Breakdown of fractal dimension invariance in high monomer-volume-fraction aerosol gels

EDITOR:

• Matti Maricq

Aerosol gels are volume-spanning, semi-rigid networks of solid nanoparticles possessing ultralow density and high surface area (Dhaubhadel et al. 2007; Chakrabarty et al. 2014). The current consensus is that aerosol gels have a hybrid morphology with equal mass (D_m) and surface (D_s) fractal dimensions ∼2.5 at super-micron length scale and  ∼ 1.8 at submicron length scale, respectively (Sorensen et al. 2003; Dhaubhadel et al. 2006; Kim et al. 2006; Chakrabarty et al. 2014; Liu et al. 2015). In this study, we show that this consensus is partially complete and not universal. Gels, when formed under high monomer-volume-fraction (f_v) conditions, have a D_s ∼ 2 implying solid-like surfaces while retaining their D_m ∼ 2.5. In other words, the conventional assumption of dimensional invariance (D_m = D_s) breaks down for monomer-dense gels.

Using our static light scattering (SLS) apparatus, which is similar to Wang et al. (2015), we studied the structure factor S(q) in the angular range – of carbon aerosol gels produced from a buoyancy-opposed flame (BoF) aerosol reactor (Chakrabarty et al. 2014). The details of the experimental setup are available in the accompanying online supplementary information (SI). The hybrid fractal morphology of the gel particles was probed by plotting the scattered intensity as a function of the magnitude of scattering wave vector q, where with λ the wavelength of incident light and θ the scattering angle. The scattering wave vector has units of inverse length, thus q⁻¹ represents a probing length scale (Sorensen 2001). In our experiment, the monomers composing the gel particles are assumed to only scatter the incident light with no interaction with the neighboring monomers scattered light (also known as the Rayleigh–Debye–Gans [RDG] limit). Under this assumption, the scattered intensity measured at the detector is proportional to S(q) which in turn scales with a power law exponent of − (2D_m−D_s) (Sorensen 2001).

Figure 1a shows the S(q) measured for our gel particles, in which the (2D_m−D_s) exponent was found to be 3.0 ± 0.2 at q < 2 µm⁻¹ accompanied by a crossover to 1.8 ± 0.2 at larger values of q. Note that the uncertainties presented here and throughout this article are maximum ranges of the power-law fits. This crossover indicates the change in the gel’s fractal structure at sub-to-super-micron length scale. An enhanced backscatter signal is also apparent in Figure 1a. Theoretical work which considers only one aggregate at a time has never shown an enhanced backscattering. We speculate the enhanced backscattering is due to the natural property of an ensemble of aggregates: the scattered light from one aggregate gets reflected in the backward direction by other aggregates. This enhanced backscattering behavior has been observed previously for D_m = 1.8 soot aggregates (Heinson 2016).

Figure 1. Experimental results. (a) Structure factor S(q) of the carbon gel particles produced from our BoF aerosol reactor. S(q) scales with a power law exponent of − (2D_m−D_s) with values of −3.0 ± 0.2 and −1.8 ± 0.2 indicating the hybrid morphology at different length scales. The white bars in the inset of electron microscopy images of the carbon gels represent 10 µm length scale. (b) The number of monomers N vs. radius of gyration normalized by monomer radius, R_g/a of these gels is shown by diamonds (experimental work) (Liu et al. 2017) and circles (simulations using our diffusion limited cluster-cluster aggregation (DLCA) model). Both indicate a D_m = 2.5 ± 0.1. From (a) and (b), a D_s ≈ 2 (smooth and solid surface) is implied for these particles.

Recently, Liu et al. (2017) demonstrated that the number of constituent monomers (N) in an aggregate could be experimentally determined by processing its two-dimensional microscope image after applying proper correction factors to account for projection artifacts. Here, we use the previously obtained N datasets from Liu et al. (2017) for the carbon gels to evaluate their mass fractal scaling relationship. Aggregates have a fractal structure with a mass scaling dimension D_m that describes N with the aggregate’s linear size quantified as [1] where R_g is the root-mean-squared radius of the aggregate, a is the monomer radius (a mean of 30 nm for our carbon gel), and k₀ is a prefactor. Based on our measurements, shown by diamonds in Figure 1b, we find the average D_m = 2.5 ± 0.1 for our gel particles, which is consistent with previous findings (Sorensen et al. 2003; Dhaubhadel et al. 2006; Kim et al. 2006; Chakrabarty et al. 2014; Liu et al. 2015). The low monomer-packing density, observed in Liu et al. (2017) to be 0.24–0.002 for aggregates corresponding to R_g = 0.8–200 µm, respectively, suggests small values for the Lorentz–Lorenz factor implying minimal cross-talk between the scattered light of the monomers, thus, justifying our use of the RDG approximation. Combining the knowledge of D_m with S(q) measurements, we retrieve D_s = 2.0 ± 0.4 implying a smoother or more solid surface for the gel particles.

