3556
Ind. Eng. Chem. Res. 1995.34, 3556-3567
Unsteady-State Diffusion in Block Copolymers with Lamellar Domains Niloufar Faridi, J. L. Duda,* and Ilyess Hadj-Romdhanet Center for the Study of Polymer-Solvent Systems, Department of Chemical Engineering, The Pennsyluania State University, University Park, Pennsylvania 16802
An unsteady-state model has been developed to predict the diffusion coefficients of small molecules in copolymers having a lamellar morphology. The model incorporates simplifications which are valid when the diffusion in one of the components or phases of the polymer is much faster than in the other phase. This behavior is realized in many systems such as block copolymers in which styrene constitutes one segment of the polymer molecule attached to a rubber segment such as polybutadiene. Three solutions for the model equations are presented an analytical solution valid for the initial stage of the sorption process, a numerical solution of the complete model equations, and a simple approximate model which will be shown to be valid when there is essentially a dynamic equilibrium between the two phases. This lamellar copolymer model is then evaluated with experimental diffusivity data obtained with the capillary column inverse gas chromatography method and the sorption data obtained by Rein et al. (1990, 1992). The most significant result of this work is that sorption behavior in many block copolymer systems with lamellar structure can be described by a simple model which assumes dynamic equilibrium between adjacent lamellae. Introduction Polymeric materials have gained importance in many transport applications including membranes in gas and liquid separations and barriers in packaging. A wide variety of processes and products involve the transport of solvents through polymer blends and copolymers, which combine two or more polymer components with widely different transport properties into a microphaseseparated system with advantageous morphological characteristics. Knowledge of the relationships between the block copolymer morphology and the transport process is essential for successful manufacturing and utilization of these heterogeneous polymers. Various transport models have been developed for predicting effective steady-state diffusion coefficients in polymer blends and block copolymers. The steady-state diffusion coefficient for a composite material of sheets in parallel has been described by a linear combination of the components’ steady-state diffusion weighted by their respective volume fractions (Crank, 1975). Sax and Ottino (1983) have developed models to describe steady-state transport of small molecules in several morphologies exhibited by block copolymers. In their development,they assume that the block copolymer has an ordered microstructure and a disordered mamstructure with three possible different morphologies (lamellar, cylindrical, and spherical). For some perfectly ordered morphologies, it is possible to obtain exact expressionsfor the effective diffusion coefficients(Barrer and Petropoulos, 1961; Charrier, 1975). For disordered composites, the effective transport coefficients are predicted by a model based on the effective medium theory (EMT) (Landauer, 1952; Davis, 1977). In heterogeneous media and composites, the effective unsteady-state diffusion coefficient is not necessarily equal to the steady-state diffusion coefficient of the
* To whom correspondence should be addressed. JLDW PSUADMIN (BITNET). ’Current address: 3M Center, Building 42-1E-01, P.O.Box 33331, St. Paul, MN 55133-3331.
Figure 1. Transmission electron micrograph of the triblwk copolymer (styrene-butadiene-styrene) having lamellar rnorphalOW.
material and therefore cannot be related to a combination of the diffusion coefficients in the separate phases. A description of unsteady-state diffusion in random blends has been presented by Ottino and Shah (1984) and Shah et al. (1985). Their simulations are based on a two-dimensional hexagonal tessellation arrangement. In their model, they solved the equations of conservation of mass in all the cells in the tessellation representing the blend or composite to obtain the concentration distributions of the permeants in the various phases. Unlike random polymer blends, block copolymers often exhibit a local order in the structure with longrange disorder. For example, Figure 1is a transmission electron micrograph of a typical lamellar triblock copolymer system (Kraton D-1101) which consists of polystyrene rich phases (white regions) and butadiene rich phases (black regions). As this figure shows, the phases appear to be bicontinuous and consist of parallel lamella which twist and undulate on a large scale. Any model developed to describe transport in such block copolymers should incorporate this nonrandom charac-
0888-5885/95/2634-3556$09.00/00 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995 3557 teristic. Rein and co-workers (1990) have been able to produce sheets of block copolymer in which orientation of the lamellae is perpendicular to the surface of the sheets throughout their thickness. These investigators have modeled this well-ordered morphology by solving the two dimensional unsteady-state diffusion equations for the case in which a permeant is diffusing in the direction of the lamellae and equilibrium of the penetrant exists in the two phases at the interface of the lamellae. From the comparison of their model predictions with the experimental results, these investigators concluded that the diffusion coefficients of gases in the polybutadiene regions of the copolymer is lower than in the homopolybutadiene. This effect was considered to be a result of the restrictions on chain motions of the polybutadiene blocks that are connected to the polystyrene blocks. The objective of this study is t o develop an unsteadystate model that describes sorption of small molecules in a block copolymer with a lamellar morphology such as that illustrated in Figure 1. The proposed model is based on this characteristic and the notion that copolymers with lamellar morphology have small-scale order and large-scale disorder. A n analytical solution of the model equations is developed for the initial stages of the sorption. The solution of this lamellar copolymer model is then extended to cover the complete sorption process by a finite-difference solution of the governing equations. Finally, a very simple approximate solution is presented which is valid when there is essentially local equilibrium of the penetrant between aqjacent lamellae at all positions in the sample. The model is then evaluated utilizing two sets of diffusivity measurements. Diffusion of toluene, methyl ethyl ketone (MEW, and heptane in a styrene-butadiene triblock copolymer (SBS) obtained with the capillary column inverse gas chromatography (CCIGC) method are correlated by the use of a n adjustable tortuosity parameter. Finally, the ambiguity associated with this adjustable parameter is eliminated by predicting the effective diffusivities obtained by Rein and co-workers (1990,1992)for diffusion of several gases in a styrene-butadiene diblock copolymer (SB) in which the tortuosity can be assumed to be one since the lamellae are oriented parallel to the direction of the diffusion throughout the polymer sample.
