In order to account and compensate for the dissipative processes contributing to the aging of cathodic surfaces protected by impressed current cathodic protection (ICCP) systems, it is necessary to develop the proper modeling and numerical infrastructure that can predict aging associated with quantities affecting the controller of these systems. In the present work, we describe various approaches for developing cathodic surface aging models (CSAMs) based on both data-driven and first principles-based methodologies. A computational ICCP framework is implemented in a manner that enables the simulation of the effects of cathodic aging in a manner that allows the utilization of various CSAMs that affect the relevant potentiodynamic polarization curves of the cathodic materials. An application of this framework demonstrates the capabilities of this system. We introduce a data-driven CSAM based on a loft-surface approximation, and in response to the limitations of this approach, we also formulate a first principles-based multiphysics and thermodynamic theory for aging. Furthermore, we discuss the design of a systematic experimental task for validating and calibrating this theory in the near future.

Introduction

The utilization of impressed current cathodic protection (ICCP) systems for protecting against corrosion of marine or underground metal structures is a very widely applicable practice for both commercial and defense applications. On the commercial side, oil-platforms and hulls of commercial shipping vessels are typical examples. Similarly, in the case of defense relevant naval applications surface ship hulls are also utilizing ICCP systems. Typically, ICCP systems consist of anodes connected to a DC power source, often a transformer–rectifier connected to AC power. The purpose of these systems is to inhibit or stop the corrosion of exposed metal surfaces (cathodes) by achieving the proper electric potential via the electrical current flowing from the anodes to these cathodic sites. The electrical current that is continuously regulated and monitored by the ICCP system helps prevent the electrochemical mechanism of galvanic corrosion prior to its attack [14].

Although advanced versions of such systems tend to account for the potentiodynamic polarization behavior of the exposed cathodic surfaces, they generally do not account for the variations of the polarization behavior caused by the aging of the cathodic surfaces, simply because such data are not readily available.

The goal of the present work is to describe the initial progress of an effort aiming to develop a multiphysics framework that integrates cathodic surface aging models (CSAMs) within an ICCP infrastructure that will be first implemented and validated computationally. A secondary goal of the followed approach is to identify aging processes that contribute toward polarization curve variability in a manner that allows the development of CSAMs that are aware of these processes such as they can be integrated with the computational ICCP framework.

The contents of this paper expand on work presented at the ASME 2017 IDETC/CIE conferences [5]. The approach followed involves the design and implementation of a computational framework capable of simulating the electric far-field response of a vessel equipped with an ICCP system. This framework is described in Sec. 2. Subsequently, this framework is utilized to establish the sensitivity of the variability of the polarization response relative to the far-field electric response of an immersed hull in Sec. 3. Section 4 will discuss the data-driven CSAMs and will focus on a description of the loft surface method (LSM) for representing typical experimentally acquired potentiodynamic polarization data. The shortcomings of the data-driven approach are discussed and provide motivation for Secs. 5 and 6, in which we present the initiation of developing an cathodic surface aging theory based on first principles and continuum thermodynamics. Section 7 describes an overview of the experimental methodology we have defined for constructing potentiodynamic polarization curves for HY-100 steel coupons. Finally, the paper concludes with a section summarizing the findings of this work and describing the path forward.

Computational Impressed Current Cathodic Protection Framework

Impressed current cathodic protection systems are used on large ships among other exposed marine structures. ICCP system anodes are made mostly of platinum-coated wires or rods that are mounted on the hull inside an insulated housing. The anodes are wired to power supplies located internally in the hull that actively drive current into the seawater, turning the hull cathodic once again. The voltage at the ICCP anodes must be constantly regulated to ensure that sufficient current is flowing to protect the ship from corroding, while preventing excessive current from entering the hull that might cause hydrogen embrittlement thereby weakening it. Electrodes called reference cells are mounted at several positions on the hull to monitor the effects of the anode current and to regulate it accordingly. Reference cell electrodes are often made of silver–silver chloride (Ag/AgCl) and this is the assumption made in this work.

A ship's ICCP system automatically adjusts its anode currents until the reference cells measure a specified potential relative to the hull, called the set potential. Generally, the set potentials for naval ICCP systems range from about −800 to −850 mV with respect to the hull versus Ag/AgCl.

Every time the coating/paint on the hull is deteriorating and exposes the metal below to the seawater, then this location becomes a cathodic site that is expected to be protected by the ICCP system. The condition of the newly exposed sites as a function of calendar time does not remain the same due to many processes present beyond the traditional electrochemical ones. Knowing the behavior of the cathodic surface in terms of potentiodynamic polarization curves enables the optimal protection of the respective hull via proper incorporation of these curves into the ICCP system. Here the term “calendar (or service) time” signifies that the physical process considered are evolving over long durations extending from t>102 hours, in order to manifest as material aging.

It should be noted that efforts for simulating the function of ICCP systems without the utilization of variable and aging-based polarization behavior can be found in the open literature [68]. The use of finite element methods for the simulation of ICCP systems is also discussed by [9]. Boundary element methods have also been so employed [1012], and the work of Yan et al. [13] is particularly noteworthy because of the inclusion of nonlinear polarization curves. Additional work on the modeling of ICCP systems has focused on underground pipeline applications [1417], underground tanks [18], and oil well casements [19]. The work of Yan et al. [20] is particularly interesting, as it incorporates a portion of the relevant multiphysics related to the buildup of minerals on ICCP protected surfaces.

