Hydrogen diffusion in nano-sized materials

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Diplomarbeit

Hydrogen diffusion in nano-sized materials

investigated by direct imaging Andreas Bliersbach

Rheinische Friedrich – Wilhelms University Bonn Department of Physics and Astronomy Helmholtz – Institut für Strahlen– und Kernphysik

Uppsala University

Department of Physics and Astronomy Materials Physics

July 6, 2011

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The presented thesis work was conducted in the Materials Physics Divi- sion of the Department of Physics and Astronomy of Uppsala University, Sweden. It was carried out under supervision of Prof. Dr. Björgvin Hjörvars- son and Dr. Gunnar Karl Pálsson during the period of July 07, 2010 through July 06, 2011.

The Diplomarbeit (diploma thesis) will be submitted at the Department of Physics and Astronomy of the Rheinische Friedrich – Wilhelms Univer- sity Bonn, Germany, and is part of the compulsory requirements for award- ing a Diplom (diploma) in physics. The work will be presented in two public examinations (Colloquia). The first presentation is at Ångström laboratory, Lägerhyddsvägen 1, Building 6, floor 1, Uppsala, Sweden, Monday, June 13, 2011 at 15:00. The second presentation will take place at the Helmholtz – In- stitut für Strahlen– und Kernphysik, Seminarraum 1, Nussallee 14-16, Bonn, Germany, Thursday 16, 2011 at 09:00.

1. Gutachter: Prof. Dr. Björgvin Hjörvarsson Universität Uppsala

2. Gutachter: Prof. Dr. Manfred Fiebig Universität Bonn

Abstract

The kinetics of interstitial hydrogen are of great interest and importance for metal-hydride storage, purification, fusion and fission reactor technology, material failure processes, optical sensors for hydrogen gas and many other technologies [19, 26, 29, 31]. In particular nano-sized materials motivate fascinating applications and scientific questions.

If hydrogen is absorbed in vanadium it alters the band structure around the Fermi energy. These modifications of the band structure lead to a change in the absorptance of vanadium which are in first order approximation proportional to the concentration. We present a method to quantify chemical diffusion of hydrogen in nano-sized materials.

The induced changes in the absorptance of vanadium hydride (VHx) thin-films are observed visually and in real-time as a function of position.

Concentration profiles and their evolution in time, during chemical diffusion, were measured down to a hydrogen content corresponding to just a few effective monolayers, randomly distributed within VHx. For concentrations reached via phase transitions distinct diffusional behavior was found, where a diffusion-front, a strong concentration gradient, migrates in the direction of the diffusive hydrogen flux. The results show that decreased size strongly influences the energy landscape and reveal different rate limiting steps for absorption and desorption.

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1. Introduction

One of the greatest challenges mankind has to face in the 21st century is to still an ever growing hunger for energy. The demands on an energy source, to suffice the requirements of a modern society, have grown, as environ- mental issues, hazardous waste and global warming gain a more and more important role in discussion. A sustainable and pollution free hydrogen economy is thought by many to be one piece in a broad strategy to tackle this looming energy crisis. Storing hydrogen, however, displays a challenge in itself. Hydrogen is gaseous at room temperature, highly combustible and therefore difficult to store safely and convenient.

Solid storage materials like metal-hydrides, complex hydrides, carbon structures e.g. carbon nanotubes or graphene and many others offer a safe and interesting alternative to gaseous or liquid hydrogen [31]. Great scientific endeavors have been devoted to the tailoring and provision of high volumetric density, fast and reversible charging cycles at applicable temperatures, low self-discharge and the stability of sophisticated hydrogen storage materials [19, 26, 29]. The kinetics of hydrogen are of great interest and importance not only for metal-hydride storage but for purification, fusion and fission reactor technology, material failure processes, optical sensors for hydrogen gas and many other technologies.

This work presents investigations on diffusional properties of hydrogen interstitials conducted on thin-films of vanadium of 10 and 50 nm thickness. A new observational technique, applicable for nano-sized materials and sensitive even at low concentrations, is introduced. To observe hydrogen concentration the hydrogen induced altering of the band structure around the Fermi energy is utilized. These modifications of the band structure lead to a change in the absorptance of vanadium which are in first order approximation proportional to the concentration.

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2. Theory

2.1 Hydrogen in vanadium

Hydrogen is, with almost 74% of the total barionic matter [27], the most abundant chemical element in our Universe. Despite its abundance, most of earth’s hydrogen is bound in chemical compounds. These compounds are generally defined as hydrides and may be classified by four types of chemical bonds they display. With some exceptions the type of bond in a binary compound is closely related to the difference in electronegativity of both constituents. Figure 2.1 shows a periodic table of possible hydrides and the electronegativity of the respective element-hydrogen bonds.

Chapter 2: Concepts of Hydrogen in Metals

The electronic structure of the host metal

Not all metals form hydrides (fig. 2.5). In those that do, it is found that the form in which the hydrogen is bound can be very different. To the left side of the periodic system, ionic binding occurs. Since the electronegativity of the alkali and the alkali earth metals is very low, they give their electron(s) to the hydrogen and the two form a binding of ionic nature. Elements that form covalent chemical hydrides are found on the right of the periodic system, where the hydrogen atom is bound closely to one host atom and they share the additional electron.

Figure 2.5: The chemical state of hydrogen with other elements. In the middle of the periodic system one can find those, that don’t form hydrides at standard conditions (STP) [16].

In the middle of the periodic system, the metal that form hydrides can be found, which we will focus on. In this group of hydrides, the hydrogen sits in interstitial sites of the metal host lattice. The hydrogen s-electron is forming a hybridised state with the metal d-band. Hence, electrons from the metal host are occupying the new formed s-d band, causing a weaker binding of the metal atom with the neighbouring metal atoms and consequently an increase of the lattice parameter.

As mentioned in section 2.1, the hydrogen uptake arises from the difference in chemical potential. As seen in equation (2.8) the heat of solution ∆H, or binding energy of one hydrogen atom is an experimentally determined property which allows to compare different metals. For the transition metals it is instructive to look at fig. 2.6, which shows the trend of the binding energy in the periodic system.

The trend of weaker binding energy when going to the right in the periodic system can be explained from the electron density at interstitial sites. Since the amount of valence electrons increases from the left to the right, the trend holds as well for the electron density.

Figure 2.1: Periodic table of possible binary hydrides, classified by the type of chem- ical bond they display. Elements that do not form compounds with hydrogen under standard conditions are highlighted in yellow. Below the symbols of the particular hydrides the Allred–Rochow electronegativities of the constituents are denoted.

The figure has been adapted from [28].

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1. Ionic bond: Alkali and alkaline earth metals are, with respect to hydrogen, strongly electropositive. They form ionic bound hydrides called saline hydrides.

2. Polymeric covalent bond: Elements of the Groups 11 and 13 of the periodic table, with an electronegativity comparable to hydrogen, form polymeric covalent hydrides.

