Surface and Bulk Fluctuations Surface and Bulk Fluctuations

of the Lennard-Jones Clusrersof the Lennard-Jones Clusrers

D. I. Zhukhovitskii

Joint Institute for High Temperatures

Russian Academy of Sciences

Moscow Russia

LiquidLiquid――vapor boundaryvapor boundary::

gradual transitiongradual transition oror layered structure layered structure??

Gas

Liquid

Gas

Liquid

Intermediate phase

Investigation methodsInvestigation methods::

1. 1. analytical;analytical;

2. 2. experimentalexperimental ( (x-ray and neutron reflectivityx-ray and neutron reflectivity measurementsmeasurements));;

3. 3. numerical simulation numerical simulation ((molecular dynamicsmolecular dynamics and Monte Carloand Monte Carlo))..

Shortcomings of the capillary wave model:

1. What is a microscopic wave surface? How should the wave overhangs be treated ?

2. The problem of capillary and bulk fluctuations separation.

3. What kind of the surface tension should be associated with the microscopic capillary fluctuations?

Cluster particles are divided in three groups: we call particle 1 with the radius vector internal if there exists at least one particle 2 with radius vector belonging to the same cluster that forms more than four bonds such that the conditions

are satisfied. Here, is the number density of bulk liquid. The cluster particles that are not internal and form more than four bonds are called surface particles. The cluster particles that are not internal and form less than five bonds are assigned to the virtual chains.

22 2 1 2

1 2 1 2 2 2 / 31

( ) 1,

3r r

r n

r rr r

1r2r

n

Proposal #1: separate the pivot particles for the wave (fluctuation) surface and virtual chains of particles loosely bound to the cluster. Thus, the maximum surface curvature is limited.

The slice snapshots of

clusters formed by 1592

(1) and 2320 (2) particles.

Solid circles are internal

particles, hatched circles

are surface particles, and

open circles are virtual

chains. The surface

particles form a

monolayer highly curved

by thermal fluctuations.

For the internal particles,

the number of bonds is

no less than 10. The

particles with less than

five bonds form virtual

chains (overhangs).

0 4 8 12 16

b

0

100

200

300

400

F(b

)

1

2

3

0 4 8 12 16

b

0

100

200

300

400

F(b

)

1

2

3

0 4 8 12 16

b

0

100

200

300

400

F(b

)

1

2

3

0 4 8 12 16

b

0

100

200

300

400

F(b

)

1

2

3

0 4 8 12 16

b

0

100

200

300

400

F(b

)

1

2

3

0 4 8 12 16

b

0

100

200

300

400

F(b

)

1

2

3

0 4 8 12 16

b

0

100

200

300

400

F(b

)

1

2

3

0 4 8 12 16

b

0

100

200

300

400

F(b

)

1

2

3

Isolate the surface particles of a slice passing through the cluster center of mass. The polar coordinates of particles are the values of a continuous function expanded in the Fourier series:

max max0

1 1

( ) cos sin .2

k k

k kk k

P k k

The slice spectral amplitudes were calculated by averaging over both the cluster configurations and Euler angles corresponding to the rotation of each configuration:

1 2

2 2

, .cs k k

kcs

gS

g

0.0 0.5 1.0 1.5k/gcs

0.0

0.2

0.4

0.6

k Sk

2

43

1

_

Slice spectral amplitudes for the clusters formed by 150 (1), 1000 (2), 3000 (3), and 24450 (4) particles at the temperature 0.75 of the interparticle potential well depth (MD simulation). Dots represent theoretical calculation for a 24450 particles cluster.

Proposal #2: The bulk fluctuations are generated primarily by the discontinuity in the spatial distribution of cluster internal particles. Therefore, such fluctuations can be simulated by the surface particles of the cluster truncated by a sphere smaller than the cluster. The total spectral density is , where

k k kS Q R max 2

1/ 3

1

.12

k

kk

R g

0 20 40 60 800.0

0.2

0.4

0.6

0.8

k

kk

1

2

3 4

Different components of the fluctuation spectral amplitude for the surface of a cluster comprising 30000 particles at the temperature 0.75 of the interparticle potential well depth. (1), bulk, k = Rk and (2), capillary fluctuations, k = Qk ; (3), total spectral amplitude, k = Sk ; (4), total spectral amplitude including the virtual chains.

