Phase Transitions in Vycor Studied by Optical Techniques

Volltext

(1)Bachelor Thesis. Phase Transitions in Vycor Studied by Optical Techniques. author: Yannick Dupuis. Conducted at the Institut Néel, CNRS. Submitted to the Department of Physics of the University of Constance. first supervisor: Prof. Dr. Georg Maret (University of Constance) second supervisor: Prof. Dr. Laurent Saminadayar (Institut Néel). Konstanz, june 2014. Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-0-410650.

(2)

(3) This thesis is dedicated to the memory of my grandfather Alfred Nagel *14th of October 1920 - †3rd of June 2014. I. Yannick Dupuis.

(4) Yannick Dupuis. II.

(5) Summary Wetting fluids condense in porous media for pressures smaller than their saturated vapor pressure. Unlike in bulk conditions, the transition from gas to liquid is strongly hysteretic. Drying the material requires to reduce the pressure well below the condensation pressure. Understanding the origin of this hysteresis, and the underlying evaporation mechanisms, is an active field of research. In my Bachelor Thesis I examine this hysteretic cycle for n-hexane in the porous environment of Vycor. For that purpose, I built a new experimental set-up which allows simultaneously to obtain precise isotherms and to optically probe the sample. Specific attention was brought on the kink of the evaporation branch and small-angle light scattering was used to measure the fluid correlations at this point. The results show a recurring heterogeneity in the sample upon evaporation. The techniques which I introduced will continue to be used by the group and developed further.. Zusammenfassung Benetzende Flüssigkeiten kondensieren in einem porösen Medium unterhalb des Dampfdrucks. Anders als in einer freien Umgebung, beschreibt die Kondensation darin eine Hysterese. Um das Material zu trocknen, muss der Druck weit unter den Kondensationsdruck gesenkt werden. Den Ursprung dieser Hysterese und den ihr zugrunde liegende Verdampfungsmechanismus zu verstehen, ist ein aktives Forschungsgebiet. Meine Bachelorarbeit befasst sich mit diesem Hysteresezyklus für n-Hexane in dem porösen Material Vycor. Dafür habe ich einen experimentellen Aufbau erstellt, der es ermöglicht genaue Isothermen zu messen, sowie die Probe optisch zu untersuchen. Ein besonderes Augenmerk habe ich auf den Knick auf der Verdampfungskurve gerichtet. An dieser Stelle habe ich Kleinwinkelstreuung angewandt, um die Korrelationen in dem Fluid zu untersuchen. Das Ergebnis ist eine wiederkehrende Heterogenität in der Probe bei Verdampfung. Die eingeführten Messtechniken werden weiterhin von der Gruppe verwendet und weiter entwickelt werden.. III. Yannick Dupuis.

(6) Yannick Dupuis. IV.

(7) Acknowledgments I carried out my Bachelor Thesis at the Institut Néel (CNRS) in Grenoble, France. I am thankful to the laboratory for providing the equipment and funding the research, in particular for this Thesis. This Bachelor Thesis was part of my year abroad with the student exchange programm ERASMUS. I am therefore thankful to the European Union who gives young people the possibility to study abroad. I would like to thank Pierre-Étienne Wolf for providing me the opportunity to carry out my Bachelor Thesis in his team at the Institut Néel. He has worked with me and helped me in countless situations, constantly providing input and new ideas. I thank Panayotis Spathis who has been an excellent tutor, teaching me many things during my stay here. Thank you very much for your patience and the time you spent in discussions and problem solving. I would like to thank the entire team HELFA for the good working atmosphere and the continual exchange of ideas. During this Thesis I created a new experimental set-up and depended therefore very much on the help of the technicians at the institute. I would like to thank Emilio Barria, Pierre Chanthib, Gilles Pont, Anne Gerardin, Gregory Garde and Yannick Launay of the cryogenic department for their technical support. A special thanks goes to Jérôme Debray for polishing the Vycor samples. I am thankful to Georg Maret, my supervisor at my home university (University of Constance), who established the contact with Pierre-Etienne Wolf and the Institut Néel.. V.

(8) Yannick Dupuis. VI.

