$q$-independent slow-dynamics in atomic and molecular systems

TL;DR

This study reveals that in supercooled water and silica, the collective density correlation time becomes wavevector independent at large wavelengths, a feature not observed in all systems like Lennard-Jones mixtures.

Contribution

It demonstrates the wavevector independence of collective density correlation times in atomic systems at large scales, linking it to slow dynamics observed in soft matter.

Findings

Wavevector independence observed in supercooled water and silica.

Lennard-Jones mixtures do not show this wavevector independence.

The phenomenon is linked to the nature of particle diffusion and slow dynamics.

Abstract

Investigating million-atom systems for very long simulation times, we demonstrate that the collective density-density correlation time ($\tau_{\alpha}$) in simulated supercooled water and silica becomes wavevector independent ($q^0$) when the probing wavelength is several times larger than the interparticle distance. The $q$-independence of the collective density-density correlation functions, a feature clearly observed in light-scattering studies of some soft-matter systems, is thus a genuine feature of many (but not all) slow-dynamics systems, either atomic, molecular or colloidal. Indeed, we show that when the dynamics of the density fluctuations is due to particle-type diffusion, as in the case of the Lennard Jones binary mixture model, the $q^0$ regime does not set in and the relaxation time continues to scale as $\tau_{\alpha} \sim q^{-2}$ even at small $q$.

Tables

Table 1. Table 1: Values for quantities related to concentration fluctuations and interdiffusion for the BMLJ and BKS systems. D A subscript 𝐷 𝐴 D_{A} , D B subscript 𝐷 𝐵 D_{B} , Λ Λ \Lambda and Λ D subscript Λ 𝐷 \Lambda_{D} are expressed in MD units for the BMLJ system and nm 2 /ps for BKS, whereas S 𝑆 S is a dimensionless number.

System	$D_A$	$D_B$	$\Lambda$	${\Lambda }_D$	$S$
BMLJ	$5.83\cdot {10}^{-5}$	$1.7\cdot {10}^{-4}$	$1.4\cdot {10}^{-4}$	$1.48\cdot {10}^{-4}$	0.95
BKS 2.36 g/cm3	$1.17\cdot {10}^{-4}$	$1.80\cdot {10}^{-4}$	$4.17\cdot {10}^{-5}$	$1.38\cdot {10}^{-4}$	0.33
BKS 2.48 g/cm3	$2.07\cdot {10}^{-4}$	$3.08\cdot {10}^{-4}$	$1.07\cdot {10}^{-4}$	$2.41\cdot {10}^{-4}$	0.44
BKS 2.60 g/cm3	$3.54\cdot {10}^{-4}$	$5.05\cdot {10}^{-4}$	$1.99\cdot {10}^{-5}$	$4.04\cdot {10}^{-4}$	0.49

Equations

$$F(q,t)=⟨{\rho }_{\mathbf{q}}^{\ast }(t){\rho }_{\mathbf{q}}(0)⟩,{\rho }_{\mathbf{q}}(t)\equiv \frac{1}{\sqrt{N}}\sum _{i=1}^N{e}^{i\mathbf{q}\cdot {\mathbf{r}}_i(t)}$$

$$\frac{d^2{\varphi }_q}{d{t}^2}+{\Omega }_q^2{\varphi }_q+{\int }_0^t{M}_q(t-t^{\prime })\frac{d{\varphi }_q}{dt}d{t}^{\prime }=0$$

$$M_q(t)={\gamma }_0{q}^2\delta (t)+M_q^s(t),M_q^s(t)\equiv A_s{q}^2\exp [-(t/{\tau }_s{)}^{\beta }]$$

$$\frac{d^2{\varphi }_q}{d{t}^2}+{\Omega }_q^2{\varphi }_q+{\gamma }_0{q}^2\frac{d{\varphi }_q}{dt}+{\int }_0^t{M}_q^s(t-t^{\prime })\frac{d{\varphi }_q}{dt}d{t}^{\prime }=0.$$

$$F_c(t)=(1-f)\exp (-(t/{\tau }_f{)}^2)\cos (\omega t)+f\exp (-(t/{\tau }_{\alpha }{)}^{\beta })$$

$$S_{CC}(q)=(1-x{)}^2{S}_{AA}(q)+x^2{S}_{BB}(q)-2x(1-x)S_{AB}(q),$$

$$D_{AB}=\Phi \Lambda$$

$$\Phi =\frac{x(1-x)}{S_{CC}(q=0)},$$

$$g_{\mathrm{int}}(t)\equiv {\left(1+\frac{m_{A}x}{m_B(1-x)}\right)}^{2}Nx(1-x)⟨{\left[{\vec{R}}_A(t)-{\vec{R}}_A(0)\right]}^2⟩$$

