Reliably measuring hydrogen uptake in storage materials

7 Reliably measuring hydrogen uptake in storage materials E. MacA. G R A Y, Griffith University, Australia

7.1

Introduction

The most fundamental driver of hydrogen storage research is the need to achieve performance that makes the technology viable. The goal of operating automobiles from an on-board hydrogen storage tank has led to a focus on the reversible capacity of the storage system, which must respect constraints on available mass and volume. For this and other reasons, the need to authoritatively measure the hydrogen uptake of a new storage material in the laboratory is paramount. The long-standing controversy over the hydrogen storage properties of nanostructured carbons can almost certainly be ascribed in part to the special problems associated with measuring small samples with high surface area, prepared by nanoscience routes: experience with metals and complexes prepared by metallurgy does not carry over completely to the low-density materials of current interest. Thus the new world of nanostructured materials has brought new challenges requiring careful design of both apparatus and experiment to produce reliable results for hydrogen uptake. The technical aspects of measuring hydrogen uptake are addressed in this chapter, including methodology and apparatus. Because the storage context implies end-use requiring hydrogen gas, electrochemical techniques are not covered. The most direct and authoritative approaches to measuring hydrogen uptake are discussed, rather than indirect measures such as lattice parameter or resistivity that require calibration of the concentration scale. The focus is on the most common techniques, which are based on measurement of the hydrogen pressure in a closed system (manometry) or of the mass of the sample and its loading of absorbed or adsorbed hydrogen (gravimetry). A problem affecting both classes of technique, that of the influence of inaccurate knowledge of the volume occupied by the sample itself on the reliability of the data, is given prominence. A new approach to measuring hydrogen uptake, the variable-volume hydrogenator, is proposed as the best solution to this all-pervading problem. 174

Reliably measuring hydrogen uptake in storage materials

175

Generally the hydrogen concentration is represented here by the hydrogento-host atomic ratio: H = nH = mH / M H X nX mX / MX

[7.1]

where X denotes the host, nH and MH are the number of absorbed moles and molar mass of hydrogen, and nX and MX are the number of moles of host atoms in the sample and their molar mass. The mass fraction of hydrogen in the sample is:

fH =

mH = mX + mH

H/ X M H/ X + X MH

[7.2]

Unless specified, ‘absorption’ generally includes adsorption and ‘hydrogen’ generally includes protium (1H or H and H2) and deuterium (2H or D and D2).

7.2

Compressibilities of hydrogen and deuterium

A direct measurement of hydrogen uptake by a material requires knowledge of the density of the surrounding hydrogen gas. This requirement is obviously intrinsic to those techniques in which the number of moles of hydrogen taken up by the sample is calculated from a change in pressure or volume. Even where the measurement is of a change in the mass of hydrogen in the sample or in the sample cell, the hydrogen gas density is needed to apportion hydrogen between the sample and the surroundings, or to correct for buoyancy if the sample is weighed. Considering the absolute accuracies to which pressure, volume, temperature and mass may be practically measured, hydrogen may safely be taken to be ideal above room temperature and at pressures below a few bar. At higher pressures or lower temperatures, the need to account for its non-ideality becomes apparent. Perhaps the simplest approach, and the one used exclusively here, is to define the compressibility, Z, of the gas by: pVm =Z RT

[7.3]

where Vm is the molar volume. Thus Z = 1 refers to an ideal gas. The density of the gas is then: ρ=

Mp ZRT

[7.4]

where M is the molar mass (formula weight) of the gas. The origin of the non-ideality is interactions between the gas atoms or molecules, which may

176

Solid-state hydrogen storage

be crudely modelled by the modified equation of state for a fluid of van der Waals:  a   p + 2  ( Vm – b ) = RT  Vm 

[7.5]

where a is a measure of the attractive part of the interaction and b is a measure of the excluded volume of the gas, i.e. of the repulsive core of the interaction. While a and b may be represented by constants in a sufficiently restricted space of experimental parameters, their dependence on p and T makes the general application of Eq. (7.5) impractical without modification. The same comments apply to virial equations of low order. Tabulated data of good accuracy (say better than 0.1%) for some or all of Vm, ρ and Z are available in a wide range of pressures and temperatures for protium and, to a lesser extent, deuterium. A comprehensive tabulated source is Vargaftik (1975). Other sources are discussed by Hemmes et al. (1986). The most recent, most comprehensive and generally most accurate source is contained in the commercially available National Institute of Standards and Technology (USA) Reference Fluid Thermodynamic and Transport Properties database (REFPROP, URL http://www.nist.gov/srd/nist23.htm), in which are embedded advanced equations of state for protium (Jacobsen et al., 2007) and deuterium (McCarty, 1989). From a practical perspective, access to software that calculates Vm, ρ or Z as a function of p and T greatly facilitates the calculation of hydrogen uptake, especially if it can be incorporated in or called by a spreadsheet that performs the hydrogen uptake calculations for an entire experiment. The REFPROP package satisfies these criteria, albeit at the cost of some complexity owing to its very comprehensive capabilities. Alternatively, a calculator is available for just the compressibilities of protium and deuterium, based on numerical solution of the modified van der Waals equations developed for protium by Hemmes et al. (1986) and for deuterium by McLennan and Gray (2004). This calculator reproduces the tabulated data of Michels et al. (1959) and is of comparable accuracy to REFPROP for 273 < T < 423 K and p < 1000 bar. The accuracy is limited at cryogenic temperatures by the uncertainty in the original data in Michels et al. (1959). This calculator is available free from the author on request. Figure 7.1 demonstrates the importance of using compressibility in the calculation of hydrogen uptake at even modest pressures by contrasting the calculated absorption isotherms for the LaNi5–D2 system at room temperature with and without correction for the deuterium compressibility. Even at pressures of a few bar the error is significant over the breadth of the absorption plateau. Once pressures of several hundred bar are reached, the uncorrected isotherm is completely misleading.

Reliably measuring hydrogen uptake in storage materials

177

700

Pressure (bar)

600 500 400 300 200 100 0 0.0

0.2

0.4

0.6

0.8 H/M

1.0

1.2

1.4

1.6

7.1 Isochoric isotherms for the LaNi5–D2 system recorded at 25 °C with (filled circles) and without (open circles) correction applied for the compressibility of the deuterium gas. Even at a pressure of a few bar, the error is accumulating noticeably, becoming extreme at pressures of hundreds of bar.

7.3

Measurement regimes

The most widely used representations of hydrogen uptake characteristics are pressure-composition isotherms (pcT), notionally isobaric (constant-pressure) temperature scans, of which thermal desorption spectroscopy (TDS) is an example, and notionally isoplethic (constant-concentration) temperature scans, the latter usually being displayed on a van’t Hoff diagram for the purpose of determining the enthalpy and entropy changes associated with the reaction. The isochoric (constant-volume) regime corresponds to the constraint of a closed system in which a fixed total amount of hydrogen is exchanged between the gas atmosphere and the sample. Hydrogen is added to the system in aliquots and then the system is closed while the sample and gas react under isochoric conditions. These regimes are contrasted in the pcT diagram in Fig. 7.2, in which paths from an isotherm at temperature T1 to an isotherm at T2 arrive at the same point, P, in the pressure–composition space from very different starting points, according to the constraint. Pressure–composition isotherms may be constructed under an isochoric constraint by adding hydrogen aliquots to the system and measuring the hydrogen absorption or desorption stepwise, or under an isobaric constraint by actively controlling the pressure over the sample to a constant value while the sample absorbs or desorbs.

