ON THE DEPENDENCE OF THE VISCOSITY OF SOLUTIONS ON THE MOLECULAR WEIGHT OF THE DISSOLVED SUBSTANCE

V. S. SMORODINOV

(Presented by the scientific-methodological seminar of the Chemical-Technological Faculty)

Usually, in equations for the dependence of viscosity on the composition of a solution, concentration is expressed in mole fractions \[1, 2\]. When the latter are expressed in weight units, a dependence of the viscosity of the solution on the molecular weight of the components of the solution is established. This circumstance may be used for the theoretical development of a method for determining the molecular weight of a dissolved substance on the basis of viscosity measurements of solutions of various classes, for which the laws governing viscosity are known. Such calculations for ideal mixtures of liquids are based on the law of additivity of the logarithm of viscosity \[3\]. In the present work, some regularities of the viscosity of regular solutions are considered.

The dependence of viscosity on the composition of a binary regular mixture of liquids is expressed by the equation \[2\]:

\[ \eta = \eta_1 N_1^2 + \eta_2 N_2^2 + 2a_{12}N_1N_2, \tag{1} \]

where \(\eta_1\), \(\eta_2\) and \(N_1\), \(N_2\) are, respectively, the viscosity and mole fractions in the solution of components 1 and 2; \(a_{12}\) is the coefficient of “mutual” viscosity, independent of the composition of the solution. From equation (1) we obtain:

\[ a_{12}=\frac{\eta-\eta_1N_1^2-\eta_2N_2^2}{2N_1N_2}. \tag{1a} \]

Let us consider on the viscosity isotherm two points satisfying the condition: \(N_2' + N_2'' = 1\) (Fig. 1). Taking into account that for such two points the relation \(N_1'N_2' = N_1''N_2''\) is valid, from equations (1a) for the two mixtures, by eliminating \(a_{12}\), the expressions can be obtained:

\[ \frac{\eta''-\eta'}{N_2''-N_2'}=\eta_2-\eta_1=\tg\alpha; \tag{2} \]

\[ \eta''-\left[\eta_1+N_2''(\eta_2-\eta_1)\right] = \eta'-\left[\eta_1+N_2'(\eta_2-\eta_1)\right]; \quad \Delta\eta_{\mathrm{add}}''=\Delta\eta_{\mathrm{add}}', \]

\[ \Delta\eta_{\mathrm{add}}=N_1N_2\left[2a_{12}-(\eta_1+\eta_2)\right]. \]

Hence, for regular solutions the following conclusions may be drawn:

1. The isotherm \(\Delta\eta_{\mathrm{add}} - N_2\) is symmetric, i.e., an extremum is observed at the stoichiometric composition.

2. With a significant deviation from additivity, an extremum is observed on the viscosity isotherm; moreover, the maximum is shifted toward the more viscous component, and the minimum toward the less viscous component.

To determine the molecular weight \(M_2\) of a dissolved liquid from the viscosity of its solutions, it is necessary to select such a value of the coefficient \(a_{12}\) that satisfies equality (1) for various mixtures. As an initial estimate we choose values of \(a_{12}\) close to the conditions of an ideal mixture [4]:

\[ Q_{12}^{\mathrm{id}} = 2\sqrt{\eta_1\eta_2} - \frac{\eta_1+\eta_2}{2} \tag{3} \]

and the corresponding values of \(N_2\) from equation (1) and \(M_2\) from the formula:

\[ M_2 = M_1 \frac{\left(\dfrac{1}{N_2}-1\right)} {\left(\dfrac{100}{g_2}-1\right)}, \]

where \(g_2\) is the weight percent of component 2. The true values of \(M_2\) and \(a_{12}\) correspond to the coordinates of the point of intersection of the curves \(a_{12} - M_2\) for different mixtures.

The possibility of such a calculation is shown for the system benzene–carbon tetrachloride, which, on the basis of the properties of its components, may be classed as regular solutions [5].

Fig. 1. Viscosity isotherm of a regular system.

Fig. 2. Graphical determination of the coefficient \(a_{12}\) and \(M_2\) for the system \(\mathrm{C_6H_6}(1) — \mathrm{CCl_4}(2)\) from the viscosity of solutions at \(20^\circ\). Curves 1 and 2 correspond to \(g_2\) of 38.9 and 64.1%, respectively.

