THE DYNAMICS OF THREE-PHASE TRIPLE JUNCTION AND CONTACT POINTS DONG WANG * , XIAO-PING WANG † , AND YA-GUANG WANG ‡ Abstract. We use the method of matched asymptotic expansions to study the sharp interface limit of the three-phase system modelled by the Cahn-Hilliard equations with the relaxation boundary condition. The dynamic laws for the interfaces, the triple junction, and the contact points are derived at different time scales. In particular, we show, at O(t) time scale, the dynamic of the triple junction is determined by the balance of the chemical potential gradient along the three interfaces meeting at the triple junction. At faster time scale O(t), the motion of the triple junction is controlled by the contact point motions and geometric constraints. Key words. Cahn-Hilliard equations, three-phase, triple junction, contact point, sharp interface limit AMS subject classifications. 76T30, 35Q35, 41A60 1. Introduction. The phase field model has been widely used to describe the multiphase problems. The basic idea is to represent the interface implicitly by an order parameter φ (e.g. scalar function to model two-phase problems or vector valued functions to model three-phase problems) which varies continuously over thin interfa- cial layers and is mostly uniform in the bulk phases. The evolution of φ is driven by the gradient of a total free energy of the system. The total free energy is then a sum of three terms: a bulk free energy which is usually taken as a multi-well potential func- tion, an interface energy term depending on the gradient of φ, and a surface energy on the solid boundary. Consider a three-phase system in a two dimensional domain Ω with a closed and smooth solid boundary ∂ Ω, the total free energy functional can be written as F (φ)= Z Ω 2 |∇φ| 2 + 1 F (φ) dΩ+ Z ∂Ω l(φ)d∂ Ω (1.1) where φ =(φ 1 ,φ 2 ,φ 3 ) is a vector valued order parameter which typically represents the relative concentration of the three phases with φ 1 + φ 2 + φ 3 = 1 and φ i represents the mass fraction of the i th phase; measures the interface thickness between the two phases; F (φ) is a triple well potential function with three equal minima at ~a i (i = 1, 2, 3) that denote the three different phases (e.g. a 1 = (1, 0, 0),a 2 = (0, 1, 0), and a 3 = (0, 0, 1)); and l(φ) gives the surface energy density on the solid surface. The evolution of φ of the system is then described by the Cahn-Hilliard equation φ t = ∇· (M (φ)∇μ) in Ω, (1.2) where M (φ) is the diffusion mobility factor and can take various form in different physical situations (for example, in [2], M (φ) is chosen as M (φ)= Mφ with M being the constant mobility). μ := δF δφ = -Δφ + 1 f (φ) is the chemical potential. There are many studies of (1.2) with general mobility factor, see, for example, [8, 9, 10, 11] and * Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China. ([email protected]). † Corresponding author. Department of Mathematics, Hong Kong University of Science and Tech- nology, Clear Water Bay, Kowloon, Hong Kong, China. ([email protected]). ‡ School of Mathematical Sciences, MOE-LSC and SHL-MAC, Shanghai Jiao Tong University, 200240, Shanghai, China ([email protected]). 1
Transcript
THE DYNAMICS OF THREE-PHASE TRIPLE JUNCTION ANDCONTACT POINTS
DONG WANG ∗, XIAO-PING WANG † , AND YA-GUANG WANG ‡
Abstract. We use the method of matched asymptotic expansions to study the sharp interfacelimit of the three-phase system modelled by the Cahn-Hilliard equations with the relaxation boundarycondition. The dynamic laws for the interfaces, the triple junction, and the contact points are derivedat different time scales. In particular, we show, at O(t) time scale, the dynamic of the triple junctionis determined by the balance of the chemical potential gradient along the three interfaces meeting atthe triple junction. At faster time scale O(εt), the motion of the triple junction is controlled by thecontact point motions and geometric constraints.
1. Introduction. The phase field model has been widely used to describe themultiphase problems. The basic idea is to represent the interface implicitly by anorder parameter φ (e.g. scalar function to model two-phase problems or vector valuedfunctions to model three-phase problems) which varies continuously over thin interfa-cial layers and is mostly uniform in the bulk phases. The evolution of φ is driven bythe gradient of a total free energy of the system. The total free energy is then a sum ofthree terms: a bulk free energy which is usually taken as a multi-well potential func-tion, an interface energy term depending on the gradient of φ, and a surface energyon the solid boundary. Consider a three-phase system in a two dimensional domainΩ with a closed and smooth solid boundary ∂Ω, the total free energy functional canbe written as
F(φ) =
∫Ω
(ε
2|∇φ|2 +
1
εF (φ)
)dΩ +
∫∂Ω
l(φ)d∂Ω (1.1)
where φ = (φ1, φ2, φ3) is a vector valued order parameter which typically representsthe relative concentration of the three phases with φ1 +φ2 +φ3 = 1 and φi representsthe mass fraction of the ith phase; ε measures the interface thickness between the twophases; F (φ) is a triple well potential function with three equal minima at ~ai(i =1, 2, 3) that denote the three different phases (e.g. a1 = (1, 0, 0), a2 = (0, 1, 0), anda3 = (0, 0, 1)); and l(φ) gives the surface energy density on the solid surface. Theevolution of φ of the system is then described by the Cahn-Hilliard equation
φt = ∇ · (M(φ)∇µ) in Ω, (1.2)
where M(φ) is the diffusion mobility factor and can take various form in differentphysical situations (for example, in [2], M(φ) is chosen as M(φ) =Mφ withM beingthe constant mobility). µ := δF
δφ = −ε∆φ+ 1ε f(φ) is the chemical potential. There are
many studies of (1.2) with general mobility factor, see, for example, [8, 9, 10, 11] and
∗Department of Mathematics, Hong Kong University of Science and Technology, Clear WaterBay, Kowloon, Hong Kong, China. ([email protected]).†Corresponding author. Department of Mathematics, Hong Kong University of Science and Tech-
nology, Clear Water Bay, Kowloon, Hong Kong, China. ([email protected]).‡School of Mathematical Sciences, MOE-LSC and SHL-MAC, Shanghai Jiao Tong University,
the references therein. In this paper, for mathematical simplicity, we only considerthe case when M(φ) = 1 and focus on the dynamics of the contact angles, contactpoints, and the triple junction. The boundary conditions are given by
∂µ
∂~n= 0 on ∂Ω,
where ~n is the outward normal direction of ∂Ω. The variation of (1.1) leads to theboundary condition for φ (see [7] and the references therein)
ε∂nφ+∇φl(φ) = 0 on ∂Ω.
