Longitudinal viscosity of 2D Yukawa liquids

6.7 MB·English

by  Yan Feng

Longitudinal viscosity of 2D Yukawa liquids
Yan Feng,∗ J. Goree, and Bin Liu
Department of Physics and Astronomy, The University of Iowa, Iowa City, Iowa 52242
(Dated: January 11, 2013)
The longitudinal viscosity η is obtained for a two-dimensional (2D) liquid using a Green-Kubo
l
method with a molecular dynamics simulation. The interparticle potential used has the Debye-
Hu¨ckelorYukawaform,whichmodelsa2Ddustyplasma. Thelongitudinalη andshearη viscosities
l s
arefoundtohavevaluesthatmatchveryclosely,withonlynegligibledifferencesfortheentirerange
oftemperaturesthatisconsidered. Thebulkviscosityη isdeterminedtobeeithernegligiblysmall
b
or not a meaningful transport coefficient, for a 2D Yukawa liquid.
3
PACSnumbers: 52.27.Lw,52.27.Gr,66.20.-d,83.85.Jn
1
0
2
I. INTRODUCTION thermal conduction and shear viscosity [5, 12, 13]. This

n method of measuring bulk viscosity is so difficult that
a
J The longitudinal viscosity ηl is a transport coefficient it has been used only for water and a handful of exotic
0 of interest for fluids [1]. It is the counterpart to the liquids [5], and even for these substances the results for
1 better-knowntransverseviscosity[2],whichismorecom- thebulkviscosityhavelargeuncertainties. Incontrastto
monly called the shear viscosity η . The latter charac- this difficult experimental situation, however, longitudi-
s
] terizes the momentum flux perpendicular to a velocity nalviscosityandbulkviscosityarementionedmoreoften
h
p gradient. These viscosities are theoretically predicted to inthetheoreticalliterature,whereithasbeencalculated
- be related by [3, 4] for example using the Green-Kubo relation [2, 14–28],
m
using the hydrodynamic limit [29], or derived using a
s d−1 Chapman-Enskog approach [30]. In this paper, we will
η =2 η +η , (1)
a l d s b makeuseoftheGreen-Kuboapproach,whichwepresent
l
p inSec.II.WewillusetheGreen-Kubomethodtoobtain
wheredisthedimensionalityofthesystem,andη isthe
. b
s bulk viscosity. The bulk viscosity η [5] is also called the ηl andηs,andforcomparisonwewilluseEq.(1)tostudy
c b η .
i volumeviscosityorexpansiveviscosity; itisaparameter b
s
y for liquids as well as molecular gases [6]. Both the shear Dusty plasma [31–35] is partially ionized gas contain-
h and bulk viscosities appear in the Navier-Stokes equa- ing micron-size solid particles, also called dust particles.
p tion [7] of a fluid. However, as compared with the shear Thesedustparticlesarehighlychargednegativelywithin
[
viscosity η , the longitudinal viscosity η and the bulk the plasma by absorbing more electrons than ions, since
s l

2 viscosity ηb are studied less often. negatively charged electrons have a higher temperature
v Physically, these kinds of viscosity characterize energy than positively charged ions. Due to the shielding pro-
4 dissipation in a fluid. Bulk viscosity is for energy dissi- vided by free electrons and ions in the plasma, the inter-
5
pation due to compression and rarefaction of a fluid, for action between dust particles in a plasma can be mod-
7
exampleinshockwavesandhigh-frequencysoundwaves. eledusingaYukawaorDebye-Hu¨ckelpotential[36],sim-
1
. Shear viscosity, on the other hand, is for energy dissipa- ilar to charged particles in a colloidal suspension [37].