3. Volatile covalent bond: With increasing electronegativity the chem- ical polarity inverts and volatile covalent hydrides form. Metallic elements donate their electron to bind hydrogen as H⁻whereas non- metallic elements bind acidic H⁺.

4. Metallic bond: A wide range of elements show metallic bonding in- cluding transition metals, the Lanthanides and Actinides. They have comparable or lower electronegativities than hydrogen.

Transition metals like vanadium, tantalum or niobium have incomplete d sub-shells and form metal-bonded compounds with hydrogen. One of the features of most metal-hydrides is that they are interstitial alloys. In interstitial alloys hydrogen occupies tetrahedral or octahedral sites inside the metal lattices, which remains metallic under absorption. Hydrogen is extremely mobile in these metal lattices and the compounds shows often non-stoichiometric composition, taking up large amounts of hydrogen in a variety of different phases. It is remarkable that the hydrogen density inside these metal-hydrides can rise beyond that of liquid hydrogen, e.g.

the hydrogen density in TiH2is 113 % larger compared to liquid hydrogen [28]. While TiH2exhibits the highest hydrogen density within the transition metals similar or even higher concentrations can be found in other binary hydrides.

Among hydrides transition metal-hydrides are of supreme interest in todays search for a future energy carrier and various other technological problems including purification of hydrogen, hydrogen sensors or in the nuclear industry as powerful neutron moderators [10, 28, 30].

At sufficiently low concentration hydrogen interstitials in transition metals can be described by the lattice gas model, known from statistical me- chanics. Exhibiting one of the few close to reality systems to test the lattice gas model, thus raising large interest from the theoretical point of view.

In this work vanadium hydride VHx is used and a separate section will be dedicated to it below.

2.1.1 Vanadium hydride

Vanadium is a transition metal known to form mono- and dihydrides while exposed to a H2atmosphere. It undergoes an exothermic dissolution process at temperatures between 573-1373 K below which hydrogen has to

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overcome a dissociation barrier before it can be adsorbed [16]. Once H is adsorbed it diffuses rapidly inside the host lattice where it can occupy either the interstitial octahedral (O) or tetrahedral (T) sites. As seen in figure 2.2 the distinction between the two sites is due to the symmetry of the nearest metal atoms of the interstitial. For the interstitial O-site the nearest metal atoms form an octahedron with the interstitial at its center. Corre- spondingly the interstitial T-site is in the center of a tetrahedron, formed by the nearest metal atoms.

Figure 2.2: The left column shows BCC unit cells with tetrahedral (black diamonds, upper image) and octahedral (black dots, lower image) interstitial sites respectively.

The corresponding panels to the right illustrate the tetrahedron and octahedron, formed by the nearest metal atoms around an interstitial site, marked in red, in its center.

Throughout this work we will denote the hydrogen concentration in terms of the atomic ratio CH=^N_N^H_V, where NH/Vis the number of hydrogen or vanadium atoms respectively.

For the 6 T-sites, per vanadium atom, a maximum concentration of 6 [H/V] is expected, yet, experiments observe a maximum at CH = 1

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[9]. This might be a result of a repulsive H-H interaction blocking the occupancy of nearest neighbor sites.

Upon hydrogen uptake vanadium will undergo volumetric and structural changes, which will be described in more detail in section 2.1.4. However, it is worth mentioning that for reasons of simplification one may perceive the phase diagram of VHxas one of interstitials inside an unchanged host lattice. It is understood that hydrogen forms gas-(α), liquid-¡α⁰¢ and solid- like¡

β¢ phases while absorbed in vanadium [1, 21]. Just as in other gas-like

Figure 2.3: Phase diagram of bulk VHx. Greek letters denote the different hydride phases.This phase diagram has been adapted from [18].

phases of binary compounds, hydrogen does not show any long range order in theα-phase. Mainly T-sites are occupied and the atoms move freely. At an elevated temperature and higher concentration hydrogen condensates to form a liquid-likeα⁰-phase inside the host lattice. Approaching the β regime hydrogen abruptly changes site occupancy, to reside in O-sites, and freezes to a solid-like state. It is generally understood that the initial type of O-sites, corresponding to the different geometrical directions (x, y and z), determines the occupancy of additional hydrogen atoms. Hydrogen in the β-phase shows long range order and at concentration CH= 0.5 every other (110) plane of O-sites is occupied. However, there is strong evidence that for confined systems, like the ones used throughout this work, site occupancy and phases are decoupled. It is therefore a priori unknown where hydrogen resides.

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Besides the three mentioned phases in the VHx phase diagram there are various other, like the mixed phases¡

α + β,α + β⁰,β + β⁰etc.¢ orδ and γ, with the first similar to β but different long range order and the latter showing a fcc-type structure. A VHxphase diagram ranging from 0 - 593 K is depicted in figure 2.3.

If vanadium is exposed to oxygen it rapidly forms oxide layers which are less permeable for hydrogen. To prevent oxidation vanadium is commonly covered with a thin layer of palladium (≈ 5 nm). Since palladium has a small heat of solution compared to vanadium it is a valid assumption that just minor amounts of hydrogen will be absorbed by it.

2.1.2 Thermodynamic properties

If a piece of vanadium is introduced to a hydrogen atmosphere it will spontaneously absorb hydrogen atoms with sufficiently high energy to overcome the surface energy barriers.

The driving forces behind absorption as well as desorption are thermodynamic potentials. The extremal principle predicts a minimization of these potentials in order to reach thermodynamic equilibrium. In case of fixed absolute temperature T , pressure p and particle number N of the hydro- gen gas, the Gibbs potential or Gibbs free energy G is given by

G = H − T S, (2.1)

where H is the system’s enthalpy and S the entropy.

Hydrogenation and dehydrogenation is accompanied by a change in the Gibbs potential of the system

∆G = ∆HH− T ∆SH, (2.2)

where∆HHand ∆SH denote the enthalpy and entropy of solution for hydrogen in vanadium respectively. The sign of∆G determines wheather a process is exothermic (∆G < 0) or endothermic (∆G > 0), i.e. if energy is released or absorbed during the reaction. In general this energy is in the form of heat, explaining why the condition of spontaneity (∆G < 0) depends strongly on the temperature of the system.

However, G and∆G are macroscopic quantities and the more interesting quantity is often the Gibbs free energy for a single hydrogen atom. This is the chemical potential and it is given by

µH= µ ∂G

∂NH

¶

p,T,N_H

, (2.3)

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with NH, the number of hydrogen atoms. Only if there is no difference between the chemical potentials of all present hydrogen phases will the system assume its equilibrium state. In the course of minimizing the Gibbs potential hydrogen, will transit from states of higher energy to states of lower energy. This does not solely include phase transitions, equalizing the chemical potentials

µ1= µ2= . . . = µi= 0, ∀i-phases, (2.4) but also causes chemical diffusion until

∂µ

∂C = 0, (2.5)

2.1.3 Kinetics of hydrogen uptake

While the prior chapter focused on how the system reaches thermodynamic equilibrium, the kinetic processes behind hydrogen uptake remain unknown.

