A Technique for Calculation of Shell Thicknesses for Core-Shell-Shell Nanoparticles from XPS Data

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.

Introduction

Nanoparticles are a topic of increasing interest for applications across many disparate fields, including medicine^1–5, materials^6–9, and opto-electronics^10–15. 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 methods^16–19. 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.

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.

X-ray Photoelectron Spectroscopy (XPS) is a useful technique for characterisation of nanometre scale overlayers on materials^20–22, 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 non-destructive 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.

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 photoelectrons^25–27. 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 intensities^27,29. 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

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 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 T_NP 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 A_i,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

$$A_{i,j}=\frac{I_i{I_j}^{\infty }}{I_j{I_i}^{\infty }}$$ (1)

where I_i is the measured XPS intensity from material i, and $I_i^{\infty }$ 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 B_i,j and C_i,j to refer to the following ratios of attenuation lengths L_x,y for photoelectrons arising from material y travelling through material x.

$$B_{i,j}=\frac{L_{i,i}}{L_{i,j}}$$ (2)

$$C_{i,j}=\frac{L_{i,i}}{L_{j,i}}$$ (3)

Thus B_i,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, C_i,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 B_i,j can be obtained from (E_i/E_j)^0.872 where E is the electron kinetic energy³⁴, thus we can also state that B_2,0 = B_2,1B_1,0. A practical estimate of C_i,j can be obtained from (Z_j/Z_i)^0.3 where Z is the number-averaged atomic number³⁴, and thus C_2,0 = C_2,1C_1,0. 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.

$$B_{i,j}{C}_{i,j}=\frac{L_{i,i}}{L_{j,j}}$$ (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: B_2,1, B_2,0, C_2,1 and C_2,0.

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.

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 A_1,0 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 A_1,0 for an equivalent particle with no outer shell, A*_1,0. The T_NP equation²⁷ may then be applied to A*, using the core radius R, and the appropriate attenuation length ratios B_1,0 and C_1,0, 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 T_NP formula.

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.

Figure 2.

Schematic describing the use of the straight-line approximation for simulation of a nanoparticle.

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 L_2,2), T₁ and T₂ between 0.1 and 10 (in units of L_2,2), 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(L_2,2).

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 A_1,0 plotted against the thickness of the outer shell, T₂.

Figure 3.

Ratio of A*_1,0 to A_1,0 for a particle in which all attenuation lengths are equal, with a radius of 10(L_2,2) and an inner shell thickness of 2(L_2,2). Simulated data are shown alongside a fit. The term n is an arbitrary constant that results in the optimal fit for the function.

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.

Next, the change to this correction term A*/A due to variation in attenuation lengths may be considered. The terms B_2,1 and B_2,0, 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 B_2,1 and B_2,0 with other attenuation parameters held constant is shown in Figure 4 for a range of values of R.

Figure 4.

Data points are simulated values for the ratio of A^*_1,0 to A_1,0 for varying values of B_2,1 (b, d and f) and B_2,0 (a, c and e) for particles of radii R(L_2,2) = 5, 50, and 500. Lines represent a fit to the simulated data of the same colour.

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, B_2,1 and B_2,0.It is clear from simulation that the change in intensity due to the second shell increases exponentially with the product of T₂ and B_2,1, and decreases exponentially with the product of T₂ and B_2,0. 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 (B_2,1) and would decrease with an increase in the equivalent parameter for electrons from the core (B_2,0). 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.

$$\frac{{A^{*}}_{1,0}}{A_{1,0}}=\{1+n[\mathit{\text{ln}}(T_2+1)]\}{e}^{[(B_{2,1}-B_{2,0})T_2]}$$ (5)

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, C_2,1 and C_2,0 respectively. From simulations, it is apparent that the term denoting the opacity of the core, C_2,0, has little to no effect on the ratio of A*_1,0 to A_1,0, and can be neglected. The opacity of the inner shell, denoted by C_2,1, while not as significant as the penetration terms B_2,1 and B_2,0, 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 C_2,1 is shown in Figure 5, and while imperfect, it serves to minimise the error acceptably (to within the ~10% error in the estimation of attenuation lengths) across the full range of reasonable physical parameters.

Figure 5.

Intensity ratios plotted against outer shell thickness for multiple values of C_2,1. Each of the three graphs shows the effect of C_2,1 for a different set of values for the penetration terms B_2,1 and B_2,0.

