Understanding the kinetics of catalysed reactions in microheterogeneous thin film electrodes

Article history: Received 8 January 2020 Received in revised form 14 April 2020 Accepted 4 May 2020 In this paper which is dedicated to Professor Richard Compton for his 65th birthday, we examine the problem of describing the transport and kinetics of catalytic reactions in which the catalyst is immobilized within a support matrix such as for example redox enzymes immobilized in a polymer matrix, to form a chemically modified electrode. We examine amathematical procedurewhich enables a full analytical solution to theMichaelis-Menten kinetic rate equation when coupled to Ficksian diffusion in thin bounded film. This governing reaction/diffusion equation is non-linear and a full analytical solution has, up until very recently, not been developed. Analytical solutions valid for low and high substrate concentrations have been previously reported. This solution for the amperometric steady state current is accurate whatever the value of substrate concentration. The analysis is applied to diffusion/reaction in a planar slab. General analytical solutions valid for steady state conditions for both the amperometric and potentiometric sensor response are provided. We then extend this useful analysis to consider the effect of concentration polarization in the solution, and to consider the effect of competitive inhibition on the amperometric current response. Finally, the analysis is extended to a polymer modified electrode when a redox mediator is used in the polymer film. General expressions for the current are obtained which were valid for any value of the substrate concentration. Kinetic case diagrams are developed and nine approximate limiting expressions for the amperometric response at steady state when the catalytic matrix is either conducting or insulating are developed.


Introduction
The problem of quantitatively describing the transport and kinetics of reactants within bounded thin polymeric films (aka chemically modified electrodes) is very challenging and various approaches have been developed over the last 30 years [1,2]. The early work of Saveant et al. [3] and Albery and Hillman [4] is seminal. This early work had a focus on describing diffusion coupled with bimolecular reaction or diffusion coupled with substrate pre-activation within a thin film. This topic remains of interest due to the fact that it enables the technologically important areas of chemical/bio-sensing and surface catalysis within fuel cell and electrolysis devices to be described in a mathematically precise manner [5,6]. Using this approach an analytical solution to a well defined reaction/diffusion problem can be developed which will describe how the concentration of the reactant (or substrate) will vary through the catalytic layer as a function of distance and time. It also enables an expression for the net reaction rate or reaction flux to be derived. In electrochemical systems the reaction flux is usually expressed as a current. This can be related in a definite way to real system parameters such as catalyst loading, catalyst concentration, substrate concentration and film thickness. The mathematical model should make simple predictions on how the reaction rate depends on each of the latter parameters and thereby enable the rational design of a modified electrode in which substrate detection/catalysis is optimized. We have recently summarised advances in modelling the mechanism of mediated electron transfer at redox active surfaces where the binding interaction between surface site and substrate can be complex [7].
In the present paper we focus attention on amperometric chemical sensing via surface immobilized redox active catalytic species (such as redox enzymes) which are embedded in a polymeric support matrix. In amperometric detection the target analyte diffuses through the solution phase, partitions into the catalytic film, diffuses within the film and subsequently is oxidized at the catalyst surface within the layer. The oxidized active form of the reduced catalyst is regenerated via application of an oxidizing potential at the support electrode. Communication between the underlying support electrode is accomplished in one of two ways. If the matrix containing the catalytic species is electronically conductive then the active form of the catalyst may be regenerated at a polymer strand which is in direct electronic communication with the support electrode surface and hence responds the applied potential input. Conversely if the polymer matrix in non conducting a small molecule redox mediator may be used to shuttle charge between catalyst and support electrode thus facilitating catalyst regeneration and turnover. The redox mediator (in its reduced form)

Nomenclature
General Michaelis Menten Problem s (mol cm −3 ) Substrate concentration in film. s L = κs ∞ (mol cm −3 ) Substrate concentration at outer region of layer which in absence of substrate polarization effects is given by the product of the substrate partition coefficient and the bulk substrate concentration κs ∞ .
x (cm) distance variable L (cm) layer thickness.
Compares the flux for catalytic reaction with that of diffusion of species X. Can refer either to substrate S or mediator B. Termed reaction/diffusion parameter. D X (cm 2 s −1 ) Diffusion coefficient of species X. X can refer to substrate or charge transfer mediator. Also can refer to inhibitor species if inhibition effects are considered.
Φ ¼ ffiffi ffi γ p ¼ L X K Thiele Modulus. Describes ratio between film thickness and reaction layer thickness X K which is the distance into the film the reactant species travels before it is destroyed by reaction with catalyst. This will be important when the Thiele modulus is large and catalytic reaction will be much more rapid than reactant diffusion.
(cm) Reaction layer thickness. Can refer either to substrate or mediator depending on system investigated.
Composite kinetic parameter derived via AGM procedure.
Normalised current defining sensor response when operating in amperometric detection mode. n, z Number of electrons transferred.  Potentiometric detection with Michaelis-Menten kinetics w ¼ p κs ∞ Normalised product concentration. w 0 normalised product concentration at electrode surface. p (mol cm −3 ) Product concentration. E, E 0 (V) Electrode potential and standard electrode potential respectively.
Reaction/diffusion with competitive inhibition Michaelis constant taking competitive inhibition into account.
K I Dissociation constant for catalyst/inhibitor complex.
Ratio between inhibitor concentration in polymer film to the dissociation constant for catalyst / inhibitor complex.
Modified reaction/diffusion parameter with competitive inhibition present.
Enzyme catalysis with small molecule mediation a, b (mol cm −3 ) Reduced and oxidized mediator concentrations. a ∞ (mol cm −3 ) bulk concentration of reduced mediator.
ter. Compares substrate/enzyme reaction flux to that for substrate diffusion.
Mediator reaction/diffusion parameter. Compares substrate/enzyme reaction flux with that of mediator diffusion. κ S , κ M Partition coefficient of substrate and redox mediator.
