Kinetics of Hydrogen Storage on Catalytically-Modified Porous Silicon

Mawla Boaks1 and Peter J. Schubert2

Abstract— Porous silicon has been demonstrated as a hy-drogen storage media with surface-bound hydrogen contentas high as 6.6% by weight. Hydrogenated porous silicon isreadily synthesized by electrochemical etching in a solution ofhydrofluoric acid. Hydrogen gas can be released thermally attemperatures starting at 280 C. It has been proposed thata suitable catalyst at the pore mouth can both reduce thedesorption temperature and facilitate gaseous recharge of thesilicon matrix. This work presents a detailed kinetic studyusing density functional theory (DFT) of a reversible hydrogenstorage system based on porous silicon via the mechanismsof dissociation, spillover, and bond-hopping of hydrogen atoms.For each of these steps, activation energy values and vibrationalfrequency has been determined. Using these activation energiesalong with vibrational frequency values evaluated from themicro level DFT study, the kinetic performance of catalytically-modified porous silicon as a potential hydrogen storage materialhas been completed for the first time. The energy differencebetween full and empty charge is computed at the atomicscale and compared to macroscopic calculations, showing closeagreement. These results show the potential for rapid rechargeat 8 bar at temperatures commensurate with waste heat froma proton-exchange membrane fuel cell.

I. INTRODUCTION

Efficient and reversible storage is the primary challengeto a Hydrogen Economy. Hydrogen is 6.9 times lighter thanLithium per unit electric charge providing a fundamentaladvantage over electrochemical battery technology. Of coursehydrogen storage must be paired with a fuel cell to comparedirectly to a battery, so the ultimate theoretical advantageover lithium metal batteries will be moderated, perhaps anadvantage of a factor of 3.

Ultra-high pressure storage of hydrogen in gaseous formincurs a heavy penalty in parasitic energy loss needed forcompression. Widespread gaseous use implies large centralhydrogen generation plants with vast networks of specializedpipelines. The enormous capital expense of such infrastruc-ture, needed in advance of market adoption of hydrogen-fueled vehicles and home fuel cell applications presentsa dilemma from which there is no obvious emergence.Cryogenic storage in liquid form requires energy-intensiverefrigeration and substantial insulation to reduce boil-off.While useful for large rockets, liquid storage is impracticalfor vehicles or for diffuse distribution. Solid state storagein metal hydrides is handicapped by the mass of the metalsrequired, and by the highly exothermic nature of the recharge

1M. Boaks graduated with a MS degree from IUPUI and currently pur-suing Ph.D. at Electrical and Computer Engineering Department, BrighamYoung University, Provo, UT 84602, USA. mboaks@byu.edu

2P. J. Schubert is a professor at the Department of Electrical andComputer Engineering, Indiana University-Purdue University Indianapolis,Indianapolis, IN 46202, USA pjschube@iupui.edu

process demanding either slow absorption or complex cool-ing mechanisms.

A novel method of solid state hydrogen storage usingporous silicon has been proposed and studied [1], and is nowthe subject of four US patents. Silicon bonds with hydrogenatoms at energies lower than does carbon, and hydrogenhas been shown to bond-hop along silicon surfaces. Poroussilicon can be synthesized with very high surface area. Thegreatest energy barrier in such a system is the dissociationof gaseous dihydrogen (H2) into atomic hydrogen, whichcan be mediated by the introduction of a catalyst. Strategicplacement of a catalyst such as palladium at pore mouthsprovides a reversible pathway from dihydrogen gas to dis-sociated hydrogen on clusters of catalyst which transfers tothe silicon surface via spillover and then bond-hops alongthe surface to form a solid state hydrogen storage reservoir.This system is charged by gas pressure and discharged bytemperature. An additional discharge mechanism is availablevia infra red light which passes through lightly-doped siliconbut is absorbed by the Si-H bond.

Herein the kinetics of this reaction pathway are studiedusing Density Functional Theory (DFT). In this way theenergy barriers for each of the three steps can be studied indi-vidually, namely: dissociation, spillover, and bond-hopping.In the composite the charging rate of catalytically-modifiedporous silicon can be estimated. These detailed calculationsare compared with first-order macroscopic kinetic calcula-tions to provide support for the feasibility of the results.

