kBT Ln Research Articles

Reversible Hydrogen Electrode (RHE) Scale Dependent Surface Pourbaix Diagram at Different pH.

In the analysis of electrocatalysis mechanisms and the design of catalysts, the effect of electrochemistry-induced surface coverage is a critical consideration that should not be overlooked. The surface Pourbaix diagram emerges as a fundamental tool in this context, providing essential insights into the surface coverage of adsorbates generated via electrochemical potential-driven water activation. A classic surface Pourbaix diagram considers the pH effects by correcting the free energy of H+ ions by the concentration-dependent term: -kBT ln(10) × pH, which is independent of the reversible hydrogen electrode (RHE) scale. However, this is sometimes inconsistent with the experimentally observed potential-dependent surface coverage at an RHE scale, especially under high-pH conditions. Here, we derived the pH-dependent surface Pourbaix diagram at an RHE scale by considering the energetics computed by density functional theory with the Bayesian Error Estimation Functional with van der Waals corrections (BEEF-vdW), the electric field effects, the derived adsorption-induced dipole moment and polarizability, and the potential of zero-charge. Using Pt(111) as the typical example, we found that the surface coverage predicted by the proposed RHE-dependent surface Pourbaix diagram can significantly minimize the discrepancy between theory and experimental observations, especially under neutral-alkaline, moderate-potential conditions. This work provides a new methodology and establishes guidelines for the precise analysis of the surface coverage prior to the evaluation of the activity of an electrocatalyst.

Dissipation during the Gating Cycle of the Bacterial Mechanosensitive Ion Channel Approaches the Landauer Limit.

The Landauer principle sets a thermodynamic bound of kBT ln 2 on the energetic cost of erasing each bit of information. It holds for any memory device, regardless of its physical implementation. It was recently shown that carefully built artificial devices can attain this bound. In contrast, biological computation-like processes, e.g., DNA replication, transcription and translation use an order of magnitude more than their Landauer minimum. Here, we show that reaching the Landauer bound is nevertheless possible with biological devices. This is achieved using a mechanosensitive channel of small conductance (MscS) from E. coli as a memory bit. MscS is a fast-acting osmolyte release valve adjusting turgor pressure inside the cell. Our patch-clamp experiments and data analysis demonstrate that under a slow switching regime, the heat dissipation in the course of tension-driven gating transitions in MscS closely approaches its Landauer limit. We discuss the biological implications of this physical trait.

Overcoming Boltzmann's Tyranny in a Transistor via the Topological Quantum Field Effect.

The subthreshold swing is the critical parameter determining the operation of a transistor in low-power applications such as switches. It determines the fraction of dissipation due to the gate capacitance used for turning the device on and off, and in a conventional transistor it is limited by Boltzmann's tyranny to kBT ln(10)/q. Here, we demonstrate that the subthreshold swing of a topological transistor in which conduction is enabled by a topological phase transition via electric field switching, can be sizably reduced in a noninteracting system by modulating the Rashba spin-orbit interaction. By developing a theoretical framework for quantum spin Hall materials with honeycomb lattices, we show that the Rashba interaction can reduce the subthreshold swing by more than 25% compared to Boltzmann's limit in currently available materials but without any fundamental lower bound, a discovery that can guide future material design and steer the engineering of topological quantum devices.