Composite kinetic parameter obtained via AGM technique in redox catalysis problem.
Competition parameter relating rate constant for direct oxidized mediator regeneration at electrode surface with that of redox mediator diffusion.
diffuses from the solution, partitions into the film, diffuses to the catalytic site, reacts there and subsequently a fraction of the oxidized mediator diffuses to the electrode where it is regenerated via oxidation to its reduced form to continue the catalytic cycle. Some oxidized mediator diffuses in the opposite direction and is lost from the layer. These scenarios are presented in Fig. 1. Furthermore the substrate and mediator concentration profiles through the catalytic layer depend both on distance and time and on the nature of the chemical rate law operating. If we ignore the time dependence then the steady state reaction/diffusion equation is given by: In this expression we have ignored a complicating factor such as product inhibition. The value of the parameter α is dependent of the type of modified electrode system studied. For reaction/diffusion within a slab then α = 1. For reaction/diffusion within a hemispherical film deposited on an inlaid disc microelectrode support then spherical coordinates are used and α = 3. Finally for reaction/diffusion within a coated wire electrode we use cylindrical polar coordinates and set α = 2. For planar diffusion within a slab we set ξ = x, whereas for spherical and diffusion we note that ξ = r and ξ = ρ respectively.
Typically for enzymatic and metal oxide catalysts the mechanism involves a binding interaction forming an adduct between the catalytic site and the substrate which subsequently decomposes to product. This is the Michaelis-Menten mechanism. In this case the form of the rate equation is: This expression is non linear with regard to substrate concentration s. The thermodynamics of binding are expressed by the Michaelis constant K M which is a measure of the affinity of the substrate for the catalyst site, and the turnover rate by the catalytic rate constant k c . The net catalytic efficiency is formally expressed by the ratio k/K M . Typically the reaction rate or current varies with substrate concentration in a non linear manner, usually hyperbolic. Substrate transport follows the Fick diffusion equation. In the following discussion we restrict attention to the case where the catalyst is immobilized within a conducting matrix and so mediator transport and reaction kinetics may be neglected. Substrate diffusion in solution will be neglected and only substrate diffusion within the surface layer will be considered and quantified by the diffusion coefficient D S . Combining eqs. 1 and 2 we obtain the following non linear differential equation for the diffusion and reaction of substrate S within the thin layer: In this paper we discuss analytical solutions of eq. 3 which are valid for all values of substrate concentration and use this solution to derive analytical expressions for the current response under steady state conditions. We considersimple linear reaction/diffusion within a slab, and analytical expressions valid for all values of substrate concentration for the steady state current response for this system will be derived. In the next paper non linear reaction/bounded diffusion within a thin hemispherical film deposited on an inlaid disc electrode and within a film coated microwire electrode will be discussed.

Planar reaction diffusion in a bounded slab
We initially consider the simple case of planar diffusion and reaction of substrate S within a thin slab of thickness L immobilized on a support electrode containing a homogeneous distribution of catalytic particles of concentration c Σ . Hence the governing reaction diffusion equation admits the following form: In this expression we introduce the pseudo first order rate constant k = k c c Σ /K M where c Σ denotes the total catalyst concentration (molcm −3 ) respectively. This equation must be solved subject to the following boundary conditions: Here κ denotes the partition coefficient of substrate and s ∞ is the bulk concentration of substrate in solution. Hence the product κs ∞ represents the reactant concentration at the layer solution interface. The latter boundary condition implicitly assumes that concentration polarization of substrate in the solution may be neglected.
We introduce the following dimensionless quantities: Where u, χ represent the dimensionless concentration and distance parameters respectively. Furthermore α denotes a saturation parameter and γ defines a reaction/diffusion parameter. The saturation parameter compares the value of the substrate concentration in the layer to the Michaelis constant. When this parameter is small the catalytic kinetics is unsaturated and the rate is first order with respect to substrate concentration. When it is large the kinetics are saturated and zero order kinetics pertain. The reaction /diffusion parameter compares the rate of reaction between substrate and catalyst moiety and the rate of substrate diffusion in the layer and is directly related to the Thiele modulus via the following expression: denotes a characteristic reaction layer thickness which is a measure of the distance travelled by the substrate in the film before it reacts with the immobilized catalyst particle.
Hence eq. 4 transforms to: which must satisfy the following boundary conditions Now the net amperometric current corresponding to the rate of substrate reaction in the layer is given by the following equivalent expressions Diffusion competition parameter. Compares substrate diffusion rate to that of redox mediator in layer.
current response for reaction in a conducting polymer payer and in a non conductive polymer layer respectively. relating the reaction flux: Note that in eq. 9 we have noted that the pseudo first order rate constant is of the Michaelis-Menten form given by: k = k c c Σ /(K M + κs ∞ ) and that (ds/dx) x=0 = 0. Both definitions of normalised current presented above give rise to the same result. We use the differential expression based on the Fick diffusion expression outlined above in our presentation.It is a characteristic of Michaelis Menten reaction kinetics that the pseudo rate constant for reaction between catalyst and substrate depends on substrate Chemical reaction between substrate and catalyst and between catalyst and mediator, and mediator and support electrode also occurs. The net reaction rate therefore reflects on the balance between diffusive transport and reaction kinetics. (b) Schematic representation of the reaction and diffusion of substrate and redox mediator within a porous polymeric matrix. In this case the matrix is not conducting and the mediator reacts at the underlying support electrode.
concentration and in particular, on the relationship between the bulk substrate concentration and the value of the Michaelis constant for the substrate/catalyst reaction. When κs ∞ << K M then the pseudo first order rate constant is given by k = k c c Σ /K M .
Consequently we can introduce a normalised steady state current or reaction flux y as follows: Hence the problem reduces to evaluating an analytical expression for u which will be valid for all values of γ and α. Once this is achieved an analytical expression for the normalised flux of the amperometric sensor can be readily derived via eq. 10.
