- Email: [email protected]
Wetting problems for coatings on windshields
Applied Surface Science 142 Ž1999. 375–380
Wetting problems for coatings on windshields M. Bostrom ¨ ) , Bo E. Sernelius Department of Physics and Measurement Technology, Linkoping UniÕersity, S-581 83 Linkoping, Sweden ¨ ¨
Abstract We have calculated the wetting angle as function of doping concentration for water on ITO ŽIn 2 O 3:Sn. and determined the critical concentration for spreading. The calculation relies on the dielectric properties of water and the doped semiconductor. We have modelled these properties. For water we have used experimental optical data and the temperature dependence of the experimental surface tension as input to the modelling. The modelling of the doped semiconductor is based on a model by Penn wD.R. Penn, Phys. Rev. 128 Ž1962. 2093x, extended to take into account the contribution from the free carriers. q 1999 Elsevier Science B.V. All rights reserved. PACS: 68.10.Cr; 68.45.Gd; 73.20.Mf; 77.90.q k Keywords: Indium oxide; ITO ŽIn 2 O 3 :Sn.; Water; Surface wetting; Surface energy
1. Introduction The doped oxide semiconductor ITO ŽIn 2 O 3 :Sn. can be used in many applications. It is transparent in the visible range of the spectrum and reflecting in the infrared w1,2x. Films of ITO are ideal coatings for energy efficient windows since they combine the desired radiative performance with good chemical inertness and strong adherence to glass w3,4x. The wavelength where the material changes from having high transmittance to having high reflectance can be changed continuously by varying the dopant concentration. This means that one may tailor the optical properties of the films w5–8x. Another very useful combination of properties possessed by ITO is optical transparency and metallic conductivity. This means that the films can be used as transparent, )
Corresponding author. Tel.: q46-13-288948; Fax: q46-13137568; E-mail: [email protected]
metallic contacts in, e.g., electro-optical devices. In the present work we are concerned with another pair of properties held by ITO, viz., optical transparency and good thermal conductivity. During cold and clear nights, objects facing the cloud-free sky radiate heat through the optical window of the atmosphere in the infra-red region of the spectrum. The side of the object in the direction of the sky is cooled due to its radiation. Other sides facing other objects or buildings will not be affected as much. These sides are emitting and reabsorbing approximately equal amount of energy. There is almost an equilibrium set up between the objects. If the object is a good heat conductor, heat is transferred from its warmer sides to its cooled side, and the whole object will keep a temperature close to the one of the surroundings. If the object instead is a poor conductor, the side facing the sky will be colder than the surroundings and water will condense and freeze on its surface. This is what we often, to our
0169-4332r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 4 3 3 2 Ž 9 8 . 0 0 6 8 2 - 5
376
M. Bostrom, ¨ B.E. Serneliusr Applied Surface Science 142 (1999) 375–380
annoyance, experience on the windshields of our cars. Sometimes the windshield is icy while the rest of the car is free of ice. One has tried to overcome this problem by coating the outer surface of the windshield with ITO. Unfortunately, it turns out that one runs into another problem. The new problem is to get water off the windshield on rainy days. The water has a tendency to stick to the coating. The material wets too much w9x. We have made a theoretical study of the wetting properties of ITO as function of doping concentration and determined the critical concentrations for spreading. The surface energies for water–air, ITO– air and water–ITO were calculated. These surface energies determine the wetting properties. Lifshitz w10x presented a theory that can be used to calculate the free energy of attraction between macroscopic objects. If the dielectric functions of two interacting objects are known, we can calculate the vdW Žvan der Waals. energy between them for different separations. Using dielectric functions that have both frequency and wave vector dependence makes it possible to calculate this vdW energy at zero separation between the objects. This energy is shared among two surfaces and constitutes an important source of surface tension. In Section 2 we present the theory needed to calculate the surface energy and the wetting angle. Section 3 is devoted to the study of the dielectric function of water. We use the experimental values of the refractive index and the temperature-dependent surface tension to determine this. In Section 4 we describe a qualitative model for the dielectric function of ITO. In Section 5 we present the calculated wetting angles as functions of doping concentrations for different temperatures and finally in Section 6 we summarise our result. 2. Theory Since the water droplet and the ITO film are of macroscopic sizes, we can consider the interfaces as planar when we calculate the surface energy. Parseigan and Ninham w11x investigated the temperature-dependent vdW forces within the Lifshitz theory w12x and found that for some materials, e.g., water, it is important to take temperature effects into account.
