Equilibrium and non-equilibrium states of polymer blends investigated by small-angle neutron scattering

Physica B 174 (1991) 159-163 North-Holland

Equilibrium and non-equilibrium states of polymer blends investigated by small-angle neutron scattering D. Schwahn, T. Springer and S. Janssen Forschungszentrum Jiilich GmbH, lnstitut fiir Festkrrperforschung, Posffach 1913, W-5170Jiilich, Germany The polymer blends PS/d-PB, PS/d-PS and PVME/d-PS were investigated by small-angle scattering of neutrons. From the structure factor S(Q) the Flory-Huggins interaction parameter has been determined. Very close to the critical temperature a transition from a meanfield to an Ising-like behaviour occurs with 3' = 1.22. Furthermore, the time dependence structure factor has been investigated after the application of a temperature step. The results were compared with the exponential Cahn-Hilliard-Cook theory for the time dependent structure factor S(Q, t).

1. Introduction

crossover

of

the

critical exponent 3' for from 1.0 to 1.25. (ii) The non-equilibrium region was studied by the time dependent structure factor S(Q, t), if a rapid step between two temperatures has been applied, either with both temperatures in the homogeneous region, or with one of them in the unstable region inside the miscibility gap.

S(Q---~O), namely The measurement of the structure factor for concentration fluctuations in polymer blends S(Q) ( Q -- scattering vector) by small-angle scattering of neutrons (SANS) is a method to investigate the thermodynamic equilibrium properties of blends. Furthermore, transitions between equilibrium states or between an equilibrium to a non-equilibrium state can be studied. A number of such experiments have been carried out in the range of wavevectors Q = 10 -1 to 10 -5/~-1 on several polymer blends, part of which will be reported here (see refs. [1, 2]). (i) The structure factor in the homogeneous equilibrium region of a blend yields the FloryHuggins interaction parameter x(~b, T) as a function of concentration of one of the components, ~b, and of temperature, T; this function is related to the free energy of mixing for the two components of the blend. The Flory-Huggins parameter is the basis of a quantitative understanding of many properties of blends in the meanfield approximation, in particular the phase diagrams (cf. refs. [3, 4]). In a region very close to the critical temperature T c we observe a transition of the critical scattering S(Q ~ 0) from a meanfield behaviour- as usually observed in high molecular b l e n d s - to an Ising behaviour with a typical

2. Theoretical basis Using the random phase approximation [5, 6], and expanding the structure factor S(Q) to powers Q2, one obtains

2 S-I(Q) = s-l(o)..1_ 1Q2[ ~Al'~'wA'---~R~ + (~BvwBRgB J]" (1) RgA,B are the radii of gyration of the chain molecules of the two components A and B, and ~'~wA,B are the corresponding weight averages of the molecular volume per mol. 4)A, ~bB are the corresponding concentrations. The well-known relation between S(Q--*O) with the thermodynamic potential is S-t(0)

= (0 2 AGm/O~b2)/RT.

0921-4526/91/$03.50 © 1991- Elsevier Science Publishers B.V. (North-Holland)

(2)

160

D. Schwahn et al. / Equilibrium and non-equilibrium states of polymer blends

In the meanfield approximation the Gibbs free energy of mixing can be written as

6A In CA AGm/RT=

~']'wB

6B In 6B q-

9w~-~- + 6A6BX ,

(3) where X is the so-called Flory-Huggins interaction parameter. It is considered as a 'segmental' or 'local' quantity which does not depend on the molecular conformation or on Vw. In addition to the energetic interaction part, it normally includes an entropic part, assumed to be caused by the free volume of the blend. X may depend on 6A.B and on T (beyond the trivial factor 1/RT from the definition in eq. (3)). We introduce the effective Flory-Huggins parameter (with 6A = 6, 6B = 1 - 6 ) F = [02)(6(1 - 6 ) / 0 6 2 ] = x -

(a - 26)

OX

02X -

}6(1

-

6)

a62

(4) •

If X does not depend on the concentration, one has F = X. Finally, one gets S-~(0) = 2[Fs - F(6, T)] [ =

1 1 ]-2F(6, 617"wA+ (1 -- 6)lT"wa

T)

