Formation of hollow nanoshells in solution-based reactions via collision coalescence of nanobubble–particle systems

Nanobubbles help in synthesizing hollow nanoshells. Such shells are spherical and have narrowly distributed radii R (figure 1). It seems as if bubbles never change their radius or ever collide. This suggests pre-existing, free nanobubbles. The collision coalescence of bubble–particle systems (BPSs) seems obviously mistaken. It is a great challenge to observe the formation of the shells. The high pressures and temperatures during hydrothermal synthesis make the reaction system a black box. In this work, we will demonstrate that the collision coalescence of BPSs can explain the formation of nanoshells, while no simpler theory is compatible with the empirical observations.

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The number density n of shells (and thus of bubbles during growth) is proportional to the mass density of the available shell material. For example, palladium concentration C_Pd is multiplied by atomic weight m_Pd and divided by a shell's mass. The latter equals shell volume ${V}_{{\rm{s}}}=4\pi ({R}^{3}-{(R-h)}^{3})/3$ times density ρ_Pd = 12.02 g cm⁻³ divided by a porosity parameter p. In general: $n={C}_{{x}}{m}_{{x}}/[{V}_{{\rm{s}}}({\rho }_{{x}}/p)].$ The specific surface of smooth shells is 4πR²/[V_s(ρ_x/p)], being for example only 6 m² g⁻¹ for dense (p = 1) Pd shells. A typical product (figure 1(a)) has instead a five times larger BET surface area of 30 m² g⁻¹. A rough surface is defined to be one where the depressions are still wider than they are deep; such that they can triple surface areas. Our fivefold value proves true porosity (holes deeper than wide), and indicates a porosity factor of p = 1.5 to 2. However, the porosity p cannot be far above unity (consider the shells' structural integrity), and a log-normal distributed R gives only a 5% relative error to n. One can therefore estimate n for many reports that claim the involvement of free, pre-existing bubbles. SiO shells [1] imply up to 10 bubbles per μm³, which is similar to the number of bubbles derived from observations of ice fractures with about 20 bubbles per μm³ [2]. Pd shells [3], with similar radii, imply between 10⁻² and 10⁻³ (figure 1(c)). ZnSe microshells [4] imply only 3 × 10⁻⁴ down to 5 × 10⁻⁵, almost a million times less than 20/μm³ [2]. Where are the missing bubbles if they pre-existed? Worse, shell thickness h can stay constant when R doubles [1], although the shell material spreads over the bubble surface (∼R²), implying that n decreased to only 25%. If diffusive gas exchange removes 75% of bubbles before shell growth starts, the size distribution would Ostwald ripen [5] into a sub-population of bubbles that shrink away completely and a second sub-population of bubbles that grow to about 10 μm, at which point they reach the surface within one minute. This speaks against pre-existing bubbles, and indeed, shell-synthesis usually involves gas production during the assembly of the shells. Even the SiO shells blamed entirely on already present gases [1] involve a gas producing decomposition (SiO → Si + O). While n varies by over six orders of magnitude among reports, the bubbles' gas molecules are usually on the order of the number of atoms in the shell. There are about 20 times more Pd atoms in the Pd shells [3] (5 to 60 million) than gas particles in the bubbles. The report on hydroxyapatite microspheres [6] similarly calculates 30 to 90 million CO₂ molecules in its average bubbles. Thus, the hypothesis that bubbles and shells come into existence together and that their numbers are closely correlated may yet arise, but the issue is not this trivial. With the Pd shells, n increases with the concentration of the Pd that turns formic acid into gas (figure 1(c)), as intuitively expected. However, a greater gas supply at constant C_Pd decreases n. If the grains which build the shells immediately generate and maintain bubbles, collision coalescence (figure 1(d)) of such BPSs is inevitable. Every single grain is likely immediately the particle of a small BPS. Details of the collision statistics explain the otherwise counterintuitive relations.

