Macromolecules 1994,27, 4960-4967
4960
Static and Dynamic Light Scattering from Polystyrene/ Poly(dimethylsiloxane)/Poly(methyl methacrylate)/Toluene Solutions+ Claude Strazielle and Michel Duval* Zmtitut Charles Sadron, CRM, 6 rue Boussingault, 67083 Strasbourg C&der,France
Mustapha Benmouna Mar-Planck-Zmtitut fur Polymerforshung, Postfach 3148,D-55021 Mainz, Germany Received February 15, 1994; Revised Manuscript Received June 14, 1994.
ABSTRACT The static and dynamic scattering properties of polystyrene (PS)/poly(dimethylsiloxane) (PDMS)/poly(methyl methacrylate) (PMMA)/toluene solutions are investigated in the conditions of homogeneoussingle-phasemixtures. The casesof PS/PDMS/tolueneand PS/PMMA/toluenewere investigated previously in our laboratory,and the purpose of this work is to see the effects of a third polymer in the mixture. The parameters are chosen in such a way that the so-called optical 9 condition is fulfilled for the visible componenta, and the scattered light probes directly the composition fluctuations of the PS/PDMS blend in a matrix of toluene + PMMA. The data show that the presence of PMMA does not change dramatically the qualitative behavior of the scattering properties and affects essentially their values. The results which are obtained independently from static and dynamic measurements are self-consistent. The quasi-elastic data suggest that the hydrodynamic interaction is screened out in the range of concentrationsconsidered in this work. The theoretical framework used to analyze the data is based on the random-phaseapproximation and k found to be sufficient within the conditions of our experiments as long as the mixtures remain stable in a state of a single homogeneous phase. 1. Introduction
In this paper, we report static and dynamic light scattering data obtained from polystyrene (PS)/poly(dimethylsiloxane) (PDMS)/poly(methyl methacrylate) (PMMA)/toluene solutions in the homogeneousone-phase region. Four systems are studied in order to characterize the effects of molecular weight, concentration, and composition of the mixtures. Two quaternary mixtures differing by the molecular weight of their polymer components are considered. They are referred to as systems I and 11. The static structure factor S(q), the dynamic scattering function S(q,t),the eigenmodes I'i(q), and the first cumulant Q ( q ) of the latter quantity are measured as a function of the scattering angle 9 and the volume fraction cp of the PS + PDMS blend. Here, t represents the time and q the amplitude of the wavevector which is related to the wavelength of the incident radiation A, the index of refraction R , and 9 by the known relati~nship:l-~ q = (In/X)nsin(U2)
(1)
The concentration is denoted c if expressed in g/cm3,and cp = CD when expressed in volume fraction, 0 being the partial specificvolume of the polymer. The above mixtures were chosen because of their interesting optical properties expressed by the incrementa of refractive indices ( d ~ l d c ) . These quantities satisfy approximately the following conditions:
Extensive measurements of static and dynamic properties in wide ranges of the scattering angle and the polymer concentration have shown that these systems can be
* To whom correspondence should be addressed.
+ Dedicated to the memory of Latifa Odd-Kaddour.
Abstract published in Advance ACS Abstracts, July 15, 1994.
