Jeronimo Agrisuelas
email: jeronimo.agrisuelas@uv.es

Claude Gabrielli

J J García-Jareño

Hubert Perrot

Ozlem Sel

F Vicente

Polymer dynamics in thin p-type conducting films investigated by acelectrogravimetry. Kinetics aspects on anion exclusion, free solvent transfer, and conformational changes in poly(o-toluidine)

Keywords: Intrinsically conducting polymer, Electrochemical quartz crystal microbalance, Electrochemical impedance spectroscopy, ac-Electrogravimetry, Conformational changes

  Polymer dynamics in thin p-type conducting films investigated by ac-electrogravimetry. Kinetics aspects on anion exclusion, free solvent transfer, and conformational changes in poly(o-toluidine)

Introduction

In order to convert conceptual and experimental findings to practical and relevant technological devices based on intrinsically conducting polymers (ICPs), several parameters should be optimized (molecular structure or solvent-electrolyte medium among others) [START_REF] Van Dyke | Electrochemical investigations of electronically conductive polymers. 4. Controlling the supermolecular structure allows charge transport rates to be enhanced[END_REF][START_REF] Heinze | Electrochemistry of Conducting Polymers-Persistent Models and New Concepts[END_REF]. The conduction mechanism and the electrolyte as a counter ion source have received substantial attention [START_REF] Novak | Electrochemically active polymers for rechargeable batteries[END_REF][START_REF] Kaiser | Electronic transport properties of conducting polymers and carbon nanotubes[END_REF]. Inevitably, ICPs immersed in aqueous or non-aqueous solutions entail the transfer of solvent molecules in permselective conditions. This aspect is taken into account either qualitatively or quantitatively but the studies are commonly restricted to overall electrochemical process. Specially, Hillman and coworkers have extensively studied the mobile species transfer processes with the intention of achieving a deeper understanding on the redox dynamics of ICPs (spatial distributions, viscoelasticity, solvent effect, mechanisms) [START_REF] Bruckenstein | Consequences of Thermodynamic Restraints on Solvent and Ion Transfer During Redox Switching of Electroactive Polymers[END_REF][START_REF] Hillman | Dynamic separation of mobile species transfer processes at polymer modified electrodes using the electrochemical quartz crystal microbalance[END_REF][START_REF] Brown | Manipulation of electroactive polymer film viscoelasticity: the roles of applied potential and frequency[END_REF][START_REF] Hillman | Spatial distributions of polymer and mobile species in poly(o-toluidine) films[END_REF][START_REF] Hillman | Solvent effects on the electrochemical p-doping of PEDOT[END_REF][START_REF] Bruckenstein | Ion and solvent transfers accompanying redox switching of polypyrrole films immersed in divalent anion solutions[END_REF].

Solvent inside the films acts as a plasticizer, lubricating the inter-molecular motions and relaxing the film. The inner solvent matrix of a polymeric film can be divided in two kinds of populations: molecules intimately bound to polymeric structure and free molecules. The amount of free molecules changes during the charge/discharge process of the polymer by the transfer between the polymer and solution. The extent of solvent transfer depends on the solution composition and it does not require to have integer stoichiometry coefficients [START_REF] Bruckenstein | Consequences of Thermodynamic Restraints on Solvent and Ion Transfer During Redox Switching of Electroactive Polymers[END_REF]. The influence of solvent transfer requires a good understanding and quantification in technological applications where the polymer film is in contact with a liquid phase. For example, the driven volume variation in biomimetic gel motors depends on the number of counterions and solvent molecules exchanged during the electrochemical reactions [START_REF] Otero | Artificial Muscles: A Tool To Quantify Exchanged Solvent during Biomimetic Reactions[END_REF].

Solvent transfer can be due to different electrochemical-driving forces in

ICPs [START_REF] Bruckenstein | Consequences of Thermodynamic Restraints on Solvent and Ion Transfer During Redox Switching of Electroactive Polymers[END_REF][START_REF] Hillman | Transport of Neutral Species in Electroactive Polymer-Films[END_REF][START_REF] Bruckenstein | Solvent and Salt Transfer Upon Redox Switching of Electroactive Polymers[END_REF]: i) Donnan equilibrium to maintain equal chemical potentials inside and outside of the film, ii) changes of the volume in the hydration matrix inside the film owing to ion movements or iii) changes in the macromolecular structure of the reactive polymer, conformational motions. Although there is no a priori relationship with the injected charge and the non-charged molecules, neutral solvent transfer can be expected to be controlled by a non-ideal thermodynamic behavior (non-linear film solvation with charge state) [START_REF] Hillman | Ion, solvent and polymer dynamics in polyaniline conducting polymer films[END_REF][START_REF] Hillman | Modelling mobile species dynamics within electroactive films under mixed thermodynamic and kinetic control[END_REF]. As an example, this control can take place in films where slower electronic transfer along dynamic chains introduces a slow conformational change with slow noncharged species transfer. However, under specific experimental conditions or for very thin films, the transfer of neutral solvent can be kinetically controlled (although "indirectly") because both insertion/expulsion of ions and movements of polymeric backbones (both kinetically controlled) have a "direct" effect on the solvent transfer [START_REF] Agrisuelas | Kinetic and Mechanistic Aspects of a Poly(o-Toluidine)-Modified Gold Electrode. 2. Alternating Current Electrogravimetry Study in H2SO4 Solutions[END_REF][START_REF] Agrisuelas | Effects of anion size on the electrochemical behavior of H2SO4-structured poly(o-toluidine) films. An ac-electrogravimetry study in acid solutions[END_REF]. In the view of that, the transfer of neutral solvent, or in other words, the hydration extent of ICPs could be modulated varying the oxidation state via an external potential.

The main difficulty in the study of solvent transfers during electrochemical reaction in ICPs is the neutral character of these molecules. For these reasons, complementary techniques should be used together with electrochemical techniques like quartz crystal microbalance to detect mass changes in the polymeric film or probe beam deflection to detect concentration gradients of species transfer close to the polymer|solution interface [START_REF] Vilas-Boas | A combined electrochemical quartz-crystal microbalance probe beam deflection (EQCM-PBD) study of solvent and ion transfers at a poly[Ni(saltMe)]-modified electrode during redox switching[END_REF][START_REF] Hillman | Kinetics and mechanism of the electrochemical p-doping of PEDOT[END_REF][START_REF] Pruneanu | Electrochemical quartz crystal microbalance study of the influence of the solution composition on the behaviour of poly(aniline) electrodes[END_REF][START_REF] Inzelt | Simultaneous chronoamperometric and quartz crystal microbalance studies of redox transformations of polyaniline films[END_REF].

In this paper, ac-electrogravimetry is used to characterize ions and solvent transfer in the polymer|solution interface. This technique was clearly proven to be helpful in polymer studies as an easy and fast identification methodology for cation and anion transfers involved in the charge compensation process together with a noticeable kinetic information when electrochemical and mass impedance spectroscopy are discussed together [START_REF] Gabrielli | Ac-electrogravimetry study of ionic exchanges on a polypyrrole modified electrode in various electrolytes[END_REF][START_REF] Rubin | Electrochemical and electrogravimetric behaviors of conducting polymer. Theoretical aspects and application to co-polymer films based on juglone[END_REF][START_REF] Kim | Proton transport properties in hybrid membranes investigated by ac-electrogravimetry[END_REF][START_REF] Agrisuelas | Ionic and Free Solvent Motion in Poly(azure A) Studied by ac-Electrogravimetry[END_REF][START_REF] Agrisuelas | Identification of Processes Associated with Different Iron Sites in the Prussian Blue Structure by in Situ Electrochemical, Gravimetric, and Spectroscopic Techniques in the dc and ac Regimes[END_REF]. To separate among anion, cation and solvent contributions from the global electrogravimetric response, it is necessary to find experimental conditions where the time constants of these three transfers are fairly different [START_REF] Gabrielli | Ions and solvent transport across conducting polymers investigated by ac electrogravimetry. Application to polyaniline[END_REF]. Experimentally, this situation is not always achieved, especially in complex electroactive systems.

A different strategy is to find an appropriate model and procedure for analyzing the experimental results. If that is found, isolated kinetic studies of solvent transfer can be performed by ac-electrogravimetry. As a first approximation, partial electrogravimetric responses can be mathematically obtained from the global response.

An anion-solvent transfer can be analyzed if the cation contribution is eliminated. In a similar manner, if the anion contribution is eliminated, a partial cation-solvent transfer can be obtained [START_REF] Gabrielli | Ions and solvent transport across conducting polymers investigated by ac electrogravimetry. Application to polyaniline[END_REF][START_REF] Gabrielli | Charge compensation process in polypyrrole studied by ac electrogravimetry[END_REF]. Recently, the electrogravimetric response of the global free solvent transfer was separated from the electrogravimetric response of ions transfer [START_REF] Agrisuelas | Kinetic and Mechanistic Aspects of a Poly(o-Toluidine)-Modified Gold Electrode. 2. Alternating Current Electrogravimetry Study in H2SO4 Solutions[END_REF][START_REF] Agrisuelas | Effects of anion size on the electrochemical behavior of H2SO4-structured poly(o-toluidine) films. An ac-electrogravimetry study in acid solutions[END_REF]. Now, a refined procedure for the interpretation of ac-electrogravimetry results is proposed to study the anion p-doping of ICPs. Here, the anion exclusion effect is taken into account [START_REF] Hillman | Transport of Neutral Species in Electroactive Polymer-Films[END_REF][START_REF] Bund | Effect of the solvent and the anion on the doping/dedoping behavior of poly(3,4-ethylenedioxythiophene) films studied with the electrochemical quartz microbalance[END_REF][START_REF] Plieth | The role of ion and solvent transport during the redox process of conducting polymers[END_REF]. The anion insertion during the charge balance excludes a specific number of solvent molecules equivalent to the anion volume, which are expelled from the film during the oxidation. During the reduction, the free volume liberated by anions is rapidly occupied by a number of free solvent molecules equivalent to the anion volume.

Therefore, other mass variations during the experiment should be due to the free solvent transfer owing to the formation/creation of free volume inside the film by chain motions or by accommodation of the charged species in the reactive polymer. Consequently, the kinetics of this free solvent transfer should be a "fingerprint" to the kinetics of conformational changes in the films. A priori, this theoretical approach allows both processes to be decoupled.

A perusal of the literature reveals that very few studies have directly addressed the kinetics of conformational changes of polymers; partly due to the difficulties reported about the appropriate techniques available. Recently, Otero and coworkers emphasized that conformational and structural magnitudes induced by electrochemical reactions are not included in the chemical kinetic model of ICPs [START_REF] Otero | Biomimetic intracellular matrix (ICM) materials, properties and functions. Full integration of actuators and sensors[END_REF][START_REF] Otero | Structural and Biomimetic Chemical Kinetics: Kinetic Magnitudes Include Structural Information[END_REF]. They show an intense experimental procedure to explore the change of the intermolecular forces during the electrochemical reactions of polymers. The investigation of the global behavior took into account all the electrochemical processes involved between the modified electrode and the solution and the global behavior was explained through structure and thermodynamic phenomena (structural chemical kinetics).

The motivation of this work is to show a good approximation of the most physically realistic description of the system where ICP changes its volume to store inserted ions or solvent molecules by conformational arrangements during the electrochemical transitions [START_REF] Kaneto | Contribution of conformational change of polymer structure to electrochemomechanical deformation based on polyaniline[END_REF][START_REF] Lizarraga | Quasi-Equilibrium Volume Changes of Polyaniline Films upon Redox Switching. Formal Potential Distribution and Configurational Modeling[END_REF][35][START_REF] Lee | Conformations of Polyaniline Molecules Adsorbed on Au(111) Probed by in Situ STM and ex Situ XPS and NEXAFS[END_REF]. A representative p-type ICP were chosen to test the applicability of the proposed model to a dynamic molecular system of technological importance in aqueous media and to explore quantitatively the dynamics of thin films.

