Simulation of diblock copolymer surfactants. III. Equilibrium interfacial adsorption

Monte Carlo simulations are used to study adsorption of highly asymmetric diblock copolymers to a polymer-polymer interface, and the results compared to self-consistent field theory (SCFT) predictions. The simulation model used here is a bead-spring model that has been used previously to study equilibrium and kinetic properties of spherical micelles [J. A. Mysona et al., Phys. Rev. E 100, 012602 (2019); Phys. Rev. E 100, 012603 (2019); Phys. Rev. Lett. 123, 038003 (2019)]. Interfacial copolymer concentration $\mathrm{\Gamma }$ and interfacial tension $\gamma$ are measured as functions of bulk copolymer concentration at concentrations up to the critical micelle concentration over a range of values of the Flory-Huggins $\chi$ parameter. The dependence of interfacial pressure $\mathrm{\Pi }$ = ${\gamma }_0-\gamma$ on $\mathrm{\Gamma }$ (where ${\gamma }_0$ is the interfacial tension in the absence of copolymer) is found to be almost independent of $\chi$ and to be accurately predicted by SCFT. The bare interfacial tension ${\gamma }_0$ and total interfacial tension $\gamma \left(\mathrm{\Gamma }\right)$ can also be accurately predicted by SCFT using an estimate of $\chi$ obtained from independent analysis of properties of symmetric diblock copolymer melts. SCFT predictions obtained with this estimate of $\chi$ do not, however, adequately describe the thermodynamics of the coexisting bulk copolymer solution, as a result of contraction of the strongly interacting core block of dissolved copolymers. Accurate predictions of the relationship between bulk and interfacial properties can thus only be obtained for this system by combining SCFT predictions of the interfacial equation of state with a fit to the measured equation of state for the bulk solution.

Figure 1

Interfacial copolymer concentration $\mathrm{\Gamma }$ computed using the contact algorithm (open squares) and the surface excess method (open circles) plotted vs. exchange chemical potential $\mathrm{\Delta }\mu$, for systems with $\alpha =10$. In this and all subsequent plots, all quantities are plotted using simulation units in which $\sigma =k_{B}T=1$. The predicted exchange chemical potential $\mathrm{\Delta }{\mu }_c$ at which the free copolymer concentration in the A-rich phase is equal to the CMC is indicated by a vertical dashed line.

Figure 2

Interfacial concentration $\mathrm{\Gamma }$ vs. exchange chemical potential $\mathrm{\Delta }\mu$ for $\alpha =10$ (a) and $\alpha =16$ (b). Open circles show simulation results. The solid line in each plot is the result of the fit to Eq. (8). The dotted line in (b) shows a fit of the dilute region to the two-dimensional ideal gas equation of state given in Eq. (7).

Figure 3

Interfacial concentration $\mathrm{\Gamma }$ plotted vs. mole fraction ${\varphi }_1$ of dissolved copolymer in the coexisting bulk $A$-rich phase for $\alpha =10$ (a) and $\alpha =16$ (b). Open circles show simulation results while solid lines show the result of the fit to Eq. (8). Both simulation results and this fit are plotted here by using Eq. (4) to convert $\mathrm{\Delta }\mu$ to corresponding values of ${\varphi }_1$.

Figure 4

Interfacial tension $\gamma$ vs. interfacial concentration for $\alpha =10$ (open circles) and $\alpha =16$ (open squares). Solid lines show values predicted by integrating the Gibbs adsorption equation, using the measured adsorption isotherm and the bare interfacial tension ${\mathrm{\Gamma }}_0$ as inputs. Each solid line ends at the critical value ${\mathrm{\Gamma }}_c$ corresponding to the CMC.

Figure 5

Interfacial pressure $\mathrm{\Pi }$ vs. interfacial concentration $\mathrm{\Gamma }$ for multiple different values of $\alpha$. All of the data shown here collapse onto a single curve, indicating that the interfacial pressure is independent of $\alpha$.

Figure 6

Comparison of the value of $\chi N_B$ calculated through two different methods of estimating $\chi \left(\alpha \right)$, plotted vs. $\alpha$. Squares represent the value ${\chi }_m$ obtained in Refs. [33, 34]—the copolymer melt method. Circles represent values of ${\chi }_d$ found by fitting the dilute solution equation of state the dilute solution method—the dilute solution method.

Figure 7

SCFT predictions for $\mathrm{\Pi }$ vs. $\mathrm{\Gamma }$ obtained using both ${\chi }_d$ and ${\chi }_m$ at all four values of $\alpha =$ 10, 12, 14, and 16.

Figure 8

Comparison of simulation results (open circles) and SCFT predictions (lines) for interfacial pressure $\mathrm{\Pi }$ plotted vs. $\mathrm{\Gamma }$ for $\alpha =10$ (a) and $\alpha =16$ (b). The vertical dashed line indicates the critical interfacial concentration ${\mathrm{\Gamma }}_c$ at the CMC. SCFT predictions computed using the estimate of ${\chi }_d$ obtained from dilute solutions are shown by solid lines, while predictions computed using the value ${\chi }_m$ obtained from diblock copolymer melts are shown by dashed lines. These two lines are generally indistinguishable, except for the case of $\alpha =10$ at values of $\mathrm{\Gamma }$ somewhat greater than ${\mathrm{\Gamma }}_c$. The straight dotted line extending from the origin is the 2D ideal gas prediction $\mathrm{\Pi }=\mathrm{\Gamma }{k}_{B}T$, which is valid at very low values of $\mathrm{\Gamma }$.

Figure 9

Simulation results (open circles) and SCFT predictions (lines) for interfacial tension $\gamma$ plotted vs. interfacial concentration $\mathrm{\Gamma }$ for $\alpha =10$ (a) and $\alpha =16$ (b). SCFT predictions computed using ${\chi }_d$ are shown by solid lines, and predictions computed using ${\chi }_m$ are shown by dashed lines. Vertical dashed lines indicate critical value ${\mathrm{\Gamma }}_c$ at the CMC.

Figure 10

Interfacial concentration $\mathrm{\Gamma }$ vs. bulk free copolymer mole fraction ${\varphi }_1$ for $\alpha =10$ (a) and $\alpha =16$ (b). Open circles show simulation results. Solid and dashed lines show SCFT predictions computed using ${\chi }_d$ and ${\chi }_m$, respectively. Vertical dashed lines indicate the CMC.

Figure 11

Simulations results (symbols) and predictions for interfacial tension $\gamma$ as a function of bulk copolymer mole fraction ${\varphi }_1$ for all four values of $\alpha$ considered in simulations. Dotted lines show the hybrid SCFT prediction, in which the relationship between $\mathrm{\Delta }\mu$ and ${\varphi }_1$ is computed using Eq. (4) and the dependence of $\gamma$ on $\mathrm{\Delta }\mu$ is predicted using SCFT with $\chi ={\chi }_m$. Straightforward SCFT predictions computed using ${\chi }_d$ and ${\chi }_m$ are shown by solid and dashed lines, respectively. Vertical dashed lines indicate the CMC.