Stochastic models of dense or hollow nanoparticles and their scattering properties

A versatile model of disordered nanoparticles is proposed for the purpose of analysing small-angle scattering data. The approach generalizes the classical clipped Gaussian field models of random structures by allowing the clipping function to be space dependent.

Abstract

A family of stochastic models of disordered particles is proposed, obtained by clipping a Gaussian random field with a function that is space dependent. Depending on the shape of the clipping function, dense or hollow particles can be modelled. General expressions are derived for the form factor of the particles, for their average volume and surface area, and for their density and surface-area distributions against the distance to the particle centre. A general approximation for the form factor is also introduced, based on the density and surface-area distributions, which coincides with the Guinier and Porod expressions in the limits of low and high scattering vector magnitude q. The models are illustrated with the fitting of small-angle X-ray scattering (SAXS) data measured on Pt/Ni hollow nanoparticles. The SAXS analysis and modelling notably capture the collapse of the particles’ porosity after being used as oxygen-reduction catalysts.

1. Introduction  

Small-angle scattering (SAS) of X-rays (SAXS) or neutrons (SANS) is a very versatile experimental method that enables one to characterize the structure of nanomaterials over length scales that range from 1 to 100 nm, in a wide variety of chemical and physical environments (Glatter & Kratky, 1982 ▸; Stribeck, 2010 ▸). Moreover, this can often be done in a time-resolved way, both for synchrotron SAXS (Narayanan et al., 2001 ▸; Grillo, 2009 ▸) and for SANS (Pipich et al., 2008 ▸; Carl et al., 2020 ▸). However, a significant difficulty with SAS is the data analysis. There are generally a variety of structures that are compatible with a given scattering pattern (Gommes et al., 2012 ▸), so that converting scattering data into meaningful structural information requires models (Pedersen, 1997 ▸).

When developing models to analyse SAS data from disordered materials, one has always to find a trade-off between conceptual simplicity and realism. The former is needed for the robust analysis of scattering patterns with as few parameters as possible. The latter ensures that as many structural characteristics as possible are accounted for in the material’s description. This is particularly challenging in the case of disordered materials, the structure of which is partially random. For these types of materials, stochastic models offer a practical way through the almost antagonistic requirements of geometric realism and mathematical simplicity (Gommes, 2018 ▸).

In this paper we introduce a family of stochastic models of either dense or hollow particles, obtained by clipping a Gaussian random field with a function that is space dependent. The well known Gaussian random-field model of biphasic materials (Quiblier, 1984 ▸; Berk, 1987 ▸; Teubner, 1991 ▸; Chen et al., 1996 ▸; Levitz, 1998 ▸) is obtained as a particular case of the present general family of models for constant clipping functions. We derive general analytical expressions for the volume and surface area of the particles, for their density and area profiles, and for their form factors. The derivations are limited here to a material with uniform scattering contrast, but the modelling procedure can be generalized to multiple phases as well. We illustrate the practical use of these models for SAXS data analysis with the case of hollow PtNi nanoparticles supported on carbon, used as oxygen-reduction catalysts.

2. Experimental  

The theoretical developments of the present paper were motivated by the SAXS analysis of hollow PtNi nanoparticles supported on Vulcan XC72 porous carbon. Details of the preparation of the PtNi nanoparticles and the membrane electrode assembly (MEA) sample can be found in the work of Chattot et al. (2018 ▸) and Chattot et al. (2020 ▸), respectively.

A proton-exchange membrane fuel-cell accelerated stress test was conducted in a dedicated single 5 cm² cell, controlled by a test station (FCS-4M-100W, LeanCat Fuel Cell) and a Biologic SP-400 potentiostat. First, the cell was operated at 353 K under H₂/air at 150 s.c.c.m./350 s.c.c.m. flow rate (atmospheric pressure) for the anode/cathode and the humidification rate was 100%. The MEA was quickly conditioned for about 1 h by maintaining a cell voltage of 0.6 V. Then, the gas at the cathode was switched to N₂, and 5000 square-wave potential cycles between 0.6 and 1.1 V (3 s–3 s) were applied. Since the anode consisted of Pt nanoparticles in equilibrium under H₂, the cathode cell potential versus the reversible hydrogen electrode can be considered equal to the overall cell voltage. We hereafter refer to the samples before and after the accelerated stress test as the fresh and spent catalysts, respectively.

Fig. 1 ▸ shows transmission electron microscopy (TEM) images of the catalysts, before and after the stress test. The fresh catalyst contains a large number of relatively well defined hollow nanoparticles, as shown in Fig. 1 ▸(a2). The material is, however, very heterogeneous, as these well defined nanostructures coexist with much smaller compact objects [Fig. 1 ▸(a1)] as well as with larger and very distorted hollow structures [Fig. 1 ▸(a3)]. The nanoparticles in the spent catalyst [Fig. 1 ▸(b)] are also extremely distorted, but they are mostly dense and slightly larger.

Figure 1.

Transmission electron micrographs of the hollow PtNi nanoparticles, in (a) the fresh and (b) the spent catalyst. The nanoparticles are the smaller black objects and the larger grey objects belong to the carbon support. The insets (a1) to (a3) are three magnified views of nanoparticle types coexisting in the fresh catalyst; the scale bars in the insets are 40 nm long.