Recent works clearly prove that the poly(o-toluidine) (POT) is a good candidate [START_REF] Agrisuelas | Kinetic and Mechanistic Aspects of a Poly(o-Toluidine)-Modified Gold Electrode. 2. Alternating Current Electrogravimetry Study in H2SO4 Solutions[END_REF][START_REF] Agrisuelas | Effects of anion size on the electrochemical behavior of H2SO4-structured poly(o-toluidine) films. An ac-electrogravimetry study in acid solutions[END_REF].

POT is a polyaniline derivative where -CH3 group in the polymer backbone causes greater electron localization and interchain separation than polyaniline [START_REF] Jozefowicz | X-Ray Structure of the Polyaniline Derivative Poly(ortho-Toluidine) -the Structural Origin of Charge Localization[END_REF]. This fact favours the formation of two time-stable conducting forms with polarons or bipolarons.

Moreover, literature clearly exhibits their increasing technological interest for this polymer in electrochromic devices [START_REF] Kumar | Poly(o-toluidine) polymer as electrochromic material[END_REF], biofuel cell-based devices [START_REF] Raoof | A novel, effective and low cost catalyst for methanol oxidation based on nickel ions dispersed onto poly(o-toluidine)/Triton X-100 film at the surface of multi-walled carbon nanotube paste electrode[END_REF] or biosensors [START_REF] Borole | Glucose oxidase electrodes of poly(o-anisidine), poly(o-toluidine), and their copolymer as biosensors: A comparative study[END_REF].

Ac-electrogravimetry: Theory and background for anion exclusion in dynamic polymers

During the redox reaction of a conducting polymer in aqueous or non-aqueous media, transfer of ionic species can compensate the excess or defect of charge inside a polymer film, eq [START_REF] Van Dyke | Electrochemical investigations of electronically conductive polymers. 4. Controlling the supermolecular structure allows charge transport rates to be enhanced[END_REF], the universally accepted concept of coupled electron/ion transfer.

Most conducting polymers 〈P〉 lose electrons when the neutral chains are submitted to anodic potentials generating positive charges with consequent insertion of charge balancing anions from the solution (p-doping).

Often, the amount of free solvent molecules (not structural) inside a polymeric matrix is a function of the oxidation state of the polymer to satisfy osmotic and volume constraints [START_REF] Bruckenstein | Ion and solvent transfers accompanying redox switching of polypyrrole films immersed in divalent anion solutions[END_REF]. The insertion of ionic molecules to balance the charge of the oxidized film involves the displacement of inner solvent molecules. In the simplest situation when the volume of the holes in the ionic channels remains constant like in rigid polymers, the amount of, for example, water molecules which can be expelled out from the film during the anion insertion depends on the volume of anions (Figure 1), therefore,

𝑉 A -= 𝑛𝑉 H 2 O .
In a reverse process, free water molecules from solution can occupy the space left by anions during the anion expulsion. This fact is known as exclusion [START_REF] Hillman | Transport of Neutral Species in Electroactive Polymer-Films[END_REF][START_REF] Plieth | The role of ion and solvent transport during the redox process of conducting polymers[END_REF][START_REF] Hillman | Mobile Species Uptake by Polymer-Modified Electrodes[END_REF].

〈P, 𝑛H 2 O〉 + A - 𝑘 e - ′ ,𝑘 A - ← 𝑘 e -,𝑘 A - ′ → 〈P + , A -〉 + e -+ 𝑛H 2 O (1)
where 〈P,𝑛H 2 O〉 is the hydrated polymer and 〈P + ,A -〉 is the polymer matrix doped with anions A -, 𝑛 being the number of free water molecules excluded during A -insertion. The parameters, 𝑘 A -and 𝑘 e -are the kinetic constants of anion and electron expulsion from polymer, respectively, whereas, the parameters 𝑘 A - ′ and 𝑘 e - ′ are the kinetic constants of anion and electron insertion to polymer, respectively.

By assuming that anions transport through very thin films and in electrolytes with high ionic strength are fast enough, the kinetic constants only depend on the resistance of anions to cross the polymer|solution interface [START_REF] Gabrielli | Ac-Electrogravimetry Study of Electroactive Thin Films. I. Application to Prussian Blue[END_REF][START_REF] Gabrielli | Ac-Electrogravimetry Study of Electroactive Thin Films. II. Application to Polypyrrole[END_REF]. Then, the kinetic constants of both electron and anion transfer are equal as follows 𝑘 e -= 𝑘 A - ′ and 𝑘 e - ′ = 𝑘 A -.

Gabrielli et al. have extensively described that for very thin polymer films [START_REF] Gabrielli | Ac-electrogravimetry study of ionic exchanges on a polypyrrole modified electrode in various electrolytes[END_REF][START_REF] Gabrielli | Ions and solvent transport across conducting polymers investigated by ac electrogravimetry. Application to polyaniline[END_REF][START_REF] Gabrielli | Ac-Electrogravimetry Study of Electroactive Thin Films. I. Application to Prussian Blue[END_REF][START_REF] Gabrielli | Ac-Electrogravimetry Study of Electroactive Thin Films. II. Application to Polypyrrole[END_REF][START_REF] Gabrielli | ac Electrogravimetry on conducting polymers. Application to polyaniline[END_REF][START_REF] Gabrielli | Charge transport in electroactive thin films investigated by ac electrogravimetry[END_REF], the change of the concentration of the anions, 𝛥𝐶 A -, when a small sine wave potential perturbation, 𝛥𝐸, is applied as a function of perturbation frequency is:

𝛥𝐶 A - 𝛥𝐸 (𝜔) = -𝐺 A - 𝑗𝜔𝑑 𝑓 + 𝐾 A - (2)
where 𝑗 = √-1, 𝜔 is the angular frequency, which equals to 2𝜋𝑓, 𝑓 is the perturbation frequency, 𝑑 𝑓 is the thickness of the polymer film,

𝐾 A -= 𝑘 A -+ 𝑘 A - ′ 𝐶 A - 𝑠𝑜𝑙 (3)
and

𝐺 A -= -𝑏 A -𝑘 A -(𝐶 A --𝐶 A - 𝑚𝑖𝑛 ) + 𝑏 A - ′ 𝑘 A - ′ (𝐶 A - 𝑚𝑎𝑥 -𝐶 A -)𝐶 A - 𝑠𝑜𝑙 (4)
where, 𝐶 A -is the concentration of A -in the polymer, 𝐶 A - 𝑚𝑎𝑥 is the maximum concentration of the sites available for the insertion in the polymer, 𝐶 A - 𝑚𝑖𝑛 is the minimum concentration of the occupied sites and 𝐶 A - 𝑠𝑜𝑙 is the concentration of species in the solution. The parameters, 𝑘 A -and 𝑘 A - ′ are the above mentioned kinetic rate constants activated by potential,

𝑘 A -= 𝑘 A -00 𝑒𝑥𝑝(𝑏 A -(𝐸 -𝐸 0′ )) and 𝑘 A - ′ = 𝑘 A -00 𝑒𝑥𝑝(𝑏 A - ′ (𝐸 -𝐸 0′ )),
respectively where 𝐸 0′ is the formal potential and 𝑘 A -00 is a preexponential factor of the kinetic rate constants independent of applied potential 𝐸.

The parameters, 𝑏 A -and 𝑏 A - ′ are the Tafel coefficients related to the number of electrons (𝑛 e -) involved in the faradaic process and the symmetry of direct and reverse transition with respect to the formal redox potential [START_REF] Agrisuelas | Innovative Combination of Three Alternating Current Relaxation Techniques: Electrical Charge, Mass, and Color Impedance Spectroscopy. Part I: The Tool[END_REF][START_REF] Agrisuelas | Innovative Combination of Three Alternating Current Relaxation Techniques: Electrical Charge, Mass, and Color Impedance Spectroscopy. Part II: Prussian Blue reversible arrow Everitt's Salt Process[END_REF]:

𝑏 A --𝑏 A - ′ = 𝐵 A -= 𝑛 e -𝐹 𝑅𝑇 ( 5
)
where 𝑅 is the gas constant and 𝑇 is the absolute temperature.

In a more sophisticated and realistic model for anion p-doping of ICPs, electrochemical reactions induce conformational changes such as swelling, shrinking, compaction or relaxation processes owing to repulsive/attractive interactions inside the film or formation/destruction of molecular covalent bonds [START_REF] Otero | Structural and Biomimetic Chemical Kinetics: Kinetic Magnitudes Include Structural Information[END_REF]. The reactive film could act like a molecular pump during a sinusoidal potential perturbation. Accordingly, a constant thickness of films cannot be considered since it may change depending on the polarization potential [START_REF] Kaneto | Contribution of conformational change of polymer structure to electrochemomechanical deformation based on polyaniline[END_REF][START_REF] Lizarraga | Quasi-Equilibrium Volume Changes of Polyaniline Films upon Redox Switching. Formal Potential Distribution and Configurational Modeling[END_REF][35][START_REF] Kim | Ions/Solvent Exchanges and Electromechanical Processes in Hexasulfonated Calix[END_REF]. Eq (2) can be slightly modified in terms of surface concentration considering only a bidimensional surface [START_REF] Agrisuelas | Kinetic and Mechanistic Aspects of a Poly(o-Toluidine)-Modified Gold Electrode. 2. Alternating Current Electrogravimetry Study in H2SO4 Solutions[END_REF][START_REF] Agrisuelas | Effects of anion size on the electrochemical behavior of H2SO4-structured poly(o-toluidine) films. An ac-electrogravimetry study in acid solutions[END_REF]:

𝑑 𝑓 𝛥𝐶 A - 𝛥𝐸 (𝜔) = 𝛥Γ A - 𝛥𝐸 (𝜔) = -𝐺 A - 𝑗𝜔 + 𝐾 A - ′ (6)
where 𝐾 ′ = 𝐾 𝑑 𝑓 ⁄ and, then, 𝑗𝜔𝑑 𝑓 changes to 𝑗𝜔.

Based on this model, eq [START_REF] Hillman | Dynamic separation of mobile species transfer processes at polymer modified electrodes using the electrochemical quartz crystal microbalance[END_REF] allows the theoretical electrochemical impedance, 𝛥𝐸 𝛥𝐼(𝜔) ⁄ to be calculated only for the involved anion transfer. Thus,

𝛥𝐸 𝛥𝐼 (𝜔) = 𝑅 𝑢 + 1 𝑗𝜔𝐶 𝑑𝑙 + 1 𝑍 𝑓 (7)
where 𝑅 𝑢 is the uncompensated solution resistance, 𝐶 𝑑𝑙 is the double layer capacity and 𝑍 𝑓 is the faradaic impedance defined as:

𝑍 𝑓 = 1 𝑗𝜔𝑆𝐹𝑧 A -[ 𝐺 A - 𝑗𝜔 + 𝐾 A - ′ ] (8)
where 𝑧 A -is the valence of the anion, 𝐹 is the Faraday constant, and 𝑆 is the active electrode surface area. Eq ( 8) is close to classical Randless equivalent circuit where the charge transfer resistance and the Warburg element were replaced by a simple 𝑅,𝐶 circuit.

