Karl B Glasner

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Journals/Publications

• Glasner, K. (2023). Segregation and domain formation in non-local multi-species aggregation equations. Physica D: Nonlinear Phenomena, 133936.
• Glasner, K. B. (2023). Data-driven learning of differential equations: combining data and model uncertainty. Computational and Applied Mathematics, 42(1), 36.
• Glasner, K. B. (2021). Optimization algorithms for parameter identification in parabolic partial differential equations. Computational and Applied Mathematics, 40(4), 1--22.
• Glasner, K. B. (2019). Theoretical prediction of morphological selection in amphiphilic system. Physical Review E, 100(6), 062501.
• Glasner, K. B. (2019). Evolution and competition of block copolymer nanoparticles. SIAM J. Appl. Math., 79(1), 28--54.
• Glasner, K. B., & Orizaga, S. (2018). Multidimensional equilibria and their stability in copolymer--solvent mixtures. Physica D, 373, 1--12.
• Glasner, K. B. (2017). Multilayered Equilibria in a Density Functional Model of Copolymer-solvent Mixtures. SIAM J. Math. Analysis, 49(2), 1593–1620..
• Glasner, K. B., & Allen-Flowers, J. (2016). Nonlinearity saturation as a singular perturbation of the nonlinear Schrodinger equation. SIAM J. Appl. Math., 76(2), 525-550.
• Glasner, K. B., & Orizaga, S. (2016). Improving the accuracy of convexity splitting methods for gradient flow equations. Journal of Computational Physics, 315, 52-64.
• Glasner, K. B., & Orizaga, S. (2016). Instability and reorientation of block copolymer microstructure by imposed electric fields. Phys. Rev. E, 93.
• Glasner, K. (2015). Hexagonal phase ordering in strongly segregated copolymer films. Physical review. E, Statistical, nonlinear, and soft matter physics, 92(4), 042602.
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  Strongly segregated copolymer mixtures with uneven composition ratio can form hexagonally ordered thin films. A simplified model describing the size and position of micellelike clusters is derived, allowing for investigation of much larger domain sizes than in previous studies. Simulations of this model are performed to study the generation of large scale order and evolution of pattern defects. We find three temporal regimes exhibiting different scaling laws for orientational correlation length and defect number. In the early stage, topological defects are rapidly eliminated by pairwise annihilation. A slower intermediate stage is characterized by the migration of grain boundaries and the elimination of small grains. In the final stage, grain boundaries become pinned and the evolution halts. A scaling law for defect interaction is proposed which is consistent with the crossover between the first and second stages.
• Lindsay, A. E., Lega, J. C., & Glasner, K. B. (2015). Regularized Model of Post-Touchdown Congurations in Electrostatic MEMS: Interface Dynamics. The IMA Journal of Applied Mathematics, doi: 10.1093/imamat/hxv011, 29.
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  Interface dynamics of post contact states in regularized models of electrostatic-elastic interactions are analyzed. A canonical setting for our investigations is the field of Micro-Electromechanical Systems (MEMS) in which flexible elastic structures may come into physical contact due to applied Coulomb forces. We study the dynamic features of a recently derived regularized model (A.E. Lindsay et al, Regularized Model of Post-Touchdown Configurations in Electrostatic MEMS: Equilibrium Analysis, Physica D, 2014), which describes the system past the quenching singularity associated with touchdown, that is after the components of the device have come together. We build on our previous investigations of steady-state solutions by describing how the system relaxes towards these equilibria. This is accomplished by deriving a reduced dynamical system that governs the evolution of the contact set, thereby providing a detailed description of the intermediary dynamics associated with this bistable system. The analysis yields important practical information on the timescales of equilibration.
• Allen-Flowers, J., & Glasner, K. B. (2014). Transient behavior of collapsing ring solutions in the critical nonlinear Schr"odinger equation. Physica D: Nonlinear Phenomena, 284, 53--61.
• Lindsay, A., Lega, J., & Glasner, K. (2014). Regularized model of post-touchdown configurations in electrostatic MEMS: Equilibrium analysis. Physica D: Nonlinear Phenomena, 280, 95--108.
