Polymer Solution Thermodynamics:

Transcription

1 Polymer Solution Thermodynamics: 3. Dilute Solutions with Volume Interactions Brownian particle Polymer coil Self-Avoiding Walk Models While the Gaussian coil model is useful for describing polymer solutions at the Θ temperature, it gives a poor representation for most polymer-solvent systems due to its failure to account for volume interactions, or long-range interactions between segments spaced far apart on the polymer backbone. In particular, whereas a particle in random flight is able to cross its own trajectory, polymer chains have excluded volume which disallows such configurations, giving rise to an effective swelling of the polymer coil compared to ideal chain dimensions. Experimentally, the radius of gyration is found to scale not as.5 but as R g ~ ν, where ν is approximately 3/5. Although this might seem like a minor difference, for large values of the coil dimensions change dramatically: Ex: For = 1, b = 6 Å ν ν b.5 19 Å.6 38 Å 31

2 Once long-range interactions are considered, an exact description of the polymer chain statistics becomes impossible. Different numerical and analytical approaches have been taken to assess the value of ν and approximately account for the system excluded volume (see D.S. McKenzie, Physics Reports 27, 1976 for a review). Exact enumeration: The polymer molecule is modelled as a self-avoiding walk of steps on a lattice. All possible configurations are calculated for a finite sized chain, and the results are extrapolated to the large limit. ex: For a plane square lattice (z=4): # config. (SAW) # config. (ran. walk) ,146,397,316 ~ : : : const z γ-1 z 32

3 ote as increases, the number of possible configurations becomes many orders of magnitude lower for the SAW compared with a random walk on the same lattice. For a SAW, the effective number of nearest neighbors z is lattice dependent and always less than the number of nearest neighbors for a random walk on the equivalent lattice (z = for a square lattice), whereas γ is a universal exponent that depends on the chain dimensionality (γ = 4/3 for two dimensions). The total number of SAW configurations Ω for large cannot be calculated explicitly, but is obtained by extrapolation: Setting Ω = const z γ-1 and Ω +1 = const z +1 (+1) γ-1 we can define z = Ω +1 /Ω = z [(+1)/] γ-1 or ln z = ln(ω +1 /Ω ) = ln z + (γ -1) ln(1+1/) Using a Taylor series expansion for, ln (1 + x) = x - x 2 /2 + x 3 / for x ln(ω +1 /Ω ) = ln z + (γ -1)/ By plotting ln(ω +1 /Ω ) vs. 1/, ln(ω +1 /Ω ) one can obtain z and γ from the y-intercept and slope, respectively. 1/ 33

4 Denoting Ω (A) as the number of SAWs of steps from the origin to a point A on the lattice, and R (A) as the cartesian distance of A from the origin, the mean-square end-to-end distance is calculated as: <R 2 > = Ω -1 R (A) 2 Ω (A) A The scaling exponent ν for the rms end-to-end distance of a SAW of steps is calculated by analyzing the asymptotic behavior of the function: <R 2 > const 2ν As becomes large: <R +1 2 >/<R 2 > = [(+1)/] 2ν 1 + 2ν/ such that one can approximate ν ν = /2 [<R +1 2 >/<R 2 > -1] or plotting <R +1 2 >/<R 2 > vs. 1/ furnishes a slope of 2ν. LATTICE d z z γ ν linear square /3 3/4 triangular /3 3/4 simple cubic /6 3/5 b.c.c /6 3/5 f.c.c /6 3/5 Monte Carlo methods: An alternative approach to exact counting of all configurations for small values is to use Monte Carlo methods to generate a statistical sample of SAW configurations, and calculate the properties of the system from ensemble averages. This approach is 34