Inside the vanadium structure hydrogen interstitials are atomic while the surrounding gas is molecular. Consequently the molecular hydrogen has to dissociate for subsequent adsorption, absorption. The energy barriers corresponding to dissociation of hydrogen, adsorption and absorption at the surface layer and subsequent diffusion within palladium and vanadium can be summarized in the following way:

1. Dissociation of H2and chemisorption of the subsequent H atoms The energy required for dissociation is the binding energy of the hy- drogen molecule Ediss = 4.52 eV/H2 [5]. The heat of chemisorption on a palladium surface, that covers vanadium to prevent oxidation, is

∆Hchem= −0.401 eV/H [4]. ∆Hchemis comparably low and will lead to a high sticking coefficient and consequently high surface occupation of hydrogen on top of the palladium surface.

2. Diffusion through the surface layer of palladium

The energy barrier dividing surface from sub-surface sites is E_sub.surf > 0.401 eV/H and the heat of solution of the sub-surface sites is∆Hsub.surf< −0.65 eV/H [4].

3. Diffusion through palladium and vanadium

For infinitly diluted hydrogen the energy barrier for diffusion in palladium is E^diff_Pd = 0.23 eV/H while the heat of solution is

∆HPd = −0.1 eV/H [2, 10]. For vanadium the energy barrier for diffusion is E^diff_V = 0.05 eV and the corresponding heat of solution is

∆HV= −0.316 eV/H [2, 20].

The physisorption of molecular hydrogen on the palladium surface, that means the physical adsorption of H2, requires near to no activation en-

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ergy. Including this and a large impinging rate of the gas molecules the physisorption can be neglected [7]. Subsequent dissociation is facilitated by the corresponding catalytical activity of palladium. Is a hydrogen atom adsorbed it will, due to thermal activation and the non-vanishing probability, be absorb and occupy sub-surface sites. Jumping back into the energetically favorable surface site is often hindered by succeeding hydrogen atoms, blocking these sites, so that the sub-surface sites will be filled up with hydrogen comparably fast. Jumps from the sub-surface sites into palladium show behavior equaling transitions from surface to sub-surface sites. This will lead to a continues flux of hydrogen atoms into the palladium structure. Inside palladium hydrogen can diffuse laterally, driven by the gradient in concentration, to finally jump into the energetically favorable interstitial sites of vanadium. Figure 2.4 illustrates the energy landscapes of a palladium covered vanadium piece at infinitely diluted hydrogen concentration.

Figure 2.4: Energy landscape of an ideal, palladium covered, vanadium single crys- tal.

It is important to know that the activation energy for diffusion and the heat of solution for hydrogen in vanadium is concentration dependent. For theα-phase at infinite dilution, the activation energy for self diffusion has been deduced from Gorski-effect measurements to E^α_act= 0.045 eV [2]. At higher concentrations, in theβ- and β⁰-phase, the activation energy, for self diffusion, increases to E^C_act^x = 0.41 − 0.29 eV with 0.486 ≤ x ≤ 0.736 H/V [13]. Figure 2.5 illustrates the heat of solution for hydrogen in vanadium as

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a function of the hydrogen concentration C . Figure 2.6 shows a sketch of the activation energy for diffusion, again as a function of C .

Figure 2.5: Heat of solution for hydrogen in vanadium. The sampling points are taken from [20] and marked with black circles (No error bars are provided by the source). Greek letters denote the different phases of vanadium hydride.

Figure 2.6: Activation energy for hydrogen diffusion in vanadium. The sampling points are taken from [2] and [13] and marked with black circles (No error bars are provided by the sources).

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The energy landscape shown above in figure 2.4 will deform with increas- ing concentration. Combining the changes in Eactand∆H for a certain concentration will give the related energy landscape. Figure 2.7 illustrates an energy landscape with a linear concentration gradient. It is important to notice that the concentration axis is discrete for the depicted black line and only shows the concentration for the local minima and maxima. Bearing that in mind it is easily conceivable that the diffusional properties of hydrogen in vanadium are concentration dependent.

The temperature dependence of the chemical potential, which is the driving force behind chemical diffusion, was not included in figure 2.7.

Figure 2.7: Change in the heat of solution and the activation energy for diffusion of hydrogen in vanadium as a function of the hydrogen concentration. The oscillatory behavior of the solid black line is inspired by the energy landscape of hydrogen at a certain concentration. For the black line the concentration axis is discrete and just applicable for local minima and maxima. The dashed red line corresponds to E_act and the solid red line to∆H. The zero point of the activation energy axis was arbi- trarily chosen to intersect with the heat of solution of infinitely diluted hydrogen in vanadium.

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2.1.4 Volume changes and self-trapping

The absorption of hydrogen into vanadium leads to the hybridization of the 1s-state of hydrogen and one of the host lattices 3d -states [11]. They form a bonding and an anti-bonding state. While the first lies below the mentioned d -band, the latter lies above the Fermi energy E_F of vanadium.

Several competing theories account for this hybridization and the inter- ested reader is referred to [3] and [14]. The explanation Fukai et. al. offer in [11] states that the number of states below EF does not change with the hybridization, hence, hydrogen has to donate its electron to a state above E_F. The bonding state is strongly localized on the hydrogen atom and by at- tracting d -electrons from the host metal, hydrogen lowers the interatomic interaction of nearest-neighbor vanadium atoms and leads to a dilatation of the host-lattice [12]. The interstitial hydrogen induces a local strain field, decaying quadratically with the distance to hydrogen [6]. A competition between the elastic repulsive lattice expansion and the lowering of the ground state of hydrogen leads to an equilibrium sate called the self-trapped state.

Additional hydrogen atoms will favor regions with an already expanded lattice and their combined strain field will create an even larger attraction for further hydrogen atoms. It has been mentioned in section 2.1.1 that experimentally a maximum concentration of hydrogen in vanadium is found at CH= 1. This repulsive interaction stands in contrast to the long ranged, host mediated, attractive interaction caused by the strain field. However, additional hydrogen atoms at nearest-neighbor sites lead to a further increase in the local electron density and at some point it will cost more energy to expand the lattice than the hydrogen atoms gain by occupying the sites.

This short ranged repulsive interaction needs further investigation and the mechanisms behind it are not yet fully understood.

2.2 Chemical Diffusion

The term diffusion is today generally understood as the process by which matter is moved due to random motion. This is strictly speaking not entirely correct and we distinguish between two sorts of diffusion, self-diffusion and chemical diffusion.

