1. Introduction
Isotropic and nematic liquid crystals (LC), each representing distinct phases of matter, have been focal points of extensive research due to their diverse applications in optics, electronics, and display technology [1–5]. While the integration of colloidal particles into nematic liquid crystals has garnered considerable attention [1, 6–8], this manuscript takes a novel approach by focusing on the theoretical exploration of colloidal particle interactions within isotropic liquid crystals. In the nematic phase, the interaction of colloidal particles is strongly influenced by the anisotropic nature of the liquid crystal environment [9–12], and the long-range nature of these interactions is primarily attributed to the alignment of the liquid crystal molecules along a preferred direction known as the ‘director'. As a result, colloidal particles within the nematic phase tend to align with the local director, leading to a far-reaching influence that extends beyond the immediate vicinity of the particles.
Surface effects exert a significant influence on the interactions between colloidal particles in liquid crystals. The orientation of liquid crystal molecules at interfaces, dictated by boundary conditions such as planar, homeotropic, or hybrid anchoring, directly impacts colloidal alignment and the overall director fields [7, 10, 13–15]. This, in turn, in the nematic phase affects anisotropic interactions, potentially leading to the creation of topological defects [16–19]. The elastic deformations induced by colloidal particles in their neighborhoods extend over distances, influencing the stability of structures formed within the liquid crystal medium [20–23]. Moreover, boundary conditions play a crucial role in determining how colloidal particles respond to external fields, such as electric or magnetic fields, and can modulate the rheological properties of the system [6, 13–15, 24]. Researchers frequently leverage specific boundary conditions to engineer desired structures and functionalities, making it a pivotal aspect in the study of these complex materials [7, 10].
In contrast, the isotropic liquid crystal phase, marked by a lack of long-range molecular order and a random orientation of molecules, results in colloidal particle interactions that are typically considered short-range [12, 25, 26]. In this phase, the absence of a preferred molecular alignment limits the range over which colloidal particles influence each other, and interactions are predominantly governed by short-range forces and steric effects. However, the short-range interactions of colloidal particles are important for understanding the phenomenon in living cells. The cytoskeleton is a dynamic network of protein filaments within the cell, providing structural support, facilitating cellular movement, and participating in various cellular processes [27–31]. The major components of the cytoskeleton include microtubules, actin filaments, and intermediate filaments. Microtubules, which are tubular structures formed by polymerized tubulin subunits, can exhibit nematic or polar order in certain contexts. Nematic order refers to the parallel alignment of these filaments in a specific direction, reminiscent of the orientation seen in certain liquid crystal phases. Actin filaments, along with associated proteins, can form dynamic networks that display properties of a viscoelastic liquid. The continuous rearrangement and flow of actin networks are comparable to the behavior of liquid crystal materials. Both microtubules and actin filaments are polar filaments in the sense that one end of the filaments is different from the other. In the polar phase, the parallel alignment is different from anti-parallel alignment while, in the nematic phase, there is no distinction between the two types alignments.
Understanding and characterizing these distinctive features of colloidal interactions in both isotropic and nematic liquid crystal environments are pivotal for tailoring the properties of materials and for the design of advanced functionalities. In this work, we aim to unravel the intricacies of short-range phenomena, contributing valuable insights into the understanding of fundamental principles that govern the self-assembly, structural transitions, and dynamic behavior of colloidal particles in isotropic liquid crystal environments [20, 21, 32–34].