In addition to TEM, the collapse of the nanostructures during the stress test was confirmed electrochemically by CO stripping, which provides an independent measurement of the surface area of metal nanoparticles (Chattot et al., 2020 ▸). The area thus obtained, combined with the known metal loading of 20.4 wt%, yields a specific surface area a = 45 ± 7 m² per g_Pt in the fresh catalyst. The metal surface area per unit mass of the entire material (carbon and metal) decreases by 54% during the stress test. In the meantime, the specific metal loading, estimated using scanning transmission electron microscopy/X-ray energy-dispersive spectroscopy, increases to about 30 wt% as a consequence of support oxidation. The specific surface area of the metal particles is therefore difficult to estimate accurately in the spent catalyst. As an order of magnitude, however, half of the particle surface is lost during the stress test.

SAXS signals from the various sample powders – contained either in a Kapton capillary (fresh catalyst) or on Kapton tape (spent catalyst) – illuminated with a 70 keV X-ray beam were collected on beamline ID31 at the ESRF with a Dectris Pilatus CdTe 2M detector positioned 6.5 m behind the sample. The signal was protected from air scattering by a flight tube under mild vacuum. The energy, detector distance and tilts were calibrated using a standard silver behenate powder and the 2D scattering patterns were reduced to 1D curves using the pyFAI software package (Ashiotis et al., 2015 ▸). The scattering by the empty carbon support was measured on a slightly different setup (Gommes et al., 2019 ▸). In all cases, the scattering data were corrected for the contribution of the empty sample holder. The SAXS patterns of the porous support and of the fresh and spent catalysts are shown in Fig. 2 ▸.

Figure 2.

Small-angle X-ray scattering patterns of the fresh and spent catalysts, and of the unloaded porous support. The solid black line is the model used to extrapolate the support contribution over the same angular range as that of the catalysts.

3. The particle model  

3.1. General procedure  

We develop a general model of particles aimed at capturing the various types of disordered structures present in the fresh and spent catalysts (see Fig. 1 ▸). The central mathematical concept in the model is that of a Gaussian random field (GRF) (Quiblier, 1984 ▸; Berk, 1987 ▸), which assigns to any point of space x a stochastically defined value Y(x). An isotropic GRF can be thought of as the following sum (Berk, 1991 ▸; Levitz, 1998 ▸):

where q _n is a vector with random orientation and modulus q = |q|, drawn from a prescribed isotropic distribution with probability density f _Y(q)4πq ²dq, and φ_n is a random phase uniformly distributed in [0, 2π). In equation (1), the sum is over N contributions; in the limit of N → ∞ the values of Y(x) are Gaussian distributed with a mean of zero, and the factor (2/N)^1/2 ensures that the variance is equal to one. The GRF is therefore comprehensively characterized by the function f _Y(q), which we refer to as its power spectral density.

A characteristic of a GRF that is mathematically equivalent to f _Y(q) is its correlation function, defined as g _Y(r) = 〈Y(x)Y(x + r)〉, where the angle brackets stand for the ensemble average, i.e. the average evaluated over different realizations. The correlation function is related to the power spectral density f _Y(q) via the following Fourier transform (Berk, 1991 ▸):

For reasons that will be clear shortly, we consider only Gaussian fields with a correlation function that is quadratic for small values of r, which enables us to define the characteristic length l _Y through the relation

close to r = 0. Because the functions g _Y(r) and f _Y(q) are Fourier transforms of each other, the length l _Y can also be obtained from the power spectral density. The relation is

Qualitatively, the length l _Y can be thought of as the distance over which the values of Y(x) change significantly.

Examples of Gaussian fields are provided in Fig. 3 ▸. Those fields are obtained with a correlation function of the type

with l _Y = 15 nm [Fig. 3 ▸(a)] and l _Y = 5 nm [Fig. 3 ▸(b)]. The specific function in equation (5) is a particular case of a more general function used in earlier work (Gommes & Roberts, 2008 ▸, 2018 ▸); it behaves as an exponential over large distances but it is also quadratic at the origin, as imposed by equation (3). The corresponding power spectral density is

as obtained from a Fourier transformation of g _Y(r).

Figure 3.

Modelling of disordered dense nanoparticles, with Gaussian random fields Y(x) having characteristic lengths (a) l _Y = 15 nm and (b) 5 nm, clipped with two different radial functions α(r) [panels (c) and (d)]. The resulting particles are shown in (a/c), (a/d), (b/c) and (b/d), and in each case four realizations are shown to highlight the variability of the particles within one given model. The coloured areas in panels (c) and (d) highlight the regions corresponding to the solid.

A classical procedure for modelling disordered co-continuous structures – such as porous materials, emulsions, phase-separated copolymers etc. – consists of clipping a Gaussian field (Quiblier, 1984 ▸; Berk, 1987 ▸; Teubner, 1991 ▸). With those models, a given phase of a material is modelled as the points of space where a GRF takes values larger than a user-defined threshold. The particle models we discuss here are based on a generalization of this procedure, by which the classical constant threshold is replaced by a space-dependent function α(x) (Gommes & Pirard, 2009 ▸). In mathematical terms, the indicator function of the particles – taking the value 1 if point x is in the particle and 0 otherwise – is defined as

where H(x) is Heaviside’s step function (equal to one for x > 0 and zero otherwise), Y(x) is a Gaussian random field and α(x) is any user-specified space-dependent function.

The procedure is illustrated in Fig. 3 ▸ with a Gaussian field having a power spectral density of the type in equation (6) and a linear clipping function of the type

where r is the distance from the origin, and l _α and α₀ are two parameters of the model. As shown in Fig. 3 ▸, a variety of morphologies can be modelled with this procedure, depending on the values of the parameters.

