Three-body critical Casimir forces

Within mean-field theory we calculate universal scaling functions associated with critical Casimir forces for a system consisting of three parallel cylindrical colloids immersed in a near-critical binary liquid mixture. For several geometrical arrangements and boundary conditions at the surfaces of the colloids we study the force between two colloidal particles in the direction normal to their axes, analyzing the influence of the presence of a third particle on that force. Upon changing temperature or the relative positions of the particles we observe interesting features such as a change of sign of this force caused by the presence of the third particle. We determine the three-body component of the forces acting on one of the colloids by subtracting the pairwise forces from the total force. The three-body contribution to the total critical Casimir force turns out to be more pronounced for small surface-to-surface distances between the colloids as well as for temperatures close to criticality. Moreover, we compare our results with similar ones for other physical systems such as three atoms interacting via van der Waals forces.

Figure 1

Cross section of three parallel cylindrical colloidal particles of radii $R_i\left(i=1,2,3\right)$ with their axes parallel to the $y$ axis, immersed in a near-critical binary liquid mixture (not shown). The three colloidal particles with BCs $\left(a_i\right)$ are located at surface-to-surface distances $L_{ij}$, with $i,j\in \{1,2,3\}$ and $i\ne j$. In the case of four spatial dimensions the figure shows a cross section of the system, which is invariant also along the fourth direction, so the cylinders correspond to parallel hypercylinders with two translationally invariant directions, i.e., the $y$ direction and the fourth dimension.

Figure 2

Three configurations considered for three parallel cylindrical colloids: (a) isosceles triangle: $L_{13}=L_{23}$; (b) right-angled triangle: ${(L_{23}+2R)}^2={(L_{12}+2R)}^2+{(L_{13}+2R)}^2$; (c) line: $L_{23}=L_{12}+L_{13}+2R$. All macroscopically extended cylindrical colloids are aligned parallel to the $y$ axis (see Fig. 1).

Figure 3

Normalized scaling function ${\overline{K}}_{(+,+,-)}^{(1,x)}({\Theta }_{12}=L_{12}/{\xi }_{+},{\Lambda }_{12}=L_{12}/R,{\Lambda }_{13}=L_{13}/R)$ of the CCF acting on colloid (1) along the $x$ direction (i.e., of the total horizontal force on (1)) for ${\Lambda }_{12}=1$ in (a) and ${\Lambda }_{12}=1.25$ in (b) with the centers of the colloids forming an isosceles triangle ($L_{13}=L_{23}$). The scaling function is shown for $t>0$ as function of the scaling variable ratio $R/{\xi }_{+}={\Theta }_{12}/{\Lambda }_{12}=\left(\right|T-T_c{|/T_c)}^{\nu }R/{\xi }_0^{+}$ for four values of the scaling variable ${\Lambda }_{13}=L_{13}/R_3:{\Lambda }_{13}=1$ (black lines), ${\Lambda }_{13}=1.25$ (red lines), ${\Lambda }_{13}=2$ (green lines), and ${\Lambda }_{13}=\infty$ (blue lines), while ${\Lambda }_{23}={\Lambda }_{13}$ for all curves in (a) and (b). For fixed $R$, each curve corresponds to a different vertical position of colloid (3) along the perpendicular bisector of $L_{12}$ (dotted double-headed arrow), with the colloids (1) and (2) fixed in space. As expected, upon increasing ${\Lambda }_{13}$ the scaling function approaches the blue lines, which correspond to the scaling function of the pairwise force between the colloids (1) and (2), i.e., if colloid (3) is infinitely far away from the colloids (1) and (2). ${\overline{K}}_{(+,+,-)}^{(1,x)}<0(>0)$ implies that colloid (1) is attracted to (repelled from) colloid (2) along the $x$ direction. Note that according to our choice of the coordinate system the positive (negative) $x$ direction points away (towards) colloid (2) (see Figs. 1 and 2). The circles and arrows in panel (a) are schematic representations of the colloids and of the surface-to-surface distances between them; accordingly they are not drawn to scale.

