Adam Craig Finnefrock was born in Long Beach, California in 1970. Before the age of 18, his family had moved over a dozen times. In spite of this emotional trauma (or perhaps, because of it) the young preppie left New England to study in Houston, Texas in 1987. He graduated from Rice University with bachelor degrees in Physics and Mathematical Sciences in May 1992. He matriculated to Cornell University and joined Professor Joel Brock's research group in late 1992. He will be taking a postdoctoral position with Professor Sol Gruner, where he will attempt to overcome his crushing ignorance of all things biological. The culpability for his questionable choice of career resided on his parents' bookshelves, teeming with optimistic science fiction from the '60s and early '70s.
The results presented in this dissertation would not be possible without the assistance of many people.
First, I have worked in the laboratories of Joel Brock for the past six years. Alternately serving as teacher, taskmaster, advisor, and friend, he has provided the motivation and means for all of the work described herein. He has been extremely generous with his time, and has taught me most of my experimental skills, and what I know of x-ray scattering.
Professor Abruña has been my unofficial mentor. A primary collaborator, he has also given me excellent advice throughout and helped me to find the ``larger picture''. He has also provided courage (and food, occasionally) in the face of hardship, on and off of the beamline. His advice on the entire academic process was invaluable.
Lisa Buller has been my counterpart in the research group of Professor Abruña. Her assistance was crucial, especially during the earliest stages of this project, where hours were long and data were few. She has provided for the practical aspects of the electrochemical results described herein.
Kristin Ringland has been an enormous help throughout this project. She has been present for virtually all of the data acquisition and has assisted in many other ways too numerous to detail. Throughout the rigors of grim and intense synchrotron runs, she has uttered not even one complaint. As has been remarked, she is still probably ``the perfect graduate student''. Arthur Woll has been a constant friend in the lab, and I have appreciated his good humor throughout. He and I have had many discussions on x-ray scattering from surfaces, and have discussed the need for a comprehensive treatment that we can understand. I hope to see more about this in his dissertation. Emma Sweetland, the first student to graduate, helped me through the earliest years. She is also responsible for much of the laboratory infrastructure that we often take for granted.
Samantha Glazier was the newest member of this collaboration. Her dauntless enthusiasm and relentless curiosity have been refreshing for us veterans. I would also like to thank two other x-ray electrochemists. Mike Toney of IBM gave me some early encouragement and practical advice on cell design and x-ray measurements. Ben Ocko of NSLS has also given me lots of advice, and acted as the most steadfast critic. Jean Jordan-Sweet at IBM has been responsible for the beamline where this x-ray data was taken. Supervising the steady stream of users, ensuring that the beamline is ready for use, and repairing the broken/altered equipment afterwards is a thankless job. I would like to thank her for it.
I would like to thank the members of my special committee, who have taken the time to read this dissertation, and are likely to be the ones to ever do so. (Though if you are reading this, who knows?) I really appreciated their comments and careful readings; they have made this dissertation far better and more readable that I could have done alone. I would particularly like to thank the Chair, Carl Franck, for his encouragement, support, and advice (particularly on the choice of postdoctoral appointments) throughout most of my graduate studies. I also would like to thank him for hosting the Easy Physics seminars, which have been an interesting staple of my physics diet.
This would not be possible without my family, who got me to this stage in the first place and gave me the tools to continue. Jennifer Mass, my fianceé, has helped me through this dissertation from the other side of a Ph.D. I could not have made it without her patience and support throughout the entire process. I would simply be lost without her.
The U.S. taxpayer has provided generously, if somewhat unwittingly, for this research. This work was supported by Cornell's Materials Science Center (NSF Grant No. DMR-96-32275). Additional support was provided by the NSF (Grant Nos. DMR-92-57466 and CHE-94-07008) and the Office of Naval Research. The x-ray data were collected at the Cornell High Energy Synchrotron Source (CHESS), which is supported by the National Science Foundation (Grant No. DMR-93-11772), and at the IBM-MIT beam line X20A at the National Synchrotron Light Source (NSLS), Brookhaven National Laboratory. NSLS is supported by the U.S. Department of Energy, Division of Materials Sciences and Division of Chemical Sciences. (Contract No. DE-AC02-76CH00016).
The electrodeposition of a metal adsorbate onto a solid surface is a key aspect of important technological processes such as electroplating and corrosion inhibition. In a number of cases, metal overlayers can be electrodeposited onto a dissimilar metal substrate at a potential that is less negative than the Nernst potential (that required for bulk deposition). Experimentally, this ``underpotential deposition'' (UPD) provides a precise means for quantitatively and reproducibly controlling coverage in the submonolayer to monolayer (and in some cases multilayer) regime [89,7,128].
The initial stages of adsorption/deposition, along with the growth mechanism, dictate the structure and properties of the deposit. UPD is an important experimental technique for investigating the early stages of deposition, and the diverse factors that influence it, for several reasons. First, in contrast to vacuum-surface experiments, the electrochemical interface provides direct control over the chemical potential of adsorbed species. This has been recently exploited by Ocko and coworkers [105] to study two-dimensional Ising lattice dynamics. Second, the charged double-layer (section 2.3) produces enormous (up to 107 V/cm) electric fields, capable of driving surface rearrangements [106]. Third, UPD is generally reversible. Thus, it is possible to perform repeated measurements of a deposition/desorption transition using the same sample and systematically varying the control parameters.
The strongest interaction in a UPD process is between the metal to be deposited and the substrate [89,6,51]. Thus, UPD is usually restricted to the deposition of one monolayer prior to the onset of bulk deposition; in some systems, however, up to three atomic layers can be deposited. Although the metal-substrate interaction usually dominates, other interactions can also be important. For example, strongly adsorbing anions in the electrolyte are particularly important as both anion-metal and anion-substrate interactions significantly affect UPD processes. Furthermore, the adsorbed species rarely loses its charge completely during the early stages of deposition [119,120,149,129,94,155,156]. Rather, it becomes completely reduced only when the applied potential is close to the Nernst potential. This variable charge state alters the electrostatic interaction between the deposit and the anions. At more positive potentials, there is a strong attractive electrostatic interaction that disappears as the metal is discharged. This attractive interaction can produce a metal-anion bilayer on the electrode surface at intermediate potentials [156,123,91,133,131].
In addition to the surface coverage, both the presence of other adsorbates, especially anions, and the surface structure of the substrate can profoundly affect the structural and electronic characteristics of the deposit [90,149,94,150,71]. Although there is a great deal of existing work on UPD lattice formation, the early stages of deposition are not well-understood [121,33]. In much of this earlier work, the structure of a UPD overlayer was determined by transferring the electrode into an ultra-high vacuum (UHV) chamber and employing established surface science techniques such as low-energy electron diffraction (LEED). However, such measurements are inherently ex situ and cannot provide information on the kinetics of deposition.
Recently, in situ probes such as scanning tunneling microscopy (STM) [92,64,76], atomic force microscopy (AFM) [93], and surface x-ray scattering (SXS) [101,132,133,139,140,107] have been applied to UPD systems. In addition to eliminating the ambiguity of ex situ measurements, they offer the possibility of studying the kinetics of deposition. Kinetic studies are crucial for identifying the rate-limiting steps in the electrochemical growth of not only metals but also of technologically relevant materials such as GaAs [136] and CdTe [135]. Such studies can also provide important tests of the large body of theoretical work on nonequilibrium statistical physics, especially on the kinetics of growing surfaces and interfaces [17].
The UPD of metal overlayers onto single crystal electrodes provides an excellent family of experimental systems for studying fundamental aspects of materials growth. In particular, Cu UPD on Pt(111) has been extensively studied by a variety of techniques. The process is very sensitive to the presence of anions and appears to be kinetically controlled. The exact structure and nature of the overlayer, particularly at intermediate coverages, has been the subject of some controversy. Based on LEED studies, Michaelis et al. [102] identified the intermediate overlayer as a 4 ×4 structure. However, more recent in situ anomalous x-ray diffraction measurements of the overlayer structure as a function of potential by Tidswell et al. [131] indicate that the intermediate overlayer structure is a more complicated incommensurate CuCl bilayer.
Surface x-ray scattering techniques have been previously applied to UPD. For example, Toney and coworkers have studied Pb, Tl, and Cu UPD on Ag and Au surfaces [101,132,133]. In addition, Ocko and coworkers have studied a variety of equilibrium surface structures as a function of both the solution concentration of the adsorbate (especially anions) and the surface charge, with emphasis on gold substrates [139,140,107]. However, all of these studies have been static in nature and have not addressed the kinetics of adlayer formation. This is due, in part, to the severe experimental challenges that such measurements present.
Time-resolved surface x-ray scattering represents a nearly ideal probe for studying the time evolution of the overlayer structure during UPD. X rays in the 0.5 Å to 1.5 Å region are not significantly absorbed by aqueous solutions allowing for the in situ study of the electrode/solution interface. In addition, the line shape of the scattered x rays can be interpreted simply in terms of well-known correlation functions, allowing direct tests of theory. Using signal averaging techniques, transient structures with lifetimes as short as a few microseconds can be studied [127].
In this dissertation, I report the first time-resolved surface x-ray scattering measurements of metal electrodeposition. The specific system chosen system is the UPD of Cu2+ onto Pt(111) in the presence of Cl- anions. Some of the results have been already published [65,78,5,66]; inclusion of these results in this dissertation is with the written permission of these journals.
To my knowledge, these are the only time-resolved x-ray measurements of any UPD process. This is not surprising, because these measurements are extremely difficult to perform. UPD is extremely sensitive to contaminants, requiring special protocols and rigorous cleanliness throughout the preparations of the sample, the solutions, and the electrochemical cell. To observe the scattering from only a single monolayer, a synchrotron x-ray source is necessary. Additional scattering from the solution and the film that contains it can easily overwhelm the signal of interest. These considerations imply that static x-ray scattering measurements from the UPD layer are quite difficult. Compounding this by performing time-resolved measurements of the nonequilibrium UPD system adds another challenge. The signal to noise ratio must be sufficiently high in each time bin to obtain useful data. This ratio can be improved by depositing the UPD layer under voltage control, and then pulling out most of the solution. This is the traditional method for studying UPD structures in situ. However, this configuration completely prevents further manipulation of the UPD layer; the contact between the sample face and the other electrochemical electrodes is diminished. The kinetics of the UPD formation or dissolution are then completely inhibited.
Because of all these difficulties, the conventional wisdom was that time-resolved x-ray scattering measurements of UPD were not possible. To resolve these challenges, we had to develop and successively improve several aspects of our experiment. The first and most dramatic improvement came in the observation that annealing the sample at high temperature for up to an hour dramatically improved the crystal mosaic (an indication of the size and relative alignment of domains within the crystal). Similar behavior has been observed in single noble metal crystals in UHV [69]. The next limitation was the misorientation of our samples' faces with respect to the crystal axis. Ultimately, this imposed a constraint on the maximum terrace size. Following Joel Brock's suggestion, I designed a crystal polishing apparatus that could orient the sample in any arbitrary direction with a precision of 0.001°. The sample could be repolished along this axis, producing a crystal face with the desired orientation. (Other experimental groups in Clark Hall have adopted our design and procedures to obtain dramatic improvements in sample quality.) Lisa Buller had a electrochemical cell built, based upon Mike Toney's original design. The final challenge was to interface the potentiostat to the rest of our timing hardware and software. This involved months of programming and testing.
After overcoming these challenges, we were able to obtain very useful and interesting data. We have studied in situ the ordering kinetics of the two-dimensional Cu-Cl crystal electrodeposited on a Pt(111) surface. We simultaneously measured high-resolution time-resolved x-ray scattering and chronoamperometric (current vs. time) transients. Both measurements were synchronized with the leading edge of an applied potential step that stimulated the desorption of Cu and subsequent ordering of the Cu-Cl crystal. In all cases, the current transient occurred on a shorter time-scale than the development of crystalline order. The time-dependent x-ray intensity data (2×104 data points) were well fit by an Avrami-like function with only three parameters. By performing a series of voltage-step experiments, we demonstrated that the ordering time diverged with applied potential f as t ~ exp[ 1 / (f- f0) ], consistent with the nucleation and growth of two-dimensional islands. Monitoring the time-dependent widths of the x-ray peak, we observed a narrowing corresponding to the growing islands.
This dissertation is organized into chapters as follows. Chapters 2 and 3 are introductory in nature. Chapter 2 is an introduction to electrochemistry, specifically oriented to the phenomenon of UPD. It is aimed at a physicist who may be unfamiliar with electrochemical phenomena, and the presentation is from a fundamental perspective. Wherever possible, I have made analogies to examples familiar to most physicists. Chapter 3 is a derivation of some x-ray phenomena, starting with the classical x-ray scattering from an electron.
The experimental apparatus and procedures are documented in chapter 4. These include sample preparation and data acquisition procedures. Static x-ray measurements and their subsequent analysis are in chapter 5. Kinetic (time-resolved) x-ray and chronoamperometric (current vs. time) measurements are found in chapter 6. These data are analyzed in terms of a nucleation and growth model. Finally, conclusions are presented in chapter 7. Long derivations and discussions are relegated to the appendices.
In this chapter, I will introduce and discuss some of the rudiments of electrochemistry from a physics perspective. The first section introduces the electrochemical potential. The second section concerns the nature of the electrode - solution interface, and discusses several models for the electric charged double-layer. Then, the behavior and transport of ions in solution is discussed. The following sections describe bulk deposition and underpotential deposition. Finally, specific adsorption is explained and adsorption isotherms are examined.
What is electrochemistry? I like the definition with which Schmickler [114] begins his recent book:
Electrochemistry is the study of structures and processes at the interface between an electronic conductor (the electrode) and an ionic conductor (the electrolyte) or at the interface between two electrolytes.
