THE VISCOSITY OF DILUTE SOLUTIONS OF LONG-CHAIN MOLECULES. I1bz MAURICE L. HUGGINS Kodak Research Laboratories, Rochester, New York Received July 1, 1988 INTRODUCTION
This paper is a report of an attempt to develop a satisfactory theory, heretofore lacking, of the viscosity of dilute solutions of long-chain compounds. For solutions or suspensions of incompressible, spherical particles which are large in comparison with the molecules of the solvent, Einstein (3, 4) has derived the relationship
For solutions or suspensions of particles consisting of spheres rigidly held together t o form a rod-like molecule (the distance between sphere centers being twice the sphere diameter), Kuhn (15, 16) deduced the equation
I n the derivation he assumed that the Brownian motion is large in comparison with the velocity gradient in the solution. For ellipsoidal particles Eisenschitz (5, 6) derived the equation
neglecting the Brownian motion, and (4)
for very long particles, with a large Brownian motion effect. Presented a t the Fifteenth Colloid Symposium, held a t Cambridge, Massachusetts, June 9-11, 1938. A preliminary report of this work was presented a t the Ninety-fourth Meeting of the American Chemical Society, held a t Rochester, Kew York, September, 1937. Contribution No. 647 from the Kodak Research Laboratories. 911
912
MAURICE L. HUGGIKS
In these equations,
is the fractional increase in viscosity of the solution over that, of the pure solvent, c/V- is the volume of solute per unit volume of solution, fi is the ratio of the length of the rod-like molecule t o the diameter of thespheres of i\-hich it is composed, and fi is the axial ratio of the ellipsoid. Enipirically Staudinger (26,27,28) showed that, for yery dilute solutions of several types of long-chain compounds, the specific viscosity divided by the concentration in subnioles (“Grundmole!’) per liter is approximately
.I2 11 10
08 .07
.oa OS .04
oa DZ
0I 0
0
J
10
20
15
x)
Fro. 1. Experimental and previous theoretical values of of parafins i n carbon tetrachloride a t 20°C.
35
7.Jc
4011
for dilute solutions
proportional to the number of subwzolecules (“Grundniolekule”, e.g., CH2 groups in a paraffin chain) in the chain: 9LP = C
k, n
(5)
This empirical relationship has been much used by Staudinger and others
deducing approximate average molecular weights of high polymers. The experimental viscosity data of Meyer and van der Wyk (22) and of Staudinger and Staiger (28) for solutions of normal paraffins are compared with t h e theoretical values calculated from equations 1, 2, 3, and 4 in figurc 1. Since, in the writer’s opinion, the inclusion of an Einstein incomiri
presihilitytern1:rni
(iG)
in equation 2 for rod-like molecules isvery question-
ahle. n curve representing the equation
i; a1.o included in the figure.
VISCOSITY O F SOLUTIONS O F LONG-CHAIN MOLECVLES
913
It is evident that none of the theoretical curves is satisfactory. The experimental points lie approximately on a straight line, but (from Meyer and van der Wyk’s data a t least) not one that passes through the origin. Aside from this comparison, there is considerable other evidence favoring the supposition that long-chain molecules, such as the paraffis, are very much kinked in solution, rather than straight. The rigidity of such kinked molecules is more questionable, although there is considerable evidence (10, 11, 12, 13, 14, 23, 24, 25; cf., however, 1) for the existence of potential energy humps of the order of magnitude of 3000 cal. (Le., about 5 times R T ) per mole tending to prevent rotation about each single bond in a paraffin chain. If so, intermolecular collisions will only very rarely have sufficient energy to produce rotation over these energy humps, but rotation through a few degrees will take place much more readily. In view of the foregoing, a theoretical calculation of the viscosities of dilute solutions of rigid, randomly kinked chain molecules has been made,
FIG.2. Illustrating the orientation of reference axes, etc.
with some progress in extending the results to non-rigid, kinked chain molecules. Only an outline of the methods and results will be presented here; the details will be given in a later paper. DERIVATION O F A GENERAL EQUATIOK FOR RIGID MOLECULES
We first consider a rigid molecule in dilute solution and, following an elaboration and extension of Kuhn’s procedure (15, 16, 17), calculate the relation between the work done on this molecule by the solution as a function of the velocity gradient of the solution, the viscosity of the solution, and the coordinates of the atoms in the molecule. The origin is taken at the center of moments of the molecule, with the axes so oriented that the surrounding liquid moves in the Z direction with a velocity, qx, which is proportional to 5 . The atoms are numbered 1, 2, 3 . . . . i.. . . n and their cobrdinates designated as z,,yt! z , (see figure 2).