$$\Lambda =\underset{t\to \infty }{\lim }\frac{g_{\mathrm{int}}(t)}{6t}.$$

$${\Lambda }_D=(1-x)D_A+x{D}_B.$$

$$S\equiv \frac{\Lambda }{{\Lambda }_D}.$$

Full text

††preprint: APS/123-QED

The wave-vector ($q$) dependence of the dynamics of atomic, molecular, and colloidal systems close to dynamic arrest has been the focus of intense research Mezei et al. (1987); Van Megen and Pusey (1991); Kob and Andersen (1995a); Horbach and Kob (2001a); Sciortino et al. (1997); Tölle (2001); Torre et al. (2001); Eckert and Bartsch (2002); Bandyopadhyay et al. (2004); Bhattacharyya et al. (2008); Roldán-Vargas et al. (2017). In particular, the $q$-region that corresponds to the nearest neighbour distance in glass- and gel-forming systems has revealed a series of interesting phenomena Binder and Kob (2011); Donth (2013): (1) a two-step relaxation for both self and collective density correlation functions, indicating a faster intra-cage motion and a slower structural relaxation ($\alpha$-relaxation), respectively; (2) a significant stretching of the $\alpha$-relaxation, which originates from the coupling between distinct modes Götze (2008); Balucani and Zoppi (1995); (3) a deviation from the diffusive $q^{-2}$ behavior of the self correlation time; (4) oscillations in the $q$-dependence of the collective relaxation time, often in phase with the oscillation of the structure factor; (5) a faster decay of the self dynamics compared to the collective one, indicating that relative particle displacements play an important role in the decorrelation of the system.

The region at very small $q$, where the wavelength is much larger than the inter-particle distance, has also been thoroughly characterized. Here, conservation laws in one-component systems determine a three-mode decay of the collective correlation functions Hansen and McDonald (2013): two modes associated with damped sound waves (the Brillouin peaks) and one to the damped decay of the heat diffusion (the Rayleigh peak). In all three cases, the damping time follows a $q^{-2}$ dependence. In the case of glasses (where the $\alpha$-relaxation time is longer than the experimental observation), a clear cross-over has been identified between the region where the system can be considered an elastic continuum and the region where an excess of vibrational states Schirmacher et al. (1998); Monaco and Giordano (2009) is superimposed to the Debye density of states Tanguy et al. (2002); Wittmer et al. (2002); Zaccone and Scossa-Romano (2011).

In some colloidal systems, where the size of the particles provides access to smaller ratios between the wavelength of the probe radiation and the interparticle distance, a $q$-independent ($q^{0}$) relaxation mode has been reported. From the early measurements in polymer melts in which entanglement induces an effective transient network Brochard (1983); Adam and Delsanti (1985); Li et al. (2010), evidence of a $q^{0}$ mode has been presented for rodlike micelles Nemoto et al. (1995), semidilute polymers with bonding agents Nemoto et al. (1996); Michel et al. (2000a), telechelic ionomers Johannsson et al. (1995), microemulsion droplets in solution with telechelic polymers Michel et al. (2000b) (where the latter provide transient links between distinct droplets). More recently, a $q^{0}$ mode has been reported for equilibrium gels of DNA tetrafunctional nanostars Biffi et al. (2015); Nava et al. (2017) in which a short self-complementary DNA sequence provides a temperature-controllable link between different particles. This $q^{0}$ relaxation dynamics has been interpreted as originating from local elasticity fluctuations propagating through the system Brochard (1983); Nava et al. (2017). A recent simulation study of a particle model for vitrimers, binary-mixture networks in which the microscopic dynamics is slaved to a bond-swapping process Rovigatti et al. (2018), has also reported a clear $q^{0}$ dependence for the collective dynamics extending over more than one order of magnitude in $q$.

