Abstract
This review of modern theories of viscous flow addresses the question of the nature of the liquid state as developed by Bernal and Eyring, and gives the greatest attention to the theories of viscous flow constructed by them and their followers, first, because they are prominent representatives of two directions in the development of the problem of the liquid state, and second, because they proceed from more elaborately developed concepts of the nature and structure of liquids than had been done previously.
Full Text
Modern Theories of Liquid Viscosity
A. A. Leont’eva, Moscow
Introduction
The study of the nature of the liquid state, which at present is of such urgent importance, is developing mainly along two lines. One of them, relying on the results of X-ray analysis, regards the liquid state as being close in nature to the solid state and as representing a “spoiled” picture of an ideally ordered crystalline structure.1
The other line proceeds from the conception of a liquid as a real gas. In this case the theoretical treatment of the question of the nature of a liquid is based on the methods of statistical mechanics developed for the gaseous state.
Recently the first line, apparently, has been gaining an ever larger number of adherents. This is reflected in the fact that most of the theories of the viscous flow of a liquid proposed in recent times proceed from conceptions of the liquid state as a state close to the solid.
For the construction of theories of viscous flow it is highly essential what conceptions of the nature of the liquid the author of the theory proceeds from, since the times of a purely phenomenological treatment of the question have already passed, and before every author of a theory there arises the question of what constitutes the elementary process of flow and what basic characteristics of the liquid state must be specified in order to create a theory capable, in good agreement with experiment, of giving a quantitative expression for the variation of viscosity with temperature, pressure, etc.
In the present review of modern theories of viscous flow, the question of the nature of the liquid state will be touched upon as it is developed by Bernal and Eyring,1 and the greatest attention is devoted to the theories of viscous flow constructed by them and by their followers, first, because they are vivid representatives of the two directions in the development of the problem of the liquid state, and second, because they proceed from more fully developed conceptions of the nature and structure of liquids than had been done earlier.
Turning to a review of theories of viscosity, we shall begin with the theory of Ya. I. Frenkel, one of the first exponents of those views on the nature of the liquid state that bring it closer to the solid state.
THEORY OF Ya. I. FRENKEL
In B. V. Bak’s review “The Present State of the Theory of Viscosity”² the theory of Ya. I. Frenkel is only mentioned; therefore a more detailed exposition of it is given here. As is known, in deriving the formula for viscosity, Ya. I. Frenkel proceeded from the conception of a liquid as a “distorted” crystal lattice, in which the atoms, having broken away from their places, have passed into the “interstitial space” and oscillate about new unstable positions of equilibrium. The elementary process of flow consists in the spontaneous displacement of a particle from one temporary equilibrium position to another. In each of these random equilibrium positions the particle performs a certain number of oscillations, determined on average by the quantity \(e^{\frac{U}{kT}}\). Here \(U\) is a certain activation energy required for the displacement of a particle from one random equilibrium position to another. (In what follows, instead of \(U\) we shall use the notation \(E_\eta\).) The order of magnitude of this energy is the same as that of the heat of fusion.
In the first version of the theory³ the formula for viscosity is derived as follows. If the displacement of a particle from one equilibrium position to another occurs by a jump, then it may be assumed that the rate of change of equilibrium position is equal to the magnitude of the displacement divided by the residence time of the particle in a given equilibrium position. Let the displacement of the equilibrium position be equal to \(\delta\), and the period of the particle’s natural oscillations be \(\tau_0\). Then the velocity with which the particle “crawls” from one equilibrium position to another is equal to:
\[ v=\frac{\delta}{\tau_0\cdot e^{\frac{E_\eta}{kT}}} =\frac{\delta}{\tau_0}\cdot e^{-\frac{E_\eta}{kT}}. \tag{1} \]
The mechanism described by this equation is nothing other than the mechanism of diffusion; therefore \(v\) and \(\delta\) are directly related to the diffusion coefficient \(D\) by the Clausius equation:
\[ D=\frac{1}{3}v\cdot\delta =\frac{\delta^2}{3\tau_0}\cdot e^{-\frac{E_\eta}{kT}}. \tag{2} \]
According to Einstein’s equation¹) \(fD=kT\), where \(f\) is the coefficient of friction, equal to the ratio of the force to the velocity it causes, i.e.
\[ f=\frac{kT}{D}. \tag{3} \]
¹) Considering together Einstein’s diffusion equation and the equation determining the change in the viscosity of a solution upon addition of one solute molecule, S. I. Vavilov⁴ determines the effective value of the particle radius and the molecular viscosity, which differ from the macroscopic viscosity and the gaskinetic radius of the particle.
Substituting the values of \(D\) from (2) and \(f\) from Stokes’ formula (assuming the particles to be spherical, \(f=6\pi\eta a\)), we obtain an equation for determining the viscosity coefficient \(\eta\), namely:
\[ \eta=\frac{1}{6\pi a}\cdot f=\frac{kT3\tau_0}{6\pi a\delta^2}\cdot e^{\frac{E_\eta}{kT}} =\frac{\tau_0}{a\delta^2}\cdot \frac{kT}{2\pi}\cdot e^{\frac{E_\eta}{kT}} . \tag{4} \]
In the second variant of the theory\(^5\), the diffusion coefficient \(D\) is expressed by the Smoluchowski–Einstein formula, \(D=\dfrac{\delta^2}{6\tau}\), where \(\delta\), as before, is the displacement of the particle and \(\tau=\tau_0\cdot e^{\frac{E_\eta}{kT}}\) is the relaxation time of the liquid.
Combining this expression for the diffusion coefficient with Stokes’ equation, Ya. I. Frenkel obtains the viscosity coefficient \(\eta\), which is equal to:
\[ \eta=\frac{\tau_0 kT}{\pi a\delta^2}\cdot e^{\frac{E_\eta}{kT}} . \tag{5} \]
Denoting the coefficients multiplying \(T\cdot e^{\frac{E_\eta}{kT}}\) by \(B\), we obtain formulas (4) and (5) in the form:
\[ \eta=B\cdot T\cdot e^{\frac{E_\eta}{kT}} . \tag{6} \]
This formula, as Ya. I. Frenkel indicates, agrees less well with experiment than the exponential dependence of the form \(\eta=A\cdot e^{\frac{E_\eta}{kT}}\), into which the absolute temperature does not enter as a multiplier. In order to reduce formulas (4) and (5) to the form of a simple exponential dependence on temperature, Ya. I. Frenkel assumes \(\delta^2\) to be proportional to \(T\).
It is necessary to note that the question of the viscous flow of liquids is considered by the author in connection with the question of electrical conductivity\(^6\); Frenkel’s theory indicates a connection between the processes of electrical conductivity and viscosity.
This theory was tested by K. S. Evstropiev\(^7\) on molten salts and glasses, and for associated liquids the Walther formula proved more suitable, while for non-associated liquids—the simple exponential formula, to which, as was already said, Ya. I. Frenkel’s formula may also be reduced.
Recently E. Seddon\(^8\) published a survey-type work in which, in addition to energy curves, he calculates the coefficient of \(T\cdot e^{\frac{E_\eta}{kT}}\), equal according to Ya. I. Frenkel’s theory to \(\dfrac{\tau_0}{a\delta^2}\). This coefficient, assuming equality of \(a\) and \(\delta\) and constancy of \(\tau_0\), may be taken as a characteristic of the number of molecules or atoms in a unit volume. Seddon calculates \(E_\eta\) and \(\dfrac{\tau_0}{a\delta^2}\) for simple liquids (mercury), a homologous series of alcohols, paraffins, and organic and inorganic glasses. It turned out that unstabilized glasses give energy curves with a maximum in the annealing interval, whereas for stabilized glasses a smooth curve is obtained.
The quantity \(\frac{\tau_0}{a\delta^2}\), associated with the reciprocal magnitude of the sizes of aggregates, increases rapidly with decreasing temperature for water, alcohols, and glasses. For paraffins, mercury, benzene, and carbon tetrachloride this quantity remains almost constant, which indicates constancy of association in them. These calculations cannot, generally speaking, give absolute values of aggregate sizes. For mercury alone it was possible to compare the results of calculating \(\frac{\tau_0}{a\delta^2}\) from the viscosity curve with the calculation of the same quantity from \(\tau_0\) and \(a\), known for this liquid from other data; in this case a good agreement of the calculation results was obtained.
Fig. 1. Energy curves for glass of the system Na\(_2\)O — CaO — SiO\(_2\).
I — unstabilized, II — stabilized glass
Fig. 2. Dependence of the characteristic quantity of molecular aggregates on temperature for the same glass.
I — unstabilized, II — stabilized glass
In Fig. 1, borrowed from Seddon’s work, activation-energy curves \((E_\eta)\) are shown as a function of temperature for glass of the system Na\(_2\)O — CaO — SiO\(_2\). For stabilized glass (curve II) a smooth, steeply falling curve is obtained; for unstabilized glass (curve I) a maximum appears in the annealing interval. The course of the quantity \(\frac{\tau_0}{a\delta^2}\) as a function of temperature for the same glasses is presented in Fig. 2, also taken from Seddon. Curve I, with a minimum, corresponds to the unstabilized glass.
ANDRADE’S THEORY
Andrade’s theory\(^9\) is set forth in detail in the above-mentioned article by B. V. Bak, and therefore we shall confine ourselves here only to a brief reminder of its basic propositions. This theory also proceeds from the notion of the closeness of the liquid state to the solid state; therefore the mechanism of transfer of momentum from layer to layer is effected not by the passage of molecules from one layer into another, as in gases, but by means of temporary combinations of molecules, or their aggregates, oscillating about their equilibrium positions, which are displaced slightly. The interaction between the molecules of individual layers is effected only by means of normal forces between them; forces tangential to the surfaces of the aggregates are absent. The transfer itself
the transfer of momentum can occur only under a favorable situation of the surrounding particles, i.e., at definite values of their potential energy. Let us recall Andrade’s formula for the temperature dependence of viscosity in its general form:
\[ \eta \cdot v^{\frac{1}{3}} = A \cdot e^{\frac{C}{vT}}; \]
this formula contains an exponential factor characterizing the energy relations of the process, as well as the specific volume. Verification of this formula for various liquids (for the most part organic ones) gives very good agreement with experiment.