(9) Contents 1. 2. Theoretical Background 1.1 Phase Transition in a Porous Environment 1.1.1 Isotherm . . . . . . . . . . . . . 1.1.2 Equilibrium . . . . . . . . . . . . 1.1.3 Hysteresis in a Single Pore . . . . 1.1.4 Pore Blocking . . . . . . . . . . 1.2 Vycor . . . . . . . . . . . . . . . . . . . 1.3 Hexane, C6 H14 . . . . . . . . . . . . . . 1.3.1 Saturated Vapor Pressure . . . . . 1.3.2 Gas Density . . . . . . . . . . . . 1.3.3 Liquid Density . . . . . . . . . . Experiment 2.1 Experimental Set-up 1 . . . . . . . . 2.1.1 Vase . . . . . . . . . . . . . . 2.1.2 Trap . . . . . . . . . . . . . . 2.1.3 Tank . . . . . . . . . . . . . . 2.1.4 Tubing . . . . . . . . . . . . 2.1.5 Cell . . . . . . . . . . . . . . 2.1.6 Volumes . . . . . . . . . . . . 2.1.7 Hexane Transfer . . . . . . . 2.2 Characterization of the Vycor Sample 2.2.1 Porosity . . . . . . . . . . . 2.2.2 Number of Mole - Sample 2 . 2.3 Attempted Isotherms . . . . . . . . . 2.4 Experimental Set-up 2 . . . . . . . . 2.5 Final Isotherms . . . . . . . . . . . . 2.6 Experimental Apparatus . . . . . . . 2.6.1 Pressure Gauge . . . . . . . . 2.6.2 Temperature Sensor . . . . . 2.6.3 Room Thermalization . . . . 2.7 Relaxation Time . . . . . . . . . . . . 2.7.1 Relaxation through Capillary . 2.7.2 Relaxation in a Vycor Disc . . 2.7.3 Experimental Data . . . . . . 2.8 Conclusion . . . . . . . . . . . . . . VII. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. 3 4 4 6 7 8 10 11 11 13 13. . . . . . . . . . . . . . . . . . . . . . . .. 15 15 16 17 17 18 18 18 20 20 21 22 22 25 26 28 28 28 29 30 30 33 34 35.

(10) CONTENTS. 3. Optical Measurements 3.1 Optical set-ups . . . . . . . . . . . 3.1.1 Parallel Light Transmission 3.1.2 Small-Angle Scattering . . . 3.2 Results . . . . . . . . . . . . . . . . 3.2.1 Adsorption . . . . . . . . . 3.2.2 Desorption . . . . . . . . . 3.3 Conclusion . . . . . . . . . . . . .. Yannick Dupuis. VIII. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. 37 37 37 38 41 41 43 47.

(11) Motivation I am writing my Bachelor Thesis at the "Institut Néel" (France, Grenoble) which is part of the "Centre national de la recherche scientifique" (CNRS). My colleagues work in the field of low temperature physics and study, among other things, phenomena taking place when fluid condensates and evaporates in a porous environment. Currently they confine fluid in Vycor (porous silica, porosity 28%) or aerogels (silica strands, porosity 99.9%). Evaporation in porous material is a longstanding problem which continues to raise numerous questions on the precise mechanisms involved in this process. In a porous environment with interconnected pores, two theories are generally proposed: cavitation, a local process resulting from the nucleation of vapor bubbles, and a collective effect, the percolation of vapor starting from the surface of the sample. Our team has shown that with helium evaporation evolves from a collective process at low temperatures to a more local process as the closure temperature of the hysteresis loop is approached (published by Bonnet et al. in [3]). The first evidence for percolation has been inferred by the experiments of Page et al. [19], which demonstrates that the particular structural factor deduced from small-angle light scattering was linked to the fractal correlations of the empty pores when hexane was evaporated. This optical signature obtained by Page et al. in hexane could not be observed in the experiment by Bonnet et al. where helium was used as the fluid. Therefore, the objective of my Bachelor Thesis is to bridge the gap between the experiments at low temperature conducted by my colleagues and the experiments at ambient temperature. To this end, I examine the adsorption and desorption of n-hexane in Vycor with a similar experimental set-up as Page et al. in 1995. I will draw parallels between the results. Nobody in the team has ever worked with hexane before, therefore I cannot rely on their experience and I have to investigate and test how to manipulate hexane myself. This thesis will lay the groundwork for verifying the reproductibility of the published results in hexane and to improve and extend the measurement techniques used in helium. I will use the transmission of parallel light and the scattering of a laser light sheet to give an optical overview of the effects taking place in the Vycor sample. Furthermore I will introduce a scattering measurement at angles below 12◦ to the measuring methods of the team. This method is sensitive to spatial correlations of the fluid distribution. In the future, additionally acoustic measurement methods should be added to the hexane set-up, which can then also be used in the experiment with helium.. 1.

