DIFFUSION OF UONODISPERSE POLYSTYRENE IN MIXEDSOLVENTS
good agreement with the present calculations is that of Kirkwood and Nuller. Salemz3has noted that this formula gives an upper bound to the dispersion energy, and hence it gives a lower bound to the area which can be calculated from a dispersion energy constant. A good case can be madez4for reducing both the SlaterKirkwood and London estimates by a multiplicative factor of 2-' which would bring them into more reasonable agreement with the Kirkwood-Muller values, but would still leave them 20-3070 above the area obtained in the present paper. At least it appears 31
1135
that Henry's law data in conjunction with the Kirkwood-Muller formula will yield areas in which one can have considerable confidence.
Acknowledgment. Financial support from the National Research Council and the Research Fund of the University of British Columbia is greatfully acknowledged. (23) L. Salem, Mol. Phys., 3, 441 (1960). (24) J. R. Sams and R. Yaris, J . Phys. Chem., 6 7 , 1931 (1963).
Multicomponent Diffusion Involving High Polymers. I.
Diffusion of
Mon 4 s p e r s e Polystyrene in Mixed Solvents
by E. L. Cussler, Jr.,* and E. N. Lightfoot Department of Chemical Engineering, University of Wisconsin, Madison, Wisconsin (Received September 16, 1964)
JIulticoniponent diffusion coefficients at three compositions of the ternary system polystyrene-toluenecyclohexane were measured at 28.00" with a Gouy interferometer. The strong concentration dependence of the diffusion coefficients was removed by repeating all experiments at the same average concentration but different concentration differences and extrapolating the results to zero concentration difference. That the polymer behaves as a single species is shown by binary data. The multicomponent diffusion coefficients measured show the largest deviations from Fick's law yet observed. Because of these large deviations, the data offer the opportunity of evaluating multicomponent flux equations. The two most promising forms are briefly discussed, stressing the restraints on the various coefficients. Experimentally, they are compared as to accuracy of measurement, variation with composition, and molecular significance.
Diffusion in some multicomponent systems deviates sharply from the predictions of Fick's law, and more general flux equations are required to describe the concentration profiles within experimental error. The purpose of this paper is to seek a liquid system where these deviations are large and to choose the best set of flux equations to describe these effects. Large deviations have previously been reported for gases of very and for aqueous Systems different mOleCUhr
of electrolytes whose solutes have radically different m~bility.~ Since large deviations in physical size and mobility appear to cause large deviations from Fick's *
Department of Physical Chemistry, University of Adelaide, South (1) J. B. Duncan and H. L. Toor, A.1.Ch.E. J.. 8,38 (1962). (2) W. E. Stewart and R. Prober, I d . Eng. Chem. Fundamentals, 3 , 224 (1964). (3) R. P. Wendt, J . Phya. Chem., 66, 1279 (1962).
volume 69,Number 4
April 1965
E. L. CUSSLER, JR.,AND E. K . LIGHTFOOT
1136 -
law, the system polystyrene-cyclohexane-toluene was chosen for experimental investigation. The results were used to determine the set of multicomponent diffusion coefficients which are most nearly independent of concentration, which can be best estimated from binary data, and which represent known physical occurrenoes. A wide variety of multiconiponent flux equations have been proposed either on the basis of a physical model or from irreversible thermodynamic arguments. Although each set of flux equations defin& a set of multicomponent diffusion coefficients, these different sets are i n t e ~ ~ e l a t e d . ~The - ~ differences in the sets originate in the choice of the driving force, in the specification of a reference velocity, and in the treatment of the restraints on the system. The two most promising forms are the generalized Stefan-R/laxwell equations and the generalized Fick’s law equations. The generalized Stefan-Maxwell equations have the form n
V+i =
C +i+$ti(vj
- vt>
3=1
(1)
where 4%is the volume fraction of species i and v, is the velocity of species i. The coefficients R,, of this type equation are independent of reference velocity and are similar to the “generalized Stefan-Maxwell coefficients” and to the frictional coefficients defined by other investigator^.^-'^ In the special case of an
Rij
=
