Stable Prenucleation Calcium Carbonate Clusters Define Liquid–Liquid Phase Separation

Abstract Liquid–liquid phase separation (LLPS) is an intermediate step during the precipitation of calcium carbonate, and is assumed to play a key role in biomineralization processes. Here, we have developed a model where ion association thermodynamics in homogeneous phases determine the liquid–liquid miscibility gap of the aqueous calcium carbonate system, verified experimentally using potentiometric titrations, and kinetic studies based on stopped‐flow ATR‐FTIR spectroscopy. The proposed mechanism explains the variable solubilities of solid amorphous calcium carbonates, reconciling previously inconsistent literature values. Accounting for liquid–liquid amorphous polymorphism, the model also provides clues to the mechanism of polymorph selection. It is general and should be tested for systems other than calcium carbonate to provide a new perspective on the physical chemistry of LLPS mechanisms based on stable prenucleation clusters rather than un‐/metastable fluctuations in biomineralization, and beyond.

dust-free tissue paper. Calcium potential and pH were recorded using a polymermembrane-based calcium ion selective electrode (ISE, Metrohm 6.0508.110) and glass electrodes (Metrohm Unitrode flat membrane 6.0256.100) with internal reference, respectively. The internal reference system of the pH electrode was also used as reference for the calcium ISE. Added volumes and potentials were automatically recorded every 10 seconds during the titration experiments. The calcium ISE was calibrated by titration of 10 mM calcium chloride solution into ultrapure water set to the desired temperature and pH (previously adjusted by addition of NaOH) while a gentle stream of nitrogen was flushed over the calibration sample to avoid CO2 uptake. A three-point calibration of the pH electrodes was performed using standard pH buffer solutions from Mettler-Toledo with the product numbers: pH = 4.01: 51302069; pH = 7.00: 51302047; pH = 9.21: 51302070.
Potentiometric assessment of the spinodal regime. The spinodal regime was probed via direct mixing of 50 mL calcium chloride solution (0.4 M) with 50 mL sodium carbonate solution (0.4 M). The Carbonate solution in the vessel was rigorously stirred and initially adjusted to pH 11.0. Calcium chloride solution was added directly into the vessel. The calcium potential and pH upon mixing were measured as described above. Due to the formation of a gel, the magnetic stirrer was replaced by a vertical mixer (Metrohm 2.804.0040) with a propeller of 94 mm in diameter (Metrohm 6.1909.010). The stirring speed was adjusted to assure formation of a vortex and optimal mixing but avoid the formation of bubbles affecting the electrode signals. During the experiments, the pH was kept constant via automatic counter titration of sodium hydroxide (1M) and hydrochloric acid (1M). Calibration of the free calcium concentration was performed in ionic strength adjusted environment as described in the literature and activity effects due to increased ionic strength were considered using the Davies equation. 1 3 The data evaluation of titration experiments was carried out as described in the literature 1,2 to calculate the IAP values and microscopic and macroscopic binding equilibrium constants (supplementary discussions 1 and 2).
Solid state ATR-FTIR: ATR-FTIR spectra of precipitated ACC were recorded on a Perkin Elmer spectrometer 100 equipped with a diamond ATR crystal from 760 to 4000 cm -1 with a spectral resolution of 4 cm -1 allowing the detection of the characteristic calcium carbonate bands noted n1, n2, n3 and n4 corresponding to the symmetric stretch, out-of-plane bending, asymmetric stretch and in-plane bending vibrational modes, respectively.
Liquid state ATR-FTIR. Time-resolved IR measurements provide insights into kinetic reaction mechanisms as we have shown previously with various applications. [3][4][5] In this study we performed rapid-scan measurements using a Bruker vertex 80V FTIR spectrometer equipped with a photoconductive mercury cadmium tellurium (MCT) detector (Kolmar Technology) and an ATR diamond single reflection unit (Golden Gate TM Heated Diamond ATR, Specac GS10540) and ZnSe lenses, restricting the spectral range at a lower wavenumber of 800 cm -1 , thus the carbonate n4 spectral region was not accessible. A custom-made ATR-FTIR stoppedflow with a mixing cell 50 µL (TgK Scientific SF-61/FT-IR) was mounted on top of the heated diamond window. The temperature was kept constant at 25˚C during the experiments using a built-in temperature controller. A Lauda thermostat (Ecoline E300) was used for preheating reactants within the umbilical tubes linking a syringe pump to the ATR-FTIR stoppedflow mixing cell on top of the ATR-FTIR unit. Calcium chloride, CaCl2, and sodium carbonate (Na2CO3) (0.4M, 0.3M, 0.2M, 0.1M, 0.08M, 0.06M) were mixed at a 1:1 volume ratio within the ATR-FTIR stopped-flow mixing cell. After reactant mixing 100 scans were recorded in rapid-scan mode between 800-1900 cm -1 with a scanner velocity of 320 Hertz and a spectral resolution of 4 cm -1 and averaged. A waiting time of one second was set in-between each of the scans. This procedure was repeated 5000 times. Averaged spectra were calculated and off-set corrected between 889-891 cm -1 , where no absorption band occurs. The transients were extracted at 869 cm -1 corresponding to the second derivative minimum of the n2 carbonate vibrational band. In order to compare the reaction kinetics, all transients were normalised to their maximum plateau values reached at the end of each experiment.
Fitting of ATR-FTIR time transients. The equations 6 used for fitting of the transients consist of a combination of at least one exponential growth and one logistic growth regimes as follows: where Ifit(t) is the fitted IR intensity over time; α the inverse of the exponential growth amplitude factor; t the time constant; Ki the logistic growth saturation value in arbitrary units for level i in this regime, i.e. the heights of each single plateau once the growth rate is minimum; Fi accounting for an evolution of the saturation time (∆Ri in seconds) from 10% to 90% for level i defined as according to ref. 6 ; Bi the mid-point of growth for level i in seconds; and y0 the offset.
