PHYSICAL REVIEW E 67, 056311 共2003兲
Tolman’s nonlinearity of capillary waves Anton V. Dolgikh, Dmitry L. Dorofeev, and Boris A. Zon* Voronezh State University, Voronezh 394693, Russia 共Received 6 August 2002; published 22 May 2003兲 A nonlinear theory of nanometer capillary waves is developed that takes curvature dependence of the surface tension coefficient 共Tolman’s nonlinearity兲 into account. Estimations are given that indicate the importance of Tolman’s nonlinearity for thermocapillary waves. DOI: 10.1103/PhysRevE.67.056311
PACS number共s兲: 68.03.Cd, 68.03.Kn
I. INTRODUCTION
The ordinary theory of capillary waves on the surface of a liquid is based on the assumption that the surface tension is a constant, independent of the surface curvature 关1兴. However, as it is well known, this assumption becomes incorrect for large curvatures. Taking into account only two leading terms, the dependence of the surface tension coefficient on the curvature radius R can be taken in the form 关2兴
冉
␣ ⫽ ␣ 共 ⬁ 兲 1⫺
冊
2␦ , R
共1兲
where ␦ is the Tolman length and ␣ (⬁) is the surface tension of the flat surface (R→⬁). Thus, the change in the surface tension becomes essential when R lies in the range of several ␦ . Estimation of the Tolman length was carried out in a number of theoretical and experimental works 共see, for example, Refs. 关3– 6兴 and references therein兲. It can be both negative and positive, and its absolute value is about the width of the monomolecular liquid-vapor interfacial layer and lies in the nanometer range. In the present work, we consider the influence of dependence 共1兲 on the characteristics of capillary waves. This dependence is most pronounced when R amounts to several nanometers. The proposed approach is built on the frame of the model of continuous liquid, the molecular microstructure of the medium is neglected. Therefore, the wavelength is assumed to be sufficiently large in comparison with the intermolecular distance. It is also assumed that the wave period is larger than the characteristic relaxation time 共the time of settled life of a molecule in the well of an intermolecular potential 关7兴兲. For shorter periods, the medium could not be treated as a liquid but rather as a solid one and, correspondingly, the capillary waves must be substituted by the Rayleigh ones, as treated by Frenkel 关7兴. Obtained results may be of interest for the theory of thermocapillary waves, widely investigated in the past years, both experimentally 关8 –10兴 and by molecular dynamics simulations 关11,12兴. It was assumed in Ref. 关8兴 that an enhancement of the effective surface tension for the shortest thermocapillary waves may explain some unexpected experimental results obtained for an x-ray reflection on the metha-
nol surface. Thus, in our work we discuss a specific mechanism that may lead to such a change in the effective surface tension, and we call it Tolman’s nonlinearity. This nonlinearity may also be essential for some applications in nanotechnology, for example, in creating wavelike periodic microstructures on a surface 关13兴 and in the problem of electrohydrostatic instability 关14兴. II. MAIN FORMALISM
It is assumed that the velocity field in a liquid possesses the property of potentiality, ⌬⌽⫽0,
where ⌽ is velocity potential. The axis z is directed vertically up, and the uperturbed liquid occupies the region z⬍0. Capillary waves propagate along axis x. Since the wave surface (x,t) is not spherical, we must generalize Eq. 共1兲 for nonspherical surfaces. For this purpose, we substitute the factor ⫺1/R in Eq. 共1兲 by the mean curvature of the surface. With assumption / xⰆ1, it yields ⫺
1 2 1 → . R 2 x2
共3兲
Then the Laplace equation, determining pressure under the curved surface of the liquid „z⫽ (x,t)…, in account of Eqs. 共1兲 and 共3兲, takes the form
冋 冉 冊册
2 2 ⌽ ⫽␣共 ⬁ 兲 ⫹ ␦ t x2 x2
2
,
共4兲
where is the density of the liquid. Expressing the velocity of the liquid in terms of potential ⌽, we get a condition that connects Eqs. 共2兲 and 共4兲:
⌽ 兩 . ⫽ t z z⫽ (x,t)
共5兲
Following the standard procedure 关1兴, we assume here that amplitude A of the wave is small in comparison with its length , i.e., ⫽A/Ⰶ1.
