Introduction to Polymer Theory

Gert Jan Vroege

Van ’t Hoff Laboratory for Physical and Colloid Chemistry Utrecht University The Netherlands Introduction

chemical details vs. universal properties

A polymer is a statistical mechanical system, for which the role of entropy is very important Programme

1. Ideal polymers: • conformations: Gaussian coil • in an external field • in a Self Consistent Field (SCF) 2. Non-ideal polymers • excluded volume • attractions 3. Concentrated solutions: • Flory-Huggins theory • scaling theory (semi-dilute solutions) Polymer conformations

segment

End-to-end vector

bond length bond vector

d N d End-to-end vector: R = ƒ ri d i=1 R = 0 Polymer conformations d N d End-to-end vector: R = ƒ ri i=1

d d d N N d d 2 Y Y R = R R = ƒƒ ri rj i=1 j=1 N d N d d 2 Y = ƒ ri + ƒƒ ri rj i=1 i=1 j≠i ∝ N ∝ N smaller if j-i larger d R2 ∝ N d R2 ∝ N 1/ 2 Chain models (1) Freely jointed chain:

d d d 2 Y 2 ri = ri ri = b d d Y ri rj = 0 (i ≠ j)

d N d N d d 2 2 Y 2 R = ƒ ri + ƒƒ ri rj = Nb (+ 0) i=1 i=1 j≠i Chain models (2) Freely rotating chain: d d d 2 Y 2 ri = ri ri = b

d d Y 2 1 ri ri+1 = b (cosθ )

d d Y 2 2 ri ri+2 = b (cosθ )

d d Y 2 j−i ri rj = b (cosθ ) Chain models (3) d N 2 2 » 1 2 3 ?ÿ 2 1+ cosθ R = ƒb 1+ (cosθ ) + (cosθ ) + (cosθ ) + ⁄ = Nb i=1 1− cosθ 1+ cosθ = Nb 2 with b ≡ b eff eff 1− cosθ

General result when NO INTERACTION between segments d 2 2 R = Nbeff with more general beff End-to-end distribution (1) d d P(R, N) : probability of finding R after N segments?

recursion relation: d d d P(R, N) = P(R − rN , N −1) d rN

d d 2 2 Taylor expansion (N 1 and R rN ): d d d ∂P ∂P d P(R − rN , N −1) ≈ P(R, N) + (−1)+ ƒ (−rN ,α ) ∂N α =x, y,z ∂α 2 d d 1 ∂ P ? + ƒ ƒ (−rN ,α )(−rN ,β )+ 2 α =x, y,z α =x, y,z ∂α∂β End-to-end distribution (2)

d ? d d apply r : rN r = 0 N d dN d d d d d rNα rNβ = rNα r rNβ = 0 (α ≠ β ) rN N rN

d 2 d 2 d 2 1 2 rNx d = rN y d = rNz d = b rN rN rN 3 End-to-end distribution (3) d d ∂P 1 ≈ ∂2 ∂2 ∂2 ’ P(R, N) ≈ P(R, N) + (−1)+ ∆ + + ÷P +? ∂N 6 «∂x2 ∂y2 ∂z2 ◊ ∂2 ∂2 ∂2 With the definition of the Laplacian ∆ ≡ + + ∂x2 ∂y2 ∂z2 d we thus find that P(R, N) is the solution of: d ∂P(R, N) b2 d = ∆P(R, N) ∂N 6 d cf. the diffusion equation for c(R,t) : d ∂c(R,t) d = D∆c(R,t) ∂t End-to-end distribution (4) Cf. one diffusing colloidal particle (Einstein):

d R2 = 6Dt

b2 N

An (ideal) polymer is like the trajectory of a diffusing particle! End-to-end distribution (5) d ∂P(R, N) b2 d The solution of = ∆P(R, N) is (by analogy): ∂N 6 d 3/ 2 d ≈ 3 ’ ≈ 3R2 ’ P(R, N) = ∆ ÷ exp∆− ÷ «2π Nb2 ◊ « 2Nb2 ◊ Central Limit Theorem Consider the sum of N independent, stochastic variables (N is large). This sum has a normal (= Gaussian) distribution with σ 2 ∝ N.

d N d d 2 d Ω d2 2 2 R = ƒ ri σ R = (R − 0) = R ∝ N (×beff ) i=1 d Variation in R2 : d d 2 2 d 2 2 2 σ 2 = (R − R ) = Nb R 3

A Gaussian coil is a strongly fluctuating object!

