Absolute thermodynamic properties of moltensalts using the two-phase thermodynamic (2PT)superpositioning method
Jin Wang, Brahmananda Chakraborty and Jacob Eapen*
We show that the absolute thermodynamic properties of molten salts (mixtures of KCl and LiCl) can be
accurately determined from the two-phase thermodynamic (2PT) method that is based on superpositioning
of solid-like and gas-like (hard-sphere) vibrational density of states (DoS). The 2PT predictions are in
excellent accordance with those from the thermodynamic integration method; the melting point of KCl
evaluated from the free energy and the absolute entropy shows close conformity with the experimental/
NIST data. The DoS partitioning shows that the Li+ ions in the eutectic LiCl–KCl molten mixture are largely
solid-like, unlike the K+ and Cl� ions, which have a significant gas-like contribution, for temperatures
ranging from 773 K to 1300 K. The solid-like states of the Li+ ions may have practical implications when
employed for chemical and nuclear reprocessing applications.
1. Introduction
There is an engaging interest in determining the absolutethermodynamic properties such as free energy, enthalpy,entropy and specific heat in the liquid and disordered states.1–7
The term absolute is used in the sense that the thermodynamicproperties can be evaluated directly through the partition func-tion for a given state in the canonical ensemble; there is however,an arbitrary reference energy which is associated with thepotential energy of the system. Knowing the absolute propertiesis particularly useful for optimizing the back-end of the nuclearfuel cycle, which entails the crucial step of separation of actinidesfrom the spent nuclear fuel, followed by the development of newfuel through recycling. The advantage of having a closed fuelcycle includes a reduction of high-level nuclear waste and theoptimal utilization of the fissionable isotopes. While aqueous-based (wet) processes are well-established there is a currentinterest for high temperature pyrometallurgical processes8–10
that utilize molten salts such as LiCl or KCl. A eutectic mixtureof LiCl and KCl is particularly favored because of its lowermelting point (626 K) relative to that of the constituent salts(883 K for LiCl and 1043 K for KCl). Originally developed forspent metallic fuel from the fast reactors, pyro processes canalso be used for spent fuel from the current light water reactors(LWRs) using electroreduction, electrorefining, and pyropartitioning;they are also regarded to be more proliferation-resistant thanthe traditional wet processes.
Reprocessing of nuclear fuel involves a large number offissionable actinides8,11 as well as non-fissionable fission products;the separation processes typically depend on the chemical activityof the species, which in turn depends on the relative free energy.10
Thermodynamic integration (TI), which is the method of choicefor a variety of applications,3,6,12–15 especially with moleculardynamics (MD) simulations, may not be the most optimal choicefor nuclear reprocessing applications given the inordinate numberof species that are targeted for separation. In several instances,TI can be limited by the optimal choice of the integration path thatis not known a priori, and is also constrained in having a reversepath that implicitly excludes a first order phase change. Anotherchallenge arises in proscribing appropriate reference systems,which can allow computation of absolute thermodynamic pro-perties. While TI can successfully predict free energy differentialswith actinide transmutation,10 integration paths typically requiretrajectories that span over 1 ns, which is expensive by currentstandards. A problem of a different nature may arise with actinidedissolution in molten salts – depending on the volume fraction,the interaction strength and the thermodynamic state, the dis-solved species may form glassy states or even undergo a first-orderliquid–liquid phase transition,16–20 which is somewhat poorlyunderstood currently. Very strong attractive interactions have beenshown to cause an amorphous transition in a model nano-colloidal system with a volume fraction as low as 5%.21 Thus itwould be profitable to identify a methodology that can obviatesome or all of the aforesaid possible drawbacks for systems withlarge number of species in complex liquid forms.
