ABSORPTION OF ULTRASONIC WAVES IN LIQUIDS AND THE MOLECULAR MECHANISM OF BULK VISCOSITY

I. G. Mikhailov and V. A. Solov'ev

INTRODUCTION

Numerous experimental studies of the absorption of ultrasonic waves in liquids, carried out in recent years, have found that for most liquids the measured absorption coefficient greatly exceeds that calculated from the Stokes–Kirchhoff hydrodynamic theory.

At present it may be regarded as established that this “excess over Stokes” part of the absorption is caused by losses arising under compression, i.e., by bulk viscosity.

There are no simple and direct methods for measuring bulk viscosity. Therefore measurement of the absorption of ultrasonic waves is still the only method for measuring this viscosity.* In contrast to the ordinary shear viscosity, bulk viscosity has as yet been studied very little. This is explained by the fact that ultrasonic methods began to be applied comparatively recently and are still relatively complex, which limits their wide dissemination. Nevertheless, the experimental and theoretical studies already available indicate that the existence of bulk viscosity is connected with a number of molecular processes that are of considerable interest from the standpoint of existing theories of the liquid state.

The present article gives a review of a number of works concerned with elucidating the molecular mechanism of bulk viscosity.

The number of experimental and theoretical works on excess-over-Stokes absorption of ultrasonic waves is very large. We have not attempted to give an exhaustive review of these works, but have considered only those of them that are of greatest interest from the standpoint of elucidating the molecular mechanism of bulk viscosity.

* If one does not count the recently published method for measuring bulk viscosity by means of acoustic wind.¹

Recently a review article by Markham et al.^2 was published; it discusses in detail various phenomenological theories of relaxation processes. This article contains a fairly complete bibliography and may be used for further familiarization with the question of sound absorption in liquids.

§ 1. PHENOMENOLOGICAL RELAXATION THEORY

The equation of motion of the particles of a liquid during the propagation of a plane sound wave has the form

\[ \rho \frac{\partial^2 u}{\partial t^2}=-\frac{\partial p}{\partial x}, \tag{1} \]

where \(\rho\) is the density of the liquid, \(u\) is the displacement of the particles of the liquid, and \(p\) is the pressure. To solve this equation it is necessary to substitute into it an explicit expression for the pressure as a function of the density. If the liquid is ideal, then for small sound amplitudes

\[ p=p_0+\frac{\partial p}{\partial \rho}\Delta\rho =p_0+\frac{\partial p}{\partial \rho}\rho\Delta s, \tag{2} \]

where \(p_0\) is the equilibrium hydrostatic pressure, and \(s=\dfrac{\Delta\rho}{\rho}=-\dfrac{\partial u}{\partial x}\) is the strain (compression). The derivative \(\dfrac{\partial p}{\partial \rho}\) should be evaluated at constant entropy, since the process of sound propagation at the ordinarily used sonic and ultrasonic frequencies is, to a first approximation, adiabatic.

Taking the derivative of (2) with respect to \(x\) and substituting into (1), we obtain the ordinary wave equation

\[ \rho \frac{\partial^2 u}{\partial t^2} =K\frac{\partial^2 u}{\partial x^2}, \tag{3} \]

where \(K=\rho\left(\dfrac{\partial p}{\partial \rho}\right)_S\) is the adiabatic bulk modulus.

For a harmonic process, the solution of equation (3) will be

\[ u=u_0 e^{i\omega\left(t-\frac{x}{a}\right)}, \]

where \(\omega\) is the frequency and

\[ a=\sqrt{\frac{K}{\rho}} \tag{4} \]

is the speed of sound.

Equation (3) describes an undamped wave. The attenuation of sound is caused by the viscosity and thermal conductivity of the medium. The thermal conductivity of liquids plays a relatively small role; therefore, in what follows, in studying sound absorption, we shall restrict ourselves only to taking viscosity into account.

Taking viscosity into account, equation (2) takes the form:

\[ p=p_0+Ks+\left(\eta+\frac{4}{3}\mu\right)\frac{\partial s}{\partial t}, \tag{5} \]

where \(\mu\) is the ordinary shear viscosity, and \(\eta\) is the so-called bulk viscosity, determining that part of the pressure which depends on the rate of all-round compression. In a plane sound wave there occurs not an all-round, but a one-sided deformation of compression, also including shear. It is precisely for this reason that, in addition to the bulk viscosity, the shear viscosity also enters formula (5).

In ordinary hydrodynamic calculations (viscous flow) it is assumed that \(\eta=0\). Stokes put forward the assumption that, in studying the propagation of sound as well, the influence of bulk viscosity may be neglected.

The question of the validity of Stokes’s hypothesis has been discussed repeatedly in the literature. Only in 1937 did L. I. Mandelstam and M. A. Leontovich\(^3\) subject this question to a detailed investigation. They pointed out the necessity of taking bulk viscosity into account when studying the absorption of sound, and also established the connection between bulk viscosity and relaxation processes in polyatomic gases and liquids\(^*\).

Substituting (5) into equation (1), we obtain the wave equation, whose solution will already be a damped wave:

\[ u=u_0 e^{i\omega\left(t-\frac{x}{a}\right)-\alpha x}, \tag{6} \]

where \(\alpha\) is the amplitude attenuation coefficient.

For the convenience of subsequent calculations one may introduce the operator

\[ \mathcal{K}=K+\left(\eta+\frac{4}{3}\mu\right)\frac{\partial}{\partial t} =K+\eta'\frac{\partial}{\partial t}, \]

where \(\eta'\) is the “effective” viscosity. By substituting this operator, the wave equation for the case of a damped wave is reduced to the form (3).

For a harmonic process, \(\dfrac{\partial}{\partial t}\) must be replaced by \(i\omega\). Then, instead of \(K\), equation (3) will contain \(\mathcal{K}=K+i\omega\eta'\). In this case the velocity of sound will prove to be complex:

\[ \widetilde{a}=\sqrt{\frac{\mathcal{K}}{\rho}}, \]

\(^*\) Foreign authors, when introducing bulk viscosity, usually refer to Tisza,\(^4\) who in 1942 published a paper that to a considerable extent independently repeated Mandelstam and Leontovich’s article. It should be noted, however, that Tisza’s work is distinguished by much less rigor and completeness of the derivations.

where

\[ a=\frac{1}{\operatorname{Re}\left(\frac{1}{\tilde a}\right)} \quad \text{and} \quad \alpha=\operatorname{Im}\left(\frac{\omega}{\tilde a}\right). \]

Substituting the expression for the complex velocity, we have:

\[ a=\sqrt{\frac{K}{\rho}}\, \sqrt{ \frac{ 2\left(1+\omega^{2}\eta'^{\,2}/K^{2}\right) \left(\sqrt{1+\omega^{2}\eta'^{\,2}/K^{2}}-1\right) }{ \omega^{2}\eta'^{\,2}/K^{2} } }, \]

\[ \alpha=\omega\sqrt{\frac{\rho}{K}}\, \sqrt{ \frac{ \sqrt{1+\omega^{2}\eta'^{\,2}/K^{2}}-1 }{ 2\left(1+\omega^{2}\eta'^{\,2}/K^{2}\right) } }. \tag{7} \]

If \(\omega\eta' \ll K\), we obtain, retaining terms no higher than first order of smallness,

\[ a=\sqrt{\frac{K}{\rho}} \quad \text{and} \quad \alpha=\frac{\omega^{2}\eta'}{2\rho a^{3}}. \tag{8} \]

When higher powers of \(\dfrac{\omega\eta'}{K}\) are taken into account, dispersion of the velocity is obtained, and \(\alpha\) is no longer proportional to \(\omega^{2}\). P. A. Bazhulin and M. A. Leontovich\(^5\) pointed out that, for large values of \(\dfrac{\omega\eta'}{K}\), formulas (7) must be used with caution. The condition \(\omega\eta' \simeq K\) corresponds to very large absorption per wavelength (for \(\omega\eta'=K\), \(a\lambda \simeq 2.5\)). For gases in this frequency range, the equations of hydrodynamics generally become inapplicable, since the wavelength proves comparable with the mean free path of the molecules. In liquids and in amorphous solids these equations are usually applied also when \(\dfrac{\omega\eta'}{K}\) is comparable with unity. However, one must not forget that they are based on thermodynamic relations pertaining to equilibrium processes. Meanwhile, in the case of large damping, i.e. large energy losses, the indicated thermodynamic relations cannot be applied. In addition, when terms of second order with respect to \(\dfrac{\omega\eta'}{K}\) are taken into account, one should at the same time also take into account possible deviations from proportionality between the viscous stress and the rate of deformation. Of course, even in the case of very viscous liquids, when \(\omega\eta'\) is comparable with \(K\), the dependence of viscous forces on the rate of deformation may be linear (or nearly linear); in that case this last objection falls away. But the question of the linearity of the equations of viscous friction can be decided only by experiment. Hydrodynamics itself does not claim to give a more exact description than is given by for-

ABSORPTION OF ULTRASONIC WAVES IN LIQUIDS

...mulas (8). The calculation of the subsequent approximations can always give an error of the same order as the correction obtained.

In the form set forth above, the theory of sound absorption was developed as early as Stokes. However, he erroneously believed that, in studying the propagation of sound, just as in solving ordinary hydrodynamic problems, the volume viscosity could be set equal to zero. Thus the absorption coefficient according to Stokes is equal to

\[ \alpha=\frac{2\omega^{2}\mu}{3\rho a^{3}}. \tag{9} \]

However, Stokes’ theory, as is known, does not agree with experimental data. Absorption in polyatomic gases and liquids proves, as a rule, to be greater than that calculated from formula (9). In addition, in a number of gases and liquids a nonquadratic dependence of absorption on frequency and dispersion have been observed. To explain these phenomena, the assumption was introduced that the establishment of thermodynamic equilibrium takes place comparatively slowly, which leads to excess absorption.

As such a relaxation process Einstein\(^6\) considered the establishment of chemical equilibrium in a partially dissociated gas. Kneser\(^7\) considered the establishment of an equilibrium distribution of energy between the external and internal degrees of freedom in a polyatomic gas.

Then Mandelstam and Leontovich\(^3\) gave a general thermodynamic theory of relaxation processes, applicable both to gases and to liquids. They also showed that these processes can be taken into account by introducing a relaxing volume viscosity. Later Kneser\(^ {10}\) published a theory essentially identical with that of Mandelstam and Leontovich. His work differs only in that he did not give an explicit thermodynamic expression for the parameters entering into the equations, and therefore his theory is completely unsuitable for concrete calculations. It is characteristic that, although Kneser had in mind only the relaxation of the establishment of the equilibrium distribution of energy (and of chemical equilibrium in dissociation), he was unable to indicate any way of calculating the parameters of his equations from molecular considerations.

On the basis of the general results of relaxation theory, it is possible, in a purely phenomenological way, without going into consideration of the molecular mechanism, to investigate the influence of relaxation processes on the propagation of sound. The development of the general phenomenological theory was carried out mainly by M. A. Isakovich\(^8\) and Ya. I. Frenkel and Yu. N. Obraztsov\(^9\).

Macroscopically, relaxation manifests itself in the fact that, for example, deformation may be composed of an instantaneous part and a delayed part, occurring with a finite velocity. Here various...

combinations of “elastic” and “viscous” terms in the equation relating stress and deformation. In doing so, however, one should remember that such combination is a purely formal operation, and the resulting equations must be justified by molecular considerations.

One of the simplest possible combinations of elasticity and viscosity is represented by equation (5), according to which the stress \(f=(p-p_0)\) is composed of an elastic \((Ks)\) and a viscous \((\eta' \dot{s})\) part; the latter is the force of internal friction. For clarity, such equations are illustrated by models consisting of springs, whose stiffness is equal to the corresponding moduli of elasticity, and pistons, whose coefficients of friction are equal to the corresponding coefficients of viscosity. The deformation of such a model will be described by the same differential equation as the deformation of our liquid, if the force acting on the model is compared with the stress.

Equation (5) is represented by a parallel connection of a spring and a piston (Fig. 1). This figure also shows the behavior of such a model under a constant load. The compression-modulus operator corresponding to this model has the form:

\[ \mathcal{K}=K+\eta'\frac{\partial}{\partial t} =K\left(1+r\frac{\partial}{\partial t}\right), \]

where \(r=\dfrac{\eta'}{G}\) is the retardation time—that is, the time during which, under constant stress, the deformation reaches \(\left(1-\dfrac{1}{e}\right)\) of its equilibrium value.

Fig. 1.

The second simplest method of combining elasticity and viscosity consists in the deformation being composed of an elastic (instantaneous) deformation and a viscous flow. Such deformation is observed in many amorphous bodies, which are solid under short-term loading but “flow” under prolonged loading. This also applies to ordinary low-viscosity liquids, which, under sufficiently rapid action, must exhibit the properties of a solid body.

A model of such an elastic-viscous body was proposed by Maxwell. It is shown in Fig. 2, where its behavior under constant stress and under constant deformation is also shown. The equation describing such deformation has the form:

\[ \dot{s}=\frac{1}{G}\dot{f}+\frac{1}{\mu}f, \tag{10} \]

Absorption of Ultrasonic Waves in Liquids

and the modulus operator

$$ \mathcal{H}=\frac{\mu \dfrac{\partial}{\partial t}}{1+\tau \dfrac{\partial}{\partial t}}, \quad \text{where } \tau=\frac{\mu}{G} $$

is the relaxation time, i.e., the time during which the stress at constant deformation falls (“relaxes”) to \(1/e\) of its initial value.

Fig. 2.

Fig. 3.

This equation can apply only to shear deformation, since it permits unlimited flow, which, of course, is impossible in the case of deformation of all-round compression.

Further, one may assume that the deformation is composed of an instantaneous one (of the type of equation (2)) and a delayed one (equation (5)). The same result is obtained if one assumes that the stress is composed of an equilibrium part (2) and a relaxation part (10). The corresponding models are shown in Fig. 3. The parameters of these models are: the equilibrium modulus \(K_0\) and the equilibrium compressibility \(\beta_0=1/K_0\); the instantaneous modulus \(K_\infty\) and the instantaneous compressibility \(\beta_\infty=1/K_\infty\); the relaxation modulus \(K_1=K_\infty-K_0\) and the delayed compressibility \(\beta_1=\beta_0-\beta_\infty\); the relaxing viscosity \(\eta\) and the “viscosity” (having no clear physical meaning)

$$ \eta_r=\frac{K_\infty^2}{(K_\infty-K_0)^2}\eta; $$

the relaxation time \(\tau=\eta/K_1\), and the delay time \(r=\eta_r\beta_1=\dfrac{K_\infty}{K_0}\tau\).

