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Far-From-Equilibrium Kinetic Processes

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Far-From-Equilibrium Kinetic Processes DOI 10.1515/jnet-2015-0059 Ë J. Non-Equilib. Thermodyn. 2015; 40 (4):275–281 Research Article J. Miguel Rubí and Agustin Pérez-Madrid* Far-from-equilibrium kinetic processes Abstract: We analyze the kinetics of activated processes that take place under far-from-equilibrium condi- tions, when the system is subjected to external driving forces or gradients or at high values of anities. We use mesoscopic non-equilibrium thermodynamics to show that when a force is applied, the reaction rate de- pends on the force. In the case of a chemical reaction at high anity values, the reaction rate is no longer constant but depends on anity, which implies that the law of mass action is no longer valid. This result is in good agreement with the kinetic theory of reacting gases, which uses a Chapman–Enskog expansion of the probability distribution. Keywords: Far-from-equilibrium, kinetic processes, mesoscopic non-equilibrium thermodynamics ËË J. Miguel Rubí: Departament de Física Fonamental, Facultat de Física, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain, e-mail: [email protected] *Corresponding author: Agustin Pérez-Madrid: Departament de Física Fonamental, Facultat de Física, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain, e-mail: [email protected] 1 Introduction Processes that need a minimum amount of energy to proceed are usually referred to as activated processes, which in many cases take place under the inuence of external driving forces. The presence of these forces modies the free energy barrier that the system has to surmount, which leads to changes in activation rates. Many examples can be found in physicochemical and biological systems. Mechanical forces may provide the energy that reactants need to transform into products [1, 2] and can also induce chemical changes in polymeric materials [3]. Forces may induce activation of covalent bonds [4]. They can also aect the kinetics of single-molecule reactions [5]. The amount of tension applied at the ends of an RNA molecule progressively breaks the bonds, giving rise to new congurations of the molecule [6]. The kinetics of nucleation processes is also aected by the presence of gradients imposed to the metastable phase [7, 8], which results in changes in nucleation rate. Activation kinetics is governed by the law of mass action, i.e. the rate of a chemical reaction is propor- tional to the product of the masses of the reactants. This law enables one to formulate a set of coupled dif- ferential equations for all components whose solution leads to the evolution in time of the concentrations. Although the law can be applied systematically to chemical reactions, biochemical cycles, and genetic net- works, we will show in this article that its validity is limited to suciently small values of the anities [9]. In this article, we present a general formalism to treat activated processes under far-from-equilibrium conditions under high thermodynamic or applied forces. In our approach, at short time scales, the system is in mesoscale local equilibrium [10, 11] and undergoes a diusion process through a potential barrier [12, 13]. This irreversible process entails dissipation due to changes in the conformation of the system, which can be measured through the entropy production computed in the space of the reaction coordinate. From the probability current, one obtains the law of mass action [14], for which the current is proportional to the dierence between the fugacities at the initial and nal states of the process. The presence of a constant driving force is incorporated in the chemical potential through the work made by this force. The eect of this force is to tilt the energy landscape, and thus, detailed balance principle is preserved. In activated processes taking place at higher anity values, the diusion coecient in the space of the reaction coordinate is not a constant. This fact leads to corrections to the law of mass action. Brought to you by | Universitat de Barcelona Authenticated Download Date | 3/31/17 6:50 PM 276 Ë J. M. Rubí and A. Pérez-Madrid, Far-from-equilibrium kinetic processes The article is structured as follows. In Section 2, we present the mesoscopic non-equilibrium thermo- dynamics approach to activated process, showing how the law of mass action can be derived from the entropy production of the system. Section 3 is devoted to the study of the eect of an applied force on the activation kinetics. In Section 4, we show how the law of mass action is not valid at high anity values. Finally, in Section 5, we present our main conclusions. 