J. Phys. Chem. B 1997, 101, 175-181
175
Novel “Flow Injection” Channel Flow Cell for the Investigation of Processes at Solid-Liquid Interfaces. 1. Theory J. Justin Gooding, Barry A. Coles, and Richard G. Compton* Physical and Theoretical Chemistry Laboratory, Oxford UniVersity, South Parks Road, Oxford OX1 3QZ, United Kingdom ReceiVed: April 9, 1996; In Final Form: October 14, 1996X
A novel “flow injection” channel flow cell is described which enables transient phenomena at the solidliquid interface to be investigated. In the cell the solution reactant is injected into the main channel of a flow cell upstream from, and through the opposite wall from, the solid substrate. A detector electrode is positioned downstream of the latter so that current-time transients can be used to infer the kinetics and mechanism of processes at the solid-liquid interface. The performance of the cell was theoretically modeled using FLOTRAN. It was shown that the “injection” of reactant causes only very minor disturbance to the main channel flow. This facilitates the assumption of Poiseuille flow in the channel so as to enable the numerical interpretation of the current-time transients.
Introduction Reactions between solids and liquids are of widespread synthetic, industrial, and environmental importance. At the fundamental level they involve a coupled sequence of mass transport, adsorption/desorption phenomena, heterogeneous reaction, chemical transformation of intermediates, etc., whose identification, separation, and kinetic quantification are all necessary if the mechanism of the process is to be fully understood and described. Rotating ring-disk electrodes (RRDE) have been applied to such problems1 where the reaction may be controlled by electrochemical parameters. However, in the case of insulating solids, the direct use of electrochemical techniques is precluded, and for such reactions,2-6 the channel flow cell has been shown to be well-suited. The application of the channel flow cell method for investigating heterogeneous reactions is shown schematically in Figure 1a. The solid substrate is located flush with the surface in one wall of the flow cell, through which the solution based reactant flows under laminar flow conditions. Reaction takes place at the solid/liquid interface, and the products are swept downstream by the flow into the zone of the detector electrode. The detector electrode response is measured as a function of mass transport (solution flow rate). The well-defined hydrodynamics in the channel, together with a suitable model for the heterogeneous reaction and any homogeneous solution chemistry, enable the theoretical prediction of the detector electrode response. In this way the theoretical responses for different candidate mechanisms can be derived and compared with the experiment to elucidate the process which physically takes place. Conventional channel flow cells and the RRDE are restricted to steady-state processes. Although many reactions can be investigated in this way, there are some cases such as reactions possessing rapid kinetics where the reaction may be complete before steady state is achieved. For this reason the concept of the double-channel electrode was extended to the investigation of heterogeneous processes7-10 so that transient experiments could be conducted by electrogenerating a solution phase reactant at an electrode upstream of a solid substrate and using a second electrode, downstream of the solid, to record current transients so as to infer the nature of the interfacial process(es). The ability of the double-channel electrode to investigate * To whom correspondence should be addressed. X Abstract published in AdVance ACS Abstracts, November 15, 1996.
S1089-5647(96)01063-2 CCC: $14.00
Figure 1. Schematic diagram of (a) a channel flow cell configured for solid substrates and (b) the flow injection channel flow cell with the injection slit located on the opposite wall to the detector electrode.
heterogeneous reactions was further enhanced by locating the generator electrode on the far wall from the detector.10,11 With both geometries the time the reaction commences can be precisely controlled by means of the flow rate and the time at which the upstream electrolysis is initiated. Note that this use of the double-channel electrode technique is distinct from the analytical applications of double-electrode channels,18 since in this case an essential factor is the precise knowledge of the hydrodynamic conditions and the geometry connecting the electrodes and the substrate. However, the variety of solid-liquid interfacial reactions amenable to study using a double-channel electrode is restricted by the range of reactant species that can be generated at the upstream electrode. If a double-channel electrode is used in a transient study of a heterogeneous process, the main carrier solution must be inert to the solid substrate prior to the potential step. The potential step must introduce the solution reactant into the flow stream, and therefore, the solution reactant is restricted either to the product of anodic stripping of the electrode8 or the electrolytic conversion of the inert carrier solution.9 The aim of the work presented in this paper, and in the sister publication,12 is the generalization of the generator function so © 1997 American Chemical Society
176 J. Phys. Chem. B, Vol. 101, No. 2, 1997
Figure 2. Mechanism by which injection is switched on and off by changing the volume flow rate. δD represents the limit of the injection solution diffusion layer.
