Shear Viscosity in Two-Dimensional Dipole Systems N. E. Djienbekov1N. Kh. Bastykova1A. M. Bekbussyn1T. S. Ramazanov1and S. K. Kodanova1 1Institute for Experimental and Theoretical Physics

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Shear Viscosity in Two-Dimensional Dipole Systems

N. E. Djienbekov,1N. Kh. Bastykova,1A. M. Bekbussyn,1T. S. Ramazanov,1and S. K. Kodanova1, ∗

1Institute for Experimental and Theoretical Physics,

Al-Farabi Kazakh National University, 71 Al-Farabi ave., 050040 Almaty, Kazakhstan

The results of modeling shear ﬂows in classical two-dimensional dipole systems are presented.

We used the method of non-equilibrium molecular dynamics to calculate the viscosity at various

shear rates. The coeﬃcients of shear viscosity are given in the limit of low shear rates for various

regimes of interparticle correlation from a weakly correlated gaseous state to a strongly non-ideal

liquid state near the crystallization point. The calculations were carried out for bare (unscreened)

dipole systems, as well as for dipole systems in a polarizable medium that provide screening of the

dipole–dipole interaction. The eﬀect of shear thinning in 2D dipole systems is reported at small

values of the coupling parameter. In addition, it is shown that dipole systems can become both less

and more viscous due to the presence of a screening medium, depending on the degree of interparticle

correlation. The optimal simulation parameters are discussed within the framework of the method

of nonequilibrium molecular dynamics for determining the shear viscosity of two-dimensional dipole

systems. Moreover, we present a simple ﬁtting curve which provides universal scaling law for both

bare dipole - dipole interaction and screened dipole-dipole interaction.

I. INTRODUCTION

Two-dimensional systems governed by a repulsive

dipole-dipole pair interaction are relevant for various sys-

tems. For example, the repulsive dipole-dipole interac-

tion is used to describe two-dimensional colloidal sys-

tems [1–3]. In complex plasmas, the interaction between

charged dust particles can be modiﬁed due to external

ﬁelds and ﬂuxes of ions and electrons [4–11]. It was shown

that a repulsive dipole-dipole interaction is realized in

complex plamsas at certain conditions [6,12–19]. Fur-

thermore, a system of polar molecules [20] and a dipole-

like excitonic phase state (created by bound electron-hole

excitons) can be described using a model of classical 2D

system of dipoles [21,22].

Aforementioned examples have motivated studies of

various properties of classical two-dimensional systems

using the repulsive dipole-dipole potential [23]. For ex-

ample, Khrapak et al [6] investigated thermodynamic and

dynamic properties of a classical 2D system of dipoles.

Earlier, the characteristic oscillation modes of particles

in the 2D dipole system were analyzed by Golden et

al [21,22]. In Refs. [21,22], it was demonstrated

that a dipolelike excitonic phase state created by bound

electron-hole excitons in semiconductors can be described

using model of a classical 2D system of repulsive dipoles.

These works on oscillation modes in 2D dipole systems

were continued by the study of the dumping of the trans-

verse excitations in the long wave length domain [24,25].

More recently, Aldakul et al [17] investigated melting,

freezing, and the liquid-crystal phase transition point of

classical 2D dipole systems. In this work, we extend these

studies of 2D dipole systems by modeling shear viscos-

ity and shear ﬂows in classical 2D systems with repulsive

dipole interaction across coupling regimes.

∗kodanova@physics.kz

In addition to a standard dipole-dipole interaction, in

this work we use screened dipole-dipole interaction. In

the latter case screening can be due to a polarizable

medium surrounding 2D dipole system [16,17,26]. For

example, regarding aforementioned a dipolelike excitonic

phase state, it was recently shown that screening due to

excess charges modiﬁes electron-hole excitons [27]. In

complex plasmas, the stream of ions creates a focused

ion cloud near a charged dust particle in downstream di-

rection due to attraction of ions by a negative charge

of a dust particle and the inelastic collision of ions with

atoms [28,29]. The focused ion cloud together with the

charged dust particle create an compound particle with

zero total charge and non-zero dipole moment [16]. Ad-

ditionally, hot electrons—with the electron Debye length

much larger than both the ion Debye length and the size

of the compound particle— provide screening of ion and

dust particle charges at long distance [16,30]. This leads

to the formation of the screened dipole-dipole interaction

between compound particles. The impact of screening on

the structural properties, oscillation modes, and thermo-

dynamic characteristics of 2D dipole systems has been

discussed in Ref. [17].

To compute the shear viscosity of 2D systems one

can use the reverse nonequilibrium molecular dynam-

ics method (NEMD)[31], [32]-[33]. This method was

used previously to investigate shear ﬂows in classical 2D

Yukawa systems [33]. It was shown that the NEMD

allows to determine shear viscosity in a good agree-

ment with experimental observation [34]. Moreover, the

NEMD allows one to study a non-Newtonian ﬂuid behav-

ior, i.e., when shear viscosity vary with the velocity gra-

dient. One of the peculiar properties of non-Newtonian

ﬂuids is decrease of the viscosity as shear is increased.