The first electrochemical experiments were also some of the first biophysical experiments. These are the famous studies of electrified frog legs, performed by Luigi Galvani [68,67]. Since then, experimental science has fragmented into a multitude of disciplines and spawned many industries. Presently, electrochemical processes are crucial to a wide variety of commercial processes. These include batteries, which are of great importance in the quest for low-emission electric vehicles. Corrosion is an electrochemical process under active study, especially in industry. Electroplating, for either the prevention of oxidation, or coating one metal with a more precious one (such as in jewelry) is another process of importance. Recently, specific multilayer semiconductor structures have been electrochemically synthesized.
Two more examples illustrate the importance of electrochemistry. Electroanalytic processes alone account for $68.2 billion worldwide [134]. The primary products include chlorine, aluminum, copper, and sodium hydroxide. Electrolysis of water is still used in Europe to produce high purity hydrogen and oxygen. The global production of aluminum consumes the same amount of electricity as 10% of the United States electricity sales. Electrochemistry is also crucial to electrochemical biosensors, which are now used in medical settings. They monitor the concentrations of various gases dissolved in the blood (such as carbon dioxide, oxygen, pH) or electrolyte levels (sodium, potassium, calcium, chloride). These sensors are portable and give continuous real-time results at the patients' bedside. Previously, the alternative was to periodically take blood samples and send them to the hospital lab for analysis.
In analogy to the chemical potential mi, let us define the electrochemical potential of species i with charge qi in an electric potential f [22]:
| (1) |
For convenience, chemists often separate the chemical potential into a concentration-dependent term and a concentration-independent term.
| (2) |
This separation in (2.2) is reminiscent of the isolation of the pressure-dependent terms in gas mixtures:
| (3) |
| (4) |
| (5) |
| (6) |
Returning to the electrochemical potential (2.1), the definition of chemical activity (2.2) can be incorporated as
| (7) |
A complete model of the electric double layer [23,39,115] was given by Bockris, Devanathan, and Müller [34]. This is illustrated in figure 2.1. It contains positively charged species adsorbed onto the electrode, polar solvent (water) molecules, and solvated species both near and far from the electrode surface. We will take these components in turn, and gradually build up to this complex arrangement.
If we apply a negative charge onto our electrode surface, then it will attract positive ions from solution. This will have the effect of making the charge of the electrode appear to be less negative to a test charge deep in the bulk solution. This ``charge screening'' is exactly analogous to the screening of point particles described by Debye, which is present in a broad range of physical contexts. Charge screening is discussed again in section 2.4.
Throughout this section, the potential far from the electrode (in the ``bulk solution'') is set to zero. Relative to this potential, the electrode is at an electric potential f0.
The earliest model of the double layer was proposed by Helmholtz [137,138] in 1879. He considered just the a layer of positively charged ions, tightly bound to the negatively charged electrode surface. The centers of these ions were postulated to lie on a single ``Helmholtz'' plane at a distance zH from the electrode surface. The resulting potential is identical to that within a capacitor, and is a linear interpolation between the electrode and bulk potentials, as shown in 2.2.
| (8) | |||||||||||||||||
An entirely opposite approach was undertaken by Gouy in 1910 [73] and Chapman in 1913 [54]. They proposed that none of the ions were tightly bound to the surface. Each ion is not constrained to lie in a tight double layer, but is sensitive to the electric potential formed by the other ions. In this way, the positions of the ions are not predetermined, but are the result of a statistical equilibrium with respect to this potential. This is only a mean-field model; individual ions are expected to react only to the overall field produced by all the other ions.
This mean-field model can be analyzed with some rigor. First, the electric potential f depends on the charge density r as stated by Poisson's equation (in Gaussian units),
| (9) |
The charge density is the sum of the charges from species i
| (10) |
| (11) |
| (12) |
| (13) |
| (14) |
| |||||||||||||||||||
| (17) |
In order to continue and keep the equations manageable, it is helpful to restrict the discussion to a ``symmetrical'' electrolyte. These are also called z:z electrolytes, because they consist of only one cationic and one anionic species, of equal charge magnitude (often denoted z). The electrolyte used in these experiments, HClO4 (which dissociates into H+ and (ClO4)-), is an example of a symmetrical electrolyte. Then (2.18) becomes
| (18) |
| (19) |
| (20) |
| (21) |
| (22) |
| (23) |
| (24) |
| (25) |
In the limit that f0 ® 0, we obtain
| (26) |
For aqueous solutions, e = 78.49 [24] at 25°C, and then k = 3.29 ×107 (q/e) c¥, where k is given in cm-1 and c¥ in M (moles/liter). Our electrolyte is 0.1 M HClO4, so k = 3.3 ×106 cm-1 or 3.3 ×10-2 Å-1. This constitutes an enormous voltage gradient, on the order of 106 to 107 V/m.
The Gouy-Chapman model is an improvement over the Helmholtz model, but it does not take into account the finite size of ions. There must be a plane of closest approach, just as predicted by Helmholtz. The minimum distance of this plane from the electrode surface is the ionic radii. If the ions are solvated, then they will not even be able to approach that closely. Stern realized this in 1924 [125] and proposed a model to incorporate this. Essentially, it is a combination of the two previous models.
Call the distance of closest approach zOHP. (OHP stands for ``Outer Helmholtz Plane''. The ``Inner'' plane will be defined shortly.) The Helmholtz description applies for z £ zOHP, and the Gouy-Chapman description applies for z ³ zOHP. At this boundary, we require continuity in the potential fOHP and its derivative. In the z £ zOHP region, we have the Helmholtz linear potential drop
| (27) |
| (28) |
If we choose zOHP, then we can find fOHP by applying (2.20) and (2.28) at z = zOHP and requiring the continuity of df/dz there. Numerically solving these self-consistent equations produces a value for fOHP. The Stern model adds only this one additional parameter, zOHP.
We will discuss the following models only qualitatively. As we add more components to the model, additional variables are added that are difficult to measure. But it is important to keep the additional components in mind, if only to appreciate the difficulty of predicting exact quantitative behavior.
The Grahame model [74] (1947) includes the possibility of ions specifically adsorbed (section 2.7) on the electrode surface. This specific adsorption is chemical in nature, and cannot be explained simply by electrostatic arguments. These ions can have either positive, negative, or no charge. We expect, however, that their ionization state and degree of attraction to the electrode will be influenced by the electrode's potential. The closest approach of these adsorbates defines the ``Inner Helmholtz Plane''.
The Bockris-Devanathan-Müller model includes (polar) solvent molecules that bring us to our complete picture shown earlier 2.1. In retrospect, all of this model-building may seem somewhat ad hoc. A contrasting approach has been put forth recently by Borukhov et al. [36]. The authors start with the Poisson-Boltzmann equation but include the contributions to the free energy from the finite size of the ions. This remediates some of the defects of the Gouy-Chapman model and is in agreement with experiments they cite where large multivalent ions are adsorbed onto a charged Langmuir monolayer.
Having discussed the electrode surface and its ionic neighborhood, we turn to the charged ions in bulk solution. First, the various modes of transport of charged ions to the electrode surface are discussed and compared. These play an important role in the kinetics of deposition at that interface. Second, the importance of a supporting electrolyte in electrochemical experiments is described.
In any deposition/growth system, the transport of particles to the surface is an important consideration. Often, the evolution of the surface morphology is determined by the relative rates of transport to the surface and reactions at the surface. Two familiar limiting cases are diffusion-limited aggregation (DLA) [153,] and kinetic-limited growth (such as the KPZ model [86]). All electrochemical reactions take place only at the electrode surface, so transport of ions to that interface is of paramount importance.
We expect the current of species i to be proportional to its respective electrochemical potential gradient,
| (29) |
| (30) |
| (31) |
| (32) |
| (33) |
Diffusion refers to the random, Brownian motion of particles that follow Fick's first and second laws of diffusion:
| |||||||||||||||||
The diffusion constant can be solved for by using a Fourier transform (appendix B)
| (36) |
The second mode, electromigration, refers to the motion of charged ions under the influence of a The final mode of ionic transport is due to the transport of the solution itself. Formally, this is termed hydrodynamic transport, though it is usually known as convection. Because of the extremely small solution volume in our x-ray cell, hydrodynamic transport is not a consideration. However, it is an important consideration in larger electrochemical cells. When the solution is stirred or the electrode rotated, this will give rise to hydrodynamic effects. Rotating ring-disk techniques [50] make use of this effect, for instance. Convection must also be taken into account for sensitive measurements that take place over several minutes and where significant depletion occurs near the electrode.
In addition to the species of interest, most electrochemical experiments incorporate a supporting electrolyte. This is either the solution or is dissolved in the solution at a high concentration with respect to the species of interest. For instance, in our experiments the primary solution was H2O, and the supporting electrolyte was 0.1M HClO4. A brief and practical discussion of supporting electrolytes can be found in Brett and Brett [40].
There are several advantages to using a supporting electrolyte. First, the double-layer does not extend far into the solution; the majority of the potential drop is very close to the electrode (section 2.3). Second, ions are well-screened. As described by Debye-Hückel theory (see, for example, McQuarrie [98]), a charge in solution tends to attracts charges of opposite sign. The gives rise to an effective ``ionic atmosphere'' that diminishes the effective net charge felt by a test charge some distance away. Third, because there are far more charged ions in solution, the overall resistance of the solution is much diminished. Fourth, most of the current is carried by the electrolyte, not the dilute ions. This has implications on the dominant mode of transport.
In any electrochemical system, it is important to determine what fraction of the measured current is derived from diffusion as opposed to electromigration. When both effects are present, the analysis becomes complicated. However, electrochemical experiments are generally carried out with a large concentration of a supporting electrolyte relative to the concentrations of active species. The supporting electrolyte does not take part in the reaction at the electrode, but does carry the majority of the current through the solution. The electrolyte serves to screen the ions, making the ``ideal gas'' approximation more realistic. Hence, the deposited ions arrive mostly via diffusion, and electromigration effects can be neglected [26].
In this section I will discuss the deposition of ``bulk'' amounts of material onto an electrode surface. In the next chapter I will turn to ``underpotential'' deposition, which occurs at voltages closer to the rest potential, and is sometimes the precursor to bulk deposition.
As a prelude to understanding underpotential deposition, it is necessary to understand something about bulk deposition. First I will discuss the Nernst equation, which determines the onset of bulk deposition. Then I will introduce the Cottrell equation, which is a simple realization of bulk deposition. Both of these are covered in the more analytical electrochemistry texts.
In discussing chemical activities (section 2.2) and and electrochemical potentials (section 2.2), we have already developed the necessary machinery to write down the Nernst equation. The following treatment parallels Bard and Faulkner [27].
Using the relation between chemical potentials and chemical activities (2.2), the Gibbs free energy is
| (37) |
| (38) |
| (39) |
It is also common to write the Nernst equation in terms of concentrations, which are easily measured. (The activities are often not known.) Using activity coefficients (2.5) ai = gi ci,
| |||||||||||||||||||||||
Bulk deposition is just an example of these reversible reactions we have been discussing. Consider the deposition of copper ions from solution onto an inert electrode. This is written as
| (42) |
Consider a deposition experiment where the potential is abruptly shifted from above the Nernst potential (no deposition) to below the Nernst (bulk deposition). Before t = 0, the system is in equilibrium with a mean concentration of c¥ everywhere; c(z,t £ 0) = c¥. At t = 0, the voltage is altered such that deposition occurs. The electrode surface is assumed to be perfectly adsorbing so that it is a perfect sink for the adsorbing ions; c (z = 0+, t > 0) = 0. Each of these adsorbed ions transfers n electrons to/from the electrode. Furthermore; the solution container is semi-infinite and hence inexhaustible. Far from the electrode, the bulk solution concentration will be maintained; c (z ® ¥, t) = c¥. We must solve the linear diffusion equation
| (43) |
| |||||||||||||||
The standard method [28,] is to Laplace transform (2.44) and the initial condition:
| (46) |
| (47) |
| (48) |
| (49) |
| (50) |
| (51) |
| (52) |
The concentration profile and current density are plotted in figure 2.6 for D = 9 ×10-6 cm2 / s, typical of aqueous solutions. Note that the current is arbitrarily large at early times (limited by extrinsic factors), and decays away as a power-law. The concentration profile begins as a steep distribution, but broadens at later times as diffusion makes more ions available.
Imagine that we start at the rest potential and slowly sweep the potential in a negative direction. In contrast to the cartoon of bulk deposition (figure 2.5), small peaks in the current response can be observed. This phenomenon is not ubiquitous. These peaks are only observed for particular species deposited onto particular electrodes. In addition, this phenomena is very surface-sensitive. For instance, the peak positions and heights when Cu is deposited onto Pt vary depending upon the particular Pt crystal face: (111), (100), or (110). This process is termed underpotential deposition, because the deposition takes place at potentials ``under'' the Nernst potential (closer to the rest potential). A potential applied beyond the Nernst potential is often termed the overpotential, as in (6.2).
Although underpotential deposition (UPD) is a complex process, we can qualitatively justify this behavior. Presuming that a Cu ion has a greater affinity to bond to a Pt atom than it does to another Cu atom, then we can imagine that the underpotential deposition situation shown in the top panel of figure 2.8 would be favored at some potentials for which bulk deposition, shown in the bottom panel, would not. In the top panel, each deposited Cu is directly in contact with the Pt surface. In the bottom panel, the subsequent Cu layers are only in contact with the prior Cu layers.
In the next section, I present simple models of ``specific adsorption'', of which underpotential deposition is a particular example.
In the discussion of the double layer (section 2.3) we considered the electrostatic interaction of charged ion species with the charged electrode. In our most sophisticated model, we assumed that no ion could approach closer than the radius of its solvation sphere. Ions that do lose their solvation spheres and penetrate within the outer Helmholtz plane are said to be specifically adsorbed. Their interaction is more than electrostatic, and is comparable to a chemical bond. Bard and Faulkner [29] make the analogy:
The difference between nonspecific and specific adsorption is analogous to the difference between the presence of an ion in the ions atmosphere of another oppositely charged ion in solution (e.g., as modeled by the Debye-Hückel theory) and the formation of a bond between the two solution species (as in a complexation reaction).
