Slow Logarithmic Coarsening in Crystalline 2D Systems

DANIEL A. VEGA; LEOPOLDO R. GÓMEZ

Mar del Plata - Argentina

Congreso; Pasi 2006; 2006

During the last four decades a great deal of effort was committed to understanding the mechanisms of pattern formation, ubiquitous in a number of diverse contexts [1-7]. With a few exceptions, it has been shown through numerous studies that different systems show a coarsening process satisfying scaling at long times [1]. In this case the dynamics can be characterized by a length scale x(t) that grows in time t as a power law (x ~ ta) [6]. Particularly, it has recently been found through simulations [4] and experiments [5] that in sphere-forming block copolymers thin films the orientational and translational correlation lengths grow in time according to different kinetic exponents. The difference in kinetic exponents has been attributed to a preferential annihilation of dislocations located along small angle grain boundaries [6]. Scaling in time has also been observed experimentally in thin films of block copolymers in the smectic phase [8-11]. In this case, it was shown that the orientational correlation length grows in time as x ~ t1/4 and that the dynamics is led by the annihilation of multipoles of disclinations. In order to understand the ways coarsening proceeds, several mechanisms have been proposed and studied: curvature driven grain growth, annihilation of topological defects and, to a lesser extent, grain rotation are examples of such mechanisms. More than forty years ago, Lifshitz predicted the possibility of formation of a stable lattice of domains on a system with p-fold degenerate equilibrium states. According to this picture, this lattice can be formed during the coarsening process as a consequence of the dynamic frustration to reach equilibrium [12]. As consequence of the relaxation driven by the curvature of grain boundaries, bounded regions where three grains meet can become pinned to their positions, slowing down the dynamics. Once the system becomes trapped into this dynamically stable state, the only path to induce further coarsening is through fluctuations or driving forces large enough to unlock the system from the local traps. Although this grain structure would not minimize the total free energy of the system, it was shown that it could be kinetically stable. The first steps to introduce Lifshitz’s ideas in the coarsening process quantitatively was made by Safran [13]. It was found that the domains grow according to a power law in time for p < d + 1 (d is the special dimensionallity), but logarithmically in time in the case p > d + 1. Although a few systems have been found where the growth of the correlation length is logarithmic [14-15], to the best of our knowledge, there are no systems clearly verifying the Lifshitz-Safran predictions at present. In this work the kinetic of ordering of a 2D hexagonal system in the region close to both, spinodal and order-order transitions was investigated. In this region of the phase diagram the dynamics is leaded by the triple points and Pierls-like forces are surmounted by the line tension. We found configurations of domains with the same features as those originally proposed by Lifshitz [16]. As a consequence, different correlation lengths grow logarithmically in time, in excellent agreement with the predictions of coarsening at low temperatures proposed by Safran. References: [1]       C. Bowman and A. C. Newell, Rev. Mod. Phys. 70, 289 (1998). [2]       D. R. Nelson and B. I. Halperin, Phys. Rev. B19, 2457 (1979). [3]       J. Toner and D. R. Nelson, Phys. Rev. B 23, 316 (1981). [4]       B. I. Halperin and D. R. Nelson, Phys. Rev. Lett. 41, 121 (1978). [5]       A. J. Bray, Adv. Phys. 43, 357 (1994). [6]       M. R. Evans, J. Phys.: Condens. Matter. 14, 1397 (2002). [7]       E. J. Kramer, Nature, 437, 824 (2005). [8]       C. K. Harrison, D. H. Adamson, Z. Cheng, J. M. Sebastian,S. Sethuraman, D. A. Huse, R. A. Register, P. M. Chaikin, Science 290, 1558 (2000). [9]       Harrison C. K., Z. Cheng, S. Sethuraman, D. A. Huse, P. M. Chaikin, D. A. Vega, J. M. Sebastian, R. A. Register, D. H. Adamson, Phys. Rev. E66, 011706 (2002). [10]     D. A. Vega, C. K. Harrison, D. E. Angelescu, M. L. Trawick, D. A. Huse, P. M. Chaikin, R. A. Register, Phys. Rev. E71 061803-1 (2005). [11]     C. K. Harrison, D. E. Angelescu, M. Trawick, Z. Cheng, D. A. Huse, P. M. Chaikin, D. A. Vega, J. M. Sebastian, R. A. Register, D. H. Adamson, Europhys. Lett. 65, 800 (2004). [12]     L. M. Lifshitz, Sov. Phys. JETP 15, 939 (1962). [13]     S. A. Safran, Phys. Rev. Lett. 46, 1581 (1981) [14]     J. D. Shore, M. Holzer, J. P. Sethna, Phys. Rev. B46, 376 (1992) [15]     D. Boyer, J. Viñals, Phys. Rev. Lett. 89, 055501 (2002). [16]     Gómez et al, Phys. Rev. Lett., in press.