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Dissociative Adsorption, Dissolution, and Diffusion of Hydrogen in Liquid Metal Membranes. A Phenomenological Model Pei-Shan Yen, Nicholas D. Deveau, and Ravindra Datta Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b03933 • Publication Date (Web): 31 Dec 2017 Downloaded from http://pubs.acs.org on January 1, 2018

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Industrial & Engineering Chemistry Research

Dissociative Adsorption, Dissolution, and Diffusion of Hydrogen in Liquid Metal Membranes. A Phenomenological Model Pei-Shan Yen, Nicholas D. Deveau and Ravindra Datta

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Fuel Cell Center, Department of Chemical Engineering, Worcester Polytechnic Institute, Worcester, MA 01609.

Abstract

There is only limited experimental data and theoretical treatment available in the literature on hydrogen sorption and diffusion in liquid metals, in stark contrast with that in solid metals. This paper utilizes our predictive phenomenological model requiring minimal input, the Pauling Bond Valence-Modified Morse Potential (PBV-MMP) model, for estimating the thermodynamic and kinetic parameters of hydrogen solution and diffusion in liquid metals treated as quasicrystalline. The PBV-MMP model is a refinement of the Unity Bond Index-Quadratic Exponential Potential (UBI-QEP) model for estimating the energetics of solid metal surface reactions. The sequential kinetic steps of hydrogen dissociative surface adsorption on the feed side, sub-surface penetration, and atomic interstitial diffusion in the bulk, followed by these steps in reverse on the permeate side, are thus treated via the PBV-MMP model within Eyring’s transition-state theory framework, while the entropic changes are evaluated via Eyring’s free volume model. Our predictions agree with experimental results for different liquid metals reported in the literature, as well as with our results for hydrogen sorption and permeation in our recently reported sandwiched liquid metal membrane (SLiMM). Keywords: Sieverts’ law, interstitial diffusion, liquid metals, bond energy, hydrogen membrane

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Corresponding author: [email protected] 1 ACS Paragon Plus Environment

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Introduction The so-called “hydrogen economy” envisions widespread generation, storage, and usage of hydrogen as a clean and efficient energy vector.1,2 The metal-hydrogen (M-H) system3 is of great scientific and technological interest in this regard for containment, storage, and separation of hydrogen. This paper stems from our interest in the use of dense metal membranes for hydrogen separation and purification for fuel cells and other applications. Those based on Pd and its alloys are the most advanced, and are now at the pilot plant stage.4 However, Pd membranes are expensive, subject to poisoning, and wanting in durability. Consequently, many other metals are being investigated,5 but so far there has been no practical alternative to Pd. We have recently developed an entirely novel class of sandwiched liquid metal membranes (SLiMM) (Figure 1).6 It involves a thin film of a low melting metal or alloy sandwiched between two porous and inert supports. It was proposed on the hypothesis that the solubility as well as diffusivity of hydrogen in a liquid metal is higher.6,7 Thus, we have been able to show that a SLiMM based on a liquid Ga film sandwiched between inert porous SiC and/or grahite supports possesses a permeability at 500 ºC that is roughly 35 times greater than that of Pd.6 However, there are a number of technical problems that still need to be resolved for practical application of SLiMM, including investigation of liquid metal interaction with other gaseous components in the reformate, and selection of robust and inert porous supports with improved liquid metal wettability and stability. Further, much work is needed to develop a fundamental understanding of the liquid M-H system, which is thus the goal of this paper. The focus in the literature so far has been on solid M-H systems.3 Of course, improved understanding of solution and interstitial diffusion of small atomic solutes in liquid metals is vital for a number of other applications as well. For instance, transport of 2 ACS Paragon Plus Environment

