mutual diffusion and thermodynamics in the blends of polystyrene and tetramethylbisphenol-a...

Mutual Diffusion and Thermodynamics in the Blends of Polystyrene and Tetramethylbisphenof-A Pofycarbonate

EUGENE KIM,'* EDWARD J. KRAMER,'+ JOHN 0. OSBY? and DAVID J. W A S H 4

'Department of Chemistry, Baker Laboratory, and 'Department of Materials Science and Engineering and the Materials Science Center, Cornell University, Ithaca, New York 14853; 3 D ~ ~ Chemical USA, BIdg. B-I 471, Freeport, Texas 77541; 4Du Pont Company, Sabine River Works, Building 10, Orange, Texas 77631-1089

SYNOPSIS

A study was made of miscible polymer blends of deuterated polystyrene (d-PS) and tetramethylbisphenol-A polycarbonate (TMPC). The Flory interaction parameter X was obtained from the relation between mutual and tracer diffusion coefficients, ~ and D", which were measured by forward recoil spectrometry. The temperature dependence of diffusion at PS weight fractions w of 0.25 and 0.5, and the composition dependence at temperatures 45°C above the glass transition temperature, Tg, were investigated. A stronger dependence of x on both temperature (at w = 0.5) and composition was observed in comparison with other miscible binary polymer blends involving PS. Analysis using the generalized lattice-fluid model of Sanchez and Balazs' showed that the incorporation of a significant specific interaction is needed to explain the temperature dependence of X. The diffusion coefficients obtained in the one-phase region were extrapolated to the two-phase region, and these were compared with the effective diffusion coefficient extracted from phase separation dynamics measured by light scattering.' A significant discrepancy between the extrapolated and effective diffusion coefficients was observed. 0 1995 John Wiley & Sons, Inc. Keywords: mutual diffusion coefficient specific interaction compressibility effect generalized lattice-fluid model forward recoil spectrometry

INTRODUCTION

Since Shaw first reported3 on the miscibility between PS and TMPC and on their lower critical solution temperature ( LCST ) behavior, the thermodynamic nature of this blend has been studied in various ways: using small-angle neutron scattering (SANS) as a function of composition4 and also as a function of temperature; using glass transition measurements; 'v7 using light scattering techniques; and using pressure-volume-temperature ( PVT ) measurements? Diffusion measurements would also provide valuable information, not only about the in- trinsic mobility of each chain that can be obtained from the tracer diffusion measurements, 9~10 but also

* To whom correspondence should be addressed. ' Present address: Department of Chemical Engineering, Princeton University, Princeton, NJ 08544. Journal of Polymer Science: Part B: Polymer Physics, Vol. 33,467-478 (1995) 0 1995 John Wiley & Sons, Inc. CCC 0887-6266/95/030467-12

about the Flory interaction parameter X, which can be obtained from the relation between the mutual and tracer diffusion coefficients." We previously studied the tracer diffusion lo of this blend system. Here we report on the thermodynamic aspects of the diffusion studies and how the interaction parameter x is modified by changing temperature and composition. We also used the information obtained in the one-phase region to calculate the phase diagram as well as the diffusion coefficients in the two- phase region that control the phase separation process.

Thermodynamics of Mixing

The diffusion and scattering experiments probe the second derivative, with respect to composition, of the free energy of mixing that is equivalent to the inverse of the equilibrium structure factor S, at the thermodynamic limit (at the wave vector q = 0):

467

468 KIM ET AL.

where AFmix is the free energy of mixing for a mole of lattice sites. The interaction parameter X is de- noted as x,, where sc represents scattering. The interaction parameter a t spinodal point X , is

x =-(-+-) 1 1 1 2 4lNl 4zNz

where Ni is the degree of polymerization, and 4i is the volume fraction of component i.

The Flory-Huggins theory of polymer blends is strictly based on a regular solution model with the same lattice cell size for all monomers and, in the context of the theory, the interaction parameter is independent of composition. However, the X,, has been observed by neutron scattering experiments to be, in general, a function of composition in various miscible polymer blend^.'^-'^ Recently, microscopic theories that take into account features beyond the Flory-Huggins approximation have been developed by Schweizer and Curro,2'-'' Freed et al.,23*24 Bates et al.,25 and Liu and Fredrickson?6

Assuming that the heat capacity of mixing AC, is independent of temperature, the temperature dependence of x a t given composition may be approximated asz7

B B x ( T ) = A + - + C l n T = A + -

T T (3)

where A , B, and C are constants: the noncombinatorial entropic contribution is embedded in A and C, and enthalpic contribution is contained in B. The contribution of the term containing C is not very sensitive to temperature and it is usually ignored.

