Guiding Principles for Geochemical/Thermodynamic Model Development and Validation in Nuclear Waste Disposal: A Close Examination of Recent Thermodynamic Models for H⁺—Nd³⁺—NO₃⁻(—Oxalate) Systems

Abstract

Development of a defensible source-term model (STM), usually a thermodynamical model for radionuclide solubility calculations, is critical to a performance assessment (PA) of a geologic repository for nuclear waste disposal. Such a model is generally subjected to rigorous regulatory scrutiny. In this article, we highlight key guiding principles for STM model development and validation in nuclear waste management. We illustrate these principles by closely examining three recently developed thermodynamic models with the Pitzer formulism for aqueous H⁺—Nd³⁺—NO₃⁻(—oxalate) systems in a reverse alphabetical order of the authors: the XW model developed by Xiong and Wang, the OWC model developed by Oakes et al., and the GLC model developed by Guignot et al., among which the XW model deals with trace activity coefficients for Nd(III), while the OWC and GLC models are for concentrated Nd(NO₃)₃ electrolyte solutions. The principles highlighted include the following: (1) Principle 1. Validation against independent experimental data: A model should be validated against experimental data or field observations that have not been used in the original model parameterization. We tested the XW model against multiple independent experimental data sets including electromotive force (EMF), solubility, water vapor, and water activity measurements. The results show that the XW model is accurate and valid for its intended use for predicting trace activity coefficients and therefore Nd solubility in repository environments. (2) Principle 2. Testing for relevant and sensitive variables: Solution pH is such a variable for an STM and easily acquirable. All three models are checked for their ability to predict pH conditions in Nd(NO₃)₃ electrolyte solutions. The OWC model fails to provide a reasonable estimate for solution pH conditions, thus casting serious doubt on its validity for a source-term calculation. In contrast, both the XW and GLC models predict close-to-neutral pH values, in agreement with experimental measurements. (3) Principle 3. Honoring physical constraints: Upon close examination, it is found that the Nd(III)-NO₃ association schema in the OWC model suffers from two shortcomings. Firstly, its second stepwise stability constant for Nd(NO₃)₂⁺ (log K₂) is much higher than the first stepwise stability constant for NdNO₃²⁺ (log K₁), thus violating the general rule of (log K₂–log K₁) < 0, or $\frac{K_1}{K_2}>1$. Secondly, the OWC model predicts abnormally high activity coefficients for Nd(NO₃)₂⁺ (up to ~900) as the concentration increases. (4) Principle 4. Minimizing degrees of freedom for model fitting: The OWC model with nine fitted parameters is compared with the GLC model with five fitted parameters, as both models apply to the concentrated region for Nd(NO₃)₃ electrolyte solutions. The latter appears superior to the former because the latter can fit osmotic coefficient data equally well with fewer model parameters. The work presented here thus illustrates the salient points of geochemical model development, selection, and validation in nuclear waste management.

1. Introduction

In nuclear waste disposal, a long-term performance assessment (PA) of a geological repository generally relies on a series of conceptual/mathematical models to predict geochemical conditions in the near and far field [1,2,3,4,5,6]. Among them, a source-term model (STM) for radionuclide solubility calculations is key to a PA of a repository [1,2,3,4,5,6,7]. Consequently, the validity and the applicability of an STM for nuclear waste disposal are of paramount importance to the PA. In the near field of a geological repository, there are multiple layers of engineered barriers, called the engineered buffer system (EBS), including waste forms, waste packages, and clay-based buffer layers surrounding each waste package. The dissolved concentrations of radionuclides are primarily controlled by the solubility limits of corresponding radionuclide mineral phases to be formed under repository conditions. Given the sparingly soluble nature of these minerals, the dissolved concentrations of radionuclides such as actinides in the near field are limited to trace levels. Depending on the types of host rocks, the ionic strengths of aqueous solutions in the near field of repositories could vary greatly [8,9,10,11,12,13,14]. In this paper, we focus on high-ionic strength environments, such as those in salt rock formations. In this sense, for an STM, we are concerned with the thermodynamic behavior of trace-level radionuclides in concentrated background electrolyte solutions, that is, an STM describes the trace activity coefficients of radionuclides. As a good example, Xiong and Wang [15] established an STM (termed as the XW model hereafter) describing the interactions of trivalent actinides with oxalate at the expected trace concentration levels of these species in geological repositories. This model does not involve the Nd(III)-NO₃ association.

For concentrated Nd(NO₃)₃ electrolyte solutions with high Nd(III) concentrations up to ~6.3 mol·kg⁻¹, there are two recent models developed separately by Guignot et al. [16] (termed as the GLC model) and by Oakes et al. [17] (termed as the OWC model). As discussed below, such high Nd(III) concentrations are unlikely to be present in a geological repository. The GLC and OWC models use the same data set from literature for their model fitting. The difference is that the GLC model uses five parameters without the introduction of the Nd(III)-NO₃ association, whereas the OWC model uses nine parameters with the explicit addition of the Nd(III)-NO₃ association.

In most countries over the world, nuclear waste management programs are regulated by independent agencies to ensure regulatory compliance. Development of a defensible STM is thus critical for a compliance certification application of a repository. In this article, we will highlight some key guiding principles that have been used for STM development and validation in nuclear waste management programs. To our knowledge, such principles have not been fully elaborated in the literature. We will then illustrate these principles by closely examining the three aforementioned models for aqueous H⁺—Nd³⁺—NO₃⁻(—oxalate) systems. In this paper, all supporting calculations were performed using the computer code EQ3/6 version 8.0a [18,19].

