Solvent coarsening around colloids driven by temperature gradients

Using mesoscopic numerical simulations and analytical theory, we investigate the coarsening of the solvent structure around a colloidal particle emerging after a temperature quench of the colloid surface. Qualitative differences in the coarsening mechanisms are found, depending on the composition of the binary liquid mixture forming the solvent and on the adsorption preferences of the colloid. For an adsorptionwise neutral colloid, the phase next to its surface alternates as a function of time. This behavior sets in on the scale of the relaxation time of the solvent and is absent for colloids with strong adsorption preferences. A Janus colloid, with a small temperature difference between its two hemispheres, reveals an asymmetric structure formation and surface enrichment around it, even if the solvent is within its one-phase region and if the temperature of the colloid is above the critical demixing temperature $T_c$ of the solvent. Our phenomenological model turns out to capture recent experimental findings according to which, upon laser illumination of a Janus colloid and due to the ensuing temperature gradient between its two hemispheres, the surrounding binary liquid mixture develops a concentration gradient.

Figure 1

Spherical Janus colloid of radius $R$ with reduced temperatures ${\tilde{T}}_r$ and ${\tilde{T}}_l$ on its right and left hemisphere, respectively. The azimuthal angle $\phi$ is measured from the $x$ axis in the horizontal $\widehat{x}\widehat{y}$ midplane of the colloid, the polar angle $\theta$ is measured from the $z$ axis, and $r$ is the radial distance from the center. The initial temperature in the whole binary solvent is ${\tilde{T}}_i={\tilde{T}}_r$.

Figure 2

Temperature-gradient induced demixing around a homogeneous colloid immersed in a binary solvent with off-critical concentration ${\psi }_0=0.1$ and undergoing slow cooling below $T_c$. The results in (a), (b), (d), and (e) correspond to $L=100,R=10,\alpha =0.5,h_s=1,\mathcal{D}=50$, and $\nu ={10}^{-4}$. (a) Temperature profile in the midplane $z=L/2$ at an early time $t=10$, exhibiting a strong gradient. Initially, the solvent is hot and the colloid is cold. (b) Coarsening patterns in the midplane of the colloid. Close to the colloid, disconnected concentric circular structures form; away from the colloid, spinodal-like patterns prevail. As time progresses, the shells propagate into bulk via the formation of new layers. As expected, in the long-time limit the system forms a planar interface trapping the colloid. (c) Comparison of the approximate analytic prediction of the OP profile without any temperature gradient $(•)$ and numerical data $(\blacksquare )$ with a temperature gradient, for $R=10,{\tilde{T}}_i=1,\alpha =0.01,h_s=0.1$, and $\nu ={10}^{-4}$. (d) Nonmonotonic time dependence of the number of concentric shells $N_s$; the decrease is much slower than the increase. (e) Radial two-point equal-time correlation function $C(\zeta =r-R,t)$ vs reduced distance $\zeta /R$ for three values of $t$. $C(\zeta ,t)$ decays spatially fast at early times, while with increasing time it develops multiple minima corresponding to concentric shell-like layers around the colloid. Data have been averaged over 10 independent initial configurations.

Figure 3

Angularly averaged temperature profile $\tilde{T}\left(\zeta \right)$ around a homogeneous colloid. The temperature gradient decreases with time. The results correspond to the same parameters as used in Fig. 4.

Figure 4

Angularly averaged order-parameter profiles $\psi \left(\zeta \right)$ around a homogeneous colloid at two times $t$. At the very early time $t=2$, surface demixing prevails while in the bulk $\psi \left(\zeta \right)$ attains its initial value ${\psi }_0=0.1$. Over the course of time, new layers form and the maximum value of $\psi \left(\zeta \right)$ increases. The lines interpolate the data points. The parameter values are those used in Fig. 3.

Figure 5

(a) Snapshots of a cooling binary solvent $({\psi }_0=0.1)$ around an adsorption-neutral, homogeneous colloid. At early times, the phase $\psi >{\psi }_0$ is formed near the colloid surface until at $t\simeq 600$ the phase near the surface is replaced by the one with $\psi <{\psi }_0$. Results correspond to $L=100,R=10,{\tilde{T}}_1=-1={\tilde{T}}_l={\tilde{T}}_r,\alpha =0,h_s=0$, and $\nu ={10}^{-4}$. (b) Plot of the crossover time ${\tau }_0$ vs system size $L$ for $R=5$. The solid line drawn through the rightmost data points refers to the finite-size critical relaxation time $\propto L^z;z=4$ (see the main text).

Figure 6

Coarsening of a binary solvent with ${\psi }_0=0.1$ around a Janus colloid. Results correspond to $L=100,R=10,\alpha =0.5,h_{s,l}=1,h_{s,r}=0.5$, and $\nu ={10}^{-5}$. (a) Evolution patterns in its midplane $\left(\theta =\pi /2\right)$. The left and right hemispheres are gray and yellow, respectively. Initially, both the solvent and colloid are at $\tilde{T}=1$. From $t=1$ to 400, the system evolves at constant temperature everywhere. At $t=401$, the temperature of the left hemisphere is quenched to ${\tilde{T}}_l=0.7$. The corresponding stationary configuration is shown at $t=1400$. Although both hemispheres are maintained above $T_c$, structure formation is observed. (b) Temperature distribution $\tilde{T}(r,\phi )>0$ of the solvent in the midplane around a Janus colloid. Dependence (b) on the azimuthal angle $\phi$ for two radial distances $r$ from the colloid center and (c) on $r$ for two opposite angles $\phi$ corresponding to the right and to the left hemisphere, respectively. The shaded region in (c) corresponds to the space occupied by the colloid. Inset of (c): as in the main figure, but for the homogeneous colloid discussed in Fig. 2 for which there is no dependence on $\phi$.