2. Model introduction
We consider a model in which two circular/spherical particles are suspended in a 2D/3D liquid crystal medium, as shown in figure 1. The centers of the two particles are separated by a distance of 2H. The liquid crystal is assumed to be made of polar molecules that carry a unit vector di, which represents the polarization direction. The average orientation p = ⟨di⟩ is used as an order parameter of the system. Note that p is not necessarily a unit vector. The magnitude of p represents the polarization strength and the direction of p represents the average orientation of the molecules. When ∣p∣ = 0, the liquid crystals are considered isotropic. The free energy density per unit area/volume of the liquid crystal reads
$\begin{eqnarray}\begin{array}{l}g=\displaystyle \frac{1}{2}\alpha {\left|{\boldsymbol{p}}\right|}^{2}+\displaystyle \frac{1}{2}{\kappa }_{1}{\left({\rm{\nabla }}\cdot {\boldsymbol{p}}\right)}^{2}+\,\displaystyle \frac{1}{2}{\kappa }_{2}{\left[{\boldsymbol{p}}\cdot ({\rm{\nabla }}\times {\boldsymbol{p}})\right]}^{2}\\ +\displaystyle \frac{1}{2}{\kappa }_{3}{\left|{\boldsymbol{p}}\times ({\rm{\nabla }}\times {\boldsymbol{p}})\right|}^{2}.\end{array}\end{eqnarray}$
The first term (named the polarization energy gp) implies that the liquid crystal is in an isotropic phase, such that ∣p∣ = 0 minimize the free energy. Here, we neglect the fourth-order term, which only leads to quantitative differences. The last three terms (named elastic energy ge) are analogized to the Frank–Oseen elastic energy that accounts for the deformation of splay, twist, and bend of the liquid crystal molecules, respectively [5, 12]. We adopt the one-constant approximation, where κ1 = κ2 = κ3 = κ, and introduce the attenuation length ${L}_{0}=\sqrt{\kappa /\alpha }$, representing how far the boundary anchoring effects can reach. We stress that the liquid crystal considered in this work is made of polar molecules. Mathematically, it means p and −p represent different states. This is different from the typical liquid crystal in which the director does not make a difference when the sign is flipped.
Figure 1. A schematic representation of the particles suspended in isotropic liquid crystal (2D). |
The interaction of colloidal particles in the liquid crystal arises from the surface-anchoring effect, which means the colloidal particles are able to induce polarization at the particle surface. This effect is phenomenologically described by a surface energy density gs, which depends, in principle, on the order parameter p of the liquid crystal. We assume the polarization induced by the colloidal particle surface is along its normal direction n (‘homeotropic' orientation), which is a unit vector pointing towards the outside of the surface. This effect is described by the Rapini–Papoular-type surface energy density [1, 35–37]
$\begin{eqnarray}{g}_{{\rm{s}}}=W{\left({\boldsymbol{p}}\cdot {\boldsymbol{n}}-{P}_{0}\right)}^{2},\end{eqnarray}$
where P0 is the preferred order parameter at the LC-particle interface. A positive/negative P0 represents that the polarization tends to point outward/inward. The coefficient W represents the strength of the uniform surface anchoring. In this study, we only consider strong anchoring conditions, such that P0 = ± 1 and the first type of boundary condition p = ± n is strictly imposed.The total free energy G can be obtained by the integration
$\begin{eqnarray}\begin{array}{l}G={\displaystyle \int }_{{\rm{\Omega }}}g{\rm{d}}V,\end{array}\end{eqnarray}$
where the integral region Ω is the space filled with LC, which is a functional of the order parameter p and its spatial derivatives. At equilibrium, the order parameter p tends to minimize the total free energy G. A variational method is used to obtain the equations for the order parameter p. The equations are numerically solved in COMSOL for two particles separated at difference distances D. We therefore obtain the total free energy G(D) as a function of D. The effective force between the two particles as a result of the surface-induced polarization field p is given by $\begin{eqnarray}F=-\displaystyle \frac{{\rm{d}}G}{{\rm{d}}D}.\end{eqnarray}$
3. Results
3.1. Interaction between colloidal particles in 2D geometry
We first consider a 2D case in which two circular particles with a radius of R are separated by a distance D = 2H and placed in a box, as shown in figure 1. The box is large enough such that results with larger boxes do not make a difference. The polar order parameter of the 2D liquid crystal reads ${\boldsymbol{p}}=\left[{p}_{x}\left(x,y\right),{p}_{y}\left(x,y\right)\right]$ and the total energy equation (1 ) becomes
$\begin{eqnarray}\begin{array}{l}g=\displaystyle \frac{1}{2}\alpha \left({p}_{x}^{2}+{p}_{y}^{2}\right)+\displaystyle \frac{1}{2}{\kappa }_{1}{\left(\displaystyle \frac{\partial {p}_{x}}{\partial x}+\,\displaystyle \frac{\partial {p}_{y}}{\partial y}\right)}^{2}\\ +\displaystyle \frac{1}{2}{\kappa }_{3}\left({p}_{x}^{2}+{p}_{y}^{2}\right){\left(\displaystyle \frac{\partial {p}_{x}}{\partial y}-\displaystyle \frac{\partial {p}_{y}}{\partial x}\right)}^{2}.\end{array}\end{eqnarray}$
We perform the variational method against px and py with equation (5 ) to obtain two variational equations.
