• Q. Deng, F. Ahmadpoor, W. Brownell, P. Sharma. "The collusion of flexoelectricity and Hopf bifurcation in the hearing mechanism." Journal of the Mechanics and Physics of Solids (2019).











  • (*Equal contribution) Zelisko, M.*, Ahmadpoor, F.*, Gao, H., & Sharma, P. "Determining the Gaussian Modulus and Edge Properties of 2D Materials: From Graphene to Lipid Bilayers. " Physical Review Letters , 119(6), 068002, 2017.

  • Fatemeh Ahmadpoor, Peng Wang, Rui Huang and Pradeep Sharma. "Thermal Fluctuations and Effective Bending Stiffness of Elastic Thin Sheets and Graphene: A Nonlinear Analysis" Journal of the Mechanics and Physics of Solids (IF: 4.2), 107, 294-319, 2017.

  • F. Ahmadpoor and P. Sharma, “A Perspective on the Statistical Mechanics of 2D Materials”, Extreme Mechanics Letters , 14 (2016): 38-43

  • Fatemeh Ahmadpoor, and Pradeep Sharma. "Thermal Fluctuations of Vesicles and Nonlinear Curvature Elasticity—Implications for Size-dependent Renormalized Bending Rigidity and Vesicle Size Distribution." Soft Matter 12.9 (2016): 2523-2536.

  • Fatemeh Ahmadpoor, and Pradeep Sharma. "Flexoelectricity in two-Dimensional Crystalline and Biological Membranes." Nanoscale  7.40 (2015): 16555- 16570.

  • Fatemeh Ahmadpoor, Liping Liu, and Pradeep Sharma. "Thermal Fluctuations and the Minimum Electrical Field that Can be Detected by a Biological Membrane."Journal of the Mechanics and Physics of Solids  78 (2015): 110-122.

  • Fatemeh Ahmadpoor, Qian Deng, Liping Liu and Pradeep Sharma. "Apparent Flexoelectricity in Lipid Bilayer Membranes due to External Charge and Dipolar Distributions." Physical Review E  88.5 (2013): 050701.

  • Shodja, H. M., F. Ahmadpoor, and A. Tehranchi. "Calculation of the Additional Constants for FCC Materials in Second Strain Gradient Elasticity: Behavior of a Nano-size Bernoulli- Euler Beam with Surface Effects." Journal of Applied Mechanics 79.2 (2012): 021008.


How do the weak sound waves get amplified in a cochlea? This deceptively simple question has attracted a fair amount of attention and several creative mechanisms have been proposed that purport to understand how the inner ear’s hair cells actively collude to achieve the requisite sensitivity, frequency selectivity, range and nonlinear amplification. Some of the proposed mechanisms target the nature of the mechanoelectric transduction mechanism while others adopt a more dynamical systems approach and focus on the fact that stereocilia of the hair cells operate on the verge of an instability phenomenon—the so-called Hopf bifurcation. In this work, we propose a physics-based model to understand how flexoelectricity, a universal electromechanical coupling that exists in all dielectric substances, facilitates the mechanics of the active motion of hair bundles. A key feature of our model is that we eschew a “black-box” approach, and all parameters are well-defined physical quantities such as membrane bending modulus, geometrical characteristics and others. Furthermore, the model is derived from the well-accepted principles of mechanics and soft matter physics. While the role of flexoelectricity in the hearing mechanism has been noted before, we show for the first time that flexoelectricity is an essential ingredient in inducing the Hopf bifurcation state considered responsible for several highly nonlinear and peculiar features of the hearing mechanism. We find that the biomembranes’ bending modulus and the intracellular charge concentration (which for instance could represent K+ or Ca2+) are the two key control parameters that significantly impact the stability of the system and hence the hearing mechanism. Our work highlights the importance of flexoelectricity, confirms earlier assertions that the cochlea amplifies the acoustic stimuli through its exceptional electromechanical energy conversion property, and provides insights into how physical properties such as biomembranes’ bending modulus impact the performance of the hearing system.​

