T H È S E. Development and improvement of the experimental techniques for fluid examination. Docteur en Sciences de l Ingéneur. Prof.

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1 UNIVERSITÉ LIBRE DE BRUXELLES ÉCOLE POLYTECHNIQUE DE BRUXELLES T H È S E présentée en vue de l obtention du Grade de Docteur en Sciences de l Ingéneur Présentée et soutenue par Viktar YASNOU Development and improvement of the experimental techniques for fluid examination Thèse dirigée par: Supervision scientifique: Prof. Frank DUBOIS Dr. Valentina SHEVTSOVA Prof. Yury BOKHAN Année académique : soutenue le 23 Octobre 2014

2 Abstract The aim of the thesis is the development and improvement of the experimental techniques for fluid examination. The thesis consists of two parts and both examine heat and mass transfer in liquids using the optical methods and thermal analysis. The first part deals with the measurement techniques for studying flow patterns and their stability in systems with gas/liquid interface, in particular, in a liquid bridge system. The second part is aimed at the improvement of the existing experimental techniques to study the heat/mass transfer in the mixtures with Soret effect, enclosed in a container. Part A is motivated by preparation of the experiment JEREMI (The Japanese- European Research Experiment on Marangoni Instability) to be performed on the International Space Station (ISS). One of the objectives of the experiment is the control of the threshold of an oscillatory flow in the liquid zone by the temperature and velocity of the ambient gas. The developed set-up for a liquid bridge allows to blow gas parallel to the interface at different temperatures and investigate the effects of viscous and thermal stresses on the stability of the flow. The present study reports on isothermal experiments with moving gas and non-isothermal experiments with motionless gas when the cooling of the interface occurs due to evaporation. The discussion concerning the experimental observations is based on two sources: an interface shape measured optically and the records on thermocouples giving an indication of how temperature and frequency evolve over time. Part B is related to ground-based studies in course of preparation and realization of the microgravity experiment DCMIX (Diffusion Coefficient in MIXtures). DCMIX project is a series of experiments aimed at measuring of the Soret coefficients in liquid mixtures on the ISS which involves a wide international group of scientists. Two experiments have been recently completed and the third one is under preparation In the course of this thesis all the aspects of the previously existing set-up for measurements of the Soret (thermal diffusion) and diffusion coefficients in binary mixtures were studied, uncertainties were identified and improvements were done to obtain reliable results. The final design has been validated by measuring coefficients in three binary benchmark mixtures and waterisopropanol. The obtained results agree well with literature data.

3 A note of thanks I would like to acknowledge everyone who has assisted me throughout my doctoral studies over the years. First, I would like to express my deep gratitude to my scientific adviser, Dr. Valentina Shevtsova, for the interesting topics, professional guidance, inspiration, invaluable advice, patience, friendly supervision, constant support and encouragement. I also would like to thank Prof. Yury Bokhan, for its activity as second supervisor of my work. My greatest appreciation and gratitude to Aliaksandr Mialdun, this thesis would not have been possible without his support and assistance! I am also grateful to other members of our group Non-linear Phenomena in Liquids Yuri Gaponenko, Jean-Claude Legros, Denis Melnikov, for their advice, interesting discussions and helpful suggestions. I would like to thank Professor Frank Dubois creating favourable conditions for efficient and fruitful work in Service Chimie-Physique E.P. of ULB and for his advice and care. Many thanks to all other colleagues of Service Chimie-Physique E.P. for their kindness, support and inexhaustible readiness to help. A special thank to my family for their endless support throughout my studies. Without their constant assurance and assistance, completion of this work would have not been possible. I want to thank my lovely wife for her patience, support and motivation in the completion of my PhD degree. Thank you all.

4 Content 1. Part A: Measurement techniques for studying heat/mass transfer in a liquid bridge....1 A1. Introduction A1.1. Thermocapillary convection in a liquid bridge A1.2. Deformation of interface....6 A1.3. Stability of flow under the influence of different factors....7 A Interfacial heat flux....7 A Shape of interface...11 A1.4. Experimental methods of observation.. 13 A Observation of flow..13 A Measurements of temperature field...14 A2. Experimental set-up.. 16 A2.1. General overview..16 A2.2. Fluids management..18 A2.3. Volume control..20 A2.4. Optical system..20 A2.5. Control unit..22 A3. Validation of the new set-up...24 A4. Shape detection..30 A4.1. Preliminary shape detection with pixel-size accuracy..30 A4.2. Determination of the threshold for sub-pixel shape detection A4.3. Final shape detection with sub-pixel accuracy..35 A5. Measurements of volume, evaporation and dynamic surface deformation A5.1. Measurements of volume...36 A5.2. Measurements of evaporation...36 A5.3. Detection of dynamic deformation...41 A6. Stability of convection driven by thermocapillary and buoyant force...45 A6.1. Variety of experimental procedures A Scanning by T at constant volume and T mean A Scanning by volume at constant T and T mean A Scanning by mean temperature at constant T and constant volume.57

5 A6.2. Analysis and comparison of the instability regimes...58 A6.3. Stability window.. 63 A6.4. Effect of the mean temperature on the flow stability Part B:Measurements technique for heat/mass transfer in mixture with Soret effect.70 B1. Introduction. 70 B2. Set-up.. 73 B2.1. Initial configuration..73 B2.2. Cell optimization.. 76 B2.3. Final cell design elaborated in the work..80 B2.4. Interferometer modification.. 82 B2.5. Control unit.. 84 B3. Image processing..86 B3.1. Fringe analysis for phase-measuring interferometry...86 B3.2. Subtraction of reference image B3.3. Experimental limitations and precision of the method...92 B3.4. Beam deflection problem B4. Data extraction..95 B4.1. Governing equations B4.2. Fitting equation for full path...96 B5. Results..99 B5.1. Water Isopropanol (IPA) B Contrast factors B Summary on Soret, diffusion and thermodiffusion coefficients..100 B Region with negative Soret effect; water rich mixture, C> B Intermediate concentration regime, 0.2<C< B Region with low water content C< B Comparison with benchmark values 105 B Error estimation B Discussion 107 B5.2. Binary couples in tetralin isobutylbenzene n-dodecane system B Contrast factors. 108 B Transport coefficients.109 B Discussion 112

6 3. Conclusions Outlook List of Figures List of Tables Bibliography List of publications

7 Part A Measurement techniques for studying heat/mass transfer in a liquid bridge A1. Introduction A1.1. Thermocapillary convection in a liquid bridge Surface tension of liquids is usually a decreasing function of temperature, and any temperature variation along an interface generates a surface flow transporting warm fluid towards cooler regions. A fluid motion driven by surface tension differences along a liquid-gas interface is called thermocapillary convection or Marangoni convection. Such flows are very common in nature and in numerous industrial applications, in which, e.g., evaporation, melting or welding are taking place. Figure A1.1. Floating zone technique of crystal growth. Thermocapillary flows and their hydrodynamic stability have received some particular attention being motivated by crystal growth technology. In particular, the containerless methods were suggested to avoid contamination from crucible walls to meet the needs of electronic industry for ultra-pure homogeneous materials. The containerless growth of semiconductor crystals or the so-called floating zone technique is a technological process to grow crystal and an efficient purification technique. It was 1

8 developed by [1] for zone refining of Germanium, and further used for silicon by Keck and Golay [2] and by Emeis [3]. The floating zone method shortly can be described as a process when a long hard rod of semiconductor is slowly pulled through a ring heater (see Fig. A1.1) and, at first, is locally melted and then re-crystallized, while the impurities concentrate at the end of the rod. On industrial level, floating zone method was used for the first time by SIEMENS AG. The floating zone is kept between two solid parts melting and freezing ones. Being contactless to crucibles, and thus preventing potential sources of contamination, the method makes it possible to grow monocrystalline silicon with the highest purity which is important for a number of electronic and optoelectronic applications. Floating zone monocrystalline starts to grow from a high purity, small diameter seed crystal. This seed crystal is prepared in the right crystalline direction in order to grow pure silicon with no crystalline defects. (a) (b) Figure A1.2. Sketch of floating zone method (a) and of a liquid bridge (b). Liquid bridge, representing a bottom half of a floating zone is a model used to describe physical phenomena occurring in the real technological processing. Using the floating zone technique, industry obtains silicon with unique properties due to small amount of crystal defects, such as vacancies or interstitial agglomerates. Float zone silicon can be grown with values of resistivity exceeding Ohm-cm because of the intrinsic process purity. Perhaps the most important feature of the floating zone technology is the ability to exactly control the resistivity of the crystal. This is particularly 2

9 important for applications using the bulk of silicon wafers for manufacturing devices. There are two practical ways of obtaining very good resistivity control. One is by doping the crystals when they are pulled by introducing controllable amounts of gaseous dopants into the growth chamber. The most common dopants are phosphorus and boron for n- and p-type, respectively. This technique is called in-situ doping or gas phase doping. The other technique is by doping the crystal after it has been pulled, it is called ex-situ doping, and it is done in neutron irradiating reactors. The starting material for ex-situ doping is high resistivity silicon that after being irradiated with a controllable dose of neutrons changes its resistivity by transforming silicon atoms into dopant atoms. Crystals produced by this method in a terrestrial environment are not large in size due to the weight of the melt which tends to destroy the liquid zone held by surface tension. Silicon crystals are presently industrially grown with diameters up to 150 mm, weighing more than 35 kg. One of the reasons for the non-homogeneity of the final crystals was attributed to the presence of buoyancy during the technological process. The non-steadiness of the flow generates variations of the properties of the produced materials. Among the crystal defects are voids, inclusions, distortion, strain, dislocations, inhomogeneities, striations [4]. In order to improve homogeneity, studies of convection in the melting zone, its stability and dependence on various factors were initiated. In the early days of spaceflights, it was believed that crystals of exceptional quality could be grown from the melt in the microgravity environment due to the absence of the undesired buoyant convection. However, as the ring heater creates temperature gradients along the free surface of the melt, Marangoni convection inevitably occurs in the melt independently of buoyant convection. Many microgravity experiments have been performed since the early 70s. Between the Apollo missions and the year 1995 at least 77 experiments were successfully conducted. A liquid bridge model represents half of a floating zone and is widely used to mimic the process of crystal growth. The idea to replace the whole liquid zone by its half is straightforward. The ring heater used to melt a rod is situated in the middle of the melted zone. Thus, the temperature reaches its maximum at the mid-height of the zone. Spreading from the hot to the cold areas along the free surface, two sets of convective rolls (above and below the heater) are developing. Hence, for the sake of simplicity, only a half-zone can be considered. In the liquid bridge, a temperature difference T (the control parameter of the system) is imposed between the supporting flat concentric disks. When 3

10 experimenting on ground, only a small liquid volume may be held between the rods. The typical radius of disks is several millimeters. At small values of the imposed temperature difference a two-dimensional state is observed. When the temperature difference exceeds the critical value, this basic state becomes unstable and gives rise to a new spatial pattern (3D steady or oscillatory). Pioneer experiments on oscillatory thermocapillary flows were conducted in model floating zones by Chun [5]. Under microgravity conditions the first experiments were performed independently by D. Schwabe and C.H. Chun at the same missions (TEXUS3A 1980, partly successful) and (TRXUS 3b, fully successful), which showed that thermocapilalry forces provide a very strong convection [6], [7]. (a) (b) Figure A1.3. (a) - Temperature field obtained via CFD simulations at the central cross section (top picture) and at unrolled interface of a liquid bridge. Red and blue colours show higher and lower temperature, respectively. The temperature field clearly shows a flow structure with wave number m=2. (b) experimentally recorded temperature field in case of a welldeveloped travelling wave with m=1. The liquid bridge is in the centre, while the two side images are its reflections in the mirrors put behind. The Prandtl number Pr (ratio of kinematic viscosity to thermal diffusivity) of the liquid is one of the key parameters of the flow structure above the threshold of instability. If Pr is small, e.g., for liquid metals, the flow above the critical point is stationary although 4

11 3D. For Pr >0.07, the supercritical flow is oscillatory. A hydrothermal instability in a high Pr number LB emerges through a supercritical Hopf bifurcation as either a travelling wave in the azimuthal direction or a standing wave. The magnitude of the critical temperature difference, T, is related to the physical properties of the liquid, to the geometrical constraints and is sensitive to the ambient conditions. The supercritical flow is characterized by an azimuthal wave number m, which describes the azimuthal periodicity of the thermocapillary flow and may change with the aspect ratio Γ = d/r (ratio height d to radius R of LB) of the liquid bridge, the temperature difference T, the gravity. A flow in a LB has to comply with the azimuthal periodicity, consequently, m has to be an integer. The wave number m is equal to the total number of warm (or cold) spots in the temperature field (e.g., in the developed external interface or in a transverse crosssection). The azimuthal wavenumber m of a slightly supercritical flow was found to be primarily determined by the aspect ratio. The first empirical correlation between the azimuthal wave number m and the aspect ratio Γ near the critical point was suggested by Preisser et al. [8] as m 2.2/Γ. Computer simulations in absence of buoyancy provided the same relation but with a slightly different coefficient 2.0 for Pr < 7 [9]. When including buoyancy forces, corresponding to the terrestrial conditions, it results in a different value of the coefficient [10], [11]. Furthermore, the existence of stable m = 0 mode, a hydrothermal mode running along the interface in the axial direction, was reported [12], [13], [14]. Previously, the axisymmetric oscillatory solution was predicted by [15] in an infinitely long column. The mechanism of instability of thermocapillary convection in the form of a hydrothermal wave was first suggested by Smith and Davis [16]. They employed linear theory to investigate the stability of liquid films of small depth for two different planar geometries whose basic-state solutions are referred to as the linear flow solution (infinite liquid layer) and the return-flow solution (two-dimensional slot). The authors considered the effect of three-dimensional disturbances for a fluid with constant physical properties and a surface tension linearly dependent on temperature set into motion by a horizontal thermal gradient imposed along the interface. The free-surface is regarded as nondeformable, the dynamics of the gas phase are neglected, the liquid is bounded from below by an adiabatic rigid plane and there are no body forces. Heat transfer across the interface is controlled through the Biot number B = hd/k, where h is the heat transfer coefficient, d is the layer thickness, and k is the liquid thermal conductivity. Smith and Davis [16] found 5

12 two classes of thermal-convective instabilities, namely, stationary longitudinal rolls and propagating hydrothermal waves (hereafter referred to as HTWs). Riley and Neitzel [17] conducted a series of experiments in a laterally heated rectangular geometry, considering very thin liquid layers of 1cSt silicone oil (d = mm; Pr = 13.9). They reported observing pure HTWs for d 1.25 mm presenting a good agreement with the results from the linear theory of Smith and Davis [16]. A definitive proof for the existence of the HTWs in a liquid bridge was presented by Kuhlmann [18]. As the distance from threshold of instability is increased, secondary instabilities may occur when a primary pattern undergoes transitions to other states, e.g., chaos [19], in a possible succession of different patterns as the system moves further from the critical point. The thermocapillary convection cannot be totally suppressed, but the oscillatory regime can be weakened. Most of the works aimed at suppressing the oscillations use methods of altering the steady state and thus decreasing the effective Marangoni number to attenuate the fluctuations. Among the methods the most popular are based on using magnetic field in a floating-zone of electromagnetically active melt (Dold et al. [20], and Cröll et al. [21]). Other approaches were based on creating a counter flow of the ambient gas (Dressler and Sivakumaran [22]), on imposing vibrations of the end-walls (Anilkumar et al. [23]), or on using surrounding gas at a certain temperature and pressure that decreases the surface tension (Azumi et al. [24]). Another set of attempts to weaken hydrothermal wave is based on rotating the whole system. Among evident drawbacks of these methods aimed at decreasing the effective Marangoni number is that the weakening of the basic state enhances macro-segregation of chemical compositions because of the weakening of the global mixing. There exist more sophisticated methods consisting in heating locally the free surface by actuators and using a feedback control algorithm defined by the oscillations in the liquid bridge themselves. The inputs of the algorithm are signals from local sensors (Petrov et al. [25]). A1.2. Deformation of interface As a result of the spreading of a hydrothermal wave, the position of the interface fluctuates [26]. The net interface deformation is composed of three contributions: the static deformation of the interface defined by Young-Laplace equation, subcritical deformation 6

13 due to stationary convection (base state), and supercritical dynamic deformation due to oscillatory convection. The theoretical studies of the stability of the liquid bridge with deformed interface by Shevtsova et al [27], [28], [29], Kuhlmann et al. [30], Nienhueser et al. [31] preceded the experimental ones. The points of interest in the experimental works by Montanero et al. [32], Shevtsova et al. [33], Ferrera et al. [34], [35] were focused on measurements of the amplitudes of deformations, the fundamental frequency and their impact on Marangoni flow. These efforts were directed at both the static and the dynamic deformation of thermocapillary liquid bridges. One of the results is that the subcritical deformations grow linearly with increasing strength of the basic Marangoni flow. Another source of dynamic fluctuations of interface is the gas flow around the LB. Herrada et al. [36] performed calculations of an isothermal liquid bridge with a straight cylindrical interface in a coaxial gas flow. The conditions of a fully developed flow were prescribed for gas at both ends of a liquid zone. They reported that the maximum magnitude of the free surface deformation depends almost linearly on the gas velocity. A similar study was performed by Gaponenko et al. [37]. They were interested in deformations of an isothermal interface caused by a co-axial gas flow entering from the top. Matsunaga et al. [38] measured on ground the dynamic fluctuations of an interface in an isothermal LB of 3 mm in radius caused by a shear-driven flow. The experiments were conducted with 5cSt silicone oil and nitrogen as the outer fluid. The gas was blown parallel to the interface. They obtained that the magnitude of the dynamic deformation varies from 1 to 15 microns, that is between 0.03 and 0.5% of the radius of the liquid bridge, depending on the gas velocity and the volume ratio. The dynamic deformations are larger when gas enters from the top, from the hot side. A1.3. Stability of flow under the influence of different factors A Interfacial heat flux The important role of the heat transport through the liquid-gas interface on the stability of the thermocapillary flow in LB has been reported since the 80s. Heat flux modelled by Newton s law of cooling relates the rate of the change of the temperature to the difference between the temperature of the medium T and the ambient temperature T amb : λ l dd dd = h(t T aaa), 7 (A1.1) where λ l is the thermal conductivity of the liquid, h is the heat transfer coefficient.

