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Photoelectrochemical imaging—part I. Background and theory
EktracW Act& Vol. 38, No. 1. pp. 51-69, 1993 Printedin Great Britain.
0013-4686193$5.00+ om Pcqpmoll~Ltd.
PHOTOELECTROCHEMICAL IMAGING-PART BACKGROUND AND THEORY
I.
D. E. WILLIAMS,*A. R.J.KUCERNAKand R. PIZAT AEA Industrial Technology, Building 552, Harwell Laboratory, Didcot, Oxon OX1 1 OBA, U.K. (Received 12 Nooember 1991)
Abstract-Imaging of spatial variability of electrode processes using signals stimulated by a focused light spot is described. The signals include photovoltage and photocurrent, photoacoustic and photothermal effects, as well as the reflected light intensity. The theory of contrast in the photocurrent image is presented: methods utilizing intensity modulated light are compared with those in which the spot is scanned rapidly without intensity modulation. Effects of variation of intensity modulation frequency and spot scan speed are calculated; blurring, streaking and shadowing effects are explained. For systems which are not photoactive, the feasibility of an imaging procedure based upon the small thermal effect induced by the focused spot on the current for an electrode process is amessed. Methods of imaging utiliaing potential variations at constant (usually zero) current are compared with those involving the measurement of current variations at constant potential. Key words: photoelectrochemistry, imaging, photocurrent, photothermal, theory.
INTRODUCTION
Photoelectrochemical imaging involves the mapping of the photocurrent or photovoltage developed at an electrode-electrolyte interface as a function of the position of an illuminating spot. The electrode may be a metal, usually carrying a passivating film, or a semiconductor. The technique produces positionsensitive information by locally perturbing the surface and observing the macroscopic change in the system response. Deviations from the unperturbed (“background”) state are measured as a function of the position of perturbation: since the excitation is local&d, any deviation in response can be attributed to effects within that localized region. Responses that can be measured are illustrated in Fig. 1. Thus, spectroscopic information concerning the local region is in principle available through the photoacoustic Illupnfng
I
Optical
4,
Photothermal
Fig. 1. Signals and effects that can be used to form an image under illumination by a scanned, focused light spot.
* Present address: Department of Chemistry, University College London, 20 Gordon St, London WClH OAJ,U.K.
effect[l] and also through photocurrent spectroscopy. An additional component of the current across the interface dependent upon the local reaction dynamics could arise as a consequence of the small heating effect caused by the light, referred to here as the photothermal effect. Much the largest additional current is usually a consequence of absorption of light and production of mobile charge carriers, for example by excitation across the semiconductor band gap, or by photoemission processes. One way of thinking about the system is to view the electrolyte as a transparent Schottky contact; charge carriers excited as a consequence of absorption of light are separated by the electric fields present in the vicinity of the interface and are transported across the interface by an electrochemical reaction. This reaction can result, particularly in the case of semiconductor materials, in the etching of the electrode. It is clear that there is a number of factors which might give rise to spatial variation of the photocurrent measured in an external circuit: carriers must be generated and separated and may recombine, and image contrast can result from spatial variations in all of these. Speci6c reasons for contrast could include the presence of defects in the solid intersecting the surface, variations in the composition of the solid in the near-surface region and the presence of particular catalytic centres at the interface. Other workers in this field[2-7j have utilized for imaging what may be termed a “step-scan, lock-in” method, in which either the illuminating beam or the sample is moved from one imaging point to the next, the illumination is intensity modulated, and the signal is derived by phase-sensitive detection referenced to the intensity modulation. The image acquisition time is necessarily limited in this method by the need to rest at each image point for the time required for a sufilcient number of ilh&nation
58
D. E. WILLIAMS et al.
cycles to give satisfactory detection of the signal and noise rejection. In electrochemical systems, an image .of 512 x 512 points typically takes at least 30min to acquire using this method. A method in which the beam or specimen is scanned continuously but slowly in comparison with the illumination modulation frequency is an obvious extension. As we shall see below, the key characteristic of this method is the use of a fixed modulation frequency for the illumination We have presented in recent publications[&121 a method in which the light beam is scanned rapidly across the specimen surface and the variation in the total current flowing across the interface, as a function of the spot position, forms the image. The frame acquisition time is kept short enough that any background current remains constant over the frame, and the background current may then be rejected by a high-pass tilter: it is the variations in current within each frame which are imaged. The total (filtered) current is sampled at each image point by a frame store. In this method, the illumination intensity is not modulated and noise reduction is obtained, if necessary, by averaging of successive frames of the image. For convenience, we refer to this as the %ontinuous-scan, direct detection” method. The image acquisition time is generally much less with this method than with the “step-scan, lock-in” method and, indeed, time variations in an image of 512 x 512 points can be tracked on the scale of a few seconds, although attention to detail is required in the design of the experimental apparatus, to minimize noise pick-up yet retain an adequate response bandwidthC13, 143. New features arise in the image, particularly blurring, streaking and shadowing effects, which are a consequence of the transient illumination of each image point by the flying spot and which can be related to the local kinetics of reaction of the photogenerated charge carriers at the interface. We have recently described the use of numerical simulation to explore these effects[l2]. The “continuous-scan, direct detection” method is equivalent to illumination of each image point by a light beam with a spectrum of intensity modulation frequencies: we will develop this interpretation further in the present paper. The purpose of the present paper is to develop formally the theory of image contrast in both the “stepdirect “continuous-scan, lock-in” and scan, detection” methods, to relate the image contrast to spatial variations in basic parameters describing the photoelectrochemical processes occurring at the interface and to show how complementary information can be obtained from the two techniques. For systems which are not photoactive, an assessment is given of the feasibility of an imaging method based upon the small thermal effect of the light beam. Application to imaging the photoelectrochemistry of iron and stainless steel is presented in following papers[14]. A FREQUENCY DOMAIN APPROACH TO THE STUDY OF IMAGE CONTRAST The idea of a photocurrent transfer function was introduced by Peter[lS] to quantify the response of
a system to sinuosoidally intensity modulated light: in the case where the photocurrent is linearly dependent upon the light intensity, the photocurrent response to such illumination is also sinusoidal but varying in amplitude and relative phase as the modulation frequency is changed over ranges in which the time constants of the photocurrent relaxation are comparable with the period of the intensity modulation. That is, the stimulus is an intensity modulated light, the response is a photocurrent, and the transfer function is the frequency domain ratio of the two. Thus the response of the photocurrent density as a function of the angular frequency of the excitation (angular frequency of the intensity modulation) is given by i&e) = H(o). W),
(1)
where H(o) denotes the excitation function (the component of the light intensity varying at angular frequency o) and Y(w) denotes the photocurrent transfer function. Peter et aZ.[lS] have derived this function for a variety of assumed mechanisms of the electrochemical reactions at the interface involving the photogenerated carriers, and have developed the technique of intensity modulated photocurrent spectroscopy (IMPS) to measure it. A common form of the photocurrent transfer function is the “snail” (Fig. 2), involving a single relaxation of the photoresponse, attenuated by the equivalent RC network of the electrochemical cell: Y(o) = G,F(o)Y,(w)
= G,F(o
,
(2) where F(o) = l/(1 + jar,) denotes the attenuation introduced by the cell. Here G, denotes the carrier generation rate, go gives the steady-state photocurrent response, g1 denotes the maximum response, rl is the time constant for recombination of photogenerated carriers and r1 is the relaxation time of the measurement cell, 71 = CR,, where C denotes the electrode interface capacitance and R, the series resistance (predominantly the solution resistance) through which this capacitance is charged. The image contrast arises as a result of the spatial variation of the parameters G, , go, gr and rr. THEORY OF IMAGE CONTRAST IN THE “STEP-SCAN LOCK-IN” METHOD In this method the light intensity is modulated and the beam scanned slowly enough that the response from any given position can be extracted oia a lock-in amplifier. A sinusoidal excitation function is conveniently employed, H(w) = I,[1 + AZ,,,sin@)], where I, denotes the mean light intensity and AZ,,, the depth of modulation. The sensitivity to spatial variations of any of the parameters depends upon the frequency chosen for the measurement. This conclusion can be expressed quantitatively using the photocurrent transfer function; consideration of the “snail”, for example, indicates immediately that if the frequency were too high (o > l/r,), then the amplitude of any signal varia-
59
tlmels 0.6 0.4 0.2 Yimag
0 -0.2 -0.4
-0s;
0.2 ’
’
0.4
’
0.6 ..
’
’
0.8
1
’
1.2
Y real
Fig. 2. (a) Photocurren t response to a step in light intensity for a process with a simple 6rst order relaxation attenuated by the experimental cell, and (b) the corresponding photocurrent transfer (parameters: 7’1= Sms, 72 = 1 mq go = 0.2, g1 = 1).