II. METHODS

A. Chemical Kinetics

In this proposed storage system gaseous hydrogen goesthrough a series of equilibrium reactions that form differenttypes of compounds that bond silicon and hydrogen. At thebeginning, molecular hydrogen (dihydrogen) is dissociatedinto atomic hydrogen by the palladium catalyst.

H2 → 2H+Pd

These hydrogen ions are mobile and can spillover onto thedangling or vacant bonds available on the silicon surface,creating the Si-H.

H+Pd + Si∗ → SiH

The net chemical reaction, mediated by the catalyst, is asfollows:

H2 + 2Si∗ → 2SiH

____________________________________________________

This is the author's manuscript of the article published in final edited form as: Boaks, M., & Schubert, P. J. (2019). Kinetics of hydrogen storage on catalytically-modified porous silicon. Journal of Catalysis, 371, 81–87. https://doi.org/10.1016/j.jcat.2019.01.033

From the neighborhood of a catalyst atom or cluster thosehydrogen atoms which have moved onto the silicon sur-face by spillover can migrate across the surface by bond-hopping. Such surface diffusion is treated in series with thecombination of dissociation and spillover. Dissociation fromgaseous dihydrogen is modeled to first order by mass transferaccording to:

−rd = kb(Cb − 2Ci)

Here, rd, kb, Cb, Ci represent mass transfer rate by diffusionfrom gas to surface, mass transfer coefficient from thebulk dihydrogen gas, concentration of hydrogen in the bulkgas and atomic hydrogen concentration at interface withthe catalyst, respectively. The first order reaction from theheterogeneous catalyst to the silicon is governed by:

−rv = kiCi

Here, rv , ki, and Ci are spillover reaction rate, rate constant,and the concentration of H respectively. During steady statecharging the reaction rate of dissociation and spillover areequal so -rd=-rv , thus:

kb(Cb − 2Ci) = kiCi

After simplifying it becomes:

Ci =kb

ki + 2kbCb

The overall rate, driven by the concentration and hencepressure of the recharging molecular hydrogen gas will begiven by:

−r =kikb

ki + 2kbCb (1)

In general, rate constant is given by:

k = ν ∗ exp

(− EaRT

)(2)

Here, ν is the vibrational frequency and Ea is the activationenergy. Both of these quantities are evaluated using DFT forhydrogen dissociation, spillover, and bond hopping process.

B. Nanoporous Silicon Matrix

Silicon has a diamond cubic crystal structure with a latticeconstant of 5.43 A. The conventional silicon unit cell consistsof 8 atoms. In our nanoporous silicon matrix, 3.5nm porediameter is considered based on experimental evidence. Thisrequires at least 8 unit cells in both x and y directions toallocate a complete pore. Four-fold symmetry was used toreduce computational effort. A silicon supercell consistingof 4× 4× 4 conventional unit cells was constructed. All theatoms lying within the pore radius of 1.75nm were removedto imitate the electrochemical etching procedure done inlaboratory to produce nanoporous silicon. The pore depthwas extended all the way to the bottom of the supercell.This resulted in a hexagonal shape pore with edges along(100) and (110) planes.

All the DFT computations were done in open sourceDFT tool ABINIT. ABINIT default LDA: new Teter (4/93)

Fig. 1. Nanoporous silicon matrix accommodating one quarter of a porein a 4× 4× 4 silicon supercell: top view (left) and side view (right).

with spin-polarized option was used as exchange-correlationfunctional throughout our DFT computation. In ABINIT,periodic boundary condition is applied in all three dimen-sions that creates interaction between electron densities at theperiodic boundaries. Literature survey suggested a vacuumspace of 15 A[3] to eliminate interaction between periodicimages. Therefore, the system was expanded by another 4conventional unit cells along z-dimension whereas x and ydimensions were kept same as before resulting in a 4×4×8silicon supercell. In this newly formed supercell, the upperhalf was kept empty and there were only atoms in lower halfof the supercell along z-direction. The 1st Brillouin zonewas initially integrated by 2 × 2 × 2 Monkhorst-Pack grid.The introduction of vacuum corresponds to a decrease inreciprocal space by a factor of 2. Number of k-points alongz-dimension was reduced accordingly resulting in 2 × 2 ×1 Monkhorst-Pack grid. Troullier-Martins pseudopotentialswere used for all the elements in our system. A cutoff energyof 435 eV was used for all the DFT computations. Tolerancevalue of 0.00002 eV was used. All the computations werecarried out in a fast research supercomputer called Big RedII. KGB parallelization was used to further increase thecomputational speed where standard data partitioning wasdone over processors corresponding to k-points, block ofbands, as well as Fourier space of coefficients over planewave basis set[4].

