Scattering from colloidal cubic silica shells: Part I, particle form factors and optical contrast variation

scattering study on dilute dispersions of cubic silica shells (


Introduction
The thermodynamic properties of cube fluids, such as the equation of state, have not yet been measured experimentally.Current insight is based upon only a few theoretical [1] and simulation [2,3] studies.In part II of this research [4] we present the experimental equation of state and static structure factors of colloidal cubic silica shells (CSS), determined from static light scattering (SLS) data.In order to study stable dispersion of colloidal cubes with SLS, particles with a size in the range of 100-200 nm are desired.Recently we developed a preparation method for CSS with average sizes between 100 and 150 nm [5].These shells are expected to be suitable model particles to study the thermodynamic properties of concentrated cube fluids.
In literature, colloids with a rounded cubic shape are often described as superballs with a specific shape parameter m.Superballs are a mathematical shape described by [6]: where R el is the edge length of the superball.For m ¼ 2 and m ¼ 1, Eq. ( 1) describes a perfect sphere and a perfect cube, respectively.
For intermediate m values, Eq. ( 1) describes a rounded cube.Scattering measurements on concentrated dispersions of cubes, requires knowledge of single particles properties, such as the internal refractive index profile, details on the particle shape, and their corresponding form factor.The complex shape of the cubic silica shells makes it difficult to calculate these properties ab initio, so one has to rely on experiments.Therefore, the goal of this paper is to characterise the scattering properties of dilute dispersions of CSS in detail.Nowadays, well defined colloidal cubes can be prepared in different sizes and in various materials [7][8][9][10][11][12][13].These cubic particles are of interest since they form different crystal structures compared to spherical particles [14][15][16].Understanding how cubic particles respond to shear [17,18] and assemble in confinement [19,20,15,21] or on interfaces [22,23] is therefore a current topic of research, in light of the envisioned applications of cubic colloids in sensing [24], catalysis [8] and energy storage [25].
Cubic silica shells with sizes ranging between 100 and 150 nm are promising for applications in thin film optical coatings [26].The thickness and refractive index profile dictate the properties and performance of these optical coatings [27].Optical coatings are often used to decrease reflection [28], and on glass surfaces are often prepared from porous materials [29].Using anisotropic particles to produce optical coatings is limited to needle shaped particles [30] and little is known on the properties of coatings prepared from cubic colloids.Detailed characterisation of cubic silica shells is required to evaluate the applicability of silica shells in optical films.
Scattering techniques such as SLS and dynamic light scattering (DLS), small angle X-ray scattering (SAXS) and small angle neutron scattering (SANS) are often employed to characterise particulate colloids [31].SLS for instance, allows for detailed characterisation of the size and optical properties of particles [32,33], as well as probing the interactions and structure of the dispersion [34].Most experiments and theory regarding SLS on colloids focused on spheres [35,36], although the scattering of anisotropic particles like rod [37] and disk-like [38] particles are also well understood [39,40].Silver bromide cubes have been studied with SLS [41], and also multiple reports of SAXS and SANS on dilute [14,[42][43][44] and concentrated [45,46,14] dispersions of cubic particles have been reported.Scattering on stable dispersions of cubic shells is, to the best of our knowledge, still lacking.Combining light and X-ray scattering allows for a detailed characterisation of submicron particles because the wave vector ðKÞ range in both techniques complement each other (typically for SLS K 6 0:05 nm À1 and SAXS K P 0:05 nm À1 ).
In this paper we first report measurements of hollow particle form factors with both SLS and SAXS.These experimental form factors are subsequently compared to theoretical form factors of poly-disperse cubic particles and polydisperse spherical particles and results from transmission electron microscopy.Finally, we report a contrast variation study to probe the refractive index profile of the silica shells.

