analysis of transient hydrogen uptake by metal alloy particles

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Analysis of Transient Hydrogen Uptake by MetalAlloy Particles

Wenlin ZhangTexas A & M University - College Station

Supramaniam SrinivasanUniversity of South Carolina - Columbia

Harry J. PloehnUniversity of South Carolina - Columbia, [email protected]

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Publication InfoJournal of the Electrochemical Society, 1996, pages 4039-4047. Te Electrochemical Society, Inc. 1996. All rights reserved. Except as provided under U.S. copyright law, this work may not bereproduced, resold, distributed, or modied without the express permission of Te Electrochemical Society (ECS). Te archival

version of this work was published in the Journal of the Electrochemical Society.hp://www.electrochem.org/Publisher's link: hp://dx.doi.org/10.1149/1.1837333DOI: 10.1149/1.1837333
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J.Electrochem. Soc.,Vol. 143, No. 12, December 1996O The Electrochemical Society, Inc. 403910. J. S . Gill, L. M. Callow, and J. D. Scantlebury, Corro-sion, 39, 61 (1983).11. S. Sathiyanarayanan and K. Balakrishnan, Brit. Cor-ros. J., 29, 152 (1994).12. L. Meszaros and J. Devay, Acta Chim. Hung., 129, 79(1992).13. L. Meszaros and J . Devay, ibid., 129, 83 (1992).14. M. Sluyters-Rehbach and J. H. Sluyters, in Electro-analytical Chemistry, Vol. 4 , A. J. Bard, Editor, p.1,Marcel Dekker, Inc., New York (1970).15. S. K. Rangarajan, J. Electranal. Chem. 1, 396(1959/60).16. A. J. Bard an d L. R. Faulkner, Electrochemical Meth-ods, Fundamentals and Applications, p. 326, JohnWiley & Sons, Inc., New York (1980).17. M. Abramowitz and I. A. Stegun, Handbook of Mathe-matical Functions, Dover Publications, New York(1968).18. w.'H. skyer, CRC Stand ard Mathematical Tables, CRCPress, Boca Raton, FL (1981).19. H. W. William. S . A. Teukolskv. W. T. Vette rlin ~. nd B.

I? ~la me ry, ' umerical ~e"c' i~es,. 340, %-~d ed.,Cambridge University Press (1992).20. MathCad 4.0 User's Guide, 4th ed., Cambridge, USA(1993).21. J. 0' M. Bockris and A. K. N. Reddy, Modern Electro-

chemistry, p. 737, Plenum Press, New York (1970).22. A. J. Bard a nd L . R. Faulkner, Electrochemical Meth-ods, Fundamentals and Applications, p. 511, JohnWiley & Sons, Inc. New York (1980).23. J. A. L. Dobbelaar. Ph.D. Thesis. Universitv of Delft.Netherlands (1990).24. J. A. L. Dobbelaar an d J. H. W. de Wit, This Journal,137, 2038 (1990).25. LabVIEW for Windows Version 3.0, National Instru-ments corporation (1993).26. G. W. Johnson, LabView, Graphical Programming:Practical Application in Instrumentation and Con-trol, McGraw Hill, New York (1994).27. L. P. B. M. Janssen and M. M. C. G. Warmoeskerken,Transport Phenomena Data Companion, DUM;Delft (1987).28. V ~o va nc ice 6c nd J. O'M. Bockris, This Journal, 33,1797 (1986).29. J. 0' M. Bockris and S. U. M. Khan, Surface Electro-chemistry, A Molecular Level Approach, p. 333,Plenum Press, New York (1993).30. K. Juttner, W. J. Lorenz, M. W. Kendig, and F. Mans-feld, This Journal, 135, 332 (1988).31. K. Juttner, K. Manandhar, U. Seifert-Kraus, W. J.Lorenz, and E. Schmidt, Werkst. Korros. 37, 377(1986).

Analysis of Transient Hydrogen Uptake byMetal Alloy Particles

Wenlin Zhang,*-sc Supramaniam Srinivasan,**rband Harry J. Ploehnb"Center for Electrochemical Systems and Hydrogen Research, Texas Engineering Exper iment Station,Texas A&M University System, College Station, Texas 77843-3402, USAbDepartmentof Chemical Engineering, University of South Carolina, Swearingen Engineering Centel;Columbia, South Carolina 29208, USA

