Critical dynamics of capillary waves in an ionic liquid: XPCS studies

1. Critical dynamics of capillary waves in an ionic liquid: XPCS studies Eli Sloutskin Physics Department Bar-Ilan University Israel Now at SEAS, Harvard

2. Introduction & motivation: • Ionic liquids and their surface structure. • Experimental technique: surface XPCS. • Spontaneous heterodyne - homodyne switching. • Results: • Propagating and overdamped regimes of thermal CW. • Critical behavior of thermal CW excitations. • Future directions: • Can XPCS measure surface viscoelasticity ? OUTLINE

3. Classical Salts: NaCl Tm= 1074 K MgCl2 Tm= 1260 K YBr3 Tm= 1450 K Room Temperature Molten Salts • Ionic liquids: • Solvent free electrolytes. • “Green” industry. • 100s synthesized since ’97. Butylmethylimidazolium tetrafluoroborate Tm = -71 oC hRT = 101 cP

4. The static surface structure of ILs Surface induced ordering is required to fit the reflectivity. Sloutskin et al., JACS 127, 7796 (2005) r (e/Å3) z(Å) X-ray reflectivity studies Molecular dynamics • Oscillatory surface electron density profile • Surface mixture of cations and anions R.M. Lynden-Bell and M. Del Pópolo, PCCP 8, 949 (2006). B.L. Bhargava and S. Balasubramanian, JACS 128, 10073 (2006). Sloutskin et al., J. Chem. Phys. 125, 174715 (2006).

5. The surface dynamics of ILs Tc (???) Surface dipole density Dynamic light scattering V. Halka, R. Tsekov, and W. Freyland, PCCP 7, 2038 (2005). • Modified elastic term in CW spectrum. • Possible ferroelectric order-disorder phase transition at T≈380K. • XPCS yields the S(q,w) of T-excited surface capillary waves. • No contribution from bulk modes. • Higher k-vectors can be probed. • To date no XPCS for any ionic liquid.

6. X-ray photocorrelation spectroscopy (XPCS) z qx a < ac kout qx qz q kin b = a Specular reflection b a I L~ 50-100 A o Troïka (ESRF) Overdamped mode G(t) G(t) t t(sec) b ≠ aDiffuse scattering by CW A. Madsen et al., website of ESRF. Intensity autocorrelation G(t) at a given wave vector qx is related to the dynamics of surface excitations for the same wave vector. qx Propagating mode t t(sec)

7. Experimental G(qx,t) functions: qualitative description. w w=ck k Propagating T=134 oC qx=24 mm-1 G(qx,t) overdamped T=40 oC t(s) • Capillary wave excitations with shorter wavelengths have higher frequencies. • Damping increases for shorter wave lengths. Viscous dissipation is due to velocity gradients: h∆v term in the Navier-Stokes equation. • At high qx and low T, capillary waves are overdamped.

8. Theoretical description For incompressible fluid: divv = 0 Navier-Stokes: h temporal damping Assume: liquid density and viscosity remain constant up to the dividing surface Boundary conditions at the interface: Stress tensor: ideal interface (i) Young-Laplace: (ii) Surface tension Dispersion relation for the capillary waves: E.H. Lucassen-Reynders and J. Lucassen, Adv. Colloid Interface Sci.2, 347 (1969)

9. Transition from propagating to overdamped waves: hydrodynamical theory Dispersion relation w Propagating k=24 mm-1 G(qx,t) W Damping log(w), log(W) t(sec) “ “ - Byrne and Earnshaw (1979) Use independently-measured (T), (T) and (T) log(y) Experimental Tc is higher than 40 oC !!! Solve y(Tc)= 0.145 Tc=35o C @ k=24 mm-1

10. Analytical approximations are inapplicable in the critical damping regime ! propagating overdamped S(k,w) S(k,w) propagating w w(kHz) k XPCS measures the actual population of ripplon energy levels at a given T, not the energy levels allowed by hydrodynamics. Linear response theory (Jäckle & Kawasaki, 1995) Spectrum of surface ripplons: Fluctuation-Dissipation k=const T=const Tc=45o C @ k=24 mm-1 overdamped The calculated Tc seems now to be correct. Can we use JK for full shape analysis of our XPCS data ? J. Jäckle and K. Kawasaki, J. Phys.: Condens. Matter7, 4351 (1995)

11. Heterodynevs.homodyne G(t) spontaneous switching t(sec) laser Grating J.C. Earnshaw (1987) PMT Homodyne Beating against a reference beam Who ordered a reference beam for XPCS ?