Rji
(2)
ideal gas, the Rtl are ~ y m m e t r i c e and ~ ’ ~are ~~~ identically equal to the binary diffusion coefficients.15 I n all cases, they reduce to the binary limit a t infinite dilution Iim
+
(4%
41)
-
Rif
= -
1
1
(3)
Lf
where st, is the binary diffusion coefficient. In general, however, the R,, need not be positive, although the eigenvalues of the coefficient matrix must be posit i ~ e . ~Because ,~ of their simple limits, the R,, are often easily estimated from binary data. For the solution of practical problems, the generalized Fick’s law equations relative to constant volume are the best set n-1
-j,
= - p r ( v i - yo> =
C Dt,vpf
3=1
(4)
where pi is the mass density of species i. I n many experimental cases, the volume average velocity is zero. The Dii are in general not reciprocal, so that in the absence of activity data, four coefficients are reThe Journal of Physical Chemistry
quired in a ternary experiment. The D,, have the following binary limits 16, l7
D,,
lini
(P,
+
pn)
-
+
= D1,
lim D 1 , = PI
(6)
P
lirn D t 3 ( 3 2 ,=) 0 PZ
-
+
0
v1 lim DI3(,#3) = = Pl
(3
P
‘v,
P
[a, - %I
(7)
(8)
Again, the eigenvalues of the diffusion coefficient matrix must be positive. For a ternary system, this is equivalent to
Dn
+
(0x1- D22I2
0 2 2
>0
+ 4DizD21 > 0
(91
(10)
Thus, there is no theoretical reason for any szngle coefficient to be positive. However, for a wide class of dilute electrolytes, the terms D,ici+3) are small relative to the D,, and may be neglected.’*
Experimental Materials. The polystyrene used in the experiments, designated S-109, was supplied by Dr. H. W. McCorniick of the Dow Chemical Co. It had a numberaverage molecular weight of 182,000 and a weightaverage weight of 193,000. It‘contained as an impurity 300 p.p.m. of acrowax, a lubricant which had been added in earlier experiments by other investigators. Acrowax is a complex nitrogen derivative of the higher fatty acids manufactured by the Glyco Products Co. New York, K, Y.19 Whether this con(4) s. R. de Groot and P. Mazur, “Non-Equilibrium Thermodynamics,” North-Holland Publishing Co.. Amsterdam, 1962. (5) C. Truesdell, J . Chem. Phys., 37, 2336 (1962). (6) E. N. Lightfoot, E. L. Cussler, Jr., and R. I,. Rettig, Chem. Eng. (Tokyo),28, 480 (1964). (7) E. N. Lightfoot, E. L. Cussler, Jr., and R. L. Rettig, A . I . C h . E .J . , 8 , 708 (1962). (8) L. Onsager, Ann. N . Y . Acad. Sci., 46, 241 (1945). Acta Chem. Scand., 11, 362 (1957). (9) 0. La”, (10) R. UT. Laity, J . Phys. Chem., 63, 80 (1959). (11) R. J. Bearman, ibid., 65, 1961 (1961). (12) P. J. Dunlop, ibid., 68, 26 (1964). (13) L. Onsager, Phys. Rev.,37,405 (1931). (14) L. Onsager, ibid., 38, 2265 (1931). (15) C. F. Curtiss and J. 0. Hirshfelder. J . Chem. Phys., 17, 550 (1949). (16) I. J. O’Donnell and L. J. Gosting in “Structure of Electrolytic Solutions,” W. J. Hamer. Ed., John-Wiley and Sons, New York, N. Y., 1959. (17) F. 0. Shuck and H. L. Toor, J . Phys. Chem., 67, 540 (1963). (18) R. L. Baldwin, P. J. Dunlop. and L. J. Gosting, J . Am. Che Soc., 77, 5235 (1955).
DIFFUSIOKOF R~ONODISPERSE POLYSTYRENE IN MIXEDSOLVENTS
1137
Non-Aq grease. The Non-Aq grease was washed six centration of impurity is significant must be determined times in toluene and six times in cyclohexane to remove experimentally. Reagent grade cyclohexane and toluany impurities which would dissolve in the solvents ene were used in all experiments. during the experiment. Before being filled the cell and Solutions. The desired solution concentrations for each experiment were calculated by assuming the cell holder were heated to the experimental temperature densities were additive. Densities were taken to be of 28". 1.040, 0.77148, and 0.85821 for polystyrene, cycloThe temperature was measured by a niercury-inhexane, and toluene, respectively. After two experiglass thermometer calibrated against a standard thermometer belonging to Professor L. J. G o ~ t i n g . ~ ~ ments were made, approximate values of the partial specific volumes were calculated. These were used to It was found to read 28.044' at a true 28.000". est iinate the remaining compositions. General Experimental Procedure. In each experiAll solutions were prepared by weight on a B-5 ment with the Gouy interferometer, diffusion occurs across a sharp initial boundary between the less dense llettler balance, always against a sealed tare. All upper solution and the more dense lower solution. weights in air were corrected to those under vacuum. A definite order for making the ternary solutions