Compacton formation under Allen–Cahn dynamics
We study the solutions of a generalized Allen–Cahn equation deduced from a Landau energy functional, endowed with a non–constant higher order stiffness. We analytically solve the stationary problem and deduce the existence of so–called compactons, namely, connections on a finite interval between the two phases. The dynamics problem is numerically solved and compacton formation is described.
pacs:64.60.Bd, 68.03.g, 64.75.g
Phase–field models describe physical systems that can exhibit different homogeneous phases. The state of the system on the volume is coded into a so called phase–field depending on the space and time variables and , respectively. Two values of the phase–field, say and , represent the two homogeneous phases.
These models play a crucial role, for instance, in the study of phase reordering (Bray, ; Langer, ; Eyre, ): a system is quenched from the homogeneous high temperature phase into a broken–symmetry one (a ferromagnet or a gas abruptly cooled below their critical temperature) and the evolution of the phase–field describes the process of separation of the two phases.
associated with the Landau energy functional
where associates an energy with the phase–field and the squared–gradient term of the phase–field variations is weighted by the energy cost , called higher–order stiffness. According to the usual physical interpretation the energy has to be chosen as a double well function with the two minima corresponding to the two phases and and .
If the higher–order stiffness is a constant positive number and no constraint to the total value of the field is imposed, it is possible to compute the gradient of the Landau functional in the Hilbert space to get the standard Allen–Cahn equation
with normal derivative of the phase–field on the boundary equal to zero. Analogously the Allen–Cahn equation endowed with Dirichlet or mixed boundary conditions could be derived specifying a–priori the proper essential boundary condition in the definition of the Hilbert space in which the gradient of the Landau functional should be computed.
The standard Allen–Cahn equation, also called the time–dependent Ginzburg–Landau equation, was introduced in (Allen, ) to describe the motion of anti–phase boundaries in crystalline solids. In this context represents the concentration of one of the two components of the alloy and is proportional to the squared interface width.
In this paper we consider the case in which the higher–order stiffness is not constant, but it is a sufficiently regular positive function of the field, namely, such that for any . This situation has been considered, for instance, in BSBS () where the authors studied a similar model to describe glass–like relaxation in binary fluid models. A closely connected problem, in which a not constant higher–order stiffness is used, is the study of the gas–liquid interface in capillary tubes FJ (). In this case, the gradient equation (1) provides the generalized Allen–Cahn equation, see Appendix A,
again with suitable conditions on the boundary .
We focus on the one–dimensional case and study the solutions of the Allen–Cahn equation (4) in the case in which the higher–order stiffness coefficient vanishes at the phases, namely, . As we shall discuss in the following section, in such a “pathological” case there exist stationary solutions connecting the two phases on a finite interval of length . It is well known that this is not possible in the standard constant higher–order stiffness case, in which connections can only be considered on infinite domain AF ().
These solutions appeared in the scientific literature in different contexts, see, e.g., BSBS (); FJ (); Witelski (), and have been called compactons, in order to underline the property of being localized within a domain of finite measure. Our main goal, here, is to study the behavior of the solutions of the evolution equation (4) and, in particular, to describe the process leading to the formation of a compacton on the finite interval . We shall discuss both the interface Dirichlet boundary conditions
and the homogeneous Neumann boundary conditions
We shall respectively refer to these two cases as to the (D) and (N)–boundary conditions.
Let us summarize our main result. Compactons can be used to construct stationary solutions of the Allen–Cahn equation performing many excursions between the two phases, whose total number is bounded by . In the standard Allen–Cahn stationary problem, i.e., when the higher–order stiffness coefficient is constant, stationary profiles oscillating between the two phases are not allowed when (D)–boundary conditions are imposed. On the other hand, it is possible to construct profiles oscillating between two values of the phase–field “close” to the two pure phase values in the (N)–boundary condition case. These solutions, in the conservative mechanics equivalent model language in which the stationary Allen–Cahn model can be immediately recasted, correspond to the periodic motions of the system with total (kinetic plus potential) energy slightly smaller than zero.
In the (N)–boundary condition case single interface and periodic profiles are proven to be unstable BBB (); FH (). See also Gurtin (), where it is shown that, in presence of a global constraint, the periodic profiles are not even local minimizers of the Landau functional (2) defining the model, or in other words the corresponding second variation of the Landau functional is strictly negative at some perturbation of them.