that the substrate may be exposed to any solution species at a precise instant for a defined period of time. This could in principle be achieved by using mechanical means, e.g., a twoway valve to switch between an inert solution and one containing reactant, or a valve to control the addition of a reactant solution through a mixer arrangement, or by switching an injection loop into the flow. However, during the transient changeover between solutions, the conditions may not be welldefined in mechanically controlled systems due to such factors as opening behavior and dead volume of the valves or the variable and flow-rate dependent path through mixing devices. Switching in an injection loop may involve momentarily halting the main carrier flow which will destroy the Poiseuille flow structure and possibly cause turbulence, and the flow pattern must then be re-established from an uncertain initial condition. It would therefore be difficult to achieve precise and reproducible concentration profiles during the transient onset and the cutoff of exposure of the substrate to the reactant. We have designed a flow cell in which inert carrier and reactant solutions flow continuously, with substrate exposure being controlled by varying the flow rate. This system allows a precise and reproducible “on” and “off” control of the exposure of the substrate to the reactant, as well as control of the concentration, and avoids the limitations imposed by mechanical valves. In particular, it offers the advantage that the behavior throughout the transient period between the “on” and “off” conditions is clearly defined and may be calculated if required. In concept the cell is related to the double-channel electrode, by replacing the generator electrode with a slit located in the wall of the channel opposite to the detector electrode (Figure 1b). Reactant solution continuously enters the channel flow cell from the slit and joins a main flow of inert carrier solution. At slow flow rates the reactant diffuses across the channel and reacts with the solid substrate before being swept out of the channel. The flow rate, the flux of material entering the injection slit (Figure 1b), and the rate of the interfacial reaction will define the amount of reactant which passes over the solid of interest to be detected at the downstream electrode. An important objective is for the active compound to be introduced into the flow stream without significant disruption of the main carrier flow, thus allowing the solution of the convective diffusion equations using well-established computational methods.3,4,13-15 The experimental current-time transients recorded at the detector electrode may then be interpreted in terms of the processes occurring at the solid-liquid interface under study. For transient measurements the exposure needs to be turned “on” and “off”. The switching was achieved by altering the flow rate while allowing injection solution to continually enter the channel. As shown in Figure 2, at sufficiently fast flow rates, no reactant can reach the opposite wall before being swept out of the channel. The solid substrate was exposed to the reactant by reducing the flow rate. Exposure could then be turned off again by increasing the flow rate. The advantages of being able to introduce reactant solution in this way are the investigation of rapid kinetics, the production of labile solution reactants in situ, and the investigation of virgin
Gooding et al. surfaces. This system will clearly allow the investigation of fast heterogeneous kinetics comparable to that achieved by the opposed double-electrode channel cell11 (where it was shown that an adsorption rate constant k0 ) 1 cm s-1 gave a clearly defined response) but with the benefit of allowing a far wider range of reactants. The analogy of such a cell with flow injection analysis (FIA)16 is obvious, and hence, the new cell is called a “flow injection” channel flow cell. By definition FIA involves sample determination under non-steady-state conditions, where a quantity of dissolved sample is injected upstream of the detector into a carrier or reagent stream which flows through the system,16 and dispersion may be employed to obtain a continuous gradation in the sample/reagent ratio.17 However, the similarity with FIA is limited, in that reactant solution is injected continuously into the new cell without any on-off control and is present in the cell at all times. In this paper the flow injection cell is investigated theoretically to ascertain the influence of the injection flow on the main carrier flow and to enable the determination of the reaction mechanism and kinetic parameters. In the sister publication12 the experimental development and characteristics, together with some illustrative results, are presented. The flow profiles within the cell were calculated by adapting the fluid dynamics program FLOTRAN, designed originally for solving heat transfer problems, for this application. The calculated injection responses for the reduction of potassium ferricyanide,
Fe(CN)63- + e- h Fe(CN)64-
(1)
in the absence of a solid substrate, were compared to those obtained by solving the convective diffusion equations in the channel using the well-established backward implicit finite difference (BIFD) method3,4,13-15 with appropriate boundary conditions to describe the flux of injection material into the cell. The resulting theoretical framework provides the basis for the interpretation of experimental current-time transients in terms of the processes occurring at the solid-liquid interface under investigation. Theory Modeling of the flow injection channel flow cell is required to determine the hydrodynamics in the injection slit and in the main channel and hence the mass transport to the detector electrode. With two flow streams meeting at the injection slit the velocity profile within the cell is less well defined than in the conventional channel flow cell where Poiseuille flow exists throughout the region of interest. In the first instance, the cell hydrodynamics and the detector electrode response, once steady state is achieved, will be investigated as a function of the primary channel flow rate, when there is no reactive solid substrate present. To determine the hydrodynamics throughout the cell requires solving the equations governing the dynamics of a fluid. If we consider a solution as consisting of particles, then within a volume of solution the flux through the surface must equal the decrease of mass within the volume. The flux through this volume will depend on the momentum of the solution particles and the presence of any energy gradients (e.g., temperature or concentration gradients). Thus the fluid’s dynamics can be described by equations which consider the conservation of mass, momentum, and energy.19-21 FLOTRAN solves these equations (see Appendix) to calculate the flow profile throughout the channel and slit,19 with the aid of appropriate boundary conditions.
