Analysis of the partial molar excess entropy of dilute hydrogenin liquid metals and its change at the solid-liquid transition

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Citation Caldwell, Andrew H. and Antoine Allanore. "Analysis of the partialmolar excess entropy of dilute hydrogen in liquid metals and itschange at the solid-liquid transition." Acta Materialia 173 (July2019): 1-8.

As Published http://dx.doi.org/10.1016/j.actamat.2019.02.016

Publisher Elsevier BV

Version Original manuscript

Citable link https://hdl.handle.net/1721.1/131149

Terms of Use Creative Commons Attribution-NonCommercial-NoDerivs License

Detailed Terms http://creativecommons.org/licenses/by-nc-nd/4.0/

Analysis of the Partial Molar Excess Entropy of DiluteHydrogen in Liquid Metals and Its Change at the

Solid-Liquid Transition

Andrew H. Caldwella, Antoine Allanorea,∗

aDepartment of Materials Science and Engineering, Massachusetts Institute of Technology,77 Massachusetts Avenue, Cambridge, MA, 02139

Abstract

A systematic change in the partial molar enthalpy of mixing (∆hmixH ) and partial

molar excess entropy (∆sexH ) for dilute hydrogen-metal systems at the solid-

liquid transition is reported. Expressions for ∆hmixH and ∆sexH are derived from

the Fowler model of hydrogen solubility, and the change in ∆sexH at melting

is bounded. The theoretical bound is in agreement with measured data. A

connection is made between the change in ∆sexH and short range order in the

metal-hydrogen system.

Keywords: Liquids, Hydrogen, Solubility, Statistical mechanics,

Thermodynamics

1. Introduction1

Metals processing invariably requires the handling of metals in the liquid2

state. Such operations rarely occur in inert atmospheres. The mole fraction3

of dissolved gases, in particular hydrogen (H), are typically in the range of4

10−6 to 10−2 for liquid metals, and therefore degassing procedures are routinely5

employed in process metallurgy. This is done to prevent degradation of the6

mechanical properties of the solidified product, and such effects are well-studied7

[1, 2]. The severity of these effects depends on the concentration of H in the8

metal M, which can be determined from the solution thermodynamics of the M-9

H system. Calculating and predicting H solubility, defined here as its equilibrium10

∗Corresponding authorEmail address: allanore@mit.edu (Antoine Allanore)

Preprint submitted to Elsevier January 25, 2019

concentration in the metal, is therefore of considerable importance for metals11

processing.12

Prior reports have compiled existing data on the solubility of H in liquid13

metals, along with dissolved oxygen, sulfur, and nitrogen [3, 4, 5]. These data14

reveal a correlation between the two key quantities describing the mixing ther-15

modynamics: the partial molar excess entropy (∆sexX ) and the partial molar16

enthalpy of mixing (∆hmixX ). ∆hmix

X is defined as17

∆hmixX = hX − 1

2h0X2

(1)

where hX is the partial molar enthalpy, and 12h

0X2

is the standard state enthalpy18

of X2(g). ∆sexX is the defined as19

∆sexX = sX − 12s

0X2− sidX (2)

where sX is the partial molar entropy, s0X2is the standard state entropy of X2(g),20

and sidX is the ideal partial molar entropy of mixing. The correlation between the21

partial molar excess entropy (∆sexX ) and the partial molar enthalpy of mixing22

(∆hmixX ) for the liquid state can be rationalized from the chemical reactivity of23

the metal-gas system, by considering the relative strength of the solute-metal24

chemical bonding.25

In the present work, specific attention is drawn to the solution behavior26

of H in solid and liquid metals near Tfus and its statistical thermodynamic27

description. Statistical thermodynamic treatments of H dissolved in a metal are28

reported by a number of investigators[6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].29

These studies derive ∆hmixH and ∆sexH from the principles of statistical mechanics.30

Together, these quantities define the activity coefficient (γH) as a function of31

temperature and consequently the hydrogen concentration that is at equilibrium32

with a given partial pressure of H2 gas:33

xH =p

12

H2

γH= p

12

H2exp

[−∆hmix

H

RT+

∆sexHR

]. (3)

Eq. 3 is one form of what is referred to as Sieverts’ law [18], which states that34

the concentration of a gas solute X in a metal is proportional to the square-root35

of the partial pressure of X2 above the metal. Sieverts’ law is valid in the dilute36