By subtracting the non-faradaic contributions, the electrical charge/potential transfer function, 𝛥𝑞 𝛥𝐸(𝜔) ⁄ , can be calculated as:

𝛥𝑞 𝛥𝐸 (𝜔) = 1 𝑗𝜔 1 𝛥𝐸 𝛥𝐼 (𝜔) -𝑅 𝑢 -𝐶 𝑑𝑙 (9)
and by using the eq ( 7) and ( 6), this theoretical transfer function is equal to:

𝛥𝑞 𝛥𝐸 (𝜔) = 𝑆𝐹𝑧 A -[ 𝐺 A - 𝑗𝜔 + 𝐾 A - ′ ] (10)
Here, non-charged species (free H 2 O) do not participate in the charge balance but the insertion/expulsion of these molecules involves effective mass changes of the reactive film. Therefore, considering eq [START_REF] Hillman | Dynamic separation of mobile species transfer processes at polymer modified electrodes using the electrochemical quartz crystal microbalance[END_REF], the mass/potential transfer function, 𝛥𝑚 𝛥𝐸(𝜔) ⁄ , is easily deduced to:

𝛥𝑚 𝛥𝐸 (𝜔) = 𝑆 (𝛿 A -𝑀 A -[ 𝐺 A - 𝑗𝜔 + 𝐾 A - ′ ] + 𝑛𝛿 H 2 O 𝑀 H 2 O [ 𝐺 H 2 O 𝑗𝜔 + 𝐾 H 2 O ′ ]) (11)
where 𝑀 i is the molar mass of the 𝑖 species involved (A -or H 2 O) and 𝐾 H 2 O

′

and 𝐺 H 2 O have a similar meaning as eq (3) and ( 4), respectively. The parameter 𝛿 A -is +1 due to the increase of mass when the polymer is oxidized, eq [START_REF] Van Dyke | Electrochemical investigations of electronically conductive polymers. 4. Controlling the supermolecular structure allows charge transport rates to be enhanced[END_REF]. Accordingly, the 𝛿 H 2 O is -1 due to the decrease of mass resulting from the expulsion of free water molecules induced by the insertion of anions (exclusion).

In a rigid film, kinetics of free H 2 O transfer are governed by kinetics of anion transfer to satisfy volume constraints (Figure 1) [START_REF] Bruckenstein | Ion and solvent transfers accompanying redox switching of polypyrrole films immersed in divalent anion solutions[END_REF]. Consequently, 𝐺 A -= 𝐺 H 2 O and

𝐾 A - ′ = 𝐾 H 2 O
′ can be easily assumed and, then, eq [START_REF] Otero | Artificial Muscles: A Tool To Quantify Exchanged Solvent during Biomimetic Reactions[END_REF] is simplified to:

𝛥𝑚 𝛥𝐸 (𝜔) = 𝑆(𝛿 A -𝑀 A -+ 𝑛𝛿 H 2 O 𝑀 H 2 O ) [ 𝐺 A - 𝑗𝜔 + 𝐾 A - ′ ] (12)
In a reactive polymer with a constant thickness in all polarization potentials considering only the exclusion effect, the capacitive response would show a loop owing to a single electrochemical transfer (Figure 2a), in this case with charge balancing anions.

On the contrary, the global electrogravimetric response depends on anion mass and anion volume. As an example, a loop in the first quadrant of the global electrogravimetric response would appear because the mass of one anion (in this case 80 g mol -1 ) is higher than the mass of four (𝑛 = 4) displaced water molecules (in this case

𝑛𝑀 H 2 O =4•18=72 g mol -1
) due to the anion volume (Figure 2b).

For dynamic polymers, 𝛥𝑞 𝛥𝐸(𝜔) ⁄ has a similar response to the one observed for rigid polymers. However, electrogravimetric responses, during the same anion transfer, should depend on the change of film thickness associated to the conformational changes and the number of H 2 O transferred out of exclusion effect (Figure 3). A general theoretical expression for 𝛥𝑚 𝛥𝐸(𝜔) ⁄ considering both free water transfers from pure anion exclusion and free water transfer from induced conformational movements can be expressed as:

𝛥𝑚 𝛥𝐸 (𝜔) = 𝑆(𝛿 A -𝑀 A -+ 𝑛𝛿 H 2 O 𝑀 H 2 O ) [ 𝐺 A - 𝑗𝜔 + 𝐾 A - ′ ] + 𝑆(𝑛 ′ 𝛿 H 2 O' ′ 𝑀 H 2 O ) [ 𝐺 H 2 O' 𝑗𝜔 + 𝐾 H 2 O' ′ ] (13)
where 𝛿 In the situation (a) in Figure 3, the pure exclusion effect can be enhanced due to the contraction of film (-𝛥𝑑 𝑓 ) during the insertion of anions, so,

𝛿 H 2 O' ′ = 𝛿 H 2 O = -1.
The response leads to a net loss of mass during the oxidation of polymer (loop in the third quadrant of Figure 4) [START_REF] Agrisuelas | Innovative Combination of Three Alternating Current Relaxation Techniques: Electrical Charge, Mass, and Color Impedance Spectroscopy. Part I: The Tool[END_REF]. In the situation (b) of Figure 3, the pure exclusion effect is balanced due to the expansion of film thickness (+𝛥𝑑 𝑓 ). Therefore, 𝛿 H 2 O' ′ = 𝛿 H 2 O = 0, 𝑛 = 𝑛 ′ = 0 and the electrogravimetric response correspond to a pure anion transfer (loop in the first quadrant of Figure 4). Finally, a large expansion of film thickness (+2𝛥𝑑 𝑓 ) creates holes occupied by free H 2 O from solution as situation (c) of Figure 3 shows. Here, 𝛿 H 2 O = 0, 𝑛 = 0 and the electrogravimetric response appears in the first quadrant of ′ in eq [START_REF] Bruckenstein | Solvent and Salt Transfer Upon Redox Switching of Electroactive Polymers[END_REF]. Thus, if this procedure is assumed, the kinetic constants of free water molecules experimentally obtained from ac-electrogravimetry, 𝐺 H 2 O' and

𝐾 H 2 O' ′ , could
approximate the kinetics of conformational changes of the polymer.

Experimental

Film preparation

The electrochemical polymer deposition and characterization were performed by cyclic voltammetry using an AUTOLAB potentiostat-galvanostat (PGSTAT302) and a lab-made oscillator acted as an Electrochemical Quartz Crystal Microbalance at room temperature (about 295±2 K). The three-electrode cell involved a reference electrode (Saturated Calomel Electrode or SCE, Tacussel XR 600) and a platinum counter electrode. The working electrode was a gold electrode (𝑆 = 0.3 cm 2 ) patterned on a 9

MHz quartz crystal resonator (TEMEX, France). All the solutions were prepared with deionized and double-distilled water. The potential of the working electrode is referred to the SCE reference electrode in all measurements.

POT thin films were synthetized for ac-electrogravimetry experiments on a gold/quartz electrode. The polymerization solution was 0.5 M H2SO4 (Fisher scientific, for trace analysis) and 0.2 M o-toluidine (Fluka). All chemicals were used as received.

POT was formed through 100 voltammetric cycles between -0.1 V and 0.9 V at 100 mV s -1 scan rate (which equals to ~ 40 µg cm -2 deposited film mass). The films generated following this procedure show a more open structure owing to the properties of bisulfate anions [START_REF] Zotti | Cyclic Potential Sweep Electropolymerization of Aniline -the Role of Anions in the Polymerization Mechanism[END_REF][START_REF] Elmansouri | Influence of the counter ion on the properties of poly(o-toluidine) thin films and their Schottky diodes[END_REF].

Film conditioning

POT film was 10 times cycled between -0.1 V and 0.9 V in 0.5 M H2SO4 at 100 mV s -1 until a stable cyclic voltammogram was obtained avoiding significant polymer degradation by hydrolysis [START_REF] Agrisuelas | Effects of anion size on the electrochemical behavior of H2SO4-structured poly(o-toluidine) films. An ac-electrogravimetry study in acid solutions[END_REF]. Afterwards, ac-electrogravimetry studies were performed at steady-state potentials in the range of 0.55 V to 0.1 V in the same solution.

ac-Electrogravimetry set-up

For ac-electrogravimetry studies, a fast quartz crystal microbalance was coupled with a four-channel frequency response analyzer (FRA, Solartron 1254) and a potentiostat (SOTELEM-PGSTAT Z1). The modified working electrode was polarized at a given potential and a sinusoidal potential perturbation with a small amplitude (10 mV rms) was superimposed between 65 kHz and 0.01 Hz. The microbalance frequency change, 𝛥𝑓 𝑚 and the alternating current response, 𝛥𝐼, were sent to the four channels FRA, which allowed the electrogravimetric transfer function, 𝛥𝑚 𝛥𝐸(𝜔) ⁄ and the electrochemical impedance, 𝛥𝐸 𝛥𝐼(𝜔) ⁄ , to be simultaneously obtained. Finally, the experimental data were fitted with the theoretical expressions given in Section 2 by means of Levenberg-Marquardt method [START_REF] Macdonald | Comparison and application of two methods for the least squares analysis of immittance data[END_REF].

Results and discussion

Kinetic model for POT films in acidic aqueous media

In acidic solutions, the voltammetric response of POT films has two successive quasi-reversible redox process independent from the existing anion (inset of Figure 5a) [START_REF] Henderson | Chronoamperometric resolution of ion and solvent transfers at a poly(o-toluidine) modified electrode by combined electrochemical quartz crystal microbalance (EQCM) and probe beam deflection (PBD)[END_REF][START_REF] Agrisuelas | Kinetic and Mechanistic Aspects of a Poly(o-toluidine)-Modified Gold Electrode. 1. Simultaneous Cyclic Spectroelectrochemistry and Electrogravimetry Studies in H2SO4 Solutions[END_REF][START_REF] Agrisuelas | Effects of anions size on the redox behavior of poly(o-toluidine) in acid solutions. An in situ vis-NIR cyclic spectroelectrogravimetry study[END_REF]. In the oxidation direction, the first redox process involves the transfer of anions (A -) to keep the electroneutrality during the leucoemeraldine-polaron (LE⇄P)

transitions. The second redox process corresponds to bipolaron-pernigraniline transitions (BP⇄PN) with the transfer of H 3 O + which is necessary for protonation/deprotonation of electroactive sites.

By cyclic spectro-electrogravimetry and ac-electrogravimetry [START_REF] Agrisuelas | Kinetic and Mechanistic Aspects of a Poly(o-Toluidine)-Modified Gold Electrode. 2. Alternating Current Electrogravimetry Study in H2SO4 Solutions[END_REF][START_REF] Agrisuelas | Kinetic and Mechanistic Aspects of a Poly(o-toluidine)-Modified Gold Electrode. 1. Simultaneous Cyclic Spectroelectrochemistry and Electrogravimetry Studies in H2SO4 Solutions[END_REF], a third reaction step was elucidated corresponding to the polaron-bipolaron (P⇄BP) transitions with specific and independently measurable electrochemical properties. The electrochemical mechanism can be summarized as:

LE + A - 𝑘 1 ← 𝑘 1 ′ → P + e -+ 𝑥 1 H 2 O (14)
P + A - 𝑘 2 ← 𝑘 2 ′ → BP + e -+ 𝑥 2 H 2 O ( 15
)
BP 𝑘 3 ′ ← 𝑘 3 → PN + e -+ H 3 O + ( 16
)
where 𝑥 𝑖 represents the net effective number of water molecules exchanged during the electrochemical transition and depends on the nature of the anion transferred [START_REF] Agrisuelas | Effects of anion size on the electrochemical behavior of H2SO4-structured poly(o-toluidine) films. An ac-electrogravimetry study in acid solutions[END_REF].

