Van der Waals interaction between surface and particle with giant polarizability
We show that a nanoparticle with a “giant” polarizability α (i.e., with the polarizability volume α′ = α/4πε₀ significantly exceeding the particle volume) placed in the vicinity of a surface experiences a strongly increased van der Waals force at distances comparable or smaller than the characteris...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2016
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| Цитувати: | Van der Waals interaction between surface and particle with giant polarizability / K.A. Makhnovets // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2016. — Т. 19, № 2. — С. 162-168. — Бібліогр.: 14 назв. — англ. |
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irk-123456789-1215552017-06-15T03:03:55Z Van der Waals interaction between surface and particle with giant polarizability Makhnovets, K.A. We show that a nanoparticle with a “giant” polarizability α (i.e., with the polarizability volume α′ = α/4πε₀ significantly exceeding the particle volume) placed in the vicinity of a surface experiences a strongly increased van der Waals force at distances comparable or smaller than the characteristic scale R₀ ∝ (α′ )¹/³. At distances close to R₀, the oscillation mode of the particle dipole moment softens, so nonlinear polarizability must be taken into account to describe the particle-surface interaction. It is shown that a proper treatment of nonlinear effects results in the van der Waals force that is free of divergences and repulsive contributions. 2016 Article Van der Waals interaction between surface and particle with giant polarizability / K.A. Makhnovets // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2016. — Т. 19, № 2. — С. 162-168. — Бібліогр.: 14 назв. — англ. 1560-8034 DOI: 10.15407/spqeo19.02.162 PACS 34.35.+a, 42.65.Pc http://dspace.nbuv.gov.ua/handle/123456789/121555 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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We show that a nanoparticle with a “giant” polarizability α (i.e., with the polarizability volume α′ = α/4πε₀ significantly exceeding the particle volume) placed in the vicinity of a surface experiences a strongly increased van der Waals force at distances comparable or smaller than the characteristic scale R₀ ∝ (α′ )¹/³. At distances close to R₀, the oscillation mode of the particle dipole moment softens, so nonlinear polarizability must be taken into account to describe the particle-surface interaction. It is shown that a proper treatment of nonlinear effects results in the van der Waals force that is free of divergences and repulsive contributions. |
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Makhnovets, K.A. |
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Makhnovets, K.A. Van der Waals interaction between surface and particle with giant polarizability Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Makhnovets, K.A. |
| author_sort |
Makhnovets, K.A. |
| title |
Van der Waals interaction between surface and particle with giant polarizability |
| title_short |
Van der Waals interaction between surface and particle with giant polarizability |
| title_full |
Van der Waals interaction between surface and particle with giant polarizability |
| title_fullStr |
Van der Waals interaction between surface and particle with giant polarizability |
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Van der Waals interaction between surface and particle with giant polarizability |
| title_sort |
van der waals interaction between surface and particle with giant polarizability |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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2016 |
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http://dspace.nbuv.gov.ua/handle/123456789/121555 |
| citation_txt |
Van der Waals interaction between surface and particle with giant polarizability / K.A. Makhnovets // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2016. — Т. 19, № 2. — С. 162-168. — Бібліогр.: 14 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT makhnovetska vanderwaalsinteractionbetweensurfaceandparticlewithgiantpolarizability |
| first_indexed |
2025-07-08T20:06:51Z |
| last_indexed |
2025-07-08T20:06:51Z |
| _version_ |
1837110625163018240 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2016. V. 19, N 2. P. 162-168.
doi: 10.15407/spqeo19.02.162
© 2016, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
162
PACS 34.35.+a, 42.65.Pc
Van der Waals interaction between surface
and particle with giant polarizability
K.A. Makhnovets
Institute of High Technologies, Taras Shevchenko National University of Kyiv,
4-G, prosp. Glushkova, 03022 Kyiv, Ukraine, e-mail: kotjono4ek@gmail.com
Abstract. We show that a nanoparticle with a “giant” polarizability α (i.e., with the
polarizability volume 04πεα=α′ significantly exceeding the particle volume) placed
in the vicinity of a surface experiences a strongly increased van der Waals force at
distances comparable or smaller than the characteristic scale ( ) 31
0 α′∝R . At distances
close to R0, the oscillation mode of the particle dipole moment softens, so nonlinear
polarizability must be taken into account to describe the particle-surface interaction. It is
shown that a proper treatment of nonlinear effects results in the van der Waals force that
is free of divergences and repulsive contributions.
