Ionic Liquids: evidence of the viscosity scale-dependence

Ionic Liquids (ILs) are a specific class of molecular electrolytes characterized by the total absence of co-solvent. Due to their remarkable chemical and electrochemical stability, they are prime candidates for the development of safe and sustainable energy storage systems. The competition between electrostatic and van der Waals interactions leads to a property original for pure liquids: they self-organize in fluctuating nanometric aggregates. So far, this transient structuration has escaped to direct clear-cut experimental assessment. Here, we focus on a imidazolium based IL and use particle-probe rheology to (i) catch this phenomenon and (ii) highlight an unexpected consequence: the self-diffusion coefficient of the cation shows a one order of magnitude difference depending whether it is inferred at the nanometric or at the microscopic scale. As this quantity partly drives the ionic conductivity, such a peculiar property represents a strong limiting factor to the performances of ILs-based batteries.


Modeling the Molecular scale dynamics (ps-ns/Å-nm)
Due to the large incoherent neutron scattering cross-section of the hydrogen element, protonated samples scatter mainly incoherently. In this case, the QENS experimental intensity is directly related to the incoherent dynamical structure factor S inc (Q, ω), the Fourier transform over space and time of the self-correlation function G s (r,t) of the nuclei in the system 1 : S inc (Q, ω) = G s ( r,t)e i( Q r−ωt) d rdt (S1) If these particles do not experience perfect ergodic dynamics, G s (r,t = ∞) is non-zero, so that S inc (Q, ω) shows an elastic component: where A(Q), the Elastic Incoherent Structure Factor (EISF), is the Fourier transform of the G s (r,t = ∞) loci. In the case of a particle diffusing in a reduced volume of space, such as a sphere or a cylinder, A(Q) is simply the form factor of this confining volume. L(Q, ω) is related to the time dependence of the particle auto-correlation function. This term is generally described by a sum of Lorentzian lines 1 and is in practice usually well accounted for, by a single Lorentzian line with a Half-Width-at-Half-Maximum (HWHM), Γ(Q, ω).

Modeling the side-chains dynamics
In the reference frame of the IL cation, along with free and/or jump rotation of the methyl group carried by the imidazolium ring, the side-chain methylene group ( Fig.1) experiences rapid (ps) and local (1-2Å) dihedral reorientations known to significantly contribute to the QENS signal. Within the statistics of a QENS experiment, it is not possible to finely describe the details of such a complex contribution. In order to limit the number of parameters in our model, we have therefore made the choice to account for the side-chains dynamical contribution by a single average dynamical structure factor: where A sc (Q) is the EISF of the relaxation, while L sc (Q, ω) is a Lorentzian function with Γ sc HWHM (the sc subscript stands for side-chain). As discussed later-on in this paper, we have taken advantage of specific deuteration of the side-chain to check the validity of this approximation. Next to the side-chain protons, the imidazolium protons experience no dynamics. In its own reference frame, S(Q, ω) CF inc (CF stands for Cation Frame), the total dynamical structure factor of an IL cation is: where p is the fraction of the sole side-chains protons. In the case of fully hydrogenated BMIM p = 12/15 = 0.8. As the neutron incoherent cross-section of deuterium is negligible compared to the one of hydrogen (σ (D) = 2 barns << σ (H) = 80 barns), in the case of a BMIM cation with a fully deuterated alkyl side-chain, p = 3/6 = 0.5. 1/9

Diffusion within aggregates
Burankova et al. 2 have proposed that a confinement space could account for a localization of the molecules within the IL nanometric aggregates. This development is based on the so-called Gaussian model developed by Volino et al. 3 , which describes the translational motion of particles confined in spaces with soft boundaries. At small Q, the HWHM of this line tends to a plateau Γ loc (Q → 0) =hD loc /σ 2 loc , while for large Q, it recovers the jump diffusion law until a saturation related to τ, the residence time between two successive jumps: To reduce the complexity, our model use a single Lorentzian function to describe the diffusion within aggregates (with a single fitting parameter: Γ loc ). The corresponding dynamical structure factor is: where L loc (Q, ω) is a Lorentzian function: and A loc (Q) is the EISF 3 : Γ loc and A loc (Q) are then fitted with the Gaussian dynamical structure factor and the Gaussian EISF, respectively ( Fig.3, see SI.3 for details). The total isotropic distance visited by the confined particle is of the order of 6.σ loc (± 3 standard deviations of the Gaussian EISF function).