regarded as pseudoternary mixtures of the blend PS + PDMS in an effective solvent. The third polymer component (PMMA) modifies the thermodynamic and viscous properties of the solvent. For a symmetric mixture, one finds that the optical properties reduce to a single effective increment of refractive index (d~ldc),ffand the thermodynamic properties can be characterized by a single effective interaction parameter xett4which will be denoted x for simplicity. By letting the composition of PS in the mixture PS/PDMS be VZ,i.e., z = cpps/(cpps + WDMS),one achieves the so-called optical 8 conditionw where the scattering radiation probes directly the fluctuations of the relative concentration of PS with respect to PDMS. Furthermore, in order to compare the results with those obtained in the ternary solution of PS/PDMS/toluene, we have repeated the same experiments in the absence of PMMA (systems I' and 11'). We find that both static and dynamic data show similar tendencies and hence can be described within the same theoretical framework. The addition of PMMA essentially modifies the numerical values of the measured quantities. This justifies the procedure of treating the system in the presence of PMMA as a pseudoternary mixture and allows for the analysis of data using the simple and known formulas of ternary mixtures in solution under optical 6 conditions. This procedure is commonly used in polymer systems because it is attractive for its simplicity and avoids the introduction of cumbersome and irrelevant parameters! It focuses on the essential physical properties of the system under investigation. The analysis of the data is based on the use of the random-phase approximation (RPA)Bfor the static properties and the Rouse model'0-12 for the dynamical properties. These models are commonly used in the interpretation of the scattering data for multicomponent polymer systems either in bulk or in s01ution.l~ This paper is organized as follows: In section 2, we present a brief background of the theoretical formalism needed to analyze the experimental data. In particular, we describe the static RPA for multicomponent systems
QQ24-9297/94/2227-496Q$Q4.5Q1Q0 1994 American Chemical Society
Macromolecules, Vol. 27,No. 18, 1994
Light Scattering from PS/PDMS/PMMA/Toluene 4961
and display the explicit equations in the case of a ternary mixture of polymer A/polymer B/solvent. These equations are useful since the solution of PMMA + toluene is regarded as an effective solvent in which the blend PS/ PDMS is dissolved. The dynamical formalism based on the Rouse model is also introduced. It is found that the hydrodynamic interaction using the Oseen tensor is not needed in the description of the scattering data. This means that in the concentration range considered in these experiments, the long-range hydrodynamic interaction is completely screened out, consistent with the observations made p r e v i ~ u s l y . ~The J ~ sample preparation, the experimental setup, and the data analysis are described in section 3. In section 4, the results are discussed within the framework of the theoretical model described in section 2. In particular, a comparison between the data of PS/ PDMS/PMMA/toluene and PS/PDMS/toluene is presented in order to identify the effects of PMMA. 2. Theoretical Background
The RPA has been used extensively to analyze the static properties of polymer blends A + B far from the conditions of phase ~ e p a r a t i o n . ~ JIts ~ Jgeneralization ~ to an arbitrary number of components is straightforward and can be implemented using a matrix formulation. The case of solutions where one component is a low molecular weight solvent has raised some questions due to the fluctuations, especially in the dilute range where the volume fraction of solvent is high and its quality of solvatation is good. The renormalization group theory has been used to correct for these fluctuations and led to new critical exponents which cannot be predicted by the classical RPA.3JklB Nevertheless, in spite of its shortcomings, the latter approximation is found to be quite useful and successful in describing the qualitative features of the scattering properties from multicomponent polymer solution~.6-8J~J9-~3 Within this approximation the static structure matrix S ( q ) can be expressed as a sum of the bare structure matrix So(q) and the excluded-volume matrix v. The former describes the effective structural properties of single chains, and the latter reflects the apparent excluded-volume interactions between monomers in the mixture. The general relationship for an arbitrary mixture has been given by Benoit1124 in the classical form: s-'(q) = S,-'(q)