C. Vibrational Frequency

Since the hydrogen atom is considered to be vibrating onthe silicon surface, the interaction between hydrogen atomand silicon atoms may impact the associated vibrationalmodes. Let us consider N atoms in the system and theirCartesian coordinates are defined as a single vector, r =(r1,....,r3N ). We further consider the local energy minima islocated at r0. New coordinate is defined as x = r-r0. Takingthe Taylor expansion along the energy minima at r0 ignoringterms with order greater than 2 yields:

E = E0 +1

2

3N∑i=1

3N∑j=1

[∂2E

∂xi∂xj

]x=0

xixj

The first derivative term in Taylor expansion is zero becauseit was evaluated at the energy minimum. Let us consider aHessian matrix consisting of 3N × 3N elements defined by,

Hij =

[∂2E

∂xi∂xj

]x=0

If we define the atoms as classical particles that followsNewton’s law of motion, then we will have, Fi = mi

(d2xi/dt2). Here, the force and mass associated with ithcoordinate are Fi and mi respectively and (d2xi/dt2) isthe acceleration of this coordinate.Rewriting the equation ofmotion in matrix form gives:

d2xdt2

= −Ax

This is the mass-weighted Hessian matrix whose elementsare defined as, Aij = Hij /mi. This matrix form equation hasa special set of solutions associated with the eigenvectors ofA called the normal modes. The eigenvector of this matrixand their eigenvalues satisfy following equation,

Ae = λe

Here, λ and e are eigenvalues and eigenvectors of matrixA respectively. In general, A has 3N eigen vectors andcorresponding eigenvalues. The 3N normal mode frequenciesthen can be defined using the 3N eigenvalues as follows,

νi =

√λi

2π(3)

Now, the frequency calculation of hydrogen atom willresult in 3N = 3 eigenvalues due to the fact that hydrogenatom has 3 (three) degrees of freedom along three spatialcoordinates. The Hessian matrix in our case will be a 3x3matrix defined as follows:

Hij =

δ2Eδx2

δ2Eδxδy

δ2Eδxδz

δ2Eδyδx

δ2Eδy2

δ2Eδyδz

δ2Eδzδx

δ2Eδzδy

δ2Eδz2

(4)

From finite difference approximation the diagonal elementsof the Hessian matrix are defined as, for example:

δ2E

δx2≈ E(x+ δx, y, z)− 2E(x, y, z) + E(x− δx, y, z)

δx2(5)

However, the non-diagonal elements are defined as followingaccording to finite difference approximation:

δ2Eδxδy ≈

E(x+δx,y+δy,z)−E(x+δx,y−δy,z)−E(x−δx,y+δy,z)+E(x−δx,y−δy,z)4δxδy ;

(6)But since the H atom movement is along the (110) plane,we need to rotate the x and y axis by an angle of θ = 45degrees. The following rotation matrix is used to serve thepurpose:

R =

(cos θ − sin θsin θ cos θ

)The elements of the Hessian matrix were modified ac-

cordingly. It is worth mentioning that all the terms presentin the numerator at the right hand side of above equationsare evaluated using DFT. The mass-weighted Hessian matrixA will have 3 eigenvectors e1,e2, and e3 and corresponding3 eigenvalues λ1, λ2, and λ3. For these eigenvalues, corre-sponding normal mode frequencies will be calculated usingequation (3). Discrete distances used in equations (5) and (6)were computed with movements about the energy minimum

of 0.04 A. Finally, when electron-rich catalyst atoms wereincluded, those silicon atoms more than 4 unit cells remotefrom the pore wall were omitted in order to reduce systemmemory.

