HHS Public Access Author manuscript Author Manuscript
Surf Interface Anal. Author manuscript; available in PMC 2017 May 01. Published in final edited form as: Surf Interface Anal. 2016 May ; 48(5): 274–282. doi:10.1002/sia.5923.
A Technique for Calculation of Shell Thicknesses for Core-ShellShell Nanoparticles from XPS Data David J. H. Cant1, Yung-Chen Wang2, David G. Castner2, and Alexander G. Shard1 1National
Physical Laboratory, Hampton Road, Teddington, Middlesex, TW11 0LW, United
Kingdom 2Departments
Author Manuscript
of Bioengineering & Chemical Engineering, National ESCA & Surface Analysis Center for Biomedical Problems, University of Washington, Seattle WA
Abstract This paper extends a straightforward technique for the calculation of shell thicknesses in core-shell nanoparticles to the case of core-shell-shell nanoparticles using X-ray Photoelectron Spectroscopy (XPS) data. This method can be applied by XPS analysts and does not require any numerical simulation or advanced knowledge, although iteration is required in the case where both shell thicknesses are unknown. The standard deviation in the calculated thicknesses vs simulated values is typically less than 10%, which is the uncertainty of the electron attenuation lengths used in XPS analysis.
Author Manuscript
Introduction Nanoparticles are a topic of increasing interest for applications across many disparate fields, including medicine1–5, materials6–9, and opto-electronics10–15. Many nanoparticle systems studied are increasingly complex – core-shell/core-multi-shell systems are now routinely synthesised and investigated. Such particles are commonly characterised using electron microscopy, optical absorption/luminescence spectroscopies, and various light scattering and sedimentation methods16–19. Shell thickness, however, can be difficult to measure using these methods; electron microscopy is limited to materials that exhibit good contrast between the core and shells, absorption and luminescence changes due to nanoparticle shells are often the subject of study, but are not useful as a reliable measurement technique, and light scattering or sedimentation based methods typically provide little or no information regarding particle composition.
Author Manuscript
Knowledge of shell thicknesses is important to adequately understand how the shell affects the properties of the nanoparticle. In nanoparticles intended for electronic and optical applications, shell thickness can be important when considering excited state and electron transport properties; in medical or biotechnological applications, an understanding of the effects of shell thickness on reactivity and interaction with the local environment is typically required. For nanoparticles bearing multiple shells, measurement of shell thicknesses is not a trivial issue and, even in supposedly single shell systems, the presence of capping ligands, adsorbed molecules, and contaminants may result in a second overlayer that will affect the measurement.
Cant et al.
Page 2
Author Manuscript
X-ray Photoelectron Spectroscopy (XPS) is a useful technique for characterisation of nanometre scale overlayers on materials20–22, as it can provide quantitative information on the amount of each element present in the surface region (approx. < 10 nm) of a sample, distinguish between the different chemical states of elements that are present, and is nondestructive to most samples. The primary issues with the use of XPS for analysis of nanoparticles are sample preparation and data interpretation. The issue of sample preparation is centred around the requirement of XPS to have a clean, dry sample, mounted on a substrate that does not produce spectra that could obfuscate the spectra produced by the sample. This can be difficult to achieve with nanoparticle samples, due to the prevalence of multi-component solutions used for nanoparticle synthesis and processing. In particular, a nanoparticle suspension must be free of non-volatile solutes as these may dry onto the particles and form a contaminant layer. Baer et al.23 describe in detail the issues faced in nanomaterial surface characterisation, and discuss the factors that must be considered when removing nanoparticles from solution. The effects of environment and sample preparation technique on the data obtained from surface characterisation techniques are also discussed in the ISO TC201 (surface chemical analysis) technical report ISO/TR 1418724.
