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Liquid Crystal Mediated Interactions Between Nanoparticles in a Nematic Phase V. Tomar,† T. F. Roberts,† N. L. Abbott,† J. P. Hernández-Ortiz,*,‡ and J. J. de Pablo*,† †

Department of Chemical and Biological Engineering, University of Wisconsin−Madison, Madison, Wisconsin 53706, United States Departamento de Materiales, Universidad Nacional de Colombia, Sede Medellín, Medellín, Colombia

‡

ABSTRACT: A continuum theory is used to study the interactions between nanoparticles suspended in nematic liquid crystals. The free energy functional that describes the system is minimized using an Euler−Lagrange approach and an unsymmetric radial basis function method. It is shown that nanoparticle liquid-crystal mediated interactions can be controlled over a large range of magnitudes through changes of the anchoring energy and the particle diameter. The results presented in this work serve to reconcile past discrepancies between theoretical predictions and experimental observations, and suggest intriguing possibilities for directed nanoparticle self-assembly in liquid crystalline media.

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INTRODUCTION The directed self-assembly of small particles is of considerable technological interest for the development of advanced materials and devices. In recent years, liquid crystals (LCs) have been considered as potentially controllable media to mediate the assembly of micrometer colloidal particles.1−32 In such systems, particle assembly is driven by the elastic deformation caused by the inclusion of colloidal particles, and the subsequent reorganization of the liquid crystal and particles to minimize the free energy. Past studies of colloids in LCs have largely focused on large particles (diameters greater than 1 μm). Experimental1,2,8,16,25,31,33 and theoretical7−13,16−18,32,34−38 works have reported strong, LC-mediated attractive energies between such particles on the order of hundreds of kBT. For smaller nanoparticles, available predictions using continuum theories and molecular simulations also indicate that attractive forces between are of order 100 kBT.9,13,14,39 Recent experiments, however, suggest that interactions between 150 nm diameter gold nanoparticles are much weaker than that, in the range of only a few kBT.40 The actual magnitude of the LC-mediated forces between nanoparticles is central to their use for applications; assembly of strong attractive particles would be irreversible and difficult to control, whereas weaker interactions would be much more amenable to external manipulation and directed self-assembly. The aim of this work is to identify the source of the discrepancy between past predictions and recent experiments and, in doing so, provide a theoretical framework to better understand the LC-mediated assembly of nanoparticles. In this work, we use numerical simulations based on a continuum theory to analyze the interactions between nanoparticles suspended in a liquid crystal. In particular, we investigate the effects of particle size and anchoring energy at the particle surface. Our analysis reveals that nanoparticle LC© 2012 American Chemical Society

mediated interactions are highly sensitive of surface anchoring, an issue that was widely overlooked in past investigations of colloids in LCs where strong anchoring was implicitly or explicitly assumed. This work is organized as follows: in Section II the theoretical model and the thermodynamic description of the system are presented. Results are then shown for particles with homeotropic anchoring followed by the nondegenerative planar anchoring. We conclude with a summary of the most important results.

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THEORETICAL MODEL The description of the thermodynamics of phase transitions in liquid crystals adopted here, requires that an internal structural parameter be used to characterize the degree of alignment. The average directional cosine between a particular molecular direction u and the director n is not a convenient option because its average will vanish for most situations. We therefore use the second moment of n given by the following:41,42 Q(x , t ) = MII(x , t ) −

δ 3

(1)

where δ is the 3 × 3 identity tensor and the second moment MII is given by the following: MII(x , t ) =

∫ nnψ(n, x, t )d n

(2)

where ψ(n,x,t) is the configurational distribution function of orientation n at point x and time t. According to this Received: October 20, 2011 Revised: February 3, 2012 Published: March 12, 2012 6124

dx.doi.org/10.1021/la204119p | Langmuir 2012, 28, 6124−6131

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definitions, tr(Q) = 0 while tr(MII) = 1. The tensor order parameter Q can be diagonalized in terms of its eigenvalues, i.e., ⎛ 2S 0 0 ⎜ ⎜ 3 ⎜ η−S 0 Q=⎜0 3 ⎜ ⎜ −η − ⎜0 0 ⎝ 3

