## Abstract

We use a multi-scale analysis to derive a sharp interface limit for the dynamics of bilayer structures of the functionalized Cahn–Hilliard equation. In contrast to analysis based on single-layer interfaces, we show that the Stefan and Mullins–Sekerka problems derived for the evolution of single-layer interfaces for the Cahn–Hilliard equation are trivial in this context, and the sharp interface limit yields a quenched mean-curvature-driven normal velocity at *O*(*ε*^{−1}), whereas on the longer *O*(*ε*^{−2}) time scale, it leads to a total surface area preserving Willmore flow. In particular, for space dimension *n*=2, the constrained Willmore flow drives collections of spherically symmetric vesicles to a common radius, whereas for *n*=3, the radii are constant, and for *n*≥4 the largest vesicle dominates.

## 1. Introduction

The functionalized Cahn–Hilliard (FCH) equation has been proposed as a model for interfacial energy in phase-separated mixtures with an amphiphilic structure [1]. Of particular interest are polymer electrolyte membranes in which hydrophobic polymers are functionalized by the addition of acid-tipped side chains that greatly modify their solubility [2–4]. In particular, the solvation energy of the tethered acid groups drives the mixture to *increase* surface area so as to facilitate the access of the side chains to the solvent phase whose screening effect serves to lower the overall electrostatic energy [1]. In addition, there is considerable interest in the properties of bilayer structures within the biological community, where issues such as endocytosis, vesicle budding and the opening of pores are of interest [5,6].

The FCH energy is a natural test bed for the study of bilayer structures. In contrast to models based on single-layer or heteroclinic interfaces with no intrinsic bending energy, the FCH naturally produces stable bilayer, or homoclinic, interfaces with an intrinsic width which is resistant to external forces. Moreover, unlike sharp interface approximations such as the Canham–Helfrich energy [7,8], the FCH naturally accommodates merging and pinch-off events that are dominant mechanisms for formation of networks.

The CH energy, introduced in Cahn & Hilliard [9], characterizes a binary mixture by a phase field function *u* that maps *Ω*⊂**R**^{n} into mixture values [−1,1]. It models the free energy as a balance between entropic effects, which seek to homogenize the species, and the mixture potential, *W*, which assigns energies to blends
1.1where the parameter *ε*≪1 controls the width of the inner structures. Motivated by the study [3], we consider a class of double-well potentials, *W*, which describes the energy of the mixture *u*, with two *unequal depth* local minima at *b*_{−}<*b*_{+}, for which *W*(*b*_{−})=0>*W*(*b*_{+}) and *W*^{′} has precisely three zeros, at *b*_{−}<*b*^{0}<*b*_{+}. The phase *u*=*b*_{−} with the higher self-energy is the majority phase, whereas the *u*=*b*_{+} phase is the minority phase (amphiphilic surfactant or lipid). The well-tilt, or difference in self-energies *W*(*b*_{−})−*W*(*b*_{+})>0 is a significant bifurcation parameter for network morphologies [4].

The single-layer interfaces of the CH free energy are natural minimizers of for untilted wells *W*, i.e. *W*(*b*_{−})=*W*(*b*_{+}). Solutions generated from single-layer interfaces are well known to *Γ*-converge to interfacial surface area as , [10,11]. That is, for *ε*≪1, minimizing sequences *u*_{ε} which converge to a limit in *L*^{1}(*Ω*) localize their gradients on an interface *Γ*⊂**R**^{n}, whereas tends to a value which is proportional to the interfacial surface area. The FCH remaps this paradigm, balancing the square of the variational derivative of the CH energy against a small multiple of itself,
1.2The term functionalization is borrowed from synthetic chemistry where it refers to the addition of hydrophilic (functional) groups to a hydrophobic polymer to modify its solubility. Mathematically, ‘functionalization’ is a systematic reformulation of the original energy. Indeed for the *η*_{2}=0 problem, *all* critical points of , that is the solutions of , render and hence are global minimizers of . The parameter *η*_{2} unfolds this highly degenerate situation: crucially, for *η*_{2}>0, the unfolding term *favours* the critical points of with more surface area. For the particular form of the CH energy, the FCH takes the form,
1.3

There is an extensive literature that uses single-layer interfaces to describe a wide range of physical phenomena, including image segmentation, phase transitions, multi-phase flows, crystallization and other phase transitions. Higher-order energies, which resemble the FCH with *η*_{2}<0 and an untilted well *W*, have been proposed, see in particular equations (1.5) of Loreti & March [12] and (3.16) of Torabi *et al*. [13]. Indeed, the De Giorgi conjecture, which concerns the *Γ* limit of the FCH energy for *η*_{2}<0 with an untilted well has been established [14]. Extensions of these models to address deformations of elastic vesicles subject to volume constraints [15,16], and multi-component models that incorporate a variable intrinsic curvature have been investigated [17,18]. However, the single-layer interface forms the essential underpinning of each of these models. Conversely, it is easy to see that, for *η*_{2}>0, the FCH energy does not have an *ε* independent lower bound over configurations with prescribed volume fraction. In this regime, the FCH has no traditional *Γ*-limit, and the natural tendency to view the FCH energy as a diffuse interface regularization of a Canham–Helfrich [7,8] sharp interface energy of the form
1.4is potentially misleading. The identification of the FCH with a Canham–Helfrich-type sharp interface energy is predicated on the assumption that the underlying structures are of co-dimension 1 and free of defects, such as end-caps and junctions. Over , the FCH free energy supports co-dimension one bilayer interfaces, whose evolution we study here, as well as and a wide range of stable co-dimension 2 and co-dimension 3 morphologies [4,19] described below, in addition to many locally stable defect structures. The structure of the problem, and the physically motivating examples, change fundamentally and dramatically with the sign of *η*_{2}. For these reasons, the FCH merits a distinct name, see Promilsow & Wetton [2], which evokes the amphiphilic nature of functionalized polymers.

