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Calculation of the van der Waals force between a spherical particle and an infinite cylinder
Advances in Colloid and Interface Science 104 (2003) 311–324
Calculation of the van der Waals force between a spherical particle and an infinite cylinder V.A. Kirsch Institute of Physical Chemistry of the Russian Academy of Sciences, Leninsky Prospekt, 31, Moscow 119991, Russia
Abstract Formulae for the van der Waals attraction energy and force between a spherical particle and an infinite cylinder are derived by the method of additive summation of the pair interactions described by the potential of the general form Umsyamrym. The formula of Rosenfeld and Wasan for the non-retarded vdW force between a sphere and a cylinder (ms 6) is confirmed and the compact expression for the retarded force (ms7) is obtained. The comparison is given for the forces of the retarded vdW interaction between a sphere and a cylinder, another sphere, a row of spheres and a half-space. Also, the compact formulae for the energy of the vdW interaction of a point-like particle (atom, molecule) with a sphere and a cylinder are derived for the case of arbitrary m. 䊚 2003 Elsevier Science B.V. All rights reserved. Keywords: Van der Waals interaction; Pair potential; Additive approach; Retardation effect
1. Introduction Many problems in physics, physical chemistry and biology deal with the van der Waals (vdW) interaction w1x of fine spherical particles with bodies of cylindrical shape, like nanowires, nanotubes, fibres. In accordance with the general theory of Lifshitz w2,3x the van der Waals forces appear between neutral bodies (atoms, molecules, macrobodies) as a result of the correlated interaction of their instantaneous electrical and magnetic dipole moments through the fluctuating electromagnetic field. The van der Waals forces include the Keesom and Debye forces which are related to the presence of the permanent dipole moments, and dispersion forces, E-mail address: [email protected] (V.A. Kirsch). 0001-8686/03/$ - see front matter 䊚 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0001-8686Ž03.00053-8
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acting between both polar and non-polar molecules. The role of the latter is the most significant for many systems, excluding only highly-polar molecules. The first direct experiments on the vdW forces between macroscopic bodies were performed by Derjaguin and his colleges w4x. A two-particle potential of the dispersion interaction of neutral atoms (molecules) in vacuum was found by Casimir and Polder in Ref. w5x. Of special interest are the asymptotes of this potential, obtained in the limit of small and large interparticle separations r relative to the characteristic wavelength of radiation in the spectra of interacting atoms, l0s2pcvy1 0 , where v0 is the atomic frequency, c is the light velocity in vacuum. At distances a
a6 , r6
(1)
where a is the constant of interaction. The Keesom and Debye interaction potentials obey the same dependence on the separation distance. At the distances r'l0 the dispersion interaction is no longer instantaneous and is determined by the finite time of propagation of the signal from one temporary dipole to another, 2rcy1, which results in the retardation of the vdW interaction. In the limit of r4l0 the London forces do not exist. In this case, the vdW interaction is fully retarded and is described by the following asymptotic potential w5x: U7sy
a7 r7
(2)
The retarded vdW forces sometimes are termed the Casimir forces, as they are related with the quantum Casimir effect w7x. The full expression for the Casimir and Polder potential, valid for all separations r4a, is given by a cumbersome integral w5x for which it is convenient to use a simple analytical approximation. Schenkel and Kitchener w8x suggested the following approximation: Usa6Žy2.45pyr7q2.17p2 yr8y0.59p3 yr9., for r)0.5p,
(3)
where psl0 y2p. The calculation of the vdW force between macrobodies within the framework of the general theory involves the solution of the boundary value problem for the Maxwell equations. It should be emphasised that the BVP for the Maxwell equations possesses an analytical solution only for the systems with a high symmetry, like a single sphere, infinite cylinder, symmetrical shell or void, two half-spaces separated by a gap w9x. In the case of the interaction between the sphere and cylinder the general theory of Lifshitz gives no analytical answer, since the variables in the Maxwell equations cannot be separated for the geometry considered. Therefore, in
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the present work an approximate additive approach w10x will be invoked, which is valid for a pair of uniform and isotropic bodies of arbitrary geometry. Following this approach, the vdW interaction energy between a spherical macroparticle and a cylinder is found by summation of the pair interactions of the constituent point particles (atoms or molecules):
| | U dV dV sn | w dV sn | w dV
ESC m snCnS
m
VC VS
C
S
S m
C
VC
C
C m
S
S
(4)
VS
where nk is the concentration of the point particles, dVk the volume element, wkm the energy of interaction of the point particle with the macrobody, indices c and s denote the cylinder and sphere, correspondingly. Here Um is the two-particle attractive potential: Umsy
am , m)0. rm
(5)
The van der Waals interaction force is found from: fSCsy
≠ESC , ≠d
(6)
where d is the separation distance (gap) between the sphere and cylinder. For characterising the interaction properties of macrobodies the so-called Hamaker constant w10x is introduced: Amsp2amnCnS Although the use of the additive summation method yields in the slightly overestimated Hamaker constant compared to the Lifshitz ‘constant’ (up to 20%), it gives a correct dependence of the van der Waals energy on the geometrical parameters of the interacting bodies. For the case of the sphere–cylinder system the additive approach has been employed previously by Rosenfeld and Wasan for deriving the unretarded (ms6) vdW force w11x. In the present work, this approach is used to derive relatively compact expressions for the vdW energy and force for the sphere and cylinder by integrating the attractive pair potential of the general power–law form, given by Eq. (5). A general solution for the arbitrary exponent m will facilitate the use of analytical approximation of Schenkel and Kitchener (3) for the full Casimir and Polder potential, as well as the derivation of the expression for the fully retarded force (ms7).