Diffusion in Miscible Polymer Blends

Tracer diffusion, which is a measure of the mobility of a single-probe molecule, is driven purely by the combinatorial entropy of mixing. The excess free energy of mixing (exothermic or endothermic en- thalpy of mixing plus noncombinatorial entropy of mixing) is not directly linked to it. When the mutual diffusion coefficient fi is measured and x is nonzero (and therefore the nonideal free energy of mixing becomes an additional driving force), the diffusion is thermodynamically enhanced or The mutual diffusion coefficient 6 is given by a product

of the thermodynamic term 2 ( X , - Xsc) and the in- trinsic mobility term (q51~zDn the Onsager transport coefficient),

The transport coefficient DT is predicted to be a function of the tracer diffusion coefficients of the two We use the results of the so called "fast t h e ~ r y , ~ ~ ~ ~ ~ ~ ' which has been verified by most of the carefully conducted experiment^,^'-^^ viz:

Thus, D can be used to extract the interaction parameter x,, from the measured tracer diffusion coefficients and polymerization indices of the two species.

The Generalized Lattice-Fluid Model by Sanchez and Balazs'

Two entropic origins have been proposed to explain the LCST behavior (i.e., phase destabilization by heating) often found in binary polymer blends. One is the compressibility e f f e ~ t ~ ~ - ~ ' that was originally formulated under the assumption of random mixing, and the other is the specific interaction4' that, to the contrary, assumed the blend was incompressible. In the former, the volume change upon mixing is the origin of the change in miscibility, and in the latter, the spatial rearrangement (e.g., orientational alignment) of unlike chains that is associated with the attractive interaction is responsible. The specific interaction contribution that accompanies the entropy loss becomes smaller a t higher temperatures due to thermal agitation, and eventually phase separation occurs. Sanchez and Balazs' combined the idea of specific interaction in an incompressible binary blend previously proposed by ten Brinke and Karasz*' with the original compressibility theory by Sanchez and Lacombe (Lattice-Fluid m~del )?~ .~ ' They showed that both the phase diagram and X , (as the measured by SANS) in PS and poly- (vinylmethylether) (PVME) blends could be explained successfully-which was impossible if one used only one of the theories. We briefly introduce their model, which is utilized extensively to analyze our data.

They derived the equation of state as follows:

DIFFUSION/THERMODYNAMICS IN PS/TMPC BLENDS 469

where F is the reduced density of the mixture. The parameter representing the size of the chain, r, was obtained from the following relation:

where ri is obtained from

(7)

In the above relation, T f , P," , and p' are the equation of state parameters and Mi is the molecular weight of species i. The total mixing interaction energy 8 ; was also given as

where el2 is the nonspecific mer-mer interaction between mer 1 and mer 2 and the additional energy attributed to the specific interaction fspecific was obtained

* (10) l + q

fspeciAc = BE - kT In [ 1 + 4 exp(g,l The specific interaction energy, BE, is the energy difference between specific (strong) interaction and nonspecific (weak) interaction. The interaction energies in the Sanchez/Balazs model are given schematically in Figure 1. Larger positive numbers of e12 and BE mean more attractive interactions and, as the contribution of the specific interaction in- creases, 6c becomes larger. The value q is the number of ways (degeneracy) that the nonspecific interaction can occur and there is no degeneracy assigned to the specific interaction. Their expressions for AFmix and x,, are

Specific interaction -7 (more fovored orientation) T E

Non-specific interaction (q degenerate orientotions)

I

Figure 1. Schematic illustration of the nonspecific and specific interaction energies in the model of Sanchez and Balazs.'

- TS,,, - FE; (11)

d2( m m i x I Scomb)

1 kT k

where (-Scomb) represents the term in the bracket on the right-hand side of eq. (11). The theory is described in more detail in the original paper by San- chez and Balazs.' The characteristic parameters (P*, p*, and T*) for pure components that are obtained from the PVT relation enter into the last two terms in eq. (11). For given q, c12 and BE, is computed from eq. (6) , and then x,, is calculated from eq. (12). There are virtually no free parameters* with the exception of q, because e12 and BE can be obtained by fitting the temperature dependence of Xsc. How- ever, as noted by Sanchez and Balazs in the theory, adding specific interactions does not influence the composition dependence of X,, which could only be generated by the compressibility effect. This may not be strictly true because x, is expected to be af- fected by the extent of formation of the specific interaction, which would be composition dependent. The details concerning the composition dependence of X,, will be discussed below.

EXPERlMENJAL SECTlON

The weight-average molecular weight and polydispersity index of each of the polymers used for the

measurements and PVT measurements are listed in Table I. The PVT relation was measured for blends of hydrogenated PS and TMPC at PS weight fractions w = 0, 4 , , and 1 as follows. First, densities were measured at 25OC and atmospheric pressure using an autopycrometer ( Micrometritics ) . The changes in density as a function of temperature (up to ca. 280°C by 8-10°C increment) and pressure (up to 200 MPa by 1 MPa increment) were measured using a PVT apparatus. The absolute accuracy of the device is - 2 X cm3/g; however, volume changes as small as - 2 X cm3/g can be resolved. The details of the procedure have been fully described el~ewhere.~~ The equation-of-state parameters for each pure component were obtained

* We fixed a lattice coordination number z to be 12 and didn't treat it as another parameter following the scheme of Sanchez and Balazs,' which is in principle related to q.

