Abstract
We consider the mathematical treatment of a system of nonlinear partial differential equations based on a model, proposed in 1972 by J. Newman, in which the coupling between the Lithium concentration, the phase potentials and temperature in the electrodes and the electrolyte of a Lithium battery cell is considered. After introducing some functional spaces well-adapted to our framework, we obtain some rigorous results showing the well-posedness of the system, first for some short time and then, by considering some hypothesis on the nonlinearities, globally in time. As far as we know, this is the first result in the literature proving existence in time of the full Newman model, which follows previous results by the third author in 2016 regarding a simplified case.
1 Introduction
Let us suppose we have a mathematical system of differential equations involving, after a homogenization process (or other averaging methods), two different scales, that we may call macro and micro, for simplicity. A macro-scale domain is given by
Let us considered, motivated by the modeling of charge transport in Lithium batteries (as we will show below) the following (relatively) abstract framework (which can be generalized to other cases).
Our system has five unknowns functions:
where
This system is coupled with the following system of micro-scale diffusion equation: For almost each point
In the equations written above,
Finally, we add the following initial value problem:
where the dependence of f on
The question that arises is the following. Under which conditions can we prove existence and/or uniqueness of a local (in time) solution
We study here a particular interesting case arising in the modeling of Lithium batteries. Lithium-ion batteries are currently used extensively for storing electricity in mobile devices, from phones to cars. Nevertheless, they are still many drawbacks for them, as their reduced charge capacity, the long time needed to charge them, thermal runaways, etc. Therefore, an intensive research is being done in order to improve these devices. A good mathematical model of the transport mechanisms within batteries is very important in that research in order to understand the physics involved in the processes and to allow quick numerical experiments. But this models need to be well understood from a mathematical point of view, so that conclusions extracted from them are more rigorous. The well-posedness of the mathematical models being used is, therefore, one of the key points to be considered. This is, indeed, the goal of this paper.
In [31] a full model for Lithium ion batteries was presented, based on the well-known J. Newman model (see [27]), and partial results for the well-posedness were given. In this paper, we intend to complete those well-posedness results and study the regularity of the solutions. Let us write the complete system as presented in [31] but using in the electrolyte the electric potential measured by a reference Lithium electrode (
A typical Lithium-ion battery cell has three regions: a porous negative electrode, a porous positive electrode and an electro-blocking separator. Furthermore, the cell contains an electrolyte, which is a concentrated solution containing charged species that move along the cell in response to an electrochemical potential gradient.
Let L be the length of a cell of a battery, let
A schematic representation of a cell is given in Figure 1.
Schematic representation of a battery. Dark gray disks represent negative electrode particles, light gray disksrepresent positive electrode particles, the dotted region represents the presence of electrolyte. On both sides of the cell the lined region represents the current collectors.
The unknowns in the system of equations that we study are:
the concentration of Lithium ions in the electrolyte
the concentration of Lithium ions in the electrodes
the electric potential
the electric potential measured by a reference Lithium electrode in the electrolyte,
the temperature
The system of equations is given by (1.5)–(1.9) below.
The transport of Lithium through the electrolyte can be modeled by the following macro-scale system of equations (conservation of Lithium in the electrolyte):
where
The transport of Lithium through the electrodes can be modeled, at almost every macroscopic point
where
The electric potential in the electrodes,
and the conservation of charge in the electrodes
where
Finally, we consider the temperature
where
In [31], instead of equations (1.5), (1.7) and (1.8) the author considers the equations
where
We simplify the domain notation by introducing the following sets:
In fact, we will consider
![Figure 2
Domains
J
δ
{J_{\delta}}
(spatial domain of definition of
c
e
,
φ
e
,
ϕ
s
{c_{\mathrm{e}},\varphi_{\mathrm{e}},\phi_{\mathrm{s}}}
) and
D
δ
{{{D_{\delta}}}}
(spatial domain of definition of
c
s
{c_{\mathrm{s}}}
). Notice that, since we are using a radial coordinate, every segment
{
x
}
×
[
0
,
R
s
(
x
)
]
{\{x\}\times[{0,R_{\mathrm{s}}({x})}]}
in
D
δ
{{{D_{\delta}}}}
represents a ball
{
x
}
×
B
R
s
(
x
)
{\{x\}\times B_{R_{\mathrm{s}}({x})}}
in
{
x
}
×
ℝ
3
{\{x\}\times\mathbb{R}^{3}}
.](/document/doi/10.1515/anona-2018-0041/asset/graphic/j_anona-2018-0041_fig_0002.png)
Domains
The analytical expression of
where U is the open circuit potential and η the surface overpotential of the corresponding electrode reaction.