This will impose the equilibrium contact angle condition (i.e. Young’s equation)[26, 24, 25] in the sharp interface limit. To allow a dynamic condition for the contactangle, one can use the following relaxation boundary condition:
φt = −δ[ε∂nφ+∇φl(φ)] on ∂Ω,
where δ ∼ O(1) determines the relaxation time of the interface intersecting with theboundary.
Then, the dynamics of a three-phase system on a solid surface can be modelledby the Cahn-Hilliard equation with the relaxation boundary condition as follows:
φt = ∆µ, in Ω× (0,+∞)
µ = −ε∆φ+ 1ε f(φ) in Ω× (0,+∞),
φt = −δ[ε∂nφ+∇φl(φ)− 13 (∇φl(φ) · ~e)~e] on ∂Ω× (0,+∞),
∂nµ = 0 on ∂Ω× (0,+∞),
φ = φ0 on Ω× 0.
(1.3)
Because of the constraint φ1+φ2+φ3 = 1, we can choose f(φ) = ∇φF (φ)− 13 (∇φF (φ)·
~e)~e to ensure the consistency condition µ1 + µ2 + µ3 = 0. This can be realized byprojecting ∇φF (φ) onto the plane Σ = φ ∈ R3|φ · ~e = 0 where ~e = (1, 1, 1) (moredetails can be found in [27, 3]). In the relaxation boundary condition, l(φ) is alsoprojected in a similar way.
The above system is a special case of a more general diffusive interface modelfor the multiphase flow consisting of a coupled Cahn-Hilliard-Navier-Stokes systemwith the generalized Navier boundary conditions (GNBC) introduced in [20, 21, 23]to model the moving contact line problem. In the slow dynamics, one can neglect theeffect of the flow and the system is reduced to (1.3) which enables us to study theevolution of the interface along the solid boundary and the dynamic contact angle[7].
For the two-phase system, this problem will be simplified to the classical Cahn-Hilliard equation for the phase field order parameter [6]. The sharp interface limit isstudied by Pego in a classical paper [19] using the method of multiscale expansionsfor the Neumann and Dirichlet boundary conditions. In [7], Chen, Xu and Wanganalyze the Cahn-Hilliard equation with relaxation boundary condition and they alsoderived the dynamics of the contact angle when the interface has contact with thesolid boundary.
For the three-phase system modelled by vector-valued Ginzburg-Landau Equa-tion, asymptotic behavior of the solution and dynamics of the interface motion wasstudied by Rubinstein, Sternberg and Keller [22] in the sharp interface limit. Owen,
The dynamics of three-phase triple junction and contact points 3
Rubinstein and Sternberg[18] studied the behaviour of the contact line when theboundary condition is of the Neumann type or Dirichlet type. They showed that theinterface meet the boundary of domain with a π/2 angle when the boundary conditionis of the Neumann type, and when the boundary condition is of the Dirichlet type, theangle should depend on the boundary condition and potential. Lia Bronsard and Fer-nando Reitich [4], presented a formal asymptotical analysis of the Ginzburg-Landauequation and studied the behavior around the triple junction.
For the three-phase system modelled by Allen-Cahn / Cahn-Hilliard system [5],Cohen [17] obtained that the limiting behaviour of the triple junction is governed tothe lowest order by Young’s law, a mass flux balance, and a condition on the sum ofmean curvatures. Several studies of the Cahn-Hilliard equations have also appeared,see for example Alt and Pawlow [1], Eyre [14], Elliott and Luckhaus [12], Elliott andGarcke [13], Garcke and Novick-Cohen [15], Bronsard [4] et al. .
The main purpose of this paper is to study the sharp interface limit of the three-phase system in a two-dimensional bounded domain, modelled by the Cahn-Hilliardequations with relaxation boundary condition on the solid boundary. The methodsin [19, 4] will be generalized to derive the dynamic laws for interfaces, contact pointsand the triple junction at different time scales. We will focus on an ideal case wherethe solid boundary is a unit circle. First, at O(t) time scale, by using the multi-scaleanalysis and the matched asymptotic expansions, we obtain that, in the sharp interfacelimit, the dynamics of interfaces, triple junction and contact points is described bya Hele-Shaw type equation for the free boundaries, with the angles between threeinterfaces at the triple junction obeying the Herring condition. We also show that thecontact points do not show noticeable motion at O(t) time scale, and the interfaceswill approach circular curves as t → +∞. We then study the dynamics at the fasttime scale O(εt). Under the assumption that all three interfaces are circular curves, weshall derive the dynamics of the contact points, contact angles and the triple junctionat the fast time scale.
The outline of the paper is as follows. In Section 2, we firstly use the methodof matched asymptotic expansions to derive the sharp interface limit of the problem(1.3) at O(t) time scale and derive the dynamics of the triple junction. In Section 3,combining the property of the system derived before with volume preserving of eachphase, we study a fast time scale motion of the system and derive a ODE system togovern the dynamics of the contact points, contact angles and the triple junction.
2. The behaviour of the solution at O(t) time scale. In this section, wewill first study the sharp interface limit of the problem (1.3) at the O(t) time scaleusing the method of matched asymptotic expansions. We then study the asymptoticbehaviour of contact points and interfaces as t→ +∞.
2.1. Sharp interface limit. In this subsection, we will use the method ofmatched asymptotic analysis to study the sharp interface limit of the problem (1.3).We will perform outer and inner expansions in three regions and layers as describedin Fig.2.1.