*Email address:
[email protected] 1063-651X/2003/67共5兲/056311共5兲/$20.00
共2兲
67 056311-1
共6兲
©2003 The American Physical Society
PHYSICAL REVIEW E 67, 056311 共2003兲
DOLGIKH, DOROFEEV, AND ZON
This allows us to substitute condition z⫽ (x,t) in Eqs. 共4兲 and 共5兲 by z⫽0. Instead of x we introduce ⫽x⫺ut, where u is the velocity of a capillary wave. This gives us the Laplace equations with nonlinear boundary condition:
2⌽ z
冋
2
⫹
2⌽
2
⫽0;
冋
⫺u
⌽ ⫽ z
兺
p⫽1
⬁
共7兲
; z⫽0
冉 冊册 2
⫺2c m s n sin共 mk 兲 cos共 nk 兲 ⫹s m s n cos共 nk 兲 cos共 mk 兲兴 . 共10兲
. z⫽0
⌽ 共 ⫹ 兲 ⫽⌽ 共 兲 .
共8兲
Obviously, the ratio between s 1 and c 1 depends only on the choice of initial . For simplicity, we set s 1 ⫽c 1 and introduce dimensionless values
Boundary condition 共8兲 allows us to seek the potential in the form ⌽ 共 ,z 兲 ⫽
兺 c ne n⫽1
兺 m 2 n 2 k 3 关 c m c n sin共 mk 兲 sin共 nk 兲 m,n⫽1
2
As a second boundary condition 共over ), we take the periodic one
⬁
共 u 3 p⫺ ␣ 共 ⬁ 兲 u p 2 k 兲关 c p sin共 pk 兲 ⫺s p cos共 pk 兲兴
⫽␣共 ⬁ 兲␦
⌽ ⌽ ⌽ ⫺␣共 ⬁ 兲␦ ⫽␣共 ⬁ 兲u z z 2
u 3
册
⬁
˜c n ⫽c n /c 1 ,
⬁
nkz
cos共 nk 兲 ⫹
˜s n ⫽s n /c 1 ,
u 0 ⫽„␣ 共 ⬁ 兲 k/ …1/2,
˜u ⫽u/u 0 ,
兺 s n e nkz sin共 nk 兲 . n⫽1 共9兲
Taken in this form the potential satisfies Eq. 共2兲; k⫽2 / is the wave number. Substituting this expansion into Eq. 共7兲 gives
˜c 1 ⫽s ˜ 1 ⫽1,
and the nonlinearity parameter
␥ ⫽ ␦ k 2 c 1 /u. Then Eq. 共10兲 becomes
⬁
兺
p⫽1
˜ p cos共 pk 兲兴 p 共 ˜u 2 ⫺p 兲关˜c p sin共 pk 兲 ⫺s ⬁
⫽␥
兺
m,n⫽1
␥ ⫽ 2
˜ m˜s n sin共 mk 兲 cos共 nk 兲 ⫹s ˜ m˜s n cos共 nk 兲 cos共 mk 兲兴 m 2 n 2 关˜c m˜c n sin共 mk 兲 sin共 nk 兲 ⫺2c
⬁
兺
m,n⫽1
˜ n˜s n 兲 cos共 m⫺n 兲 k ⫹ 共˜s n˜s n ⫺c ˜ m˜c n 兲 cos共 m⫹n 兲 k ⫺2c ˜ m˜c n 共 sin共 m⫺n 兲 k ⫹sin共 m⫹n 兲 k 兴 . n 2 m 2 关共˜c m˜c n ⫹s
˜ p ⫽0 ᭙p⭓3, First approximation: ˜c p ⫽s
Equalizing the coefficients at corresponding harmonics in the left-hand and right-hand sides yields
␥ ⫺ p 共 ˜u 2 ⫺p 兲˜s p ⫽ 2
˜ 2 ⫽4 ␥ 共˜c 2 ⫹s ˜ 2 兲 , 2 共 2⫺u ˜ 2 兲˜s 2 ⫽0, 1⫺u
⬁
兺
n⫽1
n 2 兵 关共 n⫹ p 兲 2˜c n⫹p ⫹ 共 n⫺p 兲 2
˜ 2 ⫽4 ␥ 共˜c 2 ⫺s ˜2兲, 1⫺u
˜ p⫺n 兲兴˜c n ⫹ 关共 n⫹ p 兲 2˜s n⫹p ⫻ 共˜c n⫺p ⫺c
gives the solution
˜ p⫺n 兲兴˜s n 其 , ⫹ 共 n⫺ p 兲 2 共˜s n⫺p ⫹s
␥ ˜c 2 ⫽ , 2
⬁
⫺p 共 ˜u 2 ⫺p 兲˜c p ⫽ ␥
兺
n⫽1
˜ 2 兲˜c 2 ⫽ ␥ 2 共 2⫺u
n 2 关共 n⫹ p 兲 2˜c n⫹p
˜s 2 ⫽0,
˜u ⬇1⫺ ␥ 2 .
