Multi-particle collision dynamics

Multi-particle collision dynamics (MPC), also known as stochastic rotation dynamics (SRD),^[1] is a particle-based mesoscale simulation technique for complex fluids which fully incorporates thermal fluctuations and hydrodynamic interactions.^[2] Coupling of embedded particles to the coarse-grained solvent is achieved through molecular dynamics.^[3]

Method of simulation

The solvent is modelled as a set of $N$ point particles of mass $m$ with continuous coordinates ${\displaystyle {\vec {r}}_{i}}$ and velocities ${\displaystyle {\vec {v}}_{i}}$. The simulation consists of streaming and collision steps.

During the streaming step, the coordinates of the particles are updated according to

${\displaystyle {\vec {r}}_{i}(t+\delta t_{\mathrm {MPC} })={\vec {r}}_{i}(t)+{\vec {v}}_{i}(t)\delta t_{\mathrm {MPC} }}$

where ${\displaystyle \delta t_{\mathrm {MPC} }}$ is a chosen simulation time step which is typically much larger than a molecular dynamics time step.

After the streaming step, interactions between the solvent particles are modelled in the collision step. The particles are sorted into collision cells with a lateral size $a$. Particle velocities within each cell are updated according to the collision rule

${\displaystyle {\vec {v}}_{i}\rightarrow {\vec {v}}_{\mathrm {CMS} }+{\hat {\mathbf {R} }}({\vec {v}}_{i}-{\vec {v}}_{\mathrm {CMS} })}$

where ${\displaystyle {\vec {v}}_{\mathrm {CMS} }}$ is the centre of mass velocity of the particles in the collision cell and ${\displaystyle {\hat {\mathbf {R} }}}$ is a rotation matrix. In two dimensions, ${\displaystyle {\hat {\mathbf {R} }}}$ performs a rotation by an angle ${\displaystyle +\alpha }$ or $-\alpha$ with probability $1/2$. In three dimensions, the rotation is performed by an angle $\alpha$ around a random rotation axis. The same rotation is applied for all particles within a given collision cell, but the direction (axis) of rotation is statistically independent both between all cells and for a given cell in time.

If the structure of the collision grid defined by the positions of the collision cells is fixed, Galilean invariance is violated. It is restored with the introduction of a random shift of the collision grid.^[4]

Explicit expressions for the diffusion coefficient and viscosity derived based on Green-Kubo relations are in excellent agreement with simulations.^[5][6]

Simulation parameters

The set of parameters for the simulation of the solvent are:

• solvent particle mass $m$
• average number of solvent particles per collision box ${\displaystyle n_{s}}$
• lateral collision box size $a$
• stochastic rotation angle $\alpha$
• kT (energy)
• time step ${\displaystyle \delta t_{\mathrm {MPC} }}$

The simulation parameters define the solvent properties,^[1] such as

• mean free path ${\displaystyle \lambda =\delta t_{\mathrm {MPC} }{\sqrt {kT/m}}}$
• diffusion coefficient ${\displaystyle D={\frac {kT\delta t_{\mathrm {M} PC}}{2m}}{\Bigg [}{\frac {dn_{s}}{(1-\cos(\alpha ))(n_{s}-1+\exp ^{-n_{s}})}}-1{\Bigg ]}}$
• shear viscosity $\nu$
• thermal diffusivity ${\displaystyle D_{T}}$

where $d$ is the dimensionality of the system.

A typical choice for normalisation is ${\displaystyle a=1,\;kT=1,\;m=1}$. To reproduce fluid-like behaviour, the remaining parameters may be fixed as ${\displaystyle \alpha =130^{o},\;n_{s}=10,\;\delta t_{\mathrm {MPC} }\in [0.01;0.1]}$.^[7]

Applications

MPC has become a notable tool in the simulations of many soft-matter systems, including

• colloid dynamics^[3][8][9]
• polymer dynamics^[10][11]
• vesicles^[12]
• active systems^[7]
• liquid crystals^[13]

References

1. G. Gompper, T. Ihle, K. Kroll and R. G. Winkler "Multi-Particle Collision Dynamics: A Particle-Based Mesoscale Simulation Approach to the Hydrodynamics of Complex Fluids", Advanced Computer Simulation Approaches for Soft Matter Sciences III, Advances in Polymer Science 221 p. 1 (2009) [arXiv: http://arxiv.org/abs/0808.2157]
2. A. Malevanets and R. Kapral "Mesoscopic model for solvent dynamics", Journal of Chemical Physics 110 pp. 8605-8613 (1999)
3. A. Malevanets and R. Kapral "Solute molecular dynamics in a mesoscale solvent", Journal of Chemical Physics 112 pp. 7260-7269 (2000)
4. T. Ihle and D. Kroll "Stochastic rotation dynamics. I. Formalism, Galilean invariance, and Green-Kubo relations", Physical Review E 67 066705 (2003)
5. T. Ihle, E. Tüzel and D. M. Kroll "Resummed Green-Kubo relations for a fluctuating fluid-particle model", Physical Review E 70 035701 (2004)
6. T. Ihle, E. Tüzel and D. M. Kroll "Equilibrium calculation of transport coefficients for a fluid-particle model", Physical Review E 72 046707 (2005)
7. J. Elgeti "Sperm and Cilia Dynamics" PhD thesis, Universität zu Köln (2006)
8. J. T. Padding and A. A. Louis "Hydrodynamic and Brownian Fluctuations in Sedimenting Suspensions", Physical Review Letters 93 220601 (2004)
9. M. Hecht, J. Harting, M. Bier, J. Reinshagen and H. J. Herrmann "Shear viscosity of claylike colloids in computer simulations and experiments", Physical Review E 74 021403 (2006)
10. K. Mussawisade, M. Ripoll, R. G. Winkler, and G. Gompper "Dynamics of polymers in a particle-based mesoscopic solvent", Journal of Chemical Physics 123 144905 (2005)
11. M. Ripoll, R. G. Winkler and G. Gompper "Hydrodynamic screening of star polymers in shear flow", The European Physics Journal E 23 pp. 349-354 (2007)
12. H. Noguchi and G. Gompper "Dynamics of fluid vesicles in shear flow: Effect of membrane viscosity and thermal fluctuations", Physical Review E 72 011901 (2005)
13. K.-W. Lee and Marco G. Mazza. "Stochastic rotation dynamics for nematic liquid crystals". Journal of Chemical Physics. 142 164110 (2015).