was This boundary was formed by sharpening with a single followed. After the polymer was weighed, the approxiprong 23-gage stainless steel capillary.32 Because of mate amount of cyclohexane was added. The sample the high viscosity of the polymer solutions, an aspirator was placed in an oven at 40" and stirred slowly with operating through a 5-1. surge tank was required to a magnetic stirrer for at least 24 hr. Then the sample obtain the desired flow of 1-2 ml./min. through the was brought up to the desired weight with cyclohexane capillary. All tubing in this apparatus was of polywhich had been warmed to 35". After the desired ethylene. This technique came at least as close to weight of toluene was added, the experiment was iminfinite sharpening as that previously used. mediately begun. In this way, volatility losses were Three groups of reference photographs are required minimized and polymer precipitation avoided. These establish the posiin the Gouy proced~re.~5,~3 Solution densities were measured in 25-nil. short neck tion of the cell relative to the reference channel and Pyrex pycnometers, which had been calibrated at allow determination of the fractional number of fringes. The integral number of fringes is found from one or two least three times with air-saturated water. The density of the water at 28" was taken to be 0.99626 Rayleigh photographs taken during the experiment . 3 3 , 3 4 g . / c n ~ ~ Because . of the limited amount of polymer Reference photographs were taken with Kodak Tri-X available only one sample of each solution was measured. Panchromatic, Kodak Super-ortho Press, or Kodak The samples were weighed against a sealed tare with Spectroscopic plates. the balance described above. Partial specific volumes During the diffusion between the two solutions, 7-20 were found by the usual method.20 photographs of the Gouy interference fringes were Apparatus. In these experiments, free diffusion was taken on Kodak Super-ortho Press or Kodak Spectrostudied using the Gouy interferometer belonging to Professor L. J. Gosting. A general description of the (19) A. B. Warth, "The Chemistry and Technology of Waxes," apparatus is available elsewhere.21 The wave optical Reinhold Publishing COT^., New York. N . Y . , 1956. theory has been highly developed, and tables of the (20) P.J. Dunlop and L. J. Gosting, J . Phys. Chem., 63, 86 (1959). theoretical constants used are readily ~ b t a i n e d . ~ ~ -(21) ~ ~L. J. Gosting, Adaan. Protein Chem., 11, 476 (1956). The particular apparatus used in these experiments has (22) G. Kegeles and L. J. Gosting, J . Am. Chem. SOC.,69,2516 (1947). (23) L. J. Gosting and M.S. Morris, ibid., 71, 1998 (1949). been described in previous publication^.^, 25-27 (24) L. J. Gosting and L. Onsager, ibid., 74, 6066 (1952). The diffusion cell, manufactured of fused silica by the (25) L. J. Gosting, E. M. Hanson, G. Kegeles, and M. S. Morris, Pyrocell Manufacturing Co., is similar to the Tiselius Rea. Sci. Instr., 20, 209 (1949). cells described 28-30 but has a reference (26) P. J. Dunlop and L. J. Gosting, J . Am. Chem. Sac., 75, 5073 channel parallel to the diffusion channeL31 In order (1953). (27) P.3. Dunlop and L. J. Gosting, ibid., 77, 5238 (1955). that ordinary Rayleigh photographs may be taken (28) A. Tiseluis, Trans. Faraday Sac., 33, 524 (1937). during the experiment, the channel is filled with the (29) L. G. Longsworth, Chem. Rev., 30, 323 (1942). less dense of the two diffusing solutions. The internal (30) J. St. L. Philpot and G. H . Cook, Research, 1 , 234 (1948) diameter a of the cell along the optical path is 2.4960 (31) It. L. Rettig, Ph.D. Thesis, University of Wisconsin, 1964. cm. ; the optical lever arm b measured from the center (32) D. S. Kahn and A. Polson, J . Phys. Colloid Chem., 51,816 (1947). of the cell to the photographic plate is 304.99 cm. (33) L. J. Gosting, J . Am. Chem. SOC.,72, 4418 (1950). The flanges of the cell were greased with an outer (34) K. P. Wendt and L. J. Gosting. J . Phys. Chem., 63, 1287 band of Lubriseal grease and an inner band of Fisher (1959). Volume 69, Number 4
April 1.966
E. L. CUSSLER, JR.,AND E. K.LIGHTFOOT
1138
