Habib Ammari
email: habib.ammari@ens.fr

Pol Grasland-Mongrain
email: pol.grasland-mongrain@ens-cachan.fr

Pierre Millien
email: pierre.millien@ens.fr

Laurent Seppecher
email: laurent.seppecher@ens.fr

Jin-Keun Seo
email: seoj@yonsei.ac.kr

A mathematical and numerical framework for ultrasonically-induced Lorentz force electrical impedance tomography *

Keywords: Mathematics Subject Classification (MSC2000): 35R30, 35B30 electrical impedance tomography, hybrid imaging, ultrasonically-induced Lorentz force, optimal control, orthogonal field method, viscosity-type regularization

We provide a mathematical analysis and a numerical framework for Lorentz force electrical conductivity imaging. Ultrasonic vibration of a tissue in the presence of a static magnetic field induces an electrical current by the Lorentz force. This current can be detected by electrodes placed around the tissue; it is proportional to the velocity of the ultrasonic pulse, but depends nonlinearly on the conductivity distribution. The imaging problem is to reconstruct the conductivity distribution from measurements of the induced current. To solve this nonlinear inverse problem, we first make use of a virtual potential to relate explicitly the current measurements to the conductivity distribution and the velocity of the ultrasonic pulse. Then, by applying a Wiener filter to the measured data, we reduce the problem to imaging the conductivity from an internal electric current density. We first introduce an optimal control method for solving such a problem. A new direct reconstruction scheme involving a partial differential equation is then proposed based on viscositytype regularization to a transport equation satisfied by the current density field. We prove that solving such an equation yields the true conductivity distribution as the regularization parameter approaches zero. We also test both schemes numerically in the presence of measurement noise, quantify their stability and resolution, and compare their performance.

Introduction

Ultrasonic imaging is currently used in a wide range of medical diagnostic applications. Its high spatial resolution, combined with a real-time imaging capability, lack of side effects, and relatively low cost make it an attractive technique. However, it can be difficult to differentiate soft tissues because acoustic impedance varies by less than 10% among muscle, fat, and blood [START_REF] Goss | Comprehensive compilation of empirical ultrasonic properties of mammalian tissues[END_REF]. In contrast, electrical conductivity varies widely among soft tissue types and pathological states [START_REF] Foster | Dielectric properties of tissues and biological materials: a critical review[END_REF][START_REF] Morimoto | A study of the electrical bio-impedance of tumors[END_REF] and its measurement can provide information about the physiological and pathological condition of tissue [START_REF] Ammari | Spectroscopic imaging of a dilute cell suspension[END_REF]. Several techniques have been developed to map electrical conductivity. The most well known is electrical impedance tomography, in which electrodes are placed around the organ of interest, a voltage difference is applied, and the conductivity distribution can be reconstructed from the measurement of the induced current at the electrodes [START_REF] Ammari | An Introduction to Mathematics of Emerging Biomedical Imaging[END_REF][START_REF] Ammari | T-Scan electrical impedance imaging system for anomaly detection[END_REF][START_REF] Cheney | Electrical impedance tomography[END_REF]. This technique is harmless to the patient if low currents are used. However, the ill-posed character of the inverse problem results in lower spatial resolution than that achieved by ultrasound imaging, and any speckle information is lost.

The Lorentz force plays a key role in acousto-magnetic tomographic techniques [START_REF] Roth | The role of magnetic forces in biology and medicine[END_REF]. Several approaches have been developed with the aim of providing electrical impedance information at a spatial resolution on the scale of ultrasound wavelengths [START_REF] Ammari | Mathematical models and reconstruction methods in magneto-acoustic imaging[END_REF][START_REF] Grasland-Mongrain | Lorentz force electrical impedance tomography[END_REF][START_REF] Li | Imaging electrical impedance from acoustic measurements by means of magnetoacoustic tomography with magnetic induction (MAT-MI)[END_REF][START_REF] Mariappan | Magnetoacoustic tomography with magnetic induction: Bioimepedance reconstruction through vector source imaging[END_REF][START_REF] Montalibet | Etude du couplage acousto-magnétique: détection des gradients de conductivité électrique en vue de la caractérisation tissulaire[END_REF][START_REF] Roth | The role of magnetic forces in biology and medicine[END_REF][START_REF] Roth | Ultrasonically-induced Lorentz force tomography[END_REF][START_REF] Wen | Hall effect imaging[END_REF]. These include Hall effect imaging, magneto-acoustic current imaging, magneto-acoustic tomography with magnetic induction, and ultrasonically-induced Lorentz force imaging. Acousto-magnetic tomographic techniques have the potential to detect small conductivity inhomogeneities, enabling them to diagnose pathologies such as cancer by detecting tumorous tissues when other conductivity imaging techniques fail to do so.

In ultrasonically-induced Lorentz force method (experimental apparatus presented in Figure 1) an ultrasound pulse propagates through the medium to be imaged in the presence of a static magnetic field. The ultrasonic wave induces Lorentz' force on the ions in the medium, causing the negatively and positively charged ions to separate. This separation of charges acts as a source of electrical current and potential. Measurements of the induced current give information on the conductivity in the medium. A 1 Tesla magnetic field and a 1 MPa ultrasonic pulse induce current at the nanoampere scale. Stronger magnetic fields and ultrasonic beams can be used to enhance the signal-to-noise ratio [START_REF] Grasland-Mongrain | Lorentz force electrical impedance tomography[END_REF].

This paper provides a rigorous mathematical and numerical framework for ultrasonically-induced Lorentz force electrical impedance tomography. We develop two efficient methods for reconstructing the conductivity in the medium from the induced electrical current. As far as we know, this is the first mathematical and numerical modeling of the experiment conducted in [START_REF] Grasland-Mongrain | Lorentz force electrical impedance tomography[END_REF] to illustrate the feasibility of ultrasonically-induced Lorentz force electrical impedance tomography. Earlier attempts to model mathematically this technique were made in [START_REF] Ammari | Mathematical models and reconstruction methods in magneto-acoustic imaging[END_REF][START_REF] Kunyansky | A mathematical model and inversion procedure for magneto-acoustoelectric tomography[END_REF].

The paper is organized as follows. We start by describing the ionic model of conductivity. From this model we derive the current density induced by an ultrasonic pulse in the presence of a static magnetic field. We then find an expression of the measured current. The inverse problem is to image the conductivity distribution from such measurements corresponding to different pulse sources and directions. A virtual potential used with simple integrations by parts can relate the measured current to the conductivity distri-bution and the velocity of the ultrasonic pulse. A Wiener deconvolution filter can then reduce the problem to imaging the conductivity from the internal electric current density. The internal electric current density corresponds to that which would be induced by a constant voltage difference between one electrode and another with zero potential. We introduce two reconstruction schemes for solving the imaging problem from the internal data. The first is an optimal control method; we also propose an alternative to this scheme via the use of a transport equation satisfied by the internal current density. The second algorithm is direct and can be viewed as a PDE-based reconstruction scheme. We prove that solving such a PDE yields to the true conductivity distribution as the regularization parameter tends to zero. In doing so, we prove the existence of the characteristic lines for the transport equation under some conditions on the conductivity distribution. We finally test numerically the two proposed schemes in the presence of measurement noise, and also quantify their stability and resolution.

The ultrasonically-induced Lorentz force electrical impedance tomography investigated here can be viewed as a new hybrid technique for conductivity imaging. It has been experimentally tested [START_REF] Grasland-Mongrain | Lorentz force electrical impedance tomography[END_REF], and was reported to produce images of quality comparable to those of ultrasound images taken under similar conditions. Other emerging hybrid techniques for conductivity imaging have also been reported [START_REF] Ammari | Electrical impedance tomography by elastic deformation[END_REF][START_REF] Ammari | Mathematical models and reconstruction methods in magneto-acoustic imaging[END_REF][START_REF] Ammari | Resolution and stability analysis in acoustoelectric imaging[END_REF][START_REF] Capdeboscq | Imaging by modification: numerical reconstruction of local conductivities from corresponding power density measurements[END_REF][START_REF] Gebauer | Impedance-acoustic tomography[END_REF][START_REF] Seo | Magnetic resonance electrical impedance tomography (MREIT)[END_REF][START_REF] Widlak | Hybrid tomography for conductivity imaging[END_REF].

Electric measurements from acousto-magnetic coupling

Let a physical object to be imaged occupy a three-dimensional domain Ω with a smooth boundary ∂Ω. Assume that this body is placed in a constant magnetic field B in the direction e 3 where {e 1 , e 2 , e 3 } denotes the standard orthonormal basis of R 3 . We are interested in recovering the electrical conductivity of this body σ ∈ L ∞ (Ω) with the known lower and upper bounds:

0 < σ ≤ σ ≤ σ < ∞.
An acoustic transducer sends a short acoustic pulse from y ∈ R 3 in the direction ξ ∈ S 2 , with S 2 being the unit sphere, such that ξ • e 3 = 0. This pulse generates the velocity field v(x, t)ξ with v(x, t) taking the following form:

v(x, t) = w z -ct A z, |r| , (1)
where

z = (x -y) • ξ and r = x -y -zξ ∈ Υ ξ := {ζ ∈ R 3 : ζ • ξ = 0}.
Here,

w ∈ C ∞ c R , supported in ] -η, 0[, is the ultrasonic pulse profile; A ∈ C ∞ R × R + , supported in R + × [0, R],
is the cylindrical profile distribution of the wave corresponding to the focus of the acoustic transducer; and R is the maximal radius of the acoustic beam.

The ionic model of conductivity

We describe here the electrical behavior of the medium as an electrolytic tissue composed of ions capable of motion in an aqueous tissue. We consider k types of ions in the medium with charges of q i , i ∈ {1, . . . , k}. The corresponding volumetric density n i is assumed to be constant. Neutrality in the medium is described as i q i n i = 0.

(

) 2
The Kohlrausch law defines the conductivity of such a medium as a linear combination of the ionic concentrations

σ = e + i µ i q i n i , (3)
where e + is the elementary charge, and the coefficients µ i denote the ionic mobility of each ion i. See, for example, [START_REF] Montalibet | Etude du couplage acousto-magnétique: détection des gradients de conductivité électrique en vue de la caractérisation tissulaire[END_REF][START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF].