(12)

(13) Chapter 1 Theoretical Background Introduction The team I worked in at the Institut Néel has to date done research on helium in porous environments, hence the need for low temperatures. The key advantage of helium is its small scattering power, avoiding multiple scattering effects on the entire isotherm. An isotherm is a curve on which the temperature stays constant, while the pressure around the porous material varies. Depending on the effects taking place in the sample, the amount of the mass uptake differs. Our team has studied several of these effects, including percolation and cavitation, with high-resolution isotherms during the condensation and evaporation of helium in Vycor. The publication of Bonnet et al. [3] shows that with helium in Vycor there is a transition from the percolation to the cavitation regime when increasing the temperature (figure 1.1). This crossover from one regime to the other was deduced from light scattering, the behavior on premature desorption, and a theoretical model.. Figure 1.1: Measured isotherms of helium in Vycor Page et al. found evidence for a percolation effect by the use of small-angle scattering [19] at the kink of the isotherm (red square in figure 1.2-a). This is currently the only report for percolation in Vycor. This result was inferred from the dependence of the small angle light scattering : Figure 1.2-b displays the characteristic of the fractal distribution of vapor expected for a percolation process. When low angle scattering was attempted at low temperature with helium, it appeared that the scattering power of helium was too small and the majority of the signal came from the roughness of the sample’s surface which could be sufficiently polished. Therefore, our team would like to work with hexane to be able to observe the same phenomenon and 3.

(14) CHAPTER 1. THEORETICAL BACKGROUND. observe whether percolation evolves indeed to a more local process at higher temperature. For this reason my Bachelor Thesis deals with the condensation and evaporation of hexane in Vycor. a.). b.). Figure 1.2: a.) Isotherm from Page et al. [19] with hexane in Vycor; b.) Characteristic of fractal distribution for small angle scattering [19]. This chapter will discuss the origin of the hysteretic behavior of isotherms in a porous environment. Furthemore, I will introduce the properties of Vycor and n-hexane, the fluid which I used for my experiments.. 1.1. Phase Transition in a Porous Environment. In this experiment we study the condensation and evaporation of hexane in the porous environment of a Vycor sample.. 1.1.1. Isotherm. One experimental tool used to study the condensation and evaporation are isotherms such as the one presented in figure 1.2-a. This allows me to monitor the quantity of adsorbed liquid as a function of pressure. The quantity of fluid in the porous material can be measured gravimetrically or volumetrically. In the gravimetric measurement, the mass uptake of the sample is directly measured using for example a precision laboratory scale. In the volumetric measurement, a known volume is connected to the sample’s cell. The pressure in that volume is measured before and after it equilibrates with the sample and the condensed amount of fluid is inferred from the pressure difference. I will use the volumetric method in my experiment. This requires to infer the mole quantities from the experimentally measured pressure and temperature in a known volume. In the following, I discuss the validity of the perfect gas approximation compared to the Van der Waals equation. The perfect gas law states : PV = nRT Yannick Dupuis. 4.

