Analysis of hydride formation for hydrogen storage: Pressure–composition isotherm curves modeling

Intermetallics 15 (2007) 934e944www.elsevier.com/locate/intermet

Analysis of hydride formation for hydrogen storage:Pressureecomposition isotherm curves modeling

Ch. Lexcellent*, G. Gondor

Laboratoire de Mecanique Appliquee RC, FEMTO-ST UMR CNRS 6174, Universite de Franche-Comte,24 chemin de l’Epitaphe, 25000 Besancon, France

Received 24 August 2006; received in revised form 6 November 2006; accepted 10 November 2006

Available online 24 January 2007

Abstract

A thermodynamic model of a partially open two-phase system initiated by Schwarz and Khachaturyan is extended for the hydrogen storageinvestigation. A particular attention is paid to the modeling of the evolution of hydrogen content absorbed in the solid solution as a function ofthe pressure of hydrogen gas (for isothermal external conditions), e.g. PCI curves. The extension concerns the description of anhysteretic PCIcurves with plateau or slope. Finally, the experimental data including anhysteretic behavior and also hysteretic one are fairly described by ourmodeling.� 2006 Elsevier Ltd. All rights reserved.

Keywords: B. Hydrogen storage; B. Phase transformations

1. Introduction

The present paper deals with hydrogen storage, which isnowadays a challenging subject in energetic and industrial ap-plications developed by Schlapbach, Sandrock and Dantzer[1e3]. Traditionally, hydrogen has been stored and transportedmainly as a compressed gas. Recently, an alternative techniquehas been developed by exploiting the possibility of manymetals to absorb hydrogen. This latter solution seems to pres-ent some advantages in terms of safety, global yield and longtime storage. Hence, hydrogen can be reversibly stored in a lotof crystalline solids. The hydrogen gas H2 under pressure dis-sociated into two atoms of H which enter as interstitials intothe intermetallic lattice frame. At low hydrogen concentration,the hydrogen atoms usually form a dilute interstitial solid so-lution a. Increasing the hydrogen content causes a part of thesolid solution to precipitate into a b phase of larger interstitialconcentration and lower density [4,5]. This b phase is calledhydride one. The two phases a and b form spatial lattice

* Corresponding author. Tel.: þ33 381 666 052; fax: þ33 381 666 700.

E-mail address: [email protected] (Ch. Lexcellent).

0966-9795/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.intermet.2006.11.002

frames. For example, by powder metallurgy a hexagonal unitcell of the Mg2Ni (a phase) is obtained and hydrogenationleads to the formation of monoclinic phase Mg2NiH4, whichconstitutes the b phase [6].

In the present work, a special attention will be devoted tothe understanding and the modeling of the experimental curvescalled PCI where P represents the hydrogen gas pressure im-posed on the boundary of the sample, C the content of hydro-gen atoms absorbed by the intermetallics under isothermalexternal conditions (I ).

Two approaches are used for modeling PCI curves. On oneside in 1937, Lacher [7] found some mathematical expressionsbased on the interstitial site occupation to fit isothermal curvesof the logarithm of hydrogen gas pressure as a function of thehydrogen concentration in the metal. To extend this work,Beeri et al. [8], Samsum et al. [9], Feng et al. [10] and Lotot-sky et al. [11] gave more efficiency expressions to fit betterthese PCI curves. In 2003, Beeri et al. [12] extended theirstudies to the case of double plateau, which corresponds totwo consecutive phase transformations and is outside of ourpurpose.

On another side, the PCI curves are considered as phasetransition ones between the a and b phases. Fang et al.

mailto:[email protected]

http://www.elsevier.com/locate/intermet

935Ch. Lexcellent, G. Gondor / Intermetallics 15 (2007) 934e944

Nomenclature

a crystal lattice parameteraaa, aab, abb parametersA Nv0GS1þ n=1� n32

0

Ap proportionality coefficientb positive constantbaa, bab, bbb parametersc hydrogen contentcmax maximum hydrogen contentc�aðTÞ hydrogen content at the beginning of the

phase transitionc�bðTÞ hydrogen content at the beginning of the

phase transitioncm hydrogen content at the plateau midpointex thermal evolution exponential parametersFa(ca) ga(ca)þ Aca(1� ca)Fb(cb) gb(cb)þ Acb(1� cb)fs constant slopega, gb free energies (per host lattice site) of the

unconstrained a and b phases in theirsingle-phase states

Gcoh Gibbs free energy for the two-phase region(aþ b phases)

