Shearing Janus Nanoparticles Confined in Two-Dimensional Space

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Letter

Shearing Janus Nanoparticles Confined in Two-Dimensional Space: Reshaped Cluster Configurations and Defined Assembling Kinetics Zihan Huang, Pengyu Chen, Ye Yang, and Li-Tang Yan J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.6b00724 • Publication Date (Web): 10 May 2016 Downloaded from http://pubs.acs.org on May 10, 2016

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The Journal of Physical Chemistry Letters

Shearing Janus Nanoparticles Confined in Two-Dimensional Space: Reshaped Cluster Configurations and Defined Assembling Kinetics Zihan Huang, Pengyu Chen, Ye Yang, and Li-Tang Yan* Advanced Materials Laboratory, Department of Chemical Engineering, Tsinghua University, Beijing 100084, P. R. China Corresponding Author * Li-Tang Yan, email: [email protected]

ACS Paragon Plus Environment

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ABSTRACT: The self-assembly of anisotropic nanoparticles (ANPs) possesses a wide array of potential applications in various fields, ranging from nanotechnology to material science. Despite intense research of the thermodynamic self-assembly of ANPs, elucidating their nonequilibrium behaviors under confinement still remains an urgent issue. Here, by performing simulation and theoretical justification, we present for the first time a study of the shear-induced behaviors of Janus spheres (the most elementary ANPs) confined in two-dimensional space. Our results demonstrate that the collective effects of shear and bonding structures can give rise to reshaped cluster configurations, featured by the chiral transition of clusters. Scaling analysis and numerical modeling are performed to quantitatively capture the assembling kinetics of dispersed Janus spheres, thereby suggesting an exotic way to bridge the gap between anisotropic and isotropic particles. The findings highlight confinement and shearing engineering as a versatile strategy to tailor the superstructures formed by ANPs towards unique properties. TOC GRAPHICS

KEYWORDS anisotropic nanoparticle, spatial confinement, shear flow, reshaped cluster configuration, defined assembling kinetics


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Thermodynamic self-assembly of anisotropic nanoparticles (ANPs), which plays a crucial role in discovery and design of new materials, has always been the research focus since the advent of novel synthesis strategies brought spectacular varieties of nano-sized building blocks.13

By rationally designing and changing the surface patterns of ANPs, novel structures such as

kagome lattice,4 supracolloidal helices,5 and one-component icosahedral quasicrystals,6 have been realized through experiments and simulations. These remarkable findings not only possess significant potential to generate materials with unique properties,7-10 but also make ANP a potential candidate to reveal fundamental principles of anisotropic interactions.11-14 However, the behaviors of ANPs at nonequilibrium state (particularly, under external fields) still remain less known, leading to a great demand for corresponding physical descriptions. Meanwhile, as these nano-sized particles enable real-space experiments to provide particle-level insight into atomic systems,15-16 it is essential and attractive to focus considerable attention on this urgent issue. As the simplest and most elementary ANP, Janus sphere whose two surface domains have distinct properties has been extensively studied theoretically and experimentally.17 Numerous investigations of its directed self-assembly have been reported in both twodimensional (2D)18-21and three-dimensional (3D)22-25 geometries. These previous excellent works indicate that Janus sphere could be the optimal choice to explore the nonequilibrium features of ANPs. To implement this strategy, external electric26 or magnetic27 fields have been applied to study the corresponding nonequilibrium behaviors of Janus particles, resulting in unique phenomena that follow new criteria rather than conventional equilibrium thermodynamics. In particular, extensive experimental studies of Janus particles focus on the 2D geometry,4, 18, 20, 28-30 as a consequence of the confinement generated by gravity4,

20

or buoyancy.18,

28-30

In stark

contrast to 3D space, such a spatial confinement may lead to novel structures with unique properties,4, 29-35 especially when considering the key role of the collective effects of translational and rotational motions of Janus spheres in their dynamic behaviors. This aspect is particularly


