van der Waals energies of cylindrical and spherical single layer
systems
N. S. Witte
Citation: J. Chem. Phys. 99, 8168 (1993); doi: 10.1063/1.465644
View online: http://dx.doi.org/10.1063/1.465644
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van der Waals energies of cylindrical and spherical single layer systems
N. S. Witte
School of Physics, University of Melbourne, Parkville, Victoria 3052 Australia
(Received 29 March 1993; accepted 19 July 1993)
The van der Waals energies for cylindrical and spherical single layer systems are obtained, in the
nonretarded limit, from the heuristic procedure of summing the normal surface electromagnetic
mode frequencies. Both the zero and finite temperature results are presented. The geometrical
dependence of the energies for the cylindrical and spherical systems is shown to satisfy
inequalities involving simple model dielectric configurations. The exact energies have a
first-order correction to the planar case, for a large inner radius in comparison to the film
thickness, which is proportional to the mean curvature with a new Hamaker coefficient. It is
conjectured that this applies to arbitrarily shaped smooth films with small curvature. In the
two-body summation approximation the energies factorize into a geometrical factor, for which
exact analytical forms are found, and the standard Hamaker constant. The effect of geometry on
the van der Waals energy is shown to be important, in that the effective Hamaker constants for
curved films of inner radius r and layer thickness w rises rapidly at r~ 2.6w for the cylindrical
and at r~ 1.8w for the spherical system.
I. INTRODUCTION
The interplay of the geometry and scale of neutral
macroscopic bodies interacting with each other as a result
of long-range electromagnetic van der Waals forces is of
importance in the stability of colloid systems, and in biological assemblies to name a few examples. The thermodynamic effects of the van der Waals interaction in systems of
interfaces and films in planar, cylindrical, and spherical
geometries, including full retardation has been investigated
by the group of Belosludov, Nabutovskii, and Korotikh.
Their method utilizes the finite temperature Green's functions for the fluctuating electromagnetic fields and their
boundary relations across the various interfaces, first developed by Dzyaloshinski, Lifshitz, and Pitaevski 1 (DLP).
Their work also provides a more detailed description in
each case, by providing the chemical potential density, the
density of mass variation, and the tangential components
of the electromagnetic stress-energy tensor, as well as the
normal stress components which are simply related to the
free energy. Their work is reviewed in Ref. 5. While this
work is more general in respect of the effects of a finite
propagation speed for the electrodynamical interaction,
and thus is correct in describing the van der Waals forces
at larger separations, d~ 10- 6_10- 7 m, the overlap with
the present work is quite minimal.
van Kampen et af. 2 showed that a simpler theory of
summing allowed normal mode frequencies, which was
identical to that obtained from the Green's function procedure of DLP. The advantage of employing the "simple"
or "heuristic" method of summing the zero-point energies
of the fundamental oscillators is that the total free energy
of more complex geometries and configurations can be explored. The disadvantage is that one finds only a total free
energy for the global system considered as a whole compared to the completely diassembled layers, and this
8168
method can provide no insight into the individual discontinuities in the normal stress components across a given
interface, for example. The problem of van der Waals energies of cylindrical and spherical films was initially undertaken using this method in Parsegian and Weiss,14 but we
believe these results are in error and seek to present the
corrected version.
The purpose of this paper is to derive the energies
associated with single layer systems of dielectric media due
to van der Waals interactions, or dispersion forces, and to
provide a thorough analysis of these results. The single
layers (denoted by SL) separate two distinct dielectric media in three geometrical configurations: the plane parallel,
the cylindrical, and the spherical geometries. The single
layers themselves will be composed of three distinct dielectric media. The calculations will be made initially at zero
temperature, but this can be fully generalized to finite temperatures. The expressions for the van der Waals energies
for the three geometries are derived in Sec. II, where numerous notations and definitions are set out. IneqUalities
for the general energies are developed in Sec. III, which are
related to sets of simpler dielectric model configurations.
The large radius expansions are discussed in Sec. IV, and
first-order corrections in the radius of curvature of the
nonplanar geometries are found. In addition, it is shown
that the nonplanar results map smoothly into the planar in
the limit. Small radius expansions are investigated in Sec.
V, which provide some simple expressions for the single
layer energies. Because the appropriately scaled energies
depend only on the ratio of the inner radius to the width of
the intermediate layer the above two limiting behaviors
covers all the possible cases. The two-body summation approximation (TBSA) of the van der Waals energies, ,arising from just the pairwise interactions, is investigated in
Sec. VI. This will lead to useful analytical results for effec-
J. Chern. Phys. 99 (10). 15 November 1993 0021-9606/93/99(10)/8168/15/$6.00 @ 1993 American Institute of Physics
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N. S. Witte: van der Waals energies
tive Hamaker constants that are valid for all sizes and
thicknesses of the media components.
II. DISPERSION ENERGIES FOR THE THREE
GEOMETRIES
8169
Medium
2
.
_ _---.J'--_---'-_ _..... X
The macroscopic theory of dispersion forces for condensed media has a solid foundation that is adequately
discussed elsewhere, and it is only necessary to point out
the two essential limitations to the theory presented here.
One is the fact that all the systems predict divergent behavior as the inner layer thickness, w ..... o, which is due to
the neglect of the spatial dispersion of the atomic and molecular polarizabilities. At the same time, however, large
thicknesses and radii greater than approximately the order
of 500 ..... 1000 A are beyond the validity of the calculations
made here, and require the incorporation of full retardation effects. This upper limit derives from the wavelength
of the highest-frequency peak of the integrand appearing in
Eq. (2.19), etc., and usually occurs in the ultraviolet. Numerical calculations4 for the retarded PSL and Plane Triple Layer systems clearly reveal the limitations of the nonretarded approximations. In presenting the nonretarded
theory here we are not implying that the inclusion of full
retardation is insignificant, or unnecessary, but that it
should be seen as the next step following these investigations and that because of the complexity of the expressions
involved a step by step approach needs to undertaken. Furthermore, as will be seen later, we believe that many of the
results presented here will survive the inclusion of retardation effects, and the appearance of these new results in a
simpler model serves to promote understanding of their
origin.
z
~
r::::
MedIum
1
3
r,-->
I<-w---"
.
-----~
,-. --- --.
--.-~--
.
."
z
Medium
The three geometries, the plane, the cylindrical, and
the spherical, and coordinates of the interfaces are defined
in Figs. l(a)-l(c).
Following van Kampen,2,9 one seeks solutions for the
surface electrostatic modes, with no free charges or currents, i.e., in an isotropic homogeneous dielectric media
with no spatial dispersion characterized by a single dielectric function, which depends only on the frequency and, of
course, thermodynamic parameters,
VXE=O.
