Temperature and concentration dependence of hydrogen diusion in vanadium measured by optical transmission

Temperature and concentration dependence of hydrogen diusion in vanadium measured by optical transmission

Stefan Book November 2014

Abstract

Hydrogen diusion is investigated in a 50 nm lm of vanadium and a vanadium su- perlattice. Diusion constants for three dierent temperature and pressure pairs are determined for the 50 nm lm. The diusion constants for the temperature and pres- sure pairs are determined to be 4.5 ± 0.1 ·10⁻⁵ cm⁻² at 463 K and 0.05 H/V, 5.6

± 0.1 ·10⁻⁵ cm⁻² at 463 K and 0.12 H/V and 8.0 ± 0.2 ·10⁻⁵ cm⁻² at 493 K and 0.05 H/V. The temperature and concentration dependence of the diusion constants are determined. A concentration dependence of the diusion constant is found with a higher rate of diusion for a higher hydrogen concentration. The activation energy of chemical diusion is determined to be 0.38 ± 0.03 eV.

Contents

1 Introduction 2

2 Theory 3

2.1 Interstitial sites and phases . . . 3

2.2 Hydrogen absorption . . . 4

2.3 Diusion and Fick's law . . . 5

2.4 Thermodynamic inuence of connement . . . 8

2.4.1 Finite size . . . 8

2.4.2 Strain . . . 8

2.4.3 Electronic boundary conditions . . . 9

2.5 Electronic structure and transmittance . . . 9

3 Experimental approach 10 3.1 Experimental setup . . . 10

3.2 Sample design . . . 12

3.2.1 50 nm vanadium thin lm . . . 12

3.2.2 Iron vanadium superlattice . . . 13

4 Results 14 4.1 Thin lm results . . . 14

4.2 Superlattice results . . . 15

5 Discussion 17

6 Conclusions 18

7 Acknowledgements 18

8 Appendix 21

1 Introduction

Many of our current energy sources have adverse eects on the environment and climate.

With the demand for energy constantly on the rise, the search for sustainable energy solutions is becoming increasingly important. This constitutes a challenge, not only in producing green energy but also in nding suitable energy carriers.

A promising candidate for an environmentally sound energy carrier is hydrogen. The benet of using hydrogen as fuel is that the emission from its combustion is water vapour and that it can be generated in an environmentally friendly way by electrolysis of water.

An important property of all fuel and a crucial drawback of hydrogen is energy density.

The stored energy per unit of weight or volume is of great importance in determining the aptness of a fuel for a given application. The conventional way of storing hydrogen is as a compressed gas, but this suers from low volume- and weight density. A possible way to increase the energy density is to store the hydrogen in a metal. The discovery that there are metals capable of absorbing hydrogen was made in 1863 by French chemists Louis Joseph Troost and Henri Étienne Sainte-Claire Deville^[1]. Since then the absorption of hydrogen by metals has been a subject of extensive research.

Generally for metal hydrides the volume density of hydrogen is good, up to 125 g/L as compared with 25 g/L for highly compressed hydrogen (350 bar)^[2]. However the weight density is still low 1-7%^[3] and comparable to a typical value of 6% for a tank of highly compressed hydrogen (350 bar)^[4]. This is a problem for mobile applications, such as fuel cells in vehicles. However it is not a problem for stationary applications such as bulk hydrogen storage for power plants, where power generated by e.g. solar or wind farms during periods of low energy demand could be stored for future use.

A metal capable of absorbing hydrogen will only do so at high enough temperatures and hydrogen pressures. For the use of metal hydrides to be economically viable it is important that the loading and unloading of hydrogen can be done quickly and under reasonably low temperatures and pressures. The rate at which hydrogen is absorbed for a given temper- ature and pressure is determined by the thermodynamic properties of the metal. These thermodynamics properties can for example be inuenced by the dimensions of the metal.

For instance the surface to volume ratio for a thin lm is higher than for bulk materials so surface eects become of greater importance as the dimensions of the material is reduced.