Returning to dimensional values,
˜ n⫺p 兲兴˜s n . ⫹ 共 n⫺ p 兲 2 共˜c p⫺n ⫺c
c 1 ⫽s 1 ,
˜ p ⫽0 ᭙ p⭓2 gives ˜u ⫽1, i.e., Zero approximation ˜c p ⫽s u⫽u 0 , which coincides with a familiar relation for capillary waves with constant surface tension 关1兴.
c 2⬇
␥ c 2 1
s 2 ⫽0,
u⬇ 共 1⫺ ␥ 2 兲 u 0 .
共11兲
The surface profile ( ) can be represented in the form of the expansion similar to Eq. 共9兲:
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TOLMAN’S NONLINEARITY OF CAPILLARY WAVES
FIG. 2. Velocity of capillary wave as a function of ␥ .
FIG. 1. Wave forms with nonlinearity parameter ␥ ⫽⫾0.5. The dashed line presents the first harmonic. ⬁
共 ,t 兲 ⫽
兺
n⫽1
⬁
a n cos共 nk 兲 ⫹
兺
n⫽1
b n sin共 nk 兲 .
Then the second relation in Eq. 共7兲 gives a n ⫽s n /u,
b n ⫽⫺c n /u,
and, taking into account relations 共11兲,
共 ,t 兲 ⫽ 冑2a 1 cos共 k ⫹ /4兲 ⫺ 共 ␥ a 1 /2兲 sin共 2k 兲 . Finally, we rewrite this expression for an arbitrary initial phase
冉
共 ,t 兲 ⫽A cos关 k 共 ⫺ 0 兲兴 ⫹
␥ 2 冑2
cos关 2k 共 ⫺ 0 兲兴
冊
共 r兲 ⫽
and express the parameter ␥ in terms of wave amplitude A and length ,
␥⫽
␦ k 2A
冑2
⫽
共 2 兲2 ␦A
冑2
2
.
positive and negative ␦ . Such a deformation of the wave form and the change in velocity are similar to the wellknown Stokes wave 关1兴. However, in the case of Stokes waves, the cause of nonlinearity is finite amplitude of the wave in comparison with its wavelength, i.e., finiteness of parameter , defined in Eq. 共6兲. On the contrary, the cause of, considered here, Tolman’s nonlinearity is the finiteness of the curvature radius in comparison with the Tolman length, i.e., finiteness of parameter ␥ , defined in Eq. 共12兲. Note that Ⰶ1 does not necessarily imply ␥ Ⰶ1, because ␥ depends not only on A/ but also on ␦ / and multiplied by a rather large factor. Thus, for nanometer capillary wave, the described Tolman’s nonlinearity, associated with ␥ , can be at least equally important as the familiar Stokes’ nonlinearity, associated with . In particular, Tolman’s nonlinearity can be essential for thermally induced capillary waves on the surface of the liquid. Below, we estimate the characteristic value of ␥ for these waves using the method described in Refs. 关15,16兴. Let us assume the surface of the liquid to be a square in the x-y plane with side L, and require ⫽0 at the boundary. Thermally induced fluctuations of surface at an arbitrary moment of time can be represented in the form of an expansion over harmonics of capillary waves:
共12兲
Note that factor (2 ) 2 / 冑2⬇27.9 is rather large, hence nonlinear effects may be significant even for wavelengths much larger than the Tolman length ␦ .
