Patterning Soft Materials at the Nanoscale:

Transcription

1 Lecture 9 Patterning Soft Materials at the Nanoscale: Phase-field methods Recommended reading: S.C. Glotzer and W. Paul, Annual Reviews of Materials Research 32 ( ), 2002 and references therein (e.g. Castellano and Glotzer, Lee, Douglas and Glotzer, Glotzer, DiMarzio and Muthukumar) Larson book, pp389 1

2 Phase Field Methods Phase field methods are continuum methods. Partial differential equations solved by finite difference methods. Key variable is one or more fields that depend on space and time (like a concentration field or density field). Typically based on a free energy formalism where free energy is a function of the field variable and where free energy is minimized to find equilibrium values of the field. First phase field method: TDGL/Cahn-Hilliard. 2

3 Phase separation and nanotechnology We have seen that patterning materials on the nanoscale can often be achieved by exploiting the process of phase separation. Templating surfaces for positioning of nanodevices. Creating nanostructured surfaces for catalysis. Creating photonic crystals. Manipulating arrays of nanoparticles into nanowires, etc. Producing layered nanostructures of various shapes and sizes. 3

4 Phase Separation ( Unmixing ) When quenched to the immiscible region of the phase diagram, a mixture will separate into two coexisting phases with equilibrium concentrations given by the values of φ 1 and φ 2 on the binodal. The dynamics of the separation process is controlled by The final quench temperature T f the overall composition φ the rate of the quench dt/dt the viscous or viscoelastic properties of the phases formed and the interfacial tension Γ between the two phases 4

5 Phase Separation Mechanisms Nucleation and growth Metastable - stable with respect to small amplitude variations in composition, but unstable with respect to large variations. Mixture will phase separate via nucleation and growth. Phase separation occurs only after a nucleus of one phase forms that is large enough to grow. Spinodal decomposition Unstable with respect to small amplitude variations in composition. Mixture will phase separate via spinodal decomposition. Phase separation begins immediately if variations are characterized by a long enough wavelength. 5

6 Phase Separation Mechanisms f(φ) B A C single well double well T > T c T < T c In metastable region, 2 f/ φ 2 > 0. Thus, if a small patch of homogeneous fluid at A suffers a weak compositional fluctuation, thereby producing two patches with compositions corresponding to B and C, then the ave free energy will increase, which is unfavorable, so fluid will remix. φ 1 φ s1 φ s2 φ 2 φ 6

7 Phase Separation Mechanisms f(φ) E D F single well double well T > T c T < T c In unstable region, 2 f/ φ 2 < 0. Thus, if a small patch of homogeneous fluid at D suffers a weak compositional fluctuation, thereby producing two patches with compositions corresponding to E and F, then the ave free energy will decrease, which is favorable. Thus weak fluctuations grow spontaneously. φ 1 φ s1 φ s2 φ 2 φ 7

8 The Kinetics of Phase Separation by Spinodal Decomposition Recall Fick s Laws Fick s first law relating flux to a concentration gradient: Continuity equation: j i (x,t) = "D i #$ i "# i (x,t) "t = $% & j i (x,t) Fick s second law = Fick s first law + continuity: "# i (x,t) "t = D i $ 2 # i 8

9 The Cahn-Hilliard Cook Model The CHC model is a non-linear uphill diffusion or mass balance equation for an immiscible mixture involving the free energy of the mixture. Main steps in derivation: 1. Generalized transport equation relating flux (= mass current density) to gradient in chemical potential for each species. 2. Continuity equation relating the spatial and time dependence of the concentration of each component in the mixture to the flux that expresses the conservation of mass. 3. Combine the two. 9

10 The Cahn-Hilliard Cook Model The mass current is related to the chemical potential µ k through: n"1 j i (x,t) = " $ M ik #µ k + j T (x,t) where M ik is the mobility of species i due to k and j T (r,t) is the mass current arising from thermal noise. Conservation of mass requires: k "# i (x,t) "t = $% & j i (x,t) 10

11 The Cahn-Hilliard-Cook Model The chemical potential µ k is related thermodynamically to the free energy functional F{φ i (x,t), φ j (x,t), φ k (x,t),, φ n-1 (x,t)} by: µ i = "F "# i where n is the total number of species and the derivative is taken subject to the constraint φ i + φ j + φ k + φ n = 1. For an incompressible, binary A/B mixture where φ A + φ B = 1, there is only one independent concentration φ, one independent chemical potential µ, and one independent continuity equation. 11

