= = = = = = = = = = = =
= = = =
= = = =
lower limit in eq 29 upper limit in eq 29 transformation matrix such t h a t TTHT = D temperature lower temperature limit upper temperature limit scaled, dimensionless temperature diagonal matrix of weighting factors elements of T t ’ n-dimensional vector of variables elements of z x a t a base point candidate optimum x scaled x vector search vector elements of Ax transformed variables elements of y y vector a t the end of the previous iteration elements of ,yo
G R E K KLETTERS = scalar parameter in one-dimensional search cy0 = optimal value of a along the search vector p = maximum steepest descent weight factor relative to unit’y y = steepest descent distance factor 6 = steepest descent threshold factor 6yi = perturbation in variable i for first derivative calculations 6‘yi = perturbation in variable i for second derivative calculations 6lky = n-rector with the kth element containing 6’yk and all others zero 6’k,2y = n-vector with the kth element = 6‘yk, the Ith element = 6’yl, and all other elements = zero LY
=
e
discrimination factor for distinguishing small eigenvalues
SUPERSCRIPTS T = vector or matrix transposition operator ( i ) = refers to iteration i Literature Cited
Bard, Y., Malh. Comp. 22,665 (1968). Booth, A. D., “Numerical Methods,” Butterworths, London, 14.66 -V1l.
Davidon, W. C., Atontic Energy Commission R. and D . Report, ANL-6990 (Rev.) (1959). Fariss, R. H., Law, V. J., Prepared Lecture Kotes, Chemical Engineering Department, Tuhne University, Ken- Orleans, La., 1968. Fletcher, R., Powell, AI. J. D., Computer J . 6 , 163 (1963). Greenstadt, J., Math. Computation 21, 360 (1967). Kowalik, J., Osborne, 11. R., “Methods for Gnconstrained Optimization Problems,” Elsevier, Sen- York, Tu’. Y., 1968. Law, V. J., Prepared Lecture Notes, Chemical Engineering Department, Tulane University, Kew Orleans, La., 1967. Perlis, S., “Introduction to Algebra,” Blaisdell, Waltham, Mass., 1966. Powell, 11.J. D., ComputerJ. 7 , 155 (1964). Rosenbrock, H . H., Storey, C., “Computational Techniques for Chemical Engineers,” Pergamon Press, New York, S . Y., 1966. Shah, B. V., Buehler, R. J., Kempthorne, O., J . S I A M 12, 74 (1964). Wilde, 1). J . , “Optimum Seeking Llethods,” Prentice-Hall, Englewood Cliffs, Ii.J., 1964. RECEIVED for review June 20, 1968 RESUBMITTED July 15, 1971 ACCEPTED November 27, 1971
Diffusion in Dilute Polymeric Solutions Herman R. Osmers*’ and Arthur B. Metzner University of Delaware, Xewark, Del.
The mutual diffusivity of three liquid solutes in two dilute aqueous polymeric solutions was measured as a function of polymer concentration with an optical wedge. The experimental data obtained yielded solute diffusivities which never deviated from that in the pure solvent b y more than 20% and were always smaller. This i s in contrast to expectations raised in the prior literature that mass transfer of solutes in solvents might increase significantly on addition of small amounts of polymer. The data are correlated by a thermodynamic model in which the easily measured excess volume of mixing polymer and solvent i s the principal parameter. Data for the concentration dependence of the diffusivity of polyacrylonitrile in dimethylformamide are presented. These suggest the need for inclusion of additional molecular parameters in polymer diffusion theories and development of a phenomenological theory.