This surprising and previously undescribed observation was further investigated using the diffusion-limited cluster–cluster aggregation (DLCA) algorithm under varying f_v conditions. Sub-micron size aggregates made with the DLCA algorithm have a D_m = 1.8 and on average k₀ = 1.35 (Meakin 1985; Heinson et al. 2012). These aggregates possess invariant D_m = D_s = 1.8 and their S(q) scales with a power-law exponent of −1.8 (see Figure S6 of SI). Since D_m is less than the spatial dimension d = 3, these aggregates, if allowed to grow beyond this size range, will eventually form gel particles with D_m = 2.5 (Heinson et al. 2017). The crossover to the gel phase is marked by the ideal gel size R_g,G defined as: [2] where f_v is the primary parameter controlling this transition for DLCA aggregates. Aggregates with sizes greater than R_g,G are space-filling gels with a D_m = 2.5 and are composed of sub-micron size aggregates of D_m = 1.8 (Fry et al. 2004; Heinson et al. 2017). A general description of the DLCA algorithm is available in the SI.

Numerous experimental studies have shown that DLCA accurately predicts the formation of real-world aggregates (Jullien and Botet 1987; Sorensen et al. 1992; Cai et al. 1993). In the BoF reactor, the formation of carbon gels primarily occurs where the aerosol flow reverses direction, as shown in Figure S4 of SI. Here, the soot particles focus on high f_v regions, where they aggregate together to form gel particles. We postulate that when DLCA simulations are run under high f_v conditions, they produce aggregates that capture the essential morphological features of our carbon gel particles. Resuspension of the gel particles and subsequent transport to the scattering volume of our SLS apparatus does involve fragmentation of the millimeter-sized gels into tens of micron-sized particles. Pictorial examples of nascent and fragmented gel particles, shown in Figure S5 of SI, visually highlight the scale-invariant fractal nature of the particles over several orders of magnitude in size. Since the maximum probing length of our SLS apparatus is ∼3 µm, which is much smaller compared to the size of our fragmented gel particles, there arose no necessity to implement the phenomenon of fragmentation in our DLCA simulations. The simulated aggregates corresponding to f_v = 0.1 (hereafter, monomer-dense gel; see Figure 2b inset) represent the best match to the carbon gel particles produced from our flame reactor. Note that both the simulated DLCA aggregates shown in the insets of Figure 2 possess a hybrid morphology corresponding to D_m≈2.5 and 1.8 at super- and sub-micron length scales, respectively. The difference in their morphology, however, stems from their distinct surface features – the less dense gel particles corresponding to f_v = 0.003 (referred to as monomer-dilute gel in Figure 2a) has a more open and porous surface compared with that of Figure 2b.

Figure 2. Monomer pair correlation function g(r) for gel particles and their projected surfaces. Pair correlation vs. intermonomer distance r normalized by monomer diameter d scales as r^D-3, where D = D_m or D_s depending on the type of analysis (volume or surface). (a) For the less dense gel particles corresponding to f_v = 0.003 (Monomer-Dilute Gel), both volume g(r) and surface g(r) follow a D_m = D_s = 1.8 trend (solid line) until crossing over to the super-micron size regime where D_m = D_s = 2.5 ± 0.2. (b) The “Monomer-Dense Gel” corresponding to f_v = 0.1 follows the D_m = D_s = 1.8 trend at small length scales ∼ r/d = 4. At scales larger than this, surface g(r) follows a D_s = 2.0 ± 0.1 while g(r) maintains a D_m ≈ 2.5. Both experience an exponential cutoff at large r/d due to the finite size of the particles.

