Materials Science and Engineering A327 (2002) 24 – 28
www.elsevier.com/locate/msea
Measurement of the Gibbsian interfacial excess of solute at an
interface of arbitrary geometry using three-dimensional atom
probe microscopy
Olof C. Hellman, David N. Seidman *
Department of Materials Science and Engineering, Northwestern Uni6ersity, 2225 N. Campus Dr., E6anston, IL 60208 -3108, USA
Abstract
We show how the Gibbsian interfacial excess of solute can be calculated from three-dimensional atom probe data, even in the
case of irregularly shaped interfaces. Standard treatments of interfacial thermodynamics implicitly define a one-dimensional
geometry for an interface by assuming a planar interface. Of course, many real systems exhibit non-planar interfaces, and these
treatments are difficult to apply. We show how our treatment derives from Gibbs’ original approach and how it is used to derive
real thermodynamic quantities. The technique can be applied to any interfacial excess quantity. © 2002 Elsevier Science B.V. All
rights reserved.
Keywords: Gibbsian excess; Segregation; Interface; Atom probe microscopy
1. Introduction
One of the applications of three-dimensional atom
probe (3DAP) microscopy [1,2] is the measurement of
the chemical compositions of interfaces, such as grain
boundaries [3] or heterophase interfaces [4]. The segregation of a solute species to such a boundary is
quantified by the Gibbsian interfacial excess of solute,
Ys, a rigorously defined thermodynamic property [5].
Atom probe microscopy produces a discrete count of
the atoms in the vicinity of an interface thus allowing
for a direct measurement of Ys.
Gibbs outlined an approach for quantifying the interfacial excess, which assumed an interface in a medium
with a continuous concentration profile, and involved
the definition of a dividing surface [6]. Cahn refined this
treatment to avoid the necessity of choosing a dividing
surface, and in the process allowed for the composition
at an interface to be expressed in numbers of atoms of
each species, rather than a particular concentration [5].
Cahn’s treatment is not only more elegant, but allows
for more direct application to 1D atom probe field ion
microscopy, where the raw data is in the form of
individual atoms: i.e. the local densities in the material
need not be considered. Both of these treatments, however, assume that the interface is planar, and that linear
profiles of composition across the interface can be
expressed in 1D form. 3DAP produces data for which
this assumption is invalid.
We present a straightforward treatment that extracts
the interfacial excess from a region of analysis that
includes an interface of any arbitrary geometry, maximizes the statistical accuracy, and is insensitive to decisions made during the analysis concerning the
placement of the interface. At no point in the analysis
is the measurement of the area of the interface ever
required, and thus there is no error associated with its
measurement. In addition, the method can accumulate
data from more than one interface as measured by
3DAP, thus allowing for improved statistics to be acquired from multiple samples or multiple regions of a
single sample.
2. From 1D to 3D
* Corresponding author. Tel.: + 1-847-491-4391; fax: +1-847-4672269.
E-mail address: [email protected] (D.N. Seidman).
There are a number of subtleties in the extraction of
the interfacial excess when the interface is not planar.
0921-5093/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 1 - 5 0 9 3 ( 0 1 ) 0 1 8 8 5 - 8
O.C. Hellman, D.N. Seidman / Materials Science and Engineering A327 (2002) 24–28
One challenge exists simply in finding the interface and
estimating its shape. The second challenge exists because of the uncertainty in calculating the area of the
interface; if the interface is curved, the choice of its
position changes the area of that interface. Another
challenge exists when dealing with a non-sharp segregant distribution. A broad peak in the segregant profile
makes defining a single interface inappropriate for a
rigorous calculation, because the effective area of the
interface changes with each point on the segregation
profile. Still another challenge is finding the direction
perpendicular to the interface if the interface is curved,
so that an appropriate profile can be produced from the
raw data. We address each of these concerns here.
3. Defining the interface
One of the key advantages of Cahn’s formalism for
interfacial excess is the fact that the position of the
interface is not specified a priori. However, for a nonplanar interface, it is necessary to specify the shape of
the interface. This is because a 2D surface in three
dimensions can be arbitrarily complex: there are an
unlimited number of degrees of freedom for defining
such a surface.
Although not formally specified, Cahn’s approach
also requires decisions to be made about the morphology of the interface: the shape is taken to be a plane,
and the orientation of the plane must be specified.
Thus, even in Cahn’s formulation, two degrees of freedom must be fixed to specify the orientation of the
interface. What is not required is the exact position of
the interface perpendicular to the chosen planar orientation, although real interfaces may actually have a
position, which constitutes one of the microscopic degrees of freedom.