178

Solid-state hydrogen storage

Isopleth

psys

Isochore

Isobar P

T1 T2

H/X

7.2 Delineation of isochoric, isobaric and isoplethic approaches to the same point P on a pressure–composition isotherm at temperature T2, starting from an isotherm at a higher temperature T1.

A point that has not been directly addressed in the literature is whether the outcomes of a change in hydrogen content of the sample via an isochoric, an isobaric or an isoplethic path are equivalent. The only evidence known to the author suggests that they are not equivalent. An isobaric measurement of the Pd–H2 system by Benham and Ross (1989) produced a quite different isotherm from that familiar from isochoric measurements. In particular, an isobaric study of the effects of small changes in hydrogen pressure on the hydrogen content of LaNi5 by Gray (1992) suggests that isochoric measurements produce an isotherm at an apparently lower pressure than the real isotherm pressure measured isobarically, i.e. at constant chemical potential. Figure 7.3 shows the approach to equilibrium via an isochoric path along with actual gravimetric data for the LaNi5 system published by Gray (1992). Considering first the experimental data, when the system pressure was slowly increased then decreased by a few kPa, the sample did not switch abruptly from conversion from the α to the β phase to desorption of a fixed proportion of the α and β phases: a smooth corner is apparent where the hydrogen content of the sample initially continues to increase while the gas pressure decreases. Assuming that the isobaric measurement best represents the true state of the sample and that the state of the sample does not depend on the measurement regime, changing to isochoric conditions would force the system to approach apparent equilibrium along the isochore, stopping when the isochore is tangent to the real pcT curve. At least in the case of LaNi5, then, isochoric isotherms

Reliably measuring hydrogen uptake in storage materials

179

26 24 Isochore 22

Relative pressure (kPa)

20

Isobar

e ic isoth

rm

18 16

Isocho

er ric isoth

m

14 12 10 8 6 0.72

0.74

0.76

0.78 H/M

0.80

0.82

0.84

7.3 Isochoric approach to equilibrium compared to a real isobaric isotherm for the LaNi5–H2 system in which the gas pressure was increased then decreased by a few kPa under active pressure control to within a few Pa. The point at which the isochore is tangent to the isobaric curve defines the isochoric isotherm, which therefore lies below the isobaric isotherm. The experimental data are from Gray (1992).

appear to actually record points at which the sample is close to pure desorption of the dilute and concentrated phases, rather than absorption. A fourth regime worthy of consideration is constant molar flow, particularly for investigating the kinetics of hydrogen absorption and desorption. In this case a flow controller could be used to control and measure the flow of gas to the sample cell (but see also Section 7.7) with the pressure in the sample cell used as a measure of the chemical potential required to force the hydrogenation reaction at the measured rate.

7.4

Measurement techniques

There are numerous techniques available for measuring the amount of hydrogen absorbed or desorbed by the sample under the desired constraint(s). These generally involve measurements of some combination of hydrogen pressure, component volumes, hydrogen flow and sample mass in a gas handling

180

Solid-state hydrogen storage

system consisting of a hydrogen source, a hydrogen sink, a gas manifold and a sample cell. Because flow-based measurements are more difficult for small samples, and practically rather limited in the range of available pressure ratings for mass-flow transducers, we will focus on measurements based on pressure (manometric) and sample weight (gravimetric). The constant-volume system is ubiquitous, but we will also mention a variable-volume approach in Section 7.6, particularly in connection with the problem of sensitivity to the volume (density) of the sample itself (Section 7.5). A manometric system of constant volume is known as a Sieverts (or colloquially pcT) apparatus and is universally operated under isochoric conditions, although isobaric conditions are possible with modification, as explained below.

7.4.1

Sieverts technique

Figure 7.4 shows a generic Sieverts hydrogenator. The measurement of hydrogen uptake is made step-wise. Suppose that, at the end of the k–1th step, a pressure psys, the system pressure, of hydrogen is present throughout the hydrogenator. The gas in the reference volume is at temperature Tref and the gas in the sample cell is at Tcell. The connecting valve, S, is closed to isolate the sample cell, which has an empty volume Vcell. A new pressure pref at temperature Tref is established in the reference volume, Vref. S is then opened and, after a suitable settling time, a new value of psys is measured, along with new values of Tref and Tcell. Following Blach and Gray (2007), the number of moles of hydrogen atoms absorbed or desorbed by the sample in the kth step, ∆n Hk is then calculated from the change in the pressure measured when S is opened:

Pressure transducer

P

Vacuum pump Hydrogen source

S

Vref

Vcell

7.4 Minimal Sieverts (isochoric) hydrogenator. To record an absorption isotherm, gas prepared at measured pressure and temperature in Vref is admitted to the sample cell via valve S. The absorbed amount is calculated from the final pressure in the system and the temperatures of the gas in Vref and Vcell using Eq. (7.6).

Reliably measuring hydrogen uptake in storage materials k k   psys p ref ∆n Hk = 2   – k k k k k k   Z ( p ref , Tref ) RTref Z ( psys , Tref ) RTref

181

  Vref 

k k –1    psys psys – –  ( Vcell – VX )  k k k k –1 k k –1 –1   Z ( psys , Tcell ) RTcell Z ( psys , Tcell ) RTcell 

[7.6]

where Z is again the compressibility and VX = mX/ρX. Note that the mass of the sample, mX, and its density, ρX, depend in general on its hydrogen content. Equation (7.6) simply accounts for all the gas in the system. The total change in the hydrogen content of the sample after N steps is: N

n HN = Σ ∆ n Hk

[7.7]

k =1

As the measurement relies on changes in the pressure in the system owing to absorption or desorption of hydrogen by the sample, a figure of merit which relates to the system resolution and accuracy is a helpful design parameter. Consider the evolution of the pressure in the hydrogenator during the kth step, and momentarily omit the index k for convenience. Once the connecting valve S has been opened and the ideally instantaneous change in pressure owing to the larger volume sampled by the pressure transducer has occurred, the system is isochoral (constant volume) and the number of moles of hydrogen contained as gas and in the sample is constant: Vj 2 p Σ + n H = constant sys j Z ( psys , T j ) T j R

[7.8]

where the sum runs over all the volumes in the hydrogenator, Vref and Vcell in the simplest case under consideration. The change in the hydrogen content of the sample is therefore reflected in a change in the system pressure which is given, according to Eq. (7.8), by: Vj ∆ n H = – 2 ∆ psys Σ j Z ( psys , T j ) T j R

[7.9]

Using the definition of H/X, the isochoric constraint may then be expressed in terms of the time evolution of the system pressure as the hydrogen is absorbed or desorbed during the kth step according to the kinetics of the sample:

( )

k k psys ( t ) = psys (0) – s k ∆ H X

[7.10]

where psys(0) is the system pressure immediately after the valve S has been opened, before any change in nH has occurred, and –sk is the slope of the isochore for the kth step:

182

Solid-state hydrogen storage

sk =

nX R Vj 2Σ j Z ( pk , T k) T k sys j j

[7.11]

sk indicates the sensitivity of the system to changes in H/X, as measured by changes in the system pressure, and so helps to quantify the performance of the hydrogenator. Figure 7.5 shows the actual locus of the system during an isochoric absorption step. If we ignore Z and assume that the measurement is isothermal, sk is a constant. The inclusion of Z bends the isochore, increasing the hydrogenator sensitivity when Z > 1, with potentially beneficial effect when measuring at high hydrogen pressures. Blach and Gray (2007) propose a figure of merit, η, for the sensitivity of the Sieverts hydrogenator:

psys

k p sys(0)

Isochore slope –sk k p sys k –1 p sys

nXk –1

nXk

H/X

7.5 Real locus of the system in pressure–composition space during an isochoric step from the k−1th to the kth point on the recorded isotherm (Eqns 7.10, 7.11). The excursion of the pressure in the k occurs when the valve S (Fig. 7.4) is opened hydrogenator to p sys(0) instantaneously. The system then approaches equilibrium according to the kinetics of the sample and the thermal relaxation time of the sample/cell sub-system. The pressure excursion is lessened if S is opened slowly, so that absorption commences while psys is still rising. The slope of the isochore is constant only if the compressibility is constant.

