Short-time current response to the curved edge of the electrode.

Artjom V. Sokirko and Keith B. Oldham

Department of Chemistry, Trent University, Peterborough, Ontario, Canada K9J 7B8

 

Introduction

In the recently published paper [1] the short-time response to the electrode of an arbitrary shape has been analysed. The main attention has been delivered to calculation of correction to the Cottrelian current [2] (normally used for describing current to the flat surface) arising due to the electrode curvature. The results of the original paper [3] discussed in detail in [1] show that the short-time current to the electrode with sharp edges differs significantly from that with similar geometry but smooth edges. Two new studies have been performed to clarify the importance of electrode vertexes on the partial examples of the vertex of cube [4] and the vertex of cone [5].

In the present work we address the problem of current contribution to the curved edge of the electrode. This paper can be considered as a logical continuation of [1-5], because together with [4] and [5] it gives some answers to the problems marked as "unsolved" in Ref.[1].

 

Theory.

A curved electrode edge is sometimes mistakenly identified with a smoothed edge. The smoothed edge appears when one large surface smoothly adjoins another surface. According to our terminology, the "smoothed" edge is not an edge, because formulas for a curved surface can be continuously applied to both surfaces and the junction between them. In this paper we will only discuss a sharp edge, for example, a cub's edge or an edge of a very thin disc.

There is one significant difference between the two above mentioned examples. Cub's edge is a straight line between two planes. An edge of a very thin round electrode is a circle separating two flat surfaces. We will call the edge of such kind "curved edge". Another example of the curved edge is a junction between conducting plane and a hemispherical drop, worn on the plane. The problem of a steady-state current for such system has been recently developed in [6]. In this case the curved edge is a circular line between flat and curved surfaces.

The general consideration of the curved-edge problem is rather complicated. All edges can be subdivided in two categories. The first category includes the edges whose section, perpendicular to the direction of the edge, contains more electrolyte than the electrode, so called "wedge" geometry. Certainly, the current to the "wedge" is higher than that to the flat surface. The second category of edges shows more electrode material than electrolyte in the perpendicular section and can be described as "groove". The current to the surface with "groove" edge is less than that to a flat surface.

Very much alike the fact that every curved surface can be considered approximately flat for a very short time, every edge can be considered "straight" in first approximation. According to [2], the short-time electric current to the straight edge depends only on the edge cross-section angle. Therefore, we may expect that the short-time response to the curved edge consists of the two parts: the main component, equal to that for the straight edge and a relatively small correction due to the edge curvature.

The first study of the curved-edge correction for the "wedge" geometry was performed in [1] on the example of an infinitely thin round disc electrode. One of the conceptual disadvantages of our theoretical approach to the "wedge" geometry is that it is limited to electrochemical reactions without deposition of any solid on the electrode surface. If deposition occurs, the sharp edge gets smoother, and, as it was mentioned above, ceases to be an edge in our terms. One can say that "wedges" have no shape-preserving feature.

In the present work we will concentrate on the "groove" geometry. Because the electric current to the very edge is formally absent, the "groove" tends to qualitatively preserve its main features for any type of electrochemical reaction. Of course, its quantitative characteristics may drift with time. In case of the metal deposition, the effective angle geometrically characterising the amount of electrolyte near the edge decreases with time, and we can say that "grove is closing".

Two examples of the curved edge will be considered in details. The first is the junction between the bottom and the curved surface inside the semi-infinite round conducting cylinder (see Fig. 1). This example is particularly useful for a deep pit modelling, which often appears as a result of corrosion. Another example is the "groove" between conducting cylinder and an infinite plane (Fig. 2). This is a setup model for various types of stifts inside an electrochemical reactor, which often appears in the reality. Another example, taken from electroplating, is an object, suspended in the electrolytic cell and having a junction with a conducting wire very similar to the Fig. 2.

 

Curved edge inside the semi-infinite cylinder.

