Block-Copolymer Ordering with a Spatiotemporally Heterogeneous Mobility August W. Bosse, Jack F. Douglas, Brian C. Berry, Ronald L. Jones, and Alamgir Karim Polymers Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8541, USA (Received 7 June 2007; revised manuscript received 7 August 2007; published 20 November 2007) Motivated by recent zone annealing measurements on stripe-forming block-copolymer ﬁlms [B. C. Berry et al., Nano Lett. 7, 2789 (2007)], we study block-copolymer ordering with a spatiotemporally heterogeneous mobility. Speciﬁcally, we implement a time- and space-dependent mobility ﬁeld in the relaxation of a diblock copolymer self-consistent ﬁeld theory. The model includes a gradient in the local mobility and intrinsic nanoscale mobility variations characteristic of glass phenomenology. The simula- tions demonstrate that a spatiotemporally heterogeneous mobility can have a signiﬁcant inﬂuence on microdomain ordering in block-copolymer systems, and that nanoscale dynamic heterogeneities asso- ciated with glass formation can impact the structure of the ordered block-copolymer microphase. DOI: 10.1103/PhysRevLett.99.216101 PACS numbers: 81.16.Rf, 64.70.Pf, 81.16.Dn, 81.40.Gh There is considerable scientiﬁc and technological inter- est in using microphase-separated block-copolymer (BCP) ﬁlms as lithographic masks in the fabrication of next- generation nanostructured devices (e.g., see Refs. [1,2]). However, if BCP thin ﬁlms are to evolve into a commer- cially viable nanofabrication material, one must have sig- niﬁcant control over microdomain ordering. Recently, there have been efforts in using time- and space-dependent thermal ﬁelds to improve order in BCP materials — a process termed zone reﬁnement or zone an- nealing. This approach was pioneered by Hashimoto and co-workers in bulk BCPs [3 – 5]. In addition, Angelescu et al. [6] and Berry et al. [7] have shown that zone anneal- ing can also be a powerful tool for controlling order in BCP ﬁlms. Perhaps the most unexpected aspect of this new work is the ﬁnding by Berry et al. that signiﬁcantly enhanced microdomain ordering can be obtained for glassy BCP ﬁlms zone annealed over a temperature range well below the microphase separation transition (MST) temperature T MST of the BCP melt. This observation suggests that dynamics — in particular, glassy dynamics — and the phys- ics of glass formation may have a signiﬁcant impact on BCP microdomain ordering. The observation of Berry et al. prompted our investiga- tion of how time-dependent thermal gradients could inﬂu- ence BCP ordering under conditions where the temperature is always below T MST and where the physics of glass formation is prevalent. Previous computational studies of directional ordering in BCPs have focused on temperature gradients that heat the system to temperatures above T MST [8,9]. However, these studies do not directly apply to the ordering of block copolymers under conditions where T< T MST . Speciﬁcally, we need to address physical aspects of glass formation that are relevant to BCP ordering for T< T MST . Accordingly, we explore the inﬂuence of a time- and space-dependent mobility ﬁeld associated with an applied time- and space-dependent thermal ﬁeld. We also incorpo- rate local mobility heterogeneities intended to model the intrinsic dynamic heterogeneities of glass-forming poly- mer liquids (see Ref. [10], and references therein). Previous work has indicated that this kind of heterogeneity can inﬂuence the degree of polycrystalline growth during crystallization [11], and we anticipate a similar effect for BCP ordering. We base our modeling and simulation on a standard polymer self-consistent ﬁeld theory (SCFT) where an ap- plied thermal gradient is modeled as a mobility gradient in the SCFT relaxation. The virtue of this technique, in com- parison to Ginzburg-Landau, phase-ﬁeld, and other simu- lation methods of BCP ordering, is that we can introduce dynamic heterogeneities in the mobility at a physically relevant scale. In this Letter, we show that gradients in the mobility ﬁeld have a signiﬁcant effect on the ordering process of the BCP, and that mobility heterogeneities associated with glass formation can indeed dramatically impact the structure of the resulting ordered BCP microphase. The polymer thin ﬁlm is modeled using a standard 2D SCFT for an incompressible melt of monodisperse AB diblock copolymers in a volume V , each with an index of polymerization N (all spatial variables are speciﬁed in units of R g [9], the unperturbed radius of gyration of a diblock copolymer). The fraction of the A block along the copolymer is denoted f. Individual diblocks are modeled as a Gaussian thread (i.e., a continuous bead-spring), and local A-segment –B-segment interactions are parametrized by a Flory  parameter [9]. The segment-segment inter- actions are decoupled to yield a formally equivalent ﬁeld theory in terms of two ‘‘chemical potential’’ ﬁelds W  and W  that are conjugate to the local total density and the local composition, respectively [9]. The average local vol- ume fractions of A segments,  A , and B segments,  B , are calculated as functional derivatives of the canonical parti- tion function with respect to W  . The mean-ﬁeld phase behavior of this model is fully determined by the values of f and the product N [9]. We use a saddle point approximation to calculate the mean-ﬁeld values of W  . This approximation is carried out PRL 99, 216101 (2007) PHYSICAL REVIEW LETTERS week ending 23 NOVEMBER 2007 0031-9007= 07=99(21)=216101(4) 216101-1