Ion deviation by Lorentz force

We embed the medium in a constant magnetic field B with direction e 3 , and perturb it mechanically using the short, focused, ultrasonic pulses v defined in [START_REF] Ammari | An Introduction to Mathematics of Emerging Biomedical Imaging[END_REF]. The motion of the charged particle i inside the medium is deviated by the Lorentz force

F i = q i vξ × B. (4)
This force accelerates the ion in the orthogonal direction τ = ξ × e 3 . Then, almost immediately, the ion reaches a constant speed given by v τ,i = µ i |B|v at the first order. See [START_REF] Montalibet | Etude du couplage acousto-magnétique: détection des gradients de conductivité électrique en vue de la caractérisation tissulaire[END_REF][START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF] for more details. Finally, the ion i has a total velocity

v i = vξ + µ i |B|vτ.
The current density generated by the displacement of charges can be described as follows:

j S = i n i q i v i = i n i q i vξ + i n i µ i q i |B|vτ.
Using the neutrality condition (2) and the definition of σ in (3), we get the following simple formula for j S :

j S = 1 e + |B|σvτ, (5)
which is in accordance with the formula used in [START_REF] Ammari | Mathematical models and reconstruction methods in magneto-acoustic imaging[END_REF]. This electrolytic description of the tissue characterizes the interaction between the ultrasonic pulse and the magnetic field through a small deviation of the charged particles embedded in the tissue. This deviation generates a current density j S orthogonal to ξ and to B, locally supported inside the domain. At a fixed time t, j S is supported in the support of x → v(x, t). This current is proportional to σ, and is the source of the current that we measure on the electrodes placed at ∂Ω. In the next section, a formal link is substantiated between j S and the measured current I.

Internal electrical potential

Because the characteristic time of the acoustic propagation is very long compared with the electromagnetic wave propagation characteristic time, we can adopt the electrostatic frame. Consequently, the total current j in Ω at a fixed time t can be formulated as

j = j S + σ∇u, ( 6
)
where u is the electrical potential. It satisfies

∇ • (j S + σ∇u) = ∇ • j = 0. (7)
Figure 2 shows the configuration under consideration. Let Γ 1 and Γ 2 be portions of the boundary ∂Ω where two planner electrodes are placed.

Denote Γ 0 = ∂Ω \ (Γ 1 ∪ Γ 2 ).
As we measure the current between the two electrodes Γ 1 and Γ 2 , the electrical potential is the same on both electrodes, and can be fixed to zero without loss of generality. Further, it is assumed that no current can leave from Γ 0 . The potential u can then be defined as the unique solution in H 1 (Ω) of the elliptic system

     -∇ • (σ∇u) = ∇ • j S in Ω, u = 0 on Γ 1 ∪ Γ 2 , ∂ ν u = 0 on Γ 0 . (8)
Throughout this paper ∂ ν denotes the normal derivative. Note that the source term j S depends on the time t > 0, the position of the acoustic transducer y ∈ R 3 , and the direction ξ ∈ S 2 . The electrical potential u also depends on these variables. The measurable intensity I is the current flow through the electrodes. Integrating (8) by parts gives

Γ 1 σ∂ ν u + Γ 2 σ∂ ν u = 0,
which is the expression of current flow conservation. We define the intensity I by

I = Γ 2 σ∂ ν u. (9)

Virtual potential

In order to link I to σ, we introduce a virtual potential U ∈ H 1 (Ω) defined as the unique solution of

         -∇ • (σ∇U ) = 0 in Ω, U = 0 on Γ 1 , U = 1 on Γ 2 , ∂ ν U = 0 on Γ 0 . ( 10
)
Then we multiply (8) by U and integrate by parts. Assuming that the support of v does not intersect the electrodes Γ 1 and Γ 2 , we obtain

- Ω σ∇u • ∇U + Γ 2 σ∂ ν u = Ω j S • ∇U.
From the property of U in [START_REF] Gebauer | Impedance-acoustic tomography[END_REF] and the definition of I in (9), the above identity becomes

I = Ω j S • ∇U.
The above identity links the measured intensity I to an internal information of σ using the expression of j S in (5):

I = |B| e + Ω v(x, t)σ(x)∇U (x)dx • τ.
According to (1), v depends on y, ξ, and t, so does I. We define the measurement function as

M y,ξ (z) = Ω v(x, z/c)σ(x)∇U (x)dx • τ (ξ) (11)
for any y ∈ R 3 , ξ ∈ S 2 and z > 0. We assume the knowledge of this function in a certain subset of R 3 × S 2 × R + denoted by Y × S×]0, z max [. We will discuss later the assumptions we have to impose on this subset in order to make the reconstruction accurate and stable.

Construction of the virtual current

For simplicity, let us restrict ourselves to the two dimensional case where both the conductivity σ and the virtual potential U do not change in e 3 -direction. For convenience, the same notations will be used as in the three dimensional case.

In order to obtain the information of σ contained in M y,ξ , we need to separate the contribution of the displacement term v from this measurement function. Using the cylindrical symmetry of this integration we write for any z ∈]0, z max [,

M y,ξ (z) = R Υ ξ w(z -z )(σ∇U )(y + z ξ + r)A(z , |r|)drdz • τ (ξ), = R w(z -z ) Υ ξ (σ∇U )(y + z ξ + r)A(z , |r|)drdz • τ (ξ), = (W Φ y,ξ ) (z) • τ (ξ), (12)
where W (z) = w(-z), denotes the convolution product, and

Φ y,ξ (z) = Υ ξ σ(y + zξ + r)A(z, |r|)∇U (y + zξ + r)dr.
As will be shown in section 6, through a one dimensional deconvolution problem that can be stably solved using, for instance, a Wiener-type filtering method, we get access to the function Φ y,ξ • τ (ξ). Now the question is about the reconstruction of σ from Φ y,ξ • τ (ξ). We can notice that Φ y,ξ is a weighted Radon transform applied to the virtual current field σ∇U . The weight A(z, |r|) is critical for the choice of the method that we can use. Closer this weight is to a Dirac mass function, better is the stability of the reconstruction. In this case, if the field σ∇U does not have too large variations, we can recover a first-order approximation; as discussed in the rest of this section.

In order to make the reconstruction accurate and stable, we make two assumptions on the set of parameters Y × D×]0, z max [. For any x ∈ Ω, we define

S x = ξ ∈ S : ξ = x -y |x -y| for some y ∈ Y .
The first assumption is

(H1) ∀x ∈ Ω, ∃ ξ 1 , ξ 2 ∈ S x s.t. |ξ 1 × ξ 2 | = 0,
and the second one reads

(H2) ∀x ∈ Ω, ∀ξ ∈ S x , ∃ unique y ∈ Y s.t. ξ = x -y |x -y| .
From the assumption (H2), we can define a distance map |x -y| as a function of x and ξ. We will denote d Y (x, ξ) = |x -y|. By a change of variables, we rename our data function Σ as

ψ(x, ξ) = Φ y,ξ d Y (x, ξ) • τ (ξ) = Υ ξ (σ∇U )(x + r)A d Y (x, ξ), |r| dr • τ (ξ). ( 13
)
Now if we denote by

γ(x, ξ) = Υ ξ A d Y (x, ξ), |r| dr τ (ξ), (14)
then we expect that ψ(x, ξ) ≈ (σ∇U )(x) • γ(x, ξ), provided the supp(A) is small enough and σ∇U does not vary too much. The following lemma makes this statement precise.

Lemma 3.1 Consider a fixed direction ξ ∈ S and consider the domain covered by the pulses of direction ξ defined by Ω ξ = {x ∈ Ω : ξ ∈ S x }. Suppose that the virtual current σ∇U has bounded variations, then

ψ(•, ξ) -σ∇U • γ(•, ξ) L 1 (Ω ξ ) ≤ cR σ∇U T V (Ω) 2 ,
where R is the maximum radius of the cylindrical support of the envelope A and c > 0 depends on the shape of A. Here, T V (Ω) 2 denotes the total variation semi-norm.

Proof. For a.e. x ∈ Ω ξ , we have

|ψ(x, ξ) -(σ∇U )(x) • γ(x, ξ)| ≤ Υ ξ |(σ∇U )(x + r) -(σ∇U )(x)| A d Y (x, ξ), |r| dr,
and so

ψ(•, ξ) -σ∇U • γ(•, ξ) L 1 (Ω ξ ) ≤ Υ ξ Ω ξ |(σ∇U )(x + r) -(σ∇U )(x)| A d Y (x, ξ), |r| dxdr ≤ σ∇U T V (Ω) 2 Υ ξ |r| sup 0<z<zmax A(z, |r|)dr ≤ 2πR σ∇U T V (Ω) 2 R+ sup 0<z<zmax A(z, ρ)dρ.
Note that in the most interesting cases, σ∇U has bounded variations. For example, if σ has a piecewise W 1,∞ smoothness on smooth inclusions, then σ∇U has bounded variations. This also holds true for σ in some subclasses of functions of bounded variations. In the following, we make the assumption, as in Lemma 3.1, that σ∇U has bounded variations.

In conclusion, our data approximates the quantity (σ∇U )(x) • γ(x, ξ) for any x ∈ Ω, ξ ∈ S x where the vector γ(x, ξ) is supposed to be known. To get the current (σ∇U )(x), we simply consider data from two linearly independent directions. Using assumption (H1), for a fixed x ∈ Ω, there exist ξ 1 , ξ 2 ∈ S x such that det(ξ 1 , ξ 2 ) = 0. We construct the 2 × 2 invertible matrix

Γ(x, ξ 1 , ξ 2 ) = γ(x, ξ 1 ) ⊥ γ(x, ξ 2 ) ⊥ ,
and the data column vector

Ψ(x, ξ 1 , ξ 2 ) = ψ(x, ξ 1 ) ψ(x, ξ 2 ) .
We approximate the current σ∇U (x) by the vector field

V (x, ξ 1 , ξ 2 ) = Γ(x, ξ 1 , ξ 2 ) -1 Ψ(x, ξ 1 , ξ 2 ).
Indeed, for any open set Ω ⊂ Ω ξ 1 ∩ Ω ξ 2 , the following estimate holds:

V (•, ξ 1 , ξ 2 ) -σ∇U L 1 ( Ω) 2 ≤ sup x∈ Ω Γ(x, ξ 1 , ξ 2 ) -1 L(R 2 ) 2 i=1 ψ(•, ξ i ) -σ∇U • γ(•, ξ i ) L 1 (Ω ξ i ) 1/2 ≤ cR σ∇U T V (Ω) 2 .
It is worth mentioning that if more directions are available, then we can use them to enhance the stability of the reconstruction. The linear system becomes over-determined and we can get the optimal approximation by using a least-squares method.

Recovering the conductivity by optimal control

In this section we assume that, according to the previous one, we are in the situation where we know a good approximation of the virtual current D := σ∇U in the sense of L 1 (Ω) 2 . The objective here is to provide efficient methods for separating σ from D.

For a < b, let us denote by L ∞ a,b (Ω) := {f ∈ L ∞ (Ω) : a < f < b} and define the operator F : L ∞ σ,σ (Ω) -→ H 1 (Ω) by

F[σ] = U :          ∇ • (σ∇U ) = 0 in Ω, U = 0 on Γ 1 , U = 1 on Γ 2 , ∂ ν U = 0 on Γ 0 . ( 15
)
The following lemma holds.

