Anal. Chem. 1993, 65, 1537-1545
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Iontophoretic Transport through Porous Membranes Using Scanning Electrochemical Microscopy: Application to in Vitro Studies of Ion Fluxes through Skin Erik R. Scott+and Henry S. White* Department of Chemical Engineering and Materials Science, University of Minnesota] Minneapolis] Minnesota 55455
J. Bradley Phipps Alza Corporation, 8295 Central Avenue NE, Spring Lake Park, Minnesota 55432
Scanning electrochemical microscopy (SECM) is used to map localized iontophoretic fluxes of electroactive species through porous membranes. A method is described that allows both the rate of transport of species from a microscopic pore and the pore's diameter to be measured. SECM images and analyses of synthetic porous membranes (track-etchedpolycarbonate and mica membranes) and hairless mouse skin are reported. Preliminary analysis of SECM images of the mouse skin indicates that a significant percentage of the iontophoreticflux occurs through pores associated with hair follicles. I. INTRODUCTION Research and clinical practice of electrically enhanced administration of ionic drugs across the skin, commonly referred to as iontophoresis, has increased dramatically in the last decade.' In order to exploit the skin as a medium for introduction of drugs into the body, it is important to understand the mechanism of ion transport through the skin under the influence of electric fields and concentration gradients. A key question in iontophoretic drug delivery is whether skin appendages (pores),e.g., hair follicles and sweat ducts, contribute significantly to the total mass transport and electrical conductance of skin. Hair follicles have diameters on the order of tens of microns and could present shunt pathways for ion permeation across the highly resistive outermost layer (stratum corneum) of the epidermis. Previous experiments suggest that the electrical conductance of skin conductance at these sites is significantly higher than that of bulk skin.2-4 While quantiative average mass-transfer rates across skin samples are measured readily? the direct quantitative measurement of individual pore transport rates has not previously been reported. In this report, we describe measurements of localized ion transport rates through skin pores using scanning electrochemicalmicroscopy (SECM).4,611The capabilitiesof SECM
* To whom correspondence should be addressed Department of Chemistry, Henry Eyring Building, University of Utah, Salt Lake City,
make it an ideal method for studying mass transport across microporous membrane s a m p l e ~ . ~In J ~this application, a microelectrodetip is rastered across the surface of a membrane mounted in a diffusion cell, as depicted in Scheme I. An electroactivespecies,e.g., Fe*, is dissolved in the lower (donor) compartment and is allowed either to freely diffuse across the membrane or is driven acrossthe membrane by an applied electric field. Once the redox molecules enter the receptor compartment, they are transported away from the membrane by diffusion and/or migration. The SECM tip in the receptor compartment is poised at a potential such that the electroactive speciesis oxidized or reduced at a mass transport limited rate. In Scheme I, Fe3+ is reduced to Fe2+. The magnitude of the detected faradaic current, measured as the tip is rastered across the membrane surface, is proportional to the local concentration of redox species (vide infra). Since the concentration of the species is greater at regions where the flux is large, the resulting SECM images provide a direct visualization of local ion pathways in the membrane. In this paper, we establish a basis for the quantitative application of this technique. A model is presented by which SECM data can be used to quantitatively determine the mass transport rate and size of a single membrane pore. In the Results and Discussion section, SECM analyses are used to measure the mass transport rates and sizes of two control samples: a microdisk electrode and a micropore in a mica membrane. We also report SECM analysis of iontophoretic transport across hairless mouse skin. The agreement of the concentration contour shapes, observed for the microdisk electrode and for mica and skin samples, justifies the use of a single, simple model for analysis of all three sample types.
11. EXPERIMENTAL SECTION Scanning Electrochemical Microscopy. The SECM used in our studies4J2is depicted in Figure 1. Membrane samples to be imaged were mounted in a custom-built Teflon vertical diffusion cell. The receptor compartment is open at the top to allow the tip to access the sample. Porta in the donor and receptor compartments allow access by the Ag/AgCl current driving electrodes and reference electrodes. The iontophoretic current (or in the case of the Pt disk electrode sample, faradaic current) was controlled by a commercial potentiostat/galvanostat(ModelRDE-4, Pine Instruments,Grove
UT .- 84112. .
Present address: Medtronic, Inc., 6700 Shingle Creek Parkway, Brooklyn Center, MN 55430. (1)Guy, R. H., Ed. Adu. Drug Rev. 1992,9 (2/3). (2)Burnette, R.R.;Ongpipattanakul, B. J.Pharm. Sci. 1988,77,132+
137.
(3)Cullander, C.; Guy, R. H. J.Inuest. Dermatol. 1991,97,55-64. (4)Scott, E. R.;White, H. S.; Phipps, J. B. Solid State Ionics 1992, 53-56, 176-183. (5) Phipps, J.B.; Padmanabhan, R. V.; Lattin, G.A. Solid State Zonics 1988,1778-1783. 0003-2700/93/0365-1537$04.00/0
(6)Lee, C.; Kwak, J.; Anson, F. C. Anal. Chem. 1991,63,1501-1504. (7)Kwak, J.; Bard, A. J. Anal. Chem. 1989, 61,1794-1799. (8)Wipf, D. 0.;Bard, A. J. J. Electrochem. SOC. 1991,138,L4. Bard, A. J. J. Electrochem. SOC. 1991,138,469-474. (9)Wipf, D. 0.; (10)Bard, A. J.; Denault, G.;Lee, C.; Mandler, D.; Wipf, D. 0. Acc. Chem. Res. 1990,23,357. (11)Bard, A. J.;Fan, F.-R. F.; Pierce, D. T.;Unwin, P. J.; Wipf, D. 0.; Feimeng, Z.Science 1991,254, 68-74. (12)Scott, E. R.;White, H. S.; Pipps, J. B. J. Membr. Sci. 1991,58, 71-87. 0 1993 American Chemical Society
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ANMYTICAL CHEMISTRY, VOL. 8.5. NO. 11, JUNE 1. 1993
Scheme I. Schematic of SECM Detection of Local Ion Flux
donor cornpamen!
City, PA). The scanned tip was a 8-pm-diameter carbon diber microdisk electrode, electroplated with Pt on the end and insulated on the sides with a -1-pm-thick layer of em-linked poly[oxy(allyl)phenylenel ?,I3 A custom-built, low-current, potentiostat was used to control the tip potential with respect to a saturated calomel reference (SCE) electrode. All potentials quoted in this paper are with respect to SCE. The potentiostat incorporated an ultralow input bias current operational amplifier (AD-549 Analog Devices, N o r w d , MA) and a T-configuration feedback network in the current followerI4to minimize offset current, noise, and drift. Tip current could be measured with a precision of i 2 PA. Positioning of the electrode in the X-Y plane, with a repeatability of 1pm, was done with piezoelectric inchworm microtranslation stages (Model TSE-75, Burleigh Instruments, Fishers. NY). Z axis positioning was done with a dc motor-driven state (Model 850, Newport Corp., Fountain Valley, CA). The absolute height above the sample z was determined by lowering the tip until it contacted the sample and then raising it in increments of -3 Fm, until the tip could move freely in the X-Y plane. The Z axis origin was defined at this point. Dataacquisitionand X-Ypositioningwerecontrolled by an IBM-XT microcomputer,using a program written in BASIC. The position of the tip above the skin was monitored by a video camera using a closeup zmm lens. SECM images were constructed by plotting the current measured at the scanned tip i, vs position as the tip is rastered in the X-Y plane. The tip scan rate was 10or 20 rmls. For each x scan line, SIN data points were acquired; the average of every 4 of these points was calculated, and these 200 averages were stored. Datawasplotted inreal timeusingalineplotformatand stored on hard disk. Stored data were replotted as 16-levelgrayscale images using a Macintosh 11-ci computer. The contrast in the images can be directly interpreted as maps of the local concentration of the species of interest. Concentration coutours in the X-Z plane were measured by firat locating the center of the feature of interest (e.& a pore) and subsequently measuring the tip current as a function of x pmitions. for a set of z positions above the feature of interest. Scan lines in the x direction were made at 15-20 values of z, ranging between approximately 0 and 150 rm. Reagents. K,Fe(CNk (Mallinkrodt, Paris, KY). anhydrous FeCI, (Sigma, St. Louis, MO), NaCI, K3Fe(CN)B,NaH2P0, (Baker,Phillipsburg,NJ). and Na,HPO, (Fisher,Fair Lawn, NJ) were used as received and were ACS reagent grade or higher. 18 MRan H20. obtained from either a Water Prodigy (Labconco, Kansas City, MO) or E Pure (Barnestad, Dubuque, IA) purification system, was used throughout. Polycarbonate Membrane. ApproximatelyO.1-pm-diameter pores in a 15-pm-thickpolycarbonate membrane were produced by an irradiation/chemical etch process.Is The exposed membrane area was 3.1 X em'. Polycarbonate and mica membranes were mounted between the centers of two 5 em x 5 cm X 1.9 mm glass plates. A hole in the center of each glass plate exposed a disk-shaped area of the membrane. Grease (type H, (13) Potje.Kamloth, K.; Janata, J.: Mira. J. Ber. Bumenfes. Phys. Chem. 1989,93, 1480-1485. (14) Stelzner. R. W . Chem. Imtnrm. 1969,2,216247. (15) Quinn, J. A,; Anderson, J. L.; Ho,W. S.: Petmy, W .J. Biophys. J. 1972, f2.9!%-1001.