In this thesis the temperature and concentration dependence of diusion in vanadium is investigated by means of optical transmission. An attempt is made at investigating eects from reduced dimensions by measuring diusion in a stack of nano-sized vanadium lms called a superlattice.

It is worth noting that hydrides are interesting for a number of reasons besides hydrogen storage applications. The tendency of hydrogen to alter the electronic structure of the host material presents many possible applications ranging from rechargeable batteries^[5]

to switchable mirrors.^[6]

2 Theory

2.1 Interstitial sites and phases

Vanadium is comprised of a body-centered cubic crystal lattice of vanadium atoms. That is to say that when looking at a piece of pure crystalline vanadium there is a three dimensional periodicity of the material that can be described as a structure of repeating cubes where the vanadium atoms are situated at all vertices and each center of these cubes. A unit cell of a body centered cubic lattice is shown in gure 1.

Fig. 1: A unit cell of a body-centered cubic lattice

If the atoms at the vertices and centers are approximated as spheres, voids will exist in the lattice since the spheres can only occupy part of the volume (68%). These voids are called interstitial sites and in the body centered cubic lattice there exist two types of interstitial sites, tetrahedral and octahedral. The names are referring to the shapes that are made up by the atoms surrounding the site, or rather the shape that they would make out if one drew lines connecting them. The interstitial sites of the body centered cubic lattice is shown in gure 2.

Fig. 2: Interstitial sites in a body-centered cubic lattice, the octahedral site is marked with a blue cross and the tetrahedral with a red circle.

The hydrogen occupies these interstitial sites and can undergo diusion by performing discrete jumps between neighbouring sites. Depending on the concentration of hydrogen and the temperature of the system, the interstitial hydrogen takes on dierent phases. For a given temperature the concentration of hydrogen is a function of the external hydrogen pressure. Plotting the hydrogen concentration versus the hydrogen pressure for a given

temperature yields an isotherm for hydrogen in vanadium. In an isotherm the dierent phases of hydrogen become apparent as an abrupt change in concentration at a certain pressure.

Fig. 3: Schematic diagram of two vanadium hydride isotherms for temperatures T1 and T2. (T1 < T2)

These phases form as a result of the competing inuences of the hydrogen-hydrogen interac- tion and the principle of entropy maximization. At low concentrations the H-H interaction is attractive; it is host mediated and is the result of a local expansion of the host lat- tice caused by an interstitial hydrogen atom. The expansion causes neighbouring sites to become energetically favourable, making the interaction attractive. However the maximiza- tion of entropy favours an even distribution of hydrogen atoms. At low concentration and high temperatures entropy dominates and the hydrogen is in the disordered α-phase.

The α-phase is a gas-like phase were the hydrogen shows no long range order and moves freely. At higher concentrations hydrogen forms a mixed α+β-phase. Going to even higher concentrations a solid-like β-phase is reached where the hydrogen exhibits long range order and low mobility. It is generally thought that, for bulk vanadium, hydrogen occupies the tetrahedral site in the α-phase and octahedral site while in the β-phase.

2.2 Hydrogen absorption

The material considered in this section will be vanadium covered by a thin palladium layer in an atmosphere of hydrogen.

Vanadium can absorb a certain amount of hydrogen at given temperatures and pres- sures. The absorption process can be described by considering the path of hydrogen; from molecules in the surrounding atmosphere to atomic hydrogen diusing within the vana- dium.

A hydrogen molecule that is impinging on the palladium surface has a chance of sticking to it by way of the Van der Waal interaction. This process is known as physisorbtion, the

molecule sticks to the surface without being chemically altered. A physisorbed molecule may subsequently be chemisorbed, meaning that the hydrogen molecule dissociates and the individual hydrogen atoms bind to the palladium. A reason for having the palladium layer on top of the vanadium is that palladium serves as a catalyst for this dissociation process. The individual atoms adsorbed to the surface can diuse into the palladium and subsequently into the vanadium. The binding energy of hydrogen to vanadium is larger than that of hydrogen to palladium so hydrogen prefers occupying vanadium sites

The process is depicted in gure 4 below.