Theory: static light scattering
SLS yields the angular scattering intensity profile IðKÞ of an ensemble of particles as a function of scattering wave vector K, defined as: Here k 0 is the wavelength of light in vacuum, n m is the refractive index of the solution and h is the scattering angle with respect to the incident beam.In a typical scattering experiment, the intensity of the scattered light is measured at different scattering angles, yielding a IðhÞ or IðKÞ profile.In the Rayleigh-Gans-Debye approximation, the normalized excess scattering intensity from particles over the that of the solvent is given by the Rayleigh ratio RðKÞ.
For sufficiently dilute dispersions, RðKÞ can, for vertically polarized light, be written as [36,33]: Here, PðKÞ is the form factor, v c is the volume of the particle, q is the particle number density, and n c ; n s , and n m are the refractive index of the particle, solvent and dispersion, respectively.The form factor reflects the internal structure of the particle, which is defined as [32]: Here FðKÞ is the particle scattering amplitude of the particle: and is obtained by integrating over all scattered waves (e i KÁr ) within the particle.In Eq. ( 5) n c ð r !Þ is the local refractive index of the particle at position r? inside the particle.An analytical expression of PðKÞ is available for several particle shapes, such as thin rods and spherically symmetric particles [40].For a cubic shell with a homogeneous refractive index profile, the form factor amplitude FðKÞ is given by [40,47]: Here R 1 and R 2 are, respectively, halve of the edge-length of the outer cube and the inner void, and K x;y;z are projections of the scattering vectors on the corresponding axes depicted in Fig.
A.1 (in the supplementary information), and V c ðR i Þ is the volume of cube with edge-length R i .To obtain the form factor, the modulus square of Eq. ( 6) should be orientationally averaged, which can be performed numerically (Eq.A.15 in the supplementary information).Without numerical fitting of the data, however, some important particle properties can already be obtained from the scattering profiles RðKÞ in the small K regime where the Guinier approximation holds [35]: Determining the slope from the form factor by: d lnðPðKÞÞ yields R g , the radius of gyration of the particles, a measure of the scattering contrast distribution around the particles centre of mass.
For optically homogeneous particles, R g is defined as: where R is the distance from the centre of mass of the particle.In the SI (Appendix C) it is shown that for a solid cube and a cubic shell, the radii of gyration are respectively given by [48]: and In addition to the radius of gyration, the average refractive index of the particles and optical inhomogeneities can be obtained from SLS [33,49].Following Philipse et al. [49], we write for an optically inhomogeneous particle the refractive index profile n c ðrÞ of the particle as the average refractive index plus the local deviation Dn c ðrÞ: One can then split Eq. ( 5) into two parts: where F 0 ðKÞ is the scattering amplitude for a cubic shell with homogeneous refractive index n c and the form factor amplitude FðKÞ defined by Eq. ( 5).The term ðKÞ arises from the inhomogeneities in the refractive index ðDn c ðrÞÞ and can be written as: In the limit K !0 only the average particle refractive index contributes to the zero angle scattering since ðKÞ reduces to zero [49].To obtain the particle refractive index Eq.( 3) should be rewritten to: with: where d is the number of particles per unit mass (kg À1 ) and c is the particle concentration in kg m À3 .Eq. ( 15) implies that when the scattering intensity vanishes, the particles are perfectly index matched, and the refractive index of the particles is equal to the refractive index of the solvent.At scattering wave vectors K -0, heterogeneities in the refractive index profile of the particle (Eq.( 14)) contribute to the total scattering intensity.These heterogeneities will result in different apparent refractive indices at different K values [49].Determining the changes in apparent refractive index at different angles will assess the homogeneity of the silica shells.

Preparation of cubic silica shells (CSS)
Cubic silica shells (CSS) were prepared by using a template of cubic Cu 2 O nanoparticles.The Cu 2 O particles were coated with a silica shell using the PVP-assisted Stöber method [50,51] to obtain cubic core shell particles (CCS).cubic silica shells (CSS) were then obtained by dissolving the core in nitric acid.Experimental details are reported in [5].The dimensions of the particles used in this research are listed in Table 1.Transmission electron microscope samples were prepared by adding a drop of particle dispersion to 0.5 mL water.A single drop of this diluted dispersion was then deposited on a carbon coated copper grid and dried under a heating lamp for 1 h.Micrographs are presented in Fig. 1.

Static light scattering
For the analysis of particles with static light scattering (SLS), a homebuilt setup was used.A schematic overview of the setup is depicted in Fig. D.2 in the SI.In the setup a mercury lamp is used as a light source.The light passes several optical filters to control the intensity, wavelength, polarisation and beam width/height.The scattered light is collected by a rotating detector that scans between an angle of 20°and 140°.The sample is prepared by redispersing the particles in solvents filtered with 0:2 lm PTFE filters and transferring the dispersion to a dust-free cuvette.The cuvette is then placed in a toluene bath to perform the experiment (see Table 3).