Theory
Model Formulation. To develop a model to describe diffusion in a block copolymer consisting of lamellar domains such as that illustrated in Figure 1,the unique morphology should be considered along with the fact that in most systems, the diffusion in one phase is much faster than in the other phase. For example, in Kraton D-1101 which is shown in Figure 1, diffusion in polybutadiene rich phases (black regions) is much faster than diffusion in polystyrene rich phases (white regions). Figure 2 is a schematic diagram of one region of this undulating lamellar morphology. Based on the visualized morphology illustrated in Figure 2 and the large difference between the diffusion rates in the two phases, a simplified model of solvent or penetrant sorption into a copolymer can be envisioned. At the beginning of the sorption process, the solvent molecules would diffise relatively quickly along the axial direction ( 2 ) of the tortuous polybutadiene rubber (BR) phase. Due to the fast diffusion in the rubber and retardation of the solvent flux into the low-diffusivity PS phase, the gradient of the solvent concentration in the x direction
Figure 2. Visualization of one lamellar region in a copolymer.
of the rubber phase will be negligible. In contrast, the axial flux of solvent in the polystyrene (PS)phase can be neglected in the overall sorption process and solvent equilibrium will be maintained at the interfaces between the two phases. The assumptions utilized in the model are as follows: (1)The system is isothermal. (2) Diffusion coefficients and distribution coefficients are concentration independent. (3) The thickness of the phases is constant. (4) Axial diffusion in the PS phase is negligible. (5) Flux between parallel rubber lamellae through the PS phase is negligible. (6) The influence of the interfacial zone between the PS and BR phases on the sorption process can be neglected. (7) The copolymer morphology is not affected by the solvent concentration. (8)Since diffusion in the rubber phase is fast compared to the other phase, the details of the distribution of solute in the x direction become unimportant and a description of the longitudinal dispersion of solute in the rubber phase in terms of a local mean concentration (i.e., averaged in x direction) will suffice. In the following formulation, the prime ('1 symbol on variables and parameters will refer to the polymer phase in which the diffusion process is slow (the polystyrene phase in the case of Kraton). Nonprimed symbols will refer to the phase in which molecular transport is fast (the rubbery polybutadiene phase in the Kraton case). Utilizing the above assumptions, the continuity equations for the rubber and glassy phases may be written as
where C and C' are the concentration of the solvent in the fast and slow diffusing phase, respectively. Similarly, D and D are diffusion coefficients of the solvent in the two phases. Appropriate initial and boundary conditions for a sorption process with a step change in the concentration of the solvent a t one surface (C= 0)
3558 Ind. Eng. Chem. Res., Vol. 34,No. 10, 1995
and no flux condition at the other surface of the specimen (z = zz') are
at t = 0, z > 0
C(x,z,t)= C'(x,z7t)= 0 C(x,z7t)= C ,
at z = 0
(3)
that exists in the fast diffusing phase at early times simplifies the problem, and the inverse Laplace of these equations for initial times can be taken and an analytical solution is attained:
(4) (2n - 1)n ( 5 - 1) cos
(13)
2
(7) In these boundary conditions, C, is the equilibrium concentration of the solvent that is introduced to the polymer sample, K is the ratio of the partition coefficient of solvent in the two phases ( K = K'lIO, z' is the thickness of the copolymer sample, z is the tortuosity factor, and 2L and W are the thicknesses of the two phases, respectively. This formulation can now be put into dimensionless form by introducing the following variables 4 = c/c,
4' = C'/Kc,
x = ( X - L)lL'
5 = ZhZ'
F(8')= 16
A, = KL'IL
"
-z c
(-l)n+m-l
A32n=1m = l
(e-A1(2n-1)2n20/4
where
+ F(0') (14)
where F(8') is defined as
8' = Dt/L2
(9)
(x - 1)
2
(2m - 1)
The governing transport equations (eqs 1and 2) can now be written in dimensionless form as
A, = (L/zz')2
(2n-1) n cos
cos
(2n - 1)
(2m - 112 (2n - 1 ) 2
-
- e -A3(2m-1)2n20/4)
A3
ix
X
(2n - 1)n ( 5 - 1) (2m - 1)n (2 - 1) cos 2 2
(15)
Having solved for q(8') and q'(8') (dimensionless concentration of solvent in the two phases), we can now obtain the amount of the solvent in each phase. Thus, by adding the amount of solvent in two phases, the total sorption in the sample can be expressed as a function of time for the solvent-copolymer system:
A, = D L 2 / D c 2
I. Analytical Analysis. The model equations (eqs 8 and 9) are coupled linear equations and can be solved using the Laplace transform technique. The concentration of the solvent in the two phases in the Laplace domain becomes
1 1 1+A2 64
1 Y
m=l
X
(2772 - 1)2 (2n - 1)2 (2m - 1)2 1
where
Obtaining the inverse Laplace transform of the above equations is difficult analytically. However, as was mentioned previously, D >> D , hence As is a small number. Utilizing this assumption and also considering that the amount of solvent that diffises into the slow diffusing phase does not affect the amount of solvent
A3
As the ratio D'lD gets smaller, the analytical solution becomes more accurate and is valid for a longer time period. 11. Numerical Analysis. In order to numerically solve eqs 1 and 2 with the appropriate boundary conditions, it is convenient to define the dimensionless time as 8 = DtM2 while the other dimensionless variables are maintained as defined in the analytical analysis. The dimensionless forms of the transport equations used in the numerical analysis become
Ind. Eng. Chem. Res., Vol. 34,No. 10,1995 3559
0'25
I
0.20
Here
B = 1/22
B, = KL'/L
B, = Dd2/DLf2
i I t
0.10
where
B , = A,L2I.d2
0.15
B2 =A2
B, = A,L2/d2 0.05
