# Dissipative particle dynamics

Dissipative particle dynamics (DPD) is an off-lattice mesoscopic simulation technique[1] which involves a set of particles moving in continuous space and discrete time. Particles represent whole molecules or fluid regions, rather than single atoms, and atomistic details are not considered relevant to the processes addressed. The particles' internal degrees of freedom are integrated out and replaced by simplified pairwise dissipative and random forces, so as to conserve momentum locally and ensure correct hydrodynamic behaviour. The main advantage of this method is that it gives access to longer time and length scales than are possible using conventional MD simulations. Simulations of polymeric fluids in volumes up to 100 nm in linear dimension for tens of microseconds are now common.

DPD was initially devised by Hoogerbrugge and Koelman[2][3] to avoid the lattice artifacts of the so-called lattice gas automata and to tackle hydrodynamic time and space scales beyond those available with molecular dynamics (MD). It was subsequently reformulated and slightly modified by P. Español[4] to ensure the proper thermal equilibrium state. A series of new DPD algorithms with reduced computational complexity and better control of transport properties are presented.[5] The algorithms presented in this article choose randomly a pair particle for applying DPD thermostating thus reducing the computational complexity.

## Equations

The total non-bonded force acting on a DPD particle i is given by a sum over all particles j that lie within a fixed cut-off distance, of three pairwise-additive forces:

${\displaystyle f_{i}=\sum _{j\neq i}(F_{ij}^{C}+F_{ij}^{D}+F_{ij}^{R})}$

where the first term in the above equation is a conservative force, the second a dissipative force and the third a random force. The conservative force acts to give beads a chemical identity, while the dissipative and random forces together form a thermostat that keeps the mean temperature of the system constant. A key property of all of the non-bonded forces is that they conserve momentum locally, so that hydrodynamic modes of the fluid emerge even for small particle numbers. Local momentum conservation requires that the random force between two interacting beads be antisymmetric. Each pair of interacting particles therefore requires only a single random force calculation. This distinguishes DPD from Brownian dynamics in which each particle experiences a random force independently of all other particles. Beads can be connected into ‘molecules’ by tying them together with soft (often Hookean) springs. The most common applications of DPD keep the particle number, volume and temperature constant, and so take place in the NVT ensemble. Alternatively, the pressure instead of the volume is held constant, so that the simulation is in the NPT ensemble.

## Parallelization

In principle, simulations of very large systems, approaching a cubic micron for milliseconds, are possible using a parallel implementation of DPD running on multiple processors in a Beowulf-style cluster. Because the non-bonded forces are short-ranged in DPD, it is possible to parallelize a DPD code very efficiently using a spatial domain decomposition technique. In this scheme, the total simulation space is divided into a number of cuboidal regions each of which is assigned to a distinct processor in the cluster. Each processor is responsible for integrating the equations of motion of all beads whose centres of mass lie within its region of space. Only beads lying near the boundaries of each processor's space require communication between processors. In order to ensure that the simulation is efficient, the crucial requirement is that the number of particle-particle interactions that require inter-processor communication be much smaller than the number of particle-particle interactions within the bulk of each processor's region of space. Roughly speaking, this means that the volume of space assigned to each processor should be sufficiently large that its surface area (multiplied by a distance comparable to the force cut-off distance) is much less than its volume.

## Applications

A wide variety of complex hydrodynamic phenomena have been simulated using DPD, the list here is necessarily incomplete. The goal of these simulations often is to relate the macroscopic non-Newtonian flow properties of the fluid to its microscopic structure. Such DPD applications range from modeling the rheological properties of concrete[6] to simulating liposome formation in biophysics[7] to other recent three-phase phenomena such as dynamic wetting.[8]

The DPD method has also found popularity in modeling heterogeneous multi-phase flows containing deformable objects such as blood cells[9] and polymer micelles.[10]

The full trace of the developments of various important aspects of the DPD methodology since it was first proposed in the early 1990s can be found in "Dissipative Particle Dynamics: Introduction, Methodology and Complex Fluid Applications – A Review".[11]

The state-of-the-art in DPD was captured in a CECAM workshop in 2008.[12] Innovations to the technique presented there include DPD with energy conservation; non-central frictional forces that allow the fluid viscosity to be tuned; an algorithm for preventing bond crossing between polymers; and the automated calibration of DPD interaction parameters from atomistic molecular dynamics. Recently, examples of automated calibration and parameterization have been shown against experimental observables. Additionally, datasets for the purpose of interaction potential calibration and parameterisation have been explored.[13] [14] Swope et al, have provided a detailed analysis of literature data and an experimental dataset based on Critical micelle concentration (CMC) and micellar mean aggregation number (Nagg).[15] Examples of micellar simulations using DPD have been well documented previously.[16][17][18]