The first sort, the so called tracer diffusion or self-diffusion, describes the unprompted mixing of molecules in the absence of a gradient in concentration or chemical potential. It can therefore, in contrast to chemical diffusion, take place in equilibrium. The name tracer diffusion arises from the observation of a tracer isotope inside the material of interest. Tracer diffusion, in the absence of strong isotope effects, is assumed to be equal to the self-diffusion.

The second sort of diffusion is chemical diffusion and it takes place in the presence of a gradient in concentration or the chemical potential. Thus,

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the system is in a non-equilibrium state striving for thermodynamic equilibrium. Chemical diffusion is one of the processes equalizing the system, ending in

∂µ

∂C = 0.

In a remarkable paper from 1855 the German physiologist Adolf Eugen Fick published his law of diffusion for use both in physiology as well as physics. He established his equations from a macroscopic point of view, in direct analogy to Ohm’s law and Fourier’s mathematical equation of heat conductivity. Fick, however, was unable to explain why a gradient in concentration or the chemical potential should act as a driving force for diffusion. Half a century later in 1905 Albert Einstein argued in his PhD thesis from a microscopic, atomistic point of view and was able to relate Fick’s phenomenological diffusion equations to Brownian motion. With a well established view of atoms, molecules and their motion, we will take a shortcut to Fick’s laws and derive them in a more modern and intuitive way.

2.2.1 Fick’s first and second law

To derive Fick’s phenomenological equations we take a somewhat different approach than Fick himself.

Figure 2.8: Element of volume.

Considering an element of volume, as seen in figure 2.8, we seek an expression for the flow of parti- cles through an area F during time t as a function of the particle con- centration C . F can be chosen in such a way that it is intersecting a point P (x, y, z) and is parallel to the faces, ABC D and A⁰B⁰C⁰D⁰, of our parallelepiped. The total flux through F is, in case of particle conservation, the sum of particles

moving through from left to right minus particles moving through from right the left.

• Number of molecules in a finite volume left of F : FδxC³

P −^δx₂, t´

• Number of molecules in a finite volume right of F : FδxC³

P +^δx₂ , t´ If we chooseδx small enough we can assume that C is constant within δx. Without forces acting on the molecules they will just undergo Brown- ian motion. Half the particles will move from left to right, while the same

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happens from right to left. Hence the flux through F duringδt is:

J (P,δt) = 1 Fδt

µ 1 2 µ

FδxC µ

P −δx 2 , t

¶¶

−1 2 µ

FδxC µ

P +δx 2 , t

¶¶¶

= δx² 2δt





³C³

P −^δx₂, t´´

−³ C³

P +^δx₂, t´´

δx





===⇒δx→0 J (P,δt) = −D∇C(P,t) (2.6)

Equation (2.6) is called Fick’s first law, where J is the particle flux, C (P, t ) is the particle concentration as a function of time t and position P. D is a second order tensor. The law states that the particle flux is directed from regions of high concentration to regions of low concentration with a pro- portionality constant D, which we will from now on call diffusion constant or diffusivity. The diffusivity is proportional to the square of position over time and is usually temperature dependent. Its range spreads over mag- nitudes between diffusion in different media like gases¡≈ 10⁻¹£cm²/s¤¢, liquids¡≈ 10⁻⁵£cm²/s¤¢ and diffusion in solids ¡≈ 10⁻⁹£cm²/s¤¢. Above an analogy to Ohm’s law and Fourier’s mathematical equation of heat conductivity was mentioned. The close relation lies between the particle current density J, the heat fluxφHand the electric current density j.

Heat flux: φH = −σH· ∇T (Fourier’s law) Electric current density: j = −σel· ∇ψ (Ohm’s law) And even fluid flux: ΦV = −κ · ∇p

All these processes are in non-equilibrium conditions driven by a gradient in the chemical potential. Utilizing the continuity equation ^∂ρ_∂t+ ∇j = 0 and equation (2.6) we receive Fick’s second law:

∂C(x,t)

∂t = ∇D∇C (x, t ) (2.7)

Fick’s second law is a partial differential equation of first order in t and of second order in x. It describes the flux of particles, due to a concentration gradient, as a function of the position x and time t .

In case of an isotropic medium, e.g. a crystal structure with cubic sym- metry, the tensor D can be expressed as a scalar diffusion coefficient and equation (2.7) simplifies to:

∂C(x,t)

∂t = D∆C (x, t ), (2.8)

which is the most common form found in literature.

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To solve a parabolic partial differential equation, like equation (2.8), we need spacial boundary conditions, of either Dirichlet or Neumann type and initial conditions at t = 0.

Figure 2.9: A simple system of boundary and initial conditions. Two boxes are di- vided by an initially impermeable border at x = 0. The black dots inside the left box correspond to particles, which are held at a constant concentration C0. The right box is empty for times t ≤ 0.

Figure 2.9 depicts a simple, two dimensional problem. At time t ≤ 0 the area with x > 0 is empty and not connected to the concentration reservoir to its left. At t = texthe dividing wall gets permeable and particles can diffuse into the initially empty area. The initial and boundary conditions are:

Partial Differential Equation: ^∂C_∂t = D^∂_∂x²^C2, x > 0, t > tex

Boundary Condition I: C (0, t ) = C0, ∀t > tex

Boundary Condition II: C (∞, t) = 0, ∀t Initial Condition: C (x, 0) = 0, ∀x > 0

Solutions to the simplified diffusion equation (2.8) have usually one of two standard forms. They can be comprised of a series of error functions or related integrals, or they take the form of a trigonometrical series, converg- ing for large values of t [8]. In our case the concentration distribution C (x, t ) follows a complementary error function. A thorough derivation is provided and can be found in the appendix on page 61. The solution is given by

C (x, t ) = C0· erfc µ x

2p D t

¶

, (2.9)

with erfc denoting the complementary error function. Figure 2.10 shows the evolution of equation 2.9 over time, with D and C₀= 1. Following intuition the previously empty area with x > 0 will gradually fill until it reaches the same concentration C0as in the concentration reservoir.

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Figure 2.10: Form of the complementary error function for D = C0= 1 after different times t > t0.

2.2.2 Diffusion of hydrogen interstitials in vanadium

As mentioned above hydrogen will occupy two kinds of interstitial sites within vanadium, namely tetrahedral and octahedral sites (see figure 2.2).

The mechanisms behind hydrogenation and dehydrogenation of vanadium have been discussed in section 2.1.3. As soon as there is a gradient in the hydrogen concentration or the chemical potential inside the vanadium structure, which can be equalized by migration of the hydrogen atoms, the gradient will lead to chemical diffusion.

Not much is known about the possible mechanisms for diffusion of interstitial hydrogen atoms but it is understood that for different temperature regions, different mechanisms are predominantly responsible for mass transport. Figure 2.11 shows a sketch of the sectioning of these possible mechanisms as a function of temperature [2].

