Interplay of internal stresses, electric stresses and surface diffusion in polymer films
We investigate two destabilization mechanisms for elastic polymer films and put them into a general framework: first, instabilities due to in-plane stress and second due to an externally applied electric field normal to the film’s free surface. As shown recently, polymer films are often stressed due to out-of-equilibrium fabrication processes as e.g. spin coating. Via an Asaro-Tiller-Grinfeld mechanism as known from solids, the system can decrease its energy by undulating its surface by surface diffusion of polymers and thereby relaxing stresses. On the other hand, application of an electric field is widely used experimentally to structure thin films: when the electric Maxwell surface stress overcomes surface tension and elastic restoring forces, the system undulates with a wavelength determined by the film thickness. We develop a theory taking into account both mechanisms simultaneously and discuss their interplay and the effects of the boundary conditions both at the substrate and the free surface.
The stability of polymer thin films is an important research subject in polymer physics and materials science. On the one hand, the aim may be to obtain a stable film, as in coatings and lubrification. On the other hand, soft films are used for microstructuring, where they are destabilized to yield well-designed patterns that are used e.g. as a mould for further microfabrication processes. In both cases it is crucial to understand the stabilizing and destabilizing mechanisms that prevail in polymer films, which can be either internal (Van der Waals forces due to reduced dimensions, internal stresses, decomposition in mixtures) or external (external stresses, external fields).
In recent studies on spin-coated polymer films it became apparent that thin films are prone to store residual stresses Croll (1979); Reiter et al. (2005). Such stresses are created due to the fast evaporation process of the spin-coating process: as evaporation is fast, the polymer chains do not have the time to reach their equilibrium configurations and in the final, glassy state the film has frozen-in non-equilibrium configurations that give rise to stresses. If these stresses are not relaxed, e.g. by ageing or tempering the films, they influence the film stability as recently shown in dewetting experiments and discussed theoretically Reiter et al. (2005); Bodiguel and Fretigny (2006); Vilmin and Raphaël (2006a); Damman et al. (2007); Ziebert and Raphaël (2009). There it has been shown that stresses increase the initial dewetting velocity and also strongly influence the long time dynamics of the dewetting films. In case the film does not dewet, the stresses may still lead to destabilization Vilmin and Raphaël (2006b), as they should give rise to an Asaro-Tiller-Grinfeld instability Asaro and Tiller (1972); Grinfeld (1986, 1993). This mechanism has been proposed for stressed solids in contact with their melt or for solids which evolve via surface diffusion. Its origin is the fact that the solid can relax stress and lower its energy by creating surface undulations. For polymer thin films the interplay between residual stresses and other, e.g. externally applied, destabilization forces constitutes an interesting question of importance for all further manipulations of freshly spin-coated films.
In this work we reformulate the energy approach usually used to describe the Grinfeld instability in a way that highlights the connection with other known instabilities in thin films. We use the bulk elastostatic equations together with a time-dependent kinematic boundary condition at the free interface. A direct coupling term between the height of the polymer film and the displacement field arises, which has not been discussed before as it is less relevant in atomic solids. In polymer films, however, this coupling should be present and important. Moreover this term establishes the connection to other elastic instabilities, namely to a buckling-like instability under compressive stress and, in the case of an externally applied field, to the elasto-electric instability investigated by Sharma et al. Shenoy and Sharma (2001); Mönch and Herminghaus (2001); Shenoy and Sharma (2002); Sarkar et al. (2008). Finally the growth rate of the height of the polymer film is derived in case of simultaneous action of stress and external field. This result is briefly compared to recent experiments concerning the electrohydrodynamic instability of very viscous (high molecular weight) spin-coated thin polymer films heated above the glass transition Barbero and Steiner (2009); Schäffer et al. (2001).
The work is organized as follows: First, in section II we recall the classical, energy-based formulation of the Grinfeld mechanism. In section III, we start from a nonlinear elastic theory, derive the bulk elastic equations and investigate in section IV the stability under stretch/compression. In section V we show that by allowing surface diffusion via a kinematic boundary condition for the height of the film, the Grinfeld result is regained in a well-defined limit. The coupling between height and displacement via the kinematic boundary condition can influence the classical Grinfeld instability for intermediate stresses. In section VI we add the external electric field to our description. We regain the instability discussed in Sharma et al. Shenoy and Sharma (2001) in a certain limit. Moreover the full growth rate of the film height is calculated and its consequences for experiments are briefly discussed.