In this Letter we explore the possibility that a $q^{0}$ dependence of the collective relaxation time is much more common than previously thought, being a generic feature of systems with slow dynamics, including systems composed of atoms and molecules. The $q^{0}$ mode can arise in a range of $q$-values intermediate between the hydrodynamic and the nearest neighbour regions when the collective relaxation is not associated with single particle diffusion. We simulate slow-dynamics systems that are large enough (up to a million atoms) to allow the numerical evaluation of the collective relaxation over wavelengths corresponding to distances up to 50 times larger than the typical nearest-neighbour distance. Here we mostly focus on supercooled water whose dynamics has been extensively investigated experimentally with different scattering techniques Monaco et al. (1999); Bencivenga et al. (2007); Torre et al. (2004); Taschin et al. (2006); Arbe et al. (2016). For water, a comparison between the ultrasound data Davis and Jarzynski (1972) and the lowest $q$ explored with inelastic x-ray scattering Monaco et al. (1999) suggests that a region with $q^{0}$ dispersion exist just below the experimentally accessible window. Using simulations, we now provide clear evidence that, indeed, there exists a wide range of wave vectors in which the collective dynamics is $q$-independent. We also show that a model based on the the Mori-Zwanzig formalism properly describes the $q$-independence of the dynamics. Finally, we present results for two more systems, both well-known binary models of glass-formers: BKS silica Van Beest et al. (1990) and a binary mixture Lennard Jones (BMLJ) Kob and Andersen (1995b). In agreement with the explanation we provide, we observe a $q^{0}$ mode only in BKS silica.

Methods: For TIP4P/2005 Abascal and Vega (2005) and BKS Van Beest et al. (1990) we perform NVT simulations utilizing GROMACS 5.1.4 Van Der Spoel et al. (2005) with a velocity-Verlet integrator and a timestep of $\Delta t=1$ fs. The temperature is controlled using a Nosé-Hoover thermostat Nosé (1984); Hoover (1985). The Coulomb interactions are included with a particle-mesh Ewald treatment Essmann et al. (1995) with a Fourier spacing of 0.1 nm. For both the Lennard-Jones and real space Coulomb interactions the same cut-off $r_{\text{\rm cut}}$ is used ($r_{\text{\rm cut}}=0.9$ nm and 1 nm for TIP4P/2005 and BKS, respectively). Lennard-Jones interactions beyond $r_{\text{\rm cut}}$ are taken into account assuming a uniform fluid density. The TIP4P/2005 system consists of $2.5\cdot 10^{5}$ molecules (1M interacting sites) in a cubic box. We investigate three densities (0.90, 0.95 and 1.00 g/cm3 corresponding to box lengths of 20.26, 19.89, and 19.56 nm, respectively) at two temperatures (240 and 250 K). The molecule constraints are maintained using the LINCS (Linear Constraint Solver) algorithm Hess (2008). The BKS system consists of 287712 particles (1/3 Si and 2/3 O) in a cubic box. We study three densities (2.36, 2.48 and 2.60 g/cm3, corresponding to box lengths of 15.95, 15.69 and 15.45 nm) at 3500 K. As a binary-mixture Lennard-Jones (BMLJ) glass-former, among the different alternatives Pastore et al. (1988); Roux et al. (1989); Wahnström (1991), we choose the Kob-Andersen model Kob and Andersen (1995b). We use the RUMD package Bailey et al. (2017) to simulate $10^{6}$ particles at $\rho^{*}=1.2$ and $T^{*}=0.466$, where the ∗ superscript indicates reduced units: the unit of length is the diameter of the large particles, $\sigma_{AA}$, the unit of mass is the particle mass $m$ and the unit of energy is the interaction strength between large particles, $\epsilon_{AA}$. Differently from Ref. Kob and Andersen (1995b), the unit of time is $\sqrt{m\sigma_{AA}/\epsilon_{AA}}$. The total duration of the simulations is 20 ns for TIP4P/2005 and BKS (20 to 100 times the slowest collective relaxation time in the system) and $6\times 10^{7}$  MD time steps for the BMLJ system. The relaxation times and non-ergodicity factors obtained from fitting the correlation functions of the TIP4P/2005 and BKS systems in the whole $q$-range are reported in the Supplemental Material (SM).

Results — TIP4P/2005: Fig. 1 shows the (oxygen-oxygen) collective density correlation functions $F(q,t)$ for several $q$ values in the window $0.3<q<3$ nm*-1*, corresponding approximatively to wavelengths 7 to 70 times larger than the OO nearest neighbour distance (0.28 nm). $F(q,t)$ is evaluated as

