VISCOUS AND THERMODYNAMIC PROPERTIES OF HIGH-POLYMER SUBSTANCES*)

R. Powell and H. Eyring

In statistical thermodynamics it is shown that, starting from definite ideas about the structure of a substance, one can derive all its fundamental thermodynamic properties.^22 To the same extent it proves possible, starting from a definite law of interaction between molecules and using statistical methods of calculation, to find the rates of chemical reactions and other kinetic properties of a substance.

Recently statistical and kinetic methods have been applied to high polymers—substances with long molecules, which are of great technical interest. The behavior of these substances in solutions, their dielectric properties, and the laws of flow have been investigated.

The aim of the present article is to give a brief account of the method of the activated complex and of its applications to simple systems and, in particular, of its extension to the case of large molecules.

1. THERMODYNAMICS OF LIQUIDS WITH SIMPLE MOLECULES

A satisfactory physical model of a liquid must possess such properties that from it one could derive not only the equilibrium properties of the liquid, but also its nonequilibrium properties, which determine the rates of the processes taking place. Therefore, for example, the study of viscosity or diffusion should lead to a more perfect model of the liquid state, which in turn will make it possible to find the values of the heat capacity or of the saturated-vapor pressure.

It would be interesting to show that models intended for finding the rates of processes in simple liquids and in solutions lead to identical thermodynamic properties.^95

From an analysis of data on diffusion and viscosity one may conclude that simple liquids must possess the following properties:

1. Diffusion or displacement of molecules in a liquid occurs

*) Advances in Colloid Science, vol. 1, 183, 1942, translated by V. G. Levich.

by the transition of molecules into neighboring unfilled equilibrium positions, or “holes.” In associated liquids the presence of such holes is the principal factor determining their flow. Thus a liquid may be regarded as a binary mixture of molecules and holes.

1. For normal liquids the heat of activation for diffusion or viscous flow is approximately equal to one third of the heat of vaporization. Since the heat of vaporization is expended in creating voids of dimensions equal to the dimensions of the molecules of the liquid, it is clear that the “holes” are considerably smaller than the molecules. The true volume involved in the formation of new equilibrium positions, i.e., the volume of the holes, can be calculated from the dependence of the viscosity of the liquid on pressure. In this way it was found that in normal nonmetallic liquids the volume of the holes is approximately equal to one seventh of the volume of the molecules.

2. If the expansion upon melting (amounting to about \(10\%\) in typical cases of nonmetallic substances) is due to the appearance of new equilibrium positions in the solid crystal, then upon melting the substance acquires about 0.7 mole of new equilibrium positions for each mole of molecules. The distribution of a mole of molecules among 1.7 moles of equilibrium positions leads to an increase of the entropy of melting by 2 CGS units, which is observed for many simple substances.

Starting from data obtained in the study of the rates of processes in liquids, Walter and Eyring\(^{95}\) constructed a distribution function for the liquid state. The authors assume that in a liquid \(N\) molecules are distributed at random among \(N\) initial equilibrium positions that existed in the solid, and \(n_k\) new equilibrium positions. Some of the molecules of the liquid oscillate about equilibrium positions, as in a solid; the other part moves chaotically, as in a gas. Of the total number \(3N\) of translational degrees of freedom of \(N\) molecules, the fraction \(V_s/V\) is accounted for by vibrational degrees of freedom, and the remainder, \((V - V_s)/V\), by degrees of freedom of chaotic translational motion. Here \(V\) denotes the volume of 1 mole of liquid and \(V_s\) the volume of 1 mole of solid at the melting point.

It should be noted that this model differs from the usual model of a liquid, in which it is assumed that all molecules are in equivalent equilibrium positions and that each molecule has a definite average free volume, with each molecule moving in some average potential field. The Eyring and Walter model also differs from the model proposed by Lennard-Jones and Devonshire\(^{52}\), who assume that the number of new equilibrium positions \(n_k\) arising upon melting is exactly equal to the number of molecules \(N\) in the body. In the theory of Walter and Eyring the number of new equilibrium positions \(n_k\) is considered a linear function of \(V - V_s\), increasing with increasing \(V\) and tending to no limit.

In the theory it is assumed that the distribution function for molecules in crystal-like equilibrium positions has the form

\[ f_s e^{E_s/RT} = \left( \frac{e^{-\theta/2T}}{1-e^{-\theta/T}} \right)^3 e^{E_s/RT}, \tag{1} \]

where \(E_s\) is the potential energy of a molecule in the solid and \(\theta = 3/4\,\theta_{\mathrm{Deb}}\), where \(\theta_{\mathrm{Deb}}\) is the Debye temperature.

For molecules in gas-like equilibrium positions, the distribution function will be

\[ f_g e^{-\delta E_s/RT} = \left\{ \frac{(2\pi mkT)^{3/2}}{h^3}\,\frac{V_s}{n} \right\} e^{-\delta E_s/RT}, \tag{2} \]

where \(\delta E_s\) is the energy of interaction of the molecules, and \(n\) is the ratio of the volume of a molecule to the volume of a “hole.”

Before the distribution function for gas-like states can be applied to calculations, it is necessary to choose the proper form of the function \(\delta\) and a numerical value of \(n\).

If \(N\) molecules of a liquid are distributed at random among \(N+n_h\) equilibrium positions, the factor \((N+n_h)!/N!n_h!\) will enter the distribution function, so that the complete distribution function will have the form

\[ f= \left\{ \left(f_s e^{E_s/RT}\right)^{V_s/V} \left(f_g e^{-\delta E_s/RT}\right)^{(V-V_s)/V} \right\}^{N} \frac{(N+n)!}{N!\,n!}. \tag{3} \]

The distribution function \(f\) passes into the corresponding distribution function for a solid or for a gas if \(\delta \to 0\), and \(V \to \infty\).

Calculations based on the distribution function (3) were checked for the cases of liquid argon, nitrogen, and benzene. From the observed values of the change in volume and entropy on melting, the value of \(n\) was found to be 8.35 for all three liquids. The simplest form of the function \(\delta\), which leads to excellent agreement between calculated and experimental data, is

\[ \delta=\frac{(V_s/V)^2}{2\gamma+0.18} \]

for argon, where

\[ \gamma \equiv \frac{n}{N}=\frac{V-V_s}{V_s/n}. \]

For nitrogen and benzene, \(\delta\) has an analogous form.

Proceeding from the indicated distribution function, the authors calculated the values of the latent heats of fusion, critical constants, volumes and elasticities of the vapor over the whole interval of the liquid state, heat capacities, compressibilities, and second virial coefficients. The results of the calculations agree with the experimental data. Table 1 gives the calculated and observed values of some of these quantities for argon.

Table I

Observed and calculated values of various quantities for argon

T°K	Volume in cm³ — calculated	Volume in cm³ — observed	Vapor pressure, atm — calculated	Vapor pressure, atm — observed
83.85	28.27	28.03	0.660	0.674
87.44	28.81	28.69	1.000	1.000
89.95	29.17	29.07	1.310	1.323
97.71	30.42	30.15	2.735	2.671
111.87	32.9	32.63	8.02	7.40
122.34	35.9	35.08	15.04	13.58
137.59	43	41.02	33.14	28.29
147.93	52	51.68	48	43.19

Melting point — calculated	Melting point — observed	Critical constant — calculated	Critical constant — observed
$T_m$ 82.9° $V_m$ 3.14 cm³ $S_m$ 3.40	83.85° 3.05 cm³ 3.35	$T_c$ 154.2° $V_c$ 78.7 cm³ $P_c$ 59.4 atm	150.66° 75.26 cm³ 48.00 atm

II. THERMODYNAMICS OF HIGH POLYMERS

As in the case of liquids with simple molecules, the experimental study of the rates of processes in liquid polymers can help to elucidate their equilibrium thermodynamic properties.^69

Thus, the experimental work of Flory,^19 who measured the viscosity of liquid polyesters, enabled Kausmann and Eyring^40 to conclude that, in the flow of liquids with long molecules, the displacement occurs of individual mobile units of the molecule, containing only about 25 atoms of the chain, and not of the molecule as a whole. This same model, in which the molecule is regarded as consisting of separate mobile “segments,” was then used to study the thermodynamic properties of liquids with long molecules. The assumption that a long molecule consists of smaller “kinetic units” had also been put forward earlier for a qualitative description of osmotic pressure^54 and surface tension.^55 However, this assumption was criticized, since the existence of such kinetic parts did not follow from any direct...

experimental data and seemed to have been invented ad hoc for the description of thermodynamic properties[^36].

Therefore, the fact that investigations in a completely different field have led directly to the segmental model of linear macromolecules appears very favorable for this hypothesis.

1. Melting

As we saw in the theory of simple liquids, the process of melting may be considered, from the statistical point of view, as the appearance of new equilibrium positions in a solid body. From this point of view, the latent heat of melting is expended on deformation and on overcoming the cohesive forces between molecules and on the formation in the body of additional free space for new equilibrium positions; the entropic part of the free energy, on the other hand, increases upon melting partly because of the random distribution of molecules among old and new equilibrium positions, and partly because of the additional possibilities for molecular rotation that arise upon melting.

If long macromolecules consist of segments moving almost independently of one another, then the new positions arising upon melting represent equilibrium positions of these segments. The thermodynamic properties of the segments are determined by the value of the heat of melting \(\Delta H_f\), by the change in entropy upon melting \(\Delta S_f\), and by the melting temperature, equal to the ratio \(\Delta H_f/\Delta S_f\).

Fig. 1. Melting points of a homologous series.

The melting temperature for homologous series of compounds with macromolecules first increases with chain growth and then tends to a constant limit. The value of the limiting melting temperature proves to be the same for all homologous series. In Fig. 1 are presented the experimental values of the temperatures

melting. The limiting melting temperature is found at \(122^\circ\mathrm{C}\), or \(395^\circ\mathrm{K}\). Since the increase in the melting temperature upon addition of each \(\mathrm{CH}_2\) group is about \(20^\circ\) (for the lower homologues), the average length of the segments responsible for melting is \(395/20\), or about 20 carbon atoms.

Fuller, Baker, and Pape\(^{27}\) studied polyamides at temperatures lying below their melting points by means of X-rays, and also measured their elastic constants. These measurements showed that if a specimen is annealed or quenched at a temperature close to the melting point, the X-ray pattern of the specimen becomes sharper and at the same time less elastic. These authors concluded that at high temperatures the segments of polymer molecules in the solid state can rotate, and an amorphous specimen can be transformed into a crystalline one.

