DOI

1

Defects at the Nanoscale Impact Contact Line Motion at all Scales - Perrin, Hugo and Lhermerout, Romain and Davitt, Kristina and Rolley, Etienne and Andreotti, Bruno

PHYSICAL REVIEW LETTERS 116, (2016)

Abstract : The contact angle of a liquid drop moving on a real solid surface depends on the speed and direction of motion of the three-phase contact line. Many experiments have demonstrated that pinning on surface defects, thermal activation and viscous dissipation impact contact line dynamics, but so far, efforts have failed to disentangle the role of each of these dissipation channels. Here, we propose a unifying multiscale approach that provides a single quantitative framework. We use this approach to successfully account for the dynamics measured in a classic dip-coating experiment performed over an unprecedentedly wide range of velocity. We show that the full contact line dynamics up to the liquid film entrainment threshold can be parametrized by the size, amplitude and density of nanometer-scale defects. This leads us to reinterpret the contact angle hysteresis as a dynamical crossover rather than a depinning transition.

PHYSICAL REVIEW LETTERS 116, (2016)

LPS

Abstract : The contact angle of a liquid drop moving on a real solid surface depends on the speed and direction of motion of the three-phase contact line. Many experiments have demonstrated that pinning on surface defects, thermal activation and viscous dissipation impact contact line dynamics, but so far, efforts have failed to disentangle the role of each of these dissipation channels. Here, we propose a unifying multiscale approach that provides a single quantitative framework. We use this approach to successfully account for the dynamics measured in a classic dip-coating experiment performed over an unprecedentedly wide range of velocity. We show that the full contact line dynamics up to the liquid film entrainment threshold can be parametrized by the size, amplitude and density of nanometer-scale defects. This leads us to reinterpret the contact angle hysteresis as a dynamical crossover rather than a depinning transition.

DOI

2

Structure and Permeability of Porous Silicon Investigated by Self-Diffusion NMR Measurements of Ethanol and Heptane - Puibasset, J. and Porion, P. and Grosman, A. and Rolley, E.

OIL \& GAS SCIENCE AND TECHNOLOGY-REVUE D IFP ENERGIES NOUVELLES 71, (2016)

Abstract : The adsorption and phase transitions of confined fluids in nanoporous materials have been studied intensely because of both their fundamental interest and their crucial role in many technologies. Questions relating to the influence of the confinement of fluids, and the disorder or elastic deformation of porous solids on the liquid-gas phase transition are still under debate. Model systems are needed to understand the adsorption phenomenon. In this context, Porous Silicon (PoSi), which is a single crystal obtained by etching a (100) silicon wafer is an excellent candidate. Indeed, it consists of non-connected tubular pores running parallel to the [100] axis perpendicular to the wafer surface, with transverse sections with a polygonal shape of nanometric size whose areas are widely distributed. Once detached from the wafer, free PoSi membranes can be considered a nanoscale disordered honeycomb. Adsorption/desorption experiments have been performed to characterize the structure: they have shown that evaporation occurs collectively, an intriguing observation generally associated with a disordered pore structure with many interconnections through narrow necks. The characterization of fluid mobility inside the pores should give complementary information about the pore structure and topology. This paper focuses on the dynamics of a fluid confined inside the structure of porous silicon, and in particular the self-diffusion measurements (pulsed field gradient spin echo Nuclear Magnetic Resonance (NMR)). The results show a strong anisotropy of the self diffusion tensor, as expected in this highly anisotropic structure. However, a non-zero self-diffusion in the directions perpendicular to the pore axis is observed. In order to interpret these puzzling results, molecular and Brownian dynamics calculations are underway.

OIL \& GAS SCIENCE AND TECHNOLOGY-REVUE D IFP ENERGIES NOUVELLES 71, (2016)