III. RESULTS

A. Catalyst

In as-synthesized nanoporous silicon (npSi) based solidstate hydrogen storage systems, palladium is used as catalystfor its ability to reduce the substantial gaesous dihydrogen(H2) dissociation energy barrier. In this npSi, Si-H and Si-H2 are most prominent, with the presence of a small numberof Si-H3 bonds[5] where the silicon atom is bonded toonly one neighboring silicon atom and are therefore thestructurally weakest spot. These weak spots are supposed togive away dihydrogen gas first during dehydrogenation[6].In this catalyst deposition method, storage media is initiallyheated to a temperature for partial dehydrogenation where Si-H3 bonds give away two hydrogen atoms. The nanoporoussilicon matrix is then cooled from this temperature and Pdcatalyst atom is deposited in a solution[7]. Catalyst atoms inclusters of at two or more atoms are expected to facilitategaseous hydrogen dissociation and the spillover process[8].As the size of the cluster increases, multiple H2 moleculesare adsorbed to the cluster[10]. For palladium, the tetrahedralstructure consisting of 4 (four) atoms is found to be lowestin energy in a prior DFT study[9] and thus most stable.Therefore, we have evaluated this tetrahedron structure ofcatalyst atoms at three different locations as possible strategiclocations for catalysis. Palladium clusters were placed on andadjacent to the intersection of (100) and (110) planes at thepore edge near the pore mouth. These locations are depictedin figures 2, 3, and 4. Dark blue spheres represent siliconatoms in a nanoporous silicon matrix whereas silver ballsrepresent Pd catalyst atoms deposited at strategic location.

Fig. 2. Cluster of 4 Pd atoms placed on a quarter pore of porous siliconadjacent to the intersection of (100) and (110) plane: top view (left) andside view (right).

Fig. 3. Cluster of 4 Pd atoms placed on a quarter pore of porous siliconalong (110) plane: top view (left) and side view (right).

Fig. 4. Cluster of 4 Pd atoms placed on a quarter pore of porous siliconat the intersection of (100) and (110) plane: top view (left) and side view(right).

Two different Pd-Pd bond lengths of 2.60 and 2.69Angstrom have been reported for a Pd tetrahedron structuresupported on a substrate[9]. Therefore, we computed groundstate total energy for Pd clusters located at the three pro-posed locations for bond lengths of both 2.60 and 2.69 A.Naturally, the location with lowest ground state total energyrepresents the most stable location. In our DFT calculations,a Pd tetrahedron cluster with a Pd-Pd bond length of 2.69A located adjacent to the intersection of (100) and (110)plane was found to be the most stable location. Subsequentdissociation and spillover studies were performed with thissite as the strategic location for the 4-atom Pd catalyst cluster.

B. DissociationIn a study of dissociation, the silicon supercell was defined

with vacuum volume along the z-coordinate, representingthe gaseous headspace above the porous silicon surface. Asingle gaseous H2 was sited 15 A above from the surface,and centered above the pore void. The location of the H2

molecule was positioned such that its wave function doesnot overlap with the silicon atoms, thus ensuring sufficientseparation between periodic images to facilitate more accu-rate plane wave solutions[3]. Moc et al. have demonstratedthat when H2 is dissociated on a Pd4 cluster the individualhydrogen atoms bridge those Pd-Pd edges on opposing sidesof the tetrahedron[12]. We configured Pd4H2 structures suchthat the dissociated H atoms are placed at equal distances of1.66 A from the Pd atoms. There are two such Pd-Pd edgeswhich do not share any Pd atom. For our nanoporous siliconmatrix, three such configurations were studied. In two ofthem, the dissociated hydrogen atom gets overlaps a Si-Hbond length and were discarded leaving one configurationwhich is illustrated in figure 5, where the hydrogen atomsare shown as small red spheres.

Fig. 5. H2 dissociation on Pd tetrahedron cluster: top view (left) and sideview (right).

The ground state total energy was computed for before andafter hydrogen dissociation phase. The difference between

these two resulted in a dissociation energy barrier of 0.47eV, a result which is significantly lower than previous workreported by the researchers.

C. Spillover

Hydrogen uptake in a storage medium is significantlyincreased by spillover from supported heterogeneous metalcatalyst[8][11] where dissociated hydrogen atoms move fromthe catalyst to nearby low valence silicon surface atoms.Because the Pd atoms forming the tetrahedron lands on someof the lower coordination silicon atoms at the surface, therewere silicon atoms with coordination number 2 and 3 avail-able for spillover within a close proximity to the dissociatedhydrogen atoms. Among the three available silicon atoms,two had coordination number 2 whereas the other atom hada coordination number of 3. Figures 6 and 7 demonstratethe state of the matrix after hydrogen spillover to these twosilicon atoms respectively. Corresponding ground state totalenergy for both was computed using DFT.

