T H E VISCOSITY OF DILUTE SOLUTIONS OF LONG-CHAIN MOLECULES. 11'
MAURICF L. FUGGINS Kodak Reeearch Laboratories. Rochester, New York Received Augvst 37,1038
In the first paper of this series (4) there were given, without detailed derivation, equations relating the specific viscosity, van, of a dilute solution to the concentration and the coordinates and sizes of the atoms in the solute molecules. The derivations will now be supplied. DERIVATION OF THE GENERAL EQUATION
The procedure is a n extension of that used by Kuhn ( 5 ) in deriving an equation for the viscosity of a rod-like molecule. Considering any rigid solute molecule, let the coordinates of the n component atoms, measured relative to rectangular axes passing through the center of moments, be designated as xi, yi, zi. The surrounding liquid is assumed to be moving in the z direction with a velocity (relative to that a t the molecule center) qx, which is proportional to x (figure 1). The velocity gradient causes the molecule (if not spherical) to rotate around the Y-axis. In spite of such rotation, however, each atom is, in general, not moving with the same velocity as the liquid immediately surrounding it. After calculating this relative velocity, one aasumes Stokes' law to obtain the work done by the atom on the liquid, and, after suitable summation and averaging, the increase in viscosity resulting from the velocity gradient. Defining Ri, ~ iand , O i as indicated in figure 1, xi = ri COS ei
(1)
zi = ri sin ei
(2)
(3) and (4)
The velocity of atom i relative t o the immediately surrounding liquid is Ui = 1
+&
dUfsj
Communication No. 690 from the Kodak Research Laboratories. 439
(5)
440
hfAURICE L. WOOINS
where u+.i =
(6)
-2idi
and U*,i = zi (e - q) (7) The force acting on this atom, according to Stokes’ law, has the direction - ui and the magnitude
1 Ki I = 6 rqaiui
(8)
Its components in the X and 2 directions are: K,,i
(9)
= Gnqaizidi
and
K,,i
= -Gnqaizi(di
- q)
(10)
FIQ.1. Illustrating the orientation of reference axes, etc. 7
is the viscosity of the solution and ai is the “effective radius” of atom i.
This effective radius is smaller than the true radius if the atoms in the molecule are not well separated from each other. (Concerning the estimation of ai, see the discussion at the end of this paper.) The moment acting on atom i, tending to rotate the molecule, is M i = K,,i z i - K,,i z i = 67rqai(r:di - z:p,
(11)
For no acceleration or deceleration of the rotation, ZMi = 0
(12)
441
LONQ-CHAIN MOLECULES
the summation (here and elsewhere in this paper unless otherwise specified) being over all the n atoms of the molecule. Hence
z a((r:di - xIq)
o
(13) For a rigid molecule 8; is the same for all the atoms, hence the subscript can be dropped. Solving for 8, =
The energy loss in unit time, due to atom i, is
by substituting into equations 5, 6, 7, and 8 . The energy loss per unit of time for the whole molecule is E
=
&i =
6rqq2F:.
(16)
where
22. 2ai.z: - ai rq cos2ei . ai r2 sin2oi F I, = zai Zair: zai r2 If all the atoms in the molecule have the same radius, a, E
F,,
= 6rqq2aF,,
ZX:.ZZ: = --
(17)
(18)
- z r : cos2o,.zr: sin'
ei
(19)
Zrr Zr: The energy loss due to all the solute molecules, per cubiccentimeter of solution, is E2 = C'NZ= 6irNc'p2qF~, (20)
-
-
N is Avogadro's number, c' is the concentration in moles per cubic centimeter, and F:. is the average of FLZ over all the solute molecules. The number of solvent molecules per cubic centimeter of solution (assuming that the average volume of a solvent molecule is the same in the solution as in the pure solvent and that the density of the solvent is the
- times the number of solvent ( ) ; molecules per cubic centimeter of pure solvent. V is the volume of the same as that of the solution) is 1 -
solution and u is the volume occupied by solute molecules. The energy transformed to heat per cubic centimeter of pure solvent is q%o, where 90 is the viscosity of the pure solvent. Therefore the energy loss due to the solvent molecules in 1 cc. of solution is
442
MAURICE L. HUQGINS