$\begin{eqnarray}\begin{array}{l}{p}_{x}\left[\alpha +{\kappa }_{3}{\left(\displaystyle \frac{\partial {p}_{x}}{\partial y}-\displaystyle \frac{\partial {p}_{y}}{\partial x}\right)}^{2}\right]-{\kappa }_{1}\left(\displaystyle \frac{{\partial }^{2}{p}_{y}}{\partial x\partial y}+\displaystyle \frac{{\partial }^{2}{p}_{x}}{\partial {x}^{2}}\right)\\ -{\kappa }_{3}\left({p}_{x}^{2}+{p}_{y}^{2}\right)\left(\displaystyle \frac{{\partial }^{2}{p}_{x}}{\partial {y}^{2}}-\displaystyle \frac{{\partial }^{2}{p}_{y}}{\partial x\partial y}\right)\\ -2{\kappa }_{3}\left({p}_{x}\displaystyle \frac{\partial {p}_{x}}{\partial y}+{p}_{y}\displaystyle \frac{\partial {p}_{y}}{\partial y}\right)\left(\displaystyle \frac{\partial {p}_{x}}{\partial y}-\displaystyle \frac{\partial {p}_{y}}{\partial x}\right)=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{p}_{y}\left[\alpha +{\kappa }_{3}{\left(\displaystyle \frac{\partial {p}_{x}}{\partial y}-\displaystyle \frac{\partial {p}_{y}}{\partial x}\right)}^{2}\right]-{\kappa }_{1}\left(\displaystyle \frac{{\partial }^{2}{p}_{x}}{\partial x\partial y}+\displaystyle \frac{{\partial }^{2}{p}_{y}}{\partial {y}^{2}}\right)\\ +{\kappa }_{3}\left({p}_{x}^{2}+{p}_{y}^{2}\right)\left(\displaystyle \frac{{\partial }^{2}{p}_{x}}{\partial x\partial y}-\displaystyle \frac{{\partial }^{2}{p}_{y}}{\partial {x}^{2}}\right)\\ +2{\kappa }_{3}\left({p}_{x}\displaystyle \frac{\partial {p}_{x}}{\partial x}+{p}_{y}\displaystyle \frac{\partial {p}_{y}}{\partial x}\right)\left(\displaystyle \frac{\partial {p}_{x}}{\partial y}-\displaystyle \frac{\partial {p}_{y}}{\partial x}\right)=0.\end{array}\end{eqnarray}$
They are numerically solved with the general form of partial differential equations (PDE) in COMSOL. Two commonly used anchoring conditions are adopted here: (i) symmetric homeotropic anchoring. The surfaces of two colloidal particles induce the same polarization that is perpendicular to the surface. In particular, we set pleft = n and pright = n; (ii) anti-symmetric homeotropic anchoring. The surface polarizations are in opposite directions, pleft = n and pright = − n. Hereafter, we use $\bar{F}$ to represent the non-dimensionalized variable of F. The dimensionless variables are listed in table 1.Table 1. A list of dimensionless physical quantities. Here, ${\bar{G}}^{(2)}$ and ${\bar{F}}^{(2)}$ represent the free energy and the effective force in the 2D case, and ${\bar{G}}^{(3)}$ and ${\bar{F}}^{(3)}$ represent the free energy and the effective force in the 3D case. |
| Dimensionless variables | Expression | Dimensionless variables | Expression |
|---|---|---|---|
| $\bar{x}$ | x/L0 | $\bar{y}$ | y/L0 |
| $\bar{z}$ | z/L0 | $\bar{r}$ | r/L0 |
| $\bar{H}$ | H/L0 | $\bar{R}$ | R/L0 |
| ${\bar{G}}^{(2)}$ | $G/\alpha {L}_{0}^{2}$ | ${\bar{G}}^{(3)}$ | $G/\alpha {L}_{0}^{3}$ |
| ${\bar{F}}^{(2)}$ | F/αL0 | ${\bar{F}}^{(3)}$ | $F/\alpha {L}_{0}^{2}$ |
The distribution of p around two particles with symmetric homeotropic anchoring conditions is shown in figure 2 (a, b). The black arrows indicate the direction of p and the color codes indicate the magnitude. Note that we set ∣p∣ = 0 at the simulation box boundary. Although there are arrows near the box boundary, their magnitudes are almost zero (white color). When the two particles are far from each other, the order parameter p is radially distributed around the particles and the magnitude quickly decays to zero with distance. When they get closer, the distribution of p is strongly distorted, particularly in the middle of the lines connecting the center of the two particles, where the order parameter p is almost zero, similar to the