1028
ANITA F. SCHICK A N D S. J. SIiYGER
ON T H E COSCESTRATION DEPENDENCE OF T H E RATES OF DIFFUSION OF MACROMOLECULES I N SOLUTION AN INVESTIGATION OF FRACTIONS OF POLYSTYREXE~ AKITA F. SCHICK
AND
S. J. SIXGER2
Department of Chemistry, Polytechnic Institute of Brooklyn, Brooklyn, S e w York Received October 87, 1949 I. INTRODUCTION
In investigations of the properties of asymmetric protein and polymer molecules in dilute solution, one is constantly made aware of the strong dependence upon concentration which most physical properties exhibit. Whether these are equilibrium properties, such as osmotic pressure, or kinetic properties, such as sedimentation rate, the appropriate parameters must be extrapolated to infinite dilution to be interpreted as constants characteristic of the single molecules. It has therefore been somewhat puzzling to note that while sedimentation rates are often strongly concentration dependent, diffusion rates are often not particularly so; and that while sedimentation rates always decrease with increasing concentration, as is easily understood, diffusion rates usually increase, but with certain systems decrease, with increasing concentration. The basis for explaining these facts was established by Beckmann and Rosenberg (2), who showed that the general diffusion equation developed by Onsager and Fuoss (18),when interpreted for the systems under consideration, contains the answer to the problem. Two factors contribute to the concentration dependence of the diffusion rate of macromolecules in solution: the t,hermodynamic factor, which usually, but not always, tends to increase it, and the hydrodynamic factor, which always decreases it, with increasing solute concentration. The relative magnitudes of these factors determine the variation of the diffusion rate. This hypothesis was tested experimentally by Singer (24) with a series of fractions of cellulose acetate in acetone by means of cooperative osmotic pressure, sedimentation, and diffusion measurements, and was found to hold within the experimental errors encountered. I n order to verify the hypothesis more generally and contribute to a better understanding of its molecular basis, it is desirable to extend these investigations to more flexible molecules than cellulose derivatives, and to study the effects of different solvents on the various factors involved. We have therefore investigated a series of four polystyrene fractions in butanone and carbon tetrachloride and another fraction in six solvents, by diffusion, viscosity, osmotic pressure, and sedimentation experiments. 1 Taken in part from the thesis presented by A . F. Schick in partial fulfillment of the requirements for the Ph.D. degree at the Polytechnic Institute of Brooklyn, June, 1948. Presented before the 115th Meeting of the American Chemical Society, which was held in San Francisco, April, 1949. * Present address: Gates and Crellin Laboratories, California Institute of Technology, Pasadena, California.
RATES OF DIFFUSION OF MACROMOLECULES I N SOLUTION
1029
11. THEORETICAL
For a detailed account of the hypothesis under consideration, the reader is referred to the papers by Beckmann and Rosenberg (2) and Singer (24). It was shown that the diffusion constant in dilute solutions of large molecules can be given by the equation:
in which Do is the diffusion constant and fo the molar frictional coefficient at infinite dilution; B is the coefficient of the quadratic term in the power series expressing the osmotic pressure as a function of Concentration; M is the molecular weight of the solute (number-average for a polydisperse system), and c is its concentration; k , is the empirical constant obtained from sedimentation experiments (equation 2 below), and k D is the empirical constant obtained from diffusion measurements. Three assumptions were made in deriving equation 1: (a) that the gradient of the free energy of dilution of the system is the driving force of the diffusion process; ( b ) that the molar frictional coefficient at a finite concentration in a given system is the same in sedimentation and diffusion; and (c) that the frictional coefficient at the concentration c is given by the equation:
The first two assumptions are certainly reasonable, and can be tested by the accuracy with which equation 1 represents the data. Justification was found for the assumption of a linear increase of f with c in dilute solutions from the fact that much sedimentation data on polymer solutions seemed to indicate it, and from the theory of Burgers ( 5 ) predicting such a linear dependence in dilute solutions of spheres. Recently, however, Bueche (4) has developed a theory indicating that in very dilute solutions of large molecules f is proportional to cl’* and, at somewhat higher concentrations, to c. The exact nature of this dependence, while of considerable interest, is not vital to an experimental test of the quantitative dependence of k D on the variations of osmotic pressure and frictional ooefficient with concentration. Since the sedimentation data reported in this paper seem to satisfy equation 2 in the concentration range studied, we have continued to use the relation although we do not necessarily attach theoretical significance to it. The value of B in solutions of macromolecules has been the subject of many theoretical investigations in the past few years. This parameter measures the departure of the free energy of dilution of the system from its ideal value. In the case of protein solutions away from the isoelectric point, electrostatic interactions and differential salt distributions contribute markedly to its magnitude (19). With nonionic polymer solutions in which there are small heats of mixing, the relatively large values of B which are often observed are due mainly to large entropies of dilution, which in turn are related to the large volume fractions
1030
ANITA F. SCHICK AND S . J. SISGER
occupied by the molecules in solution (8, 28). As is indicated in equation 1 , in order to interpret k , theoretically we must know the relation between B and k s . For these polymer solutions, however, there has not yet been developed a comprehensive theory correlating B with the molecular dimensions of the solute and, in turn, with kinetic parameters such as intrinsic viscosity and k a . Lacking this theory, we can say little more concerning the anticipated variation of k , with B , hence k o , with a change in the solvent other than that the two parameters should vary in the same direction. I t develops that B increases somewhat more rapidly than k , in the polystyrene solutions which x e have investigated. TABLE 1 Characterzzatzon of polystyrene fractions* uox
FPAC'IION
D-8 D-2 B-72-16 B-72-9
PSAF
0.29 0.44 0.80 1.09 1.22
0.45 0.74 1.46 2.07 2.25
9.8 13.3 20.0 25.0 2.73t: 9.56
M.