2
"2
A. Mode-summation derivation
V· D=O,
3
(2.1 )
The electrostatic potential <f; introduced as a solution to
Laplace's equation is subject to the continuity conditions
across interfaces j and j + 1,
<f;/Xj) =<f;Hl(X),
(2.2a)
E'JIl' V<f;/Xj) =€Hln. V<f;HI (x).
(2.2b)
One then decomposes the most general solution into the
individual modes appropriate to the geometry. Treating the
energy contribution from such individual modes separately
one sums these at the end of the calculation.
One conventionally defines a transfer matrix, M, which
relates the set of coefficients, A and B, of the eigenfunction
3
FIG. 1. Plane single layer PSL system with coordinates XI' X2' and
distance w. Cylindrical single layer CSL system with radii rl' r2' and
distances rand w. Spherical single layer SSL system with coordinates rl'
r2' and distances rand w.
expansion for the electrostatic potential in one layer (layer
+ 1), such that
j) to those of a neighboring layer (layer j
(~) HI =Mjj+l(X)(~) /
(2.3)
Use will be made of the definitions of the dielectric parameters'
8t=I+€/Ej,
8ij=I-€/€j.
One has, for the plane parallel case,
J. Chern. Phys., Vol. 99, No. 10, 15 November 1993
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(2.4)
N. S. Witte: van der Waals energies
8170
1 (Oh+l
M jj +l(Xj)="2 0- ~ . +2kx·
jj+le
J
f) '
2kx
Oll+le0+
. jj+l
(2.10)
(2.5)
.
where k is the magnitude of the transverse wave number in
the x-y plane, i.e., ~=~+TS and ranges from zero to
infinity.
For the cylindrical case, one obtains
_! (Oh+l
+oll+l~m
+s;,-:-."...J
M j j+l(rj)-2
uJJ+l"-'jm
s;,
rll
Ujj+l-ull+l"-'jm
MC/SSL=M23(r2)M12(rl),
where the three cylindrical functions defined as
(2.12)
and·
gSSL (a> ) = ( 1- 21
0m=2P/m(Pj)I'",(pj),
cfm=2p jK m(pj)K'",(pj)'
have the arguments P= Iklr. The 1m and Km denote the
modified Bessel functions of the first and second kinds with
index m. The dash denotes a derivative with respect to the
argument. It is understood that the above transfer matrix
corresponds to the multipole of order m appearing in the
Bessel functions, and often this index will be suppressed for
the sake of simplicity. The one-dimensional eigenvalue k
ranges over the entire real line.
For the spherical geometry the transfer matrix for multipole 1'>0 is simply
(2.8)
The total transfer matrix is then formed from the product of all the interfacial transfer matrices. The 11 component of the total transfer matrix relates the coefficient of
the unbounded eigenfunction (say as r-O or x-- ~ (0) on
one side of the multilayer system to the coefficient of the
bounded eigenfunction on the other side of the system (in
this case as r-- 00 or x __ + (0). Naturally the former coefficient must vanish while the latter remains finite. Because the dielectric functions are taken to be functions of
frequency alone, this places a constraint on the frequency,
leading to electromagnetic surface mode frequencies given
by the dispersion relation g(a» = (M T ) 11=0. The multilayer energies, at zero temperature, are found in the semiclassical approximation by summing over the zero-point
energies of these modes.
B. Single layer energies
and leads to
(2.9)
(2.13)
The g( a», being meromorphic functions, enables one
to use the relation, which gives the sum of the zeros minus
the poles of g(a» on the right-half complex plane, oy
L ([Zeros
=-
1
21Ti
of g(a»] - [Poles of g(a»]}
f+iOO
-ioo
g'(a»
a> g(a» da>.
(2.14)
The summation over the allowed spectrum of frequency
modes is further replaced by an integral along the imaginary frequency axis, over 5= - ia>, as shown in Ninham
and Parsegian. 3
These SL energies are, however, formally infinite and
meaningless without suitable subtraction of a zero-energy
reference, and the appropriate, and only choice of the zero
occurs when the media 1 and 3 layers are separated by an
infinite distance. This then leads to the requirement on the
renormalized dispersion function G( a> ), which enters into
the expression for the contribution to the total energy from
a particular multipole that and limw~oo G(a» =1. For the
PSL geometry this prescription for the renormalized G(a»
is given simply by
GpSL lim gPSL
.
w-oo
1+.!l21.!l32e-2kw.
gPSL
(2.15)
Here the dielectric reflection coefficient, .!lij' is defiiled as
.!l .. = €i-€j
lj-
(2.16)
€i+€j
However, in the case of cylindrical multilayer systems
some care needs to be exercised. A satisfactory renormalized dispersion function is the ratio of the second term of
Eq. (2.12) to its first term, i.e.,
GcsL=l
For a single layer system with a composition of media
types 1, 2, 3, then one has for the planar case (PSL),
~ 1 °i2 ) ( 1- 21 ~ 1 023 )
+(~I~1 0i2f t 21+1) (;;:\ 023r2 21- 1).
(2.7)
~m= -pAl'",(p)Km(p) +lm(pj)K'",(p)],
MpSL=M23(X2)M12(Xl) ,
(0i'2+0i2Q) (8i3+023~) -oi2023cfcf
,
(2.6)
(2.11)
which leads to
gcsda» =
-oll+lcfm)
s;,+
while in the cylindrical (CSL) and spherical (SSL) cases,
one has
.!l32.!l21
cfcf
;;0;;0
(1 +.!l32"-'i)( 1+.!l21"-'i) .
(2.17)
One can easily show that this approachs unity as w tends to
infinity. The spherical single layer dispersion function can
be found using the method as in the plane case, and
is given by
J. Chern. Phys., Vol. 99, No. 10, 15 November 1993
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N. S. Witte: van der Waals energies
r
G -1
__
SSL- + ( r+w
)2/+1
41(1+1)il 32il 21
(21+1+il 32 )(21+1+il21 )'
(2.18)
The energy contribution per unit area, denoted by ~,
from all of the individual modes in each of the three geometries are summed or integrated over all the spectra of
eigenvalues appropriate to the geometry to give the total
interaction energy. For the planar case, one has
=-
fz
32ff2w2
f+oo
_ 00 ds g3 ( -
(2.19)
.:i 32.:i21 ),
where g3 (z) is the trilogarithm function defined in Appendix B. The results for the cylindrical and spherical layer
systems are
fz
~csL(r,w)=~
8171
m=-oa
f
+OO
-00
dk
f+oo
-00
dsln GCSL
(2.20)
(2.23)
for any appropriate function f
The result for the retarded stress tensor for a CSL
system can be found in Korotkikh and Nabutovskii, 8 although they present no result for the nonretarded limit,
while a few other specific limiting cases are investigated.
