Supplemental

In the main article, the USAXS data was fit using a two-level unified equation that assumes that the domains are statistically isotropic (refer to Fig. 810(c)). In this supplementary, the domain anisotropy was accounted for by considering the plates and domains as oriented cylinders with some distribution in the orientation [1] as shown in Fig. 8(d). This approach, however, was complicated by the fact that the USAXS data are slit-smeared with scattering that is different from the scattering about the vertical scatting vector, qz. In this supplementary, we discuss the anisotropic model that was obtained from this study using a minimum number of fit parameters as possible. These fit parameters were then compared with the parameters obtained from the unified equation.

USAXS GeometryThe Bronse-Hart setup used in this investigation has a few advantages over a traditional pinhole-SAXS instrument. These advantages include larger beam sizes and a wider q-range. To maximize the X-ray flux, however, the scattered X-rays that were collected were slit-smeared. For the instrument used in this investigation, the scattered X-rays measured the qz, shown in Fig. S1, via the analyzer crystals. These scattered X-rays, however, also contain a qy component at the detector that makes the measured intensity “slit-smeared”. When the scattering population is isotropic where I(qz) = I(qy) = I(q), then this correction can be made by either smearing the model or de-smearing the data. For anisotropic systems, however, the model must be appropriately smeared. Any vector in reciprocal space, qR, can be represented by its unit vectors by the equation:

qR=qx i+q y j+qz k (S1)Where

qz=4 π sinθ /2λ (S2)

Where the coordinates x, y, and z are defined in Fig. S1, q is the magnitude of the measured scattering vector, θ is the angle of measurement, and λ is the X-ray wavelength. Since there is a component of qy in qz, the intensity from any model must be smeared across the slit-length, qslit, at each increment in θ. Here, the data are smeared by the equation:

I smr ( qz )=K∫0

q slit

I model (qR (q y , qz )) d q y (S3)

Where K is the scaling constant and the scattering vector, qR is calculated from qy in qz via each respective angles of ψ and θ following

qx=2π (cos−θ cosψ−1 )

λ(S4)

q y=2 π (sin ψ )

λ(S5)

qz=2 π (sin−θ cosψ )

λ(S6)

Where θ is calculated from Equation S2 and ψ varies between 0 and the valued defined by qslit. Equation S3 is generally applicable to any small-angle scattering model but it is computationally arduous, especially when the scattering is anisotropic.

Fig. S1. Schematic showing the geometry and coordinates that were used to develop the domain model.

Model Development

The small-angle scatting (SAS) model was based on the observed zinc morphology from the SEM imaging where the zinc deposition appeared to consist of crystal domains that are composed of stacked plates. For the SAS model, these plates were considered as disks with a specific thickness, tp, and radius, Rp, which are stacked to form cylindrically shaped “domains”, shown in Fig. 8, with a height of 2RAr where Ar is the aspect ratio of the domain. For a first approximation, the scattering from the domains was modeled as the summation of the SAS from the entire domain and the small portion of its plates that did not merge and remained discrete. This approximation makes the following assumptions:

1. There is a size distribution of domains that accounts for both domains with a different number of plates and domains with different degrees of coalescence between the plates.

2. The orientation distributions with respect to θ and ψ, and the size distribution are Gaussian.

3. The plates or space between the plates have a height of 5 nm which was measured from the AFM images (refer to the main article).

4. The degree of fusing was sufficiently high such that the interference scattering between the plates can be ignored. This assumption arises from the observation of the relative scale between the Guinier knees in the USAXS data. Therefore, the coalesced plate volume fraction. Vmerged, should be near one.

5. The orientation mode, θo, was varied and a value of σθ = 20° was assumed. We note that the data can be fit to a model that assumes θo = 90° and σθ varied, which results in the increase in σθ with monolayer addition. Similarly, θo = 0° can be assumed, but the chi-square is slightly higher and the fit parameters were not consistent with the qualitative trend observed by AFM imaging (Fig. 3). From these experiments, there are more domains with an orientation of 90° (parallel to the substrate) with 400 monolayers. As the monolayer number increased to 2800, more domains perpendicular to the surface (θ° > 0) were observed.

The scattered intensity from a size distribution of domains, I ODP (qR (q y , qz )), was calculated by the following equations:

I ODP (qR (q y , qz ))= ∑R=0.1R

R=10 R

V cyl(R , RAr) [ IOD (|qR|, R , α )+ I OP (|qR|, R , α ) ]Pv ( R , R , σ ) Δ R (S7)

I OD (q ,R , α )= [ ρD−ρsol ]2 Fcyl

2 (q , α ,R , RAr ) (S8)

I OP (q , R ,α )=[ ρp−ρspace ]2 ν p F cyl2 (q ,α , R , t p ) (S9)

F cyl (q ,α ,t p , R )=2[ sin q t pcos αq t p cosα ][ J 1 (qR sin α )

qR sin α ] (S10)

where ρ space is the electron density of the space between the plates, R is the mean radius, Pv is the volume distribution that was assumed to be Gaussian with a standard deviation of σ, α is the angle between the cylinder axis and qR, J1 is the first order Bessel function, and ρ is the scattering length density. Equation S8 and Equation S9 are connected by two parameters, νp and ρD and the scattering from the plates and domains are calculated using the normalized scattered intensity of an oriented cylinder, Fcyl. Both parameters, νp and ρD, are calculated using the coalesced volume fraction, νmerged, by the equations:

νp=N p

V plate

V cyl=[1−νmerged ] [1−νspace ] (S11)