2. Key Guiding Principles for the Development of a Defensible STM

In the following, we summarize the key guiding principles for the development of a defensible STM, based on our own long-term experience in the field.

The first guiding principle is that an STM should be validated by comparison of model predictions with independent experimental results or field observations that have not been not used in model parameterization. As a general rule, comparing model predictions with independent observations and/or experimental results is good scientific practice [20]. Such practice is common in geochemical modeling. For instance, Lacroix et al. [21] developed a model for groundwater pH control in bioremediation, and the model was later validated by independent experimental results [22]. Similarly, the model for the hydromagnesite dissolution reaction in [23] was later validated by independent experimental results [24] and field observations [25]. As the buffer assemblage of brucite and hydromagnesite controls the fugacity of CO₂(g) in a repository [13], the validation of the hydromagnesite dissolution model enhances the confidence in PA calculations and reduces the associated uncertainty. This guiding principle is essentially a gate keeper for model development and validation.

The second guiding principle is that a model should be cross-examined among different variable(s), especially for those having a direct high impact on radionuclide solubility calculations. This is exemplified in Xiong and Wang [26]. In that work, a previously developed model was cross-checked against pH measurements, revealing a deficiency of that model. Bear in mind that a model parameterized on one variable does not guarantee that the model would give a reasonable prediction of another variable.

The third guiding principle is that models developed must honor the known physical constraints. For instance, one of the physical constraints regarding the stepwise stability constants of aqueous complexes is that there is a well-known rule: (logK₂–logK₁) < 0, or ${\displaystyle \frac{K_1}{K_2}}>1$. This rule has a theoretical basis and experimental affirmations, and appears in the standard textbooks and early literature (e.g., [27,28,29,30,31,32,33,34]). To honor this rule is important to striking a balance between species complexation and weak inter-species interaction in modeling multiple-component concentrated electrolyte solutions. Numerous Pitzer models honor this rule (e.g., [8,35,36,37,38]).

The fourth guiding principle is that a model should minimize the number of f parameters to be fitted. All STMs up to date are empirical to various degrees. This principle will guide us to reduce the associated empiricism of a specific model, therefore improving model predictability. Similarly, in model selection, a model with fewer parameters is generally preferred over a model with more parameters, provided that both models can adequately fit a given data set.

In the following, the above key guiding principles are illustrated by a close examination of the three models identified in Section 1.

3. Determination of Thermodynamic Properties of Trace Components in Concentrated Electrolyte Solutions

In an STM, the accurate representation of trace activity coefficients of radionuclides in a concentrated multicomponent electrolyte solution is required [39,40,41]. For instance, Novak et al. [40] investigated Np(V) and Np(VI) at trace concentrations ranging from ~10⁻⁷ to 10⁻⁵ mol·dm⁻³ for Np(V) and 10⁻⁴ to 10⁻³ mol·dm⁻³ for Np(VI) in multi-component WIPP brines with ionic strengths ranging from 0.844 to 7.78 mol·dm⁻³. In the development of the XW model [15], solubility measurements of Pr₂(C₂O₄)·10H₂O and Nd₂(C₂O₄)·10H₂O in HNO₃ or HNO₃-H₂C₂O₄ solutions were performed. Pr(III) and Nd(III) are used as analogs to Pu(III) and Am(III)/Cm(III), respectively. Regarding the solubility measurements with Nd₂(C₂O₄)·10H₂O, the equilibrated systems contain Nd(III), H⁺, NO₃, and C₂O₄, representing the mixed electrolyte system HNO₃ + Nd(NO₃)₃ + Nd₂(C₂O₄)₃ + H₂C₂O₄ + H₂O with Nd(III) concentrations ranging from 10⁻² to 10⁻⁴ mol·kg⁻¹ (limited by the solubility-controlling phase, Nd₂(C₂O₄)·10H₂O). The experimental matrices for the long-duration (151 days) tests in Xiong and Wang [15] are recaptured in

Table 1. The experimental matrices for the experiments with the longest durations of 151 days in Xiong and Wang [15].

Exp #	m (Nd(III))	m (HNO3)	m (H2C2O4)	Total Stoichiometric Ionic Strength, m
Nd-0.1	2.65 × 10−4	0.100	0	0.102
Nd-0.5	2.01 × 10−3	0.500	0	0.515
Nd-1.0	5.02 × 10−3	1.00	0	1.04
Nd-2.0	7.73 × 10−3	1.94	0.017	2.05
Nd-3.0	1.31 × 10−2	2.72	0.033	2.92
Nd-4.0	1.37 × 10−2	3.22	0.057	3.49

. In calculation of total stoichiometric ionic strengths in Table 1, Nd(III) is assumed to be an Nd₂(C₂O₄)₃ electrolyte. At such low levels of Nd(III) concentrations, the traditional isopiestic technique is apparently not applicable. In this context, the interaction parameters for Nd³⁺—NO₃⁻ consistent with the existing well-established oxalate and nitric systems from the literature [42,43] are evaluated. The Pitzer model developed is intended to describe a trace/minor activity coefficient of Nd³⁺.