In recent years, as is evident from Andrade’s report, read at the conference on viscosity organized by the Royal Society of London in 1937,^10 no substantial changes or additions have been introduced into the theory. Andrade concluded this report by pointing out the need to accumulate factual material; moreover, in his opinion, simple liquids should be studied, in particular metals, whose melts in many cases are monatomic, closely packed liquids. Andrade’s laboratory is at present engaged in studying the viscosity of melts of alkali metals.
THEORY OF S. E. HAIKIN
In his theory of the viscosity of liquids, S. E. Haikin^11 uses Stewart’s conception of the structure of liquids. Stewart assumes that a liquid consists of small submicroscopic aggregates retaining a crystalline structure. The interaction of these aggregates, located in different layers of the liquid between which there is a velocity gradient, leads to the appearance of the force of viscosity. With regard to the elementary forces of interaction between two aggregates, S. E. Haikin, proceeding from the fundamental premise of his theory concerning the closeness of the liquid and solid states, assumes that the character of the forces of interaction between aggregates must be the same as in a solid body.
As is known, in a solid body, when one layer of molecules slides relative to another, tangential forces arise which, according to S. E. Haikin’s assumption, cannot disappear even when the solid passes into the liquid state. Without denying the normal forces of interaction between molecular aggregates, which must exist, at least in the form of cohesive forces, S. E. Haikin introduces the arbitrary assumption that viscous flow is determined only by tangential interactions. In this respect his theory is the opposite of Andrade’s theory, in which the bond between molecular complexes leading to the transfer of momentum from layer to layer is effected only by normal forces. The elementary forces arising between molecules or their complexes in S. E. Haikin’s theory are taken to be constant in magnitude, but directed opposite to the relative velocity of displacement of neighboring aggregates. The relative velocity of the aggregates arises already because they participate in Brownian motion. If, however, the liquid flows, then
to the relative velocity of Brownian motion is added the relative velocity caused by the presence of a velocity gradient in the laminar flow. Thus it is obtained that the resultant of all the forces acting between layers of aggregates must be proportional to the velocity gradient of the laminar flow and directed opposite to it. With respect to the elementary process of interaction of aggregates it is further assumed that the relative velocity of two aggregates, depending on their participation in Brownian motion, is the same for all aggregates. Their size is also taken to be the same for all, and they all have one and the same cubic form. Under these assumptions, calculation of the resultant of all elementary forces and comparison of it with the usual expression for Newton’s viscosity force lead to the following expression for the coefficient of viscosity:
\[ \eta=\frac{f_{0}\sqrt{\rho}\,\Delta l^{\frac{5}{2}}}{8\sqrt{kT}} . \tag{7} \]
Here \(\Delta l\) is the size of the aggregate, \(\rho\) is the density of the liquid, \(f_{0}\) is the force acting on a unit surface, and \(k\) and \(T\) are, as usual, Boltzmann’s constant and the absolute temperature.
Regarding the magnitude \(f_{0}\), which is generally speaking unknown, it is assumed that it is of the same order as in a solid. This assumption is based on the consideration that the degree of order in a liquid cannot be greater than in a solid, and therefore \(f_{0}\) in a liquid cannot be smaller than the corresponding force in a solid. In a solid, by \(f_{0}\) one must understand the critical tangential stress in the absence of thermal motion (\(f_{0}\) does not depend essentially on temperature for solids). Knowing the order of magnitude of \(f_{0}\) and the experimental values of \(\eta\) for various liquids, one can estimate from formula (7) the linear dimensions of the aggregates. It turned out that they lie within the limits from \(3\cdot 10^{-7}\) to \(1.5\cdot 10^{-6}\) cm, which, apparently, corresponds to the real state of affairs.
Formula (7) already contains \(T^{-1/2}\), but when the temperature is raised it is necessary to take into account the change in the sizes of the aggregates. On the average their sizes should decrease with increasing temperature. Here one has to abandon the initial assumption of equality of the aggregate sizes and to accept that the decrease in the sizes of aggregates occurs, on the average, through the disintegration of the larger among them. If one further assumes that the heat capacity of the liquid in the first approximation does not depend on temperature and that the number of broken bonds grows linearly with temperature, then one may write the following equality between the number \(N_T\) of bonds broken at temperature \(T\) and the number \(N_n\) of bonds broken on melting (\(T_n\) is the melting temperature):
\[ N_T=N_n\left(1+\frac{T-T_n}{\tau}\right). \tag{8} \]
The quantity \(\tau\) must be determined, and this is done as follows. If \(\vartheta\) is the latent heat of fusion, then \(\tau\cdot \Delta c=\vartheta\). Thus, \(\tau\) is that temperature increment over which there accumulates a heat of dissociation of aggregates equal to the heat of fusion.
The equation (8) is rewritten in the following form:
\[ N_T=\frac{N_n}{\tau}\cdot (T-T_0), \tag{9} \]
where \(T_0\) is a constant which, like \(\tau\), can be determined from the heat capacity and the heat of fusion.
Assuming further that the sizes of the aggregates \(\Delta l\) are proportional to \((T-T_0)^{-1}\), and denoting all coefficients independent of temperature by \(A\), we obtain the following temperature dependence of viscosity:
\[ \eta=\frac{A}{\sqrt{T\cdot (T-T_0)^5}}. \tag{10} \]
S. E. Khaikin tested this formula for mercury, bismuth, water, lead, and bromine, calculating the constant \(A\) and knowing in advance \(T\) from the heat capacity and the heat of fusion. It turned out that, for temperatures close to the melting temperatures, agreement with experiment is satisfactory; for higher temperatures the discrepancy with experiment reaches 20–30%. These discrepancies are explained by the incorrect choice of \(T_0\), and S. E. Khaikin proposes to correct the calculated values of viscosity by choosing \(T_0\), or more precisely, by choosing that part of the excess heat capacity of liquids which goes to the dissociation of aggregates.
Fig. 3. Experimental verification of S. E. Khaikin’s formula
A successful method for checking S. E. Khaikin’s formula was proposed by A. Ya. Modestov2. The applicability of this formula was controlled by the presence of a rectilinear dependence between the absolute temperature \(T\) and \(\eta^{-\frac{2}{5}}\cdot T^{-\frac{1}{5}}\). Modestov applied S. E. Khaikin’s formula to water, a number of alcohols, and other organic associated and non-associated liquids. With this method of verification it turned out that either a linear dependence is obtained, or the entire temperature interval in which viscosity measurements were made is divided into two sections, in each of which S. E. Khaikin’s formula agrees well with experiment. This means that between the temperature \(T\) and the produ—
by introducing \(\eta^{-2/5}\cdot T^{-1/5}\), in each such segment there is a linear dependence with special coefficients peculiar to each segment.
Fig. 3, borrowed from Modestov’s article, illustrates well all that has been said above using the example of water—1, methyl—2, ethyl—3, propyl—4, isopropyl—5, butyl—6, and amyl—7 alcohols. Water behaves here in a special way, since for it the vertex of the angle of intersection of the rectilinear segments is turned in the opposite direction than for the alcohols. The same, but with less good agreement with experiment, was obtained by A. A. Leont’eva\(^{13}\) for melts of sodium and potassium borates. In this latter case, Andradé’s formula nevertheless satisfies the experiment better.
BERNAL–UORD THEORY
If Andradé, in constructing a theory of the viscosity of liquids, leaves aside the question of the details of the molecular structure of liquids, then for Bernal\(^{14}\) the principal problem is the question of the connection of viscosity with the molecular structure of the liquid.
Bernal’s theory of the structure of a liquid proceeds from those geometrical or configurational relationships between the molecules of a liquid which can be detected by means of X-ray analysis. The basic characteristics of a liquid are cohesion and fluidity, which depend on the irregularity of the distribution of atoms and molecules and on the ability of this distribution to change under the influence of heat and mechanical stresses. In constructing his theory, Bernal restricts himself to monatomic or quasi-monatomic liquids and assumes that at ordinary temperatures and pressures a liquid may be regarded as molecularly homogeneous. This means that density fluctuations in it are small and that there are no large regions with a regular quasicrystalline structure. The meaning of this last restriction is that, around any molecule, the others are distributed statistically in the same way; in other words, the probability of finding a molecule at a given distance from some other one does not depend on the position of each of them in the liquid. This last assertion leads to the simplest method for characterizing internal configurations in a liquid. Configurations in a liquid, as in a solid, are determined by the distribution function, \(G(r)\)—a measure of the statistical density of molecules at a distance \(r\) from the given molecule. The distribution function for liquids is derived from X-ray scattering curves with the aid of Prince’s equation\(^{15}\):
\[ G(s)=\int_{0}^{\infty}4\pi r^{2}G(r)\,dr\cdot\frac{\sin(sr)}{r\cdot s}, \tag{11} \]
where \(G(s)\) is the part of the incident beam of X-rays of wavelength \(\lambda\) scattered at an angle \(\vartheta\cdot s=\dfrac{\pi\cdot\sin\frac{\vartheta}{2}}{\lambda}\), and \(4\pi r^{2}G(r)\,dr\) is the probab-
…probability of finding a molecule at a distance \(r, r+dr\) from the given one. The distribution function is not an analytic function and is not readily amenable to treatment.
For liquids, Prins obtained the distribution function by distorting the theoretical scattering curves for the corresponding crystalline substances by introducing a certain function. This device corresponds to a long-observed fact: the scattering curves for liquids are close to the curves of the corresponding crystalline forms for small scattering angles (i.e., for short-range order). Thus, by specifying the number of molecules and the intermolecular distances characteristic of the given crystalline system, Prins obtained for liquids a distribution function \(G(r)\) of the form:
\[ G(r)=\sum_s G_s(r)=\sum_s \frac{n_s}{4\pi r_s^2}\cdot \sqrt{\frac{a_s}{2\pi a r_s kT}}\cdot e^{-\frac{a^2(r-r_s)^2}{2a r_s kT}}, \tag{12} \]
where \(n_s, r_s\) are coefficients depending only on the type of crystalline structure, \(a\) is a constant characterizing the physical nature of the liquid, and \(a\) is the compressibility.
