domingo, 21 de marzo de 2010

Cinetica de los materiales!

Samer Elatrache, V-17810600 estudiante de CRF

Macroscopic Transport
Whenever a material system is not in thermodynamic equilibrium, driving
forces arise naturally to push it toward equilibrium. Such a situation can occur,
for example, when the free energy of a microscopic system varies from point
to point because of compositional inhomogeneities. The resulting atomic
concentration gradients generate time-dependent, mass-transport effects that
reduce free-energy variations in the system. Manifestations of such processes
include phase transformations, recrystallization, compound growth, and degradation
phenomena in both bulk and thin-film systems. In solids, mass transport
is accomplished by diffusion, which may be defined as the migration of an
atomic or molecular species within a given matrix under the influence of a
concentration gradient. Fick established the phenomenological connection
between concentration gradients and the resultant diffusional transport through
the equation
dC
dx
J = -D- (1-21)
The minus sign occurs because the vectors representing the concentration
gradient dC/& and atomic flux J are oppositely directed. Thus an increasing
concentration in the positive x direction induces mass flow in the negative x
direction, and vice versa. The units of C are typically atoms/cm3. The
diffusion coefficient D, which has units of cm2/sec, is characteristic of both
the diffusing species and the matrix in which transport occurs. The extent of
observable diffusion effects depends on the magnitude of D. As we shall later
note, D increases in exponential fashion with temperature according to a
Maxwell-Boltzmann relation; Le.,
D = Doexp - E,/RT, ( 1-22)
where Do is a constant and RT has the usual meaning. The activation energy
for diffusion is ED (cal/mole) .
Solid-state diffusion is generally a slow process, and concentration changes
occur over long periods of time; the steady-state condition in which concentrations
are time-independent rarely occurs in bulk solids. Therefore, during
one-dimensional diffusion, the mass flux across plane x of area A exceeds
that which flows across plane x + dx. Atoms will accumulate with time in the
volume A dx, and this is expressed by
dJ dJ dc ( dx ) dx dt
J A - J + - d x A = - - A d x = - A d x . (1-23)
Substituting Eq. 1-21 and assuming that D is a constant independent of C or x
gives
ac( X , t ) a2cX( , t )
=D
at a x2
( 1-24)
The non-steady-state heat conduction equation is identical if temperature is
substituted for C and the thermal diffusivity for D. Many solutions for both
diffusion and heat conduction problems exist for media of varying geometries,
constrained by assorted initial and boundary conditions. They can be found in
the books by Carslaw and Jaeger, and by Crank, listed in the bibliography.
Since complex solutions to Eq. 1-24 will be discussed on several occasions
(e.g., in Chapters 8, 9, and 13), we introduce simpler applications here.
Consider an initially pure thick film into which some solute diffuses from
the surface. If the film dimensions are very large compared with the extent of
diffusion, the situation can be physically modeled by the following conditions
C(x,O) = 0 at t = 0 ( 1 -25a)
C(o0, t ) = 0 at x = 03 for t > 0. (1-25b)
The second boundary condition that must be specified has to do with the nature
of the diffusant distribution maintained at the film surface x = 0. Two simple
cases can be distinguished. In the first, a thick layer of diffusant provides an
essentially limitless supply of atoms maintaining a constant surface concentration
Co for all time. In the second case, a very thin layer of diffusant provides
an instantaneous source So of surface atoms per unit area. Here the surface
concentration diminishes with time as atoms diffuse into the underlying film.
These two cases are respectively described by
c(0, t ) = c,
lmcx(, t )d x = so
is a tabulated function of only the upper limit or argument x / 2 f i .
Normalized concentration profiles for the Gaussian and Erfc solutions
are shown in Fig. 1-15. It is of interest to calculate how these distributions
spread with time. For the erfc solution, the diffusion front at the arbitrary
concentration of C(x, t)/C, = 1/2 moves parabolically with time as x =
2merfc-'(1/2) or x = 0 . 9 6 m . When becomes large compared
with the film dimensions, the assumption of an infinite matrix is not valid and
the solutions do not strictly hold. The film properties may also change
appreciably due to interdiffusion. To limit the latter and ensure the integrity of
films, D should be kept small, which in effect means the maintenance of low
temperatures. This subject will be discussed further in Chapter 8.

Atomistic Considerations
Macroscopic changes in composition during diffusion are the result of the
random motion of countless individual atoms unaware of the concentration
gradient they have helped establish. On a microscopic level, it is sufficient to
explain how atoms execute individual jumps from one lattice site to another,
for through countless repetitions of unit jumps macroscopic changes occur.
Consider Fig. 1-16a, showing neighboring lattice planes spaced a distance a,
apart within a region where an atomic concentration gradient exists. If there
are n, atoms per unit area of plane 1, then at plane 2, n2 = n, + (dn /dx)

where we have taken the liberty of assigning a continuum behavior at discrete
planes. Each atom vibrates about its equilibrium position with a characteristic
lattice frequency v, typically lOI3 sec -'. Very few vibrational cycles have
sufficient amplitude to cause the atom to actually jump into an adjoining lattice
position, thus executing a direct atomic interchange. This process would be
greatly encouraged, however, if there were neighboring vacant sites. The
fraction of vacant lattice sites was previously given by eCEflkT(s ee Eq. 1-3).
In addition, the diffusing atom must acquire sufficient energy to push the
surrounding atoms apart so that it can squeeze past and land in the so-called
activated state shown in Fig. 1-16b. This step is the precursor to the downhill
jump of the atom into the vacancy. The number of times per second that an
atom successfully reaches the activated state is ve-'JIkT, where ci is the
vacancy jump or migration energy per atom. Here the Boltzmann factor may
be interpreted as the fraction of all sites in the crystal that have an activated
state configuration. The atom fluxes from plane 1 to 2 and from plane 2 to 1
are then, respectively logarithm of the rate is plotted on the ordinate and the reciprocal of the
absolute temperature is plotted along the abscissa. The slope of the resulting
line is then equal to - ED / R , from which the characteristic activation energy
can be extracted.
The discussion to this point is applicable to motion of both impurity and
matrix atoms. In the latter case we speak of self-diffusion. For matrix atoms
there are driving forces other than concentration gradients that often result in
transport of matter. Examples are forces due to stress fields, electric fields,
and interfacial energy gradients. To visualize their effect, consider neighboring
atomic positions in a crystalline solid where no fields are applied. The free
energy of the system has the periodicity of the lattice and varies schematically,
as shown in Fig. 1-17a. Imposition of an external field now biases the system
such that the free energy is lower in site 2 relative to 1 by an amount 2 AG. A
free-energy gradient exists in the system that lowers the energy barrier to
motion from 1 -+ 2 and raises it from 2 -+ 1.

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