H₂

Physisorption Free hydrogen

Dissociation Chemisorption

Palladium

Fig. 4: Illustration of hydrogens path from a free molecule to an absorbed atom.

2.3 Diusion and Fick's law

Hydrogen diuses by making discrete jumps between neighbouring interstitial sites. These diusion jumps are thermally activated and the rate of diusion has a strong temperature dependence. Thermally activated processes can be described by the Arrhenius equation which relates the rate of the process with its activation energy and the temperature. The rate of diusion may be expressed as:

D = D₀exp−E_a k_BT

 (1)

The prefactor D0 can be thought of as an 'attempt frequency' while the exponential can be seen as a success rate. Ea is the activation energy i.e. the height of the potential barrier between interstitial sites, which is the energy that is needed to be overcome to perform a diusion jump and kB is Boltzmann's constant.

The diusion jumps do not have a preference in direction; each hydrogen atom performing a random walk. However hydrogen atoms can not jump to a site that is already occupied. The result is that hydrogen atoms will, on average, move towards regions of lower concentration since that is the direction least likely to be obstructed by other hydrogen atoms.

This process can also be seen as the maximization of entropy of the system. A homogeneous distribution of particles corresponds to the greatest statistical weight possible and as such the maximum value of the entropy.

The random walk diusion performed by a single particle is called self-diusion whereas the collective motion of all particles due to a concentration gradient is called chemical diusion. The diusion constant of interest in this thesis is that of chemical diusion and will hereafter be referred to as just 'diusion constant'.

Chemical diusion is described macroscopically by Fick's rst law:

J = −D∂c

∂x

Where J is the particle ux, c is the particle concentration and D is the diusion constant.

However we have no means of obtaining the actual ux of particles J experimentally. Thus, for our purposes Fick's rst law does not serve us. An, for our purposes more useful, expression can be derived from the the above equation.

Consider a cylinder of length ∆x parallel to the x-axis. With diusion taking place in the x-direction and c(x,t) being the concentration of particles.

X₁ X₂

ΔX

J(x₁) J(x₂)

The total number of particles in the cylinder at a given time is given by integrating the concentration over the length of the cylinder.

Z x2

x1

c(x, t)dx ≈ c(x, t)(x2− x₁)

Where the approximation above is made for a constant concentration over a suitably short cylinder. The change of particles in the cylinder in a time ∆t is then:

∆c(x, t)(x2− x₁)

∆t

But this is the same as the dierence between ingoing and outgoing particle ow:

∆c(x, t)(x₂− x₁)

∆t = J (x₁) − J (x₂) ⇒ ∆c(x, t)

∆t = J (x₁) − J (x₂) x2− x₁ As ∆x and ∆t approaches zero we get:

∂c(x, t)

∂t = −∂J

∂x Inserting the expression for J from Fick's rst law we get:

∂c(x, t)

∂t = − ∂

∂x



−D∂c(x, t)

∂x



We assume that the concentration dependence of D is negligible for low concentrations.

Thus we can write the equation:

∂c(x, t)

∂t = D∂²c(x, t)

∂x²

This expression is known as Fick's second law. It is a second order partial dierential equation which needs two spatial and one temporal boundary condition to be solved.

These boundary conditions have a clear physical interpretation and will be stated without further motivation.

1. The initial hydrogen concentration in the sample is zero.

c(x, 0) = 0

2. The hydrogen concentration is constant at the palladium window.

c(0, t) = c0

3. No hydrogen may pass through the end of the sample.

∂c(x, t)

∂x    x=L

= 0

The solution to this dierential equation has the form of a series of complementary error functions given the specic boundary conditions.^[8]

c(x, t) = c0

∞

X

k=1

herfc2(k − 1)L + x 2√

Dt



+erfc2kL − x 2√

Dt



(−1)^k−1

i (2)

This equation describes the hydrogen concentration as a function of position and time and will serve as our theoretical model for the diusion of hydrogen within vanadium. Fitting this model to our experimental data gives the diusion constant for the given measurement.

