Impact of distributions on the photocatalytic performance of anatase nanoparticle ensembles

Abstract

While it has been established that the size and shape of nanoparticle photocatalysts determines their performance, producing monodispersed samples is not always economically efficient. Most samples contain a distributions of sizes and shapes, and so predictions based on single-particle properties often fail to identify which feature is more important than the other. Using simple statistical approaches and ensembles of structures it is possible to explore the impact of reducing different types of polydispersivity, and thus identify realistic targets for synthesis at the industrial scale. In the case of anatase, photocatalytic performance can be significantly increased without requiring monodispersivity, provided some of the particles are small.

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Titania nanostructures are of considerable commercial interest, due to their superior photochemical properties.^1–3 The crystalline polymorph and the morphology are known to be critical parameters^4–9 in determining the overall performance^7,10 and general suitability for specific applications.¹¹

While rutile is the thermodynamically stable phase of titania under ambient conditions at the macroscale,¹² synthesis of nanocrystalline titania consistently resulted in anatase nanocrystals,¹³ and it is now well established that anatase is the thermodynamically stable phase at the nanoscale.^14,15 It has previously been demonstrated that the morphology of anatase nanostructures can be engineered with remarkable control,^16,17 but this requires very specific synthesis conditions and preserving this degree of control may not be commercially viable for large scale production. Most samples typically contain distributions of sizes and shapes, and so detailed predictions of the performance of polydispersed samples of anatase (which go beyond conventional single-particle structure/property relationships) are needed to provide a basis for commercial decisions. These decisions include, but are not limited to, the degree and type of monodispersivity required, and the cost effectiveness of achieving it. Should synthesis efforts be targeted toward producing anatase nanoparticles of a specific size and/or shape, and what is the fault tolerance for persistent polydispersivity?

Both size and shape are critical to the performance of a photocatalysis, since the efficiency of reactions typically depends on the crystallographic orientation of the surface facets, the total available surface area, and the type of reaction.¹⁸ In the case of anatase, the {001} facets are the most photoactive,^17,19,20 and this can be experimentally estimated by measuring the products of reactions at the surfaces, using a variety of techniques.²¹ Among the reaction products, the OH˙ radical is extremely important, as it is related to the photocatalytic oxidation of water.^22,23 Moreover, it has been established that addition of hydrogen peroxide in the titania photocatalytic system accelerates the OH˙ formation, thereby improving the activity.²⁴ In this case, OH˙ is formed on reduction of hydrogen peroxide with a conduction band electron, e_cb⁻ (which is the primary reaction for the production of OH˙), or by O₂˙⁻. Alternatively, hydrogen peroxide can be oxidized to O₂˙⁻ by a valence band hole, h_vb⁺ (which is the primary reaction for O₂˙⁻ production), or by OH˙.²⁵

Theoretically the efficiency can be estimated via the in-plane density of Ti cations on the anatase {001}, which is 6.99 × 10¹⁸ m⁻², and the total area of the {001} facets (which depends on the size and the shape).²⁶ To begin to explore the impact of structural polydispersivity on photocatalytic performance we must first establish a virtual sample of anatase nanoparticles, and calculate their relative stability. While it is possible to generate a large set of explicit virtual samples (that contain a considerable variety of shapes and sizes) to study anatase nanoparticle ensembles using explicit computer simulations, this represents a very large undertaking (particularly if we wish to use ab initio simulation methods) and considerable time and resources will be spent describing structures that have a very low probability of observation and contribute little to the properties of the ensemble. It is more efficient to use an analytical model to generate the sample, rather than to attempt to simulate thousands of individual nanoparticles. This can be done in a straightforward manner using a model²⁷ parameterized by density functional theory,²⁸ where it is a simple matter to compare the free energy of formation of any input morphology²⁹ in an appropriate thermochemical bath using ab initio thermodynamics.³⁰

Previously it has been established that a phenomenological nanomorphology model, designed to compare the thermodynamic stability of isolated (unsupported) nanoparticles, is ideally suited to this task. The model used here provides the geometric summation of the Gibbs free energy with contributions from the particle bulk and surfaces, such that:

where M is the molar mass, ρ is the mass density, q is the surface-to-volume ratio, T is the temperature, and γ_i(T) is the specific surface free energy of facet i. These components converge to the Wulff shape at large sizes. In addition to this, the surface stress σ_i(T) and the external pressure P_e produce a volume dilation, the magnitude of which depends on the bulk modulus B₀. These components are not present in a Wulff construction, and are significant at small sizes. The weighting factor is a fractional area, so , and (to be consistent with the Laplace–Young formalism).

