# Advances in Permeation of Solutes into Hair: Influencing Factors and Theoretical Models
## 4 Permeation Models of Solutes in Hair
### Two-Part/Two-State Model  
Considering the influence of the hair structure and composition on the hair permeation and mechanical properties, [PERSON] et al. [62] observed the position of absorption of erythrosin B dye in hair and developed a two-part/two-state model based on heat-setting and relaxation experiments and transmission electron microscopy experiments.  
Based on the transition temperatures of the hair components, the model [47] divides the hair into two structural parts, which are called the high-temperature parts (high-Tc parts) and low-temperature parts (low-Tc parts). The corresponding transition temperatures of the two parts are 70 and 0 \({}^{\circ}\)C. The high-Tc parts are mainly composed of macrofibrils and exo-cuticles consisting of keratinous proteins; the low-Tc parts are mainly composed of inter-macrofibrillar materials, CMC, and endo-cuticles of non-keratinous proteins and lipids. For any part, when the temperature is higher than the transition temperature, the structures become soft and plastic; when the temperature is lower than the transition temperature, the structures are in a hard and elastic solid state. The high- and low-temperature transition temperatures are caused by temperature-induced breakage of S-S bonds, hydrogen bonds, and ionic bonds.  
The transition temperature is related to the hair's moisture content and the solvent's pH. The transition temperature is higher for lower hair moisture content and solvent pH. The moisture content of the hair and the pH of the solvent affect the permeation and absorption properties of the solute in the hair by affecting the transition temperature.  
To calculate the diffusion coefficient (\(D\)) of a solute in hair, the model assumes that the high-Tc parts are discretely distributed in the continuous phase composed of the low-Tc parts without considering the action of the cuticle. The diffusion coefficient of a solute in hair is calculated using the following equation [47]:  
\[D=\frac{(X_{H}D_{H}+X_{L}D_{L})D_{L}}{X_{H}D_{L}+X_{L}(X_{H}D_{X}+X_{L}D_{L})} \tag{1}\]  
where \(X_{\text{H}}\) and \(X_{\text{L}}\) are the volume fractions of the high-Tc parts and low-Tc parts, respectively, and \(X_{\text{H}}+X_{\text{L}}=1\). \(D_{\text{H}}\) and \(D_{\text{L}}\) are the diffusion coefficients of the solute in the high-Tc parts and low-Tc parts, respectively:  
\[D_{H}=n_{H}exp(-\Delta E_{H}/RT) \tag{2}\]  
\[D_{L}=n_{L}exp(-\Delta E_{L}/RT) \tag{3}\]  
where \(\Delta E_{\text{H}}\) and \(\Delta E_{\text{L}}\) are the activation energies of the high-Tc parts and low-Tc parts, respectively (\(\Delta E_{\text{H}}>\Delta E_{\text{L}}\)). \(n_{\text{H}}\) and \(n_{\text{L}}\) are the diffusion constants of the high-Tc parts and low-Tc parts, respectively. In the model, the diffusion coefficient increases with increasing temperature.  
Because the high-Tc parts are mainly composed of high-sulfur proteins, the disulfide bonds between the proteins form a net-like structure inside the region, which limits the permeation of macromolecules in the high-Tc parts. The critical size of molecules that can penetrate the network structure of the high-Tc parts is 1 nm. When the size of the solute molecule is less than 1 nm, the molecule can diffuse in both the high- and low-Tc parts. When the molecular size is larger than 1 nm, the low-Tc parts are its only diffusion channel [47].  
The two-part/two-state model considers the effects of the hair structure and composition, solvent temperature and pH, and hair moisture content on the hair permeation characteristics based on the phase-transition temperatures of the hair components. However, the physicochemical properties of the solute and the hair charge characteristics are not considered, and the effect of the cuticle layer is ignored when calculating the diffusion coefficient of the solute in hair.