Lattice-Gas Model for Energy Storage Materials: Phase Diagram and

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Lattice-Gas Model for Energy Storage Materials: Phase Diagram and Equilibrium Potential as a Function of Nano-Particle Size Alexander V. Ledovskikh, and Marnix Wagemaker J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b00914 • Publication Date (Web): 11 Apr 2016 Downloaded from http://pubs.acs.org on April 26, 2016

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Lattice-Gas Model for Energy Storage Materials: Phase Diagram and Equilibrium Potential as a Function of Nano-Particle Size A.V. Ledovskikh*, M. Wagemaker**

Department of Radiation Science and Technology, Delft University of Technology Mekelweg 15, 2629 JB Delft, The Netherlands

ABSTRACT: Insertion reactions are of key importance for Li/Na–ion batteries and hydrogen storage materials. Nano-sizing of these energy storage materials has been shown to have fundamental impact on the storage properties. Predicting these properties based on rather simple thermodynamic grounds is of high importance for fundamental understanding, achieving the optimal performance of nano materials, as well as for the practical ability to manage battery systems. Here we report on the development of a new thermodynamic lattice gas model based on the equation of state of the energy carrier that is able to describe the impact of particle size on fundamental physical-chemical characteristics, such as the phase diagram and equilibrium potentials of energy storage materials that exhibit a first order phase transition upon Li or H insertion. The model is based on the first principles of chemical and statistical thermodynamics and takes into account complex structural changes taking place in energy storage materials and because of its general nature can be adapted to describe the influence of any state variable (particle size, temperature, etc.). The model is applied and validated using experimental data on different particle sizes of the LiFePO4 battery electrode material resulting in excellent agreement. The model can be used to simulate phase diagrams and predict equilibrium potential isotherms with respect to the electrode nano-particle size. The relative simplicity of the model allows easy prediction of material properties as required by for instance advanced battery management systems.

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INTRODUCTION

Energy Storage Materials (ESM) play very important role in modern society which needs clean, renewable and efficient energy storage devices including Li-ion and NiMH batteries, gas-phase hydrogen storage in metal-hydride materials and fuel-cells

1

. An important class of ESM is

insertion materials where the energy carrier (Li, H, etc.) is, either from the gas phase or from an electrolyte, stored in the voids of the ESM. Typically ESM are complex alloys or compounds of various transition metals as for instance employed in the electrodes of batteries and hydrogen storage devices

1-3

. The class of the ESM considered in the present paper is characterized by a

continuous first order phase transition initiated by the charge carrier insertion process. In electrochemical battery systems phase transitions taking place in the electrodes (plateau region) have large practical importance determining the voltage profile of batteries and potentially charge transport kinetics and cycle life, key parameters determining the battery performance 4. One of the typical ESM used in modern Li-ion batteries as a positive electrode is LiFePO4. Schematic representation of the insertion of Li ions into LiFePO4 is shown Fig.1.a. During the energy carrier insertion process (absorption process can be also considered in the same way as insertion 1-4) at low values of the normalized concentration of the charge energy carrier ( x ) a solid solution is formed, which is generally denoted by the -phase. After the concentration of the energy carrier (for example, Li ions) in the ESM reaches a certain critical value ( x ) a phase transition is initiated and the -phase transforms into the -phase (Fig.1.a). After the -phase has been fully converted into the -phase at x  , the solid solution is again formed which is denoted by the -phase. Between concentrations x and x  system stays in the two-phase coexistence region which is also called as “plateau region” 2-3, 5-8. The equilibrium potential in the two-phase coexistence region (miscibility gap) is characterized by a plateau

9-11

. The equilibrium

electrochemical electrode potential curve (red) and phase diagram (blue) are schematically represented in Fig.1.b. The lengths and height of the two-phase coexistence plateau region depends on the external state variables including temperature, particle size, thin film thickness, alloy composition, etc. 8-9, 12-30

. Changing of state variables is the origin of changing of microscopic and macroscopic

properties of the material and, as a result, changing the plateau region. A well-known example is the increase of temperature of two phase (ESM) materials that results in a shrinkage of the miscibility gap vanishing completely above the critical temperature where no phase transition 2 ACS Paragon Plus Environment

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occurs and the system is in a homogenous uniform state

6-8, 12, 31

. However, the influence of the

other state variables on the phase diagram is much less known and investigated. Especially, the impact of the nano-particle size appears very important for the properties of the ESM. When particle sizes approach the nanoscale, the influence of surfaces and interfaces can no longer be neglected in the rationalization and prediction of material thermodynamic and kinetic properties 18, 32-35

. The impact of nano-particle size is highly relevant for Li and H insertion

35

, where

experiments indicate a decrease of the miscibility gap upon particle size reduction, suggesting a critical particle size where the phase transition is absent during decrease of the nano-film thickness composition of the ESM alloy

6-8, 27-30

9, 24-26

13-23

. Similar effect has been observed

, changing of the (non-)stoichiometric

and joint influence of the alloy composition and nano-

18

particle size . Existing mathematical models describing energy storage systems are often based on statistical thermodynamics, allowing simulating macroscopic characteristics by using microscopic parameters, such as atomic interaction energies, etc. Early semi-empirical model have been proposed by Lacher 36. A different approach has been adapted by Flanagan et al. 5 who described the thermodynamics of the hydrogen storage system in terms of chemical potential of reactants and reaction products. A simple absorption model has been proposed by McKinnon et al. 37. This model is based on a Lattice Gas LG), statistical mechanics approach taking into account the interaction between absorbed atoms. According to the LG approach 38, the bulk of the ESM can be considered as a host-guest system in which guest energy carrier atoms (lattice gas) can occupy the available host sites. Another LG model based on first principles of statistical and classical thermodynamics has been recently proposed successfully describing complex thermodynamics and equilibrium kinetics of gas phase and electrochemical hydrogen storage systems 6-11. Recent phase field modeling based on the free energy functional theory of Cahn and Hillard

39

was

extended towards battery electrode materials including the impact of the two-phase microstructure, coherency strain, elastic energy, particle shape and nucleation effects

40-43

. Applying a relatively

simple regular solution model was successfully implemented in phase field modeling by Han et al.

44

. Phase field models successfully predicted both the impact of size and temperature on the

physical characteristics of LiFePO4 17, 22. However, the complexity of phase field models leads to time-consuming calculations. In the present work, a new lattice gas model is developed based on general thermodynamic considerations independent of the detailed material characteristics, allowing fast and easy prediction of the impact of state variables such as temperature and particle size on the phase 3 ACS Paragon Plus Environment


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diagram, equilibrium potential and other important fundamental physical-chemical characteristics of energy storage materials that exhibit a first order phase transition upon insertion of charge carriers. The specific case of LiFePO4, an intensively studied electrode material for Li-ion batteries, is simulated and compared with experimental observations. The final equations are derived within the basic theory of chemical and statistical thermodynamics without empirical and heuristic assumptions. The relative simplicity of the model makes it possible to predict material properties based on easy to measure parameters and is of large relevance for future application and implementation in advanced battery management systems.

MODEL DESCRIPTION

The basic elements of a solid 45 ESM are unit cells in which guest charge carrier atoms (Li, H, etc.) can be inserted in voids of the host lattice which we refer to as host sites. For these materials it can be assumed that the total number of host sites is constant and that, according to the LG approach

6-11, 38

, one host site is either occupied by only one charge carrier or empty. Assuming

the linear changing of the phase fractions during the phase transition 5, 46-47 the number of host sites in α- and β-phase ( N and N  , respectively) as a function of the total normalized concentration of the guest energy carrier atoms ( x ) can be written for three crystallographic regions as 6-11

M d ,   x x  N   M d   x x     0, 

x  x   , x  x  x  ,  x  x

0,   x  x  N   M d  x x     M d , 

x  x   , x  x  x   x  x

(1)

where M is the total number of unit cells which was assumed constant during insertion process,

x and x  are the concentrations in phase transition points; d is the number of available host sites per one unit cell. The number of guest energy carrier atoms n and n can be expressed as 6-11


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 xnmax ,   x x  n   x nmax   x x     0, 

x  x   , 

x  x  x  , x  x

0,   x  x  n   x  nmax  x x      xn ,  max

x  x   , x  x  x  (2)  x  x

where nmax is the total number of available host sites. The entropy of the LG ( Si ,  J K 1  ) can be described by Boltzmann equation 6, 38, 48-50

Si  k ln Wi

(3)

where k ,  J K 1  is Boltzmann constant. The thermodynamic probability Wi (the number of microstates consistent with the given macrostate) expressing the distribution of ni LG atoms in

Ni host sites (Eqs.1,2) in phase i can be written as 6, 38, 48-50

Wi 

Ni ! ni ! Ni  ni !

(4)

Applying Stirling’s approximation 38, 48-49 to Eq.4 and substituting the result in Eq.3 the entropy of the LG is obtained 6, 8

n n  n   n  Si   kNi  i ln i  1  i  ln 1  i    Ni Ni  Ni   Ni  

(5)

In thermodynamics gases are often described by the equations of state which express the relationship between state variables. More specifically, the equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions

49, 51

. Nowadays,

many equations of state are known describing the thermodynamics of real gases, for example, the 5 ACS Paragon Plus Environment


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well-known van der Waals equation of state 38, 49, 51-52. Typically the equation of state is represented in the form where the gas pressure is a function of the other state variables such as gas volume, temperature, concentration of the gas, etc. As in the classical van der Waals equation 38, 49, 52 it has been assumed that the pressure of the LG in phase i can be expressed as a differential logarithm of the thermodynamic probability described by Eq.4 with respect to the gas volume at constant number of gas atoms, temperature and internal energy leading to the lattice gas pressure ( Pi ,  Pa  ) of phase i:

Pi   ln Wi  k i  T  V ni ,T

(6)

where T ,  K  is temperature; V i ,  m3  is the volume of the LG which can be expressed as

V i  Ni

Vi d

(7)

3 where Vi ,  m  is the volume of the single unit cell. Expressing the thermodynamic probability

(Eq.4) in terms of the entropy (Eq.3), substituting Eq.7 into Eq.6 and assuming Vi and d independent on the number of host sites and constant at the considering temperature, one obtains

 S  d Pi  T  i   Ni ni ,T Vi

(8)

Comparing this thermodynamic system with a normal gas one may conclude that the number of host sites ( Ni ) plays role of gas volume in the LG system. Assuming that the number of the LG atoms ( ni ) is independent of the number of host sites and differentiating the entropy (Eq.5) according to Eq.8 the equation of state expressing the pressure of the LG in solid solution regions as a function of unit cell volume and normalized concentration of the LG can be obtained


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Pi  

RT ln 1  xi  bi

(9)

where xi is so-called phase-normalized concentration of the LG atoms in phase i 7; bi ,  m3 mol 1  is the molar volume of available host sites

bi 

N AVi d

(10)

where N A ,  mol 1  is Avogadro’s constant. Important to note is that parameter bi (Eq.10) has the same dimension and meaning as the volume parameter b in the classical van der Waals equation where it defines the volume occupied by gas molecules per molar volume or volume excluded by a mole of gas particles 38, 49, 51-52. The phase-normalized concentration of the LG atoms is defined as 7

xi 

ni . Ni

(11)

where the number of host sites ( Ni ) and guest energy carrier atoms ( ni ) are defined by Eqs.1,2. Obviously the phase-normalized concentration remains constant during the phase transition and equal to the normalized concentration in the corresponding transition point ( xi  7

ni  x i  const ) Ni

. Substituting the entropy Eq.5 into Eq.8 followed by differentiating with respect to the number

of host sites Ni the following LG pressure equation of state for two phase coexistence region was derived

Pi  

RT 0 Si bi


(12)


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where Si0  x i ln x i  1  x i  ln 1  x i 

6, 8

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. As expected the pressure of the LG in the two-phase

coexistence region is constant. Calculation of the LG pressure by Eqs.9 and 12 is shown in Fig.2. The calculation is performed for the “bulk mode” of LiFePO4 with unit cell volumes V  0.272 nm3 and

V  0.291 nm3 (particle radius is 140 nm) 17 taking into account that a maximum of four Li ions ( d  4 ) can be stored per orthorhombic unit cell 53. It is observed that in phase transition points the partial LG pressures in the α- (blue) and in the β-phase (red) are discontinuous. However as the concentration of the LG in both α- and β-phases is assumed continuous (Eq.11) the LG pressure should be also continuous in the phase transition points. Further it was assumed that the van der Waals (or “volumetric”) interaction energy forces between LG atoms act in the two phase coexistence region characterizing the influence of the gas volume and the amount of interacting atoms on the LG pressure. This interaction forces can be defined as the van der Waals interaction term which is added to the plateau pressure equation of state (Eq.12)

RT 0 Pi   Si bi

ai i2 (V i )2

(13)

where ai ,  J m3 mol 2  is the average van der Waals interaction energy 38, 49, 51-52 which can be defined with respect to the mean field theory 50 and  i ,  mol  is the number of moles of the LG in phase i

i 

Ni xi NA

(14)

In the solid solutions the distribution of charge carrier atoms was assumed as uniform without van der Waals interactions between LG atoms. The sign of the van der Waals term

ai i2 in Eq.13 (V i ) 2

characterizes the direction of inter-atomic forces (“–“) indicating attractive forces in the -phase (the plateau pressure has to be lower to achieve the continuity with solid solution region, Fig.2) 8 ACS Paragon Plus Environment

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and repulsion forces (“+“) dominate in the -phase. In the classical van der Waals equation only long-range attractive forces are taken into account because in real gases the distance between gas molecules is relatively large making short-range repulsion negligibly small

38, 49, 52

. However, in

the LG the distance between atoms is small (less than the unit cell dimensions) making repulsive forces significant. The interaction term ai is directly related to the molar volume of host sites ( bi ) and the molar depth of the Lennard-Jones potential 51, 54 well binding energy (  i ,  J mol 1  ) 38, 52

ai   i bi

(15)

The final set of equations of state expressing the pressure of the LG as a function of corresponding unit cell volumes and normalized LG concentration in all crystallographic regions taking into account Eqs.7,14,15 can be written as  RT ln 1  x  , x  x   0 2  RTS    x , x  x  x  ,  x  x  0,

(16.1)

0, x  x  1  P    RTS 0    x 2 , x  x  x  , b  x  x   RT ln 1  x  ,

(16.2)

1 P   b

The direct physical meaning of the “LG pressure” in Eqs.16 is the internal pressure of the energy carrier inside of the ESM. To preserve the continuity of the LG pressure, the continuity conditions have to be applied to the set of equations of state (Eqs.16) in the phase transition points x and

x  RT  S0  ln 1  x      x2  0     0 2  RT ln 1  x    S    x   0


(17)


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The continuity conditions Eqs.17 preserve continuity of the LG pressure in the phase transition points ( x and x  ). On the other hand knowing the binding energy (  i ) as a function of considering state variable, the phase transition points and, consequently, the phase diagram can be directly calculated from the set of Eqs.17. The equation of state (16) results in equations for the Gibbs free energy and equilibrium electrode potential of the electrode through the general thermodynamic algorithm shown in Fig.3. According to general thermodynamics, an increase of the gas volume decreases the Helmholtz free energy; the rate of decreasing being higher for larger pressures 50-51, 54. Mathematically, the partial derivative of the Helmholtz free energy with respect to the gas volume at a constant temperature is equal to the negative pressure. In terms of the present model this can be written as

 AiLG    Pi  i   V ni ,T

(18)

where AiLG ,  J  is the Helmholtz free energy of the LG in phase i , which by definition can be expressed in terms of the internal energy ( U ,  J  ) and entropy ( S ,  J / K  ), AiLG  U i  TSi 54

50-51,

. Substituting bi and xi (Eqs.10,11) into equation of state (16) taking into account the phase

volumes V i (Eq.7) and integrating the result according to Eq.18 results in the Helmholtz free energy of phase i

ALG

 n  n n U  kTN   ln   1     N N  N   N   U    x2   kTN S0 , NA  0,  

  n  ln 1    N

   , x  x  x  x  x  , x  x


(19.1)

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ALG

  0,  N   U     x 2   kTN  S 0 , NA     U   kTN  n ln n  1  n    N N  N    

x  x x  x  x  ,   n  ln 1    N

(19.2)

   , x  x   

where U i, U i ,  J  are integrating constants depending on the concentration of the LG. Obviously

n n  n the terms kNi  i ln i  1  i  Ni Ni  Ni

  ni  ln 1    Ni

 0   and kNi Si in Eqs.19 represent the entropy of the 

LG. It has been assumed that the Helmholtz free energies (19) are continuous in the phase transition points x and x  (mathematically, lim ALG  lim ALG and lim ALG  lim ALG ). Taking x x

x x

x x

x x

into account that the number of host sites in both α- and β-phases in solid solution regions remains constant and equal to the total number of host sites ( nmax ), a level constant which is numerically equal to the van der Waals interaction energy term (Eqs.13,15) at the corresponding phase transition points can be introduced in the solid solution equations to keep continuity of the Helmholtz free energy

U i  U i   i x i2

nmax NA

(20)

In the other words, it was assumed that the total internal energy consists of two parts, van der Waals interaction term (  i x i2

Ni ) and concentration dependent internal energy ( U i ). NA

To describe the concentration dependent internal energy ( U i ) the general approach 6-11 has been used according to which various energy contributions have to be taken into account. Let define Ei ,  eV  as the energy of the individual LG atom located in phase i and assume that the LG atom at a particular host site can interact with LG atoms at any other sites with certain



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interaction energy ( U ii ,  eV  ). The mean-field approximation 48, 50 has been applied according to which the interaction energy between occupied sites does not depend on their location and distance. In the other words, the nearest-neighbor influence is not taken into account. LG atoms were assumed to be randomly distributed along the host ESM (Bragg-Williams approximation, one of the basis of the mean-field theory)

38, 48-49

. Assuming inter-phase interactions to be

negligibly small the total concentration dependent internal energy ( U i ) can be described by the following energy Hamiltonian

  U U i  e  Ei ni  ii ni2  2nmax  

(21)

where e is the elementary charge ( e  1.6e19 C ). By definition the Gibbs free energy of the LG in phase i ( GiLG ,  J  ) can be written as

GiLG  Hi  TSi

51, 54

. For the considered nano-particle system it was assumed that the volume

expansion of the ESM is rather small and changing of the concentration of the energy carrier leads only to the increasing of the internal energy. Thus the internal energy was assumed to be equal to the enthalpy ( H i  U i ) and the Gibbs free energy to be equal to the Helmholtz free energy ( GiLG  AiLG ). The total Gibbs free energy of the bulk of phase i ( Gi ,  J  ) consists of the Gibbs free

energy of the LG and the Gibbs free energy of the crystal lattice ( GiL ,  J  )

Gi  GiLG  GiL

(22)

Assuming that the contribution of each single unit cell to the total lattice energy is Li ,  eV 

6-11

and taking into account that the entropy of the crystalline host material is a small constant 51, the Gibbs free energy of the crystal lattice is represented by

GiL  eLi M i


(23)

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where M i is the number of unit cells in phase i , M i 

Ni d

6-8

. Combining Eqs.19-23 the following

equation describing the total Gibbs free energy can be written as   e  L      G  e  L   0,      0,    G  e  L    e  L     

n   N U n n n   n  E n   n2    x2 max  kTN   ln   1    ln 1   N d 2nmax  NA N N N          N U N  E n   n2    x2   kTN S0 , d 2nmax  NA

   , x  x 

(24.1)

x  x  x  x  x

x  x N d N d

 E n   E n 

U  2nmax U  2nmax

N  n     x 2   kTN  S 0 , NA  n  n n  n n2     x 2 max  kTN    ln   1    N  N N N A      

(24.2)

x  x  x 

2

  n  ln 1  N   

   , x  x   

The chemical potential of phase i of the ESM can be calculated as a derivative of the Gibbs free energy (Eqs.24) with respect to the corresponding amount of moles of the LG ( i , Eq.14) 51, 54

 Gi   Gi    NA     i T , Pi  ni T , Pi

i  

(25)

If the system remains in electrochemical equilibrium the equilibrium potential Eieq , V  can be calculated from the chemical potential (Eq.25) 51, 55 as

Eieq  

1 i zF

(26)

where z is the number of electrons involved in one-step charge-transfer energy storage process;

F , C mol 1  is Faraday constant, F  eN A .



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Page 24 of 50

Differentiating of the total Gibbs free energy Eq.24 according to Eqs.25,26, taking into account the assumption that the number of host sites ( Ni ) in the two phase coexistence region depends on the number of guest LG atoms ( ni ) the equilibrium potentials of α- and β-phases as a function of the total normalized concentration ( x ) of the LG are obtained

 RT  x  x  x  E  U x  F ln  1  x  ,     L  x RT S0 Eeq      E  U x x     , x  x  x  , x d F F x    0, x  x  

(27.1)

 0, x  x  0  L  x RT S  Eeq      E  U  x  x     , x  x  x  , x d F F x     RT  x   E  U  x  ln  x  x , F  1 x  

(27.2)

where x 

x  x x   x

and x 

x  x . Eqs.27 derived for one-electron charge-transfer energy x   x

storage processes z  1. Important to note that in both solid solution regions the equilibrium potentials in Eqs.27 are described by exactly the same equations as previously presented for the description of thermodynamics of the hydride-forming materials 6, 9-11. According to the general condition of phase equilibrium the chemical potentials of α- and β-phases are equal (    ) 51, 54. Taking into account Eq.26 in electrochemical equilibrium the equilibrium potentials of α- and β-phases also have to be equal in two phase coexistence region

Eeq  Eeq

(28)


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The extra continuity conditions should be applied to the equilibrium potential Eq.27 making it continuous in the phase transition points x and x  (mathematically, lim Eeq  lim Eeq and x x

x x

lim Eeq  lim Eeq ). Combining the continuity of the equilibrium potential in the phase transition x x

x x

points with general condition of α- and β-phases equilibrium Eq.28 some model parameters in Eq.27 can be identified expressing, for example, the lattice energy parameters ( Li ) and energy of the individual LG atom in the α-phase   RT  x    x RT S0  ln    L    x d  F F x   F  1  x     RT  x     x  RT S 0   L  ln     x d     1  x  F F F x          RT  x   RT  x   E  E  U x  ln  ln    U x         F 1  x F 1  x       

(29)

More details about model parameters identification is considered in the Results and Discussion section. The equilibrium potential represented by Eqs.27 expresses the so-called “configuration potential” 32-34 derived for isolated bulk material. To be able to compare the simulation results with experimental electrochemical measurements the equilibrium electrode potential ( E eq , V  ) can be calculated by adding the energy level defined by the constant reference electrical potential  ref

32-

34, 51, 55

E eq  Eieq  z ref

(30)

The proposed model provides a simple general thermodynamic algorithm for simulation of phase diagrams, equilibrium potentials and other important physical characteristics of the ESM. It can be adapted to specific materials and processes by adjusting the model parameters based on experimental observations or theoretical estimations by other statistical methods.



