Ind. Eng. Chem. Res. 1992,31,218-228
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Literature Cited Alexander, G. B. Silica organosols. U.S.Patent 2,921,913,1960. Billmeyer, F. W. Textbook of Polymer Science, 3rd ed.; Wiley: New York, 1984,p 478. Broge, E. C. Surface-modified siliceous particles. US. Patent 2;739,078, 1956. Feigin, R. I.; Napper, D. H. Stabilization of Colloids by Free Polymer. J. Colloid Interface Sei. 1980, 74, 567-571; Depletion Stabilization and Depletion Flocculation. Zbid. 1980, 75, 525-541. Iler, R.K. The Chemistry of Silica; Wiley: New York, 1978; Chapter 6. Kito, T.; Yoshinaga, K.; Hatanaka, N.; Emoto, J.; Yamaye, M. Novel Polyalcohols with Hydroxymethyl Side Chains from Base-Catalyzed Polycondensation of Diols. Macromolecules 1985, 18, 846-850.
Koberstein, E.; Lakatos, E.; Voll, M. Ber. Bunsen-Ges. Phys. Chem. 1971, 75, 1104-1114. Lange, K. R. The characterization of molecular water on silica surfaces. J. Colloid Sei. 1965, 20, 231-240. Ogihara, T.; Shimizu, T. Manufacture of silica sols dispersed in organic solvents. European Patent 372,124,1990. Pluta, L. J.; Vossos, P. H. Making silica organosols. U.S.Patent 3,699,049,1972. Sato, T. Physical-Chemical Properties of Suspension (I). Stability of Dispersions. Shikizai 1986, 59, 682. Stossel, E. Colloidal dispersions of SiOz hydrogels in polyols. US. Patent 3,004,921,1961. Received for reuiew April 10, 1991 Revised manuscript received August 15, 1991 Accepted August 27, 1991
Size Effects on Solvent Diffusion in Polymers Dominique Arnouldt and Robert L. Laurence* Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 01003
Capillary column inverse gas chromatography (CCIGC) has been used for the measurement of diffusion coefficients in polymer-solvent systems a t conditions approaching infinite dilution of the volatile component. In a study of the effect of penetrant size and configuration on solvent diffusion in two polymers, poly(methy1 methacrylate) (PMMA) and poly(viny1acetate) (PVAc), measurements of diffusion coefficients were made a t temperatures ranging from Tgto 70 K above Tg.Over 30 solvents were evaluated: alkanes, alkenes, aromatic hydrocarbons, and aliphatic esters. The effect of solvent size on the activation energy of diffusion, ED,is examined in the limit of zero mass fraction of solvent. The data allow discrimination between the conflicting theories describing the variation of EDwith solvent size, the ceiling-value hypothesis and the hypothesis based on free-volume theory, and suggest that the Vrentas-Duda free-volume theory offers a better description. It was shown that the flexibility and compactness of the diffusing species have a profound influence on the diffusive behavior. The results of the study of the effect of penetrant size and configuration on diffusion indicate that free-volume theory must be reexamined or replaced with a model which would provide an improved accounting of differences in the solvent geometry and flexibility.
Introduction The free-volume theory of transport developed by Vrentas and Duda (1977a,b) presents an expression for the self-diffusioncoefficient of the solvent, D1, in the limit of zero mass fraction:
v2*.