Moreover, the electrochemical transitions of eq [START_REF] Hillman | Ion, solvent and polymer dynamics in polyaniline conducting polymer films[END_REF] and [START_REF] Hillman | Modelling mobile species dynamics within electroactive films under mixed thermodynamic and kinetic control[END_REF] are kinetically faster than the electrochemical transitions of eq [START_REF] Agrisuelas | Kinetic and Mechanistic Aspects of a Poly(o-Toluidine)-Modified Gold Electrode. 2. Alternating Current Electrogravimetry Study in H2SO4 Solutions[END_REF].

Accordingly, LE⇄P (eq [START_REF] Hillman | Ion, solvent and polymer dynamics in polyaniline conducting polymer films[END_REF]) and P⇄BP (eq [START_REF] Hillman | Modelling mobile species dynamics within electroactive films under mixed thermodynamic and kinetic control[END_REF]) transitions are good candidates to test the anion exclusion model for dynamic polymers. Thus, ac-electrogravimetric experiments were carried out in the potential range of 0.55 V to 0.1 V in 0.5 M H2SO4 acid solution. In the potential range of 0.45 and 0.1 V, LE⇄P and P⇄BP transitions are expected as it was described elsewhere [START_REF] Agrisuelas | Kinetic and Mechanistic Aspects of a Poly(o-toluidine)-Modified Gold Electrode. 1. Simultaneous Cyclic Spectroelectrochemistry and Electrogravimetry Studies in H2SO4 Solutions[END_REF]. The experimental charge/potential transfer function results represented in a complex capacitance plane, 𝛥𝑞 𝛥𝐸(𝜔) ⁄ , show a typical loop at 0.25 V associated to one charge transfer balanced by HSO 4 -(Figure 5a). Around 0.45 V, the partial charge transfer associated with the H 3 O + transfer must be taken into account and be separated. Moreover, the quasi-vertical branch observed at low frequencies and at potentials close to 0.1 V points to a slight contribution of a parasitic reaction which can be removed using a simple 𝑅 𝑝 , 𝐶 𝑝 circuit.

In accordance with previous works [START_REF] Agrisuelas | Kinetic and Mechanistic Aspects of a Poly(o-Toluidine)-Modified Gold Electrode. 2. Alternating Current Electrogravimetry Study in H2SO4 Solutions[END_REF], the theoretical equation of 𝛥𝑞 𝛥𝐸(𝜔) ⁄ for POT films in 0.5 M H2SO4 acid solution can be proposed from eq [START_REF] Bruckenstein | Ion and solvent transfers accompanying redox switching of polypyrrole films immersed in divalent anion solutions[END_REF] considering

𝑧 HSO 4 -= 𝑧 H 3 O + =
The non-ideality of impedance spectra (deviation from a perfect semicircle) can be mathematically corrected using a constant-phase element, (𝑗𝜔) 𝛼 . This fact is frequently observed in intercalation systems [START_REF] Gabrielli | Impedance Investigation of the Charge Transport in Film-Modified Electrodes[END_REF][START_REF] Mathias | An Alternating-Current Impedance Model Including Migration and Redox-Site Interactions at Polymer-Modified Electrodes[END_REF]. Here, (𝑗𝜔) 𝛼 is considered as a mathematical resource to achieve the best fittings and it does not cause any change in the physical meaning of the kinetic parameters [START_REF] Agrisuelas | Kinetic and Mechanistic Aspects of a Poly(o-Toluidine)-Modified Gold Electrode. 2. Alternating Current Electrogravimetry Study in H2SO4 Solutions[END_REF].

Eq [START_REF] Agrisuelas | Effects of anion size on the electrochemical behavior of H2SO4-structured poly(o-toluidine) films. An ac-electrogravimetry study in acid solutions[END_REF] was fitted to experimental 𝛥𝑞 𝛥𝐸(𝜔) ⁄ until convergence to obtain valuable kinetic information of both ions transfers. Solid line Figure 5a shows a good agreement between the experimental and simulation data using eq [START_REF] Agrisuelas | Effects of anion size on the electrochemical behavior of H2SO4-structured poly(o-toluidine) films. An ac-electrogravimetry study in acid solutions[END_REF]. Later, 𝐺 HSO 4 -,

𝐾 HSO 4 - ′ , 𝛼 HSO 4 -, 𝐺 H 3 O + , 𝐾 H 3 O + ′
and 𝛼 H 3 O + parameters are used for subsequent electrogravimetric analysis. 𝛼 HSO 4 -and 𝛼 H 3 O + obtained are close to the ideal system for both involved species (0.94±0.04).

Figure 5b shows the corresponding electrogravimetric response of POT film at 0.25 V. A single loop in the first quadrant indicates the increase of mass during the oxidation. Therefore, insertion of anions is expected to balance charge excess. For the multiparametric fitting process, eq (13) was used with similar modification as it is commented above in section 2. At more anodic potentials, the partial contribution to

+ (𝛿 H 2 O' ′ 𝑀 H 2 O ) [ 𝑛 ′ 𝐺 H 2 O' (𝑗𝜔) 𝛼 H 2 O' + 𝐾 H 2 O' ′ ] + (𝛿 H 3 O + 𝑀 H 3 O + ) [ 𝐺 H 3 O + (𝑗𝜔) 𝛼 H 3 O + + 𝐾 H 3 O + ′ ] (18)
In the present form, this function is over-parameterized. For this reason, to reduce the number of parameters, some of them were fixed. Thus, 𝛿 HSO 4 -𝑀 HSO 4 -is +97 g mol -1 , Determination of 𝒏 ′ . At this point, eq [START_REF] Agrisuelas | Effects of anion size on the electrochemical behavior of H2SO4-structured poly(o-toluidine) films. An ac-electrogravimetry study in acid solutions[END_REF] and ( 18) can explain properly the electrochemical behavior of POT films, as shown by the fits obtained (solid lines in Figure 5). However, 𝑛 ′ is still an unknown parameter and an apparent 𝑛 ′ 𝐺 H 2 O' was obtained. In this form, 𝑛 ′ 𝐺 H 2 O' cannot provide a valuable information on the conformational kinetics of the film but it may be refined.

𝛿 H 2 O 𝑀 H 2 O = -
As it was mentioned in section 2, 𝑛 ′ represents the number of water molecules transferred by changes of free volume inside the films owing to conformational changes during the electrochemical reactions. By taking into account the schemes provided in (eq (6)) as [START_REF] Agrisuelas | Kinetic and Mechanistic Aspects of a Poly(o-Toluidine)-Modified Gold Electrode. 2. Alternating Current Electrogravimetry Study in H2SO4 Solutions[END_REF][START_REF] Agrisuelas | Effects of anion size on the electrochemical behavior of H2SO4-structured poly(o-toluidine) films. An ac-electrogravimetry study in acid solutions[END_REF]:

𝑛 ′ 𝐺 H 2 O' 𝐾 H 2 O' ′ 𝐾 HSO 4 - ′ 𝐺 HSO 4 - = 𝑛 ′ 𝑑𝛤 H 2 O' 𝑑𝐸 | 𝜔⟶0 𝑑𝐸 𝑑𝛤 HSO 4 - | 𝜔⟶0 = 𝑛 ′ 𝑑𝛤 H 2 O' 𝑑𝛤 HSO 4 - | 𝜔⟶0 = 𝑛 ′ (19)
Figure 6 shows the number of water molecules excluded per anion transfer, 𝑛, as a reference and the number of transferred water molecules owing to the conformational changes of film during the electrochemical reaction of POT, 𝑛 ′ , calculated from eq [START_REF] Hillman | Kinetics and mechanism of the electrochemical p-doping of PEDOT[END_REF] between 0.45 V and 0.1 V. As it was commented above, 𝑛 is a constant parameter at all polarization potentials (4.7), however, 𝑛 ′ , changes between 1 and 2 water molecules owing to the conformational changes of film with a maximum around 0.25 V. This means that the free volume inside the film where H 2 O can be inserted owing to conformational changes depends on the oxidation state of film induced by the applied potential, or in other words, it depends on film structure and the amount of anions inside the film. As an example, the film is more oxidized (BP form) and anion doped at 0.45 V than at 0.1 V where POT is more reduced (LE form) and undoped of anions (Table 1).

Once

and ∆𝛤 HSO 4 -= 𝛤 HSO 4 - 𝑚𝑎𝑥 -𝛤 HSO 4 - 𝑚𝑖𝑛 is the increment between the minimum surface concentration of the sites occupied by anions in the polymer (𝛤 HSO 4 - 𝑚𝑖𝑛 ) and the maximum surface concentration of the sites available for anion insertion (𝛤 HSO 4 - 𝑚𝑎𝑥 ) during the redox conversion, and 𝐸 𝑝,HSO 4 -is the peak potential where 𝑑𝛤(𝐸) 𝑑𝐸 ⁄ is higher. Therefore, valuable information about 𝐵 HSO 4 -, ∆𝛤 HSO 4 -and 𝐸 𝑝,HSO 4 -can be obtained for each transition process [START_REF] Agrisuelas | Effects of anions size on the redox behavior of poly(o-toluidine) in acid solutions. An in situ vis-NIR cyclic spectroelectrogravimetry study[END_REF].

The peak at 0.124 V corresponds to LE⇄P and the peak at 0.384 V to the P⇄BP.

About two LE⇄P (28 nmol cm -2 ) per one P⇄BP (14 nmol cm -2 ) occur in the POT film during the potential scan. This difference could be due to two possible formation of bipolaronic forms (BP) inside the film: i) electrochemically by oxidation of an electroactive site next to a formed polaron (eq ( 22)) or ii) by electronic reconfiguration from two electrochemically formed polarons (eq [START_REF] Rubin | Electrochemical and electrogravimetric behaviors of conducting polymer. Theoretical aspects and application to co-polymer films based on juglone[END_REF]) [START_REF] Genies | Polyaniline films. Electrochemical redox mechanisms[END_REF][START_REF] Albery | Spectroscopic and electrochemical studies of charge transfer in modified electrodes[END_REF]:

Consequently, the electronic reconfiguration in POT would reduce the amount of BP, which can be formed electrochemically by a superimposed potential.

Ease of species transfer at the polymer|solution interface. From an electrical point of view [START_REF] Benito | An electrochemical impedance and ac-electrogravimetry study of PNR films in aqueous salt media[END_REF], kinetic parameters, 𝐺 𝑖 , are analog to the inverse of a charge transfer resistance at the polymer|solution interface, 1 𝐹𝐺 ⁄ = 𝑅 𝑐𝑡 . Therefore, it refers to the ease of species transfer during the electrochemical reactions of electroactive films.

For anions and H 2 O′ transfers, two peaks of 𝐺 𝑖 are expected because both species are involved in two different electrochemical processes, LE⇄P and P⇄BP. However, anions and solvent transfers show only one peak of 𝐺 𝑖 in Figure 8 centered at ~ 0.32 V where both transitions take place simultaneously (see Figure 7). This particular behavior was previously explained considering that one specie (anion or water) can be inserted in two structurally different sites of the film (P or BP sites) [START_REF] Agrisuelas | Kinetic and Mechanistic Aspects of a Poly(o-Toluidine)-Modified Gold Electrode. 2. Alternating Current Electrogravimetry Study in H2SO4 Solutions[END_REF].