14 number Bi: The local rate of heat transfer through the interface is characterized by the Biot BB = hr λ l. (A1.2) The value of the heat transfer depends on a number of factors. The most important ones are convective mechanism of heat transfer, evaporation and thermal radiation: h=h c + h e + h r. While the radiative flux is difficult to estimate, the others can be estimated. Melnikov et al. [39] have shown how to estimate the convective heat transfer coefficient h c due to the presence of a moving gas surrounding the interface of a liquid bridge. Their approach is based on a work by Kays et al. [40]: h c = 0.664RR 1/2 1/3 ggg Pr λ ggg ggg d, (A1.3) where ν is the kinematic viscosity, Re=V i d/ν gas is the Reynolds number in gas phase, the subscript gas denotes the ambient gas. V i is the velocity of the liquid at the interface. It enters into the definition of Re as it is assumed that the ambient gas is entrained by the moving interface, and thus its characteristic velocity equals to that of the liquid. The Reynolds number of the moving gas is calculated via computer modelling for the experimental conditions. Evaporation is characterized by the mass loss dm/dt through the interface, and consequently, by the related loss of energy. Knowing the evaporation rate and neglecting the thermal diffusivity and kinetic energy of the surrounding air, one obtains from the energy balance: λ = L v dd S dd, (A1.4) where L v is the latent heat of evaporation, S=2πRd is the area of the evaporating surface. Hence, L v h e = S(T i T ggg ) dd dd, (A1.5) where (T i -T gas ) is the difference in temperature between the interface and surrounding gas. The mass loss may be experimentally evaluated by monitoring the liquid volume change with time. Dressler et al. [22] performed experiments in a silicone oil liquid bridge. They used a vertical jet of air blown tangentially over the free surface for producing a viscous shear 8

15 drag opposing the Marangoni shear at the free surface. An average reduction of 66% of the Marangoni velocities was successfully obtained during their experiments. Velten et al. [41] carried out ground experiments on liquid bridges formed by sodium nitrate (Pr = 9), potassium chloride (Pr = 1,) and tetracosane (Pr = 49). They studied both positive and negative temperature gradients. The authors somewhat confirmed the observations of [22] that the air motion around the liquid column, mainly caused by buoyancy due to the heating and cooling arrangement of the experiment, has a strong effect on the onset of an oscillatory flow. This work reported that values of T cr were higher in the heated-from-below cases. The difference was attributed to alterations in the flows of gas surrounding the zone, which was confined by a larger quartz cylinder playing the role of shielding. In the heating-from-above case, this gas flow exhibits a pair of counter-rotating tori which, in turn, modify the radial heat transfer. Kamotani and co-workers [42], [43] and [44] experimentally measured the effect of surface heat loss/gain for 2 and 5 cst silicone oils with high Prandtl numbers (Pr = 24-49). They performed ground-based experiments on LB using 2 and 3mm diameter rods and reaching Γ= d/r = A wide range of ambient temperatures was studied by placing the experimental set-up into an oven. The authors estimated the Biot number to be about unity or smaller. It was obtained that even at such moderate values of Bi the enhanced heat loss from the surface significantly destabilized the Marangoni flow. Changing the air temperature alters the critical temperature difference by a factor between two and three. The airflow analysis showed that even though the heat loss was relatively small, the critical temperature difference was affected appreciably. Analyzing a very high sensitivity of the critical conditions to the heat transfer through the free surface reported in [43], the authors made as a possible conclusion that this effect could be due to some changes of dynamic free surface deformation [44], which, in turn, is small and is itself strongly affected by the heat loss. Shevtsova et al. [45] and Mialdun et al. [46] experimentally investigated different thermal conditions around the interface of a liquid bridge of aspect ratio Γ = 1.2 made of 5 and 10cSt silicone oils. The experiments were carried out under terrestrial conditions for a wide range of liquid volumes. They varied temperature profile in the air performing experiments with and without external shielding. They reported a remarkable influence of the temperature distribution in the air on the stability of the thermocapillary flow. 9

16 Schwabe [47] performed microgravity experiments onboard the sounding rocket MAXUS-4 in 2cSt silicone oil (Pr=28) LB of very large aspect ratio Γ= 5 (the length d = 15.0 mm and the radius R = 3.0 mm). The opposite direction of the hydrothermal wave with respect to the theoretical predictions of [15] was attributed to strong surface cooling, as the estimated Biot number was Bi = 3.5. The authors stated that the high heat loss at the interface promotes instability waves travelling counter to the surface flow. In the last several decades, many numerical studies of the thermal convection in liquid bridges with interfacial heat transfer have been performed. Xu et al. [15] considered an infinitely long liquid bridge under weightlessness. Having kept constant the ambient gas temperature and chosen only Bi = 0 and 1 they predicted an increase of the critical Marangoni number when increasing the Biot number. Sim and Zebib [48] studied the effect of free surface heat loss on the critical conditions by means of direct numerical simulations in LB with unity aspect ratio and Pr = 30. For that study, a uniform temperature in the gas was chosen equal to the temperature of the cold disk. It was obtained that cooling the free surface stabilizes the flow. Increasing the Biot number from 0 up to 20 led to a monotonous increase of T cr. A liquid bridge with high Prandtl number fluid, Pr=18, surrounded by an ambient gas of constant temperature was studied by [49] for a large aspect ratio, Γ=1.8. They reported that increasing the Biot number from zero up to 1.8 does decrease the critical Marangoni number by 33%. Recently, the effect of interfacial heat exchange on thermocapillary flow in a cylindrical liquid bridge of 1 cst silicone oil (with Prandtl number about 16) with aspect ratio 1.8 has been investigated in absence of gravity by a linear stability analysis [50]. With both constant and linearly distributed ambient temperature, the computed results predicted that the stability curve for the thermocapillary flow exhibits a roughly convex trend with variation of the Biot number. Summarizing the aforementioned, it seems likely that no general trend could be found so far between the heat transport through the interface and the dynamics of the thermocapillary flow. Only one conclusion could be made, that for the thermocapillary flow in high Pr fluids cooling changes the radial temperature gradients and, as a consequence, influences the critical conditions according to the mechanism of the instability described by [9]. 10

17 The influence of heat loss (or gain) on the stability of the thermocapillary flow in LB is strongly influenced not only by the Biot number and the ambient temperature in the gas, but also by the parameters governing the liquid flow, i.e., the Prandtl number, the aspect ratio, gravity, and the flow in the gaseous phase [45], [51]. The recent experimental benchmark [52] demonstrated large scattering (~15-20%) of the results for the critical parameters between different experimental groups working with the same liquid. Any comparison between different results could be made only if they were obtained under identical conditions. In some theoretical works, the Biot number was varied with large increments, and thus, possible local changes in the dynamics of the flow were disregarded. One should perform the study carefully by varying the Biot number by small steps, especially between 0 and 1, as at high heat loss the behaviour of the pattern is more or less the same because the cooling prevails any other factor. Experimenting with a liquid bridge of 10 cst silicone oil, Shevtsova et al. [53] experimentally demonstrated that the convective mode may change when the temperature of the cold rod of the liquid bridge varied while keeping its volume constant. This discovery has a direct connection with the influence of heat transfer on a thermocapillary flow. Changing the cold temperature (the environmental thermal conditions stay the same) one manipulates the heat flux through the interface. Along with the mode change, the twodimensional flow destabilizes ( T cr drops by almost 25%) while the temperature at the cold rod increases from 10 to 30 C. A Shape of interface Another important factor with respect to the stability of a two-dimensional flow is the volume of the liquid (denoted as Vol) contained in the bridge and, correspondingly, the static deformation of interface. The magnitude of the deformation is determined by the ratio of the hydrostatic to the capillary pressure, i.e., by the so-called static Bond number BB = ρgd2 σ, (A1.6) where ρ is the liquid density, g is the gravity acceleration, σ is the surface tension. At zero-gravity, the Bond number is zero, and the interface of a liquid bridge with volume of the corresponding cylinder equal to πr 2 d is straight. On the contrary, on ground Bo 0 and the free surface is always deformed. The greater is the Bond number, the greater is the deformation of the interface. 11

18 Both experimental and theoretical results on the influence of the liquid volume on the stability of a thermocapillary flow are usually presented in the form of a curve T cr versus volume ratio V r, which is the ratio of the volume of the liquid bridge to the volume of the corresponding cylinder, i.e., VV = VVV πr 2 d. (A1.7) One of the first experimental evidences is that the critical temperature difference depends strongly on the liquid volume. It was obtained by Cao et al. [54], Hu et al. [55], Masud et al. [56] and Hirata et al. [57]. It was also found experimentally, that the wave number m at the onset of instability changes with the volume of the liquid bridge. A nonmonotonic response of the stability curve to increasing the liquid volume was obtained for high-prandtl liquids. Shevtsova et al. [58] showed for high Prandtl numbers under terrestrial conditions that the stability diagram (ΔT cr vs. Volume) consists of two branches with different oscillatory modes. In a LB of 10 cst silicone oil of aspect ratio Γ = 4/3 they identified two branches corresponding to different azimuthal wave numbers. In slim liquid bridges with a concave interface the stable mode corresponds to m = 1, but it is m = 2 when the interface is convex. In the region around Vr=1 the stability curve has a local maximum, meaning a strong stabilization of the flow. These results were confirmed later in [59]. Performing experiments with n-decane in a wide range of Vr [0.7;1.05] Melnikov et al. [39] measured the critical temperature difference and confirmed the existence of two branches with a gap at around Vr=0.9. The numerical results have shown that these two branches of T cr are not associated with a change of mode. When the interface is not straight, the relation between the mode m and the aspect ratio should be revised. Considering a LB with small Prandtl numbers, Pr = 0.01, no buoyancy, Lappa et al. [60] suggested a modified empirical relation, which re-defines the aspect ratio via the radius of the liquid column at the mid-height: m 2 Γ, Γ d = h(z = d 2 ) (A1.8) where h(z) is the radial coordinate of the interface, which depends on the height z. Performing linear stability analysis, Nienhüser et al. [31] have confirmed this formula and validated it up to Pr = 4. 12

19 With respect to the stability of deformable liquid bridges, the theoretical developments have not progressed too much. The reason is mainly that modelling a flow in non-straight geometries becomes technically difficult and CPU time-consuming. The first calculations on steady thermocapillary convection in a floating zone with a deformable free surface were performed in the limit of small deformations by Kozhoukharova et al. [61]. It is worth mentioning a few numerical simulations of two-dimensional thermocapillary flows in liquid bridges with strong deformation of the free surface performed by Shevtsova et al. [27], [28], Sumner et al. [62], Tang et al. [63]. They studied the influence of g-jitter on thermocapillary convection in an axisymmetric LB with deformation of the free surface. Lappa [64] made a study on the influence of liquid volume and gravity on the flow instability in a floating zone. Being aware of the difficulties related to modelling a three-dimensional thermocapillary flow in a liquid bridge with a non-cylindrical interface a benchmark was performed, where 9 research groups were participating. A goal was to analyze the validity of the models of the flow and to predict the onset of hydrothermal instability in a liquid bridge with Pr=0.02 [29]. A1.4. Experimental methods of observation A Observation of flow The use of non-intrusive methods like optical methods is very efficient and avoids or at least minimizes their influence on the flows. The investigations need to visualize the flow at the surface and in the bulk. Visualization of a bulk flow often involves the use of tracers, e.g., different kinds of particles. Tracer particles, however, must be very small and have a density close to that of the working liquid. To properly visualize the flow by one of these methods, an appropriate lighting is required. Depending on the type of illumination, a flow structure can be observed either in a plane or in the entire volume. A common technique is to use light sheets. The use of a diffused illumination with the help of a source of incoherent light results in a view integrated over the depth. Due to the specific nature of a thermocapillary flow - it is driven by a temperature variation its illumination must be done carefully. The optical methods for flow visualization in non-homogeneous liquids could be shadowgraphy, schlieren and interferometry. 13

20 A Measurements of temperature field The temperature field is measured by two main methods: contact thermometry and radiation (contactless) thermometry. Contact thermometry is carried out with a sensor, e.g., a thermocouple, or a Platinum resistance thermometer, which always remains in contact with the device under test. The principle of any contact thermometer is that it is designed to have some physical parameter changing in a well-known way with temperature. A contactless thermometer is put into a contact with the tested object by means of electromagnetic radiation, e.g., infrared radiometer. Radiation thermometry measures the radiation of the device under test without contact, by means of an infrared sensor. The thermal radiation emitted by an object is not only defined by the temperature of the object, but also by the emissivity ε of its surface. The power output of an object with surface temperature T s surrounded by a medium with temperature T amb is given by the Stefan Boltzmann law: P = εσσ(t 4 4 s T aaa ), (A1.9) where S is the radiating surface area, σ= W m 2 K 4 is the Stefan Boltzmann constant. The emissivity deserves special attention. It is not a universal constant, but a dimensionless number between 0 (for a perfect reflector) and 1 (for a perfect emitter). Non-metallic and non-transparent objects are generally good radiators with an emissivity larger than 0.8. The emissivity of metals can vary between 0.05 and 0.9. Shiny, highly reflective metal surfaces will have lower emissivities. The emissivity of a surface depends not only on the material but also on the nature of the surface, its temperature, wavelength and angle. For example, a clean and polished metal surface will have a low emissivity, whereas a roughened and oxidised metal surface will have a high emissivity. Radiation thermometers are generally calibrated using blackbody reference sources that have an emissivity close to 1. Unfortunately, the emissivity of a material surface is often very difficult even to estimate. It must either be measured or modified in some way, for example, by coating the surface with high emissivity black paint, to provide a known emissivity value. When working with liquid surfaces, this approach is obviously not possible. Thus, one needs to calibrate the thermometer before using it for reliable experimental temperature records. There are other methods of contactless thermometry. It is worth mentioning acoustic (based on the dependence of the velocity of sound on temperature) and laser- 14

21 based methods (e.g., laser-absorption radiation thermometry or Rayleigh-scattering thermometry). Contactless thermometry has both advantages and disadvantages as compared to contact thermometry. One does not need to wait for establishing thermal equilibrium between the object and the thermometer. It is, thus, better suited for the measurement of fast temperature changes. It is always better to use the radiation thermometry as any direct contact with a flow inevitably generates disturbances. IR cameras allow high spatial resolutions of surface temperatures. Their response to temperature changes is immediate. On the other hand, records of a thermocouple are much more accurate than those from a reasonably priced IR camera. Temperature measurements that are taken with thermocouples, however, have a very limited spatial resolution. Another drawback of the radiation method of temperature measurements is that one cannot access the interior of the tested domain. The signal to be measured by contactless thermometry may be perturbed on its way from the object to the thermometer, or the signal may merely correspond to an average temperature (averaged over the line of sight, for example). To determine the mode of the supercritical flow, one needs to use either an IR camera or a set of thermocouples. This work is related to ground-based studies in course of preparation of the experiment JEREMI (The Japanese-European Research Experiment on Marangoni Instability) to be performed on the International Space Station (ISS) with a launch date of One of the objectives of the experiment is the control of the threshold of an oscillatory flow in the liquid zone by the temperature and velocity fields in the ambient gas. In the present work the ambient gas is motionless but the cooling of the interface occurs due to evaporation. The study performed in this thesis will extend our knowledge about flow stabilization and its control. 15

22 A2. Experimental set-up A2.1. General overview Considering the numerous requirements to be met by the set-up, its design appears to be quite complex (Yasnou et al. 2012) [65]. First, it has to allow blowing gas axially in the duct around the liquid bridge in both directions with the possibility to control the temperature of the gas. Second, independent and precise temperature control of both top and bottom metal rods and their precise alignment are needed. The temperature inside the liquid has to be measured at different points. As there is no direct access to the liquid as it is surrounded by a glass tube of the air gas channel, all sensors and supplying pipes have to be driven through the supporting rods. The general view of the set-up is shown in Fig. A (a) (b) 9 Figure A2.1. a) View of the set-up, b) Simplified sketch of the set-up 1 - lower plate, 2 - lower clamp, 3 - upper plate, 4 - upper clamp, 5 - middle plate, 6 - cylinder guides, 7 - screw pairs, 8 - toothed belt, 9 - liquid bridge position The set-up is a structure consisting of a lower plate with a clamp for holding the lower rod, an upper plate with an auto-centering clamp for holding the upper rod and a moving middle plate on which the optical system is integrated. The middle plate moves vertically along cylindrical guides. This displacement is driven by screw pairs synchronized by a toothed belt. The clamp of the lower rod has the possibility of precise two-axis positioning as well as precise control of verticality. The exact adjustment of the liquid bridge height is performed by the upper rod driven by a micro-screw. 16

23 The geometry of the liquid bridge is shown in Fig. A2.2. The working parts of the rods 3mm in radius are made of brass. The minimal distance between them is h min =0.5 mm, which is conditioned by the space necessary for the thermocouples installed in the upper rod. The maximum distance between the rods is determined by the hydrostatic stability of the liquid bridge at normal gravity and is approximately h max =5-7 mm. The rods are placed inside a glass tube with inner radius R ext =5mm. Gas can be injected between the internal walls of the glass tube and the rods. For producing a laminar gas flow after the gas inlet, the rods have a system of six azimuthally symmetric wide ducts (marked 9 in Fig. A2.2). To additionally homogenize the gas flow, a relatively long distance of L=25 mm is left between the ducts, where the gas is injected, and the liquid bridge. According to direct numerical simulation of a gas flow in such geometry [66], one can expect formation of a pure parabolic profile in the gas channel. The gas supplying tube can be easily linked either to the upper or to the lower rod depending on the gas flow direction which is chosen for the investigation L h d D Figure A2.2. Rods structure 1 - planting cylinders, 2 - cavity for heat-carrying liquid, 3 - tubes, 4 - thermistors, 5 - thermocouples, 6 - capillary for injecting working liquid, 7 - heat-insulating tubes, 8 - pneumoconnectors, 9 - ducts, 10 - side windows, 11 - connectors, 12 - plastic disk. To maintain the required temperature of the end parts of the rods which are in contact with the test liquid, the cavity is equipped with tubes through which heat-carrying liquid is pumped from the heat exchanger. To reduce the heat exchange between the working parts of the rods and the bases of the rods, the former are fixed through heatinsulating and mechanically rigid composite plastic tubes. Low thermal conductivity of the 17

