batteries have been the focus of intense scrutiny in recent years .this research is driven by specialized requirements for energy storage , e.g. , capabilities for high current for automotive purposes , high capacity batteries to buffer disparities in supply and demand for the power grid , and high energy density for batteries used in portable electronic devices. much effort is directed toward studying rechargeable lithium batteries, as that is the preferred base material due to the high energy densities achievable , its good cyclability properties , a high working voltage , and its abundance in the earth s crust .numerical modeling flanks experimental work in the goal to optimize batteries .traditionally , macro - homogeneous modeling uses diffusive processes through porous material to describe batteries on a mesoscale ( i.e. , smaller than the electrode , larger than a molecule ) , and adds charge and mass balance equations as well as transfer kinetics across the boundary surfaces. such simulations very successfully reproduce the macroscopic behavior of batteries , and can be used to optimize parameters such as electrode thickness .mesoscopic porous electrode models form the basis for the li - ion battery module for instance in the commercial comsol multi - physics simulation package . however, some of the underlying assumptions of this approach can be considered `` uncertain at best.''  mesoscale modeling requires constitutive equations .it is possible to input a great many effects based on parameterized experimental data into such descriptions .those include , but are not limited to , ionic and electronic conductivities , specific surface areas , tortuosities , porosities , activity coefficients , transference numbers , concentrations , diffusion coefficients , current densities , electrochemical potential , reaction rate constants , solid electrolyte interface transport processes , contact resistances between active material and current collector , and mechanical properties such as strain tensors , elastic moduli or fracture strengths. unfortunately , this parameterizability which makes the approach suitable to describe real batteries also limits its general predictive power , especially on the nanoscale .  in recent molecular dynamics ( md ) works , on the other hand ,much effort has been devoted to dealing with certain aspects of battery design in detail .examples are plentiful and include improving the intercalation of lithium into graphite , or calculating the transport properties of lithium through various electrode materials . in contrast to mesoscale models , md simulations tackle micro- and nano - physical aspects of the problem ( typically only considering half - cells ) , but can not make macroscopic predictions on , for instance , how the voltage during discharge changes under the influence of various control parameters such as temperature , discharge current , or changes of cell geometry .standard md methods based on either conventional _ ab initio _ density functional theory ( dft ) or traditional _ charge equilibration _ ( qe )  can not be used to model a battery as a whole , because those seek to equalize the chemical potential . however , the difference in the chemical potential between two electrodes is precisely what drives charge transport in a battery .furthermore , both approaches are ill - suited to model history - dependent effects .the reason is that they carry out a unique energy minimization based on instantaneous nuclear positions . the electron transfer process during a redox reaction brings about a quasi - discontinuous change of the electronic state , modifying all molecular orbitals, while the atomic configuration remains virtually unaltered .  time - dependent dft ( tddft )  can elucidate history dependence to some degree .however , this is expensive computationally , and , more severely , conceptual difficulties remain .they pertain to ( a ) setting up a meaningful initial state with the correct voltage between anode and cathode , and ( b ) reproducing correct level hopping due to an overestimation of the long - range polarizability in current dft schemes . the application of tddft , and similarly _ ab initio _md , to electrochemical processes with regards to batteries has been limited.  recently , it was proposed that the limitations of md calculations can be mitigated by introducing the oxidation state as a time - dependent variable , which needs to be subjected to dynamics. in addition to exchanging partial charges as in the standard _split charge equilibration _ ( sqe )method, atoms can then change their ionization state ( i.e. , participate in a redox reaction ) by swapping integer charges across a bond ( integer charge transfer hereafter , the method is referred to as the redox split - charge equilibration ( redoxsqe ) method .  in a previous work, we applied redoxsqe to case studies of contact electrification between two clusters of ideal metals and ideal dielectrics , respectively .if two initially neutral clusters with differing electron affinities are brought into contact , they will exchange charge . after separation, some portion of the transferred charge does not flow back , generating a remnant electric field that was not present before the contact .neither conventional qe methods nor ( non - time - dependent ) dft can capture this history - dependence .redoxsqe in contrast , successfully produces charge hysteresis effects during approach and retraction , despite identical atomic positions .  in this paperwe apply the redoxsqe method to a more complex problem .we want to bridge the gap between the mesoscale and highly accurate ( dft , accurate force field ) approaches , and model a nano - battery .if properly parameterized , redoxsqe can be used to model the microphysics at both electrode - electrolyte interfaces , including their structural evolution and changing morphology , as well as battery performance degradation . at this stage ,the simulations are meant to serve as proof - of - concept , rather than emulate any real system or produce new quantitative insights . however , even at its present qualitative level , our model reproduces generic features of battery discharge .we believe that redoxsqe can be parameterized to describe real materials quantitatively , because unmodified sqe combined with rebo ( reactive empirical bond - order ) force fields has yielded good agreement with experimental and dft results ( heats of formation of isolated molecules , radial distribution functions for water and ethanol , and energies of oxygenated diamond surfaces ) for systems in which each element had a well - defined oxidation state.  this paper is structured as follows .we outline the method in sec .[ sec : method ] below .we also introduce the additional parameters and procedures not covered in ref . , which describes the method in greater detail .[ sec : setup ] covers the setup of the specific simulations in this work . in sec .[ sec : results ] of this paper , we present the results attained by varying both internal and external parameters , and compare the outcome to generic properties of macroscopic batteries .we close with a discussion and summary of our findings in sec .[ sec : discussion ] .this section briefly outlines the numerical methods ( redoxsqe ) used in this study , and the parameters involved . for a more detailed descriptionwe refer the reader to ref . . for comparisons with dft - based resultssee the work by verstraelen _et al._, whose sqe+q is similar in spirit to redoxsqe , but applies charge constraints to fragments of molecules rather than to alter the oxidation state of individual atoms .we implement molecular dynamics with a long - range potential due to fractional charges ( `` split charges '' ) , as discussed in ref . we add in the modification proposed in ref .:  in this expression , represents the short - ranged potential ( see below ) , is the standard coulomb potential , while the are the electronegativities , and the term involving is due to the atomic hardness ( as in the standard qe ) .the total atomic charges , , are the sum of an integer charge ( in increments of the elementary charge ) on an individual atom , as well as partial charges that are shared between any two bonded atoms ( see secs .[ subsec : bondhard ] and [ subsec : sqe ] ) .the fractional charges are antisymmetric in their indices , i.e. , .single subscripts refer to quantities on individual particles ( e.g. , total charges or atomic properties ) , while double subscripts refer to quantities shared between two particles , such as a split charge , or a bond property .  the last term of eq .( [ eq : sqe_potential ] ) describes the effect of the bond hardness ( as also used in the _ atom - atom charge transfer _( aact ) framework  ) , and is discussed in detail below .parameterizing the potential in terms of _ both _ atomic and bond properties alleviates most issues that methods only containing one or the other suffered from ( see ref . and references therein , for a summary of sqe s advantages over other _electronegativity equalization methods _ , which we do not repeat here ) .the equations of motion are solved with a conventional velocity - verlet algorithm , in a dedicated md code .we use a langevin thermostat  combined with stochastic damping , with a damping constant of after the initial equilibration .the coulomb interaction is effected in a nave direct - sum approach ( see sec .[ subsec : limitations ] below ) .  for simplicity, we use the `` 6 - 12 '' lennard - jones ( lj ) potential without cutoff for the short - range interactions .the electrolyte is a kob - andersen - like mixture, in order to prevent crystallization .we use six atom types , two for the electrolyte ( positively and negatively charged ) , one for each electrode in its neutral state , and one for each ionized electrode species .electrode atoms have twice the mass of solvent ( electrolyte ) atoms .our choice of parameters is summarized in table [ table : lj_parameters ] .the unit system is explained in sec .[ subsec : units ] .   the bond hardness between particles and , which are a distance apart , is parameterized by the following piecewise function : where ( ) and are short and long cutoff radii , respectively .the symbols and denote bond parameters , constant for each bond type . in this paper the plateau value , because we are primarily concerned with metallic contacts .the functional form of the bond hardness is similar to that of mathieu. however , while we work with two variable critical radii , that work parameterized only the inner threshold with a variable scaling factor .note that mathieu shortly simplifies as universal , i.e. , not only as bond - independent but also atom - independent .the other parameter in ref . is a multiplicative factor .this effectively incorporates our .mathieu uses the van - der - waals radius of a given atom instead of a variable outer threshold .our parameterization of the bond hardness smoothly approaches at the lower threshold , while it diverges at the upper threshold , where the bond breaks . at both thresholds ,the force brought about by the distance - dependence of the bond hardness has a cusp , which may lead to very small drifts in the total energy .this is discussed in detail in the previous work.   prior to calculating the coulomb force and the md step , in the so - called equilibration step , we update all split charges on `` active '' bonds for a fixed atomic configuration .a bond is classified as `` active '' if its bond hardness is zero or finite ( but not infinite ) , i.e. if the bond length .inactive bonds do not carry partial charges .we minimize the potential energy with respect to the split charge distribution by solving the homogeneous linear system of equations with a steepest - descent solver. the potential is that of eq .