(Invited) Suspended Graphene Monolayers for Low Voltage Switches

Since the dawn of electronics, the energy efficiency of computation has been a driving force for change, motivating disruptive technology advances from vacuum tubes to discrete semiconductor devices to ultra-large-scale integrated circuits [1]. Modern society’s reliance on the electronics behind digital hardware will continue to grow, with accelerating demand for computing power driven by advances such as artificial intelligence and machine learning. Improving energy efficiency while maintaining functional density of integrated electronics is widely acknowledged as the most important technological challenge for electronics in the 21st century [2-7]. The thermodynamic limit to irreversible computation at room temperature is ~ kBT ln 2 = 18 meV per bit of information erasure, corresponding roughly to a limit of 18 meV per logical operation in a transistor based logic. By contrast, the energy consumed in a modern integrated circuit per logical operation is of the order 104-105 kBT ~ 0.6-6 keV, as a result of the energy consumed to transmit information between transistors by charging interconnect capacitance up to the operating voltage of the transistor.The mean energy consumed per transistor per clock-cycle in a digital circuit [7]: E = CV 2 [α + (I off/I on)β] where C ~ 10 fF is the average load capacitance dominated by interconnects, V ~ 1 V is the transistor operating voltage, α ~ 1% is the fraction of transistors that switch per clock-cycle, I on/I off ~ 106 is the on/off current ratio of the transistor, and β ~ 100-1000 is the ratio of clock period (~1 ns) to the intrinsic transistor switching delay time (~1 ps). Improving the energy efficiency of a transistor circuit can thus be achieved by reducing operating voltage V while maintaining the on/off current ratio I on/I off . Transistors are limited by the sub-threshold swing of current versus voltage, S = ∂VG/∂log(ID), which is typically subject to the thermionic limit ~(kT/e)×ln10 ≈ 60 mV/decade, although tunneling transistors can modestly surpass this limit but at the cost of a significant loss of current drive [2-7]. Switches with improved sub-threshold swing S are known as steep-slope devices.We propose a new approach to energy efficient electronics, consisting of nanomechanical suspended graphene steep-slope switches. The mechanical properties of graphene are ideally suited to nanoelectromechanical systems (NEMS). Graphene’s Young’s modulus Y ~ 1 TPa [8] is higher than any other material, but the atomic thinness of graphene t = 0.34 nm results in an elastic modulus for deflection of Ee = Et/(1−υ) ~ 400 N/m for an unstrained membrane. Graphene monolayers are the most flexible membranes known, and thus require less voltage than any other membrane for electrostatically actuated deflection. Adopting graphene as a membrane material can in principle achieve a ~10 fold reduction in operating voltage to ~0.1 V and thus a ~100 fold reduction in switching energy, as a direct result of suspended monolayer graphene’s uniquely low deflection modulus [9]. Experimental realization of such low-voltage operation poses challenges in device fabrication and scaling.

Reminiscent capillarity in subnanopores

Fluids in large and small pores display different behaviors with a crossover described through the concept of critical capillarity. Here we report experimental and simulation data for various siliceous zeolites and adsorbates that show unexpected reminiscent capillarity for such nanoporous materials. For pore sizes D exceeding the fluid molecule size, the filling pressures p are found to follow a generic behavior kBT ln p ∼ γ/ρD where γ and ρ are the fluid surface tension and density. This result is rationalized by showing that the filling chemical potential for such ultra-small pores is the sum of an adsorption energy and a capillary energy that remains meaningful even for severe confinements. A phenomenological model, based on Derjaguin’s formalism to bridge macroscopic and molecular theories for condensation in porous materials, is developed to account for the behavior of fluids confined down to the molecular scale from simple parameters.

Perspective: Identification of collective variables and metastable states of protein dynamics.

The statistical analysis of molecular dynamics simulations requires dimensionality reduction techniques, which yield a low-dimensional set of collective variables (CVs) {x i } = x that in some sense describe the essential dynamics of the system. Considering the distribution P( x ) of the CVs, the primal goal of a statistical analysis is to detect the characteristic features of P( x ), in particular, its maxima and their connection paths. This is because these features characterize the low-energy regions and the energy barriers of the corresponding free energy landscape ΔG( x ) = -k B T ln P( x ), and therefore amount to the metastable states and transition regions of the system. In this perspective, we outline a systematic strategy to identify CVs and metastable states, which subsequently can be employed to construct a Langevin or a Markov state model of the dynamics. In particular, we account for the still limited sampling typically achieved by molecular dynamics simulations, which in practice seriously limits the applicability of theories (e.g., assuming ergodicity) and black-box software tools (e.g., using redundant input coordinates). We show that it is essential to use internal (rather than Cartesian) input coordinates, employ dimensionality reduction methods that avoid rescaling errors (such as principal component analysis), and perform density based (rather than k-means-type) clustering. Finally, we briefly discuss a machine learning approach to dimensionality reduction, which highlights the essential internal coordinates of a system and may reveal hidden reaction mechanisms.

Molecular Dynamics Simulations and the Landauer’s Principle

Landauer’s principle is a fundamental link between thermodynamics and information theory, which implies that the erasure of information comes at an energetic price. In the present contribution we analyze to what extend the usual molecular dynamics (MD) simulation formalism can handle the Landauer’s bound kBT ln 2 in the simplest case of one particle treated classically. The erasure of one bit of information is performed by adiabatically varying the shape of a bistable potential in a full cycle. We will highlight the inadequacy of either the microcanonical or canonical ensemble treatments currently employed in MD simulations and propose potential solutions.

