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Deformation of Soft Elastomeric Layers by Periodic Interfacial Tension Gradients Satish Kumar* Department of Chemical Engineering and Materials Science, University of Minnesota, 151 Amundson Hall, 421 Washington Avenue SE, Minneapolis, Minnesota 55455 Received August 20, 2002. In Final Form: December 2, 2002 Gradients in interfacial tension at a solid-air or solid-solid interface exert tangential stresses which may deform the solid if it is sufficiently soft. We report a theoretical study of this issue by considering the case of spatially periodic interfacial tension gradients that have been patterned onto planar elastomeric layers. Assuming that the gradients are weak, we apply regular perturbation theory, Taylor series, and the theory of linearized elasticity to determine the leading order correction to the interfacial deformation. For solid-air interfaces, is found that the maximum vertical interfacial displacement occurs when the characteristic wavelength of the gradients is of the same order of magnitude as the solid thickness. For solid-solid interfaces, the maximum vertical interfacial displacement occurs when the characteristic wavelength of the gradients is on the order of the thickness of the thinner solid. The effects of varying the thickness and modulus ratios of the solids are also reported. The results suggest that patterned interfacial tension gradients may serve as a novel route for the creation of topographically patterned surfaces and interfaces in soft materials.
1. Introduction Gradients in surface tension at a liquid-air interface can drive a flow in the liquid. These gradients, which may arise due to temperature differences or variations in the concentration of surface-active species, exert tangential stresses at the interface that are balanced by tangential stresses from the liquid flow.1 Gradients in surface tension at a solid-air interface also exert tangential stresses, but the solid must respond by deforming since it cannot flow. For solids of relatively high shear modulus, such as metals, this deformation will be negligible. But for solids of relatively low shear modulus, such as polymer gels and rubbers, this deformation could be discernible. The present manuscript reports a theoretical study of this issue by considering the model case of spatially periodic interfacial tension gradients. There are two main motivations for conducting the study reported here. The first is to gain a fundamental understanding of interfacial phenomena at solid-air and solidsolid interfaces. While there has been a considerable amount of work on liquid flows driven by surface or interfacial tension gradients, there have not been, to our knowledge, prior investigations (either experimental or theoretical) of solid deformations driven by these forces. The second is to determine the extent to which the interface deforms normal to itself. If such normal deformations do occur and can be replicated periodically in space, then interfacial tension gradients could serve as a way of creating topographically patterned interfaces. Such interfaces may have a number of uses, serving as coatings, adhesion modifiers, or microfluidic channels, and the importance of surface topography in practical applications involving soft materials was recently highlighted by Assender, Bliznyuk, and Porfyrakis.2 The technology * Phone: (612) 625 2558. Fax: (612) 626 7246. E-mail: kumar@ cems.umn.edu. (1) Deen, W. M. Analysis of Transport Phenomena; Oxford University Press: New York, 1998. (2) Assender, H.; Bliznyuk, V.; Porfyrakis, K. Science 2002, 297, 973.
Figure 1. Interface between two elastomeric layers having thicknesses H (bottom) and H ˆ (top). Without any interfacial tension gradients, the interface is flat and located at z ) 0. The presence of such gradients will produce tangential stresses at the interface, which may then deform.
required to create spatially varying patterns of interfacial tension (equivalently, interfacial energy3) on solid surfaces exists and is actively being applied to manipulate polymer nanostructures,4 direct microfluidic flows,5 and regulate interfacial self-assembly.6 In this work, the effect of spatially periodic interfacial tension gradients on solid deformation is studied. To make analytical progress, the gradients are assumed to be sufficiently weak so that regular perturbation theory, Taylor series, and the linearized elasticity equations can be applied. Both solidair and solid-solid interfaces are considered. The governing equations and their solution are described in section 2, results are presented in section 3, and a discussion and conclusions are given in section 4. 2. Governing Equations and Solution The problem geometry is shown in Figure 1. A solid layer of thickness H shares an interface with another solid layer of thickness H ˆ . If there are no gradients in the interfacial tension, the interface will be flat and located at z ) 0. The surfaces at z ) -H and z ) H ˆ are assumed to be bonded to a rigid material so that they cannot deform. (3) deGennes, P. G. Rev. Mod. Phys. 1985, 57, 827. (4) Peters, R. D.; Yang, X. M.; Wang, Q.; de Pablo, J. J.; Nealey, P. F. J. Vac. Sci. Technol., B 2000, 18, 3530. (5) Zhao, B.; Moore, J. S.; Beebe, D. J. Science 2001, 291, 1023. (6) Srinivasan, U.; Liepmann, D.; Howe, R. T. J. Microelectromech. Syst. 2001, 10, 17.
