ferrite magnetic materials Ni-Zn ferrite
Why does the grain size increase with Sn concentration in Ni-Zn ferrtie (Ni0.6-x/2Zn0.4-x/2SnxFe2O4)?
In polycrystalline materials under the isochoric and isothermal conditions that means system is isolated mechanically from the environment the global Helmholtz free plays the major role for the establishment of the stationary equilibrium states. That means combination of the bulk and the surface Helmholtz free energies as whole should be minimized with respect to any possible variations in the size and the shape of the individual grains.
The grain size is closely related with the grain boundary (GB) Helmholtz specific free energy, which is in turns depends upon the chemical composition of GB or the segregation of doping elements such as Sn whether they reduce or increase the specific GB free energy without to much affecting the bulk volumetric free energy. It seems to me that the segregation of excess Sn concentration results in your case increase in the specific surface free energy of GB which may be compensated by the total bulk free energy variation of the grains through the spontaneous decrease in the surface area to volume ratio by switching towards the larger grain sizes, namely; S/V = 3/R about.
There is another possibility, which occurs very rarely, namely doping with Sn doesn't affect very much on the surface free energy but rather reduced the free energy of the bulk regions. In this case large grain size again is preferred to attain the minimum global Helmholtz free energy state.
Reduction in surface specific Helmholtz free energy may result fragmentation of the large particles into small ones. Especially this may be reason of the observed formation of vortex lines structure in type II super-conductors due to the negative surface specific free energy as an extreme case.
Some oxides could bring down the melting point. SnO is an example So for the same sintering T, addition of Sn helps to sinter better or it is equivalent to raising the sintering T for the ferrite. So grain growth is favoured.
One write down the following global Helmholtz free energy which includes surface free energy associated with grain boundaries (GB).
Global F = Vol.8{ Fs 3/R + Fb } where Vol= Np (4 Pi R^3 /3) Specimen Volume. Np Number of grains, R mean grain size (radius), Fs specific surface energy of GB. Fb is volumetric Bulk Helmholtz free energy, which includes stored stain energy as well if any.
Increase in Sn concentration results segregation of Sn at the grain boundaries, which may cause increase in the specific surface Free energy of grain boundaries, which may be compensated according to the above equation by increasing the grain size R to keep the global free energy invariant.
The mostly there might be an upper limit for the dopant concentration above which segregation may be in the form second phase particles, which doesn't affect the specific surface energy any more. That is the case for Ni doping in ZnS system.
Dear Tarık Ömer Oğurtanı, thanks for your ans. also for your valuable time.
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