X-RAY SPECTROMETRY, VOL. 23, 195-202 (1994)

Analytical Electron Microscopy of Thin Segregated Layers

R. G. Faulkner and T. S. Morgan Institute of Polymer Technology and Materials Engineering, Loughborough University of Technology, Loughborough, Leicestershire LE11 3TU, UK

E. A. Little Materials Performance Department, Harwell Laboratory, Didcot, Oxfordshire OX1 1 ORA, UK

Analytical convolution methods used to explain the interaction of a fine electron beam with a thin segregated layer in a typical electron microscope specimen are described. Expressions are derived which quantify the shape of the measured concentration profile determined in the microscope by scanning the beam acrass the segregated layer. The expression contains parameters concerned with the beam profile and the thickness of the segregated layer. The method is applied to grain boundary segregation studies of neutron-irradiated femticmartensitic steels.

INTRODUCTION

This paper describes mathematical attempts to relate the segregation composition profiles measured in modern analytical electron microscopy (AEM) to actual profiles. The difficulty is that the beam size for analysis often approximates that of the segregated layer. The problem can be overcome by the use of convolution techniques' or Monte Carlo Simple mathe- matical analytical equations assuming single electron scattering can be used to quantify the beam speading which occurs when electron beams interact with thin

but these lack the precision offered by the other methods for segregated layer studies.

The deconvolution approach has been developed as outlined in this paper and it is found that reasonably accurate predictions are possible and, at the same time, more information about the electron beam intensity profile becomes available. The method is applied to studies of segregated layers of P, Cr and Mo in a ferritic-martensitic steel, FV448 (see Table 1 for composition), in the neutron-irradiated and non- irradiated condition.'

ELECTRON PROBESEGREGANT INTERACTION MODEL

One of the characteristics of equilibrium segregation is its spatial distribution; it is confined only to the mono- layer immediately adjacent to the boundary plane. The widths of the segregated layers as measured by the full widths at half maximum of the profiles generated by the FEGSTEM are 3-4 nm. Although narrow, this width is

clearly greater than the width of one monolayer either side of a disordered boundary region. This arises because of the finite width of the incident electron probe. If the segregant is truly the result of equilibrium segregation, then the total width of the enriched zone will be less than 1 nm. In comparison with this, the elec- tron probe diameter in a FEGSTEM for typical EDX conditions has been estimated at 3.5 nm, as that diam- eter containing 80% of the probe current.

This is further supported by previous data" which show that measured x-ray source diameters are about 9 nm for specimen thicknesses of 100 nm. Since this work involves specimens of 40 nm thickness, a beam diameter of 3.5 nm is assumed. This is based on the assumption that beam spreading is linearly related to specimen thickness and that the beam diameter is 1 nm at the point of impact of the electron beam with the specimen.

As a consequence, the width of the effect to be mea- sured is less than the width of the measuring device. The acquired solute profile is thus the convolution of the segregant and electron probe current distributions, where the composition at any point in the profile is the value averaged over the volume of the material excited by the incident electron beam. Hence not only is the apparent width of a segregant distribution altered, but so too is its apparent maximum. The use of a finite probe size smooths out a discontinuous solute distribu- tion; the greater the probe diameter, the more the true distribution becomes smoothed.

Attempts to quantify and deconvolute the true solute distribution from an acquired concentration profile can be approached in two ways. The first method is to con- volute an assumed incident electron probe current dis- tribution with an assumed segregant distribution and neglect beam-broadening effects caused by high-angle inelastic scattering of electrons as they negotiate the

Table 1. Chemical composition of FV448 martensitic steel

Element Fe C Si Mn S P Cr Ni MO v Nb

Wt.% Balance 0.10 0.38 1.01 0.016 0.006 10.65 0.64 0.64 0.16 0.30

CCC 0049-8246/94/050195-08 0 1994 by John Wiley & Sons, Ltd.

Received 8 December 1993 Accepted 20 April 1994

196 R. G. FAULKNER, T. S. MORGAN AND E. A. LITTLE

thin foil. The second method is to incorporate the segre- gant and electron probe current distributions into a numerical Monte Carlo simulation which models the beam-broadening events in thicker foils.

To deconvolute accurately the segregant distribution from the generated profile requires a precise knowledge of the probe current distribution, a large number of closely spaced analysis points across the interface and, for the application of a Monte Carlo simulation, explicit knowledge of the local foil thickness. An example of the application of the second technique for a binary alloy situation (Fe-Si) is shown in Fig. 1." By varying the width of the segregated layer, the model predicts the change in Fe/Si x-ray intensity ratio. The model allows for changes in foil thickness, beam energy and matrix- segregant combination.