ABSTRACTThis paper describes a new approach to solving the equations comprising the shrinking core model for diffusion andreaction of a chemical species in a solid spherical particle. The reactant adsorbs on the partic le surface, diffuses into theparticle's interior, and reacts with the particle to form a solid product. The shrinking core model assumes a fast reactionrat e compared to reactant diffusion so tha t the reaction is localized in the interfacial zone between the unreacted solidcore and the sur round ing shell of reacted product. Analytical solutions of the governing conservation equations usuallyinvoke the pseudo-steady sta te (PSS) approximation which neglects the transient mass accumulation a nd diffusion-induced convection terms in the continuity equation for the diffusing reactant. However, small particle radii and slowreactant diffusion cast doubt on the validity of the PS S approximation. Dimensional analysis reveals an approximationtha t is less restrictive than PSS, yet enables a semi-analytical solution for the diffusing reacta nt distribution and inter-face velocity. For sufficiently large values of t he sur face mass fraction of the diffusing reactant, the PSS approximationleads to serious errors in the time dependence of the interface position and fractional conversion. However, our estimateof the surface mass fraction of hydrogen in LaNi, particles suggests the validity of the PSS approximation for hydriding ofmetal alloy particles. The shrinking core model thus enables an estimate of hydrogen diffusivity in metal alloy particles.

IntroductionSeveral rare earth intermetallic compounds of the gener-al composition AB, can absorb and release large amounts ofhydrogen (H) and th us have potential u tility for H storage.For example, various forms of LaNi, and its alloys havebeen extensively investigated and used in practical applica-t ions. '~~mprovements in performance can be achieved bypartia l substitution of certain elements for either La or Niin LaNi,.3 Fundamental understanding of hydriding mech-anisms and H transport are leading to better mathematicalmodels that can help guide efforts to synthesize new substi-tuted metal alloys.The kinetics of the hydriding process have been investi-

gated e~tensively ?~he hydriding process begins with theentry of H a t the surface of the metal alloy particle, eitherfrom the gas phase or electrochemically from the liquid* Electrochemical Society Student Member.* * Electrochemical Society Active Member.' Present address: Schlumberger Perforating and TestingCenter, Rosharon, Texas 77583-1590 .

phase. Gas-phase and electrochemical hydriding processesdiffer in their initial step. In the gas-phase hydridingprocess, the firs t step is the dissociative adsorption of ahydrogen molecule

where 2M represents two neighboring adsorption sites onthe alloy surface. In the electrochemical hybriding process,an external circuit supplies electrons to the alloy particles.Hydrogen atoms adsorb i n a charge-transfer stepH,O + M + e MH,, + OH- Dl

followed by subsequent processes that may include electro-chemical recombination or other "parasitic" phenomena aswell as H diffusion into the particle. Here, we assume tha tthe subsequent hydriding process in the interior of the par-ticle is the same regardless of whether the H enters h m hegas phase o r from an electrolyte solution.

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4040 J.Electrochem. Soc.,Vol. 143, No. 12, December 1996O he Electrochemical Society, Inc.Hydriding of the interior of the metal alloy begins withrapid diffusion of a small amount of H through microcracksor pores into the interior of the metal alloy to produce adilute solid solution denoted as the a-phase (Fig. 1).Experiments have shown tha t this process does occur and israpid,' but the resultant H mass fraction in the a-phase isquite small and may be neglected. At the same time, tran-sient surface hydriding occurs as absorbed H reacts withthe surface layer of metal alloy to form a surface layer of the

metal hydride according toXH + M * MH, PI

The product metal hydride is denoted as the p-phase.Recent experiments3 have characterized the stoichiometriclimit for hydriding of LaNi, and related partial ly substitut-ed metal alloys. In most cases, a stoichiometric coefficient xnear six has been found for these materials. Here, we treatx a s a parameter tha t is constant throughout the P-phase.If the reaction rate in the a-phase is fast compared to therate of H transport through the P-phase, then the reaction,Eq. 3 , will be limited to an interfacial zone between the a-and p-phases. After the initial transient hydriding of thesurface (not considered in this work), the a /@ nterface andthus the reaction zone move into the interior of the particle.Figure 1 schematically depicts this process. The hydridingreaction in the particle interior creates a concentration gra-dient tha t drives H transport through the metal hydride P-phase. When H reaches the a /p interface, it is incorporatedin the growing P-phase.The P-phase thus contains "free" (nonstoichiometric) Hmoving through the MH, toward the a /P interface.Although upper limits on the mass fraction of free H in theP-phase have not been established, observations of Huptake indicateR hat it never becomes very large.The mechanism of H transport thought the P-phase hasnot been fully elucidated. Bulk diffusion of H through a net-work of pores has been ruled out because intermetallicmetal hydride mater ials are, in fac t, alloys with crystallinestructure and few pores. Instead, H transport in the P-phaseis probably similar to a solid-state electron hoppingprocess.' Nuclear magnetic resonance ~ tu di es ' "~ ~ 'ave iden-tified a t least two types of motion, including localizedshort-range motion associated with M-H bond vibrations,and long-range multistep jumps tha t are primarily respon-

Unhydrided a hase1H2

. . - -(1 Phase or and Phase

Fi 1. Schematic representation of h e shrinki core model for the"fptat e of hydrogen by a d i d metal alloy p rt ic e.

