A simple model for block-copolymer layer morphology formation

Transcription

1 A simple model for block-copolymer layer morphology formation A. J. Wagner and A. Croll, Department of Physics, North Dakota State University, Fargo, ND Prato Workshop, February 12, 2016 A big thanks to the organizers Burkhard and Ravi for this wonderful school and workshop and a special thanks to Burkhard for putting my mind to rest on the correctness of the hydrodynamics description of the fluctuating LB method yesterday.

2 Outline Andrew s experimental system Experimental results Simple model for phase-separation: φ 4 Coupled model for multiple layers Matching experiment and simulation Outlook 1

3 Block co-polymers Block copolymers are made of different large repeating molecules chemically bound together. The different kinds of repeating molecules typically don t like each other and will try to stay out of each other s way. When many of those molecules are brought together, they will arrange in regular structures. If the two molecules of a di-block co-polymer are about the same size they will form lamella. The presence of a surface will typically induce layers parallel to the plates. Block-copolymer made of two different polymers. Block-copolymers arranging themselves next to a surface. 2

4 Incomplete toplayers: holes and/or islands When there is not enough material to make a full top layer, there will be a partial top layer. This top layer has defects, as shown schematically on the right. Schematic view of hole in top layer. This means that the top-layer will not have holes or islands, as is shown in this image of a thin film using an optical microscope. Experimental result for the height of a thin film. 3

5 Temperature depencence of layer height The configuration of the coils of the block co-polymer depend on temperature. Above a temperature of 180 C the coils will mix, and no layers are formed. At lower temperatures the diblock de-mixes and it will form lamella. However, the height of the lamella is slightly temperature dependent, and as the tendency to de-mix increases at lower temperatures, so will the height of lamella. Two blocks mixing Two blocks de-mixing 4

6 Using layer height to change the top morphology If we change the height of the lamellar layers, the amount of material needed for form a perfect layer changes. Near bi-continuous holes This allows us to take a lamella system with a perfect top-layer and by changing the temperature either open up holes or grow islands. Smaller holes 5

7 Similarity to phase-separation When you are familiar with phase-separation you will notice that these patterns are very similar to two-dimensional phase-separation patterns. And even though this system is clearly three dimensional, the similarity of the patterns is uncanny Furthermore it has been shown that a large variety of experimental systems can be described by just a handful of universal models. And these systems are categorized as belonging to one of these universality classes. So it appears reasonable to investigate if a simple phaseseparation model can be used to model this complex system

8 Simplest model for phase-separation (φ 4 model) The simplest model that shows phase-separation relies on a very simple free energy: ( A F = dx 2 φ2 + B 4 φ4 + κ 2 ( φ)2 (1) and in absence of detailed knowledge of the underlying free energy of your system this is a good place to start. For A < 0 this is a double-well potential, and minimizing the free energy can be accomplished by pushing the system into the minima of the free energy. ) F φ Free energy A The κ term accounts for the fact that like regions like to sit next to each other and that there is a cost for interfaces φ Phase Diagram 7

9 Phase-separation dynamics Chemical potential: µ = δf δφ = Aφ+Bφ3 κ 2 φ (2) Simplest Dynamics: model A (e.g. magnetic systems): t φ = Mµ+ξ (3) Next simplest Dynamics: model B (Conserved order-parameter, e.g. binary melt) t φ = (M φ+ ξ) (4) With Hydrodynamics: model H (e.g. binary fluid): t φ+ (φu) = (M φ+ ξ) (5) t u+u u = p+ (σ + ξ) (6) and ξ is a random noise term to allow for fluctuations. 8

10 Numerical method - lattice Boltzmann We will use a lattice Boltzmann method that separates the ideal gas part of the free energy from the non-ideal part. This is a little artificial for the φ 4 model which has equilibrium values of ±φ 0 = ± A/B. The holes, corresponding to φ 0 have a total density of ρ = 0, and the layer has a finite density, say 2φ 0. If we now identify φ = ρ φ 0 then the φ 4 free energy gives these two densities ρ = 0 and ρ = 2φ 0 as equilibrium densities. 9

11 Numerical method - lattice Boltzmann - cont There are many good ways to discretize this equation. Our method of choice is the lattice Boltzmann method: f i (x+v i,t+1) = f i (x,t)+ j Λ ij [f 0 i (φ(x,t)) f i(x,t)]+ξ i +F i (7) where φ(x,t) = i f i (x.t) (8) and {v i } = {(0,0),(1,0),( 1.,0),(0,1),(0, 1)} and if and i f 0 i = ρ; i i F i = 0; f 0 i v i = 0; i i f 0 i v iαv iβ = ρθδ αβ (9) F i v iα = m α µ nid (10) where µ nid is the non-ideal part of the chemical potential. The ξ i are noise terms. This is exactly the same model I derived on Tuesday, but with an added forcing term. 10

