Xiaofeng Ren, Ph.D. - US grants

Ren 9703727 Martensite transformations are phase transformations that produce a change of shape and a change of crystal symmetry. Shape-memory materials are materials that are extremely malleable in the martensite phase below a transformation temperature, but that return to a `remembered' original shape when heated above the transformation temperature. Coherent phase transitions of crystalline solids lead to mixtures of distinct phases or phase variants with characteristic fine-scale structures. The main issues addressed in this project are the characteristic scales, the generation and the propagation of the microstructures. A nonlocal theory involving integro-differential evolutionary equations is proposed to answer these questions. This theory lies between the traditional microscopic quantum theory which uses Schodinger's equation and the macroscopic elasticity theory which uses elliptic and hyperbolic partial differential equations. In the study of the characteristic scales, a periodic solution of a stationary nonlocal equation is needed. But the equation seems to have multiple solutions. The `right' periodic solution should be selected by a suitable least energy principle. Another intriguing question is to find a `generalized' traveling wave solution of a dynamic nonlocal equation, which describes the propagation of microstructures. This solution should have an oscillatory structure at one end, constant value at the other end, and as time increases the uniform-to-oscillatory transition region should advance in a periodic manner. Such a solution can also be viewed as a heteroclinic orbit in an infinitely dimensional function space connecting a periodic stationary solution to a constant stationary solution. Materials and processing are critical to the success of industries such as the aerospace, automotive, biomaterials, chemical, electronics, energy, metals, and telecommunications industries. This project is concerned with shape memory materials and martensite transformations. The mathematical theory in this project is aimed at understanding the characteristic scales, the generation and the propagation of the microstructures during shape memory materials' phase transformations. The study of stationary periodic solutions ald uniform-to-oscillatory waves of the integro-differential evolutionary equations used in this nonlocal mathematical theory is at the cutting edge of mathematical analysis and computation.

Morphology patterns that appear in complex materials are
intimately related to their physical properties. Examples include
lamellar, cylindrical, and spherical patterns in block copolymers,
neutral and charged Langmuir monolayers, and smectic liquid
crystal films. The investigator and his collaborators analyze and
predict morphology patterns based on mathematical models derived
from first principles of statistical physics. They develop
techniques to deal with nonlocal constitutive relations, singular
perturbations, and critical eigenvalues. These mathematical
phenomena are proved to be the reasons behind self-organization
and pattern formation. The investigator finds effective reduction
methods that accurately simplify the original infinite-dimensional
problems to manageable finite-dimensional ones. The investigator
also studies the dynamic phenomenon of pattern nucleation in
materials by finding and characterizing some unstable solutions to
the Euler-Lagrange equations of the free energy. The project
expands our knowledge of singularly perturbed variational
problems. It enhances classical theories such as Gamma
convergence.

Complex materials, such as block copolymers, are used
everywhere. The polyurethane foams used in upholstery and bedding
are composed of multi-block copolymers known as thermoplastic
elastomers that combine high temperature resilience and low
temperature flexibility. Common box tapes use triblock copolymers
to achieve pressure-sensitive adhesion. Block copolymers are
blended with asphalt in road construction to reduce pavement
cracking and rutting at low and high temperature extremes. This
project seeks to deepen our understanding of the mathematical
theories underlying these systems. It produces effective methods
that characterize and predict the mechanical, optical, electrical,
ionic, barrier, and other physical properties of these materials.
They help today's synthetic chemistry technologies to produce
exquisitely structured materials to meet an ever rising demand
from civil infrastructure and manufacturing.

Ren
DMS-0907777

A diblock copolymer molecule is a linear chain of an
A-monomer block grafted covalently to a B-monomer block. Because
of the repulsion between the unlike monomers, the different type
sub-chains tend to segregate, but as they are chemically bonded
in chain molecules, segregation of sub-chains cannot lead to a
macroscopic phase separation. Only a local micro-phase
separation occurs: micro-domains rich in A monomers and
micro-domains rich in B monomers emerge as a result. A
nano-sized pattern formed from micro-domains is known as a
morphology phase. The investigator develops singular limit
methods to study morphology phases that concentrate on points,
curves, and surfaces in space, and that might not be found by
existing free energy methods. These methods reduce complicated
nonlinear, nonlocal, variational, and partial differential
equation problems to simpler geometric problems. They analyze
the saturation phenomenon: a process of elongation, deformation,
and breaking off of a small number of large objects to form a
large number of small objects. They also explain defects in
morphological phases caused by local-nonlocal competition or
topological constraints. In the case of block copolymer vesicles
the investigator studies the bending rigidity in the free energy.
The singular limit techniques are extended to problems without
variational structures, such as the Gierer-Meinhardt system for
biological morphogenesis in development.

