Abstract
Although metal hydrides are considered promising candidates for solidstate hydrogen storage, their use for practical applications remains a challenge due to the limitation imposed by the slow kinetics of hydrogen uptake and release, which has driven the interest in using metal nanoparticles as advanced materials of new hydrogenstorage systems since they display fast hydrogenation and dehydrogenation kinetics. Nevertheless, the understanding of the adsorption/release kinetics requires the investigation of the role played by the stress which appears to accommodate the misfit between the metal and hydride phases. In this paper, we present a continuum theory capable of assessing how the misfit stress affects the kinetics of hydride formation and growth in metallic nanoparticles. The theory is then applied to study the kinetics of adsorption/release in spherical particles. This work extends Duda and Tomassetti (2015, 2016) by considering stressdependent hydrogen mobility.
Keywords
diffusioninduced stress; configurational forces; hydrogenstorage systems; phase transformation
1 INTRODUCTION
When immersed in a hydrogen gas atmosphere, metals such as palladium and magnesium, can soak up hydrogen like a sponge by forming metal hydrides. In fact, as first reported by Graham (1866)Graham, T. (1866). XVIII. On the absorption and dialytic separation of gases by colloid septa. Philosophical transactions of the Royal Society of London, 156, 399439., ‘’at room temperature and atmospheric pressure, palladium can absorb up to 900 times its own volume of hydrogen. That means, if you were to pump hydrogen into a bottle, it would take enormous pressure to store the same amount easily absorbed in a palladium bed of the same volume” (Wolf and Mansour, 1995Wolf R. and Mansour K. (1995). The amazing metal sponge: soaking up hydrogen. The Projects in Scientific Computing Archive. www.psc.edu/science/Wolf/Wolf.html.
www.psc.edu/science/Wolf/Wolf.html...
). For this reason, metal hydrides are considered promising candidates for solidstate hydrogen storage, although their use for practical applications remains a challenge due to the limitation imposed by the slow kinetics of hydrogen absorption and desorption (Sakintuna et al., (2007)Sakintuna, B., LamariDarkrim, F., and Hirscher, M. (2007). Metal hydride materials for solid hydrogen storage: a review. International journal of hydrogen energy, 32(9), 11211140., Jain et. al. (2010)Jain, I. P., Lal, C., and Jain, A. (2010). Hydrogen storage in Mg: a most promising material. International Journal of Hydrogen Energy, 35(10), 51335144., Rusman and Dahari (2016)Rusman, N. A. A., and Dahari, M. (2016). A review on the current progress of metal hydrides material for solidstate hydrogen storage applications. International Journal of Hydrogen Energy, 41(28), 1210812126.).
The kinetics of hydrogen absorption and desorption by a storage metallic material is the outcome of a sequence many steps. For instance, the steps involved in hydrogen absorption includes: surface adsorption and dissociation of hydrogen molecules, and surface penetration of hydrogen atoms; diffusion of hydrogen atoms within the host metal; phase transformation from a lowhydrogencontent phase (
It is well known that hydrogen absorption/desorption is accompanied by the generation of stress. In fact, during hydride formation/decomposition, the
In this paper, we present a continuum theory capable of assessing how the stress affects the kinetics of hydrogen uptake and release in metals under the assumption that the
The theory is applied to study the kinetics of adsorption/release of hydrogen in a spherical particle, a problem that is amenable to analytical treatment. In this case, the spherical particle is partitioned in the
The paper is organized as follows. In Section 2, we reprise and extend the theory presented in Duda and Tomassetti (2015Duda, F.P. and Tomassetti, G. (2015). Stress effects on the kinetics of hydrogen adsorption in a spherical particle: an analytical model. Int. J. Hydrogen Energy, 40: 1700917016., ^{2016}Duda, F.P. and Tomassetti, G. (2016). On the e_ect of elastic distortions on the kinetics of diffusioninduced phase transformations. J. Elasticity, 122: 179195.) for the problem of the soluteinduced diffusion and phase transformation in elastic solids. In Section 3, the general theory is specialized to describe the kinetics of the
Throughout this paper, the standard notation of continuum mechanics is adopted (Gurtin (1981)Gurtin, M. (1981). An Introduction to Continuum Mechanics, Academic Press.) and symbols are defined the first they appear.
2 THE CONTINUUM MODEL
Let ℬ be the reference domain for a solid solution composed of a host elastic solid and an interstitial solute. The domain ℬ is the stage of two interdependent processes taking place at two different scales, namely a macroscopic (mechanical) one due to the deformation of the host solid and a microscopic (chemical) one due to solute diffusion through the interstices of the host solid.
As Figure 1 indicates, ℬ is separated by the moving interface
Here and henceforth, when writing
The remaining of this section follows closely Gurtin and Voorhees (1993)Gurtin, M. E., and Voorhees, P. W. (1993). The continuum mechanics of coherent twophase elastic solids with mass transport. Proc. R. Soc. Lond. A 440 (1909): 323343. and Duda and Tomassetti (2015Duda, F.P. and Tomassetti, G. (2015). Stress effects on the kinetics of hydrogen adsorption in a spherical particle: an analytical model. Int. J. Hydrogen Energy, 40: 1700917016., ^{2016}Duda, F.P. and Tomassetti, G. (2016). On the e_ect of elastic distortions on the kinetics of diffusioninduced phase transformations. J. Elasticity, 122: 179195.). See also Fried and Gurtin (1999)Fried, E. and Gurtin, M. (1999). Coherent solidstate phase transitions with atomic diffusion: a thermomechanical treatment. J. Statistical Physics, 95: 13611427..