jumping units per mole of jumping units, In other words, 5 is the ratio of the size of the hole required for a solvent molecule to jump to the size of the hole needed for the movement of a polymer jumping unit. In the early version of the theory, the entire solvent molecule was assumed to be able to perform the jump and the quantity 6 was defined as 5= = Q1*M1/V2*Mj2 (3)
v1*/v2*
The parameters Dol and (7Q2*5)/K12depend on the properties of the polymer-solvent system, the parameter K22- Tg2depends on the polymer properties alone, and T i s the temperature. The work described here sought to investigate the influence of the solvent on the parameters iTol-ana(YP2*5)/K12. The parameters (7V2*)/KI2 and K22are related to the Williams-Landel-Ferry (WLF) constants, C l g and C2g, of the polymer:
The quantity 5 is defined as the ratio of the critical molar volume of solvent, V1*,to the critical volume of polymer
* To whom correspondence should be addressed. 'Current address: GE Plastics, Inc., Beauvais Plant, B. P. #1 60134 Villers-St. Sepulchre, France. 0888-588519212631-0218$03.00/0
where Vl* and Q2* are the specific critical hole-free volumes (Vrentas and Duda, 1977b) of the solvent and the polymer, Ml is the molecular weight of the solvent, and is the molecular weight of a polymer jumping unit. k i t h the assumption that all solvent molecules move as single units, the apparent activation energy for diffusion in the limit of zero solvent concentration, E D , defined as
should increase indefinitely as the size of the solvent molecule increases. This prediction of the free-volume theory does not agree with experimental data (Chen and Ferry, 1968;Fujita, 1961; Kokes and Long, 1953; Meares, 1965) which suggested that the activation energy approaches a limiting value as the size of the solvent molecule increases. Another interpretation of these experiments (Fujita, 1961; Kokes and Long, 1953; Meares, 1965) is that the ceiling value is reached for solvent molecules as large 0 1992 American Chemical Society
Ind. Eng. Chem. Res., Vol. 31, No. 1,1992 219
as the polymer jumping unit. For solvent molecules larger than the polymer jumping unit, the activation energy becomes independent of the solvent size. This interpretation is based on the belief that for these large molecules the diffusion p r m is controlled by the motion of the polymer molecules. A number of researchers (Brown et al., 1973; Chen and Ferry, 1968; Prager et al., 1953; Prager and Long, 1951; Rhee et al., 1977) suspected the possibility of segmental motion of solvent molecules in amorphous polymers. Vrentas and Duda (1979) modified eq 3 to account for segmental motion of solvent molecules. Assuming that such molecules have one jumping unit (of a particular size) with a much greater probability for jumping than any other size jumping unit, eq 3 becomes (5) 5 = V1*/V2* = v1*Mj1/p2*Mj2 where and Mjl are now defined as the specific critical hole-free volume and molecular weight of the solvent jumping unit. Clearly, the jumping unit of sufficiently flexible solvent molecules will be only a portion of the solvent molecule. Thus, an increase of the size of the solvent molecule will not lead necessarily to an increase in the value of the parameter [ (or ED). The experimentauy observed leveling of the activation energies would be explained by a leveling of [ due to segmental motion. Another interpretation of the leveling of the parameter E with the size of the solvent molecule is given by Ferguson and von Meerwall(1980). These authors suggest that the leveling of the parameter 5 is not due to segmental motion of the solvent molecules but is the result of changes in the size of the polymer jumping unit (or V2*).These changes would be the result of an adjustment of the polymer jumping unit to the size of the solvent molecule. Vrentas and Duda (1979) reexamined the diffusion data for poly(methy1acrylate) (Burgess et al., 1971; Fujita, 1968; Kishimoto et al., 1960) and polyisobutylene (Blyholder and Prager, 1960; Kokes and Long, 1953; Van Amerongen, 1946; Wong et al., 1970). Their analysis suggests that these data, used in the literature to justify a ceiling value for the activation energy, could also be explained by a segmental motion of the large solvent molecules considered. However, they conclude that more diffusion data are necessary to be able to chose between these two interpretations of the leveling of the activation energies with solvent size. Vrentas et al. (1986) reported values for the parameter 5 for four members of the n-alkane series (C5,Cg, Cl0, and C12)and a much larger solvent, 1,3,5-triisopropylbenzene, and in polystyrene. The 5 values for the n-alkane series are almost the same ([ z 0.42), whereas 1,3,5-triisopropylbenzene has a 5 value of 1.37. These data indicate that there is no leveling of 5 when the size of the entire solvent molecule is increased. Furthermore, the leveling of 5 observed for the n-alkane series can only be explained by segmental motion. It cannot be due to an adjustment of the size of the polymer jumping unit to the size of the solvent since the largest value of [ was obtained with the largest solvent molecule. This experimental study examines diffusion of solvent molecules in amorphous polymers and attempts to confirm the result of Vrentas and Duda (Vrentas et al., 1986,1980). Another objective of the study is to understand what parameters influence the displacement mechanism. If solvent molecules are effectively capable of segmental motion, what characteristics of the solvent molecule influence the parameter E? Recent work (Mauritz and Storey, 1990, Mauritz et al., 1990) has proposed a modification of the free-volume theory to account for molecular shape. Another paper
vl*
(Ehlich and Sillescu, 1990) presents an additional interpretation of [ in the Duda-Vrentas analysis. This work supplies a large amount of data which could be used toward evaluation of these proposals.
