Solubility of sparingly-soluble electrolytes–a new approach

Abstract

The technique of solid titration was developed to explore the true solution equilibria for complex solids and solutions, and in particular for biominerals. It has been shown that the conventional method of equilibration over a large excess of solid is inappropriate for the investigation of solubility when there is incongruent dissolution. The new method shows great sensitivity, reliability and reproducibility, and can be applied to many systems in medical and dental contexts, as well as general chemistry, geology, and the pharmaceutical industry.

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The solubility of a substance is of fundamental importance in a large number of contexts as it determines the direction and rate of many processes such as dissolution, precipitation, hydrolysis, corrosion and phase transformation, as well as concentration-dependent reactions. Solubility also has a key role in biological processes which involve the formation and resorption of hard tissues as well as of pathological calcifications. Furthermore, and particularly in a biological environment, the activity and availability of a water-soluble drug depends at least in part on its solubility. However, the conditions in a supernatant solution can vary considerably depending on the character of the solute and the solvent. For strong electrolytes, the assumption of complete dissociation may be a satisfactory approximation, even though it is known that ion pairs, for example, must be present in low concentration in any system. However, for complex systems where this is not even approximately true, and in particular for sparingly soluble electrolytes, computational difficulties lie in the fact that the solution will contain a variety of known (major) species, but also (almost certainly) a number of unreported ion pairs and complexes, for all of which formation constants are required if a full accounting is to be made. Even worse is the formation of new or unexpected phase(s), whether due to metastable equilibrium, as is shown in particular by calcium phosphates, Ostwald succession, or simply that the conditions are conducive to a novel outcome. However, and strangely, in some fields the calculated solubility product (SP) is normally based on a great simplification of the solution speciation; often, the presence of complexes and ion pairs is completely ignored, or only the major such species taken into account. Thus the true equilibrium of such systems cannot be accurately expressed through such a crudely approximate SP. As a case in point, little agreement has been reported for any of the calcium phosphates.¹ In addition, the equilibrium solid phase, that is, in contact with the solution, requires confirmation, although this is seldom done. However, thin layers of transformed material (i.e. second phases) would not be detectable by the usual X-ray powder diffraction methods.

Prompted by such problems, a novel technique of solid titration was developed. In contrast with the conventional method of adding a large excess of solid to a given solvent or solution in one portion, solid titration is based on the concept of nucleation of the stable phase very close to the saturation point, that is, with minimal supersaturation. The titration consists of the addition of a series of very small powdered solid increments, monitoring the presence and dissolution of that titrant with a very sensitive laser-scattering system. Below saturation, all solid dissolves, and the scattering signal returns to the baseline; above saturation, either there is undissolved solid, when the signal does not diminish, or the most stable solid is expected to form as a new precipitate, irrespective of the solid added, resulting in the continuous dissolution of the titrant as the new material forms. This is equivalent to the more usual solution titration, but without the solvent, and so is free of dilution problems. The titrant solid also provides (temporary) nuclei for crystal growth. A further clear advantage of this approach in comparison with any electrochemical method is that it is easy to detect unambiguously the end-point of the titration. Dissolution and reprecipitation can be clearly seen and followed semi-quantitatively in the laser-scattering signal. Although the rate of dissolution may be very low, and thus the time per increment long (from an hour or so to a few days), no solid increment is to be made until the previous addition has been completely dissolved or leaves some (small) remnant. There are thus no assumptions concerning solution speciation, and the analytical concentrations of all components are known precisely from what has been weighed and added, limited only by weighing precision. Solubility can therefore be expressed directly in terms of total dissolved solid even if the solution equilibria are not fully understood. This then provides an exact input statement for developing speciation models, if desired, as no analytical technique is required for the solution (avoiding the uncertainties and inaccuracies that these must entail): the vanishingly small amount of solid at the end-point hardly disturbs the summation of increment masses.