In earlier work [8] we have proposed analytical solutions to eq. 9 which are valid for the limiting cases of low and high saturation parameter values, and have proposed a solution based on the reasonable assumption that the non -linear kinetic term u 1 þ αu can be approximated by the linear expres- Hence the reaction/diffusion equation transforms to: We have shown that this approximation is valid only for certain values of α and u. Specifically the approximation pertains for all values of u where the Michaelis-Menten kinetics are unsaturated (when α < 1). For α > 1 the approximation becomes inaccurate if significant depletion of substrate occurs within the film, if u falls to less than 0.8 at any point in the film. This will occur when the parameter γ is large. In short our strategy was to transform the non linear reaction/diffusion equation into a linear equation which can be readily integrated.
In recent years Rajendran and co-workers [9,10] have used the variational iteration method (VIM) to model the response of a potentiometric and amperometric enzyme sensor in which linear diffusion is coupled to non linear Michaelis-Menten kinetics. This technique produces solutions to the boundary value problem in terms of convergent series requiring no linearization or small perturbation. The analytical results valid for all saturation parameter values were compared with those earlier limiting cases proposed by Lyons et al. [8] and were found to be in good agreement. More recently Malvandi and Ganji [11] developed a variational iteration method coupled with Padé approximation (VIM-Padé) to obtain analytical expressions involving rational functions for substrate concentration profiles for bounded catalytic systems with non linear Michaelis-Menten Kinetics. Rajendran et al. [12][13][14] outlined how the method of homotopy perturbation could be used to derive an analytical expression for the substrate concentration profile within a thin layer when the reaction kinetics exhibit Michaelis-Menten kinetics. Finally Dharmalingam and Veeramuni [15] applied the Akbari-Ganji method (AGM) to develop an expression for the amperometric current response to non linear reaction/diffusion in an electroactive polymer film.
In this paper we further develop the AGM to examine steady state non linear reaction diffusion in bounded thin films of planar slab geometry with a particular focus on amperometric detection. Hence we solve eq. 7 subject to the conditions outlined in eq. 8 to obtain approximate closed form analytical expressions for the substrate concentration profile and the normalised reaction flux which are valid for all values of the saturation parameter α and defined values of the reaction diffusion parameter γ. We will compare the approximate solution with the numerical solution obtained using the NDSolve facility in Mathematica 12 to determine the parameter set where goodness of fit between the simulated and closed analytical solution is optimized. We do this for the slab geometry. The cases of both spherical and cylindrical diffusion coupled with non linear Michaelis-Menten reaction kinetics will be discussed in a subsequent paper.
We follow the recent excellent work of Dharmalingam and Veeramuni [15] and assume that a suitable solution for the reaction/diffusion presented in eq. 7 will have the following form: We can readily show using the boundary conditions presented in eq. 8 that A = sech β and B = 0 and we obtain: We now use the approach of Dharmalingam and Veeramuni [15] and we use the Akbari-Ganji method (AGM) to evaluate the unknown parameter β. This is a powerful semi-analytic approach for solving non linear ordinary differential equations. In this method a solution function consisting of unknown constant coefficients is assumed satisfying the target differential equation. This solution is substituted into the latter to generate one or more algebraic equations. Finally, the unknown coefficients are computed using these algebraic equations in which a relevant boundary condition is inserted. The literature describing this method is not at all clear in discussing the general validity of this approach. However the closed form approximate analytical solutions obtained using AGM are in good agreement with numerically simulated results and with other approximate methods of solution such as VIM or HPM [9,10,[12][13][14]. To follow AGM we introduce the following function: Hence substituting eq. 13 into eq. 14 we obtain: This will only be true provided χ = 1 and so And specifically from eq. 8 we note that: Consequently we obtain that: Hence the solution to eq. 6 is: Furthermore the substrate concentration at χ = 0 is given by When we have unsaturated catalytic kinetics and α << 1, then eq. 18 reduces to: This is the same as eq. 10 in our initial 1996 paper [8]. Alternatively for saturated catalytic kinetics α >> 1 and eq. 18 reduces to: Eq. 20 can be simplified further. If the argument in the hyperbolic cosine functions is small then we can Taylor expand the functions to give: and so the substrate concentration profile when the catalytic kinetics are saturated is alternatively given by: This expression for the concentration profile is valid when γ 2α < 1. The expression in eq. 20 is the same as that presented in eq. 11 of our initial 1996 paper [8].
When the reaction/diffusion parameter γ is large then the catalytic reaction kinetics are much faster than substrate diffusion through the film, then we note that cosh and so the normalised substrate concentration profile takes the following form: Physically this expression corresponds to an exponential decay in concentration from an initial value of u = 1 at χ = 1 with a time constant of ffiffiffiffiffiffiffiffiffiffiffi γ 1 þ α r in a direction going in to the film from the outer surface. Hence there is considerable concentration polarization of substrate in the layer. Alternatively when the reaction/diffusion parameter is small corresponding to the case where catalytic reaction kinetics are more sluggish than substrate diffusion through the layer corresponding to for γ << 1 we note that: Hence under these circumstances there is little concentration polarization of substrate within the film. The unsaturated catalytic kinetics are much more sluggish than substrate diffusion, and a uniform substrate concentration with little depletion is expected in the layer.