Experimental Section Materials. The poly(methy1 methacrylate) (PMMA) used in this study was a commercial grade (PRD-41) manufactured by the Rohm and Haas Co. The PMMA had a weight average molecular weight of 200000 and a polydispersity of 2.35 (determined by gel permeation chromatography). The glass transition temperature of the PMMA was 389-393 K, as determined by differential scanning calorimetry. The density of the PMMA was given by 2200 PPMMA = 0.8478T~+ 1641.8 where T K is the temperature in kelvin. The WilliamsLandel-Ferry (WLF) constants, C l g and C2g, of PMMA used in this study were Clg = 14.8 K and C2g = 80.0 K (Liu et al., 1980). The poly(viny1 acetate) (PVAc), obtained from Aldrich Chemical Co., had a weight average molecular weight of 227 670 and a polydispersity of 2.64. Its density was 1.191 g/cm3, and its glass transition temperature was 305 K. The WLF constants of PVAc used were Clg = 15.6 K and C2g = 46.8 K (Liu et al., 1980). All the solvents used in this work were spectroscopic grade products obtained from Aldrich. Equipment. Presented here is a brief description of the general equipment used to obtain the basic data, that is, the elution curves. A more complete description of the apparatus of capillary column inverse gas chromatography (CCIGC) is given by Pawlisch (1985) and in earlier papers (Arnould and Laurence, 1989, Pawlisch et al., 1988,1987). The chromatograph used was a Varian 3700 (Varian Instrument Division) equipped with a flame ionization detector (FID), splitless capillary injector, and circulating air oven. Dry-grade air and hydrogen were used for the FID detector. The temperature in the oven was measured with four copper-constantan thermocouples placed at different locations in the vicinity of the column to measure any spatial variations in temperature. The carrier gas used was helium. Its flow rate was regulated with a Tylan mass flow controller (Tylan Corp.). The pressure at the inlet of the column was measured with a Setra Systems pressure transducer (Setra Systems Inc.). The marker gas used for this work, methane, was injected into the column with a fixed-volume 10-pL syringe (Pressure-Lok Mini-Injector, Precision Sampling Corp.). The solvents were injected in the column as liquids with a variable-volume 0.5-bL syringe (S.G.E. microliter syringe type B, Scientific Glass Engineering Pty. Ltd.). A Macsym 120 (Analog Devices) microcomputer was used to record, display, and store the detector output signal. The data files were stored on floppy disks for subsequent analysis. Preparation of the Capillary Columns. A poly(methyl methacrylate) (PMMA) column used in this work was prepared by Bric (Pawlisch et al., 1988) using the procedure described by Pawlisch (1985). The same procedure was used to prepare a poly(viny1 acetate) (PVAc) column. Although this procedure is not described here in its entirety, the principles of the static coating procedure are given below. A blank Pyrex column is filled with a dilute solution of the polymer to be coated. One end of the column is sealed, and a vacuum is drawn on the other end of the column at constant temperature. As the solvent evaporates, a polymer film is deposited on the wall. The drying rate for
220 Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992
PVAc in tetrahydrofuran at 28 "C was 0.1 m/h. Columns had lengths of about 20 m, an inside diameter of 0.8 mm, and an outside diameter of 1.2 mm. The thicknesses of the polymer films,estimated from the concentration of the polymer solution used to coat the column, were 5.3 pm for the PPMA column and 5.0 pm for the PVAc column. The PMMA column had an inner radius of 370 pm and a length of 17.8 m. The PVAc column had an inner radius of 380 pm and a length of 18.3 m.