(15) 1.1. PHASE TRANSITION IN A POROUS ENVIRONMENT. while the Van der Waals equation gives: RT a − 2 Vm − b Vm V with Vm = n P=. For hexane, the Van der Waals equation is considered to give a precise value, while the perfect gas law is an approximation. However, the use of the perfect gas law considerably facilitates the calculations of the matter transfer, which is why both equations are crucial. The maximal error in the data range of this experiment is 0.71% (at 200mbar and 292K), more values are visualized in figure 1.3 (range: 292K ≤ T ≤ 298K, 0mbar ≤ P ≤ 200mbar and V = 30cm3 ).. 0.6 0.4 D n in % 0.2. n. 0.0 20. 292 294. 15 296 T in K. 298. 10 P in mbar*10. 5 300. Figure 1.3: The difference of the van der Waals equation to the perfect gas law in %, calculation input for Mathematica in Appendix C, figure 3.13; van der Waal constants from [20], see table 3.2. nVycor nVycor,full 0.8. ò ò ò. 0.6. ò. 0.4. ò ò ò. 0.2 æ. æ æ. æ. ò. 50. ò. æ. òæ. ò. æ æ. ò. æ æ. æ æ æ æ æ æ. æ æ. æ. 100. 150. P in mbar. Figure 1.4: Isotherm 6 plotted with the perfect gas law (red) as well as with the Van der Waals equation (blue) in the same graph. The adsorption is marked with dots, while the desorption is shown by triangles 5. Yannick Dupuis.

(16) CHAPTER 1. THEORETICAL BACKGROUND. I conclude that the perfect gas law can be used as a good approximation, nevertheless I will calculate the isotherms with the Van der Waals equation. An isotherm in a porous environment has a characteristic form which can be seen in figure 1.1. It always describes a hysteresis between the adsorption and the desorption phase. Since hexane wets Vycor it decreases the saturated vapor pressure, thus the Vycor is filled at a lower pressure.. 1.1.2. Equilibrium. The mechanical and thermodynamic equilibrium must be considered in order to explain the effects in the porous material. Mechanical Equilibrium Consider the mechanical equilibrium for a drop. The equilibrium for a drop is reached at −Pint. dS dVint dVext − Pext +σ =0 dR dR dR. while dVint dVext =− dR dR Vint and Vext are the volumes inside and outside the drop, S is the surface of the drop and σ is the surface tension. The result of which is the Laplace’s equation Pint − Pext = σκ with the curvature κ =. 2 R. for a spherical and κ =. 1 R. (1.1). for a cylindrical interface.. Thermodynamic Equilibrium The shift in the saturated vapor pressure due to the interaction between the porous material and the gas (hexane wetting Vycor) can be described by an attractive potential U. This potential depends on the distance to the wall z (Van der Waals potential close to the wall: U(z) ∝ z13 ). The chemical potential in the pore µg (T, Pg ) is then described by µg (T, Pg ) = µg (T, P∞ g ) + U(z). (1.2). with the experimentally measured pressure P∞ g outside the pore far away from the wall. A thermodynamic equilibrium is reached if µg (T, Pg ) = µl (T, Pl ). (1.3). The approximation for an incompressible liquid and a perfect gas (integration of GibbsDuhem relation dµ0 |T = vdP|T ), give us Pl − Psat (1.4) µl (T, Pl ) = µl (T, Psat ) + %l ! Pg µg (T, Pg ) = µg (T, Psat ) + Rgc T ln (1.5) Psat while %l is the liquid density (in volume per mass or volumes per mole, depending on the choice for µ) and Rgc the gas constant. Yannick Dupuis. 6.