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ S⎟ ⎟ ⎠

The elastic contribution to the free energy functional, assuming a uniaxial system, can be written in terms of the tensor order parameter Q as follows:63,64 1 ∂Q ij ∂Q ij 1 ∂Q jk ∂Q jl L1 + L2 ∂xk ∂xl 2 ∂xk ∂xk 2 ∂Q ∂Q kl 1 1 ∂Q jk ∂Q jl + L3Q ij kl + L4 2 ∂xi ∂xj 2 ∂xl ∂xk

fE =

(3)

where we have used Einstein index notation and the Li constants are related to the elastic moduli and the scalar order parameter (or maximum eigenvalue) by the following:

where the maximum eigenvalue, 2S/3 and (η−S)/3, is associated with the scalar order parameter S(x). The secondlargest eigenvalue,( −η − S)/3 + η, corresponds to the biaxiality η(x). The eigenvector corresponding to the maximum eigenvalue is the director n, and n′ is the eigenvector for the second-largest eigenvalue. These eigenvectors define an orthogonal coordinate system that allows the tensor Q to be written as (eq 3). A general form of the tensor in terms of its eigenvalues and eigenvectors can be defined as follows:43−46 ⎡ 1 ⎤ Q = S⎢nn − δ⎥ + η[n′n′ − (n × n′)(n × n′)] ⎣ 3 ⎦

L1 =

∫ d3x[fLdG (Q) + fE (Q)] + ∮ d2 xfS (Q)

L3 =

6S 2

(k33 − k11 + 3k22)

1 2S3

(k33 − k11)

1 L4 = 2 k24 S

(4)

(8)

(9)

(10)

(11)

The kij constants are nonvanishing elastic moduli that describe the independent modes of deformation, namely: splay (k11), twist (k22), bend (k33), and saddle-splay (k24). In the present study, we adopt a one-elastic constant approximation: k11 = k22 = k33 and k24 = 0; such an assumption has been widely used in the literature9,15,26,57,65−67 based on the geometric and long-range conditions of the system of interest. The final term in (eq 5) describes the interaction of the liquid crystal with the colloidal surfaces. We adopt a RapiniPapoular type term of the form,14,24,26,68,69 fS =

1 W (Q − Q0)2 2

(12)

where W represents the strength of the uniform surface anchoring and Q0 is the tensor order parameter preferred at the LC-particle interface. Two commonly observed anchoring conditions are considered in this study: (i) homeotropic anchoring and (ii) nondegenerative planar anchoring. Given the fact that we are interested in homogeneous surfaces and equilibrium conditions, we assume that the preferred planar anchoring comes from a nondegenerative orientation at the particles′ surface, observed in several experiments.70−72 Other studies of particles have used the Rapini-Papoular potential for homeotropic anchoring,9,10,13,62,66,73,74 which facilitates comparison of our findings to literature results. The degenerative planar anchoring case can be included through the fourth order Fournier-Galatola potential,75 which includes projections of the order parameter tensor at the surface. For completeness, we also performed simulations using the Fournier-Galatola potential, with essentially the same results as those obtained by the Rapini-Papoular potential. Planar anchoring conditions are thus taken into account by computing a preferred tensor Q0 following the definition in (eq 4), i.e.,

(5)

The short-range free energy is represented by a Landau-de Gennes expression of the form,57−61 fLdG =

1

1 L 2 = 2 (k11 − k22 − k24) S

Homogeneous nematic configurations are uniaxial, and the biaxiality is zero. However, in situations where Q is allowed to vary, the biaxiality can be nonzero.41−43 Including η in a numerical analysis increases its complexity. Although several past efforts in the literature have considered biaxial systems,44−49 uniaxiality is a common simplification which is used even to study phases that are biaxial, such as Smectic C phases.50−56 We therefore assume the system to be uniaxial, in which case the tensor order parameter Q simplifies and it contains all of the necessary information to describe the system thermodynamically: its maximum eigenvalue defines the scalar order parameter S and the corresponding eigenvector describes the director field n. The thermodynamic description of the system starts with the definition of a free energy functional F(Q). This free energy functional is divided into a term responsible for the phase transitions f LdG (short-range), a term that penalizes the nonhomogeneous distortions, disclinations and deformations f E (long-range) and a term that describes the surface-LCs interactions f S. The free energy is then defined using a Q dependent functional of the form,41,42,57 F(Q) =