107t
9.6 5.1 3.3 2.5 0 .jot 0.965
82,000
140,000 480,000 1,100,000 770.000
95,000 240, 000 560,000 910,000 680,000t 1,140, oOo5
* The results in the columns headed [?]but, and M,, except for those obtained with fraction PSAF, were determined by Goldberg (9) or Goldberg, Hohenstein, and Mark (10). The viscosity measurements on the first four fractions in butanone were performed a t 40.3"C., and in toluene a t 30°C. PSAF viscosity determinations in butanone and toluene were made a t 27°C. t The sedimentation results were corrected to 27°C. by the usual procedure (271, while the diffusion constants were all determined a t that temperature. The diffusion measurements of the first four fractions were performed in butanone solution. t. Determined in decalin solution. Determined in ethylbenzene solution.
.
111. EXPERIMENTAL
A. Mcterials Polystyrenes D-8 and D-2 are fractions of a solution-polymerized material which was prepared as follows: one hundred fifty grams of styrene which had been alkali-washed and vacuum-distilled was polymerized a t 60'C. in the presence of 0.5 per cent benzoyl peroxide and 150 g. of toluene for 70 hr. The degree of conversion was 72 per cent and the intrinsic viscosity of the polymer in toluene a t 28°C. was 0.45. The polymer was fractionated by Dr. A. Goldberg according to the procedure described by Goldberg, Hohenstein, and Mark (10). Polystyrenes B-72-18 and B-72-98 are fractions of an emulsion-polymerized material, the preparation and fractionation of which have been reported by Goldberg, Hohenstein, and Mark. Polystyrene P S h F is a somewhat more polydisperse fraction, refractionated from a solution-polymerized polystyrene 3
These fractions are designated e and c , respectively, in the paper by Goldberg, Hohen-
etein, and Mark (10).
RATES OF DIFFUSION O F MACROMOLECULES IN SOLUTION
1031
fraction by the same methods used to obtain the polymers discussed above. The data characterizing these fractions are recorded in table 1. The solvents were distilled from C.P. grade materials, and their boiling points, densities, and refractive indices agreed with the best recorded values (20).
B. Molecular weight determinations The number-average molecular weights, M,, of fractions D-8 and D-2 were determined by Goldberg (9) and those of fractions B-72-18 and B-72-9 by Goldberg, Hohenstein, and Mark (10) from osmotic pressure data obtained with the osmometer designed by Zimm and Myerson (29). That of fraction PSAF in butanone was determined in a similar manner. The osmotic pressure data for this fraction are graphed in figure 1.
r
2
I
c
l
:
i
l
b
b
+
Q
9
ib
C. ~/100CS
FIG.1: Osmotic pressure data for fraction PSAF in butanone at 27°C. grams per square centimeter.
r
is expressed in
The molecular weights, M,,obtained by combining sedimentation velocity and diffusion measurements were calculated by means of the Svedberg equation:
in which s and D are the sedimentation and diffusion constants extrapolated to infinite dilution, is the partial specific volume of the solute, and d is the density of the solvent. The sedimentation experiments were performed in an air-driven ultracentrifuge of the Beams-Pickels type, the operation of which has been described elsewhere (24, 25). C. The diffusion apparatus The diffusion experiments mere performed in all-glass cells, described by Stern, Singer, and Davis (26). In operating this cell the bowdary between solvent and solution is created at a stopcock, and this boundary is then carefully raised by hydrostatic pressure into the optical chamber of the cell. The scale method of Lamm (16), in conjunction with a conventional optical system, was
1032
ANITA F. SCHICK AKD S. J. SINGER
utilized for the observation of the refractive-index gradients in the diffusing systems. The experiments were carried out in an apparatus designed to permit the simultaneous performance of several lengthy experiments. The thermostat bath in which three diffusion cells mere mounted could be moved in a direction perpendicular to the stationary optical system so that one cell at a time could be moved into the optical path. The movement of the bath was designed to be as free of vibration as possible, and independent experiments on solutions of mannitol in water indicated that the motion of the bath did not disturb the diffusion process to any significant extent. A detailed description of the apparatus and its operation will be published elsewhere.
