V. S. SMORODINOV

ON THE THEORY OF VISCOSITY OF BINARY LIQUID MIXTURES WITH ALLOWANCE FOR COMPLEX FORMATION. RATIONAL SYSTEMS

(Presented by the scientific seminar of the Department of Physical Chemistry)

A vast amount of experimental material has been accumulated on the viscosity of liquid mixtures. This is explained by the fact that viscosity reflects well changes in the molecular structure of a liquid and often makes it possible to detect a transformation not captured by other methods of physicochemical analysis. In viscometric studies, the type of curves expressing the dependence of the viscosity of mixtures on composition is established, and a conclusion is drawn about the chemical process occurring in the system, leading to the formation of definite chemical compounds. The tendency to establish only the type of curve has in many cases led to a decrease in the accuracy of measurements.

Comparison of the results of investigations of the most diverse systems shows that the form of a physicochemical diagram cannot give any definite indications of the character of the interaction in the system, that is, determine whether chemical or intermolecular forces are involved in the interaction. The capacity of components for intermolecular interaction is determined from the deviation of the viscosity from the law of ideal mixtures.

Most authors consider that in mixtures close to ideal, additivity of the logarithm of viscosity is observed. Thus, Pospekhov [1] found four systems in which strict additivity of the logarithm of viscosity is observed when composition is expressed in mole fractions; moreover, by a number of other properties these systems belong to the ideal ones. In some of these systems the viscosities of the components differ by a factor of 2–3. Pospekhov notes that the signs of formation of new compounds in a system are:

a) isotherms of the logarithm of viscosity that are concave toward the composition axis and S-shaped;

b) rectilinear isotherms, isotherms concave toward the composition axis, and S-shaped isotherms of viscosity. Rectilinear or convex-toward-the-composition-axis isotherms of the logarithm of viscosity are a sign of the absence of chemical interaction in the system.

Assuming that additivity of the logarithm of viscosity is preserved in ternary systems, Usanovich [2] proposed considering systems with the formation of unstable compounds as three-component systems, in which the concentrations of the components and of the new compound are related to the equilibrium constant of the reaction in the system by the law of mass action. As

as the author notes, the equation obtained in this way describes all types of viscosity isotherms of mixtures. It is assumed that the viscosity coefficient of the compound entering into the equation obtained remains constant for mixtures of different concentrations.

Zhdanovskii [3], on the basis of the well-known Poiseuille formula, showed that in ideal mixtures the kinematic fluidity is an additive quantity when the composition is expressed in volume fractions. However, as Deitch correctly notes, Zhdanovskii’s formula has no advantages over Arrhenius’ formula as applied to mixtures of different concentrations.

The viscosity method, in combination with other methods of physicochemical analysis, makes it possible to solve a number of problems in creating a unified theory of solutions. One such problem is the establishment of the equilibrium composition and the interaction reactions in a solution that lead to this composition.

In an earlier work by Stromberg [4], the possibility was shown of taking into account the formation of intermolecular compounds in establishing the dependence of the viscosity of binary liquid mixtures on composition and temperature. Deviations of viscosity from the law of ideal mixtures are explained in [4] on the basis of Andrade’s theory, which takes into account the different interaction energies between like and unlike molecules, but does not take into account the formation of intermolecular compounds that are sufficiently long-lived (in comparison with the period of oscillation). In this case it is assumed that the transfer of momentum between two molecules in two neighboring layers of liquid occurs at the moment of approach of the molecules at the extreme deviation from the equilibrium position during oscillation. On the basis of the fact that the viscosity of liquids decreases with increasing temperature, it is accepted that the cause of this is a decrease in the fraction of oriented molecules, determined by the Boltzmann statistical law.