It is crucial to emphasize the distinction between single-layer interfaces, which separate two dissimilar phases across a co-dimension one interface, and bilayers that separate two identical phases by a thin region of a second phase. *Significantly, the single-layer framework cannot support perforation of the interface.* In many biological processes, it is essential to understand the opening and closing of pores within a vesicle, or the roll-up of a bicelle into a closed vesicle [20]. Single-layer models treat the inside and outside of a vesicle as distinct phases: they cannot be merged. By contrast, the *η*_{2}>0 unfolding of the tilted-well FCH model supports stable, strongly incompressible bilayers that admit not only the opening of perforations, but the roll-up of the bilayer into a solid filament or a solid micelle, in a manner that naturally accounts for the competition between these morphologically distinct structures for a scarce surfactant phase. We do not fully address this competition within this work, rather we demonstrate that the sharp interface limit of the FCH energy for bilayer structures has fundamental distinctions with the sharp interface limit derived for single layers, see (1.8) and the following discussion.

The essential feature of functionalization is that it greatly increases the possible collection of stable interfacial structures. For the CH energy, the quasi-stable structures are dominated by non-self-intersecting, smooth, co-dimension one, closed interfaces, *Γ*⊂*Ω*. The normal and tangential spaces of *Γ* form a coordinate system in a neighbourhood about *Γ*, in which at leading order the Cartesian–Laplacian becomes where *z*=*z*(*x*) is the inner variable, often presented as the *ε*-scaled, signed distance to the interface. Subject to a total mass constraint, the co-dimension one critical points of the CH energy are constructed from inner structures solving the second-order differential equation
1.5Choosing the Lagrange multiplier *λ* so that the modified well *W*(*s*)+*λs* is ‘un-tilted’, i.e. taking equal values at its two minima, then (1.5) supports a heteroclinic or single-layer connection, *U*_{s} between the two minima. The interface *Γ* is then ‘dressed’ with *U*_{s}, see (2.11), yielding an approximate critical point of that takes distinct constant values on the regions on the opposite sides of the interface *Γ*. It is significant that the squared-variational term within the FCH prevents the Lagrange multiplier associated with the mass constraint from ‘un-tilting’ the well *W*, rather the multiplier serves to shift the location of the two minima, particularly that of the majority phase to *b*=*b*_{−}+*O*(*ε*), see (5.10) and [3]. As a consequence, there is a bilayer solution *U*_{b}=*U*_{b}(*z*;*λ*) of (1.5) with *λ*=0, which is homoclinic to the shifted majority phase *b*, potentially stable [19].

For a co-dimension one interface, *Γ*, the dressing of *Γ* with *U*_{b} yields to a bilayer interface with an *O*(*ε*) width which is potentially stable, coherent structure [19]. Moreover, stable structures can also be generated near higher co-dimensional interfaces. For example in **R**^{3}, co-dimension two structures are formed by dressing a one-dimensional filament *Γ* with a radially symmetric solution of
1.6where *R*>0 is the scaled distance to the centre line of *Γ*. The end result is a ‘spaghetti’-like structure with an *O*(*ε*) unscaled radius, but a spatially extended length. The FCH also possesses co-dimension three micellular structures in **R**^{3} as well as defect states such as end-caps and multi-junctions that play an essential role in network formation, see [4,21].

This paper presents a formal reduction in the *H*^{−1} gradient flow of the functionalized energy , called the FCH equation,
1.7subject to periodic or zero-flux boundary conditions on *Ω*⊂**R**^{n}. Here, *μ* is the chemical energy. This flow has the distinction of decreasing while preserving the masses of the majority and minority (surfactant) phases via a flux law that depends locally on the values of concentration *u*. One may anticipate that solutions of the FCH equation quickly converge towards a slow-manifold comprised approximate critical points of and, modulo self-intersection or pinch-off events of the underlying interface, subsequently flow slowly along the manifold until arriving at a morphology of approximate critical points of whose surface area is maximal in comparison with other approximate critical points of .

