Effects of Magnetic Field Directed Orthogonally to Surfaces on

Apr 3, 2007 - Enhanced detection of quantum dots by the magnetohydrodynamic effect ... on a chip through controlled-current redox magnetohydrodynamics...
2 downloads 0 Views 229KB Size
Download PDF
6024

J. Phys. Chem. C 2007, 111, 6024-6032

Effects of Magnetic Field Directed Orthogonally to Surfaces on Electrochemical Processes Oleg Lioubashevski, Eugenii Katz, and Itamar Willner* The Institute of Chemistry, The Hebrew UniVersity of Jerusalem, Jerusalem 91904, Israel ReceiVed: December 31, 2006; In Final Form: February 13, 2007

The study provides a model for the effect on electrochemical processes of a magnetic field that is directed perpendicular to a planar electrode surface. The analysis assumes the formation of a hydrodynamic boundary layer of the fluid motion that is generated above a semi-infinite electrode as a result of magnetic force action. For a uniform magnetic field, the model predicts the decrease of the diffusion boundary layer thickness upon increasing either the magnetic field strength, B, or the reagent bulk concentration, C, with the power dependence correspondingtoC 4/3B 2/3.Thelimitingcurrentdensity,iL,isformulatedasiL ≈0.5χm1/3(µoF)-1/3R5/3D5/9ν-2/9nFC 4/3B 2/3. This permits semiquantitative interpretation of the enhanced mass transfer at electrode surfaces for electrochemical reactions and predicts a magnetic field effect on mass transfer-limited heterogeneous reactions of magnetic species. The results imply that the perpendicular magnetic field will affect any mass transport limited transformation involving paramagnetic species at flat surfaces.

Introduction The effects of magnetic fields on electrochemical systems are well documented and attract substantial scientific attention. The term “magnetoelectrolysis” reflects the different modes by which magnetic fields influence the performance of electrochemical processes.1 The interest in the interactions of magnetic fields with electrochemical processes originates from the possibility of controlling the morphology of deposits on the electrodes or the mass transport of reactive components toward the electrode surface, and thus controlling the reaction rate at the electrode/electrolyte interface.2 Different types of forces may be exerted on electroactive species in a magnetic field, such as forces originating from the motion of charged species and forces resulting from magnetic field gradients or paramagnetic species concentration gradients. Such forces may contribute to field-driven flow of the solution and thereby influence the mass transport and the reactions rates at the electrode interfaces.3 It was established that the magnetic field effects on electrochemical processes originate from fieldinduced convection at the electrode surface that decreases the diffusion boundary layer thickness and thereby increases the mass transport at the electrode and the resulting limiting electrical current.4 The flow of solution generated by the momentum transfer from the magnetic field-driven ions to neighboring solute molecules was suggested as the ground principle that controls the electrochemical reactions by the magnetic field. That is, the overall magnetic field effect on the dynamics of the system may be described as a magnetic body force that acts on a unit volume of solution element through which the current passes, rather than in terms of the microscopic interaction of the magnetic field with the discrete ions that generate the current. The analogy between magnetic fields and electrode rotation as a means of generating convection has been noted previously.5,6 Persuasive evidence of the hydrodynamic nature of the induced convection is that a similar enhancement can be achieved by electrode rotation, and the effect of the * To whom all correspondence should be addressed: Tel: 972-26585272. Fax: 972-2-6527715. E-mail: [email protected].

magnetic field is reduced by increasing the viscosity of the electrolyte.7 The effect of a magnetic field on the electrochemical processes has been investigated experimentally with respect to its strength and orientation relative to the electrode surface. Substantial efforts were directed to relate the limited current, IL, with experimental parameters that affect the electrochemical processes in the systems studied where the magnetic field was applied parallel to the electrode surface. The study of Aogaki et al.,4b on the electrodeposition of copper, and of Aaboubi et al.,5,8 on the redox reactions of the ferricyanide-ferrocyanide electroactive couple, showed that the steady-state limiting current, IL, was proportional to C4/3B1/3. The dependence of IL on different parameters of the system under magnetic field applied parallel to the electrode surface was systematically studied by Leventis et al.7a using a range of compounds and solvents. It was found that the limiting current could be expressed in terms of a simple power law, given by eq 1

IL ∼ n3/2A3/4Dν-1/4C 4/3B1/3

(1)

where ν is the viscosity of the solution, C and B correspond to the bulk concentration of the electroactive species and the strength of the magnetic field, respectively, D is the diffusion constant of the electroactive species, and A and n are the electrode surface area and the number of electrons per molecule participating in the electrochemical reaction, respectively. Subsequently Legeai et al.7b studied in detail the oxidation reactions of hexacyanoferrate(II) and hydroquinone and obtained the following dependence on the system parameters (eq 2)

IL ∼ nA5/6Dν-2/3-7/4C 4/3B1/3

(2)

where  and A are the dielectric constant of the solution and the electrode surface area, respectively. Thus, for the field applied parallel to the electrode surface, most of the studies correlated the limiting current, IL, with the strength of the magnetic field, B, and the concentration, C, as IL∼ C 4/3B1/3. In a previous study, a quantitative theoretical model that accounts for the effect of the homogeneous magnetic field,

10.1021/jp069055z CCC: $37.00 © 2007 American Chemical Society Published on Web 04/03/2007

