Numerical Studies on Phase Field Diffusion and Flow Solvers

Transcription

1 Numerical Studies on Phase Field Diffusion and Flow Solvers Author: E. Man Supervisors: Prof.dr.ir. C.R. Kleijn, M.Sc. H.G. Pimpalgaonkar May 7, 2014 Delft University of Technology Facaulty of Applied Sciences Department of Chemical Engineering Research group: Transport Phenomena

2 ABSTRACT The goal of this project is to validate and evaluate developed numerical solvers that solve both the Cahn-Hilliard diffusion equation and the Cahn-Hilliard-Navier-Stokes coupled equations. The solvers are written for the implementation in the open source CFD software platform, OpenFOAM. In Chapter 1, we give a brief description of the phase field theory which we have implemented in our solvers. This theory is used to study phase separation in multiphase flows. We also discuss the physics behind the phase separation. We also give a short derivation of Cahn-Hilliard equation and the coupled Cahn-Hilliard-Navier-Stokes equation. In chapter 2, we validate our phase field diffusion solver and diffusion-reaction solver. We examine the temporal evolution of the characteristic length R(t) and we show that it follows classical Lifshitz-Slyozov law. To validate diffusion reaction solver, we simulate Liesegang pattern inducing reaction systems and validate our results with the available results in literature. In chapter 3, we validate our Cahn-Hilliard-Navier-Stokes coupled solvers for the velocity field using test cases for which analytical solution is available. Herein, we evaluate the effect of the phenomenological parameters on some of the simulation characteristics, such as parasitic current and phase field parameter overshoot. We show that these characteristics also follow the expected behavior. 1

3 Contents 1 Phase Field Model and Theory Cahn-Hilliard(CH) Equation Navier-Stokes(NS) Equation Scope of the Project Validation of Cahn-Hilliard Diffusion and Diffusion-Reaction Solver Case: Length Scale Study Case: Diffusion-Reaction Solver Validation of Cahn-Hilliard-Navier-Stokes coupled Solver Case: Square Droplet Case: Lid Driven Cavity Conclusions and Recommendations Conclusions Recommendations APPENDIX 25 2

4 Chapter 1 Phase Field Model and Theory In the Fickian diffusion description, molecules move in the direction of the gradient of concentration. Despite of its general applicability, sometimes it cannot describe some of the common transport phenomenona, like the separation of the components of an immiscible mixture, e.g. an oil/water mixture, separating into oil and water layer. We start with a well dispersed mixture of immiscible components, say oil and water, where the concentration is homogeneous across the whole system, Fick s diffusion law would predict this as a stable system, but the experiments show that this is an unstable system. As time progresses, the system evolves into two separate water-rich and oil-rich layers. The physical explanation of such phase separation can be understood from the principle of the minimum energy: for a closed system, the total free energy must be minimized. For a binary system, like oil/water mixture, there are 3 different possible free energy contributions: the interaction of the same molecules: oil-oil or water-water, and the interaction of two different molecules; oil-water interaction. Both oil-oil and water-water interactions must be more energetically favorable than the oil-water interactions for phases to separate. Thermodynamically, the system will evolve to minimize the number of interactions between different kind of molecules and replace it with the interaction of same kind of molecules, which in turn is equivalent to minimization of the contact surface of the two fluids. This leads eventually into the aforementioned separation. The theory that can correctly predict and describe this phenomenon is the Phase Field Theory, which was developed by John W. Cahn and John E. Hilliard [1]. In the following section we will explain the Cahn-Hilliard equation and present the simplified version of the derivation of the equation that is done in the paper of Cahn and Hilliard. 1.1 Cahn-Hilliard(CH) Equation Phase field theory also describes a diffusion process. It can be regarded as an extension of the Fick s Diffusion law, a more complex variant of it. As we know the Fick s second law of diffusion (1D), c t = D 2 c x 2 (1.1) 3

