Materials Science and Engineering A327 (2002) 24–28

Measurement of the Gibbsian interfacial excess of solute at aninterface of arbitrary geometry using three-dimensional atom

probe microscopy

Olof C. Hellman, David N. Seidman *Department of Materials Science and Engineering, Northwestern Uni�ersity, 2225 N. Campus Dr., E�anston, 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 thecase of irregularly shaped interfaces. Standard treatments of interfacial thermodynamics implicitly define a one-dimensionalgeometry for an interface by assuming a planar interface. Of course, many real systems exhibit non-planar interfaces, and thesetreatments are difficult to apply. We show how our treatment derives from Gibbs’ original approach and how it is used to derivereal thermodynamic quantities. The technique can be applied to any interfacial excess quantity. © 2002 Elsevier Science B.V. Allrights reserved.

Keywords: Gibbsian excess; Segregation; Interface; Atom probe microscopy

www.elsevier.com/locate/msea

1. Introduction

One of the applications of three-dimensional atomprobe (3DAP) microscopy [1,2] is the measurement ofthe chemical compositions of interfaces, such as grainboundaries [3] or heterophase interfaces [4]. The segre-gation of a solute species to such a boundary isquantified by the Gibbsian interfacial excess of solute,�s, a rigorously defined thermodynamic property [5].Atom probe microscopy produces a discrete count ofthe atoms in the vicinity of an interface thus allowingfor a direct measurement of �s.

Gibbs outlined an approach for quantifying the inter-facial excess, which assumed an interface in a mediumwith a continuous concentration profile, and involvedthe definition of a dividing surface [6]. Cahn refined thistreatment to avoid the necessity of choosing a dividingsurface, and in the process allowed for the compositionat an interface to be expressed in numbers of atoms ofeach species, rather than a particular concentration [5].Cahn’s treatment is not only more elegant, but allowsfor more direct application to 1D atom probe field ion

microscopy, where the raw data is in the form ofindividual atoms: i.e. the local densities in the materialneed not be considered. Both of these treatments, how-ever, assume that the interface is planar, and that linearprofiles of composition across the interface can beexpressed in 1D form. 3DAP produces data for whichthis assumption is invalid.

We present a straightforward treatment that extractsthe interfacial excess from a region of analysis thatincludes an interface of any arbitrary geometry, maxi-mizes the statistical accuracy, and is insensitive to deci-sions made during the analysis concerning theplacement of the interface. At no point in the analysisis the measurement of the area of the interface everrequired, and thus there is no error associated with itsmeasurement. In addition, the method can accumulatedata from more than one interface as measured by3DAP, thus allowing for improved statistics to be ac-quired from multiple samples or multiple regions of asingle sample.

2. From 1D to 3D

There are a number of subtleties in the extraction ofthe interfacial excess when the interface is not planar.

* Corresponding author. Tel.: +1-847-491-4391; fax: +1-847-467-2269.

E-mail address: d-seidman@northwestern.edu (D.N. Seidman).

0921-5093/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved.PII: S0 9 21 -5093 (01 )01885 -8

O.C. Hellman, D.N. Seidman / Materials Science and Engineering A327 (2002) 24–28 25

One challenge exists simply in finding the interface andestimating its shape. The second challenge exists be-cause of the uncertainty in calculating the area of theinterface; if the interface is curved, the choice of itsposition changes the area of that interface. Anotherchallenge exists when dealing with a non-sharp segre-gant distribution. A broad peak in the segregant profilemakes defining a single interface inappropriate for arigorous calculation, because the effective area of theinterface changes with each point on the segregationprofile. Still another challenge is finding the directionperpendicular to the interface if the interface is curved,so that an appropriate profile can be produced from theraw data. We address each of these concerns here.

3. Defining the interface

One of the key advantages of Cahn’s formalism forinterfacial excess is the fact that the position of theinterface is not specified a priori. However, for a non-planar interface, it is necessary to specify the shape ofthe interface. This is because a 2D surface in threedimensions can be arbitrarily complex: there are anunlimited number of degrees of freedom for definingsuch a surface.

Although not formally specified, Cahn’s approachalso requires decisions to be made about the morphol-ogy 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 free-dom must be fixed to specify the orientation of theinterface. What is not required is the exact position ofthe interface perpendicular to the chosen planar orien-tation, although real interfaces may actually have aposition, which constitutes one of the microscopic de-grees of freedom.

For an interface that is not planar, we can reproducethe condition that the specific position of the interfaceneed not be specified: i.e. although an initial choicemust be made for the shape and location of the inter-face, we can construct a formalism such that the resultis insensitive to moving the interface in the directionperpendicular to the initial reference. That is, for theinterface represented by a spherical particle, we need tospecify that the interface is a sphere centered at aparticular point, but the result must be independent ofthe choice of the radius of the sphere.

This can be generalized for surfaces of arbitraryshape. We assume that any surface is composed of aclosed set of polygons, each of which has a sign defin-ing the direction of the surface (i.e. the sides of thepolygon are polar). Given any arbitrary surface of thisform and a scalar displacement distance, a secondsurface can be generated consisting of all the points inspace whose minimum distance to the original surface is

the displacement. Positive and negative values of dis-placement correspond to different sides of the surface.

