fact that the influence of volume effects should increase with increasing degree of polymerization, Kuhn proposed using formula (6.1), putting in it:

\[ A = A_0 \cdot N^{\varepsilon}, \]

where $\varepsilon$ is some small positive number. Then we obtain

\[ \overline{h^2}=NA^2=N^{1+2\varepsilon}A_0^2. \tag{6.2} \]

Here $A_0$ must depend on the volume of the links. Below we shall see that a formula of the type (6.2) must indeed hold; however, Kuhn gave no prescriptions for calculating $\varepsilon$ and $A_0$ (and did not even show that $\varepsilon$ is in fact different from zero), and therefore his formula (6.2) was still, to a considerable extent, speculative in character.

Further steps in this direction were undertaken in the works of Benoit$^{31,38}$ and Sadron$^{25,39}$. The authors of these works assumed that, because of volume effects, a part of the circumference $A'A''$ (Fig. 7) is forbidden for the atom under consideration. Then allowance for volume effects is equivalent to allowance for the hindrance of internal rotation, and in this case

\[ \eta=\frac{\sin B}{B}, \]

where $2B$ is the arc of the circumference not forbidden by spherical effects. If $B=0$, then $\eta=1$ (a rigid chain); if $B=\pi$, then $\eta=0$ (free rotation). However, in the papers considered the question of the relation of $B$ to the molecular constants and to the degree of polymerization remains completely open; therefore the formulas obtained give practically nothing. It should be noted that, long before the appearance of the papers by Benoit and Sadron, the idea of a region on the circumference forbidden by volume effects appeared in the work of Laskowski and Berg$^{78}$. However, in that work no dependence of $\overline{h^2}$ on $B$ was given, nor any prescription for calculating $B$ for long chains. In general it must be said that all these works are applicable rather to the accounting of spherical obstacles caused by the interaction of neighboring atoms than to volume effects in the sense that interests us.

Some attempts to substantiate the possibility of taking volume effects into account while retaining the Gaussian function and even the relation $\overline{h^2}\sim N$ were undertaken by Montroll$^{21}$, and also by Frisch, Collins, and Friedman$^{79}$. They took into account the impossibility of overlap of several neighboring links of the chain. It is clear that, since allowance for any interaction between neighbors does not prevent the breaking of the chain into independent statistical elements, we thereby preserve the Gaussian function $W(h)$ and the proportionality between $\overline{h^2}$ and $N$. However, since in doing so we in fact do not take into account the volume effects as an interaction of all the links of the chain with all the others, this result, contrary to Montroll’s opinion$^{21}$, has no relation to the properties of real polymer molecules. We note that Montroll’s mathematical apparatus was also criticized$^{80}$ (see also$^{81}$).

The most serious attempt to take account of volume effects while retaining the Gaussian distribution \(W(h)\) belongs to Flory\(^{82}\). Flory proceeds from the fact that the relative number of chain configurations corresponding to a given value of \(h\) is expressed by the Gaussian function \(W_0(h)\), multiplied by the fraction of configurations not forbidden by volume effects, which we shall denote by \(F(h)\):

\[ W(h)=W_0(h)\cdot F(h). \tag{6.3} \]

The problem consists in calculating \(F(h)\). It is clear that volume effects influence the dimensions of the chain precisely because \(F(h)\) has different values for different \(h\) (the larger \(h\) is, the larger they are). However, Flory immediately replaces \(F(h)\) by \(F=\overline{F(h)}\), ignoring the circumstance that volume effects influence different configurations to different degrees.

Flory next proceeds to calculate the quantity

\[ F=\overline{F(h)}. \]

The method he applied is easily generalized so that it can also be used for calculating the function \(F(h)\). A small region of the space occupied by the chain is considered, the segment density*) in which may be regarded as constant. The region is divided into \(\nu_j\) cells and the question is considered of the distribution in it of \(n_j\) segments.

The probability that the \((k+1)\)-st segment will collide with one of the preceding ones is equal to \(\dfrac{k}{\nu_j}\). Hence it is easy to see that the probability that no pair of segments will collide is equal to

\[ \overline{f_j}=\prod_{k=0}^{n_j-1}\left(1-\frac{k}{\nu_j}\right)\simeq e^{-\frac{n_j^2}{2\nu_j}}. \tag{6.4} \]

Let the region we have considered be a spherical layer of radius \(s_j\) and thickness \(\Delta s_j\), whose center coincides with the center of inertia of the molecule. Then

\[ n_j=nW(s_j)\Delta s_j, \]

\[ \nu_j=\frac{4\pi s_j^2\Delta s_j}{v}, \tag{6.5} \]

where \(W(s_j)\) is the probability that the distance from some

*) The segments referred to here do not coincide with the Kuhn segments mentioned above. Their dimensions are smaller than the dimensions of Kuhn segments and are rather close to the dimensions of a monomeric unit.

segment to the center of gravity of the molecule is equal to \(s_j\); \(n\) is the total number of segments and \(v\) is the volume of one cell.

Assuming that \(W(s_j)\) is a Gaussian function and choosing its parameter so that

\[ \overline{s_j^2}=\overline{R^2}=\frac{1}{6}\varkappa^2 N A^2, \]

where the factor \(\varkappa\) shows how many times the dimensions of the chain have changed owing to volume effects, we obtain:

\[ \ln F=\sum_j \ln \overline{f_j} = -\frac{27}{2^{5/2}\pi^{3/2}}\, \frac{v n^2}{(\varkappa^2 N A^2)^{3/2}}. \tag{6.6} \]

Flory assumes that the segment length \(l' = v^{1/3}\) and that \(n l' = N A\). Then equation (6.6) becomes

\[ \ln F=-\frac{C\sqrt{N}}{\varkappa^3}, \tag{6.7} \]

where

\[ C=\frac{27}{2^{5/2}\pi^{3/2}}\cdot\frac{l'}{A}\simeq \frac{l'}{A}. \tag{6.7a} \]

The quantity \(C\) cannot be calculated within the framework of Flory’s theory; therefore he did not carry out quantitative calculations using formulas (6.6) and (6.7a).

