VISCOSITY O F SOLUTIONS
25
the order alcohol, dioxane, benzene, ether, and acetone. The general problem of stability of sols in non-aqueous media appears to be very interesting and should merit further investigation. It is hoped that more work may be done in this laboratory to determine changes and precipitation characteristics. SUMMARY
1. Suspensions of several metals in ether, dioxane, and acetone have been prepared by reduction methods. 2. Colloidal suspensions of metallic sulfides in the same solvents have been prepared by double decomposition reaction with hydrogen sulfide. 3. As reported in previous work on methyl alcohol and benzene, protective materials were necessary to ensure stability. Cellulose nitrate was used with dioxane and acetone, while rubber was the best agent with ether. 4. A comparison with results published in a previous paper shows that the suspensions decrease in stability in the order alcohol, dioxane, benzene, ether, acetone. REFERENCE (1) VERNON,ARTHUR.4.,AND NELSON,HARRISON A , : J. Phys. Chem. 44, 12 (1940).
T H E INFLUEKCE OF BROWNIAN MOVEMEKT ON T H E VISCOSITY OF SOLUTIOPTS'
R. SIMHA Department of Chemistry, Columbia L'niversity, New York c i t y , New Y O T ~ Received M a y 1.9, 1999 INTRODUCTION
The study of the internal friction of pure liquids is of great technical as well as scientific interest. From the latter point of view it has become especially important, since a more intensive study of the liquid state has been undertaken. The viscosity and its dependence on other quantities have been shown to furnish substantial information about the structure of the medium considered. No less important is the study of the viscosity of solutions, especially in the study of the high polymers. Besides technical problems this Presented a t the Ninety-seventh Meeting of the American Chemical Society, held a t Baltimore, Maryland, April, 1939.
26
R. BIMHA
question arose: On the basis of the viscosity of a solution, to what extent is it possible to draw quantitative conclusions as to the shape and size of particles suspended therein? Although it is not yet possible to give a comprehensive answer, much progress has been made both by experiment and theory. A detailed representation of these problems is found elsewhere (7, 15; also the last paper mentioned in reference 3). Einstein (2) was the first to calculate the viscosity of a suspension of spheres on the basis of the phenomenological hydrodynamic equations. Here a necessary assumption is that the suspended particles are la.rgc compared with the molecules of the solvent. It is further assumed that no gliding between solid and liquid occurs. Jeffery (11) treated the same problem for ellipsoids. Both authors restrict themselves to dilute solutions and rigid particles and obtain proportionality between the specific viscosity and the concentration, c: P ! ! = PO
VC
(110 is the viscosity of the solvent; p is the viscosity of the solution.) The factor v has the value 2.5 for spheres and is a function of the axial ratio for ellipsoids. We shall refer to this paper. Recently Eirich, Guth, Simha, and coworkers (3,7) tried to extend the theory and a t the same time to investigate the validity of the underlying assumptions by experiments on models of known characteristics.
THE PURELY HYDRODYNAMIC CABE
Before considering the influence of Brownian movement on the behavior of suspended particles, such of Jeffery's results as are used in this paper may be briefly discussed. He calculates the motion of an ellipsoid suspended in a liquid with a velocity uI0).' ujo)is composed by a rotation and a distortion, as a pure translation of the fluid is not disturbed by the presence of the ellipsoid, if inertia forces are neglected. One must therefore find a solution of Stokes' equations -OVzui
?
= axi
which gives at infinite distance from the particle u'3 -- ujo)= ai&, f
[kxl
- [lxk
* In the following i, k, E follow from each other by cyclic permutation; summations are performed only on twice occurring indices r , 3.