Reliably measuring hydrogen uptake in storage materials

η=

sk δp

183

[7.12]

where δp is the usable absolute resolution of the pressure transducer, and recommend η ≥ 100 as a rule of thumb to obtain data of high quality, assuming that other errors (in the volumes, temperatures and compressibilities in Eq (7.6) do not limit the hydrogenator performance. While the universal practice is to join the end-points of each isochoric step to make an isotherm, two caveats must be acknowledged: (i) the system undergoes excursions to pressures that may be far from the isotherm and only approaches the drawn isotherm as the free energy drive approaches zero owing to the falling (absorption) or rising (desorption) hydrogen pressure; (ii) the value of ∆nH calculated for a given step is a global average value for the entire sample and is not representative of the local change in hydrogen content unless the sample is homogeneous over a sufficiently small length scale (where ‘homogeneous’ comprises a multiphase system with spatially invariant phase proportions). Large-aliquot effect These intrinsic features of the isochoric technique can combine to adversely affect the quality of the measurement of a sample that exhibits pressure hysteresis owing to the so-called ‘large-aliquot effect’ reported by Park and Flanagan (1983) and incorrectly ascribed to high interface velocity. This effect arises when a large step in hydrogen concentration (high ∆nH) is made in a system with low total volume, necessitating a big excursion in psys (Eq. 7.9). Pons and Dantzer (1993) predicted from combined heat- and masstransfer modelling that temperature gradients in a two-phase system exhibiting pressure hysteresis cause a severe spatial inhomogeneity of the phase proportions. Gray et al. (1994) explained the large-aliquot effect as a consequence of the pressure excursion (∆psys) when a transforming twophase hydride system does so in a range of pressures, meaning that the plateaux in the absorption and desorption pressure–composition isotherms are sloping, owing for instance to a distribution of interatomic potentials at H sites. Considering a step across the absorption plateau, a large increase in system pressure may provide sufficient free energy drive to transform all parts of the sample. As isochoric absorption begins it simultaneously heats the absorbing regions and reduces the pressure. psys thus falls below the transformation pressure of some parts of the sample and these may actually go into desorption. The result is that while the sample globally follows the isochore, local regions are far from it. This effect is greatly exacerbated by poor heat sinking of the sample, which promotes severe temperature gradients owing to the enthalpy of the transformation, as predicted by Pons and Dantzer

184

Solid-state hydrogen storage

(1993). Kisi and Gray (1995) demonstrated the occurrence of inhomogeneous phase proportions with X-ray and neutron diffraction. Figure 7.6 shows an extreme example of the large-aliquot effect in LaNi5. The comparison is between two neutron powder diffraction patterns from the same sample recorded at very nearly the same value of H/X, with the difference that one was arrived at in small pseudo-isobaric steps and the other in a single, large isochoric step requiring a very high excursion of the system pressure. The lattice parameters of LaNi5Dx depend strongly on the phase proportions and state (absorption or desorption) of the transforming α and β phases (Kisi et al. 1994) (plus γ phase if present), and the diffraction pattern is an average over the volume (several cm3) of the large, poorly heat-sinked sample. Therefore only a homogeneous sample will yield a well-defined diffraction pattern. In Fig. 7.6 the more sharply defined pattern (dotted line) implies that the phase proportions are uniform throughout the sample. The second pattern (solid line) is unintelligible beyond the fact that the phase proportions are far from uniform. The corresponding system pressure was half-way between the absorption and desorption plateau, confirming the mixture of absorbing and desorbing regions. 50 000

Al

Neutrons/ms

40 000

30 000

20 000

10 000 95

100

105 Time of flight (ms)

110

115

7.6 Portions of neutron powder diffraction patterns recorded on the high resolution powder diffractometer (HRPD) instrument at ISIS (UK) from LaNi5 charged with deuterium in situ to approx. D/M = 0 to 0.6 in the α + β two-phase region. Dotted line: after multiple pseudoisobaric absorption steps. Solid line: after a single isochoric absorption step from D/M = 0. The latter data are uninterpretable except that they obviously represent regions of sample with widely distributed lattice parameters. The highest peaks come from the aluminium sample cell and demonstrate that the only difference between the two measurements is the state of the sample.

Reliably measuring hydrogen uptake in storage materials

185

While the large-aliquot effect was explored using measurements on LaNi5– H2, it is just as relevant to materials of current interest based on Li complexes, as these exhibit pressure hysteresis, have high enthalpies of transformation, poor thermal conductivity owing to small particle size and sloping pressure plateaux (e.g. Luo and Rönnebro, 2005). Isobaric operation of a Sieverts hydrogenator Figure 7.7 shows an in-principle solution to the problem of the large-aliquot effect through a simple modification to the Sieverts isochoric hydrogenator to permit isobaric operation. The principle of determining H/X remains the same as in Eq. (7.6), but the pressure in the sample cell is controlled at a constant value by feeding the hydrogen to or from the sample through a needle valve (NV) under automatic control. Figure 7.8 depicts the locus of the system during a pseudo-isobaric absorption step in which the pressure in the sample cell is controlled at pset until the isobar crosses the isochore, at which point NV is fully open and the final hydrogen content of the sample can be calculated in the usual way. The addition of a second pressure transducer on the sample cell is necessary for pressure control and also allows H/X to be calculated at all times, albeit with degraded accuracy since two pressure readings are involved. Because the pressure over the sample never rises above (absorption) or sinks below (desorption) the set value during an ideally

Pref

Pcell NV

Vacuum pump

S Hydrogen source

Vref

Vcell

7.7 Sieverts hydrogenator modified from the standard configuration (Fig. 7.4) for isobaric operation by the addition of a needle valve (NV) and a second pressure transducer. The needle valve feeds gas to or from the sample under feedback control such that Pcell is constant. In addition to providing the feedback signal, the second pressure transducer allows the hydrogen content of the sample to be calculated at all times.

186

Solid-state hydrogen storage

psys

Isobar k p set k p sys

k –1 p sys

nXk –1

nXk

H/X

7.8 Real locus of the system in pressure–composition space during a pseudo-isobaric step from the k–1th to the kth point on the recorded isotherm in the modified Sieverts hydrogenator. The excursion of the k pressure in the hydrogenator is limited to p set by the controlled opening of the needle valve (NV in Fig. 7.7).

isobaric step, temperature gradients in the sample cannot cause compositional inhomogeneity (at least, not with the normal exothermic absorption and endothermic desorption). Of course the initial pressure in the reference volume must be chosen judiciously relative to the set pressure for the isobar, or else the isobar may meet the global system isochore well before it meets the isotherm. Even so, the potential for inhomogeneity is greatly reduced in this pseudo-isobaric approach to equilibrium. Isobaric isotherms have rarely been recorded, perhaps owing to the extra complexity of the apparatus, despite the physical good sense of measuring isotherms under a constraint of constant chemical potential. An example from a gravimetric apparatus is given by Benham and Ross (1989). If the isotherm has very low slope then a pure isobaric approach cannot obtain data points in the two-phase region, but a mixed isobaric/isochoric approach will do this successfully and restrict the excursion of the pressure over the sample. It is perhaps a moot point that the large-aliquot effect should not occur anyway without a sloping plateau, but the different results found for Pd–H2 by Benham and Ross using the isobaric rather than the usual isochoric conditions make further exploration of isobaric methodologies worthwhile.