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hydrogen and its isotopes in liquid metals is of interest beacuse of their use as coolants in fast neutron reactors.8 The solubility and diffusion of hydrogen in liquid metals is also of great importance in metal casting practiced on a vast industrial scale in the production of aerospace, automotive, and structural components.9 A limited amount of information on hydrogen solubility and diffusivity in liquid metals is available in the literature, specifically in liquid Ni, Cu, Ag, Sn,7 Fe,10,11, Li12 and in Li alloys.8 From this it is apparent that both the solubility13 and diffusion coefficient14 of hydrogen in liquid metals are high and, in fact, increase sharply upon melting of a metal.6 Further, there is available only a limited amount of theoretical research on the modeling of solution and interstitial diffusion of hydrogen in liquid metals,15-18 and on the reasons for the sharp increase at the melting point of a metal.17 The objective of this work, thus, is to develop a convenient semiempirical model, requiring minimal input, for the prediction of kinetic and thermodynamic parameters involved in the surface dissociative adsorption, dissolution, and atomic diffusion of H2, or indeed for other diatomic gases such as O2 or N2, in a liquid metal. We limit ourselves to the most common case of a dilute solution, when thermodynamic nonideality is not significant.19 The molecular framework adopted here is that of Deveau et al.,19 involving a sequential 5-step mechanistic model of hydrogen permeation in Pd. For the kinetic and diffusion steps, we follow the transition-state theory (TST) framework of Eyring and coworkers, utilizing our recently developed semi-empirical approach20 for estimation of reaction step thermodynamics and activation barriers. This is the so-called Pauling Bond Valence-Modified Morse Potential (PBVMMP) approach for bulk metals,20 an adaption of the well-known Unity Bond Index-Quadratic Exponential Potential (UBI-QEP) approach21 for predicting energetics of metal surface catalysis. The PBV-MMP methodology is further coupled with Eyring’s free volume model to estimate

3 ACS Paragon Plus Environment


step entropic changes for predicting pre-exponential factors in the Arrhenius and the van’t Hoff equations. A part of this framework, namely, the Pauling Bond Valence (PBV) method,22-24 based on a set heuristics20 promulgated by Pauling, has of late proved to be useful in structure elucidation of ionic solids.25,26 Our methodology extends the PBV approach to apply to structural and energetic considerations for bulk metals, by combining it with a Modified Morse Potential (MMP) for twocenter bond energy, and is hence called the PBV-MMP approach.20 The MMP part is similar in spirit to the UBI-QEP method,21 although there are some differences in our approach. The UBIQEP model involves a “quadratic-exponential” potential energy model, a modified form of Morse potential, in terms of a “bond-index” that sums to unity. The PBV-MMP model, on the other hand, adopts a modified Morse potential in terms of the bond valence that sums over all its bonds to atomic valence, a well-accepted rule widely used in the bond valence theory.25 Further, while the UBI-QEP model is silent on the bond-valence (BV) bond-length (BL) relationship, being concerned mainly with the extrema on the energy landscape, the PBV-MMP model is capable of predicting the individual bond lengths and the bond valences along with bond energies, based on the BV-BL correlation. Additionally, the entropic changes are estimated based on the free-volume theory of Eyring and coworkers.20

Theory Sequence of molecular steps in a liquid metal membrane As shown schematically in Figure 1, the mechanism of hydrogen permeation through a dense liquid (or solid, for that matter) metal membrane involves the following sequential steps, sρ :19 s1) the H2 molecules on the feed side (denoted by the subscript “f”) undergo dissociative


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adsorption on the liquid metal surface sites S, forming adsorbed H atoms, H ⋅ S f ; s2) which then infiltrate the bulk metal lattice, occupying the sub-surface interstitial sites X as H ⋅ X f ; s3) the interstitial H atoms then diffuse across the membrane via hopping; s4) on eventually reaching the permeate side (denoted by the subscript “p”), the interstitial H atoms H ⋅ X p then egress from the bulk metal to its downstream surface; and finally s5) the adsorbed H atoms H ⋅ S p associatively desorb as H2 molecules on the permeate side. These steps can be described by the following microkinetic sequence: sρ

σρ

s1:

H 2, f + 2S f D 2H ⋅ S f

+1

s2 :

H ⋅Sf + X f D H ⋅ X f + Sf

+2

s3 :

H ⋅ X f + Xp D H ⋅ Xp+ X f

+2

s4 :

H ⋅ X p + Sp D H ⋅ Sp + X p

+2

s5 : OR :

2H ⋅ S p D H 2, p + 2S p

(1)

+1

H 2, f D H 2, p

The surface site, in general, may require the H atoms binding to nΣ surface liquid metal atoms, i.e., S ≡ nΣ M , while the interstitial site, X may require its binding to n metal atoms, i.e., X ≡ nM . The stoichiometric number σ ρ in the column on the right is the number of multiples of a step

sρ needed to add up to the overall process, or overall reaction (OR),27,28 i.e.,

OR ≡ ∑ σ ρ sρ , by ρ

canceling out all the intermediate species. The OR in this case simply involves transfer of H2 from the feed (f) to the permeate (p) side (Figure 1). The combination of steps 1 and 2, i.e., surface adsorption followed by sub-surface ingress, further, amounts to hydrogen dissolution into the bulk liquid metal, while step 3 describes the overall hydrogen diffusion process across the liquid film (Figure 1), written in Eq. (1) as H



