CO M M UN I CA,TI ONS
COMPOSITION DEPENDENCE OF T H E VISCOSITY OF BINARY LIQUID SYSTEMS The statistical mechanical theory of transport in liquid mixtures is used to develop a predictive relationship for the variation of the shear viscosity of binary liquid systems based on previous work on the binary diffusion Coefficient. The relationship is expected to be valid for systems which are approximately regular in the thermodynamic sense. Comparison with experimental data indicates excellent agreement for a variety of binary systems.
s
progress has recently been made on the problem of the variation of liquid-phase binary diffusion coefficients with composition. Vignes (1966) has had remarkable success in correlating the concentration dependence of the binary diffusion coefficient with thermodynamic properties for a wide variety of nonassociating liquid systems. Cullinan (1 966) has pointed out the theoretical implications of the Vignes expression by deriving this result from absolute rate theory. T h e theoretical foundations of the predictive relationship (which gives the binary diffusion coefficient at any point in the composition field in terms of a thermodynamic factor and the diffusion coefficient at the two composition extremes) coupled with the similarities of cliffusion and viscous flow from the viewpoint of absolute rate theory imply the existence of similar relationships for the viscosity of binary liquid systems. T h e work of Vignes (1966) and Cullinan (1966) has shown that, for nonassociated liquid systems, the activation energy for diffusion varies linearly with mole fraction between the infinitely dilute extremes. I n view of the similarities mentioned above, the activation energy for viscous flow might be expected to vary in a siinilar linear manner. However, this is not the case. I n fact, ISyring (Glasstone et al., 1941) has shown that the activation energy for viscous flow exhibits a departure from linearity in mole fraction roughly in proportion to the excess free energy of mixing. I n other words, even though the processes of diffusion and viscous flow are similar and the diffusion coefficient and viscosity are interrelated, this relationship depends on the thermodynamic behavior of the system at hand. This is not surprising, as the diffusion process comes about by reason of a thermodynamic driving force (a chemical potential gradient), whereas viscous flow occurs by reason of a mechanical driving force (shear stress). The statistical mechanical theory of transport in liquid systems (Bearman, 1961) does provide a direct relationship between the diffusion coefficient and viscosity for a class of solutions roughly equivalent to the thermodynamic regular solution classification (excess entropy change on mixing is zero). This relationship may bie combined directly with the results of Cullinan (1966) and Vignes (1966) to yield a prediction of the viscosity of a binary liquid mixture in terms of the viscosities of the pure components. I n this paper this relationship is derived and tested with data for some typical systems. IGNIFICANT
Theory
With associated systems the apparent sole exception, the binary mutual diffusion coefficient is given by (Vignes, 1966)
This relationship is consistent with an activation energy which varies linearly with mole fraction between the two composition extremes (Cullinan, 1966). A more restrictive and generally less useful relationship is given by Bearman (1961) for solutions which exhibit no volume change on mixing and whose radial distribution functions are independent of composition:
where for this class of solutions the self-diffusion coefficients are related by
DiVi
=
DzVz
(3)
Equation 2 is far more restrictive than Equation 1. O n the basis of limited experimental evidence (Vignes, 1966), Equation 2 is closely followed by regular solutions, but wide deviations occur in strongly nonideal systems such as acetone-carbon tetrachloride, which still obey Equation 1. If attention is restricted to those systems for which Equations l and 2 are equally valid (presumably, those systems for which Equation 2 is valid), then Equations 1, 2 , and 3 may be combined to give
(5) is the arithmetic mean molar volume (the reciprocal of the actual molar concentration as the volumes are presumably additive). VOL. 7
NO. 1
FEBRUARY 1968
177
According to the statistical mechanical theory (Bearman, 1966) the product of viscosity and self-diffusion coefficient is constant for a given system, so
Combination of Equations 4 and 6 yields the following expression for the mixture viscosity: B =
VAq2
(rdx1
(7)
v2
Evaluation of Equation 7 at the limit x1 = 1 shows that
Finally, substitution of Equation 8 back into Equation 7 gives the desired predictive equation:
is the geometric mean molar volume. Comparison of Equation 9 with Equation 1 shows that the (d In ?I/ thermodynamic correction factor for diffusion, 1 d In X I ) , is analogous to the viscosity correction factor, V A / V G . Because the arithmetic mean always exceeds the geometric mean, this correction factor is always greater than unity. Under this restriction it can be shown that any extremum of Equation 9 between the composition extremes must be a maximum. Thus the theory rules out any minima for the viscosity of the binary systems under consideration. I t is interesting to compare these results with the basic theory of Eyring (Glasstone et al., 1941). T h e fundamental equation for the viscosity of a binary mixture with additive volumes is (Glasstone et al., 1941)
+
Test of Theory
T h e system, benzene-carbon tetrachloride, is a typical thermodynamically regular system. The viscosity has been measured over the composition range at 25’ C. by Grunberg (1954). Density data of Cullinan (1965) were used with the pure component viscosities to give the predicted curve in Figure 1. The agreement in this case is excellent. This is not surprising, since it has been demonstrated (Vignes, 1966) that this system follows both Equation 1 and Equation 2. Both chlorobenzene and bromobenzene form essentially ideal binary systems with benzene. I n Figure 2 the predicted curve for benzene-chlorobenzene is compared with the experimental values of Linke (1941) using density data of Poltz (1936). There is some scatter in the data because of temperature variations, but the agreement is very good. I n Figure 3, the predicted curve for benzene-bromobenzene is presented with the data of Yajnik et al. (1925), using the density data of Martin and Collie (1933). I n this case, the predicted curve is about 5% below the experimental points. Figures 4, 5, and 6 present similar comparisons for the systems, heptane-2,2,4-trimethylpentane, octane-2,2,4trimethylpentane, and diisopropyl-diallyl (Toropov et al., 1955). The agreement is generally good. The excellent agreement with these thermodynamically well behaved systems should not be taken to imply universal agreement with Equation 9 for all nearly ideal and regular systemsfor example, the viscosity of the regular system, cyclohexanebenzene (Timmermans, 1959), goes through a minimum in the composition field. As mentioned previously, Equation 9 cannot predict a minimum. T h e implication is that although this system is regular in the thermodynamic sense, it does not satisfy the requirements of the “regular” classification of Bearman (1961). On the other hand, agreement is still fairly good for a strongly nonideal system such as acetone-benzene (Fischler, 1913), as indicated in Figure 7 .
‘“1
PREDICTED
0
EQ. 9
G R U N B E R G (1954)
0.9 W
If the activation energy for viscous flow, AG, is assumed to vary linearly with mole fraction between the composition extremes, then an expression similar to Equation 9 is obtained but with the volume ratio on the right side inverted. If the right sides of Equations 9 and 11 are equated, the following mixing rule is indicated:
AG = xlAG1
+ x ~ A G+~ 2 RT In VQ V.4
(12)
where the last term on the right represents the departure from the linear mixing rule. Since Eyring (Glasstone et al., 1941) empirically found this departure to be proportional to the excess free energy of mixing, this present result might then be used to estimate the heats of solution for the class of mixtures under consideration. The analogy between Equation 1 and Equation 9 should not be overemphasized, because Equation 1 is a more general result. Since the exact macroscopic classsification of solutions which follow the theory of Bearman (1961) is not clear, a sampling of systems for comparison with Equation 9 is in order. 178
l&EC FUNDAMENTALS
2 0 P
1
0.8 -
I2 W V
0.7
>
t rn
0
u 2 >
i
1 c
0 . 6 v
1 0.5
-
0.4
-
F
-
0.3 0
0.2 Xc,
MOLE
0.4 0.6 0.8 FRACTION CCg4
1.0
Figure 1. Viscosity of benzene-carbon tetrachloride at 25’ C.
0.85 -
-0
-PREDICTED
PREDICTED EQ.9 LINKE
T=20°C
A 0
T=400C T=60°C
0.50
w
5
0.80
I
TOROPOV ET AL.(1955
w
-
v,
0
0
I-
I-
&
5
z W V
z w u
i
>
.
0.45
0.40
c
t
$
EQ.9
0.75 -
v)
0
0
5 >
-
6
6
0.35
> 0.30
0.70 0.25 0.4 0.6 0.8 1.0 X c , MOLE FRACTION CHLOROBENZENE
0.6
0.8
0.2
X,,
MOLE FRACTION HEPTANE
0.2
0
0.4
0
Figure 4. penta ne
Viscosity
1.0
of heptane-2,2,4-trimethyI-
Figure 2. Viscosity of benzene-chlorobenzene at 19.7" i 0.2" C.
.I .o
0
PREDICTED EQ.9 YAJNIK ET AL.(1925)
- PREDICTED
J
EQ.9
T=60°C
0.9
0.55
5
0
Ez
W
0.8
v)
5 0.50
w
a_
0
i
I-
0.7
z w
*
k v)
i
0.6
c 8
II >
6
0.45
v)
0.5
0.40
I?
>
1
0.4
I
6
0.35
0.3
0.2 0.4 0.6 0.8 1.0 Xg, MOLE FRACTION BROMOBENZENE Figure 3.
30" C.
Viscosity of benzene-bromobenzene
0.30 0
0.2
0.4
0.6
0.8
1.0
Xo, MOLE FRACTION OCTANE
at Figure 5.
Vicosity of octane-2,2,4-trimethylpentane VOL. 7
NO. 1 F E B R U A R Y 1 9 6 8
179
- PREDICTED
Conclusions
EQ.9
T h e predictive relationship presented here is valid for binary mixtures which fall into the “regular” classification of Bearman (1961). Unfortunately, this classification apparently does not correspond precisely with any convenient thermodynamic classification, although it probably constitutes a class only slightly different from the class with ideal entropy of mixing. The corresponding theory for diffusion (Cullinan, 1966) gives an equation for the binary friction coefficient which is similar to Equation 9 with the viscosities replaced by the appropriate friction coefficients. Thus, the ratio of viscosity to friction coefficient varies geometrically with mole fraction for the present class of solutions. This must have some significance, since the friction coefficient is essentially a proportionality factor between a thermodynamic force and flux, while viscosity is a proportionality between corresponding mechanical quantities.
0.55
0.50
0.45
, 0.40
0.35 Nomenclature
binary mutual diffusion coefficient of system 1-2 limiting value of D~~ when species i is infinitely dilute self-diffusion coefficient of species i = activation energy for viscous flow in binary mixture = activation energy for viscous flow in pure species i = Planck constant = Avogadro’s number = ratio of limiting diffusion coefficients = gas constant = absolute temperature = arithmetic mean molar volume = geometric mean molar volume = partial molar volume of species i = mole fraction of species i = activity coefficient of species i = viscosity of binary mixture = viscosity of pure species i
= = =
0.30
0.25
0.2
0
0.6
0.8
1.0
FRACTION DI ISOPROPY L
X D , MOLE Figure 6.
0.4
Viscosity of diisopropyl-diallyl
- PREDICTED
0.60
0
T VA VC vi
EQ. 9 FISCHLER (1913)
Xi
Yi ll lli
0.55 Literature Cited
Bearman, R. J., J. Phys. Chem. 65,1961 (1961). 5,281 (1966). Cullinan, H. T.,IND.ENC.CHEWFUNDAMENTALS Cullinan, H. T., Ph.D. thesis, Carnegie Institute of Technology, 1965. Fischler, J., Z. Elektrochem. 19, 126 (1913). Glasstone, S. K.,Laidler, K. J., Eyring, H., “Theory of Rate Processes,” McGraw-Hill, New York, 1941. 50, 1293 (1954). Grunberg, L., Trans. Faraday SOC. Linke, R., Z. Phys. Chem.A188,17 191 (1941). Martin, A.R., Collie, B., J . Chem. SOC. London 1933,1413. Poltz, H., Z. Phys. Chem. B32,243 (1936). Timmermans, J., “Physico-Chemical Constants of Binary Systems in Concentrated Solutions,” Vol. I, Interscience, New York,
0.50
0.45
0.40
1959.
Toropov, N. A., Airapetova, R. P., Kiryukhin, V . K., Zh. Obshch. Khim. 28 (87), 1315 (1955). Vignes, A., IND.END.CHEM.FUNDAMENTALS 5,189 (1966). Yainik. N. A.,Bhalla, M. D., Talwar, R. C., Soofi, A,, Z . Phys. 6h:hem’.A118,’305 (1925).
0.35
HARRY T. CULLINAN, JR.
0.30 0 X,, Figure 7.
180
0.2
0.4
0.6 0.8 1.0 MOLE FRACTION ACETONE
Viscosity of acetone-benzene at
I I E C FUNDAMENTALS
25’ C.
Department of Chemical Engineering State University of New York at Bufalo Buffalo, N . Y . 14274
RECEIVED for review May 22, 1967 ACCEPTED September 5, 1967
