Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 15.2.3 - Molecular Dynamics Study of Vapor-liquid Equilibrium Condition for Nanodroplet
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 Material Information
Title: 15.2.3 - Molecular Dynamics Study of Vapor-liquid Equilibrium Condition for Nanodroplet Micro and Nano-Scale Multiphase Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Yaguchi, H.
Yano, T.
Watanabe, M.
Fujikawa, S.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: molecular dynamics
thermodynamics
equilibrium condition
argon
vapor
droplet
chemical potential
 Notes
Abstract: Molecular dynamics (MD) simulations of equilibrium system of single argon nanodroplet and its surrounding argon vapor are carried out to address a fundamental issue whether the thermodynamic description is applicable to the nanoscale inhomogeneous system. The profiles of both density and pressure in the transition layer are computed. Numerical results show that the chemical potentials of liquid and vapor phases are not equal when a droplet is so small that the number of molecules consisting the transition layer may be comparable to that in the droplet. The phase equilibrium condition in thermodynamics fails in such a case, and therefore, all thermodynamics relations based on this condition, such as Kelvin equation connecting vapor pressure to droplet radius, do not hold.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Molecular Dynamics Study of Vapor--liquid Equilibrium Condition for Nanodroplet


H. Yaguchi *9


T. Yano F, M. Watanabe and S. Fujikawa *


Division of Mechanical and Space Engineering, Hokkaido University, Sapporo 060-8628, Japan

t Department of Mechanical Engineering, Osaka University, Suita 565-0871, Japan
Sh-yaguchi~,mech-me.eng. hokudai.ac~jp
Keywords: molecular dynamics, thermodynamics, equilibrium condition, argon, vapor, droplet, chemical potential




Abstract

Molecular dynamics (MD) simulations of equilibrium system of single argon nanodroplet and its surrounding argon
vapor are carried out to address a fundamental issue whether the thermodynamic description is applicable to the
nanoscale inhomogeneous system. The profiles of both density and pressure in the transition layer are computed.
Numerical results show that the chemical potentials of liquid and vapor phases are not equal when a droplet is so
small that the number of molecules consisting the transition layer may be comparable to that in the droplet. The phase
equilibrium condition in thermodynamics fails in such a case, and therefore, all thermodynamics relations based on
this condition, such as Kelvin equation connecting vapor pressure to droplet radius, do not hold.


Introduction


/Inter~face



Liquid film



Interface


Droplet


A single-component vapor-liquid two-phase system
consisting of a droplet and its surrounding vapor in an
equilibrium state is a fundamental system in various
fields of science and technology, such as ultrafine par-
ticle formation (Adachi et al. 2003), cloud formation
(Chuang 2006), uptake of chemical species into aerosols
(Morita & Garrett 2008), and semiconductor cleaning
(Watanabe et al. 2009). According to thermodynamics,
the equilibrium condition of the vapor-liquid system is
described as


Figure 1: (a) Nanodroplet of Rs=3.91 nm and (b) pla-
nar liquid film in vapor-liquid equilibrium state at 85 K.


radius Rs such that Eq. (1) holds.


Method of Molecular Dynamics Simulations

Figures 1 shows typical snapshots of the vapor-liquid
equilibrium systems at 85 K for (a) a nanodroplet and
(b) a planar liquid film. The periodic boundary condi-
tions are imposed for both cases in all three directions of
the simulation cells. The simulations of all nanodroplets
are performed in cubic simulation cells with dimensions
L x Lx L:; L for each simulation is given in Table 1.


where <, and <, are chemical potentials of bulk vapor
and liquid, respectively, and the temperature should be
uniform in the whole system. Equation (1) is of fun-
damental importance in thermodynamics of phase equi-
librium; in fact, all thermodynamics relations in the
phase equilibrium, such as Clausius-Clapeyron relation,
Kelvin equation, and so on, can be derived from Eq. (1).
However, thermodynamics is a macroscopic or contin-
uum theory for equilibrium systems. The continuum
theory implicitly assumes that the system concerned
comprises sufficiently large number of molecules. Our
aim of this study is to investigate the validity of thermo-
dynamics for an argon nanodroplet in equilibrium with
argon vapor by molecular dynamics (MD) simulations.
More precisely, we obtain the lower bound of droplet
