The diusion constant multiplied by the time will later be used as a tting parameter.

This equation describes the hydrogen concentration as a function of position and time and will serve as our theoretical model for the diusion of hydrogen within the vanadium.

Fitting our experimental data to this theoretical gives diusion constant for the sample of interest. The diusion constant multiplied by time in the error functions will later be used as a tting parameter, b.

Dt ≡ b

2.4 Thermodynamic inuence of connement

If the thickness of a piece of vanadium is reduced and approaches that of a two dimensional

lm, as for example in a thin lm or superlattice, the inuence of connement and boundary eects become of increasing importance. There are three important eects that inuence the thermodynamic nature of thin lm vanadium; these are eects from nite size, strain and electronic boundary conditions.

2.4.1 Finite size

The interaction between the interstitial hydrogen atoms is mediated by a distortion of the host metal lattice. This interaction is long ranged which lends the interaction sensitive to the geometry of the sample. Since the driving force behind phase transitions is the hydrogen-hydrogen interaction, a change in boundary conditions for this interaction leads to a change in the phase diagram. Such a change was shown experimentally in vanadium thin lms by Pálsson^[9].

2.4.2 Strain

There is often a mismatch in the lattice parameters of the dierent materials in the layers of a thin lm or superlattice. For instance, in the superlattice the lattice parameters of vanadium and bcc iron are 3.02 and 2.87 Ångströms respectively. This is a relative dierence of about 5 percent, causing a strain in the vanadium when tted to the iron, leading to a tetragonal distortion of the unit cell in the vanadium. This sort of mismatch induced strain is known as epitaxial clamping. The resulting distortion induces a change in electron density at the interstitial sites of the lattice.^[9] Such a change in electron density would change the potential for hydrogen at the interstitial sites and in turn aect the thermodynamics of the interstitial hydrogen. For instance experiments have shown that the interstitial occupancy can change from octahedral to tetrahedral.^[10]

2.4.3 Electronic boundary conditions

At the iron-vanadium interface of the superlattice the thermodynamic properties of the vanadium may change due to the electronic boundary conditions imposed on the vana- dium by the iron. Zones of reduced hydrogen concentration, or depletion zones have been observed to occur at these interfaces.^[11] The depletion zones are interpreted as being caused by a charge transfer between the two metals. Such a charge transfer would cause an increase in electron density in the vanadium at the sites near the interface, causing these sites to become less energetically favourable. This eect becomes more signicant for thinner lms since the depletion zone has a xed thickness.

2.5 Electronic structure and transmittance

A free atom will absorb an incoming photon if the energy of that photon corresponds to an allowed energy transition for one of its electrons. For atoms in a crystal it could be asked what the allowed energies are for the electrons and thus what energy range of photons that the material can absorb. For atoms in a crystal the allowed energy levels form almost continuous bands called energy bands. The energy bands are separated by gaps of disallowed energy levels named band gaps. The band gaps are due to interference between the periodicity of the crystal lattice and the wave nature of the electrons.

The introduction of hydrogen into the vanadium crystal introduces another electron into the system. Soft x-ray emission spectroscopy measurements performed by Fukai^[12]showed that the hydrogen bonds to the vanadium by sd³ hybridisation of hydrogens 1s and vana- diums 3d electrons.

The hybridisation of the vanadium 3d electrons shifts them towards lower binding energies.