[TABLE]

where $N$ is the total number of molecules, ${\bf r}_{i}$ the position of the $i$-th oxygen atom, and the average is performed over different initial times along the MD trajectory and over all ${\bf q}$ vectors with the same modulus $q$. $F(q,0)$ coincides with the structure factor $S(q)$. It is immediately clear from Fig. 1 that in this $q$ window (extending the window of $q$ explored with high-resolution inelastic x-ray scattering measurements Monaco et al. (1999); Bencivenga et al. (2007)), all correlation functions decay at long time with the same characteristic time scale. Fig. 1 also shows the self-correlation function $F_{\rm self}(q,t)$, plotted with dashed lines. In this small-$q$ region the Gaussian approximation holds true, that is, $F_{\rm self}(q,t)=\exp(-q^{2}\left\langle r^{2}(t)\right\rangle/6)$, where $\left\langle r^{2}(t)\right\rangle$ is the mean-squared displacementHansen and McDonald (2013) (comparison not shown). It is important to note that the self correlation functions decay at much longer times compared to $F(q,t)$, indicating that the decorrelation of the collective dynamics over the probed length-scales does not require particle diffusion.

To further support the idea that a single ($q$-independent) characteristic time controls the decay of the density fluctuations in this low-$q$ region we compare the numerically calculated correlation functions with predictions based on the Mori-Zwanzig (MZ) formalism. For any system, the normalised collective correlation function $\phi_{q}\equiv F(q,t)/S(q)$ formally satisfies the equation Hansen and McDonald (2013)

[TABLE]

where $\Omega^{2}_{q}=\frac{k_{B}T}{M}\frac{q^{2}}{S(q)}$.

The memory function, which is the autocorrelation of the stochastic force Hansen and McDonald (2013), and $\Omega^{2}$ completely specify the dynamics. Guided by the $q$ independence of $\tau_{\alpha}$ we approximate $M_{q}(t)$ as

[TABLE]

where $\gamma_{0}q^{2}\delta(t)$ describes the damping associated with the fast microscopic dynamics and $M^{s}_{q}(t)$ models the structural relaxation. This function is characterised by the three parameters $A_{s}$, $\tau_{s}$ and $\beta$, defining respectively the amplitude and the time-dependence of $M^{s}_{q}(t)$. All four parameters ($\gamma_{0}$, $A_{s}$, $\tau_{s}$ and $\beta$) are in principle functions of $q$. Consequently,

[TABLE]

In deriving Eq.(4) we have neglected the coupling between density and temperature fluctuations and the thermal diffusion contribution which is (in the hydrodynamic limit) proportional to $C_{v}/C_{p}-1$ Stanley (1971). For water this is a reasonable approximation, since at the temperature of maximum density $C_{v}=C_{p}$ Stanley (1971).

We solve the time dependence of Eq. (4) numerically (in Fourier space), searching for the values of the parameters which minimize the differences with the correlation functions evaluated from the simulated trajectories. We find that, in the range $0.3<q<3$ nm*-1*, it is possible to represent all correlation functions in the entire time window by using $q$-independent values for $\gamma_{0}$, $A_{s}$, $\tau_{s}$ and $\beta$. Fig. 2 compares the $q$-independent model predictions with the numerical data for two different state points. Such excellent agreement provides strong evidence that a collection of local relaxation processes (as indicated by the value $\beta=0.5$) fully describes the $q$-independent decay of the collective dynamics at small $q$. The prediction of Eq. (4) for $q$ smaller than the ones numerically accessed (black lines in Fig. 2(b)) visually demonstrate the cross-over to the hydrodynamic limit (having neglected the $T$-fluctuations, the limit coincides with the damped harmonic oscillator model with adiabatic sound speed $v_{s}=\Omega_{q}/q$ Hansen and McDonald (2013); Balucani and Zoppi (1995)). In general, hydrodynamics sets in when the time scale of the non-structural component becomes lower than $\tau_{\alpha}$. Selecting the period of the sound wave as the typical time, the wave-vector below which the relaxation time loses its $q^{0}$ character is reached when $q\ll 2\pi/(v_{s}\tau_{\alpha})$. Therefore, upon approaching the glass transition, as $\tau_{\alpha}$ becomes longer and longer, the region of $q$ in which a $q^{0}$ mode is expected becomes larger and larger, provided that $S(q)$ does not vary significantly.