( [ eq : sqe_potential ] ) .typically , the minimization requires only a handful of iterations .however , following an integer charge transfer ( see sec . [subsec : ict ] ) , up to several thousand iterations can be necessary to find the split charge distribution minimizing the energy .next , we update the total charge on each atom according to eq .( [ eq : q_from_sqs ] ) , based on the atomic integer charges carried , and the bond charges , and proceed with the normal md calculation .  for the battery simulations shown below , we only allow split - charge ( and integer - charge ) exchange between electrode atoms .electrolyte atoms are modeled as fixed - charge particles .the novel feature of redoxsqe is that it allows for integer charge transfer ( ict ) besides the exchange of partial charges across dielectric bonds. this section briefly describes the implementation , while sec .[ subsec : ictm ] explains how we treat charge transfer across metallic bonds .  at each time step , we select all `` dielectric '' bonds with , i.e. , any pair of atoms that is sufficiently close together to share a split charge , but not close enough to have a vanishing bond hardness ( `` metallic '' bonds ) .we also exclude electrolyte atoms from participating in icts because they are assumed to be unreactive with the electrode for maximum battery efficiency. furthermore , our nave implementation sets hard limits on the oxidation state for each atom type in the simulations presented below we do not allow double ionization .more sophisticated rules are conceivable , but left for future work .  for each eligible bond , we draw a random number between zero and one , uniformly distributed .if it is smaller than a certain threshold ( somewhat arbitrarily chosen ) , an ict is attempted .this is to approximate the electron transfer rate . in more realistic simulations, this rate must be determined from quantum - chemical calculations . for a trial ict ,we increment or decrement the integer charge ( i.e. , the oxidation state ) of each participating atom by one elementary charge , with the algebraic sign the same as the sign of the split charge between the two atoms .then we re - equilibrate the partial charges , and calculate the system s total potential energy . if the charge transfer has lowered the energy ( i.e. , the system now evolves on a landau - zener level with strictly lower energy ) , the move is accepted , otherwise it is rejected and the original state restored .a modification in future code implementations will be be to accept icts according to some metropolis - type condition instead. then , the energy can also increase with a certain probability during an ict , fulfilling the principle of detailed balance , and producing the correct equilibrium distributions . in our current modelwe expect that the metropolis algorithm mainly changes the dynamics near the transition state , i.e. , the reorganization of the solvent might take longer due to back jumps .however , the final state will not be altered because fluctuations of solvated ions to become neutral are extremely rare events .  besides changing the oxidation state, an ict also changes the atom type .this is necessary because an ion may have different atomic properties ( such as radius and interaction parameters ) as well as bond characteristics from its neutral counterpart .the type change necessitates also taking into account the short - range interaction energy for the ict .an electrode atom only changes its type if it gains the `` correct '' charge .the opposite charge is absorbed by , and distributed across , metallic bonds , e.g. , among connected remaining electrode atoms .for the anode , this means that an atom is stripped of a negative integer charge ( i.e. , one or more electrons ) , and becomes a cation ( i.e. , it changes from atom type 1 into type 3 ) , while the negative charge remains on the anode .it serves to compensate for the positive charge accumulated by sending negative charge through the external resistor .while the net positive charge is localized on the cation , the remnant negative charge is distributed across the entire anode as split charges instantaneously , even though _ formally _ there is still _ one particular _ anode atom that carries the charge via its oxidation state , for book keeping .conversely at the cathode , adsorbed cations receive negative integer charges transferred from the anode and are neutralized ( they change from atom type 4 to type 2 ) , as their surplus net positive charge is absorbed .in addition to the procedure described above , we draw another random number and only proceed with the trial ict if this exceeds some threshold ( for instance , to attempt an ict only every tenth md step ) , in order to alleviate a bias introduced by the order in which we query bonds .this results in two trial icts per atomic oscillation period , on average . in future implementations we will randomize the order of bonds for which we attempt an ict , and fully eliminate the bias .all in all , the maximum number of attempted icts per md time step is , where is the number of redox - active atoms , is their average coordination number , and is the fraction of icts that passed picking the second random number ( e.g. ) .lastly , , where is the fraction of dielectrically bonded atoms .the factor is for atoms dissolved in redox - inert solvent ( e.g. , in the electrolyte ) , and in clusters of redox - active material .   for a `` metallic '' bond with , the backflow of partial chargeexactly compensates the transfer of integer charge .such a move would always be accepted because the energy is unchanged , but would still cause an expensive yet unnecessary re - equilibration of split charges .in addition to the integer charge transfer across a `` dielectric '' bond with finite bond hardness , we therefore implement a second mode of ict that applies to metallic bonds .we call such an operation ictm .we implement ictms such that we draw a random number for each metallic bond between two atoms otherwise eligible of an ict .if this number exceeds a threshold , an integer charge is swapped , and immediately compensated by an equal split charge transfer in the opposite direction .together , those transfers are energetically neutral moves in a metal .no sqe needs to be performed , and no type change occurs , so no further computations are needed .ictms thereby allow for `` oxidation state diffusion . ''if the integer charge get transferred onto the `` front atoms '' in the anode ( the atom connected to the wire , see sec .[ subsec : ext_resistor ] below ) , we do not allow it to move away anymore .i.e. , the total charge after adding the `` external '' split charge ( connected via resistor to the other electrode ) .] as a consequence , all free negative integer charges ( which could be interpreted as electrons ) eventually migrate to the front atom , and are sent across the external resistor , in accordance with the real physical process .similarly , negative integer charges emanate from the cathode s front atom and diffuse toward cations adsorbed to the electrode surface .ictm happens as next - neighbor hopping .it would be more meaningful if a metal cluster as a whole had an excess of integer charge ( positive or negative ) , rather than individual atoms in a metal cluster being assigned an oxidation state .this would also reproduce realistic physics more faithfully by allowing an immediate transfer of integer charges between any two atoms connected to a metallic cluster .however , for bookkeeping and domain decomposition reasons , we stick to the current procedure , eliminating the overhead of a cluster analysis .we emphasize that during an ictm , only negative integer charges can diffuse through the electrodes .it is _ not _possible for two initially neutral metal atoms to assume the configuration + 1/-1 , in contrast to the ict in dielectrics , because one of the two atoms changes its type in such a case .  also note that the random number is necessary to reduce ( albeit not fully avoid ) spurious directed motion .we perform the scan for the ictm deterministically ( e.g. , atom is always queried before atom ) , and therefore would introduce a preferred transfer order if every move was accepted .the configuration of a simulation with 1194 atoms at an intermediate time is qualitatively illustrated in fig . [fig : snapshot ] .( our default setup is somewhat smaller , and described in sec .[ subsec : setup ] . ) the total charge on a given atom is encoded in its color , blue being a positive charge and red a negative charge . for visual distinction, atoms are displayed at sizes that do not reflect their lj radii .both metallic and ionic species are visualized as much bigger than electrolyte atoms are .the latter are + 1/-1 fixed - charge particles , but their charge coloring is halved , again for better contrast .the medium - sized particles are cations , while the largest particles are metallic atoms .those may or may not carry a negative oxidation state , as negative integer charges can hop freely across metallic bonds in an ictm ( see sec .[ subsec : ictm ] ) .the anode in this simulation initially held 5 layers of atoms ; all atoms of the rightmost layer have undergone redox reactions , each donated a negative integer charge to the anode , and subsequently dissolved as a cation ( roughly 45 atoms in total in this picture ) .the other layers are still in the middle of this process , and are only partially dissolved .the anode is positively charged , partly as a result of polarization charges as consequence of the layer of anionic electrolyte particles that wet its surface , and partly because insufficient atoms have dissolved as cations to carry away its excess charge . as expected for a metal , the charges reside primarily on the bulk surface  .the cathode started out with only the single fixed layer at the beginning of the simulation , but in this snapshot , about 75 pre - dissolved cations in the right half - cell have already adsorbed , donated their charge , and become part of the metallic cathode .this process is not homogeneous , the accretion occurs in clusters and produces an uneven surface with fractal - like features .if redoxsqe is properly parameterized for real materials , it can be useful in the quest to understand the detailed morphology and structure of the surface layer , as well as phenomena such as dendrite formation, because it resolves the electrode - electrolyte interface .it may also be useful for intercalation studies of lithium into graphite .  besides some minor polarization charges , the cathode is neutral overall because adsorbed cations have contributed sufficient charges to compensate for the negative charge that has traveled along the connection through the external resistor ( see sec .[ subsec : ext_resistor ] ) .the external load is implemented as a dedicated partial charge between an atom in the fixed layer of either electrode .this split charge between cathode and anode is not updated in the sqe step ( see sec .[ subsec : sqe ] ) but receives its value according to ohm s law ( if the switch is closed ) : where is the constant external resistance , and is the instantaneous driving voltage between the two connected front atoms .the separator only allows electrolyte particles to pass through , all others experience a repulsive force ( see sec .[ subsec : separator ] ) if they move in between the two thin lines .