Chapter III: Computational Electrohydrodynamic Lithography of Dielectric Films 44
3.2 Thin Film Model of Electrohydrodynamic Lithography (EHL)
The molten polymer layer is modeled as an incompressible Newtonian fluid. Electro- hydrodynamic flow within the melt is driven by the jump in Maxwell stress across the gas/liquid interface due to the contrast of dielectric constant in each medium. In this chapter, we exclusively focus on the dynamics of perfect dielectric liquid films. The slen- der geometry of lithographic system allows us to directly apply the thin film equation (2.37) in its most general form derived from the classical lubrication theory in Section 2.2. In this section we provide a brief review on the derivation of the interfacial pressure P and tangential traction (if any)Fk present in the thin film equation (2.37). The for- malism of interfacial Maxwell stresses adopts the approach oulined in the review article by Saville (1997) on electrohydynamics which however does not include the lubrication limit. Although as mentioned in Section 3.1 there are articles on the linear stability analysis of electrohydrodynamic thin film, to the best of author’s knowledge, the first appearance of the full nonlinear evolution equation for perfect dielectric films under a flat electrode is due to Schäffer et al. (2001). Shortly after, Pease and Russel (2003) and Pease and Russel (2004) generalized the lubrication model to describe the effect of charge transport in leaky dielectric films. The equation was then immediately extended to the case of patterned electrodes (Wu and Russel,2005; Wu, Pease, and Russel,2005;
Verma et al.,2005). The interested reader should consult these references for detailed discussions on the scaling argument used in lubrication approximation.
Electrohydrodynamic thin film equation
Assuming a perfect dielectric medium with no bulk free charges , the Maxwell equations simplify to the electrostatic (Laplace) equation in free space,
∇ ·ε0ε∇ψ= 0, (3.2)
where ε is the homogeneous isotropic dielectric constant of the medium and ε0 is the vacuum permitivity. Interface conditions for electromagnetic fields (Jackson, 1999) require continuity of electric potentialψand displacement field−ε0ε∇ψin the direction normal to the interface,
ψair =ψliq,
n·ε0εair∇ψ=n·ε0εliq∇ψ,
at z=h. (3.3)
We introduce the nondimensional electric potential, Ψ = ψ−ψbot
∆ψ , ∆ψ=ψtop−ψbot, (3.4)
where ∆ψ is the voltage difference between the top electrode (ψtop) and ground sub- strate (ψtop) at the bottom. Under our previous choice of nondimensionlization employed in the slender limit, the electrostatic equation (3.2) subject to interface boundary con- dition (3.3) is identical to the heat transfer problem (2.49) within a slender gap in the thermocapillary model. As a result, the electrostatics problem is dominated by electric conduction along the vertical direction,∂2Ψ/∂Z2, which yields a similar solution to the heat problem,
Ψ =
Z−(1−κ)H
D−(1−κ)H if H ≤Z ≤D(X, Y), κH
D−(1−κ)H if 0≤Z ≤H,
(3.5)
where the permittivity contrast parameterκ=εair/εliqis defined alike. Note in the limit κ → 0, the solution (3.5) to electric potential corresponds to a perfectly conducting material, e.g., a liquid metal film.