2

limit and is generally observed for many M-H systems for PH2≤ 1 atm, in both37

the solid and liquid state. In Eq. 3, the proportionality constant is γH. In this38

analysis, γH, and by extension ∆hmixH and ∆sexH , are defined according to the39

following criteria:40

1. The concentration coordinate for the system of a gas solute H in liquid41

metal M is the mole fraction, defined as xH ≡ nH/(nM + nH).42

2. The standard state for the dissolved gas H is the pure diatomic gas 12H243

at 1 atm pressure.44

3. The solution thermodynamics are treated in the limit of infinite dilution.45

These criteria yield the definition shown in Eq. 3. The analysis presented46

herein restricts itself to the concentration and temperature regimes in which47

Eq. 3 is true1. Thus, this analysis focuses on the dilute solution behavior of H48

in liquid metals, as this is the concentration regime most frequently encountered49

in metallurgical processes. “Dilute” here refers to concentrations for which H50

solute self-interaction is a negligible contribution to the H activity coefficient.51

In this paper, we propose to compare H solubility data in liquid and solid52

metals near Tfus. Specifically, we report on a systematic shift in ∆hmixH and53

∆sexH at melting for a number of M-H systems. While not universal, the shift54

is sufficiently compelling as to suggest a common mechanism for the change55

in solution behavior that may be used to quantitatively predict the solution56

properties of H in liquid metals from solid-state data. A connection is made57

between the change in ∆sexH at melting and short range order in the liquid58

through a derivation of ∆hmixH and ∆sexH from the proton gas model of Fowler59

and Smithells [6].60

1At approximately 2500 K (2227 C) the equilibrium partial pressure of monatomic hy-

drogen gas H(g) is on the order of a few percent and is therefore non-negligible. Above this

temperature, H dissolution is no longer completely described by Eq. 3. Since the melting

temperatures of the majority of elemental metals are less than 2500 K, Eq. 3 suffices for our

discussion of the solution thermodynamics presented herein. Note, then, that for refractory

metals (e.g., W, Mo) in the vicinity of Tfus there will necessarily be some error in the use of

Eq. 3. For a perspective on H(g) dissolution, see Gedeon and Eagar [19].

3

2. Change in ∆hmixH and ∆sexH at the Solid-Liquid Transition61

∆hmixH and ∆sexH are the change in the partial molar enthalpy of mixing62

and partial molar excess entropy, respectively. A partial molar property is the63

change in the integral property per addition of one atom of, for example, H to64

a liquid M-H solution. At the equilibrium solid-liquid transition we define the65

quantities ∆∆hmixH and ∆∆sexH :66

∆∆hmixH = ∆hmix

H,liq −∆hmixH,sol (4)

∆∆sexH = ∆sexH,liq −∆sexH,sol (5)

The values of ∆hmixH and ∆sexH for M-H systems for which solid- and liquid-67

state data exist are listed in Table 1. For each M-H system in Table 1 a quantity68

Tsol/Tliq is defined to be the ratio of the maximum temperature for which the69

values ∆hmixH,sol and ∆sexH,sol are valid to the minimum valid temperature for70

∆hmixH,liq and ∆sexH,liq. For most of the M-H systems, Tsol/Tliq is close to unity, in-71

dicating that ∆∆hmixH and ∆∆sexH solely reflect changes in the solution behavior72

of dissolved H due to melting. It is important to recognize that, as a result of73

the weak temperature dependence of both ∆hmixH and ∆sexH [20], the tempera-74

ture range over which reported values of these properties quantitatively describe75

the solution behavior is usually several hundreds of degrees Celsius. Thus, it is76

reasonable to treat ∆∆hmixH and ∆∆sexH as capturing the change in ∆hmix

H and77

∆sexH across the solid-liquid transition for systems with Tsol/Tliq close to unity.78

For systems with Tsol/Tliq far from unity, there is greater uncertainty with this79

assumption.80

Table 1: Values of ∆hmixH and ∆sexH for H in elemental metals

∆hmixH ∆sexH

System kJ (mol H)−1 J K−1 (mol H)−1 Tsol/Tliq References

Ag(l) 76.3 −29.5 [21]

Ag(s) 62.1 −48.5 0.98 [21]

Ag(2)

(s) 68.9 −42.5 1.0 [22], see also: [4]

Continued on next page

4

∆hmixH ∆sexH

System kJ (mol H)−1 J K−1 (mol H)−1 Tsol/Tliq References

Al(l) 51.5 −36.2 [23]

Al(s) 58.2 −56.7 1.0 [23]

Al(2)

(s) 56.8 −54.0 1.0 [24]