This is partly so because in water an important contribution to the free energy is the orientation correlation of the polar molecules. This effect, in the microwave region, is neglected in the usual zero temperature approximation w13x. In the non-retarded limit, the free energy of attraction between media 1 and 2 across a gap Žwith dielectric function ´ 0 . can be written as: G Ž d,T . k BT ( 2p
` X
Ý ns0
`
Hj '´ n
0
rc
d qq ln Ž 1 y D10 D20 ey2 q d .
Ž 1. where
Di0 s
´ i Ž q,i j n . y ´ 0
;
´ i Ž q,i j n . q ´ 0
j n s Ž 2p nk B T . r"
Ž 2.
The prime on the summation indicates that the term n s 0 is to be taken with a factor 1r2. Here 2p " is the Planck’s constant, c is the velocity of light, T is the absolute temperature and k B is the Boltzmann constant. In the limit of zero separation, this energy is minus twice the vdW contribution to the surface energy. The contact angle can be calculated through the following expressions w14x:
sw cos u q sws s ss ; W s sw q ss y sws
Ž 3.
where W is the work to separate water and ITO to infinity and sw , sws and ss are the surface energies of the interfaces between water–air, water–ITO and ITO–air, respectively. We use these expressions to calculate the contact angle u in Section 5. 3. Dielectric function of water We use the refractive index Ž n. and extinction coefficient Ž k . for water w15x to calculate the dielectric function, for real frequencies, in the long wavelength limit, at 208C. In the calculation of surface energies, we use the dielectric function for imaginary frequencies,
a Ž i vX . s a Ž j . s
`
H0
dv
p
ž
v Xa 1 Ž v . q va 2 Ž v . v 2 q vX 2
/
.
Ž 4. Once this is evaluated it is straightforward, with the use of some elemental knowledge of the excitation
M. Bostrom, ¨ B.E. Serneliusr Applied Surface Science 142 (1999) 375–380
channels of water, to construct a model dielectric function that fits this quite well:
´ H 2 O Ž 0, j ,T . T293
a rot Ž 0, j .
q a vib Ž 0, j . q a eabs Ž 0, j . T . 1 q b Ž T . Ž T y T293 . Ž 5.
s1q
The denominator is added to compensate for the volume expansion of water w14,16x. The second temperature correction Žin the numerator. takes into account the inverse temperature dependence of the Debye rotational relaxation w17x. The first contribution to the dielectric behaviour is due to this rotational alignment of dipoles; the other two parts are due to the vibrational modes and electronic absorption in the region from 7.2 eV up to 45 eV, respectively. We find: 68.98 j
a rot Ž 0, j . s 1q
5.6 j
q 1q
1.06 = 10 11
;
1.0 = 10 12
0.67
a vib Ž 0, j . s
j2 1q
Ž 1.5 = 10 13 .
2
j 1q
q
PH
Ž q,i v . s
2 V PH 2 v TO q v 2 q G PH v
u Ž K B .Z .y q . .
Ž 7.
;
j2
Ž 1.906 = 10 14 .
0.82
j2 1q
The complex dielectric function of heavily doped semiconductors can in many cases be separated into different contributions w18x. The first part that we take into account is the polar optical phonon. In 2 O 3 has a C-type rare-earth structure with 80 atoms in the ˚ We unit cell and a lattice parameter of 10.117 A. make a spherical approximation for the primitive cell to estimate the Brillouin wave vector, K B.Z., and find it is around 3.85 = 10 9 my1 . The phonon polarizability, a PH , existing only below K B.Z., for one single resonance frequency can be described by:
13 2
0.35
a eabs Ž 0, j . s
4. Model dielectric function of ITO
2
Ž 4 = 10 .
1q
coupling fades away depends on the system. For phonons it is on the scale of the Brillouin wave vector; for metals the proper scale is the inverse Thomas Fermi screening length; for atomic excitations Žmolecular vibrations. it should be of the order of the inverse size of the atom Žmolecule.. We treat the contributions as constants up to a cut-off. We use the experimentally found surface tension of water and its temperature dependence as a guide when we choose the wave vector cut-off. Our choices are discussed further in Section 5.
a
0.7 q
377
.
2
Ž 6.