Apiezon Corp., London) was uaed to provide a water-tight seal between theglagsandsyntheticmembranes.Butylrubbero-rings were used to seal the glass platelmembrane sandwich between the diffusion cell compartments. Mica Membrane. The mica membrane was 10f 5 r m thick. Prior to mounting the mica membrane between the plates, a single pore was created in it by setting it on top of a Teflon block and carefully lowering the tip of an electrochemicallysharpened tungsten wire with a motorized micromanipulator until the tungsten pointpiercedthe mica. An opticalmicroscopewas used to characterize the mica hole. It was found to be circular, with a radius of I =5 1 pm. Pt Microdisk Electrode. A Pt microdisk electrode was fabricated from 25-pm-diameter Pt wire (Aldrich Chemical, Milwaukee, WI) by placing the wire inside a 3-pL micropipet and then filling the pipet with epoxy. After curing, the end of the pipet was polished to expose a Pt microdisk in the center of the pipet barrel. Electrical contact was made to the Pt wire with copper wire, and the electrode assembly was mounted. facing upward, in the receptor compartment of the diffusion cell. In this experiment the donor compartment was not used. Hairless Mouse Skin. Skin was removed from the backand sides ofa freshly sacrificed male hairless mouse (age 8-13 weeks, Strain SKH-1;Charles River, Portage, MI). The loosely attached subcutaneous brown fat layer was removed by gentle rubbing with a damp gauze sponge. The sample was placed between layers of saline-soaked gauze and stored in arefrigeratorfor 1-48 h prior to use. (Variations in storage time did not affect the results.) A 2 cm x 2 cm square piece of mouse skin was sandwiched between twoglassplates. The plates had 8.0-mm-diameter holes drilled through the centers to expose a 0.50-em2disk of skin. Silicone/nylon composite gaskets were used to provide a mechanically compliant, water-tight seal between each glass plate and the skin. Gaskets were formed hy impregnating nylon mesh (mesh size 0.5 mm, Model CMNdOO, Small Parts, Inc.) with siliconerubber (Dow Corning). An 8-mm-diameter bole was cut into the center of the upper gasket. The porous nylon disk in the center of lower gasket mechanically supported the skin. Siliconewas allowed to cure at least 12 h a t 50 "C or for 24 h at room temperature before use. The square of skin was placed, epidermal side facing upward, onto the lower plate. The edges of the skin were gently pulled outward with tweezers in order to position thesampleand tostraighten foldsintheskin. Adhesion between thedermalsideoftheskinandthecuredsiliconerubber caused the skin to temporarily stay in place. Several drops of cyanoacrylateadhesive ( D mQuick Gel, LoctiteCorp., Cleveland, OH) were used to tack the skin edges and corners to the lower plate, An additional bead of adhesive was sparingly applied to the lower face of the upper plate, encirclingthe plate's hole. The upper plate was placed upon the skin and a 300-g weight was placed on the upper plate for approximately 10 min while the adhesive cured at room temperature. The adhesive prevented the plates and skin from slipping against each other while the mount assembly was being handled.
111. THEORY OF SECM ANALYSIS OF PORE TRANSPORT
Concentration Maps. Since SECM images provide a means of visualizing the local concentration field of an electrmctive species, the relationship between the membrane structure and transport behavior can be quantified. For instance,ldconcentrationpeaksinSECMimagesofporous membranes indicate the presence of high ionic flux at discrete 'pore" sites. However, the relationship between the local concentration and the membrane structure can be complex. If transport occurs not only through membrane pores but also across the 'homogeneous" (nonappendageal) matrix, it will not be obvious from concentration images whether all, or even a large fraction, of total ion transport takes place through such sites, Scheme 11. A method to determine the ratio of the pore flux to the homogeneous flux might be to compare the concentration
ANALYTICAL CHEMISTRY, VOL. 65. NO. 11. J W E 1. 1993
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1.