2.1.1. Outer expansions. We seek formal expansions in the following form,away from the interfaces and the triple junction,
Fig. 2.1. Outer region is away from the contact points, triple junction and interfaces, interiorregion is around the interface but far away from the triple junction and contact points, inner regionis just around the triple junction. When we match between different regions, we must match frominner region to interior region and then outer region.
Substituting these expansions into the equations in (1.3), we have
(φ0)t + h.o.t. =∆µ0 + h.o.t.,
µ0 + h.o.t. =1
ε
(∇φ0
F (φ0)− 1
3(∇φ0
F (φ0) · ~e)~e)
(2.1)
+ (φ1 · ∇φ0)
(∇φ0F (φ0)− 1
3(∇φ0F (φ0) · ~e)~e
)+ h.o.t..
Denote by ~a1 = (1, 0, 0),~a2 = (0, 1, 0),~a3 = (0, 0, 1) the three phases 1, 2, 3 respec-tively.
Collecting all the terms at the O( 1ε ) order, we get,
∇φ0F (φ0)− 1
3(∇φ0
F (φ0) · ~e)~e = 0,
which implies that φ0 = ~ai, i = 1, 2, 3, since F (φ) is a potential function with threelocal minima at ~ai, i = 1, 2, 3. This means that at the leading order, three differentphases generate locally.
Without loss of generality, we assume that φ0 divides the whole domain into 3regions and we let Γ1 divides phase 2 and phase 3, Γ2 divides phase 3 and phase 1,Γ3 divides phase 1 and phase 2.
At the O(1) order , from (2.1) we have (φ0)t = ∆µ0, which implies
∆µ0 = 0.
To solve the leading order outer solution, we need the boundary conditions at theinterfaces, which will be derived from the inner expansions near each interface of twophases.
The dynamics of three-phase triple junction and contact points 5
Remark 2.1. Note that µ = −ε∆φ+ 1ε f(φ), where f(φ) = (f1(φ), f2(φ), f3(φ))T=
Hence, for the leading order µ0(x, t), we also have
µ10 + µ2
0 + µ30 = 0. (2.2)
Also, if φi ≡ 0, it is easy to see, from the definition of the chemical potential µ, thatµi is a constant.
2.1.2. Inner expansions at each interface. Without loss of the generality,we will consider inner expansion around the interface Γ1 but away from the triplejunction. The expansions at Γ2,Γ3 will lead to similar results. Introduce the re-scaled coordinate:
z =1
εd(x, t)
where d is the signed distance function to Γ1 and |∇d| = 1 with ∇ denoting thegradient operator in the spatial variable x. Consider the inner expansion as followswith φi(z, x, t) and µi(z, x, t) decaying fast as z → ±∞:
Substitute these expansions into (1.3), it follows
(−m′(t)
ε)∇ηφ0 + h.o.t.
=1
ε2∆ηµ0 +
1
ε∆ηµ1 + h.o.t.,
µ0 + h.o.t. (2.18)
=1
ε
(∆ηφ0 −
(∇φ0
F (φ0)− 1
3(∇φ0
F (φ0) · ~e)~e))
+∆ηφ1 −((
φ1 · ∇φ0
)(∇φ0
F (φ0)− 1
3(∇φ0
F (φ0) · ~e)~e))
+ h.o.t..
At the O( 1ε2 ) order of (2.18), we have:
∆ηµ0 = 0, (2.19)
which implies that µ0 is independent of η by using Liouville’s theorem.The O( 1
ε ) order of (2.18) gives rise to
∆ηφ0 −(∇φ0
F (φ0)− 1
3(∇φ0
F (φ0) · ~e)~e)
= 0, (2.20)
−m′(t)∇ηφ0 = ∆ηµ1. (2.21)
Next, we will use (2.20) and (2.21) to derive the angle condition at the triple junctionand dynamic of the triple junction.
2.1.4. The Herring angle condition at the triple junction. Let T be atriangle with sides perpendicular to Γ1,Γ2,Γ3 and containing the triple junction m(t)as the circumcentre of the triangle (here we only consider the case that angles betweenthree interfaces are between π
2 and π). Denote by h the height of the triangle (SeeFig.2.2). Now, we use a new coordinate ηi = (ξi, ζi), where ζi is the tangent directionof Γi at m(t), and ξi is perpendicular to ζi. We have the following matching conditionfor ξi fixed:
limζi→∞
φ0(ξi, ζi) = φ0(ξi) (2.22)
We now consider (2.20) in the η1 variables. Multiply (2.20) by ∂ξ1 φ0 and integrateover T to have: ∫ ∫
T
(∇φ0
F (φ0)− 1
3(∇φ0
F (φ0) · ~e)~e)· ∂ξ1 φ0dη1
=
∫ ∫T
∂2ξ1 φ0 · ∂ξ1 φ0 + ∂2
ζ1 φ0 · ∂ξ1 φ0dη1.
The dynamics of three-phase triple junction and contact points 9
Fig. 2.2. Constructed triangle for the integration so that to match between different expansionlayers. Notations with different colors denote the corresponding quantities with the same color.
which implies∫ ∫T
∂ξ1(F (φ0) +1
2|∂ζ1 φ0|2 −
1
2|∂ξ1 φ0|2)dη1 =
∫ ∫T
∂ζ1(∂ξ1 φ0 · ∂ζ1 φ0)dη1. (2.23)
By using the divergence theorem in (2.23), we get∫∂T
[F (φ0) +1
2|∂ζ1 φ0|2 −
1
2|∂ξ1 φ0|2]υ1ds =
∫∂T
[∂ξ1 φ0 · ∂ζ1 φ0]υ2ds, (2.24)
where υ = (υ1, υ2) is the outward normal vector to ∂T .