5 is a combination of the first diffusion law and the continuity equation: J = D c x and, c t + J = 0 (1.3) where D is the diffusivity and c is the concentration. We start the derivation of phase field equations with same continuity equation. In absence of a source, it can be stated that any changes of the concentration can only be caused by the inflow or the outflow of the particles from a control volume due to diffusion flux. We replace the concentration with a more convenient phase field parameter φ (a function of the time and spatial coordinates), which determines the composition of both phases. The volume fraction, y, of a species can have values between 0 and 1, while total volume fraction in a control volume must add up to 1. We can then take a transformation of the volume fraction with, φ = 2y 1. Even after the transformation the continuity equation does not change. φ will takes values between -1 & 1, for a equimolar composition, φ = 0. (1.2) From a conventional Fickian analogy, the flux is proportional to the gradient of concentration, which serves to be the driving force of the diffusion. However this analogy fails to model phase separation. Clearly, there is some other force that drives the diffusional flux. In the paper of Cahn and Hilliard [1], authors proposed and proved that for an isothermal system, the driving force of the diffusion flux should be the gradient of the chemical potential, µ. Therefore J can be express as: J = M µ (1.4) where M is the mobility of the particles, analogous to the diffusion coefficient. Conventionally, chemical potential is defined as a derivative of Gibbs free energy with respect to the concentration, F c. But just like the continuity equation, in phase field equation we would like to express all parameters in terms of the phase field parameter φ. It can be seen that for F c = F φ, from here we can reformulate eq. (1.3) as the Cahn-Hilliard Equation (CH equation): φ t M 2 µ = 0 (1.5) According to Cahn and Hilliard [1], the total free energy of a nonuniform system, F, can be evaluated as an integral of the local free energy function, f, over the whole region. f can then be expressed as a sum of two contributions, which yields the following equation: F = [f(φ 0 ) + κ( φ) 2 ]dv (1.6) Physical explanation of each term is as follows: R 1. The homogeneous free energy, f(φ 0 ) represent the free energy contribution due to local composition at equilibrium. This term solely depends on the local phase composition, it takes form of a double-well potential [7]. One commonly used function to defines double-well potential is: f(φ 0 ) = 1 4 (φ2 1) 2 (1.7) 4

6 where the 2 minima represent the composition of two pure phases which is thermodynamically stable(see fig. 1.1). 2. The gradient free energy, κ( φ) 2. This is the energy penalty due to the changes in concentration in the vicinity of given point. The only part of the system that has a gradient in concentration is the interface surface, this is then also called the Interface energy. The constant κ serves as a phenomenological parameter is which related to the interface thickness. If we plug in the definition of chemical potential using the total free energy function of Cahn and Hilliard as of the free energy of Gibbs, µ becomes: µ = F φ = f (φ) κ 2 φ = φ 3 φ κ 2 φ (1.8) and combine eq. 1.5 and eq. 1.8, we derive the Cahn-Hilliard Equation: φ t = M 2 µ µ = φ 3 φ κ 2 φ φ t = M 2 (φ 3 φ κ 2 φ) (1.9) With this result, we see that CH equation is a 4 th order nonlinear differential equation. Figure 1.1: Double well potential function, f(φ 0 ) = 1 4 (φ2 1) Navier-Stokes(NS) Equation In the previous section, we described a diffusion only system. We assumed that any changes of the concentration can only be caused by diffusion flux. But in a dynamic fluid system, concentration can also be influenced by the convection flow of the fluid. Therefore we need to reformulate the species continuity equation as: φ t + ( uφ) = M 2 µ (1.10) 5

7 where u is the velocity field of the system. Since we need to take fluid motion into account, the system also has to satisfy the law of mass conservation and momentum conservation. For the incompressible fluid the mass conservation law gives: and the standard Navier-Stokes equation is modified as: u = 0 (1.11) ρ( u t + ( u ) u) = p + τ + F s + ρg (1.12) where F s is the surface tension force that is added into the NS equation to extend its applicability to the two phase flow and τ the stress tensor. Based on the principle of conservation of energy, the change of the kinetic energy that is induced by surface tension force has to be compensated with the change in free energy of the system. Below is the expression for the change of the kinetic energy from Jacqmin[10]: u F s dv (1.13) then here follow the expression for the change of free energy: F F F t = t dv = φ φ t dv = µ φ dv (1.14) t Using chain rule, we can express the change of free energy in terms of the chemical potential µ and the rate of change of φ, if we plug in the expression of eq only consider the convection flux term: F t = µ ( uφ)dv = µ(φ u + u φ)dv = µ u φdv = uφ µdv (1.15) If we equate eq and 1.15, we get an expression of the surface tension force that is in terms of chemical potential and the phase field parameter: F s = φ µ (1.16) Combining equations 1.10, 1.11, 1.12, we have the Cahn-Hilliard-Navier-Stokes couple equation that govern the velocity field in a two phase diffusion/convection system. 1.3 Scope of the Project In this poject, we would like to evaluate and validate the developed numerical solvers with some standard test cases. The solvers we want to test include: Cahn-Hilliard diffusion solver (equation 1.4), diffusion-reaction solver (see chapter 3.2) and Cahn-Hilliard-Navier-Stokes coupled solver (equation 1.12). For diffusion and diffusion-reaction solver we plan to qualitatively validate solvers using test cases from literature. For flow solvers, we wish to perform a girdsize dependence study and a simulation parameters dependence study. Different conservative and non-conservative solvers were used to simulate flows. We would also like to study the sensitivity of the parasitic current on the different solvers. For Ca hn-hilliard-navier- Stokes coupled solver, we would like to obtain velocity profiles for simple cases such as flow between plates for which an analytical solution is available. Then, we would like to evaluate our solvers for parasitic currents and interface width as done by Hoang et al. [11]. We would also like to reproduce analytical functionality between the phase field parameter shift and the Cahn number. 6