Recent’ly, a number of investigators have examined the problem of diffusion of solutes through solutions consisting of a polymer dissolved in a suitable solvent. An examination of the data reported by these authors (Biancheria and Kegeles, 1957; Boss, et al., 1967; Hoshino and Sato, 1967; Li, 1967; Li and Gainer, 1968; McCabe, 1967; Metzner, 1965; Quinn and Blair, 1967) clearly indicat’esthat, for those syst’ems in which 1
To whom correspondence should be sent at the Department
of Chemical Engineering, University of Rochester, Rochester,
K.Y . 14627.
the diffusion coefficient decreases as polymer is added to the solvent, the decrement in the solute diffusivity is a t least an order of magnitude less than the increment in the solution viscosity, which is kiioan t,o increase drastically on addition of polymer. This implies that the macroscopic uiscosity is not a suit,able iiidex of resistance t o molecular motion, but rather that there are regions wit,hiii the solution in which the transport resistance must be nearly equal to that’ in the pure solvent. Even more striking are those data which show an increase in the solute diffusivity as the polymer concentrat,ioii Ind. Eng. Chem. Fundam., Vol. 11, No. 2, 1972
161
increases (Li and Gainer, 1968; Metzner, 1965). Theories developed to predict diffusion coefficients of solutes in ordinary liquids cannot be applied to polymeric solutions because they predict a strong, though different, dependence on the macroscopic viscosity, which appears not to be a relevant variable. Clough, et aZ. (1962), and Wang (1954) have proposed models for the polymeric solution which envision the solution as consisting of two distinct regions: a polymer region including the polymer and a portion of the solvent, and a pure solvent region. Additionally, relations have been developed for the analogous problems of thermal or electrical conductivity in two phases (Metzner, 1955; Woodside and Messmer, 1961). Without making restrictive assumptions, these developments cannot be generally applied t o polymeric solutions because there is no means b y which to evaluate the diffusivity in the “phase” encompassing the polymer in solution. Li and Gainer (1968) have developed a n expression for application to polymeric solutions, but its usefulness is mitigated by t h e difficulty of accurately determining a “viscosity-increase” function, which is a critical parameter in their analysis. Their approach is of interest, however, and will be exploited. The objectives of this investigation were to investigate the phenomena of diffusion in polymeric media and to develop a relation suitable for predicting diffusion coefficients of liquid solutes in dilute polymeric solutions. T o be practically useful, it was felt that the expression developed should contain only parameters which either are readily available in the literature or are far more easily measured than the diffusivity itself, and t h a t it should have few, if any, adjustable parameters. theory
Qualitatively, the addition of a polymer to a solvent can be seen to have several effects on the diffusivity of a solute. First, if one considers the polymer to exist as a structured network within the solvent, we could anticipate a “blocking effect.” Secondly, from purely geometrical considerations, the packing of the solvent in the immediate vicinity of the polymer will be different from that in the pure solvent. This will also change the apparent diffusivity, but we cannot say a priori whether it will increase or decrease. Finally, the polymer-solvent intermolecular forces can be expected to be different from the sblvent-solvent intermolecular forces, with attendant changes in the diffusion coefficient observed in the polymer solution. I n principle, it may be possible to calculate a transport property using statistical mechanical theories once a potential function has been determined (Hirschfelder, et al., 1967). While it also may be possible to determine the required potential function applicable to the many-body problem of int’erest for the liquid state, the computational complexities involved in using the potential function are such as to preclude the calculation of a transport property a t present, except for very simple molecules (Barker, 1963). However, one does know t h a t , for a given system, the calculation of a transport property is primarily dependent upon the temperature and intermolecular spacing (Hirschfelder, 1967), which in turn are related variables. A corollary of this is to state that if a property is known under given conditions (e.g., the diffusivity of the solute is known a t one temperature) and if then the intermolecular spacing is altered (as by changing the temperature or by addition of polymer to the solvent), a new “altered temperature” may be defined and used to evaluate the property under the new conditions, assuming t’he temperature dependence of the property is known. This implies that we can estimate the effect of the polymer on the solvent b y considering the change in the intermolecular spacing of the solvent 162 Ind. Eng. Chem. Fundam., Vol. 11, No. 2, 1972
