A Straightforward Method For Interpreting XPS Data From Core−Shell NanoparticlesAlexander G. Shard*
National Physical Laboratory, Hampton Road, Teddington, Middlesex, TW11 0LW, United Kingdom
ABSTRACT: This paper describes a simple and direct method to calculatethe shell thickness of spherical core−shell nanoparticles from X-rayphotoelectron spectroscopy data. In contrast to existing methods, it is notiterative and involves a simple forward calculation that is accurate to atypical error of 4%. The method is applicable to any core−shell materialpair, but the accuracy becomes worse when the kinetic energy ofphotoelectrons arising from the core and the shell are widely separated.Application of the method to two example systems from the literature isdemonstrated: silicon oxide on silicon and carbon on gold. In both cases,accuracy in shell thickness that is significantly better than an atomic diameter is demonstrated. An accurate direct equation tocalculate the thickness of overlayers on planar samples is also provided.
■ INTRODUCTION
Engineered nanoparticles are increasingly used in innovativeproducts, and the careful control of size, shape, and chemistryare vital to their function. Electron microscopy is routinelyemployed to measure the size and shape of particles, andtechniques based upon the mobility of particles in fluidsprovide results that are affected to varying extents by the size,shape, or density of the particles. These methods are wellestablished and have a strong theoretical underpinning, andalgorithms for the interpretation of data are available.The measurement of nanoparticle chemistry, in particular the
chemistry at the surface of nanoparticles, is of increasingconcern.1−3 The surface chemistry of nanoparticles is importantfor both the application and processing of nanoparticles, since itaffects the manner in which the nanoparticles interact with thesurrounding environment. The dispersion and aggregation ofparticles critically depend upon their surface chemistry, and fora number of particles, a surface passivation layer is required toinsulate the core from the environment. In both biotechnologyand nanoparticle toxicology, the intentional or adventitiousattachment of organic compounds to the exterior of nano-particles is of great importance, as these mediate the manner inwhich the particles bind to other molecules and affect livingorganisms. Quantitatively measuring the amount of material atthe surface of a nanoparticle is therefore of major importancefor understanding the behavior of nanoparticles and ensuringconsistency in manufacture.Of the many readily accessible methods by which this
quantitative measurement could be achieved, X-ray photo-electron spectroscopy (XPS) is an appropriate and widely usedmethod.4−22 The specific advantages of XPS are that it isquantitative, chemically specific, has an information depthsimilar to the size of nanoparticles, and, in comparison toelectron beam methods, does not significantly damage theanalyzed material. It is therefore no surprise that XPS has long
been a mainstay for the analysis of supported heterogeneouscatalysts,23,24 which are usually in nanoparticulate form.The two major problems in XPS analysis of nanoparticles are
the preparation of samples for analysis and the interpretation ofdata. The preparation of samples for analysis is not the topic ofthis paper, but should not be trivialized since it is a majorbarrier to the application of XPS in this regard, particularly onnanoparticles that are prepared and used in liquid suspension.The application of XPS requires that the nanoparticles be in adry solid form without significant surface contamination and ona substrate that does not produce signals coincident with thoseof the nanoparticles. Within this paper, it is assumed that theseissues have been overcome, and we concentrate on the issue ofdata interpretation. For spherical nanoparticles at submono-layer surface coverage, XPS analysis provides a single spectrumwith intensities for the photoemission lines of elements presentin the near surface (∼10 nm). In contrast to planar surfaces,angle-resolved XPS analysis of spherical nanoparticles providesno additional information of any significant use. Interpretationrequires, as a minimum, knowledge or assumption of the corechemistry and size of the particle. For large (>10 nm) particles,it should be possible to use the shape of the energy lossbackground to infer which signals arise from the core, followingthe method of Tougaard.25 However, for small particles, thecontribution of signal from the shell chemistry under the coremay make distinction by this method difficult. Knowledge orassumption of which signals arise from the shell and whichfrom the core enables the amount of material in the shell to bedetermined. The essential problem is the conversion of therelative XPS intensities, usually expressed as a ratio of theintensity of photoelectrons arising from the shell to theintensity of photoelectrons arising from the core, into a shell
Received: May 30, 2012Revised: July 12, 2012Published: July 16, 2012
thickness. A straightforward and direct method for this has notbeen reported, although careful modeling and calculation hasbeen shown to be effective. A simple method for performingthis conversion would be helpful to many analysts who do nothave the time or expertise to undertake detailed modeling; thismethod would also be helpful in demonstrating the generalfeatures of the problem, assessing the sensitivity of XPS analysisto various parameters, and providing a simple route to theevaluation of uncertainties.It is not surprising that a simple method for this conversion
has not been reported, since a direct method of converting XPSdata to thickness for uniform, planar overlayers has not beenreported. Currently, an iterative or graphical method isrequired.26 This paper reports a direct and accurate empiricalmethod for converting XPS intensities into overlayerthicknesses, with a particular emphasis on spherical nano-particles. The variable input to the equations is the ratio ofnormalized XPS intensities, denoted here by the variable A, andthe relative electron attenuation lengths denoted by B and C.Although the equations are relatively straightforward and easyto implement, they are not quite as simple as A, B, C, since avariable R, the radius of the particle core, is also required.