Co(l) 41.0 −33.9 [4], see also: [25]

Co(s) 32.2 −45.7 1.0 [4]

Cu(l) 43.5 −35.3 [26], [25]

Cu(s) 34.5 −46.2 0.94 [26]

Cu(2)

(s) 49.0 −41.0 1.0 [22]

Fe(l) 36.4 −35.3 [25]

Feα 22.2 −53.6 0.65 [26], see also: [27]

Fe(2)α 24.3 −53.6 0.65 [22], see also: [27]

Feγ 27.0 −46.7 0.92 [26], see also: [27]

Fe(2)γ 29.9 −45.6 0.92 [22], see also: [27]

Feδ 28.8 −46.4 1.0 [22], see also: [27]

Mg(l) 29.6 −26.2 [28]

Mg(s) 19.3 −40.4 0.70 [28]

Mn(l) 24.9 −32.4 [29]

Mnα 11.3 −46.9 0.65 [30]

Mnδ 13.4 −42.5 1.0 [30]

Nb(l) −31.0 −47.3 [4]

Nb(s) −36.1 −49.6 0.18 [31]

Nb(2)

(s) −39.6 −65.3 0.18 [22]

Nb(3)

(s) −35.3 −58.0 0.37 [32]

Ni(l) 20.0 −39.7 [25]

Ni(s) 16.6 −48.7 0.88 [33]

Pd(l) −16.4 −48.2 [34]

Pd(s) −14.1 −53.8 0.99 [34]

Pd(2)

(s) −10.3 −51.3 0.19 [22]

Ti(l) −41.1 −44.5 [35]

Ti(s) −40.2 −43.3 0.38 [36]

Ti(2)

(s) −59.8 −49.9 0.71 [20]

Continued on next page

5

∆hmixH ∆sexH

System kJ (mol H)−1 J K−1 (mol H)−1 Tsol/Tliq References

U(l) 11.2 −33.7 [4]

U(s) 43.4 −42.9 1.0 [4]

The data in Table 1 are plotted in Fig. 1 where ∆∆hmixH R−1 is the ordinate81

and ∆∆sexH R−1 is the abscissa (R is the gas constant). The data point labels82

indicate the solid phase of the metal considered in the calculation of ∆∆hmixH83

and ∆∆sexH . For example, “Feα” refers to the alpha phase of Fe. The superscript84

numbers distinguish multiple values of ∆hmixH and ∆sexH reported for the solid85

phase. A single set of values is used for the liquid phase data for each M-H86

system.87

0

1

2

3

ΔΔsHex·R

-1

-2 -1 0 1 2 3

ΔΔhH

mix·R

-1/ 10

3K

Figure 1: Partial molar excess entropy (∆∆sexH , see Eq. 5) variation at melt-

ing versus the corresponding variation in the partial molar enthalpy of mixing

(∆∆hmixH , see Eq. 4) for M-H systems. Point labels denote the solid phase consid-

ered in the calculation of ∆∆hmixH and ∆∆sexH . The shaded regions are contours,

at different temperatures, of the surface defined by 1 ≤ rH ≡ (xH,liq/xH,sol) ≤ 3

(see Eq. 6).

6

Together, ∆∆hmixH and ∆∆sexH reflect changes in the equilibrium mole frac-88

tion of dissolved H. From Eqs. 3, 4, and 5, and for a constant pH2, we define the89

liquid-to-solid H solubility ratio, rH, as90

rH =xH,liq

xH,sol= exp

[−∆∆hmix

H

RTfus+

∆∆sexHR

]. (6)

A plot of ∆∆sexH versus ∆∆hmixH therefore defines a surface of the change in91

equilibrium H solubility for a given temperature. In Fig. 1 the shaded regions92

are contours of this surface. Each contour defines the inequality 1 ≤ rH ≤ 3 at93

a specific value of Tfus. Five contours are shown, from 1000 K (727 C) to 300094

K (2727 C) in steps of 500 K. This range of temperatures bounds Tfus for the95

majority of metals.96

Most of the data in Fig. 1 lie within the contours of the 1-to-3 times increase97

in solubility over the Tfus range of 1000–3000 K. This observation reflects the98

consensus in metallurgy that H solubility increases slightly upon melting [5].99

Furthermore, it is clear that the distribution of ∆∆hmixH and ∆∆sexH is non-100

random. Values of ∆∆hmixH and ∆∆sexH are within 0–2000R (J mol−1 H) and101

0–2R (J K−1 mol−1 H), respectively. There is a clustering of the data at ap-102

proximately 1000R for ∆∆hmixH and 1.3R for ∆∆sexH . The fact that chemically-103

dissimilar metals, such as Fe, Ag, U, and Nb, exhibit similar values of ∆∆hmixH104