For all measurements, the same fitting procedure was performed using the built-in Curve Fitting app in Matlab TM (using the bi-square weight method of nonlinear least-squares regression). It first consisted of fitting Ifit(t) for n=1 and i=1. If fitting procedure did not converge or reach the maximum iteration number set at 10 000, a second level of logistic growth was added to the fit equation -such as n = 1 and i = 2. Addition of a second level was considered successful if the Pearson product-moment correlation coefficient (r 2 ) increased by at least 10%. 4 the ATR-FTIR stopped-flow mixing cell. After reactant mixing 100 scans were recorded in rapid-scan mode between 800-1900 cm -1 with a scanner velocity of 320 Hertz and a spectral resolution of 4 cm -1 and averaged. A waiting time of one second was set in-between each of the scans. This procedure was repeated 5000 times. Averaged spectra were calculated and off-set corrected between 889-891 cm -1 , where no absorption band occurs. The transients were extracted at 869 cm -1 corresponding to the second derivative minimum of the n2 carbonate vibrational band. In order to compare the reaction kinetics, all transients were normalised to their maximum plateau values reached at the end of each experiment.
Fitting of ATR-FTIR time transients. The equations (6) used for fitting of the transients consist of a combination of at least one exponential growth and one logistic growth regimes as follows: " #$% (') = α • , t . + 0 1 1 + , -F i •(t-B i ) + 7 8 where Ifit(t) is the fitted IR intensity over time; α the inverse of the exponential growth amplitude factor; 9, the time constant; Ki the logistic growth saturation value in arbitrary units for level i in this regime, i.e. the heights of each single plateau once the growth rate is minimum; Fi accounting for an evolution of the saturation time (∆Ri in seconds) from 10% to 90% for level i defined as ∆; $ = ln (81) @ $ A according to ref. (6); Bi the mid-point of growth for level i in seconds; and y0 the offset. the ATR-FTIR stopped-flow mixing cell. After reactant mixing 100 scans were recorded in rapid-scan mode between 800-1900 cm -1 with a scanner velocity of 320 Hertz and a spectral resolution of 4 cm -1 and averaged. A waiting time of one second was set in-between each of the scans. This procedure was repeated 5000 times. Averaged spectra were calculated and off-set corrected between 889-891 cm -1 , where no absorption band occurs. The transients were extracted at 869 cm -1 corresponding to the second derivative minimum of the n2 carbonate vibrational band. In order to compare the reaction kinetics, all transients were normalised to their maximum plateau values reached at the end of each experiment.
Fitting of ATR-FTIR time transients. The equations (6) used for fitting of the transients consist of a combination of at least one exponential growth and one logistic growth regimes as follows: " #$% (') = α • , t . + 0 1 1 + , -F i •(t-B i ) + 7 8 where Ifit(t) is the fitted IR intensity over time; α the inverse of the exponential growth amplitude factor; 9, the time constant; Ki the logistic growth saturation value in arbitrary units for level i in this regime, i.e. the heights of each single plateau once the growth rate is minimum; Fi accounting for an evolution of the saturation time (∆Ri in seconds) from 10% to 90% for level i defined as ∆; $ = ln (81) @ $ A according to ref. (6); Bi the mid-point of growth for level i in seconds; and y0 the offset.   8 Figure S4. ATR-FTIR spectra of pc-ACC formed for different addition rates, 10 µmol min -1 (red) and 0.1 µmol min -1 (black). The n1, n2, n3 and n4 vibrational modes of the carbonate ion correspond to the symmetric stretching, out of plane bending, asymmetric stretch vibrations and in plane bending, respectively, and the nO-H mode to the O-H water stretching vibration.
The proto-structure remains unaffected by the addition rate, 9 but the water content increases. profiles present a strong increase right after mixing and rapidly decrease to stabilise towards an ACC solubility plateau characteristic of the spinodal limit. After an induction time, the IAP drops to a second plateau due to the complete transformation of the ACC precursor. The corresponding temperature dependent data is compiled in Table S3.    with activity coefficient and molar concentration of species i, g(i) and c(i), respectively, and standard concentration c 0 =1mol/L. For the dimerization equilibrium of ion pairs (eq. 2), the law of mass action yields; eq. 7

Supplementary Tables
The activity of the ion pair in the denominator can be expressed by re-arranging and inserting eq. 5, and we can re-write eq. 7 as; 13 with activity coefficient and molar concentration of species i, γ(i) and c(i), respectively, and standard concentration c 0 =1mol/L. For the dimerization equilibrium of ion pairs (eq. 2), the law of mass action yields; The activity of the ion pair in the denominator can be expressed by re-arranging and inserting eq. 5, and we can re-write eq. 7 as; with activity coefficient and molar concentration of species i, γ(i) and c(i), respectively, and standard concentration c 0 =1mol/L. For the dimerization equilibrium of ion pairs (eq. 2), the law of mass action yields; The activity of the ion pair in the denominator can be expressed by re-arranging and inserting eq. 5, and we can re-write eq. 7 as; 13 with activity coefficient and molar concentration of species i, γ(i) and c(i), respectively, and standard concentration c 0 =1mol/L. For the dimerization equilibrium of ion pairs (eq. 2), the law of mass action yields; The activity of the ion pair in the denominator can be expressed by re-arranging and inserting eq. 5, and we can re-write eq. 7 as; with activity coefficient and molar concentration of species i, γ(i) and c(i), respectively, and standard concentration c 0 =1mol/L. For the dimerization equilibrium of ion pairs (eq. 2), the law of mass action yields; The activity of the ion pair in the denominator can be expressed by re-arranging and inserting eq. 5, and we can re-write eq. 7 as; 14 eq. 8 The activity of the dimer of ion pairs in the numerator can be expressed by re-arranging and inserting eq. 7, and we obtain from eq. 8; eq. 9 All subsequent equilibria for the formation of higher associates are of the form; eq. 10 where M is the monomeric ion pair and P is a polymer of ion pairs, including the dimer of ion pairs, and the polymer of ion pairs grown by one ion pair MP. Thus, the equilibrium constants K(3), K(4), K(5), ... can be defined as; eq. 11 In this notation, the activity of the ion pair M can be calculated from eq. 5 according to; eq. 12 whereas the activity of the polymer of ion pairs grown by one ion pair MP is obtained by; eq. 13 where K(P-1) is the equilibrium constant for the previous association step. By inserting eqs.