■ THEORYAll calculations and theory in this paper do not account for theeffects of the elastic scattering of electrons in detail but assumethese can be neglected or compensated.27 The “straight line”approximation is assumed with effective attenuation lengths, L,to describe the diminution of photoelectron intensity withdistance through a material according to a simple exponentialdecay. Nanoparticles are assumed to be spherical with auniform shell thickness.Terminology. To simplify the equations in the paper, the
important inputs are combined into dimensionless terms. TheXPS experimental result is denoted by A, which is a ratio of thenormalized integrated intensities of a unique signal from thenanoparticle shell to that of a unique signal from the core.Thus,
=∞
∞AI II I1 2
2 1 (1)
where Ii is the measured XPS intensity and Ii∞ is the measured
or calculated intensity for the pure material of uniquephotoelectrons from the shell (overlayer), i = 1, and the core(substrate), i = 2, respectively.Using the notation from a recent paper on the effects of
topography on XPS analysis of overlayer thickness,28 all of thelengths are described as a ratio to the attenuation length ofelectrons arising from the unique overlayer signal in theoverlayer material. If Li,j is the attenuation length ofphotoelectrons arising from material i traveling throughmaterial j, where j = a represents the shell and j = b representsthe core, the following definitions are used to simplify laterexpressions:
=BL
L1,a
2,a (2)
=CL
L1,a
1,b (3)
The core radius of the particle, R, and shell thickness, T, areexpressed in units of L1,a.
A practical estimate29 of B can be found from (E1/E2)0.872,
where Ei are the photoelectron kinetic energies. Similarly, auseful estimate of C is given by (Zb/Za)
0.3, where Zj is thenumber-averaged atomic number of the material; this requiressome knowledge of the core and shell stoichiometry. Suchinformation may be found from a more detailed analysis of theXPS data, although materials that contain hydrogen will beproblematic in this regard. For organic materials, Z = 4 is areasonable estimate.29 These estimates of B and C provideuseful support for the assumption that the fourth attenuationlength of importance here can be estimated from the otherthree. This assumption is given in eq 4 and is used throughoutthis paper:
=L
LBC1,a
2,b (4)
Figure 1 schematically demonstrates the influence of theterms B and C on photoelectron intensities arising from the
core and shell of small nanoparticles. It can be seen that theintensity from the underside of the shell is influenced by theterm C, which describes the relative opacity of the core. Theintensity from the core of the particle is also influenced by thisterm, but more strongly by B, which describes the relativepenetration length of photoelectrons from the two materials. Itis clear from these figures that when both B and C are large, theeffect of the finite depth of the core is small and the situation issimilar to that for large particles.
Planar Samples. An essential step in providing a directequation for nanoparticles is a demonstration that a similarequation can be found for the much simpler case of planarsamples. The “Thickogram” equation26 for the iterative orgraphical calculation of film thickness is given in eq 5 using theterminology in this paper,
= − −
−A1 e
e
T
BT
planar
planar (5)
in which the photoelectron emission angle is assumed to benormal to the surface. If this is not the case, then the value ofTplanar that results from this analysis can simply be correctedthrough multiplication by cos θ, where θ is the electron
Figure 1. A schematic illustration of relative XPS intensities for core−shell nanoparticles showing the regional contributions to the XPSsignal. Contributions to the XPS signal are represented on a gray scale(white representing the largest contribution) and calculated using R =1 and T = 0.5. The boundaries between phases are marked in black.