If the solute molecules are very similar in size and shape to the solvent molecules, the viscosity of the solution is the same as that of the pure solvent. The energy loss due to the solute molecules in 1 cc. of solution is, in this case,
If, however, the solute molecules are large and spherical, their contribution is
to give agreement with Einstein’s viscosity law, which requires
In general, one may write
where k~ is a constant, depending on the relative sizes and shapes of solvent and solute molecules. For chain molecules in which the size of the units linked together (e.g., the -CH2- units in a normal paraffin) is of the same order of magnitude as the size of the solvent molecules, kE would be expected to have a value close to zero, especially if the chain is somewhat flexible. Its sign might be either positive or negative,-the latter for relatively large solvent molecules. The viscosity of the solution is given by the equation
Hence, kg
7 qit = -
1 =
V
+ &Nc’Z*
(27)
1 - 6irNc’Z
70
-
The next problem is to calculate F i E , making use of equation 17 for Fi,. We put
+
ei = e: e, (28) the 0; values being constants, measured relative to a set of axes (X’, 2’) rotating with the molecule. These axes are so chosen as to make t
/
B air: cos ei sin ei = o
(29)
443
LONG-CHAIN MOLECULES
Therefore,
ei sin' ei
COS*
+ sin' e: sin' e,, = sin' e: cos' e,, + cos' e: sin' eo = cos'
e:
cos'
e,
(30) (31)
Substituting in equation 17,
-
F:, =
[Zairi E2 e, + sin' e,) + (cos'
("
")
Zar:
cos'
eo sin' e,]
(32)
where (33)
and (34)
Let p be the average fraction of the total number of molecules having a value of 0, between eo and 80 de0 . Alternatively, p can be defined as the fraction of the time, on the average, that 80 for any given molecule is between e, and e, deo. The following relation then holds:
+
+
The integration takes care of the averaging over all values of 60 but not over all orientations of the molecules relative to the XZX'Z' plane. The function p depends on the relative magnitude of the velocity gradient q, which tends to favor orientation of elongated molecules with their long axes parallel to the Z-axis, and the Brownian motion, which tends to make all orientations equally probable. If the Brownian motion is of negligible importance, as compared with q, p is inversely proportional to 8, as given by equation 14. The proportionality constant is deduced from the relation
lzr pdOo = 1
(36)
making use of equation 30 and the integral (3) (37)
one obtains f l S = 2 ? r ( cos' ~ eo
+ s sin2 eo)
(38)
For a negligible velocity gradient q or very strong Brownian motion, all values of eo are equally probable and P =
T1T
(39)
444
MAURICE L. W G G I N S
To obtain a second approximation, for the case of strong Brownian motion, one may follow Kuhn (reference 5,pages 11, 12) in calculating a “rotational difTusion constant,” obtaining
in which k is Boltmnann’s constant. Then, considering the superposition of the rotation due to the velocity gradient and the M u i o n resulting from the Brownian motion, one deduces pd
dP - --.D = constant
de
(41)
Substituting the value ob d as given by equations 14 and 30 leads to the equation
$+
sinl eo = constant
(4.2)
where
Following Boeder (l), one then finds for p : +
---
q 2 (CO;~%~
2T
1 =-[l+a 2T
cos 40~
64
+ terms in higher powers of a]
(44)
+ terms in higher powers of a]
(45)
sin Bo cos eo 2
Substituting equation 38 into equation 35 and integrating gives
for the case of negligible Brownian motion. Substituting equation 45 into equation 35 leads to
+a[ I =(2 -W - L.
2!& .
c$i)am] + -
terms in higher powers of a (47)
445
LONG-CHAIN MOLECULES
for strong Brownian motion. Ln the limit, when a is zero,
If one does not choose the X'-and 2'-axes so as to make equation 29 true, one obtains, in place of equation 48,
Equations 27,46,47, and 48 will now be applied to certain special cases. ROD-LIKE CHAIN MOLECULE8
Consider a hypothetical rigid, rod-like chain molecule, such as represented in figure 2. Let X'-and Z'-axes be so chosen that the molecular axis is
FIG.2. Illustrating the orientation of sxea, etc., for a ri
'd rod-like molecule
in the X'Y plane. All the 8: values are then either 0 or 29, therefore, holds. Also
c = ZGr: cos' e:
= ZQr:
T.