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™ Fig. 1. Interaction of the point particle P with the cylinder (sphere). Here rz is the r projection on the cross-sectional plane.
2. Energy of van der Waals interaction of a point particle with a sphere and a cylinder A compact formula for the energy of the vdW interaction of a point-like particle (atom, molecule) and a spherical particle (Fig. 1) was derived by us previously using the method of additive summation in w12x:
wSmsy
4pamnS R3S B my1 m 5 B RS E2E FC , ; ;C F F, 3 ym D 2 2 2 D y G G
(7)
where RS is the sphere radius, ysRSqh the closest separation between the centre of the sphere and point particle, F(a,b;c;z) the hypergeometric function w13x. The expression for the vdW energy of a point particle and an infinite cylinder is given by: B my1 E
GC
wmCsyp3y2amnC
D
2
BmE GC F D2G
F G
R2C B my1 my1 B RC E2E FC , ;2;C F F, D y G G ymy1 D 2 2
(8)
where ysRCqh is the closest distance between the point particle and the cylinder axis, RC is the cylinder radius, G(z) is the Gamma function. The detailed derivation of Eq. (8) is given in Appendix 1. If the approximate additivity principle is assumed, then the energy of the vdW interaction of a point particle with a spherical or cylindrical shell with the inner radius j can be found as follows: k k wshell m (Rk,j)swm(Rk)ywm(j), ksc,s
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In another particular case when the cylinder length is finite we have not succeeded in derivation of the compact analytical expression for the energy. The hypergeometric function can be reduced to standard functions in some cases w13x. For ms6 from Eqs. (7) and (8) we obtain the expressions for the vdW energies of the non-retarded interaction of a point particle with a sphere and cylinder:
w6Ssy
4pa6nS RS3 , 3 Žy2yR2S.3
w6Csy
3p2a6nC R2C B 5 5 B RC E2E FC , ;2;C F F, 8 y5 D 2 2 D y G G
(9)
(10)
which agree with the rarefied limits from the vdW free energies of the sandwichlike coaxial and concentric systems obtained in terms of the Lifshitz theory in Ref. w14x. Analogously, in the limit of the fully retarded interaction we arrive at the formulae for the energies for ms7: 2 2 4pa7nS R3S Ž5y qRS. w sy , 15 y Žy2yR2S.4 S 7
w7Csy
(11)
2 2 8pa7nC 2 ŽRCq2y . RC 2 , 15 Žy yR2C.4
(12)
Consider next the limiting cases. If the hypergeometric functions cannot be converted to standard functions, the following expression for the analytical continuation of the hypergeometric series w13x can be used: FŽa,b;aqbyc;z.s
GŽaqbyc.GŽc. GŽa.GŽb.
Ž1yz.yc
For the normalized potential w˜ kmsŽpamnk.y1wkm the limits RS™`, RC™`, h™ 0 give the known vdW potentials for a point particle and a flat plate (half-space). The first two terms of the expansions are:
w˜ S6fy
1 1 1 2 q , w˜ S7fy q , 6h3 4RSh2 10h4 15RSh3
w˜ C6 fy
1 1 1 1 q , w˜ C7 fy q . 3 2 4 6h 12RCh 10h 15RCh3
Of some interest are the limiting cases of the ultra-fine sphere RS™0 and fibre
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RC™0. Below are the first terms of the corresponding expansions for the potentials of the unretarded interaction given by Eqs. (9) and (10)
w˜ 6Sfy
4 R3S 3p R2C C ˜ , w fy , 6 3 h6 8 h5
and for the retarded potentials given by Eqs. (11) and (12):
w˜ 7Sfy
4 R3S 16 R2C C ˜ , w fy . 7 3 h7 15 h6
The latter expressions coincide with the asymptotes of the potentials of the retarded vdW interaction of a molecule with a sphere and cylinder which were obtained within the framework of the general theory of vdW forces in w15x. 3. Interaction of a spherical macroparticle with an infinite cylinder The van der Waals interaction energy of a sphere and a cylinder of infinite length can be found by integrating the vdW energy of a point particle and a sphere over the cylinder volume (Fig. 2): arccos((r2qc2yRC2) y2rc)
cqRC
SC m
|
E snC
`
|
rdr
db) y2rc)yarccos((r2qc2yRC2) y2rc)
cyRC
|
df
wSmdz,
(13)
y`
where csdqRSqRC. After integrating with respect to z and f we obtain the following expression for the vdW energy: cqRC
|
B r2qc2yR2 E
gmŽr.arccosC
SC Em sy2Am
C
2rc
D
cyRC
Frdr, G
(14)
where B my1 E
gmŽr.s
GC
2
D
4R3S
BmE
3yp
GC F D2G
F G
1
r
B my1
FC my1 D
2
,
my1 5 R2S E ; ; F. 2 2 r2 G
(15)
From Eqs. (6) and (15) the formula for the vdW force is derived: cqRC
SC m
|
f syAm
cyRC
gmŽr.xŽr.dr,
(16)
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Fig. 2. Interaction of a sphere and a cylinder.
where
xŽr.sy
2Žr2yc2yR2C.
µŽr2yŽcqRC.2.Žyr2qŽcyRC.2.∂1y2
.
(17)
3.1. Unretarded vdW interaction The unretarded vdW force between a sphere and a cylinder is found from Eq. (16) for ms6:
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318
cqRC
f6SCsyA6
|
g6Žr.xŽr.dr, g6s
cyRC
R3S 2t2yr2yR2S
.
For even m the integral in Eq. (16) is evaluated analytically. The final formula is: f6SCsy
A6RS3 µŽ4q2sqs2.EŽ1yp.ysŽ1qs.KŽ1yp.∂, 24c5y2RC3y2s2p3y2
(18)
where ps1qsy2, ss((dqRS)2yR2S)y2cRC K and E are the complete elliptic integrals defined as w13x: py2
|
EŽz.s
py2
|
y1yzsin2udu, KŽz.s
0
1y y1yzsin2udu
0
Eq. (18) coincides with the formula for the unretarded vdW force between a sphere and a cylinder obtained by Rosenfeld and Wasan in Ref. w11x. The same problem was considered also in w16,17x. It is worth noting that the results of the computations performed by the formulas reported in w16x and w17x disagree significantly with those by Eq. (18). This is explained by the following: in the work w16x the vdW energy in the sphere–cylinder system was calculated by means of the five-fold integration of the pair potential instead of the six-fold (double volume) integration, while in the work w17x the authors used an approximate procedure of expansion of the integrand function into series when deriving the formula for the energy of interaction between a point particle and a cylinder. In the limit of small distances d™0 the first term of expansion of Eq. (18) y1y2 y2 d , coincides with the Derjaguin approximation, fSC 6 fyŽA6 y6.RSŽ1qRS yRC. CS 2 3 y6 and at the big separations d™` the force falls off as f6 fyŽ5pA6 y2.RCRCd . In the large cylinder radius limit RC ™` Eq. (18) reduces to the Hamaker formula 3 w10x for the force between a sphere and a flat plate fSP 6 syŽ2A6 y3.RSŽdq y2 y2 2RS. d . 3.2. Retarded vdW interaction The retarded vdW force between a sphere and a cylinder is found from Eq. (16) at ms7: cqRC
fSC 7 syA7
|
cyRC
g7Žr.xŽr.dr,
(19)
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Fig. 3. Dimensionless vdW force f *7sf 7RC2 A7y1 between a sphere, interacting with a plate (p curve), cylinder (c), row of spheres (r) and another sphere (s), plotted against the dimensionless gap between them: R1*s2, R2*s1, Ns3.
where g7s
S W B R ET Žr4y8r2RS2y8RS4. T 1 S URŽr2q14R2 .y X. C F arcsin S T D r GY 15pŽr2yR2S.3 TV yr2yR2S
This formula is more compact than that published by us earlier in w12x, which was expressed by the double integral. In Fig. 3 the curves for the absolute values of the retarded vdW force between a sphere and a cylinder, two spheres of radii R1 and R2, sphere and a plate, sphere and a row of equal spheres were plotted as functions of the separation distance. Here the cylinder radius was chosen as the length scale. The formulae for the retarded vdW forces for two spheres, the ‘sphere–flat plate’, and ‘sphere–row of spheres’ systems are given in Appendix 2. Although Eq. (19) does not allow us to obtain analytical asymptotes, the limiting cases can be investigated graphically. The limit of the large cylinder radius RC™` is illustrated in Fig. 4. The next figure depicts the case when d
yRC
B
Cy
yRSqRC D
1 A7 E F. 10p d3 G
(20)
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Fig. 4. Large cylinder radius limit: Rss0.25 mm, ds0.25 mm.
In Fig. 5 the comparison is given for the retarded forces for the sphere and bodies of different shape plotted against the separation distance. The curves were computed using the approximation in Eq. (20) and the full formula in Eq. (19). In Fig. 6 the
Fig. 5. Dimensionless vdW force in the sphere–plate system (p curve), sphere-cylinder (c) and sphere– sphere (s) systems, calculated against the dimensionless gap: R2*s1, R1*s1. Dotted line—Derjaguin’s approximation.
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Fig. 6. On the range of validity of the Derjaguin approximation for the sphere–cylinder system: Rs,Rcs 0.25–5.0 mm, ds0.1 mm. cs curve for the ratio fcs 7 yfD is shown as a function of the sphere and cylinder radii at the constant separation. These figures serve as an illustrative example of the range of validity of the Derjaguin approximation. In the limit of small separation and (or) in the limits of large radii of a sphere and a cylinder the forces computed by Eqs. (19) and (20) tend to coincide (Fig. 6).
3.3. Interaction at intermediate distances For the description of the vdW interaction in the whole range of separations between a sphere and a cylinder, including the intermediate distances rGl0, the approximation of Schenkel and Kitchener (3) can be used. For this case, the expression for the vdW force will assume the following form: 3 ˜ SC fSCsA6Ž2.45pf˜ 7SCy2.17p2f˜ SC 8 q0.59p f9 . for r)0.5p
(21)
SC y1 SC where f˜ SC m sfm Am . Here the fully retarded force f 7 is defined by Eq. (19), while SC SC f 8 and f 9 are found by means of the general formula of Eq. (16). In the latter case, the corresponding functions gm are given by:
g8s
RS3Ž5r2q2RS2. 12t4yt
,
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S W B R ET Žr6y18r4R2Sy72r2R4Sy16R6S. 1 TU S 4 2 2 4 X, C F g9s RSŽr q68r RSq36RS.y arcsin T D r GY 70pt5 TV yt
where tsr 2yRS2. 4. Concluding remarks In the present work the formulae for the energy and force of the van der Waals interaction of a spherical macroparticle (ball) and an infinitely long cylinder were derived by integrating the pair potentials of the general power–law form Umsy amrym. The formulae obtained are valid for any ratio of the sphere and cylinder radii and for a wide range of the separation distances. These formulae account for the effect of retardation of the vdW interaction (ms7). The compact expressions were derived also for the vdW energies of attraction of a point-like particle with a sphere and a cylinder. The results obtained can be used in different problems that require the computation of the interaction of submicron and nanoparticles with cylindrical fine fibres. Acknowledgments The author acknowledges Prof. V.I. Roldughin and Dr A.L. Chernyakov for their helpful comments. Appendix 1: The energy of the vdW attraction of the point particle P with a cylinder is found by integrating the potential given by Eq. (4) over the cylinder volume (Fig. 1): RC
|
wCmsyamnC
`
2p
dz m r y`
| |
hdh
0
du
0
2
2
2
where rs(y qh qz y2hycos(u))1y2 , ysRC qh. Integration with respect to z and u results in the following expression: B my1 E
GC C m
w yamnCsy2p
F
2
D
1y2
G
BmE
GC F D2G
RC
|
p
|Ž
hdh
0
0
du 2
2
y qh y2hycosu.