Effects of Magnetic Field

J. Phys. Chem. C, Vol. 111, No. 16, 2007 6025

parallel to the surface, on electrochemical reactions at a planar semi-infinite electrode, has been formulated.9 Basically, following Levich,10 we coupled the thickness of diffusion boundary layer, δD, with the hydrodynamic flow of the solution at the electrode interface induced by the magnetic field. The developed model predicted that the diffusional layer thickness, δD, should be proportional to the (nCB)-1/3. The shrinkage of the diffusion boundary layer, thus, enhances the supply of the electroactive reactant to the electrode and results in the augmented limiting currents in the presence of the applied magnetic field. This positive feedback process is limited by viscous shear in the boundary layer that defines a steady-state value of the flow velocity. Our model implies that the limiting current is proportional to (nC)4/3B1/3, and it was applied successfully to analyze several experimental reports.11,6,8 Magnetic Field Perpendicular to the Electrode The effect of a vertically oriented magnetic field on electrochemical processes was investigated as a function of the strength of the magnetic flux density, B, and other essential parameters of the system using cyclic voltammetry, chronoamperometric measurements, and atomic force microscopy.12,13 The magnetic field in this configuration is parallel to the current lines, and thus the Lorentz force, FL ) iL × B, which corresponds to the vector product of the net current density, iL, and the magnetic field strength, B, is equal to zero. There are, however, numerous reports of an influence of the magnetic field perpendicular to the electrode on the electrochemical processes that contain electroactive para- or diamagnetic species.13-15 During an electrochemical reaction, the magnetic susceptibility of the Nernst diffusion layer may become different from the bulk value because the electroactive species are depleted from the layer. Thus, the origin of the magnetic field influence can possibly be explained by gradients of the external magnetic field itself or/and gradients of the molar magnetic susceptibility of the solution, which is proportional to the substrate concentration. Thus, we should consider two cases: the case of the uniform magnetic field, acting only on the concentration gradient of the electroactive species, and the case of nonuniform magnetic field, acting on electroactive magnetic species in the electrolyte solution. We take into account only studies providing dependence of the limiting current or the reaction rate on the magnetic field strength and the substrate concentration. Perpendicular Uniform Magnetic Field The effect of the uniform magnetic field, perpendicular to the surface of a plane circular electrode of diameter d, on electrochemical processes involving paramagnetic species (Cu2+ and ferri-ferrocyanide) was studied in detail by Rabah et al.13a The diffusion-limiting current enhancement was shown to be expressed by eq 3 (where d is the diameter of the electrode)

IL ∼ d7/4nFD2/3C 4/3B2/3

(3)

In the case of electrochemical deposition of copper, under a broad range of parameters, eq 3 was supported well, whereas for the ferri-ferrocyanide system, within a similar range of concentrations, this same relation is only partially fulfilled. For the electrochemical deposition of cobalt (reduction of Co2+) on flat electrodes, Krause et al.14 observed the current increase along the magnetic field strength, and Uhlemann et al.13b found that the diffusion-limiting current dependence follows the expression given in eq 3 for this system.

Perpendicular Nonuniform Magnetic Field The influence of vertical nonuniform magnetic fields on the rate of the dissolution of copper sulfate pentahydrate crystal in water was examined quantitatively by Aogaki et al.12c and was found to be proportional to the (B3B)1/3C4/3. The subsequent study15 on the redox reactions of the ferricyanide-ferrocyanide electroactive couple on the microelectrode showed that the steady-state limiting current density was proportional to the (B3B)1/3. The enhanced mass transfer in the applied magnetic field was explained in terms of magnetoconvection via formation of vortex cells, and the mass flux was predicted to increase with the (B3B)1/3C 4/3.14 For the electrodeposition of copper (reduction of cuprous ions, Cu2+), however, the steady-state limiting current density was reported by the same research group12c to be a function of the first power of the magnetic flux density, B. White et al.12a investigated the action of the force generated in nonhomogeneous magnetic fields on the electrolyte solution using the disk-shaped Pt microelectrodes. It was concluded that the gradient magnetic force that arises in a nonuniform magnetic field is dependent on the magnetic field strength and gradient and on the magnetic properties of the redox-active molecules. Numerical estimates of the Lorentz force and field gradient forces indicate that these forces are of comparable magnitude. These experiments allow the two forces to be separated and individually investigated by variation of the field homogeneity and the electrode orientation. The effect of the gravitational field on the electrochemical reaction on microelectrodes was studied by White et al.16a The dependence of the steady-state limiting currents on the electrode orientation relative to the gravity was measured for the ferriferrocyanide redox system in 0.2 M KCl solutions at Au and Pt microdisk electrodes (radius ) 6.4-25 µm). The difference between the fluid density of the bulk solution and that of the diffusion layer (0.08% for the studied system16b) creates a buoyant force that generates natural convective flow. The results demonstrated that steady-state currents originated from both diffusional and natural convection mass-transfer fluxes and that the latter one accounted for up to 15% of the total current at 25-µm-radius microdisk electrode. Similar effects that show the influence of the applied magnetic field were previously documented in convective systems containing para- or diamagnetic fluid components. The driving force for natural thermoconvection is usually the density difference between hot and cold regions of the fluid. If the fluid has a magnetic susceptibility that varies with temperature, then magnetic forces, rather than buoyancy, can be used to drive convective motion. Recent studies have shown that magnetic convection can be initiated in a homogeneous magnetic field and enhanced in a field gradient in para- or diamagnetic fluids.17 Accordingly, the magnetic field was used to enhance as well as suppress buoyancy-driven convection in a solution of gadolinium nitrate, in gaseous oxygen as well as in ferrofluid, with the sign of the effect depending on the relative orientation of magnetic field and temperature gradients.18 Thus, magnetic field can play for the magnetoconvection the same role as the gravity field for the natural convection. Theoretical Model Previous attempts to model the parallel magnetic field effect on the limiting current of electrochemical processes at the electrode surfaces were reported by Fahidy,3b Aogaki,4 and our laboratory.9 The fact that the Lorentz force acting on the current passing through the electrolyte yields a transverse flow of the solution parallel to the electrode surface allowed Fahidy to