According to Hamberg w19x, three main phonon resonances dominate. We assume that the total phonon polarizability can be written as a sum of these Žsee Table 1.. Penn w20x derived a model dielectric function for an isotropic semiconductor. We use this to describe, in an approximate way, the wave vector dependence of the background screening due to the polarizability of the valence electrons:
4.6 = 10 32
In the calculations we also need the wave vector dependence of these contributions. These are much more difficult to find from experiments since optical measurements only probe the long-wave-length limit. In general the wave vector dependence is an effect of the fact that the system cannot respond to fields that have a very fast spatial variation. How fast the
´` y 1
´ penn Ž q,0 . s 1 q
q
2 2
;
ž ž // 1q
2 kF
ž / qc
2
s
4 EF Eg
)
1y
qc
Eg 4 EF
1 q 3
Eg
ž / 4 EF
2
.
Ž 8.
M. Bostrom, ¨ B.E. Serneliusr Applied Surface Science 142 (1999) 375–380
378
Table 1 The In 2 O 3 data for polar optical phonons in Ivar Hamberg’s thesis Phonon resonance
v TO
G PH
V PH
v1 v2 v3
412 cmy1 f 51.0 meV 365 cmy1 f 45.2 meV 330 cmy1 f 40.9 meV
5 cmy1 f 0.62 meV 12 cmy1 f 1.49 meV 16 cmy1 f 1.98 meV
330 cmy1 f 40.9 meV 600 cmy1 f 74.4 meV 450 cmy1 f 55.8 meV
In the long wavelength limit, Hamberg found that ´` s 3.95. The optical bandgap, Eg , is taken to be 3.75 eV w19,21x. We make the following ansatz for the dielectric function: a ´ Ž q,i v . ( 1 q 2 . Ž 9. v q v 12 Ž q . Demanding that it follows Penn’s approximation and obeys the f-sum rule finally yields:
v p2
a VE Ž q,i v . f v2q
v p2 ´` y 1
q
2 2
.
Ž 10 .
ž ž // 1q
qc
We use the bare electron mass when we calculate the Fermi energy and the plasmon frequency. For the number of electrons that participate, we use the partial density of states calculated by Albanesi et al. w22x. We estimate that roughly three electrons per In 2 O 3 will contribute. This means a valence electron density of 4.64 = 10 28 my3 , which gives v p s 1.22 = 10 16 radrs and qc s 1.044 = 10 10 my1 . With experimental data for the surface energy of ITO, it would be possible, within the model, to estimate this more properly. The free carrier contribution finally, both intrinsic and due to doping with tin, is assumed to be given by the random-phase approximation:
a FC Ž Q,iW . s
y
1
2p Q 2
=ln W y Q
½
1q
" v s 4 EFW ;
EF s
" 2 k F2 2 mUn
.
Ž 12 .
We use mUn s m e w0.3 q 0.06599Ž nr10 20 .1r3 x w23x. Alternatively, one can use 0.35m e for simplicity, with quite similar results.
5. Results and discussion The surface energy of water is strongly temperature-dependent ŽFig. 1. and we use this to find a dielectric function that gives approximately the same result as the experiment. All model parameters except the cut-off wave numbers are determined by the fit to the experimental optical constants. The cut-off wave numbers were estimated from the fitting to the temperature-dependent surface tension of water. For the vibrational modes, we used qc s 7.8 = 10 9 my1 and for the electronic absorption qc s 6.4 = 10 9 my1 . Thus, both have a magnitude of the order of the inverse mean separation between molecules. Water has an anomalous dielectric function. In fact if one treats v s 0 Ženters the n s 0 term in Eq. Ž1.. and
4Q 3 2
W 2 q Q2 Ž 1 y Q.
2
qtany1
q s 2 k F Q;
W 2 q Q2 y Q4
W 2 q Q2 Ž 1 q Q.
tany1
where
QŽ1 q Q. W
QŽ1 y Q. W
5
Ž 11 .
Fig. 1. The surface energy of water as function of temperature. The circles correspond to the experimental values w16x and the solid line to the calculated surface energy.