Selectivity and Desorption Free Energies for Methane–Ethane Mixtures in Covalent Organic Frameworks

We show how expanded Wang–Landau simulations can be used to study the adsorption of methane–ethane mixtures in COF-102, COF-105, and COF-108. This approach has several advantages. First, a single simulation run is performed to determine key thermodynamic properties such as the adsorption isotherms and selectivity. Second, the combination of the expanded method with the Wang–Landau sampling in the grand-canonical ensemble provides direct access to the grand potential Ω = −kBT ln[Θ(μ1,μ2,V,T)] via a numerical evaluation of the grand-canonical partition function. From there, we calculate several thermodynamic quantities of adsorption, including the Gibbs free energy, enthalpy, and entropy, which give important insights into the mechanism of adsorption for the methane–ethane mixtures in covalent organic frameworks (COFs). In particular, using a solution thermodynamics approach, we identify a direct correlation between the separation efficiency (selectivity) of a given COF and its energetic efficiency (through...

Intramolecular Interactions Overcome Hydration to Drive the Collapse Transition of Gly15

Simulations and experiments show oligo-glycines, polypeptides lacking any side chains, can collapse in water. We assess the hydration thermodynamics of this collapse by calculating the hydration free energy at each of the end points of the reaction coordinate, here taken as the end-to-end distance (r) in the chain. To examine the role of the various conformations for a given r, we study the conditional distribution, P(Rg|r), of the radius of gyration for a given value of r. The free energy change versus Rg, -kBT ln P(Rg|r), is found to vary more gently compared to the corresponding variation in the excess hydration free energy. Using this observation within a multistate generalization of the potential distribution theorem, we calculate a tight upper bound for the hydration free energy of the peptide for a given r. On this basis, we find that peptide hydration greatly favors the expanded state of the chain, despite primitive hydrophobic effects favoring chain collapse. The net free energy of collapse is seen to be a delicate balance between opposing intrapeptide and hydration effects, with intrapeptide contributions favoring collapse.

Beating Landauer’s bound by Memory Erasure using Time Multiplexed Potentials

The Landauer’s Principle, proposed by Rolf Landauer in 1961, states that the logically irreversible operation of erasing a single bit of information requires atleast kbT ln 2 amount of energy, kb and T being Boltzmann’s constant and temperature respectively. However, Landauer’s bound holds only for erasure mechanisms that are perfect. In this article we investigate the effect of imperfections in erasure mechanisms. If the proportion of successful erasures is quantified by p, we show that the minimum energy needed to erase a bit of information is given by kbT[ln 2 + p ln p + (1 - p) ln (1 - p)], also known as the Generalized Landauer bound. Furthermore, we provide a mechanism for realizing a memory bit by multiplexing an optical trap rapidly and propose a mechanism of erasure, for various success proportions p. Using our framework, we show using Monte Carlo simulations that heat dissipation lower than the Landauer’s bound is achievable by reducing p. Thus, we establish an independent method of beating Landauer’s bound by resorting to partially successful erasures.

Thermodynamics of force-dependent folding and unfolding of small protein and nucleic acid structures.

In this paper, we outline the theoretical framework for understanding the equilibrium force-dependent folding and unfolding transitions of protein domains and small nucleic acid structures, both having small rigid folded structures and highly flexible unfolded polymeric chain conformations. A complete statistical description of the state described by the probability function ρ(ξ)(n,x), is obtained, where n is an index denoting the structural state, and x is the extension of the molecule. ξ denotes an external constraint applied to the molecule, which is either a constant force or a harmonic spring attached to one end of the molecule. The extension probability distribution regardless of the structural state: , the free energy landscape: -kBT ln(ρ(ξ)(x)), and the probability of the states regardless of the extension: , are analyzed using the force-dependent structural transitions of the classic titin I27 domain as an example. The impact of different external constraints is also discussed.

Absolute Organic Crystal Thermodynamics: Growth of the Asymmetric Unit into a Crystal via Alchemy.

The solubility of organic molecules is of critical importance to the pharmaceutical industry; however, robust computational methods to predict this quantity from first-principles are lacking. Solubility can be computed from a thermodynamic cycle that decomposes standard state solubility into the sum of solid–vapor sublimation and vapor–liquid solvation free energies ΔGsolubility° = ΔGsub° + ΔGsolv°. Over the past few decades, alchemical simulation methods to compute solvation free energy using classical force fields have become widely used. However, analogous methods for determining the free energy of the sublimation/deposition phase transition are currently limited by the necessity of a priori knowledge of the atomic coordinates of the crystal. Here, we describe progress toward an alternative scheme based on growth of the asymmetric unit into a crystal via alchemy (GAUCHE). GAUCHE computes deposition free energy ΔGdep° = −ΔGsub° = −kBT ln(Vc/Vg) + ΔGAU + ΔGAU→UC as the sum of an entropic term to account ...