12 The Cahn-Hilliard-Cook Model Thus for an incompressible, binary mixture, we have: j(x,t) = "M#µ + j T (x,t) "#(x,t) "t = $% & j(x,t) µ = "F "# (1) (2) (3) 12

13 The Cahn-Hilliard-Cook Model Combining the three equations gives: "#(x,t) "t ( = $% & $M% 'F '# + j + * T (x,t)- ), µ which is usually written as: j "#(x,t) "t = $ % M$ &F &# + '(x,t) Cook term 13

14 The Cahn-Hilliard-Cook Model The noise term satisfies the conditions "(x,t) = 0 and "(x,t)" (x',t') = #2Mk B T$ 2 %(x # x')%(t # t') where the average is performed over the thermal fluctuations in the system. 14

15 The Cahn-Hilliard-Cook Model If the mobility M is independent of concentration φ, then: "#(x,t) "t = M$ 2 %F %# + &(x,t) If the free energy functional takes a particular form, this equation is known as the Cahn-Hilliard-Cook equation. Often, the noise term is neglected, since it does not affect scaling behavior of domain growth, in which case the equation is called the Cahn-Hilliard equation. 15

16 Form of CH Free Energy Functional The Cahn-Hilliard free energy functional originally proposed for binary mixtures has the following form: Bulk free energy of mixing F CH ["(x)] k B T % = dx f ( " ) k B T + # 2 $"(x) ( + ' 2 * & ) Square gradient term : describes the energy penalty for interfaces between A-rich and B-rich domains. 16

17 Form of CH Free Energy Functional For small molecule mixtures, the interfacial energy parameter κ arises due to increases in potential energy at an interface and is related to the interaction energies by κ = χλ 2, where λ is the effective interaction distance between molecules, and " = z % ' k B T # $ 1 AB 2 # + # AA BB & ( ) For polymer blends, κ has an additional contribution due to entropy penalties at the interface. ( * ) 17

18 Landau Theory: The TDGL Model Recall, the Helmholtz free energy of mixing of an incompressible binary mixture is f = U - TS: f (") k B T where = " ln" + (1# ")ln(1# ") + $"(1# ") $ = z & ( k B T % # 1 AB 2 % + % AA BB ' ( ) ) + * 18

19 Taylor Expansion of Bulk Free Energy Close to φ c = 1/2, the free energy expression can be approximated via a Taylor expansion by the form: ψ is the order parameter f (") k B T # $ r 2 " 2 + g 4 " where " % & A $ & B = & $ (1$ &) = 2& -1, r = 1 2 (' $ ' c) and g = 1 3 Note that this form (with only even terms) holds only for symmetric mixtures. With higher order terms neglected, this ψ 2,ψ 4 form of the free energy is called the Landau free energy. 19

20 Landau Free Energy Note that when χ - χ c changes sign, f(ψ) changes form: f (") k B T # $ r 2 " 2 + g 4 " where " % & A $ & B = & $ (1$ &) = 2& -1, r = 1 2 (' $ ' c) and g = 1 3 Thus the form of f will be different below and above χ c. 20

21 Free Energy ψ 1, ψ 2 defined by: f(ψ) single well χ < χ c $ "f ' $ & ) = "f ' & ) %"# (# 1 %"# ( # 2 This schematic depicts the free energy for a symmetric mixture. double well χ > χ c ψ s1, ψ s2 defined by: 2 f/ ψ 2 = 0 For every χ > χ c, there exists unique values of ψ 1, ψ 2, ψ s1, and ψ s2. ψ 1 ψ s1 ψ s2 ψ 2 ψ 21

22 Free Energy f(ψ) χ < χ c ψ 1, ψ 2 defined by: $ "f ' $ & ) = "f ' & ) %"# (# 1 %"# ( # 2 This schematic depicts the free energy for an asymmetric mixture. single well double well χ > χ c ψ s1, ψ s2 defined by: 2 f/ ψ 2 = 0 For every χ > χ c, there exists unique values of ψ 1, ψ 2, ψ s1, and ψ s2. ψ 1 ψ s1 ψ s2 ψ 2 ψ 22

23 Example of Asymmetric Free Energy f (") k B T = " N A ln" + (1# ") N B ln(1# ") + $"(1# ") " c = # c = N B ( N A + N ) B ( N A + N ) 2 B 2N A N B 23

