STUDY OF LINEAR POLYMERS USING AN ULTRACENTRIFUGE

S. Ya. Frenkel

1. INTRODUCTION

Most of the mechanical properties of technical polymers are determined by two statistical parameters—the average molecular weight and the degree of polydispersity—and by two structural parameters—the degree of branching and the flexibility of the chains. The influence of these parameters on such properties as mechanical strength, elasticity, frost resistance, etc., has been considered in sufficient detail, for example, in the well-known monographs of V. V. Korshak^1 and P. P. Kobeko^2. A comprehensive study of polymers provides, as far as possible, for the simultaneous investigation of the four listed parameters and their comparison with the mechanical properties of the original samples. It is easy to see that such a comprehensive study is possible only if the polymer is broken down into individual molecules. In the work of V. A. Kargin and co-workers^3,4,5 it was shown that polymer solutions are true solutions, i.e., single-phase systems; therefore they constitute an ideal object for comprehensive study. Methods for determining average molecular weights are now well known^1. The degree of polydispersity is determined by the nature of the distribution with respect to molecular weights; the use of some kind of mass spectrometer would make it possible to obtain a practically continuous spectrum of the molecular weights of a given polymer. The flexibility of chains can be judged, for example, in the following way. Suppose that the molecular weight of a narrow fraction of molecules of practically identical weight has been measured and that, in this way, the degree of polymerization \(Z\) has been found. From the known value of \(Z\), on the basis of spectral-analysis data, one can find the length of the extended molecular chain. After this, the actual dimensions of the molecule are determined; in solution it is coiled into a more or less compact coil.

Obviously, the smaller the effective dimensions assumed by such a coil, the greater the flexibility of the chain. (In this, of course, one must also take into account thermodynamic factors, namely the binding energy of the polymer with the solvent.)

In a similar way, the branching of a chain can also be established: even in so-called “good” solvents, the effective dimensions of a branched molecule will be smaller than those of an unbranched chain of the same molecular weight and chemical composition. Branching also has a substantial effect on the value of the so-called second virial coefficient \(B_2\) of the osmotic equation

\[ \frac{\pi}{RT}=\frac{1}{M}c+B_2c^2+\ldots, \]

namely, for a branched isomer \(B_2\) is always smaller than for a linear one.

The purpose of the present review is to show how the ultracentrifuge can be used for the comprehensive investigation of polymers. In doing so, the solution of the problems formulated above is not always carried through to the end, since this would take too much space. For example, in the first part of the review it is shown how hydrodynamic and thermodynamic parameters (in particular, the dimensions of individual molecules) are calculated, i.e. all the quantities necessary for judging the flexibility or branching of chains; however, the details of the calculations are omitted. The necessary information on this part can be found in the works of M. V. Vol’kenshtein and collaborators \(^{6,7}\) and of V. N. Tsvetkov \(^{8}\). By contrast, methods for calculating molecular-weight distributions are presented in full, since in this respect the ultracentrifuge is a unique instrument.

As is known, the ultracentrifuge, designed by Svedberg \(^{9}\) in 1925 (the idea of the ultracentrifuge was contained in an earlier work by A. V. Dumanskii \(^{10}\)), was initially used exclusively for the study of proteins—monodisperse substances with rigid symmetric molecules (globules). All hydrodynamic parameters of protein molecules could be calculated by means of simple geometric models (spheres or ellipsoids).

The application of the ultracentrifuge to the study of linear polymers was for a long time hindered by the absence of clear ideas about the nature of threadlike molecules in solutions. The use of “rigid” models (Simha \(^{11,12}\)), by analogy with the treatment of protein globules as ellipsoids, and attempts to describe the hydrodynamic behavior of molecular coils in terms of the “axial ratio” (i.e. the ratio of the long and short axes of an ellipsoid) were in contradiction with experimental data and, in particular, did not in any way explain the empirical depend—

dependence of the intrinsic viscosity and of the sedimentation and diffusion constants on molecular weight. A substantial step forward was the theory of W. Kuhn[^13], based on the concept of an “immobilized” solvent. In this theory it was initially assumed that all the solvent enclosed within the molecular coil moves with it as a single whole. The friction coefficient of molecules in viscous flow, or in sedimentation and diffusion, can then be calculated by analogy with the Stokes coefficient of a rigid sphere and proves to be proportional to the mean-square radius of the coil, i.e. \(M^{1/2}\) (\(M\) is the molecular weight). In fact, however, the friction coefficient increases with molecular weight proportionally to \(M^a\), where \(a \gg \dfrac{1}{2}\). The new theories of Kirkwood[^14,^15], Debye[^16,^17], and Flory[^18] (see § 3) make it possible not only to explain these deviations from proportionality to \(M^{1/2}\), but also to determine—on the basis of the magnitude of these deviations—the mean-square radius of the coils \(\langle r_0 \rangle^{1/2}\). According to Debye[^17] and Kirkwood[^14], the deviation from proportionality is connected with purely hydrodynamic effects, which appear in calculating the friction coefficient by summing the disturbances introduced by each separate link (“segment”) of the polymer chain. According to Flory[^18], on the other hand, the concept of a completely immobilized solvent is, at least in a first approximation, correct, but for thermodynamic reasons the dimensions of the coils increase faster than \(M^{1/2}\).

The exposition of these theories and of their experimental consequences will be the subject of the first part of the review. In conclusion to the first part it will be shown how the ultracentrifuge can be used to study the thermodynamic properties of solutions of high polymers.

The second part of the review is devoted to the problem of studying molecular-weight distributions. For this purpose the ultracentrifuge was first used by Gralén[^19] in 1944. The early work of Signer and Gross[^20] and of Kraemer and co-workers[^21] was not systematic in character and solved this extremely important problem only in particular respects. The real development of work in this direction took place in Sweden in Svedberg’s laboratory (the investigations of Gralén, Jullander, Chinnel, and Rånby), in the USA in Williams’s laboratory (the work of Wales carried out on an equilibrium ultracentrifuge, and the work of Baldwin, Williams, and Gosting carried out on a sedimentation ultracentrifuge), and in the USSR in S. E. Bresler’s laboratory, where the problem of using a large sedimentation ultracentrifuge was solved in the most complete way.

The necessary information on the ultracentrifuge is available in the well-known monograph by Svedberg and Pedersen[^9], and can also be found in a number of articles and reviews in Russian[^22,^23,^24,^25]. We therefore considered it useful to collect all this information in onehiqizo

in the introductory paragraph (§ 1), where the basic concepts and formulas are given—mostly without derivation. For the reader insufficiently familiar with the fundamentals of statistics, § 2 gives an elementary exposition of the theory of simple distributions, specifically as applied to molecular-weight distributions.

The allocation of all this information to an introductory part also seemed advisable in order to relieve the main part of the review from sometimes necessary references and derivations and to preserve the logical integrity of the exposition. The formulas of the introductory part, in contrast to the formulas of the main text, are numbered with Roman numerals.

§ 1. BASIC RELATIONS FOR THE ULTRACENTRIFUGE

a) Method of sedimentation velocities

The sedimentation of particles in the field of an ultracentrifuge occurs under the action of the centrifugal force

\[ M(1 - V\rho)\omega^2 x, \]

which, in steady motion, is balanced by the frictional force

\[ f\left(\frac{dx}{dt}\right). \]

Here \(M\) is the gram-molecular weight, \(V\) is the partial specific volume of the dissolved substance, \(\rho\) is the density of the solvent, \(\omega\) is the angular velocity of the ultracentrifuge rotor, \(x\) is the distance of the particles from the axis of rotation, and \(f\) is the molar friction coefficient. The condition of steady motion is thus written in the form

\[ \frac{M(1 - V\rho)}{f} = \frac{1}{\omega^2 x}\frac{dx}{dt} = \frac{1}{\omega^2}\frac{d\ln x}{dt}. \tag{1} \]

The left-hand side of this equation contains all the quantities characterizing the given system; thus this quantity is a molecular characteristic—it is called the sedimentation constant and is denoted by \(s\):

\[ s = \frac{M(1 - V\rho)}{f}. \tag{II} \]

If \(M\) is expressed in dimensionless units, then in the CGS system \(s\) will have the dimension cm/sec·dyne. The practical unit of the sedimentation constant is 1 svedberg \((S)\), equal to \(10^{-13}\) cm/sec·dyne.

In very strong centrifugal fields (of the order of \(10^5 g\) and greater; \(g = 981\ \mathrm{cm/sec^2}\)) a boundary is established between the pure solvent and the solution; this boundary moves toward the bottom of the cuvette when \(V\rho < 1\), or toward the meniscus when \(V\rho > 1\). The motion of the boundary is not quite uniform, for at different distances from the axis the centrifugal field is different.

Integrating expression (1) gives a working formula for calculating the sedimentation constant:

\[ s=\frac{\ln \dfrac{x_2}{x_1}}{\omega^2(t_2-t_1)}, \tag{III} \]

where \(x_1\) and \(x_2\) are the positions of the boundary at the times \(t_1\) and \(t_2\). Usually this formula is supplemented by corrections accounting for changes in viscosity, density, and partial specific volume with temperature, reducing \(s\) to the standard temperature \(20^\circ\mathrm{C}\).^9

Fig. 1. Sedimentation diagram obtained by the method of superposed scales.^9

The motion of the boundary is now observed by various refractometric methods. We shall briefly describe the method of superposed scales,^9,26,27 which plays an important role in the study of polydispersity. A light beam passes through a small scale, the image of which is obtained on a photographic plate. Between the scale and the plate there is a cuvette in which a boundary has already arisen. Since at low concentrations the gradients of concentration and refractive index differ only by a constant factor, the deflection of each mark of the scale from the undistorted image \(Z\) will be proportional to the concentration gradient in the corresponding section of the solution column. The undistorted image is obtained analogously, but for the pure solvent. By superposing both images with the aid of a microcomparator, it is possible to construct (Fig. 1) a diagram of the values of the concentration gradient, or of the quantity proportional to it, \(Z\) (in microns), as a function of the ...

at various distances from the axis of rotation. Usually the value \(\dot{x}_m\) corresponding to the maximum \(Z\) is taken as the position of the boundary. In other methods\(^{28,29}\) the concentration-gradient curve is obtained directly on the photographic plate; here the values of \(x_m\) are determined practically as accurately as in the method of superposed scales; however, the exact form of the curve

\[ \frac{dc}{dx}=f(x), \]

needed in calculations of polydispersity, is blurred.

In order to determine the molecular weight from formula (II), it is necessary to find, from an independent experiment, the value of the molar coefficient of friction \(f\).

Assuming that the frictional coefficients in sedimentation and in translational diffusion are the same, and using Einstein’s equation

\[ D=\frac{RT}{f}, \tag{IV} \]

where \(D\) is the diffusion coefficient, \(R\) is the universal gas constant, and \(T\) is the absolute temperature, we obtain, combining (II) and (IV):

\[ M=\frac{s}{D}\,\frac{RT}{1-V\rho}. \tag{V} \]

This is Svedberg’s first formula\(^{9}\) for calculating molecular weight. Of course, \(s\) and \(D\) are reduced to the standard temperature.

b) Method of sedimentation equilibrium

If the ultracentrifuge field is comparatively weak (\(\leq 10^4 g\)), no boundary arises, but at every point of the cuvette an equilibrium is established between the centrifugal force and the force of diffusion, which is simply the gradient of the chemical potential \(\mu\) in the direction \(x\). This gradient, as is known, is equal to

\[ \nabla\mu=\frac{\partial\mu}{\partial c}\,\frac{dc}{dx} =\frac{1}{c}\,\frac{\partial\pi}{\partial c}\,\frac{dc}{dx}, \]

where \(\pi\) is the osmotic pressure.

The condition of sedimentation equilibrium, therefore, can be written in the form

\[ \frac{1}{c}\,\frac{d\pi}{dx}=(1-V\rho)\,\omega^2 x. \tag{VI} \]

If the solution satisfies the van’t Hoff law, then

\[ \pi=\frac{RTc}{M} \]

and (VI) is transformed into

\[ \frac{1}{c}\,\frac{dc}{dx} =\frac{M}{RT}(1-V\rho)\,\omega^2 x. \tag{VII} \]

Integration of this expression leads to Svedberg’s second formula

\[ M=\frac{2RT}{1-V\rho}\,\frac{\ln \dfrac{c_2}{c_1}}{\omega^2\left(x_2^2-x_1^2\right)} . \tag{VIII} \]

Here \(c_1\) and \(c_2\) are the concentrations at the points \(x_1\) and \(x_2\).

From the differential formula (VII) it follows that

\[ \frac{dc}{dx}=\frac{2x}{A}\,c, \]

where

\[ A=\frac{2RT}{1-V\rho}, \tag{IX} \]

i.e.,

\[ \frac{c_2}{c_1} = \frac{\left(\dfrac{dc}{dx}\right)_2 x_1}{\left(\dfrac{dc}{dx}\right)_1 x_2} = \frac{z_2 x_1}{z_1 x_2}, \]

since the ordinates of the sedimentation diagrams \(z\) are proportional to \(\dfrac{dc}{dx}\). This makes it possible to obtain one more important formula:

\[ M=A\,\frac{\ln \dfrac{z_2x_1}{z_1x_2}}{\omega^2\left(x_2^2-x_1^2\right)} . \tag{X} \]

§ 2. DISTRIBUTION FUNCTIONS

The numerical distribution function with respect to molecular weights — \(q_n(M)\) — is defined by the known relation

\[ dn=q_n(M)\,dM, \tag{XI} \]

where \(dn\) is the relative number of molecules in the interval of weights from \(M\) to \(M+dM\).

The normalization condition for the distribution function for the whole polymer is

\[ \int_0^\infty q_n(M)\,dM=1, \tag{XII} \]

and for the \(i\)-th fraction of this polymer

\[ \int_0^\infty q_n(M)\,dM=n_i, \tag{XIII} \]

where \(n_i\) is the number fraction, i.e. the ratio of the number of molecules within fraction \(N_i\) to the total number of molecules in the polymer \(N\).

In an analogous manner, the weight distribution function with respect to molecular weights \(q_w(M)\) is defined:

\[ dw=q_w(M)\,dM, \tag{XIV} \]

where \(dw\) is the weight fraction of molecules in the interval from \(M\) to \(M+dM\). The normalization is carried out so that, for the entire polymer,

\[ \int_0^\infty q_w(M)\,dM=1, \tag{XIIa} \]

and for a fraction

\[ \int_0^\infty q_w(M)\,dM=w_i. \tag{XIIIa} \]

If the weight of a fraction is \(W_i\), and the weight of the whole polymer is \(W\), then

\[ w_i=\frac{W_i}{W}. \]

The number and weight distribution functions are related to each other by the relation

\[ q_w(M)=Mq_n(M). \tag{XV} \]

A function \(q(M)\) is called unimodal if it has only one maximum (Fig. 2). The value of the molecular weight \(M_m\) corresponding to the maximum is called the modal, or most probable, value. If the maximum ordinate divides the entire distribution into equal parts, so that \(q_w(M+\Delta M)=q_w(M-\Delta M)\) for any \(|\Delta M|\), the distribution function is called symmetric.

Fig. 2. Unimodal distribution.

The quantities

\[ m_i=\int_0^\infty q_w(M)M^i\,dM \tag{XVI} \]

are called the moments of the distribution with respect to the origin of coordinates. In particular, the number-average weight

\[ \overline{M}_n=\int_0^\infty q_n(M)M\,dM \tag{XVII} \]

and the weight-average weight

\[ \overline{M}_{w}=\int_{0}^{\infty} q_{w}(M)\,M\,dM \tag{XVIII} \]

are the first-order moments of the corresponding distributions \(q_{n}(M)\) and \(q_{w}(M)\). For symmetric functions the modal and mean values coincide.

The physical meaning of the various average weights is fairly well known \(^{1,2}\).

Let us now suppose that the origin of coordinates has been shifted to the point \(\overline{M}_{w}\). The quantity

\[ \int_{-\infty}^{+\infty} M^{2}q_{w}(M)\,dM=m_{2}^{0}=\mu^{2} \tag{XIX} \]

is called the second moment of the distribution \(q_{w}(M)\) relative to the mean value, or the dispersion of this distribution, while the quantity

\[ \mu=\left|\sqrt{m_{2}^{0}}\right| \tag{XX} \]

is the standard deviation of this distribution. The dispersion and the standard deviation are measures of the width of the distribution, i.e., the principal characteristics of polydispersity.

It is easy to show that

\[ m_{2}^{0}=m_{2}-m_{1}^{2}=m_{2}-\overline{M}_{w}^{\,2}, \tag{XXI} \]

where

\[ m_{2}=\int_{0}^{\infty} M^{2}q_{w}(M)\,dM. \]

Let us rewrite equation (XXI) in the form

\[ \mu^{2}=\frac{m_{2}}{m_{1}}\,m_{1}-m_{1}^{2}. \]

The quantity

\[ \frac{m_{2}}{m_{1}}= \frac{\displaystyle\int_{0}^{\infty} M^{2}q_{w}(M)\,dM} {\displaystyle\int_{0}^{\infty} Mq_{w}(M)\,dM} =\overline{M}_{Z} \tag{XXII} \]

is called the \(Z\)-average weight and is obtained in the calculation of molecular-

molecular weight of a polydisperse substance by formula (X) (whence the name “\(Z\)-weight” also arises).

Thus, from the weight-average and \(Z\)-average weights one can find the variance of the distribution \(q_w(M)\):

\[ \mu^2=\overline{M}_z\,\overline{M}_w-\overline{M}_w^2 . \tag{XXIII} \]

Analogously, the variance of the distribution \(q_n(M)\) is

\[ \mu_n^2=\overline{M}_n\,\overline{M}_w-\overline{M}_n^2 . \tag{XXIIIa} \]

It is sometimes useful to compare the distributions of different quantities, say \(s\) and \(M\). For this purpose an absolute measure is introduced—the coefficient of dispersion \(\Delta\), independent of the choice of units and equal to the ratio of the standard deviation to the mean value, i.e., in the present case

\[ \Delta_M=\frac{\mu}{\overline{M}_w}. \tag{XXIV} \]

In addition to the width of a distribution, it is useful to know its asymmetry. According to Jullander\(^{30}\), the asymmetry \(\lambda\) is equal to the ratio of the areas into which the maximum ordinate divides the distribution:

\[ \lambda= \frac{ \displaystyle\int_{M_m}^{\infty} q_w(M)\,dM }{ \displaystyle\int_{0}^{M_m} q_w(M)\,dM }. \tag{XXV} \]

Positive asymmetry corresponds to \(\lambda>1\), negative asymmetry to \(\lambda<1\). Jullander’s criterion, while very graphic, does not, however, exhaust the possible details of the distribution, and for characterizing asymmetry it is better to use higher moments \(m_i\): \(m_3, m_4\), etc.

Let us now introduce the concept of the “\(q\)-average” weight\(^{31}\):

\[ \overline{M}_q=\frac{m_q}{m_{q-1}}. \tag{XXVI} \]

Since \(i\), as in formula (XVI), may take arbitrary values, including negative ones, and since, according to the normalization condition,

\[ m_0=\int_{0}^{\infty} q_w(M)\,dM=1, \]

for the first four \(q\)-averages one may write in the system \(q_w(M)\)

\[ \left. \begin{aligned} \overline{M}_0 \equiv \overline{M}_n &= \frac{m_0}{m_{-1}} = \frac{1}{\displaystyle \int_0^\infty \frac{q_w(M)}{M}\,dM};\\[6pt] \overline{M}_1 \equiv \overline{M}_w &= \frac{m_1}{m_0};\\[6pt] \overline{M}_2 \equiv \overline{M}_z &= \frac{m_2}{m_1};\\[6pt] \overline{M}_3 \equiv \overline{M}_{z+1} &= \frac{m_3}{m_2} = \frac{\displaystyle \int_0^\infty M^3 q_w(M)\,dM} {\displaystyle \int_0^\infty M^2 q_w(M)\,dM}, \end{aligned} \right\} \tag{XXVII} \]

and in the system \(q_n(M)\)

\[ \overline{M}_n \equiv \overline{M}_1;\qquad \overline{M}_w \equiv \overline{M}_2 \ \text{etc.} \]

The method of model functions, which is considered in detail in § 9, is based on the measurement of \(q\)-average weights.