The relaxing viscosity \(\eta\) is related to the delay time by the following relation:

$$ \eta=\frac{\beta_1}{\beta_0^2}\,r. $$

I. G. MIKHAILOV AND V. A. SOLOV'EV

The deformation of such a model obeys the equation

\[ \dot f+\frac{1}{\tau}f=\frac{K_0}{\tau}s+K_\infty \dot s, \tag{11} \]

and the operator of the bulk modulus has the form:

\[ \mathcal{K}=K_0+\frac{\eta \dfrac{\partial}{\partial t}}{1+\tau \dfrac{\partial}{\partial t}} = \frac{1}{\beta_\infty+\beta_1\dfrac{1}{1+\tau \dfrac{\partial}{\partial t}}}. \tag{12} \]

Replacing \(\dfrac{\partial}{\partial t}\) by \(i\omega\), we decompose it into its real and imaginary parts:

\[ \mathcal{K}=K(\omega)+i\omega\eta(\omega) = K_0+\frac{K_1\omega^2\tau^2}{1+\omega^2\tau^2} +i\omega\frac{\eta}{1+\omega^2\tau^2}. \tag{13} \]

The simplest relaxation theory, taking into account only one relaxation process, leads, as will be shown below, precisely to such an operator of the bulk modulus.

To calculate the velocity and absorption of sound, to (13) one must add another term, accounting for the shear viscosity, \(-i\omega \dfrac{4}{3}\mu\).

Substituting into (7), we obtain an expression for the velocity and absorption of sound. If the condition

\[ \omega\eta'(\omega)=\omega\eta(\omega)+\frac{4}{3}\omega\mu \ll K(\omega), \]

is satisfied, then we obtain the same expressions as before (formula (8)), but now they contain the frequency-dependent modulus \(K(\omega)\) and viscosity \(\eta'(\omega)\):

\[ \left. \begin{aligned} a^2&=\frac{1}{\rho}\left\{K_0+\frac{K_1\omega^2\tau^2}{1+\omega^2\tau^2}\right\};\\ \alpha&=\frac{\omega^2}{2\rho a^3}\left\{\frac{\eta}{1+\omega^2\tau^2}+\frac{4}{3}\mu\right\}. \end{aligned} \right\} \tag{14} \]

It is easy to see that for \(\omega\tau \simeq 1\) there is an increase in the velocity and a decrease in the quantity \(\dfrac{\alpha}{\omega^2}\) with frequency. For \(\omega\tau \ll 1\) and \(\omega\tau \gg 1\), \(\dfrac{\alpha}{\omega^2}\) and \(a\) are constant, but for \(\omega\tau \ll 1\), \(\dfrac{\alpha}{\omega^2}\) is determined by the quantity \(\eta'=\eta+\dfrac{4}{3}\mu\), whereas for \(\omega\tau \gg 1\), \(\dfrac{\alpha}{\omega^2}=\dfrac{2\mu}{3\rho a^3}\), i.e., it has the value obtained from Stokes’ theory. Thus, relaxation-

... theory gives, at least, a qualitative explanation of the discrepancies indicated above between the results of classical hydrodynamic theory and experiment.

In the literature one often encounters expressions for the “super-Stokes,” i.e. volume-viscosity-dependent, part of the absorption of sound at low frequencies. We give expressions for this quantity in terms of the parameters of both models shown in Fig. 3:

\[ a_0=\frac{\omega^2}{2a}\frac{K_\infty-K_0}{K_0}\tau =\frac{\omega^2}{2a}\frac{\beta_0-\beta_\infty}{\beta_0}\tau . \tag{14'} \]

It should be noted that the dispersion is correctly described by equation (14) only in the case when \(\omega^2\tau^2 K_1 \ll K_0\). If this condition is not fulfilled, the value of \(a\) should be calculated from equation (7) with greater accuracy, since in this case terms of the form \(\omega^2\eta'^2\) have the same order of magnitude as \(\omega^2\tau^2K_1\), and they too must be taken into account. This circumstance has not always received attention*).

The phenomenological relaxation theory also permits further generalizations. First of all, one should take into account the relaxation of the shear viscosity, since under a sufficiently rapid action all liquids must exhibit shear elasticity. This generalization can be carried out by connecting one more Maxwell element in parallel to the model of Fig. 3. Then, instead of equations (14), we obtain

\[ \left. \begin{aligned} a^2&=\frac{1}{\rho}\left\{K_0+\frac{K_1\omega^2\tau^2}{1+\omega^2\tau^2} +\frac{\frac{4}{3}G\omega^2\tau_\mu^2}{1+\omega^2\tau_\mu^2}\right\},\\[6pt] \alpha&=\frac{\omega^2}{2\rho a^3}\left\{\frac{\eta}{1+\omega^2\tau^2} +\frac{\frac{4}{3}\mu}{1+\omega^2\tau_\mu^2}\right\}, \end{aligned} \right\} \tag{15} \]

where \(G\) is the shear modulus, \(\tau_\mu=\dfrac{\mu}{G}\) is the relaxation time of the shear viscosity, and \(\tau\) is the relaxation time of the volume viscosity.

In some very viscous liquids relaxation of the shear viscosity is indeed observed. Thus, according to P. A. Bazhulin\(^{12}\),

*) For example, Goh\(^{11}\) did not take this additional dispersion into account, although in the case he considered (water) \(K_1>K_0\), and therefore this part of the dispersion is even greater than that which he took into account. Moreover, in this same region the hydrodynamic equations cease to be valid, and the remarks made above on the limits of applicability of the hydrodynamic equations come into force. This may lead to still larger errors in calculating the dispersion.

\(\alpha/\omega^2\) in castor oil at frequencies from \(4\ \text{MHz}\) to \(16\ \text{MHz}\) has a value below the Stokes value and decreases as the frequency is increased. The existence of relaxation of the shear viscosity is also indicated by the results of a study of the fine structure of the Rayleigh scattering line\({}^{13}\).

Ya. I. Frenkel and Yu. N. Obraztsov also considered the relaxation shear elasticity (Fig. 4, a). It is essential to note that all schemes obtained by further generalizations of this kind can be represented by a model of the type shown in Fig. 4, b, with a sufficient number of parallel Maxwell elements. The corresponding operator of the elastic modulus in this case has the form:

Fig. 4.

\[ \mathcal{K}=K_0+\sum_i \frac{\eta_i \dfrac{\partial}{\partial t}}{1+\tau_i \dfrac{\partial}{\partial t}} . \]

In the most general form it can be proved that any conceivable connection of any number of springs and pistons can be reduced to this form by decomposing the operator of the modulus describing this system into partial fractions (the modulus operator is an algebraic fraction in \(\dfrac{\partial}{\partial t}\)). From this, formulas for the dispersions and for the dependence of the absorption coefficient on frequency may be obtained. When the previously indicated conditions for the velocity and absorption are fulfilled, equations of type (15) are obtained, but with a larger number of terms.

Consideration of a large number of relaxation times turns out to be necessary, for example, in the case of high polymers, where very many different molecular mechanisms of deformation operate. It should be noted, however, that a fully rigorous justification of the applicability of such a general expression for the modulus operator has not yet been given.

§ 2. GENERAL THEORY OF RELAXATION PROCESSES

(THE WORK OF MANDELSHTAM AND LEONTOVICH)

Mandelshtam and Leontovich \(^{3}\) assumed that, in the equation of state of a liquid, besides the pressure \(p\), density \(\rho\), and temperature \(T\), there also enters a certain parameter \(\xi\), which in some way determines the internal structure of the liquid. This may be, for example, the concentration of excited molecules, the concentration of one of the reacting components in chemical equilibrium, the degree of short-range order, etc. Then the equation of state may be written in the form \(\rho = R(p,T,\xi)\)*.

At equilibrium, \(\xi\) assumes the value \(\xi_0=\xi_0(p,T)\), depending on the pressure and temperature, and then the equation of state has the usual form: \(\rho=R(p,T,\xi_0)=\rho(p,T)\). During the propagation of a sound wave, the pressure and temperature, and consequently also \(\xi_0\), change. We shall assume that the establishment of the equilibrium state corresponding to the given values of \(p\) and \(T\) occurs not instantaneously, but with a certain finite rate. The equation describing the process of establishing equilibrium is derived from the condition that equilibrium corresponds to a minimum of the thermodynamic potential

\[ \left(\frac{\partial \Phi}{\partial \xi}\right)_{\xi=\xi_0}=0. \]

Introducing further the assumptions of linearity of the equation and of the smallness of the deviation from equilibrium (i.e. of the smallness of the amplitude of the sound wave), one may conclude that, with a suitable choice of the variable \(\xi\), the desired “reaction equation” has the form:

\[ \dot{\xi}=-k'\left(\frac{\partial \Phi}{\partial \xi}\right)_{p,T}, \]

where \(k'\) is a constant. If \(p\) and \(T\) are constant, this is obvious. If, however, \(p\) and \(T\) vary, then the change of \(\xi\) may be associated, generally speaking, not only with its departure from the equilibrium value, but also directly with the change of \(p\) and \(T\)**. In this case the part of \(\dot{\xi}\) depending on the deviation from equilibrium may be represented in the form \(\dot{\xi}-a\dot{p}-b\dot{T}\). But, taking this quantity as the new variable \(\dot{\xi}\), we again obtain an equation of the same form.

* Mandelshtam and Leontovich wrote this equation in the form \(\rho=\rho(p,T,\xi)\), i.e. they took as independent variables \(p,T,\xi\). We shall set forth their theory in the variables \(\rho,T,\xi\), since in this form it is more convenient to compare it with other works. The scheme of the calculations is not changed thereby, so that the subsequent exposition essentially repeats exactly the work of Mandelshtam and Leontovich (with some abridgments).

** Mandelshtam and Leontovich give the convincing example of a dependence of \(\xi\) on density, when \(\xi\) at a given point changes simply because of the flow of matter. Probably examples of this kind of dependence on other variables can also be given.

Expanding \(-\dfrac{\partial \Phi}{\partial \xi}\) in a series and restricting ourselves to the first powers of the small quantities \(\Delta p, \Delta T, \Delta \xi\), we have:

\[ \ddot{\xi}=-k'\left(\Phi_{\xi\xi}\Delta \xi+\Phi_{\xi p}\Delta p+\Phi_{\xi T}\Delta T\right), \tag{16} \]

where \(\Delta \xi, \Delta p, \Delta T\) are deviations from the unperturbed (in the absence of sound) values, and the derivatives are calculated for the unperturbed state.

To solve the problem of sound propagation it is also necessary to use the law of conservation of energy, which in the variables \(p, T\) has the following form (for an adiabatic process):

\[ \dot H-\frac{1}{\rho}\dot p=0, \tag{17} \]

where \(H\) is the enthalpy, referred, like \(\Phi\), to unit mass. Performing the differentiation, we have:

\[ H_\xi\dot{\xi}+H_T\dot T+\left(H_p-\frac{1}{\rho}\right)\dot p=0. \tag{18} \]

The partial derivatives in equation (18), just as in (16), refer to the unperturbed, i.e. equilibrium, state (if quantities of second order of smallness are neglected). To calculate these derivatives one may therefore use the general thermodynamic relations:

\[ H=H(p,T,\xi)=\Phi+TS=\Phi-T\Phi_T, \]

\[ \rho=R(p,T,\xi)=\frac{1}{\Phi_p}. \]

Using these equations, as well as the equilibrium condition \(\Phi_\xi=0\), we have:

\[ \Phi_{\xi T}=-\frac{H_\xi}{T}; \]

\[ \Phi_{\xi p}=-\frac{1}{\rho^2}R_\xi; \]

\[ H_p=\frac{1}{\rho}+\frac{TR_T}{\rho^2}. \]

The derivatives \(H_T=C_p\) and \(R_T\) are calculated at constant \(\xi\). Further on we shall need their relation with the equilibrium heat capacity \(c_p\) and the equilibrium coefficient of expansion \(\beta_T\). These quantities must be calculated at a constant (and equal to zero) value of the quantity \(\Psi=\Phi_\xi\):

\[ C_p=H_T=\left(\frac{\partial H}{\partial T}\right)_{\Psi} +\left(\frac{\partial H}{\partial \Psi}\right)_T\frac{\partial \Psi}{\partial T} =c_p+\Phi_{\xi T}\frac{H_\xi}{\Phi_{\xi\xi}} =c_p-\frac{H_\xi^2}{T\Phi_{\xi\xi}}. \]

In exactly the same way one calculates

\[ R_T=\rho_T-\frac{H_\xi R_\xi}{T\Phi_{\xi\xi}} \quad \text{and} \quad R_p=\rho_p-\frac{R_\xi^2}{\rho^2\Phi_{\xi\xi}} . \]

Substituting the calculated values of the partial derivatives into equations (16) and (18), we obtain:

\[ \dot{\xi}=\frac{1}{r'}\left(\Delta \xi-\frac{1}{\rho^2}\frac{R_\xi}{\Phi_{\xi\xi}}\Delta p-\frac{H_\xi}{T\Phi_{\xi\xi}}\Delta T\right); \]

\[ T R_T p-C_p\rho^2\dot{T}+\rho^2 H_\xi\dot{\xi}=0. \]

Here \(r'=\dfrac{1}{k'T\Phi_{\xi\xi}}\) is the retardation time\(^*\), characterizing the rate of establishment of equilibrium at constant pressure and temperature.