Block copolymers are soft condensed materials that in
contrast to crystalline solids, are characterized by fluid-like
disorder on the molecular scale and a high degree of order on a
longer length scale. An almost unlimited number of molecular
architectures can be designed by modern nano-technologies to
produce materials with particular mechanical, electric, barrier,
ionic and other physical properties. Common box tapes use
triblock copolymers to achieve pressure-sensitive adhesion.
Block copolymers are blended with asphalt in road construction to
reduce pavement cracking and rutting at low and high temperature
extremes. In this project the investigator studies pattern
formations within block copolymers that are related to changes in
the morphology phase of the material, and hence to larger-scale
material properties. The project includes graduate students, who
develop skills and knowledge in both mathematics and materials
science.

This project investigates several important binary and ternary systems with self-organizing properties, with an emphasis on ternary systems and the longer ranging confinement mechanism through nonlocal interaction or inhibitor variables. One intriguing feature that sets ternary systems apart from binary systems is the triple junction phenomenon. The three constituents of the system may meet at a point in the two dimensional case or at a curve in the three dimensional case. The PI proposes to show the existence of a double bubble assembly pattern, where the triple junction phenomenon occurs in each double bubble. Two novel techniques, restricted perturbation classes and internal variables, will be used in the proof. The second feature in ternary systems is the complexity of the long range interaction manifested in a two by two matrix of parameters. For instance it will be shown that the core-shell pattern appears only if the 2-2 entry is greater than the 1-2 entry of the matrix. When multi-constituent inhibitory systems appear in a curved space, such as a vesicle of lipid membranes, the role played by the Riemann curvature will also be investigated. Patterns and their possible defects will be shown as a balance and compromise of growth, inhibition, and curvature.

Exquisitely structured patterns arise in many multi-constituent physical and biological systems as orderly outcomes of self-organization principles. Examples include morphological phases in block copolymers, animal coats, and skin pigmentation. Common in these pattern-forming systems is that a deviation from homogeneity has a strong positive feedback on its further increase. On its own, it would lead to an unlimited increase and spreading. Pattern formation requires in addition a longer ranging confinement of the locally self-enhancing process. This project derives geometrical patterns from self-organization principles in various binary and ternary physical and biological systems. Analysis of these geometric structures is a fundamental step in understanding the mechanical, optical, electrical, ionic, barrier and other properties of these systems. This project supports a culturally diverse environment that fosters critical and independent thinking both in classroom and research settings in the ethnically diverse metropolitan area of Washington, DC. It produces mathematics of depth and beauty, and offers profound insights into the natural world.

Pattern formation is an orderly outcome of self-organization principles. Morphological phases are complex geometric structures that form as structured patterns in physical and biological systems. Examples include morphological phases in block copolymers, animal coats, and skin pigmentation in cell development. Common in these pattern-forming systems is that a deviation from homogeneity has a strong positive feedback on its further increase. In addition, pattern formation provides a longer ranging confinement of the locally self-enhancing process. Studying pattern formation reveals the mechanical, optical, electrical, ionic, barrier, and other properties of these systems. In this project, the investigator studies morphological phases in biological and physical systems having multiple constituents. Often these phases have been observed in experiments where no current theory or models predict them. The investigator aims for a rigorous analysis that not only predicts the existence of these structures but also their particular properties. Morphological phases are important in modern materials such as block copolymers, which are commercially used as thermoplastic elastomers: wine bottle stoppers, outer coverings for optical fibre cables, adhesives, bitumen modifiers, or in artificial organ technology. Graduate and undergraduate students participate in the work of the project.

The investigator studies morphological phases in physical and biological systems. Systems of particular interest are the Ohta-Kawasaki, FitzHugh-Nagumo, Gierer-Meinhardt, and tri- and tetrablock copolymers. Typically a morphological phase consists of micro-domains separated by narrow interfaces. Often, narrow-interface morphological phases can be approximated by their sharp interface limits, where constituents are fully separated by interfaces with no thickness. In the sharp interface case this project investigates assemblies of tori, Hopf links, double bubbles, and triple bubbles. Analytical methods are developed to use sharp interface morphological phases as skeletons to reconstruct morphological phases with narrow interfaces. Through renormalization and an infinite-dimensional reduction technique, where Coulomb forces are treated as long-range interactions, free energies of lower-dimensional structures are derived. Graduate and undergraduate students participate in the work of the project.