Equation (50)
and
s = zair: sin' el = o
(51)
For no Brownian motion, from equation 46,
-= 0
FA
(52)
446
U U R I C E L. HUQGINS
and, from equation 27, 7.p = ks
V
(53)
All the solute molecules would, for this limiting case, align themselves with their axes parallel to the Z-axis, their rotational velocity then being zero. (This result would not have been obtained if the difference in velocity of the solvent on opposite sides of each atom had been allowed for. One would still find, however, that FL3 = 0.) (54) Following Staudinger (7) in defining c as the concentration in submoles (“Grundmole”) per liter (identical in the present instance with gramatoms per liter), c = 10OOnc’
(55)
and v
CMO
71oooP Mo is the “molecular weight” of a submole (in this case the atomic weight) and p is the density of the pure liquid solute in grams per cubic centimeter. Substituting in equation 53,
of
For strong Brownian motion or zero velocity gradient, Le., for small values CY (equation 43), substitution in equation 47 gives
-
F:,
=
(Z&)8ve.
(18 + -5f+ terms in higher powers of a) 1024
where 4 is the angle the molecule axis makes with the XZX‘Z’ plane. all orientations are equally probable - 2 cos2 4 =
3
(58)
If (60)
Assuming for simplicity that the chain contains an odd number (n = 2m 1) of atoms,
+
LONQ-CHAIN
MOLECULES
447
Hence, (62)
18
and, from equations 27, 55, and 56,
Approximately, 'I.p c
=:
rNl'an2 24000
(64)
If, following Kuhn (5), one takes the volume of the molecule as that of a cylinder of length nE and diameter 2a, equation 64 can be transformed to %P
nzZ v = 24a -* V -
For the special case treated by Kuhn, in which I = 4a, this becomes VIP
( )v
1 nZ2v =24 2a
(66)
which is equivalent to his equation 38 except for a factor, 2/3, which has been introduced here (see equation 60) to allow for molecular orientations other than in the XZ plane, and except for his inclusion of the Einstein 5v
term, - 2V'
If one takes the volume of the molecule as the s u m of the volumes of n spheres of radius a, equation 64 leads to
RIGID, RANDOMLY KINKED CHAIN MOLECULES
Consider next a rigid chain molecule which is centrosymmetric and which has constant equal bond angles and bond distances, but is otherwise randomly kinked (figure 3). The average value of the square of the distance from the atom a t the molecular center (the origin) to any other atom, i, according to a relationship derived by Eyring (2), is
-
Ri = Bil'i
(68)
448
bfAURICE L. HUOQINS
where
... +2f'
& = 1 + 2 ( ! ) t . + 2 ( - ) t 'i +- 22t + ) ( ~ + 2
2
(69)
and
E
= cosff
(70)
The included angle between adjacent bonds is (T - CY). The expression for B i can also be put in the form Bi =
24'+' 1 I f € - -.-2E 1 + -._ (1 - E ) 2 i 1-f (1 - f ) 2 i
z- +
1 - f
f
for large i
(71)
(72)
a
FIQ. 3. Representing a randomly kinked, centrosymmetric molecule
For random orientation with respect to the angles 4 i , with all ai values equal] - air:)ave. = a cos26, R: (73) = ,2% a c R: (74)
(c
-
-3 ta
2 ~~i
(75)
1
the summation being over half of the chain only. Performing the summation, one obtains 212aB' n z ~= Z2aB'(n2- 2n 1) W:)Em. = (76) 3 6 with
(c
+
To calculate the other quantities needed for substitution into equations 46 and 48, one chooses the X'- and 2'-axes so that the projections on the
449
LONQ-CHAIN MOLECULES
XZX'Z' plane of the end atoms of the chain lie on the X'-axk.