GS shear modulusk(1) function for a/b phase transformationk(2) function for b/a phase transformationka, kb free energy parametersN total number of interstitial sitesp hydrogen gas pressurep0 pðc�aÞp1 pðc�bÞp0 standard pressurepa/b forward phase transformation pressurepb/a reverse phase transformation pressurepeq equilibrium pressurepg total pressure applied on the solidp0

g standard total pressure applied on the solid

pm pressure at the midpointR universal gas constantT temperatureTc critical temperatureV volumev0 V/NVH partial molar volume of hydrogen potential

Greek lettersa phase solid solutionb phase hydrideg activity coefficient of hydrogenDH0 enthalpy contribution [11]DH0 observed standard enthalpy [11]DHf enthalpy of hydride formationDHS solution heat of hydrogenDS0 entropy contribution [11]DS0 observed standard entropy [11]DSf entropy of hydride formation30 concentration dependence of the crystal lattice

parameter a ð30 ¼ ð1=aÞ � da=dcÞlx thermal evolution parametersmH hydrogen in metal chemical potentialmH2

hydrogen gas molecule chemical potentialmg 1=2mH2

m0g standard hydrogen gas chemical potential

mr interstitial chemical potential in the reservoirn Poisson ratioq c/cmax

q0v plateau midpoint [11]Pf thermodynamical force associated with the

progress of the phase transformationj(1) function for a/b phase transformationj(2) function for b/a phase transformationu atomic fraction of b phase(1�u) atomic fraction of a phaseU Gibbs free energy of partially open system

[13,14] gave an equation to describe the single-phase absorp-tion parts and another one for the phase transition domain. Useis also made of the tools developed for shape memory alloys(SMA). Khachaturyan [15], Muller and XuOn [16] andRaniecki et al. [17] established the thermodynamics of a closedtwo-phase system. The solid phase transition is between themother phase called austenite and the produced phase calledmartensite.

For the partially open two-phase system (only H atoms areexchanged), corresponding to hydrogen storage Schwarz andKhachaturyan [4,5] chose a non-convex free enthalpy expres-sion taking into account the hysteresis. In the present paper,their theory is extended in order to fit other experimentalPCI curves also, e.g. curves without hysteresis but with pla-teau or slope associated with the phase transition domain. Inthis situation a convex free enthalpy expression is chosen. A

particular attention is paid to the choice of the interactionenergy expression between the two phases. The modeling ofexperimental PCI curves ends this paper including the casewith hysteresis.

2. Pressureecomposition isotherm(PCI) curves description

As a lot of intermetallic compounds under hydrogenationcan generate hydrides, there are many experimental PCIcurves in the literature [18,19]. The choice of these getter ma-terials is mainly made for technological reasons (e.g. value ofthe yield pressure for the phase transformation (p.t.), yieldvalue of the temperature for the p.t., maximum of the contentratio H/M attainable, cyclic behavior, weight, price.).

936 Ch. Lexcellent, G. Gondor / Intermetallics 15 (2007) 934e944

As explained by Schwarz and Khachaturyan [5], the hydro-gen atoms will occupy an interstitial position in the a phaselattice. First part of the curve (ln p, c) corresponds to thehydrogenation of a phase until a saturation value equals toc�aðTÞ at the T temperature (c stands for the hydrogen contentin the solid).

(1) For 0 � c � c�aðTÞ: domain I.

Intermetallic þ hydrogen # solidsolutionða phaseÞ

Mþ xH#MHx ð1Þ

(2) For c�aðTÞ � c � c�bðTÞ: domain II corresponds to the c(H/M) interval inside which the phase transformation occurs.

Solid solutionða phaseÞ þ hydrogen # hydrideðbphaseÞ

MHx

�c¼ c�a

�T��þ�

h� x�

H#MHh

�c¼ c�b

�T��

ð2Þ

(3) For c � c�bðTÞ: domain III corresponds to the hydrogena-tion of the b phase.

The observation of PCI curves obtained from different al-loys shows us the monotonous increase of hydrogen contentc (H/M) with the external pressure p corresponding to domainsI and III. Three different characteristic behaviors can beobserved during the phase transformation (p.t.) (domain II).