The quantity \(F\) calculated in this way does not depend on \(h\), and, consequently, with its help one cannot directly take into account volume effects using equation (6.3). Flory attempted to avoid this difficulty by introducing consideration of configurations of a system of \(G\) molecules. The total number of such configurations corresponding to a given distribution \(h_i\) is equal to:

\[ \Omega=G!\prod_i \frac{\omega_i^{G_i}}{G_i!}. \tag{6.8} \]

Here \(G_i!\) is the number of molecules with distance between the ends \(h_i\), and \(\omega_i\) is the number of configurations of one molecule corresponding to this distance.

Denoting the total number of configurations of a molecule by \(\omega_0\), and the total number of configurations of the system by \(\Omega_0\), and taking into account that

\[ \omega_i=\omega_0\cdot W_0(h_i)\cdot F, \tag{6.9} \]

we obtain (using Stirling’s formula):

\[ \ln \frac{\Omega}{\Omega_0} = G\cdot \ln(F\cdot G) + \sum_i G_i \ln \frac{W_0(h_i)}{G_i}. \tag{6.10} \]

The quantity \(F\) was defined above; it remains to estimate the second term. To estimate the latter Flory assumes that, since without taking into account

volume effects: \(G_{i0}=G\cdot W_0(h_i)\), then, taking them into account, one may put

\[ G_i=G\cdot W_0\left(\frac{h_i}{x}\right), \tag{6.11} \]

where \(x\) has the same meaning as in equations (6.6) and (6.7). Substituting (6.7) and (6.11) into (6.10), it is not difficult to obtain

\[ \ln\frac{\Omega}{\Omega_0} = G\left\{-\frac{C\sqrt{\overline{N}}}{x^3} +3\ln x-\frac{3(x^2-1)}{2}\right\}. \tag{6.12} \]

The equation for \(x\) is obtained from the condition for the maximum of this expression, corresponding to the maximum entropy,

\[ x^5-x^3=C\sqrt{\overline{N}}. \tag{6.13} \]

It is easy to see, however, that the result obtained by Flory is paradoxical. Indeed, the number of configurations of a single molecule, as follows from formulas (6.9) and (6.7), is a monotonic function of \(x\), whereas the number of configurations of a system of \(G\) noninteracting molecules (equation (6.12)) has a maximum at \(x\) satisfying condition (6.13).

This obvious contradiction is explained by the fact that Flory took the influence of volume effects into account in a nonuniform manner. If one starts from equation (6.9), then, as Grimley pointed out\(^{83}\), instead of (6.11) we should have used the expression

\[ G_i=G\frac{W_0(h_i)\cdot F(h_i)} {\sum_j W_0(h_j)\cdot F(h_j)}. \tag{6.14} \]

Substitution of (6.14) into (6.10) gives, instead of (6.12):

\[ \ln\frac{\Omega}{\Omega_0}=G\ln F. \tag{6.15} \]

This expression is a monotonic function of \(x\), and from it we cannot obtain any condition analogous to (6.13).

Replying to Grimley, Flory\(^{84}\) wrote that the substitution (6.14) is illegitimate, since it immediately restricts us to the case of the most probable distribution. However, this objection is unfounded, since Flory himself also substitutes into (6.10) the most probable distribution (6.11), but not the one that should have followed from his formula (6.9).

Recently several attempts have been made to depart from the Gaussian function, but the crude approximations made in doing so have led to the fact that the formulas obtained in these works are devoid of

of physical meaning. The indicated attempts belong to Hermans1, Hadviger2, Hermans, Klamkin and Ullman3, and Grimley4. The authors of the first three papers proceed from a recurrence relation between the distribution functions for the distances of the \(k\)-th and \((k+1)\)-st atoms from the origin:

\[ W(\mathbf{h}+\mathbf{s},\, k+1)=\int W(\mathbf{h},\, k)\psi(\mathbf{s})\,ds, \tag{6.16} \]

where \(\psi(\mathbf{s})\) is the probability that the link joining the \(k\)-th and \((k+1)\)-st atoms is characterized by the vector \(\mathbf{s}\). The main difficulty consists in finding a suitable expression for the function \(\psi(\mathbf{s})\), which, generally speaking, depends on the positions of all the other atoms of the chain. The authors of the papers under consideration assumed that the function \(\psi(\mathbf{s})\) is determined by the mean density of the remaining links at the point considered. If this density were equal to zero, then

\[ \psi(\mathbf{s})\,ds=\frac{1}{2}\sin\vartheta\,d\vartheta, \]

where \(\vartheta\) is the angle between the vectors \(\mathbf{h}\) and \(\mathbf{s}\). In order to take account of the presence of other links, this expression must be multiplied by the probability that the link joining the \(k\)-th and \((k+1)\)-st atoms, being directed at an angle \(\vartheta\), does not collide with any other link. Instead of calculating this latter probability, the authors essentially specified \(\psi(\mathbf{s})\) in a completely arbitrary way, as a result of which all of them obtained incorrect results.

