arXiv subjects

Xiaoming Mao

Publications and source records attributed to Xiaoming Mao.

At least 73 records · Page 4Linked to original sources

arXiv2015

Finite-temperature buckling of an extensible rod

Thermal fluctuations can play an important role in the buckling of elastic objects at small scales, such as polymers or nanotubes. In this paper, we study the finite-temperature buckling transition of an extensible rod by analyzing fluctuation corrections to the elasticity of the rod. We find that, in both two and three dimensions, thermal fluctuations delay the buckling transition, and near the transition, there is a critical regime in which fluctuations are prominent and make a contribution to the effective force that is of order $\sqrt{T}$. We verify our theoretical prediction of the phase diagram with Monte Carlo simulations.

cond-mat.soft↗

arXiv2015

Finite temperature mechanical instability in disordered lattices

Mechanical instability takes different forms in various ordered and disordered systems. We study the effect of thermal fluctuations in two disordered central-force lattice models near mechanical instability: randomly diluted triangular lattice and randomly braced square lattice. These two lattices exhibit different scalings for the emergence of rigidity at $T=0$ due to their different patterns of self stress at the transition. Using analytic theory we show that thermal fluctuations stabilize both lattices. In particular, the triangular lattice displays a critical regime in which the shear modulus scales as $G \sim T^{1/2}$, whereas the square lattice shows $G \sim T^{2/3}$.

cond-mat.soft↗

arXiv2015

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 $z<z_c \approx 2d$, where $d$ is the spatial dimension, frames are unstable with respect to internal deformations. This pedagogical review focuses on properties of frames with $z$ at or near $z_c$, which model systems like randomly packed spheres near jamming and network glasses. Using an index theorem, $N_0 - N_S = dN - N_B$ relating the number of sites, $N$, and number of bonds, $N_B$, to the number, $N_0$, of modes of zero energy and the number, $N_S$, 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 $z=z_c$ 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.

cond-mat.soft↗

arXiv2014

Rigidity percolation by next-nearest-neighbor braces on generic and regular isostatic lattices

We study rigidity percolation transitions in two-dimensional central-force isostatic lattices, including the square and the kagome lattices, as next-nearest-neighbor bonds ("braces") are randomly added to the system. In particular, we focus on the differences between regular lattices, which are perfectly periodic, and generic lattices with the same topology of bonds but whose sites are at random positions in space. We find that the regular square and kagome lattices exhibit a rigidity percolation transition when the number of braces is $\sim L\ln L$, where $L$ is the linear size of the lattice. This transition exhibits features of both first order and second order transitions: the whole lattice becomes rigid at the transition, whereas there exists a diverging length scale. In contrast, we find that the rigidity percolation transition in the generic lattices occur when the number of braces is very close to the number obtained from the Maxwell's law for floppy modes, which is $\sim L$. The transition in generic lattices is a very sharp first-order-like transition, at which the addition of one brace connects all small rigid regions in the bulk of the lattice, leaving only floppy modes on the edge. We characterize these transitions using numerical simulations and develop analytic theories capturing each transition. Our results relate to other interesting problems including jamming and bootstrap percolation.

cond-mat.soft↗

arXiv2014

Alignment and Nonlinear Elasticity in Biopolymer Gels

We present a Landau type theory for the non-linear elasticity of biopolymer gels with a part of the order parameter describing induced nematic order of fibers in the gel. We attribute the non-linear 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.

cond-mat.soft↗

arXiv2014

Mechanical instability at finite temperature

Many physical systems including lattices near structural phase transitions, glasses, jammed solids, and bio-polymer gels have coordination numbers that place them at the edge of mechanical instability. Their properties are determined by an interplay between soft mechanical modes and thermal fluctuations. In this paper we investigate a simple square-lattice model with a $ϕ^4$ potential between next-nearest-neighbor sites whose quadratic coefficient $κ$ can be tuned from positive negative. We show that its zero-temperature ground state for $κ<0$ is highly degenerate, and we use analytical techniques and simulation to explore its finite temperature properties. We show that a unique rhombic ground state is entropically favored at nonzero temperature at $κ<0$ and that the existence of a subextensive number of "floppy" modes whose frequencies vanish at $κ= 0$ leads to singular contributions to the free energy that render the square-to-rhombic transition first order and lead to power-law behavior of the shear modulus as a function of temperature. We expect our study to provide a general framework for the study of finite-temperature mechanical and phase behavior of other systems with a large number of floppy modes.

cond-mat.soft↗

arXiv2014

Self-assembly of three-dimensional open structures using patchy colloidal particles

