Introduction

A temperature gradient in a solution can induce mass flux, which is a phenomenon termed thermodiffusion (or the Soret effect1, named after Charles Soret, who formally described it in 1879, or thermophoresis in the context of colloidal and biological science). Soret’s experiment investigated the migration of NaCl and KNO3 in water under a temperature gradient, building on earlier observations by Carl Ludwig2 of NaSO4. Thermodiffusion is ubiquitous in different systems, including gases3, hydrocarbon mixtures4, biofluids5, colloidal suspensions6, and ionic solutions7. Consequently, this has led to important applications in air purification8, assessment of oil reservoirs4, uranium enrichment9, and in determining ligand-protein binding affinities10,11. Thermodiffusive water treatment has also recently been proposed12,13, being the first thermal desalination method to operate in a single phase without membranes. Though often understood via an analogy to Fickian diffusion, thermodiffusion is more complex in the observed behaviour. In Fickian diffusion, species transport always opposes the concentration gradient (with a positive mass diffusion coefficient D), whereas species transport in thermodiffusion under non-isothermal conditions can occur towards the hot (thermophilic) or the cold (thermophobic) regions. Thus, the thermodiffusion coefficient DT can be positive or negative14. The temperature at which the transition between thermophilic and thermophobic behaviour occurs is termed the inversion temperature, T0. A prediction of T0 is important for designing thermodiffusion-based applications.

Although thermodiffusion was discovered more than 150 years ago2, the underlying mechanisms are not fully understood. Unlike Fickian diffusion, which can be largely explained with Brownian motion, there are different contributing factors to thermodiffusion, and the mechanisms for different types of systems are distinctively different6. Some hypothesise that thermodiffusion in aqueous electrolyte solutions is driven by changes in solvation and ionic shielding entropies with temperature, creating an entropic force15,16. Such forces are hard to quantify, but it has been argued that structure formation at low temperatures minimises the free energy, whilst structure breaking due to larger translational entropic gains minimises the free energy at high temperatures17. In the case of aqueous solutions, structural changes in the hydrogen bond network could play a role in the change in entropy and in the addition of enthalpic contributions18, and these effects differ with the ion types and solvents19. In contrast, other studies on ionic solutions stress the creation of local and global electric fields, reducing this to an electrophoretic effect20. T0 is a characteristic of thermodiffusion, and studying this could shed light on the competing effects at play21,22. T0 has only been used to analyse the thermodiffusion mechanism of associated binary mixtures23 through its correlation with the ratio of the vaporisation enthalpy of pure components. However, to the best of our knowledge, T0 has never been used to study thermodiffusion in aqueous alkali halide solutions, in part because of challenges associated with the measurement and modelling, both of which are addressed in this study.

Thermodiffusion in aqueous salt solutions could be influenced by a combination of ion–water and ion–ion enthalpic and entropic contributions, as well as the influence of the salt on the surrounding water–water interactions and structure. We are interested in exploring a theory proposed by Eastman24 and further examined by Duhr and Braun15 stating that the hydration entropy (Fig. 1a, left panel), Shyd, and the entropy of ionic shielding (Fig. 1a, right panel), Ssh, are two major contributors to thermodiffusion. In terms of the Soret coefficient ST(T)

$${S}_{{{{\rm{T}}}}}(T)\propto -\left[{{{{\rm{S}}}}}_{{{{\rm{hyd}}}}}(T)-{{{{\rm{S}}}}}_{{{{\rm{sh}}}}}(T)\right].$$
(1)

By investigating a large concentration range, the effect of decreasing ion–ion interactions should manifest. In Fig. 1a, at infinite dilution, the entropy of ionic shielding is zero, and the effect of lone ion hydration entropy can be investigated in isolation (left panel). On the basis of this theory, because of the negative ion hydration entropy, the ions should be thermophobic at infinite dilution. That is, as C → 0, Ssh(T) → 0 and Shyd < 0, thus [Shyd(T) − Ssh(T)] < 0 and the ions are always thermophobic (ST > 0). This was also predicted by Eastman almost 100 years ago24, who similarly linked thermodiffusion to entropic contributions. We aim to clarify the relationship between T0 and concentration C in aqueous alkali halide solutions (LiCl, NaCl, KCl, NaF, NaI) as a means of testing the implication of the theory, i.e. as C → 0, the ions are always thermophobic (ST > 0) because Shyd < 0, and therefore no T0 can be measured.

Fig. 1: Contributors to thermodiffusive behaviour.
Fig. 1: Contributors to thermodiffusive behaviour.
Full size image

a Hydration entropy (lone ion and solute interaction) and ionic shielding (ion–ion interactions and entropy) contribute to the thermodiffusive behaviour of ions. At infinite dilution, ionic shielding is either absent or minimal in case of a lone contact ion-pair, and the thermal preference of the thermodiffusive transport is only determined by the hydration entropy. Since the hydration entropy is generally negative, the ionic species are thermophobic and there is no inversion temperature T0. b Bulk MD simulations predicting T0 for NaCl in water as a function of concentration. The error bars are standard deviations of the goodness of fit for each of the replicates in the fitting of T0. At low concentrations in the highlighted region, the NEMD methodology becomes computationally expensive to obtain suitable statistics and has significant variance (Supplementary Fig. 1). c The simulation result for the data point at C = 0.055 m, i.e. mean molar fraction of 0.00106. The simulation box (adapted from Xu et al.12 with permission of Nature Communications) measures 50 × 50  × 200 Å3 and the hot (red) and cold (blue) thermostats of width 10 Å are placed at z = 50 and 150 Å. The grey arrows indicate the direction of mass flux. The concentration profile within the highlighted square is plotted below the simulation box and is the cumulative concentration averaged over time between tini = 50 ns and tend = 320 ns. The error bars in the concentration profile represent the temporal fluctuation of the concentration in the replicates. The horizontal red bar is the error bar in the T0(C) plot and represents the standard deviation of different T0 values calculated for each replicate where a good fit to the theoretical function26 is possible.