Author Manuscript Author Manuscript
The issue of XPS data interpretation as applied to nanoparticles and nanoscale structures in general can be relatively complex, due to the topographic effects becoming more important as the sample dimensions approach that of the attenuation length of photoelectrons25–27. Previous work in this area has typically focussed on the simple case of core-shell particles, either by direct comparison to simulated intensities28 or by development of an empirical formulae that provides an approximation to simulated intensities27,29. Such approaches are useful, and have been shown to give consistent results30 However, an empirical method for dealing with the case of core-shell-shell systems has not yet been described - detailed simulation can deal with the effects of additional layers31 but requires specialized software or expertise. Therefore it is important to consider whether a simplified scheme available to any analyst can be found. In this work we demonstrate that it is possible, but it is not as straightforward as the core-shell problem.
Theory
Author Manuscript
The assumptions made for the calculations within this paper are as follows: it is assumed that particles are spherical, and that both shells consist of an even distribution of material across the surface of the particle; the photoelectron intensity arising from the particle is assumed to follow the ‘straight line’ approximation, whereby all photoelectrons are considered as having travelled along a straight line from within the particle, with the decrease in photoelectron intensity due to transport through the particle following a simple exponential decay with distance, characterised by the effective attenuation lengths (L) within each material. Elastic scattering of electrons is not considered in detail, rather it is assumed that the effect of elastic scattering can be compensated for or neglected32. It is assumed that the relative photoelectron intensities arising from a sample of nanoparticles can be considered as equivalent to that arising from a single nanoparticle, as justified by Werner et al.33. The attenuation lengths used within this paper are calculated using the method presented by Seah34, originally intended as an empirical means of obtaining effective attenuation lengths for use in determining the film thicknesses of planar overlayers on a flat
Surf Interface Anal. Author manuscript; available in PMC 2017 May 01.
Cant et al.
Page 3
Author Manuscript
substrate. These attenuation length values have been found to be useful in previous work on overlayer thicknesses in core-shell systems27 and are used here. Shard,27 in his equation (11), presents a quick, straightforward calculation of shell thickness that can be readily applied by non-specialists and general users, with an error that is typically less than the expected error in estimated attenuation lengths34. This is henceforth referred to as the TNP equation.
Terminology As with the previous work on core-shell systems27, formulas will be given in terms of a set of dimensionless parameters, using an extended but similar terminology for consistency. The experimental output is denoted by Ai,j and represents the ratio of normalised integrated intensities of a unique signal from material i to that from j, as shown in equation. 1
Author Manuscript
(1)
where Ii is the measured XPS intensity from material i, and is the measured or calculated intensity for a planar sample of pure material. The core, inner shell, and outer shell of a particle will be denoted by i and j values of 0, 1, and 2 respectively. In considering attenuation lengths, we will similarly use the terms Bi,j and Ci,j to refer to the following ratios of attenuation lengths Lx,y for photoelectrons arising from material y travelling through material x.
Author Manuscript
(2)
(3)
Author Manuscript
Thus Bi,j is the ratio of attenuation lengths in material i of electrons arising from material i to electrons arising from material j, and thus describes the relative penetration lengths of electrons arising from materials i and j. Similarly, Ci,j represents the ratio of the attenuation length of electrons travelling from material i through material i to the attenuation length of electrons travelling from material i through material j, and thus describes the relative opacity of materials i and j. A practical estimate of the value of Bi,j can be obtained from (Ei/Ej)0.872 where E is the electron kinetic energy34, thus we can also state that B2,0 = B2,1B1,0. A practical estimate of Ci,j can be obtained from (Zj/Zi)0.3 where Z is the number-averaged atomic number34, and thus C2,0 = C2,1C1,0. Finally, we make the assumption - shown in equation 4 - that the ratio of the attenuation lengths of electrons travelling through the material they originate from can be estimated from a combination of B and C.
Surf Interface Anal. Author manuscript; available in PMC 2017 May 01.
Cant et al.
Page 4
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(4)
These assumptions effectively reduce the number of independent parameters that must be considered from the nine that result from electrons travelling from and through 3 different materials, to four: B2,1, B2,0, C2,1 and C2,0.