As a result of the low mass of hydrogen quantum tunneling is likely to be observed for low temperatures. With increasing temperature it is possible that hydrogen migrates through incoherent tunneling, facilitated by scat- tering with thermal phonons, thermally activated jumps over the energy barriers or finally fluid or gas like diffusion.

The chemical diffusion of hydrogen interstitials is in any of these cases much faster than the self-diffusion of vanadium atoms and it is therefore often assumed that hydrogen moves in an almost constant vanadium lattice. However, hydrogen induces an expansion of the vanadium lattice which is accompanied by a change in the chemical potential. The former was discussed in section 2.1.4 whereas the latter was reviewed in

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Figure 2.11: Sketch of the possible mechanisms behind diffusion predominantly performed within different temperature regions.

section 2.1.3. These alterations will lead to a deviation from the picture of hydrogen in a constant vanadium lattice. It is important to regard that the induced changes in the chemical potential, which is the driving force behind chemical diffusion, will certainly alter the diffusional properties of the corresponding region. Because the hydrogen induced changes are proportional to the concentration of hydrogen the diffusional behavior is consequently anisotropic for regions with varying concentration. This effect was visualized in figure 2.7 where the change in the energy landscape is shown as a function of concentration.

In general it can be said that with rising concentration the energy of hydrogen is lowered and the diffusion is slowed down. A minimum of the heat of solution in vanadium is at around 0.48 H/V whereafter the heat of solution increases again.

2.3 Optical absorption and transmission

In the late 1960s, hydrogen was understood to donate its electron to the conduction band of the host lattice under absorption while changing the shape of the band marginally. The rigid band model was established, stat- ing that the electronic density of states of an alloy could be inferred from that of the host [15]. Short of a century later, Fukai et al. used soft x-ray emission spectroscopy, on hydrogenated vanadium, to observe a substan- tial change in the density of states located 7 eV below the Fermi energy [11].

They interpreted the observed changes as the hybridization of the 1s-state of hydrogen and one of the host lattices 3d -states. Accompanied by this hybridization is a change in the density of states in the d-band, leading to an increased Fermi level, consistent with the behavior used to motivate the rigid band model. This is shown in figure 2.12.

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Figure 2.12: Soft x-ray emission spectra of vanadium V (solid, black) and vana- dium deuteride V_C_D_=0.734(dashed, red). The black dotted curve shows the density of states for vanadium calculated by the augmented plane-wave method. Note the bulging-out at 7 eV below the Fermi energy and both humps on the high and low energy site of the d-band peak. The data in this figure has been adapted from [11].

The absorption of light by matter is strongly correlated to the available states the electrons of the absorbant can be excited to, the occupation and clearly the photon energy. Neglecting high photon energies, one can distinguish between inter- and intraband transitions. The interband transition rate W_{i f}, from a given state | i 〉 to a final state | f 〉, can be calculated via Fermi’s golden rule. Fermi’s golden rule describes the transition probability between two unperturbed states caused by a perturbation ˆV e.g. the absorp- tion of a photon, and is given by:

W_{i f} =2π

~

¯

¯〈 f | ˆV | i 〉¯

¯

2ρf. (2.10)

In the wavelength region of visible light the matrix element for the inter- action 〈 f | ˆV | i 〉 =¯

¯M_{i f}¯

¯corresponds to the optical transition matrix and ρf is the density of final states. However, not all electrons are in bonded states and quasi-free electrons in the conduction band have a continuous energy distribution. These intraband transitions are commonly modeled by a series of damped harmonic oscillators and today incorporating frequency dependent dampening.

As shown above, in figure 2.12, the density of states changes under hydrogenation and thus leads to a change in optical absorption. Calculating the

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Figure 2.13: Changes in the transmittance over wavelength due to hydrogenation at 150°C and 665 mbar hydrogen pressure, where N_xmbar/0stands for the number count in the spectrometer for hydrogenated and empty vanadium respectively. For wavelength below ≈ 390 nm it was found that the spectrometers results where in- accurate. This figures have been adapted from [22].

alteration of the electronic structure is however fairly complicated. Changes in the absorption coefficientγ showed frequency dependence and even a change in sign was achieved by simply decreasing the thickness of vanadium from 50 nm to 10 nm, as shown in fig. 2.13. To derive a relation between hydrogen concentration and the transmittance of photons we can, however, use that the flux of photons through absorbing material decays exponentially. This is described by Beer-Lambert’s law and is given by

I = I0e^γ(λ)L, (2.11)

with the incident flux I₀and the traveled distance L through the material.

Hydrogenated vanadium exhibits a different absorption coefficient and the relative change in transmitted light constitutes to

IH

I = e^−γ^H⁽^λ)L^H^+γ(λ)L. (2.12) LH= L + ∆L incorporates the changes in volume caused by hydrogenation which where discussed in subsection 2.1.4.

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Taking the natural logarithm one obtains ln

µIH

I

¶

=¡−γH(λ) − γH(λ)∆L + γ(λ)¢L. (2.13)

From our collection of data and previous research [23] we deduce contin- uous and linear dependence between C_H and ln³_I

H I

´ for a certain wave- length. A scaling for CH∝ ln³_I

H I

´can be estimated either by comparison with bulk data, if the assumption of comparable thicknesses holds, or by measuring the concentration ex-situ.

During the cause of this work two samples have been measured, a 10 nm and a 50 nm vanadium single crystal thin-film. The proportionality constants are

CH = α · ln µIH

I

¶

(2.14)

α10 = 3.4 (2.15)

α50 = −1.38. (2.16)

Both proportionality constants have been derived by Prinz et. al. [22]. For α50 Prinz compared the isotherms of the 50 nm sample with bulk data, which overlapped remarkably after scaling. Since 10 nm vanadium can not be viewed as bulk-like, this scaling is not applicable and in turn the points of maximum order where superimposed at c = 0.46 [H/V] to derive α10. (The order was determined by 4-probe resistance measurements)

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3. Experimental setup

To quantify hydrogen diffusion in thin films changes in the transmitted intensity are measured as a function of position and time. Consequently an important part of the experimental setup will be the optical apparatus, designed to measure the transmission at each position of the sample. To ensure clearly defined and reproducible conditions, i.e. temperature, hydrogen pressure and surface cleanness, the sample is placed in a sample chamber which is bakeable up to 523 K and connected to an ultra high vac- uum (UHV) system with a base pressure of p0 ≈ 10⁻⁷ Pa. In addition to the mentioned features the UHV-system is used to minimize the amount of water surrounding the sample. Water, present in form of humidity, can condensate on the surface, at elevated temperatures deteriorate the sample through oxidation, block dissociation of H2and is therefore undesirable. By surrounding the sample with UHV the condensation rate is decreased to approximately three monolayers of H₂O per day. For a heated system the condensation rate is lowered additionally.

