J. Phys. Chem. 1993,97, 6696-6709
6696
Surface Diffusion of Silver at the Ag( 11l)/Liquid-Water Interface from Electrocrystallization Measurements John D. Porter' and Timothy 0. Robinson+ Department of Chemistry, University of California. Berkeley, and Lawrence Berkeley Laboratory, Berkeley, California 94720 Received: March 18, 1993
We observed rhythmic Ag cluster nucleation that was coupled to surface mass transport, and we used this phenomenon to measure Dsfor silver adatoms. Low coverages (8 = 0.002) of Ag(ad) were generated on a surface that was initially adatom-free by reduction of Ag+(aq) at sparse, redox-active sites. The adatoms diffused from the generation sites across step-free Ag( 111) surfaces of known geometry. Diffusing adatoms were trapped at redox-silent sites that were potent catalysts for the nucleation of 2D clusters. The f m t supercritical cluster to form at a nucleation site swept rapidly over the Ag( 1 11) surface by catalytic reduction of Ag+(aq) at the reactive step edge. The expansion of the monatomic step from the nucleation site yielded a brief charge pulse, a fresh, adatom-free surface, and renewal of the cycle. Regular trains of charge pulses were observed in steady-state experiments. Analysis yielded the locations of the nucleation sites and the following values of Ds:(1.4 f 0.2) X 10-6 cmz/s at 20.0 O C , (3.3 f 1.0) X 10-6 cm2/s at 35.0 O C , and (4.0 f 0.9) X 10-6 cmz/s at 50.0 OC (f2u). These are the same as values predicted by molecular dynamics simulation for Ag(ad) diffusion on Ag( 111) in ultrahigh vacuum, implying that there is little net effect of the interfacial water on Ag(ad) diffusion rates. The mean residence time for Ag(ad) species at the Ag( 11l)/water interface is >4 s at these temperatures, corresponding to a standard exchange rate P < 40 nA/cmZ between Ag(ad) and Ag+(aq). In extreme contrast, we measure P = 1300 A/cmZ, more than 10 orders of magnitude larger, for the same exchange reaction at the edge of an atomic step on Ag( 111).
reflection electron mi~roscopy4~ to infer the existence of surface self-diffusion and to calculate diffusion rates. The self-diffusion of metal atoms on surfaces is important in We are particularly interested in the following issues related crystal growth and in sintering, faceting, recry~tallization,l-~ tometal/liquid interfaces: How does coordinationof theadsorbed surface reconstruction, and catalyst ageing? Self-diffusion is metal atom and the metal surface by the solvent change the also of fundamental importance because it is one of the simplest potential energy surface for the system? Is there a significant dynamic processes that can occur at a metal surface.s9 It can change in frequency factors and adatom trajectories compared provide information about bonding sites at the surface and the to uhv? Does solvent adsorption on the surface lead to steric form of the potential energy surface experienced by adsorbates. hindrance of surface diffusion, and is displacement of solvent Self-diffusionhas also been studied extensively through molecular necessary for adatom diffusion? Does local structuring49in the dynamicsI0-l2(MD) and Monte C a r l ~ I ~ (MC) - ~ ~simulations. solvent affect surface diffusion rates? Are surface diffusion Materials that are well-suited to experiment in ultrahigh constants for metal adatoms higher or lower than in the absence vacuum (uhv) have been studied the most. Several direct, in situ of solvent? Our results provide some insight into several of these experimental methods have been developed to study the physics of surfacediffusionin uhv. For example,field-ion micro~copy,2~-~~ fundamental issues, and they indicate that the silver/liquid-water system is well-suited to this type of investigation. field-emission microscopy,2s28 high-resolution transmission elecIn section I1 we collect and review published thermodynamic tron m i c r ~ s c o p y , ~laser-based ~-~~ desorption method^,^^-'^ moand kinetic data for the silver/water interface that are relevant lecular-beam ~ c a t t e r i n gand , ~ ~dynamic optical second-harmonic to the present investigation, although the silver/water system has generati~n~',~~ have all been used to study the motion of adsorbates not been applied directly to the study of self-diffusion in the past. on metal and other surfaces. Indirect, ex situ and sampling From these data we calculate adatom residence times and selfmethods have also been used to study self-diffusion, and they diffusion coefficients to compare with our measured values. In include radiotracer t r a n ~ p o r t , B dynamic . ~ ~ * ~ sintering and surface section 111, we outline the simple time-of-flight experimental profiling,*scanning tunneling microscopy (STM),4I4* reflection method that we used in the present study. We also discuss electron micro~copy,~~ and dynamic work function measurediagnostic criteria that can be. used to select appropriate menk46 experimental data for analysis by the time-of-flight method. In Immersing a metal surface in a reactive liquid such as water section IV, we present the experimental methods that were used introduces several new and interesting issues related to metal to prepare and characterize our samples. Our methods differ in atom self-diffusion, including whether or not the process even some important ways from those that have been reported takes place. Metal/liquid interfacesare of fundamentalrelevance previously in the literature. A description of the experimental to many important interfacial phenomena, including electroresults is presented in section V. In section VI, we discuss the crystallization, corrosion, and heterogeneous catalysis, but study results of kinetic measurements and derive surface diffusion of self-diffusionin thesesystemsisdifficult. Todate, experimental coefficients for silver adatoms from our results. We compare our studies at the metal/liquid interface have used indirect methods values of Ds with the estimates calculated from previous such as dynamic, macroscopic surface-roughness measureexperimentsand with values obtained from models of self-diffusion m e n t ~ , ~dynamic ' ~ ~ ~ STM surface pr~filing,"~~~ and ex situ in uhv. In section VII, we discuss briefly the implications of our results to important, previously unresolved issues in metal Present address: Director, College of Chemistry Graphics Laboratory, University of California, Berkeley, Berkeley, CA 94720. electrocrystallization. Section VI11 summarizes our findings.
