Dynamics and Structure ofJanus Particles

Kyoto UniversityDepartment of Chemical Engineering

Okura Tatsuya

2016

Kyoto UniversityDepartment of Chemical Engineering

Dynamics and Structure of Janus Particlesby Okura Tatsuya

Abstract

We apply direct numerical simulations to investigate the behavior of Janus particles undershear flow. The Janus particle is modeled as a sphere composed of two distinct hemispheres.The self-assembly of Janus particles has been investigated so far both in computer simulationsand in experiments, however, detailed understanding of even this simplest case is still missing.In this research, we investigate the structure and dynamical properties of Janus particlesdispersions, as a function of shear rate, temperature, volume fraction and interaction strength.

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Symbols

C Anisotropic interaction strengthkB Boltzmann constantφfp Body forcex Direction of shear flowrij Distance between particle i and jρf Fluid densityvf Fluid velocityη Fluid viscoisityFH

i Hydrodynamic forceNH

i Hydrodynamic torqueλ Interaction rangeMi Mass of particle iIi Moment of inertia of particle iξ Interface thicknessUjanus(rij, qi, qj) Janus potentialQi Orientation matrixqi Normalized orientaion vectorN Number of particlesσ Particle diametera Particle radiusϕ Particle volume fractionPe Peclet numberRi Position of particle iU(rij, qi, qj) Potentialg000(r) Radial distribution functiong110(r) Radial distribution function taken into account the relative orientationg101(r) Radial distribution function taken into account the relative orientationUrepulsion(rij) Repulsion potentialΩi Rotational velocityγ Shear rateφ(x) Smooth profile functionσf Stress tensorT Temperaturet Timeϵ Truncated LJ potential’s energy unitI Unit tensor

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Contents

Abstract i

Symbols ii

1 Introduction 21.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Previous Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Simulation Method 3

3 Potential Model 43.1 Repulsion Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Janus Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4 Results and Discussion 64.1 Binary Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4.1.1 Simulation Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 64.1.2 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4.2 Multi-particle Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.2.1 Simulation Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 74.2.2 Structure Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.2.3 Numerical Analysis of the Structure . . . . . . . . . . . . . . . . . . . 8

4.3 Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5 Conclusions and Future Work 13

References 14

Acknowledgements 15

Appendix 16

1

1 Introduction

1.1 Overview

A Janus particle (named after two-faced Janus god) is modeled as a sphere composed oftwo symmetric hemispheres, characterized by different surface properties. According to thesurface characteristics, these particles can form a variety of clusters such as micelles, vesicles,or layers. Such a system would be useful for the design of nano-materials which have potentialapplications as drug delivery system or emulsion stabilizers.The self-assembly of Janus particles has been investigated so far both in computer simulationsand in experiments, however, detailed understanding of even this simplest case is still missing.In this research, we investigate the dynamical properties and the structure of Janus particlesdispersions, focusing on the process of cluster forming under shear flow, as a function of shearrate, temperature, volume fraction and interaction strength.

1.2 Previous Research

Previous research[1] has studied what happens to Janus particles under shear flow.

Fig. 1: Binary simulation

(a) Initially, the tails of both particles are facing each other, resulting in maximum attraction.(b) Then, shear makes particle move and rotate, decreasing the mutual attraction.(c) Finally, the attractive contribution (in Ujanus) becomes sufficiently small so that shearcan break the pair up.

On the other hand, multi-particle simulations suggest that at low shear rate, shear canhelp to breakup and reform aggregates which have energetically unfavorable configurations.Furthermore, shear flow increases the mobility of the dispersed particles and this improves theprobability of merging free particles and unstable aggregates. However, this process cannotbe sustained indefinitely, and clusters rapidly decay when shear rate is increased further.

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2 Simulation Method

In order to investigate the behavior of Janus particles, we applied direct numerical simulationsof spherical particles, using the smoothed profile (SP) method[4]. In the SP method, theboundary between the colloidal particle and the host fluid is replaced with a continuousinterface by assuming a smoothed profile, as shown in Fig.2 and 3 . This simple modificationenabled us to accurately characterize the hydrodynamic interactions.The motion of the particles is obtained by solving the Newton-Euler equations of motion fora given particle position Ri, translational velocity Vi and rotational velocity Ωi:

Ri = Vi (1)

MiVi = FHi + F other

i (2)

Ii · Ωi = NHi (3)

Qi = skew(Ωi) ·Qi (4)

where

skew(Ω) =

0 −Ω3 Ω2

Ω3 0 −Ω1

−Ω2 Ω1 0

(5)

Mass and moment of Inertia are denoted as Mi and Ii, respectively, and Qi is the orientationmatrix. The hydrodynamic torque and force exerted by the solvent on the particle arerepresented as NH

i and FHi , respectively.

