GALAMOST
GPU-Accelerated Large-Scale Molecular Simulation Toolkit
Station: Research
1. Introduction
Anisotropic colloidal building blocks have brought an almost unbelievable revolution in materials science. Recently, a vast collection of ordered and disordered supracolloidal structures have been achieved in experiments and simulations through the self-assembly of surface-anisotropic and/or shape-anisotropic particles. However, to date, how to create novel desired structures by rational design of anisotropic colloidal building blocks remains a significant challenge in materials science. For most anisotropic colloidal particles, their sizes typically range from about $10$ nm to $1000$ nm, and even up to the micrometer scale; the length scales of their aggregated structures will be much larger. Thus, the aggregation phenomena of anisotropic colloidal building blocks involve both the microscopic and mesoscopic scales. It is an extremely challenging task and also a more feasible strategy to gain insight into the aggregation behavior and mechanism of anisotropic colloidal building blocks via computer simulation.

For a large class of complex polymer colloids with surface-anisotropy and/or shape-anisotropy, Dr. Zhan-Wei Li have developed a series of effective and general mesoscale models and simulation methods[1-10]. At present, these mesoscale models can be used to simulate the aggregation behavior of five major kinds of anisotropic colloids, including soft Janus particles, soft patchy particles, ellipsoids, Janus ellipsoids, and patchy ellipsoids. In order to improve the computational efficiency of these mesoscale models in dynamics simulations, ZWL and coworkers have built the corresponding highly optimized GPU-accelerated simulation toolkit ANISOMAST (ANIsotropic SOft MAtter Simulation Tookit), which runs with GALAMOST (GPU-accelerated large-scale molecular simulation toolkit)[11] or MUSES (Multiscale simulator especially for soft matter) as a backend. ANISOMAST presents a set of simulation strategy applicable to the investigation of aggregate behavior (e.g. self-assembly, crystalline, liquid crystal, quasicrystal, glass transition, and so on) of complex anisotropic colloids. With the aid of ANISOMAST, ZWL and coworkers have studied and elucidated the formation and control mechanisms for the aggregate structure of different types of shape-anisotropic and/or surface-anisotropic polymer colloids, and revealed the design principles and regulation rules for creating various supracolloidal structures using anisotropic polymer colloids. These research will provide conceptual and theoretical guidance for the design and development of high-performance supracolloidal functional materials. In the future study, ZWL and coworkers will continue to focus on the aggregation behavior of anisotropic soft matter systems, enrich and enhance the functions of ANISOMAST, and hope it will become a mast for the ship of anisotropic soft matter simulations.

2. Model and simulation methods

2.1 Soft Janus particle models (SJPMs)

In our SJPMs[5-8], the deformable and anisotropic characteristics of soft Janus particles are described via a single-site soft anisotropic attractive potential

\begin{align}
U_{ij}=\left\{ \begin{array}{ll} \frac{\alpha_{ij}^R}{2}\left(1-\frac{r_{ij}}{r_c}\right)^2-
f^\nu\frac{\alpha_{ij}^A}{2}\left[\frac{r_{ij}}{r_c}-\left(\frac{r_{ij}}{r_c}\right)^2\right] &  r_{ij}\leq r_c(\equiv1.0) \\  0  & r_{ij}> r_c.
\end{array} \right.
\end{align}

In this potential, the anisotropic factor $f$ is mainly responsible for describing the anisotropic feature of soft Janus particles, which is different for different SJPMs. The magnitude of $\alpha_{ij}^R$ controls the strength of repulsion, $\alpha_{ij}^A$ controls the strength of attraction between the attractive patches, and $\nu$ controls the angular width of the attraction. Thus, both $\alpha_{ij}^A$ and $\nu$ control the flexibility of Janus particle aggregates. The direction of the attractive patches on particle $i$ is specified by unit vectors $\mathbf{n}_{i}$, and the size of the attractive patch is described by Janus balance $\beta$, which is half of the opening angle of the attractive patch. In the soft one-patch Janus particle model[5], $f$ is given as