I. Introduction
0022-365419312097-6696$04.00/0 0 1993 American Chemical Society
The Journal of Physical Chemistry, Vol. 97, No. 25, 1993 6697
Diffusion of Ag at the Ag( 11l)/Water Interface
II. Silver ElectrocrystallizPtion Efficiency, Detection Limits,and Reaction Rates. The process of silver electrocrystallization takes place in aqueous solution through the following net half-cell reaction: Ag+(aq)
+ e- e Ago(surface);
EO(l)= 0.7991 V (SHE,298 K) (1) The equilibrium potential for reaction 1 under nonstandard conditions, EO(l), can be calculated from Eo( 1) using the Nernst equation. The net extent of reaction 1 can be monitored in the steady state using Faraday's law of electrolysis, and the instantaneous net rate of the reaction can be monitored using the differential form of the relation:
(3) where N is the number of moles of silver deposited, Q is the charge passed (incoulombs), &' is the fractional current efficiency, ( ) denotes temporal mean value, z = 1 is the charge number of the standard reduction half-cell reaction, F is the Faraday constant (96485.31 C/mol), Z is current (in amperes), and t is time (in seconds). There are very few reductions that can take place at significant rates in parallel with reaction 1 because EO(l)is so high. This includes the reduction of oxygen and the reduction of trace metalion impurities. Yet Eo( 1) is not so positive that the oxidation of water will occur when E = Eo( 1). Hence, the fractional current efficiency for reaction 1 can approach 1 exactly when the reaction is performed at low net rates at potentials near Eo( 1). For these reasons, reaction 1 was the de facto standard process used to define the Faraday constant for many ~ e a r s . 5The ~ kinetics of reaction 1 can be studied over a very wide dynamic range of current densities, Le., net interfacial reaction rates. Steady-state experimental conditions were defined here by applying a constant potential E = EO(1) to the silver electrode. This minimized the non-Faradaic capacitive currents that result from dynamic redistribution of charge near the interface. Both physical and chemical interferences were reduced to nearly insignificant levels by this strategy, and Q H 1 was a very good approximation, as we demonstrate in section V. Currents were measured to a precision of f 2 4 pA at sampling rates up to 10 kHz. This correspondsto an uncertainty of f2.4 fC in measured chargeper sampled point, or f2.5 X mol or f l 5 000 atoms of silver. Hence, electrochemical kinetic measurements are capable of extremely high sensitivity and precision in this system under optimum conditions. For example, the progress of a single atomic step over the surface of the silver electrode was easily detected and studied in detail, as shown in section V. However, it is also clear from these technological bounds that our electrochemical kinetic measurements were not capable of detecting the formation or removal of individual adsorbed silver atoms. Also, because no net rearrangement of charge occurs during the process, our electrochemical kinetic measurements could not be used to measure directly the motion of adatoms in the plane of the metal surface. In section I11 we outline timeof-flight methods that get around these limitations. Reaction 1 has an extremely high standard exchange rate at the surface of polycrystallinesilver electrodes. The value reported by Gerischer and Tischer,sl P = 24 f 5 A/cmZ, is typical of high-quality data for this reaction. By definition,s' P is the value of the forward and reverseexchangerates at standard temperature, with pure materials in their standard states and solution species at 1 mol/L concentration. This makes reaction 1 one of the fastest, most electrochemically-reversibleinterfacial redox pro-
cessea known,when it is carried out at polycrystallinesurfacc3.5Z-54 The extremely high rate of exchange for reaction 1 also makes it a popular choice for reference electrode systems in electrochemical studies. Capillary-Grown Ag Surfaces. In 1966, in a landmark experiment in the field of electrocrystallization, Budevski and mworkers r e p ~ r t e dthe ~ ~formation .~~ of ideal, 'dislocation-free", bulk-terminated Ag( 100) surfaces. Currents for reaction 1 measured near &(l) in concentrated aqueous silver nitrate solutions were 3 or 4 orders of magnitude less at the ysingular"57 surfaces than at polycrystalline surfaces. These experiments established that the thermodynamics and the kinetics of reaction 1 are extremely sensitiveto local environment at the silver/water interface. The sharply-reduced rate of reaction 1 at singular faces of s i l ~ e r implies ~ ~ . ~ that ~ polycrystalline silver surfaces contain sites with extremely high rate constants for reaction 1, and these sites are absent from the surfaces produced by the capillary-growth methodss84 used by Budevski et al. H e m , the apparent simplicityof eqs 1-3 belies the complexityof reaction 1 at the microscopic level. Their experiments made it clear that macroscopic experimental observables measured for this system are weighted mean values. The integration for each observable is performed over a time-dependent ensemble of surface sites of