The temperature-dependent free energy ΔG(T) may be determined by parameterizing with γ_i(T) and σ_i(T). This may be done using any suitable computational method, provided that it is used consistently (to avoid uncertainties associated with the mixing methods) and has sufficient accuracy to distinguish between different facets and provide the right energetic ordering. From experiment, it is usually found that titania surfaces include TiOH surface hydroxyl groups,^32–36 and under neutral pH conditions the surfaces are terminated with H₂O (either as molecular H₂O or as dissociated OH⁻ + H⁺) or OH.^37,38 For this reason the temperature-dependent specific surface energies and isotropic surface stresses for H₂O– and OH–terminated surfaces have been included, as described in ref. 30. These properties were originally calculated using ab initio thermodynamics, based on simulations performed using density functional theory (DFT) within the generalized gradient approximation (GGA),²⁸ and have been shown to support excellent agreement with experimental observations of photocatalytic activity.²⁶ In the present study we have averaged over H₂O– and OH–terminated surface results, as we aim to study particles that will occupy the coexistence zone of the phase diagram.³⁰

Once parameterized, the models above can be used to compare the relative stability of each shape (n), and predict their probability of observation (p) as a function of the probe temperature T. This can be easily obtained using a Boltzmann distribution:

p(n) = e^−G_n(T)β (2)

where β is reciprocal of the thermodynamic temperature of the system, calculated at the temperature of exposure to 532 nm light. This can in turn be used to calculate the population of each n in an ensemble containing N particles (each with a specific size and shape), using statistical mechanics:where the denominator is the canonical partition function.

Using the size- and shape-dependent relative stability G_n we can compute the probabilities p_n∀n ∈ N, and then determine the populations for a range of equilibrium and non-equilibrium (metastable) geometries. Irrespective of the sizes and shapes included in the ensemble, the populations for non-equilibrium structures are always non-zero at finite temperatures, though they are of course low for the least stable morphologies. The ensemble average of the density of active sites (X) can then be computed for any ensemble containing N particles using:

where x_n is the density of active sites on particle n.

In this study a virtual sample of nanoparticles from 3 nm to 60 nm in average diameter has been used (spanning the range characteristic of typical commercial samples²¹), with shapes containing a fractional area of the active {001} facets ranging from 10% to 75%. These shapes are illustrated in Fig. 1, where we can see that as we increase the area of the {001} facets the particles become platelets, and accurately replicate the morphologies targeted by many groups.^17,20

Fig. 1 Anatase nanoparticle morphologies included in this study, with (a) 10%, (b) 15%, (c) 20%, (d) 25%, (e) 30%, (f) 35%, (g) 40%, (h) 45%, (i) 50%, (j) 55%, (k) 60%, (l) 65%, (m) 70%, (n) 75%, of {001} surface area.

If we apply resolution of 1 nm to the size distribution and a 5% resolution to the shape distribution, then there are 812 possible monodispersed size/shape combinations. This size/shape resolution is trivial to achieve theoretically, but complicated to achieve computationally, and challenging to achieve experimentally. For each of these monodispersed combinations a theoretical estimate of the photocatalytic performance, based on the number density of cations on the {001} facets, is shown as a histogram in Fig. 2. Here we can see that the activity will increase with both increasing {001} area and decreasing particle size (or equivalently, increasing the effective molar area), though the absolute values resulting from this type of prediction should be used with caution (as the efficiency of real samples may not be ideal). While including all active sites will undoubtedly be an overestimation, the relative differences between different ensembles (computed consistently) can be used with confidence. With this is mind, it is easy to see the scope for improving the photocatalytic performance if this degree of monodispersivity and size/shape selectivity could be achieved.

Fig. 2 Photocatalytic performance as represented by the density of active cationic sites, comparing the results for specific monodispersed configurations (bars), with different size/shape configurations in a polydispersed ensemble where all configurations are present and weighted by their relative probability of observation (surface).

However, if no attempt were made to create monodispersed samples then the virtual sample would be an ensemble containing all of these 812 combinations, and the prevalence of each configuration will be determined by the probability of observation, which is related to the relative thermodynamic stability.³¹ Computing the probability of each configuration and applying it as a weighting factor to each structure, as illustrated by the overlaid surface-plot in Fig. 2, we can begin to see the impact of polydispersivity. At small sizes the maximal activity is reduced by ∼50% since these particles are less thermodynamically stable and therefore less likely.

