**3.3 Van Hove function of salty water**

by adding *<sup>S</sup>*(*Q*) exp.(�*D*(*Q*)*Q*<sup>2</sup>

shows a two-step decay,

function [14].

water [14, 23].

**Figure 4.**

**52**

**3.2 Self-diffusion**

*t*), which is justified for the self-correlation

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

�ð Þ *<sup>t</sup>=τ*<sup>1</sup> *<sup>γ</sup>*<sup>1</sup>

change in molecular bond. Earlier through molecular dynamics (MD) simulations, it was found that the time scale of losing one nearest neighbor, τLC, is directly related to viscosity through *τ*LC = *τ*<sup>M</sup> = *η*/*G*∞, where *τ*<sup>M</sup> is the Maxwell relaxation time, *η* is viscosity, and *G*<sup>∞</sup> is instantaneous shear modulus [22]. By relating *τ*<sup>2</sup> to *τ*LC through simulation (for water *τ*<sup>2</sup> = *τ*LC), this relationship was proven for

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

*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).*

The first term (*τ*<sup>1</sup> = 0.32 ps) describes the ballistic motion of the atom, whereas the second term with the temperature-dependent *τ*<sup>2</sup> describes the

þ *A*2*e*

�ð Þ *<sup>t</sup>=τ*<sup>2</sup> *<sup>γ</sup>*<sup>2</sup>

*:* (11)

*A t*ðÞ¼ *A*1*e*

*Inelastic X-Ray Scattering and X-Ray Powder Diffraction Applications*

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

#### **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 RO2*� *+ RCl*�*. The dashed line at R = 2.42 Å shows the RO2*� *+ RNa+. The dash-dotted line at R = 2.8 Å shows the RO2*� *+ RO2*�*. The range between the dotted lines (R1* <sup>0</sup> *and R1* <sup>00</sup> *) 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].*

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,

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

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 inter-

where *Fres*(*Q*, *t*) is the Fourier transformation (Eq. (3)) of the energy resolution function. However, when *Fres*(*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

*G r*ð Þ¼ *; t ww*�*wGw*�*<sup>w</sup>*ð Þþ *r; t ww*�*sGw*�*<sup>s</sup>*ð Þ *r; t ,* (12)

*F Q*ð Þ¼ *; t Fobs*ð Þ *Q; t =Fres*ð Þ *Q; t ,* (13)

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 corre-

*Time evolution of peak height at around R = 3.21 Å. the solid line shows the result of fitting using two*

lation, *G*w-s,

**Figure 8.**

on dynamics is local.

**55**

**4. Limitations of the method**

*(compressed) exponential functions (Eq. (12)) [25].*

*Atomic Dynamics in Real Space and Time DOI: http://dx.doi.org/10.5772/intechopen.88334*

mediate function for resolution,

#### **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].*

#### **Figure 7.**

*One-dimensional profiles of Gw*�*<sup>s</sup>*ð Þ *r;t 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 (RO2*� *+ RCl*�*), R = 2.42 Å (RO2*� *+ RNa+), and R = 2.8 Å (RO2*� *+ RO2*�*), respectively.*

**Figure 8.**

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

*Inelastic X-Ray Scattering and X-Ray Powder Diffraction Applications*

*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*

*One-dimensional profiles of Gw*�*<sup>s</sup>*ð Þ *r;t 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 (RO2*� *+ RCl*�*), R = 2.42 Å (RO2*� *+ RNa+), and R = 2.8 Å*

*linear combination of time evolution for m = 0 and 2.26 mol/kg [25].*

**Figure 2** can be collected in 12 h.

**Figure 6.**

**Figure 7.**

**54**

*(RO2*� *+ RO2*�*), respectively.*

*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].*

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,

$$G(r,t) = w\_{w-w}G\_{w-w}(r,t) + w\_{w-s}G\_{w-s}(r,t),\tag{12}$$

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.

### **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,

$$F(Q, t) = F\_{abs}(Q, t) / F\_{res}(Q, t), \tag{13}$$

where *Fres*(*Q*, *t*) is the Fourier transformation (Eq. (3)) of the energy resolution function. However, when *Fres*(*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.