The approximate analytical solution outlined in eq. 18 is directly compared with the numerical integration of the non-linear reaction/diffusion equation presented in eq. 7. This was achieved using the NDSolve capability in Mathematica 12. The results are presented in Fig. 2 for the case of unsaturated catalytic kinetics and in Fig. 3 for saturated catalytic kinetics. We note that the correspondence between simulated and closed form analytical solutions are excellent when the reaction kinetics are unsaturated. However in Fig. 3B which corresponds to the situation where saturated kinetics pertains, the agreement between simulated values and those derived via eq. 21 is very good. The agreement with the more general solution presented in eq. 18 (orange curve) is less good. There is excellent agreement between the simulated profile and eqs. 18 and 21 in the region where χ = 1. We compute the normalised current from the concentration gradient at that point in the analysis so the expression derived for the normalised current is in excellent agreement with that derived via numerical simulation. When reaction between the substrate and the catalyst is very fast then eq. 18 is seen to underrepresent the substrate concentration through the film when compared with the result of numerical simulation. Deviation from the simulation arise typically when the saturation parameter is large and when the reaction/diffusion parameter is large. This is typically in region IV of the kinetic case diagram (see later discussion) when the outer region of the film is partially saturated and the inner region unsaturated. This case pertains for 1 < α < γ/2 Under these circumstances eq. 21 is the more appropriate expression to adopt for the concentration profile of substrate in the layer. In Fig. 4 we examine the general case where the saturation parameter is close to unity. Here we choose α = 1 an compare simulation results with the closed form analytical solution under conditions where the balance between catalytic kinetics and substrate diffusion is varied. Again very good agreement is observed. The largest divergence is observed in panel C where substrate depletion is significant which occurs at large γ values. These results suggest that the general solution is of most use is the scenario where there is balance between catalytic reaction kinetics and substrate diffusion and where the substrate concentration in the film is close to the Michaelis constant of the catalytic reaction.
The normalised current response y is obtained via eq. 10. We may readily show that: This expression is termed the general case and will pertain to the situation where the catalytic kinetics are neither unsaturated nor saturated when α is close to unity. In Fig. 5 the variation of the normalised current computed via eq. 24, with saturation parameter α, is presented for values of the reaction/diffusion parameter γ in the range 0.05 to 15. This is in effect a normalised calibration curve, which depicts the variation of current response with substrate concentration. In Fig. 6 the normalised current is plotted as a function of reaction/diffusion parameter γ, for various defined values of the saturation parameter ranging from 0.1 to 10. Limiting values for the normalised amperometric current response valid for all γ values are readily derived in the limits of: α << 1 and α >> 1 respectively. Furthermore other limiting expressions are obtained in the limit of γ << 1 and γ >> 1 for all α values. Eq. 24 defines the general case.
In our earlier 1996 paper [8] we quoted an empirical expression constructed by Albery and co-workers [16] for immobilized enzyme electrodes which could be adapted to describe reaction/diffusion in electroactive thin films. Indeed we fitted our experimental data to this expression. The Albery equation is: In Fig. 7 we compare eqs. 24 and 25 derived using AGM proposed by Dharmalingam and Veeramuni [15] directly for a fixed value of γ = 15 and for a range of saturation parameter values between 0 and 100. Both normalised current response curves exhibit a similar development but the Albery expression over estimates the normalised flux by a significant amount. In Fig. 8 we present the variation of the steady state flux ratio Y = y/y A as a function of saturation parameter α values over a wide range from 0 to 1000. Each curve presented corresponds to a set value of . In panel A the catalytic kinetics are much slower than substrate diffusion, whereas in panel B substrate diffusion is much slower than the catalytic kinetics and substrate depletion is significant. Here the best fit with the simulation (blue curve) is the profile corresponding to eq. 21 (green curve) rather than that predicted from eq. 18 (orange curve).     the reaction/diffusion parameter γ ranging from 0.1 to 1000. This figure suggests that when γ values are less than 0.1 the normalised flux ratio is essentially unity over the entire range of α values. However as the magnitude of γ increases corresponding to more favourable catalytic reaction kinetics in the layer, the normalised flux ratio Y initially decreases with increase in α value, to a broad minimum located within a specific α value range, and then increases again as the saturation parameter value is increased still further to approach a value of unity in the limit of large saturation parameter values. Furthermore, the location of the flux ratio minimum varies with γ value, being located at increasingly larger α values as the catalytic kinetics become more rapid. So the prediction of normalised current according to Albery (eq. 25) and the present work (eq. 24) are very similar when the catalytic kinetics are sluggish over a wide range of substrate concentration, but when the kinetics are more rapid the Albery expression over predicts the normalised flux by a factor of 20-35% over a significant range of α values. As previously noted [8] we can identify four limiting cases of eq. 24 and also of eq. 25. The behaviour of the system can be well described in terms of a kinetic case diagram which is a plot of log γ versus log α. This case diagram is outlined in Fig. 9. One limiting case arises when the catalytic kinetics are saturated. Hence α << 1 ∀ γ and we note that the normalised flux reduces to: This expression combines two limiting cases. Case I pertains when the catalytic kinetics are slower than substrate diffusion and γ << 1. Hence tanh ffiffi ffi γ p ≅ ffiffi ffi γ p and eq. 26 reduces to: y≅αγ ð27Þ Translating back into dimensioned quantities we obtain: Hence the amperometric current response is first order with respect to substrate concentration, catalyst concentration, and layer thickness. The reaction occurs uniformly throughout the film. On the other hand when γ >> 1 the reaction kinetics are rapid and we note that tanh ffiffi ffi γ p ≅1: and eq. 26 reduces to: This corresponds to case II. Under these circumstances L >> X K the layer thickness is much greater than the kinetic length and so reaction occurs in a thin reaction layer at the outside of the film. This will pertain when the reaction kinetics between substrate and catalytic sites occur rapidly. Hence the current response is given by: Here the current again is first order in substrate concentration, first order with respect to catalyst concentration, and occurs in a thin reaction layer of thickness X K . Note that eq. 26 joins the two cases I and II. When α >> 1 then s >> K M and the catalytic kinetics are saturated. Hence the normalised current adopts the following form which is valid ∀γ: When γ is small then ffiffiffiffiffiffiffi ffi γ=α p << 1 and tanh½ ffiffiffiffiffiffiffi ffi γ=α p ≅ ffiffiffiffiffiffiffi ffi γ=α p and so the normalised current reduces to: y≅γ ð32Þ This expression defines case III. Here the catalytic kinetics are more sluggish than substrate diffusion and the substrate concentration is the film is larger than the Michaelis constant for the catalytic reaction. Hence the current response for case III is: Hence the rate determining step involves the decomposition of the catalyst/substrate adduct to form products. This occurs throughout the entire thickness of the immobilized film. The current is zero order in substrate concentration and first order with respect to catalyst and layer thickness. Finally when γ >> 1 and the catalytic kinetics are rapid then eq. 31 reduces to: This defines case IV where both the catalytic kinetics are rapid and the substrate concentration in the layer is larger than the Michaelis constant. Under these circumstances the current response is given by: In case IV the current is half order with respect to substrate concentration in the layer, is first order with respect to catalyst and occurs in a thin reaction layer adjacent to the film/solution interface. Hence cases II and IV are connected via eq. 31. In the kinetic case diagram cases II and IV are separated by the line γ = α located in the top right hand quadrant. Case IV pertains when one has thick catalytic layers and when the kinetics are saturated when s > > K M . Under such circumstances one may expect that the outermost regions of the film will be completely saturated whereas the inner regions of the film are unsaturated.
