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Dissipative Particle Dynamics: Foundation, Evolution, Implementation, and Applications

  • Z. Li
  • X. Bian
  • X. Li
  • M. Deng
  • Y.-H. Tang
  • B. Caswell
  • G. E. KarniadakisEmail author
Chapter
Part of the Advances in Mathematical Fluid Mechanics book series (AMFM)

Abstract

Dissipative particle dynamics (DPD) is a particle-based Lagrangian method for simulating dynamic and rheological properties of simple and complex fluids at mesoscopic length and time scales. In this chapter, we present the DPD technique, beginning from its original ad hoc formulation and subsequent theoretical developments. Next, we introduce various extensions of the DPD method that can model non-isothermal processes, diffusion-reaction systems, and ionic fluids. We also present a brief review of programming algorithms for constructing efficient DPD simulation codes as well as existing software packages. Finally, we demonstrate the effectiveness of DPD to solve particle-fluid problems, which may not be tractable by continuum or atomistic approaches.

Keywords

Coarse-Graining Computational biology Fluctuating hydrodynamics Fluid mechanics Lagrangian approach Mesoscopic method Multiscale simulation Particle-based method Soft matter Stochastic simulation Thermostat 

MSC2010:

76Z05 76V05 74F10 80A32 92C35 74F25 80A30 92C45 92C40 

References

  1. 1.
    E. Abu-Nada, Natural convection heat transfer simulation using energy conservative dissipative particle dynamics. Phys. Rev. E 81(5), 056704 (2010)Google Scholar
  2. 2.
    Y. Afshar, F. Schmid, A. Pishevar, S. Worley, Exploiting seeding of random number generators for efficient domain decomposition parallelization of dissipative particle dynamics. Comput. Phys. Commun. 184(4), 1119–1128 (2013)CrossRefGoogle Scholar
  3. 3.
    M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids (Clarendon Press, Oxford, 1989)zbMATHGoogle Scholar
  4. 4.
    M. Anand, K. Rajagopal, K.R. Rajagopal, A model incorporating some of the mechanical and biochemical factors underlying clot formation and dissolution in flowing blood. J. Theor. Med. 5(3–4), 183–218 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    M. Arienti, W.X. Pan, X.Y. Li, G. Karniadakis, Many-body dissipative particle dynamics simulation of liquid/vapor and liquid/solid interactions. J. Chem. Phys. 134(20), 204114 (2011)Google Scholar
  6. 6.
    J.B. Avalos, A.D. Mackie, Dissipative particle dynamics with energy conservation. Europhys. Lett. 40(2), 141–146 (1997)CrossRefGoogle Scholar
  7. 7.
    J.A Backer, C.P Lowe, H.C.J Hoefsloot, P.D Iedema, Poiseuille flow to measure the viscosity of particle model fluids. J. Chem. Phys. 122(15), 154503 (2005)Google Scholar
  8. 8.
    R.W. Balluffi, S.M. Allen, W.C. Carter, Kinetics of Materials (Wiley, Hoboken, 2005)CrossRefGoogle Scholar
  9. 9.
    G. Besold, I. Vattulainen, M. Karttunen, J.M. Polson, Towards better integrators for dissipative particle dynamics simulations. Phys. Rev. E 62(6), R7611–R7614 (2000)CrossRefGoogle Scholar
  10. 10.
    X. Bian, S. Litvinov, R. Qian, M. Ellero, N.A. Adams, Multiscale modeling of particle in suspension with smoothed dissipative particle dynamics. Phys. Fluids 24(1), 012002 (2012)Google Scholar
  11. 11.
    A.L. Blumers, Y.-H. Tang, Z. Li, X.J. Li, G.E. Karniadakis. GPU-accelerated red blood cells simulations with transport dissipative particle dynamics. Comput. Phys. Commun. 217, 171–179 (2017)CrossRefGoogle Scholar
  12. 12.