Modelling To make progress toward a method for calculating shell thickness in core-shell-shell nanoparticles, it is helpful to consider the two shells separately.
Author Manuscript
To calculate the thickness of the inner shell, we can consider the core and inner shell system as a basic core-shell nanoparticle and determine the change in A1,0 due to the outer shell as a function of the outer shell thickness, T2, and attenuation length ratios. We can then use an expression of this function to calculate the value of A1,0 for an equivalent particle with no outer shell, A*1,0. The TNP equation27 may then be applied to A*, using the core radius R, and the appropriate attenuation length ratios B1,0 and C1,0, to calculate the inner shell thickness. Likewise, to calculate the thickness of the outer shell, we can consider the nanoparticle as a core-shell particle consisting of the outer shell and a core formed by the combination of the core and inner shell. We thus require a method by which to obtain the effective A, B and C values from the ‘combined’ core for use in the TNP formula.
Author Manuscript
Calculation of the inner shell Simulations were carried out using the straight-line approximation to calculate the photoelectron intensity arising from each material. A 2-dimensional cross-section of half a nanoparticle is considered, as shown in Figure 2. The intensities from individual straight line columns of this cross section are then calculated, which represent hollow cylindrical cuts of the particle. The resulting intensities are weighted for the cross sectional area of the cylinder and summed, and the output given as ratios of these total intensities, which are equivalent to that of a complete nanoparticle. The script used to simulate core-shell-shell nanoparticles is included within the supporting information, alongside a more detailed explanation of the calculation.
Author Manuscript
Simulations of core-shell-shell nanoparticles and equivalent core-shell particles without the outer shell were conducted across a range of physically reasonable values for the various attenuation lengths and particle dimensions - specifically, R between 0.1 and 1000 (in units of L2,2), T1 and T2 between 0.1 and 10 (in units of L2,2), B values between 1/2 and 2, and C values between 1/3 and 3. Intensity ratios produced by these simulations were compared to derive the terms of the A*1,0 adjustment function. As particle dimensions will often be considered in terms of a specific attenuation length, this will be reported alongside the dimensions where given, e.g. R(L2,2).
Surf Interface Anal. Author manuscript; available in PMC 2017 May 01.
Cant et al.
Page 5
Author Manuscript
Initially, we can consider the simple situation of a particle consisting of identical materials in the core and both shells (i.e. attenuation length ratios B and C equal to one for all materials) and consider solely the effect of the increased attenuation from the presence of the outer shell. Figure 3 shows the ratio of A*1,0 to A1,0 plotted against the thickness of the outer shell, T2. The curve shown in Figure 3 may be readily approximated by a function of the natural logarithm of T2. More accurate fits may be obtained by increasingly complex functions, however when extended to consider systems in which attenuation lengths may vary between materials, this results in an impractical increase in the complexity of the function required.
Author Manuscript
Next, the change to this correction term A*/A due to variation in attenuation lengths may be considered. The terms B2,1 and B2,0, representing the relative attenuation lengths of electrons from the inner shell and core respectively, exhibit the greatest effect on the relative signal from the inner shell and the core. The term A*/A, plotted against T2 for varying values of B2,1 and B2,0 with other attenuation parameters held constant is shown in Figure 4 for a range of values of R.
