Method to Study the Primary Nucleation for Solid Solution : Application to Uranium-Neodymium Oxalate Coprecipitation

Actinides co-precipitation is currently investigated in order to synthesize solid solutions of actinides mixed oxalates. This paper deals with the thermodynamic and kinetic study of the precipitation of uranium-neodymium oxalate system. Based on an analysis of the theories developed in the literature, a new expression for the determination of the supersaturation ratio for the solid solutions is presented. An experimental study of the nucleation kinetics was performed on the mixed uranium-neodymium oxalates. Homogeneous and heterogeneous primary nucleation laws are obtained using a specific stopped flow apparatus. The experimental results are consistent with the classical behaviour of nucleation phenomena. The values of the kinetic parameters for the solid solution point out that the formation of the uraniumneodymium mixed oxalates is kinetically favoured compared with the simple uranium and neodymium oxalates.


Introduction
Current concepts for future nuclear systems aim at improving the fuel cycle with the co-management of actinides in order to enhance the fuel performance and to reduce the proliferation risk.The actinide co-conversion processes, such as COEX TM process [1], play an important role by producing mixed actinide compounds used as starting materials for fuel re-fabrication.The co-conversion of actinides into oxide can be achieved by different processes such as oxalic co-precipitation, thermal co-denitration or co-gelation.The Oxalic co-precipitation is one investigated way to synthesize solid solutions of actinide mixed oxalates which have to meet strict requirements [2].Thus to support the process development, the determination of mechanisms and kinetic laws associated with the co-precipitation appears to be a key issue.This article focuses on the kinetic study in order to determine the primary nucleation rate for the uranium-neodymium oxalate system, which simulates the uranium-actinide system behaviour.
Compared with simple salts, mechanisms involved in the formation of mixed crystals are much more complex and remain poorly understood.The phenomenological theories are not still clarified in the literature.Equilibrium conditions for a solid solution-aqueous solution (SS-AS) system can be described using the approach developed by Lippmann who proposes an extension of the solubility product concept [3].Several both experimental and theoretical studies have recently focused on the thermodynamics of mixed crystals [4][5][6][7][8][9][10][11][12][13] contrary to the kinetic studies which are reported in very few papers only [14,15].The key point in the precipitation of solid solutions is the definition of the driving force.All authors agree to describe the difficulty in determining a rigorous expression of the supersaturation which is a fundamental crystallization parameter.The difficulty arises from the need to express supersaturation using the chemical potentials of components in both solid and aqueous solutions which are mutually independent.The way that the driving force should be defined is still controversial [16,17].Shtukenberg et al. [11] propose to evaluate the supersaturation using Lippmann's approach, whereas Astilleros et al. [8] developed two formulations, the first one based on the stoichiometric saturation concept [18] and the second one based on the actual activity concept [8].
This study focuses on the uranium(IV)-neodymium(III) oxalate solid solutions described by the following precipitation reaction, where can vary from 0 to 0.5 [2]: The solid solutions composed of U(IV)-Nd(III) oxalates are complex as the stoichiometric coefficients are different for each component.Solid solutions usually described in the literature have simple formula such as (Ba, Sr)SO 4 or (Ba, Sr)CO 3 [19][20][21] where cations and anions have the same charge, while in our case (see Equation ( 1)) the formation of the solid solution is much more complex and needs a more profound investigation not previously encountered in the literature.Thus, the aim of this work is to study experimentally the formation of this kind of complex solid solutions and to find adequate nucleation rate expressions in order to use them in modelling of precipitation processes in nuclear industry.The first part of this paper is concerned with the discussion of the Lippmann model in the case of complex solid solutions.A new suitable expression for the relative supersaturation ratio is then presented.Section 3 describes the experimental study of the nucleation kinetics performed on the mixed uranium-neodymium oxalate.The experimental results obtained are presented in section 4 and are confronted with the new model of the supersaturation.