Conclusion: many (global) properties of polymers do not depend on the (local) details of the model. Entropy of a Gaussian coil d S(R) = k lnW B d

= cst + kB ln P(R) 3k d = cst − B R2 2Nb2 d 3k T d A(R)(= −TS )= cst + B R2 ENTROPIC SPRING 2Nb2 Conditional probability

probability conditional probability d d d d

P(R, N) = GN (R | 0) GN (R | R′)

d d d d independent end points: G (R′′ | R′) G (R | R′′) ν d d N −ν Ω integrate over R′′ GN (R | R′) (OK for Gaussian chains) Additional polymer models

Gaussian bond model: d ≈ 3 r 2 ’ ∆ i ÷ Every single bond Gaussian ∝ exp∆ - 2 ÷ « 2 (1)b ◊ Bead-spring model: 3k T spring constant: B 2(1)b2

(used in the Rouse/Zimm models for polymer dynamics) Continuous model: permits the use of path integrals A polymer in an external field (1) d d assume a segment at position R has energy ϕ(R)

the recursion relation now changes to: d d d d ≈ ϕ(R) ’ P(R, N) = P(R − rN , N −1) d exp∆− ÷ rN « kBT ◊ Taylor expansion: d d d ≈ d ’ ≈ ’ ∂P ? ϕ(R) ? P(R − rN , N −1) ≈ ∆P(R, N) + (−1)+ etc.÷×∆1− + ÷ « ∂N ◊ « kBT ◊ d ∂P b2 ϕ(R) = ∆P − P ∂N 6 kBT cf. diffusion in an external field A polymer in an external field (2) d d similarly for the conditional probability GN (R | R′) d ∂G b2 ϕ(R) d d d − = − ∆G + G R′ is a parameter, but: R ↔ R′ ∂N 6 kBT ∂ψ Z2 d cf. − iZ = − ∆ψ +V (R)ψ ∂t 2m d QM: time-dependent Schroedinger equation for ψ (R,t)

linear, partial differential equations; solution method:

separation of variables Separation of vadriables ∂G b2 ϕ(R) − = − ∆G + G ∂N 6 kBT d assume G can be written as: G = f (N)dψ (R) d ∂f (N) b2 d ϕ(R) d −ψ (R) = − f (N)∆ψ (R) + f (N)ψ (R) ∂N 6d kBT divide by f (N)ψ (R) : d d 1 ∂f (N) b2 ∆ψ (R) ϕ(R) − = − d + = λ f (N) ∂N 6 ψ (R) k T dB f (N) = c exp(−λN ) b2 d ϕ(R) d d − ∆ψ (R) + ψ (R) = λψ (R) 6 k T eigenvalue equation Ω B d

eigenvalues λn , complete set (orthogonal) eigenfunctions ψ n (R) d

G =ψ n (R)exp(−λn N ) or any linear combination for different n A polymer in an external field (3) d d d

linear combination: GN (R | R′) = ƒcnψ n (R)exp(−λn N ) d d n using R ↔ R′ d d d d

bilinear expansion: GN (R | R′) = ƒψ n (R)ψ n (R′)exp(−λn N ) d n b2 d ϕ(R) d d where − ∆ψ n (R) + ψ n (R) = λnψ n (R) 6 kBT 1) continuous spectrum of eigenvalues d d d R Ω d eik YR d 1 b2k 2 example: ϕ( ) = 0 ψ k = and λk = 6 here we need all eigenfunctions Ω Gaussian coil A polymer in an external field (4) d d d d

bilinear expansion: GN (R | R′) = ƒψ n (R)ψ n (R′)exp(−λn N ) n

2) discrete spectrum of eigenvalues Ω lowest eigenvalue λ0 dominates for large N GROUND STATE DOMINANCE d d d d - GN (R | R′) ψ 0 (R)ψ 0 (R′)exp(−λ0 N ) chain ends are decoupled! A polymer in an external field (5) d segment density c(R) ?

