⇦Chinese Physics Letters, 2020, Vol. 37, No. 5, Article code 050501 Self-Assembly of Dimer Motors under Confined Conditions * An Zhou (周安)1, Li-Yan Qiao (乔丽颜)1, Gui-Na Wei (魏瑰娜)2**, Zhou-Ting Jiang (姜舟婷)3, Ye-Hua Zhao (赵叶华)1** Affiliations 1Department of Physics, Hangzhou Dianzi University, Hangzhou 310018 2Departments of Renal Medicine, The Second Affiliated Hospital of Zhejiang University College of Medicine, Hangzhou 310009 3Department of Physics, China Jiliang University, Hangzhou 310018 Received 21 January 2020, online 25 April 2020 *Supported by the National Natural Science Foundation of China (Grant Nos. 11674080, 11974094, and 21873087).
**Corresponding author. Email: yhzhao@hdu.edu.cn; weiguina@163.com Citation Text: Zhou A, Qiao L Y, Wei G N, Jiang Z T and Zhao Y H et al 2020 Chin. Phys. Lett. 37 050501    Abstract Chemically synthetic nanomotors can consume fuel in the environment and utilize the self-generated concentration gradient to self-propel themselves in the system. We study the collective dynamics of an ensemble of sphere dimers built from linked catalytic and noncatalytic monomers. Because of the confinement from the fuel field and the interactions among motors, the ensemble of dimer motors can self-organize into various nanostructures, such as a radial pattern in the spherical fuel field and a staggered radial pattern in a cylindrical fuel field. The influence of the dimer volume fraction on the self-assembly is also investigated and the formed nanostructures are analyzed in detail. The results presented here may give insight into the application of the self-assembly of active materials. DOI:10.1088/0256-307X/37/5/050501 PACS:05.40.Jc, 82.70.Dd, 82.40.Ck © 2020 Chinese Physics Society Article Text Active particles that convert locally stored energy in the environment for their directed propulsion have the ability to execute a delivery task. Bacteria and other micro-organisms swim to obtain food or to respond to stimuli. As typical active matter, motile bacteria exhibit various phase structures and collective properties, and they provide an ideal system to study fundamental principles in nonequilibrium physics.[1–4] For example, experiment showed weak synchronization and large-scale collective oscillation in dense bacterial suspensions.[1] The emerging behavior arises from the fact that motile bacteria carry out self-propelled motion in the environment with strong fluctuations. It is critical that the underlying mechanism of diverse phenomena may inspire new strategies to control the self-organization of molecular machines and swarming nanomotors. Synthetic motors are regarded as beneficial supplements to the bacteria to study the collective dynamics of active matter because they offer platforms that are fabricated with simpler and more controllable interactions.[5–7] In addition, such synthetic motors show great potential applications, such as nanoscale assembly, collective oscillations, nanoactuators, etc.[8–13] These chemosynthetic self-propelled nanomotors can consume fuel in the environment and utilize the self-generated concentration gradient to influence the motor dynamics. For example, chemically self-propelled motors use asymmetric catalytic activity to produce self-generated gradients that lead to directed motion.[6] Chemically synthetic motors comprise catalytic colloids and noncatalytic colloids, which is already implemented in laboratories, such as bimetallic nanorods consisting of catalytic side (Pt) and noncatalytic side (silica).[11] One may conclude that a requirement for propulsion is that there is a heterogeneous catalytic reaction at the surface of the motor.[13] In the biological realm, the collective behavior of molecular motors plays an important role in the basic process of cell life. The collective dynamics of the chemosynthetic motors have been studied in the laboratory, such as the self-assembly of chemically active Janus colloidal particles.[14] The interaction between different motors through forces that arise concentration gradients and direct intermolecular forces.[15] In this Letter, we investigate the self-assembly behavior of chemically powered sphere dimer motors in prescribed fuel fields with different configurations.
cpl-37-5-050501-fig1.png
Fig. 1. The local steady state density of F particles $n_{\rm F}(r)$ around the center of the spherical fuel field with $R_{\rm F}=10.0$.