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important for the systems under shear flow, which directly affects the transport of particles and may open doors to extensive applications of nanoparticles at nonequilibrium state.36 However, the shear-induced behaviors of Janus spheres in 2D confinement have not yet been addressed, leaving a critical and urgent task to be accomplished. In this Letter, the landscape of Janus-sphere systems subjected to linear shear flow is constructed by systematic computer simulations. We investigate the influence of shear on the dynamics of clusters formed by Janus spheres [i.e., Janus clusters (JCs)], and the assembling kinetics of dispersed Janus spheres. In sharp contrast to 3D space, Janus spheres under 2D confinement are more likely to form small micelles, straight chains and 120° kinks due to the restricted geometry,18,

19

thereby making the cluster morphologies and assembling kinetics

significantly different from those in 3D space.37, 38 We investigate the shear dynamics of these clusters, and propose a fundamental mechanistic account for the shear-induced disassembly of JCs from the perspective of energy and bonding structures. In particular, the collective effects of shear and bonding structures are found to lead to unique dynamic behaviors and reshaped cluster configurations, such as the chiral transition of kinked JCs. Thus, a potential way to tailor the assembled structures of ANPs is provided by using external fields. Moreover, the shear-rate dependence of assembling kinetics of dispersed Janus spheres is examined, and three regimes representing distinct effects of shear are identified upon increasing shear rate for dilute suspensions. Particularly, a power-law relation describing this shear-rate dependence is found in the intermediate regime, where the exponent is nearly independent of the Janus balance. Further, we find that the assembling kinetics can be captured by the virial expansion of clustering rate. Thereby an equation that can be harnessed to predict the clustering rate of Janus spheres based on that of isotropic particles is established, suggesting an exotic way to bridge the gap between ANPs and isotropic particles. Our findings offer a fundamental and novel insight into the


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nonequilibrium ANP systems, and highlight the versatile strategy harnessing confinement and external fields to tailor the superstructures formed by ANPs towards unique properties.

Figure 1. (a) Representative Janus-sphere system under linear shear flow. Small blue beads represent the fluid particles of MPCD solvent. (b) The model of Janus spheres. The blue domain is hydrophobic (HPB) while the orange domain is hydrophilic (HPL). (c) The calculated velocity distributions of pure solvent when γ& = 0.2, 0.5, 1.0 τ-1 respectively. The solid lines represent the theoretical values. The inset schemes the meanings of ux and y. (d)-(f) Phase diagrams of the dynamic behaviors of small micelles with various γ& and εatt: dimer (d), trimer (e) and tetramer (f), where kB is the Boltzmann constant. Regimes of diffusion (marked in light blue), tumbling (TB), reversible alignment (RA) and breakup (B) are included in the phase diagrams, where the insetting images scheme the details. Trimer disassembles into one dimer and one Janus sphere in Regime B1, and three individual Janus spheres in Regime B2. Regime C in (f) is not subdivided due to the complicated behaviors of tetramer in this regime. Here we utilize a hybrid simulation method, which combines multiparticle collision dynamics (MPCD)39-41 for solvent with molecular dynamics (MD) for Janus spheres, to investigate the dynamics of JCs under shear (See Supporting Information for details). This


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method considers both hydrodynamic interactions and thermal noises. The linear shear flow in x direction with shear rate γ& , is generated by moving two no-slip hard walls (Figure SI.1a), and the validity of this approach is demonstrated in Figure 1c. The mass, length and time unit of system are m0, a0 and τ respectively, and the temperature of the system is T. The model of Janus sphere is given in Figure 1b, with radius R = 1.5a0 and hydrophobic range depicted by angle θmax. The interaction potential between Janus spheres is presented in Supporting Information. In our simulations, individual JC is sheared to avoid the interplay among different clusters, which may complicate the dynamics of the system.42 Indeed, in 2D confinement, even small micelles (dimer, trimer and tetramer) show complex nonequilibrium features when applying a linear shear. Figure 1d-f present different dynamic behaviors of these JCs in response to various potential depths of the hydrophobic range, εatt, and shear rates γ& , where the insetting images scheme the details. For extremely weak shear ( γ& ~ 10-3 τ-1), the diffusion dominates over the shear, making the motion of JC similar with that at equilibrium state. As γ& increases, the effect of shear becomes more prominent, and JCs thereby undergo a periodic tumbling motion. Trimer or tetramer with triangular bonding can rotate as a whole group where constituent particles change their positions and orientations synchronously (Figure SI.2). In contrast, shearinduced orientation takes place for dimer owing to its single bond. Further, the disassembly of a cluster is expected when γ& is high enough. Indeed, dimer with only one bond breaks up directly, while the triangular bonding makes the breakup of trimer and tetramer more complicated. Upon the fracture of only one bond, the constituent particles can reconfigure into a deformed cluster, leading to unique shear responses such as reversible alignment for trimer (top inset of Figure 1e). All the detailed dynamic behaviors are displayed in Movies I – III. Moreover, these observed behaviors also hold for the quasi-2D systems when the confinement is strong (Figure SI.3).