Needless to say, these interactions will be very complex. Models of these processes need to combine the ionization of charges near surfaces, solvation, chemical bonding, charge-screening, and surface phenomena such as work functions.
Even in the absence of any adsorption, there would be some concentration (equal to the bulk concentration) of ions of species i in a region near the electrode surface. The surface excess concentration Gi [116,30] is defined to be the concentration of species i in excess of the bulk concentration, normalized by the area of the electrode. The ``coverage'' q is defined to be the surface excess normalized by its saturation value, q = G/ Gsat, so that 0 < q < 1. The definition of this region ``near'' the electrode surface is somewhat arbitrary. In principle, it can include the diffuse double-layer. However, the Gi in the diffuse double layer will have little effect if we have a supporting electrolyte. First, most of the the ions drawn into the diffuse double layer will be from the supporting electrolyte, not the species i. Second, the width of the diffuse double-layer narrows exponentially as the concentration of ions increases.
In the limit that the adsorbates on the surface to do not interact, we can write down a simple model for the adsorption onto a surface. This is very similar to standard the simplest lattice-gas models of adsorption discussed in introductory statistical mechanics courses. I also assume that there is no interaction between the species and the solution itself, and that there is no species-species interaction in the solution. If concentrations are not too high, then the supporting electrolyte screens these charges.
The rate of adsorption is proportional to the concentration of ions in solution c, the number of sites available for adsorption (1 - q), and a Boltzmann factor involving the Gibbs free energy of the activated complex G¢ (see figure 2.9) and the Gibbs free energy of the ion in solution Gsol. Isotherms, by definition, are equilibrium measurements. Hence we usually assume that the system has come to equilibrium such that the concentration near the electrode is equal to that in the bulk solution, c¥. We are also implicitly neglecting the diffuse double layer (which is described by the Poisson-Boltzmann distribution).
| (53) |
Similarly, the rate of desorption is proportional to the concentration on the surface q and another Boltzmann factor involving G¢ and the Gibbs free energy of the adsorbed ion Gad.
| (54) |
| (55) |
To make a slightly more realistic model, we assume that there is some interaction between the adsorbates. Using a mean-field approach, let DG = DG0 + gq. If the adsorbates attract one another, then g > 0. If they repel, then g < 0. In the case that g = 0, we recover the Langmuir result. Typically this isotherm is written as
| (56) |
| (57) |
The choice of which isotherm to use depends upon experimental conditions. The Langmuir isotherm is an accurate description for small coverages (q), or equivalently, small concentrations. In this regime the adsorbates are sufficiently sparse that they do not interact. Because of the approximation used to derive it, the Temkin isotherm is only used for 0.2 < q < 0.8 and g not approaching zero. The Langmuir and Frumkin isotherms are virtually indistinguishable as q® 0, and both are linear in that regime. These two points become apparent upon expanding (2.57) in this limit and keeping terms to second order,
| (58) |
This chapter provides an introduction to x-ray scattering, suitable for a first-year graduate student studying physics, chemistry, or a related field. It covers a broader range of material than is necessary simply to interpret the results in chapters 5 and 6. The reader may safely decide to skip ahead and return back to the specific sections that are referenced in those chapters.
Early x-ray experiments were performed with x-ray tubes, and today most still are. Presently there are at least two other available sources, both superior to tubes. Rotating anode sources, while expensive, can easily fit within a room. Synchrotron sources are extremely large multi-user facilities, of which only a handful exist in the world. The benefits include an enormous gain in flux and angular collimation. Both of these sources were used for this dissertation, and I will discuss them in turn.
``Conventional'' x-ray sources [57] work on the same principle as Röntgen's original apparatus; electrons are accelerated into an block of material (the anode), generating x-rays. While the x-ray tube of Röntgen used electrons ionized from gas, today the electrons are produced by a high-current filament. These electrons are then accelerated by an electric field into the anode. When struck by the electrons, the anode produces a broad, continuous spectrum of x-rays, due to the electron deceleration within the anode. This is typically called bremsstrahlung from the German ``braking radiation''.
The more useful spectral components are the ``characteristic'' radiation lines, which arise from electronic transitions within the anodic atoms. If an electron kicks out an electron from an atom, the atom will be in an excited, ionized state. Subsequently, one of the remaining atomic electrons will fall into the unoccupied state, releasing an x-ray photon and conserving energy. Due to the quantized energy levels, the resultant x-ray spectrum is also discrete, and characteristic of the atomic element. The wavelengths are labeled according to the energy transition. For instance, the Ka lines correspond to transitions from L (n = 2) to K (n = 1), the Kb from M (n = 3) to K, and La from M to L. The principal quantum number here is denoted by n. These characteristic lines can be exceedingly narrow ( < 0.001 Å), so it is possible to have nearly monochromatic radiation for an x-ray experiment. Sometimes it is even possible to resolve the lines even further. The Ka, for instance, can split into the Ka1 and Ka2. These correspond to transitions from L states with slightly different energies (the fine structure).
The intensity (per dl) in the characteristic lines is higher than the bremsstrahlung by a few orders of magnitude. Nevertheless, the flux per solid angle is low because the radiation is spread isotropically into all directions. The overall intensity can be boosted by using the highest electron beam current possible. In practice, this requires both water-cooling the anode and rotating it to prevent a single focus spot from overheating. These rotating anodes provide the highest flux presently available in a ``bench top'' laboratory setting.
For the work presented in this dissertation, a Rigaku (Model RU200) rotating anode was used primarily for orientation of samples, training, and as a testing bed for the experiments. This instrument has a tungsten filament that can support up to 200 mA current. The electrons are accelerated over potentials as large as 60 kV into a rotating, water-cooled copper anode.
A synchrotron x-ray source begins with an ultra-high vacuum (10-9 Torr) storage ring. Within the ring are electrons circulating at near-light speeds. Whenever a charge is accelerated (for instance, if constrained to a circular path) it emits radiation. In doing so, it loses energy. To keep the electrons moving in stable orbits, energy in the radio frequency range is added at intervals synchronized with the electron ``bunches''. While electromagnetic radiation is produced for any acceleration, it is advantageous to place additional accelerating devices at specific locations. In our experiments, these were simple ``bending magnets'' that sharply steer the electron beam. More sophisticated devices, such as ``wigglers'' and ``undulators'' cause the electron beam to be accelerated up and down several times within a narrow spatial region. This leads to a corresponding increase in the intensity of the overall x-ray beam delivered.
A brief, if dated (1979), review of synchrotron radiation can be found in [56]. An even shorter overview is presented in [87]. A thorough and very recent (not yet in print) account of synchrotron radiation and related devices is [77]. The remainder of this section will use a few results derived in [130].
The most striking feature of synchrotron sources is the high degree of collimation (unlike conventional sources, which radiate into all 4p solid angle). The ``opening angle'' for the radiation is peaked sharply forward and determined by the speed of the electrons. The full-width at half-maximum is [130]
| (59) |
| (60) |
A second feature is that synchrotron radiation has different polarization characteristics from the unpolarized conventional sources. The ratio of power in the parallel polarization (in the plane of the synchrotron ring) to that in the perpendicular polarization is
| (61) |
The third feature is that synchrotron x-ray radiation has a broad, continuous energy spectrum. The upper value is limited by the electronic velocity, but the range is also dependent upon the characteristics of the beamline acceleration device and ``optics'' within the beamline itself. Some experiments make use of the entire multifrequency (``white'') beam. The majority (including the ones described herein) use a monochromator to select a comparatively narrow range of frequencies. A typical monochromator consists of one or more Bragg diffractions (3.67) from single crystals or specially engineered multilayer structures.
Although x-ray scattering is inherently a quantum phenomenon, many important features can be correctly derived from a classical treatment [141]. A quantum mechanical treatment can be found in [46,112]. We will also take the nonrelativistic limit and use Gaussian units.
Following the general treatment in [82] assume a ``free'' charge of magnitude q and mass m. This is subject to an incident electromagnetic plane wave of frequency w, wavevector k1 = k n1, electric field amplitude E1, and polarization e1
| |||||||
By the Lorentz force law and Newton's second law, a free point charge q will then accelerate as
| (64) |
We know that accelerating charges emit radiation. Following (14.18) from [83], the radiation field observed from a distance R along a unit vector n2 is
| ||||||||||||||||||||||||
| (67) |
The instantaneous energy flux is described by the Poynting vector,
| (68) |
| (69) |
| (70) |
Now think of this as a scattering process, and so define a scattering cross-section. For the usual particle-scattering situation, this is
| (71) |
| (72) |
| (73) |
| (74) |
| (75) |
| (76) |
Take n1 and n2 to define the ``scattering plane'', and define the angle 2 q to lie between them:
| (77) |
| (78) |
For an incoming wave with the electric field polarized perpendicularly to the scattering plane (a|| = 0), this polarization factor is unity. This case is applicable (3.3) to synchrotron radiation (section 3.4), where the scattering plane is usually perpendicular to the synchrotron ring. This takes advantage of the high resolution along that direction, determined by the opening angle (3.1). For a parallel polarization (a|| = 1), this factor becomes cos2 (2 q). Conventional sources (section 3.3), such as rotating anodes, produce randomly polarized radiation that has the factor 1/2 (1 +cos2 (2 q)).
The intensity of scattered peaks also is proportional to sin(2q). This factor is a consequence of the Jacobian between angle space and reciprocal space; a volume element in reciprocal space is smaller than its generating volume in angle space by a factor sin(2q). These polarization factors P(2 q) are sometimes bundled into the ``Lorentz-polarization'' factor, which is just P(2 q) / sin(2q).
In this section, I will discuss scattering from multiple objects. By considering the phase difference between scattered waves from spatially separated objects, I introduce the structure factor. Then I discuss the connection between the structure factor and the correlation function.
I assume the reader is familiar with the use of Fourier transforms. This section merely defines the Fourier transform as I will use it, as there is some variety in the normalization and sign conventions in the literature.
Throughout this dissertation, the Fourier transform of a function f(r) in d dimensions is defined to be
| (79) |
| (80) |
| (81) |
| (82) |
| (83) |
Take the cross section derived in (3.15) and let the charge q equal the charge of an electron, -e. Defining the classical electron radius, r0 = e2 / mc2 = 2.818 × 10-5 Å, we have
| (84) |
If we have a collection of N identical scatterers at various positions ri, then the phase factors in the electric field amplitudes (3.4) and (3.9) become important. The ratio E2 / E1 contains the phase factor ei (k2 - k1) ·r. For the collection,
| (85) |
The square of the sum of the phase factors is called the structure factor, and is defined as
| (86) |
It is common to define the ``momentum transfer'' q = k2 - k1, in regards to the momentum that is transferred to the scattered charge. Keep in mind that q has dimensions of inverse length (true momentum would require a factor of (h/2p)). For elastic scattering, the wavevector of the incident and scattered waves have an identical magnitude k = [(2 p)/( l)]. From the simple vector addition in figure 3.2,
| (87) |
| (88) |
If the scatterers form a continuous body rather than discrete point particles, we can rewrite (3.28) as an integral
| (89) |
We can separate the integrals in the structure factor so that
| (90) |
| (91) |
| (92) |
| (93) |
We can also define a modified structure factor with the forward scattering (q = 0) removed.
| (94) |
In this treatment of scattering from multiple objects, we have taken the first Born approximation. In our language of classical continuum fields, the scattered field does not interfere with itself. In the quantum description, this is equivalent to each x-ray photon scattering from, at most, one electron. The assumption of negligible ``multiple scattering'' is usually valid because the numerical value of the classical electron radius (2.818 × 10-5 Å) is so small. This ``kinematic'' approximation fails when the observed scattering becomes large. One instance of this is small-angle scattering (q ® 0). Another is when there is a coherent superposition of fields (Bragg diffraction).
I will define the two-point correlation function (also known as the pair correlation function) as
| (95) |
For an translationally invariant system, ár(r1,r2) ñ = ár(r1 -r2) ñ, implying g(r1,r2) = g(r1 - r2). For a homogeneous system, the one-particle densities are equivalent: r(r1) = r(r2) = r. In general, for large enough r = r1 - r2, we expect that the particles will be uncorrelated:
| (96) |
From (3.36), we see that the correlation function is just proportional to the Fourier transform of the modified structure factor. We can shift the coordinate system to place r2 at the origin. Then the Fourier transform of rg(r) is
| ||||||||||||||||||||||||||||||||||||||||||||||||||
Because it is a convenient abbreviation that I will use in section 3.10, let me mention one more relation.
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Previously (section 3.5), we discussed the scattering from a free electron. In this section, we consider the x-ray scattering from an atom, based upon references [2,,,,,]. As shown in (3.15), a particle's cross section depends on its mass m as 1/m2. Since the proton/electron mass ratio is over 1800, the x-ray scattering from the nucleus is miniscule, and is usually neglected.
Ultimately, we would like to have a function, the atomic form factor, which accounts for all the subatomic structure, and tells us how the scattered amplitude is modified (compared with the case of a single free electron). Because this is a common desire, these functions are tabulated in standard references [152,1,2].
If we consider just the electron probability density cloud r(r) surrounding the nucleus and approximate each of the Z electrons as being free, then we arrive at the standard form factor
| (104) |
However, the free-electron assumption is only an approximation, and fails most noticeably near ``adsorption edges''. When the x-ray energy is tuned close to an absorption edge, it can eject a core-level electron from the atom. (This is related to the electron-induced ionization discussed in section 3.3).