and ∆∆sexH , in the vicinity of 1000R and 1.3R, respectively, suggests that a com-105

mon mechanism is responsible for the change in solution behavior of H across106

melting. We later adopt ∆∆hmixH = 1000R and ∆∆sexH = 1.3R as characteristic107

values.108

The characteristic values of ∆∆hmixH and ∆∆sexH may not be universally109

applicable. As Fig. 1 illustrates, outliers such as the Al-H system exist. In addi-110

tion, uncertainties in the values of reported partial molar mixing quantities, like111

∆hmixH and ∆sexH , tend to be large [37], and so drawing more precise correlations112

from the ∆∆hmixH and ∆∆sexH data should be done with some caution. In the113

absence of reported experimental uncertainties for ∆hmixH and ∆sexH , recourse114

should be made to reported uncertainties for other liquid metal-gas systems115

(e.g., M-O) and to liquid metal alloys in general to guide our understanding116

of the degree of scatter in the data shown in Fig. 1. Partial molar excess en-117

tropies and enthalpies of mixing are known to within about 1 J mol−1 K−1 and118

7

1 kJ mol−1, respectively [3, 38]. An approximate uncertainty for the data points119

in Fig. 1 may then be represented by error bars of ±0.2. This uncertainty does120

not change the systematic trend illustrated in Fig. 1.121

Nevertheless, a characteristic change in ∆hmixH and ∆sexH across melting may122

not be entirely surprising. Among the gas solutes frequently encountered in123

metallurgy—H, O, S, and N—hydrogen is the least reactive, with the smallest124

electronegativity difference. Dissolved H, particularly at high temperatures in125

the dilute regime (where the number of available H sites is much greater than the126

number of H atoms), is therefore more “gas-like” than, for example, dissolved127

O. It may be surmised, then, that changes in the quantities that define the128

interaction between M and H, i.e., ∆hmixH and ∆sexH , due to melting are the result129

of structural changes in the metal itself, more so than the result of changes in130

the specific chemical interaction. “Structural” here entails properties such as131

coordination number, molar volume, and ordering. Since the structural changes132

that occur during melting of most crystalline metals are often considered similar133

(see, for example, similarities in the entropies of fusion [39]), it can be argued134

that ∆∆hmixH and ∆∆sexH should be similar as well for different M-H systems.135

3. Derivation of ∆∆hmixH and ∆∆sexH from a Statistical Model of H136

Solubility137

In the previous section it was found that the change in ∆hmixH and ∆sexH at the138

solid-liquid transition is around 1000R and 1.3R, respectively, for most of the139

M-H systems in Table 1, constituting a characteristic shift in the equilibrium140

H concentration. A rationalization for the magnitude of these quantities is141

provided in the following using the statistical mechanical model of H solubility142

proposed by Fowler and Smithells [6]. This is the “proton gas” model, in which143

the metal, either solid or liquid, is treated as a potential field through which H144

atoms move.145

The grand canonical partition function (Γ) of dissolved H in a metal M of146

volume V may be written as :147

Γ = exp(lHλHωe

−χs/kT). (7)

8

The derivation of this form of Γ is given in Appendix A. lH (Eq. 8) is the148

molecular translational partition function, λH is the absolute activity of H, ω is149

the H nuclear spin statistical weight, and χs is the ground state energy of H in150

the metal, referenced to the state of infinite separation outside the metal.151

lH =

(2πmHkT

h2

)3/2

VH (8)

In Eq. 8, VH (Eq. 9) may be thought of as the effective volume in the metal over152

which H atoms behave “gas-like”. It is the classical configuration integral.2 VH153

is critical for connecting ∆∆sexH to short range ordering.154

VH =

∫V

e−W/kT dV (9)

In Eq. 9, W is the configurational potential energy of the H assembly referenced155

to the lowest energy state of H in the metal and is a function of the positional156

coordinates of the H atoms.157

The equilibrium number of dissolved H, NH, is158

NH = kT

(∂ ln Γ

∂µH

)= lHλHωe

−χs/kT (10)

The derivation of ∆sexH follows from the definition of the partial molar en-159

tropy as the temperature-derivative of the chemical potential. It is convenient160

to first convert the number of H atoms into the atomic ratio3 of H atoms and161

M atoms, as shown in Eq. 11.162

NH

NM=lHVλHω

MM

ρMNAe−χs/kT (11)

The chemical potential of dissolved H may then be written as:163

µH = kT lnλH = kT ln

(NH

NMθ(T )

)(12)

2The quantity VNHH /NH! is sometimes referred to as the configurational potential energy

partition function [40].3In the limit of infinite dilution, the atomic ratio is equivalent to the atomic fraction, and

so this conversion introduces no error with regards to the experimental data, which is defined

with respect to atomic fraction.

