Atomic Dynamics in Real Space and Time

Abstract

Atomic and molecular dynamics in strongly disordered matter, such as liquid, cannot be fully described in terms of phonons, because they are marginalized and often overdamped. Their dynamic and transport properties depend on local atomic rearrangements which are strongly correlated. To describe such local dynamics, the usual representation in momentum (Q) and energy (E) space in terms of the dynamic structure factor, S(Q, E), is not effective. We discuss an alternative approach in real space (r) and time (t), with the van Hove function, G(r, t), and show how this approach facilitates understanding of real-space local dynamics of liquids and other disordered systems in the length scale of Å and time scale of pico-second.

1. Introduction

In crystalline solids, phonons are the elementary excitations of lattice dynamics. They can be observed with well-defined dispersions in the dynamic structure factor, S(Q, E), where Q is the momentum exchange and E (= $\mathrm{\hslash }$ω) is the energy exchange in scattering. The S(Q, E) can be measured by inelastic X-ray scattering (IXS) as discussed in this book or by inelastic neutron scattering (INS). However, in strongly disordered materials, shortwave phonons have very short mean free path and lifetime. Only long-wave phonons, for which the atomic structure is irrelevant and the material acts as a continuum elastic body, propagate over some distance. However, the total spectral weight of long-wave phonons is small, because the phonon density of states increases as E². For this reason, a majority of phonons are overdamped, and some of them are localized, particularly in liquid.

Atomic dynamics in liquid and soft matter is usually studied by quasi-elastic scattering (QES) at low Q, where the QES intensity is dominated by the self-correlation and the auto-correlation of the same atom. The energy width of QES is proportional to DQ², where D is the diffusion constant. For hydrogen, which has a very large incoherent cross section, the value of D determined by QES at low Q is the isotopic diffusivity, D_i, determined by the diffusion profile of isotopes measured by sectioning. The chemical diffusivity, D_c, which characterizes the flow of elements responding to concentration gradient, is usually different from D_i, because of the backflow of diffusing atoms. The ratio, f = D_c/D_i, is known as the correlation factor and describes how diffusive atomic jumps are correlated.

This example illustrates the importance of studying the distinct atomic correlation and the dynamic correlation among different atoms. The importance of knowing the distinct terms of the correlation function is beginning to be recognized [1, 2], but the difficulty of measurement delayed advances. Only recently the progress in IXS instrumentation [3, 4] and the advent of pulsed neutron sources [5] made it feasible to measure S(Q, E) over wide ranges of Q and E in a reasonable time and opened the possibility of garnering information on distinct atomic correlations.

Conversely, even though the substantial progress in instrumentation allowed us to collect a much larger amount of data which contain vastly richer information, the IXS data are mostly processed in the same way, just by focusing on the phonon dispersion and its width [6]. Similarly, usually only the diffusivity is obtained from the QES data. What is missing is the analysis of the diffuse IXS intensity which is usually discarded as background. This situation is a perfect analog of the structural analysis of disordered crystals. The conventional methods of crystallographic analysis focus only on the Bragg peaks, and the diffuse scattering intensity is processed separately to determine the short-range order. However, by combining the Bragg and diffuse intensities as the total scattering intensity and Fourier-transforming it into the atomic pair distribution function (PDF), precise local structural information can be obtained [7]. In the same manner, by including the diffuse IXS intensity, valuable information on local dynamics can directly be obtained.

2. Van Hove function

2.1 Definition

After the correction for absorption and normalization, the IXS intensity, I(Q, E), is reduced to S(Q, E) [8],

$$S\left(\mathbf{Q},E\right)=\frac{1}{2\pi \mathrm{\hslash }N}\sum _{i,j}\int ⟨exp\left[\mathbf{Q}\cdot \left({\mathbf{r}}_{\mathbf{i}}\left(0\right)-{\mathbf{r}}_{\mathbf{j}}\left(t\right)\right)\right]⟩exp\left(-\frac{E}{\mathrm{\hslash }}t\right)\mathit{dt},$$E1

where r_i(t) is the position of the atom i at time t. It is useful to separate it into the self-part (i = j), S_s(Q, E), and the distinct-part (i ≠ j), S_d(Q, E).

$$S\left(\mathbf{Q},E\right)=S_s\left(\mathbf{Q},E\right)+S_d\left(\mathbf{Q},E\right).$$E2