6026 J. Phys. Chem. C, Vol. 111, No. 16, 2007 propose3b that the diffusion boundary layer thickness, under an applied magnetic field, δD(B), is smaller than the diffusion boundary layer without a magnetic field, δD(0), and accordingly, this results in the enhancement of the limiting current, IL. These theoretical models and different experimental studies suggested that the limiting currents relate to the concentration of the electroactive reactant and the magnetic field strength by the dependence IL ∼ C 4/3B1/3. The case of the homogeneous perpendicular magnetic field was discussed by Waskaas and Kharkats for different electrochemical systems and cell geometries.19 They introduced the concentration gradient force that is operative only near the electrode, where it induces an ion transport. Theoretically, it was shown that the vertical magnetic field tends to cause an additional convective transfer of all components of the solution. The suggested mechanism was coined magnetoconVection, and to obtain any magnetic field effect, a gradient of a paramagnetic ion concentration from the solution to the electrode surface should exist. Furthermore, both the experimental results and the suggested mechanism show that the magnetic field effect increases with increasing magnetic flux density and magnetic susceptibility of the solution. The dependence of the enhanced limited current on the system parameters was not derived, however. In a nonuniform Vertical magnetic field, a volume magnetic force exerted on the solution that contains the electrochemically active magnetic species is proportional to the magnetic field gradient, B3B. This force is operative in the whole volume of the electrolyte where the magnetic field gradient exists. The mass transfer of an electrodeposition process was theoretically examined12c for a nonuniform magnetic field vertically imposed on the electrode, and it was suggested that in the case of unstable conditions the magnetic force could induce numerous convection cells in the solution. It was predicted that the mass transfer of the solute is thus accelerated, and the mass flux is increased following the relation (B3B)1/3C 4/3. As far as we are aware, there is no theoretical model that quantitatiVely accounts for the vertically directed homogeneous magnetic field effects on electrochemical processes at the electrode surfaces. The present paper describes the detailed derivation of a simplified model that quantitatively accounts for the effect of the vertical magnetic field and its consistency with some available experimental results and presents a dependence of the enhanced limited current on system parameters. Formulation of the Model In the following section, we formulate a quantitative model that describes the perpendicular magnetic field effect on the mass transport of the electroactive species to the electrode and as a result the augmentation of the limiting current density. The model is developed using the boundary layer approximation for the forced Navier-Stokes equation (NSE). The incorporation of the external magnetic force into the NSE allows us to relate the bulk flow velocity to the magnetic body force acting on the solution in order to obtain the expression for the diffusion layer of the electroactive species, δD, as a function of the applied magnetic field. We do not consider any other type of possible forces because our simplified model deals with a homogeneous magnetic field imposed on a diamagnetic solution and a nonmagnetic electrode at a constant temperature. For a detailed review on the possible forces in the system, the reader is referred to the paper of Hinds et al.2 We will regard a planar semi-infinite electrode, placed horizontally, with the magnetic field applied perpendicular to

Lioubashevski et al. the surface, and consider the electrochemical reaction given in eq 4, that occurrs at an electrode of radius R (not a microelectrode).

Az + ne- h Bz-n

(4)

The evolution of the concentration profiles of the redox-active species in the absence of migration (i.e., in the presence of a large excess of supporting electrolyte) is given by the solution of the general convective-diffusion equation, eq 510

∂C/∂t ) D∇2C - U∇C

(5)

where U is the fluid velocity and D and C are the diffusion coefficient and the concentration of the electroactive species in bulk, respectively. According to the Nernst approximation, a concentration gradient of an electroactive reactant exists only in the diffusion boundary layer of thickness δD, while in the rest of the solution the concentration is equal to C, and the mass flux of the redoxactive species toward the electrode is given by eq 6

j ) D(C - Cel)/δD

(6)

where C is the bulk and Cel is the surface concentration of the electroactive species (reactant) at the electrode, respectively. For sufficiently high current densities, achieved upon application of an appropriate electrode potential, Cel tends to zero because for mass-transport limited reactions the reaction rate at the electrode surface is substantially higher than the reactant supply. Therefore, the electrochemical boundary conditions for the masstransport limited current can be written as C ) 0|y)0 and C ) C|yf∞, where y notes the direction normal to the electrode surface and its positive half-axis is aligned into the solution. The current produced by the redox reaction in the masstransport limited regime is controlled by the diffusional flux of the electroactive species to the electrode surface (eq 6) that is directed orthogonal to the electrode surface. The limiting current resulting from the Faradaic process of the redox-active species, IL, is given by eq 7

IL ) nAFj|y)0 ) ADnFC/δD

(7)

where n is the number of the electrons involved in the Faradaic process, A is an electrode surface area, and F is the Faraday constant.20 Because the electron transfer at the electrode surface requires charge compensation, a flow of ions is generated in the electrolyte solution, and this flow could be influenced by the external magnetic field of appropriate orientation. There are two possible sources of the magnetic force acting on the electrochemical system and responsible for the observed magnetic field effects. The first is related to the movement of charged species and the second is related to the magnetic properties of the electrochemically active substrate. The first type of magnetic body force is the Lorentz force, FL, acting on the unit volume of charge-carrying ions and resulting in a momentum transfer to the solvent that could result in the fielddriven solution flow. In our previous study,9 we described the dependence of the limited current on magnetic field strength for the parallel field orientation. In the present report, we consider a system where the magnetic field as well as the net current are directed perpendicular to the electrode surface, and the resulting Lorentz body force is equal zero. For the fluid that contains electroactive para- or diamagnetic species, the magnetization, M, induced by the field, B, depends on the local value of the applied magnetic field as well as on

Effects of Magnetic Field

J. Phys. Chem. C, Vol. 111, No. 16, 2007 6027

the molar magnetic susceptibility, χm, of these species, which is proportional to their concentration. A concentration gradient of the magnetic species, created as a result of the heterogeneous electrochemical reaction, produces a spatial variation in the magnetization through the molar magnetic susceptibility, and therefore results in the spatially nonuniform magnetic body force density. This magnetic body force can promote or inhibit convection in a manner similar to the gravitational body force. The density of the magnetostatic energy of the electrolyte containing magnetic species in the magnetic field (no currents taken into consideration) is given by the following equation (SI units)21 (eq 8)

Emag ) -(1/2µ0)MB ) -(1/2µ0)χmCB2

(8)

where M ) χmCB is the magnetization induced by the field B, µ0 is the magnetic permeability of free space, and χm is the molar magnetic susceptibility of the species involved. To calculate the force associated with this magnetic energy, we have to take a spatial derivative of the energy, eq 9

Fmag ) -∇E ) (1/2)∇(χmCB2)

(9)

Thus, the related magnetic force acting on the electrolyte, that contains magnetic species, due to the magnetic energy gradient, is given by eq 10

Fmag )

χmB2∇C χmCB∇B + 2µ0 µ0

(10)

where ∇B is the magnetic field gradient. This force includes two terms: the concentration gradient force, FC (Kelvin body force) and the field gradient force, FB. The concentration gradient force, FC, is directed toward areas with higher concentration of the paramagnetic species, eq 11a

FC )

χmB2∇C 2µ0

(11a)

and the field gradient force, FB, is directed toward areas with higher values of the magnetic field strength, eq 11b

FB )

χmCB∇B µ0

(11b)