(5)

where Fs is F at the spinodal Ts(6 ) of the phase diagram, where S(0)---~. The interaction parameter will be separated into an enthalpic (Fh) and entropic (F,) part, namely F=Fh(6)/ T - F~(6). The compensation and the spinodal temperature are defined by F = 0 and S - 1 ( 0 ) = 0, respectively, or Tcomp =

['h/F~, and

r. T s ( 6 ) - Fs + F~ "

(6)

Applying the Landau-Ginzburg criterion [5, 6] one expects a crossover between the meanfield and an Ising behaviour in a region where the concentration fluctuations become strong; the crossover temperature T* is given by

1 Tc

1 Tc

1 T*

1 Lomp '

(7)

N is the degree of polymerization. Except in cases, where T c is n e a r Tcomp , the crossover region has a width of about 1/N. Beyond that region, the meanfield behaviour holds with exponent 7 = 1. A transition from one fluctuation state to another can be written as [5, 8] ~rr(Q, t ) = ~rr(Q) + ( 1 - ~rr(Q) ) e x p { - 2 t / r ( Q ) } ,

(8)

O'r(Q, t) = ST(Q, t)/Sro(Q ) and O'r(a) = Sr(O)/Sro(Q ). Here Sr0(Q ) and s r ( a ) are

with

the equilibrium structure factors at the initial and final temperatures TO and T, respectively. The relaxation rate is "c-I(Q)=Q2A(Q)S~I(Q), where A(Q) is a nonlocal Onsager coefficient. The chemical diffusion constant is obtained for Q--->0, namely Dcou=A(O)ST.I(o). In the unstable region, formally one has S ~ 1 ( 0 ) < 0 and the rate z-1 becomes negative for Q < Qc where Q2=c -1/ST(O)A2; A2 is the square bracket in eq. (1). This leads to an exponential growth of fluctuations. At larger times, coarsening occurs, namely the growth of precipitates with phase boundaries; there appears a strong increase of scattering intensity and a shift of the diffraction peak in S(Q) [1].

3. Experiments

3.1. Samples The investigated polymers are summarized in table 1. The blend PS/d-PB (2 and 6) was produced without solvent. The isotope blend d(eutero)-PS/PS (1 and 3) and the blend d-PS/ PVME (3, 4 and 5) were dissolved in toluene and then dried in vacuum. Finally, the isotope mixture was annealed at 160°C for several days, and the d-PS/PVME blend at 100°C, both in an argon atmosphere. All samples were kept under argon in a niobium container during the SANS experiments.

D. Schwahn et al. / Equilibrium and non-equilibrium states of polymer blends Table 1 The molecular weights are determined with GPC of the investigated polymers. No.

Polymer

1 2 3 4 5 6

PS PS d-PS d-PS PVME d-PB

M,(10 ~ dalton) 1030 2.01 980 232 60 1.72

Vw(103 cm3/mol)

Mw/M"

950 1.91 870 206 59 1.73

1.09 1.09 1.14 1.08 3 1.10

161

I

15

I

d-PSIPVNE • = 0.13

"

i

r=o /~

~ lO

"i"

5

, ~ " - ~ ,,,t(~"

A 1272"C Tc =132"(

3.2. The S A N S experiments

0

I

I

t

1

2

3

eLz ilO-'.,~-z ]

The SANS experiments were carried out with the instrument KWS I at the Jfilich research reactor FRJ-2 (thermal flux 2 x 1014 neutrons/ s cm z) with a resolution of 1 0 - 3 ~ -1 up to 0.1/~-1. Figure 1 presents the inverse equilibrium structure factor versus Q2 in the homogeneous region, i.e. below Tc for d-PS/PVME (3 and 5, demixing occurs at high T in this blend). The slope yields the combined radii of gyration (see eq. (1)) and the intercept as a function of concentration and temperature gives Fs and F~. Similar measurements have also been carried out on the other systems. The results for F, Fs and F~ as obtained from the scattering data as in fig. 1 are summarized in table 2. Obviously, the interaction parameter for the isotopic blend is about 100 times below the values for blends which differ chemically. From eq. (8) one recognizes that mixing occurs if F h / ( Fs + F ~ ) > 0 . One can show that, if