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These results show that phase transitions of JCs are expected when shear is applied. In particular, our findings suggest that these phase transitions may be first-order-like, and very small hysteresis effects can be identified in the transition dynamics (Figure SI.4). 104

6 9 12

7 10 13 15

(b) 40

8 11 14 16

γ = 0.24 τ-1 γ = 0.22 τ-1 γ = 0.20 τ-1

B

L

φ

d TP

γ = 0.18 τ-1 γ = 0.16 τ-1 γ = 0.13 τ-1

30

Ek (kBT)

(a)

TP (τ)

20

D

10

TB

1.5

103

ϕ (rad)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.0 0.5 0

2000

10

t (τ)

4000

0.0 6000

-3

-1

κγ (τ )

10-2

0 100

200

300

400

t (τ)

Figure 2. (a) The log-log plots of TP versus κ γ& where N changes from 6 to 16. The red line is the theoretical prediction made by eq 1. The right inset shows the definitions of the length L, width d and orientation angle φ of rodlike JC, the aspect ratio p is defined by p = L/d. The bottom inset is the time dependence of φ when N = 8 and γ& = 0.10 τ-1. (b) The kinetic energy of the tail particle (marked in red) Ek versus time at different γ& when N = 8. Three dynamic behaviors, namely tumbling (TB), deformed cluster (D) and breakup (B) are schemed by the insetting snapshots. The two dashed lines represent the threshold values of Ek. Next we turn to large JCs which present extended structures, including straight chains and 120° kinks, in 2D confinement18 (Figure SI.5). In view of the obvious geometric differences, we consider the influence of shear on both these cluster structures respectively. A straight chain of JC can be seen as a rod [i.e., rodlike Janus cluster (RJC)], and the Brownian translational and rotational diffusions of them do be similar (Figure SI.6). Thus its shear dynamics may be described by Jeffery orbits, which are traced out by the periodic tumbling motion of a single rod under linear shear.42-44 To rationalize this intuitively, we compare the tumbling periods of RJC,


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consisting of N Janus spheres, between this theory and simulation results. For this purpose, we perform a series of simulations where N and γ& are changed from 6 to 16 and from 0.03 to 0.10 τ1

, respectively. As shown in the bottom inset of Figure 2a, the tumbling motion of RJC is

characterized by the time dependence of orientation angle φ. Average period of tumbling TP can thereby be measured through averaging the periods of φ. TP can also be theoretically determined based on the Jeffery orbits,44 that is TP = 2π / κγ& ,

(1)

where κ is a dimensionless coefficient depending on the aspect ratio of RJC, p (See the definition in the caption of Figure 2a), and can also be calculated from simulations (Supporting Information, Figure SI.7). The simulation and theoretical results of the dependence of TP on κ γ& are summarized in Figure 2a. An excellent agreement can be found between the points from simulations and the red line determined by the theory, corroborating that the shear dynamics of RJCs can be captured by Jeffery orbits. Similar with small micelles, RJCs also suffer reconfiguration and disassembly when shear is strong enough, as displayed by the insetting snapshots of Figure 2b where N=8. A close observation of the dynamics reveals that the disassembly of RJCs starts from the breakaway of the tail particle marked in red in the snapshots, since the tail particle has the least number of bonds contacting to its neighbors in contrast to other constituent particles. Further, our intensive studies manifest that the disassembly can be effectively identified by examining the kinetic energy of the tail particle, Ek. Figure 2b shows the representative plots of Ek where the RJC undergoes a typical tumbling upon various shear rates. Clearly, the disassembly of JC, initiating from the escape of tail particle, is most likely to occur when Ek reaches the maximum (φ = π/2, Figure SI.8). Particularly, our simulations indicate there exist two consequent threshold values of