( following ohm s law ) completes the circuit . ]we briefly describe the initial configuration of our default setup , and the start of a simulation .* we work in a two - dimensional configuration , to reduce the computational costs associated with large particle numbers .we do not apply periodic boundary conditions : our finite system is confined by fixed walls on all sides .* our default system contains 358 atoms overall .we have also carried out simulations with roughly double and quadruple that number .* of the total number of atoms , 118 are electrode atoms . of those , half are attached to the anode and are arranged in an hexagonal lattice with three ( up to five , in larger simulations ) layers .only 20 atoms are connected to the cathode in one layer , another 39 ionized cathode atoms are dissolved in the electrolyte .this represents a crude approximation to solubility equilibrium . in a historic voltaic cell, the dissolved particles would be cu ions . *the electrolyte is distributed spatially randomly in the two half cells , under the constraint that each side is initially electrically neutral . *we choose our md time steps such that we sample a typical thermal oscillation of an atom with steps .* initially , we allow the electrolyte to equilibrate in each half - cell , barring all atoms from passing the separator . during this stage ,the electrode particles are held stationary , and no icts / ictms are carried out . * after md time steps , the electrolyte has assumed a liquid glass - like state .we insert the separator instead of the impassable wall between the two half - cells , and allow both icts and ictms .* we equilibrate for another md time steps with all electrolyte atoms free to move , but damped at , a factor more strongly than our default value . * finally, the electrode atoms are released , and the damping is set to its normal , low value . only the last row of each electrode remains fixed in place .this limits the amount of charge transferable to roughly 66% of the electrode .* we measure during the following md steps .   in order to isolate the two half - cells of our battery , a simple model of a _ separator _is inserted . in galvanic cells ,this component ( called `` salt bridge '' in that context ) is a membrane often made of filter paper , or consists of a u - shaped glass tube filled with ( possibly gelified ) inert electrolyte. it allows ionic species to pass through , and thus to complete the circuit with the external resistor , but prevents intermixing of dissolved electrode ions , which is often undesired .  in a present - day ,commercial batteries , the two half cells are kept apart by a solid separator which is permeated by the electrolyte . in the ideal case ,no electrons are allowed to pass through , but the ion conductivity is large .the separator also needs to be mechanically resistant to abuse , chemically stable in a concentrated alkaline environment ( for alkaline batteries ) , as well as not participating in redox reactions that occur in the cell.  we implement a mathematical separator such that dissolved electrode atoms and ions feel a repulsive force when they approach the barrier , but electrolyte particles are unaffected ( in our model they carry charge , and can complete the circuit ) .if electrode particles were allowed to mix , they could exchange split charges and form salts , or even adsorb to the opposing electrode and thus create a short circuit .notice that the energy barrier posed by the separator is not infinitely high . as a thermally - activated process that occurs with a probability of , individual ions are expected to still pass the barrier .we choose the separator s repulsive energy in dimensionless units ( ) .this means that an ion sitting at the separator has a probability to pass through of at our default temperature .our separator is a crude idealization of its real - world counterpart , even though their properties are similar . in a more realistic and properly parameterized simulation, the separator is a crucial component in its own right , and needs to be implemented carefully .one atom of the fixed layer on each electrode is chosen as the connecting point of the `` wire '' to connect to the opposite electrode through an external resistor .we refer to those atoms as `` front atoms . ''they serve as endpoints for the dedicated split charge that models the external resistor .we investigate several different modes of operation of the external circuit :  1 .* switch open , no electrical connection . * the `` external split charge '' is constant .this mode is used for equilibration runs , as well as for aging tests of our battery .* switch closed , constant resistance , discharging . * in this case , the current is determined by the instantaneous difference in chemical potential ( i.e. , the voltage ) between the electrodes , divided by a fixed resistance .initially , the voltage is approximately given by the difference of the electrode s electronegativities , modified by the electric field effected by the instantaneous charge distribution .if charge transfer continued in this manner indefinitely , the transferred charge would set up an opposing electric field after some time , resisting further charge transfer .however , this electric field causes a charge separation in the electrolyte , and ions rearrange to compensate it .recall that the separator allows free exchange of electrolyte particles across the half cells .there still is a charge buildup in the electrodes , making it energetically favorable for the electrodes to shed some of that charge .this is achieved by oxidizing surface atoms on the anode , and releasing them into the solution .analogously , at the cathode ions dissolved in the electrolyte adsorbing to the surface are reduced .this way , the electrodes are neutral again , and the voltage returns ( or approaches ) that of the initial state .this process ends when there are no further ions to be dissolved ( and/or adsorbed ) .* switch closed , constant resistance , charging .* this is the same setting as in the discharge case , except that we add an external voltage opposing ( and overpowering ) the discharge voltage .we investigate to which extent the electrodes return to their previous state , and observe the battery s hysteresis .this allows to study , for instance , surface passivation .* constant power or constant current . * for brevity ,we only present data for discharge under constant resistance , and not under constant power or constant current , even though it is possible to model those discharge modes as well .the _ constant power mode _ is a good approximation to numerous real - world applications , as many electronic devices need a minimum power throughput to function properly . in the _ constant current mode _, charge can continue to flow even beyond the point when the voltage drops to zero . at that pointthe anode and cathode reverse their roles .this setting allows studying of over - discharging behavior achievable when multiple batteries with differing remaining capacities are connected in series . in that case, the voltage of the cells that still have capacity remaining can drive the empty ones into pole reversal ..normalized model parameters in our default system.^_^ [ cols="<,^,^,^,^",options="header " , ]   ^_^ see sec .[ subsec : units ] for details on the normalization used . ^_^the parameter is explained in sec .[ subsec : bondhard ] .electrolyte ( solvent ) atoms do not exchange partial charges in our model , neither among themselves nor with other species , and therefore those bonds do not have a assigned to them .  throughout this work we use dimensionless parameters . in order to facilitate the interpretation of the presented data , this section provides ballpark estimates for representative values of real materials .the unit of charge can be associated with the elementary charge = 1.6\times 10^{-19}~\mathrm{c} ] .this is to approximate the lennard - jones parameter used for cu - cu interactions in the literature. the value for zn is comparable in magnitude albeit slightly larger .we define the unit of mass as the atomic mass of copper , \approx 64~\mathrm{amu} ] .again , this is so that the interaction between metallic electrode particles is comparable to values for (cu - cu ) in the literature .  together ,length , charge , energy , and mass specify a complete set of units for our purposes .derived units are the unit of current \approx 350~\mathrm{na} ] as resistance normalization , \approx 160~\mathrm{mv} ] as unit of time .our battery demonstrator operates at a temperature , comparable to a liquid - salt battery.  with these choices of units , the default value for the electronegativity difference between anode and cathode is  v , which is close to many standard cells , e.g. , alkaline ( 1.5 v ) or nimh ( 1.2 v ) batteries . in the default parameterization ,we use atomic hardnesses of  ev , which is much smaller than typical values , e.g. ,  v / e and  v / e. moreover , unlike real systems , our standard values for and do not reproduce a neutral dissociation limit of dimers , because , as can be seen from eq .( [ eq : sqe_potential ] ) .we made this choice of parameters to accelerate the generation of ions .at the same time , we ensured in selected simulations that the qualitative features of the discharge curves remained unchanged for much larger values of and smaller values for ( see sec .[ sec : results ] ) .  in order to compare our results with macroscopic systems , our model battery would have to be scaled up by a factor of roughly . in principle, each spatial dimension can have a different scaling factor , however , for simplicity , one may assume the same factor of in each direction .an inherent ambiguity of how to scale the direction normal to the interfaces remains .one way to scale the simulation is to take our nano - battery as an electrical element and connect of them in series , and in parallel .this method yields an overall open - circuit voltage ( ocv ) of for our default choice of parameters ( ) . connecting the external resistors in a similar fashion as the batteries leads to a scaled resistance of ( microscopic system : in dimensionless units ) .consequently , the nominal macroscopic discharge current would be  ka .discharge now takes times longer ( about  s ) than in the microscopic case , because the total number of transferrable charges increases by , while the current only increases by .alternatively , one can scale both battery and resistor _ as a whole _ in each spatial dimension . this way one retains the value for the macroscopic resistance of .in contrast , the voltage in this case still has its microscopic value of .the resulting nominal macroscopic discharge current is  ma , which is not very much lower than real - world currents . now the discharge would take times longer than in our microscopic model ( about 14 d ) .we stress that the discharge characteristics of such a scaled - up version of our model battery would be different from the ones presented in this work because the electrode surface - to - volume ratio would be much reduced in the macroscopic battery , among other reasons .  