The new complication in electrohydrodynamic thin film is the additional interfacial trac- tion f which arises when the (dimensional) Maxwell stress tensor
t =εe⊗e−1
2|e|2I (3.6)
suffers a discontinuity across the liquid/air interface (Saville, 1997),
f = (tair−tliq)n, (3.7)
where e is the dimensional electric field. For an electrostatics system, the jump in Maxwell stress gives rise to normal traction forces only. To see this claim, let’s consider the tangential traction expressed in terms of electric field components,
sαtn=sαεe⊗e−1
2|e|2In=ε(sα·e)(n·e), (3.8)
whereεand ecan be the dielectric constant and electric field in the medium on either side of the interface (sα=1,2 is the unit tangent vector of the interface). But we know displacement field is continuous across the interface, i.e. −n·εaireair =−n·εliqeliq, as well as the tangential components of electric field e. Hence tangential stress (3.8) evaluated on either side of the interface immediately suggests sαtairn= sαtliqn and therefore zero tangential traction. This leaves us with normal jump of the Maxwell stress only,
n(tair−tliq)n= εair 2
h(n·eair)2− X2 α=1
(sα·eair)2i−εliq 2
h(n·eliq)2− X2 α=1
(sα·eliq)2i. (3.9) Nondimensionlizing the normal traction (3.9) with the lubrication scales and the ex- panding electric field E, normal vector N and tangent vectors Sα as usual yield
N(Tair−Tliq)N = ε0∆Ψ2 2[L]2[P]
εair1
∂Ψair
∂Z 2
−εliq1
∂Ψliq
∂Z 2
+O(1). (3.10) Substituting the leading order solution to Ψ found in (3.5) leads to the leading order approximation to the normal traction F =FNN produced by Maxwell stress in terms of reference pressure[P]and length scale [L]defined previously,
FN = 1 2
εairε0∆Ψ2
[L]2[P] 1−κ
2[D(X, Y)−(1−κ)H]2. (3.11) To gain more insights into the normal traction (3.11), we define[E]to be the charac- teristic electric field strength between the top electrode and the bottom substrate,
[E] = ∆Ψ
[H], Ec = 1
ε0εair[E]2
σ/[L] , (3.12)
whereEc is the (rescaled) electric-capillary number. The total pressureP at film inter- face is then given by
P =− 1
Ca∇k2H+ Bo
Ca∇k2H− Ec Ca
1−κ
2[D−(1−κ)H]2. (3.13) In micro- and nanofabrications where electrohydrodynamic thin film model is meant to apply, the lateral length scale[L]∼O(micron)or even smaller and the effect of gravity soon becomes negligible compared to other interfacial forces. Substituting interfacial pressureP from (3.13) into the conservative form (2.37) of the thin film equation with gravity term dropped yields the electrohydrodynamics of a viscous thin liquid film,
∂H
∂τ +∇k· 1
3CaH3∇k
∇k2H+ Ec1−κ
2 1
[D−(1−κ)H]2
= 0. (3.14)
Partial differential equation (3.14) is identical to the previously derived equation for perfect dielectric films (Schäffer et al., 2001; Pease and Russel, 2004; Wu and Russel, 2005; Verma et al.,2005; Nazaripoor et al.,2016). We further apply transformations,
H∗= H
[H∗], D∗ = D
[H∗], X∗ = X
[X∗], τ∗ = τ
[τ∗], (3.15)
similar to the ones for the thermocapillary equation (2.62), where the scalings are given by
[H∗] = Dref
1−κ, [X∗] = 1 1−κ
s 2
EcDref3/2, [τ∗] = 12 1−κ
Ca
Ec2Dref3 . (3.16) Here Dref represents some reference height of the electrode topography D(X, Y), for instance, the spatial average or the maximum/minimum height of D(X, Y) over a periodic domain in the (X, Y)-plane. In what follows we drop the (·)∗ subscription on all the transformed variables and recast equation (3.14) into a parameter-free form of gradient flow,
∂H
∂τ =∇k·nM(H)∇kh−∇k2H−Π(H, D)io (3.17) with the usual mobility coefficientM=H3 and the effective electrostatic pressure
Π(H, D) = 1
(D−H)2 (3.18)
which is inversely proportional to the distance squared between the local height of the electrode topography D(X, Y) and the local thickness of the instantaneous film profileH(X, Y, τ). Although in the rescaled model (3.17), the physical electrode is now positioned at Z = D(X, Y)/(1−κ), we still refer to the virtual singularity D(X, Y) as the electrode topography. This is motivated by the consideration that, unlike the thermocapillary problem in Chapter2, we are less concerned with making a distinction between the virtual singularity and the physical electrode as the goal is to achieve controlled pattern growth in the early and intermediate stages of the evolution rather than to rely on the runaway process of localized cusp-like pillar growth attracted by the virtual singularity.