+v
(3)
In the case of homopolymers (Le., in the absence of copolymers), the matrix So(q)is diagonal and its elements are
structure factors Sij can be obtained from the solution of the matrix equation (3). The result ism
sab(q)
where the denominator A ( q ) is A(q) = [I + U,S,O(q)I[1
where ~i is the partial specific volume of polymer i and V, is the molar volume of the solvent. In the case of a ternary mixture of polymer A/polymer Blsolvent, the partial
+ ubbs:(q)l
Ua;S:(q)
S,,?q) ( 6 ~ )
can be obtained from eq 6a by interchanging the indices a and b, and the off-diagonal elements are equal (Le., Sat,(@ = S b ( q ) ) . The bare structure factors Sjo(q) are defined by eq 4. In the case of the systems investigated here, the polymer components have approximately the same molecular weights and radii of gyration which means that the form factors are equal to P(q) and the degrees of polymerization to N , i.e.: Sbb(q)
Pa(q)
= Pb(q) = P(q); N a = Nb = N
(7)
Moreover, the solvent quality in the presence of PMMA can be described by effective excluded-volumeparameters satisfying the following equalities:
u, = ubb = U = U,b - x
(8)
where x is ameasure of the effectivedegree of compatibility between A and B polymers in the solvent. Although crude, these assumptions allow us to have a reasonable and simplified description of the experimental data. In the zero-average contrast conditions, one has direct accees to the structure factor SI(^), reflecting the fluctuations in the relative composition of polymer A in the mixture A + B.13 SI(q)is given by the followingcombination of Sij(q): sI(q)/(P2= S,(q)/(p,2
+ &(q)/%2 - 2sab(q)/(da'pb
(9)
+
where cp = cpe + cpb is the total A + B (or PS PDMS) volume fraction. Combining eqs 6-9 yields SI(Q) which is written in the reciprocal form as follows: Sil(q) = [x(l - x)(PNP(q)I-l- 2%
(10)
where x is the mean composition of polymer A in the mixture A + B; i.e., x = cpdcp. Equation 10is a special limit of the more general case where the sizes of polymers A and B are different:g sf'(q)
where the superscript zero is introduced to ease the notations and designates the bare structure factors. The parameters pi, Ni, and Pi(@ are the volume fraction, the degree of polymerization, and the apparent form factor of polymer i, respectively. The excluded-volumeparameters U i j are expressed in terms of the Flory interaction parameters xij and the solvent volume fraction cps as:2k27
(6b)
S,O(q)/A(q)
= -UabS:(q)
= [(PaNaPa(q)l-l
+ [cpbNbPb(q)l-' - 2 x
(11)
This analysis can be extended to the dynamical properties. First, one notes that the measured intermediate scattering function SI(q,t)can be fitted with a relatively good accuracy to the single-exponential function:
SI(q,t) = SI(@ exp[-r,(q) tl
(12)
where, at t = 0, the scattering signal as expected coincides with the static structure factor SI(^). The decay rate defined by the frequency rI(q1is related to the structure factor and the mobility MI by:12 rI(d = q2hBTMI(q)Si1(q)
(13a)
kg represents the Boltzmann constant and T the absolute temperature. In the Rouse model MI is a constant
Macromolecules, Vol. 27, No. 18, 1994
4962 Strazielle et al.
Table 1. Molecular Characteristics of Polymer Samples I = Mw/MN R, (A) A2 X 104 (mol.mL/g2)
sample Mw X 10-8 system I 1.10 PSlO 0.135 1.80 PDMS1" 0.130 1.80 PMMAlb 0.128 system I1 PS2' 1.40 1.30 PDMSBa 1.05 1.20 PMMA2b 0.96 1.20 a Light scattering in toluene. b Light scattering in THF.
XC
136
5.6
152 115
4.1 4.9
660
2.8
0.39
500 470
2.5 2.4
0.61
x 102 (cm3/crn3)
0.45 0.55 0.73
0.17
Composition in PS xps = rpps/(rpps + ~ D M S and ) XPDW = 1 - xpg.
of the data and makes t h e identification of t h e effects of PMMA in t h e matrix somewhat easier.