The first method described above, the simple convol- ution of the two distributions neglecting beam- broadening effects, will provide a measure of the best profile that could be generated by the geometric inter- action of electron probe and segregant distributions. The interfaces analysed in this study are partly chosen as being sited in very thin areas of the foil, where beam- broadening effects may not be significant. Thus the first approach is applied here to model analytically the inter- action; if beam broadening is significant it will show in the comparison of the results with the modelling.

The geometry of the interaction is modelled as shown in Fig. 2. The electron probe current is assumed to be radially normally distributed, with a mean of zero [i.e. centred on (0,O)l and a variance given by a'. This

-. 0 20 LO 60 80 100

l a y e r thickness, rm

Figure 1. (a) Output from model showing trajectories of elec- trons at 100 kV in an Fe foil of thickness 2000 A, with a 20 nm layer of Si. Si/Fe ratio = 0.76. (b) Graph showing the relationship between layer thickness and the Si/Fe counts ratio predicted by the model.

C o n c e n t r a t i o n C o n c e n t r a t i o n A A -

xd D i s t a n c e

C = C o + m ( x - ( x d + j d , ) 1

Figure 2. Geometry of the electron probe-segregant interaction assumed for analytical modelling. The electron beam distribution is Gaussian and the slab segregant has a concentration profile of the form shown in either (i) or (ii).

assumption has been used by several other workers, e.g. Doig and F l e ~ i t t . ~ The beam intensity at a point (x,y) from the beam centre is then given by

I=-exp[- 1 (XZ + YZ) ] a& 2aZ

The proportion of the total number of electrons inci- dent on a segregated layer of width do whose mid-point is a distance xd from the incident electron probe centre is thus given by

1

2a2

Evaluating the denominator:

- - jm exp($) dx Jrn exp(*) 20' dy a& - m - m

= a&

ANALYTICAL ELECTRON MICROSCOPY OF THIN SEGREGATED LAYERS 197

Evaluating the numerator

Hence the proportion of the total number of electrons incident on a layer of width do situated a distance xd from the beam centre is

For a solute distribution shown by (i) in Fig. 2, of concentration Cb in the segregant layer and C , in the adjacent matrix, the derived concentration, C, generated from an electron probe incident at a distance xd from a segregant layer of width do is thus

c = CO + i (cb - CO)

For a fixed width of segregated layer, the derived con- centration profile is determined by the standard devi- ation of the probe distribution, o. The probe diameter is usually defined in terms of a diameter containing a certain proportion of the total probe current, usually 80%. Hence we need to find the correspondence between the beam diameter containing, say, 80% of the current and the standard deviation of the normal approximation to the electron distribution.

The electron distribution can be written equivalently in terms of radial coordinates as follows:

= o&

The radius, I , , , containing 80% of the current, I , , , is solved by

I , , = 0.8o& = lff Lao exp($)r dr d6

U

Hence

0.8 = - [ exp (;>) - - 11

and 4 0 In 0.2 = - - 202

j r , , = 1.794~

Thus, for a beam diameter specified by &,, the stan- dard deviation defining the normal approximation to the electron distribution is given by

s = 0.2794,, (2) Data analysis Equation (1) can be used in conjunction with the con- version defined by Eqn (2) to model analytically what the derived concentration profile will be for a concen- tration of solute C, = O.O%, a boundary concentration c b = 3.0%, a segregant layer width of 1 nm and a nominal beam diameter of 3.5 nm: the situation for monolayer phosphorus segregation in FV448 analysed by the FEGSTEM. The analytically derived concentra- tion profile is as shown in Fig. 3. The mean data obtained experimentally in five separate analyses from both sides of austenite grain boundaries of material aged at 465 “C is also plotted for comparison. The error bars indicate the mean standard deviation, divided by the square root of the number of observations for each point. It can be seen that whilst the derived profile has been fitted to the maximum of phosphorus concentra- tion detected at the interface, the apparent width of the derived profile is narrower than that observed experi- mentally. This indicates that one or more of the assumptions made concerning the two distributions are not strictly valid.

First, the segregant is assumed to be a top-hat func- tion of width 1 nm to approximate monolayer equi- librium segregation either side of a disordered boundary plane. The segregant may extend to greater than one monolayer either side of the interfacial plane, and if so it is likely to have considerable concentration gradient ‘tails’ rather than be purely a top-hat profile of an absorbed layer. Other information about the segrega- tion obtained by relating the experimental information to theoretical models indicates that the segregation is equilibrium type, i.e. monolayer. Therefore, segregant layer widths can be assumed to be up to a maximum of 1 nm; we have modelled with this maximum value. Second, and more likely, the electron probe diameter may effectively be greater than 3.5 nm. Even if the nominal incident beam diameter is 3.5 nm, beam- broadening effects may not be negligible, the boundary may not be perfectly aligned with the electron beam and some specimen drift may occur. Each of these factors may contribute to make the effective probe size greater than the nominal incident beam diameter.