Reliably measuring hydrogen uptake in storage materials

187

While isobaric conditions will prevent compositional inhomogeneity where the plateaux are sloping, the free energy drive is low owing to the small pressure excursion, stretching the kinetics in time significantly compared with the isochoric approach, in which the free energy drive remains relatively high until the isochore approaches the isotherm. Isobaric conditions also have a place in measuring the true kinetics of absorption and desorption by sweeping the hydrogen content of the sample through its full range in a single step under constant chemical potential.

7.4.2

Gravimetric technique

Figure 7.9 shows a generic gravimetric measuring system in which the gas atmosphere is managed by a manifold essentially the same as the basic Sieverts hydrogenator in Fig. 7.4. The entire balance mechanism is pressurised and the tare weight is thus surrounded by the same pressure of hydrogen gas as the sample. This is a significant point of differentiation between this more traditional configuration and that in which the sample cell is isolated from the balance mechanism by a magnetic suspension, so that the balance mechanism remains at one atmosphere. The single-sided balance is a particular case of that depicted in Fig. 7.9, so the mathematical formalism by which the

Gas / vacuum

T

S

7.9 Minimal gravimetric hydrogenator, based on a symmetric balance with both the sample (S) and the tare weight (T) suspended in the hydrogen gas.

188

Solid-state hydrogen storage

hydrogen content of the sample is calculated is developed for the more general case. The principle of the measurement is to determine the mass of adsorbed or absorbed hydrogen, mH, by weighing the sample in hydrogen. The sample mass, ms = mX + mH, is approximately balanced by a tare mass, mt. Every component of mass m and density ρ immersed in the gas experiences a buoyancy force equal to the weight of gas at density ρg displaced by its volume v. The total force on the balance may then be written as: F=g

ρ

 g, j  Σ k ( m j – v j ρg, j ) = g j = Σ k m j 1 – j =s,t,… s,t,… ρj  

[7.13]

where s refers to the sample side of the balance, t to the tare side and k = +1 for components on the sample side or –1 for components on the tare side. ρg,j is the density of the gas in the region of the jth component. The term in parentheses expresses the buoyancy effect on every component. The central problem of the gravimetric technique is to extract ms from Eq. (7.13). Clearly it is advantageous to use materials of the same density on the sample and tare sides of the balance to minimise the resultant buoyancy contribution to Eq. (7.13) through subtraction of terms with k = ±1. For the most general case of a beam balance with tare weight and sample suspended in the same working gas at different temperatures, Eq. (7.13) becomes: F = m – m – Mp  v s ( Ts , p ) – v t ( Tt )  s t R  Z ( Ts , p ) Ts g Z ( Tt , p ) Tt  + ∆m b –

v bt ( Tt )  Mp  v bs ( Ts ) – R  Z ( Ts , p ) Ts Z ( Tt , p ) Tt 

+ ∆m h –

Mp R

+ ∆m 0 –

Mp ∆v 0 RZ ( T0 , p ) T0

 v hs ( Ths ) v ht ( Tht )   Z (T , p) T – Z (T , p) T  hs hs ht ht   [7.14]

where M is the formula weight of the working gas, vs is the volume of the sample that is not accessible to the working gas, vt is the volume of the tare weight, which is assumed to be impervious to hydrogen, vbs and vbt are the volumes of the buckets (if any) on the sample and tare sides of the balance, ∆mb is the difference in the masses of the buckets, vhs and vht are the volumes of the hangdown (suspension) components, ∆mh is the difference in the masses of the hangdown components on the sample and tare sides, Ths and Tht are effective temperatures for the hangdown components, which typically traverse zones in a range of temperatures between Ts and T0, the temperature at the balance beam, ∆v0 is the difference in the effective volumes of the two

Reliably measuring hydrogen uptake in storage materials

189

sides of the balance beam and ∆m0 is the difference in the effective masses of the two sides of the balance beam. Equation (7.14) assumes that the balance beam is isothermal and that all components except the sample are impervious to the working gas. Several cases of Eq. (7.14) correspond to commonly encountered practical gravimetric systems and are considered in Section 7.5.2. Gravimetric measurement regimes A gravimetric system is subject to many of the same concerns expressed about the Sieverts technique. The gas may be introduced to the sample region in pure isochoric or isobaric modes, although the pressure shock of strict isochoric operation may damage the balance mechanism. A more likely scenario is that to protect the balance mechanism, gas flows to the balance via a control valve that sustains a pressure difference between the gas handling system and the sample environment. If the sample (say) absorbs hydrogen very quickly, then the actual operating regime may be constant flow, but the pressure over a very slowly absorbing sample will quickly exceed the equilibrium absorption pressure, making the regime effectively isochoric. This variability may cause a degree of irreproducibility in the measured isotherms, whose position may depend on the measuring regime (Fig. 7.3). For this reason it is desirable that the gas flow to and from the balance be actively controlled in a reproducible way, such as to ensure that the pressure over the sample changes monotonically throughout the execution of an isotherm, or in small isochoric steps, or in pseudo-isobaric steps (Fig. 7.8). Depending on the means by which the hydrogen gas pressure in the balance is changed, gravimetry is also susceptible to the large-aliquot effect. The author’s experience is that while the sample temperature excursion owing to a change in the hydrogen content may be larger than in a Sieverts apparatus, owing to the rather high thermal resistance of the gas between the sample and the walls of the pressure cell, the temperature gradient within the sample is relatively small for the same reason, thus mitigating the large-aliquot effect.

7.5

System characterisation

An accurate model of the system consisting of the apparatus and the sample is essential for extracting reliable values for the uptake of hydrogen from Eq. (7.6) (Sieverts technique) or Eq. (7.14) (gravimetric technique).

7.5.1

Sieverts apparatus

Calibration of the volume of each component of the Sieverts hydrogenator is fundamental to its accuracy. The system model must include the temperature

190

Solid-state hydrogen storage

profile of the system. For this reason it is very desirable that the manifold and reference volume(s) be temperature controlled and isothermal. While measuring the temperature of the gas in each major volume is straightforward, the sample cell and its pipework (to the right of S in Fig. 7.4) require careful modelling because the sample temperature may be very different from that of the rest of the hydrogenator. The characterisation of an isothermal hydrogenator, such as one totally immersed in a temperature-controlled bath, is very straightforward. For instance, an accurately known calibration volume may be connected to the hydrogenator, filled with a measured pressure of calibrating gas, and then sequentially opened to the previously evacuated hydrogenator component volumes. These are then calculated based on the fixed quantity of gas in the system. The compressibility of the calibrating gas must of course be accounted for. The same methodology is the basis for characterisation of the system when the temperatures of the reference volume and sample differ. A likely scenario is that Vref is known and Vcell or Vcell – VX (the cell dead volume) in Eq. (7.6) is not known. Beginning with the sample cell evacuated and a pressure p0 of calibrating gas in Vref, the number of moles of gas in the system is fixed at n0 = (p0Vref)/[Z(p0, T 0)RT 0], where T 0 is the initial temperature of the reference volume, by closing the system. This gas is then admitted to the sample cell via S and the system pressure settles to a value psys(Tref, Ts) that depends on the temperature distribution in the sample cell and its associated pipework. Supposing that all components to the right of S in Fig. 7.4 are isothermal at temperature Ts, then the cell volume (or dead volume) may be straightforwardly calculated from n0 =

psys ( Tref , Ts )  Vref Vcell   Z ( psys , Tref ) Tref + Z ( psys , Ts ) Ts  R  

[7.15]