    J. Bonet, T.S.L. Lok, Variational and momentum preservation aspects of smooth particle hydrodynamic formulations. Comput. Methods Appl. Mech. Eng. 180(1–2), 97–115 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    H. Bow, I.V. Pivkin, M. Diez-Silva, S.J. Goldfless, M. Dao, J.C. Niles, S. Suresh, J. Han, A microfabricated deformability-based flow cytometer with application to malaria. Lab Chip 11, 1065–1073 (2011)Google Scholar
  14. 14.
    H.B. Callen, Thermodynamics and An Introduction to Thermostatistics (Wiley, New York, 1985)zbMATHGoogle Scholar
  15. 15.
    Z.H. Cao, K. Luo, H.L. Yi, H.P. Tan, Energy conservative dissipative particle dynamics simulation of natural convection in eccentric annulus. Int. J. Heat Mass Transf. 65, 409–422 (2013)CrossRefGoogle Scholar
  16. 16.
    H.-Y. Chang, X.J. Li, H. Li, G.E. Karniadakis, MD/DPD multiscale framework for predicting morphology and stresses of red blood cells in health and disease. PLOS Comput. Biol. 10, e1005173 (2016)CrossRefGoogle Scholar
  17. 17.
    S. Chien, S. Usami, J.F. Bertles, Abnormal rheology of oxygenated blood in sickle cell anemia. J. Clin. Invest. 49, 623–634 (1970)CrossRefGoogle Scholar
  18. 18.
    A. Davtyan, J.F. Dama, G.A. Voth, H.C. Andersen, Dynamic force matching: a method for constructing dynamical coarse-grained models with realistic time dependence. J. Chem. Phys. 142(15), 154104 (2015)Google Scholar
  19. 19.
    M.G. Deng, Z. Li, O. Borodin, G.E. Karniadakis, cDPD: a new dissipative particle dynamics method for modeling electrokinetic phenomena at the mesoscale. J. Chem. Phys. 145(14), 144109 (2016)Google Scholar
  20. 20.
    M.G. Deng, W.X. Pan, G.E. Karniadakis, Anisotropic single-particle dissipative particle dynamics model. J. Comput. Phys. 336, 481–491 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    R. Erban, S.J. Chapman, Reactive boundary conditions for stochastic simulations of reaction-diffusion processes. Phys. Biol. 4(1), 16–28 (2007)CrossRefGoogle Scholar
  22. 22.
    P. Español, Hydrodynamics from dissipative particle dynamics. Phys. Rev. E 52(2), 1734–1742 (1995)MathSciNetCrossRefGoogle Scholar
  23. 23.
    P. Español, Dissipative particle dynamics with energy conservation. Europhys. Lett. 40(6), 631–636 (1997)CrossRefGoogle Scholar
  24. 24.
    P. Español, Fluid particle model. Phys. Rev. E 57(3), 2930–2948 (1998)CrossRefGoogle Scholar
  25. 25.
    P. Español, M. Revenga, Smoothed dissipative particle dynamics. Phys. Rev. E 67, 026705 (2003)CrossRefGoogle Scholar
  26. 26.
    P. Español, P. Warren, Statistical mechanics of dissipative particle dynamics. Europhys. Lett. 30(4), 191–196 (1995)CrossRefGoogle Scholar
  27. 27.
    X.J. Fan, N. Phan-Thien, N.T. Yong, X.H. Wu, D. Xu, Microchannel flow of a macromolecular suspension. Phys. Fluids 15(1), 11–21 (2003)CrossRefzbMATHGoogle Scholar
  28. 28.
    D.A. Fedosov, B. Caswell, G.E. Karniadakis, A multiscale red blood cell model with accurate mechanics, rheology, and dynamics. Biophys. J. 98(10), 2215–2225 (2010)CrossRefGoogle Scholar
  29. 29.
    D.A. Fedosov, H. Lei, B. Caswell, S. Suresh, G.E. Karniadakis, Multiscale modeling of red blood cell mechanics and blood flow in malaria. PLoS Comput. Biol. 7, e1002270 (2011)MathSciNetCrossRefGoogle Scholar
  30. 30.
    D.A. Fedosov, W.X. Pan, B. Caswell, G. Gompper, G.E. Karniadakis, Predicting human blood viscosity in silico. Proc. Natl. Acad. Sci. USA 108(29), 11772–11777 (2011)CrossRefGoogle Scholar
  31. 31.
