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Reimagining Third Phase Formation as the Miscibility Gap of a MolecularSolutionMichael Servis, David T. Wu, Jenife Shafer, Aurora Clark
Submitted date: 28/10/2019 • Posted date: 30/10/2019Licence: CC BY-NC-ND 4.0Citation information: Servis, Michael; Wu, David T.; Shafer, Jenife; Clark, Aurora (2019): Reimagining ThirdPhase Formation as the Miscibility Gap of a Molecular Solution. ChemRxiv. Preprint.
Liquid/liquid phase transitions are inherent to multicomponent solutions, which often contain a diversity ofintermolecular interactions between their molecular constituents. In one such example, a phase transition isobserved in liquid/liquid extraction where the nonpolar organic phase separates into two phases undersufficiently high metal and acid extraction by the amphiphilic extractant molecule. This deleteriousphenomenon, known as third phase formation, complicates processing and limits efficiency. While empiricallywell documented, the molecular origin of this phenomenon is not understood. The prevailing conceptualizationof the organic phase treats it as a microemulsion where extractant molecules form reverse micelles thatcontain the extracted aqueous solutes in their polar cores. Yet recent studies indicate that a microemulsionparadigm is insufficient to describe molecular aggregation in some solvent extraction systems, implying thatan alternative description of aggregation, and explanation for third phase formation, is needed. In this study,we demonstrate that the formation of a third phase is consistent with crossing the liquid-liquid miscibility gapfor a molecular solution rather than a Winsor II to Winsor III transition as presumed in the microemulsionparadigm. This insight is provided by using a graph theoretic methodology, generalizable to other complexmulticomponent molecular solutions, to identify the onset of phase splitting. This approach uses connectivityobtained from molecular dynamics simulation to correlate the molecular-scale association of extractants andextracted solutes to the solution phase behavior using percolation theory. The method is applied to investigatea solvent extraction system relevant to ore purification and used nuclear fuel recycling: tri-n-butylphosphate/uranyl nitrate/water/nitric acid/n-dodecane. In analogy to a molecular solution, immediatelypreceding the liquid-liquid coexistence curve from the single phase region, the metal-ligand complexespercolate. This demonstrates that describing this solution with microemulsion chemistry is neither applicablenor broadly required to explain third phase formation. Additionally, the method developed herein can predictthird phase formation phase boundaries from simulation for this and potentially other solvent extractionsystems.
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Reimagining third phase formation as the
miscibility gap of a molecular solution
Michael J. Servis,∗,† David T. Wu,‡,¶ Jenifer C. Shafer,‡ and Aurora E. Clark∗,†,§
†Department of Chemistry, Washington State University, Pullman, WA
‡Department of Chemistry, Colorado School of Mines, Golden, CO
¶Department of Chemical and Biological Engineering, Colorado School of Mines, Golden,
CO
§Pacific Northwest National Laboratory, Richland, WA
E-mail: [email protected]; [email protected]
1
Abstract2
Liquid/liquid phase transitions are inherent to multicomponent solutions, which3
often contain a diversity of intermolecular interactions between their molecular con-4
stituents. In one such example, a phase transition is observed in liquid/liquid extrac-5
tion where the nonpolar organic phase separates into two phases under sufficiently6
high metal and acid extraction by the amphiphilic extractant molecule. This deleteri-7
ous phenomenon, known as third phase formation, complicates processing and limits8
efficiency. While empirically well documented, the molecular origin of this phenomenon9
is not understood. The prevailing conceptualization of the organic phase treats it as10
a microemulsion where extractant molecules form reverse micelles that contain the11
extracted aqueous solutes in their polar cores. Yet recent studies indicate that a mi-12
croemulsion paradigm is insufficient to describe molecular aggregation in some solvent13
extraction systems, implying that an alternative description of aggregation, and expla-14
nation for third phase formation, is needed. In this study, we demonstrate that the15
1
formation of a third phase is consistent with crossing the liquid-liquid miscibility gap for16
a molecular solution rather than a Winsor II to Winsor III transition as presumed in the17
microemulsion paradigm. This insight is provided by using a graph theoretic method-18
ology, generalizable to other complex multicomponent molecular solutions, to identify19
the onset of phase splitting. This approach uses connectivity obtained from molecu-20
lar dynamics simulation to correlate the molecular-scale association of extractants and21
extracted solutes to the solution phase behavior using percolation theory. The method22
is applied to investigate a solvent extraction system relevant to ore purification and23
used nuclear fuel recycling: tri-n-butyl phosphate/uranyl nitrate/water/nitric acid/n-24
dodecane. In analogy to a molecular solution, immediately preceding the liquid-liquid25
coexistence curve from the single phase region, the metal-ligand complexes percolate.26
This demonstrates that describing this solution with microemulsion chemistry is nei-27
ther applicable nor broadly required to explain third phase formation. Additionally,28
the method developed herein can predict third phase formation phase boundaries from29
simulation for this and potentially other solvent extraction systems.30
Introduction31
Complex multicomponent solutions can exhibit liquid-liquid phase transitions whose charac-32
terization requires mapping the phase boundaries as a function of different system variables.33
Examples can be drawn from the formation of biological membranes1,2 to separations sci-34
ence.3 Fundamentally, a number of significant challenges exist toward understanding and35
characterizing these phenomena, either through identifying the most appropriate conceptual36
model that describes the phase behavior, or elucidating the influence of molecular-scale spe-37
ciation and organization upon the macroscopic phase transition. A significant need exists38
for generalized methods that can provide such insights.39