8 Chapter 2 Validation of Cahn-Hilliard Diffusion and Diffusion-Reaction Solver In this chapter, we will validate the CH diffusion solver developed within the Transport Phenomena group by reproducing results from literature. First, we will validate our solver qualitatively by verifying that the solver reproduces the Lifshitz-Slyozov law. Next, we will test our diffusion-reaction solver for simulation of formation of precipitation pattern. 2.1 Case: Length Scale Study In this validation study, we will estimate an important simulation property, namely the Characteristic Length (R(t)) of the system. As explained before in the previous chapter, when free energy exhibits a double-well potential function, the homogeneous mixture will spontaneously separate into two pure phases and begin the process of coarsening. The size of single (pure) phase region will grow gradually, until the system reaches its chemical equilibrium of minimum free energy. If we closely examine fig. 2.1, we can observe growth of the morphological patterns at different times. It has been shown that if we rescale the rate of growth of the patterns by the characteristic length, R(t), growth of microstructure are essentially the same [8]. That means the morphology of the system is self similar and only depends on time via R(t). Many studies [2][3][4] came to a conclusion that this particular characteristic length, R(t), follows a specific temporal growth law: R(t) n R n 0 = Kt (2.1) where R(t) and R 0 are respectively the characteristic length of the single phase domain at a given time and at t = 0, and K is the coarsening constant. In previous studies, they find a power law relation of R(t) n R0 n t with n 3 for the stage right after the initial coarsening, and n 4 for later stage [3]. This particular growth model of R(t) with n = 3 is referred to as the Lifshitz-Slyozov(LS) law [5]. Lifshitz and Slyozov had derived this law for the phenomenon called Ostwald Ripening: A large droplet will grow in size by the condensation of diffused material through the evaporation of smaller droplets. This is called evaporation- 7

9 (a) (b) (c) (d) Figure 2.1: Morphological pattern of the phase separation at time: (a) t = 0s, (b) t = 0.001s, (c) t = 0.01s, (d) t = 1s condensation mechanism, which is the dominant mechanism in the initial stages of coarsening. Our goal in this study is to confirm the validity of the CH solver by qualitatively reproducing the same functionality as that of Lifshitz-Slyozov law. Scaling Function It has been shown in the studies that there are two methods to estimate the characteristic length R(t). It can be done either via Fourier transformation and calculation of the structure factor [2][3][4], or via autocorrelation of the phase field function [8]. In our studies we have chosen the latter approach. It is less complex mathematically, and also requires less postprocessing procedures. Structure function studies are more important when dealing with real experiment. Suppose we have a two dimensional discrete N by N phase field function φ(m, n), with 0 m N 1 and 0 n N 1. From this phase field function we can take the standard 2D auto-correlation function: C(k, l) = N 1 N 1 X X φ(m, n)φ (m k, n l) m=0 n=0 8 (2.2)