Our central results confirm this intuition, however, there are several surprises. Indeed, for the single-layer interfaces of the CH equation, it was shown [22] that the sharp interface reduction yields a spinodal decomposition at *O*(*ε*), a Stefan-problem, at *t*=*O*(1), and a Mullins–Sekerka problem, at *t*=*O*(*ε*^{−1}), for the outer chemical potential. By contrast, for the choice of scaling in (1.7), the flow is slower, but more importantly the chemical potential problem *uncouples from the free surface problem*; the chemical potential converges to a spatial constant, and the free-surface problems reduce to Ricci-type flows. Indeed, on the *O*(*ε*^{−1}) time scale, in the place of the Mullins–Sekerka problem, we find a quenched, mean-curvature-driven flow, in which the normal velocity couples to an evanescent, spatially constant, chemical potential, *B*_{1}(*t*_{1}),
1.8and
1.9where *κ*_{j}, introduced in (2.8) is the sum of the *j*+1-st powers of the curvatures, whereas the *W*-dependent constants *σ*_{1},*ρ*_{1}>0 are surface tensions and decay rates.

While the exponential decay of *B*_{1} quenches the mean-curvature flow, the sign of *B*_{1}, see (5.30), is of significance. The equilibrium value, *b*_{−}, of the majority phase, i.e. the left minima of the double well *W*, is associated with a mixture with a small, positive concentration of the minority phase (typically amphiphilic surfactant or lipid). The case *B*_{1}>0 corresponds to an initial mass of the minority phase in excess of that required to construct the prescribed initial interface *Γ*_{0} with *u*=*b*_{−} far from the interface. That is, the mass constraint is such that the adjusted background state *b*>*b*_{−}. Conversely, *B*_{1}<0 corresponds to a deficit in the initial mass of the minority phase: to construct the prescribed initial interface *Γ*_{0}, the value *b* of *u* far from the interface satisfies *b*<*b*_{−}. An analysis of the FCH shows that the excess surfactant configurations are strongly unstable, as growth of interface leads not only to a lower energy through the *η*_{2} term, but drives the background concentration towards equilibrium. On the other hand, the deficit situation, *B*_{1}<0, is a more stable balance, with growth of interface limited by the expense of further reductions in the far-field value, *b*, of *u*.

This dichotomy is consistent with the normal velocity given in the sharp interface reduction (1.8). For *B*_{1}>0, the mean-curvature flow is ill-posed as an evolution problem, leading to uncontrolled growth of interface, whereas for *B*_{1}<0, it is well-posed in time with possible finite-time singularities associated with ‘contraction’ of isolated vesicles as their surfactant phase is absorbed into the background state and possibly redistributed to other structures. For *B*_{1}<0, the exponential decay of *B*_{1} to zero quenches this fast transient. For *B*_{1}>0, the quenching is even faster, however, the ill-posedness of the curvature flow removes us from the framework of the formal analysis presented herein. This dichotomy has no analogy within the single-layer models, but is compatible with experimental observations, particularly [5] in which the addition of a small background component of oleate (lipid) into a suspension of phosphate containing vesicles lead to the spontaneous destabilization of the vesicles to an uncontrolled filamentation process. Indeed, the authors of that work propose the destabilization as a mechanism for cell division in primitive vesicles.

In the deficit situation, *B*_{1}<0 approaches zero, initiating the slower *O*(*ε*^{−2}) time scale. Here, the chemical potential again uncouples from the interface *Γ*, which evolves under the *interface-preserving* normal velocity
1.10where *Π*_{Γ}, introduced in (6.31) is a zero-average, curvature-weighted projection and *σ*_{2}>0 is a surface tension depending only on *W*. The interface-preservation property is a direct consequence of the incompressibility of the bilayer structure and the mass constraint that fix the total interfacial width. This is distinct from sharp interface reductions in single-layer models. Indeed, the normal velocity generated by an *H*^{−1} gradient flow of the CH energy (1.1) couples to a Mullins–Sekerka problem for the chemical potential [22], whereas the *L*^{2} gradient flow of an FCH type energy yields a Willmore type flow, but for which the quantity of interfacial material, the minority phase of the FCH, is not conserved [12]. In particular, the interface-preserving Willmore flow generated by bilayer structures yields markedly different evolution in space dimension *n*=2, *n*=3 and for *n*≥4, see (6.35).

## 2. The whiskered coordinates and inner-expansions

The FCH equation possesses many regimes, and a global analysis of the PDE is not reasonable. We assume an initial state that starts within a neighbourhood of a bilayer interface. More specifically, we assume that we have a smooth, co-dimension one initial interface , which divides *Ω*=*Ω*_{+}∩*Ω*_{−} into an interior *Ω*_{+} and an exterior *Ω*_{−}, given parametrically, at least locally, by
2.1where is smooth. We will describe the geometric evolution of the bilayer interfaces as a flow in time *t* on this space, yielding a curve *Γ*(*t*) parametrized by *ϕ*(*s*,*t*) over a set *Q*(*t*). The bilayer solutions we consider are expressed in the normal-tangential coordinate system in a neighbourhood of *Γ*(*t*).

For simplicity, we choose the parametrization so that *s*_{i} corresponds to arc length along the *ith* coordinate curve, and the coordinate curves are lines of curvature. In this setting, the vectors defined by
2.2form an orthonormal basis for the tangent space to *Γ* at *ϕ*(*s*,*t*). Denoting the outer normal vector of *Γ* pointing towards *Ω*_{−} by **n**(*s*,*t*)=(*N*_{1},…,*N*_{n}), we have the relations
2.3where *k*_{i}=*k*_{i}(*s*) are principal curvatures of *Γ* at *ϕ*(*t*,*s*).

