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We study a nonlinear boundary value problem arising from electrochemistry. The essential difficulties are due to the strong nonlinear nature of part of the boundary condition. This part of the boundary condition is of an exponential type and is normally in the corrosion literature associated with the names of Butler and Volmer. We examine the questions of existence and uniqueness of solutions to this boundary value problem. In a numerical example we compare the behaviour of the solutions to the nonlinear problem with the behaviour of the solutions to a corresponding linearized problem. In contrast to earlier studies we put a major emphasis on studying parameter values that may be relevant for the case in which part of the boundary is in a transition to passivity—in practice most likely because it is nearly covered by an oxide layer. 0. Introduction. In this paper we shall study a nonlinear boundary value problem arising from galvanic corrosion. Before proceeding to a mathematical analysis of this problem we shall devote this first section to a brief explanation of some background material. Consider the battery cell illustrated in Fig. 1 (see p. 480). It consists of two strips of metal (one of silver and one of zinc) partially emerged in a saltwater solution. It is well known that chemical reactions at each metal strip will result in a current passing between the two. At the silver strip, hydrogen gas is produced, 2H20 + 2e~ -> H2 T +20H". Here the reactant gains electrons, i.e., one says that a reduction takes place, and the strip is defined to be cathodic. Simultaneously, at the zinc strip, zinc is dissolved into the solution, Zn —> Zn2+ +2e~. The reactant loses electrons, an oxidation takes place, and the strip is defined to be anodic. In electrochemical cells, oxidation and reduction always occur simultaneously. As a result of the chemical reactions, the metal strips change shape. In the voltaic cell Received December 1, 1995. 1991 Mathematics Subject Classification. Primary 35J20, 35J65, 73M99. 479 ©1998 Brown University 480 MICHAEL VOGELIUS and .IIAN-MING XU Fig. 2. A cell formed by rust on iron illustrated above, the driving force for the oxidation-reduction reactions is the chemical potential difference between silver and zinc, and the chemical reactions produce a current. One may conversely use current generated by an outside power source to cause chemical reactions (involving electron transfer) at the metal surfaces. This alternate process is called electrolysis. The quantitative relationship between the electric current and the mass of substances consumed and produced by the chemical reactions has been studied intensively going back to the early work of Faraday. Today, the use of electrochemical reactions plays a significant role in industry. Aside from electrochemical energy conversion, examples can be found in the production of chlorine, aluminum, and other chemicals, and in electroplating and electromachining. On the other hand, electrochemical phenomena also cause great damage, primarily in the form of corrosion. Corrosion is defined as the deterioration of a metal by chemical or electrochemical reaction with its environment. Most corrosion processes are electrochemical. In aqueous media, the action is similar to that taking place in the battery cell shown in Fig. 1, where the zinc electrode is corroded, i.e., metallic zinc is converted into hydrated zinc ions Zn"+. Figure 2 illustrates a local cell on a piece of iron when it becomes rusted [16]. It is predominantly from the point of view of corrosion detection and corrosion control that we are interested in the problems associated with electrochemistry. To justify the NONLINEAR ELLIPTIC BOUNDARY VALUE PROBLEM 481 commercial interest in these subjects it may be relevant to mention that a study conducted by the National Institute of Standards and Technology, under the request of the U.S. Congress, showed that the effect of corrosion cost the U.S. over 200 billion dollars in 1989, which was approximately 4.2% of the U.S. gross national product of that year [2]The idealized electrochemical system we consider consists of a domain 0, containing an ionic solution (electrolyte). A part of the boundary of fl is electrochemically active, the rest is electrochemically inactive (but may transmit an imposed current). For simplicity we restrict our attention to two-dimensional domains. We emphasize that the entire boundary is in contact with the same bulk solution, i.e., there are no membranes involved. One may imagine that outer circuits carry flows of electrons to and from the electrochemically active boundary parts, but exactly how this is done technically is of no importance for the analysis that we present here. Our main focus is to study the mathematical characteristics of a boundary value problem which incorporates a quite realistic model for the currents on the electrochemically active boundary parts. Since Faraday's law states that the rate of change of the boundary at any point is proportional to the normal current flux across the boundary at the point [5], a realistic modeling of the boundary fluxes thus indirectly provides a good understanding of the shape changes (corrosion) of the boundary. We shall work entirely within the so-called potential model; we refer to [5], [12], [17] and the references therein, for a detailed derivation and justification of this model. Briefly stated the transport processes in the electrolytic solution are modeled by a dilute solution theory, in which the concentrations of the various species of the electrolytic solution are considered spatially constant away from the boundary. In this fashion one obtains a single elliptic equation for the electric potential, , V • (—«V(^>) = 0 in fl, (1) where k, the conductivity of the solution, is given by k = F2^zfuiCi. (2) i In (2), Zi is the charge number for the species i, Ui is its mobility, and a its concentration. Since the concentrations c, are assumed to be constant, so is the conductivity k. Equation (1) therefore simply expresses that cf) is harmonic in fl. Let us now take a closer look at the interface between the electrochemically active boundary part and the electrolyte, where the oxidation-reduction reactions occur. As an aide to understanding the physical situation we consider for a moment a single strip of metal (anodic, say) placed in an electrolytic solution. After some period of time a state of equilibrium has been reached, at which point there is a discontinuity in the electrical potential across the metal-electrolyte interface (free electrons have gathered on the metal strip, metal ions have been released into the solution). The exact value of this potential jump depends on the specific metal and the specific electrolyte. Note that no net current is flowing between the metal and the electrolyte as a result of this potential jump. Since the difference (metai '/'solution) is not directly amenable to measurements, one introduces a reference device, such as a normal hydrogen electrode (NHE), to make a 482 MICHAEL VOGELIUS and JIAN-MING XU Active / \ Transition

A nonlinear elliptic boundary value problem related to corrosion modeling