on the measurement of the χ-parameter under the zero average contrast condition

On the Measurement of the X-Parameter Under the Zero Average Contrast Condition

M. BENMOUNA, E. W. FISCHER, B. EWEN, and M. DUVAL

Max-Planck-lnstitut fur Polymerforschung Postfach 31 48, 6500 Mainz, FRG

SYNOPSIS

Several methods for measuring directly the interaction parameter x as a function of polymer concentration are presented. These methods are applied to symmetric ternary mixtures of homopolymers and copolymers in solution under the zero average contrast condition. They can be used for an arbitrary concentration from dilute solution to the bulk and at any value of the scattering wavenumber q. Possible combinations of static and dynamic measurements are presented. 0 1992 John Wiley & Sons, Inc. Keywords: Flory interaction parameter in ternary polymer-solvent systems, concentration dependence of X , light scattering from ternary mixtures of solvent with two polymers, zero average contrast

I NTRO DUCT10 N

T o our knowledge, the most reliable measurements of the Flory interaction parameter x for ternary polymer-1 /polymer-2 solvent mixtures were reported by Inagaki et al.’-4 using static light scattering under optical theta conditions. Later Ould-Kaddour and Strazielle used the same technique to measure x for a series of ternary mixture^.^-^ All these measurements were made for dilute solutions far below the overlap concentration c* and for relatively low molecular weights satisfying the condition qR, = 0, where Rg is the radius of gyration of the polymer with higher molecular weight. Therefore, only a limited information can be gained under these conditions. The purpose of this paper is to show that one can easily extend the Inagaki method to conditions where the measurement of x becomes possible for arbitrary values of concentration and molecular weight provided the system is a disordered homogeneous mixture with either homopolymers or copolymers. This gives the possibility of combining both light scattering and neutron scattering techniques. In the final part of this paper, we suggest

Journal of Polymer Science: Part B: Polymer Physics, Vol. 30,1157-1164 (1992) 0 1992 John Wiley & Sons, Inc. CCC 0887-6266/92/01001157-0S$04.00

complementing static measurements with dynamic scattering data using either quasielastic light scattering (QELS) or neutron spin echo (NSE) techniques. Of course, the measurements of the x-parameter for polymer blends ( in the bulk) have been reported quite extensively by use of other techniques. It is not our intention here to review the data obtained by these techniques, but it is worth mention- ing that one should make a distinction, as pointed out by Sanchez’ and Han and co-worker~ ,~ among different types of interaction parameters. In particular, the x-parameters in the expressions for the free energy, the chemical potential, the static structure factor S ( 4 ) , and the dynamic structure factor S (9 , t ) or its characteristic relaxation rates.” Here we are interested in the X-parameter measured either from S ( q ) by static scattering or from S ( q , t ) by dynamic scattering, assuming that these two techniques lead to the same value of X. Furthermore, under conditions where hydrodynamic interaction can be neglected ( the Rouse limit), one has direct access to the X-parameter from S ( q , t ) and its relaxation rate. If hydrodynamic backflow effects must be included, a more detailed analysis is required and one may have to introduce a q-dependent x-parameter because of the long-range nature of hydrodynamic interaction, as will be shown below.

1157

1168 BENMOUNA ET AL.

STATIC SCATTERING INTENSITY: THE GENERAL RESULT AND THE INAGAKI EQUATION

The static screening intensity for a multicomponent mixture in the limit of zero scattering angle was obtained long ago by Stockmayer" using a purely thermodynamic approach. This was the basic result used by experimentalists to evaluate the X-parameter by light scattering for ternary mixture^.'^-'^ Re- cently a more general expression was obtained within the framework of the random phase approximation,'6-'g which in the case of a ternary Polymer- 1 /Polymer-2/solvent mixture shows that the static screening intensity is proportional to I ( q ) :

where in light scattering, al and a2 are replaced by ( and and in neutron scattering, they represent the difference of scattering length between a monomer and a solvent molecule assuming that they occupy the same volume. The vij are the usual excluded-volume parameters, which can be expressed in terms of the solvent concentration dS and the interaction parameters:

q is the wavenumber and S:(q), S;( q ) are given in terms of polymer concentrations 4i, the degrees of polymerization Ni, and the form factors Pi ( q ) ( i = 1, 2) as follows:

The result in eq. ( 1 ) is general in the sense that one can use it under arbitrary conditions of temperature, concentration, wavenumber, and molecular weight, provided the system remains homogeneous in a disordered state and does not undergo strong fluctua- tions. Letting q = 0 in eq. ( 3 ) gives

Substituting this limit into eq. ( 1 ) gives the following result which is identical with Stockmayer's equation:

Inagaki et al. derived their result by taking the low- concentration limit of eq. (5) and keeping only the terms of orders or less.