From the implicit function theorem, for each *x*_{0} on *Γ*, there exists a neighbourhood of *Γ*, with the property that the map *x*↦(*s*,*r*) defined by
2.4is locally and smoothly invertible for each fixed time *t*. In particular, in this neighbourhood, we have the functions *s*=*s*(*x*) and *r*=*r*(*x*) that relate the local coordinates to the Cartesian ones. We introduce the scaled coordinate *z*=*r*/*ε* and the ‘whiskers’
2.5which correspond to line segments of length ℓ, in unscaled distance, emanating from *Γ* in the normal direction. From the implicit function theorem, for ℓ sufficiently small, these whiskers do no self-intersect locally in *s*. We say that an interface *Γ* is far from self intersection, if there exists ℓ>0 such that none of the whiskers of length ℓ intersect each other or ∂*Ω*. We introduce the neighbourhood
2.6of *Γ*, which comprises all points *x*∈*Ω* that are within a distance ℓ of *Γ*.

The proof of lemma (2.1), which summarizes the properties of the coordinate system, can be found in earlier studies [3,22–24].

### Lemma 2.1

*Let Γ=Γ(t) be a smooth interface of the form (2.1) with curvatures* *uniformly O(1). The normal velocity V* _{n} *of Γ at ϕ(s,t) is given by −∂*_{t}*r(s,t), and the tangential coordinates (r,s) enjoy the properties
**and
**In particular, in the scaled (z,s) coordinates, we may expand the Cartesian–Laplacian in terms of the Laplace–Beltrami operator Δ*_{s} *and the curvatures
*2.7*where κ*_{j} *is related to sums of jth powers of the curvatures
*2.8*The Jacobian* *J**of the transformation x↦(z,s) takes the form
*2.9*and* *satisfies
*2.10

### Definition 2.2

For a function that tends to a constant values at *O*(1) exponential rates as we say that we *dress* the interface *Γ* with *ψ*, obtaining the *Γ*-extended function
2.11where ℓ>0 is the minimal (unscaled) distance of *Γ* to the compliment of its neighbourhood and is a fixed, smooth cut-off function which is one on while *η*(*s*)=0 for *s*≥2.

### Definition 2.3

Let *Γ* be far from self-intersection. We say that *f*∈*L*^{1}(*Ω*) is localized on *Γ* if there exist constants *M*,*ν*>0, independent of *ε*>0, such that
2.12for all *x*∈*Γ*_{ℓ}.

### Lemma 2.4

*If Γ is far from self-intersection and f is localized on Γ, then we have the following integral formula
*2.13*where J is the Jacobian associated with the immersion* *.*

We assume that the initial data *u*_{0} of (1.7) is close to a bilayer interface, that is, for some interface *Γ*, *u*_{0} is close to the *Γ*-extension *U*_{Γ} of the homoclinic solution *U* of (1.5) with Lagrange multiplier *λ*=0. We track the evolution of the interface *Γ*=*Γ*(*t*) and which induces *U*(*t*)=*U*_{Γ(t)}. In our formal analysis, we do not construct a map from *U* back to *Γ*. We record the existence of the homoclinic solution and the properties of the associated linearization in lemma 2.5.

### Lemma 2.5

*Let U be the solution of (1.5) which is homoclinic to b*_{−}*, corresponding to λ=0, and even in z, then U attains it maximum value U*_{M} *at z=0, where U*_{M} *is the unique zero of W in (b*_{−}*,b*_{+}*). Moreover, there exists ν>0 such that the linearization,
*2.14*operating on H*^{2}(**R**) *has spectrum that satisfies
*2.15*where λ*_{0}*<0 is the ground-state eigenvalue. The corresponding eigenfunctions are ψ*_{0}*≥0 and ψ*_{1}*=U*_{z}*. Moreover, we record that
*2.16*and
*2.17*and introduce the functions* *for j=1,2 which are the solutions of
*2.18*which are orthogonal to the kernel of* *.*

### Proof.

The existence of the homoclinic solution is immediate from phase-plane considerations. Indeed, writing (1.5) as a second-order dynamical system in (*U*,*U*_{z}), there are saddle points at (*b*^{±},0). Because *W*(*b*_{−})=0>*W*(*b*_{+}), one deduces that the unstable manifold of (*b*_{−},0) crosses the horizontal axis at a point that we label (*U*_{M},0). By reversibility, this orbit returns to (*b*_{−},0) along the stable manifold, forming the homoclinic orbit. The linearization is a Sturm–Liouville operator which has simple, real eigenvalues which are enumerable by their number of nodal points. Taking ∂_{z} of (1.5), we see that , and because *U*_{z} has one nodal point, it is the first eigenfunction, *ψ*_{1}, above the ground state *ψ*_{0} whose eigenvalue *λ*_{0}<0. The remainder of the spectrum is real and an *O*(1) distance to the right of 0. The relation (2.16) follows from a direct calculation, whereas (2.17) follows from taking of (1.5). The functions *Φ*_{1} takes the form , where is the *L*^{2}(**R**) solution to
2.19Because at an exponential rate as the right-hand side is in *L*^{2}, and even about *z*=0, hence orthogonal to the kernel of . The existence of *Φ*_{2} follows from a similar argument. □