10.1021/la020733s CCC: $25.00 © 2003 American Chemical Society Published on Web 02/04/2003
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Kumar
In the presence of interfacial tension gradients, the interface will deform and the interfacial tension can be described by the function
γ(x, z ) ζ(x)) ) γ0 + γˆ (x, z ) ζ(x))
(1)
Here, γ0 is the mean interfacial tension, γˆ is a function which describes the gradients, and is a dimensionless parameter whose magnitude controls the gradient strength. The function z ) ζ(x) describes the location of the interface. To illustrate the basic concepts, it is sufficient to consider two-dimensional deformations. The effects of lateral boundaries are neglected, which is appropriate for interfacial tension gradients that vary on a length scale much smaller than the width of the solid layers. Given a particular functional form of the interfacial tension gradients, we would like to find the function z ) ζ(x). In general, as we shall see, this is a nonlinear problem because boundary conditions need to be applied at an interface whose location is unknown. However, analytical progress can be made by considering the limit of weak gradients, , 1. This allows three important simplifications. First, regular perturbation theory can be used to expand all variables in terms of power series in . Second, the boundary conditions can be applied at the undeformed interface, z ) 0, with the help of Taylor series. Third, the equations of linearized elasticity7 will be appropriate for describing the solid deformation since the deformation gradients will be small when , 1 (there is no deformation when ) 0). We will use F and G to denote the density and shear modulus of the lower solid and use Fˆ and G ˆ to denote the corresponding quantities in the upper solid. If we scale length with H and pressure and stress with G, the nondimensional linearized elasticity equations are
-∇p + ∇2u ) 0
(2)
∇·u ) 0
(3)
G ˆ -∇pˆ + ∇2u ˆ )0 G
(4)
∇·u ˆ )0
(5)
where u and u ˆ are the displacements in the lower and upper solids, respectively. The solids are assumed to be incompressible, as reflected by the mass conservation equations ((3) and (5)), and this gives rise to pressure terms, p and pˆ , in the momentum conservation equations ((2) and (4)). Body forces, which have a strength proportional to FgH/G in the lower solid and Fˆ gH/G in the upper solid, where g is the magnitude of the gravitational acceleration, are assumed to be negligible. At the upper and lower surfaces, which are fixed to a rigid material, we have
and the forces must be in balance:
n·σˆ - n·σ + 2H T + ∇sT ) 0
(9)
where n is the unit vector normal to the interface which points into the top solid, H is the mean curvature of the interface, and the stress tensors are given by
σ ) -pI + ∇u + (∇u)T σˆ ) -pˆ I +
(10)
G ˆ [∇u ˆ + (∇u ˆ )T] G
(11)
with I being the identity tensor. The surface gradient operator is denoted by ∇s ) (I - nn)‚∇, and T represents the dimensionless interfacial tension:
ˆ (x, z ) ζ(x)) T(x, z ) ζ(x)) ) T0 + T
(12)
where T0 ) γ0/GH and T ˆ ) γˆ /GH. The normal (n) and tangential (t) vectors to the interface, as well as the mean curvature, can all be expressed in terms of ζ(x):1 2 -1/2
[ (∂x∂ζ) ] (- ∂x∂ζe + e ) ∂ζ ∂ζ t ) [1 + ( ) ] (e + e ) ∂x ∂x
n) 1+
x
z
2 -1/2
x
2 -3/2∂2ζ
[ (∂x∂ζ) ]
2H ) 1 +
z
∂x2
(13) (14) (15)
where ex and ez are unit vectors in the x- and z-directions. It is clear from these expressions that the boundary conditions are a nonlinear function of the unknown variable ζ, which itself is related to the z-component of the interfacial displacement through the implicit expression ζ(x) ) uz(x, z ) ζ(x)). For , 1, the interfacial deformation will be small, meaning that we can perform a Taylor expansion of all variables that need to be evaluated at the interface. This will allow us to apply the boundary conditions at z ) 0 rather than at the unknown location z ) ζ(x). Furthermore, regular perturbation theory can be applied to develop an approximate solution in a systematic way. This involves expanding all variables in powers of :