Ii Grinfeld instability - classical way of calculation; effects of boundary condition at the substate
To start with we briefly review the classical treatment of the Grinfeld instability of an elastic medium under uniaxial stress Noz (); Cantat et al. (1998). Usually a semi-infinite solid is investigated, but in view of the thin film geometry we allow for a finite thickness of the film. The known results for the semi-infinite case can then be obtained by performing the limit towards infinity. As the dynamics of this instability is energy-driven – the system can lower its energy by creating surface undulations – all the information needed to describe the system is contained in the (free) energy of the system, which has an elastic part, , and a surface part, .
We chose the coordinate system in such a way that the free surface is at , see the sketch of the geometry in Fig. 1.
For simplicity we assume a plane strain situation Maugis (2000) where the uniaxial prestress is taken along the -axis. Consequently, we consider an undulation of the surface along given by
We assume either an infinite system or periodic boundary conditions in -direction. is the wave number of the perturbation, its amplitude and a small book-keeping parameter used in the following when dealing with expansions.
The elastic energy of an linearly elastic solid can be written via the stress field as Landau and Lifshitz (1986)
with the elastic or Young’s modulus. Summation convention is implied for indices occurring twice (). Using a plane strain approximation, one gets
where now () and and . Assuming incompressibility, i.e. a Poisson’s ratio of , one gets , . We also will use the shear modulus later on and note the known relations, . As the system is invariant in -direction, has units of energy per unit length.
The second energy in the problem is the surface energy
where is the surface tension and is measured with respect to the state of a flat surface.
To evaluate the elastic energy, one has to solve the elastostatic problem. The prestress is uniaxial along the -axis and given by , and . Note that holds for the case of a compressive stress and in case of a tensile stress. Undulations of the surface will give rise to an additional relaxational stress . The total stress has to fulfill the Cauchy equilibrium equation
and the compatibility equation
The equilibrium equation is then automatically fulfilled and the compatibility reduces to . This biharmonic equation has to be solved with the following boundary conditions (BC).
At the free surface , the normal-normal component of stress has to balance the surface tension, while the shear stress has to vanish. With and denoting the unit vectors normal and tangential to the surface, respectively, the BC at the free surface read
At the bottom surface , where is the film thickness, we impose vanishing normal displacement
meaning that the film is not allowed to detach from the substrate. As the second BC, we study two possibilities, depending on the preparation of the system: First, to study the case of possible slippage at the lower interface, one prescribes
implying vanishing shear stress at the bottom (or equivalently a vanishing force on the lower surface of the film in -direction, i.e. no traction force). This condition will be called ’slip BC’ in the following. A second relevant situation, applying to the case where the polymer film is rigidly attached to the lower surface, will be referred to as ’fixed BC’,
We will see that these two different BC, slip vs. fixed, have a qualitative influence on the instabilities discussed in the following.
Slip BC at the bottom: The solution of the elastostatic problem with the slip BC at the bottom, Eqs. (10, 11), is the Airy stress function Eq. (72) given in appendix A. The coefficients occurring therein have to be determined by the BC at the free surface: one calculates the stresses via Eqs. (7), evaluates them at the free surface and expands in powers of . From the BC at the free surface, Eqs. (9), one then determines the coefficients in the Airy stress function at order , yielding Eqs. (73).
The problem is now solved at linear order in the undulation, and we can study the corresponding energy of the system. The elastic energy will change due to the undulation-induced relaxational stress . This change, , explicitly reads
With the Airy stress function determined, the stress field can be evaluated. The integrations in Eq. (13) have first to be performed over the film thickness, . Then one usually averages over , assuming periodic boundary conditions: by writing it is understood that one has averaged like . Note that due to this averaging the contribution in vanishes. To leading order one calculates
The surface energy is directly calculated from Eq. (4) and yields in order
Note that the averaged quantities, , have units of energy per unit area.
Let us now briefly discuss the obtained result. To regain the classical limit of an semi-infinite elastic half space one performs the limit . The change in total energy, , then reduces to
As becomes apparent, the prestress leads to an decrease of the energy. The stress enters quadratically, thus both compressive and tensile stress trigger the instability which makes it different from buckling instabilities Landau and Lifshitz (1986), see also section IV. The surface tension acts against the instability and stabilizes high wave numbers, see the second term in Eq. (16).