Lemma 4.1 The operator F is Fréchet differentiable and for any σ

∈ L ∞ σ,σ (Ω) and h ∈ L ∞ (Ω) such that σ + h ∈ L ∞ σ,σ (Ω) we have dF[σ](h) = v :      ∇ • (σ∇v) = -∇ • (h∇F[σ]) in Ω, v = 0 on Γ 1 ∪ Γ 2 , ∂ ν v = 0 on Γ 0 . (16)
Proof. Let us denote by

w = F[σ + h] -F[σ] -v.
This function is in H 1 (Ω) and satisfies the equation

∇ • (σ∇w) = -∇ • (h∇(F[σ + h] -F[σ]))
with the same boundary conditions as v. We have the elliptic global control:

∇w L 2 (Ω) ≤ 1 σ h L ∞ (Ω) ∇(F[σ + h] -F[σ]) L 2 (Ω) . Since ∇ • (σ∇(F[σ + h] -F[σ])) = -∇ • (h∇F[σ + h]),
we can also control

F[σ + h] -F[σ] with ∇(F[σ + h] -F[σ]) L 2 (Ω) ≤ 1 √ σ h L ∞ (Ω) ∇F[σ + h] L 2 (Ω) .
Then, there is a positive constant C depending only on Ω such that

∇F[σ + h] L 2 (Ω) ≤ C σ σ .
Finally, we obtain

∇w L 2 (Ω) ≤ C √ σ σ 2 h 2 L ∞ (Ω) .
We look for the minimizer of the functional

J[σ] = 1 2 Ω |σ∇F[σ] -D| 2 . ( 17
)
In order to do so, we compute its gradient. The following lemma holds.

Lemma 4.2 For any σ ∈ L ∞ σ,σ (Ω), dJ[σ] = (σ∇F[σ] -D -∇p) • ∇F[σ],
where p is defined as the solution to the adjoint problem:

     ∇ • (σ∇p) = ∇ • (σ 2 ∇F[σ] -σD) in Ω, p = 0 on Γ 1 ∪ Γ 2 , ∂ ν p = 0 on Γ 0 . (18)
Proof. As F is Fréchet differentiable, so is J.

For σ ∈ L ∞ σ,σ (Ω) and h ∈ L ∞ (Ω) such that σ + h ∈ L ∞ σ,σ (Ω), we have dJ[σ](h) = Ω (σ∇F[σ] -D) • (h∇F[σ] + σ∇dF[σ](h)).

Now, multiplying (18) by dF[σ](h), we get

Ω σ∇p • ∇dF[σ](h) = Ω (σ 2 ∇F[σ] -σD) • ∇dF[σ](h).
On the other hand, multiplying ( 16) by p we arrive at

Ω σ∇p • ∇dF[σ](h) = - Ω h∇F[σ] • ∇p,
and therefore,

dJ[σ](h) = Ω h(σ∇F[σ] -D -∇p) • ∇F[σ].
Lemma 4.2 allows us to implement a numerical gradient descent method in order to find σ. A regularization term can also be added to J[σ] in order to avoid instability. As we are seeking discontinuous σ with smooth variations out of the discontinuity set, a good choice would be the minimization of the regularized functional:

J ε [σ] = 1 2 Ω |σ∇F[σ] -D| 2 + ε||σ|| T V (Ω) , (19)
where ε > 0 is the regularization parameter.

The orthogonal field method

In this section, we present an alternative direct method to optimal control for reconstructing the conductivity σ from the internal data σ∇U . It is based on solving a transport equation. The following approach may be extended to the three dimensional case. However, several proofs would need to be revisited. Given a vector field D = σ∇U which is parallel to ∇U everywhere, we may construct the vectorial field F = (D 2 , -D 1 ) which is everywhere orthogonal to D. The flow of F may define the level sets of U . Assuming that the variations of the conductivity σ are far enough from Γ 0 , we can assume that U (x) = x 2 on this boundary part. Then U is a solution of the following transport equation:

F • ∇u = 0 in Ω, u = x 2 on ∂Ω. (20)
In the case where ( 20) is well posed and can be solved, we can reconstruct the virtual potential U . The conductivity σ is deduced from U and D by the following identity

σ = D • ∇U |D| 2 . ( 21
)
Despite to its very simple form, this first-order equation is really tricky. Existence and uniqueness are both difficult challenges in the general case. Our main difficulty here is due to the fact that F is discontinuous. As the function U that we are looking for is a natural solution of this equation, we are only concerned here with the uniqueness of a solution to (20).

Uniqueness result for the transport equation

The uniqueness of a solution to [START_REF] Montalibet | Etude du couplage acousto-magnétique: détection des gradients de conductivité électrique en vue de la caractérisation tissulaire[END_REF] is directly linked to the existence of outgoing characteristic lines defined by the dynamic system:

X (t) = F (X(t)), t ≥ 0, X(0) = x, x ∈ Ω, (22)
which usually needs the continuity of F . As σ is in general not continuous, F is not continuous, which makes the classical existence results useless. Nevertheless, under some assumptions on σ, we can insure the existence of the characteristic lines.

Definition 5.1 For any k ∈ N, α ∈]0, 1[, for any curve C of class C 1,α such that Ω \ C is a union of connected domains Ω i , i = 1, 2, • • • n, we define C k,α C Ω to be the class of functions f : Ω -→ R satisfying f | Ω i ∈ C k,α Ω i ∀i = 1, • • • n. Definition 5.2 A conductivity σ is said to be admissible if there exists a constant α ∈]0, 1[ and a curve C of class C 1,α such that σ ∈ C 0,α C Ω ∩ L ∞ σ,σ (Ω) and inf Ω\C σ∇F[σ] • e 2 > 0.
If σ is admissible and belongs to C 0,α C Ω , then the solution U of (10) belongs to C 1,α C Ω and the field F = (σ∇U ) ⊥ satisfies

F ∈ C 0,α C Ω and inf Ω\C F • e 1 > 0.
Moreover, as F is orthogonal to σ∇U , we can describe the jump of F at the curve C. Defining the normal and tangential unit vectors ν and τ and also the local sides (+) and (-) with respect to ν, we can write F on both sides as

F + = σ + ∂ ν U + τ + σ + ∂ τ U + ν, F -= σ -∂ ν U -τ + σ -∂ τ U -ν with the transmission conditions, σ + ∂ ν U + = σ -∂ ν U -and ∂ τ U + = ∂ τ U -. Finally, we characterize the discontinuity of F by [F ] = [σ]∂ τ U ν,
where [ ] denotes the jump across C.

With all of these properties for the field F , we can prove the existence of the characteristic lines for [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF].

Theorem 5.1 (Local existence of characteristics) Assume that F ∈ C 0,α C Ω with C of class C 1,α for α ∈]0, 1[. Assume that the discontinuity of F on C satisfies F + = f τ + σ + gν, F -= f τ + σ -gν
with f, g, σ + , σ -∈ C 0,α (C) where σ + , σ -are positive and g is locally signed. Then, for any x 0 ∈ Ω, there exists T > 0 and X ∈ C 1 [0, T [, Ω such that t → F (X(t)) is measurable and

X(t) = x 0 + t 0 F (X(s))ds, ∀t ∈ [0, T [. Proof. If x 0 /
∈ C, then F is continuous in a neighborhood of x 0 and the Cauchy-Peano theorem can be applied.

If x 0 ∈ C, then we choose a disk B ⊂ Ω centered at x 0 . The oriented line C separates B in two simply connected open domains called B + and B -. For ease of explanation, we may assume that C ∩ B is straight line (since we can flatten the curve using a proper C 0,α -diffeomorphism).

Assume that g(x 0 ) > 0. Up to rescaling B, we can assume that g(x) > 0 for all x ∈ C ∩ B. We extend F | B + to a continuous field F ∈ C 0 (B) by even reflection. The Cauchy-Peano theorem insures the existence of T > 0 and X ∈ C 1 [0, T [, Ω such that X(0) = x 0 and X (t) = F (X(t)) for all t ∈ [0, T [. As g(x 0 ) > 0, we have X (0) • ν(x 0 ) > 0 and X(t) ∈ B + in a neighborhood of 0. Thus, for a small enough t, X (t) = F (X(t)). If g(x 0 ) < 0, then we apply the same argument by interchanging B -and B + .

Suppose now that g(x 0 ) = 0. The field F is now tangent to the discontinuity line. If f (x 0 ) = 0, then X(t) = x 0 is a solution. We assume here that f (x 0 ) > 0. As g is assumed to be locally signed, we can suppose that g ≥ 0 in a small sub-curve of C satisfying (x -x 0 ) • τ (x 0 ) > 0. Again, we extend F | B + to a continuous field F ∈ C 0 (B) by even reflection and use the Cauchy-Peano theorem to show that there exists T > 0 and X ∈ C 1 [0, T [, Ω such that X(0) = x 0 and X (t) = F (X(t)) for all t ∈ [0, T [. In order to complete the proof, we should show that X(t) belongs to B + for t small enough. If not, there exists a sequence t n 0 such that X(t n ) ∈ B -. By the mean value theorem, there exists tn ∈ (0, t n ) such that F (X( tn )) • ν(x 0 ) = X ( tn ) • ν(x 0 ) < 0. Thus, X(t) belongs to B + and X (t) = F (X(t)) for t small enough.

Note that the local monotony of g is satisfied in many cases. For instance if C is analytic and σ is piecewise constant, then ∇U is analytic on C and hence, g is locally signed.

It is worth mentioning that existence of a solution for the Cauchy problem [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF] has been proved in [START_REF] Bressan | On discontinuous differential equations[END_REF] provided that F •ν > 0 on C. Here, we have made a weaker assumption. In fact, we only need that F • ν is locally signed.

inf Ω\C F • e 1 ≥ c,
where c is a positive constant. Then for any x 0 ∈ Ω there exists 0 < T < T max where

T max = 1 c diam(Ω) and X ∈ C 0 [0, T [, Ω satisfying
X(t) = x 0 + t 0 F (X(s))ds, ∀t ∈ [0, T [, lim t→T X(t) ∈ ∂Ω.
This result means that from any point x 0 ∈ Ω, the characteristic line reaches ∂Ω in a finite time.

Proof. Let x 0 ∈ Ω and X ∈ C 0 [0, T [, Ω a maximal solution of [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF]. Using F • e 1 ≥ c we have that X (t) • e 1 ≥ c and so X(t) • e 1 ≥ x 0 • e 1 + ct and as X(t) ∈ Ω for all t ∈ [0, T [, it is necessary that T < T max . As F ∈ C 0,α C (Ω), F is bounded, X is Lipschitz, and the limit of X(t) when t goes to T exists in Ω and is called X(T ). Let us show that X(T ) ∈ ∂Ω. Suppose that X(T ) ∈ Ω, then applying Theorem 5.1 at X(T ), we can continuously extend X on [T, T +ε[ for some positive ε which contradicts the fact that X is a maximal solution.

