Ind. Eng. Chem. Res. 1995,34, 2536-2544
2536
Prediction of PolymerBolvent Diffusion Behavior Using Free-Volume Theory Seong-Uk Hongt Department of Materials Science and Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802
Free-volume parameters for various solvents are estimated using viscosity-temperature data. Using these parameters in conjunction with the Vrentas-Duda fkee-volume theory, diffusion behavior is then predicted for several polymer/solvent systems over wide ranges of concentration and temperature. The prediction results are comparable with experimental data for some polymer/solvent systems. For other systems, however, the theory overpredicts diffusion behavior. To understand the discrepancy between the experiment and prediction, a sensitivity analysis is shown for each parameter in the theory. The predictions can be improved by modifying one of the parameters.
Introduction The phenomenon of small molecule migration in polymeric materials controls the effectiveness of polymerization reactors, as well as the characteristics of the polymer produced. Other polymer processing operations affected by molecular transport include (1)devolatilization, (2) mixing of plasticizers (or other additives), and (3)formation of films, coatings, and foams. Molecular diffusion behavior is also essential to polymer products such as barrier materials, membranes for separation processes, and controlled drug delivery systems. The fundamental physical property required to design and optimize processing operations is the mutual diffision coefficient,D. This quantity is known to reach a maximum value due to the opposing contributions from the self-diffusion coefficient and the thermodynamic factor (Vrentas et al., 1982). Various diffusion models based on free-volume concepts have been proposed (Vrentas and Duda, 1977a,b; Fujita, 1961; Cohen and Turnbull, 1959). In order to describe diffision behavior over wide ranges of temperature and concentration, accurate estimates of the freevolume contributions of both the polymer and the solvent must be available. Although the Vrentas-Duda free-volume theory (Vrentas and Duda, 1977a,b) has been successful in correlating and predicting the diffusion coefficients of some solvents in polymers, limited data on solvent free-volume parameters (Zielinski and Duda, 1992)have restricted our ability to correlate and predict diffusion behavior. Since all transport processes are assumed t o be governed by the same free volume (Blum et al., 19861, it is possible to determine the solvent free-volume parameters using data for viscosity as a function of temperature. Since the viscositytemperature relationship shows its greatest curvature a t reduced temperatures, low-temperature data are essential to estimate solvent free-volume parameters accurately. The purpose of the present work is to determine solvent free-volume parameters using liquid viscosity data and to predict polymerlsolvent diffusion behavior using these parameters.
Theory According to the free-volume diffusion model developed by Vrentas and Duda (1977a,b) the solvent self~~~~
~~~~
Current address: Department of Chemical Engineering, The Pennsylvania State University, University Park, PA 16802-4400. E-mail:
[email protected]. +
diffusion coefficient, D1, is given by eq 1, and the polymer/solvent binary mutual diffusion coefficient, D, is expressed by eq 2, with subscripts 1 and 2 referring to the solvent and polymer, respectively, in each:
-E
D,= D, e x p ( z ) exp x
I
- Tgl+ T ) + w2(-K1-2)(K22 - Tg2+ T )
Y
D = D,(1 - 91)2(1- 2x$4
(2)
Here, DOis a pre-exponential factor, E is the critical energy which a molecule must possess to overcome the attractive forces holding it to its neighbors, and y is an overlap factor which is introduced because the same-free volume is available to more than one molecule. is the specific hole free volume of component i required for a diffusion jump, ai is the weight fraction of component i, and 6 is the ratio of the molar volume of the jumping unit of the solvent to that of the polymer. KIIand K21 are free-volume parameters for the solvent, while K12 and K22 are those for the polymer, 41 is the solvent volume fraction, and x is the polymer-solvent interaction parameter. Although there are 13 independent parameters in eq 2, grouping some of them means that only 10 parameters ultimately n q d to be evaluated; Kdy,K21 - Tg1, K ~ YK22 , - Ta, x, DO,E , and 6. x is not required in estimating the solvent selfdiffusion coefficient. Of the 10 parameters, 6 are pure component properties and should be estimated a priori to correlate the diffusion data. In addition, x can be determined by correlating solubility data with the Flory-Huggins equation (1953). Finally, the three E,and 6) can be evaluated, remaining parameters (Do, using measured diffusion data, by nonlinear regression analysis. In order t o predict diffusion behavior, however, all parameters need to be determined without using any diffusivity data. Methods for estimating freevolume parameters to predict diffusion behavior are discussed in the following section.
e,E,
Estimation of Free-Volume Parameters fl and The two critical volumes, and can be estimated as the specific volumes of the solvent and
E.