12 and 13 into eq. 11, we obtain; eq. 14 We recall the assumption that all association steps are equal and independent, as originally introduced in the first publication on PNCs. 2 In the meantime, the validity of this assumption has been corroborated based on computer simulations, 10 and can now also be rationalized mechanistically: since ion association is driven by the entropy gain due to the release of ionic The activity of the dimer of ion pairs in the numerator can be expressed by re-arranging and inserting eq. 7, and we obtain from eq. 8; All subsequent equilibria for the formation of higher associates are of the form; eq. 11 In this notation, the activity of the ion pair M can be calculated from eq. 5 according to; a(M)=K(1)·aECa aq 2+ F·aECO 3,aq 2-F eq. 12 whereas the activity of the polymer of ion pairs grown by one ion pair MP is obtained by; a(MP)=K(P-1)·a(P)a(M) eq. 13 where K(P-1) is the equilibrium constant for the previous association step. By inserting eqs.
12 and 13 into eq. 11, we obtain; We recall the assumption that all association steps are equal and independent, as originally introduced in the first publication on PNCs (7). In the meantime, the validity of this assumption has been corroborated based on computer simulations (2), and can now also be rationalized mechanistically: since ion association is driven by the entropy gain due to the release of ionic hydration waters upon calcium and carbonate ions binding together (10), the The activity of the dimer of ion pairs in the numerator can be expressed by re-arranging and inserting eq. 7, and we obtain from eq. 8; All subsequent equilibria for the formation of higher associates are of the form; eq. 11 In this notation, the activity of the ion pair M can be calculated from eq. 5 according to; a(M)=K(1)·aECa aq 2+ F·aECO 3,aq 2-F eq. 12 whereas the activity of the polymer of ion pairs grown by one ion pair MP is obtained by; a(MP)=K(P-1)·a(P)a(M) eq. 13 where K(P-1) is the equilibrium constant for the previous association step. By inserting eqs.
12 and 13 into eq. 11, we obtain; We recall the assumption that all association steps are equal and independent, as originally introduced in the first publication on PNCs (7). In the meantime, the validity of this assumption has been corroborated based on computer simulations (2), and can now also be rationalized mechanistically: since ion association is driven by the entropy gain due to the release of ionic hydration waters upon calcium and carbonate ions binding together (10), the The activity of the dimer of ion pairs in the numerator can be expressed by re-arranging and inserting eq. 7, and we obtain from eq. 8; All subsequent equilibria for the formation of higher associates are of the form; eq. 11 In this notation, the activity of the ion pair M can be calculated from eq. 5 according to; a(M)=K(1)·aECa aq 2+ F·aECO 3,aq 2-F eq. 12 whereas the activity of the polymer of ion pairs grown by one ion pair MP is obtained by; a(MP)=K(P-1)·a(P)a(M) eq. 13 where K(P-1) is the equilibrium constant for the previous association step. By inserting eqs.
12 and 13 into eq. 11, we obtain; We recall the assumption that all association steps are equal and independent, as originally introduced in the first publication on PNCs (7). In the meantime, the validity of this assumption has been corroborated based on computer simulations (2), and can now also be rationalized mechanistically: since ion association is driven by the entropy gain due to the release of ionic hydration waters upon calcium and carbonate ions binding together (10), the The activity of the dimer of ion pairs in the numerator can be expressed by re-arranging and inserting eq. 7, and we obtain from eq. 8; All subsequent equilibria for the formation of higher associates are of the form; eq. 11 In this notation, the activity of the ion pair M can be calculated from eq. 5 according to; a(M)=K(1)·aECa aq 2+ F·aECO 3,aq 2-F eq. 12 whereas the activity of the polymer of ion pairs grown by one ion pair MP is obtained by; a(MP)=K(P-1)·a(P)a(M) eq. 13 where K(P-1) is the equilibrium constant for the previous association step. By inserting eqs.
12 and 13 into eq. 11, we obtain; We recall the assumption that all association steps are equal and independent, as originally introduced in the first publication on PNCs (7). In the meantime, the validity of this assumption has been corroborated based on computer simulations (2), and can now also be rationalized mechanistically: since ion association is driven by the entropy gain due to the release of ionic hydration waters upon calcium and carbonate ions binding together (10), the The activity of the dimer of ion pairs in the numerator can be expressed by re-arranging and inserting eq. 7, and we obtain from eq. 8; All subsequent equilibria for the formation of higher associates are of the form; P+M⇌MP eq. 10 where M is the monomeric ion pair and P is a polymer of ion pairs, including the dimer of ion pairs, and the polymer of ion pairs grown by one ion pair MP. Thus, the equilibrium constants K(3), K(4), K(5), ... can be defined as; eq. 11 In this notation, the activity of the ion pair M can be calculated from eq. 5 according to; whereas the activity of the polymer of ion pairs grown by one ion pair MP is obtained by; a(MP)=K(P-1)·a(P)a(M) eq. 13 where K(P-1) is the equilibrium constant for the previous association step. By inserting eqs.
12 and 13 into eq. 11, we obtain; We recall the assumption that all association steps are equal and independent, as originally introduced in the first publication on PNCs (7). In the meantime, the validity of this assumption has been corroborated based on computer simulations (2), and can now also be rationalized mechanistically: since ion association is driven by the entropy gain due to the release of ionic hydration waters upon calcium and carbonate ions binding together (10), the The activity of the dimer of ion pairs in the numerator can be expressed by re-arranging and inserting eq. 7, and we obtain from eq. 8; All subsequent equilibria for the formation of higher associates are of the form; P+M⇌MP eq. 10 where M is the monomeric ion pair and P is a polymer of ion pairs, including the dimer of ion pairs, and the polymer of ion pairs grown by one ion pair MP. Thus, the equilibrium constants K(3), K(4), K(5), ... can be defined as; eq. 11 In this notation, the activity of the ion pair M can be calculated from eq. 5 according to; whereas the activity of the polymer of ion pairs grown by one ion pair MP is obtained by; a(MP)=K(P-1)·a(P)a(M) eq. 13 where K(P-1) is the equilibrium constant for the previous association step. By inserting eqs.