For isotropic matter such as liquid, we use the spherical average,

$$S\left(Q,E\right)=\frac{1}{4\pi }\int S\left(\mathbf{Q},E\right)d\mathrm{\Omega },$$E3

where$Q=\left|\mathit{Q}\right|$ and Ω is the solid angle in Q space. Upon the Fourier transformation into the time-domain, we obtain the intermediate scattering function [8],

$$F\left(Q,t\right)=\int S\left(Q,E\right)exp\left(i\frac{E}{\mathrm{\hslash }}t\right)\mathit{dE},$$E4

which has been widely used in the analysis of soft matter dynamics [9, 10, 11]. Another step of the Fourier transformation, this time from momentum space to real space, leads to the van Hove function [12],

$$\begin{array}{ll}G\left(r,t\right)& =1+\frac{1}{2{\pi }^2{\rho }_{0}r}\int F\left(Q,t\right)sin\left(\mathit{Qr}\right)\mathit{QdQ}\\ & =\frac{1}{4{\mathit{\pi \rho }}_0{\mathit{Nr}}^2}\sum _{i,j}\delta \left(r-\left|{\mathbf{r}}_i\left(0\right)-{\mathbf{r}}_{\mathbf{j}}\left(t\right)\right|\right).\end{array}$$E5

Again it is useful to divide it into the self-part, G_s(r, t), and the distinct-part,

G_d(r, t). Although the van Hove function has been known for a long time, its experimental determination has rarely been done [13] because it requires S(Q, E) to be known over wide ranges of Q and E. Only recently it became practical [14, 15] because of the progress in instrumentation as noted above and discussed below.

In the regular X-ray diffraction measurement, the energy resolution is of the order of 1 eV, far greater than the phonon excitation energies. Therefore what is measured is the energy-integrated intensity,

$$S\left(Q\right)=\int S\left(Q,E\right)\mathit{dE},$$E6

which leaves only the same-time (t = 0) contribution in Eq. (1). Therefore its Fourier transform, the PDF,

$$g\left(r\right)=1+\frac{1}{2{\pi }^2{\rho }_{0}r}\int \left[S\left(Q\right)-1\right]sin\left(\mathit{Qr}\right)\mathit{QdQ},$$E7

is equal to G(r, 0). In other words the van Hove function describes how the PDF, the snapshot correlation function, decays with time.

2.2 Evolution with time

At a short time scale (∼0.1 ps), atomic motions are ballistic, but after atoms leave the neighbor cage, they become diffusive. Then the self-term of the van Hove function in the diffusive regime should be

$$G_s\left(r,t\right)=\frac{1}{{\rho }_0}{\left(\frac{1}{4\pi D_{i}t}\right)}^{\frac{3}{2}}exp\left(-\frac{r^2}{4D_{i}t}\right).$$E8

Therefore the self-diffusion coefficient, D_i, can be determined from the self-part of the van Hove function. The double Fourier transform of Eq. (8) is

$$S_s\left(Q,\omega \right)=\frac{D_i{Q}^2/\pi }{{\omega }^2+{\left(D_i{Q}^2\right)}^2},$$E9

which can be measured by QES and is routinely used for determining D_i. In the case of quasi-elastic neutron scattering (QENS) from hydrogen, because the incoherent cross section of hydrogen is so large, the scattering intensity is totally dominated by S_s(Q, ω), and the measurement readily yields the value of D_i. In general, however, there can be some contributions from the distinct-part to the low-angle QES, which can make the measurement inaccurate. On the other hand in the van Hove function the self-part is cleanly separated, at least initially, resulting in more accurate determination of D_i.

The decay of the first peak of the PDF with time describes how the nearest neighbor shell of an atom, known as the first-neighbor cage, disintegrates with time. The van Hove function depicts this decay nicely and can relate the time scale of decay to the topological relaxation time and to viscosity as shown below.

The early prediction on the distinct-part of the van Hove function was that it could be expressed by the convolution of the PDF by the self-part (Eq. (8)) [16]. Then the QES width should be equal to DQ², by Eq. (9). But de Gennes noted that QES becomes anomalously narrow in the vicinity of the first peak in S(Q) [17]. He suggested that this phenomenon, now known as the de Gennes narrowing, was due to the collective nature of the dynamics represented by the first peak in S(Q). Since then it became customary to equate the observation of the de Gennes narrowing to the confirmation of collective excitations. A recent study [18], however, showed otherwise. It was found that even in high-temperature liquid, in which atomic motions are uncorrelated, the decay time of G(r, t) depends linearly on distance. In general,

$$\tau \left(r\right)={\tau }_0+{\tau }_r{\left(\frac{r}{r_1}\right)}^{\chi },$$E10

where r₁ is the position of the nearest neighbor peak of the PDF. As shown in Figure 1, the exponent χ depends on dimensionality d approximately as χ = (d-1)/2; thus χ = 1 for three dimensions. This is because at large r, each PDF peak describes not just one atomic distance but many. Therefore its decay with time does not correspond to the single atom dynamics. The number of pairs of atoms in each peak, N_r, is proportional to the surface area, 4πr^d−1. Then its fluctuation is proportional to $\sqrt{N_r}\sim r^{\left(d-1\right)/2}$; therefore χ = (d-1)/2. Now the first peak of S(Q) represents the medium-range part of the PDF, beyond the first peak [19], so its decay is slow, reflecting the behavior of the PDF beyond the first peak. This argument proves that the de Gennes narrowing does not necessary imply collective excitations but can be just the natural consequence of geometry.