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exchange between terminal intersitital sites X f and X p , and results from an integration of the Fick’s law across the membrane thickness, as described by Deveau et al.19 To utilize the above microkinetic description for characterizing a liquid metal membrane, we need the requisite thermodynamic, kinetic, and diffusion parameters for the elementary steps in Eq. (1), assuming that mass-action kinetics describe the elementary reaction steps, while Fick’s law describes the interstitial diffusion of H atoms in the liquid metal. Of course, steps 4 and 5 are simply the reverse of steps 2 and 1, respectively, so that the kinetic, thermodynamic, and transport parameters are needed only for the first three steps, and are the focus of the theoretical model below. It may be recalled, further, that the equilibrium constant K ρ of the elementary kinetic step

"

sρ

!

and its forward and reverse rate constants are interrelated, i.e., K ρ = k ρ / k ρ ,27 where the rate constants are described by the Arrhenius relation ! ! ⎛ Eρ ⎞ ⎛ Eρ ⎞ ! ! ! ! ⎟⎟ ; kρ = Λ ρ exp ⎜⎜ − ⎟⎟ kρ = Λ ρ exp ⎜⎜ − ⎝ RT ⎠ ⎝ RT ⎠

(2)

and the frequency factors from the thermodynamic transition-state theory (TTST) ! ! ⎛ ΔSρ‡,o ⎞ ⎛ ΔSρ‡,o ⎞ ! ! k BT k BT ⎟⎟ ; Λ ρ = κ ⎟⎟ Λρ = κ exp ⎜⎜ exp ⎜⎜ h h ⎝ R ⎠ ⎝ R ⎠

(3)

while the entropy changes of activation and activation energies are related to step ! ! ! ! thermodynamics via the Hess’s type relations: ΔSρo = ΔSρ‡,o − ΔSρ‡,o and ΔH ρo = Eρ − Eρ .

In the following, we further adopt the “free volume” model of Eyring for the liquid metal for estimating the entropic changes, for both stable and transition states, while the reaction enthalpies and activation energies are predicted via our PBV-MMP model.20 A brief synopsis of


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the PBV-MMP model is provided in the Supporting Information, while more details are provided in our recent paper.20 Enthalpy of dissociative hydrogen adsorption To evaluate the enthalphy change ΔH ad = ΔH1 , that of dissociative hydrogen adsorption on the liquid metal surface, i.e., for step s1: H2 + 2S D2H ⋅ S , we consider the following Hess’s thermodynamic cycle: sρ

ΔH ρo

σρ

si :

H 2 (g) D H(g) + H(g)

o + DH-H

+1

sii :

H(g) + nΣM D H⋅ M n Σ

− QH⋅MnΣ

+2

OR :

H 2 + nΣM D 2H⋅ M n Σ

(4)

o ΔH1o = DH-H − 2QH⋅MnΣ

involving gas-phase dissociation of the H2 molecule, followed by surface adsorption of the € o resulting gas-phase H atoms. Here DH-H is the binding energy of H2(g), while QH-MnΣ is the

binding energy for the H atom and the surface site, S = nΣ M . The binding site is defined by the

€ number nΣ of contiguous surface metal atoms, e.g., nΣ = 1 for ontop (t), nΣ = 2 for bridge (b), nΣ = 3 for fcc (111) hollow (h), nΣ = 4 for fcc (100) hollow, nΣ = 5 for bcc (100) hollow, as

shown schematically in Figure 2. In other words, the heat of dissociative hydrogen adsorption of molecular H2 is ΔH1o = DHo −H − 2QH-Mn

(5) Σ

For determining the binding energy of an atom H on a metal surface, QH-MnΣ , we use the PBVMMP model briefly described in Supporting Information and in our recent publication.20 Assuming, further, that the nΣ H-M bonds on the surface are all equivalent (Figure 2), then the



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normalized bond valence, xij ≡ vij / Vi , between H and M atoms, xHM, i = 1 / nΣ , and the total bond energy of the adsorbed H for all the nΣ nearest neighbor bonds21 o HM