Deviations of the measured form factor f from f0 are known as as ``anomalous dispersion''. Typically, this modification to the amplitude is separated into a real term f¢(E,Z) and an imaginary term f¢¢(E,Z). The latter term allows for a change of phase in the scattered beam and is manifested as absorption. Physically, the anomalous dispersion arises from the resonance of the incident x-ray with differences between atomic energy levels.
Summarizing the various terms that comprise the atomic form factor f(q,E,Z):
| (105) |
Absorption of x-rays is caused by an incident x-ray striking an atom and causing a core level electron to be ejected [61]. This is simply the photoelectric effect, which is usually presented in the context of ultraviolet photons incident upon the outer shells. Historically, this was one of the most dramatic experiments leading to the quantum paradigm. After the photoelectron is ejected, the atom is in an excited state. Just as described in section 3.3, an electron will fall into the vacated state and emit a characteristic x-ray; this process is called ``fluorescence''. Because the electron falls from an atomic level and not from the vacuum, the fluorescence energy is always less than the energy of the absorbed x-ray.
Empirically, the absorption of x-rays by matter is observed to be
| (106) |
| (107) |
The absorption coefficient mi peaks dramatically near energies that correspond to the absorption edges. Following each edge is a branch of the absorption curve following the form [62]
| (108) |
In order to minimize absorption, we used an incident x-ray energy of 8800 eV, near the bottom of the absorption branch, but sufficiently far from the Cu K (8979 eV) edge. This choice of energy also reduced fluorescence from the Cu atoms that occurs at or above the absorption edge.
A Bravais lattice is the set of points that can be reached from a single point by applying translation vectors. These translation vectors are called ``basis vectors'' and are equal to the number of dimensions of the space of the lattice. For real crystals described by Bravais lattices, there are three basis vectors. Calling these basis vectors ax, ay, az, the Bravais lattice consists of the set of vectors {R},
| (109) |
Any Bravais lattice also has a reciprocal lattice [10], which is the set of plane wave vectors {q} that have the same periodicity as the Bravais lattice. That is, ei q ·r = ei q ·(R+r), or
| (110) |
Consider an infinite array of point charges
| (111) |
| (112) |
| (113) |
| (114) |
The calculation for a three-dimensional crystal is just a trivial extension of the one-dimensional case. Take a Bravais lattice as
| (115) |
| |||||||||||||||||||||||||||||||||||
| |||||||||||||||||||||
| (121) |
| (122) |
| |||||||||||||||
Take d as the projection of a given R along q. This d is the distance between two planes of the lattice. From q ·R = q d = 2 n p and the definition of q (3.30), we recover Bragg's law:
| (125) |
In this dissertation, we will be most concerned with the structure of platinum. This is a face-centered cubic crystal, which can be visualized as a cubic crystal with an extra atom at the center of each of the cubic faces. Some crystal structures are not Bravais lattices1, for example, silicon and diamond. They can, however, be described as a face-centered cubic crystal with another face-centered cubic crystal superimposed upon it. The second lattice has a relative displacement of 1/4 along the cubic body diagonal. The ``diamond structure'' is described by vectors that run over the Bravais lattice, but also include the basis vector for this displacement.
Although the face-centered cubic structure is a Bravais lattice, it is convenient to describe it as a simple cubic lattice with a basis. The basis vectors in this case describe the atoms on the faces. For a simple cubic crystal with basis vectors a [^x], a [^y], and a [^z], the displacement basis vectors that generate the face-centered cubic crystal are
| |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| |||||||||||||||||||||||||||
This section is based upon [11,,,,]. Consider an ideal crystal as in (3.53),
| (132) |
| (133) |
To consider an inelastic scattering process, take the dynamical structure factor [8]
| ||||||||||||||||||||||||||||||
| (135) |
| (136) |
| (137) |
| (138) |
| (139) |
| (140) |
The extra intensity goes into the other modes, which are the higher-order terms in the expansion of the exponential. The scattering from these modes is called ``thermal diffuse scattering''. The successive terms in the expansion consists of zero, one-phonon, two-phonon, etc. processes. While the zeroth-order term is elastic, all the others are not. The first-order thermal diffuse scattering has a discrete energy spectrum, while the second-order and higher terms are smooth functions of scattered x-ray energy [12]. The second-order thermal diffuse scattering has the same intensity as Compton scattering, and is often ignored.
It is often difficult to separate out the elastic from the inelastic terms, because of the poor relative energy resolution in x-ray detection. The incident x-ray energy is on the order of keV, while the thermal excitations are on the order of kB T, 1/40 eV at room temperature. In practice, we are accepting such a broad range of energies that all energy transfers are selected. This is equivalent to performing an integration of (3.76) over all energy transfers w
| ||||||||||||||||||||||||||||
X-ray scattering from surfaces is usually presented in either of two ways. In section 3.11.1, I present the method of truncating an infinite crystal, popularized by Robinson [111]. The following section (3.11.2) describes the more traditional approach from classical electrodynamics. The final section (3.11.3) establishes the connection between these two approaches.
This section builds upon the treatment of the infinite crystal in section 3.9, and extends it to semi-infinite and finite crystals.
In analogy with (3.54), consider an ideal semi-infinite crystal. One face of the crystal is presumed to be truncated at x = 0 while the opposite ``end'' stretches to infinity. As before, we begin with the one-dimensional case, which illustrates the relevant behavior.
| (142) |
| (143) |
Now, we truncate the crystal at both ends, so that it is a one-dimensional crystal containing N scatterers.
Using the relation
| (144) |
| (145) |
| (146) |
The structure factor here is just like the diffraction intensity from a diffraction grating with N slits (shown in figure 3.5). Near the Bragg diffraction peaks (q a ® 2 pn), the limit limx ® 0 [sinN x/ sinx] = N. Because the crystal is finite, the structure factor maxima are now N2, not infinite. The minimum value is zero, which is also in contrast with the semi-infinite crystal, but identical to the infinite crystal.
The extension to three dimensions is straightforward [111]. Consider a three-dimensional crystal with Nx, Ny, Nz scatterers in the x, y, z directions. The scattering amplitude is then
| (147) |
| (148) |
| (149) |
Taking the Nx ® ¥, Ny ® ¥ limit but holding Nz finite, we obtain a physically reasonable depiction of a crystal surface.
| (150) |
As mentioned in section 3.9.1, absorption and extinction limit the x-ray scattering intensity. Because of this finite penetration depth, even a truly semi-infinite crystal will scatter like a finite crystal. A reasonable order-of-magnitude for Nz is 1000, given a penetration depth » 1 mm [111]. Figure 3.6 plots (3.92), proportional to the intensity along a CTR.
An alternative treatment is to consider x-ray scattering from a surface as an example of the more general problem of reflection and refraction at a boundary between two dielectric media. Since x-rays are just electromagnetic radiation, this should be perfectly valid. For the moment, however, we neglect the atomistic nature of the sample and assume it to be a smooth, continuous structure. The atomic periodicity can be added in later (section 3.11.3).
Consider a smooth interface between air (n = 1) and a block of amorphous material (n < 1 for x-rays). This ties into the classical electrodynamic treatment described by Jackson [84], with n¢® n, n ® 1, and m = m¢. Jackson uses qi, the angle between the incident beam and the surface normal. For consistency with the rest of the dissertation, I write results in terms of its complementary angle a between the incident beam and the surface.
Then the component of the electric field perpendicular to the plane of reflection are
| ||||||||||||||||||||||||||||||
| ||||||||||||||||||||||||||||||
The Fresnel reflection and transmission coefficients are plotted in figure 3.8. If n < 1, as we will show is true for x-rays incident on most materials, then there exists a critical angle ac such that n = cosac. For a < ac, cosa > n, so the square roots become imaginary and the magnitude of Er / E0 is unity for both polarizations. This is termed total external reflection.
While a long derivation of the index of refraction can be found in Warren [147], a simpler and more illuminating treatment can be found in Jackson [85]. I will not repeat the entire model here, but just connect the results to our discussion.
For large enough photon energies ((h/2p) w), the dielectric constant approaches the ``plasma limit'' and
| (155) |
| (156) |
| (157) |
| (158) |
The critical angle is defined by n = cosac. By expanding the cosine to second order in the critical angle (which should be small), we obtain
| (159) |
As an example, the critical angle for platinum is calculated in this section.
The atomic weight of platinum, W, is 195.078 g/mol, its atomic number Z = 78, and the mass density rm = 21.090 g/cm3. Hence, the density of electrons in platinum is
| (160) |
For a Cu K a emission line, l = 1.54 Å, so d = 5.401 ×10-5 by (3.100) and ac = 1.04 ×10-2 (radians) by (3.101). Note that d << 1 as claimed previously. This is true even for this extreme example of a high Z, high mass-density material.
There are some apparent disparities between the results of the CTR theory (section 3.11.1) and the classical Fresnel reflectivity (section 3.11.2). The former predicts Bragg peaks connected by crystal truncation rods. Given no adsorption and a truncated infinite crystal, these Bragg peaks are predicted to have infinite intensity. The Fresnel formulae have no Bragg peaks or truncation rods, and the intensity maximum saturates at unity below the critical angle.
The Fresnel treatment cannot predict Bragg peaks, because the scattering media is assumed to be a solid block of constant density. The CTR treatment assumes a perfect crystalline lattice. If we extend the CTR treatment to consider continuous, homogeneous media, then integrals will take the place of summations. The scattering amplitude is
| (161) |
| (162) |
| (163) | |||||||||||
| ||||||||||||||||||||||||||||||||||||
| (167) |
| (168) |
| ||||||||||||||||||||||||||||||||||||
The result S(qz) µ 1/qz2 can be obtained in two other ways. Taking the continuum limit a ® 0 of the semi-infinite structure factor (3.85) yields this directly. The second method is to follow [124] and consider the structure factor (3.31) again:
| (170) |
| (171) |
| (172) |
Throughout this chapter, I have implicitly used the ``kinematic'' theory, which assumes that the first Born approximation holds. Because x-rays interact very weakly with matter, the approximation is valid for most points in reciprocal space. However, at Bragg points (where Bragg's Law is satisfied) and near the (000) reciprocal space point, the Born approximation breaks down. That is why the CTR treatment fails to predict the existence of the critical angle or the finite intensity for a semi-infinite crystal.
However, we can show the equivalence of the two approaches far from (000), in the large-a limit. Starting with (3.94), rewrite cos2a = 1 - sin2 a. Since n2 - 1 is very small, we can approximate the terms within the square roots by
| (173) |
| (174) |
To incorporate the Fresnel coefficients into simple kinematic x-ray scattering, multiply each amplitude, both incoming and outgoing, by Et/E0 as in (3.95) or (3.93). Because n is so close to unity, it hardly matters which polarization is assumed. From (3.29), qz = 2 k sina and then defining (qz)c = 2 k sinac, we can simplify (3.93) to be
| (175) |
In this chapter, the specific procedures and apparatus used in these experiments are documented. First, the sample preparation protocol is described. The next section details the electrochemical apparatus and procedures. The following section describes the x-ray scattering apparatus, and the timing apparatus is discussed thereafter. Finally, suggestions for future improvements are made.
Samples (nominally Pt(111)) were obtained from the Materials Science Center growth facility in Bard Hall. These samples were oriented through Laue back reflection, and then cut to the desired orientation by electrical discharge. Then, they were polished with SiC paper and Al2O3 powder down to a grit size of 0.25mm until a mirror-like surface was obtained.
In principle, this procedure should produce crystals with well-oriented faces. To allow large terraces to form on metal crystals, it is desirable to reduce the miscut angle between the crystallographic axis (e.g., (111)) and the surface normal. However, miscut angles as large as 2° were measured in our lab by a combination of laser reflection and high-resolution Bragg diffraction. These can be traced to the Materials Science Center crystal mounting apparatus, which was insufficiently rigid to ensure a miscut smaller than a few degrees.
In this section, the angle q = (2q)/2 is the Bragg diffraction angle, while f is a rotation angle about the surface normal. On a miscut crystal, the Bragg diffraction peak will not be coincident with the surface normal. The angle between them is defined to be g.
The surface normal was aligned with the f axis as follows. By reflecting a laser beam from the mirror-like face of the crystal, a tight spot was cast onto a far wall or ceiling. Rotating the crystal about f caused the laser beam to trace out a cone, causing the spot to trace out a corresponding ellipse on the wall. By adjusting the tilt stages on the sample goniometer, the ellipse could be narrowed until the spot did not move with f. Then, the surface normal was well-aligned with the f axis.
Moving the f angle to some fiduciary value, such as 0°, the Bragg diffraction angle q was recorded. Then, f was set to 180° and a different q was found. These two angles differ because the diffraction peak traces out a cone, similarly to the laser beam. The projection of the miscut along this one axis is (see figure 4.1)
| (176) |
Likewise, by taking measurements at f = 90° and f = 270°, the orthogonal projection of g is measured,
| (177) |
| (178) |
After the miscut of each platinum crystal was measured, it was mounted onto the orienting/polishing apparatus shown in figure 4.2. The apparatus consists of a cylindrical barrel (E), with three dowels mounted on it (C). (Only two dowels are shown in the figure.) These form one half of the ``kinematic mount''; the other half is a thick disk (B), into which the dowels press. Opposite the first dowel is a circular depression (G), opposite the second is a groove (H), and opposite the third is just the flat surface of the disk. One of the dowels is fixed; the other two can be raised and lowered my means of small adjustment screws running through the barrel. These permit the disk to be oriented by a few degrees in any direction with respect to the barrel axis. A long screw (D) attached to a spring and knurled knob (F) runs through the barrel axis and passes through the clearance hole (I). When tightened, the orientation is securely fixed. Finally, a mushroom-shaped tip (A) fastens to the kinematic disk (B) with three screws. The sample fits on to the end of this tip.