(i) The reversible behavior with ‘‘plateau’’: the p.t. corre-sponds to a ‘‘plateau’’ (Fig. 1). It is known as the idealcase reproduced by Schhlapbach [1]. The behavior isanhysteretic, which means that the absorption curvecorresponding to the p increase is the same as that ofthe desorption one corresponding to the p decrease.Moreover, if one calls peq (T ) the value of the pressureof the plateau at the temperature T, its evolution withthe temperature is driven by an equation called theVan’t Hoff one [18,20]:

ln�peq

�¼ DHf

RT�DSf

Rð3Þ

Historically, this expression comes from RankineeCalendar[21] delivered for the phase transformation between liquid and

Fig. 1. Ideal case of hydrogen absorption/desorption. Van’t Hoff law [1].

gas for the water. It seems more suitable than Duperrayempiric formulae where peq is given as a four-power functionof temperature T (in degree centigrade).

(ii) The reversible behavior with ‘‘hardening’’ or ‘‘slope’’:in this case as exhibited in Figs. 2 and 3 associated toLaNi4.7Al0.3þ 6H # LaNi4.7Al0.3H6 [10], the desorp-tion curve is very slightly under the absorption curveso that the hysteresis can be considered as negligible.But the phase transformation curve (domain II) exhibitsa p increase with the hydrogen content c. Feng et al.[10] explain that this ‘‘hardening’’ case is due to HeH atomic interaction.

(iii) In the irreversible behavior, as shown in Fig. 4 corre-sponding, for example, to Mg1.9Al0.1NiþH # Mg1.9

Al0.1NiH, the desorption curve is seriously under theabsorption one and the curve exhibits consequenthysteresis [22,6].

Moreover, some intermetallics can exhibit two consecutivephase transformations [11,12] which are outside of our presentpurpose.

3. PCI curves modeling

3.1. A survey of literature and an extensionof Schwarz and Khachaturyan theory

The first model for describing the PCI curves was proposedby Lacher [7] for the palladiumehydrogen system becausepalladium is able to absorb a lot of hydrogen gas (H2).

The model was built on the principal assumption of a defi-nite number of interstitial sites in the lattice for hydrogenoccupation free to move in the whole volume of the crystal.

The assumption of equality of the chemical potential in gasphase 1=2H2 with one of the H in solid solution gives:

c

Ln

(p

/p

_a

tm

)

1st segment (AB)

2nd segment (BC)

3rd segment (CD)

A

B

C

D

Fig. 2. Schematic representation of PCI curves of hydrogen-absorbing alloys

showing the three segments [10].


1

2mH2¼ mH0 ln

�p

p0

�¼ cte

RT� bq

RT� 2 ln

�1� q

q

�ð4Þ

Two models were recently proposed to describe the PCI ofreal system hydrogen storage materials. One of the modelswas proposed by Fang et al. [13,14] and the second by Lotot-sky et al. [11].

As in their precedent presentation, Fang et al. separated thePCI curves into three domains. They assumed that both thefirst and the third segments are controlled by the solubilityof the hydrogen atoms in the a and b phases, respectively.The second segment is controlled by the phase transitionfrom the a phase to the b phase (see Fig. 2).

For the first and third domains they consider c as a functionof p and T:

Fig. 3. The absorption and desorption PeC isotherms of LaNi4.7Al0.3 alloy

from Feng et al. [10]. The points are experimental data.

Fig. 4. Pressureecomposition isotherms at 553 K of the Mg2Ni and

Mg1.9Al0.1Ni prepared by hydriding combustion synthesis [22].

c¼ ApðpÞg=2exp

�� gVHp

RT

�exp

�� gDHs

RT

�ð5Þ

For the domain II, the relation between the pressure p and cat temperature T can be given as follows:

ln p¼ DHf

RT�DSf

Rþ fs

�c� cm

�ð6Þ

with:

fs ¼ln p� ln pm

c� cm

ð7Þ

where cm (H/M) and pm are the hydrogen content and pressureat the midpoint of second segment (domain II) of a PCI curve,respectively.