The viscosity of a mixture of liquids consisting of two kinds of molecules, as a function of composition, is expressed by the equation [4]

\[ \eta = \sum_{i,j}^{2} a_{i,j} N_i N_j, \tag{1,a} \]

where \(a_{i,j}\) is the coefficient of “mutual” viscosity, independent of composition but dependent on temperature;

\(N_i, N_j\) are the analytical mole fractions of the components of the mixture.

\[ a_{i,j} = C_{i,j} e^{\frac{E_{i,j}}{RT}}, \]

where \(C_{i,j}\) is a coefficient independent of the composition of the mixture and of temperature;

\(E_{i,j}\) is the interaction energy of like and unlike molecules, cal/mol;

\(R\) is the universal gas constant;

\(T\) is the absolute temperature.

Formula \((1,a)\), after transformations, takes the form

\[ \eta = b_2 N_1^2 + b_1 N_1 + b_0, \tag{1,б} \]

where

\[ \begin{aligned} b_2 &= (a_{11} + a_{22}) - 2a_{12};\\ b_1 &= 2a_{12} - 2a_{22};\\ b_0 &= a_{22}. \end{aligned} \]

The dependence of viscosity on the composition of the mixture is expressed by a second-degree equation, which well describes convex viscosity isotherms—

convex and concave, with an extremum but without inflection points. Investigation of this equation makes it possible to determine the conditions for the appearance of an extremum and its position relative to the composition axis of the mixture, as well as its displacement and disappearance with a change in temperature. The maximum on the viscosity isotherm of a mixture is explained by a stronger interaction of unlike molecules in comparison with like molecules (even in the case where no intermolecular compound is formed). To construct the viscosity isotherm of such systems it is necessary to know the values of the coefficients \(a_{11}\), \(a_{22}\), \(a_{12}\). The coefficients \(a_{11}\), \(a_{22}\) are the viscosities of the first and second components, respectively. To determine the coefficient \(a_{12}\), it is also necessary to have an experimental value of the viscosity of one mixture.

Assuming that in an ideal system the additivity of the logarithm of viscosity is obeyed, one can determine the value of the coefficient \(a_{12}\) satisfying the condition of an ideal system from the expressions:

\[ \lg \eta = N_1 \lg \eta_1 + N_2 \lg \eta_2 \quad \text{or} \quad \eta = \eta_1^{N_1}\cdot \eta_2^{N_2} \tag{2} \]

and

\[ \eta = a_{11}N_1^2 + a_{22}N_2^2 + 2a_{12}N_1N_2, \tag{1, в} \]

where \(a_{11}=\eta_1\) and \(a_{22}=\eta_2\). By calculation it can be shown that if the isotherm of the logarithm of the viscosity of the system is rectilinear (that is, the system is ideal), then the values of the coefficient \(a_{12}\), calculated from equation \((1, в)\), remain constant for mixtures of different composition. Equating the right-hand sides of equations \((1, в)\) and (2), we obtain

\[ \eta_1^{N_1}\eta_2^{N_2} = \eta_1N_1^2 + \eta_2N_2^2 + 2a_{12}N_1N_2. \tag{3} \]

Taking into account that \(a_{12}\) does not depend on the composition of the mixture, we determine the value of \(a_{12}\) from equation (3), taking, for example, \(N_1=N_2=0.5\)

Table 1

Viscosity isotherms of an ideal system, calculated from equations \((1, в)\) and (2)

\[ a_{12}=2\sqrt{\eta_1\eta_2}-\frac{1}{2}(\eta_1+\eta_2), \tag{4} \]

that is, the coefficient of mutual viscosity \(a_{12}\) in an ideal system is equal to the difference between twice the geometric mean and the arithmetic mean of the viscosities of the components.

Calculation of the viscosity isotherm of the system by equations \((1, в)\) and (2), with substitution of the value \(a_{12}=2.40\) obtained from expression (4), gives very close results (Table 1). It is assumed that the deviation of the system from ideality may be associated with deviation of the quantity \(a_{12}\) from the value calculated from equation (4).