tions a&oss the surface would be small. It will also be evident from the following that the resultant image contrast depends upon whether the magnitude of the resultant photocurrent or the components in phase or in quadrature with the excitation signal are used to form the image. The magnitude of the photocurrent is used in the following analysis. The contrast within a given area is conveniently detined as the relative variance of the photocurrent over the area, aZ(Z h)/(Z,,,>‘, or alternatively as the variance of the piotocurrent measured over that area, a2(Z,,). The relative variance is a convenient measure where images are scaled such that the mean image density falls in the centre of the brightness range available but is invalid for noisy images (ie ones in which the scale of random fluctuations is greater than the signal): in these cases the large photocurrent variance would imply a great degree of contrast, where in fact only noise exists. Conclusions concerning the effects on contrast of varying the excitation frequency can conveniently be arrived at by assuming that the parameters giving rise to the contrast vary independently over the surface l&l = G,Z,AZ,IJW)l
* IW4l
(3)
function
Equation (4) gives the first conclusion: that contrast (expressed as the relative variance) due to variation of the carrier generation rate, G,, is not dependent upon the intensity modulation frequency. Of course, the photocurrent does vary with the modulation frequency and at high enough frequency tends to zero as a result of the attenuation by the cell-hence contrast expressed purely as the variance of the photocurrent would vary with w. The second term in Equation (4) can be written in terms of the variation of the parameters over the surface (again assuming that these vary independently of one another)
dlr,l)2
( >
=GziY
1
D. E. WILLLUISet al.
60
Substitution of the expression for Y0 [from Equation (211 leads to the conclusion that contrast (expressed either as the variance or as the relative variance) due to the variation of the parameter g,, is maximum in the limit of low frequency and minimum (but not zero) in the limit of high frequency. Contrast due to variation of the parameter g1 is maximal in the limit of high frequency and zero in the limit of low frequency, and finally contrast due to variation of the parameter 71 is zero in the limits of both low and high frequency, and maximal when w = l/7, : the results for the photocurrent contrast in the step-scan, lock-in method are summarized below : Variation in contrast (w-a) (6a) (a + 0)
@+a) (a-*0)
(6b)
tation spectrum at any particular point depends upon the distance of the point off the centreline of the beam scan. The time-dependent photocurrent resulting from the transient illumination of a particular point on the surface could in principle be obtained by substitution into Equation (1) of the photocurrent transfer function peculiar to that point and inverse transformation. As has been pointed out before, the image intensity at any position is the total photocurrent flowing through the specimen when the beam is located at that position: this is obtained by integration of the photocurrent resulting from all previously illuminated points. A blurring of the image comes about as a result of the slow recovery of the signal following the transient illumination. The objective is to calculate the contrast variation expected as a result of the variation over the surface of some parameter of the photocurrent transfer function: we denote the parameter of interest a. At some position x0 along the line of the beam scan, the magnitude of the photocurrent (which gives the image density at that point) would be
V6Y
THEORY OF IMAGE CONTRAST IN THE “CONTINUOUS-SCAN, DIRECT DETECTION” METHOD This method of image formation contrasts with the step-scan, lock-in method in that each point receives a spectrum of excitation frequencies. The excitation function, H(w), is determined by the intensity profile within the focused light spot and the velocity of the spot across the surface. The intensity distribution for a focused spot is illustrated in Fig. 3a and from this the excitation function resulting from scanning the spot over a surface may be obtained by Fourier transformation (Fig. 3b): note that the exci-
-a,-
s x0/v
_!PL= iv,@, a) dt, 0
where i, denotes the photocurrent density (a function of the time after the illumination and of the parameter a, through the photocurrent transfer function) from an element of the surface a position vt behind the current spot position (v is the spot scan velocity) and the integration proceeds back to the edge of the specimen which is presumed to lie within the image frame. The result depends upon the shape of the spot: here, a rectangular shape of width 6y perpendicular to the scan direction has been assumed for simplicity. The photocurrent density as a function of time is the inverse transform of the product of the excitation function, H, and the photocurrent transfer function, Y: it is convenient for this purpose to use the Laplace transformation. That is
iv&, a) = Y- ‘CH(s)Y(s, a)],
v&
(s-1)
Fig. 3. (a) Intensity distribution in a focused light spot; the spot radius, a,, is taken as the radius of fhe second minimum in the intensity distribution. (b) Excitation functions of points underneath the movmg spot, evaluated by Fourier transformation along lines parallel to the scan direction at successively greater distances from the centre of the spot; v denotes the spot scan velocity.
61
Photo&etrochemieal imaging where O- 1 denotes the inverse Laplace transform with s the Laplace parameter (the substitution s =ja, gives the frequency domain functions) and G,, denotes the carrier generation rate. Now suppose that the variations over the surface of Y(s, a) as a result of variations of a can be expanded as small variations about a mean value Y(s, a) = Yds, ac) +
aY aa
6a
then '[H(s)Yc(s, a,,)] dt h = x”‘“9’VSY s 0 L
+ 6ap_l(H(s)
g)
dt.
(10)
The tirst integral is constant over the surface and gives the mean photocurrent whereas the second gives the variation about the mean caused by small variations over the surface of the parameter a. It is apparent that one can conveniently define the contrast in thii case in terms of a type of transfer function, -=$l-‘(Zf,(s)~)dr v 6y aa
(11)
which can be used to assess the transformation of spatial variations in any parameter a of the photocurrent transfer function Y into spatial variation of the photocurrent (image density): for example, a sinusoidal spatial variation of a would be transformed into a damped and phase-shifted spatial variation of I,. It will be useful now to calculate the contrast resulting from spatial variations in the parameters characterizing the photocurrent transfer function. A “snail” form of the photocurrent transfer function [Equation (2), Fig. 23 is used: the spatially variable parameters are g,, gl, and rl and zs is a characteristic of the cell, In order to derive the key features of the response in a reasonably clear way, a rectangular intensity profile for the beam is used: this makes an
analytical treatment tractable. The beam intensity is 1, and width 2u,, ie H(s) = I,s(l
- e-‘7;
T = 2+/v.
(12)
A step function spatial variation of the parameter of interest is assumed. The spatial coordinate is conveniently transformed so that the step occurs at x = 2~6, since there would be no photocurrent variation with position until the edge of the beam hit the parameter step position. It will be apparent later that the photocurrent contrast resulting from any arbitrary spatial variation of the parameter can be derived by approximating the variation as a sum of step function changes. The results of the calculation of the image contrast resulting from step changes in the parameters go, gl and rl are presented in Figs 4-6. Details of the calculations are given in Appendix A. One point to note is that there is a characteristic time scale detined by the ratio of the spot radius to the scan velocity. A streaking effect-an overshoot followed by a relaxation of the respon se-occurs in images of step changes in the parameters g, and ri and is qualitatively different in the two cases. A streaking effect does not, however, occur in images of the parameter go, only a blurring caused by the finite rate of rise of the signal to its new level following the step variation in the parameter. The spatial variation of parameters has so far been represented as a single step. Although this is a very simple function, it allows for the synthesis of more complex spatial variations in parameters by the scaling, shifting and addition of multiple steps. This is illustrated in Fig. 7 for the simple case of a square pulse spatial variation of the parameter, gl. A pulse change in the value of a parameter can conveniently be synthesized by the simple addition of two identical steps of opposite sign, shifted relative to one another in position. The image is simply the sum of the response to the two steps, that to the second being shifted and inverted with respect to that of the tirst. Two different sets of parameters were used in calculations, corresponding to a lo-fold alteration in the beam scanning velocity. This alteration of the
Fig. 4. Variation of photocurrent (image contrast function) with Position for a square spot, width &, moving with velocity v across a step in parameter go, indicated in (ah for dilkent values of the dimensionkss time constant ‘i’, =I r2 v/(2@, (b) 0.1. (c) 0.213, (d) 0.457, (e) 1.0, (0 2.13, (g) 4.57.
D. E. WILLLU.Bet 01.
62
0.6
'0
'0
12
3
4
12
3
4
5
5
6
6
7
7
Fig. 5. Variation of photocurrent (image contrast function) with position for a square spot, width 2u,, moving with velocity v across a step in parameter g, (step position at z = 1, as indicated by the arrow), for different values of the dimensionless time constants Ys = r,v/(2a,): (a) 0.1, (b) 1, (c) 10, and Y, = r,v/(2a,): (i) 0.1, (ii) 0.32, (iii) 1, (iv) 3.2, (v) 10.
scan speed produces a very interesting result for a pulse spatial variation of g, : the position dependence of the current for the slower scan speed (Fig. 7b), shows a dark streak following the feature; however, a lo-fold increase in speed (Fig. 7a), results in the virtual disappearance of this streak in the calculated image. Another parameter has of course been introduced, and that is the length of the feature being imaged. For a response which eventually decays to zero (images of variations in g1 and or here), it should be recognized that a full range of different types of response is possible, simply by varying the feature length. Thus, if the transient has peaked and returned almost to zero before the beam has traversed the full length of the feature, then when the
beam passes off the feature, there would be a dark streak following the feature which would be. almost a mirror image of the response when the beam passed on to the feature. For shorter feature length the undershoot would not be as great. This point is illustrated in Fig. 8. The variation of response has thus far been confined to the consideration of 1-D variations in one of the parameters of the photocurrent transfer function. By moving to two dimensions, and allowing variations in transfer function parameters across a plane, it is possible to generate maps of the variation of photocurrent across a surface and thus simulate the experiments performed with the microscope. The extension to two dimensions is quite simple. Each successive “line” in the picture does not interact with
Photoelectrochemical imaging
63
Fig. 6. Variation of photocurren t (image contrast function) with position for a square spot, width 2a,, moving with velocity v across a step in parameter z, (step position at z = l), for different values of the dimensionless time constants Y, = z2 v/&J: (a) 0.1, (b) 1, (c) 10, and Y, = s,v/(2a0): (i) 0.1, (ii) 0.32, (iii) 1, (iv) 3.5 (v) 10.
the surrounding lines, and may be calculated in the manner described above. Results are illustrated in Fig. 9 for the expected image contrast resulting from particular spatial variations of the parameters go, g1 and zi. The simulation assumed that the laser beam was scanning from left to right. Images were calculated for two values of the dimensionless time constant, corresponding to a N-fold variation in the scan speed. At the faster scan speed, there is a signit% cant amount of blurring, shadowing and streaking, especially for the cases involving variation of the g1 and ~~ parameters. At the higher scan rate, the image of the gi parameter variation still shows a peaked response once the beam scans onto the altered zone whereas the image of the z1 parameter shows no peak in its initial response, only a slow rise; the cor-
responding dark-transient when the beam scans off the feature is greatly extended in length. The image of the go parameter variation is slightly rounded only. Responses very similar to these have indeed been observed[&14]. Depending on which parameter is spatially variable and upon the feature size and spot scan speed, the images may be similar to those produced using the step-scan, lock-m method or may approach a spatial derivative of them. BEAM
EFFECTS
The continuous-scan, direct detection method in particular involves the use of an intense, focused and rapidly scanning spot, and it is appropriate to consider effects specifically associated with the power
D. E. WILLLUS et al.
64 0.6 0.4
(a)
0.2
[4 gIG,L, 1
0
-0.2 -0.4
__...._....._
-0.6
4
6
6
10
____ _J 12
14
Fig. 7. Image contrast for a spot moving over a square pulse spatial variation of the parameter gi, depicted as trace (c), for different values of the dimensionless time constants Y’, and Y, , corresponding to a IO-fold change in spot scan velocity: (a) Y, = IO, Y’, = 3.2;(b)YI = 1,Yy,= 0.32.