Cation center-of-mass long-range diffusion
The dynamical structure factor of this mode is simply the Fick's law in the reciprocal space 1 : where Γ lr , the HWHM of this Lorentzian, is directly proportional to D lr , the long-range self-diffusion coefficient: The Equation S9 is the so-called DQ 2 law.

Derivation of the total dynamical structure factor
Three contributions describe the dynamics of an IL cation: side-chain motions, local diffusion within aggregates and long-range diffusion. As they occur in different time windows, we suppose that they are independent. S (Q, ω) T inc , the total dynamical structure factor, is therefore a convolution of the dynamical structure factor Eq.S4, S5 and S8 related to these individual modes: where L x+y (Q, ω) is a Lorentzian line of HWHM Γ x+y = Γ x + Γ y and The three dynamical contributions of equation S11 take place on different time ranges. The side-chain reorientational and dihedral motions are faster than the local diffusion witch is itself expected to be faster than that the long range one so that Γ sc Γ loc Γ lr . Equation S11 can then be simplified: S (Q, ω) cation inc ≈ I 1 (Q) L lr (Q, ω) + I 2 (Q) L loc (Q, ω) + I 3 (Q) L sc (Q, ω) (S16)

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with: In conclusion the dynamical structure factor proposed by this model is composed by three Lorentzian relaxations whose each HWHM are linked to a unique dynamical mode while the intensities are combination of the different EISF.
In the time domain as directly accessed by NSE Eq.1 can be rewritten as: where τ i =h/Γ i with i=loc, sc and τ lr = (D lr Q 2 ) −1 . D lr is the long-range tranlational diffusion coefficient inferred by the neutron methods at the molecular scale (ps-ns / 0.1-10 nm).

Theoretical aspects
In the framework of the Sears approximation 4 , I(Q,t) R s , the self (incoherent) intermediate scattering function of an isotropic rotation over a sphere of radius b writes: The time-dependent term F l (t) is the first order rotational auto-correlation function: where α(t) is the angle between u(t), accounting for the particle orientation at time t = 0 and its orientation at time t later. P l is the Legendre polynomial of degree l. For times longer than few ps, F 1 (t) et F 2 (t) Eq.S22 simplifies by just considering two correlation times 3τ 1 et τ 1 : The correlation times related to the BMIM translational diffusive motions, within an aggregate or long-range, and the tumbling of the whole molecule are supposed to be significantly different, so that can one consider these modes as being independent. From Eq.S11, in ToF QENS, the total dynamical structure factor is then: where F T denotes the Fourier Transform over time. For NSE, from Eq.S20, in the time domain the equation above is:

Numerical application in the case of BMIM
The dynamics of bulk BMIM-TFSI has recently been investigated by Nuclear Magnetic Resonance Relaxation Dispersion (NMRD) and PFG-NMR over a wide range of temperatures 5 . At 298 K, Seyedlar et al. measure the room temperature Rotational correlation times: τ R =400 ps.
Tokuda et al. propose radius of the BMIM molecule: R BMIM = 3.3Å. This later quantity is estimated from the radius of an equivalent sphere matching the 3D structure of the cation as deduced from ab-initio calculation. For the sake of consistency of our QENS/NSE, NMR and DLS data, we use R BMIM = 2.3Å. This 30% difference with the value proposed by Takuda et al. seems acceptable as the structure of the BMIM cation (Fig.1a) is far from spherical. At the highest ToF QENS energy resolution (see the Methods section) used in this study, the maximum correlation accessible time is of the order of few tens of ps. A correlation time of τ R = 400 ps is therefore detected as a extremely narrow Lorentzian line whose HWHM is negligible compared to the other Lorentzian contributions. The total dynamical structure factor of BMIM then reduces to Eq.S11. As shown in Fig.S1, within the statistics of a NSE experiment, also in the time domain, the molecule tumbling can be neglected and the BMIM intermediate scattering function reduces to Eq.S20.

Implementation of the Gaussian model
To determine D loc , σ loc and τ, L loc (Q, ω) is fitted with the Gaussian model in the time domain. The intermediate scattering function of the Gaussian model is: and the intermediate scattering function corresponding to the diffusion within the aggregates is: where τ loc =h/Γ loc .
(1 − A loc (Q))L(Q,t) loc inc is then fitted with I(Q,t) loc inc − A loc (Q) to obtain the Gaussian parameters D loc , σ loc and τ.

Inferring the molecular long-range translational diffusion at the molecular scale from the NSE data
Staring from Eq.S20, the long-range tranlational diffusion coefficient at the molecular scale (ps-ns / 0.1-10 nm) is directly inferred from the NSE data corrected by the local short times contributions as measured by ToF QENS according to: