Determination of Interfacial Free Energy of Goethite Nanoparticles using Diffraction-based Method

Goethite nanoparticles with equivalent crystal sizes of ~ 9 - 38 nm were synthesized via reactions of ferric nitrate with potassium / sodium hydroxide in aqueous solutions. Silicon internal standard was used in determination of the lattice parameters of the synthesized samples at ambient conditions. The lattice parameters were derived from Rietveld analyses. Synchrotron high-pressure XRD (HPXRD) was used to determine the bulk modulus of selected goethite samples. Size-dependent bulk modulus is used for deduction of size-dependent surface stress and surface free energy. A sample was loaded into a diamond anvil cell and compressed to a required pressure (up to tens of GPa) for XRD at ALS 12.2.2 synchrotron beamline station. Data were collected and analyzed to get the lattice parameters of nano-goethite at different pressures. From fits to the Birch-Murnaghan equation of state, the bulk modulus of nano-goethite was obtained. Ambient XRD shows that as the particle size decreases, the goethite unit cell (Pnma space group) contracts in the two long sides a and c, but expands in the short side b. Overall, the unit cell volume decreases with decreasing particle size. The higher surface stress exerting on smaller nanoparticles causes the larger shrink in the unit cell, which is used for derivation of the surface stress. HPXRD measurements showed that the bulk modulus of nano-goethite increases as the particle size decreases, probably due to pressure-induced enhancement of dislocation contents in fine nanoparticles. Intervene of dislocations near nanoparticle surface and gain boundaries can make the nanoparticles stiffer. From the determined unit cell volume and bulk modulus as a function of particle size, both the surface stress and the surface free energy of goethite nanoparticles as a function of particle size were derived using equations derived previously (Zhang et al., Phys. Chem. Chem. Phys. 2009, 11, 2553). The two quantities exhibit maximum at a certain size. Above 10 nm, the surface free energy of goethite is ~ 0.43 - 0.53 J/m2, close to the average surface enthalpy (0.60 J/m2) determined from calorimetry (Mazeina and Navrotsky, Clays and Clay Minerals 2005, 53, 113). The decreases of the surface free energy at small sizes predicts a smaller critical nucleus than that predicted using the classical nucleation theory (which assumes a constant surface free energy as the bulk crystal). This indicates that nanoparticle nucleation is not as difficult as it would be based on the classical nucleation theory.