density and geometry of the focused spot. There are also heating effects, effects of scattered light producing spurious signals, and effects of nonlinearity of the response to variation of the illumination intensity. Finally, high local concentrations of charge carriers might be produced and there may be specific consequences of this, too, in the chemistry of the reactions taking place at the solution interface. A high flux of photogenerated charge carriers at the surface may cause intense dissolution or, in the case of passive metals, further film growth or alteration of film stoichiometry. Under sufficiently high illumination intensities heating effects may come into effect with the possibility of oxide thickening[16], changes in the kinetics of surface reactions or, in extreme cases, melting of the substrate. Another potential ditficulty is that because of the high current densities observed at the laser spot (milliamps to amps per square centimetre vs. microamps per sauare centimetre for conventional Dhotoelectrochemistry) the potential at the illuminated spot
might deviate significantly from the applied potential as a consequence of the resistive voltage drop in the solution and specimen in the vicinity of the spot. In this connection, as far as transport processes around the spot are concerned, the scanning focused spot may be considered to be a scanning microelectrode. In the substrate, the accumulation of a bigh local concentration of charge carriers at the solution interface would change the electric field distribution there, with consequent possible effects on transport rates and reaction kinetics. Virtually all photoeletrochemically active systems at high enough light intensity show a decrease in photocurrent conversion efficiency, the photocurrent tending to saturate with further increase of tight intensity. Exceedingly high illumination intensities (w 102z-1023 photon s-l cmP2) are possible for moderate laser beam powers (of the order of milliwatts) focused to a spot of the order of micrometres in diameter: one consequence of photocurrent saturation with increasing light intensity can then be that
0.4
0.2
II41 4.&oIo
0
-0.2 -0.4 t
0
I
4
I
6
12
16
20
24
26
Fig. 8. Image contrast for a spot moving over a rectangular pulse spatial variation in the parameter gi, which starts at the position marked (a) and proceeds to one of the following arrowed positions, giving the contrast variations shown by the curves (b)-(f). The contrast for traversal over a step variation in 8, is that represented by curve (g). Dimensionless time constants, Y, = 10, Y, = 3.2.
65
Photoelectrochemical imaging (a>
t
Fig. 9. Simulation of the image contrast expected for some 2-D variations in the parameters go (top, parallelogram), g1 (middle, triangle) and T~(bottom, circle). Both grey-scale (a, c) and surface plots (b, d) are given, with the surface plots rotated for clarity. The second set of images (c, d) corresponds to a IO-fold increase in soot scan velocitv over that for the first set fa. bl: (a. b), Y, = 1, Y, = 0.1; Y, = 10, Y, = 1 and [ = 5.’
an increase in beam diameter (lowering the intensity but keeping the total photon flux constant) results in an increase of the photocurrent. Another consequence is that disproportionately large currents in comparison with those stimulated by the main beam can arise from light scattered back onto the surface-from the coverslip or specimen mount, for example[ll, 133.
PHOTOTHERMAL EFFECTS AS THE BASIS FOR AN IMAGING PROCEDURE. Processes of absorption and scattering of the incident beam at the surface under study result in the degradation of some of the incident energy into heat. There are two sorts of photothermal effect to be distinguished. Firstly, there is the photoacoustic effect, which forms the basis of photoacoustic microscopy. It occurs when the beam is intensity modulated and refers to the conversion of the resulting thermal wave
CC,d),
into a sound wave of frequency equal to that of the intensity modulation. The intensity of the sound wave is proportional to the absorption coefftcient of the light and hence the wavelength dependence of the effect can be used to measure the absorption spectrum of the local point. Phenomena of scattering and interference of sound waves as a consequence of interaction with subsurface defects give rise to additional contrast in the image. Secondly, there is a direct thermal effect on the current for an electrode process. It is the direct thermal effect on the current for an electrode reaction which is of particular interest for the present paper. Thermal effects of a focused light source. on electrode processes at the mercury-solution interface were noted by Benderskii[lfl and Barker[18], and exploited by Bender&ii and Velichko[19] to study relaxation in the electrical double layer. Thermal effects of a focused source on an anodic oxide were noted by Schultze and co-workers[16, 203. A thermal voltammetry experiment was described by
66
D. E. WILLLUG etal.
Miller[Zl]. The formalism given by previous authors can be used to derive rough estimates of the temperature excursion in a scanning laser microscope experiment of the type envisaged here. The conclusion (Appendix B) is that with moderate beam powers (N 1 mW) and spot radius less than about lo*, temperature excursions under the spot of up to about 10K might be expected. This local temperature rise would cause an increase in the local rate of electrochemical reaction. There would also be as a consequence some enhanced convection in the adjacent solution with again a possible local increase in the rate of the electrochemical reaction at the interface. The following calculations consider only the thermal activation effect on the local electrochemistry and not any effect of local enhancement of mass transport. An assessment of the feasibility and information content of a photothermal electrochemical microscopy experiment can be made starting with the assumption of a simple form for the local current density at constant potential: i = i, exp[AEJ(RT)]
(13)
in which BE, denotes the activation energy. The effect of a temperature variation is then simply Si = (6T)i AEJ(RT’).
(14)
If a denotes the area of the focused spot, then the variation in the total current, detected and imaged in the experiment, is 61 k: (6T)i AE,a/(RT’)
(1%
and the contrast in the image is
a’(dZ)= d(i)[yy
In systems where the dark current was very small, the first term in Equation (16) would be the most important. The image would show the spatial variation of the dark current: given the parameters leading to Equation (17), for example, then a spatial variation in 61 on the scale of tens of picoamps (small but certainly measureable) would result from a spatial variation in i on the 10RAcm-2 scale, as might perhaps occur for a well passivated surface with local points of breakdown, or incipient breakdown of passivity. In systems where the dark current was large, with the image processing system coupled into the measurement through a high-pass filter so that only changes in the current were recorded, then the image would display any regions where the spatial variation of current density was comparable in magnitude with the dark current density: for example, dead spots, where the dark current density was much lower. Spatial variations in the terms BE, and 6T, since these are weighted by the value of the dark current density, should also show up. The success of the imaging would depend on the effectiveness of the frame averaging or intensity modulation procedures in diminishing the influence of random temporal variations in the current. It is evident that, if & denotes the specimen area, then 6111 N (a/d) AE, 6T/(RT2)
(18)
so that, if random temporal fluctuations in Z scaled directly with I, then the ratio of signal (photoinduced effects correlated with position) to noise (random fluctuations uncorrelated with position) would scale with the ratio a/d and the problem of averaging out the noise would become greater the larger the electrode being examined.
+ o’(dT)[$]
+
qsr)
a2@E3E [
1 2
(16)
since one could expect spatial variations of the current density and activation energy of the electrode process and also of the temperature change, as a consequence of spatial variations in the absorption coefficient of the light. Which of the terms had the dominant effect on the contrast would depend on the dark current. It is also evident that there is a tradeoff between the resolution of the method and the size of the signal to be expected, since the signal increases directly with the illuminated area for a given temperature rise. The size of the signal and the resolution would also be sensitively affected by the thermal conductivity of the specimen and electrolyte, since a larger temperature excursion would be achieved with lower thermal conductivity, but the relaxation of temperature after the beam had passed would also be slower, resulting in blurring and streaking effects of the type described above for the direct photocurrent image. The size of the effect to be expected is indicated from Equation (15): if AE, N 18kcalmol-‘, T = 300K and 6T = lOK, then AE, 6T/(RT2) - 1; then with spot radius 10 pm, (61/A) = (i/Acmd2)(3 x 10-6cm-2).
(17)
COMPARISON OF PHOTOCURRENT PHOTOVOLTAGE IMAGING
AND
In the study of electrochemical systems, the measurement of current at controlled potential is usually preferred, since it provides a measurement of rate of reaction at constant driving force. However, measurement of potential at controlled current can be preferable in practical situations, such as in the study of corrosion processes where the behaviour at open circuit (zero net current) is of importance. Another circumstance in which the measurement of potential could be preferable is that in which the rate of the electrode processes is very low-passive systems, for example, since in such systems very small current changes can give rise to rather large potential changes. In such circumstances, the photovoltage signal at constant current could be much larger than the photocurrent signal at constant potential, and hence a photovoltage image might be rather easier to obtain. The disadvantage might be that the photovoltage image is rather more difficult to interpret. At zero current, for example, the electrode potential would often be a mixed potential, determined by the balance of anodic and cathodic processes which are not necessarily the reverse of one another. A photopotential change would result from
Photeelectrochemicalimaging the stimulation of an additional component of the current which could be either anodic or cathodican anodic photocurrent would result in a cathodic shift in potential and vice versa. The magnitude of the potential change, however, would depend not only on the magnitude of the photostimulated effect, but also on the dependence of this effect on the potential (which changes as a result of the stimulation) and upon the potential dependence of the anodic and cathodic components of the dark current. The foregoing arguments relate the steady-state photovoltage change. The transient photovoltage change is similarly determined by the balance between transient photocurrent, transient dark current and capacitive charging current. The resulting photovoltage transient could be quite complicated, and the resultant streaking and shadowing effects in the image difficult to interpret other than qualitatively. Photovoltage imaging of the corrosion inhibition of copper by benzotriazole and of subsequent inhibition breakdown has been reported [S, 13}, and illustrates both the possibilities and difIlcult.ies in interpretation with this method.
CONCLUSION Imaging of electrochemical systems using signals stimulated by a focused, scanned light spot can be performed in a variety of ways, delivering information primarily on the spatial variability of reaction kinetics. The theoretical concept that provides the link between reaction kinetics and photocurrent is that of the photocurrent transfer function. The images can be correlated with spatial variations of composition and microstructure, measured in situ and at the same time. Acknowledgements-This work was funded by the Underlying Science programme of the U.K. Atomic Energy Authority. The authors would like to acknowledge useful discussions with Dr L. M. Peter, University of Southampton.