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emission angle with respect to the surface normal. In the limitof small thicknesses, Tplanar = A, and in the limit of largethicknesses, Tplanar = ln(A)/B. The advantage of theseapproximations is that at intermediate thicknesses, the formeroverestimates the actual thickness and the latter underestimatesit, as shown in Figure 2b and c. One may envisage that, using arelatively simple weighting, these may be spliced together toprovide an accurate and direct estimate of Tplanar. In fact, asimple weighting factor using A alone, as given in eq 6 andshown in Figure 2d, can approach this for values of B close to 1.
≈ ++
−T
A A B AA
ln( ) 22planar
2 1
2 (6)
However, if B = 1, eq 5 is no longer transcendental, anddirect inversion using Tplanar = ln(1 + A) is better, as shown inFigure 2a. A refinement of the weighting method results in eq7, which is plotted in Figure 2e and provides Tplanar accuratelyto within 0.04 (i.e., less than the diameter of an atom) over therange 0.5 < B < 2 for all practical values of Tplanar.
= ++
− −T
A A B ABA
ln( ) 21.9planar
2.2 0.95 0.42
2.2 (7)
This equation may be implemented when a direct forwardcalculation of thickness from XPS data is required or to providean initial estimate for further refinement by an iterativecalculation. In practice, however, further refinement isunnecessary, since the error provided by this equation isinsignificant compared with other sources of uncertainty.Equation 7 is clearly in error for very small and very largevalues of A, and if Tplanar < 0.1, the estimate provided by Tplanar= ln(1 + A) is better. The large values of A where the error ineq 7 becomes significant are not of practical importance, sincein these cases, the XPS signal from the substrate will be tooweak to permit an accurate analysis. The importance within this
work is that this accurate direct equation suggests that a directequation may be found to accurately convert XPS data fromnanoparticles into a shell thickness.
Microscopic Spherical Particles. The determination ofoverlayer thickness on nonplanar samples has previously beenaddressed in detail, and a practical approach is to calculate anequivalent planar thickness, as described above, and multiply bya geometrical correction term, or Topofactor,28 to calculate theconformal thickness. The Topofactor depends upon thetopography of the sample and the values of A and B. Formicroscopic spherical particles, when R is much larger than 1but not so large that X-ray shadowing effects become significant(i.e., R is less than ∼1000), under these conditions, a constantTopofactor of ∼0.67 is acceptable at 10% error. The sphericalTopofactor reduces to 0.5 for infinitesimally small values of T,and a more accurate expression to 1% error is also available.28
As described later, it is important to be able to directly calculateT from A in the case R→∞. This is possible by introducing eq7 into the accurate spherical Topofactor expression, but resultsin a cumbersome equation. A simpler approach to introducethe Topofactor is by retaining the form of eq 7 and optimizingthe parameters within the equation to provide the conformalthickness of the overlayer on a large sphere, TR→∞. Thisintroduces some additional error, providing T accurately towithin 0.05, as long as T < 3, but retains a form that is relativelysimple. The result optimized for large spheres is given in eq 8.
= ++→∞
− −T
A A B ABA
0.74 ln( ) 4.28.9R
3.6 0.9 0.41
3.6 (8)
The prediction of eq 8 is shown in Figure 2f, wheredeviations are evident at high and low values of T. Thisamounts to ∼5% error as T approaches 3, and for values of Tsmaller than 0.1 where the average error is more than 10%, abetter prediction is given by using the planar estimate and the
Figure 2. A comparison of direct estimates for Tplanar and TR→∞ for values of B, the ratio of electron attenuation lengths, ranging from 0.5 to 2. B = 1is shown by a bold line, and the common practical limits of B = 0.7 and B = 1.4, by dashed lines. In parts a−d, values of B larger than 1 result in Tplanarbeing larger than Tactual and provide the lines above the bold line. In part e, the prediction of eq 7 for Tplanar is shown, and in part f, the prediction ofeq 8 for TR→∞ on spheres.