B my1 E
GC D
sy2p
3y2
2
F G
BmE y GC F D2G
1 my1
RC
|
0
B my1
FC
D
2
,
my1 h2 E ;1; 2 Fhdh 2 y G
my1 2
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The integral with respect to the angle coordinate u is non-trivial. It was evaluated using the following expression w13x: p
B sin2my1xdx 1E B 1 1 E sBCm, FFCn,nymq ;mq ;a2F, 2 v D 1q2acosxqa . 2G D 2 2 G
|Ž 0
where B is the beta function, ±a±-1, Rem)0. The hypergeometric function can be expanded into hypergeometric series w13x: `
FŽa,b;c;z.s 8
GŽaqk.GŽbqk.GŽc.zk
ks0
GŽa.GŽb.GŽcqk.k!
,
and integrated term-to-term. This gives the final expression for the vdW attractive energy between a point particle and an infinite cylinder: B my1 E
GC
wmCsyp3y2amnC
D
2
BmE GC F D2G
F G
R2C B my1 my1 B RC E2E FC , ;2;C F F. D y G G ymy1 D 2 2
Appendix 2: The formula for the retarded vdW force between two spheres with dimensionless radii R1 and R2 can be found from the corresponding expression for the vdW energy obtained by Bouwkamp w18x by differentiating the latter by the separation distance. The result is: fSS 7 sy
4 Bw E A7R1R2 ST 1 2 U C F ln q ey2 j q 30C2 TV R1R2 D w1 G 8 js1
4
2C8ejy3y js1
16C2 ŽC4yŽR12yR22.2.∂ Žw1w2.2
(A.1)
e1sCq(R1qR2), e2sCy(R1qR2), e3sCq(R1yR2), e4sCy(R1yR2), CsdqR1qR2, w1se1e2, w2se3e4. Similarly, using the results of w18x, for the sphere–plate configuration we have: sp 7
f sy
8A7R3SŽdqRS. 15d3Ždq2RS.3
(A.2)
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The vdW attraction force of a sphere of radius R1 to a row (chain) of 2Ny1 spheres of equal radii R2 is computed as the geometrical sum of the forces between the given sphere and every sphere in the row: N SS f7SRsfSS 7 ŽC1.q2 8 f7 ŽCk.cosxk,
(A.3)
ks2
where C1sdqR1qR2 is the distance between the centres of the given sphere and the closest sphere in a row, CksyC12qŽ2Žky1.R2.2, cosxksC1 yCk and k is the number of the sphere in the row. Note that C1 is perpendicular to the row direction. When considering an infinite row of spheres there is no necessity of summing over infinite number of spheres, as at a certain separation there is a finite radius of interaction for a given particle. Thus, one should increase the number N until the value of force is constant with the prescribed degree of accuracy. References w1 x w2 x w3 x w4 x w5 x w6 x w7 x w8 x w9 x w10x w11x w12x w13x w14x w15x w16x w17x w18x
B.V. Derjaguin, N.V. Churaev, V.M. Muller, Surface Forces, Plenum, New York, 1987. L.D. Landau, E.M. Lifshitz, Statistical Physics (Pt. 2), Pergamon, Oxford, 1980. Yu.S. Barash, V.L. Ginzburg, Usp. Fiz. Nauk 143 (3) (1984) 1, Sov. Phys. Usp. 27 (1984) 467. B.V. Derjaguin, I.I. Abrikosova, E.M. Lifshitz, Quart. Rev. Chem. Soc. 10 (1956) 295, London. H.B.G. Casimir, D. Polder, Phys. Rev. 73 (4) (1948) 360. F. London, Z. Phys. 63 (1930) 245. V.M. Mostepanenko, N.N. Trunov, The Casimir Effect and its Applications, Clarendon Press, Oxford, 1997. J.H. Schenkel, J.A. Kitchener, Trans. Faraday Soc. 56 (1) (1960) 161. Bo.E. Sernelius, Surface Modes in Physics, Wiley-VCH, Berlin, 2001. H.C. Hamaker, Physica 4 (10) (1937) 1058. J.I. Rosenfeld, D.T. Wasan, J. Colloid Interface Sci. 47 (1) (1974) 27. V.A. Kirsch, Colloid J. 62 (6) (2000) 714. I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series and Products, Academic Press, New York, 1994. N.S. Witte, J. Chem. Phys. 99 (10) (1993) 8168. V.M. Nabutovskii, V.R. Belosludov, A.M. Korotkih, JETP 2 (8) (1979) 700, in Russian. Y. Gu, D. Li, J. Colloid Interface Sci. 217 (1) (1999) 60. S.W. Montgomery, M.A. Franchek, V.M. Goldschmidt, J. Colloid Interface Sci. 227 (2) (2000) 567. C.J. Bouwkamp, Physica 13 (8) (1947) 501.