27
VISCOSITY OF SOLUTIONS
(aik and [ k are constant components of distortion and rotation, respectively.) The x k refer to axes h e d in the particle, which will perform a rotation with the angular velocitieswd, so that on the surface of the ellipsoid
ui = W E X l
- WZXk
The resultant couple acting on the particle must vanish, as inertia forces were neglected. This gives the values of w i . Finally Jeffery considers the special case of a motion between two flat plates, where the one moves uniformly against the other, as approximately realized in a Couette apparatus: t
U1
=
/ U2
= 0,
U: = KX:
The primed quantities refer to a system fixed in space. On introducing the Eulerian angles 9, 6,$, one obtains for an ellipsoid of revolution with thelsemi-axesful and a2
=)
( B = a1 - a2 a2
+ a:
the equations of motion of the immersed particles:
} (2)
and therefore
For t = 0, 9 = 0. k is an integration constant determined by the initial value of 6 which, however, is unknown. It is a measure for the angle of the cone described by the axis of revolution about the normal to the plane of the undisturbed motion. Every k value is consistent with the equations 1. At the present state of the theory, where inertia and the non-stationary terms in the hydrodynamic equations are neglected, it is not possible to eliminate this undeterminateness without additional assumptions. This lack is felt not only in problems of double refraction of flow, but also here, as k enters in the expression for the dissipated energy and therefore determines the viscosity. For the increase of energy dissipated per unit time and due to the presence of the particle, one obtains
+-cos2$ a:a:
2 sin26 +
a:(a::
+ a:)
28
R. SIMHA
In equation 3 8 has to be expressed by 9 and k with the aid of equation 2 ; the ai, a:’ are functions of al and a2 defined in Jeffery’s paper. dW/dt is a function of the time t. The average value is found by computing the integral )
Jeffery assumes that the particles are orientated in a manner which corresponds to a minimum of dW/dt. This gives for prolate ellipsoids k = Q),6 = 0, i.e., the axis of revolution is normal to the plane of flow, ?r
corresponds to a maximum, and vice versa for oblate 2 spheroids. Experiments on models show that long particles strike into the plane of flow with increasing axial ratio and there orientate themselves parallel to the direction of flow in contradistinction to the minimum h y p ~ t h e s i s . ~Using the method of small vibrations, the author (18) was able to show that the more elongated the particles, the more stable becomes the position in the plane of flow. whereas k = 0,6 =
-
PARTICLES WITH BROWNIAN MOVEMENT
With decreasing particle size and velocity gradient and increasing temperature, the influence of the rotatory Brownian movement must be considered. The heat motion of thesolvent moleculesby means of collisions with the solute particles tends to bring about an isotropic angular distribution of the ellipsoids. The ratio KID(D = rotational diffusion constant of the spheroids) is a measure for this influence, which shows a certain analogy to the orientation of dipoles in an external field. Also in this case the distribution function F(p) of all possible directions is determined by a generalized diffusion equation. With the aid of F the average value dW/dt can be calculated. In the purely hydrodynamic case characterized 1 by + m the distribution function is proportional to - as is evident from (b’
equation 3a. We shall not go here into details concerning this problem of so-called “partial Brownian movement.” For a critical discussion see references 3 and 7. The p - 8 coupling and therefore the undeterminateness mentioned before persists so long as there is any orientation tendency a J. M. Burgers in the Second Report on Viscosity and Plasticity (Amsterdam, 1938) remarks that the orientation for long particles parallel to the undisturbed velocity gives practically the same value for the viscosity as Jeffery’s minimum value. It is, however, much smaller than that found in the experiments mentioned. This fact seems t o be due essentially t o concentration effects (cf. page 33).
VISCOSITY O F SOLUTIONS
28
effective in the flow. This is no longer the case for complete Brownian movement. Here the gradient K is so small and the thermal motion of the particles is so intensive that the ratio KIDtends toward zero. The macroscopic flow is no longer able to bring about a regular rotation of the ellipsoids. Their axes on the average are uniformly distributed through the liquid. At the same time, however, a change must be made in expressing the dissipated energy as a function of the orientation. The hitherto used formula 3 was derived with the aid of the equations of motion 2 . It therefore implicitly involves a coupling of the angles cp and 6. Xoiv if the particles have no regular motion within the plane of flow,4 then the same must hold for the motion referring to 6, which would in any case induce an orientation tendency. Therefore no mutual dependence of the angles which described the gyroscopical motion of the ellipsoids in the hydrodynamical case exists. In this way two modifications in the former considerations occur: (a) The problem to be dealt with depends on two independent variables cp and 9. The distribution function F therefore equals N / 4 r ( N = number of suspended ellipsoids). (6) One must assume vanishing spins w i and calculate the energy increase due to ellipsoids a t rest, orientated under all possible angles 6, cp against the direction of the undisturbed flow. For this purpose we can use Jeffery's solution of equations 1 in inserting W , = 0 in his formulas 22, 24, and 26. In order to calculate dW/dt one describes as usual a sphere of finite radius R around the particle, but so that R 9 a l , a2. Then the rate a t which work is done by the hydrodynamical stresses on the surface of the sphere is computed. As explained in Jeffery's paper, one considers for this purpose a motion which assumes on the boundary of the sphere the velocities uj0)of the undisturbed motion. As R is finite, to the solution of equation 1 used hitherto a second motion has to be added, so that the resultant solution fulfills the condition mentioned before. The calculation gives:
* This motion tends to drive elongated ellipsoids in the direction of flow (q = ~ / 2 ) , as in this position 8 reaches its minimum.