In the limit of infinitely large values of l _α, the clipping function reduces to a constant α₀ and one recovers the classical clipped Gaussian field model (Levitz, 1998 ▸; Gommes, 2018 ▸). It is also interesting to note that the opposite limit, i.e. small values of l _α, correspond to a polydisperse sphere model [see Fig. 3 ▸(a/c)]. Spherical particles are indeed obtained whenever the Gaussian field Y(x) is almost constant over distances typical of α(r), which corresponds to the asymptotic limit l _Y/l _α ≫ 1. In that limit, the points of space where Y(x) > α(x) make up a sphere with radius

where Y is the local value of the field, which is spatially constant in the limit l _Y ≫ l _α. Because Y is Gaussian distributed with a mean of zero and a variance of one, the particle model reduces to spheres with Gaussian-distributed radii with average 〈r〉 = −l _αα₀ and variance = = .

When the field Y(x) is variable over distances comparable to l _α the model yields morphologies more complex than spheres. These morphologies include smoothly deformed particles [Fig. 3 ▸(a/d)], and particles with local bulges protruding out of the surface [Fig. 3 ▸(b/c)]. Extremely complex, and occasionally disconnected, structures are obtained for shallow clipping functions, that is for α₀ ≃ 0 and l _α > l _Y [Fig. 3 ▸(b/d)].

It is useful to stress here that the SAS patterns of polydisperse spherical particles and of distorted particles with a Gaussian surface are expected to be distinctly different. Indeed, the two types of structures might lead to similar average density profiles, but the surface area of distorted particles is necessarily larger. As a consequence, the low-q part of the scattering patterns might be similar in both cases, but the Porod scattering in the high-q regions will be different.

3.2. Hollow particles  

The model described in Fig. 3 ▸ can only produce dense particles because the clipping function takes its smallest value α = α₀ at r = 0 [see Fig. 4 ▸(a)]. From now on we refer to that first model as model A, and we introduce two generalizations towards hollow structures. The first generalization is sketched in Fig. 4 ▸(b). Using a clipping function with large values for r = 0 makes the particle hollow. The corresponding clipping function has two additional parameters α_c and α_s for the centre and the shell, respectively, in addition to l _α and α₀. Formally, the clipping function of model B is defined as

where R _s = l _α(α_s − α₀) can be thought of as the radial position of the shell, i.e. where α(r) takes its lowest value α_s. On the other hand, the parameter α_c controls the probability of the centre of the particle being dense or hollow. Realizations of model B are shown in Fig. 5 ▸(b), in which the hollow centres are indeed visible.

Figure 4.

Sketches of the three clipping functions α(r) used to model dense (model A) and hollow (models B and C) particles. The coloured areas highlight the regions corresponding to the solid.

Figure 5.

Realizations of the particle models A, B and C corresponding to the clipping functions shown in panels (a), (b) and (c), obtained with a Gaussian field having a characteristic length l _Y = 10 nm. The density profiles S ₁(r) are shown in (a1), (b1) and (c1), and the specific surface profiles a _V(r) are shown in (a2), (b2) and (c2). The right-hand axes display the cumulative volumes V(r) and areas A(r) as a function of distance from the centre (in blue), with horizontal lines calculated from equation (14) for the volumes and equation (19) for the areas.

In some circumstances it might be desirable to have a model that ensures that the inner cavity of the particle is closed. This can be achieved by modifying the clipping procedure as sketched in Fig. 4 ▸(c). In that case, two clipping functions, say α(r) and β(r), are used and the particle is modelled as the points of space where the Gaussian field is intermediate between the two functions. In the simple case shown Fig. 4 ▸(c), the two functions are linear with the same slope, which can be written as

so that the model has a total of three parameters: α₀ and l _α control the size and shape of the outer surface of the particle, and α₁ − α₀ controls the thickness of the shell. Realizations of this model are shown in Fig. 5 ▸(c). The indicator function of model C is formally defined as

Here, the second equality results from assuming α(x) ≤ β(x) everywhere, which is the case for equation (11). To understand why this procedure leads to hollow particles with a continuous shell, one should note that the clipping functions α(r) and β(r) define the outer and inner surfaces of the particles. The condition β(r) > α(r) therefore ensures that the inner and outer surfaces never touch each other, so that there is no hole in the shell.

3.3. Particle average volume and area  

The volume and surface area of the particles defined in Figs. 3 ▸ and 5 ▸ are random variables, because they depend on the particular realization of the Gaussian field Y(x). This does not preclude one from calculating average values for the particles’ volume and area. For example, the volume of a particle is defined as the integral over the entire space of the indicator function, namely

Because the ensemble averaging commutes with the space integration, the average volume is obtained as

where we have used the classical notation = for the one-point probability function (Torquato, 2002 ▸). The latter function is the probability of a given point of space being in the solid phase of the particle.

In the case of models A and B, with the indicator function defined in equation (7), the one-point probability function S ₁(x) is calculated as the probability of a Gaussian variable taking values larger than the threshold function α(x) at that specific position. Because the threshold is a radial function, this can be written as

where we use the notation (Roberts & Knackstedt, 1996 ▸)

Y is a Gaussian variable and H(x) is Heaviside’s step function. In terms of the error function erf(x), the function Λ₁ can be calculated as

In the case of model C, the one-point probability function is calculated as

which results from equation (12) with β(r) ≥ α(r). Equation (18) is the probability of a Gaussian variable being larger than α(r) and smaller than β(r).