For a harmonic process \(\dfrac{\partial}{\partial t}=i\omega\), and we obtain:

\[ (1+i\omega r')\Delta \xi-\frac{H_\xi}{T\Phi_{\xi\xi}}\Delta T = \frac{R_\xi}{\rho^2\Phi_{\xi\xi}}\Delta p, \tag{19} \]

\[ \rho^2 H_\xi\Delta \xi+C_p\rho^2\Delta T=-T R_T\Delta p. \tag{20} \]

The complex adiabatic compressibility is

\[ \frac{1}{\mathcal{K}} = \frac{1}{\rho}\frac{\Delta p}{\Delta \rho} = \frac{1}{\rho}\left\{ R_p+R_\xi\frac{\Delta \xi}{\Delta p} +R_T\frac{\Delta T}{\Delta p} \right\}. \tag{21} \]

Next it is necessary to determine \(\dfrac{\Delta T}{\Delta p}\) and \(\dfrac{\Delta \xi}{\Delta p}\) from equations (19) and (20) and substitute into this equation. The result, by elementary algebraic transformations, is brought to the form:

\[ \frac{1}{\mathcal{K}}=\beta_\infty+\frac{\beta_0-\beta_\infty}{1+i\omega r}, \tag{22} \]

where

\[ \beta_0=\frac{1}{\rho}\rho_p\frac{C_V}{c_p}, \qquad \beta_\infty=\frac{1}{\rho}R_p\frac{C_V}{C_p}, \qquad r=r'\frac{C_p}{c_p}. \]

In the derivation the well-known thermodynamic relation is used

\[ C_p-C_V=\frac{T R_T^2}{\rho^2 R_p}, \qquad c_p-c_v=\frac{T\rho_T^2}{\rho^2\rho_p}. \]

Let us also give the explicit

\(^*\) The distinction between relaxation time and retardation time, justified in mechanical processes, generally speaking has no meaning here: both \(r'\) and \(\tau'\), introduced below, are relaxation times (in the thermodynamic sense). However, we call \(r'\) the retardation time in order to avoid confusion in passing to mechanical quantities.

expression for the delayed compressibility:

\[ \beta_1=\beta_0-\beta_\infty=\frac{1}{\rho}\left(\rho_\rho\,\frac{c_v}{c_p}-R_\rho\,\frac{C_V}{C_p}\right)= \]

\[ =\frac{1}{\rho^3}\left\{\frac{T R_T^2}{C_p}-\frac{T p_T^2}{c_p}+\frac{R_\xi^2}{\Phi_{\xi\xi}}\right\}. \tag{23} \]

Thus, the theory of Mandelstam and Leontovich, in the form presented by us, gives for the compressibility operator an expression corresponding to the model of Fig. 3, a.

From the expressions given it is evident that the “mechanical” delay time \(\tau\) is not equal to the “molecular” time \(\tau'\). The point is that \(\tau'\) characterizes the rate of establishment of equilibrium at constant temperature (and pressure), whereas \(\tau\) determines the rate of compression at constant entropy and pressure, i.e., at constant enthalpy. The relation between \(\tau\) and \(\tau'\) is formally analogous to the relation between \(\tau\) and \(\tau'\) in the models of Fig. 3. Here temperature plays the role of stress, enthalpy the role of deformation, and the heat capacities \(C_p\) and \(c_p\) correspond to the compressibilities \(\beta_\infty\) and \(\beta_0\).

As already indicated, Mandelstam and Leontovich took \(\rho\) and \(T\) as independent variables. Then the characteristic thermodynamic function will be not the thermodynamic potential \(\Phi\), but the free energy \(F\). It is convenient to write the law of conservation of energy in the form

\[ \dot E-\frac{p}{\rho^2}\dot\rho=0 \]

(where \(E\) is the internal energy per unit mass), and the reaction equation

\[ \dot\xi=-k''(\Delta \xi+F_{\xi T}\Delta T+F_{\xi \rho}\Delta\rho) =-\frac{1}{\tau'}\left(\Delta\xi+\frac{F_{\xi T}}{F_{\xi\xi}}\Delta T+\frac{F_{\xi\rho}}{F_{\xi\xi}}\Delta\rho\right), \]

where the relaxation time \(\tau'=\dfrac{1}{k''F_{\xi\xi}}\) characterizes the rate of establishment of equilibrium at constant density and temperature.

A calculation entirely analogous to that given above then gives

\[ \mathcal K=K_0+\frac{i\omega\tau\,(K_\infty-K_0)}{1+i\omega\tau}. \tag{24} \]

Here

\[ K_0=\rho p_\rho\,\frac{c_p}{c_v}=\frac{1}{\beta_0}; \qquad K_\infty=\rho P_\rho\,\frac{C_p}{C_V}=\frac{1}{\beta_\infty}; \]

\[ K_\infty-K_0=\frac{1}{\rho}\left\{\frac{T P_T^2}{C_V}-\frac{T p_T^2}{c_v}+\frac{P_\xi^2}{F_{\xi\xi}}\right\}; \tag{25} \]

\[ \tau=\tau'\,\frac{C_V}{c_v}. \]

The relation between the derivatives at constant \(\rho\) and at equilibrium is given by the formulas:

\[ P_\rho=\left(\frac{\partial p}{\partial \rho}\right)_\xi =p_\rho+\frac{P_\xi^{2}}{\rho^{2}F_{\xi\xi}}\,^{*}), \]

\[ C_V=c_V-\frac{E_\xi^{2}}{TF_{\xi\xi}}, \]

\[ P_T=p_T-\frac{P_\xi E_\xi}{TF_{\xi\xi}}. \]

Thus, the theory of Mandelstam and Leontovich, set forth in the variables \(p, T, \xi\) or \(\rho, T, \xi\), leads to one and the same result, but the form of the expressions obtained corresponds in one case to the model of Fig. 3,a, and in the other to Fig. 3,b. The equivalence of these models and the relations between their parameters given in § 1, of course, do not require such a proof, since these models correspond simply to different forms of writing one and the same equation. We have presented different formulations of the Mandelstam–Leontovich theory in order to show more clearly the connection between these models and to facilitate comparison of different theories, since many authors use the variables \(p\) and \(T\).

The relation between \(r'\) and \(\tau'\) can be obtained from the phenomenological relation

\[ \frac{r'}{\tau'}=\frac{K_\infty}{K_0}. \]

Substituting here the expressions obtained above for \(K_0\) and \(K_\infty\), we have:

\[ \frac{r'}{\tau'}=\frac{C_V}{c_V}\cdot\frac{c_p}{C_p}\cdot\frac{r}{\tau} =\frac{P_\rho}{p_\rho}. \]

This result can also be obtained directly from consideration of the models of Fig. 3 for the case of an isothermal process.

Let us also note that equation (24) proves the existence of bulk viscosity, which in slow processes is equal to \(\eta=(K_\infty-K_0)\tau\), and in sufficiently rapid ones relaxes, i.e. decreases with frequency according to the formula

\[ \eta(\omega)=\frac{\eta}{1+\omega^{2}\tau^{2}}. \]

Mandelstam and Leontovich in their work also gave a direct derivation of the expression for \(\eta\), considering slow uniform expansion. We shall not reproduce this proof separately here, since it is carried out in complete analogy with the derivation of formula (24). In view of the fact that in the linear approximation the replacement of \(i\omega\) by \(\dfrac{\partial}{\partial t}\) is always permissible, the derivation presented may be regarded as a fully rigorous proof of the existence of bulk viscosity and of its connection with relaxation processes.

\[ \text{*) In this formula there is a misprint in Mandelstam and Leontovich.} \]

In their work, Mandelstam and Leontovich indicate a number of possible relaxation mechanisms in liquids. In particular, they note the possibility of relating structural relaxation to ordinary shear viscosity. They also indicate possible ways of calculating the parameters entering into their equations.

To apply the theory to particular liquids it is necessary to know (in the variables \(\rho, T\)) the quantities \(E_\xi, p_\xi, F_{\xi\xi}\) and the relaxation time

\[ t'=\frac{1}{k' F_{\xi\xi}} . \]

The remaining quantities pertain to equilibrium processes, and they may be regarded as known. In the following paragraphs we shall examine in detail methods for calculating the unknown parameters in various particular cases.

The method of Mandelstam and Leontovich also makes it possible to draw certain conclusions about the limits of applicability of relaxation theory. Some authors (Bauer\(^{14}\), Kneser in his 1950 paper\(^{15}\)) express doubt as to whether thermodynamic relations may be applied in considering the nonequilibrium process of sound absorption. The method of Mandelstam and Leontovich makes it possible to express explicitly the condition for the applicability of these relations: it is easy to see that they are used only for processes that are essentially equilibrium ones (very rapid processes—at constant \(\xi\)—are also equilibrium ones), and therefore their use is entirely legitimate. However, in deriving twice (in the “reaction equations” and in the law of conservation of energy) an expansion in powers of the deviations of the parameters \(\rho, T, \xi\) from their unperturbed (in the absence of sound) values is performed, with only the first powers of these deviations retained. This means, first, that \(\Delta p, \Delta T, \Delta \xi\) are regarded as small (i.e. the amplitude of the sound wave is small), and, second, that these quantities have the same order of smallness (more precisely, that \(\Delta \xi\) has an order of smallness not lower than the largest of the remaining quantities). In practice this condition reduces to that indicated by Leontovich and Bazhulin and Leontovich\(^{5}\): the absorption over a wavelength \(\alpha\lambda\) must be much less than unity, i.e. \(\omega t' \ll K\) (see above). Thus, the theory does not claim to describe very strongly absorbing liquids, and the question of its applicability in this case can be settled only by experiment. It turns out that the predictions of relaxation theory are well justified at fairly high absorptions, but in very viscous liquids this theory, apparently, is indeed inapplicable\(^{16}\). It is possible, however, that this is connected not with the fundamental difficulties indicated above, but with the appearance in this region of some effects of another kind, not taken into account by this theory. Indeed, even in weakly absorbing solids (glass, metal), where these difficulties do not exist, relaxation theory also proves inapplicable (unless an excessively large number of relaxation times is introduced, which can explain any observed depend-

…dependence \(\alpha\) on the frequency, but in these cases it cannot be reasonably justified.

In the work of Mandelstam and Leontovich, the case of a large number of relaxation processes is also considered. The formula obtained in the general case is complicated and requires special investigation. It should be expected that it reduces to a formula of the form (15) with a large number of terms in brackets. We cannot dwell on this question here. Let us note only that, in those cases where several relaxation processes may be expected, many authors introduce a certain mean relaxation time.

In the above-mentioned review article by Markham\(^{2}\), his own theory of relaxation processes is also presented. This theory is fully equivalent to the theory of Mandelstam and Leontovich, but the author presents his results in a form less convenient for calculations. We shall not analyze this work in detail.

§ 3. RELAXATION OF THE PROCESS OF MOLECULAR EXCITATION

Kneser, in his recent work\(^{15}\), attempted to give a general relaxation theory of the absorption of ultrasonic waves in liquids. He assumes that the relaxation process is the process of excitation of molecules (or, in general, of some degrees of freedom). Such an interpretation assumes that these degrees of freedom do not interact, so that one may speak of their separate excitation. This is undoubtedly true if the question concerns the transfer of energy to intramolecular degrees of freedom (analogously to Kneser processes in gases). However, Kneser and a number of other authors used this hypothesis also in those cases where the relaxation process is, for example, a rearrangement of the molecular structure of a liquid. It is possible that in some particular cases this method may give a more or less reasonable approximation, but, of course, it cannot be regarded as universal.

Moreover, the result obtained by Kneser is incorrect, since in the derivation he made a number of fundamental errors. Using the general result of the theory of Mandelstam and Leontovich, one can give a correct theory of the processes considered by Kneser.

As the parameter \(\xi\) we shall take, following Kneser, the relative concentration of excited molecules (as Mandelstam and Leontovich indicated, with such a choice of \(\xi\) equation (16) always applies). We shall further assume that the excitation of molecules requires an expenditure of energy \(E_\xi = Q/M\) (\(M\) is the molecular weight; \(E\) is, as before, referred to unit mass, and \(Q\) to one mole) and that upon excitation each molecule receives the additional volume \(v/N\) (\(N\) is Avogadro’s number), i.e.,

\[ R_\xi = -\rho^2 \frac{v}{M}. \]

Hence one can immediately calculate

\[ H_{\xi}=E_{\xi}-\frac{p}{\rho^{2}}R_{\xi}=\frac{Q+pv}{M}. \]

To calculate \(\Phi_{\xi}\) (here it is more convenient to use the variables \(p, T\)), it is necessary to know also the increase of the entropy \(S\) upon excitation. This increase can be found from the following considerations. In the distribution function for the case when \(N\xi\) molecules are in the excited state, and \((N-N\xi)\) in the ground state, there enters the factor

\[ \frac{N!}{(N\xi)!(N-N\xi)!}, \]

which takes into account the number of physically distinct states under the possible permutations. In addition, if the statistical weights of the ground and excited states for one molecule are not the same, then the distribution function will contain also the factor \(w^{N\xi}\), where \(w\) is the statistical weight of the excited state relative to the unexcited one. The corresponding part of the entropy will be:

\[ -k\ln\{(N\xi)!(N-N\xi)!\}+k\ln N!+kN\xi\ln w \simeq \]

\[ \simeq R\{(1-\xi)\ln(1-\xi)+\xi\ln\xi\}+k\ln N!+\sigma\xi, \]

where \(\sigma=R\ln w\) is the part of the entropy (per 1 mole) which takes into account the statistical weight of the excited state. Here we have used Stirling’s formula

\[ x!\simeq \left(\frac{x}{e}\right)^x. \]

Differentiating, we have:

\[ MS_{\xi}=-R\{\ln\xi-\ln(1-\xi)\}+\sigma, \]

\[ MS_{\xi\xi}=-R\frac{1}{\xi(1-\xi)}. \]

From the first equality it follows that

\[ \Phi_{\xi}=H_{\xi}-TS_{\xi}=\frac{1}{M} \left( Q+pv+RT\ln\frac{\xi}{1-\xi}-\sigma T \right). \]

Equating \(\Phi_{\xi}=0\), we obtain the equilibrium condition for \(\xi\) at constant pressure and temperature:

\[ \frac{\xi_{0}}{1-\xi_{0}} = e^{-\frac{Q+pv-\sigma T}{RT}} = e^{-\frac{\Delta\Phi}{RT}} \tag{26} \]

This relation simply expresses Boltzmann’s distribution law, and it could have been written down at once, as is usually done, but here it has been obtained automatically. In the exponent here stands the quantity \(Q+pv-\sigma T=\Delta\Phi\), the increase of the thermodynamic potential upon excitation of an individual molecule (calculated per mole).