sible for H transport. These results suggest tha t a s H atomsvibrate and hop around in the crystalline hydride, theyremain closely associated with the metal atoms.Nevertheless, the stochastic nature of the jumps suggeststha t the H transport in the p-phase can be modeled as a dif-fusion process with low effective diffusivity. Values of H dif-fusivity are critical for assessing metal alloy performanceand for battery design.The physical picture of hydriding, as depicted in Fig. 1,can be expressed mathematically as the shrinking coremodel'' which has been widely used to describe processesinvolving diffusion and reaction in spherical particles. Mostprevious analytical solutions invoked the pseudo-steadystate (PSS) approximation1' which simplifies the mathe-matics by eliminating the transient and diffusion-inducedconvection terms in the differential mass balance for thereactant. The validity of the PSS approximation has beentested through comparisons with solutions derived throughperturbation rneth~ds'"'~nd numerical methods."-'Wnerecent study'' compared results from an improved pertur-bation analysis, a moving boundary finite element numeri-cal solution, and the classic PSS solution. However, thismodel poses a finite reaction rate at the a/ P interface whichmay not be consistent with the neglect of reactant con-sumption elsewhere in the p-phase. Nonisothermal effectsand nonlinear kinetics have also been explored.IgPrevious studies agree on the validity of the PSS approx-imation when the concentration of the diffusing reactant issmall compared to the solid concentration. Only Lindmanand Simons~ on'~ecognized tha t the neglect of diffusion-induced convection might be a cause for concern. None ofthe previous studies developed rigorous boundary condi-tions based on jump mass balances." Only Bowenl"andBischoff15considered the implications of dimensional analy-sis for assessing the validity of the PSS approximation.The shrinking core model with the PSS approximationhas been used previously to describe H uptake by metalalloy particle^.^'." However, the validity of the P SS approx-imation in this application has not been established.Several factors motivate us to address this point. First,dimensional analysis suggests that , because of the low Hdiffusivity in the P-phase, the transient accumulation termmay have a magnitude comparable to that of the diffusionterm in the differential mass balance for H. Furthermore, noone has assessed the validity of neglecting the diffusion-induced convection term. Finally, errors introduced by thePSS approximation may prevent accurate determination ofH diffusivities from hydriding experiments.This paper re-examines the shrinking core model asapplied to the generic case of transport and reaction of onechemical species ("reactant") inside a solid spherical parti-cle. Differential mass balances for total mass and reactantgovern the mass-average velocity and the distribution ofreactant in the growing P-phase. Jump balances for totalmass, reactant, and solid at the a/P interface combine torelate the a/P interface velocity to the flux of reactant to theinterface. Dimensional analysis of the reactant continuityequation (mass balance) shows that the diffusion-inducedconvection and homogenous reaction terms can be neglect-ed when the a- and P-phases have similar densities and thereaction rate in the p-phase is small.However, slow reactant diffusion in the P-phase impliestha t transient accumulation term and the diffusion term inthe reactant continuity equation have the same magnitude.This necessitates the development of a solution that is lessrestrictive than the classic PSS approximation. A semi-ana-lytical "transient" solution of the reactant continuity equa-tion expresses the spatial and temporal variations of thereactant mass fraction and the velocity of the a /@ nterface.Numerical integration of the latter fixes the spat ial positionof the a/P interface a s a function of time, allowing evalua-tion of the reactant mass fraction distribution and the frac-tional conversion. The results obtained from this transientmodel are compared with those calculated under the PSSapproximation, enabling us to assess the validity of the PSSsolution under different conditions. Finally, we use the



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J. Electrochem. Soc., Vol. 143, No. 12,December 1996O The Electrochemical Society, Inc. 4041generic model to consider the particular case of hydrogenuptake by metal alloy particles.

TheoreticalModelTransient shrinking core model.-The generic shrinkingcore model treats the general case of reactant (H) transport-ed through a spherical shell of reacted product (MH,)towards the core of unreacted solid (M ) at the center. Each

internal phase has uniform internal crystalline structure.The core (a-phase) contains only unreacted solid (M). Theshell (P-phase) contains reacted product (MH,) and freereactant (H). Although the species labels suggest hydridingof metal alloys, the model should be viewed as a generic onefor reactant uptake via diffusion through and reaction witha solid spherical particle.Spherical symmetry eliminates all angular velocity com-ponents and the dependence of the remaining variables onangular coordinates. The effective reactant diffusivity in theP-phase, D, is assumed to be constant. Likewise, the bulkdensities of the a- and P-phases, pa and pP,are approximat-ed as constant but unequal. We also assume that the parti-cles remain a t constant temperature.In spherical coordinates, the continuity equation forphase i(i = a or P) reduces to

