## Solvent Miscibility

### Quick Start

The Solvent Blend app shows how a mix of two solvents can give solubility performance neither can achieve on their own. But if you try that blend you might find that they aren't miscible at the desired ratio. Here we show that miscibility can, to some extent, be predicted from the HSP values, though relative solvent size plays a surprisingly large effect. However, currently there is no solvent software on the planet that can reliably predict miscibility - it is a curiously difficult problem.

To start, choose toluene and methanol to see what immiscibility looks like, then swap methanol for DMF, then DMF for acetone to see what increasing miscibility looks like.

### Solvent Miscibility

Predicting the miscibility of solvents is surprisingly hard. This is because the thermodynamics of mixing are rather subtle. Sanchez-Lacombe (SL) theory^{1} does a good job at predicting miscibility via some fundamental pure solvent values and a single cross-term for which there is, at present, no predictive tool. An "appified" version of SL theory is under development and it is hoped that this powerful theory will become more widely known and used.

Fortunately, Lacombe^{2} has derived, using the same fundamental thermodynamic approach, a rather simple formula based on solubility parameter (SP) theory and this app implements the Lacombe formula, replacing SP with HSP. Of course this simplified version suffers from limitations (discussed below) not present in SL theory.

Where the volume fractions of the solvents are φ_{1} and φ_{2}, the molar volumes are MVol_{1} and MVol_{2}, r is their ratio MVol_{1}/MVol_{2}, the HSP Distance squared is D^{2} and R is the gas constant then the temperature, T, at which the solvents become miscible is given by:

T=φ_{1}*φ_{2}*2*MVol_{1}*D^{2} / [4R(r*φ_{1}+φ_{2})]

As most of us are not interested in immiscibility below -50°C or above 100°C the graph is plotted between those temperatures, with the minimum and maximum forced to those values so you can see whether the solvents are miscible to at least -50 or are immiscible to at least 100°C.

As Lacombe points out, there are some drastic assumptions being made here. The first is that the volume of the liquid is independent of temperature. The second is that the entropy of mixing is a simple value, independent of temperature. The first assumption says that the thermal expansion coefficient for pure solvents and the mixture is 0, whereas we know it is small but significant. The second means that predictions of heat capacities will be very poor as the entropy term is independent of temperature. Sanchez-Lacombe theory fully accounts for both these terms and the effects are significant. So our miscibility formula is limited. But it still captures three essential features.

- Miscibility is greater at small and large φ
_{1}. Although this is obvious, it is good to have it calculated. - Unlike solvents (large HSP Distance) are less likely to be miscible at room temperature. Again this is obvious, but it is good to have a working formula.
- Absolute and relative MVols play a significant role through the MVol
_{1}and r factors in the formula. That is why naive HSP Distance calculations of miscibility have not been successful.