When the kinetics are sluggish then γ is small then tanh Þ and ∀α the general eq. 24 reduces to: This expression can be seen to join cases I and III. This is just a normalised form of the Michaelis Menten equation. It will be valid for thin films where there is very little concentration polarization of the substrate in the film and where the reaction kinetics are rate determining.
Finally, we consider the situation where the catalytic kinetics are rapid compared with substrate diffusion. Hence γ >> 1. Hence ∀α such that α < γ This expression defines the join between cases II (α << 1) and IV (α >> 1). We present a summary of the key kinetic cases and equations in Table 1 and a reaction order summary in Table 2.
We note that the Albery equation presented in eq. 25 reproduces the limiting kinetic case expressions for the normalised current for cases I, II and III. However for case IV corresponding to α >> 1, 0 < γ > 2α, the limiting current is predicted to be: Hence we see that the expression for the normalised current response in these circumstances is y≅y A = ffiffi ffi 2 p or approximately 0.71 times that predicted by the Albery model.

Concentration polarization in the solution
We have neglected the situation of concentration polarization in the solution. Under such circumstances the boundary conditions take the following form: The first boundary condition is the same as that previously noted. The second differs from that previously adopted since we have replaced s ∞ the bulk concentration of substrate with s L the concentration of the substrate at the film/solution interface. We also need to introduce a third boundary condition that of flux balance at the film/solution interface. In this we differentiate between D F the substrate diffusion coefficient in the film and D S that in the solution. Furthermore X D represents the diffusion layer thickness. We introduce the Biot number which compares the rate of substrate diffusion in the solution with that of substrate diffusion in the film: We can readily transform the boundary conditions to the following form: Assuming that the solution of eq. 6 is of the form outlined in eq. 12 then using the boundary conditions in eq. 41 we can readily show that: Also the normalised current y is given by: Note that: Hence we see that the addition of concentration polarization of substrate in solution makes a clean evaluation of the β parameter difficult. This is because concentration polarization in the layer and in solution are intrinsically coupled. We note that when the Biot number is large corresponding to substrate diffusion in solution being larger than that in the layer υ −1 → 0 and β≅ ffiffiffiffiffiffiffiffiffiffiffi γ 1 þ α r which we obtained previously.
Using the 'magic' approximation which leads to eq. 11 and solving using the boundary conditions outlined in eq. 41 we can show that: Where We note that when the Biot number is very large u 1 ≅ 1 as it should. Finally, the steady state current response is given by: We note that for large Biot numbers eq. 47 reduces to This expression reduces to eq. 27 for case I, eq. 29 for case II, eq. 32, and for case III provided that ffiffi ffi γ p =α << 1 and α >> 1. If both γ and α are both   Hence for case IV the 'Magic' approximation over-estimates the current response by a factor of typically ffiffiffi α p .

Potentiometric sensor response
We can readily derive an expression for the response of the sensor when it is operating in potentiometric mode. Early work on this topic was reported by Mell and Maloy [17] for amperometric detection and by Carr et al. [18,19] for potentiometric detection. In this case the pertinent differential equations for substrate and product are given by: Here we have introduced the normalised product concentration as w = p/κs ∞ . The pertinent boundary conditions are: Furthermore we note that: We have previously indicated that: Hence the normalised product concentration is: In particular the potentiometric sensor device measures the product concentration at χ = 0 so: Again in the low α limit the product concentration at the electrode surface is Now if the catalytic reaction kinetics are fast then we use: And eq. 54 reduces to: Hence the surface product concentration (with respect to the bulk substrate concentration) reduces rapidly in an exponential manner as the ratio L/X K increases. Whereas in contrast when saturated kinetics pertains: Hence the surface product concentration under these conditions of high substrate concentration is given by p 0 = k c c Σ L/2D F /L. Hence we note that the limiting expressions for the product surface concentration at χ = 0 are in excellent agreement with those previously reported by Carr [18,19]. The potential is related to the product concentration by the Nernst equation: The steady state concentration profiles of substrate and product within the immobilized layer are outlined in Fig. 10 for a low value of the saturation parameter when the kinetics are first order and unsaturated and for various values of reaction/diffusion parameter in the range 0.1 to 100.
These results mirror nicely the computed results of Brady and Carr [19]. The variation of w 0 versus α is outlined in Fig. 11 for various values of γ, whereas a plot of w 0 versus γ for typical values of α are presented in Fig. 12. We note that w 0 decreases rapidly with increasing substrate concentration in the layer, and the rate of decrease is more prevalent the larger The product concentration at the electrode surface will only begin to increase significantly with respect to the bulk substrate concentration in the layer only when the rate of the catalytic kinetics becomes quite rapid with respect to substrate diffusion in the film.