We then expect that any time dependent solution of the Allen–Cahn evolution equation, for any choice of the initial profile, will never tend in the long time limit to one of these oscillating stationary solutions. In other words, the standard Allen–Cahn evolution cannot create an alternating profile and, indeed, such an equation is used to model domain coarse–graining in phase separation.
The question we pose in this paper is the following: in presence of compactons, can the Allen–Cahn evolution describe the alternating profile formation? In this paper, by means of a numerical computation in the framework of a specific model, we shall give a positive answer to such a question. In particular we shall show that the alternating compacton profile formation is possible with both (D) and (N)–boundary conditions.
In our study we shall use the following techniques: the stationary solution of the Allen–Cahn equation (4) will be studied analytically and the “usual” qualitative Weierstrass study will allow the construction of the phase portrait which will provide a thorough description of the structure of the stationary profiles. On the other hand, the time–dependent solutions will be studied numerically and a code based on the finite element method will be adopted.
In order to perform the numerical study a particular choice of the functions and will be done. We borrow those functions from FJ () where a model describing the gas–liquid interface in a capillary tube has been proposed. It is worth noting that we shall not discuss the evolution equations proposed in FJ (), but the Allen–Cahn equation with stationary profiles coinciding with the ones in the FJ () model. Indeed, our main interest is that of understanding the Allen–Cahn evolution in presence of compactons and to this aim we have chosen, as a prototype model, the one in FJ () whose stationary solutions has a clear physical interpretation. Moreover, this model allows to study analytically the compactons, whose behavior can be expressed in terms of special functions. This will provide us with an effective analytical control of our numerical results.
One of the main results in FJ () is the possibility to describe the existence of local, non–spreading, and compactly supported bubbles in a capillary tube. In that paper the model was studied numerically. Here we solve analytically the equation giving the stationary states of the system and explain some of the features of the compacton solutions presented in FJ ().
The paper is organized as follows: in Section II we discuss under quite general hypotheses the existence of compactons for the stationary Allen–Cahn equations. In Section III we consider the model introduced in FJ () to study the gas–liquid interface in capillary tube and, in such a context, we find explicitly (in terms of special function) the compacton solutions and discuss their main physical properties. In Section IV we study numerically the solutions of the Allen–Cahn equation with higher–order stiffness and energy as in FJ (). Section V is devoted to some brief conclusions.
We consider the Allen–Cahn problem (4) on the one–dimensional space . The equation for the stationary solutions then becomes
Here, and in the following, the prime will always denote the derivative with respect to the natural argument, whereas space (time) derivatives will be written explicitly as or with a subscript ( or with a subscript ).
We assume , with , and such that for , and . The choice of the potential energy with two isoenergetic minima models the existence of two coexisting phases. Moreover, we assume that tends to in and at least as a power law, namely, there exists such that
The two last assumptions are crucial for the compacton existence111We have chosen the Duffing potential energy for simplicity. The same discussion can be repeated for very general double well potential energies, but the condition on the higher–order stiffness coefficient have to be chosen accordingly., see (7) and the discussion which follows, as well as the related arguments in BSBS (); FJ ().
It is very important to remark that any regular solution of (17) is such that the conservation law
Note that the problem is similar, see also CI (); CIS2010 (); CIS2012 (); CIS2013 (), to that of an holonomic conservative mechanical system with Lagrangian coordinate , not constant mass matrix , and potential energy of the conservative force , once the space variable is interpreted as time. A lot of care has to be used when one wants to exploit this analogy, since the mass matrix is not positive defined, but it is equal to zero in the pure phases and .
The aim of this model is that of describing a compact interface (or connection) between the pure phases and , namely, we look for a solution of (5) equal to zero on a finite (say left) space interval, equal to one on a finite (say right) space interval, and continuously joining these two pure phases on an intermediate “finite” space interval. This intermediate interval will be the compact interface (or connection) between the two pure phases.
In standard cases, i.e., when the higher–order stiffness coefficient is constant, an interface with zero derivative at the boundary can only be achieved on an infinite space interval (heteroclinic problem). This property is very general and is connected to the uniqueness of the solution of a Cauchy problem which is ensured if the differential equation describing the interface is sufficiently regular. In the model we are studying here, this regularity of the equation is not satisfied due to the presence of the not positive definite mass matrix . This is the key peculiarity of the model that gives rise to the existence of compacton solutions.
First of all we note that the constant functions and trivially satisfy (5). So that we can imagine to construct a solution of this equation such that for all and for all , with given. The problem, now, is that of finding the interface joining the two pure phases on the “finite” interval . Note that the pure phases fix the value of the constant of motion (6) to zero; hence, the interface we are looking for has to satisfy
By separation of variables we get the implicit solution
Since we assumed to vanish in and at least as a power, we have that the integral above is convergent on the interval . Hence, we have proven the existence of the compacton and we can also conclude that
expresses its width.