PART I

§ 3. HYDRODYNAMIC PARAMETERS

AND DIMENSIONS OF CHAIN MOLECULES IN SOLUTION

In the further exposition we shall proceed from the basic equations (II) and (IV):

\[ s=\frac{M(1-V\rho)}{f} \quad \text{and} \quad D=\frac{RT}{f}. \]

The coefficient of translational friction \(f\), entering into these expressions, is a complicated function of the dimensions and shape, as well as the “solvation” of the molecules. At one time the opinion was expressed that, owing to orientational effects (rotation and stretching of a chain molecule in the direction of the field), the coefficient \(f=f_s\) in sedimentation may differ somewhat from the coefficient \(f=f_D\) in diffusion. However, calculations and direct measurements carried out by C. Zimmmer \(^{32a}\) showed that if such a difference exists, in any case it does not exceed 1%. It is also essential to point out that \(f\) depends strongly on concentration. One of the following paragraphs will be devoted to the question of the concentration dependence of the friction coefficient; here we shall note that the characte-

...characterize individual molecules can only be the values of \(s\), \(D\), and \(f\) extrapolated to infinite dilution, for at finite concentration the translational motion of molecules is complicated by the hydrodynamic interaction between them. This interaction is so large that at a certain concentration (usually of the order of \(0.5\%\)) all dependence of the sedimentation constant on molecular weight is lost, and the solution begins to sediment as a whole, like a swollen gel. The region in which the dependence of \(s\) on \(M\), following from formula (II), arises is conventionally called the region of free sedimentation. In fact, here too mutual retardation is substantial, and only at extreme dilution is absolutely free sedimentation realized; in this case the molecules move completely independently of one another. We shall denote the corresponding values of the hydrodynamic parameters by \(f_0\), \(s_0\), and \(D_0\); instead of formulas (II), (IV), and (V) we shall now have

\[ s_0=\frac{M(1-V\rho)}{f_0}, \tag{3,1} \]

\[ D_0=\frac{RT}{f_0}, \tag{3,2} \]

\[ M=\frac{s_0}{D_0}\,\frac{RT}{1-V\rho}. \tag{3,3} \]

To calculate the hydrodynamic parameters it is necessary to start from a definite model of the polymer molecule in solution. The statistical theory of the configuration of linear molecules, proposed by Mark, Guth, and W. Kuhn \(^{33—36}\), developed by Ya. I. Frenkel and S. E. Bresler \(^{37,38}\) and, more recently, by M. V. Vol'kenshtein and co-workers \(^{39,40}\), predicts that, owing to the thermal motion of individual segments, an isolated linear molecule consisting of \(Z\) segments assumes the most probable configuration of a statistically coiled coil; moreover, the mean square distance between the end segments, \(\langle r_0^2\rangle\), is proportional to the degree of polymerization and to the square of the effective length \(b\) of a segment:

\[ \langle r_0^2\rangle=b^2Z. \tag{3,4} \]

The effective length \(b\) is not the true distance between neighboring nodal atoms of the polymer chain, although it is proportional to this distance; the proportionality coefficient is a function of the valence angles and electrical factors that restrict the freedom of rotation of the segments.

At one time it was assumed that the coefficient of translational friction \(f_0\) is directly related to the intrinsic viscosity

\[ [\eta]=\lim_{c\to 0}\frac{\eta_r-1}{c}, \tag{3,5} \]

where \(\eta_r\) is the relative viscosity, and \(c\) is the concentration in \(g/cm^3\). On this basis the conclusion was drawn that \(f_0\) and \([\eta]\) are completely analogous functions of the molecular weight and of the shape of the molecule, and that one of them may be expressed in terms of the other. However, a rigorous hydrodynamic examination of the question shows that this is incorrect. The conditions of viscous flow in a viscometer differ radically from the conditions of free sedimentation and diffusion. Viscous flow is characterized by the presence of a velocity gradient in a direction perpendicular to the direction of the liquid flow; this causes rotation of deformable chain molecules, which is entirely absent in sedimentation or diffusion.

Since this question is of fundamental importance, it should be considered in greater detail. V. Kuhn \(^{33}\) first showed that the intrinsic viscosity \([\eta]\) is associated not with the translational, but with the rotational motion of flexible chain molecules. Since any interaction between these molecules is excluded when the relation

\[ \frac{\eta_r - 1}{c} \]

is extrapolated to infinite dilution, the value \([\eta]\) depends only on the work performed by the flowing solvent on isolated molecular coils. This work must be a function of the velocity gradient, the viscosity of the solvent, and the shape of the coil; the latter is specified in a coordinate system usually associated with the center of gravity of the molecule. The specific methods of calculation, of course, depend on the choice of the hydrodynamic model for the coil. For greater clarity, let us represent the macromolecule as beads (monomeric “submolecules”) strung on a flexible thread; this necklace then coils in such a way as to form a spherical coil of radius \(r_s\), in which the beads are uniformly distributed over the volume. Such a model was first applied to calculations in 1938 by Huggins \(^{41a}\), and later—for a more thorough analysis—by Brinkman \(^{41b}\) and Debye \(^{16}\). Figure 3 gives a section in the diametral plane of such a coil. Let us choose rectangular coordinates so that the \(x\)-axis coincides with the direction of flow of the liquid, and the \(z\)-axis with the direction of the velocity gradient \(\nabla v = \mathbf{G}\). The \(y\)-axis is directed perpendicular to the drawing. The center of gravity

Fig. 3. Schematic representation of a rotating molecular coil.

molecule \(O\), coinciding with the center of the sphere, moves in the direction \(x\) with velocity \(v_0\). If we now transfer the origin of coordinates to \(O\), without changing the direction of the axes, then, as follows from the figure, \(v'_0 = 0\), and the velocity of the flow near beads \(a\) and \(b\), symmetrically situated along the diameter, will be \(v'_a = r_s G\) and \(v'_b = -r_s G\). If the coefficient of translational friction of each bead is \(\zeta\), then on beads \(a\) and \(b\) there acts a rotating moment \(M_{ab} = 2r_s \times r_s G\zeta = 2r_s^2 G\zeta\), tending to rotate the coil about the \(y\)-axis. If we further assume that the molecule is freely washed by the solvent, an analogous calculation can be carried out for each pair of beads situated symmetrically with respect to the axis of rotation; in this case \(M = 2r^2 G\zeta \cos \widehat{(r,z)}\), where \(r\) is the current coordinate. Summation over all pairs of beads gives the rotating moment \(M\), acting on the molecule as a whole. Thus, the presence of a velocity gradient, characteristic of viscous flow, causes rotation of the coils, in the course of which additional frictional losses arise and ultimately determine the value of \([\eta]\). The same conclusion—though by a less graphic route—can also be obtained with the aid of other models. The magnitude of the moment \(M\) also depends on whether all elements of the chain (or beads) interact with the solvent, or only some of them do. This question will be considered in detail below (immobilization of the solvent, or shielding).

Thus, the coefficient of translational friction of the macromolecule as a whole \(F\) does not enter into the expression for \([\eta]\), whereas the friction coefficients \(\zeta\) of the individual elements of the chain enter as factors in the components of the moment \(M\). As for the coefficient \(F\), it must be directly made up of the coefficients \(\zeta\); in particular, for an absolutely transparent molecule (Fig. 3), simply \(F = \zeta Z\), whereas in the general case \(F < \zeta Z\), since not all beads interact with the solvent.

The characteristic viscosity is directly related to the coefficient of rotational friction \(f_r\), which can be determined from experiments on the measurement of rotational diffusion, \(\theta\). By analogy with (3,2),

\[ \theta=\frac{RT}{f_r}. \tag{3,6} \]

It can further be shown that

\[ [\eta]=\frac{f_r}{2M\eta_0}, \tag{3,7} \]

where \(\eta_0\) is the viscosity of the solvent. Combination of the last two for-

\(\chi\gamma\lambda\) makes it possible to calculate the molecular weight from the formula

\[ M=\frac{1}{[\eta]\theta}\,\frac{RT}{2}, \tag{3,8} \]

which coincides with Svedberg’s formula.

In this paragraph we shall consider four variants of hydrodynamic theories relating the parameters \(f\) and \(f_r\) (and consequently \(s_0\), \(D_0\), \([\eta]\), and \(\theta\)) to the dimensions and configuration of chain molecules. Although these theories proceed from different concrete models representing statistically coiled threadlike molecules, in the final analysis they lead to entirely analogous dependences of \(f\) and \(f_r\) on molecular weight.

a) Immobilization of the solvent according to W. and G. Kuhn\(^{13}\)

In W. Kuhn’s theory\(^{33}\) the molecular coil is regarded as consisting of \(N_m\) statistical elements (“segments”) of hydrodynamic length \(A_m\) and thickness \(d_h\). In calculations it is approximately assumed that the segments are elongated ellipsoids of revolution with axes \(A_m\) and \(d_h\) (usually \(A_m \gg d_h\)). The segments may contain a fairly large number of monomeric units; if \(l\) is the length of one monomer, then \(A_m=ln\), where \(n\) is a constant number. Neighboring segments rotate freely about valence bonds; as a result of thermal motion there arises a coil configuration with a mean-square distance between the terminal segments

\[ \langle r_0^2\rangle^{1/2}=A_m N_m^{1/2}(=bZ^{1/2}). \]

During translational motion of such a coil in a liquid of viscosity \(\eta_0\), it experiences a resistance

\[ \mathbf{L}=-\mathbf{u}F, \]

where \(\mathbf{u}\) is the velocity, and \(F\) is the friction coefficient \(\left(F=\frac{f_0}{N};\ N\text{ is Avogadro’s number}\right)\). In calculating \(F\), it is first of all necessary to determine how the solvent moves inside the coil. W. and G. Kuhn assume that at least part of the solvent is so firmly retained in the space between the segments (is immobilized) that it moves as one whole with the coil. Another part of the solvent may flow through the coil. The degree of immobilization obviously depends on the compactness of the coil. In one limiting case the coil may be regarded as completely impermeable; then the solvent located inside it is fully immobilized. If, on the contrary, the coil is very strongly extended, it turns out to be transparent to the solvent, and no immobilization occurs. W. and G. Kuhn propose to characterize the degree of immobilization by the shape parameter \(\lambda_t\); if one introduces the length of the molecule, i.e., the distance between the ends of the extended chain

\(\bar h=A_mN_m\), then \(F\) can be expressed by a formula analogous to Stokes’ formula:

\[ F=6\pi\eta_0 \bar h \lambda_t . \tag{3,9} \]

A theoretical calculation was carried out only for two limiting cases. For an absolutely transparent coil, \(F\) is simply made up of the friction coefficients of the individual segments; in doing so, it will only be necessary to take into account all possible orientations of the segments with respect to the vector \(\mathbf u\). The corresponding averaging gives:

\[ F=N_m\frac{3\pi\eta_0A_m}{\ln \dfrac{2A_m}{d_h}}=\operatorname{const} Z. \tag{3,10} \]

If the coil is completely impermeable to the solvent, the calculation of \(F\) may be performed as for a rigid sphere of radius \(\bar r=0.46\langle r_0^2\rangle^{1/2}\), i.e.

\[ F=8.7\eta_0(N_m)^{1/2}A_m=\operatorname{const} Z^{1/2}. \tag{3,11} \]

Thus, in the two limiting cases, the following values are obtained for the parameter \(\lambda_t\):

\[ \lambda_t= \begin{cases} \dfrac{8.7}{(N_m)^{1/2}}=\dfrac{\operatorname{const}}{Z^{1/2}} & \text{(complete immobilization)},\\[1.2em] \dfrac{3\pi}{\ln \dfrac{2A_m}{d_h}} & \text{(absolutely transparent coil)}. \end{cases} \tag{3,12} \]

For intermediate cases the parameter \(\lambda_t\) cannot be calculated so simply. B. and G. Kuhn determined it experimentally on macroscopic models, making use of the well-known theorem of hydrodynamic similarity, according to which, when all linear dimensions of the system are changed by a factor \(\alpha\), the relation

\[ \frac{L'}{u'}=\alpha\frac{L}{u}=6\pi\eta_0R'\lambda_t, \tag{3,13} \]

holds, where \(L'\) and \(u'\) are the force and velocity in the enlarged model system. The molecular models were made of aluminum wire of diameter \(1\) mm; the length of the “segments” varied from \(1\) to \(25\) mm. For greater accuracy, for each value of \(N_m\) from 6 to 10 models were constructed, corresponding to different statistical configurations.

The fall of these models in a viscous medium with \(\eta_0=60\) poise was then observed. Obviously, \(L'=P(1-V\rho)\), where \(P\) is the weight of the “molecule,” \((1-V\rho)\) is the Archimedean factor already encountered by us, and \(u'\) is found from experiment. Since \(\dfrac{L}{u}=6\pi\eta_0\bar h\lambda_t\), the problem of finding \(\lambda\)

already presented no difficulty. It turned out that \(\lambda_t\) is a linear function of the molecular dimensions, i.e., \(N_m^{1/2}\) or \(Z^{1/2}\):

\[ \frac{1}{\lambda_t}=0.16\ln\frac{A_m}{d_h}+0.02+0.1(N_m)^{1/2}. \tag{3.14} \]

There are sufficient grounds to assume that, at very large values of \(N_m\), the solvent enclosed in the inner layers of the coil is so strongly screened by the outer segments from the liquid washing the molecule that the immobilization may be regarded as complete. If we turn to (3.14), it is seen that in this case the first two terms may be neglected, i.e.,

\[ \frac{1}{\lambda_t}=0.1(N_m)^{1/2}, \tag{3.15} \]

which, to within the coefficient, coincides with (3.12). Conversely, at very small \(N_m\) (for example, at \(N_m \ll 1\), which would correspond to the configuration of a rigid rod or an elongated ellipsoid) the molecule is completely washed by the liquid, there is no immobilization whatever, and the last term in (3.14) may be neglected. This gives for \(\lambda_t\)

\[ \frac{1}{\lambda_t}=0.16\ln\frac{A_m}{d_h}+0.12, \tag{3.16} \]

which again practically coincides with the corresponding value of (3.12). Thus, the empirical formula (3.14) is in agreement with the theoretical formulas for two limiting configurations. V. and G. Kuhn believe that it is also valid for all intermediate configurations. Substitution of \(\lambda_t\) into (3.9) leads to the expressions

\[ D_0=\frac{(a_D+b_D Z^{1/2})}{Z} \tag{3.17} \]

and

\[ s_0=a_s+b_s Z^{1/2}, \tag{3.18} \]

where \(a_D\), \(b_D\), \(a_s\), \(b_s\) are readily computable constants depending on \(A_m\) and \(d_h\).

In an analogous way, with the aid of torsion balances, for the models there was found the coefficient of rotational friction

\[ F_r=\frac{\overline{R}}{18}\eta_0 \overline{H}_1^{\,2}\lambda_r, \tag{3.19} \]

where \(\overline{H}_1=1.4\langle r_0^2\rangle^{1/2}\), and \(\lambda_r\) has, for rotation, the same meaning as \(\lambda_t\) for translational motion. Substitution of \(F_r\) gives for the characteristic viscosity and rotational diffusion the second

a pair of equations of the form

\[ \left. \begin{array}{rl} \theta &= -\,\dfrac{a_\theta+b_\theta Z^{1/2}}{Z^2},\\[6pt] \text{and}\quad [\eta] &= \dfrac{Z}{a_\eta+b_\eta Z^{1/2}}, \end{array} \right\} \tag{3,20} \]

from which the molecular weight can also be found.

We shall postpone discussion of the theory until the next section, and here limit ourselves to a single remark. As follows from all that has been set forth, the physical meaning of the concept of immobilization reduces to the fact that immobilization is a function of molecular weight: it increases with increasing \(M\). The question of the degree of compactness and the degree of immobilization as a function of the nature of the solvent remains open (or, more precisely, must be decided only experimentally). This is an undoubted defect of the theory, since it is known that in good solvents some polymers (for example, cellulose acetate) have a very extended configuration even at very high molecular weights.

In one of the recent works B. Kuhn, H. Kuhn, and Moning \(^{42}\) somewhat modified the old theory, replacing the “hinged” model of the coil by a “rope” one, in which the segments are represented by arcs of a circle of equal length but different curvature. These arcs pass smoothly into one another and rotate freely about the valence bonds; upon averaging over all mutual rotations and over all radii of curvature, an expression analogous to (3,4) is obtained, but with somewhat different numerical coefficients. If one proceeds from the rope model, then in the formulas of this section one should replace \(N_m\) by \(N_\sigma=1.532N_m\) and \(A_m\) by \(B_\sigma=A_m/1.532\). This will only insignificantly change the coefficients in expressions (3,17), (3,18), and (3,20).

b) Kirkwood–Riseman Theory \(^{14,15}\)

Here it is assumed that the dimensions of the segment and of the monomer coincide. The coil is represented in the form of “wire” beads: \(2n+1\) “skeletal” atoms C, or beads, numbered from \(-n\) to \(+n\), are located at the positions of the “hinges” of the Kuhn model and are connected by \(2n\) bond vectors \(\mathbf{b}_l\), directed from atom number \(l-1\) to atom number \(l\). The length of each such vector is \(b_l\). The internal configuration of the chain is described by the angles \(\varphi_l\) between the planes of successive pairs of vectors \((\mathbf{b}_{l-1},\mathbf{b}_l)\) and \((\mathbf{b}_l,\mathbf{b}_{l+1})\). Internal rotation about the vectors \(\mathbf{b}_l\) is hindered by a moment \(m_l\), having the plane of symmetry \(\varphi_l=0\).

For large \(n\), \(2n \simeq Z\). The mass of the monomer \(m_0\) is considered to be concentrated in a bead; \(Z=\dfrac{M}{m_0}\). In calculating \(F\) or \([\eta]\), one must determine the perturbations in the flow of the surrounding liquid caused by the individual links. To each bead there is assigned a coefficient of friction \(\zeta\). For the subsequent calculation it is necessary to know the mean-square distance \(\langle r_{ls}^{2}\rangle\) between beads with numbers \(l\) and \(s\). It is given by the usual formula

\[ \langle r_{ls}^{2}\rangle = |l-s|\,b^{2}, \tag{3.21} \]

where the effective length of a link is

\[ b=\left(\frac{1+\langle \cos\varphi\rangle}{1-\langle \cos\varphi\rangle}\right) \left(\frac{1-\cos\gamma}{1+\cos\gamma}\right)b_{0}. \tag{3.22} \]

Here \(\langle \cos\varphi\rangle\) is the mean value of \(\cos\varphi\) over the whole coil, and \(\gamma\) is the valence angle. The value \(\langle \cos\varphi\rangle=0\) corresponds to an extended chain; the maximum value of \(\langle \cos\varphi\rangle\) occurs for free rotation.

A force \(\chi_l=-\zeta \mathbf{w}_l\) acts on one bead. The relative velocity \(\mathbf{w}_l\) of this bead is equal to \(\mathbf{v}_l-\mathbf{u}_l\), where \(\mathbf{v}_l\) is the velocity of the liquid at point \(l\) in the absence of element \(l\), and \(\mathbf{u}_l\) is the velocity of this element. Thus, in the expression for \(\chi_l\) there is taken into account both the shielding of point \(l\) by the surrounding beads \((\mathbf{v}_l)\) and the direct interaction of element \(l\) with the liquid \((\mathbf{u}_l\zeta)\).

The further problem consists in calculating the perturbation introduced by the molecule as a whole at the point \(\mathbf{R}\) (i.e., at a distance \(R\) from the center of gravity). If this perturbation has been calculated, it remains to set \(R\) equal to the outer radius of the coil.