where v; s the mass-average radial velocity at each point inphase i. This equation has the solution

in the a-phase, since the velocity must be finite at r = 0.Likewise, for the P-phase, we find

indicating that the product ?v! in the P-phase is at most afunction of time.The a /P interface is located at r = r,. The overall jumpmass balance a t the a /@nterface

can be derived from the general transport theorem for bod-ies containing a phase interface (Ref. 20, p. 25). Physically,Eq. 7 is a mass conservation statement th at relates the nor-mal components of the mass flux on each side of a phaseinterface moving at a radial velocity s. Using Eq. 5, Eq. 7simplifies to

which relates the mass-average velocity of the P-phase tothe a / P interface velocity.With the constitutive equation for mass flux given byFick's first law of binary diffusion (Ref. 20, p. 481), the con-tinuity equation for the reactant can be written in terms ofmass fractions (o) as

where R is the rate of consumption of the reactant in the p-phase due to homogeneous reactions. The initial condition

follows from the assumption of negligible reactant ini tiallyin the particle, and surface boundary condition is

which is appropriate when we neglect mass transfer resist-ance at the particle surface. The quantity o,, representsthe mass fraction of free reactant in the solid at the sur-face, r = R.

The jump balance for free reactant at the a/p interface(Ref. 20, p. 451)

provides the other boundary condition. Physically, Eq. 12 isa mass conservation statement that relates the normal com-ponents of the mass flux of the reactant a t the moving inter-face to rate of consumption of reactant at the interface dueto chemical reactions. Equation 10 implies that p; - aw",0 so that Eq. 12 becomesThe basic premise of the shrinking core model and theneglect of reactant consumption elsewhere in the particlenecessitate large values of the reaction rate R a t he inter-face. Since both sides of Eq. 13 must be of the same magni-tude and p i is not large, we require v i >> s. That is, a fastreaction rate necessitates that the velocity of reactant at theinterface must be greater than the velocity of the interfaceitself. Together with the definition of the mass flux

Eq. 13 becomes

which, in physical terms, states that the rate of consump-tion of reactant a t the a/p interface equals the reactantmass flux to the interface.The jump mass balance for unreacted solid (M) at the a/pinterface, analogous to Eq. 12, reduces to

after assuming tha t v& = 0 (motionless solid in the a-phase)and pb, = 0 (no unreacted solid in the p-phase). We assumethat stoichiometry of the reaction requires the consumptionof 1mole of solid and x moles of reactant for each mole ofproduct produced. Thus

relates the consumption rates of reactant and solid. Fick'sfirst law of binary diffusion becomes

if we assume zero mass flux of product in the P-phase andsmall reactant mass fraction at the interface. Equations 15-18 combine to produce

giving the interface velocity in terms of the reactant massfraction a t the interface. Equation 8 becomes

to be used as a boundary condition for Eq. 6.The convection term in Eq. 9 can now be evaluated. First,we express the integration constant in Eq. 6 in terms of theparticular values of ra n d v! at the interface

Equation 20 then provides

everywhere in the a-phase. The continuity equation forreactant, Eq. 9, finally becomes



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J. Electrochem. Soc ., Vol. 143, No. 12, December 1996OThe Electrochemical Society, Inc. 4043

1= -[I - 3(1 - x)'I3 + 2(1 - x ) ] [43]6 w *

for the interface position y or fractional conversion x = 1-y3 as functions of time. Equation 43 is identical to the solu-tion obtained via the PSS approximation.13Hence the sum-mation terms in Eq. 42 should be viewed as transient cor-rections to the PSS solution.Equation 42 is a first-order nonlinear differential equa-tion that relates the dimensionless interface position andtime. We integrated Eq. 42 numerically using the Adam-Mouton methodz5 o determine ~ (7 ) . sing this, evaluationof Eq. 41 determined the dimensionless reactant mass frac-tion in the p-phase as a function of position and time.

Results and DiscussionComparison of transient and PSS solutions.-Equation42 suggests that the transient correction to the PSS solutionfor interface velocity may be important under some circum-

stances. However, the summation term in Eq. 42 must betruncated after a finite number of terms to enable numeri-cal integration. Figure 2 shows the dimensionless interfaceposition as a function of dimensionless time for differentlevels of truncation of the summation in Eq. 42. For the rel-atively large value of w * = 10, we see a significant differencebetween the PSS solution (zero terms in the summation)and the transient solution which includes one term in thesummation. This result indicates the well-known inaccura-cy of the PSS approximation for large values of w * . Thetransient solution with two summation terms differs fromthe one-term solution, but higher numbers of summationterms make only small contributions. Allsubsequent resultsfrom the transient model reported here include only twoterms in Eq. 42.Predictions of the time dependence of the interface posi-tion for various values of w * are shown in Fig. 3. For largevalues of w * , the transient and PSS solutions differ consid-erably, but the difference becomes negligibly small as w *decreases. This result agrees with previous analyses14J8hatindicate the validity of the PSS approximation for smallvalues of the surface concentration of the diffusing species.