M. Bostrom, ¨ B.E. Serneliusr Applied Surface Science 142 (1999) 375–380
v ) 0 Ženters the n / 0 terms in Eq. Ž1.. on equal footing one could find, contrary to experiment, that the surface energy increases with temperature. To get the right temperature dependence, one has to favour the higher frequencies. In order to find the best agreement with experiment, we chose different cutoffs for the v s 0 Ž qc s 5.6 = 10 9 my1 . and v ) 0 Ž qc s 2.9 = 10 11 my1 . rotational contributions. Another way to achieve a similar effect would be to increase the electronic absorption cut-off. That way, however, gives less accurate temperature dependence. The reason for our difficulties could possibly be that we are neglecting some other temperature dependence in the dielectric function; more experimental data can clarify this. We have furthermore studied how the surface energy of the model semiconductor depends on the doping concentration. We find for undoped In 2 O 3 a surface energy of 0.227 Nrm. The work needed to separate water and In 2 O 3 is 0.125 Nrm at room temperature. We note that if this work becomes at least twice the surface energy of water there will be spreading. The corresponding values for a doped sample with a carrier density of 8 = 10 26 my3 are 0.253 Nrm and 0.134 Nrm, respectively. Knowledge of the experimental values of the surface energies and the wetting angles can be used as a test for any dielectric function. The proper dielectric functions of glass and ITO would enable the calculation of the vdW contribution to the adhesion strength. It is suggested to influence the measured pull-off strength between glass and ITO w24x. The wetting angle ŽFig. 2. decreases as temperature or doping concentration increases. We find that over a critical doping concentration the spreading of water will be complete, making it difficult to remove the water. This critical concentration decreases with temperature. The value of the surface tension and the critical doping concentration depend on the dielectric functions of the interacting materials. Within the model dielectric function for ITO, the critical parameter is the density of electrons contributing to the background screening. It is straightforward to extend the formalism to include a thin layer covering the ITO surface. We studied the effect of applying a thin layer of undoped indium oxide over an ITO surface. We find that for a ˚ ., or thicker, layer one lattice parameter thick Ž10 A
379
Fig. 2. The wetting angle for ITO as a function of doping concentration. The wetting angle decreases with doping concentration and with increasing temperature.
the surface energy will be dominated by the thin layer. In the calculation we use the bulk dielectric properties to represent the thin film. This might be debatable. According to Agranovich w25x, it is safe ˚ . much larger than the for transition layers Ž10–100 A lattice constant. Our layer is thinner, but it still suggests that it might be possible to produce an ice-free and non-wetting windscreen by covering the ITO surface with a thin film.
6. Conclusions The goal of this study has been to investigate the wetting properties of heavily doped indium oxide used as coating on windshields. Our result that doped indium oxide wets too much is in good agreement with experiment. We suggest that it might be possible to produce an ice-free and non-wetting wind screen if the ITO surface is covered with a thin film of, e.g., undoped indium oxide. This film must not be too thick. Thermal equilibrium between different parts of the window should easily be established. Our treatment relies on many uncertain parameters that should be determined from experiments. We hope that our work will inspire further experimental and theoretical research in this area.
Acknowledgements Financial support from the Swedish Natural Science Research Council is acknowledged.
M. Bostrom, ¨ B.E. Serneliusr Applied Surface Science 142 (1999) 375–380
380
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x
C.G. Granqvist, Appl. Opt. 20 Ž1981. 2606. C.G. Granqvist, Proc. SPIE 401 Ž1983. 330. Z.M. Jarzebski, Phys. Status Solidi A 71 Ž1982. 13. K.L. Chopra, S. Major, D.K. Pandya, This Solid Films 102 Ž1983. 1. I. Hamberg, C.G. Granqvist, K.-F. Berggren, B.E. Sernelius, L. Engstrom, ¨ Phys. Rev. B 30 Ž1984. 3240. I. Hamberg, C.G. Granqvist, K.-F. Berggren, B.E. Sernelius, L. Engstrom, ¨ Proc. SPIE 502 Ž1984. 2. I. Hamberg, C.G. Granqvist, K.-F. Berggren, B.E. Sernelius, L. Engstrom, ¨ Solar Energy Mater. 12 Ž1985. 479. I. Hamberg, C.G. Granqvist, K.-F. Berggren, B.E. Sernelius, L. Engstrom, ¨ Vacuum 35 Ž1985. 207. C.G. Granqvist, private communication. E.M. Lifshitz, Sov. Phys. JETP 2 Ž1956. 73 ŽEngl. Transl... V.A. Parseigan, B.W. Ninham, Biophys. J. 10 Ž1970. 664. I.E. Dzyaloshinskii, E.M. Lifshitz, L.P. Pitaevskii, Adv. Phys. 10 Ž1961. 165. V.A. Parseigan, B.W. Ninham, J. Chem. Phys. 52 Ž1970. 4578.