Because of a typographical error (?) for the expression
8~~8 in their Eq. (10), there is some ambiguity surrounding
their final result in Eq. (15). Equation (15) of their work
sets out the form for ilia' (RI,d) and il 2a' (R 2 ,d), which is
of direct relevance to this work. The full retarded stress
tensor for a SSL system was found in Refs. 6 and 7, and
their limiting case of rl' '2 ~1L0 yields an expression for the
nonretarded ilaLz (w). Both of these quantities are the contributions to the jump discontinuity in the normal stress
component across the interf~ces (1 ..... 2, 2 ..... 3 respectively),
due to interactions across the sandwich layer, medium 2.
As is evident from their Eqs. (22) and (23) (apart from a
typographical error) this is identical to that formed from
the expressions for the total energy 41Tri~ SSL(rl,r2) stated
here, via the relations
and
(2.24)
fz
00
~ssdr,w)=~ L_ (21+1)
lorr 1-0
f+oo dslnG
-00
ssL ·
(2.21 )
The energies of single and multilayer systems are often
expressed in terms of the effective Hamaker constant, A =
-121Tw2~, as well. The energy used here is an energy per
unit area of the inner interface.
At this point an initial check of the results for the
cylindrical renormalized dispersion function can be made
in the limit as the inner radius tends to infinity, while the
layer thickness w and the multi pole order m remains finite.
This corresponds to using the asymptotic expansions of the
cylinder functions for large argument, as given in Eq.
(Al). One finds that only the product of CI at the inner
radius and of cK at the outer radius survive to give
GCSL ..... GpSL ' The wave number eigenvalue k in the cylindrical geometry naturally maps over to the restricted wave
number magnitUde in the planar geometry. Also, in the
spherical case a direct check can be made in the same limit
as above with one difference. As well as letting the inner
radius r=rl tend to infinity, one must let the angular momentum number I also tend to infinity, such that x=l/r is
constant, and it is this later quantity that corresponds to
the planar wave number. Using the limiting definition of
the exponential function,
(2.22)
one finds GSSL ..... GpSL after the identification of x with k.
All the above results can be easily generalized to finite
temperatures by the simple replacement of the frequency
integration with the summation over the Matsubara frequencies, Sn=21TkBTn/fz,
For the cylindrical case the corresponding connection with
the total van der Waals energy, expressed in terms of
WCSdrl,r2), would be
a
-ari (21Trl ~ csd/21Tri=ilaj (Ri,d),
i= 1,2.
(2.25)
Also, in Parsegian and Weiss,14 one finds an identical
expression for the energy per unit area for the SSL. Although this work treats the large radius limit of the CSL
energy no general form for the energy is derived or used.
III. EXTREMAL DIELECTRIC MODELS
In their study, Parsegian and Weiss 14 briefly investigate a simple model case for the cylindrical single layer,
without approximation in the relative radii of layers, or
making any expansion concerning the differences in the
dielectric media. In this section this idea is extended and
systematized in order to provide strong limits on the behavior of an arbitrary cylindrical or spherical single layer
system. As noted earlier, the dielectric difference functions
ilij' or physically the "transmission coefficients" are always real and greater than equal to - I and less than or
equal to + 1 for all imaginary frequencies is. In what fol-
TABLE I. Dielectric configurations for the extremal single layer systems.
£1
00
1,00
0,1
°
£2
£3
0,1
1,00
I
1,00
0,1
00
°
- P=A 21
-1
-I
+1
+1
q=A32
Realization
-1
+1
-1
+1
£1>£2>£3
J. Chern. Phys., Vol. 99, No. 10, 15 November 1993
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£1' £3>£2
£2>£1'£3
£3>£2>£1
N. S. Witte: van der Waals energies
8172
lows we take a single-frequency term from the finite temperature Matsubara sum and investigate all possible "maximal" or "minimal" extreme cases of the dielectric
behavior, where Aij= ± 1. For the single layer one has the
following cases enumerated in Table I.
An example of an arrangement of dielectric layers
where such a case may arise from the above is when only
the zero Matsubara frequency contributes as
4,0
CSL
3,0
2.0
::;. 1.0
. \ ..........
L
.....
F
...
0
...., ....
\\
........
~\
I-
........
.......
\,
~-1.0
\
-\
.Q
\,
-2.0
"',T-'-
......
\,
"\
-3,0
A. Cylindrical layers
For the above four cases, one finds that the free energy
per unit area contribution of a single term is simply expressible in terms of one function,
2
Tp,q(O)
+",
=~;. m=~oo
f+'" du
»)
T+'+\..
0
0,5 1.0 1.5 2.0 2.5 3.0
loglC( w/r )
SSL
(3.1)
I<j!)(uO-1)K<;J(u) ,
2,0
where the superscripts a=(1+p)/2 and (3=(1-q)/2 of
the modified Bessel functions show the order of the derivative with respect to the argument, and O=r1/r2' The T
functions defined here are related to a single term of the
Matsubara sum for the energy per unit area If a by
I
------Tp,q
Ifa
k B T-161Tr
..........
4,0
_",
1<;:) (u)K~) (UO- 1
Xln ( I
-2.0 -1.5 -1.0 -0.5
.........
T-'+
'\. T+'-
-4.0
(a)
......
\
-
N
1.0
L
.....
o
]-10
.2
.
-2,0
All four of these functions are displayed in Fig. 2(a) over
the full range of their arguments. The two mixed forms are
positive with T- 1, + 1> T+ I, -I > 0, while the remaining
two are negative, 0> T+ 1,+ I> T-1,-I.
From the general properties of the cfJ, cT, and cK
functions outlined in Appendix A, and of the Aij constants,
it is possible to establish that
1<GCSL(rl ,r2.i A,A')
cfcf
=G- 1,-I(r"r2),
in the case where AA' > 0. For the remaining case of
AA' <0, one has to consider the two further subcases of
this, namely, A32 > 0, A21 < and A32 < 0, A21 > 0, because
of a fundamental assymetry between them. In these two it
is straightforward to show that
°
A2IAdl-C?) (1 +~) + (1+A 21 C?) (1 +A32~»0,
(3.2a)
(3.2b)
-3,0
-4,0
-2.0 -1.5 -1,0 -0.5
(b)
0
0,5
1.0
1.5
2.0
loglO( w/r )
FIG. 2. (a) Exact cylindrical single layer energies for extremal model
dielectrics labeled (+ 1,+ 1), (+ 1,-1), (-1,+ 1), and (-1,-1) vs
w/r. (b) Exact spherical single layer energies for extremal model dielectrics labeled (+1,+1), (+1,-1), and (-1,-1) vs w/r.