24 Phase Diagram T~1/χ spinodal one phase (miscible) unstable UCST binodal or coexistence curve two-phase (immiscible) metastable For each χ > χ c, plot ψ 1, ψ 2, ψ s1, and ψ s2. This defines boundaries between (i) a miscible and immiscible region and (ii) a metastable and unstable immiscible region. ψ 1 ψ s1 ψ s2 ψ 2 ψ 24

25 Time-dependent Ginzburg-Landau Equation Recall, the CHC equation is: "#(x,t) "t F CH ["(x)] k B T = M$ 2 %F CH %# + & (x,t) % = dx f ( " ) k B T + # 2 $"(x) ( + ' 2 * & ) The TDGL equation is the CH (or CHC) equation with the Ginzburg-Landau form of the free energy functional: F GL ["(x)] k B T & = dx # r 2 " 2 + g 4 " 4 + $ 2 %" 2 ), ' ( * + 25

26 TDGL The functional derivative δf/δψ is given by: "F GL [#(x)] "# = $r# + g# 3 $%& 2 # Then, the TDGL equation becomes Usually set to zero "#(x,t) "t = M ( $r% 2 # + g% 2 # 3 $&% 4 #) + '(x,t) 26

27 Dimensionless TDGL Through the following change of variables: it s possible to rewrite the TDGL equation in a dimensionless form: "#(x,t) "t x " ( r #) 1 2 x t " ( Mr 2 #)t $ " ( g r ) 1 2 $ = $% 2 # + % 2 # 3 $ % 4 # + &'(x,t) Depth of quench controlled entirely by noise term. 27

28 Dimensionless TDGL In the dimensionless equation, the noise term satisfies the following conditions: "(x,t) = 0 and "(x,t)"(x',t') = #$ 2 %(x # x')%(t # t') Then the only parameters left to specify the dynamics are the conserved average concentration ψ 0 = <φ> - φ c and the dimensionless noise strength parameter ε: ε usually set to zero. ( ) " = 2g # d 2 ( r 4$d ) 2 χ and χ c are contained in κ and r! 28

29 Three Dynamical Regimes of Phase Separation Early-time (linear) regime Immediately following a quench to the unstable, two-phase region Concentration fluctuations of wavelength λ grow or decay exponentially in time at constant λ. Intermediate-time ( late -time) regime When average domain size is larger than the interfacial width Domains coarsen and average size grows as a power law in time Very late time (hydrodynamic) regime (fluids only) At very late times hydrodynamic interactions dominate the coarsening process Domain size grows linearly with time 29

30 Dimensionless TDGL From the dimensionless TDGL, studies have shown there is only one fixed point for this model at T=0. That is, all scaling behavior describing how the domains grow in time is T-independent (χ-independent). True both in d=2 and d=3. R " t 1/ 3 The parameters that were scaled out of this equation describe the speed at which the process will occur at a given T (or χ), but this affects only amplitudes in front of scaling laws. R = A(",M,#)t 1/ 3 30

31 Solving the TDGL Equation Often, the noise term is neglected in studies of SD, so that the TDGL is: "#(x,t) "t = $% 2 # + % 2 # 3 $ % 4 # This equation follows the time evolution of the order parameter everywhere in space following a quench from the one-phase to the two-phase region. To solve it numerically, we must turn the differential equation into a finite difference equation, and numerically integrate the resulting equation of motion for ψ. 31

32 Finite Difference Equation for TDGL First, space is discretized by replacing the continuous space of position vectors x by a simple square (2d) or cubic (3d) grid with L 2 or L 3 sites, respectively, and mesh size δx. We ll consider 2-d for simplicity; generalization to 3-d is straightforward. The order parameter field (or composition difference field) ψ(x,t) then becomes ψ ij m at the m th time step at grid point i,j. Time is discretized by replacing the continuous time variable t by a series of m discrete time steps of duration δt. 32

33 Finite Difference Equation for TDGL The TDGL may be numerically integrated using a firstorder Euler finite difference scheme: m " +1 ij =" m ij + #t $" m ij $t ( ) m " +1 ij =" m ij + #t %& 2 " m ij + & 2 ( m " ) 3 ij % & 4 m " ij Now must discretize these spatial derivatives 33