3.1 The optical setup

The optical setup could in principle consist of a monochromatic light source and a position sensitive camera. It would still contain every essential feature, since the recorded light contains all necessary information of the hydrogen induced changes. The performance of this setup can, however, be drastically improved by adding supplementary equipment to maximize stability. Therefore a beamsplitter is mounted between the light source and the sample. One beam is directed through a diffusor before exposing the sample, to ensure that the sample is homogeneously illuminated. The other beam will be recorded in an amplified Si detector and can later be used to account for fluctuations of the light source. For reasons of long term stability, low heat dissipation and monochromaticity an incoherent light emitting diode was chosen as a light source. The light emitting diode is driven by a stable current (∆I ≈ 0.02%) and passively cooled by a heatsink to further improve stability. In section 2.3 it was mentioned that for different samples, different wavelengths are desirable. The 50 nm sample shows a minimum of ^I_I^H

0 for 590 nm and the 10 nm sample shows a maximum at 405 nm. Corresponding light emitting

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diodes are used to maximize the change in transmittance. To record the transmitted light we use an 8 bit grayscale charged coupled device (CCD) which might in the future be exchanged against a camera with active cooling and a larger bit rate. The final design is shown in figure 3.1. The setup is enclosed in a black-walled box to minimize reflections from the inner walls and shut out external light.

Figure 3.1: The left and right part of this figure show a technical drawing and a schematic of the experimental setup respectively.

The sample chamber consists of a custom made stainless steel CF-16 sample holder, shown in figure 3.2, and two CF-16 fused silica viewports.

The holder is connected to a vacuum system via a bellow, to reduce transport of vibrations from the rest of the system. The vacuum system consists of pressure gauges ranging from 10⁻⁷ Pa up to 10⁶Pa, a backing pump and a turbo molecular pump, providing a base pressure of 10⁻⁷Pa, a metal-hydride cartridge filled with ultra pure hydrogen, a metal-deuteride cartridge providing ultra pure deuterium and a buffer volume of total

≈ 1.10 L (liter). Ultra pure in this context means that the gas purity exceeds 99.9999%. To achieve this the cartridges are refilled from gas bottles with a purity of 99.9999%. The gas is then first purified in a getter gas purifier and filtered again by entering the metal-hydride/-deuteride. The overall purity exceeds four trailing nines by far.

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Figure 3.2: Technical drawing of the sample holder. Units are in mm and degrees.

The sample lies inside the 10.30 mm cutout, held up by 0.50 mm remaining steel on each side. The cutout in form of an apex-truncated square pyramid, below the sample, is designed to reduce back reflection. The ring shaped guides on top and bottom are fitted for copper gasket seals that seal the space between viewport and holder.

Depending on the type of measurement different magnifications were used. To acquire the resolution d of the given setting, a patterned sam- ple was moved into focus without changing the focal point of the lens system. The patterned sample is covered with equidistant cubes, each 80µm apart. By determining the number of pixels between each cube the scaling can be derived and the error estimated. Resolutions between d = 2.42 and 10.52µm/pixel have been used and the errors are well below

∆d ≈ ±0.05µm/pixel. (3.1)

3.1.1 Temperature and pressure at the sample

The temperature T of the sample is controlled from the outside by a tightly wrapped heating wire around the sample chamber and a thermocouple connected to it. The sample holder is a solid, single piece of stainless steel and with a conducting area of roughly 13% the sample is considered in good thermal contact with the sample holder. For measured temperatures between 300 to 600 K a temperature gradient can be estimated to 4 K, giving an overall temperature uncertainty of

∆T ≈ +1/ − 9 K. (3.2)

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The pressure p is regulated by needle valves leading to the hydride/deuteride cartridge and to the backing- and turbo molecular pump. A major drawback is that the entire UHV system has a relatively large surface area compared to its volume, making it exceedingly difficult to keep the pressure constant over a long period of time (t > 2 h). The pressure decrease is asymptotic and can be caused by absorption of the inner walls of the UHV system and leaking valves to the pumps and hydride cartridge. By feeding with a flow of hydrogen from the hydride cartridge the pressure decrease can in theory be reduced. Permanent hydrogen feeding however requires constant observation and was therefore not frequently used. The hydrogen pressure is measured by a variety of differently ranged capacitive pressure gauges, while the UHV system pressure is monitored by a cold cathode gauge and a residual gas analyzer.

In addition to the asymptotic decay in pressure a correlation between pressure and the room temperature was found. The temperature in the room is influenced by ventilation, population in day and night cycles and the outdoor temperature (through windows) and shows fluctuations of approximately ±1 K. This leads to thermal expansion inside the system.

The amplitude of these cycles is however, much smaller than the change caused by the asymptotic decay and can therefore be neglected. The uncertainty of the measured pressure is solely driven by the accuracy of the used pressure gauges. The manufacturer, INFICON, gives an accuracy of 0.2% of the reading.

3.1.2 Uncertainty in the measured concentration

Regulating the concentration, CH, inside the lattice is achieved by tuning the sample-temperature and the pressure of the surrounding hydrogen gas.

The right conditions for T and p were taken from isotherms measured on samples with the same geometry. The isotherms were acquired using si- multaneous transmission and resistance measurements. Prinz et. al. [22]

claim an uncertainty in concentration of 5%. This 5% is, however, not the error for our measured concentration but the error from scaling the trans- mission data to CHand hence our accuracy. The precision of the measured concentration is in general much better and is estimated by the maximal observed standard deviation of CHin position and time. This gives

∆C_H¹⁰≈ ±0.01H/V (3.3)

for the 10 nm sample and

∆C_H⁵⁰≈ ±0.001H/V (3.4)

for the 50 nm sample.

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3.2 Sample design

The samples that we use are exclusively thin film vanadium samples grown by magnetron sputtering. The substrates are 10 × 10 × 1 mm³or 10 × 10 × 0.5 mm³, monocrystalline double side polished magnesium oxide pieces for the 10 nm or 50 nm sample respectively. Magnesium oxide is transpar- ent and has a lattice parameter comparable to the (110)-plane of vanadium.

By using the right conditions during growth we can consequently achieve epitaxy, i.e. we can grow 45° rotated monocrystalline vanadium on top of magnesium oxide. The difference between the lattice parameters of magnesium oxide and the (110)-plane of vanadium leads to strain in the vanadium crystal. The lattice parameter will be reduced in plane, while the out of plane lattice parameter increases.

With the growth systems we operate it is possible to deposit a single monolayer or up to several hundred atomic layers of high crystalline quality. The quality of each batch of thin films is inspected by X-ray reflectivity and diffraction measurements. For the samples used during this work the coherence length, a measure of the long range order of a crystal, is of the order of the film thickness.

As mentioned in section 2.1.1, vanadium oxidizes rapidly. To protect vanadium the films are covered by a layer of polycrystalline palladium.

Palladium prevents vanadium from undergoing oxidization and makes it possible to handle it outside of UHV systems. The amount of hydrogen absorbed by palladium compared to vanadium is negligible due to a higher heat of solution. Furthermore the activation energy for diffusion inside palladium is significantly larger than in vanadium reducing the risk of accidentally measuring the diffusion inside palladium.