A step in temperature occurring at the valve S between Tref and Ts is an unrealistic assumption. Two simple thermal models are now considered, followed by a variation with application in high-pressure systems. Equivalent volume model e If in Eq. (7.15) Vcell/Ts is replaced by Vcell ( Ts )/ Tref then an effective cell e volume Vcell may be calculated:

 n 0 RTref  Vref e Vcell =  –  Z ( psys , Ts )  psys ( Tref , Ts ) Z ( psys , Tref ) 

[7.16]

This volume depends on the sample temperature through psys and on Tref. If Ts is scanned through the range of working temperatures then a representation of the effective cell volume as a function of Ts is obtained. The weakness of

Reliably measuring hydrogen uptake in storage materials

191

this approach is that Tref must be reasonably constant throughout the measurement of hydrogen uptake based on this equivalent volume, say to within about 1 K. It is otherwise robust, as long as the compressibility of the calibrating gas is included in the calculation so that the equivalent volume does not become dependent on pressure as well as sample temperature. Divided volume model The simplest possible thermal model of the real temperature distribution in the sample cell and its associated pipework is a step function in temperature at a position which should be determinable through a suitable calibration procedure. Figure 7.10 shows the sample cell divided into volumes V1 at temperature Tref and V2 at the temperature of the sample, Ts, with the constraint that Vcell = V1 + V2 is the real volume of the system closed off by the valve S. The basis of the calibration in this divided-volume approach is to start with the sample temperature at some base value and change Ts through the full working range while recording the system pressure. Once again, n0 = (p0Vref)/[Z(p0, T 0)RT 0] moles of calibrating gas are introduced to the reference volume. When this gas is admitted to the sample cell via S, the system pressure settles to a value psys(Tref, Ts) that may easily

S

V1

Vcell = V1 + V2

V2

7.10 Divided-volume model of the temperature distribution in the sample cell and associated pipework. V1 is assumed to be at the same temperature as the reference volume. V2 is assumed to be at the same temperature as the sample.

192

Solid-state hydrogen storage

be shown to depend on the temperature of the region around the sample through:

Rn 0 Z ( psys , Tref ) Tref Z ( psys , Tref ) Tref – Vref = V1 + V2 psys ( Tref , Ts ) Z ( psys , Ts ) Ts

[7.17]

If Ts is scanned and a graph is constructed of the left side of Eq. (7.17) against [Z(psys, Tref)Tref]/[Z(psys, Ts)Ts], the intercept will be V1 and the slope will be V2. The divided-volume model is generally more robust than the equivalentvolume model but nevertheless has some weaknesses. First, Tref must be constant during the calibration procedure. Second, the position at which the step function in temperature between Tref and Ts is assumed to occur must be independent of Ts, so that V1 and V2 are constants. If the sample cell is at Ts and the transition between V1 and V2 occurs in the connecting pipework, this condition is fairly easy to satisfy. Third, to accurately delineate V1 and V2, the coefficient of V2 in Eq. (7.17) must be a fairly strong function of temperature, either through a wide range of scanned values for Ts or through a rapidly varying compressibility. Volume calibration based on compressibility A related approach that relies on a non-linear variation of compressibility with temperature is useful in high-pressure Sieverts systems and can be used, in fact is best used, with hydrogen as the calibrating gas. Re-casting Eq. (7.17) so that psys is a function of the measured variables, psys ( Tref , Ts ) =

Vref Z ( psys , Tref ) Tref

Rn 0 V1 V2 + + Z ( psys , Tref ) Tref Z ( psys , Ts ) Ts [7.18]

the right side of Eq. (7.18) is fitted to the measured pressure with V1 and V2 as fitting parameters. This approach does not require Tref to be constant and makes no assumption about the compressibility but still assumes that V1 and V2 are constant in the temperature range scanned.

7.5.2

Gravimetric apparatus

Equation (7.13) shows that the accuracy of the value of the sample mass depends absolutely on accurate knowledge of the volumes or densities of the suspended components. Methodologies for measuring these in situ are now considered for three common realisations of a gravimetric system.

Reliably measuring hydrogen uptake in storage materials

193

Two-sided balance with symmetric temperature distribution If the system is perfectly balanced by all components of the balance itself having the same masses and volumes on the sample and tare sides, only the difference between the sample and tare weights contributes to the total measured force. For this reason it is very advantageous to maintain the temperature environments around the sample and tare weight as nearly identical as possible, by, for example, a two-tube furnace for measurements above room temperature. For the realistic case of symmetric temperature but asymmetric masses, Eq. (7.14) reduces to

Mp F =m –m – [ v ( T , p ) – v t ( Ts )] s t g Z ( Ts , p ) RTs s s + ∆m b –

Mp [ v ( T ) – v bt ( Ts )] Z ( Ts , p ) RTs bs s

+ ∆m h –

Mp [ v hs ( Ths ) – v ht ( Ths )] Z ( Ths , p ) RThs

+ ∆ m0 –

Mp ∆ v0 Z ( T0 , p ) RT0

[7.19]

While the mass imbalances between the sample and tare sides of the balance (∆m0, ∆mh, ∆mb) are constant and can usually be electronically tared (offset) when the sample is loaded, the volume imbalances require quantification. This can be accomplished by first stripping back the balance suspension to the bare beam and recording the resultant force as a function of gas pressure with the entire balance at or near T0, thus allowing the value of ∆v0 to be extracted by fitting the data to the last line of Eq. (7.19). The suspension components pose a more complicated problem, owing to the temperature distributions that may exist along their lengths. If the suspension is light and of dense material, the buoyancy forces on the suspension components will be very small, probably negligible, and changes in their volumes with sample temperature may certainly be neglected. Thus a measurement made as a function of gas pressure with the hangdown components in place and the entire system at T0 will yield a sufficiently reliable value for ∆v h ( Th ) ≈ ∆vh(T0) = vhs(T0) – vht(T0) if one is required. If the suspension consists of, say, thick quartz filaments, then a measurement of the resultant balance force at roughly constant pressure in a wide range of temperatures may be fitted to the third line of Eq. (7.19) to obtain effective values for ∆vh, e.g. referred to Ts by using Th = Ts in Eq. (7.19). Now adding the sample and tare buckets and again recording the resultant force as a function of gas pressure at Ts = T0 = Tt will allow the value of ∆vb(T0) = vbs(T0) – vbt(T0) to be