The exponential dependence at potentials far from the peak potential agrees with the proposed model [START_REF] Agrisuelas | Usefulness of F(dm/dQ) function for elucidating the ions role in PB films[END_REF]:

𝐺 𝑖 = 𝐺 𝑖 𝑚𝑎𝑥 𝑐𝑜𝑠ℎ [ 𝐵 𝑖 2 (𝐸 -𝐸 𝑝,𝑖 )] (24)
where 𝑖 is HSO 4 -or H 2 O', 𝐺 𝑖 𝑚𝑎𝑥 represents the maximum value for this function at 𝐸= 𝐸 𝑝,𝑖 , and 𝐸 𝑝,𝑖 represent the peak potential of 𝐺 𝑖 .

By taking into account eq [START_REF] Bruckenstein | Consequences of Thermodynamic Restraints on Solvent and Ion Transfer During Redox Switching of Electroactive Polymers[END_REF], for the direct transitions (oxidation half-reactions):

𝐵 𝑖 2 = 𝑛 e -𝛼𝐹 𝑅𝑇 (25)
and for the reverse transitions (reduction half-reactions):

𝐵 𝑖 2 = 𝑛 e -(1 -𝛼)𝐹 𝑅𝑇 (26)
are obtained. Values of 𝛼=0.5 imply that the variation of 𝐺 𝑖 on the applied potential presents a symmetry with respect to the formal redox potential, and values for 𝐵 𝑖 are the same for the direct and reverse transitions. In any other case, 𝐵 𝑖 values are different, and the dependence of 𝐺 𝑖 on the applied potential should be asymmetrical.

At 298 K, 𝐵 𝑖 2 ⁄ = 39 V -1 for 𝑛 e -=2 and 𝛼=0.5, one electron for LE⇄P transitions which takes place mainly between 0.1 and 0.32 V and one electron for P⇄BP transitions which takes place mainly between 0.35 V and 0.55 V [START_REF] Agrisuelas | Effects of anion size on the electrochemical behavior of H2SO4-structured poly(o-toluidine) films. An ac-electrogravimetry study in acid solutions[END_REF]. Thus, it is possible to obtain an estimation of this parameter for the reverse and forward reactions from the slope of log(𝐺 H 2 O' ) versus 𝐸 plot (Figure 8).

Values obtained for the different slopes are close to 39 V -1 for HSO 4 -and H 2 O' as expected theoretically. Therefore, about 20 V -1 correspond to each of the transitions.

These results are in agreement with the 𝐵 𝐻𝑆𝑂 4 -values obtained from fittings of 𝑑Γ HSO 4 -(𝐸) 𝑑𝐸 ⁄ in Figure 7 where 𝐵 HSO 4 -for LE⇄P is 13 V -1 and 𝐵 HSO 4 -for P⇄BP is 18 V - 1 .

Previous studies of POT films immersed in different acids show relatively lower values as the anion interacts more strongly with films [START_REF] Agrisuelas | Effects of anion size on the electrochemical behavior of H2SO4-structured poly(o-toluidine) films. An ac-electrogravimetry study in acid solutions[END_REF]. Lateral interactions between randomly distributed electroactive sites during the electrochemical reaction of films reduce the theoretical value of 𝐵 𝑖 considering the Laviron model deduced from thermodynamic basis of the Frumkin-type isotherm [START_REF] Laviron | The use of linear potential sweep voltammetry and of a.c. voltammetry for the study of the surface electrochemical reaction of strongly adsorbed systems and of redox modified electrodes[END_REF]. Therefore, lateral interactions are more likely to occur during LE⇄P than during P⇄BP. This fact can be due to the extent of doping in the film, see Table 1. Higher amount of anions inside the film (as it occurs during P⇄BP) can separate the molecular chains by repulsion, as a result reduces the strength of lateral interactions. On the contrary, stronger lateral interaction can be expected at lower doped states of polymer as at potentials where LE⇄P takes place, reducing the distance between charged electroactive sites (inter-chain or intra-chain).

The evolution of 𝐺 HSO 4 -and 𝐺 H 2 O' with respect to the potential is very similar; therefore, they show a similar ease to cross the polymer|solution interface. This fact agrees with the proposed model, which assumes that non-charged species transfer can be kinetically controlled by charge injection in polymeric backbone to induce conformational changes. At 0.32 V, the formation of the polaronic structure of POT films which twists the structure (coil structures owing to the anion-POT interactions) results in more water molecules that are able to be ejected from the film owing to conformational changes (𝑛 ′ close to 2 in Figure 6) [START_REF] Petrova | Fully Doped Oligomers of Emeraldine Salt: Polaronic versus Bipolaronic Configuration[END_REF][START_REF] Huang | Optical-Properties of Polyaniline[END_REF]. Indirectly, this agrees with the expansion/contraction of film owing to the single/double bond transitions and to the accommodation of anions inside the film.

Rate of species transfer at the polymer|solution interface.

In general, the most important feature is that all transfer processes could be kinetically controlled by injected charge in the polymer, in other words, it can be modulated depending on the applied potential. Figure 9 shows 𝐾 H 2 O' ′ and 𝐾 A - ′ values obtained by fittings. Although similar rates are reached for both species, interestingly some differences were found completing mechanistic details of POT electrochemistry. Between 0.1 and 0.3 V where mainly LE⇄P takes place, the conformational changes of film allows a set of water molecules to be transferred faster than anions. The lower amount of anions inside the films could be the cause for this. On the contrary, the exclusion effect caused by anions is faster than the conformational changes when POT film is doped with anions. This takes place between 0.3 and 0.45 V where mainly P⇄BP occurs.

The maximum of transfer rate takes place in both species around 0.32 V, as it is the case in Figure 8. As above, the creation of polaronic forms is a serious structural perturbation which can involve conformational movements of the molecular chains making the non-charged species transfer faster [START_REF] Petrova | Fully Doped Oligomers of Emeraldine Salt: Polaronic versus Bipolaronic Configuration[END_REF][START_REF] Huang | Optical-Properties of Polyaniline[END_REF]. considering a pure exclusion of effect like in Figure 1 between 40 KHz and 1 mHz of frequency perturbation. The parameters used in eq [START_REF] Bruckenstein | Ion and solvent transfers accompanying redox switching of polypyrrole films immersed in divalent anion solutions[END_REF] and ( 12) are 𝐺

CONCLUSIONS

A -= 1x10 -5 mol cm -2 V -1 , 𝐾 A - ′ = 70 s -1 , 𝑧 A -= 1, 𝑆 = 0.3 cm 2 , 𝛿 A -𝑀 A -= +80 g mol -1 , 𝛿 H 2 O 𝑀 H 2 O = -18 g mol -1 . 𝑛 = 4,

  H 2 O' ′ indicates the direction of transfer and the number of molecules of H 2 O associated with conformational movements (𝑛 ′ ) can be equal or different than 𝑛. 𝐺 H 2 O'and 𝐾 H 2 O' ′ are kinetic parameters of free water transfer from induced structural movements. The method of elimination of the pure anion exclusion from electrogravimetric results to consider the contribution of other free water transfers is close to the method used by Bruckenstein and Hillman as a diagnostic for species interference[START_REF] Bruckenstein | Consequences of Thermodynamic Restraints on Solvent and Ion Transfer During Redox Switching of Electroactive Polymers[END_REF][START_REF] Hillman | Transport of Neutral Species in Electroactive Polymer-Films[END_REF][START_REF] Bruckenstein | Solvent and Salt Transfer Upon Redox Switching of Electroactive Polymers[END_REF].

Figure 4 .

 4 Figure 4. Simplistically, the electrogravimetric responses of three situations consider similar kinetics of anion transfer and conformational changes, 𝐺 A -= 𝐺 H 2 O' and 𝐾 A - ′ = 𝐾 H 2 O'

Figure 3 ,

 3 Figure 3, one electron transfer involves one monovalent anion transfer, the transfer of one

Figure 2 .

 2 Figure 2. Simulation in the complex plane of 𝛥𝑞 𝛥𝐸(𝜔) ⁄ (a) and 𝛥𝑚 𝛥𝐸(𝜔) ⁄ (b)

  therefore the volume of one anion exclude 4 water molecules (𝑉 A -= 4𝑉 H 2 O ). Black squares are global ac-electrogravimetric response, red circles are the partial ac-electrogravimetric response of anion and blue triangles are the partial acelectrogravimetric response of free H 2 O.

Figure 3 .

 3 Figure 3. Scheme of the electrode|polymer|solution electrode during p-doping of a polymer by anions (A -) considering the contraction of polymer (a), the expansion of polymer (b) and (c) together with the transfer of water molecules. 𝑘 A -and 𝑘 e -are the kinetic constants of anion and electron expulsion from polymer, respectively. 𝑘 A - ′ and 𝑘 e - ′ are the kinetic constants of anion and electron insertion to polymer, respectively. ∆𝑑 𝑓 is the film thickness increment. The different color of water molecules allows H 2 O from film and solution to be distinguished.

Figure 4 .= 1 ,

 41 Figure 4. Simulation in the complex plane of 𝛥𝑚 𝛥𝐸(𝜔) ⁄ considering the exclusion of effect between 40 KHz and 1 mHz of frequency perturbation together with changes of polymer thickness like in Figure 3. The parameters used in eq (13) are 𝐺 A -= 𝐺 H 2 O' = 1x10 -5 mol cm -2 V -1 , 𝐾 A - ′ = 𝐾 H 2 O' ′

Figure 5 .

 5 Figure 5. Experimental and theoretical charge/potential transfer function, 𝛥𝑞 𝛥𝐸(𝜔) ⁄ (a), and mass/potential transfer function, 𝛥𝑚 𝛥𝐸(𝜔) ⁄ (b), of POT film in 0.5 M H2SO4aqueous solution at 0.25 V. Cyclic voltammetry of a POT film between -0.1 V and 0.9 V at 100 mV s -1 in the same solution in the inset of (a).

Figure 6 .

 6 Figure 6. Number of water molecules (H 2 O) excluded per anion transfer, 𝑛, and number of transferred water molecules (H 2 O') owing to conformational changes, 𝑛 ′ , calculated from eq (19) between 0.45 V and 0.1 V of POT film in 0.5 M H2SO4 aqueous solution.

Figure 7 .

 7 Figure 7. Evolution of the derivative of the insertion law of HSO 4 -, 𝑑𝛤 HSO 4 -𝑑𝐸 ⁄ , at different polarization potentials of POT film in 0.5 M H2SO4 aqueous solution. Solid lines are simulations with eq (20) and (21) using the reached parameters by fitting of experimental results (red circles) where LE⇄P and P⇄BP transitions can be separated. The parameters are included inside the plot for each transition.

Figure 8 .

 8 Figure 8. 𝐺 HSO 4 -and 𝐺 H 2 O' evolution with respect to potentials perturbation obtained by fitting 𝛥𝑞 𝛥𝐸(𝜔) ⁄ and 𝛥𝑚 𝛥𝐸(𝜔) ⁄ of POT film in 0.5 M H2SO4 aqueous solution.

Figure

  Figure 9. Rate transfer of exclusion effect represented with values of 𝐾 HSO 4 -

  Figure 1

Figure 9

 9 Figure 3

  18 g mol -1 and 𝛿 H 3 O + 𝑀 H 3 O + = -19 g mol -1 . The parameters 𝐺 HSO 4 -, 𝐾 HSO 4 - 𝛼 HSO 4 -, 𝐺 H 3 O + , 𝐾 H 3 O +

′ , ′ and 𝛼 H 3 O + are fixed from results of fitted Δ𝑞 Δ𝐸(𝜔) ⁄ function at a given polarization potential. The parameter 𝑛 was calculated considering 𝑉 HSO 4 -= 47 Å 3 and 𝑉 H 2 O = 10 Å 3 obtained from the Connolly Solvent Excluded Volume in the commercial software ChemBio3D Ultra v. 12.0 ChemBioOffice 2010, thus 𝑛 = 4.7. Initially, 𝛿 H 2 O' ′ 𝑀 H 2 O could be +18 g mol -1 or -18 g mol -1 because the conformational changes of film are unpredictable. For this reason, both values were used to perform the fittings. The better convergence between the experimental and theoretical values were obtained using 𝛿 H 2 O' ′ 𝑀 H 2 O =+18 g mol -1 and, for this reason, this parameter was fixed as such (solid line in Figure 5b). Finally, only three parameters (𝑛 ′ 𝐺 H 2 O' , 𝛼 H 2 O' and 𝐾 H 2 O' ′ ) are kept free during multiparametric fittings. Like above, 𝛼 H 2 O' obtained is close to the ideality at all polarization potentials (0.95±0.02).

