Droplet Formation in Binary and Ternary Stochastic Systems

Transcription

1 Droplet Formation in Binary and Ternary Stochastic Systems Thomas Wanner Department of Mathematical Sciences George Mason University Fairfax, Virginia 22030, USA Work supported by DOE and NSF. IMA Workshop on Interactions among Localized Patterns in Dissipative Systems June 4, 2013

2 Phase Separation in Alloys Binary systems with a miscibility gap: Quenching (rapid cooling) of homogeneous binary alloys may lead to phase separation on the atomic scale. A B: Shallow quenches lead to nucleation. A C: Deep quenches lead to spinodal decomposition. Phase Diagram [Novick-Cohen (1985)]

3 Nucleation Shallow quenches lead to nucleation and subsequent growth: Certain alloys appear to be in a stable state after quenching, maintaining the homogeneous mixture of the components. After a while, droplets appear at random locations in the alloy. The droplets grow in time and merge, giving rise to a larger scale phase separation.

4 Outline 1. Stochastic Binary Nucleation Nucleation Results due to Bates & Fife The Cahn-Hilliard-Cook Model Large Deviations and Domain Exit 2. Nucleation in Ternary Alloys The Cahn-Morral Model Nucleation Statistics Boundary Effects and Scaling 3. Equilibria and Nucleation Pathways Numerical Path-Following Interior Droplet Equilibrium Solutions Morse Decompositions and Nucleation Pathways 4. Degenerate Forcing and Droplet Size Forcing with Degenerate Noise Saddle-Node Two-Parameter Continuation Maximal Droplet Counts

5 The Cahn-Hilliard Model Deterministic model due to J.W. Cahn & J.E. Hilliard (1958) u t = (ε 2 u + f (u)) in Ω, u ν = u ν = 0 on Ω u = u(t, x) is an order parameter ε > 0 models interaction length Ω is a bounded domain f (u) = u u 3 is a cubic nonlinearity

6 Basic Properties Homogeneous equilibrium: The constant solution u(t, x) µ is unstable if f (µ) > 0 and stable if f (µ) < 0, leading to spinodal decomposition and nucleation, respectively. Mass conservation: Ω u(t, x)dx remains constant for t 0. Energy functional: Solutions decrease the energy E ε [u] = Ω ( ) ε 2 2 u 2 + F (u) dx F denotes the bulk free energy, which is given by a double-well potential such as F (u) = (u 2 1) 2 /

7 Structure of the Equilibrium Set For understanding the dynamics of nucleation in the Cahn-Hilliard equation, precise knowledge of its equilibria is crucial. They solve the nonlinear elliptic system ε 2 u + f (u) = c in Ω u = 0 on Ω ν 1 Ω u dx = µ Ω For convenience, we introduce the new parameters c = 1 Ω f (u) dx and λ = 1 Ω ε 2. Trivial constant solutions: u µ and c = f (µ).

8 Grinfeld, Novick-Cohen (1995): Equilibria in 1D Complete description of the set of equilibria in one dimension, depending on the total mass µ. Proof uses phase plane analysis combined with transversality arguments. Only limited information about the attractor dynamics. µ 2 < 1 5 spinodal decomposition

9 Equilibria in 1D 1 5 < µ2 < 1 3 spinodal decomposition 1 3 < µ2 < 1 nucleation [Figures from Grinfeld, Novick-Cohen (1995)]

10 Nucleation in the Deterministic Model in 1D Bates, Fife (1993): Nucleation in one space dimension Nucleation is triggered by a spatially localized perturbation called canonical nucleus. The canonical nuclei are saddle equilibria with index 1 on the boundary of the basin of attraction of ū 0 µ. One half of the unstable manifold converges to a global minimizer of the energy. (a) local minimizer µ (b) canonical nucleus (c) global minimizer

11 Results and Limitations: Deterministic Nucleation The equilibrium u µ is locally stable, but it is not the global energy minimizer, if the total mass µ lies in the metastable region: f (µ) < 0. Nucleation is initiated by a spatially localized perturbation of µ, called canonical nucleus or spike. Available results describe both the later stage dynamics and the canonical nuclei: Bates & Fife (1993), Wei & Winter ( ), Bates, Dancer & Shi (1999), Bates & Fusco (2000),... Limitations: No dynamical description of the initial nucleation stage. Deterministic model fails to explain gradual transition from spinodal decomposition to nucleation.