Next, we separate the integral of (2.24) into three parts along three sides of the tri-angle and parametrize these line intergrals in (ξi, ζi) coordinates on the correspondingsides. Notice that we have the following relationship
∂ξ1 = − sin γi∂ξi + cos γi∂ζi
∂ζ1 = − cos γi∂ξi − sin γi∂ζi
where 0 ≤ γi ≤ 2π are the angles between Γi and the ξ1 axis (See Fig.2.2). We thenhave
|∂ζ1 φ0|2 − |∂ξ1 φ0|2
=− cos(2γi)|∂ζi φ0|2 + 4 sin γi cos γi∂ξi φ0 · ∂ζi φ0 + cos(2γi)|∂ξi φ0|2
∂ξ1 φ0 · ∂ζ1 φ0
=− cos γi sin γi|∂ζi φ0|2 − cos(2γi)∂ξi φ0 · ∂ζi φ0 + cos γi sin γi|∂ξi φ0|2.
Using the matching condition (2.22) and the fact that lim|ζi|→∞
|∂ζi φ0| = 0, and by
10 D.WANG, X.WANG, Y.WANG
taking the limit as h→∞ on both sides of (2.24) (see also [4]), we get:
LHS = limh→∞
∫∂T
[F (φ0) +1
2|∂ζ1 φ0|2 −
1
2|∂ξ1 φ0|2]υ1ds
=
∫ +∞
−∞[1
2|∂ξ2 φ0|2 cos(2γ2) + F (φ0)] cos(γ2)dξ2
+
∫ +∞
−∞[1
2|∂ξ3 φ0|2 cos(2γ3) + F (φ0)] cos(γ3)dξ3
RHS = limh→∞
∫∂T
∂ξ1 φ0 · ∂ζ1 φ0υ2ds
= −∫ +∞
−∞|∂ξ2 φ0|2 cos(γ2) sin(γ2) sin(γ2)dξ2
−∫ +∞
−∞|∂ξ3 φ0|2 cos(γ3) sin(γ3) sin(γ3)dξ3.
Again denote the interface tension Φi =∫ +∞−∞ |
∂φ0
∂ξi|2dξi and combining (2.8) with
LHS = RHS, we have:
cos(γ2)Φ2 = − cos(γ3)Φ3.
Since γ2 + π2 = Θ3 and γ3 − γ2 = Θ1, where Θi is described as in Fig.2.2, then we
have:
sin(Θ3)Φ2 = sin(Θ2)Φ3.
If we rotate T so that its base is around Γ2, the same argument will lead:
sin(Θ2)Φ1 = sin(Θ1)Φ2.
Thus, we have the following Herring angle condition at the triple junction:
sin(Θ1)
Φ1=
sin(Θ2)
Φ2=
sin(Θ3)
Φ3, (2.25)
which gives the force balance at the triple junction.
2.1.5. Leading order behaviour. We now summarize the leading order be-haviour at the O(t) time scale. In the leading order, the domain Ω is divided into threedisjoint regions Ωi (i = 1, 2, 3) such that Ω = Ω1∪Ω2∪Ω3. The dynamics of the phasevariable φ0, the chemical potential µ0, interfaces and triple junction is described bythe following Hele-Shaw type equations with free boundaries in (x, t)|x ∈ Ω, t > 0,
φ0 = ~ai, in Ωi, i = 1, 2, 3
∆µ0 = 0, in Ωi, i = 1, 2, 3
∂µ0
∂~n = 0, on ∂Ω
µ0 · [φ0]Γi = −Φi · κi, on Γi (i = 1, 2, 3)
[φ0]Γidt = [D~n(µ0)]Γi , on Γi (i = 1, 2, 3)
sin(Θ1)Φ1
= sin(Θ2)Φ2
= sin(Θ3)Φ3
, at Γ1 ∩ Γ2 ∩ Γ3
(2.26)
The dynamics of three-phase triple junction and contact points 11
where [.]Γi represents the jump along Γi interface (defined in Section 2.1.1), κi thecurvature of Γi, Φi the interface tension of Γi, and Θi the angles at the triple junctionbetween interfaces as shown in Fig. 2.2.
Remark 2.2. The dynamic of triple junction can be implicitly determined bysolving the above problem (2.26) from the dynamic of interfaces and the Herring anglecondition at the triple junction. On the other hand, we can also use the matchedasymptotic expansion to derive an explicit system to describe the dynamic of triplejunction in the next subsection.
2.2. Motion of the triple junction. To derive the dynamic law of the triplejunction point explicitly, we integrate equation (2.21) over T with three verticesP1, P2, P3 (See Fig.2.2), we have
−m′(t) ·
∫T
∇η1 φ0dη1 =
∫T
∆µ1dη1. (2.27)
Use the Green theorem, the left hand side is converted into a boundary integral onthree sides of the triangle. We then parameterize these line integrals into the localcoordinates (ξi, ζi). Calculate the integral of ∇η1 φ0 in each component, we have∫
T
∂φ0
∂ξ1dη1 =
∫ P1
P3
φ0(ξ2) sin(Θ3)dξ2 −∫ P2
P1
φ0(ξ3) sin(Θ2)dξ3 (2.28)∫T
∂φ0
∂ζ1dη1 =
∫ P3
P2
φ0(ξ1)dξ1 +
∫ P1
P3
φ0(ξ2) cos(Θ3)dξ2
+
∫ P2
P1
φ0(ξ3) cos(Θ2)dξ3. (2.29)
For the right hand side of (2.27), we apply the divergence theorem to get∫T
∆µ1dT =
∫∂T
∂µ1
∂~nds
=
∫ P3
P2
∂µ1
∂ζ1dξ1 +
∫ P1
P3
∂µ1
∂ζ2dξ2 +
∫ P2
P1
∂µ1
∂ζ3dξ3, (2.30)
where ~n is the outward normal vector of the boundary of the triangle.Note that lim
ξi→∞limζi→∞
φ0 is a constant vector. The integrals in (2.28) and (2.29)
are divergent. To calculate the limit as the length h of the height of T tends to∞, we divide h on both sides of (2.27). Since Θ1,Θ2,Θ3 are determined by (2.25),we have |P1P2| = h
sin Θ2, |P3P1| = h
sin Θ3, |P2P3| = − h
tan Θ2− h
tan Θ3(we always assume
Θ1,Θ2,Θ3 ∈ (π2 , π)). Using the matching condition (2.22), rescaling (2.28) by dividingthe characteristic length h (i.e. the height of the T as in Fig. 2.2) on both sides, andusing first mean value theorem for definite integrals, it is easy to see that, the scaled(2.28) can be calculated as the follows:
limh→∞
(1
h
∫ P1
P3
φ0(ξ2) sin(Θ3)dξ2 −1
h
∫ P2
P1
φ0(ξ3) sin(Θ2)dξ3
)
= limh→∞
(|P3P1| sin(Θ3)
hφ2 − |P1P2| sin(Θ2)
hφ3
)=φ2 − φ3, (2.31)
12 D.WANG, X.WANG, Y.WANG
where φ2 is a vector representing the mean value of the integral of φ0 with respect to
ξ2 from P3 to P1 (φ3 and φ1 are also similarly defined). Note that these mean valuesare determined based on the profile of φ0 through the corresponding interface.