2.1 Bulk equations
We now introduce the field equations of the theory that hold in bulk, it means, away from the interface
where div is the divergence operator,
In addition to the aforementioned balances, one should consider as basic an energy imbalance whose local version in bulk admits the representation
where
We refer the reader to Fried and Gurtin (1999)Fried, E. and Gurtin, M. (1999). Coherent solidstate phase transitions with atomic diffusion: a thermomechanical treatment. J. Statistical Physics, 95: 13611427. for a detailed derivation of the foregoing equations.
2.2 Interface and boundary conditions
We assume that the
• Continuity of the displacement and chemical potential
where
• Force and solute content balances
where
• Maxwell relation
To describe the motion of the sharp interface an extra condition should be added: the interface motion is dissipationless, this implies that
giving the socalled Maxwell condition.
We consider
where
where
2.3 Constitutive equations
Guided by the dissipation inequality (2), we introduce constitutive equations for
The response function
We now present a constitutive specialization suitable for describing situations involving small deviations of the transition chemical potential
We assume the additive decomposition of the total strain
observes that
where
The constitutive response for the grand canonical potential is assumed to have the form
where
From (8)_{2,3,4}, the response function
where
where
From (8)2 and (12) we have
thereupon the solute content density
with
From (8)_{4} and (13) the solute flux
A final remark concerning the determination
where
2.4 Summary of the governing equations
We now summarize the governing equations of the theory under consideration for the moving boundaryinitial value problem. The unknown quantities are:
• Bulk equations
with
• Interface conditions
with
• Boundary conditions
Notice that the problem described by equations (20)  (22) is a “Stefan Problem” (e.g. Vuik (1993)Vuik C. (1993). Some historical notes about the Stefan problem, Nieuw Archief voor Wiskunde, 11: 157167.) due to the presence of a phase boundary that can move with time. In this case, the socalled Stefan condition is provided by (21)3.
3 SPHERICAL GEOMETRY
In this section, we specialize and solve the equations for spherical particles. In this case, it is worth mentioning that the equations describing the lattice strain in the coreshell particle were earlier derived by other authors. See, e.g., a brief review in the Supporting Information for and Syrenova et. al. (2015)Syrenova S. et al. (2015). Hydride formation thermodynamics and hysteresis in individual Pd nanocrystals with different size and shape. Nature Materials, 14: 1236{1244..
Let us now consider that
The concentric sharp interface and the normal velocity are
The stress, strain and chemical potential fields are obtained from the knowledge of the current position of the interface as will be discussed below. Thanks to spherical symmetry of the problem
and
where
According to the equations (16), (17) and (26), the stress components are then given by
Taking (27) into account, the force balance described in (1)1 can be rewritten as
which general solution is given by
From conditions (3)_{1}, (4)_{1} and (6), we have
where
We now consider the determination of the chemical potential. From (27)3 and (29), it follows that
the general solution of which is given by
From conditions (3)_{2}, (7)_{1} and (5), we have
where
Notice that (32) and (33) imply that
4 THE EVOLUTION EQUATION OF THE INTERFACE
We now obtain the evolution equation of the interface. From (4)_{2}, (18) and (34) we have
with
After using (14), (27)_{3}, (29), and (30), we rewrite equation (36) as
where
with
We are looking for solutions of (37) such that
When the specimen is saturated at the beginning of the process, i.e.,
We now use equation (37) to determine two salient features of the
• It remains in the single
• The
On the other hand, if the solid sphere is initially in the single
• It remains in the single
• The
Therefore, the absorption/desorption cycle displays a hysteresis, with loop amplitude given by
where
It is worth noticing that for situations in which
which is identical to the equation obtained by Schwarz and Khachaturyan (2006)Schwarz, R.B. and Khachaturyan, A.G. (2006). Thermodynamics of open twophase systems with coherent interfaces: application to metalhydrogen systems. Acta Materialia, 54: 313323.. Notice that these authors measure solute content in number of solute atoms per interstitial sites, whereas here we use solute number density.
We now derive expressions for the times
and using (37), it follows that
Figure 2 depicts the time dependence of the volume fraction of hydride phase
during the
5 CONCLUSIONS
A continuum theory aimed at describing stress effects on the kinetics of hydrogen uptake and release in metals was presented in this paper. The theory was built upon on the following simplifying assumptions: the
Acknowledgements
Fernando P. Duda gratefully acknowledges financial support by CNPq and FAPERJ. Angela C. Souza would like to point out that this work has been carried out despite the economic difficulties of the author’s country. The authors gratefully thank the referees for their constructive comments which helped improving this paper.
References
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Available Online: April 03, 2018
Publication Dates

Publication in this collection
2018
History

Received
30 July 2017 
Reviewed
24 Feb 2018 
Accepted
21 Mar 2018