9

where θ(T ) =(lHV ω

MM

ρMNAe−χs/kT

)−1. Taking the derivative of µH with respect164

to temperature at constant P and NH, the total partial molar entropy is:165

sH = −k ln

(NH

NM

)− k

(T∂ ln θ(T )

∂T+ ln θ(T )

). (13)

The ideal configurational component is −k ln (NH/NM). Substituting for θ(T ),166

one finds for the excess component:167

sexHk

=3

2+

1

kT

∂ lnVH∂T

+3

2ln

(2πmHkT

h2

)+ln

(VHV

)+lnω+ln

(MM

ρMNA

)(14)

As discussed in the Introduction, the chosen standard state of dissolved H is168

the 12H2 gas standard state. The “relative” partial molar excess entropy, ∆sexH ,169

is then sexH − 12s

0H2

. Upon subtracting the entropy of H gas (see Appendix B),170

one finds:171

∆sexHk

=1

4

(ln

(m3

H

16πA2kh2T

)− 1

)+ln

(MM

ρMNA

)+

1

2lnP+ln

(VHV

)+

1

kT

∂ lnVH∂T

(15)

The relative partial molar enthalpy, ∆hmixH , can be derived following the172

same steps taken for Eqs. 13–15, where instead the Gibbs-Helmholtz relation173

defines the enthalpy from the chemical potential. For concision, we omit these174

steps and present the expression for ∆hmixH in Eq. 16.175

∆hmixH

kT=χs + 1

2χd

kT− 1

4+

1

kT

∂ lnVH∂T

(16)

It is important to note here that the terms in the expressions for ∆hmixH /kT176

(Eq. 15) and ∆sexH /k (Eq. 16) that are of the same order in T cannot be dis-177

tinguished as “enthalpic” or “entropic” if, experimentally, ∆hmixH and ∆sexH are178

calculated from an Arrhenius fitting of temperature-composition data. Conse-179

quently, it is prudent to define “effective” values of ∆hmixH and ∆sexH that are the180

result of allowing like-terms in the quantities ∆hmixH /kT and ∆sexH /k to com-181

bine. The effective values of ∆hmixH and ∆sexH may then be identified directly182

from Eq. 3. In doing so, a valid comparison can be made between the theoretical183

and measured values of ∆hmixH and ∆sexH . Eqs. 17 and 18 define the effective184

values of ∆hmixH and ∆sexH , respectively, in the proton gas model. For clarity, we185

will continue to refer to these quantities as ∆hmixH and ∆sexH .186

10

∆hmixH

kT=χs + 1

2χd

kT(17)

∆sexHk

=1

4ln

(m3

H

16πA2kh2T

)+ ln

(MM

ρMNA

)+

1

2lnP + ln

(VHV

)(18)

At this point, it is valuable to compare these results with experimental data.187

Fig. 2 is a plot of reported values of ∆sexX /R versus 1/T for M-X systems,188

where X is H, O, S, or N. The data point labels indicate the elemental liquid189

metal, M. The temperature coordinate of each data point is the lower bound190

of the temperature range for which the measured solution property, here ∆sexX ,191

is valid. For most of the M-X systems shown in Fig. 2 this temperature is in192

the vicinity of Tfus, though obvious exceptions exist (e.g., Al-O). Of primary193

interest here is the shaded band, which is the theoretical ∆sexH (Eq. 18) for194

values of VH/V between 0.1 (lower curve) and 1.0 (upper curve). The density195

(ρM) and molar mass (MM) are 104 kg m−3 and 0.1 kg mol−1, respectively,196

chosen as representative values for the elemental liquid metals.197

The measured values of ∆sexH are well-described by the results of the proton198

gas model within the bounds of 0.1 ≤ VH/V ≤ 1.0. This corroborates a simi-199

lar observation by Fowler and Guggenheim for solid transition metal-hydrogen200

systems in the dilute limit [40]. Given that the majority of the M-H systems201

in Fig. 2 fall within 0.1 ≤ VH/V ≤ 1.0, one may argue that values of ∆sexH202

well outside this range are suspect. The status of Au-H and Cr-H as outliers is203