This shift changes the density of states around the Fermi energy since the 3d electrons have binding energies close to the Fermi energy. The change in density of states changes the absorption of optical photons since close to the Fermi energy there are unoccupied states into which electrons can be exited. This allows for indirect measurement of the hydrogen concentration in vanadium from optical transmission. The hydrogen concentration can be related to the change in transmittance through Beer's law:^[13]

C_H = α(c) · ln(I_H I₀)

Where I0 is the transmitted intensity through the non-hydrogenated sample, IH is the transmitted intensity through the hydrogenated sample and α is an unknown factor which can be determined using neutron scattering or nuclear reaction analysis.

The change in the prefactor α is assumed to be negligible for the small changes in concen- tration investigated in this thesis. Measurements performed by Prinz^[13] determined that the value of α for the 50 nm thin lm is −1.38.

3 Experimental approach

The method used to observe the hydrogen diusion is to measure the change in trans- mittance induced in the vanadium by the presence of interstitial hydrogen atoms in its lattice. This can be done in a very straightforward way by the use of a camera and a light source. From such a setup the transmitted intensity as a function of position and time can be measured by continuously taking images of the illuminated sample. By measuring the change in transmittance the hydrogen concentration can be deduced as a function of position and time. Using Beer's law from 2.6 the measurements of change in transmittance can be translated into changes in hydrogen concentration. The theoretical model can be

tted to the measured changes in concentration using Fick's second law, or more speci- cally equation (2). From this procedure the diusion constant can be acquired for the given temperature and hydrogen concentration of the experiment.

3.1 Experimental setup

The experimental setup provides a controlled environment in which the diusion of hydro- gen in a sample can be analyzed. During experiments the sample is kept in vacuum which serves to prevent surface contamination of the sample and provides an environment for producing a high purity hydrogen atmosphere.

The sample is situated in a sample chamber that is connected to the vacuum chamber. The vacuum chamber is equipped with a turbo molecular vacuum pump backed by a secondary pump, a metal-hydride capsule for storing hydrogen and several pressure gauges spanning dierent pressure ranges.

Before performing a measurement the chamber is pumped and heated to 180^◦C to evap- orate any water or other contaminates adsorbed to the chamber walls and sample. This helps to produce a high vacuum (∼ 10⁻⁸ mbar) with a minimal amount of contaminates such as oxygen and water. To control the quality of the vacuum the partial pressures of contaminates is measured using a mass spectrometer.

When the a vacuum of ∼ 10⁻⁸ mbar has been reached the valve connecting the vacuum chamber to the pumps is closed and the system can be lled with hydrogen. The hydrogen used is introduced into the chamber from a metal-hydride capsule. The capsule has in turn been loaded with 99.9999% pure hydrogen which is puried upon loading using a gas purier and is further puried by being loaded into the metal-hydride since the metal- hydride does not absorb elements other than hydrogen.

When doing the experiment the hydrogen absorption is to be performed at a given temper- ature and pressure. The sample chamber is kept at a constant temperature using a heating element. The temperature of the sample is approximated to that of the sample chamber since they are considered to have good thermal contact.

The uctuations in temperature of the sample was approximated by Bliersbach^[16] to be

±1K. The hydrogen pressure at the sample is continuously recorded using a pressure gauge.

During measurement the sample is contained within the sample chamber, the top and bottom of which feature two temperature resistant windows to let light through. The light from an LED light source is lead both to the sample chamber and to the light detector using a beam-splitter. The light which is not absorbed by the sample is transmitted to a 8 bit CCD camera. The optical setup is illustrated in gure 5. The wavelength of the LED is 590 nm and was chosen to maximize the sensitivity to hydrogenation. The choice was based on measurements of change in transmittance from hydrogenation for dierent wavelengths of light performed by Prinz.^[13]

Fig. 5: Schematic drawing of the optical setup.

The CCD camera, intensity detector and pressure gauges are connected to the measure- ment computer where the measurement data is stored. The communication between these instruments and the computer of the experimental setup is facilitated by a custom written LabVIEW-program.

The LabVIEW-program is used for performing the setup of the camera such as choosing exposure time and dening the 'region-of-interest' of the sample. Regions-of-interest being user dened areas of the sample that are used to provide information of the diusion progress during measurements.