The coarse-grained model for the system contains both the motors and fluid environment, and the molecules in solution are modeled by a large number $N_{\rm T}=N_{\rm S}+N_{\rm F}+N_{\rm P}$, typically $10^{6}$, of point-like molecules with identical masses $m=1.0$. The self-propel nanomotors complete the self-assembly process in a system of volume $V$, with $V=L_x\times L_y\times L_z=60\times 60\times 60$, and the system is constructed by both a specific fuel field that is full of fuel particles (F) and an inactive region that consists of solution particles (S). When S particles diffuse into the chemical active region full F, they are converted to be F. As F particles diffuse out of the active region, they decay to S with reaction ${\rm F}\stackrel{k_{\rm cat}}{\longrightarrow}{\rm S}$, where the rate constant $K_{\rm cat}$ is 0.08. Figure 1 shows the radial distribution of the local steady state concentration of F particles in a spherical fuel field. In our simulation, self-propelled sphere-dimer nanomotors are constructed from C and N spheres linked by a rigid bond with length $R=4.6$. The molecules are randomly distributed in the cells. Each cell contains about $n$ solution particles obeying the Poisson distribution. The evolution of the entire system is carried out using a hybrid scheme that combines molecular dynamics (MD) and multiparticle collision dynamics (MPCD).[16,17] Molecular dynamics is carried out by integrating the Newton equation with the MD time step $\tau_{\rm MD}$ ($\tau_{\rm MD}=0.01$). The reaction and collision between particles in solution is expressed as MPC. In each time interval $\tau_{\rm MPC}(\tau_{\rm MPC}=0.5)$, MPCD takes place among F, S, and P particles. The system is subdivided into cubic cells with length $a_0 = 1$. Multiparticle collisions are performed independently in each cell, and the postcollision velocity of particle $i$ in cell $\xi$ is given by ${{\boldsymbol v}}_i(t+t_{\rm M})={{\boldsymbol v}}_{\rm cm}(t)+\hat{\omega}_{\xi}(\alpha)({{\boldsymbol v}}_i(t)-{{\boldsymbol v}}_{\rm cm}(t))$, where $\omega_{\xi}(\alpha)$ is the rotation matrix and ${\boldsymbol v}_{\rm cm}$ is the center-of-mass velocity of that cell. Random grid shifts are applied in each direction of the simulation box to ensure Galilean invariance. In our simulation, we constructed spherical and cylindrical pattern models of active fuel particles. In order to better visualize the structural behavior of the ensemble, we initially put those nano-motors in the fuel pattern and observed their self-assembly behavior over time. The catalytic (C) and noncatalytic (N) spheres in the dimer motors have radius $\sigma_{\rm C}=\sigma_{\rm N}=2.0$. An irreversible chemical reaction takes place in the surface of C monomer, F+C $\rightarrow$ P+C once F encounters the C sphere. These point-like species (S, F and P) interact with the nanodimer monomers through repulsive L–J potentials, $V_{ab}(r)=4\varepsilon_{ab}[(\sigma_{b}/r)^{12}-(\sigma_{b}/r)^6+1/4]$, $r < r_{\rm c}$, where $\varepsilon_{ab}(a$ = F, S, P and $b$ = C, N) is the energy parameter and $r_{\rm c}=2^{(1/6)}\sigma_b$ is the cutoff distance. All the particles in the system have the same energy parameters with dimer sphere ($\varepsilon_{ab}=5.0$ except for $\varepsilon_{\rm PN}=0.1$). In order to maintain system in a steady state, the product particle P has to be converted back to fuel particle F or solution particle S when they diffuse far enough away from the C monomer and the conversion cutoff radius is chosen to be $R+2\sigma_{\rm N}$. In our simulations, all quantities are reported in dimensionless L–J units based on energy $\varepsilon$, mass $m$ and distance $\sigma$: $r/\sigma\rightarrow r$, $t(\varepsilon/m\sigma^2)^{1/2})\rightarrow t$, $KT/\varepsilon\rightarrow T$. The velocity Verlet algorithm is used to integrate Newton's equations of motion. The system temperature is kept at $T=1/6$ and the rotation angle is fixed at $\alpha=\pi/2$. We investigate the self-assembly dynamics behavior of chemically powered sphere-dimer motors in a system that has a fuel field with a specific shape. The evolution of the system is based on a mesoscopic hybrid molecular dynamics-multiparticle collision (MD-MPC) dynamics scheme, and there is no force that is calculated among the solvent particles and follows the basic conservation laws, such as conserving mass, momentum and energy.[18] The self-assembly behavior has a strict requirement on the values of the microscopic parameter: MPCD parameters for the solvent, energy parameter between molecular and Boltzmann temperature. It is useful to discuss self-assembly behavior.
cpl-37-5-050501-fig2.png
Fig. 2. (a) Initial configuration of $N_{\rm D}=8$ dimers comprising catalytic (red) and noncatalytic spheres (blue) that are randomly distributed in the spherical fuel field (yellow). (b) Instantaneous configuration after the ensemble reaching a steady state.
