Chapter II: Cuspidal Formation in Thermocapillary Thin Liquid Films
2.4 Stability Considerations by Analogy to Gradient Flows
Some elementary properties of equation (2.64) can be examined in the linear limit.
Consider an initially uniform film of thicknessH0 (i.e. base state) subject to a periodic perturbationH=H0+δH0exp (βτ) exp(iK·X)where|δH0| 1is the infinitesimally small amplitude and K is the two-dimensional wave vector. The resulting expression for the growth rateβ as a function of wave number K=|K|is
β(K) =−H03K4+ H02
(1−H0)2K2. (2.65)
The critical wave number, designated as Kcrit for which β(Kcrit) = 0, sits at the boundary between the band of growing wave numbers0< K≤Kcrit and the decaying band 0 < K ≤Kcrit. In other words, there always exist exponentially growing distur- bances of wavelengthΛ > Λcritfor an initially flat film of thicknessH0 (disturbances of 0< Λ < Λcrit decay instead), among which the fastest growing (or the most unstable) wavelengthΛmax corresponds to the maximum of β(K) in equation (2.65),
Λmax=√
2Λcrit, Λcrit= 2πpH0(1−H0). (2.66) If we convertΛmax back to dimensional form, we recover λmax in (2.60), which is the lateral length scale eventually used to scale the system.
0.0 0.2 0.4 0.6 0.8 1.0 -3
-2 -1 0 1
Φ(H) dΦ/dH d4Φ/dH4 Φch(H)
divergence
Figure 2.4: Plots of Φ(H), 0.2×dΦ/dH and 0.005×d4Φ/dH4 for the thermocapillary equation. Magnitudes have been rescaled to accommodate all curves on a common scale.
Φch(H) = 200((H−0.4)2−0.05)2−1.5 is an example of the double-well potential in Cahn-Hilliard theory of phase separation.
be rewritten in Cahn-Hilliard form described by
∂H
∂τ =∇k·
M(H)∇kδF[H]
δH
, (2.67)
known more generally as gradient flow (Giacomelli and F. Otto,2003). The thermocap- illary model described by equation (2.64) posed on the in-plane domain Ω can also be written in this form where the free energy functional is given by
F[H] =Z
Ω
1 2
∇kH2+Φ(H) dΩ (2.68) with mobility coefficient
M(H) =H3 (2.69)
and potential function
Φ(H) =H[ln(1−H)−lnH], (2.70) such that evaluating the (unconstrained) variational derivative of the free energy
δF
δH =−∇k2H+ dΦ
dH (2.71)
recovers the original governing equation (2.64). The curves in figure 2.4 show that, in contrast to the conventional double-well potential Φch(H) in the Cahn–Hilliard theory of binary phase separation (J. E. Taylor and Cahn,1994),Φ(H)has no global minimum (and that Φ(H), dΦ/dH and d4Φ/dH4 all diverge at the virtual singularity H = 1), mimicking an infinitely sharp potential well at virtual singularity H= 1.
We first show that for any periodic domainΩ, functional F[H]is indeed a free energy, i.e.,dF/dτ ≤0. We evaluate the quantitydF[H]/dτ for the free energyF[H]defined in equation (2.68) by applying Leibnitz’s rule for differentiation over a fixed periodic domainΩ:
dF[H]
dτ = d dτ
Z
Ω
1 2
∇kH2+Φ(H) dΩ=Z
Ω ∇kH·∂∇kH
∂τ + dΦ dH
∂H
∂τ dΩ. (2.72)
Interchanging the order of operators∇kand∂/∂τ followed by application of the Green’s first identity to the first integral in equation (2.72) gives
dF[H]
dτ =Z
Ω
−∇2kH+ dΦ dH
∂H
∂τ dΩ, (2.73)
where continuity of H and higher order derivatives ensures that the boundary term proportional to ∇kH vanishes identically. Substitution of the term ∂H/∂τ in equation (2.73) by the relations given in equation (2.67) and (2.68) yields
dF[H]
dτ =Z
Ω
−∇2kH+ dΦ dH
∇k·
M(H)∇k−∇2kH+ dΦ dH
dΩ. (2.74)
Application of Green’s first identity subject to the vanishing boundary term yields the desired inequality
dF[H]
dτ =− Z
ΩM(H)∇k
−∇k2H+ dΦ dH
2 dΩ≤0. (2.75)
The proof for infinite (lateral) domain simply requires that the integrand in equation (2.68) be augmented by the term Φ[H(X→∞, τ)], but otherwise proceeds similarly.
First and second variations of free energy
Due to the conservative (divergence) form of the evolution equation, the total volume of the liquid film over a periodic domain is a conserved quantity. The energy analysis is only instructive to the dynamics if it is carried out for film states of identical total volume.