If the segregant distribution, a top-hat function of width 1 nm, is assumed to be valid, but now the derived concentration profile is fitted to match the experimen- tally determined values by altering the probe diameter for the model, the second curve in Fig. 3 is obtained. Here the effective probe size is increased to 5.4 nm and the boundary concentration of segregant, Cb , is corre- spondingly increased to 4.5%.

198 R. G. FAULKNER, T. S. MORGAN AND E. A. LITTLE

1.5 -~

Theoretlcel convolutlons:

1.0

. 5

n n -.I

-12 -10 -8 -6 -4 -2 0 2 4 6 a 10 12

Distance f r o m PAGB (nm)

Figure 3. Fitting of experimental data for mean phosphorus concentration across prior austenite grain boundaries in FV448 aged at 465 "C by theoretical convolutions of monolayer-type segregation with electron probe diameters of (i) 3.5 and (ii) 5.4 nm.

An indication as to whether the effective probe diam- eter is an accurate assessment of the physical situation can be gained by applying the analytical model to the experimentally determined data for other solutes. The effective probe diameter should remain constant for each solute since the effects of boundary tilt, specimen drift and beam broadening affect the acquisition of the whole x-ray spectrum. The derived concentration pro- files modelling the segregation of molybdenum are shown in Fig. 4. The first profile is derived assuming a beam diameter of 3.5 nm and a 1 nm wide boundary concentration, C,,, of 30.85%. In this case also, it is too narrow to fit the experimental data for prior austenite grain boundaries in material aged at 465 "C. The second profile is obtained using an effective probe size of 5.4

nm and a boundary concentration of 46%, i.e. 0.46 monolayer coverage. The experimental data are matched very well by the analytically derived concen- tration profile using the same effective probe diameter as for phosphorus above. This consistency suggests that the analytical model is describing the physical situation accurately, but that the incident electron beam diameter is effectively slightly 'broadened' in its interaction with the specimens.

Hence the apparent grain boundary enrichment detected in the aged FV448 material is consistent with it being the result of equilibrium segregation. The discrep- ancy in absolute values is accounted for by the electron probe-specimen interaction. Theoretical calculations regarding the equilibrium segregation of phosphorus in

-12 -10 -6 -6 -4 -2 0 2 4 6 8 10 12 Oistance f r o m PAGB lnml

Figure 4. Fitting of experimental data for mean molybdenum concentration across prior austenite grain boundaries in FV448 aged at 465°C by theoretical convolutions of monolayer-type segregation with electron probe diameters of (i) 3.5 and (ii) 5.4 nm.

ANALYTICAL ELECTRON MICROSCOPY OF THIN SEGREGATED LAYERS 199

isolation, without the interactions of other solutes, predict a monolayer coverage of 9.8 at.% at grain boundaries in equilibrium at 465 “C. The detected value of 1.16 wt.% is the concentration averaged over the interaction volume of the electron probe. Using a con- volution of a Gaussian-shaped electron distribution and a top-hat function solute concentration profile of width 1 nm to fit the experimentally determined data yields a monolayer coverage of 4.5 wt.%.

EXTENSION OF THE MODEL FOR ACCOMPANYING DEPLETION AROUND THE SEGREGATED LAYER

We can extend the analytical model for the electron probe/interface to approximate to more complicated concentration profiles. The electron beam current is assumed to be radially normally distributed as before. Now, however, the concentration profile is assumed to be of the form shown as (ii) in Fig. 2. The boundary monolayer has a composition c b and thickness do with an adjacent matrix concentration of Coy but which increases linearly with increasing distance from the interface with a concentration gradient, m, such that

This approximates the observed depletion of chromium towards the boundary plane. The apparent concentra- tion, averaged over the volume of the foil excited by the electron current distribution at a distance xd from the centre of the interface, is given in this case by

c = CO + m[x - (xd + $do)]

4 x 2 + y2)

X { -m[X - (xd - $do)] + co} dx dy

1 -(x2 + y’)

X {m[X - (xd -I- +do)} + co) dx

By inspection, the factor SYrn exp(-y2/202) dy and the denominator double integral are equal and thus cancel. Hence, the apparent concentration is given by the sum- mation of the three integrals:

/ - x2\

200

13

12

1 1

10 -60

R. G. FAULKNER, T. S. MORGAN AND E. A. LITTLE

__ ~nsoreticai convolution

Ei Experlmontsl dnts -

-

I I I I I

-40 -20 0 20 40 60

I Tnsoretlcal

convolution

L-7 Experlmontsl dot8

-60 -40 -20 0 20 40 60

Dis tance f rom l a t h boundary (nm)

Figure 5. Fitting of experimental data for mean chromium concentration across lath boundaries in FV448 irradiated at 400°C by the theoretical convolution of a solute distribution as shown in Fig. 5.4(ii) with a Gaussian electron probe of 5.4 nm diameter.