lnagaki Equation

It is easy to verify, by expanding eq. (5) and neglecting all terms of orders 43 and higher than that final result can be put in the form:

where we have used eq. (2) to obtain:

The so-called optical theta condition of Inagaki et al. is written:

Substituting the latter condition into eq. ( 6 ) yields the simple Inagaki equation:

The interesting feature of this result is that it does not depend on the polymer-solvent interaction parameters and moreover, it gives direct access to the polymer-polymer X-parameter from the slope of I ( q = O ) / @ as a function of 4. Equation (9) can also be written in a slightly different form by fac- toring the sum of the first two terms and rearranging the equation as follows:

with

ZERO AVERAGE CONTRAST MEASUREMENT OF X PARAMETER 1159

This result is obtained by assuming that monomers 1 and 2 occupy the same volume as a solvent molecule. If this assumption is relaxed, and v l , v2, and us are the volumes per monomer 1, monomer 2 , and a solvent molecule, one obtains the result of Inagaki et al. (see eqs. (11) to (13) of Ref. 1):

with

and the optical theta condition reads:

with a slight but obvious difference in notations. Several ternary mixtures were investigated by this

method. We shall give examples of systems which satisfy the optical theta condition as given by eq. ( l l c ) or eq. (8) together with the values of ( a n / & ) in mL g-' measured with a Brice-Phoenix differ- ential refractometer at wavelength A 0 = 546 nm (see ref. 5 ) (1 denotes the first polymer, 2 the second)

( i ) Compatible mixtures PS/PMVE/s tyrene

(an lac ) , = 0.065

(dn /dc ) z = -0.072

PS / PVME lo-dichlorobenzene

(an /&) , = 0.064

(an /&) , = -0.081

PS / PVME /chlorobenzene

(g)l = 0.09

(g), = -0.052

(i i) Incompatible mixtures PS/PMMA/bromobenzene3

( & / & ) I = 0.048

(an/&), = -0.05

PS / PDMS/toluene

(an/&), = 0.111

(an /ac ) , = -0.094

PMMA/ PDMS / toluene

( a n / d c ) , = 0.016

(an /&) , = -0.094

P S / PVACIstyrene

(an/&), = 0.065

(an/&), = -0.065

PS /PDMS /cyclohexane

(an/&), = 0.168

(an/&), = -0.025

PS / PDMS /chloroform

(an /&) , = 0.157

(an /ac ) , = -0.045

PMMA/PDMS/chloroform

(an/&), = 0.063

PMMA/ PDMS /CC1,

(an lac ) , = 0.142

(an/&), = -0.052

PVME/ PDMS /CC1,

(an /&) , = 0.02

One observes that PDMS has a negative value for the increment of refractive index in several classical good solvents which makes it suitable for such a study. One also notices that several mixtures satisfy a condition which is approximately (dn/ac)l N - ( d n / d ~ ) ~ . This is also particularly useful for the following discussions.

THE SCATTERING INTENSITY FOR A SYMMETRIC MIXTURE UNDER THE ZERO AVERAGE CONTRAST CONDITION

Let us consider a symmetric mixture with the following properties:

1160 BENMOUNA ET AL.

( i ) The two polymers have equal sizes

( i i ) The two polymers have the same thermodynamic properties with respect to the solvent

This means that the solvent quality is the same for both polymers which may present a certain degree of incompatibility defined by x.

(iii) The composition is 50/50 which means that

(1%) x = 1. 2 , 41 = 42 = + / 2

( iv) In this case, the optical theta condition defined by eq. (8) becomes

These four conditions may appear to be very constraining, but the examples of ternary mixtures given above show that it is possible to meet either exactly or approximately condition ( iv) , which is perhaps the most constraining.

Substituting eqs. (12a) to (12d) into eq. (1) one can easily show after some straightforward manipulations that eq. ( 1 ) reduces to the simple form: *'

This result is obtained rigorously under the conditions defined by eqs. (12a) to (12d) and without further assumptions or approximations. In particular, there is no apparent limitation on the concentration, molecular weight, or q values, as long as the mixture is in a disordered state and homogeneous, or in other words, as long as the RPA which was used to derive eq. ( 1) remains valid and sensible. One notes that in the dilute regime the Inagaki equation is recovered [ eqs. (9) and ( 10) ] and in the bulk state, the standard RPA result is ~b ta ined . '~ Therefore eq. ( 13) may be used to evaluate the vari- ation of the X-parameter from the dilute regime to the bulk state.