� It corresponds to the case (ii) of PCI with ‘‘hardening’’where the constant slope is fs.� It permits to extend the Van’t Hoff law (3) delivered for

case (i) with a plateau peq as:

ln pm ¼DHf

RT�DSf

Rð8Þ

If p0 ¼ pðc�aÞ and p1 ¼ pðc�bÞ, one obtains:

pm ¼ffiffiffiffiffiffiffiffiffip0p1

p ð9ÞIn the spirit of Lacher model, the theoretical model

proposed by Lototsky et al. [11] describes PCI curves as:

ln p¼ ln p0þ 2 ln

�q

1� q

�� 27

2

Tc

Tqþ 2

�q

1� q

�ð10Þ

This expression takes into account the asymmetry betweenthe curve of the beginning and end of the p.t.

The relation between DS0, DH0

ln p0 ¼�DS0

RþDH0

RTð11Þ

and observed standard entropy and enthalpy (DS0 and DH0)calculated from Van’t Hoff dependencies of plateau pressureversus temperature is expressed as:

DH0 ¼ DH0�27

2RTcq0v ð12aÞ

DS0 ¼ DS0� 2R

ln

�q0v

1� q0v

�þ q0v

1� q0v

ð12bÞ

As a synthesis of their investigation, the authors claim thatthis model allows experimental PCI data fitting compared toLacher model, especially for the non-ideal isotherms.

This investigation of Lototsky et al. is not really convincingbecause it does not distinguish the phase transformation itself.Moreover the results of the simulation are not in goodagreement with experimental measurements (Fig. 5, Lototskyet al. [11]).


At last, in our opinion, for modeling hydride formation, sig-nificant progresses have been made about the thermodynamicsof partially open two-phase systems with coherent interfacesby Schwarz and Khachaturyan [4,5]. Moreover, they applytheir general theory to metalehydrogen systems.

In detail, they develop a thermodynamic theory for theabsorption/desorption of interstitial atoms e.g. hydrogen H,by a solid crystalline host forming a coherent mixture oftwo phases (a and b) with different contents c of H atoms.

Their investigation is devoted to the absorption/desorptionof H atoms by a metal that is free to exchange with an externalsource (for instance a reservoir of hydrogen gas). In this situ-ation, the system is considered as open because the solid isfree to exchange interstitial H atoms with the reservoir.

The chemical potential for the interstitials in the reservoir ismr. Their natural assumption is that the reservoir is very large,so that any exchange of interstitials between the solid and res-ervoir does not change mr. The thermodynamic potential of theopen system is the Gibbs free energy at constant mr:

Uðc;mrÞ ¼ Gcoh � mrc ð13Þ

with Gcoh representing the chemical potential for the two-phase region of the coherent phase diagram for c�a � c � c�b:

Gcoh

�c�¼�

1�u�

ga

�c�a

�þugb

�c�b

�þAc

�1� c

�ð14Þ

the chemical free energies ga( p, T, c) and gb( p, T, c) are de-fined at the constant total pressure p produced by all externaldevices and at temperature T. A is the material constant.

Furthermore knowing the value of c�a, c�b and c, one candetermine the equilibrium value for the coherent b phase:

u¼

c� c�ac�b� c�a

!ð15Þ

From

c¼�

1�u�

c�a þuc�b ð16Þ

Fig. 5. Simulation of Lototsky model for LaNi4.8Sn0.2 [11].

It results in the identity

c�

1� c�¼�

1�u�

c�a

�1� c�a

�þuc�b

�1� c�b

�þ�

c�b� c�a

�2

u�

1�u�

ð17Þ

and

Ucoh

�u;mr

�¼�

1�u��

ga

�c�a�þAc�a

�1� c�a

�� c�amr

�þu

�gb

�c�b

�þAc�b

�1� c�b

�� c�bmr

�þA�

c�b� c�a

�2

u�

1�u�

ð18Þ

Let call Fa(ca) and Fb(cb):

FaðcaÞ ¼ gaðcaÞ þAcað1� caÞ ð19aÞ

Fb

�cb

�¼ gb

�cb

�þAcb

�1� cb

�ð19bÞ

Gcoh ¼�1�u

�Fa

�ca

�þuFb

�cb

�þAðcb � caÞ2u

�1�u

�ð20Þ

Ucoh can be written from the three domains:

We assume here that the reservoir contains a diatomic gasH2 at the pressure pg and at the temperature T. The dissocia-tion of this gas gives an interstitial monoatomic gas which isin equilibrium with the diatomic one. pg is considered as thetotal pressure applied on the solid.