For this system \(a_{12}^{\mathrm{id}} = 0.757\) cP. The results of the calculation are presented in Table 1 and in Fig. 2.

As is seen in Fig. 2, the calculated molecular weight \(M_2\) is equal to 154 and agrees with reference data, which confirms the unambiguous dependence of the viscosity of regular solutions on the molecular weight of the dissolved liquid and the adopted structure of regular solutions [2].

If one of the components of the solution is a solid substance with unknown molecular weight \(M_2\), then in equation (1) there proves to be

unknowns: three coefficients—$M_2$, $a_{12}$, $\eta_2$ (the viscosity of the supercooled liquid), and the calculation becomes more complicated.

The simplicity of the method for determining viscosity has led to the appearance of a method for determining the molecular weight of polymers from the viscosity of their dilute solutions. In this, empirical equations are used which express the dependence of the characteristic viscosity on molecular weight and which have been established with the aid of some absolute method for determining molecular weights. The empirical relation for a polymer of a given structure is derived for the given solvent and temperature.

Table 1

Determination of the molecular weight of carbon tetrachloride from the viscosity of its benzene solutions

The viscosity of polymer solutions depends not only on the concentration of the solutions and the molecular weight, but also on the character of the interaction of the solvent with the polymer and on the structure of the polymer, i.e., on the factors determining the volume occupied by the macromolecule in solution.

The quantitative influence of these factors on viscosity has not been established. Therefore viscosity cannot be used as an absolute method for measuring molecular weights.

In the present work an attempt is made to apply the regularities of the viscosity of regular solutions to athermal systems, in which the orientation of molecules is insignificant. To determine the molecular weight of polymers, the viscosity of dilute solutions is usually used, in which the interaction of the chains with one another is weakened. In the calculation the following assumptions are adopted. All polymer molecules are flexible and, in dilute solutions, coil up into coils washed by the solvent; solvate shells of solvent are absent [6, p. 420]. The effect of long-range intramolecular interaction (interaction between units located far from one another along the chain but brought close together as a result of coiling of the chain) on the volume of the molecular coil is not taken into account.

For dilute polymer solutions, equation (1) can be transformed by setting $N_2=\dfrac{C_2M_1}{d_1M_2}$ and $\eta_2 \gg a_{12} \gg \eta_1$,

\[ \eta=\eta_1+2a_{12}\left(\frac{C_2M_1}{d_1M_2}\right)+\eta_2\left(\frac{C_2M_1}{d_1M_2}\right)^2, \]

where $d_1$ is the density of the solvent, g/ml; $C_2$ is the concentration of the solution, g/ml.

The expression for the reduced viscosity takes the form:

\[ \eta_{\mathrm{red}.}=\frac{\eta-\eta_1}{\eta_1 C_2} = \frac{2a_{12}}{\eta_1}\left(\frac{M_1}{d_1M_2}\right) + \frac{\eta_2}{\eta_1}\left(\frac{M_1}{d_1M_2}\right)^2 C_2 \tag{4a} \]

which coincides in form with the empirical equation [6, p. 419]

\[ \eta_{\mathrm{red.}}=[\eta]+kC_2, \tag{46} \]

where \([\eta]\) is the intrinsic viscosity, depending on the nature of the solvent, the molecular weight of the polymer, and its interaction with the solvent; \(k\) is a constant depending on the nature of the solvent and the molecular weight of the polymer and not depending on its interaction with the solvent. From equation (4a) it follows that determination of the three unknown coefficients—\(M_2\), \(a_{12}\), \(\eta_2\)—from the experimentally known \([\eta]\) and \(k\) is impossible even for simpler systems, and it is not possible to express \(a_{12}\) through \(\eta_1\) and \(\eta_2\), as in the case of ideal systems. However, for determining \(M_2\) it is more convenient to use the coefficient \(k\) than \([\eta]\), since for polymers the empirical dependence is known [6, p. 213]:

\[ \eta_2=BM_2^{3.4}, \tag{5} \]

where \(B\) is a constant quantity that is a function of temperature. This dependence is valid over a broad range of molecular weights (from 530 to \(1.5\cdot10^5\)). The expression for the coefficient \(k\), taking equation (5) into account, assumes the form:

\[ k=B\left[\frac{1}{\eta_1}\left(\frac{M_1}{d_1}\right)^2\right]M_2^{1.4}\sim M_2^{1.4}. \tag{6} \]