5 advantageous in that 1) longer chains may be generated, and 2) off-lattice models can be used, thereby avoiding lattice artifacts. From Monte Carlo calculations on chains with excluded volume, the generally accepted result for ν in 3 dimensions is ν =.588. Analytical Approaches Long-range interactions between two segments are influenced by steric effects, van der Waals attractions, and the nature of solvent. Here we are interested in the properties of the system at longer length scales, and can thus express the excluded volume interaction as a short-range (point) potential: vkt δ(r n - R m ) where v has the dimensions of volume, and can be identified with the binary cluster integral (see section 3.1 of Polymer Conformations notes): v = dr [ 1 exp[-u(r)/kt] Model U(r) = U hc (r) + U att (r) U(r) U hc (r) = r < b = r > b Expand: exp[-u att (r)kt)] 1 U att (r)/kt r v = v o [ 1 - Θ/T] (or v = b 3 [1-2χ] in F-H model) Here, v o is identified with a hard core segmental volume, and Θ is the theta temperature. For T = Θ (χ =.5), attractive and repulsive contributions to the interaction potential cancel, so that the polymer chain distribution obeys Gaussian statistics. For T > Θ, the segment-segment interaction is repulsive (v is positive) and the chain swells. 35

6 The total interaction energy for a single chain with volume interactions is: U/kT = v/2 dn dm δ(r n - R m ) The conformational distribution function consequently becomes (pp. 21-2): Ψ[R n ] = const exp[-3/2b 2 dn ( R n/ n) 2 - v/2 dn dm δ(r n - R m )] ote that addition of the excluded volume term results in non-markovian properties, so that the distribution function for the end-to-end vector is no longer the solution to a diffusion-type equation. Another cause for concern our distribution function as written accounts only for second order interactions. Is this a justifiable approximation? We can estimate the number of binary collisions, in the case where T = Θ as: φ coil ~ 1/2 while the number of three-body interactions is substantially lower: φ coil (φ coil ) ~ and only important, therefore, when T < Θ. Analysis of the chain statistics when volume interactions are included requires further approximation of the conformational distribution function. Various expansions of Ψ[R n ] have been performed with some success in characterizing polymer coils with volume interactions in different limits. The most significant analytical approaches to the problem of excluded volume, however, were put forth by P.G. degennes, S.F. Edwards and P.J. Flory, summarized briefly in the following pages. 36

7 Renormalization Group theory P.G. DeGennes first applied the techniques developed by K. Wilson for describing critical phenomena to the problem of excluded volume. While a full development of RG theory as applied to polymers is beyond the scope of this class, its importance merits a discussion of the basic principles involved. ear a critical point (e.g., at the liquid-gas transition or the Curie point for magnetic systems), the size correlations in the fluctuations (i.e., density or magnetic spin), denoted as ξ, follow the relationship: ξ ~ [(T - T c ) / T c ] -ν T T c ξ where T c is the critical (or Curie) temperature, and ν is a universal exponent (ν = 2/3 for a magnetic ordering transition.) As T approaches T c, the length scale of fluctuations is on the order of the size of the system and ξ diverges. If we consider any small region of the system, the magnetic spins, say, are highly correlated, so that this region will have a mean magnetic moment. M(T) ξ T>>T c T >T c T c T The system of elementary spins can thus be cast into a new system of blocks of spins, described by a Hamiltonian similar to the initial system but with coupling constants between spin fields rescaled by some power of the new characteristic length. Successive transformations can be made until one reaches a stationary point in the iteration procedure, from which universal exponents can be deduced. 37

8 For polymer chains the analogy to ξ is the coil dimension <R 2 > 1/2, which also exhibits fluctuations on the scale of the coil itself for large : <R 2 > 1/2 = b ν ~ (1/) ν In analogy to the system of magnetic spins, DeGennes proposed to divide the polymer chain into /g subunits (g segments per subunit), from which one can define a new segment length parameter: b 1 = b g 1/2 [ 1 + h(g,u) ] where the function h(g,u) is obtained through a perturbation expansion of Ψ[R n ], and u is a dimensionless coupling constant, u = v/b 3 (or u = v/b d for arbitrary dimension). Since the coil is swollen compared to a Gaussian coil, b 1 > b g 1/2. The corresponding rms end-to-end distance can be rewritten in terms of the rescaled chain: <R 2 > 1/2 = b 1 (/g) ν and the rescaled excluded volume becomes: v 1 = v g 2 [ 1 - p(g,u) ] where v 1 < v g 2, since subunit repulsion limits their interpenetration. 38