Fig. 6. After spillover from Pd cluster: split hydrogen forms bond withlow valence (2) silicon located at nearby surface; top view (left) and sideview (right).

Fig. 7. After spillover from Pd cluster: split hydrogen forms bond with lowvalence (2) silicon located at alternate nearby surface locations; top view(left) and side view (right).

The spillover energy barrier was computed by comparingthe ground state total energy of before and after spilloverphases. The computed spillover energy barrier values forabove configurations were 0.60 and 0.49 eV respectively.Even though hydrogen atom has to overcome a lowerspillover barrier of 0.49 eV in second configuration, thedangling silicon atom is located almost 1 unit cell away fromthe Pd cluster which makes it less likely to occur. The moreconservative choice is a resultant spillover barrier of 0.60 eV.

D. Bond Hopping

In the bond hopping process, the hydrogen atoms ter-minated in surface silicon atoms hop to the next availablesilicon atoms one atom further down the adjacent siliconunit cell along the z-direction. This corresponds to a bond

hopping distance of 5.43 A. Initially we considered bondhopping along z direction only. We kept the hydrogen atomfixed at the initial and final locations and then took 9(nine) equidistant intermediate points in a straight line. Wecalculated ground state total energy for all those 11 locations,known as ”images”, using DFT. The difference in energybetween the images plotted against the hopping distanceresulted in following graph:

Fig. 8. Energy difference (eV) vs Bond Hopping Distance (A) for bondhopping in a straight line along z-direction.

The peak of the energy barrier for z-direction hop ofan entire unit cell is 1.63 A from the initial site. Thehydrogen atom must overcome a substantial energy barrierof 2.12 eV in order to bond hop to the next silicon cell.Next we explored alternate two-hop vectors whereby thehydrogen atom moves along diagonal x and y pathways. Thismovement is evaluated within a prismatic cylinder along thestraight-line path having a radius of 0.10 A thus forming acloud of possible locations for hydrogen atom. At least 10points were investigated for each intermediate location givinga total of 120 DFT runs forming the whole 3D cloud. Groundstate total energy was computed for all 120 points, which areplotted in figure 9. The size of the points increases with theincreasing value of energy. Therefore, it is likely that thebond hopping will progress through a pathway connectingcircles having smaller radii.

Next, we identified the lowest ground state energy of allcloud points. When we plotted the ground state total energyvs bond hopping distance for these points, the graph revealedone additional local minima as can be seen in figure 10.

Even though the bond hopping energy barrier got reducedsignificantly from 2.12 eV to 1.54 eV, but this doesn’trepresent the actual bond hopping pathway. But the presenceof an additional local minima provided with the possibilityof a quasi stable state between the initial and final image. Inthat case, the hydrogen atom may cover a hopping distanceof 5.43 A in two serial hops having lower energy barriercompared to the straight line hop. It may hop from the initialstate to the quasi stable state at a distance of around 2.50 Ain a diagonal direction along the crystal face and finally thesecond hop from the quasi stable state to the final state at adistance of 2.93 A.

In order to find the actual bond hopping pathway weutilized the Nudged Elastic Band (NEB) method. According

Fig. 9. 3D cloud of manually created possible location of H atom in bondhopping pathway

Fig. 10. Energy difference (eV) vs Bond Hopping Distance (A) calculatedby taking minimum energy values for each point along H atom’s bondhopping pathway from a cloud of manually created points in a 3D space.

to Jonsson et al. convergence to a minimum energy pathis guaranteed in NEB method if a sufficient number ofimages are considered[13]. We initiated the NEB method byallowing the hydrogen atom to relax at the initial and finalconfigurations ensuring two local minima. Improved tangentNEB method was used in our transition state investigation.ABINIT default tolerance value was used as the convergencecriteria. Both single and double bond hopping pathways wereconsidered. For alternate pathway consisting of two hops,the hopping distance changed to 4.50 A after hydrogen atomrelaxation between which the first hop takes place. For thesecond hop, the distance is 2.35 A. Therefore, we took 7intermediate images for the first hop and 3 intermediate

Fig. 11. Bond Hopping Alternate Pathways Consisting of Single Hop(left)and Double Hop(right).