widely-varying intrinsic reactivity. Single-crystal surfaces containing a tractable number of different types of surface site can be prepared using the capillarygrowth technique Budevski et al. introduced to this field. This reduction in complexity permits macroscopic observables for the silver/liquid-water system to be interpretedin terms of microscopic models. We have taken advantage of this approach in the present study. Budevski and -workers have published several summaries and reviews of their findings61d4 since the first report of 'dislocation-free" silver/water interfaces. They used Nomarski optical microscopy, laser interferometry, and electrochemical kinetic measurements to establish that the capillary-grown surfaces were free of step bunches. Recently, they used in situ and exsitu STM t0confirm~~9~~ that the capillary-grown surfaces are nearly free of atomic steps under equilibrium conditions at potentials near EO(1). Adatom Coverages .nd Exchange Rates. Vitanov et 01. measured67dgthe exchange current density for the formation of silver adatoms on (100) silver surfaces at 45 OC in 6 mol/L silver nitrate to be Z0,~(100)= 66 mA/cm2. Allowing for Ag+(aq) concentration differences, this corresponds to an exchange rate that is about 3 orders of magnitude less than the mean value of P for polycrystalline silver reported by Gerischer and T i s ~ h e r . ~ ~ The equilibriumcoverage of silver adatoms has been estimated by Vitanov et al.69 to be in the range I'o(l00) = (2-5) X 10-lz mol/cm2 for Ag(100) surfaces at 45 OC in aqueous 6 mol/L silver nitrate at potentials near Eo(1). The Ag+(aq)/Ag(ad) exchange rate on Ag( 100) surfaces, lor( 100). can be combined with the equilibrium coverage of adatoms, I'o(lOo), to estimate a mean residence time for adatoms on the (100) metal surface under their experimental conditions: (4) where TOis the mean residence time (in seconds). We calculate ~o(100)= 6 ps from the estimated values of I'o(100) and Z0,~(100)given above for silver in 6 mol/L silver nitrate at 45 OC. The adatom residence time estimated from the data of Vitanov et al. is much shorter than the sampling interval used in the present experiments. Their results suggest that the dynamics of
6698 The Journal of Physical Chemistry, Vol. 97, No. 25, 1993 surface diffusion at the Ag( 100)/water interface should not be measurable in real time under the experimental conditions used here. Adatom Surface Diffmion. Bostanov et al. measured the shape of growing, bunchedatomicsteps at theAg( 100)/water interface using an electrochemical method?O They calculated a mean spacing of 16 nm between atomic steps from the inferred geometriesof themacrosteps. They assumed that thestepspacing was controlled by the surface-diffusion penetration length, &(loo), defined as the root-mean-squared displacement of an adatom during its finite residence time on the surface. They calculate &( 100) = 8 nm under their experimental conditions of 6 mol/L silver nitrate and 45 OC. As a rough approximation, we assume that the adatom concentration is zero on the (100) terraces between the parallel, advancing steps in their experiment. Further, we assume that the adatoms simply undergo a random walk in one dimension to the step edge, with motion parallel to the step being irrelevant to their measurement. An upper bound for the self-diffusion coefficient of silver adatoms at the Ag(100)/water interface can becalculated from their data using the Einsteinrelation,as follows: (C2) = 2rDsr
(5) where (C2) is the mean-squared displacement of the adatom in the direction normal to the step (in cm2) at time t (in seconds), tis the dimensionalityof the random walk, and Dsis the diffusion coefficient of the adatom (in cm2/s). Substituting the value of &( 100) reported by Bostanov et ai. for t and the value of TO(100) calculated from the data of Vitanov et al. for t in eq 5, then we calculate Ds(lOO) < 6 X 1 W cmz/s. If the adatoms in their experiments were migrating in a finite surface concentration gradient, then Dscould be signifieantly lower than this value. 2DClmter Nucleation and Step Growth. Budevski and coworkers have also used capillary-grown silver surfaces to study clustering of adatoms, classical two-dimensional nucleation, and the growth of monatomicsteps. They describes phenomenological "critical" threshold potential, negative of EO(l), for the homogeneous nucleation of a critical 2D cluster of silver atoms on "singular" surfaces. Nucleation rates are reported to be vanishinglysmallat potentialsmorepositivethan thecritical potential, consistent with a model of classical nucleation of adatom clusters. They report values for the critical potential that are as little as 5-8 mV negative of Eo(1) to as much as 12 mV negative of Eo(1)?14 From these and other measurements, Budevski et al. estimate64 a mean specific edge energy for a monatomic step to be about 2 X lo-" J/cm in 6 mol/L silver nitrate at 45 OC.The nucleation preexponential factors they calculate from their data lie between 1010 and 1015cm-2 s-1. Using classical nucleation theory? we estimate from their data that rates of homogeneous, steady-state nucleation in excess of about 106cm-2s-I at 45 "Crequireoverpotentials q = (E-&( 1)) 5-25 mV at singular silver/water interfaces. According to the Nernst equation,this driving force correspondsto a criticaladatom surface coverage I'*(q) of about 2.6 times the equilibrium value at Eo(l), i.e., r*(-25 mV) = (2.6)I'0, or to an adatom supersaturation of about 160%. Since the condition E = EO(1) defines the adatom supersaturation for the singular surface to be approximately zero, no homogeneous 2D-clusternucleation should take place at all near EO(1). The observation of 2D-cluster nucleation at potentials near EO(1) implies nucleation at active sites. We assume that active sites bind and stabilize subcritical 2D clusters, effectively increasing the local value of EO(1 ) >~ EO(1) at the active site. We also assume that nucleation rates at an active site scale with locally-defined supersaturation according to classical nucleation theory. In other words, if we observe nucleation rates around 106 s-I for applied potentials very near EO(l), we assume ro= (2.6)I'*~,i.e.,B* = (I'*A/h) = 0.38,where r*Ais theequilibrium
Porter and Robinson adatom coverage at the active site at its appropriate, local EO(1 ) ~ro . is the equilibrium coverage at the macr08copic EO(l), and 8* is the critical surface coverage ratio at the active site. The stochastic nature of 2D-cluster nucleation has been characterized in severalstudies by Budevski and co-workers using capillary-grown silver electrodes. They have interpreted their results in terms of classical models of homogeneous nucleation on singular surfaces. In agreement with classical models, the waiting times between the appearanceof supercritical nuclei have been reported to be random according to Poisson s t a t i s t i ~ a . ~ ' - ~ ~ Locations of nucleation sites on the singular crystal faces have also been reported to be random and ~niform.72.~5 However, experimental results obtained by Budevski and coworkers have also been interpreted in terms of nucleation at active ~ i t e s , 7and ~ , ~this ~ is the interpretation that is most consistent with the resultsof the present study. Active sites appear to beespecially prevalent at the boundary of the crystal where it meets the glass maps75.78of nucleation sites reveal ~ a p i l l a r y . ~Published ~.~~ evidence of the decoration of persistent line defects75 and passivated screw dislocation^.^^ Some defects that are active toward nucleation appear to propagate with the crystal as it Thesignatureof this typeofdefect is a strongcorrelation in the ( x y ) coordinates of sequential nucleation sites, persisting over the deposition of many monolayer^?^.^^^^^ At potentials near Eo(l), monatomic steps have been found to spread across the silver/water interface at a constant rate of u (in cm/s), that is proportional to overpotential, q. Potential dependences of D have been measured70to be (&/ &) = -( 1.00 i 0.05) cm s-l V-I for monatomic steps on both Ag( 100) and Ag( 111) surfaces under steady-state conditions, or about -1.9 cm s-1 V-l on "activated" surfaces.70From thesevalues of u, Vitanov et al. estimate69the exchange current density for reaction 1 at an atomic step to be = 200 A/cm2 in 6 mol/L Ag+(aq) at 45 OC. Allowing for concentration differences, this is about twice the mean value reported by Gerischer and Tischer for P at polycrystalline surfaces. Step expansion is selfperpetuating in the steady state in the presence of one or more screw dislocation^.^^ Step propagation is facile for all potentials negative of EO(l), even those more positive than the critical potential for 2D nucleus formation on the singular crystal faces.55*56 This important fact hasbeenused toprove that therearenoactiveatomicstepsexposed to the electrolytesolution on the electrochemically-inert,capillarygrown interfaces. The geometry of the perimeter of an advancing monatomic step has been debated, but it is generally accepted to be either a p p r o x i m a t e l y c i r ~ ~ l a r ~a ~polyhedralasaembly ~ ~ ~ ~ ~ ~ ~ ~ o r of circular segment^.^^^*^^*^ The expansionof an atomic step across a surface provides information about the geometry of the boundary of the surface, through the time dependence of the length of the moving step. Under favorable conditions, growth transients for monatomic steps can be deconvolutedto extract geometricinformation about the shape of the surface and the location of the nucleation site.55,56,70.75,76,8~2
III. Time-of-F'hght Metbod for Studying Self-Dlffision The time-of-flight method is based on an analysis of the timedependent current passing across the silver/water interface, under conditions of steady-state deposition of silver at constant applied potential. The most useful situation for the timesf-flight method is the nucleation and subsequent growth of a monatomic step over a crystal facet. The surface area swept out by the expanding monatomic step can be calculated from eq 2, by integrating the current transient to obtain the charge, Q. Simplediagnosticteats are used to recognize when it is appropriate to apply the timeof-flight method. The tests rely upon the use of IQI as a direct measure of the area of the facet and the use of as a measure of the length of the facet.
