Bare Surface Tension and Surface Bare Surface Tension and Surface Fluctuations of Clusters withFluctuations of Clusters with

Long–Range InteractionLong–Range Interaction

D.I. Zhukhovitskii

Joint Institute for High Temperatures, RAS

LiquidLiquid――vapor interface structurevapor interface structure::

smooth smooth or or stratified stratified ??(van der Waals)(van der Waals) ( (GibbsGibbs))

Gas

Liquid

Gas

Liquid

Intermediate phase

22 2 1 2

1 2 1 2 2 2 / 31

( ) 1,

3r r

r n

r rr r

We define three particle types: internal and surface particles and virtual chains.

-2 -1 0 1 2r – Re

0.0

0.2

0.4

0.6

0.8

(r)

T = 0.67

< >

Average configurations yield smooth density distribution inside the transitional region:

Aim of researchAim of research::

1. 1. Development of a theory of surface fluctuations for Development of a theory of surface fluctuations for

clusters with long–range interaction.clusters with long–range interaction.

2. 2. Working out a proper method for MD simulation of Working out a proper method for MD simulation of

such clusters in vapor environment.such clusters in vapor environment.

3. 3. Calculation of slice spectra.Calculation of slice spectra.

4. Estimation of fission threshold. 4. Estimation of fission threshold.

Cluster particles are assumed to interact via the pair additive potential

( ) ( ) ( ),short longu r u r u r

12 6

12 6

( ) ( ), ,( )

0, ,

( ) 4 , 2.5 ,

c cshort

c

c

v r v r r ru r

r r

a av r r a

r r

where

and the long–range component

0 1( ) .longu r

gn r

System under consideration

Probability of cluster fluctuation is defined by corresponding change in the Gibbs free energy

0 0[ ( , )] [ ( , )] ,U U

,

( , ) ( , ), .lm lml m

a Y l m l where

Assuming small fluctuation amplitudes we have derived

20

2

2 ( 1)( 1)( 2) ,

2 2 1

l

lml m l

ll l a

l

Theory of cluster capillary fluctuations

20 0 0 04 , ,

5R U

where 0 is the bare surface tension. Based on the equipartition theorem we arrive at the amplitudes of fluctuation modes

2

0

2 1.

2 ( 1) (2 1)( 1)( 2)B

lm

l k Ta

l l l l

Formation of virtual chains limits the local curvature of the fluctuation surface:

2( , ) 0.548.

20/ 1 / 2 This allows one to write and to find the spectrum cutoff number

1/ 2

200 02 , if

B

Rk T

and otherwise. If we introduced a common cutoff2 1/ 40(2 ) 1/ 3,Rn

then we would arrive at failure of the capillary wave theory: at sufficiently high temperature (T = 0.95), when

2 / 3

,8

Bk Tn

there is no non-negative solution for 0. This difficulty is removed in proposed theory.

By definition, the bare surface tension 0 refers to a flat (nonperturbed) interface. Due to the parachor considerations, it depends on the surface density, which is independent on the field strength (field pressure vanishes on the surface). Therefore, 0 is field independent. The quantity

2( , ) 0.548

Is also field independent by definition. Due to the relation

2

0

1 ,2

the ordinary surface tension proved to be field independent as well.

2 222 2

2

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

4 8 7c lml

R Tl a R

1. The case = 0. The interface variance

is reached at kmax= (02/8)1/4. Divergence of interface variance at R → ∞ is removed:

22

0

ln 1 .4

Bc

k T

and proportional interface width diverge with cluster size.

In the case of gravitational attraction, the interface variance vanishes with the increase in R:

2. The case > 0 (pseudogravitation). The maximum of spectral slice amplitude

2 1/ 2 1/ 2 1/ 20 0

2 1 1arctan arctan

2B B

kl k

k Tk k T kkQ k

l

2 2 322 0

2 20

3 1 4.

4 3gr

cgr

n M R

M n R

at

0 10 20 300.0

0.2

0.4

0.6

kQk

k

Theoretical CF slice spectrum for different

3. The case < 0 (Coulomb-like repuilsion). The surface variance is

2 2and .lm ca

2

20

355 10, so that 10 11.5 10.

