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Do stresses modify wetting angles?
Acta mater. 49 (2001) 1005–1007 www.elsevier.com/locate/actamat
DO STRESSES MODIFY WETTING ANGLES? D. J. SROLOVITZ1† and S. H. DAVIS2 1
Princeton Materials Institute and Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA and 2Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208, USA ( Received 17 July 2000; received in revised form 21 November 2000; accepted 21 November 2000 )
Abstract—The wetting angles at triple lines in solid–solid systems give rise to singularities in the elastic field. This paper presents a transparent analysis of the effect of stress on the wetting angles that describe such equilibrium three-phase junctions. 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Thin films; Thermodynamics; Theory & modeling—surfaces
The angles describing the intersection of three interfaces, separating different material phases are known as wetting angles. This term has its origins in the fluid-mechanics literature [1], where it is used to describe the angle at which an isolated, liquid island meets the solid substrate on which it rests (see Fig. 1). In equilibrium, the wetting angle is determined by energy minimization at the triple line (i.e. the corner). In analyses of islands and other three-phase systems, the wetting angle provides the key boundary condition in determining three-phase interface morphologies. This same approach is routinely used to describe morphologies where intersections of solid–solid interfaces occur, e.g. a solid island on a substrate. When the solids are elastic,
Fig. 1. Illustration of the intersection of a solid or liquid island with a solid substrate. q is the wetting (Young–Dupree) angle and gsv, gis, and gvi are the interfacial tensions associated with substrate–vapor, island–substrate and vapor–island interfaces, respectively.
† To whom all correspondence should be addressed. Tel.: ⫹1-609-258-5138; Fax: ⫹1-609-258-5877. E-mail address: [email protected] (D.J. Srolovitz)
the linear elastic stress fields diverge at the corner of the wedge formed near the triple line. In this report, we examine the question of whether these singular elastic fields are capable of modifying the equilibrium wetting angles. Consider an island on a planar, rigid substrate. In the absence of elastic stresses, interfacial thermodynamics defines [1] the equilibrium angle q in the configuration of Fig. 1 to satisfy cos q ⫽ (gsv⫺gis)/gvi (the Young–Dupree equation), where gvi, gsv and gis are the vapor–island, substrate–vapor and island–substrate interfacial energies, respectively (Ref. 2 discusses alternate approaches). The angle chosen serves as a boundary condition on the shape of the interface for equilibrium configurations even in dynamic situations, as long as the departure from equilibrium is sufficiently small. If r measures the distance from the corner, one is interested in the behavior for r→0. When elastic deformation is present, the local stresses and strains near corners of wedges of arbitrary opening angles are known [3, 4]. The stress- and strain-field singularities are of the form s苲Ar⫺a, where A and a are constants that depend on the material’s properties, geometry and far-field loading. The exponent a is found as a solution to a transcendental equation and is shown versus q in Fig. 2 for the geometry of Fig. 1; a increases from 0 to 1/2 as the angle q increases from 0 to p. The elastic energy density w in the material is proportional to the product of the stress and strain and hence behaves with distance from the corner as r⫺2a. In order to determine whether the elastic fields change the equilibrium wetting angle from that determined by interfacial tension alone, we analyze
1359-6454/01/$20.00 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 0 1 ) 0 0 0 0 4 - 0
1006
SROLOVITZ and DAVIS: WETTING ANGLES
Fig. 3. Illustration of the intersection of Fig. 1 for either a rigid, non-planar substrate or a compliant substrate. The dashed lines represent tangent planes. Fig. 2. Plot of the elastic field exponent a versus the wetting angle for the case where the island and substrate have the same elastic constants [3, 4].