Diffusion of Ag at the Ag(lll)/Water Interface
Necessary ConditiosS. A time-of-flight method for measuring surface diffusion requires the following: (a) a source of adatoms located at a known position on the surface, (b) a detator of adatoms located at a known distance from the source, and (c) a method of determining the instant that adatoms are released from the source and the instant that they arrive at the detector. At present, it is not possible to engineer these features deliberately into the silver/water system, but we have found thatit is possible for a naturally-evolving system to contain the necessary elements. By definition, a potent sourceof adatoms is a site with a locallyhigh value of exchange current density for reaction 1, P,compared to the value of P on the rest of the surface. We have found that facet edges are potent sources of adatoms that are well-suited to time-of-flight experiments. Facet edges have the additional advantages that they are very distinctivegeometric features, they are relatively straightforward to locate and characterize, and, typically, they form at least one of the boundaries of activelygrowing crystal faces. A recent studygsof shape oscillations during silver growth on Ag( 111) in uhv revealed facet edges to be potent sources of adatoms when growing atomic steps were present on the Ag( 111) facet. Although electrochemicaltechniquescannot detect the motion of adatoms parallel to the silver/water interface, they can be used to measure reactions at the detector site. The time-of-flight method uses silver electrocrystallization as a chemical amplifier to detect the presence of adatom surface diffusion. The key is to perform experimentsunder circumstanceswhere the nucleation of a supercritical 2D cluster is conditional upon the diffusion of adatoms from the potent source. Therefore, the best detectors for the time-of-flight method are potent nucleation sites where supercritical clusters can form at a (surface) diffusion-limited rate. Although we can determine the location of these sites at the silver/water interface, their structures and the mechanism of their action remain unknown. After nucleation of a supercritical cluster, step propagation is facile and rapid at all potentials negative of Eo( 1). Step growth results in a measurable electrochemical signal that is used to mark the arrival of adatoms at the nucleation site. The new surface that is formed in the wake of the propagating step is initially free of adatoms since the expanding step is a perfect sink for those species. The growing step expands approximately isotropicallyfrom its point of origin, and, in particular, one segment travels back toward the adatom source. Along its way, She step engulfs adatoms that are still diffusing from the source. The arrival of the expanding monatomic step at the adatom source effectively resets the diffusion clock. Hence, step growth also defines the instant that diffusion begins from the adatom source for the next sequence of diffusion, nucleation, and growth. Expansion of a monatomic step from a nucleation site is relentless, and the step continues across the available surface until expansion is halted by the perimeter of the surface at every point. The time-dependent expansion of this reaction front can be deconvoluted to yield important geometric parameters. Specifically, the stepgrowth events are analyzed using geometric models to determine the location of the nucleation site and the size and geometry of the surface into which the step expands. This geometric information is then used to determine the distance between the adatom source and the nucleation site. TiOf-Flight Model. As a first approximation, weconsidered generation of adatoms at a one-dimensional line source on the surface, followed by surface diffusion and capture at a point nucleation site. This is a crude model for nucleation at an active site on a faceted single crystal. The model describes onedimensional diffusive flux in a surface concentration gradient, and it can be solved by applying Fick‘s second law under appropriate boundary conditions. We make the following simplifyingassumptions: (a) The rates of generation,trapping, and removalof adatoms on the free surface
The Journal of Physical Chemistry, Vol. 97, No. 25, 1993 6699 are insignificant. (b) The nucleation site is located a distance d from the adatom source within the onedimensional diffusion zone, Le., between the generation site at one end ( x = 0) and a reflective end farthest from the adatom source (x = h). (c) The presence of the nucleation site does not affect adatom concentrations or transport for concentrations below the critical value. (d) Nucleation occurs in a probabilistic fashion after the local adatom concentration exceeds the critical value for the active nucleation site. The boundary conditionsfor the solution arc (a) the local surface concentration of adatoms at the source remains at the equilibrium value, (b) the initial concentration of adatoms is zero in the diffusion zone, and (c) the adatom flux through the end of the diffusion zone is zero. Expanding the terms of the solution in appropriate Maclaurin series, truncating, and simplifyingyields the approximate results:
where h is the length of the one-dimensional diffusion zone (in centimeters), d is the distance of the nucleation site from the source (in centimeters), B* is the critical ratio of adatom surface coverage at the active site with respect to the rest of the surface, e* = ( P A / & ) , t d is the time interval (in seconds) between the start of adatom diffusion from the source and the instant the local surface coverage ratio exceeds 6* at the nucleation site, tn is the stochastic waiting time (in seconds) for nucleation of the supercritical cluster after the critical adatom concentration is exceeded, and ti is the total induction time (in seconds) between the beginning of diffusion and nucleation. The expansions used in deriving eqs 6a and 6b are most valid for 6’ = 0.5. Nucleation at Active Sites. The time-of-flight method requires the existence of a potent nucleation site with the following two properties: (a) For a given surface coverage of adatoms, the nucleation rate at the active site is much greater than the rate over the rest of the surface. For example, on a surface 10 pm X 10 pm, enhancement of the rate constant for nucleation at the active site might have to be more than a factor of lo8 or so compared to the intrinsic value on the singular crystal surface. (b) The standard exchange rate for reaction 1 at the nucleation site is no greater than the rate at other surface sites in thediffusion zone. There is convincing experimentalevidence for the existence of active