There is no common agreement on the structural assumptions of U. Some papers use a fitting function either polynomial [36] or exponential [6], whereas other authors propose an improved version with logarithmic behavior close to the limit cases [32, 35]. We will start working in a general framework, which contains as a particular important example the Butler–Volmer flux (see [27, 31, 35]), in which the functions are taken as
where
The function U was later proposed in [32] as
where
The following particular choice of
where
We point out that T has been assumed to be uniform across the whole cell. Nonetheless, a more complete model (specially one dealing with temperature blow-up) with a heat diffusion equation could be used. The study of blow-up in this type of equations has been largely studied (see, for example, [2, 12]).
By using this model, the voltage at time t of the cell can be estimated (see [31]) as
The model, as written above, presents a number of difficulties: pseudo-two-dimensional (P2D) model, elliptic equations coupled with parabolic equations, discontinuities in space of some of the involved functions, the fact that
In order to state the results, we will introduce the following definitions, which we introduce as a mathematical tool:
Notice that
which is a system simpler than (1.7).
First we will state four general results for a large class of functions
Theorem 1 (Well-posedness with general flux and nature of the possible blow-up).
Let Assumptions 2.1–2.3 and 2.5 hold. Then, if
We remark that without Assumption 2.13 existence for potentials
In the particular case of
Theorem 2 (Well-posedness for the Butler–Volmer flux and nature of the possible blow-up).
Let us suppose that Assumptions 2.1, 2.2, 2.5, 2.19, 2.20, 2.21 and 2.24 hold.
Then, if
Furthermore,
In the last part of this paper (see Section 2.6), we give an ad hoc modification of the system for which we can state a global existence theorem by removing, in two steps, the impediments to global existence of solutions given by (1.27). Actually, since the blow-up conditions (1.27) do not correspond to the physical intuition, it is likely that the solutions do not, in fact, develop this kind of behavior.
In a first stage we will find a bound for the flux with respect to
Theorem 3 (Blow up behavior when
j
Li
is bounded with respect to
ϕ
s
-
φ
e
,
Li
).
Let Assumptions 2.1, 2.2, 2.5, 2.19, 2.20, 2.24 and 2.27 hold. Then, if
Finally, by assuming some additional conditions (see Assumptions 2.27 and 2.29), we will obtain a bound for the temperature T and prove what can be considered a first global existence result in the literature for this system:
2 Mathematical framework
2.1 Regularity assumptions of the nonlinear terms and initial data
Our most general formulation in this paper concerns the case of considering the following regularity conditions on the data:
Assumption 2.1.
Let us assume the data:
where
Notice that this implies that the lateral limits
Assumption 2.2.
We have
The key ingredient of our approach is to choose some suitable convex subsets of some appropriate functional spaces. The choice of spaces well adapted to the system is a delicate matter, and errors or loose approaches to this might result in incorrect results (for an explanation of this philosophy see, e.g., [9]). Let us define the following open convex sets where we expect to find the solutions:
where
the natural space in which we will look for the pair
Instead of narrowly focusing on (1.12)–(1.16), we shall state an assumption (sufficient to prove Theorem 1) satisfied by a broader family of functions:
Assumption 2.3.