\begin{align}
f = \left\{ \begin{array}
\cos\frac{\pi\theta_i}{2\beta}\cos\frac{\pi\theta_j}{2\beta} & \textrm{if $\cos\theta_i\geq \cos\beta$ and $\cos\theta_j\geq \cos\beta$}\\
0 & \textrm{otherwise}.
\end{array} \right.
\end{align}
In the soft ABA-type triblock Janus particle model[6], $f$ is given as
\begin{align}
f = \left\{ \begin{array}
\cos\frac{\pi\theta_i'}{2\beta}\cos\frac{\pi\theta_j'}{2\beta} & \textrm{if $|\cos\theta_i|\geq \cos\beta$ and $|\cos\theta_j|\geq \cos\beta$}\\
0 & \textrm{otherwise}.
\end{array} \right.
\end{align}
In the soft BAB-type triblock Janus particle model[7], $f$ is given as
\begin{align}
f = \left\{ \begin{array}
\cos\frac{\pi}{2}\left(\frac{\pi/2-\theta_i'}{\pi/2-\beta}\right)\cos\frac{\pi}{2}\left(\frac{\pi/2-\theta_j'}{\pi/2-\beta}\right) & \textrm{if $|\cos\theta_i|\leq \cos\beta$ and $|\cos\theta_j|\leq \cos\beta$}\\
0 & \textrm{otherwise}.
\end{array} \right.
\end{align}


Figure 2.1. The uniaxial disklike, rodlike and Janus models

2.2 Soft patchy particle models (SPPMs)

In our SPPMs[8,9], the deformable and anisotropic characteristics of soft patchy particles are also described via a single-site soft anisotropic potential analogous to the one used in SJPMs. This anisotropic potential is expressed as

\begin{align}
U_{ij}=\left\{ \begin{array}{ll} \frac{\alpha_{ij}^R d_{ij}}{2}\left(1-\frac{r_{ij}}{d_{ij}}\right)^2-
\sum\limits_{\kappa=1}^{M_i}\sum\limits_{\lambda=1}^{M_j} f^\nu \left(\mathbf{n}_{i}^{\kappa}, \mathbf{n}_j^{\lambda},  \mathbf{r}_{ij}\right) \frac{\alpha_{ij}^A d_{ij}}{2}\left[\frac{r_{ij}}{d_{ij}}-\left(\frac{r_{ij}}{d_{ij}}\right)^2\right] &  r_{ij}\leq d_{ij} \\  0  & r_{ij}> d_{ij},
\end{array} \right.
\end{align}