We would like to note here that in contrast to the Lorentz force expressions for the forces FC and FB do not include a current. That means that no feedback is expected between the magnetic field and the electrochemical current in the system. The concentration gradient force, FC, is defined by the product of the gradient of the magnetic ion concentration and their magnetic susceptibility, and results in a redistribution of velocities in the diffusion layer. An induced ion transport direction depends on the sign of the product of the magnetic susceptibility and the concentration gradient, defined by which electroactive component possesses magnetic properties, the substrate or product of the electrochemical reaction. The negative sign means that the flow directed toward the electrode surface and the positive sign means the outward flow. The direction of the magnetic field is irrelevant to the direction of FC because the magnetic flux density in eq 11a relates to the second power of the strength of the magnetic field, B 2. The field gradient force, FB, has the direction defined by the product

of the magnetic field and the gradient of the magnetic field and the magnetic susceptibility of the ions. As a result of the action of the magnetic force, an additional convective transport of all of the components of the solution is generated. A typical value of FC for paramagnetic ions with χm ) 1 × 10-8 m3 mol-1 in 0.1 M (102 mol m-3) solution and a diffusion layer thickness δ ∼ 1 × 10-4 m (100 µ) is on the order of 4000 N m-3 for B ) 1 T. The force acting on diamagnetic ions has an opposite sign and will be an order of magnitude smaller; thus, it should be regarded solely in the absence of paramagnetic ion concentration gradients. Recently, it was experimentally demonstrated that the concentration gradient force, FC, is a body force proportional to B 2 and acts in the direction of the concentration gradient of electrogenerated paramagnetic radicals.22 It was shown to compete and even reverse the effects of another body force, gravity, that causes the density-gradient-driven natural convection at electrodes. Thus, FC can balance gravity, holding volumes of solution in the vicinity of an electrode where the diffusional mass transfer continues to take place as was manifested by the measured current. The flow velocity profile for the parallel magnetic field configuration has been obtained recently by flow imaging,7a,23 and it was found to have a pronounced maximum at an angle equal to 90° relative to the magnetic field direction. Thus, the flow generated by the Lorentz force has mainly unidirectional velocity tangential to the surface.9 Alternatively, the perpendicular magnetic field applied to microelectrodes produce cyclonic magnetoconvective flow owing to radial currents, while voltammograms are always sigmoidal.12e Millelectrodes or macroelectrodes, in contrast, generate linear diffusion resulting in a peak-current response;12f thus, we can discard the cyclonic flow from our consideration. The effect of the perpendicular magnetic field on the solvent and on the fluid motion generated within the diffusion boundary layer is depicted schematically in Figure 1. While a diffusion boundary layer of thickness δD is formed due to the electrochemical process in the absence of the magnetic field, Figure 1A, the magnetic field yields an electrolyte flow in the vicinity of the electrode surface that is characterized by a hydrodynamic boundary layer of thickness δ0, Figure 1B. We assume here that the natural convection induced by the density variations due to electrochemical reactions on the surfaces of electrodes does not play a major role. To determine the changed concentration profiles of the redoxactive species near the electrode for electrochemically induced diffusion (eq 5) coupled to generated magnetoconvective flow, one has to determine the velocity profile of the fluid, U(x,y). The velocity profile can be obtained by solving the forced Navier-Stokes equation (NSE), eq 12, and the continuity equation, eq 1310,24

F

(∂U∂tB + (UB∇)UB) ) -∇P + η∆UB + BF ∇U h )0

i

(12) (13)

The Navier-Stokes equation expresses Newton’s second law of motion for a unit volume of an incompressible Newtonian fluid of constant density, F, and dynamic viscosity, η. The left side of the NSE represents the acceleration of the fluid volume element as a result of the action of the volume forces presented on the right side. The velocity vector of the unit volume element of fluid is denoted as U, and, assuming a steady flow, we obtain ∂U/∂t ) 0. The first term on the right-hand side of the NSE is

6028 J. Phys. Chem. C, Vol. 111, No. 16, 2007

Lioubashevski et al.

Figure 1. Schematic illustration of the mechanism of enhanced masstransport of a diamagnetic electroactive species under a perpendicularly applied uniform magnetic field. (A) Diffusion fluxes, JS for the substrate and JP for the product. (B) Magnetically driven fluid flow. Vector presentation of fluid velocities, U(y), near an electrode. (C) Streamlines (or flows) presentation of magnetoconvection (for illustration only, two convective cells shown).

the force due to the hydrodynamic pressure gradient, ∇P, through the system, and the second term, η∆U, expresses the frictional forces between the fluid unit volume element and the surrounding fluid or the electrode surface. The third term, Fi, refers to any external volume forces acting on the unit fluid element. For the system under consideration, the external volume force is the concentration gradient force, FC, (eq 11a) coupled to the reactant concentration gradient. Evidently, eqs 12 and 13, which represent a hydrodynamic problem, should be coupled to eq 5 via the concentration C(x,y) of the electroactive species, which is controlled by the convection-diffusion equation and by the electrochemical boundary conditions. This is in contrast to the hydrodynamic problem of the rotating disk electrode, which has been solved for the flow velocity profile independent of the electroactive species concentration profile.10 Obtaining the velocity profiles generated by a magnetic field is a substantially more complicated task, and an analytical solution, analogous to the Levich equation for the rotating disk electrode, still does not exist. The entire zone of inviscid fluid motion above the electrode may be roughly subdivided into two regions: a region of inviscid motion and a boundary layer region in which viscosity plays a crucial role. The latter region is known as the hydrodynamic boundary layer, and its thickness, denoted as δ0, is given10 by eq 14

δ0(x) ≈ 5.2(νX/U)1/2

(14)

where X is the distance from the plate upstream edge, ν is the solvent viscosity, and U is the flow velocity. (It should be noted that this equation is applicable only for laminar flow.) The properties of the hydrodynamic boundary layer define the hydrodynamic behavior of the fluid flow near the electrode. According to the Nernst model, the fluid within the diffusion boundary layer, having thickness δD, is assumed to be quiescent, and the transport of electroactive species is accomplished by molecular diffusion. Upon application of the vertical magnetic field, the concentration gradient force, FC, induces a flow of the solution inside the diffusion boundary layer, mainly normal to the electrode surface and directed according to the sign of the product of the magnetic susceptibility of ions and the concentration gradient, χm∇CP , where CP is the concentration of the reaction product. Usually, if the reactant in a redox reaction is diamagnetic, then the product after exchanging a single electron is converted to paramagnetic; thus, we should consider paramagnetic species in our model because the

magnetic susceptibility of diamagnetic ions is an order of magnitude smaller. In the case of a paramagnetic product in solution (χm > 0, χm3CP < 0), the gradient force FC results in a generated flow directed toward the electrode surface. Far from the electrode the fluid moves toward the electrode, while in a thin layer immediately adjacent to its surface the orthogonally directed flow displaces a fluid near the electrode, and the displaced fluid is diverted into a laterally moving fluid. The radial velocity of the fluid increases as the surface of the electrode is approached, until the characteristic velocity of the fluid in the boundary layer is attained. Upon reaching the lateral walls of the container, the flow attains the upward direction and thus results in the formation of a convection cell. The above-described picture is similar to the flow distribution for the natural convection. Thereby, the generated flow is eventually converted into entirely tangential flow (see Figure 1C with streamlines for illustration25) that can be regarded as unidirectional flow along the surface. This tangential flow leads to the formation of a hydrodynamic boundary layer of thickness δ0, eq 14. The well-known relationship between the thickness of the diffusion layer, δD, of the reactant (or product) species and the thickness of hydrodynamic boundary layer of the fluid moving over the plate, δ0 was derived by Levich,10 eq 15

