Xiaoming Mao

Criticality and isostaticity in fibre networks

In fibre networks, mechanical stability relies on the fibres’ bending resistance—in contrast to rubbers, where entropic stretching is the key. The extent to which the mechanics of fibre networks is controlled by bending is, however, an open question. The study of a general lattice-based model of fibrous networks now reveals two rigidity critical points, one of which controls a rich crossover from stretching-dominated to bending-dominated behaviour.

Aluminum nanoscale order in amorphous Al92Sm8 measured by fluctuation electron microscopy

Fluctuation electron microscopy (FEM) measurements and simulations have identified nanoscale aluminum-like medium-range order in rapidly quenched amorphous Al92Sm8 which devitrifies by primary Al crystallization. Al92Sm8 amorphized by plastic deformation shows neither Al nanoscale order, nor primary crystallization. Annealing the rapidly quenched material below the primary crystallization temperature reduces the degree of nanoscale Al order measured by FEM. The FEM measurements suggest that 10–20A diameter regions with Al crystal-like order are associated with primary crystallization in amorphous Al92Sm8, which is consistent with the quenched-in cluster model of primary crystallization.

Entropy favours open colloidal lattices

Burgeoning experimental and simulation activity seeks to understand the existence of self-assembled colloidal structures that are not close-packed. Here we describe an analytical theory based on lattice dynamics and supported by experiments that reveals the fundamental role entropy can play in stabilizing open lattices. The entropy we consider is associated with the rotational and vibrational modes unique to colloids interacting through extended attractive patches. The theory makes predictions of the implied temperature, pressure and patch-size dependence of the phase diagram of open and close-packed structures. More generally, it provides guidance for the conditions at which targeted patchy colloidal assemblies in two and three dimensions are stable, thus overcoming the difficulty in exploring by experiment or simulation the full range of conceivable parameters.

Surface phonons, elastic response, and conformal invariance in twisted kagome lattices

Model lattices consisting of balls connected by central-force springs provide much of our understanding of mechanical response and phonon structure of real materials. Their stability depends critically on their coordination number z. d-dimensional lattices with z = 2d are at the threshold of mechanical stability and are isostatic. Lattices with z < 2d exhibit zero-frequency “floppy” modes that provide avenues for lattice collapse. The physics of systems as diverse as architectural structures, network glasses, randomly packed spheres, and biopolymer networks is strongly influenced by a nearby isostatic lattice. We explore elasticity and phonons of a special class of two-dimensional isostatic lattices constructed by distorting the kagome lattice. We show that the phonon structure of these lattices, characterized by vanishing bulk moduli and thus negative Poisson ratios (equivalently, auxetic elasticity), depends sensitively on boundary conditions and on the nature of the kagome distortions. We construct lattices that under free boundary conditions exhibit surface floppy modes only or a combination of both surface and bulk floppy modes; and we show that bulk floppy modes present under free boundary conditions are also present under periodic boundary conditions but that surface modes are not. In the long-wavelength limit, the elastic theory of all these lattices is a conformally invariant field theory with holographic properties (characteristics of the bulk are encoded on the sample boundary), and the surface waves are Rayleigh waves. We discuss our results in relation to recent work on jammed systems. Our results highlight the importance of network architecture in determining floppy-mode structure.

Isostaticity, auxetic response, surface modes, and conformal invariance in twisted kagome lattices

Model lattices consisting of balls connected by central-force springs provide much of our understanding of mechanical response and phonon structure of real materials. Their stability depends critically on their coordination number z. d-dimensional lattices with z = 2d are at the threshold of mechanical stability and are isostatic. Lattices with z < 2d exhibit zero-frequency “floppy” modes that provide avenues for lattice collapse. The physics of systems as diverse as architectural structures, network glasses, randomly packed spheres, and biopolymer networks is strongly influenced by a nearby isostatic lattice. We explore elasticity and phonons of a special class of two-dimensional isostatic lattices constructed by distorting the kagome lattice. We show that the phonon structure of these lattices, characterized by vanishing bulk moduli and thus negative Poisson ratios (equivalently, auxetic elasticity), depends sensitively on boundary conditions and on the nature of the kagome distortions. We construct lattices that under free boundary conditions exhibit surface floppy modes only or a combination of both surface and bulk floppy modes; and we show that bulk floppy modes present under free boundary conditions are also present under periodic boundary conditions but that surface modes are not. In the long-wavelength limit, the elastic theory of all these lattices is a conformally invariant field theory with holographic properties (characteristics of the bulk are encoded on the sample boundary), and the surface waves are Rayleigh waves. We discuss our results in relation to recent work on jammed systems. Our results highlight the importance of network architecture in determining floppy-mode structure.