in many figures we show discharge curves , i.e. , plots showing the instantaneous voltage vs. the transferred relative charge . in those plots ,the voltage is normalized by the electrode atom s difference in electronegativities , because that is the difference in chemical potential in absense of any charge effects or electric fields .the transferred relative charge is the total charge flown across the external resistor , normalized by the total number of atoms in either of the electrodes .we consider not only the dissolvable layers but also the fixed electrode atoms .each atom can change its oxidation state by one : the mobile layers can desorb as cations , while the fixed layer s charge is balanced through the formation of a double layer from the electrolyte .note that the charge through the external resistor is fractional because we implement it as a dedicated ( and not - equilibrated ) split charge .we chose to do this instead of sending across only integer increments in order to get smooth curves .this can be considered in implicit average over many time steps .   at this stage, our model only contains some rudimentary approximation to chemistry , and the ability to model redox reactions .we only consider two - body forces , and leave dihedral and torsional interactions for future work .moreover , at short distances we do not screen the coulomb interaction , even though the wave functions of atoms overlap in such a situation , and the point - charge approximation breaks down .our current implementation can be made to run much more efficiently .particular points to note are the following .we do not cut off the lennard - jones interaction , which makes its complexity .we also compute the coulomb interactions with an expensive direct - sum algorithm , which limits the system sizes we can currently study .we could save computing time with a more efficient approach .we plan an implementation into the open - source code lammps , which will alleviate this limitation .even with such improvements the solution of the large linear system is more expensive than the setup of the system matrix , for which the coulomb term is calculated .two ways to make this cheaper come to mind : the first is to use a more efficient algorithm for the solution of the linear system , for example using a conjugate gradient method. second , we could update only the split charge in vicinity of the ict . at this developmental stage ,we re - equilibrate _ all _ partial charges after an ict .instead , one could implicate only the partial charges within some cutoff radius .this would mean introducing a finite signal speed for the sqe , as a split charge could be transferred a certain distance in one time step .the instantaneousness of the update would be lost , making it harder to model metals .the advantage is that it would make the operation instead of .the resulting error can be estimated and is bounded : the change in the split charges , and thus the error as a result of restricting the update distance , drops off exponentially with increasing cutoff distance.  lastly , in future implementations we will randomize the order of bonds for which we attempt an ict and for which an ictm is carried out , and eliminate the slight ordering bias currently present in the code .all of the optimizations described above are left for future work ; even without them ( albeit for small systems ) the method yields encouraging results .in this section we describe the results gleaned from our simulations .we emphasize again that our model is more a proof - of - concept rather than a faithful representation of a real battery . as such, the parameters have not been chosen appropriate for any specific material .our intent is to demonstrate that redoxsqe nevertheless reproduces generic properties of macroscopic batteries without further input .the model necessarily has a number of parameters , albeit not nearly as many as some mesoscopic porous electrode simulations. some are microphysical and chemical parameters , for instance the lj properties of the materials , the atomic hardness , or the electronegativities . those are in principle readily parameterizable or even measurable quantities for real materials , but we elect to use representative values ( see sec .[ subsec : units ] ) , rather than to model specific materials .the parameters associated with the bond hardness are not as easily measured , but can in principle be fit to values found from measurements and _ ab initio _ quantum - chemical dft simulations , such as esp ( electrostatic potential ) partial charges , hirshfeld - i charges , and dipole moments of molecules. other parameters are implementation choices such as number of atoms to model , the electrode setup , and the geometry of our cell . finally , we vary parameters that have a documented and experimentally accessible impact on battery performance , such as the temperature , and whether the battery is discharged continuously or in pulses .the dependence on this last set of parameters results naturally and self - consistently with our method , and does not have to be put in implicitly or explicitly .   in this section ,we explore the dependence of the discharge characteristics on internal model parameters , while sec .[ subsec : results_external ] focuses on external parameters . in sqe, the difference in electronegativity between atoms of two metals determines the open - circuit voltage ( ocv ) . in all following plots ,the voltage is normalized to this value .  separating a diatomic molecule adiabatically results in neutral products for each pair of stable elements .however , if , it is energetically favorable that negative integer charge ( i.e. , an electron ) remains on the more electronegative partner. for a multi - atom system , the expression for the neutral dissociation limit is not as simple anymore , because the atomic hardness is reduced in an ensemble. in our default system , we choose and , in order to facilitate the formation of ions at the anode , even though these values lead to a violation of the neutral diatomic dissociation limit .however , fig .[ fig : discharge_comparison_chi ] shows that this does not have a large impact on the discharge curve : even for and , satisfying the neutral dissociation limit , the qualitative picture remains .   and which violates the dissociation limit : dissociating a diatomic molecule will produce ionic instead of neutral products . the runs with not have this flaw , and still exhibit similar discharge behavior , even though the likelihood of anode atoms to dissolve as ions is reduced .the scatter is larger for smaller since the theoretical voltage is reduced by a factor of , and thus the signal - to - noise ratio halves , even though in this case we average over 8 different realizations . ]we note that the large scatter stems from the small number of atoms in our simulations .if we dissolve of the total number of anode atoms , we have 30 cations in solution , and a statistical uncertainty of , i.e. , .the stochastic error reduces by a factor of two by averaging over 4 independent realizations .  in sec .[ subsec : units ] we reported representative values as normalization for the dimensionless parameters used in this work , and noted that our standard value for the atomic hardness was rather low compared with values for real materials . in fig .[ fig : discharge_comparison_kappaatomic ] , we show the discharge behavior of our nano - battery for and , again satisfying the neutral diatomic dissociation limit . in this case ,ion formation ( i.e. , redox reactions at the electrode surface ) takes much longer , and the external resistor needs to be scaled up in order to get similar behavior .a factor of 2.5 in is approximately compensated by a factor of 25 in resistance .in addition , the plateau is far less pronounced for the larger , and the discharge proceeds faster .note that the initial voltage in this case is also higher .the reason is that some pre - dissolved cations adsorb to the electrode immediately , and cause a greater difference in chemical potential , and therefore ocv . in order to have the curves overlap for better visual comparability, we scaled down the results for by a factor of 1.3 , which compensates for the larger ocv .note that the simulation with and takes very long , for the reasons described above , and was terminated before all charge was transferred .( approximating real atomic hardnesses more closely than our standard value ) require much higher external resistances in order to achieve similar discharge behavior as for the default value of . for larger values ,the plateau is much less pronounced .the runs are scaled down by a factor of 1.3 so that they overlap the curves better ( see text ) . ]  in fig .[ fig : discharge_comparison_epsis ] , we exemplify the influence of the lj parameters on the discharge curve .we vary between electrode ions and the electrolyte .a higher value means that it is favorable for an ion to surround itself with electrolyte atoms , as opposed to other atoms with which its is lower .the discharge curve has a higher and more extended plateau for a smaller , caused by the reaction on the cathode , where a lower value means that it is more likely that an ion is adsorbed to the electrode . at the anode, the opposite should be the case . there , a_ greater _ should make it more likely for an ion to be dissolved into the electrolyte .however , the test case with changes the plateau only marginally .we conclude that the cathode reaction is more important in this respect . if the method in implemented into more sophisticated software ( e.g. , lammps ) , it can be used with more realistic force fields for particle - particle interactions than what is used herein ( i.e. , simple two - particle lj interaction ) .   between electrode ions and the solvent ( electrolyte ) .the data represents an average over 4 independent runs , and each point is averaged over many md time steps .the solid lines are inserted to guide the eye .this figure exemplifies the optimization potential that lies in picking optimal materials for the battery : a smaller affinity between electrolyte and cations boosts battery performance significantly .the anode half - cell reaction seems less important in this respect , as optimizing it does not yield a comparable additional improvement . ]  while the lj parameters in principle can be deduced from experiments , pinning down the parameterization details of the bond hardness so that they match experiments or _ ab initio _ results is harder to accomplish . fortunately , those parameters do not affect the results very strongly , as evidenced in fig .[ fig : discharge_comparison_cutoffs ] . a change by in the cutoffs for , given in eq .( [ eq : bondhard ] ) , does not make a big difference : all curves nearly overlap with those of our default model in the practically relevant regime ( until a discharge of ) . in fig .[ fig : discharge_comparison_kappa ] we scale the parameter up and down by a factor of .the results do not depend sensitively on this choice , either .are not fully independent . as shown in eq .( [ eq : bondhard ] ) , all three have a scaling effect on . ]these results indicate that the detailed form of is not of great importance .the only criterion is that the next - nearest neighbor should not be connected with a dielectric bond .this translates into the `` long '' cutoff be substantially smaller than the distance to the next - nearest neighbors .otherwise a great number of additional icts will be attempted .this does not change the result , either , but the computations will be slowed down tremendously .the simulation shown by the yellow curve in fig .[ fig : discharge_comparison_cutoffs ] shows this case , and we terminated it before all charge had transferred .by does not significantly alter the results : all curves resemble our default model .this indicates that the detailed form of is not of great importance . ]  .the data represents an average over 4 independent runs , and each point is averaged over many md time steps .the solid lines are inserted to guide the eye .we vary the parameter by a factor of to greater and smaller values .the results do not depend sensitively on this choice . ]we tested the effect of various other simulation parameters .for brevity we only describe the results without including additional figures .