ρD=νmerged ρp+ [1−νm ] [ νspace ρspace+ [1−ν space ] ρp ] (S12)

Where νspace is the volume fraction of space between plates, ρ space is the electron density between plates, and νmerged is the volume fraction of the stack that coalesced. Equation S11 assumes that the space between the plates is poorly defined and therefore does not scatter. An example of such a system would be stacked plates that exhibits roughened surfaces resulting in a region between the plates that exhibits a mean electron density that is lower than within the plates. Accounting for the anisotropy of the domains, the scattering model, I model (q ), is calculated by the equation:

Imodel ( qR (q y , qz))=KS (q , ξ , p )∫0

π

∫0

π /2

Pθ (θ , θ0 , σθ ) Pφ ( φ ,φ0 , σφ ) ΙODP (qR ( qy , qz ))sin θ d θ d φ+b (S13)

Where the function, S (q , ξ , p ), is a structure factor that accounts for a distribution of mean distance, ξ, between domain centers, and some degree of crystallinity that is accounted for in the packing factor, p. In Equation S13, Pθ and Pφ are Gaussian functions that assign the probability of a specific orientation defined by Fig. S1 [1] From the SEM imaging shown in the main article, the function Pφ is one while Pθ is defined by the standard deviation, σ θ, and the mean, θ0. The double integral in Equation S13, combined with the integral in Equation 3 were computationally expensive and typical calculation times were between 15 minutes and 40 minutes using a computer with a 3.6 GHz Intel Xeon E5 Quad-Core processor and 32 GB of memory.

Model Fitting

Equation S13 is a scattering model expected based on the SEM imaging. From the AFM measurements, the thickness of the plates, tp, was estimated to be 5 nm and was used in the model fitting. While this parameter can be varied during the fit, it is correlated, to some degree, with σ θ and θ0. The parameters ξ, p and b found by least-squares fitting of each USAXS data set using the unified equation (Equation 2) were used in the model fitting of Equation S13. These parameters could be varied but result in longer fitting times and do not affect the portion of the USAXS data that corresponds to the individual domain and plate morphologies. One of the advantages of using Equation S13 is that the evolution in plate and domain orientation can be extracted from the model fitting, since the plate thickness was taken from AFM data and was known a priori. This can be accomplished by either assuming θo equals 90° and fitting σθ or by assuming a reasonable value for σθ and fitting θo. In order to communicate the evolution in a clear way, σθ was assumed to be 20° and θo was fit to the data; in all cases, the initial value of θo was 45°, and varied to achieve a minimum χ2, given by the equation

χ2=∑ [ Ires (q ) ]2 (S14)

I res (q )=[I model (q )−Idata (q ) ]

e (q)(S15)

Where e(q) is the error obtained from the instrument, Idata(q) is the measured intensity from the Zn deposit and Ires(q) is the standardized residual. There are a total of 6 fit parameters that are varied to minimize the chi-square statistic, using the Lmfit package [2] (Non-Linear Least-Squares Minimization and Curve-Fitting) for Python. These parameters are: K, R, σ, θo, νmerged and tp. The model fits to the data and standardized residuals are shown in Fig. 9 in the main article for both Equation S13 (anisotropic model) and Equation 2 (unified model).

The results of the anisotropic model (Equation S13) can be compared to the results obtained from the unified model (Equation 2). In the main article, the radii of gyrations of the domains and plates were extracted from the unified model fits. These parameters depend on both orientation and shape and therefore it is useful to compare them with the size distributions obtained from the model fitting shown in Fig. S2. Similar to the RgD evolution shown in the main article (Fig. 5a), the size of the domains increases with increasing monolayer addition. The degree of coalescence, Fo, calculated from Equation 5 in the main article was also calculated from the fits of Equation S13 by:

Fo=[ ρD−ρsol ]

2

[ ρp− ρspace ]2 ν p

(S16)

The evolution in the degree of plate coalescence, Fo, from Equation S16 is compared to Equation 5 from the main article in Fig. S3. In both models, the same increase in Fo was observed with monolayer addition and domain growth. From the model fits of the anisotropic model, the deposit was most aligned within a few monolayers (Fig. S4). With increasing growth, the domains take on more orientations, which suggested that new domains were nucleated and grown; this is also apparent in Fig. S2, where the widths of the size distributions increase with monolayer addition. Therefore, the anisotropic model can be used to extract the same information as the unified model, as well as some additional information about the orientation and size distribution. The main advantage of Equation S13 is that the plate thickness, tp, obtained by AFM, can be used as an input parameter to obtain the evolution in orientation and aspect ratio.

Fig. S2. Plot of the size distribution of the domain radii obtained from the model fit to Equation S13.

Fig. S3. Plot of the degree of coalescence, Fo, obtained from Equation S13 to the slit smeared USAXS intensity collected for the USAXS scans at 400, 1200, 2000, 2800 and 3600 Zn ML of charge passed.

Fig. S4. Plot showing the evolution in the standard deviation,σ θ, and the aspect ratio, Ar, obtained from Equation S13 to the slit smeared USAXS intensity collected for the USAXS scans at 400, 1200, 2000, 2800 and 3600 Zn ML of charge passed.

References

1. Guinier, A. and G. Fournet, Small-angle scattering of X-rays. Structure of matter series. 1955, New York,: Wiley.

2. Newville, M., et al., LMFIT: Non-Linear Least-Square Minimization and Curve-Fitting for Python. 2014.