An implicit finite-difference method was employed (Carnahan et al., 1969) to solve the set of coupled equations, and the details of this analysis are presented by Faridi (1995). For illustrative purposes, the numerical analysis results which follow are based on input parameters which approximate the conditions for diffision of toluene in the triblock copolymer Kraton D-1101at 80 "C. The details of this case are presented in the Experimental Section. Consequently, unless stated otherwise, the numerical analysis results which follow are based on the following input parameters: L'IL = 0.6,T = 3, DID' = 4 x lo6,K = 0.94,z' = 0.001,B1 = 0.11,B2 = 0.56,and B3 = 7.0. To check the validity of the numerical solution, it is compared with the analytical solution for early times of the sorption process in Figure 3. This figure shows that a t early times of the sorption process, the numerical solution matches the analytical solution, which gives some confirmation of the numerical technique. As one would expect, the two analyses deviate at longer values of time and Figure 3 gives some indication of the time interval for which the analytical solution is applicable. Several system parameters are incorporated into the lamellar copolymer model. However, four key parameters which control the dynamics of the sorption process are DID',L'IL, z, and K. Figures 4 -7 illustrate how these key parameters influence the dynamics of the sorption process. In all cases, the fractional approach t o equilibrium, MtlMm is presented vs the square root of the dimensionless time and, unless otherwise stated, the values of the input parameters are those presented earlier. Figure 4 shows the influence of the overall process of decreasing the diffision rate in the phase which has a low diffusivity. Not only does the overall sorption process slow down, but also the shape of the sorption curve starts to deviate from the behavior which is characteristic of sorption in a homogeneous media with a constant diffusivity. Since the diffusion coefficient in the fast phase was kept constant in these three cases, Figure 4 shows that at early times there is a strong correspondence between the three cases, but a broad distribution in behavior at high values of time is noticed where equilibrium has essentially been attained in the fast diffusing phase, but solvent continually diffuses into the phase with the low diffusivity. Similarly, Figure 5 shows the influence of the ratio of the thicknesses of the two phases on the overall sorption process. Again, this figure shows that as the thickness of the phase with the low diffisivity increases, the diffision process slows down and the sorption curve deviates from a typical homogeneous sorption process. In sorption studies with homogeneous polymers, an inflection point in the sorption curve, such as illustrated
0.00 0.0
0.1
0.2
0.3
0.4
0.5
0.6
sqrt (Dt/Y2)
Figure 3. Comparison of the early time analytical solution and the numerical analysis of the lamellar copolymer model. 1 .oo
0.80
i L
0.60
t
0.40
0.20
0.0
1.0
2.0
3.0
4.0
5.0
6.0
sqrt (Dt/Y2) Figure 4. Influence of the D/D' ratio on the model predictions of sorption behavior for the SBS-toluene system.
in the lowest curve in Figure 5, is usually interpreted as non-Fickian d f i s i o n behavior (Enscore et al., 1977). In homogeneous polymers, such a behavior is associated with coupling between polymer relaxation and molecular diffision. As Figure 5 indicates, such "non-Fickian" behavior can be exhibited by Fickian diffusion in the two phases of a heterogeneous media. This model suggests that this S-shaped sorption curve would be observed in block copolymers in which the lamellae of the phase with the characteristic of slow diffusional behavior are thick. In other words, in Kraton this would correspond to a large polystyrene segment on the block copolymer relative to the butadiene segment. The influence of the tortuosity factor on the sorption process is illustrated in Figure 6. There is certainly nothing unexpected from this result, and all three of these curves could be superimposed if the tortuosity was incorporated in the dimensionless time. Finally, the influence of the partition coefficient between the phases is demonstrated in Figure 7. As one would anticipate, as the solubility in the phase with the low diffisivity increases, the overall sorption process slows down.
3660 Ind. Eng. Chem. Res., Vol. 34,No. 10,1995 1 .o
1 .00
0.8
0.80
0.60
0.6
i I
I' I I
I
0.4
0.40
0.2
0.20
0.00
0.0 0.0
2.0
6.0
4.0
sqrt (Dt/z'2)
Figure 5. Influence of the L'/L ratio on the model predictions of sorption behavior for the SBS-toluene system.
0.00
1.00
3.00
2.00
4.00
5.00
sqrt (Dt/L2) Figure 7. Influence of K on the model predictions of sorption behavior for SBS-toluene system.
1.o
0.8
0.6
2'
I
I
0.4
0.2
0.0 0
1
2
3
4
5
6
0.0
sqrt (Dt/zI2)
Figure 6. Influence of t on the model predictions of sorption behavior for the SBS-toluene system.
In order to provide more insight concerning diffision through lamellar block copolymers, the concentration profiles in each phase at different time steps are illustrated in Figures 8 and 9. Figure 8 shows the concentration distribution in the axial direction in the phase with the high diffusivity. As one would expect at low values of time, there is a very sharp gradient in the concentration a t the surface of the sample exposed to the solvent. As time increases, the gradient decreases and the concentration builds up at the other surface which has the no flux boundary condition. In Figure 9, the concentration distribution in the x direction is shown in the polymer phase with the low diffisivity. This figure indicates that aRer a period of time, this gradient becomes quite small. The results in Figure 9 suggest that for most of the sorption process, there is essentially solvent equilibrium between the two phases a t every point along the axial direction. Although, the rate of diffision in the one phase is very slow compared to the other phase, the thickrws of the lamella compared to
0.3
0.5
0.8
1.o
t Figure 8. Concentration distribution in the axial direction in the fast diffusing phase.
the overall thickness of the sample is so small that this condition of local equilibrium can essentially be maintained. The behavior demonstrated in Figure 9 suggests that the lamellar copolymer model can be simplified for some conditions. For example, in the case of Kraton, Figure 9 would suggest that at every point along the butadiene lamella, the solvent quickly diffises through into the thin polystyrene lamella, and at every point the polystyrene phase is in equilibrium with the butadiene phase. Because of the large difference in the diffisivities of the two phases and the large difference between the thickness of the lamella and the sample thickness, it is not unreasonable that one can neglect the axial diffision in the polystyrene phase while at the same time impose the simplification that the polystyrene phase is in equilibrium with the butadiene phase at every point in the axial direction. Consequently, as an approximation, the details of the diffision in the polystyrene phase which is the phase with the low diffusivity can be essentially neglected and the model reduces to:
Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 3561 1.00 -
1
Dt/z“ = 4
0.80
-=
0.80
1
0.60 ,
0.60
i
I
4
0.40 -
0.20
.oo
0.40
-
0.20
0.0
0.2
0.6
0.4
0.8
0.00 0.0
1.0
Figure 9. Concentration distribution in the x direction in the slow diffusing phase, at 5‘ = 0.2.
