Chapter 31 – STRUCTURE FACTORS FOR POLYMER SYSTEMS Up to now, this book has focused on infinitely dilute systems only. Such systems are non- interacting and require solely the calculation of the form factor P(Q) for isolated particles. More concentrated (or interacting) systems require the calculation of the structure factor S(Q). Structure factors for fully interacting polymer systems are considered here. These apply to semi-dilute and concentrated polymer solutions and polymer blend mixtures in the homogeneous phase. 1. SCATTERING FROM INCOMPRESSIBLE SYSTEMS Consider a system consisting of N “particles” of scattering length bP occupying the sample volume V. The following would still hold if the word “polymers” were substituted for the word “particles”. The scattering cross section is proportional to the density-density correlation function as follows: dΣ(Q) 2 1 N r r 2 1 = bP ∑< exp()− iQ.rij >= bP < n P (−Q)n P (Q) > . (1) dΩ V i,j V Here nP(Q) is the fluctuating particle density in Fourier space. The cross section for particles in solution is given by: dΣ(Q) 1 1 = b 2 < n (−Q)n (Q) > +b 2 < n (−Q)n (Q) > dΩ P V P P S V S S 1 + 2b b < n (−Q)n (Q) > . (2) P S V P S The subscripts P and S stand for particle and solvent respectively. For the sake of convenience, the following scattering factors are defined: v 2 S (Q) = P < n (−Q)n (Q) > (3) PP V P P v 2 S (Q) = S < n (−Q)n (Q) > SS V S S v v S (Q) = P S < n (−Q)n (Q) > . PS V P S The specific volumes vP and vS and scattering length densities ρP = b P v P and ρS = bS vS are defined for the polymer and the solvent respectively. To clarify, vP is the 1 monomer volume and vS is the volume of the solvent molecule. The scattering cross section becomes: dΣ(Q) = ρ 2S (Q) + ρ 2S (Q) + 2ρ ρ S (Q) . (4) dΩ P PP S SS P S PS Most scattering systems are incompressible. It is often convenient to make the following incompressibility assumption: v P n P (Q) + vSn S (Q) = 0. (5) This introduces the following simplification: 2 2 v P < n P (−Q)n P (Q) > = vS < n S (−Q)n S (Q) > (6) = −v P vS < n P (−Q)n S (Q) > . In other words: SPP (Q) = SSS (Q) = −SPS (Q) = −SSP (Q) (7) This simplifies the cross section to the following form: dΣ(Q) = (ρ − ρ ) 2 S (Q) = Δρ2S (Q) . (8) dΩ P S PP PP This is reasonable since the contrast factor Δρ2 is always calculated relative to a “background” scattering length density value. Here, the solvent’s scattering length density is taken to be that reference value. 2. INTER-PARTICLE INTERACTIONS Consider a system consisting of N polymers of contrast factor Δρ2 occupying volume V. Each polymer comprises n monomers of volume v each so that the polymer volume is vP = nv. Let us separate out the intra-polymer and the inter-polymer terms in the scattering cross section as follows: 2 Nn Nn dΣ(Q) 2 v ⎡ r r r r ⎤ = Δρ ⎢ ∑∑< exp()− iQ.rαiβj > + ∑∑< exp ()− iQ.rαiβj >⎥ . (9) dΩ V ⎣αβ=β i,j α≠ i,j ⎦ The indices α and β run over the polymer chains and the indices i and j run over the monomers in a specific polymer chain. Consider a pair of polymer coils (called 1 and 2) and sum over all pairs. 2 2 n n dΣ(Q) 2 v ⎡ r r r r ⎤ = Δρ ⎢N∑ < exp()− iQ.r1i1j > + N(N −1)∑< exp ()− iQ.r1i2 j >⎥ . (10) dΩ V ⎣ i,j i,j ⎦ Note that this formalism holds if the word “particles” were to be substituted for the word “polymers” assuming (of course) that the particles have internal structure (think monomers). coil 1 r S 1i r r1i2 j r R12 r r S2 j r r1i R1 r coil 2 r2 j r R 2 Figure 1: Schematic representation of the coordinate system showing a pair of scatterers that belong to two different polymer coils. r The inter-distance between the scattering pair r1i2 j can be expressed as r r r r r1i2 j = −S1i + S2 j + R12 and the inter-particle average can be split into the following parts: r r r r r r r r < exp(− iQ.r1i2 j )> = < exp(iQ.S1i )>< exp(− iQ.S2 j )>< exp(− iQ.R12 ) > . (11) The first two averages are within single particles and the third average is across particles. The summations become: n r r r r n r r n r r ∑ < exp()− iQ.S1i2 j > =< exp(− iQ.R12 )> ∑ < exp(iQ.S1i )()>∑ < exp − iQ.S2 j > .