Similarly, for (2.29), we also have
limh→∞
1
h
(∫ P3
P2
φ0(ξ1)dξ1 +
∫ P1
P3
φ0(ξ2) cos(Θ3)dξ2 +
∫ P2
P1
φ0(ξ3) cos(Θ2)dξ3
)
=− (1
tan Θ2+
1
tan Θ3)φ1 +
1
tan Θ3φ2 +
1
tan Θ2φ3. (2.32)
To calculate the limiting behaviour of (2.30) as h → +∞, the matching condi-tions for the inner and outer expansions of the chemical potential along Γi must bedeveloped. In the coordinate ηi = (ξi, ζi), when εζi is between O(ε) and o(1), weexpand
where τi is the tangent direction of the interface Γi at the triple junction. To matchthis expansion to the inner expansion at the triple junction, one requires that
of ξi (as ζi → +∞). (2.30) can then be calculated as the following (for convenience,we use Dτi µ0 to represent Dτi µ0(m(t), t) in follows):
limh→∞
1
h
(∫ P3
P2
∂µ1
∂ζ1dξ1 +
∫ P1
P3
∂µ1
∂ζ2dξ2 +
∫ P2
P1
∂µ1
∂ζ3dξ3
)
=− (1
tan Θ2+
1
tan Θ3)Dτ1 µ0 +
1
sin Θ3Dτ2 µ0 +
1
sin Θ2Dτ3 µ0. (2.36)
Combining (2.27) (2.31) (2.32) and (2.36), we have follows:
−m′(t) ·
(φ2 − φ3
−( 1tan Θ2
+ 1tan Θ3
)φ1 + 1tan Θ3
φ2 + 1tan Θ2
φ3
)
=− (1
tan Θ2+
1
tan Θ3)Dτ1 µ0 +
1
sin Θ3Dτ2 µ0 +
1
sin Θ2Dτ3 µ0. (2.37)
It is easy to see that φ1 = (0, 12 ,
12 ), φ2 = ( 1
2 , 0,12 ), φ3 = ( 1
2 ,12 , 0). Denote by Υ =
(Υi,j)2×3 and Ξ = (Ξi,j)1×3 with
Ξ = (− 1
tan Θ2− 1
tan Θ3)Dτ1 µ0 +
1
sin Θ3Dτ2 µ0 +
1
sin Θ2Dτ3 µ0
Υ =
(0 − 1
212
12 tan Θ2
+ 12 tan Θ3
− 12 tan Θ3
− 12 tan Θ2
).
The dynamics of three-phase triple junction and contact points 13
Then the system (2.37) is written as:
−m′(t) ·Υ = Ξ. (2.38)
If we write µ0 in components (µ10, µ
20, µ
30) and use the fact that
Υ1,1 + Υ1,2 + Υ1,3 = Υ2,1 + Υ2,2 + Υ2,3 = 0,
Ξ1,1 + Ξ1,2 + Ξ1,3 = 0, (2.39)
we then have
m′(t) ·
(12 − 1
21
2 tan Θ3
12 tan Θ2
)=− (
1
tan Θ2+
1
tan Θ3)(Dτ1 µ
20, Dτ1 µ
30) +
1
sin Θ3(Dτ2 µ
20, Dτ2 µ
30)
+1
sin Θ2(Dτ3 µ
20, Dτ3 µ
30). (2.40)
Denote Ψ =
(− cos Θ2 sin Θ3 − sin Θ2 sin Θ3
cos Θ3 sin Θ2 − sin Θ2 sin Θ3
)and µ
′
0 = (µ20, µ
30), we can solve (2.40)
to give
m′(t) =
2
sin Θ1 sin Θ2 sin Θ3
(sin Θ1Dτ1 µ
′
0 + sin Θ2Dτ2 µ′
0 + sin Θ3Dτ3 µ′
0
)Ψ, (2.41)
which gives the dynamics of the triple junction explicitly. We note that this is abalance of the chemical potential flux (along three interfaces) at the triple junction.
2.3. Asymptotic behaviour. In this section, we study the asymptotic be-haviour of the system at O(t) time scale. We show that the interfaces approachcircular curves as t → ∞ and the contact points do not have noticeable motion atthis time scale.