    D.A. Fedosov, B. Caswell, S. Suresh, G.E. Karniadakis, Quantifying the biophysical characteristics of Plasmodium-falciparum-parasitized red blood cells in microcirculation. Proc. Natl. Acad. Sci. USA 108, 35–39 (2011)CrossRefGoogle Scholar
  32. 32.
    C. Gardiner, Handbook of Stochastic Methods: For Physics, Chemistry and the Natural Sciences, 3rd edn. (Springer, New York, 2004)CrossRefzbMATHGoogle Scholar
  33. 33.
    H. Grabert, Projection Operator Techniques in Nonequilibrium Statistical Mechanics, vol. 95 (Springer, Berlin/Heidelberg, 1982)Google Scholar
  34. 34.
    R.D. Groot, Applications of Dissipative Particle Dynamics. Lecture Notes in Physics, chapter 1, vol. 640 (Springer, Berlin/Heidelberg, 2004), pp. 5–38Google Scholar
  35. 35.
    R.D. Groot, P.B. Warren, Dissipative particle dynamics: bridging the gap between atomistic and mesoscopic simulation. J. Chem. Phys. 107(11), 4423–4435 (1997)CrossRefGoogle Scholar
  36. 36.
    J.Y. Guo, X.J. Li, Y. Liu, H.J. Liang, Flow-induced translocation of polymers through a fluidic channel: a dissipative particle dynamics simulation study. J. Chem. Phys. 134(13), 134906 (2011)Google Scholar
  37. 37.
    J.-P. Hansen, I.R. McDonald, Theory of Simple Liquids, Fourth Edition: With Applications to Soft Matter (Academic, Amsterdam, 2013)zbMATHGoogle Scholar
  38. 38.
    C. Hijón, P. Español, E. Vanden-Eijnden, R. Delgado-Buscalioni, Mori–Zwanzig formalism as a practical computational tool. Faraday Discuss. 144, 301–322 (2010)CrossRefGoogle Scholar
  39. 39.
    P.J. Hoogerbrugge, J.M.V.A. Koelman, Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics. Europhys. Lett. 19(3), 155–160 (1992)CrossRefGoogle Scholar
  40. 40.
    M.X. Huang, Z.Q. Li, H.X. Guo, The effect of janus nanospheres on the phase separation of immiscible polymer blends via dissipative particle dynamics simulations. Soft Matter 8(25), 6834–6845 (2012)CrossRefGoogle Scholar
  41. 41.
    J.H. Irving, J.G. Kirkwood, The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics. J. Chem. Phys. 18(6), 817–829 (1950)Google Scholar
  42. 42.
    D.K. Kaul, H. Xue, Rate of deoxygenation and rheologic behavior of blood in sickle cell anemia. Blood 77, 1353–1361 (1991)Google Scholar
  43. 43.
    T. Kinjo, S.A. Hyodo, Equation of motion for coarse-grained simulation based on microscopic description. Phys. Rev. E 75(5), 051109 (2007)Google Scholar
  44. 44.
    J.M.V.A. Koelman, P.J. Hoogerbrugge, Dynamic simulations of hard-sphere suspensions under steady shear. Europhys. Lett. 21(3), 363–368 (1993)CrossRefGoogle Scholar
  45. 45.
    J. Kordilla, W.X. Pan, A. Tartakovsky, Smoothed particle hydrodynamics model for Landau-Lifshitz-Navier-Stokes and advection-diffusion equations. J. Chem. Phys. 141, 224112 (2014)CrossRefGoogle Scholar
  46. 46.
    L.D. Landau, E.M. Lifshitz, Fluid Mechanics (Volume 6 of A Course of Theoretical Physics) (Pergamon Press, New York, 1959)Google Scholar
  47. 47.
    A.W. Lees, S.F. Edwards, The computer study of transport processes under extreme conditions. J. Phys. C: Solid State Phys. 5(15), 1921–1929 (1972)CrossRefGoogle Scholar
  48. 48.