w14x S. Brunauer, L.E. Copeland, in: E.V. Condon ŽEd.., Handbook of Physics, McGraw-Hill, New York, 1967, pp. 5–92. w15x M.R. Querry, D.M. Wieliczka, D.J. Segelstein, Handbook of Optical Constants of Solids II, Academic Press, New York, 1991, p. 1059. w16x I.S. Grigoriev, E.Z. Meilikhov, Handbook of Physical Quantities, CRC Press, New York, 1997, pp. 10.3 and 14.1. w17x P. Debye, Polar Molecules, Chemical Catalog, New York, 1929. w18x G.D. Mahan, Many-Particle Physics, 2nd edn., Plenum, New York, 1993. w19x I. Hamberg, PhD dissertation, Chalmers University of Technology, 1984. w20x D.R. Penn, Phys. Rev. 128 Ž1962. 2093. w21x Bo.E. Sernelius, M. Morling, Thin Solid Films 177 Ž1989. 69. w22x E.A. Albanesi, S.J. Sferco, I. Lefebvre, G. Allen, M. Lannoo, Solid State Commun. 86 Ž1993. 27. w23x A.K. Kulkarni, S.A. Knickerbocker, J. Vac. Sci. Technol. A 14 Ž1996. 1709. w24x W.F. Wu, B.S. Chiou, Thin Solid Films 293 Ž1997. 244. w25x A.A. Agranovich, in: V.M. Agranovich, D.L. Mills ŽEds.., Surface Polaritons, North-Holland, Amsterdam, p. 189.
Wetting problems for coatings on windshields M. Bostrom ¨ ) , Bo E. Sernelius Department of Physics and Measurement Technology, Linkoping UniÕersity, S-581 83 Linkoping, Sweden ¨ ¨
Abstract We have calculated the wetting angle as function of doping concentration for water on ITO ŽIn 2 O 3:Sn. and determined the critical concentration for spreading. The calculation relies on the dielectric properties of water and the doped semiconductor. We have modelled these properties. For water we have used experimental optical data and the temperature dependence of the experimental surface tension as input to the modelling. The modelling of the doped semiconductor is based on a model by Penn wD.R. Penn, Phys. Rev. 128 Ž1962. 2093x, extended to take into account the contribution from the free carriers. q 1999 Elsevier Science B.V. All rights reserved. PACS: 68.10.Cr; 68.45.Gd; 73.20.Mf; 77.90.q k Keywords: Indium oxide; ITO ŽIn 2 O 3 :Sn.; Water; Surface wetting; Surface energy
1. Introduction The doped oxide semiconductor ITO ŽIn 2 O 3 :Sn. can be used in many applications. It is transparent in the visible range of the spectrum and reflecting in the infrared w1,2x. Films of ITO are ideal coatings for energy efficient windows since they combine the desired radiative performance with good chemical inertness and strong adherence to glass w3,4x. The wavelength where the material changes from having high transmittance to having high reflectance can be changed continuously by varying the dopant concentration. This means that one may tailor the optical properties of the films w5–8x. Another very useful combination of properties possessed by ITO is optical transparency and metallic conductivity. This means that the films can be used as transparent, )
Corresponding author. Tel.: q46-13-288948; Fax: q46-13137568; E-mail: [email protected]
metallic contacts in, e.g., electro-optical devices. In the present work we are concerned with another pair of properties held by ITO, viz., optical transparency and good thermal conductivity. During cold and clear nights, objects facing the cloud-free sky radiate heat through the optical window of the atmosphere in the infra-red region of the spectrum. The side of the object in the direction of the sky is cooled due to its radiation. Other sides facing other objects or buildings will not be affected as much. These sides are emitting and reabsorbing approximately equal amount of energy. There is almost an equilibrium set up between the objects. If the object is a good heat conductor, heat is transferred from its warmer sides to its cooled side, and the whole object will keep a temperature close to the one of the surroundings. If the object instead is a poor conductor, the side facing the sky will be colder than the surroundings and water will condense and freeze on its surface. This is what we often, to our
0169-4332r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 4 3 3 2 Ž 9 8 . 0 0 6 8 2 - 5
376
M. Bostrom, ¨ B.E. Serneliusr Applied Surface Science 142 (1999) 375–380