f (x, z) ) f (1)(x, z) + 2f (2)(x, z) + O(3)
(16)
(8)
where f is a given variable. (The deformations and pressures are assumed to vanish when ) 0.) The smaller is, the more accurate the perturbation solution is expected to be. These expressions can then be put into the governing equations and boundary conditions, and the Taylor expansions discussed above can be performed. Collecting terms of like powers in will produce linear problems which can be solved to yield the functions f(1)(x, z), f (2)(x, z), and so forth. The assumption , 1 may be appropriate in many cases, and we can estimate as γ*/GH, where γ* provides a measure of the magnitude of the interfacial tension gradients. Taking γ* ) 0.01 J/m2, G ) 103 N/m2, and H ) 10-3 m gives ) 0.01. The chosen values of G and H could be obtained with polymer gels,8 while self-assembled monolayers could be used to modulate the interfacial tension.9
(7) Landau, L. D.; Lifshitz, E. M. Theory of Elasticity, 3rd ed.; Butterworth-Heinemann: Oxford, 1986.
(8) Kumaran, V.; Muralikrishnan, R. Phys. Rev. Lett. 2000, 84, 3310. (9) Genzer, J.; Efimenko, K. Science 2000, 290, 2130.
u)0 u ˆ )0
at at
z ) -1
(6)
H ˆ z) H
(7)
At the interface (z ) ζ(x)), the displacements must be continuous:
u)u ˆ
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In this work, we determine only the first terms in the perturbation expansions. Since the governing equations are linear, the governing equations for the leading order displacements and pressures are simply (2)-(5) with a superscript “(1)”. Similarly, a superscript “(1)” can be appended to the boundary conditions (6)-(8), and (8) can be applied at z ) 0. It is convenient to separate the force balance, (9), into its normal and tangential components. The tangential component at O() is
(
) (
)
(1) ∂uˆ (1) ∂u(1) ∂u(1) ∂T ˆ (x, z ) 0) G ˆ ∂uˆ x z x z + + )0 + G ∂z ∂x ∂z ∂x ∂x (17)
while the normal component has the form (1) ∂2u(1) ∂u(1) z z G ˆ ∂uˆ z + p(1) - 2 + T0 2 ) 0 (18) -pˆ (1) + 2 G ∂z ∂z ∂x
Note that the force balance is now linear and can be applied at z ) 0. We consider spatially periodic interfacial tension gradients and take T ˆ (x, z ) 0) ) sin(kx), where we refer to k as the pattern wavenumber. More complex periodic patterns could easily be studied by applying the principle of superposition since the governing equations and boundary conditions are now linear. Substitution of this expression into (17) shows that the horizontal displacements must be proportional to cos(kx) and the vertical displacements must be proportional to sin(kx). The normal force balance, (18), then implies that the pressures are proportional to sin(kx). Thus, we seek solutions of the form
u(1) x (x, z) ) ux(z) cos(kx)
(19)
u(1) z (x, z) ) uz(z) sin(kx)
(20)
(1)
p (x, z) ) p(z) sin(kx)
(21)
uˆ (1) x (x,
z) ) uˆ x(z) cos(kx)
(22)
ˆ z(z) sin(kx) uˆ (1) z (x, z) ) u
(23)
(1)
pˆ (x, z) ) pˆ (z) sin(kx)
(24)
Once (19)-(24) have been substituted into the governing equations, it is fairly straightforward to show that
uz(z) ) C1ekz + C2e-kz + C3zekz + C4ze-kz
(25)
C3zekz - C4ze-kz (26) p(z) ) 2C3ekz + 2C4e-kz
(27)
ˆ 1ekz + C ˆ 2e-kz + C ˆ 3zekz + C ˆ 4ze-kz uˆ z(z) ) C
(28)
ˆ 1 + k-1C ˆ 3)ekz + (-C ˆ 2 + k-1C ˆ 4)e-kz + uˆ x(z) ) (C C ˆ 3zekz - C ˆ 4ze-kz (29) G ˆ (2C ˆ 3ekz + 2C ˆ 4e-kz) G
conditions at the interface provide another four equations. Thus, these eight equations can be solved to determine the eight unknown constants. Doing this over a range of the problem parameters allows us to study how the interfacial displacements depend on the pattern wavenumber, k, the mean interfacial tension, T0, the elastic modulus ratio, G ˆ /G, and the thickness ratio, H ˆ /H. 3. Results
ux(z) ) (C1 + k-1C3)ekz + (-C2 + k-1C4)e-kz +
pˆ (z) )
Figure 2. Interfacial displacements for solid-air interfaces versus the pattern wavenumber, k, at various values of the mean interfacial tension T0: (a) horizontal displacement, ux(z ) 0), and (b) vertical displacement, uz(z ) 0). The solid line corresponds to T0 ) 0, the dashed line to T0 ) 1, and the dasheddotted line to T0 ) 10.
