Thermal 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
Aim of research:
1. Working out a proper method for MD simulation of
such clusters in vapor environment.
2. Calculation of slice spectra.
3. Estimation of fission threshold.
4. Development of a theory of surface fluctuations for
clusters with long–range interaction.
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
.
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
i ri  Fi  0, where Fi  y 3
y
 2
y 3
1
y1
y 2 
y 1  Fi ,
1 
y1, y2, 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


 (F y  F x ) 
i Fzi yi
i ( Fxi xi  Fyi yi ) 
yi i 
  xi i
 i

 Fxi yi
 Fxi zi
Simulation cell: a cluster in equilibrium vapor environment
1.3
1.2
g = 200
g(rb)/ge
100
1.1
450
1.0
0.9
1.2
1.4
1.6
1.8
rba
2.0
2.2
2.4
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.
0.1
4p 3
4p 3
Re n l +
(R - Re3 )n v
3
3
= G (R ), ge = (Re / rl )3,
0.0
(g - ge)/g
rl = (3 / 4p n l )1/ 3 .
-0.1
-0.2
-0.3
0
200
400
600
g
800
1000
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 there
exists at least one particle 2 with radius vector r2 belonging to the same cluster that forms
more than four bonds such that the conditions
(r1  r2 )2
1
r1  r2  r , r 
 2/3
2
r1
3n
2
1
are satisfied.
2
2
400
300
F(b)
3
200
2
100
1
0
0
4
8
12
b
16
10
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
We isolate the surface particles (pivot 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.
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 Sk  Qk  Rk ,
kmax
R
k 1
k
2
  g 1/ 3 .
12
13
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.
0.8
kk
3
0.6
0.4
2
0.2
1
4
0.0
0
20
40
60
k
80
14
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 , x = (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
Precursor stage of a supercritical cluster
21
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
–
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
2
   0  3kmax
/16 ,
where  0 is the bare surface tension.
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
 (q)   0   q 2 ,
where  is the bend rigidity. Unfortunately, results obtained by different researchers
are inconsistent. Thus, Mecke (1999) obtained a decreasing dependence  (q ); some
derived more complicated dependences.
25
Theory of cluster capillary fluctuations
Probability of cluster fluctuation is defined by corresponding change in the Gibbs
free energy
  [x ( , )]  U [x ( , )]  0  U 0 ,
x (, )   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 R 2 , 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
Limitation of the maximum surface curvature by formation of a virtual chain
27
Formation of virtual chains limits the local curvature of the fluctuation surface:

x ( , )
 0.548.
2
This allows one to write  /  0  1   2 / 2 and to find the spectrum cutoff number
1/ 2

  0 
 0  2 R 

k
T
 B 
, if

 02
and  (2 0 ) otherwise. If we introduced a common cutoff    Rn1/ 3 ,
then we would arrive at failure of the capillary wave theory (critical point paradox): at
sufficiently high temperature (T = 0.95), when
2 1/ 4

 kBTn2 / 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

x ( , )
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.
Bulk fluctuations
Bulk fluctuations are characterized by the radial distribution of surface particles
ìï dr
ïï ,
R - D < r < R,
D
pr (r ) dr = í
ïï
ïïî 0, r £ R - D or r ³ R ,
and the distribution of their number
pg (gcs ) =
é (g - g )2 ù
1
cs
ú.
exp êê- cs
ú
2p gcs
2gcs
êë
ú
û
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
2kBTk 1  1
kQk 

 0 2 l k   l 2

  
 k 
k
arctan

arctan
 1/ 2 
 1/ 2  
 0 1/ 2 
 
  
kBT
is reached at kmax= (02/8)1/4. Divergence of interface variance at R → ∞ is removed:
k BT   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


2
0.6
0.4
0.2
0.0
0
10
20
k
30
Surface variance 2 as a function of cluster size at   
æ z
n é
(1) r (z ) = l ê1 - erf ççç
è 2s
2 êë
öù
÷
ú
÷
÷
øú
û
(2) s 2 =
2
T = 0.75
1.1
1
gs
æ1
2
åi = 1 ri - çççèg
gs
s
2
ö
åi = 1 ri ø÷÷÷÷
gs
k max
(3) s = (1/ 2)å S k
2
k= 1
1 – dens. prof.
2 – direct
3 – spectrum
0.9
1
0.7
2
3
0.5
102
103
104
g
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.
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!
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