As for boundary conditions, in this work we restrict ourselves to periodic boundary condition on bothHandDfor a periodic domainΩin the(X, Y)-plane. This is justified by the practical purpose of soft lithography which is to achieve efficient fabrication of massive arrays of identical structures at micro- and nanoscales. Though other types of boundary conditions such as flux injection rate of liquid material through the boundaries of a non-periodic domain are certainly possible and left for future studies.
Some rough estimates on the regime where the electrostatic-hydrodynamics thin film equation (3.17) is expected to operate can be obtained from the electric-capillary number Ecdefined in (3.12). For a length scale[L]∼O(10 micron), surface tensionσ ∼0.078 N/m (e.g., water-air interface) and a relative permittivity εair ∼ 1, we have Ec ∼ 10−5−3(∆Ψ/volts)2. The electrohydrodynamics thin film equation (3.14) is only an accurate description of liquid motion ifEcisO(1). For aspect ratiosranging from0.05 to0.25, the required voltage difference between the ground substrate and the proximate electrode, separated by a few microns, is roughly between 3.5 to 40 volts, which is experimentally feasible (Schäffer et al.,2000).
Unstable film growth under flat electrode
In previous work (Schäffer et al.,2001), the classical linear stability analysis of equation (3.14) is examined for the early time behavior of sinusoidal perturbations to a polymer film of uniform thickness. The result described by the electrohydrodynamics model quantitatively agrees with experimental data without any adjustable parameter. Here we present the same result only with the dimensionless equation (3.17). Adopting the linear stability analysis in the thermocapillary model for a flat film of thickness H0 and a flat electrode of height D0, the resulting expression for growth rate is a function of wave numberK,
β(K) =−H03K4+ 2H03
(D0−H0)2K2, (3.19) where the critical wavelength and the most unstable wavelength are given by
Λmax=√
2Λcrit, Λcrit=√
2π(D0−H0)3/2. (3.20) IfΛmaxis transformed back to its dimensional formλmax(not shown), we indeed recover expression (3.1). It is clear from (3.20) that the wavelengthΛmaxof the fastest growing perturbation rapidly decreases as the gap width D0−H0 diminishes. This is reflected in free energy functionalF[H, D]of the electrohydrodynamic thin film equation (3.17) as well. Consider the free energy functional
F[H, D] =Z
Ω
1
2∇kH· ∇kH+Φ(H, D(X, Y)) dΩ, (3.21) where the potential function
Φ=Z −Π(D, H) dH=− 1
D(X, Y)−H (3.22) is inversely proportional to the local separation between patterned electrode and local film thickness. NotedF[H, D]/dτ ≤0 irrespective of whether electrode D(X) is uni- form or patterned. For a flat electrode such that D0 = 1 after transformation (3.15), the fourth-derivative test (2.88) reveals a similar unstable nature as in the thermocap- illary film: no stable equilibrium film profile can be reached under the balance between capillary and Maxwell stresses.
In fact the situation is even worse. If we perform self-similar analysis near the virtual singularity D= 1, to the leading order we arrive at the truncated PDE
∂H
∂τ +∇k2
∇k2H+ 1 (1−H)2
+O((1−H)−1) = 0, (3.23)
which implies the local scaling balance
1−Hapex∼X2/3 ∼(τs−τ)1/6 (3.24) as the film apex Hapex approaches the virtual singularity. The leading order self-similar solutionW1 = (1−H)/(τs−τ)1/6 has an asymptotic behavior∼η2/3 asη→ ∞where
N4 N7 N3
N8 N9 N6
N1 N5 N2
N8N4
N1
N5
N2N6
N3
N9N7
ξ1 ξ2
Figure 3.3: Lagrange polynomial basis {N1(ξ1, ξ2), ... ,N9(ξ1, ξ2)} and nine correspond- ing nodes for bi-quadratic interpolation in the canonical domain[−1,1]×[−1,1].