9 Rescaling also introduces a new coupling constant: u 1 = v 1 /b 1 d = u g 2-d/2 [ 1 - k(g,u) ] where k = 1 - (1-p)/(1 +h) d. The stationary or fixed point is reached upon repeating this decimation procedure until the coupling constant tends to a finite limit, u*. Physically, we reach a length scale whereby the excluded volume scales precisely with the subunit volume. The subunits behave as impenetrable spheres, in a similar sense as two distinct polymer coils in dilute solution. Hence for larger length scales, the properties of the system are self similar, in an analogous fashion to the ideal chain model studied earlier. The value of u* is obtained by setting u m = u m-1 = u* in u m = u m-1 g 2-d/2 [ 1 - k(u m-1 )] Hence, u m /u m-1 = 1 = g 2-d/2 [ 1 - k(u*)] Correspondingly, we find: b m /b m-1 = g 1/2 [ 1 + h(u*)] = µ (const) The value of the critical exponent is obtained from <R 2 > 1/2 = b m (/g m ) ν = b m-1 (/g m-1 ) ν ν = ln µ/ln g Following this formalism degennes obtained ν =.598. RG methods have been applied to numerous other problems in polymer solutions. More discussion of this technique and its applications can be found in the Chapter XI of degennes and in the recent text Polymers in Solution by J. des Cloizeaux and G. Jannink. A similar rescaling approach will be employed later to characterize semidilute polymer solutions. 39

10 Steepest-descent model (Flory theory) A more simplistic, but still very useful approach for qualitatively describing many of the properties of polymer chains with volume interactions was originally proposed by P.J. Flory and is based on the method of steepest descent. The free energy of a single chain with fixed end-to-end vector R is just given by F(R) = - kt ln Φ(R,) Recalling that the partition function for a chain is the sum over all distributions of the end-to-end vector: Z = dr Φ(R,) = dr exp[-f(r)/kt ] The total free energy can be calculated from F = - kt ln Z. Using the steepest descent approximation for F, F min F(R) = F(R eq ) We begin by expressing F(R) in terms of entropic (elastic) contributions and excluded volume interactions: F(R) = F el (R) + F int (R) Recalling our previous results for ideal chains in tension and compression, F el (R) can be approximated by: F el (R) = 3/2 kt (R 2 /b 2 + b 2 /R 2 ) tension compression where R = R. The excluded volume contribution is calculated by first recasting the interaction potential for the chain U in terms of c(r): 4

11 F int (R) = U/kT = v/2 dn dm δ(r n - R m ) = v/2 dr dn dm δ(r - R n) δ(r - R m ) = v/2 dr c(r) 2 Approximating c(r) as the mean segment density within the polymer coil c coil = /R 3, and integrating over the volume of the coil, one obtains: F int (R) = const vkt c coil 2 R 3 or including ternary interactions for generality: F int (R) = const vkt c coil 2 R 3 + const wkt c coil 3 R 3 Summing the contributions of F el (R) and F int (R) and minimizing F(R) in R we obtain, neglecting all prefactors: (F(R)/kT)/ R = α 2-1/α 2 - (v 1/2 /b 3 ) α -3 - (w/b 6 ) α -6 = α 5 - α = v 1/2 /b 3 + (w/b 6 ) α -3 (3.1) where α is the reduced coil dimension α = R eq / 1/2 b. For the highly swollen coil, α >> 1 and the second terms on both sides of the eq. (3.1) are neglected. The scaling result for R eq is α 5 = v 1/2 /b 3 R eq = R F 3/5 b (v/b 3 ) 1/5 41