electric-field diagram of two positive charges (figure 2(a)). The total free energy ${\bar{G}}_{\mathrm{tot}}$ monotonically decreases with the distance $\bar{H}$ between the two particles, which is primarily caused by the decrease in the elastic energy (figure 2(c)). The polarization energy ${\bar{G}}_{{\rm{p}}}$ remains almost constant. The effective repulsive force $\bar{F}$ can be very strong at small distances $\bar{H}$, and decays exponentially with $\bar{H}$ at large distances (figure 2(d)). The radii of the circular particles have a significant impact on the interaction. In general, larger particles have stronger repulsive forces. The relationship between the force $\bar{F}$ and the distance $\bar{H}$ for different particle radii $\bar{R}$ can be fitted with $\bar{F}\propto {{\rm{e}}}^{{A}_{0}\bar{R}}{{\rm{e}}}^{{M}_{0}\bar{H}}$, where A0 = 3.5, M0 are dimensionless constants. This is reflected in figure 2(e), where $\mathrm{ln}(\bar{F})-{A}_{0}\bar{R}$ versus $\bar{H}$ is plotted and the three different curves are found to collapse into a single master curve at large distance $\bar{H}$.
Figure 2. The interaction between two circular particles with symmetric homeotropic anchoring conditions in a 2D plane. (a, b) Distribution of the polar order parameter p around the two particles. The black arrows indicate the orientation of p, and the color codes represent the magnitude of p. The radius of the particles is $\bar{R}=0.8$, and the distance is $\bar{H}=1$ in (a) and $\bar{H}=3$ in (b). (c) The total free energy ${\bar{G}}_{\mathrm{tot}}$ (black line), elastic energy ${\bar{G}}_{{\rm{e}}}$ (red line), and polarization energy ${\bar{G}}_{{\rm{p}}}$ (blue line) are treated as functions of the distance $\bar{H}$ between the two particles. The dots in (c) indicate the corresponding points calculated in (a) and (b). (d) The relationship between the effective force $\bar{F}$ and the distance $\bar{H}$. The radii of particles are 0.8 (black line), 1.0 (red line), and 1.2 (blue line), respectively. (e) The relationship between $\mathrm{ln}(\bar{F})-{A}_{0}\bar{R}$ and $\bar{H}$. Here, A0 = 3.5. The radii of particles are 0.8 (black line), 1.0 (red line), and 1.2 (blue line), respectively. |
We next consider the interaction of two circular particles with anti-symmetric homeotropic anchoring conditions. When the particles are distant from each other, the polar order parameters p are radially pointing outward around the left particle and pointing inward around the right particle (figure 3(b)). When they get close, the distribution of p is similar to the electric field of two particles with opposite charges (figure 3(a)). It is found that the total free energy ${\bar{G}}_{\mathrm{tot}}$ is a non-monotonic function of the distance $\bar{H}$, as shown in figure 3(c). At very short distances, the total energy ${\bar{G}}_{\mathrm{tot}}$ decreases with the distance due to the decrease in the elastic energy ${\bar{G}}_{{\rm{e}}}$, while the polarization energy ${\bar{G}}_{{\rm{p}}}$ is increasing. At long distances, the total energy ${\bar{G}}_{\mathrm{tot}}$ slightly increases with $\bar{H}$ due to the increased polarization energy. To see how the force scales with the particle radii at long distances where the force is negative (attractive), we plot $\mathrm{ln}(-\bar{F})-{A}_{0}\bar{R}$ as a function of the distance $\bar{H}$ and find that the three curves of different radii collapse to a single master curve. This again implies that the force scales with the particle radii as $\bar{F}\propto {{\rm{e}}}^{{A}_{0}\bar{R}}$ (figure 3(e)). The constant A0 = 3.5 is the same in the symmetric conditions.