470 KIM ET AL.

Table I. Weight-Average Molecular Weight and Polydispersity Indices of Deuterated Polystyrene and Protonated Polystyrene and Tetramethylbisphenol A-Polycarbonate

MW Mw/Mn

d-PS" 489 000 PSb 253 000 TMPC".~ 42 200

1.1

2.88 - 2

a Used for mutual diffusion measurements. Used for PVT relation measurements.

by fitting the PVT data for w = 0 and w = 1 to eq. ( 6) using a nonlinear least-square-fitting procedure.* Table I1 lists the parameters obtained from the data for the melt in the pressure region of 0-10 MPa. We assumed that the parameters were the same when PS was replaced by the d-PS except that p* was corrected for d-PS by multiplying 112/104 to account for the deuteration. The equation-of-state parameters were in reasonable agreement with the values obtained by Kim and Paul8 except for the deuteration correction.

To measure the mutual diffusion coefficients, a bilayer with each layer of thickness of 300-400 nm was prepared. The base film was prepared by spin- casting the polymer dissolved in anisole (methox- ybenzene) onto a silicon wafer. The top film was floated onto deionized water and then picked up by the base layer. The diffusion profile was obtained by forward recoil spectrometry (FRES) .*' Because of the strong composition dependence of B, the difference in concentration between the two layers had to be made small (10% 1. In this way a single diffusion coefficient at the average composition of the two layers could be obtained. These procedures for mutual diffusion measurements in polymer blends have been explained in more detail elsewhere." The measurements of the tracer diffusion coefficients D* in the same blend are described in our previous re- ports.l0 The D*s of both species diffusing into ma-

* The PVT properties of the blends at various compositions ( w = 0, 4, 3, and 1 ) were also studied using the equation-of-state theory of Flory, Orwoll, and Vrij and that of the cell model in the same way as Ougizawa et al. did in their recent article on thermodynamic studies in PS and poly(viny1 methyl ether) (PVME) blends. [ T. Ougizawa, G. T. Dee, and D. J. Walsh, Muc- romolecules, 24,3834 (1991) .] Those values obtained by the fitting procedure showed the same trends that were found in PS/ PVME using both theories; P* being higher than average and T* lower than average for the blends, leading to the disagreement that the interaction parameter predicted from P* is negative and that from T* is positive value. Ougizawa et al. suggested that this may be due to a large temperature dependence of interaction, which is what we found in these studies.

Table 11. Equation-of-State Parameters

Temperature T* P* P* Range ("C) (K) (atm) (g/cm3)

d-PS 160-270 737 3870 1.18 TMPC 190-275 771 3950 1.18

~~ ~ ~

p* for d-PS was multiplied by 112/104 to account for deuteration.

trices of sufficiently high molecular weight ( includ- ing the molecular weight range studied here) were shown to be independent of the molecular weight of the matrix and to decrease as MF2 , as expected from reptation, where Mi is the molecular weight of each tracer chain. Diffusion coefficients, both D and D*, were thus obtained as a function of temperature at w = 0.5 and 0.25, and also as a function of composition at 45°C above TY (Tg = 197.6 - 124.2 w + 36.97 w2 + 5.5132 w 3 "C, Fig. 2).

The binodal corn position^^^-^^ at different temperatures above the LCST (241-271OC) were measured by FRES. Bilayers of about 200-300 nm pure d-PS and pure TMPC were prepared and annealed under vacuum to allow the molecular transport to take place. The d-PS species has the lower surface energy46 and it was chosen for the top film. These are then annealed in the two-phase region (so that molecular transport can take place across the in- terface) until the composition of each layer reaches the corresponding equilibrium coexisting ( binodal )

I 0.0 0.2 0.4 0.6 0.8 I 0

Weight fraction of PS. w

Figure 2. Glass transition temperature, TY (0), and the experimental temperature of diffusion measurement as a function of weight fraction w of PS (broken line). The solid lines represent the region at which the temperature dependence experiments were carried out at w = 0.5 and 0.25. The dotted-broken line is the polynomial fit to the Tg.


value. The typical profiles before and after annealing are given in Figure 3. The profile was simulated with two step functions that correspond to the coexisting concentrations that were convolved with a Gaussian with a full width at half-maximum of 80 nm to account for the limited depth resolution of the technique. The composition of each layer was monitored as a function of time. The layers were taken to be equilibrated when there was no further change in the profiles (Fig. 4).

