D.2.1 A semi-implicit quasi-spectral solver
We numerically solve the dynamic equations of the continuum model of active droplets introduced in section 3.1. This includes the equations for the concentration field (3.1) with fluxes (3.2), the chemical potential (3.5) with (3.6), stress balance (3.11) with incompressibility (3.12) and the chemical reaction (3.13) with (3.14).
For this we use a spectral method in a 3d rectangular box. This has the advantage that in a spectral decomposition, the spatial operators become simple multiplications with the wavenumber Chen and Shen (1998). However, our equations contain a number of nonlinear functions, which are easier to evaluate in real space. We therefore transform forward and back in each time step.
To calculate the next time step ti from the fields found in time step ti−1, we
use a semi-implicit Runge-Kutta method Ascher et al. (1997) (method (2,3,3)) for the concentration field. This evaluates the gradient term in µ¯, Eq. (3.5), implicitly, while evaluating the rest of µ¯as well as the advection term of the fluxes, vc, explicitly. This effectively means that the terms related to the interfacial profile are calculated implicitly, which allows for larger time steps as an explicit scheme.
For the concentration field, we choose no-flux boundary conditions (∂nc = 0,
where the derivative is in a direction normal to the simulation box), which leads to a decomposition in cosine functions in the spectral description. The Laplacian then
D.2. Numerical solution of the continuum model
is−k2for a mode with wave vector k. The Stokes equation can also be solved using
spectral methods. Here, no-flux conditions lead to vn= 0. Additionally we enforce
incompressibility using a reprojection method. For this, the velocity field calculated by neglecting the partial pressure, Pp = 0, can be split into two parts (Helmholtz
decomposition),
v = vψ+ vϕ=∇ × ψ − ∇ϕ (D.6)
with vector field ψ and scalar field ϕ, and velocity parts vψ =∇×ψ and vϕ=−∇ϕ.
With this, we find
∇ · v = ∆ϕ (D.7)
and thus, using incompressibility,∇ · v = 0, we can calculate ϕ. We thus find the incompressible part of the velocity field
vψ = v− ∇ϕ . (D.8)
We can evaluate this in Fourier space using a spectral method. For a rectangular box aligned with the coordinate system, we thus find that each velocity component vα
is decomposed by sines in one direction and cosines in the other direction. Spatial derivatives convert a sine-description into cosines, and vice versa.
We normalize concentration, length, time and energy by ∆c = c(0)− − c(0)+ , w = 2(κ/b)1/2, t0= w2/Dand eˆ0 = κw(∆c)2/3, respectively.
D.2.2 Numerical details for simulations with hydrodynamics
For the simulation with hydrodynamic flows, Fig. 7.5, we employed the method de- tailed in D.2.1. We choose c(0)+ /∆c = 0, kt0 = 10−2, νt0= 2·10−3and η wˆ3/(t0ˆe0) =
2. Additionally, we use as box-length L/wˆ = 100in all 3 dimensions, number of grid- points in one direction N = 128 and simulation time T /t0 = 4· 103. For the time
step, we start with a time step of ∆t/t0 = 10−4, and double the time step to a final
step size of ∆t/t0= 0.01.
We start with initial conditions R = R0(1 + ϵY2,0). The concentration field at
positions r is initialized by the function
c(r) = c (0) + + c (0) − 2 + c(0)+ − c(0)− 2 tanh d(r) w . (D.9)
where d(r) is the oriented distance of r to the nearest point on the ellipsoid. The value of d(r) is negative for points inside the droplet and positive for points outside.
D.2.3 Numerical details for simulations without hydrodynamics
We employed two methods to solve the dynamic equations without hydrodynamic flows. For the results in Fig. 5.2, Fig. 5.1 and Fig. 5.4, we solved the dynamic equa- tions for the case v = 0 numerically using the xmds2 software package, Dennis et al.
D. Details on the continuum droplet model
(2013) (version 2.2.2), with an adaptive Runge-Kutta scheme of order 4/5, with tol- erance 10−5. For the results in Fig. 5.3, we used the semi-implicit method described in D.2.1, with v = 0. We tested that both methods yield similar results and con- verge for vanishing stepsizes. In both cases, the Laplace operator was evaluated by a spectral method, while the chemical rates were evaluated in real space. Numerical calculations were performed in a finite volume with no flux boundary conditions.
We normalize concentration, length and time by ∆c = c(0)− − c(0)+ , wˆ and t0 = wˆ2/D, respectively, where the characteristic length scale is wˆ = 2(κ/b)1/2. The relevant dimensionless model parameters are c(0)+ /∆c, k±t0, ν±t0/∆c and c±c /∆c.
In all numerical calculations, we chose c(0)+ /∆c = 0and k±t0 = 10−2.
D.2.3.1 Stability diagram
Using three dimensional calculations in Cartesian coordinates, we observed that droplet configurations during the division of isolated single droplets were approximately ax- isymmetric. To determine the stability diagram shown in Fig. 5.2 we therefore per- formed calculations in cylindrical coordinates imposing axisymmetry. We used an axisymmetric cylindrical box with length 60wˆ and radius 30wˆ, discretized with 120 and 60 points, respectively.
The initial conditions were given by a concentration profile that corresponded to a droplet geometry of a slightly prolate ellipsoid with unequal half axes of length
R/wˆ − 0.1 and R/wˆ + 0.1, centered at the box center. The initial droplet size was chosen close to the stationary size in the continuum model. As an estimate for the stationary size we typically chose R/wˆ = 0.9R¯s/w. Here, R¯sis the stationary radius
calculated in the effective droplet model and w = 6β+γ/∆c, see Section 3.2. The
concentration field at positions r was initialized by the function
c(r) = c∞+ c (0) − 2 + c∞− c(0)− 2 tanh d(r) wˆ . (D.10)
where d(r) is the oriented distance of r to the nearest point on the ellipsoid. The value of d(r) is negative for points inside the droplet and positive for points outside. The concentration far from the droplet is c∞= ν+/k++ c(0)+ .
We calculated the dynamics of the concentration field over a time interval T /t0=
104, for different values of ν±t0/∆c. The parameters c±c related to the chemical re-
action were chosen as c+
c /∆c = 0.25and c−c/∆c = 0.75. Because close to the
shape instability the dynamics slows down, we may slightly overestimate the region of stability, since we cannot detect the exact instability with the finite time intervals simulated.
D.2.3.2 Calculations for multiple divisions
Several subsequent divisions break cylindrical symmetry. The calculations shown in Fig. 5.4 were therefore performed in three dimensions using Cartesian coordinates.
D.3. Comparison of droplet deformation in the continuum model and the effective droplet