In the development of the XW model, the concentrations of rare-earth elements and trivalent actinides, represented by Pr(III) and Nd(III), are limited to trace levels by sparingly soluble solid phases that are similar to the natural metal oxalates and other metal-bearing phases (e.g., [44,45,46,47,48]) such as:

• Coskrenite-(Ce) [(Ce,Nd,La)₂(SO₄)₂(C₂O₄)·8H₂O];
• Deveroite-(Ce) [(Ce_1.01Nd_0.33La_0.32Pr_0.11Y_0.11Sm_0.01Pb_0.04U_0.03Th_0.01Ca_0.04)_2.01(C₂O₄)_2.99·9.99H₂O];
• Gasparite-(La)[(La_0.65Ce_0.17Nd_0.07Ca_0.06Mn_0.05Pr_0.02)_1.02(As_0.70V_0.28P_0.02)_1.00O₄];
• Gasparite-(Ce) [(Ce_0.43La_0.24Nd_0.15Ca_0.11Pr_0.04Sm_0.02Gd_0.01)_1.00(As_0.99Si_0.03)_1.02O₄];
• Levinsonite-(Y)[(Y,Nd,Ce)Al(SO₄)₂(C₂O₄)·12H₂O];
• Zugshunstite-(Ce) [(Ce,Nd,La)Al(SO₄)₂(C₂O₄)·12H₂O].

Note that all of these natural mineral phases contain Nd(III). The presence of these low-solubility phases in natural environments clearly indicates that the concentrations of trivalent actinides and lanthanides, Nd(III) in particular, are limited to trace levels in natural environments, including the near field of a geological repository. In Xiong and Wang [15], the natural systems with trace levels of lanthanides and actinides were discussed in some detail.

In high-level liquid nuclear waste (HLLW), actinides and lanthanides are also present at a trace level (e.g., [49,50,51,52]). For actinides, their concentrations range from 1.4 × 10⁻⁴ mol·dm⁻³ to 1.2 × 10⁻³ mol·dm⁻³ [49,50,51,52]. The concentrations for lanthanides in HLLW range from 1.0 × 10⁻³ mol·dm⁻³ to 4.3 × 10⁻² mol·dm⁻³ [49,50,51]. Consequently, in combination with the above-mentioned expected conditions in geological repositories, actinides and lanthanides at minor- and trace-level concentrations, not at a concentrated-level, are of high and practical interest in the field of nuclear waste management. Accordingly, accurate knowledge of the activity coefficients for actinides and lanthanides at a trace level is required for developing an STM in nuclear waste management.

For electrolytes at concentrations of trace levels, Malatesta’s group has demonstrated that models based on isopiestic methods in the concentrated region cannot accurately describe the activity coefficients of trace electrolytes. For instance, Malatesta and Carrara [53] noted that while salts such as [Co(en)₃]₂(SO₄)₃ (en—ethylendiamine) were investigated by isopiestic methods in the concentrated region, and assumed to behave in a Debye–Hückel Theory (DHT) way at lower concentrations, the absolute values of the activity coefficients calculated in such a way could deviate by 100–200% or more. As Malatesta [54] pointed out, knowledge of the dilute solution is of vital importance, “otherwise the activity coefficients of concentrated electrolytes remain biased by an unidentified error in their turn” (italics were Malatesta’s). The work from Malatesta’s group motivated Xiong and Wang [15] to rely on their model (the XW model) development on solubility measurements. In the following, the XW model is tested against various experimental data to see how the XW model performs for its intended use.

4. Model Validation Against Independent Data Sets

Malatesta et al. [55] experimentally determined the mean activity coefficients of La(NO₃)₃ electrolyte solutions in the diluted region including those in high dilutions. They determined the mean activity coefficients of La(NO₃)₃ electrolyte solutions based on electromotive force (EMF) measurements using liquid membrane cells. As both La(III) and Nd(III) belong to the light rare-earth element group, the mean activity coefficients of La(NO₃)₃ and Nd(NO₃)₃ electrolyte solutions are expected to be very similar. Accordingly, the mean activity coefficients of Nd(NO₃)₃ predicted by the XW model are first compared with the high-quality data from Malatesta et al. [55]. As illustrated in Figure 1, the mean activity coefficients predicted by the XW model are in good agreement with the experimental data from Malatesta et al. [55]. Note that the experimental data of Malatesta et al. [55] were not used in the XW model parameterization. Hence, this good agreement serves as the first example of model validation for the XW model.

Using Ce(III)–oxalate as an analog, Chang [56] studied an industrial process for recovering actinides. The typical operating conditions for the precipitation of actinide oxalates include the equilibrium concentrations of oxalic and nitric acids at 0.25 mol·dm⁻³ and 0.92 mol·dm⁻³, respectively [56]. Ce(III) and Nd(III) are expected to have similar chemical behaviors in solution since both belongs to the light rare-earth element group. To test the performance of the XW model, solubilities of Nd₂(C₂O₄)·10H₂O predicted by the model are compared with the experimental solubility data of Ce(III)– and Nd(III)–oxalates under the similar conditions from Chang [56] and Kim et al. [49]. As shown in Figure 2, the head-to-head comparisons indicate that the XW model predictions are again in good agreement with the experimental data, which are again independent from the XW model parameterization. In these head-to-head validation tests, the concentrations of oxalic and nitric acids range from 0.02 mol·kg⁻¹ to 0.27 mol·kg⁻¹, and from 0.5 mol·kg⁻¹ to 2.15 mol·kg⁻¹, respectively. Those comparisons clearly validate the XW model. In the above comparisons, the original molar concentrations in Chang [56] and Kim et al. [49] were converted into molal concentrations, according to the method in [57] for the calculations of the densities of mixed electrolyte solutions, based on the densities of HNO₃ from [58] and of H₂C₂O₄ from [59].

