Bare Surface Tension and Surface
Fluctuations of Clusters with
Long–Range Interaction
D.I. Zhukhovitskii
Joint Institute for High Temperatures, RAS
Liquid―vapor interface structure:
smooth
(van der Waals)
Gas
or
stratified ?
(Gibbs)
Gas
Intermediate phase
Liquid
Liquid
We define three particle types: internal and surface particles and virtual chains.
2
(
r

r
)
1
2
2
1
2
r1  r2  r1 , r2 
 2/3
2
r1
3n
Average configurations yield smooth density distribution inside the transitional region:
<
>
0.8
T = 0.67
(r)
0.6
0.4
0.2
0.0
-2
-1
0
r – Re
1
2
Aim of research:
1. Development of a theory of surface fluctuations for
clusters with long–range interaction.
2. Working out a proper method for MD simulation of
such clusters in vapor environment.
3. Calculation of slice spectra.
4. Estimation of fission threshold.
System under consideration
Cluster particles are assumed to interact via the pair additive potential
u(r )  ushort (r )  ulong (r ),
where
v(r )  v(rc ), r  rc ,
ushort (r )  
0,
r  rc ,

 a12 a 6 
v(r )  4  12  6  , rc  2.5a,
r 
r
and the long–range component
ulong (r )  
 0 1
gn r
.
Theory of cluster capillary fluctuations
Probability of cluster fluctuation is defined by corresponding change in the Gibbs
free energy
  [ (, )]  U[ (,)]  0  U0 ,
 ( , )   almYlm ( , ),  l  m  l.
where
l ,m
Assuming small fluctuation amplitudes we have derived
0
2
 2 (l  1)
 l
   
 (l  1)(l  2)   alm ,
2 l 2  2l  1
 ml


0  4 0 R2 , U 0   0 ,
5
where 0 is the bare surface tension. Based on the equipartition theorem we arrive
at the amplitudes of fluctuation modes
alm
2

2l  1
k BT
.
2 (l  1)  (2l  1)(l  1)(l  2)  0
Formation of virtual chains limits the local curvature of the fluctuation surface:

This allows one to write
 ( , )
2
 0.548.
 /  0  1   2 / 2 and to find the spectrum cutoff number
1/ 2
  0 
 0  2R 
 , if
 kBT 
and 

02
(202 )1/ 4 otherwise. If we introduced a common cutoff    Rn1/ 3 ,
then we would arrive at failure of the capillary wave theory: at sufficiently high
temperature (T = 0.95), when

 k BTn 2 / 3
8
,
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
1 ,
0
2
the ordinary surface tension  proved to be field independent as well.
1. The case  = 0. The interface variance
R2
 
4
2
c

 (2l  1)
l 2
alm
2
(2   2 )T (2  1)(2  5)
ln
8
7
ln R 2
and proportional interface width diverge with cluster size.
2. The case  > 0 (pseudogravitation). The maximum of spectral slice amplitude
2k BTk 1  1
kQk 

 0 2 l k   l 2

  
 k 
k
arctan

arctan
 1/ 2 
 1/ 2  
 0 1/ 2 
 
  
k BT
is reached at kmax= (02/8)1/4. Divergence of interface variance at R → ∞ is removed:
kBT   2 
 
ln 1 
.
4 0 
 
2
c
In the case of gravitational attraction, the interface variance vanishes with the increase
in R:
2
2 3
2

n
M
R
3


1
4

gr
0
 c2 
at


.
2 2
4  gr M n R
3
0
Theoretical CF slice spectrum for different 
kQk
  
  
  
0.6
0.4
0.2
0.0
0
10
20
k
30
3. The case  < 0 (Coulomb-like repuilsion). The surface variance is
 c2 
k BT  12.5
(2  1)(2  5) 

ln
 .
4 0    10
27
The maximum value  = –10 corresponds to singularity of
alm
2
and  c2 .The cluster
becomes unstable with respect to fission. The classical fission threshold [Bohr and
Wheeler (1939), Frenkel (1939)] supposes greater charge:
3 Q2

5 5 R 2  10, so that   10
4 R
0
11.5  10.
Molecular dynamics simulation
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
1
y


r

F

0,
where
F

i i i
i
 3
y
 2
y 3 y 2 
1
y 1  Fi ,
y1
1 
y1, y, and y3 are the Euler angles. They are solutions of equation set

  ( Fyi yi  Fzi zi )
 i

i Fyi xi


 Fzi xi

i




(
F
z

F
y
)
zi i 
 y    yi i
i
i
i
1
  

i ( Fxi xi  Fzi zi )
i Fyi zi  y 2    i ( Fxi zi  Fzi xi ) .
  

y
3
i Fzi yi
i ( Fxi xi  Fyi yi )     i ( Fxi yi  Fyi xi ) 
 Fxi yi
 Fxi zi
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
P( ) 
0
2
kmax
kmax
k 1
k 1
   k cos k    k sin k .
The slice spectrum are
defined as the averages both
over configurations and over
the Euler cluster rotation
angles:
Sk 
2
2
g



 cs k k
g
y 1 ,y 2
.
cs
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
0.5
kQk
CF,
CF,
CF,
BF,
0.4
simulation
theory
rough estimate
simulation
0.3
0.2
0.1
0.0
0
20
40
60
k
CF spectral amplitudes for clusters comprising 20000 particles at  = 10, T = 0.75:
theory, simulation. BF amplitudes are shown for comparison
0.5
kQk
CF, simulation
CF, theory
BF, simulation
0.4
0.3
0.2
0.1
0.0
0
20
40
60
k
CF spectral amplitudes for clusters comprising 20000 particles: theory, simulation. BF
amplitudes are shown for comparison
kQk
 = 0,
T = 0.75
0.6
0.4
0.2
0.0
0
20
40
60
k
80
CF spectral amplitudes for clusters comprising 20000 particles at  = –4.96, T = 0.75:
theory, simulation. BF amplitudes are shown for comparison
kQk
1.2
CF, simulation
CF, theory
BF, simulation
0.8
0.4
0.0
0
20
40
60
k
Second spectral amplitude for clusters comprising 20000 particles as a function of 
2Q2
8
theory, m = 0,1,2
theory, m = 0
simulation
curve fit
precursor stage
6
4
2
0
5
6
7
8
9
–
10
Deformation parameters of clusters comprising 20000 particles ,  = (c/a)2/3 – 1,
at T = 0.75
2.0
  1
  

1.5
1.0
0.5
0.0
0
5000
10000
15000
t, MD units
20000
25000
Fission of a supercritical cluster
S0 /Smax
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
1.0
intermediate
minimum
0.8
0.6
0.4
0.2
0.0
0
5000
10000
15000
t, MD units
20000
25000
Autocorrelation function and correlation decay time for the second slice spectral
amplitude for different 
1.0

– = 6.89, 7.87, 8.86, 9.40
Autocorrelation function
0.8
3000
0.6
2000
0.4
1000
0.2
0
0.0
0
1000
2000
3000
4000
5000

4
5
6
7
8
9
–
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