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Analysis of the Partial Molar Excess Entropy of Dilute Hydrogen in Liquid Metals and Its Change at the Solid-Liquid Transition

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Analysis of the Partial Molar Excess Entropy of Dilute Hydrogen in Liquid Metals and Its Change at the Solid-Liquid Transition Analysis of the Partial Molar Excess Entropy of Dilute Hydrogen 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 mix A systematic change in the partial molar enthalpy of mixing (∆hH ) and partial ex molar excess entropy (∆sH ) for dilute hydrogen-metal systems at the solid- mix ex liquid transition is reported. Expressions for ∆hH and ∆sH are derived from ex the Fowler model of hydrogen solubility, and the change in ∆sH at melting is bounded. The theoretical bound is in agreement with measured data. A ex connection is made between the change in ∆sH and short range order in the metal-hydrogen system. Keywords: Liquids, Hydrogen, Solubility, Statistical mechanics, Thermodynamics 1 1. Introduction 2 Metals processing invariably requires the handling of metals in the liquid 3 state. Such operations rarely occur in inert atmospheres. The mole fraction 4 of dissolved gases, in particular hydrogen (H), are typically in the range of −6 −2 5 10 to 10 for liquid metals, and therefore degassing procedures are routinely 6 employed in process metallurgy. This is done to prevent degradation of the 7 mechanical properties of the solidified product, and such effects are well-studied 8 [1, 2]. The severity of these effects depends on the concentration of H in the 9 metal M, which can be determined from the solution thermodynamics of the M- 10 H system. Calculating and predicting H solubility, defined here as its equilibrium ∗Corresponding author Email address: [email protected] (Antoine Allanore) Preprint submitted to Elsevier January 25, 2019 11 concentration in the metal, is therefore of considerable importance for metals 12 processing. 13 Prior reports have compiled existing data on the solubility of H in liquid 14 metals, along with dissolved oxygen, sulfur, and nitrogen [3, 4, 5]. These data 15 reveal a correlation between the two key quantities describing the mixing ther- ex 16 modynamics: the partial molar excess entropy (∆sX ) and the partial molar mix mix 17 enthalpy of mixing (∆hX ). ∆hX is defined as ∆hmix = h − 1 h0 (1) X X 2 X2 1 0 18 where h is the partial molar enthalpy, and h is the standard state enthalpy X 2 X2 ex 19 of X2(g). ∆sX is the defined as ∆sex = s − 1 s0 − sid (2) X X 2 X2 X 0 20 where s is the partial molar entropy, s is the standard state entropy of X , X X2 2(g) id 21 and sX is the ideal partial molar entropy of mixing. The correlation between the ex 22 partial molar excess entropy (∆sX ) and the partial molar enthalpy of mixing mix 23 (∆hX ) for the liquid state can be rationalized from the chemical reactivity of 24 the metal-gas system, by considering the relative strength of the solute-metal 25 chemical bonding. 26 In the present work, specific attention is drawn to the solution behavior 27 of H in solid and liquid metals near Tfus and its statistical thermodynamic 28 description. Statistical thermodynamic treatments of H dissolved in a metal are 29 reported by a number of investigators[6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. mix ex 30 These studies derive ∆hH and ∆sH from the principles of statistical mechanics. 31 Together, these quantities define the activity coefficient (γH) as a function of 32 temperature and consequently the hydrogen concentration that is at equilibrium 33 with a given partial pressure of H2 gas: 1 2 mix ex p 1 −∆h ∆s x = H2 = p 2 exp H + H : (3) H H2 γH RT R 34 Eq. 3 is one form of what is referred to as Sieverts' law [18], which states that 35 the concentration of a gas solute X in a metal is proportional to the square-root 36 of the partial pressure of X2 above the metal. Sieverts' law is valid in the dilute 2 37 limit and is generally observed for many M-H systems for PH2 ≤ 1 atm, in both 38 the solid and liquid state. In Eq. 3, the proportionality constant is γH. In this mix ex 39 analysis, γH, and by extension ∆hH and ∆sH , are defined according to the 40 following criteria: 41 1. The concentration coordinate for the system of a gas solute H in liquid 42 metal M is the mole fraction, defined as xH ≡ nH=(nM + nH). 