2.3.1. Energy estimates. To derive the energy estimate, we multiply the equa-tion φt −∆µ = 0 by µ on both sides and integrate over the whole domain to have
0 =
∫Ω
µ(φt −∆µ) dΩ
=
∫Ω
−µ∆µ+
(−ε∆φ+
1
ε
(∇φF (φ)− 1
3(∇φF (φ) · ~e)~e
))φt
dΩ
=
∫∂Ω
−µ∂µ
∂n− ε∂φ
∂n· φtdΓ +
∫Ω
|∇µ|2 + ε∇φ · ∇φt +1
ε∇φF (φ) · φt
dΩ
=
∫Ω
|∇µ|2 + ε∇φ · ∇φt +
1
ε∇φF (φ) · φt
dΩ +
∫∂Ω
φ2t
δ+ σ(x)∇φl(φ)φt
dΓ
=∂
∂t
∫Ω
1
εF (φ) +
ε
2|∇φ|2dΩ +
∫∂Ω
σ(x)l(φ)dΓ
+
∫Ω
|∇µ|2
dΩ +
∫∂Ω
φ2t
δ
dΓ,
14 D.WANG, X.WANG, Y.WANG
where we have used the boundary conditions given in (1.1).Let
E(φ) =
∫Ω
1
εF (φ) +
ε
2|∇φ|2dΩ +
∫∂Ω
σ(x)l(φ)dΓ,
we then have
−∂E(φ)
∂t=
∫Ω
|∇µ|2dΩ +
∫∂Ω
φ2t
δdΓ.
Integrate with respect to t, we have∫ +∞
0
∫Ω
|∇µ|2dΩ +
∫∂Ω
φ2t
δdΓ
dt ≤ −E(φ)|t=+∞ + E(φ)|t=0
≤ E(φ0), (2.42)
which implies that∫ +∞
0
∫∂Ω
φ2t
δ dΓdt is a bounded function. We also have that∫
Ω|∇µ|2
approaches to 0, when t→ +∞. Consequently, we have |∇µ|2 → 0 as t→ +∞. Thismeans µ approaches to a constant when t → +∞. It follows from (2.17) that theinterface becomes circular since the curvature of three interfaces become a constant.
2.3.2. The behaviour of contact points. Consider one of the contact pointswith position in the polar coordinates (r, θ) = (1, θ(t)) since the solid boundary ∂Ωis a unit circle we defined before, and from t = 0 to t = T , the contact point movesfrom θ(0) = 0 to θ(T ) = b > 0. For each x1 ∈ (0, b), denote by t±(x1) the time atwhich φε(1, x1, t
+ε (x1)) = (1
3 ,23 , 0),φε(1, x1, t
−ε (x1)) = (2
3 ,13 , 0). Obviously, we have
(−b3,b
3, 0) = lim
ε→0
∫ b
0
[φε(1, x1, t
+ε (x1))− φε(1, x1, t
−ε (x1))
]dx1
= limε→0
∫ b
0
∫ t+ε (x1)
t−ε (x1)
φε,t(1, x1, t)dtdx1. (2.43)
Denote by φε(1, x, t) = (φε,1(1, x, t), φε,2(1, x, t), φε,3(1, x, t))T . From the second com-
ponent of the relation (2.43), the boundedness of∫ +∞
0
∫∂Ω
φ2t
δ dΓdt, and the use ofCauchy-Schwarz inequality, we have
b
3= limε→0
∫ t+ε (x1)
t−ε (x1)
∫ b
0
(φε,2)t√δ
√δdx1dt
≤ limε→0
[∫ t+ε
t−ε
∫∂Ω
(φε,2)2t
δdΓdt
] 12
δ12 [A(Dε)]
12 (2.44)
≤ limε→0
[∫ +∞
0
∫∂Ω
(φε,2)2t
δdΓdt
] 12
δ12 [A(Dε)]
12
≤ limε→0
C0[A(Dε)]12 ,
where (φε,2)t represents∂φε,2(1,x1,t)
∂t , C0 denotes the upper bound of
√δ
[∫ +∞
0
∫∂Ω
(φε,2)2t
δdΓdt
] 12
,
The dynamics of three-phase triple junction and contact points 15
and A(Dε) is the area of region:
Dε :=
(x, t)|x ∈ ∂Ω, 0 ≤ t ≤ T, φε,2(1, x, t) ∈ (
1
3,
2
3)
.
Obviously, Dε has thickness O(ε) so limε→0|Dε| = 0. Then, from (2.44), we get b = 0
as ε → 0, which implies that in the limit ε → 0, the contact points do not shownoticeable motion.
3. Fast time motion. In this section, we will study the behaviour at a fast timescale s = εt. In the previous section, we have known that when t→ +∞, the interfacesapproach circular and the contact points do not move in the leading order. Note thats ∈ [0, 1] is equivalent to t ∈ [0, 1/ε]. Hence we may assume that the interfaces arecircular initially at this fast time scale (as shown in Fig.3.1). We will concentrate onthe behavior around contact points on the solid boundary to derive the dynamics ofthe contact angles and contact points. Firstly, we introduce some constraints whichshould be kept in this fast time scale. Then we take asymptotical expansions aroundthe contact points to study the behavior around the contact points. Finally, we obtainan ordinary differential system to describe the motion of the whole system at this fasttime scale.
Fig. 3.1. Left: The initial profile of the system where the unit circle is the solid boundary withthree circular interfaces inside. Right: A chord with S as chord length and β as the angle of thechord. Notations with different colors denote the corresponding quantities with the same color.
3.1. Volume conservation. Integrating both sides of (1.3) in Ω and using theboundary condition for µ, we have
∂
∂t
∫Ω
φdΩ =
∫Ω
∂φ
∂tdΩ =
∫Ω
∆µdΩ =
∫∂Ω
∂µ
∂ndΓ = 0.
This shows that the volume of three phases are conserved in the system. To calculatethe volume of three phases in Fig.3.1, we note that the area of the chord with angleβ and length S is given by
A = (S
2 sinβ)2β −
(S2 )2
tanβ=S2(β − sinβ cosβ)
4 sin2 β.