A quintessential example found within separations science is liquid/liquid extraction40
(LLE), which is the predominant industrial technique for the targeted recovery of met-41
2
als.4 Liquid/liquid extraction commonly utilizes a neutral solvating extractant molecule in a42
nonpolar organic solvent to selectively extract metal cations with charge neutralizing anions43
from an acidic aqueous media. Metal-ligand complex aggregation affects their organic phase44
solubility and impacts the limiting organic concentration (LOC) of the metal before the onset45
of phase instability.5 In so-called “third phase formation,” the organic phase splits into two46
phases: a solute-rich “heavy” organic phase and a solute-poor “light” organic phase. This47
undesirable phase transition is a processing impediment that limits the efficiency of solvent48
extraction systems. Much work on understanding third phase formation has focused upon49
LLE applications that pertain to the nuclear fuel cycle. Here, the industry standard Plu-50
tonium and Uranium Reduction EXtraction (PUREX) liquid/liquid extraction process uses51
the ubiquitous amphiphilic extractant molecule tri-n-butyl phosphate (TBP) to selectively52
extract uranium U(VI) from a nitric acid aqueous media into an aliphatic solvent in the form53
(TBP)2(UO2)(NO3)2.6
54
Interpreting organic phase aggregation from experimental or simulation data mandates55
a choice of modeling paradigm between a molecular solution or a microemulsion, which we56
compare and contrast in Figure 1. Aggregation in the organic phase has been typically57
conceptualized as a microemulsion consisting of reverse micelles composed of extractant58
molecules solvating a polar core containing the extracted aqueous solutes.3,7–10 Small angle59
neutron (SANS) and X-ray (SAXS) scattering have been used to study the aggregation of60
extracted TBP/uranyl nitrate adducts in organic solvents.11–14 In those studies, extracted61
metal-ligand complexes, or stoichiometric solvates, are treated as reverse micelles. Under this62
description of the organic phase, third phase formation is analogous to a transition from a63
two phase water-in-oil micellar system (Winsor II) to a three phase system (Winsor III) with64
the bicontinuous phase corresponding to the heavy organic phase.3,10 The reverse micelles of65
water-in-oil microemulsions have been modeled as colloidal particles, where micelle-micelle66
interactions are described by a necessarily simplistic potential energy function, U(r), such as67
that between hard spheres or ellipsoids with surface attraction. Those models are typically68
3
characterized by a hard-core length scale that is assumed to be much larger than the particle-69
particle interaction and solvent length scales. For micromeulsions, solutions are characterized70
by properties such as the critical aggregation concentration, interfacial tension and shape71
factor of the extractant Va0lc
, where V is the hydrocarbon volume, a0 the area of the polar72
headgroup and lc the maximum length of the hydrocarbon tail group.9,1573
Figure 1: For association of metal-ligand complexes (this study), or the formation of hydro-gen bonded clusters,16,17 two modeling paradigms for organic phase solutions are contrasted:microemulsions (left) and molecular solutions (right). Differences in model length scalesare illustrated in the top row. The middle row highlights properties which inform model-ing for microemulsions (reverse micelle U(r), critical aggregation concentration, interfacialtension and extractant packing factor15) and molecular solutions (vertex connectivity, bondprobability and cluster distribution). In the bottom row, the microemulsion Winsor II toWinsor III description of third phase formation is contrasted with the proposed percolationtransition tied to a liquid-liquid phase transition of a molecular solution.
4
In contrast, for an associating molecular solution, aggregation and phase behavior may74
be interpreted using a graph theoretic approach in which vertices (whether molecules or75
metal-ligand complexes) are connected by edges which represent intermolecular interactions.76
As discussed further below and demonstrated for simple liquids, the phase transition for77
the formation of a dense phase may then be predicted as occurring approximately at a per-78
colation transition for appropriate vertex and connectivity definitions.18–20 We hypothesize79
here that third phase formation can likewise be approximately identified with a percolation80
transition for extracted species. Different solvent extraction systems may require different81
mappings of molecules and their intermolecular interactions to graphical representations. For82
example, molecular connectivity has been used to determine connectivity in the formation83
of a system-spanning hydrogen bonded network in the absence of metal ions.16 Such molec-84
ular solutions are described by discrete, directional interactions occurring over length scales85
comparable to that of their molecular constituents. Furthermore, upon the formation of86
solvate species, the interaction of such solvates may result in solution structure approaching87
mesoscale lengths. This hierarchical organization of solvates is contrasted with the mesoscale88
structure of microemulsions where micellar or bicontinuous solutions are characterized by89
a polar phase stabilized within a nonpolar phase, or vice versa, by an amphiphile. The90
atomistic resolution provided by MD simulations when combined with a graph theoretical91
approach can distinguish between the microemulsion and molecular solution paradigms. For92
example, the hydrogen bonded network of a solution of water-in-oil reverse micelles would93
show highly connected clusters of a characteristic size related to the reverse micelle length94
scale. The absence of such features would, instead, indicate a normal molecular solution.16,1795
Recent studies have shown that the treatment of organic phases as microemulsions, and96
the associated colloidal modelling paradigms, has led to counterintuitive or seemingly un-97
physical conclusions. These include a significant size dependence of the reverse micelles on98
system variables (e.g., TBP volume fraction or acid concentration) and fitted hard sphere99
surface attraction values that correspond to the existence of a percolated network of those100
5
complexes even at low metal concentrations.21 These conclusions are presumed to stem from101