Keywords: van der Waals interaction, nonlinear polarizability, giant polarizability, hard
mode, soft mode.
Manuscript received 21.01.16; revised version received 28.04.16; accepted for
publication 08.06.16; published online 06.07.16.
1. Introduction
Although van der Waals forces are usually too small to
play significant role in interactions between
macroscopically large objects, they can become
dominant at nanoscale. For example, van der Waals
forces are responsible for self-assembly of molecules
and nanoparticle arrays, which have been actively
studied in recent years [1-4]. In this work, we revisit the
well-known problem of the van der Waals interaction
between surface and nanoparticle, focusing on the
previously unexplored case of a particle with giant
polarizability. Here, the term “giant” means that the
polarizability volume is significantly larger than the
effective volume of the nanoparticle. Lately, giant
polarizability has been observed in Na14F13 molecular
clusters and in a number of other alkali-halide clusters
with MnXn–1 composition [5, 6]. The polarizability
volume of these clusters may be up to 30 times larger
than the effective volume of the cluster. Another
example could be nanoparticles made of ferroelectric
materials, under the conditions close to the ferroelectric
transition. We will show that the van der Waals force
between a surface and a nanoparticle with a giant
polarizability α increases significantly comparing to the
standard result when the distance between the particle and
surface becomes smaller than the certain characteristic
scale ( ) 31
00 εα∝R , where 0ε is the vacuum
permittivity. The reason for this behavior is softening one
of the dipole oscillation modes, which occurs at distances
close to R0. Within the description including only the
linear part of the polarizability, this softening signals an
instability (and would lead to a divergence in the van der
Waals force). This implies the necessity to take into
account the nonlinear polarizability, which would lead to
stabilization of the system.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2016. V. 19, N 2. P. 162-168.
doi: 10.15407/spqeo19.02.162
© 2016, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
163
A similar setup, considering interaction of a
nonlinearly polarizable ellipsoidal nanoparticle with a
surface or with another nanoparticle has been recently
studied in several articles [7-9]. According to the results
of Refs. [7, 8], nonlinearity generates repulsive
contributions to the van der Waals force that become
dominant at small distances and lead to the emergence of
a minimum in the total van der Waals potential at
distances about R0. However, nonlinear contributions in
Refs. [7, 8] have been taken into account only
perturbatively. In our previous paper [9], we have
considered interaction between two point-like
nanoparticles with giant polarizability, and have
concluded that a careful treatment of nonlinearity does
not lead to repulsive forces: on the contrary, the van der
Waals force is strongly enhanced at distances less or
about R0, remaining purely attractive at the same time. In
this paper, we show that essentially the same conclusion
remains valid in the case of a particle interacting with a
surface. We argue that the appearance of the repulsive
component in the van der Waals force, reported in [7, 8],
is an artifact of the weak coupling perturbation theory.
2. Model and its analysis in harmonic approximation
We consider an isotropic point-like particle placed in a
medium with the relative permittivity mε at distance R
from the surface of a substrate (infinitely thick) with the
relative permittivity sε (see Fig. 1). The case of a
metallic substrate can be obtained by formally setting
−∞=εs .We assume that the particle has a linear
polarizability α and third-order nonlinear polarizability
β− . We further assume that the particle has an inversion
center, so the second-order nonlinear polarizability
vanishes, and we set 0>β to ensure stability, so the
response of the dipole moment d
r
of the particle to the
local field E
r
is described by the expression
EEEd
rrr
2β−α= .