    H. Lei, G.E. Karniadakis, Quantifying the rheological and hemodynamic characteristics of sickle cell anemia. Biophys. J. 102, 185–194 (2012)CrossRefGoogle Scholar
  49. 49.
    H. Lei, G.E. Karniadakis, Probing vasoocclusion phenomena in sickle cell anemia via mesoscopic simulations. Proc. Natl. Acad. Sci. USA 110(28), 11326–11330 (2013)CrossRefGoogle Scholar
  50. 50.
    H. Lei, B. Caswell, G.E. Karniadakis, Direct construction of mesoscopic models from microscopic simulations. Phys. Rev. E 81(2), 026704 (2010)Google Scholar
  51. 51.
    H. Lei, D.A. Fedosov, B. Caswell, G.E. Karniadakis, Blood flow in small tubes: quantifying the transition to the non-continuum regime. J. Fluid Mech. 722, 214–239 (2013)CrossRefzbMATHGoogle Scholar
  52. 52.
    H. Lei, X. Yang, Z. Li, G.E. Karniadakis, Systematic parameter inference in stochastic mesoscopic modeling. J. Comput. Phys. 330, 571–593 (2017)MathSciNetCrossRefGoogle Scholar
  53. 53.
    B. Leimkuhler, X.C. Shang, On the numerical treatment of dissipative particle dynamics and related systems. J. Comput. Phys. 280, 72–95 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    B. Leimkuhler, X.C. Shang, Pairwise adaptive thermostats for improved accuracy and stability in dissipative particle dynamics. J. Comput. Phys. 324, 174–193 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    H. Li, G. Lykotrafitis, A coarse-grain molecular dynamics model for sickle hemoglobin fibers. J. Mech. Behav. Biomed. 4(2), 162–173 (2011)CrossRefGoogle Scholar
  56. 56.
    H. Li, G. Lykotrafitis, Two-component coarse-grained molecular-dynamics model for the human erythrocyte membrane. Biophys. J. 102(1), 75–84 (2012)CrossRefGoogle Scholar
  57. 57.
    X.J. Li, A.S. Popel, G.E. Karniadakis, Blood-plasma separation in Y-shaped bifurcating microfluidic channels: a dissipative particle dynamics simulation study. Phys. Biol. 9(2), 026010 (2012)Google Scholar
  58. 58.
    Z. Li, Z. W. Zhou, G.H. Hu, Dissipative particle dynamics simulation of droplet oscillations in AC electrowetting. J. Adhes. Sci. Technol. 26(12–17), 1883–1895 (2012)Google Scholar
  59. 59.
    H. Li, V. Ha, G. Lykotrafitis, Modeling sickle hemoglobin fibers as one chain of coarse-grained particles. J. Biomech. 45(11), 1947–1951 (2012)CrossRefGoogle Scholar
  60. 60.
    Z. Li, G.H. Hu, Z.L. Wang, Y.B. Ma, Z.W. Zhou, Three dimensional flow structures in a moving droplet on substrate: a dissipative particle dynamics study. Phys. Fluids 25(7), 072103 (2013)Google Scholar
  61. 61.
    Z. Li, X. Bian, B. Caswell, G.E. Karniadakis, Construction of dissipative particle dynamics models for complex fluids via the Mori–Zwanzig formulation. Soft Matter 10(43), 8659–8672 (2014)CrossRefGoogle Scholar
  62. 62.
    Z. Li, Y.-H. Tang, H. Lei, B. Caswell, G.E. Karniadakis, Energy-conserving dissipative particle dynamics with temperature-dependent properties. J. Comput. Phys. 265, 113–127 (2014)CrossRefzbMATHGoogle Scholar
  63. 63.
    X.J. Li, Z.L. Peng, H. Lei, M. Dao, G.E. Karniadakis, Probing red blood cell mechanics, rheology and dynamics with a two-component multiscale model. Phil. Trans. R. Soc. A 372, 20130389 (2014)CrossRefzbMATHGoogle Scholar
  64. 64.
    Z. Li, X. Bian, X.T. Li, G.E. Karniadakis, Incorporation of memory effects in coarse-grained modeling via the Mori–Zwanzig formalism. J. Chem. Phys. 143(24), 243128 (2015)Google Scholar
  65. 65.