Experimental Procedures Data Collection. After the GC had reached steadystate operation (i.e. stable temperature and carrier gas flow rate), the output of the FID detector, which had a range of 0-10 volts, was set to a base-line value just above zero. Methane was then injected. The quantity injected was chosen in order to obtain a maximum output signal (i.e. peak height) of 8-9 V. The magnitude of the signal with respect to the base line was proportional to the concentration of the methane at the outlet of the column, given constant carrier gas flow. The resulting elution curve (Le. detector signal versus time) was recorded both on a chart recorder and on the Macsym 120 microcomputer. After the methane run, the elution curve of the solvent was obtained in the same fashion and was followed by a second methane run. Methane runs were used to determine the carrier gas velocity and to ensure that the flow was constant. Other basic data recorded during the experiments were the inlet carrier gas pressure and the four temperatures in the oven. The two primary improvements derived from the use of the Macsym 120 microcomputer were the following: the number of data points collected in a single run was extended from 1024 points to 15000 points, and the maximum sampling frequency was increased from 5 to 20 sec-'. The data acquisition code was rewritten to allow changes in the sampling frequency during the experiment. Model. A model of the capillary chromatograph has been presented in earlier papers (Pawlisch et al., 1988, 1987). The solution in the LaPlace transform domain for the dimensionless exit concentration, Y(s),is given by the following equation: Y(s)= exp(1/2y) exp[-(1/2y)(l
+ 4 r J , ( ~ ) ) ' / ~(7) l
where +(s) = s
2s'/2 +tanh .P
The dimensionless model parameters, a,8, and y are defined by
The parameter a! is a thermodynamic parameter; @ and y represent the polymer- and gas-phase transport properties. The parameter, K, is the distribution coefficient between the phases, V is the mean velocity of the mobile (gas) phase, L is the length of the column, 7 is the film thickness, and D, and D, are the diffusivities in the stationary and mobile phases, respectively. Data Analysis. The f i t phase of the data analysis was done on the Macsym 120. It consisted of subtracting the value of the base-line signal from the raw output signal, determining the range of the elution peak, and finally integrating the elution peak to obtain the normalized first and second moments. The values of these moments were used to estimate the parameters a and /3 with a moment fitting procedure. This first phase of the analysis was
achieved with an interactive graphics routine, rewritten for the Macsym 120. Graphics subroutines were added to improve the visual estimation of the range of the elution peak. The second step of the analysis, the Fourier domain fitting, was also done on the Macsym 120 microcomputer instead of the larger mainframe computer used earlier (Pawlisch, 1985). A graphics subroutine was added to allow a visual estimate of the quality of the fit between the experimental and theoretical elution curves in the time domain. The values of a and /3 determined by the moment analysis were used as initial estimates. The partition coefficient and the diffusion coefficient can be estimated from a single elution curve (Pawlisch et al., 1988,1987). However, in order to improve the precision of the estimates, the results of three sets of experimenta carried out at three different flow rates were used. The parameter a is independent of the carrier velocity. Thus, an average value of a! was used to calculate the partition coefficient. A plot of versus carrier velocity is linear, with the slope inversely proportional to the diffusivity. A least-squares fitting was used to estimate this slope.
o2
Experimental Results for Poly(methy1 methacrylate) (PMMA) Influence of the Temperature on the Diffusion Process. Free-volume theory (Vrentas and Duda,1977a,b) describes the influence of the temperature on the diffusion process a t temperatures above the glass transition temperature by
For temperatures near the glass transition temperature of the polymer, the amount of hole-free volume in the liquid is relatively small, and the self-diffusion process is free-volume-driven (Vrentas and Duda, 1977a). Over that temperature interval, the second exponential (the freevolume term) is much smaller than the energy term and the energy term can be treated as a constant. Equation 10 can be written as eq 1, where Do, is treated as a constant. Clearly, the temperature interval over which this approximation is correct will depend on the polymersolvent system. This interval will be smaller for a polymer + T - Tg2) with a large amount of free volume (i.e. Kl2(KZ2 large) and/or for a solvent with a small jumping unit (i.e. 5 small). A similar approximation can be made for temperatures much higher than the glass transition temperature. For such temperatures, the amount of free volume in the system is large, and the energy term in eq 10 is much smaller than the free-volume term. In that case, the diffusion process is energy-driven, the free-volume term can be absorbed in the preexponential factor, and eq 10 becomes the frequently used Arrhenius equation. These two approximations are exhibited in Figure 1. In region 111, the diffusion process is free-volume-driven, and it can be satisfactorily described by eq 1, where
In region 11, the contributions of the free-volume and energy terms are both significant. The diffusion process is described by eq 7 and
Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992 221 T ("C) I70
Region I
:
Region I1
j
160
140
Is0
10
Region I11
PMMA / Methyl acetate] PMMA / Ethyl acetate
h
fi
10
(K~+T-T~-' 10
Figure 1. Free-volume description of the temperature dependence of the solvent self-diffusion coefficient. T 170
I60
I50
10'/(K zz+T-Tgz) ( O K . '
)
Figure 3. Free-volume correlation of the diffusivity data for the
W) 130
140
acetate series in PMMA.
120
10 4
0
PMMA / Methanol PMMA / Acetone
10
.
F l PMMA / Benzene
.'
PMMA I Ethylbenzene
I]
v)
"8
v
P 10
10
7
8
9
10
1 0 % z~ z + ~ - ~ g z )
I1
ti
