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Using subdivision potential approach and mean-field theory for a ferromagnetic cluster, we obtained nanothermodynamic properties for a ferromagnetic nanocluster in the presence and also the absence of the magnetic field. The subdivision potential and the magnetic field both makes Gibbs and Helmholtz free energies of the ferromagnetic nanocluster stand at a lower level compared to those of the ferromagnetic cluster. Our main conclusion is that the presence of the magnetic field leads to decrease in the amount of specific heat capacity for the ferromagnetic cluster. On the other hand, this effect leads to increase in the amount of specific heat capacity for the ferromagnetic nanocluster.

Science and technology at nanoscale are of great importance in the twenty-first century [

Nanostructural materials which their constitutive structural units’ size, are at a few nanometers scale, compared to ordinary materials have much advanced, preferable and completely different mechanical, thermal, electronic and optical properties. For example, electrical resistance of ferromagnetic materials such as iron and nickel consisting of nanostructural fine-graineds, is 55% and 35% higher than the corresponding values for the same materials but consisting of microstructural fine-graineds, respectively. The use of nanostructures in devices at nanoscale, requires a deep and comprehensive understanding of their stability and mechanical, electrical and thermal performance in interactive mode. Using the application of thermodynamics for big devices (classic thermodynamics), works have been done in the field of energy studies and thermal properties of these structures.

Given the close correlation between properties of nanostructures and their size, naturally one cannot expect that classic thermodynamics can offer a correct modeling for thermal properties of the nanostructures. For example, calculation of melting point in structures at nanoscale, has been encountered with some problems which in turn are caused by the application of classic thermodynamics relations developed for thermodynamic limit [^{23}, this condition is well satisfied. Basically at this limit, the intensive properties of the device become size independent [

For the first time in the early sixties, American chemist, Terrell Hill, through publishing a paper [

Statistical standpoint of Evans fluctuation theory [

In this paper, using the partition function obtained from mean-field theory for ferromagnetic nanocluster and the application of nanothermodynamic relations, transition energy from classical thermodynamics to nanothermodynamics is calculated. Then, the effects of this energy on Helmholtz and Gibbs free energies, chemical potential and specific heat capacity are investigated in the presence and the absence of the magnetic field.

Nanothermodynamics can explain thermodynamic properties of small enough devices [

In thermodynamics, a macroscopic device’s energy is given by following relations:

Generally, in classical thermodynamics, except internal energy, one can enumerate seven thermodynamic potentials that all of them are obtained through Legendre transform of E with respect to corresponding variables [

Generally, in nanothermodynamics, relations such as

Terrell Hill approach, in investigation of thermodynamics of small devices, is called subdivision potential approach. In investigation of conductive thermodynamics of small devices, we only consider devices that satisfy two conditions: First, they are so small that macroscopic thermodynamics is not applicable on them. Second, they are so big that their properties can be assumed to be continuous.

In this approach, first as a principle, we accept that without any fundamental changes we can use common statistical mechanics for small devices. The basis of our work is as follows: we divide a macroscopic device into B independent subdevices. B represents the number of small subdevices resulting from subdividing a big device. If B is big, then we are sure that each subdevice is small enough and the effects of their size become important. Common thermodynamic relations are not valid on each individual small device, but we can take all of these small devices as a macroscopic device and therefore we can use classical thermodynamic relations [

where the index t represents the corresponding properties on all small devices.

N_{t} represents the number of all particles in the macroscopic device and B represents the number of particles in a small subdevice.

The energy of this particular macroscopic device also is a function of B; therefore, the correct form of Equation (1) is as follows [

Using Equation (3), we have:

This means that the subdivision potential,

Since Equation (3) is written for a macroscopic device, then using Euler’s theorem [

One can obtain Helmholtz and Gibbs free energies for nanomaterials in nanother- modynamic state as:

Finally, if we define nanothermodynamic chemical potential as the common relation

Through a series of calculations, we obtained subdivision potential in terms of Helm- holtz free energy as:

Given that

From Equation (8), one can obtain:

Consider a physical device consisting of N clusters of particles. Each particle can be aligned in either of two

where

where

In this limit, thermodynamic potential is very big and we have

where:

If we take partition function as Equation (11), all thermodynamic properties are vanished above the critical temperature. Note that for this partition function, subdivision potential is always vanished. The critical temperature at macroscopic state is defined as

In the case of a limited cluster, canonical partition function is obtained as [

For partition function mentioned above, one can obtain thermodynamic properties such as internal energy and entropy as follows:

The temperature dependence per particle of internal energy as well as the entropy for the ferromagnetic cluster is shown in

In addition, in the presence of the magnetic field, there is a sharp decline in the level of internal energy. The presence of the magnetic field makes the spins to be polarized in the same direction and leads to decrease in the internal energy of the system.

Above the critical temperature, the entropy of the ferromagnetic cluster is vanished, but in the presence of the magnetic field for small N, the entropy is vanished just after a short time interval. Since degeneration depends on all up and down spin states, then we have

Also, chemical potential is:

where

(Direct working with derivatives of the digamma function or the factorial function is inappropriate, but taking natural logarithm from the factorial and also from derivative of the factorial function is well-defined [

Gibbs and Helmholtz free energies and chemical potential for a ferromagnetic cluster are presented in

As can be seen from

If we divide this ferromagnetic cluster into B independent subdevices, subdivision potential is obtained as:

This is the energy which is important in dividing a big device into small pieces. The temperature dependence per particle of this quantity is shown in

According to

potential has a nonzero value. By using a large sample, i.e. by increasing the number of particles, this potential is vanished.

Finally, by using the Hill’s approach, the temperature dependence of nanothermo- dynamic properties such as chemical potential, Gibbs and Helmholtz free energies, per ferromagnetic nanocluster particle are shown in

For more detailed analysis, we have found that the chemical potential variations compared to the initial state in the subdivided device (

As shown in

The last term includes the size effects which is absent in the thermodynamic state (It exists only in the nanothermodynamic state). Temperature dependence of the specific heat capacity per particle in the case of N divided ferromagnetic nanoclusters is shown in

We study the properties of the ferromagnetic nanocluster by dividing a cluster of ferromagnetic materials into a set of small independent devices. When the size effects in each small device become important, a series of particular energies are generated in each small device. The diagram of this energy per particle for the ferromagnetic cluster in subdivision potential approach is shown in

Using mean-field theory that provides us by the canonical partition function of the ferromagnetic cluster, we obtain the variations of Gibbs and Helmholtz free energies for the ferromagnetic cluster in terms of temperature. On the other hand, the size effects that are developed at nanostate, are not considered in mean-field theory. Therefore, it does not give us any information about the ferromagnetic cluster at nanoscale that is subdivided by taking into account the size effects.

Finally, we study the effect of the subdivision potential on the specific heat capacity of ferromagnetic and ferromagnetic nanoclusters. As shown in

Simiari, M., Mogaddam, R.R., Ghaebi, M., Farshid, E. and Zomorrodian, M.E. (2016) Assessment of Nanothermodynamic Properties of Ferromagnetic Nanocluster in Subdivision Potential Approach and Magnetic Field. Journal of Applied Mathematics and Physics, 4, 2233- 2246. http://dx.doi.org/10.4236/jamp.2016.412216