Let us consider the semi-infinite conducting cylinder with radius r=a, filled by electrolyte with uniform concentration Cb. Assume, that at the moment t=0 the potential was changed up to a value high enough to assure that the electrochemical reaction on the electrode proceeds much faster than the diffusion transport in the system. It will cause the immediate drop of the surface concentration to a practically zero value. Further we will use mostly dimensionless concentration u=C/Cb, which will allow us to use directly formulas developed in the theory of thermoconductivity [7]. The initial condition becomes:

u=1 at t=0 (1)

and the boundary conditions at the electrode surface appear as

u=0 at z=0 (2)

u=0 at r=a (3)

Time-dependent diffusion problem can be described by the second Fick's law:

 = D u + D u (4)

where D is diffusion coefficient, and are differential components of Laplace operator in the cylindrical coordinate system:

=  r   =  

Our system has an axial symmetry with respect to rotation around the axe z=0, therefore all derivatives in the direction of axial angle j vanish and we can drop the term D from the right part of equation (4). In order to have a complete set of boundary conditions we have to add two quite obvious relations:

u is finite at z->- (5)

 =0 at r=0 (6)

Solution of problem (1)-(6) for the semi-infinite cylinder can be very easily found by de-composition method. Let us consider the diffusion problem inside an infinite cylinder:

 = D u (7)

with boundary and initial conditions (1), (3) and (6). The solution for that problem is well known [7,8]:

u = v(t,r)   (8)

where J0 and J1 are Bessel's functions, ak are the positive roots of the equation J0( aa )=0.

Diffusion in z-direction can be described simply as a diffusion to an infinite plate:

 = D u (9)

with boundary and initial conditions (1), (2) and (5). The solution for that simple problem is described [7,9] by an ordinary error function F:

u = (10).

Now the solution for the original problem (1)-(6) can be presented in form:

u = v(r,t) (11)

which can be obtained by direct substitution of (10) into (3) and applying equalities (7) and (9).

Current density j1 towards the curved surface of the cylinder can be found for any time t by taking derivative from equation (11):

j1 = - F n D Cb  r=a =  f(t) (12)

where f(t) is

f(t) = a  r=a (13)

F is Faraday number and n is number of electrons, transferred in the electrochemical reaction. However, because we are interested in the effects of disturbance due to the presence of the edge, we have to compare the total current to the electrode with that without the edge, i.e. with the current to the part z>0 of an infinite cylinder. The current density j2 to the surface of the infinite cylinder is also defined by the first equality (12), but function u(r,t) should be substituted from equation (8):

j2 = - F n D Cb  r=a =  f(t) (14)

The difference I1 in electrical currents to the semi-infinite cylinder and upper part of the infinite cylinder is given by the surface integral:

I1 =  =  = - 4 F n D Cb f(t)   (15)

Equation (14) together with equations (13) and (8) allows us to calculate current density to the surface of the cylindrical electrode, valid for arbitrarily long time. However, these equations are rather complicated. For a short time much simpler asymptotic expansion is available [1,7]:

j2 = F n D Cb   (16)

Comparison of (16) and (14) gives:

f(t) ª a   (17)

The local current to the electrode surface is normally described by current density, measured in Am m-2. The total current to the electrode is measured in Am. As it was noticed in [1], the most logical way to describe the current to the edge is to use linear current density i, that is electric current per unity of edge length and measured in Am m-1. Linear current density towards the curved surface of the cylinder due to the edge is:

i1 =  = F n D Cb  

We have calculated the linear current density towards the curved part of the edge. However, eqn. (18) is not a final result, because the linear current density should include a contribution from the bottom of the cylinder as well. The current density j3 to the bottom of the cylinder is

j3 =- F n D Cb  z=0 =  v(t,r) (19)

Cottrelian current density to the infinite plane z=0:

j4 =   (20)

In order to find the difference I2 due to the edge presence we integrate over the surface of the bottom disc:

I2 =  =  = - F n D C2p   (21)

Calculation of the integral in the right part of equation (21) can be performed by using the Gauss theorem. Note, that 2p  is the amount of matter in a slice of an infinite cylinder with radius a and thickness equal to unity. As the total amount of the substance in this slice changes only due to diffusion to the walls at r=a, we can conclude that the equality

2p  = 2p   (22)

is always held. We used relation (13) for space derivative of v(t,r). Substitution of (22) into (21) simplifies the integral calculation greatly. According to the initial condition (1), the integral in the right part of (21) is equal to zero at t=0, which can also be obtained by putting const of integration equal to zero:

I2 = - 2p F n D2 C   (23)

Equation for linear current density can now be calculated from (23) and (17):

i2 =  = F n D Cb  

The total linear density i is

i = i1 + i2 = F n D Cb  

 

The linear current density to the junction of a cylinder and a plane.