* we increased the damping by a factor of 10 without observing any qualitative nor quantitative changes in the discharge behavior .this means that the langevin damping only has the desired effect to limit the battery heating up as a consequence of energy release , but is not strong enough to influence the dynamics much .* a variation in the lj parameters between electrode metals and their ions by also did not alter the characteristics significantly .our choice stems from the idea that ions are bound less tight to the electrodes than neutral metal atoms .this again aids the release of cations into solution .* we modified the relative importance of the different contributions to the potential energy .a decrease of the effect of the coulomb energy by up to ( and thereby a corresponding increase in relative importance of the other effects ) did not influence the discharge curve qualitatively . * similarly , neither scaling the battery in one direction , nor doubling the amount of electrolyte without adding more electrode atoms , nor changing the electrode surface area modifieded the results significantly .we conclude that we are not hampered by a lack of electrolyte . however , reducing the number of electrolyte atoms to half its default value will limit the number of ions that can be dissolved , and deteriorate our battery demonstrator s performance .an ideal battery should retain its theoretical voltage until the active material has been used up , that is , until the anode is completely dissolved , or all free cations have been reduced and adsorbed at the cathode surface . only at this pointshould the voltage drop to zero. in reality , batteries have an internal resistance , and both the electrolyte and the electrode are polarizable .the former reduces the actual voltage drop across the battery , while the latter is responsible for forming helmholtz double layers at the electrode surfaces , and thus depleting some of the battery s capacity .such effects cause both the actual working voltage as well as the usable capacity to be reduced from their theoretical limits .additionally , a battery s voltage also depends on the discharge current , such that a higher discharge current will decrease the discharge voltage . in fig .[ fig : discharge_comparison_resistance ] we show curves demonstrating this behavior . at high external resistance ( i.e. , low discharge current ) , the voltage stabilizes at of its theoretical voltage until about of the available charge has been transferred , at which point it drops quickly . at medium resistance ( our default model ) , the voltage does not feature such a pronounced plateau , but still has of its voltage at discharge .in contrast , the voltage for a discharge at high currents decreases much more steeply .these curves are reminiscent of those presented in refs . , the discharge curves of panasonic s zinc carbon batteries, and those of duracell s alkaline batteries.    if the internal resistance of a battery exceeds the external resistance , we effectively have a short circuit , and the battery will discharge as a capacitor , with an initial exponential decay of the voltage .a factor that is of crucial importance to real batteries is the temperature at which they operate .electric vehicles need to be able to reliably operate at temperatures ranging from all the way to .however , low temperatures decrease both the actual voltage as well as the battery s capacity. figure [ fig : discharge_comparison_temp ] shows the temperature dependence of our battery demonstrator . as in macroscopic batteries , the voltage is closer to the theoretical voltage for high temperatures , while it is significantly reduced for lower temperatures .the battery s capacity the area under the curve decreases by if the temperature is lowered by , and increases by if the temperature is raised by from our default temperature .we note that our temperature of is quite large , comparable to liquid - salt batteries, but greatly exceeds room - temperature .we once more emphasize the qualitative nature of our findings , not their realism .  ), the battery s performance deteriorates dramatically , while it improves at a higher temperature ( ) . ]  in fig .[ fig : aging ] we present aging studies of our model battery .it is common in some electronic devices to intersperse recuperation periods with discharge periods . during this time ,polarization effects are reduced and some of the initial voltage can be recovered. in order to examine the effects of intermittent discharge on our model battery , we discharge it until a certain amount of current has been drained , and then open the switch and let the system age . were it an electrochemical capacitor , no recovery of voltage would be expected .however , we see a nearly full recuperation of the initial voltage ( with some fluctuations ) . this voltage is held for time steps . after some time, ions manage to pass through the separator ( which is a thermally activated process , see sec .[ subsec : separator ] ) .this causes a voltage drop that contributes to self - discharge . in a macroscopic batterythis process will not take place as quickly as in our nanoscale device , but one mechanism for is elucidated in our model .the use of redoxsqe furthermore allows us to study the morphology changes that the electrodes have undergone , how much surface material is passivated and other microphysical parameters of interest .all reactions occurring in our system are micro - reversible , therefore our nano - battery is a secondary , rechargeable cell .figure [ fig : charging ] shows what happens if we recharge our battery demonstrator .we discharge our default system at the default constant resistance ( ) .after of the total capacity ( 35 out of 59 integer charges ) have been transferred through the external circuit , a charging current is switched on , in opposite direction as the discharge current .we choose its magnitude approximately three times the `` average '' discharge current ( averaged over the complete discharge of the same model ) .we consider two cases : in the first , the charging current is switched off again when the voltage reaches times the ocv . in this case , the discharge curve in the second cycle has deteriorated compared with the initial discharge ; the battery has degraded .the reason is that the electrodes do not fully return to their initial state during the charging , but merely the electrolyte reconfigures to balance the dissolved cations in either half cell .if one were to let the system relax after charging , some dissolved cations would return to the anode , and some material deposited on the cathode would also dissolve again. a realistic all - atom simulation using redoxsqe can be useful in the investigation of dendrite formation in this process which can short - out and destroy li - ion batteries. it could also help to study the cycling behavior of batteries , and their degradation .  in the second case ,all charge is transferred back .then , the voltage is much higher than the initial ocv .this behavior is also seen in real batteries, and stems from a buildup of a polarization layer opposite that which forms during discharge . in this case , the second discharge is not very dissimilar from the first one .   of the total chargehas been transferred through the external resistor . at this point , a constant external charging current is switched on ; its magnitude is chosen approximately three times the ( average ) discharge current .this charging current is switched off again when the voltage reaches the ocv ( middle , green triangles ) , or when all charge has been transferred back ( top , red circles ) . in the latter case ,the voltage is much higher than the initial ocv . ]in this work , we demonstrated that the redoxsqe method  can model redox reactions in an atomistic molecular dynamics setting , and used it to simulate a nanoscale battery demonstrator . even though we did not use a parameterization describing any real energy materials , we reproduced generic discharge curves of macroscopic batteries .for example , lower operating temperatures reduce the effective capacity of a battery .higher discharge rates have the same effect , but the voltage recuperates when the battery is aged ( e.g. , discharged in pulses ) . upon recharging , the battery performance degrades slightly , and the electrode surface morphology changes during the battery s operation .some internal model parameters are not fully accessible experimentally , such as the bond hardness , which is an ad - hoc parameter arrived at in a top - down fashion , by describing the bond - breaking behavior . in a theoretical work ,verstraelen _ et al . _ connect the bond hardness to parameters computed with atom - condensed dft , i.e. , derive it in a bottom - up fashion , and may help to motivate this parameter and ascertain its value quantitatively .we showed that , for the battery demonstrator , the results neither depend strongly on the detailed implementation , nor on the precise value of the bond hardness .the _ atomic _ hardness plays an important role in determining the time scales of ion formation at the electrode interface , and thus determines partly the internal resistance of the battery , while the electronegativity difference sets the open - circuit voltage . butchanging those two parameters does not alter the qualitative discharge picture .rather , the battery s behavior is predominantly defined by external quantities such as temperature , rates of discharge .we obtain results similar to the intermittent discharge mode, and see self - discharge when the battery is aged too long .  in a recent previous paperwe applied the same technique to case studies in contact electrification between two clusters of ideal metals and ideal dielectrics , respectively , showcasing its ability to simulate history - dependence. redoxsqe reproduces charge hysteresis effects during approach and retraction .one shortcoming of our current implementation is that the electrolyte is modeled with fixed - charge particles that do not participate in split charge exchange , nor in icts .it is an ideal insulator for electrons , and the lack of electronic conduction leaves only penetration of the separator by ions as self - discharge mechanism .this idealization will be abandoned in future work .further development effort will need to be expended on optimizing the method ( see sec .[ subsec : setup ] , and ref . ) to make multi - million atom simulations possible .an implementation into lammps is planned . in order to simulate specific materials or battery setups ( such as alkaline batteries , or li - ion rechargeables ), much chemically - specific parameterization will need to be done. furthermore , more realistic empirical many - body force fields are necessary for realistic all - atom simulations .we point out that the model in its current implementation can best describe non - directed interactions , as they are prevalent for instance in alkali batteries , with their isotropic reactions of s - orbitals .  