22 C:ommunications in Nonlinear Science & Numerical Simulation 1997

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The Properties of Concentrations in a Polymer Blend’

Yun HUANG & Delu ZHAOt (Department of Physics, Peking University, Beijing 100871, China) t(Institute of Chemistry, Academia Sinica, Beijing 100080, China)

Abstract: For the flexible polymer molecule the concepts of two concentrations, namely, segmental and molecular concentrations have been proposed in this paper. The former is equivalent to the volume fraction. The latter can be defined as the number of the gravity centers of macromolecules, or as the number of the chainheads per unit volume. The two concentrations should be correlated each other by the conformational function of the poly- mer chain and could be used in different thermodynamic equations. On the bases of these concepts it has been proved that the Flory-Huggins mixing entropy should be the result of the mixing “ideal chainhead gases”. The general correlation between the mixing free energy and the scattering function (structural factor) of polymer blends has been studied based on the general thermodynamic fluctuation theory. When the Flory-Huggins mixing free energy was adopted, de Gennes scattering function of polymer blend can be derivated. Keywords: segmental concentration, molecular concentration, mixing free energy, scat- tering function

1. Introduction

A fundamental feature of polymer systems is the connectivity of segments consisting the macromolecules. It brings an important influence on the thermodynamic properties of polymer solution. Flory and Huggins have proposed a formula of the mixing free energy of nolvmer blendsl’l

where lc is the Boltzmann constant, T is the absolute temperature, @I, es, Nr and N2 are the volume fractions and the chain lengths for the two species components in a polymer blend. x is the Flory-Huggins interaction parameter. But this formula is subjected many

‘The paper was received on Jan. lOth, 1997

HUANG, et al.: The Properties of Concentrations in a Polymer Blend . . . 23

critisms due to the mean field assumption, and some improved theories are proposed[2-61. Therefore, a important task of the theoretical studies is to overcome the frame of the mean field method of molecules in other statistical mechanics. In this work we try to propose a new physical concept in a polymer system.

2. Segmental and molecular concentration in a polymer blend

As well known, for a small molecule with no internal structure, only a transitional free- dom degree have been considered. But for a complex molecules which free energy will com- prise a transitional part and an internal part. For a polymer blend, the two thermodynamic variables are necessary: one describes the transitional freedom degree of the gravity centre of the macromolecule, and one is conformational freedom degree describing the flexibility of a polymer chain. Which are called segmental and conformational concentrations.The segmental concentration can be defined as the number of the segments in the unit volume. For the convenience the segmental volume is taken as the unit, so the segmental is exactly equivalent to the volume fractions @I and Qr2 for polymer components 1 and 2 in a blend respectively.

Then a molecular concentration will be introduced. Generally the gravity center of a macromolecule can be taken as the position of this molecule. However, it is most con- venient for analytical and simulating calculation that, equivalently, the first segment of macromolecule, which is called as chainhead, can be used as the molecular position too. We regard that the distribution of chainhead per unit volume can represent a molecular concentration in polymer blend, which we denoted as Cl and CZ respectively. Since the chainhead itself is a segment, so the molecular concentrations (chainhead concentration) is a continual function of spatial position as same as segmental concentration.

The both of t,wo concentrations are very important. In fact, in a different thermody- namic equation or phenomena only one concentration can be used and can’t be the other. For example, in the scattering experiments, the segments will act as the scattering center, therefore the scattering function should be the average square fluctuation of the segment concentration; the impressibility condition which represented the exclusive volume effect should be satisfied by the segment concentration too, the mixing heat is proportional to the contact number between the segments of the two component polymers, it should be proportional to the product of the two local segment concentrations. In other hand, poly- mer binary blend is a mixture of two groups of different macromolecule and does not be a mixture of two groups of dispersive segments, therefore the mixing entropy and free energy should be a function of the molecular concentrations of the two polymer components; the spinodal decomposition is a phase separation behavior of two polymers, therefore it should be described by t,he molecular concentration too. So, we should carefully distinguish the kind of the concentration in the thermodynamic equation.