There are two types of regions, single and multiple. The single region of interest integrates the transmitted intensity over a large part of the sample. This gives information of the total hydrogen concentration in the sample and is used to determine when equilibrium is reached during experiments.

The multiple-region-of-interest is similar but discretises the sample in the direction of diusion. This provides a one dimensional transmittance prole which can be used to follow the diusion front during measurements.

The LabVIEW-program is also used for monitoring the values measured by the pressure gauges and the intensity detector during measurements.

The data acquired from an experiment is in the form of a series of images captured by the CCD camera and the transmission proles for each image. The data analysis is performed using a MATLAB-program that ts a series of complementary error functions to these transmittance proles.

3.2 Sample design

3.2.1 50 nm vanadium thin lm

The thin lm sample consists of a 50 nm vanadium layer on top of a magnesium oxide substrate. To achieve epitaxial growth the vanadium is deposited at a 45 degree angle with the magnesium oxide substrate. Meaning that the (110) direction of the vanadium is in the (100) direction of the substrate. This is done because the discrepancy in lattice parameter between the substrate and the vanadium is close to√

2.The lattice parameter is the distance between atoms in the (100) direction, see gure 6 below.

MgO (100)

V (100) V (110)

Fig. 6: Geometrical illustration for epitaxial growth of vanadium on magnesium oxide

A ve nanometer layer of palladium on top of the vanadium layer prevents oxidation of the vanadium and also to serves as a catalyst for the dissociation of hydrogen molecules.

The sample is covered by a ve nanometer layer of aluminium oxide with the exception of a window at the far end of the sample.

The oxide layer prevents the absorption of hydrogen ensuring that hydrogen uptake only occurs at the window. This allows for a controlled hydrogen absorption with a distinct diusion in one direction: from the oxide window and laterally through the sample. The orientation of the oxide window ensures that the diusion takes place along the (110) direction of the vanadium.

MgO 50 nm Vanadium

Window 5 nm AlO3

Fig. 7: Cross section of the 50 nm vanadium thin lm

3.2.2 Iron vanadium superlattice

The superlattice sample consists of a substrate of magnesium oxide on which alternating layers of vanadium and iron have been deposited. The layers of vanadium and iron are approximately fourteen and two atomic layers thick respectively. There are a total of 25 layers of vanadium and iron. To achieve epitaxial growth the stack of iron and vanadium layers are deposited at a 45 degree angle with the magnesium oxide substrate just as in the thin lm sample. The palladium cap, aluminium oxide and oxide window are also the same as for the thin lm.

Fig. 8: Cross section of the vanadium-iron superlattice

4 Results

4.1 Thin lm results

The data used for the thin lm sample is from measurements performed by Andreas Bliers- bach. ^[16]

We consider three measurements performed at two dierent temperatures and at hydro- gen pressures resulting in two dierent hydrogen concentrations. The temperatures and hydrogen concentrations for the dierent measurements are presented in the table below:

Temperature [K] Hydrogen concentration [H/V]

493 0.05

463 0.05

463 0.12

From these measurements we can investigate both the temperature and concentration dependence of the diusion rate. For the temperature dependence of the diusion rate an Arrhenius behaviour is predicted as shown in equation (1).

To investigate these dependencies we rst need the diusion constants from the measure- ments. The constants are determined using a custom written MATLAB-program. The program ts the series of error functions (sect. 2.3) to the transmission proles from a measurement. Every t of the transmission data gives a value for the parameter b ≡ Dt.

From the b values as a function of time we can determine the diusion constant. This is done by tting a linear function to the values of b since the time dependence of b is linear.

b = Dt + C

The value of b as a function of time for the three experiments are plotted in gures 10-12 in the appendix.

The diusion constants from the measurements and reference values from literature are presented in the table below.