In addition, Moiseev et al. [60] measured the water vapor pressures of H₂O-HNO₃-REE(NO₃)₃ (REE = Pr, Nd, Sm) systems at 25 °C. Some of their vapor pressure measurements for the H₂O-HNO₃-Nd(NO₃)₃ systems were performed at Nd(III) concentrations and ionic strengths similar to the experimental matrices for the XW model development. Hence, their data are ideal for independent validation of the model. Table 2 lists the experimental results and the values predicted by the model. The results show that the XW model predictions are in excellent agreement with the experimental measurements.

The XW model was also compared with the measured water activity data of H₂O-HNO₃-Nd(NO₃)₃ from O’Brien and Bautista [61] and Lalleman et al. [62], and the H₂O-HNO₃-Ce(NO₃)₃ systems at 25 °C from Rodriguez-Ruiz et al. [63], based on the parameters for H₂O-HNO₃-Nd(NO₃)₃ from the XW model. Again, those data are independent from the model development. Table 2 and Figure 3 show that the model predictions are again in excellent agreement with independent experimental data on the ternary mixtures. Based on our evaluations, the aforementioned experimental data are of good quality. Other researchers also found that those data are of high quality. For example, Lassin et al. [64] used data from Moiseev et al. [60] for the model validation tests.

Table 2. Validation tests for the XW model based on experimental vapor pressure measurements in the H₂O-HNO₃-Nd(NO₃)₃ system at 25 °C from Moiseev et al. [60], experimental water activity measurements in the same system 25 °C from Lalleman et al. [62], and the H₂O-HNO₃-Ce(NO₃)₃ system at 25 °C from Rodriguez-Ruiz et al. [63].

x(Nd(NO3)3)	x(HNO3)	x(H2O)	m(Nd(NO3)3)	m(HNO3)	Total ionic strength, mol·kg−1	P, kPa, exp.	P, kPa, XW Model	Error, |%|
0.0026	0.0454	0.9520	0.15	2.65	3.56	2.825	2.779	1.63
0.0057	0.0323	0.9620	0.33	1.86	3.84	2.847	2.824	0.808
x(Nd(NO3)3)	x(HNO3)	x(H2O)	m(Nd(NO3)3)	m(HNO3)	Total ionic strength, mol·kg−1	aw, exp.	aw, XW Model	Error, |%|
0.0085	0.0141	0.9775	0.48	0.80	3.68	0.929 ± 0.005	0.927	0.215
0.0098	0.0160	0.9740	0.56	0.92	4.28	0.921 ± 0.005	0.910	1.17
x(Ce(NO3)3)	x(HNO3)	x(H2O)	m(Ce(NO3)3)	m(HNO3)	Total ionic strength, mol·kg−1	aw, exp.	aw, XW Model	Error, |%|
0.0018	0.0018	0.9964	0.10	0.10	0.70	0.9873	0.9907	0.342
0.0089	0.0018	0.9893	0.50	0.10	3.1	0.9657	0.9587	0.722
0.0018	0.0089	0.9893	0.10	0.50	1.1	0.9763	0.9763	0.000
0.00885	0.00885	0.9823	0.50	0.50	3.5	0.9497	0.9394	1.08
0.0018	0.0177	0.9806	0.10	1.0	1.6	0.9610	0.9572	0.395

Table 2. Validation tests for the XW model based on experimental vapor pressure measurements in the H₂O-HNO₃-Nd(NO₃)₃ system at 25 °C from Moiseev et al. [60], experimental water activity measurements in the same system 25 °C from Lalleman et al. [62], and the H₂O-HNO₃-Ce(NO₃)₃ system at 25 °C from Rodriguez-Ruiz et al. [63].

x(Nd(NO3)3)	x(HNO3)	x(H2O)	m(Nd(NO3)3)	m(HNO3)	Total ionic strength, mol·kg−1	P, kPa, exp.	P, kPa, XW Model	Error, |%|
0.0026	0.0454	0.9520	0.15	2.65	3.56	2.825	2.779	1.63
0.0057	0.0323	0.9620	0.33	1.86	3.84	2.847	2.824	0.808
x(Nd(NO3)3)	x(HNO3)	x(H2O)	m(Nd(NO3)3)	m(HNO3)	Total ionic strength, mol·kg−1	aw, exp.	aw, XW Model	Error, |%|
0.0085	0.0141	0.9775	0.48	0.80	3.68	0.929 ± 0.005	0.927	0.215
0.0098	0.0160	0.9740	0.56	0.92	4.28	0.921 ± 0.005	0.910	1.17
x(Ce(NO3)3)	x(HNO3)	x(H2O)	m(Ce(NO3)3)	m(HNO3)	Total ionic strength, mol·kg−1	aw, exp.	aw, XW Model	Error, |%|
0.0018	0.0018	0.9964	0.10	0.10	0.70	0.9873	0.9907	0.342
0.0089	0.0018	0.9893	0.50	0.10	3.1	0.9657	0.9587	0.722
0.0018	0.0089	0.9893	0.10	0.50	1.1	0.9763	0.9763	0.000
0.00885	0.00885	0.9823	0.50	0.50	3.5	0.9497	0.9394	1.08
0.0018	0.0177	0.9806	0.10	1.0	1.6	0.9610	0.9572	0.395

