Supporting information for: A multi-fluid model for microstructure. formation in polymer membranes

Transcription

1 Electronic Supplementary Material ESI for Soft Matter. This journal is The Royal Society of Chemistry 2017 Supporting information for: A multi-fluid model for microstructure formation in polymer membranes Douglas R. Tree, Kris T. Delaney, Hector D. Ceniceros, Tatsuhiro Iwama, and Glenn H. Fredrickson, Materials Research Laboratory, University of California, Santa Barbara, CA Department of Mathematics, University of California, Santa Barbara, CA Asahi Kasei Corporation, 2-1 Samejima, Fuji, Shizuoka Japan Materials Research Laboratory, Materials Department and Chemical Engineering Department, University of California, Santa Barbara, CA drt@mrl.ucsb.edu Obtaining Transport Equations from the Rayleighian In this section, we show how the Rayleighian specified in the main manuscript produces the transport equations in Equations Substituting Equations 11, 12 and 17 into Eq. 6 gives the fully specified Rayleighian for the ternary multi-fluid model, R[{v i }] = [ p,n,s dr i ζ i 2 v i v m σv : v p,n p v + φ i v i µ i ]. i 1 S1

2 Component force balances are given by stationary values of the functional derivatives of the Rayleighian i.e., the Euler-Lagrange equations, δr = R R = 0 2 δv i v i v i where R is the integrand of the Rayleighian. Taking these functional derivatives and substituting Equations 4 and 14 to eliminate v and v m yields, δr = ζ pζ n δv p ζ δr = ζ pζ n δv n ζ δr = ζ pζ s δv s ζ v s v p + ζ nζ s ζ v s v n φ s σ v + φ s p. 5 v p v n + ζ pζ s ζ v p v s φ p σ v + φ p p + φ p µ p 3 v n v p + ζ nζ s ζ v n v s φ n σ v + φ n p + φ n µ n 4 The total momentum equation is readily obtained by summing Equations 3 through 5, p,n 0 = p + σ v φ i µ i. 6 This equation can be more compactly written by introducing the osmotic stress tensor, S1 S4 p,n Π = φ i µ i 7 which gives 0 = p + σ v Π. 8 i i Component diffusion equations can be obtained by combining Equations 3 through 5 with the conservation of mass expression in Eq. 2. We define the diffusive flux of component i as, j i = φ i v v i 9 S2

3 which allows us to re-write Eq. 2 as, φ i t + v φ i = j i. 10 Substituting the definition of the friction coefficient from Eq. 15 into Equations 3 5 and using the definition of diffusive flux gives, [ ] j p = v 0 ζ0 1 φp σ v + φ p p + φ p µ p [ ] j n = v 0 ζ0 1 φn σ v + φ n p + φ n µ n j s = v 0 ζ 1 0 [ φs σ v + φ s p ] 13 To isolate the diffusive flux, we eliminate the pressure terms from Equations by performing the operations, j p φ n + φ s φ p j n + j s 14 j n φ p + φ s φ n j p + j s 15 and use the fact that i becomes, p,n,s φ i = 1 and ii i p,n,s j i = 0. With these manipulations, Eq. 10 i φ p t + v φ p = v ] 0 [φ p 1 φ p µ p φ p φ n µ n ζ 0 φ n t + v φ n = v [ ] 0 φ p φ n µ p + φ n 1 φ n µ n ζ which are more compactly written matrix notation as φ i t + v φ i = p,n M ij µ j j=1 18 S3

4 where we have introduced the component mobility matrix M pp = v 0 ζ 0 φ p 1 φ p 19 M pn = M np = v 0 ζ 0 φ p φ n 20 M nn = v 0 ζ 0 φ n 1 φ n. 21 The gradient of the chemical potential appears in both Equations 8 and 18. Taking the requisite functional derivatives per the definition in Eq. 10 yields, µ i = k BT v 0 f0 κ i 2 φ i φ i 22 where f 0 / φ i is the derivative of the homogeneous free energy in Eq. 8 with respect to volume fraction of component i. Taking the gradient and using the chain rule gives a chemical potential gradient with explicit volume fraction terms, µ i = k BT v 0 p,n j [H ij φ j K ij 2 φ j ] 23 where H pp = N p φ p 1 + N s φ s 1 2χ ps 24 H pn = H np = N s φ s 1 + χ pn χ ns χ ps 25 H nn = N n φ n 1 + N s φ s 1 2χ ns 26 is the Hessian matrix, and K pp = κ p 27 K pn = K np = 0 28 S4