In this way the consequent inequalities hold,
1>Gcsdr"r2i A21 < 0,A32 > 0) >G- 1,+I(rl,r2),
1>GcsL (rl,r2i A21 > O,A32 <0) >G+ 1,-I(rl,r2)'
Thus, in summary, it is clear that anyone term of the
Matsubara sum has a lower bound of T-1,-1 when
A32A21 > 0, while each term with A21 > 0, A32 < has an
upper bound of T+ 1, - " and that with A21 <0, A32 >0 has
an upper bound of T- 1,+ I. If each term in the Matsubara
sum satisfies one of these bounds, then the complete sum
must. For the completely general set of media dielectric
dispersion different cases may arise at various frequencies
so that the above bounds may not be of great use numer-
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°
N. S. Witte: van der Waals energies
ically, but in some sense an average of the bounds may
apply. But certainly they will give useful bounds if not
rigorous ones.
B. Spherical layer
In an entirely analogous way model or extremal forms
can be derived for the spherical layer case also. For the
four cases above, one finds the simple models to be
00
TM(e)=
L
1=1
( 2 1 + I-~(p+q) 2/ 1)
(2/+I)ln I+pq
e +
2/+ I +Hp+q)
Clearly one has T- I .- I > T+ I .+ I >0 and 0> T- I .+ I
= T+ I.-I. In this case the T functions defined here are
related to a single term of the Matsubara sum for the energy per unit area 'if! 0 by
'if! 0
I
-----TM
.
Note that there is a difference of a factor of 2 with the CSL
definition. A real difference in this case with the cylindrical
one is that one has a symmetry in the two mixed cases
being identical. All three cases are plotted in Fig. 2(b).
The general G SSL will also satisfy inequalities similarly.
From the observation that
0< (2/+2)-2
«2/+ l+a32 )-1(2/+ l+a21)-1
«2/) -2,
and if il32a21 > 0, then
1<GSSL(rl,r2Iil32il21 > 0) <G- 1.- 1(rl ,r2).
Noting that
(2/+ 1+il) (2/+ 1+il') +4/(1+ l)ilil';;;.O,
when ilil' <0, one can directly show that
G-l.+l(rl,r2)<Gssdrl>r21
1
and spherical cases due to a dependence of the form w- 2 •
As in the case of the plane layer systems, this divergence is
a direct consequence of the neglect of nonlocal contributions arising from the individual atoms or molecules in the
media. The spatial dispersion occurring in real systems introduces a length scale, of atomic dimensions, that this
continuum model lacks, and the interaction energy would
then have a finite limit.
•
(3.3 )
k BT-S1T?
8173
a 32a 21 <0) <1.
1
Thus, in summary, T- .+ is a lower bound for
T(il 32 ,a21 ) when il32il21 <0, whereas T- I .- I is an upper
bound for T(a 32 ,a21 ) when a 32a 21 > O.
A. Cylindrical layers
As the inner radius of cylindrical multilayer systems
approaches infinity, successive individual terms of the integrand In G in the muItipole sum differ little from each
other, and these merge together into a continuum with
only a weak dependence on the multipole order. Thus the
dominant contribution to the sum comes from a large
range of m values, including large absolute values of m,
and so the sum over m tends into an integral over a new
variable, Y'=.m/r, written in the limiting form as
+00
f+oo dy ;~~ F(yr,r,w), as- r-->
rI m~oo
F(m,r,w)
-+
-00
00,
( 4.1)
where F is an arbitrary function of m, r, and w. An expansion form for this plus first- and second-order correction
terms is given later in Eq. (4.3). The consequence of this is
that one must -find the asymptotic expressions for the integrands for large argument and large order, but carefully
retaining the subtle differences between factors evaluated at
slightly different radii. The appropriate expansions of the
modified Bessel functions, a variant of the Debye expansions, which are asymptotic expansions with respect to r or
m and uniform with respect to k, y, and d, can be found in
Eqs. (A2) of the Appendix A.
Considering the CSL case in detail one finds that the
substitution of these expansions equations (A3) leads to
the cancellation of prefactors in c! and cK, namely
1i1Te2m7] and 1Te- 2m7], and thus the intermediate result to
order O(r-I),
IV. LARGE RADIUS EXPANSION
In this section the relationship between the planar on
the one hand, and the cylindrical and spherical layer systems on the other, is explored more fully in order to provide a check on the energies presented here, and to indicate
how the latter two geometries map into the first in the
limits of large inner radius. The case of the sandwich layer
vanishing, w-->O, with finite values of r is identical to just
this case, as there are no other length scales in the problem
in which the effective Hamaker constant only depends on
r/w. This would only be true in the nonretarded limit. The
interesting fact that emerges in the limit as the inner layer
vanishes is that the energy diverges for both the cylindricAl
( I + il 21
C?) (1 + a32i1)
= _e- 2w (k2+I) 112
(4.2)
Continuing along this direction the next step is to take out
the entire zeroth order or r= 00 factor and expand the
logarithm of the remaining factor in an asymptotic series
up to and including second order in r-I. The other part of
the asymptotic development in large radius involves the
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N. S. Witte: van der Waals energies
8174
use of the Euler-Maclaurin summation formula to perform
the summation over m, also taken up to second order in
1/r,
CSL
3.0
r-
1
mX", f(m,r,w) =2 fo'" dy f(yr,r,w)
·2.0
-~ r- %"Iy=o +... .
2
(4.3)
One combines these two asymptotic expansions, again up
to second order. By transforming from Cartesian coordinates (y,k) to polar coordinates (e,q), one can analytically
perform the angular and radial integrations and arrive at
the final result,
fz
'iff CSL= 'iff PSL - 64-n2rw
~
"~
l-
e
-1.0
-2.0
(4.4)
Large radius expansions for the CSL have been reported in Parsegian and Weiss 14 and their result, which can
be compared to our first-order correction, differs from
ours. We believe their result is incorrect for two reasons.
First, even assuming their method to be correct there is an
omission of a factor of r (their definition not ours) from
the second term in the brackets on the right-hand side of
their Eq. (21). This error propagates through their analysis thereafter. However, even taking this omission intoaccount the result would still be incorrect. The essence of the
error lies, in that they have only used an exponential approximation to the modified Bessel functions, which is
completely incapable of giving rise to an expansion in w/r.
Thus the method is satisfactory to reproduce the zerothorder or plane parallel limit, but not for any higher orders.
In fact, the exponential form they have taken is quite inappropriate as the basis for the development of higherorder corrections.