Bernal attempts to construct the distribution function on more general grounds, and emphasizes its profound physical meaning.
If one plots the graphical dependence of \(G(r)\) on \(r\), where \(r\) is the distance from some arbitrarily chosen molecule or atom—the choice of which in a true liquid is immaterial—one obtains a curve with a series of maxima of different heights. The first of them is the highest. If, from the given molecule, one describes spheres with radii \(r_1, r_2,\ldots\), corresponding to the positions of the first, second, etc. maxima of the distribution function, then the entire volume of the solid or liquid is divided into a series of coordination spheres. The radius of the first sphere, \(r_1\), is the mean distance from the given molecule to its nearest neighbors, and so on. Each maximum of the distribution function corresponds to a definite coordination sphere. The influence of the first sphere on the distribution function is considerably greater than the influence of the others, and the first maximum is therefore the highest. Thus Bernal, like Prins, represents the distribution function in the form of a sum of a series:
\[ G(r)=G_1(r)+G_2(r)+\cdots+G_s(r). \]
Each function \(G_s(r)\) represents the action of the \(s\)-th coordination sphere. In order to pass to liquids proper, it is necessary, just as Prins did, to “spoil” the sharpness of the maxima by introducing a certain normalizing factor. Here there arises the general question of how to characterize a continuum of configurations (which, from a purely geometrical point of view, a liquid may be regarded as being) by the smallest number of variables.
Bernal showed that three such variables can be chosen: the mean distance \(r_1\) between the nearest neighbors and the molecule taken as the origin; the mean number \(N\) of neighbors nearest to it (the coordination number of the liquid); and the quantity \(\lambda\), characterizing the irregularity of the distribution characteristic of the liquid. The distribution function, as a function of these three variables, has the form:
\[ G(r)=\sum_s \frac{N_s}{4\pi R_s^2\cdot r_1^2}\cdot \sqrt{\frac{1}{\pi \lambda^2 r_1}}\cdot e^{-\frac{R_s}{r_1^2}\frac{(r-R_s r_1)^2}{R_s r_1}} . \tag{13} \]
Here \(R_s\) is the ratio of the radius of the \(s\)-th coordination sphere to \(r_1\).
Since the distribution function depends on these three quantities, any possible distribution of molecules in a liquid is consequently expressed through them; here \(r_1\) and functions of it are constant scales, and thus the character of the configuration, in the sense of the magnitude of the coordination number and of the irregularity of the distribution, is characterized by \(N\) and \(\lambda\), which are independent of one another. In some regions of values of \(\lambda\) no \(N\) are possible, but within the permissible values \(\lambda\) and \(N\) may vary independently.
The distribution function constructed in this way satisfies any homogeneous-irregular distribution of molecules. In order that \(r_1\), \(N\), and \(\lambda\) characterize a real monatomic liquid, it is necessary to introduce the properties of the molecules composing it. This is most easily done on the basis of energy relations. The potential energy of two symmetrically or quasi-symmetrically surrounded molecules at a distance \(r_{1,2}\) is expressed by the power series:
\[ U_m(T)=\sum_q \frac{a_q}{r_{1,2}^q}. \tag{14} \]
The coefficients of the series \(a_q\) are characteristic for a definite kind of molecule and may be known either empirically from the crystal structure or computed from quantum theory. Assuming such a law of interaction between molecules, one can express the potential energy of the liquid through the above-mentioned three parameters \(r_1\), \(N\), and \(\lambda\), which characterize the geometry of the liquid. Bernal gives the following expression for the potential energy of the liquid:
\[ U(r_1,N,\lambda)=4\pi \sum_q a_q \int_0^\infty r^{2-q}\cdot G_c(r)\,dr, \tag{15} \]
where \(G_c(r)=\Gamma G(r)\).
The nearest neighbors have the greatest significance in this sum. The quantity \(U(r_1,N,\lambda)\) is very sensitive to changes in \(r_1\), so that, to a first approximation, the quantities \(\lambda\) may be neglected when determining those equilibrium values of the variables which give the minimum values of \(U\). Setting \(\lambda=0\), from the condition of the minimum of \(U(r_1,N,\lambda)\) one can calculate \(r_1\), and, of course, they coincide with those obtained for the corresponding crystals. For not very large \(\lambda\), i.e., for not very great disorder of the particles, \(r_1\) should not differ strongly from the crystalline \(r_1\). Therefore Bernal subsequently takes for \(r_1\) the corresponding values of the equilibrium distances in crystals. To determine \(N\) and \(\lambda\), it is necessary to have additional equations. Such equations are the expressions for the free energy and the condition for its minimum. The expression for the free energy is obtained on the basis of the following considerations. The expression for the energy of a liquid differs from the expression for the energy of a crys-
became that the coefficients at powers of \(r\) are not constant, once and for all determined by the structure, but variable and depend, in particular, on temperature. Therefore, schematically, a liquid may be treated as a solid body with a certain number of different crystalline phases.
At the point of transition from one phase to another, the condition of equality of the free energies of both phases must be satisfied. In the case of a liquid the number of phases is infinitely large, and each temperature is a transition temperature. The condition of equilibrium, consequently, consists in the free energy having a minimum value for all possible configurations at any temperature.
In order to put concrete content into this condition, it is necessary not only to reduce the number of parameters determining the configuration to the smallest possible number, but also to express the free energy \(F\) through these parameters. For the free energy Bernal gives the expression:
\[ F = U - \varphi T + 3kT \lg \frac{h\nu}{kT}, \tag{16} \]
where \(\varphi\) is the entropy inherent only in the configuration of the liquid. To find its exact expression in terms of \(N\) and \(\lambda\) appears difficult. The most probable assumption is that \(\varphi\) is proportional to the square of \(\lambda\). Generally speaking, on the basis of these equations one can derive only a general idea of those \(N\) and \(\lambda\) and of their interrelations which correspond to equilibrium conditions. To simplify the reasoning, the expression for the mutual potential energy of the molecules is taken in the form: \(U_r = \dfrac{a_m}{r^m} + \dfrac{a_n}{r^n}\), and \(r_1\) in the equilibrium position is equal to:
\[ r_1=\left(-\frac{n a_n}{m a_m}\right)^{\frac{1}{n-m}}. \]
As is known, \(m\) and \(n\) characterize the liquid.
A qualitative examination of the equations for determining the equilibrium coordination numbers \(N\) and the factor of structural irregularity \(\lambda\) leads to the following conclusions. At \(\lambda=0\) the quantity \(N_{\text{equil}}\) depends, mainly, on \(m\). For small \(m\), \(N_{\text{equil}}\) must be small; for large \(m\) it reaches the greatest possible values: 12–14. Lower values of the coordination number are possible also for larger \(m\), but only in the case when directing forces are present. With increasing \(\lambda\), \(N_{\text{equil}}\), generally speaking, decreases. Both \(N_{\text{equil}}\) and \(\lambda_{\text{equil}}\) are functions of temperature, but the character of this dependence is difficult to establish. For \(\lambda\), in agreement with Prins, Bernal assumes that \(\lambda^2_{\text{equil}} = kTL^2\), where \(L\) does not depend on temperature and characterizes only the physical type of the liquid. Using this equation and determining \(N_{\text{equil}}\) from \(\dfrac{\partial F}{\partial N}=0\), he obtains an expression for the total energy of the liquid at high \(T\) and then \(C_v\). Further one can obtain \(C_p\), the specific heat of a gram-mole at constant pressure, the expression for which breaks up into three terms:
\[ C_p = C_N + C_\lambda + C_c, \tag{17} \]
where \(C_N\) and \(C_\lambda\) represent changes of energy under equilibrium
coordination and disorder, and \(C_c\) is the heat-capacity value for the corresponding crystal. Bernal calls the sum \(C_N + C_\lambda\) the configurational specific heat and considers the introduction of this quantity necessary for explaining the observed large specific heats of simple liquids in comparison with crystals.
Further, the values of the coordination number \(N_{\mathrm{равн}}\) and \(\lambda_{\mathrm{равн}}\) should, generally speaking, depend on pressure. Qualitatively one may assert that an increase in pressure and the resulting decrease in volume should bring the liquid to a greater regularity of structure, with \(\lambda\) and \(r\) decreasing and \(N\) increasing. From the point of view of the configurational theory of the liquid state, one may consider two kinds of compressibility of a liquid: normal compressibility, due to a decrease in \(r\), which should be approximately the same as for crystals, and configurational compressibility, depending on the change of \(N\) and \(\lambda\) with pressure. Analogously, thermal expansion must also contain a configurational term.
Up to now all the arguments have concerned simple liquids. Qualitatively they can also be extended to more complex cases, but in doing so one must take into account the shape of the molecules and the presence of directional forces between molecules or their parts.
The presence of directional forces in the form of homeopolar, strongly polarized ionic bonds, and weaker hydroxyl and hydrogen bonds tends, generally speaking, to lower the effective coordination of the liquid, except in special cases where the bonds can form closed groups of quasimolecules, for example in fatty acids. The lower coordination caused by the existence of bonds affects the thermal properties and the viscosity of liquids. It is obvious that in this case changes of configuration caused by the intensification of thermal motion or by shear stress must require a greater expenditure of energy than in the case of simple liquids, and this is expressed in an increase of the configurational specific heat and viscosity. Such liquids should show a tendency toward glass formation, glass being understood as a supercooled liquid whose viscosity is so great that nonequilibrium configurations are preserved in it. In this work, devoted to the geometry of the liquid state, Bernal confines himself to the above general considerations concerning the influence of coordination on viscosity.
Eyring \(^{16}\), on the basis of Bernal’s theory, arrives in the following way at the derivation of an exponential dependence of viscosity on temperature. The coordination of an ion or molecule of a liquid can change with time under the action of local stresses arising as a result of the Boltzmann distribution of rotational and vibrational energy. On the other hand, changes of configuration may arise under the influence of external stresses; moreover, changes of configuration leading to the dissipation of shear are “encouraged.” The mechanism of transformation of shear deformation into configurational transformations is unknown, but Bernal \(^{10}\) assumes that it is composed of elementary processes consisting of exchange with neighbors, as shown in Fig. 4. Each such process includes intermediate states in which
molecules move apart, and this requires a certain activation energy. As Eyring pointed out (see below), one may assert in this sense that viscosity is essentially connected with chemical kinetics.