η =X/(τs−τ)1/4 is the self-similar coordinate. However unlike the conic cusp found in the thermocapillary system, spatial derivative d(·)/dX of η2/3 becomes singular as η→0. Therefore near the singular point Xs, local film profile is expected to converge to a genuine cuspidal shape |1−H| ∝ |X −Xs|2/3 asHapex →1. In contrast to the thermocapillary system where the slope of film profile remains finite even in the late stage of the self-similar runaway process, surface slope in the apical region blows up indefinitely for electrohydrodynamic thin film, which is a clear violation of the slender assumption that lubrication models all rely on. These peculiar features of electrohydrodynamic thin films, if not properly controlled, would lead to multiple sites of localized cuspidal blow-up.
Finite element discretization
In this section we outline the numerical scheme for simulating the thin film equation.
There are many classical references on the general theory and practice of finite element method such as treatments on elliptic problems (Ciarlet, 2002), parabolic problems (Thomée,2006), implementation details (Zienkiewicz, R. L. Taylor, and Zhu,2013) and Navier-Stokes equation (Girault and Raviart, 1986). Owing to the vast literature, here we do not aim to give a comprehensive review of the finite element method. Interested readers should consult the aforementioned references. The presentation on the finite element approximation follows the exposition in the review article (Becker et al., 2002) on the numerical method for fourth order nonlinear degenerate diffusion problems.
We clarify here all finite element computations in this chapter are implemented in C++
from scratch with the help of high performance linear algebra library EIGEN (Guen-
nebaud, Jacob, et al., 2010). The reason for a bottom-up implementation is that, applying adjoint method to solve the inverse problem in EHL requires a highly cus- tomized matrix assembly procedure that is not readily available in most closed-source commercial software. The tutorial of finite element method presented in this section aims to prepare for a correct implementation of the adjoint method. Accuracy of our finite element code is discussed at the end of this section.
Let Qh be the quadrilateralization of a periodic rectangular domain Ω. The grid sizeh is the maximal edge length over all elements. LetO be the canonical element [−1,1]× [−1,1] with canonical coordinates ξ = (ξ1, ξ2). In this work, we restrict ourselves to straight elements Qe ∈ Qh obtained only by translating and rescaling of the canonical element O,
X ∈Qe ϕ−1e
−−−* )−−−ϕe
ξ ∈O, (3.25)
whereϕe is some bijectively affine map between element Qe∈Qh in the (X, Y)-plane and the canonical square elementO in the (ξ1, ξ2)-plane.
The set of all bi-quadratic functions in the canonical space is spanned by any linear- independent combination of the polynomial basis {1, ξ1, ξ12} ⊗ {1, ξ2, ξ22}. In finite ele- ment method, it’s standard to use the Lagrange polynomial basis{Ni}9i=1,
N1(ξ) = 1
4(ξ12−ξ1)(ξ22−ξ2), N2(ξ) = 1
4(ξ21+ξ1)(ξ22−ξ2), N3(ξ) = 1
4(ξ12+ξ1)(ξ22+ξ2), N4(ξ) = 1
4(ξ21−ξ1)(ξ22+ξ2), N5(ξ) = 1
2(1−ξ12)(ξ22−ξ2), N6(ξ) = 1
2(ξ21+ξ1)(1−ξ22), N7(ξ) = 1
2(1−ξ12)(ξ22+ξ2), N8(ξ) = 1
2(ξ21−ξ1)(1−ξ22), N9(ξ) = (1−ξ12)(1−ξ22),
(3.26)
so that any bi-quadratic function can be represented by its value at the nine local nodes shown in figure3.3. There are nine nodes in the canonical square element with indices {1,2,3,4}for the ones at four corners, {5,6,7,8} for the mid points of four edges and 9 for the central interior point, hence the name “Q9” element. As shown in figure 3.3, the Lagrange basis is designed in such way that, Ni(ξ) = 1 at the i-th local node and
= 0 at all other nodes. Since only translation and scaling are involved in the mapping ϕefor each elementQe, any Lagrange basis function after being mapped to the physical space,
Nei(X) =Ni(ϕ−1e (X)), (3.27) is also a bi-quadratic function in the (X, Y)-plane.