notwithstanding those necessary improvements ,it is encouraging that the method already reproduces generic features of batteries .mesoscale battery models require many assumptions and intimate knowledge of the materials in question , and can not answer fundamental microphysical questions .dft / md methods , on the other hand , have been used for highly detailed and isolated problem aspects , but need to stay away from the electrode - electrolyte interface where redox reactions take place .arguably , this is the most interesting region , as it determines not only the ultimate cell performance , but also is where cell degradation takes place .et al . _ write `` _ _ the ability to predict cell degradation remains a challenge because so many unaccounted for and seemingly unrelated micro - scale degradation mechanisms have been identified or postulated . [ ... ] without ... theoretical analysis , cause - and - effect relationships between observation and degradation pathway can be difficult to demonstrate . _ _ '' redoxsqe is a first step toward filling this gap .it allows to model all aspects of a ( microscopic ) battery in one simulation , and gather insights into the processes happening at the electrode - electrolyte interface .besides modeling an entire all - atom battery , redoxsqe could also serve as part of hybrid multiscale schemes , where bulk phenomena inside an electrode or within the electrolyte are computed with a mesoscopic model while the electrochemical activity is tackled by redoxsqe .daniel s handbook of battery materials  states as requirement for a generic life estimation model that it `` _ _ must relate the measured cell performance at any given time to a combination of ... effects ._ _ '' in conjunction with a more realistic force field and with a proper parameterization of the materials , redoxsqe holds promise to enable the study of battery degradation and the optimization of battery performance .65ifxundefined [ 1 ] ifx#1 ifnum[ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1'''' [ 0]secondoftwosanitize [ 0 ] + 12$12 & 12#1212_12%12[1][0] link:\doibase 10.1038/35104644 [ * * ,  ( ) ] link:\doibase 10.1038/451652a [ * * ,  ( ) ] _ _ ( , , ) _ _ ( , , ) link:\doibase 10.1016/j.jpowsour.2009.11.048 [ * * ,  ( ) ] link:\doibase 10.1149/1.2425269 [ * * ,  ( ) ] link:\doibase 10.1149/1.2127445 [ * * ,  ( ) ] link:\doibase 10.1149/1.2221597 [ * * ,  ( ) ] link:\doibase 10.1149/1.2054684 [ * * ,  ( ) ] link:\doibase  10.1149/1.1836921 [ * * ,  ( ) ] link:\doibase 10.1149/1.1838376 [ * * ,  ( ) ] link:\doibase 10.1016/s0013 - 4686(00)00340 - 6 [ * * ,  ( ) ] link:\doibase  10.1149/1.1634273 [ * * ,  ( ) ] link:\doibase 10.1149/1.1785012 [ * * ,  ( ) ] link:\doibase 10.1016/j.jpowsour.2005.05.070 [ * * ,  ( ) ] link:\doibase 10.1021/jp074490u [ * * ,  ( ) ] link:\doibase 10.1149/1.2778258 [ * * ,  ( ) ] link:\doibase 10.1149/1.2759840 [ * * ,  ( ) ] link:\doibase 10.1016/j.cplett.2009.12.033 [ * * ,  ( ) ] link:\doibase 10.1016/j.jpowsour.2008.12.032 [ * * ,  ( ) ] link:\doibase 10.1149/2.074206jes [ * * ,  ( ) ] link:\doibase 10.1039/b920970k [ * * ,  ( ) ] link:\doibase 10.1039/b922656 g [ * * , ( ) ] link:\doibase 10.1021/cm801262x [ * * ,  ( ) ] link:\doibase 10.1021/cm902696j [ * * ,  ( ) ] link:\doibase 10.1098/rsta.2010.0070 [ * * ,  ( ) ] link:\doibase 10.1021/cm200949v [ * * ,  ( ) ] link:\doibase 10.1021/cm200683n [ * * ,  ( ) ] link:\doibase 10.1021/jp2122467 [ * * ,  ( ) ] * * ,  ( ) link:\doibase 10.1021/cm900373u [ * * , ( ) ] link:\doibase 10.1021/ja00290a017 [ * * ,  ( ) ] _ _ ( , , ) link:\doibase 10.1103/physrevlett.52.997 [ * * , ( ) ] * * ,  ( ) link:\doibase 10.1016/0009 - 2614(96)00440-x [ * * ,  ( ) ] link:\doibase 10.1016/j.cpc.2005.03.059 [ * * ,  ( ) ] link:\doibase 10.1088/1674 - 0068/25/02/161 - 168 [ * * , ( ) ] * * ,  ( ) * * ,  ( ) link:\doibase 10.1140/epjb / e2013 - 40047-x [ * * ,  ( ) ] * * ,  ( ) * * ,  ( ) * * ,  ( ) link:\doibase 10.1063/1.4791569 [ * * , ( ) ] link:\doibase 10.1021/j100161a070 [ * * , ( ) ] link:\doibase 10.1063/1.480198 [ * * ,  ( ) ] link:\doibase 10.1103/physrevb.17.1302 [ * * , ( ) ] link:\doibase 10.1103/physreve.51.4626 [ * * ,  ( ) ] * * ,  ( ) _ _ ( , , ) * * ,  ( ) link:\doibase  10.1016/j.ssc.2003.12.033 [ * * ,  ( ) ] link:\doibase 10.1021/ja209759s [ * * ,  ( ) ] link:\doibase 10.1002/qua.22202 [ * * , ( ) ] * * ,  ( ) in _ _ ( , , ) * * ,  ( ) http://industrial.panasonic.com/www-data/pdf/aae4000/aae4000ce2.pdf [ `` , '' ] ( ) ,  http://media.duracell.com/media/en-us/pdf/gtcl/technical_bulletins/alkaline%20technical%20bulletin.pdf [ `` , '' ] ( ) ,  link:\doibase  10.1038/nmat2764 [ * * ,  ( ) ] _ _ ( , , )

Abstract: batteries are pivotal components in overcoming some of today s greatest technological challenges . yet to date there is no self - consistent atomistic description of a complete battery . we take first steps toward modeling of a battery as a whole microscopically . our focus lies on phenomena occurring at the electrode - electrolyte interface which are not easily studied with other methods . we use the redox split - charge equilibration ( redoxsqe ) method that assigns a discrete ionization state to each atom . along with exchanging partial charges across bonds , atoms can swap integer charges . with redoxsqe we study the discharge behavior of a nano - battery , and demonstrate that this reproduces the generic properties of a macroscopic battery qualitatively . examples are the dependence of the battery s capacity on temperature and discharge rate , as well as performance degradation upon recharge .


mechanical expression ( hydraulic pressing ) is widely used in the processes of solid - liquid separation for extraction of fruit juices and vegetable oils , dewatering of fibrous materials , etc .( schwartzberg , 1983 ) .efficiency of this process can be increased by raw material plasmolysis , cellular damage or permeabilization prior to its expression .different methods are traditionally used to increase the degree of raw material plasmolysis : heating , osmotic drying or freezing dehydration , alkaline breakage , enzymatic treatment , etc .( rao & lund , 1986 ; aguilera & stanley , 1999 ; tsuruta , ishimoto & masuoka , 1998 ; ponant , foissac & esnault , 1988 ; jones , 1988 ; barbosa - cnovas & vega - mercado , 1996 ) .earlier on , the method of electric field treatment ( both d.c . and a.c . )was also proposed for cellular material plasmolysis ( known as electro - plasmolysis ) .the methods of electro - plasmolysis were shown to be good for juice yield intensification and for improving the product quality in juice production ( scheglov , koval , fuser , zargarian , srimbov , belik et al . , 1988 ; mclellan , kime & lind , 1991 , bazhal & vorobiev , 2000 ) , processing of vegetable and plant raw materials ( papchenko , bologa , berzoi , 1988 ; grishko , kozin , chebanu , 1991 ) , food stuffs processing ( miyahara , 1985 ) , winemaking ( kalmykova , 1993 ) , and sugar production ( gulyi , lebovka , mank , kupchik , bazhal , matvienko et al ., 1994 ; jemai , 1997 ) .but all these electric field applications are usually restricted by the high and uncontrolled increase in cellular tissue temperature and product quality deterioration because of electrode material electrolytic reactions , etc .recently , a variety of new high and moderate pulsed electric field ( pef ) applications were successfully demonstrated for liquid and solid foods ( barbosa - cnovas , pothakamuri , palou & swanson , 1998 ; wouters & smelt , 1997 ; knorr , geulen , grahl & sitzmann , 1994 ; knorr & angersbach , 1998 , barsotti & cheftel , 1998 ) .the pef application provides a possibility of fine regulation of electric power input and may result in effective permeabilization of cellular membranes ( zimmermann , 1975 ; chang et al ., 1992 ; weaver & chizmadzhev , 1996 ) without significant temperature elevation ( barbosa - cnovas et al . , 1998 ) .one of emerging and promising method is the combined pef and pressure application , which demonstrates significant yield intensification for juice extracted from apples and beets and clarification of the extracted juice ( vorobiev , bazhal & lebovka , 2000 ; gulyi et al . , 1994 ) .but the major problem arising from simultaneous application of mechanical expression and pef treatment is the choice of optimal modes of treatment .the mechanism of solid / liquid expression from cellular materials is rather complex and may include many different phases of consolidation process ( lanoiselle , vorobiev , bouvier & piar , 1996 ). the electric breakdown of a cellular system can influence consolidation phases and change drastically the expression curves .unfortunately , up to now there are no accepted mechanism of electric breakdown in the cellular systems and reliable criteria for choosing optimal parameters of electric field treatment ( lebovka et al . , 1996 , lebovka , bazhal & vorobiev , 2000 ) .another problem is the poor reproducibility of the experimental data , which is typical for objects of biological origin .the properties of cellular materials influence significantly the the electrotreatment efficiency .the electrometry can be used for characterization of changes in tissue properties under the influence of external factors ( electric field , pressure ) .this is a simple method , as far as the electrical conductivity , , reflects a degree of a water saturated tissue permeability ( sahimi , 1995 ) .but the general dependence between the structure of a cellular material and may be rather complex , because the conductivity of a biological tissue may be influenced by a number of processes , such as resealing of membranes in cells ( heinz , angersbach & knorr , 1999 ) , diffusional redistribution of moisture inside the samples , etc .the objective of this study is the optimization of moisture expression from biological raw materials under simultaneous pressing and pef treatment .the liquid expression from fine - cut cellular tissue after pef application for different modes and durations of precompression has been investigated .apple was used as the example of cellular material .the useful method of data treatment , which allows to reduce data scattering caused by differences in quality of the samples is described .discussion of the consolidation kinetics before and after pef treatment is also given .the reasons for the simultaneous application of mechanical expression and pef treatment are as follows .the excessive quantity of extraparticle liquid and absence of contacts between solid particles increases electrical energy losses .so the effectiveness of the pef treatment is restricted by the uniform and tight packing of raw material between electrodes and previous removing of extraparticle air and excessive liquid ( from cells destroyed by cutting ) .the method of raw material compact formation is its pre - consolidation . moreover ,effectiveness of the pef treatment with respect to the water - saturated cellular materials is restricted by the low values of moisture content , .the pef treatment is ineffective for a water - saturated system ( which is the case for the fine - cut apple raw material ) because of electric breakage and uncontrolled increase of current flow through the system .the initial steps of consolidation remove an excess liquid from the extracellular volume .therefore , we can expect increase of the pef treatment efficiency after pre - consolidation of the raw material .freshly harvested apples of golden delicious variety were selected for investigation and stored at until required .the moisture content of apples was within 80 - 85% .the fine - cut apple pieces ( 3 - 5 mm diameter ) were prepared from an apple pap using rasp .figure [ f1 ] is a schematic representation of the experimental set - up .all experiments were carried out using laboratory filter - press cell equipped with an electrical treatment system .the polypropylene frame had a cylindrical cavity compartment ( 20 mm thick , 56 mm in diameter ) .the cavity compartment of frame was initially filled up with apple slices and then was tightly closed from both sides by the steel plates .one of the plates , covered by a filter cloth , was used as a stationary electrode .the other plate was attached with an elastic rubber diaphragm .a mobile wire gauze electrode was installed between the diaphragm and the layer of apple slices .pressure was applied to the layer of apple slices through the elastic diaphragm using the hydraulic pressure controller gds standard ( gds instruments ltd , uk ) with water as a pressure fluid .the pressure controller provided a constant pressure from 1 to 30 bars .the yield of liquid was controlled by balance pt610 ( sartorius ag , germany ) .  