12 The Stochastic Cahn-Hilliard-Cook Model H. Cook (1970), J. Langer (1971): u t = (ε 2 u + f (u)) + σ ξ in Ω, u ν = u ν = 0 on Ω The noise process ξ = ξ(t, x) is defined on an abstract probability space (P, A, P) and it is the generalized derivative of a Q-Wiener process (W (t)) t 0 in a Hilbert space H given by the expansion W (t) = α k β k (t) ϕ k with Qϕ k = αk 2 ϕ k 2 H ϕ k k=1 The operator Q is the covariance operator of W, the sequence (β k ) k N consists of independent real-valued standard Brownian motions over (P, A, P).

13 Nucleation in the Cahn-Hilliard-Cook Model Blömker, Gawron, W. (2010): Using a large deviations approach, one can show that for small noise intensities σ, the dynamics of nucleation is connected to the deterministic attractor structure: Due to the additive noise in the equation, solutions will leave the domain of attraction of µ. The most likely exit points will be critical points of the energy, i.e., unstable saddle equilibria. The most likely exit paths will be heteroclinic connections traversed in the opposite direction. ] [ Energy E " ¹

14 Specific Noise Properties We consider the one-dimensional case Ω = (0, 1) with α k = kπ: W (t) = kπ β k (t) ϕ k with ϕ k (x) = 2 cos(kπx). k=1 The basic underlying scale of Hilbert spaces is given by H α = { u = u k ϕ k : u 2 α = k=1 } (kπ) 2α uk 2 < Then W is a Q 2 -Wiener processes in H 2 with trace-class covariance operator Q 2 = x 2, since αk 2 ϕ k 2 H = (kπ) 2. 2 In other words, the noise is white in time and colored in space, it is the spatial derivative of space-time white noise. k=1

15 Functional-Analytic Framework Follow Da Prato & Zabczyk to obtain a domain exit result. Employ dynamical systems methods by viewing the model as evolution equation in C = {u C[0, 1] : 1 0 udx = 0} via du dt = A εu + G(u) + σ ξ, u(0) = µ with A ε u = ε 2 4 x u and G(u) = 2 x f (u). The stochastic convolution is defined as W Aε (t) = t 0 e (t τ)aε dw (τ) It has a version which is mean-square continuous in H 2 and α-hölder continuous with respect to (t, x), for α [0, 1/8).

16 Functional-Analytic Framework Consider only a globally Lipschitz continuous nonlinearity f, with stable homogeneous state h 0 0 for σ = 0. Interpret the Cahn-Hilliard-Cook model as an integral equation in H 2. The mild solution is given by t u u 0 σ (t) = e taε u 0 + σ W Aε (t) + e (t τ)aε G(u u 0 σ (τ)) dτ 0 For initial conditions u 0 C = {u C[0, 1] : 1 0 udx = 0} there exists a unique mild solution in C([0, ), C). For σ = 0 and initial conditions u 0 H α, with α N, there exists a unique mild solution in C([0, ), H α ). These solutions coincide for appropriate u 0 and σ = 0.

17 Small Noise Dynamics on a Bounded Interval For small noise intensities σ, the solutions to the Cahn-Hilliard-Cook model tend to follow the deterministic solutions of the Cahn-Hilliard equation. One can show that P ( sup u u 0 σ (t) u u 0 0 (t) > δ 0 t T ) C δ,t e D δ,t /σ 2 This estimate uses that our noise process is Gaussian. Implications for Nucleation: If u 0 lies in the domain of attraction of the homogeneous state h 0 = µ, then u u 0 0 converges to this state exponentially fast. Thus, with probability close to 1 the stochastic solution u u 0 σ quickly converges to the homogeneous state as well. Why do we even observe nucleation??

18 Large Deviations: Nucleation Heuristics Key Observations: For small noise intensity σ, the solution u u 0 σ will spend most of its time in a neighborhood of the homogeneous state. Nevertheless, large deviations from this state do occur with positive probability. The probability for such excursions cannot exceed Ce D/σ2. Related Heuristic: Rolling Dice When rolling a die once, the probability of rolling a 1 or a number divisible by three is small. If I am allowed to roll the die infinitely often, each of these two events will eventually happen with probability 1. Typically, the second event is going to occur earlier than the first event.