To test this theory, a combination of experimental and modelling methods should be used: T0 should be experimentally measured for different concentrations to obtain the trend T0(C); since the experimental resolution becomes insufficient when C → 0, modelling is required for a system containing the minimum number of ions to complement the experiment. In this work, the methods used include phase-shifting interferometry (PSI) measurements25, non-equilibrium molecular dynamics (NEMD) modelling26, and MD analysis of free-energy perturbation (FEP)27. Direct measurement of T0 in a concentration range down to 0.02 m using digital interferometry25 was achieved. The Gibbs free energy of hydration ΔΔG for lone ions (1 ion per 883 water molecules equating to a concentration of 0.063m) is computed by simulations of free energy perturbation (FEP) to understand the energetic and structural changes that occur at different temperatures. Through experimental results, we found that at a moderate concentration, alkali halide solutions tend to have the highest T0. When C → 0, i.e. infinite dilution, T0 drops, making thermophobic behaviour more dominant. Still, when extrapolating T0(C) on the linear axis, T0 seems to have a finite value above 0 °C as C → 0. However, since the entropy of a system is proportional to the number of ions under dilute conditions, T0 is plotted on a basis of \(\log (C)\), and the tested salts tend to a purely thermophobic behaviour. When it comes to modelling, FEP results show that only thermophobic behaviour could occur at C = 0.063 m for these alkali halides, as ΔΔG shows a monotonic drop with decreasing temperature. One exception seems to be I− which shows non-monotonic ΔΔG change with temperature. However, when considering its counterion Na+, FEP of NaI shows no inversion. This suggests that Ssh can still be a factor contributing to thermodiffusion even at ultra-dilution, and counterion contributions should be carefully considered for ions with a small ΔShyd. Furthermore, utilising principles from statistical mechanics, we generated an entropic model of these aqueous alkali halide solutions based on microscopic configurations and interactions of particles, which qualitatively represent the trends in T0 when changing the concentration or salt identity. Our results clarify the underpinnings of thermodiffusion in alkali halide solutions across concentration ranges, especially towards infinitely dilute conditions.

Results

“Sampling issue” with bulk molecular dynamics (MD) simulations

MD simulations are often used to understand the thermodiffusion of different systems at the molecular level. Most prior MD simulations examine thermodiffusion by modelling a relatively large number of molecules under a temperature gradient and analysing the time-averaged local concentration along that temperature gradient26,28,29. In Fig. 1b, we present NEMD data for T0(C) in aqueous NaCl, following the same method as in Hutchinson et al.26 that identified TIP3P-FB as the “best performing” water model for predicting ST. It is believed that under a “moderate” temperature gradient, local equilibrium still exists in the thermodiffusive systems. Experimentally, gradients are generally about 10−7 K Å−1 or smaller (e.g. approximately 5 K across 5 mm in measurements based on interferometry). In contrast, large temperature gradients (on a scale of roughly 1 K Å−1) are often used in NEMD to have a computationally functional system size with experimentally relevant temperature ranges. However, to quantify the valid range for assuming a “moderate” temperature gradient, temperature changes should occur gradually over distances much greater than molecular scales. This assumption has been shown to hold for 10−1 K Å−1 and even 102 K Å−1 in supercritical fluids30,31.

Reasonable statistics of ion distribution across the simulated temperature range were obtainable at high concentrations with less than five replicates. However, at low concentrations, NEMD required extremely long timescales and many replicates to reliably average the subtle concentration differences of a small number of ions, which we refer to as the “sampling issue”. For example, in Fig. 1c, the local ion concentration distribution across a temperature range in a 0.055 m NaCl solution is presented. The level of variance and noise is significant at low concentrations, as seen in Supplementary Fig. 1. The concentration range with the sampling issue in our simulations is highlighted in Fig. 1b, and the error bars are standard deviations of the goodness of fit for each of the replicates in the fitting of T0. While the NEMD results suggest that there is no T0 greater than 0 °C when C = 0.055 m, the results are subject to considerable uncertainty, limiting the strength of any definitive claims, and investigating other salts becomes computationally expensive. One seemingly simple solution is to extend the simulation time, but this requires significant computing resources. For these reasons, we looked beyond NEMD to model thermodiffusion at ultra-dilution and instead utilised FEP as a means of determining how ions thermodynamically favour each temperature condition. Although FEP is an equilibrium measure of Gibbs free energies, they have previously been correlated with thermodiffusive properties in isothermal titration calorimetry experiments of protein–ligand systems11, and in the measurement of the Soret and Seebeck coefficients of salt aqueous solutions27. The success of utilising equilibrium processes in explaining thermodiffusion, a non-equilibrium phenomenon, has been discussed multiple times15,27,32.

