Thermal Surface Fluctuations Thermal Surface Fluctuations of Clusters withof 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

Aim of researchAim of research::

1. Working out a proper method for MD simulation of 1. Working out a proper method for MD simulation of such clusters in vapor environment.such clusters in vapor environment.

22. . Calculation of slice spectra.Calculation of slice spectra.

3. Estimation of fission threshold. 3. Estimation of fission threshold.

4. Development of a theory of surface fluctuations for 4. Development of a theory of surface fluctuations for

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

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

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

Definition of a cluster: a particle belongs to the cluster if it has at least one neighbor particle at the distance less than rb, which belongs to this cluster. The problem is, how to define rb.

1.2 1.4 1.6 1.8 2.0 2.2 2.4

rba0.9

1.0

1.1

1.2

1.3

g(r b) /

g e

g = 2 0 0

1 0 0

4 5 0

3 3 3

3

1/ 3

4 4( )

3 3( ), ( / ) ,

(3/ 4 ) .

e e v

e e

R n R R n

G R g R r

r n

p p

p

+ -

= =

=

l

l

l l

0 200 400 600 800 1000

g

-0.3

-0.2

-0.1

0.0

0.1

(g -

ge) /

g

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.

Particle 1 with the radius vector r1 that belongs to the cluster will be called internal if thereexists at least one particle 2 with radius vector r2 belonging to the same cluster that formsmore than four bonds such that the conditions

are satisfied.

10

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

-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:

We isolate the surface particles (pivot 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.

13

Bulk fluctuations arise from randomicity of particles location. Hence, they coincide with fluctuations of the surface particles of a cluster truncated by a sphere. The total spectral density is

max 21/ 3

1

.12

k

kk

R g

,k k kS Q R

140 20 40 60 800.0

0.2

0.4

0.6

0.8

k

kk

1

2

3 4

Different components of the surface fluctuations spectral amplitudes for a cluster comprising 30000 particles at = 0 and the temperature of 0.75 interparticle potential well depth. (1), bulk fluctuations, k = Rk ; (2), capillary fluctuations, k = Qk ; (3), the total spectral amplitude, k = Sk ; (4), the total spectral amplitude without isolation of the virtual chains.

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

21

Precursor stage of a supercritical cluster

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

–

25

20 max3 /16 ,k

Gibbs and Smoluchowski treated the liquid–vapor interface as a more or less abrupt change of the density and predicted that this interface is perturbed by thermal fluctuations. Mandelstam (1913) and Buff, Lovett, and Stillinger (1965) obtained

The interface width diverges due to short-wavelength fluctuations. A simple cutoff at the interparticle distance leads to the critical point paradox. A way to overcome it is introduction of the bend rigidity (Helfrich, 1973). This yields the wave vector dependent bare surface tension

20( ) ,q q

where is the bare surface tension.0

( )qwhere is the bend rigidity. Unfortunately, results obtained by different researchers are inconsistent. Thus, Mecke (1999) obtained a decreasing dependence ; some derived more complicated dependences.

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

27

Limitation of the maximum surface curvature by formation of a virtual chain

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 (critical point paradox): 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 relation2

0

1 ,2

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

Bulk fluctuations

Bulk fluctuations are characterized by the radial distribution of surface particles

or

, ,( )

0, ,r

drR r R

p r drr R r R

ìïï - D < <ïï D= íïï £ - D ³ïïî

and the distribution of their number

( )21( ) exp .

2 2cs cs

g cscs cs

g gp g

g gp

é ù-ê ú= -ê úê úë û

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

Surface variance 2 as a function of cluster size at

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

(1) ( ) 1 erf2 2n z

r zs

é ùæ ö÷çê ú= - ÷ç ÷çê úè øë ûl

2

2 2

1 1

1 1(2)

s sg g

i ii is s

r rg g

s= =

æ ö÷ç ÷= - ç ÷ç ÷çè øå å

max2

1

(3) (1/ 2)k

kk

Ss=

= å

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

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.

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