2 Interparticle Interactions: Theory and Mesoscopic Modeling
45
of the nanoparticle, so they are dimensionless. Therefore, the normalized reduced
anisotropy constants are k c for the core, k shell for the shell anisotropy or k srf for the
surface anisotropy. The effective exchange coupling constants between the core spin
and the shell or the surface spins are j c1 , j c2 , j shell or j srf and between the neighboring
particles in contact j inter . In what follows, we denote by H the reduced magnetic field,
by g the reduced dipolar strength, by T and T B the reduced values of the temperature
and the blocking temperature respectively.
The calculation of the zero-field cooled (ZFC)/Field Cooled (FC) magnetization
curves is a three steps process. First, we cool the system at a constant temperature
rate T from a high temperature, well above the ordering temperature, down to a
very low temperature, close to zero temperature, at zero applied field. Then, we heat
the sample from the low temperature to the high one at the same constant temperature
rate applying a small magnetic field and we calculate the ZFC magnetization curve.
Finally, we cool the system down to the lowest temperature, in the presence of the
applied magnetic field, and we calculate the FC magnetization curve. The hysteresis
loops are calculated after a field cooling procedure.
The Monte Carlo simulations results for a given temperature and applied field
were averaged over 60–80 samples with various spin configurations, realizations of
the easy-axes distribution and different spatial configurations for the nanoparticles.
2.2.1.2 Results and Discussion
(a) Dipolarly interacting assemblies of ferrimagnetic nanoparticles with
core/surface morphology
Maghemite (γ-Fe 2 O 3 ) nanoparticles of ∼12.7 nm size covered by polyacrylic acid
(PAA) surfactant were produced with volume fraction x = 0.47 [27]. Our mesoscopic
model simulates this system as N identical spherical ferrimagnetic NPs of diameter
d = 13 nm being located randomly at the nodes of a simple cubic lattice with
lattice constant, a, inside a box of edge length L = 10 measured in units of a.
The total number of NPs is N = p × (10 a × 10 a × 10 a), where p = 0.47 is
the concentration of the particles in the model. In accordance with the experimental
findings, we consider core/surface morphology for each nanoparticle and we describe
it by three spins with a common anisotropy easy axis. However, this axis is assumed
to be randomly oriented from particle to particle. In addition, the parameters of the
surface anisotropy (K S ) and magnetic moment (m) per particle were rationally chosen
upon consideration of (a) the morphology of the NPs that is not completely spherical
and (b) the approximate thickness of the surfactant layer, which appears to influence
the degree of the defected surface coordination environment; the thicker the surface
coordinating organic layer is, the lower the disorder of the uncompensated surface
spins becomes [39]. Effectively, we consider a common anisotropy easy axis for
spins in each nanoparticle resulting to a higher saturation magnetization, M s than the
corresponding bulk value [40] as observed experimentally [27].
45
of the nanoparticle, so they are dimensionless. Therefore, the normalized reduced
anisotropy constants are k c for the core, k shell for the shell anisotropy or k srf for the
surface anisotropy. The effective exchange coupling constants between the core spin
and the shell or the surface spins are j c1 , j c2 , j shell or j srf and between the neighboring
particles in contact j inter . In what follows, we denote by H the reduced magnetic field,
by g the reduced dipolar strength, by T and T B the reduced values of the temperature
and the blocking temperature respectively.
The calculation of the zero-field cooled (ZFC)/Field Cooled (FC) magnetization
curves is a three steps process. First, we cool the system at a constant temperature
rate T from a high temperature, well above the ordering temperature, down to a
very low temperature, close to zero temperature, at zero applied field. Then, we heat
the sample from the low temperature to the high one at the same constant temperature
rate applying a small magnetic field and we calculate the ZFC magnetization curve.
Finally, we cool the system down to the lowest temperature, in the presence of the
applied magnetic field, and we calculate the FC magnetization curve. The hysteresis
loops are calculated after a field cooling procedure.
The Monte Carlo simulations results for a given temperature and applied field
were averaged over 60–80 samples with various spin configurations, realizations of
the easy-axes distribution and different spatial configurations for the nanoparticles.
2.2.1.2 Results and Discussion
(a) Dipolarly interacting assemblies of ferrimagnetic nanoparticles with
core/surface morphology
Maghemite (γ-Fe 2 O 3 ) nanoparticles of ∼12.7 nm size covered by polyacrylic acid
(PAA) surfactant were produced with volume fraction x = 0.47 [27]. Our mesoscopic
model simulates this system as N identical spherical ferrimagnetic NPs of diameter
d = 13 nm being located randomly at the nodes of a simple cubic lattice with
lattice constant, a, inside a box of edge length L = 10 measured in units of a.
The total number of NPs is N = p × (10 a × 10 a × 10 a), where p = 0.47 is
the concentration of the particles in the model. In accordance with the experimental
findings, we consider core/surface morphology for each nanoparticle and we describe
it by three spins with a common anisotropy easy axis. However, this axis is assumed
to be randomly oriented from particle to particle. In addition, the parameters of the
surface anisotropy (K S ) and magnetic moment (m) per particle were rationally chosen
upon consideration of (a) the morphology of the NPs that is not completely spherical
and (b) the approximate thickness of the surfactant layer, which appears to influence
the degree of the defected surface coordination environment; the thicker the surface
coordinating organic layer is, the lower the disorder of the uncompensated surface
spins becomes [39]. Effectively, we consider a common anisotropy easy axis for
spins in each nanoparticle resulting to a higher saturation magnetization, M s than the
corresponding bulk value [40] as observed experimentally [27].