This eﬀect is referred to as shear thinning. For example,

following original studies on simple liquids by Evans et

al [35], such behavior has been reported in dusty plas-

mas [36]. Additionally, we compere results from the

NEMD simulations with the data for the shear viscos-

arXiv:2210.05123v1 [physics.plasm-ph] 11 Oct 2022

ity computed using the Green-Kubo relation connecting

the shear viscosity and the shear stress autocorrelation

function.

The paper is organized as the following: In Sec. II we

present the used pair interaction potentials. In Sec. III

we discuss the computation method and provide simula-

tions details. The results are presented in Sec. IV. The

paper is concluded by summarizing main ﬁndings.

II. BARE AND SCREENED DIPOLE-DIPOLE

INTERACTIONS

In this work, we present the results of the NEMD sim-

ulations of 2D systems with the bare dipole-dipole inter-

action potential:

βV (r) = ΓD

r3,(1)

and with the screened dipole-dipole interaction [17,26]:

βV (r) = ΓD

r3(1 + κr) exp(−κr),(2)

where ris in the units of the mean inter-particle distance,

β= 1/(kBT)is the inverse value of a thermal energy, κis

screening length, and ΓDis the parameter characterizing

coupling (correlation) strength [21,22].

The bare repulsive dipole-dipole pair interaction po-

tential (1) has been used to model two-dimensional

colloidal systems [1–3] and dipolelike excitonic phase

state of bound electron-hole excitons in semiconductors

[21,22]. The screened repulsive dipole-dipole pair inter-

action Eq. (2) provides description of dipole-dipole inter-

action in the presence of highly mobile polarizable back-

ground such as electrons in complex plasmas [15,16,26]

and electrolyte screening ﬁeld of charged colloids [37].

The coupling parameter corresponding to the melting

(crystallization) point in the 2D system with bare poten-

tial (1) is Γm'67±4[17]. The main eﬀect of screening is

to change the pair interaction from quasi-long-range po-

tential to short range potential. As the result, the liquid-

crystal phase transition point shifts, e.g., to Γm'86 ±6

at κ= 1 and to Γm'163 ±13 at κ= 2 [17]. Naturally,

we report the shear viscosity results for ΓD<Γm.

III. COMPUTATIONAL METHOD AND

SIMULATION DETAILS

A. The NEMD method for generating shear rate

Let us start with a brief description of the essence of

the NEMD method for the computation of shear viscos-

ity. The key is to use the deﬁnition of shear viscosity

in terms of a linear relationship between momentum ﬂux

and velocity gradient [38]:

jx(px) = −η∂vx

∂y ,(3)

FIG. 1. Screenshot from a NEMD simulation after a certain

amount of time after the selection of the vertical bar of parti-

cles (marked with blue), ΓD= 30, κ= 2. A horizontal shift

in the position of the particles can be observed due to the

presence of two oppositely directed ﬂows generated in slabs A

and B. The length is given in units of the mean-inter particle

distance (see Sec. III C).

where momentum ﬂux per unit length jx, momentum px,

and shear rate ∂vx/∂y are considered to be induced by

two oppositely directed streams along xaxis.

In order to calculate shear viscosity, point-like classi-

cal particles in a simulation box with side length of L

are simulated with periodic boundary conditions. In the

simulation box, we deﬁne two horizontal slabs at the lev-

els y=L/4and y= 3L/4(see Fig. 1). Let us desig-

nate these slabs as A and B. From these slabs, according

to the NEMD method, the particles with the maximum

and minimum values of vxare identiﬁed and simulta-

neously swapped with certain frequency (i.e. their mo-

menta are interchanged without changing their coordi-

nates). In other words, the algorithm ﬁrst selects the

fastest particle moving to the right in the slab A and the

fastest particles moving to the left in the slab B, and,

then, swaps the velocity values of these particles. As the

result, the mean velocity of the particles in the slab A is

directed in one direction and that of in the slab B in the

opposite direction. Thus, this exchange of particle veloc-

ities conserves energy and mimics two currents ﬂowing in

opposite directions. This is illustrated in Fig. 1, where a

snapshot from a NEMD simulation is shown.

To ﬁnd the shear viscosity from Eq. (3), ﬁrst of all,

the dependence of the xcomponent of the mean velocity,

vx, on the coordinate yis computed. Then, the value of

derivative dvx/dy in the space between two slabs is cal-

culated using the linear regression method to ﬁnd vx(y)

dependence.

Second, the momentum ﬂux is computed using the fol-

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ShearViscosityinTwo-DimensionalDipoleSystemsN.E.Djienbekov,1N.Kh.Bastykova,1A.M.Bekbussyn,1T.S.Ramazanov,1andS.K.Kodanova1,1InstituteforExperimentalandTheoreticalPhysics,Al-FarabiKazakhNationalUniversity,71Al-Farabiave.,050040Almaty,KazakhstanTheresultsofmodelingshearowsinclassicaltwo-dimensionald...

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Shear Viscosity in Two-Dimensional Dipole Systems N. E. Djienbekov1N. Kh. Bastykova1A. M. Bekbussyn1T. S. Ramazanov1and S. K. Kodanova1 1Institute for Experimental and Theoretical Physics

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