Originally developed by Lin, Blanco and Goddard,22 the two-phase thermodynamic (2PT) model has recently gained wide
Department of Nuclear Engineering, North Carolina State University, Raleigh,
attention and success in predicting absolute thermodynamicproperties for several types of fluids.1,2,4,5,7,23–26 Motivated bythe approach of evaluating the partition function in the solidstate using idealized harmonic oscillators, the central ideabehind the 2PT methodology is to construct the partitionfunction of a complex liquid state through a superposition ofsolid-like and gas-like vibrational density of states (DoS). Anexact representation of the idealized states can be derived bytreating the gas-like state through hard-sphere interactionswith weighting functions provided by the Carnahan–Starlingequation of state,27 and assuming the solid-like state to be acollection of independent harmonic oscillators. The thermo-dynamic properties of a liquid system thus can be shown to bea combination of the properties of a set of harmonic oscillators(solid-like) and a set of hard spheres (gas-like). Since the 2PTmethod thus far has not been employed to evaluate the pro-perties of molten salts and mixtures, its efficacy or accuracy iscurrently unknown.
In this investigation, we show that the absolute thermo-dynamic properties of LiCl/KCl molten salts can be accuratelydetermined from the 2PT method. The 2PT free energy is inexcellent agreement with that determined from the thermo-dynamic integration (TI) method, with relative errors of 1% orless, typically. The melting point of ionic KCl (1025 K) comparesfavorably with the experimental data (1041 K); the entropy ofthe melt phase also shows good conformity with the recom-mended data from NIST. We then determine the absolute freeenergy and entropy which are currently unknown for a eutecticmixture of LiCl and KCl for temperatures ranging from 773 K to1300 K. From the partitioning of the DoS, which is uniquelypossible in the 2PT method, we show that the Li+ ions in theLiCl–KCl mixture are dominantly solid-like even at tempera-tures as high as 1300 K. For chemical and nuclear applicationsthe solid-like states can potentially impose practical limits,for example, to the amount of dissolved species that can bepractically accommodated during reprocessing.
2. The 2PT methodology
The thermodynamic properties in the canonical ensemblecan be evaluated from the canonical partition function (Q);for example, the internal energy (E), entropy (S), specific heat atconstant volume (Cv), and the Helmoholtz free energy (A) can beevaluated as28
E ¼ kBT2 @
@TðlnQÞ
� �(1a)
S ¼ kB T@
@TðlnQÞ þ lnQ
� �(1b)
Cv ¼@
@TkBT
2 @
@TðlnQÞ
� �� �(1c)
A = �kBT(ln Q) (1d)
where kB is the Boltzmann constant and T is the absolutetemperature. The thermodynamic properties are seldomevaluated through the partition function as it is known exactlyonly for idealized systems such as harmonic oscillators andideal gas. Using a normal mode analysis, a solid state canbe approximated as a system of non-interacting harmonicoscillators (HO); for this case, the total canonical partitionfunction (QHO) can be expressed as29
QHO ¼Y3Ni¼1
qi (2)
where qi is the partition function of the ith harmonic oscillatormode, and N is the total number of oscillators. The totalpartition function can be shown to be related to the densityof states as29
ln QHO� �
¼ð10
dnGðnÞ ln½qðnÞ� (3)
where G(n) is the density of states, which is given by22
GðnÞ ¼ 4
kBT
ð10
dtCðtÞe�i2pnt (4)
where C(t) is the mass-weighted velocity autocorrelationfunction,28 which can be directly computed from a moleculardynamics (MD) simulation (ab initio or classical). Since an exactquantum-mechanical expression is known for q(n), the totalpartition function can be evaluated using eqn (3) and (4), andthe thermodynamic properties such as given by eqn (1a) to (1d)can be assessed for the solid states with reasonable accuracy.29
For a liquid state, the system cannot be considered as acollection of harmonic oscillators and a direct application ofthe above approximation will lead to non-physical results. The2PT method22 extends the above idea to the liquid state byassuming a superposition of two idealized states – hard spheres(HS) that correspond to a gas-like state, and harmonic oscillatorsthat correspond to a solid-like state. First it is hypothesized thatthe total density of states of the system that is now regarded tobe a non-interacting mixture of hard spheres and harmonicoscillators can be decomposed as
Gk(n) = GHSk (n) + GHO
k (n) (5)
where k denotes the different species in the mixture. A key stepin the 2PT method is in defining and evaluating a fluidicityparameter ( f ) given by the following expression for partitioningthe system into the idealized states.