1
tion due to a gradient in the flow velocity. In the latter Because of the high particle charge, dust particles in
0
3 case, the energy dissipation rate is proportional to both plasmas are strongly coupled (i.e., the potential energy
1 the shear viscosity and the square of the velocity gra- between neighboring particles is larger than its kinetic
: dient [8, 9]. In the case of a periodic density perturba- energy), so that the collection of dust particles exhibits
v
i tion, the energy dissipation rate is proportional to both propertiesofliquidsorsolids. Inlaboratoryexperiments,
X
the bulk viscosity and the square of the rate of density dust particles can be in two-dimensional (2D) or three-
r change [10, 11]. dimensional (3D) suspensions, depending on the experi-
a
Unlikeshearviscosity,longitudinalandbulkviscosities mental conditions. In 2D experiments, all dust particles
are in general difficult to measure experimentally [5, 12]. are confined in a horizontal plane, with negligible out-
This is because the hydrodynamic effects of bulk vis- of-plane motion due to strong confining potentials in the
cosity are significant only for rapid time variations, un- vertical direction. The dust particles are immersed in
like shear viscosity which affects flows in easily observed a rarefied gas, which applies a much weaker friction to
ways,evenundersteadyconditions. Ultrasoundattenua- moving dust particles, as compared with the case of a
tionhasbeendescribedastheonlyexperimentalmethod colloid. The size of dust particles allows imaging them
availableformeasuringbulkviscosityofafluid[5,12]. In directlyandtrackingtheirmotion,sothatvarioustrans-
this method, one can obtain the bulk viscosity after sub- port mechanisms can be studied experimentally at the
tracting the ultrasound attenuation contributions from particle level. Transport mechanisms that have been
2
studiedfordustyplasmasincludediffusion[38],shearvis- stress tensor, P (t), can be used to calculate the shear
xy
cosity [39], and thermal conduction [40]. For 2D dusty viscosity [41, 44]. Unlike P (t) which fluctuates around
xy
plasmas,viscosityisgenerallyattributedtodustparticle zero, however, P (t) fluctuates around a constant level
xx
scattering arising from interparticle interactions, while P (t).
xx
scattering due to the molecules of rarefied gas is negligi- Second, we calculate an autocorrelation function for
ble, as explained in [41]. Shear viscosity has been widely the fluctuation of P (t) using
xx
studiedfordustyplasmas,firstinsimulationsfor3Dsys-
tems [21, 42, 43], then later in experiments [39, 41] and Cl(t)=(cid:104)(Pxx(t)−Pxx(t))(Pxx(0)−Pxx(t))(cid:105). (3)
simulations [44, 45] for 2D systems, as well as 3D ex-
Here C (t) is the stress autocorrelation function. The
periments [46, 47]. The longitudinal viscosity and the l
brackets (cid:104)· · ·(cid:105) indicate an average over an equilibrium
bulk viscosity have been quantified for classical 3D one-
ensemble, which in practice we replace by an average
componentplasmas(OCP)[2,14,21,22]usingmolecular
over different initial conditions.
dynamics(MD)simulationsandgluonplasmas[48],how-
Third, weintegratethestressautocorrelationfunction
ever,ithasuntilnownotbeenquantifiedforclassical2D
over time to yield the longitudinal viscosity η [18, 51]
dusty plasmas, to the best of our knowledge. l
In this paper, we will report a determination of the 1 (cid:90) ∞
longitudinal viscosity for a 2D Yukawa liquid. We will ηl = Ak T Cl(t)dt, (4)
also report an unusual finding regarding the bulk viscos- B 0
ity: it is either negligibly small or it is not a meaningful where A is the area of the 2D system and T is its tem-
transport coefficient, for a 2D Yukawa liquid. perature. (For a 3D system, A would be replaced by
the system volume V.) These equations represent the
Green-Kubo relation for the longitudinal viscosity in 2D
II. GREEN-KUBO RELATIONS FOR η AND η
l b systems. To improve statistics, we calculate η twice,
l
using P as shown above and also using P , and we
xx yy
Green-Kubo relations are often used to calculate var- average the resulting values of η .
l
ious transport coefficients, such as diffusion [49], shear In addition to the longitudinal viscosity, we can also
viscosity [41, 44], and thermal conductivity [50]. Green- calculatetheshearviscosityη [41,44,50]for2Dsystems
s
Kubo relations are for equilibrium conditions; they use using
microscopic random motion of particles to determine
 
transportcoefficientswithoutanymacroscopicgradients. N N
The longitudinal and bulk viscosities can also be calcu- Pxy(t)=(cid:88)mvixviy− 21(cid:88)xirjyij ∂Φ∂(rrij), (5)
lated using the Green-Kubo relations [2, 14–28] using i=1 j(cid:54)=i ij ij
similar equations as the shear viscosity. The required
inputs for calculating longitudinal and bulk viscosities
C (t)=(cid:104)P (t)P (0)(cid:105), (6)
include time series of particles’ positions, velocities, and s xy xy
interparticle forces.
and
Now we review the three steps for calculating the lon-
gitudinal viscosity using the standard Green-Kubo re- 1 (cid:90) ∞
η = C (t)dt. (7)
lation [2, 15–20, 22–28]. These Green-Kubo relations, s Ak T s
B 0
which were originally developed for three dimensions
(d = 3), are adapted here for two dimensions (d = 2) The bulk viscosity [2, 16, 18, 21] for 2D systems can be
by setting the velocity and coordinate in the z direction calculated similarly, using
to be zero.