10 Since C(k, l) is considered to be a rotational invariant function, all points with the same radius r = m 2 + n 2 share statistically the same function value of C. By taking circular average of this function we can simplify this 2D autocorrelation function into a direction invariant function C( r): C( r) = G C( x)ds G ds = 2π 0 C(θ)rdθ 2πr (2.3) where numerator represents a line integral of the function C( x) along a circular close path G, and denominator is the length of the path G (circumference of the circle). But for our auto-correlation function, we only know the discretized function value at each grid point, not the function itself. To make eq. 2.3 still useful for our analysis, we first define an array of coordinates on the discretized path G with radius R 1 in polar coordinate: G R1 : p 0 = (R 1, 0), p 1 = (R 1, θ), p 2 = (R 1, 2 θ),..., p n = (R 1, n θ) (2.4) then mapping these polar coordinate back to cartesian coordinates in order to perform 2D interpolation on those points. After finding out the interpolated function value of each point, F (p i ), we can substitute into the following expression that is analogical to eq. 2.3: C( r) = F (p 0) + F (p 1 ) + F (p 2 ) F (p n ) n = 1 n F (p i ) n i=0 (2.5) Qualitatively speaking, C( r) is considered to represent the probability that two randomly chosen points separated by distance r, have the same phase field value, i.e. C(r) = P [φ(r1 ) = φ(r 2 )]. Using this definition of C( r), we can predict that: for r is small, both points are likely to be within the same phase region C( r) 1. As r grows larger, C( r) decreases exponentially into a minimum and eventually leading into an asymptotic value of overall phase composition φ 0. The distance λ between the origin and the first minimum of the function is roughly corresponding to the separation distance of the morphological pattern [8]. For λ describes the typical length of the domain, we can use this as the characteristic length, R(t). According to the LS growth law, λ must follows eq Simulation In our simulations, we consider a 2D square discretized domain that contains 256 by 256 grid points. We then simulate the homogeneous initial state by randomly assigning the phase field parameter φ = 1 or φ = 1 in each grid cell. Because of the randomness, the global phase composition statistically approaches zero i.e. φ 0 0. As time progresses, the phase field φ of each cell will evolve according to the CH equation. To prove the independence of the growth law with the initial state, we generated 8 different sets of initial condition, no two initial condition have the same distribution of φ, yet the φ 0 s are all approximately equal to 0. This was archieved by using different seed values in a random number generator. To validate the LS law, we calculated the characteristic length R(t) of every time step based on the auto-correlation function. According to eq. 2.1, we would expect a straight line from the plot of R 3 (t) versus t, yet fig. 2.3a shows otherwise. We can clearly see that the graph begins relatively linear, then around t = 0.02 the slope flatten out as time increases. We can 9

11 (a) (b) Figure 2.2: (a) Morphological pattern of the phase separation at time t = 0.007s, (b) 1D circular average of the autocorrelation function, C(r). Distance λ indicate the first minimum of the function that can be associate with the length of the microstructure. conclude that the LS law can only be applied at the initial stages of the coarsening, after that another growing mechanism must dominate the system s evolution. This is indicated in the logarithm plot of the data, see fig. 2.3d. The slopes in the two time intervals (t = [0, 0.2] and t = [0.05, 0.2]) clearly have two different values. A steeper slope (t = [0, 0.2]) indicates a faster growth relationship and shallower slope (t = [0.05, 0.2]) indicates slower growth. After the initial stage of coarsening, which is described by LS law and evaporation-condensation mechanism, most of the single phase regions have grown into a layer-like shape (see fig. 2.1c), where most of the small droplets have already been absorbed into the layer-structure. From now on, the dominant way to reduce the free energy, i.e. reduce the interface between region, is to rearrange the molecules within single phase region to smooth out the curvature. It can be seen from µ in eq. 1.4 that within a single phase region, there can be no flux. Diffusion only takes place where particles can move, i.e. along the interface. Particles will diffuse from places with high curvature into places of low curvature in order to flatten out the interface which means less contact area. This curvature-driven diffusion mechanism is much slower and has the corresponding exponent of the growth law of n = 4[3]. The statistical value of R 2 = of the linear fit on R(t) 3 = Ax + b from 0s < t < 0.02s (fig. 2.3b) strongly suggests that our data does fit into the LS growth law in the initial stage of the coarsening. To find out the actual growth constant, we can take the nonlinear fit of our data using the following approximation, R(t) A + Bt m, which is a result from the binomial expansion of eq. 2.1 (Appx A). After analyzing our data, we found that for 0s < t < 0.02s, m = 0.26 which means our growth exponent is slightly smaller than the theoretical value of m = 1/3. It has been reported in literature that this exponent asymptotically approaches the LS law [13]. Hence a simulation with more than 256x256 space grids will help in taking this number closer to 1/3. Before we discuss the result of t > 0.02, we need to mentioned that all R(t) values are the mean values of the averaging over all 8 dataset. As for the error analysis, 10