fk ¼Ð10 dnGHS
k ðnÞÐ10 dnGkðnÞ
¼ 3NHSk
3Nk¼ NHS
k
Nk(6)
where NHSk is the effective number of hard sphere atoms or ions
of the kth component of the mixture. Note that the integral ofthe total density of states over all frequencies is simply 3Nk.Thus the 3Nk degrees of freedom of the kth component isassumed to be composed of 3NHS
k or equivalently, 3fkNk hardsphere or diffusive degrees of freedom, and 3(Nk � NHS
k ) orequivalently, 3Nk(1 � fk) solid-like or non-diffusive degrees offreedom.25 With this approximation, the thermodynamic
where W is a weight function for the appropriate property,and E0
k is the reference energy which is given by ref. 25
E0k ¼ EMD
k � 3NkkBT 1� fk
2
� (8)
where EMDk is the total energy from MD simulations for the kth
component. Thus each extensive thermodynamic property, foreach mixture component, comprises of two contributions – onefrom diffusive (hard spheres) degrees of freedom, and the otherfrom vibrating (harmonic oscillators) degrees of freedom. Thedistinguishing part of the 2PT method is in deriving an exactexpression for GHS
k given by ref. 22.
GHSk ðnÞ ¼ Gkð0Þ 1þ pGkð0Þn
6fkNk
� 2" #�1(9)
Thus the density of states for the hard sphere phase can becalculated as a function of the fluidicity parameter and Gk(0)which is just the zero-frequency value of the system (total)density of states. MD simulations are employed to calculateGk(n) from the velocity autocorrelation function. With a knownGHS
k , the density of states of the HO phase is then evaluated asGHO
k (n) = [Gk(n) � GHSk (n)]. The weight functions are known
exactly for the HO and HS idealized states, and are given by5,25
WHOE ¼ bhn
2þ bhnebhn � 1
(10a)
WHOS ¼ bhn
ebhn � 1� ln 1� e�bhn
� �(10b)
WHOCv¼ ðbhnÞ
2ebhn
ebhn � 1ð Þ2(10c)
WHOA ðnÞ ¼ ln
1� e�bhn� �e�bhn=2
� �(10d)
WHSEkðnÞ ¼WHS
Cv;kðnÞ ¼ 1
2(11a)
WHSSkðnÞ ¼ sHS
k ðnÞ3kB
(11b)
WHSAkðnÞ ¼ WHS
EkðnÞ �WHS
SkðnÞ
h i¼ 1
2� sHS
k ðnÞ3kB
� (11c)
where b�1 = kBT, h is the Planck constant and sHS is the entropyof the hard sphere phase (per atom/ion), which in the 2PTmodel is evaluated as
sHSk ¼ sIGk þ kB ln zk ykð Þ½ � þ yk 3yk � 4ð Þ
1� ykð Þ2
( )(12)
The first term of the above equation denotes the ideal gas entropy,however, weighted only for the HS phase. It is given by ref. 5:
sIGk ¼ kB5
2þ ln
2pmkkBT
h2
� 3=2Vk
fkNk
� " #( )(13)
where mk and Nk are the mass and the total number of ions/atoms of the kth species, respectively. The term zk � PVk/(NkkBT)in eqn (12) delineates the compressibility of the hard spherephase, which is determined from the Carnahan–Starling equa-tion given by
zk ykð Þ ¼1þ yk þ yk
2 � yk3
1� ykð Þ3(14)
where yk is the weighted hard sphere packing fraction (by thefluidicity parameter) for the kth component, and is given byyk = fkyk, where yk denotes the packing fraction which isexpressed as
yk ¼prk sHS
k
� �36
(15)
where rk is the number density and sHSk is the hard sphere
diameter of the kth species that in turn influences the fluidicityparameter (fk). In the 2PT methodology it is assumed that thefluidicity parameter is simply a ratio of the self-diffusivity (Dk)to the hard sphere diffusivity at zero pressure (D0HS
k ), which canbe written as
f ¼ D T ;Pð ÞD0HS
k T ;P; sHSð Þ(16)
The self-diffusivity can be evaluated from MD simulationsas the time integral of the velocity autocorrelation, and isexpressible as
Dk T ;Pð Þ ¼ kBTGkð0Þ12mkNk
(17)
