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Computational Materials Science

journal homepage: www.elsevier.com/locate/commatsci

A physics-based mesoscale phase-field model for predicting the uptakekinetics of radionuclides in hierarchical nuclear wasteform materials

Y.L. Lia, B.D. Zeidmana, S.Y. Hua,⁎, C.H. Henager Jr.a, T.M. Besmannb, A. Grandjeanca Pacific Northwest National Laboratory, Richland, WA 99352, USAbDepartment of Chemistry and Biochemistry, University of South Carolina, Columbia, SC 29208, USAc CEA, DEN, Univ Montpellier, Research Department on Enrichment, Dismantling and Waste Technologies, SEAD/LPSD, Marcoule, BP 71171, 30207 Bagnols-sur-Cèzecedex, France

A R T I C L E I N F O

Keywords:Phase-field modelIon exchangeHierarchical zeolitePorous solidsNuclear wasteforms

A B S T R A C T

Multiscale, hierarchical, porous materials are promising nuclear wasteform materials with potential for effi-ciently adsorbing and sequestering radionuclides. However, fundamental understanding of such complex micro-and meso-structures on radionuclide diffusion and uptake kinetics is lacking, which is essential for designingadvanced nuclear wasteform materials. In this work we develop a microstructural-dependent diffusion modelthat accounts for three-dimensional (3D) microstructural features, including heterogeneous thermodynamic andkinetic properties, to predict radionuclide uptake kinetics in complex porous structures. Na+ and Sr2+ ionicexchange in LTA zeolite multiscale materials in batch tests is taken as a model system to demonstrate the modeland its application to hierarchical materials with multiscale porosity. It is found that the reduction of ionicmobility associated with chemistry, structure, and/or phase changes has more profound impact on the uptakekinetics in a large 3D porous particle than that in a small one. Uptake kinetics in large particles are limited bytwo diffusion processes: bulk diffusion and surface layer transport. Decreasing particle size and increasingmesoscale pore volume fraction changes the uptake kinetics from two diffusion processes to a single process anddramatically increases the uptake kinetics. Predicted uptake kinetics and the effect of microstructures on theeffective capacity of radionuclides and uptake efficiency are consistent with results observed in Na+/Sr2+ ex-change experiments. The results demonstrate that the developed model should be adaptable to transport pro-blems in other hierarchical material systems.

1. Introduction

A number of advanced nuclear wasteform materials have been de-veloped to selectively extract radionuclides from different wastestreams (liquid or gaseous) [1,2]. For example, a porous silica or glass-based material (ferrocyanide compounds, zeolitic structures, and silver-nitrate phases), surface functionalized nanoparticles (hexacyanoferrate(HCF) nanoparticles, and multi-metallic nanoparticles) salt inclusionmaterials (SIMs), and metal-organic frameworks (MOFs), are currentlythe most widely studied materials to capture and stabilize specificradionuclides. However, nuclear waste management generally over-looks nuclear wasteform materials with high performance (i.e., radio-nuclide capacity, uptake efficiency, and structure stability) that can beused directly as raw materials for a specific containment stream. Ex-periments show that the radionuclide capacity and efficiency of theuptake process strongly depend on wasteform multi-scale hierarchical

structures [3–7]. A number of factors, including inhomogeneous che-mical potentials and mobility of radionuclides in different phases andon interfaces or grain boundaries, the reduction of ionic mobility as-sociated with ionic exchange-induced structure distortion, chemistrychange and phase transition, and the effect of multiscale pore structureson fluid field and concentration field, may affect ionic diffusion, hence,wasteform material performance. To optimize wasteform structures forhigh capacity and efficient uptake, it is crucial to understand the me-chanisms of ionic diffusion and to predict the effect of hierarchicalstructures and multi-physics interactions on ionic diffusion, sorptioncapacity, and ion uptake kinetics.