30
R . SIMHA
The rate of work done is given by the integral
IIu!O’p,d S extended over the surface of the sphere. Here the p , are the components of the stress acting normal to the surface:
and
Inserting from equation 4 one obtains for the heat increase due to an ellipsoid of revolution a2 = a3, dW dt
2 - 16m.4( ala11 2
3
I
l I +
2azala2
+ a& + 2a& 2a: a:
+
2(& a G
+ aid
+
a$
+ - &(a: - d)(a13[2 - alZt3) + [&(a: + a:) + 2ayl(a:2 a:[sa:a:a: + (a: + a:)a;], d 3 )
The terms in the first line are Jeffery’s. ABexpected, an increase of energy results from the assumption that the particles do not follow the motion of the fluid. For the Couette streaming one obtains:
[=]
where dw
is given by equation 3. Now as the average value
Jefi.
E = E [‘/’‘ dt 47r o o
(6,(0) sin 9 d9 drp
31
VISCOSITY OF SOLUTIONS
In the limit f % 1one gets : Rods: Disks:
f2
jz
v
=
v
16 _ f_ =15 tang-’ f
15(log 2f
- 9)+ 5(log 2f - 3)
+-, 14 15
j,a-! (6d
DISCUSSION
The first part in the first line of expression 6a was derived by Eisenschitz (8) in starting from the energy formula 3 for the case without heat motion. From our assumptions results a rather large increase in the calculated viscosity. The approximately quadratic dependence on the axial ratio f for rigid rods, and only on f, however, is preserved. Whatever assumptions may be made concerning the orientation in the purely hydrodynamic case, one obtains always approximate proportionality with the first power off for rods. The disorientation tendency of the thermal agitation therefore increases the viscosity in the solution as compared with the viscosity without Brownian movement, as expected. These results are valid for every motion with constant components of rotation and’distortion, as was shown by the author (19). Although only proved in using expression 3, one can assume that the result will continue to be valid in this case. However, the only streaming of that kind which has been produced is the laminar Couette flow previously considered. 5 Note added in proof: Polson (Kolloid-Z. 88, 51 (1939)) measured the viscosity of solutions of various proteins, whose molecules had axial ratios varying from about 3 to 10. Using an empirical equation Y
=
4.0
+ 0.098.P
and diffusion data, he obtained values for the molecular weights agreeing satisfactorily with those found by ultracentrifugal measurements, being slightly greater, while the theoretical expressions used always gave values that were too low. Our expressions 6 and 6a on the other hand give values for the molecular weights slightly lower than those resulting from the empirical formula, and in this way satisfying agreement with the experimental data.
32
R . SIMHA
I n many cases viscosity measurements were carried on in a capillary tube viscosimeter, so that our formulas should not be directly applied. Theory predicts (20) the same viscosity values for dilute solutions in both types of flow without Brownian movement influence and neglecting inertia. The latter effect presumably lowers the viscosity, according to experiments by Eirich and Goldschmid (4)and some preliminary calculations. When the heat motion enters into play in the solution, fundamental differences between the results obtained in the two cases are to be expected, as the distribution function F and thus also the dissipated energy will depend on the velocity gradient which varies from point to point in the tube. In approaching the limit of complete Brownian movement, however, one should get equal viscosity values for dilute solutions in Couette and Poiseuille, since inertia forces will be of much less importance for small particles than for the big ones in the hydrodynamic case. Results of a direct quantitative comparison of these considerations with experimental data do not appear to be very illuminating. Staudinger’s measurements (21), for instance, for polystyrene solutions combined with formula 6a would provide values for the chain length in fair agreement with those found by other methods. These results, however, do not differ enough from those found with the aid of Eisenschitz’s expression to make a decisive conclusion concerning the basic assumptions in the derivation of our viscosity formula. It appears interesting to evaluate Meyer and van der Wyk’s (16, 17) very exact measurements on paraffin solutions in carbon tetrachloride in the light of results furnished by the hydrodynamic theory. The values found for the factor v are in several cases lower than that calculated by Einstein for spheres, in disagreement with the theory, so that we must assume with Burgers (1) that the addition of the solute changes the solvent, or that the hydrodynamic treatment is quite inadequate for these cases. On the other hand, discrepancies between the theoretically calculated dependence of the viscosity increase on the dimensions of the solute particles and that deduced by Staudinger from his measurements clearly show that flexibility of the chains must be taken into account.6 This effect not only changes the relation between molecular weight and chain length according to the theory developed by Kuhn and Mark (10, 12, 13, 14), but also reduces the energy dissipated in the flow. For flexible particles do not disturb the distortional part of the viscosimeter flow to the same extent as rigid ones. This follows also from recent experiments by Eirich and Sverak (5), who found a remarkable viscosity decrease with increaahg e For an interesting treatment of flexibility using Kuhn’s (Z. physik. Chem. A161, 1 (1932)) approximations, see M. L. Huggins (J. Phys. Chem. 42, 911 (1938); 43, 439 (1939)).