Figure 1.

The r-dependent relaxation time τ(r) for model liquid iron in 2D (red triangle) χ = 0.66, 3D (black circle) χ = 1.04, and 4D χ = 1.45, beyond the first peak. The data points are shown in the form of log{[τ(r)-τ₀]/τ_r} versus log(r/r₁) to highlight χ from the expected power law dependence. The short dashed lines serve as guides to the eye [18].

3. Local dynamics of water and aqueous solution of salt

3.1 Van Hove function of water

Figure 2 shows the S(Q, E) of water at room temperature, determined by the IXS experiment at the beam line XL35 of the SPring-8 facility [14]. Earlier IXS experiments to observe phonons did not cover the Q space much beyond 1 Å⁻¹ [6, 20, 21]. The S(Q, E) is dominated by QES, and as is given it is not easy to garner useful information without extensive modeling. Converting the data into the van Hove function makes local dynamics directly visible as shown in Figure 3. Because hydrogen is almost invisible to X-rays, the van Hove function is dominated by oxygen–oxygen correlation. To minimize the termination error for stopping the integration by Eq. (4) at a maximum Q value, F(Q, t) can be extended to large Q by adding S(Q) exp.(−D(Q)Q²t), which is justified for the self-correlation function [14].

Figure 2.

The S(Q, E) of water at room temperature, determined by the IXS experiment at the beam line XL35 of the SPring-8 facility [14].

Figure 3.

The van Hove function of water [14].

In Figure 3 the data at t = 0 is the snapshot PDF which can be obtained by the conventional diffraction measurement. At t = ∞ G(r, ∞) = 1, so that G(r, t) - 1 describes the correlation. The decay of the PDF to G(r, ∞) = 1 is not uniform, with each peak behaving in different ways. In particular the first peak moves away, while the second peak moves in, indicating that the local dynamics is highly correlated. As the nearest neighbor moves away, the second neighbor comes in to take its place to maintain the coordination unchanged. The area of the first peak above G(r, t) = 1 shows a two-step decay,

$$A\left(t\right)=A_1{e}^{-{\left(t/{\tau }_1\right)}^{{\gamma }_1}}+A_2{e}^{-{\left(t/{\tau }_2\right)}^{{\gamma }_2}}.$$E11

The first term (τ₁ = 0.32 ps) describes the ballistic motion of the atom, whereas the second term with the temperature-dependent τ₂ describes the change in molecular bond. Earlier through molecular dynamics (MD) simulations, it was found that the time scale of losing one nearest neighbor, τ_LC, is directly related to viscosity through τ_LC = τ_M = η/G_∞, where τ_M is the Maxwell relaxation time, η is viscosity, and G_∞ is instantaneous shear modulus [22]. By relating τ₂ to τ_LC through simulation (for water τ₂ = τ_LC), this relationship was proven for water [14, 23].

3.2 Self-diffusion

The portion of the van Hove function near r = 0 describes the self-correlation, G_s(r, t). Indeed it follows Eq. (8) quite well for water as shown in Figure 4 [24].

Figure 4.

The self-part of the van Hove function for water at (A) 285 K, (B) 295 K, (C) 310 K, and (D) 318 K. (circles) experimental data and (dashed line) the result of fitting by Eq. (8).

However, the values of diffusivity determined from Eq. (8) vary from the values obtained by other methods [24]. The origin of this discrepancy is yet to be determined.

3.3 Van Hove function of salty water

About 70% of the earth is covered by salty water, and 80% our body is also made of salty water. Therefore it is important to know how salt affects the properties of water, such as viscosity. We studied the local dynamics of aqueous solution of NaCl up to 2 mol/kg by IXS [24] using the BLX-43 beam line of SPring-8 which has as many as 24 analyzer crystals. With this setup a dataset similar to that shown in Figure 2 can be collected in 12 h.

As shown in Figure 5, the height of the first peak of the van Hove function is reduced by salt. The time dependence of the area of the first peak above G(r, t) = 1, shown in Figure 6, demonstrates that the addition of salt increases the slow decaying component. Furthermore it is possible to decompose the van Hove function to that of the water–water correlation, G_w-w, and that of the water-salt correlation, G_w-s,

Figure 5.