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Ek for the disassembly of RJC, i.e., E1 = 14.8 kBT and E2 = 30.2 kBT, as denoted by the two dashed lines in Figure 2b. The insetting snapshots scheme the distinct motions of tail particle with different maximums of Ek. Considering that tail particle has two bonds, the quantitative relation of the threshold values, i.e., E2 ≈ 2E1, directly reveals that the dissociation of one bond occurs once the kinetic energy surpasses the bonding energy of a bond pair. Therefore, the key role of bonding structure is featured in regard to the reconfiguration of RJCs under shear.

Figure 3. The shear responses of kinked Janus clusters (KJCs). (a) All the dynamic behaviors of KJCs (NA = 6, NB = 5) under shear with two kinds of chirality, cis (+) and trans (-). The short yellow lines represent the existing bonds in the cluster, and a defect of triangular bonding can be identified at the kink. The trans-KJC undergoes a chiral transition (CT) when shear rate is moderate. I: tumbling as a whole group; II: reversible shear alignment; III: disassembly into 2 RJCs; CT-I: tumbling as a whole group after chiral transition; CT-II: reversible shear alignment after chiral transition. (b) The definitions of cis- and trans-forms of KJCs. Rotate the KJC to make the cusp particle (marked in red) at the lowest position of the cluster. If the cusp particle


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belongs to the left RJC, then the KJC is the cis-form, otherwise it is the trans-form. (c) Phase diagram of the cis- and trans-KJCs with various shear rates γ& and NB when NA = 6. Such bonding-structure dependence is more evident for the dynamic behaviors of kinked Janus clusters (KJCs). As shown in Figure 3a, KJC presents more complicated shear dynamics than RJC due to the triangular bonding defect at the kink. To gain an in-depth insight into the influence of the kink, we consider the basic KJC with a single kink under shear. This kind of KJC is made up of two RJCs, where the numbers of their constituent particles are NA and NB respectively. Importantly, there exist two configurations to link these two RJCs (Figure 3b), resulting in two kinds of chirality, namely cis-KJC and trans-KJC respectively (See detailed definitions in the caption of Figure 3b). As expected, KJCs with different chirality show distinct responses to shear (Figure 3a). As demonstrated by the phase diagram with various γ& and NB when NA = 6 (Figure 3c), KJC takes the periodic tumbling motion when γ& is small and breaks up into two RJCs when shear is strong enough. Intriguingly, in the intermediate regimes of γ& , the bond at the kink breaks reversibly, leading to reversible shear alignment that is similar with the trimer (See Movies IV and V). In particular, the trans-KJC undergoes a chiral transition when shear rate is moderate (See Movie V), and the asymmetry of the phase diagram suggests that the trans-KJC is more likely to transform into cis-KJC, which cannot be observed in the thermodynamic self-assembly of Janus spheres. Such chiral transition also exists for quasi-2D systems under strong confinement. This highlights that shear can be harnessed to reshape the configuration of KJCs, providing a potential means to regulate the assembled structures of ANPs through using external fields. In striking contrast to the shear-induced disassembly of a formed cluster, external shear may turn to generate complicated aggregation behaviors for dispersed Janus spheres. To