In order to be able to exercise potentiodynamic polarization curve data in the context of an ICCP system, we implemented a general computational framework that implements an ICCP system for an arbitrary ICCP architecture and for an arbitrary vessel topology.

The main specification requirements considered for such a computational framework are as follows:

  1. (1)

    The framework must be able to represent any ICCP deployment topology (i.e., any location and shape of anodes and cathodes and reference electrodes).

  2. (2)

    The framework should be able to encapsulate any geometrical shape of a structure to be protected.

  3. (3)

    The framework should be able to encapsulate the complete and correctly implemented physics.

  4. (4)

    The framework should be able to emulate the conditions of the anodes being set at an electric potential level of choice relative to a reference electrode and the cathodic potential in a manner that accounts for the proper polarization behavior via a CSAM that accounts for cathodic aging.

  5. (5)

    Finally, the framework should be able to enable computation of electric field distribution in near and long fields with practicable computational efficiency.

To achieve satisfaction of these requirements, we selected the “comsolmultiphysics2 software suite as the proper computational substrate due to its flexibility in encapsulating arbitrary physics and its efficiency in defining and discretizing geometrical domains that are relevant to the universe of ICCP applications. In Fig. 1, we see a generalized depiction of the involved computational domains and associated boundaries.

The appropriate physics implemented at the early stage of this effort can be represented by the current density J continuity equation 
·J=0
(1)
along with the generalized Ohm's law 
J=σE+JE
(2)
while the electric field vector E can be expressed in terms of the electric potential V according to 
E=V
(3)
In Eq. (2), the quantities σ and JE represent the electric conductivity and any external current density vector contributions respectively. The system of Eqs. (1)(3) is describing the electric conduction physics in the saline water surrounding the hull of interest (i.e., domain Ω in Fig. 1). To enable satisfaction of requirement 4 in the requirements list above, we have implemented an optimization scheme as shown in the architecture described in Fig. 2. An initial guess for the values of the electric potential for the anodes (Vanodes) and the cathodes of the system model are selected. The forward problem encapsulated by Eqs. (1)(3) is solved by finite element analysis (FEA). Then an optimizer is used to minimize the objective function 
f=(VrefVset)2+(VcathodesVp(jc,t))2
(4)
where the cathodic current density is computed by 
jc=Γcn·JcdΓΓc1dΓ
(5)

and Jc is the global current density passing through the surface of the cathodes as computed by the FEA. Through either direct polarization data or an equivalent CSAM, we can evaluate the corresponding polarization potential Vp(jc,t) corresponding to jc and time of exposure t. If the objective function is not minimal, a new pair of (Vanodes,Vcathodes) is selected. When the objective function reaches a minimum, we use the corresponding optimal pair (Vanodes,Vcathodes) to evaluate the electric field in the solution domain or any subset of it.

Evaluating Polarization Sensitivity Via Impressed Current Cathodic Framework

In order to establish an understanding of the effect of polarization variability due to aging on the far field electric field distribution that is of concern for electric signature applications, we have utilized the framework described in Sec. 2, to simulate the electric field response of a submerged cylindrical hull made out of HY80 steel. The hull is configured with four anodes (two on each side) and three cathodes at the bottom generator of the cylinder. Figure 3 depicts a far-field view of the FEA model of this hull and the surrounding water as well as the detail of one of the cathodes.

To exercise the ICCP framework, we utilized the HY80 polarization curves from Ref. [21] that are depicted in Fig. 4. A typical result of the full (near and far) field distribution of the logarithm of the electric field magnitude Log(||E||) is displayed in Fig. 5(a). The implications of the variation of the potentiodynamic polarization curves depicted in Fig. 4 are easily observable if we select a line parallel to the axis of the cylindrical vessel at a mid-range depth (see dashed line in Fig. 5(a) and plot the corresponding electric field magnitude ||E|| along this line as shown in Fig. 5(b). Similarly, the contour plots of the distribution of the electric field magnitude ||E|| on a plane 1m below the axis of the cylinder are depicted in Fig. 6 

The results in Figs. 5 and 6 indicate that the sensitivity of the electric field to the polarization variability is very high for periods less than 30 days but for periods longer than 30 days the electric field becomes less sensitive.

Two additional observations can be made from the field distribution along the dashed line in Fig. 5(b). The first one is related to the fact that in the areas far away from the ends of the cylinder the values of the ||E|| cluster together near each other, as expected. The second observation is that the trends of the average field for the early time (1 h and 1 day) seem to be higher than those at the ends of the domain, while the average for the long durations seems to bundle up at an average value below the far field values. Clearly, this reflects not only a magnitude-related change but also a shift in the behavior from rapid changes to rapid changes combined with slow changes.

These results clearly indicate that there may be a strong reason to adjust the parameters of the relevant ICCP system controller as a function of calendar time in order to compensate for this variability. This way the controller will take care of the aging effects automatically. It is therefore imperative to establish the most accurate representation of how the potentiodynamic polarization curves are changing as a function of calendar time due to various aging processes that are associated with them.

Data-Driven Cathodic Surface Aging Models

When it comes to representing cathodic surface aging behavior, two main approaches are naturally lending themselves based on availability of experimental data. First, when experimental polarization data are available, one can attempt to build a CSAM based on these data by using any of the traditional machine learning methods.