34 Finite Difference Equation for TDGL There are several ways of differencing a derivative. We use central differences: " 2 #( x,t) $ # m i+1, j % 2# m i, j m +# i%1, j m + # i, j +1 (&x) 2 % 2# m m i, j +# i, j%1 (&x) 2 i,j+1 i-1,j i,j i+1,j i,j-1 34

35 Finite Difference Equation for TDGL For notational convenience, define the following symbol for any function f ij : # f " f + f + f + f $ 4 f ij i+1, j i$1, j i, j +1 i, j$1 i, j nn Then the finite difference equation is finally: " ij m +1 =" ij m + #t #x % nn & ( ' ( ) 2 $" ij m + " ij m ( ) 3 $ 1 #x % nn m ( ) 2 " ij ) + * 35

36 Choosing the time step and mesh size When we discretize any differential equation, we change the model. The solution to the discretized finite difference equation converges to the solution of the differential equation only in the limit of δt and δx going to zero. As with any method, the solution must be independent of the choice of δt and δx. 36

37 Choosing the mesh size and time step Morphology Evolution t = 2000dt t = 8000dt t = 40,000dt 37

38 Choosing the time step and mesh size Choice of time step δt Too large a time step can cause instabilities and spurious solutions. The time step is limited by the mesh size. Choice of mesh size δx For the solution to any finite difference equation to converge to the true solution of the differential equation, δx must be smaller than the smallest length being resolved in the system. For this model, the smallest length is the interfacial width, which decreases in time until the latest stages of unmixing. 38

39 Choosing the mesh size Too large a mesh size causes unphysical pinning N = 32, dx = 1, t = 100 N = 32, dx = 1, t = 200 Here, interface width is equal to the mesh size (pixel size). 39

40 Choosing the mesh size Reducing the mesh size below the interfacial width allows demixing N = 32, dx = 1, t = 200 N = 32, dx = 0.5, t = 200 Here, mesh size was decreased by half to be smaller than the interface width. 40

41 Choosing the mesh size Reducing the mesh size below the interfacial width allows demixing Δx = 2.0, t = 15,000 Δt Δx = 1.0, t = 15,000 Δt Mesh size of system at right was decreased to half its value in the system on left, to be smaller than the interface width. 41

42 Why does pinning occur from a too-large mesh size? Consider the interfacial profile in 1-d. ψ ij goes from one bulk value, ψ 1, at x ij - δx to the other bulk value ψ 2 at x ij + δx across an interface. For R to increase, the interface must move, which means ψ ij must change from ψ 1 to ψ 2. The driving force for this is the interfacial (square gradient) term in F, which must overcome the barrier between the wells in the double-well bulk free energy for φ ij to change. ψ 1 ψ 2 1 ( m +# m m i$1, j $ 2# ) i, j ("x) # 2 i+1, j 42

43 Why does pinning occur from a too-large mesh size? But, numerator in Laplacian term is bounded above by ψ 1 + ψ 2-2 ψ ij = constant, and thus increasing δx reduces the Laplacian while the local term remains unchanged. ψ 1 ψ 2 Thus, for δx sufficiently large, the interfacial term cannot overcome the bulk term, and pinning or unphysical slowing of the growth occurs. 1 ( m +# m m i$1, j $ 2# ) i, j ("x) # 2 i+1, j 43

44 Analyzing Microstructure Development If left to its own devices, an unstable mixture that is demixing via spinodal decomposition will eventually phase separate to two bulk coexisting phases with uninteresting morphologies. Along the way, extremely interesting morphologies (microstructures) can arise. These morphologies can be trapped in by quenching and other schemes and used for different purposes important for nanotechnology. Two methods are typically used to monitor morphology: Light or neutron scattering Laser scanning confocal microscopy 44

45 Experimental Microstructures from SD Laser scanning confocal microscopy of PS-PB/PB blend Hashimoto s group, Kyoto University 45

46 Monitoring Growth Kinetics: The Structure Factor The static structure factor S(q) is given by the Fourier transform of the pair correlation function g(r). S(q) is obtained from light, x-ray and neutron scattering experiments and provides a convenient way of monitoring the growth kinetics following a quench to the unstable two-phase region. Early-time growth of concentration fluctuations Intermediate and late-stage scaling behavior of the domain size S(q) is easily calculated in simulations. 46

