Background and Theoretical Formalism

we write down the free energy of a three-component mixture of water (W), diblock surfactant (D) and particle (P). Each diblock surfactant consists of a hydrophilic (H) block and lipophilic (L) block. Following the TGMB mean-field formalism, we write  

The particles are, in a sense, described by three interrelated densities center probability function, pp(r) local volume fraction, ( )p(r) and the smoothed density function, pp(r). The relationship between these three functions is as follows  

Note that the spatial integration is in dimensionless coordinates scaled by the diblock radius of gyration, Rgi, = a(Nc/6) In introducing this scaling, we follow the formaUsm of Drolet and Fredrickson [97], whose real-space method of free energy minimization is adopted both in the TGMB report and in this study. The power exponent d in Equation 10.4b is space dimensionality, and 0 is the Heaviside step function (Q(x) = 1 if % 0 and 0 otherwise). 

the propagator q(T, s) (where 0 s 1 is the index denoting the position along the diblock chain) and its counterpart qf(r, s) are given by the modified diffusion equations  

t(s) = Lifs f, and H otherwise. Equations 10.8a and 10.8b must be solved subject to boundary conditions q t, 0) = q j i. 1) = 1-Equation 10.1 determines the overall free energy of the system as a function of the local densities [( )a(r) for nonparticle species and pp(r) for the particles], local chemical potentials wjj), and the incompressibility constraint field (r). Minimization of Equation 10.1 with respect to all of these functions results in a set of self-consistency equations these equations are then solved on a lattice using an iterative procedure until the specified convergence is achieved. The self-consistency equations are as follows  

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