(19)
c=KC
(20)
As in the previous development, C and C“ are the concentrations of solvent in the two phases. y is the ratio of the volumes of the two phases (V’/V), and K is the partition coefficient. This equation describes the process in which as the concentration in the axial direction changes in the fast diffusing phase, a corresponding equilibrium concentration is maintained in the thin adjacent lamella whose phase is characterized by a relatively low diffusion coefficient. This formulation is equivalent to that presented by Smith and Keller (1985) t o describe simultaneous diffusion and sorption in porous materials, and the analysis of the diffusion of dyes in polymers by Weisz (1967). By combining eqs 19 and 20, this equilibrium approximation model reduces to the following equation and the associated boundary conditions:
at
C(z,t)= C ( z , t )= 0 C(z,t)= C,
-=_- o az
a2(c+yC‘>
1+YK
1.o
1.5
2.0
sqrt (D,,t/z’*)
X
a(c+y e ) -- D
0.5
az2 at t = 0, z > 0
at z = 0 atz=zz’
(21) (22) (23) (24)
It is obvious that this set of equations is comparable to diffusion in a homogeneous media with an effective diffusivity defined as
when the dimensionless time is defined as Dtld2. This analysis suggests that at longer sorption times, when the concentration gradients in one phase disappears, the sorption curve should approach a conventional “Fickian” sorption curve which is characteristic
Figure 10. Influence of B3 on the model predictions of sorption behavior.
of diffusion in a homogeneous media with an effective diffusion coefficient defined by eq 25. The key question is, under what conditions is this “equilibrium model” a good approximation t o the complete lamellar copolymer model? Analysis shows that if the fractional approach to equilibrium, MtlMm,is determined as a function of a dimensionless time based on the effective diffusivity, then the controlling parameter is B3. This dimensionless group is the ratio of the characteristic time for diffusion across the lamella of the slow diffusing phase to the characteristic time for axial diffusion through the polymer sample in the fast diffusing phase. The range of validity of the approximate equilibrium model is illustrated in Figure 10. The upper curve in this figure corresponds to the approximate equilibrium model. This analysis shows that the equilibrium model is a reasonable approximation to the sorption process when B3 > 3.
Experimental Section Measurements of diffusion coefficients and partition coefficients of toluene, n-heptane, and methyl ethyl ketone (MEK)in polybutadiene rubber (BR), polystyrene (PS),and styrene-butadiene triblock copolymer (SBS) were determined utilizing the capillary column inverse gas chromatography (CCIGC) technique a t solvent infinite dilution. The chromatograph used in this work was a Varian Model 3400 equipped with a flame ionization detector (FID), an on-column injector, and a circulating air oven. Details of the experimental technique are presented by Hadj Romdhane (1994) and Arnould and Laurence (1992). Polystyrene used in this work was supplied by Pressure Chemical Co. and has a M,, = 13 500 and MdMn = 1.04. Polybutadiene was obtained from Goodrich Co. and has a Mooney viscosity = 34-46 and 95% cis content. 3M Co. ES&T Division has supplied the SBS triblock copolymer (Kraton D-1101). This copolymer consists of 31 weight % of polystyrene and a number average molecular weight of 110 000 with MwlMn=1.26. The morphology of the SBS sample was documented by means of transmission electron microscopy (TEM)and is shown in Figure 1. The dark regions correspond to the BR microdomains selectively stained by 0,04, and the bright regions are the PS micro-
3562 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 Table 1. Partition and Diffision Coefficients of Toluene, Methyl Ethyl Ketone, and Heptane in Polybutadiene at Infinite Dilution Obtained by Capillary Column Inverse Gas Chromatomaphv Method toluene MEK heptane 10+D 10-6D 10-6D T("C) K (cm2/s) K (cm2/s) K (cm2/s) 3.1 3.97 71.1 35.0 194 2.67 80 51.9 3.76 26.9 4.77 90 138 3.33 38.6 4.47 20.9 5.65 101 4.09 100 29.2 5.25 16.5 6.59 110 74.7 4.93 6.09 7.6 22.3 5.85 13.1 120 56.2 ~
~~
~~
Table 2. Partition and Diffision Coefficients of Toluene, Methyl Ethyl Ketone, and Heptane in Polystyrene at Infinite Dilution Obtained by the Capillary Column Inverse Gas Chromatography Method toluene MEK heptane 10-120' 10-120' 10-12D' T("C) K' (cm2/s) K' (cm2/s) K' (cm2/s) 0.727 70 281 3.65 55.3 76.9 95.9 116 80.7 0.946 5.70 40.5 75 222 149 68.0 2.65 7.37 31.1 80 183 49.6 5.60 28.5 222 90 135 21.1 36.3 50.1 28.1 756 100 104 118 428 19.2 1370 19.8 290 110 80.8 14.4 1380 19.7 6210 120 63.7a 1600a a Values of D and K for the PS-toluene system a t 120 "C were obtained by Pawlisch et al. (1988).
domains. This micrograph clearly indicates that the SBS sample consists of a lamellar morphology and the lamellae are not ordered over a very long range. Because PS constitutes about 28% by volume of the polymer, and on the basis of study of Bates (1991) with polystyrene-polyisoprene block copolymers, a morphology consisting of cylindrical domains of PS embedded in a continuous BR matrix was anticipated. However, the lamellar morphology in this case might be attributed to the existence of a homopolymer in the system which was blended with the copolymer. Leary and Williams (1973,1974)proposed a microstructural thermodynamic model which predicts a lamellar morphology for a SBS triblock with an overall fraction of PS equal to 24.5%. They also validated their model predictions with micrographs taken for a sample of Kraton 1101. Similarly, Caneba et al. (1983-84) have also documented a lamellar morphology for a SBS triblock copolymer with 29 weight % polystyrene content and a molecular weight of 110 000. The solvents used in this study were reagent grade materials supplied by Thomas Scientific Co. and were used without further purification. Chromatographic data analysis was performed in terms of the elution model developed by Pawlisch et al. (1987, 1988). The method of calculation of diffusion coefficient and the partition coefficient of solvent in the polymers is described in details by Hadj Romdhane (1994) and Faridi et al. (1994).