Note that the experiments of O’Brien and Bautista [61] were conducted at three total ionic strengths of 0.6 m, 1.5 m, and 3.0 m as a function of the ionic strength fractions of Nd(NO₃)₃ at Y_B = 0.2, 0.4, 0.6, 0.8, and 1.0. Therefore, the experiments of O’Brien and Bautista [61] are up to 0.5 m for Nd(NO₃)₃ concentrations. The above validation tests using independent experimental data indicate that the XW model is valid and accurate up to ~0.5 mol·kg⁻¹ Nd(NO₃)₃ with total ionic strengths up to ~4 mol·kg⁻¹ in the H₂O-HNO₃-Nd(NO₃)₃ system.

Nuclear waste management programs, especially the disposal of nuclear waste in geological repositories, are regulated by independent regulatory agencies in most countries over the world. Therefore, rigorous quality assurance is essential for model development. Best practices for model development and validation in nuclear waste management include the following: (1) a model must be validated against independent experimental data (i.e., the data not used in model parameterization), and (2) a validated model can only be used for its intended uses within its validated parameter ranges. Testing a model against independent experimental data can be viewed as the last “door keeper” for model quality assurance. For instance, Fanghanel et al. [65] developed a model for Cm(III)–carbonate in NaCl solutions from 0 to 6 mol·kg⁻¹ based on spectroscopic studies, and their model was then validated by independent solubility measurements of NaAm(CO₃)₂ in NaCl solutions at 5.6 mol·kg⁻¹ from [66,67]. In another example, the model for the interactions of Pb(II) with carbonate from Xiong [68] is independently validated by a solubility measurement study of PbCO₃(cr) in HClO₄ + NaClO₄ solutions as a function of pH at constant partial pressures of CO₂(g), and in NaHCO₃ + Na₂CO₃ + NaClO₄ solutions as a function of pH from Bilinski and Schindler [69].

A further example is in Xiong and Wang [15]. In that work, the XW model for a H-Nd-ox-NO₃ electrolyte system had been validated by numerous solubility measurements of actinide oxalates from various research groups [70,71,72,73,74,75,76] (see figures 7–12 in Xiong and Wang [15]). All those solubility measurements of actinide oxalates are independent from the model parameterization in Xiong and Wang [15].

The GLC and OWC models were developed for the concentrated range of Nd(NO₃)₃ solutions, and no validation tests against independent experimental results have been performed by those original authors.

5. Intended Use and Validity Range

The warning about using a model within its validity range is particularly important for an empirical model. After all, all Pitzer models are empirical in nature. Thus, any attempt to extrapolate a Pitzer model beyond its validity range should be performed cautiously. As a bad example of model misuse, Oakes et al. [17] extrapolated the XW model to Nd(NO₃)₃ concentrations up to 1 mol·kg⁻¹ with ionic strengths up to 6 mol·kg⁻¹ (figure 6 in their publication), far beyond the model validity range. Such an extrapolation is misleading and certainly not advised.

Furthermore, in nuclear waste disposal, the intended uses are generally focused on aqueous speciation and solubility calculations for trace-level radionuclides in concentrated background electrolyte solutions (e.g., [39,40,41]). Oakes et al. [17] asserted that Pitzer mixing parameters such as the θ_{H⁺, Nd³⁺} parameter in the XW model must be evaluated at high solute concentrations, based on the assumption that osmotic coefficients can be measurable for all solutes at high concentrations. This is not true for a solute with a sparing solubility-controlling mineral phase, such as Nd(III). It is well known that osmotic coefficients based on isopiestic measurements are normally inaccessible for solute molalities below 0.1 mol·kg⁻¹ (e.g., [77,78,79]). Researchers thus frequently evaluate Pitzer mixing parameters for solutes present at trace concentrations based on data obtained using experimental techniques such as solubility measurements other than isopiestic measurements. For instance, Felmy and Rai [80] evaluated the θ_{H⁺, Th⁴⁺} parameter, based on solubility measurements. In this case, Th(IV) concentrations are below 0.1 mol·kg⁻¹. In other examples, Felmy et al. [81] modeled the θ_{ClO4⁻, Th(CO3)5⁶⁻} parameter, also based on solubility studies in which the solute Th(IV) is 10⁻³ mol·kg⁻¹. In Novak et al. [41], the evaluation of the θ_{CO32⁻, NpO2(CO3)3⁵⁻} parameter is based on solubility measurements in which the solute Np(V) is well below 10⁻³ mol·kg⁻¹. Similarly, the evaluation of the θ_{Cl⁻, NpO2(OH)2⁻}, θ_{Cl⁻, NpO2(CO3)⁻}, θ_{Cl⁻, NpO2(CO3)2³⁻}, and θ_{Cl⁻, NpO2(CO3)3⁵⁻} parameters is also based on solubility measurements in which the solute Np(V) is well below 10⁻² mol·kg⁻¹ in Fanghanel et al. [8]. These classical examples demonstrate that Oakes et al.’s assertion is erroneous, as it is devoid of the understanding that concentrations of source-term radionuclides are limited by sparingly soluble phases to trace levels, and the accurate knowledge of trace activity coefficients that are inaccessible by isopiestic measurements is required in an STM.