Multiscale modeling and simulation, which have emerged as valu-able approaches for the development of advanced materials [8–11],have been applied to both framework and porous materials. DensityFunctional Theory (DFT) can be used to analyze structures and bondingof cations and water within typical framework molecules, such as

https://doi.org/10.1016/j.commatsci.2018.11.041Received 12 October 2018; Received in revised form 20 November 2018; Accepted 21 November 2018

⁎ Corresponding author.E-mail address: shenyang.hu@pnnl.gov (S.Y. Hu).

zeolites [12,13]. Molecular Dynamics (MD) simulations are used tostudy free energies, cation thermodynamics, and kinetic properties[14–16]. Monte Carlo (MC) techniques allow calculation of cation se-lectivity and equilibrium ionic exchange properties [17–19], and ther-modynamic calculations can assess the effect of porosity on the freeenergy and phase stability of porous structures [20–22]. With knowl-edge from atomic-level simulations and thermodynamic calculations ofphase stabilities, the mesoscale phase-field (PF) method offers a pro-mising modeling tool to predict the effects of 3D microstructures andprocessing parameters on microstructure evolution and material per-formance in radionuclide uptake processes [10,23].

In this work we use the PF method to develop a microstructuredependent diffusion model for ion exchange in hierarchically porouszeolite structures. The model accounts for 3D hierarchically porousmicrostructures and inhomogeneous thermodynamic and kineticproperties, which allows the study of the radionuclide uptake kineticsin these complex structures. PF models of spinodal decomposition andgrain growth are used to computationally generate hierarchicallyporous structures as our model structures. Na+ and Sr2+ ion exchangein Na-LTA (Sodium Linde Type A) zeolite monoliths in batch uptaketests are taken as model systems to demonstrate the model and its ap-plications.

2. Description of multi-species diffusion in hierarchical materials

LTA zeolite monoliths typically exhibit a hierarchical porousstructure consisting of (1) a SiO2 skeleton with micron-scale macro-scopic pores; (2) aggregates of zeolite crystalline particles with nano-sized mesoscale pores; and (3) zeolite crystals with 4 Å microscalechannels [24]. Fig. 1 shows a typical LTA zeolite microstructure.

When a liquid waste stream containing radioactive ions flowsthrough a porous structure ion exchange can occur. In the case of a Na-LTA zeolite monolith and a Sr-containing waste stream, Na+ inside thezeolite crystals diffuses out while Sr2+ in the waste stream diffuses intothe zeolite crystals. This work mainly focuses on the effects of micro-and meso-structures and heterogeneous thermodynamic and kineticproperties on the Sr2+ (or other ion species) uptake kinetics during anion exchange process. For simplicity, we consider a batch test situationso that the effects of fluid flow in the multi-scale porous structure onSr2+ uptake kinetics are ignored for this work, although the PF methodcan be used to describe fluid flow via the Navier-Stokes equations[27,28]. It is also assumed that diffusion of the respective ions (Na+

and Sr2+) is charge neutral. The charge neutrality is enforced by che-mical potentials ensuring that one Sr2+ replaces two Na+ in the zeolitecrystals. With these assumptions, the microstructure and the in-homogeneous thermodynamic and kinetic properties of the system as

well as the ionic diffusion and microstructure evolution can be de-scribed by two sets of PF variables (see Fig. 2). One is the order para-meter field =η i, ( 1, 2, 3),i which describe zeolite, SiO2, and solutionphases, respectively. The order parameter field either represents theatomic structures of different phases or different crystal orientations ofa phase [23,29]. Variable =η i( 1, 2, 3)i is, respectively, equal to 1 in-side phase i and 0 outside phase i, and varies smoothly from 1 to 0across the interfaces between different phases or the grain boundariesbetween different zeolite crystalline particles. The other variable is theconcentration field =c i, ( 1, 2),i which describe the spatial distributionsof Na+ and Sr2+, respectively, within each phase.

The exchange of Sr2+ with Na+ or diffusion of Na+ and Sr2+ in thehierarchical structure is described by the following equations:

∂ ∂ = ∇ ⎛⎝ ∇ ⎞⎠ = =+ +rc tt

D µRT

i i( , ) · ( ) , 1, Na ; 2, Srii

i 2(1)

where Di is the diffusivity and µi is the chemical potential of species i. Rand T are ideal gas constant and absolute temperature, respectively.∇ = ∂∂ ∂∂ ∂∂( ), ,x y z and r=(x, y, z) is the spatial coordinate and t is time.Under given boundary conditions and initial conditions, the spatial andtemporal evolution of concentration ci are obtained by solving Eq. (1).