independent of q:
3. Experimental Section where 5 is the friction coefficient per monomer. Collecting these results yields the relaxation frequency:
It is customary t o approximate t h e form factor P(q) for Gaussian chains by P ( q ) = (1 + q2R,2/2)-l
(14b)
where the factor l/3 which is usually used in t h e low q range has been replaced by '/zin order t o fit a larger qR, range." Substituting eq 14b into 14a yields
The diffusion coefficient DI is obtained from t h e q = 0 limit:
D, = I'I(q)/q21,,o = D,H- 2xx(1 - x)cpNl
(15a)
D, is the single-chain diffusion coefficient in the Rouse model:
Combining eqs 14c and 15a yields the frequency I'I(q):
I n the presence of PMMA, the system becomes a quaternary mixture and its rigorous description requires the resolution of a matrix equation of order 3. The experimental data, however, show that this system can be treated as a pseudoternary mixture of two polymers (i.e., PS + PDMS) in a solvent (i.e., toluene PMMA). This means that the formulas derived above are still valid and that x is an effective interaction parameter. Note that t h e concentration of the polymer mixture (PS PDMS) cp is only a fraction of t h e total concentration m which means t h a t cp = y m , and (1- y ) = ( P P M M A / is ~ t h e composition of PMMA. One should keep in mind that t h e single-chain diffusion coefficient D, is modified upon addition of PMMA since the friction coefficient depends on t h e amount of this polymer added t o t h e mixture. T h e treatment of the quaternary system as a pseudoternary mixture simplifies substantially t h e theoretical analysis
+
+
3.1. Sample Preparation. Polystyrene samples (PS1 and PS2) were anionicallyprepared in the ICs laboratory. The other polymers are commercialsamples fractionated before use except for poly(dimethylsi1oxane)(PDMS1) and poly(methy1 methacrylate) (PMMA1) which are crude polymers. The weightaverage molecular weight MW and the radius of gyration R, of the samples were determined by light scattering measurements in toluene for PS and PDMS and in tetrahydrofuran (THF)for PMMA. The weight-to-number-averagemolecular weight ratio Mw/MNwas measured by gel permeation chromatography. The sample characteristics are listed in Table 1. The concentrations were made in volume fraction, and the solutions were clarified by centrifugation at 18 x 109 rpm during 2 h. Reagent-grade toluene was used throughout. Several PSI PDMS/toluene and PS/PDMS/PMMA/toluene solutions were prepared and studied at different PS + PDMS polymer concentrations in the range 0.4cp* < cp < 1.6 cp* for system I and 0.2cp* rp < 4cp* for system 11. It should be noted that the PS/PDMS/ toluene system has been investigatedextensively in our laboratory and the relationship between the critical concentration (PK and the molecular weight MW has been examined.sbPBa According to this work, the phase separation of systems I' and 11' investigated and 1.1X in this study is predicted to occur at (PK = 4.04 X (cm3/cm3),respectively. These values should be compared with cp = 3.4 X le2and 0.86 X 10-2 (cm3/cm3) which are the highest concentrations used here for these two systems, respectively. 3.2. Static and Dynamic Light Scattering Experiments. The static light scattering measurements were performed on a FICA 4200 photogonio-diffusiometer fitted with a He-Ne laser source of 4 mW at 632-nmwavelength vertically polarized. The measurements were made at several angles in the range 30-150' at a temperature of 25.0 i 0.1 OC. The dynamic light scattering measurements were performed using a home-built apparatus. The optical and mechanical parts of the apparatus have been described elsewhere.30The scattered light of a vertically polarized 488-nm argon laser (Spectra Physics 2020) was measured in the angular range 20' I 6 I150' at a temperature of 25.0 f 0.1 'C. The full homodyne correlation function of the scattered intensity defined on 192 channels was obtained by using the ALV-3000 (ALV-Langen,FRG) autocorrelator in its multi-7 mode. In this mode the correlation functions cover 7 decades in delay times going from 1 ps to 63 s. 3.3. Data Analysis. A Zimm analysis of the static scattering signals of all four systems was made. All the Zimm plots were found to be similar to the one obtained for system I1 which is shown in Figure 1. It can be observed that the inverse of the static scattering intensity varies linearly with the square of q without distortion even for the highest molecular weight and in spite of the presence of PMMA and the fact that ita anlac i s not rigorously zero. One notes that the light scattered by the PMMA + toluene solution in system I1 is about 12% higher than the light scattered by pure toluene. In the treatment of the static data, this intensity has been subtracted from the total intensity scattered by the PS/PDMS/PMMA/toluene quaternary mixture. The experimental intermediate scattering function S(q,t),related to the measured homodyne intensity autocorrelation function G(z)(q,t) by the Siegert relation,31 was analyzed using the
Macromolecules, Vol. 27, No. 18, 1994
Light Scattering from PS/PDMS/PMMA/Toluene 4963 4 r
a
I 1.75
3.50
5.25
7.00
sin2 8 / 2 + k O . 1 O 3
0 ’ 0
Figure 1. Classical Zimm plot obtained for system I1 (see eqs
2
1
3
10 and 16).