The geometry of this problem is shown on Fig. 2. From the mathematical point of view, this problem is very similar to the one, discussed in the previous section. The distribution of concentration is described by equation (4) with boundary and initial conditions (1)-(3) and (5). Instead of condition (6) on the cylinder axis we should now use the condition at infinity:

u->1 at z-> (26)

De-composition method also allows to find function u in form (11) as a simple product of two partial solutions: the diffusion to the infinite plate (10) and the diffusion to the surface of infinite cylindrical wire. The concentration distribution for the latter is given by integral [7]:

u = w(r,t) =   (27)

Current density to the curved surface of the cylinder is still given by (12) and (14) with appropriate substitution of function  instead of f(t):

 = - a  r=a (28)

The minus sign in the right part of (28) reflects the fact that although transport fluxes inside and outside cylinder are opposite to each other, the current has, obviously, the same sign in any situation. Equation (15) is also valid with the change f(t) ->  . There is no similarity in the mathematical forms of function f(t) and  - one of them is an infinite sum, another is a rather complicated integral of Bessel's functions. However, they show a remarkable similarity at the short time limit [1]:

 ª a   (29)

The only difference between (17) and (29) is the minus sign by the second term. Equation (18) for linear current density i1 is also valid for the outer problem after the change of sign by the second term in brackets.

Calculation of current to the plane surface z=0 looks slightly different. Unlike equation (21), where integration was performed over the finite disc (0<r<a), this integration has to be performed over the infinite region (a<r<). However, the consideration of the matter conservation, expressed in form (22), is still valid with the change of integration limit O by infinity in the left-hand side. In a similar way, we can find that the value of const in the right-hand side of (22) is zero. Equation (23) is valid with the change f(t) ->  and equations (24) and (25) with the change of sign by the second term in brackets.

 

Discussion.

Equations for the linear current density at the curved junction of the cylinder and the plane (25) can be presented for both above described situations (outside and inside cylinder) by a single equation:

i = - F n D Cb  

where parameter a has the meaning of the surface curvature of the electrode in the point that is in contact with the electrolyte. For convex surfaces, such as cylindrical wire (Fig. 2), a should be considered as a positive value [1]. For the diffusion problem inside the semi-infinite cylinder the electrode surface is concave and a is the cylinder radius with the minus sign.

Remarkably enough, the first term in eqn.(30) does not depend on surface curvature a. Comparison with Ref.[1] shows that this term is the same as for the straight edge with the angle between two conduction surfaces (in electrolyte face) equal to 90 degree. That could be seen directly from (30) - for the straight edge the radius of curvature is infinite and all terms, except for the first one, vanish. Note, that the main term is negative, because effective current to the electrode is described by the "grove" geometry. The linear current density to the curved surface i1 (eqn. (18)) and to the plane surface i2 (eqn. (24)) contributed equally to the first term in eqn. (30), because the "straight" edge is formed by two equally important semi-infinite planes.

Positive curvature of the surface gives us a positive increase in the current density [1]. This fact can be easily visualised by a simple geometrical consideration: convex is more exposed to the influence of the "outside world". Surprisingly, the positive curvature of the cylinder gives us a negative contribution to the linear current density. To explain that, one has to bear in mind that the linear current density i has the physical meaning of a correction to the electrical current due to the presence of the curved edge. In the transition zone between the cylinder surface and the plane, the curvature decreases from positive value to zero, therefore the curvature-related "prompt" current becomes lower with gradual movement from the cylinder to the plane. The similar consideration can be applied in order to explain the positive influence of curvature on the linear current density inside the semi-cylinder.

Let us discuss the general features of the curved edges of the electrode. The straight edge is characterised by the only parameter - angle in the solution phase q. The curved edge can be described by an infinite set of parameters changing along the edge length. Similarly to the case of the current density toward the curved surface, one can expect that the curvatures of the edge have primer influence on the value of i, which can be generally described by three parameters: space curvature of the edge line and curvatures of the two surfaces "to the left" and "to the right" from the separation line. Sometimes it is more convenient to present curvature features not in terms of mathematical curvatures, but in terms of curvature radii (see Fig. 3), where Re is the curvature radius of the space curve corresponding to the edge, and Rl and Rr are curvature radii in the direction perpendicular to the edge - to the left and to the right. Three numbers (Re, Rl, Rr) can be any real numbers - positive, negative or infinity, but not zero. Their combinations give us equations for both mean curvature H and total curvature K for both sides of the edge.