In the model system, there are three phases: SiO2, zeolite, and

Fig. 1. SEM images of microstructure in LTA-monolith particles. (a) SiO2 skeleton created from spinodal phase separation [25], (b) Aggregation of LTA zeolitecrystalline particles on the SiO2 skeleton [26].

Fig. 2. Schematic drawing showing the phases of the considered microstructureand PF variables representing the model system.

Y.L. Li et al.

liquid. The free energies of the different phases in terms of Na+ andSr2+ concentrations and temperature can be assessed by thermo-dynamic calculations [18,30]. The chemical free energy of species inear equilibrium can be approximated by a parabolic function of Na+

and Sr2+ concentrations. Thus, the chemical potential of species i in themultiphase system can be described as:

∑= − ==µ A α c c α η i( )( ( )) , 1, 2iα

i i ieq

α1

3

(2)

where c α( )ieq (i=1, 2) are the equilibrium concentrations of +Na and+Sr2 in phase α, respectively. A α( )i is the free energy coefficient of

phase α, with α =1, 2, and 3, respectively, corresponding to zeolite,silica, and liquid phases. The equilibrium concentrations c α( )i

eq andcoefficients A α( )i can be determined once the free energies of thesethree phases are known.

The diffusivities of Na+ and Sr2+ in different phases are treated asseparate quantities. The diffusivity along interfaces and grain bound-aries is usually much larger than in the bulk. Since Na+ and Sr2+ dif-fuse through porosities in the form of channels and only occupy certainlattice sites in zeolite, the occupation concentration of Na+ and Sr2+

inside the zeolite crystals changes the local chemistry, which may resultin lattice distortion and possible phase transitions. As a result, the dif-fusivity may strongly depend on the concentrations of Na+ and Sr2+.Therefore, the diffusivity of Na+ and Sr2+ are spatial/phase and con-centration dependent. In order to reflect the spatial/phase and con-centration dependency, we have constructed the following expressionfor the diffusivity:

∑= ⎧⎨⎩ + + + + − ⎫⎬⎭=D D d φ c η η d η d η d η1 (1) ( ) (2) (3) (1 )i i i i i gbi

i0 2 1 1 2 31

32

(3)

where =d α α( ), 1, 2, 3i , and dgb are the model parameters which can bedetermined by the bulk diffusivity and interface diffusivity, respec-tively. For instance, when = =η η 01 3 and =η 1,2 = +D D d{1 (2)}i i i0 isthe diffusivity of species i in SiO2. φ c η η( , )2 1 1 is not equal to zero insidezeolite particles. It describes the Sr2+ concentration dependence ofdiffusivity. In this work, φ is taken as a kind of Fermi-Dirac function soits value varies from 0 to 1, and has the form of

⎜ ⎟= ⎛⎝− ⎡⎣⎢⎛⎝ ⎞⎠ − ⎤⎦⎥⎞⎠ +φ c η

k λ( ) 1

exp 1c ηc

2 1

(1)

2eq2 1

2 (4)

where k and λ are coefficients to adjust the profile of φ as illustrated inFig. 3. For a negative value of d (1)i in the zeolite phase it can be seen

from Eq. (3) that the diffusivity of species i decreases from Di0 to+D d(1 (1))i i0 with the increase of the Sr2+ concentration. Howstrongly the diffusivity depends on Sr2+ concentration is determined bythe two coefficients k and λ which could be fitted with DFT calculationsor experimental data of the diffusivity.

3. Generation of hierarchical microstructures

The simulation requires initial microstructures that preferentiallymatch the experimental materials in terms of particle size, distribution,and porosity. The PF method can directly read data obtained from ex-perimental images such as those shown in Fig. 1(b). However, it is verychallenging to obtain 3D images from experiments. In this work we usethe PF method to generate model microstructures for use in the ionexchange model. The SiO2 skeleton can be considered as a type of 3Dconnected porous structure obtained by spinodal decomposition be-tween PEO and TEO [31]. Therefore, a PF model of spinodal decom-position [32,33] is used to generate an SiO2 skeleton for the ion ex-change model. Fig. 4 shows the microstructure evolution of one of thetwo phases during a typical PF model of spinodal decomposition. Theobtained microstructures are very similar to the SiO2 structures shownin Fig. 1(a) and are therefore used to mimic an experimental SiO2

skeleton. Adjusting the two-phase volume fraction and interfacial en-ergy in this PF model of spinodal decomposition creates structures withdifferent macro-pore sizes and skeleton wall thicknesses.