Temperature [K] Hydrogen concentration Diusion constant [10⁻⁵ cm⁻²] Measured values

463 0.12 5.6 ± 0.1

463 0.05 4.5 ± 0.1

493 0.05 8.0 ± 0.2

Literature values

493 0.02 9 ± 2 ^[17]

493 0.11 5.3 ± 0.8 ^[17]

The uncertainties of the diusion constants in the above table were estimated from the change that was found by varying the time interval used. Thus it can be seen as a lower bound since they are only the uncertainties from the linear tting and do not include the uncertainties in the values of b.

The uncertainties in temperatures are small.^[16]. The length of the sample, denoted as L in the series of error functions is determined self-consistently by choosing the value that yields a linear relationship between b and t.

An investigation can also be made on how temperature aects the diusion rate for a given concentration. This is determined by the activation energy of diusion jumps which is found by plotting the logarithm of the diusion constant versus inverse temperature.

The reason for doing so is that the slope of the plotted curve can be easily related to the activation energy.

By tting a linear function to the diusion constants the slope of the curve was acquired.

With this method the activation energy for chemical diusion was found to be: 0.38 ± 0.03 eV. The plot is shown gure 13 in the appendix.

4.2 Superlattice results

The measurement performed with the superlattice sample did not yield any result due to a mistake made in the design of the experiment. The mistake consisted of choosing too low of a hydrogen pressure for the chosen temperature of the experiment. The required hydrogen pressure was approximated since the isotherms relating the hydrogen pressure and hydrogen concentration for a given temperature where not available to the author at the time of the experiment. For this reason the isotherm for the thin lm vanadium was used to estimate the required pressure. The isotherms for the vanadium thin lm and the iron vanadium superlattice is shown below. The temperature and pressure chosen for the experiment were 180^◦C and 0.5 mbar.

The chosen pressure is indicated in the gure with a dashed red line. The isotherms available

for the superlattice are 170^◦C and 190^◦C and it is assumed that the isotherm for 180^◦C would lie between these curves. It can be seen that the isotherms for the superlattice does not behave as the 50 nm sample for low pressures and the transmitted light intensity decreases with hydrogen pressure. This behaviour of the superlattice is not well understood and could explain why no hydrogen concentration could be seen during the measurement.

The isotherms for the superlattice sample are approximations since α is unknown for the superlattice. The isotherms are scaled to reach the same hydrogen concentration as the thin lm for the highest applied pressure.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.001 0.01 0.1 1 10 100 1000

αln(I_H/I

0)

P [mbar]

Isotherms for 50nm thin film and Fe/V superlattice

50 nm 180^oC Fe/V 170^oC Fe/V 190^oC

Fig. 9: The isotherms for the vanadium thin lm measured at 180^◦C marked with blue and isotherms for the Fe/V superlattice at 170^◦C and 190^◦C marked with green and pink respectively.

The hydrogen concentration as a function of distance within the sample is plotted in gure 14 in the appendix, where it can be seen that the values of concentration is noise centered around zero concentration.

5 Discussion

A higher rate of diusion was found for a higher hydrogen concentration. This concentration dependence could be caused by the local expansion of the host lattice that is produced by interstitial hydrogen atoms. This eect could enable hydrogen to diuse faster through local regions of expansion caused by other interstitial hydrogen atoms. Such an eect would account for the higher rate of diusion at higher concentrations. However since the uncertainties in the diusion constants are low estimates and the trend is seen from the slope of a line between only two data points it is not clear that this is an actual trend.

For the temperature dependence of the diusion rate at a given concentration a higher rate is found for a higher temperature as expected. If the diusion jump is modeled as having an Arrhenius behaviour the activation energy for the jump diusion is 0.38 ± 0.03 eV.

Compared with earlier ndings^[16] there are some discrepancies in the diusion constants and consequently also the activation energy. These discrepancies could stem from dier- ences in the models used. The earlier analysis used a single complementary error function to model the hydrogen concentration as a function of position and time in contrast with the series of complementary error functions used here. Furthermore the boundary conditions used for the models were dierent.