No. T IN L Rs e pe p, Pe p, T
p,RT
(K) (nm) (kg/m3) (MPa) (N/m)
1 84.8 8000 21.0 3.91 20.6 1412 6.21 6.6 0.106 0.0127 0.96
2 84.9 4000 16.5 2.99 18.9 1416 6.77 8.4 0.115 0.0124 0.96
3 85.0 2000 13.0 2.27 17.0 1420 7.55 10.6 0.126 0.0119 0.95
4 85.1 1000 10.5 1.70 14.9 1427 8.58 12.8 0.144 0.0107 0.95
5 85.0 730 10.0 1.43 13.9 1429 9.19 14.8 0.154 0.0104 0.95
6 85.3 515 9.5 1.16 12.0 1428 10.72 16.2 0.177 0.0093 0.93
7 85.0 450 11.0 0.84 10.3 1430 12.46 17.7 0.205 0.0073 0.93
8 84.9 515 12.0 0.74 9.5 1421 13.42 18.8 0.219 0.0069 0.92
9 84.9 500 12.0 0.63 8.8 1414 14.55 16.7 0.234 0.0052 0.91
10 84.0 515 12.0 0.51 8.0 1407 16.07 15.9 0.254 0.0040 0.90
11 85.4 700 13.0 0.41 6.8 1371 18.83 12.3 0.298 0.0025 0.90
12 84.6 515 11.5 0.27 6.6 1379 19.33 11.9 0.303 0.0016 0.89


T Zs e pe p p, TopR
(K) (nm) (kg/m3) (MPa) (N/m)
84.9 3.44 26.9 1394 (1410) 4.76 (4.59) 0.083 (0.079) 0.0136 (0.0131) 0.98


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010



Table 1: The parameters and numerical results of vapor-liquid equilibrium states of an argon nanodroplet and its
vapor.


Table 2: The numerical results of vapor-liquid equilibrium state of an argon planar liquid film and its vapor. The
values in parentheses are experimental ones at 85 K (Tegeleret al. 1999; Miller et al. 1976).


The simulation of planar liquid film is in rectangular cell
with L, x L, x L,; L, L,=5 nm and L,=30 nm. In
preparatory computations, we have confirmed that sta-
tistical properties such as densities, mass fluxes, etc. are
not affected by the size of the simulation cells, the liner
dimension of which is several multiples of droplet ra-
dius.
The number of all molecules N, the total energy E in
the system, and the volume V of simulation cell are kept
constant during the present MD simulation (NVE sim-
ulation). The following 12-6 Lennard-Jones potential is
used as intermolecular potential;


( 2

where rij is a distance between the centers of a molecule
i and a molecule j, E and a are potential parameters. As
is well known, different values of e and a give prop-
erties of different monoatomic materials. In this study,
we choose argon to obtain physical perspectives from
the results. They are, respectively, given aS E/k=119.8 K
(k is Boltzmann constant) and a=0.3405 nm for argon
(Michels et al. 1949).


Equations of molecular motion are integrated by the
leapfrog method with the time step of 5 fs. First,
preparatory computations are performed with the tem-
perature control by using the velocity scaling method
for more than 100 ns to realize vapor-liquid equilibrium
at 85 K, and then, without the temperature control for
200 ns to eliminate influence of velocity scaling. The
criterion for the realization of the equilibrium states is
that the temperature and droplet size become steady and
the velocity distribution function becomes Maxwellian
everywhere in the quiescent system. After the care-
ful confirmation of the equilibrium state, these config-
urations in the system are adapted as the initial states:
thereafter the main computations are performed during
201.6 ns without the temperature control.
The instantaneous shapes of smaller nanodroplets are
far from a spheres, because the shapes are fluctuating
with the time. Nevertheless, it is confirmed that time-
averaged spatial-distributions of macroscopic quantities
such as density, mass fluxes, velocity, and etc., are spher-
ically symmetric. These quantities are averaged with re-
spect to I and as well as the time t and are expressed as
functions of the radial distance r alone in the spherical
coordinate system (r, 8, ~) in which the origin coincides











Isnn. . . . . . . .