Supersaturation Concept in SS-AS Systems
In precipitation, the supersaturation concept describes in a quantitative way the deviation of a given aqueous solution from the equilibrium state.For a pure solid of the form BA, the solubility product is defined as the product of activities of species in solution at equilibrium conditions: , , , where K s is the solubility product of BA, a A and a B the activity of the ions A z− and B z+ in the liquid phase, a AB the activity of BA in the solid phase and z the positive and negative charge of ions, eq means at equilibrium state.The driving force of the nucleation process, the relative supersaturation ratio, can be calculated by the wellknown general expression: where S is the supersaturation ratio, the product of the ionic activities and υ i the stoichiometric number of ion i.
Regarding the solid solutions, the application of the supersaturation concept (for example in the case of the coprecipitation of BA and CA compounds, see Equations ( 4) and ( 5)) requires first to define the equilibrium conditions.

Lippmann's Approach
Lippmann [3] proposed to extent the solubility product concept to the solid solution based on an analogy to liquid-vapour systems.We consider a solid solution-aqueous solution system where the cations B z+ and C z+ and the anions A z− are presented in the aqueous phase and react to lead to the solid solutions     formed by the pure end-members BA and CA.The equilibrium conditions between the solid solution and the aqueous solution are given by two expressions provided by the following reaction schemes: , , where K 1 and K 2 are the solubility product of the endmembers, X 1 and X 2 the molar fractions of endmembers in the solid solution, a A , a B and a C the activities of the ions A z− , B z+ and C z+ in the aqueous solution, a BA and a CA the activities of BA and CA in the solid solution, C A , C B and C C the concentrations of the ions A z− , B z+ and C z+ , γ 1 and γ 2 the activity coefficients in the solid phase and γ ±1 and γ ±2 the mean activity coefficients in the liquid phase.These reactions are coupled only through the mass balance that implies: We note BA the endmember 1 and CA the endmember 2 with conventionally 1 2 . According to Lippmann, in the same way as the total vapour pressure of a liquid mixture is the sum of the partial vapour pressure of each component, the total solubility product can be defined as the sum of the partial solubility products of each endmember of the solid solution: where is the total solubility product.Using the Lippmann approach, the driving force can be expressed as follows [11]: where S is the supersaturation ratio depending on X.
In the literature, this approach described by Lippmann was applied by several authors for bivalent cations and anions and the considered systems can thus be described by Equations ( 4) and (5) where 2 z z     .The classical systems studied were (Ba, Sr)SO 4 , (Ba, Sr)CO 3 , (Ca, Mg)CO 3 , etc.There are however more complex solid solutions in which the endmembers present different stoichiometric coefficients: this specially applies to the nuclear industry where different actinide solid solutions are formed by coprecipitation [22][23][24].
For solid systems with endmembers of different stoichiometry, the Lippmann formalism needs to be adjusted.To illustrate these cases, we consider now a simple SS-AS system where the cations B + and C 2+ and the anion A 2− are presented in the aqueous phase and react to lead to solid solutions formed by the pure end-1 2