24 plastic (0.2 W/m/K) also reduces heat exchange between the ends of the rods and the blowing gas. The temperature of the surfaces which are in contact with the test liquid (the tips of the rods) is controlled by means of thermistors placed at a distance of 0.3 mm from them. Three thermocouples (with wires diameters of 24 µm) for measuring the temperature of the liquid at different positions of the bridge are installed on the upper rod; their junctions are located at a distance of 0.3 mm from the surface. The injection of liquid to establish the liquid bridge is performed through a capillary placed in the lower rod. A2.2. Fluids management The temperatures of the rod tips are maintained by pumping heat-carrying liquid through the cavities by means of a hydraulic system. The principle of the operation of this system is shown in Fig. A PID Filter Thermal bath damper peristalti c pump damper Figure A2.3. Hydraulic system 1 - heat exchangers, 2 - filters, 3 - dampers, 4 - peristaltic pump, 5 - copper blocks, 6 - Peltier elements, 7 - thermal bath, 8 - heat-insulating casing, 9 - PID-controller The hydraulic system consists of heat exchangers, filters, dampers and a peristaltic pump. The heat exchangers are a set of three copper blocks between which Peltier elements are placed. Heat-carrying liquid passes through the central copper block, and 18

25 liquid coming from the thermal bath is pumped through two external blocks. The whole assembly is placed in a casing in order to establish a temperature isolating barrier. Such a structure is necessary for maintaining a stable and precise temperature. The peristaltic pump is connected between two dampers smoothing pressure pulsations and eliminating vibrations transmission to the liquid bridge. Due to the fact, that liquid is supplied to the working part of the rods through thin capillaries, filters are installed to prevent plugging of the latter. The tubes supplying liquid to the rods are heat-insulated. Temperature is controlled by a PID-controller. The temperature of the surface test liquid is read out from the PID-controller and kept in the computer. The temperature stability of the rods ensured by the system is better than 0.1 K (rms value). PID Controller PC Thermal bath ath Figure A2.4. Pneumatic system 1 - gas bottle, 2 - mass flow controller, 3 - heat-exchanger, 4 - aluminium blocks, 5 - Peltier elements,6 thermoregulated bath, 7 - heat-insulating casing, 8 - PID-controller The gas flow and its temperature are managed by the pneumatic system shown in Fig. A2.4. A heat exchanger produced on the same principle as in the hydraulic system is used in the pneumatic system. It differs in that it is made of aluminum blocks of a much larger area and has a turbulator installed in the centre of the gas channel. Working gas flows from the bottle into the heat exchanger through a pressure regulator and a computerdriven mass flow controller, which allows maintaining the gas injection rate with a great precision. As in the hydraulic system, the temperature is controlled by a PID-controller. The data on the gas temperature are read out from the PID-controller and stored in the computer. 19

26 A2.3. Volume control To establish a liquid bridge and to keep its volume constant, the liquid evaporation is compensated, as shown in Fig. A2.5. PC Figure A2.5. System for compensation of liquid evaporation 1 - tank for discharging liquid, 2 - tap, 3 - syringe pump, 4 - computer, 5 - thermal insulation To establish a liquid bridge, the liquid is injected between two rods by means of a syringe pump. The shape of the liquid bridge is controlled by an optical system. The volume of the liquid is calculated in real-time, a command is sent to the syringe pump to inject or pump out a calculated volume of liquid. The volume stabilization is done according to PID law, which allows maintaining the volume with a precision of 0.1%. To minimize the effect of ambient air temperature change (thermal expansion of liquid) on the liquid bridge volume, the syringe pump and supplying capillaries are insulated. A2.4. Optical system The optical system assembled on the middle plate consists of two identical optical couples (illuminator camera) located at a right angle to each other. Each couple operates at its own and different wavelength (using corresponding LEDs and filters) to avoid crossreflections. We use green 520 nm and red 655 nm LEDs in combination with long pass and short pass filters with a cut-off wavelength of 600 nm. These filters are mounted between the lens and the camera. 20

27 LED CCD Figure A2.6. Optical system 1 - collimated light source, 2 - glass tube, 3 - cylindrical lens, 4 - liquid bridge, 5 - compensating cylindrical lens, 6 - objective lens, 7 - camera, 8 - filter The structure of one of the optical modules is shown in Fig. A2.6. A collimated light source is used for obtaining a sharp image of the liquid bridge. Since the glass tube surrounding the liquid bridge is acting like a diverging cylindrical lens (with an estimated focal length of -65 mm), a pair of compensating converging cylindrical lenses (of 150 mm focal length) is introduced into the optical path (#3 in Fig. A2.6) to make the beam lighting the liquid bridge perfectly parallel. Another (#5 in Fig. A2.6) compensating cylindrical lens, necessary for compensating bridge geometry distortion, is placed between the liquid bridge and the camera. Calibration of the overall optical system has been done by use of a square grid of 0.1 mm step mounted inside the same glass tube in the meridional plane of the liquid bridge. The procedure was basically the same as the one applied by Gaponenko et al. [66] and Matsunaga et al. [38]. The high-speed cameras can record the changes in the shape of the liquid bridge surface with a frequency of 500 frames per second. Using two cameras with perpendicular views enables to record four surface profiles simultaneously, which allows obtaining the tomographic scanning of the liquid bridge surface. 21

28 A2.5. Control unit. The control unit diagram of the set-up is shown in Fig. A2.7. CONTROL UNIT PERISTALTIC PUMP HEAT-EXCHANGER OF UPPER ROD PSU 0-13V USB-RS232 TEMPERATURE SENSOR OF UPPER ROD PID- CONTROLLER PSU 15V 10A CONTROL COMPUTER HEAT-EXCHANGER OF LOWER ROD PID- CONTROLLER PSU 15V 10A TEMPERATURE SENSOR OF LOWER ROD PID- CONTROLLER PSU 15V 16A DATA LOGGER THERMOCOUPLES HEAT-EXCHANGER FOR GAS LED DRIVERS PSU 5V 5A TEMPERATURE SENSOR OF GAS PSU 15V 0.6A DRIVER OF MASS FLOW CONTROLLER MASS FLOW CONTROLLER COLLIMATED LIGHT 1 COLLIMATED LIGHT 2 FILTERS SYRINGE PUMP COMPUTER COMPUTER THERMAL BATH SOCKET CAMERA 1 CAMERA 2 PSU PSU Figure A2.7. Control unit diagram of the set-up The set-up is operated by a computer by means of a control unit. The volume of the liquid bridge, the volume of the injected liquid, the area of the interface, the temperature of 22

29 the rods, the consumption and temperature of gas as well as the temperature of the liquid in three points of the liquid bridge are recorded by the computer. Two other computers are used for capturing images from the high-speed cameras. The control unit is a shielded modular structure designed and produced in our laboratory which contains the main electronic blocks of the set-up. The photo of the control unit is shown in Fig. A2.8. Figure A2.8. Photo of the control unit. In order to minimize the electromagnetic interactions between different blocks of the control unit, they are powered from individual power supplies. Each power supply is switched to the power circuit through a filter which decreases the effect of noise on the power units. In order to minimize electromagnetic interference for connecting the sensors with the control unit and transmitting the data into the control computer, shielded wires are used. Special attention is paid to the interference protection for measuring and regulating accuracy. For calibration of all the sensors, we have used in-house made devices. To achieve precise regulation, a multistage setting of PID-controllers has been performed. 23

30 A3. Validation of the new set-up Professor D. Schwabe from Giessen University offered to our laboratory a liquid bridge set-up, which he had used for many years to investigate instabilities in liquid bridges and particle accumulation phenomena (PAS). This set-up is shown in Fig. A3.1. Figure A3.1. Photo of Schwabe's set-up. To validate the new set-up, we have performed comparable experiments for determination of the onset of instability using both set-ups. The data on critical T obtained on Schwabe s set-up [67] are in a favorable agreement with the series of experimental data obtained with our new set-up. The experiments on both set-ups were conducted for different volumes of the liquid bridge with n-decane as the working liquid. The results of the experiments are summarized in Fig. A3.2. ΔT cr Schwabe's set-up new set-up V/V 0 Figure A3.2. Stability map: values of critical T as a function of relative liquid bridge volume measured on our new set-up and Schwabe's set-up. V 0 is the volume of the straight cylinder. 24

31 As a general trend, the critical T in Schwabe s set-up is lower than in the new one. A detailed analysis has been carried out to understand why there is some discrepancy between data from different set-ups. One possible justification for this might be that in the new set-up the mean temperature of the test liquid is constant, T mean =25K, while in Schwabe s set-up it varies with T. Another reason is attributed to various thermal conditions around liquid bridges because they have different structural features. The new set-up was initially designed for studying the effect of blowing gas at different flow rates on hydrodynamic behaviour of the fluids. The photos of both set-ups are presented in Fig. A3.3. Photo of Schwabe s set-up Photo of the new set-up Sketch of Schwabe s set-up Sketch of the new set-up Figure A3.3. Appearance and schematic view of Schwabe's set-up and the new set-up. 25

32 In spite of the fact, that the principal geometry and dimensions of the liquid bridge are identical (the rods radius is 3 mm, the aspect ratio is Γ=H/R 0 =1), the materials used to produce the rods and their fixations are different. In Schwabe s set-up the lower rod is made of brass and fixed in a massive heat exchanger, see the upper photo in Fig. A3.3. The upper rod, also fixed in a massive heat exchanger, is made of sapphire to follow particles displacement. In our set-up only the tips of the rods are heated/cooled while in Schwabe s set-up the whole rods as well as the plates above and below are heated/cooled. The lateral sides of the rods in our set-up are produced from thermal insulating material. In this case, the effect of the rod temperature on ambient gas is weak and the temperature of the plates installed above the upper rod and below the lower rod (see Fig. A3.3) has no influence on the gas temperature. The chambers around the liquid bridge have a different geometry and are produced from different material. Note that for conducting experiments without gas blowing in the set-up the external glass tube has been substituted with a thermal insulating pipe of a larger diameter (D=70mm) with two transparent windows for a light source and a camera, see the photo in Fig. A3.3. This was done because n-decane is an evaporating liquid and when attempting to use the glass tube of a small diameter without a gas flow, n- Decane was condensed on its inner surface. As a result, it was impossible to get high quality images for measuring the volume of the liquid bridge. 26

33 Figure A3.4. Computed temperature (a) and flow (b) fields in the two set-ups at T=7 C in two phases: n-decane and air. The left part of the plots presents distribution in Schwabe s set-up and the right part presents fields in the new set-up. The geometry of calculations is presented in correct scale: the radius of rods is R 0 =3mm, the external radius of the new and Schwabe s set-ups are Rext=15mm. The total heights of chambers filled with air are 21mm and 9mm for the new and Schwabe s set-ups, respectively. To shed light on the role of the set-up geometry on the thermal and flow fields, numerical simulations were performed by Dr. Yu. Gaponenko (MRC-NLF group) using commercial software Fluent, see for details Refs. [68], [69]. The full Navier-Stokes equations were calculated in two phases for both geometries of liquid bridges using physical properties of n-decane listed in Table A3.1. Figure A3.4 (a) shows the isotherms of a temperature field and Fig. A3.4 (b) the isolines of the stream function in both set-ups. The distributions of fields are presented in the most convenient way for comparing: two images are turned to each other by liquid zones. It follows from Fig. A3.4 (a) that the nearinterface region is warmer and less uniform in the new set-up. The thermal fields in gas phase are even more different: in Schwabe s set-up the temperature varies linearly with the height of the liquid bridge, while in the new set-up the temperature is almost constant slightly away from the free surface and is close to the mean temperature. The flow field, shown as isolines of the stream function in Fig. A3.4 (b), is similar inside the liquids in both set-ups but differs in the gas phase. Significant difference in the distributions of the fields between the two set-ups may lead to discrepancy in critical T. 27

34 Indeed, one of the driving forces, i.e., the thermocapillary force which is proportional to the temperature gradient ~dt/dz, acts on the liquid/gas interface. The temperature profiles at the interface are shown in Fig. A3.5 (a) in both cases. The temperature in the new set-up is fraction higher and, more importantly, is almost constant at the major part of the interface. This provides a smaller effective temperature gradient than in Schwabe s set-up. Consequently, the interface velocity (see Fig. A3.5 (b)) is larger in Schwabe s set-up and it may lead to the destabilization of the flow as observed in Fig. A3.2. (a) (b) Figure A3.5. Computed temperature (a) and axial velocity (b) along the interface filled with n-decane at T=7 C Geometries of the liquid bridges correspond to set-ups design. In the presence of evaporation, one of the key parameters for the flow instability is the heat flux through the liquid/gas interface. The local heat flux q(z) through the free surface area is determined as: q(z) = k lll r=r 0 (A3.1) where k is the thermal conductivity of the liquid, R 0 is the radius of the liquid bridge rods. The heat flux q is positive when the liquid locally loses heat and negative when heat is gained. The distribution of the local heat flux q along the interface and the velocity at the interface are shown in Fig. A3.6 for both set-ups at the same T as a result of numerical simulations of the full Navier-Stokes equations in two phases using constant evaporation rate determined experimentally. Determination of the evaporation rate is described in Section 5.2 for the new set-up and in the Ref. [67] for Schwabe s set-up. Figure A3.6 28

35 shows that the heat flux is positive and the interface loses heat over the entire length. The heat flux in the central part of the interface grows almost linearly towards the cold side in both set-ups, but in the new set-up the slope is smaller. Furthermore, the heat lost in the new set-up is smaller than in Schwabe s set-up. Consequently, the interface in Schwabe s set-up is cooling faster and non-uniformly and it may contribute to the increase of the interface velocity and, consequently, to the destabilization of convective flow. Figure A3.6. Computed distribution of the local heat flux q along the interface n-decane/air (see Eq.A3.1 for definition). The geometry of calculations corresponds to the geometry of the set-ups. Table A3.1. Physical properties of n-decane [70], [71] Density, ρ kg/m3 730 Specific heat capacity, cp J/(kg*K) 2190 Thermal conductivity, λ w/(mk) Dynamic viscosity, μ Pa*s 9.05E-04 Kinematic viscosity, ν m2/s 1.24E-06 Surface tension, σ N/m 2.39E-02 Variation of surface tension with temperature, dσ/dt N/(m*K) 1.18E-04 Thermal expansion coefficient, β /K 1.06 Latent heat of evaporation Ws/kg 3.503E+05 Thermal diffusivity, κ m2/s E-08 29

36 A4. Shape detection A4.1. Preliminary shape detection with pixel-size accuracy The preliminary shape detection is done by a very robust algorithm of edge detection, the so-called Canny method. The robustness of this algorithm is conditioned by the fact that it utilizes a double threshold and finds both strong and weak edges and connects them, if necessary. Therefore, the processing can hardly be affected by noise. Previously, the somewhat similar technique was used in joint research by MRC, ULB and University Extremadura, Spain [32], [33], [34], [35]. This preliminary detection is done by the edge function built in the Image Processing Toolbox of Matlab. Since the function finds any edge in the image, the result has necessarily to be sorted for useful and useless edges (see Fig. A4.1). (a) (b) Figure A4.1. Raw image (a) and the result of preliminary edge detection (b). With the aim of tracking only the necessary edges, an additional procedure is employed which selects only the edges formed by the sidewalls of the liquid bridge and of the supporting rods. The procedure consists in selecting edge pixels somewhere at the top of the image (the region that is normally free of parasitic edges) and then following this edge row by row but only within some predetermined margins around the location of the previous pixel. In such a way, all parasitic edges are excluded from the final result. The idea is sketched in Fig. A

37 tracking direction R 0 = 3.0 mm d = 3.0 mm Figure A4.2. Selection of true edges from the total edge detection result. After such a selection, only the coordinates of the sidewalls of the LB and supporting rods are left, as shown in Fig. A4.3. Figure A4.3. Result of sidewall edge pixels detection. After detecting the sidewalls, the necessary step is to cut off the edges of the supporting rods, since only the region of liquid is of interest for the present study. To do so, the bottom of the groove cut around the end of each supporting rod is chosen as a reference point. The exact vertical distance between the bottom of the groove 31

38 and the end of the rod is carefully measured using a snapshot of an empty liquid bridge (supporting rods without any liquid). Then, the measured distance in pixels is calculated on an individual snapshot to find the exact location of the liquid interface. Figure A4.4 illustrates this procedure. Figure A4.4. Selection of the region of liquid. Although the result of liquid interface detection looks rather smooth in this lowresolution figure, plotting it with magnification immediately visualizes all pixel steps, as shown in Fig. A4.5. Figure A4.5. Left side of the liquid bridge (magnified). 32

39 So, this result of interface detection with pixel accuracy is acceptable for the raw detection of the surface position and some estimation of the liquid bridge volume, but is definitely not acceptable for precise detection of small static and dynamic deformations of the free surface. It necessitates improvement of the above procedure by adding a sub-pixel detection step. A4.2. Determination of the threshold for sub-pixel shape detection All techniques for edge detection typically utilize a threshold approach. To find the necessary threshold, a method based on the analysis of an image histogram was used. It is possible to use the histogram of a complete image, but very often an image is wasted by smooth black-to-white transitions and is essentially diffused. To avoid this problem, the histogram was calculated for an artificially created auxiliary image. This image consists of pixels of two stripes cut out of the original image. The shape of these stripes follows the preliminary determined edges with margins of ±30 pixels to the left and to the right of these edges, as shown in Fig. A4.6. Figure A4.6. Selection of the regions of the original image for histogram calculation (the selected regions are in between the dashed lines). 33

40 A typical horizontal intensity profile within this stripe is shown in Fig. A4.7. Figure A4.7. Intensity profile over the region selected for histogram calculation. It is evident, that the auxiliary image created in such a way will show a histogram with two sharp peaks corresponding to the dark and bright regions. With such an improved histogram, it becomes very easy to detect the above-mentioned intensity peaks and subsequently to determine the threshold. A typical histogram is plotted in Fig. A4.8. Figure A4.8. Histogram of the selected region of the image and the way of finding the threshold. 34

41 A4.3. Final shape detection with sub-pixel accuracy To find the exact shape of the liquid bridge with sub-pixel accuracy, one needs to combine and use all the data obtained in the previous steps, namely, the threshold value, the coordinates of edge pixels and the original image itself. To correct the edge position, a part of horizontal intensity profile with margins of ±10 pixels around the edge pixel detected by Canny method is taken from the original image row by row. Then, the corrected position of the edge is found for this profile using the predetermined threshold value by a simple linear interpolation (see Fig. A4.9). The socalled global threshold is used for sub-pixel detection; it means that the threshold value is the same for all the edges in each particular image. Figure A4.9. Way of sub-pixel detection. The result of sub-pixel detection is much smoother (see Fig. A4.10), which allows evaluating the free surface deformations with a sensitivity of a micron. Figure A4.10. Comparison of the result of sub-pixel and pixel-level edge detection. 35

