1
|
García-Valladares G, Plata CA, Prados A. Buckling in a rotationally invariant spin-elastic model. Phys Rev E 2023; 107:014120. [PMID: 36797953 DOI: 10.1103/physreve.107.014120] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2022] [Accepted: 12/22/2022] [Indexed: 06/18/2023]
Abstract
Scanning tunneling microscopy experiments have revealed a spontaneous rippled-to-buckled transition in heated graphene sheets, in absence of any mechanical load. Several models relying on a simplified picture of the interaction between elastic and internal, electronic, degrees of freedom have been proposed to understand this phenomenon. Nevertheless, these models are not fully consistent with the classical theory of elasticity, since they do not preserve rotational invariance. Herein, we develop and analyze an alternative classical spin-elastic model that preserves rotational invariance while giving a qualitative account of the rippled-to-buckled transition. By integrating over the internal degrees of freedom, an effective free energy for the elastic modes is derived, which only depends on the curvature. Minimization of this free energy gives rise to the emergence of different mechanical phases, whose thermodynamic stability is thoroughly analyzed, both analytically and numerically. All phases are characterized by a spatially homogeneous curvature, which plays the role of the order parameter for the rippled-to-buckled transition, in both the one- and two-dimensional cases. In the latter, our focus is put on the honeycomb lattice, which is representative of actual graphene.
Collapse
Affiliation(s)
| | - Carlos A Plata
- Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, E-41080 Sevilla, Spain
| | - Antonio Prados
- Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, E-41080 Sevilla, Spain
| |
Collapse
|
2
|
Baldovin M, Vulpiani A, Puglisi A, Prados A. Derivation of a Langevin equation in a system with multiple scales: The case of negative temperatures. Phys Rev E 2019; 99:060101. [PMID: 31330599 DOI: 10.1103/physreve.99.060101] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/20/2019] [Indexed: 11/07/2022]
Abstract
We consider the problem of building a continuous stochastic model, i.e., a Langevin or Fokker-Planck equation, through a well-controlled coarse-graining procedure. Such a method usually involves the elimination of the fast degrees of freedom of the "bath" to which the particle is coupled. Specifically, we look into the general case where the bath may be at negative temperatures, as found, for instance, in models and experiments with bounded effective kinetic energy. Here, we generalize previous studies by considering the case in which the coarse graining leads to (i) a renormalization of the potential felt by the particle, and (ii) spatially dependent viscosity and diffusivity. In addition, a particular relevant example is provided, where the bath is a spin system and a sort of phase transition takes place when going from positive to negative temperatures. A Chapman-Enskog-like expansion allows us to rigorously derive the Fokker-Planck equation from the microscopic dynamics. Our theoretical predictions show excellent agreement with numerical simulations.
Collapse
Affiliation(s)
- Marco Baldovin
- Dipartimento di Fisica, Università di Roma Sapienza, Piazzale Aldo Moro 2, I-00185 Rome, Italy
| | - Angelo Vulpiani
- Dipartimento di Fisica, Università di Roma Sapienza, Piazzale Aldo Moro 2, I-00185 Rome, Italy
| | - Andrea Puglisi
- Istituto dei Sistemi Complessi - CNR and Dipartimento di Fisica, Università di Roma Sapienza, Piazzale Aldo Moro 2, I-00185 Rome, Italy
| | - Antonio Prados
- Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, E-41080 Sevilla, Spain
| |
Collapse
|
3
|
Ruiz-Garcia M, Bonilla LL, Prados A. Bifurcation analysis and phase diagram of a spin-string model with buckled states. Phys Rev E 2018; 96:062147. [PMID: 29347384 DOI: 10.1103/physreve.96.062147] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/11/2017] [Indexed: 11/07/2022]
Abstract
We analyze a one-dimensional spin-string model, in which string oscillators are linearly coupled to their two nearest neighbors and to Ising spins representing internal degrees of freedom. String-spin coupling induces a long-range ferromagnetic interaction among spins that competes with a spin-spin antiferromagnetic coupling. As a consequence, the complex phase diagram of the system exhibits different flat rippled and buckled states, with first or second order transition lines between states. This complexity translates to the two-dimensional version of the model, whose numerical solution has been recently used to explain qualitatively the rippled to buckled transition observed in scanning tunneling microscopy experiments with suspended graphene sheets. Here we describe in detail the phase diagram of the simpler one-dimensional model and phase stability using bifurcation theory. This gives additional insight into the physical mechanisms underlying the different phases and the behavior observed in experiments.