As an assumption, in the case of hydrogen storage

mr ¼ mg

pg;T

!¼ m0

gþ1

2RT ln

pg

p0g

!ð22Þ

so p is equal to pg. As in the theory of phase transition forshape memory alloys developed for example by Ranieckiet al. [17], a special attention is paid to the determination ofthe thermodynamical force Pf associated to the progress ofthe phase transformation:

Pf ¼�vUcoh

vuð23Þ


Pf¼ 0 corresponds to the equilibrium state which can bestable or unstable.

0 Pf ¼�

Fa

�c�a

��Fb

�c�b

��þ mr

�c�b� c�a

��A�

c�a � c�b

�2�1� 2u

�ð24Þ

The pressure associated to the forward phase transforma-tion a/b initiation is called as pa/b and the reverse p.t. ini-tiation is called pb/a.

Hence

Pf�pa/b;u¼ 0

�¼ 0 ð25aÞ

Pf�pb/a;u¼ 1

�¼ 0 ð25bÞ

delivers

ln

�pa/b

pb/a

�¼

4A�

c�b � c�a

�RT

ð26Þ

The value of A has been given by Schwarz and Khacha-turyan [4,5] with some considerations about strain inducedinteraction between dilatation point defects in elastic isotropicsolution (Eshelby [23]).

A¼ Nv0GS

1þ n

1� n32

0 ð27Þ

The term A represents the energy barrier which is propor-tional to the macroscopic volume V and thus cannot besurmounted by the thermally assisted nucleation of the newphase.

Eq. (26) gives the thermodynamic hysteresislnðpa/b=pb/aÞ caused by the macroscopic barrier A. In thiscase Pf¼ 0 represents the unstable line of equilibrium asshown in Fig. 6.

Moreover, we will demonstrate that the case A¼ 0 can rep-resent the case (i) of anhysteretic PCI curves with plateau andthe case A� 0 the anhysteretic PCI curves with slope or‘‘hardening’’ (case (ii)).

3.2. PCI curves modeling

The modeling of single-phase behavior of a or b is verysimple. The functions ga(ca) and gb(cb) can be given in thequadratic form:

ga

�ca

�¼ 1

2ka

�ca� c0a

�2 ð28aÞ

gb

�cb

�¼ 1

2kb

�cb� c0b

�2

ð28bÞ

as the equilibrium values of ca, cb, and u (at constant p, T, mr)can be found by minimizing Ucoh with respect to these threeparameters. It comes from the derivation of Eq. (21a) or (21c):

vUcohðca;mrÞvca

¼ vgaðcaÞvca

þA

�1� 2ca

�� mr ¼ 0 ð29Þ

0 m0g

�T

�þ 1

2RT ln

�p

p0

�¼ ka

�ca � c0a

�þA

�1� 2ca

�ð30Þ

0 ln

�p

p0

�¼ aaa

�T

�ca þ baa

�T

�for 0� ca � c�a

ð31Þ

Ln p

c

Case (ii)

Πf=0

A<0

Ln p

c

Case (iii)

Πf=0

A>0

Ln p

c

Case (i)

Πf=0

A=0

Fig. 6. Schematic representation of PCI curves for A¼ 0, A< 0 and A> 0.


and in the same way

ln

�p

p0

�¼ abb

�T

�cbþ bbb

�T

�for cb � c�b ð32Þ

with the precedent choice of the ga and gb functions, the log-arithm of the hydrogen gas pressure depends linearly on thehydrogen content in the two single phases a and b which isin agreement with the experimental observations.

The analysis of the behavior during phase transition c�a � c� c�b is a little more complicated. Whatever be the domain (I, II,III) the same equilibrium (stable or unstable) can be found bythe minimization of Ucoh with respect to c (hydrogen content).

Let us examine the three theoretical possibilities:

(i) For A¼ 0, there is no interaction energy between thea phase and the b phase:

Ucoh ¼�

1�u��

Fa

�c�a�� c�a

�mr

��þu

�Fb

�c�b

�� c�b

�mr

��ð33Þ

Pf ¼�vUcoh

vu¼�

Fa

�c�a

��Fb

�c�b

��þ�

c�a� c�b

�mr ¼ 0

ð34Þ

0 m0gþ

1

2RT ln

�p

p0

�¼

Fa

�c�b

��Fb

�c�a

�c�b� c�a

ð35Þ

It means that for constant temperature T: p¼ peq duringideal phase transformation.