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appropriate Topofactor for thin overlayers, which approaches0.5 for very thin films. The expression TR→∞ = (0.5 + 0.1AB−2)ln(1 + A) provides an accuracy of better than 10% in thisregion.Infinitesimally Small Particles. For very small values of R
and T, where attenuation of photoelectron intensities can beneglected, a limiting relationship among A, B, C, T, and R isstraightforwardly obtained.28 This is arranged to provide T0 ineq 9, which is valid when the product ABCR approaches zero.
= + −T R ABC[( 1) 1]01/3
(9)
In most practical situations, this limit is not reached;however, the expression provides better than 5% error forparticles smaller than ∼1 nm diameter.Nanoscopic Spherical Particles. When the size of the
nanoparticle is of the same order of magnitude as theattenuation lengths, eqs 8 and 9 are not appropriate, andanother approach is required. An important relationship toinvestigate is the behavior of T as a function of R for fixedvalues of A, B, and C. This describes the sensitivity of thedesired result (T) on the size of the core (R), given a particularexperimental result, and is important in establishing, forexample, the uncertainty of the result on the basis of that ofthe core size measurement and the effect of size dispersity. Toaddress this problem, more than 6000 numerical calculations of
T were performed with R varying from 0.05 to 10 000; A, from0.03 to 300; and B and C, from 0.5 to 2. The method used wasto begin with a trial value of T and to calculate A given B, C,and R; the value of T was then iteratively refined until Amatched the desired value. Some (∼10%) of the calculatedvalues are shown in Figure 3 as data points.The relationship between T and R for moderate values of R
and fixed values of A, B, and C is described very well by thesimple empirical relationship shown in eq 10,
α=
+∼→∞T
T RRR
R1 (10)
where α can be found by fitting the numerical data and is afunction of A, B, and C. It is straightforward to see that α relatesto the value of R at which T is half that of TR→∞. In the limit ofR ≪ α, the form of eq 10 is not inconsistent with eq 9.However, the value of α cannot be found by making thisconsistency an identity, and the description becomes poor atlow values of R. The accurate value for infinitesimally smallparticles given in eq 9 can be combined with eq 10 to provide abetter estimate of shell thickness over a greater range of R,
ββ
=+
+∼T
T T1
RNP
1 0
(11)
Figure 3. Results of numerical calculations to find T as a function of A, B, C, and R. A selection of data are shown that represent the normalexperimental range. The values of B and C are provided in the top left corner of each part of the figure with T on a linear scale plotted against R on alogarithmic scale. Numerical data are plotted as points: ◇, A = 300; ▲, A = 100; □, A = 30; ⧫, A = 10; Δ, A = 3; ■, A = 1; ○, A = 0.3. Solid lines arethe TNP predictions of eq 11.
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where β is a weight that is a function of A, B, C, and R andincreases as these become small. Simple expressions for α and βin terms of A, B, C, and R can be found and optimized byminimizing the error in predicting the numerically calculatedvalues. These expressions are given in eqs 12 and 13.
α =A B C
1.80.1 0.5 0.4 (12)
β α=R
0.13 2.5
1.5 (13)
The solid lines in Figure 3 demonstrate that, using theseexpressions, the TNP of eq 11 is capable of providing anexcellent estimate of T for core−shell nanoparticles. More than95% of the numerical values of T are predicted to within either10% or 0.1. Figure 4 plots a greater range of calculations in the
same format as Figure 2 for four subsets of data defined byindividual values of R ranging from R = 0.5 to R = 32 (∼2.5 nmto ∼160 nm diameter cores). These illustrate the degree of biasand scatter in TNP, which can be seen to be small. For largerparticles, the value of C becomes unimportant, and the 441points in the subset converge into 63 points defined only by theparameters A and B. For very large particles, the graph isidentical to that shown in Figure 2f for eq 8.The global mean relative error for TNP over the range used
here is approximately 4%, which, when compared to the ∼10%error in estimating attenuation lengths, demonstrates that useof these equations will usually not be the most significantsource of error in finding the shell thickness of nanoparticlesfrom XPS data. However, there are certain regimes in which theprediction is poor. Figure 5 plots the mean error over subsets ofthe numerical data set grouped by input variable. Thisillustrates that the prediction is worse at extreme values of B.The usual range of B is between 0.7 and 1.4; in this regime, themean error is ∼3%. The prediction also appears poor at smallvalues of A, which relates to very small T; however, the mean
absolute error in this regime is lower than 0.01, which is aboutone tenth of the diameter of an atom. Interestingly, thepredictive quality of eq 11 improves as the core size of theparticle, R, decreases. This is a remarkable result and impliesthat an important limiting factor is the ability to predict TR→∞through eq 8. The mean relative error of eq 11 is largelyunconnected to the value C, since this becomes important onlyat small R, where the prediction is, in most cases, excellent.Figure 6 provides an overview of where the equations
provided in this paper may be used: TNP from eq 11 is best inall cases when accuracy of better than 10% is required. Formicroparticles, X-ray shadowing must be accounted for, and
Figure 4. Comparison of TNP with T from numerical calculations forfour different values of R. Each graph plots 441 individual calculations,shown as points, and a line describing the relationship TNP = T.
