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D. Toneian, G. Kahl, G. Gompper, R. G. Winkler:
"Multiparticle Collision Dynamics Simulations of Viscoelastic Fluids";
Vortrag: Vienna Young Scientist Symposium, TU Wien; 25.06.2015 - 26.06.2015.

Complex fluids, such as they commonly appear in nature and industry, typically consist of a multitude of components that differ greatly in their characteristic length and time scales. For example, polymers dissolved in water often are orders of magnitude larger than the water molecules they interact with. Yet, the comparatively slow dynamics of polymers is strongly influenced by the interaction of their constituents with the fast-paced water molecules.

To efficiently capture the important dynamical aspects of such systems in computer simulations, the particle-based simulation approach denoted as Multiparticle Collision Dynamics (MPC) has been developed [1], [2].
There, the fluid is modeled as a collection of point-particles (each representing a large number of real molecules) which undergo subsequent dynamical streaming and collision steps.
In the streaming phase, all particles move ballistically without interacting with each other.
In the collision phase, the simulation volume is partitioned into cells that contain about 10 particles on average, defining the collisional environment. For each individual cell, the center-of-mass velocity of the contained particles is calculated, and then their relative velocities are rotated around a randomly oriented axis.
This algorithm conserves mass and linear momentum, so that a Navier-Stokes type of dynamical behavior is achieved.

A linear polymer dissolved in a simple fluid is modeled as a chain of particles which are linked via a suitable interaction potential. For these particles, the streaming phase is carried out as in Molecular Dynamics Simulations (i.e. numerically integrating Newton's equation of motion) instead of analytically computing a ballistic motion. During the collision phase, the polymer particles are treated as heavy fluid particles and their velocities are rotated as indicated above. This yields an exchange of momentum between the fluid and the polymer and establishes fluid-mediated correlations between the polymer particles. The simplicity of the algorithm ensures a high degree of parallelism, and hence is computationally extremely efficient when executed on GPUs.

Simple fluids and dilute polymer solutions exhibit viscous effects, but no elastic phenomena arise. However, in many biological and technical applications, elastic properties of fluids play a crucial role. In order to capture both aspects, i.e., to address viscoelastic fluids, extensions of the MPC approach have recently been proposed [3], [4]. Here, in the simplest version, the fluid is composed of dumbbells: every fluid particle is connected to exactly one other particle via a massless spring. In subsequent studies, we have then further extended the MPC model to longer polymer chains, retaining the simple connecting interaction.

Important insight into the dynamical behavior of such fluids can be gained by analyzing the center-of-mass velocity-autocorrelation function of the polymer chains. We have considered this correlation function for dumbbells (polymers consisting of N=2 linked MPC particles), trimers (N=3), and decamers (N=10). In Fourier space, the transverse velocity-correlation function C^T(k,t) ~ <v^T(k,t) * v^T(-k,0)> exhibits oscillations superimposed with an exponential decay, where the oscillation frequency is related to the polymer relaxation time, and the damping to the bare fluid's viscosity. From the known storage and loss moduli of dumbbells, an analytical expression for the correlation function in the case N=2 is derived, which is in quantitative agreement with the simulation results. Closed expressions for the Laplace-transform of C^T(k,t) are derived for the case N>2, and their numerical Laplace-inversions again reproduce simulation data accurately. Further results for the real-space correlation functions are presented, and the long-time behavior is discussed.

Having gained a quantitative understanding of this fluid model, its properties can be tuned so as to study real-world complex systems, e.g., polymer melts or viscoelastic fluids containing colloids or self-propelling microswimmers (e.g., E. coli bacteria), as they arise in engineering, biology, and medicine, among other fields.

[1] A. Malevanets and R. Kapral, "Mesoscopic model for solvent dynamics," J. Chem. Phys., vol. 110, pp. 8605-8613, 1999. [Online]. Available: http://dx.doi.org/10.1063/1.478857

[2] G. Gompper, T. Ihle, D. M. Kroll, and R. G. Winkler, Multi-Particle Collision Dynamics: A Particle-Based Mesoscale Simulation Approach to the Hydrodynamics of Complex Fluids, ser. Adv. Polym. Sci.
Springer, 2008. [Online]. Available: http://link.springer.com/chapter/10.1007%2F12_2008_5

[3] Y.-G. Tao, I. O. Götze, and G. Gompper, "Multiparticle collision dynamics modeling of viscoelastic fluids," J. Chem. Phys., vol. 128, p. 144902, 2008. [Online]. Available: http://dx.doi.org/10.1063/1.2850082

[4] B. Kowalik and R. G. Winkler, "Multiparticle collision dynamics simulations of viscoelastic fluids: Shear-thinning Gaussian dumbbells," J. Chem. Phys., vol. 138, p. 104903, 2013. [Online]. Available: http://dx.doi.org/10.1063/1.4792196

Projektleitung Gerhard Kahl:
DFS