THE NON-EQUILIBRIUM CHARGE SCREENING EFFECTS IN DIFFUSION-

DRIVEN SYSTEMS V.N. KuzovkovV.N. Kuzovkov11, , E.A. KotominE.A. Kotomin11 and M. Olvera de la Cruz and M. Olvera de la Cruz22

ERAF project Nr. 2010/0272/2DP/2.1.1.1.0/10/APIA/VIAA/08811Institute of Solid State Physics, University of Latvia, 8 Kengaraga, LV-1063 Riga, LatviaInstitute of Solid State Physics, University of Latvia, 8 Kengaraga, LV-1063 Riga, Latvia22Dept. of Mat. Science and Eng., Northwestern University, Evanston, IL 60208, USA

AbstractAbstract

Introduction. Introduction. Debye–Hückel theoryDebye–Hückel theory Model and methodsModel and methods

The effects of non-equilibrium charge screening in mixtures of oppositely charged interacting molecules on surfaces are analyzed in a closed system. The dynamics of charge screening and the strong deviation from the standard Debye-Hückel theory are demonstrated via a new formalism based on computing radial distribution functions suited for analyzing both short-range and long-range spacial ordering effects. At long distances the inhomogeneous molecular distribution is limited by diffusion, whereas at short distances (of the order of several coordination spheres) by a balance of short-range (Lennard-Jones) and long-range (Coulomb) interactions. The non-equilibrium charge screening effects in transient pattern formation are further quantified. It is demonstrated that the use of screened potentials, in the spirit of the Debye-Hückel theory, leads to qualitatively incorrect results.

ResultsResults

ReferencesReferences

Qualitative picture

The main effect: each charged molecule is surrounded by a cloud of oppositely charged molecules whose concentration exceeds the average molecule concentration. At distances larger than the cloud size the Coulomb interaction can be neglected.

Self-consistent solution of the Poisson equation: Finding the electrostatic potential φ for a spatial distribution of particles (which in turn depends on the potential). r

e

r

r

1

For d=3 the screening factor:

κ-1 – Debye screening length.

rerQ )( This factor is time-independent Positive value smaller than unity at short

distances Asymptotically striving to zero at long

distances

Poisson-Boltzmann equation

The most common formulationfor dimensionless potential ψ [1] :

[1] D.A. Walker et al , Nanoscale. 3, 1316 (2011). [2] V.N. Kuzovkov et al , J.Chem.Phys. 135, 034702 (2011). [3] S.M. Loverde et al , Phys. Rev.Lett. 98, 237802 (2007). [4] V.N. Kuzovkov et al , Phys.Rev. E. 82, 021602 (2010).[5] E.A. Kotomin and V.N. Kuzovkov, Modern Aspects of Diffusion-Controlled Reactions: Cooperative Phenomena in Bimolecular Processes, Elsevier: North Holland, Amsterdam (1996).

)sinh(22 2/1

022 )/2( Tknze B

The system of classical particles with purely electrostatic interactions is unstable: the equation has a singularity as r→0. As a result, the self-consistent non-linear equation for the potential ψ shows divergence and has no physical solution.

Typical tricks: linearization homogeneous distribution of one type of charge

)sinh(

2

)1()sinh(

e

Electrolytes or condensed matter: the particles have a non-zero size and/or short-range interactions. Physical solution of equation?

The non-equilibrium effects

Ideal gases: charges (electrons or ions) are very mobile and rapidly equilibrate.Boltzmann distribution:

Tkrezi

Binern /)()(

Soft and condensed matter: the molecules are characterized by partial diffusion coefficients, DA and DB, which could differ by orders of magnitude. Molecular diffusion is quite slow and thus molecular distribution is usually far from equilibrium [2].

Typical diffusion time for reproduction of equilibrium behaviour at the length R.

D

Rt

2

We study the kinetics of pattern formation and phase separation in a system of two types of oppositely charged molecules with competing short- and long-range interactions on surfaces/interfaces. The molecular ordering occurs on the background of the Ostwald ripening and thus is strongly non-equilibrium. We demonstrate how initial random distribution of molecules is changed for loose similar-molecule aggregates, with further reorganization into dense macroscopic domains of oppositely charged molecules. This pattern formation process is characterized by the correlation length ξ(t) [3,4].

Short-range van der Waals interactions (U0,r0):without Coulomb interactions A and B molecules repel each other whereas similar molecules attract each other and thus could aggregate.Long-range Coulomb interactions: Stabilizations of the system and pattern formation.Diffusion: coefficients DA and DB.

Dimensionless parameters: Temperature Relative contribution of the Coulomb potential withrespect to the Lennard-Jones one Asymmetry in the particles' diffusion coefficients The diffusion time unit

0/TT BkUT /00

00

2

Tkr

e

B

BA

A

DD

D

BA DD

rt

20

0

Many-point densities of a number of particles:

The exact set of the coupled kinetic equations is similar to the BBGKY(Bogoliubov-Born-Green-Kirkwood-Yvon) hierarchy in the statistical physics. The equations describe the particle diffusion drift in potentials of mean force that are, in turn, functionals of the correlation functions [4,5].

The relationship between the calculated radial distribution functions (joint correlation functions) and the molecular distribution in a real space is not obvious and far from trivial. The Reverse MC (RMC) method is used to visualize the system spatial structure.

Strong deviation from the DH theory: overcharging (charge amplification) at small r, charge reversal as r increases, oscillations of screening factor. The non-equilibrium chargescreening factors are determined by both process: kinetics (anomalously long ordering times at large

length scales) andshort-range interactions (lamellar-like patterns).

The appearance of several structural scales: the size of the super-particles (A and B large

aggregates), the size of a recurring (repeating) element of

the pattern (consisting of several super-particles),

the average size of the whole pattern.

Two extrema and two different length scales: the short-range length λsr, the long-range length λlr .

At distances exceeding the average distance between dissimilar aggregates the screening factor changes its sign since a probe molecule now is surrounded with aggregates of opposite sign.

More mobile molecules at a given time t establish the intermediate order at the larger distances as compared to less mobile molecules.This work has been supported by ERAF 010/0272/2DP/2.1.1.1.0/10/APIA/VIAA088