δD ≈ 0.62Pr-1/3δ0

(15)

where Pr is Prandtl number, Pr ) ν/D, a dimensionless parameter that characterizes the regime of convection. Thus, for Pr in the order of 103, we obtain the value of δD that is about a tenth of δ0. Chronoamperometric experiments have indicated that the magnetic field does not affect the diffusion coefficient of electroactive species.2,7 Alternatively, the magnetic field results in augmentation of the diffusion-limited current that implies that the concentration gradient, approximated by ∇C ) C/δD, must increase in the presence of the magnetic field, according to eq 6, which leads to the conclusion that the diffusion boundary layer thickness, δD, should decrease in the presence of the magnetic field. Taking into account the correlation between δD and δ0, eqs 14 and 15 (where the distance X is assumed to be equal to R, electrode radius), we obtain the next relationship that links δD and the velocity of the generated flow, U0, eq 16

δD ≈ (ν1/3D2/3R)1/2U0-1/2

(16)

This relation shows that the diffusional flow is a function of the characteristic fluid velocity, U0. Thus, to find the changes in a mass-transport limited current caused by an applied magnetic field we should obtain the dependence of the generated flow velocity U0 on B and the other essential parameters of the system affecting the limited current. Replacing δD with eq 16 in eq 6, we get a relation (eq 17) for the diffusional flux to the surface of an electrode subjected to a flowing fluid, obtained originally by Levich:10

j ≈ 0.34R-1/2ν-1/6D2/3CU01/2

(17)

We will consider the unidirectional steady laminar flow, streaming parallel to the electrode surface inside the boundary layer, thus U ) Ux(y), where x is directed along the surface (that means flow velocity has only one component, Ux), see Figure 1. To simplify the NSE within the boundary layer, we can utilize the fact that the thickness of this layer is very small compared to its length along the body (L . δ0). Under viscous

Effects of Magnetic Field

J. Phys. Chem. C, Vol. 111, No. 16, 2007 6029

flow conditions, all velocity components must become zero at the surface of a solid body immersed in a fluid. This is known as the “no slip” condition U|y)0 ) 0, which along with U|y)∞ ) U0 at the outer edge of the boundary layer comprise the boundary conditions for the hydrodynamic flow in the boundary layer.24 Hence, there is a region immediately adjacent to the solid surface where the unidirectional flow velocity, Ux(y), changes rapidly from zero to its value in the bulk stream, U0, in the direction perpendicular to the wall (y direction), while its tangential rate of change is comparatively small. In the boundary layer approximation for the flow above a plate, the pressure does not change in the direction normal to the surface but it remains equal to the pressure outside the boundary layer; thus, ∂P/∂y ) 0.24,10 On the outer edge of the boundary layer, the fluid velocity is constant and equal to U0; hence, from the Bernoulli equation it follows that the pressure in the outer region is also constant. As a result, the term containing the pressure gradient in eq 16 can be omitted. Under these conjectures, the forced Navier-Stokes equation for steady unidirectional laminar flow within the boundary layer can be simplified in the form of eq 18 (for the detailed derivation of the relation see ref 9 and its appendix).

∂Vx ∂2Vx FVx ) η 2 + FC ∂x ∂y η(∂2V

(18)

/∂y2)

The term can be determined using a further x consideration: within the boundary layer, viscous shear between the fluid flow and the electrode causes the fluid velocity to become zero at the electrode surface. Thus, all changes in tangential fluid velocity from U0 to the zero value occur within δ0; therefore, the term η∆V can be rewritten approximately as η(U0/δ20). Our choice of δ0 as the characteristic length in the system and U0 as the characteristic Velocity is equivalent to stating that the dominant dissipation of the energy of the system is the shear between the fluid flow and the electrode plate. This dissipation is balanced against the external force, FC, or the driving term, which could lead to the setup of the fluid motion over the electrode surface. To produce the flow, the driving force, FC, that is proportional to χmB2∇C/2µ0 (eq 11a), should at least be equal to or overcome the dissipative force term η(U0/ δ20); thus, using a simplified expression for the gradient of the concentration,∇C ) C/δD, we can equalize these two force terms and obtain the relation given by eq 19.

η

U0 δ0

≈ 2

χmCB2 2µ0δD

(19)

It should be noted that using of the simplified expression for ∇C (∇C ) C/δD, provided that the surface concentration of the electroactive species Cel ) 0) in eq 19, we lose the sign of the concentration gradient. Thus, further on we use the modulus of the product |χm∇C|, and the sign of the magnetic susceptibility, χm, does not have any meaning in the following equations. Using the relation between δ0 and δD (eq 15), we obtain the expression for the velocity of the generated flow, U0, eq 20.

U0 ≈ Wδ0, where W ) 0.83 χmPr1/3B2C (µ0η)-1 (20) Substitution of the derived velocity U0 into the expression for the hydrodynamic boundary layer thickness δ0 (eq 14), where the distance X is assumed to be equal to R, electrode radius, leads to eq 21.

U0 ≈ 27νR/δ02

(21)

Then excluding U0 from both eq 20 and eq 21 we obtain the expression for δ0 given by eq 22.

δ0 ≈ 3(νR/W)1/3

(22)

Thus, after the substitution of eq 22 into eq 20 we obtain an expression for the flow velocity that is generated by the magnetic force, given by eq 23

U0 ≈ 3.4(χm/Fµ0)2/3R1/3ν-1/9D-2/9(CB2)2/3

(23)

Therefore, the diffusion boundary layer thickness δD (using eq 16) can be expressed by eq 24.