LPS

Abstract : The adsorption and phase transitions of confined fluids in nanoporous materials have been studied intensely because of both their fundamental interest and their crucial role in many technologies. Questions relating to the influence of the confinement of fluids, and the disorder or elastic deformation of porous solids on the liquid-gas phase transition are still under debate. Model systems are needed to understand the adsorption phenomenon. In this context, Porous Silicon (PoSi), which is a single crystal obtained by etching a (100) silicon wafer is an excellent candidate. Indeed, it consists of non-connected tubular pores running parallel to the [100] axis perpendicular to the wafer surface, with transverse sections with a polygonal shape of nanometric size whose areas are widely distributed. Once detached from the wafer, free PoSi membranes can be considered a nanoscale disordered honeycomb. Adsorption/desorption experiments have been performed to characterize the structure: they have shown that evaporation occurs collectively, an intriguing observation generally associated with a disordered pore structure with many interconnections through narrow necks. The characterization of fluid mobility inside the pores should give complementary information about the pore structure and topology. This paper focuses on the dynamics of a fluid confined inside the structure of porous silicon, and in particular the self-diffusion measurements (pulsed field gradient spin echo Nuclear Magnetic Resonance (NMR)). The results show a strong anisotropy of the self diffusion tensor, as expected in this highly anisotropic structure. However, a non-zero self-diffusion in the directions perpendicular to the pore axis is observed. In order to interpret these puzzling results, molecular and Brownian dynamics calculations are underway.

DOI

3

A moving contact line as a rheometer for nanometric interfacial layers - Lhermerout, Romain and Perrin, Hugo and Rolley, Etienne and Andreotti, Bruno and Davitt, Kristina

NATURE COMMUNICATIONS 7, (2016)

Abstract : How a liquid drop sits or moves depends on the physical and mechanical properties of the underlying substrate. This can be seen in the hysteresis of the contact angle made by a drop on a solid, which is known to originate from surface heterogeneities, and in the slowing of droplet motion on deformable solids. Here, we show how a moving contact line can be used to characterize a molecularly thin polymer layer on a solid. We find that the hysteresis depends on the polymerization index and can be optimized to be vanishingly small (< 0.07 degrees). The mechanical properties are quantitatively deduced from the microscopic contact angle, which is proportional to the speed of the contact line and the Rouse relaxation time divided by the layer thickness, in agreement with theory. Our work opens the prospect of measuring the properties of functionalized interfaces in microfluidic and biomedical applications that are otherwise inaccessible.

NATURE COMMUNICATIONS 7, (2016)

LPS

Abstract : How a liquid drop sits or moves depends on the physical and mechanical properties of the underlying substrate. This can be seen in the hysteresis of the contact angle made by a drop on a solid, which is known to originate from surface heterogeneities, and in the slowing of droplet motion on deformable solids. Here, we show how a moving contact line can be used to characterize a molecularly thin polymer layer on a solid. We find that the hysteresis depends on the polymerization index and can be optimized to be vanishingly small (< 0.07 degrees). The mechanical properties are quantitatively deduced from the microscopic contact angle, which is proportional to the speed of the contact line and the Rouse relaxation time divided by the layer thickness, in agreement with theory. Our work opens the prospect of measuring the properties of functionalized interfaces in microfluidic and biomedical applications that are otherwise inaccessible.

DOI

4

Adsorption-induced strain of a nanoscale silicon honeycomb - Grosman, A. and Puibasset, J. and Rolley, E.

EPL 109, (2015)

Abstract : We report on systematic measurements of both adsorption and anisotropic mechanical deformations of mesoporous silicon, using heptane at room temperature. Porous Si obtained from highly doped (100) Si can be thought of as a nanoscale random honeycomb with pores parallel to the [001] axis. We show that strains is an element of(parallel to) and is an element of(perpendicular to) measured along and transversely to the pore axis exhibit a hysteretic behavior as a function of the fluid pressure, which is due to the hysteresis in fluid adsorption. The pressure dependence of the strains together with the independent measurement of the transverse stress, allows us to determine the biaxial transverse modulus and to estimate the longitudinal Young's modulus of porous Si. We argue that the value of these constants implies that Young's modulus of the 6 nm thick walls of the honeycomb is about 5 times smaller than that of bulk silicon, striking evidence of finite-size effects. Copyright (C) EPLA, 2015

EPL 109, (2015)

LPS

Abstract : We report on systematic measurements of both adsorption and anisotropic mechanical deformations of mesoporous silicon, using heptane at room temperature. Porous Si obtained from highly doped (100) Si can be thought of as a nanoscale random honeycomb with pores parallel to the [001] axis. We show that strains is an element of(parallel to) and is an element of(perpendicular to) measured along and transversely to the pore axis exhibit a hysteretic behavior as a function of the fluid pressure, which is due to the hysteresis in fluid adsorption. The pressure dependence of the strains together with the independent measurement of the transverse stress, allows us to determine the biaxial transverse modulus and to estimate the longitudinal Young's modulus of porous Si. We argue that the value of these constants implies that Young's modulus of the 6 nm thick walls of the honeycomb is about 5 times smaller than that of bulk silicon, striking evidence of finite-size effects. Copyright (C) EPLA, 2015