molecules. Therefore, if we know the change in the solvent intermolecular distances due to addition of polymeric materials, we can specify the “temperature” of the solvent which would be necessary in order t o obtain the new spacing of the polymer-solvent system. The solvent properties could thus be evaluated a t this altered temperature. lt7e shall use this approach to determine the equivalent state of the solvent in the polymeric solution. Furthermore, the fact that the polymer is soluble in the solvent indicates that the polymersolvent bonds which impede diffusion are not grossly different from the solvent-solvent bonds. Therefore, we assume the equivalence of polymer-solvent and solvent-solvent bonds so that the polymeric solution may be considered to consist of solvent only, but in a state consistent with its altered specific volume. I n other words, the polymer influence on the diffusion process is reflected in altering the state of the solvent, which is then assumed to comprise the entire solution. The intermolecular spacing in this new state will be related to the excess volume of mixing solvent and polymer, Av, as given by AV =
1 -
P
-
tk + 2)
Here, W B and WP denote the respective solvent and polymer weight fractions, and p , and pop are the respective densities .of the polymeric solution, pure solvent, and pure polymer. Consistent with the earlier discussion regarding the mutual dependence of the intermolecular spacing and temperature, we postulate that an appropriate manner in which t o characterize the solvent in the polymeric solution is to determine the altered temperature, T*, necessary to accommodate the change in specific volume given by eq 1. Then, for the diffusion process, the properties of the solvent may be evaluated at the altered temperature, T*. The temperature and specific volume of a liquid are related by the thermal expansion coefficient, p. Available data indicate that the excess volume of mixing is small relative to the specific volume of the solvent, VOB, which implies the temperature difference T* - T is also small. Therefore, the coefficient of expansion IS assumed constant and the altered temperature of the solvent may be estimated by the relation
T*=T+-
Av
/%OB
The postulate that the dilute polymeric solution a t temperature T may be modeled as pure solvent a t a n altered temperature T* t h a t is necessary to accommodate the excess volume of mixing polymer and solvent is somewhat analogous to the principle of corresponding states, but cannot be derived from it. Thomaes and Van Itterbeck (1959) have developed a theory based on this principle that enables calculation of the diffusivity of a solute in a solvent from knowledge of the diffusivity of a second solute in the same solvent for the case of very simple liquids. Their theory is not directly applicable here, but it does exploit a similar idea. For the diffusion process, we have proposed t h a t the solute properties should be evaluated a t the altered temperature, T*, and the solute properties a t the process temperature, T . Based on a consideration of the hydrodynamic theories (Bird, et al., 1960; Glasstone, 1946) and the Eyring theory (Glasstone, et al., 1941) of diffusion, it may be concluded (Osmers, 1969) that the temperature which appears explicitly in formulas predicting diffusion coefficients is to be associated with the temperature of the process, T , and not with the temperature t h a t determines the solvent properties. We are now in a position to derive an explicit relation for predicting the solute diffusion coefficient, D*,in the polymeric
solution. Xctually, we shall derive two approximations for
D * : the first-order approximation shall involve no solute properties, which will make it possible to estimate apparent diffusivities for solid, liquid, and gaseous solutes; the second approximation will incorporate the solute properties explicitly, thus giving, presumably, far better estimates. It will, however, also iiecessit’at’etedious, though st’raightforward calculations and will be applicable to liquid solutes only. The solute diffusivity in the pure solvent, D , mill be assumed known or predictable from available analyses (Gainer, 1966; Gainer and lletzner, 1965; Wilke and Chang, 1955). h s a first-order approximation for the solute diffusion coefficient in the polymeric solution, we can use t h e WilkeChang correlation (earlier, theoretical relations would yield the same result (Bird, et al., 1960)) to obtain
and Metzner for the difference in activation energies is
E ~ ,B ED,AB
in which intermolecular distances are defined by
(7) rAB
(3)