We close this section by noting that, by means of the conservation law, it is possible to describe the structure of all the solutions of the stationary equation (5). Indeed, (6) ensures that any regular solution satisfies the equation
for some .
The structure of the solution of the equation (9) lying in the interval is as follows. For the constant, and , solutions and combination of them with compactons are found. Note that, since we assumed a power law behavior of for , we have that the space derivative of the profile can be zero, finite, or divergent in the phases and . For , since vanishes in and , the profile must have divergent derivative in and . For the solution is bounded to the region in which ; since in such a region , we find a classical oscillating solution. Finally, for , the unique solution is the constant .
These results are summarized in the figure 1 in which the three points represent the constant solutions , , and , the dotted lines represent the compactons, the lines diverging in and are the solutions for , and, finally, the closed loops are the solutions for . Note that in the figure we have depicted the compacton line finite at the phases, but, as we discussed above, it can happen that close to the phases the line tends to zero or diverges.
Recalling that denotes the length of the compactons solution, note that the length
of profiles connecting the two phases and and corresponding to is such that
On the other hand, for we let be the two solutions of the equation lying in the open interval . Hence, the length of a single interface connecting to is given by
This analysis on the phase space trajectories allows us to state the following results about the existence of solutions of the stationary problem. The stationary equation (5) with (D)–boundary conditions has a unique solution corresponding to a phase line with if , has the unique compacton solution if , and has infinite solutions if which can be constructed by gluing compactons and pure phase constant segments.
The stationary equation (5) with (N)–boundary conditions has always the two pure phase constant solutions and . If is large enough, so that for some one has , the problem can have single connection or oscillating solutions connecting two points and corresponding to the phase lines with . Moreover, if the problem has also solutions which can be constructed by gluing compactons and pure phase constant segments.
Iii Bubbles in a capillary tube
A one dimensional model is adopted for describing the spatial distribution, in a capillary tube, of the liquid and the gaseous phases regarding the mixture as a non–uniform fluid, which means, according to CH_I (), a system having a spatial variation of one of its intensive scalar properties. In particular following FJ () one can assume this property, say the phase–field introduced in section II, to be the density of the gas with respect to the volume locally available. In the specific case of a capillary tube with a constant section, the phase–field is the fraction of the cross–sectional area of the tube occupied by the gaseous phase , per unit length of the tube.
Apparently the gas saturation can be related to the volume density of the liquid phase keeping in mind the obvious constraint
According with the general formulation presented in section II, a Landau energy functional is introduced whose density per unit volume is the sum of a bulk contribution, prescribed in terms of a double well potential, , and an energy penalty for gradients of the gas saturation , affected by the current value of . In order to characterize the admissible equilibrium configurations of the system we refer from now on to the constitutive model given in FJ (). Assuming the equilibrium between the gaseous and the liquid phase to be controlled only by capillary forces and therefore by the adjustment of the contact angle between the gas–liquid and the liquid–solid interfaces, see deGennes (), the double well potential is prescribed following FJ () by
being the surface energy relative to the gas–liquid interface, and the radius of the capillary tube. Following FJ () the higher–order stiffness will be written in terms of
see (16), with and . In FJ () a dimension argument is given for the definition of , moreover, it is remarked that the peculiar expression of plays a key role in the existence of compact interfaces, see also BSBS ().
The derivative of the double well potential (11) specifies the difference between the chemical potential of the gas and the chemical potential of the liquid or, analogously, the negative chemical potential of the liquid, once that of the gas has been fixed to zero, as a reference value. Its value at the pure phases, , the gas, and , the liquid, is the same, say
so that, according with classical Maxwell’s rule, the non–uniform fluid exhibits coexistence of the two phases at equilibrium only when the chemical potential is uniformly equal to over the whole spatial domain. Requiring this condition to be verified corresponds to find out the solutions of the minimization problem
which admits two solutions at and . Due to the additional linear term, the two phases correspond, now, to two isopotential minima of the function .
The regularization provided by the energy penalty proportional to the squared–gradient term via the higher–order stiffness , see equations (12)–(13), implies, at coexistence, the conservation law (6) to be rewritten as follows:
which therefore reads as a specialization of the Allen–Cahn equation when a non–uniform fluid is placed into a capillary tube.
iii.1 The compact interface problem
From now on we shall simplify the notation by letting and rewrite equation (16) as
where we have set
In order for the physical dimensions of the quantities introduced above to be consistent with the notation of Section I a suitable viscosity parameter must be introduced so that and .