42 A5. Measurements of volume, evaporation and dynamic surface deformation A5.1. Measurements of volume Knowing the exact profile R(z) of the surface, the volume of liquid can be precisely determined as: H V = π R 2 (z)dd (A5.1) and the surface of the interface is: H 0 S = 2π R (z) 1 + [R (z)] 2 dd (A5.2) 0 Here, R(z) is the dependence of LB radius upon vertical coordinate and R (z) is its derivative. A5.2. Measurements of evaporation Evaporation of the liquid may significantly change the interface temperature and, consequently, the stability of liquid bridge. A forced gas flow parallel to the interface may also considerably affect the evaporation. To shed light on these effects, the experiments in liquids with different volatile properties were conducted at different gas velocities. The developed technique allows determining an evaporation rate and its dynamics measuring the change of the interface shape. The first set of the experiments were conducted at the isothermal conditions. 36

43 a b c (a) (b) c) Figure A5.1. Decrease of LB volume with time due to evaporation for ethanol and a few shape snapshots taken at times (a) 0 min, (b) 3 min and (c) 6 min. Experimental conditions: the radius is R 0 =3mm, the height is d=3mm and T amb =22 C. The images were acquired at a frequency of 1 frame per second during 14 minutes. This final time is close to the critical moment of the thinning and breaking of the bridge for the most volatile liquid, i.e., ethanol. The curves of volume variations of an ethanol liquid bridge with time are shown in Fig. A5.1 for different gas velocities. Plots (a), (b), (c) illustrate the evolution of the liquid bridge shape with time. Figure A5.2. LB volume change with time for different liquids; experiments are performed when evaporation is enhanced by a gas stream parallel to the interface with velocity U g =47 cm/s. 37

44 Regardless of the presence of a forced gas stream, the evaporation rate depends on the physical properties of liquids and gas. The evolution of the relative volume of the liquid bridge with time at gas flow rate Q=24 ml/s, U g =47 cm/s is shown in Fig. A5.2 for three different liquids: silicone oil 5 cst, n-decane and ethanol. Figure A5.2 shows that the liquid bridge filled with silicone oil 5 cst displays a slower evaporation and the volume is approximately constant during the characteristic time of flow stabilization (10-15 min). The situation is a little worse in the case of n-decane. During the experiment the relative volume decreased from 1 to Note that the decrease of LB volume with time is almost linear for n-decane. For a highly volatile liquid such as ethanol, the curve exhibits a nonlinear behaviour and, consequently, the volume loss rate is not constant during the experiment. One of the explanations of the non-linear behaviour may be related to reduction of the surface area (S) of the liquid bridge with a decrease in volume. Furthermore, our experiments have shown that mass evaporation rate per units of the surface area defined as EE = ρ S dd dd (A5.3) provides a value independent of time and can be used for comparison of different experiments. Here ρ is the density of liquid. Figure A5.3. Mass evaporation rate defined by Eq. A5.3 as a function of gas velocity for three different liquids. A summary of the experiments is presented in Fig. A5.3 where mass evaporation rate ER is shown as a function of gas velocity for different liquids. The evaporation rate ER of silicone oil changes only marginally even for large gas velocities. In the case of n- Decane, ER grows with increase of gas velocity although the slope is relatively small in 38

45 comparison with ethanol. The evaporation rate ER of ethanol increases almost linearly at small gas velocities and then achieves saturation at V g ~60cm/s. Figure A5.4. Comparison of the evaporation rates of n-decane with and without gas blow at room temperature. Our current experiments on the stability of a liquid bridge use n-decane as a test liquid and, correspondingly, we have studied this liquid in more detail. The comparison of the evaporation rate of n-decane liquid bridge with and without gas blow shown in Fig. A5.4 demonstrates an essential increase of liquid loss when a gas flow is applied. These experimental results showed the necessity of dynamic adjustment of the liquid volume during the experiment with n-decane. Thus, we have to impose experimental requirement to compensate the volume of liquid even for moderately volatile liquid when gas is blown around. Figure A5.5. Measured mass evaporation rate per units of the surface area (see Eq.A5.3) in n-decane as a function of the surface temperature. 39

46 Various types of non-isothermal experiments to be described below were performed during the study. Consequently, to analyze the dynamics of evaporation, in addition to the isothermal ( T=0) experiments described above we have conducted nonisothermal experiments ( T 0) with n-decane. Non-isothermal experiments were performed at different mean temperatures, T mean =22 C and T mean =25 C and various T. The experimental points are presented in Fig. A5.5 by different symbols. We found that the most efficient way to present the evaporation rate for isothermal and non-isothermal experiments together is a function of the surface temperature. The importance of the surface temperature during evaporation has been recently discussed for the geometry of droplets [72], [73]. The non-linear simulations of Navier-Stokes equation in two phases without and with a small evaporation rate demonstrate that the interface temperature can be approximated as T suuu T cccc T (A5.4) To illustrate this, we presented the temperature profile on the interface in a dimensionless form Θ=(T-T cold )/ T in Fig. A5.6, which supports the suggestion that the temperature at the major part of the interface can be described by relation (A5.4). Figure A5.6. Results of numerical simulations; variation of the dimensionless temperature Θ=(T-T cold )/ T along the height of a liquid bridge. Furthermore, all the experimental results which presented evaporation as a function of the surface temperature follow a similar trend, which is shown by the dashed line in Fig. A5.5. This trend can be described by polynomial of 5th order: R = T 5 s T 4 s T 3 s T 2 s T s The important message which follows from Fig. A5.5 is that the evaporation rate increases with the increase of the mean temperature (compare the location of green and blue 40

47 symbols). We would like to draw attention that Fig. A5.5 comprises data from the experiments with and without Marangoni/buoyant flow. This plot presents experimental evidence of the general trend unifying experiments with and without convective flows. It gives hint that convective flow (or moving interface) does not provide dominant contribution to the evaporation rate. A5.3. Detection of dynamic deformation In addition to bulk flows, a gas blow causes mechanical disturbances and dynamic deformations of the free surface. These deformations do not have azimuthal symmetry, and a thorough analysis requires the analysis of the deformations on the entire free surface. Tests of the optical system developed for 3D mapping of liquid bridge surface deformation caused by the mechanical effect of a gas flow were conducted with the following parameters: relative bridge volume 1.0, gas velocity 260 cm/s with an upward flow direction, i.e., against gravity. The working fluids were 5 cst silicone oil and nitrogen. These experiments were conducted at room temperature. Two perpendicular views of the liquid bridge, as shown in Fig. A5.7, were recorded simultaneously by two cameras. The acquisition frequencies of both cameras are identical and equal to 100 fps and exposure time 5 ms, thus, giving a time step of 10 μs between the consecutive images. Each camera provides a set of 818 images. Figure A5.7. Schematics of observation. After processing each particular image, two profiles (left and right) of a liquid bridge shape of approximately 300 pixels in height were extracted with sub-pixel accuracy. 41

48 The profiles were filtered to remove optical noise of high spatial frequencies and then rescaled to have the same 101 points over the liquid bridge height for all of them. Each profile from two sides and two views was averaged over the 818 shapes and the dynamic deformation was calculated as a deviation of the profile at a particular time instant from its average state. The four profiles that came from the equally numerated images of the front and side cameras are shown in Fig. A5.8. They point out that for such a large gas velocity, U g =260m/s at the entrance, the symmetry between two profiles at the same view is broken and all profiles are different. Creative fusion of four profiles at the same time instant provides a complete reconstruction of dynamic surface deformation at any cross-section. Figure A5.8. Snapshot of dynamic surface deformation (deviation from the mean value) at different azimuthal positions of the LB when U g =260cm/s. Figure A5.9 demonstrates a pattern of the dynamic surface deformation in a horizontal cross-section of the liquid bridge at the mid-height. The deformation pattern is radially and azimuthally non-symmetric, it is oblate from the lateral side at φ=0 and φ=180 but oblateness is larger at the side φ=180. Figure A5.9. Azimuthal map of surface deformation at the mid-height of the LB. The solid dashed curve corresponds to zero deformation, two other concentric curves to -10μm and +10μm. 42

49 To provide a smooth view of dynamic deformations over the full surface, an interpolation was done at 101 height levels of the LB. The initial four points with a step of π/2 radians were transformed into 40 points with a step of π/20 radians by spline interpolation. Then, the map of surface deformation was built for each time instant as shown in Fig. A5.10 (a). (a) (b) (c) Figure A5.10. (d) Reconstruction of the map of dynamic surface deformation over the entire unrolled free surface of the liquid bridge (right plots) by stereoscopic high-speed observation from four equidistant profiles in azimuthal direction (left plots). (a) t=0ms, (b) t=10ms, (c) t=20ms, (d) t=30ms. 43

50 A series of consecutive snapshots of the reconstructed map of deformations made with a time step of 10-3 s is shown in Fig. A5.10. The acquisition frequency in the given experiment was not sufficient for producing a continuous time sweep of the process but allowed obtaining a series of snapshots of the bridge gas-liquid interface. An estimation of the natural frequency of surface waves was done with the assumption that the liquid bridge length (height) can accommodate from half to one surface wave. According to this estimation, the used acquisition frequency can allow getting from 1 to 4 images per period of the wave, which is surely not enough. Due to this fact, it is too early to draw definite conclusions about the types of vibrational modes caused by the gas flow. To increase the acquisition frequency up to 500 snapshots per second (the maximum capacity of the used cameras), more powerful light sources have to be used. In addition, experiments with a smaller gas velocity may provide a smaller frequency of surface waves. These tests have shown that the developed optical technique allows reliable identification of the liquid bridge surface deformation with a magnitude of less than 1 micron (1/10 of pixel size). The maximum deformations of the bridge in the presented experiment are achieving ± 10 microns. In the previous experiments, sub-micron deformations of the free surface by a system with high magnification and a small field of view were measured by Ferrera et al. [34]. That system needed multiple scanning in the vertical direction and matching of images to cover one complete profile of the liquid shape. The present set-up allows instant tracking of four profiles over the full LB height with a minor reduction of accuracy. To check the consistency of interface tracking, the volume of the liquid bridge was integrated over four reconstructed profiles at every time instant. The calculated volume demonstrates some minor frame-to-frame oscillations of 0.04% magnitude, which roughly corresponds to uncertainty of shape detection. Thus, it has been demonstrated that the new set-up can track and visualize tiny oscillations of the free surface of a liquid bridge due to mechanical stresses caused by a coaxial gas flow and can potentially do the same for deformations due to the instability of Marangoni convection. 44

51 A6. Stability of convection driven by thermocapillary and buoyant forces A6.1. Variety of experimental procedures In ground experiments with non-uniformly heated interface the convective flow is driven by the combined effects of buoyancy and thermocapillary forces. Experimental evidence that the critical conditions for the onset of oscillatory flows in ground conditions strongly depend on the interface shape was suggested by Hu et al. [55] who carried out experiments with 10 cst silicone oil (Pr 108). The stability diagram ( T cr vs Volume) consists of two branches which formally can be assigned to small and large volumes or slender and fat liquid bridges. For a liquid volume roughly corresponding to a cylindrical interface the critical T has a peaked maximum indicating a very stable flow. In experiments this "peak" is transformed into a "gap", the width of which depends on how large values of T can be achieved in the experiments. For silicone oil with viscosity 5cSt and higher, the peak occurs above T~60K [59], [74] and due to experimental constraints the "peak" is transformed into an open "gap". The experiments with silicone oils of 1-2 cst [74] showed that the peak maximum between branches drops down to T 10K or less with decreasing of the viscosity. However, these silicone oils are volatile and the previous experiments did not compensate the loss of a liquid volume. The evaporation of the liquid from the interface, even weak, may modify the instability threshold. The purpose of this section is to study the stability of weakly evaporating n-decane under strict control of liquid volume change due to evaporation. The physical properties of n-decane are listed in Table A3.1. To do so, the images of the liquid bridge were captured every second and the volume of the liquid bridge was also calculated every second. These data were used by the program of PID-control for compensation of the volume of evaporating liquid (stabilization of the liquid bridge volume). This program is commanding the syringe pump regulating the volume of liquid. If not stated otherwise the mean temperature of the liquid was kept at T mean =25 C. Each experiment consisted of the following steps: The rods are carefully cleaned with fresh working liquid and, if necessary, coated with anti-wetting agent. A liquid bridge of the desired volume is formed between the supporting rods by injecting liquid from the push syringe. 45

52 A starting temperature difference T 4-5K, which is below the critical one, is applied between the supporting rods as described in Section 2.2, and waiting time is allowed for the system until the rods temperature reaches prescribed values. The temperature evolution in time was recorded by three thermocouples installed on the upper rod. The photo of the thermocouples location and the precise geometry are shown in Fig. A6.1. The temperature of the working surfaces of the rods (tips) and the temperature of the air around the liquid bridge were recorded by thermistors. During the experiment images of the liquid bridge were acquired at constant intervals of time. The liquid bridge volume, the surface area and the amount of the injected fluid are calculated in real time from the images and recorded. The images themselves were not stored. The design of the new set-up allows not only obtaining data at fixed parameters such as T, volume, etc., but also receiving data while gradually changing one of the parameters (scanning). Three experimental procedures were used over the range of experimental runs: (1) scanning by T at constant T mean ; (2) scanning by T mean at constant T; (3) scanning by volume. It is worth noting another novelty of the set-up the mean temperature can be kept constant. Figure A6.1. Photo of the tip of the upper rod with thermocouples and the geometry of the arrangement of thermocouples. 46

53 Until now, all the experimental apparatus were not able to keep the mean temperature constant. Usually, one of the rods (more often the cold rod) was kept at constant temperature and the temperature of the other one was changed, for example [19], [75]. In these instruments the mean temperature of the system was continuously changing. The discussion concerning the experimental observations will be considered in two approaches: via records on thermocouples giving an indication of how temperature and frequency develop over time; via stability maps showing the critical values of the flow parameters that define different regimes of the advancement of instability. A Scanning by T at constant volume and T mean The experiments were conducted either increasing or decreasing T. The temperature of the upper rod is gradually increased, and the temperature of the lower rod is equally reduced (to increase T or way up ). Or, alternatively, the temperature of the upper rod is reduced and the temperature of the lower rod is increased (to decrease T or way down ). The typical example of the recorded data is shown in Fig. A6.2 for the experiment with relative volume V r =V/V 0 =0.75, here V 0 =πr 2 d is the volume of straight cylinder and d is the length of liquid zone. We have started discussion from this small volume as it was shown previously [45] that the stability of liquid bridges of small volumes is less affected by ambient temperature. Figure A6.2. The raw data recorded in the course of the experiment while scanning by T when V r =0.75 (way up). The ramping rate C/min, (15/01/2013). 47

54 Figure A6.2 illustrates that the volume of the liquid bridge is kept constant with a good accuracy although n-decane is evaporating. Another message from the upper plot is that the variations of ambient temperature (green curve) are small. The ramping rate of the temperature on rods is about dt/dt~0.044 C/min or dt/dt~ C/s. A series of terrestrial experiments with various ramping rates of the temperature difference were conducted by Kawamura &Ueno for liquid bridges of different Pr and different diameters [76]. They reported that the guideline for the quasi-steady limit of the ramping rate is dt dt empirical Tcr κ < 0.1, = R R R * * 0 d (A6.1) Here k is the thermal diffusivity, R 0 and d are the radius and height of a liquid bridge. For our set-up, even for the smallest Tcr 6.5K, this value is (dt/dt) emp = C/s. Thus, our ramping rate dt/dt~ C/s is at least 15 times smaller than the recommended limitation. It is important to note that the temperature shown by the thermocouples installed very close to the hot rod rapidly diverges from the temperature of the hot rod as soon as an oscillatory regime sets in (see Fig. A6. 2). Figure A6.3. Readings from the thermocouples and results of Fourier spectrum, Vr=0.75, T=8.2, (15/01/2013). 48

55 Temperature signals and their Fourier spectra for the same experiment as in Fig. A6.2 are presented in Fig. A6.3 according to the readings from the thermocouples. To perform FFT, we have used 2 n =1024 points which were selected not far from the threshold of instability. The Fourier amplitude A f is plotted in a logarithmic scale amplitude and the numbers in the plot just indicate the exponent, i.e., -8 corresponds to The spectrum is rather clean, only fundamental frequency and its first harmonics are present. Performing Fast Fourier Transform for each of the successive 1024 points (~50s) over the duration of the experiment (~3h) the variation of the fundamental frequency with time has been obtained and recalculated as a function of imposed temperature difference. The evolution of the fundamental frequency with T is shown on the upper plot in Fig. A6.4 for the same relative volume, V r =0.75. As the frequency on all thermocouples is the same, the results from one of them are shown. The frequency weakly changes over the considered range of parameters and displays tendency to increase when T >8K. The important point to note is that the fundamental frequency is diminishing near the threshold of instability. Furthermore, with the increase of the liquid bridge volume V r this decrease of the near-threshold frequency becomes more and more pronounced. Figure A6.4. (Upper plot) Summary of FFTs over the experiment: fundamental frequency as a function of T. (Lower plot) The amplitude of temperature oscillations versus T determined as root mean square over 2048 s. V r =0.75, (15/01/2013). 49

56 For example, our experiments showed that for V r =1, at the distance from critical point ( T T = T cr ) 0.1 cr ε (A6.2) the fundamental frequency drops down by 15%. To the best of our knowledge, in all previous experiments with silicone oils, the continuous increase of the fundamental frequency was reported. However, the decrease of the fundamental frequency near threshold was observed in the recent experiments with n-decane [66]. We can assume that this is due to a weak volatility of n-decane. In addition to the frequency, we have calculated the amplitude of the temperature oscillations on the thermocouples. The temperature amplitude was calculated as root mean square A T = 1 n n i= 1 ( T i T ) 2, T = 1 n n i= 1 T i (A6.3) The mean value was calculated over 102s (i.e., number of points n=2048). Temperature amplitudes for all three thermocouples determined in this way are shown in the bottom plot in Fig. A6.4 as a function of T. The amplitudes from two thermocouples coincide over the entire interval of T (green and blue curves), while the amplitude shown by the red curve slightly differs. We attribute this difference to a little different radial location of one of the thermocouples, see Fig. A6.1. All the amplitudes are decreasing in the region near T 11 C but the decay of oscillations of the periodic flow does not happen. 50