in which t h e asterisk denotes the diffusion of the solute in the altered solvent which, as discussed earlier, is hypothesized to be equivalent to the polymeric solution. Both of the temperatures are to be associated with the process temperature and so are equal, the solvent molecular weight JIB is constant, and the solute property, as reflected in its molar volume, PA,is unchanged, so t h a t these terms cancel identically. R e also assume t h a t the association parameter, #B, of the solvent is constant over the temperature interval T* - T . We then have
(4) in which p~ is the solvent viscosity t o be evaluated a t t h e * the solvent viscosity to be process temperature, T , and ~ g is evaluated a t t h e altered temperature, T*, as obtained from eq 1 and 2 . I n accord with the findings of Gainer and lletzner (1965) for diffusion in pure solvents, we would expect eq 4 to be most useful for diffusion in polymeric solutions in which the solvent viscosity is low. X second-order approximation may be obtained by using the Eyring absolute rate theory (Glasstone, et al., 1941) t o yield
exp
[
]
(E*,,B - E*D,AB) - ( E ~ ,-B ED,AB) -~ RT
(5)
in which those parameters with an asterisk are t o be evaluated a t tlie altered temperature T*. Here Vg is t h e solvent molar volume, R is the universal gas constant, and E,,B and ED.AB denote the activation energies necessary to overcome t h e viscous and diffusion energy barriers, respectively. The exponential in eq 5 may be evaluated using a relation developed by Gainer and Netzner (1965) for predicbing liquid-liquid diffusion coefficients. This has been shown to be comparable to the Kilke-Cliang correlation for low viscosity solvents, but far superior for viscous systems. It has the further advantage of being based, in principle, entirely on theoretical considerations and involves no experimental curve-fitting parameters. T h e r e a s tlie Wilke-Chang correlation may be in error by more than a n order of magnitude for viscous substances, the Gainer-AIetzner expression resulted in a mean deviation of 18% bet,ween theory and experiment for the systems investigated. The expression developed b y Gainer
=
TAA
f 2
TEE
and y denotes the fraction of the total jumping activation energy to be associated with hydrogen bonding, 1 - y being the fraction of the energy associated with dispersion forces. They further assumed that the ratio y was equal to the ratio of the heat of vaporization due to hydrogen bonding, 4H\BP,,-H,to the total heat of vaporization, AH,,,,,. T h a t is
AHvax,z-~ AHwp.z
Yz = __-
(8)
where the separate parts may be estimated using the approach of Bondi and Simkin (1957). .Use, [A is a parameter which describes the geometrical configuration of the diffusing molecule to its nearest neighbors (which can be interpreted as the number of “bonds” formed between the diffusing molecule and its neighboring molecules) and A‘ is Avogardro’s number. Gainer and Metzner found that the geometric parameter 4 could be evaluated from self-diffusion data and that it is approximately equal to 6 for all compounds except methanol and ethanol, for which it is equal to 8. The viscous activation energy, E,, can be determined from the theory of rate processes (Glasstone, et al., 1941)
E,
=
R T In
[
pV”34E,,p (1.09 X 10-3)llb1’2Ta12
The number in eq 9 assumes a cubic packing of the liquid and has dimensions which require t’hat the viscosity, p, be expressed in g/cm-sec, t h e molar volume, V , in cm3/mole, and the internal energy of vaporization, AE,,,, in cal/’mole. We may use eq 6 directly in eq 5, but first’ we will make several approximations in order to obt’ain a more tractable relation. We are to determine the solvent properties a t the altered temperature, T * , or, equivalently, the change in the solvent properties over the temperature interval T* - T . For any reasonable temperature difference the change in t’he ratio TAATBB/?-ABz appearing in eq 6 may be neglected. Furthermore, we also assume that, over the same temperat’ureinterval, the geometric parameter of the solvent, [B, and tlie ratio of the heats of vaporization, YB, are both constant. With these approximations, eq 5 and 6 may be combined to yield, after considerable manipulation
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 2, 1972
163
0.8
cI
a D = 0 94 x
sq cm / sec
0.6 I
I IO
n2
I
0
08-
racies within 12%. It was anticipated that the systems studied in this investigation would exhibit trends that would not be masked by an error of 12% or less. For this reason and t h e apparent ease with which an experiment could be conducted, the optical wedge was selected. The details of the use of this technique in the present study and a discussion of the limitations of the procedure may be found elsewhere (Osmers, 1969). Duda, et al. (1969), have recently presented a comprehensive and detailed analysis of the method. The analysis of the data obtained in the micromethod technique was based on Fick’s second law which may be written as
b D = I 19 x lV5 sq cm / sec
?!