If intermolecular compounds are formed in a solution, sufficiently long-lived in comparison with the period of molecular vibrations in the liquid, then in the process of transfer of momentum between neighboring layers of molecules in the liquid these compounds must be regarded as independent particles. These theoretical considerations are developed in the present work for the case of formation of stable and unstable compounds.

In the formation of compounds in a liquid it is necessary to consider

the presence of three kinds of molecules. In two parallel layers of liquid, nine combinations of molecules are possible:

\[ E_{11}\quad E_{22}\quad E_{33}\quad E_{12}\quad E_{21}\quad E_{13}\quad E_{31}\quad E_{23}\quad E_{32} \]

In this case, three pairs of combinations of molecules are equiprobable.

The expression for the amount of motion transmitted to the neighboring layer of molecules in the liquid perpendicular to the direction of viscous flow of the liquid is

\[ P=P_{11}+P_{22}+P_{33}+2P_{12}+2P_{13}+2P_{23}. \]

The expression for the viscosity of a binary mixture is

\[ \eta=\sum_{i,j=1}^{3} a_{i,j}X_iX_j = a_{11}X_1^2+a_{22}X_2^2+a_{33}X_3^2+ \]

\[ +\,2a_{12}X_1X_2+2a_{13}X_1X_3+2a_{23}X_2X_3, \tag{5} \]

where \(X_i, X_j\) are the equilibrium, or true, mole fractions of the components and of the compound.

Rational Systems

If a stable intermolecular compound is formed in a binary system (rational, or singular, systems), then such systems may be regarded as consisting of two binary systems (Fig. 1). On the property diagrams of such systems there is a singular point, as the point of intersection of the property isotherms of two simplest binary systems. The position of the singular point along the composition axis does not depend on temperature and is determined by the composition of the compound.

Often the singular maximum on viscosity isotherms is sharply expressed, that is, \(a_{11}\ll a_{33}\gg a_{22}\). In such systems the formation of associated compounds is assumed [5, p. 159]. The expression for the dependence of viscosity on the composition of the mixture of such systems can be obtained theoretically from equation (5), using the basic propositions of the metric of the equilibrium chemical diagram [5, p. 838]. The problem consists in calculating the true mole fractions \((X_i)\) of the components and of the compound in a mixture in which an irreversible reaction proceeds,

\[ aA+bB\to (A_mB_n)f. \]

Fig. 1. Viscosity isotherm with singular point \(a_{33}\).

A calculation has been carried out of the true mole fractions of component \(A\) and of the compound \((A_mB_n)_f\) for the right-hand part of the diagram, starting from 1 mole of the mixture taken. The number of moles of each kind of molecule in 1 mole of mixture is

\[ n_2=0;\qquad n_3=\frac{N_2/n}{f}=\frac{1-N_1}{nf}; \]

\[ n_1=N_1-mf\cdot\frac{1-N_1}{nf} =\frac{N_1(m+n)-m}{n}, \]

where \(N_i\) are the analytical mole fractions of the components. The true mole fractions of component \(A\) and of the compound in the mixture are

\[ X_1=\frac{f[N_1(m+n)-m]} {(1-mf)+N_1(mf+nf-1)}; \]

\[ X_3=\frac{1-N_1} {(1-mf)+N_1(mf+nf-1)}. \]

Substituting the expressions for \(X_1\) and \(X_3\) into equation (5) and putting \(X_2=0\), we obtain for the right branch of the viscosity isotherm

\[ \eta'= \frac{ a_{11}\{f[N_1(m+n)-m]\}^2+2a_{13}f[N_1(m+n)- } {[(1-mf)+N_1(mf+nf-1)]^2} \]

\[ \frac{ -m]\{(1-N_1)+a_{33}(1-N_1)^2\} } {[(1-mf)+N_1(mf+nf-1)]^2}. \tag{6} \]