The entropy increase \(\sigma\) entering into (26) is, as a rule, a completely unknown quantity. For Kneser processes,

when the nature of the excited state is known from spectroscopic data, it can sometimes be estimated theoretically. In all other cases there is apparently no such possibility, but one should expect that the statistical weights of the states should not differ too greatly, i.e. \(\sigma\) should be close to zero. Many authors (in particular, Knezer) do not take into account the term with \(\sigma\), assuming

\[ \frac{\xi_0}{(1-\xi_0)} = e^{-\frac{Q+pv}{RT}} = e^{-\frac{M H_\xi}{RT}} . \]

Introducing the notation

\[ \frac{Q+pv-\sigma T}{RT} = \frac{\Delta \Phi}{RT} = x, \]

we have

\[ \xi_0=\frac{1}{1+e^x}. \tag{27} \]

Taking into account that \(E_{\xi\xi}=H_{\xi\xi}=0\), we have:

\[ \Phi_{\xi\xi}=F_{\xi\xi}=-T S_{\xi\xi} = \frac{1}{M}RT\frac{1}{(1-\xi)\xi} = \frac{1}{M}RT\frac{(1+e^x)^2}{e^x}. \]

In this interpretation the lag time \(r'\) is connected with the equation of the excitation reaction:

\[ -\dot{\xi}=k_2\xi-k_1(1-\xi), \]

where \(k_1\) and \(k_2\) are the probabilities of excitation and of the reverse transition. At equilibrium \(\dot{\xi}=0\), i.e. the ratio of the equilibrium probabilities is

\[ \frac{k_{20}}{k_{10}}=\frac{(1-\xi_0)}{\xi_0}=e^x. \]

When sound passes through, the equilibrium is disturbed. Varying the “reaction equation,” we have:

\[ -\Delta\dot{\xi} = (k_{20}+k_{10})\Delta\xi + \xi_0\left( \Delta k_2-\frac{(1-\xi_0)}{\xi_0}\Delta k_1 \right) \simeq \]

\[ \simeq (k_{20}+k_{10})\Delta\xi + \xi_0 k_{20}\Delta\ln\left(\frac{k_{20}}{k_{10}}\right) = \]

\[ = (k_{20}+k_{10}) \left\{ \Delta\xi + \xi_0\frac{k_{20}}{k_{10}+k_{20}} \Delta\ln\frac{k_{20}}{k_{10}} \right\} = \]

\[ = \frac{1}{r'} \left\{ \Delta\xi + \frac{e^x}{(1+e^x)^2}\Delta x \right\}. \tag{28} \]

where

\[ r'=\frac{1}{k_{20}+k_{10}}. \]

The quantity \(k_{10}=k_{20}e^{-x}\) is often neglected. By substituting the explicit expression for \(x\), one can show that the reduced—

Here the “reaction equation” of Kneser is a special case of the equation from which Mandelstam and Leontovich proceeded.

Using the expressions obtained for the derivatives, we have

\[ \beta_{\infty}=\frac{1}{\rho}R_p\frac{C_v}{C_p} = \left(\beta_0\frac{c_p}{c_v}-\frac{\rho v^2}{M\Delta\Phi_{\xi}}\, \frac{c_v-C_{pi}\dfrac{E_{\xi}^{2}}{H_{\xi}^{2}}}{c_p-C_{pi}}\right)^{*}. \tag{29} \]

Here the notation has been introduced

\[ C_{pi}=c_p-C_p=\frac{H_{\xi}^{2}}{T\Phi_{\xi\xi}} = \frac{M}{RT^2}\, \frac{H_{\xi}^{2}e^x}{(1+e^x)^2} \]

—the heat capacity of the “relaxing” degree of freedom at constant pressure. Let us emphasize that in the general case it is not equal to the corresponding heat capacity at constant volume:

\[ C_{vi}=c_v-C_v=\frac{E_{\xi}^{2}}{TF_{\xi\xi}} = C_{pi}\frac{E_{\xi}^{2}}{H_{\xi}^{2}}. \]

Usually it is assumed that \(C_{pi}\simeq C_{vi}=C_i\), which is correct under the condition \(pv\ll Q\). This condition is probably always fulfilled.

Let us also write down an explicit expression, for example, for \(C_{pi}\) in the form in which it is often used:

\[ C_{pi}= \frac{M}{RT^2}\, \frac{H_{\xi}^{2}}{2\left(1+\operatorname{ch}\dfrac{\Delta\Phi}{RT}\right)} = \frac{R}{M}\, \frac{\left(\dfrac{Q+pv}{RT}\right)^2} {2\left(1+\operatorname{ch}\dfrac{\Delta\Phi}{RT}\right)}. \tag{29a} \]

If \(\Delta\Phi\) is large (i.e. \(\xi\ll 1\)), then this formula simplifies:

\[ C_{pi}\simeq \frac{R}{M} \left(\frac{Q+pv}{RT}\right)^2 e^{-\frac{\Delta\Phi}{RT}}. \]

We note that this expression has, at

\[ \frac{Q+pv}{RT}=2.4, \]

a maximum equal to \(0.44\,\dfrac{R}{M}e^{pv/R}\). Thus the relaxation heat capacity is always a small quantity, since \(c_p\) for liquids is of the order of \(10\,\dfrac{R}{M}\)–\(15\,\dfrac{R}{M}\).

We shall now briefly set forth Kneser’s work and point out the principal errors of his calculation.

Kneser assumes that the quantity \(\beta_0-\beta_{\infty}\) can be represented in the form

\[ \beta_1=\beta_0-\beta_{\infty} = -\frac{1}{V} \left(\frac{\partial V}{\partial \xi}\right)_S \left(\frac{\partial \xi_0}{\partial p}\right). \tag{30} \]

\[ \text{*) Formula (29) is published for the first time.} \]

Immediate calculation gives:

\[ \Delta \xi_0=\frac{C_l M}{RTx}\left(\Delta T-\frac{v}{xR}\Delta p\right) \tag{31} \]

\[ \left(x=\frac{MH_\xi}{RT},\ \text{since Kneser does not take into account the term with } \sigma\right). \]

To calculate \(\left(\dfrac{\Delta V}{\Delta \xi}\right)_S\), it is necessary to use the law of conservation of energy:

\[ E_\xi \Delta \xi+C_V\Delta T+(E_V+p)\Delta V=0, \]

or, using the usual thermodynamic transformations,

\[ E_\xi \Delta \xi+C_V\Delta T+TP_\tau\Delta V=0. \tag{32} \]

Kneser writes the law of conservation of energy differently:

\[ H_\xi \Delta \xi+C_V\Delta T+p\Delta V=0. \tag{32a} \]

It is easy to see that in doing so he makes two errors. First, he counts twice here the work of the change in volume upon excitation (in the term \(p\Delta V\) and in \(H_\xi\Delta \xi\), since \(H_\xi=\dfrac{Q+pv}{M}=E_\xi+p\dfrac{V}{M}\)). Second, he does not take into account that the internal energy of a liquid depends on the volume (the term \(E_V\Delta V\) vanishes only for an ideal gas). Therefore, in the subsequent calculation one should use formula (32), and not (32a), given by Kneser.

From equations (31) and (32) it is also necessary to eliminate the temperature change \(\Delta T\). In equation (31), \(\Delta T\) is the increment of temperature by the moment equilibrium is established, i.e. it is equal to \(\left(\dfrac{\partial T}{\partial p}\right)_S\Delta p\); to calculate this quantity one must use the equilibrium equation of state and the equilibrium law of conservation of energy. As for equation (32), it describes the transition from instantaneous to equilibrium compression, and \(\Delta T\) in this equation is the difference between the equilibrium and instantaneous temperature increments. If one takes into account that the process described by equation (32) occurs at constant pressure, then \(\Delta T\) may be eliminated using the nonequilibrium equation of state. Kneser, however, calculates \(\Delta T\) in equation (32) by means of the equilibrium equation of state.

As a result of these calculations Kneser obtained the formula

\[ \beta_0-\beta_\infty = \frac{C_{pi}}{C_V}\cdot \frac{1}{1-\dfrac{p_\tau \rho v}{\rho C_V}} \left( \frac{c_p-c_V}{c_V} + \frac{\rho_\tau v}{\rho \rho_0 R x} \right), \]

which gives the correct result only for Kneser processes \((v=0)\) in a polyatomic ideal gas. It is easy to verify,

that it does not coincide with Herzfeld’s formula (see § 4) for Kneser processes in liquids, although Kneser himself asserts the opposite. The situation is still worse in the case of isothermal structural relaxation, observed, for example, in water, where at \(4^\circ\mathrm{C}\), \(c_p=c_V\) and \(\rho_T=0\). In this case Kneser’s formula gives the plainly absurd result \(\beta_1=0\). Kneser “circumvents” this difficulty by carrying out a separate calculation for this case. It is obvious, however, that a truly general formula could not give an incorrect result in any particular cases.

Undoubtedly, with a correct calculation by Kneser’s method we would arrive at equation (29), which is a special case of the equation of Mandelstam and Leontovich. However, such a conclusion would give nothing fundamentally new. Calculations by this method are no simpler than by the method of Mandelstam and Leontovich. At the same time, Kneser’s calculation, even if the errors he made are corrected, is not sufficiently rigorous. In particular, in such a derivation the distinction between \(r\) and \(r'\) drops out. This error is practically small, but, like Kneser’s other errors, it arises because of the general lack of clarity in the physical meaning of the equations he used.

Quite recently Andrae and Lamb\(^{27}\) made yet another attempt to derive a “general” relaxation formula, using Kneser’s method described above. They write equation (30) for the relaxation compressibility in the following form:

\[ \beta_1=\beta_0-\beta_\infty=-\frac{1}{V}\left(\frac{\partial V}{\partial \xi}\right)_y\left(\frac{\partial \xi_0}{\partial p}\right). \tag{30a} \]

Above we have already discussed the meaning of the derivative \(\dfrac{\partial V}{\partial \xi}\) in this equation. It describes the process of transition from the instantaneous (at constant \(\xi\)) adiabatic compression

\[ -\frac{\Delta V_\infty}{V}=\beta_\infty \Delta p \]

to the equilibrium one

\[ -\frac{\Delta V_0}{V}=\beta_0 \Delta p. \]

Thus, the derivative \(\left(\dfrac{\partial V}{\partial \xi}\right)\) must be calculated at constant pressure and entropy. The authors write \(\left(\dfrac{\partial V}{\partial \xi}\right)_y\), and define the variable \(y\) purely formally, through the instantaneous compressibility \(\beta_\infty\):

\[ \beta_\infty=-\frac{1}{V}\left(\frac{\partial V}{\partial y}\right)_\xi\left(\frac{\partial y}{\partial p}\right)_S. \]

With such a definition, the physical meaning of the process remains completely unclear. As a result Andrae and Lamb, as will be shown

below, incorrectly compute \(\dfrac{\partial V}{\partial \xi}\). In addition, they, like Kneser, did not take into account the difference between \(r\) and \(r'\).

For \(\xi_0\), Andrews and Lamb adopted the Boltzmann expression

\[ \frac{\xi_0}{1-\xi_0}=e^{-\Delta H/RT}, \]

whereas they represented the heat of excitation \(\Delta H\) in the form

\[ \Delta H=\Delta E+(p_i+p)v=c_v\Delta T+(p_i+p)v . \]

Here \(p_i=\left(\dfrac{\partial E}{\partial V}\right)_T\) is the “internal pressure,” characterizing the force of intermolecular interaction, \(\Delta E\) is the internal energy of excitation (\(ME_\xi\) in our notation), \(v\) is the additional molecular volume. This formula is devoid of physical content. The heat of excitation \(\Delta H\) at constant pressure and constant temperature is composed of the work against the external pressure \(pv\) and the change in internal energy \(\Delta E\), while separating from \(\Delta E\) the work performed against the forces of intermolecular interaction \(p_i v\) has no meaning here. The quantity \(\Delta T\) by no means denotes any temperature increment upon excitation. The authors evidently wrote the entire formula for \(\Delta H\) using a formal analogy with the usual thermodynamic expression at constant pressure:

\[ dH=c_v dT+(p_i+p)dV . \]

However, in the present case the point is not this formula, but the increment of enthalpy at constant pressure and temperature:

\[ dH=H_\xi d\xi=\frac{\Delta H}{M}\,d\xi=E_\xi d\xi-\frac{p}{\rho^2}R_\xi d\xi =\frac{1}{M}(\Delta E\,d\xi+pv\,d\xi). \]

The authors compute the quantity \(\left(\dfrac{\partial \xi_0}{\partial p}\right)_S\), as does Kneser, by formula (31). To compute \(\left(\dfrac{\partial V}{\partial \xi}\right)_y\), the authors write the following equation for the heat of the excitation reaction:

\[ T\,dS=\Delta H\,d\xi =T\left(\frac{\partial S}{\partial V}\right)_p dV +T\left(\frac{\partial S}{\partial p}\right)_V dp, \]

whence

\[ \left(\frac{\partial V}{\partial \xi}\right)_y = \frac{\Delta H}{T\left(\dfrac{\partial S}{\partial V}\right)_p} - \frac{\left(\dfrac{\partial S}{\partial p}\right)_V} {\left(\dfrac{\partial S}{\partial V}\right)_p} \left(\frac{\partial p}{\partial \xi}\right)_y . \]

The meaning of the equation for \(T\,dS\) is not very clear. Apparently, the authors here consider the process of excitation of molecules at the expense of an external source of heat, and assume that the “external,” i.e., not affected by the relaxation process, degrees of freedom of heat receive no ...

Such a process can probably be realized, but \(\dfrac{dV}{d\xi}\) must, of course, be calculated not from such an equation, since it has no relation to the adiabatic and isobaric process of transition from instantaneous to equilibrium compression which interests us. Denoting \(\left(\dfrac{dp}{d\xi}\right)_V=\Delta p\), the authors finally obtain:

\[ \frac{\beta_1}{\beta_0} = \left\{(\gamma-1)-\gamma\beta_0(p+p_i)\frac{V\Delta p}{\Delta H}\right\} \times \]

\[ \times \left\{ \frac{\gamma}{\gamma-1}\frac{(p+p_i)v}{\Delta H}-1 \right\} R\left(\frac{\Delta H}{RT}\right)^2 \frac{e^{-\frac{\Delta H}{RT}}}{\left(1-e^{-\frac{\Delta H}{RT}}\right)^2}. \]

The authors then show that Hall’s result (see § 6) is obtained from this formula if one assumes that \(\dfrac{v}{\Delta p}=\left(\dfrac{\partial V}{\partial p}\right)_T\) and \(\Delta E=0\). The physical meaning of the first assumption is difficult to understand; the authors themselves do not attempt to explain it. As for the second assumption, we note that Hall, in his calculation, took into account the excitation energy \(\Delta E\).

The authors obtain Herzfeld’s result (see § 4) from their formula for \(v=0\) and \(\dfrac{\Delta p}{\Delta T}=2\left(\dfrac{\partial p}{\partial T}\right)_V\). This latter equality seemed suspicious even to the authors themselves, and they attempted to suppose that

\[ \frac{\Delta p}{\Delta T} = \left(\frac{\partial p}{\partial T}\right)_V \]

or

\[ \frac{\Delta p}{\Delta T} = \left(\frac{\partial p}{\partial T}\right)_S \]

(incidentally, in view of the absence of physical meaning in the quantity \(\Delta T\), these formulas are no more reasonable than any others). The first assumption concerning the quantity \(\dfrac{\Delta p}{\Delta T}\) gave \(\beta_1=0\). Under the second assumption, a result was obtained which did not coincide with Herzfeld’s formula. This is convincing proof of the untenability of the result of Andrus and Lamb. Indeed, Herzfeld’s formula for Knudsen processes in a liquid is derived by means of simple and rigorous calculations*) and is quite reliable. The assumptions on which it is based are a special case of the initial assumptions of Andrus and Lamb. Therefore, if their general formula were correct, then with the single assumption \(v=0\) it would reduce to Herzfeld’s formula.

\[ \text{*) By the method of Mandelstam and Leontovich, or by the simplified method mentioned in § 4.} \]

In view of the fact that, in deriving their formula, the authors made a number of errors, their attempt to compare it with experimental data is of no interest.