where
\begin{align}
f\left(\mathbf{n}_{i}^{\kappa}, \mathbf{n}_j^{\lambda},  \mathbf{r}_{ij}\right) = \left\{ \begin{array}{ll}
\cos\frac{\pi\theta_i^{\kappa}}{2\theta_{m}^{\kappa}}\cos\frac{\pi\theta_j^{\lambda}}{2\theta_{m}^{\lambda}} & \textrm{if $\cos\theta_i^{\kappa}\geq \cos\theta_{m}^{\kappa}$ and $\cos\theta_j^{\lambda} \geq \cos\theta_{m}^{\lambda}$}\\
0 & \textrm{otherwise}.
\end{array} \right.
\end{align}
Here, the normalized quaternion $\mathbf{q}_{i}=\left(q_{i,0}, q_{i,1}, q_{i,2}, q_{i,3}\right)$ is introduced in order to describe the orientation of patchy particle $i$, $M_i$ and $M_j$ are used to describe the number of the attractive patches of particles $i$ and $j$, the directions of the attractive patches $\kappa$ ($\kappa=1,\cdot\cdot\cdot, M_i$) and $\lambda$ ($\lambda=1,\cdot\cdot\cdot, M_j$) on particles $i$ and $j$ are specified by patch vectors $\mathbf{n}_{i}^{\kappa}$ and $\mathbf{n}_{j}^{\lambda}$, respectively. $\theta_i^{\kappa}$ is the angle between $\mathbf{n}_{i}^{\kappa}$ and the interparticle vector $\mathbf{r}_{ji}=\mathbf{r}_{j}-\mathbf{r}_{i}$, and $\theta_j^{\lambda}$ is the angle between $\mathbf{n}_j^{\lambda}$ and $\mathbf{r}_{ij}$ ($\mathbf{r}_{ij}=-\mathbf{r}_{ji}$), and then $\cos\theta_i^{\kappa}= -\mathbf{n}_i^{\kappa} \cdot\mathbf{r}_{ij}/r_{ij}$ and $\cos\theta_j^{\lambda}= \mathbf{n}_j^{\lambda} \cdot\mathbf{r}_{ij}/r_{ij}$. Thus, $\theta_i^{\kappa}=\arccos(\cos\theta_i^{\kappa})=\arccos(-\mathbf{n}_i^{\kappa} \cdot\mathbf{r}_{ij}/r_{ij})$, and $\theta_j^{\lambda}=\arccos(\cos\theta_j^{\lambda})=\arccos(\mathbf{n}_j^{\lambda} \cdot\mathbf{r}_{ij}/r_{ij})$. The sizes of the attractive patches $\kappa$ and $\lambda$ are described by $\theta_{m}^{\kappa}$ and $\theta_{m}^{\lambda}$, which are half of the opening angle of the attractive patches (i.e. the semi-angular widths of the patches). It is very easy and convenient to obtain a wide range of patchy particle models with different anisotropies, simply by changing the number $M_i$, size $\theta_{m}^{\kappa}$, direction $\mathbf{n}_{i}^{\kappa}$, and geometrical arrangement of the patches of particle $i$.
The force between two neighboring patchy particles $\mathbf{F}_{ij}$ is given by the derivation of potential,
\begin{align}\label{eqn:force}
\mathbf{F}_{ij}=&-\frac{\partial U_{ij}}{\partial \mathbf{r}_{ij}}\nonumber\\
=&\ \alpha_{ij}^R\left(1-\frac{r_{ij}}{d_{ij}}\right)\frac{\mathbf{r}_{ij}}{r_{ij}}+
\sum\limits_{\kappa=1}^{M_i}\sum\limits_{\lambda=1}^{M_j}\Bigg\{\alpha_{ij}^Af^\nu\left(\mathbf{n}_{i}^{\kappa},\mathbf{n}_j^{\lambda},\mathbf{r}_{ij}\right)\left(\frac{1}{2}-\frac{r_{ij}}{d_{ij}}\right)\frac{\mathbf{r}_{ij}}{r_{ij}}
\nonumber\\&-\frac{\alpha_{ij}^A}{2}\left[\frac{r_{ij}}{d_{ij}}-\left(\frac{r_{ij}}{d_{ij}}\right)^2\right]\nu f^{\nu-1}\left(\mathbf{n}_{i}^{\kappa},\mathbf{n}_j^{\lambda},\mathbf{r}_{ij}\right)\Bigg(\frac{\pi}{2\theta_m^{\kappa}}\sin\frac{\pi\theta_i^{\kappa}}{2\theta_m^{\kappa}}\frac{\partial \theta_i^{\kappa}}{\partial \cos\theta_i^{\kappa}}\nonumber\\&\frac{\partial \cos\theta_i^{\kappa}}{\partial \mathbf{r}_{ij}}\cos\frac{\pi\theta_j^{\lambda}}{2\theta_m^{\lambda}}+\frac{\pi}{2\theta_m^{\lambda}}\sin\frac{\pi\theta_j^{\lambda}}{2\theta_m^{\lambda}}\frac{\partial \theta_j^{\lambda}}{\partial \cos\theta_j^{\lambda}}\frac{\partial \cos\theta_j^{\lambda}}{\partial \mathbf{r}_{ij}}\cos\frac{\pi\theta_i^{\kappa}}{2\theta_m^{\kappa}}\Bigg)\Bigg\}.
\end{align}
The torque $\mathbf{\tau}_{ij}$ acting on patchy particle $i$ due to its neighboring particle $j$ is given by
\begin{align}\label{eqn:torque}
\mathbf{\tau}_{ij}=&\sum\limits_{\kappa=1}^{M_i}-\frac{\partial U_{ij}}{\partial \mathbf{n}_{i}^\kappa}\nonumber\\
=&\sum\limits_{\kappa=1}^{M_i}\sum\limits_{\lambda=1}^{M_j}\frac{\pi\alpha_{ij}^Ad_{ij}}{4\theta_m^{\kappa}}\left[\frac{r_{ij}}{d_{ij}}-\left(\frac{r_{ij}}{d_{ij}}\right)^2\right]\nu f^{\nu-1}\left(\mathbf{n}_{i}^{\kappa},\mathbf{n}_j^{\lambda},\mathbf{r}_{ij}\right)\sin\frac{\pi\theta_i^{\kappa}}{2\theta_m^{\kappa}}\frac{\partial \theta_i^{\kappa}}{\partial \cos\theta_i^{\kappa}}\cos\frac{\pi\theta_j^{\lambda}}{2\theta_m^{\lambda}}\frac{\mathbf{r}_{i}}{r_{ij}}.
\end{align}


Figure 2.2. The patchy particle models

2.3 Patchy ellipsoidal particle models (PEPMs)

In our PEPMs[10], we describe shape-anisotropic and surface-anisotropic characteristics of patchy ellipsoidal particles via a single-site shape- and surface-anisotropic potential
\begin{equation}
U ( \mathbf{A}_i, \mathbf{A}_j, \mathbf{r}_{ij} ) = U_{GB} (
\mathbf{A}_i, \mathbf{A}_j, \mathbf{r}_{ij})+ U_{P} (
\mathbf{A}_i, \mathbf{A}_j, \mathbf{r}_{ij})U_{GB} (
\mathbf{A}_i, \mathbf{A}_j, \mathbf{r}_{ij}),
\end{equation}

where $U_{GB} (\mathbf{A}_i, \mathbf{A}_j, \mathbf{r}_{ij})$ is the shape-anisotropic component describing the interaction between two like or unlike patchy ellipsoidal particles, and $U_{P}(\mathbf{A}_i, \mathbf{A}_j, \mathbf{r}_{ij})$ is the surface-anisotropic component describing the directional interaction between the patches on patchy ellipsoidal particles defined by patch vectors. Each patchy ellipsoidal particle is characterized by its semiaxis lengths $a_i$, $b_i$, and $c_i$, and well depths $\epsilon_{ia}$, $\epsilon_{ib}$, and $\epsilon_{ic}$ for side-to-side, face-to-face, and end-to-end interactions. The normalized quaternion $\mathbf{q}_{i}=\left(q_{i,0}, q_{i,1}, q_{i,2}, q_{i,3}\right)$ is introduced to describe the orientation of patchy ellipsoidal particle $i$.