Comment on “Probing the Thermodynamics of Competitive Ion Binding Using Minimum Energy Structures”

Comment on “Probing the Thermodynamics of Competitive Ion Binding Using Minimum Energy Structures”

Calculation of Partition Functions and Free Energies of a Binary Mixture Using the Energy Partitioning Method: Application to Carbon Dioxide and Methane

It is challenging to compute the partition function (Q) for systems with enormous configurational spaces, such as fluids. Recently, we developed a Monte Carlo technique (an energy partitioning method) for computing Q [ J. Chem. Phys. 2011 , 135 , 174105 ]. In this paper, we use this approach to compute the partition function of a binary fluid mixture (carbon dioxide + methane); this allows us to obtain the Helmholtz free energy (F) via F = -k(B)T ln Q and the Gibbs free energy (G) via G = F + pV. We then utilize G to obtain the coexisting mole fraction curves. The chemical potential of each species is also obtained. At the vapor-liquid equilibrium condition, the chemical potential of methane significantly increases, while that of carbon dioxide slightly decreases, as the pressure increases along an isotherm. Since Q is obtained from the density of states, which is independent of the temperature, equilibrium thermodynamic properties at any condition can be obtained by varying the total composition and volume of the system. Our methodology can be adapted to explore the free energies of other binary mixtures in general and of those containing CO(2) in particular. Since the method gives access to the free energy and chemical potentials, it will be useful in many other applications.

Fundamental limits of energy dissipation in charge-based computing

According to Landauer’s principle, dissipation of energy is only necessary when information is erased, suggesting that vastly more efficient logical switches than transistors are possible. However, an influential analysis of binary switching suggests that representing information with electric charge is the root of the problem, that Landauer’s principle is fundamentally flawed, and that any movement of charge, such as charging a capacitor, must dissipate at least kBT ln(2). Here, using a RC circuit, an energy loss of much less than kBT ln(2) is demonstrated while delivering energy of 100 kBT ln(2) to the capacitor. This shows that there is no fundamental lower limit to energy dissipation in moving charge.

Magnetization reversal of two-dimensional superlattices of Mn3O4 nanocubes and their collective dipolar interaction effects

Magnetic properties of two-dimensional (2D) paddy-field like superlattices of Mn3O4 cubic nanoparticles have been investigated by magnetization measurements. The 2D ordered structure extends over several microns in size. Each nanocube is of single-crystal about 6 nm in size. The magnetic properties are investigated with the powders dispersed in nonmagnetic n-eicosane to “dilute” the dipolar interaction. By accounting for the temperature variation effect of the magnetocrystalline anisotropy, Kmag(T), the temperature dependent coercivity, HC(T), can be well described by the equation, HC(T)=H0kmag(T)/mS(T){1−[kBT ln(t/t0)/E0kmag(T)]3/4}, in which kmag(T)=Kmag(T)/Kmag(0) is the reduced temperature dependent magnetocrystalline anisotropy and mS(T)=MS(T)/MS(0) is the reduced saturation magnetization. The effects of collective dipolar interaction on the magnetic properties are also studied with the as-prepared powder sample. The apparent magnetic anisotropy is seriously reduced with the presence of dipolar interaction. The switching volume is determined by the analysis on the magnetic measurements both with and without the dipolar interaction effect. There is a discrepancy in the value of switching volume determined by the two different analysis methods. Possible reasons are discussed.

The thermodynamics of writing a random polymer

The notion that information has physical, and in particular, thermodynamic, content can be traced to the paradox of Maxwell's demon, a sly creature who observes the microscopic motions of gas particles on both sides of a partition (1). By controlling a trap door the demon segregates fast particles from slow ones to create a temperature difference across the partition, seemingly without expending any work. Generations of physicists have scratched their heads over this apparent violation of the second law of thermodynamics (2–5). The resolution that has eventually emerged acknowledges that a real-life Maxwell's demon—say, a nanoscale machine designed for the task—collects information as it operates, and work must be expended to erase this information, otherwise the demon's memory banks fill up. The minimum work required is kBT ln 2 per bit of information, precisely what is needed to rescue the second law from the paradox. In this issue of PNAS, Andrieux and Gaspard (6) analyze the flip side of the thermodynamic cost of information erasure; namely, the cost of information acquisition. The setting of their analysis is not a demon and a gas, but rather a process essential to living organisms: copolymerization, in which a chain-like molecule grows by the addition of chemically distinct units (monomers). The most celebrated example is the replication of DNA, by which genetic information is copied at the molecular level, ultimately to pass down the generations of a family tree.