15000



500 o-

( a) oo


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


loC o




( a) o


_L1


S100


,x50


r (nm)


r (nm)


Figure 2: Profiles of (a) density and (b) pressure in the
nanodroplet-vapor system at 85 K: Rs = 2.99 nm.




with the gravity center of nanodroplet.
In the simulation of the argon planar liquid film and
its vapor system, the computational and data processing
methods are almost the same as those in the preceding
study by Ishiyama et al. (21li14. The total number
of molecules is N=4000 and the temperature is set at
85 K. The preparatory computations to realize the equi-
librium state are performed as well as in the simula-
tions of nanodroplets. Cartesian coordinates (x, y, z),
the origin of which coincides with the gravity center
of the liquid film, are adopted. The physical quanti-
ties are averaged with respect to x: and y as well as the
time t. There are two interfaces in the cell as shown in
Fig. 1(b); hence physical quantity profiles from liquid to
vapor are expressed as functions of the absolute value of
z-coordinate.
In this study, we calculate macroscopic physical quan-
tities, such as the pressure and surface tension, from
the molecular motion on the basis of mechanical argu-
ment. The detailed information of definition and cal-
culation method for these macroscopic physical quanti-
ties is available on the textbooks (Rowlinson & Widom
1982; Allen & Tildesley 1987) and our previous paper
(Yaguchi et al. 2010).


Figure 3: Profiles of (a) density and (b) pressure in the
nanodroplet-vapor system at 85 K: Rs = 1.43 nm.



Results and Discussion

Figures 2(a) and 3(a) are the density profiles in the r-
direction for nanodroplets, and Fig. 4(a) is the den-
sity profiles as functions of the absolute value of z-
coordinate for planar liquid film. The high density re-
gions on the left-hand side of profiles are the bulk liquid
phase whose density is pe, whilst the low density regions
are the bulk vapor phase whose density is p,. The region
with abrupt changes in density and pressure is the tran-
sition layer.
Figures 2(b) and 3(b) show the profiles of the pres-
sures pr, po, and pe, which are acting on the surfaces
normal to r, 8, and directions for nanodroplets. The
pressure ps and pp are identical to the pressure pr in
the bulk vapor and liquid phase, sufficiently outside the
transition layer. Thus, we define the pressure in the bulk
liquid as the liquid pressure pe and the pressure in the
bulk vapor as the vapor pressure p,, respectively. The
liquid pressure pe is defined as the pressure at r = 0, cal-
culated by using the cubic spline interpolation with zero-
gradient at r = 0, shown as solid curves in Fig. 2(b) and
3(b). As shown in Fig. 2(b) and 3(b), the liquid pressure
pe is larger than the vapor pressure p, due to the surface
tension ys.











141111. . . . . . . .


-1 A o p,



-20 -
(b)

| z| (nm)

Figure 4: Profiles of (a) density and (b) pressure in the
liquid film-vapor system at 85 K.



We adopt the radius of surface of tension as the
droplet radius Rs. The surface of tension is defined as
the surface on which the surface tension acts. The ra-
dius Rs is determined simultaneously with the surface
tension ys from the pressure profiles in the transition
layer. The detailed calculation method and results of ys
and Rs are discussed in our previous paper (Yaguchi et
al. 2010).
Figure 4(b) shows the pressures p, r and pe,which
are acting on the surfaces normal to .r, y, and z direc-
tions for planar liquid film. The pressure p, and r are
identical to the pressure p, in the bulk vapor and liquid
phase. The saturated vapor pressure p, is defined as the
pressure in the bulk vapor phase. Except for the transi-
tion layer, the pressure p, is distributed with uniformity
in bulk vapor and liquid because contribution of surface
tension y, vanish in the case of planar surface. The sur-
face of tension Zs is also determined with y, for liquid
film as well as nanodroplet.
Table 1 shows these numerical results of vapor-liquid
equilibrium states of an argon nanodroplet and its va-
por. The qualitative tendency agrees with the results of
preceding studies (Thompson et al. 1984: Nijmeijer et
al. 1992: Vrabec et al 2006). For reference, the mean
free path I = m/( Jyro pe,) of vapor molecules and the
compressibility factor, p /(pe,RT), are shown, where at
is mass of one molecule and R is gas constant. Table 2
also shows result of planar liquid film. The values of