To investigate the surface dimension and hybrid morphology of the simulated gels, we studied their monomer pair correlation function g(r), which is the probability that another monomer will be present at a distance r from a given monomer. g(r) is proportional to the average number of monomers in a shell of radius r with thickness dr and thus scales as r^D−3, where D is D_m or D_s depending on the aggregate volume or surface analysis, respectively. The g(r) was calculated for the simulated gels for their entire volume (circles) and their projected surfaces (open squares), and are plotted versus r normalized by the monomer diameter d to obtain independent measurements of D_m and D_s (see Figure 2). Both the surface and volume g(r) follow a D_m = D_s=1.8 trend (solid line) until the crossover R_g,G length is reached at r/d ∼ 30 and 4 for the monomer-dilute and monomer-dense gels, respectively (Heinson et al. 2017). At length scales beyond R_g,G, the mechanism responsible for the correlation cannot be explained by classical DLCA-type aggregation. Beyond these crossovers, the monomer-dilute gels follow a D_m=D_s = 2.5 ± 0.2 trend (Figure 2a). For the monomer-dense gel (Figure 2b), the surface g(r) yields a D_s=2.0 ± 0.1 while g(r) for the entire volume follows a D_m=2.6 ± 0.2. The g(r) gets truncated with exponential-like cutoffs, indicating the outer bounds of the particles.

Figure 3a shows the forward normalized S(q) of the monomer-dense gel and monomer-dilute gel plotted versus q. The S(q) of the simulated aggregates is calculated using the following equation: [3] where r_i is the monomer coordinate. Based on the RDG approximation, S(q) is approximately equal to the scattered intensity of the gel particles (Sorensen 2001). For the monomer-dilute gel, S(q) scales with a power-law of −2.5 ± 0.2 at small q and crosses over to a −1.8 ± 0.1 slope. For the monomer-dense gel, S(q) scales with −3.0 ± 0.1 at small q similar to our experimentally measured S(q) after which it crosses over to a −1.6 ± 0.2 slope. In Figure 3b, the ensemble N vs. R_g/a data is plotted for both f_v = 0.003 and 0.1 DLCA systems. Both sets of aggregates, irrespective of the differences in their surface appearances, have fractal morphologies that scale with a D_m ≈ 2.5 at super-micron length scale. Given that S(q) scales as − (2D_m−D_s), the only way to reconcile the differences in the −3.0 and −2.5 slopes in Figure 3a is by assigning D_s=2 and D_s=D_m for monomer-dense and monomer-dilute gel, respectively.

Figure 3. Discrepancy in scaling of S(q) and D_m. (a) Forward normalized S(q) of monomer-dense (circles) and monomer-dilute gel (squares). For monomer-dilute gel, S(q) scales with a power-law of −2.5 ± 0.2 at small q and crosses over to a −1.8 ± 0.1 scaling. This scaling dependence is consistent with previous findings. For monomer-dense gel, S(q) scales with −3.0 ± 0.1 in the small q region and then crosses over to a −1.6 ± 0.2. (b) The corresponding N vs. R_g/a plots for both aggregate types scale with a constant D_m=2.5 ± 0.2 via Equation (1). For monomer-dense gel with D_m=2.5, their S(q) slope of −3 can only be explained provided D_s=2.0.

The porosity of gels is determined by the monomer-volume-fraction f_v. Since the monomer-dense gels have very small pores, their surface will appear smooth, i.e., the fluctuations in the surface become negligible at small correlation length scales leading to a D_s=2. The D_s=2 surface fractality of the dense gel particles is consistent with their appearance in the electron microscopy images, having a solid impermeable looking outer facade, while the more dilute gels (corresponding to low f_v values) still maintain their porous fractal-like appearance. In summary, this work has shown that super-micron length-scale aerosol gel particles, formed via the dense DLCA mechanism, do not follow the classical D_m=D_s=2.5 convention; instead, they assume hybrid fractal dependencies which can be uncovered experimentally using light-scattering analysis. Our observation of this fractal scale dependence occurring under monomer-dense conditions has significant implications for the synthesis of materials with tunable porosity, extremely low density and refractive index, and high surface area per unit volume, and accurate estimation of climate forcing by super-micron size soot aggregates from large-scale combustion processes.

Additional information

Funding

This work was supported by the US National Science Foundation (NSF) grants [AGS-1455215] and [CBET-1511964].