7 EQUATION OF STATE AND STRESS TENSOR IN INHOMOGENEOUS COMPRESSIBLE COPOLYMER MELTS This chapter was previously published as: 'Equation of state and stress tensor in inhomogeneous compressible copolymer melts. Dynamic mean- eld density functional approach'. N.M. Maurits, A.V. Zvelindovsky, J.G.E.M. Fraaije, Journal of Chemical Physics, 108 6, 1998, p. 2638-2650. 7.1 Summary We have derived an expression for the global stress in inhomogeneous complex cop olymer liquids. We apply the principle of virtual work to the free energy as de ned in the dynamic mean- eld density functional metho d. This metho d automatically provides the full stress tensor devi- atoric and isotropic parts and hence an equation of state for inhomoge- neous compressible cop olymer melts. The excluded volume interactions and cohesive interactions b etween chains have b een explicitly taken into account. Therefore the expressions for the stress and thermo dynamic pressure have a wide range of validity. The connectivity of the chains is automatically accounted for and the free energy adapts very well to changes in the molecule prop erties. In the limiting case of homogeneous systems it simpli es to known results. In order to study rheological prop- erties of cop olymer melts and npT -ensemble simulations, the pressure and stress comp onents have to be calculated at any given moment in time. Weshowhow the pressure and stress can b e numerically evaluated during simulations using a Green propagator algorithm, instead of having to cal- culate the time-dep endent con guration distribution function explicitly from a Smoluchowski equation. We provide illustrativenumerical results that indicate how the pressure changes during microphase separation. STRESS TENSOR 105 7.2 Intro duction General The dynamic mean- eld density functional metho d is a metho d for nu- merically calculating the phase separation dynamics of cop olymer liquids in 3D. It predicts morphologies and has recently been successfully used 34 to predict part of a Pluronic/water phase diagram. The metho d is based up on a generalized time-dep endent Ginzburg-Landau theory for 19, 21, 22 conserved order parameter of the following general form: Z Z X @ r I D r; r r dr = IJ 1 J 1 1 @t V J =1 Z Z X D r; r IJ 1 1 dr + r;t ; 7.1 1 I r V J 1 J =1 D r; r = r r; r r ; 7.2 IJ 1 r IJ 1 r 1 with particle concentration elds r I = 1;:::;Z, Onsager kinetic I co ecients , intrinsic chemical p otentials F = r F is the IJ I I 1 free energy, = k T and noise elds r;t. The noise has a Gaus- B I sian distribution with moments dictated by a uctuation-dissipation the- 22{ 24 orem. The goal of mesoscopic mo deling is to obtain a theory of ordering phe- nomena in p olymer liquids, based on a 'molecular' description. We use a free energy functional for a collection of Gaussian chains in a mean- 37, 132 eld environment. In this approach we try to retain as much as p ossible of the underlying molecular detail, such as the architecture and comp osition of the chain molecules. To this end, we do not use a phe- nomenological expansion of the free energy in the order parameters, as 16{21 is commonly done in Ginzburg-Landau mo dels, but de ne an intrin- sic free energy functional based on a density distribution function that 28 results from a minimization criterion. This intro duces external p oten- tials as Lagrange multiplier elds. The density particle concentration elds and external p otential elds are coupled one-to-one through the density functional. In this approach the free energy immediately re ects changes in the molecular prop erties which enables practical application of the metho d. Some results of numerical calculations of phase separa- tion in incompressible blo ck cop olymer melts and concentrated surfac- tant solutions were already discussed in Refs. 28 and 82. The metho d 109 has been extended to include compressible melts and nonlo cal kinetic 133 co ecients. In this pap er we have derived an expression for the global stress in inhomogeneous complex cop olymer liquids which, in the limiting case of homogeneous systems, is analogous to the expression for the stress tensor 37 that was derived by Doi and Edwards. This expression was obtained from a con gurational average of microscopic stresses where the distri- bution function results from a Smoluchowski equation in the presence of homogeneous ow. To this end, we employ the principle of virtual work 106 STRESS TENSOR on the free energy as de ned in the dynamic mean- eld density functional metho d. This metho d provides an expression for the total stress tensor and hence from its