RESULTS AND DISCUSSION

Temperature and Composition Dependence of x Figure 5 shows the fi and D* from ref. 10 in the blend of w = 0.5 as a function of temperature using Vogel-Fulcher plot47 with T, = 83°C. At T < 24OoC, fi is larger ( X < 0 ) than either D* because of the

I .o

0.8 c 0 L C 0.6

5 0.4

LL

Ql

- 3 0.2

0.0

-I00 0 100 200 300 400 500 600 Depth (nm)

I .o

0.8 c .- .- 8 0.6 I=

5 0.4 (Y

- 3 0.2

- - 0.0 - ,, , , , ~ , I

-100 0 100 200 300 400 500 600 Depth (nm)

Figure 3. Interdiffusion profile as a function of depth for binodal composition measurement in d-PS/TMPC blend. The bilayer composed of top d-PS layer with the thickness of 200 nm and the bottom TMPC layer with the thickness of 280 nm was annealed at 251°C for 10 min. The d-PS volume fraction (deuterium peak) (0) and the TMPC volume fraction (hydrogen peak) (0) , (a) before and ( b f after annealing, are shown. The initial hydrogen content (ca. 3% ) in d-PS layer was corrected for in determining the TMPC concentration.

t o'6 t $ 0 - 4 0 2 I

--------------- o--- 0 t --o---Q ------ 0 -----

0.0 0 I

0 10 20 3 0 40 50 60 Annealing lime ( m i d

Figure 4. The change in composition of each layer was monitored as a function of time at 241 (0 and 0) and 251°C (m and 0). The compositions at the binodal were determined to be 0.11 and 0.68 at 241°C and 0.03 and 0.81 at 251°C in this way.

segment interaction enhancement. The D shows maximum at 240°C and then goes down at higher T due to the increasingly positive X,, as X,, ap- proaches its value of the spinodal X,. We calculated x,, as a function of temperature a t w = 0.5 and also at 0.25 from eqs. ( 4 ) and ( 5 ) . Because the monomer sizes of the constituent components are different in PS and TMPC, the degree of polymerization of

?knperoture ("C) 240 220 210 200 190 180

1619 1 ,j+ , , , , 1 6.0 7.0 8.0 9.0 10.0

I/(T-T~) ( x 10-3)

Figure 5. Temperature dependence ( Vogel-Fulcher plot) of the tracer diffusion coefficients" D* of d-PS (0) and d-TMPC (0) scaled to weight-average molecular weights of 489,000 and 42,000, respectively, and that of the mutual diffusion coefficients D of d-PS/TMPC blends of the same molecular weights ( 0 ) . In the tracer experiments the matrix film of w = 0.5 was used and in the mutual diffusion experiment the average weight fraction of d-PS of two layers was 0.5 with the w of them before diffusion being 0.45 and 0.55. The value for T, was 83OC.

472 KIM ET AL.

component i, Ni , was defined as the molar volume of polymer i divided by the hypothetical common monomer volume, uo , which is taken as the geometric mean of the volume of the structural monomer unit of d-PS and TMPC, (ref. 4). The thermal expansion of uo was also taken into account.

In Figure 6 X,, so determined is plotted as a function of 1 /T for the two d-PS/TMPC blends, w = 0.5 and w = 0.25, together with those for other miscible polymer blends of d-PS: with poly (xylenylether ) (PXE) , l1 with poly (vinylmethylether) (PVME) , l5

and with PS.48 The corresponding values of A / u o and B/uo are tabulated in Table 111. To eliminate any factors that come from the different lattice cell size, the interaction parameter per unit volume (not per lattice site) , xsc/u0, was used where all uos were calculated from the geometric mean of the corresponding monomer volumes. The temperature dependence that represents the enthalpic contribution was found to be largest for d-PS/TMPC blend with w = 0.5. The significantly smaller temperature dependence was observed at w = 0.25. It is interesting to note that in all cases the ratios of B to A are pretty close, as shown in Table 111. We also realize that the magnitude of the positive coefficient B for the isotopic (d-PS/PS) blend is smaller by more than two orders of magnitude than that for the other blends composed of chemically dissimilar components. It might be expected that only a negligible change in the interaction due to the substitution of hydrogen by deuterium is caused in the chemically

I Y I 1 .. .. I

0 * ̂0 x

- I - - ” E - 9

X ‘%-2

-3

1.9 2.0 2.1 2.2 2.3 2.4

I /T ( ~ - 1 ) ( X 10-3)

Figure 6. The interaction parameter per volume X,/ uo as a function of 1/ T for various miscible polymer blends. The common component of these blends is d-PS, so that [O, solid line] corresponds to w = 0.25 with TMPC, [ 0, solid line] w = 0.50 with TMPC, [broken line] w = 0.55 with PXE, [ broken-dotted line] w = 0.50 with PVME,I5 and [ broken-dotted-dotted line] w = 0.45 with PS.48 Refer to Table I11 for details.

dissimilar polymer blends, and the corresponding contribution to the overall X is small. However, Yang et al?’ found an increase of the LCST by about 40°C in PS/PVME blends when PS is deuterated. As will be seen below, no such noticeable change was found in PS/TMPC blends.

