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SIMU Challenges in Molecular Simulations: Bridging the Length- and Timescales gap Volume 4
Abstract and Figures
The European Science Foundation SIMU programme aimed at building cooperation across Europe in the field of computational physics and chemistry of condensed matter, with particular emphasis on the development of novel computational techniques to perform multiscale molecular simulations. Articles in this issue include:
1. European Collaboration in ab-initio Computer Simulation, by Volker Heine; 2. Interview of Berni J. Alder, by Donal Mac Kernan and Michel Mareshal; 3. Dissipative Particle Dynamics Revisited, by Pep Espanol; 4.
Conformation-Dependent Sequence Design: a Review of the Method and Recent Theoretical and Computer Simulation Results, by Alexei R. Khokhlov, Victor A. Ivanov, Alexander V. Chertovich and Pavel G. Khalatur; 5. Atomistic Computer Simulations of Friction between Solids , review article by Martin H. Muser and Mark O. Robbins; 7. The Quasicontinuum Method Revisited, by Jens Jørgen Mortensen,
Jakob Schiøtz and Karsten W. Jacobsen; 8. Cambridge University Centre for Computational Chemistry, laboratory, Jean-Pierre Hansen
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... The twofold requirement is, first, that cells should be densely populated by particles and, second, that the fields so derived should vary slowly both in space and time with respect to the mesoscopic space and time scales L m and T m . 3 The required smoothness of the macro-scopic fields plays a pivotal role in proving that the local streaming fields have a first-order contact with the macroscopic velocity field at the respective cell centres (cf. Sect. ...
... Consider, for instance, a three-dimensional body whose characteristic linear size is L = λL M , with λ largertypically, much larger-than 1. A fully molecular model of it would comprise about 3 (169) 37 This issue is discussed at length in [2,34,37]. To cure the non-uniqueness of binary representations of the cluster expansion (238) for n > 4, Torres-Sánchez, Vanegas and Arroyo [37] introduce a connection on the shape manifold, and compute the corresponding covariant derivative of the interaction potential, thus skirting its non-unique extension to an ambient linear space. ...
... The multiscale method based on the present theory would sample the molecular level only at scattered locations, the typical distance between them being L M . Each mesoscopic space cell would contain about (L m /d) 3 particles, and there would be (L/L M ) 3 = λ 3 cells. With respect to a fully molecular model, the number of particles explicitly handled by this method would therefore be reduced by a factor (L M /L m ) 3 , i.e., by many orders of magnitude. ...
In this paper, I determine the minimum amount of information that continuum mechanics needs to obtain from Newtonian molecular dynamics, in order to avail itself of stress-strain responses uniformly valid for a vast range of macroscopic regimes, being quantitatively determined by microscopic physical properties. Described from the opposite, bottom-up point of view, the procedure I put forward uses the basic kinematic and dynamical machinery of continuum mechanics to upscale molecular dynamics to the macroscopic level in a practicable and efficient way.
The paper traces the early stages of Berni Alder’s scientific accomplishments, focusing on his contributions to the development of Computational Methods for the study of Statistical Mechanics. Following attempts in the early 50s to implement Monte Carlo methods to study equilibrium properties of many-body systems, Alder developed in collaboration with Tom Wainwright the Molecular Dynamics approach as an alternative tool to Monte Carlo, allowing to extend simulation techniques to non-equilibrium properties. This led to the confirmation of the existence of a phase transition in a system of hard spheres in the late 50s, and was followed by the discovery of the unexpected long-time tail in the correlation function about a decade later. In the late 70s Alder was among the pioneers of the extension of Computer Simulation techniques to Quantum problems. Centered around Alder’s own pioneering contributions, the paper covers about thirty years of developments in Molecular Simulation, from the birth of the field to its coming of age as a self-sustained discipline.
The stochastic differential equations corresponding to the updating algorithm of Dissipative Particle Dynamics (DPD), and the corresponding Fokker-Planck equation are derived. It is shown that a slight modification to the algorithm is required before the Gibbs distribution is recovered as the stationary solution to the Fokker-Planck equation. The temperature of the system is then directly related to the noise amplitude by means of a fluctuation-dissipation theorem. However, the correspondingly modified, discrete DPD algorithm is only found to obey these predictions if the length of the time step is sufficiently reduced. This indicates the importance of time discretisation in DPD.
We critically review dissipative particle dynamics (DPD) as a mesoscopic simulation method. We have established useful parameter ranges for simulations, and have made a link between these parameters and χ-parameters in Flory-Huggins-type models. This is possible because the equation of state of the DPD fluid is essentially quadratic in density. This link opens the way to do large scale simulations, effectively describing millions of atoms, by firstly performing simulations of molecular fragments retaining all atomistic details to derive χ-parameters, then secondly using these results as input to a DPD simulation to study the formation of micelles, networks, mesophases and so forth. As an example application, we have calculated the interfacial tension σ between homopolymer melts as a function of χ and N and have found a universal scaling collapse when σ/ρkBTχ0.4 is plotted against χN for N>1. We also discuss the use of DPD to simulate the dynamics of mesoscopic systems, and indicate a possible problem with the timescale separation between particle diffusion and momentum diffusion (viscosity).