Hermans1 did not calculate the function \(W(h)\) with account taken of volume effects and confined himself to the calculation of \(\overline{h^2}\), which in his treatment turned out to be

\[ \overline{h^2}=NA^2\left(1+\frac{0.78\cdot v_0}{\sqrt{N}\,A^3}\right), \tag{6.17} \]

where \(v_0\) is the volume of a link.

According to formula (6.17), the influence of volume effects weakens as the degree of polymerization \(N\) increases. This conclusion is plainly erroneous. In fact, the influence of volume effects, measured by the ratio \(\overline{h^2}\) to \(NA^2\), must evidently be determined by the number of collisions per unit volume of the chain. If the chain consists of \(N\) links, then the number of collisions of each link with all the others is proportional to the mean concentration of links, i.e.

\[ \frac{Nv_0}{(NA^2)^{3/2}}. \]

Thus, the total number of collisions of all links with all links must be proportional to

\[ \frac{N^2v_0}{(NA^2)^{3/2}}=\frac{\sqrt{N}\,v_0}{A^3}. \]

This quantity

\[ \frac{\sqrt{N}\,v_0}{A^3} \]

should characterize the role of volume effects in a given chain. Thus, it is clear that volume effects play a greater role the larger the number and volume of the links and the smaller \(A\), i.e., the greater the flexibility of the chain. Obviously, the equation (6.17) obtained by Hermans contradicts the above obvious condition and is therefore erroneous, which is not surprising in view of the arbitrariness of his initial assumptions.

An equation differing from (6.17) only by a numerical coefficient in the second term was also obtained by Grimley\(^{122}\), who applied a somewhat different method. For the reasons given above, Grimley’s work is also erroneous. A common shortcoming of the works\(^{85}\) and \(^{122}\) is that they do not in fact consider collisions of all links with all, but only collisions of all the remaining links with the last link. The number of these collisions is proportional not to

\[ \frac{\sqrt{N}\,v_0}{A^3}, \]

but to

\[ \frac{v_0}{\sqrt{N}\,A^3}. \]

It is clear that this leads to an underestimation of volume effects.

In a later work Hermans and his coauthors\(^{121}\) somewhat improved their method, taking into account that the density of the links of a polymer chain in the vicinity of the link under consideration is greater than their average density, since all the links are connected with one another. This changed the final result in such a way that the influence of volume effects proved to be independent of the degree of polymerization

\[ \overline{h^2}=NA^2\left(1+1.72\,\frac{v_0}{A^3}\right). \tag{6.17a} \]

From what has been said above it follows that this formula too is incorrect. Hadwiger\(^{86}\), making a different, but likewise arbitrary, assumption concerning the form of the function \(\psi(s)\), obtained the following distribution function:

\[ W(h)h^2dh=\mathrm{const}\cdot e^{-\frac{3}{2NA^2}\left(h-\frac{\alpha NA}{3}\right)^2}h^2dh, \tag{6.18} \]

where \(\alpha\) is a certain factor connected with the proper volume of the links and tending to zero if volume effects are absent. Hadwiger’s theory does not make it possible to relate \(\alpha\) in any way to molecular quantities; it follows from it only that \(0\leq \alpha <1\). From (6.18) we obtain that the most probable value of \(h\) is

\[ h_0=\frac{1}{6}\left(\alpha+\sqrt{\alpha^2+\frac{24}{N}}\right)NA. \tag{6.19} \]

If \(\alpha=0\), then \(h_0=\sqrt{\frac{2}{3}NA^2}\), as it should be (cf. (2.6a)).

However, if \(\alpha \ne 0\), then for large \(N\) it follows from (6.19) that

\[ h_0=\frac{1}{3}\alpha NA. \tag{6.20} \]

This result means that, as \(N\) increases, the chain begins to resemble a trans-chain, for which \(h_0 \sim N\). Obviously, this is incorrect, since with increasing degree of polymerization the chain should become more and more coiled, thereby deviating from the extended trans-state.

For the same reason one must regard as untenable Simha’s\(^{87}\) attempt to apply to the problem of volume effects equation (2.25)

\[ \overline{h^2}=c_1N+c_2N^2, \]

based on the absence of spherical symmetry of the function \(\tau_k(l_k)\). It is clear that in this case as well we arrive at an absurd result—the gradual approach of the chain to the trans-state as the degree of polymerization increases.

Such are the principal attempts available in the literature to take volume effects into account. We shall not dwell in detail on the mainly qualitative works of Taylor\(^{49}\) and Simha\(^{88}\), on Orr’s purely phenomenological work\(^{89}\), or on Bryant’s unsuccessful attempt\(^{90}\), which the author himself subsequently abandoned\(^{91}\).

The most recent attempt to take into account the influence of volume effects on the dimensions of polymer molecules in solution belongs to the authors of the present article\(^{92}\).

Our method is based on equations (6.3), (6.4), and (6.5), proposed by Flory; however, unlike that author, we did not replace the fraction of configurations not forbidden by volume effects by its mean value.

Thus we have:

\[ W(h)h^2dh=W_0(h)F(h)h^2dh, \tag{6.21} \]

where

\[ W_0(h)=\mathrm{const}\cdot e^{-\frac{3h^2}{2NA^2}}. \tag{6.21a} \]

Following Flory, let us consider a volume element \(x, x+dx;\ y, y+dy;\ z, z+dz\), and divide it into \(\nu\) cells of the size of one segment; then

\[ \nu=\frac{dx\,dy\,dz}{v_0}, \]

where \(v_0\) is the volume of a segment.