Experimentally, a SiO2 microstructure is classically transformed intothe structure shown in Fig. 1(b) using a pseudomorphic transformationby a dissolution-precipitation process that occurs when silica is mixedwith a solution containing NaOH and an aluminum salt (such asNaAlO2) [34]. Controlling the synthesis process parameters (aging time,crystallization temperature, etc.) leads to diverse Na-LTA morphologies[3]. With the obtained SiO2 skeleton microstructure in Fig. 4, a PFmodel of grain growth [29,35] is then employed to add an aggregationof zeolite particles into the skeleton microstructure. Controlling thenucleation densities, nuclei sizes, and growth rates, a 3D hierarchicalstructure with different macro- and meso-scale pores, particle sizes, andvolume fraction of the zeolite particle phase can be obtained as shownin Fig. 5. Therefore, for given microstructural features, such as themacro- and mesoscale pore sizes, zeolite particle size distributions, andvolume fractions of SiO2 and zeolite phases, this PF modeling capabilityallow us to generate 3D microstructures that reasonably capture ex-perimental microstructures.

4. Results of ion exchange

The effects of Na-LTA microstructures on ionic exchange kinetics inbatch tests where 100-mg Na-LTA monolithic particles are added to100-mL Sr2+ solution were studied. The particle radius is about 200-µm. We consider the particles having the following microstructures,respectively: (1) large zeolite particles without any pores shown inFig. 5(d); (2) SiO2 supported aggregation of dense micrometer-sizedzeolite particles with macroscale pores but without mesoscale poresshown in Fig. 5(c); and (3) SiO2 supported aggregation of loose mi-crometer-sized zeolite particles with macroscale and mesoscale poresshown in Fig. 5(b). Each simulation only considers one such particlemicrostructure in the simulation domain, which has the dimension of× ×l l l256 256 2560 0 0. Periodic boundary conditions are applied in alldirections. The fast Fourier spectral method [36] for spatial dis-cretization and a forward Euler scheme for time derivatives are em-ployed to solve Eq. (1). Table 1 gives the initial +Na concentration in theNa-LTA zeolite crystals and the initial +Sr2 concentration in the solu-tion. The Na+ concentration in the zeolite material corresponds to theformula NaAlSiO4, which is a representative Na-LTA. The +Sr2 con-centration in solution corresponds to the amount of +Sr2 needed for acomplete exchange between +Na from the solid and +Sr2 from the so-lution with a ratio of 2 to count for the charge difference. The +Na

Fig. 3. Function =⎜ ⎟⎡

⎣⎢⎢− ⎛⎝⎜⎛⎝ ⎞⎠ − ⎞⎠⎟⎤⎦⎥⎥ +φ c( )

k cc

λ

1

exp0

21

varies with parameter k and λ when

c0 =0.5.

Y.L. Li et al.

concentration in Table 1 refers to the zeolite solid phase. The con-sidered zeolite has a cation +Na exchange capacity of 5.6mmol/g and adensity of 0.76 g/cm3 [3]. It equals 4.256mol/L using the same units as

Sr2+ in the solution phase. Thus, =c0 4.256mol/L is used to normalizethe concentration variables.