nucleation sites at the silver/water interface. However, the existence of a site that has a high activity for nucleation but not for charge transfer has not been considered or reported previously in published studies of metal electrocrystallization. It is reasonable to assume that enhancement of nucleation rates at an active site is associated with stabilization of adatoms at the site, i.e., with a shift in reaction 1 to the right caused by an overall lowering of the free energy of the products. For a simple, outer-sphere, one-electron-transfer process, lowering the free energy of the product is expected to increase the rate of the interfacial reaction because of proportional lowering of the activation barrier for the pr0cess.5~-~~ However,reaction 1 occurs via a complex, multistep mechanism that also involves atom transfers, and there is no reason, a priori, for the activation free energy of the rate-determining step to be lowered by lowering the free energy of the product. The extreme variation in the rate of reaction 1 from site to site on the surface could also be true for other trpes of surface reactions, for example, adatom trapping, that do not necessarily involve charge transfer. Hence, the rates of nucleation and charge transfer are not necessarily correlated. The type of coupled surface mass-transport and cluster nucleation considered here has not been described before for silver electrocrystallization. The existence of diffusion-controlled
6700 The Journal of Physical Chemistry, Vol. 97, No. 25, 199'3 nucleation at active sites can be tested by applying diagnostic criteria to experimental data obtained under steady-state conditions. We assume a Frank-van der Merwegrowth mechanism. Dhgnostic Criteria for tbe Timesf-Fligbt Mctbod. The diagnostictests described here can be used to determine whether or not experimental conditions are appropriate for the timaofflight analysis. The characteristic event essential to the timeof-flight method is the nucleation of a supercritical 2D cluster, but not all types of nucleation and growth event yield information about surface diffusion. In order to distinguish one type of nucleation law from another,it is necessary to analyze an ensemble of nucleation events that are recorded as a time series under conditions of steady-stategrowth. Thedata used for this purpose are the set of [Ia(j),t,u)], where is the absolute charge of the jth nucleation and growth event and two>is a convenient, empirical "waiting" time for the event, defined here to be the time interval between the (j- 1) nucleation event and thejth one. Note that tw(j)1ti(i). It is recommended that data be recorded for a number of different surface areas, i.e., a number of different values of while otherwisemaintaining the same set of imposed experimentalconditions. The natural evolution of a sampleduring the capillary-growth method actually makes this situation rather easy to achieve under typical experimental conditions. The empirical diagnostic tests are based upon the degree of correlation between and two>, when the data are plotted in the (la,tW)plane, and upon the descriptive statistics of t,(Ia). The tests are not speciiic enough to differentiate between all possible types of nucleation laws. In terms of correlation between IQI(j) and t,(j), there are three distinguishable cases to consider: C+, significant positive correlation between t, and C-,significant negative correlation CO,no significant correlation between t, and between t, and In terms of the descriptive statistics of the waiting time at a particular value of tw(lQI), there are two important cases based upon the relative magnitudes of the mean, (tw)Q,and variance, var(t,)Q = s2(tw)Q:Sda, S2(tw)Q M(hkl)r( where A, is thesurfadiffusion penetrationlength for the adatom, the area of the adatom capture zone around the active nucleation site is independent of the charge la. The time needed to achieve local supersaturation at the nucleation site is then simply proportional to 1/IoL. Hence, thb setof circumcltanceo results in condition (CO,&), i.e., no correlation between twand 14 and low normalized variance for long induction times. Condition (Q,Sno) would result if the rate of nucleation were also low at the active site. Conditions ( G S d ) and (Q,$,) are not suited to a time-of-flight analysis. At the other extreme, if the surface should contain a potent nucleation site but be missing a potent adatom source and < M(hkl)r(XIz), the entire surface is within thecapture m e of the nucleation site. For adatoms taking a bounded random walk, the probabilityof adatom capture at the nucleation site is proportional to the ratio of the capture cram section of the site divided by the total area of the surface, i.e., inversely proportional to On the other hand, the rate of appearonceof adatoms is proportional to the total area of the surface, Le., proportional to IQI. The product of those terms is independent of I Q so the waiting time between nucleation events would be independent of the size of the surface for small surfaces. This is a h condition (G&) or (GSsw).but it occurs for all values of IQI and it is not appropriate for a timaof-flight analysis. The lack of a potent source of adatomsgenerates behavior that is equivalent to a fmed *induction time" for nucleation with the induction time being independent of the size of the facet. For those systems where nucleation is coupled to surface mass transport, it should be possible to observe a transition from behavior (C+,&), when d is small, to (Q,Sd*), when d exceeds A,, Although (CO,Sb) is not a useful condition by itself, the location of the transition point from (C+,Saot) to (G&]is an extremely valuable piece of information. The value of 1, at the transition point is a direct estimate of the mean residence time of the adatoms, TO, which can be used to calculate Ios through eq 4. This analysis would also be valid in the presence of background currents due to parallel processes and would allow
la
-
[a.
Diffusion of Ag at the Ag( 11l)/Water Interface
L
CE
H 5" Figwe 1. The 'spark-plug" apparatus is shown in schematiccro88section. A threaded PTFE cap (T)supports the silver wire reference electrode (RE), silver wire counter electrode (CE),and shielded working electrode (WE)contact,the precision-bore Pyrexcapillary(c), and thesilversinglecrystal sccd (x). The entire assembly is threaded into a small, c l o d Pyrex tube (10 mL total volume) filled with 5 M AgNO,(aq). The sealed apparatus is mounted in a thermostat coil for temperature
regulation. the calculation of los under conditions where direct current measurements are not useful. The value of d at the transition point is a direct estimate of X, and, hence, of D,.