annoyance, experience on the windshields of our cars. Sometimes the windshield is icy while the rest of the car is free of ice. One has tried to overcome this problem by coating the outer surface of the windshield with ITO. Unfortunately, it turns out that one runs into another problem. The new problem is to get water off the windshield on rainy days. The water has a tendency to stick to the coating. The material wets too much w9x. We have made a theoretical study of the wetting properties of ITO as function of doping concentration and determined the critical concentrations for spreading. The surface energies for water–air, ITO– air and water–ITO were calculated. These surface energies determine the wetting properties. Lifshitz w10x presented a theory that can be used to calculate the free energy of attraction between macroscopic objects. If the dielectric functions of two interacting objects are known, we can calculate the vdW Žvan der Waals. energy between them for different separations. Using dielectric functions that have both frequency and wave vector dependence makes it possible to calculate this vdW energy at zero separation between the objects. This energy is shared among two surfaces and constitutes an important source of surface tension. In Section 2 we present the theory needed to calculate the surface energy and the wetting angle. Section 3 is devoted to the study of the dielectric function of water. We use the experimental values of the refractive index and the temperature-dependent surface tension to determine this. In Section 4 we describe a qualitative model for the dielectric function of ITO. In Section 5 we present the calculated wetting angles as functions of doping concentrations for different temperatures and finally in Section 6 we summarise our result. 2. Theory Since the water droplet and the ITO film are of macroscopic sizes, we can consider the interfaces as planar when we calculate the surface energy. Parseigan and Ninham w11x investigated the temperature-dependent vdW forces within the Lifshitz theory w12x and found that for some materials, e.g., water, it is important to take temperature effects into account.
This is partly so because in water an important contribution to the free energy is the orientation correlation of the polar molecules. This effect, in the microwave region, is neglected in the usual zero temperature approximation w13x. In the non-retarded limit, the free energy of attraction between media 1 and 2 across a gap Žwith dielectric function ´ 0 . can be written as: G Ž d,T . k BT ( 2p
` X
Ý ns0
`
Hj '´ n
0
rc
d qq ln Ž 1 y D10 D20 ey2 q d .
Ž 1. where
Di0 s
´ i Ž q,i j n . y ´ 0
;
´ i Ž q,i j n . q ´ 0
j n s Ž 2p nk B T . r"
Ž 2.
The prime on the summation indicates that the term n s 0 is to be taken with a factor 1r2. Here 2p " is the Planck’s constant, c is the velocity of light, T is the absolute temperature and k B is the Boltzmann constant. In the limit of zero separation, this energy is minus twice the vdW contribution to the surface energy. The contact angle can be calculated through the following expressions w14x:
sw cos u q sws s ss ; W s sw q ss y sws
Ž 3.
where W is the work to separate water and ITO to infinity and sw , sws and ss are the surface energies of the interfaces between water–air, water–ITO and ITO–air, respectively. We use these expressions to calculate the contact angle u in Section 5. 3. Dielectric function of water We use the refractive index Ž n. and extinction coefficient Ž k . for water w15x to calculate the dielectric function, for real frequencies, in the long wavelength limit, at 208C. In the calculation of surface energies, we use the dielectric function for imaginary frequencies,
a Ž i vX . s a Ž j . s
`
H0
dv
p
ž
v Xa 1 Ž v . q va 2 Ž v . v 2 q vX 2
/
.
Ž 4. Once this is evaluated it is straightforward, with the use of some elemental knowledge of the excitation
M. Bostrom, ¨ B.E. Serneliusr Applied Surface Science 142 (1999) 375–380
channels of water, to construct a model dielectric function that fits this quite well:
´ H 2 O Ž 0, j ,T . T293
a rot Ž 0, j .
q a vib Ž 0, j . q a eabs Ž 0, j . T . 1 q b Ž T . Ž T y T293 . Ž 5.
s1q
The denominator is added to compensate for the volume expansion of water w14,16x. The second temperature correction Žin the numerator. takes into account the inverse temperature dependence of the Debye rotational relaxation w17x. The first contribution to the dielectric behaviour is due to this rotational alignment of dipoles; the other two parts are due to the vibrational modes and electronic absorption in the region from 7.2 eV up to 45 eV, respectively. We find: 68.98 j
a rot Ž 0, j . s 1q
5.6 j
q 1q
1.06 = 10 11
;
1.0 = 10 12
0.67
a vib Ž 0, j . s
j2 1q
Ž 1.5 = 10 13 .