We discuss mostly hollow Pd shells, but the counterintuitive behaviors have been observed in many other experiments. Hollow Pd shells were obtained in a synthesis [3] where formic acid at initial concentration C_F reduces ${{{\rm{PdCl}}}_{4}}^{2-}$ ions and supplies gases. A 'typical' synthesis is defined by the vector (r, C_F, C_Pd, T) being (0.64, 2.88 M, 0.15 mM, 100 °C). The autoclave filling ratio r describes the initial height of the fluid column, but r = 0.8 becomes r_100°C = 0.9, halving the space above the expanded fluid and increasing gas pressures. Vapor pressures depend only on T and Antoine equations predict [7] that, even at 190 °C and r only 0.2 (r_190°C = 0.223), 98% of water is still in the liquid phase. Nevertheless, below r = 0.53, mostly irregular aggregates result, and networks result at r = 0.8. The strong r-dependence implies gas release, which makes many parameters time-dependent.

With C_F at 0.72 M, only nanoparticle aggregates are obtained. From C_F = 1.44 M to 5 M one observes R = 43 to 180 nm, all ±15% (figure 2). R increases with C_F. However, a larger C_F should provide more gas pressure and smaller radii, a common experimental observation [6] with nanobubbles. The Laplace pressure ΔP = 2γ/R dominates the bubble pressure and is upheld by the oversaturation ΔC_gas according to Henry's law (ΔP = ΔC_gas K_H). Hence, $R\propto 1/{\rm{\Delta }}{C}_{{\rm{gas}}}$ and a straightforward $R\propto 1/P$ has indeed been reported [1]. An increase of R with gas concentration is predicted for non-interacting bubbles in a closed volume [8] (similar to shielding [9]). However, those are meta-stable states [5]. Neither the models' numbers nor slopes fit the data, and n would need to adjust before the shell growth, as if anticipating a future equilibrium.

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Large shells' surfaces are incomplete (figure 2(b)), hence, yet higher C_F cannot produce stable shapes. The respective shell thicknesses h are 13, 21, 16, 12, 10, and 7 nm (±20%). One may expect $h\propto {R}^{-2}$ because the Pd is distributed over the shell surface, but R triples while h decreases by a factor of three from 21 to 7 nm. Therefore, although C_F increases the gas supply, this counterintuitively decreases n (figure 1(c)). This suggests collision growth: size R depends on the amount of Pd the BPSs carry; and larger capture cross sections deplete smaller BPSs due to differing relative mobility. C_F increases R, and the collision cross section is then larger early on, thus fewer shells result. Early on, n may be proportional to C_Pd, but the final R is stabilized dynamically by the number of grains on the BPS and local gas concentrations, so R becomes independent of C_Pd. In the cited, meta-stable [5] many-bubble scenarios [8], R would depend crucially on C_Pd. The experimental data however could not be clearer: R stays constant when varying C_Pd. Keeping C_F at 2.88 M, C_Pd = 0.15, 0.25, and 0.4 mM results in h = 16, 23, and 28 ± 5 nm, when irregular grain aggregates start to appear. R does not change. This is confirmed impressively with C_F = 5 M and C_Pd = 0.4 mM: R is 180 nm again, but h adjusted to a thick 18 nm. Also n is thus almost as before, because the two effects compensated each other. C_F decreases n. C_Pd increases it, because the number of grains, which is the initial n, increases proportionally. C_Pd does not, but C_F does tune R. This suggests dynamic stability from ongoing acid conversion at a rate that increases with C_F. More Pd increases the total rate, but at the BPS, the local concentration is important. Diffusion calculations show that global flow equilibrium is reached and can therefore determine the final R. More Pd can increase gas production only very early on. C_Pd is irrelevant for the later growth stages because the initial collision rates are proportional to the initial density squared, which removes more grains faster.