D. Calculation of the d i f u s i o n constant The standard methods of calculation of the diffusion constants from free diffusion measurements as described by Lamm (16) were followed. For comparison, two values of the constant, the area-maximum ordinate value, D,, and the second-moment value, D2, o, were determined. These were obtained from the equations:
A* F 2 D, = 4rtH2
(4)
In these equations A is the area, H the maximum ordinate, and u the standard deviation of the diffusion curve at the time t after the start of diffusion. F is an apparatus constant, determined by various optical distances in a given experiment. The second-moment diffusion constant, under certain circumstances a weight-average value (11, 14), \vas utilized in calculating molecular weights by means of equation 3. The value of t was in most cases a corrected value obtained in the manner followed by Stern, Singer, and Davis (26). (A,")* and U* mere plotted against t', the time after the formation of the boundary between solvent and solution. Straight lines drawn through these points usually intercepted the time axis a t some negative value, and the absolute magnitude of this time correction was then added to t' to give t. As has been observed frequently, the process of moving the solvent-solution boundary from the position of its formation into the optical part of the cell results in a slight disturbance of the boundary, which can be corrected for in the above-mentioned manner. Typical plots for the zero-time corrections are shown in figure 2 . The best criterion for the reliability of a given experiment is obtained when the experimental diffusion curves are transformed to a set of normal coordinates, as was suggested by Lamm (16). The normal coordinates X and Y are derived
RATES O F DIFFUSION O F MACROMOLECULES IS SOLUTION
1033
from the experimental coordinates z and Z, which are obtained from the scale photographs by the following equations:
x = -2az
y=-
5uz /Zdr
This transformation eliminates the variables of time, concentration, and optical constants, and in a perfect experiment the experimental points obtained a t different times will fall on the same normalized curve. The choice of the normalized
t.10-5
I
FIG.2 : Zero-time correction plots for (a) fraction D-8, (b) fraction D-2, and (e) fraction B-72-18, all in butanone.
ideal curve is arbitrary. We have followed the procedure of Beckmann and Rosenberg (2) in choosing a normal curve of 2.5 square units and maximum ordinate of 2. Examples of the normalized diffusion curves which we obtained are given in figure 3, in which the dashed curves represent the normalized ideal curves. E . The evaluation of k , from diflusion data The deviation of the experimental normalized curve from the ideal serves as a measure of the concentration dependence of the diffusion constant. Qualitatively, a positive concentration dependence is manifested by a displacement or skewness of the experimental curve toward the solvent side (to the left of the original
1034
AKITA F.
SCHICK AR'D S. J . SIR'GER
boundary in figure 3), while the curves with a negative concentration dependence are displaced toivard the solution side. Quantitatively, the concentration dependence of the diffusion constant was calculated by two methods. The procedure suggested by G r a b (11) assumes a
X
FIG. 3: Normalized diffusion curves for (a) fraction PSAF and (b) fraction B-72-9 in butanone.
linear dependence of D upon c, while that of Beckmann and Rosenberg (2) does not introduce this assumption. The value obtained by the former method is given in terms of the normal coordinates by:
DO=
Da.0
(1 f
2)
RATES OF DIFFUSION O F MACROMOLECULES Ih' SOLUTION
1035
where X,,, is the value of X at which Y has its maximum value, Y,,,,,, and co is the initial concentration of the solution. The Beckmann and Rosenberg procedure involves plotting D,/Dn,o as a function of 2c/c0, where
in which Yo is the ordinate of the ideal normalized curve and AY = Y - Yo. The curves so obtained in our experiments were usually linear near the value 2 c / ~= 1, and k D was evaluated from this portion of the curves. The values sf k D obtained independently by these two methods checked closely, indicating that the concentration dependence of the diffusion constant is essentially linear in the systems studied.
F . T h e calculation o j k o f r o m sedimentation and osmotic pressure data The indirect calculation of k , from sedimentation and osmotic pressure data according to equation 1 requires the knowledge of the parameters B , M , and k,. The values of B for the systems studied were obtained from the osmotic pressure data of M. Schick (21). The values of M used were the number-average molecular weights recorded in table 1. k , was evaluated by plotting the reciprocal of the sedimentation constant against concentration (11); the fact that linear relationships were usually obtained from such plots justifies the use of equation 2 for the purposes of these experiments.
G. Viscosity determinations In order to aid in characterizing the systems investigated, viscosity measurements were made on solutions of fraction PSAF in the solvents used in the diffusion studies. Ostwald viscosimeters (5-ml. capacity) were employed, and the small kinetic energy corrections were neglected. IV. RESULTS
The data obtained from the analysis of the diffusion curves are listed in table 2. In figure 2 are shown several graphs from which the zero-time corrections were obtained, while in figure 3 are represented several normalized diffusion curves. Comparison of k D values obtained by the G r a l h and the Beckmann and Rosenberg methods is made in tables 3 and 5 . I t may be seen that the two values generally agree quite well. In figure 4 we have reproduced a few examples of plots according to the Beckmann and Rosenberg procedure for evaluating kD. Sedimentation experiments were pwformed with fractions D-8, D-2, B-72-18, and B-72-9 in butanone, and with fraction PSAF in ethylbenzene and decalin. The results of these measurements are given in table 4,and are plotted in figure 5 . Measurements of the partial specific volume were made pycnometrically
1036
A,?'ITA F. SCHICK AND S. J. SINGER
TABLE 2 D d a obtained f r o m the analysis of the diffusion curves
0.3918 g./100 cc.; F = 0.7456; zero-time correction = 20,oOO 8ec.