Kim and Paul’ also measured the PVT properties of the pure components, PS and TMPC, and the interaction parameter was calculated by fitting the phase boundary to the original lattice fluid model by Sanchez and Lacombe, which did not take the specific interaction contribution into account. They justified the use of this model from the fact that the electrostatic or polar interactions are not thought to be strong, so that the equation-of-state effect rather than specific interactions might explain the destabilization of the mixture with increasing temperature. They calculated the temperature dependence of X, to be very small* compared to our results, which would suggest that the compressibility effect makes only a minor contribution to the temperature dependence of xSc. To account for the strong temperature dependence of x,,, we included specific interactions in our analysis using the model of Sanchez and Balazs. For a given value of q , the temperature dependence of X,/V~ was used to determine e12 and BE by a curve-fitting procedure (Fig. 7). A large contribution of the specific interaction for w = 0.5 was required to fit the data (e.g., 6 c / k = 138 K and q 2 / k = 112 K for q = 10). One may come up with the same spinodal temperature considering only the equation-of-state effect, but a mis- leading value of X,, in the one-phase region results. The shape of the X,, vs 1 / T curve predicted from the Sanchez and Balazs model in Figure 7 is concave downward, resulting from the fact that the contribution of the specific interaction becomes smaller at higher temperatures. If 8~ was equal to zero, the temperature dependence of X,, would indeed be negligible (the dotted line in Fig. 7), especially compared to the very strong temperature dependence of x,, observed experimentally. The slope B in the plot of X, vs 1/ T is proportional to the following expres- sion that contains the specific interaction term fspeeific [ineq. ( lo ) ] ,

* The value of B / u o at w = 0.5 was calculated to be -0.104 from the temperature dependence estimated by Kim and Paul (Fig. 13 in ref. 8), which is 13 times smaller than our value of -1.34 (Table 3 ) .

The magnitude of the size parameter ri was about 10 time larger than Ni and the corresponding discrepancy was corrected by comparing the energetic per unit volume (X,, was divided by uo which was 17 (cm3/mole) and 175 (cm3/mole) using ri and Ni respectively).


Table 111. and Small-Angle Neutron Scattering Measurementsd

Values of Xsc/uo for Various Blends in A/uo + (B/u0)/T Form Obtained Using

A/uo (mol/cm3) B/uo (mol K/cm3) B / A

50% d-PS with TMPC" 25% d-PS with TMPCb 55% d-PS with PXE' 50% d-PS with PVMEd 45% d-PS with PS'

0.00259 0.00063 0.00103 0.0011

-0.0000030

-1.34 -0.31 -0.57 -0.48

0.0021

-517 -492 -553 -436 -700

~~

The values of uo (cm3/mol) used for diffusion measurement data are 175,8'b 109," and 106; respectively. (In SANS experiment: x,/

a Obtained in our experiments. Obtained in our experiments. Excerpted from ref. 11. Excerpted from Table I1 in ref. 15. Excerpted from ref. 48.

uo is directly measured without requiring uO.)l3

B [k(TT + TX) - ~ ~ 2 1 - Zfspeciflc. (13)

It was observed that as 6c becomes close to zero (i.e., fspecific = 0) B continuously decreases, whereas the intercept A was determined mostly by the equation- of-state parameters and remained nearly constant. The small difference in the equation-of-state parameters P*, T*, and p* for the two pure components given in Table I1 implies that the compressibility effect contributes only in a limited way. The values of e I 2 / k and 6 c / k for various values of q in the blends of w = 0.25 and 0.5 are tabulated in Table IV, and are also compared with the values for PS/ PVME blends that were obtained by fitting to the

0 f P

7 - I

P 0

,I - - E -

-2

-3 I .9 2 0 2 1 2.2

I / T (K-') ( X

Figure 7. The temperature dependence of X,, at w = 0.5. The parameters in the generalized lattice-fluid model by Sanchez and Balazs,' eI2 and be, were determined by a curve-fitting procedure using q = 10 (solid line). The values of e I 2 / k and 8 e / k obtained were 112 and 138. Dotted line shows a temperature dependence in the case of 8e = 0, which represents the case where there is no specific interaction contribution, only the compressibility effect.

spinodal data over all compositions.' These quite different values of q (and corresponding values of c I 2 / k and 6 e / k ) gave equivalently good fits to our data. The different numbers of the parameters needed for the different compositions might show the limitations of the model, but a t the same time it tells us that specific interaction is the dominating form of interaction and that the extent of it is a function of composition. Now, we illustrate the details of the composition dependence of x,, over the composition range of 0.15 < w < 0.75 at a constant temperature above Tp .