Author Manuscript
The fit lines shown in Figure 4 are calculated from a combination of the logarithmic fit mentioned previously, with the apparent exponential decay of signal with T2 due to the two attenuation parameters, B2,1 and B2,0.It is clear from simulation that the change in intensity due to the second shell increases exponentially with the product of T2 and B2,1, and decreases exponentially with the product of T2 and B2,0. This seems intuitively reasonable, as it would be expected that the ratio of signal from the inner shell to the core would increase with an increase in the penetration length of electrons from the inner shell in the outer shell (B2,1) and would decrease with an increase in the equivalent parameter for electrons from the core (B2,0). A slight change in intensity ratio was observed with changing particle radius, but this change is insignificant and may be neglected with little effect on calculated thicknesses. The most accurate fit to simulation data was obtained by replacing the constant n used previously in the logarithmic term with a function of the attenuation parameters – the form of this term is detailed later. The resulting form of this fit is shown in equation 5. (5)
Author Manuscript
Finally, we can consider the effect of the attenuation parameters as a function of the relative opacity of the inner shell and core materials to the outer shell, C2,1 and C2,0 respectively. From simulations, it is apparent that the term denoting the opacity of the core, C2,0, has little to no effect on the ratio of A*1,0 to A1,0, and can be neglected. The opacity of the inner shell, denoted by C2,1, while not as significant as the penetration terms B2,1 and B2,0, has a noticeable effect on the observed intensities. This effect, however, is non-trivial to fit, and appears to be significantly interdependent with the other attenuation terms. It cannot be accounted for with as great a degree of accuracy as the penetration terms while maintaining a practical level of simplicity in the resulting equation. A simple approximation for the effect of C2,1 is shown in Figure 5, and while imperfect, it serves to minimise the error acceptably Surf Interface Anal. Author manuscript; available in PMC 2017 May 01.
Cant et al.
Page 6
Author Manuscript
(to within the ~10% error in the estimation of attenuation lengths) across the full range of reasonable physical parameters. When combining the terms to account for all relevant attenuation lengths, and fitting across a wide range of physically sensible values for each, the final correction factor for the intensity ratio A1,0 is shown in equation 6. (6)
where n is a function of the parameters B2,1, B2,0 and C2,1 given in equation 7. (7)
Author Manuscript
These values may then be inserted into the TNP equation, remembering that the units of length are L1,1. (8)
Calculation of the outer shell thickness When calculating the outer shell thickness, we consider the nanoparticle as a core-shell particle in which the ‘core’ is formed from both the core and inner shell, while the ‘shell’ consists of the outer shell alone. In this case, we must first determine the effective core-shell intensity ratio for such a system, Aeff, which is given by equation 9.
Author Manuscript
(9)
We must also determine effective values for the attenuation parameters used in the TNP equation. We may assume that the effective values for these terms, Beff and Ceff, are a combination of the equivalent terms for each of the inner materials, B2,1 and B2,0, or C2,1 and C2,0. The simplest approximation of which would be a linear combination of the two terms, adjusted by some weighting factor w, as shown in equations 10 and 11. (10)
Author Manuscript
(11)
While a form for w cannot be determined directly from fitting, some reasonable assumptions can be made. For example, we can assume that the weighting function is related to the relative amounts of material present in the core and the shell. We would expect that w holds a maximum value of 1 at the point where the majority of the material present forms a part of the first shell, i.e. in the limit of very large A1,0, as in this case the effective value of B would Surf Interface Anal. Author manuscript; available in PMC 2017 May 01.
Cant et al.
Page 7
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be expected to be equivalent to B2,1. Thus, we may consider a w of the form shown in equation 12 (12)
where p is a function of the two other relevant parameters, the attenuation ratios between the inner shell and the core, B1,0 and C1,0. After comparing simulated results with calculated ones for a selection of w functions, equation 13 was found to give useful results for T2 across the full parameter space. (13)
Author Manuscript
The values Aeff, Beff, and Ceff can thus be used in the TNP formula for calculation of shell thicknesses for a core-shell nanoparticle, using the value of (R+T1) as the ‘effective’ RNP, to calculate the value of the outer-shell thickness T2 with the units of length L2,2. (14)
Applicability
Author Manuscript
With the methods detailed above it is possible to calculate both T1 and T2 for core-shellshell nanoparticles, without the use of simulation or complex graphical methods. Initially it may seem that a large amount of fore-knowledge is required for either of these calculations – in order to calculate T2, you require T1, and vice versa. However in reality, only the value of R is required. Values for T1 and T2 may be estimated initially, and by iteration between the inner shell and outer shell methods we improve the accuracy of these estimates to within the formula’s average innate accuracy of
Surf Interface Anal. Author manuscript; available in PMC 2017 May 01. Published in final edited form as: Surf Interface Anal. 2016 May ; 48(5): 274–282. doi:10.1002/sia.5923.