2. Osmotic Pressure

The osmotic pressure of ideal dilute solutions is described by van ’t Hoff’s law

\[ \frac{\pi V_1}{RT}=N_2 . \tag{4} \]

If long molecules move segment by segment, the osmotic pressure of their solution must be determined by some effective quantity \(n_2^{*}\) such that

\[ \frac{\pi V_1}{RT}=N_2^{*}=\frac{n_2^{*}}{n_1+n_2^{*}}, \tag{5} \]

where \(n_1\) is the number of solvent molecules and \(n_2^{*}\) is the effective number of molecules in the solution. The effective number of molecules in the solution can be expressed in terms of the true number by means of the following reasoning.

If a segment of a polymer molecule is surrounded by other polymer molecules, it can move into any of the voids of sufficiently large size. If, however, this segment is surrounded by small solvent molecules, then the voids will with high probability be filled by the latter. Therefore, it may be assumed that the probability of displacement of a segment of a polymer molecule depends on the nature of its surrounding neighbors and is a linear function of the fraction of the total volume occupied by polymer molecules.

If each polymer molecule, when surrounded only by polymer molecules, behaves as if consisting of \(Q\) separate segments, then

\[ n_2^{*}=n_2\left[1+(Q-1)\varphi_2\right], \tag{6} \]

where \(n_2\) is the total number of dissolved molecules and \(\varphi_2\) is the fraction of the volume filled by the polymer.

Finally, the expression for the osmotic pressure takes the form

\[ \frac{\pi V_1}{RTN_2}=1+(Q-1)\varphi_2, \tag{7a} \]

\[ \frac{\pi V_2}{RT\varphi_2}=1+Q\varphi_2, \tag{7b} \]

\[ \frac{\pi M_2}{RTc_2}=1+\frac{Qc_2}{d_2}, \tag{7c} \]

where \(N_2\) is the number of moles of dissolved polymer, \(\varphi_2\) is the fraction of the solution volume occupied by the polymer, \(c_2\) is the concentration in grams per cubic centimeter, \(M_2\) is the molecular weight, \(V_2\) is the molar volume, and \(d_2\) is the density of the polymer. Terms containing powers of \(\varphi_2\) higher than the first have been omitted in these formulas.

Equation (7) reduces to the van’t Hoff law at low concentrations and leads to a linear increase of the reduced osmotic pressure \(\pi/c\) with the solution concentration \(c\). An equation of this type satisfactorily describes the experimental data for many polymer solutions. As an example, Fig. 2 gives a series of curves for solutions of cellulose acetate of different molecular weights \(^{83}\). The slope of the curves determines the dimensions of the mobile segments, and the intercept on the ordinate axis gives the average molecular weight. As can be seen from Fig. 2, the slope of all the curves is approximately the same. From the slope of the curves of the dependence of \(\pi/c\) on \(c\), the dimensions of the mobile segments have been determined for a large number of polymers.

In polystyrenes and polyethylene oxides the length of the mobile segments is about 20–30 atoms; in cellulose nitrate, cellulose acetate, and methylcellulose, about 9–14 atoms; in rubber the length of the segments ranges between 42 and 460 and varies strongly from sample to sample. Any branchings and lateral bonds make the polymer molecules less mobile, so that the apparent chain length proves to be greater. Probably in most cases the change in the length of the mobile segments in rubber and in certain other types of polymers is connected with differences in the degree of branching of the molecules. The solvent in concentrated polymer solutions has an anomalously high value of the saturated vapor pressure, as though a substance of considerably lower molecular weight were dissolved in it. Thus, for example, in a solution of carbon tetrachloride, oleyl oleate has a molecular weight of 242 instead of 532.5; a rubber sample with molecular weight 270,000 in a solution of toluene had an average molecular weight of 417.

Such behavior of polymers in solutions is readily explained from the standpoint of the model of the polymer molecule adopted by us,

consisting of mobile segments with a chain length of about 20 atoms. An analogous thermodynamic treatment may also be proposed to explain the features of the surface tension of high polymers.

Table 2

Surface tension of higher normal hydrocarbons of the paraffin series

Hydrocarbons	Surface tension, 20°C
$n$-$\mathrm{C}_6\mathrm{H}_{14}$	18.43
$n$-$\mathrm{C}_8\mathrm{H}_{18}$	21.80
$n$-$\mathrm{C}_{10}\mathrm{H}_{22}$	23.91
$n$-$\mathrm{C}_{19}\mathrm{H}_{38}$	$\sim 28.1$
$n$-$\mathrm{C}_{28}\mathrm{H}_{54}$	$\sim 29.4$
$n$-$\mathrm{C}_{60}\mathrm{H}_{122}$	$\sim 29.1$

As can be seen from Table 2, the surface tension of high-molecular normal paraffins, for a sufficiently long chain, ceases to depend on its length, which indicates that the dimensions of the mobile parts of the molecules responsible for the surface tension are independent of the dimensions of the molecules themselves.

The actual sizes of the mobile segments can be estimated from the temperature coefficient of surface tension with the aid of the equation

\[ \frac{d}{dT}\left(\sigma V^{2/3}\right)=2.12. \]

With such estimates, the average molecular weight of compounds with a true molecular weight of 350–900 proves to be about 150–200.

Thus, the estimate of the sizes of the mobile segments from the osmotic pressure of solutions (18–23 for polystyrenes and polyethylene oxide) and from melting temperatures (20–25 for hydrocarbons) is in agreement with the estimate based on measurements of the viscosity of polymers of similar structure (20–25 atoms for hydrocarbons, 28–34 atoms for polyesters).

Fig. 2. Osmotic pressure of cellulose acetate solutions.

3. Swelling (filling)

Measurements of the viscosity of pure polymers show that in polymers there are free equilibrium positions—voids of sufficiently large size, capable of being filled by mobile segmen-

pores of molecules. When a polymer swells, some of these voids are filled by small solvent molecules. From this one can draw three qualitative conclusions.

1. During swelling there must occur a decrease in volume.
2. The size and number of voids, and consequently the swelling as well, should not depend on the degree of polymerization.
3. The process of swelling is accompanied by a decrease in entropy, since the mobile segments of the polymer molecules lose the possibility of moving into the voids filled by the solvent.

All these conclusions are in agreement with experimental facts: a decrease in the volume of a polymer upon swelling has been observed^24; when cellulose nitrate is filled with acetone, the swelling pressure for a given amount of acetone does not depend on the molecular weight^75; finally, a decrease in entropy of the order of several CGS units has been observed in the dissolution of gases in synthetic rubber and in the filling with water of agar, casein, keratin, and cellulose^8,25,79,91.

III. METHOD OF THE ACTIVATED COMPLEX

The method of the activated complex has been successfully applied to many chemical processes—gas reactions, reactions in solutions, heterogeneous reactions^32. It has proved to be an equally powerful method of investigation in the study of a number of physical processes, such as, for example, viscous flow, diffusion in solids and liquids, or dielectric relaxation. This method has so many possible applications that it seems appropriate to us to set forth here once more its basic relations in the most convenient form.

Let us consider two molecules capable of entering into a chemical reaction with one another. Initially these molecules are separated from each other. As they approach one another, the potential energy of the system increases. At the top of the potential barrier the two molecules form a quasi-molecule, or activated complex. This activated complex differs from an ordinary molecule only in that it is unstable with respect to one degree of freedom corresponding to completion of the reaction. Therefore the system crosses the potential barrier, and the activated complex decomposes into the final reaction products. This process is shown in Fig. 3, which schematically represents the course of a reaction of the type

\[ \mathrm{H}_2+\mathrm{I}_2 \to 2\mathrm{HI}. \]

The method of interpretation itself does not depend on the number of molecules reacting or formed as a result of the reaction, nor on the physical or chemical nature of the transformation process.

The reaction rate \(k'\) is related by the following relation to the concentration of activated complexes \(c_a\), the average rate of their motion through the barrier \(u\), and the length of the reaction path \(\delta\):

\[ k'=\frac{c_a u}{\delta} \tag{8} \]

The average velocity of motion of activated complexes over the barrier is

\[ \bar{u}= \frac{\displaystyle \int_{0}^{\infty} u e^{-m u^{2}/2kT}\,du} {\displaystyle \int_{-\infty}^{\infty} e^{-m u^{2}/2kT}\,du} = \left(\frac{kT}{2\pi m}\right)^{1/2} \tag{9} \]

To find the concentration of activated complexes, it is necessary to know only the concentration of the reactants and the equilibrium constant (the law of mass action) between the activated and the initial states of the system. Then

\[ \frac{c_a}{c_0}=K_a= K^{*}(2\pi mkT)^{1/2}\frac{\delta}{h}, \tag{10} \]

where \(h\) is Planck’s constant. The factor \((2\pi mkT)^{1/2}\dfrac{\delta}{h}\), taken out of the constant \(K_a\), represents the statistical integral for one translational degree of freedom of the activated complex, corresponding to the completion of the reaction. Therefore the reaction rate can be represented in the form

Fig. 3. Energy course for typical reactions.

\[ k' = c_0 \frac{kT}{h}K^{*} \tag{11} \]

and the specific reaction rate, referred to a reactant concentration equal to unity:

\[ k'_0=\frac{k'}{c_0}=\frac{kT}{h}K^{*}. \]

Thus, the rate of decomposition of the activated complex is the same for all reactions. It decomposes with a frequency equal to \(\dfrac{kT}{h}\), or numerically \(\sim 10^{13}\) per second at room temperature. The concentration of the activated complex is determined by the equilibrium constant \(K^{*}\).

The equilibrium constant can be represented in several equivalent ways. It can be defined as the ratio of the statistical sums of the activated and normal states. In this case the rate of the process can be represented in the form

\[ k'=\frac{kT}{h}\frac{F_a}{F_0}. \tag{12} \]

Further, the equilibrium constant can be expressed in terms of thermodynamic quantities. With this notation, the rate of the process can be written as

\[ k'=\frac{kT}{h} e^{-\Delta F^{*}/RT} \tag{13a} \]

or

\[ k'=\frac{kT}{h} e^{\Delta S^{*}/R} e^{-\Delta H^{*}/RT}, \tag{13b} \]

where \(\Delta F^{*}\), \(\Delta H^{*}\), and \(\Delta S^{*}\) are the free energy, entropy, and heat of activation. The activated-complex method can be applied directly if the reaction mechanism is known, i.e., if the heat of activation can be determined from potential-energy curves, while the statistical sums for the normal and activated states can be calculated from the known moments of inertia and force-law constants of the reacting molecules. Such calculations have been carried out for a number of gas reactions.

However, this is not always feasible and, in any case, always leads to complicated calculations. Therefore, a simpler approach consists in taking the values of the heats of activation from experiment, while the pre-exponential factors are calculated from the known properties of the molecules.

Another frequently used way of applying equation (13) consists in finding the value of the free energy from experimental data on reaction rates. If the latter measurements are made at two different temperatures, the free energy of activation can be separated into the heat of activation and the entropy term.