12 ote that we recover the SAW exponent, ν = 3/5. The calculation can be performed for a chain of arbitrary dimensions taking c coil = /R F d and replacing R 3 by R d in the interaction term. For dimension d, the equilibrium R is given by R F d+2 ~ v 3 b 2 from which we extract for the exponent ν the result: d ν ν = 3/(d + 2) /4 3 3/5 ote that this result is in precise agreement with exact enumeration studies! It has been noted that the agreement is somewhat fortuitous, as the overestimate of F el (R) is cancelled by an overestimate of F int (R) (see DeGennes, pg. 46). However, similar scaling models have proven to be highly successful in capturing the global properties of polymer systems, and we will employ them often during this class. ow let s consider the opposite limiting case, where interactions between segments are attractive, v < (T < Θ), and consequently α << 1. eglecting the left hand side of eq. (3.1) yields: -v 1/2 /b 3 = (w/b 6 ) α -3 R eq = R col ~ 1/3 (-v/w) 1/3 The scaling relationship 1/3 indicates a highly collapsed configuration. For the segment concentration in the coil: c coil = /R col 3 ~ 42

13 indicating a uniform, condensed state, very unlike the Gaussian coil model. Such a state is referred to as globular, and is often adopted by proteins and other biopolymers. MC simulation of swollen vs. globular states = 626 (from Grosberg and Khokhlov, pg. 86) Self-Consistent Field Theory Another approach that has been very successful in describing polymer systems with segmental interactions is the self-consistent mean field (SCF) model first developed by Edwards for characterizing chains with excluded volume. Here we describe the main features of the SCF approach. In the next section, we discuss how it has been applied to predict the segment concentration profiles of polymers at surfaces and interfaces. In the SCF formalism, the chain conformational distribution function Ψ[R n ] is once again approximated by replacing the potential term with a mean- 43

14 field potential, U(R n ) = U MF ( R n ) The value of U MF (R n ) is self-consistently determined from the mean local segment density: <c(r)> = n= < δ(r - R n ) > The mean field potential is then given by: U MF (R n ) = vkt <c(r n )> Physically speaking, the potential U MF (R n ) experienced by a chain segment at r = R n is simply proportional to the mean segmental density at r. Substituting this new expression into the conformational distribution function gives: Ψ MF (R n ) = const exp[-3/2b 2 dn ( R n/ n) 2 - v dn <c(r n)>] from which we can understand the advantage of this approach: replacing the true local potential by U MF (R n ) restores the chain s Markovian nature! What are the implications? ow we can once again obtain the Green s function for the chain from the solution to the diffusion equation: [ / n - b 2 /6 R 2 + v <c(r)>] G(R,;n) = > 44

15 The local average concentration <c(r)> is in turn calculated by: <c(r)> = 1/G dr n dn δ(r - R n ) G(R,R n ;-n) G(R n,;n) = 1/G dn G(R,r;-n) G(r,;n) Here, the normalization factor 1/G is the reciprocal of the partition function for a chain with ends fixed at R and, G = G(R,;) = dr n G(R,R n ;-n) G(R n,;n) ********************************************************************************** ote that the calculation of <c(r)> requires a solution to the diffusion-type equation for G(R,;n), which in turn depends on U MF (r=r) and hence the mean local density. The basic premise of the SCF model is to solve for U MF (r) (and therefore <c(r)>) and G(R,;n) in a self-consistent fashion. ********************************************************************************** The analytical solution derived by Edwards for a polymer coil in dilute solution is mathematically involved and will not be attempted here (see F.W. Wiegel, Physics Reports 16, 1975, pp for a detailed review). His formulation leads to the result <R 2 > ~ 6/5 b 2, in agreement with exact enumeration, and the Flory model results. 45