By utilizing the HY-80 steel polarization data from Ref. [21] as depicted in Fig. 4, we evaluated a number of methods from the perspective of their efficiency defined as the least deviation from the actual values versus the relative ease of computational implementations. The examined methods were

  • Multivariate polynomials

  • Bezier surfaces

  • Nonuniform rational B-splines

  • Support vector machines

  • Artificial neural networks

  • Graph-based expression model discovery

  • Multivariate adaptive regression splines

  • LSM

The detailed comparison of all of these methods for the required needs falls outside the scope of the present paper and will not be discussed here. However, we have established that the LSM [22] created the quickest and most accurate encapsulation of the data. An example of representing the cluster of curves in Fig. 4 with a continuous surface by using the LSM is shown in Fig. 7. This representation shows the electric current density contours that now is presented in terms of the electric potential on the horizontal axis and the physical time on the vertical axis. The white area on this graph represents the absence of experimental data in the original potentiodynamic polarization curves.

The computational implementation of the LSM has the following characteristics:

  • It was implemented as a C++ library.

  • It was developed for cross-platform deployment.

  • It is compatible with a wide variety of central processing unit architectures.

  • It is extremely light-weight: can be used in low energy embedded microcontroller systems.

  • It is very fast: can be used to execute millions of queries per second as required by large computational models.

  • It is enabled for both forward and inverse evaluation (i.e., can provide current from voltage and voltage from current).

  • It can be interfaced with practically any computational framework including comsolmultiphysics.2

A disadvantage of any data-driven machine learning based CSAM is the fact that one has to use experimental data and pay the cost of developing them. Furthermore, one must live with the understanding that the derived CSAM will always be valid within the bounds of the observed data for every distinct material system tested. To overcome these limitations, we embarked on an effort to develop a bottom-up theoretical CSAM that is based on first principles as described in Sec. 6.

Phenomenological Identification of Parameters Capable of Accounting for Cathodic Surface Aging

Prior to developing a first principles-based CSAM that models the aging of cathodic effects in a manner that accounts for the change of the relevant polarization curves, we must specify the list of objectives that the to be developed approach must satisfy. These objectives were defined to be

  • Capture features of potentiodynamic polarization curve variability in terms of aging.

  • Identify aging processes and their features and effects.

  • Establish a measure of aging.

  • Develop a first-principles based theory that allows the development of models capturing polarization curve variability as a function of aging.

  • Validate developed theory and establish bounds of its applicability.

  • Demonstrate its applicability to electric field applications.

In order to establish the features of the potentiodynamic polarization curves that seem to encapsulate the effect of the cathodic surface aging according to the first objective, and after careful examination of Fig. 4 we can observe two striking features: the first involves the horizontal shift (along the electric potential axis) of the common point between the cathodic and the anodic branches of the curves (that correspond to the equilibrium potential as it is known on the corrosion science community) for 1 h versus for those for longer times; the second involves the spreading along the vertical (current density) axis for the areas away from the equilibrium point. A third and very weak effect is the shift of the equilibrium points along the vertical axis as a function of calendar time. These observations suggest that any physics-based model must contain parameters and the variation of which can lead to these types of responses.

An effort to establish how an existing first-order model could reflect the identified aging features is the Butler–Volmer theory and the associated model. According to this strictly electrochemical model, only electrical charge transfer at the electrode (and not by the mass transfer to or from the electrode surface from or to the bulk electrolyte) is responsible for the relation between electric potential and current density. It accounts for the simultaneous anodic (oxidation) reaction and cathodic (reduction) reaction on the same electrode surface. The utility of the Butler–Volmer equation in electrochemistry is wide, and it is often considered to be central in the phenomenological electrode kinetics [23]. Its form is given by 
j=j0aeαaZFηRTj0ceαcZFηRT
(6)
where j0a,j0c,z,F,R,T,αa,αc stand for the anodic and cathodic exchange current densities, the number of electrons involved in the electrode reactions, Faraday's constant, the universal gas constant, the anodic and the cathodic charge transfer coefficients, respectively. The quantity η is referred to as the activation overpotential defined as 
η=VVeq
(7)
where V,Veq represent the electrode potential and the equilibrium electrode potential. Furthermore, the charge transfer coefficients are related through the relationship 
αa=1αc
(8)

Figure 8 shows the plot of Eq. (6) for a representative case of an electrode made from platinum or palladium for an initial current j0a=j0c=j0=1mA/cm2=10A/m2 and for αc=0.1,0.3,0.5 and for Veq=0.00,0.01Volts. A direct comparison between Figs. 4 and 8 indicates that the main features that are attributable to aging for Fig. 4 also appear in Fig. 8. In fact, the spread among both the anodic and the cathodic branches seems to be captured by the variation of αc. The horizontal shift of the equilibrium point where the anodic and the cathodic branches meet is a consequence of the equilibrium potential Veq. The vertical shift of the same point can be attributed to both the variation of the equilibrium potential Veq and of course the initial current j0.

It is therefore conceivable that one way to address the cathodic aging problem from a modeling perspective would be to discover how the three quantities αc,Veq,j0 may depend on calendar time. If enough systematic data were available, then it would be possible to establish phenomenological relationships between calendar time and these quantities by employing data-driven machine learning approaches. These are approaches that we plan to follow after we complete our data-gathering program via the experimental plans that will be described later in this paper.