δD ≈ 1.84 (µ0FRν2/3D4/3)1/3 (χmCB2)-1/3

(24)

The latter expression provides the dependence of δD on the parameters controlling the electrochemical reaction. Accordingly, the limiting current, IL) AiL, for a disk electrode of area A ) πR2, could be written as expressed by eq 25.

IL ≈ 0.5πR5/3χm1/3(µ0F)-1/3ν-2/9D5/9nFC 4/3B2/3

(25)

We support our derivations by conducting dimensional analysis of the obtained relation, eq 25. Upon substitution of dimensions for every parameter in this equation, we attain the value of ampere in the right-hand side. In our previous work,9 we have shown that for the magnetic field parallel to the electrode surface the coupling between the electrochemical current and the applied magnetic field via Lorentz force defines an expressions for δD and for the limiting current, iL, that are derived to be proportional to (nFCB)-1/3 and ∼(nFC)4/3, respectively. For a magnetic field perpendicular to the electrode surface, the gradient force, FC, is acting on the volume element containing a concentration gradient ∇C of magnetic ions (eq 11a), and thus the current is not included directly in the expression for the force. Therefore, the expressions for FC and δD do not include the term nF, and as a result, the enhanced current is proportional to (nFC)C1/3. It should be noted that in this case the derived power dependence of n and F is not consistent with the derived power of the concentration, C 4/3, as follows from eq 25. This reflects the different nature of external forces applied on the electrochemical solution upon directing the magnetic field parallel or perpendicular to the electrode surface. Now we will consider the nonuniform magnetic field that is directed perpendicular to the electrode surface. The case of vertical nonuniform magnetic field differs from the previously addressed uniform field by the expression of the force in the simplified NSE, eq 18. The action of the field gradient body force was experimentally demonstrated on Nd-Fe-B permanent magnet microelectrodes,12a,b,d where magnetization created a field gradient in the local vicinity of the electrodes. We regard the situation when χmB∇B < 0, and thus the magnetic force is directed toward the electrode surface. To produce the flow, the driving force, FB, proportional to χmCB∇B/µ0 (eq 11b), should be at least equal to or higher than the dissipative force term ηU0/δ02, and we equalize the two terms to obtain the following relation for U0, analogous to eq 20 (eq 26).

U0 ≈ Wδ02, where W ) (χm/µ0Fν)CB∇B

(26)

Finally, we obtain the velocity of the generated flow given

6030 J. Phys. Chem. C, Vol. 111, No. 16, 2007

Lioubashevski et al.

by eq 27

U0 ≈ 5.2(νRCB∇B)1/2(χm/µ0Fν)1/2

(27)

and then we get the expression for the diffusion boundary layer thickness, eq 28.

δD ≈ 1.4(µ0FR)1/4ν1/6D1/3(χmB∇BC)-1/4

(28)

Accordingly, the substitution of δD into eq 10 for the limiting current density, iL, yields eq 29.

iL ≈ 0.7(χm/µ0FR)1/4ν-1/6nFD2/3C5/4(B∇B)1/4

(29)

It should be noted that this equation describes a sole contribution from the gradient of the magnetic field. In the experiments it is impossible, however, to separate it from the contribution of the magnetic susceptibility gradient of the solution near the electrode surface caused by the heterogeneous reaction. Thus, we expect that the experimental power dependence of iL as a function of B will be higher than the derived value of 1/4. We would like to note that in contrast to the Lorentz force, the concentration gradient force, FC, and the field gradient force, FB, do not include the electrochemical current in their expressions (eqs 11a and b). The results obtained in the present work for these forces suggest that one does not need an electrochemical current in order to enhance mass-transport limited chemical process. The necessary conditions for the enhanced masstransport are a presence of magnetic species (para- or dia-) involved in the chemical reaction and the existence of at least one of the two gradients: a magnetic species concentration or a magnetic field. This allows us to expand the class of reactions, which could be affected by externally applied magnetic field to fast heterogeneous nonelectrochemical systems involving magnetic species, provided that these reactions are masstransport limited. The mass flux of the redox-active species toward the electrode subjected to the perpendicularly applied magnetic field is given by eq 30, which is obtained by the substitution of the expression for the thickness of the diffusion layer, δD, eq 24, into eq 6.

j ≈ 0.5 (χm/µ0FR)-1/3ν-2/9D5/9C4/3B2/3

(30)

The derived dependence of the mass flux on the system parameters defines the corresponding reaction rate of heterogeneous mass-transport limited reactions. Analysis of Experimental Results by the Theoretical Model The formulated model implies that the limiting current, IL, under a magnetic field applied perpendicular to the surface of a plane electrode relates to the different system parameters in the form of simple power dependence (eq 25). It should be realized that the coupling of the electrogenerated diffusion boundary layer to the hydrodynamic flow, induced by the magnetic field, leads to the C 4/3B 2/3 power dependence of the limiting current. We will now demonstrate the validity of this model with available experimental results, discuss some experimental deviations from the model, and suggest future experiments. In Figures 2 and 3, we show the results obtained by Rabah et al.13a that correspond to the electrochemical process of paramagnetic species (Cu2+) and a para-/diamagnetic couple (ferri-ferrocyanide) for the circular electrode of diameter d,

Figure 2. Ratio between the cathodic limiting current, IL, and the concentration of the electroactive species, C, vs the product B 2/3C1/3d7/4 for the electrodeposition process of copper on the electrode. Experimental conditions: Copper sulfate concentrations in 0.15 M sodium sulfate solution, pH ) 3. (O) d ) 8 mm and C ) 0.1 M; (2) d ) 5 mm and C ) 0.5 M. IL, C, B, and d are expressed in SI units; d is the electrode diameter. (Adapted from ref 13a, Figure 5, with permission.)