As for the spherical fuel field, we study the dynamics behavior with $N_{d}=8$ and $N_{d}=20$ sphere dimer motors which are initially randomly distributed in the spherical fuel field. Figure 2(a) shows one of the initial instantaneous configuration of the ensemble. Figure 2(b) shows the steady configuration in which $N_{d}=8$ dimer motors assemble into radial pattern where all of the N spheres lie in the inner region of the fuel field and the C spheres exist in the outer region. At the beginning, these dimers lie in the fuel field randomly. Chemical reaction that occurs on the surface of C sphere converts reactants to product particles. Chemical gradients play an especially important role in the self-assembly process and in the resulting cluster dynamics. The long-range heterogeneous concentration field created by motors in the vicinity of a given motor will affect its propulsion properties, because the dimer will respond its local concentration gradient as well as that of neighboring motors.[19] Consequently, the dimer motors will move up the gradient and tend to be propelled toward neighboring motors. Once motors approach closely, they typically tend to align because motor configurations with the N end toward the C end of a neighboring motor will result in a small concentration gradient across the N monomer. This phenomenon is similar to the chemotactic behavior in bacteria. All the C spheres of the motors will reach the edge of the fuel field with a portion of the C monomer entering into the fuel-poor area. Meanwhile the C sphere will also exist in the gradients of product particles when it reaches the boundary. Because we chose the energy character of solution particles to produce particles with a catalytic sphere to be $\varepsilon_{\rm S} > \varepsilon_{\rm P}$, the energy asymmetry will tend to restrict the motors in the fuel-rich domain. Therefore, the confinement from the fuel field also shapes the configuration of the cluster. By utilizing the shape of the fuel field, these motors can successfully self-assemble into transient but long-lived radial patterns. The characteristic structural features of the cluster configuration after the ensemble reaches a steady state can be identified in the steady-state radial distribution in the spatial motion. Figure 3(a) shows a plot of catalytic monomers, noncatalytic monomers and the center of mass of the dimer with the center of fuel field radial distribution function, respectively. We have $$ g_{\alpha}(r)=\frac{1}{2\pi{rn}}\langle{\sum_{i=1}^N\delta(|r_{\alpha,i}|-r)}\rangle,~~ \tag {1} $$ where $r_{\alpha,i}$ is the position of monomer $i$ of species type, represents catalytic monomer (C), noncatalytic monomer (N) and the mass of center of the motor (Z), respectively. $r_{\rm C}$ is the position vector of the center of the spherical fuel field, and n is the number density of dimers in the space. There are well defined peaks at different specific place for each $g_{\alpha}(r)$ function. It is noticeable that the $g_{\rm C}(r)$ function has a large peak at $r=7.95$, which corresponds to the relation $r=R_{\rm F}-\sigma_{\rm C}=7.8$. Thus, the C monomers are distributed on the edge of the radical configuration. The values of $g_{\rm C}(r)=R_{\rm F}-\sigma_{\rm C}-R/2=5.7$ and $g_{\rm Z}(r)=R_{\rm F}-\sigma_{\rm C}-R=3.7$ are consistent with the positions of corresponding peaks in the radial distribution function, which indicates the configuration with the majority of motors aligned in the radial direction. The configuration can also be observed in Fig. 2(b).
cpl-37-5-050501-fig3.png
Fig. 3. (a) Radial distribution function $g_{\alpha}(r)$ of the C, N and Z for the system. (b) Spatial orientational correlation function $\psi(r)$.
