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Grzetic DJ, Delaney KT, Fredrickson GH. Field-Theoretic Study of Salt-Induced Order and Disorder in a Polarizable Diblock Copolymer. ACS Macro Lett 2019; 8:962-967. [PMID: 35619489 DOI: 10.1021/acsmacrolett.9b00316] [Citation(s) in RCA: 19] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
We study a salt-doped polarizable symmetric diblock copolymer using a recently developed field theory that self-consistently embeds dielectric response, ion solvation energies, and van der Waals (vdW) attractions via the incorporation of segment polarizabilities and fixed dipoles. This field theory is amenable to direct simulation via the complex Langevin sampling technique and, thus, requires no approximations beyond the phenomenology of the underlying molecular model. We measure the shift in the order-disorder transition (ODT) of a diblock copolymer with salt-loading in field-theoretic simulations and observe rich behavior in which solvation, dilution and charge screening effects compete to determine whether the ordered or disordered phase is stabilized. At low salt concentrations, the salt behaves as a selective solvent, localizing into the high-dielectric domains and stabilizing the ordered phase. At high salt concentrations, however, the salt localization vanishes due to charge screening effects, and the salt behaves as a nonselective solvent that screens vdW attractions and stabilizes the disordered phase.
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Tripathi AK, Kumar D, Puri S. Coarsening dynamics in the Swift-Hohenberg equation with an external field. Phys Rev E 2019; 99:022136. [PMID: 30934234 DOI: 10.1103/physreve.99.022136] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/24/2018] [Indexed: 11/07/2022]
Abstract
We study the Swift-Hohenberg equation (SHE) in the presence of an external field. The application of the field leads to a phase diagram with three phases, i.e., stripe, hexagon, and uniform. We focus on coarsening after a quench from the uniform to stripe or hexagon regions. For stripe patterns, we find that the length scale associated with the order-parameter structure factor has the same growth exponent (≃1/4) as for the SHE with zero field. The growth process is slower in the case of hexagonal patterns, with the effective growth exponent varying between 1/6 and 1/9, depending on the quench parameters. For deep quenches in the hexagonal phase, the growth process stops at late stages when defect boundaries become pinned.
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Affiliation(s)
- Ashwani K Tripathi
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi - 110067, India.,Center for Soft and Living Matter, Institute for Basic Science, Ulsan 44919, Republic of Korea
| | - Deepak Kumar
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi - 110067, India
| | - Sanjay Puri
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi - 110067, India
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Grzetic DJ, Delaney KT, Fredrickson GH. The effective χ parameter in polarizable polymeric systems: One-loop perturbation theory and field-theoretic simulations. J Chem Phys 2018; 148:204903. [DOI: 10.1063/1.5025720] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Douglas J. Grzetic
- Materials Research Laboratory, University of California, Santa Barbara, California 93106, USA
| | - Kris T. Delaney
- Materials Research Laboratory, University of California, Santa Barbara, California 93106, USA
| | - Glenn H. Fredrickson
- Materials Research Laboratory, University of California, Santa Barbara, California 93106, USA
- Departments of Chemical Engineering and Materials, University of California, Santa Barbara, California 93106, USA
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Carilli MF, Delaney KT, Fredrickson GH. Nucleation of the lamellar phase from the disordered phase of the renormalized Landau-Brazovskii model. J Chem Phys 2018; 148:054903. [DOI: 10.1063/1.5003150] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/24/2022] Open
Affiliation(s)
- Michael F. Carilli
- Department of Physics, University of California, Santa Barbara, California 93106, USA
| | - Kris T. Delaney
- Materials Research Laboratory, University of California, Santa Barbara, California 93106, USA
| | - Glenn H. Fredrickson
- Materials Research Laboratory and Department of Chemical Engineering, University of California, Santa Barbara, California 93106, USA
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5
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Riesch C, Radons G, Magerle R. Scaling properties of ageing orientation fluctuations in stripe phases. Interface Focus 2017; 7:20160146. [PMID: 28630676 PMCID: PMC5474038 DOI: 10.1098/rsfs.2016.0146] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
We investigate the non-equilibrium dynamics of an ordered stripe-forming system free of topological defects. In particular, we study the ageing and the coarsening of orientation fluctuations parallel and perpendicular to the stripes via computer simulations based on a minimal phase-field model (model B with Coulomb interactions). Under the influence of noise, the stripe orientation field develops fluctuations parallel to the stripes, with the dominant modulation length λ*∥ increasing with time t as λ*∥ ∼ t1/4 and the correlation length perpendicular to the stripes ξ⊥θ increasing as ξ⊥θ ∼ t1/2. We explain these anisotropic coarsening dynamics with an analytic theory based on the linear elastic model for stripe displacements first introduced by Landau and Peierls. We thus obtain the scaling forms and the scaling exponents characterizing the correlation functions and the structure factor of the stripe orientation field. Our results reveal how the coarsening of orientation fluctuations prevents a periodically modulated phase free of topological defects from reaching equilibrium.