(17) 1.1. PHASE TRANSITION IN A POROUS ENVIRONMENT. 1.1.3. Hysteresis in a Single Pore. In this part we consider a single pore with two free interfaces on adsorption and desorption. The adsorption is the process when the pressure is increased, forcing the condensation of a film of liquid on the Vycor pore walls which grows thicker until the pore is filled as illustrated in figure 1.5. The curvature of the liquid in that pore is κ = 1r , while r = R − z is the distance between the center of the pore and the liquid layer (R is the radius of the pore and z the thickness of the layer).. Figure 1.5: Illustration of the adsorption process in one pore The condensation does not proceed at equilibrium but is unstable when r approaches zero because it would imply a negative compressibilty of the system. At a critical distance zc , an instability appears, which forces the pore to condensate entirely (Saam & Cole [21]). By starting at an equilibrium state (equation 1.3), inserting equations (1.1), (1.2), (1.4) and (1.5) and using that Psat − Pg σr for small r, we obtain the Kelvin equation !! P∞ 1 σ g = exp − + U(z) (1.6) Psat Rgc T %l (R − z) Equation 1.6 is correct for values upto the critical distance zc , above which a negative dP∞ g compressibility dz < 0 results (forbidden branch, figure 1.6-a). The result becomes non-physic, hence the pore must be full. a.) 0.8. b.) 140. 0.7. 120. zc. 0.6. density (g/L). Pg∞ / Psat. 0.5 0.4 0.3 0.2. 100 80 60 40. 0.1. adsorption forbidden branch. 0.0 0. 5. 10. 15. 20. adsorption desorption forbidden branch. 20 25. 30. 0. 35. 0.0. z (Å). 0.2. 0.4 Pg∞ / Psat. 0.6. 0.8. 1.0. Figure 1.6: Graphs from [4] at 3.16K with a pore radius of 35Å for helium a.) Graph of equation 1.6; for the critical distance zc the pore is filled up to 82%, b.) Isotherm with the effects on adsorption and desorption discussed in this chapter For the pressure Pg in the cell, equation 1.6 simplifies to: Pg − σ = e Rgc T %l r Psat. (1.7). During desorption the pressure is reduced, emptying the pore from both free interfaces at thermodynamic equilibrium (figure 1.7). The pore drains at equilibrium without an instability. 7. Yannick Dupuis.

(18) CHAPTER 1. THEORETICAL BACKGROUND. Figure 1.7: Illustration of the desorption process in one pore. The different shape of the meniscus results in the curvature κ = equation [9]. Pg − 2σ = e Rgc T %l r Psat. 2 r. and we obtain the Kelvin. (1.8). From the two Kelvin equations 1.7 and 1.8, we conclude that a higher pressure is needed on adsorption than on desorption (considering that the liquid layer is thin or non-existing at the beginning of adsorption), which explains the appearance of the hysteresis, as shown in figure 1.6-b. The pressure during this evaporation process is determined by the curvature of the liquid interface. While the interface is progressing in the pore, the curvature remains constant, hence the pressure does not vary while the pore is drained (figure 1.6-b). For a smaller pore radius, due to the Kelvin equation, PPsat decreases. For a pore size distribution (no uniform size of pores), the isotherm gets wider. The isotherm is very sensitive to the shape and distribution of the pores. Therefore slight differences in the pores of two samples can result in noticeable changes to the sorption isotherm. There are limits to this method because it assumes that the liquid is not compressible and neglects the interaction between several pores. However this collective effect plays a major role, as introduced and elaborated by Mason in [16].. 1.1.4. Pore Blocking. In the approach of an interconnected network of pores, the adsorption process stays unchanged, while on desorption pore blocking affects the process. As illustrated in figure 1.7, a filled pore drains at its equilibrium pressure if it has access to vapor. In an interconnected pore network, this requires one of the neighboring pores to be empty. Otherwise, the pore cannot be emptied because it needs at least one free interface. The other possibility is the nucleation of a vapor bubble inside this pore (this effect is considered later in this chapter). Let us consider a domain of interconnected pores with a distribution of different sizes. At a certain pressure some large pores (site A, figure 1.8) could already be empty if they had a free interface. However, the smaller bonds between the sites are not yet ready to be drained because they need a lower pressure, limiting the penetration of the interface in the porous material. This effect is called pore blocking. Yannick Dupuis. 8.

(19) 1.1. PHASE TRANSITION IN A POROUS ENVIRONMENT. Figure 1.8: The concept of interconnected pores by Mason [16]. When the pressure is lowered, the number of pores, that could be empty if they had a free interface, increases. When the evaporation pressure inside the porous material is higher than the external pressure, this can lead to a percolation effect (collective effect). This explains the sudden change from a plateau on the isotherm to a vertical fall (see figure 1.9-b).. a.). b.). Figure 1.9: Theoretical isotherm calculated by Ball & Evans [1] a.) For the independent pore model, b.) For the network model; Γ on the y-axis is an arbitrary unit directly proportional to the adsorbed liquid. The percolation is a correlation at long distance, forming structures much larger than the wavelength of visible light, the result of which is small-angle scattering. This scattering effect was detected by Page et al. and I will try to do so as well by measuring the smallangle scattering. Another effect considered in the approach of an interconnected network of pores is cavitation. Cavitation is the formation of vapor cavities in a liquid. In a porous environment this effect can occur during pore blocking if the pressure surrounding the porous sample is lowered enough as to make the thermal nucleation of a vapor bubble possible. This effect occurs, when the critical radius for a bubble formation is larger than the radius of the bonds connecting the pores. Otherwise these bonds drain before the cavitation pressure is reached, leading to the evaporation of the fluid inside the cavity. 9. Yannick Dupuis.