Let the magnitude of the energy necessary for this process be equal to \(B\) kg cal/mole. According to Boltzmann’s law, the number of ions or molecules possessing this energy is proportional to \(e^{-\frac{B}{RT}}\). As their number increases, shear deformations will dissipate more rapidly. Obviously,
Fig. 4. Scheme of rearrangement of molecules during flow (from Bernal’s report)
the viscosity of the liquid must then decrease, and consequently the value of the viscosity must be inversely proportional to the number of molecules having energy \(B\). Thus \(\eta\) may be taken to be equal to \(A \cdot e^{\frac{B}{RT}}\), i.e. one obtains for \(\eta\) the usual exponential dependence on temperature, found empirically by a number of authors, beginning with Guzman \(^{17}\), that is, \(\eta = A \cdot e^{\frac{B}{RT}}\).
In the general case \(A\) and \(B\) are functions of temperature. The activation energy \(B\), moreover, must also depend on pressure, since it depends essentially on the coordination of the molecules of the liquid.
Bernal believes, on the basis of Bridgman’s results for mercury, that with increasing pressure \(B\) should increase. At constant coordination, \(B\) remains constant in the case of many liquids and over wide temperature ranges. The value of \(B\) for a given liquid is usually determined graphically from curves of the dependence of the logarithm of viscosity on the reciprocal of the absolute temperature. Ward calculated \(B\) from experimental data on the viscosity of many liquids of different types and correlated the magnitude \(B\) with the type of liquid. The classification of liquids is based on the magnitude of the forces of interaction between particles and on the character of the dependence of these forces on the distance between them. Thus liquids are subdivided into ionic, homeopolar, metallic, and molecular. A narrower subdivision within these classes of liquids can be made by taking into account the presence or absence of directive forces. Thus ionic liquids can be divided into two subgroups, depending on how strongly their bonds are polarized. If the polarization of the bonds is high, then they have a directional character. The homeopolar class has only directional bonds. Metals in the liquid state are divided into metals with a close-packed structure and with a structure of complex-
...according to the electron theory of metals. In the class of molecular liquids one can distinguish liquids with dipole, hydroxyl, and hydrogen bonding. In these cases one must also take into account a third criterion—the presence or absence of permanent groups in the molecule. In view of the interest presented by Ward’s tables, we reproduce them here almost in full1.
Table 1
Ionic liquids without directing forces
| $T_{m}$ | $T_{boil}$ | $\Delta$ | $L_{m}$ | $L_{boil}$ | $B$ | $A$ | $\eta_{m}$ | |
|---|---|---|---|---|---|---|---|---|
| NaCl | 1 077 | 1712 | 0,59 | 7,22 | 44,3 | 9,10 | 2,11 | 14,6 |
| NaBr | 1 028 | 1666 | 0,62 | — | 38,6 | 8,00 | 2,85 | 14,3 |
| KCl | 1 045 | 1690 | 0,62 | 6,41 | 40,5 | 7,40 | 4,29 | 15,1 |
| KBr | 1 000 | 1649 | 0,64 | — | 38,2 | 7,96 | 2,92 | 15,7 |
| AgCl | 728 | 1827 | 1,51 | 3,05 | 44,3 | 5,30 | 7,64 | 29,7 |
| AgBr | 703 | — | — | 2,37 | — | 4,85 | 11,7 | 37,8 |
| AgJ | 825 | — | — | — | — | 5,15 | 15,8 | 36,5 |
| NaNO$_3$ | 581 | — | — | 3,69 | — | 3,68 | 12,2 | 29,44 |
In this table and in all the following ones, $\Delta$ denotes the quantity $(T_{boil}-T_m)/T_m$, which may be taken as the “degree of liquid state” of the liquid; $L_m$ and $L_{boil}$ denote the heats of fusion and vaporization in kg cal/mole; $B$ denotes the activation energy in kg cal/mole; $A$ is the coefficient in the exponential formula for viscosity in poises; $\eta$ is the viscosity at the melting point in millipoises.
This first table shows that the value of $B$ for ionic liquids is rather large and is of the order of the heat of fusion. Evidently, in this case the configurational changes during shear are connected with the destruction and rearrangement of the coordination of an ion by ions of the opposite sign, a process analogous to melting. Below are given the indices for liquids of other types.
Table 2
Ionic liquids with directing forces
| $T_m$ | $T_{boil}$ | $\Delta$ | $L_m$ | $L_{boil}$ | $B_1$ | $B_2$ | $\eta_m$ | |
|---|---|---|---|---|---|---|---|---|
| B$_2$O$_3$ | 294 | — | — | 3 | — | 75,0 | 21,7 | 8,10$^{2}$ |
| LiNO$_3$ | 528 | — | — | 6,10 | — | 4,1 | — | 68,4 |
Table 3
Metallic liquids with a close-packed structure
| $T_{пл}$ | $T_{кип}$ | $\Delta$ | $L_{пл}$ | $L_{кип}$ | $B$ | $A\cdot 10^4$ | $\eta_{пл}$ | |
|---|---|---|---|---|---|---|---|---|
| Na | 370,5 | 1153 | 2,10 | 0,61 | 25,0 | 0,96 | 21,5 | 7,9 |
| K | 335,3 | 1033 | 2,08 | 0,57 | 21,0 | 1,15 | 9,75 | 5,5 |
| Cu | 1356 | 2573 | 0,90 | 2,75 | 116,0 | — | — | 37 |
| Ag | 1233 | 2223 | 0,80 | 2,63 | 59,5 | 4,87 | 56,9 | 41,4 |
| Zn | 692,4 | 1180 | 0,71 | 1,74 | 24,0 | 2,92 | 41,4 | 33,4 |
| Cd | 593,9 | 1040 | 0,75 | 1,48 | 26,0 | 1,585 | 66,2 | 25,3 |
| Hg | 234,2 | 630 | 1,68 | 0,57 | 14,0 | 0,598 | 55,5 | 20,1 |
| Sn | 504,8 | 2533 | 4,03 | 1,73 | 78,0 | 1,603 | 41,3 | 20,3 |
| Pb | 600,5 | 1893 | 2,14 | 1,31 | 46,0 | 2,32 | 40,8 | 28,5 |
| Sb | 903,5 | 1653 | 0,83 | 2,92 | 45,0 | 2,92 | 28,8 | 14,6 |
| Bi | 544 | 1723 | 2,17 | 2,70 | 46,0 | 1,715 | 38,2 | 18,6 |
Table 4
| $\mu\cdot 10^{18}$ | $T_{пл}$ | $T_{кип}$ | $\Delta$ | $L_{пл}$ | $L_{кип}$ | $B$ | $A$ | $\eta_{пл}$ | |
|---|---|---|---|---|---|---|---|---|---|
| Molecular nonpolar liquids | |||||||||
| A | 0,0 | 84,0 | 87,5 | 0,042 | 0,27 | 1,50 | 0,524 | 1,24 | 2,83 |
| N$_2$ | 0,0 | 63,4 | 77,4 | 0,221 | 0,17 | 1,34 | 0,468 | 0,762 | 3,11 |
| CO | 0,1 | 66,2 | 81,2 | 0,227 | 0,22 | 1,41 | 0,463 | 0,951 | 3,21 |
| CH$_4$ | 0,0 | 89,2 | 118,2 | 0,254 | 0,24 | 2,200 | 0,740 | 0,347 | 2,25 |
| O$_2$ | 0,0 | 54,8 | 90,2 | 0,654 | 0,106 | 1,67 | 0,406 | 1,945 | 8,09 |
| C$_2$H$_4$ | 0,0 | 103,8 | 169,4 | 0,632 | — | — | 0,739 | 2,01 | 7,24 |
| Molecular dipolar | |||||||||
| C$_2$H$_5$Br | 1,85 | 154 | 311 | 1,02 | — | 6,6 | 1,12 | 6,25 | 25 |
| CH$_3$I | 1,65 | 206,9 | 315,6 | 0,66 | — | 6,5 | 1,12 | 7,11 | 11 |
| C$_2$H$_5$I | 1,65 | 164,5 | 345,2 | 1,10 | — | 7,25 | 1,25 | 7,08 | 35 |
| C$_3$H$_7$I | 1,65 | 161,6 | 375,4 | 1,32 | — | — | 1,41 | 6,54 | 55 |
| Molecular with a hydroxyl bond | |||||||||
| H$_2$O | 1,87 | 273 | 373 | 0,37 | 1,43 | 9,65 | 3,05 | 0,59 | 0,18 |
| CH$_3$OH | 1,67 | 175,2 | 337,5 | 0,92 | 0,53 | 8,4 | 1,84 | 2,49 | 50 |
| C$_2$H$_5$OH | 1,70 | 155,7 | 351,5 | 1,24 | 1,14 | 9,4 | 2,34 | 2,11 | 500 |
| C$_3$H$_7$OH | 1,66 | 146,0 | 370,8 | 1,54 | — | 9,9 | 3,16 | 0,97 | 5500 |
| C$_4$H$_9$OH | 1,66 | 183,2 | 390,7 | 1,13 | 2,22 | 10,4 | 3,46 | 0,72 | 1000 |
| Molecular with a hydrogen bond | |||||||||
| HCOOH | 1,45 | 281,4 | 373,5 | 0,33 | 2,70 | 5,8 | 2,49 | 2,44 | 24 |
| CH$_3$COOH | 1,04 | 289,6 | 391,1 | 0,35 | 2,68 | 5,8 | 1,93 | 4,22 | 12 |
Within the limits of one and the same class of liquids, $B$ changes noticeably, but the ratio of $B$ to $L_{пл}$ remains more or less constant. Instead of calculating $\dfrac{B}{L_{пл}}$, Bernal proposed calculating the ratio
\(B\) to \(T_{\mathrm{m}}\), and showed that within one and the same class of liquids it maintains constancy rather well. Thus, calculating this ratio, which may be called the viscous entropy change, Bernal obtains its variations within the following limits: for ionic and simple molecular liquids from 6.2 to 8.4, for metals—from 2.6 to 4.2.