Figure 8 shows D and D * lo as a function of composition at T - Tg = 45°C. The broken line represents the prediction of D for an athermal blend ( x = 0) whereas the dotted-broken line is D for a composition independent X,, = 0.453 - 235/ T (the value determined at w = 0.5). Note that in TMPC-rich mixtures (w = 0.15) the interaction is very small (the D is close to the value predicted for the athermal blend) and for w > 0.25 a strong thermodynamic enhancement of diffusion is evident. For d-PS-rich blends, D becomes larger than that predicted from X,, ( T , w = 0.5). Figure 8 clearly illustrates that there is an attractive interaction between d-PS and TMPC that depends strongly on composition, being weak for TMPC-rich blends and much stronger for d-PS-rich ones.

For certain isotopic polymer blends16 and polymer-solvent solutions, 50 x can be approximated by a sum of two independent terms, one representing the temperature dependence and one the composition dependence, that is, X ( T , 4) = X ( T ) + X ( 4) . Such a form implies that the enthalpic contribution of the interaction parameter remains the same for different compositions. However, it cannot be the

474 KIM ET AL.

Table IV. Obtained by Fitting the Temperature Dependence of X, , at w = 0.25 and 0.5, and for the PS/PVME Blend by Fitting Spinodal Data Over All Compositions (Unit: K)

Sanchez/Balazs Interaction Energy Parameters Divided by k at Various qs for PS/TMPC Blends Were

d-PS/TMPC PS/PVME

w = 0.25 w = 0.5 All Compositions

C12lk 6c/k c12/k 6c/k ~ 2 / k 6c/k

q = 2 110 46 94 89 q = 6 117 61 108 115 110 44 q = 10 119 71 112 138 111 53

case in our system, * because a different temperature dependence at w = 0.25 and 0.5 was observed. Therefore, the dependence of the specific interaction on the environment (composition) may be one of the possible origins of the strong composition dependence. It is interesting to note that decreasing the amount of PS was observed to increase the ten- dency for crystallization of TMPC (i.e., decreasing the amount of d-PS tends to reduce the formation of the specific interaction).

We then tried to find an empirical formula that could fit the temperature dependence of X,, at w = 0.5 and 0.25 and the composition dependence at T = Tg + 45°C. The free energy of mixing could be constructed using x,, values within the temperature and composition ranges accessible and we tried to determine a function to represent x,, in the T - 4 plane.+ The following empirical functional form was chosen that basically allows A and B in eq. ( 3 ) to be composition dependent:

X - - C . (14) (h 1 The coefficients determined from fitting the data are given in Table V. The x,, were calculated as a

* In addition, this form of X., gives rise to an unrealistic phase diagram that is far from the one observed by experiments because instability region was reached at too low temperature in TMPC- rich compositions.

The ranges were limited due to the accessible diffusion coefficient, ca. 1 X lo-'' - 1 X and due to the uncertainty in X,,, especially in the blends of w > 0.5, which is due to the fact that D is much larger than D* and thus the small error in D is magnified in X, [ eqs. ( 4 ) and ( 5) 1. The temperature dependence of X in this range of composition is purely from the extrapolation of the trend found in small w portion of the eq. ( 14).

function of composition at Tg using eq. (14), which are shown in Figure 9 (solid line) together with those experimentally measured at Tg + 45OC. These were compared with the results obtained from SANS measurement^.^ The values obtained by SANS measurements tend to be higher. This may be attributed to the fact that the concentration fluctua- tions of blends fall out of equilibrium at temperatures higher than Tg.51 The structure factor and X,, of the mixture can be calculated using the diffusion data and the cooling rate during the temperature change from where it was in equilibrium to below Tg. Such calculations are illustrated e1sewhe1-e.~~

The other important polymer that was found to be miscible with PS over the entire range of composition was poly (xylenylether) (PXE) .53 The

Weight fraction of PS, w

Figure 8. The diffusion coefficients in the blends of PS and TMPC as a function of composition at T - Tg = 45°C. The tracer diffusion coefficients D* of d-PS (0 ) and d- TMPC (0)" scaled to the molecular weights of 489,000 and 42,000 and the mutual diffusion coefficients fi ( 0 ) in the d-PS/TMPC diffusion couples of the same molecular weights are drawn and compared with the 6 prediction with x,, = 0 (broken line) and with X, , = 0.453 - 235/T (the composition independent X ) (broken-dotted line).