As we are predominantly interested in polymer thin films, let us perform the opposite limit, . This amounts to saying that the modulation wavelength is large compared to the film thickness. In this ’thin film’ limit, the change in total energy reads
Again, the prestress is destabilizing, independent of its sign. However, there is no wave number dependence of the destabilizing term to leading order. The same result was recently obtained in the framework of a lubrication approximation Vilmin and Raphaël (2006b).
To cross-check, in the classical limit of a semi-infinite elastic half space one again recovers Eq. (16). This is expected as for a half space the BC at the bottom should not be important. In contrast, in the thin film limit one gets
Note that the prestress still lowers the energy, but now has the same wave number dependence as the contribution from surface tension. Thus only above a threshold,
the prestress can destabilize the system.
We have seen that the total energy of the system can be lowered by surface undulations in all the cases discussed above. To establish these favorable undulations, it needs a mechanism that allows rearrangements to occur. In the classical case of a solid in contact with its vapor, this is achieved by melting-crystallization processes at the surface. This results in a velocity of the boundary Grinfeld (1986), where is a mobility.
A second possible mechanism – on which we would like to focus here in view of polymers – is surface diffusion. If atoms or vacancies (in case of a solid) or polymer chains (in case of polymer films) feel an inhomogeneous chemical potential at the surface, they will diffuse. As a result, the boundary will move with a velocity , with a mobility coefficient Asaro and Tiller (1972). Note the second order spatial derivative stemming from the diffusion process and reflected in the -dependence of .
Much more is known about the Asaro-Tiller-Grinfeld-instability, for which we refer to the literature. For the nonlinear evolution beyond the instability, see Nozières (1993); Kohlert et al. (2003) for analytical work and Kassner et al. (2001) for phase-field modeling. Concerning experiments, very clean realizations of this instability have been observed in Helium crystals Torii and Balibar (1992) and single crystal polymer films Berréhar et al. (1992).
Iii Stretched elastic solid: nonlinear bulk formulation
To properly describe finite stresses in a thin polymer film, one has to use a nonlinear elasticity formulation. Let us assume that the film was originally in a stress-free state, described by coordinates . Then we stretch (or compress) the film, for simplicity uniaxially in the -direction by a factor (). This state will be described by coordinates and considered as the base state. This state will be under uniaxial stress , see below. Finally the film is brought in close contact with the substrate (either still permitting for slip, or perfectly fixed to it, see the two BCs discussed in the last section) and we let it evolve. This current state will be described by coordinates . Note that we discuss here only the simple situation where the film is attached after the stretch. The situation where the polymers attach to the substrate while the film is stretched (which probably better corresponds to the situation during spin-coating) is more involved as the uniaxiality is broken due to the presence of the substrate, cf. Ref. Sekimoto and Kawasaki (1989).
The total deformation gradient from to reads
describes the stretching (compression) of the film by a factor (). Note that this step must be described in the nonlinear regime, as stresses are finite. The second tensor (with the identity),
introduces the usual linear displacement gradient tensor in the current state with respect to the stretched state. As we are only interested in the stability of the base state, here a linearized theory is enough for our purposes. As usual we denote with and the left and right Cauchy-Green tensors. As is in Eulerian frame we adopt it for the stresses. is in Lagrangian frame and is more convenient for the energy definition. Using a Neo-Hookean elastic solid Macosko (1994) , the Cauchy stress tensor is defined as
It describes the stress after a deformation in the current configuration. is a Lagrangian multiplier (an effective pressure having units of ) that ensures the incompressibility condition. In the base state, from Eq. (24) one directly gets except for
This establishes a connection between the stretch factor and the prestress .
Now, let us consider again a plane deformation with respect to the prestretched base state. Evaluating the Cauchy stress tensor in linear order in the displacement gradient, imposing plane strain and using incompressibility, one arrives at the bulk equations
Note the asymmetry introduced by , i.e. the prestretch. All quantities can be expressed either in the base state or in the current state - as deformations are small, they amount to the same expressions. For one regains the classical elastostatic equation for an incompressible solid, where is the displacement field.
The elastic energy density, , for the Neo-Hookean elastic solid reads
Here we did not include the pressure as a Langrangian multiplier (giving rise to a term ), as incompressibility is imposed when solving the bulk equations, see the next section. Note that for plane strain and small deformations one regains Eq. (3) to second order in displacement gradients, i.e.