In conclusion, let us note that calculations according to a scheme analogous to Kneser’s scheme have been successfully carried out for the case of isothermal processes. In particular, such a method was applied by Landau and Lifshitz \(^{18}\) in a simplified exposition of the Mandelstam and Leontovich theory. According to the same scheme, calculations of isothermal structural relaxation were made by Ya. I. Frenkel \(^{19}\), Gierer and Wirtz \(^{20}\), and Hall \(^{21}\). In an isothermal process \(\Delta T=0\), and therefore the difficulties indicated above disappear. In this case such a calculation scheme is indeed the simplest and most transparent. Instead of formula (30) one may write:

\[ \beta_0-\beta_\infty=-\frac{1}{V}\left(\frac{\partial V}{\partial \xi}\right)_p \frac{\partial \xi_0}{\partial p}. \tag{33} \]

In this formula all derivatives are calculated directly from the nonequilibrium equation of state and the explicit expression for \(\xi_0\).

§ 4. EXCITATION OF INTRAMOLECULAR DEGREES OF FREEDOM

In liquids relaxation processes are possible which are fundamentally no different from the so-called Kneser processes in polyatomic gases. These processes, as is known, consist in an exchange of energy between external and internal degrees of freedom. Owing to the much stronger interaction between molecules of a liquid, the relaxation time of such processes is several orders of magnitude smaller than in gases. Therefore the condition \(\omega\tau \simeq 1\) can be fulfilled only at very high frequencies. It is known that in those liquids in which the exchange mechanism has the principal influence on absorption, it has not been possible to observe relaxation of the volume viscosity even at the highest ultrasonic frequencies*). This indicates that the relaxation time of these processes must be no greater than \(10^{-9}\) sec. This conclusion is also confirmed by the existence of the fine structure of the Rayleigh scattering line. In order for this phenomenon to be observed, the relaxation time of the volume viscosity must have a value of the order of \(10^{-9}\)—\(10^{-11}\) sec.

In calculating the influence of Kneser effects on the propagation of ultrasound, one usually introduces as a parameter \(\xi\) the nonequilibrium energy of the intramolecular degrees of freedom. Then the internal degrees of freedom must be assigned a complex heat capacity

\[ \widetilde{C}_i=\frac{C_i}{1+i\omega\tau}, \]

and then the complex adiabatic

*) The only exception is apparently \(CS_2\), in which relaxation is observed at frequencies \(\simeq 10^8\) cycles.

the compression modulus

\[ \mathcal K=\rho\left(\frac{\partial p}{\partial \rho}\right)_T \frac{\widetilde C_p}{\widetilde C_V}, \]

where the complex heat capacities are

\[ \widetilde C_V=C_V+\widetilde C_i,\qquad \widetilde C_p=C_p+\widetilde C_i. \]

We can obtain the same result by using formula (22) (or directly from the Mandelstam–Leontovich formula). To do this one must put \(v_i=0\) in formula (29), since the excitation of the internal degrees of freedom occurs, of course, without a change in the molecular volume. Then \(H_i=E_i\), \(C_{pi}=C_{Vi}=C_i\), \(R_p=\rho_p\). Introducing the notation \(c_p-c_V=\Delta\), after simple transformations we obtain the formula proposed by Herzfeld\({}^{22}\):

\[ \eta=K_0\,\frac{c_i\Delta}{c_p\widetilde C_V}\,\tau =K_0\,\frac{c_i\Delta}{c_p c_V}\,\tau', \tag{34} \]

Herzfeld himself derived this formula with the aid of the complex heat capacity.

Let us note that in such a process \(\tau'=r'\), \(\tau=\dfrac{\widetilde C_V}{c_V}\tau'\), and \(r=\dfrac{C_p}{c_p}r'\), of course, are different. Contrary to Kneser’s opinion, his “general” formula (see § 3) for the case of liquids is not equivalent to this formula. In the case of gases Kneser’s formula indeed coincides with this formula (then \(\Delta=R\)), if one assumes that its “relaxation time” is in fact the retardation time

\[ r=\frac{C_p}{c_p}\tau'. \]

All the quantities entering formula (34), except for the relaxation time, can be determined experimentally. The heat capacities \(c_p\) and \(c_V\) are calculated from thermodynamic data, while the internal heat capacity \(C_i\) can be found from spectroscopic measurements by formula (29a). In doing so one usually assumes \(\sigma=0\), although in some cases spectroscopic data on the nature of the excited state apparently make it possible to calculate this quantity. In his work\({}^{22}\) Herzfeld calculated by this method, from the super-Stokes part of the absorption

\[ \alpha_0=\alpha-\frac{2\omega^2}{3\rho a^3}\,\mu, \]

the relaxation times for benzene, carbon tetrachloride, methyl alcohol, and water. He obtained for \(C_6H_6\), \(\tau'=3\cdot10^{-10}\) sec, for \(CCl_4\) — \(1\cdot10^{-10}\) sec, for \(CH_3OH\) — \(1.8\cdot10^{-10}\) sec, for \(H_2O\) — \(2.1\cdot10^{-8}\) sec. The last result shows that absorption in water, in any case, cannot be ascribed to such a mechanism, since the relaxation region would then fall in a well-studied frequency range and would undoubtedly have been found experimentally. In addition, Herzfeld pointed out that at \(4^\circ\) C there can be no Kneser absorption in water at all, since at this temperature \(\Delta=c_p-c_V=0\). A later experimental investigation by Fox and Rock\({}^{23}\) showed that the absorption in water, upon lowering the temperature, monotonically

increases, and in the vicinity of \(4^\circ\mathrm{C}\) no peculiarity of the temperature behavior is observed.

Recently Herzfeld \(^{34}\) calculated the relaxation time for Kneser processes, taking into account the interaction between intramolecular and intermolecular vibrations in a liquid. In doing so he adopted a Lennard-Jones model for the structure of the liquid and a Lennard-Jones form for the potential of intermolecular interaction. A numerical calculation gave for benzene \(\tau = 1.2\cdot 10^{-10}\) sec, i.e., three times smaller than that calculated from ultrasonic data. In view of the rough approximations made in the derivation, the agreement with experiment should be regarded as good. However, such an estimate cannot explain the peculiarities in the behavior of different liquids.

Kittel \(^{24}\) derived a formula identical with Herzfeld’s formula, using one of the equations of state of a liquid based on the “free-volume” model (Tonk’s equation). However, this derivation is of no interest, since the expression for the volume viscosity associated with Kneser effects is entirely independent of the particular form of the equation of state. Kittel does not take other absorption mechanisms into account. Kittel’s considerations on the connection between the magnitude of Kneser absorption and the structure of molecules are noteworthy. He observes that the molecules of the three most strongly absorbing liquids of low viscosity (\(\mathrm{CS_2}\), \(\mathrm{CCl_4}\), benzene) are nonpolar. This is explained by the fact that, owing to the weakness of the electrostatic interaction of the molecules, the coupling between external and internal degrees of freedom is insignificant. Therefore the relaxation time (a quantity inverse to the transition probability) is large in this case and, consequently, the volume viscosity is large.

Conversely, in the case of strongly associated liquids, intermolecular interaction increases the probability of energy exchange between external and internal degrees of freedom, i.e., decreases the Kneser part of the absorption.

In the case of solids, where the separation of vibrations into intra- and intermolecular ones becomes no longer quite definite, one should expect still shorter relaxation times. Kittel gives for solids a value of the order of \(10^{-12}\) sec, obtained from thermal-conductivity data. Kittel’s considerations on the temperature dependence of the absorption coefficient are also interesting, but we shall not present them, since this question was examined much more thoroughly by Bauer.

The most interesting work on Kneser effects in absorption is Bauer’s article \(^{14}\). Bauer proceeded from the same assumptions as did the other authors who considered Kneser effects, without stating them explicitly. Bauer’s general formula corresponds exactly to Herzfeld’s formula derived by us above, and the derivation of the “reaction equations” repeats Kneser’s derivation given in § 3. Taking into account the possibility of excitation of several vibrational levels, Bauer

introduces the total oscillatory heat capacity and the mean relaxation time. By the same method as Herzfeld, he calculated these mean relaxation times for a number of liquids, obtaining, as did Herzfeld, values \(\simeq 10^{-10}\div 10^{-11}\) sec. The most interesting part of the work is the theoretical estimate of the relaxation times. For gases the relaxation time is determined by the collision frequency and by the probability of transition of the molecule from the excited state to the normal one (and by the probability of the reverse transition, but it is small if the excitation energy is greater than \(kT\), and is usually neglected). The transition probabilities for gases were calculated by Zener[^25] and Landau and Teller[^26]. Bauer derived a formula for the transition probability in the case of liquids at room temperature, solving the corresponding quantum-mechanical problem under the assumption that the attraction between molecules of the liquid can be represented by a constant potential, i.e. one independent of the distance between the molecules. Such an approximation is reasonable, since attraction depends on distance much more weakly than repulsion. The formula obtained gives for the transition probability a value \(\simeq 10^{-2}\div 10^{-3}\), and since the collision frequency in a liquid is \(\simeq 10^{12}\ \mathrm{sec}^{-1}\)1, for the relaxation frequency \(\dfrac{1}{2\pi\tau}\) the correct order of magnitude, \(10^9—10^{10}\) Hz, is obtained.

Bauer also considers sound absorption in binary mixtures. It was known that in some mixtures (for example, benzene—toluene) the absorption drops sharply even at small concentrations of the more weakly absorbing component. Bauer shows that this phenomenon can be explained if one assumes that collisions of the type \((\mathrm{A}^*—\mathrm{B})\), \((\mathrm{B}^*—\mathrm{B})\), \((\mathrm{A}—\mathrm{B}^*)\) are much more effective (i.e. the transition probability is much greater) than collisions of the type \((\mathrm{A}^*—\mathrm{A})\) (here A is the molecule of the strongly absorbing substance, B the molecule of the weakly absorbing one, and the asterisk denotes an excited molecule). In this case it is obvious that the addition of even a small number of B molecules will sharply increase the total effectiveness of collisions, i.e. will reduce the relaxation time and, consequently, the absorption coefficient. Bauer’s more rigorous calculation gave a result in excellent numerical agreement with the experimental data on absorption in the benzene—toluene mixture.

The temperature dependence of the absorption coefficient is also of interest. From Herzfeld’s formula (34) it follows that

\[ \alpha=\frac{\omega^2}{2a}\,\frac{\Delta}{c_p c_V}\,C_i\tau'. \]

If several internal degrees of freedom are excited, then \(C_i\) is the total internal heat capacity, and \(\tau'\) is the mean

time of relaxation. Hence the temperature coefficient \(\alpha\):

\[ \frac{1}{\alpha}\frac{d\alpha}{dT} = -\frac{1}{a}\frac{da}{dT} + \frac{1}{C_i}\frac{dC_i}{dT} + \frac{1}{\tau'}\frac{d\tau'}{dT} + \frac{1}{\Delta}\frac{d\Delta}{dT}. \]

The heat capacity of the internal degrees of freedom is approximately \((\xi \ll 1)\) represented by the formula

\[ C_i=\frac{Q^2}{MRT^2}e^{-Q/RT}, \qquad \text{i.e.} \qquad \frac{d\ln C_i}{dT} = \frac{1}{T}\left(\frac{Q}{RT}-2\right). \]

If several degrees of freedom are excited, then \(Q\) is some mean excitation energy. The dependence of \(\tau'\) on temperature is determined by the collision frequency \(\frac{kT}{h}\) and by the probability of transition from the excited level to the ground one. For gases, as experience shows, the transition probability increases with temperature approximately as \(T^n\), i.e. \(\tau'\) decreases as \(T^{-n-1}\), where \(n \simeq 3\text{–}4\). For liquids \(n\) should be smaller, of the order of \(1\text{–}2\). Finally, \(\Delta=\frac{T\beta_T^2}{\rho^2\beta_p}\sim T\). Thus we have:

\[ \frac{1}{\alpha}\frac{d\alpha}{dT} = -\frac{1}{a}\frac{da}{dT} + \frac{1}{T}\left(\frac{Q}{RT}-2-n\right). \]

This equation makes it possible to determine \(n\) from experimental data. Bauer shows that for a number of liquids \(n\) turns out to be much less than unity, and in many cases a negative quantity, which contradicts Bauer’s equation for the transition probability. He explains this discrepancy by the fact that the relaxation times for the excitation of different excited states are different; namely, higher levels correspond to larger relaxation times (smaller transition probabilities). Then the introduction of an average relaxation time reduces the role of the high levels, since their heat capacities are small, and, since the temperature coefficient of heat capacity is high for them, the calculations give a lower value of the mean quantity \(\frac{1}{C_i}\frac{dC_i}{dT}\). Bauer obtained good agreement with experiment in the case of liquid \(\mathrm{CS_2}\), when he introduced three relaxation times for the levels \(397\ \mathrm{cm}^{-1}\), \(656.5\ \mathrm{cm}^{-1}\), and \(1523\ \mathrm{cm}^{-1}\), putting \(n=1\).

§ 5. STRUCTURAL RELAXATION

The existence of relaxation processes connected with the rearrangement of the intermolecular structure of a liquid was already indicated by Mandelstam and Leontovich\(^3\). Later, Frenkel and Obraztsov\(^9\), relying on the temperature dependence of absorption, showed that the relaxation processes in water cannot be explained by the Kneser ones.

effects, but must be connected with the intermolecular structure. However, in spite even of Hertsfeld’s work, which showed that Knésor’s processes in water cannot occur, the majority of foreign authors continued to connect relaxation phenomena only with these processes. The connection of relaxation with the structure of a liquid was first considered by Ya. I. Frenkelʹ[^19].