12 we used the following formula to determined the confidence interval: X n ± A S n n (2.6) where X n is the mean value, A is statistical value of the student s T-distribution value(99%), S n the sample variance and n the sample size. The errors of the interval 0s < t < 0.02s are so small that we decide not to plot it. As for the growth constant at t > 0.02s, we plotted R 4 (t) versus time, we found for the linear fit R 2 = 0.986, which is a decent indication of the validity of n = 4. On the other hand the error bars are relatively large, this shows that for a reliable fit and conclusion, we need to increase our sample size. And for the nonlinear fit of eq. 2.1, we only obtain an unphysical value that is not even close to m = 1/4, which is expected given the large error bars. (a) (b) (c) (d) Figure 2.3: The cubic of the characteristic length versus time, (a) t = 0s to t = 0.3s, (b) Logarithm plot, (c) Only the linear part of the (a) i.e. t = 0s to t = 0.02s and the linear fit of R 3 (t) = Ax + b with R 2 = 0.992, (d)the fourth power of the characteristic length versus time t = 0.05s to t = 0.2s and linear fit R 4 (t) = Ax + b with R 2 =

13 2.2 Case: Diffusion-Reaction Solver Next, we wish to validate the diffusion-reaction solver. One way to test it is to simulate precipitation patterns. One specific case is called the Liesegang Pattern. Depending of the geometry of the domain, the Liesegang pattern can be bands, rings or shells. These alternating patterns are the alternating single phase regions that are created during precipitation of reactions due to phase separation. One of the determining factors of the patterns is the instability condition. Unlike the previous case, in which the whole domain is considered to be thermodynamically unstable, Liesegang pattern formation is controlled by a guiding field of instability condition, this guiding field can be temperature, PH-level and in case of precipitation, the concentration of reactant. The source of instability often starts at one end of the domain, then gradually spread over the whole region, and the same goes with the formation of the pattern. (a) (b) Figure 2.4: 1D Liesegang pattern at two different time (a) t = 0.6s, (b) t = 3s. Modeling of the governing equation We follow here the approach given by Antal et al[6] to simulate Liesegang pattern formation. Let us consider the standard CH equation (eq. 1.9). To incorporate a guiding field into our simulation, we need to slightly modify the equation to satisfy the instability condition. Based on the equation, we see that the start of the phase separation is only determined by the sign of the first two terms of the equation. We can simply insert a parameter ɛ to control activation of phase separation. If we consider a 1D domain and rescale all parameters to dimensionless the CH equation can be written as: φ t = 2 x 2 (ɛφ φ3 + κ 2 φ x 2 ) (2.7) The parameter ɛ represent the presence of the source of instability. Depending on the initial condition ɛ can always rescale between 1 and +1. When ɛ changes sign the separation starts. Unless ɛ changes sign, phase separation does not initiate. Since ɛ is related to the concentration profile of the reactant, we can use the standard Fick s equation of diffusion to model the evolution of ɛ: ɛ t = D 2 ɛ x 2 (2.8) where D is the dimensionless diffusion coefficient of the system. For a 1D domain that the reaction starts at one end, the corresponding initial conditions are: at t = 0, ɛ(x > 0) = 1 and ɛ(x = 0) = 1, and also at then other end x = L there is no mass flux, ɛ x = 0. 12

14 Result In our simulations, we are able to reproduce similar results of Antal et al[6]. First we can confirm that the ɛ profile in time indeed follows Fick s theory of mass transfer. According to the penetration theory, which is derived from the Fick s law, the following relationship of penetration depth and time must hold, x p t. After performing a power fit on the data we found the exponent of 0.48 ± 0.18, which is very close to the theoretical value of 0.5. (a) (b) Figure 2.5: (a) ɛ profile of t = 0.05s(blue),t = 0.5s(red),t = 1s(green) and the penetration depth calculated as tangent of the ɛ profile (dashed line), (b) Power fit of the penetration depth vs time, y = ax b with b = 0.48 ± In fig. 2.6, we can see the details of the formation of the alternating pattern. By comparing our result(left) with the result of literature [6] (right), we can conclude that we qualitatively have the same result. The peaks and lows of the pattern follow precisely the squared root function of the guiding field, also the spinodal line is indicated in the figure. These two lines are the direct result of the altering of the first two terms of the CH equation. This way, the 13