12 and 13 into eq. 11, we obtain; We recall the assumption that all association steps are equal and independent, as originally introduced in the first publication on PNCs (7). In the meantime, the validity of this assumption has been corroborated based on computer simulations (2), and can now also be rationalized mechanistically: since ion association is driven by the entropy gain due to the release of ionic hydration waters upon calcium and carbonate ions binding together (10), the 14 The activity of the dimer of ion pairs in the numerator can be expressed by re-arranging and inserting eq. 7, and we obtain from eq. 8; All subsequent equilibria for the formation of higher associates are of the form; P+M⇌MP eq. 10 where M is the monomeric ion pair and P is a polymer of ion pairs, including the dimer of ion pairs, and the polymer of ion pairs grown by one ion pair MP. Thus, the equilibrium constants K(3), K(4), K(5), ... can be defined as; eq. 11 In this notation, the activity of the ion pair M can be calculated from eq. 5 according to; whereas the activity of the polymer of ion pairs grown by one ion pair MP is obtained by; a(MP)=K(P-1)·a(P)a(M) eq. 13 where K(P-1) is the equilibrium constant for the previous association step. By inserting eqs.
12 and 13 into eq. 11, we obtain; We recall the assumption that all association steps are equal and independent, as originally introduced in the first publication on PNCs (7). In the meantime, the validity of this assumption has been corroborated based on computer simulations (2), and can now also be rationalized mechanistically: since ion association is driven by the entropy gain due to the release of ionic hydration waters upon calcium and carbonate ions binding together (10), the hydration waters upon calcium and carbonate ions binding together, 11 the thermodynamic driving force should not differ if, e.g., a carbonate ion binds to a single calcium ion, or to a calcium ion that is the end member of a larger, chain-like PNC (the structural form of PNCs is a chain-like, dynamically-ordered liquid-like oxyanion polymer, DOLLOP 10 ). In other words, it can be assumed 2,10 that the values of all equilibrium constants K(1), K(2), K(3),... are equal; eq. 15 Combining eq. 5, eq. 9 and eq. 14, we can thus write with eq. 15; eq. 16 This shows that as per the assumption made, i.e., the values of all equilibrium constants are equal, also the form of the law of mass action becomes identical for each association step.
Hence, for the situation that all equilibrium constants of consecutive associations of ion pairs have the same equilibrium constant (eq. 15), the ion pairing equilibrium cannot be distinguished from the formation of higher associated states, namely, PNCs. We can thus reformulate eq. 16, writing; eq. 17 where K(cluster)=K(1)=K(2)=K(3)=...=K(P-1)=K(P). That is, PNC formation with equal equilibrium constants for all association steps of ion pairs can be described by the generic equilibrium; eq. 18 In the experiments with the calcium-ion selective electrode (ISE), the free calcium concentration upon constant addition of dilute calcium solution into dilute carbonate buffer is monitored, yielding a linear increase in free calcium in the pre-nucleation stage. The 15 thermodynamic driving force should not differ if, e.g., a carbonate ion binds to a single calcium ion, or to a calcium ion that is the end member of a larger, chain-like PNC (the structural form of PNCs is a chain-like, dynamically-ordered liquid-like oxyanion polymer, DOLLOP (2)). In other words, it can be assumed (2,7) that the values of all equilibrium constants K(1), K(2), K(3), ... are equal; Combining eq. 5, eq. 9 and eq. 14, we can thus write with eq. 15; This shows that as per the assumption made, i.e., the values of all equilibrium constants are equal, also the form of the law of mass action becomes identical for each association step.
Hence, for the situation that all equilibrium constants of consecutive associations of ion pairs have the same equilibrium constant (eq. 15), the ion pairing equilibrium cannot be distinguished from the formation of higher associated states, namely, PNCs. We can thus reformulate eq. 16, writing;  (2)). In other words, it can be assumed (2,7) that the values of all equilibrium constants K(1), K(2), K(3), ... are equal; Combining eq. 5, eq. 9 and eq. 14, we can thus write with eq. 15; This shows that as per the assumption made, i.e., the values of all equilibrium constants are equal, also the form of the law of mass action becomes identical for each association step.
Hence, for the situation that all equilibrium constants of consecutive associations of ion pairs have the same equilibrium constant (eq. 15), the ion pairing equilibrium cannot be distinguished from the formation of higher associated states, namely, PNCs. We can thus reformulate eq. 16, writing;  (2)). In other words, it can be assumed (2,7) that the values of all equilibrium constants K(1), K(2), K(3), ... are equal; Combining eq. 5, eq. 9 and eq. 14, we can thus write with eq. 15; This shows that as per the assumption made, i.e., the values of all equilibrium constants are equal, also the form of the law of mass action becomes identical for each association step.
Hence, for the situation that all equilibrium constants of consecutive associations of ion pairs have the same equilibrium constant (eq. 15), the ion pairing equilibrium cannot be distinguished from the formation of higher associated states, namely, PNCs. We can thus reformulate eq. 16, writing;  (2)). In other words, it can be assumed (2,7) that the values of all equilibrium constants K(1), K(2), K(3), ... are equal; Combining eq. 5, eq. 9 and eq. 14, we can thus write with eq. 15; This shows that as per the assumption made, i.e., the values of all equilibrium constants are equal, also the form of the law of mass action becomes identical for each association step.
Hence, for the situation that all equilibrium constants of consecutive associations of ion pairs have the same equilibrium constant (eq. 15), the ion pairing equilibrium cannot be distinguished from the formation of higher associated states, namely, PNCs. We can thus reformulate eq. 16, writing; While the cluster with n constituents forms also in the PNC model, eq. 23 does not account for the convolution of the ion binding with all other association equilibria. Thus, eq. 23 does not invalidate eq. 20. The convolution of ion binding in a series of coexisting clusters yields a linear binding for equal equilibrium constants for each association step (eq. 20).