Let us denote the number of segments in the volume element under consideration, under the condition that the distance between the ends of the chain is equal to \(h\), by

\(n(x, y, z; h)=N\cdot W(x, y, z; h)\). Then, using equations (6.4) and (6.5), we obtain:

\[ \ln F(h)=-2\pi N^{2}v_{0}\int_{0}^{\infty} W^{2}(s;h)\,s^{2}ds, \tag{6.22} \]

where \(W(s;h)\) is the probability that, in a molecule whose end-to-end distance is equal to \(h\), some segment is at a distance \(s\) from the center of gravity of the chain.

We shall regard the function \(W(s;h)\) as Gaussian, with a parameter depending on \(h\)*:

\[ W(s;h)=\left(\frac{\beta(h)}{\sqrt{\pi}}\right)^{3} e^{-\beta^{2}(h)\cdot s^{2}} . \tag{6.23} \]

We determine the parameter \(\beta(h)\) from the condition that \(\overline{s^{2}}\) must be equal to the mean-square radius of inertia of a chain with a fixed end-to-end distance \(h\) (see Section 3).

For a Gaussian chain with fixed end-to-end distance, from (3.20) we have:

\[ \left(\overline{R_{h}^{2}}\right)_{0}=\frac{1}{12}NA^{2}+\frac{1}{12}h^{2}. \tag{6.24} \]

Volume effects must increase both \(h\) and \(R\) by the same factor \(\varkappa\). Therefore, taking volume effects into account,

\[ \overline{R_{h}^{2}}=\frac{\varkappa^{2}}{12}NA^{2}+\frac{1}{12}h^{2}. \tag{6.25} \]

From the condition

\[ \overline{s^{2}}=\frac{3}{2\beta^{2}}=\overline{R_{h}^{2}}=\frac{\varkappa^{2}}{12}NA^{2}+\frac{1}{12}h^{2} \]

we obtain:

\[ \beta^{2}(h)=\frac{18}{\varkappa^{2}NA^{2}+h^{2}}. \tag{6.26} \]

Hence

\[ \ln F(h)=-\left(\frac{3}{\sqrt{\pi}}\right)^{3}\frac{N^{2}v_{0}}{2}\cdot \frac{1}{(\varkappa^{2}NA^{2}+h^{2})^{3/2}} . \tag{6.27} \]

* It can be shown \(^{123}\) that, when volume effects are neglected, the distribution function of segment density relative to the center of gravity is approximately Gaussian.

Assuming that it retains this character also when volume effects are taken into account, we do not make a large error, since the result depends almost not at all on the form of this function \(^{82}\).

and

\[ W(h)h^2\,dh=\mathrm{const}\cdot e^{-\frac{3h^2}{2NA^2}} -\left(\frac{3}{\sqrt{\pi}}\right)^3 \frac{N^2v_0}{2}\, \frac{1}{(x^2NA^2+h^2)^{3/2}}\,h^2dh . \tag{6.28} \]

From (6.28) we obtain, for the most probable value of \(h\), the following equation (valid for \(x\gg 1\)):

\[ h_0^2=NA^2\left\{ \left[ \frac{1}{2}\left(\frac{3}{\sqrt{\pi}}\right)^3 \frac{\sqrt{N}\,v_0}{A^3} \right]^{2/5} -x^2 \right\}. \tag{6.28a} \]

On the other hand,

\[ h_0^2=x^2NA^2. \tag{6.28b} \]

Comparing (6.28a) and (6.28b), we obtain that

\[ x^2=\frac{1}{2} \left[ \frac{1}{2}\left(\frac{3}{\sqrt{\pi}}\right)^3 \frac{\sqrt{N}\,v_0}{A^3} \right]^{2/5}. \tag{6.29} \]

Thus, the mean-square distance between the ends of the chain, which, owing to the sharpness and symmetry of the maximum of the distribution function (6.28), practically coincides with the square of its most probable value, is equal to

\[ \overline{h^2}=NA^2\cdot\frac{1}{2} \left[ \frac{1}{2}\left(\frac{3}{\sqrt{\pi}}\right)^3 \frac{\sqrt{N}\,v_0}{A^3} \right]^{2/5}, \tag{6.30} \]

where

\[ A=l\left(\frac{1+\cos\alpha}{1-\cos\alpha}\frac{1+\eta}{1-\eta}\right)^{1/2}. \]

This equation is valid only for sufficiently large volume effects; therefore it should not be surprising that, as \(v_0\to 0\), \(\overline{h^2}\) tends to zero, and not to \(NA^2\).

For smaller volume effects we obtain:

\[ \overline{h^2}=x^2NA^2;\qquad x^5-x^3=\frac{1}{2^{7/2}} \left(\frac{3}{\sqrt{\pi}}\right)^3 \frac{\sqrt{N}\,v_0}{A^3}. \tag{6.30a} \]

As \(v_0\to 0\), this formula gives \(x\to 1\) and \(\overline{h^2}\to NA^2\). For \(x\gg 1\) it passes into formula (6.30).

Let us note that formula (6.30a) can also be obtained from Flory’s equations (6.13) and (6.7a), if the constant \(C\) entering them is slightly modified.

According to Flory,

\[ C=\frac{27}{2^{5/2}\pi^{3/2}}\cdot\frac{v}{A^3}\left(\frac{n}{N}\right)^2 . \tag{6.31} \]

Flory introduced the segment length \(l' = v^{1/3}\) and set \(n l' = N A\), as a result of which he obtained from (6.31):

\[ C=\frac{27}{2^{5/2}\pi^{3/2}}\cdot\frac{l'}{A}. \tag{6.7a} \]

In view of the uncertainty of the quantity \(l'\), this formula does not permit any quantitative calculations.