For the flux function we assume
such that
for all
Remark 2.4.
In particular, it follows from Assumption 2.3 (in particular due to (2.2) and (2.4) applying the Mean Value Theorem) that there exists a positive continuous function
for all
Finally, on the temperature term
Assumption 2.5.
We have
2.2 Definition of weak solution
We introduce the natural space for radial solutions
with the norm
and the space
Remark 2.6.
Even though we will always present the variables as
Definition 2.7.
We define a weak solution of (1.5)–(1.9) as a quintuplet
such that
for all
for all
and
2.3 Definition of weak-mild solution
Dealing directly with the weak formulation is technically very difficult. On the other hand, solutions in the classical sense (having all the necessary derivatives) may not exist. There is an intermediate type of solutions, known as “mild solutions”. As a general rule (and in particular this applies to our problem), any classical solution is a mild solution and any mild solution is a weak solution.
Let us introduce this kind of solutions in the simplest case: the heat equation. Consider the problem
in a bounded, smooth domain
by considering the decomposition of
This kind of solution is known as a “mild solution”. As in [17], one can define the “Green operator” for problem (2.8) as the function
In our problem we will need to work with a suitable Green operator associated to each of the equations. Assuming Assumptions 2.2, we will define several Green operators.
For any
as the solution of the problem
For system (1.6) we will need to do some extra work due to the fact that the equation is only “pseudo-two-dimensional”. First we define the solution of problem (1.6) for every x fixed,
by solving the corresponding problem
The next step is to consider the dependence on x. Therefore, we construct the Green operator associated to problem (1.6) collecting all
given by
Finally, we consider the Green operator for system (1.9) as the function
defined as
This operator is well defined and of class
It is well known (see, e.g., [21]) that these operators are locally Lipschitz continuous and
Definition 2.8 (Weak-mild solution).
We define a “weak-mild solution of problem (1.5)–(1.9)” as a quintuplet
- (2.10)
Definition 2.9 (Piecewise weak-mild solution).
We define a “piecewise weak-mild solution” as a quintuplet
Remark 2.10.
It is well known that for problems of type (2.8), any piecewise weak-mild solution is a weak solution.
Definition 2.11.
Given a solution (in any of the previous senses)
We say that the extension is “proper” if
Notice that the contribution of
where
In this sense we can rewrite (2.10) in terms of the restriction
Proposition 2.12.
In Definitions 2.9 and 2.11, condition (2.10) is equivalent to the following property:
such that
Proof.
It is trivial that (2.10) implies (2.11). Let us consider a quintuplet
Define
Since
Furthermore, since
Therefore,
2.4 Assumptions and results regarding Theorem 1
It was first shown in [31] that the uniqueness of solutions
Assumption 2.13.
As in (1.24), we define
and assume that
This can be done because
Remark 2.14.
We recall that
Remark 2.15.
Another alternative to get the uniqueness of solution is to use the condition
The idea of the proof of Theorem 1 (which is done in Section 5) is the following. First we will show (see Proposition 2.16 and its proof in Section 3) that we can solve (1.7) and (1.8) to obtain
Proposition 2.16.
Let
Hence we have uniqueness up to a constant. In particular, there exists a unique solution
Due to this proposition we know that the following functions are well defined:
where
where
In Section 3 we will prove that:
Proposition 2.17.
Let Assumptions 2.2–2.3 hold. Then the operator
Remark 2.18.
Since we will allow for charge and discharge cycles, we allow for I to be piecewise continuous, and this is why we define the piecewise weak-mild solution (see Definition 2.9).
The proof of the local existence of solutions will be based on finding a unique fixed point, in
2.5 Assumptions and remarks regarding Theorem 2
In Theorem 1 one of the reasons of a finite existence time could be that
Assumption 2.19.
There exists a constant
Assumption 2.20.