Although this choice does not make air: cos 0; sin 0: equal to zero, as required by equation 29, it can be shown that CS = 6 m ( x a&
cos 8: sin
(78)
The error introduced in using equation 48 rather than equation 49 is thus negligible. One would likewise expect the inexactness of equation 46 resulting from this choice of axes to be negligible. Kuhn (6) has calculated statistically the probability that the middle atom in a randomly kinked chain of (m 1) atoms is a distance S, away
+
t
FIQ.4. Illustrating the notation used
from the straight line joining the end atoms. Extending his treatment, one can show that the probability that the ith atom in such a chain is a distance Si from the line joining the ends is given by the equation
Here BJ is the function given by equation 69. Averaging in the usual manner one obtains
+,,
If one designates by the angle the molecular axis (through the center and end atoms) makes with the X'-axis and, by + i , the angle which the
450
MAURICE L. HUGGINS
perpendicular from the ith atom to the molecular axis makes with the X'Y plane (see figure 4), 7: COS*
= (R:
- 83 COS'
All values of
+i
el = [(E: - s:)'" +m
+ S: cos2
COS +m
- si COS +i sin 4m12
(81)
sin2 4,,,
+i
are equally probable, hence
-
cos +i = 0
1 cos2+i = 2
The probability of a given value of $m is proportional to cos $m, assuming random orientation of the molecules with regard to orientation relative to the XZ plane. Taking this into account, COS2
$Im
2 3
=-
(86)
and sin2+, =
1
3
(87)
Substituting in equation 83 gives
Similarly,
Summing, one obtains
(92)
3
(93)
LONG-CHAIN MOLECULES
45 I
and (94) (95)
3 where B' is given by equation 77 and
Making the approximutions that (dE3)ave.
=
v'F~
(97)
and
c.s one now substitutes equations 76, 93, and 95 into equations 46 and 47, For negligible Brownian motion,
452
MAURICE L. HUGGMS
For strong Brownian motion or small velocity gradient,
Both B' and B" approach the value
as m becomes large (see figure 5). Introducing the function
Equation 99 can be written
= 0.0311Z2a(n2- 2n
+ 1)B-p'
= 0.031112anZB, for large n
(104) (105)
Likewise, putting
p = 9 B' 7 B,
2 B't2 7 B'B,
and making the approximation in the small 'YC term that B' = B" = PB,
(107)
equation 100 becomes
-t
7 F,, = - lzam2B,p 54
(108)
= 0.032412a(n2- 2n
+ 1)B-0 (1 + 4-2 )
(109)
For large n and small CY this reduces to =
0.032412an2B,
(110)
Both p and p' approach unity as m increases, as shown in figure 5. The axes of rotation for randomly kinked chain molecules which are not centrosymmetrical do not, in general, pass through the midpoints of the chain. is, therefore, smaller than for otherwise similar molecules which do have centers of symmetry. It is desired to find the average for non-symmetrical kinked molecules, the coordinates of the atoms in each molecule being relative to a
LONG-CHAIN MOLECULES
453
Y-axis so situated as to make the (and hence Z) for that molecule a minimum. There seems to be no easy direct way of calculating this average. A reasonable approximation, however, can be obtained in the following manner: Let a quantity G' be defhed by the equation
G' = ZaiMiri
(111)
in which ai is the radius of the ith atom, Mi its mass, and ri the distance from the origin to its projection on the XZ plane. For a long, straight, 1 like, equally spaced atoms, rod-like molecule, composed of n = 2m all in the X Z plane, laMonZ G'=2laMoxi= (112) 1 4
+
FIG.6. Illustrating notation for a bent rod-like molecule Comparing this with equations 105 and 110, one notes that G' for this long rod-like molecule and F:, for a large centrosymmetric kinked molecule become identical if we put
M o = 0.1244B,1 or 0.1296B-1 (113) The angle between the straight lines joining the middle of a randomly kinked chain molecule with its two ends may have any value from 0 to A. For a sufficiently long chain, all relative orientations in space of these two lines are equally probable, and the probability of any given included angle is proportional to the sine of that angle. It seems reasonable to assume that the desired average for centrosymmetrical kinked mole-
xz
454
MAURICE L. HUQQXNS
cules is related to for unsymmtrical kinked molecules in the same way as G' for a straight rod-like molecule is related to the G' for a rod-like molecule of the same length bent in the middle (figure 6), taking the probability of each bending angle as proportional to the sine of that angle and measuring the T values (for each bending angle) relative to such an origin as leads to the minimum value of G'. This origin is obviously on the line bisecting the two legs of the molecule. One chooses this line as the X-axis and calls the distance from the joint to the origin Boml. Then, neglecting the contribution of the atom at the bend, m