67
12. A. R. Kucemsk, R. Pest and D. E. Williams, J. elecno&em.Sot. 138,1645 (1991). 13. A. R. Kucemak, Thesis, Southampton University (1991). 14. A. R. Kucemak, R. Peat and D. E. Williams, Electm chim. Acta 38, 71-87 (1993); and in Critical Factors in Locaked Corrosion (Edited by G. S. Frankel and R. C. Newman) Electrochemical Society, Pennington, NJ. (1992). 15. L. M: Peter, Chem. Rev. 90,753 (1990). 16. J. W. Schultz and J. Thietke, Electrochim. Acta 34, 1769 (1989). 17. V. A. Benderskii, Elektrokhimiya4,499 (1968). 18. G. C. Barker. J. electroanal.Chem. 140, l(l982). 19. V. A. Benderskii and G. I. Velichkb, j. ekkroanal. them. 140,1(1982). 20. K. Leitner and J. W. Schultxe, J. phys. Chem. 92, 181 (1988). 21. B. Miller, J. electrochem.Sot. 130, 1639 (1983).
APPENDIX A: CALCULATION OF CONTRAST IN CONTINUOU!+SCAN, DIRECT-DETECTION PHOTOCURRENT IMAGING This calculation takes the “snail” form of the photocurrent transfer function [main text, Equation (2), Fig 23 and considers effects in the image resulting from step spatial variations in each of the parameters characterixing the snail, namely go, which describes the xero frequency, or steady-state photocurrent conversion efficiency, gl, which describes the instantaneous photocurrent response, and r,, which describes the photocurrent relaxation. If a denotes the parameter with whose spatial variation we are concerned, then the function calculated is [aI aa] as described in the main text. A step variation in ts/c parameter, a, a rectangular intensity profile for the beam, and a square spot shape are assumed. The results of the c&ulations are presented in Figs 4-6. (a) Variations in the steady-state parameter, g,, From Equation (2) aY(s)
1
ag,==+ Substituting into Equation (II), using Equation (12) for the excitation function, and integrating for the two time periods before and after the beam edge reaches the parameter step, respectively, yields:
1. U. Sander, H.-H. Streblow and J. K. Dohrmann, J. phys. Gem. 85,477 (198 1). 2. M. A. Butler, J. electrochem. Sot. 131,218s (1984). 3. D. Shukla and U. Stimming, Werkst. Korros. 40, 43 (1989). 4. M. R. Koxlowski, P. S. Tyler, W. H. Smyrl and R. T. Atanasoski, Surf: Sci. 1% 505 (1988). 5. M. R. Kozlowski, W. H. Smyrl, L.~Atanasoska and R. Atanasoski Electrochtm. Acta 34.1763 (1989). 6. K. Leitner. and J. W. Schultxe,. Ber. kmknges. phys. Chem. 92,181(1988). 7. G. Vercruysse, W. Rigole and W. P. Gomes, Solar Energy Mater. 12, 157 (1985). 8. D. E. Williams, A. M. Riley and R. Peat, SPIE Proc., ScanningXmagingTechnology,Vol. 809 (1987). 9. A. R. Kucernak, R. Peat and D. E. Williams, SPIE Proc., Vol. 1028, p. 202 (1988). IO. R. Peat, A. M. Riley, D. E. Williams and L. M. Peter, .I. electrochem. Sot. 136,3352 (1989). 11. R. Peat, A. R. Kucernak, D. E. Williams and L. M. Peter, Semicond. Sci. Technol. 5,914 (1990).
!!e 1,x0
x0 (
1 2a, 2%vC&
x0
>
&,
2%_
= -
-
[ % z2
~~(1 -
e-*‘wa)(AZa)
v
e-xO~n2(ezadn2-
1).
V
These equations are characterised by one dimensionless time parameter. Y’, = rs v/(2u,,) and a dimensionless distance scale, z = x&u,,). A dimensionless contrast function can thus be written as: 1 4& C,J, x,>2u,=l
!?!Pk 1=z-
*,(I - e-‘flz) (A3a)
X,i2tJ,---[
at7s
- Y, timyelma - 1).
(A3b)
This is the function plotted in Fig 4. For the case of variations in go, a blurring of the image results as a consequence of the characteristics of the measurement cell (7&.
68
D. E. WILLLUS et al.
(b) Variations in the high-frequency eter, gl From Equation (2) T
response param-
= (-&-x-+).
(A4)
so that the dimensionless contrast function can be written as
= x[[(xY 1Y2 +
z +
Y ,)fPpl - ,yY: emrPC1 + Y’, - Y ,]
xe > 2a,
Substitution and integration as above yields
= x[bYi e-m*(e’pPz - 1) X,d2a,
=
-
- (xY,Y, + z + Yl)e-zpyl(elP1 - 1) + ecl-z)lul].
q_71e-kPl)
b=A.7,
-
+
‘TV -
This is the function plotted in Fig. 6.
T*)
x0
cc-
52’
e-(c’r2) +
T2
6454
V
x0 > 2a, = b[rie-chl(edhl - 1) - r2 e-c’rz(ed’rl - l)] d=2a,.
WW
V
equations are characterized by a dimensionless distance as before, and by two dimensionless time constants, Y, = r,v/(2u,) and Y, = r2 v/(2a,). We relabel:
These
b=t’=--vX 51 - =2
Yl Yl -y*
and then write the dimensionless contrast function:
+ Y2e-zp1 + Y’, - Y2)
= X(-Y,e-z~i x0
>
2a, = xfYle-zIyl(elPL
Wa)
- 1) - Y2e-~fla(e1P1 - l)]. (A6b)
This is the function plotted in Fig. 5. Increase, by increase of the scan speed, v of the parameter Y, causes blurring, and of Y 1streaking and shadowing, in the image. (c) Variation ofthe relaxation From Equation (2)
time parameter, T~
s (s + Til)(S
+
(‘47)
7;y
and, as above, x0 d
2a,
1
2a, vG, I,
1
+c+
>
71)e-c/‘L
-
bT:eTc’” +
r2
_
1) + d e-(c-d)lrt
7J
1
d2. V
Again, dimensionless parameters are defined
VW
T,]
(Aga)
V
2a, = bg,[br:e-c’rz(ed’T* - 1) - (br,r, + c + x e-r/yed/ri
-
c=- x0
72 -7,’ x,,
The calculation of the increase in surface temperature AT, upon transient illumination may be accomplished in a manner similar to that of Benderskii and Velichko[19]. Light from a HeCd laser, wavelength 44Onm, is assumed. The amount of light absorbed by the metal and the overlying oxide film can be calculated from the reflectivity of the metal and from the thickness and absorption coefficient of the oxide. For copper and iron, as representative cases (passive oxide thickness -Snm, oxide absorption coeffcient -2 x 10s cm-‘, metal reflectivity -OS), about 9% of the light would be absorbed in the oxide and 40-50% in the metal. In a hrst approximation, the temperature rise induced by light absorbed in the oxide is ignored. Another approximation is to treat the illumination of each image. point as being a pulse and to ignore the interaction between the image points: that is, the analysis is approximate to an experiment in which the spot is moved from one image point to the next, a pulse illumination is applied at each point, and the temperature is allowed to relax before the next point is illuminated. The local temperature rise in a continuous beam scan experiment would be larger as a consequence of heat transfer laterally across the surface. When the size of the metallic electrode is much larger than the size of the illuminating spot the temperature produced by a laser pulse incident on a metal-solution interface is
-
= bg,[(br,t, b=-
APPENDIX B: APPROXIMATE CALCULATION OF THE TEMPERATURE RISE TO BE EXPECTED DURING SCANNING OF A PASSIVE METAL SURFACE BY A FOCUSED LASER SPOT
WV
where K, c, p and a,, ci, p, are the heat conductivity, specific heat, and density of the metal and solution respectively, L the light intensity on the electrode surface, R the light reflection coefficient andf(t) describes the laser pulse timedependent shape. For a pulse duration of t,, the maximum temperature achieved would be AT, = $-$$
[I + rEy’*]-‘t;/*.
(B2)
The temperature at times after the pulse is calculated to be independent of the metal and has the form AT/AT, = (to/t)“*
t B t, .
(B3)
Substituting values for the known constants, for both iron, ATJ(L#*) is equal to and copper O.O7Kcm*W-‘s-I’*.
Photoelectrochemical The calculation can be used to give a rough estimate of the temperature excursion in two ditierent kinds of experiment. Fiiy, for a step-scan, lock-in type of experiment, in which the fight is modulated with a square intensity profile by a chopper at lOkHr, ryz = O.OO~S~~~, so that with a beamof1mWpowerfocusedtoaspot5~indiameter,to a first approximation the expected rise in temperature is 25 K. Scondly, for a continuous-scan, direct detection type of experiment, for a region on the electrode of the same sire as the iRuminating spot, several factors would influence AT, upon passage of the laser over this region. Foremost is the sixe of the spot, as the power density (and thus ATd scales with the square of the radius, next is the beam power (with which AT, varies linearly), and finally the period of 1
4
7
10
69
imaging
illumination. The period of illumination is dependent not only on the scan-line time, but also the scan-line length and, indirectly, the spot sixe; AT, varies with the square root of these factors. Given in Fig Bl is a diagram displaying AT,‘,for variations in spot sire and scan-line length. A 1 mW laser beam of wavelength 442 nm was assumed, as was a lOmsline_’ scan time (these are typical values which might be used in experiments). An alternative approach in which all the light is assumed to be absorbed in a surface tilm has been presented by Leitner and .Schultxe[20], for the specific case of an anodic oxide tihn on titanium. Results similar to the ones presented above were obtained for similar systems (10K temperature jump for a 1 mW, 6pm diameter laser spot).
13
16
10
1900 1700
Z”“” g z
1300
6 x
1100
‘0 f
p
900 700
Y 500 300 100
Fig. Bl. Calculated contour map of variation in AT, with ditferent diameters for the illuminating spot and ditTerent scan-line lengths. A 1 mW laser beam of wavelength 44Omn was assumed, as was a lOms/ line scan time.