914
MAURICE L. HUGGINS
Because of the velocity gradient in the liquid, the molecules rotate about the Y-axis. We represent the length of the vector from the origin to the projection of the atom i onto the XZ plane by ri and the angle measured counterclockwise from the X-axis t o this vector by Oi. The work expended on atom i in unit time is deduced to be
The work expended on all n atoms, if they all have the same radius a , is c
= 6irvq2aFzr
(8)
where E X : . ZZ: - ~ r cos2 2 Oi. Zr: sin' Oi F " = T Zr:
(9)
From this is derived the relationship:
in which N is Avogadro's number and orientations. It can be shown that, if
is the average of F,, over all
6nqaq(Zr: cos20; - Zr? sin20:) kT is small compared to unity ( k being the Boltzmann constant and T the absolute temperature) , then
-
Fa, =
2%
1 2R; sin2 0 ; .2R: cos2 0; ZR:
12 3. ij
[
1 (ZR: sin 0: cos 0;)' -3 2R:
[
(11)
The angles Oi are constants, being measured relative t o an arbitrary set of X - and Z-axes rotating with the molecule. APPLICATION TO RIGID CHAIN MOLECULES
For linear rod-like molecules equation 11 reduces to -
F,.
=
t ( n 5- n)
___ 144
(12)
VISCOSITY O F SOLUTIONS OF LONG-CHAIN MOLECULES
915
With large n and small concentrations, from equation 10, qL? =z ?rNa12n2 ___
24000
c
We next consider a rigid, randomly kinked, centrosymmetric chain molecule. For the average value of R: we make use of a relationship, derived by Eyring (7) ,
-
R: = lZBi
(14)
where
B =1
+2
( 5 9 t
9 0
+ 2(-&i - 2
+ 2(-)t3 i - 3 2
+ ... + 2
p (15)
l
By an extension of Kuhn's (18) statistical methods, it is possible to show that the average of the square of the distance of the ithatom of the chain molecule from the straight line joining the origin (at the center of the molecule) to the end (mth) atom is
Also,
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YACRICE L. HCGGINS
Making the approximations
Z R sin' ~ 0: .ZR? cos2 O:
ZRZ:
~
_
_
_
ZR: sin' 0,.212;cos20, .-. ZR4
_
I
(20)
and performing the summations, we obtain
-
F,, = 0.0324fiB,Z2a(nZ- 2n
+ I)
(22)
in which is a complicated function of n which approaches unity as n increases (see figure 4 ) .
By a complicated procedure, involving an analogy with a rod-like molecule bent in the middle, with all angles of bending equally probable, for randomly a factor of 4/5 has been calculated for the conversion of for ot,herwise similar molecules kinked centrosymmetric molecules t o which do not necessarily have centers of symmetry. For this more general case, then,
zs
= 0.0259fiB,12a(n2- 2n f 1) VSP -
c
=
4.84 X 10-4NflB,12a(n - 2) 1 - 4.84 X 10-4NpB,12a(n - 2)c
(24) (25)
For large values of n and small values of c equation 25 reduces to C
= 4.84 X 10-4NB,12an
(26)
We thus have a theoretical derivation of Staudinger's empirical viscosity law (26).
VISCOSITY O F SOLUTIONS O F LONG-CHAIN MOLECULES
917
EXTENSION TO NON-RIGID CHAIN MOLECULES
An &tempt has been made to determine the effect of introducing flexibility (resulting from free or restricted rotation about the bonds in the chain) into the molecular model. The problem is a complicated one, which will be dealt with in detail a t another time. For the present a few comments will suffice. In general, the viscosities should be nearly the same for flexible longchain, kinked molecules as if they were rigid. The solvent exerts a stretch2 between 7r and 3 ~ / 2and a coming force for @, between 0 and ~ / and pression force during the rest of the rotation. Tending to counteract this is the entropy change, which can be calculated approximately (2, 8, 9, 18, 19, 21). As a net result it can be shown that the qs,/c values, as calculated
FIG. 5
for a rigid molecule, should be multiplied by a “flexibility factor,” H , which varies with
approximately as indicated in figure 5. In addition, a small constant term should be added to qap/c to allow for rotation of units other than a t the ends of the chain and a term of the form const./n to cover rotation of the atoms and groups near the ends. COMPARISON WITH EXPERIMENTAL DATA
The best experimental data now available for testing these theoretical results are for solutions of the paraffins (22, 28). We use equation 25 for rigid, randomly kinked molecules, rough calculations indicating that the corrections for flexibility would not make much difference in the result.
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MAURICE L. HUGGINS
In agreement with x-ray and electron diffraction data we take
I
= 1.54 X 10-Ecm.
(28)
and assume tetrahedral bond angles, hence
B,
= 2
(29)
The effective radius, a, must be of the order of magnitude of cm. Certain qualitative considerations suggest that it should approximately equal the radius of the sphere having the same surface area as the surface, per unit segment, of the molecule, considered as cylindrical. If this is assumed we calculate, from the density of liquid paraffins, a = 1.28 X 10-8cm.
0
s
IO
IS
(30)
n
FIQ.6. Experimental and theoretical values (from equation 25) of vap/c for dilute solutions of paraffins in carbon tetrachloride at 20°C.