Fig. i. SANS results for the structure factor, for a d-PS/ PVME blend, in Zimm representation at different temperatures; Q = scattering vector. At the compensation temperature S(Q) coincides with the dashed line ( F = O) (see eqs. (1, 2)).

both nominator and denominator of this equation are negative, one obtains a lower critical solution temperature, i.e. a miscibility gap at high temperature. This is the case for the system d-PS/PVME. In the inverse case (nominator and denominator positive) the blend has an upper critical solution point, as in the case PS/d-PB. From these data, phase diagrams have been calculated and verified by experiments, but the results are not shown here (see refs. [1, 2]). The results confirm that the meanfield concept as presented before works well. Very close to the spinodal, however, the meanfield concept is expected to break down

Table 2 Generalized Flory-Huggins parameter F for the investigated blends. System

component

F (mol / cm 3)

d-PS/PVME A (to = 0.13)

(3:5)

7.79

d-PS/PVME B (4) = 0.2)

(4:5)

9.18 x 10 -4

~'( 1 0 - 4 - -

0.31 T(K) 0.375 T(K)

d-PS/PS (to = 0.48)

(I :3)

-4.88 x 10 -6 + 0.28 x I0 -~

T(K) PS/d (tO = 0.53)

(2:6)

-1.45 x 10 -4 + 0.47 T(K)

D. Schwahn et al. / Equilibrium and non-equilibrium states of polymer blends

162

because of fluctuations (see first paragraph). The predicted transition has been actually observed as shown in fig. 2 for the system PS/d-PB, with a crossover T * - T ~ = 6 K above T~, and with a clear change of y from 1.0 to 1.22. Such a crossover has also been found previously for d-PS/PVME, where T * - T~ was 2 K below T c [7]. The observations agree, within order of magnitude, with eq. (7). Within a region of IT*T~[, the concept of a spinodal loses its sense; this region gets broader as the molecular weight decreases. Figure 3 shows the typical relaxation behaviour of the normalized structure factor as defined in eq. (10), o-(Q, t), for the isotope blend d-PS/PS (3 and 1 in table 1) at 4~ = 0.48. The temperature was changed rapidly from 204 to 170°C such that a transition between two equilibrium states occurs. It has been tried to fit the data with the relaxation function eq. (8). However, the resulting transition function could not be interpreted by the exponential relaxation. Agreement was achieved by a modified function, replacing t/r in eq. (8) by (t/~') ~, where the 'stretching exponent' a was supposed to depend on Q. for Q ~ 0, the expected behaviour with a = 1 has been obtained by extrapolation. Using this limiting situation, the collective diffusion

10 PS/d-PB ¢~= 0.53

L" 8 Et.a

]=1.22

r~ t, O

~=I ~

"7

IITcM F

t/'l

a

.

'~Tc,

2.60

265 T-1 [10-3K-11

2.70

Fig. 2. S ~(0) vs. l I T for the PS/d-PB blend at the critical concentration. A deviation from meanfield to Ising critical behaviour is observed near T c.

1.~

I

I

I

I

I

To=20/+°C--T =170°C (Vw = 0.91.106cm31 mot)

1.6 ]

1

'

hl~.

, t = 20rain

~

lilt=

~.~ 1./~

~

1.2

"

1.0

1

lh

equilibrium of t=oo

~

2

½

/,,

5

(3, [10-3 A-1 ] Fig. 3. The time dependent and normalized structure factor (eq. (10)) after a temperature step downwards between 170°C and 204°C after 20min, 1 h and 3h. d-PS/PS The structure factor is normalized to S(Q) at the initial temperature (sample at equilibrium). The abscissa is tr(Q, t) = 1 fol t = 0. The molecular volume is 0.89 × 1 0 6 cm3/mol.