the asymptotic behavior of the interface-tension and strain-energy singularities. To this end, we calculate these two contributions to the energy within a circle of radius R centered at the corner:
冕
Eelastic苲 wr dr苲
1 R2(a⫺1)
冕
Einterface苲 (gvi ⫹ gsv ⫹ gis) dr苲R
(1a)
(1b)
In the limit that R→0, the ratio of these energies is
lim
冉
R→0
冦
⬁ a⬍1/2 Einterface ⫽ 1 a⫽1/2 Eelastic 0 a>1/2
冊
(2)
Equation (2) implies that the order of the singularity in the energy at the triple line is dominated by the interfacial energy for a⬍1/2 and the contribution from the elastic energy singularity is negligible. The two are equally important when a ⫽ 1/2 and the elastic energy is dominant when a>1/2. Since a is constrained to lie in the range 0ⱕaⱕ1/2, the elastic singularities are never solely dominant. Apart from the special value of the wetting angle of q ⫽ p where a ⫽ 1/2, discussed below, we find that the singularity associated with elastic fields is weaker than that due to interfacial energy and hence cannot modify the wetting angle. This shows that the common usage of the Young–Dupree wetting angle in determining morphologies in multiphase solid systems is justified. We return briefly to the special case of a ⫽ 1/2, which occurs when q ⫽ p in Fig. 1. This geometry corresponds to the case of a crack (i.e. a slit across which there is no bonding). In linear elastic fracture mechanics, a crack will propagate if and only if the strain energy release associated with growing the crack is larger than the resultant increase in surface
energy associated with the new surface created. In other words, the fact that the singularities in the elastic fields is of the same order as that associated with the interfacial energy in the crack geometry, derived here, is the foundation of fracture mechanics. An analysis of the finite angle crack root is given in Ref. 5. In all of the above, the substrate has been assumed to remain planar. If the substrate is not planar, then Fig. 3 shows an example of the local structure. In a sufficiently small neighborhood of the triple line, the geometry between tangent planes is precisely that of Fig. 1, as long as all interfaces are smooth on the scale of observation. One has a local wedge of angle q within which one calculates the elastic energy and the above conclusions hold. If the substrate undergoes significant deformation, then Fig. 3 again shows a view of the local geometry. The situation is now more complex. If, as usual, the vapor–island interfacial energy is concentrated (rather than diffuse), the triple line is the site of a delta function in displacement for the linearly elastic substrate. If, however, one relaxes the interface into a thin interfacial region, or one utilizes non-linear elasticity, one can estimate the displacement [6]. It turns out to ˚ , even for relatively soft materials. Thus, be about 1 A in continuum mechanics, the deformation is negligible, and the case of a planar substrate is effectively regained. If the substrate is so soft that a substantial ridge is created, one should then solve for the elastic energies in the two solid wedges by methods similar to those described above. Finally, we note that if matter flow can occur in the substrate (e.g. by interfacial, surface or bulk diffusion or by plastic flow), then the Young–Dupree relation represents normal, as well as tangential balances. If matter flow occurs on a small scale, this can give rise to ridges created by mass transport that also modify the classic Young–Dupree relation [7]. In conclusion, elastic effects in solids are incapable of modifying the classic wetting angle, determined by interfacial tensions, except in crack-like geometries. Acknowledgements—The authors gratefully acknowledge enlightening discussions with Prof. Zhigang Suo. This work was supported by the (DJS) US Department of Energy through DE-FG02-99ER45797 and by (SHD) NASA, Microgravity Sciences and Applications Program.
SROLOVITZ and DAVIS: WETTING ANGLES REFERENCES 1. Dussan, E. B., Ann. Rev. Fluid Mech., 1979, 11, 371. 2. Graf, D. and Riegler, H., Langmuir, 2000, 16, 5187. 3. Bogy, D. B., J. Appl. Mech., 1971, 38, 377.
4. 5. 6. 7.
1007
Dundurs, J. and Lee, M. -S., J. Elasticity, 1972, 2, 109. Yu, H. H. and Suo, Z., Acta Mater., 1999, 47, 77. Dussan, E. B., private communication (1971). Saiz, E., Tomsia, A. P. and Cannon, R. M., Acta Mater., 1998, 46, 2349.