A Technique for Calculation of Shell Thicknesses for Core-ShellShell Nanoparticles from XPS Data David J. H. Cant1, Yung-Chen Wang2, David G. Castner2, and Alexander G. Shard1 1National
Physical Laboratory, Hampton Road, Teddington, Middlesex, TW11 0LW, United
Kingdom 2Departments
Author Manuscript
of Bioengineering & Chemical Engineering, National ESCA & Surface Analysis Center for Biomedical Problems, University of Washington, Seattle WA
Abstract This paper extends a straightforward technique for the calculation of shell thicknesses in core-shell nanoparticles to the case of core-shell-shell nanoparticles using X-ray Photoelectron Spectroscopy (XPS) data. This method can be applied by XPS analysts and does not require any numerical simulation or advanced knowledge, although iteration is required in the case where both shell thicknesses are unknown. The standard deviation in the calculated thicknesses vs simulated values is typically less than 10%, which is the uncertainty of the electron attenuation lengths used in XPS analysis.
Author Manuscript
Introduction Nanoparticles are a topic of increasing interest for applications across many disparate fields, including medicine1–5, materials6–9, and opto-electronics10–15. Many nanoparticle systems studied are increasingly complex – core-shell/core-multi-shell systems are now routinely synthesised and investigated. Such particles are commonly characterised using electron microscopy, optical absorption/luminescence spectroscopies, and various light scattering and sedimentation methods16–19. Shell thickness, however, can be difficult to measure using these methods; electron microscopy is limited to materials that exhibit good contrast between the core and shells, absorption and luminescence changes due to nanoparticle shells are often the subject of study, but are not useful as a reliable measurement technique, and light scattering or sedimentation based methods typically provide little or no information regarding particle composition.
Author Manuscript
Knowledge of shell thicknesses is important to adequately understand how the shell affects the properties of the nanoparticle. In nanoparticles intended for electronic and optical applications, shell thickness can be important when considering excited state and electron transport properties; in medical or biotechnological applications, an understanding of the effects of shell thickness on reactivity and interaction with the local environment is typically required. For nanoparticles bearing multiple shells, measurement of shell thicknesses is not a trivial issue and, even in supposedly single shell systems, the presence of capping ligands, adsorbed molecules, and contaminants may result in a second overlayer that will affect the measurement.
Cant et al.
Page 2
Author Manuscript
X-ray Photoelectron Spectroscopy (XPS) is a useful technique for characterisation of nanometre scale overlayers on materials20–22, as it can provide quantitative information on the amount of each element present in the surface region (approx. < 10 nm) of a sample, distinguish between the different chemical states of elements that are present, and is nondestructive to most samples. The primary issues with the use of XPS for analysis of nanoparticles are sample preparation and data interpretation. The issue of sample preparation is centred around the requirement of XPS to have a clean, dry sample, mounted on a substrate that does not produce spectra that could obfuscate the spectra produced by the sample. This can be difficult to achieve with nanoparticle samples, due to the prevalence of multi-component solutions used for nanoparticle synthesis and processing. In particular, a nanoparticle suspension must be free of non-volatile solutes as these may dry onto the particles and form a contaminant layer. Baer et al.23 describe in detail the issues faced in nanomaterial surface characterisation, and discuss the factors that must be considered when removing nanoparticles from solution. The effects of environment and sample preparation technique on the data obtained from surface characterisation techniques are also discussed in the ISO TC201 (surface chemical analysis) technical report ISO/TR 1418724.