Pre-nucleation Ion Association Evaluated based on a Multiple Binding Model
As demonstrated previously, 2 the binding of ions in PNCs can be evaluated in more detail based on a model that was originally derived for assessing protein-ligand binding equilibria. 12 The basic assumption is that upon binding calcium ions to a carbonate ion, all possible binding events are equal and independent. This assumption mirrors eq. 15 (see section 1). eq. 25

Pre-nucleation Ion Association Evaluated based on a Multiple Binding Model
As demonstrated previously (7), the binding of ions in PNCs can be evaluated in more detail based on a model that was originally derived for assessing protein-ligand binding equilibria (11). The basic assumption is that upon binding calcium ions to a carbonate ion, all possible binding events are equal and independent. This assumption mirrors eq. 15 (see section 1).

Pre-nucleation Ion Association Evaluated based on a Multiple Binding Model
As demonstrated previously (7), the binding of ions in PNCs can be evaluated in more detail based on a model that was originally derived for assessing protein-ligand binding equilibria (11). The basic assumption is that upon binding calcium ions to a carbonate ion, all possible binding events are equal and independent. This assumption mirrors eq. 15 (see section 1).

20
Experimentally and computationally, N=2 within experimental accuracy, which is consistent with a chain-like structural form of PNCs called a 'dynamically-ordered liquid-like oxyanion polymer' (DOLLOP). 10 The corresponding Gibbs standard free energy of PNCs can thus be obtained by; eq. 29 with absolute temperature T and the universal gas constant R.
Similarly, Gibbs standard free energy corresponding to the microscopic binding of calcium and carbonate only, without PNC formation, can be obtained by; eq. 30 The corresponding Gibbs standard free energy of PNCs can thus be obtained by; ΔG 0 (cluster)=-RT•lnK(cluster) eq. 29 with absolute temperature T and the universal gas constant R.
Similarly, Gibbs standard free energy corresponding to the microscopic binding of calcium and carbonate only, without PNC formation, can be obtained by; ΔG 0 (micro)=-RT•lnK mb eq. 30 The corresponding Gibbs standard free energy of PNCs can thus be obtained by; ΔG 0 (cluster)=-RT•lnK(cluster) eq. 29 with absolute temperature T and the universal gas constant R.
Similarly, Gibbs standard free energy corresponding to the microscopic binding of calcium and carbonate only, without PNC formation, can be obtained by; where DG 0 (i) is the standard free energy of species i, R is the universal gas constant, and T the absolute temperature. As per eq. 16, the individual standard free energies of all associated states are equal, and this value thus represents an average over all associated states (eq. 16), ΔG 0 (cluster By re-arranging and inserting eqs. 5 and 7 in the denominator and numerator of eq. 33, respectively, we realize with eq. 17; K(2 direct )=K(1)·K(2)=K(cluster) 2 eq. 34 and for the corresponding standard free energy of the dimer of ion pairs; ΔG 0 E[CaCO 3 ] 2,aq 0 F=-2RT·lnK(cluster) eq. 35 Due to eqs. 11-17, we can generally express the standard free energy of a specific PNC consisting of m ion pairs as; 22 eq. 36 This shows that the PNCs become, considerably, more and more stable with increasing cluster size m. Owing to the properties of the law of mass action, however, the larger clusters will still be less and less abundant than the smaller ones, but only at low IAPs. A direct consequence is that the interconnected PNC equilibria break down at a specific IAP, at which the activity of the species of a particular higher association step becomes larger than that of the previous one. In this case, every higher associated state draws the available substance from all previous association equilibria and phase separation will occur spontaneously, based on ion association. This is spinodal liquid-liquid demixing. Because all equilibria subsequent to the first association steps (eqs. 1-4) have the same form (eqs. [11][12][13][14][15][16][17], this is the case for; eq. 37 Re-arranging eq. 5 and inserting the activity of the ion pair in the denominator of eq. 8 yields with eq. 16; eq. 38 Solving eq. 38 for IAP yields; eq. 39 We define the specific IAP, above which eq. 37 becomes true, as IAP(spinodal), and with This shows that the PNCs become, considerably, more and more stable with increasing cluster size m. Owing to the properties of the law of mass action, however, the larger clusters will still be less and less abundant than the smaller ones at low IAPs. A direct consequence is that the interconnected PNC equilibria break down at a specific IAP, at which the activity of the species of a particular higher association step becomes larger than that of the previous one. In this case, every higher associated state draws the available substance from all previous association equilibria and phase separation will occur spontaneously, based on ion association. This is spinodal liquid-liquid demixing. Because all equilibria subsequent to the first association steps (eqs. 1-4) have the same form (eqs. [11][12][13][14][15][16][17], this is the case for; Re-arranging eq. 5 and inserting the activity of the ion pair in the denominator of eq. 8 yields with eq. 16;  This shows that the PNCs become, considerably, more and more stable with increasing cluster size m. Owing to the properties of the law of mass action, however, the larger clusters will still be less and less abundant than the smaller ones at low IAPs. A direct consequence is that the interconnected PNC equilibria break down at a specific IAP, at which the activity of the species of a particular higher association step becomes larger than that of the previous one. In this case, every higher associated state draws the available substance from all previous association equilibria and phase separation will occur spontaneously, based on ion association. This is spinodal liquid-liquid demixing. Because all equilibria subsequent to the first association steps (eqs. 1-4) have