1
First,wecalculateadiagonalelementofthestressten- P(cid:103)(t)= (P (t)−P (t)+P (t)−P (t)), (8)
2 xx xx yy yy
sor P (t), which is defined as
xx
 
N N
Pxx(t)=(cid:88)mvixvix− 21(cid:88)xirjxij ∂Φ∂(rrij). (2) Cb(t)=(cid:104)P(cid:103)(t)P(cid:103)(0)(cid:105), (9)
ij ij
i=1 j(cid:54)=i and
Here i and j indicate different particles which all have 1 (cid:90) ∞
the same mass m, N is the total number of particles, ηb = Ak T Cb(t)dt. (10)
B 0
r = (x ,y ) is the position of particle i, x = x −x ,
i i i ij i j
y =y −y ,r =|r −r |,andΦ(r )istheinterparticle We will use the same simulation data as the inputs in
ij i j ij i j ij
potentialenergy. Thepositionsandvelocitiesofparticles calculations of η and η .
l s
in Eq. (2) vary with time, and this accounts for the time Ithasbeenquestionedtheoreticallywhethertransport
dependence of P (t). The off-diagonal element of the coefficients are meaningful for 2D liquids. This question
xx
3
has been studied theoretically, starting with a 2D hard
disk system [52] and then liquids with other interparti-
cle potentials [50]. A transport coefficient is deemed to
be not meaningful if the corresponding autocorrelation
function has a long-time tail that decays as slowly as
1/t, so that the Green-Kubo integral does not converge.
For a 2D Yukawa liquid, the validity of transport coeffi-
cients has been discussed in detail in [41, 44, 50, 53, 54].
In Sec. IV, we will present our autocorrelation functions
and discuss whether they have a long-time tail.
Equations (2-10) are presented in physical units, al-
though we will perform simulations using dimensionless
units. Some of the parameters we will use when making FIG. 1: The temperature fluctuation during about one half
quantities dimensionless include the area A of the sim- ofthe106 stepsusedfordataanalysisforΓ=20andκ=0.5.
ulated system, the areal number density n, the particle Here, Γ−1 is a dimensionless temperature.
mass m, the Wigner-Seitz radius a ≡ (nπ)−1/2, a char-
acteristicplasmafrequency[55]ω =(Q2/2π(cid:15) ma3)1/2,
pd 0
and the particle kinetic temperature T. Here, Q is the We use a thermostat only for the initial equilibrium
particle charge. of our simulation, and not for the data used to calculate
η and η . For each simulation run, we first integrate
l s
105 steps using a Nos´e-Hoover thermostat to approach
III. SIMULATION METHOD equilibrium at a desired temperature [44] under steady
conditions. Wethenturnoffthisthermostattointegrate
another 106 steps. Only the data in the latter 106 steps
To model 2D dusty plasmas, we perform equilibrium
will be used to calculate the viscosities. We use a suffi-
MD simulations using a binary interparticle interaction
cientlysmalltimestepsothattheenergyconservationis
with a Yukawa potential [36]. We integrate the equation
(cid:80) adequatelyobeyedduringthesimulationrun,aswehave
of motion m¨r = −∇ φ for all particles. This equa-
i ij
verified for our simulation data. We measure T, which
tion of motion does not include any friction term or any
can differ slightly from the desired temperature, using
Langevinheatingterm. Particlesareconstrainedtomove
the mean square velocity fluctuation.
only within a single 2D plane. Our simulation includes
N = 1024 particles in a rectangular box with periodic Figure1showsthetimeseriesofthemeasuredtemper-
boundary conditions to model an infinite system. The ature from one of our simulation runs. The temperature
Yukawa potential is φ = Q2exp(−r /λ )/(4π(cid:15) r ), fluctuates about a steady level during the 106 steps sim-
ij ij D 0 ij
where λ is the screening length. We truncate the ulation interval. The temperature fluctuations are due
D
Yukawa potential at distances beyond a cutoff radius of to the finite simulation size. The absence of a general
24.76 a; this truncation has been justified in [44]. This upwardordownwardtrendinthetemperatureasafunc-
simulated system is essentially the same as a Yukawa tion of time is due to our choice of an adequately small
OCP, except that we constrain the particle to move only integration time step. When we report a value for Γ, we
on a single plane at z =0. use a temperature that was averaged over the 106 steps,
for a given run.