Results — BKS and BMLJ: We next investigate two of the most commonly-studied binary mixture glass-former models. BKS silica Van Beest et al. (1990) generates a structure in which the (positively charged) Si atoms form a prevalently tetrafunctional open network in which the (negatively charged) O atoms mediate the Si-Si bonds. As expected for charged systems, there are essentially no large wavelength concentration fluctuations Stillinger Jr and Lovett (1968). By contrast, BMLJ Kob and Andersen (1995b) generates a dense structure in which each particle is surrounded by particles of both species. BMLJ particles are electrically neutral and concentration fluctuations do not vanish in the small $q$ limit. This crucial difference is evident in the Bhatia-Thornton concentration-concentration structure factors Bhatia and Thornton (1970) (reported in the SM). In the small-$q$ region explored in the simulation, the decay of the collective total-density fluctuations in the two binary systems is completely different. While BKS (Fig. 3) behaves as TIP4P/2005 water and displays an almost $q^{0}$ dependence for all three investigated densities, the same function in BMLJ manifests a strong $q^{-2}$ dependence (see main panel and inset of Fig. 4(a)). We also show the (non-normalised) partial correlation functions for $q\sigma_{\rm AA}=0.4$ in Figure 4(b). All the correlation functions decay similarly in the whole $q$-range considered, and the associated relaxation times always display a $q^{-2}$ dependence. Thus, in contrast to the one-component water case and the two-components silica case, the small-$q$ $\alpha$-relaxation process in BMLJ does not acquire a $q$-independent behaviour.

Discussion and conclusions: The reported numerical investigation of the low-$q$ collective dynamics in atomic and molecular systems exhibiting slow dynamics provides indisputable numerical evidence of the existence of a large region of $q$, corresponding to wavelengths larger than the inter-particle distance, in which the relaxation time is $q$-independent. These findings demonstrate that a $q^{0}$ mode is not a prerogative of soft-matter systems, but it is a genuine feature of many slow-dynamics systems, either atomic, molecular or colloidal. The $q$-independence of $F(q,t)$ can be equivalently expressed as a $q$-independence of the memory function within the MZ formalism.

Interestingly, from the point of view of the mode coupling theory (MCT), which neglects crystallisation, the difference between the $q$-dependence of the memory functions associated with the structural relaxation of disordered one-component systems and binary mixtures can be traced back to the conservation of the overall momentum of the system Götze (2008); Weysser et al. (2010). In one component MCT, consistent with what we have found, $M_{q}$ is predicted to be $q$ independent at small $q$. In the case of binary mixtures, momentum conservation is not observed by each component, since only the sum of all partial momenta is conserved. As a result, the MCT relaxation times might display a $q^{-2}$ dependence Götze (2008); Weysser et al. (2010). Therefore, within this framework, our results seem to indicate that binary mixtures for which the large-wavelength concentration fluctuations are suppressed behave, at large length-scales, as effective one-component systems. A comparison between the kinetic and thermodynamic contributions to the interdiffusion constant of the BMLJ and BKS systems clearly demonstrate the presence of strong cross correlations only in the latter (see SM) Horbach et al. (2007); Zausch et al. (2009); Hansen and McDonald (2013). These predictions have never been carefully tested due to the numerical difficulties of simulating very large systems over very long periods of time. Our results will definitively stimulate novel theoretical analysis to properly frame the $q^{0}$ phenomenon. In particular, it will be interesting to understand in more depth under which conditions binary mixtures do or do not exhibit a $q^{0}$ mode. The reported results will also stimulate new small-angle neutron and x-ray experiments, where a confirmation of the predicted behaviour can possibly be achieved. The results presently available for water Monaco et al. (1999); Bencivenga et al. (2007) (reaching a lowest $q$ of 1 nm*-1*) are consistent with the numerical findings discussed here.

Although the memory function formalism provides a deep formal understanding of the $q$ independence, it does not offer a physical picture of the local processes responsible for probing all different available microstates at small length scales. Different systems will relax locally in different ways, possibly via rototranslational motion (as proposed for water Saito et al. (2018)), bond-swapping or bond flickering in networks. Despite these differences, the effect of such local decorrelations on the small $q$ dynamics is system independent. The local changes of elasticity brought in by these local events possibly propagate via the vibrational modes of the system Nava et al. (2017) resulting in the decay of the collective density ($N/V$) fluctuations at large distances controlled by changes not in the number of particles $N$ but in the volume $V$.