The shape-anisotropic component $U_{GB} (\mathbf{A}_i, \mathbf{A}_j, \mathbf{r}_{ij})$ has the most general form of a Gay-Berne potential for dissimilar biaxial ellipsoidal particles
\begin{equation}
U_{GB}( \mathbf{A}_i, \mathbf{A}_j, \mathbf{r}_{ij} ) = U_r(
\mathbf{A}_i, \mathbf{A}_j, \mathbf{r}_{ij}) \eta(
\mathbf{A}_i, \mathbf{A}_j) \chi( \mathbf{A}_i, \mathbf{A}_j, \mathbf{r}_{ij}),
\end{equation}
where $U_r(\mathbf{A}_i, \mathbf{A}_j, \mathbf{r}_{ij})$ controls the shifted distance dependence of the interaction, based on the distance $h_{ij}$ of closest approach between the two ellipsoidal particles and the shift parameter $\gamma$
\begin{equation}
U_r(\mathbf{A}_i, \mathbf{A}_j, \mathbf{r}_{ij}) = 4 \epsilon \left[\left( \frac{\sigma}{ h_{ij} + \gamma \sigma}\right)^{12} - \left(\frac{\sigma}{ h_{ij} + \gamma \sigma}\right)^6\right].
\end{equation}
In general, the exact calculation of $h_{ij}$ is nontrivial, an approximation given by Perram et al. is usually employed together with the Gay-Berne potential,
\begin{equation}
h_{ij}( \mathbf{A}_i, \mathbf{A}_j,
\mathbf{r}_{ij} ) = r_{ij} - \sigma_{ij} ( \mathbf{A}_i, \mathbf{A}_j,
\mathbf{\hat{r}}_{ij} ),
\end{equation}
and
\begin{equation}
\sigma_{ij} ( \mathbf{A}_i, \mathbf{A}_j,
\hat{\mathbf{r}}_{ij} ) = \left[ \frac{1}{2} \hat{\mathbf{r}}_{ij}^T
\mathbf{G}_{ij}^{-1}( \mathbf{A}_i, \mathbf{A}_j) \hat{\mathbf{r}}_{ij}\right]^{ -1/2 }.
\end{equation}
The symmetric overlap matrix $\mathbf{G}_{ij}( \mathbf{A}_i, \mathbf{A}_j)$ is defined in terms of the diagonal shape matrices $\mathbf{S}_i=\mbox{diag}(a_i, b_i, c_i)$ and $\mathbf{S}_j=\mbox{diag}(a_j, b_j, c_j)$ as
\begin{equation}
\mathbf{G}_{ij}( \mathbf{A}_i, \mathbf{A}_j) = \mathbf{A}_i^T \mathbf{S}_i^2 \mathbf{A}_i +
\mathbf{A}_j^T \mathbf{S}_j^2 \mathbf{A}_j.
\end{equation}
$\eta(\mathbf{A}_i, \mathbf{A}_j)$ and $\chi( \mathbf{A}_i, \mathbf{A}_j, \mathbf{r}_{ij})$ control the anisotropic interaction strength based on the relative orientation and position of the ellipsoidal particles,
\begin{equation}
\eta(\mathbf{A}_i, \mathbf{A}_j) = \left[ \frac{ 2 s_i s_j }{\det \left( \mathbf{G}_{ij}( \mathbf{A}_i, \mathbf{A}_j) \right)}\right]^{
\upsilon / 2 } ,
\end{equation}
with
\begin{equation}
s_i = (a_i b_i + c_i c_i)(a_i b_i)^{ 1 / 2 },
\end{equation}
and
\begin{equation}
\chi( \mathbf{A}_i, \mathbf{A}_j, \mathbf{r}_{ij}) = [2 \hat{\mathbf{r}}_{ij}^T \mathbf{B}_{ij}^{-1}( \mathbf{A}_i, \mathbf{A}_j)
\hat{\mathbf{r}}_{ij}]^\mu,
\end{equation}
with
\begin{equation}
\mathbf{B}_{ij}( \mathbf{A}_i, \mathbf{A}_j) = \mathbf{A}_i^T \mathbf{E}_i \mathbf{A}_i +
\mathbf{A}_j^T \mathbf{E}_j \mathbf{A}_j,
\end{equation}
where $\mathbf{E}_i=\mbox{diag}
((\epsilon_0/\epsilon_{ia})^{1/\mu}, (\epsilon_0/\epsilon_{ib})^{1/\mu}, (\epsilon_0/\epsilon_{ic})^{1/\mu})$ and $\mathbf{E}_j=\mbox{diag}
((\epsilon_0/\epsilon_{ja})^{1/\mu}, (\epsilon_0/\epsilon_{jb})^{1/\mu}, (\epsilon_0/\epsilon_{jc})^{1/\mu})$ are the diagonal interaction matrices, and the parameters $\mu$ and $\nu$ are empirical exponents that can be used to tune the potential.
The surface-anisotropic component $U_{P}(\mathbf{A}_i, \mathbf{A}_j, \mathbf{r}_{ij})$ is expressed as
\begin{equation}\label{Eq:Up}
U_{P}(\mathbf{A}_i, \mathbf{A}_j, \mathbf{r}_{ij})= \sum\limits_{\kappa=1}^{M_i}\sum\limits_{\lambda=1}^{M_j} \gamma_{\epsilon}f^\alpha \left(\mathbf{A}_i, \mathbf{A}_j, \mathbf{r}_{ij}\right),
\end{equation}
and
\begin{align}
f\left(\mathbf{A}_i, \mathbf{A}_j, \mathbf{r}_{ij}\right) = \left\{ \begin{array}{ll}
\cos\frac{\pi\theta_i^{\kappa}}{2\theta_{m}^{\kappa}}\cos\frac{\pi\theta_j^{\lambda}}{2\theta_{m}^{\lambda}} & \textrm{if $\cos\theta_i^{\kappa}\geq \cos\theta_{m}^{\kappa}$ and $\cos\theta_j^{\lambda} \geq \cos\theta_{m}^{\lambda}$}\\
0 & \textrm{otherwise},
\end{array} \right.
\end{align}
with
\begin{align}
\begin{array}{lll}
\theta_i^{\kappa}=\arccos(-\mathbf{A}_i^T{\mathbf{n}_{i}^{\kappa}}^b\cdot\hat{\mathbf{r}}_{ij})& \textrm{and} &\theta_j^{\lambda}=\arccos(\mathbf{A}_i^T{\mathbf{n}_{i}^{\lambda}}^b\cdot\hat{\mathbf{r}}_{ij}).
\end{array}
\end{align}