We perform a multi-scale analysis of the solution *u* and chemical potential *μ*. At a time scale *τ*, we have the inner spatial expansions
2.20and
2.21Using (2.7), we convert the Cartesian–Laplacian of *u* into local coordinates. Collecting orders of *ε*, we find
2.22From (1.7), we write the chemical potential where the prefactor *P*:=−*ε*^{2}*Δ*+*W*^{′′}(*u*)−*ε*^{2}*η*_{2} acts on the CH residual Expanding these, in turn, we find
2.23and
2.24Combining these expansions, we group the orders of the chemical potential
2.25
2.26
and
2.27
2.28Moreover, we may expand the Cartesian–Laplacian of the chemical potential as
2.29

## 3. Fast times: relaxation to the *U*_{Γ} bi-layer

We first consider the fast-time evolution, which leads to a relaxation towards the bilayer profile for initial conditions that are sufficiently close.

### (a) Time scale *T*_{2}=*t*/*ε*^{2}: outer expansion

Away from the interface, the outer expansion for the density and chemical potential takes the form
3.1and
3.2where each *u*_{i} and *μ*_{i} depends upon *x* and *T*_{2}=*t*/*ε*^{2}. The first two orders of the chemical potential, defined in (1.7), have the form
3.3and
3.4For the outer expansion, the time derivative ∂_{t}=*ε*^{−2}∂_{T2} yields the expression
3.5whereas the chemical potential takes the simple form
3.6Applying these to the FCH equation, (1.7), and matching orders of *ε*, we find
3.7In the *T*_{2} time scale, the outer solution *u* does not evolve to *O*(*ε*^{2}).

### (b) *T*_{2}=*t*/*ε*^{2}: the inner expansion

We recall the inner expansions (2.20)–(2.21) for the concentration *u* and chemical potential *μ*, with *τ*=*T*_{2}. We also expand the normal distance *r*=*r*_{0}+*εr*_{1}+*O*(*ε*^{2}), obtaining
3.8Using (3.8) to expand the left-hand side of the FCH equation (1.7), and (2.29) on the right-hand side, we find at *O*(*ε*^{−3})
3.9which implies that ∂_{T2}*r*_{0}=0 because is not constant in *z*. The interface *Γ*(*t*) does not move to leading order on the *T*_{2} time scale. At *O*(*ε*^{−2}), equation (1.7) reduces to
3.10Recalling the form of from (2.25), and our assumption that the initial data is at leading order of the form *u*_{0}=*U*_{Γ}(*x*), where *U* is the homoclinic solution of (1.5) with *λ*=0, which corresponds to the boundary conditions
3.11it follows that and *u*_{0}=*U*_{Γ} is an equilibrium solution of (3.10). A similar analysis applies to the time scale *T*_{1}=*t*/*ε*, with the conclusion that *U*_{Γ} is an equilibrium solution of (1.7) on this time scale. For brevity, we omit the details.

## 4. The time scale *t*: a gradient flow

Applying the outer expansions (3.1) and (3.2) to (1.7) on the *t* time scale and collection *O*(1) terms, we obtain a nonlinear diffusion equation for *u*_{0},
4.1Performing the inner expansions (2.20) and (2.21) for *τ*=*t*, we have the leading-order inner expressions
4.2Matching (4.2) and (2.29), the *ε*^{−2} and *ε*^{−1} terms give
4.3and
4.4From (4.3), it is easy to verify that is an equilibrium solution to ; however, to derive, the sharp-interface is stable within a larger framework. To this end, consider a more general form for ,
4.5where *δ* is sufficiently small, but independent of *ε*. A regular perturbation expansion of (2.25) about *U*_{z} shows that (4.5) leads to an inner solution
4.6where is homoclinic to *b*_{−}+*δB*_{0}(*s*,*t*)*α*^{−2}_{−} along the whisker *w*(*s*), and *Φ*_{2} is defined in (2.18). Within this framework, (4.4) simplifies to
4.7Integrating (4.7) in *z* from to , and remarking that ∂*r*/∂*t* is independent of *z* while is homoclinic in *z*, yields the key equalities,
4.8

### (a) Inner–outer matching

The derivation of the interface evolution at this and subsequent time scales requires a matching of the inner and the outer expansions. We follow the procedure of Pego [22]. Fixing *x*∈*Γ*, we require
4.9when *εz* is between *O*(*ε*) and *o*(1). Expanding the left-hand side around *x* as , we have
4.10where ∂_{n} is the directional derivative along **n**, and
4.11We can obtain a similar expansion as . The matching condition (4.9) gives
4.12
4.13
4.14
and
4.15A similar expansion yields the matching conditions for *u*,
4.16
4.17
and
4.18