57 Figure A6.5. Fouirier map of the experiment with V r =0.75 while T increases with ramping rate C/min (15/01/2013). The small plot on the right is given to help understanding of a stability map; it is Fouirier spectrum for a single point T=8.2 C (shown by the white dotted line in the left plot). The arrow shows development of the experiemnt in time (direction of the scanning). Having in hands spectra for each T provides an opportunity to build a complete Fouirier map for the selected volume of the liquid bridge, as shown in Fig. A6.5, which can be also seen as the stability map. The small plot on the right side is shown to simplify understanding of his map, it is the Fouirier spectrum at T=8.2 C (see the white vertical line). The colours show the Fourier amplitude of the fundamental frequency and its harmonics. Figure A6.5 exhibits a periodical regime over the considered range of T; even if the spectrum becomes noisy at T>12K, the fundamental frequency has the largest amplitude. This map also allows getting a general impression about the temporal evolution of flow states. 51

58 Figure A6.6. Fouirier map of the experiment with Vr=0.75 while T decreases with ramping rate K per min. The arrow shows the direction of the scanning (way up) (15/01/2013). In addition to the regular repetition of the experiment for statistical reasons on the way up, we have conducted experiment with V r =0.75 while T was decreasing with the same ramping rate C/min (way down). The overall view of the stability map in Fig. A6.6 is similar to that in Fig. A6.5, although a more precise look reveals the difference. When decreasing temperature difference (way down), the instability disappears down at T cr = 8.12K which is larger than T compared with the case on the way up, T up cr =7.25K. Another observation in Fig. A6.6 is that at the beginning of the experiment the oscillations are aperiodic within 13.6< T<14K. Note, that our experience showed that very close to the threshold of instability, no matter whether we decrease or increase temperature, the flow exhibits somewhat aperiodic character due to growing fluctuations. Logically, at larger T these fluctuations are larger and we observe clearly multiple frequencies in Fig. A6.6 close to T<14K while in Fig. A6.5 still a single frequency dominates at this T. The experiments with the increase and decrease of T have been carried out by Frank & Schwabe in NaNO 3 liquid bridge [75] and they reported about partly different flow states on the way up and down. A Scanning by volume at constant T and T mean In this series of runs the liquid bridge of the largest possible volume is established and is allowed to evaporate while prescribed T is imposed at an initial time moment and 52

59 kept constant over the experiment. The loss of liquid due to evaporation is not compensated and the volume of the liquid bridge decreases continuously. The variation of volume with time is shown in the bottom plot in Fig. A6.7. The optical system records the shape change and the software registers the volume variation and the evaporation rate, respectively. Figure A6.7. Data recorded in the course of the experiment while scanning by volume at T=7 C and T mean =25 C (09/01/2013). The temperature records on the thermocouples provide information how the volume change affects the stability of the liquid bridge. Results are processed in the same way as described in Section A The Fouirier maps for different T are shown in Fig. A6.8. The upper plot ( T=7K) in Fig. A6.8 demonstrates that liquid bridges with the smallest volume lose stability first. The experiment began with the volume V/V at T=7K but instability appeared only when the volume was decreased to V/V 0 0.9; however, the periodic regime persisted only within a very limited volume range, as the stability window appeared starting from V/V Beyond this range, when V/V 0 <0.834, the periodic regime exists for all volumes accessible in the experiment. Experiments with a slightly larger temperature difference, in particular for T>7.5K, did not display interruption of the oscillatory regime. For example, the second plot in Fig. A6.8 at T=8K shows smooth evolution of the fundamental frequency and its harmonics over all the volumes. However, it is possible to notice a spectral line at one-half of the fundamental frequency in the region (V/V ) where previously decay of oscillations in the upper plot was observed. 53

60 Figure A6.8. Fouirier maps of the experiments at constant T=7K, T=8 and T=9K, when volume decreases due to evaporation (07/01/2013 and 09/01/2013). 54

61 Usually, the presence of subharmonic at half the driving frequency indicates a period doubling bifurcation, which is one of the main precursors of the onset of chaotic motion. Indeed, at larger T ( T=9K), the third plot in Fig. A6.8 shows that the spectrum becomes more complex with multiple peaks, in particular, around V/V If the ratio of two different independent frequencies in the spectrum is irrational, the temporal evolution is called quasi-periodic. Figure A6.8 at T=9K exhibits two incommensurate frequencies but each of them has a satellite with one-half of frequency. Note, that previous experimental [19], [75], [76], [41] and numerical [10], [77] investigations of convective flows in liquid bridges have reported about non-linear states far above the critical point, 1< ε <3, here these phenomena are observed when the system is slightly above the critical point ε 0.2. There are two additional factors which were not controlled/analysed in the previous works: liquid bridge volume and evaporation. Obviously, the system exhibits rich dynamics of the flow field and its analysis is non-trivial due to the sensitivity to small changes of ambient conditions and volume. Possible temporal convective flow states are shown in Fig. A6.9 for T=9K when LB volume decreases from a large (laminar flow state) to a small one. The flow in LB of a large volume exhibits periodic oscillations (case 1 in Fig. A6.9) although the main Fourier peak is not sharp and hardly contains distinguishable additional frequencies on each side. One may speak either about a period flow or about a quasi-periodic flow with three incommensurate frequencies. The small frequency may be also a result of some slowly varying ambient conditions or simply this is because of the always present noise in the reality. Then the additional frequency will be the difference between the fundamental and a very small frequency. T=9 C TC signals Fourier spectra Phase plane 1 V r =1.14, t=18min 2 V r =1.06, t=49min 55

62 3 V r =0.983 t=80min 4 V r = t=97min 5 V r =0.899 t=115min 6 V r = t=140min 7 V r = t=160min 8 V r = t=175min 9 V r = t=190min 10 V r= t=205min Figure A6.9. Flow regimes at different volumes for the same T=9K. The instability sets in at T 7K. Column 1 identifies the relative volume and the case number shown in Fig. A6.8 (bottom plot); column 2 shows time series of the temperature on thermocouple 1; column 3 - its Fourier spectrum, and column 4 - the phase plane. 56

63 Case 2 corresponds to the period-doubling bifurcation and this flow state persists for the large range of volumes 1.0<V/V 0 <1.08. The following state, case 3, can be classified as a periodic state but a careful analysis reveals that Fourier peaks are split. These flow states were not reported in literature on the convective flows. Further diminishing of the volumes (starting from case 4) drives the system to more complex dynamics when a quasi-periodic state co-exists with period doubling and then to an aperiodic state, cases 7-8. Finally, arriving at the small volumes V/V 0 <0.73 the system displays the same states as for the largest volumes period doubling and periodic flow states (cases 9-10). It is important to reiterate that in the present case the imposed temperature gradient is constant and the gradients which drive this instability are a natural consequence of the evaporating process. Thus, the conclusion is that the convective flow in a liquid bridge provides a unique model to study non-linear dynamics. A Scanning by mean temperature at constant T and constant volume The third type of the experiment is a scanning by mean temperature. We stabilize the volume of the liquid bridge, impose T, and gradually increase or decrease the temperature of the upper and lower rods simultaneously and equally, thus changing the mean temperature of the liquid. Results processing is done in the same way as described in Section The temperature recordings from thermocouples and thermistors are shown in Fig. A6.10 when T=6 C and V/V 0 =0.95 and mean temperature increases from 23 C until 30 C. The instability sets in at T cr mean =28 C. The new set-up also allows investigating the flow states when the mean temperature decreases with time. The results of the similar experiment with decreasing of the mean temperature have shown that T cr mean =26.4 C. Figure A6.10. Readings from thermocouples (TC i ) and thermistors during the experiment with increasing T mean while T=6K and V/V0=0.95. The instability sets in at T mean =28.5K (02/10/2012). 57

64 It should be noted that the temperature on the thermocouples (TC i ) effectively evolves parallel to the temperature of the hot rod even when instability sets in. We should remind, that in Fig. A6.2 for the same T the temperatures of the hot rod and recorded by the thermocouples strongly deviated. Because T=6 C is below critical for all experiments at T mean =25 C, we may suggest that in the present case instability is primarily caused by the evaporation, and provides a different pattern of the oscillatory flow. The analysis of viscosity variation with temperature has shown that the change of T mean by 1 C provides variation of dynamic viscosity (μ) by 1% [78] and much less for kinematic viscosity (ν= μ/ρ). Essentially parallel behaviour of curves for T hot and TC i in Fig. A6.10, prior to the onset of the instability, indicates that variation of viscosity in this range of T mean is not a dominating mechanism of instability. A6.2. Analysis and comparison of the instability regimes As it was mentioned above, for the case of high Prandtl numbers in terrestrial conditions the stability diagram (ΔT cr vs. Volume) consists of two different oscillatory instability branches. For the case of 5 and 10 cst silicone oil and aspect ratio d/r 0 ~1 two branches correspond to different azimuthal wave numbers [53]. Our experiments with n- Decane also showed that the critical temperature difference is very sensitive to the liquid bridge volume. Experiments connected with Marangoni convection are known to have poor repeatability as they are strongly dependent on surface tension, which, in turn, depends on the purity of the liquid used. During the experiment, n-decane can adsorb impurities from ambient gas, so it was decided to carry out two types of experiments long- and short-duration. In addition, the temperature of ambient gas is an important parameter. The stability diagram in Fig. A6.11 was obtained on the basis of measurements in two successive days when scanning by ΔT was applied. Similar symbols correspond to experiments on the same day. An example of the short-duration experiment is shown as a Fourier map on the right side of Fig. A6.11, while the experimental procedure for the longduration runs can be seen in Figs. A6.5-A

65 An example of the short experiment Figure A6.11. Experimental stability diagram ΔT cr vs. relative volume at T mean =25 C when scanning was made by T in course of a series of the short-duration experiments. The example of the short experiment is shown on the right side (22/01/ /01/2013). Figure A6.12. Experimental stability diagram: critical frequency vs. relative volume at T mean =25 C when scanning was made by T (22/01/ /01/2013). Critical frequency, corresponding to these experiments, displays the similar trend of two branches and is shown in Fig. A6.12. However, these stability diagrams are not complete as they are obtained for particular conditions. The experiments in Figs. A6.11 and A6.12 were short in time: at the initial step we create ΔT just below threshold, increase temperature and stop the experiment. Perhaps, in this way evaporation does not influence instability significantly and results are similar to the case of silicon oils, it is a 3D oscillatory instability in the form of hydrothermal waves. Perhaps, some particular azimuthal mode is excited. This question needs further investigation. 59

66 Figure A6.13. Evolution of critical frequency in the experiments with decreasing volume at T=7.5K: the circles show the frequency at the onset of an instability (09/01/2013). Two branches indicate the instability, shown in Fig. A6.12. Having the ability to track the behaviour of the system in time, we conducted a set of experiments with small steps in volume (V/V 0 ). Thus, comparing these long-duration experiments with the results in Figs. A6.11 and A6.12 for short-duration experiments, we may analyze the role of evaporation on the flow stability. Indeed, performing experiments with evaporative decrease of volume we have found a strong destabilization of the flow which could be attributed to the loss of heat and evaporation at the interface. To shed light on the flow organization, we will analyze the variation of the critical frequency at the threshold of instability. Figure A6.13 compares the critical frequency from the short-duration experiments (from Fig. A6.12) and from the experiments at ΔT=7.5K while the initial volume progressively decreases due to evaporation. For a better overview we show the data only for the limited range of volumes, where the changes are the most powerful. The frequency in the experiments with evaporation (yellow circles) displays a continuous smooth behaviour throughout all the volumes. A typical jump (discontinuity), observed at V~1 in the short-duration experiments, is absent. The frequency value is smaller for all volumes. Keeping in mind the analogy between the behaviour of T and f (Figs. A6.11 and A6.12) we may expect essential destabilization of a flow in these experiments at T=7.5K in comparison with the short-duration experiments. 60

67 Figure A6.14. Comparison of the evolution of critical frequency with volume at T=7.5K and T=8.0K (07/01/2013). Two branches indicate the instability from the short experiments, shown in Fig. A6.12. Figure A6.15. Evolution of the critical frequency with T at the constant volume V r =1.05 (18/01/2013). The comparison of the critical frequencies at T=7.5K and T=8.0K in Fig. A6.14 shows the progressive drop of the frequency at T=8.0K in the region of large volumes. Our experiments with scanning by temperature difference for a liquid bridge with a large volume, V/V 0 =1.05, have confirmed these observations as shown in Fig. A6.15. We have found that the flow states are, generally, periodic in the region of T where the frequency decreases. Perhaps, this may be due to the 2D oscillatory instability, which would be interesting to investigate in future numerical work. 61

68 Figure A6.16. Evolution of critical frequency in the experiments with decreasing volume at T=9K: the circles show the frequency at the onset of instability (09/01/2013). Two branches indicate the instability, shown in Fig. A6.12. Figure A6.17. Evolution of critical frequency in the experiments with decreasing volume at T=10K: the circles show the frequency at the onset of instability (09/01/2013). Two dashed curves indicate the trend of the instability, shown in Fig. A6.12. Figure A6.16 demonstrates that increase of T up to T =9K does not change the behaviour of the critical frequency at large volumes. This is in line with profile f cr ( T) in Fig. A6.15 which displays a plateau for 8.5< T<10. While the frequency in the limit of large volumes obtained in the long-duration experiments is always smaller than that measured in the short-duration experiments, for the small volumes it may be the opposite. The further development is even more intriguing. Figure A6.17 shows the critical frequency at a larger temperature difference, T=10K. Observations show that oscillations decay in a certain region of liquid bridge volumes, 0.93<V/V 0 <

69 The finding concerning the flow stabilization in this region does not contradict with the results of short-duration experiments, as the experimental points were absent in this region and only the trend line is shown. The stabilization of the flow in this region of volumes is also confirmed by the experiments with scanning by T. A6.3. Stability window The transition from a steady flow to oscillatory convection is an important result when studying thermocapillary convection. Figure A6.15 demonstrated that at a certain diapason of volumes the flow becomes stable, although T seems to be above critical. Here, we focus on the analysis of localized stabilization of the flow, because it has never been reported in the literature on liquid bridges. A series of experiments were carried out in the range of the volumes of a liquid bridge where stabilization was expected. T = 7.53K T = 10.93K T = 11.86K T = 12.45K T = 13.60K Figure A6.18. Fourier map with stability window. Experiments are conducted at constant volume, V/V 0 =0.95, when T increases. The low plot presents the phase plane at T shown by arrows (31/01/2013). Figure A6.18 presents the Fourier map for the experiment at V/V 0 =0.95, when the temperature difference T=6K is initially imposed between the rods and then is gradually increased. It can be seen clearly in Fig. A6.18 that stabilization occurs in a limited range of T. Instability arises at T=7.4K but persists only until T=9.51 and disappears to return 63

70 again at a higher T. A narrow stability window occurs between T=9.51K and T=9.98K. The evolution of the fundamental frequency and its harmonics with T can be traced directly on the Fourier map. Starting from the instability threshold and up to the stability window, the fundamental frequency is subjected only to small changes. After the stability window the main frequency jumps up; it may indicate either the change of an azimuthal wave number or transition between a standing to travelling wave. The system trajectories in phase planes are shown for the selected ΔT on the plot below the stability map. Arrows point out the values of T for which the phase portraits are shown. It is worth noting that before the stability window the flow above T cr is periodic while after the stability window the flow exhibits a quasi-periodic and chaotic behaviour. Note, that these aperiodic flow states are observed when the distance from the critical point is not very large, ε <1. Figure A6.19. Illustration of the stability window: (upper plot) Fourier map of the experiment at constant T=10K when volume decreases due to evaporation, T mean =25 C; (lower plot) the amplitudes of the temperature oscillations on three thermocouples (02/04/2012). To identify more precisely the extension of the stability window, we have performed a series of experiments with scanning by the volume. A large stability window can be clearly seen in Fig. A6.19 at the experiment with T=10K. The Fourier map (upper 64

71 plot) shows that on the side of small volumes outside the stability window, the frequency spectrum is noisy with multiple harmonics. However, the amplitude of the temperature oscillations presented on the lower plot in Fig A6.19 presents a similar value on all three thermocouples. On the contrary, on the side of large volumes outside the stability window, the frequency spectrum exhibits less noise, but the amplitudes of the temperature oscillations are different on all the thermocouples. It may suggest that the flow organization in the sense of azimuthal wave numbers is more complex on the side of large volumes and most probably we deal with a standing wave. Figure A6.20. Stability diagram. The horizontal lines indicate the experiments aimed at searching for the boundaries of the stability window. A few arrows indicate development of the experiment in time. Figure A6.21. Complete stability diagram as a summary of all experimental results. 65

72 To identify the stability window in large parameter space, we conducted a series of experiments at several constant T when tracking the system stability with decreasing of volume. The results are collected in Fig. A6.20 where each horizontal line corresponds to the experiment at constant T and T mean =25 C. The dashed curves and points around them correspond to short-duration experiments. Data on the diagram also include a few experimental points at a constant volume, V/V , but for clarity of the diagram the vertical lines are not shown. The experiment at T=9K was repeated 4 times and the stability window was observed in two of them. We assume that T=9K presents the lower boundary of the window. The widest part of the window occurs at T=10K and it comprises the range of the volumes 0.91<V/V 0 < 0.95 which is slightly floating depending on the experimental procedure. For example, performing an experiment with increasing and decreasing T exhibits a stability window at a slightly different range of volumes. The local stabilization of the flow is, once again, highlighted by the summary of all experimental data presented in the final stability diagram in Fig. A6.21. Although the stability window occupies a small part of the parameter space this finding paves the way to further studies and especially on the numerical side. A6.4. Effect of the mean temperature on the flow stability If the liquid on the interface is warmer, then its molecules have a higher average kinetic energy, and evaporation will be faster. Our set-up provides the possibility to change the mean temperature of the liquid and, consequently, the temperature of the interface. The role of evaporation in the stability of a thermo-capillary flow is quite complex: on the one hand, it stabilizes the flow by removing heat from the interface; on the other hand, it destabilizes it by inducing higher thermal gradients along the interface. Figure A6.22. (Numerical results) Comparison of the dimensionless (a) interface temperature and (b) velocity between cases with and without evaporation. The results were obtained in the geometry of the new set-up and using a measured evaporation rate when T=7 C, T mean =25 C and V/V 0 =1 (straight cylinder). 66