the same form (eqs. [11][12][13][14][15][16][17], this is the case for; Re-arranging eq. 5 and inserting the activity of the ion pair in the denominator of eq. 8 yields with eq. 16;  This shows that the PNCs become, considerably, more and more stable with increasing cluster size m. Owing to the properties of the law of mass action, however, the larger clusters will still be less and less abundant than the smaller ones at low IAPs. A direct consequence is that the interconnected PNC equilibria break down at a specific IAP, at which the activity of the species of a particular higher association step becomes larger than that of the previous one. In this case, every higher associated state draws the available substance from all previous association equilibria and phase separation will occur spontaneously, based on ion association. This is spinodal liquid-liquid demixing. Because all equilibria subsequent to the first association steps (eqs. 1-4) have the same form (eqs. [11][12][13][14][15][16][17], this is the case for; Re-arranging eq. 5 and inserting the activity of the ion pair in the denominator of eq. 8 yields with eq. 16;  This shows that the PNCs become, considerably, more and more stable with increasing cluster size m. Owing to the properties of the law of mass action, however, the larger clusters will still be less and less abundant than the smaller ones at low IAPs. A direct consequence is that the interconnected PNC equilibria break down at a specific IAP, at which the activity of the species of a particular higher association step becomes larger than that of the previous one. In this case, every higher associated state draws the available substance from all previous association equilibria and phase separation will occur spontaneously, based on ion association. This is spinodal liquid-liquid demixing. Because all equilibria subsequent to the first association steps (eqs. 1-4) have the same form (eqs. [11][12][13][14][15][16][17], this is the case for; Re-arranging eq. 5 and inserting the activity of the ion pair in the denominator of eq. 8 yields with eq. 16; where the parameter Y is a (unknown) function f of the actual IAP in the mother liquid, f(IAP). As shown in previous work, at the liquid-liquid binodal limit, the barrier for the direct nucleation of crystals is formidable, 8 and hence, the probability p(polymorph) is essentially zero; eq. 44 Given the values of Ksp(polymorph), with regards to the function Y, we can thus conclude that at the liquid-liquid binodal limit; eq. 45 In analogy with eq. 41, we can express the probability for liquid-liquid demixing as; eq. 46 23 the same time, it is a question of the probability for the direct formation of the different polymorphs, p(polymorph), versus the probability of liquid-liquid demixing, p(demix). Hence, both parameters have to be assessed as a function of the IAP of a given solution. From the viewpoint of Boltzmann statistics, we may write; p(polymorph) ∝ exp(-ΔG 0 (polymorph)/RT) eq. 41 where DG 0 (polymorph) is the standard free energy of calcite, aragonite, or vaterite that can be calculated from the corresponding solubility Ksp(polymorph) according to; ΔG 0 (polymorph)=-RT•ln s 1 K sp (polymorph) t eq. 42 Combination of eqs. 41 and 42 yields; where the parameter Y is a (unknown) function f of the actual IAP in the mother liquid, f(IAP). As shown in previous work, at the liquid-liquid binodal limit, the barrier for the direct nucleation of crystals is formidable 25 , and hence, the probability p(polymorph) is essentially zero; Given the values of Ksp(polymorph), with regards to the function Y, we can thus conclude that at the binodal limit; eq. 45 In analogy with eq. 41, we can express the probability for liquid-liquid demixing as; p(demix)∝exp(-ΔG 0 (l-l)/RT) eq. 46 where DG 0 (l-l)=-RTlnK(l-l) is the standard free energy of liquid-liquid demixing with corresponding equilibrium constant K(l-l). As noted previously (9), the fact that there is no where the parameter Y is a (unknown) function f of the actual IAP in the mother liquid, f(IAP). As shown in previous work, at the liquid-liquid binodal limit, the barrier for the direct nucleation of crystals is formidable 25 , and hence, the probability p(polymorph) is essentially zero; Given the values of Ksp(polymorph), with regards to the function Y, we can thus conclude that at the binodal limit; eq. 45 In analogy with eq. 41, we can express the probability for liquid-liquid demixing as; p(demix)∝exp(-ΔG 0 (l-l)/RT) eq. 46 where DG 0 (l-l)=-RTlnK(l-l) is the standard free energy of liquid-liquid demixing with corresponding equilibrium constant K(l-l). As noted previously (9), the fact that there is no where the parameter Y is a (unknown) function f of the actual IAP in the mother liquid, f(IAP). As shown in previous work, at the liquid-liquid binodal limit, the barrier for the direct nucleation of crystals is formidable 25 , and hence, the probability p(polymorph) is essentially zero; Given the values of Ksp(polymorph), with regards to the function Y, we can thus conclude that at the binodal limit; eq. 45 In analogy with eq. 41, we can express the probability for liquid-liquid demixing as; p(demix)∝exp(-ΔG 0 (l-l)/RT) eq. 46 where DG 0 (l-l)=-RTlnK(l-l) is the standard free energy of liquid-liquid demixing with corresponding equilibrium constant K(l-l). As noted previously (9), the fact that there is no where the parameter Y is a (unknown) function f of the actual IAP in the mother liquid, f(IAP). As shown in previous work, at the liquid-liquid binodal limit, the barrier for the direct nucleation of crystals is formidable 25 , and hence, the probability p(polymorph) is essentially zero; Given the values of Ksp(polymorph), with regards to the function Y, we can thus conclude that at the binodal limit; eq. 45 In analogy with eq. 41, we can express the probability for liquid-liquid demixing as; p(demix)∝exp(-ΔG 0 (l-l)/RT) eq. 46 where DG 0 (l-l)=-RTlnK(l-l) is the standard free energy of liquid-liquid demixing with corresponding equilibrium constant K(l-l). As noted previously (9) where the parameter