In viscous flow in the direction \(x\), the molecule, for the reasons indicated above (cf. Fig. 3), begins to rotate with angular velocity \(\omega\), while the internal links tend to change their mutual orientation to a greater extent the more weakly they are shielded. It can be shown that in this case

\[ \mathbf{u}_l=\omega\,(\mathbf{e}_z \times \mathbf{R}_{0l}), \tag{3.23} \]

where \(\mathbf{e}_z\) is the \(z\)-component of the external rotating moment, and \(\mathbf{R}_{0l}\) is the distance of the \(l\)-th bead from the center of gravity, found from the condition of a Gaussian distribution of the bead density in the coil (counting from the center of gravity).

Further summation of the perturbations from each link will ultimately lead to the expression

\[ [\eta]=\frac{N\zeta b^{2}}{36\eta_{0}m_{0}}\,Z\Phi(\lambda_{0}Z^{1/2}), \tag{3.24} \]

where

\[ \lambda_0=\frac{\zeta}{(6\pi^3)^{1/2}\eta_0 b} \tag{3.25} \]

and

\[ \Phi(x)=\left(\frac{6}{\pi^2}\sum_{k=1}^{\infty}\frac{1}{k^2}\frac{1}{1+\dfrac{x}{\sqrt{k}}}\right). \tag{3.26} \]

In translational motion the calculation of the perturbations is carried out in an analogous way, but the final result is determined only by the screening effect, since there is no rotational motion of the sphere as a whole (internal rotations still take place). The final result has the form:

\[ F=\frac{Z\zeta}{1+\dfrac{8}{3}\lambda_0 Z^{1/2}} . \tag{3.27} \]

We can again consider two limiting cases. For very small \(Z\), which corresponds to absolute free drainage (or permeability),

\[ F\simeq Z\zeta, \tag{3.27a} \]

which agrees with formula (3.10) and with Staudinger’s well-known empirical rule

\[ [\eta]=\operatorname{const} Z;\qquad F=\operatorname{const}_1 Z. \tag{3.28} \]

For very large \(Z\) (complete immobilization according to Kuhn)

\[ F=\operatorname{const} Z^{1/2}, \tag{3.27б} \]

again in agreement with the theory of W. and H. Kuhn.

If, further, we write the equations for \(s_v\) and \(D_0\):

\[ D_0=\frac{kT}{Z\zeta}\left[1+\frac{8\lambda_0}{3}Z^{1/2}\right], \tag{3.29} \]

\[ s_v=\frac{m_0(1-V\rho)}{\zeta}\left[1+\frac{8\lambda_0}{3}Z^{1/2}\right], \tag{3.30} \]

then the analogy of the final result of the two theories considered will become still more obvious.

If now, in the coordinates \(s_v, Z^{1/2}\), one reproduces (3.30), then for a series of polymer homologues a straight line should be obtained; from its slope the effective length of a segment \(b\) will be found, and from the point of intersection with the ordinate axis—the friction coefficient \(\zeta\) of a single bead.

In the theory of B. and G. Kuhn an analogous construction yielded quantities \(A_m\) and \(d_h\) having practically the same meaning, but somewhat less definite. Equations (3.18) and (3.30), as we shall see from the following paragraph, are very well satisfied over a very wide range of \(Z\).

In a similar way Kirkwood and Riseman calculated the coefficient of rotational diffusion^{15}, related to \([\eta]\) by relation (3.8).

c) The Debye–Bueche Theory

In this theory the real configuration of the coil is replaced by the “equivalent hydrodynamic sphere” shown in Fig. 3, whose motion is completely equivalent to the motion of the coil. As has already been said, this sphere is composed of \(Z\) beads strung on an absolutely flexible thread; to each bead are assigned a friction coefficient \(\zeta\) and the weight of a monomer \(m_0\); the density distribution in the sphere is taken to be constant, and therefore its effective radius \(r_s\) is approximately equal to one half of the root-mean-square radius of the real coil; more precisely,

\[ \frac{r_s^2}{10}=\frac{\langle r_0^2\rangle}{36}. \tag{3.31} \]

Following B. and G. Kuhn, Debye and Bueche introduce a parameter of transparency of the coil for the solvent, or the screening length \(L\). If \(\nu\) is the density of beads, then

\[ \frac{1}{L^2}=\frac{\nu\zeta}{\eta_0}. \tag{3.32} \]

The quantity \(L\) characterizes the penetration of the flow into the coil and its retardation by the beads with which it collides there. In other words, it characterizes the screening of the internal parts of the coil from the hydrodynamic velocity field. The depth at which the (relative) velocity of the liquid decreases by a factor of \(e=2.71828\ldots\) is the screening length \(L\).

In the first version of the theory^{16} Debye considers absolutely transparent molecules and calculates for them the characteristic viscosity as the total sum of the losses of energy due to the friction of the individual beads against the solvent.

It can be shown that the absolute value of the angular velocity \(\omega\) with which the molecule begins to rotate under the action of the velocity gradient \(\mathbf{G}\) is equal to \(\dfrac{G}{2}\). Direct calculation then shows that the loss of energy for a bead located at a distance \(R\) from the center of gravity is equal to \(W=\dfrac{\zeta G^2}{4}R^2\), and since at the exter-

boundary of the coil \(R^2=\mathrm{const}\,Z\), it turns out that \(W\) and \([\eta]\) are proportional to the degree of polymerization, which again agrees with the Staudinger rule and with the limiting version of Kuhn’s theory, when \(N_m\) is small and immobilization of the solvent does not occur.

In the more general case Debye and Bueche take into account partial screening of the inner regions of the coil, which can be characterized by the screening ratio

\[ \sigma=\frac{r_s}{L}. \tag{3,33} \]

Complete immobilization in the terms of this theory corresponds to complete screening, i.e. \(\sigma=\infty\), and absolute permeability to \(\sigma=0\). As in the theory of W. and H. Kuhn, screening depends on the molecular weight. This is quite understandable: at very small \(Z\), the representation of a polymer molecule as a coil is devoid of any meaning; it will simply be a thread, or a rod, freely washed by the solvent, which corresponds to \(\sigma=0\). At very large \(Z\), on the contrary, the internal regions of the coil are practically completely isolated from the surrounding liquid, i.e. \(r_s\gg L\), and \(\sigma\to\infty\). The dependence of \(\sigma\) on \(Z\) can be represented with sufficient accuracy by the formula

\[ \sigma=\mathrm{const}\,\sqrt[4]{Z}. \tag{3,34} \]

The coefficient of translational friction, by complete analogy with (3,9), can be represented by the formula

\[ F=6\pi r_s\eta_0\psi(\sigma), \tag{3,35} \]

where \(\psi(\sigma)\) is a function of the screening ratio, calculated and tabulated by Bueche. For absolutely transparent coils, obviously, \(F\) is simply composed of the friction coefficients of each individual bead, i.e.

\[ F=Z\zeta, \tag{3,35a} \]

which agrees with the limiting variants of the preceding theories. For \(\sigma\to0\), \(\psi(\sigma)\to \dfrac{2}{9}\sigma^2\), and since \(\sigma^2=\mathrm{const}\,Z^{1/2}\) and \(r_s=\mathrm{const}\,Z^{1/2}\), then for transparent coils (3,35a) is automatically obtained also from (3,35).

In the other limiting case, at \(\sigma\to\infty\), Kuhn’s formula should evidently be obtained,

\[ F=\mathrm{const}\,Z^{1/2}. \tag{3,35b} \]

Indeed, at \(\sigma=\infty\), \(\psi(\sigma)=1\), and (3,35b) is likewise satisfied automatically \(^{17}\).

All intermediate values of \(\sigma\) are covered by values of \(\psi(\sigma)\) lying between 0 and 1. Since \(\sigma\) is a weak function of molecular weight, and \(\psi(\sigma)\) is an even weaker function of \(\sigma\), one may approximately put \(\psi(\sigma)=M^\varepsilon\), where \(0\leqslant\varepsilon\leqslant0.5\). This leads to the well-known empirical dependence

\[ F=\mathrm{const}\, M^{a(\sigma)}, \tag{3,36} \]

which is satisfied over a wide, although limited, range of values of \(M\). Comparing (3,36) and (3,35), we find:

\[ a(\sigma)=\frac{1}{2}+\frac{1}{5}\frac{\sigma}{\psi}\frac{\partial\psi}{\partial\sigma}. \tag{3,37} \]

When \(M\) changes from 0 to \(\infty\), \(a(\sigma)\) changes from 1 to \(1/2\), but even within a change of \(M\) by 3 orders of magnitude (from \(10^4\) to \(10^7\)) the value \(a(\sigma)\) for a series of polymer homologues practically remains one and the same.

In an analogous way it can be shown that

\[ [\eta]=\frac{\frac{4}{3}\pi r_s^3}{M}\,\varphi(\sigma), \tag{3,38} \]

where \(\varphi(\sigma)\) is another function of the shielding ratio, also tabulated by Debye and Bueche. For small \(\sigma\), \(\varphi(\sigma)\to \dfrac{\sigma^2}{10}\), while for \(\sigma=\infty\), \(\varphi(\sigma)=2.5\), which gives for the two limiting cases

\[ [\eta]= \begin{cases} \mathrm{const}\, Z & \text{(absolutely transparent coils)},\\ \mathrm{const}_1\, Z^{1/2} & \text{(completely impermeable coils)}. \end{cases} \]

This again is in complete agreement with the results of the preceding theories. For intermediate shielding, on the basis of the same considerations from which formula (3,36) was derived, one can obtain the no less well-known empirical relation

\[ [\eta]=\mathrm{const}\, M^{\beta(\sigma)}, \tag{3,39} \]

which is also satisfied over a sufficiently wide range of values of \(M\). The coincidence of the parameters \(a(\sigma)\) and \(\beta(\sigma)\) in the limiting cases led to erroneous conclusions about their coincidence in general\({}^{12}\). In fact, as can be shown,

\[ \beta(\sigma)=\frac{1}{2}+\frac{1}{4}\frac{\sigma}{\varphi}\frac{\partial\varphi}{\partial\sigma} \tag{3,37a} \]

and always \(\beta(\sigma)\geqslant a(\sigma)\).

This excess of \(\beta(\sigma)\) over \(a(\sigma)\) reflects a simple physical fact. The principal losses of energy in the rotation of molecules in solution

occur at the peripheral units, which are the most strongly screened. Therefore the value of the rotational-friction coefficient \(f_r\) is determined mainly by the more “transparent” outer parts of the molecules, and the molecule as a whole, in rotational motion, appears “more transparent” than in translational motion.

The dimensions of coils cannot be obtained from the Debye–Bueche theory as simply as according to V. and G. Kuhn or according to Kirkwood and Riseman. Since \(\beta(\sigma)\) and \(a(\sigma)\) have been tabulated\({}^{17}\), from a series of experiments with polymer homologues the values of \(\sigma\) and \(\varphi\) or \(\psi\) can be found. Substitution of \(\varphi\) or \(\psi\) into (3.38) or (3.35) would make it possible to calculate \(r_s\). However, \(r_s \ne \langle r_0^2\rangle^{1/2}\), and expression (3.31), which relates these parameters, is evidently correct only for small \(\sigma\).

On the other hand, the Debye–Bueche theory, which makes it possible to write the sedimentation and diffusion constants in the form

\[ s_0 = K_s M^{1-a(\sigma)} \tag{3.40} \]

and

\[ D_0 = K_D M^{-a(\sigma)} \tag{3.41} \]

has other merits, which will become clear below.

d) Flory and Fox theory\({}^{18,43,44}\)

Until now, in calculating hydrodynamic parameters, the interaction of the polymer with the solvent has not been taken into account at all. Yet it is reflected very substantially in the configuration of molecular coils. In so-called good solvents there is a preferential statistical surrounding of the segments by solvent molecules, and the coil as a whole tends to assume a more extended configuration, facilitating such surroundings. As early as 1942, Mark\({}^{45}\) pointed out that in such solvents the configuration may prove to be more extended than is predicted by statistical theory (according to which \(\langle r_0^2\rangle^{1/2} = bZ^{1/2}\)). Conversely, in “poor” solvents the segments tend to enter the surroundings of other segments, which leads to contraction of the coil and to a more compact configuration. Flory\({}^{46}\) proposes to regard the interaction of the coil with the solvent as a process of “swelling.” He, like Debye, abandons the idea of a real coil, replacing it by an equivalent hydrodynamic sphere with uniform distribution of the units over the volume

\[ V_s = \frac{4}{3}\pi r_s^3 . \]

The concentration of segments in this volume may be 1000 times less than the concentration of the solvent. Upon swelling of the molecule, owing to interaction

segments with the solvent and with one another, the root-mean-square radius \(\langle r^2\rangle^{1/2}=\mathrm{const}\, r_s\) increases with \(M\) not proportionally to \(Z^{1/2}\), but faster:

\[ \langle r^2\rangle^{1/2}=\langle r_0^2\rangle^{1/2}\alpha . \tag{3,42} \]

Here \(\langle r_0^2\rangle^{1/2}\) corresponds to the “theoretical” configuration in an ideal solvent (without any interaction), while the parameter \(\alpha\) depends on the molecular weight, the entropy and heat of dissolution, and takes into account, in particular, the selectivity of the segments with respect to the solvent molecules.

In describing the hydrodynamic properties of coils, Flory and Fox, at least in the first approximation, assume that the solvent is completely immobilized inside the effective volume \(V_s\). Therefore, for \([\eta]\) and \(F\) one should obtain the limiting expressions of the theory of V. and G. Kuhn \(^{13}\), but with replacement of \(\langle r_0^2\rangle^{1/2}\) by the true value \(\langle r^2\rangle^{1/2}\). Substitution of (3,42) into Kuhn’s formulas gives \(^{47}\)

\[ [\eta]=K M^{1/2}\alpha^3 \tag{3,43} \]

and

\[ \frac{F}{\eta_0}=K_f M^{1/2}\alpha . \tag{3,44} \]

As Flory and Fox have shown,

\[ K=\Phi\left(\frac{\langle r_0^2\rangle}{M}\right)^{3/2}, \tag{3,45} \]

where \(\Phi\) is a universal constant for all polymers, while \(K_f\), according to Flory and Mandelkern \(^{47}\), is equal to

\[ K_f=P\left(\frac{\langle r_0^2\rangle}{M}\right)^{1/2}, \tag{3,46} \]

where \(P\) is another universal constant. It can be shown that \(\alpha\sim M^\varepsilon\), where \(\varepsilon\) is, depending on the magnitude of the entropy and heat of dissolution, of the order from 0 to 0.13. Thus, substitution of \(\alpha\) into (3,43) and (3,45) immediately leads to formulas (3,36) and (3,39), with maximum values \(\beta(\sigma)=0.9\) and \(a(\sigma)=0.63\).

It is easy to see that Flory’s theory makes it possible to explain deviations from Kuhn’s limiting rule \((F\sim M^{1/2}\) and \([\eta]\sim M^{1/2})\), without resorting to additional hypotheses about the motion of the solvent inside the coil. Its advantage is the direct allowance for the interaction of solvent and polymer, and its shortcoming is the neglect of the factors of “immobilization” or permeability.

But this shortcoming, as will be shown in the next paragraph, can easily be eliminated by simply replacing \(\langle r_0^2\rangle^{1/2}\) in the hydrodynamic theories by \(\langle r_0^2\rangle^{1/2}\alpha\).

S. Ya. Frenkel

§ 4. COMPARISON OF THEORIES WITH EXPERIMENT

a) Dependence of the sedimentation constant
on molecular weight

At first glance it may seem that the theories of W. and G. Kuhn and of Kirkwood–Riseman, on the one hand, and the theories of Debye–Bueche and Flory, on the other, are in contradiction, since the first two predict a dependence of the form

\[ s_0=a_s+b_s Z^{1/2}, \tag{4.1} \]

whereas the second give the dependence

\[ s_0=K_s M^{1-a}, \tag{4.2} \]

or

\[ \lg s_0=\lg K_s+(1-a)\lg M. \tag{4.2*} \]

In fact, however, formulas (4.1) and (4.2) are completely identical over a very wide range of values of \(M\), and only in the limit as \(Z\to 0\) and \(Z\to\infty\) do substantial differences appear between them. As an example one may use the data of the work of S. E. Bresler, I. Ya. Poddubnyi, and coauthors\(^ {48}\) for two similar synthetic rubbers, conventionally designated

Fig. 4. a) Dependence of \(s_0\) on \(Z^{1/2}\) for two rubbers (according to the theories of Kuhn and Kirkwood). b) Dependence of \(\lg s_0\) on \(\lg M\) for the same rubbers (according to the theories of Debye and Flory).

K-1 and K-3, in one and the same solvent—octane. In Fig. 4, a, a construction of type (4.1) is given, and in Fig. 4, b—of type (4.2*). The limits of val-

tions of \(M\) for K-1 from \(4\times 10^4\) to \(6\times 10^5\), and for K-3 from \(3\times 10^4\) to \(10^6\). We see that the experimental points lie equally well on both graphs. Thus, in practice it is immaterial which of the formulas given above is used to express the dependence of \(s_0\) on \(M\). However, when interpreting the results, differences between the theories emerge. Before turning to their consideration, let us dwell on the limiting case of very small \(M\). As was indicated above, in this case the concept of a statistical coil loses all meaning, for the obvious reason that the number of statistical elements is counted in units or in a small number of tens. The molecule then acquires the configuration of a chain or a rod freely washed by the solvent; since in this case \(F \cong Z\), the sedimentation constant ceases to depend on the molecular weight. This circumstance is taken into account by the exact formula of Bjoche[^17] \(F=6\pi r_s\eta_0\psi(\sigma)\), since

\[ \text{as } M\to 0,\quad \psi(\sigma)\to \frac{\sigma^2}{10}=\text{const } Z^{1/3}, \]

and since \(r_s Z^{1/3}\sim Z\), \(F\) in fact proves to be a linear function of the degree of polymerization. However, formula (4.2) does not take this circumstance into account and, consequently, cannot be applied at low values of \(M\). Formula (4.1) is more accurate in this respect, for as \(Z\to 0\), \(s_0\) tends to the constant limiting value \(a_s\). In fact, however, the coil configuration is lost already at sufficiently large values of \(Z\) (of the order of 50–100). Therefore formula (4.2) also cannot be correct at very low values of \(M\). In those cases where it is necessary to extrapolate the dependence \(s(M)\) down to \(M=0\) (this is necessary in calculating distribution functions—see Part II), one has to use empirical formulas of the type

\[ s_0=a'_s+b'_sM. \tag{4.3} \]

Such a dependence was found in the work cited above2 for divinylstyrene rubber at \(M<4\times 10^4\), and also for polyvinylpyrrolidone[^49] in the same range of molecular weights.

Formula (4.3) is not purely formal. Its physical meaning may be reduced to the following. The derivative \(\dfrac{ds}{dM}\), strictly speaking, must be equal to zero as \(M\to 0\), since \(s\) is then a constant quantity. If, however, one proceeds from formula (4.2), then

\[ \left(\frac{ds}{dM}\right)_{M=0}=\text{const } M^{-1/2}=\infty. \]

Formula (4.3), admittedly in a somewhat artificial way, corrects this circumstance, taking into account, on the one hand, the constancy of \(s\) at small \(M\), and on the other hand the fact that

\[ \frac{s}{dM}\to 0, \]

since \(b'_s\) is a very small quantity. (In the work mentioned, for styrene rubber \(b'_s=5.3\cdot 10^{-19}\).)

b) Configuration and dimensions of threadlike molecules in solution

The configuration of molecular coils can be judged in various ways. The simplest is to determine the degree of coiling as the ratio of the root-mean-square distance between the ends of the chain, \(\langle r^2\rangle^{1/2}\), to the hydrodynamic length \(L\) of the extended (but undeformed) molecule. Introducing the corresponding parameter

\[ \chi=\langle r^2\rangle^{1/2}/L, \tag{4,4} \]

we immediately note that, in principle, a change in the configuration of one and the same molecule in different solvents is uniquely taken into account by the quantity \(b\), or \(A_m\), proportional to \(\langle r^2\rangle^{1/2}\). Thus, a change in the slope \(b_s\) in formula (4,1) indicates a change in configuration for a given series of polymer homologues: the smaller this slope, the larger \(b\) is and the more extended a configuration the coil assumes. As was indicated in the introduction, the ability of chain molecules to assume a more or less extended configuration is connected with the “flexibility” of the chains, determined by the values of the angle \(\varphi\) in formula (3,22).