(7)

A⎛ U⎞ AU AU 2 ⎜1 − ⎟tr(Q ) − tr(Q3) + tr(Q2)2 2⎝ 3⎠ 3 4 (6)

where A and U are phenomenological coefficients that can depend on temperature (or concentration) and pressure. These parameters set an energy scale for the phase transition A, and a nematic strength U.10,11,13,42,62

⎡ 1 ⎤ Q0 = S⎢n 0n 0 − δ⎥ ⎣ 3 ⎦ 6125

(13)


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For homeotropic anchoring conditions, we have the following: ⎡ 1 ⎤ Q0 = S⎢vv − δ⎥ ⎣ 3 ⎦

along with the constraints: N

∑ αjPk(xj) = 0,

(14)

j

In (eq 13), we have: n 0(x) = n(x) − [v(x) ·n(x)]v(x)

Q

Φ(r ) = r 2m − 2 log r = r 6 log r

P3(x) = αN + 1x13 + αN + 2x23 + αN + 3x33 + αN + 4x12x2 + αN + 5x12x3 + αN + 6x22x1 + αN + 7x22x3 + αN + 8x32x1 + αN + 9x32x2 + αN + 10x12

(16)

+ αN + 11x22 + αN + 12x32 + αN + 13x1x2 + αN + 14x1x3 + αN + 15x2x3 + αN + 16x1 + αN + 17x2 + αN + 18x3 + αN + 19

∏⎜ Q

for x ∈ V and

∏⎜ Q

(24)

Details of the RBF method can be found in a previous publication from our group78 and additional literature.79−86 A domain size of 40R (where R is the particle radius) along the axis through the center of the particles and 16R along the other two axes in a full 3D geometry is adopted in all simulations (see Figure 1). We have verified that our results are

(17)

⎛ δF ⎞ ·v⎟ = 0 ⎝ δ∇Q ⎠

(23)

together with a corresponding cubic polynomial of order three:

for any B(x)∈ 93 × 93 for any x ∈ V and x ∈ σS. The free energy must satisfy the following Euler−Lagrange equations,

⎛ δF ⎞ ⎟=0 ⎝ δQ ⎠

(22)

Here, N is the number of collocation nodes and αj are real coefficients. For this work, we use a generalized thin-plate spline of order four:

(15)

where n(x) is the local director (eigenvector) from Q(x) and v(x) is the normal unit vector at any point x along the particle surface σS. To minimize the free energy functional (eq 5) a projector operator ΠQ is defined according to the natural construction of the tensor Q, which is symmetric and traceless (eq 1), 1 1 ∏ (B) = (B + BT ) − tr(B)δ 2 3

1≤k≤m−1

(18)

for x ∈ σS where v is the unit outward normal to the surface. The Volterra derivatives are defined by the following:76,77 δF ∂F ∂ ∂F = − · δQ ∂Q ∂x ∂∇Q

(19)

The solution to these equations is found by allowing the tensor order parameter Q to evolve toward equilibrium according to a Ginzburg−Landau relaxation equation of the form, ⎤ ⎡ ⎛ δF ⎞⎥ ∂Q 1⎢ = − ⎢∏ ⎜ ⎟⎥ ∂t γ ⎝ δQ ⎠ ⎦ ⎣Q