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RESULTS AND DISCUSSION

The model described in the main section gives a general thermodynamic algorithm for simulation of phase diagrams, equilibrium potentials and other important physical characteristics of the ESM. Changing of state variables results in changing of the physical characteristics of the material such as the unit cell volumes, binding and interaction energies, etc. For example, experimental observations

13-23

indicate that for a number of Li battery electrode materials,

including LiFePO4, a decrease of the nano-particle size results in an increase of the α-phase unit cell volume and a decrease of the β-phase unit cell volume. It is suggested that at a critical particle radius ( rc ) the unit cell volumes of both α- and β-phases become identical. Experimentally estimated values of rc are between 10-20 nm. For larger particle sizes the physical characteristics of the ESM, such as unit cell volumes strive to asymptotically limiting “bulk” values which in practice can be reached at particle sizes between 100-200 nm

13-23

. In the other words, the size

inducing changes of the physical-chemical characteristics of the ESM progressively occur when the particle size is reduced below 100-200 nm. In this context the following “limiting fractions” can be defined for single unit cell volumes expressing how much “free space” each volume has to be changed from the current state to its bulk value V  V min and f   min V

f 

Vmax  V Vmax

(31)

where Vmin and Vmax are respectively minimal and maximal volumes of α- and β- unit cells in the bulk mode. The changing of unit cells volumes with respect to the particle radius r in differential form can be written as  dV V  Vmin    f V    V V   V  Vmin  dr  max  dV   f V   V  V V V   V   dr Vmax 


(32)

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where V and V ,  m 1  are contraction and expansion coefficients respectively. The second equation for β-phase in Eqs.32 represents the classical Logistic Equation (LE) form:

56

in differential

V dy  V y 1  y  ; y  max . Integration of Eqs.32 with respect to the particle radius r gives dr V

the expressions for the unit cell volumes of the α- and β-phases as a function of the nano-particle radius  V0Vmin V  0 V  V0  Vmin  exp V r    V0Vmax exp V r  V  max V  V0 exp V r  1    



   

 (33)

where V0 and V0 are the unit cell volumes of the α- and β-phases respectively at infinitely small particle radius ( r  0 ). The second equation for β-phase in Eqs.31 represents the classical sigmoidal LE in integral form which is successfully employed for modelling in various scientific fields 56. The first equation reproduces so-called “anti-logistic” equation which describes inverse “S-shape” curve. Simulation of the unit cell molar volumes of the α- and β-phases as a function of nanoparticle radius, calculated by the set of Eqs.33 for standard LiFePO4 battery electrode, is presented in Fig.4. Good agreement between experimental data 17 (points) and simulation results (lines, blue

- and red -phase) is found. Unit cell volumes are continuous in the phase transition points in both α- and β-phases however being discontinuous between the two phases (the definition of a first order phase transition, according to the Ehrenfest classification of phase transitions 51). The model parameters are estimated by the non-linear least squares method. Asymptotic values of the unit cell volumes at the “bulk mode” resulted in Vmin  0.272 nm3 , Vmax  0.291 nm3 , the critical radius and unit cell volumes at which both α- and β-phases become identical rc  10.37 nm ,

Vc  0.282 nm3 and contraction and expansion coefficients V  1.56 m 1 , V  1.721 m 1 . To simulate the phase diagram binding energies (Eq.15)  i should be described. Reference values of  i can be directly calculated from experimentally measured phase transition points by continuity Eqs.17. It was assumed that, similar to the unit cell volumes, binding energies have 17 ACS Paragon Plus Environment


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limiting “bulk” values if the radius of the nano-particle is larger than ~100 nm. The binding energy goes down with decreasing nano-particle radius

57-58

and is assumed zero at the critical particle

radius and in the solid solution regions with respect to the arbitrary chosen energy level. The binding energy was described by the generalized LE 56 which was derived similarly as the unit cell volume Eq.33 with respect to some arbitrary chosen potential level  i

 i   i 

 i  imax exp   r  0



max i



i



  i0 exp   i r  1  

(34)

where  imax ,  J mol 1  is the maximal asymptotic value of the interaction potential at large values of the particle radius (“bulk mode”);   ,  m 1  are interaction potential increase coefficients;  i0 i

are values of interaction potentials at infinitely small particle radius ( r  0 );   is reference energy level parameter. As an example, a simulation of the binding energy of the β-phase as a function of the nano-particle radius by Eq.34 is shown in Fig.5 (line). Good agreement with values directly calculated from the experimentally measured phase diagram by Eq.17 (points) is achieved, in particular for small particle radii. The maximal values of the binding energy at the “bulk mode” given by the simulation are  max  117.9,  max  6.46 kJ mol 1 . Simulation of the phase diagram of LiFePO4 by Eqs.17 is shown in Fig.6. Good agreement is observed between simulation results (lines, blue - and red -phase) and previously reported experimental data 17 (points). The two-phase miscibility gap between phase transition points x and x  decreases upon particle radius reduction and finally disappears at critical particle radius

rc . Behavior of the width of the miscibility gap at bigger particle radius is similar to that of the volume of the unit cells (Fig.4) reaching a bulk limiting value (“bulk mode”). The model parameters describing unit cell volumes (Eqs.33) have been estimated simultaneously by comparing simulation results with experimental data using the non-linear least squares method. The phase transition points calculated further by Eq.17 were also compared with experimentally measured phase diagram data simultaneously with optimization the other characteristics (unit cell volumes and binding energy). Four parameters were optimized: Vi 0 , Vi max for each phase ( i   ,  ) to describe unit cell volumes as a function of particle radius. In order to estimate continuous function of the binding energy (Eq.34) as a function of the particle radius six parameters were 18 ACS Paragon Plus Environment

