Phase behavior and orientational ordering in block copolymers doped with anisotropic nanoparticles

A molecular field theory and coarse-grained computer simulations with dissipative particle dynamics have been used to study the spontaneous orientational ordering of anisotropic nanoparticles in the lamellar and hexagonal phases of diblock copolymers and the effect of nanoparticles on the phase behavior of these systems. Both the molecular theory and computer simulations indicate that strongly anisotropic nanoparticles are ordered orientationally mainly in the boundary region between the domains and the nematic order parameter possesses opposite signs in adjacent domains. The orientational order is induced by the boundary and by the interaction between nanoparticles and the monomer units in different domains. In simulations, sufficiently long and strongly selective nanoparticles are ordered also inside the domains. The nematic order parameter and local concentration profiles of nanoparticles have been calculated numerically using the model of a nanoparticle with two interaction centers and also determined using the results of computer simulations. A number of phase diagrams have been obtained which illustrate the effect of nanoparticle selectivity and molar fraction of the stability ranges of various phases. Different morphologies have been identified by analyzing the static structure factor and a phase diagram has been constructed in coordinates' nanoparticle concentration–copolymer composition. Orientational ordering of even a small fraction of nanoparticles may result in a significant increase of the dielectric anisotropy of a polymer nanocomposite, which is important for various applications.

Figure 1

Phase diagrams in terms of the monomer-monomer discrimination coupling constant $N\chi$ and the NP-monomer discrimination interaction constant $(J_A-J_B)/k_{B}T$ for different composition ratios: (a) $f=0.5$, (b) $f=0.3$, and (c) $f=0.15$. The other model parameters are ${\rho }_{\text{NP}}/{\rho }_m={10}^{-3}, N=100, r_0=0.1R, l=0.3R, s=0.1R$, and ${\rho }_m{r}_0^3=1$.

Figure 2

Phase diagrams in terms of the monomer-monomer discrimination coupling constant $N\chi$ and the reduced NP number density ${\rho }_{\text{NP}}/{\rho }_m$ for different composition ratios: (a) $f=0.5$, (b) $f=0.3$, and (c) $f=0.15$. The other model parameters are $N=100, r_0=0.1R, l=0.3R, s=0.1R, {\rho }_m{r}_0^3=1$, and $J_A-J_B=k_{B}T$.

Figure 3

Translational order parameters of the nanocomposite for negative (a) $J_A-J_B=-k_{B}T$ and positive (b) $J_A-J_B=k_{B}T$ values of the discrimination interaction constant and for ${\rho }_{\text{NP}}={10}^{-3}{\rho }_m$ NPs. The other model parameters are $N=100, r_0=0.1R, l=0.3R, s=0.1R$, and ${\rho }_m=s^{-3}$.

Figure 4

(a) and (c) Density and (b) and (d) orientational order parameter profiles of the anisotropic NPs in the (a) and (b) hexagonal and (c) and (d) lamellar phases evaluated at the points marked in Fig. 3 for negative $J_A-J_B=-k_{B}T$ and for $N=100, r_0=0.1R, l=0.3R, s=0.1R$, and ${\rho }_m=s^{-3}$.

Figure 5

(a) and (c) Density and (b) and (d) orientational order parameter profiles of the anisotropic NPs in the (a) and (b) hexagonal and (c) and (d) lamellar phases evaluated at the points marked in Fig. 3 for positive $J_A-J_B=k_{B}T$ and for $N=100, r_0=0.1R, l=0.3R, s=0.1R$, and ${\rho }_m=s^{-3}$.

Figure 6

Effect of the copolymer composition $f$ and the content of the particles ${\varphi }_R$ on the morphology of the mixture with $N=20, N_R=5, \sigma =0.2$, and $a_{AB}=70$.

Figure 7

Effect of the NR concentration ${\varphi }_R$ on the microdomain structure at $f=0.25, \sigma =0.2, N=20, N_R=5$, and (a) ${\varphi }_R=0$, (b) ${\varphi }_R=0.05$, and (c) ${\varphi }_R=0.1$.

Figure 8

Local fraction of (a) the NRs ${\varphi }_R\left(z\right)$ and (b) their orientational order parameter $S\left(z\right)$ in the lamellar phase for different values of the NR interaction selectivity $\sigma$ (specified) for $f=0.35, {\varphi }_R=0.1$, and $N_R=5$.

Figure 9

Local fraction of (a) the NRs ${\varphi }_R\left(z\right)$ and (b) their orientational order parameter $S\left(z\right)$ in the lamellar phase for different values of the copolymer composition $f$ (specified) for $\sigma =0.25, {\varphi }_R=0.1$, and $N_R=5$.

Figure 10

Local fraction of (a) the NRs ${\varphi }_R\left(z\right)$ and (b) their orientational order parameter $S\left(z\right)$ in the hexagonal cylindrical phase for different values of the copolymer composition $f$ (specified) for $\sigma =0.4, {\varphi }_R=0.1$, and $N_R=5$.