(20)

with boundary conditions given in (eq 18), and where γ is a rotational viscosity (or diffusion) coefficient.26,32,78 Here, the effects of fluid flow are neglected; they are not expected to influence the equilibrium configurations. The Ginzburg−Landau relaxation is carried out numerically using a semi-implicit Euler scheme for the time derivative, whereas for the spatial derivatives an augmented-unsymmetric radial basis function (RBF) approach is employed. Nonlinear terms are treated semi-implicitly with an explicit linearization of the cross terms that include derivatives of the corresponding variable.32,65,78,79 For the augmented RBF method, an unknown function g(x) is approximated as a linear combination of radial basis functions of order m: Φj(||x − xj||), which depend on the distances between a subset of trial centers X ⊂ 9 n, {xj ∈ X,j = 1,...,N},80−86 and a set of polynomials Pm−1 of order m−1 as follows:

Figure 1. Schematic representation of the particles suspended in a liquid crystal with two preferred orientations at the particle surface.

not affected by finite domain boundary effects. An adaptive RBF collocation point distribution was implemented, where regions close to the particles had more collocation points than regions far from them. In addition, an adaptive time step was selected in order to obtain faster equilibration during relaxation.

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The numerical values for the Landau-de-Gennes energy constant A in (eq 6) and the elastic constant k11 in (eq 7) are selected to represent a 4′-pentyl-4-cyanobiphenyl (5CB).57,87 5CB is a widely used low-molecular weight thermotropic liquid crystal. The values used here for the Landau-de-Gennes and elastic-splay free energy functionals are A = 1 × 105 J/m3 and k11 = 5−7 × 10−12 N.67,88−90 The isotropic−nematic transition is controlled by parameter U in the Landau-de-Gennes free energy functional, which sets a

N

g (x ) ≅

∑ αjΦ( j

RESULTS

x − xj ) + Pm − 1(x) (21) 6126


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value for the bulk scalar order parameter given by the following10,42,62 Sbulk =

1 3 8 1− + 4 4 3U

(25)

Bulk systems are in the isotropic phase for values of U < 2.8, and in the nematic phase for U ≥ 2.8.11,13,62 We use U = 5, which corresponds to a bulk scalar order parameter S = 0.76. This set of parameters defines a characteristic length scale, the nematic coherence length ξN = (18L1/AU)1/2, which in turn determines the range over which the elastic and Landau-deGennes contributions are similar. For 5CB, ξN = 13.4 nm. We first consider a single nanometer-sized particle suspended in a bulk liquid crystalline medium with homeotropic anchoring conditions. The particle radius is R = 25 nm, around 2 times the nematic-coherence length. Two different anchoring conditions are considered: (i) strong anchoring, with W = 10 mJ/m2, and (ii) moderate anchoring, with W = 0.3 mJ/m2. The anchoring strength also defines a characteristic length scale where the elastic contributions become comparable to the surface anchoring; the so-called extrapolation length, defined as ξW = L1/W, is ξW = 0.43 nm for strong anchoring and ξW = 14.4 nm for weaker anchoring. Figure 2 shows the equilibrium defect

Figure 3. (Color online) Three-dimensional representation of the defect structure and contour plot of the scalar order parameter S along the horizontal plane passing through the center of two nanoparticles, with R = 25 nm and strong anchoring with W = 10 mJ/m2. (a) Distance between particles r = 80 nm, (b) r = 40 nm and (c) r = 10 nm. Notice that once the separation between particles is the same order of magnitude than as nematic-coherence length (r ≈ ξN) the defects interact to form the three-ring structure. Past work has examined in detail how the defects surrounding two nanoparticles evolve and interact with each other in the case of infinite anchoring.9,13,14.

placed close to each other, under strong homeotropic anchoring conditions. In the three examples shown in the figure the Saturn rings are perpendicular to the director n; the angle between the vector joining the centers of the particles r and the normal to the director is labeled α. When r is perpendicular to n, α = 0. As shown in the figure, the two particles exhibit Saturn ring defects when separated by a large distance. Those rings interact when the particles approach each other, until a threering defect structure forms when the particles are close to each other. The three-ring defect structure is in good agreement with results from previous molecular and mesoscale studies of liquidcrystal mediated interactions between nanoparticles.9,11,13 Note that once the separation between particles is of the same order of magnitude as the nematic-coherence length (r ≈ ξN) the defects interact to form the three-ring structure. A potential of mean force (PMF) ω(r) is defined as follows: ω(r ) = F(r ) − F(r → ∞)