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 A_1,0 is shown in equation 6.

$$A_{1,0}^{*}=A_{1,0}\{1+n[\mathit{\text{ln}}(T_2+1)]\}{e}^{[(B_{2,1}-B_{2,0})T_2]}$$ (6)

where n is a function of the parameters B_2,1, B_2,0 and C_2,1 given in equation 7.

$$n=\frac{1}{20}[(2B_{2,1}-B_{2,0})(4.5+C_{2,1})+2(B_{2,0}-1)C_{2,1}+4.6]$$ (7)

These values may then be inserted into the T_NP equation, remembering that the units of length are L_1,1.

$$T_1(L_{1,1})=T_{NP}[R(L_{1,1}),A_{1,0}^{*},B_{1,0},C_{1,0}]$$ (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, A_eff, which is given by equation 9.

$$A_{\text{eff}}=\frac{\frac{I_2}{I_2^{\infty }}}{\frac{I_1}{I_1^{\infty }}+\frac{I_0}{I_0^{\infty }}}=\frac{A_{2,1}{A}_{2,0}}{A_{2,1}+A_{2,0}}$$ (9)

We must also determine effective values for the attenuation parameters used in the T_NP equation. We may assume that the effective values for these terms, B_eff and C_eff, are a combination of the equivalent terms for each of the inner materials, B_2,1 and B_2,0, or C_2,1 and C_2,0. 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.

$$B_{\text{eff}}=w{B}_{2,1}+(1-w)B_{2,0}$$ (10)

$$C_{\text{eff}}=w{C}_{2,1}+(1-w)C_{2,0}$$ (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 A_1,0, as in this case the effective value of B would be expected to be equivalent to B_2,1. Thus, we may consider a w of the form shown in equation 12

$$w=\frac{A_{1,0}}{A_{1,0}+p}$$ (12)

where p is a function of the two other relevant parameters, the attenuation ratios between the inner shell and the core, B_1,0 and C_1,0. 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.

$$w=\frac{A_{1,0}}{A_{1,0}+0.8+0.5B_{1,0}^4}$$ (13)

The values A_eff, B_eff, and C_eff can thus be used in the T_NP formula for calculation of shell thicknesses for a core-shell nanoparticle, using the value of (R+T₁) as the ‘effective’ R_NP, to calculate the value of the outer-shell thickness T₂ with the units of length L_2,2.

$$T_1(L_{2,2})=T_{\mathrm{N}\mathrm{P}}[R_{\mathrm{N}\mathrm{P}}(L_{2,2}),A_{\text{eff}},B_{\text{eff}},C_{\text{eff}}]$$ (14)

Applicability

With the methods detailed above it is possible to calculate both T₁ and T₂ for core-shell-shell 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 <10%. Initial estimates that are an order of magnitude from the true value will typically converge to a final value within two or three iterations.

It is important to note the dimensions one is operating in when using this technique to calculate nanoparticle shell thicknesses – typically, when using the original core-shell calculation given by Shard²⁷, particle dimensions (R_NP, T₁, T₂) must be given in units of the attenuation length ratio of electrons travelling from and through whichever shell is the subject of calculation. In practice, this means that T₂ values, where used, are always in terms of the attenuation length of outer shell electrons passing through the outer shell (L_2,2). R values are in terms of the equivalent attenuation length for the inner shell (L_1,1) when calculating the inner-shell thickness, and in terms of the outer-shell attenuation length (L_2,2) when calculating the outer-shell thickness. Calculated values of T₁ are in units of the inner-shell attenuation length (L_1,1) and must be converted to units of the outer-shell attenuation length (L_2,2) when used to calculate T₂. Conversion from units of L_2,2 to L_1,1 can be done by multiplying by B_2,1C_2,1 as shown in equation 4, and naturally the opposite conversion can be achieved by dividing by the same factor.