Processes at Solid-Liquid Interfaces The calculation of the mass transport to the detector electrode is greatly simplified if under steady-state conditions the primary flow is assumed to be parallel throughout the channel. This assumption infers that there is no perturbation of the primary flow due to injection from the secondary flow. Hence comparison of the FLOTRAN results with those obtained from assuming a parabolic flow, using the backward implicit finite difference method,3,4,13-15 is of interest. In the modeling it was assumed that the injection solution was 100 mM potassium ferricyanide in water with a background electrolyte of 0.2 M KCl and 0.1 M KOH and the carrier solution was an aqueous solution of 0.2 M KCl and 0.1 M KOH. All the input parameters used in the models were typical values for such an experimental system. The precise values correspond to the experimentally relevant parameters discussed in the sister publication to this paper.12 Finite Element Model FLOTRAN (ANSYS, Inc., Houston, PA) is a commercial engineering package designed for solving fluid flow and heat transfer problems. FLOTRAN is a solver only, and it must be used in conjunction with another programsin this case ANSYS (ANSYS, Inc., Houston, PA)swhich can define the model geometry, the finite element grid, and the boundary conditions. ANSYS also processed the output from the FLOTRAN program. The ANSYS preprocessor was used to set up a twodimensional model of the flow injection channel. For computational efficiency the number of mesh elements should be minimized, so a graded mesh was employed with fine divisions in the regions of high gradients graduating to a coarser mesh where gradients were low. The number of elements varied between 6000 and 7000 depending on the gap length between the injection slit and detector electrode. Convergence was obtained with element sizes of 10 µm × 10 µm (x × y dimensions) at the mouth of the slit, 10 µm × 20 µm in the main channel adjacent to the slit grading to 200 µm × 20 µm in the gap and channel outlet. Over the electrode elements were 25 µm × 20 µm. To ensure that the mesh was fine enough for correct convergence, the mesh size was varied and tested to meet the criterion that, with a further 2-fold reduction in element size in any region of the channel, the calculated current remained unchanged to three significant figures. As the energy equation used in FLOTRAN relates to heat transfer and there was no facility for including species transport, FLOTRAN must be adapted to calculate concentration. Concentration was therefore represented by temperature and diffusion coefficient by thermal conductivity, and the value computed for heat flux then represented the quantity of reactant oxidized/ reduced at the electrode. The temperature/concentration analogy used made 1° equivalent to 1 mol m-3 (i.e., 1 mM) and 1 J to 1 mol. Appropriate values for the heat capacity and thermal conductivity were entered into FLOTRAN to represent the chemical quantities in a consistent set of units. For heat capacity, which has units of J kg-1 K-1, the equivalent units were mol kg-1 (mol m-3)-1. Thus the equivalent units simplify to m3 kg-1, which is the reciprocal of density. Hence the specific heat entered into FLOTRAN was 0.001 07 J kg-1 K-1. Similarly the thermal conductivity (J m-1 K-1 s-1) was analogous to the diffusion coefficient in units of m2 s-1. The diffusion coefficient for ferricyanide22 at 25 °C in aqueous 0.2 M KCl is 7.6 × 10-10 m2 s-1. The density and viscosity of the solution used for modeling were measured values.12 Density and viscosity both varied with concentration of the reactant, and these dependencies were incorporated into FLOTRAN by entering appropriate numerical values as temperature coefficients
J. Phys. Chem. B, Vol. 101, No. 2, 1997 177 for these parameters. The density difference is particularly important because in the experimental arrangement the cell is positioned in the flow system such that the x direction is vertical with the cell inlet positioned below the outlet. While this adaptation of FLOTRAN involved the input of unusual numerical values for a number of parameters, no problems were experienced with running or convergence of the program. The solution using FLOTRAN requires the specification of the appropriate boundary conditions. The boundary conditions used, where the x direction is along the channel and the y direction defines the channel height, as in Figure 1b, were as follows: (1) Inlets: (a) Parabolic flow profiles in the main channel and the slit. The x-velocity profile at the inlet to the main channel was expressed as
[
U ) Uo 1 -
]
(y - h)2 h2
(2)
where Uo is the maximum velocity, h is half the cell height, and y is the position in the y direction. At the inlet to the injection slit the y velocity was