### (b) Sharp interface reduction: gradient flow

The matching condition (4.13) shows that which, in conjunction with the second equality of (4.8), implies that the normal derivative of the outer chemical potential is continuous across *Γ*. Similarly, the matching condition (4.16), in conjunction with the first equality of (4.8), implies the continuity of *u*_{0}. That is across the interface *Γ* we have the jump conditions
4.19This implies that *u*_{0} is a strong solution of (4.1) over the entire domain *Ω*. The evolution of the interface *Γ* *uncouples* from the evolution of *u*_{0}. Indeed, the interface motion can be calculated after the fact. Moreover, the resulting equation for *u*_{0}
4.20subject to periodic boundary conditions, is a mass-preserving *H*^{−1} gradient flow on the reduced energy
4.21where the effective potential is a triple well with equal zeros at *b*_{−}<*b*^{0}<*b*_{+}. In particular, if the outer values of the initial data are sufficiently close to *b*_{−}, then *u*_{0} will converge to a constant value that is close to *b*_{−}.

For completeness of presentation, we assume the leading-order initial value *u*_{0}(*t*=0) equals the spatial constant *b*_{−}, which is an equilibrium of (4.20), and verify that the normal velocity of the interface is indeed zero. We return to (4.7) with , where , enjoys the property as . We integrate from 0 to *z* twice, because ∂*r*/∂*t* is independent of *z*, we obtain
4.22We also integrate (4.7) in *z* from to 0, which yields
4.23To find , we simplify (2.26). Observing that so that *U*_{zz}−*W*^{′}(*U*)=0, , and we obtain
4.24From (2.15), this equation has solution only if . Recalling the form, (4.22), of the chemical potential , the solvability condition reduces to
4.25and integrating by parts on , we obtain
4.26Combining this expression with (4.23), we obtain the normal velocity
4.27where we have introduced the constants
4.28Because on *Ω*, it follows that *μ*_{0}≡0 on *Ω* and *V* _{n}=0 at this time scale.

## 5. The time scale *t*_{1}=*εt*: a quenched mean curvature flow

On the time *t*_{1}=*εt*, we obtain the first non-trivial dynamics of the interface *Γ*.

### (a) Outer expansion

For *t*_{1}=*εt*, the ∂_{t} derivative expands as
5.1Inserting (5.1) and (3.6) in the FCH equation, (1.7), and matching terms at *O*(1) and *O*(*ε*), we have
5.2and
5.3The first equation, (5.2), is consistent with our assumption that the outer-solution *u*_{0}=*U*_{Γ}=*b*_{−}. The outer equation (5.3) reduces to
5.4where the *O*(*ε*) outer chemical potential is related to the *O*(*ε*) outer concentration via the expression
5.5

### (b) Inner expansion

Recalling the inner expansion (2.20) and (2.21) at *τ*=*εt*, we expand
5.6We insert (5.6) and (2.29) into the FCH equation and match terms at the *ε*^{−2}, *ε*^{−1} and *ε*^{0} orders. The first equation is consistent with our choice , which in fact implies . With this reduction, the remaining two equations become
5.7and
5.8The matching condition (4.13) implies that as which, together with (5.7) implies the existence of independent of *z* such that However, because the expression (2.26) for simplifies to
5.9Recalling the function *Φ*_{2} introduced in (2.18), we find the solution
5.10where we assumed on each whisker *w*(*s*) because adding a term proportional to *U*_{z} merely serves to shift the front location. The equation (5.8) simplifies to
5.11As in §4, we conclude from the matching conditions that while integrating (5.11) yields
5.12and integrating (5.11) in *z* from to 0, we obtain
5.13To calculate , we return to (2.28), and recall that while from (5.10), we deduce that . With these simplifications, we have
5.14We may solve for if and only if we can invert , which requires
5.15From parity considerations, the non-zero terms in the integral are
However, using (2.17), we rewrite the last term above as
5.16so that it combines with the middle term. Moreover, using (2.16) we find,
5.17Substituting for from (5.12) and (5.13), we reduce the solvability condition to
5.18Integrating by parts and solving for the normal velocity, we obtain
5.19where we recall the constants *m* and *S*_{1} from (4.28).

### (c) Sharp interface limit: quenched curvature driven flow

We summarize the *t*_{1}=*εt* evolution in the following model,
5.20
5.21
and
5.22Indeed, (5.20) and (5.21) imply that *Δμ*_{1}=0 in *Ω*, and with periodic boundary conditions on ∂*Ω*, it follows from the maximum principle that *μ*_{1} is a spatial constant *μ*_{1}(*x*,*t*)=*B*_{1}(*t*_{1}) for all *x*∈*Ω*. Consequently, ∂_{n}*μ*_{1}=0 on *Γ*, and by continuity of the inner and outer chemical potentials, we have and the motion of the interface on the *t*_{1} time scale reduces to
5.23

The value of *B*_{1} is related to *u*_{1} through (5.5), and is determined in part via the conservation of the total mass of the minority phase,
5.24which is fixed by the initial data. This value couples to the flow, (5.23); if the evolution causes the interface *Γ* to lengthen, then either its width must compress or minority phase must be drawn from the outer region, lowering the value of *u*_{1}.