(ii) For A< 0 which seems to be a surprising hypothesis,the interaction energy Ac(1� c) is no longer a barrierto overcome but an easier way!

This concept was used for modeling phase transitionbetween cubic L21

phase and pre-martensitic rhombohe-dral phase in NieTi alloys [24]. For c�a � c � c�b:

Ucoh

c;mr

!¼

1

c�b� c�a

!h�c�bga

�c�a

��þ c�

gb

�c�b

�

� ga

�c�a

��iþAc

1� c

!� mr

ð36Þ

�vUcoh

vu¼ 0¼�

ga

�c�b

�� gb

�c�a

�c�b� c�a

!�A

1� 2c

!þ mr

ð37Þ

0 ln

�p

p0

�¼ aab

�T

�cþ bab

�T

�ð38Þ

however, in this case

ln

�pa/b

pb/a

�¼

4A�

c�b� c�a

�RT

< 0 ð39Þ

this means:

p0 ¼ pa/b < pb/a ¼ p1 ð40Þ

and

ln pH2¼�1�u

�ln p0þu ln p1 ð41Þ

(iii) The case A> 0 is naturally associated with the presenceof hysteresis. In this case Pf¼ 0 can be considered asan unstable equilibrium line. In order to specify the ki-netic equation of phase transformation, we presume thattwo functions jðiÞðPf ;uÞði˛ 1; 2 Þgf exist such as an ac-tive process of hydride formation (du> 0), the forwardtransformation can proceed only when j(1)¼ const(dj(1)¼ 0) and an active process of hydride decomposi-tion which can proceed only if j(2)¼ const0 (dj(2)¼ 0).

jð1Þ ¼Pf � kð1Þ�u�; jð2Þ ¼ �Pf þ kð2Þ

�u�

ð42Þ

In our case, as a reference of the experimental curves, thechoice is very simple.

jð1Þ ¼Pf�u;mr

�� kð1Þ

�u�¼ 0 ð43Þ

with

Fig. 7. Top: schematic of free energy curves as a function of composition of

interstitial atoms for the homogeneous a and b phases of a two-phase system

with coherent interfaces. The thin curves ga(c) and gb(c) represent the chem-

ical contribution to the free energies. Bottom: chemical potential of the inter-

stitial hydrogen atoms in the gas phase (proportional to the gas pressure) as

a function of the composition of interstitials in the solid [4].


kð1Þ ¼�

1

2RT ln

�pa/b

p0

�� 2A

�c�b � c�a

�u

��c�b � c�a

�ð44Þ

jð1Þ ¼ Fa

�c�a

��Fb

�c�b

�þ�

m0gþ

1

2RT ln

�p

pa/b

�

�A

�c�b� c�a

��c�b� c�a

�ð45Þ

Fig. 8. Schematic of chemical potential for the coherent two-phase solid. The

arrows indicate the transformation paths when the solid is in equilibrium with

a source of interstitials of constant chemical potential. No equilibrium coexis-

tence of phase is possible along these arrows [5].

The flow line a/b is ln p ¼ ln pa/b from u¼ 0 to u¼ 1or with c moving from c�a to c�b (Fig. 7).

For the reverse, the reasoning is similar:

jð2Þ ¼ �Pf�u;mr

�þ kð2Þ

�u�¼ 0 ð46Þ

with

kð2Þ ¼�

1

2RT ln

�pb/a

p0

�� 2A

�c�b� c�a

�u


�ð47Þ

jð2Þ ¼ �Fa

�c�a

�þFb

�c�b

�þ�� m0

g �1

2RT ln

�p

pb/a

�

þA

�c�b� c�a


�ð48Þ

The flow line b/a is ln p ¼ ln pb/a from u¼ 1 to u¼ 0or with c moving from c�b to c�a (Fig. 8).

Moreover, concerning the effect of temperature, the choiceof mg expression (Eq. (27)) is redundant with the Van’t Hoffequation (Fig. 7).

For the moment, dependence of c�a and c�b with the temper-ature is chosen as a power law:

c�a�T�¼ laðTÞa ð49aÞ

c�b

�T�¼ lbðTÞb ð49bÞ

Fig. 9. Case (i): ‘‘plateau’’, A¼ 0. Experimental points and modeling for La0.90Ce0.05Nd0.04Pr0.01Ni4.63Sn0.32 [19].


c�aðTÞ and c�bðTÞ constitute, respectively, the left and theright boundaries of the two-phase domain aþ b in the dia-gram (ln p, c) with an upper point corresponding to c�aðTcÞ ¼c�bðTcÞ ¼ c�ðTcÞ for p¼ pc.