Lone-ion free-energy perturbation (FEP) simulations

We conducted lone-ion molecular dynamics FEP simulations to predict thermodiffusive behaviour at ultra-dilute conditions. Based on previous works15,24,33, the Soret coefficient for a single ion was defined as:

$${S}_{{{{\rm{T}}}}}^{{{{\rm{i}}}}}=\frac{1}{{k}_{{{{\rm{B}}}}}T}\frac{\delta {G}_{{{{\rm{hyd}}}}}}{\delta T},$$
(2)

where Ghyd is the Gibbs free energy of hydration of the lone ion at local equilibrium. Therefore, using the methodology shown in Fig. 2a and detailed in the Methods and Supplementary Fig. 2, we obtained a series of ΔGhyd at various temperatures and used the value at 270 K, ΔGhyd(270 K), as a reference. ΔGhyd from these FEP results agree well with the experimental results in the literature34, as shown in Supplementary Fig. 3 and Supplementary Table 1. Calculating the Gibbs free energy difference ΔΔGhyd(Tk − Tref = 270 K) = ΔGhyd(Tk) − ΔGhyd(Tref = 270 K) (termed ΔΔG for simplicity) allows direct comparisons of the temperature dependence of each ion. From the relationship between ΔGhyd and ΔT for each ion, we can determine the sign of the Soret coefficient for the lone ion \({S}_{{{{\rm{T}}}}}^{{{{\rm{i}}}}}\). Increasing ΔGhyd with T represents thermophobic movement, while decreasing ΔGhyd with T represents thermophilic movement.

Fig. 2: Advancements in numerical and experimental methodologies.
Fig. 2: Advancements in numerical and experimental methodologies.
Full size image

a Computational method using free-energy perturbation (FEP) simulations. At infinite dilution, the temperature dependence on the ion hydration properties can be calculated from ΔΔGhyd(Tk − Tref = 270 K), which quantifies the difference in free energy between a lone ion hydrated at some temperature Tk and that at a reference temperature (Tref  = 270 K) each calculated in independent simulations. The free energy is calculated by gradually introducing (or removing) the ion into a water box with periodic boundary conditions through a series of intermediate simulations λ: 1 → 0, such that we can gather a smooth pathway that represents the free energy of solvation. b, Experimental method based on phase-shifting interferometry (PSI)25. An example is shown with an aqueous solution of NaCl C0 = 0.5 m with the phase map taken before (\({\Psi }_{0}^{{\prime} }\)) and after (\({\Psi }_{{{{\rm{end}}}}}^{{\prime} }\)) the concentration profile has developed, with the phase value representing the mean concentration removed. \(\Delta {\Psi }^{{\prime} }={\Psi }_{{{{\rm{end}}}}}^{{\prime} }-{\Psi }_{0}^{{\prime} }\) and the concentration change induced by thermodiffusion \(\Delta C=\frac{\Delta {\Psi }^{{\prime} }}{{{{\rm{OP}}}}\times {{{\rm{CF}}}}}\), where the optical path OP and the contrast factor CF are assumed constants. Due to negative CF (Supplementary Fig. 12), the region with a positive gradient of \(\frac{-\delta \Delta {\Psi }^{{\prime} }}{\delta T}\) along y represents thermophilic mass transport, while the region with a negative gradient is thermophobic. Importantly, the local maximum corresponds to T0 (obtained by a third-order polynomial fit). The error bars in T0 are obtained by using a confidence interval of 95% for at least three replicates. More experimental data are available in the Supplementary Fig. 10.

Direct experimental measurement of the inversion temperature

To directly measure T0, we developed a digital interferometry technique (phase-shifting interferometry25) that finds T0 from the measurements of the local maximum concentration in a linear temperature field, as shown in Fig. 2b. A Soret cell (Supplementary Fig. 4, similar to that in the literature25,35, was used with thermistors and Peltier modules for rapid and accurate temperature control. Water cooling enables a wide and stable range of temperatures. The homogeneous solution is placed first in the Soret cell with a cell height h of 4 or 6 mm. The Lewis number, defined as the ratio of thermal diffusivity to mass diffusivity, is very large (Le ≫ 1); thus, the temperature profile develops much quicker than the concentration profile. A positive temperature gradient is applied rapidly, and the development of the temperature profile is closely monitored with PSI, fully developing at time tbg. At this point, a mirror reflecting the test beam in the interferometer is adjusted to suppress most of the fringes caused by temperature gradients. Next, a uniform temperature was applied and maintained so that the concentration and temperature could return to their original homogeneous states. The same temperature gradient is applied again and, at tbg, the phase map Ψ0(y) is recorded as the background data representing a homogeneous concentration with a quasi-linear temperature gradient (Supplementary Fig. 5). If the background image is taken after the development of the concentration profile has already progressed significantly, the data will not reflect the actual concentration profile (Supplementary Fig. 6). Therefore, the ratio tbg/τ should be carefully considered, as discussed in more detail in Supplementary Method 1. After time \(\tau =\frac{{h}^{2}}{{\pi }^{2}D}\), the concentration profile can be considered fully developed25, while the phase map Ψend(y) representing a fully developed concentration profile is recorded at time tend ≫ τ.