LetVhbe the space consisting of continuous functions which are piecewise bi-quadratic on each elementQe∈Qh. A function inVh is uniquely defined by its values on the set
Corner hat
Edgehat Edge
hat
Interior hat
Figure 3.4: Global hat functionsVi at corner node of local indices {1,2,3,4}, at edge node of local indices {5,6,7,8}and at interior (central) node of local index 9.
of nodes{Xj}j∈J of quadrilateralized domain Qh whereJ is the index set of all nodes.
For each nodal point Xj, we associate a compactly supported hat function Vj ∈ Vh such that
Vj(Xi) =1 if i=j,
0 if i6=j. (3.28)
The set of hat functions over all the nodal points spans the finite-dimensional vector space Vh. In figure 3.4 we show a typical quadrilateral mesh where each rectangle is a Q9 Lagrange element. Depending on the type of each individual node (e.g., corner, edge or interior), the corresponding hat function may span one, two or four elements.
The projectionHh(X)of a continuous functionH(X)into the finite element spaceVh is accomplished by the nodal interpolation operator Ph,
Hh=Ph[H]≡X
i
H(Xi)Vi . (3.29)
In what follows, we identify the projected function Hh ∈ Vh with its nodal vector representationH in boldface character,
H = [H1, H2, ...]>←→Hh(X) =X
i
HiVi(X). (3.30) Composition of continuous functions can be projected intoVh in a similar fashion. For example let Π(A(X), B(X), ...) be some elementary function (possibly nonlinear) of its arguments. We then associated a nodal vectorΠ with Π(A(X), B(X), ...),
Π(A,B, ...) = [Π1, Π2, ...]> ←→Ph[Π(A, B, ...)] =X
i
Π(Ai, Bi, ...)Vi(X). (3.31)
The inner product of the finite-dimensional vector space∈Vh between two continuous functionsA∈Vh andB ∈Vh is defined as
hA, Bi ≡ Z
Ω
ABdΩ. (3.32)
An equivalent formulation in the nodal vector representation (3.30) yields
hA, Bi=A>MB, Mij =hVi,Vji, (3.33) where M is the mass matrix. The entries of mass matrix M are precisely the inner product of all possible pairs of basis hat functions. It’s evident that M is positive definite and symmetric, however not diagonal due to the overlaps of hat functions which share one identical element in the physical domain Ω (see figure 3.4). In practice, the solution of many transient problems become more efficient if the mass matrices can be diagonalized and hence inverted trivially. The process of replacing the true (consistent) mass matrix by a diagonal approximation without drastic degradation in overall accuracy is called mass lumping (Thomée, 2006; Zienkiewicz, R. L. Taylor, and Zhu,2013). The essential idea is to approximate the inner producth·,·iin (3.32) with the lumped mass product h·,·ih such that
hA, Bih≡ Z
Ω
Ph[AB] dΩ (3.34)
which yields the lumped mass matrixMh for which
hA, Bih=A>MhB, (Mh)ii=h1,Vii. (3.35) Recall the partition of the unity function, 1 = PjVj(X). The name “lumped mass matrix” comes from the observation that, every row (column) of the true mass matrix M is lumped into a sum of its entries through the lumped mass product (3.35),
(Mh)ii=h1,Vii=hX
j
Vj,Vii=X
j
hVj,Vii=X
j
Mji.