X X
B C A members B 2 A and CA.The equilibrium conditions of this binary system are as follows: B eq A eq B eq A eq C eq A eq C eq A eq According to the Lippmann proposition, the total solubility product would be obtained by adding Equations ( 9) and ( 10 This equation is not adequate as it is composed of terms with different dimensionalities: the first endmember solubility product K 1 is expressed in (mol•m −3 ) 3 , whereas the second endmember solubility product K 2 is in (mol•m −3 ) 2 .This example points out that the Lippmann model for a complex solid solution cannot be directly applicable since it fails to satisfy the fundamental dimensionality principle.To our opinion, the Lippmann theoretical approach is right, but his mathematical approach needs to be generalized.We accept the Lippmann similarity between liquid phase-vapour phase equilibrium and solid solution-aqueous solution equilibrium, but a new mathematical expression consistent with equilibrium relations can be proposed.For a liquid-vapour equilibrium, assuming a thermodynamically ideal liquid solution, the total vapour pressure is given as follows: 1, 1 2, 2 eq eq eq P P while Equation (7) proposed by Lippmann is equivalent to the following expression: which is not a correct equation.In these equations, and are the partial 1,eq P 2,eq P vapour pressure, X 1 and X 2 the molar fractions of the components and P eq the total vapour pressure of the liquid mixture.Whatever the system considered, the terms added in Equation ( 12) have always the same dimensionality as a pressure.The concentration dimensionality is directly proportional to a pressure.Therefore, to be consistent with the vapour-liquid system, the total equilibrium term should have always the same dimensionality as a concentration and should be defined as the sum of terms with a concentration dimensionality (in thermodynamics, in the domain of multi-component phase equilibrium, only terms having pressure or concentration dimensionalities in power 1 can be additional terms).
In addition, if we consider the expression of the supersaturation ratio defined by Shtukenberg et al. [11] (see Equation ( 8)), we can notice that for the pure endmembers this expression doesn't converge to the general expression of the supersaturation ratio (see Equation ( 3)).

New Expression
Based on these considerations, we propose a new expression to determine the total solubility term at equilibrium.For a system defined by Equations ( 4) and ( 5), the new formulation is: Based on equilibrium Equations ( 4) and ( 5), it follows: Therefore, the absolute supersaturation will b and the supersaturation ratio: We can also note that the Expressions ( 16) and ( 17) co see Eq (3)).s lowing ction nverge to the right expressions for the single component systems when 1 X  or 0 X  , contrary to Lippmann's approach ( uation More generally, for any kind of solid olution of the form obtained from the endmembers , 11 , 22 Using the new expression, the total solubility term and the supersaturation ratio are defined as follows:

Other Expression of the Supersaturation
a Applying the concept of the stoichiometric saturation to solid solution [18] and the Lippmann approach [3], Pina et al. [7,9,19] derived the supersaturation ratio expression based simultaneously on equilibrium relations of Reactions ( 3) and ( 4) and on the formation of the solid solution according to the single reaction: Using equilibrium Equations ( 3), ( 4) and ( 22), they pr The weak point of the Putnis et al.'s formalism [20][21][22][23][24][25] is the dualism used to obtain the equilibrium expressions.Indeed the authors accept simultaneously the solid phase as composed of two endmembers BA and CA (Reactions (3) and ( 4)) and as composed of a single compound B X1 C X2 A (Reaction ( 22)).We have compared our formalism proposed in this study with the one proposed by Putnis et al. in the case of the coprecipitation of (Ba, Sr)SO 4 where 4 SrSO 2 is three orders of magnitude higher . The calculations of the supersaturation r alues ranging from 0 to 1 (not shown atio for X 1 v here) give a maximum difference of 2.4 times higher for our expression than for the Putnis et al. expression.This difference is very sensible to the ratio 2 1 K K .When K 1 and K 2 are of the same order of magni hich is the tude (w and case in the nuclear industry), the values of supersaturation ratio obtained by the two approaches are almost the same [26].Here, we can also note that the mathematical calculations based on the Putnis et al. formalism are much more complex than those corresponding to the formalism we propose.For all these reasons, in this article, only the new expression based on the analogy be-tween liquid-vapour equilibrium and SS-AS equilibrium is used to calculate the supersaturation.