For an interface that is not planar, we can reproduce
the condition that the specific position of the interface
need not be specified: i.e. although an initial choice
must be made for the shape and location of the interface, we can construct a formalism such that the result
is insensitive to moving the interface in the direction
perpendicular to the initial reference. That is, for the
interface represented by a spherical particle, we need to
specify that the interface is a sphere centered at a
particular point, but the result must be independent of
the choice of the radius of the sphere.
This can be generalized for surfaces of arbitrary
shape. We assume that any surface is composed of a
closed set of polygons, each of which has a sign defining the direction of the surface (i.e. the sides of the
polygon are polar). Given any arbitrary surface of this
form and a scalar displacement distance, a second
surface can be generated consisting of all the points in
space whose minimum distance to the original surface is
25
the displacement. Positive and negative values of displacement correspond to different sides of the surface.
The definition of an interface shape and its position
is thus the first step in calculating the interfacial excess
for a non-planar interface. To remain true to Cahn’s
formalism, however, it is important that the following
steps not depend on the exact position of that interface,
but instead be independent of any translation of the
initial interface perpendicular to itself. We note that
this allows much flexibility in the calculation. For example, if a particular dataset includes a number of
different particles of different sizes, the surface would
trace the boundaries of each of these precipitates. There
is no restriction that the chosen surface be interconnected; indeed, there is no topological restriction beyond the requirement that the surface be closed.
We note that any isosurface derived from a 3D scalar
field will meet the restriction of having a closed surface.
This is convenient, because it is common to use 3DAP
data to calculate a 3D grid of concentration values. An
isoconcentration surface generated from this grid is
therefore a suitable reference point for calculating the
segregation profile. We have outlined the procedure for
this calculation previously [7].
4. Calculating the area
The interfacial excess of solute is expressed as a
number of atoms per unit area of interface. However,
we have also noted that the calculation needs to be
independent of the choice of the interface position.
Because the area of a curved interface changes with its
position, this would seem to be a difficult task. Effectively, this means that at each point along the distribution, there is a different interfacial area appropriate for
that point. For a radially symmetric distribution, the
appropriate area that should be used at a radius r is the
sphere surface area, 4pr 2.
In the 1D case, the area A of the interface is a
constant, and can be moved outside of the sum when
calculating the excess
Ys =
N
% (Cn − C0) /(A(1−C0)),
(1)
n=1
where the sum is over all of the atoms under consideration, Cn is the concentration of the nth atom (i.e. 1 if
it is the segregating species of interest, and 0 otherwise),
and C0 is the bulk concentration of that species. The
(1− C0) term is the necessary correction for a non-dilute solution [8].
However, if the area of the interface depends on its
position, the area must be moved inside the sum, so
that an appropriate area is used for each segregating
atom:
26
Ys =
N
O.C. Hellman, D.N. Seidman / Materials Science and Engineering A327 (2002) 24–28
(Cn −C0)
/(1 −C0).
An
n=1
%
(2)
In this case, An is the area of the interface assuming
that the area is placed at the position of the nth atom.
While this approach is rigorous, it is impractical to
calculate the area of the given interface for each individual atom for an arbitrary interface. Our approach
avoids the calculation of the area entirely, overcoming
both of these concerns. We sample a set of atoms
within a certain proximity from the reference interface.
We measure the closest distance of each atom to the
reference interface—their proximity. Atoms within the
proximity range l0 9Dl2 occupy a thin shell of thickness Dt centered on a distance l0 from the interface.
While it is tedious to measure the area of that interface,
it is a simple matter to count the number of atoms
within this range. Making the assumption that the
absolute atomic density of the material is known, the
area of the interface is given by
Al = Nl /zDl,
(3)
where Al is the effective area of the slice at proximity l,
Nl is the number of atoms in the slice, Dl is the shell
thickness, and z is the ideal atomic density. In the case
that it is not appropriate to use a constant density in all
regions of the sample, the density could also be calculated as a function of the concentrations of the various
species in the slice.
Examining the atoms within these shells provides
more than a natural mapping of the 3D concentration
distributions onto one dimension. Calculating the concentration in a series of these shells as a function of
proximity results in a plot of concentration with respect
to position, exactly the same input as is required for
Gibbs’ analysis of interfacial excess. A peak in the
concentration profile in such a plot is analogous to a
peak in a linear concentration profile. The area under
such a peak has units of distance. Multiplying by an
atomic density results in number of atoms per unit
area, which is the physical dimension of the Gibbsian
interfacial excess of solute. This is the step at which the
effective area at each shell is introduced.