Now, we introduce some notations we will use in this section (See Fig.3.1). Wedenote three contact points position, which depends on time s, as
Ci(s) = (cos θi(s), sin θi(s)), i = 1, 2, 3
16 D.WANG, X.WANG, Y.WANG
where θi(s) ∈ [0, 2π) are the polar angles of the point Ci(s). Denote the triple junctionposition asm(s) = (m1(s),m2(s)), and we then denote the three vectors between three
contact points and the triple junction as ~li(s) = Ci(s)−m(s). and αi(s) as the threeangles between the inward direction of the tangent line of interface at the contactpoint and the clockwise direction of the tangent line of the solid boundary at thecontact points. Now, we introduce the calculation of the area of the region betweenthe ith interface and the (i + 1)th interface. For instance, without loss of generality,the area for the region between the 3rd interface and the 1st interface (see black solidcurves in Fig. 3.2) can be calculated by A3 = A(D1) + A(D2) + A(D3) − A(D4)where A(·) represents the area of each region in the figure, D1 denotes the triangleregion bounded by the brown dotted lines in the figure, and Di (i = 2, 3, 4) denote
the regions of three chords in the figure. Using the notations of ~li(s), χi(s), θi(s) and
Fig. 3.2. Diagram for the calculation of area of the region of phase 3. D1 denotes the triangleregion bounded by the brown dotted lines and D2, D3, and D4 denote the regions of three chords.Notations with different colors denote the corresponding quantities with the same color.
αi(s), the formula for the area of a chord we introduced, and the area formula for atriangle in the cross product between two vectors, we can have
A(D1) =1
2|~l3(s)×~l1(s)|,
A(D2) =1
2(θ1(s)− θ3(s)− sin(θ1(s)− θ3(s))),
A(D3) =|~l3(s)|2(χ3(s)− sinχ3(s) cosχ3(s))
4 sin2 χ3(s),
A(D4) =|~l1(s)|2(χ1(s)− sinχ1(s) cosχ1(s))
4 sin2 χ1(s)
where
χi(s) =π − αi(s)−θi+1(s)− θi(s)
2
− arccos
(~li(s) · (cos θi(s)− cos θi+1(s), sin θi(s)− sin θi+1(s))
|~li(s)| × |2 sin( θi+1(s)−θi(s)2 )|
).
The dynamics of three-phase triple junction and contact points 17
Remark 3.1.1. χi can be negative and thus the area of the chord with the angle χi can be
negative. Take the region between 3rd interface and 1st interface in Fig. 3.2 forexample, χ3 is negative if D3 ⊂ D1 ∪D2 and thus A(D3) is negative. That will makethe formula A3 = A(D1) +A(D2) +A(D3)−A(D4) be general for arbitrary profiles.
2. We take i+ 1 = 1 when i = 3 in the formula.In general, we write
Ai =1
2|~li(s)×~li+1(s)|+ 1
2(θi+1(s)− θi(s)− sin(θi+1(s)− θi(s)))
+|~li(s)|2(χi(s)− sinχi(s) cosχi(s))
4 sin2 χi(s)
− |~li+1(s)|2(χi+1(s)− sinχi+1(s) cosχi+1(s))
4 sin2 χi+1(s). (3.1)
From volume conservation, we know that Ai (i = 1, 2, 3) are constants independentof s.
3.2. Angle condition. From (2.25), we rewrite the angle condition as the fol-lowing:
sin(Θ1)
Φ1=
sin(Θ2)
Φ2=
sin(Θ3)
Φ3,
where (Θ1,Θ2,Θ3) are uniquely determined by (Φ1,Φ2,Φ3). It is easy to show that
Θi = arccos(~li(s) ·~li+1(s)
|~li(s)||~li+1(s)|) + χi(s)− χi+1(s). i = 1, 2, 3 (3.2)
3.3. The constraint of curvature and interface tension. In Section 2.3.1,we showed that µ = (µ1, µ2, µ3) approaches to a constant vector. Then, using (2.17),we have
0 = µ1 − µ2 + µ2 − µ3 + µ3 − µ1
= −(Φ1 · κ1 + Φ2 · κ2 + Φ3 · κ3). (3.3)
This leads to
0 =Φ1 · κ1 + Φ2 · κ2 + Φ3 · κ3
=Φ1 ·2 sinχ1(s)
|~l1(s)|+ Φ2 ·
2 sinχ2(s)
|~l2(s)|+ Φ3 ·
2 sinχ3(s)
|~l3(s)|. (3.4)
3.4. Expansion near contact points. Now, we take expansions around threecontact points Ci, i = 1, 2, 3. Without loss of the generality, we will consider expansionaround the contact point C1 = (cos θε1(s), sin θε1(s)) but away from the triple junctionand other contact points. We use the stretched variable defined by
z1 =1− rε
,
z2 =−(x− cos θε1(s)) cos(θε1(s)− αε1(s))− (y − sin θε1(s)) sin(θε1(s)− αε1(s))
ε,
18 D.WANG, X.WANG, Y.WANG
𝛼𝑖(s)
𝑧2
𝑧1
𝐶𝑖
Fig. 3.3.
where r =√x2 + y2, and z2 is a stretched variable along the norm direction of the
interface (See Fig.3.3). We then consider the following expansions:
Substituting these expansions into (1.3), from the leading order equation and theboundary condition we get in z1 > 0, z2 ∈ R, s > 0,
∂2Φ0
∂z21+ 2 cosα1(s) ∂2Φ0
∂z1∂z2+ ∂2Φ0
∂z22− f(Φ0) = 0,
(− sinα1(s)θ′
1(s)− δ cosα1(s))∂Φ0
∂z2
= −δ ∂Φ0
∂z1− δ∇Φ0 l(Φ0) + δ(∇Φ0 l(Φ0) · ~e)~e, on z1 = 0.