Table I gives values of \(b\) for polystyrene in various solvents\(^{50}\) and for nitrocellulose\(^{51}\), according to the data of Newman and Eirich and their co-workers.

Table I

Polymer	Solvent	\(\zeta \times 10^{10}\) g/sec	\(b\) (Å), by \(s_0\)	\(b\) (Å), by light scattering\(^{52}\)
Polystyrene	Methyl isopropyl ketone	3.9	5.9	—
Polystyrene	Methyl ethyl ketone	3.6	6.4	5.8
Polystyrene	Dichloroethane	—	—	7.5
Polystyrene	Toluene	1.5	9.2	7.3
Nitrocellulose	Chloroform	0.96	11.0	—
Nitrocellulose	Ethyl acetate	—	52	—

If, on the basis of these data, one calculates \(\langle r^2\rangle^{1/2}\), it turns out that for nitrocellulose \(\chi \simeq 0.5\), whereas for polystyrene in every case \(\chi \leqslant 0.1\). Thus, nitrocellulose molecules do not even assume the configuration of a statistical coil. This indicates the extremely low flexibility of its chains. The latter property is in good agreement with the macroscopic properties of cellulose and its derivatives, in particular with their low elasticity.

It seems desirable to compare these results with independent “measurements of the dimensions” of molecules. As is known, the quantity $\langle r^2\rangle^{1/2}$, and consequently also $b$, can be determined in experiments on light scattering at various angles (see $^{52,53,54}$). Unfortunately, detailed data on light scattering are rather few and are practically limited to Zimm’s work$^{52}$ on polystyrene. The average values of $b$ obtained by him are given in the last column of Table I. As can be seen, the values of $b$ are close in order of magnitude; however, the discrepancies cannot be explained by measurement errors alone.

In the work of Stein and Doty$^{55}$, light scattering in acetone solutions of acetylcellulose was investigated. In a series of values of $M$ from 163,000 to 52,000, the quantity $\chi$ changes from $\sim 3$ to 1, which in general agrees with Newman’s data for nitrocellulose. Taking

$$ Z=\frac{M}{263} $$

($^{9}$, p. 418 and following), we find for the effective segment of acetylcellulose at $\chi>2$, $b \simeq 55\,\text{\AA}$, also in approximate agreement with Table I. If, however, $b$ is calculated by formula (3,30), using the results of S. S. Zinger$^{32}$ for four fractions of acetylcellulose of the same origin as in work$^{55}$, one obtains $b=30.2\,\text{\AA}$.

One may proceed, for verification and comparison of the various theories, in another way. Namely, from the values of $[\eta]$ one may calculate the parameter $b$ and $\varepsilon$ of the Kirkwood–Riseman theory, use them to calculate $s_0$, and compare this with the experimental values. (Comparison with the Debye–Bueche formula (3,35) is not expedient in view of a certain uncertainty in the value of $r_s$ at large $\sigma$.) The table II given for polyisobutylene is taken from the work of Flory and co-workers$^{56}$. In the fourth column of the table are given the values of $s_0$, calculated also from the experimental values of $[\eta]$, but by means of formula (4,5) (see below). Cyclohexane was used as the solvent.

Table II

$M \times 10^{-3}$	$s_0$ exper.	$s_0$ according to Kirkwood–Riseman	$s_0$ according to Flory
1420	4.45	5.6	4.7
672	3.33	3.9	3.5
127	1.94	2.1	1.9
86.7	1.49	1.6	1.4
30.9	0.925	1.1	0.90

It follows from the table that Flory’s theory agrees better with the experimental data. This might have been expected, since

Since the hydrodynamic theories of Kirkwood–Riseman and Debye–Bueche do not take into account the thermodynamic factors of the interaction of the polymer with the solvent, the degree of screening in these theories is connected only with the sizes of the molecules. But it is quite clear that the permeability of molecules also depends on their configuration. It has already been noted that the quantity \(a(5)\) depends much more strongly on the nature of the chosen solvent than on molecular weight: within a sufficiently broad series of polymer homologues \(a(5)\) practically does not change for a given solvent and may change sharply on passing to another solvent.

The point is that in a good solvent (at a large value of the interaction parameter
\[ \alpha=\left[\frac{\langle r^2\rangle}{\langle r_0^2\rangle}\right]^{1/2} \]
) the molecule assumes an extended configuration and, as a consequence, becomes more transparent to the solvent. Therefore the introduction of the parameter \(a\) into the Kirkwood or Debye formulae, i.e. the replacement of \(b\) by \((ba)\) and of \(r_s\) by \((r_s\times \alpha)\), should at once lead to a correlation of the formulae of the hydrodynamic theories with Flory’s theory. This can be shown by direct calculation, but it is simpler to demonstrate it otherwise. In corrected form the quantity \(\lambda\) in formula (3,27) will be
\[ \lambda=\frac{\zeta}{(6\pi^3)^{1/2}\eta_0ba}=\frac{\lambda_0}{a}. \]

Since \(F=Z'\zeta/[1+(8\lambda/3)Z'^{1/2}]\), then for sufficiently large \(Z\), when the second term in the denominator becomes approximately 20 times greater than unity, one may approximately put
\[ \frac{F}{\eta_0}\simeq \left(\frac{3}{8}\right)(6\pi^3)^{1/2}Z^{1/2}ba, \]
which agrees with Flory’s formula
\[ \frac{F}{\eta_0}=P\langle r_0^2\rangle^{1/2}a. \]

Thus, the Kirkwood–Riseman theory, with a slight modification, leads directly to formula (4,2). For small \(Z\), of course, one should use the complete expression for \(F\).

The constant \(P\) for linear polymers should be of the order of 5.1 (according to experimental data). But
\[ \left(\frac{3}{8}\right)(6\pi^3)^{1/2} \]
is just equal to 5.13. This may serve as an additional argument in favor of Flory’s theory.

In conclusion, let us dwell on one more consequence of Flory’s theory, which makes it possible to calculate more simply the mean dimensions of molecules of linear polymers. A combination of for—

...of the formulas of the preceding paragraph (3.43), (3.44), (3.45), and (3.46) gives

\[ \frac{\eta_0 N s_0 [\eta]^{1/3}}{M^{2/3}(1 - V\rho)} = \Phi^{1/3} P^{-1} \tag{4.5} \]

(\(N\) is Avogadro’s number). Thus, the product on the left-hand side of this expression must be a constant for all polymers, independent of the solvent. The quantity \(P^{-1}\Phi^{1/3}\) should be of the order of \(2.5 \times 10^{-6}\). As is seen from Table III below, this relation is in very satisfactory agreement with the experimental data. It is interesting to note that the equality \(P^{-1}\Phi^{1/3} \simeq 2.5 \times 10^{-6}\) is satisfied even for globular proteins, if their molecules have the configuration of elongated ellipsoids of revolution (Mandelkern, Scheraga \(^{59}\)).

Table III

Polymer	Solvent	\(P^{-1}\Phi^{1/3}\times 10^{-6}\)
Polystyrene\(^{57}\)	Methyl ethyl ketone	2.6
Polystyrene\(^{57}\)	Toluene	2.3
Acetylcellulose\(^{326}\)	Acetone	2.7
Polysarcosine\(^{58}\)	Water	2.3
Polyisobutylene\(^{56}\) (\([\eta]\) expressed in \(100\ \mathrm{cm^3/g}\))	Cyclohexane	2.5
	Average:	\(2.5 \pm 0.1\)

With the aid of formula (4.5), from one pair of measurements \(s_0\) and \([\eta]\) (if \(M\) is known; otherwise a third measurement is necessary, say, \(D_0\)) one can derive the dependence of \(s_0\) or \([\eta]\) on \(M\) for an entire series of polymer homologues and find the dimensions \(\langle r^2\rangle^{1/2}\) and \(\langle r_0^2\rangle^{1/2}\) corresponding to any degree of polymerization. At the same time the interaction parameter \(\alpha\) will also be found. Of course, one should not yet be in haste to use formula (4.5) for calculations, since it still requires further verification.

c) Investigation of Branching

The behavior of one and the same polymer in different solvents, or of different polymers in one and the same solvent, does not yet make it possible to draw a conclusion as to whether there are branches in the chains or not. The presence of branches, obviously, should pri-

lead to a decrease in the effective length of the segment \(b\). In fact, the root-mean-square distance between the ends of the chain, \(\langle r^2\rangle^{1/2}\), for a linear polymer of degree of polymerization \(Z\) will be distinctly larger than for its branched isomer (from purely geometrical considerations). Since for both isomers \(\langle r^2\rangle^{1/2}=bZ^{1/2}\), it is clear that \(b\) for the branched polymer is smaller than for the unbranched one. When working with a polymer of one composition in the same solvent, it is possible, from the change in the slope \(b_s\) (4,1) or \((1-\alpha)\) (4,2*), to establish the presence of a branched fraction. As far as we know, special studies of this kind have not been carried out up to the present.

If we again turn to the rubbers \(K\)-1 and \(K\)-3, which are in general rather similar in chemical composition (see Fig. 4, a, b), it becomes clear that in rubber \(K\)-3 the probability of branching is considerably greater than in \(K\)-1. This is also indicated by comparison of the virial coefficients of these rubbers (in octane). Zimm \(^{60}\) also showed that for a branched polymer the second virial coefficient \(B_2\) of the osmotic equation

\[ \frac{\pi}{RT}=\frac{c}{M}+B_2c^2+B_3c^3+\cdots \]

will be smaller than for its unbranched isomer. For \(K\)-1 in octane \(B_2=0.56\cdot10^{-3}\ \text{cm}^3/\text{g}\), while for \(K\)-3 it is \(0.18\cdot10^{-3}\ \text{cm}^3/\text{g}\). However, it still cannot be asserted that \(K\)-3 is necessarily a branched polymer, since it is not identical in composition to rubber \(K\)-1.

Another method for studying branching is based on the fact that the ability to orient is much more strongly expressed in unbranched molecules and, consequently, they offer greater resistance to rotation than do their branched isomers. At the same time, the difference in the coefficients of translational friction in such isomers, although present, is not so strongly expressed. Therefore the degree of branching can be calculated from the magnitude of the ratio of the coefficients of translational and rotational friction or, equivalently, of the coefficients of translational and rotational diffusion \(D_0\) and \(\theta^{8,61}\). In passing, it thereby becomes possible to calculate the sizes of the molecules.

§ 5. CONCENTRATION DEPENDENCE OF THE SEDIMENTATION CONSTANT

As was indicated, the only term in (II) that depends on concentration is the molar friction coefficient \(f\):

\[ f=f(M,c). \]

The change of \(f\) with concentration is composed of the direct interaction between macromolecules and of the disturbances introduced

or in the relative motion of the solvent. These effects depend on the size and number of macromolecules; therefore one may assume that

\[ f=f_0 \times \Gamma(M,c), \tag{5,1} \]

where \(\Gamma(M,c)\) is a perturbation function taking both effects into account. Expanding it in powers of \(c\) leads to the usual expression \(^{56,57}\)

\[ =f_0(1+k_{1s}c+k_{2s}c^2+\cdots), \tag{5,2} \]

where

\[ k_1=\left(\frac{\partial \Gamma}{\partial c}\right)_{c=0},\qquad k_{2s}=\frac{1}{2}\left(\frac{\partial^2\Gamma}{\partial c^2}\right)_{c=0}. \tag{5,2*} \]

For quite understandable reasons \(\Gamma(M,c)_{c=0}=1\). As Goldberg \(^{62}\) has shown, the expansion (5,2) breaks off at the quadratic term. From (5,2) there follows directly the extrapolation formula

\[ s=\frac{s_0}{1+k_1c+k_2c^2}, \tag{5,3} \]

which, at small \(c\), passes into the empirical relation of Gralén \(^{19}\):

\[ s=\frac{s_0}{1+k_sc}. \tag{5,3*} \]

From purely formal considerations it is easy to convince oneself that, in general form, the perturbation function must have the form

\[ \Gamma(M,c)=[1+\varphi(c)]M^{\psi(c)}, \tag{5,4} \]

where \(\varphi(c)\) and \(\psi(c)\) are power series in \(c\). This leads to the important relation

\[ s=\frac{s_0}{\Gamma(M,c)}=\frac{K_s}{[1+\varphi(c)]}M^{1-[a+\psi(c)]}=K_s(c)M^{1-a(c)}, \tag{5,5} \]

which has the same form as the equation for \(s_0\). Formula (5,5) plays a major role in the calculation of molecular-weight distributions. In Fig. 5, a, a group of straight lines \(\lg s(c)=f(\lg M)\), obtained by the author for one synthetic rubber, is presented, while in Fig. 5, b, there is a graph for \(K_s(c)\) and \(1-a(c)\). From (5,5) it follows at once that, when \(a+\psi(c)=1\), free sedimentation ceases and the solution begins to settle as a single whole. It is not difficult to show that (5,5) can also be written in a form corresponding to the Kuhn–Kirkwood theories:

\[ s(c)=A(c)+B(c)Z^{1/2}. \tag{5,5*} \]

Just as (5,5) indicates an (apparent) increase in permeability, (5,5*) indicates an (apparent) increase in the effective length of the segment.

Precise determination of the dependence of \(k_{1s}\) on \(M\) is not of great importance, in view of the existence of the more general relations (5.5) and (5.5*). Usually, by analogy with formulas (4.1) and (4.2), one assumes

\[ k_{1s}=a+bZ^{1/2}\quad \text{or}^{326}\quad k_{1s}=AM^p\ (p\leq 0.5). \tag{5.6} \]

In view of the smallness of the coefficients \(a\), \(b\), and \(A\), these formulas give practically coincident values over a very wide interval of \(M\).

Fig. 5. a) The function \(\lg s=f(\lg M)\) at different concentrations for butadiene SK\(^ {48}\). The numbers at the vertical lines denote the fraction numbers. b) Dependence of the parameters \(K_s\) and \(a\) on concentration (on the basis of the data of Fig. 5, a).

Attempts at a more rigorous derivation of the relation between \(s\) and \(s_0\) were contained in a large number of works\(^ {63}\), among which special mention should be made of the work of Bürgers\(^ {64,65}\), where, on the basis of the concept of the counterflow of the solvent in a direction opposite to the direction of sedimentation, an expression of the type (5.3*) was obtained. Lamm\(^ {66}\) characterizes the counterflow by a negative sedimentation constant \(s_1\) and shows that

\[ s_0=s+ \]

Refinement of the expression for \(s_1\) must obviously lead to formulas of the type (5.3).

We shall not dwell further on these questions, which have been discussed elsewhere, since they have no direct bearing on the subject of the present review.

§ 6. SOME THERMODYNAMIC PROBLEMS SOLVED BY MEANS OF THE ULTRACENTRIFUGE

a) True and apparent molecular weight

If the concentration dependence of the diffusion coefficient were also reducible to the form \(D = \dfrac{RT}{f} = \dfrac{RT}{f_0 \Gamma(M,c)}\), Svedberg’s formula (V) would be valid at any concentration. In fact, however, the driving force of diffusion in the direction \(x\) is the gradient of the chemical potential

\[ \nabla \mu = \frac{1}{c}\frac{d\pi}{dx}, \]

where \(\pi\) is the osmotic pressure; and since for a nonideal solution\(^{60,67}\)

\[ \frac{\pi}{c} = RT\left(\frac{1}{M} + B_2 c + B_3 c^2 + \cdots\right), \tag{6.1} \]

where \(B_2, B_3\) are virial coefficients, it is easy to show that

\[ D = \frac{RT}{f}\left(1 + 2B_2Mc + \cdots\right), \tag{6.2} \]

i.e.,

\[ D = D_0 \frac{1 + 2B_2Mc + \cdots}{\Gamma(M,c)}. \tag{6.3} \]

If the concentration is sufficiently small, in the expansion for \(\Gamma(M,c)\) one may confine oneself to the second term \((B_1sc)\), which gives approximately

\[ D = D_0(1 + k_D c). \tag{6.4} \]

This dependence was introduced empirically by Gralén\(^{19,30}\). Flory\(^{68}\) finds that it is more correct also to take into account the quadratic terms of the expansions in the numerator and denominator, which should lead to a curvature of the rectilinear dependence (6.4) (a small minimum near zero is possible). The dependence of \(D\) on concentration at very high dilutions was studied for a number of systems by V. N. Tsvetkov\(^{69}\). In any case, over a considerable range of concentrations formula (6.4) is approximately satisfied. We, however, shall be interested in another question.

Substitution of \(s\) and \(D\) into Svedberg’s formula (V) will lead to the expression

\[ \frac{s}{D}\frac{RT}{1-\bar V\rho} = \frac{s_0}{D_0}\frac{RT}{1-\bar V\rho}\, \frac{1}{1+2B_2Mc+\cdots}, \]

i.e.,

\[ M_c=\frac{M}{1+2B_2Mc+\cdots}. \tag{6,5} \]

Thus, application of Svedberg’s formula at finite concentrations leads to an “apparent molecular weight” \(M_c\). Obviously, instead of separately extrapolating \(s\) and \(D\) to \(s_0\) and \(D_0\), we may write:

\[ \frac{1}{M_c}=\frac{1}{M}+2B_2c+\cdots \tag{6,6} \]

The right-hand side of this equation coincides with Debye’s equation\(^{52}\) for light scattering

\[ \frac{Hc}{\tau}=\frac{1}{M}+2B_2c+\cdots, \tag{6,7} \]

where \(H\) is Debye’s optical constant, and \(\tau\) is the turbidity.

Equation (6,1), in principle, also differs in no way from (6,6) and (6,7), which is quite understandable, since the last two equations are derived from the first. Thus, by means of an ultracentrifuge the second virial coefficient of the osmotic equation can be measured, this coefficient being the main characteristic of the deviation of polymer solutions from ideality. In Fig. 6 are given plots of \(\dfrac{Hc}{\tau}=f(c)\) and \(M_c^{-1}=f(c)\) for four equal fractions of butadiene synthetic rubber\(^{70}\). The virial coefficient \(B_2\), generally speaking, depends on the molecular weight—but only over a very broad interval of values of \(M\) (from \(10^4\) to \(10^7\), \(B_2\) changes only by a factor of 5–10\(^{52,67}\)). In the interval from 140,000 to 500,000 (for the fractions in Fig. 6), this difference cannot be at all noticeable. We see that in fact

\[ \frac{Hc}{\tau}=f(c) \quad\text{and}\quad \frac{1}{M_c}=f_1(c) \]

give straight lines of identical slope

\[ 2B_2=1.12\times 10^{-3}\ \mathrm{cm^3/g}. \]

Of course, it is disadvantageous to carry out several experiments at different concentrations in order to find \(B_2\), when the same quantity can more simply be found from light-scattering or osmometry data. But we may proceed conversely: if \(B_2\) is known from an independent experiment, then for a series of fractions, instead of extrapolating \(s\) and \(D\) to infinite dilution, it is sufficient to find \(M_c\) at a fixed concentration and at once pass to the true weight.

according to formula^70

\[ M=\frac{M_c}{1-2B_2 cM_c}. \tag{6.8} \]

This procedure was first proposed by Schulz^71,72. S. E. Bresler and the author^70,48 successfully applied this formula to calculate the molecular weights in a mixture of 30 fractions of three synthetic rubbers of overall complexity.

Fig. 6. Comparison of the values \(M_c\), \(M_m\), and \(B_2\) obtained for butadiene rubber SK^48 by the light-scattering and ultracentrifuge methods. (Solid lines are experimental values of \(\dfrac{H_c}{\tau}\). \(\bullet\)—\(M_c\); \(\circ\)—\(M_m\) according to formula (6.8).)