Scheme 11. Iswoncentration Contours above Porous Membranes' irnprmmble p h e
srniprmable p h e
A.
n B.
,
.
C.
Transport in (A) and (B)occurs exclusively through pores in an otherwise impermeable membrane. In (C), transport occurs through both pores and the semipermeable membrane. The concentration distributionsabove structures B and Care nearly indistinguishable at moderate distances above the membrane surface.
directly over a high flux site to the concentration over the homogeneous membrane a t a large distance from the pore and to back-calculate theindividualcontributionstothe total flux from theconcentrationdata,by solving themaeatransport equations for the system. However, for complex membranes (e.&,thestratumcorneum),convectioneffects,combined with the complexitiesof overlappingdiffusion layers from several discrete sources, make the task of finding a closed-form solution difficult. The effect of overlappingconcentration fields from several discrete sources is illustrated in an SECM image of a microporouspolyearbonate membrane during iontophoretic transport of Fe3+ (Figure 2). In this experiment, the donor compartment contained 0.1 M FeCb and the receptor compartment contained 0.1 M NaCl. The average current density across the membrane was 3.2 mA/cm2. For polycarbonate membranes, the flux of Fe3+ across the homogeneous polycarbonate phase is negligibly small compared to
that through the aqueous filled pores, and discrete current peaks are seen over the pores. However, since the average spacing between pores is small (approximately 50 am on average), the diffusion layers of several neighboring pores overlap t o produce a uniformly high "background" concentration: eachcurrentpeakis typically only 15% higher than the apparent background. Were this membrane to be comprised of some material in which flux through the homogeneous material were significant, it would be difficult to discriminate, on the basis of an image such as this, the relative contributions of the porous and homogeneous pathways to the total concentration field. For membrane samples in which the pores are spaced far apart (Scheme IIA) the concentration field associated with each pore can be independently measured. For this m e , SECM data can be analyzed to determine the size of the pore openingandtherateofmmstransportofthespecieaofinterest through it, as we will show in the following section. Concentration Profiles above Pore Openings. Iontophoretic transportofanelectroactivespeciesthroughaporous membrane results from concentration and electric field gradients. Thus, transport is the result of both diffusion and migration within the membrane, as well as in the donor and receptor solutions. For aqueousfilled pores, theconductivity of the pore and the donorlreceptor solutions are comparable; however, for complex hiological samples, such as akin, the conductivity of the pores may be significantly different from that of the donor/receptorsolutions. Theanalysis wedescribe below avoids these complexitiesby considering only the flux of ions once they have emerged from the pores and are diffusing away from the pore into the receptor compartment (Scheme I). In the presence of an excess concentration nf supporting electrolyte in the receptor compartment (which is the case in all of the experiments reported here), the flux of the ionic species is dominanted by radial diffusion form the pore. Assuming, for simplicity, that the pore opening behaves aa a hemispherical source, having radius r,, an expression for the rate of transport from the pore opening, N (mol/s), can be readily obtained from the solution of the continuity equation (V*C(r) = 0). Using Fick's law and employing the boundary conditions for the iontophoretic
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10
disk
I/ hO0
200
400
0
x (pm)
Figure 2. SECM image of Fe3+emerging from a 600 pm X 600 pm region of microporous polycarbonate membrane. Current peaks indicate high Fe3+concentration due to flux from membrane pores: pore diameter, 0.09 pm; pore density, 105/cm2;donor solution,0.1 M Feci3;receptor, 0.1 M NaCI; tip potential, v, = 0.1 V. experiment, C ( r ) = 0 as r
-
and C ( r ) = C, at r = ro, yields