constrained regularization method (CONTIN) developed by Pr0vencher.a The base line of thescatteringfunction was allowed to float in the fitting procedure, and any measurement with a base line different from zero (less than 10%of the total number of measurements)was rejected. An example of the distribution function A(r/2q2)calculated by the CONTIN method is shown as the inset in Figure 3. This distributionfunctionis monomodal for all the systems investigated except for the highest concentration in system I1 where a fast mode with a low amplitude (less than 5 % of the total distribution)occurs. This fast mode could be due to the fact that the mixture does not meet exactly the zero-average contrast condition. The contribution of this mode to the scattered intensity is neglected. Moreover, the width of the distribution function of the monomodal curves is slightly larger than the width of the distributionfunctionobtained in the binary solutions PSl/toluene and PS2/toluene. For example, the analysis of the correlation functions of the scatteredintensities measured in the same experimental conditions (geometrical parameters, times of measurement,...) for a binary solution and for a ternary or quaternary solution gives a distribution width of 0.27 and 0.35,respectively. This increase in the width of the distribution function could be due to the fact that the degrees of polymerization of the two species are not rigorously identical and a small effect of polydispersity is observed. 4. Results and Discussion 4.1. Static Measurements. Two quaternary mixtures
are investigated, differing essentially by their molecular weights. They are called systems I and 11, and their molecular characteristics are given in Table 1. For each sample, the concentration of PMMA is fixed roughly at c* which is defined by (U~NPMMA = ( ~ A ~ W C ) P M=M1.A The experimental quantities A2, Mw, and c represent the second virial coefficient, the molecular weight, and the concentration, respectively. The composition of PS ( x = (ppdcp)is kept constant, and cp = (pps + WDMS is changed. This means that the composition of PMMA changes. The scattered intensity is measured as a function of the angle for several values of cp. Figure 1represents the results in the form of a Zimm plot. The same procedure was repeated systematically for eachs system described in Table 1. For shortness, we give the Zimm plot corresponding to system I1 only, noting that the others are qualitatively similar. The curves giving the variation of the inverse intensity with the concentration are straight lines having a negative slope. This showsthat the apparent virial coefficient is negative and the effective interaction between PS and PDMS components is strongly repulsive. The incompatibility of these polymers is enhanced by the presence of PMMA. A further analysis of the data calls
b
* i 0
X
I
\
0 ‘ 0.00
T
1.20
0.60
50 X I O 2 (cm3/cm3) Figure 2. (a) Inverse static scattering intensity at zero angle (i.e., q = 0) as a function of the volume fraction of the PS +
PDMS mixture. The filled triangles (A)are obtained for system I in the presence of PMMA. The filled circles ( 0 )correspond to system I’ in the absence of PMMA (see eq 16a). (b) Same as Figure 2a but for systems I1 and 11’. for the following observations: the variation of the scattered intensity, extrapolated to q = 0, as a function of the concentration is interesting because it has a direct implication to the interaction parameter x and the critical concentration a.Parts a and b of Figure 2, show the variation of the normalized inverse intensity at q = 0 as a function of cp for systems I and 11,respectively (symbol A). These figures show also the data obtained for the ternary mixture of PS/PDMS in toluene (i.e. systems I’ and 11’) and represented by the symbol (0).These results are consistent with the linear variation with Q given by eq 11. Using the standard experimental notations,lJO one writes K‘cplAI(q=O)
- 2x(p
(16a)
where the concentration cp is expressed in volume fraction and a is
4964 Strazielle et al.
Macromolecules, Vol. 27, No. 18, 1994
Table 2. Critical Concentration VK, the Interaction Parameter x, and the Quantity a Obtained from Static Measurement sa system (PK x 102 (cm3/crn3) x x IO2 a x 104 I 2.99 5.9 35.3 I' 3.92 L4.061 4.9 13.91 38.0 133.51 I1 0.91 2.3 4.21 11' 1.05 [1.11] 2.0 11.51 4.23 r3.731 a The values within the brackets are calculated using eqs 17c,d and 16b for the three quantities, respectively.