c =
1 Fraction D-8 in butanone
1 Average, . , . . , , , , , , ,
-
,
,
1
rerondr
185,060 222,380 294,680
cm
1
0 06165 0 06013 0 06200
1
., , , , , ., , , , , ,, ,
, ,
,
1
cm
cm.l
1
0 0342 0 0319 0 0272
0 5190 0 6364 0 8173
1 ,
, , , ,
I,
... . .., , ., ,..,,
7.80 7.96 7.71 7.S2
~
7.77 7.07 7.80
1
7.5,
c = 0.4168 g./l00 cc.; F = 0.7446; zero-time correction = 30,000 8ec. Fraction D-2 in butanone
'1
142,620 194,940 226,380 310,800
0 06387
1
0.0514 0 06311 0 0424 0 06358 0 0395 0 06452 1 0.0334
.. .. ........
Average.. , . , . , , . , . , .
.
~
1 I
5.05 5.07 4.95 4.95
0.2597 0 3568 0 4045 0 5550
.. ... ... ... ., ..
.
..,.I
5.01
4.78 5.01 5.05 5.30 1
5.01
c = 0.3977 e./100 cc.: F = 0.7452: zero-time correction = 65.000 sec Fraction B-72-18 in butanone
179,780 321,800 496,520 672,620
' '
0 05660
' 1
0.05715 0.05767
1
0 05861
0 0488 0 0348 0.0282 0.0245
i
, ~
3.91 3.53 3.59 3.54
3.63 3.63 3.65 3.64
3.6r
3.6,
0.2471 0.3994 0.6415 0 8579
__
Average
c = 0.4030 g./100 cc.; F Fraction B-72-9 in butanone ~
l
=
0.7441; zero-time correction = 140,000 sec.
315,380 396,680 571,880 740,240
'1
0 05351 0 05517 0 05348 0 05622
~
0 0417 0 0359 0 0284 0.0246
, 1
~
0 0 0 0
'
2980 4410 I 5652 1 7822
Average
2 3 2 2
62 15 74 93
'
1
2 3 2 3
30 30 73 11
~2 86
2 86
~~
c = 0.4062 g./l00 cc.; F = 0.7359; zero-time correction = 0
Fraction PSAF in buta-1 none 1
164,940 1 260,220 280,500 339,600
~
~
0.06052 0.06089 0.06214 0.06076
~
~
0.0615 0.0498 0.0484 0.0433
1 ~
'
0.1468 ' 2.47 2.43 0.2280 0.2563 2.53 0.3204 2.61
2.59 2.53 2.59 2.56
-1
A v e r a g e . , , , . . , , . . . . . . . . . , . . . . . , , , . , . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . ' 2.51
2.57
c = 0.4022 g./100 cc.; F = 0.7359; zero-time correction = 250,000 sec. Fraction D-8 in CCl,
1 ~
Average
406,600 499,120 586,480
1
0.03615 0.03736 0.03785
1
0.0196 0.0186 0.0180
, 1
0.5475 0.6368 0.7974
3.25 3.14 3.39
3.21 3.03 3.00
3.28
3.08
1037
RATES O F DIFFGSION OF MACROMOLECULES IiS SOLUTION
TABLE 2-Continued
c = 0.4001 g./100 cc.; F = 0.7337; zero-time correction = 300,000 8ec.
i
Fraction D-2 in CCl,
Average. .
, , ,
., ,
, , , , , ,.
,L 1 548,760
1
803,580 , , , , , ,
,
., .
cm.2
cm.
0.03875 0.03868 0.03813
0.0223 0.0207 0.0182
..
., ,
, , , , , ,
,
1
1
cm.*
1
0.4771 0.5258 0.6712
1
, , , , ,
, ,
,
,
I
2.34 2.22 2.25
~
i
2.36 2.35 2.34
_____ . .~ 2.2,
2.35
c = 0.4028 g./100 cc.; F = 0.7371; zero-time correction = 100,000 sec.
',
Fraction B-72-18 in CC1,
'
303,640 358,180 459,760 612,760
,
'
0.03998 0.03939 0.03955 0.04073
~
~
0.0408 0.0367 0.0324 0.0286
___-
Average . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c = 0.6206 g./100 cc ; F
Fraction B-72-9 in CCI,
Average. . . . . c
=
~
1
0.7370; zero-time correction
561,320 0.06250 754,880 , 0.06453 907,220 0.06377 1:099,880 0.06285 ~
~
~
~
:::::
=
..,
~
1
0.2498 0.3431 0.0416 ' 0,4039 0.0372 1 0.4727 1
1
.