19 Goal: Large Deviations: Rate Functional Find a collection A of paths A {u : [0, T ] X u(0) = u 0, u is smooth} which maximizes the probability that u u 0 σ stays close to A, and such that paths in A exit the domain of attraction of the homogeneous state. Large Deviations Theory: Provides a rate (or action) functional I u 0 T (u) for every path u in such a way that the probability of u u 0 σ staying close to u is of the order e I u 0 T (u)/σ2. Shows that for suitable u this probability is concentrated near the path u, i.e., the probability of u u 0 σ being outside a neighborhood of u is exponentially smaller.

20 Let u u 0 c The Associated Control Problem We reinterpret paths u as solutions of a control problem! denote the mild solution of the nonlinear control system t u = A ε u + G(u) + Q 1/2 2 c(t) u(0) = u 0 with u 0 C and control function c H T = L 2 ((0, T ), H 2 ), where c 2 H T = T 0 c(s) 2 2ds. Then the rate function is given by { 1 T (u) := 2 c 2 H T, if u = u u 0 c +, otherwise I u 0 for some c H T The role of the neighborhood of a path where the measure is concentrated is played by the sublevel sets K u 0 T (r) := { u H T : I u 0 T (u) r} = { u u 0 c : c 2 H T 2r }

21 Large Deviation Estimates Blömker, Gawron, W. (2010): The mild solution u u 0 σ of the Cahn-Hilliard-Cook model satisfies the large deviation estimates of Freidlin and Wentzell: Let r 0, δ, γ > 0 be arbitrary. Then there exists a σ 0 > 0 such that: For all u 0 C, 0 < σ < σ 0, and u C T with I u 0 T (u) r 0 we have P ( u u 0 σ u CT < δ) e (I u 0 T (u)+γ)/σ2, and for all u 0 C, 0 < σ < σ 0, and 0 < r < r 0 we have P ( dist CT (u u 0 σ, K u 0 T (r)) δ) e (r γ)/σ2, where C T = C([0, T ], C) and u CT = max t [0,T ] u(t).

22 The Energy as Quasi-Potential In order to maximize the probability of staying close to a path u, we have to minimize its rate functional I u 0 T (u). This leads to the following definition: For u 0, u 1 C define the quasi-potential I (u 0, u 1 ) = inf { I u 0 T (u) : u(0) = u 0, u(t ) = u 1, T > 0 } Then the following result holds: For suitable u C in the basin of attraction of the homogeneous state h 0 the quasi-potential I (h 0, u ) is given by I (h 0, u ) = 2 (E ε (u ) E ε (h 0 ) )

23 The Energy as Quasi-Potential The previous result follows from our specific choice of underlying phase space H 2, together with the fact that the right-hand side of the Cahn-Hilliard equation is the negative H 1 -gradient of the energy functional E ε. Using the identity Q 1/2 2 IT h0 (u) = 1 T 2 = 1 2 = T 0 T 0 = 1 x this gives c(s) 2 2ds = 1 T 2 Q 1/2 2 c(s) 2 1ds 0 u(s) A ε u(s) G(u(s)) 2 1ds u(s) + A ε u(s) + G(u(s)) 2 1ds + 2 (E ε (u(t )) E ε (h 0 ) ), where u = u h0 c. For the last equality we use the fact that the time derivative of E ε (u(t)) is given by ( u(t), A ε u(t) + G(u(t))) 1.

24 Exit from the Domain of Attraction This shows that in order to minimize the rate functional I u 0 T (u), the endpoint u(t ) has to be a point s e of lowest E ε -energy on the boundary D of the deterministic basin of attraction D of the homogeneous state h 0. The associated exit path u is basically the time-reversed heteroclinic orbit from s e to h 0. u 0 u 0 E s e ± D 0 h D

25 Stochastic Domain Exit Blömker, Gawron, W. (2010): For small κ > 0 let D(κ) = D \ B κ (s±,λ 1 ), where s1 ±,λ are the canonical nuclei, and let τ = inf{t > 0 : u u 0 σ (t) D(κ)}. Then for any 0 < κ < δ we have lim σ 0 P(uu 0 σ (τ) B δ (s 1 ±,λ )) = 1 ½ ± s 1 + D 0 h Most solutions exit the basin of attraction of h 0 close to the spikes s 1 ±,λ! E s 1 {