12 The noise terms Since the addition of a conservative force does not add any dissipation, the noise terms are just the same as for the simple diffusive system. For D2Q5 we get and m a i = θ θ 1θ 0 0 1θ 1θ 12θ 12θ 12θ 2θ 1 2θ 1 2θ 1 2θ 1 2θ 1 2θ 2θ 2θ 2θ 2θ (11) < ξ a ξ b > ρ = ρ 2τa 1 (τ a ) 2 (1 δa0 δ b0 )δ ab (12) 11

13 Noise current We then get for the noise current η α = τ j v iα ξ i = where N is a random variable with < NN >= 1. i ρ2θ(τ j 1/2)N (13) 12

14 Hydrodynamic limit f i (x+v i,t+1) = f i (x,t)+ j Λ ij (f 0 j f j)+ξ i +F i ( t +v iα α )f i ( t +v iα α ) 2 f i = j Λ ij (f 0 j f j)+ξ i +F i From this we get f i = f 0 i j Λ 1 ij [( t +v iα )f i ξ i F i ] (14) Putting this in the LB equation we get ( t +v iα α )[f 0 i j Λ 1 ij (ξ j +F i )] ( t +v iα α ) j (Λ 1 ij 1 2 δ ij)( t +v iβ β )(f 0 j ) = j Λ ij (f 0 j f j)+ξ i +F i (15) We now want to sum this expression over the conserved quantity ( i1 ) to get our hydrodynamic equation. 13

15 Summation This can be summed over i to give (SRT Λ 1 ij = τ δ ij ) t ρ α (η α +τm α µ nid ) (τ 1 2 ) α β (ρθδ αβ ) = 0 (16) If we define M = (τ 1/2)ρ and set m = [1 1/(2τ)]ρ we can combine this to give with t ρ = α (M α µ(ρ φ 0 )+η α ) (17) < η α >= 0; < η α η β >= 2Mθδ αβ (18) 14

16 Spinodal decomposition and nucleation If the system is nearly homogeneous and residing in the blue shaded region it will phaseseparate into values corresponding to the blue binodal line. A φ If the system is nearly homogeneous and residing in the red shaded region it will only phaseseparate with the help of sufficient noise to help it get over the nucleation barrier. 15

17 Chemical potential and nucleation F µ φ µ(φ) Equilibrium Spinodal Linearizing the equation of motion around φ = ˆφ+δφ, we get t δφ = (M(A+3Bˆφ 2 ) δφ κ 2 δφ+ η) (19) Neglecting the noise and κ we have a diffusion equation with negative diffusion constant inside the spinodal regime. Outside it is a normal diffusion equation and we need noise to get into the region with the effective negative diffusion constant to see phaseseparation φ 16

18 Numerical results - spinodal decomposition Spinodal decomposition works as expected. I won t give many detail here, as this physics is well understood

19 Numerical results - nucleation Outside the spinodal regime the noise will induce nucleation, but the nucleation rate depends on the noise, the time-scale, the spatial scaling (κ) κ=1 κ= κ = 0 shows nucleation 0.0 FFT φ κ = 1 supresses nucleation k 18

20 Scaling arguments The original φ 4 model is scale free, i.e. one can rescale time and length, as well as the order-parameter scale to get an equation of motion where only the noise-amplitude is a relevant parameter: t φ = (M (Aφ+Bφ 3 κ 2 φ)+ ξ) (20) φ = A/Bϕ (21) t ϕ = (M (Aϕ Aϕ 3 κ 2 ϕ)+ B/Aξ) (22) = A/κ ˆ (23) (κ/a 2 M) t ϕ = ˆ (ˆ (ϕ ϕ 3 + ˆ 2 ϕ)+ Bκ/A 2 ξ) (24) t = ˆtMA 2 /κ (25) ˆt ϕ = ˆ (ˆ (ϕ ϕ 3 ˆ 2 ϕ)+ Bκ/A 2 ξ) (26) This allows us to pick a length and time scale to match to the experiments from any simulation without having to worry about re-running the simulation for different parameters (except noise). 19

21 Theory: Spinodal decomposition The theory of spinodal decomposition is well understood: starting from the linearized equation of motion, we can show that inside the spinodal region the system is unstable and will phase-separate at one, well defined, spinodal lengthscale 2κ λ s = 2π A+3Bφ 2 av and the pattern will grow with and amplitude of ( m(a+3bφ 2 Amp(t) = exp av ) 2 t 4κ (This is assuming that noise is irrelevant here). ) (27) (28) 20