In the outer region *Ω*\*Γ*_{ℓ}, we have the expansion
5.25whereas in the inner region, *Γ*_{ℓ} from (5.10), the inner expansion takes the form
5.26We insert these expansions into the right-hand side of (5.24)
5.27Assuming that |*Γ*|=*O*(1), changing to whiskered coordinates in the localized integrals, and using the Jacobian expansion (2.10), we find at leading order
5.28We expand *M*=*M*_{1}*ε*+*M*_{2}*ε*^{2}+*O*(*ε*^{2}) and the surface area |*Γ*|=*γ*_{0}+*εγ*_{1}+*O*(*ε*^{2}), evaluate the integrals, and recall the definition of *m* from (4.28), to obtain
5.29This permits us to solve for *B*_{1} in terms of the length of the interface,
5.30where the negativity of *B*_{1} is a condition imposed on the initial data, see the discussion following (1.8). On the other hand, when subject to a normal velocity *V* , measured in time units *t*_{1}, the interfacial surface area grows at the rate
5.31so that, the interfaces, *Γ*(*t*) subject to (5.23) have the leading-order growth
5.32Taking ∂_{t1} of (5.30), we arrive at the ordinary differential equation
5.33In particular, *B*_{1}, and hence *μ*_{1} and *u*_{1} decay exponentially to zero on the *t*_{1} time scale. Naturally, the total mass *M*_{1}, given in (5.29), is conserved under the flow, so the equilibrium interfacial length satisfies While it is possible that portions of the interface *Γ* become singular in finite time, that is *Γ* may fail to be far from self-intersection, generically one expects the interface to move an *O*(1) amount in an *O*(*ε*^{−1}) time, as measured in the unscaled *x* and *t*, before approaching its *t*_{1} equilibrium.

For the particular case in which *Γ* is composed of *N* disjoint, hollow balls (i.e. spherical vesicles) of radii *R*_{i}=*R*_{i}(*t*_{1}), in *Ω*⊂**R**^{n}, the quenched-curvature-driven flow reduces to
5.34and
5.35where *α*_{n} is the surface area of the unit ball in **R**^{n}. The conserved quantity takes the form
5.36If *B*_{1}(0)>0, then each ball grows in radius to a finite limit. If *B*_{1}(0)<0, then it is possible individual balls may collapse to an *O*(*ε*) radius before *B*_{1} tends to zero, in which case the evolution (5.34)–(5.35) breaks down.

## 6. The time scale *t*_{2}=*ε*^{2}*t*: surface-area-preserving Willmore flow

For the *t*_{2}=*ε*^{2}*t* time scale, the outer solution has equilibrated to *u*_{0}=*b*_{−} and *u*_{1}=0. As a consequence, the outer expansion reduces to
6.1and
6.2Matching terms in (1.7) in the outer region yields
6.3

The inner expansion reduces to
6.4and
6.5so that the left-hand side of (1.7) take the form
6.6Inserting (6.6) and (2.29) in (1.7), the *ε*^{−2} and *ε*^{−1} terms give
6.7and
6.8Equation (6.7) and matching condition (4.14) imply the existence of *B*_{2}(*s*,*t*), independent of *z*, such that . In the light of (6.4), we expand (2.28), to find
6.9Because both *U*^{′′} and *B*_{2} are orthogonal to the kernel of we can solve for ,
6.10where *Φ*_{2} was introduced in (2.18), whereas *Ψ*_{j}⊥*U*_{z} satisfies .

Turning to equation (6.8), it simplifies to
6.11As in §4, we can conclude that , so that the outer chemical potential *μ*_{2} will solve (6.3) in all of *Ω*, in particular it is constant.

To derive the motion of the interface, we return to (6.11) and integrate twice from 0 to *z*, which yields,
6.12while integrating from *z* from to 0, are recalling that is constant yields
6.13To evaluate we need the *O*(*ε*^{3}) terms in the inner expansion of from (2.23) and (2.24). In general, this is cumbersome, but because and we simplify the expression for *P* and and derive the inner chemical potential
6.14Using the relation (6.10) for , we re-write this in the form
6.15where we have introduced the *t*_{2} residual
6.16The solvability condition requires that *R*_{2}⊥_{L2(R)}*U*^{′}. We examine the terms one by one. First, from (6.12) and (6.13), we have
6.17For the next two terms, because is self-adjoint, using (2.17) and (2.16), we find
6.18and
6.19where *m* is defined in (4.28) and we have introduced
6.20The fourth term yields no contribution,
6.21Inserting these results into the solvability condition, and isolating the normal velocity, we find
6.22