However, the equality \(n l' = N A\) is incorrect. In general, no relation whatsoever can be established between \(l'\) and \(A\), since the quantity \(l'\) depends on the proper volume of a link together with all its substituents, whereas the quantity

\[ A=l\sqrt{\frac{1+\cos\alpha}{1-\cos\alpha}\cdot\frac{1+\eta}{1-\eta}} \]

is determined by the length of the links of the main chain, the valence angles, and the degree of hindrance to rotation. It is clear, therefore, that these quantities must be completely independent.

If, instead of using the relation \(n l' = N A\), we introduce cells whose volume is equal to the volume of one link, \(v_0\), and consider the distribution of links in these cells, then in (6.31) we obtain

\[ v=v_0 \quad \text{and} \quad n=N. \]

Under these conditions

\[ C=\frac{27}{2^{5/2}\pi^{3/2}}\frac{v_0}{A^3}, \tag{6.31a} \]

\[ x^5-x^3=\frac{1}{2^{5/2}}\left(\frac{3}{\sqrt{\pi}}\right)^3 \frac{\sqrt{N}\,v_0}{A^3}, \tag{6.32} \]

We see that this equation differs only slightly from the equation (6.30) proposed by us.

In a subsequent paper by Flory and Fox \(^{100}\), another derivation of equation (6.30a) and its generalization, taking into account the influence of the solvent, is given. The authors consider the swelling of a chain by \(x\) times in comparison with \(\sqrt{N A^2}\) as the result of mixing of the polymer links with solvent molecules. Such mixing is accompanied by a change in the thermodynamic potential \(\Delta \Phi=\Delta H-T\Delta S\), which must, of course, be negative.

The energy term \(\Delta H\) is determined by the well-known equation of Hildebrand \(^{118,119}\):

\[ \Delta H=BV_1v_2^2, \tag{6.33} \]

where \(V_1\) is the molar volume of the solvent, \(v_2\) is the volume fraction of the polymer, and \(B \sim E_{12}-E_{11}-E_{22}\) is a constant characterizing the change in the energy of intermolecular interaction upon mixing \((E_{11}, E_{22}, E_{12}\) are the cohesive-energy densities for the solvent, the polymer, and their mixture).

If the polymer and solvent molecules possess similar intermolecular interactions, then one may assume that \(E_{12}\cong \sqrt{E_{11}E_{22}}\); then \(B\sim(\sqrt{E_{11}}-\sqrt{E_{22}})^2\) and \(\Delta H>0\), i.e., mixing is energetically unfavorable. In reality, there may be cases in which mixing is associated with a lowering of the energy of the system.

The entropy term \(\Delta S\) consists of two parts. The first part is the entropy of mixing, which, of course, is always positive and, consequently, favors mixing. Calculated per one mole:

\[ \Delta S_1=\psi Rv_2^2, \tag{6.34} \]

where \(\psi\) is an empirical parameter, constant for the given polymer–solvent pair\({}^{124}\), and \(R\) is the gas constant. The second part is the change in the configurational entropy of the chain upon its swelling. The authors still assume that, upon swelling of the chain, the distribution function remains Gaussian. Then a chain with end-to-end distance \(h\) must, upon swelling by \(x\) times, change its configurational entropy by the amount

\[ \Delta S(h)=k\ln\frac{W(xh)}{W(h)} =3k\ln x-3k\beta^2h^2(x^2-1); \tag{6.35} \]

where

\[ W(h)=4\pi\left(\frac{\beta}{\sqrt{\pi}}\right)^3 e^{-\beta^2h^2}h^2\,dh \quad \text{(see § 2).} \]

Integrating over all \(h\) and referring to 1 mole of chains, we obtain

\[ \Delta S_2=-3R\left(\frac{x^2-1}{2}-\ln x\right). \tag{6.36} \]

The equilibrium state corresponds to a minimum of the thermodynamic potential, determined from the condition \(\dfrac{\partial(\Delta\Phi)}{\partial x}=0\). To calculate the dependence of \(\Delta H\) and \(\Delta S_1\) on \(x\), the chain is divided into layers of thickness \(\Delta s_j\), located at a distance \(s_j\) from the center of gravity. The mixing process is regarded as the introduction of \(n_{1j}\) solvent molecules into the \(j\)-th layer. Then

\[ \frac{\partial(\Delta H)}{\partial x} =\sum_j \frac{\partial(\Delta H_j)}{\partial x} =-\frac{BV_1}{N_A}\sum_j v_{2j}^{\,2}\frac{\partial n_{1j}}{\partial x}. \]

and

\[ \frac{\partial(\Delta S_1)}{\partial x} = \sum_j \frac{\partial(\Delta S_j)}{\partial x} = \frac{\psi R}{N_A}\sum_j v_{2j}^{\,2}\frac{\partial n_{1j}}{\partial x}. \]

Taking into account that the volume of the \(j\)-th layer in the swollen chain is equal to \(4\pi x^3 s_j^{\,2}\Delta s_j\), and assuming the distribution of links over such layers to be Gaussian, we obtain from the condition

\[ \frac{\partial(\Delta \Phi)}{\partial x} = \frac{\partial(\Delta H-T\Delta S_1-T\Delta S_2)}{\partial x} =0, \]

\[ x^5-x^3=2C_M\psi\left(1-\frac{\theta}{T}\right)\sqrt{M}, \tag{6.37} \]

where

\[ \left. \begin{aligned} C_M&=\frac{2\pi}{2^{5/6}\pi^{3/2}}\, \frac{v^2}{N_A V_1}\left(\frac{M}{NA^2}\right)^{3/2},\\ \theta&=\frac{BV_1}{R\psi}. \end{aligned} \right\} \tag{6.38} \]

(\(v\) is the specific volume of the polymer, \(M\) the molecular weight, \(N_A\) Avogadro’s number).