Table V. Fitted Parameters of Eq. (14)

A0 A* A2 A3 C

-2.424 X 4.996 X lo-' -0.1466 5.783 X lo-' 1.98 x 10-3

Bo B1 B2 B3 B4 B6

0.000 923.8 -7903 28780 -37790 16390

structural similarity of PXE to TMPC has been pointed and the immiscibility between PS and polycarbonate based on bisphenol-A (PC) for which the repeating unit has four less methyl groups on the backbone rings than TMPC suggest the impor- tance of the four methyl groups to the specific interaction. However, it is interesting to note that in the blend of d-PS/PXE X,, at T = Tg + 60°C was found to be nearly independent of composition.'' The relatively large temperature dependence of the x,, in PS/PXE, as shown in Table 111, suggests that the contribution of specific interactions should be important in this blend system as well as in PS/ TMPC.

Phase Diagram

Spinodal (x, - x,, = 0) and binodal diagrams for the d-PS/TMPC blend were computed* from the empirical x,, ( T , 4 ) [ eq. (14) and Table V] and are shown in Figure 10. A detailed description of how the binodal compositions were obtained is given elsewhere.52 The experimental binodal compositions obtained using FRES are also plotted in Figure 10 along with other experimental phase diagrams for blends of hydrogenated PS and TMPC.2,7,8 The nondeuterated blends were of comparable molecular weights to ours (the molecular weight of TMPC was roughly 1/10 of PS so the critical composition 4, should not be significantly differentt). We note three facts. First, only a rough agreement is seen between our experimental values and our calculated binodal. Second, there seems to be little or no shift in the LCST due to the deuteration of PS, unlike the case of PS/PVME where the LCST was raised by 40°C by substituting d-PS for PS.49 Third, all

* Constructing the phase diagram using Sanchez/Balazs theory was a meaningless trial due to its inability to account for composition dependence of x,. ' The rough estimation of q5c = dlv,,,,/( dlv,, + dNTMpc) shows that the value q5c was 0.23 for our material, 0.24 for Kim and Paul, 0.29 for Guo and Higgins, and 0.26 for Illers et al.

the experimental data available show that 4, = 0.4 rather than at the much lower 4 suggested by the difference in molecular weight between d-PS and TMPC (e.g., if X were not a function of composition). In the case of PS/PVME blends, the shift of 4, toward higher concentration of the more com- pressible component ( PVME ) could be attributed to the equation of the state contributions.' In the d-PS/TMPC case, however, this effect could not help rationalize the shift of 4, toward 0.5 as well as the composition dependence because of the small effect of compressibility.

Mutual Diffusion in Two-Phase Region

Guo and Higgins used light scatterin? to investigate the initial stages of spinodal decomposition for a w = 0.5 PS/TMPC blend at temperatures less than 10°C above the LCST. The effective diffusion coefficient D,R was extracted using the linearized theory of Cahn and Hillard55 (0 in Fig. 11) by monitoring growth of the amplitude of the composition fluctu- ations as a function of time. By extrapolating X,,

0.00

- 5 -0.05 2

i 6 -0.10

c 0

u - E -0. I 5 c -

-0.20 I I

0.0 0.2 0.4 0.6 0.5 I Weight fraction of PS. w

Figure 9. The measured X,, at Tg + 45°C (0 ) with the error bar and the Xses that were calculated as a function of composition at Tg from eq. ( 14) (solid line) were plotted. These were compared with the SANS data by Yang and O'Reilly4 (0).

476 KIM ET AL.

270

260

I

." 250 e

3 ?

c 240

0

220 230 0.0 L 0.2 0.4 0.6 0.8 1.0

Volume fraction of dPS. .+ Figure 10. Computed spinodal (broken line) and binodal (solid line) curves for the blends of d-PS and TMPC of the molecular weights of 489,000 and 42,000, respectively. The symbol 0 represents binodal compositions obtained using FRES. The experimental phase diagrams were also drawn for comparison: from the Tg measurements by Illers et al. using 320k PS and 411k TMPC7 (0 ) , from the Tg measurements by Kim and Paul using 330k PS and 33k TMPCs ( A ) , and from the cloud point measurements by Guo and Higgins using 318k PS and 53k TMPC' (0).

from one- to two-phase region and D* of both species along the TMPC-rich spinodal line,* we could also estimate the negative fi at the same molecular weights of their component polymers from eq. (4) so as to compare with Deff. Our spinodal temperature is about 12°C higher than that found by Guo and Higgins.+ To make a comparison with their data we thus first made our spinodal temperature coincide with theirs by changing A in xsc( T ) = A + B/T (i.e., we maintain the same temperature dependence of xsc). Our extrapolated Dee is shown as a solid line in Figure 11. There is a large discrepancy between the temperature dependence of our extrapolated fi and the one found by Guo and Higgins.$ The origins for this discrepancy may be any of the following: 1) the extrapolation of X, into the two-phase region, or 2) our d-PS/TMPC blend is fundamentally dif-

* Based on the assumption that the slow growth of the TMPC- rich domain dominates the spinodal decomposition process, we approximate the D* (ref. 10) to be ones along the TMPC-rich spinodal line, which was obtained by simply shifting our phase diagram to match the T, of Guo and Higgins. ' The spinodal temperature of 232'C obtained by Guo and Higgins (ref. 12) in kinetic studies by light scattering technique (and extrapolated to zero heating rate) was lower by nearly 10°C than any other result reported previously in the literature.

