Structure and position-dependent properties of inhomogeneous suspensions of responsive colloids

Responsive particles, such as biomacromolecules or hydrogels, display a broad and polymodal distribution of conformations and have thus the ability to change their properties (e.g., size, shape, charge density, etc.) substantially in response to external fields or to their local environment (e.g., mediated by cosolutes or pH). Here we discuss the basic statistical mechanics for a model of responsive colloids (RCs) by introducing an additional “property” degree of freedom as a collective variable in a formal coarse-graining procedure. The latter leads to an additional one-body term in the coarse-grained (CG) free energy, defining a single-particle property distribution for an individual polydisperse RC. We argue that in the equilibrium thermodynamic limit such a CG system of RCs behaves like a conventional polydisperse system of nonresponsive particles. We then illustrate the action of external fields, which impose local (position-dependent) property distributions leading to nontrivial effects on the spatial one-body property and density profiles, even for an ideal (noninteracting) gas of RCs. We finally apply density-functional theory in the local density approximation to discuss the effects of particle interactions for specific examples of (i) a suspension of RCs in an external field linear in both position and property, (ii) a suspension of RCs with highly localized properties (sizes) confined between two walls, and (iii) a two-component suspension where an inhomogeneously distributed (nonresponsive) cosolute component, as found, e.g., in the studies of osmolyte- or salt-induced collapse or swelling transitions of thermosensitive polymers, modifies the local properties and density of the RC liquid.

Figure 1

(a) Number density profiles ${\rho }_i\left(z\right)/{\rho }_i^0$ of an ideal gas of RCs (normalized such that $N=\mathrm{const}$) confined between walls at $z=0$ and $L=15$ in a linear external potential ${\varphi }_{\mathrm{ext}}=A\sigma z$ with single-particle property distributions according to Eqs. (27) to (29). The external field parameter is $\beta A=1$. The dotted lines in (a) are the spatial property profiles ${\sigma }_i\left(z\right)$ using the same scale as for the normalized density. (b) The emerging property distribution $N_2\left(\sigma \right)$ for the Gaussian ideal gas distribution $p_2\left(\sigma \right)$, cf. Eq. (28), in the external field. (c) The property distribution $N_3\left(\sigma \right)$ for the double Gaussian ideal gas distribution $p_3\left(\sigma \right)$, cf. Eq. (29), in the external field.

Figure 2

[(a) and (c)] Number density profiles for the Gaussian ideal property distribution $p_2\left(\sigma \right)$, cf. Eq. (28), of monodisperse systems for repulsive and attractive $B_2$, cf. Eq. (54), respectively. [(b) and (d)] The corresponding emerging property distributions, $N\left(\sigma \right)$, in the interacting systems depend on the interactions: The more repulsive (attractive) the interactions, the smaller (larger) the mean property $\sigma$.

Figure 3

Artistic illustration of hard spherical RCs with highly localized properties (sizes) confined between two hard walls in distance $L_z=25{\sigma }_0$ in the $z$ direction. Sphere diameters decrease linearly from $\sigma (z=0)={\sigma }_0$ to $\sigma (z=L_z)=0.2{\sigma }_0$ [scaling factor $f=-0.8$ in the linear scaling function Eq. (58)].

Figure 4

Scaled density profiles, $\rho \left(z\right){\sigma }_0^3\pi /6$, and packing fraction $\eta \left(z\right)=\rho \left(z\right)\sigma {\left(z\right)}^3\pi /6$ of hard spherical RCs confined between two walls, cf. Fig. 3, separated by a distance $L_z=25{\sigma }_0$ from solution of the LDA equation (62). Panels (a) and (b) are for a scaling factor 0.2, i.e., sphere diameters increase from 1 to $1.2{\sigma }_0$ from left to right [see the dashed line and right scale in (a)]. Panels (c) and (d) are for a scaling factor $f=-0.8$, i.e., sphere diameters decrease from 1 to $0.2{\sigma }_0$ from left to right [see the dashed line and right scale in (c)].

Figure 5

Distributions of a two-component system (cosolutes and RCs) between two walls separated by a distance $L=8$ according to Eq. (66). A linear external field ${\varphi }_{\mathrm{ext},1}\left(z\right)=\tilde{A}z$ acts only on the nonresponsive, mutually ideal species 1, while the RCs (species 2) are interacting with species tuned by coupling parameter $C$ (legend), see text. The spatial distribution of cosolute species 1 (a) couples to the density profiles and property distributions of the RCs (species 2) shown in panels (b) and (c), respectively.