4.1. Out-bubbling

4.2. Reduction

4.3. Diffusion

4.4. BPS

Collision growth of clusters explains many phenomena such as sizes of raindrops under warm clouds [24], stable homogenization of emulsions [25], and particle size distributions in spray dryers [26]. Our calculations involve tests of algorithm stability, time optimizations, and many physical assumptions (see supplementary material). We use bins i. Their BPSs have on average ${q}_{i}=100\;i$ grains. The number densities ${n}_{i,t}$ are initially zero, except for the initial grain number density ${q}_{1}{n}_{\mathrm{1,0}}=11.7\;\mu {{\rm{m}}}^{-3}({C}_{{\rm{Pd}}}/0.15\;{\rm{mM}})$ in the first bin. We derive a typical population balance equation:

$$\begin{eqnarray}&&{n}_{i,t+\delta }={n}_{i,t}+{\rm{\Delta }}t\left(\displaystyle \sum _{k=1}^{i-1}{n}_{k,t}{v}_{k}{\sigma }_{k,i-k}{n}_{i-k,t}\right.\\ &&\quad \quad \quad \quad \;\left.-{n}_{i,t}\displaystyle \sum _{k=1}^{i\mathrm{max}}({v}_{k}+{v}_{i}){\sigma }_{i,k}{n}_{k,t}\right)\end{eqnarray} \tag{ 1 }$$

R_i may stay zero until some minimum $q\approx {q}_{1},$ and then linearly grow with i. Most importantly, it rises ever more slowly with i. This is modeled by ${R}_{i}=\tfrac{i}{i+1}{R}_{\infty -\mathrm{typical}}{\left(\tfrac{{C}_{{\rm{F}}}}{{C}_{F-\mathrm{typical}}}\right)}^{x},$ a time-independent effective radius (as if the global gas concentration does not change). It reproduces R's observed dependence on C_F if ${R}_{\infty -\mathrm{typical}}\approx 80\;{\rm{nm}}$ and x ≈ 1.3. We can reproduce the observations, including reaction times. The fits do depend on certain assumptions; however, recall that available theories of free bubbles cannot model the parameters' behavior. Collision coalescence can describe it. Two further results depend on the effective relative velocity between any two BPSs indexed by i and k, which depends on R_i of the 'projectile' (not the target), namely via ${v}_{i,k}\propto {{R}_{i}}^{-z}.$ Values of z smaller or equal to unity lead to ${n}_{i,t}\gt {n}_{i+1,t}$ and cannot produce a peaked distribution. Size selection improves with higher z. However, another result is that n increases with C_F at $z\gt 2.$ At z = 1.6, C_F = 2.2 M, C_Pd = 0.15 mM, and after t = 50 min (25 time steps), the distribution of BPSs has already peaked (figure 2(c)). The distribution over the n-bins is much broader (the long tail misleads) than the distribution of radii, which peaks at a lower bin. Thus, the average is only $R=41\pm 9\;{\rm{nm}}.$ The process size-selects effectively, because R's standard deviation decreases with t. R stabilizes about 25% below ${R}_{\infty -\mathrm{typical}}$ (figure 2(d)).

The counterintuitive behavior of important parameters in nano-shell syntheses led to a novel analysis that focuses on the number density n of the involved bubbles. The analysis was applied to several published reports, and it was found that free-bubble theories fail to account for the observed behaviors. The suggested collision coalescence between bubble–particle systems (BPSs) is consistent with all observations. The stability of the radius is partially a dynamic stability at all radii, but the collision process statistics also runs into a certain average radius R. The evolving size distributions reproduce bubble number densities n that are otherwise usually several orders of magnitude wrong. The simulations use an effective relative collision velocity that depends on a power z. Only $1\lt z\lt 2$ allows size selection as well as the counterintuitive decrease of n when increasing shell material. Also, grain-monolayer thin, incomplete and multi-metallic shells with superior catalytic activity were synthesized via a new hydrothermal route. The power z = 1.6 reproduces these experiments very well.