In the SP method, the dynamic of the fluid is described by the Navier-Strokes (NS) equationfor a given value of fluid viscosity η and density ρf :

(∂t + uf · ∇)uf = ρ−1f ∇ · (−pI + σf ) (6)

∇ · uf = 0 (7)

for the stress tensor σf , fluid velocity uf and pressure field p, under the incompressibilitycondition(∇ · uf = 0).The basic idea of the SP method is to solve the modified NS equation over the entire domain,by treating the colloids as fluid particles, where particles are represented using a smoothphase field 0 ≤ φ(x, t) ≤ 1, which removes the troublesome boundary conditions at theparticle surface. Here φ = 0 stands for the fluid, 0 < φ < 1 describes the interface and φ = 1the particle domain. The rigidity of the particles can be maintained by introducing a bodyforce φfp in the NS equation.

Fig. 2: SP method Fig. 3: SP method

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3 Potential Model

One of the simplest Janus particle models is a sphere where one half is covered with anattractive patch and the other one is covered with a repulsive patch.

Fig. 4: Janus particles model

We call the blue part the tail and orange one the head. In our simulations, we model theinteraction between two Janus particles as follows. In terms of potential energy, we take intoaccount not only distance but also orientation. The potential energy is described as a sumof Repulsion Potential (Urepulsion) and Janus Potential (Ujanus).

U(rij, qi, qj) = Urepulsion(rij) + UJanus(rij, qi, qj) (8)

where rij is the distance between particle i and j, normalized vectors qi and qj denote theorientation of particle i and j. Fig.5 shows the Janus pair potential model for four distinctconfigurations. In this model, tail to tail is the most stable configuration.

Fig. 5: Janus pair potential model for four distinct configurations

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3.1 Repulsion Potential

Urepulsion includes an isotropic contribution, depending only on the distance between theparticles. We adopted a truncated LJ potential as the repulsion potential to avoid overlappingof the particles.

Urepulsion(rij) =

4ϵ

[( σrij)2n − ( σ

rij)n]+ ϵ (rij ≤ 21/nσ),

0 (rij > 21/nσ).(9)

where rij = |Ri − Rj|, the parameters σ = 2a and ϵ = 1.0 denote the length and energyunits, respectively. Especially, we employed 38-16-truncated LJ potential (n = 16) as shownin Fig.6.

Fig. 6: Repulsion potential

3.2 Janus Potential

On the other hand, UJanus includes an anisotropic contribution, which depends not only onthe distance but also on the orientation of the particles.

UJanus(rij, qi, qj) = Φ(rij)(qj − qi) · rij (10)

Φ(rij) =Cσ

rijexp[−λ(rij − σ)] (11)

where σ is the diameter of the spheres and the parameters C and λ denote the interactionstrength and range, respectively. We set λσ = 3 to avoid getting unphysical structures.

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4 Results and Discussion

4.1 Binary Simulations

4.1.1 Simulation Conditions

At first, we conducted some simulations with two particles. We put two particles with atail to tail configuration at the center of the system. Then, we added the different value ofshear in the direction of x. In these simulations, we investigated the effect of shear rate andinteraction strength on the initial state.

Table 1: Binary simulation conditions

Box size 64 * 64 * 64Time step 100 * 300γ 10−4 - 10−2

kBT 1σ 6C 1.0 - 7.0

Fig. 7: Initial state for two particles which are in a tail-to-tail configuration

4.1.2 Phase Diagram

The phase diagram shows the stability of a pair for interacting particles as a function ofJanus attraction and shear rate as shown in Fig.8.Horizontal axis is Peclet number (Pe) and shear rate. Vertical axis is interaction strength.Pe is defined to be the ratio of the rate of advection by the flow to the rate of diffusion.Furthermore, the cluster was considered a stable state when its composition remained un-changed over the considered simulation time. When hydrodynamic force affects two particlesover Janus attraction, they cannot keep the tail to tail configuration.

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Fig. 8: Phase diagram for a pair stability

4.2 Multi-particle Simulations

4.2.1 Simulation Conditions

Next, we conducted multi-particle simulations to investigate the behavior of Janus particleswith C = 5.0 at volume fractions ϕ = 0.01, 0.2 and 0.3.

Table 2: Multi-particle simulation conditions

Box size 128 * 128 * 128Time step 300 * 500γ 0 - 0.02kBT 0.1 - 1σ 6C 5.0

4.2.2 Structure Analysis

In order to identify the structure, we employed the radial distribution functions defined as:

g000(r) =V

4πr2N2<

∑i

∑j =i

δ(r − rij) > (12)

g110(r) =V

4πr2N2<

∑i

∑j =i

δ(r − rij)qi · qj > (13)

g101(r) =V

4πr2N2<

∑i

∑j =i

δ(r − rij)qi · rij > (14)

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(i)g000(r) counts the number of neighboring particles in a shell with distance r from a particle.(ii)g110(r) measures the relative orientation of two particles, where positive values representparallel and negative values represent antiparallel orientation on the average.(iii)g101(r) also reflects the orientation of two particles. Specifically, positive values indicatetail to tail configuration. Negative values indicate head to head configuration.