The phase maps only give the relative change in concentration instead of absolute values. Here, the mean concentration Cmean can serve as a common reference frame between tbg and tend because Cmean is constant due to mass conservation. Therefore, \({\overline{\Psi }}_{0}\) and \({\overline{\Psi }}_{{{{\rm{end}}}}}\) (averaged in y) are subtracted from Ψ0(y) and Ψend(y), respectively, to obtain \(\Psi {{\prime} }_{0}(\,y)\) and \({\Psi }_{{{{\rm{end}}}}}^{{\prime} }(\,y)\). \({\Psi }_{0}^{{\prime} }(y)\) represents C = Cmean everywhere in the cell. In Fig. 2b, \(\Delta {\Psi }^{{\prime} }(y)={\Psi }_{{{{\rm{end}}}}}^{{\prime} }(y)-{\Psi }_{0}^{{\prime} }(y)\), and since \(\Delta C=\frac{\Delta {\Psi }^{{\prime} }}{{{{\rm{OP}}}}\times {{{\rm{CF}}}}}\) with OP and CF treated as constants, \(\Delta {\Psi }^{{\prime} }\propto \Delta C\) and \(\Delta {\Psi }^{{\prime} }(y)\) can represent the change in concentration under competing Fickian and thermodiffusive fluxes. More details of the conversion between \(\Delta {\Psi }^{{\prime} }\) and ΔC are available in Supplementary Method 1. Furthermore, ST of many systems including alkali halide solutions are known to increase monotonically with temperature36,37, i.e. the ions are thermophobic above T0 and thermophilic below T0, resulting in accumulation of ions at T0. To determine the location at which the concentration reaches the local maximum, which corresponds to the region at T0, we fit the profile \(\Delta {\Psi }^{{\prime} }(y)\) with a third-order polynomial approximation, with uncertainties determined from the confidence interval 95%. Fitting to \(\Delta {\Psi }^{{\prime} }(y)\) is effectively equivalent to fitting C(T), and a third-order polynomial fit for C(T) contains the underlined assumption of a second-order polynomial relationship between ST and T 38,39, as well as the assumption that the change in local concentration is negligible. Fitting based on other empirical relationships between ST and T has been proposed36,40, but the fitted results do not show a clear location for the maximum concentration due to the large noise near the boundaries. Hence, the fit based on the empirical relationship ST(T) is not physically meaningful in our case. In this study, five different alkali halide aqueous solutions of LiCl, NaCl, KCl, NaF, and NaI were experimentally tested. This gives two sets of data, one with varying cation but the same anion Cl − , the other with varying the anion but the same cation Na+.

Varying cations

When keeping the anion Cl − , T0 is experimentally measured from ultra-dilute (0.02 m) to high (>2.00 m) concentrations, as shown in Fig. 3. For all three salts, there is an initial sharp increase in T0 until a certain concentration where a maximum value of T0 is reached. For LiCl, the initial increase in T0 is evident with concentration. Between 0.5 and 2 m, LiCl is thermophilic within the temperature range 0 and 40 °C. This is in line with the previously reported inversion temperature of T0(0.5 m) = 45 °C by Lee et al.41. The complete dataset for LiCl T0, incorporating both the present work and literature data across broad ranges of temperature (T) and concentration (C), is provided in Supplementary Fig. 7. This together with the experimental results of this work, especially the measured T0(15.24 m) = 43.5 ± 1.67 °C, confirms the non-monotonic trend of T0 with increasing C. For NaCl and KCl, the initial increase is also followed by a decrease in T0, which levels off at higher concentrations. This non-monotonic trend of T0(C) has similarities to the non-monotonic relationship between ST and C at a set average temperature41,42,43. The trends of T0(C) as C → 0 seem to indicate that although thermophobic behaviour is more dominant as the concentration decreases, T0 still exists at an infinite dilution for this series of salts.

Fig. 3: Thermodiffusive behaviour in alkali halides solutions for different cations.
Fig. 3: Thermodiffusive behaviour in alkali halides solutions for different cations.
Full size image

Measured T0 for LiCl, NaCl, and KCl aqueous solutions from an ultra-dilute concentration of 0.02 m to moderate concentration ca. 2.5 m. Error bars are estimated for a 95% confidence interval based on at least three replicates. The left plot is for C ∈ [0.02, 0.50]m with C in a log scale. Linear fits were obtained with an R2 value of 0.99, 0.98, and 0.99 for LiCl, NaCl, and KCl, respectively. On the right is for C ∈ [0.02, 2.50]m with C on a linear scale, with the yellow box indicating the \(\log (C)\) plot range. Literature data is plotted in the same colour with asterisks: I.42, II.37, III.38. A complete comparison of the literature values of LiCl T0 is also available (Supplementary Fig. 7). All the ions are thermophobic above T0 and thermophilic below T0. On the left, linear fits are applied to each data set while on the right, the lines are visual guides. Due to the experimental limitation that ΔC → 0 when C → 0, measuring \(\Delta {\Psi }^{{\prime} }\) at ultra-dilution is not experimentally possible. Thus, ΔΔG for different ion types calculated from lone ion FEP is presented in the inset (1 ion per 883 water molecules equating to a concentration of 0.063 m). ΔΔG decreases monotonically with decreasing T, indicating a generic thermophobic behaviour for lone cations.