Contrast variation measurements
Particles were redispersed from ethanol in DMSO by washing through centrifugation as follows the particles were centrifuged for 15 min at 16,000 g, the supernatant was removed and the particles were redispersed in filtered DMSO by sonication and vortexing.These washing steps were done at least five times to ensure no ethanol was present in the dispersion.The particles in DMSO were then added to mixtures with known concentration of ethanol and DMSO as listed in Table D.1 in the SI.SLS samples were then prepared according to 3.3 The refractive index of these mixtures were calculated with the Bruggeman mixing formula for two components [52]: where v 1 is the mole fraction of component one, n 1 ; n 2 and n eff are the refractive indices of components one, two and the effective refractive index of the mixture, respectively.To check whether this formula holds for the used ethanol/DMSO mixtures, calculated values were compared to values obtained with an Abbe refractometer.
It was found that for all ethanol/DMSO mixtures in Table D.1 in the SI, no deviation in refractive index larger than 0.2 was observed.Therefore, we used the calculated values from Eq. ( 17) without further modification.The particle concentration was determined by drying a known volume and weighing the solid residue.The concentrations of the dispersions were 0.5, 0.6 and 1.8 mg/mL for CSS1, CSS2 and CSS3 respectively.

SAXS measurements
SAXS measurements were performed at the DUBBLE beamline BM-26B at the European Synchrotron Radiation Facility in Grenoble, France [53].Scattering of X-rays with the wavelength of 1.04 Å is recorded by a Pilatus 1 M (Dectris) detector located 4.65 metres behind the sample.An evacuated fly-pass tube is inserted between sample and detector to reduce scattering of the X-ray beam in air.Intensity of the transmitted direct beam is measured by a photodiode inside a beamstop located at the end of the fly-pass tube.SiO 2 @Cu 2 O cubic core shell and cubic silica shells were dispersed in ethanol.The dispersions were centrifuged for 1 min at 1000 g to remove dust and clusters from the dispersion.The solid content of the dispersions was 0.8 wt % for CCS particles and 2.8 wt% for the CSS particles, as listed in Table 2.

Form factor from SAXS
In Figs. 2 and 3, SAXS scattering curves of CSS and CCS cuboids are depicted.The scattering curve of the cubic SiO 2 shells shows three distinct intensity minima.The minima at K ¼ 0:0425 nm À1 and K ¼ 0:0862 nm À1 , are the first and second minima from the shape of the CSS.These minima then dampen, due to the size polydispersity of the particles (13 %).At high K-values, additional oscillations become visible at K ¼ 0:51 nm À1 , which correspond to a length of 13 nm.These oscillations agree with an average shell thickness of 14 AE 1.1 nm as determined from TEM.The experimental scattering curve was compared to numerical models of perfect cubic shells and spherical shells, both with an edge length or radius of 150 nm and an inner void of 121 nm.The SAXS data indicate that the cuboid shape is in between that of a cube and a sphere, which corresponds to superballs with an m-value of 4. Additionally, in Fig. 2 it is visible that for both models, the minima arising from the shell thickness are at slightly lower K values than for to the SAXS data (K ¼ 0:42 nm À1 , vs K ¼ 0:50 nm À1 ).This discrepancy can be attributed to polydispersity in the silica shell thickness, which is not accounted for in the models.Any roughness in the silica shell is hard to distinguish via TEM and could account for an overestimation in the shell thickness.
Similarly, in Fig. 3, the scattering curve of SiO 2 @Cu 2 O is compared to models of a cubic and spherical core-shell particle.Similar to the scattering curves of the cubic SiO 2 shells, the first two minima are located at K values slightly lower compared to the minima in the spherical model but at slightly higher K values compared to the cubic model.The experimental data, however, deviate significantly from the models at K values higher than 0.2 nm À1 and no minimum is visible arising from the uniform silica shell similar  to the cubic silica shells.This can be explained by scattering from inhomogeneities in the cuprous-oxide core.TEM-micrographs seem to indicate that the particles are not single-crystalline, which might result in density inhomogeneities inside the Cu 2 O core which cause excess scattering at high K values.These inhomogeneities within cuprous-oxide particles were earlier observed with X-ray laser diffraction [54].

Radius of gyration from the form factor
Dispersions of CSS in ethanol:DMSO (95:5, vol:vol), depicted in Fig. 1, were studied with static light scattering.Three different types of particles were prepared from the same Cu 2 O cores, but coated with silica layers of varying thickness.The different cubic shells were examined to evaluate the influence of the coating thickness on the size and the effective refractive index.
To determine the size, the radius of gyration ðR g Þ was determined by plotting lnðRðKÞÞ of the scattering intensity as a function of K 2 (Eq.( 8)).In the Guinier regime (KR g < 3) the data should follow linear behaviour with a slope of ÀR 2 g =3.The plots in Fig. 4 show that the obtained scattering curves indeed are linear for most scattering angles.Only in the low K-regime a small deviation from linearity behaviour is visible, probably caused by some residual dust particles or a small fraction of larger clusters.In Table 3, the measured gyration radii derived from the Guinier plots are presented and compared with values calculated from the average particle size obtained from TEM measurements (Table 1).The size obtained from SLS is in excellent agreement with the TEM data.It is known that for silica particles, the size obtained from TEM measurements is often significantly smaller than the size obtained from SLS and DLS [55].This reduced size from TEM is caused by shrinking of the silica matrix by the electron beam [56].Since the silica shells of our particles are thin (8-15 nm) the shrinking of the silica shell probably falls within the uncertainty of the measurements.The excellent agreement of the R g with the TEM-data suggests that the cubes are dispersed as single particles that do not cluster.