Q888-5885/95/2634-2536$09.QQIQ0 1995 American Chemical Society
e
E,
Ind. Eng. Chem. Res., Vol. 34,No. 7,1995 2637 Table 1. Group Contribution Methods To Estimate Molar Volumes at 0 K component H C (aliphatic) C (aromatic) N N (in ammonia) 0 0 (in alcohol)
F c1 Br I P S triple bond double bond 3-membered ring 4-membered ring 5-membered ring 6-membered ring OH (alcoholic) OOH (carboxyl)
Sugden (cms/mol) 6.7 1.1 1.1 3.6 0.9 5.9 3.0 10.3 19.3 22.1 28.3 12.7 14.3 13.9 8.0 4.5 3.2 1.8 0.6
Biltz (cm3/mol) 6.45 0.77 5.1
large number of polymers (Ferry, 19801, whereas polymer free-volume parameters are provided in Table 2 (Hong, 1994). x. It is easy to find polymerholvent interaction parameters for many polymers and solvents in the literature (Sheehan, 1966; Orwoll, 1977). It is also possible, using the Flory-Huggins equation (1953), to determine x from solubility data where the equilibrium volume fraction of the solvent in the polymer is known as a function of solvent vapor pressure, P I :
16.3 19.2 24.5
Pl/P,"= 41exp(42+ 2422)
16.0 8.6
where P: is the solvent saturation vapor pressure. x also can be determined from a semiempirical equation developed by Bristow and Waston (1958): = 0.35
10.5 23.2
polymer at 0 K. Molar volumes of the solvent and polymer at 0 K can be estimated using group contribution methods (Haward, 1970), and a summary of these methods is shown in Table 1. Kldy and K22 - Ta. For pure polymers, the temperature dependencies of the viscosity are usually expressed in terms of the Williams-Landel-Ferry (Williams et al., 1955) equation:
+ RT\61 - 62)2 "1
(6)
(7)
Here, is the solvent molar volume, and 61 and 62 are the solubility parameters of the solvent and polymer, respectively. 5. The parameter 6 is the ratio of the critical molar volume of the solvent jumping unit to that of the polymer jumping unit. For a solvent molecule which moves as a single unit, 6 may be defined as
v$
The free-volume parameters for polymers are simply related to the WLF constants as follows (Duda et al., 1982):
where v:(O)and are the solvent molar volume at 0 K and the molar volume of the polymer jumping unit, respectively, and M I and Mzj are the molecular weights of the solvent and the polymer jumping unit, respectively. J u et al. (1981a) assumed that the size of the polymer jumping unit is independent of the solvent and proposed a linear relationship between 6 and the solvent molar volume at 0 K, so that
K22= C z F
E = aV:(O)
(4)
(9)
(=lm$)
Values of C p F and C E F have been tabulated for a
where a is a constant which has been determined from the polymerholvent diffusion data. Once a is known for a particular polymer, the value of 6 for any solvent in that polymer can be estimated if the solvent moves as a single unit. a values have been
Table 2. Polymer Free-Volume Parameters polymer K22 - Tgz a x io3 T, % w12/y) x 104 -395.7 poly(a-methylstyrene) 0.859 5.74 445 -111.5 172 polybutadiene, cis-trans" 0.954 6.10 -101.4 161 polybutadiene, high cisb 0.954 6.12 0.732 5.64 -362.7 418 polycarbonate poly(dimethylsi1oxane) 0.905 9.32 -81.0 150 poly(ethy1 methacrylate) 0.915 3.40 -269.5 7.77 335 poly(ethy1ene-co-propylene y 1.005 8.17 -175.3 12.50 216 poly(ethy1styrene) 0.956 4.49 -286.9 355 1.004 2.51d -lO0.sd 205 poly(isobuty1ene) -208.4 262 poly(isopropy1 acrylate) 0.819 5.44 0.748 3.98 -231.0 10.38 276 poly(methy1 acrylate) poly(methy1 methacrylater 0.788 3.05 -301.0 6.76f 381 polypropylene 1.005 5.02 -205.4 253 poly(propy1ene oxide) 0.852 9.52 -174.0 198 poly@-methylstyrene) 0.860 5.18 -330.0 348 0.850 5.82 -327.0 7.19 373 polystyrene poly(viny1 acetate) 0.728 4.33 -258.2 10.23 305 205 butyl rubber 1.004 2.39 -96.4 200 natural rubber 0.963 4.64 -146.4 neoprene 0.708 3.91 -163.3 228 a Cis:trans:vinyl = 43:50:7. b Cis:trans:vinyl = 96:2:2. Ethy1ene:propylene = 56:44 (by mole). Another set of values are also possible K2z - T e = -134.6). e Atactic. f Different studies (Ju et al., 1981b) suggested a = 4.61 x (Klzly = 4.42 x so that some ambiguity regarding this value exists.