Figure 3. The interaction between two circular particles with anti-symmetric homeotropic anchoring conditions in a 2D plane. (a, b) Distribution of the polar order parameter p around the two particles. The black arrows indicate the orientation of p, and the color codes represent the magnitude of p. The radius of the particles is $\bar{R}=0.8$, and the distance is $\bar{H}=1$ in (a) and $\bar{H}=3$ in (b). (c) The total free energy ${\bar{G}}_{\mathrm{tot}}$ (black line), elastic energy ${\bar{G}}_{{\rm{e}}}$ (red line), and polarization energy ${\bar{G}}_{{\rm{p}}}$ (blue line) are treated as functions of the distance $\bar{H}$ between the two particles. The dots in (c) indicate the corresponding points calculated in (a) and (b). (d) The relationship between the effective force $\bar{F}$ and the distance $\bar{H}$. The radii of particles are 0.8 (black line), 1.0 (red line), and 1.2 (blue line), respectively. (e) The relationship between $\mathrm{ln}(-\bar{F})-{A}_{0}\bar{R}$ and $\bar{H}$. Only the parts below the x-axis are drawn. Here, A0 = 3.5. The radii of particles are 0.8 (black line), 1.0 (red line), and 1.2 (blue line), respectively. |
3.2. Interaction between colloidal particles in 3D geometry
We are now in a position to consider the interactions of two spherical particles suspended in a 3D space filled with isotropic liquid crystals. Two particles of a radius R are set apart by a distance of D = 2H along the z-axis through the center of the particles (see figure 4). We assume rotational symmetry around the z-axis and, in the cylindrical coordinate system (r, φ, z), the polar order parameter p can be simplified as ${\boldsymbol{p}}=\left[{p}_{r}\left(r,z\right),0,{p}_{z}\left(r,z\right)\right]$. The total energy equation (1 ) becomes
$\begin{eqnarray}\begin{array}{l}g=\displaystyle \frac{1}{2}\alpha \left({p}_{r}^{2}+{p}_{z}^{2}\right)+\displaystyle \frac{1}{2}{\kappa }_{1}{\left(\displaystyle \frac{{p}_{r}}{r}+\displaystyle \frac{\partial {p}_{r}}{\partial r}+\displaystyle \frac{\partial {p}_{z}}{\partial z}\right)}^{2}\\ +\displaystyle \frac{1}{2}{\kappa }_{3}\left({p}_{r}^{2}+{p}_{z}^{2}\right){\left(\displaystyle \frac{\partial {p}_{r}}{\partial z}-\displaystyle \frac{\partial {p}_{z}}{\partial r}\right)}^{2}.\end{array}\end{eqnarray}$
Figure 4. A schematic representation of the particles suspended in isotropic LC (3D). |
Similarly to what we have done for the 2D geometry, we perform a variational method with equation (8 ) against pr and pz
$\begin{eqnarray}\begin{array}{l}{p}_{r}\left\{\displaystyle \frac{{\kappa }_{1}}{{r}^{2}}+\alpha +{\kappa }_{3}\left[-{\left(\displaystyle \frac{\partial {p}_{r}}{\partial z}\right)}^{2}+{\left(\displaystyle \frac{\partial {p}_{z}}{\partial r}\right)}^{2}\right]\right\}\\ -{\kappa }_{1}\left(\displaystyle \frac{1}{r}\displaystyle \frac{\partial {p}_{r}}{\partial r}+\displaystyle \frac{{\partial }^{2}{p}_{z}}{\partial r\partial z}+\displaystyle \frac{{\partial }^{2}{p}_{r}}{\partial {r}^{2}}\right)\\ +2{\kappa }_{3}{p}_{z}\displaystyle \frac{\partial {p}_{z}}{\partial z}\left(-\displaystyle \frac{\partial {p}_{r}}{\partial z}+\displaystyle \frac{\partial {p}_{z}}{\partial r}\right)\\ +{\kappa }_{3}\left({p}_{r}^{2}+{p}_{z}^{2}\right)\left(-\displaystyle \frac{{\partial }^{2}{p}_{r}}{\partial {z}^{2}}+\displaystyle \frac{{\partial }^{2}{p}_{z}}{\partial r\partial z}\right)=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{p}_{z}\left[\alpha +{\kappa }_{3}{\left(\displaystyle \frac{\partial {p}_{r}}{\partial z}-\displaystyle \frac{\partial {p}_{z}}{\partial r}\right)}^{2}\right]-{\kappa }_{1}\left(\displaystyle \frac{1}{r}\displaystyle \frac{\partial {p}_{r}}{\partial z}+\displaystyle \frac{{\partial }^{2}{p}_{z}}{\partial {z}^{2}}+\displaystyle \frac{{\partial }^{2}{p}_{r}}{\partial r\partial z}\right)\\ +2{\kappa }_{3}\left(\displaystyle \frac{\partial {p}_{r}}{\partial z}-\displaystyle \frac{\partial {p}_{z}}{\partial r}\right)\left({p}_{r}\displaystyle \frac{\partial {p}_{r}}{\partial r}+{p}_{z}\displaystyle \frac{\partial {p}_{z}}{\partial r}\right)+\,\left({p}_{r}^{2}+{p}_{z}^{2}\right)\\ \times \left[\displaystyle \frac{{\kappa }_{3}}{r}\left(\displaystyle \frac{\partial {p}_{r}}{\partial z}-\displaystyle \frac{\partial {p}_{z}}{\partial r}\right)+{\kappa }_{3}\left(\displaystyle \frac{{\partial }^{2}{p}_{r}}{\partial r\partial z}-\displaystyle \frac{{\partial }^{2}{p}_{z}}{\partial {r}^{2}}\right)\right]=0.