The inset of Fig. 3 shows the FEP results of ΔΔG for the lone cations Li+, Na+, and K+. There is a monotonically decreasing ΔΔG with decreasing temperatures, which shows that the solvation of the ions becomes more favourable at lower temperatures, indicating thermophobic behaviour. Moreover, the electrostatic properties of the cations (e.g. charge density)44 correlate with the magnitude of their thermodiffusive behaviour quantified by the Soret coefficient for lone ions \({S}_{{{{\rm{T}}}}}^{{{{\rm{i}}}}}\) (Supplementary Fig. 8)27. For the most charge-dense cation, Li+, T0 shows the strongest dependence on C, and the slope of ΔΔG is larger, i.e. the ion has an increased preference for lower temperatures when the effect of ionic shielding is waning or absent.

Varying anions

While the cation remained as Na+ and only the anion was changed, T0 was measured at various concentrations as reported in Fig. 4a (NaCl data are the same as in Fig. 3 – kept for comparison). Again, a first increasing and then decreasing non-monotonic trend is observed with increasing concentration. Based on the trend of \({T}_{0}(\log (C))\) at low concentrations, there is only thermophobic behaviour as C → 0. Similarly, the FEP calculations suggest that lone F−  and lone Cl−  anions do not exhibit inversion of thermodiffusion at ultra-dilution. However, in the FEP results, we observe inversion behaviour for lone I− , indicated as the local minimum in ΔΔG(T). This may be an example where the thermodynamic contributions are small enough that the negligible kinetic effects15 become significant. That no inversion is observed for NaI in the experimental data, but it is observed for I− in FEP, can be explained by considering the impact of the counterion ever present in experimental conditions. Fig. 4b shows that if we include the counterion in the FEP calculations (NaI and paired NaI, biased such that the counterions remain in contact through the simulation), the inversion exhibited for the lone ion is no longer present, as the cation dominates the effective behaviour. That is, the cation displays stronger thermodiffusion, and the anion moves with it as a result of the electrostatic interaction. This indicates that I−  possesses some properties missing in Eq. (1) that do not manifest unless Shyd is small, as is the case for I−  whose Shyd value is half the magnitude of any of the other ions studied (Supplementary Table 1 from Marcus34). The inversion behaviour of I− , indicated by the non-monotonic relationship between ΔΔG and T in Fig. 4a inset, appears to originate from its large size (and hence low charge density). This is verified by manually altering the MD parameters of the ions (Fig. 4b and Supplementary Fig. 9) to change their size, charge and dispersion energy (i.e. the ϵ term in the Leonard-Jones potential). While the size and charge of the ion had a large impact on inversion (both altering charge density), changing the dispersion term had limited impact. The results in Fig. 4c from the FEP simulations also show that Shyd is near zero for lone I −  while positive Shyd is seen when its size increases.

Fig. 4: Thermodiffusive behaviour in alkali halide solutions with different anions.
Fig. 4: Thermodiffusive behaviour in alkali halide solutions with different anions.
Full size image

a Measured T0 for NaF, NaCl, and NaI aqueous solutions. The error bars are estimated for a 95% confidence interval based on at least three replicates. The left plot is for C ∈ [0.02, 0.50]m with C on a logarithmic scale. Linear fits were obtained with an R2 value of 0.99, 0.98, and 0.95 for NaF, NaCl, and NaI, respectively. On the right is for C ∈ [0.02, 3.00]m with C on a linear scale, with the yellow box indicating the \(\log (C)\) plot range. Literature data is plotted in the same colour with asterisks: I.77, II.37, III.38. All ions are thermophobic above T0 and thermophilic below T0. On the left, linear fits are applied while on the right, the lines are visual guides. FEP results are shown in the inset for lone anions (1 ion per 883 water molecules equating to a concentration of 0.063 m). b, c ΔΔG and ΔShyd from FEP results for Na+, I−  (σ = 0.488 nm), free-roaming NaI, a forced NaI pair, and I−  re-parameterised to be larger (σ = 0.600 nm) or smaller (σ = 0.350 nm). With the presence of the Na +  counterion, ΔΔG decreases monotonically with decreasing temperature for both free-roaming and paired NaI. Further parameter investigations are in Supplementary Fig. 9. More results on the counterion and ion pairing effects is in Supplementary Fig. 13. In c, the presence of small ions, such as Na+ or “Small I−”, results in a markedly negative hydration entropy Shyd, while larger ions such as I−  or “Large I− ” has positive Shyd values.

Modelling entropic concentration trends

While FEP results align with theory, experiments suggest that thermophilic behaviour persists at ultra-dilution. This discrepancy can be explained by considering the entropic basis for thermodiffusion; specifically, fitting T0 to \(\log (C)\). Following Boltzmann’s entropy formula, \({{{\rm{S}}}}={k}_{{{{\rm{B}}}}}\ln \Omega\), where the number of microstates (Ω) correlates with concentration. Initially at low concentrations, Ω increases as the number of possible ion arrangements grows. Thus, within the concentration range of C ∈ [0,02, 0.50]m, T0 is plotted against C on a logarithmic scale in the left panel of both Figs. 3 and 4a. Strong linearity can be observed with R2 > 0.95, supporting the entropic nature of thermodiffusion.