47 Monitoring Growth Kinetics: The Structure Factor On a discrete grid, the structure factor at a given time t, S(q;t), is given by: S(q;t) = "(q;t)"(q';t) = 1 % e #iq$x G(x;t) L 2 % x = 1 L 2 e #iq$x " x + x';t)" x';t The wavevectors q = (2π/Lδx)n, and n = 1, 2,, L/2. < > denotes an average over all possible origins, and over an ensemble of many statistically independent configurations. x' [ ( ( )) # " 2 ] %% x x' where the sum is performed over all L d lattice sites. 47

48 Monitoring Growth Kinetics: The Structure Factor When the morphology of the mixture or blend is isotropic, S(q;t) can be further smoothed by spherical averaging -- averaging over all wavevectors between q and q + δq. This amounts to averaging over an entire spherical shell in q-space to obtain S(q;t): Before averaging S(q;t) = $ ( q"#q / 2)< q < ( q +#q / 2) $ ( q"#q / 2)< q < ( q +#q / 2) S(q;t) The denominator is the number of grid points in a spherical shell of width δq centered around q. Ideally, δq should be taken as small as possible, e.g. δq = 2π/Lδx. 1 48

49 Monitoring Growth Kinetics: The Structure Factor Evolution of light scattering pattern following a quench from 75C to 25C. System: Nearly symmetric critical mixture (φ c = 0.486) of deuterated and protonated 1,4-polybutadiene. UCST T c = 62 C Bates and Wiltzius, JCP 91, 3258 (1989) 49

50 Monitoring Growth Kinetics: The Structure Factor S(q) at various times t following a quench from 75C to 25C. System: Nearly symmetric critical mixture (φ c = 0.486) of deuterated and protonated 1,4-polybutadiene. UCST T c = 62 C Bates and Wiltzius, JCP 91, 3258 (1989) 50

51 Monitoring Growth Kinetics: The Structure Factor S(q) at various times t following a quench from 75C to 25C. System: Nearly symmetric critical mixture (φc = 0.486) of deuterated and protonated 1,4-polybutadiene. UCST T c = 62 C Bates and Wiltzius, JCP 91, 3258 (1989) 51

52 Monitoring Growth Kinetics: The Structure Factor A key quantity used to characterize the evolving morphology of a phase separating mixture is the characteristic size of the domains, R(t). In experiments, the peak wavevector q m is often used as a measure of the characteristic inverse domain size R -1 (t). In simulation, the discreteness of the grid used to integrate the CH equation make the determination of q m difficult. Instead, one can calculate the first moment q 1 of S(q;t) given by q 1 t ( ) = " q " q q 2 S(q;t) qs(q;t) 52

53 Monitoring Growth Kinetics: The Structure Factor Because this measure of the characteristic wavevector uses data acquired over the entire range of q, it is a cleaner, more accurate measure than the peak position. Of course, the two lengths are not the same, but if all lengths scale the same with time, then we can use any definition. Early time: exponential growth of S(q) at constant q. Thus all lengths equivalent. Intermediate/late time: power law growth of R(t); self-similar patterns, all lengths scale the same. q 1 t ( ) = " q " q q 2 S(q;t) qs(q;t) 53

54 Monitoring Growth Kinetics: Pair Correlation Function Another approach to calculating the time dependence of the characteristic domain size is to calculate the first zero of the real-space, composition-composition correlation function G(x), which is the Fourier transform of S(q): G(x) = " e iqx S(q) q G(x) is then spherically averaged to obtain G(x), and then normalized to unity at x=0: g(x;t) = G(x;t) G(0;t) 54

55 Monitoring Growth Kinetics: Total Area Method A cheap way to estimate a characteristic domain size is from R = L d area of interface Here area of interface = sum of all transitions from positive to negative ψ in all spatial directions Here R = 36/

56 Linear Theory of Spinodal Decomposition Early stage kinetics: To study the behavior of concentration fluctuations immediately following a quench from the one-phase to the unstable region, we can linearize the CH equation about the initial average concentration difference ψ 0 = ψ - δψ, where δψ is a small fluctuation around ψ 0. Note that all of this can be done in terms of φ rather than ψ through a trivial change of variables since ψ = 2 φ 1. The CH equation before linearizing is: "#(x,t) "t %F = M$ 2 %# = M$ ( "f 2 "# &'$ + * 2 #- ), 56

57 Linear Theory of Spinodal Decomposition Cahn Hilliard before linearizing: "#(x,t) "t ' "f = M$ 2 "# %&$ * ) 2 #, ( + Write ψ = ψ 0 + δψ. Then: Cahn-Hilliard after linearizing about ψ 0 = ψ - δψ "#$(x,t) "t &&" 2 f = M% 2 (( ''"$ 2 ) ) + #$,-% 2 #$ * + $ 0 * < 0 in unstable two-phase region 57