73 To check validity of this point of view for our set-up, two-phase simulations were performed by Gaponenko (MRC-NLF group, ULB). The model used in computations considers that the concentration of the vapour of liquid in a gas phase is negligibly small. We should remind that the test liquid is weakly volatile and the chamber with gas around the liquid bridge is large; it justifies the application of the model. The results are shown in Fig. A6.22 and they support the idea, that even though the interface temperature slightly decreases due to evaporation, the interface velocity increases. Hereafter, we are focused on the impact of the mean temperature on the flow stability. A series of experiments were conducted at several mean temperatures in the range between 20 C and 25 C to test T mean impact on the flow stability in a liquid bridge. The increase of T mean leads to elevation of the surface temperature and growth of the evaporation rate. The change of the evaporation rate with the interface temperature was discussed in Section 5 and it can be seen in Fig. A5.5. Our experiments have found that the increase of the mean temperature destabilizes the flow. The measured data for the volume V/V 0 =0.95 are collected in Fig. A6.23 where each point corresponds to the experiment with predefined constant T mean but the temperature difference T either increases (way up) or decreases (way down). Figure A6.23. Critical temperature difference ( T cr ) as a function of the mean temperature (T mean ) in the experiments with the increase of T (rhombuses) and the decrease of T (squares) at V/V 0 =0.95. Green triangle presents the result of the experiment with scanning by T when T =6K (see Fig. A6.10). 67

74 Figure A6.23 points out a hysteresis: on the way up an oscillatory mode starts at higher T values (squares) than that T on the way down (rhombuses). The data on the plot also include a single point (green triangle), which was obtained at constant T=6K while increasing T mean (see Fig. A6.10). The experimental data can be approximated by linear dependence T= ζ T mean + γ, ζ= , γ=12.77 which is presented by the dashed line in Fig. A6.23. Note that a linear relationship may not hold in the case of a larger range of T mean. Another message of this plot is that the destabilization of flow is quite strong, T cr reduces by 15% when T mean changed by 5 C. The flow pattern in the considered system is very sensitive to variations of T, e.g., the non-linear flow states were observed in Section only 20% above the threshold. Figure A6.24. Evolution of frequency with T at constant volume V/V 0 =0.95 when the mean temperatures are: T mean =20 C (blue rhombuses), 22 C (magenta squares), 23 C (green triangles) and 25 C (brown circles). An interesting finding is that, despite the obvious change of the threshold of instability, the flow states are not essentially affected by T mean. Figure A6.24 presents evolution of the critical frequency with T for several T mean. All the data sets display a similar tendency: after the onset of instability the frequency essentially diminishes within C, then a sharp frequency skip occurs where the frequency jumps up by 20% 68

75 (>0.1Hz). With a further increase of T, the frequency grows smoothly for all T mean until the temperature difference reaches T 11 C where a sudden transition occurs from a periodic flow to a quasi-periodic and aperiodic flow. Figure A6.24 presents the main frequency from one thermocouple (i.e., the frequency with the largest Fourier amplitude as programmed by the software), but at T>11 C the multiple frequencies have a similar amplitude and it provides large scattering of the points. Figure A6.25. Fourier map at constant volume V/V 0 =0.95 and T mean =20 C when ΔT increases. In order to facilitate the identification of flow states with the increase of T, the Fourier map is presented in Fig. A6.25 when T mean =20 C and V/V 0 =0.95. We can see more clearly the tendencies outlined in Fig. A6.24. It is entirely impossible to describe all appearances of chaotic behaviour, but what is unusual in this experiment is that the flow undergoes a very sharp transition from periodic to irregular (chaotic). One can identify the incommensurate frequency which exists only in the tiny region 10.9< T< The temporal evolution of the other experiments displays a similar behaviour. A possible explanation for this could be that this region of parameters, V/V 0 =0.95 and T 10.5 C, is adjacent to the stability window discussed in Figs. A6.20-A6.21. To conclude this part, we presented experimental evidence that the flow in a liquid bridge system with weakly evaporating liquid has a fairly rich structure. Another feature, that highlights this part of work, is the development of powerful experimental techniques and procedures which allow studying this rich dynamics of the system. 69

76 1. Part B Measurements technique for heat/mass transfer in mixture with Soret effect B1. Introduction Another important part of transport processes in fluid physics is diffusive mass transport. It manifests itself in practically every situation when fluid is composed of more than one component. What is referred to as diffusion is an isothermal molecular transport of mass in mixtures, which occurs in the presence of a concentration gradient and tends to reduce concentration variations. Many real systems and technical processes operating with multicomponent fluids take place in non-isothermal conditions. On top of the flux driven by concentration nonuniformity, a mass transport of species appears caused by the temperature gradient. Such transport is known as thermodiffusion or the Soret effect [79]. Here, the term species may refer to molecules, polymers, or small particles (colloids). There are many important processes in nature and technology, where these phenomena play a crucial role. The composition of underground hydrocarbon reservoirs is significantly affected by diffusion as well as the Soret effect (due to the presence of geo-thermal gradient) [80], [81]. The effect of thermodiffusion is employed for isotope separation in liquid and gaseous mixtures as well as other separation processes [82] that involve colloids, macromolecules or nano-fluides. Other potential applications include high-pressure combustion [83], solidification processes, oceanic convection [84], biological systems [85] and CO 2 geological storage. Nowadays, both theoretical and numerical tools for the investigation of such phenomena are well developed, but these powerful instruments need precise knowledge of transport coefficients to elaborate correct predictions. Interest in reliable measurements, arising from this need, causes appearance of new approaches [86], [87], revisiting earlyestablished techniques [88] and tendency of cross-checking the results obtained by different methods [89], [90]. For this reason the Benchmark of Fontainebleau was performed [89], where several laboratories carried out a quantitative comparison between 70

77 different experimental methods actively used in to determine diffusion, thermodiffusion, and Soret coefficients of three binary liquid mixtures. The Benchmark is still in progress as recently the results from Beam Deflection technique have been introduced into the benchmark [88] and some actively used methods have not yet contributed to Benchmark data base. The existing methods can be divided into two groups. The first group employs convection arising from compositional and thermal variations in gravity field. In the Rayleigh-Bénard set-up, the data are extracted from critical parameters for the oscillatory onset of convection in binary fluid [91]. Thermogravitational Column technique [92], [93] relies on coupling between convection and horizontal thermodiffusion in a side-heated vertical slot. However, liquid sampling required in this method may disturb the diffusive process [91]. For the second group a no-convection state is crucial for the measurements. The sampling problem exists in the Standard Soret Cell [91], where liquid is placed between two differentially heated copper plates. Modern techniques based on optical methods of observation do not perturb the diffusive process. In the Beam Deflection technique, the Soret cell has transparent walls and evolution of composition is observed via deflection of a laser beam passing through the medium [94], [95]. In the Thermal Lens technique [96], [97], the temperature gradient is created by heat generation resulting from the absorption of light into the fluid. The sample behaves like an optical lens due to the change of the refractive index resulting from thermal and compositional gradients. Another important method is the Thermal Diffusion Forced Rayleigh Scattering [98], [99], [100], [101] where a grating created by the interference of two laser beams is converted into a temperature grating by the energy absorbed by a chemically inert dye. This periodic temperature field induces a periodic field of concentration due to the Soret effect. Note that the relaxation time is short but a very great number of experimental runs in the same configuration are required for obtaining reliable results [91]. Interferometry is recognized as one of the most precise methods of measuring diffusion coefficients in liquids both in the past and nowadays (see e.g. [102]). Strong points of this technique are high sensitivity and ability to monitor the concentration field in the whole cell but not in the distinctive points. For these reasons, the method is extremely promising for applying in thermodiffusion studies as well, which was demonstrated in [103]. 71

78 But at the same time, it has to be noted that application of interferometry for the measurement of the Soret effect is a very challenging task. First, very small concentration differences are supposed to be measured over the experiment cell, hardly exceeding one weight percent in total. Second, a very high long-term stability of the interferometer is needed, as a typical experiment can last a few days. And third, in the experiments of this type both temperature and concentration of a liquid sample are inevitably entering into overall refractive index variation. Moreover, refractive index variation due to concentration change (that is the quantity of interest) in most cases is order of magnitude less in comparison with refractive index variation due to the temperature gradient. So, careful separation of both factors is vital. Early attempts to use interferometry for measuring Soret coefficients date back to as early as 1957 (e.g. see [104]), but since then the method hasn t found a wide use either for the reasons listed above or due to the lack of properly developed interferogram processing techniques. The purpose of the current chapter is two-fold. First, to report improvements of the interferometer set-up built in Service Chimie-Physique E.P. of ULB for thermodiffusion studies. Second, to present results of measurements of Soret coefficients for several binary mixtures, both associated and hydrocarbon ones. By comparison of data given by interferometry with accurate and strongly cross-checked data coming from other techniques, we would like to draw a conclusion about perspectives of interferometry in this particular field. 72

79 B2. Set-up B2.1. Initial configuration In the first design of the set-up, for the measurements of the Soret effect by means of optical diagnostics, a classic thermodiffusion cell was chosen with transparent lateral walls clamped between two thermostabilized copper blocks of mm in size. A close-up view of the cell is shown in Fig. B2.1(a). Laterally, a liquid volume is enclosed in a rectangular cell, which is made of optical quality fused quartz (custom made by Hellma) with external dimensions of mm, where H = 6.3 mm is the cell height; a wall thickness of 2.0 mm leaves L = 18.0 mm for the optical path in the fluid in both horizontal directions. The glass frame is sealed between the copper blocks with two gaskets from thermal conductive rubber 0.2 mm thick (Chomerics, Cho-Therm 1674, thermal conductivity α = 1.0 Wm -1 K -1 ). The choice of the cell geometry and sealing material will be discussed in more detail in the following section dedicated to cell optimization. To prevent excess stress in the glass part caused by non-uniform clamping pressure, four equal height spacers were additionally placed between the copper blocks and the glass frame. (a) (b) Figure B2.1. (a) Cross section of the cell. (b) Scheme of the set-up. Each copper block has a channel of 1.0 mm in diameter for the cell filling. The external part of the channel has a widening to accommodate standard connectors for fluid management. The inlets of the channels into the cell are located at opposite corners. The outgassed liquid is injected through the channel in the bottom plate. To outgas the liquid, a 73

80 flow-through degasser (Systec, OEM Mini Vacuum Degasser ) was interposed between the injecting syringe and the cell. The degasser designed for outgassing solvents for HPLC applications does not affect the composition of liquid mixtures. As deep degassing requires a low flow rate, injection of liquid is done by a syringe pump (KD Scientific, model 210) with a flow rate of 0.1 ml/min. The filling procedure is sketched in Fig. B2.2. Figure B2.2. Cell filling configuration: (1) Syringe pump; (2) Flow-through degasser; (3) Cell in filling position; (4) Liquid waste vessel. To ensure proper concentration of liquid mixture and absence of impurities, the cell (as well as the whole filling circuit) is flushed with acetone before passing to another liquid system. After flushing with acetone, the system is connected to a vacuum pump to remove acetone vapor. In case when components are the same and the change of concentration between the previous and the next mixture is not big (10 20 wt.%), the flushing with acetone was not implemented, but in all the cases the cell was filled twice with the experimental mixture for short times (0.5 1 hour) before final filling. Each copper block has a hole for a sensor (a calibrated NTC thermistor Epcos, B57861S861) and the temperature is kept constant by a Peltier element (Altec, , P mmm = 34 W, I mmm = 3.9 A). The temperature of Peltier elements back sides is kept constant by water heat exchangers connected to the circulation water bath (Lauda, E200). The thermal contact of Peltier elements with the copper blocks on the working side and with the water blocks on the back sides was improved by filling the gap between the blocks and Peltier elements with thermal conductive paste. The whole structure was gently fixed by a set of screws with equally applied torque. 74

81 The temperature of each copper block is regulated independently by two computerdriven PID-controllers (Supercool, PR-59), enabling temperature stability of ±( ) K. Temperature logging is also provided by a controller with a resolution of 10 3 K. The data are recorded by a computer with an imposed sampling rate (up to 20 Hz). The concentration variation inside liquid mixtures is measured by digital interferometry. From a variety of different interferometer schemes (Rayleigh, Gouy, Fizeau, Michelson, Mach-Zehnder), the last one has been selected because it allows simultaneous observation of a fringe pattern and the object itself and can easily treat the beam deflection problem. The schematic drawing of the set-up is shown in Fig. B2.1(b). The light source was a He-Ne laser (Thorlabs, HRR020) with power of 2 mw and wavelength λ = nm. The laser beam was expanded to cover the full area of the cell and then collimated. The collimated beam is split by a 50R/50T plate beam splitter (50 50 mm in size, Edmund Optics, NT45-854) into the reference arm and into the object arm which includes the cell assembly. Then both beams are redirected by 50 mm diameter mirrors of λ/10 flatness (Thorlabs, PF20-03-G01) to the second beam splitter to interfere. The resulting interferogram is recorded by a CCD camera (JAI, CV-M4+CL) with effective pixels on a 2/3 sensor and a frame rate up to 24 fps. The camera is equipped with an objective lens of 70 mm focal length and F aperture (Schneider, TXR 2.2/ ). The field of view covered by the imaging system is around 25 mm, so the resolution is 50 pixels/mm. All optical components were originally mounted on an optical bench plate of mm, which was placed on top of the optical table but was mechanically decoupled and thermally isolated from the latter. To improve the mechanical stability of the interferometer, all the elements were mounted on low and rigid posts 25 mm in diameter. To ensure the thermal stability of the interferometer during the experiment (up to 2 3 days), the whole set-up including the bearing bench plate is placed inside a box made of 3 cm thick foam thermal insulation material (the shaded part in Fig. B2.1(b)).The box is equipped with an air-to-air cooling/heating assembly based on a Peltier element (Supercool, AA , P = 58 W, I = 3.1 A) and driven by a dedicated PID-controller of the same type as for the cell s thermoregulation. A set of shields made of the same insulation material is inserted into the box to prevent air perturbation over optical 75

82 paths. The temperature inside the box was kept equal to the mean temperature of liquid with residual fluctuations of ±0.1 K. B2.2. Cell optimization After some experience with the existing set-up it was recognized that the most important aspect of the experiment design was the experiment cell implementation. There exist very general considerations regarding the cell geometry. For example, the height of the cell should be large enough to allow easier manipulations with the probing beam. But it should not be too large since the time of the experiment is proportional to the cell height squared (τ D = H 2 /D). The length of the cell (beam path in liquid) should also be large enough since the signal measured by the instrument is directly proportional to the optical path. However, a long cell, which is advantageous for the beam deflection technique, is not the best choice for interferometry, as the set-up should be able to safely treat beam deflection. The cell geometry is important, but here we focus our attention on another factor that is really crucial for the measurement: the thermal design of the cell. Non-accurate thermal design can completely discard all advantages of interferometry, since it determines presence and intensity of residual convection inside the cell. Due to drastically different characteristic times for transport of momentum and mass, the presence of convective flows in a diffusion cell will significantly alter the concentration field. For example, it was shown that a local convective vortex with a flow speed of μμ/s can completely homogenize concentration distribution in the region where this vortex exists [103]. So, residual convection has to be reduced as much as possible. This problem has already been seriously addressed to before [105], [106], but it appears that there are still some possibilities of improvement. Before describing the improvements implemented in the present work, it is very didactically useful to compile and consider all the previous steps of the cell s optimization. First of all, it is clear that for any cell with a relatively large size (a few millimeters or more), the thermal gradient has to be directed against gravity, and only in this case the mechanical equilibrium in the cell is generally stable. But convection can easily appear at the lateral walls of the cell and in the corners where thermal perturbations are inevitable. A good example is the very first iteration of the cell probed by the interferometer which was described in [105] and sketched here in Fig. B2.3(a). The authors claimed that in a cell 76

83 with such geometry it was impossible to see any separation caused by the Soret effect due to a non-negligible heat flux over the lateral walls and resultant residual convection. After some trials and errors, a modification was found which eliminated this heat flux at least in the central layer of the cell with minimal cost. The modified cell geometry is presented in Fig. B2.3(b) and the values of lateral temperature gradient at side walls are compared for both geometries in Fig. B2.3(c). It is clear from the latter plot that at least one-third of the cell height is not any more affected by the lateral heat flux. (a) (b) (c) Figure B2.3. (a) Temperature field in the system with thick quartz walls after [105] (one quarter of the cell is shown). O-ring is located at the top of the glass in the middle (z-axis origin is shifted to the cell center); (b) Modification of the cell with thin walls; (c) Horizontal temperature gradient inside liquid close to the wall. Dashed curves 1 and 2 show numerical results in the case of thick and thin walls, respectively; curve 3 shows experimental measurements for the thin walls [105]. Finally, using the advantage of optical interferometry to monitor any motions in the entire cell during transient process, the zones affected by convection were determined and 77

84 excluded for data calculation [106]. The concentration field obtained in the cell and nonaffected layer are shown in Fig. B2.4. (a) (b) (c) Figure B2.4. (a) Sketch of the cell; (b) Concentration map; (c) Vertical profile at x = 5 mm after separation of water ethanol solution with mass fraction of water in the cubic cell. Open circles are experimental points, solid line is a theoretical profile for a given time instant and dashed lines confine the central trustworthy layer [106]. Using the approach of considering only the trustworthy layer it became possible to implement measurements having a reasonable agreement with literature data [103]. But in addition to the important drawback of the presence of uncontrolled residual convection, this cell suffered from the requirement to be sufficiently high to accommodate a reasonably thick non-disturbed layer. This requirement made experiments in this cell excessively long (one week or so). After additional considerations, supported by stationary heat transfer calculation of the supposed cell geometry with realistic boundary conditions, a completely new iteration of the Soret cell was issued [107], [106], which was free from any grooves and protrusions in the temperature controlled copper blocks. This step allowed further suppression of lateral heat fluxes which, in turn, made it possible to reduce the cell height. But this new geometry, although profitable from the point of view of thermal design, experienced some difficulties with sealing. The first attempt to use PTFE sheet 0.2 mm thick as a seal, reported in [106], though demonstrated much better performance (see Fig. B2.5), still suffered from traces of residual convection in corners. Furthermore, cell assembling with such seals was an extremely delicate task; if treated improperly, the glass frame was easily cracked. 78