33
VISCOSITY OF SOLUTIONS
degree of swelling of their models. Theory and experiment, further show (6, 9) that the linear range of the viscosity versus concentration graph decreases rather rapidly with increasing axis ratio, so that neglect of the hydrodynamic reciprocal action of the solute particles is no longer justified, although the flexibility effect may counteract this tendency. This influence of concentration must be considered also in the case of overwhelming thermal motion. On principle it must further be examined, whether the theoretical methods used are still appropriate in the case considered, where the particles of the solute sometimes are of almost molecular size, a t least in one dimension. The experimental material and the points mentioned indicate that a complete elucidation of these questions can not be expected from a purely hydrodynamic theory in the form hitherto used, because of the complex superposition of such factors as concentration, degree of dispersity, rigidity of the suspended molecules, Brownian movement, and solvation. Therefore a statistical and kinetical treatment of these phenomena similar to that started in pure liquids should parallel the phenomenological one, although the treatment in this case appears to be more complicated than in the case where only molecules of one kind are present. Further experiments, especially in respect to the temperature dependence of all operative influences, appear to be necessary also. Nevertheless it seemed desirable to discuss some aspects of the problem a t the present state of theory and experiment, in order to establish a point of departure for investigations in the directions indicated. SUMMARY
The calculation of the viscosity of dilute solutions of non-spherical particles as a function of their dimensions and of the concentration was discussed. Three different cases must be distinguished according to the magnitude of the factor KID(D = rotational diffusion constant of the particles assumed as spheroids and K = velocity gradient of the flow in the viscosimeter). KID infinite characterizes the purely hydrodynamic case, treated by Jeffery for the case of a Couette streaming. Finite KID means that the irregular thermal motion is superimposed upon the orientation tendency of the shear. This problem and its difficulties, due to the incompleteness of the hydrodynamic solution, were not further discussed, but the third case of complete Brownian movement
(6
+ 0)
was considered. Here the macroscopic flow is no longer able to
bring about a regular rotational motion of the ellipsoids against the thermal motion, which, on the average, effects an isotropic distribution of the particle axes in the fluid. The specific viscosity
' 2 was' calculated. '0
34
R . SIMHA
Possible extensions of these results to other types of flow, especially the streaming in a capillary tube, were discussed. Theoretical considerations as well as experimental results show improvements in the theory and further experiments to be necessary. The author wishes to thank the Faculty Fellowship Fund for the grant which enabled him to carry on this work. He is especially indebted to Professor V. K. La Mer, in whose laboratory the work was done. REFERENCES
(1) BURGERS,J. M . : Second Report on Viscosity and Plasticity. Amsterdam (1938). (2) EIRSTEIN, A.: Ann. Physik 19, 289 (1906);34, 591 (1911). (3) EIRICH, F., GUTH,E., SIMHA,R., AND OTHERS: Untersuchungen uber die Viskositiil von Suspensionen und Losungen, Kolloid-Z., 1936-39;Monatsh. 71, 67 (1937). (4) EIRICH, F.,AND GOLDSCHMID, 0.:Kolloid-Z. 81,7 (1937). F.,AND SVERAK, -4.:Kolloid-Z., in press. (5) EIRICH, (6) EIRICH,F., MARQARETHA, H., AND BUNZL, M.: Kolloid-Z. 76, 20 (1936). F., AND SINHA,R.: Reports on Progress in Physics 6 (1939),in press. (7) EIRICH, (8) EISENSCHITZ, R.: Z. physik. Chem. Ala, 133 (1933). (9) GOLD,0.: Thesis, Vienna, 1937. (10) GUTH,E., AND MARK,H.: Monatsh. 66,93 (1934). (11) JEFFERY, G. B.:Proc. Roy. SOC.(London) Alo1, 163 (1923). (12) KUHN,W.: Kolloid-Z. 68,2 (1934). (13) KUHN,W.:Kolloid-Z. 76, 258 (1936). (14) MARK,H.:Chem. Rev. 26, 121 (1939). (15) *MARK,H.:High Polymers. Elsevier, Amsterdam (in press). (16) MEYER,K . H., AND VAN DER WYK,A.: Helv. Chim. Acta 16, 1067 (1935). (17) MEYER,K. H., AND V A N DER WYK,A.: Kolloid-Z. 76, !278 (1936). (18) SIMHA,R.: Unpublished work. (19) SIMHA,R.: Thesis, Vienna, 1935. (20) SIMHA, R.: Kolloid-2.76, 16 (1936). (21) STAUDIXQER, H.: Die hochmolekularen Verbindungen. Julius Springer, Berlin (1932).