06 -
a
*‘0
I
4,
IO
c
\ -
c
I
05 0
V
0,8
06
D = 0.94 x lBSsq cm/sec
c
P
=
bt
bz
(.*?!)
in which c denotes the concentration of the diffusing solute, t is the time, aiid z is the spatial coordinate of the system. The boundary conditions that were experimentally satisfied are c(z,O) = c
0
zo
{
(-m,t)
=
0
c ( m , t ) = c,
velocity d
D
:
I 2 9 x IO-’
=
0 at z
= m
which yields the solution
sq cm/sec
06-
I
-
I
D* =
dc 2d7l e
(13)
~
Equation 10 clearly shows the influence of the diffusing solute and thus, in principle, is to be preferred over eq 4. Gainer and Rletzner (1965) suggested that their development for diffusion in liquids be used in preference to the WilkeChang (1955) correlation whenever the solvent viscosity level is above 5 or 6 cP. JTe should also note that even t,hough the .~, ratio of the viscosity activation energies, E * L I , ~ / E pmay be close to unity, the difference of this ratio from unity is significant because it appears in the exponent,ial. Finally, the applicability of eq 10 is limited to diffusion of liquids because the development of Gainer and Rletzner has not been extended to gaseous or solid solutes. The predict’ive relations developed herein, eq 4 and 10, involve parameters that either are available in the literature or call be measured far more easily than the apparent diffusion coefficient. Experimental Method
Sishijima and Oster (1956, 1961) have adapted an optical micromethod (frequently called the “optical wedge” or “wedge interferometer”) for studying liquid-liquid diffusion from a method earlier employed to study diffusion in solids (Robinson, 1950). The principal advantage of this technique is that the scale on which it is conducted is so relatively small that one can complete an experiment in a matter of minutes, rather than the hours, or even days, required of most other methods (Geddes and Pont’ius, 1959; Robinson aiid Stokes, 1955). The method has been used by Secor (1965) and Paul (1967, 1969) for the measurement of polymer diffusion coefficients in liquids and by Li and Gainer (1968) for diffusion of liquid solutes in polymeric solutions. They all reported accu164 Ind. Eng. Chem. Fundam., Vol. 11, No. 2, 1972
in which 7 = z / d L i s a reduced coordinate. Equation 11 does not account for the effect of nonideal excess volume of mixing the solute and the polymeric solut,ion (note t,hat this excess volume of mixing is quite distinct from and unrelated to the excess volume of mixing polymer and solvent). However, Duda and Vrentas (1965) have found that the error incurred by using Fick’s second law to determine a diffusion coefficient from a one-dimensional diffusion experiment is minimized by imposing the condition of zero velocity of the solution a t the extreme boundary a t which the solute concentration is highest. By considering the highly noiiideal ethanol-water system they found that t’he error in the diffusion coefficient is a maximum of about 5% a t an ethanol concentration of 50% and virtually zero as the ethanol concentration approaches zero. Thus, eq 13was chosen as being suitable for this work; the diffusivity a t zero concentration was obtained by ext,rapolat,ion. I n order to make a comparison between the experimental measurements of apparent diffusivity and the predictions made according to the theory, it was necessary t o have sufficient data to calculate the excess volume of mixing according to eq 1. The solvent density can be obtained from many sources generally and their is a significant amount of data on solid polymers in the literature (Brandrup and Immergut, 1966; Roff, 1956; Schildknecht, 1952). One can estimate whether the excess volume of mixing will be positive or negative using the relat’ion AV = w,(D,
- v,”)
(w, + 0)
(14)