trace an expression for the thermo dynamic pressure, i.e. an equation of state for inhomogeneous cop olymer melts. Before we explain the details of the derivation, we rst discuss previous studies on equations of state and stress tensors for inhomogeneous cop olymer melts. Equations of state Equations of state for chain molecules are usually limited to homogeneous systems and often contain severe simpli cations regarding connectivity. In Ref. 83, the connectivity is taken into accountbyallowing only a lim- ited numb er of degrees of freedom less than 3 for chain elements, by use of the so-called Prigogine parameter c. In this cell mo del, the Prigogine parameter determines the expression for the free volume p er chain. Sev- eral authors have tried to improve the cell mo del by including geometry 134 factors in the free volume expression, by using di erent expressions 135 for the mean p otential energy per mer or by including the p ossibility 87 of empty sites. The lattice- uid mo del develop ed in Ref. 90 that puts chains on a lattice in order to b e able to calculate the partition function, is improved in Ref. 92 by including molecular shap e di erences and in Ref. 91 by including entropy contributions. 136, 137 The lattice theories of Flory and Huggins are generalized to continuous space in Ref. 95. The complete disregard for chain structure is rst improved in Ref. 138 by relating the compressibility factor of an n-mer uid to compressibility factors of monomer and dimer uids at the same volume fraction. In Ref. 139 the theory in Ref. 138 is further improved by including structural information for a diatomic uid, us- ing site-site correlation functions at contact, which results in a mo di ed thermo dynamic p erturbation theory. These studies are all restricted to athermal homogeneous melts. Monte Carlo simulations of lattice mo dels for p olymer melts are em- ployed in Refs. 140 and 141 to measure the chemical p otential by a particle-insertion metho d. This enables calculation of the osmotic pres- sure by thermo dynamic integration. Stress tensor Expressions for the stress in a visco elastic material usually serve the 132, 142 purp ose of analyzing theoretical and exp erimental results. Many authors study stress-strain relationships in order to access the rheological prop erties of the material. The metho ds that are used to derive expres- sions for the stress given a Hamiltonian date back to the sixties and seventies. A few di erent approaches can b e discerned. STRESS TENSOR 107 In Ref. 21 sto chastic equations are given for low molecular weight uids. These equations or other hydro dynamic equations valid for u- 143 ids in a two-phase region can b e combined with the uid momentum balance equation. This allows for a relationship to be derived between the Hamiltonian of the system and the divergence of the stress tensor. If the Hamiltonian is a spatial average of a lo cal function, a compatible 143{ 150 lo cal stress tensor can be derived which is unique up to a diver- gence free eld. The lo cal stress tensor can be spatially averaged to 145, 151, 152 obtain an expression for the average shear stress. Similar ex- pressions are also used in inhomogeneous systems as a measure of the 147, 153, 154 spatial anisotropy of domains distorted by shear ow. The av- erage shear stress may also b e expressed in terms of the nonequilibrium 147, 151, 155 structure factor. The total stress of the material is often com- p osed of this 'anisotropy' factor, a separate viscous contribution and a 146, 147, 153, 154, 156, 157 hydro dynamic pressure contribution. Another approach to visco elasticity is to consider the visco elastic p olymeric material as a collection of particles susp ended in a viscous solvent. The stress is then comp osed of the usual viscous solvent part and a microscopic contribution due to the forces b etween interacting par- 37,126,158,159 ticles. A more thermo dynamic approachistaken in Ref. 160 for solutions of ro dlike p olymers, in Ref. 155 for homogeneous p olymer melts, in Ref. 161 for liquid crystals and in Ref. 162 for critical liquids. Here the principle of virtual work is applied to nd the stress tensor by calculating the elastic resp onse of a material to a small deformation. 37 In this pap er we generalize the expression for the stress tensor to the case of compressible inhomogeneous complex cop olymer liquids. The dynamical free energy A as considered in Ref. 37 accounts for entropy e ects of ideal b eads and energy e ects due to the p otential eld H : Z A dR k T ln + H : 7.3 B Here, is the distribution function of the b eads.