In many cases, knowledge of these quantities makes it possible to elucidate or refine the mechanism of the reaction taking place. For example, the magnitude of \(\Delta F^{*}\) for viscous flow is closely related to the thermodynamic quantity of the energy of vaporization; in the denaturation of proteins, large positive values of \(\Delta S^{*}\) mean that a large number of bonds are broken in the course of the process; a larger value of \(\Delta H^{*}\) in dielectric relaxation than in viscous flow for one and the same liquid indicates that in the former process a greater number of bonds must be disrupted. The activated-complex method is thus simple and convenient for studying the rates of processes. The following paragraphs will be devoted to some applications of this method. One may hope that the activated-complex method will also be applied to other problems that have not yet been solved.

IV. VISCOSITY OF SIMPLE LIQUIDS

The scheme of liquid flow considered earlier can be illustrated by Fig. 4. A given molecule (hatched in the figure) continuously changes its equilibrium position and, making its way past

of its neighbors (in doing so passing through the corresponding energy barrier) and enters the nearest unoccupied equilibrium position. This motion occurs equally often in all directions. If, however, some force acts on the molecule, it will more often be displaced in the direction of the acting force and more rarely in the opposite direction.

A single and the same scheme is applicable in considering several transport processes: 1) when the applied force is the force of viscous friction, viscous flow arises; 2) when the force is the gradient of the corresponding activity, diffusion occurs; 3) if the applied force is the gradient of the electric potential, the process of ionic conductivity takes place; and, finally, 4) if the applied force is the force of gravity, the process of sedimentation occurs.

Fig. 4. Model for flow processes.
a) Initial state.
b) Activated state.
c) Final state.

The expression for the viscosity of a liquid can be found without difficulty by means of the following argument. Let a viscous force \(f\) act on \(1\ \mathrm{cm}^2\) in two layers of molecules. We shall denote the distance between neighboring equilibrium positions by \(\lambda\), the distance between molecules in the direction perpendicular to the acting force by \(\lambda_1\), in the direction of flow by \(\lambda_2\), and in the direction perpendicular to both of these directions by \(\lambda_3\). Then the mobility—the quantity reciprocal to the viscosity—will be determined as the difference of velocities in two parallel layers of liquid situated at a distance of \(1\ \mathrm{cm}\), referred to the acting force \(f\), i.e.

\[ \varphi=\frac{1}{\eta}=\frac{\Delta u}{\lambda_1 f}. \tag{14} \]

The difference of velocities \(\Delta u\) is equal to the product of the frequency of transitions of a molecule from one equilibrium position to another by the path traversed in each such jump:

\[ \Delta u=\lambda(k_f-k_{bf}), \tag{15} \]

where \(k_f\) is the frequency of transitions in the presence of the acting force and \(k_{bf}\) in its absence.

If \(\Delta F^*\) is the free energy of activation for the transition of a molecule from one equilibrium position to another in the absence of an external force, then the transition occurring in the direction of the applied force occurs with an activation energy equal to \(\Delta F^*\) minus the work of this force over the path \(\dfrac{\lambda}{2}\), i.e.

\[ \Delta F^{**}=\Delta F^*-f\lambda_2\lambda_3\frac{\lambda}{2}. \]

The transition in the opposite direction has activation energy

\[ \Delta F^{**}=\Delta F^*+f\lambda_2\lambda_3\frac{\lambda}{2}. \]

Therefore

\[ k_f=\frac{kT}{h}e^{-\Delta F^*/RT}\left(e^{-f\lambda_2\lambda_3\lambda/2kT}\right) \tag{16a} \]

and

\[ k_b=\frac{kT}{h}e^{-\Delta F^*/RT}\left(e^{-f\lambda_2\lambda_3\lambda/2kT}\right). \tag{16b} \]

Combining (16a) and (16b) and expanding the second exponential factor in a series, we find the following expression for the mobility:

\[ \varphi=\frac{1}{\eta}=\frac{\lambda^2\lambda_2\lambda_3}{\lambda_1 h}e^{-\Delta F^*/RT}. \tag{17} \]

To a good approximation the first factor is equal to the volume of one molecule, so that equation (17) may be rewritten in the form

\[ \varphi=\frac{V}{Nh}e^{-\Delta F^*/RT}, \tag{18a} \]

\[ \varphi=\frac{V}{Nh}e^{\Delta S^*/R}e^{-\Delta H^*/RT}. \tag{18b} \]

Formula (18b) shows that if a graph of the dependence of \(\lg\eta\) on \(1/T\) is plotted, a curved line should be obtained. This equation will be checked below in several ways.

1. Relation to evaporation

The bonds that must be broken in order for a molecule to pass from one equilibrium position to another during the flow of a liquid have the same nature as the bonds broken during evaporation. The difference between the first and second processes consists in the fact that, during flow, the molecules do not have to perform work equal to \(RT\) against atmospheric pressure, and there is no gain in entropy \(\Delta S_{\mathrm{vap}}\) due to transition into the gaseous state with the corresponding increase in volume.

Consequently, the equality must hold

\[ \Delta F_{\mathrm{vap}}+T\Delta S_{\mathrm{vap}}-RT=\Delta E_{\mathrm{vap}}. \]

Figure 5 shows the dependence of \(\Delta F^{*}\) on \(\Delta E_{\mathrm{vap}}\) for a number of liquids. The squares in the upper part of the straight line refer to compressed nitrogen, oxygen, and argon; the circles in its middle part—to various

Fig. 5. Relation between the processes of viscous flow and evaporation.

organic liquids, and the triangles—to water and various alcohols. The equation of the straight line shown in Fig. 5 will be

\[ \eta = \frac{Nh}{V} e^{\Delta E_{\mathrm{vap}}/2.45RT}. \tag{19} \]

The heat of activation of viscous flow is often related to the energy of evaporation \(\Delta E_{\mathrm{vap}}\) \(^{15}\). For typical organic liquids the heat of activation of viscous flow is equal to one third or one fourth of the energy of evaporation.

2. Significance of Holes

After all that has been said above, a natural question arises: what is the principal factor determining the viscous flow of liquids—the presence in the liquid of unoccupied equilibrium states, i.e., holes, or the barrier that must be overcome by a molecule jumping into a new equilibrium position.

As experiment shows, the principal factor is the presence of holes. Bachinskii showed that under such changes of pressure and temperature of a liquid in which its volume remains unchanged, the viscosity of the liquid also remains almost constant \(^{4}\). Bingham’s investigations confirmed this result \(^{5}\). However, it is precisely the volume of the liquid that determines the number of holes present in it. Thus, with an unchanged number of holes, the viscosity of the liquid also remains constant. A more detailed study of the question on the basis of the “hole” model showed that for non-associated liquids the heat of activation of viscous flow at constant volume is indeed small, and pri-

led to satisfactory estimates of the viscosity values near the melting point for a number of liquids^74. These results are closely connected with the results considered in the preceding section, which pertained to the thermodynamic properties of simple liquids.

3. Sizes of holes

At high pressures the viscosity of a liquid may be represented in the form

\[ \eta_P=\frac{Nh}{V} e^{(\Delta F_0+\rho\,\overline{\Delta V^{*}})/RT}\,{}^{*}), \tag{20} \]

where \(P\) is the pressure and \(\overline{\Delta V^{*}}\) is the mean change in volume upon activation.

Strictly speaking,

\[ P\overline{\Delta V^{*}}=\int_0^P \overline{\Delta V^{*}}\,dP. \]

However, since the curve of the dependence of \(\lg \eta_p/\lg \eta_0\) on pressure is, to a good approximation, approximated by a straight line, one may assume that \(\overline{\Delta V^{*}}\) does not depend on pressure.

As experiment shows, the activation volume is equal to one sixth or one seventh of the volume of a molecule.

4. Viscosity of mixtures^74

As we have just found, the principal factor determining the viscous flow of a liquid is the presence in it of holes—unfilled equilibrium states. The formation of holes requires the expenditure of a certain energy, equal to approximately one third of the heat of vaporization. If we are dealing not with a single liquid but with a mixture of two liquids, the energy going into the formation of holes in such a liquid must have an average value between the energies of formation of holes in both liquids. For ideal solutions, therefore, the viscosity may be represented in the form

\[ \eta=\frac{Nh}{V} e^{(N_1\Delta F_1+N_2\Delta F_2)/RT}. \tag{21} \]

To a good approximation, formula (21) agrees with the empirical Arrhenius and Kendall rule

\[ \eta=\eta_1^{n_1}\eta_2^{n_2}. \tag{22} \]

If the deviation from ideality is taken into account in the free energy of vaporization, equation (21) will describe the viscosity of nonideal mixtures.

\[ \underline{\phantom{xxxxxxxxxxxx}} \]

*) The zero subscript in \(\Delta F_0\) means that at all concentrations the solution remains ideal, i.e. the activity coefficient \(\gamma\) is equal to unity. The same applies also to formulas (38)—(41), (43), and (44).

Fig. 6 illustrates the results of applying formula (21), as well as certain empirical laws, to the benzene—phenol system.

V. VISCOSITY OF HIGH POLYMERS

If the heat of activation of viscous flow of normal liquid paraffins is plotted as a function of the number of carbon atoms in the molecule, then, as the latter increases, the curve becomes saturated (Fig. 7), just as does the analogous curve for the melting temperature.

Fig. 6. Effect of concentration on viscosity in the case of simple liquids.

Fig. 7. Dependence of the heat of activation for viscous flow on chain length.

Hence it may be concluded that, when the number of hydrocarbon groups in a molecule becomes sufficiently large, the flow of the liquid is effected by displacements of separate mobile segments, whose dimensions do not depend on the total length of the molecule^40. The limiting value of the heat of activation \(\Delta H^* - 6—7\) kcal corresponds to the length of the mobile segments, amounting to 20—25 carbon atoms. Some data on the character of such flow may be obtained from Flory’s data on the viscosities of polyester melts^19.

It turned out that the heat of activation for viscosity does not depend on the molecular weight of the substance. This indicates that the dimensions of the parts of molecules displaced during flow likewise do not depend on molecular weight.

The viscosities of polyesters with mean (by molecular weight) chain length \(Z_w\) are expressed by the following empirical formula:

\[ \ln \eta=\frac{A}{R}+\frac{BZ_w^{1/2}}{R}+\frac{C}{RT}, \tag{23} \]

where the constants have the following values: \(A=-28\), \(B=0.5\); \(C=8.3\) kcal. From formula (18) for the viscosity it follows that

\[ \ln \eta=\ln \frac{Nh}{V}-\frac{\Delta S^{*}}{R}+\frac{\Delta H^{*}}{RT}. \tag{24} \]

Thus, the value of \(\Delta H^{*}\) proves to be equal to 8.3 kcal. From the known heat of vaporization of simple molecules and the relation \(\Delta H^{*}=\Delta H_{\mathrm{vap}}/4\) one may estimate the length of the segments for a number of polyesters as 28–34 carbon atoms.