The one-point probability function S ₁(r) is equal to the average density at a distance r from the origin, calculated over independent realizations of the particles. Examples of average density profiles of models A, B and C are shown in Figs. 5 ▸(a1)–5 ▸(c1). In the case of model A, the average density has a maximum in the centre. By contrast, the average density of the hollow-particle models B and C has a maximum at some intermediate distance r, corresponding to the most probable position of the shell. The particle volumes are then obtained through equation (14), which in spherical coordinates is writen as . The cumulative volumes V(r), defined as the average particle volume at a distance lower than r from the centre, are plotted in Figs. 5 ▸(a1), 5 ▸(b1) and 5 ▸(c1). The horizontal asymptotes in the same graphs are the average volumes of the entire particles.

The surface area of a particle is also a random variable. Its average value is calculated as

where a _V(x) is the average surface area in an infinitesimal volume centred on point x. A general expression for a _V was derived by Gommes & Pirard (2009 ▸), namely

where α and ∇α are the local values of the clipping function and of its gradient, respectively, and Σ(x) is the following function:

In equation (20), the first factor is the classical expression = , valid for the classical clipped GRF model (Berk, 1991 ▸; Teubner, 1991 ▸), and the factor Σ accounts for the gradient of the clipping function. The surface-area profiles of models A and B calculated from equation (20) are plotted in Figs. 5 ▸(a2) and 5 ▸(b2). In the case of model A, the value of a _V(r) is maximum at some intermediate distance r corresponding to the average position of the particle surface. For model B, a more complex pattern is observed, resulting from the existence of inner and outer surfaces. In the case of model C, there are two contributions to the local surface-area density a _V. The contribution of the outer surface is calculated through equation (20), and that of the inner surface is calculated through equation (20) with α replaced by β. In Fig. 5 ▸(c2), the two contributions cannot be discriminated because of the small thickness of the shell.

The average total surface area of a particle 〈A〉 is calculated numerically from equation (19), i.e. as in spherical coordinates, with a _V(r) calculated from equation (20) for arbitrary clipping functions α(r) and β(r). The corresponding cumulative areas A(r), corresponding to the average surface area at a distance smaller than r from the centre, are plotted in Figs. 5 ▸(a2), 5 ▸(b2) and 5 ▸(c2). The horizontal lines are the asymptotes, equal to the total surface area of the particles.

3.4. Particle form factor  

The form factor P(q) is the intensity scattered by a single particle (Sivia, 2011 ▸). In a stochastic context this is conveniently calculated in two steps, by evaluating first the scattering amplitude and then the average value of the squared modulus . In terms of the particle indicator function , the amplitude is

which is a random variable because it depends on the values of the indicator function. From equation (22), the scattered intensity is then calculated as

which results from expressing P(q) = , where the star stands for complex conjugation. In equation (23), the function S ₂(x ₁, x ₂) is the two-point probability function, defined as

This ensemble average can be interpreted as the probability that the two points x ₁ and x ₂ both belong to the solid phase of the particle (Torquato, 2002 ▸).

In the present context of clipped Gaussian field models, with the indicator function defined in equation (7), the two-point probability function is conveniently expressed in terms of the following error function:

which generalizes the function Λ₁(α) introduced in equation (16) to the case of two Gaussian variables. Equation (25) is the probability of two Gaussian variables Y ₁ and Y ₂ with correlation g ₁₂ simultaneously being larger than α₁ and α₂, respectively. In principle, this can be calculated as the two-dimensional integral of a bivariate Gaussian distribution, but it is easier to evaluate it numerically via the following one-dimensional integral (Berk, 1991 ▸; Roberts & Teubner, 1995 ▸):

With this notation, the two-point probability function of models A and B is expressed as

where r = x ₁ − x ₂ is the distance between the two points, and g _Y(r) is the correlation between the values of the Gaussian field at those two points, defined in equation (2).

It is customary in the scattering literature to define the correlation function C(r) as the intersection volume of a particle and a copy of it that has been translated by a distance r (Guinier & Fournet, 1955 ▸; Glatter & Kratky, 1982 ▸; Sivia, 2011 ▸). In a stochastic context such an intersection volume is a random variable, the average value of which is obtained as the integral of S ₂(x ₁, x ₂) over all pairs of points at a vectorial distance r from one another, namely

With this definition of C(r), the double integral in equation (23) reduces to the Fourier transform of the correlation function C(r), and one recovers the classical expression for the form factor.

In the case of models A and B the clipping functions are radial, which simplifies the evaluation of C(r). In particular, from equation (27), the two-point probability S ₂(x ₁, x ₂) does not depend on six variables (two points, each in three-dimensional space) but only on three variables, namely the radial positions of the two points r ₁ = |x ₁| and r ₂ = |x ₂| (because the clipping function α is radial), and the distance r = |x ₁ − x ₂| between them [through the field correlation g _Y(r), which is also a radial function]. We write this dependence as S ₂{r ₁, r ₂, r}, with curly brackets. The integral in equation (28) is then conveniently calculated in spherical coordinates, in which the radial distance to the origin is ρ and the polar angle θ is measured with respect to the direction of r. With these coordinates, the radial positions of the two points x + r/2 and x − r/2 are

The correlation function is then obtained as

which results from the integral in equation (28) through a change of variable . The form factor is then obtained as a Fourier transform, i.e.

which holds for any statistically isotropic particles. The correlation functions and form factors of model A are illustrated in Fig. 6 ▸ with a few examples. Although the model is formally identical to the one shown in Fig. 3 ▸, different values of the parameters α₀, l _α and l _Y were used here to demonstrate the versatility of the model.