N = 2xDC,r, (1) where D is the diffusion coefficient for the electroactivespecies and r is the radial distance measured from the center of the pore opening. The concentration at the pore opening, C,, is complex function of the iontophoretic current, the pore geometry, as well as the number of pores and is frequently unknown. In addition, the size of the pore, ro,is generally not known. Thus, eq 1 cannot be used directly to solve for N . However, integration of V 2 C ( r )= 0 provides a relationship between the concentration of the electroactive species at a distance from the pore opening, C ( r ) , and C,: r C ( r ) = rocs (2) This equivalence, valid for all C ( r )and r within the diffusion layer of a hemispherical source, allows the unknowns Csand ro in eq 1 to be replaced by values C(r) and r for any hemispherical shell that is concentric with the pore opening. Concentration contours about a hemispherical source can be experimentally measured with SECM by choosing an arbitrary value for C ( r ) and finding the locus of points above the pore opening for which the concentration equals C ( r ) . By fitting the resulting isoconcentration contour to the equation of a hemisphere, the radius r can be determined, and the product r C ( r ) substituted into eq 1 to calculate N . Applicability of Hemispherical Model to Pores. Pore openings on membrane surfaces are not hemispherical. However, because the flux lines from a microscopic pore diverge radially from the pore, the resulting concentration field will appear approximately hemispherical for values of r that are large compared to the size of the pore. To illustrate this, we now consider the quantitative applicability of the hemispherical model to a disk-shaped pore opening, since the latter geometry more closely resembles the pore structure of membranes. Saito has derived a closed-form solution for the concentration profile and flux for steady-state diffusional mass transport to a disk-shaped sink16 which can be easily modified to describe diffusion from a disk-shaped source. The rate of mass transport from a disk-shaped source of radius a, in cylindrical coordinates ( r , radial direction; t,axial) is given by
N = 4DC,a
(3)
hemisphere
Figure 3. Theoretical contours of equal concentration, projected in the x-z plane, of species subject to semiinfinlte diffusion at steady state from hemisphere (solid lines) and disk (dashed lines) sources, centered at the origin. Contours are plotted for CJ2, CJ4, and CJS, where C, is the concentration at the surface of each source. The hemisphere and disk are drawn to scale: hemisphere radius, r, = 1; disk radius, a = 7r12. Table I. Comparison of t h e Radii of Hemispherical Isoconcentration Contours Obtained by Least Squares Fits of Theoretical Contours Resulting from Diffusion from a Disk-Shaped Sourcee C,i2 c,/4
c-18
2 4 8
1.808 3.968 8.044
-9.60 -0.80 +0.55
r is the radius of a hemispherical profile resulting from a hemispherical source of radius equal to unity. r’ is the radius of a least squares hemispherical fit to profiles from a disk-shaped source of radius ri2; see text. b (1 - ( r - J ) / r ’ ) x 100%.
The concentration profile is” C(r)=
2cs
a2lI2 (4) ( ( r 2+ t 2- a 2 ) + ((r2+ t 2- a’)’ + 422a2)112)112 where C, is concentration at the surface of the pore. Equation 3 differs from the hemispherical case in that the geometric constant 2~ in eq 1is replaced by 4. Thus, the mass transport rate from a disk is equivalent to that of a hemisphere with an effective radius, reff, given by the expression K
reff= 2ala (5) Theoretical concentration contours, calculated from eqs 2 and 4 for hemispherical and disk sources are plotted in Figure 3. The two-dimensional projections of the contours for the hemispherical pore are perfect hemicircles, whereas at short distances compared to reff, the contours resulting from diffusion from a disk-shaped source are flattened in the center, due to planar diffusion at close distances above the disk. This is evident in the contour C ( r ) = C,/2. At larger distances, however, the contours of the disk more closely resemble those of the hemisphere. If the simple hemispherical transport model is employed in SECM analysis of pore transport rates from a disk-shaped source, a small error will be introduced in the calculation of N as a consequence of calculating r from isoconcentration contours that are assumed to result from transport from a true hemispherical source. Table I illustrates how the hemispherical approximation to the concentration profiles above a disk-shaped pore opening improves as the distance away from the surface increases. Even at moderate distances, e.g., -4r, the error in N using the hemispherical approxi(17) In ref 16, eq 7 has a typographical error. This has been corrected
(16) Saito, Y. R e v . Polarogr. 1968, 15, 177-187.
in eq 4 of this article.
