Density Functional Approach to the Molecular Theory of Rod-Coil Diblock Copolymers

Abstract

The general density functional approach is used to develop a molecular-statistical theory of rod-coil diblock copolymers which is valid in the case of both weak and strong segregation. The free energy of the rod-coil copolymer is expressed as a functional of the number densities of rod and coil monomers which depend on the translational and orientational degrees of freedom. The equilibrium densities are determined by minimization of the free energy functional and depend on the orientational and translational order parameters of the monomers. The order parameters are calculated numerically by minimization of the free energy taking into account the incompressibility condition within the formalism of Lagrange multipliers. Phase diagrams are obtained and the profiles of orientational and translational order parameters are presented as functions of temperature and the fraction of coil fragments. It is shown that the lamellar phase possesses strong orientational order and the stability of the phase is increasing with the increasing fraction of rod monomers.

Self Consistent Equations for the Order Parameters

The phase diagram in Fig. 4 has been calculated by direct substitution of the incompressibility condition into the free energy (51) and subsequent minimization with respect to the order parameters $\sigma$ and $S$ and the single translational order parameter $\psi$:

$$\psi = {{\psi }_{r}} = - \frac{{{{f}_{c}}}}{{{{f}_{r}}}}{{\psi }_{c}}.$$

(87)

This minimization yields the following expressions for the order parameters ψ and σ:

$$\begin{gathered} \tilde {\psi } = {{\psi }_{r}}\frac{{C_{{rr}}^{{(2)}}(C_{{rr}}^{{(2)}} - 2C_{{rc}}^{{(2)}}) - 4\beta {{J}_{2}}(C_{{rr}}^{{(0)}} - C_{{rc}}^{{(0)}} + \chi )}}{{{{{(C_{{rr}}^{{(2)}} - 2C_{{rc}}^{{(2)}})}}^{2}} - 4\beta {{J}_{2}}(C_{{rr}}^{{(0)}} + C_{{cc}}^{{(0)}} - 2C_{{rc}}^{{(0)}} + 2\chi )}} \\ \, + {{\psi }_{c}}\frac{{{{f}_{c}}}}{{{{f}_{r}}}}\frac{{2C_{{rc}}^{{(2)}}(C_{{rr}}^{{(2)}} - 2C_{{rc}}^{{(2)}}) + 4\beta {{J}_{2}}(C_{{cc}}^{{(0)}} - C_{{rc}}^{{(0)}} + \chi )}}{{{{{(C_{{rr}}^{{(2)}} - 2C_{{rc}}^{{(2)}})}}^{2}} - 4\beta {{J}_{2}}(C_{{rr}}^{{(0)}} + C_{{cc}}^{{(0)}} - 2C_{{rc}}^{{(0)}} + 2\chi )}}, \\ \end{gathered} $$

(88)

and

$$\tilde {\sigma } = \sigma - \left( {{{\psi }_{r}} + \frac{{{{f}_{c}}}}{{{{f}_{r}}}}{{\psi }_{c}}} \right)\frac{{4C_{{cr}}^{{(2)}}(C_{{rr}}^{{(0)}} - C_{{rc}}^{{(0)}} + \chi ) + 2C_{{rr}}^{{(2)}}(C_{{cc}}^{{(0)}} - C_{{rc}}^{{(0)}} + \chi )}}{{{{{(C_{{rr}}^{{(2)}} - 2C_{{rc}}^{{(2)}})}}^{2}} - 4\beta {{J}_{2}}(C_{{rr}}^{{(0)}} + C_{{cc}}^{{(0)}} - 2C_{{rc}}^{{(0)}} + 2\chi )}},$$

(89)

where the order parameters ${{\psi }_{r}},{{\psi }_{c}}$ and σ are defined by standard equations (39)–(41) as averages of the corresponding microscopic quantities with the one-particle distribution functions. One can readily see that the self-consistency of the theory is broken because it requires $\tilde {\psi } = {{\psi }_{r}} = - ({{f}_{c}}{\text{/}}{{f}_{r}}){{\psi }_{c}}$ and $\tilde {\sigma } = \sigma$ which does not follow from these equations. As a result of this inconsistency the order parameter can exceed, for example, the unit value, as can be seen in Fig. 5.

In contrast, it can be shown that the minimization of the free energy (83) with respect to the order parameters ${{\psi }_{r}},{{\psi }_{c}},\sigma$ and S taking into account the equation for λ yields the self-consistent equations for all order parameters which directly correspond to their general definitions given by Eqs. (39)–(41).