The corrected values of \(M\) obtained from (6.8) agree well with direct osmotic measurements.

It should be pointed out that in the formula

\[ M_c=\frac{s}{D}\,\frac{RT}{1-V\rho} \tag{6.9} \]

the concentration at which \(D\) is measured is not quite definite, since diffusion proceeds from a solution into the pure solvent^25. According to Schulz^72, Gralen^19, and Jullander^30, the mean concentration in diffusion measurements is approximately equal to half the initial concentration. Therefore, in calculating \(M_c\), diffusion should be measured at twice as high a concentration as \(s\). V. N. Tsvet-

proposed a method that makes it possible to observe very small concentration gradients, thanks to which diffusion can be measured from a solution of concentration \(c+\Delta c\) into a solution of concentration \(c\), with \(\Delta c \ll c\). In this way one immediately obtains the correct value of the diffusion coefficient at the given concentration \(c\).

In practice, if the molecular weight and the second virial coefficient \(B_2\) are not very large, at small concentrations the hydrodynamic and thermodynamic coefficients \((1+2B_2Mc)\) and \(\Gamma(M,c)\) in (6.3) more or less balance each other, and the diffusion coefficient is almost independent of concentration \(^{72,73,74}\). Therefore measuring it at the same initial concentration as \(M\) and \(s\) usually does not lead to errors. The measurement itself of \(s\) and \(D\) at the same initial concentration is, however, very essential in calculating polydispersity (see Part II).

Lamm \(^{66}\), as already mentioned, assumes that

\[ s=s_0-|s_1|, \tag{6.10} \]

where \(s_1\) characterizes the sedimentation of the solvent in the opposite direction. In calculating \(|s_1|\), Lamm takes into account all thermodynamic factors, and the formula proposed by him,

\[ M=\frac{s+|s_1|}{D}\,\frac{RT}{1-V\rho}, \tag{6.11} \]

upon substitution of \(|s_1|\), proves to be in fact identical with (6.8).

b) Application of the equilibrium ultracentrifuge

An even greater basis for the study of the thermodynamic properties of polymer solutions can be provided by the equilibrium ultracentrifuge. As indicated in the introductory part, the condition of sedimentation equilibrium can be written in the form

\[ \frac{d\pi}{dx}=c\omega^2x(1-V\rho), \tag{6.12} \]

and equations (VIII) and (X) were obtained by substituting \(\pi=\dfrac{RTc}{M}\).

For real polymer solutions we must use the virial expansion (6.1). This again leads to the concept of apparent weight. Anticipating somewhat, let us point out that for polydisperse substances calculation by formula (VIII) (or an analogous one—see \(^{9}\)) gives the weight-average weight at the point \(x\), \(M_{wx}\). This weight does not coincide with the “complete” weight-average weight \(M_w\), since the concentration of the heavy fractions increases in the direction toward the bottom of the cuvette. \(M_w\) will be obtained by averaging the values of \(M_{wx}\) along

over the entire length of the liquid column, i.e.

\[ M_w=\frac{\int_a^b M_{wx} x c_x\,dx}{\int_a^b x c_x\,dx}, \tag{6,13} \]

where \(c_x\) is the polymer concentration at a distance \(x\) from the axis of rotation, and \(a\) and \(b\) are the coordinates of the meniscus and the bottom of the cuvette. As Wales showed\(^{31}\), if the second virial coefficient \(B_2\) is known,

\[ M_{wx}=\frac{\dfrac{dc}{dx}}{c_x\left(2Ax-2B_2\dfrac{dc}{dx}\right)};\quad \left(A=\frac{1-V\rho}{2RT}\omega^2\right). \tag{6,14} \]

This formula was derived for sufficiently dilute solutions, when higher virial coefficients may be neglected. Taking here \(B_2=0\), we obtain the apparent weight at a given concentration

\[ (M_{wx})_{c_x}=\frac{RT\dfrac{dc}{dx}}{c_x x\omega^2(1-V\rho)}. \tag{6,15} \]

If \(B_2\) does not depend appreciably on molecular weight, averaging over all coordinates gives the “total” apparent weight\(^{62}\)

\[ (M_w)_{c_0}=\frac{M_w}{1+2B_2M_w c_0} \tag{6,16} \]

(\(c_0\) is the initial concentration), i.e., analogously to (6,6) and (6,7)\(^{75}\),

\[ \frac{1}{(M_w)_{c_0}}=\frac{1}{M_w}+2B_2c_0. \tag{6,17} \]

Thus, by calculating the weight-average weight from formula (6,15) at several concentrations, it is possible, by extrapolating to infinite dilution, to obtain the true weight and the virial coefficients. It is useful to note that the calculation by formula (6,15) can be substantially simplified by using a Kegeles cuvette\(^{76}\), which makes it possible to obtain simultaneously the distributions

\[ c=f_1(x)\quad\text{and}\quad \frac{dc}{dx}=f_2(x). \]

The value of \(B_2\) was calculated in this way for fractions of polystyrene\(^{77}\) and practically coincided with the value obtained directly (osmometrically).

In another paper, \({}^{75}\) Wales gives a method for the direct calculation of the weight-average molecular weight \(M_w\) by means of an equilibrium ultracentrifuge. The calculation itself is rather cumbersome; however, by introducing a number of tabulated correction functions, it can be reduced to a minimum. For polystyrene fractions, Wales in this way obtained an excellent correlation between \(M_w\) and the osmotic weight, as well as \(B_2\) and

\[ \frac{\Delta\left(\frac{\pi}{c}\right)}{\Delta c}. \]

Of interest is the work of Wales \({}^{78}\), in which the sedimentation equilibrium of concentrated polymer solutions was investigated. The equation of sedimentation equilibrium can then be expressed in the form

\[ \frac{\frac{d\pi}{dc}-\frac{RT}{M_w}}{c_0} = RT\{2B_2+3B_3c_0+\ldots\}, \tag{6.18} \]

where \(B_2\) and \(B_3\) are virial coefficients. Excellent agreement with osmometric data was obtained for fractions of polystyrene and polyvinyl acetate.

Thus, with the aid of an equilibrium ultracentrifuge it is possible to measure the partial molar free energy of dilution of polymers \(\overline{\Delta F}\), equal to

\[ \overline{\Delta F}=-\pi \bar v, \tag{6.19} \]

where \(\bar v\) is the partial molar volume.

In 1945 Debye \({}^{79}\) showed that in mixed solvents there is usually a peculiar solvation effect, amounting to the fact that the polymer molecules are surrounded predominantly by molecules of one kind. If a binary system is a mixture of solvent and precipitant, the solution as a whole may be represented as an emulsion, in the mixed solvent, of a homogeneous polymer solution in a good solvent. This effect can be observed by the light-scattering method. Wales and Williams \({}^{80}\) showed how, for these same purposes, an equilibrium ultracentrifuge can be used; moreover, the parameters characterizing solvation can then be calculated by a more direct route.

§ 7. SOME NEW METHODS FOR CALCULATING MOLECULAR WEIGHT AND HYDRODYNAMIC PARAMETERS

Often, when working with relatively low-molecular substances, it is not possible to apply the usual methods for calculating the sedimentation constant \(s\). This is due to the fact that, because of the low rate of sedimentation and rapid diffusion, the boundary, even at high angular velocities, is not formed. Sedimentation equilibrium is established the more rapidly, the larger the diffusion coefficient.

However, even for such comparatively short polymers the time required to establish equilibrium will still be large (obviously, on the order of several days at \(M=10^3—10^4\). Even for simple salts of the NaCl type this time is approximately 10 hours\({}^{9}\). Therefore it seems desirable to calculate the molecular weight and hydrodynamic parameters from the distribution of the concentration of the substance in the radial direction before equilibrium has been established. For this purpose one would have to solve, in general form, the transport equation in an ultracentrifuge, which in cylindrical coordinates has the form:

\[ \frac{\partial c}{\partial t} = D\left( \frac{\partial^2 c}{\partial r^2} + \frac{1}{r}\cdot\frac{\partial c}{\partial r} \right) - \omega^2 s\left( r\frac{\partial c}{\partial r}+2c \right), \tag{7,1} \]

where \(r\) is the distance from the axis of rotation. The complete solution of this equation is connected with insurmountable difficulties. A number of exact partial solutions proposed by Archibald\({}^{81,82}\) are also practically limited because of their great complexity. Recently D. Ifantis and D. Vog\({}^{83}\), with the aid of a large differential analyzer, obtained numerical solutions of (7,1) for three model systems with \(M=1,\ 2\), and \(3\cdot 10^3\). The molecules were represented in the form of rigid spheres of density 1.5, bearing a definite volume of solvation water. From the existing theoretical formulae, \(s_0\) and \(D_0\) were calculated for such molecules, and the distributions \(c(x,t)\) were found for various \(t\) from 0 to 10 hours and for the established equilibrium. In the authors’ opinion, their data can be used for recalculation to other systems differing in particle shape, amount of solvated solvent, and molecular weight. The corresponding recalculation formulae are given in their work. However, the very approach to the problem raises objections. The following two are the most essential:

1) There is no basis for modeling low-molecular polymers, including polypeptides, by rigid spheres or ellipsoids. On the contrary, there are far more grounds for believing that they (with minor exceptions) have the configuration of a flexible thread or a highly permeable coil (cf. §§ 3 and 4). Therefore the proposed recalculation methods can hardly give a correct result.

2) The concept of solvation “immobilized water” in the form of a shell around the molecule likewise is neither proven nor the most probable (cf. work\({}^{59}\)) even for proteins. Rather, in the case of solvation there occurs a process resembling the “swelling” of a Flory coil and accompanied by a simultaneous change in the shape of the macromolecule. Therefore the method based on the numerical solution of equation (7,1) for simple models still requires refinement.

Archibald\({}^{84}\) proposed a number of ingenious approximate methods of calculation for non-established equilibrium. In his work two variants are given for calculating the molecular weight of homogeneous substances, as well as a method for calculating the weight-average molecular weight \(M_w\) for

polydisperse substances. Of great interest is a method for the simultaneous calculation of the three principal parameters, \(M\), \(s\), and \(D\), from the data of a single experiment. Unfortunately, these procedures have not been experimentally verified, with the exception of one case, when C. E. Bressler and co-workers\(^{85}\), under unestablished equilibrium in a large ultracentrifuge (at \(1000\) rev/sec), determined the average molecular weight of protein proteolysis products, which turned out to be of the order of \(1000\), in good agreement with other data.

Another method of calculation under unestablished equilibrium in a large ultracentrifuge was proposed by C. E. Bressler\(^{86,87}\). It can be shown that even at speeds of \(60000\) rev/min, substances with molecular weight below \(5 \cdot 10^4\) practically do not settle to the bottom of the cuvette, but form near it a diffuse layer, the distribution of the substance in which can be described by Boltzmann’s barometric formula

\[ c(x)=\operatorname{const} e^{\frac{v}{D}x}, \tag{7,2} \]

where \(v\) is the sedimentation velocity near the bottom of the cuvette, equal to \(s\omega^2x\). The distribution obeying formula (7,2) takes place in a very narrow layer, no more than \(2\) mm from the bottom. Since the thickness of the layer \(\Delta x\) is considerably smaller than \(x\)—the mean value of \(x\) in the diffusion layer—one may put \(v=s\omega^2\bar{x}\). Therefore the height of the sedimentation diagram \(Z\) at the point \(x_i\), close to \(\bar{x}\), will be

\[ Z_i=\operatorname{const} e^{\omega^2 s\bar{x}x_i/D}, \]

and at the point \(x_j\)

\[ Z_j=\operatorname{const}\cdot \exp\left(\frac{\omega^2 s\bar{x}}{D}x_j\right), \]

or, since

\[ \frac{s}{D}=M\frac{1-V\rho}{RT}, \]

\[ M=\frac{RT}{1-V\rho}\cdot \frac{\ln \dfrac{Z_i}{Z_j}}{\omega^2\bar{x}(x_i-x_j)} . \tag{7,3} \]

This formula follows directly from (X), if there one puts \(x_1 \simeq x_2\) and \(x_2^2-x_1^2 \simeq 2\bar{x}(x_2-x_1)\). Formula (7,3) was used by C. E. Bressler for calculating the molecular weight of protein decomposition products\(^{86}\). C. E. Bressler, K. S. Makarov, and the author\(^{87}\) used it for calculating the molecular weight of two plastins. In this work it also proved possible to apply Svedberg’s formula (V) directly. Taking into account the difference in the diffusion behavior of the plastins in the ultracentrifuge and with independent measurements of diffusion, the agreement obtained between the values of the molecular weights proved quite satisfactory.

Gutfreund and Ogston\(^{88}\) proposed a method for calculating the sedimentation constant in the absence of a boundary, by the complete transfer of material through some fixed cross-section \(X\) of the cuvette. It can be shown\(^{9}\) that in this case (under the condition \(\left(\dfrac{\partial c}{\partial x}\right)_{x=X}=0\))

\[ \ln\left[1-\frac{2Q'_t}{c_0 x_0^2}\right] = -2s\omega^2 t, \tag{7,4} \]

where

\[ Q'_t=\int_{x_0}^{X} dx \int_{x}^{X} \frac{\partial c}{\partial x}\, dx . \]

With the aid of this formula, Gutfreund and Ogston determined the sedimentation constants of insulin oxidation products with molecular weights of the order of 5000. The plastins proved to be convenient substances for a direct test of formula (7,3), since in the late stage of the experiment they formed a sedimenting boundary near the meniscus. The values of \(s\), calculated from formula (7,4) and from the usual formula

\[ s=\frac{1}{\omega^2}\,\frac{\ln \dfrac{x_2}{x_1}}{t_2-t_1}, \]

agreed within 15%, and the observed discrepancies were not random and may serve as a measure of polydispersity.

It is useful to note, in conclusion, that the plastins investigated proved to be extremely polydisperse substances with average weights from 5 to 7 thousand; they therefore have nothing in common with proteins, although they are also polypeptides.

Quite recently, new experimental techniques\(^{118,119}\) have been proposed which make it possible to create artificially a sharp boundary in the middle of the cuvette even for such low-molecular-weight substances as, for example, sucrose. The idea of these new methods consists in layering, under the action of the centrifugal field, a less dense solution (in the simplest case, a pure solvent) upon a denser one (a homogeneous solution of the low-molecular-weight substance). Such a technique has long been used in diffusion experiments\(^{26,19}\). However, it is impossible to create a boundary in advance in the ultracentrifuge cuvette, since during the manipulations preceding the start of the rotor mixing will inevitably occur. Kegeles\(^{118}\) proposed a special design of cuvette with two pairs of auxiliary reservoirs, located at the upper and lower edges of the sector-shaped space of the cuvette. The latter is filled before the experiment with the (dense) solution, while the reservoirs at the bottom are unfilled; the upper reservoirs are filled with the (less dense) solvent. The difference in densities may be very small. When the ultracentrifuge is started, the excess solution, under the action of the centrifugal force, passes through narrow channels into the lower reservoirs, while the solvent passes

from the upper reservoirs into the working space, layering itself upon the solution, with the formation of a very sharp boundary. A cuvette more convenient to handle was designed by Pickels, Schachman, and Harrington[^119]. Here, instead of a system of reservoirs, a little cup of capacity \(\sim 0.25\) ml was installed above the sectorial cavity; from it, at \(2\)—\(4\) thousand rpm, solvent entered the working space through a tiny orifice, layering itself upon the solution. Schachman and Harrington[^120] also used this cuvette to investigate a number of other problems, the most interesting of which are the study of counterflow of solvent, substantially affecting the sedimentation rate at finite concentrations, measurement of the “differential sedimentation constant” from a solution of concentration \(c+\Delta c\) into a solution of concentration \(c\) \((\Delta c \ll c)\), and also the concentration anomaly that arises during sedimentation of binary mixtures. The last question, which we shall encounter again in Part II, should be considered in somewhat greater detail. About 20 years ago Pedersen[^121] and Macfarlane[^122] established that, during sedimentation of a mixture of two proteins, an increase is observed in the concentration of the slow component at the expense of the fast one (this was not difficult to establish from the change in the areas of the corresponding portions of the sedimentation diagrams). Initially it was assumed that this effect was connected with some chemical reaction; however, much later Ogston and Johnston[^123] showed that this change in relative concentrations is rather of hydrodynamic origin and in any case is not associated with any chemical interaction of the components.

According to Ogston and Johnston, in the “homogeneous region,” i.e., between the bottom and the fast boundary, where both components sediment together, sedimentation of the slow component proceeds with a smaller value of \(s\) than in the region between the two boundaries, where the slow component sediments alone and, consequently, at a lower total concentration. The presence of two sedimentation rates is by no means as obvious as it may seem at first glance, since the condition of stationarity of the sedimentation process of the given component at the bottom and at the periphery is thereby violated (see § 9); however, if such a possibility is admitted, then, as Schachman and Harrington[^124] showed, the increase in the concentration of the slow component is obtained automatically when calculating its transport through the layer between the two boundaries. Whatever the origin of this effect may be, it is almost imperceptible at low concentrations and very strong at concentrations above 1%. In mixtures of viruses Schachman and Harrington observed an almost threefold increase in the concentration of the slow component.

PART II

STUDY OF MOLECULAR-WEIGHT DISTRIBUTIONS

§ 8. THE SIGNIFICANCE OF MOLECULAR-WEIGHT DISTRIBUTIONS

The mechanical properties of technically important polymers are determined to a considerable extent by the average molecular weight. However, the average molecular weight is a statistical quantity and cannot give a complete characterization of a polymer. In complicated distributions with several maxima, the concept of an average weight loses any real meaning altogether.

It is known, for example, that the properties of cellulose are determined to a considerable extent by the presence of a maximum in the molecular-weight distribution in the region of relatively low molecular weights (the so-called hemicelluloses), although the average molecular weight of a cellulose sample may be high. In rubbers, the presence in the molecular-weight distribution of fractions with low molecular weights is far from indifferent for the strength of the resulting vulcanizates. Therefore the problem of studying the complete molecular-weight distribution of a polymer is in many cases very important.

Knowledge of the molecular-weight distribution function makes it possible to shed light on the mechanism of the processes of polymer formation, i.e., on the mechanism of polymerization and polycondensation reactions. It is known, for example, that the formation of a polymer by a polymerization reaction is governed by three rate constants or probabilities: \(k_1\)—the probability of formation of an active center initiating chain growth, \(k_2\)—the probability of chain propagation by addition of a monomer unit to the end of the growing chain, and \(k_3\)—the probability of death of the radical located at the end of the growing chain. The gross yield of polymer per unit time is proportional to \(k_1L\), where \(L\) is the chain length of the macromolecule. On the other hand, the chain length is determined by the value \(\frac{k_2}{k_3}\).

It is known that the radical located at the end of the growing chain may cease to exist in various ways: either by collision with a similar end of another macromolecule, which leads to recombination or disproportionation; or by recombination with a free radical formed during decomposition of the polymerization initiator; or, finally, by collision with an initiator molecule or with some foreign impurity. To each of the processes of death of the polymerization center there corresponds its own constant \(k_3\) and its own average chain length. Hence it follows that the total distribution function of the polymer may contain several maxima, each of which reflects a definite mechanism of death of the polymerization centers. This opens up

possibility of a detailed study of the mechanism of polymerization by determining the molecular-weight distribution of the reaction product.

For the study of the molecular-weight distributions of polymers, for many years the method of fractional precipitation was used; it consists in adding a precipitant in portions to a dilute polymer solution, as a result of which the polymer separates from the solution in the form of fractions. Theory and experiment show that in this process the most high-molecular fractions precipitate first, and the lowest-molecular fractions last.

By measuring the average molecular weights of the precipitated fractions, and also knowing their percentage yields, one can construct the so-called stepwise precipitation curve^48. In constructing this curve one has to introduce the arbitrary assumption that each precipitated fraction contains macromolecules of a definite molecular weight.

It is clear that this cannot be true; the very question of the homogeneity of the fractions long remained open. The opinion was even expressed that the distribution within each fraction is identical with the distribution in the original polymer. Therefore the question of the degree of reliability of distribution functions obtained by graphical differentiation of the integral precipitation curves \(c=f(M)\) also remained unresolved.