In this work throughout, we use the lumped mass matrix Mh for its computational efficiency when dealing with transient problems. With a slight abuse of notation, from now on we redefine M as the lumped mass matrix Mh unless stated otherwise. The standard definition of the stiffness matrixK is given by
Kij =h∇kVi,∇kVji ≡ Z
Ω(∇kVi)·(∇kVj) dΩ, (3.36) where the inner product is understood to be the sum of component-wise scalar products.
Definition (3.36) is motivated by the finite element projection of the Laplacian operator
∇k2 since hA,∇k2Bi=Z
ΩA∇k2BdΩ=Z
∂ΩAN∂Ω· ∇kBdS− Z
Ω(∇kA)·(∇kB) dΩ=−A>KB (3.37)
ξ1 ξ2
1 1
1 1
√1 3
− 1
√3
− 1
√3
√1 3
ξ1 ξ2
r3 5 0
− r3
5
− r3
5 0 r3
5 25 81 40 81 25 81
25 81 40 81 25 81
25 81 40 81 25 81
Figure 3.5: 2-point (left) and 3-point (right) Gauss quadrature rules for two-dimensional integration: quadrature weights are shown next to the nodes (dots).
where the boundary integral (N∂Ω is the in-plane normal of domain boundary ∂Ω) vanishes under periodic or no-flux boundary conditions. In the same spirit we introduce the weighted stiffness matrix W(H) for the mobility function M(·) acting on some functionH ∈Vh,
W(H)ij =h∇kVi,Ph[M(H)]∇kVji. (3.38) Integrals arising from the finite element method are often approximated by nqd-point Gauss quadrature rules. For the Q9 element used in this work, the 2- and 3-point quadrature rules are recommended (Cook et al., 2007) for approximating integral of a sufficiently continuous function A(ξ) over the canonical square element,
Z 1
−1
Z 1
−1A(ξ) dξ1dξ2≈
nqd2
X
k=1
wkA(ξk) (3.39)
where ξk and wk are the nqd-point quadrature nodes and weights for the canonical square elementO as illustrated in figure3.5.
Following Becker et al. (2002), we employ a semi-implicit discretization for time integra- tion which leads to the variational (weak) form of the thin film equation (3.17): given a known time series of film profilesH0, H1, ... , Hk∈Vh, find two functions Hk+1 ∈Vh andPk+1∈Vh such that,
hA,Dτ[Hk+1]ih+h∇kA,Ph[M(Hk)]∇kPk+1i= 0,
hB, Pk+1ih− h∇kB,∇kHk+1i=−hB, Π(Hk+1, D)ih,
(3.40) for allA, B ∈Vh. To avoid confusion in subscripts,Hkand similar objects always refer to the function in finite element spaceVh at thek-th time step whereas(Hk)i refers to thei-th nodal value of the film profileHk. The discrete time derivative operatorDτ[·] in variational formulation (3.40) is treated with either the first or second order backward
Scheme α+k α0k α−k
BDF1 1 1 0
VS-BDF2 1 + ∆k
1 + 2∆k
(1 + ∆k)2
1 + 2∆k − ∆2k 1 + 2∆k
BDF2 2
3 4
3 −1
3
Table 3.1: Coefficients of the backward differentiation formula (BDF) family. ∆k =
∆τk/∆τk−1 is the ratio between adjacent time steps.
differentiation formula (BDF) scheme (Crouzeix and Lisbona,1984),
Dτ[Hk+1] =
1
∆τk(Hk+1−Hk), (BDF1)
1
∆τk
"
1 + 2∆k
1 + ∆k
Hk+1−(1 + ∆k)Hk+ ∆2k 1 + ∆k
Hk−1
#
, (VS-BDF2) 1
∆τ 3
2Hk+1−2Hk+1 2Hk−1
, (BDF2)
(3.41) where ∆τk is the incremental time step. Note BDF1 is identical to the first order implicit Euler scheme. For BDF2 with a constant time step, we set ∆τk = ∆τ for all k. The variable step BDF (VS-BDF2) requires the ratio between current and previous time stepS, designated as ∆k= ∆τk/∆τk−1.