G' = 2aMo C ~i
(114)
Putting
i
= Xm
(117)
one gets G' = 2laMom
(A' A-
For large m, G' = 21aMom'
L
- 2BoX cos y +$)"*
(1 18)
+ j3;)"'ch
(119)
m
1'
(AZ
-
2BOAcos y
= laMomz!@
where !@ =
(1
- Bo cos Y)(l - 280 cos y
+ &)'/' + )9; cos y
as a function of B can be comFor any given value of y, values of puted and the minimum value, \kmin., obtained. This varies from 1 / 2 for y = 0 to 1 for y = a / 2 . The average value of !@min., making allowance for the sin 2y probability factor, was determined graphically to be 0.81, or (within the accuracy of the calculation) 4/5. Without the sin 2-y factor, the average was 0.80. Introducing the factor 4/5 into equation 104 gives, for negligible Brownian motion or large velocity gradient, -
F:,
=
-Z'.B,B'(n' 18 4
- 2n + 1)
455
LONG-CHAIN MOLECULES
Substituting this and equations 55 and 56 into equation 27,
~ B M+O
rNl2aB,,9' ( n - 2
3&'1000 E?= -1ooop C - 3 4 .TN 1 0 0 0 l2aBW/3'(n- 2
+ :) + ;)c
(123)
For large n, small k,, and small c, giving N the value 6.024 X lo", ' 2= 2.82 X 10mB,12an C
(124)
Similarly, for strong Brownian motion or small velocity gradient,
i=
2.94 X 1OZoB,?an
(127)
DISCUSSION
Although the chain molecules which have been considered in this paper have been purely hypothetical, consisting of single, like atoms linked together, there is nothing in the nature of the treatment which would not apply equally well to molecules consisting of groups of atoms linked together, provided, if random kinking is assumed, the interactions between .these groups do not interfere with the randomness of the kinking and provided a suitable "effective radius" a for the group is used. One must count one group per joint (not one per repeating unit), and if all the groups are not alike, as in vinyl polymers, for instance, one must use some sort of average value for a. It should be emphasized also that the derivations in this paper are based on the assumption that the contribution of each atom (or group) to the viscosity work is the same as if the flow of the immediately surrounding liquid were entirely unaffected by the presence of the neighboring atoms. This is, of course, untrue for actual chain molecules, in which the distance I between atoms or groups is not much larger (if at all) than the diameter of each. This may be corrected for by using for the eflective radius, a, of an atomic group in a chain molecule a value somewhat smaller than its true radius. In the first paper of this series it was shown that approximate agreement with the empirical value for
456
MAURICE L. HUGGINS
the paraffis was obtained by assuming that a is the radius of a sphere 1 having - t h the surface area of a cylinder having a length equal to n times n
the length per CH2 observed in solid paraffis and having the diameter required for molecular close-packing in the liquid state. No claim of accuracy or of theoretical validity is made for this method, however. Comparisons of empirically determined values of the effective radius for different series will probably lead to a better method of calculation. It should be noted that the results for randomly kinked molecules differ from those for rod-like molecules in two important respects. First, they differ in their dependence on n, Staudinger’s empirical relation (proportionality between qrp/c and n) resulting only for kinked molecules. Second, they differ in their variation with the raticl q / T . This variation is negligible for randomly kinked molecules but very great for rod molecules. SWMARY
Theoretical derivations have been given for the previously reported general equation relating the viscosity of a dilute solution to the coordinates and sizes of the atoms in the component molecules, andalso for the special equations applying to rigid chain molecules which are rodlike or are randomly kinked. The difficulty of determining theoretically the “effective radius” of a group of atoms in a chain molecule, for use in these equations, has been briefly discussed. Long, randomly kinked chains should obey Staudinger’s empirical “law.” Long rod-like molecules should show a very great dependence of viscosity on the velocity gradient in the solution. The assistance of Miss Dorothy Owen in making and checking many of the calculations for this paper is gratefully acknowledged. REFERENCES (1)
BOEDER,P.: Z. Physik 76, 258 (1932).
(2) EYRINQ,H.: Phys. Rev. 99, 746 (1932). (3) DE HAAN,D. B.: Nouvelles Tables d’IntBgrales DBfinies. P. Engels, Leida (1867). (4) HUQQXNS, M. L.: J. Phys. Chem. I,911 (1938). (5) KUHN, W.: 2. physik. Che,m. A161, 1, 427 (1932). (6) KUHN, W.:Kolloid-2. 68, 2 (1934). (7) STAUDINQER, H. : Die hoohmolekularen organisohen Verbindungen. Juliw Springer, Berlin (1932).