5 Table S1: Characteristic Scales Scale Expression Description R bn 1/2 r end-to-end distance of the reference polymer τ N 2 r η r v 0 /k B T Rouse time of reference polymer p η r /τ viscous stress scale K nn = κ n 29 is the matrix of gradient coefficients. Substituting Eq. 23 into Equations 8 and 18 gives, p + σ v = k BT v 0 p,n i,j j,k φ i Hij φ j K ij 2 φ j 30 [ φ i t + v φ i = k BT p,n ] M ij Hjk φ k K jk 2 φ k. 31 v 0 With a fully specified model, we seek to write Equations 30 and 31 in dimensionless form. We define the characteristic length scale to be R = bn 1/2 r 32 the end-to-end distance of a reference polymer of length N r. The characteristic time scale is set to the Rouse time in a solution with a reference viscosity η r, τ = N 2 r η r v 0 k B T 33 and the characteristic pressure scale is set to be p = η r /τ. 34 These characteristic scales are summarized in Table S1. S5

6 Using the definitions of R, τ and p, the momentum equation from Eq. 30 becomes, p + σ v = k BT v 0 τ η r =N r p,n i,j p,n i,j φ i H ij φj K ij R 2 2 φ j φ i [ N r H ij φ j N r R 2 K ij 2 φ j ] 35 where = R, p = p/p and σ v = σ v /p. Similarly, the diffusion equation becomes, φ i t + ṽ φ i = k BT = τ R 2 v 0 { p,n [ p,n j,k M ij H jk φk R 2 K jk 2 φ k ] [ ηr b 2 M ij N r H jk φ k ] } N r R 2 K jk 2 φ k 36 j,k where t = t/τ, ṽ = vτ/r. The non-dimensionalization procedure suggests that we define a dimensionless gradient coefficient κ i = N r R 2 κ i, a dimensionless mobility matrix M ij = η r b 2 M ij and a re-scaled Hessian matrix H ij = N r H ij, which appear in Equations Finally, note that the Peclet number, which would normally appear before the diffusion term in Eq. 36 is equal to unity. This is a consequence of the fact that all of the convective flows in the system are internally generated. If there had been boundary conditions prescribing alternate length and time scales, it would have been possible to obtain Pe 1. However, the capillary number does appear in Eq. 35. Here there are two energy scales; surface energy is characterized by k B T and viscous energy by η r R 3 /τ. This means that the capillary number Ca = N 1 r 37 is fixed by the reference degree of polymerization and is a consequence of the way that the surface forces and viscous forces scale with the reference length scale. At the monomer length scale, N r = N s 1 and Ca 1. However, zooming out to the scale of the polymer radius of gyration, N r = N p, Ca = N 1 p and the surface tension forces are much stronger. S6

7 Ternary Phase Diagram For fixed N i and χ ij, the homogeneous free energy in Eq. 8 can be used to produce a ternary phase diagram consisting of i a binodal, ii a spinodal and iii a critical point. The binodal curve is typically obtained by equating the chemical potentials of each species in both phases. Because f 0 is an intrinsic quantity, it is convenient to use an equivalent formalism where we equate the exchange chemical potentials µ i = f 0 φ i 38 and the osmotic pressure Π = f 0 i φ i µ i 39 in each phase. S5 Substituting Eq. 8 into Equations 38 and 39 and assuming pseudo-binary parameters, gives 1 φ α N ln p 1 φ α p φ α n ln + χ φ α φ β p 1 φ β p φ β n φn β = 0 40 n φ α ln n 1 φ α p φ α n ln + χ φ α φ β n 1 φ β p φ β p φp β = 0 41 n 1 φ α p φ α n ln + φ α 1 φ β p φ β p φ β p 1 1 χ φ α p φ α n φ β n N pφn β = 0 42 With four unknowns φ α p, φ α n, φ β p, φ β n, these three equations define the binodal curve in composition space for a given χ and N. The spinodal is calculated by setting the determinant of the Hessian matrix in Eq. 23 to zero, Z H pp H pn H nn H np = 0 43 S7