Only the application of the ultracentrifuge made it possible to answer these questions unambiguously and provided an absolute method for investigating polydispersity.

In the following paragraphs a brief substantiation is given of the corresponding methods of calculation and of the principal results of the investigations.

§ 9. APPLICATION OF EQUILIBRIUM ULTRACENTRIFUGATION. METHOD OF MODEL FUNCTIONS

Direct calculation from sedimentation-equilibrium diagrams can give the weight-average or \(Z\)-average molecular weight. Applying the method of mathematical induction, Wales showed^31 that any weight \(M_{qx}\) (i.e. the \(q\)-average weight at a distance \(x\) from the axis of rotation; this quantity has the same meaning as \(M_{wx}\)—cf. § 6) is related to the preceding weight \(M_{q-1,x}\) by the relation

\[ M_{qx}=M_{(q-1)x}+M_{1x}\frac{d\ln M_{(q-1)x}}{d\ln c_x}\qquad (M_{1x}\equiv M_{wx}), \tag{9,1} \]

and the total \(q\)-average weight is obtained by averaging over the whole

length of the cell:

$$ M_q=\frac{\displaystyle \int_a^b \prod_{p=1}^{q} M_{px}\,x c_x\,dx}{\displaystyle \int_a^b \prod_{p=1}^{q-1} M_{px}\,x c_x\,dx}. \tag{9.2} $$

Practically all calculations are carried out by graphical integration over intervals of \(0.5\) mm. The procedure is noticeably simplified when working with Kegeles’s cell\({}^{76}\), which makes it possible simultaneously to obtain the distributions \(c(x)\) and \(Z(x)\sim \dfrac{dc}{dx}\). The thermodynamic correction and extrapolation to infinite dilution are taken into account in the usual way.

Thus, in principle, on an equilibrium ultracentrifuge one can determine any \(q\)-average weight, or all the moments of the distribution \(q_w(M)\). The problem of reconstructing a distribution function from its moments is rather difficult. In general form it was solved in the works of Stieltjes\({}^{89}\), Herglotz\({}^{90}\), and Wales\({}^{31,91,77,78}\). The last of these formulates a theorem stating that knowledge of four consecutive \(q\)-weights is sufficient to reproduce the function \(q_w(M)\) as a whole. From the mathematical point of view the proposed calculation techniques raise no objections, but in practice they prove to be very limited and inaccurate. It is not hard to see that the sedimentation-equilibrium diagram gives the distribution function in the form of a Laplace transform:

$$ \frac{dc}{dx}=\lambda(x,M)=\int_0^\infty e^{\frac{M(1-\nu \rho)}{RT}\omega^2 x}\,q_w(M)\,dM, \tag{9.3} $$

and in the final analysis all calculation methods reduce to inversion of this transform. It is so nonlinear, and comparatively small measurement errors may, in the inversion process, grow to such an extent that a large part of the details of the distribution is blurred, and only its general features are preserved: the upper and lower limits of \(M\), the modal value \(M_m\), and the (directly calculated) mean values \(M_w\), \(M_Z\), etc. The problem is still further complicated when several maxima are present. It should be added that the errors increase almost in a geometric progression in going from lower to higher \(q\)-average weights. Wales considers it practically already impossible to compute a weight of order higher than 4 (i.e., \(M_{Z+2}\)).

A more accurate result can be obtained if the distribution function is known in general form beforehand: in this case

by the dispersion of the distribution

\[ \mu = M_z M_w - M_w^2, \tag{9,4} \]

or the relation of the \(q\)-weights it is possible to calculate the parameters determining the width and asymmetry of the function \(q_w(M)\). It is easy to see that this method too is applicable only to unimodal functions and in itself imposes a restriction, reducing the distribution in advance to a function of one class. However, for certain specific mechanisms of polymerization or polycondensation the character of the distribution function can be predicted. In general form such a model (parametric) function can be represented by the equation \(q_w(M)=f(M,\beta)\), where \(\beta\) is a parameter directly related to the moments of the distribution. Since any real distribution function must tend to zero as \(M \to 0\) and \(M \to \infty\), it is clear that the model function must contain a pre-exponential factor \(M^a\) (\(a\) is zero or any positive number) and an exponential term of the type \(\exp[-(bM^c)]\), where \(b\) and \(c\) are also positive numbers. Proceeding from these considerations and guided by the purely formal convenience of calculation, Kraemer and Lansing\(^9\) proposed in 1935 a model function of the form

\[ q_w=\frac{w}{M_n \beta \sqrt{\pi}}e^{-y^2}, \qquad y=\frac{1}{\beta}\ln\frac{M}{M_m}, \tag{9,5} \]

where \(w\) is the total weight of the polymer. The parameter \(\beta\) can readily be calculated from combinations \(M_w\) and \(M_z\) or \(M_n\) and \(M_w\). This method was applied in a series of works by Kraemer and co-workers,\(^9\) and also by Mosimann,\(^{92}\) concerned mainly with the study of cellulose derivatives. The Kraemer–Lansing distribution is neither necessary nor the most probable; moreover, it requires satisfaction of the relation \(M_q : M_{q-1} = \mathrm{const}\), which has never been observed experimentally. Therefore (9,5) is suitable only for a qualitative estimate of polydispersity. More justified is the Schulz distribution:\(^{93}\)

\[ q_w(M)= \frac{\left(\dfrac{\beta+2}{w}\right)^{\beta+2}} {\Gamma(\beta+2)} M^{\beta+1}\exp\left[-(\beta+2)\frac{M}{M_w}\right], \tag{9,6} \]

which should in fact be expected in some polymerizations. For this distribution the characteristic relation is

\[ M_q : M_{q-1}=\frac{q+\beta+1}{q+\beta}, \quad \text{i.e. } M_q-M_{q-1}=\mathrm{const}, \tag{9,7} \]

whence \(\beta\) is calculated directly. Relations of this kind have been obtained experimentally, most often with \(\beta=0\). This corresponds to the broadest, so-called “equilibrium” distribution (Flory),\(^{94,100}\) which is obtained if, at some stage of polycondensation or polymerization, there is established-

is established between the degradation and synthesis of chains (for more details see § 11). In this case

\[ M_n : M_w : M_z : M_{z+1} = 1 : 2 : 3 : 4 \tag{9.7a} \]

(we recall that in the system \(q_w(M)\), \(M_n \equiv M_0,\ M_w \equiv M_1,\ldots\)).

The introduction of new model functions\({}^{77}\), or the application of general computational procedures if relation (9.7) is not satisfied, does not lead to substantially different results. In Williams’ laboratory, separate samples of polystyrene, polyisobutylene, cellulose acetate, etc. were investigated in this way.\({}^{77,95}\) In all cases similar unimodal distributions were obtained, with a strongly pronounced positive asymmetry, close to the “equilibrium” one. Examples of such distributions are given in Fig. 7, \(a\)—\(e\). Such uniformity of the functions \(q_w(M)\) for entirely different polymers cannot reflect reality. Other works (see below) clearly show the presence of a number of distribution maxima in polystyrene (independently of the method of preparation) or in cellulose derivatives. But these important details were blurred in the process of Laplace transformation (9.3). It is also characteristic that in a number of regions in Fig. 7, \(b\), \(q_w(M) < 0\). This is completely devoid of physical meaning and is connected with general shortcomings of the method.

The method of \(q\)-average weights is more justified in the qualitative investigation of processes connected with aggregation (or crosslinking) and disaggregation (or degradation) of polymer chains in solutions under the action of various agents. The initial ratio of \(q\)-weights changes in such processes. Thus, Wales and co-workers showed that under the action of oxidizing agents in polystyrene solutions an “equilibrium” distribution is gradually established.\({}^{96}\) In an analogous way, association in cellulose acetate solutions was investigated.\({}^{97}\)

On the whole, as can be seen, the limits of applicability of the equilibrium ultracentrifuge are fundamentally restricted, and the accuracy of the distributions obtained is low. In the next paragraph we shall show that the sedimentation-velocity method can give much more, since the experimentally measured concentration gradient

\[ \frac{dc}{dx} = f(M,x) \]

is in this case a comparatively simple linear transformation of the function \(q_w(M)\).

§ 10. THE ULTRACENTRIFUGE AS A MASS SPECTROMETER

Summarizing the content of the preceding paragraph, one may say that from the data of the equilibrium ultracentrifuge moments of the molecular-weight distribution of various orders are calculated, from which the distribution itself is then reconstructed. When working with a large ultracentrifuge, it is possible directly

Fig. 7. Distributions obtained with an equilibrium ultracentrifuge1.
a) Polyisobutylene. Viscometric weight \(M_{\eta}=412\,000\); \(M_w=474\,000\); \(M_z=670\,000\); \(M_{z+1}=800\,000\).
b) Polystyrene fraction. \(M_{\eta}=90\,000\); \(M_w=130\,000\); \(M_z=145\,000\); \(M_{z+1}=200\,000\).
c) Sample of electrodialyzed acetylcellulose. \(M_n=32\,000\); \(M_{\eta}=54\,100\); \(M_w=55\,000\); \(M_z=91\,000\); \(M_{z+1}=150\,000\). Solid and dashed lines correspond to the choice of different parameters in reproducing the distributions.

directly obtain the molecular-weight distribution itself, \(q_w(M)\). A number of computational methods are possible in this case; each of them we shall briefly consider below. The general theory of the spreading boundary is given here in the compact form proposed by C. E. Breeseler and the author\(^70\). Some calculation methods (under a number of simplifying assumptions) are presented in the review by G. Schuynell and Ronby\(^98\).

As was indicated in the introductory part, a sedimentation diagram is a graphical representation of the differential distribution of the concentration \(\dfrac{dc}{dx}\) near the sedimenting boundary (Fig. 8, a). With the appropriate normalization the function \(\dfrac{dc}{dx}=f_\xi'(x)\) can be reduced to \(\dfrac{dw}{dx}=q_w(x)\)—the weight distribution of the polymer in the radial direction \(x\). The normalizing

Fig. 8. Sedimentation diagram in coordinates:
\(x,\ \dfrac{dc}{dx}\) (a) and \(\xi,\ \dfrac{dc}{d\xi}\) (b).

Fig. 9. Spreading of the sedimenting boundary; \(y\)—comparator readings by which the values of \(x\) and \(z\) are found.

factor for passing from the experimental coordinates \(Z, x\) to the coordinates \(\dfrac{dw}{dx}, x\) is usually a constant number.

As sedimentation proceeds, the boundary not only shifts but also spreads, as is seen from Fig. 9, which shows sedimentation diagrams of a narrow fraction of synthetic rubber, taken at different moments of time.

This spreading of the boundary is caused by two factors—diffusion and polymolecularity. Assuming that these factors are independent, we may consider them separately.

Let us first suppose that the polymer is characterized by a continuous distribution function with respect to sedimentation constants

\[ q_w(s)=\frac{dw}{ds}, \tag{10,1} \]

and that diffusion is so small that it may be neglected.

Let us denote by \(s_m\) the sedimentation constant corresponding to the maximum of the sedimentation diagram \(x_m\). For a symmetric distribution, \(s_m=s_w\), the weight-average sedimentation constant. According to the basic formula of the sedimentation-velocity method, in this case

\[ x_m=x_0 e^{\omega^2 t s_m}, \tag{10,2} \]

where \(x_0\) is the coordinate of the meniscus, \(t\) is the time from the start of the experiment, and \(\omega\) is the angular velocity of the ultracentrifuge rotor.

Molecules characterized by another sedimentation constant,

\[ s=s_m\pm\Delta s, \]

at the same moment will be at a distance \(x=x_m\pm \xi\) from the axis of rotation (Fig. 8, \(a\)):

\[ x=x_0 e^{\omega^2 t s}. \tag{10,3} \]

Thus the displacement \(\xi\), characterizing the broadening of the boundary, is caused by the presence of the distribution \(q_w(s)\), i.e., it is a consequence of polydispersity. The absolute magnitude of the displacement is

\[ \xi=x_m\left[e^{\omega^2 t\Delta s}-1\right]. \tag{10,4} \]

Expanding the expression in brackets in a series and making sure that one may restrict oneself to the first term (for usually \(\omega^2\leq 4\times 10^7\), \(t=10^3\text{--}10^4\), and \(\Delta s\sim 10^{-13}\)), we obtain:

\[ =x_m\omega^2 t\Delta s, \tag{10,5} \]

Taking \(x_m\) as the new origin of coordinates (Fig. 8, \(b\)), we find that the sedimentation diagram, i.e., the distribution with respect to displacements,

\[ \frac{dc}{d\xi}=q_w(\xi_s), \]

is completely equivalent to the original distribution (10,1). In fact,

\[ \frac{dc}{d\xi}=\frac{(x_m\omega^2 t)^{-1}dc}{ds}, \tag{10,6} \]

and the function \(\dfrac{dc}{ds}=f(s)\) coincides, up to a constant factor, with

\[ q_w(s)=\frac{dw}{ds}. \]

Thus, the distribution function with respect to sedimentation constants—provided that diffusion may be neglected—can be obtained from the sedimentation diagram by a simple change of coordinates. The transition from the abscissa \(x\) to \(s\) is made according to the obvious formula\(^ {98}\)

\[ s=s_m\frac{\ln\dfrac{x}{x_0}}{\ln\dfrac{x_m}{x_0}}, \tag{10,7} \]

which follows from (10,2) and (10,3).

The width of the distribution \(q_w(s)\) is characterized by the standard deviation \(\sigma\).

The distribution \(\dfrac{dc}{d\xi}\), obviously, can also be characterized by the standard deviation \(\delta\), which is found from the condition (in the general case, instead of the modal values \(x_m\) and \(s_m\) corresponding to the maximum of the distribution, one should use the weight-average values, \(s_w\) and \(x_w\)):

\[ \delta=\frac{1}{2}x_m\left(e^{\omega^2t\sigma}-e^{-\omega^2t\sigma}\right) =x_m\,\operatorname{sh}(\omega^2t\sigma)= \]

\[ =x_m\left[\omega^2t\sigma+\frac{(\omega^2t\sigma)^3}{3!} +\frac{(\omega^2t\sigma)^5}{5!}+\cdots\right]. \tag{10,8} \]

This series converges rapidly, and for sufficiently narrow fractions one may use the approximate formula

\[ \delta=x_m\omega^2t\sigma. \tag{10,9} \]

All these considerations are also valid in the case of distributions with several maxima, which may be regarded as a sum of simple distributions.

Let us now introduce diffusion into consideration. Obviously, a group of molecules with a definite value \(s=s_i\) also has a quite definite value of the diffusion coefficient \(D=D_i\), since ultimately both \(s\) and \(D\) are single-valued (for homologous series) functions of the molecular weight \(M\). In this case the distribution \(q_w(\xi_s)\) is overlaid by distributions in displacements due to diffusion \(q_w(\xi_D)_i\) (Fig. 10). The dotted line in the figure represents the distribution \(q_w(\xi_s)\), while the solid lines represent the distributions \(q_w(\xi_D)_i\). The resultant distribution proves to be broader.

Fig. 10. Schematic representation of broadening due to diffusion (solid lines), superposed on the distribution due to polydispersity (dotted line).

Assuming that the distributions in displacements due to polydispersity and diffusion are independent, we can write for the dispersion of the composite distribution

\[ M_2^0=(m_2^0)_s+\sum_{i=1}^{n}(m_2^0)_{D_i} \tag{10,10} \]

(the theorem of probability theory on the additivity of second moments\(^ {99}\)). The terms of the sum in this formula are

\[ (m_2^0)_{D_i}=w_i2D_it, \tag{10,11} \]

where \(w_i\) is the weight fraction of molecules with diffusion coefficient \(D_i\), and \(2D_i t\) is the variance of the distribution in displacements due to diffusion over time \(t\). It is easy to see that

\[ \sum_{i=1}^{n} w_i D_i = D_w \]

is the weight-average diffusion coefficient, which is usually obtained in independent measurements of \(D\), for example by Lamm’s method\({}^{26}\). Substituting now into (10,10) the value \((m_2^0)_s\), we finally obtain

\[ M_2^0 = (x_m^0 \omega^2 t \delta)^2 + 2D_w t. \tag{10,12} \]

It is important to note that in this derivation the only assumption is the assumption of independence of the displacements due to sedimentation and diffusion. Experiment confirms this assumption\({}^{48,101}\).

One can now construct, for each of the fractions under investigation, the plot

\[ \frac{M_2^0}{2t} = \Phi(x_m^2 t) = D_w + \frac{\omega^4 \sigma^2}{2}\,(x_m^2 t). \tag{10,13} \]

Here by the quantity \(M_2^0\) one should understand the second moment of the experimental curve \(\dfrac{dc}{d\xi}\). Since usually the sedimentation diagram is not normalized to unity, one should simply divide its second moment by the area and substitute it in (10,13) instead of \(M_2^0\). The points on the graph must lie on a single straight line cutting off on the ordinate axis a segment equal to \(D_w\). The tangent of the angle of inclination of this straight line is equal to \(\dfrac{\omega^4\sigma^2}{2}\). In this way it is possible to separate broadenings caused by polymolecularity and by diffusion.

Until recently, on the basis of the work of Gralen\({}^{19,98}\), it was assumed that, for a value of the diffusion coefficient \(D \ll 10^{-6}\,\dfrac{\mathrm{cm}^2}{\mathrm{sec}}\), the broadening of the diagram due to diffusion may be neglected. Formula (10,12) makes it possible to derive a strict criterion for deciding this question. Let us require that the standard deviation of the total distribution \(q_w(\xi_s,D)\) exceed the standard deviation of the distribution in displacements due to polydispersity \(q_w(\xi_s)\) by no more than 10%. Again denoting \(\omega^2 x_m \sigma t\) by \(\delta\), we obtain:

\[ 2Dt = \frac{1-0.9^2}{0.9^2}\,\delta^2, \]

whence \(2Dt \simeq 0.25\delta^2\).

Thus, if the diffusion coefficient is determined independently, and if it turns out that \(2D_w t \leq 0.25 M_2^0\), further processing of the sedimentation diagram may be carried out as though the broadening of the boundary were caused only by polydispersity. In most practically important cases, in the study of industrial polymers, the diffusional broadening of the sedimenting boundary cannot be neglected.

The method of separating the broadenings due to diffusion and to polydispersity was proposed in independent works by Baldwin and Williams \(^{102,103,104}\) and by S. E. Bresler with co-workers \(^{48,70,105}\). The theory set forth makes it possible, from the variance of the sedimentation diagram, to obtain the variance of the distribution \(q_w(s)\), and in the best case—when diffusion may be neglected—to pass directly from the sedimentation diagram to the distribution with respect to the sedimentation constants.

In doing so, however, one has to take account of concentration factors. They manifest themselves in two ways.

One should begin with the fact that, because of the sector-shaped form of the cell, during sedimentation there occurs the so-called centrifugal dilution. In the sedimentation of a homogeneous substance, the concentration at time \(t\) is equal to

\[ c(t)=c_0 e^{-2\omega^2 s t}=c_0\left(\frac{x_0}{x_m}\right)^2; \tag{10,14} \]

where \(c_0\) is the initial concentration. In the sedimentation of a polydisperse substance, the broadened boundary should be regarded as the superposition of an infinitely large number of infinitely narrow boundaries corresponding to the individual components. The concentration \(c_i\) of each such component at time \(t\) will be

\[ c_{i,t}=c_{i,0}e^{-2\omega^2 s_i t}=c_{i,0}\left(\frac{x_0}{x_i}\right)^2, \]

i.e. the initial distribution is distorted: the fast components are diluted more rapidly than the slow ones, and the mixture is enriched in slow components. This effect, however, is very small, and it is easy to take into account; usually there is even no need for this, since with a reasonable choice of the time \(t\) the slowly varying factor \(\left(\frac{x_0}{x_i}\right)^2\) is suppressed by the rapidly varying distribution function, so that it is sufficient to take into account only the overall dilution, i.e. the factor \(\left(\frac{x_0}{x_m}\right)^2\). \(^{63,98,106}\)

Very great importance is usually attached to another circumstance. The concentration along the boundary changes continuously from zero to the initial value, so that at some point \(x\)

\[ c(x)=\int_{x_0}^{x}\frac{dc}{dx}\,dx < \int_{x_0}^{x_2<x_b}\frac{dc}{dx}\,dx = c_0\left(\frac{x_0}{x_m}\right)^2, \]

where \(x_b\) corresponds to the bottom of the cuvette, and the right-hand integral is the concentration in the homogeneous region (cf. § 7), beginning at some point \(x_2 < x_b\). Therefore the lighter molecules, which lag behind in the process of sedimentation, turn out to be in a region of lower concentration; those macromolecules which sediment faster (the heavier ones) turn out to be in a region where the concentration is close to the initial one. At present there are three points of view concerning the effects caused by this circumstance.