2
j 1q
q
PH
Ž q,i v . s
2 V PH 2 v TO q v 2 q G PH v
u Ž K B .Z .y q . .
Ž 7.
;
j2
Ž 1.906 = 10 14 .
0.82
j2 1q
The complex dielectric function of heavily doped semiconductors can in many cases be separated into different contributions w18x. The first part that we take into account is the polar optical phonon. In 2 O 3 has a C-type rare-earth structure with 80 atoms in the ˚ We unit cell and a lattice parameter of 10.117 A. make a spherical approximation for the primitive cell to estimate the Brillouin wave vector, K B.Z., and find it is around 3.85 = 10 9 my1 . The phonon polarizability, a PH , existing only below K B.Z., for one single resonance frequency can be described by:
13 2
0.35
a eabs Ž 0, j . s
4. Model dielectric function of ITO
2
Ž 4 = 10 .
1q
coupling fades away depends on the system. For phonons it is on the scale of the Brillouin wave vector; for metals the proper scale is the inverse Thomas Fermi screening length; for atomic excitations Žmolecular vibrations. it should be of the order of the inverse size of the atom Žmolecule.. We treat the contributions as constants up to a cut-off. We use the experimentally found surface tension of water and its temperature dependence as a guide when we choose the wave vector cut-off. Our choices are discussed further in Section 5.
a
0.7 q
377
.
2
Ž 6.
According to Hamberg w19x, three main phonon resonances dominate. We assume that the total phonon polarizability can be written as a sum of these Žsee Table 1.. Penn w20x derived a model dielectric function for an isotropic semiconductor. We use this to describe, in an approximate way, the wave vector dependence of the background screening due to the polarizability of the valence electrons:
4.6 = 10 32
In the calculations we also need the wave vector dependence of these contributions. These are much more difficult to find from experiments since optical measurements only probe the long-wave-length limit. In general the wave vector dependence is an effect of the fact that the system cannot respond to fields that have a very fast spatial variation. How fast the
´` y 1
´ penn Ž q,0 . s 1 q
q
2 2
;
ž ž // 1q
2 kF
ž / qc
2
s
4 EF Eg
)
1y
qc
Eg 4 EF
1 q 3
Eg
ž / 4 EF
2
.
Ž 8.
M. Bostrom, ¨ B.E. Serneliusr Applied Surface Science 142 (1999) 375–380
378
Table 1 The In 2 O 3 data for polar optical phonons in Ivar Hamberg’s thesis Phonon resonance
v TO
G PH
V PH
v1 v2 v3
412 cmy1 f 51.0 meV 365 cmy1 f 45.2 meV 330 cmy1 f 40.9 meV
5 cmy1 f 0.62 meV 12 cmy1 f 1.49 meV 16 cmy1 f 1.98 meV
330 cmy1 f 40.9 meV 600 cmy1 f 74.4 meV 450 cmy1 f 55.8 meV
In the long wavelength limit, Hamberg found that ´` s 3.95. The optical bandgap, Eg , is taken to be 3.75 eV w19,21x. We make the following ansatz for the dielectric function: a ´ Ž q,i v . ( 1 q 2 . Ž 9. v q v 12 Ž q . Demanding that it follows Penn’s approximation and obeys the f-sum rule finally yields:
v p2
a VE Ž q,i v . f v2q
v p2 ´` y 1
q
2 2
.
Ž 10 .
ž ž // 1q
qc
We use the bare electron mass when we calculate the Fermi energy and the plasmon frequency. For the number of electrons that participate, we use the partial density of states calculated by Albanesi et al. w22x. We estimate that roughly three electrons per In 2 O 3 will contribute. This means a valence electron density of 4.64 = 10 28 my3 , which gives v p s 1.22 = 10 16 radrs and qc s 1.044 = 10 10 my1 . With experimental data for the surface energy of ITO, it would be possible, within the model, to estimate this more properly. The free carrier contribution finally, both intrinsic and due to doping with tin, is assumed to be given by the random-phase approximation:
a FC Ž Q,iW . s
y
1
2p Q 2
=ln W y Q
½
1q
" v s 4 EFW ;
EF s
" 2 k F2 2 mUn
.
Ž 12 .
We use mUn s m e w0.3 q 0.06599Ž nr10 20 .1r3 x w23x. Alternatively, one can use 0.35m e for simplicity, with quite similar results.