The dominant deformation behavior of two-dimensional materials (bending) is primarily governed by just two parameters: bending rigidity and the Gaussian modulus. These properties also set the energy scale for various important physical and biological processes such as pore formation, cell fission and generally, any event accompanied by a topological change. Unlike the bending rigidity, the Gaussian modulus is, however, notoriously difficult to evaluate via either experiments or atomistic simulations. In this Letter, recognizing that the Gaussian modulus and edge tension play a nontrivial role in the fluctuations of a 2D material edge, we derive closed-form expressions for edge fluctuations. Combined with atomistic simulations, we use the developed approach to extract the Gaussian modulus and edge tension at finite temperatures for both graphene and various types of lipid bilayers. Our results possibly provide the first reliable estimate of this elusive property at finite temperatures and appear to suggest that earlier estimates must be revised. In particular, we show that, if previously estimated properties are employed, the graphene- free edge will exhibit unstable behavior at room temperature. Remarkably, in the case of graphene, we show that the Gaussian modulus and edge tension even change sign at finite temperatures.


The study of statistical mechanics of thermal fluctuations of graphene—the prototypical two-dimensional material—is rendered rather complicated due to the necessity of account- ing for geometric deformation nonlinearity. Unlike fluid membranes such as lipid bilayers, coupling of stretching and flexural modes in solid membranes like graphene leads to a highly anharmonic elastic Hamiltonian. Existing treatments draw heavily on analogies in the high-energy physics literature and are hard to extend or modify in the typical contexts that permeate materials, mechanics and some of the condensed matter physics literature. In this study, using a variational perturbation method, we present a “mechanics-oriented” treatment of the thermal fluctuations of elastic sheets such as graphene and evaluate their effect on the effective bending stiffness at finite temperatures. In particular, we explore the size, pre-strain and temperature dependency of the out-of-plane fluctuations, and demonstrate how an elastic sheet becomes effectively stiffer at larger sizes. Our derivations pro- vide a transparent approach that can be extended to include multi-field couplings and anisotropy for other 2D materials. To reconcile our analytical results with atomistic con- siderations, we also perform molecular dynamics simulations on graphene and contrast the obtained results and physical insights with those in the literature.


2D materials are fascinating for numerous reasons. Their geometrical and mechanical characteristics along with other associated physical properties have opened up fascinating new application avenues ranging from electronics, energy harvesting, biological systems among others. Due to the 2D nature of these materials, they are known for their unusual flexibility and the ability to sustain large curvature deformations. Further, they undergo noticeable thermal fluctuations at room temperature. In this perspective, we highlight both the characteristics and implications of thermal fluctuations in 2D materials and discuss current challenges in the context of statistical mechanics of fluid and solid membranes.


Both closed and open biological membranes noticeably undulate at physiological temperatures. These thermal fluctuations influence a broad range of biophysical phenomena, ranging from self- assembly to adhesion. In particular, the experimentally measured thermal fluctuation spectra also provide a facile route to the assessment of mechanical and certain other physical properties of biological membranes. The theoretical assessment of thermal fluctuations, be it for closed vesicles or the simpler case of flat open lipid bilayers, is predicated upon assuming that the elastic curvature energy is a quadratic functional of the curvature tensor. However, a qualitatively correct description of several phenomena such as binding–unbinding transition, vesicle-to-bicelle transition, appearance of hats and saddles among others, appears to require consideration of constitutively nonlinear elasticity that includes fourth order curvature contributions rather than just quadratic. In particular, such nonlinear considerations are relevant in the context of large-curvature or small-sized vesicles. In this work we discuss the statistical mechanics of closed membranes (vesicles) incorporating both constitutive and geometrical nonlinearities. We derive results for the renormalized bending rigidity of small vesicles and show that significant stiffening may occur for sub-20 nm vesicle sizes. Our closed-form results may also be used to determine nonlinear curvature elasticity properties from either experimentally measured fluctuation spectra or microscopic calculations such as molecular dynamics. Finally, in the context of our results on thermal fluctuations of vesicles and nonlinear curvature elasticity, we reexamine the problem of determining the size distribution of vesicles and obtain results that reconcile well with experimental observations. However, our results are somewhat paradoxical. Specifically, the molecular dynamics predictions for the thermo- mechanical behavior of small vesicles of prior studies appear to be inconsistent with the nonlinear elastic properties that we estimate by fitting to the experimentally determined vesicle size-distribution trends and data.