DOI

5

Thermally Activated Wetting Dynamics in the Presence of Surface Roughness - Davitt, Kristina and Pettersen, Michael S. and Rolley, Etienne

LANGMUIR 29, 6884-6894 (2013)

Abstract : From simple models of thermally activated contact line dynamics far below the depinning transition, one expects the velocity to depend exponentially on the applied force and the activation area to be the size of the defects on the surface. We study contact line motion on evaporated gold films and find that the dynamics are activated, but the activation area is not straightforwardly linked to the surface roughness. Surprisingly, the activation area can be significantly smaller than any features on the surface. Furthermore, it depends strongly on the liquid. We show that this indicates that the line is close to the depinning threshold at experimentally accessible velocities. A model based on independent defects is developed and used to show deviations from the purely exponential law. The dynamics are written entirely in terms of properties of the surface and partially wetting liquid. In addition, we are able to show that the region of validity of models of thermal activation on mesoscopically rough surfaces typically corresponds to velocities of less than 1 mm/s.

LANGMUIR 29, 6884-6894 (2013)

LPS

Abstract : From simple models of thermally activated contact line dynamics far below the depinning transition, one expects the velocity to depend exponentially on the applied force and the activation area to be the size of the defects on the surface. We study contact line motion on evaporated gold films and find that the dynamics are activated, but the activation area is not straightforwardly linked to the surface roughness. Surprisingly, the activation area can be significantly smaller than any features on the surface. Furthermore, it depends strongly on the liquid. We show that this indicates that the line is close to the depinning threshold at experimentally accessible velocities. A model based on independent defects is developed and used to show deviations from the purely exponential law. The dynamics are written entirely in terms of properties of the surface and partially wetting liquid. In addition, we are able to show that the region of validity of models of thermal activation on mesoscopically rough surfaces typically corresponds to velocities of less than 1 mm/s.

DOI

6

Capillary rise and condensation in a cone as an illustration of a spinodal - Pettersen, M. S. and Rolley, E. and Treiner, J.

AMERICAN JOURNAL OF PHYSICS 79, 831-837 (2011)

Abstract : Spinodal decomposition can be observed only in systems whose dynamics are slow enough to quench through the metastable region where the phase transition occurs by nucleation. We discuss the capillary rise of a fluid in a cone inserted into a bulk fluid, with the wide end down. The rise displays a first-order phase transition with a spinodal and is easily accessible both theoretically and experimentally. (C) 2011 American Association of Physics Teachers. [DOI: 10.1119/1.3599073]

AMERICAN JOURNAL OF PHYSICS 79, 831-837 (2011)

LPS

Abstract : Spinodal decomposition can be observed only in systems whose dynamics are slow enough to quench through the metastable region where the phase transition occurs by nucleation. We discuss the capillary rise of a fluid in a cone inserted into a bulk fluid, with the wide end down. The rise displays a first-order phase transition with a spinodal and is easily accessible both theoretically and experimentally. (C) 2011 American Association of Physics Teachers. [DOI: 10.1119/1.3599073]

DOI

7

Equation of state of water under negative pressure - Davitt, Kristina and Rolley, Etienne and Caupin, Frederic and Arvengas, Arnaud and Balibar, Sebastien

JOURNAL OF CHEMICAL PHYSICS 133, (2010)

Abstract : We report on the simultaneous measurements of the speed of sound and the density in liquid water under negative pressure. Application of a focused acoustic wave to the bulk liquid is able to generate negative pressures before nucleation of the vapor phase occurs. A method for time-resolved Brillouin scattering is developed to measure the speed of sound during the passage of a 1 MHz ultrasonic wave. This is coupled with a fiber optic probe hydrophone which allows the determination of the density. Together, these methods give an ambient temperature equation of state of metastable liquid water down to the acoustic cavitation threshold. Empirical equations of state of water are based on experimental data at positive pressure; the validity of their extrapolation to negative pressures had been tested only indirectly or with very weakly metastable liquid. We provide thermodynamic data that prove the fidelity of recent equations of state down to -26 MPa. However, this raises questions regarding the nature of the cavitation threshold observed in acoustic experiments, which is far less negative than expected. (C) 2010 American Institute of Physics. [doi:10.1063/1.3495971]