Using nodal vector representation, we recast variation formulation (3.40) into a nonlinear system of equations at each time step. The discrete form of variation formulation (3.40) reads: given a known time series of nodal vectorsH0,H1, ... ,Hk, find two nodal vectors Hk+1 andPk+1 such that
MHk+1+ ∆τkα+kW(Hk)Pk+1 =M(α0kHk+α−kHk−1),
−KHk+1+MPk+1 =−MΠ(Hk+1,D),
(3.42)
where coefficientsα+k,α0k and α−k at time stamp τk are defined in table 3.1. Since the mass matrixM for Lagrange Q9 element is positive definite, we can eliminate effective pressure vectorPk+1 in (3.42) which yields a nonlinear system of equations withHk+1 being the only unknown,
Fk+1(Hk−1,Hk,Hk+1,D) =0, (3.43) where the nonlinear function
Fk+1=M(Hk+1−α0kHk−α−kHk−1) + ∆τkα+kRk+1(Hk,Hk+1,D) (3.44) contains a linear component and a nonlinear function
Rk+1(Hk,Hk+1,D) =W(Hk)M−1KHk+1−Π(Hk+1,D). (3.45)
The nonlinear system of equations (3.43) can be efficiently solved via the Newton’s Method. Starting from a “good” initial guess H(0)k+1, for example H(0)k+1 = Hk from previous time step, Newton’s iteration scheme
H(i+1)k+1 =H(i)k+1−[J(i)]−1Fk+1(Hk−1,Hk,H(i)k+1,D) (3.46) produces successively better approximationsH(i)k+1to the roots of equation (3.44) where the Jacobian matrix in (3.46)
J(i)=M + ∆τkα+kW(Hk)M−1K −∂Π(H(i)k+1,D)
∂Hk+1
(3.47) is the linearization of Fk+1 about H(i)k+1.
In practice, direct inversion of the Jacobian matrix J(i) is strictly forbidden due to its large dimensions. Instead we can take advantage of the sparse pattern inJ(i)which can be seen from the following two observations: first of all,W and M−1K are assembled element-wisely (this is explained in Section 3.4) with identical locations of non-zero entries, which means if the one-ring element neighbors of the element which the i-th node belongs to and that of thej-th node are disjoint, it must be that[W M−1K]ij = 0, hence sparse; secondly, recall from (3.30) that the action ofΠ onHk+1is an entry-wise operation, which implies∂Π/∂Hk+1 must be diagonal. Therefore it is more favorable to implement the Newton’s method (3.46) with an iterative solver (Saad, 2003) such as bi-conjugate gradient stabilized method (BiCGSTAB),
J(i)δH(i) =−Fk+1(Hk−1,Hk,H(i)k+1,D), H(i+1)k+1 =H(i)k+1+δH(i).
(3.48)
Another advantage of employing an iterative method to solve the linear system (3.48) is the control over accuracy. The ultimate goal is to drive the residue in Fk+1 below certain tolerance threshold. A perfectly accurate solution to δH(i) in (3.48) is not necessary as long as the iterative increment δH(i) helps guide Fk+1 to 0. Likewise it is not necessary to update J(i) at each Newton step which can be potentially time consuming as well. For sufficiently small time step ∆τk, we only need to reassemble J(i) every a few Newton steps.
The finite element formulation outlined in this section is very general and can be applied to all sorts of thin film models including the thermocapillary equation (2.64). Calcula- tions of the mass and stiffness matrices are only slight different for the one-dimensional line mesh and the triangular mesh used in the simulation reported in Section2.5due to different forms of Lagrange basis functions on a line and triangle element.
Film evolution under an electrode of a heart-like topography
Figure3.7represents 3D views of the discrete film profileHk of an evolving elctrostatic- driven thin film at designated time stamps respectively, obtained from finite element