Figure 6.

Correlation functions and form factors for dense particles (model A) with (a) α₀ = −2, l _Y = l _α/2; (b) α₀ = −5, l _Y = l _α/2; (c) α₀ = −2, l _Y = 2l _α; (d) α₀ = −5, l _Y = 2l _α, and the value of l _α chosen to have a volume of 8000 nm³. The coloured lines are the exact results and the black lines are the approximate form factors calculated through equation (40). For each set of parameters, three independent realizations are shown.

The mathematical expressions obtained here are valid for any radial clipping function α(r), which includes model B of hollow particles as in equation (10). However, in the case of model C with two clipping functions α(r) and β(r) a different procedure is needed. The two-point probability function of model C is calculated as

This equation results from equation (24) with the indicator function of model C taken from equation (12). The correlation function C(r) is then evaluated via equation (28), which leads to four contributions similar to equation (31). Examples of hollow particles with their correlation functions are shown in Fig. 7 ▸.

Figure 7.

Correlation functions and form factors for hollow particles obtained through model B with (a) α_c = 4, α_s = 0 and l _Y = 30 nm and (b) l _Y = 10 nm, and through model C with (c) l _Y = 20 nm and α₁ = −2 and (d) α₁ = −2.5. In all cases α₀ = −3 and l _α = 3.75 nm. The coloured lines are the exact results and the black lines are the approximate form factors calculated through equation (40). The dashed line is a 1/q ² trend, characteristic of scattering by thin films. For each set of parameters, three independent realizations are shown.

The form factors of both dense and hollow particles (Figs. 6 ▸ and 7 ▸) exhibit a flat plateau at small q and a Porod-like q ⁻⁴ scattering at high q. The shape of the particles is manifest in the intermediate q range, with occasional oscillations in P(q) in the case of more ordered structures, as in Fig. 6 ▸(b). An interesting characteristic of model C [Figs. 7 ▸(c) and 7 ▸(d)] is the presence of an intermediate range of q with a scattering proportional to q ⁻². This power law is typical of scattering by thin films (Ciccariello et al., 2016 ▸), and it accounts for the shell of the hollow particle. When the shell becomes thinner the high-q limit of the q ⁻² behaviour extends to higher values of q, as is visible when comparing Figs. 7 ▸(c) and 7 ▸(d).

4. Discussion  

4.1. Approximate expression for the form factor  

The exact expression of the form factor P(q) derived in Section 3.4 can be cumbersome to use in practice because it involves a sequence of numerical integrations. First the two-point probability function S ₂(x ₁, x ₂) is calculated as a one-dimensional integral through equation (26). Then that function is integrated over a two-dimensional domain through equation (31). Finally, the result has to undergo a Fourier transformation to evaluate P(q). Here we introduce a general approximation for the form factor of any particle in terms of the radial density and specific area profiles S ₁(r) and a _V(r) (see Fig. 5 ▸). In addition to its practical usefulness (e.g. in the context of a data-fitting procedure), the approximate expression developed here helps us validate the exact analytical results of Section 3.4.

Our approximation is based on the following two mathematical constraints that any two-point correlation function S ₂(x ₁, x ₂) must satisfy (Ciccariello & Benedetti, 1985 ▸; Torquato, 2002 ▸). First, in the case where the two points are far away from each other, the values of the indicator function at x ₁ and x ₂ are statistically independent of each other, which leads to

Second, when x ₁ and x ₂ are close to each other, the correlation decreases proportionally to the distance x ₁ − x ₂ and to the local value of the specific surface area. Specifically, using the notation x ₂ = , where is a unit vector, the following relation holds:

for small values of r (Gommes & Pirard, 2009 ▸). In this equation, the left-hand side is the average value of S ₂(x ₁, x ₂) over all points x ₂ at a distance r from x ₁, and a _V(x ₁) is the specific surface area at point x ₁. In the particular case of clipped Gaussian fields, the local surface area can be calculated through equation (20), but equation (35) applies to any two-phase structure.

The two constraints in equations (34) and (35) are valid for large and small distances r, respectively. The approximation we introduce consists of assuming that the transition between the two regimes is exponential and isotropic, namely

This general functional form ensures that S ₂(x ₂, x ₂) = S ₁(x ₂) and S ₂(x ₁, x ₂) = S ₁(x ₁)S ₁(x ₂) for |x ₁ − x ₂| ≫ l(x ₁). Moreover, if the local characteristic length that enters the exponential is chosen to be

one can check that equation (35) is satisfied for small values of r.

Evaluating the Fourier transform of equation (36) via equation (23) provides the following approximation for the form factor:

The first contribution is the Fourier transform of the particle average density, namely

which is a low-q approximation. The second contribution is a high-q correction that accounts for the possible presence of many solid/void interfaces at a scale much smaller than that of the particle as a whole. As a consequence of the exponential approximation introduced in equation (36), this is calculated here as a sum of Debye–Bueche contributions of all the infinitesimal volumes that make up the particle (Gommes, 2018 ▸).