冉 冊 冉 冊
nx sin my , L L
where n,m⫽1, . . . ,N, N⫽L/l, l is an empirical constant approximately equal to a characteristic intermolecular distance. For estimation we will neglect dependence 共1兲 of surface tension ␣ on the curvature. Then, the work W necessary for creating this fluctuation is proportional to the excess of surface area ⌬S: W⫽ ␣ ⌬S⫽
III. RESULTS AND DISCUSSIONS
Figure 1 demonstrates wave forms ( ) for different values of ␥ . The nonlinearity, considered here, leads to a deformation of the original sinusoidal wave form and produces a crest-trough asymmetry: for ␥ ⬎0, crests become narrower, and troughs wider; while for ␥ ⬍0, the deformation is opposite. The wave velocity u changes according to Eq. 共11兲, as presented in Fig. 2. Note that the velocity decreases both for
A nm sin 兺 n,m
⬇
␣ 2
冕冕
冕冕
dx dy 关共 1⫹ 2x ⫹ 2y 兲 1/2⫺1 兴
dx dy 共 2x ⫹ 2y 兲 ⫽
␣2 8
2 , 共 n 2 ⫹m 2 兲 A nm 兺 n,m
and the probability of fluctuation is proportional to Boltzmann’s factor exp(⫺W/kBT). In complete agreement with the equipartition theorem, this gives following amplitude variances:
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DOLGIKH, DOROFEEV, AND ZON
2 具 A nm 典⫽
4k B T
␣ 2 共 n 2 ⫹m 2 兲
.
In order to estimate the nonlinearity parameter ␥ , note that it presents a characteristic value of the mean curvature ( xx ⫹ y y )/2 multiplied by the Tolman length, hence
具 ␥ 2 典 ⬃ ␦ 2 具 共 xx ⫹ y y 兲 2 典 ,
共13兲
where the overline denotes averaging over x,y: 共 xx ⫹ y y 兲 2 ⫽
冉 冊兺
1 4 L
4
n,m
2 , 共 n 2 ⫹m 2 兲 2 A nm
which yields
具 ␥ 2典 ⬃ ⫽
冉 冊兺
␦2 4
L
4
n,m
2␦ 2k BT
␣L4
2 共 n 2 ⫹m 2 兲 2 具 A nm 典
n 2 ⫹m 2 ⬇ 兺 n,m
2 ␦ 2 k B T 2N 4 . 3 ␣L4
Expressing ␣ in terms of intermolecular potential depth ⑀ , ␣ ⬃ ⑀ /l 2 关6兴, finally we get
具 ␥ 2典 ⬃
冉冊
k BT ␦ ⑀ l
2
.
Then taking, for example, k B T/ ⑀ ⬃0.5, ␦ /l⬃1 obtain 具 ␥ 2 典 ⬃0.5. Obviously, these expressions give only a rough estimation, since the main contribution to 具 ␥ 2 典 is associated with wavelengths ⬃l, for which the description in terms of capillary waves becomes inadequate, as mentioned in the Introduction. Nevertheless, this estimation suggests a significant role of the considered nonlinearity in the dynamics of thermocapillary waves. In particular, it means that an effective value of surface tension ␣ e f f for thermocapillary waves can significantly differ from ␣ (⬁). Indeed, using Eqs. 共1兲, 共3兲, and 共13兲, an average change of surface tension for the whole ensemble of capillary waves can be estimated as ⌬ ␣ / ␣ 共 ⬁ 兲 ⬃ 冑具 ␥ 2 典 ,
共14兲
and thus can approach values as large as 0.7. In its turn, this change in ␣ e f f will lead to a corresponding change in the mean-square amplitude of thermocapillary surface fluctuations 关8兴:
冋 册
where max and min are some ‘‘cutoff’’ values for the wavelengths 共see discussion on their proper choice in Ref. 关16兴兲. In combination with intrinsic interfacial width p , which is of the order of the molecular radius, c provides the total roughness of the liquid surface,
2t ⫽ 2c ⫹ 2p .