6. pH as a Performance Indicator for Model Testing

For electrolyte solutions, pH conditions are one of the most important parameters. As illustrated before [26], the cross-examination of whether a thermodynamic model for an electrolyte solution can correctly predict the pH values of the system is a simple and useful methodology to assess the validity of a model.

Lanthanide oxides/hydroxides including Nd oxide/hydroxide are moderately strong bases (e.g., [82,83]), and nitric acid is a strong acid. Therefore, the pH conditions of electrolyte solutions from a Nd(NO₃)₃ salt, which is a neutralization reaction product from a relatively strong base and a strong acid, are expected to be close to neutral. Indeed, the work of Moeller and Kremers [84] experimentally demonstrated that the pH conditions of REE(NO₃)₃ electrolyte solutions are close to neutral (see the initial pH values for 0.1 mol·dm⁻³ REE(NO₃)₃ electrolytes before titration in figure 1 in [84]). Similarly, other researchers (e.g., [85,86,87]) also observed that the pH values of various LaCl₃ and NdCl₃ solutions including 0.1 mol·dm⁻³ LaCl₃ and NdCl₃ are also close to neutral. Note the hydrolysis of Ln³⁺ in the early series of lanthanides (i.e., light rare-earth elements) would not occur until a pH above 8 (e.g., [88,89,90]). For instance, Wood et al. [90] conclude that “no significant hydrolysis [of Nd³⁺] occurs until approximately pH 8.4…at 30 °C…” (also, see their figure 6 in [90]), based on their well-known experimental studies. Similarly, the experimental work by Deberdt et al. [89] also demonstrates that there is no evidence of hydrolysis of La³⁺ up to pH 9.5 at 40 °C. Therefore, hydrolysis of Nd³⁺ would not affect the pH conditions of Nd(NO₃)₃ electrolyte solutions until a pH value above 8 is attained at ambient temperatures.

For the cross-examination, we performed reaction path calculations to simulate pH_m (see definition below) values as a function of molalities of Nd(NO₃)₃ electrolyte solutions. The calculations were performed using EQ3/6 version 8.0a with three separate databases, one including all parameters from the OWC model in [17] and the relevant parameters in [91], the second one incorporating the GLC model, and the third one incorporating the XW model. A calculation is initiated with trace amounts of Nd³⁺ and NO₃⁻ at 10⁻¹⁰ mol·kg⁻¹ and 3 × 10⁻¹⁰ mol·kg⁻¹, respectively, at pH_m = 7, and then Nd(NO₃)₃ is titrated into the initial solution.

As shown in Figure 4, the OWC model does not correctly predict the pH_m conditions of Nd(NO₃)₃ electrolyte solutions. The pH_m values in Figure 4 are on the Mesmer pH scale [92], and the negative logarithms of hydrogen ion concentrations are on a molal scale. The Mesmer pH scale is appropriate for pH conditions of concentrated solutions. Figure 4 shows that the OWC model predicts acidic pH_m conditions (i.e., pH_m < 4.5 in most of the concentration range) for Nd(NO₃)₃ electrolyte solutions, far from neutral. In contrast, the XW model correctly represents the pH_m conditions of Nd(NO₃)₃ electrolyte solutions. The GLC model also predicts pH_m conditions close to neutral.

To further test the models with independent experimental measurements, the pH_m values for freshly prepared Nd(NO₃)₃ electrolyte solutions are experimentally determined at room temperature (23.4 °C). The protocols for the preparation of Nd(NO₃)₃ electrolyte solutions for the determinations of the pH_m values are detailed in the Materials and Methods section. There is no impurity in the reagent grade chemical used for preparation of Nd(NO₃)₃ electrolyte solutions (see Appendix A, Figure A1, Figure A2, Figure A3, Figure A4, Figure A5 and Figure A6).

The relation between pH_m and pH_ob is expressed in the following equation from Xiong et al. [93],

$$p{H}_m=p{H}_{ob}+A_m=p{H}_{ob}+A_M-\log \Theta$$

(1)

where pH_ob and pH_m are a pH reading and a negative logarithm of the hydrogen ion concentration on a molal scale, respectively, as mentioned above; A_m is a correction factor for hydrogen ion concentrations on a molal scale; A_M is a correction factor on a molar scale; and Θ is a conversion factor from molarity to molality for an electrolyte solution. The determined pH_m values for Nd(NO₃)₃ electrolyte solutions are tabulated in

Table 3. Experimental determinations of pH_m values in concentrated electrolyte solutions at room temperature (23.4 °C).