Substituting these values for 1, B,, and a in equation 25 and reading @ from figure 4, the values used in plotting the upper curve of figure 6 are obtained. If the slightly lower value, a = 1.18 X 10-8 cm., is used, the lower curve is obtained. The agreement is very satisfying, considering the method of estimating a and other approximations used in the derivation. The slope of the theoretical curve lies between that of the curve through Meyer and van der Wyk's values and that of the curve through Staudinger and Staiger's results. Comparisons with experimental data for other series of compounds will be made at another time. THE MC BAIN EXPERIMENTS
McBain and McBain (20) compared the velocities of fall through sucrose solutions of quartz fibers and quartz spheres of the same weight and volume, showing that the difference in frictional resistance t o movement of these particles is by no means sufficient to account for the observed high
VISCOSITY O F SOLUTIONS O F LONG-CHAIN MOLECULES
919
viscosities of dilute’solutions of long-chain compounds. They conclude, therefore, “that the chief factor is structural viscosity due to entanglement and local adherence of molecules and particles, effectively immobilizing a disproportionate amount of the solvent in comparison with the amount of the colloid itself.” It should be pointed out that the McBain experiments bear little or no relation to the problem, since in their experiments the velocity of one side of the particle relative t o the surrounding liquid is the same as that of the other side. I n any viscosity determination a velocity gradient is set up. As is shown in this paper (and has been shown previously by others), it follows from classical hydrodynamics that because of this velocity gradient frictional work is done on non-spherical particles by the liquid. This effect must exist, and the evidence presented here shows that its magnitude is such as to account practically quantitatively for the experimental data, a t least for dilute solutions of paraffins. SUMMARY
By an extension of Kuhn’s hydrodynamical treatment a theoretical equation has been derived relating the specific viscosity of a solution to the coordinates and dimensions of the atoms in the solute molecules. Applied to rod-like molecules this equation leads t o proportionality (with n large) between qsp/c and n2. Applied to randomly kinked chain molecules, it leads to proportionality between vlap/cand n, the Staudinger relationship. Theoretical expressions have been obtained for the proportionality constant and also for the deviations for small values of n. Using a reasonable value for the “effective radius” of the CH2 group, without any arbitrary constants whatever, the experimental data for solutions of paraffins are quantitatively accounted for. I n conclusion, it is a pleasure to give acknowledgment to Miss Dorothy Owen, who performed a large part of the necessary calculations during the preparation of this paper. REFERENCES” (1) BAR THO LO MI^, E., AND KARWEIL,J.: Z. physik. Chem. B39, 1 (1938). (2) DOSTAL,H.:Monatsh. 71, 144 (1938). (3) EINSTEIN,A.: Ann. Physik [4] 17, 549 (1905). (4) EINSTEIN,A.: Ann. Physik [41 19, 289 (1906). (5) EISENSCHITZ, R.: 2. physik. Chem. A168, 78 (1931). (6) EISENSCHITZ, R . : 2.physik. Chem. A l a , 133 (1933). (7) EYRING,H.:Phys. Rev. 39, 746 (1932). (8) GUTH,E., A N D MARK,H.: Monatsh. 66,93 (1934). (9) GUTE, E., AND MARK,H.: Z. Elektrochem. 43, 683 (1937). (IO) HOWARD,J. B.: Phys. Rev. 61, 53 (1937). (11) KEMP,J. D.,AND PITZER,K. S.: J. Chem. Phys. 4,749 (1936).
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MAURICE L. HUGGING
(12) KFMP,J. D., AXD PITZER, K. S.: J. Am. Chem. SOC.69, 276 (1937). (13) KISTIAKOWSKY, G. B., AND NAZMI,F.: J. Chem. Phys. 6, 18 (1938). (14) KISTIAKOWKY, G. B., AND WILSON,E. B., JR.: J. Am. Chem. SOC.60,494 (1938). (15) KUHN,W.:Z. physik. Chem. A161, 1 (1932). (16) KUHN,W.:Z. physik. Chem. A161, 427 (1932). (17) KUHS, W.: Kolloid-Z. 62, 269 (1933). (18) KUHN,W.: Kolloid-Z. 68, 2 (1934). (19) KUHX,W.:Kolloid-Z. 76, 258 (1936). (20) MCBAIN,J. W.,A X D MCBAIK,M. E. L.: J. Am. Chem. Soc. 69, 342 (1937). (21) MEYER,K. H., . ~ X DFERRI,C.: Helv. Chim. Acta 18, 570 (1935). (22) MFYER,K. H . , A N D VAN DER WYK, A,: Helv. Chim. Acta 18, 1067 (1935). K. S.: J. Chem. Phys. 6, 469 (1937). (23) PITZER, (21) PITZER, K. S.: J. Chem. Phys. 6 , 473 (1937). S. C., A X D ASTON,J. G.: J. Am. Chem. SOC.60, 985 (1938). (25) SCHUMANN, (26) STAUDINGER, H. : Die hochmolekularen organischen Verbindungen. Julius Springer, Berlin (1932). (27) S T h V D I N G E R , H.: Helv. Chim. Acta 19, 204 (1936). (28) STAUDISGER, H., A S D STAIGER, F.: Ber. 68, 707 (1935).
