## Research Focus

#### Composite Material and Sea Ice

I am interested in mathematical models that can be used to describe climate phenomena that involve advection enhanced difufsion processes, phase separation and solidification. Using analysis of the heat equation, modification of the Stefan problem and Stieltjes integrals I was able to obtain analytic bounds on the thermal conductivity in the presence of fluid flow, analytic bounds on the trapping constant and a coupled system describing the evolution of the marginal ice zone involving the ice concentration and heat diffusion.

September 2017, invited speaker, Multi-scale modelling of ice characteristics and behaviour, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK. See lecture here

#### Functionalized Cahn-Hilliard

The FCH is a higher-order free energy for blends of amphiphilic polymers and solvent which balances solvation energy of ionic groups against elastic energy of the underlying polymer backbone. Its gradient flows describe the formation of solvent network structures which are essential to ionic conduction in polymer membranes. The FCH possesses stable, coexisting network morphologies and we characterize their geometric evolution, bifurcation and competition through a center-stable manifold reduction which encompasses a broad class of coexisting network morphologies. The stability of the different networks is characterized by the meandering and pearling modes associated to the linearized system. For the $$H^{-1}$$ gradient flow of the FCH energy, using functional analysis and asymptotic methods, we drive a sharp-interface geometric motion which couples the flow of co-dimension 1 and 2 network morphologies, through the far-field chemical potential. In particular, we derive expressions for the pearling and meander eigenvalues for a class of far-from-self-intersection co-dimension 1 and 2 networks, and show that the linearization is uniformly elliptic off of the associated center stable space.

May 2015, mini-symposium organizer, SIAM: Conference on Applications of Dynamical Systems, Snowbird, UT. See lecture here

## Publications

• N. Kraitzman & K. Promislow, (2014) An Overview of Network Bifurcations in the Functionalized Cahn-Hilliard Free Energy, editors: Jean Pierre Bourguignon, Rolf Jeltsch, Alberto Pinto, and Marcelo Viana, Mathematics of Energy and Climate Change: International Conference and Advanced School Planet Earth, Springer International Publishing, (pp. 191-214).

## Under Review

• N. Kraitzman & K. Promislow, Pearling Bifurcations in the Strong Functionalized Cahn-Hilliard Free Energy, Under review (SIAM Journal on Mathematical Analysis). arXiv/1711.00396
• A. Christlieb, N. Kraitzman & K. Promislow, Competition and Complexity in Amphiphilic Polymer Morphology, Under review. arXiv/1711.00419

## Work in Progress

• N. Kraitzman, R. Hardenbrook, B. Murphy, E. Cherkaev, J. Zhu & K.Golden, Bounds on the Effective Thermal Conductivity of Sea Ice in the Presence of Fluid Convection, In preparation.
• N. Kraitzman, E. Cherkaev & K. Golden, Advection enhanced diffusion in the microstructure of sea ice, In preparation.
• N. Kraitzman, E. Cherkaev & K. Golden, Analytic bounds on the trapping constant in sea ice, In preparation.

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LCB 207
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(801) 585-7659 (office)