However, in order to establish a more inclusive understanding of what may be contributing to cathodic surface aging, we have decided to follow a first principles approach for developing a physics-based theory of aging for such surfaces.

Toward a Multiphysics Cathodic Surface Aging Model

Quantitative Metrics for Aging.

Prior to developing a CSAM, we need to establish specific metrics that can be used to quantify aging. When it comes to physical systems the term “aging” has been considered by many investigators as a synonym for “degradation.” From a technical perspective however, the term “degradation” signifies the alteration of certain systemic performance state variables of a system as a function of service time of that system. The term “aging” from an etymological perspective seems to indicate some “loss of capability” as a function of service time. It is therefore understandable why the two terms are used as synonyms.

The founders of the theory of irreversible processes (TIP) [2427] have addressed early the association between irreversibility and degradation. However, it was only in the late nineties that it was recognized in a consistent manner that degradation as a physical process must increase entropy and reduce thermodynamic energies [2830], in order to be consistent with the laws of thermodynamics. In general, it was well understood that degradation processes involve different mechanisms with distinctive features, types, rates, and sequences of dissipative processes. However, the common understanding has been that when it comes to permanent and irreversible changes to systems, at least one of the processes must be dissipative.

In fact, these views stimulated the development of a formal framework where in a seminal contribution [31] it was recognized and formally stated that all types of permanent degradation are irreversible processes, which manifest as systemic disorder and generate irreversible entropy to satisfy the second law of thermodynamics. Therefore, quite naturally, entropy can be used in a fundamental way to quantify the behavior of irreversible degradation. Consequently, it is natural that these ideas were applied to various application areas including sliding wear and fretting wear, caused by effects of friction and associated with tribological components [3133] and damage [34], and recently the development of a reliability theory that was also based on the TIP [35]. A recent review of the thermodynamic degradation paradigm and the associated degradation entropy generation theorem along with their application to formulate predictive models of wear, fatigue, and battery degradation, i.e., differential equations that govern the degradation or aging, given by Bryant in [36] is yet another very important contribution.

A measure of aging or, equivalently, degradation must be able to:

  • Have null value when there is no dissipation,

  • Be positive definite,

  • Be monotonically increasing as a function of service-time,

  • Account for all dissipative processes.

In all of the works mentioned previously, the term “entropy” as a measure of degradation refers to the so-called entropy production in a system that is best defined in the context of entropy continuity equation 
St=·JS+si
(9)

where S represents the entropy per unit volume of the system under consideration, JS represents the entropic fluxes that cross the boundaries of the system, and si represents the irreversible entropy production inside the system.

The entropy production must satisfy the second law of the thermodynamics that can be written in the form 
si0
(10)

The equality corresponds to systems in equilibrium and the inequality corresponds to systems away from equilibrium. While the second law, as stated here, automatically guarantees satisfaction of the first two requirements listed previously for a quantitative measure of aging, it cannot ensure that the other two requirements of monotonicity and all inclusive dissipative processes are also automatically satisfied from an arbitrary choice of si. Therefore, for every particular physical system of interest these two properties need to be verified specifically for this particular system.

Similar to the above-mentioned investigators, and in order to account for all dissipative processes that are relevant to the aging of cathodic surfaces, we will adopt the entropy dissipation function si as the basis for constructing measures of aging. In addition, we must develop a multiphysics theory of aging that exploits the first principles of continuous multiphysics according to TIP in order to account for all different dissipative process that contribute to cathodic surface aging.

The Polarization System Under Consideration.

The current understanding for the notion of corrosion tendency is based on thermodynamics [1]. For practical applications, however, often the main objective is the ability to predict the rates of corrosion. The fact that some metals are more reactive than others should not be interpreted as the only cause of corrosion variations. It is also well established that thermodynamic equilibria pay a significant role in the study of corrosion [37]. In fact, a fundamental but traditional approach to nonequilibrium states of the system, along with corrosion rates, begins with the primary consideration that when equilibrium has been disturbed corrosion is activated. In general, we must know the equilibrium state of the system before we can appreciate the various factors that control the rate at which the system tends toward equilibrium, that is, the rate of corrosion.

An electrode is not in equilibrium when a net current flows to or from its surface. This manifests at a microscopic scale because anode–cathode pairs appear on the same metallic surface. The measured potential of such an electrode is altered to an extent that depends on the magnitude of the external current and its direction. The direction of potential change opposes the shift from equilibrium and therefore opposes the flow of current. This is true for both the cases of externally impressed origin and of galvanic origin. The electro-chemistry community considers that the potential change (measured in volts) caused by net current to or from an electrode is the so-called polarization.

In Fig. 9(a), we can see an abstraction of an ICCP for ship hull. Sometimes a cathodic surface is created because of hull-coating deterioration, which exposes “fresh” hull material to seawater. A magnification of such a cathodic surface is presented in Fig. 9(b), along with a superposition of the evolution of the electric potential as a function of the distance from the surface at a state of equilibrium.