A hydrogen impermeable oxide is deposited over the whole surface ex- cept at a window to one side of the sample where hydrogen uptake is al- lowed. Figure 3.3 shows a technical drawing of our sample design including the palladium cover and the hydrogen barrier in form of an impermeable oxide.

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Figure 3.3: Technical drawing of the sample design. The dimensions are in mm but exaggerated for the cut through. The aluminum zirconium cover will oxidize in air and thereafter be impermeable for hydrogen.

For the 10 nm sample aluminum zirconium was used, which oxidized in air to form a hydrogen impermeable layer. The sample was grown by Pana- giotis Korelis. The sample showed good behavior at first but after a series of heating and absorption/desorption cycles the samples structure changed visibly. This can be seen in the pictures 3.4(a) and (b). The changes in the samples structure did not seem to alter the mobility of hydrogen inside the sample. This indicates that the vanadium crystal remained unchanged since any modification of vanadium would lead to an altered diffusion in the corresponding region, which was not observed. The measurements of the CCD-sensor, however, were affected by blooming i.e. the overexposure and thus overload of pixels and subsequent overflow to adjacent pixels. A masking procedure, designed to eliminate over- and underexposed pixels during measurements, which will eventually lead to information loss, has been used to increase the quality of the transmission data and reduces in addition the influence of blooming.

One 50 nm sample was covered with silicon dioxide by lithography at the Ångström Microstructure Laboratory. It was found that some of the chemicals, used to lift of the positive image, facilitated oxidization of the vanadium layers beneath the palladium and consequently vanadium oxide shapes covered numerous parts of the sample. Evaporation of silicon dioxide onto another vanadium thin film and subsequent measurements showed that the evaporated oxide layer was not fully impermeable and merely slowed the diffusion down. In this context it means that the overall

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(a) (b)

Figure 3.4: In (a) the 10 nm sample inside the sample chamber is shown after grow- ing and baking at 423 K. (b) shows the same sample two months later. The sample has been exposed to several cycles of absorption and desorption at temperatures between room temperature up to 450 K and exhibits now strong inhomogeneities over the whole sample. The bright line to the right of each picture shows the uncov- ered palladium window. Two clamp marks can be seen in the top right and bottom left in each picture.

concentration inside the sample changed almost simultaneously over time during absorption and desorption. The slight delay at positions covered with silicon dioxide indicated that the hydrogen mobility through covered regions was not as fast as at regions without coverage. Figure 3.5 shows the evolution of the concentration inside the sample over time and position. A third 50 nm sample was finally covered with 5 nm of amorphous aluminum oxide by means of magnetron sputtering. This sample shows homogeneous transmittance over the whole surface and has an impermeable aluminum oxide cover. The 50 nm samples where grown and covered by Atieh Zamani and Gunnar Karl Pálsson.

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Figure 3.5: 3D plot of hydrogen concentration over time and position. The negative positions up to zero denote the palladium window, values above zero are covered by silicon dioxide. The overall concentration inside the sample changes almost simultaneously over time. The slightly delayed hydrogen migration at positions covered with silicon dioxide indicates that the hydrogen mobility through covered regions is not as high as at regions without coverage.

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4. Data acquisition, reduction and analysis

4.1 Data acquisition and reduction

For the communication between the used equipment the digital Universal Serial Bus (USB) is preferred. All non-USB components are connected via coaxial cables to a National Instruments Data Acquisition device called NI- DAQ box. The NI-DAQ box is equipped with an analog to digital converter and the digitized signal is subsequently transferred via GPIB to the main computing unit.

Control of the instruments is achieved by using a graphical data-flow programming language from National Instruments called LabVIEW. A flow chart in figure 4.1 illustrates the different steps during data acquisition. The stored data contains information on time, pressure, the monitor detector and an 8-bit grayscale image for each time step. A video of the whole measurement is recorded, which is used to detect irregularities during the measurement. Throughout acquisition the recorded data is continuously averaged over predefined time intervals before it is finally saved. This averaging enhances the data quality by reducing the noise.

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Figure 4.1: Float chart of the simplified data acquisition program.

Figure 4.3 shows the front panel during execution, i.e. the graphical user interface (GUI) for the acquisition program. For each measurement a number of settings has to be provided before the program can start. The settings can be inserted in the left part of the GUI. Three tabs are used to navigate between the following sets of settings:

• General:

Within General settings, timing the acquisition process, are displayed. The tab contains information on the measurement like the identification tag of the sample and the initial temperature and pressure. This information will be saved in a logbook file for later identification.

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• Camera:

The camera tab is used to set the exposure time, gain level and respective modes of the camera. Usually the lowest gain and exposure values are chosen. This will minimize the noise and lead to the highest possible acquisition rate, which in turn will lead to a larger number of images to average over and therefore less noise.

• ROI:

In this tab the regions of interest (ROI) can be defined or loaded from previous measurements. ROI are used to quantify changes in an image. Two types of ROI are defined, a rectangular ROI and a Multi ROI.

The rectangular ROI spans usually over the whole visible sample area.

Each pixel inside this ROI will be averaged and the result is a measure of the overall intensity inside the ROI borders. The Multi ROI consists of a number of vertical stripes parallel to the palladium window. Each stripe is in general a rectangular ROI and gives one averaged intensity value. The Multi ROI is therefore a vector of intensity values for different horizontal positions on the sample. Figure 4.2 illustrates these regions.

Figure 4.2: Illustration of rectangular ROI borders (solid black frame) and Multi ROI borders (dashed white lines) superimposed on a picture of the 50 nm sample during absorption. The darker part to the right side corresponds to the omitted window of the oxide barrier.

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Figure 4.3: GUI snapshot of the executing data acquisition program. The user in- terface can be separated into two major parts. The settings panel to the left, below an elapsed time indicator, and the measurement presentation panel to the right. A full page view can be seen in the appendix on page 36.

To the right a large part of the GUI is reserved for the representation of the recorded data. A variety of different tabs reveal distinct information that can be used to track the changes and evaluate the current status of the ex- periment.

• Live View: Shows the most recent picture captured by the CCD cam- era. Superimposed frames in red and green show the defined regions of interest.

• Histograph: Shows the histograph of the most recently captured pic- ture. The histograph is a line-chart based on an intensity histogram, i.e. the representation of the intensity distribution with the frequency on the abscissa and the intensity on the ordinate.

• Mean Intensity of Rectangular ROI: Shows the averaged intensity of a rectangular ROI, usually the whole visible sample area, over time. This screen is very useful to determine if equilibrium has been reached.

• Mean Intensity of Multi ROI: Shows the averaged intensity of each stripe of the Multi ROI normalized to their initial values. This is espe- cially useful to observe the progress of the hydrogen filling.