JOURNAL OF CHEMICAL PHYSICS 133, (2010)

LPS

Abstract : We report on the simultaneous measurements of the speed of sound and the density in liquid water under negative pressure. Application of a focused acoustic wave to the bulk liquid is able to generate negative pressures before nucleation of the vapor phase occurs. A method for time-resolved Brillouin scattering is developed to measure the speed of sound during the passage of a 1 MHz ultrasonic wave. This is coupled with a fiber optic probe hydrophone which allows the determination of the density. Together, these methods give an ambient temperature equation of state of metastable liquid water down to the acoustic cavitation threshold. Empirical equations of state of water are based on experimental data at positive pressure; the validity of their extrapolation to negative pressures had been tested only indirectly or with very weakly metastable liquid. We provide thermodynamic data that prove the fidelity of recent equations of state down to -26 MPa. However, this raises questions regarding the nature of the cavitation threshold observed in acoustic experiments, which is far less negative than expected. (C) 2010 American Institute of Physics. [doi:10.1063/1.3495971]

DOI

8

Wetting and spreading - Bonn, Daniel and Eggers, Jens and Indekeu, Joseph and Meunier, Jacques and Rolley, Etienne

REVIEWS OF MODERN PHYSICS 81, 739-805 (2009)

Abstract : Wetting phenomena are ubiquitous in nature and technology. A solid substrate exposed to the environment is almost invariably covered by a layer of fluid material. In this review, the surface forces that lead to wetting are considered, and the equilibrium surface coverage of a substrate in contact with a drop of liquid. Depending on the nature of the surface forces involved, different scenarios for wetting phase transitions are possible; recent progress allows us to relate the critical exponents directly to the nature of the surface forces which lead to the different wetting scenarios. Thermal fluctuation effects, which can be greatly enhanced for wetting of geometrically or chemically structured substrates, and are much stronger in colloidal suspensions, modify the adsorption singularities. Macroscopic descriptions and microscopic theories have been developed to understand and predict wetting behavior relevant to microfluidics and nanofluidics applications. Then the dynamics of wetting is examined. A drop, placed on a substrate which it wets, spreads out to form a film. Conversely, a nonwetted substrate previously covered by a film dewets upon an appropriate change of system parameters. The hydrodynamics of both wetting and dewetting is influenced by the presence of the three-phase contact line separating ``wet'' regions from those that are either dry or covered by a microscopic film only. Recent theoretical, experimental, and numerical progress in the description of moving contact line dynamics are reviewed, and its relation to the thermodynamics of wetting is explored. In addition, recent progress on rough surfaces is surveyed. The anchoring of contact lines and contact angle hysteresis are explored resulting from surface inhomogeneities. Further, new ways to mold wetting characteristics according to technological constraints are discussed, for example, the use of patterned surfaces, surfactants, or complex fluids.

REVIEWS OF MODERN PHYSICS 81, 739-805 (2009)

LPS

Abstract : Wetting phenomena are ubiquitous in nature and technology. A solid substrate exposed to the environment is almost invariably covered by a layer of fluid material. In this review, the surface forces that lead to wetting are considered, and the equilibrium surface coverage of a substrate in contact with a drop of liquid. Depending on the nature of the surface forces involved, different scenarios for wetting phase transitions are possible; recent progress allows us to relate the critical exponents directly to the nature of the surface forces which lead to the different wetting scenarios. Thermal fluctuation effects, which can be greatly enhanced for wetting of geometrically or chemically structured substrates, and are much stronger in colloidal suspensions, modify the adsorption singularities. Macroscopic descriptions and microscopic theories have been developed to understand and predict wetting behavior relevant to microfluidics and nanofluidics applications. Then the dynamics of wetting is examined. A drop, placed on a substrate which it wets, spreads out to form a film. Conversely, a nonwetted substrate previously covered by a film dewets upon an appropriate change of system parameters. The hydrodynamics of both wetting and dewetting is influenced by the presence of the three-phase contact line separating ``wet'' regions from those that are either dry or covered by a microscopic film only. Recent theoretical, experimental, and numerical progress in the description of moving contact line dynamics are reviewed, and its relation to the thermodynamics of wetting is explored. In addition, recent progress on rough surfaces is surveyed. The anchoring of contact lines and contact angle hysteresis are explored resulting from surface inhomogeneities. Further, new ways to mold wetting characteristics according to technological constraints are discussed, for example, the use of patterned surfaces, surfactants, or complex fluids.