where Gk(0) is evaluated from the expression
GkðnÞ ¼4
kBT
ð10
dtCkðtÞe�i2pnt (18)
We invoke the stationary principle of classical correlationfunctions,30 and thus only the real part of the Fourier transformneeds to be used for calculating the above integral. Note thatC(t) is the mass-weighted velocity autocorrelation functiondefined as
where ck(t) is the velocity autocorrelation function of the kthcomponent. The hard sphere diffusivity in the limit of zeropressure/density is computed exactly from the Chapman–Enskog’s theory, which is given by
D0HSk T ;P; sHS
k
� �¼ 3
8rk sHSk
� �2 kBT
pmk
� 1=2
(20)
The velocity autocorrelation for HS phase decays exponentiallyin time and hence the HS diffusivity can be expressed as22
DHSk T ; fkrkð Þ ¼ kBTGkð0Þ
12mkfkNk(21)
The above expression can be related to the zero-pressure hardsphere diffusivity through the following relationship
DHSk T ; fkrkð Þ ¼ D0HS
k T ; fkrk; sHSk
� � 4ykz ykð Þ � 1½ � (22)
where yk = fky. After a few algebraic steps the following relation-ship can be established25
2Dk�9/2fk
15/2 � 6Dk�3fk
5 � Dk�3/2fk
7/2 + 6Dk�3/2fk
5/2 + 2fk � 2 = 0(23)
where
Dk ¼2Gkð0Þ9Nk
pkBTmk
� 1=2Nk
Vk
� 1=36
p
� 2=3
(24)
Eqn (24) contains the details of the physical state as well asthe zero-frequency density of states that can be evaluated fromMD simulations. Eqn (23) now can be iteratively solved to getthe fluidicity parameter (fk) for each component of the mixture.Once f is known, the density of states and the weight functionsof the HS phase can be computed, and the HO density of statescan be determined as GHO
k = (Gk � GHSk ). Thus all the properties
given by eqn (7a) to (7d) can be determined, for each compo-nent of the mixture.
In this investigation, the partial volume (Vk) in eqn (24) iscalculated from the relative ionic radii, which is given by
Vk ¼sk3Pn
k¼1xksk3
V
N
� (25)
where sk and xk are the ionic radius and mole fraction,respectively, of the kth component. The Kirkwood and Buff(KB) theory31 gives a better approximation for partial volumesbut it entails the evaluation of somewhat ill-convergent KBintegrals. We have used a fixed ionic radius of 0.152 nm,0.167 nm and 0.09 nm for K+, Cl� and Li+ ions, respectively.32
As shown later, the ionic radius approximation works quitewell for the current application. We have also observed thatthe calculated thermodynamic properties are not sensitive tomoderate changes in the ionic radii. Finally, the molar (m)thermodynamic properties of the mixture are determined as
�Em ¼
Xnk¼1
xkEmk (26a)
�Sm ¼
Xnk¼1
xkSmk � R
Xnk¼1
xk ln xkð Þ (26b)
�Am ¼
Xnk¼1
xkAmk þ RT
Xnk¼1
xk ln xkð Þ (26c)
where R is the universal gas constant, and n is the total numberof components in the mixture.
3. Molecular dynamics simulations
MD simulations are principally employed for calculating thevelocity autocorrelation function. The molecular system typicallyconsists of 1000 ions interacting through a long-ranged electro-static potential and a Born–Huggins–Mayer (BHM) short-rangedpotential. Originally developed by Fumi and Tosi,33 the rigid-ionpotential is able to predict several thermodynamic, structuraland transport properties of molten LiCl–KCl mixtures34,35 and otheralkali halides33 with reasonable accuracy. The effect of polarization,however, is important for multicomponent systems,36–39 although ittends to be small for monovalent systems.
