Page 112 - Handbook of Materials Failure Analysis

5 Fick’s Diffusion Law 107

constant of proportionality D c , presented in Equation 5.28, is called chloride diffu-
sion coefficient. In general, D c is not constant, but depends on many parameters as
the time for which diffusion has taken place, location in the concrete, composition of
concrete, among other factors. If the chloride diffusion coefficient is constant, Equa-
tion 5.28 is usually referred as Fick’s first diffusion law. If this is not the case, the
relation is referred as Fick’s first general diffusion law.
There are some cases where this simple relation should not be applied. In this
regard, it is worth to mention the cases where the diffusion process may be irrevers-
ible or has a history dependence. In such cases, Fick’s diffusion law is not valid and
the diffusion process is referred as anomalous. However, nonobservation so far indi-
cates that the chloride diffusion into concrete pores should be characterized as an
anomalous diffusion. Fick’s second law can be derived considering the mass balance
principle. Therefore,
@C @ @C
¼ D c (5.29)
@t @x @x
To apply Fick’s second diffusion law, in this form, for concrete exposed to chloride
during a long period of time, one ought to know the variation of the chloride diffusion
coefficient along time. If only few observations exist in a specific case, it is possible
to estimate upper and lower boundaries for the variation of D in time. In spite of this
dependence, an especial case can be considered where the chloride diffusion coef-
ficient is independent of location, x, time, t, and chloride concentration, C. In this
case, Fick’s second law is written in this simple form:
2
@C @ C
¼ D 0 (5.30)
@t @x 2
in which D 0 is the constant coefficient of diffusion.
The solution of the differential equation presented above, for a semi-infinite
domain with a uniform concentration at the structural surface, is given as follows:

x
Cx, tÞ ¼ C 0 erfc p ffiffiffiffiffiffiffi (5.31)
ð
2 D 0 t
in which C 0 is the chloride concentration at the structural surface supposed to be con-
stant in the time; erfc is the complementary error function.
In a physical sense, field conditions deviate significantly from the assumptions
implicit in Fick’s law. For instance, the cover is not always saturated with water,
and so chloride ions penetrate concrete by diffusion and advection provided by
the penetrating moisture front. Concrete is not homogeneous due to the presence
of microcracking and interconnected pores, then, the diffusion coefficient will
change with time as hydration proceeds. Hence, Fick’s law is not an excellent model
for this phenomenon. Nonetheless, Fick’s law is often used since in many cases the
diffusion equation provides the best approximation to laboratory or field data [45].
Clearly, predictions using such approach are valid only if best-fit parameter
values are applied to structures with similar material, environmental, and field con-
ditions. It is preferable, in using this approach that concentrations are given in terms