We have
This is purely technical, but it seems reasonable since, in empirical cases in the literature, one typically has
Let us consider the following special nonlinear terms (which, for a broader generality, include some extra constants with respect to (1.12)–(1.16)), which we will take as assumption in Theorem 2.
Assumption 2.21.
We assume that
where
where
Remark 2.22.
It is a routine matter to check that this
Possible representation of U.
Remark 2.23.
Under Assumption 2.21 we can write
Substituting (2.18) into (2.19)–(2.20), we have
On the exponents we will consider the following assumption (see Remark 2.26):
Assumption 2.24.
For all
Remark 2.25.
Notice that conditions (2.23)–(2.26) can be also written as
Theorem 2 gives a sufficient condition so that, under the structural consideration of a potential U similar to the one considered by Ramos–Please [32], should there be a blow-up of the solution, this is not caused by the non-physical behavior
Considering the parameters used by some authors (see, e.g., [36])
(where
Considering (1.17), as in [32], we have
So Assumption 2.24 translates into
Remark 2.26.
It is well known that equations of the form
with
we see that, if
2.6 Assumptions and remarks regarding Theorems 3 and 4
Tackling an example of global existence of solutions for the model seems to be a Herculean mission. However, if we assume some nicer behavior of the nonlinear term U, at least far from the natural “working” conditions of the system, then we can establish a result of global existence of solutions.
2.6.1 Truncated potential behavior
In this subsection we strive to make some small modifications to the problem so that we can show that the charge potentials do not blow up in finite time. First let us assume that for large difference of potentials the flux is constant. We slightly modify (2.21)–(2.22) to the following new hypothesis:
Assumption 2.27.
Let
and H is a bounded smooth cut-off function of the exponential:
where
This modification allows us to prove Theorem 3 (see Section 7.1).
2.6.2 Truncated temperature behavior
Due to the delicate interconnectedness of the different terms in
Assumption 2.29.
Assume that
and consider that
This assumptions allow us to prove the result of global existence of solutions given in Theorem 4 (see Section 7.2).
Remark 2.30.
The case
3 Green operator for the semilinear elliptic system (1.7)–(1.8). Proof of Propositions 2.16 and 2.17
The idea of the proof of Proposition 2.16 is to follow the idea for “low overpotentials” in [31], but replacing Lax–Milgram’s theorem by the Brézis theory of “pseudo-monotone operators” (see [8]). Under some special assumptions, this result is sometimes referred to as Minty–Browder’s theorem [26, 11] (see, e.g., [34]). Uniqueness up to a constant was already proved in [31].
Let us define, for a given
which corresponds to
Notice that
We also define
which corresponds to
Remark 3.1.
Given
Proof of Proposition 2.16.
We rewrite (2.6)–(2.7) in terms of
Adding both equations, we obtain that (2.6)–(2.7) is equivalent to
Let us define, for
Notice that
Let us define the operator
Since
for all
Therefore, for all
where
Let the operator
Then
Remark 3.2.
Of course, if
Remark 3.3.
The main part of the proof above was to apply the monotonicity of
We have so far proved that the map
Proof of Proposition 2.17.
We will apply the Implicit Function Theorem for the Banach space-valued mapping
The choice of functional spaces will be
with
We will then check that
due to the definition of
We consider the maps
Our definition of weak solution is precisely
The function
On the other hand,
Since
where, for
Let
understood as a bilinear form
is continuous and coercive in
Since
is
is
Due to the uniqueness (up to a constant) result we proved in Proposition 2.16,
is the map
Therefore, as an immediate consequence of Proposition 2.17, we have the following lemma in terms of the space of piecewise continuous functions
Lemma 3.4.