1 43 2. The standard state for the dissolved gas H is the pure diatomic gas 2 H2 44 at 1 atm pressure. 45 3. The solution thermodynamics are treated in the limit of infinite dilution. 46 These criteria yield the definition shown in Eq. 3. The analysis presented 47 herein restricts itself to the concentration and temperature regimes in which 1 48 Eq. 3 is true . Thus, this analysis focuses on the dilute solution behavior of H 49 in liquid metals, as this is the concentration regime most frequently encountered 50 in metallurgical processes. \Dilute" here refers to concentrations for which H 51 solute self-interaction is a negligible contribution to the H activity coefficient. 52 In this paper, we propose to compare H solubility data in liquid and solid mix 53 metals near Tfus. Specifically, we report on a systematic shift in ∆hH and ex 54 ∆sH at melting for a number of M-H systems. While not universal, the shift 55 is sufficiently compelling as to suggest a common mechanism for the change 56 in solution behavior that may be used to quantitatively predict the solution 57 properties of H in liquid metals from solid-state data. A connection is made ex 58 between the change in ∆sH at melting and short range order in the liquid mix ex 59 through a derivation of ∆hH and ∆sH from the proton gas model of Fowler 60 and Smithells [6]. 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 mix ex 61 2. Change in ∆hH and ∆sH at the Solid-Liquid Transition mix ex 62 ∆hH and ∆sH are the change in the partial molar enthalpy of mixing 63 and partial molar excess entropy, respectively. A partial molar property is the 64 change in the integral property per addition of one atom of, for example, H to 65 a liquid M-H solution. At the equilibrium solid-liquid transition we define the mix ex 66 quantities ∆∆hH and ∆∆sH : mix mix mix ∆∆hH = ∆hH,liq − ∆hH,sol (4) ex ex ex ∆∆sH = ∆sH,liq − ∆sH,sol (5) mix ex 67 The values of ∆hH and ∆sH for M-H systems for which solid- and liquid- 68 state data exist are listed in Table 1. For each M-H system in Table 1 a quantity 69 Tsol=Tliq is defined to be the ratio of the maximum temperature for which the mix ex 70 values ∆hH,sol and ∆sH,sol are valid to the minimum valid temperature for mix ex 71 ∆hH,liq and ∆sH,liq. For most of the M-H systems, Tsol=Tliq is close to unity, in- mix ex 72 dicating that ∆∆hH and ∆∆sH solely reflect changes in the solution behavior 73 of dissolved H due to melting. It is important to recognize that, as a result of mix ex 74 the weak temperature dependence of both ∆hH and ∆sH [20], the tempera- 75 ture range over which reported values of these properties quantitatively describe 76 the solution behavior is usually several hundreds of degrees Celsius. Thus, it is mix ex mix 77 reasonable to treat ∆∆hH and ∆∆sH as capturing the change in ∆hH and ex 78 ∆sH across the solid-liquid transition for systems with Tsol=Tliq close to unity. 79 For systems with Tsol=Tliq far from unity, there is greater uncertainty with this 80 assumption. mix ex Table 1: Values of ∆hH and ∆sH for H in elemental metals mix ex ∆hH ∆sH −1 −1 −1 System kJ (mol H) JK (mol H) Tsol=Tliq References Ag(l) 76:3 −29:5 [21] Ag(s) 62:1 −48:5 0.98 [21] (2) Ag(s) 68:9 −42:5 1.0 [22], see also: [4] Continued on next page 4 mix ex ∆hH ∆sH −1 −1 −1 System kJ (mol H) JK (mol H) Tsol=Tliq References Al(l) 51:5 −36:2 [23] Al(s) 58:2 −56:7 1:0 [23] (2) Al(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] (2) Cu(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] (2) Feα 24:3 −53:6 0.65 [22], see also: [27] Feγ 27:0 −46:7 0.92 [26], see also: [27] (2) Feγ 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] (2) Nb(s) −39:6 −65:3 0.18 [22] (3) Nb(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] (2) Pd(s) −10:3 −51:3 0.19 [22] Ti(l) −41:1 −44:5 [35] Ti(s) −40:2 −43:3 0.38 [36] (2) Ti(s) −59:8 −49:9 0.71 [20] Continued on next page 5 mix ex ∆hH ∆sH −1 −1 −1 System kJ (mol H) JK (mol H) Tsol=Tliq References U(l) 11:2 −33:7 [4] U(s) 43:4 −42:9 1.0 [4] mix −1 81 The data in Table 1 are plotted in Fig.
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