The Swedish authors believe that, because of the variable concentration, a strong narrowing of the boundary occurs, since the light components sediment with a “lead,” i.e., faster than they would sediment at the same concentration as the heavy ones. Baldwin \(^{125}\) and Williams \(^{126}\) believe that, besides this, the McFarlane effect (§ 7) plays a role, owing to which the function describing the boundary

\[ \frac{dc}{dx}=q_w(x) \]

differs somewhat from the distribution function in the homogeneous region, where all components are represented simultaneously. Baldwin \(^{125}\) assumes that the concentration of the light components on the sedimentation diagram, in connection with the anomaly mentioned earlier, is higher than in the homogeneous region.

However, the large amount of experimental material now available shows that, in the range of concentrations suitable for using the ultracentrifuge as a mass spectrometer (i.e., at \(c_0 < 3\) mg/ml), the role of the indicated effects is much smaller than one might expect at first glance, and in some cases they cannot be detected at all. S. E. Bresler and the author believe that the determining role is played by the integral concentration effect, i.e., the concentration in the homogeneous region. As will be shown somewhat below, the resolving power of the ultracentrifuge, i.e., the ratio between the minimal displacement \(\delta x\) and the minimal difference of molecular weights, is proportional to the quantity \(1-a\), where \(a\) is the exponent in the formula \(s=K_s M^{1-a}\). Since (§ 5) \(a\) increases with concentration, the displacements corresponding to identical \(\delta M=M_m-M\) become smaller the greater the concentration, and the boundary narrows. The variable concentration along the boundary should deepen this effect, but apparently there exist certain unaccounted-for circumstances which reduce this additional narrowing to a minimum. In view of the obscurity of this question, we shall not dwell on it in detail here.

It would seem that the concentration effects at an arbitrarily chosen point \(x\) of the boundary can be eliminated by extrapolating the corresponding value of \(s_x\) to infinite dilution, say,

by the formula \(\dfrac{1}{s_0}=\dfrac{1}{s}-Kc_x\), where \(c_x\) is the local concentration, i.e. \(\displaystyle\int_{x_0}^{x}\frac{\partial c}{\partial x}\,dx\). Attempts of this kind have been made and led to results sharply at variance with the results of direct extrapolation (see the next paragraph, “Graphical Fractionation”). As a rule, the concentration dependence of the distribution variance turns out to be several times smaller than might have been expected from extrapolation with respect to local concentrations. Therefore S. E. Bresler and the author believe that the additional narrowing of the boundary due to variable concentration along it may be neglected. As we shall see in the next paragraph, this opinion is confirmed by experiment. From this point of view, the sedimentation of a polydisperse substance can be described visually—though not quite rigorously—as the flow of solutions of individual components through a “porous medium” formed by the remaining components in the homogeneous region (the concept of a “porous medium” was first introduced by Ogston and Fessler\(^{58}\) for monodisperse solutions).

The position of the individual infinitely narrow boundaries of which the distribution function \(q_w(x)\) is composed is thereby uniquely determined by the density of this porous medium, i.e. by the concentration in the homogeneous region.

It follows from this that concentration effects can be taken into account directly in the transition to the distribution with respect to molecular weights. It is most convenient to use here the dependence \(s(M,c)=K_sM^{1-\alpha}\), where \(K_s\) and \(\alpha\), as was indicated, are single-valued functions of the initial concentration, found from a plot of \(\lg s=f(\lg M)\) of the type shown in Fig. 5.

In the general case

\[ q_w(M)=q_w(s)\frac{ds}{dM}=q_w(s)\frac{s}{M}(1-\alpha). \tag{10,15} \]

In the case of a narrow fraction, the distributions \(q_w(M)\) and \(q_w(s)\) can be characterized by the standard deviations \(\mu\) and \(\sigma\), the relation between which is found from the formula

\[ \sigma=\frac{1}{2}K_s\left[(M_w+\mu)^{1-\alpha}-(M_w-\mu)^{1-\alpha}\right], \tag{10,16} \]

where \(M_w\) is the weight-average weight. After simple transformations we find, for \(\mu<\dfrac{1}{2}M_w\),

\[ \sigma\simeq\frac{s_w}{M_w}(1-\alpha)\mu, \tag{10,17} \]

where this relation must be valid both at infinite dilution and at finite concentration, in accordance with what was said above. Thus, it should be assumed that the quantity

\[ \mu'=\frac{\sigma(c)}{s_w(c)}[1-a(c)]M_w \tag{10,18} \]

does not depend on concentration and is equal to the true standard deviation \(\mu\). Consequently, \(\mu\) can be calculated from the data of a single experiment. The weak dependence of this quantity on \(c\), indicated earlier\(^{48}\), as has now been established, was caused by certain inaccuracies of calculation.

In conclusion let us turn our attention to formula (10.19). By analogy with a spectral instrument, we may introduce the concept of the resolving power of the ultracentrifuge

\[ L=\frac{M}{\delta M}, \tag{10,19} \]

where \(\delta M\) is the minimum difference of molecular weights recorded by the instrument. Replacing in (10.19) \(\mu'\) by \(\delta M\) and carrying out simple transformations, we obtain:

\[ L=(1-a)\frac{\xi+\bar{x}\omega^2 t s_m}{\delta \xi}. \tag{10,20} \]

Thus, the resolving power of the ultracentrifuge depends not only on the parameters of the instrument (the effective radius \(\bar{x}\) and the maximum speed \(\omega\)), but also on the properties of the molecules being studied: the looser the structure of the coils (i.e., the larger the value \(a\)), the worse the ultracentrifuge resolves the spectrum of molecular weights. Since \(a(c)\simeq a+\beta c\) (§ 5), it is easy to see that with increasing concentration the resolving power also decreases: at a certain limiting concentration the ultracentrifuge ceases altogether to respond to a change in molecular weight.

§ 11. METHODS FOR CALCULATING SEDIMENTATION DIAGRAMS AND THE PRINCIPAL RESULTS OF MEASUREMENTS

From the preceding section there follow two possibilities for the final processing of experimental data. The first of them consists in a direct transition from the sedimentation diagram of an unfractionated sample to the distribution \(q_w(s)\) and then \(q_w(M)\). This method appears to be the most natural; however, until recently it was considered less accurate than the so-called method of summing fractions, in which the distributions \(q_w(s)\) or \(q_w(M)\) for individual frac-

tions, and the complete distribution is obtained as a result of graphical summation of the partial distributions:

\[ Q_w(s)=\sum_{i=1}^{n}[q_w(s)]_i \quad \text{and} \quad Q_w(M)=\sum_{i=1}^{n}[q_w(M)]_i, \tag{11,1} \]

or

\[ Q_w(M)=Q_w(s)(1-a)\frac{s}{M}. \tag{11,2} \]

Here \(n\) is the number of fractions, and the distributions \([q_w(s)]_i\) and \([q_w(M)]_i\) for the individual fractions are normalized in such a way that

\[ \int_{0}^{\infty}[q_w(s)]_i\,ds = \int_{0}^{\infty}[q_w(M)]_i\,dM = w_i, \tag{11,3} \]

where \(w_i\) is the weight fraction of the given fraction.

The principal advantages of the fraction-summation method include the following. First, the distribution \(q_w(s)\) within a fraction always has a single maximum and is more or less symmetrical. In this case, by introducing justified model concepts of the distribution, diffusion can easily be excluded by means of the statistical method described above. Second, the dependence \(s(M)\), necessary for the final calculations, is obtained automatically. It is important to note that in this way it is possible at once to check whether the polymer consists of one or several homologous components differing in composition or structure. Finally, the fraction-summation method makes it possible to obtain—as a by-product—information on the structure and dimensions of macromolecules by means of one of the procedures described in Part I.

The disadvantages of the method include its considerable laboriousness, as well as the unaccounted-for possibility of losses of peripheral components in the process of “sharpening” the fractions by refractionation.

Direct recalculation makes it possible to obtain the complete distribution \(Q_w(s)\) much more rapidly. However, without fractionation, here too it is impossible to pass to the distribution \(Q_w(M)\). In working by both methods, the Swedish authors often, in order to save time, restricted themselves to the distribution \(Q_w(s)\) for characterizing the polymer, which can lead to errors, since components of different composition and structure may have one and the same sedimentation constant. A corresponding example will be given below (Fig. 17). However, in the case where the recalculation function \(s(M)\) is known, for example, in systematic studies of the mechanisms of polymerization or polycondensation, when changes in the reaction conditions alter only the distribution in molecular weights and not the chemical composition or branching, the advantages of the method

of direct recalculation are beyond dispute. It is inapplicable, however, for the study of mixed polymers and copolymers containing components with different relative contents of monomers.

In a number of studies of the fractional composition, the authors limited themselves to determining the average parameters of the fractions, \(s_0\), \(D_0\), and \(M\). Of course, in this case the capabilities of the ultracentrifuge as a mass spectrometer are not used\(^{72, 74, 109, 92}\).

a) Methods of the direct reproduction of the distribution function

To recalculate a sedimentation diagram into a molecular-weight distribution it is necessary to solve two problems connected with the elimination of diffusion and concentration effects. The first problem is usually solved very simply, since most often the Gralen criterion is applicable to unfractionated polymers. Indeed, we have seen that the second moment of the distribution \(q_w(s)\) is equal to \(\omega^4 x_m^2 t^2 \sigma^2 + 2D_w t\). As a rule, in an unfractionated polymer the dispersion \(\sigma^2\) is at least an order of magnitude greater than for fractions, while \(D_w\) has practically the same order as for average fractions, owing to which the term \(2D_w t\) may be neglected in comparison with \(\omega^4 x_m^2 t^2 \sigma^2\). In practical calculations, however, this circumstance still requires direct verification. For this purpose, first of all it is necessary to find the “zero” time, since as a result of the convection disturbances that occur during acceleration of the rotor, the true time of boundary formation never coincides with the moment at which full speed is reached. By extrapolating the straight lines \(\ln x=\psi(t)\) to their intersection with the line \(\psi=\ln x_0\), parallel to the axis of abscissae, one can find the true origin of the coordinates from which time should be reckoned. It is precisely this true, or “zero,” time that enters into all the formulas of this part.

Next one proceeds as follows. Suppose that, as a result of the experiment, a sedimentation diagram of irregular form has been obtained, broadening and deforming with time. By means of formula (10.7), or the still more convenient

\[ s=\ln \frac{x}{x_0}\big/\omega^2 t \tag{11,4} \]

the abscissae \(x\) of this diagram can be transformed into abscissae \(s\), and the ordinates \(\frac{dw}{ds}\) are found by multiplying the ordinates \(Z\) by \(x_m t\) and \(\left(\frac{x_m}{x_0}\right)^2\) (if it is necessary to take account of centrifugal dilution), followed by normalization. If such a transformation is made at different values of \(t\) and leads to identical distributions \(q_w(s)\), one may consider that diffusion does not distort the process of broadening of the boundary due to polydispersity. In Fig. 11,a

Fig. 11. a) Distribution \(Q_w(s)\) for rubber K-2, calculated for different times (values of \(t\) in sec. are indicated in the figures). b) The same for one of the fractions of this rubber.

the result of such a calculation is shown for unfractionated butadiene-styrene rubber \(K\)-2 \(^{48}\), and in Fig. 11,b for one of its fractions. The difference in behavior is quite obvious. The distribution \(q_w(s)\) for the strongly polydisperse sample remains unchanged, while for the fraction it gradually narrows, since with the passage of time the relative weight of the term \((\omega^2 x_m t \sigma)^2\) increases in comparison with \(2D_w t\) (see below, Gosting’s method).

After diffusion has been excluded, it remains to take concentration effects into account; there are several ways to do this.

1) Graphical fractionation according to Gralén and other extrapolation methods

Fig. 12a. “Graphical fractionation” according to Gralén.

By averaging over several exposures (to smooth the statistical scatter of the points), one obtains the distribution \(q_w(s)\) corresponding to the given concentration. In accordance with the various points of view set forth on concentration effects, they may be allowed for in different ways. The most reliable is Gralén’s empirical method \(^{19}\), which is not based on any a priori assumptions. Gralén divides the diagram

\[ \frac{dw}{ds} = q_w(s,c), \]

say, into 10 “fractions” of equal weight. For this purpose the integral curve is constructed

\[ \int_0^s \frac{dw}{ds}\,ds = f(s), \]

which is divided by equidistant horizontal segments into 10 parts (Fig. 12a); the points of intersection of the integral curve with these segments give the conditional sedimentation constants \(s_j\), characterizing the “fractions.” As the initial concentration is lowered, the resulting curve \(q_w(s,c)\) will change its shape (see below, Fig. 14), but the subdivision into 10 fractions of equal weight will remain unchanged. By extrapolating the values \(s_j\) to infinite dilution, one thus succeeds in reproducing the “true” distribution. In Figs. 12b and 12c this procedure is illustrated for unfractionated rubber \(K\)-3. As is evident

Fig. 12b. Graphical fractionation for rubber \(K\)-3. Integral curves at four different concentrations.

from the figure, all “fractions” in the interval from 10 to 90% are extrapolated to \(c = 0\) according to one and the same law

\[ \frac{1}{s_0} = \frac{1}{s} - k''c, \]

where \(k''\) is a parameter common to all fractions of the given polymer, and \(c\) is the initial (and not local) concentration, i.e. the mean concentration in the homogeneous region. An analogous result was obtained by the author also for \(K\)-1. It then turned out that \(k''\) has the same value as for the true fractions—in independent measurements—which agrees well with the concept of a “porous medium.” Although it is premature to extend this result, which underlies the considerations set forth above, to other polymers, one may nevertheless think that the time-consuming procedure of graphical fractionation can be replaced by simple extrapolation according to the formula

\[ \frac{dw}{ds_0} = \frac{dw}{ds}\frac{ds}{ds_0}, \]

where the dependence \(s=f(s_0,c)\) can be studied on fractions. It is essential here, however, that the shape of all molecules

Fig. 12c. Extrapolation of nodal values \(s\) to infinite dilution for K-3.

be the same. Otherwise the homogeneity of the “porous medium” is violated and the law of concentration dependence may change. This was shown by Schachman and Harrington\(^{124}\) using the example of sedimentation of tobacco mosaic virus in the presence of bushy stunt virus (the molecules of the former have the shape of elongated cylinders, and those of the latter—spheres). Boldun\(^{125}\) believes that, owing to the Macfarlane effect, the distribution in the homogeneous region differs from the distribution represented by the sedimentation diagram. He also proposed a rather rapid method of extrapolation, from the description of which, however, we shall refrain, since it seems unlikely that the Macfarlane effect—whatever its origin may be—could

Fig. 13. Distribution \(Q_w(s_0)\) for polystyrene\(^{98,107}\). The dashed line was obtained by the method of graphical fractionation; the solid line—by the method of summing triangles.

play any appreciable role at low concentrations (good resolution of the ultracentrifuge is obtained only at \(2\ \mathrm{mg}/\mathrm{ml}\)—cf. Fig. 14).

Instead of extrapolating to infinite dilution, one may simply immediately recalculate the distribution \(q_w(s,c)\) at a sufficiently small concentration into the distribution \(q_w(M)\), using, for the given concentration, the corresponding values of the parameters \(a\) and \(K_s\) in the formula \(s = K_s M^{1-a}\). Such a procedure is justified by the fact that the decisive role is played not by the local, but by the initial concentration, as follows from Fig. 12c. A very similar procedure had been proposed much earlier by Jullander\(^{30}\), but he used values of the local concentration and obtained distorted results.

Fig. 14. Dependence of the distribution \(Q_w(s)\) on concentration\(^{98}\), \(w\):
a) polystyrene; b) rubber. The numbers denote \(c\) in g/100 ml.

In Fig. 13 are shown the results obtained by Gralén and Lagermalm for polystyrene\(^{117}\). In Fig. 14, a and b, data are presented

Gradlen’s data for polystyrene\(^{107}\) and the author’s data for rubber K-3 make it possible to judge the increase in the resolving power of the ultracentrifuge as the concentration is decreased.

2) Gosting’s method\(^{110}\)

In the case where diffusion cannot be neglected, the procedure becomes somewhat more complicated. Now the direct transformation of the sedimentation diagram (cf. Fig. 11, б) gives the quantity

\[ q_w^*(s)=q_w(x)\,x\omega^2 t, \tag{11,5} \]

which differs from the true distribution function in that the disturbances introduced by diffusion are attributed to polydispersity. Expression (11,5), moreover, differs from (10,6) in that \(x_m\) is replaced by the variable factor \(x\), which, strictly speaking, is more accurate (within the limits of the diagram, for a broad distribution, \(x\) changes by approximately 10–15%).

Gosting’s method is based on the physical fact that the disturbances introduced by diffusion increase with time as \(\sqrt{t}\), whereas the disturbances due to polydispersity increase proportionally to \(t\). Therefore, in the limit as \(t\to\infty\), \(q_w^*(s)\) tends to the true value \(q_w(s)\).

To find an analytical expression relating \(q_w^*(s)\) and \(q_w(s)\), Gosting, under certain simplifying assumptions, solves the differential equation of the ultracentrifuge\(^{111}\):

\[ \frac{\partial c}{\partial t} = -\frac{1}{F}\frac{\partial(FI_{s,D})}{\partial x}, \tag{11,6} \]

in which \(F\) is the cross section of the sectorial cell, and \(I_{s,D}\) is the mass of substance (characterized by the parameters \(s\) and \(D\)) transported per unit time through a unit section; \(I_{s,D}\) is found from the flux equation:

\[ I_{s,D}=-D\frac{\partial c}{\partial x}+s\omega^2xc. \tag{11,7} \]

In final form Gosting obtains:

\[ q_w^*(s)\cong [q_w(s)] + \frac{D}{\omega^4}\, \frac{g(s\omega^2 t)}{x^2t} \left[ \frac{d^2 q_w(s)}{ds^2} \right] + \]

\[ + \frac{1}{2}\frac{D^2}{\omega^8} \left[ \frac{g(s\omega^2 t)}{x^2t} \right]^2 \frac{d^4 q_w(s)}{ds^4} +\cdots, \tag{11,8} \]

where \(g(s\omega^2t)\) is a power series in \(s\omega^2t\). It can be shown that the coefficients at the derivatives of \(q_w(s)\) rapidly decrease. By \(D\) one should understand the diffusion constant assigned to the polymer as a whole. When the real distribution of diffusion constants is taken into account—

of diffusion, a similar but more complicated expression is obtained. This is not reflected in the extrapolation procedure, which reduces to the fact that the function \(q_w^*(s)\) is reproduced at several values of \(t\), the graphs are divided by equally spaced vertical chords into 10–12 parts (by analogy with Fig. 11), and the nodal values of \(q_w^*(s)\) are extrapolated to infinite time on a graph with abscissas

\[ \frac{1}{x_m^2 t}. \]

Gosting’s method has so far been applied only once, in the work of Miller and Hamm\({}^{49}\), where various samples of polyvinylpyrrolidone were studied.