Figure 4

Normalized scaling function ${\overline{K}}_{(+,+,-)}^{(1,x)}({\Theta }_{12}=L_{12}/{\xi }_{+},{\Lambda }_{12}=L_{12}/R,{\Lambda }_{13}=L_{13}/R)$ of the total horizontal CCF acting on colloid (1) for ${\Lambda }_{13}=1.25$ in (a) and ${\Lambda }_{13}=1.5$ in (b) with the centers of the colloids forming an isosceles triangle ($L_{13}=L_{23}$). The scaling function is shown for $t>0$ as function of the scaling variable ratio $R/{\xi }_{+}={\Theta }_{12}/{\Lambda }_{12}$ for four values of the scaling variable ${\Lambda }_{12}=L_{12}/R:{\Lambda }_{12}=1$ (black lines), ${\Lambda }_{12}=1.25$ (red lines), ${\Lambda }_{12}=1.5$ (green lines), and ${\Lambda }_{12}=1.75$ (blue lines). For fixed $R$, the curves correspond to different surface-to-surface distances $L_{12}$ between colloids (1) and (2). Varying $L_{12}$ under the constraint $L_{13}=L_{23}$ implies that only colloid (3) is fixed in space and the centers of the colloids (1) and (2) move, equally but in opposite directions, on an arc centered at the center of colloid (3) and with radius $L_{13}+2R$. Accordingly, in this setup ${\Lambda }_{12}$ can vary between $0\le {\Lambda }_{12}\le 2{\Lambda }_{13}+2$ where the maximum value corresponds to a linear configuration with colloid (3) in the center. ${\overline{K}}_{(+,+,-)}^{(1,x)}<0(>0)$ implies that colloid (1) is attracted to (repelled from) colloid (2) along the $x$ direction. The circles and arrows in panel (a) are schematic representations of the colloids and of the surface-to-surface distances between them; accordingly they are not drawn to scale.

Figure 5

Normalized scaling function ${\overline{K}}_{(+,-,+)}^{(1,x)}({\Theta }_{12}=L_{12}/{\xi }_{+},{\Lambda }_{12}=L_{12}/R,{\Lambda }_{13}=L_{13}/R)$ of the total horizontal CCF acting on colloid (1) in the presence of the colloids (2) and (3). The scaling function is shown for $t>0$ as function of the scaling variable ratio $R/{\xi }_{+}={\Theta }_{12}/{\Lambda }_{12}$ for ${\Lambda }_{13}=1$ in (a) and ${\Lambda }_{13}=1.25$ in (b) with the centers of the colloids forming an isosceles triangle ($L_{13}=L_{23}$). Each curve corresponds to a certain value of the scaling variable ${\Lambda }_{12}=L_{12}/R_1:{\Lambda }_{12}=1$ (black curves), ${\Lambda }_{12}=1.25$ (red curves), and ${\Lambda }_{12}=1.5$ (green curves). For fixed $R$, the curves correspond to distinct surface-to-surface distances between colloids (1) and (2). Varying $L_{12}$ under the constraint $L_{13}=L_{23}$ implies that here one has the same kind of sequences of configurations as in Fig. 4. ${\overline{K}}_{(+,-,+)}^{(1,x)}<0(>0)$ implies that colloid (1) is attracted to (repelled from) colloid (2) in the $x$ direction. The circles and arrows in panel (a) are schematic representations of the colloids and of the surface-to-surface distances between them; accordingly they are not drawn to scale.

Figure 6

Normalized scaling function ${\overline{K}}_{(+,+,-)}^{(1,x,\mathrm{TB})}({\Theta }_{12}=L_{12}/{\xi }_{+},{\Lambda }_{12}=L_{12}/R,{\Lambda }_{13}=L_{13}/R)$ of the three-body horizontal CCF acting on colloid (1) for ${\Lambda }_{12}=1$ in (a) and ${\Lambda }_{12}=1.25$ in (b) with the centers of the colloids forming an isosceles triangle ($L_{13}=L_{23}$). The scaling function is shown as function of the scaling variable ratio $R/{\xi }_{+}={\Theta }_2/{\Lambda }_{12}$. The black, red, and green lines correspond to ${\Lambda }_{13}=1,{\Lambda }_{13}=1.25$, and ${\Lambda }_{13}=2$, respectively. For fixed $R$, each curve corresponds to a different vertical position of the colloid (3) along the perpendicular bisector of $L_{12}$ (dotted double-headed arrow), with the colloids (1) and (2) fixed in space. Figures 3 and 6 provide a direct comparison between the total horizontal CCF and the corresponding three-body contribution (note the different scales of the coordinates). The circles and arrows in panel (a) are schematic representations of the colloids and of the surface-to-surface distances between them; accordingly, they are not drawn to scale.