Fig. 1. Schematic view of the system considered: a point-like
polarizable particle at the distance R from the interface
between two media with different relative permittivities.
We are primarily interested in the case of small
distances mscR ,ω<< , where ( )ms ωω are the
characteristic frequencies of the polarization
oscillations in the substrate (medium), and the
characteristic frequency of the dipole oscillations 0ω
of the particle is supposed to be much smaller than
ms,ω . Thus, the retardation effects can be safely
neglected. We further assume that nonlinear
polarizabilities of both media are small as compared to
the nonlinear polarizability of the particle and can be
neglected.
In absence of retardation, interaction between the
particle and surface can be conveniently described by the
method of image charges [10]. If the dipole moment of
particle is ( )zyx dddd ,,
r
, with z axis perpendicular to the
surface, then the image dipole situated at a distance R
below the surface can be written as
( )zyxim kdkdkdd ,,−−
r
, where
sm
smk
ε+ε
ε−ε
= . Then, the
energy of the system can be written as:
( )( )( ),3
32
1
242
),(
3
0
2
0
2
4
42
imim
m
dndndd
R
dddRdE
rrrrrr
&rrr
r
⋅⋅−⋅
επε
+
+
αω
+
α
β
+
α
=
(1)
where RRn /
rr
= is the unit vector along z axis. The
minimum of the energy determined by Eq. (1) is reached
for npd
rr
0±= , where, as we shall see below, the
“equilibrium” value p0 depends on the distance R.
Considering small harmonic fluctuations, we write
pnpd
rrr
+±= 0 , where ( )zyx pppp ,,
r
describes the
fluctuations around p0. A standard quantization
procedure yields the following Hamiltonian of the
system described by Eq. (1):
( ) ,ˆ
4
ˆˆ
2
ˆ
2
ˆ
)(ˆ
22
4
2
4
0
,,
222
0
pppp
pm
m
pEH
z
zyxi
iii
rr
α
β
+
α
β
±
±⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ ω
+
π
+= ∑
= (2)
where iπ̂ are the momenta, canonically conjugate to the
fluctuations ip̂ , ( )0pE is the “static” part of the energy
(see below), and for convenience we have introduced the
“effective mass” ( ) 12
0
−
αω=m . Further, iω are the
harmonic normal mode frequencies of the dipole
fluctuations:
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2016. V. 19, N 2. P. 162-168.
doi: 10.15407/spqeo19.02.162
© 2016, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
164
One can see that z-mode becomes soft at R = R0. It
is easy to see that in the standard case, when the
“polarizability volume” 04πεα is roughly about the
physical volume of the particle, R0 is close to the linear
dimension of the particle or even smaller, so the regime
of small distances 0RR ≤ is either completely
unphysical or cannot be considered in the approximation
of a point-like particle. In what follows, we concentrate
on the opposite case of a giant polarizability, when R0 is
significantly larger than the linear dimensions of the
particle.
The static part of the energy, denoted ( )0pE in
Eq. (3), is given by the following expression:
( ) ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+⎟
⎠
⎞
⎜
⎝
⎛
−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
β
α
=
2
0
3
0
2
0
2
0 2
11
2 mm p
p
R
R
p
p
pE (4)
where ( ) 213 βα=mp is a characteristic dipole moment
value (its physical sense is that for mpd ≅ the 3-rd
order nonlinear contribution to the energy becomes of
the same order of magnitude as the linear contribution,
and thus higher order nonlinear terms should be taken
into consideration; in other words, our approximation
that is restricted to the 3-rd order nonlinearity is, strictly
speaking, applicable only for mpd << ). It is easy to see
that for R > R0 the potential function ( )0pE has only
one minimum at p0 = 0, and has two symmetrical
minima for R < R0 (see Fig. 2).