(26)

where F(r) is the free energy of the system when the particles are separated by a distance r, and F(r → ∞) is the free energy of the system when the particles are far from each other, i.e., r ≫ ξN. Figure 4 shows the PMF as a function of separation between the particles for different anchoring strengths. The range of W was chosen to represent experimentally observed values for the strength of anchoring energy.19,91−94 Here we note that analytical expressions have been derived34,35 for the potential of mean force in the far field, for intermediate to large separations between the particles. In the far field, the free energy of interaction between two particles is anisotropic, with a minimum for angles in the vicinity of α = 40°. Figure 5 shows results for the energy of interaction between two particles calculated analytically from expressions given in ref 34, a different values of particle separation r, and an anchoring energy W = 3.7 mJ/m2. The figure also shows results from our numerical calculations. To facilitate comparison between the two sets of results, the results for the energy have been normalized according to (F − Fmin)/(Fmax − Fmin), where Fmin denotes the energy at the minimum, and Fmax is the energy at the maximum of the curve. With this normalization, it becomes easier to highlight the angular dependence of the free energy and different curves (corresponding to different separations) can be displayed in a single figure. Figure 5 shows that for large separations there is good agreement between our results and

Figure 2. (Color online) Defect structure around a particle and contour plot of the scalar order parameter S along the vertical plane passing through the center of the particle. The black lines represent the local director field n of the LC along the plane. (a) R = 25 nm and strong homeotropic anchoring W = 10 mJ/m2, the isosurface of the defect corresponds to S = 0.6. (b) R = 25 nm and modest homeotropic anchoring W = 0.3 mJ/m2, the isosurface of the distorted region corresponds to S = 0.7.

structure and a contour plot of the scalar order parameter, S, along a plane passing through the center of the particle and aligned with the director vector in the bulk LC. For strong anchoring, the particle exhibits a saturn ring defect (Figure 2(a)). In the contour plot of the scalar order parameter S, the black lines represent the local director field n of the LC along the plane. The observed defect structure and the LC orientation around the particle are in good agreement with previous observations.9,11,13 When the anchoring is weaker (Figure 2(b)), the Saturn ring becomes a stripe-like distorted region at the surface of the particle. In order to analyze liquid crystal mediated interactions we now consider a system of two nanoparticles separated by a distance, r. Figure 3 shows results for three cases: (a) two nanoparticles separated by a large distance (b) two nanoparticles at intermediate separation, and (c) two nanoparticles 6127


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particles approach each other, it reaches a maximum, and then decreases to a negative value. The particles exhibit strong attractive forces once they are close to each other. A binding energy, FB, is defined as the value of the PMF when the separation between the particles is 3 nm; this distance roughly corresponds to two liquid crystal molecular lengths. For R = 25 nm and W = 11.2 mJ/m2, the binding energy is around −68 kBT; for W = 0.14 mJ/m2 the binding energy is around −1 kBT. Clearly, the binding energy between two nanoparticles in LC depends strongly on the anchoring energy W. Also note that the LC-mediated initial repulsion and the range of interaction between the particles decreases with the decrease in the magnitude of the anchoring energy. Figure 6 shows the binding energy as a function of anchoring energy for different particle radii. The binding is observed to

Figure 4. Potential of mean force, ω, as a function of the separation between particles r, for R = 25 nm and different anchoring strengths. The results shown range from weak or intermediate anchoring, at W = 0.14 mJ/m2, to strong anchoring, at W = 11.2 mJ/m2.