QH⋅MnΣ = D

nΣ

∑(2x j=1

ij

⎛ 1⎞ o − xij2 ) = DHM ⎜2 − ⎟ ⎝ nΣ ⎠

(6)

o Thus, treating the liquid metal as quasi-crystalline, the on-top site ( nΣ = 1, QH-Mn = DHM ) is Σ

energetically the least favorable, but energy differences among the various hollow sites ( nΣ = 3, o o o ; nΣ = 4 , QH-MnΣ = 1.75DHM ; nΣ = 5 , QH-MnΣ = 1.8DHM ) become increasingly small. QH-Mn = 1.67 DHM Σ

The fcc (111) hollow site ( nΣ = 3) is commonly assumed.21 o To be fully predictive, one would require the bond energy between hydrogen and metal DHM , as

well as the Morse Potential parameters bH, and inter-nuclear distance between hydrogen and o metal, rHM (Supporting Information). However, if we focus only on the stationary points, minima

and maxima, of the potential energy surface, it becomes free of the structural parameters bH, and o o , although their estimates are available.29 Thus, only the thermodynamic parameter, DHM , is rHM

needed for our purposes, which is a characteristic of the binding solute atom and the metal atom. Activation energy of dissociative adsorption With the heat of adsorption ΔH1o as estimated above, we need to only determine the activation ! ! energy for the forward step E1 , as that for the reverse step E1 can be determined using the ! " relation ΔH1o = E1 − E1 . For this, we have to consider the mechanism of dissociative adsorption

of H2 and the corresponding energy landscape in a bit more detail as shown schematically in Figure 3. Thus, it is assumed that the first step in this dissociative adsorption process is the barrierless molecular adsorption on the surface, as a diatomic molecule on an on-top site, comprising of two adjacent metal atoms. This is then followed by its surface dissociation 8 ACS Paragon Plus Environment

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reaction via a transition state into two H adsorbed atoms in the more stable configuration of a hollow site, i.e., each H atom bound to nΣ = 3 surface atoms, as shown schematically in Figure 3. A discussion of the binding of a generic diatomic molecule AB, with the bond dissociation o energy DAB , to a surface via both A and B atoms leading to di-AB-coordination spanning a

bridge site is provided by Shustorovich,21 by Shustorovich and Sellers,30 and by Maestri and

€ Reuter.31 The resulting formula for the binding energy of the AB molecule to a two-metal atom site on the surface QHH⋅M2 is provided therein. Here, however, we are interested mainly in a homonuclear diatomic molecule (A2 e.g., H2), for which the binding energy takes the form

QHH⋅M2 =

o )2 9(DHM o o 6DHM + DHH

(7)

For the enthalpy change ΔH do of the next step of surface dissociation, from a Hess thermodynamic cycle (Figure 3) composed of AB molecular desorption, AB gas-phase molecular dissociation, and A and B atomic chemisorption onto the hollow sites, the enthalpy change for the surface dissociation reaction AB ⋅ M 2 D A ⋅ M n Σ + B ⋅ M n Σ o ΔH do = QAB⋅M2 + DAB − QA⋅MnΣ − QB⋅MnΣ

(8)

For example, for adsorbed H2 dissociation on Ga surface, ΔH do = QHH⋅M2 + DHo -H − 2QH⋅MnΣ = 188.3 kJ/mol , i.e., it is a significantly endothermic reaction.

Next, an estimate of the energy barrier for this surface dissociation reaction of adsorbed diatomic molecule AB30 ⎛ ! QA⋅MnΣ QB⋅MnΣ ⎞ ⎟ Ed = β d ⎜ ΔH do + ⎜ ⎟ Q + Q A⋅M B⋅M ⎝ nΣ nΣ ⎠


(9)


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! Now we are in a position to estimate the activation energy E1 . From Figure 3, thus,

! " ! ! o E1 = Ed − QHH⋅M2 and E1 = E1 − ΔH1 . Pre-exponential factors of dissociative adsorption

! Instead of using the transition-state theory for Λ 1 , i.e., Eq. (3), we use, as is common for gaseous adsorption,19,32 the collision theory, i.e.,

! po Λ1 = cS,t

1 2π M H2 RT

(10)

where cS,t = 1/ VM is the total concentration of the surface metal atoms, VM is the molar volume of the metal, M H2 is the molar mass of H2, and p o is the standard pressure. The frequency factor for the reverse step 1 is obtained from this estimate and a combination of Eq. (3)

" ! ⎛ ΔS1o Λ 1 = Λ 1 exp⎜⎜ − ⎝ R

⎞ ⎟ ⎟ ⎠

(11)

where the standard entropy change for step 1 is calculated, as described below. From the Hess’s thermodynamic cycle described in Eq. (4), the overall entropy change of dissociative hydrogen adsorption, ΔS1o = ΔSio + 2ΔSiio . Here, the entropy change of the gas phase dissociation of a diatomic gas A2, i.e., for A2 (g) D A(g) + A(g) , followed by adsorption of the monatomic gas A, i.e., for A(g) + S D A⋅ S

