^{1,a)}, Brian T. Gettelfinger

^{1}, Gary M. Koenig Jr.

^{1}, Nicholas L. Abbott

^{1}and Juan J. de Pablo

^{1,b)}

### Abstract

A mesoscale theory for the tensor order parameter is used to investigate the structures that arise when spherical nanoparticles are suspended in confined nematic liquid crystals(NLCs). The NLC is “sandwiched” between a wall and a small channel. The potential of mean force is determined between particles and the bottom of the channels or between several particles. Our results suggest that strong NLC-mediated interactions between the particles and the sidewalls of the channels, on the order of hundreds of , arise when the colloids are inside the channels. The magnitude of the channel-particle interactions is dictated by a combination of two factors, namely, the type of defectstructures that develop when a nanoparticle is inside a channel, and the degree of ordering of the nematic in the region between the colloid and the nanochannel. The channel-particle interactions become stronger as the nanoparticle diameter becomes commensurate with the nanochannel width. Nanochannel geometry also affects the channel-particle interactions. Among the different geometries considered, a cylindrical channel seems to provide the strongest interactions. Our calculations suggest that small variations in geometry, such as removing the sharp edges of the channels, can lead to important reductions in channel-particle interactions. Our calculations for systems of several nanoparticles indicate that linear arrays of colloids with Saturn ring defects, which for some physical conditions are not stable in a bulk system, can be stabilized inside the nanochannels. These results suggest that nanochannels and NLCs could be used to direct the assembly of nanoparticles into ordered arrays with unusual morphologies.

The authors are grateful to Orlando Guzmán for helpful discussions. This work was supported by the University of Wisconsin Nanoscale Science and Engineering Center (NSEC) on Templated Synthesis and Assembly at the Nanoscale (Research Thrust 3, Driven Nano-Fluidic Self-Assembly of Colloids and Macromolecules).

I. INTRODUCTION

II. MODELS AND METHODS

A. Details of the model systems

B. Mesoscale theory for the nematic liquid crystal

III. RESULTS AND DISCUSSION

A. Effect of the ratio particle diameter/nanochannel width

B. Effect of nanochannel geometry

C. Systems with several nanoparticles

IV. CONCLUDING REMARKS

### Key Topics

- Nanoparticles
- 69.0
- Nematic liquid crystals
- 45.0
- Colloidal systems
- 35.0
- Crystal defects
- 31.0
- Liquid crystals
- 30.0

## Figures

Scheme of the different nanochannel geometries considered in this work: (a) rectangular, (b) rectangular with two straight cuts, (c) rectangular with four straight cuts, (d) cylindrical, (e) cylindrical with two straight cuts, and (f) rectangular nanochannel for a system of three nanoparticles. For all cases, the model systems consist of a wall and a nanochannel, “sandwiching” one or several spherical nanoparticles immersed in a nematic liquid crystal (NLC). All the dimensions are in nanometers. The total length of the system in the direction is for (a)–(e) and for (f).

Scheme of the different nanochannel geometries considered in this work: (a) rectangular, (b) rectangular with two straight cuts, (c) rectangular with four straight cuts, (d) cylindrical, (e) cylindrical with two straight cuts, and (f) rectangular nanochannel for a system of three nanoparticles. For all cases, the model systems consist of a wall and a nanochannel, “sandwiching” one or several spherical nanoparticles immersed in a nematic liquid crystal (NLC). All the dimensions are in nanometers. The total length of the system in the direction is for (a)–(e) and for (f).

(Color online) Potential of mean force (PMF) as a function of the minimum distance between one nanoparticle and the bottom of a rectangular nanochannel for different colloid diameters: (a) total PMF, (b) Landau–de Gennes contribution to the total PMF, and (c) elastic contribution to the total PMF.

(Color online) Potential of mean force (PMF) as a function of the minimum distance between one nanoparticle and the bottom of a rectangular nanochannel for different colloid diameters: (a) total PMF, (b) Landau–de Gennes contribution to the total PMF, and (c) elastic contribution to the total PMF.

(Color online) 3D visualizations of the NLC defect structures (represented as the contour in red) and 2D contour maps of the scalar order parameter , superimposed with the director field in the plane, for one nanoparticle with (a)–(c) and (d)–(f), at different values of : (a) (far apart from both the top wall and the nanochannel), (b) [the maximum in the LdG PMF, Fig. 2(b)], (c) [the minimum in the total and elastic PMF, Figs. 2(a) and 2(c)], (d) (far apart from both the top wall and the nanochannel), (e) [the maximum in the LdG PMF, Fig. 2(b)], and (f) [the minimum in the total and elastic PMF, Figs. 2(a) and 2(c)]. For all cases, the particle is equidistant from the sidewalls of the channel.