Now we have established the equations for a nonlinear prestretch and a subsequent linear theory. Note, however, that the Neo-Hookean model should not be used for -values too far from . Otherwise effects of e.g. the crosslink length must be taken into account and one should use more realistic models like the Mooney-Rivlin solid Macosko (1994). We will now investigate the stability of the prestressed base state with respect to surface undulations: (i) for the purely elastic case, (ii) in the presence of surface diffusion, making a connection with the classical Grinfeld instability, (iii) in the presence of an electric field normal to the free surface, regaining and generalizing results obtained previously Shenoy and Sharma (2001); He and Qiao (2007) and finally (iv) with both surface diffusion and applied electric field.
Iv Stretched elastic solid: solutions for surface modulations
In this and the following section we solve the elastic bulk equations and show how surface diffusion can be incorporated within this approach in a generic way to regain and generalize the Grinfeld result. We use the same boundary conditions as introduced in section II, i.e. Eq. (9) for the free surface and either the slip BC or the fixed BC at the bottom.
The stability of the base state can be studied by the ansatz
where the amplitudes are small perturbations of order in height perturbations, see Eq. (32) below. Note that we allowed for a temporal dependence which will be used only in the following sections. Using incompressibility, , one obtains a single decoupled equation for given by
With this equation solved, one easily obtains from incompressibility and the pressure from . The general solution of Eq. (29) reads (for 111Note that the case is singluar as it yields only one wavenumber and additional solutions like .)
Imposing the BCs at the substrate yields
As before, we parameterize the upper free interface of the thin polymer film by a harmonic function with small amplitude of order
The normal vector of this surface reads at first order. Thus at the free interface, cf. Eqs. (8), the BCs read . They fix the remaining unknown coefficients and one obtains and
in case of the slip BC at the bottom; for the fixed BC
and , . We introduced the following abbreviations
With the general solution obtained, we can now investigate whether the base state is stable or unstable. According to Eq. (32), solutions with non-zero wavenumber, if they exist, correspond to surface undulations. The condition for nontrivial solutions to exist can be written as
or . Namely, for consistency the displacement at the surface must equal the height perturbation. An alternative formulation would have been to write down the system of BCs as a -matrix equation and looking for nontrivial solutions via the zeros of the determinant. With known, Eq. (36) can be written as
with a function of wave number . If one finds wave numbers with , periodic solutions exist; otherwise implies and the film stays flat. Explicitly, for the two considered BCs one gets
For both BCs, nontrivial solutions do not exist under tension, , as one would expect. Buckling occurs under compression, but only for non-physical values, namely for for a typical surface tension of . For such high compressions, the Neo-Hookean law is no longer a good description. Moreover, the assumption that the film stayed flat in the first step (from to , i.e. before attaching to the substrate) is not valid anymore - the film would have buckled long before. Indeed the threshold for buckling for two free surfaces should be lower than for the BC that the film does not detach from the substrate surface, Eq. (10). Thus we can conclude that the film stays flat for all reasonable values of , and . Note, however, that films can be unstable if they are swollen in the presence of the substrate, cf. Refs. Tanaka et al. (1987); Sekimoto and Kawasaki (1989).
V Adding surface diffusion - the Grinfeld instability again and corrections
In the last section we investigated the stability of the base state with respect to in-plane stresses and found that the purely elastic system is stable. Here we add the effects of diffusion of polymer chains close to the film surface due to stress relaxation-induced changes in the chemical potential. As a consequence the system can produce undulations by diffusive transport of material, in addition to possible elastic displacements. We show that one regains the Grinfeld instability in a well-defined limit. The overall result is more general as it comprises corrections to the Grinfeld mechanism, see below.
If we allow for surface diffusion, Eq. (36) has to be modified in order to allow for this dynamics. For the height modulation one can write
The first two terms on the r.h.s. stem from the standard kinematic BC at a free surface, usually written as with the height of the surface and the fluid velocity Oron et al. (1997). The second term is purely nonlinear and can be neglected in the following linear analysis.
The last term on the r.h.s. represents the surface diffusion (note that in three dimensions has to be replaced by the surface Laplacian Spencer et al. (1991)). It will smoothen gradients in the chemical potential, which is given by
is the mean curvature of the surface, given at by . is the change in elastic energy density due to the surface undulation, compared to the flat surface. The coefficient is a mobility Mullins (1959); Müller and Saúl (2004) and explicitly reads , where is the thermal energy, is a microscopic volume (of the polymer chain in our case), is the surface diffusion coefficient and is the surface density of diffusing objects. Note that in the view of recent experiments on spin-cast polymer melts, we here allow for a finite chain mobility (at least close to the free surface), although we assumed a purely elastic behavior of the film. A generalization of our approach to the more adequate viscoelastic case will be the subject of a forthcoming study.