Let
4 Regularity of the Green operators
G
c
e
,
t
and
G
c
s
,
B
,
t
The regularity of
This operator has a nice representation formula
where
A number of properties can easily be derived from this expression. For instance, the continuous dependence with respect to the data:
where ρ is an increasing, continuous function such that
The term
and
Therefore, due to the linearity of the equation, for
This solves the problem of continuity of
is Lipschitz continuous. Furthermore, it is easy to check that
We also have the following time estimate, for
By defining the vectorial Green operator for the evolutionary part
due to the previous results, we have
where ρ is a continuous function such that
5 Proof of Theorem 1
First, let us assume that I is continuous. Let us define the function
It is clear that
by
Since
is globally Lipschitz continuous. Due to the continuity of
Therefore, due to (4.1), we obtain that, for
is a contracting map. Then we can apply the Banach Fixed Point Theorem to find a unique solution of problem (5.1). If I is
By applying classical results, there exists a maximal existence time
6 Proof of Theorem 2
There are a number of papers studying the semilinear equation
Let us assume that the solution
where
Notice that, since
so that we can construct the supersolution
Using the conditions on the exponents given by Assumption 2.24, we deduce that
We also have that
where
then
We construct the subsolution
so that, by the comparison principle, we have that
since
and define
then, by defining
we arrive at the definition of the searched function
7 Proof of a global in time existence result
7.1 On the truncated potential case. Proof of Theorem 3
Assume that (1.28) does not hold as
and repeat the argument. We get good bounds for
7.2 On the truncated temperature case. Proof of Theorem 4
It is immediate to establish global sub- and supersolutions for T, which ensure (1.28) does not happen, and hence we get the global existence of solutions.
8 Final remarks
8.1 Mass conservation and derived properties
We begin by analyzing the compatibility conditions for the elliptic problems with Neumann boundary conditions. The compatibility conditions for the existence of
Hence, we deduce that
This is also the compatibility condition for system (1.8).
Let us take a look at the mass balance condition. From system (1.6) we observe that
Thus, as expected, the mathematical model satisfies that the total concentration in the electrolyte is constant. Since
In fact, if one considers the Lithium concentration in the anode and cathode, for the pseudo-two-dimensional (P2D) model considered here, one gets
and, analogously,
If
then the P2D model satisfies that any lose of Lithium in its anode is instantaneously received by its cathode and viceversa. Otherwise, there would appear to be a net variation in the total amount of Lithium in the solid phase.
The state of the charge at time t of the cell,
8.2 Model validation and identification of parameters
In order to validate the model and identify suitable parameters, it is often necessary to compare the solutions of the model with real-world data and minimize some error functional using suitable parameter identification techniques and minimization algorithms (see, e.g., [19, 22, 23]). Since internal properties of the battery, such as the spatial distribution of Lithium ions, cannot be measured during operation, it is standard to measure only the temperature T (on the outside of the battery, but it is a safe assumption that the temperature is spatially homogeneous) and the output voltage V (see, for example, [5, 37]), which can be estimated by using (1.23).
9 Conclusions
In this work, we have shown that the system of equations (1.5)–(1.9) has a unique solution (under some mild conditions). In the most general setting, only local in time existence result can be shown, and the nature of the possible blow-up is characterized. The most difficult part of the study of an eventual possible blow-up is the structure of the open circuit potential U. By considering the model of U given by (2.18), which was proposed in [32], we can rule out some non-physical behaviors as the possible causes of blow-up, by making some assumptions on the coefficients. Finally, for the sake of mathematical completeness, we provide some extra conditions that allow for global uniqueness in time and make some comments regarding the conservation of Lithium in the model and the model validation.
Funding source: Ministerio de Educación Cultura y Deporte
Award Identifier / Grant number: FPU
Funding statement: The research of the authors was partially supported by the Spanish Ministry of Economy, Industry and Competitiveness under projects MTM2014-57113-P and MTM2015-64865-P. The research of D. Gómez-Castro was supported by a FPU Grant from the Ministerio de Educación, Cultura y Deporte (Spain). All the authors are members of the Research Group MOMAT (Ref. 910480) of the UCM.
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