The constant K' is related to the optical constant of the apparatus K, the molar volume of toluene V,, and the increments of refractive indices (dnldcp) by:
K' = KV,[(dn/d&, - (dn/dcp)p,,,I*
(16~)
The relationship between (dn/dp)i and the experimental quantity (dn/dc)i involves the partial specific volume ~ i (dn/dcp)i = (an/dc)i/Di
-5
:
t 16d)
The scattering signal is obviously much higher for system I1 since it is proportional to the molecular weight. The quantities CY and x obtained from the fit with the experimental data are listed in Table 2 for systems I and 11. In this table, we have collected the values of the critical concentration at the intercept with the horizontal line. These values correspond to the concentrations at which the intensity diverges. It is the indication that the mixture has reached the critical concentration and started the process of phase separation via spinodal decomposition. The group renormalization theory predicts that M< and x vary with the molecular weight following the scaling laws: 29a
I
x = Mw-0.44
(17b)
Experimentally, Ould-Kaddour and S t r a ~ i e l l investie~~~ gated several ternary mixtures of PS/PDMS/toluene and obtained = 42.7Mw-O*59
(17~)
and
x = 7.97Mw-0.45
(174
The values obtained from these equations are given within the brackets in Table 2 for the ternary systems I' and 11'. One notes that the agreement is very good concerning a, whereas for x a slight discrepancy is observed. As expected, the quaternary mixture reaches the critical concentration at a lower value which corresponds to a shift on the order of the PMMA concentration. With regards to the interaction parameter, one observes that the values obtained for the systems I and I1 are less than 20 % larger than those correspondingto systems I' and 11',which means that the compatibility between PS and PDMS is further reduced due to the presence of PMMA. 4.2. Dynamic Measurements. Dynamic light scattering measurements were performed on the mixtures described in Table 1in wide ranges of B and cp. A careful examination of the autocorrelation functions shows a single-exponential relaxation mode. A slight deviation is
ogl t / s e d
Figure 3. Experimental autocorrelation function of the scattered field measured by QELS on system I1 at 0 = 20' and 9 = 1.8 X 109 (cmS/cm3).The solid line is the result of the fit by the CONTIN analysis. The inset representsthe diffusion coefficient distribution obtained from the CONTIN analysis.
L L ,
0
0
and
2
-2
L
5
10
,I
15
q' X 1O-'' (cm-') Figure4. Variationoftherelaxationfrequency I'dqZas afunction of q2obtained from dynamic measurements using system 11. The continuous lines represent the theoretical equation (15c). The data from top to bottom correspond to the volume fractions 9 = 0.44, 1.31, 1.75, 3.50, 5.25, and 7.00 (109 cms/cma).
found for the highest volume fraction of system I1 having the highest molecular weight. In this case a second fast mode with a relatively low amplitude is observed. Figure 3 shows a typical correlation function of the normalized scattered field given by system I1 at the volume fraction cp = 1.8 X 103(cm3/cm3)and scattering angle B = 20°. The inset in this figure represents the CONTIN treatment of the correlation function of the scattered field. It should be pointed out that the width of the monomodal distribution is found to be insensitive to the experimental conditions considered in this work (i-e.,scattering angles and concentrations). The relaxation rI(q)is measured as a function of B (or q ) and 9. The results corresponding to system I1 are illustrated in Figure 4 where I'I(q)/q2is plottedas a function of q2 for several values of cp. Except the lowest concentrations, the lines have a practically constant slope independent of cp. The solid lines represent a fit with the theoretical prediction given by eq 15c. Taking into account the D, value given in Table 3 for system I1 and the slope of the solid lines of Fi ure 4 (20 x 1P20 cm4/s), one calculates a value of 230 for the dimensions of the chains in the quaternary system. One observes that this value is somewhat small as compared to the result in the 8 solvent (R, = 320 A as calculated by interpolation of the power law obtained on PS/cyclohexane at 36 0C33*). The extrapolation of the curves of Figure 4 to the q = 0 limit
x
Light Scattering from PS/PDMS/PMWToluene 4965
Macromolecules, Vol. 27, No.18,1994
a
b
h
(I)
\
E
0
v
m
0 7
X
d 0.00
0.55
1.10
p X lo2 (crn3/cm3) Figure 5. (a) Variation of diffusion coefficient DIas a function of the volume fraction cp of PS + PDMS. The filled triangles (A) represent the results of system I, whereas the filled circles ( 0 ) represent those of system 1’. Do is the calculated diffusion coefficient of an isolated chain of this molecular weight in toluene (see eq 19a). (b) Same as Figure 6a but for systems I1 and 11’. Do is the measured diffusion coefficient of an isolated chain of this molecular weight in toluene.