58 Linear Theory of Spinodal Decomposition Now Fourier transform the linearized equation. Define δψ(q,t) = δψ(x,t) e -iqx dx "#$(q,t) "t where q 2 c ' 1 (" 2 f + * % )"$ 2 -, = [%Mq 2 ( q 2 c & q 2 )]#$(q,t) $ 0 Solution is an exponential! 58

59 Linear Theory of Spinodal Decomposition Initial concentration fluctuations thus develop in time according to the following exponential: $(q )t "#(q,t) = "#(q,0)e with the wavevector - dependent growth rate $(q) = %Mq 2 (q c 2 & q 2 ) Then the structure factor S(q) evolves with time as S(q,t) = S(q,0)e 2"(q)t Grows or decays exponentially at constant q 59

60 Linear Theory of Spinodal Decomposition $(q )t "#(q,t) = "#(q,0)e ω(q) Linear theory holds at earliest times only. $(q) = %Mq 2 (q c 2 & q 2 ) Concentration fluctuations at all q < q c grow since ω(q) > 0. Concentration fluctuations at all q > q c decay since ω(q) < 0. q c q q = 2π/λ, so long wavelength fluctuations grow, and short wavelength fluctuations decay. Spinodal decomposition arises from a long-wavelength instability. 60

61 Evolution of Structure Factor Early time Later time S(q) grows exponentially at fixed q. Fastest growth is at q max = q c / 2 Beyond linear regime: domains grow with time: S(q) peak moves to smaller q ( ) q max at "#(q) "q = " $Mq 2 (q 2 c % q 2 ) "q = 0 How to describe growth in later stages of decomposition? 61

62 Late Stage Kinetics As system unmixes, A-rich and B-rich regions grow larger and the interface between them becomes thinner and less diffuse. In this stage, dynamical scaling is observed. The scaling hypothesis: in the asymptotic scaling regime, there is only one dominant length scale. When the characteristic domain size R(t) >> interfacial width, the dynamical scaling hypothesis states that by scaling all lengths by R(t), one can find time-independent functions G[x/R(t)] and H[qR(t)] such that g(x;t) = G[x /R(t)] and S(q;t) = R d (t)h[qr(t)] 62

63 Late Stage Kinetics The scaling function for the structure factor is suggested to have the form H(y) " y 2 # /2 + y 2+# where y = qr(t) and γ = d+1 for an off-critical mixture (dispersed clusters) and γ = 2d for a critical mixture (percolating clusters), where d = dimension. At large q, the scaling function at large q follows Porod s law for large values of y: H(y) " y #4 63

64 Late Stage Kinetics In the late stages of phase separation, R(t) is the only relevant length scale in the system. In the scaling regime, R(t) and the height of the structure factor peak, S(q max ;t), will scale with time as R(t) " t # and S(q max ;t) " t $ where $ = d#. If no hydrodynamics, α = 1/3 ( d) and β = 1 (in d = 3). Lifshitz-Slyozov theory of evaporation/condensation If hydrodynamics, α = 1 (in d = 3) and β = 3. 64

65 How to create and tailor patterns? Thermodynamics drives phase separation towards bulk, coexisting phases. Kinetics determines how morphology evolves. We must add constraints or additional physics to affect kinetics and/or evolving morphologies if we want to generate specific patterns or stabilize patterns at specific length scales. vs. 65

66 TDGL/CH for Block Copolymers One way of introducing a constraint on pattern size and shape is by covalently bonding two immiscible polymers together into a single block copolymer. Finite block length prevents macroscopic phase separation, and stabilizes patterns. The physics of this is captured by adding a term to the GL free energy functional: block length F ["(x,t)] = F [ GL "(x,t)] + N #2 2M A $$ B G(x,x')"(x,t)"(x',t)dxdx' 66

67 TDGL/CH for Block Copolymers F ["(x,t)] = F [ GL "(x,t)] + N #2 2M Short-range attraction $$ G(x,x')"(x,t)"(x',t)dxdx' Long-range repulsion G(x,x ) is the Green s function for the equation: " 2 G(x,x') = #4$%(x # x') In d = 3, G(x,x ) = 1/ x - x + Y(x), where 2 Y = 0 67