4.2.3 Numerical Analysis of the Structure

When we look at the detail of the structure, we observed micelles such as icosahedronscomposed of 13 particles, including a single particle as the cluster center, at low volumefraction ( ϕ = 0.01) and high interaction strength relative to the temperature (C/kBT = 50),as shown in Fig.9.This behavior has been already reported in the literature[1].

Fig. 9: Snapshot Fig. 10: Regular icosahedron

We tried to find a steady state from the time change of the total energy per particle. Thetotal energy of the system includes potential energy and kinetic energy. We extracted thedata after the steady state (time > 8700) and then, we calculated three radial distributionfunctions.

Fig. 11: Time change of energy per particle

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Fig. 12: Radial correlation function

Fig. 13: Radial correlation function Fig. 14: Radial correlation function

We obtained three peak positions almost corresponding to an ideal icosahedron, describedas dotted line in the Fig.12-14. In this sense, our microscopic data confirms the icosahedralstructure indicated by the snapshot. From the different distribution functions, we confirmthat there is almost no head to head configuration with distance σ from every particles

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More interestingly, at higher volume fraction (ϕ = 0.2), we observed larger micelles thanicosahedrons. Then they become linked to one another in the direction of shear flow andfinally merge into huge, elongated micelles. They look like ”columns”.

Fig. 15: elongated micelles

Fig. 16: elongated micelles Fig. 17: elongated micelles

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When ϕ = 0.3, in the absence of shear (γ = 0), we did not find any well-regulated structure.But they look like gyroidal structures. As the shear rate is increased up to γ = 0.02, thestructure has changed and eventually turned into clear layers.

Fig. 18: gyro Fig. 19: tetra-layers

Fig.20-22 show radial correlation functions at γ = 0.02 and ϕ = 0.3 where we found asort of layer. From the numerical analysis, we confirm that there are a lot of side by sideconfigurations and tail to tail configurations, looking at the different g(r). Obviously, theyare not icosahedrons anymore. The structures we obtained seem to be like tetra-layers.

Fig. 20: Radial correlation function

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Fig. 21: Radial correlation function Fig. 22: Radial correlation function

4.3 Rheology

Regarding the rheology of such systems, we expect that there is a correlation between theirviscosity and large scale structure. Fig.23 shows that at low volume fraction viscosity isconstant regardless of shear rate. At high volume fraction, however, especially when ϕ = 0.3the preliminary results represent shear thickening.Although we compared normal colloids with Janus particles at the same conditions excepthaving Janus patches, we did not observed shear thickening with normal colloids as shownin Fig. . We found this behavior at the end of our research, thus we do not fully understandthe mechanism yet. In future work, we will consider this matter in detail.

Fig. 23: Viscosity

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5 Conclusions and Future Work

In conclusions, we have analyzed the structure as a function of shear rate, temperature, vol-ume fraction and interaction strength, and observed different types of aggregation dependingon these factors. Interestingly, we found shear thickening at high volume fraction which wehave not observed without Janus attraction.

However, we did not find any clear bilayers (lamellar structures) or vesicles. We suspect thatthis is because of the lack of any attractive interaction for side-by-side configurations. Infuture work, we will consider the new Janus potential model[2] described as:

UJanus(rij, qi, qj) = Φ(rij)(qj − qi) · rij +Φ(rij)

43(qi · rij)2 + 3(qj · rij)2 − 2 (15)

Φ(rij) =Cσ

rijexp[−λ(rij − σ)] (16)

Fig. 24: New potential model

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References

[1] E. Bianchi, A. Z. Panagiotopoulos, and A. Nikoubashman, Soft Matter, 11, 3767-3771(2015).

[2] Rosenthal, Gubbins, and Klapp, J. Chem. Phys., 136, 174901 (2012).

[3] Avvisati, Vissers, and Dijkstra, J. Chem. Phys., 142, 084905 (2015).

[4] Adnan Hamid, ”Direct Numerical Simulation Studies of Sedimentation of Spherical Par-ticles” (2014).

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Acknowledgements

This final thesis has been possible only because of kind cooperation by many individuals.I would like to begin by saying a special thank to my supervisor Assistant Prof. John JairoMolina for providing me with valuable insight into the nature of complex phenomena. Ishould thank Prof. Yamamoto Ryoichi and Associate Prof. Taniguchi Takashi for giving mesome good advice in the seminar.Thank you also to all the members in this laboratory for helping me to understand the basicsof chemical engineering and sharing lovely time in a daily laboratory life.Finally, I would like to thank particularly my family for their love and financial support.Without you, the work reported in this paper would not have been possible. I am reallygrateful for all your help.

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Appendix

Fig. 25: Three g(r) will give us which configuration we obtain

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