Eq. (40) for the dynamics of the surface undulation is further motivated in appendix B. The terms arising naturally from the kinematic BC are commonly not included in the treatment of the Grinfeld instability, as in the usual context one concentrates on the diffusive transport of atoms or vacancies. Taking the coupling to the displacement into account – if extended objects like polymers are diffusing – corrections to the ’classical’ Grinfeld behavior arise: the time derivative in leads to a renormalization of the growth rate of the height perturbations . In view of this, in the following we will sometimes compare the ’classical’ Grinfeld and the ’kinematic’ case.
In the previous section we have already calculated the general solution for the displacements. Thus the stress tensor is also known and using Eq. (27) one gets the changes in the elastic energy with respect to the base state
For the surface energy, as before Eq. (4) yields . Now we can proceed in two ways:
Classical calculation, nonlinear case: First we can use the classical Grinfeld argument, i.e. we integrate from to over the film thickness and average over the assumed periodic -direction to obtain . Upon averaging the linear order in vanishes. At , one gets to leading order in
Note that in leading order this is exactly Eqs. (17, 19). The correction in Eq. (46) has a slightly different prefactor as in Eq. (17), which is due to the fact that the fully linear calculation from section II is only correct for infinitesimal stresses.
Consistent calculation at order : The use of an averaging in the Grinfeld calculation seems not necessary to us. We will thus determine the growth rate of surface undulations by using
The l.h.s. and the first term on the r.h.s. are of first order in . Thus it is sufficient to determine the change of the chemical potential at this order, i.e. evaluating at the surface. The full growth rates obtained by this equation are given by Eqs. (76)-(79) in appendix A for the slip and the fixed BC, respectively. In the thin film limit , one obtains
for the slip BC at the bottom and neglecting the term at the free surface. Including the kinematic term yields
For the fixed BC we get in both cases
Let us first discuss the limit of small stresses asgain, with . Both Eqs. (49, V) yield at leading order in the stress . Except for a factor of , at leading order this is exactly Eq. (17). The same is true for the fixed BC and Eq. (19). Hence in the low-stress and low-wave number limit, our results obtained for the dynamic BC at the free surface are identical to those obtained by the energy-based calculation in section II in the following sense: are growth rates as calculated from a dynamical equation for the surface undulation. When comparing to the Grinfeld calculation, there too one has to impose a diffusion dynamics driven by the decrease in energy. One can write , with . The variational derivative with respect to yields a factor of 2. Taking into account that in the energy approach one has averaged over yields another factor of 2, which explains the differing prefactors. However, one should note that using the spatial averaging process implies a calculation order , while our method is .
Let us now discuss the effect of finite stretches and of the kinematic contribution. At leading order in the stress, both Eqs. (49, V) reduce to the Grinfeld result. However, in next order in the stress Eq. (49) yields , while the kinematic version yields . First, this shows that the symmetry with respect to the sign of the stress, i.e. whether it is due to stretch () or compression (), is broken by the elastic nonlinearity. Second, the sign of the correction is sensitive to whether the kinematic BC at the free surface is important (e.g. for diffusion of extended objects like polymers in a network) or not.
To compare to a real system, we use the following parameter values as suggested by Ref. Barbero and Steiner (2009): for the thickness of the film and for the modulus. For the surface tension we use the value for polystyrene, . Fig. 2 displays the full growth rates, Eqs. (76)-(79) in appendix A, as a function of reduced wave number . Note that we renormalized . Fig. 2a) displays the case of the slip BC at the bottom, with (solid curves) and without (dotted curves) accounting for the kinematic BC at the free surface. Finite stresses lead to a Grinfeld instability. Growth rates differ whether compression () or extension () is considered. In case of the fixed BC at the bottom, see Fig. 2b), there exists a threshold stress beyond which the system becomes unstable. For the chosen surface tension, in reduced units, the system destabilizes for and . Note that a (symmetric) threshold stress also occurred in the linear model, cf. Eq. (20). Fig. 3 displays the dependence of the fastest growing wavenumber on the prestress as obtained from by Eq. (25). One clearly sees the asymmetry with respect to compression/stretch for finite stresses.