The equilibrium dipole value p0 and the
corresponding static energy can be obtained by the
minimization of ( )0pE , which yields the following
expressions:
( )⎩
⎨
⎧
><−
<>
=
,)1(,1
,)1(,0
0
2
02
0 sRRsp
sRR
p
m
( )⎪⎩
⎪
⎨
⎧
><−
β
α
−
<>
= )1(,1
4
,)1(,0
)(
0
2
2
0
0 sRRs
sRR
pE (5)
that are valid only when p0 is small as compared to pm
(i.e., when ( ) 11 <<−s ). Far from the softening point,
anharmonic terms can be neglected, and one can obtain
the Hamiltonian in the second quantization terms in the
harmonic approximation:
∑
=
+ ⎟
⎠
⎞
⎜
⎝
⎛ +ω+=
zyxi
iiiharm aapEH
,,
0 2
1ˆˆ)(ˆ h . (6)
Here, the dipole fluctuation operators and their
canonically conjugate momenta are related with creation
and annihilation operators via the standard relations:
( )ii
i
i aa
m
p ˆˆ
2
ˆ
2/1
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
ω
= +h ,
( )ii
i
i aa
m
i ˆˆ
2
ˆ
2/1
−⎟
⎠
⎞
⎜
⎝
⎛ ω
=π +h
. (7)
The ground state energy in the harmonic
approximation takes the form
( ) ∑
=
ω+=
zyxi
iharm pEE
,,
0 2
1
h , (8)
which, for R > R0 (s < 1), transforms into the following
one:
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−+−ω= ssEharm 1
2
12
2
1
0h . (9)
At large distances R >> R0 (s << 1), one can expand
square roots in (9), and obtain the standard 31 R
dependence for the van der Waals potential, obtained by
Lennard-Jones for interaction between inert gas atom
and metallic surface [11-13]:
( )
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛−ω=−ω≈
3
0
00 3
2
13
2
1
R
RsEst hh . (10)
In the case of small distances R < R0 (s > 1),
harmonic approximation for the energy yields:
( ) ( )
2
0
2
0 ,12
2
2
2
1
2
1
αβω=λ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−++
λ
−
−ω=
h
h sssEharm (11)
where we have introduced the dimensionless
nonlinearity parameter λ for the sake of later
convenience. Both for large and small distances, the
ground state energy contains a term proportional
to 3
0
3 RR − (corresponding to the contribution from
( )
( )
( ) ( ) .1,8
,)1(,12
,)1(,1
,)1(,
2
,)1(,
2
1
3
0
31
00
0
2
0
0
2
02
0
2
0
0
2
0
22
RRsR
sRRs
sRRs
sRRs
sRRs
sm
sm
m
zyx
=
ε+ε
ε−ε
ε
=κπεκα=
⎪⎩
⎪
⎨
⎧
><−ω
<>−ω
=ω
⎪
⎪
⎩
⎪⎪
⎨
⎧
><ω
<>⎟
⎠
⎞
⎜
⎝
⎛ −ω
=ω=ω
(3)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2016. V. 19, N 2. P. 162-168.
doi: 10.15407/spqeo19.02.162
© 2016, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
165
z-mode), which leads to a divergence in the van der
Waals force REF harm ∂−∂= at 0RR → , due to
z-mode softening. However, fluctuations of the dipole
momenta become large close to the softening point, so
one can expect that nonlinear terms will play an
increasingly important role in proximity to this point.
Further, for R < R0 there are two energetically equivalent
states npd
rr
0±= , which are separated by a barrier that
vanishes at 0RR → . Therefore, when R is just slightly
smaller than R0, tunneling processes between these states
become dominant (and also determined by the nonlinear
terms). It means that the naive harmonic approximation
is not applicable in the close vicinity of the softening
point.