The definition of an interface shape and its positionis thus the first step in calculating the interfacial excessfor a non-planar interface. To remain true to Cahn’sformalism, however, it is important that the followingsteps not depend on the exact position of that interface,but instead be independent of any translation of theinitial interface perpendicular to itself. We note thatthis allows much flexibility in the calculation. For ex-ample, if a particular dataset includes a number ofdifferent particles of different sizes, the surface wouldtrace the boundaries of each of these precipitates. Thereis no restriction that the chosen surface be intercon-nected; indeed, there is no topological restriction be-yond the requirement that the surface be closed.

We note that any isosurface derived from a 3D scalarfield will meet the restriction of having a closed surface.This is convenient, because it is common to use 3DAPdata to calculate a 3D grid of concentration values. Anisoconcentration surface generated from this grid istherefore a suitable reference point for calculating thesegregation profile. We have outlined the procedure forthis calculation previously [7].

4. Calculating the area

The interfacial excess of solute is expressed as anumber of atoms per unit area of interface. However,we have also noted that the calculation needs to beindependent of the choice of the interface position.Because the area of a curved interface changes with itsposition, this would seem to be a difficult task. Effec-tively, this means that at each point along the distribu-tion, there is a different interfacial area appropriate forthat point. For a radially symmetric distribution, theappropriate area that should be used at a radius r is thesphere surface area, 4�r2.

In the 1D case, the area A of the interface is aconstant, and can be moved outside of the sum whencalculating the excess

�s=� �

N

n=1

(Cn−C0)�/(A(1−C0)), (1)

where the sum is over all of the atoms under consider-ation, Cn is the concentration of the nth atom (i.e. 1 ifit 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-di-lute solution [8].

However, if the area of the interface depends on itsposition, the area must be moved inside the sum, sothat an appropriate area is used for each segregatingatom:

O.C. Hellman, D.N. Seidman / Materials Science and Engineering A327 (2002) 24–2826

�s=� �

N

n=1

(Cn−C0)An

�/(1−C0). (2)

In this case, An is the area of the interface assumingthat the area is placed at the position of the nth atom.

While this approach is rigorous, it is impractical tocalculate the area of the given interface for each indi-vidual atom for an arbitrary interface. Our approachavoids the calculation of the area entirely, overcomingboth of these concerns. We sample a set of atomswithin a certain proximity from the reference interface.We measure the closest distance of each atom to thereference interface— their proximity. Atoms within theproximity range l0��l2 occupy a thin shell of thick-ness �t 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 atomswithin this range. Making the assumption that the

absolute atomic density of the material is known, thearea of the interface is given by

Al=Nl/��l, (3)

where Al is the effective area of the slice at proximity l,Nl is the number of atoms in the slice, �l is the shellthickness, and � is the ideal atomic density. In the casethat it is not appropriate to use a constant density in allregions of the sample, the density could also be calcu-lated as a function of the concentrations of the variousspecies in the slice.

Examining the atoms within these shells providesmore than a natural mapping of the 3D concentrationdistributions onto one dimension. Calculating the con-centration in a series of these shells as a function ofproximity results in a plot of concentration with respectto position, exactly the same input as is required forGibbs’ analysis of interfacial excess. A peak in theconcentration profile in such a plot is analogous to apeak in a linear concentration profile. The area undersuch a peak has units of distance. Multiplying by anatomic density results in number of atoms per unitarea, which is the physical dimension of the Gibbsianinterfacial excess of solute. This is the step at which theeffective area at each shell is introduced.

5. Previous work

Previously, Rozdilsky et al. have addressed the prob-lem of the concentration distribution around precipi-tates by using a radial, spherically symmetric profile [9].Their technique is based on the assumption that theinterface was in fact spherical, which is only true forideal systems, and is thus of limited practicality. As wenote above, the radial profile is exactly analogous toour approach using a spherical reference surface.

Fisher and Wortis [10] outline an alterative approachto the problem of the varying area of the interface. Thisapproach defines thermodynamic quantities with anexplicit reference parameter: i.e. the quantities aredefined with respect to a sphere, cylinder or other shapeof a specific dimension. Again, however, their treatmentis limited to the case that the morphology of theinterface can be described in very simple terms, i.e. asphere, ellipsoid, cylinder, etc. In our approach thereference surface can be arbitrarily complex, and doesnot suffer from any of the associated drawbacks.

6. Example

Fig. 1 shows an atom-by-atom view of a Cu(Mg, Ag)alloy which had been internally oxidized. The Mg andO atoms are drawn larger to emphasize the MgOprecipitates. Ag atoms are drawn as small dots, and the

Fig. 1. Atom-by-atom view of a 3D atom probe dataset of aninternally oxidized Cu (Mg, Ag) alloy. The large dots are Mg and Oatoms. The small dots are Ag atoms. Cu atoms are not shown. Thescale is approximately 16×16×144 nm.