2538 Ind. Eng. Chem. Res., Vol. 34, No. 7, 1995 Table 3. Solvent Free-VolumeParameters Including E(a-1) solvent (K&) x lo3 Kzl - Tgl Do x lo4 E(w1-1) acetic acid benzene chloroform cis-decalin
0.68 1.51 0.71 1.09
-22.18 -94.32 -29.43 -100.55
19.05 8.31 6.99 7.08
solvent ethylbenzene methyl acetate toluene
418.86 386.86 353.04 461.10
( K d y )x lo3 2.22 1.25 2.20
K21 - Tgl -100.81 -38.50 -102.72
DOx lo4
E(w1-1)
3.27 5.23 3.79
381.52 336.37 356.76
Table 4. Solvent Free-VolumeParameters solvent acetic acid acetone benzene n-butylbenzene sec-butylbenzene carbon tetrachloride chloroform cumene cyclohexane cyclohexanol cis-decalin trans-decalin n-decane di-n-butylphthalate diisobutylphthalate dimethylphthalate n-dodecane ethylbenzene ethylene glycol formic acid n-heptadecane n-heptane n-hexadecane n-hexane
0.773 0.943 0.901 0.944 0.944 0.469 0.510 0.937 1.008 0.882 0.928 0.928 1.082 0.737 0.737 0.609 1.070 0.928 0.779 0.715 1.050 1.115 1.053 1.133
0.68 1.86 1.51 2.28 2.15 0.85 0.71 2.98 3.02 0.73 1.09 0.96 1.22 0.83 1.26 1.29 1.05 2.22 0.75 1.81 0.81 1.83 0.83 1.96
-22.28 -53.33 -94.32 -126.45 -130.64 -101.93 -29.43 - 127.93 - 157.81 -166.09 -100.59 -68.43 -55.14 -155.60 -194.06 -197.71 -57.96 -100.81 -139.38 -117.94 -54.49 -55.42 -53.66 -41.08
10.20 3.60 4.47 1.48 1.74 2.52 4.07 1.00 2.01 24.66 3.40 4.57 5.22 2.94 1.12 1.02 6.13 1.54 8.82 5.10 7.25 3.43 7.33 3.50
131 124 129 129 133 125 80
188 206 111
84
TB1 0.990 0.954 0.961 0.855 0.997 0.813 1.091 1.102 1.057 1.158 0.950 0.937 0.815 0.899 1.060 0.861 1.064 0.917 1.075 1.071 1.049 1.049
1.35 2.87 1.17 1.25 0.73 0.94 1.35 1.52 0.88 2.41 2.97 2.44 0.59 1.20 0.94 1.29 0.98 2.20 1.15 2.18 1.29 0.76
-154.36 -162.46 -48.41 -38.50 59.63 -78.28 -54.72 -51.98 -56.83 -38.89 -160.38 -124.11 -144.24 -68.92 -58.99 -115.08 -57.39 -102.72 -68.15 -152.29 -53.45 41.65
11.64 0.88 8.75 5.23 16.04 4.30 5.01 3.67 6.87 3.11 1.02 1.40 31.06 5.52 6.46 2.68 6.42 1.87 6.11 8.55 4.97 37.37
108 109
85 64 128 122
117 126
70
reported for various polymers (Zielinski and Duda, 1992; Hong, 1994) and are provided in Table 2. If an a value is not available for a certain polymer, an alternative method of estimating 6 is needed. Recently, Zielinski and Duda (1992) proposed a linear correlation of the molar volume of the polymer jumping unit with the glass transition temperature. This correlation has been modified by Hong (19951, using more diffusivity data, as follows: (cm3/mol)= 0.0925Tg2(K)
solvent 2-hexanol n-hexylbenzene methanol methyl acetate methyl ethyl ketone naphthalene n-nonane n-octane n-pentadecane n-pentane n-pentylbenzene n-propylbenzene 1,2-propylene glycol styrene n-tetradecane tetralin n-tridecane toluene n-undecane water o-xylene p-xylene
+ 69.47
10-6
-cn -.