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


MD in Table 2 are in good agreement with experimental
ones shown in parentheses (Tegeleret al. 1999; Miller
et al. 1976).
Now, we discuss the chemical potential in bulk vapor
and liquid phase to verify the thermodynamics equilib-
rium condition, Eq. (1). Under a constant temperature,
i.e., clT 0, the differential of chemical potential is


10. o


(>o 9


S100


.r 50
t


where script / is v for vapor phase or I for liquid phase.
Integration of Eq. (3) with respect to Rs from oc to Rs
results in
1 (R,)


dliPi


where p' is the chemical potential both in vapor and
liquid for planar surface. Making use of Eq. (4), we can
evaluate an excess chemical potential of vapor p p'
and that of liquid p, p' at a given temperature T from
results ofMD simulations.
Figure 5(a) shows the relation between the specific
volume 1/pe, and pe, in the vapor phase, where the num-
bers shown near the symbols correspond to these of the
data in Table 1. The solid curve in Fig. 5(a) is fitting
function, 1/pe, = /pe, + B, where 24 is a fitting pa-
rameter and B is determined so that the curve passes
through the point of planar surface at pe,=0.083 MPa.
The integration of the right-hand side of Eq. (4) is cal-
culated by using this fitting function to obtain pe, p'
Figure 5(b) is 1/p, versus pe in the liquid phase. The
specific volume 1/p, is found to hardly depend on the
pressure pe: hence we introduce approximate function,
1/p, 1/, :~ where -, is the specific volume in the
case of planar surface at pe=0.083 MPa. Substituting this
approximate function into the right-hand side of Eq. (4),
we can evaluate p, p'.
Figure 6 shows the comparison betiveen pe, p' and
in. l' The vapor chemical potential pe, at 85 K mono-
tonically increases with the decrease in nanodroplet ra-
dius Rs, while the liquid chemical potential p, has a
maximum around radius of about 0.8 nm. The two
chemical potentials agree for droplets with radii larger
than 1.5 nm, and they disagree for nanodroplets with
radii smaller than 1.5 nm. That is, at 85 K, the phase
equilibrium condition in thermodynamics, Eq. (1), fails
for Rs less than 1.5 nm, and therefore, all thermody-
namics relations based on Eq. (1), such as Kelvin equa-
tion, do not hold. The number of molecules consisting
the transition laver may be comparable to that in such
a small nanodroplet. We here emphasize that, even for
nanodroplets with radius smaller than 1.5 nm, the equi-
librium state in the sense that the velocity distribution




























0 5 10 15
Liquid pressure pe (MPa)


i( u . .


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


0.0012

0.0010

0.0008

0.0006

0.0004

0.0002


-Planar surface (Liquid film)



-1 2 3 45 7 8




(b)


Vapor pressure p, (MPa)


Figure 5: Relations between specific volume and pressure in (a) vapor phase and (b) liquid phase.


function of molecules in small volume is Maxwellian,
is achieved everywhere in the system, and the mechani-
cal equilibrium, as Laplace equation, also holds.


85 K g yv po
-hO~e~



-P

-B


25000

20000

15000

10000

5000


Conclusions


Molecular dynamics (MD) simulations of equilibrium
system of single argon nanodroplets and its surround-
ing argon vapor have been performed to investigate
the validity of the equilibrium condition in thermody-
namics, Eq. (1), in the nanoscale inhomogeneous sys-
tem. The chemical potentials of bulk vapor and liquid
phase disagree for nanodroplets with radius smaller than
1.5 nm, where the number of molecules consisting the
transition layer may be comparable to that in the nan-
odroplet. Therefore, all thermodynamics relations based
on Eq. (1), such as the Kelvin equation, do not hold.


Acknowledgements

This work is supported by Japan Society for the Pro-
motion of Science, Grant-in-Aid for Scientific Research
(A)(No. 21246031) and (B)(No. 19360077).


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ical Res., Vol. 111, DO9201, 2006.


0 1 2 3
Droplet radius Rs (nm)


Figure 6: Verification of chemical potential equilibrium
between vapor and liquid.




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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


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