    Z. Li, Y.-H. Tang, X.J. Li, G.E. Karniadakis, Mesoscale modeling of phase transition dynamics of thermoresponsive polymers. Chem. Commun. 51(55), 11038–11040 (2015)CrossRefGoogle Scholar
  66. 66.
    Z. Li, A. Yazdani, A. Tartakovsky, G.E. Karniadakis, Transport dissipative particle dynamics model for mesoscopic advection-diffusion-reaction problems. J. Chem. Phys. 143(1), 014101 (2015)Google Scholar
  67. 67.
    Z. Li, X. Bian, Y.-H. Tang, G.E. Karniadakis, A dissipative particle dynamics method for arbitrarily complex geometries. arXiv preprint arXiv:1612.08761 (2016)Google Scholar
  68. 68.
    X.J. Li, E. Du, H. Lei, Y.-H. Tang, M. Dao, S. Suresh, G.E. Karniadakis, Patient-specific blood rheology in sickle-cell anaemia. Interface Focus 6(1), 20150065 (2016)Google Scholar
  69. 69.
    Z. Li, H.S. Lee, E. Darve, G.E. Karniadakis, Computing the non-Markovian coarse-grained interactions derived from the Mori–Zwanzig formalism in molecular systems: application to polymer melts. J. Chem. Phys. 146(1), 014104 (2017)Google Scholar
  70. 70.
    M. Lisal, J.K. Brennan, Alignment of lamellar diblock copolymer phases under shear: insight from dissipative particle dynamics simulations. Langmuir 23(9), 4809–4818 (2007)CrossRefGoogle Scholar
  71. 71.
    M.B. Liu, G.R. Liu, L.W. Zhou, J.Z. Chang, Dissipative particle dynamics (DPD): an overview and recent developments. Arch. Comput. Meth. Eng. 22(4), 529–556 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    Z.Y. Lu, Y.M. Wang, An Introduction to Dissipative Particle Dynamics. Methods in Molecular Biology, chap. 24, vol. 924 (Humana Press, New York, 2013), pp. 617–633Google Scholar
  73. 73.
    K. Lykov, X.J. Li, H. Lei, I.V. Pivkin, G.E. Karniadakis, Inflow/outflow boundary conditions for particle-based blood flow simulations: application to arterial bifurcations and trees. PLoS Comput. Biol. 11(8), e1004410 (2015)Google Scholar
  74. 74.
    A.D. Mackie, J.B. Avalos, V. Navas, Dissipative particle dynamics with energy conservation: modelling of heat flow. Phys. Chem. Chem. Phys. 1(9), 2039–2049 (1999)CrossRefGoogle Scholar
  75. 75.
    C.A. Marsh, J.M. Yeomans, Dissipative particle dynamics: the equilibrium for finite time steps. Europhys. Lett. 37(8), 511–516 (1997)CrossRefGoogle Scholar
  76. 76.
    C.A. Marsh, G. Backx, M.H. Ernst, Fokker-Planck-Boltzmann equation for dissipative particle dynamics. Europhys. Lett. 38(6), 411–415 (1997)CrossRefGoogle Scholar
  77. 77.
    C.A. Marsh, G. Backx, M.H. Ernst, Static and dynamic properties of dissipative particle dynamics. Phys. Rev. E 56(2), 1676–1691 (1997)CrossRefGoogle Scholar
  78. 78.
    Z.G. Mills, W.B. Mao, A. Alexeev, Mesoscale modeling: solving complex flows in biology and biotechnology. Trends Biotechnol. 31(7), 426–434 (2013)CrossRefGoogle Scholar
  79. 79.
    J.J. Monaghan, Smoothed particle hydrodynamics. Rep. Prog. Phys. 68(8), 1703–1759 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  80. 80.
    H. Mori, Transport, collective motion, and Brownian motion. Prog. Theor. Phys. 33(3), 423–455 (1965)CrossRefzbMATHGoogle Scholar
  81. 81.