Competitive inhibition effects
Up until now we have neglected that inhibition effects may be important, especially in biocatalytic systems. The effect of substrate inhibition on the amperometric current response has been reported by Kulys and Baronas [20] and by Rajendran et al. [21]. The former utilized a digital simulation to obtain the predicted current response whereas the latter utilized the Adomian Decomposition Method (ADM) to obtain the steady state amperometric current response. In the present communication we discuss the general situation of competitive inhibition where an inhibitor molecule I will compete with the substrate S for a binding site on the catalyst. We assume that the inhibitor binds to the active site of the catalyst and does not affect catalysis since once the catalyst / substrate complex is formed catalysis occurs. The form of the rate equation assuming Michaelis-Menten kinetics and competitive inhibition is given by: where w denotes the inhibitor concentration and K I is the dissociation constant for the inhibitor/catalyst complex. Hence the pertinent reaction/diffusion equation is given by: From this we note that we can define an apparent Michaelis constant for the catalytic reaction given by: And so eq. 60 can we written as: We introduce the following normalised variables: Substituting these quantities into eq. 60 we obtain the following reaction/diffusion equation: This is just the same equation as we have been discussing previously (eq. 7) with modified definitions of the saturation parameter and reaction/diffusion parameter to take inhibitor concentration and binding into account. Hence the boundary conditions remain the same as outlined in eq. 8 with redefined parameters. When λω << 1 then γ′ ≅ γ α′ ≅ α and inhibition effects can be neglected.
We can readily evaluate the concentration profile of substrate by following the procedure as previously outlined. Hence we can show that: Now the normalised flux is given by: Hence the normalised current response is given by: This expression reduces to:  This defines the normalised current response for an amperometric sensor which is valid for all values of the saturation parameter α, the reaction/ diffusion parameter γ and the inhibition parameter λω. Clearly when λω << 1 eq. 68 reduces to eq. 24 as indeed it should. The variation of normalised current with fixed value of reaction/diffusion parameter γ = 15 as a function of saturation parameter α for various values of inhibition parameter λω ranging from 0.1 to 100 is outlined in Fig. 13.
Furthermore normalised current as a function of γ for a fixed α value (0.1) for various values of the inhibition factor in the range 0-100 is presented in Fig. 14. In both figures the effect of inhibition is very clear. This data refers to the situation where the catalytic kinetics are not too fast and the kinetics are unsaturated. The diminution in the normalised current response as compared with the case where inhibition does not occur with increasing inhibition is clearly evident from these figures.
Again various limiting cases can be identified. Firstly we note that 1 þ α Hence eq. 68 becomes: Now if λω/α << 1 then eq. 69 reduces to eq. 24 which pertains when the inhibitor concentration is zero. Then as before when α << 1 and λω << α then we obtain the result presented in eq. 27 which describes kinetic case I. On the other hand for α >> 1 and if λω << αthen we obtain eq. 34 corresponding to 'mixed' kinetic case IV. Here the catalytic kinetics are rapid and the substrate concentration in the layer is larger than the Michaelis constant. The reaction occurs within a distance X K in the film. As noted previously one might expect the outermost regions of the film to be completely saturated whereas the inner regions are unsaturated.
Alternatively if λω/α >> 1 then 1 þ α 1 þ λω α ≅1 þ λω and eq. 70 reduces to: Also if λω << 1 then by necessity α << 1 and eq. 70 reduces to eq. 26 derived previously corresponding to unsaturated catalytic kinetics. This defines the case I/II scenario. Alternatively if λω >> 1 then eq. 70 reduces to: This is a new expression and describes the normalised current response when competitive inhibition pertains. Two new limiting kinetic cases can be presented depending on the magnitude of the quantity γ/λω. First   Fig. 13. Variation of normalised current for an amperometric sensor exhibiting Michaelis-Menten kinetics when competitive inhibition is important. The curves were computed using eq. 69. This expression is valid for all values of the saturation parameter. The reaction/diffusion parameter is fixed and the inhibitor concentration is varied. when γ << λω then tanh½ ffiffiffiffiffiffiffiffiffiffi γ=λω p ≅ ffiffiffiffiffiffiffiffiffiffi γ=λω p and eq. 70 reduces to: This is case V. This expression transforms into the following: And we obtain case II. This is very similar to the current expression associated with case I but the current magnitude is decreased by a factor of w/ K I arising from the competitive inhibition. The reaction order with respect to layer thickness, catalyst concentration and substrate concentration is unity. The rate decreases in direct proportion to the inhibitor concentration and increases in proportion to the inhibitor/catalyst dissociation constant. This makes sense since if the affinity between inhibitor and catalyst site is low inhibition effects will not be very marked.
As in case I we have unsaturated catalytic kinetics. Second, when γ >> λω then tanh½ ffiffiffiffiffiffiffiffiffiffi γ=λω p ≅1 and eq. 71 becomes: This is case IV. The relevant expression for the current is: Again the catalytic kinetics are unsaturated but occur in a thin reaction layer of length X K . Competitive inhibition reduces the observed current by a factor of ffiffiffiffiffiffiffiffiffiffiffi w=K I p compared to that pertaining in the absence of inhibition. The current decreases in proportion to the square root of the inhibitor concentration and increases directly with the square root of the dissociation constant of the inhibitor/catalyst complex.
If λω/α << 1 then eq. 69 reduces to eq. 31 which defines the join between limiting kinetic cases III and IV. If γ/α << 1 then eq. 30 reduces as previously shown to kinetic case II defined in eq. 31 whereas if the opposite condition pertains simplification of eq. 31 produces eq. 34 characteristic of the mixed kinetic case IV.
Finally we can examine the special case of thin catalytic layers where the concentration polarization of substrate in the layer is minimal. Under such circumstances the differential equation reduces to: Under such circumstances the concentration profile is given by the direct integration of eq. 77 to obtain: This situation will be valid provided γ′/2(1 + α′) < 1 or γ 2ð1 þ αð1 þ λω=αÞÞ < 1. Furthermore the normalised current is: This defines the normalised current for a pure kinetic process valid for all values of substrate concentration taking the effect of competitive inhibition into account.
We note that eq. 79 can be cast in a Lineweaver Burk form as follows: We can readily show that the inverse reaction flux depends in a linear manner on inverse bulk substrate concentration and is given by: Hence we expect the slope of the LB plot to vary with inhibitor concentration w and the intercept to be independent of inhibitor concentration. This is a standard result in enzyme kinetics.

Statement and solution of boundary value problem
Up until now we have discussed the rather simple situation of catalysis at catalytic particles dispersed in a conducting matrix. When the matrix is not conducting a small molecule redox mediator is often utilized as illustrated in Fig. 1. This situation has been discussed previously by Lyons [20] for mediated catalysis in carbon nanotube films, by Bartlett and Pratt [21,22] and by Rajendran and Kirthiga [23] for mediated enzyme catalysis. Consequently only a brief discussion of the problem will be outlined here. Full details of the analysis can be found in the Lyons paper [20]. Here we consider catalytic particles (either redox enzymes or nanoparticles) immobilized in a highly dispersed mesh of polymer or carbon nanotube, which is in turn immobilized on a support electrode surface. A small molecule redox mediator species is used to both regenerate the reduced catalyst and to transfer electrons either to the polymer matrix chain or to the underlying support electrode surface thereby generating a current which can be measured. In this more complicated situation the pertinent transport and kinetic equations of both substrate and redox mediator must be solved in order to obtain an analytical expression for the amperometric current response.
As before we denote the bulk concentration of substrate S as s ∞ and oxidized redox mediator A (such as for example molecular oxygen) as a ∞ . We assume that both of these species diffuse through the external bathing solution and rapidly partition into the polymer/carbon nanotube matrix with partition coefficients given by κ A , κ S respectively. For ease of analysis we neglect concentration polarization of both substrate and redox mediator in solution and note that when x = L, s L = κ S s ∞ , a L = κ A a ∞ . As before we assume that the reaction between oxidized catalyst C O and substrate S proceeds via Michaelis-Menten kinetics to generate product P and reduced catalyst C R . The oxidized catalyst is regenerated via reaction of the reduced catalyst with the oxidized mediator A in the film generating the reduced mediator B. The reaction is assumed to follow simple bimolecular reaction kinetics. However it is quite possible that the reaction between oxidized mediator and reduced catalyst can involve adduct formation and thus involve a Michaelis-Menten type mechanism. Transduction occurs via one of two routes. If the support matrix is conducting the reduced mediator reacts along the length of the polymer/nanotube strand giving rise to a current. If the polymer is non conducting then the reduced mediator (e.g. H 2 O 2 ) is required to diffuse to the support electrode surface and react there with an electrochemical rate constant k′ to regenerate the oxidized mediator. In this way the catalytic cycle is repeated. When the mediator reacts along the strand we can write that the net reaction flux is: Alternatively when the mediator reacts at the underlying support electrode surface the net flux differs from that arising from the substrate We can readily show after some considerable algebra [20] that the normalised reduced mediator concentration in the layer is given by: Furthermore the mediator concentration at the electrode surface is given by: We also note that the mediator flux is given by: This flux expression is valid for all values of α, γ, κ. It is important to note that the full non linear differential equations have been solved without any approximation. In this analysis we have noted that the parameter θ which defines the balance between substrate and mediator diffusion into the film is given by: We also have noted that The variation of observed normalised flux with the substrate reaction/ diffusion parameter γ S when the reaction occurs in a conducting matrix is outlined in Fig. 15 and in a non-conducting matrix in Fig. 16. In Fig. 15 the variation of normalised substrate flux with γ S for various values of κ/α and α is outlined. A fixed value of κ = 1 was adopted in the calculation.
In Fig. 16 the variation of mediator flux with the more general parameter μ is outlined. In this figure we varied γ M over a wide range, and the loss factor was set at 0.1. In Fig. 17 the variation of mediator flux with μ is again presented but this time γ M = 20 and the loss factor was varied from 0.1 to 20. Clearly when the layer is non conducting and a redox mediator is used for amperometric detection the normalised flux varies characteristically with increasing value of the parameter μ. As shown in Fig. 13, calculations were performed for a fixed value of loss factor of 0.1. Initially a plateau type response is established for low values of the μ parameter. This is most marked when γ M is large corresponding to the case where the flux for substrate/catalyst reaction is larger than that for mediator diffusion. The flux then exhibits a significant drop over the region 0.5 < μ < 10.The magnitude of the Ψ B plateau descreses steadily with decreasing value of the parameter γ M . For μ > 50 the normalised flux is minimal for all values of the parameter γ M . A similar type of general variation of normalised flux with μ is seen in Fig. 14. In this calculation the parameter γ M = 20, and the loss factor set at a given value. Each curve corresponds to a fixed loss factor which ranges from 0.1 to 20. Hence the loss factor value has a significant effect on the normalised current response. The greater the loss factor the smaller the corresponding value for Ψ B for any value of μ.

Thin layer limit
Eqs. 98 and 101 may be simplified when the layer is thin. Under such circumstances we set u ≅ 1 in eq. 88 and note: Fig. 15. Variation of normalised substrate flux (or current) for amperometric detection in a conducting matrix (see fig. 1(a)). The curves were computed using eq. 98.  Integrating once and noting the boundary condition at χ = 0, χ and so the normalised current response is readily obtained as: This expression is valid ∀ α, κ values. A general expression for the expected current response when the layer is not conducting can be derived from eq. 89 by setting u ≅ 1 to obtain: Integrating once we note: Integrating again: The integration constants can readily be evaluated from the boundary conditions as: Consequently the normalised flux is given by: This defines the situation for a thin non conducting catalytic layer where the observed current arises via reaction of a redox mediator at the inner support electrode. Again this is valid ∀ α, κ values when diffusive effects of substrate can be neglected.