DO STRESSES MODIFY WETTING ANGLES? D. J. SROLOVITZ1† and S. H. DAVIS2 1
Princeton Materials Institute and Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA and 2Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208, USA ( Received 17 July 2000; received in revised form 21 November 2000; accepted 21 November 2000 )
Abstract—The wetting angles at triple lines in solid–solid systems give rise to singularities in the elastic field. This paper presents a transparent analysis of the effect of stress on the wetting angles that describe such equilibrium three-phase junctions. 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Thin films; Thermodynamics; Theory & modeling—surfaces
The angles describing the intersection of three interfaces, separating different material phases are known as wetting angles. This term has its origins in the fluid-mechanics literature [1], where it is used to describe the angle at which an isolated, liquid island meets the solid substrate on which it rests (see Fig. 1). In equilibrium, the wetting angle is determined by energy minimization at the triple line (i.e. the corner). In analyses of islands and other three-phase systems, the wetting angle provides the key boundary condition in determining three-phase interface morphologies. This same approach is routinely used to describe morphologies where intersections of solid–solid interfaces occur, e.g. a solid island on a substrate. When the solids are elastic,
Fig. 1. Illustration of the intersection of a solid or liquid island with a solid substrate. q is the wetting (Young–Dupree) angle and gsv, gis, and gvi are the interfacial tensions associated with substrate–vapor, island–substrate and vapor–island interfaces, respectively.
† To whom all correspondence should be addressed. Tel.: ⫹1-609-258-5138; Fax: ⫹1-609-258-5877. E-mail address: [email protected] (D.J. Srolovitz)
the linear elastic stress fields diverge at the corner of the wedge formed near the triple line. In this report, we examine the question of whether these singular elastic fields are capable of modifying the equilibrium wetting angles. Consider an island on a planar, rigid substrate. In the absence of elastic stresses, interfacial thermodynamics defines [1] the equilibrium angle q in the configuration of Fig. 1 to satisfy cos q ⫽ (gsv⫺gis)/gvi (the Young–Dupree equation), where gvi, gsv and gis are the vapor–island, substrate–vapor and island–substrate interfacial energies, respectively (Ref. 2 discusses alternate approaches). The angle chosen serves as a boundary condition on the shape of the interface for equilibrium configurations even in dynamic situations, as long as the departure from equilibrium is sufficiently small. If r measures the distance from the corner, one is interested in the behavior for r→0. When elastic deformation is present, the local stresses and strains near corners of wedges of arbitrary opening angles are known [3, 4]. The stress- and strain-field singularities are of the form s苲Ar⫺a, where A and a are constants that depend on the material’s properties, geometry and far-field loading. The exponent a is found as a solution to a transcendental equation and is shown versus q in Fig. 2 for the geometry of Fig. 1; a increases from 0 to 1/2 as the angle q increases from 0 to p. The elastic energy density w in the material is proportional to the product of the stress and strain and hence behaves with distance from the corner as r⫺2a. In order to determine whether the elastic fields change the equilibrium wetting angle from that determined by interfacial tension alone, we analyze
1359-6454/01/$20.00 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 0 1 ) 0 0 0 0 4 - 0
1006
SROLOVITZ and DAVIS: WETTING ANGLES
Fig. 3. Illustration of the intersection of Fig. 1 for either a rigid, non-planar substrate or a compliant substrate. The dashed lines represent tangent planes. Fig. 2. Plot of the elastic field exponent a versus the wetting angle for the case where the island and substrate have the same elastic constants [3, 4].