To further investigate whether an entropic model could explain both the trend towards thermophobic behaviour as C → 0 and the general trends at higher concentrations, we employed a statistical mechanics model and the results are shown in Fig. 5. The model utilises a combinatorics method combined with a two-component (cation and anion) four-layer (ion, first hydration shell, second hydration shell and free water) model to approximate the change in microstates (Ω) from bulk water at each concentration (details in the Supplementary Method 2). In general, the key factor influencing the entropic trends in this concentration range is the second solvation shell and how much the addition of each ion decreases the degrees of freedom for each of the water molecules in that layer, as quantified by the ion charge density. Following the initial increase at low concentrations, at higher concentrations, Ω decreases due to a reduction in free water molecules and the confinement of atom locations as ions crowd into a pseudo-lattice near saturation. Modelling these effects is complex due to the interplay of hydration numbers, hydration strengths (per water molecule and within each solvation sphere), and ion–ion correlations, including contact, solvent-separated, and solvent–solvent-separated pairing and clustering. Eastman24 proposed a similar three-regime model of concentric spheres (ion cavity, first hydration shell, remaining water) that follows this trend.

Fig. 5: Modelled entropy of aqueous alkali halide solutions as a function of concentration for different cations and anions.
Fig. 5: Modelled entropy of aqueous alkali halide solutions as a function of concentration for different cations and anions.
Full size image

Calculated entropy (ΔS(C) = S(C) − S(C = 0.06 m)) versus concentration for (a) varying cations of salt in aqueous solutions of LiCl (dotted line), NaCl (dashed line), and KCl (solid line), and (b) anions NaF (dotted line), NaCl (dashed line) and NaI (solid line). Given the relative nature of the microstates used to generate the entropy values, they are normalised against S(C = 0.650 m) for NaCl and have arbitrary units (a.u.). The entropy values are obtained from the model described in the text, which accounts for contributions from ions, their hydration shells, and free water molecules within a multi-layered hydration structure (Supplementary Eq. 1). Ion-specific coordination numbers, derived from experimental data, and microstate weights, based on ion charge densities, are incorporated to reflect differences in the number and strength of ion–water interactions that alter the water degrees of freedom. The model shows that entropy initially increases with concentration due to the increased possible microstates from more available system arrangements with increasing numbers of ions. An entropy decrease is then observed due to the crowding of ions reducing their translational freedom, combined with the reduction of free water molecules and changes in the hydration shells, with distinct variations observed for different ions. LiCl shows only increasing entropic values in this concentration range, as its clustering behaviour45 allows for more possible microstates through reduced ion–ion competition for water and increased cation–anion spatial configurations.

In contrast to the other salts, LiCl only shows monotonic behaviour in the modelled concentration range between 0.01 and 2.00 m (Fig. 5a). This can be explained by the formation of significant pairing and cluster formation of LiCl as evidenced by MD45, neutron and X-ray diffraction46, and ultrafast infra-red spectroscopy47. In contrast to lone ions separated by water molecules, pairing and clustering reduce ion–ion competition for water and increase the possible number of cation–anion spatial configurations; thus, it expands the concentration range over which the number of possible microstates increases. This leads to a larger concentration range over which T0 increases (Supplementary Fig. 7), and possibly contributes to the very high solubility limit of LiCl in water as well. This exemplifies specific ion effects48, where the Li+ cation behaves vastly differently from Na+ and K+ cations.

At the other end of the spectrum, NaF showed only thermophobic behaviour at 0.5 m and above. Given the low solubility limit of NaF (ca. 0.983 m) and its high charge density, the purely thermophobic behaviour for >0.5 m fits within the entropy model (Fig. 5b). That is, Ω decreases when approaching saturation. Conceivably, highly associated ions such as NaF49 and multivalent salts may be modelled even more accurately by accounting for degree of dissociation (α) analysis such as those performed by Heyovska50, but should require newly calculated α, either by using a different methodology, or using known concentration-dependent hydration numbers. Furthermore, given that this concentration-dependent entropic model was established without thermal effects, it may be extensible beyond current non-isothermal alkali halide solution to isothermal electrolyte systems, including phenomena like re-entrant behaviour51.

Discussion

Phase-shifting interferometry (PSI) measurements of the thermodiffusion inversion temperature, T0(C), indicate that T0 decreases as the concentration tends towards infinite dilution, i.e. C → 0. However, an associated T0 can still be found in dilute solutions at C = 0.02 m. Molecular dynamics simulations of lone–ion free-energy perturbation (FEP) considering only the hydration entropy reveal that alkali halide solutions are generally thermophobic at C = 0.063 m, in line with the theory. When considering entropy’s logarithmic relationships, it is conceivable that these salts will indeed be always thermophobic at infinite dilution, at least in the temperature range of liquid water. The exception, the lone I− FEP result, indicates that there may be some missing components in Eq. (1) even in the absence of ion–ion interaction. A small hydration entropy value may be overcome by some other factors, such as the usually neglected kinetic energy term described by Duhr and Braun15 (see Supplementary Fig. 8). It appears that this can manifest itself in simulations with low charge-density ions, where van der Waals forces become larger contributors to the solvation energy, and not just electrostatics52. In reality, the inversion behaviour of lone I−  is unlikely to emerge, as the stronger thermodiffusion of the counter ion overcomes this effect, as seen in our NaI simulations. Furthermore, the inversion temperature (Fig. 4) and the entropy increase monotonically (Fig. 5) for low concentrations of NaI, in line with the other alkali halides.