In this study we present the coarse-graining of one polymer chain in a melt to a single dimer. By using the projector operator formalism we derive the equation of motions for the dimer. The different forces that occur in this equation of motion are calculated from molecular dynamics simulations of the microscopic model, using constraint forces to fix the dimer configuration. The mean constraint force serves as the conserved part of the interaction, whereas the time correlation of the constraint force fluctuation leads to the nonconserved interactions: the dissipative and fluctuating forces. Using the configurational dependent coarse-grained interactions we have performed stochastic dynamics simulations of the dimer. Dimer properties of the microscopic and the coarse-grained model are shown to be in reasonable agreement. We also discuss the application of the framework to coarse-graining polymer melts into more detail, i.e., beyond the dimer.
A novel method of investigating the link between molecular features of polymer molecules and the rheological properties of dilute polymer solutions has been investigated. It applies the dissipative particle dynamics (DPD) computer simulation technique, which introduces a lattice‐gas automata time‐stepping procedure into a molecular‐dynamics scheme, to model bead‐and‐spring‐type representations of polymer chains. Investigations of static and dynamic scaling relationships show that the scaling of radius of gyration and relaxation time with the number of beads are consistent with the predictions of the Rouse–Zimm model. Both hydrodynamic interaction and excluded volume emerge naturally from the DPD polymermodel, indicating that a realistic description of the dynamics of a dilute polymer solution can be obtained within this framework, and that very efficient computer simulations are possible.
Smoothed particle hydrodynamics (SPH) is an effective numerical method
to solve various problems, especially in astrophysics, but its
applications have been limited to inviscid flows since it is considered
not to yield ready solutions to fluid equations with second-order
derivatives. Here we present a new SPH method that can be used to solve
the Navier-Stokes equations for constant viscosity. The method is
applied to two-dimensional Poiseuille flow, three-dimensional
Hagen-Poiseuille flow and two-dimensional isothermal flows around a
cylinder. In the former two cases, the temperature of fluid is assumed
to be linearly dependent on a coordinate variable x along the flow
direction. The numerical results agree well with analytic solutions, and
we obtain nearly uniform density distributions and the expected
parabolic and paraboloid velocity profiles. The density and velocity
field in the latter case are compared with the results obtained using a
finite difference method. Both methods give similar results for Reynolds
number Re = 6, 10, 20, 30 and 55, and the differences in the
total drag coefficients are about 2 ~ 4%. Our numerical simulations
indicate that SPH is also an effective numerical method for calculation
of viscous flows.
The fundamentals of the smoothed particle hydrodynamics (SPH) method and
its applications in astrophysics are reviewed. The discussion covers
equations of motion, viscosity amd thermal conduction, spatially varying
resolution, kernels, magnetic fields, special relativity, and
implementation. Applications of the SPH method are discussed with
reference to gas dynamics, binary stars and stellar collisions,
formation of the moon and impact problems, fragmentation and cloud
collisions, and cosmological and galactic problems. Other applications
discussed include disks and rings, radio jets, motion near black holes,
supernovae, magnetic phenomena, and nearly incompressible flow.
The dynamics of a surface-confined drop in a simple shear field has been studied, pursuing the dissipative particle dynamics (DPD) simulation approach. The shear field induces contact angle hysteresis in the drop, the degree of hysteresis increasing with the shear rate. At shear rates exceeding a critical value, the drop acquires the tendency to lift off the boundary, leading to its removal. In the equilibrium contact angle range, e>120°, the drop preserves its integrity as it escapes from the boundary, whereas at lower contact angles the drop assumes a distinctly elongated shape prior to its removal, which develops “necks’' at subsequent times. The drop breaks up as the necks are ruptured upon thinning, with some fragments escaping into the bulk phase and some remaining at the surface. Under certain hydrodynamic conditions the moving drop sheds a trail of tiny droplets on the surface. The simulation results are in qualitative agreement with experimental studies on the corresponding systems published in the literature.
The optimized cluster expansion methods developed in the first article of this series (I) are generalized to apply to molecular fluids. These methods make use of summations of ring and chain cluster diagrams. The summations are performed explicitly for certain classes of molecular models. The molecules in these classes contain several ``interaction sites,'' and the total interaction between two molecules is a sum of site-site potentials that depend on the scalar distances between sites on the two molecules. The principal results of this work are computationally simple techniques for calculating the thermo-dynamic properties and pair correlation functions of molecular fluids in which the intermolecular interactions are highly angular dependent. The techniques should be reliable since they arise from the same approximations that have been shown to be very accurate when applied to simple fluids.