Evaluation of the Polymer-Polymer Interaction Parameter from the

31 Evaluation of the Polymer-Polymer Interaction Parameter from the Equation of State Thermodynamic Theory

Downloaded by UNIV LAVAL on July 14, 2016 | http://pubs.acs.org Publication Date: June 1, 1979 | doi: 10.1021/ba-1979-0176.ch031

RYONG-JOON ROE Department of Materials Science and Metallurgical Engineering, University of Cincinnati, Cincinnati, OH 45221

The strength of interaction between dissimilar components greatly influence the morphology of microdomains in poly mer blends and block copolymers. In the theories of poly mer-polymer interfaces in multicomponent systems, the interaction between components usually is expressed in terms of a single parameter independent of concentration. The validity of such an approximation is examined here in the light of the equation-of-state thermodynamic theory of polymer solutions and mixtures developed by Flory and co-workers. Using the parameters evaluated by these work ers, we have computed the free energy of mixing for the pairs, polyethylene/polyisobutylene, polyisobutylene/poly styrene, and natural rubber/polystyrene. The results show that the polymer-polymer interaction parameter Λonly moderately depends on concentration and temperature.

H p h e morphology and properties of polymer-polymer blends and block copolymers depend to a large measure on the strength of interaction between the two, chemically different components i n the system. T h e degree of separation into distinct phases, the thickness of the interface between them, and the size and shape of the block copolymer domains are some of the important features which are governed largely b y the polymer-polymer interaction. There are now available a number of theoretical treatments (1-11 ) dealing with the problem of polymer compatibility and the polymerpolymer interfaces either i n polymer blends or i n block copolymers. 0-8412-0457-8/79/33-176-599$05.0070 © 1979 American Chemical Society Cooper and Estes; Multiphase Polymers Advances in Chemistry; American Chemical Society: Washington, DC, 1979.


600

MULTIPHASE

POLYMERS

These theories differ from each other greatly i n the way they evaluate the entropie effect of domain formation, i.e., the restrictions to the polymer chain conformations imposed b y the presence of the interface. However, most of them share a common approximation i n which the energetic interaction between two dissimilar polymer segments is repre sented by a single parameter, usually denoted by χ, which is assumed to be a function of temperature only for a given polymer-polymer pair. The χ parameter was originally introduced i n the polymer solution thermodynamics to represent the van Laar heat of interaction. It was soon realized, both experimentally and theoretically, that the value of χ is not a constant for a given polymer-solvent pair but depends on the concentration (and to some extent on the molecular weight) of the polymer, and also that a sizeable contribution of an entropie nature to χ has to be admitted i n order to account for its temperature dependence. When attempting to use these polymer interface theories (1-11) i n planning and interpreting experiments, we encounter two problems asso ciated with the interaction parameter i n polymer-polymer systems. First we have to find a way of determining the values of the interaction pararneters for the polymer pairs of interest. There is as yet no general experimental procedure which allows us evaluation of the χ parameter. Without reliable values of the χ parameter, comparative tests of the competing theories can only be made qualitatively. Secondly, to be able to use these theories with more confidence w e have to have some idea about the dependence of the χ parameter on concentration, tempera ture, etc. The equation-of-state thermodynamics, recently developed by Flory and his co-workers (12,13,14), has achieved a considerable improve ment i n treating the thermodynamics of polymer solutions over the classical F l o r y - H u g g i n s type solution theory. Whereas i n the latter the volume change on mixing is ignored, the newer theory takes into account the equation-of-state contribution, or the contribution b y the change i n free volume on mixing, to the free energy of mixing. In this work we evaluate the polymer-polymer interaction parameters for a few polymer pairs of interest by closely following the method of calculation suggested by the Flory equation-of-state thermodynamics and using the values of molecular parameters of component polymers evaluated by them. The results are examined to gain insight into the dependence of the interaction parameter on various physical parameters of the system such as the concentration and temperature. Evaluation of the Polymer—Polymer Interaction Parameter The strength of polymer-polymer interaction is represented here by a new parameter Λ instead of the usual χ. The "polymer-polymer inter-

Cooper and Estes; Multiphase Polymers Advances in Chemistry; American Chemical Society: Washington, DC, 1979.

31.