Calculation of the van der Waals force between a spherical particle and an infinite cylinder V.A. Kirsch Institute of Physical Chemistry of the Russian Academy of Sciences, Leninsky Prospekt, 31, Moscow 119991, Russia
Abstract Formulae for the van der Waals attraction energy and force between a spherical particle and an infinite cylinder are derived by the method of additive summation of the pair interactions described by the potential of the general form Umsyamrym. The formula of Rosenfeld and Wasan for the non-retarded vdW force between a sphere and a cylinder (ms 6) is confirmed and the compact expression for the retarded force (ms7) is obtained. The comparison is given for the forces of the retarded vdW interaction between a sphere and a cylinder, another sphere, a row of spheres and a half-space. Also, the compact formulae for the energy of the vdW interaction of a point-like particle (atom, molecule) with a sphere and a cylinder are derived for the case of arbitrary m. 䊚 2003 Elsevier Science B.V. All rights reserved. Keywords: Van der Waals interaction; Pair potential; Additive approach; Retardation effect
1. Introduction Many problems in physics, physical chemistry and biology deal with the van der Waals (vdW) interaction w1x of fine spherical particles with bodies of cylindrical shape, like nanowires, nanotubes, fibres. In accordance with the general theory of Lifshitz w2,3x the van der Waals forces appear between neutral bodies (atoms, molecules, macrobodies) as a result of the correlated interaction of their instantaneous electrical and magnetic dipole moments through the fluctuating electromagnetic field. The van der Waals forces include the Keesom and Debye forces which are related to the presence of the permanent dipole moments, and dispersion forces, E-mail address: [email protected] (V.A. Kirsch). 0001-8686/03/$ - see front matter 䊚 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0001-8686Ž03.00053-8
V.A. Kirsch / Advances in Colloid and Interface Science 104 (2003) 311–324
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acting between both polar and non-polar molecules. The role of the latter is the most significant for many systems, excluding only highly-polar molecules. The first direct experiments on the vdW forces between macroscopic bodies were performed by Derjaguin and his colleges w4x. A two-particle potential of the dispersion interaction of neutral atoms (molecules) in vacuum was found by Casimir and Polder in Ref. w5x. Of special interest are the asymptotes of this potential, obtained in the limit of small and large interparticle separations r relative to the characteristic wavelength of radiation in the spectra of interacting atoms, l0s2pcvy1 0 , where v0 is the atomic frequency, c is the light velocity in vacuum. At distances a
a6 , r6
(1)
where a is the constant of interaction. The Keesom and Debye interaction potentials obey the same dependence on the separation distance. At the distances r'l0 the dispersion interaction is no longer instantaneous and is determined by the finite time of propagation of the signal from one temporary dipole to another, 2rcy1, which results in the retardation of the vdW interaction. In the limit of r4l0 the London forces do not exist. In this case, the vdW interaction is fully retarded and is described by the following asymptotic potential w5x: U7sy
a7 r7
(2)
The retarded vdW forces sometimes are termed the Casimir forces, as they are related with the quantum Casimir effect w7x. The full expression for the Casimir and Polder potential, valid for all separations r4a, is given by a cumbersome integral w5x for which it is convenient to use a simple analytical approximation. Schenkel and Kitchener w8x suggested the following approximation: Usa6Žy2.45pyr7q2.17p2 yr8y0.59p3 yr9., for r)0.5p,
(3)
where psl0 y2p. The calculation of the vdW force between macrobodies within the framework of the general theory involves the solution of the boundary value problem for the Maxwell equations. It should be emphasised that the BVP for the Maxwell equations possesses an analytical solution only for the systems with a high symmetry, like a single sphere, infinite cylinder, symmetrical shell or void, two half-spaces separated by a gap w9x. In the case of the interaction between the sphere and cylinder the general theory of Lifshitz gives no analytical answer, since the variables in the Maxwell equations cannot be separated for the geometry considered. Therefore, in
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the present work an approximate additive approach w10x will be invoked, which is valid for a pair of uniform and isotropic bodies of arbitrary geometry. Following this approach, the vdW interaction energy between a spherical macroparticle and a cylinder is found by summation of the pair interactions of the constituent point particles (atoms or molecules):
| | U dV dV sn | w dV sn | w dV
ESC m snCnS
m
VC VS
C
S
S m
C
VC
C
C m
S
S
(4)
VS
where nk is the concentration of the point particles, dVk the volume element, wkm the energy of interaction of the point particle with the macrobody, indices c and s denote the cylinder and sphere, correspondingly. Here Um is the two-particle attractive potential: Umsy
am , m)0. rm
(5)
The van der Waals interaction force is found from: fSCsy
≠ESC , ≠d
(6)
where d is the separation distance (gap) between the sphere and cylinder. For characterising the interaction properties of macrobodies the so-called Hamaker constant w10x is introduced: Amsp2amnCnS Although the use of the additive summation method yields in the slightly overestimated Hamaker constant compared to the Lifshitz ‘constant’ (up to 20%), it gives a correct dependence of the van der Waals energy on the geometrical parameters of the interacting bodies. For the case of the sphere–cylinder system the additive approach has been employed previously by Rosenfeld and Wasan for deriving the unretarded (ms6) vdW force w11x. In the present work, this approach is used to derive relatively compact expressions for the vdW energy and force for the sphere and cylinder by integrating the attractive pair potential of the general power–law form, given by Eq. (5). A general solution for the arbitrary exponent m will facilitate the use of analytical approximation of Schenkel and Kitchener (3) for the full Casimir and Polder potential, as well as the derivation of the expression for the fully retarded force (ms7).
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™ Fig. 1. Interaction of the point particle P with the cylinder (sphere). Here rz is the r projection on the cross-sectional plane.