Results and Discussion Diffusion Coefficients. The measured diffusion coefficients and partition coefficients for the three solvents in PS, BR, and SBS over a temperature range 70-150 "C are presented in Tables 1-3. The diffusion coefficients of solvents in BR at temperatures higher than 80 "C were obtained (Faridi, 1995) by correlating the lower temperature diffusion coefficients over a temperature range using the free-volume theory of
Table 3. Partition and Diffusion Coefficients of Toluene, Methyl Ethyl Ketone, and Heptane in SBS (Kraton D-1101) at Infinite Dilution Obtained by Capillary Column Inverse Gas Chromatography Method toluene MEK heptane 1 0 - 7 ~ ~ ~ 10-IDea 10-lDetf T ( " C ) Kea (cm2/s) K e ~ (cm2/s) Keff (cm2/s) 1.86 2.60 51.8 2.25 80 199 50.7 3.07 90 147 3.88 39.5 3.45 38.4 4.41 5.51 30.7 4.79 30.4 100 110 5.98 7.65 24.5 6.31 24.3 110 84.7 8.60 10.1 19.9 8.52 120 66.1 19.6 130 53.3 12.3 15.8 16.1 17.5 13.7 13.2 21.1 14.5 17.5 16.9 140 43.2 10.4 150 37.2 27.4 33.2 12.4 28.3 Table 4. Values of Tortuosity Correlated from the Diffusion Data of the SBS-Toluene System Using the Approximate Equilibrium Model temp ("C) 10-7Deffiexp)(cm2/s) tortuosity 80 1.86 3.23 90 3.07 2.8 100 4.41 2.56 110 5.98 2.4 120 8.52 2.17
Vrentas and Duda (1977a,b). The results show that the diffusion coefficient of solvents in BR is between 3 and 6 orders of magnitude higher than the diffusion coefficient in PS and almost 1 order of magnitude higher than that observed in SBS. Tables 1 and 2 present all the solubility and diffisivity data required in the model. Furthermore, a value of L'IL = 0.4 is estimated for the SBS triblock copolymer based on the composition of Kraton D-1101. Unfortunately, an unknown parameter in the model is tortuosity, z. In order to evaluate the model in a correlative sense, the data for the toluene studies presented in Tables 1-3 were used to determine the tortuosity of SBS at different temperatures using the lamellar copolymer model. These correlative values of tortuosity were then used to predict the effective diffusivities for the other solvents. Due to the specific characteristics of solvent-SBS systems in the temperature range 80-120 "C (Bs > 7), the previous analysis indicates that the approximate model (eq 25) will be a very good approximation of the complete lamellar copolymer model after a short time. Therefore, the approximate equilibrium model was used to determine the value of the tortuosity required to correlate the SBS-toluene data as a function of temperature. The results of this correlation are shown in Table 4. These correlations indicate that the tortuosity of SBS decreases with an increase in temperature, from 3.23 to 2.17 for a temperature range of 80-120 "C. Sakurai et al. (1993) have shown that annealing time and temperature are responsible for the persistence length of the undulating lamellae. In addition, by measuring the grain size of an ordered block copolymer using the birefringence properties, Balsara et al. (1992) concluded that faster quenches will yield smaller grains. Therefore, a tortuosity which is affected by the grain size of the lamella of the block copolymer is not constant for a specific copolymer but depends upon the temperature and the thermal history of the sample. The variation of the correlating tortuosity with temperature may be due to such changes in grain size. On the other hand, since this tortuosity is a correlating parameter, it is possible that it is incorporating other complications not
Ind. Eng. Chem. Res., Vol. 34, No. 10,1995 3563 T (C)
T (C)
144
111
84
60
144
111
84
60
2.4
2.6
2.8
3.0
2.4
2.6
2.8
3.0
1 0 3 1 ~(K-Y Figure 11. Comparison of the experimental measurements and model predictions of the effective diffision coefficients for the SBSheptane system.
