Revised scattering exponents for a power-law distribution of surface and mass fractals

We consider scattering exponents arising in small-angle scattering from power-law polydisperse surface and mass fractals. It is shown that a set of fractals, whose sizes are distributed according to a power law, can change its fractal dimension when the power-law exponent is sufficiently big. As a result, the scattering exponent corresponding to this dimension appears due to the spatial correlations between positions of different fractals. For large values of the momentum transfer, the correlations do not play any role, and the resulting scattering intensity is given by a sum of intensities of all composing fractals. The restrictions imposed on the power-law exponents are found. The obtained results generalize Martin's formulas for the scattering exponents of the polydisperse fractals.

Figure 1

The dependence of the scattering exponent $\alpha$ on the power-law exponent $\gamma$ for 3D mass (black lower lines) and surface (red upper lines) fractals. (a) Results obtained in this paper given by Eqs. (5) and (6). (b) Martin's results [12] given by Eqs. (7) and (8).

Figure 2

Construction of a system with discrete power-law distribution: CMF of scaling factor ${\beta }_{\mathrm{m}}=0.33$, fractal dimension $D_{\mathrm{m}}=1.25$, and the smallest iteration $m=1$ is distributed with the exponent $\gamma =1.15$ corresponding to the scaling factor $\beta =0.3$. The number of the “block” iterations is given by $n=2$, and, besides, $f=1$. Black dots: scattering units from $0\mathrm{th}$ block (see the main text for details). Red dots: scattering units from $1\mathrm{st}$ block. An object from this block is encircled in the green circle. Blue dots: scattering units from $2\mathrm{nd}$ block. An object from this block is encircled in the orange circle. $d_0=l_2/{\beta }_{\mathrm{m}}^3$ and $d_1=l_2/{\beta }_{\mathrm{m}}^2$ are the overall lengths of CMF at iterations three and two, respectively. $L$ is the overall size of the construction, and $l_i$, with $i=0,1,2$, are the sizes of objects from the $i\mathrm{th}$ block (see the main text).

Figure 3

The normalized total (black) and structure-factor (magenta) intensities, given by Eq. (12), vs the momentum transfer (in units of $1/L$) for the structure shown in Fig. 2. $L$ is the overall size of the entire structure. Smoothed intensity (16) [shown in green (light gray)] is obtained with Eq. (15). (a) The regime $\gamma >D_{\mathrm{m}}$ and $2D_{\mathrm{m}}-\gamma >0$. (b) $\gamma >D_{\mathrm{m}}$ and $2D_{\mathrm{m}}-\gamma =0$. (c) $\gamma <D_{\mathrm{m}}$. Vertical dotted lines indicate the borders of the ranges with different exponents (see the discussion in the text). Dashed horizontal lines show the asymptotics $S_{\mathrm{as}}$ (13) and $I_{\mathrm{as}}\simeq 1/N_{\mathrm{tot}}$. Here $N_{\mathrm{tot}}$ is the total number of scattering points in the entire structure.

Figure 4

The 2D Apollonian gasket (orange) consisting from 121 disks. (a) The scaling factor $f=1$. (b) $f=0.6$. Black circles are added for better visualization.

Figure 5

The normalized total intensity (black), smoothed intensity [green (light gray)], and structure factor (magenta) vs momentum transfer (in units of the inverse overall size $1/L$) for the structure shown in Fig. 4. (a) $\gamma >D_{\mathrm{m}}$ and $2D_{\mathrm{m}}-\gamma >0$. (b) $\gamma >D_{\mathrm{m}}$ and $2D_{\mathrm{m}}-\gamma <0$. Vertical dotted lines mark the regions with different power-law exponents (see the main text for details). The dashed horizontal line represents the asymptotic values $S_{\mathrm{as}}$ (18).

Figure 6

The model based on the AG shown in Fig. 4: the disks are replaced by CMFs. The first 40 fractals are represented. The circles are imaginary and shown for better visualization. (a) $D_{\mathrm{m}}=0.9$, for which $2D_{\mathrm{m}}-\gamma >0$. The fractal iteration number inside the smallest AG disk is chosen to be $m=1$ (blue disks). Higher iterations of CMF $m=2$ and 3 are given in red and black, respectively. (b) $D_{\mathrm{m}}=0.65$, for which $2D_{\mathrm{m}}-\gamma =0$. The fractal iteration number inside the smallest AG disk is chosen to be $m=1$ (red dots). The second iteration of CMF is indicated with black dots.

Figure 7

The scattering curves for the structure shown in Fig. 6. The notations are the same as in Fig. 5. (a) $\gamma >D_{\mathrm{m}}$ and $2D_{\mathrm{m}}-\gamma >0$. (b) $\gamma >D_{\mathrm{m}}$ and $2D_{\mathrm{m}}-\gamma <0$. The lower dashed horizontal line is $I_{\mathrm{as}}=1/N_{\mathrm{tot}}$, where $N_{\mathrm{tot}}$ is the total number of scattering points.

Figure 8

(a) Dense random packing of $N=2067$ disks inside a square. The disk radii follow the power-law distribution with the exponent $\gamma =1.5$. The scaling factor $f=1$. (b) The same diagram with $f=0.6$. Black circles are added for better visualization.

Figure 9

The normalized total intensity (black), smoothed intensity [green (light gray)], and structure factor (magenta) vs momentum transfer (in units of the inverse overall size $1/L$) for the structure shown in Fig. 8. (a) $\gamma >D_{\mathrm{m}}$ and $2D_{\mathrm{m}}-\gamma >0$. (b) $\gamma >D_{\mathrm{m}}$ and $2D_{\mathrm{m}}-\gamma =0$. Vertical dotted lines mark the regions with different power-law exponents (see the main text for details). The dashed horizontal line represents the asymptotic values $S_{\mathrm{as}}$ (18).

Figure 10

The normalized total intensities without and with smoothing at various values of parameter $f$ vs momentum transfer (in units of the overall size $1/L$) when $2D_{\mathrm{m}}-\gamma >0$. The relative variance ${\sigma }_{\mathrm{r}}$ for the smoothed curves is equal to 0.2. From left to right: $f=1$, 0.4, 0.2, 0.1, and 0.05. The vertical dotted line marks the transition of intensities with exponents from $\alpha =\gamma$ to $\alpha =2D_{\mathrm{m}}-\gamma$. The intersection of the continuous horizontal line at about $2.5\times {10}^{-6}$ with scattering curves is an estimation of the position of the transition point between intensities with exponents $\alpha =2D_{\mathrm{m}}-\gamma$ and $\alpha =3$.

Figure 11

The model-dependent construction consisting from CMF of fractal dimension $D_{\mathrm{m}}$ inside a power-law distribution of disks (black) with exponent $\gamma =1.5$, at $f=1$. (a) $D_{\mathrm{m}}=0.9$ and $2D_{\mathrm{m}}-\gamma >0$. (b) $D_{\mathrm{m}}=0.75$ and $2D_{\mathrm{m}}-\gamma =0$.

Figure 12

The scattering curves for the structure shown in Fig. 11. The notations are the same as in Figs. 5 and 7. (a) $\gamma >D_{\mathrm{m}}$ and $2D_{\mathrm{m}}-\gamma >0$. (b) $\gamma >D_{\mathrm{m}}$ and $2D_{\mathrm{m}}-\gamma =0$. The lower dashed horizontal line is $I_{\mathrm{as}}=1/N_{\mathrm{tot}}$, where $N_{\mathrm{tot}}$ is the total number of scattering points.