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Affiliation(s)
- Christian Riesch
- Institut für Physik, Technische Universität Chemnitz, 09107 Chemnitz, Germany
| | - Günter Radons
- Institut für Physik, Technische Universität Chemnitz, 09107 Chemnitz, Germany
| | - Robert Magerle
- Institut für Physik, Technische Universität Chemnitz, 09107 Chemnitz, Germany
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Kim S, Li W, Fredrickson GH, Hawker CJ, Kramer EJ. Order-disorder transition in thin films of horizontally-oriented cylinder-forming block copolymers: thermal fluctuations vs. preferential wetting. SOFT MATTER 2016; 12:5915-5925. [PMID: 27334558 DOI: 10.1039/c6sm00739b] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
We present experimental and theoretical investigations of the order-disorder transition (ODT) in thin films of cylinder-forming diblock copolymers with asymmetric wetting conditions. Grazing incidence small-angle X-ray scattering (GISAXS) was implemented to determine the ODT temperatures (TODT) for poly(styrene-b-2-vinyl pyridine) (PS-P2VP) block copolymer thin films on a P2VP-preferential silicon substrate. Specifically, films consisting of multilayers of horizontally-oriented cylindrical structures (from 1- to 9-layers) were tested. We find that films with more than 2 cylindrical layers have a TODT comparable to the bulk case. However, TODT decreases as the film becomes thinner and the monolayer system has an ODT 30 °C below the bulk. Using self-consistent field theory (SCFT), we studied the ordering in corresponding thin films with asymmetric (top and bottom surface) wetting conditions. Surprisingly, SCFT is found to predict an opposite trend in TODT with film thickness than observed experimentally. Field-theoretic simulations with complex Langevin sampling were employed to resolve this discrepancy and demonstrate that thermal fluctuations in the PS-P2VP thin-film system dominate its ordering behavior in monolayer and bilayer films.
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Affiliation(s)
- Sangwon Kim
- Materials Research Laboratory, University of California, Santa Barbara, California 93106, USA. and Department of Polymer Science and Engineering, Inha University, Incheon 402-751, Republic of Korea
| | - Wei Li
- Materials Research Laboratory, University of California, Santa Barbara, California 93106, USA.