explicitly stated in the model, such as the influence of the interface between the lamellar zones on the sorption process. If the model has any validity, the tortuosity factor should be independent of the specific solvent being studied. Consequently, the semipredictive capabilities of the model can be evaluated by using the tortuosity as a function of temperature from the correlation of the toluene data to predict effective diffision coefficients of heptane and MEK in the SBS copolymer. In this process, the solubility and diffusivity data for polystyrene and polybutadiene in Tables 1 and 2 are used to predict the effective diffusivities of these solvents. Comparisons of the experimental and predictive diffusivities are presented in Figures 11and 12. It is evident from these figures that there is a reasonably good agreement between the experimental values and the predictive values of the diffision coefficients for these two solvents in SBS. A slight deviation is noticed between the experimental data and the model values at higher temperatures for the SBS-heptane system and a t lower temperatures for the SBS-MEK system. This could be due to changes in the tortuosity of the system as it is annealed in the gas chromatograph apparatus. However, it is estimated that these deviations are within the experimental error of the CCIGC technique. Although these data show that the model predictions are reasonable, the significance of this comparison is questionable because of the use of the correlating parameter, r. Fortunately, Rein and co-workers (1990, 1992) have conducted unsteady-state sorption experiments on a block copolymer in which it is reasonable to assume that r = 1 (Csernica et al., 1987, 1989). The styrenebutadiene (SB) copolymer used by Rein and associates consisted of 75% volume fraction polystyrene and had a weight average molecular weight of 187 000. The solubility and diffusivity data of Rein and coworkers for carbon dioxide (COz), argon (Ar),and methane (CHd in BR and PS are shown in Table 5. As is indicated in this table, D I D for all three gases over
1 0 3 / ~ ( ~ - 1 )
Figure 12. Comparison of the experimental measurements and model predictions of the effective diffusion coefficients for the SBS-methyl ethyl ketone system. Table 5. Solubility and Diffusion Data for Three Gases in Polybutadiene and Polystyrene Homopolymers over the TemperatureRange 20-90 "Cfrom the Sorption Experiments of Rein et al. (1990,1992) gas T ("C) 10-6D (cm2/s) 10-70' (cm2/s) K c02 90 15.5 4.37 0.80 70 10.4 2.71 0.96 50 7.50 1.31 1.28 30 4.34 0.71 1.60 20 3.17 0.48 2.15 14.1 3.99 0.67 Ar 90 10.3 70 2.25 0.86 1.33 50 6.87 1.13 0.68 1.36 25 3.93 CH4 90 12.3 1.63 0.71 70 9.77 0.70 0.86 50 5.99 0.31 1.18 25 2.98 0.13 1.8
the temperature range of 20-90 "C, is equal to or less than 2 orders of magnitude, and L'IL = 3 for this copolymer. For these specific conditions (i.e., Bs > lOOO), the lamellar block copolymer model approaches the approximate equilibrium model at very early times in the sorption process. Consequently, the simple equilibrium model was used to predict the effective diffusivities in the systems studied by Rein and coworkers (1990,1992). These predictions which utilized no adjustable parameters are compared to the Rein and co-workers measurements in Figures 13-15. These figures also include results of the model developed by Rein and co-workers (1990,1992). Figure 13 indicates that the approximate equilibrium model does an excellent job of predicting diffusion of carbon dioxide in the SB copolymer and is significantly better than the predictions of the previous model. Rein and co-workers (1990) have argued that the anomalous temperature dependence and the inability of their computer simulation to reproduce experimental diffusion coefficients is supported by the concept of a temperature dependent restriction on chain mobility in the polybutadiene regions of the block copolymer. Therefore, they introduced a /3 factor which embodies the relative reduction
3664 Ind. Eng. Chem. Res., Vol. 34,No. 10,1995 -1 1
-1 1
-12
-12
-Eo
\
5
-13
f
\
-13
v
f
5
n
/
cn
0
-c
n -14
5
-15
-15
-16 2.0
\\
-14
2.4
2.8
3.2
1 0 3 1 ~(
3.6
-16 2.0
4.0
2.4
3.2
1 0 3 / ~(
~ - 1 )
Figure 13. Comparison of the experimental measurementa and model predictions of the effective diffusion coefficients for the styrene-butadiene copolymer-carbon dioxide system.
3.6
4.0
~ - 1 )
Figure 15. Comparison of the experimental measurements and model predictions of the effective diffusion coefficients for the styrene-butadiene copolymer-methane system. T (C)
-1 1 180
-12
144
111
84
60
2.8
3.0
Experimental
- Prediction
h
cn 2
6
2.8
-I3
Y
101
P
n c
P
-14
Y
1
-15
\
-16 2.0
2.4
2.8
3.2
3.6
4.0
1 0 3 1 ~( ~ - 1 ) Figure 14. Comparison of the experimental measurements and model predictions of the effective diffision coefficients for the styrene-butadiene copolymer-argon system.
in the rate of gas diffusion through the BR regions of the copolymer compared to BR homopolymer. This ,8 factor has a value of 2-3 at room temperature but decreases to unity when the polystyrene chains become mobile at the polystyrene glass transition temperature. The simple equilibrium model not only does an excellent job of predicting the SB-CO2 diffisivities without using any immobilization factor but also predicts the activation energy for diffusion of C02 in SB which is higher than the activation energies of this gas in either of the homopolymers, as experimentally observed. Figures 14 and 15 indicate that the approximate equilibrium model predictions are reasonable for the diffision of Ar and CHI but are not as good as the predictions for the SBCO2 system. However, in all cases, the approximate equilibrium model does a better job of predicting diffusion in these SB-solvent systems than the model developed by Rein and co-workers (1990, 1992).
2.2
2.4
2.6 i 0 3 / ~( ~ - 7 )
Figure 18. Comparison of the experimental measurements and predictions of the effective partition coefficients for the SBStoluene system.
Solubility Data. The partition coefficients of solvents in BR, PS, and SBS are shown in Tables 1-3. It is of interest to compare the observed block copolymer solubility data to that predicted for measurements with the homopolymer. The partition coefficient is an equilibrium property, and for a heterogeneous system can be approximated as a linear combination of the homopolymer values (Rein et al., 1990):
K,, = K&R
+ K4ps
(26)
where ~ B and R q5ps are the volume fractions of BR and PS phases in the copolymer, respectively, and Keff is the effective partition coefficient in the SBS triblock copolymer. Figures 16-18 show comparisons between the experimental values of the partition coefficients of
Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 3565 T (C) If
144
)
111
84
60
Experimental
-
102
Prediction
i Y
IO'
Conclusions
2.2
2.4
2.6
2.8
3.0
1 0 3 1 ~(K-3) Figure 17. Comparison of the experimental measurements and predictions of the effective partition coefficients for the SBSmethyl ethyl ketone system. T (C) If
solubilities in the polystyrene and polybutadiene phases of the block copolymer. The glass transition temperature of the PS domain of the block copolymer was determined by DSC at a heating rate of 10 "C/min. The polystyrene phase of the SBS had a Tgof 55-60 "C. In contrast, the glass transition temperature of the polystyrene homopolymer is approximately 100 "C.Toi and Paul (1982) have determined that solubility of C02 decreases as the Tgof a polystyrene sample decreases. If this is true for the solubility of other solvents in polystyrene, then the R used in eq 26 should be adjusted to account for this dependency of solubility on the glass transition temperature. This adjustment would reduce the values of the predicted partition coefficients in SBS for all three polymer-solvent systems.