For comparison measurements of Kleiner, Sevilla and Cotts^[15]found the activation energy for hydrogen diusion in bulk vanadium to be 0.087 ± 0.003 eV for a hydrogen concentra- tion of 0.17 H/V. However it should be noted that their measurement was of self-diusion whereas this work deals with chemical diusion. An approximation of the self-diusion constant can be made from the chemical diusion constant from the equation below.^[18]

D_chem = D_selff_M f_M

c(^dµ_dc) kT

!

Where the factors fM and fM are related to blocking and correlated motion and are assumed to be negligible at low concentrations. Using that µ = kT ln(p) and ^dµ_dc = ^dµ_dp^dp_dc we have:

D_chem≈ D_selfc p

dp dc

Where ^dp_dc was found from the slope of isotherms for temperatures at which the diu- sion was measured. Using the above method, the activation energy of self-diusion was approximated to be 0.31 ± 0.13 eV assuming a 10% uncertainty in _dp^dc readings. The un- certainty can be seen as a lower limit since the uncertainties in the diusion constants are to be considered low estimates. Assuming a 10% uncertainty in diusion constants gives an activation energy of chemical diusion of 0.38 ± 0.13 and an activation energy of self- diusion of 0.31 ± 0.19 eV. Possible reasons for the discrepancies between the calculated self-diusion constant and the literature value are that the literature value is given for a higher concentration and measured using nuclear magnetic resonance in bulk vanadium.

6 Conclusions

For the thin lm the concentration dependence of the diusion rate was extracted for a given temperature as well as the activation energy for jump diusion at a given hydrogen concentration. The result for the concentration dependence of the diusion rate is not what is expected from theory. It is generally thought that the rate of diusion is concentration independent in the α-phase. This deviation from theoretical predictions is an interesting result and may serve as a topic further investigation.

No result was found for the superlattice and questions of connement eects on interstitial diusion in vanadium are left to future investigations.

7 Acknowledgements

I would like to thank Björgvin Hjörvarsson for introducing me to hydrogen diusion in metals and for giving me this project. Great thanks goes to my supervisor Max Wol for all of his help, patience and constantly positive outlook. Invaluable help was given to me by Wen Huang, thank you for your patience with all my questions. Thanks to Bengt Lindgren for helping me with questions of programming. I would also like to thank Anders Olsson and Vassilios Kapaklis for helping me through technical issues that were beyond me. Lastly I would like to thank Atieh Zamani for making the superlattice sample.

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8 Appendix

0500100015002000250030003500−0.02

0

0.02

0.04

0.06

0.080.1

0.12

0.14

0.16 Time [s]

b=Dt [cm ] 2

190o C | 0.05 H/V Experiment Linear data fit

Fig. 10: b as a function of t for 463 K and 0.05 H/V.

0200400600800100012001400160018002000−0.02

0

0.02

0.04

0.06

0.080.1

0.12 Time [s]

b=Dt [cm ] 2

190o C | 0.12 H/V Experiment Linear data fit

Fig. 11: b as a function of t for 463 K and 0.12 H/V.

05001000150020002500300035004000−0.05

0

0.050.1

0.150.2

0.250.3

0.35 Time [s]

b=Dt [cm ] 2

220o C | 0.05 H/V Expermient Linear data fit

Fig. 12: b as a function of t for 493 K and 0.05 H/V.

2.022.042.062.082.12.122.142.162.182.2−10.1

−10−9.9

−9.8

−9.7

−9.6

−9.5

−9.4 1000/T [K−1 ]

ln(D) [cm s 2

] −1

Diffusion constant versus inverse temperature Experiment Error line Error line Linear data fit

Fig. 13: Logarithm of diusion constant versus inverse temperature.

00.020.040.060.080.10.120.14−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03 X [cm]

C [H/V]

Hydrogen concentration versus position Experiment Data fit

Fig. 14: Hydrogen concentration as function of distance within the superlattice.