the assumptions used to fit models from experimental scattering data, including, e.g., spher-102
ically symmetric interaction potentials that allow for separation of the form and structure103
factors. However, this interpretation has been contradicted by NMR diffusometry21,22 and104
molecular dynamics (MD) simulation studies,16,17,22–26 which find that the organic phase105
TBP complexes with inorganic aqueous solutes behave as molecular solutions, governed by106
discrete electrostatic interactions and speciation, rather than as water-in-oil microemulsions.107
This molecular solution description is also consistent with extended X-ray adsorption fine108
structure (EXAFS)11,27 and UV-Vis11 spectroscopic data of the uranyl-centered complexes,109
which suggest a constant coordination environment for the uranyl ion independent of TBP110
and uranyl concentrations, and is inconsistent with the changing coordination expected in111
reverse micelles with expanding, highly interconnected polar cores. Given the direct coor-112
dination of the uranyl ions by TBP and a relatively low water concentration in the heavy113
organic phase (roughly one water molecule per three uranyl ions),13 it is reasonable to assume114
that the uranyl ions do not undergo a substantial structural change in the heavy organic115
phase such as, e.g., becoming dispersed in a bicontinuous medium. Therefore, we treat this116
systems as an associating molecular solution whose constituents can form discrete species,117
including the (TBP)2(UO2)(NO3)2 solvate.118
In this study, molecular dynamics simulation is applied to investigate the organic phase119
association of (TBP)2(UO2)(NO3)2 metal-ligand complexes at high concentration. Prior120
simulations at low uranyl concentrations identified two modes of uranyl nitrate/extractant121
complex association in organic solution: an ordered short-range pair and a more isotropic122
long-range pair.24 Importantly, the molecular structure of the extractant’s head and tail123
groups affected those modes of interactions: changing the TBP phosphate head group to a124
phosphonate group inhibited the short-range ordered dimer while changing the TBP butyl125
tails to amyl tails suppressed the long-range correlation. Shortening the solvent alkyl chain126
length from n-dodecane to n-hexane, or changing the solvent to toluene, or including ex-127
6
cess TBP and coextracted nitric acid and water all reduced complex association to different128
degrees. We extend our prior work to simulate the uranium LOC so as to understand higher-129
order organic phase aggregation and the structural origin of third phase formation for this130
metal-containing system. Organic phase aggregation of the metal-ligand complexes is quanti-131
fied using a graph theoretic representation of their self-association, in which the metal-ligand132
complexes are reduced to single vertices and their association is treated as edges between133
those vertices. A community analysis of the resulting network then identifies the morphol-134
ogy of the percolating network. We find that in the system studied here, clustering of the135
metal-ligand complexes yields a predominately linear morphology with “dense subclusters”136
of highly interconnected vertices. Further, this developed methodology is generalizable to a137
broad set of chemical systems, where it is possible to vary the definition of a node or edge138
to elucidate the percolating species responsible for phase separation.139
In contrast with colloidal or microemulsion chemistry models,3 our analysis shows a clus-140
ter size distribution at the LOC consistent with the percolation theory prediction near the141
percolation threshold. This is corroborated by the uranium-uranium radial distribution func-142
tion showing an emerging correlation length consistent with the onset of a percolation phase143
transition. Therefore, these simulations, and the new graph theoretical analysis method144
presented, provides evidence that third phase formation in this solvent extraction system is145
consistent with a liquid-liquid phase transition, where (TBP)2(UO2)(NO3)2 complexes form146
a system-spanning cluster in solution immediately preceding the organic phase splitting.147
This approach is further demonstrated to be analogous to simple binary mixtures where the148
minor component percolates with increasing concentration immediately before reaching the149
liquid-liquid coexistence curve.18,19150
In summary, this study proposes a methodology for predicting third phase formation151
via a graph theoretical formalism where the node and edge definitions enable identification152
of the percolated network at the onset of the phase transition for the appropriate vertex153
and edge definitions. This work creates a framework for elucidating the appropriate models154
7
used to describe phase transitions of multicomponent solutions and also a paradigm for155
understanding how molecular-scale detail influence macroscopic phase transition behavior.156
Methodology157
Simulation Force Fields158
The force fields used in this study are reported in detail in the Supporting Information159
of ref. 24. The modeling approach is summarized here. Simulation times of many 10s of160
ns are required to equilibrate and sample the slow dynamics of organic phase metal-ligand161
complex association, necessitating the use of classical, additive potentials. The stoichiome-162
try of the uranyl nitrate/TBP metal-ligand complexes is monodisperse and independent of163
organic phase uranium concentration.11,27 Rather than the five-fold coordination found in164
aqueous solutions, the organic phase uranyl metal center of the (TBP)2UO2(NO3)2 complex165
is six-fold coordinate with the nitrates binding in a bidentate manner. For this system, we166
thus constrained the nitrate positions to enforce the bidentate coordination. To account for167
charge transfer, partial atomic charges for the uranyl and nitrate ions were derived from168
density functional theory electronic structure calculations of the assembled complex, rather169
than using integer formal charges of separated ions. Force field parameters for n-dodecane170
and TBP were modified from the general AMBER force field (GAFF)28 to reproduce physic-171
ochemical properties of TBP/n-dodecane mixtures including TBP molecular dipole, density,172
enthalpy of mixing and volume change on mixing.29 Nitric acid was modeled with GAFF28173