Capillary fluctuations theory

20 max3 /16k

In the early study by Buff, Lovett, and Stillinger (1965), the relation

Attempts to allow for the wave number dependence of the surface tension for the fluctuations result in discrepant dependences, e.g., to the increasing one (Helfrich, 1973)

20( ) ,q q

to the decreasing dependence (Mecke, 1999), and to more complicated dependences.

was obtained, where is the coupling constant.0

( )q

where is the interface bending rigidity,

где

According to the fluctuation theory the change of Gibbs free energy of the cluster surface is

2 2220 0

,

( , ) sin ( 1)( 2),2 2 lm

l m

R Rd d a l l

,

( , ) ( , ), .lm lml m

R a Y l m l

where

Using the equipartition theorem we derive

2

20

.( 1)( 2)lm

Ta

R l l

The condition of finiteness for the cluster excess surface

2

20 0

(2 1) ( )2 2l

T TA l l

где

yields the maximum value of l

1/ 2

2

8( )

(2 )R

T

and the relation between the coupling constant and surface tension :

Here,

2

0

1 .2

2( , ) 0.548

is a universal constant.

где

The capillary fluctuation in the form of spherical harmonic Ylm

(cos )cos ,( 1)( 2)

mi lm l i i

Tr R A P m

l l

( , ) ( , )

1 2

1 2

2 21

,0 ,

( ) .l m l m

l

k cs cs k km

S l g g

where is the amplitude corresponding to ,

contributes to the 2D spectrum obtained in MD simulation as

This contribution is defined by the Fourier coefficients of slice boundary coordinates

( , )max max( , ) ( , )( , ) 0

1 1

( ) cos sin .2

l m

l m l mk k

l mk k

k k

P k k

Then the total spectral amplitude of capillary fluctuations is

2 ( )

2

1 ( ).2k k

l

Q S l

lmA2 2/ ( 1)( 2)lma T R l l

Spectral amplitudes of the capillary (1, 2) and bulk (3) fluctuations for the cluster comprising 30000 particles at the temperature 0.75 of the interparticle potential well depth. Theory (1) and MD simulation (2, 3).

0 20 40 60 800.0

0.2

0.4

0.6

3

Sp

ect

ral a

mp

litu

de

s

2

T = 0.75

1

k

где

The effective surface tension is defined as21

2

eff 2 2

(2 1)( ) ,

( 1)( 2)

l lm

lmm l lm

aT ll a

R l l a

where is independent of m . In the “classical” theory, . The total spectral amplitude of capillary fluctuations is

221

eff 02

( ) ( , ).2

l

k l kl m l

RQ l a s l m

max22

22 1eff 0

1 2

( ) ( , ) min.2

k l

k l kk l m l

Rk Q l a s l m

2

lma eff ( )l

We consider a two-parameter

( is the Heaviside step function) and a three-parameter

dependences. The parameters were determined from the best fit condition

eff 0/ ( / )[1 ( )]l

eff 0/ ( / 2 ){1 tanh[( ) / ]}l

где

Effective surface tension in the two-parameter and three-parameter approximations for the cluster formed by 30000 particles at the temperature 0.75 of the interparticle potential well depth.

0.0 0.4 0.8 1.2k/gcs

0.0

0.2

0.4

0.6

0.8

1.0

ef

f

32

_

/

Capillary fluctuations for the cluster comprising 19400 particles at T = 0.69 (1) and for the cluster comprising 30000 particles at T = 0.75 (2). Dots represent MD simulation; lines, calculation using the three-parameter effective surface tension.

0 20 40 60 800.0

0.2

0.4

0.6

1

k

2

kQk

102 103 104 g0.5

0.7

0.9

1.1

2

1

2

1 – dens. prof.

2 – direct

3 – spectrum

3

T = 0.75

Width of the liquid―vapor interface is defined by the variance2 2

22 2

2

(2 ) (2 1)(2 5)(2 1) ln ln .

4 8 7lml

R Tl a R

Interface width diverges logarithmically as the surface area increases!

-2 -1 0 1 2r – Re

0.0

0.2

0.4

0.6

0.8

(r)

T = 0.67

< >

Average of the liquid―vapor interface configurations yields smoothed dependences of system characteristics in the transition region.

Small clusters and virtual chainsSmall clusters and virtual chains

Anim.bat

Small clusters with the minimum number of bonds form the virtual chains. Respective partition function is calculated analytically:

Hence, the cluster vapor equation of state is

1( ) 3 (2)( / ) .gg

c cZ V Z

11/ 201 [exp( / ) 1] .BZ C pT D k T

Trends of research

1. Capillary fluctuations and virtual chains near the critical

point.

2. Capillary fluctuations at the surface of a liquid metal.

3. Liquid metal surface near the critical point.

4. Capillary fluctuations in strong fields.

5. Interphase boundary with a high temperature gradient.

Thank you for the attention!

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