If we consider the polydispersed sample in its entirety, the ensemble average is 43.65 sites per mMol, with a very unsatisfactory standard deviation of 50.64. To gain some appreciation of which size/shape combinations should be targeted to improve the photocatalytic performance, Fig. 3 provides the change in the micromolar density of active sites, relative to the ensemble average (which is set to zero). Here we can see that the majority of configurations perform below the ensemble average and so targeting these sizes and shapes will retard performance; it is better to retain the natural polydispersivity. Conversely we can also see that there is a significant advantage in targeting small platelets, where the performance enhancement can be has high as 44%. What we cannot discern from these results, is whether it is better to concentrate on controlling the size or the shape.

Fig. 3 Photocatalytic performance of different size/shape configurations in the ensemble, relative to the ensemble average (shown in grey).

One advantage in using a statistical approach is that we can easily construct smaller, more selective ensembles and predict how the selectivity will impact the performance (relative to the entire ensemble average). Fig. 4 provides series of results for two different types of selective ensembles. In Fig. 4(a) we can see the impact of giving priority to shape-selectivity, where each sample is monodispersed in terms of shape, but polydispersed in terms of size (including contributions from all sizes between 3 nm to 60 nm weighted by their probability of observation); the data labels refer to the figure labels in Fig. 1. In Fig. 4(b) we can see the impact of giving priority to size-selectivity, where each sample is monodispersed in terms of size, but polydispersed in terms of shape (including contributions from all shapes in Fig. 1 weighted by their probability of observation). By comparing these results it is immediately clear that a greater performance enhancement will be achieved by focussing on controlling the size; the photocatalytic activity can be increased by 2× to 5× if particles could be reduced to <7 nm, irrespective of the shapes.

Fig. 4 Change in photocatalytic performance of different selective ensembles, relative the entire ensemble average. (a) Shape-selected ensembles, containing a weighted distribution of sizes (where the labels refer to the parts in Fig. 1). (b) Size-selected ensembles, containing a weighted distribution of the shapes shown in Fig. 1.

Of course, this degree of selectivity still assumes a high degree of control is possible during synthesis, and this may be difficult (and/or expensive) to achieve at the industrial level. To assess the impact of a more moderate degree of structural control, Table 1 provides the ensemble averages and the change in the relative performance (with respect to the entire ensemble average) for a range of different restricted ensembles. In the rows the impact of smaller size-distributions are shown, and in the columns the impact of different shape-distributions; in each case eliminating some, but not all, of the polydispersivity.

Table 1 Ensemble averages (per mMol) and percentage change in relative performance (with respect to the entire ensemble average: All) for various restricted ensembles. {001}-retarded shapes have between 10%–40% {001} surface area (Fig. 1(a–g)), and {001}-enriched shapes have between 40%–75% {001} surface area (Fig. 1(g–n))

Size range	{001}-retarded	{001}-unrestricted	{001}-enriched
3–30 nm	32.72 (−25.05%)	72.91 (+67.02%)	95.02 (+117.66%)
3–40 nm	29.53 (−32.35%)	59.37 (+36.00%)	84.81 (+94.29%)
3–50 nm	25.10 (−42.50%)	50.51 (+15.70%)	72.08 (+65.11%)
3–60 nm (All)	21.68 (−50.33%)	43.65 (0%)	62.25 (+42.60%)
10–60 nm	14.53 (−66.72%)	29.79 (−31.75%)	42.60 (−2.42%)
20–60 nm	11.30 (−74.11%)	23.26 (−46.72%)	33.25 (−23.83%)
30–60 nm	9.55 (−78.11%)	19.69 (−54.90%)	28.14 (−35.54%)

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We can see from the results in Table 1 that there are gains to be made from both size and shape control, but size-control will be far more profitable. Enrichment of the samples with {001} facets is only advantageous if the size distribution contains small anatase nanoparticles. If a polydispersed sample could be restricted to less than 30 nm in size, and simultaneously limited to a {001}-enriched platelet-like morphology, these results indicate that the photocatalytic performance will double.

Although it is well known that engineering the shape of photocatalyst will enhance performance, a fact that is certainly supported by these results, the outcome of the comparisons presented here indicate that it is not as important as size. Given the time and cost associated with controlling size, shape, and the degree of polydispersivity in both, this suggests that attention and resources should be directed toward producing samples of small nanoparticles with a high fraction of {001} facets — but achieving perfect monodispersivity is unnecessary. Distributions in both size and shape are acceptable (and can even be useful), provided they are the right distributions; and they can be reproduced as required.

Acknowledgements

Computational resources for this project were supplied by the National Computing Infrastructure national facility under Mass Allocation Scheme, Grant p00.

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