• Glasner, K. B., & Lindsay, A. E. (2013). The stability and evolution of curved domains arising from one-dimensional localized patterns. SIAM Journal on Applied Dynamical Systems, 12(2), 650-673.
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  Abstract: In many pattern forming systems, narrow two-dimensional domains can arise whose cross sections are roughly one-dimensional localized solutions. This paper investigates this phenomenon in the variational Swift-Hohenberg equation. Stability of straight line solutions is analyzed, leading to criteria for either curve buckling or curve disintegration. Matched asymptotic expansions are used to derive a two-term expression for the geometric motion of curved domains, which includes both elastic and surface diffusion-type regularizations of curve motion. This leads to novel equilibrium curves and space-filling pattern proliferation. Numerical tests are used to confirm and illustrate these phenomena. © 2013 Society for Industrial and Applied Mathematics.
• Glasner, K. B. (2012). Characterising the disordered state of block copolymers: Bifurcations of localised states and self-replication dynamics. European Journal of Applied Mathematics, 23(2), 315-341.
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  Abstract: Above the spinodal temperature for micro-phase separation in block co-polymers, asymmetric mixtures can exhibit random heterogeneous structure. This behaviour is similar to the sub-critical regime of many pattern-forming models. In particular, there is a rich set of localised patterns and associated dynamics. This paper clarifies the nature of the bifurcation diagram of localised solutions in a density functional model of A-B diblock mixtures. The existence of saddle-node bifurcations is described, which explains both the threshold for heterogeneous disordered behaviour as well the onset of pattern propagation. A procedure to generate more complex equilibria by attaching individual structures leads to an interwoven set of solution curves. This results in a global description of the bifurcation diagram from which dynamics, in particular self-replication behaviour, can be explained. © 2011 Cambridge University Press.
• Glasner, K., Kolesik, M., Moloney, J. V., & Newell, A. C. (2011). Canonical and singular propagation of ultrashort pulses in a nonlinear medium. International journal of optics, 2012.
• Diniega, S., Glasner, K., & Byrne, S. (2010). Long-time evolution of models of aeolian sand dune fields: Influence of dune formation and collision. Geomorphology, 121(1-2), 55-68.
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  Abstract: Theoretical models which approximate individual sand dunes as particles that move and interact via simple rules are currently the only viable method for examining whether large dune fields will evolve into a patterned structure. We find that these types of simulations are sensitive to the influx condition and interaction function, and are not necessarily robust under common assumptions. In this paper, we review continuum dune models and how they connect to models of dune fields that approximate dunes as interacting particles with collision and coalescence dynamics. This type of simple dune field model is examined under different boundary and initial conditions. We identify different long-term behaviors depending on model parameters as well as the way in which dunes are initialized and collide. A "rule" for predicting the end state of a modelled dune field is derived, based on the statistics of a uniform influx dune size distribution and the interaction function. Possible future adjustments to the multiscale model, such as the use of a Gaussian influx dune size distribution, and their effect on the prediction rule are also discussed. © 2009 Elsevier B.V.
• Glasner, K. B. (2010). Spatially localized structures in diblock copolymer mixtures. SIAM Journal on Applied Mathematics, 70(6), 2045-2074.
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  Abstract: Above the critical temperature for the order-disorder transition, diblock copolymer melts have been observed to exhibit localized structures that exist within the homogeneous mixture. This paper uses an Ohta-Kawasaki-type density functional to explore this regime. Spatially localized peak-shaped equilibria are studied in one, two, and three dimensions, corresponding to amphiphilic bilayers, cylindrical micelles, and spherical micelles, respectively. A combination of rigorous estimates, asymptotic analysis, and numerical computation is used to characterize solutions and the regime where they exist. The interaction of superpositions of these solutions is studied by a perturbation analysis and shows how steady multipeak configurations can be achieved. Evidence is found for a secondary bifurcation slightly below the spinodal instability threshold, beyond which self-replication phenomena are observed. Dynamics in two dimensions are also illustrated, suggesting other mechanisms for instability and growth. © 2010 Society for Industrial and Applied Mathematics.
• Glasner, K., & Choksi, R. (2009). Coarsening and self-organization in dilute diblock copolymer melts and mixtures. Physica D: Nonlinear Phenomena, 238(14), 1241-1255.