The detailed process is illustrated by reference to the technique used for calcium phosphates, but many aspects can be modified as appropriate to suit other systems. Thus, a glass reaction vessel containing the test solvent or solution, in this case 600 mL of 100 mM aqueous KCl, is equilibrated at the required test temperature in a water bath in a vessel of ∼750 mL. The vessel is provided with a combination pH electrode, and to control the atmosphere a capillary gas inlet. For example, solvent-saturated nitrogen may be used to flush the system to avoid interference from CO₂, or a known CO₂ mixture to understand the effect of carbonate. The titration consists of the addition of a succession of small weighed increments of powdered solid (say, 0.5–2 mg) to the solution (via one of the lid ports), and the particles are suspended by gentle agitation with a magnetic stirrer. The dissolution process is followed by detecting light scattered at a low angle from a laser beam traversing the vessel (Fig. 1). A path length of about 10 cm provides great sensitivity, easily showing suspended solid that cannot be detected otherwise by eye (< 0.1 mg L⁻¹). The main beam is blocked, and interference from ambient light is eliminated by using a sector disc to chop the laser beam at a frequency that is not a harmonic of the local mains electricity supply frequency, using an appropriately-tuned, frequency-dependent photodiode detector circuit. Each addition of the solid, even if only 0.5 mg, therefore produces an obvious step-increase in the laser-detector output signal, which conveniently is recorded on a strip chart. With time, this signal decreases quasi-exponentially due to the dissolution of the solid. When a stable signal is obtained at or very close to the original baseline value, it is taken as indicating that the solid added has completely dissolved, although for safety some further time (double the apparent dissolution time) may easily be allowed after this point, checking that the pH is indeed stable (especially if equilibrating with a reactable gas, e.g. CO₂). The next solid addition can then be made and monitored in like fashion. The end-point of the titration is unambiguously detected by the output signal remaining higher than the original baseline irrespective of time, meaning that no more solid will dissolve or that a new solid has precipitated, or both. Of course, the precision of this end-point is only as good as the size of the last increment as an exact quantitative determination of the solid present is not possible by this system, although there must be a correlation between that solid amount and the scattering signal. Accordingly, some refinement may be made by the addition of a further increment. This confirms that the true end-point has already been exceeded by the signal stabilizing at a higher value, but this value is then used to refine the estimated saturation point by interpolation, noting that the stable presence of solid always means that the end-point has been exceeded. The interpolated end-point value for the amount of added solid is determined by extending a line through the last two scattering value points to the baseline on the working approximation of proportionality, then this value is used to interpolate on a cubic polynomial fitted to the last four pH values. Such interpolation gives a closer approach to the true end-point values, with better precision than is realizable by wet analytical methods.

Fig. 1 General scheme of solid titration apparatus. (Top) Layout of the apparatus in vertical section through the beam axis. (Bottom) Schematic diagram of the beam and collection cone geometry. A: Laser source; D: Aperture plate; F: Reaction flask; G: Beam stop; H: Plate; L: Double-walled glass water bath; J: Detector housing. The layout of the apparatus is shown in Fig. 1 (Top). A semiconductor-diode laser (1 mW CW, 194-010, RS Components, Northants, UK) was used as light source. This was chopped by a 40-slot sector disc, running at 1320 rpm, giving a chopping frequency of 880 Hz. The aperture plate (D) (Fig. 1 Bottom) was used to avoid sight from the detector of the beam path through the double-walled glass water bath (L), thus avoiding picking up light scattered from the several interfaces, and limit that from the bath water itself, which was impossible to keep dust free. Thus, this aperture plate was set as close as possible to the reaction flask (F), as well as the beam stop (G). The outer diameter of this stop was used to shield the light scattered from the last glass surface of the water bath (L), thus, scattered light would not be visible through plate D. The diameter of the second aperture in plate H was simply determined by the size of the admission cone for the detector housing (J).

