Volume 1 Paper 7
Stochastic Kinetics of Corrosion and Fractal Surface Evolution
W. M. Mullins, E. J. Shumaker and G. J. Tyler
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JCSE Volume 1 Paper 7
Originally submitted 5th March 1997, revised version submitted 12th September 1997
Stochastic Kinetics of Corrosion and Fractal Surface Evolution
W. M. Mullins, *E. J. Shumaker and #G. J. Tyler
TMCI-Wright Laboratory, Wright-Patterson AFB, Ohio, *Wright State
University, Dayton, Ohio and #University of Dayton, Dayton, Ohio.
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A kinetic model for general attack is proposed. This model predicts the
evolution of a rough surface with a Hausdorff (fractal) dimension that
approaches 2.5 as a limiting case. The model predicts a measurable critical
length scale, that can be used to determine the time of exposure. Experimental
results are shown for 2024-T3 which corroborate the model for the limiting
Fractal geometry has been found useful for quantitatively describing the
irregular shapes associated with fracture surfaces [1-4] and corrosion [5,6].
These studies, particularly those in corrosion, seek to relate the measured
"fractal" dimension of the surface to the time or severity of
exposure. Though anecdotal evidence suggests a relationship between exposure
and "fractal" dimension, and experimental results reinforce this, no
quantitative models for the formation and evolution of these surfaces have
been proposed in the common literature.
From simple rate theory, the rate of the reaction of the surface is nearly
linearly related to the chemical potential of the species on the surface,
which is related to the local surface curvature. In addition, the
electrochemical reaction requires electron transfer across the surface causing
local regions of anodic and cathodic reactions. The locations of these
reactions are random functions of time and position so that their effect on
the overall reaction can be considered as a classical "white noise"
source. The rate of surface recession can be expressed as
where y is the recession, x is a position vector on the
surface, A is the classically defined rate constant for a flat surface,
α is proportional to the surface tension, A
and β are assumed constants for this system and Wt(x)
is an uncorrelated "white noise" function with zero expectation
value . Taking the Fourier transform changes the rate model to
where ω is the (magnitude of the) wave-vector on
is a wide-sense stationary, complex,
uncorrelated random process  and δ(ω) is the Dirac delta-function. Eqn. (2) is a linear stochastic differential
equation with the general solution
where Y0(ω) is the initial
condition for the surface and is a
generalized, complex Brownian motion given by .
§4 The ω = 0 solution is the average rate of
reaction for the system. The first term in the ω ≠
0 solution is the effect of the reaction on the initial surface profile. As
expected, the effect of the reaction is to remove all surface asperities and
smooth the surface with time. Eventually all surface characteristics are
removed completely. The second term in the ω ≠ 0 solution is a stochastic (or Ito) integral  which has no closed form
solution but can be easily estimated numerically. This term generates all of
the interesting features of the system in the Fourier domain.
§5 Since experimental measurements report the power spectral density of the
surface profile, plotted on a log-log scale, it seems appropriate to calculate
the expectation of the power spectral density (psd) of eqn. (3). Neglecting
the initial surface profile (which is identically zero for an initially
polished surface anyway) the psd is
Using Ito isometry , or
where E[ ] denotes the expectation operation, for the complex
process in eqn. (3) the result is
§6 Figure 1 shows both a numerically generated solution to eqn. (4) and the
associated analytical expectation from eqn. (5) for two different times. As
can be seen from the figure, the "white noise" term in eqn. (2)
excites uniformly across the frequency domain. The high frequency terms are
damped by the reaction so that the steady state solution will approach a
straight line (of slope -2) on the log-log plot. This would correspond to a
surface with a Hausdorff dimension of 2.5.
As shown in Figure 1, a knee appears in the transient psd curve for the
system. This knee is an apparent transition from slope=0 behavior at long
length scales to slope=-2 behavior at short length scales. The position of
this knee can be determined by extrapolating the two limiting behaviors at the
extremes and equating them to give
§8 This transition length scale can be used as a measure of the time of
exposure. It must be kept in mind, however, that long exposures, or for
particularly aggressive environments, this length scale can become too long to
practically measure. In addition, errors in evaluation of the critical length
scale become large at longer exposure times.
§9 Since the system is linear, any change in the spectral content of the
"white noise" source will be reflected in the measured psd function.
Specifically, if the microstructure of the surface has any texture or
intrinsic periodicities on the scale of the measurements, then these will be
observed as characteristic periodicities in the measured recession.
§10 It should also be noted that the inclusion of surface curvature effects in
the initial kinetic equation admits the possibility of negative recession
rates (deposition) in some high curvature areas. This is considered
unrealistic, but the inclusion of asymmetric curvature effects limits the
tractability of the problem. Numerical simulations that include asymmetric
curvature effects have been performed. In all cases, these simulations behave
similarly to the model presented above and produce the same Hausdorff
dimension for short length scales.
§11 Figure 1.
Simulation (solid) and analytical expectation (dashed) of surface profile psd
for short and long time.
§12 The samples examined in this study were tensile coupons cut from 2024-T3
sheet. These were then subjected to an ASTM G44 stress-corrosion cracking
test, alternate immersion in 3.5% NaCl solution, unloaded for 3 and 6 weeks.
Following the exposure, the samples were cleaned according to the ASTM G1
procedure, cleaning in a hot H3PO4, CrO3 and
HNO3 solution followed by a rinse in room temperature HNO3.
The samples were then tested to failure in fatigue. The front and back surface
of a representative sample is shown in Figure 2.
§13 Both sides of each sample were studied using a custom-designed,
high-precision scanning acoustic microscope (HIPSAM) . The HIPSAM system
was outfitted with a 0.635cm diameter, 100MHz transducer with a 0.5cm focal
length and set to perform a time-of-flight C-scans to the top surface of each
sample. The step size of each scan was 0.05mm along the scan axis and 0.02mm
along the index axis. This resulted in a series of bit-mapped surface profile
images that could be analyzed with conventional image analysis software.
§14 Five representative 512x512 pixel regions were selected from each of the
resulting surface profile maps. The 2D Fourier transforms of these regions
were averaged and the longitudinal and transverse traces were recorded. A
representative psd plot of the traces is shown in Figure 3. As can be seen in
the figure, the psd follows the predicted trend except in the highest
frequency region. These data are for spatial resolutions on the order of a few
wavelengths of the excitation (14.9 μm) and are
likely due to defocusing and smearing of data over the frequency range.
§15 Figure 2. Representative
corrosion sample (larger images of left and right
A simple kinetic model for general attack has been described. This model
predicts the evolution of a rough surface with a Hausdorff (fractal) dimension
that approaches 2.5 as a limiting case. The model predicts a measurable
critical length scale, that can be used to determine the time of exposure.
Experimental results have been shown for 2024-T3 which corroborate the model
for the limiting case.
The corroded and fatigued samples used in this study were provided by J. N.
Scheuring and Prof. A. F. Grandt of the Purdue University School of
Aeronautics and Astronautics. A special thanks to L. L. Mann and R. W. Martin
for assistance in conversion and manipulation of the HIPSAM data. This work
was accomplished for the U. S. Air Force under contract numbers
F33615-94-D-5801 and F33615-97-5840.
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Figure 3. Power
Spectral Density (psd) of time-of-flight data for a representative sample in
longitudinal (short dash) and transverse (long dash) directions. The straight,
solid line has slope=-2.