8 and the critical point is given when both Eq. 43 and Z φ p Z φ n H nn H np = 0 44 are satisfied. S6 S8 Both equations are straightforward to generate, but too long to conveniently reproduce here. Without further approximation, the solution to all three equations require numerical methods. For this paper, we compute the phase diagram using a custom code in Python. We find the critical point by simultaneously solving Equations 43 and 44. We then solve Eq. 43 from φ s = 0 to the critical point, resolving the two branches of the spinodal. Finally, we compute the two branches of the binodals by first computing the phase co-existence point for a binary system φ s = 0, and then solve Eq marching φ α p from the polymer-rich solution of the binary problem to the critical point. Unlike Yilmaz and McHugh, S7 we did not need to resort to a least-squares method to avoid the trivial solution. Rather, we used scipy s optimize.fsolve function with an analytical Jacobian and a first-order continuation method to obtain initial guesses. Theory of a 1D interface The concentration profiles for our model can be found by minimizing the interfacial free energy, γ[φ p, φ n ] = F [φ p, φ n ] k BT V dr f eq φ p, φ n 45 of a two phase system where the α-phase {φ α p, φ α n} at the left boundary, x =, is in equilibrium with the β-phase {φ β p, φ β n} at the right boundary, x =. F [φ p, φ n ] is the free S8

9 energy functional given in Eq. 7 and f eq φ p, φ n = f 0 φ α p, φ α n + φ p φ α p µ p φ α p, φ α n + φ n φ α nµ n φ α p, φ α n 46 is the equilibrium free energy absent any interfaces. Note that the choice of the α-phase for Eq. 46 is arbitrary and the µ i that appear are the exchange chemical potentials. There are two barriers to obtaining analytical results for the interfacial profiles described by Eq. 45. The first is that γ[φ p, φ n ] describes the profiles of both φ p r and φ n r, requiring a simultaneous solution to two Euler-Lagrange equations. To avoid this, we assume that the solvent concentration is constant, φ s φ s, making the model truly binary φ = φ p, φ n = 1 φ φ s. In general, the solvent concentration is not truly constant between both phases, since the solvent may partition unequally as well as concentrate at the polymer/nonsolvent interface. However, this effect is small enough to neglect in the strong segregation limit, which we justify below. Assuming a binary system, and using the notation for the pseudo-binary parameters given above, we re-write Eq. 45 for a 1D interface as, γ[φ] = k BT b 2 dx b [ fφ + K φ ] 47 where fφ = f 0 φ f 0 φ α φ φ α µφ α 48 K φ = κ φ 2 49 and f 0 φ = φ N ln φ + 1 φ φ s ln 1 φ φs + φs ln φ s + χφ 1 φ φ s 50 S9

10 with µ = f 0 / φ. Eq. 47 has a Lagrangian form, which immediately implies that [ ] 1/2 fφ φx = 51 κ The second barrier to progress is the appearance of logarithmic terms in Eq. 50. To surmount this barrier, we assume that the phases are strongly segregated χ 1 and N 1 and use perturbation theory. S5 Expanding f 0 φ in this limit allows us to find the equilibrium concentrations, φ α and φ β. Equating exchange chemical potentials, µ α = µ β, and osmotic pressures, Π α = Π β, for the pseudo-binary model in both phases gives, 1 ϕ α χn ln 1χ 1 ϕ α ϕ ln 2 1 β 1 ϕ φ β s ϕ α ϕ β = ϕ α χ ln 1 ϕ β + ϕ α ϕ β 1 χ χn φ [ s ϕ α 2 ϕ β 2] = 0 53 where we have introduced the reduced volume fraction ϕ ν = φ ν / 1 φ s 54 such that ϕ [0, 1] for ν [α, β]. Assuming that ϕ α ɛ α and ϕ β 1 ɛ β, Eq. 52 and Eq. 53 reduce to, 1 χ 1 ln ɛ α + ɛ β + 1 ɛ α + ln ɛ β 2 1 χn χ φ s ɛ α + ɛ β 1 = 0 55 ɛ α + ln ɛ β + ɛ α + ɛ β 1 1 χ 1 χn + 1 φ s [ ɛ α 2 1 ɛ β 2] = 0 56 where we have used the Taylor expansion ln1 ɛ ɛ when appropriate. The largest term in Eq are of Oln ɛ, so we can neglect terms of Oɛ. Doing so and re-arranging S10