I
144
111
a4
60
2.8
3.0
Experimental io2
f Y
IO'
2.2
2.4
2.6
1 0 3 / ~( ~ - 1 ) Figure 18. Comparison of the experimental measurements and predictions of the effective partition coefficients for the SBSheptane system.
toluene, MEK, and heptane in the SBS block copolymer with the values predicted from eq 26. Figures 16 and 17 indicate that the predicted partition coefficients for SBS-toluene and SBS-MEK systems are lower than the experimental measurements. It could be argued that this is due to a high solubility of the solvents in the interfacial zone between the BR and PS phase, as indicated by Odani et al. (1980) and Caneba et al. (1983-84). However, the predicted values for the SBSheptane system (as shown in Figure 18)are higher than the experimental values a t temperatures above 100 "C. The prediction of solubilities in block copolymers based on eq 26 relies on the assumption that partition coefficients of the homopolymers adequately describe the
An unsteady-state model has been developed to describe the sorption of small molecules in block copolymers having an undulating lamellar morphology. It was shown that under certain conditions this lamellar block copolymer model can be approximated by a very simple model which incorporates a local equilibrium approximation. This approximate equilibrium model indicates that the effective diffusion coefficient for a solvent diffusing in a block copolymer with lamellar morphology is related to the characteristics of the system by Der = D/z2(1 yw). The tortuosity, z, is the only correlative parameter in this model that cannot be determined from the composition of the block copolymer and diffusivity and solubility measurements on the homogeneous polymer constituents. Diffisivity and solubility for three solvents in polybutadiene, polystyrene, and an SBS triblock copolymer obtained by the capillary column inverse gas chromatography technique were used to evaluate the validity of this model. Tortuosity values obtained from the correlation of the diffision data for the SBS-toluene system with the copolymer model show that the tortuosity decreases as the temperature of the system increases. Furthermore, the diffusion coefficient predictions for the other SBS-solvent systems using the tortuosity obtained form the toluene experiments were in good agreement with the experimental measurements. This indicates the capability of the model for describing the diffusion behavior of a block copolymer with lamellar morphology. The gas chromatographic experiment is a complex process consisting of many sequential sorption and desorption steps as the solvent vapor flows through the column. Consequently, the unsteady-state diffusioncoefficientfrom this experiment could be different from the coefficient which describes a simple step change sorption process which is modeled. However, it appears from the consistency of the model and the chromatographic data that this model captures the essence of the diffusion process in a lamellar block copolymer. Stronger evidence concerning the validity of the developed model was obtained by utilizing the solubility and diffusivity data of gases in BR, PS, and a styrenebutadiene block copolymer obtained by Rein and coworkers (1990, 1992). It was shown that the model (which reduces t o a simplified equilibrium model for the specific parameters of these copolymer-gas systems) does a good job in predicting the effective diffusion coefficients without the utilization of any adjustable parameter.
+
3566 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995
The final conclusion is that a very simple model can do a reasonable job of predicting the dynamics of unsteady-state sorption into block copolymers with lamellar structures on the basis of solubility and diffusivity behavior in the homopolymer component. The model does incorporate a correlative tortuosity parameter. It is also possible that this tortuosity parameter incorporates effects other than tortuosity, such as the influence of the interface regions on the sorption process. However, the predictions of the data of Rein and coworkers (1990, 1992) where r = 1 indicates that this correlating parameter is probably dominated by the tortuosity effects. It is possible that from information concerning the domain size arrangements in the block copolymer structure that the tortuosity could be predicted from random walk models (Wasow, 1951). The experimental values of the partition coefficients of solvents in SBS were found to be higher than the predicted values obtained from the homopolymer's solubility data for most cases. This trend may be evidence of the influence of the interface between the lamellae in the block copolymer structure. It is interesting that the success of the lamellar copolymer diffusion model indicates that the influence of this interface on a diffusion process is negligible.
Acknowledgment The authors would like to thank Dr. R. Spontak of North Carolina State University for his help with the transmission electron microscopy of the Kraton D-1101 triblock copolymer. The authors would also like to gratefully acknowledge Dr. R. P. Danner, Dr. R. Nagarajan, and Dr. A. Borhan for their assistance and helpful discussions throughout the course of this work.
Nomenclature A1 = dimensionless number defined as (L/rz')2 A2 = dimensionless number defined as KL'IL A3 = dimensionless number defined as DL2/DL'2 B1 = dimensionless number defined as llr2 B2 = dimensionless number defined as KL'IL B3 = dimensionless number defined as Dz'2/DL'2 C = solvent concentrationin the fast diffusingphase, g/cm3 C, = solvent equilibrium concentration, ghm3 C" = solvent concentration in the slow diffusing phase, g/cm3 D = diffusion coefficient in the fast diffusing phase, cm2/s D = diffusion coefficientin the slow diffusing phase, cm21s Den= effective diffusion coefficientin the block copolymer, cm2/s K = partition coefficient of solvent in the fast diffusing phase IC = partition coefficient of solvent in the slow diffusing phase Keff = effective partition coefficient of solvent in SBS copolymer L = half-thickness of the fast diffising phase, cm L' = half-thickness of the slow diffusing phase, cm Mt = total weight pickup at time t, g M , = total weight pick up at equilibrium time, g q = dimensionless solvent concentration in the fast diffusing phase q' = dimensionless solvent concentration in the slow diffusing phase t = time, s x = axial coordinate, cm V = volume of the fast diffusing phase, cm3 V' = volume of the slow diffusing phase, cm3