To summarize, in the last two sections we proposed a general framework that includes the Grinfeld instability as well as possible buckling. The possibility of buckling is due to the coupling of surface undulations and the displacement field via a kinemtic BC at the free surface. One gets corrections to the Grinfeld instability, as contained in the full growth rates given in appendix A. However, in the small wave-number limit and for thin films, the leading order terms are identical with the classical result. For finite stresses the -symmetry with respect to stresses predicetd by the linear Grinfeld-theory is no longer valid. In the next section we use the developed framework to study the simultaneous action of in-plane residual stress and an electric field, both acting as destabilizing factors for elastic films.
Vi Addition of external electric field
Recently the instability of polymeric liquids Schäffer et al. (2001); Barbero and Steiner (2009) and elastomers Sarkar et al. (2008) in an external electric field acting normal to the film surface has been investigated experimentally. In Ref. Barbero and Steiner (2009), it has been found that the instability is faster for freshly spin-casted films than for aged films. This suggests that stresses in the fresh films due to the nonequilibrium production process may be involved in the destabilization. In view of this we generalize the developed approach to the case where an external electric field is acting normal to the surface, in addition to the stress in -direction. The electrostatic part will be closely related to previous studies of elastic instabilities Shenoy and Sharma (2001); Mönch and Herminghaus (2001); Shenoy and Sharma (2002); Sarkar et al. (2008) due to forces normal to the surface (Van der Waals or electric field). Related studies have been undertaken in Refs. Yang (2005); Song and Yang (2006). However, there the thin film was regarded as conductive, the external stress was imposed externally (implying that the base state with applied field was fixed at rather than as in our case with the additional contribution from the electric field, see below) and the kinematic BC (i.e. the coupling of film height and displacement field) at the free surface was not taken into account.
Let us assume that the polymer film is brought into a parallel plate capacitor, see the sketch in Fig. 4. A voltage difference is applied over the distance of the two plates (the lower plate is at , the upper one at ). The gap may be filled with any dielectric. In view of the experiments in Ref. Barbero and Steiner (2009), we take (polystyrene) as the dielectric constant of the polymer film and , i.e. the gap is filled with air.
The electric field will introduce a stress at the polymer-air interface. Let us introduce the Maxwell stress tensor
where the index denotes the polymer film and the gap. The BC at the free surface, cf. Eq. (8), now reads
where we wrote only the linear order expression for the surface tension. We can put (or to a constant pressure value that is not important), and define an electrostatic ’pressure’ (strictly speaking a normal stress) by
Note that this electrostatic stress depends on the film thickness, see below. The BC finally reads .
We now have to evaluate the additional contribution from the electric field. We can again solve the problem by a perturbative method by writing , where is the first order correction due to undulations. To lowest order, one has to satisfy that the normal dielectric displacement is continuous, . Second, we have , and thus one gets , . In the next order, we have to solve Maxwell’s equations
with the BCs
These BC state that the field has to be perpendicular to the conductive electrodes and that at the film surface one has continuity in the normal displacement and the tangential field. Assuming , the system is readily solved yielding the field components given by Eqs. (81) in appendix A, in agreement with Ref. Onuki (1995).
Evaluating the normal-normal component of the Maxwell stress, for the electrostatic pressure as defined in Eq. (54) above we get to leading order
Note that both and are strictly positive, . As one has , the tangential BC at the free surface is unchanged by the electric field. In the base state, the contribution of the electric field will be an isotropic pressure Shenoy and Sharma (2001, 2002), given by . Concerning the displacements relative to the base state, the procedure is completely analogous to the one in the previous sections. Only the BC at the free surface, and the chemical potential have to be changed accordingly to include the electric stresses. In the chemical potential, Eq. (41), we have to add the contribution due to the electric stress by writing
As is a constant, its contribution to surface diffusion vanishes.
The general solution for the displacement field, Eq. (30), with the BCs at the substrate already imposed, is still valid. One only has to determine the coefficients fulfilling the new BC at the free interface. These coefficients, , and , can be obtained from the respective solutions without field by the simple substitution
This rescaling of permits to obtain also the functions , that determine the stability of the flat base state in the presence of a field, as well as the growth rates and . The obtained expressions are very general. Although unsightly they contain the physics of buckling, the elasto-electric instability, the Grinfeld-instability and surface diffusion in an applied electric field.