If we simply use D* extrapolated at w = 0.5 from one- to two-phase region not along the TMPC-rich spinodal line, the discrepancy becomes larger by nearly an additional order of magnitude over the temperatures of interest (dotted line in Fig. 11 ).

ferent from theirs, or 3) the fundamental failure of the theory. Recent studies showed that x in one- phase region could be successfully extrapolated to two-phase region to predict the (static) thermodynamic q ~ a n t i t i e s ~ ~ . ~ ~ such as the correlation length, the critical temperature of the mixture, and the dominant wave vector of spinodal decomposition at very short times. Direct comparison of X in one- and two-phase regions also seemed to be ~a1id . l~ The second possibility is that the material used by Guo and Higgins is somewhat different than ours-in view of their low spinodal temperature this possibility also seems likely. For example, it is known that methyl groups on the TMPC are very prone to oxidation. Oxidation of some of the TMPC monomers may change the thermodynamics of the blend significantly. Still, both A and B would have to be

Temperature ("C)

Figure 11. The mutual diffusion coefficients and the effective diffusion coefficients, fi and Dee, as a function of temperature in two-phase region. The negative diffusion coefficients inside the spinodal window were multiplied by -1. Dee is the values cited from the light scattering experiment2 (a), and fi was calculated using the tracer diffusion coefficients lo D * and the interaction parameter X, (= A + B / T ) as follows. (1) The solid line corresponds to the mutual diffusion coefficients both inside and outside the spinodal window that were calculated from the D* of each species on the TMPC-rich spinodal line and from the X,, with A/uo = 0.00267, and B/uo = -1.34 [X,, was shifted by changing only A in eq. ( 3 ) to make T, in accord with the one obtained by Guo and Higgins] . (2) The dotted line corresponds to the mutual diffusion coefficients inside the spinodal window that were calculated from the D* of each species extrapolated at w = 0.5 and the same X, as (1). (3) The broken line corresponds to the mutual diffusion coefficients in two-phase region that were calculated from the D* of each species on the TMPC-rich spinodal line and from the X,, with A/uo = 0.000387 and B/uo = -0.183, which was obtained by fitting to the Dee with maintaining T, unchanged.


changed dramatically to account for the decrease in the temperature dependence of Dee. The values of A/vo = 0.000387 and B/uo = -0.183 have to be used to fit (broken line in Fig. l l ) , both of which are much smaller than the value at w = 0.5 shown in Table I11 (0.00259 and -1.34, respectively). Such large shifts in both A and B with only a small change in spinodal temperature do not seem reasonable. The third possibility is that Dee is fundamentally different from fi (extrapolated) because the extrapolation ignores hydrodynamic interactions (“mode cou- pling”), which become important in the dynamics near the critical point. It was also recently reported by Kedrowski et al.57 in the isotopic blends of po- lyethylenepropylene that the fi extrapolated from the data in the one-phase region was greater by about a factor of 10 than the Dee obtained from light scattering measurements of spinodal decomposition in the two-phase region. They offered some observa- tions associated with this failure of the agreement between the two separate dynamical measurements, but no conclusion on the explanation was made, the discussion of which was deferred to the future.

CONCLUSIONS

1. The Flory interaction parameter x,, and its dependence on temperature and composition were extracted from the relation between mutual diffusion coefficient and tracer diffusion coefficients of the composing chains. A. A different temperature dependence of X,

at w = 0.5 and 0.25, as well as a strong composition dependence of x,,, was observed.

B. Based on the model of Sanchez and Bal- azs,’ a significant contribution of the specific interaction was needed to account for the temperature dependence of x,,, whereas the equation-of-state effect was found to play a negligibly small role.

2. The experimental critical composition & was approximately 0.4, which is large considering the fact that molecular weight of d-PS is about 10 times larger than TMPC and that the effect of compressibility is small.

3. The effective diffusion coefficient2 Dee that accounts for the initial stage of spinodal decomposition was considerably smaller than our calculated mutual diffusion coefficient fi extrapolated from the one-phase region.

This work was supported by the Division of Materials Research, NSF Polymers Program Grant No. DMR- 9223099, and by fellowship support from Monsanto Chemical Co. through the Polymer Outreach Program of the Materials Science Center at Cornell University.

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Received April 20, 1994 Revised August 29,1994 Accepted September 6, 1994

mutual diffusion and thermodynamics in the blends of polystyrene and tetramethylbisphenol-a...

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