2. Energy of van der Waals interaction of a point particle with a sphere and a cylinder A compact formula for the energy of the vdW interaction of a point-like particle (atom, molecule) and a spherical particle (Fig. 1) was derived by us previously using the method of additive summation in w12x:
wSmsy
4pamnS R3S B my1 m 5 B RS E2E FC , ; ;C F F, 3 ym D 2 2 2 D y G G
(7)
where RS is the sphere radius, ysRSqh the closest separation between the centre of the sphere and point particle, F(a,b;c;z) the hypergeometric function w13x. The expression for the vdW energy of a point particle and an infinite cylinder is given by: B my1 E
GC
wmCsyp3y2amnC
D
2
BmE GC F D2G
F G
R2C B my1 my1 B RC E2E FC , ;2;C F F, D y G G ymy1 D 2 2
(8)
where ysRCqh is the closest distance between the point particle and the cylinder axis, RC is the cylinder radius, G(z) is the Gamma function. The detailed derivation of Eq. (8) is given in Appendix 1. If the approximate additivity principle is assumed, then the energy of the vdW interaction of a point particle with a spherical or cylindrical shell with the inner radius j can be found as follows: k k wshell m (Rk,j)swm(Rk)ywm(j), ksc,s
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315
In another particular case when the cylinder length is finite we have not succeeded in derivation of the compact analytical expression for the energy. The hypergeometric function can be reduced to standard functions in some cases w13x. For ms6 from Eqs. (7) and (8) we obtain the expressions for the vdW energies of the non-retarded interaction of a point particle with a sphere and cylinder:
w6Ssy
4pa6nS RS3 , 3 Žy2yR2S.3
w6Csy
3p2a6nC R2C B 5 5 B RC E2E FC , ;2;C F F, 8 y5 D 2 2 D y G G
(9)
(10)
which agree with the rarefied limits from the vdW free energies of the sandwichlike coaxial and concentric systems obtained in terms of the Lifshitz theory in Ref. w14x. Analogously, in the limit of the fully retarded interaction we arrive at the formulae for the energies for ms7: 2 2 4pa7nS R3S Ž5y qRS. w sy , 15 y Žy2yR2S.4 S 7
w7Csy
(11)
2 2 8pa7nC 2 ŽRCq2y . RC 2 , 15 Žy yR2C.4
(12)
Consider next the limiting cases. If the hypergeometric functions cannot be converted to standard functions, the following expression for the analytical continuation of the hypergeometric series w13x can be used: FŽa,b;aqbyc;z.s
GŽaqbyc.GŽc. GŽa.GŽb.
Ž1yz.yc
For the normalized potential w˜ kmsŽpamnk.y1wkm the limits RS™`, RC™`, h™ 0 give the known vdW potentials for a point particle and a flat plate (half-space). The first two terms of the expansions are:
w˜ S6fy
1 1 1 2 q , w˜ S7fy q , 6h3 4RSh2 10h4 15RSh3
w˜ C6 fy
1 1 1 1 q , w˜ C7 fy q . 3 2 4 6h 12RCh 10h 15RCh3
Of some interest are the limiting cases of the ultra-fine sphere RS™0 and fibre
V.A. Kirsch / Advances in Colloid and Interface Science 104 (2003) 311–324
316
RC™0. Below are the first terms of the corresponding expansions for the potentials of the unretarded interaction given by Eqs. (9) and (10)
w˜ 6Sfy
4 R3S 3p R2C C ˜ , w fy , 6 3 h6 8 h5
and for the retarded potentials given by Eqs. (11) and (12):
w˜ 7Sfy
4 R3S 16 R2C C ˜ , w fy . 7 3 h7 15 h6
The latter expressions coincide with the asymptotes of the potentials of the retarded vdW interaction of a molecule with a sphere and cylinder which were obtained within the framework of the general theory of vdW forces in w15x. 3. Interaction of a spherical macroparticle with an infinite cylinder The van der Waals interaction energy of a sphere and a cylinder of infinite length can be found by integrating the vdW energy of a point particle and a sphere over the cylinder volume (Fig. 2): arccos((r2qc2yRC2) y2rc)
cqRC
SC m
|
E snC
`
|
rdr
db) y2rc)yarccos((r2qc2yRC2) y2rc)
cyRC
|
df
wSmdz,
(13)
y`
where csdqRSqRC. After integrating with respect to z and f we obtain the following expression for the vdW energy: cqRC
|
B r2qc2yR2 E
gmŽr.arccosC
SC Em sy2Am
C
2rc
D
cyRC
Frdr, G
(14)
where B my1 E
gmŽr.s
GC
2
D
4R3S
BmE
3yp
GC F D2G
F G
1
r
B my1
FC my1 D
2
,
my1 5 R2S E ; ; F. 2 2 r2 G
(15)
From Eqs. (6) and (15) the formula for the vdW force is derived: cqRC
SC m
|
f syAm
cyRC
gmŽr.xŽr.dr,
(16)
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317
Fig. 2. Interaction of a sphere and a cylinder.
where
xŽr.sy
2Žr2yc2yR2C.
µŽr2yŽcqRC.2.Žyr2qŽcyRC.2.∂1y2
.
(17)
3.1. Unretarded vdW interaction The unretarded vdW force between a sphere and a cylinder is found from Eq. (16) for ms6:
V.A. Kirsch / Advances in Colloid and Interface Science 104 (2003) 311–324
318
cqRC
f6SCsyA6
|
g6Žr.xŽr.dr, g6s
cyRC
R3S 2t2yr2yR2S
.