0013-4686193$5.00+ om Pcqpmoll~Ltd.
PHOTOELECTROCHEMICAL IMAGING-PART BACKGROUND AND THEORY
I.
D. E. WILLIAMS,*A. R.J.KUCERNAKand R. PIZAT AEA Industrial Technology, Building 552, Harwell Laboratory, Didcot, Oxon OX1 1 OBA, U.K. (Received 12 Nooember 1991)
Abstract-Imaging of spatial variability of electrode processes using signals stimulated by a focused light spot is described. The signals include photovoltage and photocurrent, photoacoustic and photothermal effects, as well as the reflected light intensity. The theory of contrast in the photocurrent image is presented: methods utilizing intensity modulated light are compared with those in which the spot is scanned rapidly without intensity modulation. Effects of variation of intensity modulation frequency and spot scan speed are calculated; blurring, streaking and shadowing effects are explained. For systems which are not photoactive, the feasibility of an imaging procedure based upon the small thermal effect induced by the focused spot on the current for an electrode process is amessed. Methods of imaging utiliaing potential variations at constant (usually zero) current are compared with those involving the measurement of current variations at constant potential. Key words: photoelectrochemistry, imaging, photocurrent, photothermal, theory.
INTRODUCTION
Photoelectrochemical imaging involves the mapping of the photocurrent or photovoltage developed at an electrode-electrolyte interface as a function of the position of an illuminating spot. The electrode may be a metal, usually carrying a passivating film, or a semiconductor. The technique produces positionsensitive information by locally perturbing the surface and observing the macroscopic change in the system response. Deviations from the unperturbed (“background”) state are measured as a function of the position of perturbation: since the excitation is local&d, any deviation in response can be attributed to effects within that localized region. Responses that can be measured are illustrated in Fig. 1. Thus, spectroscopic information concerning the local region is in principle available through the photoacoustic Illupnfng
I
Optical
4,
Photothermal
Fig. 1. Signals and effects that can be used to form an image under illumination by a scanned, focused light spot.
* Present address: Department of Chemistry, University College London, 20 Gordon St, London WClH OAJ,U.K.
effect[l] and also through photocurrent spectroscopy. An additional component of the current across the interface dependent upon the local reaction dynamics could arise as a consequence of the small heating effect caused by the light, referred to here as the photothermal effect. Much the largest additional current is usually a consequence of absorption of light and production of mobile charge carriers, for example by excitation across the semiconductor band gap, or by photoemission processes. One way of thinking about the system is to view the electrolyte as a transparent Schottky contact; charge carriers excited as a consequence of absorption of light are separated by the electric fields present in the vicinity of the interface and are transported across the interface by an electrochemical reaction. This reaction can result, particularly in the case of semiconductor materials, in the etching of the electrode. It is clear that there is a number of factors which might give rise to spatial variation of the photocurrent measured in an external circuit: carriers must be generated and separated and may recombine, and image contrast can result from spatial variations in all of these. Speci6c reasons for contrast could include the presence of defects in the solid intersecting the surface, variations in the composition of the solid in the near-surface region and the presence of particular catalytic centres at the interface. Other workers in this field[2-7j have utilized for imaging what may be termed a “step-scan, lock-in” method, in which either the illuminating beam or the sample is moved from one imaging point to the next, the illumination is intensity modulated, and the signal is derived by phase-sensitive detection referenced to the intensity modulation. The image acquisition time is necessarily limited in this method by the need to rest at each image point for the time required for a sufilcient number of ilh&nation
58
D. E. WILLIAMS et al.
cycles to give satisfactory detection of the signal and noise rejection. In electrochemical systems, an image .of 512 x 512 points typically takes at least 30min to acquire using this method. A method in which the beam or specimen is scanned continuously but slowly in comparison with the illumination modulation frequency is an obvious extension. As we shall see below, the key characteristic of this method is the use of a fixed modulation frequency for the illumination We have presented in recent publications[&121 a method in which the light beam is scanned rapidly across the specimen surface and the variation in the total current flowing across the interface, as a function of the spot position, forms the image. The frame acquisition time is kept short enough that any background current remains constant over the frame, and the background current may then be rejected by a high-pass tilter: it is the variations in current within each frame which are imaged. The total (filtered) current is sampled at each image point by a frame store. In this method, the illumination intensity is not modulated and noise reduction is obtained, if necessary, by averaging of successive frames of the image. For convenience, we refer to this as the %ontinuous-scan, direct detection” method. The image acquisition time is generally much less with this method than with the “step-scan, lock-in” method and, indeed, time variations in an image of 512 x 512 points can be tracked on the scale of a few seconds, although attention to detail is required in the design of the experimental apparatus, to minimize noise pick-up yet retain an adequate response bandwidthC13, 143. New features arise in the image, particularly blurring, streaking and shadowing effects, which are a consequence of the transient illumination of each image point by the flying spot and which can be related to the local kinetics of reaction of the photogenerated charge carriers at the interface. We have recently described the use of numerical simulation to explore these effects[l2]. The “continuous-scan, direct detection” method is equivalent to illumination of each image point by a light beam with a spectrum of intensity modulation frequencies: we will develop this interpretation further in the present paper. The purpose of the present paper is to develop formally the theory of image contrast in both the “stepdirect “continuous-scan, lock-in” and scan, detection” methods, to relate the image contrast to spatial variations in basic parameters describing the photoelectrochemical processes occurring at the interface and to show how complementary information can be obtained from the two techniques. For systems which are not photoactive, an assessment is given of the feasibility of an imaging method based upon the small thermal effect of the light beam. Application to imaging the photoelectrochemistry of iron and stainless steel is presented in following papers[14]. A FREQUENCY DOMAIN APPROACH TO THE STUDY OF IMAGE CONTRAST The idea of a photocurrent transfer function was introduced by Peter[lS] to quantify the response of
a system to sinuosoidally intensity modulated light: in the case where the photocurrent is linearly dependent upon the light intensity, the photocurrent response to such illumination is also sinusoidal but varying in amplitude and relative phase as the modulation frequency is changed over ranges in which the time constants of the photocurrent relaxation are comparable with the period of the intensity modulation. That is, the stimulus is an intensity modulated light, the response is a photocurrent, and the transfer function is the frequency domain ratio of the two. Thus the response of the photocurrent density as a function of the angular frequency of the excitation (angular frequency of the intensity modulation) is given by i&e) = H(o). W),
(1)
where H(o) denotes the excitation function (the component of the light intensity varying at angular frequency o) and Y(w) denotes the photocurrent transfer function. Peter et aZ.[lS] have derived this function for a variety of assumed mechanisms of the electrochemical reactions at the interface involving the photogenerated carriers, and have developed the technique of intensity modulated photocurrent spectroscopy (IMPS) to measure it. A common form of the photocurrent transfer function is the “snail” (Fig. 2), involving a single relaxation of the photoresponse, attenuated by the equivalent RC network of the electrochemical cell: Y(o) = G,F(o)Y,(w)
= G,F(o
,
(2) where F(o) = l/(1 + jar,) denotes the attenuation introduced by the cell. Here G, denotes the carrier generation rate, go gives the steady-state photocurrent response, g1 denotes the maximum response, rl is the time constant for recombination of photogenerated carriers and r1 is the relaxation time of the measurement cell, 71 = CR,, where C denotes the electrode interface capacitance and R, the series resistance (predominantly the solution resistance) through which this capacitance is charged. The image contrast arises as a result of the spatial variation of the parameters G, , go, gr and rr. THEORY OF IMAGE CONTRAST IN THE “STEP-SCAN LOCK-IN” METHOD In this method the light intensity is modulated and the beam scanned slowly enough that the response from any given position can be extracted oia a lock-in amplifier. A sinusoidal excitation function is conveniently employed, H(w) = I,[1 + AZ,,,sin@)], where I, denotes the mean light intensity and AZ,,, the depth of modulation. The sensitivity to spatial variations of any of the parameters depends upon the frequency chosen for the measurement. This conclusion can be expressed quantitatively using the photocurrent transfer function; consideration of the “snail”, for example, indicates immediately that if the frequency were too high (o > l/r,), then the amplitude of any signal varia-
59
tlmels 0.6 0.4 0.2 Yimag
0 -0.2 -0.4
-0s;
0.2 ’
’
0.4
’
0.6 ..
’
’
0.8
1
’
1.2
Y real
Fig. 2. (a) Photocurren t response to a step in light intensity for a process with a simple 6rst order relaxation attenuated by the experimental cell, and (b) the corresponding photocurrent transfer (parameters: 7’1= Sms, 72 = 1 mq go = 0.2, g1 = 1).