194

Solid-state hydrogen storage

extracted. Depending on the volumes of the buckets, it may suffice to use a thermal expansion correction to calculate ∆vb(Ts) for different sample temperatures, or a series of measurements at different temperatures can be made if necessary. The foregoing procedure accounts for all terms in Eq. (7.19) except those in the first line. While the characterisation of the balance itself can be performed with the usual working gas, although a relatively dense gas may be used advantageously to amplify the buoyancy effects, the volume difference between the sample and the tare weight must be determined with a gas that does not interact with the sample (or tare weight, obviously). Assuming that such a gas is available (but see Section 7.6.1), finally recording the resultant force as a function of gas pressure in the full range of sample temperatures to be used allows values for vs(Ts) – vt(Ts) to be extracted, including any hydrostatic pressure effect on the sample. The effect of hydrogen absorption on the mass of the sample is now able to be extracted. Two-sided balance with asymmetric temperature distribution In some circumstances it may be impractical to achieve symmetry between the sample and tare sides of a two-sided balance, e.g. if the sample is in a cryogenic environment that cannot readily be duplicated on the tare side. Equation (7.14) applies to this scenario. While measurements of ∆vh(T0) and ∆vb(T0) may be made as described in the preceding sub-section, separated values for vbs(Ts), vbt(Tt), v hs ( Ts ) and v ht ( Tt ) are required. This can be accomplished by measurements of the resultant force at roughly constant high pressure in the full range of sample temperatures to which the second and third lines of Eq. (7.14) are separately fitted subject to the constraints ∆vh(T0) = vhs(T0) – vht(T0) and ∆vb(T0) = vbs(T0) – vbt(T0). The same procedure is then applied to individually determining values for vs(Ts) and vt(Tt). Single-sided balance In a single-sided system the tare-side components are isolated from the sample side, typically by a magnetic suspension of the sample and associated components across the wall of the pressure cell. This arrangement has the advantage of also isolating the balance mechanism from the working gas and so generally facilitating a higher working pressure than is common among double-sided systems. On the down side, there is no first-order cancellation of buoyancy forces between the sample and tare sides of the balance, so the masses and volumes of the suspended components must be measured with the greatest care. For this scenario Eq. (7.14) reduces to

Reliably measuring hydrogen uptake in storage materials

F = m – Mp  v s ( Ts , p ) s R  Z ( Ts , p ) Ts g

+ mh –

195

 – m + m – Mp  v b ( Ts )  0 b  R  Z ( Ts , p ) Ts 

Mp  v h ( Th )  R  Z ( Th , p ) Th 

[7.20]

where m0 accounts for all the components outside the sample cell. As the weight and volume of the magnetic suspension components inside the sample cell are necessarily much greater than in the case of the double-sided balance, it is in principle necessary to remove the sample and sample bucket and record the resultant force at a high pressure in the full range of sample temperatures, so that a set of values can be obtained to represent vh, e.g. referred to Ts by using Th = Ts in Eq. (7.20). It is again probably advantageous to do this with a dense gas in the sample cell to amplify the buoyancy contribution. The sample bucket may now be added and an analogous set of measurements performed to obtain ∆vb(Ts). Finally, the effective sample volume may be measured with a gas towards which the sample is inert to complete the model of this system.

7.6

The sample volume problem

Whereas the need to measure temperature and pressure accurately is obvious and widely accepted, the problem of accounting for the sample volume turns out to be more subtle and potentially much more detrimental to the measurement of H/X, as recently explained in detail by Blach and Gray (2007) for the Sieverts technique. As an example, Figure 7.11 shows the disastrous effect of a ±25% change in the assumed density of a sample of potassium-intercalated graphite on the calculated isotherm for deuterium absorption by the Sieverts technique. The sample volume problem has been pointed out before (Beeri et al., 1998) in the context of high-pressure measurements, but Blach and Gray (2007) showed that the problem also occurs in an isochoric manometric apparatus at pressures of a few bar, particularly with low-density materials. Sircar (2001) gives an example of comparably wrong measurements of the Gibbsian surface excess (the total adsorbed amount less the amount of gas that would occupy the volume of the adsorbed phase) of N2 on alumina with a surface area of only 120 m2 g–1, owing to an inadequate model of the interaction of the calibrating gas (He) with the sample during measurements of its density. The Sieverts technique is sensitive to the density of the sample because the volume occupied by the sample must be subtracted from the volume of the empty sample cell in order to calculate H/X. The gravimetric technique is intrinsically sensitive to the volume or density of the sample through the buoyancy force on it. Knowledge of the sample volume is also required to

196

Solid-state hydrogen storage 700 600

Pressure (bar)

500

400 300 200 100 0 0.00

0.02

0.04

0.06 H/X

0.08

0.10

0.12

0.14

7.11 Effect of a ±25% change in assumed sample density on the apparent hydrogen uptake of C24K at room temperature, measured in a system of insufficient volume relative to the volume of sample (after Blach and Gray, 2007). Circles: ρ = 2.0 g/cm3, inverted triangles: ρ = 1.5 g/cm3, upright triangles: ρ = 2.5 g/cm3.

construct isotherms of excess hydrogen uptake (the excess over the amount of gas that would be present anyway in the absence of the sample) and, at a deeper level of investigation, to obtain information about the adsorbed phase itself via determination of the Gibbsian surface excess. The difficulty then arises of knowing the volume occupied by the sample with sufficient accuracy throughout the course of the measurement of hydrogen uptake. Taking a certain number of moles of sample atoms, corresponding to a particular achievable hydrogen uptake via H/X, the volume of the sample increases as its density decreases. As low-density samples are generally those with greatest promise for hydrogen storage at acceptable hydrogen density by mass, the sample problem is thus most significant for the very materials of greatest current interest. Whatever approach is taken to characterising a gravimetric system, comparison with deuterium uptake by the same sample is a valuable check on the measured uptake of protium.

7.6.1

Is the calibrating gas inert?

The most common way of defining the volume of the sample is to measure the effective volume of the loaded sample cell (Sieverts technique) or the sample itself (gravimetric technique) using an inert gas. The validity of this

Reliably measuring hydrogen uptake in storage materials

197

procedure is undoubted where the sample has a three-dimensional morphology with low surface area and its density does not change owing to hydrogen absorption. In reality, even He is reported to adsorb detectably onto lowdimensional active materials such as activated carbon and zeolites (Malbrunot et al., 1997), single-walled carbon nanotubes at 300 °C (Haas et al., 2005), and even relatively inert silicalite (Gumma and Talu, 2003), raising serious doubts about the reliability of this approach unless great care is taken with the determination of the skeleton density (that corresponding to the actual volume occupied by sample atoms) of the sample. When the surface area of the sample is a significant contributor to its hydrogen uptake, it is necessary to assume that the sample is equally accessible to the calibrating and working gases. As their sizes are comparable, this is less of a problem with helium and hydrogen than with, say, helium and methane. When the sample density changes with hydrogen concentration, the problem of uncertain density cannot be solved by calibration at H/X = 0 and the system should be designed to minimise its effect as far as possible. While materials that absorb H rather than adsorb H2 are expected to exhibit the greatest effects of hydrogen uptake on density, even adsorption systems may not be immune, as Dreisbach, et al. (2002) reported a swelling of activated carbon owing to He uptake. Correcting for the sample density becomes an even more crucial prerequisite when adsorption isotherms are interpreted on a model of gas plus surface adsorbed phase plus the skeleton of sample itself (e.g. Dreisbach et al., 2002). Sircar (2001) describes methodologies for both manometric and gravimetric measurements by which the Gibbsian surface excess may be determined using a simple model of the low-pressure adsorption of the calibrating gas. The assumption that the calibrating gas does not adsorb on the sample at the highest working temperature is required, however. Gumma and Talu (2003) further discuss the concept and experimental difficulty of defining the Gibbs surface that encloses the impenetrable atomic skeleton of the sample and separates it from the adsorbed phase. The ‘impenetrable volume’ of a sample is of course the volume occupied by its electrons, so even this volume in principle may vary with the strength of the interaction between the sample and adsorbent. The approach of Gumma and Talu is more realistic than that of Sircar, in that no assumption is made about the degree of adsorption of the calibrating gas and the parameters of its adsorption isotherm are fitted to experimental data in the limit of low pressure. The model of Dreisbach et al., which is based on an assumed analytical isotherm is also worthy of consideration. A critical comparison of the latest approaches to the analysis of the same data is required. While the methodologies of Sircar (2001), Dreisbach et al. (2002) and Gumma and Talu (2003) are aimed at determining the Gibbsian surface

