Network Model of the Disordered Phase in Symmetric Diblock Copolymer Melts

We present a model for the order-disorder transition of symmetric $A\text{−}B$ diblock copolymer melts in which the disordered phase is treated as a bicontinuous network, and in which self-consistent field predictions of properties of an analogous ordered network are used to estimate some properties. Such a model is shown to accurately predict the latent heat of this transition. The dependence of the location of the transition upon the invariant degree of polymerization $\overline{N}$ is shown to be consistent with a simple hypothesis that the disordered bicontinuous structure is stabilized relative to an analogous ordered network by a nearly constant entropy per network junction.

Figure 1

Plot of $\mathrm{\Delta }{g}_{\mathrm{SCF}}$ vs $\chi N$ for the single gyroid, double gyroid, and the disordered phases, where $\mathrm{\Delta }{g}_{\mathrm{SCF}}$ is the difference between the free energy per chain and that in the lamellar phase, divided by $k_{B}T$. Inset: a cubic unit cell of the SG phase.

Figure 2

Fractional change in $g^{\prime }$ at the ODT vs $\chi N$, from MD simulations (symbols) and SCFT (line). Symbols show values of $\mathrm{\Delta }{g}^{\prime }/g_L^{\prime }$ from MD simulations vs the observed value $(\chi N{)}_{\mathrm{ODT}}$. Different symbols indicate results of different simulation models, indicated by the labels H, S1, S2, and S3 used in Ref. [5]. For the SCFT prediction (line), $\mathrm{\Delta }{g}^{\prime }(\chi N)=g_{\mathrm{SG}}^{\prime }(\chi N)-g_L^{\prime }(\chi N)$.

Figure 3

SCFT prediction $\mathrm{\Delta }{G}_{\mathrm{SCF}}(\chi N,\overline{N})$ of the difference between the free energy of the SG and $L$ phases, per junction of the SG phase and normalized by $k_{B}T$, measured at the value $\chi N=(\chi N{)}_{\mathrm{ODT}}(\overline{N})$ observed in simulations. The horizontal dashed line shows a best fit value of $\mathrm{\Delta }{G}_{\mathrm{SCF}}=2.15$

Figure 4

Comparison of measured and predicted values of $(\chi N{)}_{\mathrm{ODT}}$ vs $\overline{N}$. Symbols are values measured in simulations. Solid line is a model that assumes $\mathrm{\Delta }{G}_{\mathrm{SCF}}$ has a constant value at the ODT, for which we take $\mathrm{\Delta }{G}_{\mathrm{SCF}}=2.15$. Dotted line is the empirical formula of Ref. [4].