If the solvent molecules are identical with the links of the polymer (a solution of a polymer in a hydrated polymer), then

\[ B=0 \quad \text{and} \quad \frac{M}{N}v\simeq V_1. \]

In this case, as is easy to see, equation (6.37) goes over into (6.32), which differs only slightly from our equation (6.30a). In the general case \(x\) may, as equation (6.37) shows, be both greater and less than unity, depending on whether \(T\) is greater or less than the characteristic temperature \(\theta\), which has the meaning of the critical mixing temperature of the given polymer–solvent pair at \(M=\infty\).

This means that the interaction of the segments of the polymer chain with one another and with the solvent molecules may lead not only to swelling, but also to contraction of the chain in comparison with its “ideal” dimensions \(\sqrt{NA^2}\).

A measure of the interaction between segments of polymer molecules in solution may be provided by the second virial coefficient in the expansion of the osmotic pressure of the polymer solution in a series in concentrations \(^{118}\):

\[ \frac{\pi}{RT}=\frac{1}{M}c+A_2c^2+\ldots, \tag{6.39} \]

which characterizes the deviation of polymer solutions from van ’t Hoff’s law.

The theory of the second virial coefficient shows \(^{124}\) that

\[ A_2 \sim 1 - \frac{\theta}{T}. \tag{6.40} \]

Comparison of formulas (6.37) and (6.40) leads to the conclusion that

\[ \varkappa^5 - \varkappa^3 \sim A_2 . \tag{6.41} \]

Thus, if the osmotic pressure depends on the concentration more strongly than according to the van’t Hoff law, then the molecule swells in the solution; if more weakly, it contracts. If \(A_2=0\), then \(\varkappa=1\), i.e., in solution the molecule has dimensions determined by the formula

\[ \overline{h^2}=N A^2 . \]

The derivation of formula (6.37) proposed by Flory and Fox is extremely imperfect, and almost all the objections that can be raised against Flory’s original derivation can also be raised against it.

In particular, in this case too the non-Gaussian character of the distribution function \(W(h)\) is ignored. Moreover, the treatment is to a certain extent phenomenological in character; the authors almost do not attempt to ascribe a direct molecular meaning to the individual factors influencing \(\varkappa\), nor to separate these factors, singling out, for example, the role of volume effects.

One may think, nevertheless, that equation (6.37) correctly conveys the main features of the phenomenon. In particular, for a solution of a polymer in a hydrated monomer, equation (6.37) gives results practically coinciding with ours. In this case, among all the factors influencing \(\varkappa\), only volume effects remain; these were considered by us with allowance for the non-Gaussian character of the distribution function and by a molecular-kinetic, rather than a phenomenological, method.

However, our treatment takes into account far from all types of interaction between the various parts of the chain. If one draws an analogy between the theory of polymer chains and the theory of real gases, then allowance for volume effects is equivalent to introducing the van der Waals correction \(b\). But, in addition, the dimensions of chains must also be influenced by attraction between its various parts (the van der Waals correction \(a\)). This attraction is always compensated, to one degree or another, by attraction between the links of the polymer chain and the solvent molecules.

It is evident that the total interaction energy of the chain links must be different for different configurations. Therefore, taking this interaction into account, the distribution function for the distances between the ends has the form \(^{83}\)

\[ W(h)=W_0(h)\cdot e^{-\frac{E(h)}{kT}}, \tag{6.42} \]

where \(W_0(h)\) is the distribution function without taking account of interaction, and \(E(h)\) is the total interaction energy of all the links of a chain whose end-to-end distance is equal to \(h\).

Polymers may be divided into three groups, distinguished by the form of the function \(E(h)\):

1) polyelectrolytes, in which the function \(E(h)\) is determined by the electrostatic repulsion of like-charged ions;

2) polar polymers, in which the function \(E(h)\) is determined by the induction and orientational interaction of dipoles, as well as by dispersion interaction;

3) nonpolar polymers, in which the function \(E(h)\) is determined only by the dispersion interaction of the various links.

Up to now the influence of interaction on chain configurations has been investigated theoretically only for the simplest case of polyelectrolytes\(^{24,94,95}\). In this case we are dealing not with attraction, but with repulsion of different parts of the molecule from one another, which leads to an increase in the average dimensions of the chain and, consequently, also of its intrinsic viscosity. The theoretically calculated\(^{24}\) dependence curve of the intrinsic viscosity on the degree of ionization of the chain (i.e., on the ratio of the number of ions in the chain to the total number of links) agrees well with experimental data. The methods used in the works cited above for the investigation of polyelectrolytes can, apparently, be generalized also to take account of the influence of attraction between links on the configurations of polar and nonpolar polymers. In doing so, as we indicated above, the influence of the solvent plays a particularly important role.

A somewhat special role is played by the interaction between the side groups of the chain belonging to neighboring links. Owing to this interaction, rotations about neighboring links in the chain cannot, strictly speaking, be regarded as completely independent, as we did in Section 4 when calculating \(\overline{h^2}\). Taking account of the correlation between rotations about neighboring links, of course, cannot substantially change the statistics to which a polymer chain is subject, but it must lead to some change in the magnitude of the statistical segment. For the case of a small correlation energy, such a calculation was carried out in work\(^{45}\) by the authors of the present article.