In the simulations, the zeolite particle occupies a spherical volumewith a diameter of 200 grids in the simulation cell of × ×256 256 256grids. Thus, the liquid phase volume in the simulation cell is much lessthan that in a real batch test. Considering the fact that ionic diffusion inthe solution is much faster than that in the solid phases (SiO2 andzeolite) we assume that Sr2+ and Na+ are always uniformly distributedin solution. The uniform concentration in solution varies with ion ex-change time and can be estimated as:

= = −∗ ∗ > ∗ ∗∗ ∗ ∗ > ∗ ∗ ∗∗c tc t

Vc t c

c tV

( )|( )

, ( )|( )

η ηtotal

Lη η

total

L1BC 1,

2BC

202,

3 0 3 0 (5)

The superscript * indicate the quantity is normalized. The normal-ization is as =∗r r

l0, =∗t tD

l0

02, =∗c ,i

cc

i0

=∗D D D/i i0 0 0, =∗A A RT/( )i i , wherel0, D0 are the characteristic length and diffusivity, respectively. In Eq.(5), ∗VL is the normalized liquid volume equal to 100mL in real units forthe considered solution. >η η3 0 with =η 0.50 represents the liquid so-lution phase, ∗c total1, is the total amount of +Na entering into the liquidsolution phase, and ∗c total2, is the total amount of +Sr2 entering into thezeolite particle phase, respectively. They can be calculated by∫= −∗ ∗ ∗ ∗ ∗ ∗∗ rc c V c t dV( , )total S V S1, 1

01

S (6)∫=∗ ∗ ∗ ∗∗ rc c t dV( , )total V S2, 2S (7)

where ∗VS is the normalized solid volume. Table 2 lists the dimensionlessparameters used in the simulations with a time step of = ×∗ −t∆ 1.0 10 5.

Fig. 4. Generation of SiO2 skeleton structure using a PF model of spinodal decomposition [32,33].

Fig. 5. Morphologies of a generated aggregated particle (a) and a cross-sectionslice of a particle showing the macro- and mesoscale pores (b-c) where (b) smallzeolite particles and mesoscale pores, (c) small zeolite particles but no me-soscale pores, and (d) a large zeolite particle with neither skeleton structure norpore.

Table 1Material properties and initial concentrations.

Zeolite particles Sr2+ solution

+Na concentration 4.256mol/L +Sr2

concentration2.05× −10 3 mol/L

Normalized concentration =∗c 1.010 Normalized

concentration=∗c2

0 0.482× −10 3

=c0 4.256mol/L

Table 2Dimensionless parameters used in simulations.

Zeolite, =α 1 SiO2, =α 2 Liquid, =α 3

∗c α( )eq1 1.0× −10 5 1.0× −10 5 1.0× −10 4∗c α( )eq2 0.5 1.0× −10 5 1.0× −10 5∗c α( )10 1.0 1.0× −10 5 1.0× −10 4∗c α( )20 0.0 1.0× −10 5 4.8× −10 4=∗ ∗A α A α( ) ( )1 2 1.0× 103 2.0× 103 2.0× 103=∗ ∗D α D α( ) ( )10 20 2.0 2.0 2.0=d α d α( ) ( )1 2 0.0, −0.9 and −0.999 −0.99 0.0

Y.L. Li et al.

4.1. Effect of particle size and concentration dependent diffusivity on uptakekinetics

Two particle sizes, 150 and 56 µm diameter, and two concentrationdependent diffusivities, =d α( ) 0.01 and −0.999, respectively, werestudied to determine the effect of particle size and diffusivity on uptakekinetics. Seen from Eq. (3), =d α( ) 0.01 means that diffusivity in zeoliteis independent of concentration while = −d α( ) 0.9991 means that thediffusivity becomes three orders of magnitude smaller when Sr2+

concentration in zeolite reaches the equilibrium concentration ∗c (1)eq2 .

Fig. 6(a) plots the time evolution of Na+ and Sr2+ concentrations alonga line through the center of a solid zeolite spherical particle as shown inFig. 5(d). In the simulation, coefficients of k= 20 and =λ 0.5 wereused. It shows that Na+ diffuses out of the zeolite particle and reachesan equilibrium concentration of 0.0 and Sr2+ diffuses into the zeoliteparticle and gradually reaches the dimensionless equilibrium con-centration of 0.5. The reduction of Na+ concentration causes a negativecharge − − ∗Q c(1 )1 while the increase of Sr2+ concentration causes apositive charge ∗Qc2 2 , where Q is the charge per Na+. Fig. 6(b) replotsthe evolution of scaled concentration − ∗c(1 )/21 , and ∗c2 inside theparticle. We found that − ∗c(1 )/21 is almost equal to ∗c2 , which corre-sponds to local charge neutrality. Therefore, the results demonstratethat the chemical potentials and mobility used in the simulations ensurelocal charge neutrality even though electric field equilibrium is notexplicitly considered in the current model. However, a quantitative andpredictive model needs to take the electric field into account to capturethe effect of local charge on ionic diffusion and ion exchange kinetics,especially in hierarchical nuclear wasteform materials of interest. It isstraightforward to consider charge effects as did in a previous electro-chemistry model of charge and discharge in Li ionic battery materials[37] and we intend to do so in the future.