which is a good approximation as the polymer weight fraction approaches zero. Here 0, and vpC denote, respect,ively, the part,ial specific volume and specific volume of the polymer. Partial specific volumes of a large number of polymers a t infinite polymer dilution are available in the literature (Brandrup and Immergut, 1966), but sufficient information is
Table II. Excess Volume of Mixing, Aqueous Polymeric Solutions at 25°C
Table I. Solute Diffusion Coefficients in Water D X lo5 cm2/sec
CMC
Exptl
Solute
Glycerol
r, oc 24 24 24 24
5 5 0 7
at T
0 87 1 00 0 97 0 86 (Average) 1 16
AP30
~~
at 25'C
0 1 0 0 0 1
88 01 99 87 94 19
Ethanol 24 1 Allyl alcohol 23 0 1 23 1 29 Perry, et al. (lY63). Duda and T'rentas (1965).
lit. a t 2 5 ° C wp
0 94. 1 23
X IO2
0.5 1.0 1.5 2.0
av x 103, cm3/g
wp X IO2
0.25 0.50 0.75 1.125
-0.6 -1.1 -1.7 -2.2
PVP
Av X
lo8,
AV
wp X 102
cm3/g
2.5 5.0 7.5 10.0
-0.5 -1.0 -1.5 -2.3
x
103,
cm3/g
-1.1 -2.3 -3.3 -4.2
1 190
1.0300
generally not available t'o calculate the excess volume of mixing directly a t nonzero polymer weight fractions. Though eq 14 is useful for suggesting the trend of the excess volume of mixing, and hence of t,he apparent diffusivity, the polymeric solution density over a polymer concentration interval must be known in order to obtain a prediction of the apparent diffusivity. In the present work, this was measured using a Gay-Lussac pycnometer which is ". . . capable of an accuracy of about 1 x 10-4 g/cm3" (Bauer and Lewin, 1959). This technique amounts to determining the weight of liquid occupying a known volume defined by the shape of the pycnometer.
I pi (gm/cc)
1.O25Or
I
I
I
I + o wp
PIP., O
AVG Ps DEVIATION (gm/ccl
0
099700478
00001
159
A
099700518
00001
145
O*
099730251
00002
I25
1.0200
1.0150
Results and Discussion ' 1
Three groups of data were obtained. (1) The first was selected on the basis of a preliminary investigation b y Astarita, et al. (1966), which suggested t h a t glycerol, ethanol, and allyl alcohol diffusing as solutes in aqueous CMC (sodium carboxymethylcellulose) and aqueous AP30 (a copolymer of polyacrylamide and acrylic acid) polymeric solutions would behave especially abnormal in terms of exhibiting particularly high diffusivities. Thus, four of them were studied more critically. The systems chosen were glycerol and ethanol in aqueous CXC,and glycerol and allyl alcohol in aqueous AP30. (2) Measurements of the excess volume of mixing water and PVP (polvvinylpyrrolidoiie) were made in order to compare the first-order prediction (eq 4) with the data reported by Nishijinia and Oster (1956) for the diffusion of sucrose. (3) The polymer concentration dependence of the diffusion coefficieiit, of PAN (polyacrylonitrile) in D N F (diniethylformamide) \vas studied. The reason for obtaining these data was to examine the earlier results of Secor (1965) and Paul (1967). Secor reported a sixfold increase in the diffusivity as the polymer concentration increased to approximately IS%, while Paul reported a 40y0 increase. It' was felt that this behavior is qualitat'ively similar to that of the principal subject of this investigation, and that the disparity of the two sets of data should be considered. Diffusion of Liquid Solutes in Polymeric Solutions. T h e experimenbal d a t a for t h e diffusion of t h e liquid solutes in polymeric solutions are shown in Figure 1 (complete experimental d a t a may be found elsewhere (Osmers, 1969)). T h e d a t a for diffusion in t h e pure solvent (water) are listed in Table I. These correspond to a linear estrapolation of t h e diffusivity t o zero solute concentration. T h e average change in t h e diffusivity on extrapolation t o zero solute concent,ration from t h e concentration range in which the data were obtained was ivithiii 15% of the
I
1.0000 . ._ 00
05
IO
I
15
20
w p x 102
Figure 2. Least-squares linear fit of polymeric solution densities, and densities of solvent and polymers a t 25°C: 0,CMC; A, AP30; 2, PVP. *PVP d a t a obtained with Westphal balance
