Page 23 - Nanotechnology an introduction

characteristic of ferromagnetism. This phenomenon is known as superparamagnetism.
Below a certain critical size r , the intrinsic spontaneous magnetization of a ferromagnetic material such as iron results in a single domain with all
s
spins in the same direction; above that size the material is divided into domains with different spin directions. Kittel has given a formula to estimate
the critical single domain size (diameter) in zero magnetic field:

(2.17)
where

(2.18)
is the domain wall surface energy density, J the spontaneous magnetization, E the exchange energy and K the anisotropy constant. Typical values
s
of d are 30 nm for Co, 50 nm for Fe, and 70 nm for iron oxides (hematite, Fe O , and magnetite, Fe O ). These sizes could reasonably be taken
s
3 4
2 3
to be the upper limit of the nanorealm, when magnetism is being considered. Incidentally, the magnitude of the Bohr magneton, the “quantum” of
magnetism, is given by .
Within a single domain of volume V, the relaxation time (i.e., the characteristic time taken by a system to return to equilibrium after a disturbance) is

(2.19)
where τ is a constant (~10 −11 s). Particles are called superparamagnetic if τ is shorter than can be measured; we then still have the high
0
susceptibility of the material, but no remanence. In other words, ferromagnetism becomes zero below the Curie temperature. It is a direct
consequence of the mean coordination number of each atom diminishing with diminishing size because there are proportionately more surface
atoms, which have fewer neighbors. There is thus a lower limit to the size of the magnetic elements in nanostructured magnetic materials for data
storage, typically about 20 nm, below which room temperature thermal energy overcomes the magnetostatic energy of the element, resulting in zero
hysteresis and the consequent inability to store magnetization orientation information.

Since the relaxation time varies continuously with size (note that K may also be size-dependent), it is unsuitable for determining the nanoscale; on
the other hand r could be used thus. It is known that the shape of a particle also affects its magnetization [132]. A similar set of phenomena occurs
s
with ferroelectricity [31].
2.7. Mechanical Properties
The ultimate tensile strength σ of a material can be estimated by using Hooke's law, ut tensio, sic vis, to calculate the stress required to separate
ult
two layers of atoms from one another, by equating the strain energy (the product of strain, stress and volume) with the energy of the new surfaces
created, yielding

(2.20)
where γ is the surface energy (Section 3.2), Y is Young's modulus (i.e., stiffness) and x is the interatomic spacing (i.e., twice the atomic radius, see
Table 2.1). This is basically an idea of Dupré. A.A. Griffith proposed that the huge discrepancies between this theoretical maximum and actual
greatest attainable tensile strengths were due to the nucleation of cracks, which spread and caused failure. Hence strength not only means
stiffness, but also must include the concept of toughness that means resistance to crack propagation. By considering the elastic energy W stored
σ
in a crack of length 2l, namely

(2.21)
where σ is the stress, and its surface energy

(2.22)
(ignoring any local deformation of the material around the crack), a critical Griffith length can be defined when equals :

(2.23)
if l < l the crack will tend to disappear, but if l > l then it will rapidly grow. Note the existence of accompanying irreversible processes that
G
G
accelerate the process, typically leading to microscopic failure, quite possibly with disastrous consequences. By putting l = x into formula (2.23)
G
one recovers the estimate of ultimate tensile strength, equation (2.20) (neglecting numerical factors of order unity).
If σ is taken to be a typical upper limit of the stress to which a structural component will be subjected in use, one finds values of l in the range of
G
nanometers to tens of nanometers. Therefore, as far as mechanical properties of materials are concerned, l provides an excellent measure of the
G
nanoscale. If the size of an object is less than l , it will have the theoretical strength of a perfect crystal. If it is greater, the maximum tensile strength
G
−1
will diminish ~ l (cf. equation 2.23), a result that was demonstrated experimentally by Griffith using glass fibers and by Gordon and others using
whiskers of other materials such as silicon and zinc oxide.
In reality, equation (2.23) may somewhat underestimate the critical length because of the neglect of material deformation, which causes the work of
fracture to exceed that estimated from the surface tension alone. Griffith's criterion is based on Dupré's idea, which completely ignores this
deformation. Hence Bikerman [19] has proposed an alternative hypothesis, that a crack propagates when the decrease of strain energy around a
critical domain exceeds the increase required to elastically deform this domain to the breaking point.