Fig. 7 shows the evolution of the average concentration of Sr2+

inside the particles versus the normalized time (t*1/2) for differentparticle sizes and different value of d α( )1 . The average concentration isdefined as the ratio of the total amount of Sr2+ in the particle and thetotal ideal capacity of Sr2+, which is equal to the capacity multiplied bythe particle volume. The average concentration is proportional to thetotal amount of Sr2+ because the total ideal capacity is constant forgiven particle size. The solid lines present the uptake kinetics for theparticles with concentration independent diffusivity while the solidsymbol lines are for the particles with concentration dependent diffu-sivity. As expected, for both cases, the larger the particle size, theslower the uptake kinetics (i.e. how fast the system reaches its idealcapacity) because the uptake kinetics are controlled by long-range bulkdiffusion. It can be clearly seen that at earlier stages the average con-centration and the time (t*1/2) has the same linear relationship

identified by the green lines for a given particle size. Such a linear re-lationship can only be explained by bulk diffusion control. With in-creasing Sr2+ concentration in the surface layer of the particle, thediffusivity in the surface layer decreases. From Fig. 7 it is seen that asecond linear kinetic process appears in the particles with concentrationdependent diffusivity. The second linear process is identified by theblue lines. More importantly, the uptake kinetics may be significantlyreduced if Sr2+ accumulation strongly reduces the diffusivity. The re-sults show that decreased uptake kinetics reduces the effective capacityof Sr2+ for a given time. In contrast, there is no second linear kineticphenomena for the particles with concentration independent diffusion.Many ion exchange experiments [26,38,39] show two diffusion kineticregimes in agreement with our simulation results. The experimentaland simulation results demonstrate that concentration dependent dif-fusivity plays an important role in Sr2+ uptake, and the developedmodel is able to reasonably capture these effects.

4.2. Effect of microstructure on uptake kinetics

In a microstructure with dense zeolite particles, the particles are inclose contact with each other, resulting in no mesoscale pores and noliquid penetration into the aggregation of zeolite particles. On the otherhand, in a microstructure consisting of a loose aggregation of zeoliteparticles, there are mesoscale pores that allow the solution to penetrateinto the structure. The interconnecting mesoscale pores filled with li-quid are fast channels for Na+ and Sr2+ diffusion. Figs. 8 and 9(a)

Fig. 6. (a) Time evolution of Na+ and Sr2+ concentrations in a solid zeolite particle; (b) Time evolution of scaled concentration − ∗c(1 )/21 and ∗c2 . The dashed linesare for − ∗c(1 )/21 and the symbol lines are for ∗c2 .

Fig. 7. Average concentration of Sr2+ into the particle versus time for differentparticle sizes and different parameter d (1)i , respectively.

Y.L. Li et al.

summarize the effect of microstructures with and without mesoscalepores on Sr2+ uptake kinetics. The average concentration is calculatedin the same way as in Fig. 7, i.e., it is the ratio of the total amount ofSr2+ into the aggregated particle and the total ideal particle capacity ofSr2+, which has a maximum of 1.0. In the simulations, coefficients ofk= 10.0, =λ 0.7 and =d (1)i −0.999 were used.

From the results in Figs. 8 and 9(a), several uptake kinetic trends areobserved including: (1) during the first linear kinetic process kineticsdecrease with increasing zeolite thickness; (2) the thicker the zeoliteparticle layer, the slower the overall uptake kinetics for both dense andloose zeolite particles; (3) a second linear kinetic process in the mi-crostructure with dense particles has more impact than in the micro-structure with loose particles; and (4) the second linear kinetic processbecomes less important with decreasing particle size and/or zeolitelayer thickness. These results indicate that uptake kinetics stronglydepends on the zeolite microstructure.