scopic plates. The Tri-X Panchromatic plates gave less distinct iniages and so were not used for the diffusion photographs. The intensity minima of the fringe pattern were measured with a Gaertner M 2001 RS toolmakers microscope fitted with a projection s ~ r e e n . ~Between ,~~ 16 and 20 intensity minima were measured on each photograph, including the minima numbered 0 to 6. The quantity Ct, which is the inaximuin value of refractive index gradient predicted by ray optics, is found by extrapolating Y/e-” us. Z 3 ’ j 3 to Zj2/’ = 0 where Y is the fringe displacement on the plate, equal to the derivatives of refractive index bn/bx,26r35 2,is the reduced fringe ~ i u m b e r ,and ~ ~ e-{’ , ~ ~ is found from the fringe number.23 In a binary experiment, the values of Y/t:-’* for fringes 0 to 6 were fit to a straight line in ZJZ”by least squares. This technique generally did not work for ternary experiments or for experiments where D,, is a strong function of composition. After an unsuccessful attempt a t a parabolic fit in ZlZIa,Ct was found by a manual extrapolation. Using this value of Ct, an initial value of the reduced height-area ratio %A’ for each plate was found
The true value of D A , corrected for the fact that the inii;ial boundary was not infinitely sharp, is found from
Q is found by plotting the dimensionless difference of the ideal refractive index gradient and the actual refractive index gradient vs. the reduced fringe number and then integrating over the reduced fringe member
For values of Q less than 40 X low4,the data were fit to a fifth-order polynomial and integrated by machine. For higher values of &, integrations were by Simpson’s one-third rule using intervals of 0.05 in f(1). Procedure f o r Bznary Experaments. In the binary theory of the Gouy interferometer, three basic assumptions are made. These are that the diffusion coefficient is not a function of concentration, that the refractive index is a linear function of the concentration, and that the partial specific volunies of both species are constant. If these assumptions are obeyed, then the reduced height-area ratio equals the binary diffusion coefficient and the area under the fringe deviation graph Q is zero. However, in a wide variety of systems such as those studied here, the assumptions above are not valid. For example, the assuniption that the diffusion coefficient is not a function of composition is valid only a t the e t e m p e r a t ~ r e . ~ g - ~Al limited theory for these cases has been developed by Gosting and F ~ j i t and a~~ by F ~ j i t a . They ~ ~ solved the concentration profile and the refractive index profile for the assumptions Dtj
where At is the starting time c ~ r r e c t i o n . ~ ~ ~ ~ ’ In the special case of zero polymer gradient a t the composition 5% polystyrene, 5% toluene, and 90% cyclohexane, the first four fringes were too faint to be photographed by the usual exposures. If the exposures were increased so the first four were visible, the lower minima (fringe no. 6 or greater) were completely obscured. In this case, two sets of photographs are required, one set giving the first four fringes and a second set measuring the lower portion of the pattern. Both maxima and minima of the first five fringes of the first set of pictures were measured. Ct was found by the usual extrapolation, and D A and At were calculated from the relations given above. Then, reversing this procedure, values of Ct for each of the times of the second set of photographs were calculated. In this way, both DA and Q (described below) could be experimentally determined. For each experiment, the average deviation of the refractive index profile from Gaussian shape was found as the area under the fringe deviation graph Q . 3 5 , 3 8 T h e Journal of Physical Chemistry
=
n
=
Pt
=
+ h ( P - P) + M
+ n(p)(l + Rl(P - P) + R z b - PI2 + Y,(P)(l+ - P) + Q ( P - PIZ + Dt3(P)(1
P
1 1
- P)z
(14) (15)
)
?%(P
(16) where p is the average mass concentration of the solute between the upper and lower solutions. The principal result of their work is that DA = Dtj(P)(1 f p ( A P ) 2 ~~
~
~~
+ ~~~
(17)
. ) ~~~
~
~
(35) D. F. Akeley and L. J. Gosting, J . A m . C h m . SOC.,75, 4685 (1953). (36) L. G. Longsworth, ibid., 69, 2510 (1947). (37) H. Fujita, J . Phys. SOC.J a p a n , 11, 1018 (1956). (38) H. Fujita and L. J. Gosting, J . Phys. Chem., 64, 1256 (1960). (39) V. N. Tsvetkov and S. I. Kleinin, J . Polymer Sci., 30, 187 (1958). (40) S.I. Kleinin, H. Benoit. and M . Daune, Compt. rend., 250, 3174 (1960). (41) R. Varoqui, M. Jacob. L. Freund, and M. Daune. J . chim. phys., 59, 161 (1962). (42) L. J. Gosting and H. Fujita, J . A m . Chem. SOC., 7 9 , 1359 (1957).
(43) H. Fujita. ibid., 83, 2862 (1961).
DIFFUSION OF UONODISPERSE POLYSTYRENE IN MIXEDSOLVENTS
In addition, although no exact theory is available, we postulate that
Q
= S(Ap)'
+
..