tions a&oss the surface would be small. It will also be evident from the following that the resultant image contrast depends upon whether the magnitude of the resultant photocurrent or the components in phase or in quadrature with the excitation signal are used to form the image. The magnitude of the photocurrent is used in the following analysis. The contrast within a given area is conveniently detined as the relative variance of the photocurrent over the area, aZ(Z h)/(Z,,,>‘, or alternatively as the variance of the piotocurrent measured over that area, a2(Z,,). The relative variance is a convenient measure where images are scaled such that the mean image density falls in the centre of the brightness range available but is invalid for noisy images (ie ones in which the scale of random fluctuations is greater than the signal): in these cases the large photocurrent variance would imply a great degree of contrast, where in fact only noise exists. Conclusions concerning the effects on contrast of varying the excitation frequency can conveniently be arrived at by assuming that the parameters giving rise to the contrast vary independently over the surface l&l = G,Z,AZ,IJW)l
* IW4l
(3)
function
Equation (4) gives the first conclusion: that contrast (expressed as the relative variance) due to variation of the carrier generation rate, G,, is not dependent upon the intensity modulation frequency. Of course, the photocurrent does vary with the modulation frequency and at high enough frequency tends to zero as a result of the attenuation by the cell-hence contrast expressed purely as the variance of the photocurrent would vary with w. The second term in Equation (4) can be written in terms of the variation of the parameters over the surface (again assuming that these vary independently of one another)
dlr,l)2
( >
=GziY
1
D. E. WILLLUISet al.
60
Substitution of the expression for Y0 [from Equation (211 leads to the conclusion that contrast (expressed either as the variance or as the relative variance) due to the variation of the parameter g,, is maximum in the limit of low frequency and minimum (but not zero) in the limit of high frequency. Contrast due to variation of the parameter g1 is maximal in the limit of high frequency and zero in the limit of low frequency, and finally contrast due to variation of the parameter 71 is zero in the limits of both low and high frequency, and maximal when w = l/7, : the results for the photocurrent contrast in the step-scan, lock-in method are summarized below : Variation in contrast (w-a) (6a) (a + 0)
@+a) (a-*0)
(6b)
tation spectrum at any particular point depends upon the distance of the point off the centreline of the beam scan. The time-dependent photocurrent resulting from the transient illumination of a particular point on the surface could in principle be obtained by substitution into Equation (1) of the photocurrent transfer function peculiar to that point and inverse transformation. As has been pointed out before, the image intensity at any position is the total photocurrent flowing through the specimen when the beam is located at that position: this is obtained by integration of the photocurrent resulting from all previously illuminated points. A blurring of the image comes about as a result of the slow recovery of the signal following the transient illumination. The objective is to calculate the contrast variation expected as a result of the variation over the surface of some parameter of the photocurrent transfer function: we denote the parameter of interest a. At some position x0 along the line of the beam scan, the magnitude of the photocurrent (which gives the image density at that point) would be
V6Y
THEORY OF IMAGE CONTRAST IN THE “CONTINUOUS-SCAN, DIRECT DETECTION” METHOD This method of image formation contrasts with the step-scan, lock-in method in that each point receives a spectrum of excitation frequencies. The excitation function, H(w), is determined by the intensity profile within the focused light spot and the velocity of the spot across the surface. The intensity distribution for a focused spot is illustrated in Fig. 3a and from this the excitation function resulting from scanning the spot over a surface may be obtained by Fourier transformation (Fig. 3b): note that the exci-
-a,-
s x0/v
_!PL= iv,@, a) dt, 0
where i, denotes the photocurrent density (a function of the time after the illumination and of the parameter a, through the photocurrent transfer function) from an element of the surface a position vt behind the current spot position (v is the spot scan velocity) and the integration proceeds back to the edge of the specimen which is presumed to lie within the image frame. The result depends upon the shape of the spot: here, a rectangular shape of width 6y perpendicular to the scan direction has been assumed for simplicity. The photocurrent density as a function of time is the inverse transform of the product of the excitation function, H, and the photocurrent transfer function, Y: it is convenient for this purpose to use the Laplace transformation. That is
iv&, a) = Y- ‘CH(s)Y(s, a)],
v&
(s-1)
Fig. 3. (a) Intensity distribution in a focused light spot; the spot radius, a,, is taken as the radius of fhe second minimum in the intensity distribution. (b) Excitation functions of points underneath the movmg spot, evaluated by Fourier transformation along lines parallel to the scan direction at successively greater distances from the centre of the spot; v denotes the spot scan velocity.
61
Photo&etrochemieal imaging where O- 1 denotes the inverse Laplace transform with s the Laplace parameter (the substitution s =ja, gives the frequency domain functions) and G,, denotes the carrier generation rate. Now suppose that the variations over the surface of Y(s, a) as a result of variations of a can be expanded as small variations about a mean value Y(s, a) = Yds, ac) +
aY aa
6a
then '[H(s)Yc(s, a,,)] dt h = x”‘“9’VSY s 0 L
+ 6ap_l(H(s)
g)
dt.
(10)
The tirst integral is constant over the surface and gives the mean photocurrent whereas the second gives the variation about the mean caused by small variations over the surface of the parameter a. It is apparent that one can conveniently define the contrast in thii case in terms of a type of transfer function, -=$l-‘(Zf,(s)~)dr v 6y aa
(11)
which can be used to assess the transformation of spatial variations in any parameter a of the photocurrent transfer function Y into spatial variation of the photocurrent (image density): for example, a sinusoidal spatial variation of a would be transformed into a damped and phase-shifted spatial variation of I,. It will be useful now to calculate the contrast resulting from spatial variations in the parameters characterizing the photocurrent transfer function. A “snail” form of the photocurrent transfer function [Equation (2), Fig. 23 is used: the spatially variable parameters are g,, gl, and rl and zs is a characteristic of the cell, In order to derive the key features of the response in a reasonably clear way, a rectangular intensity profile for the beam is used: this makes an
analytical treatment tractable. The beam intensity is 1, and width 2u,, ie H(s) = I,s(l
- e-‘7;
T = 2+/v.
(12)
A step function spatial variation of the parameter of interest is assumed. The spatial coordinate is conveniently transformed so that the step occurs at x = 2~6, since there would be no photocurrent variation with position until the edge of the beam hit the parameter step position. It will be apparent later that the photocurrent contrast resulting from any arbitrary spatial variation of the parameter can be derived by approximating the variation as a sum of step function changes. The results of the calculation of the image contrast resulting from step changes in the parameters go, gl and rl are presented in Figs 4-6. Details of the calculations are given in Appendix A. One point to note is that there is a characteristic time scale detined by the ratio of the spot radius to the scan velocity. A streaking effect-an overshoot followed by a relaxation of the respon se-occurs in images of step changes in the parameters g, and ri and is qualitatively different in the two cases. A streaking effect does not, however, occur in images of the parameter go, only a blurring caused by the finite rate of rise of the signal to its new level following the step variation in the parameter. The spatial variation of parameters has so far been represented as a single step. Although this is a very simple function, it allows for the synthesis of more complex spatial variations in parameters by the scaling, shifting and addition of multiple steps. This is illustrated in Fig. 7 for the simple case of a square pulse spatial variation of the parameter, gl. A pulse change in the value of a parameter can conveniently be synthesized by the simple addition of two identical steps of opposite sign, shifted relative to one another in position. The image is simply the sum of the response to the two steps, that to the second being shifted and inverted with respect to that of the tirst. Two different sets of parameters were used in calculations, corresponding to a lo-fold alteration in the beam scanning velocity. This alteration of the
Fig. 4. Variation of photocurrent (image contrast function) with Position for a square spot, width &, moving with velocity v across a step in parameter go, indicated in (ah for dilkent values of the dimensionkss time constant ‘i’, =I r2 v/(2@, (b) 0.1. (c) 0.213, (d) 0.457, (e) 1.0, (0 2.13, (g) 4.57.
D. E. WILLLU.Bet 01.
62
0.6
'0
'0
12
3
4
12
3
4
5
5
6
6
7
7
Fig. 5. Variation of photocurrent (image contrast function) with position for a square spot, width 2u,, moving with velocity v across a step in parameter g, (step position at z = 1, as indicated by the arrow), for different values of the dimensionless time constants Ys = r,v/(2a,): (a) 0.1, (b) 1, (c) 10, and Y, = r,v/(2a,): (i) 0.1, (ii) 0.32, (iii) 1, (iv) 3.2, (v) 10.
scan speed produces a very interesting result for a pulse spatial variation of g, : the position dependence of the current for the slower scan speed (Fig. 7b), shows a dark streak following the feature; however, a lo-fold increase in speed (Fig. 7a), results in the virtual disappearance of this streak in the calculated image. Another parameter has of course been introduced, and that is the length of the feature being imaged. For a response which eventually decays to zero (images of variations in g1 and or here), it should be recognized that a full range of different types of response is possible, simply by varying the feature length. Thus, if the transient has peaked and returned almost to zero before the beam has traversed the full length of the feature, then when the
beam passes off the feature, there would be a dark streak following the feature which would be. almost a mirror image of the response when the beam passed on to the feature. For shorter feature length the undershoot would not be as great. This point is illustrated in Fig. 8. The variation of response has thus far been confined to the consideration of 1-D variations in one of the parameters of the photocurrent transfer function. By moving to two dimensions, and allowing variations in transfer function parameters across a plane, it is possible to generate maps of the variation of photocurrent across a surface and thus simulate the experiments performed with the microscope. The extension to two dimensions is quite simple. Each successive “line” in the picture does not interact with
Photoelectrochemical imaging
63
Fig. 6. Variation of photocurren t (image contrast function) with position for a square spot, width 2a,, moving with velocity v across a step in parameter z, (step position at z = l), for different values of the dimensionless time constants Y, = z2 v/&J: (a) 0.1, (b) 1, (c) 10, and Y, = s,v/(2a0): (i) 0.1, (ii) 0.32, (iii) 1, (iv) 3.5 (v) 10.
the surrounding lines, and may be calculated in the manner described above. Results are illustrated in Fig. 9 for the expected image contrast resulting from particular spatial variations of the parameters go, g1 and zi. The simulation assumed that the laser beam was scanning from left to right. Images were calculated for two values of the dimensionless time constant, corresponding to a N-fold variation in the scan speed. At the faster scan speed, there is a signit% cant amount of blurring, shadowing and streaking, especially for the cases involving variation of the g1 and ~~ parameters. At the higher scan rate, the image of the gi parameter variation still shows a peaked response once the beam scans onto the altered zone whereas the image of the z1 parameter shows no peak in its initial response, only a slow rise; the cor-
responding dark-transient when the beam scans off the feature is greatly extended in length. The image of the go parameter variation is slightly rounded only. Responses very similar to these have indeed been observed[&14]. Depending on which parameter is spatially variable and upon the feature size and spot scan speed, the images may be similar to those produced using the step-scan, lock-m method or may approach a spatial derivative of them. BEAM
EFFECTS
The continuous-scan, direct detection method in particular involves the use of an intense, focused and rapidly scanning spot, and it is appropriate to consider effects specifically associated with the power
D. E. WILLLUS et al.
64 0.6 0.4
(a)
0.2
[4 gIG,L, 1
0
-0.2 -0.4
__...._....._
-0.6
4
6
6
10
____ _J 12
14
Fig. 7. Image contrast for a spot moving over a square pulse spatial variation of the parameter gi, depicted as trace (c), for different values of the dimensionless time constants Y’, and Y, , corresponding to a IO-fold change in spot scan velocity: (a) Y, = IO, Y’, = 3.2;(b)YI = 1,Yy,= 0.32.