11 gives, ɛ α = exp [ χn1 φ s ] 57 ɛ β = exp [ χ1 φ s ] 58 to leading order when χ 1 and N 1. It can be shown using a similar perturbation expansion of the phase equilibrium equations for the full ternary model, that the solvent concentration is constant at Oln ɛ, justifying the pseudo-binary assumption. Fully solving Eq. 51 for the interfacial profile requires numerical methods, but, we can obtain an expression for the interfacial width regardless. A characteristic interfacial width can be defined as, S5,S9 l = φβ φ α φx = 0 59 where we can substitute the right hand side of Eq. 51 for φ. Doing so requires an expression for f 0, which can be obtained by combining Eq. 48 and Eq. 50, f 0 φ = φ φ N ln + 1 φ φ φ 1 φ φs α s ln 1 φ α φ s N φ φ α χ φ φ α 2 60 Assuming that ϕ = 1/2 at x = 0, and using the binodal curves described by Equations 57 and 58 gives the width in Eq. 52, where terms smaller than O 1 1 and O have been χ χn neglected. S11

12 Linear Stability Analysis We begin by neglecting hydrodynamics set v = 0 everywhere and linearizing Eq. 18 about the homogeneous state { φ p, φ n }, t δφ i = j,k M ij Hjk 2 δφ k K jk 4 δφ k 61 where δφ i = φ i φ i, and M ij and H jk indicate constant matrices evaluated at φ p and φ n. Taking the Fourier transform gives, t δ ˆφ i = j,k Mij Hjk q 2 M ij K jk q 4 δ ˆφ k 62 which is useful, since the derivative operators can be simply expressed in Fourier space. Eq. 62 is a system of linear, first order differential equations, which has a formal solution in Fourier-space of δ ˆφ i q, t = k exp [A ik qt] δ ˆφ k q, 0 63 where A ik q = j Mij Hjk q 2 + M ij K jk q 4 64 From Eq. 63 it is clear that ˆφq, t becomes unstable thus precipitating spinodal decomposition when A ik q is no longer negative definite. By definition, A ik q is non-negative definite when one eigenvalue is greater than or equal to zero for some value of q. This can be made explicit by using a similarity transform of A ik, giving δ ˆφ i q, t = j,k,l X ij exp [Λ jk t] X 1 kl δ ˆφ l q, 0 65 S12

13 where Λ jk = λ λ and X ij are the eigenvalue and eigenvector matrices of A ik q. Since A ik q is rank two, there is a simple formula for the eigenvalues λ ± = T 2 ± T 2 4D 1/ where T = tr[a ik q] and D = det[a ik q]. By inspection, T is always less than zero. Therefore, λ + can only be positive when D < 0, making D = 0 the stability criteria. We can decompose this determinant into two terms: one that depends on the mobility and one that depends on the equilibrium parameters alone, det[a ik q] = det M ij q 2 det H jk + K jk q 2 68 It can be shown and it is also physically intuitive that the mobility matrix, Mij, is always positive definite, and therefore it is only necessary to consider det H jk + K jk q 2 = 0 69 for the stability analysis. This is re-assuring, since kinetics should not determine thermodynamic stability. Re-writing Eq. 69, H pp + κ p q 2 Hpn = 0 70 H np Hnn + κ p q 2 highlights its relationship with Eq. 43, proving that the preceding analysis generates the spinodal curve when κ p and κ n are zero. When κ p and κ n are non-zero, a finite size system is stabilized by the gradient free energy until a critical wavelength, 2π/q c, above which the S13