In the particular case of statistically isotropic particles, which is notably the case for all models considered in Section 3, the approximate form factor simplifies to

with

Equation (40) is plotted as solid black lines in Fig. 6 ▸ in the case of model A for dense particles and in Fig. 7 ▸ for hollow particles. Globally, the approximate form factor in equation (40) captures reasonably well the exact value for small and large values of q. In particular, the asymptotic form of equation (40) reduces to Porod’s law,

for high values of q, where = is the average total surface area of the particles. As expected, deviations between exact and approximate form factors are observed for intermediate values of q. This is notably the case for model C hollow particles, for which the q ⁻² intermediate scattering is not captured [see Fig. 7 ▸(d)].

Note that the contribution P ₁(q) to the approximate form factor in equation (40) depends only on the average density of the particles S ₁(r). That specific contribution to the scattering does not therefore depend on the characteristic length of the Gaussian field l _Y. Accordingly, the contribution P ₁(q) alone would not enable one to discriminate, say, the scattering of polydisperse spheres from that of Gaussian particles with distorted surfaces. However, the second contribution to equation (40) is calculated from the specific surface-area profile a _V(r). That term enables one to discriminate geometric distortion from mere polydispersity, based on the high-q part of the SAS signal.

4.2. Fitting the SAXS of supported hollow nanoparticles  

The Gaussian field models of particles were used to analyse the SAXS patterns of the fresh and spent catalysts presented in Section 2. For that purpose, it is necessary to account first for the relative contribution of the particles and the porous carbon support to the measured scattered intensity. The total scattering is (Gommes et al., 2019 ▸)

where ρ_s and ρ_p are the electron densities of the material that make up the solid and the particles, respectively, I _s(q) is the scattering contribution of the solid, θ_p is the average number of particles per unit volume of the entire material, and P(q) is their form factor. In general, the scattering by supported nanoparticles contains cross-correlation terms, which are ignored in equation (43). This approximation is valid in the case of low loading of the support, i.e. if the average electron density of the pore-filling phase (as if the particles were homogeneously dispersed in the pores) is much lower than that of the solid phase of the support (Gommes et al., 2019 ▸). As we shall discuss shortly, this applies to the catalysts of Section 2.

The scattering by the support is modelled as a Debye–Bueche contribution:

where ϕ_s is the volume fraction of the solid, l _D is a characteristic length of the material, and the second contribution proportional to b accounts for the turbostratic structure of carbon (Perret & Ruland, 1968 ▸). The solid contribution was fitted to the SAXS data of the empty support, as thoroughly explained by Gommes et al. (2019 ▸); the best fit is shown in Fig. 2 ▸ as a solid black line corresponding to parameters ϕ_s = 0.17, l _D = 158 Å and b = 1.85 Å. Although there is evidence that the support partially oxidizes during the stress test, we assume here that its structure remains unchanged so that the support contribution to the SAXS is the same in the fresh and spent catalysts.

The other material constants that enter equation (43) are the electron densities of the support and the particles, ρ_s and ρ_p, and the particle concentration θ_p. The electron densities are assumed to be ρ_s = 0.9 F cm⁻³ for the carbon support and ρ_p = 8.6 F cm⁻³ for the particles, assuming they are made up of pure Pt with traces of Ni (Gommes et al., 2019 ▸). The particle concentration θ_p is calculated from the known metal loading and the average particle volume 〈V〉. Assuming that the carbon making up the support has a specific mass of 2 g cm⁻³ and that the particles consist mostly of Pt with a specific mass of 21.5 g cm⁻³, the overall loading of 20.4 wt% Pt/C converts to a volume fraction ϕ_p ≃ 0.004, which means that the metal occupies about 0.4% of the entire volume of the material. Using that value, the particle concentration is then calculated as a function of the average particle volume as

Incidentally, from the volume fraction of metal ϕ_p one calculates that the average electron density of the pore-filling phase is 0.04 F cm⁻³, which is much smaller than the electron density of carbon and justifies ignoring cross-correlation effects in equation (43).

As a benchmark for our analysis, we first fitted the SAXS of the catalysts with a classical spherical-shell model (Pedersen, 1997 ▸). For the fitting, the inner radii R _i of the particles are assumed to be distributed according to a gamma distribution, having average inner radius 〈R _i〉 and shape parameter k. For large values of k the gamma distribution has a narrow peak around the average value, and the distribution broadens with decreasing k. The lowest admissible value is k = 1, for which the gamma distribution reduces to an exponential. Moreover, all particles are assumed to have the same shell thickness t independent of their inner radius. The fitting of the SAXS data with three adjustable parameters, 〈R _i〉, k and t, is shown in Fig. 8 ▸. For both the fresh and spent catalysts, the procedure converges to the lowest admissible value for the shape parameter of the distribution, k = 1, which points to extremely polydisperse structures. The values of the other fitted parameters are gathered in Table 1 ▸.

Figure 8.

Fitting of the SAXS of (a) the fresh and (b) the spent catalyst (see Figs. 1 ▸ and 2 ▸) with a polydisperse hollow sphere model. The crosses are the data and the blue line is the best fit, with the contributions of the solid support in black and of the particles in red. In each case five realizations of the particle model are shown, together with their projections.

Table 1. Values of the model parameters (spherical-shell model and stochastic models B and C) obtained from the least-squares fits of the SAXS data on the fresh and spent catalysts.

For each model, the specific surface area a, the average chord length l _C and the sphericity ψ are calculated from the fitted parameters. The statistical uncertainty on all fitted parameters is lower than 1%.