tabulated data. The data were obtained in ambient rooin temperature and adjusted t o 25" C' by assumiiig an inverse relation between the solute diffusion coefficieiit and the solvait viacosit'y over the temperature interval, the average of which was 1.6OC. similar adjustment for temperature differmew has been used previously by Zandi and Turner (1970). ('omparison of data aiid theory a t the actual tcmperaturep of the experiments yields the same result.;, but the temperature adjustment enables a more instructive graphical prcseiitatioii. The weight average molecular weight of the CXC'aiid -11'30 were approximately 2.0 X lo5 and 2.5 X lo6, respectively. The theoretical predictions were based on tlie excess volume of mixing data reported in Table 11. These n-ere deteriniiied by measuring the polymeric solution density, and tlieii, viitli the pure solvent and polymer densities obtaiiied from the literat'ure, calculating the excess volume of mixing by means of eq 1. The required density data are dion.ii i l l Figure 2. X comparison of the experimental data and tlie tlicorcticnl predictions based on the first- aiid second-order aplirosiniations (eq 4 and 10) clearly indicates a decrease in the solute diffusivity with increasing polymer concentratioii, while both approximations correlate the data within expe~iiiieiitalerror. We can anticipate that the secoiid-order approxiniatioii would be superior when used in coiinection with polymeric solutions Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 2, 1972
16.5
Table 111. liquid Properties Allyl alcohol
P, c p AH,.,,
cal/g
E PI
g/cm3
Y
M ,g/mole
Ethanol
1 216
1 094
186 0 8 0 0 850 0 6 58 1
226 5 8 3 0 785 0 6 46 1
Glycerol
954 0 174 6 1 0 92
0 0 26 58 1
Water
0 894 582 6 0 0 18
2 1 997 9 0
solutions.) Up to this polymer concentrat'ion, the differences between the data and theory are within the range of experimental error. Above this polymer concentration, there is a marked increase in the difference between the data and theory. The rapid increase in the difference is iiiterpreted to mean that polymer-polymer interactions, which have riot been considered in this analysis, are becoming more significant with regard to the potential energy barrier resisting the motion of the diffusing species. This is coiisistent with the fact that the diffusion coefficients of solutes in solid polymers are several orders of magnitude less than in normal solvents. Several investigators (Boss, et al., 1967; Sishijinia and Oster, 1956; Secor, 1965; Simha and Zakin, 1960) have proposed that' the critical concentration of a polymer in solution, C,, above which polymer-polymer interaction is appreciable and a dilute solution m-ould no longer apply is given by a relation of the form
c, =
b
[VI
0
0 00 00
5.0
100 wp x 1 0 2
150
200
Figure 3. Diffusion of sucrose in aqueous polyvinylpyr25°C; d a t a of Nishijima and Oster; -, eq 4
rolidone at
with highly viscous solveiits, tlie reason being that the secondorder relation allows for the influence of the solute force field, whereas the first-order relation does not. The importance of the satisfactory agreement between experiment' and theory is that the excess volume of mixing is seen to be the principal parameter in determining the change in the apparent diffusivity on addition of polymer to a solvent, thereby enabling estimation of the diffusion coefficieiit with information that is either generally available or relatively easy to determine experimentally. K e can also comment on the applicability of this analysis to various solute-polyniersolvent systems. The data presented are for highly polar solute and solveiit pairs, with the rat,io y (eq 8) ranging from 0.58 to 0.9 (see Table 111, which is also a listing of the other parameters used in the calculations). Siiice