This circumstance may be used in determining \(M_2\) from \(k\), depending on whether the constancy of the coefficient \(B\) is observed for polymers of different molecular weights and also in different solvents. To determine the coefficient \(B\), one experimental value of \(k\) is sufficient for any one fraction of polymer whose molecular weight has been determined by one of the absolute methods:

\[ B=\left[\eta_1\left(\frac{d_1}{M_1}\right)^2\right]\frac{k}{M_2^{1.4}}. \tag{6,a} \]

The verification of this relation is made difficult by the fact that most authors do not give experimental values of the coefficients \(k\) in their works, but only values of \([\eta]\). However, calculation shows that even in nonathermal solutions the coefficient \(B\) remains comparatively constant. Thus, for solutions of ethyl cellulose in acetone and ethyl alcohol [6, p. 426], the ratio of the coefficients \(B\) is equal to 1.13.

Fig. 3. Dependence of the reduced viscosity of dilute solutions on the concentration of polymers of different molecular weight (\(M_1<M_2<M_3<M_4<M_5\)).

From equation (4a) it follows that the intrinsic viscosity \([\eta]\) is greater in good solvents (\(a_{12}\) is large) than in poor ones, which agrees with experimental data [6, p. 426]. Equation (4a) can be transformed into the form:

\[ \eta_{\mathrm{red.}}=[\eta]+k'[\eta]^2C_2, \tag{7} \]

where

\[ \kappa'=\frac{\eta_2^2\eta_1}{4\omega_{12}^2} \quad \text{and} \quad \kappa=\kappa'[\eta]^2, \]

i.e., the tangent of the angle of inclination (Fig. 3) is proportional to the square of the characteristic viscosity. In a number of works [6, p. 427] it has been shown that the value of \(\kappa'\) is greater in poor solvents and smaller in good ones, which follows from the relations obtained.

A hydrodynamic calculation [6, p. 423] shows that, for coils streamed around by the solvent, \([\eta]\sim M_2^{0.5}\); then \(a_{12}\sim M_2^{1.5}\) and \(\kappa'\sim M_2^{0.4}\). In most cases the equation \([\eta]=K M_2^{\alpha}\) is valid, where \(\alpha=0.6\div0.8\), and \(a_{12}\sim M_2^{1+\alpha}\), or approximately \(a_{12}\sim\sqrt{\eta_2}\), i.e., \(\kappa'\) does not depend on \(M_2\) [7].

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

Conclusions

1. On the basis of A. G. Stromberg’s theory of the viscosity of mixtures, it has been shown that a criterion of a regular system is the symmetry of the diagram of deviations of the viscosity of mixtures from additivity with respect to the line of equimolecular composition.

2. A unique dependence has been established of the viscosity isotherm of regular solutions on the molecular weight of the dissolved liquid.

3. On the basis of the theory of the viscosity of regular solutions, an equation has been obtained for the dependence of the reduced viscosity of dilute athermal solutions on the concentration of a flexible polymer; with its aid, certain empirical regularities have been explained.

4. It has been suggested that the angular coefficient of the reduced viscosity is a more convenient criterion in determining the molecular weight of polymers than the characteristic viscosity, since this coefficient does not depend on the interaction of the polymer with the solvent.

References

1. S. Arrhenius. Z. phys. Chem., 1, 285, 1887.
2. A. G. Stromberg. ZhFKh, 15, 2205, 1941.
3. M. I. Usanovich. Izvestiya AN Kaz. SSR, ser. khim., no. 10, 30, 1956.
4. V. S. Smorodinov. Izvestiya Tomskogo politekhnicheskogo in-ta, 128, 80, 1964.
5. T. E. Thorpe and I. W. Rodger. J. Chem. Soc., 71, 330, 1897.
6. A. A. Tager. Physical Chemistry of Polymers, Goskhimizdat, Moscow, 1963.
7. Qian Ren-Yuan. Determination of the Molecular Weights of Polymers, Izdatinlit, p. 54, 1962.