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optimized:  i0 ,  imax ,  i for each phase ( i   ,  ). The common parameter optimized for all considering characteristics was the critical particle radius ( rc ). Other parameters e.g. Vi ,  i , etc. have been analytically calculated from the given equations. One of the differences of the present modeling approach from ones used in the other mathematical models describing ESM 6, 8, 36-37, 41-42 is that the derivation of the energy and potential equations (Eqs.24 and 27) does not start from the level of the Gibbs free energy but from the equation of state describing LG pressure (see the algorithm of model developing represented in Fig.3). Simulation of the LG pressure for particle radius 115 (blue), 25 (green) and 10.37 nm (red) by the equation of state (16) is shown in Fig.7. Partial LG pressures are continuous in the phase transition points in both α- and β-phases and discontinuous between the two phases (definition of a first order phase transition, according to the Ehrenfest classification of phase transitions 51). At the critical particle radius (red curve) the LG pressure is continuous. One may also estimate the so-called “conditional border” between the - and -phase regions (dashed line). The physical meaning of the “pressure of the LG” is different from that of normal gases. From a mechanical point of view, the pressure of the LG can be defined as bursting forces acting in the unit cells of the ESM due to presence of guest Li ions. One may suggest that pressure of the LG is the origin of the volume expansion taking place during Li ion insertion. Volume expansions can reach rather large values for hydride-forming materials 27 and Li battery materials 59. The LG pressure has two different physical meanings, firstly being the mechanical internal pressure which is origin of the volume expansion of the ESM during energy storage process  Pa  and secondly the energy density  J m 3 

60

. Using the present model the absolute energy density of the electrode can be

estimated as a function of the concentration of the energy carrier. The maximum pressure of the LG (Li ions, Fig.7), which can be obtained for LiFePO4 electrode material, was estimated to be

4 108 Pa which corresponds to the energy density 4 108 J m3 or 400 kJ / l . The advantage of the present model is the ability to predict physical characteristics of the system that cannot be measured experimentally for example, phase entropies, partial internal energies, Gibbs free energy, etc. Simulation of the partial molar entropies of the LG at the phase transition points as a function of particle radius based on Eq.5 is shown in Fig.8 (lines, blue - and red -phase). Decrease of the particle radius leads to increase of the LG partial entropies. The Sshaped entropy curves are an indication of the decrease of the miscibility gap towards the critical 19 ACS Paragon Plus Environment


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Page 30 of 50

particle radius below which the system reaches a homogeneous state. This indicates that the increase of molar entropy is the origin of the closing of the miscibility gap upon particle size reduction. This is very similar to the effect of temperature (non-)stoichiometric composition of the ESM alloy

6-8, 12, 31

6-8, 27-30

, film thickness

9, 24-26

or the

on the miscibility gap generally

observed for phase diagrams. Simulation of the equilibrium potential of the LiFePO4 electrode as a function of the normalized Li concentration by Eq.30 using Eqs.27 to calculate the configuration equilibrium potential and continuity conditions Eq.29 is shown in Fig.9: red 113 nm ( E eq  3.427 V ); green 42 nm ( E eq  3.430 V ); blue 34 nm ( E eq  3.434 V ); magenta 10.36 nm ( E eq  3.450 V ). Values of the equilibrium potential and particle radius have been chosen the same as in the reference paper of Meethong et al.

22

. As the exact shape of the equilibrium potential in solid solutions is not

completely defined by available experimental data the following simplification has been made, interaction energies were assumed as zero ( U ii  0 ) to make the number of parameters minimal for the equilibrium potential isotherm reproducing. Assuming 113 nm as the “bulk mode” the molar Gibbs free energy (see Eq.36 below) has been set to some reasonable arbitrary chosen energy level in the phase transition point x  (in the current simulation it has been set to

G0 F

 0.004 )

from which the energy of the individual Li ion in β-phase ( E ) can be found to be

E 

1 x

 G0 L U  x 2   x 2 RT 0      S   d 2 F F  F 

(35)

E can be directly calculated by Eq.27. It has been assumed that the reference constant electrical potential  ref (Eq.30) is the difference between the experimentally measured and configuration (Eq.27) equilibrium electrode potential in the “bulk mode” of the ESM. The reference potential has been found to be ref  3.423V . Remaining parameters of Eqs.27, the lattice energy ( Li ) and binding energy (  i ) have been calculated using the continuity Eqs.29 and Eq.34 respectively. For “non-bulk” particle sizes the value of the configuration potential ( Eieq ) can be found as a difference between the experimentally measured electrode potential and  ref ( Eieq  E eq   ref ) and the rest of model parameters can be calculated using Eqs.27,29. The parameter values are listed in Table 1. The limiting case, equilibrium potential at critical nano-particle radius 10.36 nm has been found 20 ACS Paragon Plus Environment

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by simple extrapolation of the experimental data as 3.450 V (magenta curve). From Fig.9 one may see that equilibrium potential of LiFePO4 increases with decrease of the particle size. The similar behavior has been observed for the thin films in the Pd-hydrogen storage system 9. However, there are some ESM which demonstrate the opposite behavior equilibrium potential decreasing together with the particle radius for example in anatase TiO2 61. Whether the equilibrium potential goes up or down upon particle size reduction depends on the competition between the enthalpy and entropy in equations (26). The advantage of the present model is a possibility of predicting of the complete electrochemical isotherm based only on the knowledge of the plateau equilibrium potential. Or if binding energies and other interaction energy values are known apriori, for example calculated by DFT, the phase diagram and equilibrium potential isotherm can predicted without input of experimental parameters. Obviously, the total Gibbs free energy of the entire bulk ESM is a summation of the free energy of phases ( G   Gi ). Taking into account Eqs.24 the molar Gibbs free energy of the ESM (at nmax  N A ) as a function of the total normalized concentration of the LG can be written as