    S.V. Nikolov, H. Shum, A.C. Balazs, A. Alexeev, Computational design of microscopic swimmers and capsules: from directed motion to collective behavior. Curr. Opin. Colloid Interface Sci. 21, 44–56 (2016)CrossRefGoogle Scholar
  82. 82.
    W.G. Noid, Perspective: coarse-grained models for biomolecular systems. J. Chem. Phys. 139, 090901 (2013)CrossRefGoogle Scholar
  83. 83.
    J.M. Ortiz de Zárate, J.V. Sengers, Hydrodynamic Fluctuations in Fluids and Fluid Mixtures (Elsevier, Amsterdam, 2006)zbMATHGoogle Scholar
  84. 84.
    H.C. Öttinger, Beyond Equilibrium Thermodynamics (Wiley-Interscience, New York, 2005)CrossRefGoogle Scholar
  85. 85.
    I. Pagonabarraga, D. Frenkel, Dissipative particle dynamics for interacting systems. J. Chem. Phys. 115(11), 5015–5026 (2001)CrossRefGoogle Scholar
  86. 86.
    W.X. Pan, I.V. Pivkin, G.E. Karniadakis, Single-particle hydrodynamics in DPD: a new formulation. Europhys. Lett. 84(1), 10012 (2008)Google Scholar
  87. 87.
    D.Y. Pan, J.X. Hu, X.M. Shao, Lees-Edwards boundary condition for simulation of polymer suspension with dissipative particle dynamics method. Mol. Simul. 42(4), 328–336 (2016)CrossRefGoogle Scholar
  88. 88.
    N. Phan-Thien, N. Mai-Duy, B.C. Khoo, A spring model for suspended particles in dissipative particle dynamics. J. Rheol. 58(4), 839–867 (2014)CrossRefGoogle Scholar
  89. 89.
    C.L. Phillips, J.A. Anderson, S.C. Glotzer, Pseudo-random number generation for Brownian dynamics and dissipative particle dynamics simulations on GPU devices. J. Comput. Phys. 230(19), 7191–7201 (2011)CrossRefzbMATHGoogle Scholar
  90. 90.
    I.V. Pivkin, G.E. Karniadakis, A new method to impose no-slip boundary conditions in dissipative particle dynamics. J. Comput. Phys. 207(1), 114–128 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  91. 91.
    I.V. Pivkin, G.E. Karniadakis, Controlling density fluctuations in wall-bounded dissipative particle dynamics systems. Phys. Rev. Lett. 96, 206001 (2006)CrossRefGoogle Scholar
  92. 92.
    I.V. Pivkin, G.E. Karniadakis, Accurate coarse-grained modeling of red blood cells. Phys. Rev. Lett. 101, 118105 (2008)CrossRefGoogle Scholar
  93. 93.
    I.V. Pivkin, Z. Peng, G.E. Karniadakis, P.A. Buffet, M. Dao, S. Suresh, Biomechanics of red blood cells in human spleen and consequences for physiology and disease. Proc. Natl. Acad. Sci. USA 113(28), 7804–7809 (2016)CrossRefGoogle Scholar
  94. 94.
    M. Praprotnik, L.D. Site, K. Kremer, Adaptive resolution scheme for efficient hybrid atomistic-mesoscale molecular dynamics simulations of dense liquids. Phys. Rev. E 73, 066701 (2006)CrossRefGoogle Scholar
  95. 95.
    R. Qiao, P. He, Simulation of heat conduction in nanocomposite using energy-conserving dissipative particle dynamics. Mol. Simul. 33(8), 677–683 (2007)CrossRefGoogle Scholar
  96. 96.
    M. Ripoll, P. Español, M.H. Ernst, Dissipative particle dynamics with energy conservation: heat conduction. Int. J. Mod. Phys. C 9(8), 1329–1338 (1998)CrossRefGoogle Scholar
  97. 97.
    D. Rossinelli, Y.-H. Tang, K. Lykov, D. Alexeev, M. Bernaschi, P. Hadjidoukas, M. Bisson, W. Joubert, C. Conti, G. Karniadakis, M. Fatica, I. Pivkin, P. Koumoutsakos, The in-silico lab-on-a-chip: petascale and high-throughput simulations of microfluidics at cell resolution, in Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, SC ’15 (Association for Computing Machinery, New York, 2015), pp. 2:1–2:12Google Scholar
  98. 98.