		Fresh catalyst	Spent catalyst
Spherical shell	〈R i〉 (Å)	10	15
	k	1†	1†
	t (Å)	21	54
	a (m2 per gPt)	51	21
	l C (Å)	3.6	8.9
	ψ	0.85	0.93
Model B	l Y (Å)	40	95
	l α (Å)	91	164
	α0	0	−0.2
	R s (Å)	85	89
	a (m2 per gPt)	53	20
	l C (Å)	3.5	9.1
	ψ	0.32	0.39
Model C	l Y (Å)	139	96
	l α (Å)	128	167
	α0	−1.18	−0.15
	α1	−0.87	2
	a (m2 per gPt)	54	21
	l C (Å)	3.5	9.1
	ψ	0.16	0.4

^†Lowest value allowed.

Assuming that the particles are made of pure Pt, the geometric surface areas can be calculated from the model parameters (Gommes, 2018 ▸), yielding the values a = 51 m² per g_Pt and a = 21 m² per g_Pt in the fresh and spent catalysts, respectively. These values compare well with those obtained from CO stripping reported in Section 2, namely 45 ± 7 m² per g_Pt in the fresh catalyst and a value approximately half of that in the spent catalyst. We also use the fitted parameter to calculate the following two global shape descriptors. The average chord length of the particles l _C is calculated as l _C = 4〈V〉/〈A〉, where 〈V〉 and 〈A〉 are the average particle volume and area, respectively. The sphericity ψ is calculated as ψ = π^1/3(6〈V〉)^2/3/〈A〉, and it takes values between 0 and 1. The value ψ = 1 corresponds to a sphere, and values smaller than 1 point to particles with a surface area much larger than a sphere of the same volume. The average chord length l _C and the sphericity ψ are distinct and complementary characteristics of the particles. The average chord length characterizes the size of solid parts of the particle. The sphericity is a dimensionless quantity that is independent of the particle size: it is a measure of how compact an object is, independent of its size. The values of ψ reported in Table 1 ▸ for the spherical-shell model are relatively close to 1. This is a consequence of the shell thickness t being larger than the average inner radius 〈R _i〉. Globally, the fitting of the SAXS with the spherical-shell model suggests that the particles become larger and more compact during the accelerated stress test.

Although the spherical-shell model fits the SAXS data fairly well, it forces the reality into a type of structure that is not representative of all particles in the samples. The spherical shell could at best describe the type of particles shown in Fig. 1 ▸(a2), which are not the only type of structure present in the fresh catalyst. Moreover, these hollow structures seem to be altogether absent from the spent catalyst [Fig. 1 ▸(b)]. We therefore also fitted the SAXS data with the stochastic models presented in Section 3.2. Model B is in general described by five parameters: the characteristic length of the Gaussian field l _Y, the two parameters l _α and α₀ that describe the outer surface of the particle (similar to those of model A), and the two parameters α_s and α_c that describe the inner hollow core. In order to reduce the number of fitting parameters to four, we impose that the absolute value of the slope of α(r) be the same on both sides of R _s (see Fig. 4 ▸). This is equivalent to assuming that the inner and outer limits of the shell are of comparable sharpness. Mathematically, we use the radial position of the shell R _s as a fitting parameter, from which one evaluates α_s = α₀ + R _s/l _α and α_c = α_s + R _s/l _α. The best fits are illustrated in Fig. 9 ▸ and the corresponding parameters are in Table 1 ▸. The SAXS data were also fitted with model C for hollow particles, and the best fits are shown in Fig. 10 ▸. Model C has a total of four parameters: the characteristic length l _Y of the Gaussian field, the characteristic length of the clipping function l _α, and the two values α₀ and α₁. The best fits are illustrated in Fig. 10 ▸ and the corresponding parameters are in Table 1 ▸.

Figure 9.

Fitting of the SAXS of (a) the fresh and (b) the spent catalyst (see Figs. 1 ▸ and 2 ▸) with stochastic model B (see Fig. 5 ▸). The crosses are the data and the blue line is the best fit, with the contributions of the solid support in black and of the particles in red. In each case five realizations of the particle model are shown, together with their projections.

Figure 10.

Fitting of the SAXS of (a) the fresh and (b) the spent catalyst (see Figs. 1 ▸ and 2 ▸) with stochastic model C (see Fig. 5 ▸). The crosses are the data and the blue line is the best fit, with the contributions of the solid support in black and of the particles in red. In each case five realizations of the particle model are shown, together with their projections.

When comparing the outcome of the SAXS fitting with models B and C, it is interesting that the two models yield very similar particle morphologies for the spent catalysts [see Figs. 9 ▸(b) and 10 ▸(b)]. The realizations and their projections shown in the figure are also qualitatively similar to those visible in the TEM micrographs [Fig. 1 ▸(b)]. Mathematically, the two models are strictly equivalent in the limit of very small values of R _s (model B) or large values of α₁ (model C), in which case they reduce to model A of dense nanoparticles. The values of the fitted parameters in Table 1 ▸ show that this is approximately the case for the spent catalyst. The average density profiles in Fig. 11 ▸ hint at slightly hollow particles for model B and dense particles for model C. These differences appear to be very minor based on the values of sphericity, which hint at relatively dense structures in the spent catalyst for both models B and C.

Figure 11.

Average density profiles corresponding to the three models fitted to the SAXS data of (a) the fresh and (b) the spent catalyst.