198

Solid-state hydrogen storage

excess adsorption, they apply equally to the determination of hydrogen uptake as an end in itself, as the problem of the interaction of the calibrating gas with the sample is common to both aims. An aspect of the inertness of the calibrating gas that receives little attention is its purity. Aside from the potential to poison the surface of the sample against hydrogen uptake, an impure calibrating gas may cause problems in the characterisation of a gravimetric system owing to preferential, reversible adsorption of heavy, reactive impurities such as O2. This gettering of the calibrating gas by the sample is much less of a problem in a manometric apparatus. In general, however, the purity of the calibrating gas should be about as good as that of the hydrogen.

7.6.2

Sieverts technique

The premise of the Sieverts technique is that accurate values for all the parameters in Eq. (7.6) are known at the end of the kth step, most fundamentally the volumes that constitute the system. Equation (7.6), which has been written so as to expose the dependence of the calculation on the volume of sample, shows that uncertain knowledge of the sample volume (via its mass and density) and cell volume affect the calculation as if an error had occurred in the calibration of the hydrogenator. As pointed out in the preceding section, expansion of the sample owing to absorption needs to be allowed for by calculation or measurement, which might not be feasible, or, preferably, by designing out the sensitivity to it. Partially differentiating Eq. (7.6) with respect to density shows that (a) the effect of a change in density on the calculated hydrogen uptake depends on ρ–2, confirming the increased sensitivity to low but uncertain sample densities, and (b) the dependence on the actual instantaneous conditions of p, V and Z is complicated and not amenable to analytic analysis. Blach and Gray (2007) therefore employed simulation to explore the consequences of inaccurately known sample density and, for comparison, compressibility and volumes. The most significant differences between the various experimental set-ups tested were in the ratio of the reference volume, Vref, to the empty volume of the sample cell, Vcell, and in the fraction of the cell volume actually occupied by sample. A first intuition, that insensitivity to the sample density will be conferred by making the sample cell volume as small a fraction of the system volume as possible, is wrong. Likewise, a small sample in a cell which is itself a small fraction of the total system volume is not completely effective. The best outcome was obtained with comparable reference and cell volumes and a sample which occupied a small fraction of the system volume. This may be rationalised as follows. The vulnerability of the Sieverts technique is that it relies on calculating the quantity of hydrogen exchanged with the sample in

Reliably measuring hydrogen uptake in storage materials

199

the kth step by the subtraction of two relatively large numbers, which are the amounts of hydrogen in the system before and after the kth step. If the cell k in Eq. (7.6), will not be much volume is very small, the system pressure, psys k , that initiated it. If different from the pressure in the reference volume, p ref Vref is fairly large, the first term in Eq. (7.6) will then be of moderate magnitude. However, the change in system pressure between the kth and (k + 1)th steps may be large if the isotherm contains a small number of points, and this difference amplifies the volume term containing the difference between the empty cell and the sample volumes, making it also of moderate magnitude. Thus there is the potential for a large effect on the calculated quantity of hydrogen exchanged with the sample owing to an error in the volume occupied by the sample, implicating the sample density. The approach of making both Vref and Vcell large compared with the notional volume of sample needs to be balanced against the figure of merit for the system (Eq. 7.12), which will degrade as the system volume becomes too large unless a pressure transducer of increased accuracy is employed. It is not possible to precisely define the optimum ratio of volumes because of the complicated dependence of the systematic error in H/X caused by a density error on p, T and Z: there is no single set-up that is optimum for all conditions of pressure and temperature, but it is certainly possible to greatly lessen the sensitivity of the results to uncertain sample density by following the rulesof-thumb proposed by Blach and Gray (2007) for the Sieverts technique: (i) ensure that the reference volume and the empty cell volume are (a) both large compared with the volume notionally occupied by the sample, by a factor of at least 100 in each case, and (b) ideally about equal; (ii) ensure a figure of merit for the hydrogenator (Eq. 7.12) of at least 100. A further important point made by Blach and Gray (2007) is that measuring a dense sample such as LaNi5 as a de facto standard to check system performance can be misleading: a perfectly satisfactory measurement of LaNi5 does not guarantee good performance with samples of much lower density.

7.6.3

Gravimetric technique

As in the case of the Sieverts technique, the appearance of the sample density on the bottom line in Eq. (7.13) implies a ρ–2 sensitivity of the result to sample density and increased buoyancy forces with samples of low density. Equation (7.14) shows that the best outcome is with matched sample and tare-weight densities as well as matched masses, thus cancelling the buoyancy force on the sample. At the opposite extreme, the single-sided balance (or a double-sided balance with grossly mismatched sample and tare-weight densities) requires careful calibration with an inert gas as discussed in Section 7.6.1.

200

Solid-state hydrogen storage

When the density of the sample varies with hydrogen uptake, Eq. (7.14) makes it clear that, because the measured value scales with sample mass and volume in the same way, nothing can be done with the design of the apparatus to lessen the effect of the changing sample density. The only recourse is to independent measurements by diffraction, for instance, or to the very detailed interpretation of a suite of data on a hopefully realistic physical model, such as that of Dreisbach et al. (2002). Independent measurements may be the only way to reliably know that the sample density is not constant anyway. It seems likely that there are published studies in which this effect has led to the wrong interpretation of the results.

7.7

The variable-volume hydrogenator

The convenience, robustness and low entry cost of a manometric hydrogenator ensure that the Sieverts apparatus is the most popular approach to measuring hydrogen uptake. The sample volume problem and the basic isochoric measurement regime are significant disadvantages. The gravimetric technique is intrinsically more flexible, owing to the decoupling of the measurement of hydrogen uptake from the choice of isochoric, isobaric, etc. measurement regimes. The weaknesses of gravimetry are the need to accurately model the volumes of all suspended components and the great difficulty of accounting for the effect of a sample density that depends on hydrogen uptake. A new manometric approach is proposed here, intermediate in cost and complexity between a manometric and a gravimetric system, in which Vref is variable.

7.7.1

Variable-volume technique

Figure 7.12 shows a minimal variable-volume system. Vref, or some substantial fraction of it, is a volume that depends on the position of a piston, ideally realised with a computer-operated motor drive: Vref (x) = V0 – αx

[7.21]

where x ∈ [0, xmax] is the position of the piston, V0 is the constant part of Vref, αxmax is the displacement volume of the variable portion and where α = – ∂Vref/∂x. Increasing x thus corresponds to compression of the gas in Vref. Within limits imposed by the size of the variable volume compared to the total system volume including Vcell, the control of x may be based on any desired constraint (through feedback) to realise conditions of constant volume (x = constant), constant pressure, constant concentration or constant molar flow. An important advantage of the variable-volume technique over the Sieverts technique becomes apparent by re-casting Eq. (7.6) for the variable-volume scenario:

Reliably measuring hydrogen uptake in storage materials Pressure transducer

201

P

Vacuum pump S

Hydrogen source

V0

Vcell

Vref = V0 – α x

0

x

xmax

7.12 Minimal variable-volume hydrogenator, constructed from a basic Sieverts apparatus by including a variable component in Vref, such as a motor-driven piston, with displacement volume αxmax.