2 Activation kinetics from mesoscopic non-equilibrium thermodynamics Processes such as cluster formation [15], protein ligand binding [16], self-assembly [17], adsorption of a single molecule on a substrate [18], and chemical reactions [19] need minimum energy to proceed. They can be modeled by a particle crossing a free energy barrier that separates two well-dierentiated states located at the minima at each side of the barrier [20, 21] and are generically referred to as activated processes [22]. Their intrinsic non-linear nature renders them untreatable by linear non-equilibrium thermodynamics [23]. Non-equilibrium thermodynamics oers a partial description of the process performed in terms of the initial and nal positions, ignoring the transient states. This results in a linear behavior of the current in terms of the anity, which only agrees with the law of mass action for small anity values [24]. If we con- sider the process at shorter time scales, the state of the system, instead of jumping suddenly from the initial to the nal state, progressively transforms by passing through successive molecular congurations. These congurations can be parametrized by a reaction coordinate . At these time scales, one may assume that the system undergoes a diusion process through a potential barrier separating the initial from the nal states. At observational time scales, the system is mostly found atã the minima of potentials and . In the quasi- 1 2 stationary limit, when the energy barrier is much higher than the thermal energy and intra-well relaxation has already taken place, the probability distribution is strongly peaked at these valuesã andã almost zero everywhere else. Under these conditions, the Fokker–Planck description leads to a kinetic equation in which the net reaction rate satises the mass action law. P(ã, t) Using Kramers’ model, an activated process in which a substance or a state A transforms into another B is assimilated to a diusion process along a reaction coordinate. The corresponding entropy production written in terms of the ux-force pair is then given by (1) 1 àì where the chemical potential is given by ò = − J , T àã B with as the Boltzmann constant and as the reaction potential. The current obtained from this expres- B sion can be rewritten in terms of the local fugacityì = k denedT ln P + alongΦ, the reaction coordinate B as k Φ(ã) z(ã) ≡ exp ì(ã)/k T B which can be expressed as 1 àz J = −k L , z àã where the diusion coecient is related to the Onsager coecientàz by means of the equalities J = −D , àã −Φ/kBT (2) B B We now assume that is constant and is equal to L andL integrate from to to obtain the non-linear kinetic D = k 0= k e . law for the average current [14]: z P D 2 D 1 2 ì2/kBT ì1/kBT 0 2 1 0 1 J ≡ X Jdã = −D (z − z ) = −D (e − e ). Brought to you by | Universitat de Barcelona Authenticated Download Date | 3/31/17 6:50 PM J. M. Rubí and A. Pérez-Madrid, Far-from-equilibrium kinetic processes Ë 277 This equation can also be expressed as A/kBT (3) 0 where and is the anity. This equation corresponds to the law of mass action 0 0 1 B 1 2 J = J (1 − e ), usually found in dierent activated processes [25–27]. TheJ scheme= D exp presented(ì /k T) reproducesA = ì − theì results of the rate theory [14, 20]. Equation (3) can be written in terms of the forward and backward reaction rates, and , as + − k− 2 k+ 1 where the reaction rates are given in terms of theJ = reactionk n − k potential:n , Φ2/kBT Φ1/kBT + 0 − 0 e e Here, for are the populationsk = D of the, minima, k = D with . 2 Φ/kBT . á á Δ Δ 1 In equilibrium, the current vanishes and one obtains detailed balance condition or, equiva- 1,eq 2,eq lently,n = P /Δ á = 1, 2 Δ = ∫ e dã + 2,eq ΔΦ/kBT z = z − 1,eq n with , showing that the forwardk and backward reaction constants cannot be chosen indepen- 2 1 = = e , dently. k n WeΔΦ can= Φ also− Φ analyze the case in which the transformation is coupled to another process , in which the concentration of the components can be controlled. The current is in this case given by A ྲ¯ B C ྲ¯ D 0 2 1 0 B D A C where the initial and nal fugacitiesJ are= −D (z − z ) = −D (z z − z z ), (ìA+ìC)/kBT (ìB+ìD)/kBT 1 A C 2 B D When both reactions are in equilibrium,z = e the detailed= z z , balance z = e condition= imposesz z . + A,eq C,eq − B,eq D,eq k n n Changes in the populations of states C and D drive= the system. away from equilibrium to a state in which k n n detailed balance is not fullled. The previous scheme can be generalized to the case of open and closed triangular reactions [28].
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