Y is a (unknown) function f of the actual IAP in the mother liquid, f(IAP). As shown in previous work, at the liquid-liquid binodal limit, the barrier for the direct nucleation of crystals is formidable 25 , and hence, the probability p(polymorph) is essentially zero; Given the values of Ksp(polymorph), with regards to the function Y, we can thus conclude that at the binodal limit; eq. 45 In analogy with eq. 41, we can express the probability for liquid-liquid demixing as; p(demix)∝exp(-ΔG 0 (l-l)/RT) eq. 46 where DG 0 (l-l)=-RTlnK(l-l) is the standard free energy of liquid-liquid demixing with corresponding equilibrium constant K(l-l). As noted previously (9) where the parameter Y is a (unknown) function f of the actual IAP in the mother liquid, f(IAP). As shown in previous work, at the liquid-liquid binodal limit, the barrier for the direct nucleation of crystals is formidable 25 , and hence, the probability p(polymorph) is essentially zero; Given the values of Ksp(polymorph), with regards to the function Y, we can thus conclude that at the binodal limit; eq. 45 In analogy with eq. 41, we can express the probability for liquid-liquid demixing as; p(demix)∝exp(-ΔG 0 (l-l)/RT) eq. 46 where DG 0 (l-l)=-RTlnK(l-l) is the standard free energy of liquid-liquid demixing with corresponding equilibrium constant K(l-l). As noted previously (9), the fact that there is no where DG 0 (l-l)=-RTlnK(l-l) is the standard free energy of liquid-liquid demixing with corresponding equilibrium constant K(l-l). As noted previously, 8 the fact that there is no discontinuity or kink in the pre-nucleation ion binding upon liquid-liquid demixing implies that the values of K(l-l)=K(cluster). We can thus rewrite eq. 46; eq. 47 where X is another unknown function g of the IAP, g(IAP). Again, previous work 8 showed that liquid-liquid demixing occurred readily upon crossing the liquid-liquid binodal limit, at least at sufficiently slow mixing rates, that is, there is no major barrier associated with this event, and at the binodal limit, p(demix)≈1. We can thus re-write eq. 47; eq. 48 Thus, at the liquid-liquid binodal limit, eqs. 48 and 44 are approximately equal; lnX + lnK(cluster) ≈ Y 1 K sp (polymorph) eq. 49 Both sides of eq. 49 are approximately zero at the binodal limit. In the following, we thus neglect a possible contribution of infinitesimals of different orders, which might cause that eq. 49 does not hold exactly. All of the following depends on this assumption. Re-arranging eq. 49 yields; eq. 50 Using eq. 45, eq. 50 can be simplified; eq. 51 A trivial solution of equation 51 is obviously lnX = -lnK(cluster). Still, the left-hand-side of eq. 51 represents the product of a (unknown) function of the IAP in the mother solution, lnX, and the solubility of the given polymorph, which represents an ion activity product itself. Eq. discontinuity or kink in the pre-nucleation ion binding upon liquid-liquid demixing implies that K(l-l)=K(cluster). We can thus rewrite eq. 46; eq. 47 where X is another unknown function g of the IAP, g(IAP). Again, previous work (9) showed that liquid-liquid demixing occurred readily upon crossing the liquid-liquid binodal limit, at least at sufficiently slow mixing rates, that is, there is no major barrier associated with this event, and at the binodal limit, p(demix)≈1. We can thus re-write eq. 47; 0=lnX+lnK(cluster) eq. 48 Thus, at the liquid-liquid binodal limit, eqs. 48 and 44 are identities; lnX+lnK(cluster)=Y 1 K sp (polymorph) eq. 49 Re-arranging eq. 49 yields; lnX•K sp (polymorph)=Y-K sp (polymorph)lnK(cluster) eq. 50 Using eq. 45, eq. 50 can be simplified; lnX•K sp (polymorph)=-K sp (polymorph)lnK(cluster) eq. 51 The left-hand-side of eq. 51 is the product of a (unknown) function of the IAP in the mother solution, lnX, and the solubility of the given polymorph, which represents an ion activity product itself. Moreover, eq. 51 is valid only at the binodal limit. Thus, we can write for the specific ion activity product defining the liquid-liquid binodal limit, IAP(binodal), irrespective of the (unknown) IAP-dependence of lnX; IAP(binodal) ∝ K sp (polymorph)•lnK(cluster) eq. 52 Thus, we finally obtain; discontinuity or kink in the pre-nucleation ion binding upon liquid-liquid demixing implies that K(l-l)=K(cluster). We can thus rewrite eq. 46; where X is another unknown function g of the IAP, g(IAP). Again, previous work (9) showed that liquid-liquid demixing occurred readily upon crossing the liquid-liquid binodal limit, at least at sufficiently slow mixing rates, that is, there is no major barrier associated with this event, and at the binodal limit, p(demix)≈1. We can thus re-write eq. 47; 0=lnX+lnK(cluster) eq. 48 Thus, at the liquid-liquid binodal limit, eqs. 48 and 44 are identities; lnX+lnK(cluster)=Y 1 K sp (polymorph) eq. 49 Re-arranging eq. 49 yields; lnX•K sp (polymorph)=Y-K sp (polymorph)lnK(cluster) eq. 50 Using eq. 45, eq. 50 can be simplified; lnX•K sp (polymorph)=-K sp (polymorph)lnK(cluster) eq. 51 The left-hand-side of eq. 51 is the product of a (unknown) function of the IAP in the mother solution, lnX, and the solubility of the given polymorph, which represents an ion activity product itself. Moreover, eq. 51 is valid only at the binodal limit. Thus, we can write for the specific ion activity product defining the liquid-liquid binodal limit, IAP(binodal), irrespective of the (unknown) IAP-dependence of lnX; IAP(binodal) ∝ K sp (polymorph)•lnK(cluster) eq. 52 Thus, we finally obtain; discontinuity or kink in the pre-nucleation ion binding upon liquid-liquid demixing implies that K(l-l)=K(cluster). We can thus rewrite eq. 46; where X is another unknown function g of the IAP, g(IAP). Again, previous work (9) showed that liquid-liquid demixing occurred readily upon crossing the liquid-liquid binodal limit, at least at sufficiently slow mixing rates, that is, there is no major barrier associated with this event, and at the binodal limit, p(demix)≈1. We can thus re-write eq. 47; 0=lnX+lnK(cluster) eq. 48 Thus, at the liquid-liquid binodal limit, eqs. 48 and 44 are identities; lnX+lnK(cluster)=Y 