\end{array}\end{eqnarray}$
They can be solved with the general form of PDE in COMSOL. We demonstrate the results for symmetric and anti-symmetric homeotropic anchoring conditions in figures 5 and 6, respectively. The distribution of p is similar to that of the 2D case. Under symmetric conditions, the two particles repel each other at all distances. Under anti-symmetric conditions, the two particles repel each other at short distances and attract each other at long distances. Under both conditions, the interaction becomes exponentially weak at long distances.
Figure 5. The interaction between two particles with symmetric homeotropic anchoring conditions in 3D LCs. (a, b) Distribution of the polar order parameter p around the two particles on a cross-section of the r-z plane. The black arrows represent the orientation of p, and the color codes represent the magnitude of p. The particle radius $\bar{R}=0.8$, and the distance $\bar{H}=1$ in (a) and $\bar{H}=3$ in (b). (c) The total free energy ${\bar{G}}_{\mathrm{tot}}$ (black line), elastic energy ${\bar{G}}_{{\rm{e}}}$ (red line), and polarization energy ${\bar{G}}_{{\rm{p}}}$ (blue line) are treated as functions of the distance $\bar{H}$ between the two particles. The dots in (c) indicate the corresponding points calculated in (a) and (b). (d) The relationship between the effective force $\bar{F}$ and the distance $\bar{H}$. The radii of particles are 0.8 (black line), 1.0 (red line), and 1.2 (blue line), respectively. (e) The relationship between $\mathrm{ln}(\bar{F})-{A}_{0}\bar{R}$ and $\bar{H}$. Here, A0 = 5.2. The radii of particles are 0.8 (black line), 1.0 (red line), and 1.2 (blue line), respectively. |
Figure 6. The interaction between two particles with anti-symmetric homeotropic anchoring conditions in 3D LCs. (a, b) Distribution of the polar order parameter p around the two particles on a cross-section of the r-z plane. The black arrows represent the orientation of p, and the color codes represent the magnitude of p. The particle radius $\bar{R}=0.8$, and the distance $\bar{H}=1$ in (a) and $\bar{H}=3$ in (b). (c) The total free energy ${\bar{G}}_{\mathrm{tot}}$ (black line), elastic energy ${\bar{G}}_{{\rm{e}}}$ (red line), and polarization energy ${\bar{G}}_{{\rm{p}}}$ (blue line) are treated as functions of the distance $\bar{H}$ between the two particles. The dots in (c) indicate the corresponding points calculated in (a) and (b). (d) The relationship between the effective force $\bar{F}$ and the distance $\bar{H}$. The radii of particles are 0.8 (black line), 1.0 (red line), and 1.2 (blue line), respectively. (e) The relationship between $\mathrm{ln}(-\bar{F})-{A}_{0}\bar{R}$ and $\bar{H}$. Only the parts below the x-axis are drawn. The constant A0 = 5.2. The radii of particles are 0.8 (black line), 1.0 (red line), and 1.2 (blue line), respectively. |
It is noted that the effective forces make almost no difference in 3D and in 2D for both symmetric and anti-symmetric anchoring conditions (figure 7, compare the solid lines and the dashed lines). In both 2D and 3D geometry, the force scales with the particle radii as $\bar{F}\propto {{\rm{e}}}^{{A}_{0}\bar{R}}$. The constant A0 = 3.5 in 2D, and A0 = 5.2 in 3D. This means that the size has a stronger impact in 3D than in 2D.