6
v
(Tg2< 295 K)
io-'
10-8
n 10-9
= o.6224Tg2(K) - 86.95
(Tg2I295 K) (10) Although these equations have not been extensively tested, diffusion coefficients can be a t least estimated from pure polymer and solvent data alone. K&, K21 - Tal, DO, and E. In 1921, Vogel proposed an empirical equation t o describe the viscosity-temperature relationship. Thirty years later Doolittle (1951) postulated that viscosity should be related to the amount of free volume in a system and derived the Vogel equation from free-volume concepts. Adopting Doolittle's expression and using the nomenclature of Vrentas and Duda leads to eq 11 for the solvent viscosity:
Hence, K d y and K21 - Tglcan be determined from a nonlinear regression of eq 11 using viscosity-temperature data. Recent NMR studies have shown that variable temperature 13C1'2 relaxation data can also be used t o estimate solvent free-volume parameters (Zielinski et al., 1992; Hong et al., 1995). In addition, DOand E can be estimated by combining
''
I
lo-'?
0.0
0.1
0.2
0.3
0.4
0.5
w,
Figure 1. Experimental data (Ju, 1981) and theoretical predictions for poly(viny1acetate)/toluene mutual diffusion coefficients.
the Dullien equation (Dullien, 1972) for the self-diffusion coefficient of pure solvents with the Vrentas-Duda freevolume equation evaluated in the limit of pure solvents. Thus,
In DoHere,
RT
-
K2, - Tgl + T
uc (cm3/mol) and M1 (g/mol) are the solvent's
Ind.Eng. Chem. Res., Vol. 34, No. 7,1995 2539 io-' - ' -ol
10-5
10-5 A
0
\
Eo
10-6 F
v
d 10-
10"
0.0
A
T'=60°C
0
T=10O0CI
10-8
0.2
0.6
0.4
0.8
0.0
1.0
a, Figure 2. Experimental data (Pickup and Blum, 1989) and theoretical predictions for self-diffision coefficients of toluene in polystyrene.
0.2
0.4
0.6
0.8
1.0
w, Figure 4. Experimental data (Kosfeld and Zumkley, 1979) and theoretical predictions for self-diffusion coefficients of benzene in polystyrene.
lo*
10-7 A
cn \
Eo
v
n
0
T=10O0C
0.6
0.8
1
10-Q
10-9
0.0
0.2
0.4
1.0
0.0
0.2
w,
0.4
0.6
a,
Figure 3. Experimental data (Zgadzai and Maklakov, 1985) and
Figure 5. Experimental data (Ju,1981) and theoretical predic-
theoretical predictions for s e l f - a s i o n coefficients of ethylbenzene in polystyrene.
tions for poly(viny1 acetate)/chloroform mutual diffusion coefficients.
critical molar volume and molecular weight, respectively, and 0.124 x is a constant which has a unit of molw3. ~1 (g/cm-s) and (cm3/g)are the viscosity and specific volume of the pure solvent, respectively, and are the only temperature-dependent parameters in the expression. Since solvent free-volume parameters have been determined previously from eq 11, with pure solvent viscosity and specific volume data as a function of temperature, DOand E (wl-cl) can be estimated from a nonlinear regression of eq 12 (Duda, 1983). According to eq 12,DOis independent of the polymer and is solvent specific. If one assumes that E does not vary much with concentration, E (wl-1) can be used over the entire concentration range. In this study, free-volume parameters for about 50 solvents have been estimated, and the results are presented in the following section.
