# A study of the evaporation of a solvent from a solution--application to writing ink aging

Forensic Sci Int. 2012 Jun 10;219(1-3):119-28. doi: 10.1016/j.forsciint.2011.12.008. Epub 2012 Jan 18.

## Abstract

When writing ink is placed on a substrate, a drying process begins. This process is dependent on the composition of the ink and of the substrate. Lociciro et al. provide an equation that describes the drying process based on models developed by earlier investigators. The work given here develops an equation for the drying process that is based on a different and rather simple model. This model considers the evaporation of a solution in an opened vertical container (e.g., a beaker) and consists of a volatile, non-hygroscopic solvent with a non-volatile solute dissolved in it. Three assumptions are made: (a) the rate of evaporation is proportional to the vapor pressure of the solution and to the solution's exposed surface area, (b) this solution vapor pressure is proportional to the solvent vapor pressure with the proportionality constant being the solvent mole fraction (Raoult's law), and (c) a small fraction of the solvent remains trapped in the solute after evaporation ceases. What results is a differential equation, which, when solved, gives the solvent weight W(t) as an implicit function. What emerges naturally from this treatment is the fact that the function W(t) can have a point of maximum acceleration. Prior to this point the drying process is fast and after this point, the drying process is slow. An approximation to W(t) is taken to be the sum of two exponential functions, one describing the fast drying region and the second describing the subsequent slow drying region. Upon including an additive constant, this approximation turns out to be similar to, but not the same as that provided by Lociciro et al. However, their equivalence is shown and then tested using the two inks examined by Lociciro et al. (the drying of a Bic and a Staedtler blue ballpoint ink). The examples of (solvent+solute) systems ("inks") given here consist of the solvent (2-phenoxyethanol) and a solute such as a dye (crystal violet) or a polymer resin such as synthetic resin SK or polyvinyl butyral. In the case of polymers, the Raoult's law assumption is replaced with one developed by the Flory-Huggins theory of solutions. Saturation and film formation, both of which slow the evaporation rate, are also discussed.