density and geometry of the focused spot. There are also heating effects, effects of scattered light producing spurious signals, and effects of nonlinearity of the response to variation of the illumination intensity. Finally, high local concentrations of charge carriers might be produced and there may be specific consequences of this, too, in the chemistry of the reactions taking place at the solution interface. A high flux of photogenerated charge carriers at the surface may cause intense dissolution or, in the case of passive metals, further film growth or alteration of film stoichiometry. Under sufficiently high illumination intensities heating effects may come into effect with the possibility of oxide thickening[16], changes in the kinetics of surface reactions or, in extreme cases, melting of the substrate. Another potential ditficulty is that because of the high current densities observed at the laser spot (milliamps to amps per square centimetre vs. microamps per sauare centimetre for conventional Dhotoelectrochemistry) the potential at the illuminated spot
might deviate significantly from the applied potential as a consequence of the resistive voltage drop in the solution and specimen in the vicinity of the spot. In this connection, as far as transport processes around the spot are concerned, the scanning focused spot may be considered to be a scanning microelectrode. In the substrate, the accumulation of a bigh local concentration of charge carriers at the solution interface would change the electric field distribution there, with consequent possible effects on transport rates and reaction kinetics. Virtually all photoeletrochemically active systems at high enough light intensity show a decrease in photocurrent conversion efficiency, the photocurrent tending to saturate with further increase of tight intensity. Exceedingly high illumination intensities (w 102z-1023 photon s-l cmP2) are possible for moderate laser beam powers (of the order of milliwatts) focused to a spot of the order of micrometres in diameter: one consequence of photocurrent saturation with increasing light intensity can then be that
0.4
0.2
II41 4.&oIo
0
-0.2 -0.4 t
0
I
4
I
6
12
16
20
24
26
Fig. 8. Image contrast for a spot moving over a rectangular pulse spatial variation in the parameter gi, which starts at the position marked (a) and proceeds to one of the following arrowed positions, giving the contrast variations shown by the curves (b)-(f). The contrast for traversal over a step variation in 8, is that represented by curve (g). Dimensionless time constants, Y, = 10, Y, = 3.2.
65
Photoelectrochemical imaging (a>
t
Fig. 9. Simulation of the image contrast expected for some 2-D variations in the parameters go (top, parallelogram), g1 (middle, triangle) and T~(bottom, circle). Both grey-scale (a, c) and surface plots (b, d) are given, with the surface plots rotated for clarity. The second set of images (c, d) corresponds to a IO-fold increase in soot scan velocitv over that for the first set fa. bl: (a. b), Y, = 1, Y, = 0.1; Y, = 10, Y, = 1 and [ = 5.’
an increase in beam diameter (lowering the intensity but keeping the total photon flux constant) results in an increase of the photocurrent. Another consequence is that disproportionately large currents in comparison with those stimulated by the main beam can arise from light scattered back onto the surface-from the coverslip or specimen mount, for example[ll, 133.
PHOTOTHERMAL EFFECTS AS THE BASIS FOR AN IMAGING PROCEDURE. Processes of absorption and scattering of the incident beam at the surface under study result in the degradation of some of the incident energy into heat. There are two sorts of photothermal effect to be distinguished. Firstly, there is the photoacoustic effect, which forms the basis of photoacoustic microscopy. It occurs when the beam is intensity modulated and refers to the conversion of the resulting thermal wave
CC,d),
into a sound wave of frequency equal to that of the intensity modulation. The intensity of the sound wave is proportional to the absorption coefftcient of the light and hence the wavelength dependence of the effect can be used to measure the absorption spectrum of the local point. Phenomena of scattering and interference of sound waves as a consequence of interaction with subsurface defects give rise to additional contrast in the image. Secondly, there is a direct thermal effect on the current for an electrode process. It is the direct thermal effect on the current for an electrode reaction which is of particular interest for the present paper. Thermal effects of a focused light source. on electrode processes at the mercury-solution interface were noted by Benderskii[lfl and Barker[18], and exploited by Bender&ii and Velichko[19] to study relaxation in the electrical double layer. Thermal effects of a focused source on an anodic oxide were noted by Schultze and co-workers[16, 203. A thermal voltammetry experiment was described by
66
D. E. WILLLUG etal.
Miller[Zl]. The formalism given by previous authors can be used to derive rough estimates of the temperature excursion in a scanning laser microscope experiment of the type envisaged here. The conclusion (Appendix B) is that with moderate beam powers (N 1 mW) and spot radius less than about lo*, temperature excursions under the spot of up to about 10K might be expected. This local temperature rise would cause an increase in the local rate of electrochemical reaction. There would also be as a consequence some enhanced convection in the adjacent solution with again a possible local increase in the rate of the electrochemical reaction at the interface. The following calculations consider only the thermal activation effect on the local electrochemistry and not any effect of local enhancement of mass transport. An assessment of the feasibility and information content of a photothermal electrochemical microscopy experiment can be made starting with the assumption of a simple form for the local current density at constant potential: i = i, exp[AEJ(RT)]
(13)
in which BE, denotes the activation energy. The effect of a temperature variation is then simply Si = (6T)i AEJ(RT’).
(14)
If a denotes the area of the focused spot, then the variation in the total current, detected and imaged in the experiment, is 61 k: (6T)i AE,a/(RT’)
(1%
and the contrast in the image is
a’(dZ)= d(i)[yy
In systems where the dark current was very small, the first term in Equation (16) would be the most important. The image would show the spatial variation of the dark current: given the parameters leading to Equation (17), for example, then a spatial variation in 61 on the scale of tens of picoamps (small but certainly measureable) would result from a spatial variation in i on the 10RAcm-2 scale, as might perhaps occur for a well passivated surface with local points of breakdown, or incipient breakdown of passivity. In systems where the dark current was large, with the image processing system coupled into the measurement through a high-pass filter so that only changes in the current were recorded, then the image would display any regions where the spatial variation of current density was comparable in magnitude with the dark current density: for example, dead spots, where the dark current density was much lower. Spatial variations in the terms BE, and 6T, since these are weighted by the value of the dark current density, should also show up. The success of the imaging would depend on the effectiveness of the frame averaging or intensity modulation procedures in diminishing the influence of random temporal variations in the current. It is evident that, if & denotes the specimen area, then 6111 N (a/d) AE, 6T/(RT2)
(18)
so that, if random temporal fluctuations in Z scaled directly with I, then the ratio of signal (photoinduced effects correlated with position) to noise (random fluctuations uncorrelated with position) would scale with the ratio a/d and the problem of averaging out the noise would become greater the larger the electrode being examined.
+ o’(dT)[$]
+
qsr)
a2@E3E [
1 2
(16)
since one could expect spatial variations of the current density and activation energy of the electrode process and also of the temperature change, as a consequence of spatial variations in the absorption coefficient of the light. Which of the terms had the dominant effect on the contrast would depend on the dark current. It is also evident that there is a tradeoff between the resolution of the method and the size of the signal to be expected, since the signal increases directly with the illuminated area for a given temperature rise. The size of the signal and the resolution would also be sensitively affected by the thermal conductivity of the specimen and electrolyte, since a larger temperature excursion would be achieved with lower thermal conductivity, but the relaxation of temperature after the beam had passed would also be slower, resulting in blurring and streaking effects of the type described above for the direct photocurrent image. The size of the effect to be expected is indicated from Equation (15): if AE, N 18kcalmol-‘, T = 300K and 6T = lOK, then AE, 6T/(RT2) - 1; then with spot radius 10 pm, (61/A) = (i/Acmd2)(3 x 10-6cm-2).
(17)
COMPARISON OF PHOTOCURRENT PHOTOVOLTAGE IMAGING
AND
In the study of electrochemical systems, the measurement of current at controlled potential is usually preferred, since it provides a measurement of rate of reaction at constant driving force. However, measurement of potential at controlled current can be preferable in practical situations, such as in the study of corrosion processes where the behaviour at open circuit (zero net current) is of importance. Another circumstance in which the measurement of potential could be preferable is that in which the rate of the electrode processes is very low-passive systems, for example, since in such systems very small current changes can give rise to rather large potential changes. In such circumstances, the photovoltage signal at constant current could be much larger than the photocurrent signal at constant potential, and hence a photovoltage image might be rather easier to obtain. The disadvantage might be that the photovoltage image is rather more difficult to interpret. At zero current, for example, the electrode potential would often be a mixed potential, determined by the balance of anodic and cathodic processes which are not necessarily the reverse of one another. A photopotential change would result from
Photeelectrochemicalimaging the stimulation of an additional component of the current which could be either anodic or cathodican anodic photocurrent would result in a cathodic shift in potential and vice versa. The magnitude of the potential change, however, would depend not only on the magnitude of the photostimulated effect, but also on the dependence of this effect on the potential (which changes as a result of the stimulation) and upon the potential dependence of the anodic and cathodic components of the dark current. The foregoing arguments relate the steady-state photovoltage change. The transient photovoltage change is similarly determined by the balance between transient photocurrent, transient dark current and capacitive charging current. The resulting photovoltage transient could be quite complicated, and the resultant streaking and shadowing effects in the image difficult to interpret other than qualitatively. Photovoltage imaging of the corrosion inhibition of copper by benzotriazole and of subsequent inhibition breakdown has been reported [S, 13}, and illustrates both the possibilities and difIlcult.ies in interpretation with this method.
CONCLUSION Imaging of electrochemical systems using signals stimulated by a focused, scanned light spot can be performed in a variety of ways, delivering information primarily on the spatial variability of reaction kinetics. The theoretical concept that provides the link between reaction kinetics and photocurrent is that of the photocurrent transfer function. The images can be correlated with spatial variations of composition and microstructure, measured in situ and at the same time. Acknowledgements-This work was funded by the Underlying Science programme of the U.K. Atomic Energy Authority. The authors would like to acknowledge useful discussions with Dr L. M. Peter, University of Southampton.