Determination of the Supersaturation Ratio Based Only on Liquid Phase Composition is ca d f endmembers BA and CA in th
As shown in Equations ( 16) and ( 17), the supersaturation lculated as a function of the composition of the soli phase X.All authors in their studies have used the molar composition X to express the supersaturation ratio.However, during the precipitation reaction, the variation of the liquid phase composition is easily measured or known, while the composition of the solid phase can be known only at the end of precipitation process.For this reason, it is interesting to express the supersaturation based only on the composition of the liquid phase.Let us illustrate it using first the simple case expressed by Equations ( 4) and ( 5) for which the supersaturation ratio is given by Equation (17).
Considering a solid solution defined by Equations ( 4) and ( 5), the mole fractions o e solid solution are given by: 2 Knowing that the conc actant A can be deduced from Equations ( 24) and (25): A eq The total solubility term expressed as a function of the liquid composition is given by Equation (15).lead to the relationship between the mole fractio solid phase X 1 and the mole fraction in n of the the liquid phase x 1 : where x 1 and x 2 are the mole fractions of the endmembers in the liquid phase at equilibrium.

same aqueous activity fr
According to the Lippmann phase diagram, the aqueous solution at equilibrium has the action as the supersaturated solution [3,8,11].Consequently, x 1 and x 2 are defined as: 27) is very important sin tion liquid and solid phase composit pr (29) ce it links the relaions at any time of ecipitation reaction.Finally, from Equation ( 27), the analytical solution of the relative supersaturation is obtained as a function of the composition of the aqueous solution only: Let us use the same methodology to find the supersaturation ratio expression in our more complex case.A series of solid solutions can be synthesized by mixing an acid solution of uranium(IV) and neodymium(III) nitrate with an oxalic acid solution [22][23][24][25][26][27].The solid solutions are formed by the following isostructural endmembers: Endmember 1: Endmember 2: The mixed oxalates can be obtained only under specific experimental conditions used in the nuclear i [2,23].Three series of mixed lanthanide(III) and actini ndustry de(IV) were studied by Chapelet-Arab et al.: the hexagonal, the triclinic and the quadratic series.The originality of these structures is the existence of a mixed crystallographic site for tetravalent actinide and for trivalent lanthanide or actinide.The charge balance depending on the Ln III Ln III + U IV mole ratio is ensured by the adjustment of single charged ions within the structure The solubility products of the endmembers 1 and 2 were experim tally determined by Costenoble under ex itions for obtaining the solid so en perimental conditions encountered in nuclear industry given in Table 1 [23][24][25][26][27][28].
The precipitation reactions of the endmembers are: Hydrazine (mol•L ) 0.10 In the case of a solid solution formed by the copre cipitation of the above endmembers, the equilibrium relations are: K a a a a As discussed before (Equation ( 20)), the total solubility term can be written as: Π i eq eq eq eq eq eq eq eq eq a a a a a a a a or: i eq eq eq eq eq eq eq eq eq eq K X K X Based on these Equation (41), the supersaturation ratio during the coprecipitation is: S X In Equation (41), the concentrations of all compounds in the liquid phase are known, while the solid phase co position X is not directly known, but must be determ of the liquid phase composition, since only this one isknown at any time of coprecipitation process.The compositions (molar fractions of endmembers) of the liquid m ined by indirect complex calculations.As noted before, the supersaturation must be expressed only as a function and solid phases are defined as follows: Number of moles of endmember 1 in the solid phase total number of moles of endmembers 1 and 2 in the solid phase Number of moles of endmember 2 in the solid phase 1 total number of moles of endmembers 1 and 2 in the solid phase X X    From Equations ( 6), ( 37) and ( 38 Nd eq eq eq eq eq eq K K i eq eq eq eq eq eq eq eq eq ,1 eq eq eq eq eq eq eq eq eq eq eq eq K K eq eq eq eq eq eq eq eq eq eq eq K K

H O N H H O Nd
U eq eq eq eq eq eq K K From the definition (42): We can obtain: i eq eq eq eq eq eq eq eq eq x K K x x and the supersaturation ratio as a function of the liquid phase composition only: In addition, we can notice that the equalization of Equation ( 50) with the right term of Equation (40) gives the cumbersome implicit equation for the determination of the solid phase composition at equilibrium with the liquid phase composition (X versus x).