1
1.41
1.40
395,000 sec
~
. . . . . . . . . . . . . . . .. . . . . . . . . . . . , . . . .
. .. .
1
1.21 0.91 1.23 1 1.10 1.20 1.12 1.17 1.12
1
~
_____ 1.20
,1
1.08
0.6602 g./lOO cc.; F = 0.7371; zero-time correction = 500,000 sec.
Fraction PSAF in CCI, ~
Average. . .
=
.
,
... .
,
666,920 860,180 1,012,940 1,205,000
0.06558 0.06643 0.06550 0.06500
., . . . . . .......
.. .
~
1
0.0514 0.0439 0.0395 0.0364
1
1
. . . . . . . .. ..
0.2787 0.3709 0.4170 0.4874 ....
1
1
1.14 1.17 1.19 1.10
. . . .~ 1.15
1 ~
1.06 1.15 1.17 1.14
'
1.1~
1
1 47 1 48 1 46 1 26
c = 0.7952 g./loO cc.; F = 0.7355; zero-time correction = 100,000 sec
Fraction PSAF In ethylbenzene
352,300 426,820 529,780 688.540
1
0 05805
0 05745 0 05663 0 05823
'
0 0530 0 1923 0 0475 0 2228 0 0422 I 0 2744 0 0365 0 3819 ~
1 1
1 48 1 41 1 40 1 50
,
1 45
~
Average c
=
0.8424 g./100 cc.; F
Fraction PSAF in toluene ~
' Average. . . . , . , . . . . . .
=
333,200 510,560 668,540 927,560 .
1 4?
0.i350; zero-time correction = 80,000 sec.
~
~
'
0.06113 0.06220 0.06160 0.06040
1, ,
1
0.0529 0.2083 0.0424 0.3314 0.0367 0.4064 0.0309 , 0.5556
. . . . . .. ... . .. . .. . .. .. . . .
~
. . ., . . .... .
.1
1.68 1.75 1.59 1.58 1.66
1 ~
1
1.72 1.81 1.76 1.77 1.77
1038
ANITA
F. SCHICK AND 5. J. SINGER
TABLE 2-Coiaclz~ded
c = 0.4040 g./lOO cc.; F = 0.7395; aero-time correction = 12,500 8ec.
Average.
1
173,780 246,500
0.06194 0.06294 0.06161
0.0665 0.0621 0.0530
0.1436 ~
ii
708,420 967,440 1,314,420
0.06906
0.06856 0.06892
~
1
0.2366
c = 0.7967 g./lOO cc.; P = 0.7347; zero-time correction
-
2.61
2.51
2.63
2.39
, 12:i 12:
0.1820
. . . . . . . . . . . . . . . . .. , . . . . . . . . . . . ..
Fraction PSAF in decalin
i 1
cm.:
Crn.
m.1
Fraction PSAF in ethyl acetate
__~
120,OOO 880.
0.1237 ::% 0.1744 0.0578 0.2157 ~
~
4.71 4.87 4.53
4.74 4.71 4.65
Average TABLE 3 V a r i a t i o n of DOa n d kD with moleciilar weight B. AND R. METEOD IPACIION
1
G R A L ~ NXEIROD
SOLVENT
Do X 10'
D-8 . . . . . . . . . . . . . . . D-2. , , , . . , . . , . . . . . B-72-18. . , . . . . . . . B-72-9. . . . . . . . . . . PSAF, . . . . . . . . . . , .
Butanone
D-8.. . . . . . . , . , , . . , D-2 . , , . . . . . . . . . . . B-72-18. . , . . . . , , , B-72-9. . ...... PSAF . . . . . . . . . , .
CClr
'
9.58 5.14 3.28 2.53 2.51
-0.86 -0.12 0.49 0.40
4.43 2.27 1.41 1.04 1.10
...
0
2.51
0
-1.31 0 0 0.50
4.40 2.27 1.41 1.06
-1.27
0 0 0.48 0.08
(27), and 7 for polystyrene was found to be 0.90 f 0.02 in butanone and 0.91 f0.02 in ethylbenzene. Signer and Gross (22) have determined = 0.91 & 0.03 for polystyrene in decalin. The sedimentation and diffusion constants extrapolated to infinite dilution, combined with the measurements of partial specific volume, were used to evaluate molecular weights by equation 3. These are the M , values recorded in table 1. Comparison of k , values obtained directly from diffusion measurements with those obtained indirectly through sedimentation and osmotic pressure measurements according to equation 1 is made in table 6. k D (calculated) was obtained by evaluating the ratio [l (2BM/RT)c]/l k,c at c = 0.5 g./lOO cc. from the experimentally determined values of B , M , and k,, and then calculating k D from equation 1.
v
+
+
ie
8
LI
10
-
I
I
I
I
I
-
-
10
5
s
Dl.0
-
I,
-
0 9
I
1
4
6
8
10 2 Cjc
I2
,(I
Le
l 111
-
0
,
l
z Cor
FIG.4
l
l
l
3
4
3
6
VIOOIC
FIG.5
FIG.4: Diffusion data obtained in butanone solution plotted according to the Beckmann and Rosenberg procedure for determining k ~ The . dashed lines are drawn through the linear portions around 2c/co = 1. FIG.5: Sedimentation data for polystyrene fractions in butanone solution corrected t o 27T. TABLE 4
Sedimentation data on volwstwrenes SOLVENT
s21
K./loo
Butanone.