(12) i,j i j The form factor amplitude is defined as: 1 n r r 1 n r r F(Q) = ∑ < exp()− iQ.S1i > = ∑ < exp(− iQ.S2 j )> . (13) n i n j 3 The single-particle form factor itself is defined as: 1 n r r P(Q) = < exp − iQ.S > . (14) 2 ∑ ()1i1j n i,j For uniform density particles, the following relation holds P(Q) =| F(Q) |2 . This is not true, however, for non-uniform density object such as polymer coils. An inter-particle structure factor is defined as: 1 N r r SI (Q) = ∑ < exp()− iQ.R αβ > . (15) N α,β The cross section can therefore be written as follows: dΣ(Q) v 2 n 2 N = Δρ2 []P(Q)+ | F(Q) |2 ()S (Q) −1 . (16) dΩ V I r r Note that the statistical average < exp(iQ.r1i2 j ) > involves integration over the following r r r probability distribution P(r1i , r2 j ,R12 ) which can be split to show a conditional probability r r r r r r r P(r1i , r2 j ,R12 ) = P(r1i , r2 j | R12 )P(R12 ) . For compact scatterers which do not interfere with r r r r r each other’s rotation P(r1i , r2 j | R12 ) is independent of R12 . P(R12 ) is the probability of r finding the centers of mass of polymer coils 1 and 2 a distance R12 apart. r r N r r r r S (Q) = N < exp()− iQ.R >= ∫ dR P(R )exp(− iQ.R ). (17) I 12 V 12 12 12 The cross section for systems in this case is given by: 2 2 2 dΣ(Q) 2 v n N ⎡ | F(Q) | ⎤ = Δρ P(Q)⎢1+ ()SI (Q) −1 ⎥ . (18) dΩ V ⎣ P(Q) ⎦ This result applies to systems with non-spherical symmetry and non-uniform density such as polymers. Polymer are, however, so highly entangled that an inter-chain structure factor SI(Q) is meaningless except for dilute solutions whereby polymer coils do not overlap. Inter-chain interactions for polymer systems are better handled using other methods described below. Uniform density scatterers (such as particles) are characterized by P(Q) =| F(Q) |2 , so that: 4 dΣ(Q) v 2 n 2 N = Δρ2 P(Q)S (Q) . (19) dΩ V I Defining a particles’ volume fraction as φ = Nnv/V, the following result is obtained: dΣ(Q) = Δρ2S(Q) (20) dΩ S(Q) = nφvP(Q)SI (Q) . This is a well-known result. It is included here even-though it does not apply to polymer systems so that the derivation does not have to be repeated when covering scattering from particulate systems later. Note that the scattering factor S(Q) and the inter-particle structure factor SI(Q) should not be confused; S(Q) has the dimension of a volume whereas SI(Q) is dimensionless. 3. THE PAIR CORRELATION FUNCTION Recall the definition for the inter-particle structure factor for a pair of particles (named 1 and 2): r r 1 N r r SI (Q) = N < exp()− iQ.R12 >= ∑< exp(− iQ.R εβ )> (21) N α,β N 1 3 r r r = ∑ ∫ d R αβ exp(− iQ.R αβ )P(R αβ ) . NV α,β r 3 r P(R αβ ) is the probability of finding particle β in volume d Rαβ a distance R αβ away given that particle α at the origin. When the self term (α = β) is omitted, this result becomes: N 1 3 r r r r SI (Q) −1 = ∑ ∫ d R εβ exp(− iQ.R αβ )P(R αβ ) (22) NV α≠β N 3 r r r r = ∫ d R exp(− iQ.R )P(R ) . V 12 12 12 r The probability P(R12 ) is referred to as the pair correlation function and is often called r g(R12 ) . Removing the forward scattering term yields the following well known result: N 3 r r r 3 S (Q) −1 = ∫ d R exp(− iQ.R )[g(R ) −1]+ (2π) δ(Q) . (23) I V 12 12 12 5 The last term (containing the Dirac Delta function) is irrelevant and can be neglected. r This last equation shows that SI (Q) −1 and g(R12 ) −1 are a Fourier transform pair. Note r that g(R12 ) peaks at the first nearest-neighbor shell and goes asymptotically to unity at r r large distances. The total correlation function is introduced as h(R12 ) = g(R12 ) −1. 4. POLYMER SOLUTIONS In the case of polymer solutions, the Zimm single-contact approximation (Zimm, 1946; Zimm, 1948) is a simple way of expressing the inter-polymer structure factor. Within that approximation, the first order term in a “concentration” expansion is as follows: 2 n v ⎛ v2 ⎞ r r ex ⎜ 2 ⎟ ∑ < exp()− iQ.r1i2 j > = − ⎜ n P(Q)⎟ + ..