a high voltage pulse generator , - ( service electronique utc , france ) provided the monopolar pulses of rectangular shape and allowed pulse duration varied within the interval of  ( to precision  ) , pulse repetition time within the interval of ms ( to precision of ms ) and number of pulses within the interval of .the conductivities were measured by contacting electrode method with an lcr meter hp ( hewlett packard , 38 mm guarded / guard electrode - a hp 16451b ) for thin apple slice samples at the frequency of hz and with conductimetre hi ( hanna instruments , portugal ) for the apple juice samples at a frequency of hz ( these frequencies were selected as optimal in order to remove the influence of polarising effect on electrodes and inside the samples ) .pulse protocols and all the output data ( current , voltage , impedance , pressure , juice yield and temperature ) were controlled using a data logger and special software hpvee v.4.01 ( hewlett - packard ) adapted by service electronique utc , france ) .high resolution scanning electron microscopy ( sem ) images were obtained using the instrument xl30 esem - feg ( philips , v=15 kv , p=3.5 torr ) .the `` wet '' chamber mode allowing observation of hydrated apple specimens in their natural state was applied .the optical absorbance of an expressed liquid was measured with photocolorimeter co75 ( wpa ltd , uk ) at the wavelength 520 nm .the characteristic absorption spectra were determined with respect to distilled water .transmittance of an expressed liquid was calculated as a ratio of filtered and nonfiltered liquid absorptions .the liquid was filtrated using a whatman 2v filter paper .all experiments were done using electric field voltages from to v , pulse duration s , pulse repetition time ms , number of pulses , constant pressure bars and total time of mechanical expression up to s. pressure value of 3 bars was accepted as the most efficient for exhibition of the effect of simultaneous pressing and pef treatment ( vorobiev et al . , 2000 ) .the experiments were repeated , at least , five times .figure [ f2 ] presents typical experimental curves of liquid yield , instantaneous flow rate , pressure and specific electrical conductivity vs. time . for convenience of presentation , here all properties are reduced to their maximal values , e.g. , , etc . , andflow rate is determined as .initially we observe a rather rapid increase of liquid yield , and decrease of electrical conductivity , .this behaviour corresponds to the layer pre - compaction ( at a constant velocity of elastic diaphragm displacement ) and expulsion of extraparticle air - liquid mixture .the maximum of instantaneous flow rate is observed approximately at s. in the absence of pef treatment ( , or ) the curve of temporarily stabilises in time interval s and by this moment the pressure reaches its maximal value of 3 bars .this behaviour correspond to the end of pre - compaction period .then , at , the curve ( ) slightly rises , which can be explained by mechanical rupture of residual cells , by the material deterioration and by the effects of the biological activity of microorganisms.  typical sem micrographs of apple tissue structure before and after pressing are presented in figs .[ f3](a ) and ( b ) , respectively .the mean size of undamaged cells is of order  m . we see , that after pressing some of the cells are destroyed , but there exist also intact cells .so , for a given mode of treatment ( bars , s ) the cellular structure is not completely disrupted after pressing and there exists some isolated cells , which remain intact during pressing period .these cells can be damaged or partially permeabilized by another methods , for example , by pef , thermal or another mode of treatment .  in analysing the results presented in figs .[ f2 ] , [ f3](a , b ) we can discern the following phases of press - cake layer consolidation process :  * phase i. initial compaction of press - cake and expulsion of air - liquid mixture , or pre - consolidation period ( ) . during this phasethe velocity of elastic diaphragm displacement is constant and a maximum of liquid flow rate is observed .* phase ii .mechanical rupture of cells and expulsion of liquid from ruptured cells ( ) .the decrease of both liquid flow rate and velocity of elastic diaphragm displacement and acceleration of the pressure increase are observed .* phase iii .final consolidation of press - cake at constant pressure , packing of press - cake and retardation of liquid flow rate ( ) .the liquid flowing from intracellular , extracellular and extraparticle volumes is expressed from the press - cake . at the beginning of this phase a minimum value of the specific electrical conductivityis observed .moisture occupies all channels and so the press - cake is said to be in an impregnated state .these phases are shown schematically at the top of fig .[ f2 ] .significant changes in kinetics of moisture expression and press - cake consolidation can be observed after pef treatment .figure [ f4](a ) presents some examples of experimental curves of liquid yield versus time .it can be seen that curves rise significantly after pef application ( applied in this case at s ) as the result of damage or partial permeabilization of intact cells and subsequent expression of liquid .  buthere , the main problem is in poor reproducibility of the experimental data .the measured curves of can deviate substantially because of differences in initial humidity of samples .this difficulty can be overcome by consideration of the normalized or reduced liquid yield .this normalization procedure was executed in two steps .we began with the first normalized form of liquid yield defined as ( see fig .[ f4](b ) ) : where is the maximal time of pressing ( here we use the value s ) , values correspond to the values of at different electric field strengths .we assume that all liquid yield curves should be equal in the time range of for identical conditions of pressing . for experiments with pef application , we take the curve for as a reference .then we calculate the area under this curve for the time period and renormalize all the curves so as to obtain the same values of for all curves .normalization coefficient is given by   the second normalized form of liquid yield curve is defined as follows ( see fig .[ f4](c ) ) :   the degree of intensification caused by pef treatment , can be determined as the following ratio of the second normalized forms for pressing with and without pef treatment : here the values of were determined in the point of the maximal time of pressing .another interesting property of the expressing is the mean characteristic time , which characterizes durability of the process .the theory developed for mechanical expression of cellular materials ( lanoiselle et al. , 1996 ) describes expression process with the set of characteristic times for the different expression phases .these are very valuable characteristics of the expression processes but , in practice , it is very difficult to find them proceeding from the expression curves .the main problem here is that we can never determine the exact value of limiting expression quantity at for vegetable stuff we are faced with the problem of high enzymatic destruction at continuous expression .so we always stop the expression process at rather long , but finite time ( in our case we choose s ) and the obtained values of are of course less then the actual values of .more general approach implies evaluation of the first moment of the function :   it is easy to show , that for the simple exponential function and  the first moment is equal to the characteristic time , for the finite but large values of we have , and at  the first moment equals to .so , in our case the value of may serve for approximate estimation of the effective characteristic time constant of the whole expression process .it is useful to use this approach for crude estimation of characteristic time or durability of expression process after pef intervention .in such a case we can treat as a mean characteristic time of liquid expulsion processes reflecting the summarized effect of all the mechanisms in a system .we can define the coefficient of pef - enhanced durability as   this coefficient shows the degree of the durability increase after the pef treatment .figure [ f5 ] presents the curves of excess normalized liquid yield versus time at different times of pef application , , and different values of electric field strength , .we have applied pef treatment in different characteristic moments :  * s ( before pressing ) ; * s ( initial phase of pressing ) ; * s ( second consolidation phase and specific conductivity of a tissue is low ) ; * s ( press - cake pressure is achieved a constant value ) ; * s and s ( final consolidation phase ) .the range of voltage values used corresponds to conditions of steady pef - application regime without any disruption of electrical treatment caused by overflow of acceptable maximal current value . for the given pulses protocol , the steady electrical treatment regimewas observed for the voltages not exceeding the following maximal values : v for s , v for s , and v for s.  as can be seen from fig .[ f5 ] the form of the liquid yield curves in the time interval of s practically does not depend on the time of pef input and the electric field strength values ( that corresponds to small values of deviations ) . only at longer time intervals in the phase iii of consolidation , we can observe a behaviour reflecting the mode of pef treatment .  at short times of pef treatment , s , the excess liquid yield increases rapidly with increase as compared with untreated sample ( see fig .[ f5](a - c ) ) .but ineffectiveness of pef treatment at short is less in comparison with the cases of later pef application .it can be explained by the influence of excess quantities of air and extraparticle liquid in the cake pores .the sample is highly water - saturated at earlier period of pre - consolidation and pef application during this period may cause dielectric breakage and uncontrolled increase of current flow through the system . at such conditions ,the intensification degree of pef treatment is rather low ( see black square for in fig . [ f6](a ) ) .when we apply the pef treatment later on , for example , at the beginning of the phase iii of consolidation , at s a liquid excess yield seems to be less dependent on ( fig . [ f5](d ) ) .pressure achieves its constant value at this time and most of cells that could be destroyed mechanically at a given pressure ( 3 bars in our experiments ) are already disrupted and most of liquid is evacuated from these cells .the residual isolated intact cells are connected with electrodes through the network of channels containing conductive moisture . at such conditions ,the transmembrane potential on intact cells should be high enough , even at low values of , and , therefore , the effective electropermeabilization of cell can be attained even at minimal values of electric field intensity ( v in our experiments ) .as we can see from comparison of typical micrographs of apple tissue structure received for pressing with and without electrical treatment ( figs .