Mechanism of Force Generation of a Viral DNA Packaging Motor

A large family of multimeric ATPases are involved in such diverse tasks as cell division, chromosome segregation, DNA recombination, strand separation, conjugation, and viral genome packaging. One such system is the Bacillus subtilis phage phi 29 DNA packaging motor, which generates large forces to compact its genome into a small protein capsid. Here we use optical tweezers to study, at the single-molecule level, the mechanism of force generation in this motor. We determine the kinetic parameters of the packaging motor and their dependence on external load to show that DNA translocation does not occur during ATP binding but is likely triggered by phosphate release. We also show that the motor subunits act in a coordinated, successive fashion with high processivity. Finally, we propose a minimal mechanochemical cycle of this DNA-translocating ATPase that rationalizes all of our findings.

Diffusion in a bistable potential. General inverse friction expansion

The derivation of the characteristic times and of the density probability distribution for the motion of a Brownian particle in a bistable potential at intermediate friction was, until now, essentially limited to low orders in the inverse frictionγ −1. On the other hand, at least for temperatures low with respect to the barrier height, the Kramers time, which is the lowest nonzero eigenvalue in the bistable potential problem, is known exactly. This paper presents a systematic approach for the determination of the solution of the Fokker-Planck equation in an arbitrary potential in the overdamped regime. This calculation includes anharmonicity corrections up to orderγ −5. One feature of this paper is to show that the problem is equivalent to replacing the original potentialφ(x) by a free energy which, for a velocity distribution at equilibrium, simply is $$\widetilde\phi $$ =φ(x) −k BT ln[g(x)], where $$g(x) = \left\{ {{{m\gamma } \mathord{\left/ {\vphantom {{m\gamma } {[2\phi ''(x)]}}} \right. \kern-\nulldelimiterspace} {[2\phi ''(x)]}}} \right\}\left\{ {1 - [{{1 - 4\phi ''(x)} \mathord{\left/ {\vphantom {{1 - 4\phi ''(x)} {m\gamma ^2 }}} \right. \kern-\nulldelimiterspace} {m\gamma ^2 }}]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } \right\}$$ For out-of-equilibrium velocity distribution an effective potential is also explicitly given. In every case the function g(x) plays a crucial role. This approach is then applied to the exact determination, in the low-temperature limit, of all the characteristic times and of the probability distribution in bistable potentials. Moreover, from the knowledge of the characteristic times and probability density distribution, it would be easy to determine the general and exact Suzuki scaling law for the relaxation from the instability point at intermediate friction.

An Approximate Differential Equation for Calculating the Electron Density in Closed Shell Atoms and in Molecules

It is well known from density functional theory that the ground-state electron density ρ(r) in an atom or molecule can be calculated exactly in principle, from a one-body potential energy V(r) into which exchange and correlation interactions are incorporated. Therefore, it is of considerable interest to set up a differential equation which will enable ρ(r) to be calculated directly from V(r) for atoms and molecules with many electrons, without recourse to wave function determination.The problem will be tackled as follows:(i) A careful discussion will be given of the differential equation in one-dimension. Its relation to local density approximations will be demonstrated and it will be shown that the linear harmonic oscillator is a case when the local density equation becomes exact.(ii) A treatment of a closed shell atom with many-electrons will be presented. Here, central field separability is utilized, and two methods are developed. In the first, a separate differential equation is to be solved for each value of the orbital angular momentum l. The second, and presently less accurate, approach solves directly for the total density ρ(r).(iii) For three-dimensional potentials V(r), with arbitrarily low symmetry, as in the general molecular problem, the route to ρ(r) via a generalization of the partition function Z(r, β) with β = (kBT)-1 is then explored. In particular, the effective potential U(rβ) which is essentially - kBT ln Z(rβ) is the tool employed. Such a route is first shown to afford an exact formulation of (i) and (ii) above, which are essentially one-dimensional problems. For a general V(r), a generalization which is approximate is proposed, based on a linearized equation for the effective potential U. This approximation can then be refined by (a) use of the lowest bound state wave function or (b) incorporating non-linear gradient terms in V(r).