DOI

9

Prewetting of Liquid Hydrogen on Rough Cesium Substrates - Rolley, E. and Guthmann, C. and Pettersen, M. S.

PHYSICAL REVIEW LETTERS 103, (2009)

Abstract : We have studied the prewetting dynamics of H(2) on rough Cs substrates obtained by low temperature deposition. The boundary between the thin and the thick van der Waals film is strongly pinned and distorted by the defects of the substrate. Comparing prewetting and wetting dynamics allows us to show that the dynamics and the geometry of the thin-thick boundary cannot be accounted for in a simple 1D model. The finite width of the boundary makes its behavior similar in many aspects to the one of a contact line.

PHYSICAL REVIEW LETTERS 103, (2009)

LPS

Abstract : We have studied the prewetting dynamics of H(2) on rough Cs substrates obtained by low temperature deposition. The boundary between the thin and the thick van der Waals film is strongly pinned and distorted by the defects of the substrate. Comparing prewetting and wetting dynamics allows us to show that the dynamics and the geometry of the thin-thick boundary cannot be accounted for in a simple 1D model. The finite width of the boundary makes its behavior similar in many aspects to the one of a contact line.

DOI

10

Height fluctuations of a contact line: A direct measurement of the renormalized disorder correlator - Le Doussal, P. and Wiese, K. J. and Moulinet, S. and Rolley, E.

EPL 87, (2009)

Abstract : We measure the center-of-mass fluctuations of the height of a contact line at depinning for two different systems: liquid hydrogen on a rough cesium substrate and isopropanol on a silicon wafer grafted with silanized patches. The contact line is subject to a confining quadratic well, provided by gravity. From the second cumulant of the height fluctuations, we measure the renormalized disorder correlator Delta(u), predicted by Functional RG to attain a fixed point, as soon as the capillary length is large compared to the Larkin length set by the microscopic disorder. The experiments are consistent with the asymptotic form for Delta(u) predicted by Functional RG, including a linear cusp at u = 0. The observed small deviations could be used as a probe of the underlying physical processes. The third moment, as well as avalanche-size distributions are measured and compared to predictions from Functional RG. Copyright (C) EPLA, 2009

EPL 87, (2009)

LPS

Abstract : We measure the center-of-mass fluctuations of the height of a contact line at depinning for two different systems: liquid hydrogen on a rough cesium substrate and isopropanol on a silicon wafer grafted with silanized patches. The contact line is subject to a confining quadratic well, provided by gravity. From the second cumulant of the height fluctuations, we measure the renormalized disorder correlator Delta(u), predicted by Functional RG to attain a fixed point, as soon as the capillary length is large compared to the Larkin length set by the microscopic disorder. The experiments are consistent with the asymptotic form for Delta(u) predicted by Functional RG, including a linear cusp at u = 0. The observed small deviations could be used as a probe of the underlying physical processes. The third moment, as well as avalanche-size distributions are measured and compared to predictions from Functional RG. Copyright (C) EPLA, 2009

DOI

11

Dripping of a crystal - Ishiguro, R. and Graner, F. and Rolley, E. and Balibar, S. and Eggers, J.

PHYSICAL REVIEW E 75, (2007)

Abstract : Dripping is usually associated with fluid motion, but here we describe the analogous phenomenon of a He-3 crystal growing and melting under the influence of surface tension and gravity. The pinch-off of the crystal is described by a purely geometric equation of motion, viscous dissipation or inertia being negligible. In analogy to fluid pinch-off, the minimum neck radius R-n goes to zero like a power law, but with a new scaling exponent of 1/2. However, for a significant part of the neck's macroscopic evolution the scaling exponent is found to be much closer to 1/3. This observation may be consistent with simulations and theoretical results showing a very slow approach to the asymptotic pinch solution, making the ``critical region'' very small, both in time and space. After pinch-off, we observe a similar 1/3-scaling for the recoil of a crystal tip, both in simulation and experiment. For very early times our experiments are consistent with an approximate theory predicting an asymptotic regime with exponent 1/2. Future experiments must show whether the transient 1/3 scaling is a universal feature of crystal melting, or perhaps an artifact of our experimental setup.