A simple step by step approach for calculating both T₁ and T₂ using this method would be as follows, with the formulas used to calculate T_NP provided below for completeness:

$$T_{R\to \infty }=\frac{0.74A^{3.6}\text{ln}(A)B^{-0.9}+4.2A{B}^{-0.41}}{A^{3.6}+8.9}$$ (15)

$$T_0=R_{NP}[{(ABC+1)}^{\frac{1}{3}}-1]$$ (16)

$$\alpha =\frac{1.8}{A^{0.1}{B}^{0.5}{C}^{0.4}}$$ (17)

$$\beta =\frac{0.13{\alpha }^{2.5}}{R_{NP}^{1.5}}$$ (18)

$$T_{R~1}=\frac{T_{R\to \infty }R}{R_{NP}+\alpha }$$ (19)

$$T_{NP}=\frac{T_{R\to \infty }+\beta T_0}{1+\beta }$$ (20)

1. Preparation:

   1. Estimate the values of B_2,1, B_2,0, C_2,1, C_2,0, and one of either L_1,1 or L_2,2 using the equations given by Seah³⁴ or another appropriate method.
   2. Determine the conversion factor B_2,1C_2,1 as described by equation (4) – multiplying by this factor will convert from units of L_2,2 to L_1,1, and vice versa for division.
   3. Measure I₀, I₁, and I₂ from XPS data.
   4. Measure or estimate I₀^∞, I₁^∞, and I₂^∞ for pure materials.
   5. Calculate A_1,0, A_2,1, and A_2,0.
2. To calculate T₁:

   1. Estimate a value of T₂, or use a value determined from (C) below, and convert into units of L_2,2.
   2. Using equation 7 and the estimated value of T₂, determine A*_1,0.
   3. Determine R in units of L_1,1.
   4. Apply the T_NP formula with B = (B_2,0/ B_2,1), C = (C_2,0/C_2,1), R(L_1,1) and A = A*_1,0 to obtain T₁(L_1,1).
3. To calculate T₂ :

   1. Determine the values of A_eff, B_eff and C_eff using equations 9–13.
   2. Estimate a value of T₁, or use a value determined from (B) above, and convert into units of L_2,2.
   3. Determine R in terms of L_2,2.
   4. Apply the T_NP formula using A = A_eff, B = B_eff, C = C_eff, and R_NP = the sum of R(L_2,2) and T₁(L_2,2), to obtain T₂(L_2,2).

These two procedures may then be alternated, replacing the estimated values for T₁ and T₂ with the calculated ones, and iterated until the deviation in result between iterations is minimised. Typically T₁ and T₂ converge after two or three iterations.

Deviation from simulated data

The overall accuracy of this method was investigated by comparison to large sets of data simulated as described previously, with randomised parameters. Nanoparticles were simulated with parameters within reasonable physical bounds; specifically, values were chosen for B between 1/2 and 2, C values between 1/3 and 3, R values between 0.1 and 1000, and T₁ andT₂ values between 0.1 and 10. Values were selected at random and independent from one another on a linear scale for attenuation parameters B and C, and on a logarithmic scale for physical dimensions R, T₁ and T₂. Simulated particles resulting in any of the three intensity ratios A_1,0, A_2,0 and A_2,1 being outside the typical detection limits of XPS (0.001 < A < 1000) were discarded. Calculations were performed first to obtain T₁, using an ‘estimated’ T₂ a tenth the size of the true value for half of the simulated particles, and using a value ten times larger for the other half. The values obtained after 5 iterations of calculation were taken as the final calculated values.

Figure 6 shows the distribution of calculated thicknesses T_CSS-1 and T_CSS-2 for a set of 5000 randomly generated particles, plotted against the thicknesses used in the simulations (T₁ and T₂).

Figure 6.

Graphs of calculated T_CSS-1 (a) and T_CSS-2 (b) values for 5000 randomly generated particles plotted against the values used in numerical simulations T₁ and T₂, shown on a logarithmic scale. The red dashed lines show the boundaries for 20% deviation in either direction from the simulated value.

It can be seen from Figure 6 that the calculated value of T₂ (T_CSS-2) is typically more accurate than the calculated value of T₁ (T_CSS-1). Both graphs however show a majority that the calculated values are within 10% of the values used in the simulations, and thus less than the ~10% error in the estimation of attenuation lengths³⁴. From these sets of data, the mean value of the ratio T₁/T_CSS-1 was 0.97 with a standard deviation of 0.091, and the mean value of the ratio T₂/T_CSS-2 was 0.99 with a standard deviation of 0.076. Figures S1 and S2 in the supporting information show the mean and standard deviation of these ratios plotted against each variable. From the deviations observed, it is clear that there are some regions where the accuracy of the calculated value worsens, particular for large values of B_2,1 and, for T₁, in a small region around T₂ ~2.5. However across the whole range of parameters the deviation of the calculated value is rarely greater than 10% of the simulated value, and thus is acceptable when compared to other sources of error.

