φ(z) Application of SCF to Surfaces and Interfaces (abridged from notes by D.J. Irvine)

Transcription

1 Application of SCF to Surfaces and Interfaces (abridged from notes by D.J. Irvine) Edwards continuum field theory reviewed above is just one flavor of selfconsistent mean field theory, but all mean field analyses share the same basic idea of replacing the complex functional U, which requires knowledge of the interactions for each possible configuration of chains in the system, with U MF, which is determined only by the average local environment around a given chain segment. Because the mean field models give a closed set of equations, some simple cases can be solved analytically. On the other hand, extremely complex systems can also be modeled and the solutions to the mean field equations found by numerical iteration this is why mean field theories are so often applied in problems of polymer surfaces and interfaces! Adsorbed/Grafted Polymers Colloids/Stabilized Particles φ(z) z Compatibilization of Blends Surface Segregation 46

2 Analytical SCF Example: Polymer brushes h d Variable Definitions: d = anchor spacing [length] σ = grafting density [# chains/area] = 1/d 2 h = brush height N = # segments/chain Questions of interest: 1. What is the equilibrium thickness h of a brush? How does it scale with N? How does it scale with σ? 2. What is the concentration profile of segments in a polymer brush? 3. What is the energy to add a chain to a polymer brush? (This will control the kinds of grafting densities achieved in practice) 47

3 We will consider the analytical approach of Milner et al (S.T. Milner, T.A. Witten, and M.E. Cates, Macromolecules, 21, 261 (1988)). To write a simple form for the free energy of the system, they assume: The grafting density is high enough and the chains are long enough that all chains can be considered strongly stretched. z Uniform concentration of segments in any given z layer...this assumption is key in simplifying the formalism. Namely, it allows us to postulate that the segment density parallel to the surface is uniform at any given distance above the substrate, thus, we need only consider the variation in the segment density perpendicular to the surface! Compare to a sparsely populated surface Concentration is nonuniform in x-y plane 48

4 For a single chain within the polymer brush, the conformational distribution function becomes: Ψ MF [z(n)] = const exp[-1/2 N dn ( z/ n)2 - v N dn <c(z(n))>] where U MF (z(n))/kt = v <c(z(n))> and z(n) is the continuum 1D path of the chain from to N. continuum! discrete dz/dn z n - z n-1 The free energy functional for a single chain in the brush becomes: N F(z)/kT = 1/2 dn ( z/ n)2 + v N dn <c(z(n))> The free energy of the brush is correspondingly the sum of all single chain contributions. Similar to the Flory model described earlier, the equilibrium distribution for the segments in the brush is assumed to have a much lower energy than all other possible configurations, so that we can neglect fluctuations from the minimum (ground state) energy configuration of the system. The equilibrium brush structure is thus obtained by minimizing F(z): F(z)/ z = 1/2 z [( z/ n)2 ] = - v [ <c(z(n))>] z 49

5 In solving this expression, Milner et al. recognized that a direct analogy could be made to the classic equation of motion: Polymer brush: -du MF (z)/dz = ½ d[( z/ n) 2 ]/dz Newton s law: F = ma -du MF /dz = d 2 z/dn 2 -du/dx = m d 2 x/dt 2 This equation of motion describes the path followed by a chain in the brush. It has the following boundary conditions: 1. Each chain starts at its free end (call this position z=ρ with the stretch, satisfying mechanical equilibrium (the chain is not under tension): dz dn z= ρ = 2. All chains, regardless of where the free end lies, must reach the grafting surface in N steps. NOTE: Our goal is to solve for U MF (z), which gives us the equilibrium concentration profile, since U MF (z) = v<c(z)>kt. Milner et al. solved the problem by comparing this equation and boundary conditions to a classical physics problem: We equate our chain s path to the path of a particle under a given potential U MF (z): The stretch derivative z/ n can be identified as the particle velocity dx/dt, meaning that our equation of motion maps into Newton s equation, where the chain segment index n corresponds to time t, and mass m=1. 5