IV. ExperiwnWMetbods Budevski, co-workers, and collaborators have published extensivelyon the silver/liquid-water system. However, there have been few detailed descriptions of the capillary-growth method for preparing singlecrystal surfaces since the initial reports55*M58-61 of the technique more than 25 years ago. Our experimental techniques differ in some significant ways from the methods described originally by Budevski and co-workers in those early publications. Although serendipity does play a role in obtaining high-quality crystal surfaces and analyzable experimental data, we found that obtaining reproducible conditions and responsks was not particularly difficult or statistically improbable. Preparing singlacrystal surfacesis straightforwardusing the methods outlined below. Apjmratus. Part of our electrochemical cell is shown schematically in Figure 1. A three-electrode experimental configuration was used here, in conjunction with a wide-band, purposebuilt potentiostat. The polycrystalline silver-wire (Johnson Matthey, 99.99%) reference electrode (RE) and counter electrode (CE) were bent near to the working-electrodecapillaryin a fashion that resembled a spark plug, as shown in Figure 1. Reference electrode potentials were found to be stable and reproducible at the microvolt level when clean, bare, polycrystalline silver wires were used for the reference electrode. We did not makea "Luggin probe" by masking the reference electrode wire inside a Teflon tube or by scaling it in soft glass. We found that this would give time-dependent, nonzero reference potentials with respect to a bare polycrystallinesilver wire in the samesolution. The frequency responseof thepotentiostaticciruit wouldalso suffer. Thesilver single-crystal working electrode was grown into a precision-bore glass capillary (c, Wilmad Glass, 76.2-fim internal diameter) that was a snug press fit in the threaded Teflon cap (Ace Glass 5803-11). The electrodes were mounted securely in the cap by swaging them in Teflon or nylon Swagelock fittings screwed into the cap, producing a gas-tight seal. The cap screwed tightly into a IO-mL threaded Pyrex cell body that was made by scaling off a sturdy liquid chromatography tube (Ace Glass 5820-16). The apparatus was cleaned by soaking it in air-saturated, methanolic KOH (both Alfa Inorganics, Ultrapure) and then in
The Journal of Physical Chemistry, Vol. 97, No. 25, 1993 6701 fresh aqua regfu (1 :3 concentrated HN03(aq):concentrated HCl(aq), Fisher), thoroughly rinsing between solutions using organiefree, deionized water ( 40 pm. interaction is weak, and the water is highly mobile at the surface. This is at least 5000 times larger than the value of &(loo) (b) There is adequate space for the silver adatoms to pass through estimated by Bostanov et Because the maximum value of naturally-occurring cavities in the water at the surface, so the d i n Figure 6 is greater than the radius of the capillary, it may magnitude of the silver-water interaction is irrelevant. (c) be necessary to conduct experimentsusing larger crystalsto obtain Solvation of the silver adatoms weakens the silver-silver intera measure of &( 111). action enough to compensate for steric interference by the water At 35 OC, the mean adatom residence time on the surface is at the surface. T ~11 ( 1) > 4 s. This at least 5 X lo5 times longer than the value Our diffusion data are too sparse to allow us to distinguish of ~o(100)we calculate from the data of Vitanov et By between these possibilities. However,data obtained for the silver/ inverting eq 4, we calculate the exchange current density for the water system in uhv suggest that the silver-water interaction formation and removal of silver adatoms on the Ag( 111) crystal may be unusually weak. The desorption enthalpy for solid water face to be lo,,( 111) < 120 nA/cm2 in 5 mol/L silver nitrate. Our adsorbed on Ag( 111) at low temperature was measured to be 48 value is at least 5 X lo5times lower than the estimate of loJ 100) kJ/mol by Klaua and M a d e ~ , 8exactly ~ the same as the enthalpy
Ag(lOO)/water system from data obtained by Budevski et al. is about 50 times smaller than the mean value of Os( 111) measured in this study. The difference could be a real one, or it could be due to systematic errors in the previous experiments. Systematic error in lo,,( loo), for example, could be due to residual currents, and systematic error in Xs(lOO) could be due to unjustified assumptionsin the macrostepgeometry method cited by Bostanov et al. The two curves labeled a in Figure 7 are the results of a recent molecular dynamics simulation by I,iu et ~ 1 . of 8 ~ silver selfdiffusion on silver(l1 l), using two slightly different potential energy surfaces and the embedded atom method. This is the first theoretical study to be performed specifically for silver. The authors noted that, where they had experimental results for other systems in uhv to compare with their calculated values, their calculated activation energies for self-diffusion on f a ( 111) surfaces tended to be too low by factors of 3-10, although the agreement of preexponential factors was better than this. The experimental values of self-diffusion coefficient obtained here for the Ag( 11l)/liquid-water interface are lower by about a factor of 20 or so from curves a. This difference could be approximately taken into account by raising the calculated activation energy of Liu et al. by a factor of 3-10. Molecular dynamics simulations of the self-diffusion of a Lennard-Jones (LJ) atom on an LJ f a ( 111) surface by Mruzik and Poundlo and Doll and McDowellll are also plotted in Figure 7 as curves b and c, respectively. The model data, Easand DO* were originally calculated by those authors in *reduced" units and were converted here to absolute values appropriate to silver by10,11,88
Porter and Robinson
6708 The Journal of Physical Chemistry, Vol. 97, No. 25, 1993
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obtained previously by Vitanov et ~ 1 . 6Roughly ~ correcting our value for differences in [A@] gives P 40 nA/cm2, which is at least 6 X 108 times lower than Gerischer and Tischer's values1 of P for polycrystalline silver. This is dramatic confinnation that the exchange rate for reaction 1 is extremely low on step free, low-index surfaces. The apparent difference between lor( 111) and ZoJ 100) could be a real one, or, more likely, it could be due to the presence of residual currents in the previouSAg( 100) experiments. The bound for ZOJ 111) estimated from the residual current in Figure 3 is about 40 times higher than the bound for lo,,( 111) calculated from the adatom residence time, indicating that most of the residual steady-state current in Figures 2, 3, and 5 is probably due to background Faradaic processes, possibly the reduction of oxygen. The adatom residence time is a much more selectiveand sensitive measure of reaction 1 than direct current measurements in the presence of parallel Faradaic processes. The interfacial impedance measurements performed on the kinetically-inert silver/water systems were used to estimate the equilibrium surface coverage of adatoms near the reversible potential. According to eq 1, the formation and removal of adatoms is coupled to charge transfer across the interface. Since the net result is the storage of charge at the interface in response to a change in potential, this process appears in the small-signal equivalent circuit of the system as an interfacial pseudocapacitanceB6 Adsorption pseudocapacitances at the reversible potential, Ca (in F/cm2), wereconverted toadatomsurfacecoverages using