(18)
1139
the multicomponent flux equations for the refractive index profile, DA and Q may be found in terms of the four diffusion coefficients and a . 3 8 , 4 5
+
l/z/K = I A SAa (19) In other words, to find the true value of the diffusion coefficient a t the average composition, the reduced height-area ratio a ) A is measured a t least two times a t the same average composition p but at different Ap. The five constants I A , SA,EO,El, and Ez are functions The various values of DA are graphed us. ( A P ) ~and a and of the four diffusion coefficients. The values of extrapolated to ( A P ) ~= 0 to obtain the true value of of these constants best fitting the data are found by the diffusion coefficient. A similar extrapolation for trial and error. Q will give a zero intercept. However, for the systems studied in this work, the If one of the species contains an impurity or a precipidiffusion coefficients are strong functions of concentratate, the limiting intercept of Q may be nonzer0.3~ tion; the refractive index varies nonlinearly with conFor a polymer, a nonzero intercept may result from the centration; the partial specific volumes are not conpolydispersity of the sample.44 Before starting ternary stant. To adapt the above theory to systems of this experiments, binary runs of each pair of components a. Because type, we must re-examine the parameter must be made to ensure that impurities and polydisof the nonlinear refractive index, the value for a is not persity are not significant. known except at the values of 0 and 1, i e . , a t the points I n summary, for a binary experiment, the critical Apl = 0 and Ap2 = 0. Thus if we consider these where parameter is Q If, at finite Ap, Q is zero, then the two points only, we can define an a for the system. reduced height-area ratio DA equals the diffusion We next assume that a t small values of Apl, DA and Q coefficient a t average composition D f j ( p ) . If Q js vary with ( A p J Z . This is analogous to the binary nonzero, the experiment is repeated three times at case but, because it has no theoretical basis, must be the same p but different ( A p ) . Then Q and DA are verified experimentally. If it is verified, then parallel extrapolated us. ( A P ) ~ . If Q is zero a t (Ap)' = 0, to the above equations, we have then DA equals DiJ(p). If Q is nonzero a t ( A P ) ~= 0 for a systeni of two solvents, then the system must 1 (21) contain an impurity or a precipitate. If Q is nonzero (A2ilZ1L 0 = IA Ap? = 0 at ( A P ) ~= 0 for a system of solvent and a polymer, the system may contain an impurity or a precipitate, or lim 1 - I A SA the polymer may be too polydisperse to be treated as a Api = 0 da>, single species. (Am)* 0 Procedure for Ternary Experiments. The basic method for ternary experiments was described in earlier w ~ r k '4s . ~in the ~ ~binary ~ ~case,~ three ~ ~basic ~ ~ ~ assumptions are made: that all four diffusion coefficients are not functions of composition, that the refractive index is a linear function of the composition of each species, and that the partial specific volumes of all We must repeat each experiment a minimum of four species are constant. Since only two independent times, twice with Ap1 = 0 and with Apz equal to two parameters, B A arid Q, are measured in each experidifferent values, and twice with Ap2 = 0 and Apl ment, it is necessary to repeat each experiment at equal to two different values. In this work, however, least twice a t the same average composition but a t each experiment was repeated six times in order to different values of Apl and Apz. check the linearity in ( A p J Z of the extrapolation of DA The choice of Apl and Apz is governed by two restricand Q. tions. First, the absolute magnitudes of Ap1 and Apz In practice, because the densities of the solutions are chosen so that the total number of fringes J is about 50-100.18 The relative values of Apl and Apz (44) E. L. Cussler, Jr., J . Phys. Chem., 69, 1144 (1965). are fixed by defining a new parameter a, the fraction of the refractive index gradient due to component 1.35,46 (45) H. Fujita and L. J. Gosting, J . Am. Chem. Soc.. 78, 1099 (1956). I n most previous work, the values of a used for each (46) L. A . Woolf, D. G. Miller, and L. J. Gosting, ibid., 84, 317 experiment were 0.0, 0.2, 0.8, and 1.0. By solving (1962).
(E)
+
-t
Volume 69,Number 4
A p r i l 1966
E. L. CUSSLER,JR.,A N D E. X. LIGHTFOOT
1140
Table I : Binary Runs for Sucrose-Water and Cyclohexane-Toluene Expt. no.&
2
2 13
1
2
a
14
1
15
2 1 2
sucrose = water = cyclohexane = toluene = cyclohexane = toluene = cyclohexane = tolune =
x
Q x
T
01
J
24.950 f 0.006
0.00739
99.560
0.5164
0.5
28.001 f 0.003
0.0815
73.393
2,256
0.9
28.003 f 0.002
0.4597
66.847
1.944
28.003 f 0.002
0.9000
64.330
1.776
System
1
DA
10s
104
-1.7 0.7
Experiments numbered chronologically.
Table I1 : Binary Diffusion Coefficients of Polystyrene” Average Experinient no.
Solvent
fraction polymer
8 ‘23 24 (Limit A ) 10 1 52 (Limit B )
Toluene Toluene Toluene Toluene Cyclohexane Cyclohexane Cyclohexane Cyclohexane
0.0500 0.0500 0.0500 0.0500 0.0500 0.0500 0.0500 0.0500
ma88
Mafraction difference
T
J
0.01571 0.01172 0.00954
28.006 f 0.002 28.006 f 0.003 28.000 f 0.002
66.543 52,040 41,246
a > (exptl.) ~ x
107
9.082 8.930 8,874
0. OOOOO
...
0.000
...
0.01154 0.00825 0,00338
27.990 f 0.002 27.995 f 0.004 28.000 f 0.003 ...
69.396 49.632 20.197
0.7932 0.871 0.9820
O.Oo0
...
0.00000
9~(calcd.)
x
107
9.078 8.935 8,871 8.752 0.794 0.878 0.987 1,010
Q (exptl.)
x
104
21.1 11.7 7.3 ..