Figure 5. Mean relative error in the TNP prediction for subsets of thenumerical data plotted against the input parameter values A, B, C, andR. Note that A, B, C, and R are plotted on logarithmic scales, and themean error is plotted on a linear scale.
Figure 6. Schematic representation of the regions in which theequations in this paper are valid for a given accuracy and particlediameter. Tplanar is given in eq 7; TR→∞, in eq 8; T0, in eq 9; TR∼1, in eq10; and TNP, in eq 11. The Topofactor of 0.72 for particles that aresignificantly larger than the X-ray attenuation length is taken fromreference 28. A dashed line provides the accuracy of attenuationlengths for comparison.
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methods to do this are described elsewhere.28 For comparison,the accuracy of attenuation lengths of ∼10% are provided, butthe accuracy of the experimental intensities and normalizationfactors encapsulated in A must also be considered. For analysesin which a precision in the value of T of better than 5% isrequired, numerical methods are necessary.The Effect of Dispersity in Core Radius on XPS
Thickness. Equation 10 demonstrates that at large R, thevalue of TNP is independent of R; therefore, dispersity in coresize does not affect the average value of TNP determined for apopulation of nanoparticles. For smaller nanoparticles, thedependence in the value of TNP upon R is approximately linear.Detailed investigations of the forward prediction of intensities28
and the backward calculation of T in this work for simulatedpopulations of nanoparticles show that, for monomodal andbimodal distributions in R, the value of TNP obtained by thismethod is robust, provided that the value of R used is the root-mean-square radius rather than the mean radius. Thiscompensates for the greater contribution of large particles tothe XPS signal, and its use is critically important only forbimodal populations.
■ APPLICATION TO LITERATURE DATATwo examples of the application of the method are providedbelow. These use data in the open literature that providesufficient detail for the method to be applied. For comparison,numerical calculations were also performed to demonstrate theaccuracy of the expressions given above. Importantly, theapplication of the equations in this paper could be performedon all data in a few minutes using standard spreadsheetsoftware.Silicon−Silicon Oxide Nanoparticles. A detailed XPS
study of the oxidation of silicon nanoparticles has beenpresented,4 from which the input variables and experimentalresults shown in Table 1 can be extracted. For this analysis, a
conservation relationship for silicon needs to be established onthe basis of the assumed densities of silicon and silicon oxidebecause in the oxidation process, the core reduces in size andthe shell increases, but the total amount of silicon is assumed tobe constant. The relationship outlined in their paper isfollowed, beginning with a trial value of the core radius, topredict the shell thickness both through eq 11 and through the
conservation relationship and iteratively to change the coreradius until the two thickness values match. Table 1 shows acomparison of the core radii and shell thicknesses found in theoriginal paper and using the method described in this paper.The values match to within the error of extracting the input
values and results from their Figures 5 and 8. This niceconcordance of the approaches used here and the one used intheir paper is encouraging, but it is not a general validation ofeither method. The match is a result of the optimization of bothmethods to essentially identical numerical calculations andtherefore confirms only that the numerical calculations areconsistent. Within the regimes in which the approximatemethods have been validated, they should be expected toperform well and identically. A more detailed examination ofthe method used,4 which is described in detail in anotherpaper30 reveals that their approach should be applied only whenboth B and C are close to 1 and when R is less than ∼10. Thefirst restriction results from a rather ad hoc treatment ofattenuation lengths for the sake of some mathematicalsimplicity. The second restriction results from an error in thetreatment of the effect of large-scale topography on effectiveoverlayer thicknesses. Their essential equation for A tends to eq5 at large R, which underestimates the true value by a factor of∼2, as explained elsewhere.28 However, within the constraintsprovided here, their iterative method is excellent, predicting thenumerical values of A generated for this paper to within a fewpercent.