To prepare a sample, the tip of the apparatus was detached from the main body, and then placed on a hot plate. A crystal-bonding compound, liquid at high temperatures, was dabbed onto the tip before the platinum crystal was added. After cooling, the compound solidified to form a rigid, yet reversible, bond. The tip was secured to the apparatus, and placed at the center of a rotation of a four-circle diffractometer. The orientation screws were adjusted until the (111) Bragg diffraction peak was constant in q for any rotation f about the barrel axis. When this was attained, the locking thumb screw was secured and the entire barrel was placed within the polishing sleeve. The great advantage of this apparatus is that it can orient the sample face to high precision, lock in that orientation, and then polish without loss of precision.
Polishing took place on a polishing wheel (Ecomet 4) run at the slowest speed (50 revolutions per minute). Each polishing step took 15 minutes, and the sample was thoroughly cleaned with water between steps. The polish began with sandpaper (Buehler 600 grit Carbimet paper discs #30-5112-600) and then successively finer (6mm, 3mm, 1mm, 0.25mm) diamond powder (Struers DP-Spray, P) on a nylon cloth (Buehler #40-7072) with some lubricant (Struers DP Lubricant Blue, HQ or Struers DP Lubricant Red, HQ). At the end of this process, a mirror finish was invariably obtained.
Thereafter, the tip was unscrewed from the apparatus and warmed on the hot plate to remove the sample. The sample was then immersed in hot nitric acid for at least four hours. This was done to remove any remaining contaminants, particularly polishing powder. Also, even the smallest powder size (0.25mm) is extremely large on the length scales that x-rays probe. To remove strain and small grooves in the crystal surface, it was annealed in a gas flame (available on tap in Clark Hall) for at least one hour. Finally, the sample was characterized and the miscut calculated as described above.
Figure 4.3 demonstrates the dramatic improvement that can take place after a sample is annealed. This sample was annealed for one hour with a propane torch. For comparison, both intensities have been normalized to yield a peak value of unity. Without this, the post-annealing peak would dwarf the pre-annealing peak. The true peak intensities differ by a factor of 23. The full-width at half-maximum was 0.60° before annealing and 0.032° after annealing, a factor of 19.
After several iterations of this orienting-polishing-annealing procedure, the bulk mosaic of the platinum crystal was » 0.018° (full-width at half-maximum) and the surface normal was oriented to within 0.027° of the (111) direction. Empirically, we have found that both the mosaic and the miscut must be small in order to observe the incommensurate overlayer. Furthermore, a high quality substrate enhances the quality of voltammetric profiles. The development of this procedure was crucial to the success of this experiment. It has also propagated to other groups (Cooper, Ho) in Clark Hall, and has enabled them to improve surface quality and signal-to-noise ratios in their own data.
Most of the solutions were prepared by Lisa Buller, and the following paragraph is paraphrased from her dissertation [52].
All solutions were prepared using water purified by a Hydro purification train and a Millipore Milli-Q system. The ionic salts were used as received and always the purest available. Perchloric acid solutions were prepared by dissolving either CuO (99.999%, Aldrich) or CuCl2 (99.999%, Aldrich) in Ultrex perchloric acid. The addition of chloride anions was achieved through the addition of CuCl2 or NaCl (99.999%, Aldrich). All solutions were bubbled for at least 15 minutes with pre-purified nitrogen, which was further purified by passage through oxygen-absorbing (MG Industries Oxisorb) and hydrocarbon (Fisher Scientific Activated Carbon 6-14 Mesh) traps to remove all traces of oxygen.
A typical well-designed electrochemical cell has three electrodes [118,43,44]. The guiding principle is to have all the interesting behavior occur at the working electrode. The other electrodes should be relatively inert and not complicate the analysis of the processes that occur at the working electrode. All potentials must be measured relative to some other reference value, which is provided by the reference electrode. The perfect reference electrode would be ``ideally nonpolarizable''. That is, its potential remains constant, regardless of the amount of current passing through it. Another purpose of the reference electrode is to ensure that an applied potential change does what we expect. Suppose we change the voltage of the potentiostat by DV. How do we know that this causes a DV at the working electrode and that part of the DV does not go into the reference electrode? If the reference electrode is ideally nonpolarizable, it maintains the same potential value, and the full DV is effected at the working electrode interface.
The counter (or auxiliary) electrode assists in this process. If the reference electrode is passing a significant amount of current, then the assumption of ideal nonpolariziability is sorely tested. It is preferable to have an alternate, low-resistance, pathway through which most of the current flows. Counter electrodes are often composed of inert metals and have large surface areas to minimize their overall resistance.
The various electrodes used in our experiments are shown in figure 4.4. The large (10 mm) electrodes used for the simultaneous in situ x-ray and electrochemical measurements are labeled by (a). The smaller (1-2 mm) ``ball'' electrodes, labeled by (b), were produced by members of the Abruña group. These were of excellent quality, and produced good electrochemical signals. However, they were too small and too difficult to orient to be of use in our x-ray measurements. A Ag/AgCl reference electrode is labeled by (c). These were constructed by Lisa Buller [52].
For electrochemical experiments on single crystals, a hanging-meniscus cell is ideal. A wire is spot-welded to the sides of the crystal face, as in figure 4.4(a,b). The face of the crystal is then dipped into the solution compartment (see figure 4.5), and pulled upwards so that only a meniscus connects the sample with the bulk of the solution. The reference and counter electrodes are placed in another compartment, connected by a frit (partially fused glass).
The advantage of this arrangement is that only the crystal face of interest in contact with the solution. Also, it is easy to use small (a few mm diameter) crystals, which are often better quality than large (10 mm diameter) crystals. The disadvantage is that the cell must remain in a fixed vertical configuration; this requirement is incompatible with most x-ray diffractometers. We used this cell only for voltammetric and current transient measurements.
To perform simultaneous electrochemical and x-ray measurements, we constructed a cell similar to the one developed by Toney and coworkers [113]. This is a reflection-geometry cell, as shown in figure 4.6. The entire sample is immersed in solution, unlike the hanging-meniscus cell. The solution is contained by 6mm polypropylene film, held in place by an O-ring.
A detailed illustration of the cell is provided by figure 4.7. The majority of the cell is Teflon (Kel-F is an alternative material with greater strength). The sample is placed in the center and held in place by two non-circular Kel-F screws that squeeze the sample laterally. The sample and screws are raised with respect to a trough, where most of the solution resides. The reference electrode is inserted from the side. The counter electrode is a platinum wire that circumnavigates the trough several times.
The x-ray reflection geometry places a limit on the in situ x-ray cell. The polypropylene film is extremely thin, and while contributing to the diffuse x-ray scattering background, does not We must incorporate the absorption of x-rays due to the layer of solution that is covering the sample.
Consider an adsorbing layer of thickness l and attenuation per unit length m. From figure 4.8, the total path length of x-rays through the solution layer will be x = l sina+ l sinb, where a is the angle of incidence, and b is the angle of reflection. For grazing incidence (small a), l is limited by the horizontal dimensions of the sample. In the specular (a = b) case, we have x = 2 l sina.
From the relation qz = [(4 p)/( l)] sina (3.30) and the absorption relation for the intensity I = I0 e-mx (3.48), then
| (179) |
For aqueous solutions, the absorption coefficient can be calculated from (3.49): m = 9.848 cm-1 for Cu Ka radiation (l = 1.542Å) and m = 1.061 cm-1 for Mo Ka radiation (l = 0.711Å). The 1/e absorption length is 1.0 mm for Ka and 0.942 cm for Mo Ka. Clearly, using high energy x-rays greatly reduces the problem of absorption.
For an typical L = 1.5, qz = 1.387Å and l = 1.542Å, even l = 1mm of solution causes an attenuation of 40%. It is therefore important to remove as much solution from the cell as possible, while still leaving enough to maintain good electrical contact between the face of the working electrode and the other two electrodes.
A potentiostat is an instrument to keep the sample under potential (voltage) control and monitors the current. (A galvanostat, in contrast, keeps the sample under current control and monitors the voltage.) The simplest possible potentiostat circuit for a three-electrode configuration is shown in figure 4.9.
The operational amplifier will supply sufficient current to keep the reference electrode at a potential -V with respect to ground (or the working electrode). Significant current will pass from the counter into the working electrode, but very little will pass through the reference electrode. This is in accordance with section 4.3.2.
The PAR 283 (Princeton Applied Research, Model 283) is a versatile instrument, which can be run as either a potentiostat or a galvanostat. It accepts commands over a GPIB (IEEE-488) interface, and has its own sophisticated, if unique, command language. During most experimental runs, we used the PAR 283 to acquire either cyclic voltammograms (described in section 5.2) or chronoamperometric transients. Cyclic voltammograms were taken of each sample while the solution layer was extended, when the solution was pulled out, and at various points during the experimental run on a given sample.
In a dilute (0.1 M) form, perchloric acid poses a minor health hazard. Contact with skin is mildly irritating, and should be rinsed off as soon as possible. Contact with the eye is more serious. For this reason, splash goggles should be worn at all times. In case of a large spill, sodium bicarbonate should be available for neutralization.
The platinum sample glows yellow-white during annealing. There is a significant ultraviolet spectral component, and the sample needs to be kept under continual supervision. To prevent permanent retinal damage, ultraviolet-resistant goggles must be worn during this process.
Before a sample is inserted into the x-ray cell, a careful protocol must be observed. UPD is extremely sensitive to chemical contaminants, especially metallic and organic ones.
X-ray scattering is a nearly ideal probe of the ordering kinetics of the two-dimensional overlayers found in UPD systems. Unlike electrons or neutrons, X-rays can penetrate through a thin solution layer, allowing the experiments to be performed in situ. X-rays provide structural information on atomic length scales without perturbing the system with mechanical probes or large fields, as scanning probe microscopes may. Finally, the extremely high flux from a modern synchrotron x-ray source, such as the National Synchrotron Light Source (NSLS), permits the weak diffraction signal from a single Cu-Cl bilayer to be studied at high resolution.
In our experiments, the white beam produced by a bend magnet on the NSLS electron storage ring was focused in both transverse directions by a total external reflection mirror. A monochromator consisting of two Ge(111) crystals was configured to select 8.80 keV x-rays. The substrate was placed in a thin film geometry x-ray cell similar to those used by Toney and coworkers [113]. The cell was placed at the center of rotation of an Eulerian cradle and two pairs of XY-slits between the sample and the detector determined the resolution of the scattered x-rays. The resolution is discussed in detail in section 5.5.
In the lab, x-rays were produced by a Rigaku (Model RU200) rotating Cu anode source. The Cu Ka1 was selected by means of either a single or triple-bounce Si(111) monochromator. Although the instrument can provide a 60 kV accelerating voltage and 200 mA filament current, the lowest power setting (20 kV, 10 mA) was usually sufficient for sample orientation.
To detect x-rays we used an integrated NaI scintillation crystal, photomultiplier, and preamplifier (Bicron 1XMP 040B-X). The resulting electrical signal was sent through a combined amplifier and pulse-height analyzer (Canberra Model 1718) for broad energy discrimination. The TTL pulses were then sent to a simple adding memory module (Kinetic Systems 3610 Hex Counter) that also received timing pulses from another module (Kinetic Systems 3655 Timing Generator).
In the lab, the signals were then acquired by a data acquisition card (DSP 6001). At the NSLS, data acquisition was handled by a CAMAC to SCSI interface module. In both places, the four-circle diffractometer (Huber) was under the control of a sophisticated software package (``spec'', by Certified Scientific Software) running on an Intel 486-based computer.
Time-resolved x-ray measurements can be accomplished in several ways.
For instance, Bergmann et al. [32] used the timing of the electron bunches around the synchrotron ring for Mössbauer experiments. This is ideal for extremely short time ranges. Very recently, Knight et al. [88] have demonstrated a prototype device etched onto a silicon wafer to study protein folding. This works by mixing two jets together (for instance, folded protein and a denaturing agent) and squirting the product through a long channel. Because the flow is lamellar, the mixing occurs by diffusion. Because the fluid volumes are extremely low (nanoliters), the diffusive length scale is extremely short, and the mixing time is on the order of microseconds. By moving the device along the x-ray beam, different times after the mixing event are examined. In this way, position and time are coupled.
In contrast, our method relies upon timing electronics to separate the x-ray signal into various time bins. This ``stroboscopic'' method was first used by our group to study charge-density wave kinetics [127]. Although the PAR 283 claims to have a trigger, it does not operate in the standard sense of the term. Normally, when an instrument (an oscilloscope, for example) is waiting for an electronic trigger, operation ceases until the trigger is detected. Then, the other operations are begun or resumed. Instead, the PAR 283 performs a variety of operations, periodically polling the input to see if the trigger signal has arrived. Only then is the specified series of actions initiated. This can lead to an unpredictable delay between the trigger input and the initiation of commands by the PAR 283. For this reason, it was decided to have the PAR 283 be the master controller and send trigger signals to the other instruments.
The control diagram is shown in figure 4.11. The potentiostat applies a voltage to the sample and continuously reads current from it. At the beginning of a voltage cycle, it sends a trigger pulse to the waveform generator (Keithley 3940 multifunction synthesizer). This sends a series of pulses to the multichannel scaling averager (DSP 2190), which consisted of a multichannel scaling module (DSP 2090) and a signal averaging memory (DSP 4101). These bin pulses both initiated the averaging memory and incremented the current memory location (time bin). These timing modules also received x-ray intensity data, which was added to the time bin. At the end of a series of voltage cycles, the memory was dumped to the computer for display and analysis. The chronoamperometric traces (current vs. time) were digitized into 5000 time bins, and collected by the potentiostat. At the end of the voltage cycle, these were also sent to the computer.
With the advent of high-energy synchrotron sources, x-ray cell geometries with a thick solution layer have become feasible. Brossard et al. [49] describe a cell very similar to ours, but without the thin solution layer constraints. The cyclic voltammetry measurements they present are not high quality; presumably, this is a function of sample preparation, and not the cell itself.