  L U x 2  2 F  E x        x  RT  x ln x  1  x  ln 1  x   ,  2    d   L U   F  L x  E x x  U x2 x2   x  E x  x   x 2 x2    2 d 2nmax  G  d 2 2 0 0    x x    x  x  RT S x  S  x ,   L U x2   F    E x       x 2  RT  x ln x  1  x  ln 1  x   ,   d 2  





x  x

(36) x  x  x  x  x

where x and x are defined in Eq.27. Simulation of the molar Gibbs free energy of the LiFePO4 electrode as a function of the normalized Li concentration is shown in Fig.10.a: red 113 nm; green 42 nm; blue 34 nm; magenta 10.36 nm. During Li insertion the Gibbs free energy goes down in all cases demonstrating the expected exothermic process upon Li-ion insertion. After the two phase coexistence the Gibbs free energy reaches a minimum and slightly goes up at the end of -phase solid solution formation. Note that the minimum of the Gibbs free energy generally does not coincide with the phase transition point x  but only the indication of zero equilibrium potential. The limiting case for the critical particle radius is also represented in Fig.10.a (curve (d), dashed 21 ACS Paragon Plus Environment


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line). Estimation of the energy of individual Li ions ( Ei , see Table 1) absorbed in LiFePO4 electrode as a function of the particle radius is shown in Fig.10.b. Parameters Ei have been calculated by Eqs.29 and 35 during simulation of the equilibrium potential. Decrease of the particle radius from the “bulk” mode (100 nm) to the critical value causes a decrease of the energy of the individual absorbed Li ions in the -phase and an increase of the energy in the -phase. The present mathematical model can be extended to describe other phenomena relevant in solid state ESM’s, like multi- first order phase transitions, hysteresis etc, leading to an even more universal description.

CONCLUSIONS A new thermodynamic lattice gas model has been developed describing the phase diagram, equilibrium potential and other important fundamental physical-chemical characteristics of energy storage materials displaying a continuous phase transition upon charge carrier insertion and demonstrated for the LiFePO4 Li-ion battery electrode material. The model allows predicting the complete electrochemical isotherm only from the plateau equilibrium potential. The model is based on the first principles of chemical and statistical thermodynamics and takes into account the complex structural changes taking place in energy storage materials upon insertion of energy carriers, Li-ions in the case of LiFePO4. Derivation of the equation of state, the Gibbs free energy and the equilibrium potential has been performed based on the lattice gas approach using the Bragg-Williams approximation, mean-field theory and including detailed description of the lattice gas entropy. Simulation and validation of the proposed model has been done using experimental data of the Li-ion battery electrode material LiFePO4 using the phase diagram, unit cell volumes and equilibrium plateau potential as function of nano-particle radius. The two-phase coexistence miscibility gap between the phase transition points decreases upon nano-particle size reduction and finally disappears at a critical radius. At the larger particle radius (>100 nm) the system has bulk properties, “bulk mode”, with physical-chemical characteristics that are independent of the particle size. The good agreement between simulation results and experimental data indicates that the proposed model provides a simple general thermodynamic algorithm for simulation of the phase diagram, equilibrium potential and other important physical and chemical characteristics of


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the energy storage materials. The general nature of the model allows predicting these characteristics with respect to any other state variables. The model allows predicting physical characteristics of the system that cannot be measured experimentally and is relatively easy to extend towards other phenomena relevant for solid state ESM’s. The relative simplicity of the model makes fast material property prediction possible as for instance demanded in advanced battery management systems.

AUTHOR INFORMATION

Corresponding Authors *

E-mail: [email protected]. Phone: +31 64 137 0063.

**

E-mail: [email protected]. Phone: +31 15 278 3800.

Notes The authors declare no competing financial interest.

ACKNOWLEDGMENTS

The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Program (FP/2007-2013) / ERC Grant Agreement no. [307161] of MW.



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FIGURE CAPTIONS

Fig. 1. a. Schematic representation of the Li insertion process into LiFePO4 electrode; b. Schematic representation of a potential-composition isotherm (red line) and phase-diagram (dashed blue line) as a function of the total normalized concentration of Li ions ( x ) in LiFePO4. Fig.2. Calculation of the LG pressure by Eqs.9 and 12 as a function of the total normalized concentration ( x ) of Li ions in LiFePO4 for unit cell volumes V  0.272 nm3 and V  0.291 nm3 (particle radius is 140 nm) 17. A maximum of four Li ions ( d  4 ) can be stored in the orthorhombic unit cell of LiFePO4 53. Fig.3. The general thermodynamic algorithm of the derivation of the Gibbs free energy and equilibrium electrode potential from the equation of state of the energy carrier. Fig.4. Simulation of the unit cells volumes of α- and β-phases as a function of particle radius, the line represents the simulation by Eq.33, points represent experimental data 17: α-phase (blue) and β-phase (red); black rhomb is the estimated critical volume. Fig.5. Simulation of the binding energy of the β-phase as a function of nano-particle radius ( r ), line is the simulation by Eq.34, points represent binding energy values calculated directly from experimental data 17 by Eq.17. Fig.6. Simulation of the phase diagram of LiFePO4 as a function of the total normalized concentration of Li ions ( x ) in LiFePO4 and nano-particle radius ( r ); line is simulation, points are experimental data 17: α-phase branch (blue) and β-phase branch (red); black rhomb is estimated critical point. Fig.7. Simulation of the LG pressure for nano-particles 113 nm (blue); 34 nm (green); critical particle radius 10.37 nm (red) as a function of the total normalized concentration of Li ions ( x ) in LiFePO4; dashed black line is conditional border between - and -phase regions. Fig.8. Simulation of the partial molar entropies of the LG at phase transition points as functions of nano-particle radius ( r ): α-phase (blue) and β-phase (red). 24 ACS Paragon Plus Environment

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Fig.9. Simulation of the equilibrium potential of the LiFePO4 electrode as a function of normalized concentration of Li ions ( x ) by Eq.30 using Eqs.27,29: 113 nm (red); 42 nm (green); 34 nm (blue); 10.36 nm (magenta). Fig.10. a. Simulation of the molar Gibbs free energy of the LiFePO4 electrode as a function of normalized concentration of Li ions ( x ) calculated by Eq.36: 113 nm (red); 42 nm (green); 34 nm (blue); 10.36 nm (magenta). b. Energy of individual Li ions inserted into LiFePO4 calculated by Eqs.29,35 (points) as a function of the nano-particle radius for α-phase (blue) and β-phase (red); the estimation line is also shown (dashed line); black rhomb is the estimated critical point.