For pH-dependent solubilities, the pH may then be adjusted by adding HCl (or KOH) solution (for example) such that all solid is (re-)dissolved, which event is readily confirmed by the scattering signal returning to the baseline. Likewise, an aliquot of some other solution may be added to dissolve solid and move to a new region of the system. Such an undersaturated solution can then be used for a further titration to determine a new end-point. For new run sequences, depending on how well the end-point is known, a relatively large addition may be made initially, say about 90% of the final value, to approach better the target value and save some time. If this is done, care must still be taken that it has all dissolved before proceeding. Overall, runs for the calcium phosphate system could take between a few days to two weeks or so, for an end-point to be found, but dissolution in this system at least is rather slow.

Dissolution rates may be increased by pulverizing the solid, introducing crystal defects (which in principle increase the solubility of that material), although these of course have no bearing on new solid that forms, whether of the reprecipitated titrant or some new phase. In some systems, bacteria may grow, compromising the light-scattering signal. The use of a UV lamp has been found helpful to suppress such growth, even for runs of over two months (which involved multiple end-point determinations), although in some systems UV-triggered reactions may occur which would be undesirable. Since the laser beam passes through the water of the bath in the embodiment described here, that water must be kept dust free to avoid baseline drift. A thermostated air space would also suffice, if with precise control.

Using this technique, the solubility of hydroxyapatite has been examined over a range of pH^2,3 and calcium: phosphate ratio values,^4–6 and in the presence of physiological CO₂,² providing data of much greater accuracy, precision and reproducibility than hitherto obtained by the ‘large excess’ method. Analysis of the end-point solid has shown which phase is actually stable, rather than relying on assumptions, for example demonstrating that hydroxyapatite is more stable than dicalcium phosphate dihydrate at low pH (< 4).³ Should there be any interference between the system of study and the glass of the reaction vessel, as indeed occurs with fluoride, a wax coating may be used as an inert barrier. As shown in Fig. 2, the titration curve of HAp in a wax-coated vessels produced the exact same result as in plain, uncoated borosilicate, thereby unambiguously showing that there is no such interference detectable. As a result, the method has been used successfully with calcium fluoride.⁷

Fig. 2 Solubility isotherm for HAp in a glass vessel coated with wax (○). Data from ref. 3 are for comparison (●).

The solid titration method has shown extraordinary reliability and reproducibility, even with a complicated solution environment⁸ with sparingly soluble electrolytes, specifically biominerals. The particular advantage is that the method is absolute: assumption- and (almost) calculation-free. No speculation concerning solution speciation is required, nor indeed does the equilibrium solid phase have to be known in advance. Solution environments can be chosen as necessary to suit the service conditions of interest. The approach may also be useful for more soluble substances, there being no fundamental limit to its applicability, although insoluble contaminants would then contribute proportionately more to background scatter and may therefore limit sensitivity.

Thus, solid titration provides a viable and sensitive approach for the study of complex systems of low solubility, especially when complications such as incongruent dissolution or non-stoichiometric phases are encountered – all of which is true for calcium phosphates, and the conventional method using a large-excess of solid fails. It is expected that this approach can be applied to many such systems in medicine, dentistry, chemistry, geology and the pharmaceutical industry.

References

1. H. B. Pan and B. W. Darvell, Cryst. Growth Des., 2009, 9(2), 639–645.
2. Z. F. Chen, B. W. Darvell and V. W. H. Leung, Arch. Oral Biol., 2004, 49, 359–367.
3. H. B. Pan and B. W. Darvell, Arch. Oral Biol., 2007, 52, 618–624.
4. H. B. Pan and B. W. Darvell, Caries Res., 2009, 43, 254–260.
5. H. B. Pan and B. W. Darvell, Caries Res., 2009, 43, 322–330.
6. H. B. Pan and B. W. Darvell, Arch. Oral Biol., 2009, 54, 671–677.
7. H. B. Pan and B. W. Darvell, Arch. Oral Biol., 2007, 52, 861–868.
8. V. W. H. Leung and B. W. Darvell, J. Chem. Soc., Faraday Trans., 1991, 87, 1759–1764.

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