Collapse
Affiliation(s)
- M Ruiz-Garcia
- Gregorio Millán Institute for Fluid Dynamics, Nanoscience, and Industrial Mathematics, and Department of Materials Science and Engineering and Chemical Engineering, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain.,Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - L L Bonilla
- Gregorio Millán Institute for Fluid Dynamics, Nanoscience, and Industrial Mathematics, and Department of Materials Science and Engineering and Chemical Engineering, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain
| | - A Prados
- Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, E-41080, Sevilla, Spain
| |
Collapse
|
4
|
Caruel M, Truskinovsky L. Statistical mechanics of the Huxley-Simmons model. Phys Rev E 2016; 93:062407. [PMID: 27415298 DOI: 10.1103/physreve.93.062407] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/11/2015] [Indexed: 06/06/2023]
Abstract
The chemomechanical model of Huxley and Simmons (HS) [A. F. Huxley and R. M. Simmons, Nature 233, 533 (1971)NATUAS0028-083610.1038/233533a0] provides a paradigmatic description of mechanically induced collective conformational changes relevant in a variety of biological contexts, from muscles power stroke and hair cell gating to integrin binding and hairpin unzipping. We develop a statistical mechanical perspective on the HS model by exploiting a formal analogy with a paramagnetic Ising model. We first study the equilibrium HS model with a finite number of elements and compute explicitly its mechanical and thermal properties. To model kinetics, we derive a master equation and solve it for several loading protocols. The developed formalism is applicable to a broad range of allosteric systems with mean-field interactions.
Collapse
Affiliation(s)
- M Caruel
- MSME, CNRS-UMR 8208, 61 Avenue du Général de Gaulle, 94010 Créteil, France
| | - L Truskinovsky
- LMS, CNRS-UMR 7649, Ecole Polytechnique, 91128 Palaiseau Cedex, France
| |
Collapse
|
5
|
Bonilla LL, Carpio A, Prados A. Theory of force-extension curves for modular proteins and DNA hairpins. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:052712. [PMID: 26066204 DOI: 10.1103/physreve.91.052712] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/22/2014] [Indexed: 06/04/2023]
Abstract
We study a model describing the force-extension curves of modular proteins, nucleic acids, and other biomolecules made out of several single units or modules. At a mesoscopic level of description, the configuration of the system is given by the elongations of each of the units. The system free energy includes a double-well potential for each unit and an elastic nearest-neighbor interaction between them. Minimizing the free energy yields the system equilibrium properties whereas its dynamics is given by (overdamped) Langevin equations for the elongations, in which friction and noise amplitude are related by the fluctuation-dissipation theorem. Our results, both for the equilibrium and the dynamical situations, include analytical and numerical descriptions of the system force-extension curves under force or length control and agree very well with actual experiments in biomolecules. Our conclusions also apply to other physical systems comprising a number of metastable units, such as storage systems or semiconductor superlattices.
Collapse
Affiliation(s)
- L L Bonilla
- G. Millán Institute, Fluid Dynamics, Nanoscience and Industrial Mathematics, Universidad Carlos III de Madrid, 28911 Leganés, Spain
| | - A Carpio
- Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
| | - A Prados
- Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, E-41080, Sevilla, Spain
| |
Collapse
|
6
|
Prados A, Carpio A, Bonilla LL. Sawtooth patterns in force-extension curves of biomolecules: an equilibrium-statistical-mechanics theory. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:012704. [PMID: 23944489 DOI: 10.1103/physreve.88.012704] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/21/2013] [Indexed: 06/02/2023]
Abstract
We analyze the force-extension curve for a general class of systems, which are described at the mesoscopic level by a free energy depending on the extension of its components. Similarly to what is done in real experiments, the total length of the system is the controlled parameter. This imposes a global constraint in the minimization procedure leading to the equilibrium values of the extensions. As a consequence, the force-extension curve has multiple branches in a certain range of forces. The stability of these branches is governed by the free energy: there are a series of first-order phase transitions at certain values of the total length, in which the free energy itself is continuous but its first derivative, the force, has a finite jump. This behavior is completely similar to that observed in real experiments with biomolecules like proteins and with other complex systems.
Collapse
Affiliation(s)
- A Prados
- Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, E-41080 Sevilla, Spain
| | | | | |
Collapse
|
7
|
Borówko M, Rżysko W, Sokołowski S, Sokołowska Z, Usatenko Z. Stretching tethered polymer chains: density functional approach. J Chem Phys 2013; 138:204707. [PMID: 23742500 DOI: 10.1063/1.4807086] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/02/2023] Open
Abstract
We propose application of density functional theory to calculate the force acting on a selected segment of a tethered polymer chain that leads to stretching the chain. The density functional allows one to determine the effects due to the presence of other chains and solvent molecules. For high and moderate solvent densities the plot of the force versus the distance of the segment from the surface exhibits oscillatory behavior that has not been predicted by other approaches.
Collapse
Affiliation(s)
- M Borówko
- Department for the Modelling of Physico-Chemical Processes, MCS University, 20031 Lublin, Poland
| | | | | | | | | |
Collapse
|