99.5 55.6 11.9
..
Q (calcd.)
x
104
21.1 11.8 7.3 -0.6 97.4 60.6 9.2 1.8
’ Experiments numbered chronologically.
could not be exactly predicted, the values of a for experiments were not exactly 0 or 1, but several thousandths off these values. The values of PA and Q must be corrected to a = 0 or 1 before the extrapolation can be made. These corrections were made 88 follows: Using the two experiments with the smallest values of Ap1 and AB, initial values of l a , SA,EO,El, and Ez were calculated. These values were used to correct the values of T)A and Q for the six experiments to a = 0 or a = 1. The corrected values were extrapolated us. the appropriate to give the actual intercepts in terms of the five constants. Using these new constants, new correction factors were calculated, and the cycle was repeated until no significant change occurred. One further point about the ternary theory requires emphasis. The labeling of the components as 1, 2, and 3 is of course arbitrary. However, in some cases, free corivection niay be caused by coupling between f l u x e ~ . ~The ~ , ~ restraint ~ that free convection does not, occur may restrict the species whose gradient is zero. IJnfortunately, this may be determined only by trial and error. The Journal of Physical Chemietry
Results Binary runs for the system sucrose-water and for three compositions of the system toluene-cyclohexane are shown in Table I. The sucrose run was made to check cell alignment and experimental procedure. Its value of P A differs from that of Gosting and Morris23 by O.l%, well within the expected error of 0.3%. The area under the fringe deviation graph Q is equal to The zero within the expected error of f 2 X three binary experiments on cyclohexane-toluene also show Q = 0 within this error, indicating that the solvents contain no significant impurities. The data required for the binary diffusion coefficients of 5 wt. % polystyrene in cyclohexane and in toluene are given in Table 11. The experimental values a)A (exptl.) and Q (exptl.) are plotted us. the square of the mass fraction difference ( A u ) ~for each trio of experiments. A straight line is drawn through the experimental points to the intercept a t ( A u ) ~= 0. This intercept is reported in Table I1 as “limit.” The ex(47) R. Wendt, J. Phys. Chem., 66, 1740 (1962). (48) G . Reinfelds and L. J. Gosting, ibid., 68, 2464 (1964).
1141
DIFFUSION OF MONODISPERSE POLYSTYRENE IN MIXEDSOLVENTS
'I)
0
h
n 4
A
d
v
W
0
A
h
3
W
0
A
I
oo N
m
8
Volume 69, Number 4
April 1966
E. L. CUSSLER, JR.,AND E. S . LIGHTFOOT
1142
Table 1V : Multicomponent Diffusion Coefficients and Partial Specific Volumes Column Solvent
Ternary pt. 0 1
w2 Vl
t2
iiJ
1 2 3
= = =
1 polystyrene cyclohexane toluene
2
3"
1 0.0500 0.0500 0.9133 1.340 1.164
Binary limit 0,0500 0.9500 ...
...
1 0.0500 0.9000 0.9133 1.164 1.340
...
4 1 = polystyrene 2 = toluene 3 = cyclohexane
5
6
3 0.0500 0.4751 0.9030 1,171 1.305
2 0.0500 0.0503 0.9464 1.162 1 ,299
Binary limit 0.0500 0.0000
1.8 -6.i -0.9 178.9
1.O l O b (0)" O.Ob (178)"
iill x 1i12 x iizl x iiz2x
107 107 107 107
8.9 -1.6 -8.9 203.1
(8.7)" (0.0)' (175)" (226)'
10.0 1.3 169.6 202.0
5.8 -6.7 14.5 171.8
R~~x R~~x R~~x R~~x
10-5 10-5 10-5 10-5
9 . 6 f 0.10 11.4 f 2 11.5 f 2 1.0 f 0.1
1 1 . 3 f O.lb
11.4 f 2 9.86 f 2 12.7 f 0 . 5 0.056 f 0.010
8.7 & 1 21.5 f 6 4 . 1 4 f 0.40 0.39 & 0.20
a
11.
... 11.3 f O . l b (0.445)'
12 f 6 57.8 f 1 -7 f 8 0.9 1
*
... ... ...
...