4

QRR

The maximum value = –10 corresponds to singularity of The cluster

becomes unstable with respect to fission. The classical fission threshold [Bohr and

Wheeler (1939), Frenkel (1939)] supposes greater charge:

2

0

12.5 (2 1)(2 5)ln .

4 10 27B

c

k T

Systems with multiple length and time scales require special integrators to prevent enormous energy drift. In the force rotation approach, an artificial torque of the long–range force components Fi arising from cluster rotation is removed by rotation of these forces. We impose the condition

and are the Euler angles. They are solutions of equation set

3 2

3 1

2 1

1

0, where 1 ,

1i i i i

i

r F F F

1

2

3

( ) ( )

( ) ( ) .

( ) ( )

yi i zi i xi i xi i yi i zi ii i i i

yi i xi i zi i yi i xi i zi ii i i i

zi i zi i xi i yi i xi i yi ii i i i

F y F z F y F z F z F y

F x F x F z F z F z F x

F x F y F x F y F y F x

Molecular dynamics simulation

Simulation cell: a cluster in equilibrium vapor environment

We isolate the surface particles situated between two parallel planes. The particle polar coordinates are the values of a continuous function

max max0

1 1

( ) cos sin .2

k k

k kk k

P k k

The slice spectrum are defined as the averages both over configurations and over the Euler cluster rotation angles:

1 2

2 2

, .cs k k

kcs

gS

g

The total spectrum is a sum of the capillary fluctuations (CF) and bulk fluctuations (BF) spectra.

CF spectral amplitudes for clusters comprising 20000 particles at = 445, T = 0.955: theory, simulation. BF amplitudes are shown for comparison

0 20 40 600.0

0.1

0.2

0.3

0.4

0.5

CF, simulationCF, theoryCF, rough estimateBF, simulation

kQk

k

CF spectral amplitudes for clusters comprising 20000 particles at = 10, T = 0.75: theory, simulation. BF amplitudes are shown for comparison

0 20 40 600.0

0.1

0.2

0.3

0.4

0.5

CF, simulationCF, theoryBF, simulation

kQk

k

0 20 40 60 800.0

0.2

0.4

0.6

kQk

k

= 0,T = 0.75

CF spectral amplitudes for clusters comprising 20000 particles: theory, simulation. BF amplitudes are shown for comparison

CF spectral amplitudes for clusters comprising 20000 particles at = –4.96, T = 0.75: theory, simulation. BF amplitudes are shown for comparison

0 20 40 600.0

0.4

0.8

1.2 CF, simulationCF, theoryBF, simulation

kQk

k

Second spectral amplitude for clusters comprising 20000 particles as a function of

5 6 7 8 9 100

2

4

6

8theory, m = 0,1,2theory, m = 0simulationcurve fitprecursor stage

2Q2

–

Deformation parameters of clusters comprising 20000 particles , = (c/a)2/3 – 1,at T = 0.75

0 5000 10000 15000 20000 25000

0.0

0.5

1.0

1.5

2.0

t, MD units

Fission of a supercritical cluster

Ratios of the second slice spectral amplitudes calculated in three reciprocally perpendicular planes, the plane of a maximum amplitude and the planes of intermediate and minimum amplitude, as a function of time for a supercritical cluster

0 5000 10000 15000 20000 250000.0

0.2

0.4

0.6

0.8

1.0intermediateminimum

t, MD units

S 0 /S

max

Autocorrelation function and correlation decay time for the second slice spectral amplitude for different

0 1000 2000 3000 4000 50000.0

0.2

0.4

0.6

0.8

1.0

Au

toco

rre

latio

n fu

nct

ion

– = 6.89, 7.87, 8.86, 9.40

4 5 6 7 8 90

1000

2000

3000

–

Conclusions

1. A leading order theory of surface fluctuations is proposed for

clusters with a long–range particles interaction.

2. CF are damped by the attractive long–range interaction; the

surface tension is independent of the field strength.

3. For the repulsive interaction, the fission threshold is defined

by the bare rather than ordinary surface tension.

4. A nonlinear theory of large fluctuations is required.

Thank you for the attension!

For more details, visit

http://oivtran.ru/dmr