A quantitative investigation of the large radius expansion can be most simply made with the extremal model
functions of Eq. (3.1). In comparison, the approximate
expression fOUIid from our large radius expansion, Eq.
( 4.4), yields the following:
-
-
0
OJ
.Q
f+'"
_'" dS[g3( -Ll32Ll21)
- (Ll32 +Ll21)gl ( -Ll 32 Ll21)] +0(r- 2 ).
1.0
T-l'-I=_~ t(3) (~r (~t(3) +In 2) (~) +0(1),
-2.0 -1.5 -1.0 -0.5
(a)
0
I0910(
0.5:
w/r }
1.0
1:5
2:0
SSL
2.0
n
0
~
"+
,
1 sl order
I--
'0
OJ-2.0
.Q
Olh order
c
c
c
-4.0
c
c
-6.0";:;'::!-'-'-';-';:-'-U:-7'-'-!::"=-,-~:-'--'-'c..::l:""""~.J...U~'-'-'-l.d
-2.0 -1.5 -1.0 -0.5 . '0'
0.5
1.0 1.5 2.0
(b)
FIG. 3. (a) Exact, zeroth- and first-order approximations to the cylindrical single layer energies for the_extremal model dielectric labeled
( - 1, + 1) vs w/r. (b) Exact, zeroth- and first-order approximations to
the spherical single layer energies for the extremal model dielectric labeled (-1,+ 1) vs w/r.
(4.5a)
T-1,+I=t(3)
(~) +~ t(3) (~) +0(1),
(4.Sb)
1
T+ ,-I=t(3)
(~r +~ t(3) (~) +0(1),
(4.Sc)
2
T+l'+I=_~ t(3) (~r + ( -~ t(3) +In 2) (~) +0(1).
(4.5d)
In Fig. 3(a) we have plotted the exact value of T- 1,+1,
as given by Eq. (3.1), as well as the zeroth-, the first-, and
the second-order scaled corrections, as given by Eq.
( 4.Sb), successively applied. This is the same function
computed by Parsegian and Weiss, 14 and the qualitative
shape of our exact result appears to be the same as theirs,
although there is a In 2 shift and a different log base along
the y axis between the two results. Clearly our first order is
a significant improvement over the zeroth order. It should
be noted that if the Parsegian and Weiss 14 result were correct then the first-order correction would vanish, and
clearly this is Imt the case (in fact, this conclusion is just as
evident from their graph). Also, in dielectric models where
medium I is identical to medium 3 so that Ll32 + Ll21 =0,
then first-order corrections still remain, so that the planar
approximation is not enhanced in this dielectric arrangement. A further observation on the merit of the inclusion
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8175
N. S. Witte: van der Waals energies
of first-order terms can be clearly seen in Fig. 2(a). Here
the T- 1.- 1 and T+ 1.+ 1 differ at the first order and their
two curves separate early, at low values of w/r, whereas the
T+ 1.- 1 and T- 1.+ 1 differ only at the second order and
their two curves separate much later, at higher values of
w/r.
The various approximations are seen, in this case, to
bracket the exact result, with the zeroth order underestimating the magnitude of the exact result and the first-order
approximation overestimating it. The range and usefulness
of the large radius approximation is also demonstrated by
Fig. 3(a). At the 10% error level the zeroth-order approximation fails when 10glO(w/r) - -0.68 or when
w/r~0.21, whereas the first-order approximation fails
when 10glO(w/r) - +0.45 or when w/r- 2.8.
B. Spherical layer
A similar limiting process operates in the case of the
SSL as it does in the CSL for large inner radius. In this case
the I summation goes over to an x integration, where
x==.l/r and both 1 and r tend to infinity, while x and w
remain finite,
?1 Lco
o
(21+ l)F(l,r,w) ->2
lco dx x lim F(xr,r,w),
r~oo
0
as r->
(4.6)
00.
Retaining only the highest-order terms in I, which ensures
cancellation between the numerator and denominator in
Eq. (2.18), and using the limiting definition of the exponential function, one can simply show the planar case is
recovered,
lim ~ssL(r,p,w)=Wpsdp,w).
Finally, making use of the integral representations of the
polylogarithms (see Appendix B), one arrives at
fz
WSSL= WPSL - 32-n2rw
f+oo
_ dS[g3 ( 00
Ll32Ll 21)
- (Ll 32 +Ll21 )gl (-Ll32Ll 21)] +0(r- 2 ).
(4.10)
In respect of the result found in Parsegian and Weiss, 14
their w- 2 and w- I terms in Eq. (41) both agree with the
zeroth- and first-order results arrived at in this work. However, their next-order term, with the In(w) dependence, is
incorrect as it stands. A sign error first appears in the
[b/(b+ l)f term of the preceding equation (40), and this
mistake propagates into Eq. (41) uncorrected. Furthermore, the logarithmic term alone is not a complete secondorder correction, as there are O( 1) or constant terms with
respect to w not included. These omitted terms, in general,
would also be of comparable magnitude numerically to any
logarithmic term. The approach taken by them to derive
Eq. (40) is not useful in finding such 0(1) terms because
all orders in the 1 expansion [in Eq. (39)] beyond third
order are required to give them.
Just as in the CSL case the large radius forms of the
Extremal SSL models have the following indicative forms,
where the expressions are obtained from the asymptotic
expansions given in Appendix C, in terms of an ascending
series in w/r,
T-I.-l=~ s(3) (~) + (~S(3) +1n 2 ) (~)
2
(4.lla)
(4.7)
In finding the r- I corrections to the spherical layer
systems, a similar procedure to that in the cylindrical case
is used. In applying the Euler-Maclaurin summation formula to the sum
(4.11b)
00
r-
2
L
(21+1)1(l,r,w)
1=0
= Jooo dx(2x+r- 1 )/(xr,r,w) +~ r- 2/(0,r,w)
1
2
-1 2 r- ( 2/(0,r,w) +
~II=J +0(r-
3
),
(4.11c)
(4.8)
and expanding the integrands, one finds that the second,
third and higher terms are all of order r- 2 • Thus,' it is only
necessary to expand the integrand of the first term up to
order r-I. In this the following expansion for large n is
used:
(4.9)
where the C, to, and t+ are pure numbers, like the coefficients, and expressions as well as values are given for
these by Eqs. (Cl). In Fig. 3(b) the exact expression for
T- 1.+ 1 as given by Eq. (3.3) and the zeroth-order, firstorder approximants to this, as found in Eq. (4.11b), are
plotted. Again it is found that that zeroth order is sufficient
to the 10% level up to w/r-0.40, and the first order is
sufficient up to w/r~0.33. While at first sight it appears
that the first order is worse than the zeroth order, this is
not a complete description of their relative merits. In fact,
for values of w/r;$ 0.20 the first order is closer to the exact
value than the zeroth order, and increasing so as w de-
creases below this.