For comparison, some experimental data from porous materialscontaining HCF are used. For example, mesoporous silica supportloaded with HCF shows very fast kinetics where equilibrium wasreached in a few minutes and was attributed to the porosity of the solidsupport [40]. However, a dense support loaded with the same kind ofHCF requires a few days to reach equilibrium [38]. The experimentalresults show that uptake kinetics strongly depend on microstructure. Itcan be seen in Fig. 9(a) that in the structure with only one large zeoliteparticle and the structure with dense small zeolite particles withoutmesoscale pores the uptake kinetics are controlled by both bulk

diffusion and surface layer diffusion. However, the overall uptake ki-netics are only controlled by bulk diffusion in the structure with loosesmall zeolite particles and meso-pores. The predicted microstructuredependence of ion exchange kinetics are consistent with experimentsshown in Fig. 9b. We have designed LTA batch tests for more accuratecomparisons. The three circles mark the average concentrations of Sr2+

corresponding to about 11%, 57% and 94% of the ideal capacity at the

time = =∗ ( )t( ) 0.15tt

1/2 1/2

0for three different microstructures. The

large difference clearly indicates that the effective capacity stronglydepends on the microstructure.

5. Summary

A 3D PF microstructure-dependent diffusion model has been de-veloped to study the effects of complex microstructures, in-homogeneous thermodynamics, and diffusion kinetics on radionuclideuptake kinetics and on the capacity of radionuclides and uptake effi-ciency, as well as highlighting known diffusion mechanisms. PF modelsof spinodal decomposition and grain growth were employed to generatehierarchical microstructures generally corresponding to experimentalmaterials for the model microstructures in the simulations. The devel-oped PF model demonstrated that uptake kinetics in large particles andaggregation of dense particles are limited by two diffusion processes:bulk diffusion and surface layer transport. The uptake kinetics changefrom two diffusion processes to a single process as the particle sizedecreases and the mesoscale pore volume fraction increases. The pre-dicted uptake kinetics and the effect of microstructures on the effectivecapacity of radionuclides and uptake efficiency agreed reasonably wellwith published ion exchange experiments in Na-LTA zeolite materials.The results demonstrate that the developed model has the capability toinvestigate effects of microstructure on uptake kinetics in complex,hierarchically-porous microstructures. However, because thermo-dynamic and kinetic properties of zeolite structures are lacking, quali-tative thermodynamic and kinetic models are used in the simulations,which means that the results are qualitative but can be made quanti-tative by acquiring the required data, either by experiment or bycomputation. The chemical potential gradient for the species in thesolution phase and the zeolite phase are the driving force for diffusion.A concentration dependent diffusivity strongly affects ion exchangekinetics. It is desired to obtain these thermodynamic and kineticproperties for more quantitative simulations. In addition, some physics,such as electric fields associated with local aggregation of ions and fluidflow in multiscale pores, may play important roles in uptake processesand will eventually be integrated into the model in the future.

Fig. 8. Average concentration of Sr2+ into the particle versus time for differentaggregated particle structures.

Fig. 9. (a) Average concentration of Sr2+ into the particle versus time for three different particle structures; (b) Experimental measurements reproduced from Ref.[40].

Y.L. Li et al.

CRediT authorship contribution statement

Y.L. Li: Methodology, Investigation, Visualization, Writing -original draft. B.D. Zeidman: Data curation, Investigation. S.Y. Hu:Conceptualization, Supervision, Writing - original draft, Writing -review & editing. C.H. Henager: Conceptualization, Fundingacquisition, Project administration, Writing - review & editing.T.M. Besmann: Conceptualization, Writing - review & editing.A. Grandjean: Data curation, Writing - review & editing.

Acknowledgements

The work described in this article was performed by PacificNorthwest National Laboratory, which is operated by Battelle for theU.S. Department of Energy under Contract DE-AC05-76RL01830. Thiswork was supported as part of the Center for Hierarchical Waste FormMaterials, an Energy Frontier Research Center funded by the U.S.Department of Energy, Office of Science, Basic Energy Sciences underAward No. DE-SC0016574.

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