The functional form of short-ranged part of the BHMpotential is given by
Fshortij ¼ Bije
�arij � Cij
rij6�Dij
rij8(27)
where rij is the interionic separation between two ions, i and j, anda, B, C and D are constants.34,35 While the first term correspondsto the electron cloud repulsion, the second and third termsaccount for the dipole–dipole and the dipole–quadrupole disper-sion interactions, respectively. The MD simulations are performedusing the LAMMPS software40 with periodic boundary conditionsat zero pressure using a timestep of 1 fs. Particle–particle–particle–mesh is used for columbic interactions, and the temperature andpressure are controlled using the Nose–Hoover algorithm. Fromthe velocity data collected from MD simulations, the velocityautocorrelation function (VACF) is constructed, typically with acorrelation time of 25 ps, using an overlapped data structure.41
From the Fourier transform of the VACF, the 2PT thermodynamicproperties are computed using an in-house computer program.
4. Results and discussion
We will first discuss (1) the numerical convergence of thefluidicity parameter with correlation time, followed by (2) abenchmark investigation of the 2PT method against publishedresults from thermodynamic integration (TI) method for KCl,(3) computation of the molar free energy and the entropychange across melting in KCl along with a comparison to NISTdata,42 and (4) an assessment of the density of states andthermodynamic properties of a eutectic mixture of LiCl–KClfor different temperatures.
4.1 Numerical convergence of the fluidicity parameter
The fluidicity parameter for each component of the mixture isiteratively solved using eqn (23). Further the partitioning of the
system into solid-like and gas-like states depends on resolvingthe zero-frequency value of the density of states. To assess theoptimal correlation time (for VACF), we have first computedfk using eqn (23) and then compared with the reconstructedfk from the equation
f k ¼Ð10 dnGHS
k ðnÞÐ10 dnGkðnÞ
¼ 1
3Nk
ð10
dnGHSk ðnÞ (28)
Insufficient sampling and numerical errors in the evaluation ofVACF and the density of states will be reflected as a differencebetween fk and fk. Fig. 1 shows the error in the reconstruction,expressed as a percentage, for the K+ ions in molten KCl at atemperature of 1100 K and at a density of 20 nm�3. At shortcorrelation times that are less than 5 ps, the numerical errorsare significant; they however, reduce almost exponentially withincreasing correlation time. It is interesting to note that the rateof decrease of the relative error is considerably weaker beyondB20 ps; in our simulations, we have used a fixed correlationtime of 25 ps, which generally results in a reconstruction errorof 3 to 4%.
4.2 Benchmarking of the 2PT method
We will now turn to benchmarking the predictions from the2PT method with published data on free energy of molten KCldetermined from the thermodynamic integration (TI) method.Absolute free energy has been computed by Rodrigues andFernandes using a coupling TI method with the BHMpotential.14 The Einstein crystal method is used to computethe absolute free energy of the solid state, while a two-stepprocedure is employed for the liquid state. First the BHMpotential is converted to a half-wing repulsive Gaussianpotential with the repulsive barrier kept close to that of theoriginal potential. Next the Gaussian potential is converted intoa null potential to transition into the ideal gas reference statewith known thermodynamic properties. For the liquid states,checks have been made to ensure that the TI paths arereversible and no phase change occurs along them.14
The free energy computed by the 2PT method (circles) andcoupling TI method14 (squares) shown in Fig. 2 confirms theaccuracy of the 2PT method against a well-established method;the relative error between the predictions is typically less than 1%.Unlike the coupling TI method, the 2PT method requires onlymodest computing time to derive the free energy with comparableprecision. Also shown in the figure is the reconstructed freeenergy (dotted line) from a thermodynamic integration approachusing the internal energy computed from the 2PT method.5 Firstthe entropy at a certain temperature (T2) is computed as
Sk T2ð Þ ¼ Sk T1ð Þ þðT2
T1
dT1
T
dEk
dT
� 2PT
(29)
where Ek is the internal energy determined from the 2PT methodas shown in eqn (7a) for the kth component of the mixture. Theinternal energy is next linearly fitted to temperature which gives aconstant derivative for Ek (the specific heat at constant volume).Then eqn (29) is integrated (numerically or exactly) to evaluatethe entropy at different states, followed by the reconstructionof the free energy as Ak = Ek � TSk. As evident from Fig. 2, thereconstructed free energy is in close agreement with the originalestimate; somewhat better agreement can be obtained from ahigher order fit to the energy variation with temperature.