Thiol Self-Assembled Monolayers on Gold Nano-particles. A detailed XPS study of gold nanoparticles withcarboxylic acid-terminated thiol self-assembled monolayers(SAMs), HS(CH2)n−1COOH (where n is the number ofcarbon atoms in the thiol), has recently been published.18 Thiswork presents sufficient data to permit analysis using themethod described in this paper. To exemplify the method, theratio of the C1s signal, which arises from the thiol shell, to theAu4f signal, representative of the core, is used. The data werepresented both in terms of elemental compositions, from whichthe “homogeneous” elemental ratios ([C]/[Au]) can be foundand also as an equivalent (or “apparent”) overlayer thickness,from which the values A can be extracted. From these data, theaverage conversion factor A = 0.73 ([C]/[Au]) can also befound. Using the values given in their paper: L1,a = 3 nm, B =0.909, and C = 2, the thickness of the carbon shell on the goldcores was then calculated using both the quick methoddescribed in this paper and using accurate numericalcalculations. The input data and results of these analyses areprovided in Table 2 and show that the estimate provided by themethod in this paper is within 0.1 nm of the numerical method.Subsequently, one of the samples was reanalyzed in greater
detail,16 along with a detailed analysis of XPS intensities. In thisanalysis, slightly different atomic ratios were used ([C]/[Au] =3.00 compared with 3.66 in the first analysis18). This is possiblya result of using an instrument different from that in theoriginal paper. A similar conversion to find A has beenperformed and is shown in Table 2. The importance of this lastresult is that it can be compared with an extensive simulationthat provided,16 among other details, a total shell thickness of1.85 nm. This includes the minor contribution of oxygen andsulfur atoms to the thickness of the SAM layer and thereforecompares very well with the values obtained in this work.From the data shown in Table 2, it is possible to establish a
relationship between the thickness determined from the XPScarbon/gold intensity ratio and the number of carbon atoms in
Table 1. Data and Variables from Reference for Core−ShellNanoparticles of Silicon−Silicon Oxidea
aX(SiO) is the concentration of oxidized Si deduced from the amountof elemental Si in a peak fit of the Si 2p region of the XPS spectra.Results of analyses using the method described in this paper are shownfor comparison.
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the SAM. This is close to linear both for the approximatemethod described here (TNPL1,a = 0.120n + 0.100, expressed innanometers) and for the numerical simulations (TL1,a = 0.117n+ 0.128, expressed in nanometers). The constant term in thelinear fits potentially arises from hydrocarbon contamination,which was previously estimated to be 0.15 nm for one of thesamples.16
A study of gold nanoparticles (RL1,a = 12.1 nm) coated withamine thiol SAMs was recently published by the same group.17
In that later study, the initial SAM consisted of n = 2 thiols, andthe time-dependent exchange with n = 11 thiols wasinvestigated. Table 3 presents the results using the method
described in this paper and using an accurate numericalapproach, making use of the conversion factor found for thecarboxylic acid SAMs to obtain A.The concordance between the accurate numerical approach
and the approximate method described here is once againwithin 0.1 nm, indicating that the method is widely applicable.The thicknesses given here may be used to estimate the averagenumber of carbon atoms in the SAM shell. Interestingly, theaccurate numerical method provides exactly the expectednumber of carbon atoms in the initial sample, which isencouraging and well within the ∼10% relative standarddeviation implied by the original data. The direct methoddeveloped in this paper provides the number of carbon atomswith a maximum error of 0.3 atoms compared with thenumerical approach. The final thickness is equivalent to ∼9
carbon atoms, indicating that complete exchange has notoccurred and that after 61 days, ∼80% of the C2 thiol has beenreplaced by C11 thiol.