Author Manuscript Author Manuscript
The issue of XPS data interpretation as applied to nanoparticles and nanoscale structures in general can be relatively complex, due to the topographic effects becoming more important as the sample dimensions approach that of the attenuation length of photoelectrons25–27. Previous work in this area has typically focussed on the simple case of core-shell particles, either by direct comparison to simulated intensities28 or by development of an empirical formulae that provides an approximation to simulated intensities27,29. Such approaches are useful, and have been shown to give consistent results30 However, an empirical method for dealing with the case of core-shell-shell systems has not yet been described - detailed simulation can deal with the effects of additional layers31 but requires specialized software or expertise. Therefore it is important to consider whether a simplified scheme available to any analyst can be found. In this work we demonstrate that it is possible, but it is not as straightforward as the core-shell problem.
Theory
Author Manuscript
The assumptions made for the calculations within this paper are as follows: it is assumed that particles are spherical, and that both shells consist of an even distribution of material across the surface of the particle; the photoelectron intensity arising from the particle is assumed to follow the ‘straight line’ approximation, whereby all photoelectrons are considered as having travelled along a straight line from within the particle, with the decrease in photoelectron intensity due to transport through the particle following a simple exponential decay with distance, characterised by the effective attenuation lengths (L) within each material. Elastic scattering of electrons is not considered in detail, rather it is assumed that the effect of elastic scattering can be compensated for or neglected32. It is assumed that the relative photoelectron intensities arising from a sample of nanoparticles can be considered as equivalent to that arising from a single nanoparticle, as justified by Werner et al.33. The attenuation lengths used within this paper are calculated using the method presented by Seah34, originally intended as an empirical means of obtaining effective attenuation lengths for use in determining the film thicknesses of planar overlayers on a flat
Surf Interface Anal. Author manuscript; available in PMC 2017 May 01.
Cant et al.
Page 3
Author Manuscript
substrate. These attenuation length values have been found to be useful in previous work on overlayer thicknesses in core-shell systems27 and are used here. Shard,27 in his equation (11), presents a quick, straightforward calculation of shell thickness that can be readily applied by non-specialists and general users, with an error that is typically less than the expected error in estimated attenuation lengths34. This is henceforth referred to as the TNP equation.
Terminology As with the previous work on core-shell systems27, formulas will be given in terms of a set of dimensionless parameters, using an extended but similar terminology for consistency. The experimental output is denoted by Ai,j and represents the ratio of normalised integrated intensities of a unique signal from material i to that from j, as shown in equation. 1
Author Manuscript
(1)
where Ii is the measured XPS intensity from material i, and is the measured or calculated intensity for a planar sample of pure material. The core, inner shell, and outer shell of a particle will be denoted by i and j values of 0, 1, and 2 respectively. In considering attenuation lengths, we will similarly use the terms Bi,j and Ci,j to refer to the following ratios of attenuation lengths Lx,y for photoelectrons arising from material y travelling through material x.
Author Manuscript
(2)
(3)
Author Manuscript
Thus Bi,j is the ratio of attenuation lengths in material i of electrons arising from material i to electrons arising from material j, and thus describes the relative penetration lengths of electrons arising from materials i and j. Similarly, Ci,j represents the ratio of the attenuation length of electrons travelling from material i through material i to the attenuation length of electrons travelling from material i through material j, and thus describes the relative opacity of materials i and j. A practical estimate of the value of Bi,j can be obtained from (Ei/Ej)0.872 where E is the electron kinetic energy34, thus we can also state that B2,0 = B2,1B1,0. A practical estimate of Ci,j can be obtained from (Zj/Zi)0.3 where Z is the number-averaged atomic number34, and thus C2,0 = C2,1C1,0. Finally, we make the assumption - shown in equation 4 - that the ratio of the attenuation lengths of electrons travelling through the material they originate from can be estimated from a combination of B and C.
Surf Interface Anal. Author manuscript; available in PMC 2017 May 01.
Cant et al.
Page 4
Author Manuscript
(4)
These assumptions effectively reduce the number of independent parameters that must be considered from the nine that result from electrons travelling from and through 3 different materials, to four: B2,1, B2,0, C2,1 and C2,0.