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  Abstract: This paper explores the evolution of a sharp interface model for phase separation of copolymers in the limit of low volume fraction. Particles both exchange material as in usual Ostwald ripening, and migrate because of an effectively repulsive nonlocal energetic term. Coarsening via mass diffusion only occurs while particle radii are small, and they eventually approach a finite equilibrium size. Migration, on the other hand, is responsible for producing self-organized patterns. We construct approximations based upon an ansatz of spherical particles similar to the classical LSW theory to derive finite dimensional dynamics for particle positions and radii. For large systems, kinetic-type equations which describe the evolution of a probability density are constructed. For systems larger than the screening length, we obtain an analog of the homogenization result of Niethammer & Otto [B. Niethammer, F. Otto, Ostwald ripening: The screening length revisited, Calc. Var. Partial Differential Equations 13-1 (2001) 33-68]. A separation of timescales between particle growth and migration allows for a variational characterization of spatially inhomogeneous quasi-equilibrium states. © 2009 Elsevier B.V. All rights reserved.
• Glasner, K., & Kim, I. C. (2009). Viscosity solutions for a model of contact line motion. Interfaces and Free Boundaries, 11(1), 37-60.
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  Abstract: This paper considers a free boundary problem that describes the motion of contact lines of a liquid droplet on a flat surface. The elliptic nature of the equation for droplet shape and the monotonic dependence of contact line velocity on contact angle allows us to introduce a notion of "viscosity" solutions for this problem. Unlike similar free boundary problems, a comparison principle is only available for a modified short-time approximation because of the constraint that conserves volume. We use this modified problem to construct viscosity solutions to the original problem under a weak geometric restriction on the free boundary shape. We also prove uniqueness provided there is an upper bound on front velocity. © 2009 European Mathematical Society.
• Glasner, K., Otto, F., Rump, T., & Slepčev, D. (2009). Ostwald ripening of droplets: The role of migration. European Journal of Applied Mathematics, 20(1), 1-67.
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  Abstract: A configuration of near-equilibrium liquid droplets sitting on a precursor film which wets the entire substrate can coarsen in time by two different mechanisms: collapse or collision of droplets. The collapse mechanism, i.e., a larger droplet grows at the expense of a smaller one by mass exchange through the precursor film, is also known as Ostwald ripening. As was shown by K. B. Glasner and T. P. Witelski ('Collision versus collapse of droplets in coarsening of dewetting thin films', Phys. D 209 (1-4), 2005, 80-104) in case of a one-dimensional substrate, the migration of droplets may interfere with Ostwald ripening: The configuration can coarsen by collision rather than by collapse. We study the role of migration in a two-dimensional substrate for a whole range of mobilities. We characterize the velocity of a single droplet immersed into an environment with constant flux field far away. This allows us to describe the dynamics of a droplet configuration on a two-dimensional substrate by a system of ODEs. In particular, we find by heuristic arguments that collision can be a relevant coarsening mechanism. © Cambridge University Press 2008.
• Glasner, K. B. (2008). Ostwald ripening in thin film equations. SIAM Journal on Applied Mathematics, 69(2), 473-493.
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  Abstract: Fourth order thin film equations can have late stage dynamics that are analogous to the classical Cahn-Hilliard equation. We undertake a systematic asymptotic analysis of a class of equations that describe partial wetting with a stable precursor film introduced by intermolecular interactions. The limit of small precursor film thickness is considered, leading to explicit expressions for the late stage dynamics of droplets. Our main finding is that exchange of mass between droplets characteristic of traditional Ostwald ripening is a subdominant effect over a wide range of kinetic exponents. Instead, droplets migrate in response to variations of the precursor film. Timescales for these processes are computed using an effective medium approximation to the reduced free boundary problem, and dynamic scaling in the reduced system is demonstrated. © 2008 Society for Industrial and Applied Mathematics.
• Glasner, B. K. (2007). The dynamics of pendant droplets on a one-dimensional surface. Physics of Fluids, 19(10).