According to Ya. I. Frenkelʹ’s theory of the structure of liquids, the molecules in a liquid, as in a solid, are located at the nodes of a crystalline lattice. In both cases this lattice is not absolutely regular: some of the nodes turn out to be empty, forming “holes” in the ideal crystalline structure, and some molecules are inserted into the intervals between nodes occupied by other molecules. In a solid, however, these irregularities occur comparatively rarely and do not disturb the basically regular character of the structure of the crystal, i.e., the concept of a crystalline lattice is fully preserved. As is said in this case, “long-range order” in the arrangement of the molecules is preserved. On the contrary, in a liquid these disturbances are so numerous that the concept of a crystalline lattice loses its meaning. In this case one can speak only of “short-range order,” i.e., of regularity in the mutual arrangement of molecules only in a small volume. The concepts of nodes and internodes here likewise have no meaning, and disturbances that violate the regularity of the structure (“holes”) must be regarded not as vacant lattice nodes, but as cracks and cavities breaking up the lattice. To speak of molecules in internodes in this case has no meaning at all.

The degree of short-range order in a crystal is conveniently defined as the ratio of the difference between the number of filled and empty nodes surrounding a given molecule to their total number (i.e., to the coordination number of the lattice \(z\)):

\[ \xi=\frac{z' - z''}{z}. \]

The structure of a liquid can also be characterized by an analogous quantity, although its meaning in this case proves to be less definite.

Frenkelʹ considers a purely isothermal process. Then the equilibrium degree of short-range order \(\xi_0\) is a function only of the density. Frenkelʹ’s calculation was carried out according to the simplified scheme of Mandelʹshtam and Leontovich for an isothermal process. We shall use the ready result (25) and somewhat modify Frenkelʹ’s presentation. Formula (25), for \(P_T=p_T=0\), gives:

\[ K_{\infty}-K_0=\frac{p_{\xi}^{\,2}}{\rho F_{\xi\xi}}. \]

Frenkel introduces the notation

\[ F_{\xi\xi}=\frac{b}{\rho}. \]

\(P_\xi=\rho^2 F_{\rho\xi}\) is calculated from the identity

\[ \left(\frac{\partial F_\xi}{\partial \rho}\right)_{\chi} =F_{\rho\xi}+F_{\xi\xi}\frac{d\xi_0}{d\rho}=0, \]

since in the unperturbed state \(F_\xi=0\) (equilibrium), and \(\left(\dfrac{\partial F_\xi}{\partial \rho}\right)_{\chi}\) denotes the derivative at equilibrium \((\chi=F_\xi=0)\). Hence

\[ F_{\rho\xi}=-F_{\xi\xi}\frac{d\xi_0}{d\rho}. \]

In view of the smallness of the amplitude of sound \(\Delta \xi_0\), it may be regarded as a linear function of \(\Delta \rho\). Denoting \(\rho \dfrac{d\xi_0}{d\rho}=-a\), we finally obtain:

\[ K_\infty-K_0=a^2 b. \]

According to Frenkel, the relaxation time of processes connected with the disruption of short-range order is of the order of magnitude of the “duration of settled life of a molecule,” and its temperature dependence has the same character. Below we shall again have occasion to use these ideas.

The only example so far of the application of the rigorous theory of Mandelstam and Leontovich to concrete relaxation processes connected with the structure of a liquid is the work of A. I. Anselm \(^{28}\). Anselm proceeded from a model of a liquid developed chiefly by Eyring and his collaborators. This model essentially represents a liquid as a strongly compressed real gas. Each molecule is regarded as moving freely within a certain limited volume. The potential energy of a molecule is taken to be the average energy of its interaction with the other molecules, and this energy is assumed not to depend on the position of the molecule within the volume made available to it. Under these assumptions the free energy per unit mass is equal to:

\[ F=-\Pi-\frac{R}{M}T\ln\frac{(2\pi mkT)^{3/2}v_f}{h^3} -\frac{R}{M}T-\frac{R}{M}T\ln J, \tag{35} \]

where \(\Pi\) is the average potential energy of a molecule in the field of the other molecules (calculated per unit mass), and \(v_f\) is the free volume per molecule. \(J\) denotes the statistical integral referring to the intramolecular degrees of freedom. Since we are interested in effects connected with the intermolecular structure of the liquid, the term containing \(\ln J\) may be disregarded. For the calculation it is necessary to have explicit expressions for \(v_f\) and \(\Pi\). Anselm starts from the assumption that a molecule can move freely

inside a sphere whose radius is equal to the distance to its nearest neighbors (molecules of the first coordination group). Then the free volume is

\[ v_f=\frac{4\pi}{3}\left(\xi v_0^{1/3}-d\right)^3, \tag{36} \]

where \(v_0\) is the total volume per molecule \(\left(v_0=\dfrac{M}{N\rho}\right)\), \(d\) is its diameter, and \(\frac{1}{2}\xi v_0^{1/3}\) is the mean distance between neighboring molecules. The parameter \(\xi\) is determined by the type of short-range order. Thus, for an ideal simple cubic lattice, as is easy to see, \(\xi=1\); for a face-centered cubic lattice \(\xi=\sqrt{2}\), etc. In general, \(\xi\), and consequently the volume actually accessible to the molecule, \(v_f\), at a given \(v_0\), is the larger, the larger the coordination number \(z\) (6 in a simple cubic lattice, 12 in a face-centered one, etc.). Anselm assumes that the disturbance of order in the arrangement of molecules in a liquid can be described by regarding the “packing factor” \(\xi\) as a variable parameter.

For the potential energy of a molecule \(\Pi\), Anselm adopts a formula which gives the equation of state of the liquid the form of the Van der Waals equation (with the modifications accounted for by formula (36)):

\[ \Pi=\frac{b}{v_0}, \]

where the constant \(b\) depends, of course, on the type of short-range order, i.e., on \(\xi\). Since nothing can be said about this dependence, Anselm expands \(b\) in a series in powers of the deviation of \(\xi\) from the equilibrium value \(\xi_0\):

\[ b=b_0+b_1(\xi-\xi_0)+\frac{b_2}{2}(\xi-\xi_0)^2+\cdots . \tag{37} \]

In the approximation adopted by Mandelstam and Leontovich, in this expansion one should restrict oneself only to the terms written here. \(b_1\) can be determined from the condition of a minimum of the free energy at \(\xi=\xi_0\); \(b_2\) remains unknown, and in numerical calculations it may be taken approximately equal to zero or determined by comparing theoretical data with experimental data (if the relaxation time is known). Let us note that formula (29) also corresponds to the approximation in which \(b_2=0\) (i.e., \(E_{\xi\xi}=0\)); more accurate concrete theories do not yet exist.

Having an explicit expression for the free energy and taking into account that the energy

\[ E=-T^2\cdot\frac{\partial}{\partial T}\left(\frac{F}{T}\right)_V \]

and the pressure

\[ p=-\left(\frac{\partial F}{\partial V}\right)_T, \]

it is not difficult to calculate all the parameters entering the Mandelstam–Leontovich formula. We do not give this calculation, nor the final

Anselm’s formula. Let us list only the molecular parameters entering into it:

1) The equilibrium packing factor $\xi_0$, determined by the quasicrystalline structure of the liquid in the near order. It can be found from X-ray data or from the structure of the corresponding crystal.

2) The “diameter” of a molecule, $d$.

3) The coefficient $b_2$, which is completely unknown but probably small. It is possible that it could in some way be determined from the equation of state of the liquid.

4) The relaxation time of the near order, $\tau'$. Anselm indicates that this time should, in order of magnitude, correspond to the “sedentary life” time of the oscillating molecules of the liquid, which can be calculated from the viscosity by Frenkel’s formula.

A. I. Anselm carried out a calculation of the absorption in monatomic mercury. If one assumes that the slight excess of sound absorption, as compared with the classical value, in liquid monatomic mercury is due to the mechanism of structural relaxation, then the calculation shows that this relaxation should be observed at a frequency of $5 \cdot 10^{11}$ Hz. This is a quite reasonable value, since it is known that in mercury, up to very high ultrasonic frequencies ($\simeq 10^9$ Hz), $\dfrac{\alpha}{\omega^2}$ retains a constant value. The relaxation time of the near order, calculated by Frenkel’s formula, is of the order of $10^{-11}$, which agrees with the preceding conclusion.

In his theory Anselm proceeded from Eyring’s formula for the free energy (35). The model of liquid structure adopted by Eyring is, in essence, closer to a strongly compressed gas than to a solid. The quasicrystalline structure of the liquid is taken into account here only indirectly: only the influence of the mean type of near order (through $\xi$) on the free volume and the mean potential energy of the molecules is considered. The true form of the potential field is not taken into consideration. Although this model makes it possible to calculate the mean thermodynamic properties of a liquid fairly well (the equation of state, etc.), it probably describes inadequately processes that are connected with a change in the detailed picture of the arrangement of the molecules, in particular structural relaxation. Moreover, this model is fundamentally unsuitable for describing the flow of a liquid. At the same time, as Mandelstam and Leontovich already pointed out, structural relaxation and ordinary shear viscosity are determined essentially by processes of the same type. Therefore the impossibility of connecting the structural absorption of sound with data on shear viscosity is a shortcoming of Anselm’s theory.

After Anselm, structural absorption was calculated by Gierer and Wirtz, Hall, Gotz and Farm, and several other authors.

Gierer and Wirtz \(^{20}\) gave a theory which, like A. I. Anselm’s theory, is in principle applicable to all liquids. They proceeded from Ya. I. Frenkel’s “hole” theory of liquids, and considered the formation of disturbances that violate long-range order as the excitation of certain “degrees of freedom.” Under this assumption one may use the results of § 2. The parameter \(\xi\) should be taken as the relative concentration of the excited “degrees of freedom,” or, simply speaking, the number of “holes” per molecule. Then \(Q\) denotes the energy of formation of “one mole of holes,” and \(v\) the “hole volume” (also per mole). In their work Gierer and Wirtz deliberately neglect the effect of the temperature change in the sound wave on the relaxation processes, believing that in the liquids they considered this effect is small (we shall discuss the validity of this assumption when analyzing the next work), and they directly calculate \(\beta_0-\beta_\infty\) from formula (33). However, we shall obtain their formula as a special case of the general equation (29), since this makes clearer the nature of the assumptions made. These assumptions reduce to the fact that both equilibrium and nonequilibrium adiabatic compression occur without a change in temperature, i.e. \(c_p=c_V\) and \(C_p=C_V\). Strictly speaking, the simultaneous fulfillment of these conditions is impossible, since if \(v\ne0\), then \(C_{pi}\ne C_{Vi}\). However, as has already been noted, usually \(pv\ll Q\), and such an approximation is permissible. Substitution in (29) gives for the relaxation compressibility:

\[ \beta_1=\beta_0-\beta_\infty=-\frac{\rho v^2}{M^2\Phi_{\xi\xi}}. \tag{38} \]

Substituting

\[ \Phi_{\xi\xi}=-\frac{RT}{M}\frac{1}{\xi(1-\xi)} \]

and assuming, as Gierer and Wirtz do, \(\xi\ll1\), we obtain:

\[ \beta_1=\frac{\rho v^2}{MRT}\,\xi. \tag{39} \]

The magnitude of the “hole volume” \(v\) may be related to the ordinary shear viscosity of the liquid \(\mu\). According to the hole theory, the molecules of a liquid oscillate with a certain frequency

\[ j_0=\frac{kT}{h} \]

near “temporary” equilibrium positions, while fluidity is determined by the frequency \(j\) of jumps from one equilibrium position to another, neighboring one:

\[ \frac{1}{\mu}=\frac{V}{6RT}\,j, \]

where \(V\) is the molar volume. The numerical coefficient in this formula, of course, cannot be taken seriously, since it is derived from very rough assumptions and gives only the order of magnitude of the viscosity.

A molecule can jump into a neighboring equilibrium position only when it is unoccupied. Therefore \(j = j_0 \xi e^{-q'/RT}\), where \(\xi\) is the concentration of holes, and \(q'\) is the activation energy for the jump of a molecule into a hole (or, equivalently, for the displacement of a hole into the place of a neighboring molecule). Again taking \(\xi \ll 1\), we have:

\[ \frac{1}{\mu}=\frac{V}{6RT}j_0 e^{-(q+pv)/RT}, \tag{40} \]

where the activation energy for the jump \(q=Q+q'\) is composed of the energy of hole formation and the activation energy for its displacement. (We do not include in \(\Delta \Phi\) a term with \(\sigma\), in view of the complete indeterminacy of this quantity.)

From (40) it follows that

\[ v=RT\frac{\partial}{\partial p}\ln \mu . \]

Substituting the expressions for \(v\) and \(\xi\) into (39), we obtain the formula of Gierer and Wirtz:

\[ \beta_1=\frac{1}{M}RT\rho\left(\frac{\partial\ln\mu}{\partial p}\right)^2 e^{-Q/RT}. \tag{41} \]

It will be shown below that this formula requires substantial refinements.

Gierer and Wirtz calculate the relaxation time \(r=r'\) from the “reaction equation.” In doing so they assume that the formation of holes can be described as the breaking of “bonds” between two neighboring molecules (analogously to the dissociation of a diatomic molecule). In such a reaction, two mutually complementary disturbances are formed simultaneously, each of which is characterized by a molecule having no nearest neighbor (with one free intermolecular “bond”). These disturbances may subsequently move independently of one another. Under such assumptions the formation of a hole—dissociation—and the reverse recombination of the disturbances are described by a second-order reaction equation. This point of view is not convincing. In a liquid, in contrast to a crystal, there are disturbances of only one kind, and, as Ya. I. Frenkel’ points out\(^{19}\), they are more like cracks and cavities disrupting the quasi-crystalline structure than empty, unoccupied sites of an ideal crystal lattice. Holes of this type can form and disappear simply through rupture or coalescence of a mass of liquid, without the participation of additional disturbances, and therefore there is no reason to regard the process of hole formation as dissociation. On the other hand, although the formation of a hole is indeed associated with excitation of at least two molecules,* such a process cannot

* Such a point of view is in general not rigorous: the appearance of a hole probably cannot be described as the excitation of individual molecules, but we shall nevertheless adhere to this approximate description.

considered as the formation of two independent disturbances. Indeed, excited molecules always surround a hole and form one disturbance; therefore the energy and the volume of a hole can always be assigned to one of the molecules surrounding it.