Bernal explains this decrease of \(\dfrac{B}{T_{\mathrm{m}}}\) for metals by the fact that, on melting metals, their volume changes much less than in all other liquids. This once again emphasizes the close connection that exists between viscous flow and change of volume, and which has long attracted the attention of researchers (A. I. Bachinskii, MacLeod, Andrade, and others).
Fig. 5. Dependence of the logarithm of viscosity on \(T^{-1}\) for boric anhydride
For ionic liquids, in the absence of directing forces, \(B\) remains constant in the measured temperature interval of viscosity; consequently, the coordination of the liquid also remains constant. If, however, we are dealing with an ionic liquid in the presence of directing forces, then with an increase in temperature and intensification of thermal motion the coordination of the liquid increases. It is possible that the transition from low to high coordination should facilitate the configurational changes occurring at the expense of the energy of shear deformation, since in this case the directed bonds are weakened. Therefore, for such a liquid one may expect a decrease of \(B\) with temperature. A good example of a liquid of this type is molten boric anhydride. For it the dependence of the logarithm of viscosity on \(T^{-1}\), from which \(B\) is calculated, is represented not by a straight line, as, for example, for NaCl, but by a broken line running more gently at high temperatures (see Fig. 5, taken from the paper of A. Leont’eva\({}^{20}\)).*
For alcohols with hydroxyl bonds we have the same character of the dependence of \(B\) on temperature. For these liquids \(B\) changes with increasing temperature because of a change in coordination, but the very character of the forces of interaction between the particles does not change. There are, however, cases where such a change is also observed. Thus, if with increasing temperature a loosening of bonds begins in weakly bound molecules, leading even to their dissociation, then the atoms or ions thereby produced must interact more strongly than when bound. This is reflected in an increase of \(B\) with increasing temperature; in other words, an anomalous dependence of \(B\) on temperature arises.
Another case in which the interaction forces also change is the case of rotation of groups previously bound by directing forces. At first, with increasing temperature, only librations arise, weakening the bonds; then—with sufficiently energetic thermal motion—the bonding forces disappear and free rotation begins. In these cases, at low temperatures \(B\) must
have a rather large value; then a transition region begins, where \(B\) depends extremely strongly on temperature and, finally, with a further rise in temperature, \(B\) becomes smaller than at low temperatures, and the course of the curve \(B(t)\) becomes normal. As an example, Eyring cites gallium for the first case—generally speaking, a diatomic substance, dissociating at high temperatures, with \(B\) increasing from \(0.85\) kg cal/mole to \(1.25\) kg cal/mole. An example of the second case is \(\mathrm{LiNO_3}\), in which the \(\mathrm{NO_3^-}\) group is bound by directed forces to \(\mathrm{Li^+}\). The same anomalous dependence of \(B\) on temperature was observed by A. A. Leont’eva\(^{20}\) for glass melts of the \(\mathrm{B_2O_3—SiO_2}\) system containing \(3—5\%\ \mathrm{SiO_2}\). For this system the dependence of the logarithm of viscosity on the reciprocal of the absolute temperature is shown in Fig. 6. For glasses containing comparatively much \(\mathrm{SiO_2}\) (12 and more wt. %), a rectilinear dependence is obtained between the indicated quantities, which permits one to suppose that in these glasses there is some firmly established coordination of cations in the melt. A glass containing \(5\%\ \mathrm{SiO_2}\), however, exhibits an anomaly in the behavior of \(B\), such as \(\mathrm{LiNO_3}\) does, since for it \(B\) at high temperatures, as determined from this graph, has a larger value than at low temperatures.
Fig. 6. Dependence of the logarithm of viscosity on \(T^{-1}\) for glasses of the \(\mathrm{B_2O_3—SiO_2}\) system
Stewart\(^{22}\) also applied the Bernal–Eyring theory to glasses. To study the processes occurring during the cooling of glass, he constructs curves of the dependence of \(B\) on temperature and comes to the conclusion that, upon cooling a glass from the melting temperature down to the softening interval, the coordination number decreases without changing the character of the bonds. The decrease in the coordination number continues down to a certain temperature, when any further change in coordination becomes impossible. On the curve \(B(t)\) a horizontal segment appears, corresponding to the transformation interval. The transformation temperature falls within this segment, but its value in each individual case depends on the conditions of measurement.
In conclusion, let us say a few words about how Bernal represents the dependence of viscosity on pressure. In his opinion, the formula of A. I. Bachinskii should be regarded precisely as an expression of the dependence of viscosity on pressure, with the volume occupied by the molecules
should be a function not only of temperature, but also of pressure. In this way one can explain the results of Bridgman’s experiments with mercury, which, it seemed, refuted all theories of viscosity constructed on changes of volume. If, however, one allows \(v_0\), the volume occupied by the molecules, to vary with pressure and to have the same compressibility as solid mercury (in fact, the same as lead, since the compressibility of mercury was not measured), then good agreement with experiment is obtained up to the highest pressures. Treating Bridgman’s results for mercury in this way, Bernal shows that the ratio
\[ \frac{v-v_0}{\eta} \]
retains a constant value rather well as the pressure is varied up to \(12000\ \mathrm{kg}/\mathrm{cm}^2\) and at temperatures of 30 and \(75^\circ\), as indeed must be the case if Bachinskii’s equation is correct. From this Bernal concludes that, so long as there is no possibility of calculating the real effective volume of a liquid, any theory of viscosity constructed on changes of volume will give less satisfactory results than a purely empirical expression for the temperature dependence of viscosity.
THE EYRING–EWELL THEORY
The theory of the viscosity of liquids, developed by Eyring and co-workers \(^{23}\), chiefly by Ewell \(^{24}\), like the Bernal–Moord theory, proceeds from definite ideas about the nature of the liquid state. Eyring approaches the question of the nature of liquids from the side of real gases and uses the methods of statistical mechanics. The distribution function for ideal gases is taken by Eyring in the form:
\[ f_g=\frac{(2\pi mkT)^{\frac{3}{2}}}{h^3}\frac{v}{N}\cdot b(T), \tag{18} \]
where \(b(T)\) is that part of the complete distribution function which is due to the vibrational and rotational energy of the molecules. Into this function one may introduce coefficients expressing the action of two, three, and so on neighboring molecules on a given one, and thereby take intermolecular forces into account, but the calculations then become extremely complicated. Eyring uses an expression for the mean potential of the molecules relative to a given one, assumes that the vibrational and rotational components of the distribution function are the same as for a gas, and obtains the distribution function for liquids in the form:
\[ f_e=v_f\frac{(2\pi mkT)^{\frac{3}{2}}}{h^3}\cdot e^{\frac{\Delta E}{RT}}, \tag{19} \]
where \(\Delta E\) is the energy of evaporation (more precisely, its value at absolute zero) and \(v_f\) is the free volume of the liquid. The determination of the free volume has very great significance in this theory. For simple cubic packing Eyring takes \(v_f\) equal to \(8(v_l^{\frac{1}{3}}-d)^3\), where \(v_l\) is the molecular volume and \(d\) is the incompressible diameter of the molecule; in the general case
\[ v_f=b^3(v_l^{\frac{1}{3}}-d)^3. \]
Eyring represents the free volume as being distributed in a liquid in the form of discrete holes, the sizes of which, in the first approximation, he considers equal to the sizes of molecules. During the process of evaporation, the concentration of holes in the liquid must be exactly the same as the concentration of molecules in the vapor, since according to Eyring the energy required to form a hole is the same as that required to remove a molecule from the liquid. The concept of holes of molecular size was greatly simplified; subsequently it became necessary to admit the existence of holes of different sizes. The question of their distribution was considered by Altar[^19]. Eyring considers the kinetics of processes occurring in liquids—in particular, the kinetics of viscous flow—as analogous to the kinetics of chemical reactions, and takes the rate constant of the “reaction” as its essential characteristic.
Of course, such a conception of the kinetics of any process is necessarily connected with the conception of the normal and activated states of the molecules, or of their complexes, participating in the reaction, and with the existence of a certain activation energy of the processes. In its development the theory used two mechanisms of viscous flow: 1) a monomolecular mechanism, in which each molecule moves independently (this motion can occur only when holes of suitable size are present; the activation energy in this case is the energy necessary for the formation of such a hole); 2) a bimolecular process, in which two molecules in adjacent layers rotate, in their relative motion, about one another through \(90^\circ\). In this case the free space necessary for such motion must be smaller than in the first case, and the activation energy must be smaller. Since Eyring regards the process of hole formation as equivalent to the process of evaporation, it is evident that in the first case the activation energy of viscous flow must simply be equal to the energy of evaporation, while in the second it constitutes some fraction of this quantity.
The derivation of an equation for determining the coefficient of viscosity \(\eta\) and the application of the theory to particular questions are the subject of Ewell’s paper[^24]. To derive the viscosity formula, Ewell uses the first, monomolecular mechanism of viscous flow.
The elementary process in this mechanism is the transition of a molecule from one equilibrium position to another, and in this transition a certain potential barrier between two equilibrium positions is overcome. The activation energy is used to form a hole into which the moving molecule could enter, or merely to increase somewhat the free space between neighboring molecules into which it can fit.
Graphically this process may be represented by Fig. 7, borrowed from Ewell’s paper. The process without shear and, consequently, without a velocity gradient is shown by the solid curve. In this case both halves of the graph are completely symmetrical, and molecules can pass equally often from right to left and from left to right. This is a simple diffusion process. The diffusion coefficient is equal to:
\[ K_1 = \frac{kT}{h}\cdot\frac{F^{*}_{a}}{F_n}\cdot e^{-\frac{\Delta E_a}{kT}} . \tag{20} \]
Here \(F_a^{*}\) is the distribution function for activated complexes having two translational degrees of freedom, \(F_n\) is the distribution function for the normal state, \(\Delta E_a\) is the activation energy, and \(h\), \(k\), and \(T\) have their usual meanings.