ro= RTC# where l'o is the equilibrium surface coverage of silver adatoms at thereversiblepotentialEo(1) (inmol/cm2),Risthegacons~t (8.314 J/(mol K)), and Tis temperature (in kelvin). Equation 12 assumes a Langmuir isotherm for the adatom species, which is likely to be reasonable at low surface coverages. We measured values of C, that were between 15 and 20 pF/cm2, corresponding to ro= 4.1 X 10-12to 5.5 X 1W2mol/cm2. These coverages are similar to the values obtained by Vitanov et al.,Q*6gand they correspond to fractional surface coverages between 0.002 and 0.003 on the (100) and (1 11) faces of silver. Because a complete set of thermodynamic and kinetic parameters has been lacking for reaction l at different types of surface site, an outstandingproblem in silverelcctrocrystallization concerns the mechanism of step expansion during deposition. Our measurements of adatom exchange rates, self-diffusion coefficients, and equilibrium adatom surface coverages can be used to calculate the rate of step advance due to (a) the adatom 'surface diffusion" mechanism and (b) the 'direct incorporation" mechanism.52The 'surface diffusion" mechanismof step advance involves the formation of adatoms by a self-exchange reaction uniformly on the surface followed by surface diffusion of the adatoms to the step in a surface concentration gradient. This mechanism is most analogous to growth from the vapor phase. The steady-state propagation velocity for an atomic step across the Ag( 11l)/water interface resulting from the surface diffusion mechanism can be calculated from@ Using the experimental data obtained in the present study, we calculate u,,j 14 X 1V cm/s. Experimentalvaluesof u calculated for the growth events in Figures 3 and 5 are all around 0.018 cm/s. Weconcludethatstepgrowthdoesnot m r bythe'surface diffusion" mechanism at the Ag( 11l)/liquid-water interface. Bostanov et al?O reach the same conclusion by inference. The 'direct incorporation" mechanism of step growth involves electron transfer to aqueous Ag(1) ions at active sites on the step and incorporation of the resulting silver atoms into the growing step. We can estimate the exchange current density for reaction
1 at the edge of a monatomic step from the measured value of V:@
where 6 a 2.5 X 10-8 cm is the width of an atomic row added to thestep. WecalculateZo# = 4200A/cm2fortheexchangecumnt density of reaction 1 at the step edge in 5 mol/L silver nitrate. Under standard conditions this correspondsto P a 1300A/cm2 for the exchange rate of reaction 1 at a growing atomic step. Our value of P is about the same as the value calculator3 by Hills et af.93for reaction 1at the surfaceof actively-growing, single nuclei of silver. This would be consistentwith the surfaceof their growing microcrystals being densely covered with atomic s t e p and with a direct incorporation mechanism of step growth. The structural dependence of the rate of reaction 1 is quite remarkable. From our data obtained in 5 mol/L silver nitrate, we estimate that the standard exchange current density for reaction 1 at the stepfree Ag( 11l)/water interface is at least a factor of 4 X 10lolower than the value at the edge of a monatomic step. This extraordinary range of activity also explains why reaction 1 appears to be so fast and reversible on polycrystalline silver: as little as a few parts in 10" of the surface need to be active toward reaction 1 in order for a significant current and net reaction to take place. Finally, the values of our kinetic parameters have very serious implications for the dynamics of steady-state homogeneous nucleation in this system. The mean residence time for adatoms, TO(11l), is numerically equal to the exponential relaxation time for the homogeneous formation of adatoms on a singular Ag( 111) surface at potentialsvery near EO(1). In order to achieve the equilibrium coverage of adatoms by self-exchange requires times in exccsll of 570. or >20 s under our experimentalconditions, beginning from an adatom-free surface. Consequently, there must be a kinetic induction time of at least this duration between monolayer nucleation and growth events when truly homogeneous adatom formation and nucleation occurs in the steady state. Yet, no such induction times have been reported before in studies of "steady-state, homogensous" 2D-cluster nucleationat silver/wata interfaces (foc. cit.). Therefore, the previous studies cannot have bten performed using truly "singular" surfaces where all sites have the same activity. At the very least, there was catalytic formation of adatoms at redox-active generation sites in those studies. Hence, it is possible that 2D-cluster nucleation was oocurring under at least partial surfacemass-transport control in the previous studies. This would help explain the apparent 'self-avoidance" of nucleationeventsthat is so obviousin publishsd current time series obtained under 'steady-state" conditi~ns."J*.~~
VIII. concln8im Measurements of the rate of elcctrocrystallization of silver from aqueous solution were interpreted using a model of surface diffusion-limited nucleation. Estimates of the surface diffusion coefficient ofsilveradatomsat theAg( 11l)/liquid-watcrintcrface were obtained. The values of D, calculated from the data are comparable to values calculated for silver adatoms on Ag( 111) in ultrahigh vacuum, indicating little net effect on the rates of adatom diffusion due to the presence of the liquid water in this case. Estimates of the standard exchange rate of the reduction of aqueous silver(1) at an atomic step and at the Ag(ll1) surface were also obtained from the data. The exchange rate at the step is 4 X 1Olotimes faster than at the planar surface. indicating a remarkable site dependence for this inner-sphere redox reaction. Adatom residence times at the Ag( 11l)/water interface are >4 sat 35 OC, and surface diffusion penetration lengths are >40pm.
Acknowledgment. This work was supported by the Director, Office of Energy Research of the U.S. Department of Energy, under Contract DE-AC03-76SF00098.
Diffusion of Ag at the Ag( 11l)/Water Interface
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