  𝑛 ′ was obtained, 𝐺 H 2 O' can be calculated for a set of water molecules transferred during the conformational changes. Later, this parameter together with 𝐾 H 2 O' ′ will be used to evaluate the kinetics of conformational changes of POT films under the experimental conditions in this work.

		𝑑𝛤 HSO 4 -𝑑𝐸	|	𝑒𝑥𝑝	=	𝑑𝛤 HSO 4 -𝑑𝐸	|	𝐿𝐸⇄𝑃	+	𝑑𝛤 HSO 4 -𝑑𝐸	|	𝑃⇄𝐵𝑃	(20)
	where for each reaction
		𝑑𝛤 HSO 4 -𝑑𝐸	=	𝐵 HSO 4 -4	𝑐𝑜𝑠ℎ 2 [	∆𝛤 HSO 4 -𝐵 HSO 4 -2 (𝐸 -𝐸 𝑝,HSO 4 -)]
	Identification of electrochemical processes. Before going into the detailed
	analysis of the kinetic parameters, the electrochemical transitions of POT can be placed
	on the potential scale by means of the derivative of the surface insertion laws of 𝐻𝑆𝑂 4 -,
	𝑑𝛤 HSO 4 -(𝐸) 𝑑𝐸 ⁄	= 𝐺 HSO 4 -𝐾 HSO 4 -′ ⁄	[16]. The peaks in Figure 7 have the same meaning as the
	theoretical derivation of a voltammetric response of electrochemical processes and, in

this case, they correspond to LE⇄P and P⇄BP transitions where the anion transfer takes place. The experimental result were fitted to

[START_REF] Agrisuelas | Kinetic and Mechanistic Aspects of a Poly(o-Toluidine)-Modified Gold Electrode. 2. Alternating Current Electrogravimetry Study in H2SO4 Solutions[END_REF][START_REF] Gabrielli | Ions and solvent transport across conducting polymers investigated by ac electrogravimetry. Application to polyaniline[END_REF][START_REF] Agrisuelas | An approach to the electrochemical activity of poly-(phenothiazines) by complementary electrochemical impedance spectroscopy and Vis-NIR spectroscopy[END_REF]

:

Table 1 .

 1 ac-Electrogravimetry has been used to explore the mechanistic aspects of electroactive POT thin films. A new electrogravimetric model is proposed which takes into account the exclusion of free water molecules by anions. Any other electrogravimetric response of the films can be related to the free water transfer owing to conformational movements of electroactive and dynamic film. Consequently, the kinetic aspects studied by ac-electrogravimetry of this kind of free water molecules act as a "fingerprint" of the kinetic aspects of polymeric movements. Here, the experimentally accessible timescale allows temporal resolution of coupled electron/anion transfer/free solvent transfer and coupled free solvent transfer/polymer structural reconfiguration. This model was applied to an intensively studied polymer with anion dopingdedoping with interesting results, specifically to POT. The experimentally accessible timescale allows temporal resolution of species transfer and polymer reconfigurations to be determined. For this film, the kinetics of conformational changes depends on the electrochemical reaction. Slightly faster free water transfers/conformational than anion insertion is observed during LE⇄P transitions. On the contrary, faster anion exclusion takes place during P⇄BP transitions. This fact has been related to the doped state of Variation of the global insertion law by integration of the simulated partial curves of 𝑑𝛤 HSO 4 -𝑑𝐸 ⁄ of POT film in 0.5 M H2SO4 aqueous solution of Figure7. Total 𝛤 HSO 4 -and 𝛤 HSO 4 -of LE⇄P and P⇄BP transitions are calculated. Scheme of the electrode|polymer|solution electrode during p-doping of a rigid polymer by anions (A -) considering the exclusion of free water molecules (H 2 O). 𝑑 𝑓 is the polymer film thickness. 𝑘 A -and 𝑘 e -are the kinetic constants of anion and electron expulsion from polymer, respectively. 𝑘 A - ′ and 𝑘 e - ′ are the kinetic constants of anion and electron insertion to polymer, respectively. The circles represent the molecule volume. In this case, 𝑉 A -= 4𝑉 H 2 O . The different color of water molecules allows the water inside film and water in solution to be distinguished.

	polymer and lateral interactions between electroactive sites. A more open structure of
	POT allows faster anion transfers than that in the case caused by the molecular
	rearrangements during P⇄BP transitions. This fact is due to the high doping levels in
	POT, which reduces the strength of lateral interactions. However, faster macromolecular
	movements are necessary before anion transfer owing to the packed state of POT during
	LE⇄P transitions. The packed state of POT is driven by a lower doping level involving
	stronger interactions.
	Although this work focuses on p-type ICPs, the proposed model provides a
	platform to extend the mechanistic interpretation of any type of ICPs by impedance
	techniques where solvent transfer and polymer reconfigurations follow the charge transfer
	steps. Using this model, they can be temporally separated, to the best of our knowledge,
	in an unprecedented detail. Therefore, it suggests that there is a considerable volume of
	under-exploited materials and experimental conditions further available for this kind of a
	study.

  9. Rate transfer of exclusion effect represented with values of 𝐾 HSO 4 -

	FIGURES
	Electrode		Polymer				Solution		Electrode		Polymer				Solution
				ionic channel			H 2 O	H 2 O						ionic channel			H 2 O	H 2 O
			H 2 O	H 2 O	H 2 O	H 2 O	H 2 O			H 2 O						H 2 O	H 2 O	H 2 O	H 2 O	H 2 O			H 2 O
			H 2 O	H 2 O	H 2 O	H 2 O	H 2 O		H 2 O	H 2 O					H 2 O	H 2 O	H 2 O	H 2 O	H 2 O		H 2 O	H 2 O
	e -	k ' e -	H 2 O H 2 O	H 2 O H 2 O	+ H 2 O H 2 O	H 2 O H 2 O	H 2 O H 2 O	k A -k ' A -	H 2 O	A -	H 2 O		e -	e -k '	H 2 O H 2 O	+ A -	H 2 O H 2 O	H 2 O H 2 O	k A -A -k '	H 2 O	H 2 O H 2 O	H 2 O H 2 O	H 2 O
		k e -								H 2 O					k e -								H 2 O
					d f													d f
												′		and rate
	transfer of conformational changes represented with the values of 𝐾 H 2 O' ′	at different
	polarization potentials obtained by fitting 𝛥𝑞 𝛥𝐸(𝜔) ⁄	and 𝛥𝑚 𝛥𝐸(𝜔) ⁄	with the
	theoretical functions of POT film in 0.5 M H2SO4 aqueous solution.

ACKNOWLEDGMENTS

Part of this work was supported by FEDER-CICyT project CTQ2011-28973/BQU.


W Thurston

Sarah Koch
email: kochsc@umich.edu

Tan Lei
email: tanlei@math.univ-angers.fr

ON BALANCED PLANAR GRAPHS, FOLLOWING

Let f : S 2 → S 2 be an orientation-preserving branched covering map of degree d ≥ 2, and let Σ be an oriented Jordan curve passing through the critical values of f . Then Γ := f -1 (Σ) is an oriented graph on the sphere. In a group email discussion in Fall 2010, W. Thurston introduced balanced planar graphs and showed that they combinatorially characterize all such Γ, where f has 2d-2 distinct critical values. We give a detailed account of this discussion, along with some examples and an appendix about Hurwitz numbers.

Introduction and Results

Let S 2 denote the topological 2-sphere, equipped with an orientation. Let f : S 2 → S 2 be an orientation-preserving branched covering map of degree d ≥ 2, let R f be the set of ramification values (or critical values) of f , and let C f be the set of critical points of f . We say that f is generic if |R f | = 2d -2; note that if f is a generic branched cover, then |C f | = 2d -2 as well. Furthermore, every critical point of a generic map f must be simple.

Let Σ ⊆ S 2 be an oriented Jordan curve running through the n := 2d -2 critical values of f , where we consider Σ to be an oriented graph with vertex set V Σ = R f ; each vertex has valence 2. Then the set f -1 (Σ) ⊆ S 2 is an oriented graph in the domain, with vertex set V f -1 (Σ) = f -1 (R f ). Among the nd -n elements of V f -1 (Σ) are:

• the vertices with valence 4, which comprise the set of critical points C f , and • the vertices with valence 2, which comprise the set f -1 (R f ) -C f , often referred to as cocritical points. Forgetting the valence 2 vertices in f -1 (Σ) gives an oriented graph Γ with vertex set V Γ = C f . We call Γ the underlying 4-valent graph of f -1 (Σ). We will refer to a connected component of S 2 -Γ as a face of Γ.

The question we address in this article is: which oriented 4-valent graphs on S 2 can be realized as the underlying graph of f -1 (Σ) for some pair f and Σ? This was asked by W. Thurston as part of a group email discussion, which he initiated in Fall 2010, and which was motivated by his visionary question: Thurston, 2010). What is the shape of a rational map?

Question (W.
Throughout the course of this discussion, he proved the following result.

Theorem 1.1. (W. Thurston, December 2010) An oriented planar graph Γ with 2d -2 vertices of valence 4 is equal to f -1 (Σ) for some degree d branched cover f : S 2 → S 2 and some Σ if and only if

(1) every face of Γ is a Jordan domain (in particular Γ is connected), (2) (global balance) for any alternating white-blue coloring of the faces of Γ, there are d white faces, and there are d blue faces, and Date: February 16, 2015.

(3) (local balance) for any oriented simple closed curve in Γ that keeps the blue faces on the left and white on the right (except at the corners), there are strictly more blue faces than white faces on the left side.

Any oriented 4-valent graph satisfying the three conditions above is called balanced. One way to construct arbitrary 4-valent graphs which fail the global balance condition in Theorem 1.1 is to pinch arcs in regions of one color only. The number of faces of that color increases without changing the number of the other faces, see Figure 1. A globally balanced but locally Outline. In Section 2, we begin by discussing Hurwitz numbers for generic branched covers S 2 → S 2 of degree d, and we then present a complete list of graphs arising in the case d = 4. We then establish the proof of Theorem 1.1 in Section 3, closely following Thurston's original proof. We have added some details (mainly for our own understanding). In Section 4, we present some of W. Thurston's work on decomposing balanced planar graphs into 'standard pieces'. In the appendix, we prove Hurwitz's classical result, that (up to isomorphism) the number of generic degree

d branched covers S 2 → S 2 is (2d -2)!d d-3 d! ,
presenting a geometric proof due to Duchi-Poulalhon-Schaeffer in [DPS].