Figure 3. Ratio between the cathodic limiting current, IL, and the electroactive species concentration, C, vs the product B 2/3C1/3. Experimental conditions: 1 M potassium chloride. Potassium ferricyanide (K3[Fe(CN)6]) concentrations: (4) 0.05 M; (0) 0.1 M; ([) 0.2 M; (b) 0.5 M. (Adapted from ref 13a, Figure 6, with permission.)

placed vertically and perpendicular to the magnetic field. In the case of copper, Figure 2, a broad range of results, where the IL/C ratio is proportional to C1/3B 2/3, can be found, whereas with similar concentrations, the same relation is valid for the ferri-ferrocyanide system, Figure 3, but with wider deviations. Uhlemann et al.13b found that dependence of electrochemical current accompanied the deposition of paramagnetic Co2+ ions (the electrolyte was a 0.01 M CoSO4 solution with addition of 0.1 M Na2SO4 as a supporting electrolyte) under the influence of high uniform magnetic fields. Figure 4 shows the double logarithmic plot of the limiting current density, iL, obtained while varying the strength of applied magnetic fields, B. Clearly, in the case of the perpendicular magnetic field and a horizontal electrode the observed line has a slope that corresponds to 0.78, which is close to the expected value of 2/3 (eq 25). Thus, the theoretically predicted linear relation between IL and R5/3B 2/3C 4/3 has been supported by three different experimental systems. We will now briefly discuss the validity of the model for the nonuniform Vertical magnetic field with some experimental results. Sugiyama et al.15 studied redox reactions of ferrocyanide-ferricyanide ions on the microdisk Pt electrodes subjected to a nonuniform magnetic field. From the experimental data shown in Figure 5, it was derived that the limiting current density, controlled by the magnetic field, is proportional to the product of the magnetic flux and the magnetic flux gradient in the power 1/3, (B∇B)1/3.15 The mass transfer of the solute is thus

Effects of Magnetic Field

J. Phys. Chem. C, Vol. 111, No. 16, 2007 6031

Figure 4. Partial current densities of the Co2+ ion reduction subjected to a uniform magnetic field using different electrode geometries and orientations. (Adapted from ref 13b, Figure 17, with permission.) Experimental conditions: (0) wall electrode, B down.

Figure 6. Plot of the mass flux density vs (B∇B)1/3 describing the dissolution of a copper sulfate pentahydrate crystal in water upon application of the nonuniform magnetic field, B. Crystal size was 8 × 8 mm2. (Adapted from ref 12c, Figure 9, with permission.)

Conclusions

Figure 5. Current densities of the ferrocyanide-ferricyanide ions redox process as a function of the magnetic field gradient. Solid and open circles imply the cathodic and anodic current densities, respectively. Solution composition is 10 mM equimolar of potassium ferricyanide (K3[Fe(CN)6]) and potassium ferrocyanide (K4[Fe(CN)6]) in 500 mM of potassium chloride (KCl) as a supporting electrolyte. Overpotential E is equal to (0.35 V, and the diameter of the working electrode is 0.16 cm. (Adapted from ref 15, Figure 7, with permission.)

accelerated by the magnetic field, and this was explained in terms of magnetoconvection via the formation of vortex cells, and the mass flux was predicted to increase with the (B∇B)1/3C 4/3. Furthermore, the dissolution rate of a copper sulfate pentahydrate crystal (paramagnetic), shown in Figure 6, was measured by Sugiyama et al.12c at a constant magnetic flux density by changing only the gradient. The concentration of the dissolved copper sulfate ions in solution was measured by the anodic stripping method. The measured material flux, proportional to the dissolution rate, appeared to be proportional to C 4/3(B∇B)1/3. We believe that the discrepancy between an observed15 and our model-predicted values of powers C5/4(B∇B)1/4 in eq 29 can be explained by an additional contribution from the concentration gradient force proportional to χmB2∇C, which results in an additional decrease of the diffusion boundary layer thickness. We surmise that the external magnetic field applied perpendicular to the surface could be used as an alternative of the rotating disk electrode in Koutecky-Levich method20 for extraction of the kinetic information. To realize this suggestion, the transport-limited heterogeneous catalytic reaction, involving para- or diamagnetic species, that takes place should be subjected to the external magnetic field applied perpendicular to the surface.

In the present study, we formulated a quantitative model that accounts for the magnetic field effect on electrochemical reactions at planar semi-infinite electrode surfaces, with the field perpendicular to the surface, as uniform as well as nonuniform. The model is based on the boundary layer approximation of the NSE associated to the Nernst layer approximation. The cornerstone of the model is the relation between the thickness of the diffusion boundary layer, δD, and the thickness of the hydrodynamic boundary layer, δ0, obtained by Levich. Basically, we coupled the thickness of the diffusion boundary layer, resulting from the electrochemical process, with the convective hydrodynamic flow of the solution at the electrode interface induced by the magnetic field, as a result of the magnetic force action. The developed model predicts that the diffusional layer thickness, δD, is proportional to (χmCB2)-1/3. The shrinkage of the diffusion boundary layer enhances the supply of the electroactive reactant to the electrode and results in the enhanced limiting currents in the presence of the applied magnetic field. Our model implies that the flow velocity is proportional to C 2/3B 4/3 and the enhanced limiting current is proportional to C 5/4B 2/3. The model was applied successfully to analyze several experimental reports. Several important conclusions can be derived from the model developed for the magnetic field applied perpendicular to the electrode surface. (i) To obtain a magnetic field effect on the electrochemical process, the electroactive species should possess the magnetic properties. (ii) The class of reactions that could be influenced by an external magnetic field can be expanded to fast heterogeneous nonelectrochemical reactions. To provide the essential concentration gradient followed by fluid magnetization gradient, one could use mass-transport limited reactions involving magnetic species. Paramagnetic species are often acting as active sites in cofactors of enzymes participating in bioelectrocatalytic transformations (e.g., hemoproteins, metalloenzymes). Thus, bioelectrocatalytic transformations for numerous biomolecules

6032 J. Phys. Chem. C, Vol. 111, No. 16, 2007 could be affected by a perpendicular magnetic field. Accordingly, we are now attempting to explore bioelectrocatalytic reactions under an applied magnetic field to correlate the results with the presented model and to use the enhanced currents for the designing of biosensors with increased sensitivities. We trust that the formulated model provides a solid background for the understanding of electrochemical transformations enhanced by means of an external magnetic field. Acknowledgment. Discussions with Dr. B. Basnar are gratefully acknowledged. This research is supported by Biomednano EC project no. 017350. Definition of Symbols Used B ∇B C IL IL0 iL A j J D F n δ ν η P F U0 Pr χm µ0 M

magnitude of the imposed magnetic flux density (T, tesla) magnetic flux density gradient (T/m) bulk electroactive species concentration (M) magnitude of the limiting current under applied magnetic field (A) magnitude of the limiting current without magnetic field (A) magnitude of the limiting current density under applied magnetic field (A m-2) area of the electrode (m2) diffusion flux of the electroactive species (mol s-1 m-2) charge flux of the electroactive species (A m-2) electrolyte diffusivity (m2 s-1) Faraday’s number (96487 C equiv-1) number of the electrons involved in the Faradaic process boundary layer thickness (m) kinematic viscosity (m2 s-1) dynamic viscosity (kg m-1 s-1) hydrostatic pressure (N m-2) fluid specific density (kg m-3) bulk flow velocity (m s-1) Prandtl number (D/ν) molar magnetic susceptibility of the species involved (m3mol-1) magnetic permeability of free space (1.257 × 10-6 N‚A-2) magnetization (A/m)

References and Notes (1) Faraday, M. Diary; Bell and Sons: London, 1933; Vol. IV 288; pp 7706-7721. (2) Hinds, G.; Spada, F. E.; Coey, J. M. D.; Ni Mhiochain, T. R.; Lyons, M. E. G. J. Phys. Chem. B 2001, 105, 9487.