Contrast variation
Cubic silica shells (CSS1, Fig. 1-B In case of low contrast, as is the case for the yellow curves in Fig. 5, the intensity increases significantly at low Kvalues, and is probably manifesting residual dust particles, whose scattering dominates the nearly index matched CSS at low K-values. The refractive index can be determined by plotting the square root of the zero-angle scattering intensity as function of the dispersion refractive index (see Eq. ( 15)).This plot should yield a straight Fig. 2. Scattering curves obtained from SAXS measurements on cubic silica shells.The cyan curve is the measured scattering profile while the blue, green, and red curves are obtained from models of, respectively, a cubic 14 nm shell, a cubic 13 nm shell and a spherical 14 nm shell, all with 13% polydispersity.Inset: visual depiction of the geometric models used to calculate the form factors. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)Fig. 3. Scattering curves obtained from SAXS measurements on SiO 2 @Cu 2 O cubic core shell particles.The cyan curve is the measured scattering profile while the blue and red curves are obtained from models of, respectively, cubic and spherical shells, both with 10% polydispersity.Inset: visual depiction of the geometric models used to calculate the form factors. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)line, where the intersection with the abscissa gives the average refractive index of the particles.In Fig. 6-A such a plot is presented for various wavelengths.From the intersects with the abscissa it follows that the particles have an average refractive index equal to 1.500, 1.494 and 1.479 for k = 365, 404 and 516 nm, respectively.
The influence of the shell thickness on the optical properties of the particles was studied, using cubic silica shells with a shell thickness of 5 nm and 15 nm (Fig. 6-B) in contrast variation studies.The results are shown in Fig. 6-C.The average refractive index of the particles does not differ significantly from the 12 nm shell particles.
The obtained values for the average refractive index are higher than literature values for silica spheres prepared by the Stöber synthesis [49].Two explanations for this deviation are as follows.The first explanation is the method we use to remove the cuprous oxide core.During the process, organic moieties in the silica matrix are very likely also removed, increasing the average refractive index of the particles.For conventional Stöber silica it is known that a significant amount of ethoxy groups is still present in the matrix [56], which reduces the refractive index.Another explanation could be the adsorption of DMSO in the silica pores.DMSO strongly solvates silica surfaces [57] and it is likely that DMSO pref-erentially adsorbs in the micropores of the silica matrix, increasing the effective refractive index of the particles.With contrast variation measurements in mixtures of DMF/DMSO, the refractive index was determined to be 1.48 AE 0.01 (Figure D.5 in the SI) for light with k = 404 nm.The refractive index of 1.48 AE 0.01 is significantly higher compared to previously reported refractive index of unmodified Stöber silica (1.4547 AE 0.001 for k = 436 nm), and significantly lower than the refractive index that was measured in ethanol/ DMSO.The contrast variation of the silica in DMF/DMSO indicate that both the preferential adsorption of DMSO and reduced carbon content in the silica matrix influence the refractive index of the CSS.
To determine whether the refractive index of the shell is homogeneous, the intensity of different K values was evaluated and compared to the zero-angle scattering intensity.According to Eqs. 10 and 12, inhomogeneities in the refractive index of the shell lead to a different effective form factor, resulting in a different apparent refractive index with respect to the average particle refractive index.Fig. 7 shows that the intensity plots at K ¼ 0; 0:025 nm À1 and K ¼ 0:04 nm À1 intersect the abscissa at virtually the same value forn s demonstrating that the shell is rather homogeneous.Moreover, the shell thickness does not seem to influence the homogeneity of the particles.