As discussed in section 4.2.1, these crystals were not ideal. The simplest course would be to procure samples from a reliable external source. If annealing is still necessary, a new method should be found. Heating with a torch sometimes produced cloudy spots in the center of the sample, where the flame was hottest. A more even annealing could be done in ultra-high vacuum and by attaching it to a heating stage.
In our experiment, the highest resolution (section 5.5) was obtained by rotating the sample about the surface normal. In this case, an area or linear detector would not be helpful. However, there may be cases in which the resolution is sufficient to simply have an area detector mounted on the end of the detector arm. A CCD (charge-coupled device) could be run in a mode such that each line is shifted down. In this case, all of the time-resolved data could be recorded on the device, which would speed up the data acquisition time by the number of q-points.
In retrospect, the PAR 283 potentiostat was difficult to program, and not as flexible as anticipated. A superior solution would be to purchase the best possible analog potentiostat (BAS is an good choice) that accepts an external line voltage. Then buy a good programmable digital-to-analog card that can be programmed easily and has sufficient time resolution.
It may also be advantageous to replace CAMAC modules with cards within the computers. At the time of these experiments, we needed to maintain compatibility with equipment at CHESS and NSLS X20A. Now, the DSP timing modules could be replaced with a multichannel scalar card (Oxford MCS, for instance). The counter/timer modules could be replaced with an integrated counter/timer card (Keithley CTM-010). These particular upgrades are already underway for the new spectrometer being set up in the Brock group laboratory.
This chapter begins the presentation of our data on the underpotential deposition (UPD) of Cu onto Pt(111) in the presence of Cl. The first section presents our cyclic voltammetry on this UPD system. Subsequent sections discuss the hexagonal coordinate system and the structure of the incommensurate UPD overlayer. Finally, our static x-ray measurements of this overlayer are presented.
Cyclic voltammetry, as the name suggests, is a measurement of the current while the voltage is being swept (usually linearly with time). The cyclic adjective refers to the fact that the voltage is swept in both directions. As a function of time, the applied voltage traces out a triangular wave (figure 5.1a).
Cyclic voltammetry is a commonly used technique in electrochemistry, with many different applications. The next section illustrates a simple example: the cyclic voltammogram from an adsorption / reduction reaction, where the adsorption follows a Langmuir isotherm. In our experiments, this technique provided information on the equilibrium phase diagram.
This section follows the theory presented in section 2.8. It may be helpful to review that section before continuing.
As suggested by Bard and Faulkner [31], consider the reduction of species O at the electrode to form species R.
| (180) |
The current density comes from the reaction (5.1), so
| (181) |
| (182) |
| (183) |
| (184) |
| (185) |
| (186) |
| (187) |
| (188) |
| (189) |
| (190) |
| (191) |
The voltammetric profile is shown in figure 5.2. These data were collected in our x-ray scattering cell at 5 mV/s with 0.1 M HClO4 as a supporting electrolyte, 1 mM Cu2+ and 10 mM Cl-. The current response exhibits two sharp and well-defined voltammetric deposition peaks centered at about +0.47 and +0.32 V (vs. a Ag/AgCl reference electrode). Upon reversing the potential sweep, the current response then exhibits two sharp stripping peaks corresponding to the reverse reactions.
A schematic of the deposition process is depicted in figure 5.3. The labels A, B, C in this figure also correspond to the potential regions in figure 5.2. At the rest potential (region A), chloride anions are adsorbed on the platinum surface in a non-ordered fashion [158,108]. As the potential is swept negatively, copper is electrodeposited onto the platinum surface at a well-defined potential [90,149,94]. The electrodeposited copper and chloride ions together form an ordered Cu-Cl bilayer structure incommensurate [131] with the platinum surface (region B). If the potential is then moved further in the negative direction, there is further copper deposition, creating a full, commensurate copper monolayer (region C) [94,90,156,157,95,72]. The copper monolayer is, in turn, believed to be covered by a disordered layer of chloride anions. On the reverse (positive-going) sweep the reverse processes take place; that is, some copper desorbs, forming the CuCl lattice structure (region B) and at more positive potentials the copper is completely stripped from the surface, leaving the disordered chloride anions adsorbed on the surface and returning the system to region A. The sharp voltammetric features seen in Figure 5.2 are the electrochemical signature of a clean and well-ordered surface.
In our experiments, we used cyclic voltammetry for three purposes. Most importantly, it served as a qualitative ``fingerprint'' of the UPD process itself. The cyclic voltammograms are extremely sensitive to contamination of the solution, poor quality of the single-crystal electrode surface, and dissolved oxygen in solution. Empirically, we found that obtaining a good cyclic voltammogram was a necessary, but not sufficient, condition to finding a well-ordered UPD layer with x-ray scattering.
Secondly, the width of the peaks tells us an important fact about this UPD process. As derived in section 5.2.1 and plotted in figure 5.1b, the full-width at half-maximum of the current peak should be close to 90.6 / n mV at room temperature. The fact that our peaks are significantly smaller than this value implies that there is significant interaction among the adsorbed ions in the UPD layer. In particular, once some ions are adsorbed/desorbed, this tends to enhance the probability that other ions will follow.
Thirdly, unlike figure 5.1b, the peak positions for the negative voltage sweep are displaced from their partners on the positive voltage sweep. This hysteresis, which is present even for very slow sweep rates (1 mV/s), is an indication that the system is kinetically limited. The reason for this, which had been unclear, is explained by our time-resolved data in chapter 6 in terms of a nucleation and growth model.
The remainder of this chapter concerns x-ray scattering from the platinum surface and the incommensurate CuCl overlayer. For cubic crystals, the basis vectors are usually defined to be mutually perpendicular and of equal length (like the x, y, and z Cartesian axes). When dealing with the (111) surface of a face-centered cubic lattice, however, it is convenient to redefine the basis vectors. The c axis is defined to be along the (111) surface normal. Because of the ABCABC... stacking, the reciprocal space (111) is mapped onto (003). The a and b real-space basis vectors, which lie in the plane of the surface, are shown in figure 5.4a. Because these basis vectors subtend 120°, these are often called ``hexagonal surface units'' [75]. The circles in the figure represent platinum atoms on the (111) surface. These real-space lattice sites are indexed in figure 5.5.
From the convention (3.64) that [^a]i ·[^q]i = dij, b* is orthogonal to a and c, and a* is orthogonal to b and c. So a* and b* must point in the directions indicated in figure 5.4b. These vectors subtend 60° and generate a triangular lattice. The reciprocal-space lattice sites are indexed in figure 5.6. Although this figure appears superficially identical to figure 5.5, the indexing is different due to the different angles subtended by the basis vectors. From this point on, Bragg peaks are indexed using these hexagonal units.
The conversion from cubic to hexagonal units is easily accomplished. Writing both q-vectors as column vectors, then the matrix product qcubic = Jh ® cqhexagonal and qhexagonal = Jc ® hqhexagonal. These transformation matrices are
| (192) | ||||||||||||||||||||||||
| (193) | ||||||||||||||||||||||||
Tidswell and coworkers [131] have characterized the incommensurate bilayer that is present for intermediate potentials. They found a triangular array of x-ray scattering rods, sharp in H and K but diffuse in L. The in-plane spacing was approximately 0.765 that of the truncation rods from the underlying platinum crystal. This corresponds to an in-plane bilayer lattice spacing 30% greater than Pt(111). Based upon their measurements of the positions and intensities of these scattering rods, they propose the model shown in figure 5.7. The Cu and Cl form a bilayer wherein the Cu atoms (small gray circles) are close to the Pt surface (large gray circles), and the Cl atoms (large empty circles) rest above the Cu, coordinated in the three-fold hollow sites of the hexagonal Cu lattice. If the Cl atoms are partially ionized toward Cl- (making them larger), it is reasonable to assume that they are in close proximity to one another and determine the incommensurate lattice spacing. Tidswell et al. claim that the spacing is near to that of close-packed spheres with the Cl- ionic radius.
It is surprising that the bilayer structure fails to follow the commensurate lattice spacing, yet preserves the orientation of the underlying Pt lattice. However, this scenario has been predicted by Novaco and McTague [103]. They hypothesize static-distortion waves (analogous to charge-density waves) in cases where the adlayer is weakly adsorbed to the substrate. Minimizing the energy leads to a preferred orientation of the adlayer with respect to the substrate. The relative orientation angle need not be zero. Shaw, Fain, and Chinn [122] experimentally observed Ar monolayers adsorbed onto graphite substrates at a range of low temperatures (32 - 52 K). Their LEED (low-energy electron diffraction) measurements demonstrated the relative orientation angle was inversely related to the Ar monolayer lattice spacing. Their previous measurements with other adsorbed noble gases find cases where the relative orientation is zero, as in our case. A complete review on this subject can be found in the Pokrovsky and Talapov [110]. Ben Ocko and coworkers [104] have studied a commensurate-incommensurate phase transition in Br UPD on Au(100). Their results are compared with theoretical predictions by Pokrovsky and Talapov [109].
Our interest is primarily in the kinetics of this system, rather than performing more detailed crystallography on the static phases. Therefore, it is necessary to find a useful parameter to monitor the emergence of order during formation of the bilayer. We chose the (0.765 0 L) rod, because it is the lowest index peak. Our choice of L = 1.5 depends upon two factors: the minimization of background scattering from solution and polypropylene film, and the L-dependent scattering from the bilayer itself. Absorption effects (section 4.3.4) are most prominent for low L, as is the diffuse background scattering. The bilayer scattering oscillates with L, as can be seen from examining the structure factor.
As mentioned above, the scattering is diffuse along the qz axis. This is a consequence of the nearly two-dimensional nature of the adsorbed layer. However, the spacing between the Cu and Cl layers causes an interference effect that is manifested in the oscillating intensity along qz. A single two-dimensional layer of hexagonally arranged atoms has a six-fold rotation axis about the surface normal. Add a second commensurate layer with the same number of atoms, by putting a chloride atom at position R + a for every copper lattice position R. Assuming that these chlorides are attracted to the copper layer, they will probably sit in the three-fold hollow sites between copper atoms. At this point, the symmetry is broken and the bilayer is only three-fold symmetric.
To derive the structure factor, we need to find the coordinates of the three-fold hollow site. Consider the location of the three-fold hollow site between (0,0), (1,0), and (1,1). Referring to figure 5.8, and recalling that the center of an equilateral triangle is 1/3 of the distance from a side to the opposite vertex, the position is
| (194) |
Now we turn our attention to the phase factors that influence the structure factor of the bilayer. We can consider the two-dimensional CuCl bilayer as though it were in isolation. The underlying Pt lattice has no fixed periodicity with respect to it, and so will not change any of the structure factors except at the specular condition H = K = 0. As shown in section 3.9.2, the structure factor due to the addition of another atom is
| (195) |
| (196) | ||||||||||||||||||||
This leads to a hexagonal lattice expanded by a factor of Ö3, and rotated by 30°, as shown in figure 5.9.
Now consider the L > 0 scattering. Due to the three-fold symmetry, the points (0 0), (1 0), and (0 1) illustrate all of the possible cases. (The (1 1) is equivalent to the (0 0).) These structure factors are plotted in figure 5.10. As expected, they follow a simple sinusoidal form, but the initial phase at L = 0 is determined by (5.17).
The careful reader will note that I have neglected the atomic form factors of Cu and Cl (that depend upon the ionization state, which is not known), and the ``Debye-Waller'' factors due to disorder within the Cu-Cl bilayer. However, in my opinion, there is a larger uncertainty that makes these considerations moot. The illustration in figure 5.9 is only one of two possibilities. The six-fold symmetry was broken by the assumption that the chloride atoms fall into the upward-pointing triangles of figure 5.5. We can equally well imagine that the chloride atoms fall into the downward-pointing triangles. This is equivalent to just a 60° rotation and changes the structure factors accordingly.
The three-fold hollow site immediately above (along the y-axis) from the origin in figure 5.9 is 1/3(1 2). (This is not shown in the figure, because we previously took the other choice.) The resulting structure factor is
| (197) | ||||||||||||||||||||
Even more likely is that there will be some combination of these two possibilities. As the incommensurate bilayer forms, different domains nucleate and grow on the surface (see chapter 6). There will be domains with both possible orientations. These domains are likely to be incoherent; that is, there will be no definite phase relationship between them. The overall intensity will be the sum of the intensities from all the domains. This intensity is shown in figure 5.11, which assumes an equal coverage for the two domain types. As can expected from the sum of two sinusoidal functions, the period of the oscillation is changed. The (0 1) and (1 0) structure factors are identical, since we are taking equal numbers from the two domain type, so H and K are identical. Of course, due to the stochastic nature of the nucleation-growth process, the distribution may be skewed toward one orientation over the other, instead of the 1:1 ratio depicted here.
Because of the conflicting reports of the CuCl overlayer structure at intermediate voltages, our first task was to make some static x-ray measurements. We have observed the overlayer structure numerous times during several experimental runs. Despite many attempts, we have never found scattering at the (0.25 m 0.25 n L) rods that can be attributed to the incommensurate overlayer, where m and n are integers up to 6. On the other hand, we have found scattering at the (0.765 0 L) and (0 0.765 L) rods that was voltage-dependent. This indicates that the measurements of Tidswell et al. [131] are correct, while the LEED measurements of Kolb [102] cannot be confirmed. Assuming that the (0.75 0 L) seen in the LEED corresponds to the (0.765 0 L) peak, then the additional peaks may be the result of multiple electron scattering. Alternatively, the ex situ experiment may change the ordering of the CuCl overlayer. This would not be surprising, as the vacuum and solution environments are very different.