| | - Glenn H Fredrickson
- Materials Research Laboratory, University of California, Santa Barbara, California 93106, USA. and Department of Chemical Engineering, University of California, Santa Barbara, California 93106, USA
| | - Craig J Hawker
- Materials Research Laboratory, University of California, Santa Barbara, California 93106, USA. and Materials Department, University of California, Santa Barbara, California 93106, USA
| | - Edward J Kramer
- Materials Research Laboratory, University of California, Santa Barbara, California 93106, USA. and Department of Chemical Engineering, University of California, Santa Barbara, California 93106, USA and Materials Department, University of California, Santa Barbara, California 93106, USA
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7
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Tripathi AK, Kumar D. Coarsening of stripe patterns: variations with quench depth and scaling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:022923. [PMID: 25768585 DOI: 10.1103/physreve.91.022923] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/30/2014] [Indexed: 06/04/2023]
Abstract
The coarsening of stripe patterns when the system is evolved from random initial states is studied by varying the quench depth ε, which is a measure of distance from the transition point of the stripe phase. The dynamics of the growth of stripe order, which is characterized by two length scales, depends on the quench depth. The growth exponents of the two length scales vary continuously with ε. The decay exponents for free energy, stripe curvature, and densities of defects like grain boundaries and dislocations also show similar variation. This implies a breakdown of the standard picture of nonequilibrium dynamical scaling. In order to understand the variations with ε we propose an additional scaling with a length scale dependent on ε. The main contribution to this length scale comes from the "pinning potential," which is unique to systems where the order parameter is spatially periodic. The periodic order parameter gives rise to an ε-dependent potential, which can pin defects like grain boundaries, dislocations, etc. This additional scaling provides a compact description of variations of growth exponents with quench depth in terms of just one exponent for each of the length scales. The relaxation of free energy, stripe curvature, and the defect densities have also been related to these length scales. The study is done at zero temperature using Swift-Hohenberg equation in two dimensions.
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Affiliation(s)
- Ashwani K Tripathi
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi-110067, India
| | - Deepak Kumar
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi-110067, India
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Riesch C, Radons G, Magerle R. Aging of orientation fluctuations in stripe phases. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:052101. [PMID: 25493734 DOI: 10.1103/physreve.90.052101] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/01/2014] [Indexed: 06/04/2023]
Abstract
Stripe patterns, observed in a large variety of physical systems, often exhibit a slow nonequilibrium dynamics because ordering is impeded by the presence of topological defects. Using computer simulations based on a well-established model for stripe formation, we show that a slow dynamics and aging occur also in stripe patterns free of topological defects. For a wide range of noise strengths, the two-time orientation correlation function follows a scaling form that is typical for systems exhibiting a growing length scale. In our case, the underlying mechanism is the coarsening of orientation fluctuations, ultimately leading to power-law spatial correlations perpendicular to the stripes. Our results show that even for the smallest amount of noise, stripe phases without topological defects do not reach equilibrium. This constitutes an important aspect of the dynamics of modulated phases.
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Affiliation(s)
- Christian Riesch
- Institut für Physik, Technische Universität Chemnitz, D-09107 Chemnitz, Germany
| | - Günter Radons
- Institut für Physik, Technische Universität Chemnitz, D-09107 Chemnitz, Germany
| | - Robert Magerle
- Institut für Physik, Technische Universität Chemnitz, D-09107 Chemnitz, Germany
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Zhang L, Wang L, Lin J. Defect structures and ordering behaviours of diblock copolymers self-assembling on spherical substrates. SOFT MATTER 2014; 10:6713-6721. [PMID: 25069010 DOI: 10.1039/c4sm01180e] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
One of the main differences of ordered structures constrained on curved surfaces is the nature of topological defects. We here explore the defect structures and ordering behaviours of both lamellar and cylindrical phases of block copolymers confined on spherical substrates by the Landau-Brazovskii theory, which is numerically solved by a highly accurate spectral method with a spherical harmonic basis. For the cylindrical phase, isolated disclinations and scars are generated on the spherical substrates. The number of excess dislocations in a scar depends linearly on the sphere radius. The defect fraction characterizing the ordering dynamics decays exponentially. The scars are formed from the isolated disclinations via mini-scars. For the lamellar phase, three types of defect structures (hedgehog, spiral and quasi-baseball) are identified. The disclination annihilation is the primary ordering mechanism of the lamellar phase.