Given in Figs. 9e11, three examples of the modeling exper-imental curves that correspond to the three cases (A¼ 0, A< 0and A> 0).

4. Discussion and conclusion

The basic foundations of thermodynamics of partially opentwo-phase system delivered by Schwarz and Khachaturyan[4,5] help us in the practical modeling of hydrogen gaspressure e H atoms’ content absorbed by the solid solutionsrelationship under external isothermal conditions.

Particularly, the treatment of a macroscopic thermodynamicbarrier (A> 0) generated by the coherency strain is natural andis associated with an hysteretic behavior. One can say thatnearly the same tools can be used for the modeling of pseudoe-lasticity of NiTi undertaking a phase transition between L21

austenite and monoclinical martensite [25].The ideal case with ‘‘plateau’’ corresponding to an anhyste-

retic behavior can be compared with certain pseudoelastic be-havior for single crystals (for instance cubic austenite deliversmonoclinic martensite for some copper based alloys).

The case of anhysteretic behavior with ‘‘slope’’ associatedwith A< 0 seems to be more heuristic. Even if it works alsofor cubic austenite transformed in rhombohedral pre-martensite

R phase for certain NiTi based alloys [24], there is nowadays nophysical explanation for this behavior.

It seems that there is an analogy between the martensitictransformation and hydride in equilibrium with a reservoirof hydrogen, especially the curve shape (stressestrain) forSMA and (ln p, c) for hydride formation with the three differ-ent situations. However, there are significant differences whichprevent from applying directly the theory to the general caseof martensite transformation.

The coherent hydrides can present the same point symme-try group for solid solution and hydride. Hence, there is nomechanism for the relaxation of elastic strain caused by Hatoms. This is why the coherent two-phase state is alwaysa stressed state. In the case of martensitic transformation,where the symmetry of the martensitic phase is differentfrom that of the parent phase, the coherent transformationproduces the structural orientation domain. Unlike the caseof hydrides, the martensitic transformation produces anassemblage of martensite variants which almost completelyeliminate the transformation-induced stress. Therefore thetransformation produces the stress-free state unlike thestressed state such as the coherent hydrides formed in thesolid solution.

The concentration c�aðc�bÞ at the start of the absorption(desorption) process is the solubility limit of the a phase (bphase) in the coherent phase diagram of the closed metalehydrogen system. In the space (ln p, c) a two-phase domain(aþ b) can be stated as it can be built for temperature-driven

Fig. 10. Case (ii): ‘‘slope’’, A< 0. Experimental points and modeling for LaNi5Sn [26].


Fig. 11. Case (iii): ‘‘hysteresis’’, A> 0. Experimental points and modeling for Na2eLiAlH6 with the addition of 2 mol% TiF3 [27].

isobaric transformations. Moreover, a special attention must bepaid to the H atoms’ absorption (desorption) rate for technicalapplications.

References

[1] Schlapbach L. Hydrogen intermetallic compounds I, vol. 63. Berlin:

Springer; 2006.

[2] Sandrock G. A panoramic overview of hydrogen storage alloys from

a gas reaction point of view. Journal of Alloys and Compounds

1999;293e295:877e88.

[3] Dantzer P. Properties of intermetallic compounds suitable for hydrogen

storage applications. Materials Science and Engineering 2002;329e

331:313e20.

[4] Schwarz RB, Khachaturyan AG. Thermodynamics of open two-phase

systems with coherent interfaces. Physical Review Letters 1995;74:13.

[5] Schwarz RB, Khachaturyan AG. Thermodynamics of open two-phase

systems with coherent interfaces: application to metal hydrogen systems.

Acta Materialia 2006;54:313e23.

[6] Janot R, Cuevas F, Latroche M, Percheron-Guegan A. Influence of crys-

tallinity on the structural and hydrogenation properties of Mg2X phases

(X¼Ni, Si, Ge, Sn). Intermetallics 2006;14:163e9.

[7] Lacher JR. A theoretical formula for the hydrogen solubility in palladium.

Proceedings of The Royal Society of London Series A 1937;161:525e45.