this ratio is related t o the extent of hydrogen bonding, it appears likely that the analysis would apply to systems in xhicli there is less hydrogen bonding. This analysis is limited to dilute polymeric solutions in which polymer-polymer bonds do not dominate, as discussed further below, Diffusion of a Solid in a Polymeric Solution. T h e dat,a reported b y Nishijima and Oster (1956j for t h e diffusion of sucrose in aqueous P V P are shown in Figure 3. T h e excess volume of mixing data necessary for calculation of the theoretical predictions based on eq 4 are listed in Table I1 and were calculated using the data in Figure 2. T h e weight, average molecular weight of t h e P V P was approximately 5.0 x 104. The predict'ioii is seen to compare favorably with the data up to a polymer colicelitration between 3 aiid 5%. (The data of Clough, et al. (1962), and Hoshino and Sato (1967) also affirm the suggestion that eq 4 may be useful, a t least qualitatively, for predicting solid solute diffusivities in polymeric 166 Ind. Eng. Chem. Fundam., Vol. 11, No. 2, 1972
where [?I is the polymeric solution intrinsic viscosity aiid b is a dimeiisioiiless constant'. However, depending upon the polymeric solution, the reported values of the constant range from 0.2 to 5.0. This suggests that a more detailed analysis involving consideration of additional molecular or solution parameters is required to develop an expression which is more generally applicable for defining the critical polymer coiicentratioii. I n any event, it seems reasonable to limit the application of a dilute solut'ion theory to polymer coiicentratioiis of only a few per cent. Consideration of Literature Data. I n view of some apparent success in relating the change in the apparent diffusivity of a solute to t h e excess volume of mixing a polymer and solvent, it is instructive t o consider d a t a available in the literature. Table IV is a compilation of all t h e liquid solute-polymer-solvent systems for Tyhich sufficient d a t a were available to calculate t h e excess volume of mixing according to eq 14. Predictions based on measuring t h e excess volumes of mixing were not attempted primarily because it was felt that, in view of the rather strong dependelice of the apparent diffusivity and the polymer partial specific volume on the polymer molecular weight (Brandrup and Inimergut, 1966; Heller and Thompson, 1951; Kishijima aiid Oster, 1956))good results could only be expected if virtually the identical polj-niers were used, and these were not available. However, it can be seen that a iiegative excess volume of mixing corresponds to a decrease in the apparent diffusivity for seven of the eight systems, which gives qualitative support to tlie model. There is an inconsistency in the data for the turpentiriePIB-heptane solution in that the diffusivity is reported to have increased while the negative excess volume of mixing indicates that tlie diffusivity should have decreased. There is some possibility of the data being subject to error as the tem is difficult to study: attempts to duplicate the experiment of Li and Gainer (1968) were entirely unsuccessful. hstarita (1965), Quiiin and Blair (1967), aiid Zaiidi and Turner (19iO) have obtained data on the diffusion of gases in polymer solutioiis (note that AP30 is also designated as ET597). hstarita found an increase in the diffusivity with iiicreasiiig polymer concentration using a laminar liquid jet, whereas Quiiin aiid Blair reported a decrease by measuring gas absorption rates into a quiescent liquid substrate. Zaiidi and Turner, also using a laniiiiar liquid jet, observed an initial decrease aiid then an increase in diffusivity with increasing polymer con-
Table IV. liquid Diffusion Coefficients in Polymeric Solutions, literature Data References
D*fD
Solute-polymer-solvent
VP
Av