parameters and water with the TIP4P model.30174
Simulation Methodology175
Although direct simulation of a two to three phase transition is desirable, it is not generally176
achievable over simulation-accessible time scales. Grand canonical ensemble simulation of a177
light organic phase in chemical equilibrium with a heavy phase is also challenging for this178
8
system due to ambiguities in the structure of the metal-ligand complex in the heavy organic179
phase. For example, Chiarizia et al. found that while the coordination geometry of the180
metal-ligand complexes is largely unchanged between the organic phase at the LOC and the181
heavy organic phase, a minority of nitrate anions change to monodentate coordination.11182
Given these practical considerations, we identify the onset of organic phase splitting from a183
monophasic simulation of the organic phase at the uranium LOC.184
The equilibrated cubic simulation box was 13.37 nm on each side, with a composition of185
320 UO2(NO3)2, 1048 total TBP (free and uranyl-bound), 4832 n-dodecane, 176 H2O and186
392 HNO3. Molecular concentrations were obtained from ref. 13. The composition gives 0.26187
M uranyl nitrate, corresponding to the LOC for 20% volume TBP in n-dodecane for uranyl188
extraction from 10 M aqueous phase nitric acid. The initial configuration was generated189
with Packmol.31 The assembled (TBP)2(UO2)(NO3)2 complexes were distributed randomly190
into the n-dodecane. “Free” TBP (not bound to uranyl nitrate) and coextracted water and191
nitric acid are also distributed randomly in the initial configuration.192
Simulations were conducted using the GROMACS 4.5.5 software package.32 Simulations193
were conducted in the isobaric isothermal NPT ensemble using periodic boundary condi-194
tions. Pressure was set to 1 bar with the Parinello-Rahman barostat33 using a 2 ps time195
constant and the temperature was set to 300 K with the Nose-Hoover thermostat34 using196
a 0.2 ps time constant. The leap-frog Verlet integrator was implemented with a 2 fs time197
step and the LINCS algorithm35 to constrain hydrogen-containing bonds. A 1.5 nm cut-off198
was used for Lennard-Jones and short range electrostatic interactions with Particle-Mesh199
Ewald summation for long range electrostatic interactions. The simulation ran for 145 ns,200
equilibrating over the first 35 ns, and sampled for analysis at 20 ps intervals.201
Graph Theoretic Analysis of (TBP)2(UO2)(NO3)2 Association202
The analysis workflow for this study, with the relationships between inputs, intermediate203
analyses and results are presented as a flowchart in Figure 2. Interactions of (TBP)2(UO2)(NO3)2204
9
complexes are first quantified using a graph theoretic approach. Each complex is defined205
as a single vertex. An edge is considered to exist between vertices if the distance between206
the uranium atoms of those corresponding complexes is no more than 1.5 nm. The cutoff207
distance was chosen from the approximate minimum in the uranium-uranium radial distri-208
bution function. Clusters are defined as connected components of the graph containing all209
of the vertices. The largest cluster, containing the most vertices, is identified at each time210
step.211
Legend: InputIntermediate
AnalysisOutput
Simulation TrajectoryConnectivity
DefinitionPercolation Theory
RadialDistribution
Function
IntermolecularNetwork
Community Partition
ConnectedComponents
NeighborOrientationProbability
Density Function
Cluster SizeDistribution
Largest ClusterIdentification
Dense SubclusterIdentificationand Analysis
Largest ClusterTopological Analysis
Phase TransitionIdentification
Figure 2: Workflow for inputs (gray nodes), intermediate analysis steps (green nodes) andresults (blue nodes) are illustrated as a flowchart.
10
Results212
Uranium-Uranium Correlation213
Aggregation of the (TBP)2(UO2)(NO3)2 complexes is analyzed through the radial distribu-214
tion function (RDF), g(r), between pairs of uranium atoms. The uranium-uranium RDF215
is plotted in Figure 3A for the 10-35 ns and 120-145 ns time ranges of the trajectory. The216
uranium-uranium correlation takes shape over the first 10 ns of the trajectory, showing the217
short-range dimer peak near 0.6 nm and the long-range peak beyond 1 nm, as previously218
reported at low uranyl concentration.24 However, the formation of extended clusters in this219
concentrated system is evidenced by the emergence of uranium-uranium correlations beyond220
those short-range and long-range pairs (Figure 3A inset). While the uranium-uranium g(r) is221
uncorrelated for distances beyond 2 nm for the 10-35 ns range, g(r) values greater than unity222
are observed after the system has equilibrated. The linear g(r) behavior in the 120-145 ns223
time range between 2 and 5 nm indicates solution heterogeneity resulting from uranium-rich224
and uranium-poor domains in solution.36225
Metal-Ligand Clustering and Percolation226
The appearance of a third phase is a result of the favorable association of solutes. Under227
appropriate conditions, that association leads to the formation of larger clusters that even-228
tually become a percolating (system-spanning) cluster. Then, at some point the heavily229
connected phase becomes thermodynamically favorable. While related, these are distinct230
events. Percolation in ideal lattices is a second-order phase transition, while the appear-231
ance of a new thermodynamically stable phase is a first-order phase transition (away from232
the critical point). For non-ideal systems with interactions, percolation has also been as-233
sociated with the spinodal.18,19 Nonetheless, it has been shown that the percolation point234
can closely track the thermodynamic phase boundary. For example, the miscibility gap235
in simple liquids can be identified with percolation theory:37 water undergoes a percola-236
11
Figure 3: In panel A, the uranium-uranium RDF is plotted for two different time rangesof the trajectory with the long-range correlation between 2 and 5 nm uranium-uraniumdistances highlighted in the inset. Dashed lines are drawn to illustrate the linear slope ofg(r) at long distances. In panel B, the (TBP)2(UO2)(NO3)2 complex cluster size distributionusing a 1.5 nm uranium-uranium distance cutoff is plotted for the 120-145 ns time range ofthe trajectory. The theoretical power law distribution at the percolation threshold givenby the Fisher exponent for three dimensions, τ ≈ 2.19, is plotted as a dashed red line forcomparison.