4.3 Free energy and entropy change across melting in KCl
A first-order phase transition occurs at the thermodynamicmelting point, which is exemplified by a slope change for thefree energy along with a discontinuity in properties such asentropy. In Fig. 3, the free energies of the solid and liquidphases are plotted as a function of temperature – a discerniblechange in the slope occurs at B1025 K, which corresponds tothe melting point of KCl. This 2PT estimate compares favorablywith the experimental melting point of 1043 K with a relativeerror of 1.5%, approximately. It may be noted that the fluidicityparameter for the solid states is negligible.
The melting point is also measurable from other propertiesfrom a MD simulation such as from the change in the structurefunction; the 2PT method however, is particularly advantageousas it can compute the properties such as entropy and specific
Fig. 1 Relative error in reconstructing the fluidicity parameter for theK+ ions in molten KCl.
Fig. 2 Comparison of absolute (molar) free energy of molten KCl fromthe 2PT and TI14 methods at a temperature of 1100 K and a density of20 nm�3.
heat which undergo a discontinuity across the phase change.The bottom inset shows an entropy jump of 0.0273 kJ(mol K)�1
across melting which agrees very well with the NIST data of0.0269 kJ(mol K)�1 (ref. 42) at standard conditions, incurring arelative error of 1.5%. However, there is a systematic over-prediction by the 2PT method leading to a relative error of2–3% between the absolute entropy values for the liquid statesand somewhat higher for the solid states; the difference canpartly arise from the inaccuracy of the interionic potential. Thedotted lines in the bottom inset correspond to the recon-structed entropy values obtained by integrating eqn (29) – the
close agreement again highlights the self-consistency of the 2PTmethodology. From the top inset, it can be observed that thefluidity parameter ( f ) shows a jump upon melting; as expected,the fluidicity values are close to zero for the solid state.
4.4 Molar free energy and entropy of eutectic LiCl–KClmixture
Lithium ions in the LiCl–KCl mixtures behave very differentlyfrom the potassium or chlorine ions; our recent investigationshows that the molar flux correlations for the Li+ group (Li+–Li+,Li+–Cl� and Li+–K+) behave very differently from the K+ group.43
Of particular interest is the formation of long lived dynamical cagefor the Li+–Li+ and Li+–Cl� interactions, even at high tempera-tures, which has been ascribed to the intermediate-ranged order-ing of the Li+ ions.35,36 An incomplete mixing of LiCl and KCl hasbeen put forward as a plausible mechanism for the intermediate-ranged ordering, which manifests as a pre-peak in the structurefunction at 1 Å, approximately. In the current study, we investigatethe density of states, particularly the partitioning into solid-likeand gas-like components, and probe the dynamical characteristicsthat determine the thermodynamic properties.
The total DoS for the K+ ions (see Fig. 4) is similar to thatobserved in typical liquids with a discernible zero-frequencycomponent that augments with increasing temperature, and asingle peak at a frequency of B2 THz. In contrast the total DoSfor the Li+ ions portrays a plateau-like peak region spanning abroad range of frequencies, which is well-preserved even at hightemperatures. Interestingly, the magnitude of the total DoS at zerofrequency for the Li+ ions is almost zero indicating a solid-likebehavior, which is further affirmed by the DoS partitions; the HOcontribution practically accounts for the total DoS with negligible
Fig. 3 Molar free energy across melting in KCl with the 2PT method.(bottom inset) Entropy jump from the 2PT method and comparison toNIST standard entropy data (open circles).42 (top inset) Change in thefluidicity parameter (f) across melting.