......
D-8
CC.
0.600 0.300 0.150
7.9 8.8 9.3 9.8'
0.43
D-2
0.501 0.251 0.125 O.Oo0
0.466 0.233 0.116
10.9 12.0 12.7 13.3'
0.48
B-72-18
0.480 0.240 0.120
0.446 0.221
13.4 16.2 17.7 20.0'
1.11
16.9 20.1 22.2 22.6 25.0'
1.75
3.55
O.Oo0
3.90 5.55 6.30 7.40 8.10 9.5:
0.400 0,202 0.076 O.Oo0
2.13 2.41 2.62 2.73'
0.73
0.110
O.Oo0 B-72-9
0.300 0.150 0.086 0.075
O.Oo0
..
Decalin. . . , , . , , . ,
101'
0.552 0.279 0.140
O.Oo0
Ethylbenzene..
x
g./lOO cc.
PSAF
PSAF
0.402 0.200 0.167 0.082 0.044
' Extrapolated. 1039
0.276 0.140 0.079 0.070
TABLE 5
k, values for fraction PSAF SOLVENT
Toluene.. . . . . . . . . . . . . . . . . . . . . . . . . . Ethylbenzene.. . . . . . . . . . . . . . . . . Carbon tetrachloride. Butanone. ...................... Ethyl acetate. Decalin.. ..........................
* p values obtained from M. Schick (21). t Interpolated in the values of Boyer and
Spencer (3) obtained from swelling pressure measurements, assuming a constant difference between p values so obtained and those derived from osmotic pressure measurements. TABLE 6
Concentration dt endence of D
1
FRACTION
, + 2 RRTM c *
SOLVENI
~ D-8. . . . . . . . . . . . . . D-2.,. . . . . . . . . . . . B-72-18.. . . . . . . . . B-72-9.. . . . . . . . . . PSAF.. . . . . . . . . . PSAF.. . . . . . . . . .
1.12 1.21 1.71 2.63 4.39 1.20
Butanone Butanone Butanone Butanone Ethylbenzene Decalin
* Calculated a t the concentration
e
-
~
1.21 1.24 1.56
~
0.92 0.97 1.10 1.40 1.58 0.88
1, 1 I
~ -0.16
-0.92 -0.12 0.55 0.64 1.3 -0.15
-0.06
0.20 0.80 1.2 -0.24
i ___i_____j
0.5 g./loO cc.
32 30
lop D.
20
17
28
3
2
2r
I
12
0
eo %P
IUTWONE
LIB IO I*
I2
ID 8 0
1
2
3
4
cme.mrot~a I W , O O I L ~
S
6
40
10
se
54 5s 109 Ms
10
bo
FIG.6 FIG.7 FIG.6: Viscosity data for fraction PSAF in the following solvents a t 27°C.: (1) toluene; (2) carbon tetrachloride; (3) ethylbenzene; (4) dioxane; (5) morpholine; (6) butanone; (7) decalin. FIG.7: Dependence of the diffusion constant on molecular weight for polystyrene fractions in butanone and carbon tetrachloride, and of the sedimentation constant on molecular weight in butanone. Curve B represents the sedimentation data. 1040
RATES O F DIFFUSION O F MACROMOLECULES I K SOLUTION
1041
I n the terminology of the theory of polymer solutions based on the quasilattice model (7, 13), the parameter B is related to the compatibility constant, p , by the equation: (0.5 - p)RT B= 1’1 di where VI is the partial molal volume of solvent, dz is the density of pure solute, and p is a constant characteristic of a given solute-solvent system. In table 5 are listed the fi and B values, calculated using the value dz = 1.096 for polystyrene a t 27’C., along with the intrinsic viscosities obtained from the viscosity data plotted in figure 6. V. DISCUSSIOK
A . Concentration dependence of the diffusion constant The results listed in table 6 indicate that in those systems for which sedimentation as well as diffusion and osmotic pressure data were obtained, the concentration dependence of the diffusion constant is given by equation 1 within the experimental errors. The only serious exception occurs with the Ion-estmolecular-weight fraction, 11-8,in butanone, in which the observed concentration dependence appears to be larger than that calculated. This may be due to some unknown peculiarity of this fraction, since a similarly large negative concentration dependence is observed in the experiment in carbon tetrachloride solution. On the whole, however, the theoretical development which was outlined earlier in this paper appears to be confirmed by our data. The occurrence of negative values of k D for some of the systems investigated, indicating that the rate of diffusion of the solute is slower in more concentrated solutions, is particularly interesting. Jullander (14), studying the diffusion of some low-molecular-weight nitrocelluloses in acetone, observed negative k D values but attributed them to experimental errors. Xeurath and Saum (17) found that a large negative k D existed with solutions of tobacco mosaic virus protein in phosphate buffer, and suggested that intermolecular interference diminished the diffusion rate. Our results establish the significance of a decrease in diffusion rate with increasing concentration, and indicate that it may be expected to occur whenever a combination of factors makes BBMIRT smaller than k..