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Table 1. Model parameters of LiFePO4 energy storage system. Lattice energy L 101 , eV

Lattice energy L 101 ,

Nm

Plateau potential, E eq , V

113 43 34 10.36

3.427 3.430 3.434 3.450

-0.105 -0.115 -0.092 0.712

Particle radius,

r,

Energy of Li ion E 10 2 ,

eV

Energy of Li ion E 102 , eV

Binding energy   10 2 ,

eV

Binding energy  , eV

4.781 3.988 3.443 0.712

6.696 4.481 3.095 -2.737

-6.558 -5.865 -5.564 -2.737

1.218 0.450 0.262 0

6.686 5.824 5.217 0


eV

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LiFePO4

-phase

Li-ions

-phase

a.

3.55

xβ

x



β r, [nm]

E, [V]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3.45

+β 3.35 0

0.2

0.4

0.6

0.8 x

1

b.

Fig. 1. a. Schematic representation of the Li insertion process into LiFePO4 electrode; b. Schematic representation of a potential-composition isotherm (red line) and phase-diagram (dashed blue line) as a function of the total normalized concentration of Li ions ( x ) in LiFePO4. 27 ACS Paragon Plus Environment


109

PLG, [Pa], (log scale)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

108

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xβ

x



β

107

+β

106

105 0

0.2

0.4

0.6

0.8

1

x

Fig.2. Calculation of the LG pressure by Eqs.9 and 12 as a function of the total normalized concentration ( x ) of Li ions in LiFePO4 for unit cell volumes V  0.272 nm3 and V  0.291 nm3 (particle radius is 140 nm) 17. A maximum of four Li ions ( d  4 ) can be stored in the orthorhombic unit cell of LiFePO4 53. 28 ACS Paragon Plus Environment

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Helmholtz free energy

PV

Gibbs free energy

A  U  TS

G  U  PV  TS  Glattice

Integrating by the gas volume

Differentiating by the number of moles

 G     T , P

 A     P  V T

 

Chemical potential   f V , P, T , ,...

Equation of state P  f V , T , ,...

Equilibrium electrode potential E eq  

1 1  G    zref      z ref F F   T , P

Fig.3. The general thermodynamic algorithm of the derivation of the Gibbs free energy and equilibrium electrode potential from the equation of state of the energy carrier. 29 ACS Paragon Plus Environment


0.295 0.29 Vi, [nm3]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.285

α+β

0.28

0.275 0.27 0

50

100

150

r, [nm]

Fig.4. Simulation of the unit cells volumes of α- and β-phases as a function of particle radius, the line represents the simulation by Eq.33, points represent experimental data 17: α-phase (blue) and β-phase (red); black rhomb is the estimated critical volume. 30 ACS Paragon Plus Environment

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8

, [kJ mol-1]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


6 4 2 0 0

50

100

150 r, [nm]

Fig.5. Simulation of the binding energy of the β-phase as a function of nano-particle radius ( r ), line is the simulation by Eq.34, points represent binding energy values calculated directly from experimental data 17 by Eq.17. 31 ACS Paragon Plus Environment


0

r, [nm]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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50

α+β 100

150 0

0.2

0.4

0.6

0.8

1

x

Fig.6. Simulation of the phase diagram of LiFePO4 as a function of the total normalized concentration of Li ions ( x ) in LiFePO4 and nano-particle radius ( r ); line is simulation, points are experimental data 17: α-phase branch (blue) and β-phase branch (red); black rhomb is estimated critical point. 32 ACS Paragon Plus Environment

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109

-phase region

108 PLG, [Pa]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


107

106

-phase region

105 0

0.2

0.4

0.6

0.8

1

x

Fig.7. Simulation of the LG pressure for nano-particles 113 nm (blue); 34 nm (green); critical particle radius 10.37 nm (red) as a function of the total normalized concentration of Li ions ( x ) in LiFePO4; dashed black line is conditional border between - and -phase regions. 33 ACS Paragon Plus Environment


6 5 Si, [J/mol K]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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4

3 2 1 0

50

100

150 r, [nm]

Fig.8. Simulation of the partial molar entropies of the LG at phase transition points as functions of nano-particle radius ( r ): α-phase (blue) and β-phase (red). 34 ACS Paragon Plus Environment

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3.55

Eeq, [V]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3.45

3.35 0

0.2

0.4

0.6

0.8

1

x

Fig.9. Simulation of the equilibrium potential of the LiFePO4 electrode as a function of normalized concentration of Li ions ( x ) by Eq.30 using Eqs.27,29: 113 nm (red); 42 nm (green); 34 nm (blue); 10.36 nm (magenta).



2

G, [kJ mol-1]

1

0

-1

-2 0

0.2

0.4

0.6

0.8

1

x

a.

0.08

0.04 Ei, [eV]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 46 of 50

α+β

0

-0.04

-0.08 0

50

100

150 r, [nm]

b.

Fig.10. a. Simulation of the molar Gibbs free energy of the LiFePO4 electrode as a function of normalized concentration of Li ions ( x ) calculated by Eq.36: 113 nm (red); 42 nm (green); 34 nm (blue); 10.36 nm (magenta). b. Energy of individual Li ions inserted into LiFePO4 calculated by Eqs.29,35 (points) as a function of the nano-particle radius for α-phase (blue) and β-phase (red); the estimation line is also shown (dashed line); black rhomb is the estimated critical point. 36 ACS Paragon Plus Environment

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Li+ ions insertion process

LiFePO4

Simulation of the phase diagram

-phase

-phase

0

Li-ions

r, [nm]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


50

α+β 100

150 0

0.2

0.4

0.6

0.8

x


1


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Lattice-Gas Model for Energy Storage Materials: Phase Diagram and

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