From just an isothermal electrochemistry perspective that does not depend on pressure and magnetic fields, only three main mechanisms can contribute or cause polarization. These are the “concentration polarization (diffusion overpotential),” “activation polarization,” and “resistance polarization” (OP). The concentration polarization is attributed to the fact that a diffusion layer adjacent to the electrode surface is formed where there is a gradient of the ion concentration. The activation polarization refers to the electrode potential change due to overcoming the energy barrier of the slowest step of the electrochemical reaction that involves accumulation of gases (or other nonreagent products) at the interface between electrode and electrolyte. The RP refers to the potential drop due to either the high resistivity of the electrolyte surrounding the electrode or an insulation effect of the film on the electrode surface formed by the reaction products or both.

Clearly, the approach that we propose must be able to deal with all three types of polarization but also account for the nonisothermal conditions as well as it should contain any mechanical (stress–strain) and magnetic field interactions that may be present.

Constituents and Chemical Reactions.

We introduce here the notation for the description of reacting mixtures in subdomains separated by planar electrochemical interfaces that was introduced in Ref. [38]. For simplicity, and without loss of generality, we first consider the planar situation where two one-dimensional regions Ω±R are separated by an interface I=Ω+Ω. For quantities defined in the bulk domains, there will often be corresponding quantities on the interface I, indicated by a subscript s.

The subsequently described definitions follow the template introduced in Ref. [38]. In each of the two domains Ω± and on the interface I we consider a mixture of several constituents. The total number of constituents in the subdomains Ω± is denoted by N + 1 and the set of constituents is M={A0,A1,,AN}, usually indexed by a α{0,1,,N}. In general, different constituents may occupy Ω+ and Ω, but for the simplicity of notation, this fact will only be indicated if necessary. We assume that each constituent of Ω± is also present on I, but there may be constituents that are exclusively present on I. Accordingly, the number of constituents on I is NSN and the set of constituents is MS={A0,A1,,ANS}.

A species constituent Aα is characterized by the (atomic) mass mα and may be carrier of charge zαe0, where zα is the charge number and e0 is the elementary charge. It is possible that chemical reactions among the constituents may occur. The M reactions in the bulk and the MS surface reactions can be represented by the general forms 
a0iA0++aNiANRbiRfib0iA0++bNiANfori{1,,M}
(11a)
 
aS0iA0++aSNiANSRbiRfib0iA0++bNiANfori{1,,M}
(11b)
The constants aai,bai are positive integers and γai=biaai denote the stoichiometric coefficients of the reactions. The reaction from left to right is called forward reaction with reaction rate Rfi>0. The reaction in the reverse direction with rate Rbi>0 is the backward reaction. The net reaction rate is defined as Ri=RfiRbi. Tt should be noted that since charge and mass have to be conserved by every single reaction in the bulk and on the surface, the following relations must hold: 
α=0Nzαγαi=0andα=0NSzαγNSαi=0
(12a)
 
α=0Nmαγαi=0andα=0NSmαγNSαi=0
(12b)

Basic Field Quantities.

The state of any metal-electrolyte system with a separating interface like the one depicted in Fig. 9(b) can be described by the intensive state variables that can vary for each point xΩ± and at any time t0. For the traditional electrochemical case, these state variables primarily refer to the electrolyte and they are the number densities nα(α=0,1,,N) of the involved species, the barycentric velocity u, the temperature T (when isothermal conditions are not assumed), and the electric potential ϕ. If there is mass transport (i.e., hydrogen ions) in the metal, then same quantities are describing the state of the metal.

Similarly, on the state of the interface I can be characterized at any t0 by the number densities of the interfacial constituents, nSα(α=0,1,,NS), the velocity w of the interface, the interfacial temperature Ts and electric potential ϕS.

The partial mass densities and mass fluxes are defined by 
ρα=mαnα,jα=ραvαandρSα=mαnSα
(13)
and the charge densities defined by 
ρ=α=0NραandρS=α=0NSρSα
(14)
The associated free charge densities can be defined as 
nF=α=0Nzαe0naandnSF=α=0NSzαe0nSa
(15)

It should be noted that if conjugate variables exist that are related through specific constitutive laws, then they can also be considered as capable of describing the intensive nature of the state of the respective systems.

For the nontraditional electrochemical aspect that accounts for the full thermodynamic perspective, the electrochemical ensemble of the system in Fig. 9(b) may be exposed to magnetic fields and mechanical loading. Therefore, in order to be all inclusive the relevant fields should be considered as well for the case of complete description of the system, especially if these factors can invoke dissipative behavior that plays an important role in systemic aging. This view would require that we account for magnetic and mechanical fields.

Interface Jump Conditions.

We define the jump of a generic function u(t,x)Ω[m] at the interface I as 
u|I+=limxΩ+Iuand[[u]]=u|I+u|I
(16)

When the function u is not defined in either Ω+ or in Ω, we set the corresponding value in Eq. (16) to zero. The normal n to the interface I points always from Ω to Ω+. In an one-dimensional setting, we have n=±1.

It has been shown in Ref. [39] that it is possible (in the purely electrochemical context only) to construct a complete model, that spatially resolves the charge layers in the vicinity of the interface, where the so-called double layer is located. According to this model, the characteristic length scale for the charge layers is 
λLref=kTε0e02nref
(17)

where k represents the Boltzmann constants, nref represents a representative value of the number density, and Lref represents the characteristic length of the system. As an example, Lref can be the distance between two electrodes and nref can be related to the anion and cation density in an electrolyte. In this case, the length Lref represents the well-known Debye length. For a solution of 0.1 mol/l, Lref1.5×1010m. The condition λ1 enables the generation of a reduced bulk model for the limit λ0, which simplifies the continuous relaxation behavior shown in Fig. 9(b) to a discontinuous behavior expressed by straight lines [39].