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08Dimensionless Time ( 7 )

Fig. 2. Effect of kuncating h e summation term in Eq. 42 on thesolutionof the transient shrinking core model (w* = 10).

In terms of the time required to reach a specified conver-sion, T(X)

quantifies the error of the PSS solution relative to the tran-sient solution. Figure 4 shows this measure of error,expressed as a percentage, for various values of w * . Asexpected, the error introduced by the PSS approximation isminor for small values of w *, especially for conversion nearunity. For values of w * > 5, the PSS solution underestimatesthe time to complete conversion by more than 50%.

General characteristics of the transient solution.-Figure3 shows that, as expected, increasing the surface mass frac-tion of reactant leads to a considerable decrease in the timeneeded to shrink the a/p interface from the particle surfaceto the center. Figure 5 depicts the corresponding curves for

D 0.05 0.10 0.15 0.20 0.25Dimensionless Time ( 7 )

Fig. 3. Dimensionless interface paltion s a function of dimension-less time for various values of w* as predicted b the hunsient (solidline] a d SS (& sW line] sol& of h eshrin6ng cored.

0.0 0.2 0.4 0.6 0.8 1OFractional Conversion (x)

Fig. 4. Percentage by which the PSS solution underestimates thetime for a given fractional conversion dative to the transient solution.



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J.Electrochem. Soc.,Vol. 143, No. 12, December 1996O The Electrochemical Society, Inc.

0.00 0.05 0.10 0.15 0.20 0.25Dimensionless Time (T)

Fig. 5. Fractional conversion as a function of dimensionless time forvarious values of o* as predickd by h e transient solution of h eshrinking core model.the time dependence of the fractional conversion. The timeneeded for complete conversion of unreacted solid to prod-uct decreases with increasing values of o*. or O(1) valuesof o *, the dimensional time required for complete conver-sion is always less than the characteristic diffusion time(i.e.,~ ~ ~ ~ ~ ~ ( 1 )1)as expected for a diffusion-limited process.Conversion profiles for various values of reactant diffu-sivities with fixed particle size and o * are shown in Fig. 6.Figure 7 shows the variation of conversion with particleradius for fixed reactant diffusivity. These results and the

Fi 6. Fractional conversion asa tnction of time for various val-ues of h e reactant diffusivity(R = 10 pm, o* = 5) .

time scaling of Eq. 24 with to= R2/D show entirely expect-ed results: increasing the reactant diffusivity or decreasingthe particle radius decreases the time needed to achieve com-plete conversion. The conversion times indicated in Fig. 6 and7 are typical of metal alloy particles used in hydriding appli-cations. Small particle radii and high reactant diffusivitiesare clearly required if complete conversion is to occur with-in practical cycle times.Figure 8 shows the distribution of free reactant inside theparticle as a function of time. During the course of the reac-tion, the reactant mass fraction inside the particle increasesmonotonically with radial position. The reactant mass frac-tion at a fixed position increases with time. At completeconversion under these con&tions, the mass fraction of freereactant near the center is about 25% of that at the particlesurface. Further reactant uptake may occur due to the con-centration gradient, but the amount can probably beneglected compared to that stored in the product.Implications for hydrogen uptake by metal alloy parti-cles.-The results of the previous section show that thetransient solution of the governing equations for the shrink-ing core model provides a realistic description of reactantuptake by a solid particle, provided the reaction is limitedto the interfacial zone between the unreacted and reactedphases. Values of the model's only parameter, o *, determinethe variat ion of the a/@ nterface position and the fraction-al conversion with time. When o * is small, the solutionreduces to that found under the PSS approximation. Largevalues of o * lead to predictions that deviate significantlyfrom those of the PSS solution.The validity of the PSS approximation for hydrogenuptake by metal alloy particles is therefore determined bythe magnitude of w * defined by Eq. 26. For LaNi, with x =6, we have MM/xMH 88.7. We also assume that the densi-ties of LaNi, and LaNi,H, are similar so that pB/p", 1. The

0 100 200 300 400 500 600 700 800Time (s)



