**3.1 Van Hove function of water**

time. The van Hove function depicts this decay nicely and can relate the time scale

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].

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

, by Eq. (9). But de Gennes noted that

*,* (10)

. Then its fluctuation is proportional to

of decay to the topological relaxation time and to viscosity as shown below.

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

*τ*ð Þ¼ *r τ*<sup>0</sup> þ *τ<sup>r</sup>*

where *r*<sup>1</sup> 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,

*d*�1

<sup>p</sup> � *<sup>r</sup>*ð Þ *<sup>d</sup>*�<sup>1</sup> *<sup>=</sup>*2; therefore *<sup>χ</sup>* = (*d*-1)/2. Now the first peak of *<sup>S</sup>*(*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

*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]/τr} versus log(r/r1) to highlight χ from the expected power law dependence. The short dashed lines serve as guides to*

*r r*1 � �*<sup>χ</sup>*

Then the QES width should be equal to *DQ*<sup>2</sup>

*Nr*, is proportional to the surface area, 4*πr*

just the natural consequence of geometry.

general,

ffiffiffiffiffiffi *Nr*

**Figure 1.**

*the eye [18].*

**50**

**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<sup>1</sup> [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*

**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].*

by adding *<sup>S</sup>*(*Q*) exp.(�*D*(*Q*)*Q*<sup>2</sup> *t*), which is justified for the self-correlation function [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,

$$A(t) = A\_1 e^{-(t/\tau\_1)^{\gamma\_1}} + A\_2 e^{-(t/\tau\_2)^{\gamma\_2}}.\tag{11}$$

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.

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

*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*

<sup>0</sup> *and R1*

*neighbor. The upper limit of this range is changed within the gray-shaded area to estimate the uncertainties [25].*

<sup>00</sup> *) was used to calculate the area, A(t), of the first*

**3.3 Van Hove function of salty water**

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

**Figure 5.**

**53**

*RO2*� *+ RO2*�*. The range between the dotted lines (R1*

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 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 water [14, 23].

#### **3.2 Self-diffusion**

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].

**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.