Conclusions
Using both small angle X-ray scattering (SAXS) and static light scattering (SLS), we have characterised cubic silica shells derived from cuprous oxide nanocubes [5] in detail, adding to the available literature on the scattering of solid cubic particles [14,47,42,44,45] form factors of cubic core-shell particles and cubic shell particles.We find that the studied cubic silica shells form stable dispersions, suitable for detailed characterisation by different scattering techniques.From SAXS measurements, the average particle size could be obtained as well as the average shell thickness.Comparison with theoretical models, confirms that the shape of the particles is intermediate between a hollow cube and a hollow sphere.This intermediate shape is as could be expected for cubes with rounded corners which were detected earlier using TEM.SLS complements the SAXS data by providing the scattering of cubic particles at low K values.The radius of gyration of the particles from the Guinier regime corresponds well to radii of gyration calculated from TEM images using a model of cubic shells, which confirms that the particles are dispersed as single, non-aggregated particles.Contrast variation measurements show that the silica shells have a homogeneous refractive index of 1.500, 1.494 and 1.479 for light with a wavelength of 365, 404 and 516 nm, respectively.These refractive indices are significantly higher than literature values [49], which is due to both a different preparation method for the silica shell and preferential DMSO adsorption on the silica surface.The contrast variation measurements show that SLS can be used to detect small variations in the average refractive index of cubic particles resulting from solvent absorption [57] and variations in preparation methods [49].
The possibility to prepare dispersions of stable cubes with known refractive index, allows for studying the scattering of more concentrated dispersions of silica cubes, which is the subject of the accompanying paper [4].Reducing the contrast between the solvent and the particles prevents multiple scattering and allows for the measurement of the structure factor at high concentrations.Further, the determined optical properties indicate the cubic silica shells are suitable for thin film anti-reflective coatings, comprising dense cube packings, which is the subject of future investigations.

Fig. 1 .
Fig. 1.TEM images of A: Cu 2 O nanocubes with an average edge length of 88 AE 11 nm.B: CSS obtained from the particles depicted in A with a 5 nm thick shell; C: CSS obtained from the particles depicted in A with a 12 nm thick shell; D: CSS obtained from the particles depicted in A with a 15 nm thick shell; E: CSS with an inner void of 121 AE 20 nm and a 20 nm thick shell; F: Cu 2 O core shell particles with a core of 112 AE 11 nm and a 10 nm thick shell.
) were dispersed in mixtures of DMSO and ethanol.The mixtures and their corresponding refractive indices at k = 365, 404 and 516 nm are shown in Table Figure D.1 in the SI.The obtained Guinier plots for light with k = 365 nm are depicted in Fig. 5, showing that most curves are linear over the probed K-range.The Guinier plots for light with ½Errormml :mathaltimg ¼ }si143:svg} > k = 404 nm and 516 nm are presented in the supporting information (Figure D.3 in the supporting information).

Fig. 4 .
Fig. 4. Left: Guinier plots of the particles depicted in Fig. 1-B measured for light with k ¼ 365 nm.Right: Guinier plots of the particles depicted in Fig. 1-B measured for light with k ¼ 404:

Fig. 5 .
Fig.5.Guinier plot of RðKÞ versus K 2 obtained from sample CSS1 for light with k ¼ 365 nm.Symbols are measured data while the lines are a first order fit to the data points.For most curves a clear linear K 2 dependence is visible over the measured K range.For curves with the lowest contrast (yellow), the intensity data slightly deviates at low K-values.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 6 .
Fig.6.A: Contrast variation experiments for CSS1 particles.The square root of the scattering intensity at K ¼ 0 plotted against the refractive index of the solvent (ns) for light with k ¼ 365 nm (blue), k ¼ 404 nm (green) and k ¼ 516 nm (red).The intersect with the abscissa yields the average refractive index of the particles, which is 1.500, 1.494 and 1.479 for 365, 404, and 516 nm light respectively.B: Similar as A but for CSS2 (Top) and CSS3 (Bottom) particles.C: Overview of the obtained np for particles with 5, 12 and 15 nm thick shell for light with k ¼ 365 nm, k ¼ 404 nm and k ¼ 516 nm, the line is to guide the eye.D: TEM micrographs of the CSS1 (Left), CSS2 (Middle), and CSS3 (Right) particles used in contrast variation.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 7 .
Fig.7.The square root of the intensity versus ns at K ¼ 0 (blue), K ¼ 0:025 nm À1 (green) andK ¼ 0:04 nm À1 (red) for 365 nm light.The common, K-independent intersection of the different curves with the abscissa indicates the silica shell is homogeneous.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 1
Overview of the prepared particles and results of size and shape from TEM F. Dekker et al. / Journal of Colloid and Interface Science 571 (2020) 419-428

Table 2
Overview of the samples measured with SAXS.

Table 3
Overview of the obtained radii of gyration from SLS and TEM. a : standard deviation originates from the spread in particles sizes based on a number count.b : standard deviation originates from the standard error in first order fitting parameters of lnðIÞ vs K 2 .