Thermal electrical noise in living cells is considered to be the minimum threshold for several biological response mechanisms that pertain to electric fields. Existing models that purport to explain and interpret this phenomena yield perplexing results. The simplest model, in which the biomembrane is considered to be a linear dielectric, yields an equilibrium noise level that is several orders of magnitude larger than what is observed experimentally. An alternative approach of estimating the thermal noise as the Nyquist noise of a resistor within a finite frequency bandwidth, yields little physical insight. In this work, we argue that the nonlinear dielectric behavior must be accounted for. Using a statistical mechanics approach, we analyze the thermal fluctuations of a fully coupled electromechanical biomembrane. We develop a variational approximation to analytically obtain the benchmark results for model fluid membranes as well as physically reasonable estimates of the minimum electrical field threshold that can be detected by cells. Qualitatively, at least, our model is capable of predicting all known experimental results. The predictions of our model also suggest that further experimental work is warranted to clarify the inconsistencies in the literature.

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In this Rapid Communication we show that the interplay between the deformation geometric-nonlinearity and distributions of external charges and dipoles lead to the renormalization of the membrane’s native flexoelectric response. Our work provides a framework for a mesoscopic interpretation of flexoelectricity and if necessary, artificially “design” tailored flexoelectricity in membranes. Comparisons with experiments indicate reasonable quantitative agreement.


In addition to enhancement of the results near the point of application of a concentrated load in the vicinity of nano-size defects, capturing surface effects in small structures, in the framework of second strain gradient elasticity is of particular interest. In this framework, sixteen additional material constants are revealed, incorporating the role of atomic structures of the elastic solid. In this work, the analytical formulations of these constants corresponding to fee metals are given in terms of the parameters of Sutton-Chen interatomic potential function. The constants for ten fcc metals are computed and tabulized. Moreover, the exact closed-form solution of the bending of a nano-size Bernoulli-Euler beam in second strain gradient elasticity is provided; the appearance of the additional constants in the corresponding formulations, through the governing equation and boundary conditions, can serve to delineate the true behavior of the material in ultra small elastic structures, having very large surface-to-volume ratio. Now that the values of the material constants are available, a nanoscopic study of the Kelvin problem in second strain gradient theory is performed, and the result is compared quantitatively with those of the first strain gradient and traditional theories.

The ability of a material to convert electrical stimuli into mechanical deformation, i.e. piezoelectricity, is a remarkable property of a rather small subset of insulating materials. The phenomenon of flexoelectricity, on the other hand, is universal. All dielectrics exhibit the flexoelectric effect whereby non-uniform strain (or strain gradients) can polarize the material and conversely non-uniform electric fields may cause mechanical deformation. The flexoelectric effect is strongly enhanced at the nanoscale and accordingly, all two-dimensional membranes of atomistic scale thickness exhibit a strong two-way coupling between the curvature and electric field. In this review, we highlight the recent advances made in our understand- ing of flexoelectricity in two-dimensional (2D) membranes—whether the crystalline ones such as dielec- tric graphene nanoribbons or the soft lipid bilayer membranes that are ubiquitous in biology. Aside from the fundamental mechanisms, phenomenology, and recent findings, we focus on rapidly emerging direc- tions in this field and discuss applications such as energy harvesting, understanding of the mammalian hearing mechanism and ion transport among others.