85 (a) (b) (c) Figure B2.5. (a) Sketch of the plain cell with PTFE seals; (b) Concentration map (c) Vertical profile at x = 9 mm after separation of water ethanol solution with mass fraction of water in the cell. Open circles are experimental points, solid line is a theoretical profile for a given time instant and dashed lines confine the central undisturbed layer [106]. As it was discussed in [107], both problems can be solved by replacing a PTFE sheet by a sheet of thermal conductive composite rubber (the design was described in the previous section). The lower rigidity of the material reduces the chance of breaking the glass frame when assembling the cell, and higher thermal conductivity (close to the conductivity of quartz glass) eliminates completely the parasitic heat flux. The result of this improvement is shown in Fig. B2.6. It is clearly seen that the distribution of concentration has a perfectly one-dimensional character here, without any trace of residual convection. The boundaries of the trustworthy layer, still marked here, confine in this case a layer which is not affected by thermal jitter of copper plates and by separation lost in the initial time of the experiment. Some of these problems can be solved and the solution will be discussed in the next sections. (a) (b) (c) Figure B2.6. (a) Sketch of the cell with thermal conductive rubber seals; (b) Concentration map; (c) Vertical profile at x = 9 mm after separation of THN nc 12 solution with 0.50 mass fraction of THN in the cell. Open circles are experimental points, solid line is a theoretical profile for a given time instant and dashed lines confine the central trustworthy layer [106]. 79

86 Such a design of the cell allowed implementing precise measurements comparable with other established techniques [107]. This was exactly the cell design with which the experiments presented in the work started. After a set of measurements and cycles of assembling-disassembling the cell it became clear that seals from thermal conductive rubber, though the best from the point of view of thermal performance, raise an extra problem: leakage. It was noted in a few unsuccessful experiments that the observed concentration field featured some anomalies in arbitrary cell corners. These anomalies were attributed to leakage of the mixture as a whole or to selective permeation of some components through the rubber seals. Feasibility of the supposed mechanism was supported by the fact that the filler of the composite rubber is tiny glass wires, which can easily form channels for the escape of liquid along disrupted filler-matrix interfaces. The alternative finding implemented in this work was, first, replacement of thermal conductive composite rubber with thermal conductive composite epoxy glue. Thermal performance of the cell based on thermal conductive composite epoxy glue was proved to be equivalent to that based on rubber, and it eliminated all anomalies of the concentration field. At the same time, the solution made the cell almost non-separable, each disassembling act required long soak of the junction in acetone and did not guarantee integrity of the glass frame. In addition, it was also found in tests with some mixtures based on very active solvents like methanol or cyclohexane that they can react with, soften and dissolve the glue-based sealing. With this drawback in mind, after some cycles of experiments, the cell design was changed again. By a new cycle of numerical simulations it was shown, that there exists a solution alternative to the previous one, which treats the problem of lateral heat fluxes equally well. B2.3. Final cell design elaborated in the work First of all, the metallic plates which lock the glass frame and liquid volume were made sectional, built-up of two pieces: a highly thermoconductive brass insert is establishing a thermoconductive bridge between a Peltier element and liquid while an external frame of stainless steel provides rigidity and dimensional precision of the structure (see Fig. B2.7(a)). Two identical plates are connected via precisely machined posts from invar which define the distance between the working surfaces of the plates with 80

87 an accuracy of a few microns. The working surfaces of the brass inserts are nickel-plated to prevent their corrosion. (a) (b) (c) Figure B2.7. (a) Sketch of the cell with indium seals; (b) Temperature distribution in the cell calculated by Comsol Multiphysics with ethanol as an experiment fluid; (с) Magnified view of the glass-seal contact, showing the reason for appearing the hidden layers. Second, to abandon the problem of solvent permeation through the rubber O-rings ab initio, the seals were replaced with ones made of indium. This material has good adhesion both to glass and to brass and also a very high thermal conductivity. In addition, indium can be used as gaskets for sealing volumes with aggressive liquids [108]. Technically, it was implemented by machining thin grooves on the working surfaces of brass inserts, filling them with indium and then clamping all the structure together with the glass frame after its precise positioning against the grooves. The design of the cell was fixed after a numerical analysis of stationary heat transfer in the supposed geometry which confirmed a very small amplitude of the heat fluxes through lateral walls (see calculated temperature distribution in Fig. B2.7(b)). Due to some small intrusion of the glass frame into the indium filler at the moment of the frame clamping, there appeared small bulges at both sides of glass-indium contact (see Fig. B2.7 (c)). On the one hand, they partially hide the thin liquid layers adjacent to the copper plates, but on the other hand, such hiding has a positive effect as it blocks reflection of a deflected beam from the plate, which otherwise can pollute the interference pattern. We have optically characterized the hidden height and 81

88 found it equal to δh = 120 μm when the free space between the plates equals to H=6.059 mm. These small invisible regions were explicitly taken into account in the fitting procedure described below. The analytical model, to be fitted to experimental data, presumes a full cell height of mm, but only values within δh z H-δh do participate in the fit. The heat transfer calculations done for the real cell geometry have drawn attention to another aspect which can slightly affect the values of measured Soret coefficients; this is temperature difference used for calculation of these coefficients. Temperature difference, which is assumed to be between the working surfaces of brass blocks, in reality corresponds to the temperature difference between the positions of sensors inside these blocks. The real position of the sensor is at least a few millimeters from the working surface of the block, and this may introduce a certain misfit between ΔΔ displayed by sensors and the real temperature difference that drives Soret separation. The temperature field presented in Fig. B2.7(b) was computed with temperature boundary conditions adjusted in such a way as to have exact values given by sensors in their proper positions (indicated by open circles in the figure). While the calculated temperature at the locations of the sensors perfectly reproduces the values provided by the measurements, the temperature difference between the working surfaces of the plates does not give the same value. On average, the temperature difference between the sensors of ΔT ssssss = 6.0 K, applied in all the experiments, corresponds to ΔΔ = K between the surfaces of the plates. This slightly reduced temperature difference was used for the calculation of the Soret coefficients in this study. B2.4. Interferometer modification In course of this work certain improvements have been introduced in the interferometer as well. First of all, the most sensitive elements of the interferometer (the beam splitters, the mirrors and the cell) have been separated and mounted on a smaller bench plate of mm (see Fig. B2.8(1)) to further improve the interferometer s thermal stability. This plate, together with the laser diode assembly (see Fig. B2.8(2)) and camera (see Fig. B2.8(3)), has been mounted on a bench plate of mm (see Fig. B2.8(4)), which, in turn, is enclosed in a temperature controlled box (see Fig. B2.8(5)). 82

89 Figure B2.8. Photo of the interferometer: 1) plate with the mounted mirrors, beam splitters and cell; 2) laser diode assembly; 3) camera; 4) bench plate; 5) temperature controlled box. With the capability of changing the working wavelength in mind, a He-Ne laser was replaced with a laser diode (LD). The choice of the wavelength for LD was affected by the requirement that the obtained results could be compared with the results of space experiments implemented with SODI instrument (see details in [109]). Thus, the first choice was LD of 670 nm (Roithner Lasertechnik, S6705MG, 5 mw). To control this diode, a laser diode driver (LDTC0520 produced by Wavelength Electronics) was used. This driver stabilizes the temperature of the laser diode by means of a Peltier element according to PID law. Also, it stabilizes the current passing through the laser diode. The stabilization of these parameters ensures stability of the emission frequency of the diode. The setting of the driver was carried out for a specific diode because electro-optical parameters of laser diodes have a fairly large scattering even in the same batch. The assembly which hosts the laser diode was designed on the basis of available commercial parts (Thorlabs). This assembly must have the capability to be quickly replaced by a similar assembly with a laser diode of a different wavelength for performing measurements with a different wavelength. 83

90 Figure B2.9. Photo of the new laser diode assembly, designed and built for the interferometer. Here, (1) is the beam redirecting mirror, (2) is the collimating lens; (3) is the grey filter to manipulate with beam intensity; (4) is the 2-D translation mount with a fixed pinhole; (5)is the beam focusing lens; (6) is the translation mount for the focusing lens; (7) is the multiaxis alignment holder; (8) is the laser diode assembly with TEM cooler and connector. Parts (4)-(6) form the so-called spatial filter to improve beam quality. B2.5. Control unit The set-up is controlled by a computer through a control unit. The control unit produced in our laboratory is a shielded system with quickly replaceable modules. This system hosts practically all the electronic blocks of the set-up excluding the laser diode assembly, sensors, Peltier elements regulating the temperature on the cell s plates and air-to-air cooling/heating assembly. The photo of the control unit can be seen in Fig. B2.10. Figure B2.10. Photo of the control unit. The modular system is convenient when switching to the measurement at another wavelength, when the laser assembly and the electronic module with a laser diode driver are replaced. 84

91 Different blocks of the control unit have been provided with their own power supplies with the aim of diminishing electromagnetic interactions. All these power supplies are switched to the power circuit through a filter, which allows reducing the effect of noise on them. Shielded wires have been used with the view of reducing the electromagnetic interference as much as possible for linking the sensors with the control unit and sending data into the control computer. To calibrate all the sensors, we have used precision calibration electronic thermometer P750 of Dostmann Electronic and have employed supplementary devices produced in our laboratory. For precision in regulation a multistep setting of PID-controllers has been done. 85

92 B3. Image processing Interferometry is a trusted and widely used optical technique for measurements of the refractivity of objects, from which related quantities like temperature or concentration can be determined [110]. For a given wavelength λ, the variation of the refractive index n(x, z) includes temperature and concentration contributions ΔΔ(x, z) = ΔΔ(x, z) + T 0,C 0,λ ΔΔ(x, z) T 0,C 0,λ (B3.1) where ΔΔ(x, z) and ΔΔ(x, z) are the temperature and concentration changes at point (x, z), here z-axis is in the direction of the temperature gradient. At the same time, ΔΔ may be obtained from the phase change ΔΔ, which is measured by interferometry: ΔΔ(x, z) = n(x, z) n 0 = λ ΔΔ(x, z) (B3.2) 2ππ Here, L is the beam path in liquid. As it follows from Eq. B3.2 the variation of the refractive index ΔΔ is equivalent to the change of the optical phase ΔΔ. Hereafter, ΔΔ will be preferably used as it is a measured value and it is an additive quantity contrary to ΔΔ. In order to increase the accuracy of phase evaluation of interference fringe patterns beyond the early fringe scanning technique, several procedures were developed, such as the Fourier transform technique [111] or the temporal phase stepping technique [112]. These procedures are well established now. B3.1. Fringe analysis for phase-measuring interferometry In optical interferometry, the change in phase between two coherent light waves (reference and object ones) is the reason for spatial intensity variation. The results of optical interference of two light waves, an interferogram or fringe pattern, are taken by a sensor as a digital picture. We will shortly address to the strategy of extracting phase information from a fringe pattern following [113] and [114], since this is fundamental for the discussed experimental technique. The intensity of two superposed waves of the same frequency is i = A r + A o 2 where A r = a r eee( iφ r (x, z)) and A o = a o eee( iφ o (x, z)) are the amplitudes of reference and object beams and φ r, φ o are the initial phase angles. Accordingly, the intensity in the fringe pattern at t = t 1 is: 86

93 i(x, z, t 1 ) = i 0 (x, z) + m(x, z) ccc[δδ(x, z, t 1 )] (B3.3) where ΔΔ(x, z, t 1 ) = φ o (x, z) φ r (x, z) is the initial phase shift, i 0 (x, z) is the background intensity distribution, and m(x, z) = 2 a r a o is the modulation function. (a) (b) Figure B3.1. (a) Typical interference pattern of 512X1024 pixels size covering full cell width with an insert of a magnified fringe pattern in the top left corner; (b) Intensity profile on a horizontal line crossing the interferogram. A typical interference pattern recorded by the camera is presented in Fig. B3.1(a). A typical intensity profile over the horizontal line z = H/2 is shown in Fig. B3.1(b). It is possible to modify the spatial frequency of the fringes, f 0 = {f x0, f z0 }, by changing the incident angle of the object and reference beams through adjusting the inclination angle of one of the mirrors. At time t 2, when the object beam goes through the diffusing media, the intensity of the fringe pattern is given by an equation similar to Eq. B3.3: i(x, z, t 2 ) = i 0 (x, z) + m(x, z) ccc[δφ (x, z, t 2 )] (B3.4) Thus, Δφ (x, z, t 2 ) = ΔΔ(x, z, t 1 ) + ΔΔ where the phase shift ΔΔ is caused by refractive index variation of the diffusing media from time t = t 1 to time t = t 2. Without violating generality we may suppose that the spatial frequency has only xcomponent (f 0 = {f x0, 0}) then Eq. B3.4 can be rewritten as i(x, z) = i 0 (x, z) + c(x, z) eee(2πf x0 ) + c (x, z) eee( 2πf x0 ) (B3.5) where c(x, z) = 0.5 m(x, z) eee[iiφ (x, z)] is the complex amplitude. The asterisk superscript ( ) denotes a complex conjugate. 87

94 a) Image after Fourier transform b) Intensity distribution in Fourier space c) Fourier domain after filtering d) Applied filtering mask (dashed curve) and results of its application (solid line) e) Coordinate shift in Fourier domain f) Intensity distribution before (dashed line) and after (solid line) spatial shift Figure B3.2. Steps of interferogram digital evaluation by 2-D Fourier transform. To calculate the phase shift Δφ, we have adopted a powerful two-dimensional Fourier transform technique. Originally developed for treating 1-D phase distributions [111], this method was later adapted for 2-D phase maps [115]. Performing the discrete Fourier transform of the array i(x, z) yields I(f x, f z ) = I 0 (f x, f z ) + C(f x f x0, f z ) + C (f x + f x0, f z ) (B3.6) The location of the terms I 0, C and C in Fourier domain and its intensity is shown in Fig. B3.2(a,b). If the introduced mirror tilt (correspondingly f x0, f z0 ) is appropriate, then these contributions are well separated in Fourier domain. Both terms C(f x f x0, f z ) and C (f x + f x0, f z ) contain equal information about the phase shift and a filter is applied in the Fourier space to keep only one of them. Filtering sets to zero all frequencies except 88

95 those belonging, for example, to the term C(f x f x0, f z ). The resulting image is shown in Fig. B3.2(c,d) while the dashed curve shows the intensity distribution of the applied mask. During this procedure all filtered out terms must be suppressed, but at the same time useful information within the remaining term C(f x f x0, f z ) should be preserved. After that, 2-D Fourier transform output is rearranged by moving the spectrum on f x0 towards the origin of Fourier domain. It eliminates the carrier frequency f x0 and mathematically gives the term C(f x, f z ), see Fig. B3.2(e,f). In performing the inverse Fourier transform one can reconstruct both the amplitude and the spatial distribution of the total phase change of the fringe pattern produced by an object beam. Accordingly C(f x, f z ) c(x, z) and Δφ II[c(x, z)] (x, z) = aaaaaa RR[c(x, z)] (B3.7) A similar procedure is applied to the reference image for the evaluation of the phase ΔΔ(x, z, t 1 ), see Eq. B3.3. The reference interferogram also determines the vector f 0. Subsequently, the phase distribution ΔΔ(x, z, t 1 ) must be subtracted from Δφ that corresponds to the fringe pattern recorded after the change of an object. Correspondingly, the required phase shift in Eq. B3.2 is ΔΔ = Δφ (x, z, t 2 ) ΔΔ(x, z, t 1 ). In this way, the method applies the holography principle and tracks only the posterior optical phase variation in the set of images following the reference image (taken at t = t 1 ). (a) (b) Figure B3.3. (c) Phase map resulting from Fourier processing: (a) wrapped 2-D distribution and (b) wrapped vertical profile; (c) unwrapped 2-D map and (d) corresponding vertical profile. (d) 89

96 The phase difference calculated by arctangent function is wrapped, which means that it belongs to the range ( π, π), see Fig. B3.3(a,b). It should be unwrapped to construct a continuous natural phase. It is a simple task for one-dimensional case, while for a two-dimensional case and noisy fringes, sophisticated unwrapping techniques are required. As in our case phase maps typically have a good quality, we adopt the following approach to phase unwrapping. The procedure starts at a pixel with a well-defined neighbourhood assuming an error-free phase there. Following a spiral path the phase unwrapping is performed by comparing a wrapped phase with previously validated neighbours. If the difference is less than π the phase remains unchanged. If the difference between two pixels is more than π, the phase equals its wrapped phase minus 2π. If the difference is less than π, the phase equals its wrapped phase plus 2π. By the end the relative phase change between two pixels is placed in the range π and π and a smooth 2- D phase map is obtained (see Fig. B3.3(c,d)). It should be noted that the processing based on Fourier transform introduces two ambiguities into the extracted phase value. The sign ambiguity is caused by the fact that the term for filtering in Fourier domain, C(f x f x0, f z ) or C (f x + f x0, f z ), is usually chosen arbitrarily. The absolute value ambiguity is naturally arising from a wrapped character of the calculated phase. These ambiguities can be eliminated by using a priori information about the system (e.g. known temperature of copper blocks and mean concentration of liquid). B3.2. Subtraction of reference image The two-dimensional phase distribution (map) contains information about many things. The wave front is distorted by: all optical elements along the beam path Δφ oooooo, non-uniform air temperature Δφ aaa, temperature distribution in glass walls Δφ ggggg, temperature distribution in liquid bulk Δφ th, and finally by concentration distribution in the liquid Δφ C. ΔΔ = Δφ oooooo + Δφ aaa + Δφ ggggg + Δφ th + Δφ C (B3.8) Each particular interference pattern is formed either by all above factors or by some of them, depending on the state of instrument. An important part of overall processing routine is the separation of contributions. The advantage, which gives vast flexibility to this technique, is that any fringe pattern stored in the computer memory can be taken as a reference depending on the aim of the current processing step. 90

97 (a) (b) Figure B3.4. Readings of temperature on sensors (left vertical axis) and phase variation measured optically (right vertical axis) with time. (a) Thermalization process at T 0 = 25 o C; (b) Non-isothermal step. Let us briefly review this approach by considering an example of typical data processing steps. The first step traces the phase distribution during the instrument thermalization process. The step lasts 4-6 hours and aims at stabilizing the temperature of the cell and interferometer at mean temperature of T 0 = 25 o C. The very first image, Reference #1 in Fig. B3.4(a), is taken as a reference and processing continues during the entire isothermal step. In this case, the reference image keeps information about the initial state of the optics only (Δφ rrf1 = Δφ oooicc ) and the following phase variation is caused by possible optical path perturbations due to different mean temperatures of the air and cell. In Fig. B3.4(a) the closed circles (see the left vertical axis) show alteration of temperature on sensors while the open circles (see the right vertical axis) show the corresponding variation of the phase Δφ z. Importance of this step is to make sure of complete interferometer stability before the experiment is started. At the end of this step, a new reference image (Δφ rrf2 = Δφ oooooo + Δφ aaa ) must be chosen to process images after applying the thermal gradient over the cell (see Fig. B3.4(b)). At the second step, with a typical duration of a few minutes, the phase variation is caused by the temperature differences in the glass wall and liquid bulk; the concentration contribution is negligibly small. Because the diffusion characteristic time, τ D, is much larger than the thermal one, τ th, thus, Eq. B3.1 can be decomposed. For benchmark liquids τ D = H 2 /D 12 h while τ th = H 2 /χ 200 s, here χ is the thermal diffusivity of 91