12
tion transition upon increasing its concentration in tetrahydrofuran immediately preceding237
the phase boundary.18,19 Previously, we connected the third phase formation phase bound-238
ary in the TBP/HNO3/H2O/n-dodecane system to the emergence of a percolated hydrogen239
bonded network. In the system studied here, there is neither a percolated hydrogen bonded240
network nor the formation of water-in-oil micelles with interconnected cores of H2O, HNO3241
and UO2(NO3)2 molecules. Rather, we apply the graph theoretical description of solute242
association to the (TBP)2(UO2)(NO3)2 complexes themselves. While the solution retains243
its molecular character, the dominant attractive interaction which drives phase separation244
occurs between (TBP)2(UO2)(NO3)2 solvates. The relative stability of that species means245
a practical description of the solution may consider interactions between discrete solvates246
rather than individual molecules. However, the short length scale of these solvates ne-247
cessitates a molecular description of their geometry and interaction, precluding a micellar248
structure where an interface forms between two distinct phases.249
The graph theoretical analysis of the phase transition allows molecular level characteri-250
zation of molecular association which can distinguish a microemulsion from a molecular so-251
lution. Additionally, it allows investigation of the percolation transition and its relationship252
to the appearance of the third phase. However, unlike lattices, the definition of continuum253
connectivity for MD simulations is not unique. Even for a well-defined interaction, such as254
a hydrogen bond, the choice of cut-off, whether energetic or geometric, is arbitrary. For255
this system, we define solvate connectivity of nearest neighbors based on a distance cut-off256
obtained from the U-U RDF.257
At the percolation threshold, the mean cluster size of the percolating component diverges258
as a cluster of infinite size exists with a finite probability. In finite systems, as is the case259
for molecular simulation, that infinite cluster manifests as a system-spanning cluster. The260
cluster size distribution transitions from an exponential distribution far from the percolation261
threshold to a power law distribution in the neighborhood of the threshold. The power law262
distribution of cluster sizes is given by ns ∼ s−τ where ns is the fraction of clusters of size s263
13
and τ is the Fisher exponent having a value of ≈ 2.19 in three dimensions.38264
Signatures of percolation, such as the power law cluster size distribution, are readily265
accessible from simulation and provide a means to identify the liquid-liquid phase bound-266
ary for systems where it may not be possible to directly simulate phase separation under267
practical simulation length and time scales. We tested the applicability of this approach268
for a binary Lennard-Jones (L-J) fluid model system with inter-particle connectivity more269
directly analogous to the solvate connectivity defined here. We observe the same qualitative270
behavior for the L-J fluid as for the water/tetrahydrofuran mixture, supporting the broader271
utility of continuum percolation as a means of approximate demarcation of the two phase272
region. Simulation details and schematic phase diagram with simulation results for the L-J273
fluid are provided in the Supporting Information.274
This method is applied to the liquid-liquid phase boundary of the uranyl nitrate solution275
at the LOC. The (TBP)2(UO2)(NO3)2 complex cluster size distributions on a log-log scale276
for the 120-145 ns time range is nearly equal to the theoretical cluster size distribution277
at the percolation threshold, τ , as shown in Figure 3B. In conjunction with the increasing278
correlation length observed in the uranium-uranium RDFs—another property which diverges279
at the percolation threshold—these results are consistent with the formation of an incipient280
“infinite” (TBP)2(UO2)(NO3)2 solvate cluster expected to precede third phase formation.281
Relative Orientation of Associating Metal-Ligand Complexes282
The spherical anisotropy resulting from the polar and aliphatic regions of the exposed sur-283
face of the (TBP)2(UO2)(NO3)2 complexes may impact the morphology of their clustering284
in solution. The relative orientation of metal-ligand complex pairs is quantified by the angle285
between P1-U1 and U1-U2 vectors of associating complexes. Here, the subscript number286
refers to the two different metal-ligand complexes. This angle, illustrated in Figure 4, is287
measured for the four phosphorus atoms of each pair of neighboring complexes. Due to the288
position of the two P atoms in one complex to be on opposite sides of the U center, and the289
14
Figure 4: The relative orientation of the metal-ligand complex, θ, is measured by the anglebetween the intra-complex phosphorus-uranium and inter-complex uranium-uranium vec-tors. The preference for values between 60◦ and 90◦ at close U-U distances results in an“equatorial band” region of the complex—illustrated by shading the rest of the sphere—which are preferentially exposed to neighboring complexes. Oxygen atoms are drawn in red,nitrogen in blue, carbon in gray, phosphorus in orange and uranium in yellow. On the right,the normalized probability density functions, P(θ), are plotted in black, blue and red lines formetal-ligand complex pairs from the 120-145 ns time range with uranium-uranium distancesof 0.9 to 1.5 nm, 0.9 to 1.2 nm, and 1.2 to 1.5 nm, respectively.