Table 3. Critical Concentration p ~the , Interaction Parameter x, and the Single-ChainDiffusion Coefficient D, Obtained from Dynamic Measurements I I’ I1 11’
3.27 3.79 0.89 1.11
5.09 4.40
2.13 1.70
2.59 3.22 0.77 0.76
yields the diffusion coefficient DI as a function of the volume cp. This procedure implies that the limit q = 0 coincides with the lower q region where rI(q)is proportional to q2. In this case, the crossover from the q2 to the q4 regions seems to be very harp.^^^?^ In parts a and b of Figure 5 DI is plotted as a function of cp for systems I and 11,respectively. We have included in these figures the data corresponding to the ternary mixture PS/PDMS/toluene in the absence of PMMA. First, it can be noted that DIdecreases as the polymer concentration increases as predicted by RPA. On the other hand, the diffusion of chains in the quaternary systems is slower, implying that the presence of PMMA in the mixture slows down the interdiffusion process. The continuous lines represent a fit to the theoretical prediction in eq 15a. One can write this equation in the standard form by introducing a Huggins-like coefficient k~:lOJl
D,((c) = D,(1 -k ~ H ’ P ) (18) where k~ is proportional to x and describes the interaction
between unlike polymers. The values of D, and x obtained from this analysis are collected in Table 3. One can make the following observations concerning the variation of the single-chaindiffusion coefficientD,with the concentration of PMMA: obviously, one would expect that the extrapolation of the curves in parts a and b of Figure 5 to (c = 0 yields a single-chain diffusion coefficient D, which is a function of the PMMA concentration. For the mixtures of lower molecular weight I and I’ this is the case since the extrapolated values are different as can be seen from Figure 5a and Table 3. The variation of D, with the polymer concentration has been observed in several other studies on interdiffusion dynamics of polymer blends.% However, for the systems of highest molecular weight, the extrapolation of the data in Figure 5b yields the same value of D,. This is not a discrepancy between the two systems I and I1 but reflects the effects of the presence of PMMA in the solvent. The effect of this polymer is more apparent in the systems with lower molecular weight I and I’ because its concentration is much higher. A more detailed investigation of the effect of the PMMA concentration will be the subject of a future paper. It would be interesting to compare the extrapolated values to the single-chain diffusion coefficient DOin the infinite dilute solution where the solvent is pure toluene. We have measured thisquantity in the cases of PSYtoluene and PDMS2holuene. The results are 1.00 X 10-7 and 1.20 X cm2/s,respectively. The former value is consistent with the empirical equation proposed by Huber et al?” which when corrected for temperature effects reads
Do = 3.59 x 104M0.678 (cm2/s)
(19a)
This predicts that, for the system of highest molecular weight, one has
In the case of the lowest molecular weight systems, eq 19a cm2/swhich is also consistent with gives DO= 3.89 X the experimental value obtained for system I’ and given in Table 3, This seems to be in discrepancy with the Rouse dynamical result given by eq 15b. Additional investigations using similar mixtures with different molecular weights have revealed that the intramolecular hydrodynamic interaction has a much more significant importance than the intermolecular interaction which is screened when the polymer concentration increases above c*. These dynamical data are in good agreement with the static data of the preceding section. If one plots on the same figure the inverse scattering intensity in the forward direction (Le., at q = 0) normalized to the value a t cp = 0 and the quantity D~(cp)/D~(v=o), as a function of (c, one finds a complete agreement for the systems investigated here (see Figure 6). These results can be fitted with the theoretical prediction 1 - 2xx(1- x)qN based upon the use of the RPA and the Rouse model, neglecting the hydrodynamic interaction. They indicate that the latter model gives a reasonable description of the normalized interdiffusion coefficient D1(cp)/DS.Although the molecular weight dependence of the single-chain diffusion coefficient as predicted by eq 15b does not seem to fit the data correctly, it was mentioned earlier that the DO values measured in this study are consistent with eq 19a, indicating a strong effect of the long-range intramolecular hydrodynamic interaction. This aspect will be examined in a future work in more detail by considering systems with different molecular weights. Furthermore, the