Lioubashevski et al. (3) (a) Fahidy, T. Z. Modern Aspects of Electrochemistry: The Effect of Magnetic Fields on Electrochemical Processes; Kluwer Academic: New York, 1999; Vol. 32. (b) Fahidy, T. Z. Electrochim. Acta 1973, 18, 607. (4) (a) Aogaki, R.; Fueki, K.; Mukaibo, T. Denki Kagaku 1976, 44, 89. (b) Aogaki, R.; Fueki, K.; Mukaibo, T. Denki Kagaku 1975, 43, 504. (5) Aaboubi, O.; Chopart, J.-P.; Douglade, J.; Olivier, A.; Gabrielli, C.; Tribollet, B. J. Electrochem. Soc. 1990, 137, 1796. (6) Lee, J.; Ragsdale, S. R.; Gao, X.; White, H. S. J. Electroanal. Chem. 1997, 422, 169. (7) (a) Leventis, N.; Chen, M. G.; Gao, X. R.; Canalas, M.; Zhang, P. J. Phys. Chem. B 1998, 102, 3512; (b) Legeai, S.; Chatelut, M.; Vittori, O.; Chopart, J. P.; Aaboubi, O. Electrochim. Acta 2004, 50, 51. (8) Aaboubi, O.; Los, P.; Amblard, J.; Chopart, J. P.; Olivier, A. J. Electrochem. Soc. 2003, 150, E125. (9) Lioubashevski, O.; Katz, E.; Willner, I. J. Phys. Chem. B 2004, 108, 5778. (10) Levich, V. G. Physicochemical Hydrodynamics; Prentice Hall: Englewood Cliff, NJ, 1962. (11) (a) Katz, E.; Lioubashevski, O.; Willner, I. J. Am. Chem. Soc. 2004, 126, 11088. (b) Katz, E.; Lioubashevsky, O.; Willner, I. J. Am. Chem. Soc. 2005, 3979, 127. (12) (a) Ragsdale, S. R.; Grant, K. M.; White, H. S. J. Am. Chem. Soc. 1998, 120, 13461. (b) Leventis, N.; Gao, X. R. J. Am. Chem. Soc. 2002, 124, 1079. (c) Sugiyama, A.; Morisaki, S.; Aogaki, R. Jpn. J. Appl. Phys., Part 1 2003, 42, 5322; (d) Grant, K. M.; Hemmert, J. W.; White, H. S. J. Am. Chem. Soc. 2002, 124, 462. (e) Pullins, M. D.; Grant, K. M.; White, H. S. J. Phys. Chem. B 2001, 105, 8989; (f) (21) Bond, A. M.; Kratsis, S.; Mitchell, S.; Mocak, J. Analyst 1997, 122, 1147. (13) (a) Rabah, K. L.; Chopart, J. P.; Schloerb, H.; Saulnier, S.; Aaboubi, O.; Uhlemann, M.; Elmi, D.; Amblard, J. J. Electroanal. Chem. 2004, 571, 85. (b) Uhlemann, M.; Krause, A.; Chopart, J. P.; Gebert, A. J. Electrochem. Soc. 2005, 152, C817. (14) Krause, A.; Uhlemann, M.; Gebert, A.; Schultz, L. Electrochim. Acta 2004, 49, 4127. (15) Sugiyama, A.; Hashiride, M.; Morimoto, R.; Nagai, Y.; Aogaki, R. Electrochim. Acta 2004, 49, 5115. (16) (a) Gao, X. P.; Lee, J.; White, H. S. Anal. Chem. 1995, 67, 1541. (b) The specific gravities of 15 mM Fe(CN)64- and Fe(CN)63- solutions are 1.0035 and 1.0026, respectively. Values at other concentrations are given in the following: Handbook of Chemisty and Physics; Weast, R. C., Ed.; CRC Press: Boca Raton, FL, 1980. (17) (a) Finlayson, B. J. Fluid Mech. 1970, 40, 753. (b) Braithwaite, D.; Beaugnon, E.; Tournier, R. Nature (London) 1991, 354, 134. (c) Nakamura, H.; Takayama, T.; Uetake, H.; Hirota, N.; Kitazawa, K. Phys. ReV. Lett. 2005, 94, 144501. (18) (a) Carruthers, J. R.; Wolfe, R. J. Appl. Phys. 1968, 39, 57185722. (b) Seybert, C. D.; Evans, J. W.; Leslie, F.; Jones, W. K. J. Appl. Phys. 2000, 88, 4347. (19) (a) Waskaas, M.; Kharkats, Y. I. J. Phys. Chem. B 1999, 103, 4876. (b) Waskaas, M.; Kharkats, Y. I. J. Electroanal. Chem. 2001, 502, 51. (20) Bard, A. J.; Faulkner, L. R. Electrochemical Methods, Fundamentals and Applications, 2nd ed.; John Wiley and Sons: New York, 2000. (21) Landau, L. D.; Lifshitz, E. M. Electrodynamics of Continuous Media; Pergamon: Oxford, 1960; Chapter 4. (22) Leventis, N.; Dass, A. J. Am. Chem. Soc. 2005, 127, 4988. (23) (a) Leventis, N.; Gao, X. R. Anal. Chem. 2001, 73, 3981. (b) Ragsdale, S. R.; White, H. S. Anal. Chem. 1999, 71, 1923. (24) Prandtl, L. Essentials of Fluid Dynamics; Blackie: London, 1952, reprinted 1954. (25) We presented two convective cells in Figure 1C for illustration only. In fact, the number of convective cells depends on the driving force (e.g., for Rayleigh-Benard instability it correlates with the Rayleigh number).