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quantitatively describe these effects of shear, we consider a dilute suspension consisting of N0 Janus spheres at various concentration ϕ. The number of discrete Janus spheres is named as Ns. Figure 4a shows a typical time evolution of Ns where the height difference between two connecting intervals, ∆Ns, may conceive possible aggregation mechanism of Janus spheres as schemed by the insetting images. To quantify the assembling kinetics, we designate tc as the characteristic time at which Ns is closest to N0/e, where tc is defined to denote the relaxation time and independent of the detailed dynamic of the system (Figure SI.9). At least 2000 independent runs are performed to obtain an average characteristic time 〈tc 〉 that significantly alleviates the influence of system fluctuation. For high computational efficiency, we employ a modified Brownian dynamics thereby neglecting the hydrodynamic interactions, which is reasonable for dilute suspensions36 (See Supporting Information for details). A detailed insight into the effect of shear on assembling kinetics can be realized by analyzing the dependence of 〈tc 〉 on γ& . As demonstrated in Figure 4b, the 〈tc 〉 − γ& relation can be divided into three regimes. For extremely weak shear with γ& < 0.003 τ-1, the effect is trivial and the shear behaves as a thermodynamic disturbance.36 At 0.003 τ-1 < γ& < 0.27 τ-1, shear becomes strong enough to accelerate the clustering process by enhancing translation and rotation of Janus spheres which allows more effective contacts among them. Particularly, in this regime, a power-law relation between 〈tc 〉 and γ& can be observed, i.e., 〈tc 〉 ~ γ& β , where the exponent β is nearly independent of θmax (Figure SI.10). However, the inset of Figure 4b shows that β increases almost linearly with the rising ϕ. This suggests that a high concentration of Janus spheres can penalize the effect of shear, which reveals the interplay between both these factors. At γ& > 0.27 τ-1, increasing γ& turns to result in large 〈tc 〉 , indicating that the aggregation of dispersed Janus spheres is suppressed at high shear rate.


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(a) 40

(b) 65o 103o 180o

SEC

32 103 ∆Ns = 1

16

N0/e

β

〈tc〉 (τ)

∆Ns = 2

TD -0.4

8

2

10

-0.6

-0.8 0.0

0 0

AS

β

24

Ns

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1500

3000

t (τ)

tc

4500

φ 0.1

10-3

0.2

0.3

10-2

-1

10-1

γ (τ )

Figure 4. (a) Typical time evolution of NS. The insets show possible aggregation mechanism of Janus spheres. The blue line is used to guide the eye and the cyan dot represents the definition of tc. (b) The log-log plots of 〈tc 〉 versus γ& with θmax = 65°, 103° and 180° where ϕ = 10.2%. Three regimes representing distinct effects of shear are shown, i.e., thermodynamic disturbance (TD), shear-enhanced clustering (SEC), and aggregation suppression (AS). The solid lines and corresponding shaded areas are theoretical values and ranges predicted by eq 5 respectively. The inset shows the dependence of β on ϕ when θmax = 91°. To gain a fundamental insight into the roles of θmax and γ& in the assembling kinetics of Janus spheres, we detail the clustering rate by utilizing a theoretical justification. For a dimensionless form, the average characteristic time of isotropic spheres at θmax = 180°, 〈tc 〉 i , is utilized to normalize 〈tc 〉 at different θmax, i.e., 〈t%c 〉 = 〈tc 〉 / 〈tc 〉 i . The clustering rate K is thereby defined by K = 1 / 〈t%c 〉 . For convenience, θmax is also normalized by λ = θmax / 180°. As indicated by the high similarities among the plots in Figure 4b, the effects of θmax (i.e. λ) and γ& on the assembling kinetics exhibit relatively weak correlation. Consequently, inspired by the equation of state for gas,45 we use virial expansion to explicitly express this function as a power series of λ where the virial coefficients ci are functions of γ& . That is,


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∞

& λ) = ∑ i =0 ci +1 (γ)λ & i. K (γ,

(2)

With boundary values K( γ& , 0) = 0 and K( γ& , 1) = 1, the third-order expansion is & λ) = c2 λ + c3 λ 2 + c4 λ 3 + O (λ 4 ) , K (γ,

(3)

where c2 + c3 + c4 = 1 (See Figure SI.11 for the reason using third-order expansion). To test the validity of eq 3, representative plots of K versus λ and their virial-expansion fits are given in Figure 5a. A good agreement can be found between simulation results and the fitting curves, implying that the virial expansion can be used to exactly reproduce the assembling kinetics. As the virial coefficients ci only depend on γ& , the effect of shear on clustering rate is characterized by the second and third virial coefficients, i.e., c2 and c3 respectively. Figure 5b presents the plots of c2 and c3 as functions of γ& , based on a series of simulations. What is striking is that both c2 and c3 evolves linearly in the semi-logarithmic γ& scale, when γ& ranges from 0.001 to 0.1 τ-1. This allows us to construct a differential equation that more directly describes the shear-rate dependence of clustering rate and reads & P (λ) / γ& , ∂K / ∂γ=