For the case of a first-order reaction, the “reaction equation” was derived above (formula (28)). In this equation the retardation time is

\[ r'=\frac{1}{k_{20}+k_{10}}=\frac{1-\xi}{k_{20}}\simeq \frac{1}{k_{20}}. \]

It is easy to see that in the case of a second-order reaction the form of the “reaction equation” does not change, but then

\[ r'=\frac{1}{2\xi k_{20}+k_{10}} =\frac{1-\xi}{(2\xi-\xi^2)k_{20}} \simeq \frac{1}{2\xi k_{20}}\,{}^{*}). \]

The recombination rate \(k_{20}\), according to Gierer and Wirtz, is determined by the frequency of molecular jumps in the presence of a hole and by the coordination number \(z\):

\[ k_{20}=\frac{1}{z}\,j_0 e^{-q/RT}. \]

This formula means that when a hole moves to the position of any one of the \(z\) molecules surrounding it, it disappears. Such an assumption makes sense if one considers that holes disappear as a result of recombination of two additional disturbances that are next to each other: one of the \(z\) possible jumps leads to their coalescence and, consequently, to closure of the hole. The coordination number \(z\) is in general difficult to determine. Gierer and Wirtz take \(z=6\), as for a simple cubic lattice. Then, in the case of a second-order reaction,

\[ r'=\frac{1}{2\xi k_{20}} =\frac{z}{2j} =\frac{z}{2j_0}e^{q/RT} =\frac{V_{\mu}}{2RT}. \tag{42} \]

This formula shows that the retardation time is, in order of magnitude, equal to the residence time of a molecule \(1/j\). For not very viscous liquids \(r' \simeq 10^{-11}\) sec. Gierer and Wirtz do not take into account the difference between \(r'\) and \(r\), and here we shall not correct this error, since it is insignificant.

Let us note that the relaxation time of the shear viscosity also has, in order of magnitude, the value of the residence time of a molecule \(1/j\). Therefore, if \(K_0\) does not differ too much from \(K_\infty\) (recall that the relaxa—

\[ \overline{\phantom{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx}} \]

*) A general formula for the relaxation time in the case of a reaction of \(n\)-th order was recently published by Meines \({}^{33}\).

ABSORPTION OF ULTRASONIC WAVES IN LIQUIDS

…relaxation of the volume viscosity \(\tau=\dfrac{K_0}{K_\infty}\,r\)), then the volume viscosity associated with processes of this kind must, according to the theory of Gierer and Wirtz, relax in the same frequency region as the shear viscosity. However, the authors themselves did not draw this obvious conclusion from their theory.

From (41) and (42), Gierer and Wirtz obtained for the volume viscosity

\[ \eta=\frac{1}{2\beta_0^2}\left(\frac{\partial \ln \mu}{\partial p}\right)^2 \mu e^{-Q/RT}. \tag{43} \]

To test this formula, they calculated, for a number of liquids, the values of \(Q\) from the experimental values of \(\eta\) and compared them with the known values \(q=Q+q'\). Such a comparison can be made because the energy of formation of a hole, \(Q\), constitutes the principal part of the activation energy for a jump, \(q\). This conclusion follows from the results of Bachinsky, who showed that the fluidity \(\dfrac{1}{\mu}\) is proportional to the difference between the volumes of the liquid and of the corresponding crystal, i.e., roughly speaking, to the number of holes in the liquid, and also from Bridgman’s data, according to which the viscosity changes only very weakly with temperature if the volume of the liquid is kept constant. Eyring’s calculations gave, for nonassociated liquids, values of \(q'\) of the order of several hundred small calories per mole, whereas \(q\) is usually \(2\text{--}3\ \mathrm{kcal/mole}\).

The values of \(Q\) obtained by Gierer and Wirtz in most cases lie between \(\tfrac{1}{3}q\) and \(\tfrac{1}{2}q\), i.e., they agree with the true values only in order of magnitude. Contrary to the authors’ opinion, this result cannot be regarded as satisfactory, since \(Q\) enters into the exponent and, therefore, even small errors in \(Q\) lead to considerable errors in \(\eta\). Such a significant deviation of \(Q\) from \(q\) also cannot be explained by the arbitrariness of the adopted value of the coordination number \(z\) and of the numerical coefficient in (40), since the uncertainty thereby introduced is too small. Thus, the calculations of Gierer and Wirtz lead to the conclusion that “hole” structural relaxation in the liquids they considered explains only a small part of the excess-Stokes absorption.

As has already been indicated, the theory of Gierer and Wirtz contains the rather implausible assumption that the formation of holes is a second-order reaction. It seems much more natural to suppose that this process is a first-order reaction. One can imagine, for example, the following mechanism for the process of disappearance of a hole. When some molecule jumps into a neighboring hole, it passes through an intermediate unstable state, the energy of which is the activation energy \(q'\). At the moment when the molecule is halfway between the old

and new equilibrium positions, the actual holes, i.e., a vacant lattice site, or an unfilled cavity between “microcrystals,” no longer exist; there is only a local rarefaction with which the stressed state of a region of the liquid is associated. This stress can disappear either through further advance of the molecule to a new equilibrium position, or through simultaneous displacement of all surrounding molecules (the mechanism of instantaneous elasticity). In the first case the hole is preserved, but is displaced to another place. In the second case it disappears. Denoting the probability of this second process by \(\chi\), we have for the reaction rate of disappearance of a hole

\[ k_{20}=\chi j_0 e^{-q'/RT}. \]

Then

\[ r' \simeq \frac{1}{k_{20}}=\frac{1}{\chi j_0} e^{q'/RT} =\frac{V_\mu}{6\chi RT}\,e^{-(Q+p v)/RT}. \tag{44} \]

Unfortunately, nothing definite can be said about the value of \(\chi\) and its dependence on temperature. In any case, formula (44) shows that the retardation time \(r'\) (and \(r\)) should probably not differ too much in its order of magnitude from the residence time of a molecule, \(\frac{1}{j}\). More definite conclusions can be drawn only with the aid of a more rigorous and complete theory of the process of formation and disappearance of holes. The retardation time can also be determined from experimental data on sound absorption. It is possible that this would make it possible to draw some conclusions about the mechanism and kinetics of the hole-formation process. For this, however, it is necessary to derive a more exact formula for the relaxation compressibility.

In this formula account must be taken not only of the change of pressure in the sound wave, but also of the change of temperature. Temperature changes were first taken into account by Goshem and Farem.\(^{29}\) However, in contrast to Girer and Wirz, they considered the change in the number of holes only as a consequence of temperature oscillations in the sound wave. It is clear that a rigorous theory must take both effects into account. To compare their relative roles it is convenient to present formula (29) in approximate form, assuming that these effects are small and approximately equal. Then, to terms of first order of smallness,

\[ \beta_1-\beta_0-\beta_\infty \simeq \frac{\rho v^2}{M^2\Phi_{\xi\xi}}\,\frac{c_V}{c_p} +\beta_0\,\frac{c_p-c_V}{c_p c_V}\,C_i . \tag{45} \]

In the theory of Girer and Wirz only the first term of this expression was taken into account. It coincides with the formula of Girer and Wirz if \(c_p=c_V\).

However, in reality this condition is almost never fulfilled. Gierer and Wirtz believed that their formula is valid for the case when \(C_i\) is small. But even in this case the expression for \(\beta_1\) contains the factor

\[ \frac{c_V}{c_p}, \]

not taken into account by Gierer and Wirtz. This factor may be several tenths less than unity.

Using the expression for \(C_i\) obtained in § 3, and assuming that \(\xi \ll 1\) and \(p v \ll Q\), we have:

\[ \beta_1= \left\{ \frac{M}{pRT}\left(\frac{v}{V}\right)^2\frac{c_V}{c_p} + \beta_0\,\frac{R}{M}\left(\frac{Q}{RT}\right)^2 \frac{c_p-c_V}{c_p c_V} \right\}e^{-Q/RT}. \tag{46} \]

The hole volume \(v\) can be determined from the dependence of the shear viscosity \(\mu\) on pressure, as was done by Gierer and Wirtz. The energy of hole formation \(Q\), as already noted above, can be calculated from the temperature dependence of \(\mu\) at constant pressure and at constant volume. One may also restrict oneself to data at constant pressure, assuming that \(Q \simeq q\). This is precisely what Ghosh and Färm did. These authors took into account only the second term of formula (46), and they did not even indicate that their expression does not exhaust the entire process. The expression for \(\beta_1\) obtained in this way was used by them to calculate the relaxation time for a number of liquids.

We used the calculations of Gierer and Wirtz and of Ghosh and Färm in order to compare the relative role of the two terms in formula (46) for the case of acetone and ethyl bromide. It turned out that in both cases these terms have the same order of magnitude. This makes it possible to conclude that the Gierer–Wirtz formula (43) is suitable only in those cases when \(c_p-c_V\) is small, while the Ghosh–Färm formula taken separately is, apparently, generally inapplicable. As already indicated, it is necessary to take into account both the direct influence of the change in pressure (the first term in (45)) and the influence of the change in temperature (the second term). Both of these terms are calculated essentially from the same ideas about the character of relaxation processes, and their simultaneous inclusion presents no difficulty*).

There is no doubt that processes connected with a change in the degree of order in the arrangement of molecules occur in all liquids and that part of the excess absorption should be connected with these processes. In all probability, these processes can be described as a change in the number of holes in the liquid, although in doing so it is necessary to take into account that the “hole theory” is only

*) We note that for practical calculations there is no need to use formula (46), since the exact formula (29) is no less convenient for calculations.

rough and still poorly developed scheme, far from taking into account all the features of the structure. Therefore, the theory of structural relaxation set forth above remains, for the time being, debatable.

First of all, it is not entirely clear whether the formation of holes can be described as the excitation of individual molecules (or “degrees of freedom”—this reservation changes nothing in essence). Further, all assumptions about the retardation time \(r'\) of the hole-formation reaction are highly uncertain. Since at present there is no theory that makes it possible to calculate the retardation time \(r'\), one has to confine oneself to determining this quantity from the experimental values of \(a\) and the calculated values of \(\beta_1\). However, such a method is suitable only in the case when all the excess X-ray absorption is associated with a single relaxation process. If, under such a determination, the values of \(r'\) turn out to be clearly overestimated, this should indicate that the given process is not the only one. However, in a whole series of cases quite reasonable values of \(r'\) are obtained, and then one can say only that the theory does not contradict experiment.

§ 6. STRUCTURAL RELAXATION OF ASSOCIATED LIQUIDS. WATER

The theory of Gierer and Wirtz applies only to non-associated liquids. For such liquids as water and, possibly, alcohols, where there are strong hydrogen bonds capable of maintaining a sterically unfavorable, loose quasi-crystalline structure, it is also necessary to consider another possible relaxation mechanism—the rearrangement of the quasi-crystalline structure from a loose to a more densely packed modification. Such a calculation for water was carried out by Hall \(^{21}\). He also regarded the rearrangement of the structure as the excitation of the corresponding molecules, i.e., he proceeded essentially from formula (29). Neglecting heating under adiabatic compression, which for water is permissible in view of the smallness of \(c_p-c_V\), we obtain formula (38), which, after substitution of the explicit expression for \(\Phi_{\xi}\), is easily reduced to the form*:

\[ \beta_1=\frac{V\left(\dfrac{v}{V}\right)^2}{2RT(1+\operatorname{ch}x)}. \tag{47} \]

On the basis of X-ray data, Hall adopted the following model of the structure of water. The “normal state” of the molecules is their tetrahedral arrangement, when each molecule is surrounded by four neighbors. This is a rather loose structure of ice, which is maintained by hydrogen bonds (three for each

* Hall himself derived this formula with the aid of relation (33).

molecule). In the “excited” state the molecules are arranged so that the quasi-crystalline structure is close to the densest packing. Hall assumes (although this is not essential) that in this state each molecule has two hydrogen bonds. For the ground state Hall takes the molar volume to be \(19.6\ \mathrm{cm}^3\), i.e. intermediate between water \((18.0\ \mathrm{cm}^3)\) and ice \((20.0\ \mathrm{cm}^3)\). He calculates the molar volume of the excited state as for densely packed spheres with a diameter equal to the smallest distance between molecules; it is \(10.4\ \mathrm{cm}^3\)*). Hall also calculates \(\beta_\infty\) independently from the following considerations: \(\beta_\infty\) must be somewhat greater than the compressibility of ice, which near \(0^\circ\mathrm{C}\) is \(10 \div 12 \cdot 10^{-12}\ \mathrm{cm}^2/\mathrm{dyne}\). The theoretical value of the compressibility, calculated with the aid of a potential function for bound molecules, is \(12\cdot 10^{-12}\ \mathrm{cm}^2/\mathrm{dyne}\), and for the case of freely rotating molecules is \(24\cdot 10^{-12}\ \mathrm{cm}^2/\mathrm{dyne}\). Hall adopted two trial values, \(\beta_\infty = 15\cdot 10^{-12}\ \mathrm{cm}^2/\mathrm{dyne}\) and \(\beta_\infty = 18\cdot 10^{-12}\ \mathrm{cm}^2/\mathrm{dyne}\). Both of these quantities gave good agreement with experiment.

Substituting the values \(\beta_1=\beta_0-\beta_\infty\), determined in this way, into formula (47), Hall found the value \(xRT \simeq 0.5\ \mathrm{kcal}/\mathrm{mole}\), which gives for the energy of a hydrogen bond (assuming that during rearrangement one bond is broken, i.e. \(1/2\) bond per molecule)

\[ \simeq 1\ \mathrm{kcal}/\mathrm{mole}. \]

For the relaxation time Hall used the formula

\[ r'=\frac{1}{k_{20}+k_{10}}=\frac{1}{k_{20}}\frac{e^x}{1+e^x}. \]

Assuming further that the transition from the excited to the unexcited state requires an activation energy \(q'\), Hall obtains:

\[ r'=\frac{kT}{h}\cdot \frac{e^{q'/RT+x}}{1+e^x}, \]

where \(\frac{kT}{h}=j_0\) is the frequency of molecular vibrations.

On the other hand, in viscous flow the molecules make a jump from one equilibrium position to the neighboring one, overcoming the same barrier \(xRT+q'\) (jumps of excited molecules, for which the barrier is \(q'\), may be neglected, although taking them into account is also not difficult). Taking into account that

\[ \mu=\frac{hN}{V}e^{q'/RT+x}, \]

we have:

\[ r'=\frac{V}{RT(1+e^x)}\,\mu. \]

*) Hall does not take thermal expansion into account.

Substituting the previously calculated value of \(\dot{\chi}\), Hall determines \(r'\) for various temperatures*). The lag time, according to his calculations, has the order of magnitude \(10^{-12}\) sec. Knowing \(r'\), he calculates \(a\) for temperatures from 0 to \(80^\circ\)C. In doing so, in order partially to compensate for the unknown dependence of \(\beta_1\) on temperature, Hall regarded the factor \((1+\operatorname{ch} x)\) in the denominator of formula (4f) as constant. The calculations gave, for both trial values \(\beta_\infty=15\cdot 10^{-12}\ \text{cm}^2/\text{dyne}\) and \(\beta_\infty=18\cdot 10^{-12}\ \text{cm}^2/\text{dyne}\), excellent agreement with experiment.