Open structures can display a number of unusual properties, including a negative Poisson's ratio, negative thermal expansion, and holographic elasticity, and have many interesting applications in engineering. However, it is a grand challenge to self-assemble open structures at the colloidal scale, where short-range interactions and low coordination number can leave them mechanically unstable. In this paper we discuss the self-assembly of open structures using triblock Janus particles, which have two large attractive patches that can form multiple bonds, separated by a band with purely hard-sphere repulsion. Such surface patterning leads to open structures that are stabilized by orientational entropy (in an order-by-disorder effect) and selected over close-packed structures by vibrational entropy. For different patch sizes the particles can form into either tetrahedral or octahedral structural motifs which then compose open lattices, including the pyrochlore, the hexagonal tetrastack and the perovskite lattices. Using an analytic theory, we examine the phase diagrams of these possible open and close-packed structures for triblock Janus particles and characterize the mechanical properties of these structures. Our theory leads to rational designs of particles for the self-assembly of three-dimensional colloidal structures that are possible using current experimental techniques.

cond-mat.soft↗

arXiv2013

Entropic effects in the self-assembly of open lattices from patchy particles

Open lattices are characterized by low volume-fraction arrangements of building blocks, low coordination number, and open spaces between building blocks. The self-assembly of these lattices faces the challenge of mechanical instability due to their open structures. We theoretically investigate the stabilizing effects of entropy in the self-assembly of open lattices from patchy particles. A preliminary account of these findings and their comparison to experiment was presented recently [X. Mao, Q. Chen, S. Granick, \textit{Nat. Mater.}, \textbf{12}, 217 (2013)]. We found that rotational entropy of patchy particles can provide mechanical stability to open lattices, whereas vibrational entropy of patchy particles can lower the free energy of open lattices and thus enables the selection of open lattices verses close-packed lattices which have the same potential energy. These effects open the door to significant simplifications of possible future designs of patchy-particles for open-lattice self-assembly.

cond-mat.soft↗

arXiv2013

Elasticity of 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 four. If lines are treated as semi-flexible polymers and crossing points are treated as crosslinks, 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 $p=p_b \approx 0.605$ that is the same for all non-zero 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 $p=p_b$, that match simulations with no adjustable parameters near $p=1$. The affine response as $p\rightarrow 1$ is identified with the approach to a state with sample-crossing straight filaments treated as elastic rods.

cond-mat.soft↗

arXiv2013

Effective Medium Theory of Filamentous Triangular Lattice

We present an effective medium theory that includes bending as well as stretching forces, and we use it to calculate mechanical response of a diluted filamentous triangular lattice. In this lattice, bonds are central-force springs, and there are bending forces between neighboring bonds on the same filament. We investigate the diluted lattice in which each bond is present with a probability $p$. We find a rigidity threshold $p_b$ which has the same value for all positive bending rigidity and a crossover characterizing bending-, stretching-, and bend-stretch coupled elastic regimes controlled by the central-force rigidity percolation point at $p_{\textrm{CF}} \simeq 2/3$ of the lattice when fiber bending rigidity vanishes.

cond-mat.soft↗

arXiv2011

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 and 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 the long-wavelength limit, the elastic theory of all these lattices is a conformally invariant field theory with holographic properties, 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.

cond-mat.soft↗

arXiv2011

Rigidity percolation on the square lattice

The square lattice with central forces between nearest neighbors is isostatic with a subextensive number of floppy modes. It can be made rigid by the random addition of next-nearest neighbor bonds. This constitutes a rigidity percolation transition which we study analytically by mapping it to a connectivity problem of two-colored random graphs. We derive an exact recurrence equation for the probability of having a rigid percolating cluster and solve it in the infinite volume limit. From this solution we obtain the rigidity threshold as a function of system size, and find that, in the thermodynamic limit, there is a mixed first-order-second-order rigidity percolation transition at the isostatic point.

cond-mat.stat-mech↗

arXiv2010

Criticality and isostaticity in fiber networks

The rigidity of elastic networks depends sensitively on their internal connectivity and the nature of the interactions between constituents. Particles interacting via central forces undergo a zero-temperature rigidity-percolation transition near the isostatic threshold, where the constraints and internal degrees of freedom are equal in number. Fibrous networks, such as those that form the cellular cytoskeleton, become rigid at a lower threshold due to additional bending constraints. However, the degree to which bending governs network mechanics remains a subject of considerable debate. We study disordered fibrous networks with variable coordination number, both above and below the central-force isostatic point. This point controls a broad crossover from stretching- to bending-dominated elasticity. Strikingly, this crossover exhibits an anomalous power-law dependence of the shear modulus on both stretching and bending rigidities. At the central-force isostatic point---well above the rigidity threshold---we find divergent strain fluctuations together with a divergent correlation length $ξ$, implying a breakdown of continuum elasticity in this simple mechanical system on length scales less than $ξ$.