The Case of a Copolymer-l-2/Solvent

The above analysis can be extended immediately to the case of a copolymer 1-2 in a solvent. The general

result for this system which was obtained under similar conditions as in the derivation of eq. ( 1 ) can be written: l7

If one denotes the composition of monomer 1 by x = N 1 / N ; N = N1 + N2 is the total degree of polymerization, and then Sy, S:, and SY2 are given by

where 4 = + 42 is the total polymer concentration. P1, P2, and P12 are the form factors of block 1, block 2, and the cross form factor due to the interferences between blocks 1 and 2, respectively. The latter is defined in terms of P1, Pa, and the total chain form factor PT by the geometrical relationship

PT = X 2 P 1 + (1 - x ) ' P ~ + 2x( 1 - X ) P ~ Z (16)

Equation ( 14) simplifies greatly when one considers a symmetric 50/50 diblock copolymer. This simpli- fication is due in part to the fact that one needs only two structure factors instead of three if the copolymer is nonsymmetric. This can be seen by noting that

and

Substituting eqs. (17a) and (17b) into eq. (14) and using the zero-average contrast condition which, in this case, is given also by eq. ( 12d), one can easily show, after some manipulations, that I ( q ) has the simple form:

As one can see, this is a general result for a symmetric diblock copolymer in the zero average contrast limit with the thermodynamic properties de-


fined by eq. ( 12b). It is valid for arbitrary concentrations and wavevectors within the limits of the RPA and as long as the system is in a disordered homogeneous state with no micelle formation. This is particularly crucial for the copolymer case since in the low-q limit ( PI,' - PT) tends to zero, making the evaluation of x very difficult if not impossible. x can be deduced from the maximum of I ( q ) since the latter increases with the value of X without shift of its position. This evaluation is classical in bulk systems and well documented in the

The determination of the X-parameter as a function of concentration for symmetrical mixtures and 50/50 diblock copolymers by this method offers a unique possibility of comparing X for these systems and checking if and how the connectivity of the two blocks affects the interaction between the two species. Another advantage of this method is that it can be extended to dynamical scattering techniques such as QELS and NSE methods. This is the subject of the next section.

DYNAMIC SCATTERING FROM TERNARY

CONTRAST CONDITION MIXTURES UNDER THE ZERO-AVERAGE

The dynamic scattering properties of multicomponent polymer mixtures have been a subject of particular interest in recent years, from both theoretical 10~18,24 and e~perimental '~, '~ points of view. In the case of a ternary mixture of two polymers and a solvent, the partial dynamic structure factors are found to relax following a two-exponential function:

where the frequencies I' and I" are the eigenvalues of the first cumulant matrix R, which is defined in terms of the mobility matrix M(q) and the static structure matrix S ( q ) by the standard equation:

To obtain simple, useful results, we limit ourselves to the Rouse model by neglecting the hydrodynamic interaction. This means that our description is valid only in the concentration range where the interaction is screened. This range starts from the overlap concentration, which is indicated approximately in our notation by $*, such that u$* N = 1. Therefore, above this concentration M becomes a diagonal matrix whose elements are independent of g. If we as-

sume that the friction coefficients per monomer are equal

and one has

Combining eqs. (20) and (21b), and assuming that the two polymers have the same sizes [eq. (12a)], yields the following eigenfrequencies:

where A S = S,, Szz - S&, and Do is the single-chain diffusion coefficient

If the mixture is symmetrical, as under the conditions defined earlier by eqs. (12b) and (12c), one obtains the following expressions for the static structure factors [see eq. ( 1 ) ] :

1 $NP[1 + v$NP/2] 2 (234 a, s,, = s2" = -

1 + v + - $NP l - - $ N P ( 2 3 ~ ) " 3 I [ : 1 Substituting eqs. (23a) to (23c) into eq. (22a), one can easily verify that the eigenfrequencies I' with the addition of the quantity under square root, and I" with its subtraction are:

DO P r /q2 = - [ l - x/x,]; x;' = $NP/2 (24a)

and

r'/q2 = Do P [ 1 + ( v + 5 ) BNP]

One notes that r is the frequency which would be obtained in the case of a pure mixture of the two polymers in the absence of solvent. It is referred to as the interdiffusion frequency.10~24~29 The second frequency I" is the one which would be obtained

1162 BENMOUNA ET AL

from a polymer/solvent binary system with an excluded volume slightly enhanced by the interaction parameter as u + x / 2 . This is referred to as the cooperative diffusion frequency. These identifica- tions can be easily understood by considering the interdiffusive and cooperative dynamic structure factors, which under the symmetrical conditions of our systems [ eqs. ( 12) and x = 4 ) and within a constant factor, are defined as:

and

One can easily verify that s I ( q , t ) and ST(q, t ) evolve following the single exponentials:

where S l ( q ) and ST( q ) are obtained as:

and

It is worth noting that if the zero-average contrast condition in eq. ( 12d) is satisfied, the measured dynamic structure factor is directly proportional to SI( q , t ) ; and from the initial decay rate r = T I , one can deduce X. Such measurements can be done as a function of concentration above the overlap $* to make sure that the hydrodynamic interaction can be neglected.