### (a) The mass constraint

As was the case for the *t*_{1} time scale, the value *B*_{2}=*B*_{2}(*t*_{2}) of the outer chemical potential, *μ*_{2}, is determined by the conservation of total mass of the minority phase, whose integral we break over the inner and outer regions
6.23In the outer region, the solution takes the form (3.1) with *u*_{0}=*b*_{−}, *u*_{1}=0 and *B*_{2}=*μ*_{2}=[*W*^{′′}(*u*_{0})]^{2}*u*_{2}=*α*^{2}_{−}*u*_{2}. In particular, *u*_{2}=*B*_{2}*α*^{−2}_{−} is a spatial constant. Because |*Γ*|=*O*(1), the outer integral reduces to
6.24In the inner domain *u* has an expansion (2.20) with , Using (2.13) to change to the whiskered coordinates and expanding the Jacobian from (2.10), the inner integral reduces to
6.25Combining the outer, (6.24), and inner, (6.25), integrals, we find the balance
6.26Scaling the minority mass as *M*=*εM*_{1} and expanding the interfacial surface area as |*Γ*|=*γ*_{0}+*εγ*_{1}+⋯ , we find that at leading order all the minority phase is located on the interface, whose surface area is fixed over the *t*_{2} evolution
6.27At second order, we obtain an expression for the outer chemical potential
6.28Recalling the relation (5.31), we have
6.29In particular, because the interface surface area is conserved to leading order, this forces the value of *B*_{2},
6.30We may express the normal velocity more succinctly by introducing the zero-average, mean-curvature-weighted projection associated with the interface *Γ*,
6.31which maps *L*^{1}(*Γ*) into itself with the property that the normal velocity *Π*_{Γ}*V* preserves surface area. The leading-order evolution of the interface *Γ*, at the *t*_{2} time scale can be written as
6.32where the constant *η*_{2} drops out because *Π*_{Γ}[*η*_{2}*κ*_{0}]=0.

### (b) Evolution of disjoint spherical vesicles

As in §5, consider an initial configuration of *N* well-separated vesicles of radii *R*_{i}=*R*_{i}(*t*_{2}) contained in a periodic domain *Ω*⊂**R**^{n}. In **R**^{n}, on the *i*th sphere each curvature , for *j*=1,⋯ ,*n*−1, which yields the relations and . The normal velocity, (6.32), can be written as
6.33where *R*(*s*):=*R*_{i} for *s* on the *i*th sphere. The integrals in the projection take the value
6.34where *R*_{c}=*R*_{c}(*t*_{2}) defines a critical radius that varies in time. So the evolution reduces to
6.35In space dimension *n*=2, the coefficient is positive and one can verify that balls with radii bigger than *R*_{c} shrink, whereas those with radii smaller than *R*_{c} grow. In space dimension *n*=3, the coefficient is zero, and the radii do not change on the *t*_{2} time scale. For *n*≥4, balls with radii larger than *R*_{c} grow, whereas those with radii smaller than *R*_{c} shrink and disappear in finite time, leading to a coarsening phenomenon and eventually a winner-take all scenario, assuming the stability of the underlying structures.

## 7. Conclusion

The FCH equation supports different classes of interfaces than the CH equation, and these interfaces manifest significantly different dynamics, in particular, collections of closed bilayers evolve on long timescales according to a surface area-preserving Willmore flow. More specifically, the long-range interaction between single-layer interfaces in the CH equation is mediated through a Mullins–Sekerka problem. For the FCH gradient flow, the long-range interaction is mediated through the spatially constant outer chemical potential—its value below equilibrium dictates the growth or decay of each bilayer.

There are many possible areas to investigate in both model development and model analysis. It is quite intriguing to consider extensions beyond binary mixtures; indeed, a preliminary discussion in this direction can be found in Gavish *et al*. [4]. For example, bilayers need not only form a barrier between the same phase, but can also separate two distinct phases. This requires a three-phase model, one for the surfactant that forms the separating membrane, and two for the distinct separated phases. The bilayer structure is still required to incorporate the competition for a scarce membrane/surfactant phase; moreover, such a ‘third-phase’ membrane can rupture, resulting in mixing of the two bulk phases, as well as support merging between distinct membrane structures.

The most challenging issues are the competition between the wide variety of possible stable morphologies supported by the FCH gradient flows. In on-going work [21], the authors have extended the analysis presented here to incorporate a co-dimension two, or closed cylindrical pore structures. As in the situation investigated in this work, the competition between coexisting bilayers and pores for scarce surfactant phase is mediated through the constant value of the outer chemical potential, this competition is quite evocative of the ‘phospholipid war’ proposed in Budin & Szostak [5] as a fundamental evolutionary force in early cell membranes. However, this analysis avoids the central issue: the role of tremendous number of locally stable ‘defect structures’ supported by the FCH free energy. These are the end-caps of open pores and the rims of open bilayers—called bicelles in the biological literature [20]. Indeed, it is tempting to conjecture that the FCH model will provide a more accurate estimation of the free energy of highly curved interfaces than the usual CH free energy. This presumption arises from the fact that for co-dimension one interfaces the squared variational derivative term in the FCH energy naturally yields a contribution proportional to , where *H* is the mean curvature of the co-dimension one interface *Γ*. However, for a strongly curved interface, the coordinate system presented in lemma 2.1 becomes singular when *H*∼*ε*^{−1} at that point the bilayer structure is likely to degenerate, quite probably resulting in the generation of a defect structure. The mathematical characterization of these defect structures, their role in merging and bifurcations and their impact on competitive geometric evolution and *Γ*-limits awaits those with the requisite curiosity.

## Acknowledgements

K.P. acknowledges support from the National Science Foundation (under grants nos DMS-0708804, DMS-093468 and DMS-1125231). This work was completed while S.D. was funded under grant no. DMS-093468.

- Received August 27, 2012.
- Accepted February 8, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.