[ f3](b ) and ( c ) ) , pef application at s ( v ) causes almost complete destruction of the material .the similar pictures are observed in the wide interval of v and micrographs allow us to identify only certain quantity of single isolated intact cells .dependencies of the coefficient of pef enhanced durability versus electric field strength , ( fig . [ f6](b ) ) substantiate conclusions set forth above .we see that the value of depends considerably on the electric field strength only at small time of pef input ( s ) .the best liquid excess yield as compared with untreated sample may be obtained at the lowest applied field when the pef is applied at an instant when the pressure in the system reaches a constant value ( s ) .figure [ f7 ] presents the experimental curves of ( a ) a specific electrical conductivity , , and ( b ) instantaneous flow rate , , versus time for the compressed layer of apple slices .the pef treatment was applied at s at different external field strengths , . as can be seen from fig .[ f7](a ) , the values begin to rise abruptly after pef treatment , and this behaviour becomes more pronounced with increasing . such rise of values corresponds to the combined effect of mechanical rupture of cells and their electrical permeabilization .we present schematically the model of pef treatment with and without pressing in fig .[ f8 ] . in the absence of mechanical pressurethe effect of the hidden ( or passive ) electrical breakdown can be rather important .the electrical conductivity of the whole system depends not only on the destruction degree of individual cells , but also on the character of their connectivity and the presence of continuously conducting channels .  at first , after pef treatment the real degree of electrical breakdown is hidden and does not affect the conductivity of the whole system .pef treatment permeabilizates cell membranes and intensifies diffusion processes .the time of electric conductivity build - up in a cellular material after pef treatment can be estimated from the time constant of diffusion processes : s , where is a mean cell dimension , and m s is a diffusion coefficient of an endocellular fluid . in general , this effect can exhibit a wide distribution of time constants because of differences in cell dimensions , diffusivity of intracellular solutions , degree of cell permeabilization , etc .moreover , it can be masked by another related phenomena of resealing process .heinz et al . ( 1999 ) observed that the insulating properties of the cell membranes can be recovered within a few seconds after the small power pef treatment and it results in decreasing of a cellular system conductivity .simultaneous effect of pressing and pef treatment( and ) can cause the primary changes .first of all , mechanical and electrical stresses can be coupled to cause membrane breakdown in cells ( akinlaja & frederick , 1998 ) .then , the external pressure enhances flowing of the fluid from destroyed cells to extracellular and extraparticle channels . all these decrease the retardation time of electric conductivity build - up after pef treatment . under simultaneous pef treatment and material compression we can also eliminate or diminish the effect of a hidden electric breakdown and to depress the cells resealing processes .so , in a general case , we should observe in final state .this conclusion is confirmed by the data on kinetics at presented in fig .[ f7](a ) by the dashed line . in this experimentwe have dropped the pressure after pef treatment .this results in considerable decrease of values as compared with the curve obtained at ( fig .[ f8 ] ) .a liquid flow rate also depends upon the degree of cellular system destruction as a result of pef treatment .but the behaviour of after pef application does not change drastically ( fig . [ f7](b ) ) .we can explain such behaviour by the fact of substantial decrease and termination of liquid expulsion from the press - cake by that time .the most pronounced peak is observed only at the highest electric field strength ( v , fig . [ f7](b ) ) , and behaviour of the curve reveals only increase of its long time tails with increase of .figure [ f9 ] presents the experimental curves of ( a ) reduced pressure values and ( b ) pressure difference versus time for different durations of pre - compression stage before the pef application .it can be seen that pef treatment diminishes tissue rigidity which corresponds to the decrease of an effective pressure in a system .electrical treatment during the pressing period up to s delays pressure increase and accelerates the liquid yield .pef application ( s ) can decrease steady - state pressure abruptly and initiate a rise of a liquid flow rate .the peaks of pressure decrease become sharper with pre - compression time increase .it corresponds to the more rapid destruction of cellular material pre - compressed during a longer time .but then , at the final stage , the mechanical properties of a press - cake reflate and pressure increases again up to the maximal level .the expressed liquid ( apple juice ) absorption ( or coloration ) and transmittance change in the course of a simple pressing process , due to filtration properties of the cellular material in the course of time . in the final phase of consolidation ( phase iii , see fig .[ f2 ] ) , the liquid coloration reduces considerable as compared with coloration of an initial portion of moisture . as we have demonstrated before ( vorobiev , bazhal & lebovka , 2000 ) , the simultaneous pressing and electric field treatment result in considerable reduction of coloration of those differential portions of liquid which were obtained after pef application .studying the temporal dependencies of optical properties in differential liquid for different modes of pressing and pef application is _ per se _ of great interest .  herewe will discuss only the optical properties of the cumulative expressed liquid , which is obtained as a result of combined pressing and pef treatment at the final stage of the process .figure [ f10 ] presents dependencies of absorption and transmittance versus electric strength for extracted liquid at different values of . on the one hand , the pef treatment decreases the liquid coloration , as can be seen from absorption curves in fig.[f10 ] .this is a positive factor of pef treatment .we can explain this phenomena by improvement of the tissue filtration properties during pressing .moreover , filtration properties of the pef treated press - cake also get improved with increase because of increase of the pressed tissue consolidation . on the other hand , transmittance of expressed liquid reduces with decrease and increase of .this is an undesirable phenomena .it can be explained by the influence of electrical treatment on the press - cake filtration properties .it is known that pef application causes many defects of the tissue .application of the pef treatment increases yield of a liquid with high contents of suspended particles .that s why it is so important to choose a proper instant for pef application allowing to obtain the cumulative liquid with low coloration and high transmittance .the pef application at a moment when pressure in the system achieves a constant value is consistent with requirement of the best quality of a juice .investigations of the moisture expression from fine - cut apple raw material under simultaneous mechanical expression and pef treatment were done .all experiments were performed using both laboratory filter - press cell and high voltage pulse generator which provided monopolar pulses of rectangular shape .the pef treatment was applied to materials that were expressed at time .then the yield of liquid was analysed in comparison with that of untreated material .the experimental results were obtained for electric field strength varying from to v , pulse duration = 100 s , pulse repetition time ms , number of pulses , constant pressure bars and maximal time of mechanical expression s.  the summary of results is as follows :  \(1 ) the data obtained allows us to conclude that all kinetics curves ( , , and ) clearly reflect three main consolidation phases in cellular material .\(2 ) the combination of pressing and pef treatment gives the most optimum results and permits to enhance significantly the liquid yield in comparison with samples untreated by pef .the pef treatment application permits to intensify pressing process whenever the pef is applied .but best liquid excess yield results at lowest value of applied field may be obtained when pef is applied after some pre - compression period .such pressure pretreatment before pef application is necessary for structuring uniformity of the press - cake , removing excess moisture and decreasing the electrical conductivity of cellular material . in our study , the pre - compression period duration of s was optimal and for which the pressure in the system reaches a constant value .the pef application in this moment of time results in the best quality of the expressed liquid ( apple juice ) , which is confirmed by its coloration and transmittance .\(3 ) the simultaneous pressure and pef treatment application reveals the passive form of the electrical damage .electrical damage under a low field without pressure application develops very slow .the pressure provokes damage of defected cells , enhances diffusion migration of moisture and depresses cells resealing processes .\(4 ) the proposed unified approach for liquid yield data analysis allows to reduce the data scattering caused by the differences in the quality of samples .the authors would like to thank the pole regional genie des procedes ( picardie , france ) and the society choquenet for providing the financial support. authors also thank dr .n. s. pivovarova and dr .a. b. jemai for help with the preparation of the manuscript .aguilera , j.m . , & stanley , d.w . ( 1999 ) ._ microstructural principles of food processing and engineering_. gaithersburg : aspen publishers .gulyi , i.s . ,lebovka , n.i . ,mank , v.v . ,kupchik , m.p . ,bazhal , m.i . ,matvienko , a.b . , & papchenko , a.y .( 1994 ) . _scientific and practical principles of electrical treatment of food products and materials_. kiev : ukrintei ( in russian ) .heinz , v. , angersbach , a. , & knorr , d. 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Abstract: influence of pulsed electric field ( pef ) simultaneous to pressure treatment on moisture expression from fine - cut cellular raw material has been investigated . dependencies of specific conductivity , liquid yield , instantaneous flow rate and qualitative juice characteristics at different modes of pef treatment are discussed . three main consolidation phases were observed in a case of mechanical expression . a unified approach is proposed for liquid yield data analysis allowing to reduce the data scattering caused by differences in the quality of samples . simultaneous application of pressure and pef treatment allowed to reveal a passive form of electrical damage . pressure provokes the damage of defected cells , enhances diffusion migration of moisture in porous cellular material and depresses the cell resealing processes . pef application at a moment when a sample specific electrical conductivity reaches minimum and pressure achieves its constant value seemed to be the most optimal .   [ cols= " < , < " , ]