First Principles.

The electrode surface region is regarded as an open thermodynamic system, fully described by its intensive state variables.

The coupled system of equations for the state variables relies on electromagnetic field equations referred to as Maxwell equations and equations of balance or conservation equations for the partial mass of the constituents, the momentum of the mixture, and the electrode. In addition, there are associated interface conditions which are derived from surface balance equations. We will follow the process that was developed for ionic polymer continua in Ref. [40] to present these equations. Due to lack of space, we will not include the extensive derivations, but we will rather focus on the final results.

Electromagnetic Field Equations.

Under the assumption of slowly varying electromagnetic fields in a medium with velocity v due to deformation (in the solid electrode) or motion due to flow (in the electrolyte), Maxwell's equations in the bulk regions and the interface reduce to 
·(ε0E)=nF+nP,ε0[[E·n]]=nSF+nSP
(18a)
 
×E=0,[[E×n]]=0
(18b)
 
×H=J+×(P×v),n×[[(uw)×P]]=0
(18c)
 
·(μ0H)+·(μ0M)=0
(18d)

This system of equations represents the laws of Gauss–Coulomb, Faraday, Ampere–Maxwell, and Gauss–Faraday and is referred to as the EHPM formulation or the Chu formulation of Maxwell's equations [41]. The quantities E,H,P,M,J represent the electric, magnetic, polarization, magnetization, and electric current density field vectors, respectively, and the constants ε0,μ0 represent the electric hyperactivity or dielectric constant of magnetic permeability of vacuum respectively.

This system of equations is often complemented from the relations expressing conservation of the free charge density nF (or sometimes referred to as continuity of electric current and can be derived from Gauss' and Ampere's laws) and polarization charges nP according to 
·J+nFt=0
(19)
and 
·P=nP
(20)
Complementary to these equations are the constitutive equations that relate the conjugate variable fields of electric displacement D and magnetic flux density B. In their linear approximation, these are known as 
D=ε·E,andB=μ·H
(21)
where ε,μ represent the second-order tensors of the electric permittivity of and magnetic permeability of the medium, respectively. For isotropic media, these tensors can be expressed as 
ε=ε1,andμ=μ1
(22)
where 1 is the unit tensor and the scalar permittivity and permeability can now be expressed as 
ε=ε0εr=ε0(1+κ),andμ=μ0μr=μ0(1+χ)
(23)
with εr,μr,κ,χ representing the relative permittivity, the relative permeability, the electric susceptibility, and the magnetic susceptibility, respectively.
The auxiliary vector fields of the polarization and magnetization are defined by 
P=Dε0EandM=μ01BH
(24)
They can be rewritten via Eq. (23) as 
P=ε0κEandM=χH
(25)
The primary constitutive Eq. (21) can now be written for the isotropic medium as 
D=ε0εrE=ε0(1+κ)E,andB=μ0μrH=μ0(1+χ)H
(26)
Balance laws of continua.

The balance laws of continuum mechanics express the conservation of mass, momentum, angular momentum, and energy.

Instead of the N + 1 partial mass balances, we use the mass balance for the total mass density of the mixture, and the remaining N mass balances serve as the basis for the respective diffusion equations for both the bulk and the interface surface as follows: 
ραt+·(ραv+Jα)=rα,forα=1,,N
(27a)
 
ρt+·(ρv)=0
(27b)
 
ρSαt+[[(ρSα(vw)+Jα)·n]]=rSα,forα=1,,NS
(27c)
 
ρSt+[[(ρ(vw))·n]]=0
(27d)
The last equation refers to the interfacial balance for the total mass density and has been generated by summing up the partial mass balances. The source term quantities rα,rSα demote the mass production rate of constituent Aα in the bulk and the surface, respectively. They are defined as 
rα=i=1MmαγαiRi,rSα=i=1MmαγSαiRSi
(28)
It should also be noted that the conservation of mass requires 
α=1Nrα=0,α=1NSrSα=0
(29)
The conservation of momentum (ρv) can be generally written as any other local conservation law in the form 
t(ρv+E×H)+·(ρvv+σ)=F
(30)
where E×H stands for the electromagnetic momentum contribution, σ represents the second-order Cauchy stress tensor, F represents a source or sink of momentum manifesting as a body force density per unit volume, and vv represents a special tensor product forming a dyad as a second-order tensor from two vectors. This equation, also known as the Cauchy momentum equation, can further be specialized to account for the different characteristic behaviors associated with the fluid electrolyte and the solid electrode. In particular, for the case of a nearly incompressible Newtonian fluid like water, and for slow flows, the momentum equation reduces to the Navier–Stokes form 
vt+(v·)vηρ2v+(pρ)=F
(31)
where the term η/ρ represents the kinematic viscosity and the term η represents the viscosity of the fluid. The stress tensor has been decomposed to the hydrostatic pressure component p and a distortional component τ according to 
σ=p1+τ
(32)
and the Newtonian fluid constitutive relation has been used according to 
τ=η(u+Tu)
(33)

where u represents the displacement vector of the medium.