Application to experimental data

XPS data were obtained for two samples of gold nanoparticles (AuNPs) functionalised with a self-assembled monolayer (SAM) of (11-Mercaptoundecyl)-tetra(ethylene glycol) (OEG). AuNPs were synthesized using the citrate reduction method and then functionalized with 10 µM of the OEG thiol under stirring for 36 hours³⁵. ImageJ analysis of transmission electron microscopy images of the two synthesized gold nanoparticles samples showed they had average diameters of 14 nm and 40 nm. The OEG functionalized AuNPs were purified by at least four cycles of centrifugation and redispersal in ultrapure water. Then a 10 µL aliquot from the purified OEG/AuNP solution was deposited onto a solvent cleaned silicon wafer and placed in a vacuum desiccator to dry. The deposition and vacuum desiccation was repeated multiple times to form a sufficiently thick and confluent layer of AuNPs for XPS analysis. A complete layer was desirable to avoid any background signal from the underlying silicon substrate. The XPS data were acquired with a Surface Science Instrument S-probe using a monochromatic Al Kα X-ray source. The photoelectron takeoff (the angle between the surface normal of the silicon substrate and the axis of the analyzer lens) was 0°. The solid acceptance angle of the analyzer lens was 30°. Data used for determination of elemental compositions was acquired at an analyzer pass energy of 150 eV. High-resolution C1s spectra were acquired at an analyzer pass energy of 50 eV. The XPS elemental compositions shown in Table 1 were determined from analyzing three spots on six replicate samples for each type of OEG/AuNP. Note that these compositions are calculated by assuming that the sample is homogenous, as is common practice, and do not represent the actual nanoparticle compositions. Representative XPS survey spectra for the two sizes of OEG/AuNPs are shown in the supplemental information. Representative high-resolution XPS C1s spectra for the two sizes of OEG/AuNPs along with the peak fits used to determine the amount of ether and hydrocarbon species are also shown in supplemental information.

Table 1.

XPS elemental compositions of 14nm and 40nm AuNPs functionalized with OEG SAMs. Total number of analyses per sample type was 18 (3 spots on 6 replicates).

Sample	XPS Atomic Percent
	Au	C	O	S
OEG/14 nm AuNPs	22.8±1.2	56.5±1.6	18.7±1.1	2.0±0.3
OEG/40 nm AuNPs	22.0±0.7	56.0±1.3	20.3±1.2	1.7±0.4

The structure of the OEG thiol is shown in Figure 7.

Figure 7.

Chemical structure of the (11-Mercaptoundecyl)-tetra(ethylene glycol).

As indicated in Figure 8, the idealised OEG SAM overlayer can be considered as two distinct shells: one containing 9 carbon atoms bound to 5 oxygen atoms, and an inner shell containing primarily hydrocarbon species, with a single sulphur atom at the surface. By considering the signals from the carbon and oxygen in the outer portion as the ‘outer shell’, and the hydrocarbon and sulphur signals as the ‘inner shell’, this can be modelled as a core-shell-shell particle, and the thicknesses of these layers calculated accordingly. This should provide a more accurate calculation of the size of the organic overlayer than simply using the core-shell model, which would treat the entire organic portion of the particle as having a uniform composition.

Figure 8.

Schematic of the functionalised gold nanoparticles depicted as a core-shell-shell system. The inner shell consists of the portion of the molecule consisting of hydrocarbon species and the sulphur, and the second shell consists of carbon bound to oxygen.

To calculate the shell thicknesses in a system such as this one, where there are multiple different atoms in each layer, we can estimate attenuation lengths using the formula from Seah³⁴ for electrons arising from each element individually and using a number-averaged Z value for the material the electrons are travelling through. In the case of organics however, a Z value of 4 provides a more reasonable estimate³⁴. Correction for the pure material intensity ratios, $I_i^{\infty }/I_j^{\infty }$ was conducted using the value reported for organic materials on gold in the supporting information of Belsey et al.³⁶. Using this method an overlayer thickness may be calculated for each element in each material; the deviation between these calculated thicknesses may then be minimised by varying the composition fractions of each element in each layer, with the assumption that the fractions of all elements present in the layer must sum to 1. In this manner an estimate of both composition and shell thickness can be achieved. Calculated values for the shell thicknesses using the method described in this paper, are given in Tables 2 and 3 alongside total thicknesses calculated using the T_NP formula.