Results and Discussion Results for several solvents, at an early stage of this study, indicated that calculated values of E(w1-1) were relatively small compared to the values of E determined from diffusion experiments and did not vary much among the solvents (see Table 3). Therefore, the assumption of negligible energy effects (E = 0) was incorporated, and only Dol was determined from eq 12 (Dol can be identified as DOwhen E is set equal t o zero in eqs 1and 12). The parameters for various solvents have been calculated by the methods described in the previous section and are provided in Table 4. The glass transition temperature of the solvents, if available in the literature (Angell et al., 1984; Dubochet et al., 1984; van Krevelen, 19761, is also reported. The pure solvent viscosity and specific volume data were obtained mostly
2640 Ind. Eng. Chem. Res., Vol. 34,No. 7, 1995 Table 5. Parameters Used in Diffusion Coefficient PredictionsD parameter
x
0.393 2.20
2.20
2.22
1.51
PVAd C l h 0.351 0.71
-102.72 4.33
-102.72 5.82
-100.81 5.82
-94.32 5.82
-29.43 4.33
-258.2 0.917 0.728 1.87
-327.0 0.917 0.850 1.87
-327.0 0.928 0.850 1.54
-327.0 0.901 0.850 4.47
-258.2 0.510 0.728 4.07
0.86 0
0.61 0
0.71 0
0.51 0
0.62 0
PVAcA'ol
Kllly x lo3 (cm3/gK) Kzl - Tgl (K) K ~ ~ 104 Y (cm3/gK) K22 - Tgz (K) 0: (cm3/g) 0; (cm3/g) D~ 104 (cm2/s)
4
E (cal/mol)
PSfI'ol
PSBB
PS/Ben
-
10-6
10") [
10-7 I 10-0
1o - ' o Y
-
a
lo-"
O lO-'z V,=0.459 10-13
Abbreviations: WAC, poly(viny1 acetate); Tol, toluene; PS, polystyrene; EB, ethyl benzene;Ben, benzene; Clfm, chloroform. a
- v;=o.9 17
lo-"
V ,= 1.834
10-'5 1O - l C
1 0.2
0.0
0.6
0.4
0.8
1.0
10-5 I
u,
Figure 7. Effect of 0: on the mutual diffusion coefficient.
. cn
N
b
10-9
,o-'o
a
'I
I
v
lo-"
0
10-2
T=400C
--.-Do, = 9.35x 10-1
10-13
I
1045
0.0
0.2
0.6
0.4
0.8
i i
1.0
i i
or (=0.432
Vj0.728
or (=OB64 Vj1.456 or (= 1.728
w,
Figure 6. Effect of Dol on the mutual diffusion coefficient. 1045 L
0.0
from the DIPPR data base (Daubert and Danner, 1994). If viscosity data a t low temperatures were not available in the DIPPR data base, data from other sources were also used (Barlow et al., 1966,1967;Liu, 1980;Ju, 1981; Zielinski, 1992). Only solvents that have viscosity data at temperatures starting from approximately the melting point of the solvent were considered in this study. If the solvent follows Arrhenius-type behavior, i.e., if the plot of In 771 versus T-l is linear, it was not considered here. Viscosity data above the normal boiling point of the solvent were not used for the calculation. Both solvent self- and mutual diffusion coefficients have been predicted and compared with experimental results for five polymer/solvent systems; poly(viny1 acetate)/toluene, polystyrene/toluene, polystyrene/ethylbenzene, polystyrenehenzene, and poly(viny1acetate)/ chloroform, using the solvent free-volume parameters determined from viscosity data in this study. These predictions are shown in Figures 1-5, while the parameters used t o generate the theoretical curves are provided in Table 5. The remaining parameters used for diffusion coefficient predictions (eqs 1 and 2)were obtained, using the methods described in the previous section, from pure
'
0.2
0.6
0.4
0.8
1.0
m,
Figure 8. Effect of 0; and 6 on the mutual diffusion coefficient.