7. EXPERIMENTAL DATA ON THE DIMENSIONS OF POLYMER MOLECULES IN SOLUTION; COMPARISON OF THEORY WITH EXPERIMENT

Above we have considered modern theoretical views on the dimensions and shape of polymer molecules in highly dilute solutions. We shall give a very brief survey of the methods for experimental determination of molecular dimensions, which will enable us to compare the theories set forth above with experiment. It is not our task here to describe the experimental techniques, which the reader can find

found in the specialized literature. We shall confine ourselves to indicating the basic principles on which one or another method is based.

The most widespread method at the present time for determining the sizes of polymer molecules in solution is light scattering. Consider the scattering of light by particles whose dimensions are comparable with the wavelength (for example, by a polymer molecule)86–99.

Fig. 18. Scattering of light by a large particle.

Let two rays \(S_1\) and \(S_2\), with identical phases, fall on such a particle (Fig. 18). It is easy to see that the intensity of the scattered light

Fig. 19. Approximate form of the indicatrix of light scattering: \(a\)—by small particles and \(b\)—by large particles.

at the point \(C_1\) must be greater than at \(C_2\), since the phases of both waves at the point \(C_1\) differ from one another less than at the point \(C_2\). Scattering “forward” is more intense than “backward” (Mie effect).

The scattering indicatrix \(J(\vartheta)\), which for small particles is symmetric with respect to the angle \(\vartheta = 90^\circ\) (\(\vartheta\) is the angle between the scattered and incident ray) (Fig. 19, \(a\)), for large particles acquires the asymmetric form shown in Fig. 19, \(b\). Obviously, the difference in the intensities of light scattered at different angles must be directly related to the sizes of the scattering particles.

Debye \(^{54}\) introduced the idea of using the asymmetry of light scattering to determine the dimensions of polymer molecules in solution. For a statistically coiled coil Debye showed that

\[ J(\vartheta)=\frac{2}{x^2}\left(e^{-x}-(1-x)\right)J^0, \tag{7.1} \]

where

\[ x=\frac{8\pi^2}{3\lambda^2}\,\overline{h^2}\sin^2\frac{\vartheta}{2}, \tag{7.1a} \]

\(\lambda\) is the wavelength of the incident light in the solution, and \(J(\vartheta)=J^0\) for \(\vartheta=0\).

It should be noted that this formula applies to highly dilute solutions, in which the influence of the polymer chains on one another may be neglected.

Usually the measurements are made at two angles, \(\vartheta=45^\circ\) and \(135^\circ\). From (7.1) it is easy to see that

\[ \frac{J_{45}}{J_{135}}=1+6.556\,\frac{\overline{h^2}}{\lambda^2}. \tag{7.2} \]

This method makes it possible to determine \(\overline{h^2}\) with fairly high accuracy; however, it has at least two shortcomings \(^{55}\). First, the asymmetry of the scattered light becomes appreciable only for sufficiently large molecular weights of the molecules under investigation (from one million and above), so that this method is applicable only to the study of the highest-molecular-weight fractions of polymers. Secondly, the light-scattering method is extremely sensitive to any admixtures and impurities in the substances investigated, which also considerably restricts the limits of its applicability.

The dimensions of polymer molecules can also be judged to some extent from their hydrodynamic behavior, namely from viscosity, diffusion, sedimentation, double refraction, etc. The theories of these effects necessarily include parameters characterizing the dimensions of the molecules. However, all these theories proceed from certain model ideas concerning the structure of the molecules and their hydrodynamic interaction with the solvent, which makes the results obtained by such methods doubtful to some extent*).

Therefore it is natural to try to find such a combination of hydrodynamic parameters that would make it possible to determine the dimensions of molecules without resorting to model assumptions. Such

*) To a certain extent the last remark also applies to the light-scattering method itself. In the very derivation of formula (7.1) the assumption is introduced that the chain obeys Gaussian statistics, which, as we indicated above (§ 6), is also connected with certain model assumptions.

combination was recently found by V. N. Tsvetkov35, who showed that the ratio of the constants of translational and rotational diffusion of a polymer molecule is, in order of magnitude, equal to the mean square radius of inertia of the molecule \(\overline{R^2}\):

\[ \frac{D_t}{D_r}\simeq \overline{R^2}=\frac{1}{6}\,\overline{h^2}. \tag{7.3} \]

A reliable method for determining the coefficient of translational diffusion of polymer molecules was recently proposed by V. N. Tsvetkov103; the coefficient of rotational diffusion is determined by measuring the angles of orientation of birefringence in a flow53, 104, 105. The values of \(\overline{h^2}\) obtained in this way agree, in order of magnitude, with the results obtained by the light-scattering method.

Recently106 the possibility was indicated of determining the dimensions of polymer chains in solution by X-ray methods; however, this method cannot yet be regarded as being developed to any appreciable extent.

At present the most commonly used method for determining the dimensions of polymer molecules is the asymmetry of light scattering. It is by this route that the greater part of the results available in the literature has been obtained. In this connection polystyrene has been studied especially thoroughly; it has been investigated over a very wide range of molecular weights and in a large number of solvents. There are considerably fewer data on polyisobutylene and polymethyl methacrylate; data on other polymers are as yet lacking.

Experimental data on the dimensions of polystyrene and polyisobutylene molecules are given in Figs. 20 and 21. The dashed line in these figures shows the dependence of \(\sqrt{\overline{h^2}}\) on \(N\), obtained from the formula

\[ \sqrt{\overline{h^2}}=\sqrt{N l^2\,\frac{1+\cos\alpha}{1-\cos\alpha}} \tag{4.15} \]

for the case of free rotation. We see that the experimental values of \(\overline{h^2}\), as a rule, exceed by several times the values calculated from formula (4.15).