ANALYTICAL CHEMISTRY, VOL. 65, NO. 11, JUNE 1, 1993
mation for a disk-shaped source is less than 1% . Although the theoretical error continues to decrease with increasing distances, practical concerns such as concentration detection limits and the possibility of competing mass-transfer mechanisms, e.g., convection,eventually limit the improvement in the accuracy of the hemispherical model. Model A. Experimental Measurement of MassTransfer Rate from Concentration Contours. Concentration contours were obtained from SECM measurements in which the SECM tip was rastered across the membrane surface a t several different vertical heights, z. From data files containing SECM tip current (it) vs position ( x , z ) , twodimensional contours of constant it(x,z)were extracted using a BASIC program. Concentration contours were extracted from it(r,z) for values of C(r)equal to C,, Cd2,and C,/4, where C, corresponds to the concentration of the innermost hemispherical shell. Absolute values of C, were obtained by calibration of the SECM tip current in solutions containing a known concentration of the electroactive species. The isoconcentration contours were approximately hemicircular, and a least squares hemicircle was fitted to each contour to determine the contour’s radius. For each contour, the value of C(r)and r were used with eqs 1and 2 to calculate the rate of transport through the pore, N. Method B. Experimental Measurements of MassTransfer Rate and Pore Radius From Concentration Profiles. Another technique can be employed which uses the hemispherical model in a slightly different manner to calculate both N and the pore radius. Let d be the radical distance from the surface of a hemisphere, r - r,. Substitution of d into eq 2 gives
Since it is directly proportional to C(r),it follows that (7)
where it, is the tip current the surface of the hemisphere. A plot of l/it vs d will be linear, with an intercept of l/it, and a slope of l/it,ro. Experimentally, one need only plot tip current along any radial line extending from the center of the source. This can be measured directly, or one can extract the appropriate data points from the same it vs the ( x , z ) data set used to generate the concentration contours. While a value for it, can be directly calculated from the intercept, a dramatic error can be introduced in the result due to uncertainty in the absolute height z of the SECM tip. It has been found empirically that direct measurement of the current i, with the tip positioned over the pore opening gives better results. Thus, by equating it, and itma, where it,, is the maximum value of the tip current measured directly over the pore opening a t the smallest possible height z, while still avoiding contact between the tip and sample (z 5 5 pm), the hemispherical pore radius can be calculated from eq 7. If the source happens to be disk shaped rather then hemispherical, the calculated value of r, is equal to reff, which is readily converted to the disk radius, a, using eq 5. As in method A, N is calculated by substituting the pore radius (reff) and C, into eq 1. Comparison of Methods A and B. As will be shown in the Results and Discussion section, both methods provide accurate measurements of N. Although method A does not directly measure re^, it is regarded as the more reliable method for measuring N, because it involves the visualization of the projected contours, allowing for easier detection of (and
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correction for) nonhemispherical concentration fields. If, for example, a pair of closely spaced pores causa a double peak in the two-dimensional contour,an algorithmcan be developed for deconvoluting the two component peaks if the positions of the pores are known. By contrast, with the llit profiles of method B, the spatial variation in concentration can only be visualized in one dimension, making it difficult or impossible to reconstruct the physical situation. Nonhemispherical concentration fields will likely manifest themselves in a nonlinear l/it vs d plot, alerting the experimenter that a simple hemispherical analysis is not sufficient. Experimentally, method B is a shortcut compared with method A. Rather than extracting contours of equal concentration from a large array of data, the tip current is measured along any radial line from the pore center. By choosing such a radial line normal to the membrane, the measurement is most convenient, and effects of “competing” contributions to the diffusion layer from neighboring sources are minimized. A practical problem of method A requires the tip to be scanned,without physical obstruction, in a linear path at small heights above the skin. In regions where obstacles, such as hairs extending from a skin sample, prevent such scanning,method A cannot be used, even though method B is still often practical. For both microporous and microdisk electrode samples, the nonzero size of the tip can lead to perturbations in the species concentration a t close approaches to the sample surface. Such perturbations result from three effecta: physical blockage of the diffusion path, consumption of species, and in the case of a metallic sample, electrochemical feedback.18 These perturbations can lead to an error in the measured value of C,, compared to a true surface concentration, C,. While an error in C, can propagate in the calculation of the pore radius using method B, an accurate value for C, is unimportant in calculation of mass transport rates using either method A or B. For both method A and method B, the deviation of a disk’s profile from that of a hemisphere will limit the accuracy of the model a t close distances. In addition, for method B, the deviation depends on the polar angle of the radial line used to plot l/it. However, it is found experimentally that such data are still reasonably linear at distances d greater than approximately 2a. SECM as a Probe of Local Concentration. Quantitative calculation of local concentration from SECM data requires that the tip is a noninteracting probe. A restatement of this criterion is that the consumption of reactant at the tip does not deplete diffusing species at a high enough rate such that the diffusion field of the sample is significantly altered. Agreement between experimental results and theoretical expectations verifies that the above assumptions are met. The criterion of a noninteracting probe can be shown to be valid if the tip is small with respect to the dimension of the spatial variation in the diffusion layer of the sample. This condition is expressed mathematically as a,