resulting symmetry around θ=90◦, values over 90◦ are reflected across θ=90◦. The resulting290
probability density functions for that angle are plotted in Figure 4 for three uranium-uranium291
distance ranges. Probability densities are normalized by sin(θ) to account for the solid an-292
gle dependence on θ. Note that the normalized curves satisfy the normalization condition293 ∫ π/20
P (θ) sin θdθ = 1, and a constant value of P (θ) would correspond to a random distribu-294
tion of orientations of one cluster relative to the direction to the other cluster.295
Solvate pairs have an increased orientation probability for θ values close to 90◦ at closer296
U-U distances, shown by the 0.9 to 1.2 nm distance range in Figure 4. This is attributed to297
the region of the exposed solvate surface with polar molecules—the uranyl apical oxygens and298
nitrates, rather than the nonpolar alkyl tails of the TBP—at that orientation, as illustrated299
in Figure 4 by the “equatorial band.” Conversely, at longer U-U distances (1.2 to 1.5 nm), the300
probability density is flatter with slightly increased probabilities for the opposite orientation:301
the nonpolar alkyl tails of the coordinating TBP molecules orient towards the other solvate.302
This could result from less efficient packing of the complexes with the solvent at that U-U303
distance and relative orientation. The observed preferential orientations of the metal-ligand304
15
complexes imply a dependence of the long-range pair and resulting clustering morphology305
on the presence of the actinyl apical oxygens and the nitrate counterions, as well as the306
extractant alkyl tail length and branching or lack thereof.307
Morphological Features of Metal-Ligand Complex Clusters308
The morphology of the extended clustering preceding third phase formation is investigated309
by considering the topology of the largest cluster, which becomes the system spanning cluster310
that emerges at the percolation threshold for a finite system. A representative largest cluster311
from time = 125 ns is depicted in Figure 5 as both a graph representation and as a snapshot.312
The complexes are drawn as spheres of diameter 1.5 nm corresponding to their connectivity313
cutoff with those not in the largest cluster drawn with partial transparency. The largest314
cluster contains, on average, 96 ± 38 of the 320 total vertices in the graph. The large315
variability of the largest cluster size stems from the dynamic equilibrium between the system-316
spanning and smaller, finite sized clusters in solution.317
Two primary characteristics of the largest cluster topology are apparent from the graph-318
ical representation: the graph is largely linear without substantial crosslinking and there are319
relatively small “dense subclusters” of highly interconnected vertices within the total cluster.320
The linear character of the large clusters could result from preferred complex coordination321
at the polar band, which provides more favorable orientations for a third complex to directly322
associate with only one of those two complexes, forming a linear triplet. The relatively weak323
association of the long-range pairs appears to not enforce a particular structural motif within324
the extended cluster beyond the dense subclusters.325
To systematically identify dense subclusters within the largest cluster at each snapshot,326
we partition the largest cluster connected component into “communities,” defined to maxi-327
mize connectivity within the communities while minimizing connectivity between them. The328
modularity optimization algorithm39 implemented to compute the community partitions is329
described in the Supporting Information. Once the community partition is determined, we330
16
Figure 5: Graphical representation of a largest cluster (top) at time = 125 ns with colorsindicating community membership (colors are reused between unconnected communities).Short-range pair edges are drawn with solid lines while long-range pair edges are drawnwith dashed lines. Of the 28 communities, 3 are identified as “dense subclusters” andcircled. Snapshots of the periodic simulation cell (bottom) from which the graph is computeddepict the metal-ligand complexes as spheres of diameter 1.5 nm. Metal-ligand complexesin the largest cluster are colored in dark gray with remaining complexes drawn with partialtransparency. The three dense subclusters are colored according to their coloring on thecommunity color-coded graph. Note that the length of edges and vertex positions do notcorrespond to any physical lengths or positions, and are chosen to aid ease of visualizing thegraph connectivity.
17
define dense subclusters as communities having a ratio of the number of edges within the331
community to the total number of vertices comprising that community of greater than 1.5.332
The distribution of number of internal edges per node for each community, given in the333
Supporting Information, showed several peaks and this criterion was chosen to isolate the334
most densely connected region of the distribution. Using this metric, we identify the dense335
subclusters which form in the largest cluster at each time step. For the time step illustrated336
in Figure 5, the three dense subclusters determined from the 28 total communities of that337
cluster are highlighted. Between 10 and 35 ns, on average there were 1.8 dense subclusters338
per snapshot with an average size of 7.5 vertices. By contrast, between 120 and 145 ns339
there were 2.2 dense subclusters per snapshot having an average of 6.9 vertices. Separately340
considering short-range pair edges from long-range pairs, as illustrated in Figure 5 by solid341
and dashed lines, respectively, the number of short-range edges per vertex within dense342
subclusters divided by the number of short-range edges per vertex over the entire largest343
cluster was 1.54 for 10-35 ns and 1.42 for 120-145 ns (as compared to that same ratio for344
all edges of 1.34 and 1.29 for those time ranges, respectively). Therefore, short-range pairs345
are found disproportionately within those dense subclusters. System variables which impact346
short-range pair formation, such as extractant head group moeity,,24 are therefore expected347
to affect the degree of dense subcluster formation within the largest cluster. Reducing the348
community definition ratio to greater than 1.0 instead of 1.5 substantially increases the total349
number of dense subclusters, but similarly shows a short-range edge ratio of 1.29 averaged350
over both 10-35 and 120-145 ns time ranges (compared to that ratio for all edges of 1.07).351
The outlined method for dense subcluster identification is flexible, both in the community352
identification algorithm, its ability to accommodate weighted graphs and in the criterion used353
to identify those subclusters from the community partition. This approach to partitioning354
solute networks can be applied to a wide range of solvent extraction organic phases where,355
generally, domains of strongly interconnected polar solutes form extended networks under356
high solute concentrations. To further characterize clustering of the complexes and their357
18
time evolution during equilibration, three standard graph topological properties qualitatively358
measuring compactness and connectivity are plotted in Figure 6 for the largest cluster at359
200 ps intervals: the number of edges per vertex, the global clustering coefficient and the360
graph link efficiency. The edges per vertex is the ratio of total number of edges to the total361
number of vertices. The global clustering coefficient is the fraction of vertex triplets which362
are closed, i.e., all vertices in the triplet are connected to all other vertices. The graph link363
efficiency is defined as one minus the mean graph distance divided by the total number of364
edges. The graph distance is the shortest number of edges connecting a given pair of vertices,365
with the mean obtained over all the vertex pairs. First or second order polynomial fits are366
overlayed on the data to highlight time evolution of these properties during equilibration (0367
to 35 ns) and over the remainder of the trajectory (35 to 145 ns). For all three properties,368
their values increase rapidly during the initial 35 ns. Then, after 35 ns, slow relaxation of369
the largest cluster results in slight decreases in all three properties.370
Figure 6: The edges per vertex (left), global clustering coefficient (center) and graph linkefficiency (right) are plotted for the largest cluster as a function of simulation time at 200ps intervals. Data is plotted in black circles. Polynomial fits are overlayed in red lines fortwo trajectory time ranges as guides to the eye to illustrate trends of the properties duringthe trajectory.