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Affiliation(s)
- Liangshun Zhang
- Shanghai Key Laboratory of Advanced Polymeric Materials, State Key Laboratory of Bioreactor Engineering, Key Laboratory for Ultrafine Materials of Ministry of Education, School of Materials Science and Engineering, East China University of Science and Technology, Shanghai 200237, China. jlin@ ecust.edu.cn
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10
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Tripathi AK, Kumar D. Stripe patterns: role of initial state and boundary conditions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:022915. [PMID: 25215807 DOI: 10.1103/physreve.90.022915] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/03/2014] [Indexed: 06/03/2023]
Abstract
This paper presents results on stripe patterns by numerical solution of the Swift-Hohenberg equation. The focus is on the role of initial state and boundary conditions. We choose initial states which generate simple defect configurations and study their evolution. Various classes of defects are identified and their motion and relaxation is studied numerically. We first study the dynamics of a straight front and present a comparison of numerical results with some analytical results. We then study the domain-wall dynamics in configurations containing two and three domains and identify some mechanisms of their relaxation. Rates of domain-wall relaxation depend on several features like incommensuration, dislocations and orientations in neighboring domains, in addition to the curvature of the walls. For a generic class of domain walls the relaxation process has an intrinsic frustration which leads to generation of dislocations. This process also generates stripe curvature thereby making relaxation nonmonotonic. We have also generated some other topological defects and studied their evolution and the effect of boundary conditions on their stability.
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Affiliation(s)
- Ashwani K Tripathi
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India
| | - Deepak Kumar
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India
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11
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Xie N, Li W, Zhang H, Qiu F, Shi AC. Kinetics of lamellar formation on sparsely stripped patterns. J Chem Phys 2013; 139:194903. [PMID: 24320351 DOI: 10.1063/1.4830396] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Chemical epitaxy based on the self-assembly of block copolymers is viewed as a promising technique to achieve ordered patterns on a large scale. Herein, we study the kinetics of lamellar formation of block copolymers under the direction of sparsely stripped patterns using cell dynamics simulations of the time-dependent Ginzburg-Landau theory. First, a scaling law is unveiled with the ordering time of lamellae, tp, with respect to the multiples between the periods of lamellae and stripe patterns, which is consistent with the power law evolution of the correlation length existing in the bulk phase of lamellae. Second, the tolerative windows of perfect order, with deviation from integer multiples, are also estimated from the aspect of kinetics. The results of the ordering time and tolerative windows are of great interest for relevant experiments or applications. Finally, a two-stage evolution is explored during the pattern formation of chemical epitaxy by probing into the evolution of defects, which is of fundamental interest for us to understand the coarsening kinetics of block copolymers under the direction of chemical patterns.
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Affiliation(s)
- Nan Xie
- State Key Laboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
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Ohnogi H, Shiwa Y. Nucleation, growth, and coarsening of crystalline domains in order-order transitions between lamellar and hexagonal phases. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:011611. [PMID: 21867186 DOI: 10.1103/physreve.84.011611] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/27/2011] [Revised: 04/04/2011] [Indexed: 05/31/2023]
Abstract
Using the numerical solution of the time-dependent Ginzburg-Landau equation, we study the entire process of transformation between the lamellar and the hexagonal phases from the early-stage nucleation and growth to the late-stage coarsening regime. The metastable crystalline structure that nucleates first is identified in terms of the mean-field theory under the single-wave-number approximation. This has been borne out by the numerically efficient preparation of single-crystal structure developed via the noise-induced self-organization. We also present results for the scaling of the late-time domain growth, which is quantified by two measures: the structure factor and the orientational correlation function. In particular, the growth exponent is shown to be robust and indifferent to conservation of the order parameter.