∈ C 0 Ω ∩ C 1 C Ω . If u is a solution of the system F • ∇u = 0 in Ω, u = 0 on ∂Ω, (23)
then u = 0 in Ω.

Proof. Consider x 0 ∈ Ω and a characteristic X ∈ C 0 [0, T [, Ω satisfying

X(t) = x 0 + t 0 F (X(s))ds, ∀t ∈ [0, T [, lim t→T X(t) ∈ ∂Ω. We define f ∈ C 0 [0, T ], R by f (t) = u(X(t)). We show that f is constant. Let us define I = X -1 (C) then f is differentiable in [0, T ]\I and f (t) = ∇u(X(t)) • F (X(t)) = 0.
Let us take t ∈ I. If t is not isolated in I, using the fact that ∂ τ u + and ∂ τ u -are locally signed, F (X(t)) is parallel to C and for an ε > 0, X(s

) ∈ B + (or B -) for s ∈ [t, t + ε[. Then, f (s) = u(x(s)) is differentiable on [t, t + ε[ with f (s) = ∇u + (X(s)) • F (X(s)).
This proves that f is right differentiable at t and (f ) + (t) = 0. By the same argument, f is left differentiable at t and (f ) -(t) = 0 and so f is differentiable at t with f (t) = 0. Finally, except for a zero measure set of isolated points, f is differentiable on [0, T ] and f = 0 almost everywhere. This is not enough to conclude because there exists continuous increasing functions whose derivative is zero almost everywhere. Since for all t, s ∈ [0, T ],

|f (t) -f (s)| ≤ sup |∇u||X(t) -X(s)| ≤ sup |∇U | sup |F ||t -s|,
f is Lipschitz and thus absolutely continuous which implies, since f = 0 a.e., that f is constant on [0, T ]. We finally have u(

x 0 ) = f (0) = f (T ) = u(X(T )) = 0.
Hence we conclude that if σ is admissible, then U is the unique solution to (20) and we can recover σ by [START_REF] Morimoto | A study of the electrical bio-impedance of tumors[END_REF].

Remark 5. [START_REF] Ammari | Spectroscopic imaging of a dilute cell suspension[END_REF] The characteristic method can be used to solve the transport problem. However, it suffers from poor numerical stability which is exponentially growing with the distance to the boundary. To avoid this delicate numerical issue, we propose a regularized approach for solving [START_REF] Montalibet | Etude du couplage acousto-magnétique: détection des gradients de conductivité électrique en vue de la caractérisation tissulaire[END_REF]. Our approach consists in forming from (20) a second-order PDE and adding to this PDE a small elliptic term of order two.

The viscosity-type regularization

In this subsection we introduce a viscosity approximation to [START_REF] Montalibet | Etude du couplage acousto-magnétique: détection des gradients de conductivité électrique en vue de la caractérisation tissulaire[END_REF]. Let ε > 0. We regularize the transport equation ( 20) by considering the well-posed elliptic problem

∇ • εI + F F T ∇u ε = 0 in Ω, u ε = x 2 on ∂Ω. (24)
The main question is to understand the behavior of u ε when ε goes to zero. Or more precisely, whether u ε converges to the solution U of the transport equation [START_REF] Montalibet | Etude du couplage acousto-magnétique: détection des gradients de conductivité électrique en vue de la caractérisation tissulaire[END_REF] for a certain topology. The following result holds.

Theorem 5.5 The sequence (u ε -U ) ε>0 converges strongly to zero in H 1 0 (Ω).

Proof. We first prove that the sequence (u ε -U ) ε>0 converges weakly to zero in H 1 0 (Ω) when ε goes to zero. For any ε > 0, ũε := u ε -U is in H 1 0 (Ω) and satisfies

∇ • εI + F F T ∇ũ ε = -ε U in Ω. ( 25
)
Multiplying this equation by ũε and integrating by parts over Ω, we obtain

ε Ω |∇ũ ε | 2 + Ω |F • ∇ũ ε | 2 = -ε Ω ∇U • ∇ũ ε (26)
and so, ũε

2 H 1 0 (Ω) ≤ Ω |∇u • ∇ũ ε | ≤ U H 1 (Ω) ũε H 1 0 (Ω) . Then ũε H 1 0 (Ω) ≤ U H 1 (Ω)
. The sequence (u ε ) ε>0 is bounded in H 1 0 (Ω) and so by Banach-Alaoglu's theorem, we can extract a subsequence which converges weakly to u * in H 1 0 (Ω). Multiplying [START_REF] Seo | Magnetic resonance electrical impedance tomography (MREIT)[END_REF] by u * and integrating by parts, we get

Ω (F • ∇ũ ε ) (F • ∇u * ) = -ε Ω ∇U • ∇u * -ε Ω ∇ũ ε • ∇u * .
Taking the limit when ε goes to zero,

F • ∇u * L 2 (Ω) = 0.
So u * is a solution of the transport equation ( 23), and by Corollary 5.3, u * = 0 in Ω. Actually, there is no need for the extraction of a subsequence to get the weak convergence result. Indeed, zero is the only accumulation point for u ε for the weak topology. Consider a subsequence u φ(ε) . It is still bounded in H 1 0 (Ω). Therefore, using the same argument as above, zero is an accumulation point of this subsequence. Now, we are ready to prove the strong convergence. From [START_REF] Tseng | The potential induced in anisotropic tissue by the ultrasonically-induced Lorentz force[END_REF] we get that

Ω |∇ũ ε | 2 ≤ - Ω ∇U • ∇ũ ε ,
and as ũε 0 in H 1 0 (Ω), the term in the right-hand side goes to zero when ε goes to zero. Hence, ũε H 1 0 (Ω) → 0.

Finally, using Theorem 5.5 we define the approximate resistivity by

1 σ ε = D • ∇u ε |D| 2 ,
which strongly converges to 1 σ in L 2 (Ω).

Numerical results

In this section we first discuss the deconvolution step. Then we test both the optimal control and the orthogonal field reconstruction schemes.

Deconvolution

In this subsection, we consider the problem of recovering Φ y,ξ from the measurements M y,ξ in the presence of noise. From [START_REF] Grasland-Mongrain | Lorentz force electrical impedance tomography[END_REF], it is easy to see that this can be done by deconvolution. However, deconvolution is a numerically very unstable process. In order to render stability we use a Wiener filter [START_REF] Mallat | A Wavelet Tour of Signal Processing: the Sparse Way[END_REF]. We assume that the signal M y,ξ (.) is perturbed by a random white noise: Σ 2 with respect to the signal-to-noise ratio.

M y,ξ (z) = M y,ξ (z) + µ(z), (27
where µ is a white Gaussian noise with variance ν 2 . Equation ( 27) can be written as

M y,ξ (z) = (W Φ y,ξ ) (z) + µ(z).
Denote by S(Σ) = R |F(Φ y,ξ )(ω)|dω the mean spectral density of Σ, where F is the Fourier transform. The Wiener deconvolution filter can be written in the frequency domain as

L(ω) = F(W )(ω) |F(W )| 2 (ω) + ν S(Σ)
.

The quotient ν/S(Σ) is the signal-to-noise ratio. So, in order to use the filter, we need to have an a priori estimate of the signal-to-noise ratio. We then recover Σ up to a small error by Σ y,ξ = F -1 F( M ) L .

Conductivity reconstructions

In the numerical simulations, we choose Ω =]0, 2[×]0, 1[. Figure 4 shows the true conductivity map in the medium. The simulations are done using a PDE solver. The data is simulated numerically on a fine mesh. For the orthogonal field method, in order to solve [START_REF] Roth | Ultrasonically-induced Lorentz force tomography[END_REF], we use a coarse mesh. Then we reconstruct an initial image of the conductivity. Based on the initial image, an adaptive mesh refinement for solving (24) yields a conductivity image of a better quality. Figure 5 shows the used meshes for solving the viscosity approximation.

The optimal control method

The minimization procedure gives a decent qualitative reconstruction. The main interfaces are easy to see, yet this method, due to its regularizing effect, fails to show details in weaker contrasts zones. Figures 6,7, and 8 show the reconstruction obtained with different measurement noise levels.

The orthogonal field method

To find the solution of problem [START_REF] Roth | Ultrasonically-induced Lorentz force tomography[END_REF], we fix ε = 10 -3 , and solve the equation on a uniform mesh on Ω. We reconstruct an approximation of σ, and adapt the mesh to this first reconstruction. We do this procedure a couple of times in order to get refined mesh near the conductivity jumps. We can see that besides being computationally lighter than the minimization method, the orthogonal field method allows a quantitative reconstruction of σ and shows details even in the low contrast zones. It is relatively stable with respect to measurement noise. Figures 9,[START_REF] Gebauer | Impedance-acoustic tomography[END_REF][START_REF] Goss | Comprehensive compilation of empirical ultrasonic properties of mammalian tissues[END_REF] show the reconstruction with different measurement noise levels. Figure 12 shows the L 2 norm of the error with respect to measurement noise, with ε fixed at 10 -3 . A smaller ε increases the noise sensibility at higher noise levels, but also improves the details and reduces the smoothing effect of the ε∆ term in [START_REF] Roth | Ultrasonically-induced Lorentz force tomography[END_REF].

Concluding remarks

In this paper we have provided the mathematical basis of ultrasonically-induced Lorentz force electrical impedance tomography. We have designed two efficient algorithms and tested them numerically. The resolution of the reconstructed images is fixed by the ultrasound wavelength and the width of the ultrasonic beam. The orthogonal field method performs much better than the optimization scheme in terms of both computational time and accuracy. In a forthcoming work, we intend to generalize our approach for imaging anisotropic conductivities by ultrasonically-induced Lorentz force [START_REF] Tseng | The potential induced in anisotropic tissue by the ultrasonically-induced Lorentz force[END_REF]. We will also propose an algorithm to find σ∇U from the data function ψ using [START_REF] Foster | Dielectric properties of tissues and biological materials: a critical review[END_REF] and correct the leading-order approximation ( 14). This will enhance the resolution of the reconstructed conductivity images. Another challenging problem under consideration is to interpret the high-frequency component of M y,ξ in terms of speckle conductivity contrasts.

Figure 1 :

 1 Figure 1: Example of the imaging device. A transducer is emitting ultrasound in a sample placed in a constant magnetic field. The induced electrical current is collected by two electrodes.

Figure 2 :

 2 Figure 2: Imaging system configuration. An ultrasonic wave propagates in a medium of electrical conductivity σ comprised between electrodes Γ 1 and Γ 2 .