Figure 7. Comparison between 2D and 3D cases. (a, b) The relationship between the effective force $\bar{F}$ and the distance $\bar{H}$ in 2D (solid line) and 3D (dash line) with symmetric homeotropic anchoring conditions in (a) and anti-symmetric homeotropic anchoring conditions in (b). The radii of particles are 0.8 (black line) and 1.2 (blue line), respectively. |
4. Discussion
In this work, we have shown that the interaction between two colloidal particles as a result of the surface-induced polarization is short-ranged. The results are obtained by numerically solving the variational equations in a large enough box. To show that the results are independent of the size of the box, we perform calculations in boxes of various sizes. Figure 8 demonstrates the free energy $\bar{G}$ as a function of the distance $\bar{H}$ in a box of the original size and a larger one. The two curves perfectly overlap such that they are visually indistinguishable, demonstrating the validity of our calculation.
Figure 8. The relationship between the free energy $\bar{G}$ and the distance $\bar{H}$ in a large box and in a small box for a particle size $\bar{R}=1.2$. (a, b) The results with symmetric homeotropic anchoring conditions in 2D (a) and 3D (b). (c, d) The results with anti-symmetric homeotropic anchoring conditions in 2D (c) and 3D (d). For 2D geometry, the small box has a length of 80L0 and a width of 40L0, and the large box has a length of 120L0 and a width of 60L0. For 3D geometry, the small box has a height of 80L0, and a radius of 20L0, and the large box has a length of 120L0 and a radius of 60L0. The small boxes in (a) and (c) are the same as in figures 2 and 3; those in (b) and (d) are the same as in figures 5 and 6. |
Estimating or measuring the short-range interaction between particles is challenging. Molecular dynamics (MD) simulations are typically used to estimate the short-range force. The repulsive pressure can be up to more than 1000 bar when the distance is smaller than 0.5 nm in MD research [38–40]. Atomic force microscopes have been used to measure the repulsive force of a silica particle when approaching a silica substrate, and the magnitude of the force is up to 2000 pN [41, 42].
Short-range interactions between colloidal particles in isotropic liquid crystals are particularly important for phenomena that occur on a local or mesoscopic scale. These interactions, which operate over relatively short distances, play a crucial role in various phenomena. For instance, these forces could lead to the aggregation and clustering of colloidal particles on a local scale, influencing the overall spatial distribution of particles. Short-range interactions contribute to the local rheological properties of the isotropic liquid crystal. The viscosity and elasticity of the material in the vicinity of colloidal particles are influenced by these short-range forces, impacting the local flow behavior. Short-range interactions also influence the response of colloidal particles to external stimuli. For example, changes in the temperature, pressure, or chemical environment can lead to local rearrangements and modifications in short-range interactions, influencing the overall response of the colloidal system.
Understanding and controlling these short-range interactions are essential for tailoring the properties of materials in isotropic liquid crystals, particularly in applications where local organization, dispersion, and dynamic behavior are critical.
5. Conclusion
In this work, we have studied the interaction between two colloidal particles in an isotropic LC. The interaction is a result of surface-induced polarization of the LC molecules, and decays exponentially with distances. We found that the short-range interactions are repulsive under symmetric homeotropic anchoring conditions. However, under anti-symmetric homeotropic anchoring conditions, the interaction is attractive at intermediate distances and repulsive at very short distances. These results hold in both 2D and 3D geometry. In both geometries, the particle radii have a strong impact on the force, which scales as $\bar{F}\propto {{\rm{e}}}^{{A}_{0}\bar{R}}$ at long distances.