The local orientational order can be described by the function $$ \psi_{\rm u}(r)=\langle\frac{1}{n(r)}\sum_{i=1}^N(\hat{u}_i\cdot\hat{u}_{\rm Z}) \delta(|(r_i-r_O)|-r)\rangle ,~~ \tag {2} $$ where $u_i$ is the unit vector along the dimer bond pointing from N sphere to C sphere, while $u_{\rm Z}$ represents the unit vector point from the center of the fuel field to the center of mass of the dimer, $n(r)$ is the number of the motors that away from the fuel center $r$. The function reaches a peak of 0.6 at $r=6.15$, which corresponds to the place where the peak of $g_{\rm Z}(r)$ appears. The structural characteristic of the ensemble is clearly analyzed in the $g_{\alpha}(r)$ and $\psi_{\rm u}(r)$ functions plotted in Figs. 3(a) and 3(b), respectively. The characteristic feature of the self-assembly structure in the process of the dynamics evolution after the ensemble reaches a stable state can also be described by the steady-state radial distribution functions in the space of the motion. Figure 4(a) shows a plot of the catalytic-noncatalytic (CN) radial distribution function, $$ P_{\rm CN}(r)=\frac{1}{N_{\rm C}^2}\langle\sum_{j < i=1}^{N_{\rm N}} \delta(|(r_{{\rm N}i}-r_{{\rm C}j})|-r)\rangle ,~~ \tag {3} $$ where $r_{{\rm C}j}$ and $r_{{\rm N}i}$ represent the position of the catalytic monomer and noncatalytic monomer, respectively, $N_{\rm C}=N_{\rm D}$ is the number of the dimer motor and the brackets denote an average over time and realizations. There is a significant peak at $r=5.0$ in this function, which corresponds to the location between C monomer and its nearest neighbor, N monomer, in the radial distribution configurations. The local orientational order can be described by $$ \psi_{\rm u}(r)=\Big\langle\frac{1}{n(r)}\sum_{j < i=1}^{N_{\rm N}}(\hat{u}_i\cdot\hat{u}_j)\delta(|(r_{{\rm N}i}-r_{{\rm C}j})|-r)\Big\rangle ,~~ \tag {4} $$ where $n(r)$ is the number of N monomer away from C sphere at $r$ and the angular brackets again signify an average over time and realization. This function is plotted in Fig. 4(b). There is a prominent positive peak in $\psi_{\rm u}(r)$ at the same position as the function $P_{\rm CN}(r)$, which indicates that the position relation of adjacent motors in the radially distributed configuration is similar to that of two adjacent radiation rays. There is also negative value to the latter, indicating that between two non-adjacent motors tend to non-parallel or opposite to each other.
cpl-37-5-050501-fig4.png
Fig. 4. (a) Plot of the probability density function $P_{\rm CN}(r)$. (b) Spatial orientational correction function $\psi_{\rm u}(r)$. The inset in penal (a) shows the configuration between two adjacent dimers contributes to the peaks in the probability density function.
cpl-37-5-050501-fig5.png
Fig. 5. (a) Initial configuration of $N_{\rm D}=20$ dimers randomly distributed in the spherical fuel field. (b) Instantaneous configuration after the ensemble reaching a stead state. The configuration is similar to that of $N_{\rm D}=8$ motors, but there are differences in internal structure under the influence of overcrowded environment.
To further investigate the self-assembly structure of the ensemble in the system with high volume fraction, the number of the dimer motors is increased from 8 to 20. The change of the volume fraction will affect the self-assembly configuration. On account of the monomers in different dimers interacting with each other through repulsive potentials, there are no cohesive intermolecular forces between dimers. They will be bounced away when they get close to each other, due to the fact that the self-assembly structure is not exactly the same as $n=8$. Because of the overcrowded environment and the repulsion between different monomers, it is difficult for some dimers to propagate to boundary of the fuel field, which will further affect the overall distribution. The configuration of the ensemble of the initial state and steady state are shown in Figs. 5(a) and 5(b), respectively.
cpl-37-5-050501-fig6.png
Fig. 6. (a) Radial distribution function $g_{\alpha}(r)$ of the C, N monomers and center of the mass of the dimer for the system, and (b) spatial orientational correlation, $\psi_{\rm u}(r)$, functions.
After the motors reach a statistically steady state, the structural feature can be confirmed by the radial distribution function $g_{\alpha}(r)$, i.e., Eq. (1). These functions are plotted in Fig. 6(a). It is apparent that each of these functions for C, N, and Z has more than one dominant peaks. Besides some peaks correspond to $N=8$ dimers, there is also a significant peak nearby $r=0$ for each function, indicating some monomers restricted in the center of the field under the influence of the overcrowded environment, and some motors are even in the opposite direction with other radially distributed motors. The feature can also be described by the function (2). It is quite evident that in addition to the same peaks as $n=8$ dimers, there is also a peak at near 0 for each of the function. The fist peak for each function perfectly illustrates that some motors are confined in the fuel center while others are radially distributed. The local orientational order can also be used to describe the architectural feature, which is plotted in Fig. 6(b). The two positive peaks indicate that the self-assembly radial structure with pointing from C monomer to N monomer and those dimers are distributed and staggered under the influence of crowded environment. Meanwhile, there is a negative peak at $r=3.7$, indicating that some of the dimers cannot overcome the crowded environment during the self-propelling process and they are in an opposite direction with other dimers. As stated earlier, changing the volume fraction leads to various assembly structure of the ensemble. Next, we investigate the self-assembly structure of the ensemble in a cylinder with fuel particles in the inner layer and solution particles in the outside. Similar to the case of spherical fuel field, we randomly distribute N dimers in the cylindrical fuel field. Figure 7(a) is the platform of the initial configuration. Those dimers that are parallel to cylinder propel in the field until they reach the edge of the junction of fuel and solvents, while others that are perpendicular to bottom propel along the axis of the cylinder. Figure 7(b) shows an instantaneous configuration of the ensemble after it reaches a steady state. It is clear that catalytic monomers in the outside and noncatalytic in the inside as for these dimers that parallel to the bottom at the initial, while some motors are perpendicular to the bottom of the cylinder.