共16兲
The values of t can be extracted from experimental data on x-ray reflection by liquid surfaces 关8 –10兴. The available accuracy is about 1–3 %, which is much better than it is needed in order to find the above change in the effective surface tension. Thus, the above theoretical analysis is, in principle, accessible for an experimental test by measuring the value of 2t and comparing it with theoretical values. However, in practice, we usually cannot clearly reveal the effect of the considered nonlinearity, since it is partially obscured by the well-known problems of the proper choice of min and p in Eqs. 共15兲 and 共16兲. Nevertheless, the discrepancy between experimental and theoretical estimations obtained for methanol in Ref. 关8兴 seems rather characteristic. The corresponding values for methanol are the following: the experimental value of t ⫽4.80⫾0.06 Å; the theoretical value for c , calculated with ␣ e f f ⫽ ␣ (⬁), is c ⫽4.74⫾0.08 Å; then Eq. 共16兲 gives for p ⬇0.7 Å, which is considerably smaller than the theoretically expected value near the molecular radius r M ⫽2.52 Å. The authors of Ref. 关8兴 suggested that such a discrepancy can be explained by a larger value of ␣ e f f in comparison with ␣ (⬁), in complete accordance with our consideration above. On the other hand, similar experimental and theoretical estimations for water and carbon tetrachloride carried out in the same work do not demonstrate any pronounced discrepancies. It is assumed in Ref. 关8兴 that a strong dimensional dependence of the surface tension and, hence, a large difference between ␣ e f f and ␣ (⬁) is a characteristic just for methanol, due to distinct anisotropy of its molecule, and is not so important for water and carbon tetrachloride with their more isotropic molecules. This last assumption is also indirectly confirmed by experimental results presented in Ref. 关17兴, which give a relatively larger value of the Tolman length for methanol, ␦ /l⬃2.7 . . . 3.5, than for water, ␦ /l⬃1.7 . . . 2.8. Thus, the above consideration shows that taking into account Tolman’s nonlinearity can be useful in the analysis of nanometer capillary wave dynamics, in particular, the thermocapillary ones. ACKNOWLEDGMENTS
共15兲
The research was carried out under the support of the CRDF and the Ministry of Education of Russian Federation 共Grant No. VZ - 010 - 0兲.
关1兴 G. B. Whitham, Linear and Nonlinear Waves 共Wiley, New York, 1974兲. 关2兴 R.C. Tolman, J. Chem. Phys. 17, 333 共1949兲. 关3兴 V.I. Kalikmanov, Phys. Rev. E 55, 3068 共1997兲.
关4兴 D.I. Zhukhovitskii, J. Chem. Phys. 110, 7770 共1999兲. 关5兴 A.E. van Giessen and E.M. Blokhuis, J. Chem. Phys. 116, 302 共2002兲. 关6兴 A. V. Dolgikh, D. L. Dorofeev, and B. A. Zon, Khim. Fiz. 共to
max k BT c⫽ 具 典 ⬇ ln , 2␣ef f min 2
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TOLMAN’S NONLINEARITY OF CAPILLARY WAVES be published兲. 关7兴 Ya. I. Frenkel, Kinetic Theory of Liquids 共Clarendon, Oxford, 1946兲. 关8兴 A. Braslau et al., Phys. Rev. A 38, 2457 共1988兲. 关9兴 H. Tostmann et al., Phys. Rev. B 61, 7284 共2000兲. 关10兴 R.K. Heilmann, M. Fukuto, and P.S. Pershan, Phys. Rev. B 63, 205405 共2001兲. 关11兴 Martin-D. Lacasse, G.S. Grest, and A.J. Levine, Phys. Rev. Lett. 80, 309 共1998兲.
关12兴 S.W. Sides, G.S. Grest, and Martin-D. Lacasse, Phys. Rev. E 60, 6708 共1999兲. 关13兴 C. Fradin, A. Braslau, and D. Luzet, Nature 共London兲 403, 871 共2000兲. 关14兴 B.A. Zon, Phys. Lett. A 292, 203 共2001兲. 关15兴 F.P. Buff and R.A. Lovett, Phys. Rev. Lett. 15, 621 共1965兲. 关16兴 J. S. Rowlinson and B. Widom, Molecular Theory of Capillarity 共Clarendon, Oxford, 1982兲. 关17兴 S.V. Stepanov, V.M. Byakov, and O.P. Stepanova, Russ. J. Phys. Chem. 74, S65 共2000兲.
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