Modelling To make progress toward a method for calculating shell thickness in core-shell-shell nanoparticles, it is helpful to consider the two shells separately.
Author Manuscript
To calculate the thickness of the inner shell, we can consider the core and inner shell system as a basic core-shell nanoparticle and determine the change in A1,0 due to the outer shell as a function of the outer shell thickness, T2, and attenuation length ratios. We can then use an expression of this function to calculate the value of A1,0 for an equivalent particle with no outer shell, A*1,0. The TNP equation27 may then be applied to A*, using the core radius R, and the appropriate attenuation length ratios B1,0 and C1,0, to calculate the inner shell thickness. Likewise, to calculate the thickness of the outer shell, we can consider the nanoparticle as a core-shell particle consisting of the outer shell and a core formed by the combination of the core and inner shell. We thus require a method by which to obtain the effective A, B and C values from the ‘combined’ core for use in the TNP formula.
Author Manuscript
Calculation of the inner shell Simulations were carried out using the straight-line approximation to calculate the photoelectron intensity arising from each material. A 2-dimensional cross-section of half a nanoparticle is considered, as shown in Figure 2. The intensities from individual straight line columns of this cross section are then calculated, which represent hollow cylindrical cuts of the particle. The resulting intensities are weighted for the cross sectional area of the cylinder and summed, and the output given as ratios of these total intensities, which are equivalent to that of a complete nanoparticle. The script used to simulate core-shell-shell nanoparticles is included within the supporting information, alongside a more detailed explanation of the calculation.
Author Manuscript
Simulations of core-shell-shell nanoparticles and equivalent core-shell particles without the outer shell were conducted across a range of physically reasonable values for the various attenuation lengths and particle dimensions - specifically, R between 0.1 and 1000 (in units of L2,2), T1 and T2 between 0.1 and 10 (in units of L2,2), B values between 1/2 and 2, and C values between 1/3 and 3. Intensity ratios produced by these simulations were compared to derive the terms of the A*1,0 adjustment function. As particle dimensions will often be considered in terms of a specific attenuation length, this will be reported alongside the dimensions where given, e.g. R(L2,2).
Surf Interface Anal. Author manuscript; available in PMC 2017 May 01.
Cant et al.
Page 5
Author Manuscript
Initially, we can consider the simple situation of a particle consisting of identical materials in the core and both shells (i.e. attenuation length ratios B and C equal to one for all materials) and consider solely the effect of the increased attenuation from the presence of the outer shell. Figure 3 shows the ratio of A*1,0 to A1,0 plotted against the thickness of the outer shell, T2. The curve shown in Figure 3 may be readily approximated by a function of the natural logarithm of T2. More accurate fits may be obtained by increasingly complex functions, however when extended to consider systems in which attenuation lengths may vary between materials, this results in an impractical increase in the complexity of the function required.
Author Manuscript
Next, the change to this correction term A*/A due to variation in attenuation lengths may be considered. The terms B2,1 and B2,0, representing the relative attenuation lengths of electrons from the inner shell and core respectively, exhibit the greatest effect on the relative signal from the inner shell and the core. The term A*/A, plotted against T2 for varying values of B2,1 and B2,0 with other attenuation parameters held constant is shown in Figure 4 for a range of values of R.