Corollary 5 . 2 (

 52 Existence of outgoing characteristics) Consider F ∈ C 0,α C (Ω) satisfying the same conditions as in Theorem 5.1 and the condition

Corollary 5 . 3 (

 53 Uniqueness for the transport problem) Consider F ∈ C 0,α C (Ω) satisfying the same conditions as in Corollary 5.2 and consider u

) 10 - 8 Figure 3 :

 1083 Figure 3: L 2 norm of the relative error Σ-Σ 2Σ 2 with respect to the signal-to-noise ratio.

Figure 4 :

 4 Figure 4: Conductivity map to be reconstructed.

Figure 5 :

 5 Figure5: Meshes for solving[START_REF] Roth | Ultrasonically-induced Lorentz force tomography[END_REF]: initial mesh (left), adapted mesh (middle), and the mesh used to generated the data (right).

Figure 6 :Figure 7 :Figure 8 :Figure 9 :

 6789 Figure 6: Reconstructed image without measurement noise.

Figure 10 :Figure 11 : 2 Figure 12 :

 1011212 Figure 10: Reconstructed image with 2% measurement noise.

* This work was supported by the ERC Advanced Grant Project MULTIMOD-267184.


El Mahdi

Rezan Demir-Cakan

Hubert Perrot

Mama El Rhazi
email: elrhazim@hotmail.com

Ozlem Sel
email: ozlem.sel@upmc.fr

Correlation between the interfacial ion dynamics and charge storage properties of poly(ortho-phenylenediamine) electrodes exhibiting high cycling stability

Keywords: poly(ortho-phenylenediamine), pseudo-capacitors, cycling stability, interfacial ion transfer, electrochemical quartz crystal microbalance, electrogravimetric impedance (acelectrogravimetry)

An integrated electrogravimetric study based on electrochemical quartz crystal microbalance (EQCM) unravels the interfacial ion transfer phenomena of the poly(orthophenylenediamine) (PoPD) thin film electrodes. Through a methodology coupling QCM with electrochemical impedance spectroscopy (ac-electrogravimetry), our work indicates that charge compensation process of PoPD in aqueous electrolytes (in acidified NaCl) occurs with the participation of multiple species, each playing a role at different temporal scales. The PoPD films are tested in a 2 electrode Swagelok cell in which Zn is used as both reference and counter electrodes and exhibit excellent stability over 8000 cycles with a relatively high specific capacitance of about 110 F•g -1 at 30 C (0.63 mA•cm -2 ) current density. The high rate capability and the excellent cycling stability of the PoPD electrodes are correlated to the electrolyte composition and the significant role of H + to the charge compensation process is unravelled, which is made possible with coupled electrogravimetric methods of our study. By determining the interfacial flux dynamics and as well as the relative proportions of species transferred at the electrode/electrolyte interface, our results contribute to the understanding of the charge-discharge process of PoPD polymer, yet underexplored but emerging as a pseudo-capacitive electrode material.

INTRODUCTION

Conducting polymers (CPs) have been extensively investigated and solicited in various fields of applications, such as in electrocatalysis, biosensors and microelectronics technology [1][2][3][4]. In energy storage, these pseudo-capacitive materials are regarded as prospective electrode component considering their salient features such as high electrical conductivity and high redox activity, as well as their relatively low cost and ease of synthesis [5][6][7][8].

Specifically, polyaniline (PANI), polypyrrole (PPy), polythiophene (Pth) and their derivatives have extendedly been studied as electrode materials due to their high specific capacitance which is substantially higher than those of the conventional carbon-based electrodes and comparable to pseudocapacitive metal oxides [9,10]. However, their cycling stability is a major impediment that limits their practical application. For example, pristine PANI, PPy and Pth based pseudocapacitive electrodes have been reported to lose substantial amount of their capacitance after about 1000 cycles and there is a lack of information about the electrode stability beyond this threshold [11].

Several methods have been used to enhance the cycling stability of conducting polymer based electrodes. It was postulated that the electrochemical performance is affected by various factors, including synthetic method, dopant ion, substrate and the morphology of the conducting polymer [5]. Therefore, the strategies to enhance the cycling stability include (i) encapsulation of the conducting polymer in inorganic networks, such as titania nanotubes [12], (ii) surface coating such as with Nafion [13] and (iii) doping with organic molecules such as para-toluenesulfonate [14]. For instance, the phytic acid (PA) doped polypyrrole (PPy) has been reported to retain its specific capacitance for about 4000 cycles on specific substrates [15]. Regarding the efforts on the morphology, PPy nanowire arrays have been shown that they retain 70 % of the initial capacitance after several hundred cycles and are advantageous over disordered nanowire networks or conventional films [16]. However, the fabrication time, the toxicity and the cost of the dopant materials limit the practical application of these synthesis methods [11]. Additionally, despite the multiple efforts, it remains still challenging to reach cycling stability more than ~1000 cycles, except from limited number of reports providing information about the long term electrode stability. For example, Li et al. has shown that the deposition of a thin carbonaceous shell (∼5 nm) onto the CP electrode drastically improves the charge/discharge cycling stability i.e., ∼95 and ∼85% capacitance retention after 10000 cycles, for PANI and PPy, respectively [11]. Similarly, the reduced graphene oxide layer has been employed as a protective layer on PANI to tackle with the cycling stability issue and 75.5% of original capacitance has been maintained up to 10000 cycles [17].

While substantial research efforts evolve in the direction of protecting the CP based electrodes especially to alleviate swelling and shrinking during cycling and thus to enhance the cycling stability, one other path is to scrutinize the structural effects by investigating the derivatives of the most researched conducting polymers. In this context, aromatic diamine polymers such as para, ortho and meta-phenylenediamine (PD) have received great attention due to the active free amino and imino groups present in their structure, which provide structural modification possibility [18]. They have been the subject of several studies in the past decade in view of their attractive (electro)chemical properties, [19,20] and also have been receiving attention as pseudocapacitive CP electrodes (ortho-, [21] and para-, [22]). In addition to their direct intervention as electrode materials, poly ortho-phenylenediamine (PoPD) polymers doped with glycine was carbonized to obtain nitrogen and oxygen doped hallow carbon spheres for supercapacitor applications [23]. Regarding the synthesis, the electropolymerization of orthophenylenediamine (oPD) in acidic, alkaline and neutral aqueous solutions has been examined and electrochemical quartz crystal microbalance (EQCM) has been used to investigate the subtleties of polymerization mechanism and the subsequent electrochemical activity of the resulting electrodes [19,24,25].

As an extension to the previous studies [19,24,25] and due to the developing interest in the PoPD type CP as pseudocapacitive electrodes, a thorough investigation of the electrochemical charge storage behaviour becomes deservedly necessary. It is also noted that, to the best of our knowledge, there is no study describing the interfacial ion flux (insertion/deinsertion and/or electroadsorption/desorption of ions) with the exact identification of the intervening (non)ionic species together with their dynamics of transfer at the PoPD film/electrolyte interfaces. It is wellknown that interfacial ion flux dynamics constitute an important part of the operating principle of super-and pseudo-capacitors, therefore the correlation of this fundamental understanding to the charge storage properties of poly(ortho-phenylenediamine) thin film electrodes is rather significant but requires specific characterization tools.

A marked improvement in understanding the charge storage mechanisms has been obtained thanks to recent progress with electrogravimetric methods (EQCM and EQCM with dissipation monitoring) [26][27][28]. Besides, the complex character of the electrode mass changes, the viscoelastic properties of the electrodes and also possibly formed solid electrolyte interface (SEI) layer can be investigated using multiharmonic EQCM with dissipation monitoring (EQCM-D) [29,30]. It can offer opportunities to identify the impact of several parameters (nature of the electrolytes, ions, binder…) on the structural change of the electrodes.

In addition to these developments in the use of EQCM based methods in the energy storage domain, here, a non-conventional complementary technique, the so-called ac-electrogravimetry has been proposed to unveil the mechanisms of the ionic exchange process [31][32][33]. It has been demonstrated that highly relevant and complementary information to the classical EQCM can be obtained: (i) kinetics and identification of species transferred between the electrode and the electrolyte, (ii) separation of the different contributions related to the charged and non-charged species involved in the electrochemical processes, (iii) identification of species transferred in opposite flux directions provided that their kinetics are sufficiently different and (iv) variation of the relative concentrations of the species inside the examined material [34][35][36][37][38]. The originality of this method is its ability to discriminate between the interfacial transfer of cations, anions and solvent molecules involved (directly or indirectly) in the charge compensation process which is ultimately related to their charge storage performance.

Thus, we report here on the investigation of the ion exchange mechanisms in PoPD pseudocapacitive electrode. The electropolymerization of oPD was carried out in acidic medium.

Thereafter, the redox transformations and the ionic flux mechanisms were investigated in aqueous NaCl electrolyte via EQCM and ac-electrogravimetry. Several parameters were considered to explain the redox behaviour of PoPD film, such as the nature and the dynamic of the ions transfer at the PoPD film/electrolyte interface. To correlate these fundamental understandings to the performance of the electrode, the cycling stability of the thin PoPD film electrode was tested in two-electrode Swagelok-type cell resulting excellent cycling stability for more than 8000 cycles with ~110 F•g (11 mF•cm 2 ) capacitance at 30C. These performances are relevant to the recent developments in the field of thin conducting polymer electrodes for micro energy storage devices [39][40][41].

EXPERIMENTAL PART

2.1. Electrode Preparation. Electrochemical polymerization of oPD was performed on the gold electrode (0.2 cm 2 ) of quartz crystal resonators (9 MHz-AWS, Valencia, Spain) and on graphite plates (~1 cm 2 ) for the electrogravimetric studies and for the tests in two-electrode Swagelok cells, respectively. Specifically, oPD monomer (1,2-diaminobenzene, Sigma-Aldrich) (5 mM) dissolved in H2SO4 solution (0.1 M) was electropolymerized using cyclic voltammetry (CV). This step was also followed through EQCM measurements i.e. when quartz crystal resonators were used as substrates. The mass of the deposited polymer was evaluated by measuring the resonant frequency of the quartz crystal (∆f), before and after the electropolymerization, which was then converted to the mass change (∆m) by the Sauerbrey equation ( where Providing that all the TFs including cross and partial TFs (detailed in Refs. [31,32,34,[36][37][38]) were fitted with the same set of parameters (where a good agreement between the experimental and the theoretical functions in terms of both the shape and the frequencies are achieved), then these parameters (Mi, Ki and Gi) are considered as unique.

Galvanostatic charge/discharge (GCD) measurements were performed by a classical twoelectrode Swagelok-type cell, where zinc metal was used as both reference and counter electrode and PoPD deposited on graphite plate as working electrode. Zn metal (-0.76 V vs SHE) is chosen among the other counterparts to be employed as electrode material in aqueous medium, as its reduction potential falls within the water stability window. The glass fiber separators (Whatman GF/C with areal weight and thickness of 4.7 mg/cm 2 and 185 µm, respectively) are used for the measurements in Swagelok cells. GCD tests were conducted in an aqueous electrolyte containing 1M NaCl (pH=2) cycled at 30 C current density that is equivalent to 0.63 mA•cm -2 .