Yukawasystemscanbedescribedbytwodimensionless
parameters: the coupling parameter Γ and the screening
parameter κ. They are defined as Γ = Q2/(4π(cid:15) ak T)
0 B
andκ≡a/λ . OnecanthinkofΓasaninversetemper- IV. RESULTS
D
ature and κ as an inverse indicator of density.
The input parameters in our simulation include κ and A. Longitudinal and shear viscosities
Γ. We choose a single value of κ = 0.5, which is typical
for2Ddustyplasmaexperiments[41]. Whenκ=0.5,the Usingtheparticles’positions,velocities,andpotentials
melting point of 2D Yukawa system is Γ ≈ 142 [56]. To from the simulation, we use Eqs. (2) and (5) to calculate
study2DYukawaliquidsoveralargetemperaturerange, thetimeseriesofthestresstensorelements. Examplesof
we choose twelve different values of Γ varying from 140 the results for the stress tensor are shown in Fig. 2. All
(corresponding to a temperature near the melting point) our calculations of η and η will be based on these time
l s
downto2(correspondingtoamuchhighertemperature). series. We note that P (t) fluctuates about a non-zero
xx
The integration time step is in a range of 0.0037 and value, while P (t) fluctuates about zero. The source of
xy
0.037 ω−1, depending on the choice of Γ, as in [44]. For thisfluctuationisthemicroscopiccompressionandshear
pd
eachvalueofΓ,weperformfourrunswithdifferentinitial motion of particles. We find that fluctuations of P (t)
xx
configurations of particles. and P (t) have comparable amplitudes and time scales.
xy
4
FIG. 2: (Color online). The fluctuations of the stress ten-
sor elements during about 5% of the 106 steps used for data
analysis for Γ=20 and κ=0.5.
We calculate the autocorrelation function of the time
series of P (t) using Eq. (3), and the result is shown
xx
in Fig. 3. Also shown is the autocorrelation function
of P (t). In panels (a-b), we see the initial decay fol-
xy
lowed by a small variation around zero. We will use the
area under these curves to calculate the viscosities, us-
ing Eq. (4) for η and Eq. (7) for η . The integral in
l s
Eq. (4) has a finite upper time limit of infinity, which
we replace with the first zero crossing [41], marked as t
I
in Fig. 3. In panel (c) we show the positive portion of
these autocorrelation functions on a logarithmic scale so
thatthetwocurvescanbedistinguished. Thetwocurves FIG. 3: (Color online). The autocorrelation functions of
are almost identical in the initial decay, which leads us thestresstensorelements,whereCl forlongitudinalviscosity
and C for shear viscosity, for Γ = 20 and κ = 0.5. The
to expect that the longitudinal and shear viscosities will s
same autocorrelation functions are shown with linear axes in
have almost the same values.
(a) and (b), and a log axis in (c). Data are normalized by
Indeed, we find that the longitudinal viscosity ηl and parametersdefinedinSec.IIandIII.Thefirstzerocrossings
the shear viscosity η have almost the same values, for in(a)and(b)aremarkedast ,whichwillbeusedtoreplace
s I
the full range of Γ that we investigate. This is seen in the upper limits in the integrals, Eqs. (4) and (7).
Fig.4(a), wherewepresentη andη determinedbyper-
l s
forming the integrals of the stress autocorrelation func-
tion in Eqs. (4) and (7). We find negligible differences
which can occur due to the absence of a thermostat, as
between them, as shown in Fig. 4(b).
discussed in Sec. III. For the vertical axis, the scatter
We also see a minimum in ηl as a function of Γ. This corresponds to the random run-to-run variation for the
minimum matches the minimum in ηs, which was previ- obtained viscosity values, i.e., random errors. In addi-
ously studied and explained as being due to a balance of tion to these random errors, there is a systematic error
kinetic and potential terms in the shear stress [44, 45]. associatedwiththechoiceoftheupperintegrallimit. By
In Fig. 4, the data have scatter in both axes. For the examining the fluctuation of the Green-Kubo integral at
horizontal axis, the scatter around each value of Γ in- longtimes[57], wedeterminedthatthissystematicerror
dicates slight differences in the measured temperature, is smaller than the random errors.
5
is negligible as compared with the shear viscosity, about
twoordersofmagnitudesmallerorevenmore. Fromthis
aspect, it seems that our results for 2D Yukawa system
that η (cid:28) η are consistent with those previous simula-
b s
tions in 3D OCP systems.