ROE

601

Polymer-Polymer Interaction Parameter

action parameter" Λ is defined by the following equation: AG

M

— RT

(^y- φ! In φχ +

γ-

φ lïi 2

φ ^ + Λ φι φ 2

(1)

2

where AG is the free energy of mixing per unit volume of the mixture, Vi and V are the molar volumes of the polymers 1 and 2, and φι and φ are the volume fractions of the two polymers in the mixture. The first term represents the usual combinatorial entropy of mixing and therefore the term containing Λ includes all other contributions to the free energy of mixing. Λ thus includes the enthalpic interaction between the two components but also the effect of entropy changes unaccounted for by the simple combinatorial term. It has the dimension of energy per unit volume. W h e n both Λ and χ are independent of concentration, Λ is equal to x R T / V . W h e n χ varies with concentration the relation between Λ and χ is a little more complex because χ i n polymer solution theories is usually defined i n terms of the excess chemical potential rather than the excess free energy of mixing given i n Equation 1. The polymer-polymer interaction parameters appearing i n various theories of interfaces (1-11 ) are usually defined with regard to the local free energy of mixing, and therefore the definition of Λ given in Equation 1 is more appropriate than the one given i n terms of the excess chemical potential. There are other advantages of the above definition of Λ , one of them being the fact that the values of χ depend on the assumed volume of a polymer segment ( usually taken equal to the volume of the solvent molecule V ) , while Λ is defined per unit volume of the mixture. Another advantage is that the values of Λ can be related approximately to the solubility parameters of the components δι and δ by: M


2

2

2

2

2

Λ —

(8i

-

δ )

2

2

(2)

In the Flory equation-of-state theory, the thermodynamic properties of a pure component liquid ( either monomelic or polymeric ) are repre sented completely by means of three characteristic constants, V i * , ΤΊ*, and p i * , which can be evaluated from the P - V - Γ and other thermo dynamic properties of the liquid. The equation-of-state thermodynamics then recognizes three types of contributions to the free energy of mixing two liquid components. The first is the one attributable to the combina torial entropy of mixing as expressed by the first term i n Equation 1. The second is the change i n enthalpy, ΔΗ , c o n t a c t , arising from creation of 1, 2 nearest neighbor contacts on mixing. The third is the free volume (or the equation-of-state) contribution resulting from the change i n Μ


602

MULTIPHASE

POLYMEiiS

volume on mixing. This latter contribution contains both enthalpic and entropie components, Δ Η , u and — T A S , f v The quantities, expressed per unit volume of the mixture, are given b y : M

M

Δ # Μ , contact =

X\2 #2 l/v

(3)

-v- )/^

(4)

2

*Hu.u = Σ

1

t=l,2 -TAS . m

+* Κ* ^

tr — Σ, S»'

1

X In [ W'

-

3

1) /

-

1) ]

(5)


i=l,2

where U i = ϋι/ϋι*, Γχ = Τ /Τι*, and the unsubscripted quantities refer to the mixture. To evaluate the expressions 3, 4, and 5 one needs, i n addition to the characteristic parameters v*, p * T* for each component, the value of X i 2 denoting the energy change on contact of 1, 2 polymer segments and the surface fraction $ defined b y : 1

{

y

2

=

( 6 )

+ 2

Φι (Si/s ) 2

where Si/s denotes the ratio of surface areas of the two types of segments ( occupying equal volumes ). The derivation of the expressions 3, 4, and 5 and more precise meaning of the various symbols can be found i n the original papers (12,13,14) by Flory and co-workers. In view of 3, 4, and 5, Λ defined i n Equation 1 can be expressed as: 2

Λ = Aj

lf

contact

-f-

v

(7)

c o n t a c t / φ ΐ Φ2

(8)

A h , f ~h A v

8 t

f

where A h , contact =

A/J

M (

A ,fv =

Α//

Μ >

h

, /φχ φ ν

(9)

2

and Λ , fv = 8

— TAS

M
0.5

25 100 150

0.30 0.28 0.26

0.12 0.11 0.11

-0.03 -0.04 -0.04

0.21 0.20 0.19

P I B / P S at wi = 0.5

0 25 100

6.53 6.39 6.00

6.52 6.38 5.99

1.02 1.10 1.31

-1.01 -1.08 -1.30

A/t

#

ΛΑ,/„

contact

be fairly independent of temperature, i n view of the dominant contribu tion to Λ by the contact enthalpy term as discussed above. The fact that Λ is positive and fairly independent of temperature for these three pairs of polymers means that as the temperature is raised, their degree of incompatibility w i l l diminish and the interfacial boundary between phases w i l l become more diffuse. I n the case of block copolymer systems the constraint imposed by the requirement that the block joints be at the domain boundaries provides additional free energy term to make the mixing more favorable. Thus, when the block lengths are fairly moderate, the first term in Equation 1 w i l l become sufficiently large and negative to render the two blocks thermodynamically miscible. Evidence of such homogenization of a styrene-butadiene-styrene block copolymers at increased temperature has been reported from rheological studies (20,21). There are currently known (22) several polymer pairs, such as P S / poly (vinyl methyl ether) (23, 24), which appear to be truly compatible thermodynamically. In most of these the compatibility probably arises from specific interactions other than dispersion forces which make Λ for the pair negative. The increase i n temperature generally w i l l weaken the specific interaction, and the increasingly unfavorable free volume effect (25) w i l l eventually make Λ positive. Thus these compatible pairs w i l l in general exhibit a lower critical solution temperature phenomenon on heating. In contrast, the three polymer pairs discussed i n this work are, at room temperature, all below their upper critical solution tem perature. The thermodynamic properties of block copolymers formed from these polymer pairs can therefore be discussed in terms of a constant polymer-polymer interaction parameter Λ without referring to the possible complications arising from the free volume effect. In this vein it seems unlikely that a styrene/a-methylstyrene block copolymer (26) w i l l exhibit enhanced domain segregation on raising the temperature. It is gratifying to see that the value of Λ for N R / P S given i n Table I is i n excellent agreement with 0.8 cal/cm which was estimated b y Meier (4) for polybutadiene/PS from the solubility parameter difference. It is, however, doubtful that the present approach of using the equation-of3