3. Effect of nonlinear polarizability
3.1. Weak coupling regime: perturbation theory
Consider first the corrections coming from the last two
terms in the Hamiltonian described by Eq. (2). It makes
sense, if the system is far from the softening point. In
this regime, all modes are “hard”, so one can simply
calculate the leading corrections using the first-order
perturbation theory in λ (for this calculation, we assume
that 1<<λ , otherwise the perturbative approach is
completely inapplicable). The corrections can be easily
obtained using the following averages:
03 == ii pp ,
i
i m
p
ω
=
2
2 h ,
2
224
2
33 ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
ω
==
i
ii m
pp h . (12)
We will restrict ourselves to the leading corrections
(first order in λ ), so the last but one term in (2)
containing odd powers of pi gives no contribution in this
order.
a) Weakly nonlinear single-well regime occurs
when λ>>− 23)1( s . This condition ensures that the z-
mode remains “hard” and can be treated perturbatively.
In this case, the potential for the dipole moment is of a
single-well type (see Fig. 3, large dashes) and the ground
state energy, with the account taken of corrections from
nonlinear terms, has the form
( ) ( )( )⎟⎟
⎠
⎞
⎜
⎜
⎝
⎛
+λ+−+−ω= sfsfssEws 210 1
2
12
2
1
h , (13)
where we have introduced the following notation:
( ) ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
ω
ω
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
ω
ω
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
ω
ω
=
xzz
sf 00
2
0
1 2
1
8
3 ,
( )
2
0
2 ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
ω
ω
=
x
sf . (14)
Fig. 2. Potential given by Eq. (4), for different values of R/R0.
Frequencies for those correction functions should
be taken from Eq. (3) for the case when )1(0 <> sRR .
Note that ωz vanishes at )1(0 →→ sRR , so ( )sf1 is
divergent at the softening point, while the function
( )sf2 remains regular. The contribution from nonlinear
terms in Eq. (13) corresponds to repulsive forces, and,
formally taken, the full expression Eq. (13) describes a
potential with a minimum (see the dash-dotted curve in
Fig. 3), similar to that obtained in Refs. [7, 8]. However,
this minimum occurs close to R = R0, which is outside
the applicability range of the perturbation theory. As we
shall see below, the proper description in the vicinity of
the softening point does not exhibit any repulsive
contributions, so this minimum should be regarded as an
artifact of the perturbation theory. Moreover, including
the perturbative corrections of higher order in λ would
add increasingly more and more singular contributions
(diverging at R = R0), with alternating signs.
b) Weakly nonlinear double-well regime occurs
when ( ) λ>>− 231s . Similarly to the previous case, this
condition ensures that z-mode remains “hard” and can be
treated perturbatively. In this case the potential for the
particle dipole moment is of a double-well type with
“deep” wells (see the short-dashed curve in Fig. 2). The
average magnitude of the dipole moment fluctuations
inside a well is roughly equal to zmω2h and is, in this
regime, much smaller than the equilibrium dipole
moment p0, which determines the distance between the
wells, so for the purpose of calculating the ground state
energy the tunneling effects can be neglected. The
ground state energy is a sum of the harmonic
approximation expression Eq. (11) and the leading
corrections in λ :
( )
( ) ( ) ( )( ) .12
2
2
2
1
2
1
21
2
0
⎟
⎟
⎠
⎞
+λ+−++
⎜
⎜
⎝
⎛
+
λ
−
−ω=
sfsfss
sEwd h
(15)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2016. V. 19, N 2. P. 162-168.
doi: 10.15407/spqeo19.02.162
© 2016, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
166
Here the frequencies for correction functions f1,2(s)
should be taken from Eq. (3) for the case when R < R0
(s > 1). Since p0 should be small comparing to the pm,
one has to require that (s – 1) << 1, so the theory is
applicable only to a small range of R values.