Hence the molar volume occupied by the segments (about \(500\ \text{cm}^3\)) and the value

\[ \Delta S^{*}=6.8-0.5Z_w^{1/2}. \]

The latter relation shows that, in flow, a segment receives an additional (rotational) entropy of 6.8 CGS units and loses in entropy an amount depending on the total length of the molecule. The second term may be interpreted as arising from the bond between segments, which must exist in order that the molecule as a whole may be displaced. The dependence of the viscosity on the chain length \(Z\) may be represented in the form

\[ \frac{1}{\eta}=Ke^{-BZ^{1/2}/R} \tag{25} \]

or in the equivalent form

\[ \frac{1}{\eta}=K\left(1-\frac{BZ^{1/2}}{Rn}\right)^n, \]

where \(n\) is a large number.

The latter relation is, in its form, identical with the expression for the probability that \(n\) events will occur successively, if \(BZ^{1/2}/Rn\) is the probability that the event under consideration will not occur.

An entangled hydrocarbon chain occupies a volume, on the average equal to \(3.4Z^{1/2}\) cubic Å. If \(n\) is the number of mobile segments, then each segment accounts for the \(1/n\) part of this volume. The chain itself occupies a volume of \(20Z\) cubic Å.

Then the probability that a segment will occupy a position unfavorable for motion is equal to

\[ 3.4Z^{1/2}/n:20Z=0.17Z^{1/2}/n. \]

Thus, the value of the parameter \(B\) proves to be equal to 0.34, whereas the value of \(B\) observed in Flory’s experiments was 0.5.

It is of interest to apply the same formulas to the viscosity of liquid sulfur. The heat of activation of the viscous flow of sulfur is about 10 kcal. From the known heat of vaporization one can estimate the size of the mobile part of the molecule at 20 atoms. Such a segment has the value \(\Delta S^{*} = -37\). If this value of \(\Delta S^{*}\) is due to the term \(BZ^{1/2}\), then, taking \(B = 0.5\) for the average chain length of sulfur, we find a value of 5500 atoms!

VI. VISCOSITY OF POLYMER SOLUTIONS\(^{{69}}\)

As the measurements of Staudinger\(^{{80}}\) and other authors have shown, the viscosity of dilute polymer solutions may be represented in the form

\[ \eta = \eta_0(1 + kZ\varphi), \tag{26} \]

where \(\eta_0\) is the viscosity of the pure solvent, \(Z\) is the chain length, \(\varphi\) is the volume concentration of the polymer, and \(k\) is a constant characteristic of the given polymer. A formula of this type can be derived from the conception of a linear polymer molecule as an aggregate of spheres connected by rigid bonds directed at definite valence angles and moving in a continuous medium\(^{{37}}\).

In the case of polymer melts the viscosity may be expressed by Flory’s formula

\[ \eta_p = \frac{Nh}{V_p}\, e^{-\Delta S_p^{*}/R}\, e^{BZ^{1/2}/R}\, e^{\Delta H_p^{*}/RT}, \tag{27} \]

where \(Z\) is the chain length, and \(V_p\), \(\Delta S_p^{*}\), and \(\Delta H_p^{*}\) are the volume, entropy, and heat of activation of an individual segment. At intermediate concentrations the viscosity of a solution can be described approximately by a formula representing a certain combination of the relations for weak solutions and for melts of pure polymers.

For both limiting cases, without loss of generality, the viscosity may be represented in the form

\[ \eta = \frac{NhF_0}{VF^{*}}, \tag{28} \]

where \(F_0\) and \(F^{*}\) are the free energies of the normal and activated states. If a polymer molecule is in a concentrated solution, then some of its segments are surrounded by polymer molecules and others by solvent molecules. In a first approximation one may assume that those segments which are surrounded by solvent move as if they were in a dilute solution, whereas segments in the environment of polymer molecules move as in a melt of the pure polymer; moreover, the interaction between different parts of one and the same molecule may be neglected. Then, with the aid of the general relation (28), the viscosity of a concentrated polymer solution can be represented as

\[ \eta = \frac{Nh}{V_{\mathrm{cp}}} \left(\frac{F_0}{F^{*}}\right)_{\mathrm{dil}}^{1-\varphi} \left(\frac{F_0}{F^{*}}\right)_{\mathrm{polymer}}^{\varphi}. \tag{29} \]

The method of averaging the slowly varying factor \(Nh/V\) is immaterial. The volume concentration of polymer \(\varphi\) is a measure of how often polymer molecules can enter the surroundings of other polymer molecules.

In averaging equation (29), explicit expressions for the viscosity in the two limiting cases have not yet been used. If, in the limiting case of a dilute solution, the viscosity is expressed by the Staudinger equation [equation (26)] and, in the limiting case of a pure polymer melt, by the Flory equation [equation (27)], the viscosity of the concentrated solution will be

\[ \eta=\left[\eta_0(1+kZ\varphi)\right]^{1-\varphi} \left[ \frac{Nh}{V_p} e^{-\frac{\Delta S_\eta^*}{R}} e^{\frac{BZ^{1/2}}{R}} e^{\frac{\Delta H_\eta^*}{RT}} \right]^\varphi . \tag{30} \]

For dilute solutions, equation (30) may be written in the equivalent form

\[ \frac{\ln \eta/\eta_0}{\varphi} = \left[ kZ+\frac{BZ^{1/2}}{R} +\frac{\Delta H_\eta^*-\Delta H_0^*}{RT} -\frac{\Delta S_\eta^*-\Delta S_0^*}{R} -\ln \frac{V_p}{V_0} \right] + \varphi\left[-\frac{k^2Z^2}{2}-kZ\right] +\varphi^2\left[\frac{k^3Z^3}{3}+\frac{k^2Z^2}{2}\right] +\ldots, \tag{31} \]

which leads to a logarithmic increase of the viscosity with concentration. On extrapolation to zero concentration, formula (31) gives the “true viscosity”\(^{46}\).

We shall now proceed to check the applicability of equations (30) and (31).

1. Dependence on molecular weight

Equation (31) makes it possible to find the dependence of the true viscosity on molecular weight. Namely, as is seen from equations (31) and (26), the Staudinger constant \(K_m\) (to within a constant depending on the choice of the system of units) will be equal to

\[ \frac{1}{Z} \left[ \frac{\ln \frac{\eta}{\eta_0}}{\varphi} \right]_{\varphi=0} = k+\frac{B}{RZ^{1/2}}+\frac{P}{Z}, \tag{32} \]

where the last three (small) terms in the first bracket of equation (31) have been combined and denoted by \(P\).

In the case of a very large molecular weight, the second and third terms in (32) may be omitted, and \(K_m\) becomes a constant independent of molecular weight. However, with decreasing molecular weight one may expect the value of \(K_m\) to shift toward larger numerical values. Experimental data on solutions of polyvinyl chlorides and polystyrenes do indeed reveal this increase of \(K_m\)\(^{81,85,86}\). Other examples may be provided by polyethylene oxide polymer\(^{21}\) and polyethers\(^{20,45}\).

2. Dependence on concentration

As is seen from equation (31), if the dependence of \(\ln \frac{\eta}{\eta_0}\) on \(\varphi\) is represented on a logarithmic scale, then there should be observed a systematic deviation from direct proportionality—namely, a decrease of

\[ \frac{\ln \eta/\eta_0}{\varphi} \]

with increasing \(\varphi\). Such a course of the curve has been noted by many investigators. It is clearly seen in Fig. 8, which shows the dependence of \(\ln \eta/\eta_0\) on \(\varphi\) for a solution of cellulose acetate in cyclohexanone\(^{44}\).

Fig. 8. Effect of concentration on the viscosity of polymer solutions (cellulose acetate in cyclohexanone).

From independent measurements of molecular weight and the experimental viscosity curve one can find the values of the parameters \(k\) and \(B\). Table 3 gives their values for several types of polymers. We see that the values of the parameter \(k\) have the same order of magnitude as the length of segments of polymer molecules determined from osmotic pressure. With increasing rigidity of the polymer molecule, both of these quantities increase. However, the inaccuracy of the data does not allow one to go beyond such a purely qualitative conclusion.

Table 3

Viscosities of high polymers

Substance	\(k\)	\(B/R\)
Nitrocellulose \(^{1,17,18,35}\)	0.00845	2.33
Cellulose acetate \(^{17,56,64}\)	0.0194	2.21
Complex polyesters \(^{19}\)	0.025	0.62
Polystyrene \(^{35}\)	0.0447	0.845
Rubber \(^{17,18}\)	0.0835	3.27

Solutions of globular macromolecules can be studied by the same methods as solutions of linear macromolecules. In this case it turns out that: 1) the viscosity of the solution is well described

by the Einstein equation^14 and by Guth and Simha,^33 derived from hydrodynamic considerations:

\[ \eta=\eta_{0}^{*}\left(1+\frac{5}{2}\varphi+\frac{109}{14}\varphi^{2}+\cdots\right). \tag{33} \]

As examples one may cite the phenol-formaldehyde sol of a resin^44 or the sernistic sol.^64 2) The numerical value of the viscosity of solutions of globular macromolecules is one or two orders of magnitude smaller than that of linear macromolecules, since in the case of globular macromolecules the expression for the viscosity lacks the large terms \(kZ\) and \(BZ^{1/2}\). Thus, for example, already 2% solutions of rubber or cellulose nitrate are extremely viscous, whereas obtaining 70% solutions of Batavian dammar resin^55 presents no difficulty. 3) A noticeable increase is observed in the viscosity of phenolic-resin solutions with increasing degree of polymerization.

3. Dependence on the Solvent

In earlier works it was repeatedly pointed out that the intrinsic viscosity of solutions is one and the same for all solvents, provided one confines oneself to good homopolar solvents.

However, it is well known that for every high polymer there exist solvents giving both viscous and low-viscosity solutions.