606

MULTIPHASE

POLYMERS

state data can ever be a practical means of evaluating Λ. A reliable value of X for a polymer-polymer pair can be estimated only when data on a number of similar polymer-solvent systems are available. There is also a theoretical difficulty arising from the unsymmetric definition of X12, so that X i 7 ^ X12. I n the treatment of polymer-solvent systems, an additional entropie term containing a parameter Q was often introduced to achieve good agreement between theory and experiment, b u t i t is difficult to estimate the error arising from omission of this term for polymer-polymer systems. Because of these uncertainties, equations 1, 2, 3, 4, and 5 may not be relied upon as a means of quantitative evaluation of A until more data for other polymer-solvent systems become available. T h e equation-ofstate thermodynamics is, however, useful in its ability to give us insight into the physical factors and their relative magnitudes which contribute to the polymer-polymer interaction parameter. The results i n this work clearly show that the dependence of A on concentration and temperature is moderate. This gives a justification as a good approximation to the use of a constant polymer-polymer interaction parameter i n the polymer interface theories where the polymer concentration encompasses the whole range w% = 0 to 1 across the phase boundary. 12

2


12

Acknowledgment This research was supported by the Office of Naval Research. Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

Roe, R. J., J. Chem. Phys. (1974) 60, 4192. Roe, R. J., J. Chem. Phys. (1975) 62, 490. Meier, D. J., J. Polym. Sci. (1969) 26(C), 81. Meier, D. J., "Block and Graft Copolymers," p. 105, Burke and Weiss, Ed., Syracuse University, Syracust, NY, 1973. Meier, D. J., Polym. Prepr. Am. Chem. Soc., Div. Polym. Chem. (1974) 15, 171. Helfand, E., Tagami, Y., J. Chem. Phys. (1971) 56, 3592. Helfand, E., Wasserman, Z. R., Macromolecules (1976) 9, 879. Nose, T., Polym. J. (1975) 8, 96. Krause, S., Macromolecules (1970) 3, 84. Boehm, R. E., Krigbaum, W. R., J. Polym. Sci. (1976) 54C, 153. Leary, D. F., Willimas, M. C., J. Polym. Sci., Polym. Phys. Ed. (1973) 11, 345. Flory, P. J., J. Am. Chem. Soc. (1965) 87, 1833. Eichinger, Β. E., Flory, P. J., Trans. Faraday Soc. (1968) 64, 2035. Flory, P. J., Discuss. Faraday Soc. (1970) 49, 7. Flory, P. J., Eichinger, B. E., Orwoll, R. Α., Macromolecules (1968) 1, 287. Eichinger, Β. E., Flory, P. J., Trans. Faraday Soc. (1968) 64, 2053. Abe, Α., Flory, P. J., J. Am. Chem. Soc. (1965) 87, 1838. Hocker, H., Shih, H., Flory, P. J., Trans. Faraday Soc. (1971) 67, 2277. Eichinger, Β. E., Flory, P. J., Trans. Faraday Soc. (1968) 64, 2061.


31. ROE Polymer-Polymer Interaction Parameter

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20. Chung, C. I., Gale, J. C., J. Polym. Sci. (1976) 14(A-2), 1149. 21. Gouinlock, Ε. V., Porter, R. S., Polym. Prepr. Am. Chem. Soc., Div. Polym. Chem. (1977) 18(1), 245. 22. Krause, S.,J.Macromol. Sci., Rev. Macromol. Chem. (1972) 7, 251. 23. Nishi, T., Kwei, T. K., Polymer (1975) 16, 285. 24. Robard, Α., Patterson, D., Macromolecules (1977) 10, 1021. 25. McMaster, L. P., Macromolecules (1973) 6, 760. 26. Robeson, L. M., Matzner, M., Fetters, L. J., McGrath, J. E., "Recent Ad vances in Polymer Blends, Grafts and Blocks," p. 281, L . H . Sperling, Ed., Plenum, New York, 1974. December 14, 1978.


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Evaluation of the Polymer-Polymer Interaction Parameter from the

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