[
V ) Vo 1 -
]
(x - l/2)2 (l/2)2
(3)
where l represents the length of the slit and thus l/2 is analogous to the channel half-height, h. The magnitude of eq 3 describes the flux of injection solution entering the main channel. This flux was incorporated into the model as a percentage of the main channel flow. (b) The transVerse Velocities were zero (i.e., U in slit and V in channel inlet). (c) The solution concentrations were 100 mM at the inlet to the injection slit and 0 mM at the inlet to the channel. (2) Outlet: Zero Pressure. (3) Walls: Solution Had Zero Velocities in Both the x and y Directions. In terms of concentration, the walls have a no flux condition. (4) At the Electrode, the Concentration Was Zero, As All the Material That Reached the Surface Was Oxidized/ Reduced. The program was run for 70 iterations to solve for flow and concentration together to obtain a converged flow pattern and a further 300 iterations for the concentration alone to obtain a converged concentration profile. The FLOTRAN modeling was computed on a Sun 670MP where calculation time was approximately 1 h. Backward Implicit Finite Difference Flux Model The steady-state injection response could also be calculated using the BIFD method provided that the primary channel flow is assumed to remain parallel throughout the channel (i.e., the injection flow did not significantly perturb the laminar flow). The greater computational efficiency of the BIFD algorithm makes it an attractive alternative to FLOTRAN for comparison with the steady-state currents from injection experiments. The BIFD method solves the convective-diffusion equations with a velocity profile described by Poiseuille flow within the main channel, with the injection process represented as a flux of material entering the channel through a “wall” at the position of the injection slit. Assuming transport by diffusion is only significant in the y direction, the steady-state convectivediffusion equation describing the mass transport of a species B within the channel is23
178 J. Phys. Chem. B, Vol. 101, No. 2, 1997
∂2[Β] ∂[Β] )0 DB 2 - U ∂x ∂y
Gooding et al.
(4)
where U is the velocity in the x direction and is given by eq 2. Equation 4 assumes that axial diffusion effects may be neglected; this is valid provided that the electrodes considered are not of microelectrode dimensions.24,25 To solve the convective-diffusion equation for the flow injection cell the appropriate boundary conditions are required. In the region of the injection slit, between 0 and x1 in Figure 1, the flow of material into the cell can be described as a flux, as discussed previously, where
D
∂[Β] )R ∂y
(5)
R is the value of the flux entering the channel as determined experimentally in the associated publication.12 The boundary conditions describing this model for the injection cell are as follows:
(i) Upstream of the slit
x ) 0;
[B] ) 0
(ii) At the injection slit
0 < x < x1; y ) 2h; D(∂[B]/∂y) ) R
(iii) At the far wall of the cell
x1 < x < x3; y ) 2h; D(∂[B]/∂y) ) 0
(iv) At the detector wall of the cell
0 < x < x2; y ) 0; D(∂[B]/∂y) ) 0
(v) At the detector electrode
x2 < x < x3; y ) 0; [B] ) 0
where the parameters x1, x2, and x3 are as in Figure 1b. The computation was conducted using a SUN 670MP with programs written in Fortran 77. For the channel dimensions of slit length ) 0.018 cm, gap length ) 1.298 cm, electrode length ) 0.432 cm, and cell height ) 0.0381 cm, the program was found to converge to five significant figures with a grid of J ) 1000 and K ) 50 000. The large value of K required for convergence is due to the narrow slit length, at least 500 divisions being needed in that region for convergence. However, because of the efficiency of the algorithm used to solve the convective diffusion equation, such a grid size required less than five minutes of CPU time. Results and Discussion The initial modeling with FLOTRAN was conducted to determine the influence of the injection slit on the flow profile in the injection cell. Subsequently it was investigated how the concentration of injection solution altered throughout the cell. The initial modeling was conducted for a slit of length 0.022 cm and a cell height, 2h, of 0.0381 cm. Figure 3 shows a streamline diagram in the region of the slit for a main flow rate of 1.48 × 10-3 cm3 s-1, where the flow of material entering from the injection slit was 0.304% of the main flow. Figure 3 is magnified 10 times in the y direction to enable the deflections in streamlines to be visible. A streamline in the diagram represents the path of a solution particle. As can be seen from Figure 3, the motion of the main solution flow past the slit results in a vortex of spinning solution being created just inside the slit entrance.