If, however, a shear from left to right is imposed on the liquid, then the symmetry of the graph (the dotted curve) is disturbed owing to a change in the activation energy. This change has the same magnitude in both its parts, but the opposite sign to the right and to the left of the potential barrier. It can be calculated as follows. Let \(f\) be the force, calculated per \(1\ \mathrm{cm}^2\), tending to displace one layer relative to another; \(\lambda\) is the distance between equilibrium positions in the direction of flow, \(\lambda_2\) is the distance between nearest molecules in the direction of flow, and \(\lambda_3\) is the distance between nearest molecules in the plane of flow along the perpendicular to the direction of flow. Then the area of a molecule in the plane of flow is equal to \(\lambda_2 \lambda_3\), the force acting on the molecule is \(f\lambda_2\lambda_3\), and if it acts over a distance \(\frac{1}{2}\lambda\), the additional energy due to the shear is equal to \(\frac{1}{2} f \lambda_2 \lambda_3 \lambda\). Obviously, in the direction of the velocity gradient (in Fig. 7 from left to right) this energy will be subtracted from the activation energy \(\Delta E_a\), while in the direction from right to left it will be added to \(\Delta E_a\), and thus the velocity in the direction of shear will become greater than in the opposite direction, and a net flow will arise. Correspondingly, for the diffusion-rate constants in the two directions one obtains the expressions:
Fig. 7. Energy profile of the motion of a molecule during shear
\[ K_1=\frac{kT}{h}\cdot \frac{F_a^{*}}{F_n}\, e^{-\frac{\left(\Delta E_a-\frac{1}{2}f\lambda_2\lambda_3\lambda\right)}{kT}}, \tag{21} \]
\[ K_2=\frac{kT}{h}\cdot \frac{F_a^{*}}{F_n}\, e^{-\frac{\left(\Delta E_a+\frac{1}{2}f\lambda_2\lambda_3\lambda\right)}{kT}}. \tag{22} \]
For brevity these expressions may be written in the form:
\[ K_1=K'\cdot e^{\frac{f\lambda_2\lambda_3\lambda}{2kT}}, \tag{23} \]
\[ K_2=K'\cdot e^{-\frac{f\lambda_2\lambda_3\lambda}{2kT}}, \tag{24} \]
where \(K'\) denotes:
\[ \frac{kT}{h}\cdot \frac{F_a^{*}}{F_n}\cdot e^{-\frac{\Delta E_a}{kT}}. \]
As is known, viscosity is hydrodynamically determined by Newton’s equation:
\[ \eta=\frac{f\lambda_1}{\Delta V}, \tag{25} \]
in which \(\lambda_1\) is the perpendicular distance between neighboring moving layers of molecules, and \(\Delta V\) is the difference of the velocities of two layers at a distance \(\lambda\), equal to \(\lambda(K_1-K_2)\). Thus
\[ \eta=\frac{f\lambda_1}{\lambda(K_1-K_2)}. \tag{26} \]
Substituting into (26) the values of \(K_1\) and \(K_2\), we obtain:
\[ \eta= \frac{f\lambda_1}{ \lambda K'\left(e^{\frac{f\lambda_2\lambda_3\lambda}{2kT}} - e^{-\frac{f\lambda_2\lambda_3\lambda}{2kT}}\right)} = \frac{f\lambda_1}{ 2K'\cdot \sinh f\frac{\lambda\lambda_2\lambda_3}{2kT} }. \tag{27} \]
Expanding \(\dfrac{\sin h\, f\lambda\lambda_2\lambda_3}{2kT}\) in a series and limiting ourselves to the first term, since usually \(\dfrac{f\lambda_2\lambda_3\lambda}{2}\ll 1\), we have:
\[ \eta=\frac{\lambda_1 kT}{\lambda^2\lambda_2\lambda_3K'}, \tag{28} \]
and then, inserting the value of \(K'\):
\[ \eta=\frac{\lambda_1 h\cdot F_n}{\lambda^2\lambda_2\lambda_3F_a^{*}}\cdot e^{\frac{\Delta E_0}{kT}}. \tag{29} \]
If, further, it is assumed that in each elementary process the molecule passes only one molecular distance, i.e. that \(\lambda=\lambda_2\), and that for an equilibrated molecule \(\lambda_1=\lambda_2=\lambda_3\), then \(\dfrac{\lambda_1}{\lambda\lambda_2\lambda_3}=\dfrac{N}{v}\), where \(N\) is Avogadro’s number and \(v\) is the molecular volume. The ratio of the distribution functions \(\dfrac{F_n}{F_a^{*}}\) is expressed in terms of the free volume \(v_f\), the mass of the molecule, and universal constants as follows:
\[ \frac{F_n}{F_a^{*}}=\frac{(2\pi mkT)^{\frac{1}{2}}}{h}\,v_f^{\frac{1}{3}}, \]
where it is assumed that only the translational degree of freedom corresponds to the flow, and that the other degrees of freedom are the same for the normal and activated states.
For the free volume, Eyring and Hirschfelder\(^{25}\) gave the general expression: \(\dfrac{1}{v_f^{3}}=\dfrac{bRT}{v^{\frac{2}{3}}\cdot N^{\frac{1}{3}}\left(p+\dfrac{a}{v^2}\right)}\) per molecule, where \(b\) is a factor depending on the packing and equal to 2 for simple cubic packing (in other cases it does not differ much from 2). If then \(\dfrac{a}{v^2}\) is considerably larger than \(p\) and if \(\dfrac{a}{v^2}\) may be taken equal to \(\dfrac{\Delta E_{\text{исп}}}{v}\), then
the formula for the free volume is considerably simplified:
\[ v_f^{\frac{1}{3}}=\frac{bRTv^{\frac{1}{3}}}{N^{\frac{1}{3}}\cdot \Delta E_{\text{vap}}}, \]
where \(\Delta E_{\text{vap}}\) is equal to the energy of vaporization per mole:
\[ \Delta E_{\text{vap}}=\Delta H_{\text{vap}}-\Delta(pv). \]
Thus, the energy of vaporization enters into the expression for \(\eta\) through the free volume. In addition, the exponential term in the formula for \(\eta\) contains the activation energy, concerning which, as has already been said, it is assumed that it is connected with the energy of vaporization. As indicated above, in the case of a monomolecular mechanism it ought to be equal to the energy of vaporization if the holes formed during flow had molecular dimensions. However, the experimental data force us to abandon the assumption that a moving molecule requires holes of a size equal to itself, and to take \(\Delta E_a\), the activation energy, equal to some fraction of the energy of vaporization \(\left(\Delta E_a=\frac{\Delta E_{\text{vap}}}{n}\right)\).
Substituting the expression for the free volume into the formula for \(\eta\), we obtain:
\[ \eta=\frac{Nh\cdot(2\pi mkT)^{\frac{1}{2}}\cdot bRT\cdot v^{\frac{1}{3}}}{v\cdot h\cdot \Delta E_{\text{vap}}N^{\frac{1}{3}}}\cdot e^{\frac{\Delta E_{\text{vap}}}{nRT}} . \tag{30} \]
or, taking \(b=2\) and substituting the numerical values of \(R\) and \(h\), we have:
\[ \eta=1{,}090\cdot 10^{-3}\cdot \frac{M^{\frac{1}{2}}T^{\frac{3}{2}}}{v^{\frac{2}{3}}\cdot \Delta E_{\text{vap}}} \cdot e^{\frac{\Delta E_{\text{vap}}}{nRT}} . \tag{31} \]
Thus, viscosity is expressed by an exponential dependence on \(T\) and on the energy of vaporization, while the coefficient before the exponential function includes the density of the liquid and the same energy of vaporization.
The formula was checked by Ewell on twenty-four normal liquids, for which determinations of viscosity, density, and heats of vaporization could be found. It turned out that not the absolute value of the viscosity, but its experimentally observed temperature coefficient is well determined by the derived formula; moreover, for liquids having spherical molecules or molecules with spherical symmetry of the field, \(n\) must be taken equal to 3, while for liquids with polar or elongated molecules, \(n\simeq 4\).
This fact led Ewell to the conclusion that, at least in the case of liquids with spherical molecules, viscous flow may be regarded as a kind of vaporization with one degree of freedom. Of all the liquids on which the theory was tested, the paper gives as an example only carbon tetrachloride; for it a graph is given, which we reproduce here (Fig. 8).
For CCl\(_4\), as for a nonpolar liquid, \(n\) must be set equal to 3, and indeed, for this value of \(n\) the theoretical curve of the temperature dependence, plotted as \(\lg \eta\) versus \(T^{-1}\) (see Fig. 8), runs parallel to the experimental curve. However, the experimental values of the viscosity are, in absolute magnitude, much closer to the theoretical ones if they are calculated for \(n=4\). Comparing the calculated viscosity values of CCl\(_4\) for \(n=3\) with the experimental ones, we find that
Fig. 8. Comparison of experimental data for viscosity with values calculated for different \(n\)
Fig. 9. Bimolecular mechanism of flow (from the work of Eyring)
the calculated values are almost three times greater than the experimental ones. In general, for liquids with \(n=3\) (this quantity is determined by the criterion of nonpolarity and the spherical symmetry of the molecule or of its force field), the ratio of the calculated viscosity values to the experimental ones ranges from 2.5 to 3.5; for liquids with \(n=4\) this ratio is approximately 2.
One explanation of this discrepancy is the assumption of the bimolecular character of the elementary process of flow. As has already been said, in this case it is assumed that two molecules in adjacent layers, moving relative to one another, rotate through \(90^\circ\), as shown in Fig. 9. The relative motion of successive layers of liquid results from the rotation of many pairs of molecules. In the course of this rotation they free the volume necessary for the process of flow, which, evidently, must be equal to
\[ \frac{v}{3} \]
or
\[ \frac{v}{4}, \]
depending on the type of molecule.