68 TDGL/CH for Block Copolymers The TDGL/CH equation for block copolymers is "#(x,t) "t = M$ 2 %F %# = M$ 2 %F GL %# &W# where W = 1/N 2, since " 2 $$ G(x,x')#(x,t)#(x',t)dxdx' = %2# 68

69 TDGL/CH for Block Copolymers To see how this extra term, arising from the long-range constraint on the maximum distance two blocks A and B on the same chain can be, stabilizes microphase-separated patterns, re-examine the linear stability analysis for the modified equation: "#(x,t) "t = M$ 2 %F GL %# &W# 69

70 Linear Theory of BCP SD Modified Cahn Hilliard before linearizing: Write ψ = ψ 0 + δψ. Then: Modified Cahn-Hilliard after linearizing about ψ 0 = ψ - δψ "#$(x,t) "t "#(x,t) "t ' "f = M$ 2 "# %&$ * ) 2 #, %W# ( + &&" 2 f ) ) = M% 2 ( + #$,-% 2 #$ ( ''"$ 2 * +,W#$ $ 0 * < 0 in unstable two-phase region 70

71 Linear Theory of BCP SD Now Fourier transform the linearized equation. Define δψ(q,t) = δψ(x,t) e -iqx dx "#$(q,t) "t where q 2 c ' 1 (" 2 f + * % )"$ 2 -, = [%Mq 2 ( q 2 c & q 2 ) &W ]#$(q,t) $ 0 Solution still an exponential 71

72 Linear Theory of BCP SD but with a modified dispersion relation! ω(q) "#(q,t) = "#(q,0)e $(q)t $(q) = %Mq 2 (q 2 c & q 2 ) &W W Linear theory holds at earliest times only. Ordinary spinodal decomposition q c q Only concentration fluctuations at q c1 < q < q c2 grow since ω(q) > 0 there. q c1 q c2 Microphase ordering of block copolymers Constraint arising from finite block length suppresses long-wavelength instability, and selects a particular pattern size scale. 72

73 TDGL/CH for Reactive Systems Another way of stabilizing patterns is to induce chemical reactions between the two components of the mixture. Example of a simple isomerization reaction: " 1 A # B " 2 Modified Cahn-Hilliard equation becomes: "#(x,t) "t %F = M$ 2 CH %# & ' 1# + ' 2 1& # A B B A ( ) 73

74 TDGL/CH for Reactive Systems Modified Cahn-Hilliard equation for isomerization reaction: "#(x,t) "t = M$ 2 %F CH %# & ' 1# + ' 2 1& # ( ) A B B A Can rewrite in terms of ψ = 2φ - 1. For Γ 1 = Γ 2 : "#(x,t) "t = M$ 2 %F %# & '# Γ = 1/N 2 Same equation as for block copolymers! 74

75 TDGL/CH for Reactive Systems Why do reactive systems behave like block copolymers? What constitutes the constraint on bulk phase separation? When A B inside a domain with equilibrium composition, B will be driven to move out of domain into B-rich domain, and vice versa. The size scale is set by the reaction rate: the slower the rate, the farther B can move; the faster the rate, the closer B can move. 75

76 TDGL/CH for Reactive Systems Γ = 1/N 2 Small Γ Large block length Large Γ Small block length 76

77 MC Simulation of Phase-Separating Mixtures with Reactions This problem may be modeled with other methods. E.g., with Monte Carlo, add reaction terms whereby A can become B and vice versa with rates Γ 1, Γ 2. Old New 77

78 Controlling patterns through competing interactions Block copolymers and chemical reactions demonstrate how pattern selection can be achieved through constraints on phase separation behavior. Competing interactions: short-range attraction driving phase separation + long-range repulsion driving demixing Zeroth order: Symmetric diblock copolymers or isomerization reaction with equal forward and backward reaction rates. (Asymmetric diblocks = unequal rxn rates: off-critical different patterns) Higher order constraints: Triblock copolymers; multiblock copolymers; non-linear block copolymers. A + B C, etc. Couple reactions to thermodynamics. 78

79 Polymer Blend Phase Separation on Nanopatterned Surfaces Modulated surface interactions can influence patterning of immiscible blends, copolymers, and other mixtures in thin films. AFM image at right shows PS/PB blend that was spuncast onto SAM substrate patterned using micro-contact printing. Elastomeric stamps of PDMS are used to stamp ink of hydrophobic and hydrophilic end-group alkanethiols onto SAM substrate to modulate surface interaction. This results in 1 micron wide stripes that attract the PB polymers. A. Karim, et al, PRE 57, R6273 (1998). 79