[8] Beeri O, Cohen D, Gavra Z, Johnson JR, Mintz MH. High-pressure stud-

ies of the TiCr1.8eH2 system statistical thermodynamics above the criti-

cal temperature. Journal of Alloys and Compounds 1998;267:113e20.

[9] Samsum BM, Fukada S, Fujiwara H. Hydrogen isotope absorption

amount and rate of PdeAl2O3 pellets. International Journal of Hydrogen

Energy 2001;26:225e9.

[10] Feng F, Geng M, Northwood DO. Mathematical model for the plateau

region of P6C-isotherms of hydrogen-absorbing alloys using hydrogen

kinetics. Computational Materials Science 2002;23:291e9.

[11] Lototsky MV, Yartys VA, Marinin VS, Lototsky NM. Modeling of phase

equilibria in metalehydrogen system. Journal of Alloys and Compounds

2003;356e357:27e31.

[12] Beeri O, Cohen D, Gavra Z, Mintz MH. Sites occupation and thermody-

namic properties of the TiCr2�xMnxeH2 (0� x� 1) system: statistical

thermodynamics analysis. Journal of Alloys and Compounds

2003;352:111e22.

[13] Fang S, Zhou Z, Zhang J, Yao M, Feng F, Northwood DO. Two mathe-

matical models for the hydrogen storage properties of AB2 type alloys.

Journal of Alloys and Compounds 1999;10e13:293e5.

[14] Fang S, Zhou Z, Zhang J, Yao M, Feng F, Northwood DO. The

application of mathematical models to the calculation of selected

hydrogen storage properties (formation enthalpy and hysteresis) of

AB2-type alloys. International Journal of Hydrogen Energy 2000;25:

143e9.

[15] Khachaturyan AG. Theory of phase transformations in solids. New York

(NY): Wikey; 1983.

[16] Muller I, Xu H. On the pseudo-elastic hysteresis. Acta Metallurgica and

Materialia 1991;39:263e71.

[17] Raniecki B, Lexcellent C, Tanaka K. Thermodynamic models of pseu-

doelastic behaviour of shape memory alloys. Archives of Mechanics

1992;44(3):261e84.

[18] Kapischke J, Hayke J. Measurements of the pressureecomposition

isotherms of high temperature and low temperature metal hydrides.

Experimental Thermal and Fluid Science 1998;18:70e81.

[19] Iosub V, Latroche M, Joubert JM, Percheron-Guegan A. Optimisation of

MmNi5�xSnx (Mm¼ La, Ce, Nd, Pr, 0.27< x< 0.5) compositions as

hydrogen storage materials. International Journal of Hydrogen Energy

2006;31:101e8.

[20] Latroche M. Structural and thermodynamical properties of metal

hydrides used for energy storage. Journal of Physics and Chemistry of

Solids 2004;65:517e22.

[21] Smith Crosbie. The science of energy. A cultural history of physics in

Victorian Britain. University of Chicago Press; 1998.

[22] Li Q, Chou KC, Jiang LC, Zheng F. Influence of the hydrogen pressure

on hydriding kinetics of the hydrogen pressure on hydriding kinetics of

the Mg2�xAlxNi (x¼ 0.01) alloys. International Journal of Hydrogen

Energy 2004;29:1383e9.

[23] Eshelby JD. Continuum theory of lattice defects. In: Solid state physics,

vol. 3. New York (NY): Academic press; 1956. p. 79e144.


[24] Lexcellent C, Tobushi H, Ziolkowski A, Tanaka K. Thermodynamical

model of reversible R-phase transformation in NieTi shape memory

alloy. International Journal of Pressure Vessels and Piping 1994;58:51e7.

[25] Raniecki B, Lexcellent C. RL models of pseudoelasticity and their spec-

ifications for some shape memory solids. European Journal of Mechanics

Solids A 1994;33(1):21e50.

[26] Sato M, Yartys A. Hydrogen absorptionedesorption characteristics of the

LaNi5Sn intermetallic compound. Journal of Alloys and Compounds

2004;373:161e6.

[27] Fossdal A, Brinks HW, Fonnelop JE, Hauback BC. Pressureecomposi-

tion isotherms and thermodynamic properties of TiF3-enhanced Na2eLiAlH6. Journal of Alloys and Compounds 2005;397:135e9.

Documents

Analysis of hydride formation for hydrogen storage: Pressure–composition isotherm curves modeling