therefore likely due to experimental uncertainty.204

A comparison may also be made with the entropy of the gas phase. The205

solid curves in Fig. 2 show 12s

0X2

as a function of temperature. If ∆sexH is taken206

to be its average value near −4.5R, the difference between ∆sexH and 12s

0X2

is207

about 5R to 8R. This represents the excess entropy arising from solute-solvent208

interaction.209

4. Discussion210

4.1. ∆sexH and Short Range Order211

The quantity VH (Eq. 9) in the proton gas model can be treated as a fitting212

parameter for interpreting experimentally measured values of ∆sexH . However,213

11

LiLa

SmFe

GaGe

In

Sb

V

Zr Au

Co

CuFe

In

Ni

Sb

Co

CuFe Mn

Ni

Nb

-8

-7

-6

-5

-4

-3

0.2 0.4 0.6 0.8 1.0

H

O

S

N

VH/V = 1.0

H

-12S H 20

-12S O

20

-12S S

20

-12S N 20

Be

Cr

Si

Au

ZnSn

Ga

AlSi

Ti

Ca

Cs

Na

Li

K

Te

Sn

PbBi Cd

Au

UMn

In

Hg

SnPb

Zn

Zr

V

TiCa

Li

Si

Na

Mo

-20

-15

-10

-5

0

5

10

sxex·R

-1

0.0 0.5 1.0 1.5 2.0 2.5 3.0

T-1

/ 103 K-1

Figure 2: Partial molar entropy of mixing (∆sexX ) as a function of 1/T for

X: H ( ), O ( ), S ( ), N ( ) in liquid metals. The shaded band is the theoretical

∆sexH (Eq. 18) derived from the proton gas model [6] for values of VH/V between

0.1 (lower curve) and 1.0 (upper curve). The majority of the data for the M-H

systems lie within this range of theoretical entropy values. The solid curves are

12s

0X2

, the standard state entropy of dissolved X, as functions of temperature.

The inset plot shows the boxed and shaded region in the full figure.

VH, the classical configuration integral, has a definite physical meaning that can214

be understood within the context of short range order in the M-H system. An215

immediate consequence of this view is that the shift in ∆sexH across the solid-216

liquid transition is the result of the local structural changes that occur during217

melting. This is the same conclusion proposed in Section 2 from a heuristic218

12

argument about chemical interaction energies, which can now be more rigorously219

treated using the proton gas model.220

The connection to short range order may be made recalling first that W is221

the potential energy of the H assembly above its ground state energy in the222

metal. W is a function of the spatial coordinates of the H atoms. In the limit223

that W → 0 everywhere in the metal, VH = V . As W increases, VH < V , and the224

effective volume in which H is well-described as a proton gas decreases. Stated225

differently, the tendency for M-H compound formation increases. VH, then, is226

a quantity that reflects the strength of the association between H and metal227

atoms. Second, the interpretation of VH as a measure of short range ordering228

requires a formal connection to the local M-H interactions. As of yet in this229

analysis, no description of the distribution of VH over the volume of the metal230

has been provided. Fowler [6] makes the connection between the proton gas231

occupying an effective volume VH and the short-range interaction between M232

and H by showing that, in the limit of small H solubility (NH αNM, where α233

is the number of H coordination sites per metal atom), it is valid to describe each234

H atom as being bound to a potential field of volume v′H = VH/(αNM), where α235

is the number of H coordination sites per metal atom. Thus, one may interpret236

VH as being uniformly distributed over αNM interstitial (or quasi-interstitial)237

sites, each with a volume v′H, associated with the metal atoms. The connection238

with short range order is now immediately clear. As W increases, reflecting239

a greater propensity for M-H compound formation, the effective volume v′H in240

which each H atom freely translates decreases.4 A localization of the H atoms241

with respect to the metal atoms is found and hence an increase in the short range242

order of the M-H system. The loss of entropy corresponding to an increase in243

W is appropriately reflected in Eq. 18, where, as VH decreases, ∆sexH decreases.244

From Eqs. 5 and 18, the change in ∆sexH across the solid-liquid transition,245

∆∆sexH , may be written as:246

4The H partition function in the proton gas model is classical. By treating H as a particle in

an infinite potential well, one can show that for v′H less than approximately 10−3 nm3, recourse

must be made to quantum statistical mechanics, as the difference between the energy states

of H is no longer much less than kT .