6 The given boundary conditions correspond to: ρ dx dt x =ρ = at t = x = at t =N The potential is analogous to an equal time potential: A particle starting at any height ρ must move to the surface in the same amount of time as all others (t = N). This is indeed the case for a harmonic oscillator, such as a simple pendulum or block and spring construction. Recall (from long ago!) that the angular frequency of a harmonic oscillator is k 2π ω = =, where k is the spring constant, m is the mass of the particle, and τ is m τ the period of oscillation. Rearranging,...The period is independent of the amplitude, i.e., the harmonic oscillator meets the equal time requirement. Moreover, the velocity fulfills the boundary condition dx/dt = at t=. Having identified what type of potential will solve our equation with the given boundary conditions, let s see what comparisons can be made between the brush and an oscillator: BRUSH τ = 2π OSCILLATOR U MF (z) =? U = (1/2)kx 2 dz i /dn v = dx/dt = - ωa sin (ωt + φ) z i x = A cos (ωt + φ) m k 51

7 We let U MF (z) take the form of a generalized harmonic oscillator: U MF (z) = -A + Bz 2 where A and B are constants to be determined. We can find B by our equal time requirement: Our harmonic oscillator has mass m = 1 and a spring constant k=2b. The path of our fictitious particle goes through one-quarter cycle as it falls to the grafting surface from its end-point in the brush: τ = 4N = π(2/b) 1/2 B = π 2 /8N 2 We know from our initial approximation that the potential is related to the concentration of segments by: U MF (z) = v kt <c(z)>, so: <c(z)> = (vkt) -1 (-A + Bz 2 ) We can determine A by computing the total amount of material in the brush, employing a conservation of mass or incompressibility condition: h Nσ = dz < c(z) > where h is the height of the brush (the maximum distance any of the chains is stretched away from the surface). Substitution for <c(z)> gives: NσvkT A = + h Bh 3 2 Using <c(z)> = at the top of the brush (z = h), we can solve for h: 52

8 h = (12/π 2 ) 1/3 (σv) 1/3 N So, by using a mean field analysis, two very powerful predictions are recovered: 1. The concentration profile <c(z)> for the brush is parabolic. 2. The brush height scales with grafting density and chain length as h~σ 1/3 N. <c(z)> h z 53

9 Numerical Lattice SCF: Scheutjens and Fleer s Basic Model for More Complex Polymer Interface Problems BULK WALL Z Lattice SCF takes Flory s original lattice model ideas and instead of analytically solving the entropy for the system, we statistically count the number of configurations available to the chains on a well-defined lattice. (Thus the entropy is exact for the model.) This places practical computational limits on the chain size. Z G(s=1,s=1,z=4) = G free (z=4) = e -u(z=4)/kt 54

10 ...the potential u(z) in the lattice model contains two parts: u(z) = u (z) + u int i(z)...u (z) is entropic (a potential which forces lattice sites to be filled, which can be thought of as a hard-sphere repulsive term), and u int i(z) is the interaction energy U MF (z) we have already seen in the other mean-field approaches. 2 1 G(s=2,s=1,z=3) Z = G free (z=3)<g(s=1,s=1,z=3)> = e -u(z=3)/kt <G(s=1,s=1,z=3)> = e u(z =3) / kt 1 6 G free(z = 2) G free(z = 3) G free(z = 4) This nearest-neighbor counting equation is simply a 1-dimensional discrete version of the diffusion equation which is used in the continuous formalism: [ N - b 2 /6 2 + dm v c(r 2 m) ]G(R,;N) = R 55

11 To see this more transparently, we can in general write the relation (see G.J. Fleer et al., Polymers at Interfaces, pp ): { λ λ λ } Gzs (, + 1) = Gz () Gz ( ls ;) + Gzs (;) + Gz ( + ls ;) 1 1 where l is the lattice parameter and λ 1 and λ = 1-2 λ 1 are the probabilities to cross to an adjacent layer or remain in the same layer, respectively (related to the number of nearest neighbors). Performing some mathematical manipulation on the above expression: { λ λ λ } Gzs+ = e Gz ls+ Gzs+ Gz+ ls u( z)/ kt (, 1) 1 ( ;) (;) 1 ( ;) { λ λ λ } Gzs+ e Gzs = Gz ls+ Gzs+ Gz+ ls Gzs u( z)/ kt (, 1) (,) 1 ( ;) (;) 1 ( ;) (,) { } λ { } Gzs+ e Gzs = λ Gz+ ls Gzs Gzs Gz ls u( z)/ kt (, 1) (,) 1 ( ;) (;) 1 (;) ( ;) This difference equation can be transformed to a differential equation by allowing the step size z = s/l: Gzs+ e Gzs Gz+ zs Gzs Gzs Gz zs s z z z u( z) s/ kt (, 1) (,) 2 1 ( ;) (;) (;) ( ;) = λ1l If we now let z, s, we obtain: Gzs uz Gzs s kt z 2 (;) () 2 (;) + Gzs (;) = λ1l 2 Replacing λ 1 l 2 by b 2 /6 and rearranging: 2 2 b u( z) + Gzs (;) = 6 2 s z kt which is the 1-d equivalent of the diffusion equation on pg