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I:'Time (s)

value of w varies with the extent of chemisorption of Hzonthe surface and depends on the gas-phase H2pressure or theapplied electric potential in an electrolyte solution.Unfortunately, experimental values of w are not available.If we suppose tha t LaNi,H, may contain as much as oneadditional H per unit at the particle surface,. hen w =0.00186, and so we estimate w * .= 0.165 as an upper limit.Thuswe conclude that the PSS approximation is reasonablefor hydrogen uptake by LaNi, particles.Extracting H diffusivities from hydriding data.-We haveused a microcalorimetric technique to study the thermody-namics and kinetics of H uptake by LaNi, and related par-tially substituted alloys. Another publicationz3 providesexperimental details and more complete results. Resultsfrom one experiment are shown here to illustrate the use of

the transient solution of the shrinking core model to quan-tify H difisivit y.A sample of activated LaNi, alloy particles was placed ina test cell and then embedded inside a microcalorimeter(Hart Scientific Model 5024). Thermopiles placed aroundthe test cell act as hea t sinks for the energy released by theexotherm of the hydriding reaction. The temperature differ-ence between the sample and the thermopiles is proportion-al to the heat flux. Admission of Hz gas to the test cellallowed the alloy sample to adsorb H; a computer recordedthe heat flux as a function of time.The measured heat flux represents a temporal convolu-tion of the actual hydriding reaction rate. A deconvolutionprocedure, discussed el~e where :~ransforms the measuredheat flux into the reaction rate as a function of time.

Assuming a constant partial molar enthalpy of the hydrid-ing reac ti~n,~ "he ransient reaction enthalpy can be inte-grated and normalized with the sample's total reactionenthalpy to find the fractional hydriding conversion, x, as afunct ion of time. The assumption of spherical particles withspecified radii allows the calculation of the dimensionlesscl/p interface position, y , via x = 1 - y3.

Fig. 7. Fractional conversion as afunction of time for various val-ues of he particle radius (Do=10-locrn2/s, o' = 5).

Figure 9 shows the a/P interface position as a function oftime for a sample of LaNi, alloy exposed to Hz gas at 7.8atm. The solid and broken curves are fits of the experimen-tal data using the transient and PSS solutions, respectively,assuming particle rad ii of 10 bm, w * = 0.165, and using theH diffusivity as an adjustable parameter. Both the transientand PSS solutions match the da ta quite well and yield D =1.70 X lo- ' cm2/s.Conclusions

The shrinking core model for reactant uptake by a solidparticle assumes that the reactant-solid reaction occursonly in a narrow interfacial zone that separates unreacted

Fig. 8. Reactant mass fraction as a function of dimensionless radi-al position and time (o* 5).



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4046 J. Electrochem. Soc ., Vol. 143, No. 12,December 1996O The Electrochemical Society, Inc.solid from reacted product. We have developed a completetransport model consisting of overall and species mass bal-ances, Fick's fi rst law for relating the reactant mass flux tothe reactant mass fraction grahent, and jump mass bal-ances that lead to correct boundary conditions. Simpli-fication of the governing equations in accord with dimen-sional analysis indicates appropriate conditions forneglecting diffusion-induced convection. The model alsoassumes tha t homogeneous reactions can be neglected in theproduct phase. Retention of the transient mass accumula-tion term in the reactant mass balance and some usefulchanges of variables enable a semi-analytical solution thatimproves upon the classic pseudo-steady s tate solution.The predictions of the trans ient model agree with the PSSsolution for small values of the su rface mass fraction ofentering reactant. However, under certain conditions, thePSS solution leads to an underestimation of the timerequired for a specified fractional conversion. Predictions ofreactant diffusivity derived from the transient and PSSsolutions could hf fe r by orders of magnitude. Thus the PSSapproximation must be used with caution when modelingreactant uptake by solid particles.For the hydrogen uptake by metal alloy particles such asLaNi,, a generous estimate of one "free" H atom per LaNi,unit cell leads to a small surface mass fraction of H becauseof the molecular weight difference between H and LaNi,.Thus we expect th at the P SS approximation will be validunder most con htio ns for H uptake by metal alloy particles.A forthcoming publicationz3will compare predictions ofthe transient shrinking core model with experimentalmicrocalorimetry data for hydrogen uptake by partiallysubstituted LaNi,. Here, we have shown that the values ofH diffusivities extracted from microcalorimetry experi-ments depend on the value of the H mass fraction at theparticle surface. Independent measurements of the surfaceH mass fract ion are needed in order to use the model to pre-dict H diffusivity in metal alloy particles. Alternately,independent knowledge of H diffusivity could be used topredict surface H mass fraction or, using a n appropriateextension of the transient model developed here, masstransfer coefficients.

Acknowledgments-This work was sponsored by the Chemical Sciences

Time (s)Fig. 9. Dimensionkss interfoce position as a function of time forLaNi , alloyparticlesbeinghydrided under Hz asat7.8 aim. Thedrcles are experimen~lata. The solid and brokencurves are the pre-didionsof the tmnsientand PSS solutions, respeetiv$y,assumingp a rticle mdiiof 10 pm ando* 0.165.