Figure 6. Binding energy FB, between two nanoparticles in a nematic liquid crystal as a function of the strength of anchoring energy W, for different particle sizes.

increase with increasing anchoring energy. Two distinct regimes can be observed. For weak or intermediate anchoring energies, the binding energy changes significantly with anchoring energy. In contrast, for strong anchoring energies (higher than approximately 2 mJ/m2), the binding energy is found to be relatively insensitive to anchoring energy and to reach an asymptotic value. The main result of our analysis, namely the dependence of LC-mediated attractive interactions between nanoparticles on anchoring energy, serves to reconcile the disagreement between past theoretical predictions (that assumed strong anchoring) and recent experimental observations. For small colloids, 150 nm in diameter, recent experiments40 reported weak attractive interactions, of the order of a few kBT. For similar sized particles, past numerical studies had predicted much stronger binding energies, on the order of 100 kBT. For the more experimentally relevant case of anchoring energies in the vicinity of 0.1 mJ/m2, we now find that the binding energy is much weaker than that. Figure 7 shows results for the binding energy, FB, as a function of W over a wide range of particle radii, R = 12.5−80 nm. For all values of W, the interparticle interactions increase with increasing size. For strong anchoring, FB increases rapidly with increasing diameter and reaches values of several dozen kBT for particles as small as 50 nm in diameter. In contrast, for weak or intermediate anchoring, FB increases much more gradually, thereby providing an effective means by which to control particle assembly in liquid crystals. Figure 8 shows results for the effect of the nematic strength (S) on the interaction between nanoparticles. We see that the magnitude of such interactions decreases with decreasing

Figure 5. Normalized interaction energy as a function of the angle between particles for three different separations: r = 59, 111, and 147 nm; R = 25 nm and W = 3.17 mJ/m2. These are center to center distances. Continuous lines come from analytical predictions,34 while dotted lines and points represent the numerical calculations. Two configurations are shown for α = 40 with the defect structure around the particles; the iso-surfaces represent a value of S = 0.47.

the analytical approximations. For short distances, however, the approximations break down and significant differences are observed between our numerical results and the far-field theories. In particular, we see that at short distances the minimum in the free energy corresponds to the case α = 0, where the vector joining the particles is perpendicular to the director. Since our aim is to calculate the binding energy between two particles, as opposed to the path followed by the particles as they approach each other, in the remainder of this work we restrict our calculations to the case α = 0, which corresponds to the strongest interactions at short distances. For strong anchoring conditions, e.g., W = 11.2 mJ/m2, in the α = 0 direction the PMF initially increases once the 6128


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Figure 7. Binding energy, FB, between two nanoparticles in a liquid crystal as a function of the radius of the particles R, for varying strengths of anchoring energy W.

Figure 9. (Color online) Defect structure around a nanoparticle, and contour plot of the scalar order parameter S along the vertical plane passing through the center of the particle. The black lines represent the local director field n of the LC along the plane. (a) R = 25 nm and strong planar anchoring W = 10 mJ/m2, the isosurface of the defect corresponds to S = 0.6. (b) R = 25 nm and weak planar anchoring W = 0.3 mJ/m2, the isosurface of the defect corresponds to S = 0.7.

opposite ends of the particle. The local alignment of the director vector along a plane though the center of the particle is shown in Figure 9(a). The two point defects on the opposite sides of the particle are observed to become less prominent for the cases with weak anchoring conditions, as shown in Figure 9(b). Figure 10 shows a 3D view of the defect structure and a

Figure 10. (Color online) Defect structure and contour plot of the scalar order parameter S along the horizontal plane passing through the center of two nanoparticles with R = 25 nm and strong anchoring with W = 10 mJ/m2. (a) Distance between particles r = 80 nm (b) r = 40 nm and (c) r = 10 nm. Notice that once the separation between particles is the same order of magnitude as the nematic-coherence length (r ≈ ξN) the defects interact to form the ring structure.