From the point of view of the polymer molecule, a “good” solvent is evidently one whose molecules tend to surround and group themselves near the polymer molecule in such a way that the latter finds itself as if in an effectively more dilute solution. In this case the polymer molecules encounter one another so rarely that their segmental nature is only weakly manifested, the osmotic pressure approaches that of an ideal solution, and the viscosity decreases. The intrinsic viscosity of rubber in benzene decreases by 40% upon addition of 15% methyl alcohol, while the osmotic pressure becomes practically equal to the pressure in an ideal solution.^30

The effective volume concentration of the polymer in a nonideal solution may be written in the form \(\gamma\varphi\), where \(\gamma\) is the activity coefficient. Then the viscosity of the solution will have the form

\[ \eta=[\eta_{0}(1+kZ\varphi)]^{1-\gamma\varphi}(\eta_{p})^{\gamma\varphi}. \tag{34} \]

For solutions of the same concentration in two different solvents, equation (34) gives

\[ \ln \frac{\eta_{1}/\eta_{01}}{\eta_{2}/\eta_{02}} = \varphi\left[ (\gamma_{1}-\gamma_{2})\ln \frac{\eta_{p}/\eta_{01}}{1+kZ\varphi} -\gamma_{2}\ln\frac{\eta_{01}}{\eta_{02}} \right]. \tag{35} \]

The second term in the brackets may be considered small in comparison

with the first. The difference \((\gamma_1-\gamma_2)\) will be positive or negative depending on which of the solvents is the better one. In a small concentration interval it may be approximately assumed that

Fig. 9. Effect of the solvent on the viscosity of polymer solutions.

the right-hand side of (35) contains a linear function of the concentration \(\varphi\), if the weak dependence of \(\gamma_1\) and \(\gamma_2\) and of the logarithmic factor on \(\varphi\) is neglected. It is possible that the length of the segments \(Z\) also changes from solvent to solvent, but this effect cannot alter the qualitative picture.

Kurtz, Garvey, and Juikin\(^ {49}\) investigated the viscosity of phenolic and natural rubbers in a number of technical solvents and obtained, for the ratios of the kinematic viscosity of one solution to the kinematic viscosity of another solution, as functions of concentration, a family of curves shown in Fig. 9.

4. Dependence on temperature

In both limiting cases of dilute solutions and polymer melts, the dependence of viscosity on temperature has the form \(\eta=\mathrm{const.}\exp(\Delta H^{\ne}/RT)\), so that the viscosity decreases with increasing temperature. For nonideal solutions, however, there is an additional effect. In the case of a good solvent each polymer molecule is surrounded by a larger number of solvent molecules than it would be in a random distribution, i.e. \(\gamma<1\). As the temperature rises, the distribution of solvent molecules approaches a random one, so that \(\gamma\) approaches unity. However, the approach of the solution to the ideal state in this case means that polymer molecules more often meet and collide with one another. Therefore the weight of the viscosity of the purely polymeric type in the total viscosity of the solution increases. As a result it may even turn out that the viscosity of the solution increases with increasing temperature.

More precisely, in this case the dependence of viscosity on temperature is determined from the relation

\[ \frac{d}{dT}\ln \eta/\eta_0 = \varphi\left[ \frac{d\gamma}{dT}\, \frac{\ln(\eta_p/\eta_0)}{\ln(1+kZ\varphi)} - \gamma\,\frac{\Delta H_p^{\ne}-\Delta H_0^{\ne}}{RT^2} \right]. \tag{36} \]

The second term in brackets gives the usual temperature coefficient

viscosity. \(\Delta H_p^{*}\) is usually about \(10\) kcal, and \(\Delta H_0^{*}\) about \(2\) kcal. If \(\gamma\) is less than unity, \(d\gamma/dT\) is positive. The numerator of the coefficient at \(d\gamma/dT\) increases rapidly with chain length, so that in polymers of large molecular weight the positive term becomes predominant.

The predicted course of the temperature dependence is illustrated by the curves in Fig. 10. Experimental data of this type have been obtained for polystyrenes and polyvinyl chlorides \(^{84,85}\). For spherical or highly branched molecules, in which the segmental type of motion plays only a small role or does not occur at all, viscosity always decreases with increasing temperature, as is seen from the lower straight line in Fig. 10.

Fig. 10. Effect of temperature on the viscosity of polymer solutions.

5. Dependence on the velocity gradient

Under the influence of viscous stresses in a liquid, intermolecular bonds are constantly being broken and changed. In the case of small molecules, restoration of the normal positions of equilibrium occurs almost instantaneously. In the case of polymer molecules, however, the situation is different. If bonds between two polymer molecules are broken, then before their restoration occurs the molecules may be carried along by the solvent and moved away from one another. During flow a certain equilibrium state is established, in which the numbers of bonds being broken and being restored are equal to one another. In this equilibrium state the number of polymer molecules surrounded by solvent molecules is somewhat greater than in a solution at rest, i.e. the sign of the derivative \(d\gamma/d\sigma\), where \(\sigma\) is the viscous stress, is negative.

Therefore

\[ \frac{d}{d\sigma}\ln \eta/\eta_0 = \varphi \left[ \frac{d\gamma}{d\sigma} \frac{\ln(\eta_p/\eta_0)}{\ln(1+kZ\varphi)} + \gamma \frac{d}{d\sigma}\ln \frac{\eta_p/\eta_0}{1+kZ\varphi} \right]. \tag{37} \]

The second term, characterizing the dependence of the limiting viscosities on viscous stresses, is apparently small. Consequently: 1) the thixotropy effect should increase with increasing solution concentration, and 2) since the numerator of the coefficient at \(d\gamma/d\sigma\) increases rapidly with increasing polymer-chain length, the thixotropy effect should increase with molecular weight.

Both of these statements are in agreement with experimental data obtained for polystyrene solutions \(^{86}\).

VII. DIFFUSION IN SIMPLE LIQUIDS

An expression for the diffusion coefficient in simple liquids can be derived from the same concepts as the expression for viscosity^88 (see Fig. 4).

Fig. 11: Model for diffusion.

The concentration and activity coefficient of the solution at various positions of the diffusing molecule are presented in Fig. 11.

The general expression for the rate of a process proceeding with activation has the form

\[ k'=\frac{kT}{h}\,e^{-\Delta F_0^*/RT}\frac{\gamma_*}{\gamma^*}. \tag{38} \]

The rates of the forward and reverse processes will be

\[ v_f=n_1\lambda\frac{kT}{h}e^{-\Delta F_0^*/RT} \frac{\gamma_1}{\gamma_1\left[1+\frac{\alpha\lambda}{\gamma_1}\frac{d\gamma_1}{dx}\right]}, \tag{39} \]

\[ v_b=\left(n_1+\lambda\frac{dn_1}{dx}\right)\lambda\frac{kT}{h}e^{-\Delta F_0^*/RT} \frac{\gamma_1\left[1+\frac{\lambda}{\gamma_1}\frac{d\gamma_1}{dx}\right]} {\gamma_1\left[1+\frac{\alpha\lambda}{\gamma_1}\frac{d\gamma_1}{dx}\right]}. \tag{40} \]

The resultant rate of the process is

\[ V=-\frac{dn_1}{dx}\lambda^2\frac{kT}{h}e^{-\Delta F_0^*/RT} \left[1+\frac{d\ln\gamma_1}{d\ln n_1}\right]. \tag{41} \]

The diffusion coefficient \(D\) is defined as

\[ V=-D\frac{dn_1}{dx}. \tag{42} \]

Combining equations (41) and (42), we have

\[ D=\lambda^2\frac{kT}{h}e^{-\Delta F_0^*/RT} \left[1+\frac{d\ln\gamma_1}{d\ln n_1}\right]. \tag{43} \]

If the molar volumes of the “solute” \(V_1\) and the “solvent” \(V_2\) do not differ very greatly from one another, equation (43) assumes the form

\[ D=\lambda^2\frac{kT}{h}e^{-\Delta F_0^*/RT} \left[\frac{d\ln a_1}{d\ln N_1}\right], \tag{44} \]

\[ D\eta=\frac{\lambda_1 kT}{\lambda_2\eta} \left[\frac{d\ln a_1}{d\ln N_1}\right]. \tag{45} \]

The close connection existing between the process of diffusion and the viscosity of flow is well confirmed by the correspondence between the heats of activation of both processes.

Figure 12 shows the dependence of \(D\eta\) and \(D\eta/(d\ln a_1/d\ln N_1)\) on concentration in the chloroform–ether system. We see that there is

good agreement with formula (45), which requires direct proportionality between \(D_\eta\) and \(d\ln a_1/d\ln N_1\)^{51}.

The absolute values of the diffusion coefficients calculated from equation (45) prove to be 2–5 times larger than those observed experimentally. This may be explained by the fact that, during flow, the molecules are oriented in such a way that \(\lambda_1\) represents their smallest dimension, while \(\lambda_2\lambda_3\) is the largest area. Indeed, in pure liquids optical anisotropy arises during flow (the Maxwell effect). The magnitude of this anisotropy can readily be calculated^{74}. Let the refractive index of a flowing molecule during flow be oriented so that its area subjected to the action of viscous stresses is \(\lambda'_2\lambda'_3\). For some average orientation in the stream, when the area of the molecule subjected to the action of viscous stresses is \(\lambda^0_2\lambda^0_3\), its refractive index will be \(n_0\). If the two orientations of the molecule differ from one another in energy and the molecules are distributed between the two orientations according to Boltzmann, then

Fig. 12. Influence of concentration on diffusion.

\[ \frac{n'}{n_0} = \exp\left[-f(\lambda'_2\lambda'_3-\lambda^0_2\lambda^0_3)x/kT\right] = 1-f(\lambda'_2\lambda'_3-\lambda^0_2\lambda^0_3)x/kT \tag{46} \]

where an additional averaging must be carried out over all values from zero to the top of the barrier. Then the experimentally measured value of the optical anisotropy will be equal to

\[ \frac{\Delta n}{n} = \frac{n_0-n'}{n_0} = 1-\frac{n'}{n_0} = f(\lambda'_2\lambda'_3-\lambda^0_2\lambda^0_3)/kT \left\{ \frac{ \displaystyle \int_0^{\lambda/2} x e^{-E/RT}\,dx }{ \displaystyle \int_0^{\lambda/2} e^{-E/RT}\,dx } \right\}. \tag{47} \]

The last expression has the same form as the experimental formula established by Maxwell,

\[ \frac{\Delta n}{n}=fM, \tag{48} \]

where \(M\) is a constant characteristic of the given liquid.

Let us estimate the value of \(M\) for benzene at \(300^\circ\mathrm{K}\). The product \(\lambda'_2\lambda'_3\) will be about \(16\cdot10^{-16}\ \mathrm{cm}^2\), and \(\lambda^0_2\lambda^0_3\) about \(11\cdot10^{-16}\ \mathrm{cm}^2\). The quantity in braces in formula (47) can be estimated by numerical integration for a barrier equal to—

of 3 kcal. It turns out to be equal to \(0.145\cdot \lambda/2\). In this case the Maxwell constant is approximately \(0.52\cdot 10^{-10}\). In Table 4, for comparison, the measured values of this constant for a number of liquids are given\(^8\).