Experimental
The nucleation kinetics is studied using a stopped flow apparatus patented by the CEA/AREVA shown in Figure 1 [29].
This apparatus is equipped with a Ha confined opposing jets m by very high micromi times of the order of one millisecond can be achieved [30].
The apparatus, made of stain ticularly appropriate to hostile environment, due to the simplicity and the experimental set up compactness.It requires a small solution volume and no unintentional release is possible.With all these safe use properties, this apparatus is well adapted to the radioactive material handling in glove boxes [31].
The block diagram of the stopped flow apparatus is represented in Figure 2. The thermostated tanks are supplied with reagents, 5 mL of each: uranium and neodymium nitrate solution and oxalic acid solution.The contact of the reagents is performed by two pistons, based on a crossbow mechanism.The pistons ensure a rapid and less steel material, is par-

si
The duration of the nucleation process is about 0.4 s.A ilution is performed at the tube outlet in a 500 mL vessel initially filled with nitric acid, oxalic acid and hydrazine solution.The role of this quench is to stop the nucleation process with a quick drop of the supersaturation.The dilution vessel is stirred by a 4-blade Rushton turbine at 1500 rpm with 4 baffles.As recommended by Nielsen [32], gelatine is introduced in a concentration of 2 g•L −1 in order to limit the agglomeration phenomena.The tube is placed in the discharging jet of the turbine to guarantee a quasi-instantaneous dilution.The solid particles formed can grow during 15 min to reach a detectable size range and the suspension is analysed by laser diffraction granulometry using a Coulter Counter apparatus.A preliminary study on the induction time [26] confirms that only nucleation takes place in the tube, as the resi nning Electron Microscopy (S multaneous injection of the reagents in the nucleation tube.The nucleation occurs in the nucleation tube with a length of 10 cm and a diameter of 2 mm.The residence time is about 10 ms.Two optical detectors linked to an electronic system allows the nucleation process time in the tube to be measured.d dence time in the nucleation tube (10 ms) is lower than the induction period for all supersaturation values.The nuclei number retained corresponds to an average between 3 analysed samples.The nucleation rate is then determined using the total number of nuclei formed and the nucleation period measured with precision in each experiment.The method is well described in [31].The crystals are observed by Sca EM) and analysed by X-Ray diffractometry in order to be certain that the desired solid solution is formed.
A tetravalent uranium nitrate solution is purified on ion exchange resins in order to separate uranium(IV) from uranium(VI).In this way, pure solution containing 99% uranium(IV) is obtained and then characterized by UV-visible spectrophotometer (Shimadzu UV 2550).Uranium(IV) ions can be oxidized by nitrous acid (HNO 2 ) which is presented in small quantities in nitric acid media.For this reason, hydrazinium nitrate is added in order to stabilize the tetravalent uranium nitrate solution for several months.Neodymium nitrate hexahydrate (99.9%,Alfa Aesar) is dissolved in the uranium nitrate solution with a determined

Analysis of Mixed Oxalate Crystals
The SEM photographs validate the formation of mixed oxalates and the absence of agglomeration (see Figure 3).
The particles are hexagonal rods which are the characteristic morphology of the solid solutions formed by the precipitation Reaction (1) [23].
In addition, careful X-Ray diffractometric analyses are performed.As it is shown in Figure 4, the X-Ray spectrum confirms the formation of pure mixed oxalate as it is consistent with the PDF card 04-11-8368 [22].
The composition of the solid phase is also analysed on different individual particles.The results lead to a homogeneous composition of individual particles.