3.2. Strong coupling expansion
When λ<<− 231 s , one may speak of the strongly
nonlinear regime, corresponding to a proximity to the
softening point R = R0, where the potential is almost flat
(dominated by nonlinear terms) for a range of dipole
moments (Fig. 2, dot-dashed and solid lines). In this case,
harmonic approximation is not a good starting point, and
perturbation theory in λ is no longer applicable. The
dipole moment fluctuations are large (comparable to p0),
so it does not make sense to consider oscillations around a
single-well minimum. Without expanding the dipole
moment d̂
r
around 0p± , the Hamiltonian corresponding
to Eq. (1) has the following form:
( )∑
=
++
α
β
+⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ ω
+
π
=
zyxi
zyx
iii ddddm
m
H
,,
2222
4
222
ˆˆˆ
42
ˆ
2
ˆˆ , (16)
⎟
⎠
⎞
⎜
⎝
⎛ −ω=ω=ω
2
12
0
22 s
yx , ( )sz −ω=ω 12
0
2 . (17)
In this regime, x- and y-modes remain “hard” and
can still be treated perturbativelly, but z-mode is “soft”.
To make use of the known results for quartic oscillator,
it is convenient to rewrite the Hamiltonian in
dimensionless variables ( ) 21
0 /ˆ~̂
hω= mdd ii and their
conjugate momenta iπ̂
~ . The Hamiltonian can be
represented as a sum of “soft” and “hard” mode parts
plus interaction terms, as follows:
( ) ( )
( )
( ) ( ) .
~̂~̂
4
ˆ,
~̂~̂~̂
2
ˆ
,
2
~̂
2
~̂
ˆ,
~̂
42
~̂
2
~̂
ˆ
,ˆˆˆˆˆˆ
2
222
int
2221
int
22
0
2
0
4
2
0
2
0
2
int
1
int000
0
⎟
⎠
⎞
⎜
⎝
⎛ +
λ
=⎟
⎠
⎞
⎜
⎝
⎛ +
λ
=
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
ω
ω
+
π
=
λ
++
π
=
++++=
ω
yxzyx
iiii
z
zzz
yxz
ddhdddh
yxidhddgh
hhhhhH
h
(18)
where sg −= 10 and λ are the quadratic and quartic
dimensionless coupling constants.
The “soft” mode is “slow” comparing to “hard
(fast)” modes, so we can perform averaging over the
“hard” dx,y modes, regarding the dipole moment dz
corresponding to the “soft” mode as a constant.
Averages of the “hard” modes can be calculated to the
first order in λ (we, as before, assume that 1<<λ ), in
the harmonic approximation according to the formulas
of the same type as Eq. (12). Such a procedure, applied
to the term ( )1
intĥ describing interaction between “soft”
and “hard” modes, leads to a renormalization of the
quadratic coupling g0:
21
110 s
sgg
−
λ+−=a . (19)
The soft z-mode can be now treated using the
strong-coupling expansion for quartic anharmonic
oscillator [14] that is a power series in the
parameter ( ) 32/4 λg . Then the ground state energy can
be obtained as follows:
( )
,4
4
2
12
2
1
0
3231
0
20
∑
∞
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
λ
⎟
⎠
⎞
⎜
⎝
⎛ λω+
+⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
λ+−ω=
n
n
n
s
gc
sfsE
h
h
(20)
where nc are the coefficients of the strong coupling
expansion (listed in Ref. [14] up to n = 22), the
correction involving ( )sf2 stems from the interaction
( )2
intĥ in Eq. (18), and the function ( )sf2 is given by
Eq. (14) with the frequencies defined by Eq. (17). From
the above results, we can see that there is actually no
singularity at 1→s (at the softening point).