O.C. Hellman, D.N. Seidman / Materials Science and Engineering A327 (2002) 24–28 27

Fig. 2. Mg 2.5 at.% isoconcentration surface of the same sample as inFig. 1.

isoconcentration surface, reflecting the fact that thereference surface is outside the actual interfaces.

The magnification effect can also have a detrimentaleffect on the calculation of the interfacial excess in thefollowing way: If the segregated species resides uniquelyin a region which has a higher or lower measureddensity than the actual expected density, the thicknessof that region will be under- or overestimated resultingin a poor estimation of the actual effective area. If asignificant amount of such density differences exist, bestresults will be obtained if the atom probe data isadjusted for this before the application of the prox-iogram technique. We are currently developing such aprocedure.

This example shows how the proxigram technique issignificantly more straightforward than a traditionalone dimensional approach. There are an infinite num-ber of different analysis directions, cylinder shapes andsizes, and binning techniques that could be applied toexamine segregation at the interfaces present. All ofthose that we tried were inconclusive in identifying astatistically significant segregation. In addition, a 1Danalysis requires the analysis of each individual particle.In contrast, the proxigram approach considers at onceall of the interfaces in the sample.

Fig. 3. (a) Proxigram of Mg, O and Ag with respect to the Mg 2.5at.% isoconcentration surface. Negative distances are outside theMgO particles. Positive distances are inside the particles. Error barsdenote one sigma statistical errors. (b) Proxigram of Ag with respectto the Mg 2.5 at.% isoconcentration surface. This is the same data in(a) but rescaled. Error bars denote one sigma statistical errors. Thepeak at the interface corresponds to 0.36 atoms nm−2.

majority species Cu are not drawn. Details of thepreparation of this sample are given by Rusing et al.[4]. Fig. 2 shows the 2.5 at.% Mg isoconcentrationsurface of the same sample. This surface roughly tracesthe morphology of the particles. The value 2.5 at.% isfar below the concentration of Mg expected for theMgO interface, and as such it estimates the surfaceslightly outside the actual interface.

Because there is a local increase in magnificationassociated with the MgO particles [11], which leads to adiffuse distribution of Mg and O atoms, a lowthreshold is necessary to identify Mg and O atomsoriginating in particles near the edge of the dataset. Fig.3a shows a proxigram of the Mg, O and Ag compo-nents with respect to this isosurface. The balance of thesample is Cu. The Ag concentration is shown by itselfin Fig. 3b. There is a pronounced peak in the silverconcentration in the vicinity of the interface, whichcorresponds to an Ag interfacial excess of 0.36atoms nm−2. The peak is on the positive side of the

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7. Conclusions

Measurement of the interfacial excess of solute orany other interfacial thermodynamic quantity is notstraightforward when an interface is curved. We havepresented a technique for evaluating such quantities forinterfaces of arbitrary geometry and which can beapplied to the rich sets of data provided by 3DAP,while remaining true to the classical approachesof Gibbs and Cahn to the thermodynamics of inter-faces.

Acknowledgements

The example data used was taken from 3DAPexperiments by Dr Jorg Rusing and Mr Jason T.Sebastian. This research is supported by the NationalScience Foundation, Division of Materials Research,Grant DMR-972896, Bruce MacDonald, GrantOfficer.

References

[1] D. Blavette, B. Deconihout, A. Bostel, J.M. Sarrau, M. Bouet,A. Menand, Rev. Sci. Instrum. 64 (1993) 2911–2919.

[2] A. Cerezo, T.J. Godfrey, S.J. Sijbrandi, G.D.W. Smith, P.J.Warren, Rev. Sci. Instrum. 69 (1998) 49–58.

[3] B. Farber, E. Cadel, A. Menand, G. Schmitz, R. Kirchheim,Acta Mater. 48 (2000) 789–796.

[4] J. Rusing, J.T. Sebastian, O. Hellman, D.N. Seidman, Microsc.Microanal. 6 (5) (2000) 445–451.

[5] J.W. Cahn, in: W.C. Johnson, J.M. Blakely (Eds.), InterfacialSegregation, American Society of Metals, Metals Park, OH,1979, pp. 3–23.

[6] The Scientific Papers of J. Willard Gibbs, vol. 1, Dover, NewYork, 1961, pp. 219–331.

[7] O. Hellman, J. Vandenbroucke, J. Rusing, D. Isheim, D.N.Seidman, Microsc. Microanal. 6 (5) (2000) 437–444.

[8] A.P. Sutton, R.W. Balluffi, Interfaces in Crystalline Solids,Oxford University Press, New York, 1995.

[9] I. Rozdilsky, A. Cerezo, G.D.W. Smith, Atomic-scale measure-ment of composition profiles near growing precipitates in theCu–Co system, in: S.P. Marsh, J.A. Dantzig, W. Hofmeister, R.Trivedi, M.G. Chu, E.J. Lavernia, J.-H. Chun (Eds.), Solidifica-tion 1998, TMS, Warrendale, PA, 1999, pp. 83–90.

[10] M.P.A. Fisher, M. Wortis, Phys. Rev. B 29 (1984) 6252–6260.[11] F. Vurpillot, A. Bostel, D. Blavette, Appl. Phys. Lett. 76 (21)

(2000) 3127.