For the case of a deformable solid like the metal electrode, the conservation of momentum Eq. (30) reduces to the well-known form 
ρvt=·σ+F
(34)
It is important to note here that the body forces are comprised by three contributions: the Lorentz FL due to the electromagnetic field-mater interactions; the sum of the forces exerted on the various particles FP and the gravitational contributions ρg according to 
F=FL+FP+ρg
(35)
where g represents the vector of the acceleration of gravity. As it was described in Ref. [40], FL can be described as 
FL=σE+J×B+(E)·P+(B)·M
(36)
where σ is the electric conductivity of the medium. The particle forces can be expressed as 
FP=α=0NραFα
(37)

where Fα represents the individual forces exerted on its particle Aα.

The conservation of energy can also be given in terms of the local form as 
Ut=·Jqσ:γ˙+J·E+E·dPdt+B·dMdt+α=0NJα·Fα
(38)

where U,Jq,γ˙ represent the scalar field of the internal energy density per unit volume of the system, the thermal flux vector, and the second-order strain rate tensor that is also equal to v.

Multiphysics implications of entropy dissipation.
By following the program of nonequilibrium thermodynamics according to the TIP, we can now derive the entropic contributions comprising the right-hand side of the entropy continuity Eq. (9) as follows: 
JS=1T[JQα=0N(zαe0mαϕ+ωα)Jα)]
(39)
and 
sI=1T2JQ·Tα=0NJα·(ωαT)+1Tσ:γ˙+1Ti=1MJiAi+1TJ·E,1TP˙·(EeqE)1TM˙·(BeqB)
(40)
where Eq. (39) shows the entropy flux resulting from heat exchange and material exchange and the presence of an external electric field E=ϕ. Equation (40) represents the additive decomposition of dissipation terms from the system that (from left to right) include heat thermal conduction, mass diffusion, mechanical heating, chemical reaction, Joule ohmic dissipation, and relaxation of the electric polarization and magnetization field dissipations. The last two terms vanish when the quantities E,B coincide with their respective equilibrium values Eeq,Beq. In fact, this equation shares common terms with the respective expression derived in each of Refs. [35], [40], and [42] for the respective physics applicable in each case. It should be noted that the primed quantities are defined as follows [26]: 
E=E+v×B,andB=Bv×E
(41a)
 
P=Pv×M,andM=M+v×P
(41b)
and ωα,JQ,Jα,Ji,Ai represent the chemical potential of substance Aα, the heat flux, the mass flux of substance Aα, the i reaction current, and the associated chemical affinity, respectively. The chemical affinity is defined as 
Ai=j=1M(γijmi)ωj
(42)
The fundamental role of Eq. (40) to nonequilibrium thermodynamics cannot be understated. However, in TIP the entropy generation sI is generally represented as the bilinear sum of products of thermodynamic forces with their associated fluxes as 
sI=i=1mXiJi
(43)

This relation indicates that there is a specific choice for each thermodynamic forces Xi along with the corresponding choice of the thermodynamic fluxes Ji and can lead to recovering the form Eq. (40).

In the linear regime of the TIP, the variation of thermodynamic forces and fluxes are small and therefore the components of the thermodynamic fluxes Ji, are assumed to be a linear combination of the components of the thermodynamic forces Xi, according to Ref. [24] 
Ji=k=1nLikXi
(44)

where Lik are the so-called phenomenological coefficients. According to the Onsager reciprocity theorem [24], the equation of motion for each individual particle is time reversible as in classical mechanics. The macroscopic implication of this assumption is that Lik=Lki. It should be stressed here that in general the phenomenological coefficients of TIP must be determined experimentally. However, the Onsager reciprocity and the conservation laws described earlier permit a reduction of the unknowns.

Due to the lack of space, we will limit our description to the construction of electric current density that is related to the overpotential as known from the electrochemical description of surface polarization and therefore it is expressed in terms of quantities that can be measured experimentally. To reflect the dissipation function contributions, we can define the electric current density to be a sum of the following terms: 
J=LEQTj=1NLEjωj+LEEELEσσ+LEBBr=1MLErAr
(45)
If we take the limit of this expression for the surface of interest and assume one-dimensional spatial dependence due to the length scale involved, then we can use the equivalence =/x=x and the surface current density will be JS=Jn|I takes the form 
JS=LEQTj=1NLEjωj+LEEELEσσ+LEBBi=1MSLEi(αJγSαizαe0)JSi
(46)
Following the process in Ref. [38], we define the overpotential ηSi for each surface reaction that leads to the expression for the surface reaction currents in the form 
JSi=Jf0,iexp(αfie0kTηSi)Jb0,iexp(αbie0kTηSi)
(47)
with transfer coefficients defined as αfi=βSiASiΓ+ if Γ+0 or αfi=βSiASiΓ+ if Γ+=0 and αbi=(1βSi)ASiΓ+ if Γ+0 or αbi=(1βSi)ASiΓ+ if Γ+=0. Here we have used the definitions 
Γ+i=αM+γαizα,Γi=αMγαizα
(48)
and the exchange rates defined by 
Jf0,i=JS0iexp(βSiAsikT[αMγαimα(ωαω¯α)]|I±)
(49)
 
Jb0,i=JS0iexp((1βSi)AsikT[αMγαimα(ωαω¯α)]|I±)
(50)

Introducing Eq. (47) in Eq. (45) yields an expression for the surface current density that accounts for all relevant physics and contains the overpotential. If we eliminate all physics except electrochemistry by setting LEQ=LEj=LEE=LEσ=LEj=LEj=0 and apply this result for one constituent this expression reduces to the traditional Butler–Volmer form as in Eq. (6) when LEB=1 provided we accept the symbolic equivalence Vϕ.