property data for the solvents and polymers. The curves in Figures 1-5 represent true predictions since none of the parameters were adjustable. For the poly(viny1 acetate)/toluene, polystyrene/ toluene, and polystyrenehenzene systems, the predictions are comparable with experimental data. The Vrentas-Duda theory can predict reasonably both temperature and concentration dependencies of diffusion coefficients. For the polystyrene/ethylbenzeneand poly(vinyl acetate)/chloroform systems, however, the predictions are poor compared to other systems. The theory overpredicts diffusion coefficients for both systems. A sensitivity analysis has been done for each parameter to check which parameters make these overpredictions. The poly(viny1 acetate)/toluene system at 40 "C was used in this analysis. The value of each parameter in Table 5 was decreased and increased by a factor of 2, and setting other parameters constant, the predictions are shown in Figures 6-13. For K21 - Tg1 and
Ind. Eng. Chem. Res., Vol. 34, No.7,1995 2541 10-5
I
\
\
0
____
T=40'C
= 2.17x IO- K,JY = K&
4 . 3 3 x 10-4
K , ~ Y= 8.66~ 1 O-'
lo-"
'
0.0
lo-" 0.2
0.6
0.4
0.8
1.0
0.0
0.2
0.8
1.0
u,
w,
Figure 9. Effect of K&
0.6
0.4
on the mutual diffusion coefficient.
10-5 E
Figure 11. Effect of K1dy on the mutual diffusion coefficient. 10-5
E
10-9
1o-jo
lo-" lo-'* 10-13
1 o-'. 1045
lo-" 0.0
0.2
0.6
0.4
0.8
1.0
m,
Figure 10. Effect of K21- Tgl on the mutual diffusion coefficient.
KZZ- Tgz, since Tgl and T@ are known constants, only Kzl and KZZwere changed. For Dol, K d y , KZI- Tgl, K~dy, and KZZ- Tgz, the prediction increases as the valGe of the parameter increases. On the other hand, for fl, and x,the prediction decreases as the value of the parameter decreases. Effects of and 6 on the prediction are the same since the product of the two parameters appears in eq 1. To see the effect of each parameter on the prediction more clearly, a sensitivity factor, F, has been introduced as follows
@,e,
E
where D(P,wl)is the prediction of the mutual diffusion coefficient using a parameter value which has been varied from a reference value, while D(P,,wl) is the prediction using a reference value of the parameter. If the F value of a certain parameter approaches 0, it
0.0
0.2
0.4
0.6
0.8
1.0
w, Figure 12. Effect of K22 - T& on the mutual diffusion coefficient.
means that this parameter is a nonsensitive parameter for predicting diffusion behavior. A negative F value means the prediction decreases by varying the value of a certain parameter, and vice versa. In this analysis, the parameter values in Table 5 were chosen as the reference values since these values can predict diffusion behavior reasonably. The F values for each parameter as a function of concentration, when the values of the parameters are reduced by half, are shown in Figures 14 and 15. For all the parameters, except for Dol, the F value varies as the concentration changes. The F value of Dol stays at -0.3 through the entire concentration range since this parameter only affects the magnitude of the prediction. The solvent free-volume parameters K d y , and KZI- Tgl)and x are nonsensitive a t 01 = 0 and become sensitive as the concentration increases. The effect of x on the prediction is negligible up to 01 = 0.2 and increases slowly thereafter, while the effect of
(E,
2642 Ind. Eng. Chem. Res., Vol. 34, No. 7,1995
0
10-0
10'"
lo"*
1
i 53
-4
-
-6
-
-8
1043
ri
io-'' .
-
-10
lo-'e
.
10-16
'
-12
0.0
0.2
0.6
0.4
0.8
1.0
0.0
0.2
Wl
0.6
0.4
0.8
1.0
a1
Figure 13. Effect of x on the mutual diffusion coefficient.
Figure 16. Concentration dependence of the sensitivity factor, F,for free-volume parameters (11).
10-5
10-6
L
a,
lo-'
LL 10-8 0000
0
0
0
0
0
0
0
0
0
-1
0.0
0.2
0.4
0.6
0.8
1.0
10-0
0.0
0.2
0.6
0.4
0.8
1.0
m, m,
Figure 14. Concentration dependence of the sensitivity factor, F,for free-volume parameters (I).
Figure 16. Experimental data (Zgadzai and Maklakov, 1985) and
fl quickly increases to a certain point
theoretical predictions for self-diffiion coefficients of ethylbenzene in polystyrene using the adjusted Dol value for ethylbenzene.