Figure 3. FLOTRAN diagram showing streamlines in the region of the injection slit. The total flow rate was 1.48 × 10-3 cm3 s-1, of which 0.304% was flow from the injection slit.
Note that the flow cell is mounted vertically into the flow system with the main channel solution flowing upward. The direction of solution flow is shown by the arrowheads in Figure 3. The streamlines show that the solution in the main channel is slightly deflected from its parallel flow and is pulled toward the injection slit. At the leading edge of the slit it appears that some of the main carrier solution is drawn into the vortex. Within the injection slit, material trapped in the vortex is thrown in an anticlockwise direction. Across the slit entrance the principal direction of flow is in the x direction (the same direction as the main channel flow) and only at the downstream edge of the slit is there evidence of convection of injection solution into the channel. The streamlines for the reactant solution in the slit in Figure 3 remain parallel at the beginning of the slit but then tend toward the lower side. This is in part due to the greater density of the injection solution relative to the carrier solution and in part due to solution motion at the slit entrance, where the vortex rotation will tend to force injection solution toward the bottom/upstream side of the slit. The main channel flow displaces the vortex downstream from the center of the slit. At the downstream edge of the slit Figure 3 shows that material is partly swept out into the channel and partly forced back up the injection slit. How far the vortex extends into the primary channel flow is probably a function of the ratio of the main flow to the injection flow. It is a slight variation of the position of this vortex which is thought to influence the decrease in measured flux with increasing flow rate.12 With a slower flow rate of 4.9 × 10-4 cm3 s-1 the vortex is closer to the main channel and hence propels injection solution further into the channel. The variation in the injection solution concentration in the region of the slit is shown in Figure 4. The contours in this figure represent a change in concentration of 2 mM. As expected the concentration changes rapidly at the slit outlet. The diagram again shows the effect of the vortex where the concentration decreases more slowly at the upstream side of the slit. Note also that the concentration is very nearly constant all the way across the slit entrance. The diffusion of the injection solution across the channel and subsequent reduction at the detector electrode is shown in Figure 5. Figure 5 is a concentration contour diagram showing contours at 0.1 mM intervals beginning at 0.1 mM. For purposes of clarity only the first 20 contours are shown and
Processes at Solid-Liquid Interfaces
J. Phys. Chem. B, Vol. 101, No. 2, 1997 179 TABLE 1: Comparison between Currents Predicted by FLOTRAN and by the BIFD Flux Model for Two Different Gapsa gap ) 1.298 cm I/µA flow rate/ 10-3 cm3 s-1 FLOTRAN 1.18 1.53 2.23 3.09
23.09 14.30 5.32 1.87
gap ) 0.376 cm I/µA BI 24.00 15.52 5.96 1.59
flow rate I/µA I/µA /10-3 cm3 s-1 FLOTRAN BI 1.25 1.75 2.51
4.91 1.44 0.21
5.07 1.29 0.15
a The slit length was 0.022 cm, electrode length was 0.432 cm, and electrode width was 0.388 cm.
Figure 4. FLOTRAN concentration contour diagram in the region of the injection slit. Conditions are as in Figure 3. Contours are at 2 mM intervals.