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N. S. Witte: van der Waals energies
8176
In the conclusion of this section one finds that the
cylindrical and spherical layer systems map continuously
into the plane layer case, and one can find r- 1 corrections
(and higher orders) to the later case. This correction for
the spherical case is twice that of the cylindrical, which can
be understood on the basis that the sum of the inverses of
the principal radii of curvature for the sphere is 2/r and
that of the cylinder 1/r. This would lead one to conjecture
that the r- 1-order corrections for the local van der Waals
energy per unit area of a smooth, but otherwise arbitrary
curved surface layer with principal radii of curvature Rl
and R2 is
.
X
(~(z)
cf(z»)
(1+a21 ) [1+a 32ig(z)] + [1+a 32Cl(z)]
+O(?),
(5.1)
and this vanishes in the limit as the inner cylinder tends to
zero. This is quite obvious, given that one remains with a
single interface separating two media in the limit, and all
such systems have a reference energy of zero.
The small radius expansions of the Extremal CSL
models take the following forms:
(5.2a)
(4.12)
(5.2b)
In general, the next-order corrections involve logarithmic
dependence on wand terms involving the dielectric properties alone, for both the cylindrical and spherical cases.
Thus, one finds the presence of nonalgebraic forms as well
as terms possibly involving squares of the mean curvature,
and the Gaussian curvature. It is significant that the scaled
total free energy w2 '11 c/ssdr,w) is riot an analytic function
about the planar limit w=O. However from just two specific cases it does not seem possible to extract the universal
form, at this order at least.
r+ 1,-I=!e3 +0(lf),
(5.2c)
r+ 1,+I= -W3 +0(lf).
(5.2d)
B. Spherical systems
For the case when the inner radius is much smaller
than the thickness of the SSL one need retain only the
leading term of the ascending expansion of the logarithm in
Eq. (2.18) and Eq. (2.21). Furthermore, multipole contributions higher than the 1= 1 term are now much smaller
than the 1= 1 dipole term (the 1=0 always vanishes), leading to
V. SMALL RADIUS LIMITS
In this section the limiting case where the inner radius
of the cylindrical and spherical layer systems vanishes is
investigated for a number of reasons. First, it furnishes a
check of the general results of this paper against those on
solid rods and spheres, from the work of Langbein. 11 Second, it provides some intermediate results, which will be of
use in understanding the interrelationships of the general
results. It will have been noticed that the interaction energies per unit surface area were defined with the inner surface rather than some other surface, such as the outer one.
This can reinforce some tendency to singular behavior as
r->O that is not otherwise present in the total energy.
Thus the energy per unit area possesses a finite limit,
namely zero, which is the correct result for a single dielectric sphere immersed in another dielectric medium.
The Extremal models for the SSL take the forms
r-1,+I=r+1,-1
(5.4b)
A. Cylindrical systems
To implement this expansion one requires the ascending expansions for the cylinder functions CJ and d, which
are given in Eqs. (A5a) and (A5b). In the CSL case the
renormalized dispersion function has an expansion of the
form GcsL =1+a?+0(r4 ), so that the logarithm of this
has a lowest-order term of ? Furthermore, the leadingorder term of C~(z) for multipoles higher than 2 is of
higher order than ~ and one can neglect these, leaving only
the m= -1,0, + 1 multipoles. The leading-order term for
the energy per unit area is
(5.4c)
VI. TWO-BODY SUMMATION APPROXIMATION
In this section an approximation will be explored that
allows the separation of geometrical effects from all others
in a nontrivial way, in contrast to the limits discussed in
earlier sections. It is often found that the general trends
exhibited in this approximation are close to the exact re-
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N. S. Witte: van der Waals energies
(6.1 )
TABLE II. Location of the maxima in the geometry factors, S(ra,rb)'
and its values at these points for CSL and SSL systems with x=r/w.
Xo=
So=
Cylindrical
Spherical
2.560 886303 1959
1.0667163992208
1.843 893 1904266
1.158 10749221689
8177
In the following material a dimensionless ratio defined as
x=r/w will prove to be useful. A convenient notation for
the Hamaker coefficients will be adopted in the form
(6.2)
suIts. This limit is found from the expansion of the normalized dispersion functions G in powers of aij up to quadratic order and the logarithm as well. This yields a clean
and simple separation of the energies into factors with the
frequency dependence, which defines a Hamaker factor,
and factors with geometry and size dependence. The results presented here will be in terms of an effective Hamaker constant, A, as well as the energy per unit area, for
two reasons. It first allows for easy comparison between
results of the different geometries, and second because it is
a scale invariant quantity-scaling all the sizes by a constant factor, does not change the effective Hamaker constant. This is defined as
with ra> rb to ensure convergence, and this result is proved
in Appendix A.
Referring to the general expression for the S factor of
the CSL in Eq. (6.5), it is clear that it is zero when x=O,
but it also rises to a single maximum and then slowly declines to a value of unity as x tends to infinity. The partie-
A. Cylindrical layers
With the CSL, one has
(6.3)
so that the effective Hamaker constant becomes
(6.4)
where the geometrical-size factor SC(ra,rb) is defined as
ular values of Xo and S~ at this point are displayed in Table
II, and the variation of SC with x is displayed in Fig. 4.
B. Spherical layers
As in the case of the CSL, this limit of the SSL is also
found from the expansion of G, as given in Eq. (2.18) in
powers of aij up to quadratic order. This yields, after expansion, to
1.20
;
,/'" ......
1.00
-------------
.'SSL
--.;.-.;.--;;,.--"---~-----_;;,.-'-
___ ! . J__ ... _
(6.6)
CSL
and the relatively simple separation of the energies into
factors with the frequency dependence, which defines a
Hamaker constant, and a factor with the geometry and size
dependence,
0.80
-0.
t..
;'0.60
t..
Ci'i
0.40
0.20
(6.7)
0
0
2.0
4.0
6.0
8.0
10.0
The summations over I can be carried out exactly, and one
.. --finally obtains
r/w
(6.8)
FIG. 4. Cylindrical and spherical single layer geometry factors SC(rZ,rl)
and SS(rZ,rl) within the two-body summation approximation versus
x=r!w.
where the S factor is now
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N. S. Witte: van der Waals energies
8178
(6.9)
The spherical geometry exhibits a strong effect, as can
be seen in Fig. 4, where SS(x) rises from zero as x increases from zero, until a maximum and then declines
slowly to asymptote at SS equal to unity as x -> 00. The
numerical values locating this maximum are also given in
Table II.
These two geometrical factors can be derived directly
from the Hamaker theory of pairwise summation applied
to concentric cylinders or spheres. The vdW potential energy of interaction at a point r in medium 3 (I r I > r2)
across medium 2 due to the r- 6 law integrated over the
inner cylinder or sphere of medium 1 is denoted by
Ep( Ir I). Then the total energy of interaction is this potential energy integrated over the outer cylinder/sphere of
medium 3, and the energy per unit hiller surface area yields
the same geometrical factor. The existence of a maxiIna in
this S factor can be understood in terms of two competing
factors within such a double integration over the two volumes. The first factor, the integral E p ' evaluated at the
outer radius of the layer, r2, is a monotonically increasing
function of the inner radius, rising rapidly from zero, and
plateaus out to a constant value as the system approaches
the planar geometry. The other factor is given approximately by the surface area of the outer radius divided by
the surface area of the inner radius. Because of the extremely rapid falloff in the vdW force law with separation
of the particles the major contribution to the total interaction energy comes from a thin shell about the inner and
outer radii, i.e., is given by the product of the above factors. This second factor is a monotonically decreasing
function of the inner radius, which also plateaus out to a
finite value, unity. Thus their product rises from zero to a
maxima and then decays slowly to a finite value. The difference between the cylindrical and spherical parameters
given in Table II can also be understood from the above
discussion. Because the cylindrical case is intermediate in
curvature between the planar and spherical cases, its maxima occurs at a greater x value, i.e., at a larger radius
relative to thickness. Also, its maximum value is slightly
suppressed in relation to that of the spherical system.
for the theory to become quantitatively predictive for a
variety of nontrivial applications. In the general case it has
been shown that the energies are bounded by various simple expressions that arise from the "extremal" dielectric
models, whose dependency on the ratio of the film thickness to inner radius was extensively explored. Furthermore, a number of useful analytical expressions have been
found for the limiting cases of large radii, of small radii,
and in the pairwise summation approximation.
As an indicator to the energy for smooth thin films of
arbitrary curvature and shape we conjecture a result that
the total energy is siInply the integral of a local energy per
unit area summed over the complete surface. Part of the
conjecture is that the energy per unit surface area of the
film include the planar contribution plus a term that is
proportional to the mean curvature with a dielectric coefficient akin to the Hamaker constant,
W(1
1)
1
1
1f=-121TW2Ao -121Tw2 2 R I +R 2 AI'
where the planar and first-order Hamaker constants are
- (.6.32 +.6.21 )gl ( - .6.32 .6.21 ) ].
The limitations of such expressions was explored for some
of the extremal dielectric models, and it was clear that they
are aCcurate for large values of r;(; ~w and r;(; 3w for the
cylindrical and spherical cases, respectively.
It was found that the pairwise summation approximation leads, naturally enough to a factorization of the energy
into -the Hamaker constant and a geometrical structure
factor, which has a compact analytic form in both cases. In
both cases this structure factor rises quickly to a maximum
and then tends slowly to unity as r/w increases from zero.
This structure factor is not simply relatable to the other
geometric functions examined here, for example, the extremal models, because the two sets are valid in complementary regions-the first when dielectric differences are
small and the latter when they assume an extreme difference. However, it is clear that one can potentially bracket
all behavior of a real system by considering both of these.
ACKNOWLEDGMENT
VII. CONCLUSION
In summary this work has extended the semiclassical
theory of van der Waals interactions amongst condensed
dielectric media in the direction of nonplanar singlelayer
geometries, namely cylindrical and spherical films. While
some of the exact and general expressions f01" the energies
have stood in the literature for some time now there were
some erroneous results, which have been reexamined here.
There have also been gaps in the full understanding of the
general properties of cylindrical and spherical films, which
this work has addressed. This work is necessary in order
The author acknowledges the Australian Research
Council Fellowship in support of this work.
APPENDIX A: BESSEL IDENTITIES AND EXPANSIONS
The product of Bessel functions defined in Eq. (2.7)
have the following general properties:
(1) c;.,(p) >0,
(2) C~(p)<O,
(3) O<~(p)<l, for all m and p.
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N. S. Witte: van der Waals energies
The Bessel function identities that have been used in
the body of the paper can be found in Refs. 16 and 17, or
simply derived from relations given therein. The asymptotic expansions for large arguments lead to the following
expressions for the products of Bessel functions of order m
appearing in the cylinder functions defined by Eq. (2.7),
8179
"..,[(
_
-2Z(
m+1 (m-1)(m+3)
l.-;;'(z) - -1Te
1 + 4z +
32~
(m-I)(m 2+2m-27)
-4 )
+
384;
. +O(z ) .
(A1c)
1. Uniform asymptotic expansions
3(m-1)
5
16; +O(z-),
1
2z
"..J
_2. 2z( _m+1
l.-;,.(z) - 1T e
1
4z
+
(Ala)
(m-1)(m+3)
32~
(m-1)(m2+2m-27)
4 )
384;
+ O(z-) ,
(Alb)
The Debye asymptotic expressions for large orders and
arguments, Eqs. (9.7.7)-(9.7.11) on p. 378 of Ref. 18 have
been used as a starting point, and these treat Bessel functions of the form ev ( vz) for large v and arbitrary z. The
essence here is that they are asymptotic expansions in v,
and uniform with respect to z. In the form that they are
required here it is necessary to derive expansions of Bessel
functions with index v and argument vx+d, i.e., for large
v but x*O and arbitrary d. This is achieved by substituting
z=x+d/v in the above-mentioned formulas and expanding the expressions for large v and then combining these
with the original asymptotic expansion. The operations described here lead to the following asymptotic expansions in
v, which are now uniform with respect to x and d,
(A2a)
(A2b)
(A2c)
(A2d)
where t=(1+x2 )-1I2 and 'l]=r l +ln[xt/(1+t)]. In this
way the various cylinder functions appearing in this work
have the following asymptotic expansions in r or m and are
uniform with respect to k, y, and I1rj'
k?C}m=2r (k?-+y2)-3/2+0(r- 2),
j=1,2,
(A3a)
2. Ascending expansions in argument
The small argument expansions can be found from angular integral definitions, to be found in Ref. 16,
2
G%(z) =- (_)m
1T
i1T/2 de cos(2me)2z cos eK (2z cos e),
0
(A4a)
x [3k?-+2r+ 12y2(k2+y2)l1r/l +O(r- 2) ),
j=l,
dfm= _1Te(-2mT/-2d/xt) X ( 1+ l~r (k?-+y2)-3/2
(A3b)
2
cfn(z) =- (_)m
. 1T
i1T/2 de cos(2me)2z sin ell (2z sin e),
0
(A4b)
X [3k?-+2y2 + 12y2(k?-+y2)l1r/l +O(r- 2) ),
j=2,
I
(A3c)
and substituting the ascending expansions for the modified
Bessel functions and integrating these term by term one
arrives at
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8180
N. S. Witte: van der Waals energies
m-
2
~(z)=SmO+ k~O
(1
(-2'Z)2k+2(2k+2)!
