Critical Casimir interactions between spherical particles in the presence of bulk ordering fields

The spatial suppression of order parameter fluctuations in a critical media produces critical Casimir forces acting on confining surfaces. This scenario is realized in a critical binary mixture near the demixing transition point that corresponds to the second-order phase transition of the Ising universality class. Due to these critical interactions similar colloids, immersed in a critical binary mixture near the consolute point, exhibit attraction. The numerical method for computation of the interaction potential between two spherical particles using Monte Carlo simulations for the Ising model is proposed. This method is based on the integration of the local magnetization over the applied local magnetic field. For the stronger interaction the concentration of the component of the mixture that does not wet colloidal particles should be larger than the critical concentration. The strongest amplitude of the interactions is observed below the critical point.

Figure 1

(a) Phase diagram of a critical binary mixture with the lower critical point and the aggregation region. (b) Schematic representation for a quasisphere on the lattice. (c) Computation of the insertion free energy: bulk system with the free energy $F_b\left(\beta \right)$, the system with fixed spins in two colloidal particles at distance $D$ with $F_{\mathrm{c}}(\beta ,H_b,D)$, the system with an external field $H_{\mathrm{c}}$ applied to spins of two colloidal particles at distance $D$ with the free energy $F_{\mathrm{h}}(\beta ,H_b,D,H_{\mathrm{c}})$. (d) Typical graphs of magnetizations $m_{\mathrm{c}}(H_{\mathrm{c}},D)$, $m_{\mathrm{c}}(H_{\mathrm{c}},D_{\max })$ as functions of “colloid” field $h_{\mathrm{c}}$ for separations $D$, $D_{\max }$. The shadowed area between curves is equal to the absolute value of the free energy difference $U\left(D\right)=\beta F_{\mathrm{c}}\left(D\right)-\beta F_{\mathrm{c}}\left(D_{\max }\right)$.

Figure 2

Numerical results for for the separation $D=3$ between two spheres of radius $R=3.5$, the value of the bulk field $H_b=-0.1$. (a) Probability distribution function $P(m_{\mathrm{c}},h_{\mathrm{c}}^j)$ of the magnetization $m_{\mathrm{c}}$ for the inverse temperature $\beta =0.2205$ and various values of the local field (from left to right) $h_{\mathrm{c}}^j=0,0.01,\cdots ,2.5$. (b) Average particle magnetization $m_{\mathrm{c}}$ as a function of the local field $h_{\mathrm{c}}$ for various values of the inverse temperature $\beta =0.1,0.1497,0.1994,0.2205,0.25,0.28$; black triangles correspond to lines from panel (a).

Figure 3

The Casimir interaction potential $U(r;H_b,D)$ as a function of the variable $r=t{(R/{\xi }_0^{+})}^{1/\nu }$ for various values of the separation $D=0,1,2,3,4,6$ for (a) zero bulk field $H_b=0,$ (b) negative bulk field $H_b=-0.05,$ and (c) negative bulk field $H_b=-0.1$. (d) The energy difference $\Delta E=E\left(D\right)-E\left(D_{\max }\right)$ as a function of the scaling variable $r$ for $H_b=-0.1$.

Figure 4

The Casimir interaction potential $U(h;\beta ,D)$ as a function of the bulk field scaling variable $h=H_b{(R/{\xi }_0^H)}^{\Delta /\nu }$ for various values of separation $D=0,1,2,3,4,6$: (a) above the critical point $\beta =0.2$, (b) at the critical point ${\beta }_c\simeq 0.221654$, and (c) below the critical point $\beta =0.24$. (d) The magnetization profile $m(x,z)$ as a function of coordinates $x,z$ for $\beta =0.25$ ($r\simeq -2.48$), $H_b=-0.1$ ($h\simeq -42.6$), and various separations $D=1,3,6$.