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Affiliation(s)
- H Ohnogi
- Statistical Mechanics Laboratory, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto 606-8585, Japan
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Kumaran V, Raman DSS. Shear alignment of a disordered lamellar mesophase. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:031501. [PMID: 21517501 DOI: 10.1103/physreve.83.031501] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/31/2009] [Revised: 11/19/2010] [Indexed: 05/30/2023]
Abstract
The shear alignment of an initially disordered lamellar phase is examined using lattice Boltzmann simulations of a mesoscopic model based on a free-energy functional for the concentration modulation. For a small shear cell of width 8λ, the qualitative features of the alignment process are strongly dependent on the Schmidt number Sc=ν/D (ratio of kinematic viscosity and mass diffusion coefficient). Here, λ is the wavelength of the concentration modulation. At low Schmidt number, it is found that there is a significant initial increase in the viscosity, coinciding with the alignment of layers along the extensional axis, followed by a decrease at long times due to the alignment along the flow direction. At high Schmidt number, alignment takes place due to the breakage and reformation of layers because diffusion is slow compared to shear deformation; this results in faster alignment. The system size has a strong effect on the alignment process; perfect alignment takes place for a small systems of width 8λ and 16λ, while a larger system of width 32λ does not align completely even at long times. In the larger system, there appears to be a dynamical steady state in which the layers are not perfectly aligned--where there is a balance between the annealing of defects due to shear and the creation due to an instability of the aligned lamellar phase under shear. We observe two types of defect creation mechanisms: the buckling instability under dilation, which was reported earlier, as well as a second mechanism due to layer compression.
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Affiliation(s)
- V Kumaran
- Department of Chemical Engineering, Indian Institute of Science, Bangalore 560012, India
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Mkhonta SK, Elder KR, Grant M. Novel mechanical properties in lamellar phases of liquid-crystalline diblock copolymers. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2010; 32:349-355. [PMID: 20680381 DOI: 10.1140/epje/i2010-10636-9] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/12/2010] [Accepted: 07/06/2010] [Indexed: 05/29/2023]
Abstract
Structural properties of flexible nematic diblock copolymers in the lamellar phase are investigated using a mean-field model. We address two complementary questions on the mechanics of the system: 1) How does the nematic order affect the elasticity of the one-dimensional solid? 2) What effect does the block copolymer microstructure has on the orientation of the nematic director? In the limit when the microstructure does not influence the nematic director orientation we predict a soft lamellar compression mode. When the microstructure does influence the nematic director orientation, small compressions lead to conventional elasticity, until a critical strain is reached, where there is then a transition to a softer response. On the other hand, we show that an identifiable lamellar symmetry provides a direction along which the nematic director prefers to align. Our model provides avenues to explore nonlinear properties of flexible diblock copolymers in which the monomers on both sides have mesogenic side groups.
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Affiliation(s)
- S K Mkhonta
- Department of Physics, University of Swaziland, Kwaluseni, Swaziland.
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15
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Song KX, Sun ZY, An LJ. Defect evolution and hydrodynamic effects in lamellar ordering process of two-dimensional quenched block copolymers. J Chem Phys 2009; 130:124907. [DOI: 10.1063/1.3090483] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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16
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Song KX, Jia YX, Sun ZY, An LJ. Lattice Boltzmann study of hydrodynamic effects in lamellar ordering process of two-dimensional quenched block copolymers. J Chem Phys 2008; 129:144901. [DOI: 10.1063/1.2981804] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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17
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Gomez-Solano JR, Boyer D. Coarsening in potential and nonpotential models of oblique stripe patterns. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:041131. [PMID: 17994960 DOI: 10.1103/physreve.76.041131] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/16/2007] [Indexed: 05/25/2023]
Abstract
We study the coarsening of two-dimensional oblique stripe patterns by numerically solving potential and nonpotential anisotropic Swift-Hohenberg equations. Close to onset, all models exhibit isotropic coarsening with a single characteristic length scale growing in time as t1/2. Further from onset, the characteristic lengths along the preferred directions x and ŷ grow with different exponents, close to 1/3 and 1/2, respectively. In this regime, one-dimensional dynamical scaling relations hold. We draw an analogy between this problem and model A in a stationary, modulated external field. For deep quenches, nonpotential effects produce a complicated dislocation dynamics that can lead to either arrested or faster-than-power-law growth, depending on the model considered. In the arrested case, small isolated domains shrink down to a finite size and fail to disappear. A comparison with available experimental results for electroconvection in nematics is presented.