(30)
ˆ i are unknown constants. To determine where Ci and C these constants, (25)-(30) are substituted into the boundary conditions. The boundary conditions at z ) -1 and z ) H ˆ /H provide four equations, and the boundary
Results are presented first for solid-air interfaces (these are obtained by simply neglecting the top solid) and then for solid-solid interfaces. In the former case, there are only two dimensionless parameters to be varied, k and T0. Figure 2a,b shows the interfacial displacements, ux(z ) 0) and uz(z ) 0), as a function of the pattern wavenumber for several different values of the mean interfacial tension. From Figure 2a, it can be seen that the horizontal interfacial displacement approaches zero for small k and 0.5 for large k. There is also an inflection point in the plot near k ) 1 that becomes more pronounced as T0 increases. From Figure 2b, it can be seen that the vertical interfacial displacement approaches zero for both small and large k and has a maximum around k ) 1. Both figures show that increasing T0 reduces the magnitudes of the displacements, and Figure 2b shows that this also decreases the value of k at which the maximum occurs.
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Figure 3. Displacement profiles for solid-air interfaces at various values of the pattern wavenumber, k, when T0 ) 0: (a) horizontal displacement, ux(z), and (b) vertical displacement, uz(z). The solid line corresponds to k ) 0.5, the dashed line to k ) 1, the dashed-dotted line to k ) 2, and the dotted line to k ) 5.
In Figure 3a,b, we plot the displacement profiles, ux(z) and uz(z), for several different values of k when T0 ) 0. From Figure 3a, we see that the largest horizontal displacements in the solid are occurring at the interface, z ) 0. From Figure 3b, we see that for relatively small values of k, the largest vertical displacements in the solid occur at z ) 0. However, for relatively large values of k (k ) 5), the largest displacements occur just below the interface. Both plots show that the largest displacement gradients occur near z ) 0 and that these tend to increase as k increases. This indicates that the largest stresses occur near the interface and that their magnitude increases as the pattern wavenumber increases. We now turn to the case of solid-solid interfaces, where two additional parameters arise: the modulus ratio, G ˆ /G, and the thickness ratio, H ˆ /H. We first fix H ˆ /H ) 1 and vary G ˆ /G and k with T0 ) 0. Figure 4a shows that the horizontal interfacial displacement, ux(z ) 0), increases monotonically from zero as the pattern wavenumber increases and plateaus at high values of k. As the modulus ratio increases, the magnitudes of ux(z ) 0) decrease for all k. Figure 4b shows the corresponding plots of the vertical interfacial displacement, uz(z ) 0). As the modulus ratio increases, the magnitudes of uz(z ) 0) decrease until they are zero for all k when G ˆ /G ) 1. For G ˆ /G > 1, uz(z ) 0) starts to take negative values, with a minimum occurring near k ) 1. This minimum becomes even smaller
Kumar
Figure 4. Interfacial displacements for solid-solid interfaces versus the pattern wavenumber, k, at various values of the modulus ratio G ˆ /G when H ˆ /H ) 1 and T0 ) 0: (a) horizontal displacement, ux(z ) 0), and (b) vertical displacement, uz(z ) 0). The solid line corresponds to G ˆ /G ) 0.1, the dashed line to G ˆ /G ) 0.5, the dashed-dotted line to G ˆ /G ) 1, and the dotted line to G ˆ /G ) 2.
