Formation of Interfaces in Incompatible Polymer Blends: A Dynamical Mean Field Study Chuck Yeung* ,† and An-Chang Shi ‡ School of Science, The Behrend College, Pennsylvania State University at Erie, Erie, Pennsylvania 16536, and Xerox Research Centre of Canada, 2660 Speakman Drive, Mississagua, Ontario, Canada L5K 2L1 Received October 22, 1998; Revised Manuscript Received March 23, 1999 ABSTRACT: The interdiffusion dynamics of a polymer interface towards its equilibrium profile is studied using a dynamical mean field method, in which the nonequilibrium chemical potentials are evaluated numerically using the full polymer mean field free energy. Our results demonstrated that the interfacial width grows as t 1/4 at early times and saturates to the equilibrium thickness at long times, in agreement with previous, more approximate analysis based on Cahn-Hilliard-type theories. Furthermore, we show that the dynamical exponent can only be obtained unequivocally via a scaling analysis. The interdiffusion of a polymer interface in the single phase regime is also studied. I. Introduction Polymer blends are used in increasing areas of ap- plication because they are cost-effective and have the potential for design of materials with properties tailored to a specific use. Because most polymer blends are composed of immiscible polymers, many blend proper- ties are determined by the characteristics of the inter- faces between the different components. To obtain good mechanical properties, good cohesion between phases is required, and different materials have to interpen- etrate into each other at the interfaces. This interpen- etration depends on the degree of miscibility of compo- nents and is generally neither easy to determine experimentally nor easy to predict theoretically. This has led to an intensive effort to understand the static and dynamic properties of polymer interfaces. The equilibrium properties of polymer interfaces are now relatively well-understood. 1 More recently interest has turned to the dynamics of polymer interfaces. In one set of experiments, a sharp interface between two polymer films was initially prepared, 2-5 and the broad- ening of the interface as the polymers interdiffuse was monitored via nuclear reaction analysis. These experi- ments found that the interdiffusion was subdiffusive with the interfacial width W growing as W(t) ∝ t R , where the exponent R ranged from about 0.25 to 0.4. It was also found that the exponent R decreases as the tem- perature is lowered. 3 All extant theoretical analysis of polymer interdiffu- sion starts from an approximate form for the coarse- grained free energy for a binary polymer blend, where φ is the volume fraction of A polymers and Λ is taken to be either a constant or, more accurately for polymers, Λ ) Λ o /[18φ(1 - φ)]. The local free energy density f o is chosen to be either of the Flory-Huggins type 6,7 or a fourth-order expansion in terms of φ(r). 8,9 For the dynamics of the system, Puri and Binder 8 and Wang and Shi 9 assumed that the dynamics are ap- proximated by the Cahn-Hilliard dynamics where Γ is the mobility. On the other hand, Harden derived a more complicated nonlocal form for the dynamics where Γ is a kernel derived from Gaussian dynamics. 7 Kawakatsu et al. has also performed a similar analysis using reptation dynamics. 12 Harden found that the interfacial width initially grows as t 1/2 followed by a t 1/4 growth regime. Puri and Binder also found that W(t) ∝ t 1/4 before saturating at the equilibrium interfacial thickness. 8 On the other hand, a similar numerical analysis by Wang and Shi found that the exponent R depended on temperature with R< 1 / 4 for all cases and R decreasing to 0.10 for lower temperatures. 9 The approximate form of the free energy used in these analysis almost guarantees that W ∝ t 1/4 because the free energy expression (eq 1) is expanded to second order in ∇φ so that the highest order gradient term in the dynamical equation (eq 2) is ∇ 4 φ. Simple dimensional analysis of eq 2 then gives t -1 ∝ W -4 or W ∝ t 1/4 . Such an expansion is appropriate close to the critical point but should not hold further away from the critical point. In general, the gradient expansion of the free energy functional must be expressed in the form of an infinite series, as shown recently by Tang and Freed. 10 Fur- thermore, it has been demonstrated that the square gradient approximation considerably overestimates the interfacial tension. 11 To understand the dynamics of the interfaces, it is desirable to develop a theory based on the exact free energy functional of the system. In this paper, the dynamics of polymer interfaces is studied using a dynamic mean field approach. Our theory adopts the numerical methods developed by Fraaije and coworkers 13-15 and by Hasegawa and Doi. 16,17 In this theory, the chemical potential δF/δφ is calculated from † Pennsylvania State University at Erie. ‡ Xerox Research Centre of Canada. F ) ∫ dr [ f o (φ) + Λ(φ) 2 |∇φ| 2 ] (1) ∂φ(r, t) ∂t ) ∇‚Γ(φ)∇ δF δφ(r′) (2) ∂φ(r, t) ∂t ) ∫ dr′ Γ(r, r′)‚∇ δF δφ(r′) (3) 3637 Macromolecules 1999, 32, 3637-3642 10.1021/ma981648n CCC: $18.00 © 1999 American Chemical Society Published on Web 05/06/1999