1 K sp (polymorph) eq. 49 Re-arranging eq. 49 yields; lnX•K sp (polymorph)=Y-K sp (polymorph)lnK(cluster) eq. 50 Using eq. 45, eq. 50 can be simplified; lnX•K sp (polymorph)=-K sp (polymorph)lnK(cluster) eq. 51 The left-hand-side of eq. 51 is the product of a (unknown) function of the IAP in the mother solution, lnX, and the solubility of the given polymorph, which represents an ion activity product itself. Moreover, eq. 51 is valid only at the binodal limit. Thus, we can write for the specific ion activity product defining the liquid-liquid binodal limit, IAP(binodal), irrespective of the (unknown) IAP-dependence of lnX; IAP(binodal) ∝ K (polymorph)•lnK(cluster) eq. 52 discontinuity or kink in the pre-nucleation ion binding upon liquid-liquid demixing implies that K(l-l)=K(cluster). We can thus rewrite eq. 46; where X is another unknown function g of the IAP, g(IAP). Again, previous work (9) showed that liquid-liquid demixing occurred readily upon crossing the liquid-liquid binodal limit, at least at sufficiently slow mixing rates, that is, there is no major barrier associated with this event, and at the binodal limit, p(demix)≈1. We can thus re-write eq. 47; 0=lnX+lnK(cluster) eq. 48 Thus, at the liquid-liquid binodal limit, eqs. 48 and 44 are identities; lnX+lnK(cluster)=Y 1 K sp (polymorph) eq. 49 Re-arranging eq. 49 yields; lnX•K sp (polymorph)=Y-K sp (polymorph)lnK(cluster) eq. 50 Using eq. 45, eq. 50 can be simplified; lnX•K sp (polymorph)=-K sp (polymorph)lnK(cluster) eq. 51 The left-hand-side of eq. 51 is the product of a (unknown) function of the IAP in the mother solution, lnX, and the solubility of the given polymorph, which represents an ion activity product itself. Moreover, eq. 51 is valid only at the binodal limit. Thus, we can write for the specific ion activity product defining the liquid-liquid binodal limit, IAP(binodal), irrespective of the (unknown) IAP-dependence of lnX; 25 51 is valid only at the binodal limit. Thus, we propose that the left-hand-side of eq. 51 can be interpreted to be proportional to the specific ion activity product defining the liquid-liquid binodal limit, IAP(binodal), and we obtain; eq. 52 Thus, we propose that the liquid-liquid binodal limit is given by; eq. 53 where A(polymorph) is a (a priori unknown) constant. We stress that eq. 53 has not been derived definitely here, and relies on the assumptions and interpretations outlined above.
Eq. 53 poses a conjecture-which has its foundation in our assessment of the probability of liquid-liquid demixing versus that of the direct formation of crystalline polymorphs-rather than an unambiguously proven equation.
The dense liquid adjusts its composition according to that of the mother liquid; the underlying equilibrium between mother liquid L1 and dense liquid L2 can be formally written as; eq. 54 With; eq. 55 Macroscopically, this equilibrium is indistinguishable from ion association yielding PNCs (eq. 17), as the dense liquid droplets formally belong to a new phase but are structurally very similar to PNCs. 10 Since no change in pre-nucleation slope is observed upon liquid-liquid separation (also see above and eqs. 46 and 47), the value of the corresponding equilibrium constant K(l-l) is identical with that of K(cluster), 13

24
The left-hand-side of eq. 51 is the product of a (unknown) function of the IAP in the mother solution, lnX, and the solubility of the given polymorph, which represents an ion activity product itself. Moreover, eq. 51 is valid only at the binodal limit. Thus, we can write for the specific ion activity product defining the liquid-liquid binodal limit, IAP(binodal), irrespective of the (unknown) IAP-dependence of lnX; IAP(binodal) ∝ K sp (polymorph)•lnK(cluster) eq. 52 Thus, we finally obtain; IAP(binodal)=A(polymorph)•K sp (polymorph)•lnK(cluster) eq. 53 where A(polymorph) is a (a priori unknown) constant.
The dense liquid adjusts its composition according to that of the mother liquid; the underlying equilibrium between mother liquid L1 and dense liquid L2 can be formally written as; Macroscopically, this equilibrium is indistinguishable from ion association yielding PNCs (eq. 17), as the dense liquid droplets formally belong to a new phase but are structurally very similar to PNCs (2). Since no change in pre-nucleation slope is observed upon liquid-liquid separation (also see above and eqs. 46 and 47), the corresponding equilibrium constant K(l-l) is identical with K(cluster) (9), K(l-l)= IAP(L2) IAP(L1) =K(cluster) eq. 56 At the critical point, IAP(L1)=IAP(L2), i.e. K(cluster)=1. The critical temperature Tcrit. can thus IAP(binodal)=A(polymorph)•K sp (polymorph)•lnK(cluster) eq. 53 where A(polymorph) is a (a priori unknown) constant.
The dense liquid adjusts its composition according to that of the mother liquid; the underlying equilibrium between mother liquid L1 and dense liquid L2 can be formally written as; Macroscopically, this equilibrium is indistinguishable from ion association yielding PNCs (eq. 17), as the dense liquid droplets formally belong to a new phase but are structurally very similar to PNCs (2). Since no change in pre-nucleation slope is observed upon liquid-liquid separation (also see above and eqs. 46 and 47), the corresponding equilibrium constant K(l-l) is identical with K(cluster) (9), K(l-l)= IAP(L2) IAP(L1) =K(cluster) eq. 56 At the critical point, IAP(L1)=IAP(L2), i.e. K(cluster)=1. The critical temperature Tcrit. can thus 26 eq. 56 At the critical point, IAP(L1)=IAP(L2), i.e. K(cluster)=1. The critical temperature Tcrit. can thus be calculated using; eq. 57 that is; eq. 58 Notably, the spinodal limit becomes IAP(spinodal)=1 at Tcrit. (eq. 40), which is a lower critical solution temperature, categorically. That is, phase separation via spinodal demixing of PNCs cannot occur below Tcrit., because higher associated states are inherently unstable in this temperature regime, where K(cluster)<1 and DG 0 (cluster)>0, for any IAP below the critical temperature. Consistently, the binodal limit vanishes at the critical temperature, where lnK(cluster)=ln1=0 (eq. 53).