67
12. A. R. Kucemsk, R. Pest and D. E. Williams, J. elecno&em.Sot. 138,1645 (1991). 13. A. R. Kucemak, Thesis, Southampton University (1991). 14. A. R. Kucemak, R. Peat and D. E. Williams, Electm chim. Acta 38, 71-87 (1993); and in Critical Factors in Locaked Corrosion (Edited by G. S. Frankel and R. C. Newman) Electrochemical Society, Pennington, NJ. (1992). 15. L. M: Peter, Chem. Rev. 90,753 (1990). 16. J. W. Schultz and J. Thietke, Electrochim. Acta 34, 1769 (1989). 17. V. A. Benderskii, Elektrokhimiya4,499 (1968). 18. G. C. Barker. J. electroanal.Chem. 140, l(l982). 19. V. A. Benderskii and G. I. Velichkb, j. ekkroanal. them. 140,1(1982). 20. K. Leitner and J. W. Schultxe, J. phys. Chem. 92, 181 (1988). 21. B. Miller, J. electrochem.Sot. 130, 1639 (1983).
APPENDIX A: CALCULATION OF CONTRAST IN CONTINUOU!+SCAN, DIRECT-DETECTION PHOTOCURRENT IMAGING This calculation takes the “snail” form of the photocurrent transfer function [main text, Equation (2), Fig 23 and considers effects in the image resulting from step spatial variations in each of the parameters characterixing the snail, namely go, which describes the xero frequency, or steady-state photocurrent conversion efficiency, gl, which describes the instantaneous photocurrent response, and r,, which describes the photocurrent relaxation. If a denotes the parameter with whose spatial variation we are concerned, then the function calculated is [aI aa] as described in the main text. A step variation in ts/c parameter, a, a rectangular intensity profile for the beam, and a square spot shape are assumed. The results of the c&ulations are presented in Figs 4-6. (a) Variations in the steady-state parameter, g,, From Equation (2) aY(s)
1
ag,==+ Substituting into Equation (II), using Equation (12) for the excitation function, and integrating for the two time periods before and after the beam edge reaches the parameter step, respectively, yields:
1. U. Sander, H.-H. Streblow and J. K. Dohrmann, J. phys. Gem. 85,477 (198 1). 2. M. A. Butler, J. electrochem. Sot. 131,218s (1984). 3. D. Shukla and U. Stimming, Werkst. Korros. 40, 43 (1989). 4. M. R. Koxlowski, P. S. Tyler, W. H. Smyrl and R. T. Atanasoski, Surf: Sci. 1% 505 (1988). 5. M. R. Kozlowski, W. H. Smyrl, L.~Atanasoska and R. Atanasoski Electrochtm. Acta 34.1763 (1989). 6. K. Leitner. and J. W. Schultxe,. Ber. kmknges. phys. Chem. 92,181(1988). 7. G. Vercruysse, W. Rigole and W. P. Gomes, Solar Energy Mater. 12, 157 (1985). 8. D. E. Williams, A. M. Riley and R. Peat, SPIE Proc., ScanningXmagingTechnology,Vol. 809 (1987). 9. A. R. Kucernak, R. Peat and D. E. Williams, SPIE Proc., Vol. 1028, p. 202 (1988). IO. R. Peat, A. M. Riley, D. E. Williams and L. M. Peter, .I. electrochem. Sot. 136,3352 (1989). 11. R. Peat, A. R. Kucernak, D. E. Williams and L. M. Peter, Semicond. Sci. Technol. 5,914 (1990).
!!e 1,x0
x0 (
1 2a, 2%vC&
x0
>
&,
2%_
= -
-
[ % z2
~~(1 -
e-*‘wa)(AZa)
v
e-xO~n2(ezadn2-
1).
V
These equations are characterised by one dimensionless time parameter. Y’, = rs v/(2u,,) and a dimensionless distance scale, z = x&u,,). A dimensionless contrast function can thus be written as: 1 4& C,J, x,>2u,=l
!?!Pk 1=z-
*,(I - e-‘flz) (A3a)
X,i2tJ,---[
at7s
- Y, timyelma - 1).
(A3b)
This is the function plotted in Fig 4. For the case of variations in go, a blurring of the image results as a consequence of the characteristics of the measurement cell (7&.
68
D. E. WILLLUS et al.
(b) Variations in the high-frequency eter, gl From Equation (2) T
response param-
= (-&-x-+).
(A4)
so that the dimensionless contrast function can be written as
= x[[(xY 1Y2 +
z +
Y ,)fPpl - ,yY: emrPC1 + Y’, - Y ,]
xe > 2a,
Substitution and integration as above yields
= x[bYi e-m*(e’pPz - 1) X,d2a,
=
-
- (xY,Y, + z + Yl)e-zpyl(elP1 - 1) + ecl-z)lul].
q_71e-kPl)
b=A.7,
-
+
‘TV -
This is the function plotted in Fig. 6.
T*)
x0
cc-
52’
e-(c’r2) +
T2
6454
V
x0 > 2a, = b[rie-chl(edhl - 1) - r2 e-c’rz(ed’rl - l)] d=2a,.
WW
V
equations are characterized by a dimensionless distance as before, and by two dimensionless time constants, Y, = r,v/(2u,) and Y, = r2 v/(2a,). We relabel:
These
b=t’=--vX 51 - =2
Yl Yl -y*
and then write the dimensionless contrast function:
+ Y2e-zp1 + Y’, - Y2)
= X(-Y,e-z~i x0
>
2a, = xfYle-zIyl(elPL
Wa)
- 1) - Y2e-~fla(e1P1 - l)]. (A6b)
This is the function plotted in Fig. 5. Increase, by increase of the scan speed, v of the parameter Y, causes blurring, and of Y 1streaking and shadowing, in the image. (c) Variation ofthe relaxation From Equation (2)
time parameter, T~
s (s + Til)(S
+
(‘47)
7;y
and, as above, x0 d
2a,
1
2a, vG, I,
1
+c+
>
71)e-c/‘L
-
bT:eTc’” +
r2
_
1) + d e-(c-d)lrt
7J
1
d2. V
Again, dimensionless parameters are defined
VW
T,]
(Aga)
V
2a, = bg,[br:e-c’rz(ed’T* - 1) - (br,r, + c + x e-r/yed/ri
-
c=- x0
72 -7,’ x,,
The calculation of the increase in surface temperature AT, upon transient illumination may be accomplished in a manner similar to that of Benderskii and Velichko[19]. Light from a HeCd laser, wavelength 44Onm, is assumed. The amount of light absorbed by the metal and the overlying oxide film can be calculated from the reflectivity of the metal and from the thickness and absorption coefficient of the oxide. For copper and iron, as representative cases (passive oxide thickness -Snm, oxide absorption coeffcient -2 x 10s cm-‘, metal reflectivity -OS), about 9% of the light would be absorbed in the oxide and 40-50% in the metal. In a hrst approximation, the temperature rise induced by light absorbed in the oxide is ignored. Another approximation is to treat the illumination of each image. point as being a pulse and to ignore the interaction between the image points: that is, the analysis is approximate to an experiment in which the spot is moved from one image point to the next, a pulse illumination is applied at each point, and the temperature is allowed to relax before the next point is illuminated. The local temperature rise in a continuous beam scan experiment would be larger as a consequence of heat transfer laterally across the surface. When the size of the metallic electrode is much larger than the size of the illuminating spot the temperature produced by a laser pulse incident on a metal-solution interface is
-
= bg,[(br,t, b=-
APPENDIX B: APPROXIMATE CALCULATION OF THE TEMPERATURE RISE TO BE EXPECTED DURING SCANNING OF A PASSIVE METAL SURFACE BY A FOCUSED LASER SPOT
WV
where K, c, p and a,, ci, p, are the heat conductivity, specific heat, and density of the metal and solution respectively, L the light intensity on the electrode surface, R the light reflection coefficient andf(t) describes the laser pulse timedependent shape. For a pulse duration of t,, the maximum temperature achieved would be AT, = $-$$
[I + rEy’*]-‘t;/*.
(B2)
The temperature at times after the pulse is calculated to be independent of the metal and has the form AT/AT, = (to/t)“*
t B t, .
(B3)
Substituting values for the known constants, for both iron, ATJ(L#*) is equal to and copper O.O7Kcm*W-‘s-I’*.
Photoelectrochemical The calculation can be used to give a rough estimate of the temperature excursion in two ditierent kinds of experiment. Fiiy, for a step-scan, lock-in type of experiment, in which the fight is modulated with a square intensity profile by a chopper at lOkHr, ryz = O.OO~S~~~, so that with a beamof1mWpowerfocusedtoaspot5~indiameter,to a first approximation the expected rise in temperature is 25 K. Scondly, for a continuous-scan, direct detection type of experiment, for a region on the electrode of the same sire as the iRuminating spot, several factors would influence AT, upon passage of the laser over this region. Foremost is the sixe of the spot, as the power density (and thus ATd scales with the square of the radius, next is the beam power (with which AT, varies linearly), and finally the period of 1
4
7
10
69
imaging
illumination. The period of illumination is dependent not only on the scan-line time, but also the scan-line length and, indirectly, the spot sixe; AT, varies with the square root of these factors. Given in Fig Bl is a diagram displaying AT,‘,for variations in spot sire and scan-line length. A 1 mW laser beam of wavelength 442 nm was assumed, as was a lOmsline_’ scan time (these are typical values which might be used in experiments). An alternative approach in which all the light is assumed to be absorbed in a surface tilm has been presented by Leitner and .Schultxe[20], for the specific case of an anodic oxide tihn on titanium. Results similar to the ones presented above were obtained for similar systems (10K temperature jump for a 1 mW, 6pm diameter laser spot).
13
16
10
1900 1700
Z”“” g z
1300
6 x
1100
‘0 f
p
900 700
Y 500 300 100
Fig. Bl. Calculated contour map of variation in AT, with ditferent diameters for the illuminating spot and ditTerent scan-line lengths. A 1 mW laser beam of wavelength 44Omn was assumed, as was a lOms/ line scan time.