and remains there. The F values of K d y and K21- Tglbehave more interestingly. These parameters are most sensitive at around 0 1 = 0.1 and increasingly less sensitive for 0 1 > 0.1. The polymer free-volume parameters Kd y , and K22 - Tg2)and 6 are very sensitive at w1= 0, but the effects on the rediction diminish as the concentration increases. K d y , and 5 are the parameters that affect the rediction over the entire concentration range. Since 1 and can easily be calculated from group contribution methods with minimal error, 5 and K d y are the most sensitive parameters in the VrentasDuda free-volume theory for this system. Since it was found that Dol was one of the least sensitive parameters and only affected the magnitude of the prediction, Dol was selected as the first parameter to modify for predicting diffusion behavior of the polystpenelethylbenzene and poly(viny1 acetate)/chloroform systems more accurately. The Dol values of ethylben-
(E,
$ E,
++
E
zene and chloroform were decreased by half, and the diffusion behavior of two systems was predicted using the new values. The results are shown in Figures 16 and 17, and the predictions are improved for both systems. For the polystyrene/ethylbenene system, since the second derivative of D1 with respect to 01 is positive in the concentration range near pure solvent limit the Vrentas-Duda theory cannot predict the diffusion behavior correctly a t this concentration range using a constant E value (Vrentas and Chu, 1987). To predict the diffusion behavior of the polystyrene/ethylbenzene system accurately over the entire concentration range, an E value as a function of concentration is needed. The prediction results with modified Dol values for the two systems suggest that if a t least one experimental diffusion coefficient is available, Dol can be an
Ind. Eng. Chem. Res., Vol. 34, No. 7, 1996 2543 I
P = varied value of a certain parameter P, = reference value of a certain parameter P1 = solvent vapor pressure (mmHg) P; = saturated solvent vapor pressure (mmHg) R = gas constant T = temperature (K) Tgl = solvent glass transition temperature (K) Tg2 = polymer glass transition temperature (K) Vc= solvent critical molar volume (cmVmo1) VI = solvent specific volume (cmVg) = specific critical hole free volume of solvent required for jump (cm3/g) = specific critical hole free volume of polymer required for jump (cm3/g) VI = solvent molar volume (cmVmo1) V$ = molar volume of polymer jumping unit (cmVmo1) V;(O) = molar volume of pure solvent at 0 K (cm3/mol) I 0.0
0.2
0.4
0.6
m, Figure 17. Experimental data (Ju, 1981)and theoretical predictions for poly(viny1 acetate)/chloroform mutual diffusion coefficients using the adjusted Dol value for chloroform.
adjustable parameter to predict the diffision behavior accurately.
Greek Letters a = polymer specific proportionality constant in eq 9 y = overlap factor which accounts for shared free volume 61 = solvent solubility parameter (callcm3)ln 8 2 = polymer solubility parameter ( c a l l ~ m ~ ) ~ ’ ~ 71 = solvent viscosity (g/cm*s) 6 = ratio of critical molar volume of solvent jumping unit
to that of polymer jumping unit
41 = solvent volume fraction
x = Flory-Huggins w1
Conclusions Three free-volume parameters ( K d y ,K21 - Tg1,and Dol) can be estimated successfully from solvent viscosity data. Determination of the free-volume parameters enables fair-to-good predictions of both self- and mutual diffusion coefficients over wide ranges of temperature and concentration when the Vrentas-Duda diffision model is used. The predictions can be improved further by adjusting Dol values for some polymer/solvent systems.
Acknowledgment The author thanks Professor J. L. Duda for illuminating discussions. The author is indebted to the Ministry of Education of Korea for the scholarship it provided.
Nomenclature A1
= constant pre-exponential factor in eq 11 (g/cm*s)
CL :F = solvent WLF parameter CELF= solvent WLF parameter (K) CzLF= polymer WLF parameter C Z ” = polymer WLF parameter (K) D = polymer/solvent binary mutual diffusion coefficient (cm2/s) DO= constant pre-exponential factor in eq 1 (cm2/s) Dol = constant pre-exponential factor when E is set equal to 0 (cm2/s) D1 = solvent self-diffusion coefficient (cm2/s) E = energy required to overcome attractive forces from neighboring molecules (callmol) F = sensitivity factor K11 = solvent free-volume parameter (cm3/gK) Kzl = solvent free-volume parameter (K) K12 = polymer free-volume parameter (cm3/gK) Kzz = polymer free-volume parameter (K) M1 = solvent molecular weight (g/mol) Mzj = molecular weight of polymer jumping unit
w2
polymer/solvent interaction parameter
= solvent weight fraction = polymer weight fraction
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IE940720C
@Abstract published in Advance ACS Abstracts, J u n e 1, 1995.