Division, Office of Basic Energy Sciences, US . Departmentof Energy, under Contract No. DE-FG03-93ER14381. Weexpress special thanks to Professor J. C. Slattery for hisvaluable suggestions. The work was carried out by one of us(W.Z.) in partia l fulfillment of the requirement for the Ph.D.degree in Chemical Engineering at Texas A&M University.Manuscript submit ted Aug. 2, 1995; revised manusc riptreceived Dec. 4, 1995.The University of South Carolina assisted in meeting thepublication costs of this article.

LIST OF SYMBOLSintegration constantreactant diffusivity, m2/selectronmolecular weight, kg/molindexreactant mass flux, kg/m2sradial position, mradial position of the a /@ nterface, mmetal alloy particle radms, mcharacteristic rate of reactant consumption in the@-phase,kg/m3 srate of reactant consumption in the @-phase,kg/m3 srat e of reactant consumption a t the a/@ nterface,kg/m%rate of M con sum ~ti on t the a/B interface. ke/mz s

, ",velocity of the ( Y / P interface, mjstime, s-characteristic process time, scoefficient in Fourier sine solutiondependent variable in Eq. 30mass-average rad ial velocity, m/srescaled reactant mass fraction, of+,number of reactant atoms per unit of product ; frac-tional conversiondimensionless radia l position of the a /@ nterfacedimensionless radial position, ( r - r,)/RGreeka unreacted solid phasep reacted product phaseq relative errorp mass density, kg/m3u interfacialT dimensionless time, t/t,o mass fractiono* dimensionless surface mass fraction of reactant,defined in Eq. 25Subscripts and superscriptsads adsorbedc at the a/ p interfaceH reactant; hydrogenHO initial value for reactantHR surface value for reactanti generic phaseM solid; metal alloyPSS pseudo-steady stater rad ial componenttrans transient solution

REFERENCES1. J. H. N. Van Vucht, F. A. Kuijpers, and H. C. A.Buruning, Philips Res. Repts., 25, 133 (1970).2. F. E. Lynch, J. Less-Common Met., 172, 943 (1991).3 . W. Zhang, Ph.D. Thesis, Texas A&M University, CollegePark, TX (1995).4. 0. Boser, J. Less-Common Met., 46, 91 (1976).5. C. N. Park an d J . Y. Lee, ibid., 83, 39 (1982).6. A. J. Goudy, D. G. Stokes, and J. A. Gazzillo, ibid., 91,149 (1983).W. Zhang, J. Cimato, and A. J. Goudy,J.Alloys Compd.,

201, 175 (1992).X.-L. Wang and S. Suda, Int. J. Hydrogen Energy, 18,139 (1992).D. Richter, R. Hempelmann, an d R. C. Bowman, Jr., inTopics in Applied Physics, Vol. 67, S. Schlapbach,Editor, Springer-Verlag, Berlin (1992).R. C. Bowman, Jr., B. D. Craft, A. Attalla, M. H.



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J. Electrochem. Soc ., Vol. 143, No. 12,December 1996 O The Electrochemical Society, Inc. 4047Mendelsohn, and D. M. Gruen, J . Less-Common Met.,73,227 (1980).E E. Spada, H. Oesterreicher, R. C. Bowman, Jr., and B.D. Craft, Phys. Rev., 30,4909 (1984).S. Yagi and D. Kunii, Chem. Eng. Sci., 16,634 (1961).K. B. Bischoff, ibid., 18, 711 (1963).J. R. Bowen, ibid., 20, 712 (1965).K. B. Bischoff, ibid., 20, 783 (1965).T. G. Theofanous and H. C. Lim, ibid., 26,1297 (1970).N. Lindman and D. Simonsson, ibid., 34, 31 (1979).G. E Carey and P. Murray, ibid., 44, 979 (1989).P. Carabin and D. Berk, ibid., 47,2499 (1992).J. C. Slattery, Momentum, Energy, and Mass Transfer inContinua. 2nd ed.. Robert E. Krieeer Publishine.v v ,~unt ington, Y 1981).J. Y Lee. S. M. Bvun. C.N. Park. and J. K. Park. J . Less-common ~et. ,"7,' 149 (1982):J. J . Reilly, in Proceedings of the Symposium on

Hydrogen Storage Materials, Batteries, and Electro-chemistry, D. A. Corrigan and S. Srinivasan, Editors,PV 92-5, p. 24, The Electrochemical SocietyProceedings Series, Pennington, NJ (1992).W. Zhang, M. P. S. Kumar, A. Visintin, S. Srinivasan, H.J. Ploehn, J . Alloys Compd., (1995).N. S. Koshlyaakov, M. M. Smirnov, and E. B. Gliner,Differenitial Equations of Mathematical Physics,North-Holland, Amsterdam (1964).C. W. Gear, Numerical Initial-Value Problems inOrdinary Diflerential Equations, Prentice-Hall,Englewood Cliffs, NJ (1971).T. B. Flanagan, W Luo, and J. D. Clewley, in HydrogenStorage Materials, Batteries, and Electrochemistry,D. A. Corrigan and S. Srinivasan, Editors, PV 92-5,p. 46, The Electrochemical Society ProceedingsSeries, Pennington, NJ (1992).