The van Hove functions around the first-neighbor correlation peak, R ∼ 2.9 Å: (A) pure water, (B) m = 0.75 mol/kg, (C) 1.5 mol/kg, (D) 2.26 mol/kg, and (E) 3.0 mol/kg. The solid lines at R = 3.21 Å show the R_O2− + R_Cl−. The dashed line at R = 2.42 Å shows the R_O2− + R_Na+. The dash-dotted line at R = 2.8 Å shows the R_O2− + R_O2−. The range between the dotted lines (${{\mathit{R}}_{\mathbf{1}}}^{\prime }$ and ${{\mathit{R}}_{\mathbf{1}}}^{\prime \prime }$) was used to calculate the area, A(t), of the first neighbor. The upper limit of this range is changed within the gray-shaded area to estimate the uncertainties [25].

Figure 6.

Temporal evolution in the area of the first-neighbor peak, A₁(t), and the enlarged view (inset): (open circles) pure water, (triangles) m = 0.75 mol/kg, (squares) 1.5 Mol/kg, (closed circles) 2.26 mol/kg, and (diamonds) 3.0 mol/kg. The shaded areas represent uncertainties of each dataset. The solid and dashed lines represent the linear combination of time evolution for m = 0 and 2.26 mol/kg [25].

$$G\left(r,t\right)=w_{w-w}{G}_{w-w}\left(r,t\right)+w_{w-s}{G}_{w-s}\left(r,t\right),$$E12

where w_w-w and w_w-s are the X-ray scattering weight for each component. The salt-salt correlation was neglected because the concentration of salt was low. If we assume that G_w-w is the same as for pure water, we can determine G_w-s from Eq. (12). As shown in Figure 7, G_w-s is almost the same for all concentrations. The decay of the area of the sub-peak at 3.2 Å, corresponding to the Cl-O distance, is also the same for all concentrations as shown in Figure 8, proving the effect of salt on dynamics is local.

Figure 7.

One-dimensional profiles of ${\mathit{G}}_{\mathit{w}\mathit{-}\mathit{s}}\left(\mathit{r},\mathit{t}\right)$ for 0 < t < 2 ps and their intensity maps. The molality of sample is 0.75, 1.5, 2.26, and 3.0 mol/kg from the left to the right. The solid lines, dashed lines, and the dash-dotted lines in the top figures represent R = 3.21 Å (R_O2− + R_Cl−), R = 2.42 Å (R_O2− + R_Na+), and R = 2.8 Å (R_O2− + R_O2−), respectively.

Figure 8.

Time evolution of peak height at around R = 3.21 Å. the solid line shows the result of fitting using two (compressed) exponential functions (Eq. (12)) [25].

4. Limitations of the method

For the determination of the van Hove function, the current setup of IXS is ideally suited to the study of local dynamics in the time scale of 0.1–2 ps and length scale up to 5 Å. The energy resolution (∼ 1.5 meV) sets the long-time limit to 2 ps. The effect of resolution is mitigated by the data analysis, by correcting the intermediate function for resolution,

$$F\left(Q,t\right)=F_{\mathit{obs}}\left(Q,t\right)/F_{\mathit{res}}\left(Q,t\right),$$E13

where F_res(Q, t) is the Fourier transformation (Eq. (3)) of the energy resolution function. However, when F_res(Q, t) becomes too small at long t, this correction is no longer sufficient. This represents a severe limitation for the IXS-derived van Hove function. To go beyond this limit, we have to resort either to neutron scattering which offers better energy resolution or to develop the method of X-ray photon correlation spectroscopy (XPCS) with free-electron X-ray laser [26]. At the moment, because the method was only recently proven to be feasible, there are many low-hanging fruits which we are busy collecting.

5. Concluding remarks

As the instrumentation for IXS is improved, it became possible to carry out many Q-E scans in a relatively short time, covering wide Q-E space. This enables us to convert the dataset of the dynamic structure factor S(Q, E) into the van Hove function G(r, t) which describes dynamics in real space and time. This new capability allows us to visualize local dynamics directly rather than through the modeling for S(Q, E). Even though the van Hove function is just the Fourier transform, in order to determine it, we have to measure S(Q, E) over a wide Q-E space, which forces us to collect much more information than we normally do. This alone brings us to a new territory. Furthermore, by visualizing it, we gain much intuition. For instance, the behavior of the first and second peaks of water shown in Figure 3 was totally unexpected. Only after seeing it, we understand right away what this means, but no one would have anticipated it before it is seen. It is fully expected that such surprises will occur when this approach is applied further to the study of local dynamics in liquids, glasses, and soft matter in general.

Acknowledgments

The results reviewed in this article were obtained through the extraordinary effort and dedication by the team members of the author, W. Dmowski, T. Iwashita, and Y. Shinohara, in particular. He is also grateful to A. Q. R. Baron for his invaluable help during the experiment at SPring-8. This work was supported by the Department of Energy, Office of Science, Basic Energy Sciences, Materials and Science and Engineering Division.

Conflict of interest

The author declares no conflict of interest, financial or otherwise.