13

∆∆sexH = R ln

(VH,liq

VH,sol

)(19)

where the entropy is per mole of H rather than per atom. The value of VH,sol247

for most metals for temperatures on the order of 103 K varies from about 0.1 to248

1.0 and is typically less than 0.5 for transition metals [40]. The magnitude of249

∆∆sexH is then bounded by making use of the following arguments: VH ≤ V and250

VH,liq/VH,sol > 1, the latter inequality being a consequence of the expectation251

that the liquid is a more disordered state than the solid (Wliq < Wsol). This252

yields a range of possible values for ∆∆sexH :253

R ln(1) < ∆∆sexH ≤ R ln(10). (20)

The lower and upper bounds are 0 and 2.3R, respectively. This range ade-254

quately accounts for the measured values of ∆∆sexH shown in Fig. 1, the average255

value of which is 1.3R.256

The change in ∆hmixH at the solid-liquid transition, ∆∆hmix

H , follows directly257

from Eqs. 4 and 17:258

∆∆hmixH = ∆χs (21)

where ∆χs is the difference between the H ground state energy (“absorption”259

enthalpy) in the liquid and solid states. Further discussion of Eq. 21 is limited,260

as the value of χs cannot be determined from the proton gas model. Other261

theoretical approaches are needed to rationalize the connection, if it exists,262

between Eq. 21 and the average measured value of ∆∆hmixH of 1000R J mol−1 H.263

It is for this reason that the present discussion primarily concerns the partial264

molar entropy.265

A connection has been made between short range order and ∆sexH . Specifi-266

cally, the change in ∆sexH at Tfus, ∆∆sexH , can be seen as the result of changes in267

short range order across melting (Eq. 20). The most direct means of experimen-268

tally validating such a connection would be neutron diffraction measurements269

of several of the metals in Table 1 above and below Tfus and at concentrations270

of dissolved H where Eq. 3 is true. Currently, the local structure is known only271

for select solid state M-H systems at relatively low temperatures [20]. Com-272

paratively more is known about the structure of pure liquid metals, and some273

14

insight may be found by examining the changes in short range order that oc-274

cur at the solid-liquid transition of the metal solvent. First, it is well-known275

that the molar volume increases between 2–5 % for most metals upon melting276

[39, 41]. In addition, small changes in coordination number have been inferred277

from X-ray and neutron diffraction measurements of a number of solid and278

liquid metals [42]. The change in coordination number at melting is crystal-279

dependent: approximately −1 for close-packed metals and +2 to +3 for BCC280

metals. Lastly, one may infer information about the short range order in metals281

from band structure-dependent electronic properties [43]. Both nuclear mag-282

netic resonance measurements [44] and photoelectron spectroscopy [45] suggest283

measurable, but small, differences in short range order between the solid and284

liquid, which largely corroborate the diffraction studies. From the above points,285

it is clear some change in short range order of the metal does occur at melting,286

though the change is not large. The increase in molar volume and changes in287

the M-M coordination number leave open the possibility of an increase in the288

number of H sites at melting, which would yield the relation ∆∆sexH > 0 implied289

in Eq. 20. It must be stressed that the relevant ordering in this analysis is that290

between the metal and dissolved H. Information about M-M short range or-291

der, such as coordination number and nearest-neighbor distance, is, on its own,292

insufficient to corroborate Eq. 20.293

5. Conclusion294

A systematic change in ∆hmixH and ∆sexH of approximately 1000R and 1.3R,295

respectively, occurs at the solid-liquid transition for a number of M-H systems296

in the dilute limit. To rationalize these changes from first-principles, expres-297

sions for ∆hmixH and ∆sexH were derived from the proton gas model of Fowler298

and Smithells [6]. Good agreement was observed between the theoretical and299

experimental values of ∆sexH for values of the classical configuration integral300

(VH) between 0.1 and 1.0. The change in ∆sexH at melting, ∆∆sexH , is directly301

related to the change in VH, from which the following bounds were derived:302

R ln(1) < ∆∆sexH ≤ R ln(10). This range of 0–2.3R reflects the range of mea-303

sured values for ∆∆sexH , the average being 1.3R. It was proposed that VH is304

a quantity which reflects short range order in the M-H system. Consequently,305

15

the measured values of ∆∆sexH correspond to a decrease in short range order306

at melting. Qualitatively, the notion that H short range order decreases in the307

liquid state is supported by investigations of structural changes in pure liquid308

metals. The statistical mechanical description of H dissolved in liquid metals309

remains a frontier for thermodynamicists. Considering the importance of H con-310

trol in process metallurgy, further studies of this topic are of significant practical311

interest.312

6. Acknowledgments313

This work was supported by the National Science Foundation (NSF) under314

grant number 1562545.315

Appendix A. The Grand Canonical Partition Function316

The general form of the grand canonical partition function of the proton gas317

model of H solubility is:318

Γ =∑NH

Q (NH, V, T ) eNHµH/kT e−NHχs/kT . (A.1)