12 e1 e2 Z We are interested in determining the composition profile in each layer away from the interface: φ(s = 7,z = 3) = C < G(s = e1, s = 7, z = 3 >< G(s = e2, s = 7,z = 3) > G free (z = 3) Again, we can compare this to the continuous description from pg. 45: N N!! 1 < c( r) > dn < δ ( r R! ) >= dng( R!!!!, r, N n) G( r,, n) n GR (,, N) The two expressions are analogous. 57

13 φ(z) Z Solving the SCF equations: 1. u(z) is unknown. Make a guess at u(z). 2. From u(z), we calculate G s; from G s we calculate concentration profiles. 3. Do the concentration profiles calculated from u(z) satisfy Σφ(z) = 1? If not, mathematically find a new guess for u(z) and repeat. SCF is solved numerically by guessing a form for the potential u(z) (the interaction field), and seeing if it provides a concentration profile (under the constraints for the system that we have imposed: connectivity of the chains, interactions between segments χ, impenetrable boundaries, etc.) which fills the lattice (i.e., there are no holes in the system). If the current u(z) profile does not fill the lattice, we mathematically determine a new guess (e.g. use a Newton-Raphson numerical procedure to guess the change in u(z) necessary to get closer to the solution) and iterate, continuing until the lattice-filling requirement is met. 58

14 We are not minimizing a free energy functional as we did for our polymer brush analysis. We allow all valid chain configurations to contribute to our ensemble, giving us a more accurate calculation of thermodynamic quantities and resulting system structures. In practice, one-dimensional calculations are fast and can be performed for very large chain lengths and lattice sizes! What Do We Get? Again, as with any of the theoretical methods we ve described in this section, we can calculate any thermodynamic properties of interest once we have the equilibrium concentration profile. For example: γ(interfacial/surface free energy) - e.g. calculate the change in interfacial tension between 2 immiscible polymers on adding a random copolymer to the blend F (total free energy) - we can compare possible equilibrium structures in a system to determine which is stable at equilibrium, e.g. block copolymer morphologies. We can also calculate H, S, etc... segment distributions - e.g. we can calculate on average how many loops or tails an adsorbing polymer has on average for a given attraction to a surface. Note that we cannot address issues of kinetics we these methods: we are applying statistical mechanics to find equilibrium properties! 59

15 Selected Bibliography for Mean Field Models in Polymer Systems overviews of mean field theory: 1. Fleer, M.A. Cohen Stuart, J.M.H.M. Scheutjens, T. Cosgrove, and B. Vincent, Polymers at Interfaces, Chapman and Hall, New York, NY (1993). continuum analytical models: 1. E. Helfand, Macromol., 8, 552 (1975). 2. E. Helfand and Z.R. Wasswerman, Macromol., 9, 879 (1976). 3. E. Helfand and Z.R. Wasserman, Macromol., 11, 96 (1978). 4. Doi and S.F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, New York, NY (1986). 5. D. Broseta, G.H. Frederickson, E. Helfand, and L. Leibler, Macromol., 23, 132 (199). continuum numerical models: 6. K.M. Hong and J. Noolandi, Macromol., 14, 727 (1981). 7. J. Noolandi and K.M. Hong, Macromol., 15, 482 (1982). 8. J. Noolandi and K.M. Hong, Macromol., 17, 1531 (1984). 9. K.R. Shull, Macromol., 26, 2346 (1993). lattice models: 1. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, NY (1953). 11. J.M.H.M. Scheutjens and G.J. Fleer, J. Phys. Chem., 83, 1619 (1979). 12. J.M.H.M. Scheutjens and G.J. Fleer, J. Phys. Chem., 84, 178 (198). 13. J.M.H.M. Scheutjens and G.J. Fleer, Macromol., 18, 1882 (1985). 14. M.R. Bohmer, O.A. Evers, and J.M.H.M. Scheutjens, Macromol., 23, 2288 (199). 15. O.A. Evers, J.M.H.M. Scheutjens, and G.J. Fleer, J. Chem. Soc. Farad. Trans., 86, 1333 (199). 6