4966 Strazielle et al.
Macromolecules, Vol. 27, No. 18, 1994
1.00
1.oo
0 50
0.50
the dynamical model used to describe the quasi-elastic data is based on the Rouse model where the long-range hydrodynamic interaction is neglected. The results show that this approximation is good only in describing the variation of the relaxation frequency with q and the concentration dependence of the interdiffusion coefficient. The results for the single-chain diffusion coefficient seem to indicate that there is a sensitive effect of hydrodynamic interactions which modifies its scaling behavior with the molecular weight.
Acknowledgment. This paper is dedicated to the 7memory of Latifa Ould-Kaddour, who left us suddenly on
0.00
0.30
0.40
0.00 0 a0
cp X :02 icm3/cm3) Figure 6. Double representationof the variation of R(B=O,cp=O)/ R(B=O) (i.e., the inverse static intensity at 6 = 0 normalized to ita value at cp = 0) and DI/DI(cp=O)(Le., the diffusion coefficient normalized to ita value at cp = 0) as a function of cp. This figure applies to all the systems considered here. X-parameter extracted by this analysis both from static and dynamic measurements is consistent with the one obtained in a previous work performed in the ICs laboratory,Bb and Figure 6 collects the results obtained from system I1 only. A similar behavior is observed for the other systems. Moreover, the critical concentration for spinodal decomposition (CK is deduced from the extrapolation of the lines through the horizontal axis. For the static data, this procedure leads to c p ~at which hR,(q=O) goes to infinity (i.e., critical opalescence), and for the dynamic data, it corresponds to the concentration at which the fluctuations are infinitely long lived (Le., critical slowing down). The latter values of (CK are listed in Table 3 for all the mixtures investigated here. One finds a good agreement between the static and dynamic data which is surprising in view of the simplicity of the theoretical model and the complexity of the multicomponent system.
5. Conclusion In this paper, we report elastic and quasi-elastic light scattering data describing the static and dynamic properties of PS/PDMS blends in a solution of PMMA and toluene. The PMMA is isorefractive with toluene, and the mixture PS/PDMS is prepared in such conditions that the so-called optical 0 requirement is met. The data obtained both from elastic and quasi-elastic measurements show that this system behaves as a ternary mixture made of a blend of PS/PDMS in a solvent whose thermodynamic properties are slightly modified as compared to those of pure toluene. A systematic comparison of the data with and without PMMA allows for the identification of the effects of this polymer on the thermodynamic and hydrodynamic properties. The static and dynamic observations are self-consistent, and the data are analyzed using a simple model based on the RPA. It is shown that the RPA provides a reasonable description of the qualitative behavior of the scattering properties of such multicomponent polymer solutions. Although we are well aware of the fact that these results should break down in the vicinity of the critical regions for phase separation, we have nevertheless examined the scaling behaviors of the critical concentration for phase separation (PK and interaction parameter x with molecular weight Mw. The data obtained on the ternay systems are in good agreement with the predictions of the renormalization group theory and with previous experimentalresults. On the other hand,
July 1992 while she had started the investigation of the static properties of these systems. Her contribution while she was among us was determinant in providing a better understanding of the static scattering properties of multicomponent polymer solutions. Her memory was always present with us during our discussions. We thank Professor H. Benoit (CRM, Strasbourg, France), Professor J. F. Joanny (CRM, Strasbourg, France), Professor A. Patkowski (MPI-P, Mainz, Germany), and Dr. J. Seils (MPIP, Mainz, Germany) for stimulating discussions and their interest in this work. M.B. expresses his gratitude to Professor E. W. Fisher for hospitality at the Max-PlanckInstitut ftir Polymerforshung (Mainz, Germany).
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