Table 4

Maxwell constant

Substance	\(M\cdot 10^{10}\)	Substance	\(M\cdot 10^{10}\)
o-dichlorobenzene	1.85	Toluene	1.04
p-xylene	1.84	Phenyl ethanol	0.67
Mesitylene	1.31	Benzene	0.64
m-xylene	1.29	Heptanol-1	0.41
Chlorobenzene	1.22	Carbon tetrachloride	0.06
o-xylene	1.21	Cyclohexane	0.03

VIII. DIFFUSION OF MACROMOLECULES

A large number of experimental studies on the diffusion of macromolecules have been carried out for globular macromolecules (for example, proteins), for which one may expect the applicability of hydrodynamic theory\(^ {50, 62, 65, 68}\). For linear macromolecules the number of studies devoted to determining diffusion coefficients is small. In Table 5

Table 5

Diffusion coefficients of high polymers

	Concentration, g/100 cm\(^3\)	\(D\cdot 10^7\), cm\(^2\)/sec	\(D\eta\cdot 10^7\)	Mol.
Methyl cellulose	0.5	4.38	4.45	14,100
Methyl cellulose	0.5	3.00	3.05	24,300
Methyl cellulose	1.0	3.07	3.11	24,300
Methyl cellulose	0.5	2.43	2.47	38,100
Methyl cellulose	1.0	2.26	2.31	38,100
Acetic-acid cellulose	0.61	10.85	3.46	20,000
Acetic-acid cellulose	1.15	10.9	3.48	20,000
Acetic-acid cellulose	0.5	4.48	1.43	53,000
Acetic-acid cellulose	1.46	4.30	1.37	53,000
Acetic-acid cellulose	0.516	3.29	1.05	90,000

are presented the results of Rulson’s measurements on the diffusion of cellulose derivatives\(^ {67}\). As is evident from this table, the diffusion coefficients, roughly speaking, are inversely proportional to the molecular

weight, but the diffusion coefficient changes somewhat more slowly than the molecular weight.

As was to be expected, the magnitude of the product \(D\eta\) for solutions of methylcellulose and cellulose acetate is approximately the same. The diffusion coefficient does not show any regular variation with the concentration of the solution, but its values in dilute solutions fall noticeably outside the range of values given in Table 5.

IX. SEDIMENTATION RATE

If a liquid is placed in a centrifuge rotating with angular velocity \(\omega\), then a force \(m\omega^2 x\) will act on a molecule moving in the liquid, where \(m\) is the difference between the mass of the molecule and the mass of the solvent in the same volume, and \(x\) is the distance from the axis of rotation. This force will assist the advance of the molecule to a new equilibrium position when it moves in one direction (i.e. over a length \(\alpha\lambda\)) and will retard it when it moves in the other direction [i.e. over a length \((1-\alpha)\lambda\)]. The rates of the forward and reverse processes are

\[ v_f = n_1 \lambda \frac{kT}{h} e^{-\Delta F^{*}/RT} \left[ e^{m\omega^2 x\alpha\lambda/kT} \right] \frac{\gamma_1}{ \gamma_1\left[1+\frac{\alpha\lambda}{\gamma_1}\frac{d\gamma_1}{dx}\right] }, \tag{49} \]

\[ v_b = \left(n_1+\lambda\frac{dn_1}{dx}\right) \lambda \frac{kT}{h} e^{-\Delta F^{*}/RT} \left( e^{-m\omega^2 x(1-\alpha)\lambda/kT} \right) \times \frac{ \gamma_1\left[1+\frac{\lambda}{\gamma_1}\frac{d\gamma_1}{dx}\right] }{ \gamma_1\left[1+\frac{\alpha\lambda}{\gamma_1}\frac{d\gamma_1}{dx}\right] }. \tag{50} \]

After simplification, the resultant velocity may be represented in the form

\[ v = \frac{n_1\lambda^2 m\omega^2 x}{h} e^{-\Delta F^{*}/RT} - \frac{dn_1}{dx}\lambda^2\frac{kT}{h} e^{-\Delta F^{*}/RT} \cdot \left[ 1+\frac{d\ln\gamma_1}{d\ln n_1} \right]. \tag{51} \]

But, as is seen from (43), the second term in equation (51) is simply the diffusion coefficient. The first term is equivalent to the usual Svedberg sedimentation equation

\[ s = \frac{dx/dt}{n_1\omega^2 x} = \frac{m\lambda^2}{h} e^{-\Delta F^{*}/RT} \tag{52a} \]

\[ = \frac{m\lambda_1}{\eta h\lambda_2\lambda_3} \tag{52b} \]

\[ = \frac{mD}{kT} \bigg/ \left| \frac{d\ln a_1}{d\ln N_1} \right|, \tag{52c} \]

where \(s\) is the sedimentation constant of the molecule1. As is seen from equation (52), this latter quantity is inversely proportional to the viscosity. In practice, the correctness of observed values is usually checked

of constant sedimentation by constancy of the product of it by the viscosity\(^ {80}\). Fig. 13 gives typical results obtained for the sedimentation velocities of polystyrenes in chloroform\(^ {76}\). On the abscissa axis is plotted the concentration of the solution; on the ordinate axis—the value of the sedimentation constant. The shaded circles represent the value of the sedimentation constant, and the unshaded circles—the value of the product \(D\eta\). The radius of a circle represents the probable error of the corresponding measurement.

Fig. 13. Sedimentation constant of polystyrene in chloroform.

We see that, within the limits of experimental accuracy, the product \(s\) by \(\eta\) indeed remains constant. It should be noted that parallelism in the dependences of the sedimentation constant and of the diffusion coefficient on the concentration of the solution is not observed.

In carrying out calculations for centrifugation it is usually assumed that \(s\) (or \(D\) and \(\eta\)) remains constant in a given experiment. However, the solution in the centrifuge experiences a hydrostatic pressure, the magnitude of which along the centrifuge chamber changes from zero to several hundreds, and in some cases even thousands, of atmospheres. At such pressures the viscosity of water increases, in comparison with the viscosity under normal conditions, by 50%, and the viscosity of organic solvents—by several times\(^8\). Consequently, the value of \(s\) proves to be variable along the chamber.

This circumstance can be detected in measurements of the sedimentation velocity at different rotational speeds of the rotor. If these measurements indicate a decrease of \(s\) with increasing speed, then in calculating the ultracentrifuge it is necessary to take into account the change of viscosity with speed [cf. equation (20)].

Measurements of the sedimentation velocity of methylcellulose, cellulose acetate, and polychloroprene indicate a weak change of the sedimentation constant under considerable variations of molecular weight\(^ {47,77}\). This fact also indicates that the motion of a polymer molecule occurs by displacement of more or less independent segments.

X. DIELECTRIC RELAXATION

If the dielectric constant of a dipolar substance is measured over a sufficiently broad frequency interval, then at some frequency a sharp decrease in the dielectric constant is observed (Fig. 14, upper curve).

For an explanation of this “anomalous dispersion,” Debye put forward the hypothesis that dipoles are oriented by the applied field and that at high frequencies this orientation ceases, since the dipoles can no longer follow the external field because of the forces of viscous friction of the medium[^12]. The well-known Debye equation reads:

\[ \alpha=\alpha_{\infty}+\frac{\mu^{2}}{3kT}\frac{1}{1+i\omega\tau}, \tag{53} \]

where \(\alpha\) is the polarizability, \(\alpha_{\infty}\) is its value at optical frequency, \(\mu\) is the permanent dipole moment of the molecule, \(\omega\) is the frequency in radians per second, \(\tau\) is the relaxation time, and \(i\) is equal to \(\sqrt{-1}\). The relaxation time represents the average interval of time required for the molecules to return to a random distribution of spatial orientations after the applied orienting field has been removed. Debye assumes that the viscous friction acting on a sphere rotating in a liquid is expressed by Stokes’ law, and obtains for the relaxation time the following expression:

Fig. 14. Anomalous dispersion of the dielectric constant.

\[ \frac{1}{\tau}=\frac{kT}{4\pi r^{3}\eta}, \tag{54} \]

where \(r\) is the radius of the sphere and \(\eta\) is the viscosity of the medium.

Among other hypotheses proposed to explain the phenomenon of anomalous dispersion, the following may be mentioned:

a) displacement of charged particles under the action of elastic and viscous forces[^19,^61],

b) polarization in a two-layer dielectric[^84],

c) high-frequency resistance at the electrode–dielectric boundary[^30],

d) inclusion of suspensions in the form of dielectric spheres in a dielectric medium[^31,^83],

e) conductivity of particles placed in a dielectric medium[^43],

f) the presence of conducting layers in a dielectric medium[^58].

It turns out that the expression for the dielectric constant derived from all the theories listed includes the frequency factor \(1/(1+i\omega\tau)\). Hence one may draw the remarkable conclusion that such a form of the dependence of the dielectric constant on frequency does not depend on the nature of the mechanism that gives rise to this dependence.

And indeed, a dependence of the dielectric constant on frequency having the Debye character has been obtained on the basis of a very general scheme of the phenomenon[^98].

Suppose that the molecule can be in only two states. Quantities pertaining to these states will be supplied with the indices 1 and 2. The rate of change of the number of molecules in the first state \((-dN_1/dt)\) due to their transition into the second state will be

\[ -\frac{dN_1}{dt}=N_1k_1-N_2k_2, \tag{55} \]

where \(N_1\) and \(N_2\) are the numbers of molecules and \(k_1\) and \(k_2\) are the specific rates (probabilities) of the forward \(1\to2\) and reverse \(2\to1\) transitions.

Let the number of molecules in both states and the transition probabilities be changed by an applied external force \(X\). Suppose that this force is so weak that the new numbers of molecules and transition probabilities can be expressed through the same quantities in the absence of the field by means of the relations

\[ \left. \begin{aligned} N_1&=N_1^0+X\frac{dN_1}{dX}, & k_1&=k_1^0+X\frac{dk_1}{dX},\\ N_2&=N_2^0-X\frac{dN_1}{dX}, & k_2&=k_2^0+X\frac{dk_2}{dX}. \end{aligned} \right\} \tag{56} \]

Since in the absence of the field the system is in equilibrium, we have

\[ \frac{N_2^0}{N_1^0}=\frac{k_1^0}{k_2^0}=e^{\Delta F/RT}, \tag{57} \]

where \(\Delta F\) is the difference of the free energies of the two states.