Electrolyte	Concentration, mol·kg−1	pH Reading (pHob) ± 2σ	Molar Scale Correction Factor, AM A	Conversion Factor from Molarity to Molality, Θ B	Molal Scale Correction Factor, Am	pHm ± 2σ
Nd(NO3)3	0.50	6.20 ± 0.08	0.395	1.117	0.345	6.55 ± 0.08
	0.75	6.05 ± 0.10	0.608	1.174	0.538	6.59 ± 0.10
	1.30	5.92 ± 0.15	1.01	1.300	0.897	6.82 ± 0.15

^A The molar scale correction factors for concentrated brines from Roselle [94] are used to approximate those for Nd(NO₃)₃ electrolyte solutions. ^B The conversion factors are calculated based on the density data for Nd(NO₃)₃ electrolyte solutions are from Spedding et al. [95].

.

Table 3 clearly indicates that concentrated Nd(NO₃)₃ electrolyte solutions have pH_m values close to neutrality, in contrast to the acidic pH_m values predicted by the OWC model in Figure 4. Thus, despite its ability to reproduce an osmotic coefficient over a wide range of ionic strength in the concentrated range, the OWC model does not appear to be valid for the intended use in nuclear management because it fails to provide a reasonable estimate for solution pH conditions, a key parameter for aqueous speciation and solubility calculations. In contrast, both GLC and XW models predict reasonable pH_m values.

7. Physical Constraints on Nd³⁺ Association with NO₃⁻

The OWC model used the Gibbs free energy of the formation of NO₃⁻ from Ref. [96] and converted it to the dimensionless standard chemical potential. However, their fitted stability constants for NdNO₃²⁺ and Nd(NO₃)₂⁺ are contradictory to those of analogous AmNO₃²⁺ and Am(NO₃)₂⁺ from Ref. [96]. Of note, Nd(III) and Am(III) are good analogs to each other, e.g., [97,98,99,100], and Oakes et al. also used this analogy in their previous work [91]. For AmNO₃²⁺, its stability constant is 1.28 in logarithmic units according to Ref. [96]. In contrast, the stability constant of NdNO₃²⁺ in the OWC model is 0.7682 in logarithmic units. For Am(NO₃)₂⁺, its stability constant is 0.88 in logarithmic units from Ref. [96]. In contrast, the stability constant of Nd(NO₃)₂⁺ in the OWC model is 1.810 in logarithmic units. Notice that the fitted values in the OWC model differ from those from Ref. [96] by about half orders of magnitude and one order of magnitude for analogous AmNO₃²⁺ and Am(NO₃)₂⁺, respectively. In other words, the stability constants of NdNO₃²⁺ and Nd(NO₃)₂⁺ in the OWC model are not consistent with the existing well-known experimental values.

Furthermore, regarding the stepwise stability constants, there is a general well-known rule: (log K₂–log K₁) < 0, or ${\displaystyle \frac{K_1}{K_2}}>1$. This rule has a theoretical basis and experimental affirmations, and appears in the standard textbooks and early literature (e.g., [27,28,29,30,31,32,33,34]). In the OWC model, the second stepwise stability constant for Nd(NO₃)₂⁺ (log K₂) is much higher than the first stepwise stability constant for NdNO₃²⁺ (log K₁), resulting in (log K₂–log K₁) = 1.0418 $\gg$ 0, or ${\displaystyle \frac{K_1}{K_2}}=0.09082\ll 1$, obviously violating the above well-established general rule. In contrast, in Ref. [96] for the stepwise stability constants for AmNO₃²⁺ and Am(NO₃)₂⁺, (log K₂–log K₁) = –0.40 < 0, or ${\displaystyle \frac{K_1}{K_2}}=2.51>1$, obeying the rule very well.

Numerous thermodynamic models with the Pitzer formulism satisfy the above well-known rule with respect to stepwise stability constants (e.g., [8,35,36,37,38,43,68]). These studies cover a wide range of stepwise stability constants for the following species: CaCl⁺, CaCl₂(aq), Am(C₂O₄)⁺/Cm(C₂O₄)⁺/Eu(C₂O₄)⁺, Am(C₂O₄)₂⁻/Cm(C₂O₄)₂⁻/Eu(C₂O₄)₂⁻, PbCl⁺, PbCl₂(aq), PbCO₃(aq), Pb(CO₃)₂²⁻, PbC₂O₄(aq), Pb(C₂O₄)₂²⁻, UO₂C₂O₄(aq), UO₂(C₂O₄)₂²⁻, NpO₂OH(aq), and NpO₂(OH)₂⁻.

Moreover, as the stability constant of Nd(NO₃)₂⁺ is very high in the OWC model, its parameterization depresses molal contributions from Nd(NO₃)₂⁺ by having abnormally high activity coefficients for Nd(NO₃)₂⁺, as described by the following equation

$$a_{Nd(N{O}_3{{)}_2}^{+}}=m_{Nd(N{O}_3{{)}_2}^{+}}\times {\gamma }_{Nd(N{O}_3{{)}_2}^{+}}$$

(2)

Since $a_{Nd(N{O}_3{{)}_2}^{+}}$ is controlled by the stability constant of Nd(NO₃)₂⁺, and is roughly constant, the increase in ${\gamma }_{Nd(N{O}_3{{)}_2}^{+}}$ will induce the corresponding decrease in $m_{Nd(N{O}_3{{)}_2}^{+}}$.

As illustrated by Figure 5, according to the OWC model, activity coefficients for Nd(NO₃)₂⁺ are abnormally high, up to ~900 as the increase in Nd(NO₃)₃ concentrations. Such extremely high activity coefficients present a severe problem for the validity of the OWC model, clearly contradictory to guiding principle 3.