    M.G. Saunders, G.A. Voth, Coarse-graining methods for computational biology. Annu. Rev. Biophys. 42, 73–93 (2013)CrossRefGoogle Scholar
  99. 99.
    T. Shardlow, Splitting for dissipative particle dynamics. SIAM J. Sci. Comput. 24(4), 1267–1282 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  100. 100.
    V. Symeonidis, G.E. Karniadakis, B. Caswell, Dissipative particle dynamics simulations of polymer chains: scaling laws and shearing response compared to dna experiments. Phys. Rev. Lett. 95, 076001 (2005)CrossRefGoogle Scholar
  101. 101.
    Y.-H. Tang, G.E. Karniadakis, Accelerating dissipative particle dynamics simulations on GPUs: algorithms, numerics and applications. Comput. Phys. Commun. 185(11), 2809–2822 (2014)MathSciNetCrossRefGoogle Scholar
  102. 102.
    Y.-H. Tang, Z. Li, X.J. Li, M.G. Deng, G.E. Karniadakis, Non-equilibrium dynamics of vesicles and micelles by self-assembly of block copolymers with double thermoresponsivity. Macromolecules 49(7), 2895–2903 (2016)CrossRefGoogle Scholar
  103. 103.
    A. Tiwari, J. Abraham, Dissipative particle dynamics model for two-phase flows. Phys. Rev. E 74, 056701 (2006)MathSciNetCrossRefGoogle Scholar
  104. 104.
    A. Vázquez-Quesada, M. Ellero, P. Español, Smoothed particle hydrodynamic model for viscoelastic fluids with thermal fluctuations. Phys. Rev. E 79, 056707 (2009)MathSciNetCrossRefGoogle Scholar
  105. 105.
    Y.X. Wang, S. Chen, Numerical study on droplet sliding across micropillars. Langmuir 31(16), 4673–4677 (2015)CrossRefGoogle Scholar
  106. 106.
    P.B. Warren, Vapor-liquid coexistence in many-body dissipative particle dynamics. Phys. Rev. E 68, 066702 (2003)CrossRefGoogle Scholar
  107. 107.
    C.M. Wijmans, B. Smit, Simulating tethered polymer layers in shear flow with the dissipative particle dynamics technique. Macromolecules 35(18), 7138–7148 (2002)CrossRefGoogle Scholar
  108. 108.
    S.M. Willemsen, H.C.J. Hoefsloot, P.D. Iedema, No-slip boundary condition in dissipative particle dynamics. Int. J. Mod. Phys. C 11(5), 881–890 (2000)CrossRefzbMATHGoogle Scholar
  109. 109.
    T. Ye, N. Phan-Thien, B. Khoo, C.T. Lim, Numerical modelling of a healthy/malaria-infected erythrocyte in shear flow using dissipative particle dynamics method. J. Appl. Phys. 115(22), 224701 (2014)Google Scholar
  110. 110.
    S. Yip, M.P. Short, Multiscale materials modelling at the mesoscale. Nat. Mater. 12(9), 774–777 (2013)CrossRefGoogle Scholar
  111. 111.
    R. Zwanzig, Ensemble method in the theory of irreversibility. J. Chem. Phys. 33(5), 1338–1341 (1960)MathSciNetCrossRefGoogle Scholar
  112. 112.
    R. Zwanzig, Memory effects in irreversible thermodynamics. Phys. Rev. 124(4), 983–992 (1961)CrossRefzbMATHGoogle Scholar
  113. 113.
    R. Zwanzig, Nonequilibrium Statistical Mechanics, vol. 54 (Oxford University Press, Oxford, 2001)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Z. Li
    • 1
  • X. Bian
    • 1
  • X. Li
    • 1
  • M. Deng
    • 1
  • Y.-H. Tang
    • 1
  • B. Caswell
    • 2
  • G. E. Karniadakis
    • 1
    Email author
  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA
  2. 2.School of EngineeringBrown UniversityProvidenceUSA

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