Acknowledgments. First and foremost, we would like to thank Bill Color the left complementary disc of Σ blue, and color the right complementary disk white. Consider Σ as a graph with V Σ = R f , where each v ∈ V Σ has valence 2. Then f -1 (Σ) is an oriented graph with vertices V f -1 (Σ) = f -1 (R f ); there are precisely 2d -2 vertices in V f -1 (Σ) with valence 4 (corresponding to the critical points of f ), and the other vertices in V f -1 (Σ) have valence 2. We label the vertices of f -1 (Σ) as follows: for each c ∈ V f -1 (Σ) , label the vertex c with the number j if f (c) = v j . In this way, there are d -1 vertices in V f -1 (Σ) labeled 1, d -1 vertices in V f -1 (Σ) labeled 2, and so on. Note that because f : S 2 → S 2 is generic, for each 1 ≤ j ≤ 2d -2, there is a unique vertex labeled j with valence 4. Color each face of f -1 (Σ) either white or blue according to the color of its image, under f , in S 2 -Σ. The result is a checkerboard pattern in the domain of f : S 2 → S 2 .

Two such labeled graphs are said to be isomorphic if there is an orientation-preserving homeomorphism φ : S 2 → S 2 sending one oriented graph to the other, preserving the labels of the vertices. According to Hurwitz, the number of isomorphic labeled graphs (of degree

d) is (2d -2)! d! d d-3
(see the appendix for further details). When d = 4, this number is 120.

2.2. The case d = 4: cataloguing all 120 quartic diagrams. We illustrate the complete list of all 120 labeled graphs arising from f : S 2 → S 2 , where d = 4. They are organized into groups, catalogued by their underlying 4-valent graphs. The data are are based on a list, computed by L. Bartholdi with Gap, of 6-tuples of transpositions in S 4 , with trivial product, modulo diagonal conjugation by S 4 . The number of such tuples is equal to the number of isomorphism classes of generic degree 4 maps f : S 2 → S 2 .

2.2.1. Constructing Γ from f : S 2 → S 2 . In the case d = 4, we recall the steps used to construct Γ, the oriented underlying 4-valent graph of the branched cover f : S 2 → S 2 :

(1) Take an oriented Jordan curve Σ ⊆ S 2 in the range running through the 6 critical values R f = {v 1 , . . . , v 6 }, and consider f -1 (Σ) ⊆ S 2 as an oriented graph in the domain.

(2) There are 6 vertices in f -1 (Σ) with valence 4 (corresponding to the critical points of f ), there are 12 vertices in f -1 (Σ) with valence 2 (the cocritical points).

(3) Each face of f -1 (Σ) has 6 vertices on its boundary. (4) The underlying oriented 4-valent graph Γ is the oriented graph f -1 (Σ) obtained by forgetting the vertices which have valence 2.

2.2.2. Constructing f : S 2 → S 2 from a diagram. For each of the diagrams below, we build a degree 4 branched cover f : S 2 → S 2 in the following way:

(1) Choose an alternating coloring of the faces (a checkerboard-coloring).

(2) Choose one 4-valent vertex (that is, a critical point) to map to the critical value v 1 ; label this critical point with a '1'. (3) Label the remaining cocritical points and critical points so that they appear in counterclockwise order on the boundary of each blue face, see Figure 10 for this labeling scheme.

(4) Define the covering f first on the vertices following the labeling, and then on the edges by homeomorphic extension, and finally on the faces by homeomorphic extension (with a little help from the Schöenflies theorem).

Remark 2.1. The recipe above for building a map f : S 2 → S 2 from the labeled diagram does not always work; for example, there is a labeled diagram that satisfies the conditions above, but there is no branched cover f : S 2 → S 2 which gives rise to the labeled graph in Figure 10, or in fact to the underlying unlabeled graph in Figure 6.

Once a diagram is realizable as an underlying oriented 4-valent graph for some f and Σ, there are

• 6 choices of the critical point to label with '1', and

• 2 choices of the set of white/blue faces. So in general, each diagram below gives rise to 12 distinct labeled diagrams. However, some of these diagrams have extra symmetry, and in these cases, there are fewer than 12 labeled diagrams (up to isomorphism).

Proof

We will prove Theorem 1.1, following the main ideas in W. Thurston's original proof. Recall the statement:

An oriented planar graph Γ with 2d -2 vertices of valence 4 is equal to f -1 (Σ) for some degree d branched cover f : S 2 → S 2 and some Σ, if and only if on the boundary, and then to reduce the problem into a matching problem in graph theory. Indeed, let Σ be an oriented simple closed curve which goes through the critical values of a degree d branched cover f : S 2 → S 2 . Consider the oriented graph f -1 (Σ); it has n vertices of valence 4, and n(d -1) -n vertices of valence 2. Split each 2-valent vertex into two 'dots', putting one into each of the two adjacent neighboring regions. Then split each 4-valent vertex (or corner) into 4 'dots', putting one into each of the four neighboring regions. Every face of f -1 (Σ) then contains precisely n dots in its interior. Grouping the dots back together at the vertices then becomes a matching problem. Now take an arbitrary oriented graph Γ with n 4-valent vertices and no other vertices. Color the faces in a blue and white checkerboard pattern. Put n men in each blue face (represented by blue fish in Figure 3), and n women in each white face (represented by red fish in Figure 3). Each is trying to find a partner from one of the neighboring faces. In addition, each corner requires one person per neighboring face (that makes two women and two men). Is there a perfect matching? The usual marriage criterion in the marriage theorem from graph theory states that for every set of N women, there are at least N men who are potential mates; this can be reduced to the global balance and local balance conditions in the theorem. Here are the details:

Necessity, condition 1: complementary Jordan domains.

Let f : S 2 → S 2 be a generic orientation-preserving branched cover of degree d, and let Σ be an oriented simple closed curve passing though the n critical values of f . Note that f restricts to a degree d covering map S 2 -f -1 (Σ) → S 2 -Σ; it follows that the connected components of S 2 -f -1 (Σ) are Jordan domains.

Necessity, condition 2: global balance. Change coordinates in the range so that Σ becomes the unit circle S 1 , and orient it so the unit disk is on the left, and color the unit disk blue. The white face is the complement. We may also assume that the point 1 is not a critical value. Now the oriented graph we are considering in the domain is f -1 (S 1 ). Each of the blue faces in S 2 -f -1 (S 1 ) contains one zero of f , and each of the white faces in S 2 -f -1 (S 1 ) contains one pole of f . The global balance condition is equivalent to the condition that the number of zeros of f equals the number of poles of f . Necessity, condition 3: local balance. Choose an alternating coloring of the complementary components of f -1 (Σ). Let Λ be a Jordan domain bounded by an oriented curve γ in f -1 (Σ) keeping blue faces on the left. Suppose that in Λ there are W white faces, B blue faces, E edges in the interior of Λ, and X edges on the boundary. The total number of women in the white faces is nW -E. The total number of men in the blue faces is nB -E -X, with X > 0. For these nW -E women, all of their potential mates are among the nB -E -X men. Necessary and sufficient conditions from the marriage lemma imply nW -E ≤ nB -E -X, from whence it follows that B > W . This establishes the local balance condition of the theorem. Sufficiency, step I: adding 2-valent vertices to Γ. Suppose that Γ is an oriented 4-valent graph which satisfies the conditions of the theorem. Every face F of Γ is a Jordan domain, so the number of edges on the boundary of F is equal to the number of corners of F . For any face with k corners, we have k ≤ n. After matching the corners, there are n -k people left in the face to match. Since the numbers of faces of the two colors are equal, the total number of men who are not matched with corners is equal to the total number of women who are not matched with corners.

In what follows, we will consider the men and women who have not been matched with corners. Let S be any subset of these women. We want to show that the set of potential mates of S is at least as large as S (satisfying the condition in the marriage theorem).

If we augment the set S with all women in every face containing an element of S, this will not increase the number of their potential mates, so we may assume S is actually the union of all women in some collection U of white faces. Let R be the closure of U with all adjacent neighboring blue faces of U . Then the men in R are exactly the potential mates for the women in S. The orientation of the boundary edges of R leaves R on the left, and leaves blue faces on the left as well.

If the interior of R is not connected, then the women in one connected component can only match with men in that same connected component. We will therefore establish that there are enough mates for the elements of S in each connected component of the interior of R, from whence it will follow that there are enough mates in R for all of the elements of S. To this end, we assume that the interior of R is connected, so that every complementary component of R in S 2 is a Jordan domain.

Denote the number of men/women inside/outside R by the following tableau: Assume at first that R is simply-connected, that is, R is bounded by a simple closed curve leaving blue faces on the left.

Denote by: • X 1 the number of boundary corners of R which are on the boundary of only one face in R (this face is necessarily blue); • X 3 the number of boundary corners of R that are on the boundary of three faces in R (necessarily two blues and one white); • v the number of vertices in the interior of R;

• W the number of white faces in R and • B the number of blue faces in R. Note that every boundary corner of R belongs to one of the two types either X 1 or X 3 , and

w = |S| = nW -(2v + X 3 ), m = nB -(X 1 + 2X 3 + 2v).
By the local balance condition, we have W ≤ B -1. Therefore, using X 1 + X 3 ≤ n, we get

w = nW -(2v + X 3 ) ≤ n(B -1) -(X 3 + 2v) = nB -(n + X 3 + 2v) ≤ nB -(X 1 + 2X 3 + 2v) = m.
This is the desired inequality. We now deal with the case where R is not simply-connected. The number of women in the complement of R is w * , and the number of men in the complement of R is m * . If we can show that m * ≤ w * , the fact that m + m * = w + w * will imply that w ≤ m, which is the desired inequality.

Let Y be a complementary component of R. It is bounded by a Jordan curve keeping Y and white faces on the left. The same curve in the opposite direction, enclosing the complement of Y , has blue faces on the left. So by the local balance condition there are more blue faces than white faces outside Y . Using the assumption that there are equal numbers of blue and white faces (the global balance condition), there are more white faces than blue faces in Y . We then apply the same argument above in the case that R is simply-connected (replacing R by Y and reversing the colors) to conclude that the set of men in Y have enough mates (necessarily in Y ); this establishes that m * ≤ w * .

We have now proved that for any set S of women, the number of potential mates is at least as large as S. By the marriage theorem, there is a perfect matching. For each matched pair, merge them into a vertex on one common boundary edge of the two faces. Make sure that the vertices are pairwise disjoint. We have now enriched Γ into a graph with n(d -1) vertices, where n are 4-valent and the rest are 2-valent. Furthermore, every face has exactly n vertices on its boundary. Sufficiency, step II: consistent labeling of the vertices.

We now want to label the enriched vertices of Γ by the set {1, . . . , n}, such that each label appears exactly d -1 times, and the natural order of the labels appears counterclockwise around each blue face (and therefore clockwise around each white face), as shown in Figure 10.

Consider V Γ × Z/nZ, where V Γ denotes the set of vertices of Γ. For every edge e, from vertex v to vertex v define h e : {v} × Z/nZ → {v } × Z/nZ by

h e : (v, x) → (v , x + 1 mod n).
The collection of h e forms a cocycle with coefficients in Z/nZ. Since H 1 (S 2 ) = 0, this is the coboundary of a function to Z/nZ.

More generally: consider a cocycle with values in S 1 = R/2πZ, and parametrize RP 1 by a circle-valued coordinate, then a particular cocycle yields exact positions for the critical values, up to rotations of the circle (constant of integration).

Sufficiency, step III: avoiding duplicate critical values. It may happen, has we see in the quartic diagrams (see Figure 10) that two critical points have the same label. In this case, we may perturb the map to obtain matchings and labels so that the n critical points have pairwise distinct labeling.

q.e.d.