Figure 5.12 shows the (0.765 0 1.5) overlayer Bragg peak at two different values of the applied potential. These data clearly demonstrate the presence of the incommensurate overlayer at 350 mV and its absence at 250 mV. The potential-independent background is due to scattering from the solution layer and the polypropylene film that contains it. By integrating for several seconds per q-point, the signal can be easily resolved above this background. In the time-resolved measurements (chapter 6), where the x-ray signal is split into many time bins, this poses a considerable experimental challenge.
The shape of the diffraction peak is well-fit by a Lorentzian line shape (the solid line in figure 5.12), and the half-width at half-maximum D corresponds to a correlation length x = 1/D » 280Å. A Lorentzian is appropriate for systems with only short range positional order. The inset indicates the location of the Bragg rods of the two-dimensional incommensurate overlayer (hollow) and the crystal truncation rods (section 3.11.1) of the Pt substrate (filled). The arrow represents the transverse scan shown in the main figure. The transverse direction is denoted by q^ and is orthogonal to (0.765 0) and at constant L.
The corresponding x-ray scans through the overlayer Bragg peak, but along the radial direction, are shown in figure 5.13. The radial direction is denoted by q|| and holds K and L constant. In this case, the peak is broader than scans through the q^ direction. At first glance, this appears to indicate that the correlation function is strongly asymmetric, with x^ > x||. This would be a surprising result. However, as the next paragraphs will show, this can be accounted for by the asymmetry of the resolution function.
The asymmetry of the resolution is due to the differing longitudinal and transverse resolutions. Figure 5.14 depicts the elements of the scattering geometry that determine the resolution. A variation in q = 1/2 (2q), the scattering angle between kf and ki, causes q to trace out the major axis of the resolution ellipse. A variation in the magnitudes ki and kf, due to a variation in the qbeam striking the monochromator, causes q to trace out the minor axis of the resolution ellipse. As shown in the figure, the longitudinal q and transverse q^ directions are not exactly coincident with the major and minor axes of the resolution ellipse, but are rotated by q with respect to it. A careful consideration of the resolution function is required for extremely high-resolution experiments [45]. For this relatively low-resolution experiment, the relative rotation is neglected.
As already used, ``perpendicular'' (q^) and ``parallel'' (q||) refer to vectors in the a*, b* plane (constant L). The term ``longitudinal'' refers a direction along the scattering vector q, while ``transverse'' is the direction orthogonal to this, but still in the scattering plane. At this point in reciprocal space, the q^ direction corresponds to the transverse direction. The q|| direction corresponds closely to the longitudinal direction, but is slightly different because of the constraint that L remain constant in q||.
The longitudinal resolution is found by differentiating the definition of q (3.30),
| (198) |
| (199) |
To confirm these calculations, we also measured x-ray intensities through the (1 0 L) crystal truncation rod (CTR) at the same L = 1.5 and q^, q|| directions as in figures 5.12 and 5.13. X-ray scans through the CTR and the overlayer rod are shown in figures 5.15 and 5.16. The top panel of each figure illustrates the intensity data, normalized to the beam monitor. The scattering from the overlayer is barely observable in comparison with the CTR scattering. This is not surprising, because the CTR intensity falls as ~ 1/(qz - q0)2, where q0 is the Bragg peak position, and L = 1 for this rod. The bottom panels illustrate the same data, but normalized to unity so that the widths may be compared. The overlayer rod is broader than the CTR in each figure.
The calculated resolutions and measured peak widths are summarized in table 5.1. Each quoted value is a full-width at half-maximum. The first and third columns are calculated longitudinal (5.19) and transverse (5.20) resolutions for the CTR and overlayer rod. The second and fourth columns are measured widths of peaks in the shown in the previous figures for the f, Dq^ and H, Dq|| directions.
It may seem surprising that the measured width dqH for the CTR is less than the calculated resolution dq. However, this is just an indication that H is not collinear with the longitudinal direction. Referring to figure 5.14, a cut through the resolution ellipse along a direction other than the major axis (the longitudinal direction) will always produce a more narrow profile.
| peak | dq (Å-1) | dqH (Å-1) | dq^ (Å-1) | dqf (Å-1) |
| Pt(111) CTR | 14.7 ×10-3 | 7.1 ×10-3 | 0.31 ×10-3 | 2.5 ×10-3 |
| CuCl overlayer | 15.0 ×10-3 | 18 ×10-3 | 0.25 ×10-3 | 7.1 ×10-3 |
As indicated, the resolution function is extremely asymmetric, with dq >> dq^. By comparing dqH and dq, the overlayer rod is seen to be resolution-limited when measured along q||. However, the overlayer rod is found to be very well-resolved along q^ by comparing dqf and dq^. From this consideration, widths from scans of the overlayer along q^ (figure 5.12) can be considered intrinsic to the CuCl overlayer itself, and a correlation length of x » 280Å can be quoted without resort to deconvolution.
The width of a CTR is related to the terrace size, but in a complicated way. For a self-affine surface (where there is no intrinsic length scale parallel to the surface), then the CTR width at the anti-Bragg position (midway between two Bragg peaks) is inversely proportional to the mean terrace size. When there is a characteristic surface length scale, the CTR width is in general a function of that length scale as well. In this case, the relationship to terrace size is specific to the correlation function which generates that length scale. As expected, the overlayer rods in our experiment are always significantly broader than the CTRs. This suggests that the terrace size is not the primary limitation on the mean island size. Without a separate and very careful surface crystallography experiment, however, this cannot be definitively proven.
In this chapter, simultaneous electrochemical and x-ray scattering measurements of the ordering kinetics of the Cu-Cl bilayer during the transition from the commensurate copper overlayer to the incommensurate bilayer are reported. First, the time-resolved data are presented. Then, a simple theory for the nucleation of the incommensurate phase is described. The subsequent section presents data to support this model. Next, the entire q-t data set is presented, followed by a theory to describe it. The data is analyzed in the context of this theory, and excellent agreement is found. The final sections concern alternate models and further theoretical explanations.
To observe the ordering kinetics during stripping, we employed a simple signal averaging technique. An example of the square-wave potential cycle that we applied is shown in figure 6.1a. At t = 0, the potential begins at 200 mV. The voltage is stepped to 350 mV at t = 10 seconds. At t = 30 seconds, the voltage is stepped back to 200 mV. This cycle repeats with a period of 40 seconds. Throughout this cycle, we simultaneously monitor both the current (figure 6.1b) and the intensity of the scattered x-rays at q^ corresponding to the peak of figure 5.12. As expected, the incommensurate scattering peak is present only for values of the potential within the incommensurate phase. Note that the rise in the intensity of the scattered x-rays in figure 6.1c is much slower than the corresponding current transient in figure 6.1b. In contrast, the scattered intensity falls on a time scale similar to that of the current transient.
The current transients describe the charge transfer at the electrode interface. These are due to two contributing processes: the capacitive charging of the double-layer, and the Faradaic charge transfer due to desorption/adsorption of ions.
Some previous chronoamperometric studies of closely related systems [79,80] exhibit distinct features in the current response that have been interpreted as evidence of nucleation. These characteristic features are not present in our data. We suspect that the geometry of the thin solution layer x-ray cell may be responsible for this difference. The capacitive effect is greater for our larger samples. This strong signal tends to mask other early features in the current response. Also, in our apparatus, as compared with hanging meniscus cells used in the other experiments, diffusion is comparatively insignificant. First, diffusion from the ``bulk'' solution is not a consideration for us, because the solution layer is so thin. Second, any diffusion that does take place will occur in one dimension, rather than three. Above the planar electrode face, the solution layer forms a very short cylinder. The ions are in close proximity to the surface and conditions are probably relatively uniform across the face, so diffusion is primarily along the surface normal.
After numerous attempts, we can definitely say that there are no features in the current response at the same time as the x-ray response. So the measured current response is ascribed to desorption/deposition into a disordered state (accompanied by charge transfer) which then gives rise to the nucleation and growth of the equilibrium ordered phase. The rise in x-ray intensity corresponds to an increased population in the incommensurate ordered phase. The desorption process can be separated from the ordering process due to the widely disparate time scales involved. Based on these data, we hypothesize a scenario wherein the abrupt positive voltage step causes a expulsion of some of the adsorbed copper ions. The remaining disordered ions gradually reorganize into a two-dimensional crystalline state with a larger lattice constant, incommensurate with the platinum substrate.
Consider the nucleation of (ordered) islands from a disordered phase, as depicted in figure 6.2. As usual, we define the Gibbs free energy of an island to be proportional to the (electro)chemical potential and to the number of particles that comprise the island. An island in contact with another phase will give rise to an excess free energy term proportional to the surface area of contact. So the existence of an island with N particles will lead to an excess of free energy
| (200) |
| (201) |
| (202) |
For an arbitrary (perhaps fractal) island of dimensionality d, then
| (203) |
| (204) |
| (205) |
Assume that there is some stochastic attempt frequency katt to make islands, only some of which are able to surmount the energy barrier. Also assume a Boltzmann distribution of energies in the attempt profile. Then the rate of islands successfully nucleated is
| (206) |
| (207) |
Let N(t) be the number of nuclei at time t. Following Schmickler, we assume ``first-order kinetics'' as follows:
| (208) |
Of course, these are only limiting cases. Although many experiments in the literature attempt to distinguish between instantaneous and progressive nucleation, we expect in general to find systems that exhibit both types of behavior, depending upon the rate of nucleation kN (which may be controllable) and the time scale of measurement.
In figure 6.4, these two cases are compared pictorially. For each column, the time axis runs downwards. In the instantaneous case (left side), all of the nucleation occurs before the first slide. As time advances, each island grows larger, but no new ones are nucleated. In the progressive case (right side), some islands have already nucleated before the first slide. However, islands continue to be nucleated even as their older siblings grow larger.
Thus, we expect that as we quench deeper and deeper into the incommensurate phase, the transition will occur ever more rapidly. To test this hypothesis, we performed a series of voltage step measurements in which the applied potential was stepped from 200 mV (within the commensurate phase) to varying potentials within the incommensurate phase. As in the previous measurement, we measured the scattered intensity at the incommensurate overlayer peak position as a function of time. This corresponds to a series of experiments similar to the one shown in figure 6.1, but varying the value of the more positive voltage. This is shown in figure 6.5.
We characterized the resulting transition time by fitting the x-ray intensity profiles (which resemble figure 6.1c) to a trapezoidal functional form, as shown in figure 6.6. While this model describes the data quite well, we ascribe no profound significance to it. Rather, we use it simply to define a characteristic time, t, which should be inversely proportional to kN. The inset to figure 6.7 plots the resulting t values. The characteristic time scale describing the ordering of the bilayer ranges varies from 50 seconds for shallow quenches to 0.7 seconds for deep ones. Figure 6.7 illustrates the exponential dependence of t on 1/h. This fits Eq. (6.6) with d = 2, over the entire phase region. The linear slope demonstrates that the growing islands are intrinsically two-dimensional (rather than three-dimensional mounds or pits on the surface) and that these islands are compact rather than fractal. The broad range of t also implies that our two-dimensional cell geometry has not inhibited the nucleation processes, but is only limited by the accessible range of voltage values. Furthermore, in all cases, t is longer than the current transient indicating that capacitive charging effects are not dominating our results.
Now we turn our attention to the development of order in the incommensurate structure formed after desorption. In order to understand the kinetics of this ordering process, we need to access the full q-t dependent x-ray scattering. We repeat the time-resolved measurement of figure 6.1 for a series of q-points linearly spaced along the same q^ direction as shown in figure 5.12. An example of such a measurement is shown in figure 6.8. The first thing to note is that the peak remains centered at a constant value of q , ruling out the possibility that the overlayer simply shifts its periodicity in response to the change in potential.
From the discussion in section 5.5, the correlation length is obtained from the width of the diffraction peak and is believed to be determined by the finite size of growing islands of Cu-Cl. The diffraction peak narrows with time, indicating that these islands are growing. The total coverage is proportional to the integrated intensity.
Ideally, we would collect a q-t data set for each f voltage transition. However, the single data set shown in figure 6.8 consumed 29 hours of synchrotron beam time. With our constraints, it was not feasible to consider collecting many data sets during that beam time allocation.
Because x-ray intensities obey Poisson counting statistics, for a measured signal intensity N, the standard deviation is ÖN. As seen in figure 6.8, the maximum signal in 1300 counts/second, while the background is 1000 counts/second. Because of this high background, we must count for long periods of time to resolve the signal.
Up to now, we have discussed the number of islands, but not the total volume comprising this phase. As shown in section 6.5, the islands are intrinsically two-dimensional. This is not intuitively pleasing, since the bilayer itself is two-dimensional. Anticipating this result, in this section we will limit ourselves to the case of two-dimensional islands.
Assume that each island grows by the incorporation of atoms into its boundary, and that this is the rate-limiting step for island growth. Then, for an island of N atoms and radius r, dN/dt = kg 2 pr. This defines kg, the rate constant for individual island growth, which has units of [length × time]-1. The area of the island is simply A = N / r. Since (for a circular island) we have A = pr2, we have two expressions for dA/dt:
| (209) |
| (210) |
Following Avrami [14,,16] consider a brief example of an area A with N circular islands, each of area a. The extended coverage is qext = N a / A. Of course, if the circles are placed randomly, then they will overlap somewhat and the true coverage q will be less than the extended coverage qext. While the true coverage is bounded by the limits q = 0 (no coverage) and q = 1 (complete coverage), the extended coverage can be infinite.
The probability that a particular point on the surface is not covered by a particular circle (1 - a / A). So the probability that it is not covered by any of N circles is (1 - a / A)N = (1 - (N a /A) / N)N. Assume that a << A. Then in the limit N ® ¥ and using limN ® ¥ (1 - x/N)N = e-x, this probability becomes exp( -Na/A ) = exp(-q). Finally, the probability that a point is covered (which is just the ``coverage'' q) is
| (211) |
In this section, I combine results from sections 6.7 and 6.8 to derive some simple expressions for the extended coverage.