(4)

where P(λ) = 0.15λ – 0.22λ2 + 0.07λ3 (See Supporting Information for details). Solving eq 4 yields K = P(λ)ln( γ& / γ& 0 ) + K0(λ), where γ& 0 is arbitrary within 0.001-0.1 τ-1 and K0(λ) is the clustering rate at γ& 0 . Rearranging terms, the expression of the solution can be transformed into −1

〈tc 〉 = 〈tc 〉 i [ P (λ) ln(γ& / γ& 0 ) + K 0 (λ) ] .

(5)

This equation suggests an approach to predict the clustering rate of Janus spheres with various hydrophobic ranges based on that of isotropic spheres under shear. To examine it, in Figure 4b we compare the simulation data of 〈tc 〉 at different θmax (blue and orange dots) with the theoretical results (corresponding solid lines) generated from the data of isotropic spheres (green dots). Indeed, the solid lines fit the dots well: the excellent agreement among the results from


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simulation and theoretical justification reveals that eq 5 does allow an effective approach to bridge the gap between Janus spheres and isotropic particles in regard to the assembling kinetics under shear. (a)1.0

(b)

-1

γ = 0.2 τ γ = 0.02 τ-1 -1 γ = 0.005 τ

0.4

3.2

0.2

c2

0.5

λ 0.0 0.0

0.5

λ

1.0

0.0 0.35

-0.51

2.4

-0.2 -0.4

c3

2.8

K

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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10-3

10-2

10-1

2.0

-1

γ (τ )

Figure 5. (a) Representative plots of clustering rate K versus λ when γ& = 0.005, 0.02 and 0.2 τ-1 respectively. The inset shows the definition of λ. The solid lines are the virial-expansion fits. (b) The plots of the virial coefficients, c2 and c3 as functions of logarithmic γ& [log10( γ& )]. The linear relationships between the coefficients and log10( γ& ) are displayed by the solid lines. In summary, the shear-induced behaviors of Janus nanoparticles in two-dimensional confinement are systematically studied for the first time. Through the investigations of formed JCs, we demonstrate that the bonding structures significantly affect the shear dynamics of JCs, and the mechanism of shear-induced disassembly is revealed from the perspective of energy and bonding structures. The collective effects of shear and bonding structures are found to generate unique dynamic behaviors and reshaped cluster configurations such as the chiral transition of kinked JCs, suggesting a potential way to tailor the assembled structures of ANPs. We also examine the assembling kinetics of dispersed Janus spheres under shear. Upon increasing shear rate, three regimes clarifying the effects of shear on the assembling kinetics are identified for dilute suspensions, that is, thermodynamic disturbance, shear-enhanced clustering, and aggregation suppression. In particular, a power-law relation describing the shear-rate dependence


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of assembling kinetics is found in the intermediate regime, where the exponent is nearly independent of the Janus balance. Further, we find that the assembling kinetics can be captured by the virial expansion of clustering rate. An equation which can be harnessed to predict the clustering rate of Janus spheres based on that of isotropic particles is thereby established, suggesting an exotic way to bridge the gap between anisotropic and isotropic particles. Our findings could also stimulate future experiments, and might offer a versatile strategy harnessing confinement and external fields to tailor the superstructures formed by ANPs towards unique properties.

Supporting Information: The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.xxx. A detailed description of the hybrid simulation method, the Brownian dynamics, and the model of Janus spheres. The study of the Brownian diffusion of rodlike Janus clusters. The calculation of coefficient κ and the derivation of eq 4. Supplemental figures and legends for movies.

ACKNOWLEDGMENTS We thank National Natural Science Foundation of China (Nos. 21422403, 51273105 and 21174080) for financial support, and Bojun Dong, Junshi Liang, Guoxi Xu, Jing Yan, and Andreas Zӧttl for helpful discussions.

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Shearing Janus Nanoparticles Confined in Two-Dimensional Space

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