(Color online) 3D visualizations of the NLC defect structures (represented as the contour in red) and 2D contour maps of the scalar order parameter , superimposed with the director field in the plane, for one nanoparticle with (a)–(c) and (d)–(f), at different values of : (a) (far apart from both the top wall and the nanochannel), (b) [the maximum in the LdG PMF, Fig. 2(b)], (c) [the minimum in the total and elastic PMF, Figs. 2(a) and 2(c)], (d) (far apart from both the top wall and the nanochannel), (e) [the maximum in the LdG PMF, Fig. 2(b)], and (f) [the minimum in the total and elastic PMF, Figs. 2(a) and 2(c)]. For all cases, the particle is equidistant from the sidewalls of the channel.

(Color online) 3D visualizations of the NLC defect structures (represented as the contour in red, top) and 2D contour maps of the scalar order parameter , superimposed with the director field in the plane (bottom), for one nanoparticle with at (a) (far apart from both the top wall and the nanochannel), (b) [the maximum in the LdG PMF, Fig. 2(b)], (c) [the first minimum in the total and elastic PMF, Figs. 2(a) and 2(c)], (d) [the minimum in the LdG PMF, Fig. 2(b)], and (e) [the second minimum in the total and elastic PMF, Figs. 2(a) and 2(c)]. The particle is closer to one of the sidewalls of the channel. (see text)

(Color online) 3D visualizations of the NLC defect structures (represented as the contour in red, top) and 2D contour maps of the scalar order parameter , superimposed with the director field in the plane (bottom), for one nanoparticle with at (a) (far apart from both the top wall and the nanochannel), (b) [the maximum in the LdG PMF, Fig. 2(b)], (c) [the first minimum in the total and elastic PMF, Figs. 2(a) and 2(c)], (d) [the minimum in the LdG PMF, Fig. 2(b)], and (e) [the second minimum in the total and elastic PMF, Figs. 2(a) and 2(c)]. The particle is closer to one of the sidewalls of the channel. (see text)

(Color online) Potential of mean force (PMF) as a function of the minimum distance between one nanoparticle and the bottom of a nanochannel for different channel geometries and two colloid diameters, and : (a) total PMF, (b) Landau–de Gennes contribution to the total PMF, and (c) elastic contribution to the total PMF.

(Color online) Potential of mean force (PMF) as a function of the minimum distance between one nanoparticle and the bottom of a nanochannel for different channel geometries and two colloid diameters, and : (a) total PMF, (b) Landau–de Gennes contribution to the total PMF, and (c) elastic contribution to the total PMF.

(Color online) 2D contour maps of the scalar order parameter , superimposed with the director field in the plane, when a nanoparticle of is close to the top wall (left), and when the particle is inside the nanochannel (at the minima of the total PMF, center). For the latter situation, 3D visualizations of the NLC defect structures are also depicted (represented as the contour in red, right). Different nanochannel geometries are represented as follows: (a) rectangular, (b) rectangular with two cuts, (c) rectangular with four cuts, (d) cylindrical, and (e) cylindrical with two cuts. The particle is always equidistant from the sidewalls of the channel.

(Color online) 2D contour maps of the scalar order parameter , superimposed with the director field in the plane, when a nanoparticle of is close to the top wall (left), and when the particle is inside the nanochannel (at the minima of the total PMF, center). For the latter situation, 3D visualizations of the NLC defect structures are also depicted (represented as the contour in red, right). Different nanochannel geometries are represented as follows: (a) rectangular, (b) rectangular with two cuts, (c) rectangular with four cuts, (d) cylindrical, and (e) cylindrical with two cuts. The particle is always equidistant from the sidewalls of the channel.

(Color online) 2D contour maps of the scalar order parameter , superimposed with the director field in the plane, when a nanoparticle of is close to the top wall (left), and when the particle is inside the nanochannel (at the minima of the total PMF, center). For the latter situation, 3D visualizations of the NLC defect structures are also depicted (represented as the contour in red, right). Different nanochannel geometries are represented as follows: (a) rectangular, (b) rectangular with two cuts, (c) rectangular with four cuts, (d) cylindrical, and (e) cylindrical with two cuts. The particle is always equidistant from the sidewalls of the channel.

(Color online) 3D visualizations of the NLC defect structures (represented as the contour in red) for different arrays of three nanoparticles with in a NLC “sandwiched” between a wall and a rectangular nanochannel: (a) nanoparticles are far apart from the nanochannel, the top wall, and each other (minimum interparticle distance ); (b) nanoparticles are close together forming a triangular array , and far apart from the nanochannel and the top wall; (c) nanoparticles are inside the nanochannel but far apart from each other ; and (d) nanoparticles are inside the nanochannel and close together , forming a linear array.

(Color online) 3D visualizations of the NLC defect structures (represented as the contour in red) for different arrays of three nanoparticles with in a NLC “sandwiched” between a wall and a rectangular nanochannel: (a) nanoparticles are far apart from the nanochannel, the top wall, and each other (minimum interparticle distance ); (b) nanoparticles are close together forming a triangular array , and far apart from the nanochannel and the top wall; (c) nanoparticles are inside the nanochannel but far apart from each other ; and (d) nanoparticles are inside the nanochannel and close together , forming a linear array.

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