Suppression of Entangled Diblock Copolymer Diffusion at and below the Order-Disorder Transition Marina Guenza, † Hai Tang, and Kenneth S. Schweizer* Departments of Materials Science & Engineering and Chemistry, and Materials Research Laboratory, University of Illinois, 1304 West Green Street, Urbana, Illinois 61801 Received February 3, 1997 Revised Manuscript Received April 7, 1997 The structure, order-disorder transition (ODT), and dynamics of self-assembling diblock copolymer melts has been intensely studied in recent years. 1,2 Microdomain scale fluctuations strongly influence equilibrium proper- ties above and below the ODT and are rather well accounted for by coarse-grained field theory (BLFH) 3 and liquid state polymer reference interaction site model (PRISM) theory. 4,5 For unentangled, short diblocks, the influence of microdomain formation on translational diffusion appears to be weak. 6,7 However, transport of entangled diblocks is significantly retarded, becoming slower as the degree of polymerization and/or segrega- tion increases, thereby indicating a strong coupling of entanglement constraints and thermodynamically- driven microphase separation. 6,8,9 We have recently developed a microscopic polymer- mode coupling (PMC) theory of diffusion in entangled blends 10 and diblock copolymer 11 liquids which is a natural extension of the successful PMC theory of homopolymer solutions and melts. 12,13 This theory provides a good description 11 of self-diffusion and tracer diffusion measurements for quenched samples 6,8 of entangled symmetric copolymer polyolefin melts above, and slightly below, the ODT. More recently, Lodge and co-workers 9 have performed measurements over a much wider range of degree of polymerization and segregation conditions for shear-aligned lamellar microstructure samples. Rather surprisingly, the measured self-diffu- sion tensor exhibited modest anisotropy. Two distinc- tive regimes of diffusion suppression were observed: a thermally activated behavior, followed by a tempera- ture-independent regime under the strongest segrega- tion conditions where a factor of =100 reduction of the diffusion constant due to microdomain formation was observed. Qualitative interpretations were advanced on the basis of “activated reptation” 9,14 and “entropic arm- retraction” motional mechanisms. 9 The primary goal of this communication is generalize and apply the PMC theory to simultaneously treat diblock copolymer self-diffusion above and well below the ODT. This requires combining the PRISM theory of diblock melts, 5 including recent extensions to estimate the location of the ODT and quantitatively describe scattering data, 15 with the PMC approach. New dy- namic scaling laws are derived, and both model calcula- tions and quantitative applications to experiments are presented. Our work represents the first general treat- ment of the entangled copolymer problem and compari- son with recent experiments. Predictions for the chain relaxation time relevant to viscoelastic and dielectric measurements, and diblock tracer diffusion, are also briefly discussed. The simplest “structurally, interaction, composition- ally, and dynamically symmetric” model is adopted. 5,11 The AB diblocks are uncrossable Gaussian chains of N segments, of equimolar f ) 1 / 2 composition, identical segment lengths σ ) d (hard core diameter), a repulsive tail potential v AB (r) is adopted corresponding to a positive bare enthalpic -parameter, and a single seg- mental friction constant  0 , which characterizes the unentangled Rouse dynamics. The compressible melt has a reduced segment density Fσ 3 , or equivalently a density screening (or “packing”) length 5  F ) 3/(πFσ 2 ). Both PRISM and PMC theories are based on an isotropic liquid description. The self-diffusion constant is D ) (N) -1 , where  ) (k B T) -1 and  is the total friction constant per segment. Although not literally true below the ODT, the isotropic model is consistent with the PMC treatment of entanglements. Moreover, it seems reasonable on the basis of the experimental observations of (i) no discontinuity of D at the ODT, 6-9,16 (ii) strong (isotropic) fluctuations near and below the ODT, 1 (iii) weak anisotropy of D in shear-aligned samples, 8,9 and (iv) similarity of D measured in some quenched and oriented samples. 8,9 PMC theory com- putes the additional friction due to time-correlated intermolecular forces felt by a tagged copolymer. Based on a projection scheme (denoted below by a superscript “Q”) that (approximately) extracts the influence of slow dynamical processes and structural constraints, the general result is 10,11 The first equality is an exact formal expression where F R (t) is the total force exerted on segment R of the tagged chain by all segments on matrix copolymers. The second line follows from the basic PMC theory ap- proximations. 12,13 The wavevector integrals quantify contributions to the friction associated with correlated dynamical processes on a length scale 2π/k, and the sums are over species type (A or B). For t ) 0 the term in braces is the medium-induced “potential-of-mean force”, W MM′ , between tagged diblock segments of type M and M′ induced by interactions with the surrounding melt. 17 The latter are described by renormalized effec- tive potentials (or direct correlation functions) C MM′ ) C 0 + (1 - δ MM′ )F -1 , where C 0 is the k ) 0 value of the repulsive (hard core) contribution and  is the effective -parameter. 5 W MM′ has both total density and concen- tration fluctuation components. 10,11,17 The fluctuating force time correlations decay via collective melt motions, as described by the (projected) dynamic structure factor S MM′ Q (k,t), and via single tagged copolymer motions, as described by the intra- molecular dynamic structure factors ω MM′ Q (k,t). In the † Permanent address: Istituto di Studi Chimico-Fisici di Mac- romolecole Sintetiche e Naturali, IMAG, National Research Coun- cil, Via de Marini 6, Genova, Italy.  )  0 + 1 N ∑ R,γ ∫ 0 ∞ dt 〈F R (0)‚F γ Q (t)〉 )  0 + ( F 6π 2  29 ∫ 0 ∞ dt ∫ 0 ∞ dkk 4 ∑ M,M′ ω MM′ Q (k,t) × { ∑ p,p′ C Mp (k)S pp′ Q (k,t)C p′M′ (k)} )  0 + ( -C 0 6π 2  29 ∫ 0 ∞ dt ∫ 0 ∞ dkk 4 ∑ M,M′ ω MM′ Q (k,t) (1) 3423 Macromolecules 1997, 30, 3423-3426 S0024-9297(97)00141-1 CCC: $14.00 © 1997 American Chemical Society