## References

1. ^ Santo, Kolattukudy P.; Neimark, Alexander V. (December 2021). "Dissipative particle dynamics simulations in colloid and Interface science: a review". Advances in Colloid and Interface Science. 298: 102545. doi:10.1016/j.cis.2021.102545. PMID 34757286. S2CID 243939224.
2. ^ Hoogerbrugge, P. J; Koelman, J. M. V. A (1992). "Simulating Microscopic Hydrodynamic Phenomena with Dissipative Particle Dynamics". Europhysics Letters (EPL). 19 (3): 155–160. Bibcode:1992EL.....19..155H. doi:10.1209/0295-5075/19/3/001. ISSN 0295-5075. S2CID 250796817.
3. ^ Koelman, J. M. V. A; Hoogerbrugge, P. J (1993). "Dynamic Simulations of Hard-Sphere Suspensions Under Steady Shear". Europhysics Letters (EPL). 21 (3): 363–368. Bibcode:1993EL.....21..363K. doi:10.1209/0295-5075/21/3/018. ISSN 0295-5075. S2CID 250913111.
4. ^ Español, P; Warren, P (1995). "Statistical Mechanics of Dissipative Particle Dynamics". Europhysics Letters (EPL). 30 (4): 191–196. Bibcode:1995EL.....30..191E. doi:10.1209/0295-5075/30/4/001. ISSN 0295-5075. S2CID 14385201.
5. ^ Goga, N.; Rzepiela, A. J.; de Vries, A. H.; Marrink, S. J.; Berendsen, H. J. C. (2012). "Efficient Algorithms for Langevin and DPD Dynamics". Journal of Chemical Theory and Computation. 8 (10): 3637–3649. doi:10.1021/ct3000876. ISSN 1549-9618. PMID 26593009.
6. ^ James S. Sims and Nicos S. Martys: Modelling the Rheological Properties of Concrete
7. ^
8. ^ Henrich, B.; Cupelli, C.; Moseler, M.; Santer, M. (7 November 2007). "An adhesive DPD wall model for dynamic wetting". Europhysics Letters (EPL). 80 (6): 60004. Bibcode:2007EL.....8060004H. doi:10.1209/0295-5075/80/60004. S2CID 67827072.
9. ^ Blumers, Ansel; Tang, Yu-Hang; Li, Zhen; Li, Xuejin; Karniadakis, George (August 2017). "GPU-accelerated red blood cells simulations with transport dissipative particle dynamics". Computer Physics Communications. 217: 171–179. arXiv:1611.06163. Bibcode:2017CoPhC.217..171B. doi:10.1016/j.cpc.2017.03.016. PMC 5667691. PMID 29104303.
10. ^ Tang, Yu-Hang; Li, Zhen; Li, Xuejin; Deng, Mingge; Karniadakis, George (2016). "Non-Equilibrium Dynamics of Vesicles and Micelles by Self-Assembly of Block Copolymers with Double Thermoresponsivity". Macromolecules. 49 (7): 2895–2903. Bibcode:2016MaMol..49.2895T. doi:10.1021/acs.macromol.6b00365. S2CID 102266045.
11. ^ Moeendarbary; et al. (2009). "Dissipative Particle Dynamics: Introduction, Methodology and Complex Fluid Applications - A Review". International Journal of Applied Mechanics. 1 (4): 737–763. Bibcode:2009IJAM....1..737M. doi:10.1142/S1758825109000381. S2CID 50363270.
12. ^ Dissipative Particle Dynamics: Addressing deficiencies and establishing new frontiers Archived 2010-07-15 at the Wayback Machine, CECAM workshop, July 16–18, 2008, Lausanne, Switzerland.
13. ^ McDonagh, James; et al. (31 May 2020). "What Can Digitization Do For Formulated Product Innovation and Development". Polymer International. 70 (3): 248–255. doi:10.1002/pi.6056. S2CID 219766018.
14. ^ McDonagh J. L.; et al. (2019). "Utilizing machine learning for efficient parameterization of coarse grained molecular force fields". Journal of Chemical Information and Modeling. 59 (10): 4278–4288. doi:10.1021/acs.jcim.9b00646. PMID 31549507. S2CID 202745539.
15. ^ Swope W. C.; et al. (2019). "Challenge to Reconcile Experimental Micellar Properties of the CnEm Nonionic Surfactant Family". The Journal of Physical Chemistry B. 123 (7): 1696–1707. doi:10.1021/acs.jpcb.8b11568. PMID 30657322. S2CID 58626650.
16. ^ Oviedo; et al. (2013). "Critical micelle concentration of an ammonium salt through DPD simulations using COSMO-RS--based interaction parameters". AIChE Journal. 59 (11): 4413–4423. doi:10.1002/aic.14158.
17. ^ Ryjkina; et al. (2013). "Molecular Dynamic Computer Simulations of Phase Behavior of Non-Ionic Surfactants". Angewandte Chemie International Edition. 41 (6): 983–986. doi:10.1002/1521-3773(20020315)41:6<983::AID-ANIE983>3.0.CO;2-Y. PMID 12491288.
18. ^ Johnston; et al. (2016). "Toward a standard protocol for micelle simulation" (PDF). The Journal of Physical Chemistry B. 120 (26): 6337–6351. doi:10.1021/acs.jpcb.6b03075. PMID 27096611.

## Available packages

Some available simulation packages that can (also) perform DPD simulations are:

• CULGI: The Chemistry Unified Language Interface, Culgi B.V., The Netherlands
• DL_MESO: Open-source mesoscale simulation software.
• DPDmacs
• ESPResSo: Extensible Simulation Package for the Research on Soft Matter Systems - Open-source
• Fluidix: The Fluidix simulation suite available from OneZero Software.
• GPIUTMD: Graphical processors for Many-Particle Dynamics
• Gromacs-DPD: A modified version of Gromacs including DPD.
• HOOMD-blue Archived 2011-11-11 at the Wayback Machine: Highly Optimized Object-oriented Many-particle Dynamics—Blue Edition
• LAMMPS
• Materials Studio: Materials Studio - Modeling and simulation for studying chemicals and materials, Accelrys Software Inc.
• OSPREY-DPD: Open Source Polymer Research Engine-DPD
• SYMPLER: A freeware SYMbolic ParticLE simulatoR from the University of Freiburg.
• SunlightDPD: Open-source (GPL) DPD software.