images for second hop. For direct hop, total number ofimages were kept at 11 as before. When this computationconverged we got a direct hop bond hopping energy barrierof 2.06 eV. Whereas in alternate bond hopping pathway, thetransition state gave an energy barrier of 1.74 eV for thefirst hop. The energy barrier value is 1.45 eV for the secondhop. These values are smaller compared to the energy barrierof direct bond hop case. Therefore, we can conclude thathydrogen atom will preferentially diffuse along the siliconsurface by bond hopping process consisting of two sequentialdiagonal hops along the crystal face in order to cover avertical bond hopping distance of 5.43 A

E. Zero-Point Energy

According to quantum mechanics, the lowest quantummechanical energy for a collection of atoms will be:

E = E0 +hν

2(7)

Here, h is the plank’s constant, ν is the classical vibrationalfrequency, and E0 is the minimum energy computed usingclassical mechanics. The difference between this quantummechanical energy and minimum energy according to clas-sical mechanics gives the zero-point energy. Now if we applythis concept to a set of atoms, there will be zero-point energycontribution for each of the individual normal modes. Theminimum energy considering all the normal modes is definedas:

E = E0 +∑i

hνi2

(8)

The bond hopping energy barrier was computed usingNEB method from classical mechanics perspective whichdoesn’t include zero-point energy corrections. Since oursystem involves light atoms such as hydrogen, inclusion ofzero-point energy correction may have significant impact onthe bond hopping energy barrier[2]. In order to find zero-point energy corrected activation energy, we will need toinclude zero-point energy corrections for both energy minimaand transition state. In any set of atoms having N degreesof freedom, there will be N-1 real vibrational modes and 1(one) imaginary mode at transition state whereas the localminimum will have N real vibrational modes. The followingequation gives zero-point energy corrected activation energyaccordingly:

∆EZP =

E† +

N−1∑j=1

hν†j2

−(EA +

N∑i=1

hνi2

)(9)

Here, E† and EA represent ground state total energy attransition state and energy minima respectively. Their differ-ence denotes the activation energy, ∆E defined in classicalmechanics. In the above equation, νi and νj defines thenormal mode frequencies at local minimum and transitionstates respectively. The above equation can be re-written as:

∆EZP = ∆E +

N−1∑j=1

hν†j2−

N∑i=1

hνi2

(10)

Using the method defined in methods section, we calcu-lated all these vibrational modes using ABINIT at both localminimum and the transition state. From equation (3) we getthe frequency values: ν†1 = 1.49×1013 s−1, ν†2 = 1.54×1013

s−1. Putting all these values in equation (10) for first hopwe get,

∆EZP = 1.58eV

This gives the zero-point corrected bond hopping energybarrier for the first hop. This value is 0.16 eV lower thanthe bond hopping barrier calculated using classical mechan-ics. The zero-point correction for the second hop did notconverge, and so no correction factor is included in thecalculations below.

F. Vibrational Frequency

The vibrational frequency was calculated following theprocedure depicted in subsection C within the Methodssection. Multiple frequency values were computed for dis-sociation, spillover, and bond hopping. We consider the ratelimiting frequency values for dissociation and spillover. Thefrequency values are 3.94 × 1013 s−1 and 3.55 × 1013 s−1

respectively. However, the direction of first bond hop closelyaligns with z-coordinate whereas for second hop it is alignedwith x-coordinate. Therefore, we choose those frequencyvalues that align with the corresponding directions. Thesevalues are 6.07×1013 s−1 and 1.61×1013 s−1 respectively.

G. Surface Diffusion

With the bond hop distance and the vibrational frequencyit is now possible to compute the diffusion coefficient for sur-face movement of ensembles of hydrogen atoms on poroussilicon. In the equation below z is the number of directionsavailable to hop, α is the bond hop length in that direction,νe is the vibrational frequency associated with that particulardirection, and Ea is the activation energy which is the peakenergy between start and end points for the hop.

D =1

zνeα

2exp(−EakT

). (11)

A tool for characterizing hydrogen carrying capacity istemperature programmed desorption (TPD). A study byRedhead [14] relates the peak temperature to the activationenergy of absorption as a function of the rate of temperatureincrease β and peak temperature Tp.