15 local free energy function is continuously changing depending on the value of ɛ. Ideally, we would like run the simulation until it reaches the stationary state, however due to the large amount of simulation time required, we stop the simulation at t = 60s. In fig. 2.7, we can see that our result (top) shows similar shape to the result of [6], we would expect that after a longer simulation time, when the system reaches the stationary state, all peaks and lows will reach the single phase phase-field value of φ = 1, +1. (a) (b) Figure 2.6: (a) Snapshot of the guiding field ɛ(x)(blue dashed), phase field φ(x) (black), spinodal line ± ɛ(x)/3 (red), local equilibria ɛ(x) (green), (b) Similar simulation result from Antal et al.[6] 14

16 (a) (b) Figure 2.7: Numerical solution of phase field parameter, (a) with L = 4000, D = 1, φ 0 = 0.05, solution at t = 60s(last time point of the simulation), (b) Stationary solution from Antal et al.[6] 15

17 Chapter 3 Validation of Cahn-Hilliard-Navier-Stokes coupled Solver 3.1 Case: Square Droplet When phase field theory is integrated with the Navier-Stokes equation to simulate multiphase flow, the surface tension force of a two-phase mixture is formulated as a function of the phase field parameter φ and the chemical potential µ. A solver has been developed in the Transport Phenomena group which solves the NS equation along with the phase field equation and continuity equation. To verify this implementation of the surface tension force into the Navier- Stokes equation, we will check our solver by performing an interface deformation simulation. Simulation In this test case, we will simulate the relaxation of a 2D square droplet in the surrounding of a different phase, where the phases of the droplet and the surrounding are the two stable states of the free energy function. Regarding the transport properties for both phases, they share the same viscosity, mobility and density, the only difference is the surface tension. Due to surface tension, the droplet will relax into the minimum energy configuration i.e. a circle. When it relaxes into a circular shape, all surface tension forces at the interface cancel out and it also reaches the minimum area of contact surface. Within this simulation, we will also evaluate some of the characteristics of the simulation. First, we will check the interface width with the value reported in literature in which a different modelling method is used to solve similar multiphase flows. Second, we will discuss two effects that often appear in the modeling of phase field theory, parameter overshoot and parasitic current. 16

18 (a) (b) (c) (d) (e) (f) Figure 3.1: Relaxation of a square droplet into a circular droplet at (a) t = 0s, (b) t = 0.01s, (c) t = 0.05s, (d) t = 0.2s, (e) t = 0.5s, (f) t = 1s. 17

19 Interface thickness In fig 3.2, we have plotted the interface thickness on 3 different meshsizes, the contour lines indicate the phase field parameter. We define interface thickness as the region between φ = 0.9 and φ = 0.9 (the inner and the outer contour line of fig. 3.2), the corresponding interface thickness is approximately between 4 to 5 grid cells. It is noteworthy to point out that the number of grid cells which the interface occupies is constant, in spite of the increasing meshsize. This means that the interface is getting smaller when we are using a finer mesh. We compare this thickness to that reported by Hoang et al [12], in their simulation. They observed that the interface is 6 grid cells for Volume of fluid (VOF) solving method, see fig (a) (b) (c) Figure 3.2: Interface of 3 mesh size, inner and outer contour line respectively: φ = 0.9 and φ = 0.9 (a) 64x64, (b) 128x128, (c) 256x256 Figure 3.3: Interface of the simulation of Hoang et al. [11] Parameter Overshoot The definition of phase field parameter φ is given as a rescaled volume fraction of the molecules. φ should be bounded in the range of [-1,1], any values beyond the range have no physical meaning. But in our simulation we have observed that φ will shift in range of [ 1 + φ, 1 + φ] with φ when using Krylov subspace solvers. Such overshoot in phase field parameter φ can affect the accuracy of the simulation, but is unavoidable unless special solvers are used. It has been reported in study of Yeu et al. [12] that this parameter overshoot is linearly related to the interface thickness, after it is rescaled into a dimensionless 18