Finally, we observe that in the $q^{0}$ regime equilibration in systems of very large size can be achieved only if simulations can be run for times longer than $\tau_{\alpha}$ (since $\tau_{\alpha}$ does not grow with decreasing $q$). This is not the case for BMLJ (and possibly other binary Lennard-Jones glass formers Pastore et al. (1988); Roux et al. (1989); Wahnström (1991)), for which doubling the system size requires four times longer equilibration times, making it essentially impossible to generate very large structurally and compositionally equilibrated configurations. Indeed, in these cases the diffusion of individual particles is required to relax frozen-in long wavelength concentration fluctuations. The observed $q^{0}$ mode in binary systems, shown here for BKS and previously for binary-mixture vitrimers Rovigatti et al. (2018), can possibly originate from the suppression of the small wavevector (large wavelength) concentration fluctuations. For BKS, this effect stems from the electrostatic nature of the Si and O interactions Stillinger Jr and Lovett (1968). By contrast, in the vitrimer system Rovigatti et al. (2018) it is the precise stoichiometry of the mixture and the bonding mechanism, which can act only between unlike particles, that prevents the occurrence of small-$q$ concentration fluctuations.

I Supplemental Material

I.1 Collective relaxation times, non-ergodicity and Debye-Waller factors

We fit the density-density collective correlation functions with the functional form

[TABLE]

where $f$, $\tau_{f}$, $\omega$, $\tau_{\alpha}$ and $\beta$ are fitting parameters.

Figure 5 shows the collective relaxation time $\tau_{\alpha}$ for all the investigated TIP4P/2005 and BKS systems.

Figure 6 shows the non-ergodicity factor $f$ for all the investigated TIP4P/2005 and BKS systems.

Figure 7 shows the partial Debye-Waller DW factors for the BMLJ and one BKS system. In the BMLJ system the DW factor differs greatly between the two species. By contrast, in BKS silicon and oxygen behaves more similarly, as their associated DW factors have similar values. We note that the values reported here are in agreement with the ones presented in Ref. Horbach and Kob (2001b).

I.2 Interdiffusion and concentration fluctuations

For a generic binary mixture composed of particles of species $A$ and $B$, the extent of the structural cross-species correlations can be estimated by computing the Bhatia-Thornton concentration-concentration structure factor $S_{CC}(q)$ Bhatia and Thornton (1970), which is defined as

[TABLE]

where $x$ is the fraction of particles of type $A$ and $S_{AA}(q)$, $S_{BB}(q)$ and $S_{AB}(q)$ are the partial structure factors. Figure 8 shows the small-$q$ limit of $S_{CC}(q)$ for the BKS (for which $A$ and $B$ are silicon and oxygen, respectively) and BMLJ systems. The curves clearly show that the BKS system exhibits exceedingly small large-wavelength concentration fluctuations with respect to the BMLJ system.

The collective transport of mass driven by concentration fluctuations can be quantified by the so-called interdiffusion Hansen and McDonald (2013); Horbach et al. (2007). The associated transport coefficient is the interdiffusion constant, which can be written as Horbach et al. (2007); Zausch et al. (2009)

[TABLE]

where $\Phi$ is a thermodynamic factor and $\Lambda$ is a kinetic factor also known as the interdiffusion Onsager coefficient. The former is defined as

[TABLE]

while the latter is related to the long-time behaviour of the mean-squared displacement of the centre of mass of one of the two species. In particular, let $\vec{R}_{A}(t)$ be the position of the centre of mass of species $A$ at time $t$ and

[TABLE]

its associated mean-squared displacement, weighted by a factor that takes into account the mass of particles of species $A$ and $B$ ($m_{A}$ and $m_{B}$, respectively), their concentration and the total number of particles, $N$. Figure 9 shows $g_{\rm int}(t)$ together with the total and partial mean-squared displacements for the BMLJ and one BKS system. In terms of $g_{\rm int}(t)$, the Onsager coefficient is given by

[TABLE]

We have already analysed the difference between the static concentration-concentration correlations as estimated by the small-$q$ behaviour of $S_{CC}(q)$ for the BMLJ and BKS system. Now we analyse the difference arising in the kinetic part. In absence of cross correlations, the Onsager coefficient can be estimated from the partial diffusion constants $D_{A}$ and $D_{B}$ through the Darken’s equation Darken (1948):

[TABLE]

Therefore, the importance of cross correlations can be quantified by looking at the ratio between the interdiffusion constant and the Darken prediction. The resulting quantity, called the Manning factor in the context of chemical diffusion in crystals Manning (1961), is defined as

[TABLE]

Table 1 shows the values for $D_{A}$, $D_{B}$, $\Lambda$, $\Lambda_{D}$ and $S$ for all the investigated binary mixtures. For the BMLJ system $S\approx 1$, which is a sign that cross correlations are very weak. By contrast, all BKS systems exhibit values of $S$ which differ significantly from $1$, demonstrating the importance of cross correlations also in the interdiffusion kinetic factor.