For even m the integral in Eq. (16) is evaluated analytically. The final formula is: f6SCsy
A6RS3 µŽ4q2sqs2.EŽ1yp.ysŽ1qs.KŽ1yp.∂, 24c5y2RC3y2s2p3y2
(18)
where ps1qsy2, ss((dqRS)2yR2S)y2cRC K and E are the complete elliptic integrals defined as w13x: py2
|
EŽz.s
py2
|
y1yzsin2udu, KŽz.s
0
1y y1yzsin2udu
0
Eq. (18) coincides with the formula for the unretarded vdW force between a sphere and a cylinder obtained by Rosenfeld and Wasan in Ref. w11x. The same problem was considered also in w16,17x. It is worth noting that the results of the computations performed by the formulas reported in w16x and w17x disagree significantly with those by Eq. (18). This is explained by the following: in the work w16x the vdW energy in the sphere–cylinder system was calculated by means of the five-fold integration of the pair potential instead of the six-fold (double volume) integration, while in the work w17x the authors used an approximate procedure of expansion of the integrand function into series when deriving the formula for the energy of interaction between a point particle and a cylinder. In the limit of small distances d™0 the first term of expansion of Eq. (18) y1y2 y2 d , coincides with the Derjaguin approximation, fSC 6 fyŽA6 y6.RSŽ1qRS yRC. CS 2 3 y6 and at the big separations d™` the force falls off as f6 fyŽ5pA6 y2.RCRCd . In the large cylinder radius limit RC ™` Eq. (18) reduces to the Hamaker formula 3 w10x for the force between a sphere and a flat plate fSP 6 syŽ2A6 y3.RSŽdq y2 y2 2RS. d . 3.2. Retarded vdW interaction The retarded vdW force between a sphere and a cylinder is found from Eq. (16) at ms7: cqRC
fSC 7 syA7
|
cyRC
g7Žr.xŽr.dr,
(19)
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319
Fig. 3. Dimensionless vdW force f *7sf 7RC2 A7y1 between a sphere, interacting with a plate (p curve), cylinder (c), row of spheres (r) and another sphere (s), plotted against the dimensionless gap between them: R1*s2, R2*s1, Ns3.
where g7s
S W B R ET Žr4y8r2RS2y8RS4. T 1 S URŽr2q14R2 .y X. C F arcsin S T D r GY 15pŽr2yR2S.3 TV yr2yR2S
This formula is more compact than that published by us earlier in w12x, which was expressed by the double integral. In Fig. 3 the curves for the absolute values of the retarded vdW force between a sphere and a cylinder, two spheres of radii R1 and R2, sphere and a plate, sphere and a row of equal spheres were plotted as functions of the separation distance. Here the cylinder radius was chosen as the length scale. The formulae for the retarded vdW forces for two spheres, the ‘sphere–flat plate’, and ‘sphere–row of spheres’ systems are given in Appendix 2. Although Eq. (19) does not allow us to obtain analytical asymptotes, the limiting cases can be investigated graphically. The limit of the large cylinder radius RC™` is illustrated in Fig. 4. The next figure depicts the case when d
yRC
B
Cy
yRSqRC D
1 A7 E F. 10p d3 G
(20)
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V.A. Kirsch / Advances in Colloid and Interface Science 104 (2003) 311–324
Fig. 4. Large cylinder radius limit: Rss0.25 mm, ds0.25 mm.
In Fig. 5 the comparison is given for the retarded forces for the sphere and bodies of different shape plotted against the separation distance. The curves were computed using the approximation in Eq. (20) and the full formula in Eq. (19). In Fig. 6 the
Fig. 5. Dimensionless vdW force in the sphere–plate system (p curve), sphere-cylinder (c) and sphere– sphere (s) systems, calculated against the dimensionless gap: R2*s1, R1*s1. Dotted line—Derjaguin’s approximation.
V.A. Kirsch / Advances in Colloid and Interface Science 104 (2003) 311–324
321
Fig. 6. On the range of validity of the Derjaguin approximation for the sphere–cylinder system: Rs,Rcs 0.25–5.0 mm, ds0.1 mm. cs curve for the ratio fcs 7 yfD is shown as a function of the sphere and cylinder radii at the constant separation. These figures serve as an illustrative example of the range of validity of the Derjaguin approximation. In the limit of small separation and (or) in the limits of large radii of a sphere and a cylinder the forces computed by Eqs. (19) and (20) tend to coincide (Fig. 6).
3.3. Interaction at intermediate distances For the description of the vdW interaction in the whole range of separations between a sphere and a cylinder, including the intermediate distances rGl0, the approximation of Schenkel and Kitchener (3) can be used. For this case, the expression for the vdW force will assume the following form: 3 ˜ SC fSCsA6Ž2.45pf˜ 7SCy2.17p2f˜ SC 8 q0.59p f9 . for r)0.5p
(21)
SC y1 SC where f˜ SC m sfm Am . Here the fully retarded force f 7 is defined by Eq. (19), while SC SC f 8 and f 9 are found by means of the general formula of Eq. (16). In the latter case, the corresponding functions gm are given by:
g8s
RS3Ž5r2q2RS2. 12t4yt
,
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322
S W B R ET Žr6y18r4R2Sy72r2R4Sy16R6S. 1 TU S 4 2 2 4 X, C F g9s RSŽr q68r RSq36RS.y arcsin T D r GY 70pt5 TV yt
where tsr 2yRS2. 4. Concluding remarks In the present work the formulae for the energy and force of the van der Waals interaction of a spherical macroparticle (ball) and an infinitely long cylinder were derived by integrating the pair potentials of the general power–law form Umsy amrym. The formulae obtained are valid for any ratio of the sphere and cylinder radii and for a wide range of the separation distances. These formulae account for the effect of retardation of the vdW interaction (ms7). The compact expressions were derived also for the vdW energies of attraction of a point-like particle with a sphere and a cylinder. The results obtained can be used in different problems that require the computation of the interaction of submicron and nanoparticles with cylindrical fine fibres. Acknowledgments The author acknowledges Prof. V.I. Roldughin and Dr A.L. Chernyakov for their helpful comments. Appendix 1: The energy of the vdW attraction of the point particle P with a cylinder is found by integrating the potential given by Eq. (4) over the cylinder volume (Fig. 1): RC
|
wCmsyamnC
`
2p
dz m r y`
| |
hdh
0
du
0
2
2
2
where rs(y qh qz y2hycos(u))1y2 , ysRC qh. Integration with respect to z and u results in the following expression: B my1 E
GC C m
w yamnCsy2p
F
2
D
1y2
G
BmE
GC F D2G
RC
|
p
|Ž
hdh
0
0
du 2
2
y qh y2hycosu.