0 0   psys Vref psys ( x ) Vref ( x )  ∆n H ( x ) = 2   –  0 0 0 Z ( p , T ) RT ( x ) Z ( p , T ) RT sys ref ref   sys ref ref 

0 psys psys ( x )    + –  ( Vcell – VX )  0 0 0 ( , ) ( ) Z p T RT x sys cell cell   Z ( psys , Tcell ) RTcell 

[7.22] where the ‘0’ superscript refers to the initial state at the start of the compression or decompression sequence in which the piston is driven in or out. The sample cell is represented in Eq. (7.22) by a single volume at temperature Tcell for clarity, but the application of a model of the temperature distribution as discussed in Section 7.5.1 is a straightforward extension. Compared with the traditional isochoric approach, it is as if a single hydrogenation or dehydrogenation step were executed. Thus the difference between the final and initial number of moles of H in the sample is computed by direct reference 0 , rather than from a cumulative stepwise sequence. to the initial state, p = psys Compared with Eq. (7.6), removal of the reference to the system pressure at the end of the preceding step removes the mechanism referred to in Section 7.6.1 for amplifying the effect of the sample volume on the calculation. As the number of isochoral steps is greatly decreased to possibly one per isotherm, the cumulative errors intrinsic to the Sieverts technique are also largely eliminated.

202

Solid-state hydrogen storage

7.7.2

Characterisation of the variable-volume apparatus

The basis of the calibration of the volumes in the hydrogenator is as discussed in Section 7.5.1. The availability of the variable reference volume greatly facilitates calibration of any other volume in the system, particularly if it is under computer control. Vref is first calibrated at enough piston positions x to accurately measure ∂Vref/∂x. The unknown volume to be calibrated, Vu, which can include the constant part of Vref, is connected and an initial pressure p0 is established. The piston is then driven in and the pressure and all relevant temperatures are recorded against x. The amount of gas in the system is constant, so assuming for clarity that all of Vref is at the same temperature and the compression sequence starts at x = 0:   V0 Vu p0  + 0 0 0 0 0 0   Z ( psys , Tref ) RTref Z ( psys , Tu ) RTu  V0 – α x Vu   = psys ( x )  –  Z ( p , T ) RT ( x ) Z ( p , T ) RT ( x ) sys sys u u ref ref  

[7.23]

which may be solved for psys(x) and fitted to the data to obtain both V0 and Vu if Tref and Tu are very different. Alternatively, if Tu = Tref then Eq. (7.23) reduces to a simple form that can be graphed to find V0 + Vu from the slope: V + Vu   p 0 / Z ( p 0, T 0 )  x=  0 1–  α   p ( x )/ Z ( p, T) 

[7.24]

In summary, the variable-volume technique promises to confer some valuable advantages: • • • • •

Great flexibility – constant volume, pressure, composition or molar flow. Mitigation of the sample volume problem. Improved overall accuracy by avoiding cumulative errors. Ability to take many data points without compromising accuracy through error accumulation. Facilitation of automated many-point volume calibrations.

At the time of writing a variable-volume hydrogenator for pressures up to 345 bar is being constructed at Griffith University. Results will be published in due course.

7.8

Summary and conclusions

The possibilities and problems of measuring the uptake of hydrogen using manometric and gravimetric apparatus have been considered. The universal dependence of the measurement on the density of the gas around the sample

Reliably measuring hydrogen uptake in storage materials

203

means that some measure of gas density beyond the ideal-gas approximation should almost always be used. The possible operating regimes (isochoric, isobaric, isoplethic, constant flow) were described. Detailed models of the Sieverts manometric hydrogenator and a generic gravimetric system based on a microbalance were presented and the characterisation of the system of hydrogenator plus sample was discussed. The problems caused by the invariable need to accurately know the effective volume of the sample were considered in some detail. While calibration with an inert gas is the in-principle solution to this problem, even helium adsorbs measurably onto many materials with large surface area, and the volume of the sample may change with hydrogen content, which calibration with He cannot accommodate. Criteria by which a Sieverts apparatus can be designed to avoid the sample-volume problem by making it insensitive to the sample volume or density were presented. The sample-volume problem is much tougher in the case of gravimetry, with no apparent design-based route available to avoid the absolute need for accurate measurements of the impenetrable volume or skeleton density of the sample. A new approach to measuring hydrogen uptake, that of the variable-volume hydrogenator, was presented and predicted to be much less susceptible to the sample-volume problem, to have excellent flexibility in choice of operating regime and to offer generally improved accuracy over the Sieverts technique.

7.9

Acknowledgements

It is a pleasure to acknowledge the colleagues with whom the work on technique and methodology has been carried out over a number of years: Tomasz Blach, Craig Buckley, Erich Kisi, Keith McLennan, Jim Webb. The data for Fig. 7.1 were kindly provided by Tomasz Blach. Thanks are due to Jim Webb for critically reading the manuscript.

7.10

References

Beeri O, Cohen D, Gavra Z, Johnson J R and Mintz M H (1998), J Alloys Cmpnds, 267, 113–120. Benham M J and Ross D K (1989), Z Phys Chem N F, 163, S25–S32. Blach T B and Gray E MacA (2007), J Alloys Cmpnds, 446–447, 692–697. Dreisbach F, Lösch H W and Harting P (2002), Adsorption, 8, 95–109. Gray E MacA (1992), J Alloys Cmpnds, 190, 49–56. Gray EMacA, Buckley CE and Kisi EH (1994), J Alloys Cmpnds, 215, 201–211. Gumma S and Talu O (2003), Adsorption, 9, 17–28. Haas M K, Zielinski J M, Dantsin G, Coe C G, Pez G P and Cooper A C (2005), J Mater Res, 20, 3214–3223. Hemmes H, Driessen A and Griessen R (1986), J Phys C: Solid State Phys, 19, 3571– 3585. Kisi E H and Gray E MacA (1995), J Alloys Cmpnds, 217, 112–117.

204

Solid-state hydrogen storage

Kisi E H, Gray E MacA and Kennedy S J (1994), J Alloys Cmpnds, 216, 123–129. Jacobsen R T Leachman J W and Lemmon E W (2007), Int J Thermophys, 28, 758–772. Luo W and Rönnebro E (2005), J Alloys Cmpnds, 404–406, 392–395. Malbrunot P, Vidal D, Vermesse J, Chahine R and Bose T K (1997), Langmuir, 13, 539– 544. McCarty R D (1989), Correlations for the Thermophysical Properties of Deuterium, Boulder, CO, National Institute of Standards and Technology. McLennan K G and Gray, E MacA (2004), Meas Sci Technol, 15, 211–215. Michels A, de Graaff W, Wassenaar T, Levelt J M H and Louwerse P (1959), Physica, 25, 25–42. Park C N and Flanagan T B (1983), J Less-Common Met, 94, L1–L4. Pons M and Dantzer P (1993), Z Phys Chem N F, 183, S225–S234. Sircar S (2001), AIChE J, 47, 1169–1176. Vargaftik N B (1975), Tables on the Thermophysical Properties of Liquids and Gases, New York, Wiley.