Moreover, the values of \(\overline{h^2}\) depend extremely strongly on the solvent: in good solvents (benzene for polystyrene, benzine and n-heptane for polyisobutylene) they are considerably higher than in poor ones (butanol for polystyrene, a mixture of n-heptane with propanol for polyisobutylene). This circumstance is well described by equation (6.37), since the better the solvent, the lower \(\theta\), i.e. the larger \(x\).

The increase of \(\overline{h^2}\) with increasing \(A_2\), predicted by the theory, is likewise qualitatively justified by experiment, since \(A_2\) is the larger, the better the solvent.

As was indicated in the preceding section, our formula and the formula of Flory and Fox predict a stronger dependence of \(\overline{h^2}\) on \(M\) than direct proportionality, whereas the theories of all the other authors lead to the conclusion that \(\overline{h^2}\sim M\). The experimental data are still too scanty to permit any definitive conclusions, but the majority of measurements

Fig. 20. Sizes of polystyrene molecules in various solvents. Curve \(a\)—formula (4.15), curve \(b\)—formula (4.12). Experimental data: ●—benzene \(^{108,109}\), ○ toluene \(^{107,113,114}\), ▲ dichloroethane \(^{107}\), △ dichloroethylene \(^{112}\), ■ butanone \(^{107,112,113}\), ◆ carbon tetrachloride \(^{108,110,111}\), □ cyclohexane \(^{107}\).

indicates that \(\overline{h^2}\sim M^\alpha\), where \(\alpha>1\). Thus, for polystyrene in dichloroethane and butanone \(^{107}\), \(\alpha=1.10\); for polystyrene in benzene \(^{108}\), \(\alpha=1.10\) (according to other measurements \(^{109}\), \(\alpha=1.00\)); for polymethyl methacrylate in acetone, \(\alpha=1.12\).

It is obvious (see Section 6) that, for determining \(\eta\), light-scattering data can be used only under such conditions when \(A_2=0\), and, consequently, \(\varkappa=1\). If \(A_2>0\), then from the experimental values of \(\overline{h^2}\) we obtain overestimated values of \(\eta\), from which no conclusions can be drawn about the energetics of internal rotation.

Unfortunately, this circumstance is sometimes forgotten: thus, Debye and Bueche \(^{116}\) attempted to obtain information about the form of the potential function \(U(\varphi)\) by combining data on dipole moments and the dimensions of polymer molecules in good solvents.

From the dimensions of polystyrene molecules \(^{107,109}\) and polyisobutylene \(^{109}\) at \(A_2=0\), we obtain for both of these polymers \(\eta=0.7\)*; thus, even at \(A_2=0\), their molecules are approximately 2.5 times more extended than in the case of free rotation. For polystyrene

Fig. 21. Dimensions of polyisobutylene molecules in various solvents. Curve \(a\) — formula (4.15), curve \(б\) — formula (4.13). Experimental data: ● benzene \(^{110,55}\), ▲ n-heptane \(^{109}\), ■ n-heptane (90%) + propanol (10%) \(^{109}\), ◆ n-heptane (80%) + propanol (20%) \(^{109}\).

this value of \(\eta\) is quite reasonable from the point of view of rotational-isomeric theory, whereas for polyisobutylene it is excessively high, which apparently indicates the important role of correlation between neighboring rotations in this polymer. In Figs. 21 and 22, the solid lines show the curves

\[ \sqrt{\overline{h^2}}=\left(Nl^2\frac{1+\cos\alpha}{1-\cos\alpha}\frac{1+\eta}{1-\eta}\right)^{1/2} \]

for \(\eta=0.7\).

* For polystyrene one should take into account the asymmetry of the side groups, owing to which the rotational isomerism in it is characterized not by one, but by two constants: \(\eta=\cos\varphi\) and \(\varepsilon=\sin\varphi\).

However, calculation shows that for \(dl\)-configurations of asymmetric side groups, which are likely to occur in polystyrene, the influence of \(\varepsilon\) on \(\overline{h^2}\) is insignificant.

Similar values (0.7 for polystyrene and 0.6 for polyisobutylene) were obtained by Flory and Fox^101 and from viscometric measurements at \(A_2=0\). By this latter method the values of \(\eta\) and \(h\) were determined for a number of other polymers: natural rubber^102 \((\eta \simeq 0.5)\), gutta-percha^102 \((\eta \simeq 0.4)\), polymethyl methacrylate^100 \((\eta \simeq 0.7)\), polydimethylsiloxane^116 \((\eta \simeq 0.4)\), cellulose tricaprylate^125 \((\eta \simeq 0.7)\), and cellulose tributyrate^124 \((\eta \simeq 0.8)\).

The theoretical value of \(\eta\) can be calculated, among these polymers, only for rubber and gutta-percha. The value \(\eta \simeq 0.3\), averaged over three unit bonds (see Table I), is in good agreement with experiment. However, as was already stated above, the viscometric determination of molecular dimensions is based to too great an extent on model concepts for the results obtained with its aid to be regarded as entirely reliable.

At present both the theoretical and the experimental study of the dimensions of polymer molecules in solution is still at an initial stage.

In essence, we do not yet possess either a reliable method for the experimental determination of these dimensions independently of any model concepts, or a molecular theory establishing the connection between the behavior of polymer molecules in solution and their chemical structure. However, what has been set forth above shows that the experimental and theoretical work of recent years has already made it possible to establish some basic facts in this field and to give them at least a qualitative theoretical interpretation.

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