20 number, the Cahn Number(Cn), as: The Cahn number is define as follows: κ Cn = L = ξ L 2 φ = Cn (3.1) 3 (3.2) where κ is the phenomenological parameter from the free energy function of the CH equation (see eq. 1.6). Since κ has dimension [m 2 ], hence we take its square root. L is the size of the domain, in our simulation it is set to L = 1. Since κ can be adjusted in our simulation, we altered Cahn number to validate equation 3.1. The result is plotted in fig It shows that φ decreases linearly with decreasing Cn, i.e. with finer mesh. Ideally, we want the deviation of φ to be as small as possible. But this also would mean the interface thickness should be very thin. To resolve this interface properly in simulation we need to use a very fine mesh. Thus, there is a trade off between the accuracy of the simulation and the available computational power. Figure 3.4: φ vs Cahn Number, Analytical solution Yue et al: φ = y = ax + b a=0.487, b=0.0009, R 2 = Cn, Linear Fit: Parasitic Current When the droplet reaches its stationary state (see fig. 3.1f), it means that the free energy is at its minimum and the driving force of any transport processes will be zero, the exact solution of the velocity should also be zero. Any non-zero velocity in the system is regarded as the parasitic current, which results due to numerical solution of the NS equation on a discretized domain. In fig. 3.5, we see the residual velocity in the domain. In this test case, we will evaluate the magnitude of the parasitic currents of the conservative and the non-conservative solvers. The conservative solvers is a MULES (Multidimensional Universal Limiter with Explicit Solution) based solver that maintains integral balance and at the same time keeps the field value (φ) offset within limit. The non-conservative solver is a Krylov subspace or iterative solver, in which neither integral balance nor field value (φ) offsets are managed efficiently. We can see from fig. 3.6 that the difference in the magnitude of the parasitic currents between the solvers is around This means that by imposing the constraints of 19

21 the phase field conservation, the parasitic currents are several order of magnitude larger than its non-conservative counterpart. Parasitic currents due to the non-conservative solvers, on the other hand, approach machine precision. Figure 3.5: Velocity profile of a relaxed droplet at t = 1s. Figure 3.6: Magnitude of parasitic currents of Conservative solver and Nonconservative solver. 20

22 3.2 Case: Lid Driven Cavity In the following case, we run a simulation where an external velocity field is applied on the system. We will check the velocity profile of the stationary state with the analytical solution and validate the solver for velocity profile. Simulation In this test case, we perform a 2D simulation of phase separation within a cavity, a standard squared domain, with both top and bottom connected to an imaginary lid that moves with the same velocity but in the opposite x-directions, v top = +1m/s, v bottom = 1m/s. Within this setup we have perform two simulations, first we assume both species to have the same transport properties, and in the second simulation one species will have a higher viscosity, µ 1 /µ 2 = 0.4. In both simulations we begin with a well dispersed mixture. In fig. 3.7, we can see the evolution of the morphological pattern; and in the last figure we find out the stationary state of the system. Strictly speaking, the system is not yet perfectly stationary, the interfaces between the phases are not a horizontal straight line yet, which means the free energy has not yet reached its minimum. For any longer simulation time, we observe a small ripple on the interface will not disappear but rather oscillate in time. So we use this last configuration as our template for the prediction of the velocity profile. It is also important to notice that this horizontal single layer configuration appears only when we apply cyclic boundary condition on the two sides of the cavity to simulate a unbound domain i.e. left wall and right wall are connected. For a actual square cavity (see Appx C), the simulation results suggest that there is no stationary state like fig. 3.7f. In fig. 3.8 we clearly see the effect of the single horizontal layer in the velocity profile. Depending on the value of the viscosity the slope of the velocity as well will change. Since viscosity determines the rate of diffusion of momentum, we expect regions with higher viscosity to show a steeper slope in velocity profile, while regions with lower viscosity should show shallower slope in velocity profile. The velocity profile obtained here is consistant with our observations. 21

23 (a) (b) (c) (d) (e) (f) Figure 3.7: Morphological pattern of lid driven cavity simulation at time: (a) t = 0s, (b) t = 0.01s, (c) t = 0.5s, (d) t = 2.5s, (e) t = 5s, (f) t = 7.5s. (a) (b) Figure 3.8: Velocity profile of the simulation with (a) same viscosity µ 1 = µ 2, (b) different viscosity µ 1 /µ 2 =