A Novel Electrolyte Solvent for Rechargeable Lithiumand Lithium-Ion Batteries

S. S. Zhang and C. A. Angell*Department of Chemistry, Arizona State University, Tempe, Arizona 85287-1 04 , USA

ABSTRACTWe describe a new type of electrolyte solvent which provides remarkable improvement in properties of electrolyte sys-tems containing it. These solvents contain acidic boron atoms, in most cases linked into heterocycles by reaction of boricacid, or oxide, with glycols. Nicknamed BEG solvents (for boric acid esters of glycol), these show grea t propensity for dis-solving salts, stabilizing alka li metals against corrosion, and in some cases, stabilizing other solvents (particularly alkenecarbonates), against anodic decomposition. In the most favorable case so far, the 1,3 propylene glycol boric acid ester(BEG-1) which contains two linked bora te groups, the mixed solvent formed by mixing 1part BEG-1 with two parts eth-ylene carbonate provides an electrochemical stability window in excess of 5.8 V (cf. 4.5 V for ethylene carbonate alonewith the same salt) , an ambient tem erature conductivity of 1.7 X S cm-' with 1m LiClO,, and enduring stability

against metallic Li, which remains sR ny on prolonged (days) immersion a t ~ooOC.We present data on various electrolytesolutions containin these components and then show their utility in devices by rubberizing them with polymers and con-structing voltaic cefis Li/LiMn,04 and C/LiMn204,which show excellent chargeldischarge characteristics a nd cyclability.Introduction

Many nonaqueous electrolytes and polymer gel elec-trolytes, compromising polar organic solvents and theirmixtures and having ambient ionic conductivities aboveSlcm, have been developed for use with rechargeablelithium and carbonaceous anode lithium-ion batteries.Unfortunately, metallic lithium reacts with almost allorganic solvents to form a low-conductivity passivationlayer on the surface of th e lithium, and repeated cyclingfrequently produces dendritic deposits which eventuallyshort-circuit the batteries. Batteries using inorganic chlo-ride electrolytes, such as Li/SOCl, and Li/SO,Cl, batter ies,suffer from voltage delay problems due to the formation ofa polycrystalline LiCl passivation layer on the lithium sur-face. As a result of such electrolyte problems, the develop-ment of rechargeable lithium batteries has been greatlyinhibited.A wide variety of approaches has been used in attem ptsto improve lithium deposition morphology and to increaselithium cycling efficiency, with litt le success so far. Typicalexamples have been the use of organic or inorganic add i-t ive~, '- '~ntroduction of new salts,13-l7Li alloy^,'^-^^ andfinally, the lithium-free carbon an ~ d e . ~ ' - ~ 'ven in the caseof carbonaceous anode lithium-ion batter ies, in which theuse of meta l lithium is completely avoided, there remainsan electrolyte problem limiting development. It is foundthat the common electrolyte solvents, such as propylenecarbonate (PC) and ethylene carbonate (EC), easily coint-ercalate into carbon layers and decompose on the surfaceof fresh carbon particles (probably because of the electro-catalytic nature of fresh carbon structure) . This leads to an

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irreversibility of the fi rst cycle, requiring extra lithium-rich cathode material (LiCoO,, LiNiO,, or LiMnO,) to beincluded in the cells in order to compensate for the con-sumption i n the initial irreversible cycle. To minimize thisirreversibility of initial lithium interca lation , the use ofc o - s ~ l v e n t s ~ ~ - ~ ~nd the modification of carbonaceouss t r u ~ t u r e ~ ~ - ~ 'ave been investigated. However, none of theabove-mentioned methods is suitable for both lithiumanode and carbon anode. Thus, a great need exists for newsolvents and co-solvents with special stability in the pres-ence of lithium metal. This in effect requires an ability ofthe solvent to react with metallic lithium i n a benign way,i.e., so as to generate a stable Li+-conducting, nonpropa-gating film on the lith ium surface. We report here on aclass of molecular substance which appears to behave inthis desirable manner and has remarkable stabilizingeffects on co-solvents a t positive potentials as well.Our original motivation in this work was to find a mol-ecular liquid which would dissolve salts by solvating theanions in order that the alkali metal transport numbercould become more favorable. To this end, we preparedinitially the boron analog of propylene carbonate, viz.

with the idea that the under-coordinated boron wouldseek a fourth ligand from the salt anion. Indeed, this liq-uid proves to be an excellent solvent for salts. However, itis a much more viscous liquid th an PC, presumably due tointermolecular interactions of the type

analysis of transient hydrogen uptake by metal alloy particles

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