The derivation of the viscosity formula according to the Eyring–Yuell theory is given here in full, as it is set forth in Yuell’s work, since it seemed of interest to present in as complete a form as possible the final
... results of this theory of the liquid state, constructed on very general foundations. The results of comparison with experiment apparently show that this theory reflects well only the qualitative side of the phenomena, giving considerable deviations (up to 300%) of the absolute calculated values of viscosity from those determined experimentally.
Ewell believes that it is necessary to clarify more precisely the question of the mechanism of the process; for the time being, one should use the viscosity formula in the form in which it was derived, but divide the calculated values by 2 for liquids with \(n = 4\), and by 3, or by a number close to 3, for liquids with \(n = 3\).
Ewell compares his expression for viscosity with the well-known exponential formula for viscosity, which he rewrites in the form:
\[ \eta = A \cdot e^{\frac{\Delta E_{\text{visc}}}{RT}}, \]
denoting the activation energy \(B\) by \(\Delta E_{\text{visc}}\). Obviously,
\[ \Delta E_{\text{visc}} = \frac{R\, d \cdot \ln \eta}{d\left(\frac{1}{T}\right)} . \]
Differentiating his viscosity formula with respect to \(\frac{1}{T}\) and equating the result to \(\Delta E_{\text{visc}}\), Ewell obtains:
\[ \Delta E_{\text{visc}} = R \cdot \frac{d \ln \eta}{d\left(\frac{1}{T}\right)} = \frac{\Delta E_{\text{evap}}}{n} - \frac{3}{2}RT + \frac{2}{3}aRT^{2} + \frac{\Delta C_{v}}{\Delta E_{\text{evap}}}\cdot RT^{2}, \tag{32} \]
where \(a\) is the coefficient of thermal expansion and \(\Delta C_{v}\) is the increment of the molar heat capacity upon evaporation. Since the first two terms decrease, while the last two increase with rising temperature, \(\Delta E_{\text{visc}}\) does not depend on temperature, which corresponds to the experimentally known fact of the constancy of \(\Delta E_{\text{visc}}\) for normal liquids. Analysis of the last formula shows, moreover, that the sum of the last two terms is approximately equal to the second, and since they have opposite signs, then
\[ \Delta E_{\text{visc}} = \frac{\Delta E_{\text{evap}}}{n}. \]
Calculations for a number of liquids showed that the ratio
\[ \frac{\Delta E_{\text{evap}}}{\Delta E_{\text{visc}}} \]
fluctuates around 3 for liquids with \(n = 3\) and around 4 for liquids with \(n = 4\). These results, in Ewell’s opinion, show that the ratio
\[ \frac{\Delta E_{\text{evap}}}{\Delta E_{\text{visc}}} \]
may be taken as a characteristic—though a rough one—of the size and shape of molecules or, more precisely, of that discrete formation which participates in the elementary process of flow and which may be called the “unit of flow.” If the ratio
\[ \frac{\Delta E_{\text{evap}}}{\Delta E_{\text{visc}}} \]
is significantly greater than 3 or 4, this means that the unit of flow is much smaller than the unit of evaporation (in most cases this is an individual molecule), and conversely. Thus, for most metals
\[ \frac{\Delta E_{\text{evap}}}{\Delta E_{\text{visc}}} \]
varies from 8 to 25. This means that the activation energy of flow is less than one third of the energy of evaporation and that the unit of flow is considerably smaller than the unit of evaporation. Since
if the unit of evaporation in the case of metals is to be taken as an atom, then, consequently, the unit of flow must be taken to be a metallic ion much smaller in size. In order to reduce the ratio \(\Delta E_{\text{evap}}/\Delta E_{\text{visc}}\) for metals to the value usually obtained for liquids (3 or 4), the following recalculation is proposed:
\[ \Delta E_{\text{visc}} = \frac{\Delta E_{\text{evap}}}{3}\cdot \frac{\text{volume of ion}}{\text{volume of atom}} = \frac{\Delta E_{\text{evap}}}{3}\cdot \left(\frac{\text{radius of ion}}{\text{radius of atom}}\right)^3 . \tag{33} \]
This relation was tested for nine liquid metals, and it turned out to hold within the limits of experimental error. Of course, this recalculation must be regarded as simplified, but the idea that metal ions without valence electrons participate in the process of flow is consistent with modern views on the structure of metals.
Thus, for normal liquids and for metals, if the indicated recalculation is carried out, the energy of viscous flow \(\Delta E_{\text{visc}}\) is equal to \(1/3\) or \(1/4\) of the energy of evaporation. In the case of liquids with directed forces it is necessary to take into account the circumstance that in them the process of flow is accompanied not only by the formation of a hole into which the unit of flow can fit, but also by the destruction of structural bonds. Therefore Ewell assumes that for liquids with directed forces \(\Delta E_{\text{visc}}\) should be equal to the sum of \(1/3\) or \(1/4\) of the evaporation energy due to nondirectional forces and the entire evaporation energy of the structural bonds. In general form this relation may be written as follows:
\[ \Delta E_{\text{visc}} = \frac{1}{n}\cdot \Delta E_{\text{evap}} \;(\text{for nondirectional forces}) + \Delta E_{\text{evap}} \;(\text{for structural bonds}). \tag{34} \]
For normal liquids the second term in this equality becomes zero. In liquids with hydrogen bonding, such as \(\mathrm{NH_3}\), \(\mathrm{H_2O}\), and all those containing the group \(\mathrm{OH}\) or \(\mathrm{H}\), the two terms are comparable in magnitude. For the structure of liquids with covalent bonding, however, the energy of bond rupture is practically equal to the activation energy, and the first term may be neglected. Ewell assigns liquid \(\mathrm{SiO_2}\) to such liquids. Of course, the energy of viscous flow \(\Delta E_{\text{visc}}\) refers to a definite unit of flow. Which grouping of atoms should be regarded as the unit of flow can be determined from consideration of the structure of liquids and the strength of the bonds in it. Flow occurs chiefly with the aid of those units of flow that have the lowest activation energy. Thus, for liquid \(\mathrm{SiO_2}\), all combinations containing one \(\mathrm{Si}\) (for example, \(\mathrm{SiO}\), \(\mathrm{SiO_2}\), \(\mathrm{SiO_3}\), and \(\mathrm{SiO_4}\)) require the rupture of four \(\mathrm{Si{-}O}\) bonds, whereas any unit of flow with two \(\mathrm{Si}\) requires the rupture of six \(\mathrm{Si{-}O}\) bonds. Since, according to thermochemical data, the energy of the \(\mathrm{Si{-}O}\) bond is \(100\ \text{kg cal}/\text{mol}\), the evaporation energy of a unit of flow in liquid \(\mathrm{SiO_2}\) must be equal to half of \(4\cdot100\ \text{kg cal}/\text{mol}\), i.e. \(200\ \text{kg cal}/\text{mol}\), and therefore it is possible that such a unit of flow is \(\mathrm{SiO_2}\) or some suitable mixture of
of the four above-mentioned combinations. In Fig. 10 a plot from Ewell’s article is given, showing the dependence of the logarithm of viscosity on the reciprocal absolute temperature for various inorganic and organic glasses. The experimental data were obtained by various authors1. The value of the activation energy of viscous flow for SiO\(_2\), from the slope of the straight line for SiO\(_2\), is found to be \(175\ \text{kg cal}/\text{mole}\), which may be regarded as good agreement with the theoretical \(200\ \text{kg cal}/\text{mole}\).
Fig. 10. Dependence of \(\lg \eta\) on \(T^{-1}\) for certain associated, very viscous, and glass-forming liquids
If Na\(_2\)O is added to SiO\(_2\) and glasses of the Na\(_2\)O—SiO\(_2\) system are obtained, then the quantity \(\Delta E_{\text{visc}}\), or the activation energy, changes, decreasing as the Na\(_2\)O content increases. This is well explained by the structure of the glasses of this system given by Warren\(^ {27}\).
Let us dwell on one more section of Ewell’s article—on the dependence of viscosity on external pressure. He treats this question as follows. For an elementary flow process it is necessary that a hole be formed in the liquid, into which the units of flow can enter. If an external pressure is imposed on the liquid, then, in order to form the hole, work must be expended both against the external and against the internal pressure. The work against the internal pressure is \(\Delta E_{\text{visc}}\); the work against the external pressure is \(pv\), and therefore the total work is equal to the sum \(\Delta E_{\text{visc}} + pv\). At small external pressures the term \(pv\) may be neglected, but at higher ones \(\Delta E_{\text{visc}}\) in equation (31) must be replaced by \(\Delta E_{\text{visc}} + pv\). Thus the equation for the temperature dependence of viscosity, taking external pressure into account, has the following form:
\[ \eta = 1{,}090 \cdot 10^{-3}\, \frac{M^{\frac{1}{2}}\cdot T^{\frac{3}{2}}} {v^{\frac{2}{3}}\cdot(\Delta E_{\text{visc}}+pv)} \cdot e^{\frac{\Delta E_{\text{visc}}+pv}{nRT}} \tag{35} \]
It must be borne in mind that \(\Delta E_{\text{visc}}\) in this equation depends on pressure, decreasing as the volume decreases. At a certain pressure value \(\Delta E_{\text{visc}}\) becomes equal to zero and then changes
…sign, so that at a sufficiently high pressure the work of transfer of a molecule from the compressed liquid into vapor can become negative. Hildebrand\(^{28}\) derived the dependence of \(\Delta E_{\mathrm{isp}}\) on pressure. Using this work, Ewell obtains the following formula for calculating the viscosity at any temperature and pressure:
\[ \eta = 1{,}090 \cdot 10^{-2}\, \frac{M^{\frac{1}{2}}\cdot T^{\frac{2}{3}}} {v^{\frac{5}{3}}\left(\dfrac{dp}{dT}\right)_v} \cdot e^{\frac{v\cdot\left(\dfrac{\partial p}{\partial T}\right)_v}{nRT}} . \tag{36} \]
This equation was checked with the aid of Bridgman’s experimental data for ether, with \(n\) taken equal to 4 and the results obtained divided by 2, as was stated above. In this way, agreement with experiment was obtained up to pressures of \(7\,000\ \mathrm{kg}/\mathrm{cm}^2\), in which Ewell sees good confirmation of the theory.