€ o − SAo (g) ΔSio ≡ 2SAo (g) − SAo 2 (g) ; ΔSiio ≡ SA⋅S

(12)

€ The gas-phase entropy (translational) of an ideal monatomic gaseous species i, in turn, is given by the Sackur-Tetrode equation33


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⎛ e 5/2 ⎞ Sio (g) = R ln ⎜⎜ 3 v f ,g ⎟⎟ ; λt = ⎝ λt,i ⎠

h 2π mi kBT

(13)

where v f ,g = kBT / p o is the free volume in the gas phase,in which the atom of i is free to move, and λt is the thermal wave-length. If it is further assumed that there is a total loss of translational entropy upon adsorption34 of a freely translating gas atom A, then ⎧ e 5/2 (2π m k T )3/2 ⎛ k T ⎞⎫ A B ΔSiio = −SAo (g) = −R ln ⎨ ⎜ B o ⎟⎬ 3 h ⎝ p ⎠⎭ ⎩

(14)

In the absence of more specific knowledge about the adsorbed species, the entropy change associated with rotational and vibrational motion is neglected in the above, relative to the change associated with the loss of translational entropy. Consequently, ΔS1o ≈ −SAo 2 (g) . CO entropy adsorption predicted using this equation, for instance, is ΔS ado = −150 J/mol K . Experiments on CO adsorption on Ir and Ru are in the range of 110 J/mol K − 163 J/mol K .35 o Furthermore, since the entropy of adsorbed species is generally small ( SA⋅S →0), the entropy

changes in surface reactions are often neglected. Thus, only in desorption step there will be significant entropy change, when the transition state would lie€between adsorbed species and a gas molecule. Hydrogen solution thermodynamics and Sieverts’ constant Before evaluting the thermodynamics and kinetics for step s 2 , the infiltration of the surface hydrogen atoms across the interface into the bulk lattice, let us first consider the thermodynamics of hydrogen solution into the bulk metal, which is a composite of these two steps:



sρ

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σρ

s1 :

H 2 + 2nΣ M D 2H ⋅ M nΣ

+ 1/2

s2 :

H ⋅ M nΣ + nM D H ⋅ M n + nΣ M

+1

OR :

1

2

(15)

H 2 + nM D H ⋅ M n

This is rewritten with altered stoichiometric numbers σ ρ , in order to obtain the standard OR for hydrogen solution normally written in the literature as:

1

2

H 2 + nM D H⋅ M n . We are interested

in the equilibrium constant for this process, KS, the Sieverts’ constant.19 For this, the Gibbs free energy change for the solution, ΔGS = (+ 1 2 )(ΔG€ . Further, using the usual form of 1 ) + (+1)(ΔG2 ) species chemical potential,19 for ideal solution (assumed here), in this provides ⎧⎪⎛ x ⎞ p o ⎫⎪ ΔGS = ΔGSo + RT ln ⎨⎜ H⋅X ⎟ ⎬ ⎩⎪⎝ 1− xH⋅X ⎠ pH2 ⎪⎭

(16)

where the standard (denoted by superscript “o”) Gibbs free energy change for hydrogen solution in bulk metal,

ΔG o − S = ln(K S ) = ln RT

(

⎛ K1 K 2 = ln ⎜⎜ ⎝

)

! k1 " k1

! k2 ⎞⎟ " k2 ⎟⎠

"

!

(17)

Here, K S = K1 K 2 , and we have also used the relation K ρ = k ρ / k ρ ,27,36. This further serves as a consistency check between the rate constants of steps s1 and s2 and the absorption equilibrium constant K S for solution. Further, with ΔGSo = ΔH So − TΔSSo , the temperature dependence of the Sieverts’ constant is given by K S = exp{ΔSSo / R − ΔH So / (RT )} , along with the esumption that the standard entropy change and the standard enthalpy change of solution are constant. At equilibrium, from Eq. (16) and Eq. (17), with ΔGS = 0 , there results the ideal absorption isotherm.


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xH⋅M =

K S pH2 / p o

(18)

1+ K S pH2 / p o

This equilibrium relation reduces to the common form of the so-called Sieverts’ law when in the term in the denominator K S pH , f / p o

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