integrate over: beginning end ν

d d 2 - c(R) N ψ 0 (R) cf. QM: bound state A polymer in an external field (6) Lifshitz entropy: derivation (for ground-state dominance) Partition function Z: # conformations d

(weighed with Boltzmann factors exp(−ϕ(R) / kBT )) d d d d d d d d G (R | R′) -ψ (R)ψ (R′)exp −λ N Z = — dR— dR′GN (R | R′) N 0 0 ( 0 ) d d 2 - λ0 N Z e (— dRψ 0 (R)) end effects free energy: - A = −kBT ln Z kBTλ0 N + end effects entropy: d use c(R) - Nψ 2 d d d 0 U − A 1 and the eigenvalue equation S = = — ϕ(R)c(R) dR − kBλ0 N T T and eliminate λ0 Lifshitz entropy: result d d d 1 2 S = NkB b —ψ 0 (R)∆ψ 0 (R) dR 6 d d 2 d d d 1 2 » ÿ Y S = −NkB b — ∇ψ 0 (R) ⁄ dR partial integration using ∆ ≡ ∇ ∇ 6 d d 2 d d 1 [∇c(R) ] 2 2 d using c(R) - Nψ S = −kB b — dR 0 24 c(R) d • independent of ϕ(R) • also valid for a collection of polymers • S decreasdes because of concentrationd gradients • S = S[c(R)] (S is a functional of c(R)) Self-consistent field method

this method incorporates inter-segment interactions:

free energy: d d d A[c(R)] = U[c(R)]−TS[c(R)] d U[c(R)] represents e.g. a non-ideal gas of segments

d ceq (R) is then obtained by functional minimization THIS APPROACH NEGLECTS FLUCTUATIONS/CORRELATIONS ! Non-ideal polymer chains d d

U[c(R)] = NkBTB c(R) B is the second virial coefficient (B > 0 repulsion) d Edwards (1965): R2 ∝ N 6/5

swelling PURE REPULSION

total number of configurations (depending on R) ∝ ≈ 3R2 ’ ∝ P(R, N) ∝ 4π R2 exp∆− ÷ « 2Nb2 ◊ but a certain fraction of these configurations is "forbidden": N (N −1)/ 2 ≈ ν ’ ≈ N 2ν ’ p(R) ≈ ∆1− c ÷ ≈ exp∆− c ÷ « R3 ◊ « 2R3 ◊ Non-ideal (repulsive) polymer chains free energy: A(R) S(R) = − = −ln[p(R)P(R, N)] kBT kBT 3R2 N 2ν ≈ cst− 2ln R + + c 2Nb2 2R3 minimize A(R) with respect to R

* 1/ 2 ν c = 0: R0 ∝ N b 5 3 ≈ * ’ ≈ * ’ R R ν c 1/ 2 ν c ≠ 0: ∆ * ÷ − ∆ * ÷ ≈ 3 N Flory «R0 ◊ «R0 ◊ b 2 * * 1/10 3/5 N 1: R ≈ R0 N ∝ N b SWELLING ! RG theory / simulations : R* ∝ N 0.588b Repulsion combined with attraction (1) Assumptions: • only attraction between nearest neighbours • coordination number: z • solvent-solvent solvent-polymer polymer-polymer

• random pair contacts: Repulsion combined with attraction (2) attractive energy within a coil:

Uattr = Nν c c(R) z (−∆ε )

(ν c c(R) is the probability to find a neighbouring segment)

U N 2 z∆ε z∆ε attr = − ν where ≡ χ (chi-parameter) k T R3 c k T k T B B B χ usually > 0 2 A(R) N ν c Compare with repulsive term in , which was: 3 kBT 2R