Gosting does not touch upon questions of concentration dependence, but it can be taken into account in the usual way. In the work of Miller and Hamm, for most of the samples investigated this dependence could be neglected.

b) Methods based on summation of fractions

3) The “half-width” method according to Gralén

In studying cellulose derivatives, Gralén\({}^{19}\) proposed taking, as a numerical measure of polydispersity, the “broadening” \(dB/dx\), in which \(B\) is the ratio of the area of the sedimentation diagram to its maximum ordinate (“half-width”). It is easy to show that, in the absence of diffusion,

\[ \frac{dB}{dx} = \frac{\displaystyle \int_0^\infty q_w(s)\,ds} {s_m\left(\dfrac{dw}{ds}\right)_{s_m}} = \frac{\Delta s}{s_m}, \tag{11,9} \]

where \(\Delta s\) is the half-width of the distribution \(q_w(s)\).

Strictly speaking, because of sectorial dilution, \(dB/dx\) is a weak function of time. Taking this effect into account, Chinell\({}^{98,112}\) proposes the expression, independent of time:

\[ \frac{dB}{dx} = \frac{d}{dx} \left[ \frac{c_0} {\left(\dfrac{x_m}{x_0}\right) \left(\dfrac{dc}{dx}\right)_{x=x_m} \ln \dfrac{x_m}{x_0}} \right] \tag{11,10} \]

and related in an analogous way to \(s_m\) and \(\Delta s\).

It is not difficult to see that the half-width method is a less rigorous variant of the statistical method of C. E. Bresler and the author, set forth in the preceding paragraph. The “half-width” \(\Delta s\) is

is a comparatively simple function of the standard deviation \(\sigma\) and is related to the “half-width” of the distribution \(q_w(M)\) by the equality

\[ \frac{\Delta s}{s_m}=\frac{1}{1-a}\frac{\Delta M}{M}, \tag{11,11} \]

which is completely identical with (10,17).

However, in the presence of diffusion the “half-width” can no longer be obtained as simply as the standard deviation \(\sigma\).

4) The triangle method of Chinchilla and Ronby\({}^{98}\)

In order to pass from the half-width \(\Delta s\) to the distribution \(q_w(s)\), it is necessary to give \(\Delta s\) a clearer geometrical meaning. Ronby\({}^{113}\) proposed approximately representing the fractions in the diagram in the form of isosceles triangles with mode \(s_m\). In this case \(\Delta s\) is simply the median line of the triangle, and summation of the fractions reduces to graphical summation of the ordinates of the triangles on the common diagram \(\dfrac{dw}{ds}, s\).

Fig. 15. Summation of fractions and the distribution \(Q_w(s)\) for nitrocellulose\({}^{98,113}\).

In Fig. 15 are shown the results of this procedure for nitrocellulose fractions\({}^{98,113}\). The height of the triangles \(H\), in the selected scale, is equal to \(\dfrac{w_i}{\Delta s}\), where \(w_i\) is the weight fraction of the given fraction. In Fig. 15, the sharp peak in the region of small molecular weights, corresponding to hemicelluloses, is noteworthy (cf. § 8).

Figure 16 shows the total distribution obtained by Chinchilla\({}^{108,98}\) for polymethyl methacrylate. This distribution correlates well with the result of direct recalculation of the sedimentation diagrams of the unfractionated polymer. The irregular shape of the distribution makes it possible to assume several parallel mechanisms of polymerization. More information

would in this respect be given by the distribution \(q_w(M)\), which, unfortunately, has not been obtained.

Fig. 16. Summation of fractions and the distribution \(Q_w(s)\) for polymethyl methacrylate \(^{98,108}\).

In the work of Gralén and Lagermalm \(^{107}\), the molecular-weight distribution of polystyrene was studied (Fig. 13). The presence of two

Fig. 17. Distribution \(Q_w(M)\) (solid line) for polystyrene \(^{107}\), obtained by graphical differentiation of the integral sedimentation curve \(W(M)\%\) (dotted line). The Roman numerals correspond to the position of \(M_m\) of the fractions in the order of their precipitation.

maxima in this distribution clearly indicates two parallel mechanisms of polymerization*).

*) In fact, there were even three of them. The deficiency of methods that are restricted to constructing the distribution \(q_w(s)\) consists precisely in the fact that the possibility of various dependences of \(s(M)\) within a single sample is not taken into account. The distribution \(q_w(M)\) for the same sample, calculated by constructing an integral step curve \(c=f(M)\) (cf. § 8), is shown in Fig. 17.

In summary, one may say that the triangle method apparently gives, on the whole, a correct notion of polydispersity; however, it is limited to regions of high molecular weights, of the order of \(10^6\), since at lower weights the effect of diffusion becomes very appreciable. In particular, in the works mentioned, neglect of diffusion is by no means always justified.

5) Application of the method of model functions

Gralén\(^{19,98}\) proposed, by analogy with the Krämer–Lansing function, representing the distribution function with respect to sedimentation constants in the form

\[ q_w(s)=K_s e^{-y^2}; \qquad y=\frac{1}{\gamma}\ln\frac{s}{s_m}. \tag{11,12} \]

The parameter of the function \(\gamma\) can here readily be related to the half-width \(B\) or \(\Delta s\).

Jullander\(^{30}\) introduced the function

\[ \frac{dw}{ds}=K_s e^{-y^2}; \qquad y=\frac{1}{\gamma}\ln\frac{1}{k}\left(\frac{s}{s_m}+k-1\right) \tag{11,13} \]

with three parameters, \(s_m\), \(k\), and \(\gamma\), which, in his opinion, makes it possible simultaneously to obtain from the experimental data the width and asymmetry of the distribution. This method of investigation was applied to various nitrocellulose samples.

6) Method of equivalent Gaussian distributions

This method was developed in the laboratory of S. E. Bresler\(^{70}\). From the formal side this method is a combination of the method of model functions with the method of summation of fractions. However, the choice of the model function is made on the basis of a definite physical hypothesis, and the correctness of this choice can be proved directly.

The principal question to be resolved is connected with elucidating the character of the molecular-weight distribution within a fraction.

The theory of equilibrium solubility of polymers predicts that the fractions should have a broad molecular-weight distribution. However, the distribution function of the precipitated polymer is not determined by equilibrium solubility. In polymers, equilibrium is established very slowly and with great difficulty. Therefore the process of fractionation during precipitation is governed not only, and not so much, by thermodynamic factors as by kinetic factors. In the experimental execution of fractionation, various accidental circumstances are of great importance: the rate of addition of the precipitant and the speed of stirring, as well as changes in temperature. Consequently, if exceptional precautionary measures are not taken,

usually not possible to reproduce completely and in all details the fractionation process of one and the same polymer. Fractions precipitating in two consecutively carried out precipitations differ somewhat from one another, although the amounts of precipitant added may be exactly the same.

On the basis of these considerations S. E. Bresler formulated a hypothesis according to which the number distribution \(q_n(M)\) within each fraction around the mean weight is random and obeys the law of errors. If such an assumption is introduced, one can immediately arrive at an analytical expression for the distribution function with respect to molecular weights within a fraction:

\[ \frac{dn}{dM}=q_n(M)=\frac{N}{\sqrt{2\pi}\,\mu}\, e^{-\frac{(\Delta M)^2}{2\mu^2}} . \tag{11,14} \]

Here \(\Delta M=M-\overline{M}\) is the deviation of the molecular weight from the most probable value \(\overline{M}\) (for the given fraction), \(N\) is the total number of particles, and \(\mu\) is the standard deviation, which, as is known, in the present case explicitly enters into the distribution function. From this hypothesis there follow four important theorems, containing the possibility of an experimental test of the hypothesis itself. Since the entire method is described in detail in the paper by S. E. Bresler and the author \(^{7}\), here we shall present these theorems without proof.

Theorem 1. If the molecular-weight distribution within a fraction is Gaussian, and the standard deviation \(\mu\) is small (\(\mu\) at least three times smaller than the most probable weight \(\overline{M}\)), then the distribution with respect to sedimentation constants, diffusion constants, displacements, etc. will also be Gaussian for each fraction.

In other words, if relation (11,14) holds, it follows from it that

\[ \begin{aligned} q_n(s)&=\frac{1}{\sqrt{2\pi}\,\sigma}\, e^{-\frac{(\Delta s)^2}{2\sigma^2}} \\ \text{and}\qquad q_n(\xi_s)&=\frac{1}{\sqrt{2\pi}\,\delta}\, e^{-\frac{\xi_s^2}{2\delta^2}}, \end{aligned} \tag{11,15} \]

where the parameters \(\sigma\) and \(\delta\) are determined by the general formulas (10,9) and (10,17).

Theorem 2. If the fractions satisfy the conditions set forth above, i.e. are normal (Gaussian) with a sufficiently small standard deviation, then all average weights for these fractions, irrespective of the method of their determination, must coincide.

It follows from elementary statistical considerations that the number-average weight \(M_n\), determined osmotically or by the end-group method, coincides with the most probable weight \(\overline{M}\).

For the weight-average molecular weight obtained in measuring light scattering, one obtains

\[ M_w=\overline{M}\left(1+\frac{\mu^2}{\overline{M}^{\,2}}\right). \tag{11,16} \]

Even in those cases when \(\mu=0.33\overline{M}\), the difference between the weights \(M_n\) and \(M_w\) does not reach \(10\%\).

The viscometric average molecular weight \(M_\eta\) lies in the interval between \(M_n\) and \(M_w^{114}\). Therefore it too must practically coincide with the two preceding average weights.

An important consequence of this theorem is the circumstance that the modal value of the molecular weight in the weight distribution

\[ q_w(M)=M q_n(M)=\frac{M}{\sqrt{2\pi}\mu}\,e^{-\frac{(\Delta M)^2}{2\mu^2}} \]

turns out to be exactly equal to the weight-average weight, as is easily verified by direct calculation, i.e.,

\[ M_m=\overline{M}\left(1+\frac{\mu^2}{\overline{M}^{\,2}}\right)=M_w, \tag{11,17} \]

where \(M_m\) is the modal value. The coincidence of the modal and average values is, generally speaking, a sign of symmetry of the distribution. Since in the present case the weight distribution differs from a Gaussian only by the slowly increasing pre-exponential factor \(M\), one can see that, in practice, multiplication of the exponential factor by \(M\) reduces to a shift of the entire distribution along the molecular-weight axis toward larger weights by the amount \(\delta M=\frac{\mu^2}{\overline{M}}\), and the actual distribution \(q_w(M)\), for a fraction that is not too broad, can be replaced by an equivalent Gaussian distribution about the weight \(M_m\). This is the content of Theorem 3.

Theorem 3. In the first approximation, not only the numerical distributions but also the weight distributions with respect to molecular weights, sedimentation constants, diffusion constants, displacements, etc., must be Gaussian, with the same values of the standard deviations as in the numerical distributions.

Theorem 4. The experimental curves representing the distribution function with respect to displacements \(q_w(\xi)\) are also Gaussian.

This follows from the general theorems of probability theory. Thus,

\[ q_w(\xi)=\frac{\mathrm{const}}{\sqrt{2\pi \overline{\xi^2}}}\,e^{-\frac{\xi^2}{2\overline{\xi^2}}}, \tag{11,18} \]

where

\[ \overline{\xi^2}=\delta^2+2Dt. \]

This circumstance considerably simplifies the calculation of the second moment of the sedimentation diagrams, which turns out to be simply equal to

\[ M_2^0=\frac{F^2}{2\pi H^2}, \qquad \overline{\xi^2}=\frac{M_2^0}{F}, \tag{11,19} \]

where \(F\) is the area, and \(H\) is the maximum ordinate of the sedimentation diagram.

The values obtained in this way are substituted into (10,13), and the broadenings due to polymolecularity and diffusion are separated. If the diffusion coefficient has been measured independently, the standard deviation \(\sigma\) can be calculated directly from the formula

\[ \sigma=\left(\frac{\overline{\xi^2}-2Dt}{\omega^4 x_m^2 t^2}\right)^{1/2}. \tag{11,20} \]

In Figs. 18, 19, and 20 some experimental confirmations of these theorems are presented. Fig. 18 shows experimental curves for a series of fractions of one rubber. On the experimental curves points are plotted for Gaussian functions that most closely reproduce the experimental data. We see that the agreement obtained is quite satisfactory.

Formula (10,13) makes it possible to verify the theory independently. Indeed, if, for a series of successive moments of time, one plots the quantities

\[ \frac{\overline{\xi^2}}{2t}, \]

relating to one and the same fraction, as a function of the argument \(x^2t\), then one should obtain a straight line with an initial ordinate equal to \(D_w\). In Fig. 19 the straight lines obtained experimentally for a series of fractions are shown. On the ordinate axis are marked the points \(D_w\), determined independently. We see that in most cases the theory agrees excellently with experiment. The values of the diffusion coefficients measured directly and those found from the sedimentation diagrams turn out to be identical\(^{48}\).

The third most important conclusion from the theory, formulated in the second theorem, consists in the equality of the different average molecular weights for the given fraction. Fig. 20 illustrates the validity of this rule. On the ordinate axis are plotted the values of the logarithms of the sedimentation constants of different fractions as functions of the logarithm of the molecular weight measured by different methods. It is seen that the points corresponding to all fractions of one and the same synthetic rubber lie well on a straight line; moreover, the scatter of the molecular weights for the same fractions with different measurement methods lies within the limits of error.

Fig. 18. Sedimentation diagrams for three fractions of butadiene rubber \(4^{s}\) in the coordinates \(Z, \xi\) or \(Z, y\) \((y=\mathrm{const}+x)\). Points are experimental values of \(Z\); circles are equivalent Gaussian distributions. \(b\) is the scale distance
\[ \left( Z=\mathrm{const}\times b\times \frac{dc}{dx}\right). \]
a) 3rd fraction; b) 6th fraction; c) 7th fraction.

measurements. Experiment does not reveal any systematic deviations from the predicted dependence.

In accordance with formulas (11.1) and (11.2), the calculation of the molecular-weight distribution may be carried out either from the directly calculated values of $\mu$, or by recalculating the total distribution $Q_w(s)$ into $Q_w(M)$. In the first case it should be remembered that the values of $\mu$ cannot exceed $0.33 M_m$; if this condition is not satisfied, the fractions cannot be considered Gaussian, and the distributions $[q_w(s)]_i$ must first be summed. In the subsequent recalculation into $Q_w(M)$, it is necessary to take into account the possibility of an anomaly in the dependence $s(M)$ at low molecular weights. The same also applies to direct calculation by the second method. These questions are considered in detail in the work of S. E. Bresler, I. Ya. Poddubnyi, and the author2. In this work the method was tested on three synthetic rubbers. In Fig. 21,a the procedure for summing fractions and the complete distribution $Q_w(s)$ for divinyl rubber K-1 are shown. (With $c_0 = 2$ mg/ml.) In Fig. 21,b the complete distributions $Q_w(s)$ for K-3 are shown, obtained by summing the fractions and by direct recalculation for the unfractionated rubber (the distributions were extrapolated to infinite dilution). The agreement of the two methods should be regarded as quite satisfactory, but it is characteristic that the distribution for the unfractionated rubber is somewhat broader; exactly the same result was obtained for two other rubbers. This suggests that, in fact, during reprecipitation of the fractions a part of the peripheral components was lost.

Fig. 19. Separation of broadenings due to diffusion and polydispersity for 5 fractions of divinyl-styrene rubber2. The divisions on the ordinate axis correspond to the values of $D_w$ (taking into account the temperature of the experiment).

In Figs. 22 and 23 the distributions $Q_w(M)$ are presented for divinyl and divinyl-styrene SK in comparison with analogous distributions obtained by the method of graphical differentiation of the integral precipitation curve.

As can be seen, the ultracentrifuge method gives a broader and more detailed picture of the distribution, although the routine method of constructing the stepwise curve as a whole also gives the correct

concept of the character of the distribution, which was disputed by some authors.

The method of equivalent Gaussian distributions was also applied to the study of polyamides ^{105}. This work very vividly

Fig. 20. Functions \(\lg s = f(\lg M)\) for three \(CK\) ^{48}.

illustrates some of the considerations expressed in the introduction to Part II.

According to Flory ^{115}, polyamides are formed in the process of “equilibrium” polycondensation, in which chain growth is accompanied by a simultaneous purely statistical decay. It is assumed here that all functional groups react identically

Fig. 21. a) Summation of fractions and the distribution \(Q_w(s)\) for divinyl rubber \(^{48}\). b) Comparison of the distributions \(Q_w(s)\) for SK-3, obtained by summation of fractions (dashed line) and by direct calculation for unfractionated rubber (solid line).

tive. If \(p\) is the degree of completion of the reaction, and \(x\) is the degree of polymerization, then

\[ q_w(x)=p^{x-1}x(1-p)^2. \tag{11,21} \]

Substituting here \(p=\dfrac{x-1}{x}\) and passing to molecular weights, it is not difficult to verify that (11,21) coincides with Schulz’s distribution (9,6) for \(\beta=0\). The same distribution, as V. Kuhn\({}^{116}\) showed, should be obtained in the case of the absolutely statistical degradation of infinitely long chain molecules.

Fig. 22. Distribution \(Q_w(M)\) for divinyl rubber\({}^{48}\).

The “equilibrium” distribution is the broadest of all model distributions, and its width must increase without bound as the degree of completion of the reaction increases. However, Flory\({}^{115}\) himself points out that experimentally such a distribution has not been observed for polycondensation products. On the contrary, from the experiments of V. V. Korshak and co-workers\({}^{117}\) it follows that even at very high degrees of completion of the reaction the distribution remains rather narrow. This is explained by the fact that, in reality, the dynamic equilibrium between the formation and degradation of chains is a more complex process than Flory assumed. In particular, Flory does not take into account the possibility that free functional groups may be incorporated into already grown chains, with subsequent degradation of these chains and a new attachment of the inserted group to one of the residues of the degraded molecule. With increasing \(p\), such interchain-exchange reactions must become more frequent, since, first, with the growth of the degree of polymerization the number of bonds capable of being destructurized increases, and, second, the amount of water liberated in the course of polycondensation increases, as a result of which the probability of hydrolysis grows. Therefore the fraction of very long chains cannot be large, and the high-molecular “tail” characteristic of the Flory distribution should not occur, but should

one should expect only a comparatively small scatter of molecular weights about the mean value. As S. E. Bresler has shown^105, in this case the distribution function must have the form

\[ q_w(x)=\frac{x(\bar{x}_n)^x}{x!}\,e^{-\bar{x}_n}\simeq \frac{1}{\sqrt{2\pi\bar{x}_n}}\, e^{-\frac{(x-\bar{x}_n)^2}{2\bar{x}_n}}, \tag{11,22} \]

where \(\bar{x}_n=\dfrac{1}{1-p}\) is the number-average degree of polymerization.

Figure 24 gives the experimental distribution curves for three samples of a mixed polyamide soluble in methanol (the reaction was stopped at different degrees of completion). The dotted lines depict the Flory distributions corresponding to the same \(p\), and the vertical strokes are the widths calculated from formula (11,22). These curves indicate a fundamental discrepancy between Flory’s theory and the experimental data. S. E. Bresler’s formula, derived on the basis of the polycondensation mechanism proposed by V. V. Korshak, agrees better with experiment; however, the theory evidently requires further development.

Fig. 23. Distribution \(Q_w(M)\) for divinyl-styrene rubber^48.

The reshuffling of units as a result of interchain exchange was also proved by another method. A mixture of two fractions with molecular...

with weights \(16\) and \(58 \times 10^3\), was heated under polycondensation conditions. The distribution function obtained was of exactly the same form as in Fig. 24, with a maximum at about \(25\,000\). Consequently,

Fig. 24. Distribution \(Q_w(M)\) for polyamides \(10^5\).
For explanations, see the text.

regardless of the starting material (monomers or a mixture of ready-made polymers), as a result of polycondensation equilibrium a single-type distribution of molecular weights is established.

In conclusion, I consider it my pleasant duty to express gratitude to Prof. S. E. Bresler for reading and criticizing the manuscript.

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1. Reference 87 as cited in the original. ↩

2. Reference 48. ↩↩↩