In the two considered examples only Re is finite, while both curvature radii Rl and Rr are infinite and should disappear from the equations for i. Analysing (30) we can conclude that "augmentative" term (i.e. the second term in the right part of (30)) is negative when the radius Re lies inside the electrode (Fig. 2) and positive when Re lies inside the electrolyte (Fig. 1). Of course, only the part near the location point is important as the far end of the vector or the centre of curvature can accidentally fall into another media.

Generalisation of this rule for different shapes is not quite straight forward. In ref.[1] the equation for the linear current density toward a very sharp edge of an infinitely thin two-sided disc electrode r=a has been obtained:

i = F n D Cb  

Such electrode represents the extreme case of "wedge" geometry - the angle measured inside solution phase q is equal to 2p. The vector Re lies inside the electrolyte and the argumentative term is positive.

Therefore, we can formulate the rule of sign: " The time-independent component (prompt current) of the linear current density and the first time-dependent (first augmentative term) correction due to the curvature of the edge line have the same sign if curvature radius lies in the electrode phase and the opposite signs if it lies in the electrolyte phase." This paper does not include a formal proof of this statement. We intend to present it in our future publications by using a numerical simulation technique for a range of angle q. Because all terms inside square brackets have to be non-dimensional, the radius and the time dependence can appear only in one possible combination  . The first and the most important time-dependent term is characterised by m=1, as in eqn. (30)-(31).

The dependence of i on Rl and Rr is another subject for future studies. We think that the main part of the dependence will give an additional term proportional to  in the right part of (30) with coefficient which depends on angle q but does not depend on value of Re (because Re has already been well accounted in the previous term). That leaves us with three parameters (Rl, Rr and  ) with dimension of length to construct one non-dimensional parameter suitable for the right part of eqn.(30). Obviously, it can be performed in infinite number of ways, but simple qualitative speculations can bring us to a highly probable answer.

First of all, this term, analogous to the other argumentative terms in (30) and (31) should be linearly proportional to  . Therefore we have Rl and Rr to construct one parameter with dimension [m-1]. Let us consider the point moving along the surfaces in the perpendicular to the edge direction from one surface to another. Obviously, the most important thing that happens to this point on the transition from one surface to another is an abrupt change of direction, which can be described as a step-wise change of the first space derivative. This fact is reflected by the first term in the i expansion. We know that the other effect on the current density is described by mean curvature. Passing the edge point, the moving point can experience an abrupt change not only of the first derivative, but of the second one as well. It means that the argumentative correction to the linear current density is likely to be proportional to the change of curvature DH across the boarder line:

DH =   (32)

The absolute value of the right side has to be taken because the necessary parameter cannot depend on our arbitrary choice of left and right. DH, as it appears in (32), is most likely to be the right thing to construct the required dimensionless value. It is reasonably based only on the scalar value of curvatures. However, one can decide that the appropriate value could be constructed with the seldom used vector of curvature, i.e. the vector with the same direction as a vector-radius of curvature R, but with the absolute value 1/|R|:

 +   (33)

Obviously, only the absolute value of vector (33) can appear as a factor in the non-dimensional combination. Vector (33) may also be responsible for the choice of the sign of that term according to a rule, similar to that for curvature Re.

To summarise, the linear current density to the curved edge can be presented in a general form as:

i = F n D Cb   (34)

where k(q), k1(q) and  1(q) are functions of angle q only. The value q=p corresponds to the smooth junction of two surfaces without an angle between them (for example, the junction a between a semi-sphere glued to the top of a cylinder with the same radius), therefore, k(p)=0. One can also hardly see how the "edge curvature " can contribute under such circumstances. However, the last term which corresponds to an abrupt change of curvature across the border line can be non-zero, because there is still a physical reason for its existence. We call such junction a "border line with the first order smoothness", because the first order derivative in the direction perpendicular to the border line has no breaks. Generally, the border line with smoothness of m-order has unbroken m-derivative and makes contributions to the linear current density, starting with the term proportional to t with coefficient  m(p). In this terminology an ordinary wedge has a zero-order smoothness, because the line itself, but not its derivatives, is continuous. Their main contribution term is proportional to t0.

Conclusion.

Literature.

1. [129]

2. []