constant has been determined by Dco. = (1 / r) Q 2 for Q - * 0 , yielding Dcoll = 10 -15 cmZ/s in good agreement with macroscopic results. The nondiffusive behaviour for larger Q or small length scales (where a = 0 . 5 ) could have the following reason: the system has a critical temperature (as calculated from F ) of 125°C; the glass transition is at about 100°C. Consequently, glass relaxation modes could influence the relaxation behaviour [9] and the relaxation is not entirely given by diffusion. So far, this behaviour of the transition in the homogeneous region is not yet quantitatively understood. Spinodal decomposition has also been investigated for the d-PS/PVME systems A and B, by up-quenching into the metastable miscibility gap. A rapid temperature step has been applied from 2 K below Tc to a temperature at about T c + 2 K which is in the unstable region. The measurements have been started immediately after the temperature has been raised, with about 31 repetitive subsequent measurements of 10s each. The results for 4 different time intervals are shown in fig. 4. The data fit very well with eq. (8) (solid curves). Consequently, the simple exponential behaviour (with fluctuation growth 1/~-<0) of eq.

D. Schwahn et al. / Equilibrium and non-equilibrium states of polymer blends I

/

•

o

I

I

°

I

2 0 [10-3A-1 ]

3

4

Fig. 4. The early state of spinodal decomposition of system B (fig. 3). ( - © - ) critical scattering at 143.8°C before the quench. Measurement time for the curves is 10 s each. ( - + - ) 10s, ( - O - ) 50s, ( - i - ) 90s, and ( - & - ) l l 0 s after the initiation of decomposition.

(10) was confirmed. An interference peak is clearly observed, whose position Qm practically does not shift with time, as expected from eq. (8). Also for system B one observes the exponential relaxation function as in eq. (8). This is in surprising contrast to the results in the homogeneous region as described before, which were non-exponential. The d-PS/PVME system has a time constant DQ 2 about 200 times smaller than the d-PS/PS system. Consequently, the transition observed for d-PS/PVME between 10-100s would occur for d-PS/PS (fig. 3) at about 5 h.

4. Summary (i) The elastic structure factor S(Q) for smallangle neutron scattering for a number of polymer blends has been investigated and interpreted in terms of the random phase approximation. The corresponding Gibbs potential of mixing was formulated in the meanfield theory. At temperatures not too close to the spinodal, this descrip-

163

tion consistently explains the experimental data for S(Q). One obtains the radius of gyration R~ and the Flory-Huggins interaction parameter of the blends which leads to a quantitative and consistent interpretation of the corresponding phase diagrams. (ii) Close to the spinodal or to the critical temperature, a crossover occurs for the temperature dependence of S ( Q ~ 0) from a meanfield behaviour, as normally observed for polymers (critical exponent y = 1.0), to an Ising-like behaviour with 3' = 1.22, as known for normal liquid mixtures; the crossover region has a width of a few K. (iii) The transition behaviour of the structure factor as a function of time, following an applied temperature step, has been investigated for the isotopic blend d-PS/PS with a glass transition close to T c. If the initial and final temperatures are both inside the homogeneous region of the blend, the behaviour does not follow the expected exponential Cahn-Hilliard-Cook prediction, except for the extrapolation Q---, 0. The extrapolated relaxation rate 1/r (Q ~ 0) yields for Dco., the interdiffusion constant in good agreement with macroscopic data.

References [1] D. Schwahn, K. Mortensen, T. Springer, H. YeeMadeira and R. Thomas, J. Chem. Phys. 87 (1987) 6078. [2] D. Schwahn, K. Hahn, J. Streib and T. Springer, J. Chem. Phys. 93 (1990) 8383. [3] D. Patterson and A. Robard, Macromolecules 11 (1978) 690. [4] M.G. Bawendi and F. Freed, J. Chem. Phys. 88 (1988) 2741. [5] K. Binder, J. Chem. Phys. 79 (1987) 6387; Colloid & Potymer Sci. 265 (1987) 273. [6] P.G. de Gennes, Scaling Concepts in Polymer Physics (Cornell Univ. Press, Ithaca, 1979). [7] D. Schwahn, K. Mortensen and H. Yee-Madeira, Phys. Rev. Lett. 58 (1987) 1544. [8] H.E. Cook, Acta Metall. 18 (1970) 297. [9] J. Jfickle and M. Pieroth, Z. Phys. B 72 (1988) 25.