From equation (57) we find

\[ \frac{dk_1}{dX} = k_2^0 e^{\Delta F/RT}\left(\frac{d}{dX}\frac{\Delta F}{RT}\right) + e^{\Delta F/RT}\frac{dk_2}{dX} = k_2^0\frac{N_2^0}{N_1^0}\left(\frac{d}{dX}\frac{\Delta F}{RT}\right) + \frac{N_2^0}{N_1^0}\frac{dk_2}{dX}. \tag{58} \]

Substituting equations (56) and (58) into (55), we obtain

\[ -\frac{dN_1}{dt} = \left[N_1^0+X\frac{dN_1}{dX}\right] \left[ k+Xk_2^0\frac{N_2^0}{N_1^0} \left(\frac{d}{dX}\frac{\Delta F}{RT}\right) + X\frac{N_2^0}{N_1^0}\frac{dk_2}{dX} \right] - \]

\[ - \left[N_2^0-X\frac{dN_1}{dX}\right] \left[k_2^0+X\frac{dk_1}{dX}\right] = \]

\[ = X\left[ k_1^0\frac{dN_1}{dX} + k_2^0\frac{dN_1}{dX} + k_2^0N_2^0 \left(\frac{d}{dX}\frac{\Delta F}{RT}\right) \right]. \tag{59} \]

If the applied field is periodic, depending on time according to the law \(X=X_0e^{i\omega t}\), equation (59) is transformed into the simple differential equation

\[ \frac{d}{dt}\left(X\frac{dN_1}{dX}\right) + (k_1^0+k_2^0)\left(X\frac{dN_1}{dX}\right) = -k_2^0N_2^0 \left(\frac{d}{dX}\frac{\Delta F}{RT}\right) X_0e^{i\omega t}, \tag{60} \]

the solution of which can immediately be written in the form

\[ - X \frac{dN_1}{dX} = \frac{k_2^0 N_2^0 \left(\dfrac{d}{dX}\dfrac{\Delta F}{RT}\right) X_0 e^{i\omega t}} {i\omega + \left(k_1^0 + k_2^0\right)} . \tag{61} \]

Equation (61) can also be rewritten in the equivalent form

\[ - \frac{dN_1}{dX} = \frac{N\left(\dfrac{d}{dX}\dfrac{\Delta F}{RT}\right)} {2\left[1+\operatorname{ch}\left(\dfrac{\Delta F}{RT}\right)\right]} \, \frac{1} {1+\dfrac{i\omega}{k_1^0+k_2^0}} . \tag{62} \]

The Debye form of the frequency dependence is thus obtained, for any mechanism of transitions from the first state to the second, for any property of the system, i.e.

\[ A = A_1 + \frac{A_2}{1+i\omega \tau}, \tag{63} \]

where \(A\) is the quantitative expression of some property of the system, \(A_1\) is its part independent of frequency, and \(A_2\) is the part depending on frequency.

The constants \(A_1\) and \(A_2\) are usually expressed through the limiting values of \(A\) corresponding to zero and infinitely large frequency. At infinitely large frequency \(A_\infty=A_1\), at zero frequency \(A_0=A_1+A_2\). Consequently,

\[ A = A_\infty + \frac{(A_0-A_\infty)}{1+i\omega\tau}. \tag{64} \]

In particular, one of such properties of the system is the magnitude of its polarizability, determining the dipole moment induced in unit volume. Therefore one may write

\[ \alpha = \alpha_\infty + \frac{(\alpha_0-\alpha_\infty)}{1+i\omega\tau}. \tag{65} \]

In experiment one measures not the polarizability, but the dielectric constant of the medium. If the Clausius–Mosotti formula is valid, relating the polarizability and the dielectric constant,

\[ 4\pi \alpha/3 = \frac{\varepsilon-1}{\varepsilon+2}, \]

then equation (65) assumes the form

\[ \varepsilon = \varepsilon_\infty + \frac{(\varepsilon-\varepsilon_\infty)} {1+i\omega\tau\left(\dfrac{\varepsilon_0+2}{\varepsilon_\infty+2}\right)} . \tag{66} \]

The quantity \(\tau(\varepsilon_0+2)/(\varepsilon_\infty+2)\) is sometimes called the dielectric relaxation time \(\tau^*\). For a number of substances the value of \(\tau^*\) somewhat exceeds the value of \(\tau\). But for water, for example, \(\tau^*=23\tau\). A number of authors have studied the applicability of the Clausius–Mosotti formula to polar liquids and have come to the conclusion that the dependence of \(\varepsilon\) on \(\alpha\) is rather close to linear \(^{10,38,66,72,73,92,99}\). If \(\varepsilon\) depends linearly on \(\alpha\), multi-

the factor \((\varepsilon_0+2)/(\varepsilon_\infty+2)\) in the preceding formula should be replaced by unity.

Separating the real and imaginary parts, equation (64) may be rewritten in the form

\[ \varepsilon=\varepsilon_\infty+ \frac{(\varepsilon_0-\varepsilon_\infty)}{1+\omega^2\tau^{*2}} -i\,\frac{(\varepsilon_0-\varepsilon_\infty)\omega\tau^*}{1+\omega^2\tau^{*2}}, \tag{67} \]

which can also be written briefly as

\[ \varepsilon=\varepsilon'-i\varepsilon''. \tag{68} \]

In Fig. 14 the upper curve shows the behavior of the real part of the dielectric constant, and the lower curve that of its imaginary part. At the frequency \(\omega=1/\tau^*\) the imaginary part of the dielectric constant has the maximum value \(\varepsilon''=(\varepsilon_0-\varepsilon_\infty)/2\), while the real part has the mean value between the two limiting values \(\varepsilon_0\) and \(\varepsilon_\infty\), i.e. \(\varepsilon'=(\varepsilon_0+\varepsilon_\infty)/2\). Measuring the variation of the real or imaginary part of the dielectric constant with the frequency of the applied field provides a convenient method for measuring the relaxation time.

The imaginary part of the dielectric constant cannot be measured directly, but is obtained from the electrical conductivity of the system in an alternating electric field.^61 The latter is determined by the relation

\[ \gamma=\frac{1}{E}\frac{dq}{dt}, \tag{69} \]

where \(E\) is the electric field and \(q\) is the surface charge density. For a parallel-plate capacitor

\[ 4\pi q=\varepsilon E \tag{70} \]

and

\[ \gamma= \frac{1}{E_0e^{i\omega t}}\, \frac{\varepsilon'-i\varepsilon''}{4\pi}\, \frac{d}{dt}E_0e^{i\omega t} = \frac{i\omega}{4\pi}(\varepsilon'-i\varepsilon'') = \frac{\varepsilon''\omega}{4\pi} +i\,\frac{\varepsilon'\omega}{4\pi}. \tag{71} \]

The last formula shows that the imaginary part of the dielectric constant is determined by the real part of the electrical conductivity.

If dielectric dispersion arises because not one but several transitions occur in the system, the dielectric constant will again be expressed as a sum of terms of type (62), but with different relaxation times. In particular, if the system has infinitely many relaxation times forming a continuous distribution, the region of dispersion of the dielectric constant will be broader, the curve of dielectric losses likewise broader, and the maximum value of the losses smaller than \(\frac{1}{2}(\varepsilon_0-\varepsilon_\infty)\). Fuoss and Kirkwood, by inverting the formulas for the dielectric constant, found from experimental data the distribution of relaxation times.^23,100

A second conclusion that can be drawn from equation (62) is that the dielectric constant and the other corresponding quantities will be small if the energies of the two states between which transitions occur differ appreciably from one another: the hyperbolic cosine in the denominator increases rapidly if the difference of the free energies \(\Delta F\) exceeds \(1\, k\mathrm{cal}\). White arrived at a similar qualitative conclusion on the basis of other considerations\({}^{96}\). This conclusion is in agreement with the fact that the dielectric constant of a pure substance in the solid state is smaller than in the liquid state.

Third, equation (62) shows that the relaxation time \(\tau\) is a quantity reciprocal to the specific frequency of transitions. Since the frequency dependence of the dielectric constant is not uniquely connected with the mechanism of relaxation, the requirement imposed on a complete theory should consist only in its being able to predict the value of \(\tau\) and \((\varepsilon_0-\varepsilon_\infty)\), and their dependence on the temperature and structure of the substance.

Thus, the Debye expression for \((\varepsilon_0-\varepsilon_\infty)\), obtained for freely rotating dipoles, has been confirmed by a large number of experimental studies for substances with polar molecules in vapors and dilute solutions. In the liquid state, however, expression (54) for the relaxation times leads to values of the viscosity of the liquid that are substantially smaller than those obtained from direct macroscopic measurements. This discrepancy ranges from ten to several thousand times.

The activated-complex method can be successfully applied to the consideration of relaxation times\({}^{16,23,87}\). The free energy of activation for dielectric relaxation can be found from the equation

\[ \frac{1}{\tau}=\frac{kT}{h}\,e^{-\Delta F^{*}/RT}, \tag{72} \]

whereas the free energy of activation for viscous flow is connected with the viscosity itself by the equation

\[ \frac{1}{\eta}=\frac{V}{Nh}\,e^{-\Delta F^{*}/RT}. \tag{73} \]

Figure 15 gives the values of both free energies for a series of liquids. We see that, for a whole series of liquids, the experimental points lie on a straight line with slope equal to unity. The agreement of the free energies indicates the existence of a close connection between the processes of dielectric relaxation and viscous flow, which is in qualitative agreement with the relaxation mechanism proposed by Debye. For comparison with this theory, equation (54) is shown by a dashed line in Fig. 15.

We see that quantitative agreement of the experimental data with Debye’s theory is absent. The conception of spherical

dipoles rotating in a continuous viscous medium leads, obviously, to a less satisfactory picture than the representation in terms of separate discrete molecules, whose motion is accompanied by rupture of bonds with neighboring molecules. The rotation of dipoles would occur much more freely than viscous flow in such associated liquids as water or glycerin, and also in solutions of polar molecules in viscous hydrocarbon solvents (for some such solutions dispersion is altogether absent; the experimental points for such solutions in Fig. 15 would lie considerably to the left of those plotted in the figure).

Fig. 15. Relation between dielectric constant and viscosity.

Data relating to solid dielectrics, such as, for example, neobutyl bromide and isoamyl bromide, are not plotted in Fig. 15, since the viscosity of solids can be estimated only in a very rough and inaccurate way. The dielectric dispersion of dilute solutions of high polymers was investigated for one of the polymers. As was to be expected, on the basis of the assumption of a segmental structure of polymer molecules, in the frequency range corresponding to the orientation of the molecule as a whole, dispersion is absent\(^7\).

The sizes of the parts of a polymer molecule that can be oriented by an external field have not yet been established. Kirkwood and Fuoss assume that each separate dipole in a polymer molecule is oriented independently, this orientation being restricted only by the directions of the valence bonds with its neighbors\(^29\).