B my1 E
GC D
sy2p
3y2
2
F G
BmE y GC F D2G
1 my1
RC
|
0
B my1
FC
D
2
,
my1 h2 E ;1; 2 Fhdh 2 y G
my1 2
V.A. Kirsch / Advances in Colloid and Interface Science 104 (2003) 311–324
323
The integral with respect to the angle coordinate u is non-trivial. It was evaluated using the following expression w13x: p
B sin2my1xdx 1E B 1 1 E sBCm, FFCn,nymq ;mq ;a2F, 2 v D 1q2acosxqa . 2G D 2 2 G
|Ž 0
where B is the beta function, ±a±-1, Rem)0. The hypergeometric function can be expanded into hypergeometric series w13x: `
FŽa,b;c;z.s 8
GŽaqk.GŽbqk.GŽc.zk
ks0
GŽa.GŽb.GŽcqk.k!
,
and integrated term-to-term. This gives the final expression for the vdW attractive energy between a point particle and an infinite cylinder: B my1 E
GC
wmCsyp3y2amnC
D
2
BmE GC F D2G
F G
R2C B my1 my1 B RC E2E FC , ;2;C F F. D y G G ymy1 D 2 2
Appendix 2: The formula for the retarded vdW force between two spheres with dimensionless radii R1 and R2 can be found from the corresponding expression for the vdW energy obtained by Bouwkamp w18x by differentiating the latter by the separation distance. The result is: fSS 7 sy
4 Bw E A7R1R2 ST 1 2 U C F ln q ey2 j q 30C2 TV R1R2 D w1 G 8 js1
4
2C8ejy3y js1
16C2 ŽC4yŽR12yR22.2.∂ Žw1w2.2
(A.1)
e1sCq(R1qR2), e2sCy(R1qR2), e3sCq(R1yR2), e4sCy(R1yR2), CsdqR1qR2, w1se1e2, w2se3e4. Similarly, using the results of w18x, for the sphere–plate configuration we have: sp 7
f sy
8A7R3SŽdqRS. 15d3Ždq2RS.3
(A.2)
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The vdW attraction force of a sphere of radius R1 to a row (chain) of 2Ny1 spheres of equal radii R2 is computed as the geometrical sum of the forces between the given sphere and every sphere in the row: N SS f7SRsfSS 7 ŽC1.q2 8 f7 ŽCk.cosxk,
(A.3)
ks2
where C1sdqR1qR2 is the distance between the centres of the given sphere and the closest sphere in a row, CksyC12qŽ2Žky1.R2.2, cosxksC1 yCk and k is the number of the sphere in the row. Note that C1 is perpendicular to the row direction. When considering an infinite row of spheres there is no necessity of summing over infinite number of spheres, as at a certain separation there is a finite radius of interaction for a given particle. Thus, one should increase the number N until the value of force is constant with the prescribed degree of accuracy. References w1 x w2 x w3 x w4 x w5 x w6 x w7 x w8 x w9 x w10x w11x w12x w13x w14x w15x w16x w17x w18x
B.V. Derjaguin, N.V. Churaev, V.M. Muller, Surface Forces, Plenum, New York, 1987. L.D. Landau, E.M. Lifshitz, Statistical Physics (Pt. 2), Pergamon, Oxford, 1980. Yu.S. Barash, V.L. Ginzburg, Usp. Fiz. Nauk 143 (3) (1984) 1, Sov. Phys. Usp. 27 (1984) 467. B.V. Derjaguin, I.I. Abrikosova, E.M. Lifshitz, Quart. Rev. Chem. Soc. 10 (1956) 295, London. H.B.G. Casimir, D. Polder, Phys. Rev. 73 (4) (1948) 360. F. London, Z. Phys. 63 (1930) 245. V.M. Mostepanenko, N.N. Trunov, The Casimir Effect and its Applications, Clarendon Press, Oxford, 1997. J.H. Schenkel, J.A. Kitchener, Trans. Faraday Soc. 56 (1) (1960) 161. Bo.E. Sernelius, Surface Modes in Physics, Wiley-VCH, Berlin, 2001. H.C. Hamaker, Physica 4 (10) (1937) 1058. J.I. Rosenfeld, D.T. Wasan, J. Colloid Interface Sci. 47 (1) (1974) 27. V.A. Kirsch, Colloid J. 62 (6) (2000) 714. I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series and Products, Academic Press, New York, 1994. N.S. Witte, J. Chem. Phys. 99 (10) (1993) 8168. V.M. Nabutovskii, V.R. Belosludov, A.M. Korotkih, JETP 2 (8) (1979) 700, in Russian. Y. Gu, D. Li, J. Colloid Interface Sci. 217 (1) (1999) 60. S.W. Montgomery, M.A. Franchek, V.M. Goldschmidt, J. Colloid Interface Sci. 227 (2) (2000) 567. C.J. Bouwkamp, Physica 13 (8) (1947) 501.