3. Relation Between Segmental and Molecular Concentrations

The relation between the average segmental and the molecular concentrations of the solution is very simple: - - - -

a1 = NlCl, a2 = N2C2, (2)

The two concentrations should be correlated by means of a conformational function G(r, R) of macromolecule chain, which is at position T and it’s chainhead being at position R.

91 (r) = J Cl (R)Gl (r, R)dRd

J (3)

a2(r) = Cz(R)G2(r, R)dRd

24 C:ommunications in Nonlinear Science & Numerical Simulation 1997

The integral should be performed over the whole space. Using those physics concepts, the mixing entropy per unit volume of a polymer blend is

Smis = -k[C1 log(C1N) + c2 log(C2~2)l (4)

where Cl and C2 are the molecular concentrations of the two components respectively. According to Flory’s theory, the local mixing heat can be regard as proportional to the product of G1 and &J. The mixing heat of the solution should be written as

Hmis - - = x@‘1@2 kT

Using Eq.(4) and Eq.(5) one can obtain the mixing free energy

- = Cl log(C1N1) + c2 log(C2N2) + x91+2 kT

It is emphasized that in Eq.(6) one is ignored the effect of the conformational entropy in the mixing process and considering only the part of the transitional entropy. As will show later, neglecting the contribution of the conformational entropy change to the mixing free energy is the shortcoming of the Flory’s mean field assume.

4. Structural Factor and de Gennes formula

A scattering function (structural factor) of a binary polymer blend is important for theoretical and experimental studies for a polymer system. we could write it from its mixing free energy formula. Since the mixing heat of a polymer blend should be a function of the segmental concentrations of the two components, and the mixing entropy should be a function of the molecular concentrations, the general form of the mixing heat and entropy should be as follows:

Hmiz = Hmis(%, +2), Liz = Smi,(G, C2) (7)

The structural factor could write as the Fourier components of the square mean fluctuation of the concentration[71. The fluctuation probability shall be proportional to the following quantity

W 0: exp[-(AH - TAS)/kT] 63)

where AS and AH are the deviation of the local concentration of the mixing entropy and the mixing heat from their mean (equilibrium) values. Expanding them in the power series and keeping to the second order, we have

(11)

HUANG, et al.: The Properties of Concentrations in a Polymer Blend . . . 25

are the differences of the concentrations from the mean values. Expanding them in the Fourier series and considering the two kinds of fluctuations in polymer blends. They are the fluctuation of molecular concentration, reflecting the random variation of spatial distribution of the chainheads, and the conformational fluctuation, reflecting the random variation in the shapes of a polymer coil. We have

AH-TAS 1 kT =ii C[

1 1

Nl‘hS~q + N2@2S2q - 2xlW,

9 Substituting it int,o Eq.(8), one get

Wocexp[-~(N is + 1

1 1 lq N&&q - 2xbw,l

(12)

(13)

2 The mean square fluctuation 6a1, should be the reciprocal of the term in the bracket, one has

1 1 1

- = Nl%Sl(q) + N&&(q) wq - 2%

Then the de Gennes scattering function g(q) for a polymer blend has written as

1 1 -=- S(Q) mq,

and 1 1 1

a= NA4, + NAS2, - 2x

The advantage of our derivation is direct connection with the general thermodynamic fluc- tuation theory. Acknowledgments This work is supported by the Chinese National Key Projects for findamental Research “Macromolecular Condensed State”, the State Science and Technol- ogy commission of China. The work of Y.H. is also supported by “Science Foundation of Peking University”.

References

[l] P.J.Flory, Principles of Polymer Chemistry, Cornell Univ. F’ress, Ithaca, New York, 1953 [2] J.M.Lu and Y.L.Yang, Science in China, A.3, 2002, 1988 [3] A.Sariban and K.Binder, Macromolecules, 21, 711, 1989 [4] J.Dudowitz,K.F.Fleed and W.G.Madder, J. Chem. Phys., 23, 4803, 1990 [5] K.S.Schweiger and J.G.Curro, J. Chem. Phys., 91, 589, 1989 [6] A.I.Pesci and K.F.Fleed, J. Chem. Phys., 90, 2003, 1989 [7] L.D.Landau and E.M.Lifschitz, Statistical Physics, Pergamon, Oxford, 1986