99.0 f 0 . 5 ... (0.564)'
Values calculated from multicomponent coefficients in column 1. Limiting values found from eq. 11, 21, and 23 and data in Table c Values estimated with eq. 21-24 from binary data in Tables I and I1 assuming p1 0 are in parentheses.
trapolation is tested by reading the smoothed values DA (calcd.) and Q (calcd.) at the known values of ( A u ) ~ , These values are also reported in Table 11. Initial data for eighteen ternary experiments at three different average compositions are given in Table 111. Parallel to the binary experiment, D A (exptl.) and Q (exptl.) are the actual experimental parameters, while DA (calcd.) and Q (calcd.) again test the extrapolation in ( A p J 2 . The limiting values of DA and Q at ( A p J 2 = 0 are included in the table as "limit." The values of the partial specific volumes for the three ternary compositions are given in Table IV. Since the partial specific volumes are not constant, these are averages across the concentration increment' and are known only to *5%. The generalized Fick's law coefficients and the generalized Stefan-Maxwell coefficients are also reported in Table IV. The diffusion coefficients of ternary point 1 considering toluene as the solvent are given in column 1; the coefficients for points 1, 3, and 2 considering cyclohexane as the solvent are given in columns 3, 4, and 5 , respectively. The binary limits of the multicomponent coefficients considering cyclohexane as the solvent are given in columns 2 and 6. Those limits which are exact are written without parentheses; those found by assuming the polymer concentration equals zero are written in parentheses.
Discussion The binary data in Table I1 show that the binary diffusion coefficient of polystyrene is a strong function The Journal of Physical Chemistry
of composition. In the system polystyrene-toluene, this results in a strong variation of DA with ( A w ) ~ . These conclusions are partially supported by literature data.49 As predicted by the theory, the effects of concentration may be eliminated by the straight line extrapolation us. (Aw)' at small values of ( A w ) ~ . The agreement with this extrapolation is very good, providing further verification of the theory, The values of D A read from the straight line differ from the observed values by less than O.l%, well within the expected error of 0.3%. In addition, the postulate that the concentration effects in the parameter Q may be eliminated by extrapolation in ( A u ) ~ is experimentally verified. The extremely good agreement is The fortuitous, since Q is known to only f 2 X value of zero for Q at ( A W ) ~= 0 indicated that in toluene the polymer behaves as a single species.44 Thus the limiting value of DAa t = 0 equals the binary diffusion coefficient at the average polymer concentration. The system polystyrene-cyclohexane also shows sharp variation of D A with ( A W ) ~ . However, the refractive index varies linearly with concentration in this system since J / A w is a constant. The variations of DA and Q , which are much larger for this system, are again removed by extrapolating us. ( A w ) ~ . However, the agreement is less good. The values of a>A read from the straight line differ by 0.6% from the (49) G. C . Park. J . chim. phys., 5 5 , 134 (1958).
1143
DXFFUSION OF MONODISPERSE POLYSTYREKE IX MIXEDSOLVENTS
experimental ones, twice the expected error. The parameter Q shows similar deviations. The concentration effects m this system are probably large enough to require that the ( A u ) ~term be included in the extrapolation. The intercept of Q x lo4 is 2 f 2. The nonzero intercept may be due to the precipitation of a small amount of polystyrene of molecular weight greater than 2Ei0,000.50 Such an impurity, which would result in a positive intercept, would be consistent with the slight cloudiness observed in these solutions. However, since the value of the intercept is equal to the expected error, this system is safely = 0 treated as a binary. Once again, 9 A at equals the binary diffusion coefficient at the average composition. The ternary data summarized in Table I11 show that both a)* and Q are strong, functions of ( A p J 2 . As one would suspect from the binary results, the effects are largest where the cyclohexane concentration is large (ternary point 2). The assumption that DA and Q vary linearly with (ApJ2 at small values of ( A p J z is verified by the results at all three compositions. The variation of a ) from ~ the straight line is in all cases less , is less than the expected error for a than 0 . 5 ~ owhich ternary experiment of 0.6%. The departures of Q x lo4from a straight line are less than the expected error of =t2 except for expt. 28, 41, and 42. In these three experiments, which show the largest deviations from Gaussian behavior yet reported, Q X lo4 is known to only +20. The cause of this large error is not known. The most probable explanation is that the variation of Q with ( A P , ) ~is significant in the extrapolation us. ( Ap,) z. = 0 provide the The values of a)A and Q at input data from which the D,, and the Ri5 are calculated. The D,,, found by the usual procedure,3s are known to *0.8 X lo-’. The Rt5 are known less accurately. Because the defining equations of the R , are nonlinear, the equations must be forced into the generalized Fick’s law form (or its inverse) and then solved.2 Thus it is the D,, which are always measured, and the R,, calculated from them, with a resultant loss of accuracy. As an example
Rlz :=
1 ~
pi72
(DiiDzz - DziDiz ) D1z
(25)