(3.5)
Let Q(z2) = limz1→+∞
Φ0(z1, z2, s). Obviously, Q satisfies the following problem:
∂2Q
∂z22
− f(Q) = 0 on R, (3.6)
Q(+∞) = (0, 1, 0), Q(−∞) = (0, 0, 1), Q(0) = (0,1
2,
1
2). (3.7)
This implies that
1
2| ∂Q∂z2|2 = F (Q(z2)). (3.8)
If we choose l, such that (see also [7])
∇Ql(Q)− (∇Ql(Q) · ~e)~e = σ1∂Q
∂z2,
then, (3.5) has a special solution:
Φ0(z1, z2, s) = Q(z2),
and θ1(s) is given by:
− sinα1(s)θ′
1(s) = δ cosα1(s)− δσ1. (3.9)
The dynamics of three-phase triple junction and contact points 19
Similarly, we get
− sinα2(s)θ′
2(s) = δ cosα2(s)− δσ2, (3.10)
− sinα3(s)θ′
3(s) = δ cosα3(s)− δσ3. (3.11)
It is easy to see that, given αi(s) (i = 1, 2, 3), one can solve the five unknownfunctions θi(s)(i = 1, 2, 3), (m1(s),m2(s)) from the five equations (3.1), (3.2), (3.4),when they are solvable. Denote the solutions as
we can write the equations (3.9)-(3.11) as − sinα1∂θ1∂α1
− sinα1∂θ1∂α2
− sinα1∂θ1∂α3
− sinα2∂θ2∂α1
− sinα2∂θ2∂α2
− sinα2∂θ2∂α3
− sinα3∂θ3∂α1
− sinα3∂θ3∂α2
− sinα3∂θ3∂α3
α
′
1(s)
α′
2(s)
α′
3(s)
=
δ(cosα1(s)− σ1)δ(cosα2(s)− σ2)δ(cosα3(s)− σ3)
, (3.14)
which can be re-written as − sinα1 0 00 − sinα2 00 0 − sinα3
∂θ1∂α1
∂θ1∂α2
∂θ1∂α3
∂θ2∂α1
∂θ2∂α2
∂θ2∂α3
∂θ3∂α1
∂θ3∂α2
∂θ3∂α3
α
′
1(s)
α′
2(s)
α′
3(s)
=
δ(cosα1(s)− σ1)δ(cosα2(s)− σ2)δ(cosα3(s)− σ3)
. (3.15)
This gives the dynamics of the three contact angles α1(s), α2(s) and α3(s). Onceαi(s) are solved, we can then determine the dynamics of the contact points and thetriple junction from (3.12) and (3.13).
Remark 3.2. Denote θY i as the Young’s angle at the ith contact point Ci. Then,we actually have σi = cos(θY i) [7]. Then, it is easy to see, only when αi = θY i, thesystem will be at equilibrium which is the static solution of system (1.3) .
3.5. Numerical example. In this example, we choose domain Ω to be a unitdisc with the boundary ∂Ω and we set σ1 = 1/2, σ2 = −1/2, and σ3 = 0 withthree static contact angles equal to π/3, 2π/3, and π/2 correspondingly. We set threeHerring angles as π
2 ,3π4 , and 3π
4 . The volumes of the three phases are A1 = π4 and
A2 = A3 = 3π8 . The initial position of the triple junction is set at (m1,m2) = (0, 0)
and the initial contact points are set at C1 : (√
22 ,√
22 ), C2 : (−
√2
2 ,√
22 ), C3 : (0,−1)
20 D.WANG, X.WANG, Y.WANG
with the initial contact angles α1 = α2 = α3 = π2 (See Fig. 3.4). We then set
δ = 1 and solve the system (3.15). The dynamics of contact angles, contact pointsare then plotted in Fig. 3.5. It is shown that when s → ∞, the triple junctionmoves toward the equilibrium position (0,−0.2602). Three contact points converge toC1 : (0.4328, 0.9015), C2 : (−0.4328, 0.9015), C3 : (0,−1) with the equilibrium contactangles α1 = π
3 , α2 = 2π3 , α3 = π
2 respectively (See Fig. 3.6).
s = 0
Fig. 3.4. Initial interface profile by keeping three interfaces as circular with initial contact
points: C1 : (√2
2,√2
2), C2 : (−
√2
2,√22
), and C3 : (0,−1), contact angles: α1 = π3
, α2 = 2π3
and
α3 = π2
, Young’s angle: Θ1 = π2
, Θ2 = 3π4
, and Θ3 = 3π4
, volume: A1 = π4
and A2 = A3 = 3π8
,and position of triple junction:(m1,m2) = (0, 0).
s=0.5 s=1 s=1.5 s=2
s=2.5 s=3 s=3.5 s=4
Fig. 3.5. The interface profile’s evolution governed by the system (3.15) at s =0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, and 4.0 respectively.
s0 1 2 3 4
0.4
0.5
0.6
0.7
0.8
0.9
1
cos(θ1)
sin(θ1)
EquilibriumEquilibrium
s0 1 2 3 4
1
1.1
1.2
1.3
1.4
1.5
1.6α
1
Equilibrium
s0 1 2 3 4
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0mEquilibrium
Fig. 3.6. From left to right: The dynamic of C1 = (cos(θ1), sin(θ1)), the dynamic of the contactangle α1, and the dynamic of m where we use (0,m) to represent the triple junction.
The dynamics of three-phase triple junction and contact points 21
4. Conclusions and further discussions. Using matched asymptotic expan-sion, we derive the sharp interface models at different time scales for the three-phaseCahn-Hilliard equation with constant mobility factor and the relaxation boundarycondition on the solid boundary. At O(t) time scale, the dynamics of the triple junc-tion is determined by the balance of the chemical potential gradient along the threeinterfaces meeting at the triple junction. At faster time scale O(εt), the motion of thetriple junction is controlled by the contact point motions and geometric constraints.These sharp interface models explicitly determine the dynamics of the triple junctions,contact angles and contact points. We remark that although we focus on the constantmobility factor case in this paper only, the method can be applied to the multi-phaseCahn-Hilliard equation with degenerate mobility or highly disparate diffusion mobil-ity. The dynamics will be quite different from what we obtained in this paper. Thesewill be investigated and reported in the future.
Acknowledgments: We thank the anonymous referees for careful reading andcritical comments. This work was supported in part by the Hong Kong RGC-GRFgrants 605513, 16302715, and NNSF of China under Grant Nos. 11631008 and91530114. The third author’s research was also supported by Shanghai Committee ofScience and Technology under Grant No. 15XD1502300.
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