RESULTS AND DISCUSSION

Electrochemical synthesis and morphological characterization of the PoPD.

Electrochemical synthesis of the PoPD thin film was performed by CV starting from the oPD monomer. Figure 1a shows the CV curves during the formation of the PoPD thin film on the gold electrode of the quartz crystal. This specific substrate was chosen to be able to perform electrogravimetric studies thereafter (Figure 1b). During the first scan, an oxidation peak with a large current appears at about +0.62 V vs Ag/AgCl, indicating the oxidation of oPD monomer. In the subsequent scans, the current value of this first anodic peak significantly decreases, whereas an increase in the current value of the redox peaks at Ecathodic = -0.16 V and Eanodic = -0.12 V vs Ag/AgCl is evident. The appearance of another redox peak is also noted at around 0 V vs Ag/AgCl. Then, these two redox peaks (in the range of 0.2V to -0.4V vs Ag/AgCl) are merged oPD (Figure 1c) starts by the oxidation of the monomers to give radical cations. These cationic radicals react chemically together to produce dimers which can be further oxidized. Then, the oligomer cations or bications propagates to give either a linear polymer chain or cyclic polymers with ladder structures through the polymerization of the oxidized products [18,19,45,46].

The frequency variations of the gold coated quartz resonator during the electropolymerization was converted to the corresponding mass variations (using Sauerbrey equation [42]) as shown in One can consider that a pure gravimetric response can be obtained as the film is acoustically thin (~150 nm) [47].

Electrogravimetric characterization of charge storage behavior.

Electrochemical Quartz Crystal Microbalance (EQCM).

Figure 2 shows the EQCM responses of PoPD thin films in 0.5 M NaCl solution with different pH values. In acidic media (pH=2), the PoPD shows a typical redox behavior between -0.4V and 0.2V vs Ag/AgCl (Figure 2a). However, in neutral and alkaline media (pH=7 and 12), the redox peaks of PoPD are almost non-existent. This has been previously described by the fact that a proton addition-elimination process participates to the charge compensation, occurring due to the electron transfer, together with the motion of electrolyte ions [46]. The corresponding mass variations were also reported as a function of potential (Figure 2b). In acidic media, a significant mass change is observed between -0.4V and 0.2V vs Ag/AgCl, which can be attributed to (i) the electroadsorption/electrodesorption and/or insertion/deinsertion of the ions and to (ii) the transfer of water molecules. On the contrary, in neutral and alkaline media, the mass responses remain almost unchanged in the explored potential range. This pH dependant phenomenon has been discussed in the literature considering the fact that (i) intrinsic electron-transfer process between adjacent electroactive sites, (ii) the proton addition-elimination process, (iii) the charge compensation by the counter ion and the solvent transfer are all involved in the charge transfer and transport process of the PoPD film. Among these concurrent processes, the protons additionelimination is considered as a prerequisite for the charge compensation process to proceed [46].

The presented electrogravimetric responses shown in Figure 2 are in good agreement with the above discussion. Based on these results, the acidic media (pH=2) was chosen for further investigation.

The CV curves in Figure 3a shows a pair of well-defined peaks indicating the electrochemical redox reaction of PoPD, with a peak-to-peak separation of 60 mV at 25 mV•s -1 . The value of peak-to-peak separation increases with the increase of the scan rate, which can be explained by a limitation in the process of charge transfer [46]. The scan rate dependence of the electrochemical response is also reflected in the corresponding mass variations. The general appearance of the Figure 3b indicates a mass increase during reduction and mass decrease during oxidation of the PoPD film. With an initial assessment, this can be considered as the characteristic of a major cation response. This behaviour is more dominant at lower scan rates (25 mV•s -1 ) but becomes more complex especially in the anodic scan of the CVs at higher sweep rates, which can lead to the idea that there is a multi-species transfer phenomena occurring with different proportions and kinetics. In-depth analysis of the EQCM data actually indicates a cation and anion mix contribution to the charge compensation process, which is more evident at higher scan rates by the presence of the slope changes during a certain potential scan direction (Figure 3b). It is noted that anions contribution appears as a mass decrease during reduction and a mass increase during oxidation (for cations, it is vice versa) [31][32][33]. The scan rate dependant mass response implies the presence of a slower cation(s) contribution which cannot be detected at higher scan rates and this/these cation(s) contribution becomes more dominant at 25 mV•s -1 .

The mass per electron (MPE = F x (Δm/Δq)) calculated from the EQCM data can provide indications of the nature of the transferred species during cycling. If only one species is involved in the charge storage process, the value of MPE would be its molar mass. The cation and anion contributions lead to negative and positive signs of MPE values, respectively [33,34]. Figure 3c presents the MPE of the species involved during EQCM at 25 mV•s -1 (calculated for oxidation and reduction), presenting a value ranging from -10 to -5 g•mol -1 , respectively. From these two values and the MPE of Na + , Cl -and H + of -23 g•mol -1 , 35 g•mol -1 and -1 g•mol -1 , respectively, it is more likely that a multi-ion contribution in the charge compensation process is present. Earlier works strengthen this idea by describing that the PoPD films can contain solvent molecules and ionic species at all oxidation states [19,25]. Therefore, this deviation between the experimental and theoretical MPE values merits a special analysis. Thus, electrogravimetric impedance spectroscopy (ac-electrogravimetry) was performed to complement the EQCM data and to get a deeper insight into the interfacial charge compensation behavior occurring in the PoPD thin film electrode, from both gravimetric and kinetic point of view. 1)

Electrogravimetric Impedance

i f i i f E G j d F I jd K          (i: ions) (Equation
( ) i f i i f q G Fd E j d K        (i: ions) (Equation 2)
where Ki represents the kinetics, Gi describes the ease or difficulty of each species' transfer at the electrode/electrolyte interface, df relates the film thickness and F is Faradays' constant. This fitting process revealed the involvement of three different ionic species which resulted in a good agreement between the experimental and the theoretical curves (Figure 4b well defined semi-circles appearing at different quadrants. It is important to note that the contributions in the first/fourth and the second/third quadrant are due to the anion and cation transfer, respectively (details of the methodology can be found in Refs. [31,32,34]). Free solvent molecules can appear in the same quadrant either with anions or with cations, depending on their flux direction, i.e., the same flux direction with anions or cations [34].

For the fitting process of the Δm/ΔE() TF, two parameters (Ki and Gi) previously obtained from impedance and charge/potential TF for each ionic species (Figure 4b and4c) are used in Equation 3:

( ) i f i i f i m G d M E j d K       
 (i: ions and neutral species) (Equation 3)

Notably, the identification of the participants in charge compensation process can be achieved thanks to the molar mass (Mi) of the species intervening in electrogravimetric TF (Δm/ΔE())

(Equation 3). The fitting of the experimental Δm/ΔE() data (measured in 0.5 M NaCl and at pH=2) using theoretical function in Equation 3 revealed a configuration involving the transfer of anions (Cl -), cations (H + and Na + ) and free H2O molecules which leads to a good agreement between experimental and theoretical data (Figure 4d and4e).

(ii) Dynamics of interfacial transfer of species: Since these measurements are frequency dependent, it is also possible to get information on the kinetics of interfacial ion transfer. At both potentials shown in Figures 4d and4e, Cl -anions were identified at high frequencies (HFs)

indicating a faster process, followed by the transfer of water molecules with an opposite flux direction. Lastly, H + and Na + cations were exchanged at lower frequencies (LFs), exhibiting also opposite flux direction to anions.

The other potentials studied in the range of "0.05 V to -0.25 V vs Ag/AgCl" indicate the persistence of multi-species contribution, with slight variations in quantity and in terms of kinetics of transfer. Form their Ki and Gi parameters, the characteristic frequency, fi, and the resistance of transfer, Rti, can be calculated using the

i i f K f d   and 1 F i i Rt G  relations,
respectively (Figure 5a and5b). In the potential range studied, the tendency of fi (Cl -) > fi (H2O) > fi (H + ) > fi (Na + ) is observed (Figure 5a) presenting the order of the kinetics of transfer of the species which is inversely proportional to their resistance of transfer, Rti, at most of the potentials shown in Figure 5b.

The multi species contribution with different interfacial transfer kinetics revealed in acelectrogravimetry is in good agreement and also strengthens the ideas discussed previously to explain the scan rate dependent mass-potential curves in EQCM (Figure 3a and3b). In a simplistic discussion, the redox behavior of PoPD thin film can be compensated either by insertion of anions or expulsion of cations during oxidation (vice versa during reduction). In the EQCM data in Figure 3b, during oxidation at high scan rates (100 mV•s -1 ), a mass increase is followed by a subsequent mass diminution. This observation may correspond to the insertion of anions which is followed by the expulsion of cations and/or water molecules by exclusion. Since ac-electrogravimetry results indicate the opposite flux direction of solvent molecules which appear at the following frequencies right after the anions (Figure 4d and4e), it is more likely that anions insertion during oxidation excludes some water molecules which results in this subsequent mass decrease during oxidation (Figure 3b). However, this cannot be the only contribution, since in the reverse scan during reduction, a mass increase is observed. The water molecules alone cannot explain the specific shape of the gravimetric response during the reduction process. Thus, there should be a contribution from the cationic species in the electrogravimetric response, as well. This behavior observed at 100 mV•s -1 becomes less and less obvious at lower scan rates (i.e. 25 and 50 mV•s -1 ) and the EQCM curves becomes dominant in a typical cationic response (Figure 3b). This is explained by the ac-electrogravimetry results showing the contribution of the H + and Na + at lower frequencies. Consequently, the contribution of cations is hardly detected at high scan rates, but starts to dominate the electrogravimetric response at lower scan rates. These results are in good agreement with the discussion of Inzelt et al. on the PoPD charge compensation behavior and provide experimental data for the disentangling the redox behavior of PoPD thin films [25].

(iii) Complementarity of ac-electrogravimetric and EQCM results -Gravimetric contribution of each species:

To further shed light on the multi-species contribution to the charge storage and their interfacial transfer, an approach benefiting from the complementarity of the EQCM and acelectrogravimetry is proposed. Specifically, the relative concentration of each species (ΔCi) with respect to the potential variation (ΔE) can be obtained from ac-electrogravimetry by using Equation 4, lead to the estimation of the relative concentration changes of each species between the different states-of-polarization through an integration process as shown in Equation 5:

( )      C G i i E j d K i f  
(Equation 4)

( ) = 0 0 0 0         C G E E i i i i C C dE dE i E E E K i  
(Equation 5)

The It can be emphasized that charge compensation process of PoPD in aqueous electrolytes (in acidified NaCl) occurs with the participation of multiple species, each playing a role at different temporal scales. Although gravimetrically not significant, the contribution of H + to the charge storage mechanism is highlighted, i.e. (i) more than ten times of Na + ion's concentration variation in the electrode during cycling (Figure 6a) and (ii) their interfacial transfer is faster than that of Na + (Figure 6b) under the conditions of our study.