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Affiliation(s)
- J R Gomez-Solano
- Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 México, Distrito Federal, Mexico
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Berry BC, Bosse AW, Douglas JF, Jones RL, Karim A. Orientational order in block copolymer films zone annealed below the order--disorder transition temperature. NANO LETTERS 2007; 7:2789-94. [PMID: 17691851 DOI: 10.1021/nl071354s] [Citation(s) in RCA: 77] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/16/2023]
Abstract
We report measurements of rapid ordering and preferential alignment in block copolymer films zone annealed below the order-disorder transition temperature. The orientational correlation lengths measured after approximately 5 h above the glass-transition temperature ( approximately 2 microm) were an order of magnitude greater than that obtained under equivalent static annealing. The ability to rapidly process polymers with inaccessible order-disorder transition temperatures suggests zone annealing as a route toward more robust nanomanufacturing methods based on block copolymer self-assembly.
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Affiliation(s)
- Brian C Berry
- Polymers Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8541, USA
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Griffith M, Dennin M. Impact of noise on domain growth in electroconvection. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:027201. [PMID: 17025568 DOI: 10.1103/physreve.74.027201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/05/2005] [Indexed: 05/12/2023]
Abstract
The growth and ordering of striped domains has recently received renewed attention due in part to experimental studies in diblock copolymers and electroconvection. One surprising result has been the relatively slow dynamics associated with the growth of striped domains. One potential source of the slow dynamics is the pinning of defects in the periodic potential of the stripes. Of interest is whether or not external noise will have a significant impact on the domain ordering, perhaps by reducing the pinning and increasing the rate of ordering. In contrast, we present experiments using electroconvection in which we show that a particular type of external noise decreases the rate of domain ordering.
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Affiliation(s)
- Mary Griffith
- Department of Physics and Astronomy, University of California at Irvine, Irvine, CA 92697-4575, USA
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Xu A, Gonnella G, Lamura A. Morphologies and flow patterns in quenching of lamellar systems with shear. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:011505. [PMID: 16907098 DOI: 10.1103/physreve.74.011505] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/22/2005] [Revised: 03/02/2006] [Indexed: 05/11/2023]
Abstract
We study the behavior of a fluid quenched from the disordered into the lamellar phase under the action of a shear flow. The dynamics of the system is described by Navier-Stokes and convection-diffusion equations with the pressure tensor and the chemical potential derived by the Brazovskii free energy. Our simulations are based on a mixed numerical method with the lattice Boltzmann equation and a finite difference scheme for Navier-Stokes and order parameter equations, respectively. We focus on cases where banded flows are observed with two different slopes for the component of velocity in the direction of the applied flow. Close to the walls the system reaches a lamellar order with very few defects, and the slope of the horizontal velocity is higher than the imposed shear rate. In the middle of the system the local shear rate is lower than the imposed one, and the system looks like a mixture of tilted lamellae, droplets, and small elongated domains. We refer to this as a region with a shear-induced structures (SIS) configuration. The local behavior of the stress shows that the system with the coexisting lamellar and SIS regions is in mechanical equilibrium. This phenomenon occurs, at fixed viscosity, for shear rates under a certain threshold; when the imposed shear rate is sufficiently large, lamellar order develops in the whole system. Effects of different viscosities have been also considered. The SIS region is observed only at low enough viscosity. We compare the above scenario with the usual one of shear banding. In particular, we do not find evidence for a plateau of the stress at varying imposed shear rates in the region with banded flow. We interpret our results as due to a tendency of the lamellar system to oppose the presence of the applied flow.