Endmembers Solubility
The solubility products of endmembers are experimentally determined under the experimental conditions re-   ported in Table 1.The first step was to synthesize the two endmembers.Then solubility measurements are per formed at room temperature in different vessels to stud the dissolution of the co-precipitates as a function of time.
When equilibrium is reached, the uranium(IV) and neodymium(III) concentrations are analysed by spectrophotometry and ICP-AES [28].In his thesis, Costenoble remarked that the solid solutions of uranium-neodymium oxalates are the most probably ideal solid solutions due to their structural similarity, so y     .On the contrary, the activity coefficients in solution were impossible to be determined, as no data are available.For instance, the individual contributions of hydrazinium and oxalic ions are not known for the calculation of activity coefficients according to the Bromley method.From Equations (35) and (36), it can be written: (51) Measuring the concentrations of all compounds in solution, Costenoble [24] has determined the values of the eq eq eq eq eq eq eq eq eq a a eq eq eq eq eq eq eq eq eq eq eq K a a a a a   4.37 10 mol L where 1 K and 2 K  are the solubility apparent constants.The supersaturation ratio is calculated by Equation (50) using 1 K and 2 K  values.In addition, the coprecipitation is performed in concentrated solutions of ions C O , so the ionic strength remains constant and then the activity coefficients in solution vary slightly during the precipitation proces this reason, the supersaturation ratio can be calculated assuming s.For

Determination of the Nucleation Kinetics
The primary nucleation kinetics of uranium and dy neomium mixed oxalates is studied for different fixed

 
Nd U Nd  ratio in solution (see Table 2) by varying their concentrations, so by varying the supersaturation ratio.In all published papers on this subject, the authors considered the primary nucleation kinetic law of solid solutions using the similar formula as for a single c pound [33]: All our experiments proved the validity of this equation.For illustration, the experimental data and Equation (55) are shown in Figure 5 for Using a non-linear regression technique on 6.40 0.4 1.86 10 exp ln  The authors gratefully acknowledge AREVA for its fihe authors are also thankful to the The val e kinetic parame t out that aqueous m tions ar of Nd and U mely favoure ompared with their si ates, which is interesting nomeno T th results obtained in this n be compared results obtained for the nucleation rates of sim anium oxalate and of simple neodymium oxa co-precipitation process.These kinetic results can be combined with a hydrodynamic model in the same way as it is done for the oxalic precipitation of uranium oxalate [34].

Acknowledgements
he data nential factors A are higher, while the exponential coefficients B are lower for solid solutions compared with simple uranium and neodymium oxalates.
Particularly interesting is the fact that the energetic barrier (proportional to B) of solid solutions is much lower than the energetic barrier of simple oxalates.All these data point out that the oxalate solid solutions are kinetically favoured compared with simple oxalates.nancial support.T Metallographic and Chemical Analysis Laboratory from CEA for the SEM photos and the DRX spectrum.
Generally, it also proves the easiness to obtain solid solutions in nuclear industry when oxalic ions added to a mixture of lanthanide and actinide ions in nitric acid media.

Conclusion
This paper describes how to obtain new expressions for the determination of equilibrium relation and supersaturation ratio for any solid solution-aqueous solution (SS-AS) system.An adequate nucleation rate expression is proposed in order to study the co-precipitation processes in nuclear industry.This approach is successfully applied to the uranium-neodymium oxalates co-precipitation.The experimental method developed to study the primary nu- 26) in Equation(15) and dividing the numerator and the denominator by 

Figure 2 .
Figure 2. Principle of the experimental setup for the nucleation kinetic study.
acid solution with a concentration of 0.7 mol•L −1 is prepared from oxalic acid dihydrate (Normapur, >99.5%, VWR Prolabo).Deionised water is used in all experiments.The experiments are performed at room temperature under experimental conditions shown in Tables1 and 2.

Figure 3 .
Figure 3. SEM photographs of uranium and neodymium mixed oxalates.
activity coefficients can be simplified in Equation (50).

Figure 5 .
Figure 5.Primary nucleation kinetics for the

Table 3
prove that the corresponding pre-expo the oxalate are cleation of solid solutions leads to satisfactory results although the expression of supersaturation ratio is complex.The experimental set up is simple and it is easy to ous-Solution E a-Thermodynamic Theory and Representa erican Journal of Science, Vol.290 No. 2, 1990, pp. 164-201.doi:10.2475/ajs.290.2.164