5. Previous work
Previously, Rozdilsky et al. have addressed the problem of the concentration distribution around precipitates by using a radial, spherically symmetric profile [9].
Their technique is based on the assumption that the
interface was in fact spherical, which is only true for
ideal systems, and is thus of limited practicality. As we
note above, the radial profile is exactly analogous to
our approach using a spherical reference surface.
Fisher and Wortis [10] outline an alterative approach
to the problem of the varying area of the interface. This
approach defines thermodynamic quantities with an
explicit reference parameter: i.e. the quantities are
defined with respect to a sphere, cylinder or other shape
of a specific dimension. Again, however, their treatment
is limited to the case that the morphology of the
interface can be described in very simple terms, i.e. a
sphere, ellipsoid, cylinder, etc. In our approach the
reference surface can be arbitrarily complex, and does
not suffer from any of the associated drawbacks.
6. Example
Fig. 1. Atom-by-atom view of a 3D atom probe dataset of an
internally oxidized Cu (Mg, Ag) alloy. The large dots are Mg and O
atoms. The small dots are Ag atoms. Cu atoms are not shown. The
scale is approximately 16 × 16×144 nm.
Fig. 1 shows an atom-by-atom view of a Cu(Mg, Ag)
alloy which had been internally oxidized. The Mg and
O atoms are drawn larger to emphasize the MgO
precipitates. Ag atoms are drawn as small dots, and the
O.C. Hellman, D.N. Seidman / Materials Science and Engineering A327 (2002) 24–28
27
isoconcentration surface, reflecting the fact that the
reference surface is outside the actual interfaces.
The magnification effect can also have a detrimental
effect on the calculation of the interfacial excess in the
following way: If the segregated species resides uniquely
in a region which has a higher or lower measured
density than the actual expected density, the thickness
of that region will be under- or overestimated resulting
in a poor estimation of the actual effective area. If a
significant amount of such density differences exist, best
results will be obtained if the atom probe data is
adjusted for this before the application of the proxiogram technique. We are currently developing such a
procedure.
This example shows how the proxigram technique is
significantly more straightforward than a traditional
one dimensional approach. There are an infinite number of different analysis directions, cylinder shapes and
sizes, and binning techniques that could be applied to
examine segregation at the interfaces present. All of
those that we tried were inconclusive in identifying a
statistically significant segregation. In addition, a 1D
analysis requires the analysis of each individual particle.
In contrast, the proxigram approach considers at once
all of the interfaces in the sample.
Fig. 2. Mg 2.5 at.% isoconcentration surface of the same sample as in
Fig. 1.
majority species Cu are not drawn. Details of the
preparation of this sample are given by Rüsing et al.
[4]. Fig. 2 shows the 2.5 at.% Mg isoconcentration
surface of the same sample. This surface roughly traces
the morphology of the particles. The value 2.5 at.% is
far below the concentration of Mg expected for the
MgO interface, and as such it estimates the surface
slightly outside the actual interface.
Because there is a local increase in magnification
associated with the MgO particles [11], which leads to a
diffuse distribution of Mg and O atoms, a low
threshold is necessary to identify Mg and O atoms
originating in particles near the edge of the dataset. Fig.
3a shows a proxigram of the Mg, O and Ag components with respect to this isosurface. The balance of the
sample is Cu. The Ag concentration is shown by itself
in Fig. 3b. There is a pronounced peak in the silver
concentration in the vicinity of the interface, which
corresponds to an Ag interfacial excess of 0.36
atoms nm − 2. The peak is on the positive side of the
Fig. 3. (a) Proxigram of Mg, O and Ag with respect to the Mg 2.5
at.% isoconcentration surface. Negative distances are outside the
MgO particles. Positive distances are inside the particles. Error bars
denote one sigma statistical errors. (b) Proxigram of Ag with respect
to the Mg 2.5 at.% isoconcentration surface. This is the same data in
(a) but rescaled. Error bars denote one sigma statistical errors. The
peak at the interface corresponds to 0.36 atoms nm − 2.
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7. Conclusions
References
Measurement of the interfacial excess of solute or
any other interfacial thermodynamic quantity is not
straightforward when an interface is curved. We have
presented a technique for evaluating such quantities for
interfaces of arbitrary geometry and which can be
applied to the rich sets of data provided by 3DAP,
while remaining true to the classical approaches
of Gibbs and Cahn to the thermodynamics of interfaces.
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Acknowledgements
The example data used was taken from 3DAP
experiments by Dr Jörg Rüsing and Mr Jason T.
Sebastian. This research is supported by the National
Science Foundation, Division of Materials Research,
Grant DMR-972896, Bruce MacDonald, Grant
Officer.