Experimental and Simulation Setup of Haghgooie

Experiments and BD simulations on similar systems in equilibrium (without driving) have been made by Haghgooieet al.[Hag04,Hag05,Hag06]. They analyzed the structure and the equilib-rium dynamics of a 2D superparamagnetic colloidal systems due to a confinement by two hard walls and compared them to the unbounded 2D system. Their channels (40µm wide and 40, 80, and 200µm long) are filled with colloidal spheres of sizeσ=2.8µm in a 5.0×TBE buffer (0.17 M ionic strength) to screen the electrostatic repulsion. A single colloidal bead has the mass density ρcolloid=1.6 g/cm³ and the magnetic susceptibilityχeff= ^4π₃ ^σ_22χ0

µ0 ≈9.0·10⁻¹²Am²/T with χ⁰≈1.

Based upon the total particle number densityn≡N/(L_xL_y)Haghgooie and coworkers use R=

√2 3n

_−1/2

(8.4) as characteristic length scale for their overdamped Brownian dynamics simulations of the equi-librium properties of the colloids confined to 2D channels. Notice, that they also used a slightly different definition for the dimensionless interaction strength

Γ_H= µ₀M² 4πk_BTR³ =

√2 3π

_3/2

Γ (8.5)

which differs only by a constant factor from definition (8.2). BD simulations returned that the system has a liquid phase for Γ_H <14.89 and a solid phase for Γ_H >15.2. These values are slightly higher than the experimental values of Zahn [Zah99] which found a hexatic phase between a dimensionless field strength of Γi=11.79 and Γm=13.59. The experiments [Hag06] were performed in the liquid phase atΓ_H≈12, whereas simulations [Hag04,Hag05] were performed in the liquid as well as in the solid phase with the number density held constant atn=0.0462.

4cf. footnote11in chapter3.

In 1827 the botanist Robert Brown (1773–1858) discovered the perpetual jiggling of (non living) particles of colloidal size suspended in water, while he was studying the microscopic life of plant pollens. This was the first observation of the so-calledBrownian motion. In the early years of the twentieth century this observation was proved to be one of the effects of molecular motion.

Albert Einstein (1879–1955) [Ein05] in his famous year 1905 published a pioneering paper on this new type of motion giving the first theoretical explanation. Brownian motion has its origin in the perpetual collisions of the Brownian particle with the molecules of the surrounding fluid (cf.

figure9.1). The Brownian particle will suffer about 10²¹ collisions per second, which cannot be resolved on colloidal time and length scales and thus lead to an erratic motion. As a consequence the theory of Brownian motion has to be formulated as astochastic theory.

Figure 9.1: Schematic drawing of the interaction of a colloidal particle with the solvent particles.

The blue (small) points symbolize the effective action of the solvent particles (not ex-plicitly drawn) exerted on the red (large) colloidal particle giving rise to the exemplary realization of a stochastic trajectory drawn on the right hand side.

Whenever a large number of degrees of freedom of a many-particle system is masked from di-rect observation the time behavior of the remaining degrees of freedom is stochastic and can be described by the theory of Brownian motion. There exists a huge number of other related phe-nomena which can be explained by this theory. Nowadays it has applications in chemistry (e.g.

chemical reactions driven by diffusion), physics (e.g. diffusion in solids, dynamic properties of macromolecules [¨Ot96], critical dynamics at phase transitions, [Cha43,Dho96,Kam05,Ris96]), mathematics (e.g. stochastic differential equations [Klo99]), engineering, biology [Sch00a], and finance (e.g. stochastic models in financial risk management [Deu04,Pau99]). In physics it has also become the simplest approximate way to treat thedynamics of nonequilibrium systems.

The process of representing a system with fewer degrees of freedom, than those actually present in the system, is calledcoarse graining. One eliminates rapidly varying degrees of freedom and keeps the coarse-grained variables with time and length scales much larger than typical molecular scales.

There doesn’t exist a unique way on how exactly to coarse-grain a system. All coarse-graining models build upon physical intuition, simplicity and inherent symmetries, one would like to pre-serve. The validity of the approximations made is usually justified a posteriori by comparison with experiment. With the help of the so-calledprojection operator technique[Zwa01,Han00] one can put the coarse-graining approximations on a more exact statistical mechanical basis. It provides a formal way to derive the Langevin equations from the microscopic Liouville equations, and was initiated by the works of Zwanzig [Zwa60,Zwa61a,Zwa61b] and Mori [Mor65a,Mor65b].

Hydrodynamics

Classical Mechanics Fokker−Planck Smoluchowski Fick

Figure 9.2: Different coarse-graining levels of a colloidal suspension.

A colloidal suspension can be described at different coarse-graining levels [Esp04] as depicted in figure 9.2. On the level of classical mechanics the microscopic state z={R_i,P_i,r_j,p_j} of the colloidal suspension is given by the positions Ri andrj and the momentaPi andpj of col-loidal and solvent particles, respectively. These can be modeled as spherical objects. The time evolution of the microstate is governed by Hamilton’s equations. At the next level of coarse-graining the solvent is treated as a continuous smooth field described by the variables of mass density, momentum density and energy density. The Navier-Stokes equations for the solvent fields are coupled with Newton’s equations for the colloids through boundary conditions, giving rise to the so-called hydrodynamic interactions. Into this level fall all those simulation techniques, which use any Navier-Stokes solver for the solvent allowing to describe thermal fluctuations and Molecular Dynamics for the colloidal particles. The next higher levels of coarse-graining neglect hydrodynamic interactions and approximate the time evolution of the colloids withz={Ri,Pi} by stochastic equations of motion (Langevin equations or equivalently Fokker-Planck equations).

For overdamped systems,i.e. systems without inertial effects, the momentum variablesP_i can be integrated out and one ends up with the time evolution given by position Langevin equations or the Smoluchowski equation. The simulation techniques on the Fokker-Planck and the Smoluchowski level will be calledBrownian Dynamics simulationsin this work. Further coarse-graining can be done if one is not interested in the actual positionsRiof the colloids but just in the number of col-loids located in a spatial region around a pointr. This amounts to the introduction of concentration fields c(r) obeying the continuity equation _∂t^∂c(r) =−∇J, where the mass fluxJhas systematic and stochastic contributions.¹ (In the dilute limit the systematic contribution j∝−∇c(r)obeys Fick’s law.) Finally at very long time scales in which the system may have arrived at equilibrium, the thermodynamic description can be used.

This chapter is organized as follows: In section 9.1 we review the mathematical description of Brownian motion, in section 9.2 we present some commonly used realizations of a Brownian dynamics simulation, and in section9.3the importance of hydrodynamic interactions are discussed as well as possible extensions to include them in a Brownian dynamics simulation. The chapter

1The mathematically equivalent Fokker-Planck equation in this case will become an equation of motion for the prob-ability functionalP[c(r),t].

closes with section9.4, which gives some detailed information on the self-written algorithm for computer simulation, the boundary conditions, and the parameters used.