In the case of the fresh catalysts, the morphologies obtained through the fitting of models B and C are distinctly different, as visible from the realizations in Figs. 9 ▸(a) and 10 ▸(a). Model B seems to yield a structure consisting of disordered parts randomly distributed in space. On a closer examination, however, the density profile [Fig. 11 ▸(a)] is seen to exhibit a clear maximum with S ₁ ≃ 0.2 at a distance of approximately 80 Å from the centre. The structure can therefore be thought of as a hollow nanoparticle with an approximate diameter of 160 Å and an 80% porous shell. With such high porosity the shell is actually made up of disconnected lumps. Such a disconnected structure is physically unrealistic. The overall quality of the fit hints at the fact that not all connected parts of the shell carry sufficient statistical weight to contribute significantly to the scattering pattern. It must also be kept in mind that the catalyst is extremely heterogeneous as it contains objects with very different sizes [see Figs. 1 ▸(a1)–1 ▸(a3)]. The fitting procedure accounts for the scattering in a statistical way by all the different types of structure in the sample.

Contrary to model B, the very construction of model C guarantees the connectivity of the particle shell. The least-squares fit copes with this structural constraint by producing objects that are significantly larger than those of model B, but very distorted and with a very thin shell [Fig. 10 ▸(a)]. It is the size of the local bulges in these larger objects that is comparable to the structures captured by model B. The different types of structure produced by models B and C are further illustrated with the density profiles shown in Fig. 11 ▸(a).

It is interesting to compare the outputs of the spherical-shell model and the two stochastic models. Some features, such as the specific surface area a and the average chord length l _C, are remarkably consistent throughout all models (see Table 1 ▸). This is expected because they can both be expressed in terms of the surface-to-volume ratio of the particles, which can in principle be obtained from a Porod analysis of the scattering patterns without any model (Ciccariello et al., 1988 ▸; Gommes, 2018 ▸). Moreover, the values are also consistent with those obtained independently from CO stripping (see Section 2).

By contrast, other characteristics are strongly dependent on the model, as is notably the case for the sphericity ψ. These differences point to the qualitatively different types of structure that are implicitly assumed when one chooses one specific model over another. In particular, based on the values of ψ, the spherical shells are relatively compact objects when compared with stochastic models B and C, and model C is the least compact structure. As a consequence, stochastic models B and C predict objects that are much larger than the spherical-shell model. This is particularly clear when comparing the density profiles in Fig. 11 ▸. Moreover, the spherical-shell model strongly underestimates the size of the particles, as is apparent when comparing the density profiles with the insets of Fig. 1 ▸ One must therefore conclude that the structures captured by the spherical-shell model are merely local features of the larger structures that are better captured by the two stochastic models.

5. Conclusions  

We have developed and discussed a family of stochastic models of disordered particles. The models generalize the classical clipped Gaussian field models of continuous structures using a space-dependent clipping function instead of a constant. Depending on the clipping procedure, a variety of morphologies can be produced. The focus of the present work was on dense and hollow nanostructures, but the same procedure can be used for other types of more complex structures. All the analytical formulae derived here for the volume [equations (14) and 15)], area [equations (19) and (20)] and correlation function [equation (31)] are valid for any radial clipping function α(r). The generalization to anisotropic structures is straightforward only numerically more cumbersome. This can also be generalized towards polyphasic structures with no more conceptual difficulties than for deterministic models. One possibility is to model the scattering length density of the material as a piecewise constant function, with each region of constant density being defined stochastically (Gommes, 2013 ▸).

Although the data analysis procedure developed in this paper produces realistic real-space illustrations of the structure, one has to keep in mind that this is achieved through modelling. Models cannot be avoided in this context because there are generally a large number of structures compatible with any given correlation function (Gommes et al., 2012 ▸). The very purpose of models is to bias the reconstruction process so as to reduce its degeneracy. In classical deterministic models, the bias is generally towards geometric simplicity. With stochastic models, the bias is towards geometric disorder. Preferring one approach over the other is a matter of modelling choice, which has to be based on the specific type of material being investigated.

In the case of stochastic models, the morphological significance of the parameters is occasionally difficult to visualize. In particular, the models capture the variability of the structures from one particle to the next, which is why several independent realizations have been systematically shown for any given set of parameters. A convenient way of visualizing the average geometry of a given particle model is through its density and surface-area profiles, for which we have obtained general expressions. We have also derived an approximation of the form factor based on the latter two profiles, which coincides with Porod’s and Guinier’s laws in the limits of high and low values of q, respectively. That approximation [equation (40)] is particularly useful in the context of data fitting because it can be evaluated much faster than the exact expression, which requires numerically evaluating a two-dimensional integral followed by Fourier transformation. In our fitting procedure, the approximation was used for coarse and manual screening of the parameters, followed by the full least-squares minimization with the exact expression.

We have used the developed models to analyse the SAXS data from two PtNi hollow nanoparticle samples, before and after being used during oxygen-reduction catalysis, which were both known from microscopy to have very disordered and heterogeneous structures. The data can be reasonably fitted with a classical spherical-shell form factor, but this procedure forces the data into a model that is geometrically too simple. In consequence, the fitted structure has little qualitative resemblance to the actual sample and the sizes are systematically underestimated. By contrast, the stochastic models yield more realistic geometries.

Funding Statement

This work was funded by Fonds De La Recherche Scientifique - FNRS grant .