An Aging Metric Based on Entropy Dissipation.

For slow degradation mechanisms, the entropy dissipation sI is positive definite because of the second law. However, this does not satisfy the requirement for a monotonic measure of aging or damage. For this reason we define the quantity ξ(t) to be a monotonically increasing cumulative entropy as it was proposed in Ref. [35] according to 
ξ(t)=0tsI[Xi(z),Ji(z)]dz
(51)
Starting at time zero from a theoretical value of zero or practically some initial entropy, ξ0, to an entropic endurance value, ξE, which makes the cathodic surface functionally nonperforming. When ξ reaches this entropic endurance, for all practical purposes, we will consider the cathode as no longer useful. It is worth noting that failure in this context is defined as surpassing a minimum operating requirement level represented by this endurance limit. Furthermore, this entropic endurance is measurable (from experiments or field observations) and may involve stochastic uncertainties due to variations in material, environmental, and operating conditions. To measure the damage in terms of the total entropy generation and the theory of damage described herein, a dimensionless damage index can be defined as follows: 
D(t)=ξ(t)ξ0ξEξ0
(52)

This index is constructed such that D(t)[0,1] and constitutes the final element of the proposed first-principles based CSAM. It should be noted that the endurance limit in this expression is reached when ξ(t)=ξE, which corresponds to D(t)=DE=1.

Experimental Approach

The planed experimental activity aims in creating the basis for both building data-driven CSAMs for a material of interest to the community and at the same time collect data to calibrate the first principles-based analytical CSAM. The analytical process based on first principles clearly established the need for determining the unknown phenomenological coefficients involved in the entropic representations of damage. The specific objectives of the experimental effort are to:

  • Establish the quantitative variability of polarization curves as a function of time for HY-100 under actual ICCP conditions,

  • Investigate the repeatability of behavior,

  • Investigate the effect of sampling rate.

To achieve these objectives, we plan to construct an array of individual cells. A single cell is shown in Fig. 10. The cell is comprised of an ICCP controller that senses the cathode's (coupon) potential with respect to a Ag/AgCl electrode, means to introduce oxygen into the solution through dripping seawater, a diaphragm, and an overflow valve.

The ICCP controller has been designed by our group and uses commercially available electronic components. The proportional–integral–derivative (PID) control of the voltage is achieved by the means of an 8 bit microcontroller operating at 16 MHz. The varying voltage output is performed using filtered pulse-width modulation, through an appropriate RC circuit. It should be noted that because of the very low power requirements, regular capacitors and resistors can be used. In addition, because of the large time-scales the system temporal response is not practically an issue.

Because of the off-the shelf components, the system can be easily parallelized and multiple experiments can be performed at once. This is depicted in Fig. 11.

Multiple cells like the one depicted in Fig. 10 will be placed in the same water bath. The temperature of the water in the bath will be kept constant using a temperature immersion circulator. The PID set point will be dialed to three different values (−0.75 V, −0.85 V, −0.95 V versus Ag/AgCl) with three systems for each value for a total of nine cells.

Six of the systems will be sampled biweekly and three weekly. By sampling, we mean switching the circuit from our ICCP PID controller to a potentiostat, performing cathodic polarization experiment and switching the circuit back to our ICCP controller. The experiment will be performed for an indefinitely long period as long as the electronic components are operational and the procedure has been automated.

Conclusions and Plans

We described the motivation, architecture, and preliminary analytical and computational framework enabling aging predictions for cathodic surfaces, in the context of their usage under ICCP conditions. This was done in order to incorporate material aging-induced changes in ICCP systems in a manner that reflects the dissipative nature of cathodic surface assemblies while enabling potential electric far field requirements. We described approaches for developing CSAMs based on both data-driven and first principles methodologies. A computational ICCP framework was implemented to account for cathodic aging in a manner that allows the utilization of various CSAMs. An application of this framework demonstrated the applicability of the implications of the variability of the potentiodynamic polarization curves as it is associated with cathodic surface aging. In addition to a data-driven CSAM based on a loft-surface approximation, we also introduced a first principles thermodynamic theory for aging and the design of a systematic experimental task for validating and calibrating this theory.

Although the reliability of the proposed first principles-based CSAM is high from a semantic perspective, because it does not ignore any physics, its quantitative reliability is limited by our ability to determine the constants as a result of proper experimentation. In fact, it is anticipated that the amount of required data may impose a very prolonged experimentation period and this may be a distinct weakness of the proposed approach. The main reason is the requirement of collecting data over long exposure durations and the need for multiple repetitions of the relevant experiments.

Future plans include the completion of the experimental infrastructure described as well as of the first-principles entropic CSAM framework. Data-driven models will also be refined to incorporate the acquired experimental data produced by exercising the proposed experimental framework.

Funding Data

  • Office of Naval Research (Grant No. N0001416WX01639).

  • U.S. Naval Research Laboratory (Core Funding).

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