cpl-37-5-050501-fig7.png
Fig. 7. (a) The top view of initial configuration of $N_d=20$ motors in a cylindrical fuel field with radius $R_{\rm F}=10.0$. (b) Instantaneous configuration after the ensemble reaches a steady state.
cpl-37-5-050501-fig8.png
Fig. 8. (a) Radial distribution function $g_{\alpha}(r)$ of the C, N monomers and the center of the mass of the dimer for the system, and (b) spatial orientational correlation, $\psi_{\rm u}(r)$.
The characteristic feature of this configuration again can be confirmed by the structure of the radial distribution function $g_{\alpha}(r)$ and the function is plotted in Fig. 8(a). It is obvious that these functions for $\alpha$ = C, N, Z have more than one peak, the first peak is almost at $r=0$, implying that there are some dimers parallel to the axis during the whole dynamical evolution. The second peak of these function indicates that others dimer are distributed along the radius after the system reaches a stable state. This structural feature can also be shown by the function (4). Unlike the spherical fuel field case, there are three positive peaks in this function, indicating that these dimers are radially staggered with catalytic monomer outward. In summary, the particle-based mesoscopic model for chemically powered self-propelled sphere-dimer motor moving in the active system provides insight into the factors that control the self-assembly configuration. Although the collective dynamic behavior of nanomotors in the fuel field is frequently investigated, here we have studied the self-assembly configuration of multiple sphere-dimer motors in the fuel field with a particular shape. With the confinement of the fuel field, the motors will assemble themselves into various configurations. During the process of self-assembly, these individual motors propel each other through the forces arisen from concentration gradient, hydrodynamic coupling, and direct intermolecular forces. We show how to utilize a prescribed fuel field to suppress random Brownian motion of self-propelled motors and assembly into a specific configuration. In addition to understanding how the chemically propelled dimer motors move in the fuel field, one can imagine situation where the prescribed shape of fuel field can be used to modify the configuration of the nanomotors for specific tasks. It is possible that such structures can be formed by themselves according to the interaction with the environment. Thus, this work should aid in the tasks that have strict requirements in regards to the configuration of motor clusters. It is well-known that species patterns ranging from stationary regular and labyrinthine patterns can arise in biochemical systems driven far from equilibrium.[20–23] Various patterns may provide the fuel fields with different shapes. Therefore, the result presented here may give insight into the dynamics of bacteria in complex systems. References Weak synchronization and large-scale collective oscillation in dense bacterial suspensionsDiffusion of Ellipsoids in Bacterial SuspensionsScale-Invariant Correlations in Dynamic Bacterial ClustersA hidden state in the turnover of a functioning membrane protein complexChemically Propelled Motors Navigate Chemical PatternsSynthetic Nanomotors: Working Together through ChemistryChemotactic dynamics of catalytic dimer nanomotorsWet Chemical Synthesis of Soluble Gold NanogapsCatalytic Nanomotors: Autonomous Movement of Striped NanorodsDream NanomachinesCatalytic Nanomotors: Self-Propelled Sphere DimersSurface Wettability-Directed Propulsion of Glucose-Powered Nanoflask MotorsMotor and Rotor in One: Light-Active ZnO/Au Twinned Rods of Tunable Motion ModesSelf-assembled autonomous runners and tumblersCollective dynamics of self-propelled sphere-dimer motorsAdvances in Chemical PhysicsAdvanced Computer Simulation Approaches for Soft Matter Sciences IIITranslocation of a forced polymer chain through a crowded channelPerspective: Nanomotors without moving parts that propel themselves in solutionSynchronization between memristive and initial-dependent oscillators driven by noiseLattice Boltzmann modeling of wall-bounded ternary fluid flowsTransport of nanodimers through chemical microchip
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