In comparison, the GLC and XW models do not have a Nd(III)-NO₃ association schema; therefore, they do not have those problems, thus exactly following guiding principle 3.

8. Minimizing the Number of Fitted Model Parameters

In the XW model, there are only two parameters that are used to describe the interactions of trace Nd(III) concentrations with NO₃⁻, i.e., β⁽⁰⁾ and β⁽¹⁾ for Nd³⁺—NO₃⁻.

The GLC and OWC models are for the concentrated region of Nd(NO₃)₃ electrolyte solutions. The GLC model was published prior to the OWC model. In the GLC model, there are five fitting parameters to describe the interactions of concentrated Nd(III) concentrations with NO₃⁻, i.e., β⁽⁰⁾, β⁽¹⁾, β⁽²⁾, C^ϕ, and α₂ for Nd³⁺—NO₃⁻. Oakes et al. [17] criticized the GLC model for two aspects: (1) α₂ as a fitting parameter, and (2) a positive value of β⁽²⁾ for Nd³⁺—NO₃⁻. Since a Pitzer model is overall empirical, as a fitting parameter, β⁽²⁾ can be either negative or positive in sign. In other publications from the GLC group, the β⁽²⁾ values for FeCl₂, FeCl₃, and Fe₂(SO₄)₃ electrolytes were –0.34439, 1.7199, and 3.07519 in [101], respectively, when the the model was fitted to the osmotic coefficients of the above electrolytes. In those cases, positive or negative values for β⁽²⁾ were determined. Other researchers, e.g., [102,103], also treated α₂/β⁽²⁾ as fitting parameters, and positive values for β⁽²⁾ were also obtained at 25 °C. Finally, Pitzer himself [104] was open to the models in which β⁽²⁾ has positive values (see his table H-1). Hence, the criticisms of the GLC model by Oakes et al. [17] seem to be questionable.

In the OWC model, there are nine fitting parameters to describe the interactions of concentrated Nd(III) concentrations with NO₃⁻, i.e., β⁽⁰⁾, and β⁽¹⁾ for Nd³⁺—NO₃⁻; β⁽⁰⁾ for NdNO₃²⁺—NO₃⁻; β⁽⁰⁾ and β⁽¹⁾ for Nd(NO₃)₂⁺—NO₃⁻; λ for NdNO₃²⁺—HNO₃⁰; λ for Nd(NO₃)₂⁺—HNO₃⁰; a stability constant for NdNO₃²⁺; and a stability constant for Nd(NO₃)₂⁺.

The five-parameter GLC model is a competing model with the OWC model, since both are for concentrated Nd(NO₃)₃ electrolyte solutions. In this sense, Oakes et al. [17] should have included the GLC model in their model comparison (their figure 6) but failed to do so. To illustrate guiding principle #4, we here examine both models for the reproduction of the osmotic coefficients of Nd(NO₃)₃ electrolyte solutions (Figure 6).

As shown in Figure 6, the five-parameter GLC model has an accuracy identical to the nine-parameter OWC model. In the OWC model, there are nine parameters related to Nd(III) plus six parameters related to NO₃⁻ with fifteen parameters in total. Note that the association constants for Nd(NO₃)²⁺ and Nd(NO₃)₂⁺ in the OWC model must be counted as fitting parameters. According to the fourth guiding principle, the GLC model is superior to the OWC model in the concentrated region because the former can fit osmotic coefficient data equally well with fewer fitting parameters, and also because the former does not have the problems with the Nd(III)-NO₃ associations assumed in the latter.

9. Summary

A source-term model for radionuclide solubility prediction is important to the performance assessment calculations for geological disposal of nuclear waste. The development of a valid source-term model is the key to the success of a nuclear waste disposal program. In this article, four key guiding principles for model development and the validation of a source-term model are presented and illustrated by a close examination of three recent models for the H⁺—Nd³⁺—NO₃⁻(—oxalate) system. The four key guiding principles are (1) validation against independent experimental results or field observations; (2) cross-checking with independent variable(s) such as pH values; (3) honoring the physical constraints; and (4) minimizing the number of fitted parameters. The close examination of the GLC and OWC models, both in concentrated Nd(NO₃)₃ solutions, indicates that the former is satisfactory, while the latter is unsatisfactory in meeting the above requirements and therefore unsuitable to being a defensible source-term model. The XW model, for the trace activity coefficients of Nd(III), is satisfactory in meeting the above requirements, and therefore can be used for a PA calculation.

• Materials and Methods

The Nd(NO₃)₃ electrolyte solutions were prepared from reagent-grade Nd(NO₃)₃·6H₂O (LOT # MKCT7713) from Sigma-Aldrich Chemicals (Sigma Aldrich Chemicals, 3050 Spruce Street, St. Louis, MO, USA) with degassed DI water at laboratory room temperature at 23.4 °C. The pH readings (pH_ob in Equation (1)) were measured using a Mettler Toledo SevenCompact pH/Ion S220 pH meter (Mettler-Toledo AG, 8603 Schwenzanbach, Switzerland). Before the measurements, the pH meter was calibrated with three OAKTON^® pH buffers at pH 4.01, pH 7.00, and pH 10.01 from Pole-Parmer Company (Pole-Parmer Company, 625 East Bunker Court, Vernon Hills, IL, USA).