Table 2.

Table of intensities, attenuation lengths and calculated thicknesses for the OEG SAM on 14 nm AuNP.

Region	Element	Intensity	Fraction of layer	Fraction of organic layer (for single shell formula)	L (in core)	L (in shell 1)	L (in shell 2)	Calculated thickness (nm)
Core	Au	440	1	-	1.32	3.79	3.79
Shell 1	C	23.3	0.91	0.73	1.17	3.32	3.32	0.8
Shell 1	S	3.61	0.086	0.02	1.27	3.61	3.61
Shell 2	C	33.2	0.61	-	1.17	3.32	3.32	0.9
Shell 2	O	45.9	0.39	0.24	0.99	2.74	2.74
		Thickness calculated from TNP single shell formula (nm) :				1.8	Total Thickness (nm)	1.7

Table 3.

Table of intensities, attenuation lengths and calculated thicknesses for the OEG SAM on 40 nm AuNPs.

Region	Element	Intensity	Fraction of layer	Fraction of organic layer (for single shell formula)	L (in core)	L (in shell 1)	L (in shell 2)	Calculated thickness (nm)
Core	Au	440	1	-	1.32	3.79	3.79
Shell 1	C	23.3	0.92	0.75	1.17	3.32	3.32	1.0
Shell 1	S	3.61	0.078	0.03	1.27	3.61	3.61
Shell 2	C	33.2	0.60	-	1.17	3.32	3.32	1.1
Shell 2	O	45.9	0.40	0.23	0.99	2.74	2.74
		Thickness calculated from TNP single shell formula (nm) :				2.2	Total Thickness (nm)	2.1

For both samples, the final values from the core-shell-shell formula for atomic fractions in the organic layers match well with the expected values given the structure of the molecule - for the inner ‘shell’, containing 1 sulphur atom and 10 carbon atoms, we would expect atomic fractions 0.91 carbon and 0.09 sulphur, for the outer ‘shell’, containing 5 oxygen atoms and 9 carbon atoms, we would expect atomic fractions 0.64 carbon and 0.36 oxygen. For comparison, if considering the organic layer as a single shell, the expected atomic fractions would be 0.76 carbon, 0.04 sulphur, and 0.20 oxygen – the compositions determined using the single-shell model thus have a greater relative error than those from the core-shell-shell model, suggesting that the assumption of a homogeneous shell increases the uncertainty in the composition. Calculated thicknesses match well with the total thickness calculated using the core-shell model. Both models determine a higher thickness for the larger nanoparticles, which should not be the case if the same monolayer is formed on both sets of nanoparticles. One possible explanation for the observed monolayer thickness difference between the two sets of samples is the degree of ordering and packing in the ethylene glycol portion of the monolayer. Typically for flat gold surfaces, the alkyl chain portion of OEG SAMs is ordered while the OEG chain is disordered³⁷. However, by varying the composition of the solution used for preparing the SAMs the packing density and ordering of the OEG chain can be controlled³⁸. A more densely packed OEG SAM is expected to have more ordered and thicker OEG chains than a less densely packed OEG SAM. Low-energy ion scattering (LEIS) can also be used to determine the thickness of SAM overlayers on AuNPs³⁹. Using an ION-TOF Qtac100 with 5 nA, 3KeV⁴He⁺ primary ions scanned over a 2 × 2 mm² area, the OEG SAMs on 40nm AuNPs were found to be 0.6nm thicker than on 14nm AuNPs³⁵, similar to the thickness differences shown in Tables 2 and 3.

Conclusions

This paper describes a technique for the calculation of shell thicknesses is core-shell-shell nanoparticles, as an extension to the method shown in²⁷ for calculation of core-shell nanoparticle shell thicknesses. Shell thicknesses can be calculated using this technique without the use of numerical simulation, or the need for any specialist expertise, and iteratively calculated values using this method converge very rapidly to provide thicknesses with a deviation that is typically less than the expected error in attenuation lengths. The method is straightforward to use, however for work where accuracy is extremely important, where the composition of the particle is uneven, or for systems with more than 2 shells, numerical methods are still preferable.

Figure 1.

Schematic of separate models for calculating inner and outer shell thicknesses of a core-shell-shell nanoparticle.