D12 may be known to only one significant figure. Thus if p1 is very small, Rlz will be poorly known. Deviations from Fick’s Law. This system shows some of the largest deviations from Fick’s law which have been observed. The cross term coefficients (D12 and D z ~ are ) large relative to the main term coefficients (Dl1 and D22). In particular, the coefficient
D12was greater than Dll for two of the three ternary points measured. Some of this deviation is due to designating one species (cyclohexane) as the solvent; but a significant part of the deviations are caused by sharp gradients in thermodynamic forces. At 28”, binary solutions of polystyrenetoluene and polystyrene-cyclohexane behave very differently. Quantitatively this difference is described by the chemical potential of the solvent5’
where ps is the chemical potential of the solvent, +p is the volume fraction of polymer, and $1 and 0 are experimental constants. For toluene, 8 = 160”K., while for cyclohexane, e = 307.2”K. Thus, at 28” polystyrene has a chemical potential of opposite sign in each of the two solvents. The result of this behavior is that these ternary solutions are nonhomogeneous: a molecule of polystyrene is surrounded by a cloud of high toluene concentration. This nonhomogeneity is responsible for a significant portion of the deviations from Fick’s law. However, the main cause of the concentration dependence of D,, is the choice of one species as the solvent. If the component designated as the “solvent” is present only in small concentrations, the cross terms can become very large. This is shown theoretically by eq. 8 and experimentally by columns 1 and 3 of Table IV. In spite of the lack of accuracy in their measurement, the R,, show smaller concentration dependence than the Dill since they do not deDend on labeling one species as “solvent.” They are pr3ferzble for problems where no single species is always present in excess. Neither the D i j or the Rij have any known molecular significance. Since the D,, are macroscopic phenolnenological coefficients, their physical significance would be expected to be complex. However, because the R,, are defined in a parallel form to the Stefan-Maxwell equations used for ideal gases, it was hoped that they would represent particular molecular interactions. Thus they would have a simple relation with binary coefficients, as is the case for ideal gases. No simple relation is found experimentally. In general, then, no fully satisfactory set of flux equations is yet known. The two most promising sets are the generalized Fick’s law coefficients D,, defined relative to volume average velocity and the generalized (50) A. R. Schultz and P. J. Flory, J . Am. Chem Soc.. 74, 4760
(1952). (51) 1’. J. Flory, “Principles of Polymer Chemistry,” Cornell University Press, Ithaca. N. Y., 1954, pp. 522, 523, 625.
Volume 69,.Vumber 4
April 1965
E. L. CUSSLER, JR.
1144
Stefan-Maxwell coefficients Ri., based on volume fraction driving force. The Dtj may be measured more accurately experimentally, but show greater concentration dependence than the Rtj. Neither set may generally be predicted from binary data. The molecular significance of neither set is known.
Acknowledgments. The authors wish to thank Professors P. J. Dunlop and L. J. Gosting for many helpful discussions of this research, and Professor Gosting for the loan of the Gouy interferometer used in the experimental work. This work was supported by National Science Foundation Grant Yo. 86-4403.
Multicomponent Diffusion Involving High Polymers.
11.
Characterization of Polydispersity from Diffusion Data
by E. L. Cussler, Jr.* Department of Chemical Engineering, University of W i s c o n s i n , Madison, Wisconein (Received September 16, 196.4)
Polydispersity may be described by average diffusion coefficients. This work develops the theory for the determination of these coefficients with the Gouy interferometer. If a particular form of the distribution function of diffusion coefficients is assumed, numberaverage and weight-average molecular weights may be calculated. Two polystyrenes of broad molecular weight distribution are studied in cyclohexane at 35'.
I f a polymer sample is sufficiently dilute, the various species diffuse independently. In this case, diffusion data can be used to measure the polydispersity; that is, the distribution of the sizes of the polymer molecules. Usually, instead of actually measuring this distribution, average molecular weights are determined by osmotic pressure, light scattering, or ultracentrifuge measurements. In a similar fashion, Daune and Benoit' suggested measurement of the average diffusion coefficients D1 and DZ D1 =
[lm lm D-':'g(D)dD]z
(1)
D-"'g(D)dD
Dz
D - "'g (D)dD where D is the diffusion coefficient and g(D) is the distribution of diffusion coefficients. Using a Jamin T h e Journal of Physical Chemistry
* Department of Physical Chemistry, University of Adelaide, Adelaide, South Australia. (1) M .D a m e and H. Benoit, J . chim. phys., 51, 233 (1955). (2) M. Daune, H. Benoit, and Ch. Sadron, J . Polymer Sei., 16, 483 (1955).
(2)
= =-
interferometer, Daune, Benoit, and co-workers2-5 evaluated D1 and D2 for several systems. This paper develops the theory of measurement of average diffusion coefficients with the Gouy interferometer and presents results for two polystyrenes. Theory. The Gouy interferometer measures the refractive index gradient for one-dimensional, isothermal diffusion.6 If the solution is sufficiently dilute, each polymer species diffuses independently. For a con-
(3) M.Daune, H. Benoit, and G. Scheibling, J . chim. phye., 54, 924 (1957). (4) R. N. Mukherjea and P. Remmp, ibid., 56, 94 (1959). (5) R. Varoqui, M .Jacob, L. Freund, and AX. Daune, ibid.. 59, 161 (1962). (6) L. J. Gosting, A d v a n . Protein Chcm.. 1 1 , 476 (1956).