These results highlight three findings. First, solvation entropy is an important driver of thermodiffusion, and the entropy of the system might govern inversion temperatures, and perhaps thermodiffusion more generally, across all concentration ranges. Second, even in the absence of ion–ion interaction, inversion may still occur when the magnitude of the solvation entropy is very small, allowing other factors to become significant. Third, counterions need careful consideration when predicting or manipulating thermodiffusion even in ultra-dilute solutions because they can drive motion via electrostatic interactions or the formation of ion pairs or clusters45 that alter entropic considerations. A revision of the theory of Eastman24, Duhr and Braun15 can be introduced based on these three considerations.

This work not only leads to a better understanding of the fundamental mechanism behind thermodiffusion in liquids but also has significant implications for a variety of applications. Despite its ubiquity, thermodiffusion is often overlooked because it is a weak effect. However, it has been shown to play a role in many natural53,54 and industrial55,56 processes. A better understanding of the thermodiffusive behaviour enables better prediction of more complex systems in a variety of fields, including fluid dynamics5, atmospheric sciences57, and geophysics58 especially concerning petroleum industry4. It also promotes the optimisation of thermodiffusion-related applications, such as in drug manufacturing processes16 and energy generation59. Recent groundbreaking developments in all-liquid thermal desalination12 and brine treatment13 based on multichannel thermodiffusion have sparked renewed interest in the thermodiffusion inversion temperature at ultra-dilution. Knowledge of inversion temperature values is key for the implementation of the first all-liquid thermal desalination method. It is also relevant to the extraction of valuable alkali metals such as lithium from lithium-rich brine deposits.

Methods

Challenges in measuring inversion temperature at ultra-dilution

In recent decades, many experimental methods for quantifying thermodiffusion have been developed60. Measurement methods can be either intrusive or non-intrusive. Intrusive methods extract samples after a concentration gradient due to thermodiffusion has been established in a convectionless environment1, a one-dimensional Poiseuille flow12, or a thermo-gravitational column (TGC)61. TGC is a popular method due to its larger concentration difference compared to convectionless methods under the combined effects of thermodiffusion and convection. However, due to approximations in the theory, including ignoring the temperature-dependent density, there are some uncertainties in the relation between the observed concentration changes and the Soret coefficient62. The reproducibility of TGC measurements at low concentration for small ST is markedly poor42,63. Other researchers37 who used convection-free methods also questioned the validity of TGC methods when measuring small concentration changes or negative ST.

In contrast, non-intrusive methods are based on the effect of concentration change on electrical conductivity, e.g. conductimetric methods62,64, or the refractive index, e.g. digital interferometry25,65 and beam deflection66. These methods are more suitable for investigating subtle concentration changes because they are highly sensitive and avoid disturbances to the concentration field during sample extraction by intrusive methods. Conductimetric methods measure the resistance ratio in each half of the solution-containing cell, thus eliminating the effect of temperature on electrical conductivity. This method has been proven to provide reproducible results for measuring positive ST39,62,64,67 or negative ST for concentrated solutions (ST inferred from the onset of thermohaline convection, e.g. finding the ΔT corresponding to the onset of Rayleigh-Bénard convection that overcame thermophilic thermodiffusion)38,68. Agar and Turner62 provided two estimates for ST for each solution: one based on the steady-state electrical conductivity ratio, which is known to derive a significantly lower ST due to convection; the other based on the initial rate of change of the ratio. The method based on the initial rate is more accurate in terms of being free from convection effects; however, it is based on only a small fraction of the total change and is extremely susceptible to small errors in the early stages. There are literature values for ST of dilute alkaline halide solutions (e.g. 0.01 m to 0.05 m). However, the validity of these small negative magnitudes ST is questionable, as they are based on the steady state ratio62,64,67 and are affected by convection effects68. Another reason for us to avoid using the conductimetric method to measure T0 is that when T0 is within the temperature range of the cell, a false positive or a false negative ST can exist. For example, in the PSI measurement for LiCl at 0.10 m between 20 and 30 °C (Supplementary Fig. 10a), we see that the average \(\Delta {\Psi }^{{\prime} }\) in the top half of the cell is significantly lower than the \(\Delta {\Psi }^{{\prime} }\) in the bottom half of the cell. Had it been a conductimetric experiment, one would only see that higher conductivity is in the bottom half of the cell and would arrive at the conclusion that there had been thermophobic transport, and in turn derive a positive ST at the mean temperature of 25 °C. However, it is clear from the PSI image that LiCl at 25 °C is close to T0 or slightly thermophilic.