Figure 2.3. The patchy ellipsoidal particle models

References

1. Zhan-Wei Li, Li-Jun Chen, Ying Zhao, and Zhong-Yuan Lu*, Ordered packing of soft discoidal system, J. Phys. Chem. B, 112, 13842-13848, 2008.

2. Zhan-Wei Li, Zhao-Yan Sun, and Zhong-Yuan Lu* Simulation model for hierarchical self-assembly of soft disklike particles, J. Phys. Chem. B, 114, 2353-2358, 2010.

3. Zhan-Wei Li, Yu-Hua Liu, Ying-Tao Liu, and Zhong-Yuan Lu*, A single-site anisotropic soft-core model for the study of phase behavior of soft rodlike particles, Sci. China Chem., 54, 1474-1483, 2011.

4. Xiao-Xi Jia, Zhan-Wei Li*, Zhao-Yan Sun, Zhong-Yuan Lu*, Hierarchical self-assembly of soft disklike particles under shear flow, J. Phys. Chem. B, 115, 3441-13448, 2011.

5. Zhan-Wei Li, Zhong-Yuan Lu, Zhao-Yan Sun*, and Li-Jia An, Model, self-assembly structures, and phase diagram of soft Janus particles, Soft Matter, 8, 6693-6697, 2012.

6. Zhan-Wei Li, Zhong-Yuan Lu, You-Liang Zhu, Zhao-Yan Sun*, and Li-Jia An, A simulation model for soft triblock Janus particles and their ordered packing, RSC Adv., 3, 813-822, 2013.

7. Zhan-Wei Li, Zhong-Yuan Lu, and Zhao-Yan Sun*, Soft Janus particles: Ideal building blocks for template-free fabrication of two-dimensional exotic nanostructures, Soft Matter, 10, 5472-5477, 2014.

8. Zhan-Wei Li, Zhao-Yan Sun, and Zhong-Yuan Lu, DOI: 10.1002/9781119113171.ch5 (pp. 109-133), John Wiley & Sons, Ltd., 2016

9. Zhan-Wei Li, You-Liang Zhu, Zhong-Yuan Lu, and Zhao-Yan Sun*, A versatile model for soft patchy particles with various patch arrangements, Soft Matter, 12, 741-749, 2016.

10. Zhan-Wei Li, You-Liang Zhu, Zhong-Yuan Lu, and Zhao-Yan Sun*, A general and efficient patchy ellipsoid model for anisotropic soft matter systems, unpublished, 2017.

11. Y.-L. Zhu, H. Liu, Z.-W. Li, H.-J. Qian,G. Milano, and Z.-Y. Lu*, GALAMOST: GPU-accelerated large-scale molecular simulation toolkit. J. Comput. Chem., 34 (25), 2197, 2013.