Fig. 4 Density of states (DoS): total (left column), solid-like (HO, middle column), and gas-like (HS, right column) for Li+ (top row) and K+ ions (bottom row)at different temperatures (773 K to 1300 K). The DoS plots for different temperatures – 773 K, 850 K, 950 K, 1043 K, 1100 K, 1150 K, 1200 K, 1300 K – areoff-set by 40 ps for visual clarity.
contribution from the hard spheres. The Li+ behavior is consistentwith the persistent and prolonged backscattering observed in themolar flux correlations,43 as well as the presence of theintermediate-ranged ordering.36 The total DoS for the K+ ions,on the other hand, is dominated by the contribution from theharmonic oscillators at low temperatures; as temperatureincreases, the contribution from the hard spheres also increasessignificantly. Interestingly, the total and partial DoS characteristicsfor the Cl� ions are very similar to those of the K+ ions; however,with a more prominent contribution from the hard sphere phase.Thus in the molten LiCl–KCl state, even at high temperatures,Li+ ions depict a predominantly solid-like characteristic, while theK+ and Cl� ions portray a more conventional fluid-like behaviorwith a significant contribution from the hard sphere phase.
Next we show the free energy and entropy of LiCl–KCl mixtureat eutectic composition for different temperatures in Fig. 5 (for1 mole of LiCl and 1 mole of KCl). With the addition of LiCl toKCl, the melting point reduces from 1043 K and attains thelowest value of 626 K at the eutectic composition (xKCl = 0.42). Asexpected the entropy increases with increasing temperature andshows a somewhat linear behavior from B1050 K, which per-haps, is related to the occurrence of melting of pure KCl aroundthis temperature (1043 K); our recent work also shows aninstability in the K+–Li+ Maxwell–Stefan (MS) diffusivity atB1100 K.43 Further investigations are needed to ascertain thephysical significance, if any, for this behavior.
5. Concluding remarks
The two-phase thermodynamic (2PT) method developed by Lin,Blanco and Goddard22 is emerging as a robust and accuratemethodology for determining the absolute thermodynamic pro-perties of fluids and fluid mixtures. Although the method hasonly been recently developed, the idea that the liquid state canbe considered as an intimate mixture of solid-like and gas-likecomponents dates back to the early work of Eyring and Ree44
who proposed that a liquid molecule briefly exhibits solid-likeproperties before showing a gas-like behavior as it jumps intoneighboring vacancies. In the 2PT method, the partition
function of a complex liquid state is constructed through asuperposition of solid-like and gas-like vibrational density ofstates (DoS). We have applied this method to calculate theabsolute free energy and entropy of molten KCl, and LiCl–KClmixture at eutectic composition. The 2PT predictions showexcellent agreement with those from a coupling thermodynamicintegration (TI) method; for comparable states, the relative erroris typically less than 1%. Further the 2PT method accuratelypredicts the melting point of molten KCl and the entropy changeacross melting with errors not exceeding few percents.
The 2PT method is less reliant on prior judgment – aspointed out,13 the TI method generally benefits from thejudicious choice of potential parameters/optimal paths fordetermining the absolute properties, which are not alwaysknown, a priori. As shown in this investigation, the 2PT methodis robust and can handle complex liquids and mixtures withrelative ease. These desirable features make the 2PT methodvery appealing for chemical and nuclear reprocessing applica-tions where the processes entail a large number of dissolvedspecies. Further there is a satisfying physical basis for the 2PTmethod, which allows the extraction of dynamical characteristicsthat determine the thermodynamic properties; as elucidatedbefore, the 2PT method is unique in this aspect. We have shownthat the Li+ ions in molten LiCl–KCl have very dominating solid-like and practically negligible fluid-like characteristics, unlikethe K+ and Cl� ions. This may have practical implications – forexample, it is easier for the dissolved species in the presence ofsolid-like ions to change the solvent characteristics, say to highlyviscous, glassy states.21 Thus there may be a trade-off in having aeutectic mixture of LiCl–KCl – on one hand, a significantly lowermelting point can be achieved that is of immense value from apractical point of view, while on the other, the presence of solid-like ions, particularly close to the melting point, may not beoptimal for reprocessing and waste recovery applications.
Acknowledgements
Funding from DOE-NEUP program is gratefully acknowledged.The authors are thankful for the useful discussions with DaneMorgan at the University of Wisconsin.
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