Optical measurements generally yield ST(Tmean), where Tmean is the mean temperature within the experimental apparatus. Through multiple experiments, ST(Tmean) can be obtained by conventional methods, and T0 is indirectly inferred by interpolation and meeting the condition ST(T0) = 037. Existing optical methods60 are most suitable when the concentration profile is quasi-linear, ignoring the temperature dependence of ST. However, there are generally few data points available and the trend of T0 as C → 0 cannot be inferred from extrapolating existing data. This shortcoming is compounded by the scarcely available data for T0 under dilute conditions less than 0.5 m. Furthermore, when C and ST are both small, the local concentration change δC ∝ STC(1 − C) can be undetectable. Attempts have been made to measure ST for aqueous alkaline halide solutions at dilute concentrations42,43,62,63,64,67 but the agreement was only fair, especially poor agreement is noted for small ST or negative ST. Furthermore, more recent optical measurements fail to replicate these data41, suggesting a lack of accuracy and precision in existing data for ST(C < 0.5 m) and highlighting the difficulty in measuring ST at dilute concentrations. In this study, we apply phase-shifting interferometry (PSI)25 to obtain a direct measurement of the inversion temperature, without the need for multiple measurements of ST and a subsequent interpolation or extrapolation of data points to estimate ST(T0) = 0. PSI is sensitive enough to distinguish T0 even when \(\Delta {\Psi }^{{\prime} }\) is 1% of the unwrapped phase difference Ψ.

Non-equilibrium molecular dynamics simulations

Following the approach we have previously developed26, we placed 47,000 molecules in the 50 × 50 × 200 Å3 simulation box. Two spatially determined thermostats, with a width of 10Å, are defined along the z axis in the range 45 Å < zcold < 55 Å and 145 Å < zhot < 155 Å, respectively (see the simulation box for ions and water in Fig. 1c). Following equilibration, the simulation was run with a constant number of molecules, volume, and energy (NVE) ensemble for 220–300 ns, for 4–18 replicates (lower concentrations required more data exclusion as shown in Supplementary Fig. 11) and the time-averaged cumulative concentration profile was obtained. Each replicate was adapted to the functional form by Hutchinson et al.26: \(C(T)={C}_{0}\exp \left\{-{S}_{{{{\rm{T}}}}}^{\infty }\left[T+\tau \exp \left(\frac{{T}_{0}-T}{\tau }\right)+k\right]\right\}\) using the SciPy curve fitting approach69, where C(T) is the local concentration at temperature T, C0 is the average concentration in the solution, T0 is the inversion temperature, \({S}_{{{{\rm{T}}}}}^{\infty }\) is the Soret coefficient at the limit of T → ∞, τ is a measure of the rate at which ST varies with T and k is a non-physical fitting parameter. T0 was then obtained from the average fit of each replicate, with outliers that had uncertainties greater than 50 °C excluded from the analysis (Supplementary Fig. 11) and the uncertainty calculated from the standard deviation of these values. Fig. 1c plots the concentration profile between the thermostats for the average of all included replicates for a 0.055 m solution.

Free-energy perturbation simulations

We used an alchemical free-energy perturbation (FEP) (Fig. 2a) as implemented in GROMACS70 to model the Gibbs free energy of solvation of ions at various temperatures. Our simulations used TIP3P-FB water parameters71 and ion parameters established by Sengupta et al.72 for monovalent ions, as these were recently shown to reproduce thermodiffusive behaviour for NaCl aqueous solutions26, both in terms of Soret coefficients and T0. The parameters for divalent ions come from Li et al.73. We investigated down to 200 K, and each condition used 20 λ values in which the electrostatics were slowly turned off for λ values between 0 and 0.5. For the λ values between 0.5 and 1, van der Waals forces were slowly turned off simulating the removal of the hydration cavity. Following an energy minimisation, constant number of atoms, volume and temperature (i.e. NVT) and constant number of atoms, pressure and temperature (i.e. NPT) equilibration for 100 ps, all production simulations were performed for a period of at least 20 ns (depending on the observed convergence, Supplementary Fig. 2) using NPT at 1bar via a Parrinello-Rahman barostat and a leap-frog stochastic dynamics integrator, which also acts as the thermostat. The energies were sampled every 20 fs and structures every 20 ps. The Bennett Acceptance Ratio (BAR) method was used to analyse all simulations. Simulations at each temperature were run independently using periodic boundary conditions with 14 Å van der Waals and particle-mesh Ewald (PME) electrostatics cutoff, and with hydrogen bond LINear Constraint Solver (LINCS) constraints; we used a 2 fs time step. A non-zero net charge can cause problems due to finite-size effects in a periodic system, as summing could lead to an infinite charge when periodic boundary conditions (PBC) are applied74. However, this can be avoided by having a sufficiently large box to make use of the electrostatic cutoff75. The validity of this approach is supported by the close agreement of simulated hydration free energies with experimentally measured values (Supplementary Fig. 3). Furthermore, given that we are interested in ΔΔGhyd(Tk − Tref = 270 K), i.e. the difference between free energies in two equivalent systems, any arising systematic errors will be cancelled out. Shyd values in Fig. 4c were calculated via a finite differences approach, as described by Kubo et al.76:

$$\Delta {S}_{{{{\rm{hyd}}}}}(T)=-\frac{\Delta {G}_{{{{\rm{hyd}}}}}\,\left(T+\Delta T\right)-\Delta {G}_{{{{\rm{hyd}}}}}\,\left(T-\Delta T\right)}{2\,\Delta T}.$$
(3)