Detrended fluctuation analysis in x-ray photon correlation spectroscopy for determining coarsening dynamics in alloys

We study the dynamics of precipitate coarsening in phase-separating alloys at late stages of phase separation by x-ray photon correlation spectroscopy (XPCS). For analyzing time series of fluctuating speckle intensities from small-angle scattering of coherent x rays, the method of detrended fluctuation analysis (DFA), which is ideal for determining power-law correlations, is applied. We discuss the application of DFA with respect to XPCS data by means of simulated time series. In particular, the effects of different signal-to-noise ratios are examined. Results from measurements of the two model systems $\mathrm{Al}\text{−}6\mathrm{at}.\%\mathrm{Ag}$ at $140°\mathrm{C}$ and $\mathrm{Al}\text{−}9\mathrm{at}.\%\mathrm{Zn}$ at $0°\mathrm{C}$ are presented. Since the DFA effectively removes adulterating trends in the data, quantitative agreement with Monte Carlo simulations is obtained. It is verified that two different coarsening mechanisms are predominant in the two systems—coarsening either by diffusion of single atoms or by movement of whole precipitates.

Figure 1

(Color online) Analysis of simulated power law-correlated data $(\gamma =0.7)$ with the classical autocorrelation function $g^{\left(2\right)}\left(t\right)$ (solid black line) and the fluctuation function $F\left(t\right)$ calculated with FA (open circles) and DFA3 (open triangles), respectively. Double-logarithmic plot. Curves are shifted along the $y$ axis for clarity. The red arrow marks the point where the autocorrelation function becomes negative for the first time. The scaling behavior on longer time scales is completely inaccessible in that approach. The dashed gray line indicates $\gamma =0.7$. A reasonable fit is hardly possible. (D)FA, on the other hand, yields the correct scaling behavior over many orders of magnitude. The dash-dotted gray lines indicate $\alpha =0.65$, compare Eq. (7), which can be perfectly fitted.

Figure 2

Illustration of the effect of Poisson noise on DFA3 curves by means of simulated random walks of phases according to Eq. (12). $I_{\max }^{\mathrm{coh}}={10}^4(●)$, ${10}^3(엯)$, ${10}^2(▾)$, ${10}^1(▵)$, ${10}^0(∎)$, ${10}^{-1}(◻)$, and ${10}^{-2}(◆)$. Perfect coherence conditions are assumed, i.e., no incoherent background is present. Double-logarithmic plot. Curves are shifted along the $y$ axis for clarity. Dotted gray lines indicate slope $1∕2$ and the dashed gray line corresponds to $\alpha =3∕2$.

Figure 3

Illustration of the effect of Poisson noise plus an additional incoherent background, $I_{\mathrm{av}}^{\mathrm{inc}}$, on DFA3 curves by means of simulated random walks of phases according to Eq. (14). $(▵):I_{\max }^{\mathrm{coh}}=10,I_{\mathrm{av}}^{\mathrm{inc}}=0$. $(▴):I_{\max }^{\mathrm{coh}}=10,I_{\mathrm{av}}^{\mathrm{inc}}=45$. Double-logarithmic plot. Curves are shifted along the $y$ axis for clarity. Dotted gray lines indicate slope $1∕2$ and the dashed gray line corresponds to $\alpha =3∕2$.

Figure 4

(Color) Typical region of interest of the SAXS-diffraction pattern of the $\mathrm{Al}\text{−}9\mathrm{at}.\%\mathrm{Zn}$ sample at $0°\mathrm{C}$. Logarithmic pseudocolor-intensity scale from 1 to 35 photons (dark to bright). For improving statistics, the sum of 10 frames ($1\mathrm{s}$ exposure each) is shown, where a dark file was subtracted from each frame before summing. Since the coarsening dynamics was very slow, the speckle structure (“graininess”) is clearly visible and not washed out.

Figure 5

DFA3 results for $\mathrm{Al}\text{−}9\mathrm{at}.\%\mathrm{Zn}$ at $0°\mathrm{C}$. Double-logarithmic plot. Dashed gray lines indicate $\alpha =1∕2$. (a) Typical fluctuation functions for three different scattering vectors: $Q∕Q_{\max }=1.04(엯)$, $Q∕Q_{\max }=2.02(▿)$, and $Q∕Q_{\max }=3.07(◻)$. (b): Typical fluctuation functions for time series consisting of concatenated time series from 10 consecutive pixels. That is, if $p$ pixels are contained in a 20-pixel-wide annulus, $p∕10$ (which is an integer) fluctuation functions of time series of length $N^{\prime }=10N$ are calculated, each consisting of 10 concatenated time series of length $N$ from consecutive pixels. The plotted $F\left(t\right)$ represents the average of the so-obtained $p∕10$ fluctuation functions. Scattering vectors are the same as in (a). Note the improved statistics in the $\alpha >1∕2$ range, which can be well fitted now. For the sake of clarity, only the fit for the middle curve is shown, full black line, yielding $\alpha \approx 0.66$.

Figure 6

(Color online) Fluctuation exponent $\alpha$ from DFA3 vs $Q∕Q_{\max }$. $\mathrm{Al}\text{−}6\mathrm{at}.\%\mathrm{Ag}$ at $140°\mathrm{C}$, full red circles. $\mathrm{Al}\text{−}9\mathrm{at}.\%\mathrm{Zn}$ at $0°\mathrm{C}$, full red triangles. Fluctuation exponents from MC simulations [5]: Coarsening via LSW mechanism with scaled intensity plus background, open black circles. Coarsening via coagulation mechanism with scaled intensity plus background, open black triangles. The agreement between the LSW and the $\mathrm{Al}\text{−}6\mathrm{at}.\%\mathrm{Ag}$ data, and the coagulation and the $\mathrm{Al}\text{−}9\mathrm{at}.\%\mathrm{Zn}$ data is good, even quantitatively. Error bars indicate fit errors.