Similarly, for the left branch of the viscosity isotherm we have

\[ \eta''= \frac{ a_{22}\{f[m-N_1(m+n)]\}^2+2a_{23}f[m-N_1(m+n)]N_1+a_{33}N_1^2 } {[mf-N_1(mf+nf-1)]^2}. \tag{6,a} \]

For calculating the viscosity isotherm of a rational system when the ratio \(m:n\) is known (it is determined from the position of the singular point on the viscosity isotherm), in addition to the viscosities of the components it is necessary to know the coefficients \(a_{13}\), \(a_{23}\), and \(f\). These coefficients can be determined from experimental viscosity values for several mixtures by solving a system of equations or by a fitting method under the following conditions:

a) absence of association of the components, or if it can be neglected;

b) absence of side reactions, except for the reaction forming the compound \((A_mB_n)_f\);

c) constancy of the composition of the compound, since only under this condition do the coefficients \(a_{33}\), \(a_{13}\), \(a_{23}\), and \(f\) remain constant.

The possibility of such a calculation was shown using as an example the system \(o\)-anisidine \((B)\)—allyl mustard oil \((A)\), in which a stable compound of equimolecular composition [6] is formed—allyl-\(o\)-anisidylthiourea \((AB)_f\)—by the reaction

\[ f\mathrm{CH_3OC_6H_4NH_2}+f\mathrm{CSNC_3H_5} \to[\mathrm{CS(NHC_3H_5)\cdot(NHC_6H_4OCH_3)}]_f. \]

Analysis of the structure of the molecules of the components of this mixture and a review of some of their properties, including the low value of the viscosity, make it possible to assume the absence of significant association of the components.

In this system two side processes take place. Thus, in the region of excess allyl mustard oil on the branch of the isotherm there is a point—

of the inflection (Fig. 2). An inflection point is also observed in other systems formed by allyl mustard oil and secondary amines. Trifonov [5, p. 159] associates its appearance with the presence of a certain interaction between the molecules of allyl mustard oil and the compound. He finds confirmation of this in the form of critical miscibility curves of the systems. Therefore, for the calculation another branch of the viscosity isotherm was used, corresponding to mixtures with an excess of amine and having no inflection point, which indicates the absence of side reactions.

Fig. 2. Viscosity isotherm of the system o-anisidine–allyl mustard oil at 70°: solid curve—experimental, dashed line—calculated branch.

On the basis of the fact that solutions of this system possess considerable electrical conductivity, while \((AB)_f\) cannot be an electrolyte, Miskidzhyan [6] assumes the occurrence of a parallel reaction with formation of allyl-o-anisidinium thiocyanate:

\[ \mathrm{C_3H_5NSC} + \mathrm{ \begin{matrix} \text{benzene ring}\\[-2pt] \mathrm{NH_2}\\[-2pt] \mathrm{O{-}CH_3} \end{matrix}} \longrightarrow \left[ \mathrm{ \begin{matrix} \text{benzene ring}\\[-2pt] \overset{+}{\mathrm{NH_2}}{-}\mathrm{CH_2{-}CH{=}CH_2}\\[-2pt] \mathrm{O{-}CH_3} \end{matrix}} \right] [\mathrm{SCN}]^{-}. \]

However, in the absence of prolonged heating this reaction proceeds slowly, and the concentration of the side product does not exceed 2.5%. Since the viscosity of the mixtures was measured immediately after their preparation, the side process may be neglected.

The calculation of the coefficients \(a_{23}\) and \(f\) consists in selecting such values of them as satisfy equality \((6,a)\) for different mixtures. Taking \(m=n=1\) for the given system, expression \((6,a)\) can be simplified to

\[ \eta'' = \frac{ a_{22}[f(1-2N_1)]^2 + 2a_{23}fN_1(1-2N_1) + a_{33}N_1^2 }{ [f(1-2N_1)+N_1]^2 }. \tag{6,b} \]