Author Manuscript
The fit lines shown in Figure 4 are calculated from a combination of the logarithmic fit mentioned previously, with the apparent exponential decay of signal with T2 due to the two attenuation parameters, B2,1 and B2,0.It is clear from simulation that the change in intensity due to the second shell increases exponentially with the product of T2 and B2,1, and decreases exponentially with the product of T2 and B2,0. This seems intuitively reasonable, as it would be expected that the ratio of signal from the inner shell to the core would increase with an increase in the penetration length of electrons from the inner shell in the outer shell (B2,1) and would decrease with an increase in the equivalent parameter for electrons from the core (B2,0). A slight change in intensity ratio was observed with changing particle radius, but this change is insignificant and may be neglected with little effect on calculated thicknesses. The most accurate fit to simulation data was obtained by replacing the constant n used previously in the logarithmic term with a function of the attenuation parameters – the form of this term is detailed later. The resulting form of this fit is shown in equation 5. (5)
Author Manuscript
Finally, we can consider the effect of the attenuation parameters as a function of the relative opacity of the inner shell and core materials to the outer shell, C2,1 and C2,0 respectively. From simulations, it is apparent that the term denoting the opacity of the core, C2,0, has little to no effect on the ratio of A*1,0 to A1,0, and can be neglected. The opacity of the inner shell, denoted by C2,1, while not as significant as the penetration terms B2,1 and B2,0, has a noticeable effect on the observed intensities. This effect, however, is non-trivial to fit, and appears to be significantly interdependent with the other attenuation terms. It cannot be accounted for with as great a degree of accuracy as the penetration terms while maintaining a practical level of simplicity in the resulting equation. A simple approximation for the effect of C2,1 is shown in Figure 5, and while imperfect, it serves to minimise the error acceptably Surf Interface Anal. Author manuscript; available in PMC 2017 May 01.
Cant et al.
Page 6
Author Manuscript
(to within the ~10% error in the estimation of attenuation lengths) across the full range of reasonable physical parameters. When combining the terms to account for all relevant attenuation lengths, and fitting across a wide range of physically sensible values for each, the final correction factor for the intensity ratio A1,0 is shown in equation 6. (6)
where n is a function of the parameters B2,1, B2,0 and C2,1 given in equation 7. (7)
Author Manuscript
These values may then be inserted into the TNP equation, remembering that the units of length are L1,1. (8)
Calculation of the outer shell thickness When calculating the outer shell thickness, we consider the nanoparticle as a core-shell particle in which the ‘core’ is formed from both the core and inner shell, while the ‘shell’ consists of the outer shell alone. In this case, we must first determine the effective core-shell intensity ratio for such a system, Aeff, which is given by equation 9.
Author Manuscript
(9)
We must also determine effective values for the attenuation parameters used in the TNP equation. We may assume that the effective values for these terms, Beff and Ceff, are a combination of the equivalent terms for each of the inner materials, B2,1 and B2,0, or C2,1 and C2,0. The simplest approximation of which would be a linear combination of the two terms, adjusted by some weighting factor w, as shown in equations 10 and 11. (10)
Author Manuscript
(11)
While a form for w cannot be determined directly from fitting, some reasonable assumptions can be made. For example, we can assume that the weighting function is related to the relative amounts of material present in the core and the shell. We would expect that w holds a maximum value of 1 at the point where the majority of the material present forms a part of the first shell, i.e. in the limit of very large A1,0, as in this case the effective value of B would Surf Interface Anal. Author manuscript; available in PMC 2017 May 01.
Cant et al.
Page 7
Author Manuscript
be expected to be equivalent to B2,1. Thus, we may consider a w of the form shown in equation 12 (12)
where p is a function of the two other relevant parameters, the attenuation ratios between the inner shell and the core, B1,0 and C1,0. After comparing simulated results with calculated ones for a selection of w functions, equation 13 was found to give useful results for T2 across the full parameter space. (13)
Author Manuscript
The values Aeff, Beff, and Ceff can thus be used in the TNP formula for calculation of shell thicknesses for a core-shell nanoparticle, using the value of (R+T1) as the ‘effective’ RNP, to calculate the value of the outer-shell thickness T2 with the units of length L2,2. (14)
Applicability
Author Manuscript
With the methods detailed above it is possible to calculate both T1 and T2 for core-shellshell nanoparticles, without the use of simulation or complex graphical methods. Initially it may seem that a large amount of fore-knowledge is required for either of these calculations – in order to calculate T2, you require T1, and vice versa. However in reality, only the value of R is required. Values for T1 and T2 may be estimated initially, and by iteration between the inner shell and outer shell methods we improve the accuracy of these estimates to within the formula’s average innate accuracy of