(20) CHAPTER 1. THEORETICAL BACKGROUND By using the grand potential Ω we can deduce the potential ∆G = 4πσr2 −. 4π 3 αr 3. with α = ∆µ · ∆% σ: surface tension of liquid ∆µ =. µg (T, P∞ g ). − µg (T, Psat ) = Rgc T ln. P∞ g. !. Psat ∆% = %l (T, Psat ) − %g (T, Psat ) ≈ %l (T, Psat ) This potential has a metastable state (maximum) at ∆Gmax =. 16π σ3 3 α2. with the critical radius rc = 2σ . α This energy level needs to be overcome with the thermal activation energy. With a thermal energy of 30kB T (resp. 70kB T ), one needs a relation of P = 0.25, (resp. 0.4) Psat For ambient temperature. As our experimental results will show: PPsat ≈ 0.5. Therefore, cavitation does not have to be considered for my experiment but only the percolation effect, which I will try to find evidence for by small-angle scattering.. 1.2. Vycor. For this experiment a disc of Vycor 7930 glass was used as porous material (diameter d = 15.8mm, height h = 1.45mm). The Vycor glass used to be produced by CORNING Incorporated, the samples used in this experiment have been sold to us by Charles J. Landry (landry@porousglass.com). In comparison to powders, which always scatter strongly, its optical behavior is suitable for my optical setup. During the condensation and evaporation process of hexane in Vycor, all states from transparent (low scattering) to opalescent (strong scattering) are reached. In a dry as well as in a filled state it is transparent while it turns opaque during the desorption process. It has a narrow poresize distribution (∼ 20%) and interconnected disordered cylindrical pores. The size of the pores are generally deducted from the characteristics of an isotherm. This approach falsely assumes the pores to be independent. This is not the case for isotherms with hexane, as well as with nitrogen or helium. However, Levitz [13] used TEM (as visualized in figure 1.10) to find a diameter size of 70 − 80Å for the pores in Vycor. A field of application for Vycor is as a gas trap or as a permeable membrane in electro-chemical electrodes. Yannick Dupuis. 10.

(21) 1.3. HEXANE, C6 H14. Figure 1.10: Digitized image by TEM of a Vycor pore network, black parts are void[12] All relevant Vycor data can be found in Appendix A, table 3.1.. 1.3. Hexane, C6H14. The fluid used for adsorption and desorption in the Vycor disc in this experiment is nhexane (or normal hexane, hereafter referred to as hexane). Hexane is an alkane of six carbon atoms, which form an unbranched isomer. The fact that it is a simple chain makes it easier to describe the behavior of the solvent. One key advantage of hexane is that hexane wets Vycor. We note that the hexane molecule is preferentially oriented with its long axis parallel to the surface of the pore walls [19]. I chose an arid version with ≤30ppm H2 O and a high purity ≥ 99% (Rotidry R Sept). The skeletal formula of the hydrocarbon hexane is shown in figure 1.11. The length of a hexane molecule is 10.3Å and its width 4.9Å [2].. Figure 1.11: Skeletal formula of n-hexane [28] All relevant hexane data can be found in Appendix B, table 3.2.. 1.3.1. Saturated Vapor Pressure. The saturated vapor pressure curve describes the coexistence of gas and liquid, which ends at the critical point. The liquid-gas transition is a first order phase transition which therefore exhibits a discontinuity in the first derivative of the free energy with respect to the density %. The difference in density from gas to the liquid state diminishes when getting closer to the critical point, at which there is no difference and the parameter to characterize the difference between the two phases (%liq − %gas ) is zero. At that point it is not possible to distinguish between gas and liquid. 11. Yannick Dupuis.