Solving the equation with respect to \(a_{23}\), we obtain

\[ a_{23}=Af+B-\frac{C}{f}, \tag{7} \]

where

\[ A=\frac{1}{2}(\eta'-a_{22})\left(\frac{1}{N_1}-2\right);\quad B=\eta'';\quad C=-\frac{(a_{33}-\eta'')}{2\left(\frac{1}{N_1}-2\right)f}. \]

Equation (7) is of the second degree with respect to \(f\), that is, in the general case each value of \(a_{23}\) corresponds to two values of \(f\). The coefficients \(a_{23}\) and \(f\) are determined graphically by a trial method from the viscosities, known from experiment, of mixtures of different composition. In doing so we assume that \(f \geqslant 1\) and represents integers, and that \(0<a_{23}<a_{33}\), since \(a_{23}\) is proportional to the size of the molecules. Choosing \(f=1, 2, 3\), etc., we determine \(a_{23}\) from equation (7).

On the basis of the calculated data of Table 2, a graph of the dependence \(a_{23}=\varphi(f)\) was constructed for various mixtures (Fig. 3). The true values

Table 2

Calculated values of \(f\) and \(a_{23}\)

of the coefficients \(a_{23}\) and \(f\) correspond to the coordinates of the point of intersection of the curves for different mixtures. It is evident from the graph that \(f \approx 1,8\) and \(a_{23}\approx 4,3\) (the curves for mixtures 3 and 4, probably because of inaccuracies in the viscosity measurements, do not pass through the point of intersection of the curves for the other three mixtures and therefore are not plotted on the graph).

Experimental data on the determination of the association coefficient for this system by another method are not known to us. However, for an analogous system (allyl mustard oil–aniline), it follows from the solubility isotherm in an indifferent solvent (naphthalene) that the degree of association for the chemical compound between allyl mustard oil and aniline is equal to two [5, p. 837].

Such agreement of the numerical value of the degree of association, obtained by two different methods for two analogous systems, may serve as sufficiently convincing proof of the correctness of the proposed method for calculating the degree of association from the viscosity of binary systems.

Choosing the mean value \(a_{23}=5,0\) with the integer value nearest to the point of intersection of the curves, \(f=2\), we calculate the left branch of the viscosity isotherm of the system by equation (6,б). The calculated and experimental

viscosity data for the mixtures are given in Fig. 2. As can be seen from the data presented, good agreement of the results is observed.

The calculated data confirm the experimental fact (for the example of hydrocarbons of one homologous series) that viscosity increases

Fig. 3. Graphical determination of the coefficients of equation (7) \(a_{23}\) and \(f\) for the system o-anisidine–allyl mustard oil at \(70^\circ\): curves 1, 2, and 3 at \(N_1\), respectively, 0.1; 0.4; 0.48.

several times faster than the molecular weight when the molecules become more complex. In the given system

\[ \frac{\eta_{(AB)_2}}{\eta_B} : \frac{M_{(AB)_2}}{M_B} = \frac{22.14}{1.14} : \frac{444.6}{123.12} = 5.4, \]

that is, the viscosity of the compound increases 5.4 times faster than the molecular weight when the molecule becomes more complex. After refinement, this rule may be used to determine the degree of association of a compound directly from the viscosity values of the components and of the compound.

The viscosity of equimolecular mixtures of other systems—secondary amines–mustard oils—exceeds the viscosity of the most viscous component by 200 times or more, and such systems are incapable of crystallization [7]. Thus, in the system diethylamine–phenyl mustard oil this ratio is equal to 26500.

These data confirm the ability of the components of the systems under consideration to form polymer-associated compounds.

On the basis of literature data [7], the quantities \(a_{23}\) and \(f\) were calculated for several other systems. Table 3 gives the results of the calculation.

The data given in Table 3 show that, at a high degree of association, the compound does not crystallize. On cooling the system, the degree of association increases. Thus, if in the first system at \(125^\circ\) \(f = 1\), and at \(100^\circ\) \(f = 1.4\), then at a lower temperature, in solubility experiments for this system in naphthalene [5, p. 837], the compound has the composition \((AB)_2\).