Figure 7

Normalized scaling function ${\overline{K}}_{(+,+,+)}^{(1,x)}({\Theta }_{12}=L_{12}/{\xi }_{+},{\Lambda }_{12}=L_{12}/R,{\Lambda }_{13}=L_{13}/R)$ in (a) and ${\overline{K}}_{(+,+,+)}^{(1,x,\mathrm{TB})}({\Theta }_{12},{\Lambda }_{12},{\Lambda }_{13})$ in (b) of the horizontal total CCF and the three-body CCF, respectively, acting on colloid (1) for a right-angled triangle, i.e., ${(L_{23}+2R)}^2={(L_{12}+2R)}^2+{(L_{13}+2R)}^2$ and ${\Lambda }_{12}=1.25$. The scaling functions are shown as functions of the scaling variable ratio $R/{\xi }_{+}={\Theta }_2/{\Lambda }_{12}$. The black, red, green, and blue lines correspond to ${\Lambda }_{13}=1,{\Lambda }_{13}=1.5,{\Lambda }_{13}=2$, and ${\Lambda }_{13}=\infty$, respectively. In (b) the horizontal dashed line corresponds to ${\Lambda }_{13}=\infty$. The circles and arrows in panel (a) are schematic representations of the colloids and of the surface-to-surface distances between them; accordingly they are not drawn to scale.

Figure 8

Normalized scaling function ${\overline{K}}_{(+,+,a_3)}^{(1,x)}({\Theta }_{12}=L_{12}/{\xi }_{+},{\Lambda }_{12}=L_{12}/R,{\Lambda }_{13}=L_{13}/R)$ of the total CCF acting on colloid (1) for two linear configurations with ${\Lambda }_{12}=1.5$ and for $(a_3=+)$ in (a) and $(a_3=-)$ in (b). The centers of the three colloids lie on the $x$ axis so $L_{23}=L_{12}+L_{13}+2R$ [see Fig. 2]. The scaling functions are shown as functions of the scaling variable ratio $R/{\xi }_{+}={\Theta }_2/{\Lambda }_{12}$. The black, red, green, and blue lines correspond to ${\Lambda }_{13}=1,{\Lambda }_{13}=1.5,{\Lambda }_{13}=2$, and ${\Lambda }_{13}=\infty$, respectively. The circles and arrows in panel (a) are schematic representations of the colloids and of the surface-to-surface distances between them; accordingly, they are not drawn to scale.

Figure 9

Ratio between the normalized scaling function ${\overline{K}}_{(+,+,+)}^{(1,x)}({\Theta }_{12},{\Lambda }_{12},{\Lambda }_{13})$ of the total horizontal CCF acting on colloid (1) and the normalized scaling function ${\overline{K}}_{(+,+,+)}^{(1,x)}({\Theta }_{12},{\Lambda }_{12},{\Lambda }_{13}=\infty )$ of the pairwise CCF between the colloids (1) and (2) for ${\Lambda }_{12}=1.25$. We consider the configuration of a right-angled triangle in (a) and of an isosceles triangle in (b). The scaling functions are shown as functions of the scaling variable ratio $R/{\xi }_{+}={\Theta }_2/{\Lambda }_{12}$. The black, orange, red, blue, and green curves correspond to ${\Lambda }_{13}=1,{\Lambda }_{13}=1.25,{\Lambda }_{13}=1.5,{\Lambda }_{13}=1.75,$ and ${\Lambda }_{13}=2$, respectively. Upon construction these ratios reduce to 1 in the limit ${\Lambda }_{13}\to \infty$. For not-too-small values of ${\Lambda }_{13}$ these ratios exhibit only a weak temperature dependence. The circles and arrows are schematic representations of the colloids and of the surface-to-surface distances between them; accordingly, they are not drawn to scale.