In general,
| (212) |
| (213) |
In the instantaneous limit, we have N¥ nuclei that have all nucleated at t = 0, so the extended coverage is
| (214) |
| (215) |
| (216) |
| (217) |
From the coverage alone, it is not possible to determine all of these parameters individually. At best, in the intermediate cases, we can find the variables kN and the ratio a º N¥ (kg / r)2. In the progressive case, we cannot even determine these two variables independently, but only their product. It is important to note that qextinstantaneous is insensitive to kN, and so fits to (6.18) will also be insensitive to kN when that limit is approached. This is reasonable, as all the nuclei have already formed before the time scale of observation.
To begin the analysis of the data in figure 6.8, we fit each time slice to a Lorentzian line shape. A Lorentzian is the lowest order approximation to the structure factor for any system with only short range positional order. Some representative time slices and fits are shown by the thin lines in figure 6.9. From these fits, we extract the half-width at half-maximum (D) and integrated intensity vs. time. These are shown as circles in figure 6.10b. As expected for growing islands, D decreases with time. At the same time, the integrated intensity (proportional to the coverage) grows monotonically.
We can continue our analysis by incorporating the simple nucleation and growth model considered previously. Instead of fitting each time-slice independently, we now want to fit the entire data set from figure 6.8 with a simple function of few parameters. This intensity function I(q,t) we choose should satisfy the following conditions.
First, I(q,t) has a Lorentzian line shape at any fixed time t. This Lorentzian function is written
| (218) |
| (219) |
| (220) |
| (221) |
Combining (6.19), (6.20), (6.21), and (6.22), the two-dimensional model function is
| (222) |
We can now re-fit the entire data set (2 ×104 points) shown in figure 6.8 to the single function (6.23). The best fit to this simple model produces c2 = 1.04. The intensities from the model and the data are compared in figure 6.11, and appear to agree. However, it is easier to compare the contours of constant intensity that are shown in figure 6.10a. The contours of constant intensity for the model and the data agree very well. The generated integrated intensity I0(t) and D(t) functions are plotted as solid lines in figure 6.10b. They agree with our previous results (plotted as circles) where each time slice was fit independently. Returning to figure 6.9, we can also compare the intensities from the two-dimensional model (thick lines) with our previous results (thin lines) and the data itself (circles). Both of the lines fit the data quite well; the minor discrepancies between them are due to a difference in the form of the background function. In sum, these kinetic data are well-described as the nucleation and growth of a two-dimensional film.
All of the fit parameters are shown in table 6.1; the physically interesting ones are summarized in the top portion. The initial time t0, against which times t are measured, was allowed to float above 10 seconds (when the voltage was stepped), but fit to 10 seconds. There seems to be no time delay before the nucleation process begins. The growth parameter N¥ (kg / r)2 is a product of various parameters from (6.21) that cannot be separated. The saturation island number N¥ is likewise coupled with a proportionality factor that cannot be isolated. From the uncertainties shown in the table, all of the fit parameters are well-determined except for kN, to which the fit is relatively insensitive. This is an indication that either the data at early times (when there is very low signal) is insufficient to fix this parameter, or that the observations are in the instantaneous limit (6.15), where q does not depend on kN.
| parameter | variable | fit value | units |
| initial time | t0 | 10.0 ±0.32 | s |
| growth parameter | N¥ (kg / r)2 | 0.0404 ±0.0069 | Å2 s-2 |
| saturation island number | CN N¥ | 1.644 ×10-5 ±3.2 ×10-7 | Å-2 |
| nucleation rate constant | kN | 1.33 ±1.33 | s-1 |
| peak intensity coefficient | CI | 4.367 ±0.035 | arbitrary |
| peak position | q^ | -0.00117 ±3 ×10-5 | Å-1 |
| background constant | b0 | 962.1 ±0.4 | counts s-1 |
| background slope | b1 | 443 ±14 | Å counts s-1 |
We have attempted to test the assumption of first-order nucleation kinetics. Rearranging (6.9), we have
| (223) |
| (224) |
Previously, we posited a voltage-dependent Gibbs free energy (). For the case of two-dimensional circular clusters,
| (225) |
In this section, an alternative route to obtaining (6.26) is demonstrated. (This treatment was initially developed by Joel Brock [48].) Consider a model in which adsorption (described by a Langmuir isotherm) is driven by the change in potential. As the potential is varied the coverage varies, and the increasing coverage then drives a conventional phase transition. This is shown by the left side of figure 6.13. This density-driven phase transition may occur well before the adsorption transition is completed, in which case the system may still be in the linear region of the Langmuir isotherm (2.59).
Upon the formation of a circular cluster with radius r, the Gibbs free energy of the system changes by DG. DG has contributions from the difference of chemical potentials of the lattice phase mlat and disordered adsorbed phase mad and from the surface energy.
| (226) |
| (227) |
For simplicity, treat the adsorbed phase as an ideal gas. Define Pr to be the pressure at which a circular nucleus of radius r is in equilibrium with the gas phase. At P¥, mad = mlat. At Pr, mad - mlat = [(g)/( rr)]. From thermodynamics we have
| (228) |
| (229) |
| (230) |
| (231) |
| (232) |
| (233) |
| (234) |
In section 2.5.2, we considered the electrode to be a perfect sink for ions: any ions that arrive at z = 0 are deposited onto the electrode irreversibly. This is the meaning of the boundary condition (2.45)
| (235) |
However, this is not a reasonable approximation for UPD. Firstly, the coverage q will not exceed unity. Secondly, the kinetics of deposition can not be neglected. As in section 5.2.1, a simple model is one with a Langmuir isotherm. We will make a further assumption, that we can linearize the Langmuir isotherm, as in (2.59),
| (236) |
The treatment follows the derivation of the Cottrell equation (section 2.5.2). We arrive at the same result as (2.48),
| (237) |
| (238) |
| (239) |
| (240) |
| (241) |
| (242) |
| (243) |
| (244) |
| (245) |
| (246) |
| (247) |
In the limit that adsorption kinetics play no role (Gsat® ¥, or b ® ¥), then the second term in (6.48) vanishes and we recover the Cottrell current (2.53). Equation (6.48) is plotted in figure 6.14 for various values of b.
The basic scenario is that the applied potential step drives copper off of the surface and the Cu-Cl grains then nucleate and grow. Upon reversal of the potential step, the Cu-Cl bilayer is destroyed as the commensurate copper layer is formed.
From the exponential dependence of the time scale on voltage (figure 6.7), it is now clear why Cu/Cl/Pt(111) cyclic voltammetry (figure 5.2) always shows hysteresis, even for extremely slow sweep rates. Incorporating the nucleation and growth model into the cyclic voltammetry formula should lead to quantitative predictions for the degree of hysteresis as a function of sweep rate. However, it is unlikely that cyclic voltammetry alone would provide sufficient evidence for nucleation and growth; there are many processes that can affect electrode kinetics and cause similar responses. Instead, electrochemists typically study nucleation with chronoamperometry [79,80].
We have carefully considered possible effects of the thin solution layer on the ordering kinetics after a potential step. If the thin-layer were an inhibiting factor, we would expect to find an upper limit to the rate at which the nucleation process could take place. This is not apparent in our data. As shown in figure 6.7, the fastest observable time scale is limited by the constraint that the voltage quench not reach into phase A, beyond the incommensurate phase B. Furthermore, there is no evidence of any roll-over in the voltage dependence of the ordering time constant. Finally, the ordering time constant is always longer than the electronic time constant.
In standard deposition-nucleation problems [117], the voltage dependence in the Gibbs free energy (6.3) is a result of the differing electric potentials in the bulk solution and at the surface of the electrode. In our problem, the incommensurate and commensurate phases are both near the electrode surface. The charge transfer from the platinum surface precedes the ordering and may be voltage dependent. In particular, how much of the charge is shared between the Cu and Cl atoms is unknown. It has been shown that the ionization state does not jump directly from Cu0 to Cu2+ [78,5]. Another possibility is that the Cu or Cl ions change their positions (especially along the z direction) as a precursor to desorption out of the ordered state. Expanded or compressed layers, which are voltage-controlled, have been observed in several UPD systems.
To resolve these underlying issues, more data on this transition must be acquired. Time-resolved in situ reflectivity, both specular and non-specular, would help to clarify the positions and occupancies of the Cu and Cl layers throughout the desorption process. X-ray standing waves, which have been applied to UPD [4,35,3], are particularly sensitive to the positions of ordered layers above the electrode surface. It may also be helpful to take data on a related UPD process: Cu on Au(111) in H2SO4 [133]. This one of the most studied systems and has been extremely well-characterized. The gold surface is much easier to work with and does not oxidize as readily as platinum, simplifying the sample transfer (section 4.3.7) and subsequent data acquisition.
Scattering from the commensurate overlayer occurs at the same H, K positions as the Pt(111) crystal truncation rods and will be difficult to observe. Thus, experimental information on the reverse reaction, formation of the commensurate layer, will be difficult to obtain.
Finally, a comparison of the electrochemical current transients in the hanging-meniscus cell and the x-ray cell is necessary. Hanging-meniscus data show distinct nucleation bumps that are absent in the x-ray cell. While capacitive charging effects (which are enhanced in the x-ray cell) may mask some features, a systematic study of each would resolve the discrepancy and clarify the voltage dependence in (6.3).
In conclusion, we have studied a system wherein desorption (rather than deposition) is followed by ordering. The charge-transfer process is much faster than the development of long-range order. The x-ray data are well described by a nucleation and growth model with only a few parameters. The potential-step experiments demonstrate that the rate of ordering agrees well with nucleation models over two decades in time, and is not limited by the thin-layer geometry.
Electrochemistry is a good model system for studying growth phenomena in general. In comparison with in vacuo systems, it has the advantage that heteroepitaxial material can be removed to recover the initial substrate. So the same deposition (or desorption) processes can be studied repeatedly, under identical conditions, and without having to change samples. Also, electrochemical systems can be simpler, and components are less expensive, than for ultra-high vacuum systems. This is more practical for traveling to distant locations (such as synchrotrons).
Despite the ``conventional wisdom'', it is possible to perform good kinetic x-ray measurements in a thin-layer electrochemical cell. The limiting rates are not specifically constrained by the cell itself. Voltammograms of ideal quality are a necessary condition to detecting the x-ray signal. Crystal quality is a determining factor in both voltammetry and the bilayer x-ray scattering. To this end, we developed a polishing / annealing procedure and apparatus that have now propagated to other research groups in Clark Hall.
We have simultaneously measured in situ x-ray scattering from the adsorbed incommensurate bilayer and current transients. This allows us to directly address the kinetics of the nonequilibrium desorption/ordering process. Upon a positive voltage step, there is Cu desorption, and the commensurate structure transforms into an incommensurate structure, with a larger in-plane lattice constant. During this process, we see a current transient and the emergence of an x-ray scattering peak. The current transients have two components: the capacitive charging of the double-layer, and the Faradaic charge transfer due to desorption/adsorption of ions. We have not yet been able to separate the two. The x-ray scattering intensity indicates the ordering of the incommensurate bilayer. The rise in integrated intensity is proportional to the increasing number density in the ordered phase. The narrowing of the peak corresponds to a increasing correlation length, which implies growing phase domains.
Using a nucleation and growth model, we can fit the entire q-t data set (2 ×104 points) to a function of a few variables. Extending these arguments, we demonstrate that the ordering time scales with voltage over the entire range, in quantitative agreement with the nucleation and growth of two-dimensional islands.
All we know about the desorption process is contained in the current transient data. One could better determine when the copper ions leave the surface by performing a similar kinetic x-ray experiment but monitoring the specular reflectivity instead of the intensity of the Cu-Cl order parameter. The information gained would be very interesting; however, the experiment would be a major undertaking, requiring a different cell design and many months of experimenting. Furthermore, the results would not affect the conclusions of the current experiment. Rather, they might shed some light on what happens before the nucleation and growth process begins.
Further studies could explore the relationship between geometry-dependent diffusion processes, charge transfer at the interface, and nucleation mechanisms. Some future experimental directions have already been detailed in section 4.6. Additional chronoamperometric measurements in a cell where the thickness of the solution layer could be systematically varied would allow one to investigate whether the desorption process is diffusion-limited in one limit and reaction-rate-limited in the other. These measurements are also well beyond the scope of the current investigation, requiring new experimental cells and, again, the results would not affect our current conclusions.
This appendix summarizes some relevant results on Gaussian distributions for the reader who may not be familiar with them.
Consider the cumulant expansion [70] for áex ñ, where x is considered small. Here, we will only expand to second order, though the extension to higher orders is straightforward. To derive the cumulant expansion, begin with
| (248) |
| (249) |
| (250) |
| (251) |
A ``Gaussian random variable'' is a random variable with a Gaussian distribution. For a Gaussian random variable x, all of the terms of order x3 and higher are identically zero. This follows from considering a Gaussian distribution of the form
| (252) |
| |||||||||||||||||||||||
| (255) |
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
I mention one more useful relation, used in section 3.10. Assuming that the Gaussian approximation is valid and áA + B ñ = 0,
| (260) |
This treatment expands upon the three-dimensional treatment by McQuarrie [99] and extends it to arbitrary dimensionality. McQuarrie actually suggests two ways to give the result (B.14); I am using the first.
The diffusion equation for G is
| (261) |
| (262) |
| (263) |
| (264) |
| (265) |
| |||||||||||||||||||||||||||||||||
| (269) |
| (270) |
| (271) |
| (272) |
| (273) |
| (274) |
1 I will not discuss quasicrystals, fascinating systems wherein the diffraction pattern exhibits (at least) five-fold symmetry. This implies that the real-space structure cannot even be even remotely described by a (three-dimensional) Bravais lattice.
2 The factor of two for the instantaneous case is required by the definition of the d-function, which runs from -¥ to ¥.