B. The effect of molecular weight on k D I n terms of the hypothesis that is being tested in this investigation, the problem of the dependence of k D on molecular weight must be resolved into the two independent problems of the effects of molecular weight on the two terms 2 B M / R T and k,. Since B is essentially independent of molecular weight ( 2 8 ) , the former of these terms varies linearly nith X. If we express k , as a function of Jf by the following empirical equation:
t,
=
constant.Jfp
(12)
1042
ANITA F. SCHICK AND 8. J. SINGER
and use the number-average molecular weights of the first four polystyrene fractions, we find from our sedimentation data in butanone that p = 0.6 =k 0.1. From equation 1 it follows that k D should increase with molecular weight, as is observed experimentally. In investigations of the diffusion properties of nitrocelluloses in acetone and amyl acetate, Jullander (14) found that k D increased gradually with molecular weight. Our studies indicate similar trends for polystyrene fractions in butanone and carbon tetrachloride. It should be emphasized, however, that for a given system the variation of k Dwith M depends entirely on the independent variation of k, with M . With the system cellulose acetate in acetone, Singer (24) found that k, varies approximately linearly with M in the molecular weight range 10,000-200,000 and although quantitative calculations of k D were not performed, it might be inferred that k D is therefore independent of M in this system. The characteristics of the dependence of k, upon M are apparently similar to those of the dependence of the intrinsic viscosity upon M . For polymer solutions it has been demonstrated experimentally and theoretically that if the [q]-M relation is expressed as follows: [q] =
KMQ
(13)
a varies from the value 0.5 in poor solvents (barring association effects) to the value 1.0 in good solvents. In the polystyrene-butanone system (10) a = 0.53, while in the cellulose acetate-acetone system (1) a = 0.82. It is therefore reasonable to expect that the exponent p in equation 12 should be larger for the cellulose acetate-acetone system than for the polystyrene-butanone system.
C . T h e effect of eolvent o n k D Our data suggest an interesting effect produced by varying the solvent, which should be tested by other diffusion measurements on a variety of systems. In table 5 are listed values of k , for fraction PSAF in various solvents. It appears that k , varies from a value around zero in poor solvents to 1.0 in good solvents for this particular polymer. Since M does not change in these systems, these. results suggest that B varies more rapidly in going from poor to good solvents than does k.. This is confirmed independently by the determination of k. by sedimentation experiments in ethylbenzene and decalin. I t is not possible at present to understand this fact in detail, owing to the absence of a suitable theory correlating k, and B. Qualitatively, we may say that as the solvent becomes poorer, Le., as the free energy of dilution for the system at a given low concentration becomes smaller, associated changes occur in the configurations of the polystyrene molecules, reducing their average lengths and penetrability to solvent. The hydrodynamic interference term, k,, may therefore be expected to decrease as the solvent becomes poorer and as B decreases. The exact nature of this decrease is a matter for future developments in the theory of polymer solutions to explain.
RATES O F DIFFUSION O F lL4CROMOLBCKLES I N SOLUTION
1043
D . Molecular weight determinations by sedimentation and diffusion measurements Although considerable research has been performed with cellulose derivatives, and molecular weights calculated by the Svedberg equation (equation 3) have been shown to apply in general within the experimental errors, little investigation has been made of the more flexible polymers such as polystyrene. Signer and Gross (22) performed equilibrium and rate sedimentation, but no diffusion, experiments on some unfractionated polystyrenes; recently, G r a l h and Lagermalm (12) have reported sedimentation rate experiments with polystyrene fractions and plan diffusion experiments with them. The molecular weights calculated by equation 3 are rather complicated average values, usually falling between the number-average and the weightaverage values (14, 23). In general the molecular weights, M E ,listed in table 1 therefore compare favorably with the number-average values, Mn. For fraction B-72-9 M , < M,,, but the difference is probably within the experimental errors for this high-molecular-weight fraction, for which the limiting osmotic pressure is difficult to determine accurately. The value of M , for fraction PSAF in ethylbenzene is 1,140,000, while M , = 770,000. Here the indication is that PSAF is more heterogeneous than the other fractions. This is confirmed by the relatively large values of [q] for this fraction in butanone and toluene (table 1). Since [q] is nearly a weight-average quantity, it reflects the presence of high-molecularweight species. The value M a = 680,000 for PSAF in decalin solution is too low, apparently owing to a value of DOwhich is somewhat too large.