The clustering topology of the metal-ligand complexes bears similarities to simpler models371
which have been applied to liquids and for which percolation phase transitions are observed:372
the cubic lattice and the Baxter adhesive hard sphere. Here, we briefly comment on the373
19
qualitative similarities and differences between the (TBP)2(UO2)(NO3)2 simulation results374
and those theoretical models. As illustrated in the Supporting Information, the largest clus-375
ter on a cubic lattice near the bond percolation threshold shows a similar linear, branching376
cluster morphology as the simulation data presented here. However, vertex clusters do not377
readily form on a cubic lattice due to the connectivity of that lattice. The dense subclusters378
observed in this study are qualitatively similar to the “dense subunits” described by Miller379
and Frenkel40 from Monte Carlo simulation of Baxter adhesive hard spheres. There, sub-380
clusters form within a percolating network to maximize the energetically favorable contacts381
between neighboring spheres. However, we expect the graph of the system-spanning cluster382
for Baxter adhesive hard spheres to be, in general, more compact and interconnected than383
the (TBP)2(UO2)(NO3)2 graph we have observed here.384
Conclusions385
Third phase formation is an empirically well-documented phenomenon in liquid/liquid ex-386
traction. However, it eludes a description linking molecular-level interactions to the resulting387
macroscopic phase behavior. Using atomistic simulation, and incorporating the experimental388
understanding of the structure of (TBP)2(UO2)(NO3)2 adducts in solution, we study associ-389
ation of those metal-ligand complexes at the uranyl LOC. We find the third phase formation390
phase transition in this system is consistent with a solvate percolation mechanism, wherein391
the complexes form a system-spanning network as the system nears the phase boundary.392
This mechanism is analogous to percolation of the minor component of a binary liquid mix-393
ture which is observed when approaching the liquid-liquid coexistence curve from the single394
phase to the two phase region.18–20,41395
Complex self-association and resulting aggregation morphology were characterized. Ori-396
entational preferences were observed between associating complexes wherein the complexes397
were more likely to present their “equatorial band” region to neighboring complexes within398
20
the long-range pairs. Clusters of associating complexes were quantified and the morphology,399
connectivity and compactness of the largest cluster, being the precursor to the percolated400
system-spanning cluster, was quantified using graph theoretical metrics. The cluster was401
partitioned into communities, with an intra-community connectivity threshold applied to402
identify “dense subclusters” within the mostly linear solvate network. Short-range complex403
pairs were found to disproportionately populate those dense subclusters, implying a sensitiv-404
ity of aggregate morphology to system variables which affect short-range pair formation.24405
Recent studies have proposed a diversity of mechanisms for aggregation and third phase406
formation across a variety of solvent extraction systems.16,17,25,42 Those studies highlight the407
role of molecular-scale interactions, often different between systems, in governing organic408
phase aggregation. The method presented here for identifying the onset of third phase forma-409
tion is independent of the specific type of molecular connectivity. The appropriate choices of410
vertices and edges may differ between systems depending on their underlying molecular-scale411
interactions. Previously, we proposed that third phase formation in the TBP/H2O/HNO3412
system in the absence of metal ions resulted from percolation of the hydrogen bonded net-413
work consisting of those molecules at high extracted water and acid concentrations.16 In this414
study, given the role of coordinating metal ions, we do not observe a percolated network415
of only hydrogen bonds, although TBP in excess of the uranyl nitrate complexes and coex-416
tracted water and nitric acid are present. Rather, the percolation is characterized directly417
in terms of (TBP)2(UO2)(NO3)2 solvate pairing.418
The analysis methodology for solvate coarse graining and cluster partitioning to quantify419
hierarchical organization within the solution could be broadly applied to complex organic420
phases. Treating third phase formation as a liquid-liquid phase transition of a multicompo-421
nent associating molecular solution, and using an appropriate graph theoretic description of422
solute interactions, has enabled simulation prediction of the phase boundary. The extension423
of this approach and its implications to other solvent extraction systems will require fur-424
ther investigation, possibly through experimental quantification of critical behavior near the425
21
phase boundary.43 The interpretation of organic phase aggregation and third phase forma-426
tion outlined in this study may explain how system variables such as solvent or extractant427
molecular structure24 promote or inhibit third phase formation. If simple models, such as428
lattice percolation,44 can sufficiently describe aggregation in solvent extraction systems, they429
could enable high-throughput phase diagram development.430
Conflicts of interest431
There are no conflicts of interest to declare.432
Acknowledgements433
This work was supported by the Nuclear Regulatory Commission Faculty Development434
Award NRC-HQ-11-G-38-0062 and by the Department of Energy Basic Energy Sciences435
Separations program grant DE-SC0001815. We acknowledge the Colorado School of Mines436
High Performance Computing center for computational resources.437
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2018, 555, 713–727.456
(10) Zemb, T.; Bauer, C.; Bauduin, P.; Belloni, L.; Dejugnat, C.; Diat, O.; Duboi, V.;457
Dufreche, J.; Dourdain, S.; Duvail, M.; Larpent, C.; Testard, F.; Pellet-Rostaing, S.458
Colloid and Polymer Science 2015, 293, 1–22.459
(11) Chiarizia, R.; Jensen, M.; Borkowski, M.; Ferraro, J.; Thiyagarajan, P.; Littrell, K.460
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Figure 7: TOC image.