It is important to remark that the interface problem (17) in the capillarity setup is an example of applications of the theory developed in Section II. Thus, as discussed in Section II, in order to ensure that the integral (7) is convergent, it is sufficient to require that the parameters in (13) are strictly positive. In other words the particular dependence of and on the contact angle discussed below (13) is not necessary to prove the existence of compactons, but, as we shall see below, affects their width . Indeed, by (8) we get
for the compacton width. Moreover, by (38) we have
Finally, recalling (12), we get
iii.2 Compacton profile
As above it is possible to write an implicit expression of the compacton profile in terms of special functions. Indeed, by performing the same computation as above, from (7) we get
Equation (37) and some simple algebra yields
By using the explicit solution given above, many interesting physics feature of the compactons discussed in FJ () can be proven analytically. For instance, in that paper it has been noted that the convexity of the interface profile for depends on whether the liquid phase has a wetting (, for instance water) or a not wetting (, for instance mercury) behavior. By means of (22) this problem is reduced to a simple computation. Indeed, recall that the compacton satisfies (17) and along the compacton the constant of motion (6) is equal to zero; thus, from (17) and (6), we get that
for any . A simple computation yields
Thus, for any we have that
Since, and , , , and are strictly positive in the open interval , we have that the profile is convex for (not wetting liquid) and not convex for (wetting liquid).
iii.3 Phase portrait
The phase portrait of the model can be deduced by solving (24) with respect to . For , , , and we find the drawing depicted in figure 2. The disks in the pictures denotes the constant solution, the line tending to zero in zero represents the compacton, closed curves are associated to the cases , the remaining lines represent the profiles in the case .
In order to find the stationary profiles one has to integrate the equation (24). For the solutions of (24) are the constant profiles and , and the compacton. For , the problem of finding the stationary profiles (in an implicit form) is reduced to the computation of the definite integral
see the figure 3. For , denoted by the two solutions of the equation lying in the open interval , the problem of finding the stationary profiles (in an implicit form) is reduced to the computation of the definite integral
see the figure 4.
Iv Approaching compactons
Once defined the admissible stationary configurations, which solve (9), in particular those describing the spatial distribution of the liquid and the gaseous phases in a capillary tube, see (22), (25), and (26) and figures 3 and 4, it is interesting to discuss which of them can be attained through the dissipative evolution described by the Allen–Cahn equation (4), endowed with (D) or (N)–boundary conditions.
In the following the solutions of the Allen–Cahn equation with (D) and (N)–boundary conditions are separately discussed when considering , say the length of the interval smaller than the length of the compacton, and . In the first case no compacton stationary profile is admissible, conversely in the second one suitable profiles, constructed gluing compactons and pure phases, are admissible solutions of the problem. The time–dependent spatial profiles are numerically captured using a finite element code which has been implemented within MATHEMATICA. Time is made dimensionless with respect to the ratio .
Let , in this case the dynamics does not tend to the compacton simply because there is not sufficient space for the compacton to arise. Assuming (D)–boundary conditions, the stationary configuration is a regular profile, see figure 5, whilst for (N)–boundary conditions the dynamics tends to one of the two pure phases, depending on the initial data, see figure 6.
It is interesting to notice that intermediate profiles of , for (D)–boundary conditions, can be obtained gluing regular profiles, of length smaller than , similar to those of figure 3, and pure phase solutions (in particular ), where the measure of the subdomain corresponding to this last partial solution fades away with increasing time. On the other hand assuming (N)–boundary conditions the evolution passes through a progressive flattening of the profiles.
Consider now , for both (D) and (N)–boundary conditions two different situations are discussed corresponding to a length of the interval larger or much larger than . In the first case only one compacton can arise, whilst in the second one more than one compacton can form, depending on the initial conditions.
Assume a linear initial profile connecting the two phases and the length of the interval close to the length of the compacton, for instance ; the dynamics is definitely similar to that in figures 5 and 6, where the stationary profile is indeed formed by the compacton and the solution corresponding to the pure phase , see figure 7.
Consider now an interval whose length is , in this case, depending on the initial conditions, one or more compactons can form in the domain so that oscillating solutions can indeed correspond to stationary states of the Allen–Cahn dissipative dynamics. In figures 8 and 9 two distinct cases are exhibited which correspond to (D) and (N)–boundary conditions.