Figure 6. Comparison between steady-state injection currents predicted by FLOTRAN (0) and by the BIFD flux model (×) for a cell of dimensions slit length 0.022 cm, electrode length 0.432 cm, electrode width 0.388 cm, gap length 1.298 cm, and cell height 0.0381 cm. Figure 5. FLOTRAN concentration contour diagram for the entire flow injection flow cell where the total flow rate was 4.9 × 10-4 cm3 s-1, of which 0.859% was flow from the injection slit. Note that the diagram is expanded 10 times in the y direction. Contours are at 0.1 mM intervals.
again the y dimension of the diagram has been expanded. In this figure the total flow rate is 4.9 × 10-4 cm3 s-1 with the injection flow comprising 0.859%. The Figure shows how, on the opposite wall to the slit, the concentration of reactant increases in the downstream direction and also shows the consumption of the reactant at the electrode. The comparatively minor distortion of the Poiseuille flow profile in the main channel, the virtually constant concentration at the slit surface, and the absence of convection of injection solution across the channel indicates that one of the original aims of the cell was achieved, the introduction of injection solution into the main channel without significant disruption of the main flow. With such minor distortion of the main carrier flow the BIFD modeling could potentially predict the steady state injection current. The BIFD model is much less computer intensive than the FLOTRAN modeling as no flow distribution has to be solved and the only difference from a double-channel electrode is the assumption of a constant flux across the surface of the slit. The predictions of the steady-state current from the flow injection cell using BIFD and using FLOTRAN were calculated separately for slit dimensions of 0.022 × 0.367 cm, electrode dimensions of 0.432 × 0.388 cm, cell height of 0.0381 cm, and gap lengths of 0.376 and 1.298 cm. The BIFD flux model currents are compared to those obtained for cells of the same dimensions using FLOTRAN and are shown in Table 1 and Figures 6 and 7.
Figure 7. Comparison between steady-state injection currents predicted by FLOTRAN (0) and by the BIFD flux model (×) for a gap length of 0.376 cm. All other cell dimensions are as in Figure 6.
As can be seen from Table 1 the predicted currents using the FLOTRAN and the BIFD flux models are very similar for both gap lengths. The similarity in predicted steady-state current using the two different models vindicates the assumption in the BIFD modeling that the injection of material into the main channel is having little significant effect on the well-defined Poiseuille flow with the injection fluxes used. The maintenance of the Poiseuille flow with the injection fluxes used in the model serves as a guide for injection fluxes to be used in experimental development of the flow injection cell.
180 J. Phys. Chem. B, Vol. 101, No. 2, 1997 Figure 6 shows that the current recorded at the downstream electrode due to the injection, the injection current, increases with decreasing flow rate. Thus at the low flow rates the injection material has sufficient time to diffuse across the channel to the detector electrode before being swept out of the cell due to the main carrier flow convection. As the main carrier flow rate is increased, less injection material is able to diffuse across the channel before being swept away and hence the injection current is lower. At the fast flow rates the detected current falls to zero. Thus the solution flow rate can be used to control the extent to which material entering the flow cell can reach the opposite side and react with the solid substrate. The dramatic increase in the amount of injection material which reaches the far wall of the channel with a minor reduction in flow rate, especially at low flow rates, would enable the investigation of an interfacial process over a wide range of reactant conditions and hence reaction rates. The influence of the gap length on the measured injection current is shown by comparing Figure 6 with Figure 7, where the same electrode and channel dimensions are used for two different gap lengths of 0.376 and 1.298 cm. The figures clearly show a large decrease in the current monitored at the detector electrode with a decrease in the gap length. In relative terms this decrease with gap length is much greater at higher flow rates. One advantage with decreasing the gap length, consistent with double-channel electrodes, is the smaller the gap between the injection slit and the detector electrode the lower the time taken for the injection transient to reach steady state. A similar