Dipolar Brush Polymers: A Numerical Study of the Force Exerted onto a Penetrating Colloidal Particle Under an External Field

Abstract

Langevin Dynamics numerical simulations have been used to compute the force profiles that dipolar polymer brushes exert onto a penetrating colloidal particle. It has been observed that force profiles are strongly influenced by externally applied fields: at large distances from the grafting surface, a force barrier appears, and at shorter distances a region with lower repulsive forces develops. Furthermore, with the right combination of polymer grafting density, polymer chain length and strength of the external field, it is possible to observe in this intermediate region both the existence of net attractive forces onto the penetrating particle and the emergence of a stationary point. The existence of these regions of low repulsive or net attractive forces inside the dipolar brushes, as well as their dependence on the different parameters of the system can be qualitatively reasoned in terms of a competition between steric repulsion forces and Kelvin forces arising from the dipolar mismatch between different regions of the system. The possibility to tune force profile features such as force barriers and stationary points via an external field paves the way for many potential surface-particle-related applications.

Keywords: Langevin dynamics; colloids; dipolar brushes; magnetic brushes; numerical simulations.

Figure 1

(a) depicts a schematic view of the system under study, (b) shows some of the chain dipolar sequences tested in this work, $N_{dip}$ and N indicate the number of dipolar monomers and the total number of monomers per chain, respectively.

Figure 2

Force profiles along the direction perpendicular to the grafting surface of the dipolar brush as a function of the distance z between the grafting surface and the penetrating particle of radius $R=5$. The chains of the brushes are fully dipolar $N=N_{dip}=20$, and no external field is applied $\alpha =0$. Several brush grafting densities are shown in each plot: ${\sigma }_g=0.04,0.0625,0.0765,0.9$. Plot (a) depicts the force profiles for brushes in good solvent $\epsilon =0$, while plot (b) depicts the force profiles for brushes in a bad solvent $\epsilon =0.25$ (sticky chains condition). Filled black symbols in each plot represent the force profile obtained for the non-dipolar brush case, $\mu =0$, at grafting density ${\sigma }_g=0.04$.

Figure 3

(a,b) shows force profiles for same brushes as in Figure 2 but now under and external moderate field $\alpha =9$. Plot (a) depicts the force profiles for brushes in good solvent $\epsilon =0$, while plot (b) depicts the force profiles for brushes in a bad solvent $\epsilon =0.25$ (sticky chains condition). (c) shows the potentials obtained from the integration of the forces shown in (b), as in our simulations $T=0.5$, $2U=U_e/\left(k_e{T}_e\right)$ (see Section 2).

Figure 4

Force profiles for same brushes as in Figure 2 and Figure 3 but now at high field $\alpha =45$. Plot (a) depicts the force profiles for brushes in good solvent $\epsilon =0$, while plot (b) depicts the force profiles for brushes in a bad solvent $\epsilon =0.25$ (sticky chains condition).

Figure 5

Force profiles representing the force perpendicular to the grafting surface as a function of the distance z between the surface and the center of the penetrating particle. Brush parameters are similar to previous figures but now the number of dipoles contained in the chain varies as $N_{dip}=0,5,10,20$, corresponding $N_{dip}=20$ to a chain where all monomers bear a dipole. The rest of cases, dipoles are equally spaced inside the monomer sequence of the chain. Plot (a) corresponds to ($\alpha =9$, $\epsilon =0$), plot (b) corresponds to ($\alpha =9$, $\epsilon =0$.25).

Figure 6

In these plots the dependence of the force profiles as a function of the radius of the penetrating particles is studied for several external fields $\alpha =0$ in plot (a), $\alpha =9$ in plot (b) and $\alpha =45$ in plot (c). Brushes are under bad solvent conditions $ϵ=0.25$, the grafting density of the brushes is set to ${\sigma }_g=0.0625$ and chains are fully dipolar $N=N_{dip}=20$.