ν c →ν ≡ν c (1− 2χ )

5 3 ≈ * ’ ≈ * ’ R R ν c 1/ 2 ∆ * ÷ − ∆ * ÷ ≈ 3 (1− 2χ )N Flory «R0 ◊ «R0 ◊ b Repulsion combined with attraction (3)

5 3 ≈ * ’ ≈ * ’ R R ν c 1/ 2 ∆ * ÷ − ∆ * ÷ ≈ 3 (1− 2χ )N Flory «R0 ◊ «R0 ◊ b CONCLUSION: • at χ = 0 swollen chain R* ∝ N 3/5 * * 1/2 • at χ =1/ 2 (θ - temperature): ideal chain R = R0 ∝ N 1 cst • right-hand side only small if χ= − i.e. abrupt change if N large 2 N χ >1/ 2 : globule (bound state, cf. QM) R* ∝ N1/3

• in general R* - b Nν Repulsion combined with attraction (4)

polystyrene in cyclohexane Concentrated polymer solutions

c*

−1/2 −4/5 N N (N ↔ N ) c* ≈ ≈ ≈ very small ! R3 (bNν )3 b3

FLORY-HUGGINS: d • concentrated systems: S(R) NO

• homogeneous systems: SLifshitz NO • random mixing (Ω places) YES Flory-Huggins theory (1)

translational entropy: Str,sp ≈ −kBΩsp lnφsp (sp = species) φ polymer: A ≈ k TΩ lnφ m, p B N solvent: Am,s ≈ kBTΩ(1−φ)ln(1−φ) »1 ÿ Am ≈ ΩkBT … φ lnφ + (1−φ)ln(1−φ) + χφ(1−φ)Ÿ N ⁄ Flory-Huggins theory (2)

1 • φc = (highly) asymmetric 1+ N 1 1 • Tc follows from χ ≈ + c 2 N

Tc →θ for large N, i.e. near the coil → globule transition • note that fluctuations are neglected! Polystyrene in methylcyclohexane Scaling theory Semi-dilute solution (good solvent) N φ* ≈ b3c* ≈ b3 ≈ N −4/5 still very small ! (bN 3/5 )3

OSMOTIC PRESSURE TYPICAL LENGTH SCALE c DILUTE Π = k T ξ (= R)= bN 3/5 B N m c ≈ φ ’ m - SEMI-DILUTE ≈ φ ’ Π kBT ∆ ÷ - 3/5 N «φ * ◊ (power law) ξ bN ∆ ÷ «φ * ◊ - c 4/5m m 3/5 4/5m m kBT N φ independent of N - bN N φ N1 m = 5/4 m = -3/4 - 5/4 - kBT 9/4 Π kBTcφ φ −3/4 b3 ξ - bφ des Cloizeaux de Gennes What is the meaning of ksi ? (1)

k T k T Flory-Huggins: Π - B φ2 - B φ ×φ b3 b3 k T k T des Cloizeaux: Π - B φ9/4 - B φ ×φ5/4 b3 b3 probability of segment × probability of contact w

probability of contact w: lower for scaling theory (correlations!)

Flory-Huggins:

des Cloizeaux: What is the meaning of ksi ? (2) on one chain: number of segments between contacts?

g monomers: g - w−1 - φ −5/ 4

3/5 5/ 4 3/5 3/ 4 bg - b(φ − ) - bφ − - ξ distance between chain contacts !

1) volume fraction within one blob: gb3 φ −5/ 4b3 - - φ Ω blobs touch ! ξ 3 (bφ −3/ 4 )3 What is the meaning of ksi ? (3)

2) a SEMI-DILUTE SOLUTION is a collection of BLOBS !

de Gennes: ξ - bφ−3/4 k T k T 3) des Cloizeaux: Π - B φ9/4 - B blobs are the osmotic units b3 ξ 3

4) ξ is also the screening length for the excluded volume

5) φ →1: ξ(- bφ −3/ 4 ) → b chains in polymer melts are ideal! (Flory)