98 liquid. Thus, the total phase variation at the observed point, affected by temperature gradient, can be written as ΔΔ = Δφ rrf2 + Δφ ggggg + Δφ th. (B3.9) Assuming that temperature distribution in the glass is linear (T = T 0 + ΔΔ[z/H 0.5]), glass thickness and temperature contrast factor for glass are well known, one can determine contribution of glass walls Δφ ggggg (z). Here, H is the cell height. Δφ ggggs (z) = T 0 + ΔΔ z 0.5 H 2πL ggggg ggggg λ where L ggggg = 4mm is the optical path in glass. Substituting Eqs. B3.2 and B3.9 into Eq. B3.1 the optically measured temperature distribution inside liquid is λ T(x, z) = T 0 + ΔΔ(x, z) Δφ rrf2 (x, z) Δφ ggggg (z) 2ππ C 0,T 0 These optical measurements of the temperature field should be done as soon as the temperature field is established, i.e., within a few thermal times, t 3 9 min. Later on, optical measurements will start deviating from sensors readings due to concentration contribution, see Fig. B3.4(b) beyond point Reference #3. This deviation point, which appears after the temperature difference is established, is the reference point for the third image processing step - thermodiffusion separation, Δφ rrf3 = Δφ rrf2 + Δφ ggggg + Δφ th. This new reference interferogram holds information about all inputs into optical phase except concentration variation. Processing the next images with respect to this one provides the phase change, from which a full 2-D map of concentration field is extracted λ C(x, z) = C 0 + ΔΔ(x, z) Δφ rrf3 (x, z) 2ππ C 0,T B3.3. Experimental limitations and precision of the method. The suggested method is based on Fourier processing, and its accuracy strongly depends upon the number of periods in the analyzed signal (i.e., the number of fringes in the interferogram). It means that the carrier fringe system has to be sufficiently dense. Small fringe spacing will also provide a distant peak in the Fourier domain, which is favourable for accurate filtering. At the same time, fringes have to be distinguished in the image. Fringe spacing of 4 6 pixels seems to be optimal for the method. Other experimental conditions that strongly influence phase accuracy are the stability of light intensity (a good quality laser is needed) and the stability of the 92

99 interferometer. The last point is especially important. Perturbations of interferometer by environmental disturbances were analyzed in detail in [103] for a similar problem and it was shown that they can affect the measurement. For this particular reason, we took extra precautions for stabilizing the interferometer as mentioned in Sections B2.1 and B2.4. We tested the interferometer stability in two cases. The first test is when only the box thermoregulation is switched on; the other one is when the box and cell are thermostabilized at the mean temperature. All the tests last at least 24 hours. The variations of phase differences with time between the opposite extremes of the field of view (the top and bottom of the cell, for example) were analyzed. The typical value of phase variation for the first test of the interferometer is ±( ) rad. This value agrees well with the common accuracy of interferometry (2π/50), which can be found in literature [114]. Phase stability in the second test is ±( ) rad. This deterioration of phase stability is not related to the interferometer itself, but to the stability of cell thermal regulation, which was also confirmed by sensors readings (e.g., see Fig 3.4(a)). The unique feature of this method is that it traces the transient path of the system in the entire cell. In this way it fits not only for the measurements of Soret coefficients but also for studying diffusive transport mechanism. B3.4. Beam deflection problem. It is well known that a light beam refracts when passing through a medium with a refractive index gradient. Both a constant temperature gradient applied over the cell and a gradually increasing concentration gradient definitely cause variations of the refractive index, which, in turn, causes deflection of the object beam. This principle is used for measurements of transport coefficients in beam deflection technique, for detailed description one can refer to [88], [95], [116]. Figure B3.5. Beam deflection problem. 93

100 In our technique it may play a negative role as the reference beam, which bypasses the cell, does not suffer from beam deflection. It means that the object beam may interfere with various regions of the reference beam at different time instants. For an ideally collimated beam with a perfect plane wave-front this point is not important. However, as the wave-front can be slightly disturbed in reality (e.g., by imperfect optical elements), it is better to exclude the problem ab origin. There is a natural way to make a correction for the beam deflection. Namely, an objective lens can return the beam refracted by an object into its initial position if it is properly focused (see Fig. B3.5.). The problem is reduced to seeking the ideal working focal plane of the objective lens. To sum up, a simple practical rule can be applied for minimizing beam deflection effect on interferometry: the imaging system has to be focused at the inner plane of the glass wall nearest to the camera, as it is drawn in Fig. B3.5. This agrees with the conclusions of [117]. 94

101 B4. Data extraction B4.1. Governing equations. Molecular and thermodiffusion processes are quantified by mass and thermodiffusion coefficients, D and D T, respectively. Due to mass conservation, there is only one independent component of a binary mixture. Let us denote the mass fraction of this independent component (solute) as C. Then the mass fraction of the dependent component (solvent) is (1 C). When pressure diffusion is negligible, the diffusive flux of the independent component is driven by concentration and temperature gradients, J = ρ D + D T, (B4.1) where ρ is the mixture mass density and T is the temperature. The transport coefficients are assumed to be constant in a sufficiently small range of temperature and concentration variations and correspond to the mean temperature T 0 and concentration C 0. Since the diffusive flux must vanish in the two dilute limits C = 0 and C = 1, the thermodiffusion coefficients can be represented as [118]. D T = C (1 C) D T (B4.2) Note that D T assumes finite values in the two dilute limits but may still depend on concentration. In the linear limit of small concentration changes the approximation C (1 C) C 0 (1 C 0 ) is used and the diffusive flux can be written as J = ρρ[ + C 0 (1 C 0 ) S T T]. (B4.3) The Soret coefficient S T = D T /D is determined as the ratio of the thermodiffusion and the mass diffusion coefficient. The signs of the Soret coefficient S T and of the thermodiffusion coefficient D T depend on the selection of the independent component. For its choice there exist two approaches in literature: one is based on the molecular mass and the other one on the density of the components. In molecular dynamics simulations the use of molecular mass is appropriate and, as a rule, the component with a higher molecular mass is chosen as the independent one. Hydrodynamic effects become important when liquid mixtures are studied on a macroscopic scale. Correspondingly, it is appropriate to choose the denser component as the independent one, and a positive Soret coefficient implies that this component migrates towards the cold side. In case of a negative Soret coefficient, even heating from above may lead to convective instabilities [119], [120] (unstable density 95

102 stratification). As an example, we have chosen water (mass fraction C) as the independent component for the analysis of water isopropanol mixtures, following the latter convention. B4.2. Fitting equation for full path. The present technique gives a unique possibility of increasing measurement accuracy by providing information about concentration distribution along the whole thermodiffusion evolution towards its steady state. In fact, the method gives a twodimensional concentration field, although the distribution itself is almost one-dimensional. This extensive information can be used for evaluation of convection, but in the case when convection is really negligible, the 1-D approach can be used for mathematical description of measurements. Each time a measurement is taken, a full concentration distribution over the thermodiffusion path is determined. The solution of thermodiffusion equations for a convection-free case can be written as follows [103]: C(z, t) = C 0 + C 0 (1 C 0 ) S T ΔΔ 1 2 z H 4 π 2 1 n 2 n,ooo ccc nnn t eee n2 H τ r (B4.4) Here, τ r = H 2 /π 2 D is the relaxation time, D and S T are the diffusion and the Soret coefficients, H is the height of the cell. The sign between the two first terms in Eq. B4.4 depends on the choice of the component which we follow: lighter or heavier. The sign is positive (+) in the case of a heavier component (as it is written) and negative (-) in the case of a lighter component. In Eq. B4.4 there are two unknown parameters, S T and τ r. The simplest procedure to find them is to compare the concentration measured at point z i at time t j with its value from Eq. B4.4 using the initial guess for S T and τ r. Then fitting is done iteratively by using the Nelder-Mead algorithm (Matlab), which minimizes the misfit function k,m δδ(z, t) = C eeeee (z i, t j ) C theee (z i, t j ) 2 (B4.5) i,j by varying the fitting parameters S T and τ r. The number of spatial pixel points in the experimental dataset is around k = 340 and the number of acquired images is varied in limits 200 < m < 800. So, the fitting is done with the matrix of a size at least k m = Thanks to such a great amount of data this approach gives satisfactory results. However, there exists a less obvious fitting parameter which allows 96

103 essential improvement of the fitting results. This fitting variable, t 0, can be called initial time. The point is that the reference image for extracting concentration distribution is not necessarily located at the very beginning of the separation step (see Fig. B3.4(b)). Note that this image has to be taken only after the temperature profile is completely established, see Section B3.2. A reliable reference image can often be found 3 9 minutes after the visible separation starts, although in the above fitting the time of the reference image is t = 0 and C eeeee (z, 0) = C 0, which is not precise. This can be solved by introducing a third fitting variable t 0. Then the theoretical anticipation for experimental concentration profile is given by C theee (z, t) = C(z, t 0 + t) C(z, t 0 ) + C 0 (B4.6) where both terms C(z, t 0 + t) and C(z, t 0 ) are calculated according to Eq. B4.4. Such correction provides much better fit to the experimental data as it is shown in Fig. B4.1(a). Note that the snapshot of data is presented at the time t = 0.16τ r, which is far from a steady state. The open circles present experimental results; the dashed curve shows C theee (z, t) when two fitting parameters were used; the solid line shows C theee (z, t) when three fitting parameters were used. We should emphasize that the margins of the new fitting parameter t 0 are known from the experiment (see Fig. B3.4(b)) with a rather good accuracy. (a) (b) Figure B4.1. (a) Fitting optimization; Experimentally measured C(z) (open circles) and its fitting curves with two (dashed curve) and three (solid curve) fitting parameters for THN-C 12 H 26 at t = 0.16τ r, (b) Ten snapshots of temperature profile for THN-C 12 H 26 display small oscillations T(z) near horizontal walls due to temperature instability of copper plates. 97

104 Finally, we will discuss a source of errors, which is less evident and more difficult to treat. It arises from the fact that the temperature of copper plates is not strictly constant; it has some minor fluctuations as the regulation is active. In this case an experimentally observed profile, considered as a pure concentration one, has some input of thermal nature. This input appears due to optical coupling of both values (i.e., Δφ th = ccccc[1 + δ(t)]) and can be either negligibly small (mmm δ(t) 0) or noticeable depending on the ratio of respective contrast factors and temperature fluctuation value. This effect can be seen in Fig. B4.1(b) where ten snapshots of temperature profile are shown for THN-C 12 H 26. Temperature fluctuations can be seen by eye near the horizontal walls where their amplitude achieves 0.5% of signal in the worst case. Our dedicated study showed that the amplitude of temperature disturbances rapidly decays and does not penetrate deeply into the liquid bulk. However, in some cases (as shown in Fig. B4.1(b)) when other uncertainties are added to the signal (e.g., due to a low contrast factor) we have to consider near-wall regions as corrupted and crop them out of consideration. For a different reason a similar approach was applied in [103]. Comparison of S T obtained on the basis of such a transient approach and from a steady state justifies the procedure. 98

105 B5. Results B5.1. Water Isopropanol (IPA) The results obtained for water IPA binary system have systematically been measured and will be given. To cross-check the correctness of measurements, an agreement has been reached between three groups: a group from Université Libre de Bruxelles (ULB), led by V. Shevtsova, a group from University of Bayreuth (UB), led by W. Köhler, and a group from University of Mondragon (UM), led by M. Bou-Ali. They have measured the system in the whole concentration range and compared the obtained results. Our group (ULB) has used optical digital interferometry (ODI) technique which allows simultaneous measurement of Soret and diffusion coefficients. The group of UB has utilized optical beam deflection (OBD) technique, the detailed description is given in [88]. The group of UM has measured thermodiffusion and diffusion coefficients independently using thermogravitational column (TGC) technique and sliding symmetric tubes (SST) techniques, respectively. The details on the instruments can be found in [121], [122]. This section presents the results provided for the benchmark by ULB, and their analysis in comparison with the results of the other groups. It is concluded by an evaluation of accuracy and applicability of ODI technique. B Contrast factors Optical digital interferometry requires precise knowledge of the contrast factor ( / ) p,t, whose accuracy directly determines the accuracy of the Soret and the thermodiffusion coefficients. From the beginning these values were calculated on the basis of data on the refractive index of the mixture water IPA taken from [123] for a temperature of 293 K and a wavelength of 589 nm. These values have been cross-checked by measurements done at ULB, see [103], using a standard Abbe refractometer and found to be a good starting point for data extraction. They are plotted in Fig. B5.1(a). The drawback of these data is that they were taken at temperature and wavelength which were slightly different from the ones used in the experiment. New data have appeared in course of the benchmark on the water IPA system [124]. The concentration contrast factors were measured at the University of Bayreuth from 288 K to 308 K and in the full concentration range for a wavelength of λ = 633 nm. The concentration derivative was determined there by refractive index measurements at different concentrations and 99

106 differentiation of an approximating polynomial fitted to a small concentration range. The measurement error was claimed to be below 0.5%. These data are presented in Fig. B5.1(b), where the symbols in the figure indicate the measured points and the curves outline the trend. (a) (b) Figure B5.1. Contrast factors ( / ) p,t versus the mass fraction of water at different temperatures in mixture water/isopropanol. Values of Soret coefficients, extracted initially with the first set of contrast factors, were corrected then with the use of the latter values. B Summary on Soret, diffusion and thermodiffusion coefficients All three coefficients S T, D, and D T are directly obtained from the experiment by performing the data evaluation based on the two independent coefficients S T and D, as described in Section and calculated D T = D S T. The results for the three coefficients are summarized in Table B5.1. Near the sign change region the measurements of S T were performed while increasing the concentration by small steps. The first point at which weak instability was observed in the cell (i.e., negative Soret) corresponds to C = The extrapolation of the curve S T (C) to the region of a negative Soret coefficient within a small range of C allowed us to determine the concentration when S T = 0 and estimate a negative value of S T for C =

107 Table B5.1. Soret, thermodiffusion, and diffusion coefficients measured by ODI at T = 298 K in mixture water/isopropanol. Shaded are data which were not accounted in benchmark as less reliable. C D/10 10 S T /10 3 D T /10 13 water m 2 s 1 K 1 m 2 s 1 K Figure B5.2. Diffusion coefficients D for water/isopropanol. This work and literature data from [125] measured by Taylor dispersion. 101

108 Figure B5.3. Soret coefficients S T for water/isopropanol. Own measurements and literature data from [126] measured by flow cell technique at T = 297 K. Figure B5.4. Thermodiffusion coefficients D T for water// isopropanol. Own measurements. The results for D, S T, and D T from Table B5.1 are plotted in Figs. B5.2, B5.3, B5.4. Some less reliable values at low water concentrations listed in the first part of the table and excluded from the data evaluation in the benchmark are shown in grey. These problematic data will be discussed in more detail below. 102

109 In addition to our measured and tabulated data, Fig. B5.2 also contains literature data for the diffusion coefficient as measured by the Taylor dispersion technique [125] and provided for the benchmark by other techniques, TGC and OBD. Figure B5.3 shows additional literature results for the Soret coefficient measured about 35 years ago by a flow cell technique [126] using a very thin fluid layer with d 0.2 mm. Both these D and S T literature data show from a reasonable to good agreement with the presented results. In particular, at intermediate and high water concentrations the agreement is good. From the first inspection it is immediately evident that the overall concentration range can be divided into three regimes. The first one is the region with negative Soret coefficients at a high water content (C > 0.75), which can not be characterised by our method, and it is fully characterized only by flow cell and OBD techniques. For a broad range of intermediate concentrations (0.20 < C < 0.75) with positive Soret coefficients there are adequate and consistent data. Finally, there is a region of low water content (C < 0.2), where the optical techniques become less reliable, whereas the TGC/SST data still show a smooth behaviour. In the following sections we will discuss these three regions in more detail. B Region with negative Soret effect; water rich mixture, C > Measuring negative Soret coefficients in the gravity field may lead to different types of instabilities, depending on the kind of experimental technique. Particularly, in the ODI instrument the separation starts to develop with a negative Soret effect until a sufficient amount of a denser component is accumulated at the top of the cell [120]. The density stratification becomes unstable and at some moment the flow is initiated in the form of large plumes, which convectively transport the heavier liquid downward. Due to the large differences between the viscous, thermal, and diffusion time scales, the system has a tendency towards these fingering buoyant instabilities. The advantage of the ODI technique is that it traces the transient path of the system in the entire two-dimensional cross-section of a cell. Thus, it allows studying both diffusive transport mechanisms and convection. Figure B5.5 shows a typical example for the development of convection in water IPA mixture containing 80% of water. 103

110 Figure B5.5. Concentration field experimentally observed by ODI during Soret separation in mixture water/isopropanol with concentration of water of C 0 = (a) 15 minutes after applying ΔΔ and beginning of separation; (b) a few hours later. Approaching the point of sign change, S T 0, the measurements have been conducted with small increments in the mass fraction. Since the ODI technique allows a reliable detection of instability, the concentration at which S T changes sign has been established rather precisely, S T = 0 at C = This value is in a good agreement with the one determined in the benchmark by the OBD technique [124]. Both Soret and diffusion coefficients measured in microgravity (IVIDIL) at C 0 = 0.9 are in a satisfactory agreement with the ones measured by OBD and the S T value coincides with that measured by a flow cell technique [126]. B Intermediate concentration regime, 0. 2 < C < This is the region where our data for all the quantities S T, D, and D T are perfectly consistent with the data from the other techniques. The Soret coefficients display a parabolic concentration dependence with a maximum at C 0.5. The thermodiffusion coefficient D T in this region depends only weakly on concentration, varying from to m 2 /(s K). The microgravity data at C = 0.5 are in an excellent agreement with all ground based measurements. Also, most literature data for D, measured 104