Additional information concerning dielectric dispersion can be obtained if the free energy is divided into the heat of activation and the entropy term. Then equation (72) can be rewritten in the form

\[ \frac{1}{\tau}=\frac{kT}{h} e^{\Delta S^{*}/R} e^{-\Delta H^{*}/RT}. \tag{74} \]

From the temperature slope of the curve \(1/\tau\), one can calculate \(\Delta H^{*}\) and, subtracting it from \(\Delta F^{*}\), obtain the value of \(\Delta S^{*}\). Table 6 gives

Table 6

Dielectric dispersion of various substances

Substances	\(t^\circ C\)	\(\Delta F^*\)	\(\Delta H^*\)	\(\Delta H^*\) visc.	\(\Delta E\) vap.	\(\Delta S^*\)	Number of rotations of molecule
Liquids
Ethyl alcohol \(^{41}\)	25	2.90	5.7	3.44	8.93	9.40	\< 1
Propyl »	25	3.57	6.1	4.53	9.1	8.50	\< 1
Butyl »	25	3.90	6.4	4.61	9.7	8.40	\< 1
Amyl »	25	4.18	6.6	6	10.1	8.11	\< 1
Hexyl »	25	4.09	6.8	6	10.5	8.19	\< 1
Nitrobenzene in spindle oil \(^{58}\)	20	3.82	5.8	—	—	6.77	1
Spindle oil \(^{71}\)	25	8.84	16.4	—	—	25.3	1
Castor oil \(^{48}\)	1	9.95	16.7	—	—	24.6	1
Resin (wood)	101	15.3	23.9	—	—	22.2	1
Solids
Cetyl alcohol in paraffin \(^{23}\)	14	4.45	12.6	—	—	28.4	1
\(dl\)-camphor \(^{101}\)	−132	4.85	10.2	—	—	38.0	2–5
3, \(x\)-dichlorocamphor	−61	7.00	20.4	—	—	63.2	2–5
Cyclopentanol	−67	6.08	9.73	—	—	17.7	∼10
Isobutyl bromide \(^{2}\)	−157	3.79	23.1	—	—	167.1	∼10
Isoamyl bromide \(^{2}\)	−145	4.42	18.0	—	—	106.1	∼10
“Galowax”-chlorinated naphthalene \(^{57}\)	−51.5	7.77	31.0	—	—	104.8	5
Phthalic glycol \(^{49}\)	30	11.21	53.5	—	—	139.0	5
Polyvinyl chloride \(^{28}\)	99	16.9	43.0	—	—	70.3	5–10
Rubber 10% S \(^{68}\)	30	9.69	27.0	—	—	57.0	2
“Permitol” — tetrachlorodiphenyl \(^{23}\)	8.5	11.85	56.5	—	—	158.6	∼5
“Permitol” — tetrachlorodiphenyl \(^{23}\)	12.5	11.22	56.7	—	—	159.1	∼5
“Permitol” — tetrachlorodiphenyl \(^{23}\)	16.7	10.57	53.2	—	—	147.0	∼5
“Permitol” — tetrachlorodiphenyl \(^{23}\)	19.9	10.00	40.8	—	—	104.8	∼5
“Permitol” — tetrachlorodiphenyl \(^{23}\)	25.3	9.50	42.0	—	—	108.9	∼5
“Permitol” — tetrachlorodiphenyl \(^{23}\)	29.8	9.05	37.5	—	—	122.0	∼5
“Permitol” — tetrachlorodiphenyl \(^{23}\)	34.5	8.67	34.6	—	—	84.4	∼5
Glycerin \(^{57}\)	−65	9.10	38.2	—	—	140	5
Glycerin \(^{57}\)	−60	8.60	32.7	—	—	113	—
Glycerin \(^{57}\)	−50	7.66	24.7	—	—	76.3	—
Glycerin \(^{57}\)	−40	6.97	21.0	—	—	60.3	—
Glycerin \(^{57}\)	−20	5.68	17.6	—	—	47.2	—
Glycerin \(^{57}\)	0	4.75	14.3	—	—	35.0	2
Glycerin \(^{57}\)	20	4.22	11.1	—	—	23.5	∼1
Ice \(^{60}\)	−45.9	9.34	14.52	—	—	22.8	7
Ice \(^{60}\)	−32.6	9.01	14.52	—	—	22.9	7
Ice \(^{60}\)	−20.6	8.85	14.52	—	—	22.5	7
Ice \(^{60}\)	−11.9	8.62	14.52	—	—	22.6	7
Ice \(^{60}\)	−8.5	8.47	14.52	—	—	22.9	7
Ice \(^{60}\)	−7.5	8.49	14.52	—	—	22.7	7
Ice \(^{60}\)	−3.9	8.40	14.52	—	—	22.7	7
Ice \(^{60}\)	−2.8	8.39	14.52	—	—	22.7	7
Ice \(^{60}\)	−0.9	8.39	14.52	—	—	22.5	7

calculated in this way, \(\Delta H^{*}\) and \(\Delta S^{*}\) for a number of liquid and solid dielectrics. We see that the heat of activation \(\Delta H^{*}\) for dielectric relaxation proves to be considerably greater than the heat of activation for viscous flow, which is usually \(2\)—\(3\) kcal and only in rare cases reaches values of the order of \(10\) kcal.

Physically this means that, in order for a molecule to be able to begin rotating, it must break a larger number of bonds with its neighbors than in flow. Thus, for alcohols the heat of activation of dipole rotation is \(2/3\) of the heat of vaporization, whereas the heat of activation of viscous flow is only from \(2/3\) to \(1/2\) of this heat (cf. Table 6).

It is interesting to note that for some substances the entropy of activation reaches enormous values, of the order of \(150\) CGS units (Table 6).

Table 7

Dielectric dispersion of tetrasubstituted benzene derivatives

Substance	Ordinary \(kc\)	\(t^\circ C\)	\(\Delta F^{*}\)	\(\Delta H^{*}\)	\(\Delta S^{*}\)
Chloropentamethylbenzene	1	—80	8,88	8,53	—1,96
Chloropentamethylbenzene	3	—80	8,90	—	—1,92
Chloropentamethylbenzene	10	—68	9,00	—	—2,30
Chloropentamethylbenzene	30	—59	8,96	—	—2,01
Chloropentamethylbenzene	100	—42	9,13	—	—2,60
1,2-Dichlorotetramethylbenzene	1	—108	7,85	7,98	0,79
1,2-Dichlorotetramethylbenzene	3	—100	7,85	—	0,75
1,2-Dichlorotetramethylbenzene	10	—90	7,89	—	0,49
1,2-Dichlorotetramethylbenzene	30	—81	7,88	—	0,52
1,2-Dichlorotetramethylbenzene	100	—69	7,91	—	0,34
1,2,4-Trichlorotrimethylbenzene	1	—85	9,09	9,01	—0,43
1,2,4-Trichlorotrimethylbenzene	3	—77	9,07	—	—0,31
1,2,4-Trichlorotrimethylbenzene	10	—64	9,19	—	—0,86
1,2,4-Trichlorotrimethylbenzene	30	—55	9,13	—	—0,55
1,2,4-Trichlorotrimethylbenzene	100	—40	9,22	—	—0,90
1,2,3-Trichlorotrimethylbenzene	1	—90	8,76	8,80	0,22
1,2,3-Trichlorotrimethylbenzene	10	—70	8,83	—	—0,15
1,2,3-Trichlorotrimethylbenzene	100	—47	8,84	—	—0,18
1,2,3,4-Tetrachlorodimethylbenzene	1	—70	9,79	9,50	—1,43
1,2,3,4-Tetrachlorodimethylbenzene	10	—46	9,95	—	—1,98
1,2,3,4-Tetrachlorodimethylbenzene	100	—21	9,94	—	—1,74
1,2,3,5-Tetrachlorodimethylbenzene	1	—78	9,51	9,27	—1,23
1,2,3,5-Tetrachlorodimethylbenzene	10	—56	9,61	—	—1,57
1,2,3,5-Tetrachlorodimethylbenzene	100	—31	9,66	—	—1,61
Pentachloromethylbenzene	1	—42	11,34	11,27	—0,30
Pentachloromethylbenzene	3	—32	11,31	—	—0,25
Pentachloromethylbenzene	10	—18	11,39	—	—0,47
Pentachloromethylbenzene	30	—6	11,38	—	—0,41
Pentachloromethylbenzene	100	11	11,48	—	—0,74

Table 8

Dielectric dispersion of plasticized polyvinyl chloride

% tricresyl phosphate	\(t^\circ\mathrm{C}\) (\(f = 60\) cycles)	\(\Delta F^*\)	\(\Delta H^*\)	\(\Delta S^*\)
0	98	16.7	40.7	64.7
10	80	15.9	29.2	37.7
20	65	15.2	22.7	22.2
30	48	14.4	17.6	10.0
40	32	13.6	14.2	1.97
50	16	12.88	12.6	\(-0.97\)
60	3	12.88	12.1	\(-0.65\)

A molecule in the activated state may be found in a considerably larger number of rotational states and accordingly have a considerably greater rotational entropy than a nonactivated molecule, which leads to large values of \(\Delta S^*\). In the case of especially large values of \(\Delta S^*\), apparently the rotation of entire groups of molecules occurs. The rotational entropy of an individual molecule can be estimated from the magnitude of the entropy of fusion or simply from the dimensions of the molecule; then, on the basis of the observed values of \(\Delta S^*\), one can find the number of rotating molecules; it is given in the last column of Table 6. For ordinary liquids only about one molecule rotates. In alcohols, apparently, not the whole molecule rotates, but only some part of it, for example the group \(\mathrm{CH_2OH}\). In solids the number of rotating molecules in a group is about 5. If polar molecules that are in the normal state can rotate freely, then upon their transition to the activated state the increase in entropy should be very small, or even equal to zero. White, Biggs, and Morgan\({}^{97}\) did in fact observe no increase in entropy upon activation of the molecules of hexasubstituted benzenes (Table 7). The increase in entropy was absent down to the lowest temperatures used in these measurements.

Fig. 16. Dielectric relaxation for plasticized polyvinyl chloride.

The assumption of free rotation is also in agreement with the results of these investigators, who observed no rotational transitions in the substances studied down to the lowest temperatures.

The magnitude of the activation entropy in dielectric relaxation makes it possible to judge quite accurately the presence of rotation in the solid state. Table 6 gives two examples—glycerin and permytol.

With increasing temperature, \(\Delta S^*\) decreases markedly, which indicates that in the normal state the freedom of rotation of the molecules increases more and more. Another example is the system of polyvinyl chloride and tricresyl phosphate (Table 8 and Fig. 16).

The value of \(\Delta S^*\) for pure polyvinyl chloride is \(65\ E V\). It decreases sharply upon addition of tricresyl phosphate, and when 40% of the latter is introduced it becomes zero. Further addition of tricresyl phosphate has no appreciable effect on \(\Delta S^*\). Thus, when 40% or more tricresyl phosphate is introduced into polyvinyl chloride, the dipoles acquire almost complete freedom of rotation.

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