Figure 8. Potential of mean force ω and binding energy FB as a function of the separation between two nanoparticles r, for R = 25 nm, suspended in a liquid crystal for varying bulk scalar order parameter S.

nematic strength. The nematic strength, however, does not influence the effect of W on interparticle interactions. Also note that the LC-mediated initial repulsion between the nanoparticles decreases as the nematic strength decreases from S = 0.76 to 0.60, and disappears as the nematic strength decreases from S = 0.60 to 0.50. The nematic strength, which in experiments can be controlled through temperature or concentration, therefore provides another means of controlling the assembly of nanoparticles in liquid crystals. For completeness, we also consider liquid-crystal mediated interactions under nondegenerate planar anchoring conditions. Planar anchoring conditions are characterized by alignment of the mesogens along the local plane of the LC-particle interface. Figure 9 shows a 3D view of the defect structure and a contour plot of the scalar order parameter S, along a plane passing through the center of the particle and aligned with the director vector in the bulk, with W = 10 mJ/m2 and W = 0.3 mJ/m2. As before, these two cases are representative of strong and weak anchoring conditions, respectively. For strong planar anchoring conditions, a nanoparticle exhibits two point defects located at

contour plot of the scalar order parameter S for a two-particle system for three cases representative of (a) two nanoparticles separated by a large distance (b) two nanoparticles at intermediate separation, and (c) two nanoparticles close to each other. As shown in the figure, a two-particle system exhibits two point defects at the opposite ends of the particles when separated by a large distance. The point defects are observed to interact with each other when the particles approach each other along the axis formed by the two point defects, followed by subsequent collapse of the two point defects into a single line defect (a ring) between the two particles when they are close to each other. Figure 11 shows the potential of mean force ω between two nanoparticles, R = 25 nm, in a liquid crystal as a function of the separation r. For strong planar anchoring conditions, e.g., W = 10 mJ/m2, the PMF increases as the particles approach each other, it reaches a maximum and subsequently decreases to a negative value. For R = 25 nm and W = 10 mJ/m2 the binding 6129


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ACKNOWLEDGMENTS This work is supported by the Department of Energy, Basic Energy Sciences, Biomaterials Program, DE-SC0004025. The original development of algorithms and methods employed here was supported by NSF, DMR-0425880. J.P.H.-O. is grateful to COLCIENCIAS, the Universidad Nacional de Colombia−Faculty of Mines and Vice-Dean of Research travel program.

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Figure 11. Potential of mean force ω as a function of the separation between the particles r, under planar anchoring conditions for varying strength of anchoring energy W. A strong homeotropic case has also been included for comparison.

energy was found to be ∼15 kBT. As before, the binding energy decreases with decreasing anchoring energy. More generally, our results for the effect of anchoring energy on the interactions between nanoparticles under planar anchoring conditions are analogous to those obtained under homeotropic anchoring conditions, shown in Figures 4−8. The main difference is that the attractive interactions between the particles, i.e., the binding energy, are considerably weaker for planar anchoring. In addition, the initial repulsion between the particles is more pronounced for planar anchoring than for homeotropic anchoring. A representative comparison between the PMF as a function of separation for the two anchoring conditions is also shown in Figure 11.

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CONCLUSIONS Our results indicate that nanoparticles suspended in liquid crystals exhibit attractive interactions whose magnitude is highly sensitive to the anchoring energy at the particles' surface. For particles of diameter 150 nm and an anchoring energy of W = 0.1 mJ/m2, the attractive energy between the particles is approximately 5 kBT, in good agreement with recent experiments. The effect of anchoring energy is more pronounced for moderate strengths than for stronger anchoring. As the particles' diameter increases, the effect of anchoring energy becomes less pronounced. We also find that the interaction between nanoparticles increases with the strength of the nematic ordering. These general observations apply to both homeotropic and parallel anchoring. The main difference between these two cases is that the LC-mediated forces between particles are considerably smaller for planar anchoring conditions as compared to homeotropic anchoring conditions. Perhaps more interestingly, the results presented in this work show that one can tune the interaction between nanoparticles suspended in liquid crystals over a wide range of magnitudes (from a few kBT to hundreds of kBT) through control of interfacial chemistry (anchoring) and size, thereby suggesting new avenues for nanoparticle self-assembly in liquid crystals.

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Corresponding Author

*E-mail: [email protected] (J.P.H.-O.); depablo@ engr.wisc.edu (J.J.d-P.). Notes

The authors declare no competing financial interest. 6130


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