Ea = RTp[ln(ATpβ

)− 3.46] (12)

A summary of 15 experimental TPD runs is shown inFig. 12 depicting a histogram of peak temperatures. Samplesof as-synthesized hydrogenated nanoporous silicon exhibit apeak near to 376 C with a standard deviation of 4.7. Samplesto which Pd catalyst were deposited exhibited slightly lowerpeak temperatures, with the two around 360 C being 3.3standard deviations away from the average. This correspondsto a reduction of about 0.1 eV in Ea.

Another TPD feature evident in the catalytically-modifiedsamples is illustrated in Fig. 13 where short duration surges

Fig. 12. Laboratory data from 14 runs of as-synthesized and catalytically-modified porous silicon using temperature programmed desorption(TPD).

in hydrogen release are observed at temperatures lower thanthe peak. Samples without catalyst are smooth and lack thesespikes.

Fig. 13. Representative TPD trace (blue) showing lower-temperaturerelease of hydrogen (spikes) plus regular peak.

IV. DISCUSSION

Experimental results from TPD suggest that zero-pointenergy is not active in the surface diffusion of hydrogen onnanoporous silicon because the derived Ea is 1.81 eV whichis just 3.8 percent greater than the DFT computed first hopenergy of 1.74 eV. Because both hops are needed for long-distance movement of hydrogen atoms the higher energybarrier will dominate the transport. Computing diffusioncoefficients from Eq. 11 yields values of diffusivity in therange of 10−16 cm2/s at 376 C which is a very lowvalue, comparable to hydrogen diffusion through metals. Theevidence from Fig. 13 suggests that when catalyst clustersare nearby hydrogen storage sites the activation energy canbe reduced. Using Eqs. (1) and (2) one obtains very highrates for dissociation and spillover. The limiting rate step isbond hopping. The Si-H bond energy is sufficiently strongthat bond hopping is a slow process at temperatures ofpractical interest. This consideration leads to a configurationwith a much higher surface density of catalyst atoms, as

shown in Fig. 14. Here, nearest neighbor silicon atoms withcoordination number 2 and 3 are within reach of the spillovereffect. If we also assume each cluster of four Pd atoms canhold 4 H atoms, then an accounting of available vacant bondsgives 9.4 hydrogens per cluster. The surface density of Pdclusters on unit cell centers and face sites averages to eightPd atoms per (100) unit cell area. The silicon matrix has64 surface atoms, including 2 backbonds per unit cell area,and 240 bulk atoms, yielding a hydrogen gravimetric storagedensity of 1.36 percent (w/w). This calculated value is withinthe bounds of experimental data, which ranged from 1.3 to2.3 percent (w/w) for catalyst-loaded npSi. While this is amodest value for this metric, the benefit is seen in Fig. 15which shows the recharge rate for the arrangement of Fig. 14.At temperatures commensurate with PEM fuel cell operation,and at pressures at or below that of a bicycle tire inflation, thefill time is on the order of 1.0 second. This further explainswhy our gaseous recharge analysis on laboratory samples didnot detect a transient because the system response time wasof a similar duration.

Fig. 14. Catalyst clustering scheme for fast recharge.

Fig. 15. Recharge dynamics versus pressure and temperature usingconfiguration of Fig. 14 with spillover into nearest neighbors.

V. CONCLUSIONS

The kinetics of recharging hydrogen gas onto catalytically-modified nanoporous silicon (npSi) has been studied witha combination of Density Functional Theory (DFT) andlaboratory testing. These results have been applied to thesequence of gaseous dissociation onto a catalyst cluster,spillover onto the npSi support or matrix, and bond-hoppingalong the interior surfaces of pores which have been createdelectrochemically within silicon crystal. The rate-limitingstep is site-to-site diffusion along the silicon surface, whichwas discovered to proceed via two discrete hops per unit celldistance. One of the two inter-cell hops has a higher energybarrier, the magnitude of which was found to agree within4 percent between DFT and lab measurements. To rechargecatalytically-modified npSi as a hydrogen storage reservoirfor practical applications requires that the surface densityof catalyst clusters is high. Instant recharge is possible usingtemperatures available as waste heat from a PEM fuel cell orother modest heat source. The penalty is the cost and densityof the catalyst. Future studies will explore non-PGM cata-lysts as the spillover energy of 0.60 eV may be lower thanrequired for practical applications. Calculations consistentwith laboratory results have provided kinetic performancecharacterization of a novel means of hydrogen storage based.

ACKNOWLEDGMENT

This work was funded in part by NSF grant 1648748.

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