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Affiliation(s)
- Aiguo Xu
- Dipartimento di Fisica, Università di Bari, via Amendola 173, 70126 Bari, Italy
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Funfschilling D, Dennin M. Early time evolution of Freédericksz patterns generated from states of electroconvection. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:057201. [PMID: 16803079 DOI: 10.1103/physreve.73.057201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/28/2005] [Revised: 01/13/2006] [Indexed: 05/10/2023]
Abstract
We report on the early time ordering in a nematic liquid crystal subjected to a sudden change in an external ac electric field. We compare time evolution for two different initial states of electroconvection. Electroconvection is a highly driven state of a nematic liquid crystal involving convective motion of the fluid and periodic variations of the molecular alignment. By suddenly changing either the voltage or the frequency of the applied ac field, the system is brought to the same thermodynamic conditions. The time ordering of the system is characterized by the evolution of features of the power spectrum, including the average wave number, total power, and shape of the power spectrum. We observe that ordering of the system occurs faster after a sudden change in frequency than it does after a sudden change in voltage.
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Affiliation(s)
- Denis Funfschilling
- Department of Physics and Astronomy, University of California at Irvine, 92697-4575, USA
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Reichhardt C, Olson Reichhardt CJ. Coarsening of topological defects in oscillating systems with quenched disorder. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:046122. [PMID: 16711893 DOI: 10.1103/physreve.73.046122] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/30/2005] [Revised: 01/09/2006] [Indexed: 05/09/2023]
Abstract
We use large scale simulations to study interacting particles in two dimensions in the presence of both an ac drive and quenched disorder. As a function of ac amplitude, there is a crossover from a low drive regime where the colloid positions are highly disordered to a higher ac drive regime where the system dynamically reorders. We examine the coarsening of topological defects formed when the system is quenched from a disordered low ac amplitude state to a high ac amplitude state. When the quench is performed close to the disorder-order crossover, the defect density decays with time as a power law with alpha = 1/4 to 1/3. For deep quenches, in which the ac drive is increased to high values such that the dynamical shaking temperature is strongly reduced, we observe a logarithmic decay of the defect density into a grain boundary dominated state. We find a similar logarithmic decay of defect density in systems containing no pinning. We specifically demonstrate these effects for vortices in thin film superconductors, and discuss implications for dynamical reordering transition studies in these systems.
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Affiliation(s)
- C Reichhardt
- Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
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23
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Qian H, Mazenko GF. Growth of order in an anisotropic Swift-Hohenberg model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:036117. [PMID: 16605608 DOI: 10.1103/physreve.73.036117] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/24/2005] [Indexed: 05/08/2023]
Abstract
We have studied the ordering kinetics of a two-dimensional anisotropic Swift-Hohenberg (SH) model numerically. The defect structure for this model is simpler than for the isotropic SH model. One finds only dislocations in the aligned ordering striped system. The motion of these point defects is strongly influenced by the anisotropic nature of the system. We developed accurate numerical methods for following the trajectories of dislocations. This allows us to carry out a detailed statistical analysis of the dynamics of the dislocations. The average speeds for the motion of the dislocations in the two orthogonal directions obey power laws in time with different amplitudes but the same exponents. The position and velocity distribution functions are only weakly anisotropic.
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Affiliation(s)
- Hai Qian
- The James Franck Institute and Department of Physics, The University of Chicago, Chicago, IL 60637, USA
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24
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Jagla EA. Numerical simulations of two-dimensional magnetic domain patterns. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:046204. [PMID: 15600492 DOI: 10.1103/physreve.70.046204] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/19/2004] [Revised: 08/09/2004] [Indexed: 05/24/2023]
Abstract
I show that a model for the interaction of magnetic domains that includes a short range ferromagnetic and a long range dipolar antiferromagnetic interaction reproduces very well many characteristic features of two-dimensional magnetic domain patterns. In particular bubble and stripe phases are obtained, along with polygonal and labyrinthine morphologies. In addition, two puzzling phenomena, namely, the so called "memory effect" and the "topological melting" observed experimentally, are also qualitatively described. Very similar phenomenology is found in the case in which the model is changed to be represented by the Swift-Hohenberg equation driven by an external orienting field.
