# Advances in Permeation of Solutes into Hair: Influencing Factors and Theoretical Models
## 4 Permeation Models of Solutes in Hair
### Heterogeneous Partition and Diffusion Model  
Based on [PERSON]'s second law, [PERSON] et al. [26] developed an equation for calculating the maximum equilibrium absorption (\(M_{\infty}\)) and effective diffusion coefficient (\(D_{\rm e}\)) of solutes in hair based on full consideration of the composition of hair:  
\[\frac{M_{\rm t}}{M_{\infty}}\approx 1-exp\left(-2.405^{2}\frac{D_{\rm e}}{R^{2} }t\right) \tag{8}\]  
where \(M_{\rm t}\) and \(M_{\infty}\) are the absorption of the solute in the hair at time \(t\) and the balance of absorption, respectively, and \(D_{\rm e}\) is the effective diffusion coefficient of the solute in the hair. Using this equation to process the experimental data, the partition coefficients and effective diffusion coefficients of five tea polyphenols in hair were calculated. Since hair mainly comprises keratin, fat, and water, the amount of solute absorbed in hair is the sum of solute absorbed in these three parts. Considering the influences of the oil-water partition coefficient, the dissociation constant of the solute, and the solvent pH on the partition coefficient, a quantitative structure-property relationship (QSPR) model of the solute partition in hair/water was developed according to linear free energy theory [64; 65; 66; 67]:  
\[PC=\theta_{\rm k}\left[f_{\rm non}\times 5.72(K_{\rm ov})^{0.35}+(1-f_{\rm non}) \times 1.17(K_{\rm ov})^{0.0019}\right]+\theta_{l}\times f_{\rm non}\times(K_{ \rm ov})^{0.7}+\theta_{\rm w} \tag{9}\]  
where \(\theta_{\rm k}\), \(\theta_{\rm l}\), and \(\theta_{\rm w}\) are the volume fractions of keratin, fat, and water in hair, respectively; \(f_{\rm non}\) is the mass fraction of the solute that has not dissociated at a specific pH; and \(K_{\rm ov}\) is the oil-water partition coefficient of the solute. [PERSON] et al. [26] hypothesized that the CMC and aqueous phase in hair are the main permeation channels of solutes in hair, the CMC is the limiting factor for permeation of charged ions in hair, and the effective diffusion coefficient of the solute in hair is not only related to the size of the solute molecules but also to the dissociation constant of the solute and its partition coefficient in hair. Based on comprehensive consideration of the above-mentioned influencing factors, they developed a QSPR model of the effective diffusion coefficient of a solute in hair (\(D_{\rm e}\)):  
\[\frac{D_{e}}{D_{w}}=exp[0.16\times(f_{non}K_{\rm ov})-10.71] \tag{10}\]  
where \(D_{\rm w}\) is the diffusion coefficient of the solute in water. The QSPR models comprehensively consider the influence of hair microstructure and composition, solute oil-water partition coefficient and dissociation constant, and solvent pH, and it can initially realize rapid prediction of the absorption amount and absorption rate of a solute in hair. However, there are also some limitations of this model:  
(1) The QSPR model can only predict the partition coefficient and effective diffusion coefficient of a solute in hair, and it cannot predict other parameters, such as the concentration distribution and cumulative absorption amount in the permeation process. (2) The dataset of this model only includes five anionic solutes and the applicability of cationic and neutral solutes cannot be determined. (3) The model fails to consider the effect of the charge interaction on the permeation and absorption properties of a solute in hair. Based on the study by [PERSON] et al. [26], [PERSON] et al. [63] studied the influencing mechanisms of the solvent pH, hair charge characteristics, solute charge characteristics, and hydrophobicity on the partition behaviors of porous hair media through absorption equilibrium experiments. Based on the influencing mechanisms of the electrostatic and hydrophobic effects, a QSPR model of hair partition was constructed according to linear free energy theory:  
\[