3.1. Further comments. It would be nice to have a better proof of sufficiency that can be turned into an algorithm to construct arbitrary rational functions whose critical values are on the unit circle. There should be some kind of cell structure whose lower-dimensional cells correspond to subdivisions of the sphere with vertices of higher even order as critical points coalesce, obtained by collapsing forests contained in the 1-skeleton of the subdivisions in the generic case. Collisions between critical points is related to the strata in Hurwitz's study.

Graph flow. One way to test whether a graph is balanced or not is to directly test a possible matching. There are good algorithms for finding matchings when they exist, or finding an obstruction when they don't exist: this is very related to graph flow, which has fast algorithms.

Here is a graph flow version of the balance condition: let Γ be a connected 4-valent graph with an alternating coloring of its faces. Give each face F a weight w(F ) equal to the number of vertices of Γ minus the number of corners of F . Define an abstract graph flow Σ whose vertices are {v(F ), F a face of Γ} and whose directed edges are of the form {(v(F ), v(F )} for every ordered pair (F, F ) of blue-white faces sharing one or more boundary edges in Γ. Add two vertices D (for departure) and A (for arrival) to Σ. Direct an edge from D to every vertex v(F ) of blue face F . This edge has capacity w(F ). Direct an edge from every vertex v(F ) of white face F to A, with capacity w(F ). Now the map is balanced if the maximal flow reaches the maximal capacity F blue w(F ).

Decompositions of balanced planar graphs

Our objective in this section is to understand the structure of balanced graphs from the point of view of decomposing them into standard pieces. 4.1. A *22 decomposition. Start with a planar 4-valent graph Γ. Find a nontrivial Jordan curve γ in S 2 that intersects Γ in only two points (here nontrivial means that the two intersection points are on different edges). This happens exactly when two faces of opposite colors have more than two common edges. Cut S 2 along γ and collapse each of the wound curves into a single point (these points will become regular points of the surgered graph). Continue this process on the two new diagrams until there are no longer any nontrivial curves with two intersection points. 4.2. Two types of tangle decompositions. Now look for a nontrivial Jordan curve that intersects Γ in four points, here nontrivial means that the curve does not go around a single vertex, and that the four points are on four distinct edges.

If the curve γ separates the 4-valent vertices of Γ into a pair of odd numbers of vertices, cut S 2 along γ and collapse each of the wound curves into a single point. The single point becomes a new 4-valent vertex of the surgered graph. See the top right dashed curve in Figure 12. Assume that the curve γ separates the vertices of Γ into a pair of even numbers of vertices, and assume that Γ has an equal number of white and blue faces. Cut the sphere S 2 along γ. Then the color imbalance on the two sides of the cut is ±1. For the side with more white (half)-faces, glue the two white edges of the cut copy of γ to connect the two white half faces into one singe white face (and fold the two blue edges into two segments), and symmetrically for the other side with more blue (half)-faces. See for example the middle two dashed curves in Figure 12 and the top left dashed curve in Figure 11.

The converse procedure of the last decomposition corresponds to Murasugi sum of knots. To do Murasugi sum of two diagrams, consider them as embedded in two spheres. Place one sphere on the left and one on the right (in 3-space). Remove a rectangle from oppositelycolored regions of the two diagrams, where the rectangles have two edges on different sides of a region and two edges interior to the region. Then glue them so as to match colors. The two spheres are merged into a single one. 4.3. Indecomposable pieces. Every diagram decomposes by these three operations into

• pieces that are isomorphic the unique quadratic diagram (2 intersecting circles -see the bottom of Figure 12), and • pieces that are not quadratic diagrams and do not have any cutting curves. These latter pieces are called hyperbolic pieces.

The first hyperbolic example is the octahedron (see the two diagrams on the right in Figure 7). The next hyperbolic example is the 3 × 4 turkshead, with 8 vertices. The 3 × n turkshead are the diagrams for a standard braid, going around in a circle with 2n crossings.

The 3 × 1 turkshead is {|z| = 1 ± ε} union an arc in the intersection of the upper half plane with the annulus, connecting 1 + ε to -1 + ε and the mirror symmetry by complex conjugation, with two vertices ±(1 + ε). The 3 × n turkshead is the inverse image of the 3 × 1 turkshead by the map z → z n . This is a lacing pattern for a drum, where a cord zigzags back and forth between a skin on the top and a skin on the bottom and closes in a loop after 2n segments. The symmetry group is the dihedral group of order 2n. d d-3 come from? This is a classical result due to Hurwitz, [H]. There are a number of proofs with different approaches, some applicable to larger classes of Hurwitz numbers. See for example [CT, ELSV, St, Va]. Here we sketch a combinatorial approach due to Duchi-Poulalhon-Schaeffer, [DPS]. Take a tree (non-embedded) with d white vertices and d -1 edges. Label the edges by 1, • • • , d -1, in blue. The number of such edge-labeled trees is known1 to be d d-3 .

For each such tree, put a blue vertex in the middle of every edge and label it by the original edge label. Now label the new set of edges by 1, . . . , 2d-2, in red. There are (2d-2)! choices.

So all together there are (2d -2)!d d-3 such bipartite edge-labeled and blue-vertex-labeled trees.

Embed (in a unique way) the tree in the plane such that around each white vertex the increasing blue labels of the incident edges appear clockwise. There are (2d -2)!d d-3 such bipartite edge-labeled and blue-vertex-labeled embedded trees.

Take a generic branched cover f : S 2 → S 2 of degree d with all critical values on the unit circle, labeled in cyclic order by 1, . . . , 2d -2, in red. Color the point 0 in blue and the point ∞ in white. Add in 2d -2 disjoint rays (in black) from 0 to ∞ so that each complementary region contains a unique critical value. Call this graph S. Now consider the pullback f -1 (S) and collapse each face containing a cocritical point to a single edge, obtaining a graph G. It has 2d -2 faces each with a red label of a critical point. This is the dual graph to underlying 4-valent graph f -1 (S 1 ).

The isomorphism class of f is uniquely determined by this face-labeled graph G. Now label the blue vertices of G by 1, • • • , d (there are d! such choices). Duchi-Poulalhon-Schaeffer show that there is a bijection from these face-labeled and blue-vertex-labeled graphs and the above bipartite edge-labeled and blue-vertex-labeled embedded trees. So the number of isomorphism classes is (2d -2)! d! d d-3 . In fact Duchi, Poulalhon, and Schaeffer take the blue vertex labeled d blue vertex as a root and find a unique spanning tree, as follows:

• Embed G in the plane so that the blue vertex with label d is accessible from the unbounded component. Orient each edge so that the incident face with the greater label is on the left. Every blue vertex has a unique incoming edge and every white vertex has a unique outgoing edge. See Figure 14. • Now repeatedly reverse orientations of clockwise cycles. This operation does not change the incoming and outgoing degrees of each vertex. A theorem of Felsner in [F] implies that after finitely many such cycle reversals, we obtain an orientation without clockwise cycles, and this orientation is unique.

• Start from the blue vertex labeled d as root and do a rightmost deep-first search in the sense opposite to the edge orientation, while erasing all other incoming edges to each visited blue vertex (including those incident to the root vertex), to get a canonical spanning tree whose edges are oriented toward the root, as done by Bernardi ([Be]). See Figure 14. Chop off the root vertex and its unique incident edge. Label in red each edge of the remaining tree by the label of the white-to-blue right side face. For visual convenience put the label on the same side as the face. Erase the orientation of the tree-edges. See Figure 14. The inverse algorithm from an embedded edge-labeled tree to f -1 (Σ) is also easy to describe; see Figure 15. Attach some half edges (hairs) to each white and blue vertex to reach the local valence value 2d -2 and to match the local cyclic order with the labels of the tree edges (counterclockwise for blue and clockwise for white). Now take a blueincident half edge whose left edge is a full edge, and walk in clockwise order, while matching each visited blue-incident half edge to the closest available white-incident half-edge in the counterclockwise direction. Put an extra blue vertex together with 2d -2 half edges in the unbounded face and match them with the remaining 2d -2 white-incident half-edges.

Figure 1 .

 1 Figure 1. If the dashed curve in the blue region is pinched to a point, the number of blue faces increases by 1, while the number of white faces remains constant.

Figure 2 .

 2 Figure 2. A globally imbalanced graph. There are 4 white faces against 2 blue ones. This figure was made by W. Thurston.

Figure 3 .

 3 Figure 3. Give the surface an orientation, and orient the graph above. There are equal numbers of blue and white faces, so the graph Γ is globally balanced; however, the right side of the graph violates the local balance condition, regardless of the orientation chosen. Note that in each face of this diagram, the number of fish plus the number of corners is equal to 8 = 2(5) -2. This figure is a slightly modified version of one made by W. Thurston.

Figure 4 .

 4 Figure 4. These six graphs represent 36 branched covers; each graph above gives rise to three distinct branched covers f : S 2 → S 2 , up to isomorphism.

Figure 5 .

 5 Figure 5. These five graphs represent 60 branched covers; each graph above gives rise to 12 distinct branched covers f : S 2 → S 2 , up to isomorphism.

Figure 6 .

 6 Figure 6. This diagram cannot be realized as the underlying diagram for any f : S 2 → S 2 . In fact, any labeling of this diagram will result in a 'duplicate critical value.' See Figure 10.

Figure 8 .

 8 Figure 8. This graph represents 6 branched covers up to isomorphism.

Figure 9 .

 9 Figure 9. These two graphs represent 12 branched covers; each graph above gives rise to six distinct branched covers f : S 2 → S 2 , up to isomorphism.

Figure 11 .

 11 Figure 11. The dashed Jordan curve on the right cuts Γ in exactly two points and they are on distinct edges; it is nontrivial. This figure was made by W. Thurston.

Figure 12 .

 12 Figure 12. The two types of tangle decompositions, the 2-valent vertices are not taken into consideration, they should be added later. This figure was made by W. Thurston.

Figure 13 .

 13 Figure 13. TurksheadAny 4-valent planar graph is a link projection: a projection of a link in the 3-ball to the boundary surface in generic positions. There is a nice theory that Martin Bridgman[Br] worked out for the structure of all possible hyperbolic diagrams, although it doesn't take into account the balancing conditions: they're all generated from 3 × n turkshead examples by choosing an arc cutting across a region between nonadjacent edges but being the first and the third edges of three consecutive edges, and collapsing the arc to a point, thus creating a new vertex. If you do this once in a region of each color, the global balance will be retained. Problem: (a) Analyze when the balancing conditions are maintained by these various operations and

Figure 14 .

 14 Figure 14. From G to the spanning tree, and then the to chopped spanning tree

Figure 15 .

 15 Figure 15. From a hairy tree and matched hairs (only two steps with dotted arcs are shown) in order to obtain f -1 (Σ).

  Thurston for starting this email discussion in 2010, engaging a large number of mathematicians in this project, and sharing his work with all of us. It is with great pleasure that we present this part of his work. Thanks to Laurent Bartholdi, Dylan Thurston, and Jerome Tomasini for helping with this manuscript in various ways. The research of Sarah Koch is supported in part by NSF grant DMS 1300315 and a Sloan Research Fellowship, and the research of Tan Lei is supported in part by ANR LAMBDA. Let f : S 2 → S 2 be a generic orientation-preserving branched cover of degree d ≥ 2. Label the critical values R f = {v 1 , . . . , v 2d-2 }, and let Σ be an oriented Jordan curve running through the critical values following the order of the labels.

2. Examples and a complete list of quartic diagrams

2.1. Hurwitz numbers.

Choose a root at a white vertex and label it by d. This root choice together with the edge labeling gives a unique labeling of the vertices by 1,

2, . . . , d -1, d (just label the far end of an edge by the label of the edge). So each root choice gives a Cayley tree and the number of such trees is known to be d d-2 .