With a large difference between the viscosities of the components and of the compound (system 3), the compound does not crystallize.

Table 3

Calculated values of \(f\alpha_{23}\) in various systems

For binary systems with a crystallizing compound, from the data of Table 3 an empirical dependence of the degree of association of the compound on viscosity can be established in the form: \(f=\varphi(D)\), where

\[ D=\frac{\eta_{(AB)f}}{\eta_A+\eta_B}. \tag{8} \]

Figure 4 presents a plot of the dependence of \(f\) on \(D\) in logarithmic coordinates, reflecting the rectilinear dependence of \(\lg f\) on \(\lg D\). The empirical formula is

\[ f=0.4\left(\frac{\eta_{\max}}{\eta_A+\eta_B}\right)^{0.6}. \tag{9} \]

From the data of Table 3 it is seen that the fivefold excess of the maximum viscosity of the mixture over the sum of the viscosities of the components indicates association of the compound being formed.

If it is assumed that, in systems with an associated compound, additivity of the logarithm of viscosity is observed, and the association coefficient of the compound is calculated by the formula

\[ f=\frac{\lg a_{33}-\lg \eta''}{\lg \eta''-\lg a_{22}}\cdot \frac{1}{\left(\dfrac{1}{N_1}-2\right)}, \tag{10} \]

we obtain numerical values of \(f\) from 0.5 to 2, differing by nonconstancy for the given system (by up to a factor of 2), including for a system with a ratio of the viscosities of the components and the compound equal to \(1:26500\), which contradicts the experimental facts.

Fig. 4. Toward the derivation of the empirical formula \(f=\varphi(D)\) for the system secondary amines—mustard oils

In conclusion I express my sincere gratitude to Prof. A. G. Stromberg for valuable advice in discussing the results of the work.

Conclusions

1. On the basis of Stromberg’s theory of the viscosity of mixtures, an equation has been derived for the dependence of viscosity on composition for binary liquid systems in which a stable chemical compound is formed.

2. A graphical method is proposed for determining the degree of association of intermolecular compounds on the basis of the derived equation.

3. It is shown that in the allyl mustard oil-o-anisidine system the degree of association of the intermolecular compound at a temperature of 70° is 1.8; in the allyl mustard oil-aniline system, at temperatures of 100 and 125°, the degree of association is respectively 1.4 and 1.0.

4. An empirical dependence has been established for determining the degree of association of crystallizing intermolecular compounds from the viscosity data of the components and the compound in binary systems—allyl mustard oil-secondary amines.

1. It has been shown that the dependence of the viscosity on the composition of ideal systems, expressed by the empirical Arrhenius formula, can be obtained as a special case from the theoretical Stromberg equation for the viscosity of binary systems at a certain value of the coefficient of “mutual” viscosity.

REFERENCES

1. D. A. Pospekhov. On the additivity of the logarithm of viscosity in binary liquid systems. ZhFKh, 30, 1, 228, 1956.

2. M. I. Usanovich. On the viscosity of liquid mixtures. Izv. AN Kaz. SSR, chemical series, no. 10, 30, 1956.

3. A. B. Zdanovskii. Regularities of viscosity in the mixing of liquids. ZhFKh, 29, 2, 209, 1955.

4. A. G. Stromberg. Toward a theory of the viscosity of binary liquid mixtures. ZhFKh, 15, 2, 205, 1941.

5. V. Ya. Anosov and S. A. Pogodin. Basic Principles of Physicochemical Analysis. Publishing House of the Academy of Sciences of the USSR, Moscow, 1947.

6. S. P. Miskidzh’yan. Electrolytic dissociation in nonaqueous systems. ZhFKh, 34, 4, 802, 1960.

7. N. S. Kurnakov. Collected Selected Works, vol. I, ONTI, Leningrad, 1938.