E. The dependence of D upon $1 Recently, theories have been developed relating the diffusion and sedimentation rates as well as the intrinsic viscosity to the molecular weight of flexible long-chain molecules. In figure 7 , log DOis plotted against log M, for the four sharp polystyrene fractions in butanone and carbon tetrachloride. In butanone the curve is represented empirically by
DO = constant.-M,-O
53
(14)
59
(15)
and in carbon tetrachloride by
Do = constant
The values of the exponents are in agreement with the values predicted by Debye and Bueche (6) and by Kirkwood and Riseman (15) and others, of -0.5 to - 1.O, where the higher iralues correspond to the more extended configurations of the solute molecules and hence to the better solvent. The experimental errors are such that the value -0.53 is not quite significantly different from -0.59, but it is interesting that thermodynamically carbon tetrachloride is a better solvent for polystyrene than is butanone. More extensive measurements are reauired, however, to evaluate the exponents more accurately.
1044
ANITA F. SCHICK AND 6. J. SINQER VI. SUMMARY
The formulation of the concentration dependence of the diffusion constant in terms of two independently measurable quantities, the increase in osmotic pressure and in the frictional coefficient with concentration, has been verified. This indicates that the basic assumptions involved in this formulation are satisfied: namely, that the gradient of the free energy of dilution is the driving force of the diffusion process, and that the frictional coefficients in sedimentation and diffusion are the same. Both positive and negative values of the concentration dependence of the diffusion rate have been found for polystyrene solutions. This has been shown to be possible, depending on the relative contributions of the hydrodynamic and thermodynamic factors to the diffusion rate. The variation of the concentration dependence of the diffusion constant with respect to molecular weight and solvent power has also been discussed. The authors wish to express their gratitude to Professor K. G. Stern, who supervised the construction of the diffusion apparatus employed in this study. The polystyrene fractions D-8, D-2, B-72-18, and B-72-9 were obtained through the kind cooperation of Dr. A. I. Goldberg. REFERENCES
(1) BADGLEY, W.,A K D MARK,H . : J. Phys. & Colloid Chem. 61, 58 (1947). C. O.,AND ROSENBERG, J.: Ann. N. Y. Acad. Sci. 46,209 (1945). (2) BECXMANN, (3) BOYER,R . F., AND SPENCER, R. 8 . : J. Polymer Sei. 3, 913 (1948). (4) BUECHE,A. M.: Personal communication. (5) BURGERS,J. M.: Proc. Acad. Sci. Amsterdam 44, 1045, 1177 (1941); 46, 9, 126 (1942). (6) DEBYE,P . , AND BUECHE,A. M.: J. Chem. Phya. 16, 573 (1948). P. J.: J. Chem. Phys. 10, 51 (1942). (7) FLORY, (8) FLORY, P. J.: J.'Chem. Phys. 13, 453 (1945). A. I.: Personal communication. (9) GOLDBERG, (10) GOLDBERG, A. I., HOHENSTEIN, w. P., AND MARX,H.: J. Polymer sci. 2, 503 (1947). (11) GRALPN,N. : Inaugural Dissertation, Uppsala, 1944. N ,. , AND LAGERMALM, G.: Festskr. J. Arvid Hedvall 1948, 215. (12) G R A L ~ N (13) HUGGINS, M.L.: Ann. N. Y. Acad. Sci. 45, 1 (1942). (14) JULLANDER, I.: Arkiv Kemi, Mineral. Geol. 2lA, No. 8 (1945). J. G., AND RISEMAN, J.: J. Chem. Phys. 16,565 (1948). (15) KIRKWOOD, (16) LAMM,0.: Nova Acta Regiae Soc. Sci. Upsaliensis 10, 1 (1937). H.AND SAUM,A. M.: J. Bioi. Chem. 126, 435 (1938). (17) NEURATH, (18) ONSAGER, L. AND Fuoss, R . M.: J. Phys. Chem. 36. 2689 (1932). (19) SCATCHARD, G.: J. Am. Chem. SOC.68, 2315 (1946). (20) SCHICK,A. F.:Ph. D . Thesis, Polytechnic Institute of Brooklyn, June, 1948. (21) SCHICK, M.J.: P h . D . Thesis, Polytechnic Institute of Brooklyn, June, 1948. (22) SIGNER,R., AND GROSS,H.: Helv. Chim. Acta 17, 59 (1934). (23) SINGER,S. J.: J. Polymer Sci. 1,445 (1946). (24) SINGER,S.J.: J. Chem. Phys. 16, 341 (1947). (25) STERN,K.G., SINGER,S. J., AND DAVIS,S.: Polymer Bull. 1,31 (1945). (26) STERN,K. G., SINGER,S. J., AND DAVIS,S.: J . Biol. Chem. 167,321 (1947). T.,AND PEDERSEN, K . 0.: The Ultracentrifuge. Oxford University Press, (27) SVEDBERG, London (1940). (28) ZIJIJI, B . H.:J. Chem. Phys. 14,164 (1946). (29) ZIMM, B . H.,A N D MYERSON, I.: J. Am. Chem. SOC.88,911 (1946).