)2k+2 (2k+2)!(m-k-2)!
00
'2z
k!(k+l)!(k+l+m)!+2(-)m k=~-1 k!(k+l)!(k+l-m)!(k+l+m)!
X [In z-~t/J(k+ 1) +~t/J(k+~) -~t/J(k+2-m) -~t/J(k+2+m)],
(A5a)
q(z) =~IF2a;2,2;~),
(A5b)
3. TBSA cylindrical geometry factor
One requires the m summation and k integration found
in the cylindrical geometry factor in Eq. (6.5), that is
zi_ Y
+00
f+oo
L
Z1Z2
dk~
m=-=-oo
-00
-
The full equation, Eq. (A6), becomes the double infinite sum,
- -~ m=-oo
f
n=max{lml-l,O}
(ZI)2n
Z;
~--
rcn+m+~)rcn-m+~)r(n+~)rcn+~)
and this can be done exactly. As a first step one uses the
addition theorem found in Ref. 15 to express the product
of 1m and I:" in terms of an integral over an angle of a
single II Bessel function, and one then integrates this term
by term to give the convergent series:
1
=-4Iklzl
00
akz 1 )2n
L
n.'( n+ 1)'.
n=max{lml-l,O}
(2n+2)!
X (n+ I-m)!(n+ 1+m)! .
X =r--=-(n-+-m-+-2=-=-)-=r:-:(-n---m-+----::c
2 ):-::r=-(:-n-+---:2:-:-)=rC"":"(n-+---:-:-1) ,
(A9)
and this can be reordered into an infinite n sum and finite
msum,
-(A7)
n+l
X
The next step involves the substitution of this product into
the k integration of the original equation so that the k
integral becomes
L
m=-n-l
r(n+m+~)rcn-m+~)
rcn+m+2)r(n-m+2) .
(AW)
The finite m sum in the above equation is simply 'IT, and
using the standard power series definition for the Hypergeometric function the final result for Eq. (A6) is
= - (n+~)z2"2n-4 Loo du u2n+2K~(u)
(All)
=_22nZ2"2n-4 r ( n+m+~)r( n-m+~)
rcn+~)rcn+~)
X
r(~n+3)
(A8)
One requires the transformation formula for the hypergeometric function suitable for expansion about Z= 1, which
can be found in Eq. (15.3.12) on p. 560 of Ref. 15,
J. Chern. Phys., Vol. 99, No.1 0, 15 November 1993
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8181
N. S. Witte: van der Waals energies
(AI2)
and keeping only terms to orders less than linear ones in
the differences of distances, Ira-rbl, i.e.,
5 3
~)
2Fl ( 2:'2:;2;~
a
1[
+ 41T In
8
r!
2";
~31T (l.-rt)2+31Tl.-?.
a
b
a
b
(,.;-~)
19] +0 (,.;-~)
1"6;r +6"
----;;- .
(A13)
and it can be verified that the accuracy of the first- or
second-order terms of the expansion in this variable is
much more accurate over a wider range of values than if
one used wlr. For the mixed Extremal model, one finds
00
(Bl)
'"
=
n!
(2w)n+l gn+l (-A),
(B2)
o dx xn In (1 +Ae-
)
=
~
(2w)n+1 gn+2( -A).
Some of the commonly used polylogarithms are
gl (z) = -In( l-z),
1
-l:1 k
02k+I
Ok (02k_1)2
In(1-0)
1
-2
11
1
= -2: ;(3)tf; -12 In tf;+so+2: tf;
which is valid for Izl <1 if m=#=1 and Izl < 1 if m= 1, and
is satisfied for all the applications presented here. The general reference is Lewin.lO Some of the integral definitions of
polylogarithms encountered in this work are
2xw
(2/+1)(1_0 21+ 1)
1=1
The general integral-index polylogarithm gm(z) is defined by
J'"
L
r.-1.+1=
APPENDIX B: POLYLOGARITHMS AND RELATED
FUNCTIONS
~
+
2k+l
k
k~1 (2k+2)! ~ ;( 1-2k)
X [(I-2k+l)B2k+2+2k+2],
where again 0=rl/r2 and So is a number given below. The
infinite summation is not a convergent one, but asymptotic
in nature. However, it is quite accurate over a wide range
of values of tf; due to the initial rapid decrease of the coefficients.
For the other two models the results are significantly
more complicated, and only the leading-order terms will be
given, although exactly
z
go(z) =-1-'
-z
z
(l-z)2 .
In addition to the polylogarithms we adopt the standard definitions of the Riemann zeta function, denoted by
;(s) and the Bernoulli numbers Bn.
r+l,+I=
f (2/+
1) (1+_1_ 021+ 1)
1=1
APPENDIX C: SPHERICAL ASYMPTOTIC
EXPANSIONS
Here we give the general asymptotic expansions for the
three extremal SSL energies, but omit the technical proof.
They are obtained by a completely independent method
from that describedjn the text of this work, that of Mellin
Transform techniques. The interested reader is referred to
Ref. 15 for the details. For large inner radii, there is a more
natural expansion variable than wlr, and this emerges from
the analysis in the above reference. This is defined as
1+ 1
3
1
="8 ;(3)tf;-2-ln 2 tf;-I +S+ -4 ln tf;+0(tf;).
A numerical value for the O( 1) coefficient So is
SO=112
(-lni-r+~;'(2)+I)
= -0.049 896 613 6,
J. Chern. Phys., Vol. 99, No. 10, 15 November 1993
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(CIa)
N. S. Witte: van der Waals energies
8182
The other two numbers, s_ and s+, are related in terms of
known numbers plus two new constants, y + and y _, and
when combined with the other ones appearing in the above
one has
5 3
3
3
1
s_= -g-4 ln 2-2ln 2- 4 y-3t'(-I) -2y +
= -1.602 053 774,
(Clb)
1
1
C=+2 ln 2+ 32 t(3) +s+,
3 3
1
..
s+=g-4 ln 2- 4 y-3t'( -1) -2y_
=0.235823291 7,
(Clc)
1
I
t+=-2 1n 2+ 32 t(3) +s+.
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J. Chern. Phys., Vol. 99, No. 10, 15 November 1993
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