3.3.

Pseudo-capacitive charge storage performances.

Thin film PoPD electrodes-half cell performance (3-electrode configuration).

The charge storage performances were measured in NaCl 0.5M at pH=2. The specific capacitance can be calculated from the CV curves using the following relation,

)

E s E C IE d E mv E E   
, where m is the mass loading of PoPD electrodes obtained Apparent gold electrode surface on the QCM is 0.2 cm 2 . by QCM, v is the scan rate, E1 and E2 are the low and high end potentials, and I(E) depicts the response in current. The PoPD thin film electrodes used in the previous sections were about 4.4 g for 0.2 cm 2 of the gold electrode of the quartz resonators (22 g•cm -2 ). The loading of the resonators was determined by measuring the frequency changes before and after the film formation.

First of all, the Cs values as a function of the scan rate used in the CV analysis are presented in Figure 7a. For this analysis, the typical loading of ~22 g•cm -2 was used, which corresponds to a film thickness of ~150 nm estimated through FEG-SEM analysis (Figure 1d).

The Cs values were decreased with an increase in the scan rate from a maximum value of 430 F•g -1 at 50 mV•s -1 to 270 F•g -1 at 200 mV•s -1 (Figure 7a). However, the Cs values at the scan rates of 10 and 25 mV•s -1 are slightly lower than that of 50 mV•s -1 and deviates from the usual scan rate dependency trend of the Cs (Figure 7a). This behavior has previously been observed for thin conducting polymer or metal/metal oxide based electrodes in aqueous electrolytes [48,49] and deserves to be further investigated.

Additionally, the Cs values obtained in our study, are in the same order of magnitude or even higher than some of the earlier reports on CPs (Table SI-2) [11,[50][51][52][53]. However, it is important to recognize that the Cs values depends on various factors, such as the synthesis route, additives, dopants, morphology, porosity, as well as the measurement configuration [7,21].

Then, in order to check the effect of the film thickness, several films were prepared by increasing the number of CV cycles during their electrodeposition. The specific capacitance (Cs) of the PoPD thin films is evaluated as a function of the loading (proportional to film thickness) at a scan rate of 50 mV•s -1 . As shown in Figure 7b However, when the loading (thickness) of the film is further increased (i.e. to ~55 g•cm -2 ) a slight capacitance fading is noticed in the range of 22 g•cm -2 -55g•cm -2 (Figure 7b). This behaviour can be related to the mass transport limitation of the ions resulting in higher overpotentials upon cycling within a fix voltage range [54].

Note that higher loadings are not investigated on the quartz resonators to be able to maintain the gravimetric regime for the analysis [25], for that reason thicker electrodes are tested in a two electrode Swagelok configuration on graphite electrodes.

Thicker film PoPD electrodes-full cell performance (2-electrode configuration).

Figure 8 presents the galvanostatic charge-discharge (GCD) performance of the PoPD electrode deposited on a graphite electrode. The mass loading and the thickness of the electrode is around 100 g•cm -2 and 700 nm, respectively. The mass loading is two times higher than that of the maximum mass deposited on Au coated quartz electrodes (Figure 7b). The current applied to the electrode is 0.63 mA•cm -2 (6.3 A•g -1 ) that is equivalent to 30 C current density. The test was performed in 2 electrode cell in which Zn is used as both reference and counter electrodes. A characteristic pseudo-capacitive behaviour is observed in which voltage-time graph has not a triangular shape (Figure 8a and8b), but instead, obvious oxidation and reduction peaks appear at around 0.85 and 0.5 V vs Zn, respectively, as depicted in the inlet figure of Figure 8c. Indeed, those well-defined peak positions are in a well agreement with the 3-electrode measurements performed with a Ag/AgCl reference electrode (Figure 3a) since the reference electrode potentials of Ag/AgCl (vs SHE) is +0.197 V and Zn is -0.760 (vs SHE). The electrode exhibited excellent cycling performances with a stable capacitance (around 110 F•g -1 ) and columbic efficiency close to 100 % without negligible capacitance fade over 8000 cycles at 0.63 mA•cm -2 current density. The gravimetric capacitance value (110 F•g -1 ) corresponds to an areal capacitance of 11 mF•cm 2 since we have a loading of ~100 µg/cm -2 . These values are relatively small compared with the Cs values obtained for the thin film PoPD electrodes in the three electrode configuration at 50 mV•s -1 (Figure 7). This could be attributed to the increased distance of electron transport from the thicker PoPD electrode to the current collector which is in agreement with the thickness dependence of the specific capacitance as shown in metal oxide or conducting polymer electrodes [55,56]. Furthermore, the post mortem FEG-SEM images of the PoPD on graphite and Zn electrode are provided in supporting information (Figure SI-4 and 5, respectively) and compared with those of the fresh electrodes. No significant morphological changes were observed even after 8000 cycles (under the conditions of our study), except the presence of residual glass fibers originating from the separator used in the Swagelok tests (identified by the EDX analysis shown in Figure SI-4, panel E). It is noted that detachment of the glass fibers is not significant enough to affect the structural integrity of the separator, thus it is not detrimental to the cell performance, at least for 8000 cycles.

In order to better position the performance of the PoPD based electrodes, they are compared with the limited number of studies exist which report on the capacitive behaviour of pristine PoPD [18,21] and PoPD composites [57][58][59]. Wang et al. described the electrochemical synthesis of PoPD in the presence of sodium dodecyl sulphate (SDS) surfactant [18]. The resulting materials were tested by charging/discharging in 1M KNO3 at the current density of 0.5 A•g -1 in the potential range of -0.2 to 0.6 V vs SCE (saturated calomel electrode) in a 3-electrode configuration. A capacitance value of about 106 F•g -1 were obtained and the electrode retained 82% of its initial capacitance at the 1000 th cycle [18]. Comparing our findings with the results of Wang et al. [18], it is important to note that pseudo-capacitive behaviour of the PoPD is clearly observed under the conditions of our study, as evidenced by the presence of the obvious redox peaks of PoPD (Figure 7b and8c inset). Similar capacitance value (around 110 F•g -1 ) to that reported in Ref. [18] is obtained, but at a higher current density (6.3 A•g -1 ) accompanied with high coulombic efficiency for 8000 cycles, indicating the high rate capability and the excellent cycling stability of the PoPD electrodes of our study (Figure 8c). Furthermore, a more recent study evaluated the capacitive properties of the PoPD electrodes and compared them with the PoPD-PANI copolymer in 1M H2SO4 using a 3 electrode configuration. In this case, bare PoPD presented the lowest Cs (8.3 F•g -1 ), PANI showed a Cs of 123 F•g -1 and PoPD-PANI copolymer exhibited the highest Cs (138.9 F•g -1 ) at 0.1 A•g -1 [21]. The low value obtained for the pristine PoPD electrode can be attributed to the fact that GCD measurements were conducted in the potential range of 0 V to 0.6V vs Ag/AgCl, thus only benefiting from the double layer capacitance of the PoPD polymer, but not from its faradaic capacitance which was enhanced in a subsequent step by copolymerization of PoPD with polyaniline [21]. In our study it is highlighted that, the potential range can be optimized to test both capacitive and pseudo-capacitive phenomena of PoPD (Figure 8). As it is widely accepted, the cycling performance is one of the bottleneck in the field of conductive polymer based electrodes [5,7,18,60], and has been tackled with, for example by the encapsulation of the CPs under a protective shell [11]. Therefore, the performances obtained in our study with the pristine PoPD polymer electrodes are quite significant, especially taking into account its very long cycling stability (Figure 8).

CONCLUSIONS

A thorough electrogravimetric study which employed EQCM and its complementary counterpart electrogravimetric impedance revealed that the charge compensation process of PoPD thin film electrodes in acidified NaCl electrolyte occurs with the participation of multiple species. This combined methodology identified that the Cl -anions contribute at high frequencies (HFs) indicating a faster process, followed by the transfer of water molecules with an opposite flux direction (excluded by the anions). Lastly, H + and Na + cations were exchanged at lower frequencies (LFs), exhibiting also opposite flux direction to anions.

Then, the pseudo-capacitive charge storage performances of the thin film electrodes were studied where the PoPD is electrodeposited on the gold-coated quartz resonators. The gravimetric capacitance values (Cs in F•g -1 ) of the thin film electrodes could be obtained thanks to the exact determination of the electrode loadings by QCM. Cs values as high as ~ 434 F•g -1 are obtained (in a half cell configuration) at a loading of 22g•cm -2 which is followed by a slight decrease (~ 400 F•g -1 ) at the loading range of 22-55g•cm -2 . These significantly high Cs values may imply an efficient interfacial charge transfer which may occur in the thicker electrode configuration.

Accordingly, the pseudo-capacitive charge storage performance of a thicker electrode (in a full cell configuration in which Zn is used as both reference and counter electrode) presented relatively high Cs values (~ 110 F•g -1 ) and a remarkable cycling stability (~ 8000 cycles) with high coulombic efficiency (~ 100%). Although gravimetrically not significant, the contribution of protons to the charge storage mechanism is highlighted since they are detected as the dominant species in terms of concentration variation in the electrode during cycling. Therefore, high rate capability, the excellent cycling stability of the PoPD electrodes accompanied with high coulombic efficiency for 8000 cycles have been correlated to the electrolyte composition and the significant role of H + to the charge compensation process has been unraveled, which was made possible with coupled electrogravimetric methods of our study. Furthermore, our results are distinguished from the limited number of reports on the charge storage performance of PoPD based electrodes, where in the present study, the experimental conditions were optimized to have the contributions from both the double layer and the faradaic capacitance. The fact that the pristine thin film PoPD polymer electrode has significant cycling stability shows its promise as an electrode or electrode component in micro energy storage devices.

  sensitivity coefficient[42,43].2.2. Electrode Characterization.The morphology and the thickness of the PoPD films were analysed using a field emission gun scanning electron microscope (FEG-SEM, Zeiss, Supra 55).

2. 3 .

 3 Electrochemical Characterization of the electrodes. Electrochemical quartz crystal microbalance (EQCM) measurements were carried out in aqueous 0.5 M NaCl solution and pH was adjusted to pH = 2 by the addition of HCl. A three-electrode cell was employed, in which, a platinum grid was used as counter electrode, Ag/AgCl (3M KCl saturated with AgCl) as reference electrode and a gold-patterned quartz substrates operating at 9 MHz as working electrode. The ac-electrogravimetry measurements were performed using a four-channel frequency response analyser (FRA, Solartron 1254) and a lab-made potentiostat (SOTELEM-PGSTAT). The QCM was used under dynamic regime, the working electrode was polarized at a selected potential, and a sinusoidal small amplitude potential perturbation was applied. The frequency range was between 63 kHz ad 10 mHz. The mass change, ∆m, of the quartz electrode was measured simultaneously with the ac response, ∆I, of the electrochemical system. The resulting signals were sent to a four-channel FRA, which permitted the electrogravimetric transfer function (TF) and the electrochemical impedance to be simultaneously obtained at a given potential. The main TFs (electrogravimetric TF ( , classical impedance ( ) and charge/potential TF ( ) (derived from impedance)) were fitted through the theoretical expressions given in Equations 1-4.