24 Chapter 4 Conclusions and Recommendations 4.1 Conclusions In this thesis, several numerical solvers are tested against the result of the other studies. In all 4 test cases we were able to reproduce qualitatively similar results, hereby validating our solver to be useful and reliable. For the four test cases, the following conclusions can be drawn: 1. The Length scale study shows that the growth relation of the morphological pattern indeed follows the Lifshitz-Slyozov law, but only at the initial stages. In order to obtain a more accurate agreement, simulations should be run with finer grid size. 2. Although the running time of the simulations of the Liesegang Pattern were not sufficient to reach the steady state, the comparison of the snapshots of both our result and that of [6] did prove the validity of the penetration theory and the diffusion-reaction solver. 3. The CH-NS coupled solver can accurately predict the deformation of a square droplet. As for numerical phenomena such as parasitic current and parameter overshoot, they are influenced by the chosen simulation parameters. Depending on the requirement of the simulation, they can be tweaked into optimum value. 4. As for the velocity profile of the lid driven cavity, the results are in agreement with the analytical solution of NS solver. 4.2 Recommendations During this project, there were several simulations that we were not able to finish. These included the simulation of a Taylor-drop experiment, which is an extension on the current lid driven cavity case. After we obtain a stable shear flow within the domain, we want to simulate droplet relaxation in such environment. In our simulation, we were not able to keep the droplet stationary. Because of the unbound domain, the droplet was carried away out of the domain by the fluid flows. Our first recommendation is to finish this simulation by fine tuning the simulation parameters, such as velocity, density, viscosty etc. to keep the droplet in place and/or employing a continuous feed back system to reposition the droplet. Next, 23

25 we would recommend to perform a sensitivity study of parasitic currents on the dependence of the meshsize. In our project we are only able to run simulations on 64x64 and 128x128. Simulations on finer meshsize should give a good indication of the parasitic current dependence on meshsize. Finally, we recommend to incorporate a better formulation of the surface tension into the solver. At the moment we are using the Weber number as input, but σ is a material property and will be a better input parameter for our simulations. 24

26 APPENDIX A Binominal Expansion Accoring to the Binominal Theorem, any power function of (x + y) n can be expanded as: ( ) ( ) ( ) ( ) ( ) n n n n n (x + y) n = x n y 0 + x n 1 y 1 + x n 2 y x 1 y n 1 + x 0 y n (4.1) n By only taking the first and last term we can approximate: B Simulation results of Length scale study Time(s) λ(r(t), Characteristic Length) Sample number Mean (x + y) n x n + y n (4.2) 25

27 C Lid driven cavity without cyclic boundary (a) (b) (c) (d) (e) (f) Figure 4.1: Phase field pattern, elapsed times are equal to (from left to right) t = 7.5s, t = 10s, t = 12.5s, t = 15s, t = 17.5s, t = 20s. 26

28 Reference [1] J.W. Cahn, J.E. Hilliard Journal of Chemical Physics 28, [2] A. Chakrabarti, R. Toral, J.D. Gunton Physical Review E 47, [3] J. Zhu, L. Chen, J. Shen, V. Tikare Physical Review E 60,3564. [4] P. Fratzl, J.L. Lebowitz, O. Penrose, J. Amar Physical Review B 44, [5] I.M. Lifshitz, V.V. Slyozov Journal of Physics and Chemistry of Solids, Vol. 19, 35. [6] T. Antal, I. Bena, M. Droz, K. Martens, Z. Rácz Physical Review E 76, [7] D. Lee, J, Huh, D, Jeong, J. Shin, A. Yun, J. Kim Computational Materials Science 81, [8] Y. Jiao, E, Padilla, N. Chawla Acta Materialia 61, [9] X. Cai, M. Wörner, O, Deutschmann th Open Source CFD International Conference, Hamburg,Germany, Oct [10] D. Jacqmin Journal of Computational Physics 155, [11] D.A. Hoang, V. van Steijn, L.M. Portela, M.T. Kreutzer, C.R. Kleijn Computers and Fluids 86, [12] P. Yue, C. Zhou, J.J. Feng Journal of Computational Physics [13] T.M. Rogers, K.R. Elder, R.C. Desai Physical Review B