However the form of the free energy (2.68) of the thin film system depends on the total liquid volume. We can enforce the constrain on the total volume by augmenting the free energyF[H]with the Lagrange multiplier constantP,
F[H, P] =Z
Ω
1 2
∇kH2+Φ(H) dΩ−P Z
Ω
HdΩ−Vol, (2.76) where the total volume
Vol =Z
ΩHdΩ . (2.77)
For H to represent an extrema of the free energy (2.76) with some fixed total volume, the infinitesimal changeδFin the free energyF[H+δH, P+δP]must vanish against all infinitesimal variations δH and δP. Evaluating the first variations of free energy (2.76) results in two expressions,
δF[H, P;δH] =Z
Ω −∇k2H+ dΦ dH
H
−P
!
δHdΩ, (2.78)
δF[H, P;δP] =− Z
Ω
HdΩ−VolδP, (2.79)
where we have used the Green’s identity RΩ∇kδH · ∇kHdΩ=−RΩδH∇k2HdΩfor a periodic domain Ω. This yields the value of the Lagrange multiplier
P =−∇k2H+ dΦ dH
H
such that Z
ΩHdΩ= Vol, (2.80)
which defines the effective surface pressure required for maintaining stationary states of constant volume. It’s straightforward to verity that the film profile H satisfying equilibrium condition (2.80) is a steady state solution to the evolution equation (2.67).
To conclude the nature (e.g., local minimum, maximum or saddle point) of these sta- tionary solutionsH obtained by solving equation (2.80), we must proceed to the second variation of the free energy aroundH,
δ2F[H, P;δH] =Z
Ω|∇kδH|2+ d2Φ dH2
H
δH2dΩ, (2.81)
δ2F[H, P;δP] = 0, (2.82) δ2F[H, P;δH, δP] =−δP
Z
Ω
δHdΩ= 0, (2.83)
where in equation (2.83) we invoke the volume constraint RΩH+δHdΩ= Vol on an admissible variationδH. Usually one needs to first numerically solve for the stationary solution H and then examines the convexity of the quadratic forms (2.81), (2.82) and (2.83) that appear under the integral of the second variation. Fortunately we can do better for the thin film equations. It has previously been shown that for a general class of thin film equations (Laugesen and Pugh, 2002) which include the form of equation (2.80), there always exist some small perturbations to the periodic stationary states H which lead to strictly negative values of the second variation whenever the potential function satisfies the relation the relationd4Φ/dH4<0over the entire range of H.
To prove this claim, let’s consider the free energy associated with a small deviation about a stationary solution H of equation (2.76) for admissible (i.e., periodic and zero total volume) perturbations δH:
F[H+δH, P+δP] =F[H, P]+δF[H, P;δH, δP]+1
2δ2F[H, P;δH, δP]+O(δH, δP)3. (2.84) By definition, the first variation of the energyδF[H, P;δH, δP]must vanish identically for any such stationary solution H. Application of Green’s first identity reduces the second variation to the form
δ2F[H, P;δH, δP] =δ2F[H, P;δH] =Z
Ω
δH −∇k2δH + d2Φ dH2 HδH
!
dΩ, (2.85) where the additional boundary integral vanishes for any periodic perturbationδH. It is now a straightforward exercise to show that there always exist admissible arbitrary per- turbationsδH such thatδ2F[H, P;δH, δP]is always strictly negative. We differentiate equilibrium condition (2.80) twice with respect to X and obtain the relation
− ∇k2∂2H
∂X2 + d2Φ dH2 H
∂2H
∂X2 =−d3Φ dH3 H
∂H
∂X 2
. (2.86)
Substituting equation (2.86) into equation (2.85) for a perturbation of the form δH = ∂2H
∂X2 (2.87)
(note RΩδHdΩ= 0 is admissible) yields δ2F[H, P;δH, δP] =−
Z
Ω
∂2H
∂X2 d3Φ
dH3 H
∂H
∂X 2
dΩ
=− Z
Ω
d3Φ dH3 H
1 3
∂
∂X ∂H
∂X 3
dΩ
= 1 3
Z
Ω
d4Φ dH4 H
∂H
∂X ∂H
∂X 3
dΩ= 1 3
Z
Ω
∂H
∂X 4d4Φ
dH4 HdΩ.
(2.88) All boundary terms from integrations by parts vanish due to periodic boundary con- ditions. For the thermocapillary model described by the potential function (2.70), its fourth derivative (as plotted in figure 2.4)
d4Φ
dH4 =−2(1−2H)2+ 4H2
H3(1−H)4 <0 for 0< H <1 (2.89) is always negative. When substituted into equation (2.88), this yields the relation δ2F[H, P;δH, δP] < 0. This inequality assures that for every (if exists) nonuniform stationary state H such that ∂H/∂X is not identically zero everywhere, there always exists a neighboring state H+δH with same periodicity as H but of strictly lower free energy. Therefore we conclude that equation (2.80) cannot therefore support any energetically stable stationary periodic states, at least not any classical smooth solutions such that H > 0 everywhere. This analysis is quite general and can be modified and applied to many other thin film systems (even volume non-conserving systems) so long as the governing interface equation can be cast into the gradient flow equation (2.67).