as G ˆ /G increases but then approaches zero for large values of G ˆ /G (Figure 5). We next fix G ˆ /G ) 1 and vary H ˆ /H and k with T0 ) 0. Figure 6a shows that the magnitudes of ux(z ) 0) increase as H ˆ /H increases and that an inflection point is apparent when H ˆ /H ) 5. Figure 6b shows that uz(z ) 0) takes negative values for H ˆ /H ) 0.5, with a minimum occurring around k ) 2. Other runs we have performed indicate that the minimum shifts to larger values of k as H ˆ /H decreases. As H ˆ /H increases, the magnitudes of uz(z ) 0) increase until they are zero for all k when H ˆ /H ) 1. For H ˆ /H > 1, the magnitudes continue to increase with the thickness ratio up to a point and then stay relatively constant. For example, the plot for H ˆ /H ) 10 (not shown) is nearly identical to that for H ˆ /H ) 5. 4. Discussion and Conclusions The results for solid-air interfaces indicate that the maximum vertical displacement at the interface occurs when the pattern wavenumber is approximately unity. Since the wavenumber has been made dimensionless with respect to H-1, this means that the maximum displacement will occur when the interfacial tension gradients have a
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would be expected to provide resistance to deformation, and this is observed since the displacements become smaller as T0 increases. The reduction in the wavenumber at which the maximum in uz(z ) 0) occurs is also consistent with the effects of a mean interfacial tension. To gain further insight into why the maximum in uz(z ) 0) occurs when k ∼ 1, we can examine the shear stress balance, (17). If we substitute (19), (20), and T ˆ (x, z ) 0) ) sin(kx) into (17) (and delete the corresponding quantities for the top solid), the following equation is obtained:
-u′x(z ) 0) - kuz(z ) 0) + k ) 0
Figure 5. Horizontal interfacial displacement for solid-solid interfaces versus the pattern wavenumber, k, at various values of the modulus ratio G ˆ /G when H ˆ /H ) 1 and T0 ) 0. The dotted line corresponds to G ˆ /G ) 2, the solid line to G ˆ /G ) 5, the dashed line to G ˆ /G ) 10, and the dashed-dotted line to G ˆ /G ) 50.
Figure 6. Interfacial displacements for solid-solid interfaces versus the pattern wavenumber, k, at various values of the thickness ratio H ˆ /H when G ˆ /G ) 1 and T0 ) 0: (a) horizontal displacement, ux(z ) 0), and (b) vertical displacement, uz(z ) 0). The solid line corresponds to H ˆ /H ) 0.5, the dashed line to H ˆ /H ) 1, the dashed-dotted line to H ˆ /H ) 2, and the dotted line to H ˆ /H ) 5.
wavelength of the same order of magnitude as the solid thickness. The presence of a mean interfacial tension
(31)
The last term on the left-hand side (LHS) of (31) represents the tangential stress due to interfacial tension gradients, and it serves as the forcing for the deformation. As it increases from zero, we would expect uz(z ) 0) to increase, and this is indeed observed at low k (Figure 2). However, for high k (strong tangential stress), we would expect the interfacial shear stress to be dominated by vertical gradients in the horizontal displacement, which are represented by the first term on the LHS of (31). However, the second term on the LHS of (31) is multiplied by k, and it corresponds to horizontal gradients in the vertical displacement. If this term is to vanish for large k, then uz(z ) 0) must vanish too. Thus, a maximum in uz(z ) 0) occurs at some intermediate value of k. The results for solid-solid interfaces indicate that increasing the modulus of the top solid relative to that of the bottom solid makes it more difficult to deform the interface. This is clearly seen in Figure 4a for the horizontal interfacial displacement and can also be seen in Figure 4b and Figure 5 for the vertical interfacial displacement when G ˆ /G < 1 and G ˆ /G > 5. However, when 1 < G ˆ /G < 5, the magnitude of uz(z ) 0) increases as the modulus ratio increases. When G ˆ /G ) 1 and H ˆ /H ) 1, we have a perfectly symmetric system and therefore might expect that the vertical displacement would vanish for all k. This expectation is confirmed by the results shown in Figure 4b. Because there is no body force in the problem, it must be recognized that the results for a given value of G ˆ /G, say C, are physically equivalent to those for G ˆ /G ) 1/C. For example, if the values of ux(z ) 0) and uz(z ) 0) for G ˆ /G ) 2 (Figure 4) are multiplied by 2 