Upon expanding terms, Γ may be written as:319

Γ =∑NH

[ΩH(T )× φH(T )NH

]eNHµH/kT e−NHχs/kT

=∑NH

[(1

NH!

∫· · ·∫e−W/kT

NH∏i=1

(dxdydz)i

)×(

2πmHkT

h2

)3NH/2

ωNH

]eNHµH/kT e−NHχs/kT

=∑NH

1

NH!

[∫V

e−W/kT dV ×(

2πmHkT

h2

)3/2

ωλH(T )

]NH

e−NHχs/kT

= exp(lHλHωe

−χs/kT).

(A.2)

ΩH(T ) is the configurational potential energy partition function. φH(T ) is the320

molecular partition function of the dissolved H without the volumetric factor.321

µH is the chemical potential of H.322

16

Appendix B. Entropy and Enthalpy of Hydrogen Gas323

The absolute activity of atomic hydrogen gas is [40]:324

λg =

(P

kT

) 12

e−χdkT

[(4πmHkT

h2

) 32 8π2AkTω2

2h2

]− 12

. (B.1)

The same approach for deriving the total partial molar entropy entropy of dis-325

solved H may be applied here to determine the molar entropy of H gas, sg.326

sg = −(∂µg∂T

)P

= − ∂

∂TkT lnλg

∣∣∣∣P

= k

(7

4− 1

2ln

(P

k

)+

7

4lnT +

3

4ln

(4πmHk

h2

)+

1

2ln

(8π2Ak

h2

)+

1

2ln

(ω2

2

))(B.2)

The molar enthalpy of H gas, hg, is327

hg =

(∂µg/T

∂(1/T )

)P

= − ∂

∂(1/T )kT lnλg

∣∣∣∣P

= k

(7

4T − χd

2k

).

(B.3)

List of Symbols

A Moment of inertia of H2(g) (4.63 × 10−34 kg m2)

aH Activity of H

h Planck constant (6.626 × 10−34 J s)

∆hmixi Relative partial molar enthalpy of mixing (J mol−1)

∆∆hmixi

Difference between ∆hmixi,liq and ∆hmix

i,sol at Tfus

(J mol−1)

k Boltzmann constant (1.38 × 10−23 J K−1)

lH Molecular translational partition function of H

MM Molecular weight of M (kg mol−1)

mH Mass of H (1.69 × 10−27 kg)

NA Avogadro constant (6.022 × 1023 mol−1)

Continued on next page

17

NH Equilibrium number of dissolved H

NM Number of M atoms

ni Number of moles of i

Pi Absolute pressure of i (Pa)

pi Partial pressure of i

Q Canonical partition function

qH Molecular partition function of dissolved H

R Gas constant (8.314 J K−1 mol−1)

ri Ratio of xi,liq to xi,sol

s0i Standard state molar entropy (J K−1 mol−1)

si Partial molar entropy (J K−1 mol−1)

∆sexi Relative partial molar excess entropy (J K−1 mol−1)

∆∆sexi Difference between ∆sexi,liq and ∆sexi,sol (J K−1 mol−1)

T Temperature (K)

Tfus Melting temperature (K)

V Volume (m3)

VH Classical configuration integral (m3)

W

Configurational potential energy of the M-H assembly

(J)

xi Mole fraction of i

yi Mole ratio of i

zi Lattice ratio of i

z′i Occupied-unoccupied site ratio: z′i = zi/α

Greek

α Number of H coordination sites per atom of M

Γ Grand canonical partition function of dissolved H

γi Activity coefficient of i

λH Absolute activity of H

µH Chemical potential of H (J mol−1)

ρM Density of M (kg m−3)

φH

Molecular partition function of H without the

volumetric factor (m−3)

χd Dissociation energy of diatomic H2(g) (J)

χs Ground state energy of dissolved H (J)

ΩH Configurational potential energy partition function

ω H nuclear spin statistical weight

18

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