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  Abstract: A sheet of liquid hanging from a solid surface is subject to the Rayleigh-Taylor instability, which leads to the development of pendant droplets. These near-equilibrium structures interact with the liquid film that connects them. The dynamics of the interaction can be rich and leads to large-scale patterning and nonlinear oscillations. We show that droplets move because of an energetically favorable response to asymmetries of the neighboring film thickness. The droplet moves so as to absorb the thicker liquid film and deposits a Landau-Levich film behind. In the case in which a source of fluid is introduced, the film between the droplets does not proceed toward rupture, but rather acts as a driving mechanism for migration and interaction with neighboring droplets. This interaction is shown to always be repulsive in the scaling regime investigated. A reduced system of droplet dynamics is derived asymptotically, and shows how oscillating behavior develops. © 2007 American Institute of Physics.
• Lu, H. -., Glasner, K., Bertozzi, A. L., & Kim, C. -. (2007). A diffuse-interface model for electrowetting drops in a Hele-Shaw cell. Journal of Fluid Mechanics, 590, 411-435.
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  Abstract: Electrowetting has recently been explored as a mechanism for moving small amounts of fluids in confined spaces. We propose a diffuse-interface model for drop motion, due to electrowetting, in a Hele-Shaw geometry. In the limit of small interface thickness, asymptotic analysis shows that the model is equivalent to Hele-Shaw flow with a voltage-modified Young-Laplace boundary condition on the free surface. We show that details of the contact angle significantly affect the time scale of motion in the model. We measure receding and advancing contact angles in the experiments and derive their influence through a reduced-order model. These measurements suggest a range of time scales in the Hele-Shaw model which include those observed in the experiment. The shape dynamics and topology changes in the model agree well with the experiment, down to the length scale of the diffuse-interface thickness. © 2007 Cambridge University Press.
• Glasner, K. B. (2006). Grain boundary motion arising from the gradient flow of the Aviles-Giga functional. Physica D: Nonlinear Phenomena, 215(1), 80-98.
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  Abstract: This paper considers the singular limit of the equation Θt=-εΔ2Θ+ε-1∇·([|∇Θ|2-1] ∇Θ). Grain boundaries (limiting discontinuities in ∇Θ) form networks that coarsen over time. A matched asymptotic analysis is used to derive a free boundary problem consisting of curve motion coupled along hyperbolic characteristics and junction conditions. An intermediate boundary layer near extrema junctions is discovered, along with the relevant nonlocal junction conditions. The limiting dynamics can be viewed in the context of a gradient flow of the sharp interface energy on an attracting manifold. Dynamic scaling of the long-time coarsening process can be explained by dimensional analysis of the reduced problem. © 2006 Elsevier Ltd. All rights reserved.
• Glasner, K. B. (2006). Homogenization of contact line dynamics. Interfaces and Free Boundaries, 8(4), 523-542.
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  Abstract: This paper considers the effects of substrate inhomogeneity on the motion of the three phase contact line. The model employed assumes the slowness of the contact line in comparison to capillary relaxation. The homogenization of this free boundary problem with a spatially periodic velocity law is considered. Formal multiple scales analysis yields a local, periodic problem whose time-averaged dynamics corresponds to the homogenized front velocity. A rigorous understanding of the long time dynamics is developed using comparison techniques. Computations employing boundary integral equations are used to illustrate the consequences of the analysis. Advancing and receding contact angles, pinning and anisotropic motion can be predicted within this framework. © European Mathematical Society 2006.
• Glasner, K. B. (2005). A boundary integral formulation of quasi-steady fluid wetting. Journal of Computational Physics, 207(2), 529-541.
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  Abstract: This paper considers the motion of a liquid droplet on a solid surface. When capillary relaxation is much faster than the motion of the contact line, the fluid geometry and its dynamical evolution can be characterized in terms of the contact line alone. This problem can be cast in terms of boundary integral equations involving a Dirichlet-Neumann map coupled to a volume conservation constraint. A computational method for this formulation is described which has two principal advantages over approaches which track the entire free surface: (1) only the curve which describes the contact line is computed and (2) the resulting method exhibits only mild numerical stiffness, obviating the need for implicit timestepping. Effects of both capillary and body forces are considered. Computational examples include surface inhomogeneities, topological transitions and cusp formation. © 2005 Elsevier Inc. All rights reserved.
• Glasner, K. B. (2005). Variational models for moving contact lines and the quasi-static approximation. European Journal of Applied Mathematics, 16(6), 713-740.
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  Abstract: This paper proposes the use of a variational framework to model fluid wetting dynamics. The central problem of infinite energy dissipation for a moving contact line is dealt with explicitly rather than by introducing a specific microscopic mechanism which removes it. We analyze this modelling approach in the context of the quasi-steady limit, where contact line motion is slower than bulk relaxation. We find that global effects enter into Tanner-type laws which relate line velocity to apparent contact angle through the role that energy dissipation plays in the bulk of the fluid. A comparison is made to the dynamics of lubrication equations that include attractive and repulsive intermolecular interactions. A Galerkin-type approximation method is introduced which leads to reduced-dimensional dynamical descriptions. Computations are conducted using these low-dimensional approximations, and a substantial connection to lubrication equation dynamics is found. © 2005 Cambridge University Press.
• Glasner, K. B., & Witelski, T. P. (2005). Collision versus collapse of droplets in coarsening of dewetting thin films. Physica D: Nonlinear Phenomena, 209(1-4 SPEC. ISS.), 80-104.
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  Abstract: Thin films of viscous fluids coating solid surfaces can become unstable due to intermolecular forces, leading to break-up of the film into arrays of droplets. The long-time dynamics of the system can be represented in terms of coupled equations for the masses and positions of the droplets. Analysis of the decrease of energy of the system shows that coarsening, decreasing the number of droplets with increasing time, is favored. Here we describe the two coarsening mechanisms present in dewetting films: (i) mass exchange leading to collapse of individual drops, and (ii) spatial motion leading to droplet collisions and merging events. Regimes where each of mechanisms are dominant are identified, and the statistics of the coarsening process are explained. © 2005 Elsevier B.V. All rights reserved.
• Glasner, K. (2003). A diffuse interface approach to Hele-Shaw flow. Nonlinearity, 16(1), 49-66.
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  Abstract: A diffuse interface model for the one-phase Hele-Shaw problem is derived from a gradient flow characterization due to Otto (1998 Arch. Rat. Mech. Anal. 141 63). The resulting dynamical model yields a generalized form of Darcy's law, and reduces to a degenerate version of the well-known Cahn-Hilliard equation. Formal asymptotics illustrate the connection to the classical Hele-Shaw free boundary problem. Some example computations are carried out to demonstrate the flexibility of the modelling framework.
• Glasner, K. B. (2003). Spreading of droplets under the influence of intermolecular forces. Physics of Fluids, 15(7), 1837-1842.
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  Abstract: The motion of fluid droplets under the influence of short and long range intermolecular forces is examined using a lubrication model. Surface energies as well as the microscopic contact line structure are identified in the model. A physically constructed precursor film prevents the usual stress singularity associated with a moving contact line. In the quasistatic limit, an analysis of the energy and its dissipation yield an ordinary differential equation for the rate of spreading. Two dimensional and axisymmetric solutions are found and compared to numerical simulations. The motion of the contact line is found to be both a function of the local contact angle and the overall droplet geometry. © 2003 American Institute of Physics.
• Glasner, K. B., & Witelski, T. P. (2003). Coarsening dynamics of dewetting films. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 67(1 2), 163021-1630212.
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  Abstract: The modelling of coarsening dynamics of dewetting films using lubrication theory for unstable thin liquid films on solid substrates was discussed. Surface tension and intermolecular interactions with the solid substrate were the dominant physical effects driving the fluid dynamics. The fluid underwent a coarsening process in which droplets moved and exchanged mass on slow time scales.
• Glasner, K. (2001). Nonlinear preconditioning for diffuse interfaces. Journal of Computational Physics, 174(2), 695-711.
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  Abstract: A method of transforming problems with diffuse interfaces is presented which leads to equations that are easier to compute accurately. Information obtained by internal layer asymptotic analysis is utilized to motivate transformations of the dependent variables. The new evolution equations which result from this change of variables can be solved numerically in a straightforward manner. Numerical experiments indicate that truncation errors can be significantly reduced in such problems, allowing a coarser grid to be used. Applications to several well-known models are presented. © 2001 Elsevier Science.
• Glasner, K. (2001). Rapid growth and critical behaviour in phase field models of solidification. European Journal of Applied Mathematics, 12(1), 39-56.
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  Abstract: Rapid solidification fronts are studied using a phase field model. Unlike slow moving solutions which approximate the Mullins-Sekerka free boundary problem, different limiting behaviour is obtained for rapidly moving fronts. A time-dependent analysis is carried out for various cases and the leading order behaviour of solidification front solutions is derived to be one of several travelling wave problems. An analysis of these problems is conducted, leading to expressions for front speeds in certain limits. The dynamics leading to these travelling wave solutions is derived, and conclusions about stability are drawn. Finally, a discussion is made of the relationship to other solidification models.
• Glasner, K. (2001). Solute trapping and the non-equilibrium phase diagram for solidification of binary alloys. Physica D: Nonlinear Phenomena, 151(2-4), 253-270.
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  Abstract: A phase field model for the solidification of binary alloys is presented and analyzed. A matched asymptotic approach is used to recover the model's leading order sharp interface motion. Equations for both the solute profile and free energy balance at the interface are derived, demonstrating solute trapping at large growth velocities and leading to a construction of the non-equilibrium phase diagram over a large range of growth conditions. A rigorous understanding of the interfacial conditions is provided, and comparisons are made to existing theories. © 2001 Published by Elsevier Science B.V.
• Glasne, K., & Glasner, K. (2000). Traveling waves in rapid solidification. Electronic Journal of Differential Equations, 2000, XXV-XXVI.
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  Abstract: We analyze rigorously the one-dimensional traveling wave problem for a thermodynamically consistent phase field model. Existence is proved for two new cases: one where the undercooling is large but not in the hypercooled regime, and the other for waves which leave behind an unstable state. The qualitative structure of the wave is studied, and under certain restrictions monotonicity of front profiles can be obtained. Further results, such as a bound on propagation velocity and non-existence are discussed. Finally, some numerical examples of monotone and non-monotone waves are provided. ©2000 Southwest Texas State University and University of North Texas.
• Glasner, K., & Almgren, R. (2000). Dual fronts in a phase field model. Physica D: Nonlinear Phenomena, 146(1-4), 328-340.
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  Abstract: We study a dual front behavior observed in a reaction-diffusion system arising initially in the context of phase field models. A precursor front propagates into a stable phase, generating a metastable "intermediate phase". This intermediate phase then decays via an oscillating front, producing a periodic structure which later coarsens. Unlike previously studied models in which dual fronts appear, the appearance of the split front is controlled not by an interchange of wave speeds, but by the existence of the precursor wave. By means of an expansion in small thermal diffusivity, we argue that this behavior is generic. © 2000 Elsevier Science B.V.

Presentations

• Glasner, K. B. (2019, Fall). Mathematical Aspects of Nanoscale Self-Assembly. University of Alabama, Applied Math seminar. Tuscaloosa, AL.
• Glasner, K. B. (2019, Fall). Morphological selection in multiphase models of amphiphilic systems. SIAM conference on partial differential equations. La Quinta, CA: SIAM.
• Glasner, K. B. (2019, Spring). Amphiphilic Structures in Multiphase Density Functional Models. SIAM conference on Dynamical Systems. Snowbird, UT: SIAM.
• Glasner, K. B. (2018, June). Competition and interaction of block copolymer nanoparticles. SIAM Conference on Nonlinear Waves and Coherent Structures. Anaheim, CA: SIAM.
• Glasner, K. B. (2017, May). Self-assembled structures in copolymer-solvent mixtures. SIAM Conference on Dynamical Systems. Snowbird, UT: SIAM.
• Glasner, K. B. (2015, Dec.). Bilayers and Multilayers in Copolymer-solvent mixtures. SIAM Conference on Analysis of Partial Differential Equations. Scottsdale, AZ.
• Glasner, K. B. (2015, May). Nonlinearity saturation as a singular perturbation of the nonlinear Schrodinger equation. SIAM Conference on Dynamical systems. Snowbird, UT.
• Glasner, K. B. (2013, 12). Interaction and non-interaction of large amplitude solitary waves in an ambient wave field. SIAM Conference on Partial Differential Equations. Orlando FL: SIAM.
• Glasner, K. B. (2012, march). Self assembly of block copolymer materials. Purdue University, Computational and Applied Math seminar. Purdue University.