and -2, respectively, we recover the results for G ˆ /G ) 0.5 (the multiplications correspond to rescaling the modulus with G ˆ and switching the direction of the z-axis). The results for solid-solid interfaces also indicate that decreasing the thickness of the top solid relative to that of the bottom solid makes it more difficult to deform the interface. Again, this is clearly seen in Figure 6a for ux(z ) 0), while the behavior for uz(z ) 0) in Figure 6b is more complicated. For H ˆ /H > 1, the vertical displacements do become smaller as the thickness ratio decreases, but for H ˆ /H < 1 they become larger. The results in Figure 6b and from other calculations we have performed suggest that when H ˆ /H < 1, the largest magnitude of uz(z ) 0) occurs at k ∼ H/H ˆ , which corresponds to a dimensional wavenumber of H ˆ -1. This means that the interfacial tension gradients need to have a wavelength on the order of the thickness of the thinner solid layer in order to get the maximum vertical displacement at the interface. Finally, we note that the appearance of an inflection point in the plot of ux(z ) 0) (Figure 6a) for H ˆ /H ) 5 is consistent with the inflection point observed for solid-air interfaces (Figure 2b), since the latter corresponds to having a top solid of infinite thickness (and zero modulus.) Similarly, increasing the thickness ratio beyond a certain point does
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not affect uz(z ) 0) very much since the boundary at z ) H ˆ /H is already far away from the interface. Again, the results for a given value of H ˆ /H ) C are physically equivalent to those for H ˆ /H ) 1/C. Indeed, if the values of k for H ˆ /H ) 0.5 are multiplied by 2 and ux(z ) 0) and uz(z ) 0) are replotted (with the latter quantity multiplied by -1), the results will fall on the curves for H ˆ /H ) 2 (the multiplications correspond to rescaling the thickness with H ˆ and switching the direction of the z-axis). The above results provide a quantitative description of the vertical displacements at solid-air and solid-solid interfaces that are produced by interfacial tension gradients. To maximize these displacements, the gradients should have a wavelength on the order of the thickness of the (thinner) solid layer. However, if the thicknesses and moduli of the two layers are closely matched, it will be difficult to create these displacements. For solid layers having similar moduli, the displacements can be enhanced by varying the relative thickness. For solid layers having similar thicknesses, increasing the modulus of one layer relative to that of the other will tend to suppress the displacements, although the relationship is not monotonic (Figure 5). An estimate of the typical size of the vertical displacements is given by Huz(z ) 0), where uz is dimensionless. If we estimate as in section 2, then Huz(z ) 0) ) (γ*/G)uz(z ) 0). Taking γ* ) 0.01 J/m2, G ) 103 N/m2, and uz(z ) 0) ) 0.1 gives a vertical displacement of 10-6 m. Thus, interfacial tension gradients at solid surfaces and
Kumar
interfaces should be capable of producing micron-scale or smaller topographical features. One way in which the predictions of our work could be experimentally confirmed would be to pattern gradients in surface energy on an elastomeric film. Below the glass transition temperature, the film will be relatively rigid and will not deform much. Above the glass transition temperature, the modulus will decrease and the displacements may become visible. The displacements would be expected to form on a time scale comparable to the longest relaxation time of the layers (G/η or G ˆ / ηˆ , where η and ηˆ are the viscosities of the bottom and top solids, respectively.) We note that for films of submicron thickness, van der Waals and electrostatic interactions between the film surface and substrate may become important and produce surface deformations.10,11 By appropriately designing gradients of interfacial tension, it may be possible to interact with these instabilities in a favorable way in order to create novel topographical patterns at elastomeric surfaces and interfaces. Acknowledgment. S.K. thanks the Shell Oil Company Foundation for support through its Faculty Career Initiation Funds program and 3M for a Nontenured Faculty Award. LA020733S (10) Shenoy, V.; Sharma, A. Phys. Rev. Lett. 2001, 86, 119. (11) Shenoy, V.; Sharma, A. Langmuir 2002, 18, 2216.