Data analysis

If this analytically derived concentration profile, Eqn (3), is now fitted to an experimentally determined chro- mium profile, a measure of the boundary concentration, Cb, can be obtained. The effective beam diameter is

(3) 1 - erf(xd + :)[ cO - cb - m(xd + d- 2

When the concentration gradient is zero, ie . m = 0,

Figure 6. Fitting of experimental data for mean chromium concentration across lath boundaries in FV448 aged at 400°C by the theoretical convolution of monolayer-type segregation with a Gaussian electron probe of 5.4 nm diameter.

ANALYTICAL ELECTRON MICROSCOPY OF THIN SEGREGATED LAYERS

2.5

2.0

.-. 1.5 x. 4J 3 - Pi : 1 . 0 U

z d

.5

0.0 -12

201

E x p e r l m m t a l dmtm: -

+Aped 400 'C

m I r r 400 bc

-

-

-

1 . 1 , 1 . 1 , 1 . ~ , ~ , ~ , ~ , , I ,

6 a 10 : -10 -8 -6 -4 -2 0 2 4

Distance f r o m PAGE (nml Figure 7. Effect of adjacent matrix concentration, C,, on the apparent boundary plane solute concentration for segregants of the same true interfacial concentration: thin curve, C, = 5.0 wt.%, C, = 0.6 wt.%; thick curve, C, = 5.0 wt.%, C, = 0.2 wt.%. Experimental data (0). for comparison are for nickel concentration across prior austenite grain boundaries of FV448 treated at 400 "C.

taken to be 5.4 nm and the width of the segregated layer 1 nm, as before. The best fit to the mean experimental data from lath boundaries in FV448 irradiated at 400°C is obtained when C, = 13.3%, C, = 9.85% and rn = 0.013% nm-'. The analytically derived profile and the experimental data are shown in Fig. 5. Thus the value determined to be a measure of the true boundary concentration, 13.3%, is actually greater than the uniform chromium concentration of 1 1.1 YO detected across lath boundaries in the as-tempered condition. This boundary plane enrichment is in spite of chro- mium being depleted from regions adjacent to the boundary plane.

In order to aid comparison, the corresponding mean experimental data from lath boundaries of material aged at 400 "C, were fitted with an analytically derived concentration profile to determine a measure of the boundary concentration, c b . Again a segregant layer width of 1 nm and an effective beam diameter of 5.4 nm are assumed for consistency. The best fit to the data is shown in Fig. 6, obtained for values of c b = 17%, c,, = 10.8% and a concentration gradient of zero.

Variations in the matrix composition can also be quantified. Figure 7 shows theoretical derived solute concentration profiles for solutes having the same true boundary concentration, Cb = 5.0%, but differing matrix concentrations of Co = 0.6 and 0.2%. The assumed geometry is as previously with an effective probe diameter of 5.4 nm and a segregated layer width of 1 nm. The apparent maximum concentration of solute detected at the plane of the boundary is then 1.75% for a matrix concentration of 0.6%, but 1.45% for a matrix concentration of 0.2%. For comparison, the mean boundary plane concentration of nickel detected

at prior austenite boundaries in FV448 aged at 400 "C is 1.74 0.36 wt.%, whilst in the material fast neutron irradiated at 400°C it is 1.36 k 0.32 wt.%. Hence, despite apparently less nickel segregation being detected in the irradiated material, the measured concentrations are actually consistent with the same true boundary concentrations in both cases.

~

CONCLUSIONS

A model based on mathematical convolution methods has been developed to describe the electron beam inter- action with thin segregated layers in transmission elec- tron microscope specimens. It is shown that if the beam intensity profile is accurately known, it is possible to quantify the shape of the segregant concentration profile. Conversely, the beam profile can be measured if the segregant profile is known. Two segregation pat- terns are considered: one where the shape is a top-hat configuration and the other a segregated layer with solute depletion in regions immediately adjacent to the layer.

Examples of applications of the model to P, Cr, Ni and Mo segregation in a ferritic-martensitic steel, FV448, have been given.

Acknowledgements

Thanks are due to AEA Technology and the University of Lough- borough for financial support for T.S.M.

202 R. G. FAULKNER, T. S. MORGAN AND E. A. LITTLE

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