cond-mat.soft↗

arXiv2010

Non-affine displacements in flexible polymer networks

The validity of the affine assumption in model flexible polymer networks is explored. To this end, the displacements of fluorescent tracer beads embedded in polyacrylamide gels are quantified by confocal microscopy under shear deformation, and the deviations of these displacements from affine responses are recorded. Non-affinity within the gels is quantified as a function of polymer chain density and cross-link concentration. Observations are in qualitative agreement with current theories of polymer network non-affinity. The measured degree of non-affinity in the polyacrylamide gels suggests the presence of structural inhomogeneities which likely result from heterogeneous reaction kinetics during gel preparation. In addition, the macroscopic elasticity of the polyacrylamide gels is confirmed to behave in accordance with standard models of flexible polymer network elasticity.

cond-mat.soft↗

arXiv2010

Coherent potential approximation of random nearly isostatic kagome lattice

The kagome lattice has coordination number $4$, and it is mechanically isostatic when nearest neighbor ($NN$) sites are connected by central force springs. A lattice of $N$ sites has $O(\sqrt{N})$ 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 (CPA) to study the mode structure and mechanical properties of the kagome lattice in which $NNN$ springs with spring constant $κ$ are added with probability $\Prob= Δz/4$, where $Δz= z-4$ and $z$ is the average coordination number. The effective medium static $NNN$ spring constant $κ_m$ scales as $\Prob^2$ for $\Prob \ll κ$ and as $\Prob$ for $\Prob \gg κ$, yielding a frequency scale $ω^* \sim Δz$ and a length scale $l^*\sim (Δz)^{-1}$. To a very good approximation at at small nonzero frequency, $κ_m(\Prob,ω)/κ_m(\Prob,0)$ is a scaling function of $ω/ω^*$. The Ioffe-Regel limit beyond which plane-wave states becomes ill-define is reached at a frequency of order $ω^*$.

cond-mat.dis-nn↗

arXiv2010

Soft random solids and their heterogeneous elasticity

Spatial heterogeneity in the elastic properties of soft random solids is examined via vulcanization theory. The spatial heterogeneity in the \emph{structure} of soft random solids is a result of the fluctuations locked-in at their synthesis, which also brings heterogeneity in their \emph{elastic properties}. Vulcanization theory studies semi-microscopic models of random-solid-forming systems, and applies replica field theory to deal with their quenched disorder and thermal fluctuations. The elastic deformations of soft random solids are argued to be described by the Goldstone sector of fluctuations contained in vulcanization theory, associated with a subtle form of spontaneous symmetry breaking that is associated with the liquid-to-random-solid transition. The resulting free energy of this Goldstone sector can be reinterpreted as arising from a phenomenological description of an elastic medium with quenched disorder. Through this comparison, we arrive at the statistics of the quenched disorder of the elasticity of soft random solids, in terms of residual stress and Lamé-coefficient fields. In particular, there are large residual stresses in the equilibrium reference state, and the disorder correlators involving the residual stress are found to be long-ranged and governed by a universal parameter that also gives the mean shear modulus.

cond-mat.dis-nn↗

arXiv2010

Soft modes and elasticity of nearly isostatic lattices: randomness and dissipation

The square lattice with central-force springs on nearest-neighbor bonds is isostatic. It has a zero mode for each row and column, and it does not support shear. Using the Coherent Potential Approximation (CPA), we study how the random addition, with probability $\mathcal{P}=(z-4)/4$ ($z$ = average number of nearest neighbors), of springs on next-nearest-neighbor ($NNN$) bonds restores rigidity and affects phonon structure. We find that the CPA effective $NNN$ spring constant $\tildeκ_m(ω)$, equivalent to the complex shear modulus $G(ω)$, obeys the scaling relation, $\tildeκ_m(ω) = κ_m h(ω/ω^*)$, at small $\mathcal{P}$, where $κ_m = \tildeκ'_m(0)\sim \mathcal{P}^2$ and $ω^* \sim \mathcal{P}$, implying that elastic response is nonaffine at small $\mathcal{P}$ and that plane-wave states are ill-defined beyond the Ioffe-Regel limit at $ω\approx ω^*$. We identify a divergent length $l^* \sim \mathcal{P}^{-1}$, and we relate these results to jamming.

cond-mat.soft↗

arXiv2006

Elastic heterogeneity of soft random solids

Spatial heterogeneity in the elastic properties of soft random solids is investigated via a two-pronged approach. First, a nonlocal phenomenological model for the elastic free energy is examined. This features a quenched random kernel, which induces randomness in the residual stress and Lame coefficients. Second, a semi-microscopic model network is explored using replica statistical mechanics. The Goldstone fluctuations of the semi-microscopic model are shown to reproduce the phenomenological model, and via this correspondence the statistical properties of the residual stress and Lame coefficients are inferred. Correlations involving the residual stress are found to be long-ranged and governed by a universal parameter that also gives the mean shear modulus.

cond-mat.dis-nn↗