the asymptotic behavior of the interface-tension and strain-energy singularities. To this end, we calculate these two contributions to the energy within a circle of radius R centered at the corner:
冕
Eelastic苲 wr dr苲
1 R2(a⫺1)
冕
Einterface苲 (gvi ⫹ gsv ⫹ gis) dr苲R
(1a)
(1b)
In the limit that R→0, the ratio of these energies is
lim
冉
R→0
冦
⬁ a⬍1/2 Einterface ⫽ 1 a⫽1/2 Eelastic 0 a>1/2
冊
(2)
Equation (2) implies that the order of the singularity in the energy at the triple line is dominated by the interfacial energy for a⬍1/2 and the contribution from the elastic energy singularity is negligible. The two are equally important when a ⫽ 1/2 and the elastic energy is dominant when a>1/2. Since a is constrained to lie in the range 0ⱕaⱕ1/2, the elastic singularities are never solely dominant. Apart from the special value of the wetting angle of q ⫽ p where a ⫽ 1/2, discussed below, we find that the singularity associated with elastic fields is weaker than that due to interfacial energy and hence cannot modify the wetting angle. This shows that the common usage of the Young–Dupree wetting angle in determining morphologies in multiphase solid systems is justified. We return briefly to the special case of a ⫽ 1/2, which occurs when q ⫽ p in Fig. 1. This geometry corresponds to the case of a crack (i.e. a slit across which there is no bonding). In linear elastic fracture mechanics, a crack will propagate if and only if the strain energy release associated with growing the crack is larger than the resultant increase in surface
energy associated with the new surface created. In other words, the fact that the singularities in the elastic fields is of the same order as that associated with the interfacial energy in the crack geometry, derived here, is the foundation of fracture mechanics. An analysis of the finite angle crack root is given in Ref. 5. In all of the above, the substrate has been assumed to remain planar. If the substrate is not planar, then Fig. 3 shows an example of the local structure. In a sufficiently small neighborhood of the triple line, the geometry between tangent planes is precisely that of Fig. 1, as long as all interfaces are smooth on the scale of observation. One has a local wedge of angle q within which one calculates the elastic energy and the above conclusions hold. If the substrate undergoes significant deformation, then Fig. 3 again shows a view of the local geometry. The situation is now more complex. If, as usual, the vapor–island interfacial energy is concentrated (rather than diffuse), the triple line is the site of a delta function in displacement for the linearly elastic substrate. If, however, one relaxes the interface into a thin interfacial region, or one utilizes non-linear elasticity, one can estimate the displacement [6]. It turns out to ˚ , even for relatively soft materials. Thus, be about 1 A in continuum mechanics, the deformation is negligible, and the case of a planar substrate is effectively regained. If the substrate is so soft that a substantial ridge is created, one should then solve for the elastic energies in the two solid wedges by methods similar to those described above. Finally, we note that if matter flow can occur in the substrate (e.g. by interfacial, surface or bulk diffusion or by plastic flow), then the Young–Dupree relation represents normal, as well as tangential balances. If matter flow occurs on a small scale, this can give rise to ridges created by mass transport that also modify the classic Young–Dupree relation [7]. In conclusion, elastic effects in solids are incapable of modifying the classic wetting angle, determined by interfacial tensions, except in crack-like geometries. Acknowledgements—The authors gratefully acknowledge enlightening discussions with Prof. Zhigang Suo. This work was supported by the (DJS) US Department of Energy through DE-FG02-99ER45797 and by (SHD) NASA, Microgravity Sciences and Applications Program.
SROLOVITZ and DAVIS: WETTING ANGLES REFERENCES 1. Dussan, E. B., Ann. Rev. Fluid Mech., 1979, 11, 371. 2. Graf, D. and Riegler, H., Langmuir, 2000, 16, 5187. 3. Bogy, D. B., J. Appl. Mech., 1971, 38, 377.
4. 5. 6. 7.
1007
Dundurs, J. and Lee, M. -S., J. Elasticity, 1972, 2, 109. Yu, H. H. and Suo, Z., Acta Mater., 1999, 47, 77. Dussan, E. B., private communication (1971). Saiz, E., Tomsia, A. P. and Cannon, R. M., Acta Mater., 1998, 46, 2349.