variation in current with flow rate and gap size was observed both theoretically and experimentally for double-channel electrodes of equivalent geometry.10,11 The variation of the detector current with both flow rate and gap size is a consequence of the necessity of the reactant to diffuse across the channel to the detector electrode before it is swept out of the channel by the convective flow. Thus, to monitor a heterogeneous reaction occurring on the opposite wall of the channel from the injection slit, the gap between where the injection solution enters the channel and the substrate must be sufficiently large or the flow rate sufficiently slow for the surface to be probed. However, as can be seen from Figures 6 and 7, only minor changes in the flow rate have quite a considerable influence on the amount of injection material reaching the detector electrode. Thus the ability of the flow injection cell to subtly probe the interfacial reaction is apparent. This has been demonstrated theoretically for a double-electrode channel of the equivalent geometry with the two electrodes on opposite walls.11 The position of the injection slit on the opposite wall of the channel from the solid substrate allows stimulation of the interfacial reaction by only tiny amounts of solution reactant. As the solid substrate is only exposed to the solution reactant when the channel flow rate is reduced, experiments could be designed where the solid is exposed to such small levels of material that the rise time of the transient was insignificant compared to the lifetime of the reaction transient. Hence, provided the initial period of the experiment is neglected, the change in detected current with time could be analyzed without requiring the calculation of the change in the concentration of injection solution with time. This is a particularly attractive approach as rapid heterogeneous reactions can be investigated without modeling the full transient behavior of the flow injection channel flow cell, which is considerably more complex and computer intensive than the steady-state case. Conclusions A novel “flow injection” channel flow cell for the study of heterogeneous kinetics at a nonconducting solid-solution
Gooding et al. interface has been described theoretically where the solution reactant enters the channel upstream ofsand through the opposite wall of the channel tosthe solid substrate and detector electrode. Modeling of the cell using the FLOTRAN program shows that provided the flux into the channel of the injected material is sufficiently low there is no major disruption of the well-defined Poiseuille flow of the main carrier stream. Theory predicts the variation in steady-state injection current with flow rate and gap length (between the injection point and the detector electrode). The current was shown to increase dramatically with decreases in flow rate or increases in gap length. Thus varying the flow rate or the gap length allows subtle probing of interfacial reactions as the amount of reactant which reaches the solid substrate on the opposite wall of the channel may be precisely controlled. The experimental realization of the “flow injection” channel flow cell is presented in an associated publication.12 Acknowledgment. We thank Zeneca for support. J.J.G. would like to thank Merton College, Oxford, for a Senior Scholarship. Appendix FLOTRAN solves the three governing equations of fluid dynamics:19-21 (1) Conservation of mass (the continuity equation). (2) Conservation of momentumsin the case of fluids, the Navier-Stokes equations. (3) Conservation of energysfor flowing fluids, this includes internal, kinetic, and potential energy. FLOTRAN mathematically expresses these three equations for an incompressible fluid as follows: (1) The continuity equation
∂ ∂ ∂ ∂F + (FU) + (FV) + (FW) ) 0 ∂t ∂x ∂y ∂z
(6)
where F is density, t is time, and U, V, and W are the velocities in the x, y, and z directions, respectively. (2) Navier-Stokes equations
x momentum: ∂(FU) ∂(FUU) ∂(FVU) ∂(FWU) + + + ) ∂t ∂x ∂y ∂z ∂ ∂U ∂p ∂ ∂U ∂ ∂U + µ + µ (7) Fgx - + µ ∂x ∂x ∂x ∂y ∂y ∂z ∂z
( )
( ) ( )
y momentum: ∂(FV) ∂(FUV) ∂(FVV) ∂(FWV) + + + ) ∂t ∂x ∂y ∂z ∂ ∂V ∂p ∂ ∂V ∂ ∂V + µ + µ (8) Fgy - + µ ∂y ∂x ∂x ∂y ∂y ∂z ∂z
( )
( ) ( )
z momentum: ∂(FW) ∂(FUW) ∂(FVW) ∂(FWW) + + + ) ∂t ∂x ∂y ∂z ∂ ∂W ∂p ∂ ∂W ∂ ∂W + µ + µ (9) Fgz - + µ ∂z ∂x ∂x ∂y ∂y ∂z ∂z
( )
( ) ( )
where Fgx, Fgy, and Fgz are the body forces which apply to the entire mass of the fluid element. Note the ∂(FUW)/∂x terms represent the change in velocity due to the pressure acting on the fluid and the (∂/∂x)(µ(∂W/∂x)) type terms encompass frictional forces.