We have considered a generalized Allen–Cahn equation deduced from a Landau energy functional with a non–constant higher order stiffness vanishing at the two pure phases. We have solved analytically the stationary problem and deduced the existence of the so–called compactons. We have also showed the possibilities of piecewise stationary solutions made of the superposition of compactons and constant pure phase profiles.
In a case of particular physical interest the compacton problem has been solved explicitly and the main physical features of such profiles connecting a liquid and a gas phase in a capillarity tube have been deduced.
The dynamics has been studied numerically and the compacton formation has been described in detail. In this framework one of the most relevant result we discussed is the possibility that, due to the presence of compactons and by choosing properly the initial condition, the dissipative Allen–Cahn evolution can result in the formation of periodic profiles connecting the two pure phases. This stationary profiles pops up as the long time limit of the dynamical problem. It is important to stress that this possibility is ruled out in the standard Allen–Cahn dynamics.
Appendix A Derivation of the Allen–Cahn equation
For completeness we sketch the derivation of the Allen–Cahn equation (4) in the case in which the higher–order stiffness coefficient is not constant.
The gradient of the Landau functional (2) in the space is a function in such a space such that
for any . In other words, the derivative of the function in any direction is equal to the scalar product of such a function with the one characterizing the direction. By (2) it follows that
For sufficiently regular, we recall the Green identity
with the boundary of and the derivative in the direction orthogonal to the boundary. We then get
Moreover, recalling the properties of the divergence operator we get
Finally, from this equality, in the Lebesgue space of functions such that the normal derivative to the boundary of vanishes, we have that
which yields the Allen–Cahn equation (4).
Appendix B Integral computations
Recall the definition of the beta function and that of the incomplete beta function
with . It is immediate to prove that
In the following we shall also need some properties of the gamma function. Recall its definition
with , and the two properties
The beta function is related to the gamma function by the equality
Let be a real such that , it is immediate to remark that
Indeed, it is sufficient to let and and recall (31).
Acknowledgements.We wish to express our thanks to R. Benzi and P. Buttà for very useful discussions.
- (1) A.J. Bray, Advances in Physics 43, 357–459 (1994).
- (2) D.J. Eyre, SIAM Journal on Applied Mathematics 53, 1686–1712 (1993).
- (3) J.S. Langer, Solids far from Equilibrium, edited by C. Godrèche, Cambridge University Press, page 297, 1992.
- (4) N. Alikakos, P.W. Bates, G. Fusco, Journal of Differential Equations 90, 81–135 (1991).
- (5) P.C. Fife, “Pattern formation in gradient systems,” in “Handbook of Dynamical Systems” vol. 2, pages 677–722, 2002, ed. B. Fielder, Elsevier Science, Amsterdam, The Netherlands.
- (6) S.M. Allen and J.W. Cahn, Acta Metallurgica 27, 1085–1095 (1979).
- (7) R. Benzi, M. Sbragaglia, M. Bernaschi, and S. Succi, Physical Review Letters 106, 164501 (2011).
- (8) L. Cueto–Felgueroso, R. Juanes, Physical Review Letters 108, 144502 (2012).
- (9) N. Alikakos, G. Fusco, Indiana University Mathematics Journal 57, 1871–1906 (2008).
- (10) T.P. Witelski, Appl. Math. Lett. 11, 127–133 (1998).
- (11) L. Bertini, S. Brassesco, P. Buttà, Arch. Rational Mech. Anal. 190, 307–345 (2008).
- (12) G. Fusco, J. Hale, J. Dyn. Differ. Equ. 1, 75–94 (1989).
- (13) J. Carr, M.E. Gurtin, M. Slemrod, Archive for Rational Mechanics and Analysis 86, 317–351 (1984).
- (14) E.N.M. Cirillo, N. Ianiro, and G. Sciarra, Journal of the Mechanics and Physics of Solids 61, 1080–1114, (2013),
- (15) E.N.M. Cirillo, N. Ianiro, and G. Sciarra, Physical Review E 81, 061121–1–9 (2010).
- (16) E.N.M. Cirillo, N. Ianiro, and G. Sciarra, Physical Review E 86, 041111 (2012).
- (17) N. Chafee and E.F. Infante, Applicable Analysis 4, 17–37 (1974).
- (18) J.W. Cahn and J.E. Hilliard, The Journal of Chemical Physics 28, 258–267 (1958).
- (19) P. G. de Gennes, Reviews of Modern Physics 57, 827–863 (1895).