80 Polymer Blend Phase Separation on Nanopatterned Surfaces A. Karim, et al, PRE 57, R6273 (1998). 80

81 TDGL/CH on Nanopatterned Surfaces Surface nanopatterning of immiscible blends can be modeled using the TDGL/CH method. A surface energy term can be added to the GL free energy of mixing: h 0 is the magnitude of the chemical potential favoring a particular component at the surface. The interaction is modulated along the surface in the x-direction with a period l p determining the stripe patterning length scale. g accounts for changes in the polymer-polymer interactions near the boundary. 81

82 TDGL/CH on Nanopatterned Surfaces Cross-section of films showing surface-modulated spinodal decomposition. Here surface pattern wavelength l p is twice the maximally unstable wavelength λ max. Thicker film Thin film A B A B A B A B A B A. Karim, et al, PRE 57, R6273 (1998). 82

83 TDGL/CH Models of Nanofilled Polymers Presence of nanoparticles in immiscible mixtures and blends affects the phase separation process since one component of the mixture will always preferentially attract the nanoparticle, even by just a small amount. Questions: How might nanoparticles be used to manipulate phase separation morphologies? How might phase separation be used to manipulate ordering of nanoparticles? 83

84 TDGL/CH for Nanofilled Polymers Preferential attraction for polymer A by the nanoparticle modeled by adding a local surface interaction energy F s [φ]= d d-1 x[hφ+gφ 2 + ] subject to boundary conditions of zero flux and local equilibrium at surface. h is a surface field that breaks the symmetry between the two phases; related to ε mf in MD simulations. Attractive interaction between nanoparticle and polymer A causes wetting of the particle by the polymer. *B.P. Lee, J.F. Douglas and SCG, Phys. Rev. E 60, 5812 (1999). 84

85 TDGL/CH for Nanofilled Polymers Spinodal decomposition of a thin film with immobilized nanoparticles. Shallow quench Deep quench B.P. Lee, J.F. Douglas and S.C. Glotzer, Phys. Rev. E 60, 5812 (1999). AFM image of PS/PVME thin film blend with dilute concentration of 100 nm silica beads. Karim, et al., Macromolecules 32, 5917 (1999). 85

86 TDGL/CH for Nanofilled Polymers Interference patterns can arise from interplay between nanoparticle spacing and spinodal wavelength Simulation image: B.P. Lee, J.F. Douglas and S.C. Glotzer, PRE 60, 5812 (1999). Phase contrast microscopy image of photocrosslinked blend. Q. Tran-Cong,

87 TDGL/CH for Nanofilled Polymer Blends If nanoparticles are allowed to move during the phase separation process, different patterns are observed. Ginzburg, et al, PRL 82, 4026 (1999) Peng, et al, Science 288, 1802 (2000) Here, MC is used to move the particles around on the grid. 87

88 Pattern Selection on the Nanoscale Again, all these examples demonstrate the rich possibilities of tailoring nanoscale patterns using soft materials. Method-wise: the TDGL/CH approach is useful, but can become cumbersome when complex geometries arise, or when hydrodynamic flow or shear or viscoelasticity is included. Then, particle-based mesoscopic methods like DPD may work better, provided you can find a good way to include the extra physics in the simulation without losing the benefits of the method. 88

89 Mesodyn TM from Accelrys Example applications of Mesodyn from Polymer blend with compatibilizers Latex seed formation Reverse micelles 89

90 Other Field Theoretic Methods Field Theoretic Polymer Simulation - Fredrickson Goes beyond mean field - includes fluctuations! Promising for predicting equilibrium mesophases of multicomponent soft materials. Examples of mesophases in ABCA tetrablock melts obtained with field theoretic simulation techniques (Drolet & Fredrickson, PRL 83, 4317 (1999)) 90

91 Bridging length and time scales Integrated in silico soft materials design Use output from one level of simulation as input in another Mesoscale Macroscale Predict bulk properties Predict morphologies Designed polymer nanocomposite First principles Specify system: quantum dot, colloidal silica, etc. in polymer Atomistic Molecular/mesoscale Predict melt structure and dynamics near nanoparticle Future goal: seamless zooming 91