5. Results and discussion The surface energy of water is strongly temperature-dependent ŽFig. 1. and we use this to find a dielectric function that gives approximately the same result as the experiment. All model parameters except the cut-off wave numbers are determined by the fit to the experimental optical constants. The cut-off wave numbers were estimated from the fitting to the temperature-dependent surface tension of water. For the vibrational modes, we used qc s 7.8 = 10 9 my1 and for the electronic absorption qc s 6.4 = 10 9 my1 . Thus, both have a magnitude of the order of the inverse mean separation between molecules. Water has an anomalous dielectric function. In fact if one treats v s 0 Ženters the n s 0 term in Eq. Ž1.. and
4Q 3 2
W 2 q Q2 Ž 1 y Q.
2
qtany1
q s 2 k F Q;
W 2 q Q2 y Q4
W 2 q Q2 Ž 1 q Q.
tany1
where
QŽ1 q Q. W
QŽ1 y Q. W
5
Ž 11 .
Fig. 1. The surface energy of water as function of temperature. The circles correspond to the experimental values w16x and the solid line to the calculated surface energy.
M. Bostrom, ¨ B.E. Serneliusr Applied Surface Science 142 (1999) 375–380
v ) 0 Ženters the n / 0 terms in Eq. Ž1.. on equal footing one could find, contrary to experiment, that the surface energy increases with temperature. To get the right temperature dependence, one has to favour the higher frequencies. In order to find the best agreement with experiment, we chose different cutoffs for the v s 0 Ž qc s 5.6 = 10 9 my1 . and v ) 0 Ž qc s 2.9 = 10 11 my1 . rotational contributions. Another way to achieve a similar effect would be to increase the electronic absorption cut-off. That way, however, gives less accurate temperature dependence. The reason for our difficulties could possibly be that we are neglecting some other temperature dependence in the dielectric function; more experimental data can clarify this. We have furthermore studied how the surface energy of the model semiconductor depends on the doping concentration. We find for undoped In 2 O 3 a surface energy of 0.227 Nrm. The work needed to separate water and In 2 O 3 is 0.125 Nrm at room temperature. We note that if this work becomes at least twice the surface energy of water there will be spreading. The corresponding values for a doped sample with a carrier density of 8 = 10 26 my3 are 0.253 Nrm and 0.134 Nrm, respectively. Knowledge of the experimental values of the surface energies and the wetting angles can be used as a test for any dielectric function. The proper dielectric functions of glass and ITO would enable the calculation of the vdW contribution to the adhesion strength. It is suggested to influence the measured pull-off strength between glass and ITO w24x. The wetting angle ŽFig. 2. decreases as temperature or doping concentration increases. We find that over a critical doping concentration the spreading of water will be complete, making it difficult to remove the water. This critical concentration decreases with temperature. The value of the surface tension and the critical doping concentration depend on the dielectric functions of the interacting materials. Within the model dielectric function for ITO, the critical parameter is the density of electrons contributing to the background screening. It is straightforward to extend the formalism to include a thin layer covering the ITO surface. We studied the effect of applying a thin layer of undoped indium oxide over an ITO surface. We find that for a ˚ ., or thicker, layer one lattice parameter thick Ž10 A
379
Fig. 2. The wetting angle for ITO as a function of doping concentration. The wetting angle decreases with doping concentration and with increasing temperature.
the surface energy will be dominated by the thin layer. In the calculation we use the bulk dielectric properties to represent the thin film. This might be debatable. According to Agranovich w25x, it is safe ˚ . much larger than the for transition layers Ž10–100 A lattice constant. Our layer is thinner, but it still suggests that it might be possible to produce an ice-free and non-wetting windscreen by covering the ITO surface with a thin film.
6. Conclusions The goal of this study has been to investigate the wetting properties of heavily doped indium oxide used as coating on windshields. Our result that doped indium oxide wets too much is in good agreement with experiment. We suggest that it might be possible to produce an ice-free and non-wetting wind screen if the ITO surface is covered with a thin film of, e.g., undoped indium oxide. This film must not be too thick. Thermal equilibrium between different parts of the window should easily be established. Our treatment relies on many uncertain parameters that should be determined from experiments. We hope that our work will inspire further experimental and theoretical research in this area.
Acknowledgements Financial support from the Swedish Natural Science Research Council is acknowledged.
M. Bostrom, ¨ B.E. Serneliusr Applied Surface Science 142 (1999) 375–380
380
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