The full solution to the boundary value problem and that derived via the thin layer approximation are related via a simple factor which depends on the parameter μ. We may readily show that: These equations are illustrated in Fig. 18 and Fig. 19.
It is interesting to note that Ψ S ≅ Ψ S, TL when μ is small. In contrast when under similar conditions we note that: And the equivalence between the full solution and the thin film approximation for the current arising for reduced mediator oxidation at the support electrode will depend both on the magnitudes of κ and α. For instance if the ratio κ/α >> 1 then Ψ B Ψ B;TL ≅1 þ κ and Ψ B ≅ Ψ B, TL when κ << 1. This condition is valid when the reaction between substrate and oxidized catalyst is much slower than the reaction between reduced catalyst and oxidized mediator. Alternatively if the ratio κ/α << 1 then Ψ B Ψ B;TL ≅1 þ α and Ψ B ≅ Ψ B, TL when α << 1. This will happen when the substrate concentration in the layer is much less than the Michelis constant.
Hence the validity of the thin layer approximation depends on the ratio κ α ¼ k c kκ A a ∞ which relates the rate of catalyst/substrate decomposition to that of oxidized mediator/ reduced catalyst reaction (active catalyst regeneration). When active oxidized catalyst regeneration is slower than catalyst/substrate complex dissociation into product and reduced catalyst then the thin layer approximation will be good when the rate of substrate reaction with oxidized catalyst is much slower than the rate of active catalyst regeneration, and f SC << f MC . If the opposite pertains and catalyst/ substrate complex dissociation into product and reduced catalyst is slow compared with active oxidized catalyst regeneration then the thin film approximation will be good provided the substrate concentration in the layer is low.

Limiting rate expressions and the kinetic case diagram
The form of the mathematical equations presented for substrate flux and observed flux suggest that the pertinent kinetic case parameters which can be used to geometrically define the problem in a kinetic case diagram are μ; α; κ; κ α and γ S . Typical expressions for the substrate flux and observed flux for several approximate kinetic cases are presented in table III. These  are derived by taking suitable limiting values of the key reaction parameters as done in previous sections of this paper.
electrode for the situation where the catalytic matrix is non conducting is given by: Again, this case VIII corresponds to modified slow rate determining substrate diffusion modified by the factor g 0 ¼ α 1 þ F l .
Here we see that the if the substrate concentration in the layer is large and much larger than the Michaelis constant, the effect of the mediator loss factor on the net reaction flux can be mitigated.
Alternatively, if κ α > > 1 then This expression valid for all values of the parameter κ joins the kinetic cases VII and IX. Case VII, termed the titration case, will be valid when κ << 1, α << 1. Under such circumstances eq. 130 is valid. When the opposite pertains κ >> 1 and the flux becomes: Here eq. 133 represents case IX which is slow substrate diffusion modified by the factor g ″ ¼ κ 1 þ F l . Here when κ >> 1 such that the substrate/catalyst rate is much greater than the mediator/catalyst rate the diminution of the reaction flux due to the influence of the loss factor can be mitigated the larger the value of κ ¼ k S κ S s ∞ k M κ A a ∞ where k S = k c c Σ /K M and k M = kc Σ are pseudo first order rate constants for substrate/catalyst reaction and active catalyst regeneration respectively. Hence when μ >> 1 we consider cases VII, VIII, and IX.
It is instructive to examine the titration case in a little more detail. Bartlett and Pratt [22] have noted that under certain circumstances the reaction kinetics may be mediator limited in one part of the film and substrate limited in another as a result of changes in substrate and mediator concentration across the layer. This type of behaviour occurs when both mediator and substrate concentrations change significantly within the film. To understand this further we may divide the film into two regions A and B separated by a demarcation line located at χ = ε. In region A for χ < ε the kinetics are substrate limited and for χ > ε corresponding to region B, the kinetics are mediator limited. When the kinetics of the catalysed reaction are fast, the process occurs in a thin reaction zone within the layer at χ = ε. For χ > ε one has no reaction since v → 0. For χ < ε there is no reaction since u → 0. In the steady state the fluxes of substrate and mediator within the reaction zone must balance so we have Simplifying we obtain: κ << 1 or κ >> 1. Considering a non conducting matrix we note that the flux is: When κ/α >> 1 the normalised mediator flux reduces to: When κ << 1 we again get case I and when κ >> 1 we get case V. We conclude that only 3 cases I, III and V pertain for thin films regardless of the conductivity of the catalytic matrix.

Concluding comments
In this paper we have examined the problem of describing the transport and kinetics of catalytic reactions in which the catalyst is immobilized within a support matrix. We have examined a mathematical procedure based of the AGM recently discussed by Dharmalingam and Veeramuni [15] which enables a closed form approximate analytical solution to the Michaelis-Menten kinetic rate equation when coupled to Ficksian diffusion in thin bounded film. We have compared this approximate closed form analytical expression with a numerical simulation obtained using the NDSolve facility in Mathematica 12 and have indicated an excellent agreement for parameter sets of (γ,α) which are appropriate. This AGM derived solution for current is accurate for substrate concentration values close to the Michaelis Constant. The analysis was applied to diffusion/reaction in a planar slab, and in a subsequent paper will be extended to reaction/diffusion a hemispherical polymer film coated on an inlaid disc electrode and in a thin layer coated on a wire electrode. Analytical solutions for both the amperometric and potentiometric sensor response were provided. We then extended this useful analysis to consider the effect of concentration polarization in the solution, and to competitive inhibition. Finally, the analysis was extended to a polymer modified electrode when a redox mediator is used in the polymer film. Kinetic case diagrams were developed and nine approximate limiting expressions for the amperometric response at steady state when the catalytic matrix is either conducting or insulating were developed.

Credit author statement
This paper was conceived and written solely by the corresponding author.

Declaration of Competing Interests
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.