Wenowexaminetheautocorrelationfunctionsusedin
calculating η , η , and η in Eqs. (4, 7 and 10) to de-
l s b
termine whether they exhibit a long-time tail. As dis-
cussed in Sec. II, if the correlation function decays more
slowly than 1/t, this long-time tail prevents the conver-
genceoftheGreen-Kubointegralsothatthecorrespond-
ing transport coefficient is deemed to be not meaningful.
In Fig. 5(a-b) we present the correlation functions C for
l
longitudinal viscosity and C for shear viscosity, and we
s
find that they do not exhibit a noticeable long-time tail
before the function becomes noisy. However, in Fig. 5(c)
thecorrelationfunctionC decaysmoreslowly,ascanbe
b
seen by comparing to the line drawn with a slope cor-
responding to a 1/t scaling. This result suggests that,
within the uncertainties that are inherent in a finite-size
simulation [50], η and η are meaningful, but η is not.
l s b
It is interesting that the signal-to-noise ratio for C , the
b
correlation function of the bulk viscosity, is still compa-
rable to that of C and C , even though its amplitude is
s l
one or two orders of magnitude smaller. Even if η were
b
meaningful, it would have a small value because C in
b
Fig. 5(c) is two orders of magnitude smaller than C and
l
FIG. 4: (Color online). (a) The longitudinal and shear vis- C .
s
cositiesfromevaluatingEq.(4)andEq.(7),respectively. (b) We cannot explain in terms of the macroscopic fluid
The difference between η and η is negligible, for all values
l s equationswhythebulkviscosityiseithernegligiblysmall
ofΓ. UsingEq.(11),thisresultindicatesthatη isnegligibly
b or not meaningful for this 2D liquid. However, in terms
small. Theminimuminη in(a)isknowntoarisefromabal-
s of microscopic motion, we can discuss some of the terms
anceofkineticandpotentialtermsintheshearstress[44,45],
and here we find that a similar minimum appears in η. of the correlation functions. The correlation function
l
for the longitudinal viscosity C involves only products
l
of P with a delayed version of itself, and likewise for
xx
B. Bulk viscosity P . Thebulkviscosityhasadifferentcharacterbecause
yy
Eq. (9) also includes cross terms like (cid:104)P(cid:103)(t)P(cid:103)(0)(cid:105).
xx yy
Thenegligibledifferencebetweenηlandηsthatwefind In fact, the correlation function for the bulk viscosity,
in Fig. 4 leads us to determine that the bulk viscosity ηb Eq. (9), can be written as the sum of two terms: Cl/2
is much smaller than either ηl or ηs. This conclusion is and a cross correlation involving Pxx and Pyy. For our
drawn from Eq. (1) for our 2D system, which is 2D Yukawa liquid these two terms almost cancel.
η =η −η . (11)
b l s
Previous simulations using the Green-Kubo approach V. SUMMARY
toobtainthebulkviscosityweremostlyfor3DLennard-
Jones interparticle potentials and soft-sphere interparti- Molecular-dynamicssimulationsofa2DYukawaliquid
cle potentials [15–18, 20, 23–28], or similar interparticle demonstratethatthelongitudinalviscosityisalmostthe
potentials [19]. In some of those simulations, the shear same as the shear viscosity, over a wide range of temper-
viscosity was also calculated [15–17, 23, 24, 28], and it ature. TheseresultswereobtainedusingGreen-Kuboin-
was found that the bulk viscosity differs from the shear tegralsoftheappropriateautocorrelationfunctions. The
viscosity, with the difference within one order of magni- very close match of the values for η and η would pre-
l s
tude. dict, using Eq. (1), that η is negligibly small or even
b
Simulations of 3D plasmas [2, 14, 21, 22] provided zero. Indeed, η might not even be a meaningful trans-
b
results for the bulk viscosity for both one-component port coefficient for the system studied here because we
plasmas (OCP) and Yukawa OCP. In these simula- found that its autocorrelation function exhibits a long-
tions [2, 14, 21, 22], it was found that the bulk viscosity timetail,sothatthecorrespondingGreen-Kubointegral
6
FIG. 5: (Color online). Correlation functions for (a) η, (b) η , and (c) η , for Γ = 140 and κ = 0.5. Here, data are shown
l s b
withlog-logaxestoallowanidentificationofanypossiblelong-timetail. WefindthatonlythecorrelationfunctionC forthe
b
bulk viscosity in (c) has a significant long-time tail, as seen by a decay that is slower than 1/t. These results, for Γ=140, are
representative of the other values of Γ studied as well.
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