■ CONCLUSIONSThis paper provides simple, direct, and accurate equations tocalculate overlayer, or shell, thickness from XPS data for flatfilms, microparticles and nanoparticles. The direct method fornanoparticles is compared with accurate numerical calculationswith example data taken from the literature and found toprovide thicknesses with an error typically better than 4%,which is smaller than the expected error in attenuation lengths.The important advantage of the method given in this paper isthat it is fast and simple to use, which is of great advantage tononspecialists and general analysts. For highly accurate workand in cases when more detail is required (such as core−shellparticles with an outer contaminant layer), numericalsimulations are still to be preferred.
NotesThe authors declare no competing financial interest.
■ ACKNOWLEDGMENTSThis work forms part of the Chemical and BiologicalProgramme of the National Measurement System of the UKDepartment of Business, Innovation and Skills (BIS). Theauthor thanks Martin Seah and Ian Gilmore from the NationalPhysical Laboratory and David Castner from the University ofWashington for helpful discussions and suggestions.
■ REFERENCES(1) Baer, D. R.; Amonette, J. E.; Engelhard, M. H.; Gaspar, D. J.;Karakoti, A. S.; Kuchibhatla, S.; Nachimuthu, P.; Nurmi, J. T.; Qiang,Y.; Sarathy, V.; Seal, S.; Sharma, A.; Tratnyek, P. G.; Wang, C. M. Surf.Interface Anal. 2008, 40, 529−537.(2) Baer, D. R.; Engelhard, M. H. J. Electron Spectrosc. Relat. Phenom.2010, 178, 415−432.(3) Baer, D. R.; Gaspar, D. J.; Nachimuthu, P.; Techane, S. D.;Castner, D. G. Anal. Bioanal. Chem. 2010, 396, 983−1002.(4) Yang, D. Q.; Gillet, J. N.; Meunier, M.; Sacher, E. J. Appl. Phys.2005, 97, 024303.(5) Tunc, I.; Suzer, S.; Correa-Duarte, M. A.; Liz-Marzan, L. M. J.Phys. Chem. B 2005, 109, 7597−7600.(6) Huang, K.; Demadrille, R.; Silly, M. G.; Sirotti, F.; Reiss, P.;Renault, O. ACS Nano 2010, 4, 4799−4805.(7) Zorn, G.; Dave, S. R.; Gao, X.; Castner, D. G. Anal. Chem. 2011,83, 866−873.(8) Abel, K. A.; FitzGerald, P. A.; Wang, T.-Y.; Regier, T. Z.;Raudsepp, M.; Ringer, S. P.; Warr, G. G.; van Veggel, F. C. J. M. J.Phys. Chem. C 2012, 116, 3968−3978.(9) Wu, W.; Njoki, P. N.; Han, H.; Zhao, H.; Schiff, E. A.; Lutz, P. S.;Solomon, L.; Matthews, S.; Maye, M. M. J. Phys. Chem. C 2011, 115,9933−9942.(10) Jang, S. G.; Khan, A.; Dimitriou, M. D.; Kim, B. J.; Lynd, N. A.;Kramer, E. J.; Hawker, C. J. Soft Matter 2011, 7, 6255−6263.(11) Guzel, R.; Ustundag, Z.; Eksi, H.; Keskin, S.; Taner, B.; Durgun,Z. G.; Turan, A. A. I.; Solak, A. O. J. Colloid Interface Sci. 2010, 351,35−42.(12) Banerjee, R.; Katsenovich, Y.; Lagos, L.; McIintosh, M.; Zhang,X.; Li, C. Z. Curr. Med. Chem. 2010, 17, 3120−3141.
Table 2. Data from Reference for Core-Shell Nanoparticlesof Gold-Carboxylic Acid SAM and Results of Analysis by theMethod Described in This Paper and Accurate NumericalCalculations
RL1,a (nm) n A TNPL1,a (nm), eq 11 TL1,a (nm), numerical
Table 3. Data from Reference 17 for Core−ShellNanoparticles of Gold−Amine SAM and Results of Analysisby the Method Described in This Paper and AccurateNumerical Calculations
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The Journal of Physical Chemistry C Article
dx.doi.org/10.1021/jp305267d | J. Phys. Chem. C 2012, 116, 16806−1681316813