However, this work contains the same error as the work of Gierer and Wirtz. Calculation of the relaxation time for rearrangement of the quasi-crystalline structure with the aid of data on the shear viscosity may give the correct temperature variation and agreement in order of magnitude, but no more, since the pre-exponential factor in the formula for the viscosity is derived from very crude model considerations and therefore is sufficiently indeterminate. Thus the excellent (to two significant figures!) agreement of Hall’s theory with experiment by no means confirms all his arguments and is rather accidental. Agreement over a wide temperature interval is not an argument in favor of the quantitative applicability of the theory, since the dependence of the absorption coefficient on temperature is determined mainly by the exponential factor in the formula for the viscosity, while in water the experimental absorption coefficient varies with temperature according to the same law as the Stokes coefficient, i.e. approximately proportionally to the viscosity. Therefore agreement at all temperatures is attained automatically if one point agrees. Let us also note that Hall’s model gives for water a negative coefficient of thermal expansion at all temperatures. Since in experiment at sufficiently high temperatures a positive coefficient of expansion is observed, it is possible that the effect considered by Hall is not predominant over the whole temperature interval. True, the increase of the velocity of sound with temperature up to \(70\)—\(80^\circ\)C seems to speak in favor of Hall’s theory.

Taking into account the remarks made, one may acknowledge that, on the whole, the calculation scheme proposed by Hall is sufficiently convincing and correctly describes the mechanism of relaxation in water. Of course, in water there must also exist, in addition, the ordinary “hole” mechanism considered by Gierer and Wirtz, and its influence on absorption should be estimated. It is quite possible that in water and in other associated liquids the influence of this mechanism is small.

After Hall, Sette\({}^{30}\) attempted to apply his interpretation to the explanation of the volume viscosity of ethyl alcohol. He regarded as the principal quasi-crystalline structure of alcohol the arrangement of molecules in chains, and as the excited one—the pair association of molecules with antiparallel dipoles. To calculate the lag time, Sette

*) In doing so Hall, like Gierer and Wirtz, does not distinguish \(r\) and \(r'\).

made use of the value of the energy difference of these structures \((xRT = -0.68\ \mathrm{kcal/mole})\) and obtained \(r' = 9 \cdot 10^{-12}\ \mathrm{sec}\). He determined the volume of the “excited” structure by assuming that in a dilute solution of alcohol in carbon tetrachloride the alcohol molecules are associated in pairs and the volumes are additive. The difference between the “normal” and “excited” volumes is insignificant, and the value of the relaxation compressibility obtained in this way proved to be much smaller than is needed to explain the observed value of the bulk viscosity. We tried to estimate \(\beta_1\), taking into account the second term of formula (46) (for alcohols, in contrast to water, \(c_p - c_v\) is not small, and this term may play a noticeable role), and likewise obtained too small a value. This indicates that the lag time adopted by Sette is incorrect, or that the principal role in the absorption is played in this case by some other process.

Comparatively recently Gosh\(^{11}\) determined the relaxation time of the bulk viscosity in water, starting from the same molecular assumptions as Hall. He, however, did not calculate the lag time independently, but used for the calculation the experimental data of Dutta and Dutta and Ghosh\(^{31}\), who discovered sound dispersion in water. According to their data, the ratio of the square of the sound velocity in water at frequency \(\nu = 3 \cdot 10^{7}\ \mathrm{cycles/sec}\) to the square of the velocity at \(\nu = 1 \cdot 10^{7}\ \mathrm{cycles/sec}\) is equal to \(1.004\) \((T = 30^\circ\mathrm{C})\). This result in itself raises great doubts, since according to all available data relaxation phenomena in water should exist at considerably higher frequencies. Relaxation theory also leads to the same conclusion.

The dependence of the sound velocity on frequency according to relaxation theory is usually expressed by formula (14) (see § 1), which may be rewritten in the following form:

\[ a^{2}=a_{0}^{2}\left(1+\beta_{\infty}\frac{\beta_{0}-\beta_{\infty}}{\beta_{0}^{2}}\omega^{2}r^{2}\right). \]

However, in the case of water \(\beta_{0}-\beta_{\infty}\) is not small in comparison with \(\beta_{\infty}\), and consequently, along with the dispersion caused by the relaxation of the bulk viscosity, one must also take into account the dispersion associated with the presence of bulk viscosity\(^*\). If this is taken into consideration, then the velocity at frequencies \(\omega \ll \dfrac{1}{r}\) is expressed by the formula (to terms of order \(\omega^{2}r^{2}\)):

\[ a^{2}=a_{0}^{2}\left\{1+\left(\frac{3}{4}-\frac{1}{2}\frac{\beta_{\infty}}{\beta_{0}}-\frac{1}{4}\frac{\beta_{\infty}^{2}}{\beta_{0}^{2}}\right)\omega^{2}r^{2}\right\}. \]

\(^*\) Strictly speaking, one should also include terms taking account of the shear viscosity, which for water has almost the same magnitude, but for simplicity we shall not make this refinement.

When \(\beta_0-\beta_\infty \ll \beta_0\), this formula becomes the preceding one. Substituting the values accepted by Goche: \(\beta_0 \simeq 50\cdot 10^{-12}\ \mathrm{cm}^2/\mathrm{dyne}\) and \(\beta_\infty \simeq 12\cdot 10^{-12}\ \mathrm{cm}^2/\mathrm{dyne}\) (Hall gives \(\beta_\infty=15\div 18\cdot 10^{-12}\ \mathrm{cm}^2/\mathrm{dyne}\), which is probably more reasonable), and taking the speed of sound at frequency \(1\cdot 10^7\ \mathrm{cps}\) to be equal to \(a_0\), which gives an insignificant error for \(r\), \(\simeq 5\%\), we obtain \(r=1.3\cdot 10^{-10}\ \mathrm{sec}\). Then the super-Stokes absorption at low frequencies and at room temperature will be

\[ \frac{\alpha_0}{\nu^2} = \frac{2\pi^2}{a_0} \left(1-\frac{\beta_\infty}{\beta_0}\right)r \simeq 1.3\cdot 10^{-14}\ \mathrm{sec}^2/\mathrm{cm}, \]

whereas the experiment gives \(17\cdot 10^{-17}\ \mathrm{sec}^2/\mathrm{cm}\). Thus, the delay time calculated from the data of Dutt and Goche proves to be incompatible with the experimental values of absorption. The dispersion observed by these authors is probably explained simply by experimental error.

Goche’s work is based on these extremely doubtful data. Moreover, Goche made a number of very gross errors (including arithmetical ones). The most substantial error by Goche, which he made in calculating the delay time, was that he used the formula

\[ \frac{\alpha_0}{\nu^2} = \frac{2\pi^2}{a_0} \left(1-\frac{\beta_\infty}{\beta_0}\right)r, \]

in which, instead of the full dispersion

\[ \left(1-\frac{\beta_\infty}{\beta_0}\right) \]

he substituted the experimental value \(0.004\), which determines the relative change of the velocity only between the frequencies 10 and 30 Mc. As a result, he obtained a greatly exaggerated delay time \(r \simeq 10^{-9}\ \mathrm{sec}\), whereas according to Hall’s calculations \(r \simeq 10^{-12}\ \mathrm{sec}\). Nevertheless, Goche for some reason believes that his results agree with Hall’s results.

§ 7. STRUCTURAL RELAXATION OF ASSOCIATED LIQUIDS. ACETIC ACID

P. A. Bazhulin\(^{12}\) discovered relaxation of the volume viscosity in acetic acid at frequencies \(\simeq 10^6\ \mathrm{cps}\) (at \(17^\circ\mathrm{C}\)). B. G. Shpakovsky\(^{33}\) discovered sound dispersion in the same region. Shpakovsky suggested that the relaxation process in this case is the disturbance of the chemical equilibrium of the dimerization reaction: in the normal state, acetic-acid molecules are joined in pairs by means of hydrogen bonds (two bonds per pair, i.e. one bond per molecule). Recently, Lamb and Pinkerton\(^{17}\) investigated sound absorption in acetic acid at frequencies from 0.5 to 67.5 Mc and the velocity in the interval 0.5–9.8 Mc at temperatures from 16 to \(60^\circ\mathrm{C}\). These measurements make possible a direct comparison with the results of the theory.

Lamb and Pinkerton assumed that the relaxation process may be regarded as excitation of acetic-acid molecules, and that this process takes place without a change in volume. Then the volume viscosity is given by Herzfeld’s formula (34); moreover, if the relative concentration of dissociated molecules \(\xi \ll 1\), then

\[ C_i=\frac{R}{M}\left(\frac{Q}{RT}\right)^2 e^{-\frac{Q-\sigma T}{RT}}, \tag{48} \]

where \(Q\) is the dissociation energy per mole of monomeric molecules (i.e., per mole of hydrogen bonds), and \(\sigma\) is the entropy of dissociation, determined by the statistical weight of the “excited” (dissociated) state. As already noted, a theoretical estimate of this quantity is difficult, since it requires knowledge of a detailed picture of the process.

The case of acetic acid is very convenient for calculations, since the entire relaxation region is accessible to direct experimental investigation. Since \(\tau' \simeq \tau\) is easily determined from the curves of the frequency dependence of absorption, and the remaining quantities in Herzfeld’s formula (34) are also known, the relaxation heat capacity \(C_i\) can be calculated from the absorption. From the temperature dependence of \(C_i\), Lamb and Pinkerton obtained \(Q=-2.32\ \mathrm{kcal/mol}\). Substituting this value into formula (48), we obtain \(\sigma=3.89\ \mathrm{cal/mol\cdot deg}\), i.e., the statistical weight of the excited state is equal to \(7.0\). This number seems unlikely.

It may be supposed that dissociation is also accompanied by a change in volume. Taking \(\sigma=0\), one can determine from formula (46) what additional volume must be in order to obtain agreement between theory and experiment. Substituting

\[ \frac{c_p}{c_V}=1.22,\qquad c_p=28\ \mathrm{cal/mol\cdot deg},\qquad M=90,\qquad \beta_0\simeq 0.5\cdot 10^{-10}\ \mathrm{cm^2/dyn}, \]

we obtain

\[ \frac{v}{V}\simeq 0.5 \]

(the sign of the volume \(v\) remains undetermined, since the value of the relaxation compressibility does not depend on which state corresponds to the larger volume). The value of \(v\) obtained by us is also not very plausible. Thus, there is apparently no complete agreement between theory and experiment here.

In the work under discussion, the temperature dependence of the relaxation time \(\tau\) was also considered. It may be assumed that the isothermal relaxation time \(\tau'\)

\[ \left(\tau=\frac{C_V}{c_V}\tau'\simeq \tau'\right) \]

depends on temperature according to the formula\(^*\)

\[ \tau'\simeq \frac{1}{k_{20}}=AT^{-m}e^{E_a/RT}, \]

\(^*\) For simplicity we do not take into account the difference between \(u'\) and \(\tau'\), i.e., we assume, as do the authors themselves, \(v'=0\).

where \(\Delta E_a\) is the activation energy for the reaction under consideration, and \(m\) is equal to zero or to a small positive number. The experimental data of Lamb and Pinkerton satisfy this formula both for \(m=0\) and for \(m=1\). In the first case \(\Delta E_a\) turns out to be equal to \(8.86\ \text{kcal/mol}\), and in the second to \(8.46\ \text{kcal/mol}\). However, as Gierer and Wirtz noted, in the case of acetic acid a second-order reaction takes place. Therefore the relaxation time must be expressed by the formula

\[ \tau' \simeq \frac{1}{2 \xi k_{20}} = \frac{1}{2 k_{20}} e^{Q/RT}. \]

Gierer and Wirtz further assume that the activation energy \(\Delta E_a\) for the rupture of the hydrogen bond is at the same time also the activation energy for the displacement of the (monomeric) acetic acid molecule, i.e. is equal to the activation energy for viscous flow. The latter is equal to \(2.6\ \text{kcal/mol}\). Thus, assuming a purely exponential dependence of \(k_{20}\) on \(T\) (\(m=0\)):

\[ \tau' \simeq \frac{1}{j_0} e^{(Q+\Delta E_a)/RT}. \]

Using the data of Lamb and Pinkerton, we obtain \(Q = 6.3\ \text{kcal/mol}\). The energy of rupture of a hydrogen bond in the gaseous state has a value of about \(7\ \text{kcal/mol}\); in a liquid it, naturally, must also be smaller. The numerical value of \(\tau\) likewise does not contradict these considerations. Indeed, \(\tau'\) can be represented in the form

\[ \tau' \simeq \frac{1}{j} e^{Q/RT}, \]

and since \(j \simeq 10^{11}\ \text{sec}^{-1}\), for \(Q\) one obtains a value of the order of \(7\ \text{kcal/mol}\), as also from the temperature dependence of \(\tau\). However, this value does not coincide with that which is obtained from the temperature dependence of \(C_l\). Thus, here too, complete agreement between theory and experiment is not obtained.

Let us also note that, according to the data of Lamb and Pinkerton, at frequencies above \(70\ \text{Mc}\) the absorption in acetic acid is still greater than it should be according to Stokes’ formula, i.e. there must exist still another region of relaxation of the bulk viscosity. The authors themselves believe that this additional relaxation is associated with the rupture of a second hydrogen bond between molecules. We find more convincing the opinion of Gierer and Wirtz, that the unrelaxed part of the bulk viscosity is associated with the relaxation of the process of hole formation, since its numerical value is of the same order of magnitude as the bulk viscosity of non-associated liquids.

CONCLUSION

The works on the theory of “super-Stokes” absorption of ultrasonic waves in liquids considered in this article lead to the conclusion that, on the question of the molecular mechanism of volume viscosity, there is still no complete clarity. The situation is especially unfavorable in this respect with the theory of structural relaxation. This is explained mainly by the absence of a satisfactory general theory of the structure of liquids. However, it can already be said that, if it were possible to develop a more successful model of structural relaxation, this would make it possible to use experimental data on the absorption of ultrasound for the study of the structure of liquids.

As for Kneser processes, their theory has been developed considerably better. The study of these effects may make it possible to obtain more complete information on the kinetics of exchange processes, which is very important for elucidating the nature of intermolecular interaction in liquids.

In order to apply experimental data to the study of the molecular processes responsible for relaxation, it is necessary first of all to be able to separate the relative influence of the various relaxation mechanisms. For this, both a further refinement of the theory and an expansion of the experimental material are required, especially in the direction of investigating the temperature dependence of absorption.

In our opinion, the further development of the theory should proceed in the direction of developing concrete molecular models on the basis of the general relaxation theory of Mandelstam and Leontovich.

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1. Strictly speaking, this concept has no meaning for liquids, but it may be used for such a crude estimate. ↩