Processes at Solid-Liquid Interfaces (3) Energy equation
∂(FCpT) ∂(FUCpT) ∂(FVCpT) ∂(FUCpT) + + + ) ∂t ∂x ∂y ∂z ∂ ∂T ∂ ∂T ∂ ∂T K + K + K + Q (10) ∂x ∂x ∂y ∂y ∂z ∂Z
( )
( ) ( )
where T is the absolute temperature, Cp is the specific heat (J kg-1 K-1), K is the thermal conductivity (J m-1 K-1 s-1), and Q is the internal heat generation rate (J m-3 s-1). The equations are solved for temperature, pressure, and velocity. FLOTRAN utilizes a finite element method to solve the above equations which involves a sequential solution algorithm with an iterative procedure. To adapt FLOTRAN to calculate concentration, concentration was represented by temperature and diffusion coefficients by thermal conductivity. Provided the appropriate boundary conditions are specified, the three sets of equations above can be solved simultaneously to give the flow, the concentration profile, and the reactant flux in the flow injection cell. References and Notes (1) Albery, W. J.; Bartlett, P. N.; Wilde, C. P.; Darwent, J. R. J. Am. Chem. Soc. 1985, 107, 1854. (2) Compton, R. G.; Wilson, M. J. Appl. Electrochem. 1990, 20, 793. (3) Compton, R. G.; Harding, M. S.; Atherton, J. H.; Brennan, C. M. J. Phys. Chem. 1993, 97, 4677. (4) Compton, R. G.; Unwin, P. R. Comp. Chem. Kinet.1989, 29, 173. (5) Compton, R. G.; Harding, M. S.; Pluck, M. R.; Atherton, J. H.; Brennan, C. M. J. Phys. Chem. 1993, 97, 10416.
J. Phys. Chem. B, Vol. 101, No. 2, 1997 181 (6) Gooding, J. J.; Compton, R. G.; Brennan, C. M.; Atherton, J. H. J. Colloid Interface Sci., in press. (7) Compton, R. G.; Sanders, G. H. W. J. Colloid Interface Sci. 1993, 158, 439. (8) Compton, R. G.; Pritchard, K. L. J. Chem. Soc., Faraday Trans. 1 1990, 86, 129. (9) Compton, R. G.; Stearn, G. M.; Unwin, P. R.; Barwise, A. J. J. Appl. Electrochem. 1988, 18, 657. (10) Compton, R. G.; Coles, B. A.; Gooding, J. J.; Fisher, A. C.; Cox, T. I. J. Phys. Chem. 1994, 98, 2446. (11) Compton, R. G.; Coles, B. A.; Fisher, A. C. J. Phys. Chem. 1994, 98, 2441. (12) Gooding, J. J.; Coles, B. A.; Brennan, C. M.; Atherton, J. H.; Compton, R. G. J. Phys. Chem., in press. (13) Compton, R. G.; Pilkington, M. B. G.; Stearn, G. M. J. Chem. Soc., Faraday Trans. 1 1988, 84, 2155. (14) Anderson, J. L.; Moldoveanu, S. J. Electroanal. Chem. 1984, 179, 107. (15) Fisher, A. C.; Compton, R. G. J. Phys. Chem. 1991, 5, 7538. (16) Hall, E. A. H. Curr. Opin, Biotechnol. 1991, 2, 9. (17) Ruzicka, J.; Hansen, E. H. Anal. Chim. Acta 1988, 214, 1 (18) Roston, D. A.; Shoup, R. E.; Kissinger, P. T. Anal. Chem. 1982, 54, 1417A (19) Flotran Theoretical Manual, ANSYS Inc., 201 Johnson Road, Houston, PA 15342-1300. (20) White, F. M. Viscous Fluid Flow; McGraw-Hill: New York, 1974. (21) Probstein, R. F. Physicochemical Hydrodynamics: An Introduction: Butterworth: London, 1989. (22) Von Stackelberg, M. V.; Pilgram, M.; Toome, V. Z. Elektrochem. 1953, 57, 342. (23) Levich, V. G. Physicochemical Hydrodynamics; Prentice Hall: Englewood Cliffs, NJ, 1962. (24) Compton, R. G.; Fisher, A. C.; Wellington, R. G.; Dobson, P. J.; Leigh, P. A. J. Phys. Chem. 1993, 97, 10410. (25) Booth, J.; Compton, R. G.; Cooper, J. A.; Dryfe, R. A. W.; Fisher, A. C.; Davies, C. L.; Walters, M. K. J. Phys. Chem. 1995, 99, 10942.