PHYSICAL REVIEW E 75, (2007)

LPS

Abstract : Dripping is usually associated with fluid motion, but here we describe the analogous phenomenon of a He-3 crystal growing and melting under the influence of surface tension and gravity. The pinch-off of the crystal is described by a purely geometric equation of motion, viscous dissipation or inertia being negligible. In analogy to fluid pinch-off, the minimum neck radius R-n goes to zero like a power law, but with a new scaling exponent of 1/2. However, for a significant part of the neck's macroscopic evolution the scaling exponent is found to be much closer to 1/3. This observation may be consistent with simulations and theoretical results showing a very slow approach to the asymptotic pinch solution, making the ``critical region'' very small, both in time and space. After pinch-off, we observe a similar 1/3-scaling for the recoil of a crystal tip, both in simulation and experiment. For very early times our experiments are consistent with an approximate theory predicting an asymptotic regime with exponent 1/2. Future experiments must show whether the transient 1/3 scaling is a universal feature of crystal melting, or perhaps an artifact of our experimental setup.

DOI

12

Dynamics and hysteresis of the contact line between liquid hydrogen and cesium substrates - Rolley, E. and Guthmann, C.

PHYSICAL REVIEW LETTERS 98, (2007)

Abstract : We have measured both the hysteresis and the dynamics of the edge of a liquid hydrogen meniscus on several solid cesium substrates. We find that the dynamics of the contact line is thermally activated. For all substrates, we find that the activation energy is of the order of the hysteresis. We show that the pinning of the contact line on mesoscopic defects of the Cs substrate is likely to control both the hysteresis and the dynamics of the contact line at low velocity, close to the depinning threshold. Such a mechanism could be relevant also for simple room-temperature systems.

PHYSICAL REVIEW LETTERS 98, (2007)

LPS

Abstract : We have measured both the hysteresis and the dynamics of the edge of a liquid hydrogen meniscus on several solid cesium substrates. We find that the dynamics of the contact line is thermally activated. For all substrates, we find that the activation energy is of the order of the hysteresis. We show that the pinning of the contact line on mesoscopic defects of the Cs substrate is likely to control both the hysteresis and the dynamics of the contact line at low velocity, close to the depinning threshold. Such a mechanism could be relevant also for simple room-temperature systems.

DOI

13

The hydraulic jump and ripples in liquid helium - Rolley, E. and Guthmann, C. and Pettersen, M. S.

PHYSICA B-CONDENSED MATTER 394, 46-55 (2007)

Abstract : We have studied the characteristics of the circular hydraulic jump using liquid helium. Surprisingly, the radius of the jump does not change at the superfluid transition. We think that the flow is still dissipative below the lambda point because the velocity exceeds the critical one. The jump radius R-j is compared with various models. In our parameter range, we find that the jump can be treated as a shock, and that capillary effects are important. Below the superfluid transition, we observed a standing capillary wave between the impact of the jet and the jump. Assuming that the superfluid flow can be described with an effective viscosity, we calculate the wave vector and thus obtain the value of the liquid thickness, which is in reasonable agreement with predictions. However, the spatial variation of the wave amplitude depends much more strongly on temperature than we calculate. (c) 2007 Elsevier B.V. All rights reserved.

PHYSICA B-CONDENSED MATTER 394, 46-55 (2007)

LPS

Abstract : We have studied the characteristics of the circular hydraulic jump using liquid helium. Surprisingly, the radius of the jump does not change at the superfluid transition. We think that the flow is still dissipative below the lambda point because the velocity exceeds the critical one. The jump radius R-j is compared with various models. In our parameter range, we find that the jump can be treated as a shock, and that capillary effects are important. Below the superfluid transition, we observed a standing capillary wave between the impact of the jet and the jump. Assuming that the superfluid flow can be described with an effective viscosity, we calculate the wave vector and thus obtain the value of the liquid thickness, which is in reasonable agreement with predictions. However, the spatial variation of the wave amplitude depends much more strongly on temperature than we calculate. (c) 2007 Elsevier B.V. All rights reserved.