Phonons and elasticity in critically coordinated lattices.

Much of our understanding of vibrational excitations and elasticity is based upon analysis of frames consisting of sites connected by bonds occupied by central-force springs, the stability of which depends on the average number of neighbors per site z. When z  <  zc  ≈  2d, where d is the spatial dimension, frames are unstable with respect to internal deformations. This pedagogical review focuses on the properties of frames with z at or near zc, which model systems like randomly packed spheres near jamming and network glasses. Using an index theorem, N0  -NS  =  dN  -NB relating the number of sites, N, and number of bonds, NB, to the number, N0, of modes of zero energy and the number, NS, of states of self stress, in which springs can be under positive or negative tension while forces on sites remain zero, it explores the properties of periodic square, kagome, and related lattices for which z  =  zc and the relation between states of self stress and zero modes in periodic lattices to the surface zero modes of finite free lattices (with free boundary conditions). It shows how modifications to the periodic kagome lattice can eliminate all but trivial translational zero modes and create topologically distinct classes, analogous to those of topological insulators, with protected zero modes at free boundaries and at interfaces between different topological classes.

Transformable topological mechanical metamaterials

Mechanical metamaterials are engineered materials whose structures give them novel mechanical properties, including negative Poissons ratios, negative compressibilities and phononic bandgaps. Of particular interest are systems near the point of mechanical instability, which recently have been shown to distribute force and motion in robust ways determined by a nontrivial topological state. Here we discuss the classification of and propose a design principle for mechanical metamaterials that can be easily and reversibly transformed between states with dramatically different mechanical and acoustic properties via a soft strain. Remarkably, despite the low energetic cost of this transition, quantities such as the edge stiffness and speed of sound can change by orders of magnitude. We show that the existence and form of a soft deformation directly determines floppy edge modes and phonon dispersion. Finally, we generalize the soft strain to generate domain structures that allow further tuning of the material.

Alignment and nonlinear elasticity in biopolymer gels

We present a Landau-type theory for the nonlinear elasticity of biopolymer gels with a part of the order parameter describing induced nematic order of fibers in the gel. We attribute the nonlinear elastic behavior of these materials to fiber alignment induced by strain. We suggest an application to contact guidance of cell motility in tissue. We compare our theory to simulation of a disordered lattice model for biopolymers. We treat homogeneous deformations such as simple shear, hydrostatic expansion, and simple extension, and obtain good agreement between theory and simulation. We also consider a localized perturbation which is a simple model for a contracting cell in a medium.

Coherent potential approximation of random nearly isostatic kagome lattice.

The kagome lattice has coordination number 4, and it is mechanically isostatic when nearest-neighbor sites are connected by central-force springs. A lattice of N sites has O(√N) zero-frequency floppy modes that convert to finite-frequency anomalous modes when next-nearest-neighbor (NNN) springs are added. We use the coherent potential approximation to study the mode structure and mechanical properties of the kagome lattice in which NNN springs with spring constant κ are added with probability P=Δz/4, where Δz=z-4 and z is the average coordination number. The effective medium static NNN spring constant κ(m) scales as P(2) for P≪κ and as P for P≫κ, yielding a frequency scale ω*~Δz and a length scale l*~(Δz)(-1). To a very good approximation at small nonzero frequency, κ(m)(P,ω)/κ(m)(P,0) is a scaling function of ω/ω*. The Ioffe-Regel limit beyond which plane-wave states become ill-defined is reached at a frequency of order ω*.

Elasticity of a filamentous kagome lattice.

The diluted kagome lattice, in which bonds are randomly removed with probability 1-p, consists of straight lines that intersect at points with a maximum coordination number of 4. If lines are treated as semiflexible polymers and crossing points are treated as cross-links, this lattice provides a simple model for two-dimensional filamentous networks. Lattice-based effective-medium theories and numerical simulations for filaments modeled as elastic rods, with stretching modulus μ and bending modulus κ, are used to study the elasticity of this lattice as functions of p and κ. At p=1, elastic response is purely affine, and the macroscopic elastic modulus G is independent of κ. When κ=0, the lattice undergoes a first-order rigidity-percolation transition at p=1. When κ>0, G decreases continuously as p decreases below one, reaching zero at a continuous rigidity-percolation transition at p=p(b)≈0.605 that is the same for all nonzero values of κ. The effective-medium theories predict scaling forms for G, which exhibit crossover from bending-dominated response at small κ/μ to stretching-dominated response at large κ/μ near both p=1 and p(b), that match simulations with no adjustable parameters near p=1. The affine response as p→1 is identified with the approach to a state with sample-crossing straight filaments treated as elastic rods.