12. Small effective sample size yields non-zero scattered intensity even for a perfectly smooth surface. • The effective sample size changes in time: • meniscus, dust particles, etc. • The scattering peaks move in time, sometimes interfering with the diffuse signal. 1Gutt et al.,PRL91, 076104 (2005) 2 Ghaderi, PhD thesis (2006) What is the origin of the spontaneous reference beam in surface XPCS ? • Interference with the reflected beam, R(qz)d(qx)d(qy) ??? • Fresnel (near field) conditions mix the low q values1. • Instrumental Dq resolution2. • Partial coherence of the beam. Why is the switching time-dependent ? Set the reference beam intensity as a free fitting parameter !

13. Full shape analysis -“static” diffuse scattering qx-2 L G(qx,t) ln(Is) • Full shape analysis with only Is and Ir being free. • The fitted Is values show the diffuse intensity scaling known from CWT (for small qz values). • Let’s calculate the experimental S(q,w) spectra… ln(qx) t(s) t(s) Ir – reference beam Unknown Detector resolution Jäckle & Kawasaki (1995) qx=24 mm-1

14. Use the fitted values of Is and Ir to evaluate the experimental S(q,w) XPCS w t qx=24 mm-1 |T-Tc|-1/2 F{..} Lucassen Stokes kout kin 313 K qrip 36 mm-1 323 K anti-Stokes w (kHz) 333 K 24 mm-1 S(q,w) S(q,w) kin kout Jäckle 343 K qrip , 24 mm-1 371 K XPCS Theory 403 K ln(T-Tc) T (oC) w(kHz) w(kHz) ln(w) • The spectra are fully described by the JK theory, assuming an unstructured interface. • No need to introduce Freyland’s modified elastic term in the CW spectrum. • No evidence for “ferroelectric” transition, at least for T<130 oC. • Does it mean that the surface of an IL is not layered ? ln(T-Tc) T=343 K A. Madsen et al., PRL 92, 096104 (2004) Let’s introduce surface viscoelasticity into the same formalism !

15. Change the boundary conditions to introduce elasticity of the surface layer ideal interface (i) ed/g S(q,w) ed/g w(kHz) T=100 oC w(Hz) q=17 mm-1 Boundary conditions at the interface: Dilational modulus Stress tensor: (i) “elastic” interface x– lateral displacement of surface element h – surface normal displacement (ii) Resonance with Marangoni waves Surface tension time = hd hd=0 The changes are too small to be detected by XPCS… D. Langevin and M.A. Bouchiat, C.R. Acad. Sc. Paris 272, 1422 (1971)

16. Future directions • Langmuir films on low-viscosity liquids ? ed=0 ed>>g ed=eres w (kHz) S(q,w)/Smax S(q,w) w(kHz) ed / g • Spontaneous surface structures: • surface freezing in alkanes, • surface layering in liq. metals… • Higher coherent flux and stable experimental conditions are needed to avoid spontaneous homo-hetero switching. Damping (kHz) w(kHz) ed / g • If switching is unavoidable, try to measure the intensity of the reference beam ! • Go to higher q-values… (Far future: check dynamics at sub-micron scales) LF on water: qx = 20 mm-1 T = 25 oC g= 50 mN/m

17. Summary • What have we learned about surface XPCS ? • Quantitative analysis, accounting for • both homo- and hetero-dyne terms, • critical damping conditions (no Lorentizan app.), • finite detector resolution • perfectly fits the experiment. • Homo-hetero switching costs extra fitting parameters. • Surface viscoelasticity is only measurable at low viscosity. • What have we learned about surface capillary waves ? • Linear response theory (JK) provides the correct description of CW dynamics in an IL. • Hydrodynamics alone (LLR) is insufficient. • Critical scaling of CW frequencies at T →Tc , even though the hydrodynamic Tc is not the actual Tc .

18. Thanks … Organizers (NSLS-II). Audience. Prof. Moshe Deutsch (Bar-Ilan University, Israel). Dr. Ben Ocko (Brookhaven, USA). Dr. Anders Madsen (ESRF, France). Dr. Patrick Huber and M. Wolff (Saarland University, Germany). Dr. Michael Sprung (APS, USA). Dr. Julian Baumert (BNL, deceased). Chemada Ltd. (Israel). German-Israeli Science Foundation, GIF (Israel).