26 Outline 1. Stochastic Binary Nucleation Nucleation Results due to Bates & Fife The Cahn-Hilliard-Cook Model Large Deviations and Domain Exit 2. Nucleation in Ternary Alloys The Cahn-Morral Model Nucleation Statistics Boundary Effects and Scaling 3. Equilibria and Nucleation Pathways Numerical Path-Following Interior Droplet Equilibrium Solutions Morse Decompositions and Nucleation Pathways 4. Degenerate Forcing and Droplet Size Forcing with Degenerate Noise Saddle-Node Two-Parameter Continuation Maximal Droplet Counts

27 Multi-Component Alloys: Cahn-Morral Systems We now turn our attention to alloys consisting of more than two components. The main focus is to understand the geometry of nucleation droplets statistics, as a function of the total concentrations of the components. Basic model for alloys consisting of N 3 components: The temporal evolution of the N-dimensional concentration vector u = (u 1,..., u N ) is described by a stochastic Cahn-Morral system: u t = (ε 2 u + f (u)) + σ ξ in Ω u ν = u ν = 0 on Ω u u N = 1 in Ω

28 Multi-Component Alloys: Cahn-Morral Systems The concentration vector u should stay in or close to the Gibb s simplex G = { v R N : } N v k = 1 and v k 0 k=1 The nonlinearity f = (f 1,..., f N ) is essentially the negative derivative of an N-well potential F where F (u) = N ( u ui α(u i 1) 2 i=1 +ui ) u2 N

29 Spinodal and Metastable Region for Ternary Alloys Deterministic stability of the equilibrium µ R 3 : For N = 3 the metastable region is depicted in red, and the spinodal region is shown in two shades of blue. µ = (ū, v, w) G

30 Nucleation in Ternary Alloys Desi, Edrees, Price, Sander, W. (2011): System with w = ū = 0.1 and v = 0.1 ū = and v = These simulations are for ε = 0.02, σ noise = 0.2, and maximum time t end =

31 Statistics for the Droplet Placement Using computational homology, one can determine the likelihood of certain nucleation locations as a function of composition. The averages are determined from 512 simulations for each mass vector, for ε = 0.05, σ noise = 0.2, and maximum time t end = µ = (0.1, 0.1, 0.8) µ = (0.15, 0.05, 0.8)

32 Droplet Type Frequency Varies with Composition The average proportion of each droplet type varies with the initial mass vector µ = (ū, v, w), while the droplet location frequencies remain basically unchanged. The horizontal axis gives ū, and v = 0.2 ū, w = 0.8.

33 Stochastic Binary Nucleation Nucleation in Ternary Alloys Equilibria and Nucleation Pathways Degenerate Forcing Suitability of the Simulation Domain Size The simulations were all performed for ε = 0.05 and lead to fairly small droplet averages. How can one be sure that the results are meaningful for smaller values of ε? Are there boundary effects? u = u = u = 0.125

34 Changing the Interaction Length Changes in the interaction length ε correspond to rescaling of the base domain. Changing the interaction length for a simulation on the unit square from ε o to ε n is equivalent to keeping the interaction length fixed, but increasing the simulation domain by a factor of ε o /ε n. "=" o "=" n d d 1 d d "=" 0 d d " o =" n d d This scaling has to affect interior and boundary droplets differently!

35 Scaling for the Boundary Droplets If one observes an average of b o boundary droplets for ε = ε o, then for ε = ε n the boundary droplet average should be b n = b o εo ε n Simulations for ū = v = 0.1 confirm this scaling: Boundary Droplet Averages ε Actual Average Predicted Average Relative Error % % % % % %

36 Scaling for the Interior Droplets The interior droplet average should scale with the area of the interior domain which disregards a boundary layer of width d, i.e., i n = i o (ε o/ε n 2d) 2 (1 2d) 2 Simulations for ū = v = 0.1 also confirm this scaling: Interior Droplet Averages ε Actual Average Predicted Average Relative Error % % % % % %

37 Scaling for the Interior Droplets The boundary layer width d was computed from the droplet averages for ε = 0.05 and ε = 0.04 as d The resulting boundary layer is indicated by a dotted line for the cases ε = 0.05 and ε = 0.02.

38 Outline 1. Stochastic Binary Nucleation Nucleation Results due to Bates & Fife The Cahn-Hilliard-Cook Model Large Deviations and Domain Exit 2. Nucleation in Ternary Alloys The Cahn-Morral Model Nucleation Statistics Boundary Effects and Scaling 3. Equilibria and Nucleation Pathways Numerical Path-Following Interior Droplet Equilibrium Solutions Morse Decompositions and Nucleation Pathways 4. Degenerate Forcing and Droplet Size Forcing with Degenerate Noise Saddle-Node Two-Parameter Continuation Maximal Droplet Counts

39 Nucleation in Spatially Extended Systems Vanden-Eijnden and Westdickenberg (2008): They consider nucleation for the stochastic Allen-Cahn equation with asymmetric potential on large domains. It is shown that one can decompose the large domain into smaller fundamental domains in which nucleation occurs almost independently. In this way, nucleation becomes likely, and will occur at random positions in the domain. These methods do not apply to the Cahn-Morral or Cahn-Hilliard models due to their global mass conservation, which leads to correlations between fundamental domains. Nevertheless, their results indicate that in order to understand nucleation on large domains it suffices to study the deterministic attractor on small domains, which is numerically feasible, at least in principle.

40 Structure of the Equilibrium Set For understanding the dynamics of nucleation in the Cahn-Morral system, precise knowledge of its equilibria is crucial. They solve the nonlinear elliptic system ε 2 u + f (u) = c in Ω u = 0 on Ω ν 1 Ω u dx = µ Ω For convenience, we introduce the new parameters c = 1 Ω f (u) dx and λ = 1 Ω ε 2. Trivial constant solutions: u µ and c = f (µ).

41 Equilibrium Continuation in the Spinodal Region Using numerical continuation techniques one can find branches of droplet structures in the spinodal region for the deterministic Cahn-Morral model. Our approach uses the continuation package AUTO, in combination with spectral and symmetry methods. ū = v = w = 0.75 Starting point in the spinodal region μ u =μ v Horizontal axis: λ = 1/ε 2

42 Equilibrium Continuation in the Spinodal Region The secondary bifurcation branches lead to three main interior droplet types, which are ordered by decreasing energy.

43 Equilibrium Continuation into the Metastable Region Each of these nontrivial branches can then be continued with respect to mass into the metastable region. A final continuation with respect to β = (ū v)/2, while keeping w fixed, uncovers droplet solution branches within the metastable region. Follow to nucleation region Starting point in the spinodal region Follow to nucleation region Asymmetry in masses Starting point in the spinodal region μ u =μ v μ u =μ v

44 Equilibrium Continuation into the Metastable Region The movie shows double droplet equilibria as a function of α for mass vectors of the form (ū, v, w) = (α, α, 1 2α).

45 Equilibrium Continuation into the Metastable Region The solutions branches of all three interior droplet types continue into the metastable region. Since we are interested in mass vectors with w = 0.8, the intersections of the solution branches with the line α = 0.1 provide the necessary droplet equilibria.

46 Equilibrium Continuation into the Metastable Region Not all of the resulting equilibrium solutions are of interior droplet type. The solutions shown below are ordered by decreasing energy. mixed quad double

47 Stochastic Binary Nucleation Nucleation in Ternary Alloys Equilibria and Nucleation Pathways Degenerate Forcing Equilibrium Energy Diagram in the Metastable Region Continuing interior droplets with respect to β = (u v )/2 yields the droplet branches over the line u + v = 0.2 and w = 0.8 in the metastable region. The relative equilibrium energies and their geometries for u = v are shown below.

48 Equilibrium Continuation within the Metastable Region Close-ups of the lower and upper part of this bifurcation diagram reveal that the double droplet equilibrium only exists for β-values between 0.02 and Moreover, the small amplitude mixed droplet acts as canonical nucleus.

49 Droplet Instability and Morse Indices: β = 0 All of the interior droplets are unstable. Their positive eigenvalues can be divided into strongly unstable directions and weakly unstable directions. The latter ones are due to the approximate translation and rotation invariance of the solutions. strongly positive weakly positive index , 1, 2, , 2, ,

50 Heteroclinic Orbits and a Droplet Morse Decomposition The droplet dynamics can be determined through numerical integration of the heteroclinic orbits between the equilibria. From this, one can deduce a Morse decomposition for the interior droplets The image shows the Morse decomposition for ū = v = 0.1.

51 Droplet Instability for Asymmetric Mass For the asymmetric mass case ū = and v = the interior droplets and their indices are indicated below. type strongly positive weakly positive index mixed mixed 2, 2, quad 2, double mixed double double mixed 0 2 2

52 An Asymmetric Droplet Morse Decomposition For large domains, global mass conservation can cause effective mass fluctuations in nearby fundamental domains. Thus it is necessary to also understand the droplet dynamics for ū v, even if the original simulation is for ū = v The image shows the Morse decomposition for ū = 0.115, v =

53 Outline 1. Stochastic Binary Nucleation Nucleation Results due to Bates & Fife The Cahn-Hilliard-Cook Model Large Deviations and Domain Exit 2. Nucleation in Ternary Alloys The Cahn-Morral Model Nucleation Statistics Boundary Effects and Scaling 3. Equilibria and Nucleation Pathways Numerical Path-Following Interior Droplet Equilibrium Solutions Morse Decompositions and Nucleation Pathways 4. Degenerate Forcing and Droplet Size Forcing with Degenerate Noise Saddle-Node Two-Parameter Continuation Maximal Droplet Counts

54 Degenerate Stochastic Forcing Blömker, Sander, W. (2013): In the ternary case, we could numerically establish a characteristic size of boundary droplets. How does the nature of the noise affect the droplet size? In the previous simulations, we used space-time white noise which acts on all Fourier coefficients of the solution. In the following, we use degenerate noise, which only acts on one or on a small collection of modes. The large deviations results no longer apply, due to the noninvertibility of the covariance operator. One still expects nucleation almost surely, and it is possible to obtain lower bounds on the nucleation timeframe.

55 Degenerate Forcing of a Single Mode The two movies below show the effects of forcing exclusively the mode mode ϕ k,k (x, y) = 2 cos(kπx) cos(kπy), in the binary Cahn-Hilliard-Cook model with ε = and µ = k = 21 and σ = 2.5 k = 31 and σ = 20.0 Only in the left case can the droplet pattern be sustained!

56 Quarter Circle Bifurcation Structure There is a minimal droplet size which can be supported for a given value of ε, or equivalently, a maximal droplet count for a given domain size. Quarter circle equilibrium: In the left diagram, the curves correspond to µ = 0.1, 0.2,..., 0.6 from top to bottom. L 2 norm µ

57 Quarter Circle Bifurcation Structure These bifurcation curves are part of a two-dimensional bifurcation surface. The green curves originate via supercritical bifurcations from the trivial solution for increasing λ. All other curves originate via saddle-node bifurcations at the smallest λ-value L 2 norm 0.5 L 2 norm µ µ In the latter cases, the droplets appear beyond the red location curve of saddle-node bifurcation points.

58 A Vast Variety of Equilibrium Geometries The quarter circle equilibrium is only one of many stationary solutions. Many of these contain multiple droplets which are packed in different ways. Does this affect the possible droplet count which can be realized for a specific λ = 1/ε 2? Sample saddle-node equilibria for µ = 0.6

59 Locations of the Saddle-Node Bifurcation Points For the eight equilibria from the previous slide the left image shows the location of saddle-node bifurcation points in the λ-µ-plane. In the right image we rescale the λ-coordinate based on the number c of droplets in the equilibrium. The scaled value is λ/c. µ a 22b µ a 22b /c (scaled) Normalized behavior coincides in metastable region!

60 Estimating the Maximal Droplet Count Denote the rescaled curve by the function λ (µ). Then at a given value λ = 1/ε 2 one would expect a maximal droplet count of λ/λ. Forcing the mode ϕ k,k tries to generate a pattern with k 2 /2 droplets. At µ = 0.59 and ε = this implies that k can be at most k = 21 and σ = 2.5 k = 31 and σ = 20.0 Droplet patterns after the onset of nucleation

61 Summary Nucleation in the Cahn-Hilliard-Cook model can be made precise using large deviation techniques. Understanding nucleation on large domains can be reduced to the study of equilibrium droplets for the associated deterministic model on small domains. For the case of three-component systems, numerical methods can uncover the droplet formation dynamics via deterministic Morse decompositions. Forcing the Cahn-Hilliard-Cook equation with degenerate noise in combination with two-parameter continuation can provide insight into maximal droplet counts. In principle, many of these open questions can be addressed via rigorous computational techniques.

62 Thank You! Collaborators: Dirk Blömker (Universität Augsburg) Evelyn Sander (George Mason University) Students: Jonathan Desi (University of Maryland, Baltimore County) Hanein Edrees (George Mason University) Bernhard Gawron (RWTH Aachen) Jonathan Price (George Mason University) Work partially supported by the US Department of Energy and the National Science Foundation.