The Case of a Copolymer l-2/Solvent

The above dynamical analysis can be readily extended to the copolymer case. For a copolymer made of two monomer species in a solvent it was shown 10~18

that Sij ( q , t ) decays following a two-exponential function, as in eq. (19). The case of a symmetric diblock copolymer leads to simple results and in particular the eigenfrequencies are:

where the static structure factors Sll = S 2 2 , and S 1 2

= Szl are obtained as:

( 5 3 ' S i 2

and

Substituting eqs. (27b,c) and (27d) into eq. yields

or, similarly,

q2 = PT Do (1 + ( v + :)$NPT) (29b)

Therefore, one obtains similar results as in the ho- mopolymer mixture. Namely, one obtains a struc- tural mode which can be identified as the interdiffusive mode and the other is the cooperative diffusion frequency. The latter is identical to the corresponding frequency in the case of homopolymers [see Eq. (24b) 1. However, one should note that the interdiffusion mode depends on the internal structure of the copolymer chain and hence describes its internal dynamics. In particular, one observes that as q --* 0 r / q 2 diverges since P1/2 - PT = 0 and one finds that r is equal to a constant a t q = 0. Similar behavior, found in charged systems, leads


to the so called plasmon frequency. The point is that the interdiffusive mode with frequency r may be much faster than the cooperative mode depending on the ranges of q's and polymer concentrations. Furthermore, if the zero-average contrast condition is fulfilled the measured dynamic structure factor is directly proportional to the interdiffusion function which relaxes following the single exponential:

with

By measuring the dynamic structure factor S,( q , t ) under the zero-average contrast condition, one can have direct access to r; and hence the interaction parameter can be easily evaluated as a function of the polymer concentration.

Hydrodynamic Effects

We have presented a method for determining the X - parameter from dynamic scattering measurements using a model based on the Rouse approximation, which means that the hydrodynamic interactions are neglected. This is a crude approximation, es- pecially in the dilute range where these interactions may become dominant. However, as the polymer concentration increases, it is known that the interaction strength is significantly reduced by the screening effect, and one introduces a correlation length above which these interactions are completely screened out.28,29 This is particularly true if one uses the light-scattering technique which scans the small wavevector domain, and indeed, scattering data have been quite successfully interpreted using the Rouse limit.25,26 In the case of neutron scattering the wavevector q is larger and the screening of hydrodynamic interactions is less effective since the radiation probes essentially single-chain dynamics in this re- gion of g . Therefore it is important to know how these interactions can be incorporated within the present description. We illustrate this generalization in the particular case of a symmetrical mixture for which r and rI and27*30

For our symmetrical mixture we have

Introducing the hydrodynamic interaction in the mobilities M ( q ) = M l 1 ( q ) = M22(q) and M ' ( q ) = M12 (4 ) = M21 ( q ) via the Oseen tensor description, one obtains: 28

where f ( x ) is given by

Combining these equations with eqs (23) yields:

with

The third term on the right-hand side of eq. (32a) is a somewhat complicated function of x and q , and it has to be evaluated numerically unless P (4 ) is modelled by a simple form such as the Ornstein- Zernicke functioa2' This result also shows that it would be possible to account for hydrodynamic interaction by allowing the parameter x to be q-dependent, which could be useful in order to have a simple interpretation of the scattering data for arbitrary values of q from either QELS or NSE techniques.

CONCLUSIONS

We have discussed several methods which can be used to measure the interaction parameter x as a function of the concentration in the zero average contrast condition. These methods represent exten- sions of the optical theta condition method suggested by Inagaki et al. in several respects. The latter has

1164 BENMOUNA E T AL.

been used only in static light scattering, in the dilute regime at q = 0, and for mixtures of homopolymers. We propose its extension to arbitrary values of the polymer concentration and the scattering wavenumber q . We show how it can be applied to copolymers and present possibilities for complementing the data by dynamic scattering measurements using either quasielastic light scattering or neutron spin echo techniques. Several examples of ternary mixtures for which such an analysis can be implemented using light scattering are given. A few experimental examples are discussed in recent paper^,^',^' and the results of a detailed study of the system PDMS/ PMMA/chloroform by QELS at several concentrations will be reported in the near future.33

M. Benmouna would like to thank the MPI-P for hospi- tality during the period when this work was accomplished. We thank the referee for bringing ref. 9 to our attention and for interesting comments on the effect of hydrodynamic interaction.

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Received August 5, 1991 Accepted February 20, 1992

on the measurement of the χ-parameter under the zero average contrast condition

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