• Control Screen: Shows in addition to the screen, already seen at Mean Intensity of Rectangular ROI, measurements from the monitor diode and two different pressure gauges.

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Measurements can take from several hours up to several days. To ensure that the desorption measurement starts under the right conditions the α-phase is measured over the whole period of time, until equilibrium is reached. In theβ-phase the mobility of a phase formation is measured and it is sufficient to record over a predefined period of time before starting desorption. This can reduce the measurement time significantly.

After acquisition the resultant data is reduced before it is finally analyzed. For data processing and the subsequent analysis MATLAB from MathWorks is used. Figure 4.4 shows a float chart of the data reduction.

The following steps are taken to reduce noise:

1. Each saved image is an average of several captured images during a predefined time window. This is done during data acquisition.

2. All images will be adjusted for fluctuations in the light intensity, recorded by the monitor detector.

3. A mask is applied to every image. The mask is chosen in a fashion that pixels, that would be over- or underexposed due to hydrogen caused changes, are eliminated. Thus, no pixel will be considered that would lose information during the measurement.

4. Each image will be cut in distinct ROI. For the rectangular ROI all pixels of the whole image are averaged and binned to a single intensity value. For multiple ROI each ROI is binned over a respective width but the whole height of the image, resulting in a vector of intensity values.

NB: The ROI during data reduction and analysis are decoupled from the ROI used in the LabVIEW program.

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Figure 4.4: Float chart of the simplified data reduction program. The loaded Start_Values file contains information regarding the measurement such as the number of mm per pixel, the scaling factor between concentration and transmission, the position of the end of the palladium window and various other constants used during data reduction.

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4.2 Data analysis

The data is analyzed in MATLAB using parametric fitting of a custom equation. Numerous models have been developed to first qualitatively and later quantitatively fit to the measured concentration data. The current model distinguishes between migration of hydrogen in theα- and β-phase where the latter is not described by regular diffusion.

Diffusion of hydrogen in theα-phase can be described by equation (2.9).

This is the solution of Fick’s second law, under distinct boundary and initial conditions, describing the form of the concentration profile as follows:

C (x, t ) = Cinit· erfc µ x

2p D t

¶ .

This is the ideal solution for a perfect system. In reality the initial concentration is reached after a certain time which is strongly dependent on the pressure, the temperature and the film thickness. Additionally desorption is the reversed process to absorption. To cope with this problem the fit al- gorithm is provided with the right C_init, and concentration offset in case of desorption, of the corresponding time. The following functions are used to fit:

y_abs(x, t ) = a · erfc µ x

2p b

¶ , ydes(x, t ) = a · erf

µ x 2p

b

¶ + c,

where yabsand ydesstand for absorption and desorption respectively, a reg- ulates the amplitude, b the slope and c the y-axis-offset of the error func- tion. Figure 4.5 represents data of anα-phase measurement on the 50 nm vanadium sample. The shape of the concentration profile is easily compa- rable to an error function and by fitting the data, for all times t , to equation (2.9) one gets a set of values for the diffusivity D. Averaging D results in one representative value of the diffusivity for a certain temperature and initial concentration.

With increasing concentration in the temperature region investigated (300-500 K), hydrogen will form a lattice solid, the β-phase. During chemical diffusion of initial hydrogen concentrations reached via phase transitions, different hydrogen phases are present simultaneously. It is not known if the mechanisms for diffusion are altered due to the presence of multiple phases. Neither exists a mathematical model to fit the data. The data analysis will thus be restricted to a semi-quantitative description.

The evolution of the concentration profile, with initial conditions corresponding to theβ-phase, shows a striking feature. A discussion of this feature will be postponed to section 5.2 but it is important to know that a

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Figure 4.5: Concentration profile for the 50 nm sample at 423 K, 465 s after hydrogen exposure. Superimposed in red is a complementary error function fit.

strong concentration gradient migrates in the direction of the diffusive hydrogen flux. In a first approach the mobility of this diffusion-front is estimated by observing the position of the front over time. This is done by fitting a Gaussian to the relative change in concentration during an arbitrary time frame. The extension of this time frame depends strongly on the speed of the migrating concentration front. Illustrated in figure 4.6 is a representative analysis of the front position with a time frame of 450 s. The slope of the squared front position over time is the mobility of the front and should follow Arrhenius behavior. Plotting the slope of the mobility as a function of the inverse temperature T⁻¹gives the enthalpy of formation for theβ-phase.

(a) (b)

Figure 4.6: The left panel shows the concentration profile for the 50 nm sample at 423 K, 3420 s after starting desorption. The front is clearly visible at about 0.17 cm.

The right panel shows the change in concentration (C (x, t )/C (x, t −∆t)) and a Gaus- sian fit, superimposed in red, to determine the front position.

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5. Results and discussion

As described in section 2.3 hydrogen will alter the band structure of vanadium upon uptake and cause a change in absorptance. By recording these changes and converting the transmitted intensity to concentration one can measure the hydrogen content within the structure over time.

A measurement of a 50 nm vanadium thin-film sample is shown in figure 5.1. Absorption and desorption behavior as well as the overall drop in transmitted intensity for different concentrations are in excellent agree- ment with previous research [23].

After the sample is exposed to a hydrogen atmosphere, hydrogen dis- sociates, enters through the surface of palladium, diffuses through it and fills the vanadium beneath until equilibrium is reached. To illustrate the changes in the transmitted intensity two images, before and after absorption, are shown in figure 5.2. The focus of the camera is on the surface plane and changes in the concentration out of plane^∂C_∂z^Hcan consequently not be detected, making it difficult to determine the diffusion constant.

Cyclic ab- and desorption do not show qualitative deviations in rates or relative transmission changes and provide first evidence of reproducibility.

It is evident that the experimental setup is sufficiently sensitive to detect hydrogen induced changes in the absorptance of a 50 nm vanadium sample of 10 × 10 mm surface area.

To observe lateral diffusion in thin-films, part of a samples surface is covered with a hydrogen-impermeable layer, excluding a window at one side of the sample. Hydrogen can spontaneously enter the metal at the omitted window and will subsequently diffuse, beneath the diffusion barrier, through the whole thin-film. The sample layout was described in the context of a detailed sample discussion in section 3.2 and a technical drawing can be seen in figure 3.3. Depicted below in figure 5.3 and 5.4 are measurements with initial conditions corresponding toα- and β-phase respectively.

The measurements were conducted on a 50 nm vanadium sample, partly covered with aluminum oxide by magnetron sputtering. The position 0 cm denotes the end of the palladium window. It is important to note the different time scales and equilibrium concentrations that both measurements exhibit.

Experiments show that absorption and desorption measurements devi- ate and that for different equilibrium concentrations distinct features in the rates can be seen. A detailed discussion of these features will follow in the sections below.

Hydrogen diffusion in nano-sized materials

Diplomarbeit