22 Theory: Phase-ordering A typical spinodal decomposition morphology is assumed to obey the Dynanmic Scaling Hypothesis, which stats that morphologies at later times look exactly like the earlier morphologies (statistically), but with a re-scaled length-scale L(t). This length-scale obeys L(t) t α (29) By looking at our equation of motion we can make the dimensional argument (neglecting noise again) φ T = 1 1 LL µ (30) φ T = 1 11 LLL φ (31) L t 1/3 (32) so that α = 1/3 for conserved order parameter systems with diffusive dynamics. 21

23 Simulation output

24 Matching simulations and experiments Firstly, we identify the orderparameter φ with the layer height. A simple shift lets us vary φ between 0 (hole) and 2 (layer at equilibrium height). The change in temperature changes the equilibrium height, and we have a layer of a height φ < 2. Depending on the depth of the quench (i.e. the value of φ) we expect to find spinodal decomposition or nucleation. There is a time involved in the initial separation of the material, then the structure will start to coarsen. 23

25 What the experiments provide Both simulation and experiment provide us with a sequence of patterns showing two distinct time-domains: the initial phase-separation and the subsequent phase-ordering. The first parameter to match is the final volume fraction. (This is difficult experimentally since the volume fraction changes due to transfer between layers). The second parameter to match is the intial length scale. The third parameter to match is the time-scale from growth. Lastly the noise amplitude should be matched. (This should be irrelevant for spinodal decomposition, but could matter for nucleation. However, since there is an exponential decrease in the nucleation rate as you move away from the spinodal, the increase in noise can be compensated for by a small change in average density.) 24

26 Matching the results to experiments This can be done for each simulation/experiment pair separately, but for this to give a predictive model, the scaling parameters should depend in a predictable way on the experimental conditions. What can be changed in the experiments: Temperature: both the initial and final temperatures have been varied. There are two limiting temperatures, the melting temperature for the top layer, which also depends on the number of layers, and the glass transition temperature below which the dynamics becomes so slow a to render experiments impractical. Number of layers: this changes the thermodynamics, as additional layers have lower melting temperatures. There is also a transfer of material between adjacent layers, which complicates the dynamics at later times. 25

27 Experiment: what can be measured? Essentially what we have is a sequence images of the phase-separating system. From this image we easily obtain a Fourier spectrum, which will show a peak at some lengthscale that growth during the decomposition stage, and during the phase-ordering phase the peak position moves to larger length-scales. More directly we can examine the growth of a single hole, where the area grows linearly in time (both in experiments and in simulations). We can also observe the nucleation rate, i.e. how many new domains are formed as a function of time. Lastly the change of the volume fraction at late times can be observed as a measure of the transfer between layers 26

28 Matching: morphology Experiment Simulation Nucleated morphologies not too far from the spinodal line compare well to the observed structures in the membrane. Requires matching of length-scale, note that this is the observed morphology after nucleation is complete. 27

29 g(r) (a.u.) Matching: Correlation function g(r) (a.u.)1.5 r ( m) r (Lattice Sites) Correlation functions show that there the droplets in the previous morphologies are uncorrelated, so they are not the result of spinodal decomposition, but of nucleation. Length-scales match, giving a somewhat independent confirmation of the choice of lentgh-scale. 28

30 Area (Lattice Sites 2 ) Area ( m 2 ) Growth of a hole Time (s) Timesteps (a.u.) Both theory and experiment show linear growth at the same rate for the area of a single hole. Even the eventual reduction in growth due to surrounding new domains is comparable. There determines the time-scale. 29

31 Num Holes/ Area ( m -2 ) Matching: Time evolution, number of domains Experiment Simulation time (s) Now that volume fraction, length- and timescales have been determined, there are no more free parameters. We can now compare the number of domains as a function of time as an independent test on whether the matching was successful. 30

32 g(r) (a.u.) g(r) (a.u.) Other Morphologies: Correlation functions Experiment r ( m) 10.5 L 6.5 L 4.5 L Theory r (a. u.) = 1.3 = 1.55 =

33 Multi-layer version So far the theoretical side has been fairly standard, although there is more work on spinodal decomposition than on nucleation and the fluctuating lattice Boltzmann method here is new. To utilize this method we want to include all the layers that have previously been neglected by writing a free energy F = ( Ai dx i 2 φ2 i + B i 4 φ4 i + κ ) i 2 2 φ i +coupling term (33) and the dynamics will contain a transfer between layers according to the difference in local chemical potentials. 32

34 Summary We introduced a new fluctuating lattice Boltzmann method for Cahn-Hiliard dynamics. I presented the matching of this simple fluctuating φ 4 model to our experimental system to the nucleation of holes in a block-copolymer multi-layers system. The match works well and verifies our assumption of simple diffusive dynamics. There are equivalent results for islands (not shown). We intend to extend this analysis to systems where several layers are involved in the dynamics, i.e. if we quench an already partially separated system. 33