Fig. 1 .

 1 Fig. 1. PoPD thin film synthesis by electropolymerization on a gold electrode of a quartz resonator in an aqueous solution of H2SO4 (0.1 M) and oPD (5 mM). (a) CV Curves, (b) corresponding mass variations (m) obtained by EQCM at a scan rate of 50 mV•s -1 , (c) proposed electropolymerization mechanism and (d) FEG-SEM image of a PoPD thin film on the gold electrode of a quartz resonator after 200 CV cycles.

Figure 1b .

 1b Figure 1b. At the beginning of the electropolymerization, during the anodic sweep, the m of the electrode is virtually constant until 0.5 V, after which a significant m increase is observed. It is important to note that this potential range corresponds to the oxidation of the oPD monomer and/or the formation of dimer and oligomers on the gold electrode of the quartz resonator. In the cathodic sweep, there is no significant mass variation until 0.1 V vs Ag/AgCl then, m increases and reaches a maximum at about -0.1V vs Ag/AgCl, followed by a decrease up to -0.4V corresponding to the redox processes of newly formed PoPD film. These observations are in agreement with earlier reports[19]. A continuous mass increase was observed (only the first 5 cycles are shown in Figure1b) which resulted in 22 µg.cm -2 of total mass deposition after typically 200 CV cycles. Figure1dshows the FEG-SEM image of a representative PoPD film deposited on the gold electrode of a quartz resonator. The figure reveals the full coverage of the gold surface with the PoPD film which has an average thickness of ~150 nm. The electrogravimetric response of the resulting PoPD film will be investigated below at the section 3.2.1. One can consider that a pure gravimetric response can be obtained as the film is

1 Fig. 2 .

 12 Fig. 2. (a) CV curves and (b) the corresponding mass variations (m) of a PoPD film electrodeposited on a gold coated quartz resonator. Measurements are performed in 0.5 M NaCl, at different pH (pH=2, 7 and 12), at a scan rate of 50 mV•s -1 .

Fig. 3 .

 3 Fig. 3. (a) CV curves, (b) the corresponding mass variation and (c) the function of PoPD

Fig. 4 .SI- 2 ,

 42 Fig. 4. (a) EQCM curve and (b to e) Ac-electrogravimetry data shown at two selected potentials.

and 4c, Figure SI- 1 and SI- 2 ). 1 .

 121 The fitting parameters are presented in Table SI-The classical impedance and the charge/potential TF provide the investigation of the charged species' contribution but do not permit their identification. Thus, Δm/ΔE() TFs were explored for tracking the flux of both charged and neutral species and to achieve their identification and the estimation of their kinetics related to the interfacial transfer.

Figure 4d and 4e

  Figure 4d and 4e present the experimental and theoretical Δm/ΔE() TFs which show several

Fig. 5 .

 5 Fig. 5. Variation of the (a) transfer resistance (Rti) and (b) characteristic frequency (fi) for PoPD film deposited on the gold coated quartz resonator, measured in 0.5 M NaCl at pH=2. The fi and Rti values are given in Table SI-1.

Fig. 6 .

 6 Fig. 6. The ac-electrogravimetric results of the PoPD film deposited on the gold coated quartz resonator. (a) The variation of the relative concentration (Ci-C0), (b) the relative mass variation of each species measured in NaCl 0.5M, pH=2 and (c) the reconstruction of total mass variation from ac-electrogravimetric results and comparison with the EQCM response at 25 mV•s -1 .

  Figure 6a is transformed into corresponding mass variations (Figure 6b), gravimetric contributions present a different behavior, anions and water molecules start dominating the m response. Furthermore, the gravimetric reconstruction can be obtained by the addition of individual mass contributions from all species concerned in ac-electrogravimetry (Figure 6c, shown in the range of "0.05 to -0.25 V vs Ag/AgCl"). A good agreement between m(global) measured with the classical EQCM (at 25 mV•s -1 ) and reconstructed m from acelectrogravimetry is achieved (Figure 6c) in the studied potential range. This comparison provides a support for the multi-species' contribution revealed by ac-electrogravimetry and further evidences the complementarity of the EQCM and ac-electrogravimetry.

Fig. 7 .

 7 Fig. 7. Variation of the specific capacitance (Cs) of PoPD films deposited on gold-coated QCM and measured in NaCl 0.5M, pH=2. Cs is plotted (a) as a function of the potential scan rate (loading of 22 g•cm -2 ) and (b) as a function of the loading of the QCM resonator at 50 mV•s -1 .

,

  Cs values presents two different trends as a function of the loading (i.e. in the range of 7.5 -22 g•cm -2 and 22 g•cm -2 -55g•cm -2 ). It starts with a markedly different behaviour of the 7.5 g•cm -2 loading (electrodeposited by 50 CV cycles), which shows the lowest Cs value. Probably 50 CV cycles are not enough to form the PoPD structure completely (and the electrode surface is not fully covered (Figure SI-3)), which is in agreement with the absence of well-defined redox peaks in the CV response shown in Figure 7b inset. Following this value, Cs shows a significant increase in the range of 7.5 -22 g•cm -2 , reaching to values as high as ~ 434 F•g -1 at 22 g•cm -2 .

Fig. 8 .

 8 Fig. 8. Galvanostatic charge/discharge curves of PoPD supercapacitor at 30 C current density. (a) and (b) are the voltage-time graphs at different time scale showing electrode's long term stability. (c) discharge capacities as a function of cycle number for the cell. Inset figures: a graph belongs to the first cycle dQ vs V curve and cross section FEG-SEM image of the PoPD coated on graphite electrode.

2 Figure SI- 2 :

 22 Figure SI-2: Electrochemical impedance diagram (ΔE/ΔI()) at 0.05V of the PoPD film electrodeposited on a gold coated quartz resonator and measured in 0.5 M NaCl solution at pH=2.

Figure SI- 3 :

 3 Figure SI-3: FEG-SEM image of the early stage of the PoPD thin film formation on the gold electrode of a quartz resonator after 50 CV cycles.

Figure SI- 4 :

 4 Figure SI-4: The FEG-SEM images of the fresh (A and B) and after 8000 cycles in Swagelok cells in 0.5M NaCl, pH=2 (C and D). The EDX spectra of the cycled electrode surface indicating the presence of the glass fiber residue originating from the separator used in Swagelok cells.

Table SI - 1 :

 SI1 Estimated values for K i (kinetics of transfer) and G i (the inverse of the transfer resistance), Rti (transfer resistance) and fi (characteristic frequency) parameters extracted from the fitting results of ac-electrogravimetry measurements of the PoPD film electrodeposited on a gold coated quartz resonator and measured at -0.05 V et -0.25 V vs Ag/AgCl, measured in 0.5 M NaCl at pH=2.

Table SI - 2 :

 SI2 Papers about Polyaniline and Polypyrrole.

	PPY	3 A.g -1	KCl (1M)		576 F.g -1		82% (1000)	4
	PPy-NW	20 mv/s	H2SO4 (1M)	110 F.g -1		̴ 25% (5000)	5
	PANI-NW 20 mv/s	H 2 SO 4 (1M)	191 F.g -1		̴ 42% (5000)	5
	PANI	50 mV.s -1	H 2 SO 4 (0.5M) 451 F.g -1			6
	pH=2 PANI PANI	Species identification 4 mA.cm -2 20mV.s -1	Mi (g.mol -1 ) NaNO 3 (1M) H2SO4 (0.5M) 192 F.g -1 Ki (cm.s -1 ) 150 F.g -1	Gi (mol.s -1 .cm -2 .V -1 ) 67% (5000)	(.cm 2 ) Rti=1/FGi	7 8	(z) fi=Ki/πdf
		(i)
	at -0.05 V
	a	Cl -	35	37.70 × 10 -5		-2.45 × 10 -7	42.30	12.01
	s	H2O	18		1.10 × 10 -5		0.22 × 10 -7	471.10	0.35
	c1	Na +	23		0.16 × 10 -5		0.02 × 10 -7	4391.64	0.05
	c2	H +	1		0.63 × 10 -5		0.78 × 10 -7	132.03	0.20
	at -0.25 V
	a	Cl -	35	53.40 × 10 -5		-12.90 × 10 -7	7.46	17.01
	s	H2O	18		2.23 × 10 -5		0.11 × 10 -7	933.58	0.71
	c1	Na +	23		1.57 × 10 -5		0.78 × 10 -7	244.38	0.30
	c2	H +	1		2.20 × 10 -5		2.20 × 10 -7	47.13	0.70
	Material	Scan Rate	Electrolyte		Specific		Capacity	retention	reference
		or Current			capacitance	(number of cycles)
		density
	PPy	50 mV.s -1	H 2 SO 4 (0.5M) 230 F.g -1		87% (1000)	1
	PPy	100 mV.s -1 H 2 SO 4		215 F.g -1			2
	PPy	50mV.s -1	H 2 SO 4 (0.5M) 250 F.g -1		82% (1000)	3
					37

ACKNOWLEDGEMENTS

This work was sponsored by MESRSFC (Ministère de l'Enseignment Supérieur et de la Recherche Scientifique et de la Formation des Cadres of Morocco) and CNRST (Centre National pour la Recherche Scientifique et Technique) (Project number PPR/2015/72). The authors thank Mrs. Stéphanie Delbrel and Mrs. Françoise Pillier for FEG-SEM analysis and Mr. Cyrille Bazin for XRD analysis (LISE, UMR8235, Sorbonne University, France). We also appreciate the Erasmus + Program for the financial support. R. Demir-Cakan acknowledges the French Embassy for the Visiting Researcher Fellowship.

ASSOCIATED CONTENT

Supporting information is available, presenting the FEG-SEM image of the early stages of the PoPD film formation, the electrochemical impedance diagrams of PoPD films and the tables presenting the fitting parameters of the ac-electrogravimetric data. hexagonal-like crystals were observed on the surface that is a typical phenomenon when Zn is electro-coated in the presence of chloride (9,10). As a whole, the amount of electrolyte matters on such dramatic morphological changes. In our study, during two electrode Swagelok cell measurements, only 0.15 ml of the electrolyte has been used and in that way we have limited the morphological changes, at least for the 8000 GCD cycles under the conditions of our study.

Highlights