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Affiliation(s)
- E A Jagla
- The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, (34014) Trieste, Italy
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25
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Boyer D. Numerical study of domain coarsening in anisotropic stripe patterns. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:066111. [PMID: 15244671 DOI: 10.1103/physreve.69.066111] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/03/2004] [Indexed: 05/24/2023]
Abstract
We study the coarsening of two-dimensional smectic polycrystals characterized by grains of oblique stripes with only two possible orientations. For this purpose, an anisotropic Swift-Hohenberg equation is solved. For quenches close enough to the onset of stripe formation, the average domain size increases with time as t(1/2). Further from onset, anisotropic pinning forces similar to Peierls stresses in solid crystals slow down defects, and growth becomes anisotropic. In a wide range of quench depths, dislocation arrays remain mobile and dislocation density roughly decays as t(-1/3), while chevron boundaries are totally pinned. We discuss some agreements and disagreements found with recent experimental results on the coarsening of anisotropic electroconvection patterns.
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Affiliation(s)
- Denis Boyer
- Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 Mexico City, Mexico.
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26
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Kamaga C, Ibrahim F, Dennin M. Dislocation dynamics in an anisotropic stripe pattern. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:066213. [PMID: 15244714 DOI: 10.1103/physreve.69.066213] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/05/2004] [Indexed: 05/24/2023]
Abstract
The dynamics of dislocations confined to grain boundaries in a striped system are studied using electroconvection in the nematic liquid crystal N4. In electroconvection, a striped pattern of convection rolls forms for sufficiently high driving voltages. We consider the case of a rapid change in the voltage that takes the system from a uniform state to a state consisting of striped domains with two different wave vectors. The domains are separated by domain walls along one axis and a grain boundary of dislocations in the perpendicular direction. The pattern evolves through dislocation motion parallel to the domain walls. We report on features of the dislocation dynamics. The kinetics of the domain motion is quantified using three measures: dislocation density, average domain wall length, and total domain wall length per area. All three quantities exhibit behavior consistent with power-law evolution in time, with the defect density decaying as t(-1/3), the average domain wall length growing as t(1/3), and the total domain wall length decaying as t(-1/5). The two different exponents are indicative of the anisotropic growth of domains in the system.
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Affiliation(s)
- Carina Kamaga
- Department of Physics and Astronomy, University of California, Irvine, Irvine, California 92696-4575, USA
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27
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Kamaga C, Funfschilling D, Dennin M. Dependence of domain wall dynamics on background wave number. Phys Rev E 2004; 69:016308. [PMID: 14995713 DOI: 10.1103/physreve.69.016308] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/25/2003] [Indexed: 11/07/2022]
Abstract
We report on the growth of domains of standing waves in electroconvection in a nematic liquid crystal, focusing on the evolution of domain walls. An ac voltage is applied to the system, forming an initial state that consists of traveling striped patterns with two different orientations, zig and zag rolls. The standing waves are generated by suddenly applying a periodic modulation of the amplitude of the applied voltage that is approximately resonant with the traveling frequency of the pattern. By varying the modulation frequency, we are able to vary the steady-state, average wave number. We characterize the evolution of the domain walls as a function of the average background wave number by measuring the total area and length of domain walls present in the system as a function of time. We find that as the background wave number is varied away from the "natural" wave number for the pattern, the evolution of the domain walls occurs at a faster rate.
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Affiliation(s)
- Carina Kamaga
- Department of Physics and Astronomy, University of California at Irvine, Irvine, California 92697-4575, USA
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28
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Qian H, Mazenko GF. Model for striped growth. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:011104. [PMID: 14995601 DOI: 10.1103/physreve.69.011104] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/31/2003] [Indexed: 05/24/2023]
Abstract
We introduce a model for describing the defected growth of striped patterns. This model, while roughly related to the Swift-Hohenberg model, generates a quite different mixture of defects during phase ordering. We find two characteristic lengths in the system: the scaling length L(t), and the average width of the domain walls. The growth law exponent is larger than the value of 1/2 found in typical point defect systems.
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Affiliation(s)
- Hai Qian
- James Franck Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637, USA
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