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Supporting Information for: Reimagining third
phase formation as the miscibility gap of a
molecular solution
Michael J. Servis,∗,† David T. Wu,‡,¶ Jenifer C. Shafer,‡ and Aurora E. Clark∗,†,§
†Department of Chemistry, Washington State University, Pullman, WA
‡Department of Chemistry, Colorado School of Mines, Golden, CO
¶Department of Chemical and Biological Engineering, Colorado School of Mines, Golden,
CO
§Pacific Northwest National Laboratory, Richland, WA
E-mail: [email protected]; [email protected]
1
Dense Subcluster Definition and Identification2
The community partition of the largest cluster was generated using the modularity opti-3
mization algorithm reported in ref. 1, where the modularity of a partition, Q, is defined4
as5
Q =1
n
∑ij
(Aij −kikjn
)δ(ci, cj). (1)
Here, A is the adjacency matrix of the partitioned connected component with matrix element6
Aij being 1 if vertices i and j are connected and 0 otherwise. The sum of all elements in the7
adjacency matrix, twice the total number of edges, is n. The number of edges from vertex i8
is ki. The community identity of vertex i is given by ci and δ(ci, cj) is a delta function with9
1
a value of 1 if i and j are in the same community and 0 otherwise. After starting with each10
vertex in its own community, trial moves are made by changing the community membership11
of vertices to that of each of their neighbors. If at least one trial move increased Q, the move12
that most increased the modularity is accepted and the community of the trial vertex is13
updated. Trial moves are made over all vertices iteratively until no increase in Q is possible.14
The dense subclusters identified in the largest cluster are sensitive to the choice of cutoff15
for the minimum number of edges within community per vertex. For all cutoffs tested,16
short-range solvate pairs disproportionately distributed into the dense subclusters. Figure 117
plots the probability density of edges within a community per vertex of that community for18
the largest cluster over the 120-145 ns time range. The probability density is not smooth19
because values are dominated by ratios of small integers. The blue vertical line indicates the20
cutoff of 1.5 chosen in this study, while the red line at 1.0 is a potential lower cutoff. Those21
candidate cutoff values were identified as natural choices due to peaks in the probability22
density at those values. Their effect on dense subcluster identification illustrated in the23
graphs in Figure 1B-C. Lowering the cutoff to 1.0 increases the number of communities24
defined as dense subclusters from 1 to 6. Over the whole 25 ns time range, the fraction of25
largest cluster vertices within dense subclusters is 0.16 using the >1.5 cutoff and 0.45 for the26
>1.0 cutoff.27
Comparison to Bond Percolation on a Cubic Lattice28
Figure 2 compares a largest cluster from a 3D cubic lattice bond percolation Monte Carlo sim-29
ulation near the percolation threshold (top) with the largest cluster of the (TBP)2(UO2)(NO3)230
metal-ligand complex network (bottom). The uranium MD simulation cluster morphology31
is similar to that of the lattice cluster, although it includes dense subclusters of metal-ligand32
complexes vertices which are not readily formed on a cubic lattice.33
2
Figure 1: In the upper left, the probability density of edges per vertex within a communityis plotted for the largest cluster over the 120-145 ns time range. The cutoff values of 1.0and 1.5 are overlayed on the distribution with a red and blue line, respectively. In graph A,the community partition for an example largest cluster is drawn with vertex color indicatingcommunity (with colors reused between nonadjacent communities). Long-range solvate pairsare drawn with dashed lines and short-ranged pairs with solid lines. In B, the vertices in thesix communities which qualify as dense subclusters using the >1.0 edges within communityper vertex are colored in gray for that same graph while the remaining vertices are red. In C,the same coloring is used for the >1.5 criteria. Differences in the graph layout representationsbetween A, B and C are due solely to the stochastic nature of the force-directed graph drawingalgorithm; the graphs are identical apart from vertex coloring.
3
Figure 2: The largest cluster from bond percolation on a cubic lattice (above) is comparedto the largest cluster from a snapshot of the (TBP)2(UO2)(NO3)2-containing molecular dy-namics simulation. The uranium solvate cluster is colored as described in Figure 1.
4
Percolation and Liquid-Liquid Phase Transition of a Bi-34
nary Lennard-Jones Fluid35
Simulations were conducted of a two component Lennard-Jones (L-J) liquid: a given fraction36
of the 1298 total particles were labeled “A” and the remaining labeled “B.” The L-J well37
depth, ε, of the pair potentials was fixed for A-B (unlike) pairs and varied for A-A and38
B-B (like) pairs while keeping εAA = εBB. The particle mass and σii values were chosen to39
represent liquid argon: 39.948 g/mol and 0.34 nm, respectively. Simulations were conducted40
in the NPT ensemble with temperature set to 120 K and pressure to 100 bar. A 1.2 nm L-J41
cutoff was applied and the simulations ran for 10 ns with a 0.2 fs time step. The remaining42
run parameters were as described for the uranyl simulations. A total of 25,960 L-J particles43
were used for the series of simulations at the εunlike/εlike ratio of 0.8 while 1,298 were used44
for the other simulations.45
Clusters of A particles were determined using a A-A distance cutoff of 0.5 nm chosen46
from the first minimum in their radial distribution function. The percolation threshold was47
determined from the cluster size distribution. Figure 3 shows simulation results with an48
overlayed schematic representation of the percolation and liquid-liquid coexistence curves49
which conform to the percolation and phase status of the simulated state points. The50
“percolation line” marks the percolation threshold of the lower number fraction component51
of the L-J fluid. Inside the percolation line, both components percolate.52
References53
(1) Blondel, V. D.; Guillaume, J. L.; Lambiotte, R.; Lefebvre, E. Journal of Statistical54
Mechanics: Theory and Experiment 2008, P10008.55
5
Figure 3: The percolation and phase status of simulations at different number fractions andrelative L-J well depths are plotted for a binary L-J fluid. Data point colors correspond tothe percolation status of each state point. Liquid-liquid coexistence and percolation curves,and the critical point, are merely schematic but are drawn consistently with the simulationresults.
6
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