As has already been said, the Eyring–Ewell theory is attractive in its generality, but comparison of the theoretical data with the experimental data shows that the mechanism placed at the basis of this theory still requires further development and refinement. In this theory the question of the free volume of liquids is of essential importance, and therefore Bernal’s remark cited above concerning theories of viscous flow constructed on volume changes may be applied to it.
In addition, calculations by Ewell’s formulas in a number of specific cases are complicated by the fact that it is necessary to know the molecular weight of the liquid. However, in the case of polymerized liquids and glasses, the indicated factor is a rather indefinite quantity, and this circumstance greatly complicates the calculations.
The Eyring–Ewell theory gave rise to a number of works devoted to its application and development. Let us mention the work of Eyrich and Simha\(^{29}\), who calculated, according to this theory, the activation energy for a homologous series of fatty acids as a function of temperature, molecular weight, and liquid structure. Applying the basic propositions of Eyring’s theory, Gvaretskii\(^{30}\) gave a temperature dependence of viscosity of a somewhat different form than Ewell. Eyring’s formula for the free volume was used by Gugel\(^{31}\), who set himself the aim of deriving the Bachinskii equation from classical kinetic theory and obtained proportionality between the viscosity coefficient and the free volume \(\left(\eta = \dfrac{C}{v_f}\right)\). Hence, if one assumes that the free volume is equal to the total volume minus the limiting volume, the Bachinskii equation is obtained. If, however, instead of \(v_f\) one substitutes its value according to Eyring for densely packed liquids, then a viscosity formula of the following form is obtained:
\[ \eta = \frac{C}{\left(v^{\frac{1}{3}} - D\right)^3}. \]
The fact that different liquids obey different equations Gugel explains by the difference in the structure and packing of the molecules of the liquid.
The critique of Eyring’s theory of liquids—more precisely, of Eyring’s derivation of the distribution function and the entropy of fusion—is the subject of a paper by Mott and Gurney^32, who, adhering to the basic propositions of Bernal’s theory, regard a liquid as a limiting state of a polycrystalline solid, in which the distinction between individual crystallites disappears.
MACLEOD’S THEORY
Macleod^33 in 1923 gave a semiempirical formula for the temperature dependence of viscosity, differing from the Batschinsky formula^34, proposed as early as 1913, only in that the free volume \((v-b)\) enters the formula to some power \(n\), characterizing the degree of association of the liquid. The formula had the form:
\[ \eta=\frac{C}{(v-b)^n}. \tag{37} \]
The quantity \(n\) for non-associated liquids is taken to be equal to unity; for associated liquids it is considerably greater than unity. Later Macleod^35 showed that \(C\) is proportional to the molecular weight of the liquid. Since it is more convenient to have in the denominator an exponent \(n\) equal to unity, Macleod transforms his formula in such a way that the dependence on the degree of association is expressed by a factor \(a\), greater than unity and decreasing with increasing temperature. The formula for viscosity then takes the form:
\[ \eta=\frac{KM_0a}{v-b}, \tag{38} \]
where \(M_0\) is the ordinary molecular weight and \(K\) is the coefficient of proportionality.
The success of Andrade’s theory, which gave a formula for viscosity containing \(v^{\frac{1}{3}}\) and an exponential factor, led Macleod to seek an equation that would include an exponential factor of the same type as Andrade’s.
For this purpose he first expresses \(v-b\) in terms of the internal pressure and considers it probable that not all molecules participate equally in the internal pressure, but that it is due chiefly to those molecules which, owing to their low velocities, associate with other molecules. The internal pressure in this case must be large and equal to the internal pressure in the solid state. Let us denote it by \(\pi_0\). Then the total internal pressure must be equal to the product of \(\pi_0\) by the number of such bound molecules, which, according to Maxwell’s law, must be expressed by an exponential function of temperature, decreasing as the latter increases. Since, moreover, a liquid can be brought into the solid state by pressure alone, Macleod further assumes that the number of these molecules must at the same time depend on the volume. Therefore, for the free volume Macleod writes the following expression:
\[ \frac{RT}{v-b}-\pi_0 e^{\frac{C}{Tv}}. \tag{39} \]
This equation contains three unknown constants, \(\pi_0\), \(C'\), and \(b\), and has a form that allows it to be solved only by approximation. It was tested by MacLeod on octane with satisfactory results.
Next, the effective molecular weight for associated liquids differs from the ordinary molecular weight, and for it MacLeod adopts the same functional dependence on temperature and volume as for the internal pressure. Therefore \(M = M_0 e^{\frac{C''}{T\nu}}\), where \(M_0\) is the ordinary molecular weight, \(C''\) is a constant. Substituting into the viscosity formula (38), in place of \(M_0\), \(M_0 e^{\frac{C''}{T\nu}}\), and in place of \(\nu-b\) its value from equation (39), we obtain:
\[ \eta = \frac{K M_0 e^{\frac{C''}{T\nu}} \cdot \pi_0 e^{\frac{C'}{T\nu}}}{RT} = \frac{B \cdot e^{\frac{C}{T\nu}}}{T}, \tag{40} \]
where
\[ B = \frac{K \cdot M_0 \pi_0}{R} \quad \text{and} \quad C = C' + C''. \]
Equation (40) shows that the viscosity is directly proportional to the internal pressure and the effective molecular weight and inversely proportional to the absolute temperature \(T\). In the form given to this equation on the right-hand side of (40), it contains only two constants: \(B\) and \(C\).
Equation (40), in the form
\[ \eta = \frac{K M_0 e^{\frac{C''}{T\nu}}}{\nu-b}, \]
was tested by MacLeod on octane, which he had studied well and for which he calculated \(\nu-b\). This test showed that formula (40) agrees with experiment no worse than Andrade’s equation. Subsequently, combining Andrade’s theory with his own, MacLeod gives, for non-associated liquids, the formula:
\[ \frac{\eta \nu^{\frac{1}{3}}}{\sqrt{T}} = \frac{B}{\nu-b}, \tag{41} \]
where \(B\) is a constant, and for associated liquids
\[ \frac{\eta \nu^{\frac{1}{3}}}{\sqrt{T}} = \frac{B}{(\nu-b)^n}. \tag{42} \]
These formulas were tested by MacLeod for organic liquids, and the agreement with experiment again proves to be no worse than in Andrade.
GOODVE’S THEORY
Among works that have appeared recently, Goodve’s work\({}^{36}\) is of interest; in it viscosity is considered in connection with the problem of thixotropy. The author believes that the thixotropy of a substance, by which he understands any isothermal decrease of viscosity with increasing shear rate, is a consequence of structure. He distin-
…distinguishes two components of the viscosity of thixotropic substances—the Newtonian and the purely thixotropic. Any stirring or shear causes stretching of bonds and even their rupture, and a series of impulses is produced, i.e., a transfer of momentum from the moving layer to the neighboring one. The magnitude of the impulse, as Goodeve shows, is inversely proportional to the rate of shear, i.e., to the velocity gradient, while the number of impulses per second is proportional to it. Their product, i.e., the total force, is independent of the rate of shear, which characterizes a thixotropic substance. If, however, the energy required to break a bond is small in comparison with the energy of thermal motion \(kT\), or if the structure of the liquid is such that shear cannot lead to a concentration of sufficient forces on the bonds, then the destruction of bonds occurs chiefly at the expense of thermal energy. In such a case the process obeys Boltzmann statistics, and the probability that, per unit time, some bond will be broken is expressed by the following equality:
\[ \frac{1}{\tau}=\nu e^{-\frac{E}{kT}} . \tag{43} \]
Here \(\tau\) is the mean duration of the existence of a bond, \(\nu\) is the frequency with which energy is redistributed between the bond and its surroundings, and \(E\) is the energy required to break the bond. The rate of shear, depending on the number of broken bonds in 1 sec in \(1\ \mathrm{cm}^{3}\) and on \(\tau\), thus contains an exponential factor, which then also enters into the expression for the total force. In this case the total force is already proportional to the rate of shear, as is observed for Newtonian liquids. Thus, for a Newtonian liquid Goodeve’s theory gives an exponential dependence of viscosity on temperature. A characteristic feature of this theory is that the principal attention is directed to the character and strength of the bonds between particles and to the duration of their existence. From this point of view the features of various particular cases are explained. Thus, for example, in the case of colloidal solutions with a comparatively small viscosity we are dealing with thixotropic substances. Their small viscosity is explained by the fact that, although the bonds between the particles are sufficiently large, the number of these bonds in \(1\ \mathrm{cm}^{3}\) is small, and there is a definite structure of the shear energy which, being concentrated on these structural bonds, destroys them comparatively easily. In glasses, on the contrary, there is a sufficiently large number of bonds with a large store of energy, but there is no structure with the aid of which the energy can be concentrated. Therefore glasses, having considerable viscosity, flow like Newtonian liquids. Goodeve supports these considerations by pointing out that in many colloidal solutions, at low rates of shear, an increase in viscosity and a decrease in thixotropic properties are observed.
Goodeve’s theory has been touched upon in this review, devoted to theories of the viscous flow of ordinary liquids, only insofar as it contains a derivation of the formula for the viscosity of a Newtonian liquid.
As can be seen from this, perhaps incomplete, review, all theories of viscosity, with the exception of the theory of S. E. Khaykin, lead to an exponential dependence of viscosity on temperature, which corresponds,
as experience shows, the actual state of affairs. With respect to the magnitude of the activation energy, there is a fundamental difference between the conceptions of Bernal–Ward–Frenkel and of Eyring–Ewell; the latter regard it as a fraction of the heat of vaporization, whereas the former authors believe that this quantity should be of the order of the heat of fusion. Ward’s tables given here show that the value of \(B\) is highly individual and depends on a whole series of factors. The question of the theory of viscous flow is still very far from a more or less definitive solution; the large number of arbitrary constants in formulas arising from different theories does not make it possible, on the basis of experimental verification, to reject decisively one or another of them. In any case, the solution of the problem of viscosity must be sought along the path of solving the more general problems of the liquid state and of the melting of matter.
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