4. Discussion and summary
It is instructive to compare the above results obtained
within different approximations. Fig. 3 shows the van
der Waals potential calculated for a particle in vacuum
( 1=εm ) near a metallic surface ( −∞=εs ), with the
nonlinear coupling constant set to 005.0=λ , and Fig. 4
shows the corresponding force. One can see that
harmonic approximation (Eq. (9) and Eq. (11), shown in
Figs. 3 and 4 as a dotted line) works well far from the
mode softening point R = R0, but obviously fails in the
proximity of R0 (note the singularity in force), because
anharmonic terms play crucial role near the softening
point. Weak-coupling first-order perturbation result
(Eq. (13) and Eq. (15), shown with a dash-dotted line in
Figs. 3 and 4) leads to a small correction to the harmonic
approximation far from the softening point, but shows
unphysical divergence in the vicinity of R0 (in the first
order by λ , this diverging contribution happens to
correspond to a repulsive force, but the sign of the
divergence alternates when higher-order corrections are
included). The reason for this behavior is the divergence
of fluctuations at 0RR → due to the vanishing harmonic
frequency of the z-mode (see Eq. (12)). At a formal
mathematical level, one can say that the actual small
parameter for the weak-coupling perturbation theory is
not λ but rather ( ) 231/ s−λ , and it is not small for
distances close to R0 even when 1<<λ . We remark that
the spurious short-distance repulsion that appears in our
first-order perturbation theory, strongly resembles the
nonlinearity-induced repulsion obtained in Refs. [7, 8]
on the basis of a quite different phenomenological
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2016. V. 19, N 2. P. 162-168.
doi: 10.15407/spqeo19.02.162
© 2016, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
167
electrodynamics approach (the essential point, though, is
that the contribution of nonlinear terms to the energy has
been computed to the first order in the nonlinear
polarizability constant).
The results of the “hybrid” approach of Eq. (20)
combining the strong-coupling expansion for the soft
mode and first-order weak-coupling corrections from hard
modes, including the renormalization of the quadratic
coupling of the soft mode, are shown in Figs. 3 and 4 with
a dashed line. One can see that this approximation
complements the weak-coupling results: it fails to
describe the situation far from the softening point, but
gives correct results close to R = R0. It is easy to see that if
Fig. 3. The potential of van der Waals interaction between a
point-like particle and metallic surface separated by the
distance R, for the model described by Eq. (1), at the value
λ = 0.005 of the quartic coupling parameter obtained in various
approximations. The solid line indicates the standard result Eq.
(10) valid at R >> R0; the dotted line corresponds to the
harmonic approximation given by Eqs (9) and (11); dash-
dotted lines show the results obtained within the weak coupling
perturbation theory for R > R0 (Eq. (13)) and R < R0 (Eq. (15)),
and the dashed line shows the result Eq. (20) obtained by
means of the strong-coupling expansion with the renormalized
quadratic coupling Eq. (19).
Fig. 4. The van der Waals force F = – ∂E/∂R between a point-
like particle and metallic surface separated by the distance R,
all notations are the same as in Fig. 3.
one takes the weak-coupling result far from the softening
point and joins it with the strong-coupling result near R0,
one ends up with a smooth monotonic curve which is free
from any singularities and describes the van der Waals
interaction that remains attractive at all distances.
Comparing the standard 1/R3 expression for the van
der Waals interaction (Eq. (10), shown with solid line in
Figs. 3 and 4) to the results obtained in other approaches,
one can see that the standard result works well at large
distances R >> R0, but strongly underestimates the van
der Waals force at distances comparable to R0 and
smaller. One can define the enhancement factor η as the
ratio of the force calculated using appropriate formulas
for each region (Eq. (13), (15), (16)) to the force
calculated from the standard result (Eq. (10)):
RE
RE
st ∂∂
∂∂
=η . (21)
This enhancement factor behaves as 31−λ∝η in
the vicinity of the softening point.
To summarize, we have shown that the van der
Waals force between a particle with a giant linear
polarizability α and a surface is significantly enhanced
at short distances of the order of ( ) 31
00 εα∝R . This
result is derived in a simplified microscopic model
assuming single-oscillator approximation for the particle
and including a stabilizing third-order nonlinear
polarizability. We also show that a careful treatment of
nonlinearity does not lead to any repulsive forces,
contrary to some recent theoretical claims [8]. This
result may be important for theoretical understanding the
van der Waals interactions in systems of alkali-halide
molecular clusters with giant linear polarizability [5, 6].
Acknowledgements
I am grateful to Dr. A. Kolezhuk and Dr. V. Lozovski
for productive discussions.
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doi: 10.15407/spqeo19.02.162
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