14 system becomes unstable. The critical wavenumber is given by solving Eq. 70, which is a bi-quadratic equation in q. Using the quadratic equation yields, Hpp qc 2 = + H [ nn Hpp ± H 2 nn + 4 H pn 2κ p 2κ n 2κ p 2κ n 2κ p Hnp 2κ n ] 1/2. 71 Once unstable, the spinodal decomposition is dominated by the fastest growing mode, q m, which is the mode that maximizes λ + q. Using its definition in Eq. 67 and the definition of the diffusivity matrix D ij and the gradient matrix B ij in Equations 56 and 28, we can get an expression for λ + in terms of D ij and B ij, λ + = q2 2 [ Dpp + D nn + B pp + B nn q 2] 72 + q 2 [ {Dpp D nn + B pp B nn q 2} D pn + B pn q 2 D np + B np q 2 ] 1/2 74 The fastest growing mode can be found by solving dλ + dq = 0 75 q=qm Unfortunately, there is no convenient closed-form expression for q m for general parameters, like there is for q c. However, Eq. 75 can be re-written in a more convenient form for numerical root-finding λ + dt dq dd dq = 0 76 which is valid so long as T 2 4D. Additionally, if we assume that the mobility and gradient matrices are scalars M ij = Mδ ij and K ij = κδ ij, then Eq. 72 is greatly simplified. In this case the fastest growing mode becomes, q 2 m = q2 c 2 77 S14

15 which is equivalent to the relation between q m and q c for binary systems. S10 Finally, note that in addition to q m, one also obtains the rate of spinodal decomposition from the preceding analysis. The rate is given by λ m λ + q m 78 and has units of inverse time. For the case of scalar mobility and gradient matrices, one obtains the simple expression λ m = Mκq 4 m 79 Dynamic scaling To ensure that the coarsening exponent is not a simulation artifact, we verify in Figure S1 that the polymer structure factor, Sq, and polymer density correlation function, gr, obey dynamic scaling at long times. Dynamic scaling requires the system to be self-similar, and therefore the structure factor and the density correlation function should both collapse to universal curves when scaled by some characteristic length, L. The structure factor has units of length squared and the correlation function is dimensionless, so L 2 SLq and gr/l should be universal at long times. Figure S1a shows that the structure factor for {φ p, φ n } = {0.3, 0.65} blue circles in Figures 6 and 7 scaled by q 1 for t 4096 collapses to a universal curve. Figure S1b shows a similar collapse for the real-space correlation function, gr/r g where R g is the location of the first zero of the unscaled gr. The inset to Figure S1b shows the product of R g and q, which approaches a steady value as t. The latter plot demonstrates that because the system is self-similar, either q or R g can be used as a characteristic length. S15

16 a q 2 sq, t t = 4096 t = 8192 t = t = t = b gr, t Rg q t q/ q r/r g Figure S1: a Polymer structure factor versus wavevector, scaled by the first moment q. b Polymer density correlation function versus 1 radial distance, r, scaled by R g, the first zero of gr. inset to b Product of R g and q as a function of time, showing the equivalence of each measure of the domain size to within a constant value at long times. Data are from the one of the sets presented in Figures 6 and 7 closed blue circles, diffusion only with parameters: N = 20, χ = 0.973, κ = 2, {φ p, φ n } = {0.3, 0.65}. References S1 M. Doi and A. Onuki, J. Phys. II France, 1992, 2, S2 M. Cromer, M. C. Villet, G. H. Fredrickson, L. Gary Leal, R. Stepanyan and M. J. H. Bulters, J. Rheol., 2013, 57, S3 S. T. Milner, Phys. Rev. E, 1993, 48, S4 H. Tanaka, J. Phys.-Condens. Mat., 2000, 12, R207. S5 D. Broseta, G. H. Fredrickson, E. Helfand and L. Leibler, Macromolecules, 1990, 23, S6 D. Broseta and G. H. Fredrickson, J. Chem. Phys., 1990, 93, S7 L. Yilmaz and A. J. McHugh, J. Appl. Polym. Sci., 1986, 31, S8 J. W. Gibbs, in The Scientific Papers of J. Willard Gibbs. Vol 1. Thermodynamics, Longmans, Green and Co., 1906, ch. On the Equilibrium of Heterogeneous Substances. S16

17 S9 J. W. Cahn and J. E. Hilliard, J. Chem. Phys., 1958, 28, S10 J. W. Cahn, J. Chem. Phys., 1965, 42, 93. S17