z = axial coordinate, cm z' = length of the copolymer sample, crn
Greek Letters ,6 = immobilizationfactor y = ratio of the volumes of the two phases ( V W 5 = dimensionless thickness in the z direction 8 = dimensionless time (Dt/d2) 8' = dimensionless time (Dt/L2) K = ratio of the partition Coefficient of solvent in the two
phases (K'IK)
b~= volume fraction of the BR phase h p s = volume fraction of the PS phase x = dimensionless thickness in the x direction r = tortuosity factor = parameter described by eq 12
Literature Cited Arnould, D.; Laurence, R. L. Size Effects on Solvent Diffusion in Polymers, Ind. Eng. Chem. Res. 1992,31,218. Balsara, N. P.; Garetz, B. A.; Dai, J. H., Relationship Between Birefringence and the Structure of Ordered Block Copolymer Materials. Macromolecules 1992,25,6072. Barrer, R. M.; Petropoulos, J. H., Diffusion in Heterogeneous Media: Lattices of Parallelepipeds in Continuous Phase. J. Appl. Phys. 1961,12,691. Bates, F. S. Polymer-Polymer Phase Behavior. Science, 1991,251,
898. Caneba, G. T.; Soong, D. S.;Prausnitz, J. M. Diffusion and Sorption of Cyclohexane Vapor in SBS and SBR Copolymers. Macromolecules 19f33-84,22,693. Carnahan, B.; Luther, H. A.; Wilkes, J. 0. Applied Numerical Methods; Wiley: New York, 1969. Chanier, J. M. Basic Aspects of Structure-Property Relationships for Composites. Polym. Eng. Sci. 1975,15,731. Crank, J. The Mathematics of Diffusion, 2nd ed.; Clarendon: Oxford, 1975. Csernica, J.; Baddour, R. F.; Cohen, R. E. Gas Permeability of a Polystyrene-Polybutadiene Block Copolymer with Oriented Lamellar Domains. Macromolecules 1987,20,2468. Csernica, J.;Baddour, R. F.; Cohen, R. E. Gas Permeability of a PolystyreneiPolybutadieneBlock Copolymer Possessing a Misoriented Lamellar Morphology.Macromolecules 1989,22,1493. Davis, H. T.,The Effective Medium Theory of Diffision in Composite Media. J. Am. Cer. Soc. 1977,60, 499. Enscore, D. J.;Hopfenberg, H. B.; Stannett, V. T. Effect of Particle Size on the Mechanism Controlling n-Hexane Sorption in Glassy Polystyrene Microspheres. Polymer 1977,18, 793. Faridi, N. PhD Thesis, The Pennsylvania State University, University Park, PA, 1995 (to be published). Faridi, N.; Hadj Romdhane, I.; Danner, R. P.; Duda, J. L. Diffusion and Sorption in Ethylene-Propylene Copolymers: Comparison of Experimental Methods. Ind. Erg. Chem. Res. 1994,33,2483. Hadj Romdhane, I. Polymer-Solvent Diffusion and Equilibrium Parameters by Inverse Gas-Liquid Chromatography Ph.D. Thesis, The Pennsylvania State University, University Park, PA, 1994. Landauer, R. The electrical resistance of a binary mixture. J . Appl. Phys. 1952,23,779. Leary, D. F.; Williams, M. C. Statistical Thermodynamics of ABA Block Copolymers. 11. J.Poly. Sci., Polym. Phys. 1973,11,345. Leary, D. F.; Williams, M. C. Statistical Thermodynamics of ABA Block Copolymers. 111. Microstructural Transitions and Model Verification. J. Polym. Sci., Polym. Phys. 1974,12,265. Odani, H.; Uchikura, M; Taira, K.; Kurata, M. Transport and Solution of Gases and Vapors in Styrene-Butadiene Block Copolymers: Absorption and Deso'rption of n-Hexane Vapor. J. Macromol. Sci., Phys. 1980,B17,337. Ottino, J. M.; Shah, N., Analysis of Transient Sorption and Permeation of Small Molecules in Multicomponent Polymer Systems. Polym. Eng. Sci. 1984,24, 153. Pawlisch, C. A.; Marcris, A.; Laurence, R. L. Solute Diffusion in Polymers. 1. The Use of Capillary Column Inverse Gas Chromatography. Macromolecules 1987,20,1564.
Ind. Eng. Chem. Res., Vol. 34,No.10,1995 3567 Pawlisch, C. A.; Bric, J. R.; Laurence, R. L. Solute Diffusion in Polymers. 2.Fourier Estimation of Capillary Column Inverse Gas Chromatography. Macromolecules 1988,21,1685. Rein, D. H.;Csernica, J.; Baddour, R. F.; Cohen, R. E.; COz. Diffusion and Solubility in Polystyrene-PolybutadieneBlock Copolymer with a Highly Oriented Lamellar Morphology. Macromolecules 1990,23,4456. Rein, D. H.; Baddour, R. F.; Cohen, R. E. Gas Solubility and Diffusion in Polystyrene-Polybutadiene Block Copolymer. J. Appl. Polym. Sci. 1992,45,1223. Sakurai, S.;Momii, T.; Taie, K.; Shibayama, M.; Nomura, S. Morphology Transition from Cylindrical to Lamellar Microdomains of Block Copolymers. Macromolecules 1993,26,485. Sax,J. E.; Ottino, J. M. Modelling of Transport of Small Molecules in Polymer Blends: Application of Effective Medium Theory. Polym. Eng. Sci. 1983,23,165. Shah, N.;Sax,J. E.; Ottino, J. M. Influence of Morphology on the Transport Properties of PolystyrenePolybutadiene Blends: 2. Modelling Results. Polymer 1985,26,1239. Smith, D. M.; Keller, J. F. Nonlinear Sorption Effects on the Determination of DiffusiodSorption Parameters. Znd. Eng. Chem. Fundam. 1985,24,497.
Toi, K.; Paul, D. R. Effect of Polystyrene Molecular Weight on the Carbon Dioxide Sorption Isotherm. Macromolecules 1982,15,
1104. Vrentas, J. S.; Duda, J. L. J. Polym. Sei., Part B: Polym. Phys.
1977a,15,403. Vrentas, J. S.;Duda, J. L. J. Polym. Sci: Part B: Polym. Phys.
1977b,15,417. Wasow, W. On the Mean Duration of Random Walks J. Res. Natl. Bur. Stand. 1951,46,462. Weisz, P. B. Sorption-diffusion in Heterogeneous Systems: Part 1. General Sorption Behavior and Criteria. Trans. Faraday. Soc.
1967,63,1801. Received for review January 20, 1995 Revised manuscript received June 12, 1995 Accepted July 17, 1995@ IE9500650
Abstract published in Advance ACS Abstracts, September 1, 1995. @