(22) CHAPTER 1. THEORETICAL BACKGROUND P. P critical point liquid. Pc. solid gas triple point. liquid. gas. T. ϱc. ϱ. Figure 1.12: General phase diagram In figure 1.12 one can see the gas-liquid coexistence curve, which ends at the critical point. In the T-P-diagram the two states coexist on a curve, the transition is continuous while in the %-P-diagram it is characterized by an inaccessible zone (gray area), forcing the system to transit from one density to another when passing through the area. To estimate the saturated vapor pressure of hexane, I will use the Lee-Kesler method [11]. This is a recommended method to calculate the curve of the saturated vapor pressure for hydrocarbons such as hexane, when not within immediate range of the critical point (values in table 3.2). Equation 1.9 given below can be used when the acentric factor ω, the critical pressure Pc and the critical temperature T c are known. 6.09648 − 1.28862 ln(T r ) + 0.169347 · T r + ln(Pr ) = 5.92714 − Tr ! 15.6875 ω · 15.2518 − − 13.4721 ln(T r ) + 0.43577 · T r (1.9) Tr P Pr = : reduced pressure Pc T T r = : reduced temperature Tc With the corresponding values for hexane (see table 3.2) we get the curve which is shown in figure 1.13.. Figure 1.13: Saturated vapor pressure curve with the Lee-Kesler-method; sector useful for this experiment on the right side The derivate of that curve is plotted in figure 1.14. We deduce that a one-degree temperature change causes a shift of pressure by several mbar. This will be relevant in the evaluation of this experiment and will be discussed in chapter 2.6.3. Yannick Dupuis. 12.

(23) 1.3. HEXANE, C6 H14. Figure 1.14: Derivative of the saturated vapor pressure curve; sector useful for this experiment on the right side. 1.3.2. Gas Density. The gas density of hexane %g can be calculated in my data range by the use of the perfect gas formula: %g =. Mmol P RT. which gives us kg P %g ≈ 0.35 3 · m 100mbar Hexane gas has a density of %gas = 0.66 · 10−3 cmg 3 at 187mbar and a liquid density of %liq = 0.6603 cmg 3 . Therefore hexane takes about 1000-times more space when evaporated to a pressure of 187mbar (as is the case in my experiment).. 1.3.3. Liquid Density. The liquid density of hexane %l can be calculated with the compressibility coefficient κ which is given in table 3.2. The definition of κ is κ=. 1 ∂% % ∂P. T. The density changes by ∂% g ≈ 1.1 · 10−7 3 ∂P cm mbar due to a pressure shift at around 1bar. In the pressure range of zero to 200mbar of our experiment, this shift is not of importance. A pressure variation of 50bar results in a density change of 0.8%. Hexane inside the pore is expected to be compressed due to the silica attraction. I have not found in the literature values for this interaction, but the resulting pressure on the pore axis cannot exceed Psat,300K ≈ 200mbar (otherwise hexane would condense around zero 13. Yannick Dupuis.

(24) CHAPTER 1. THEORETICAL BACKGROUND pressure). Using a z13 dependence (an upper bound for the cylinder geometry), 80% of the pore cross section will be below 103 · 200mbar = 20bar. 50bar is thus a reasonable upper bound to evaluate the compression effect. Due to the small shift in density, I consider the liquid incompressible.. Yannick Dupuis. 14.

(25) Chapter 2 Experiment n-Hexane is a flammable, harmful and possibly carcinogenic (in France considered highly carcinogenic) fluid which should therefore be handled with care. This is why I took special measures to make an appropriate set-up. Up to my work, the team and the lab had no previous experience with this fluid. This is why I had to solve several experimental issues. To this end, the first experimental set-up now, had to be improved later on in a second updated version. I will now describe these two set-ups while explaining the importance of relaxation effects in my experiments.. 2.1. Experimental Set-up 1. To start off, a basic experimental set-up was chosen with the objective of improving it as far as to obtain decent isotherms. In particular, the use of hexane in the experiment (detailed in chapter 2.3) made it necessary to change the set-up for safety and practical reasons. Figure 2.1 illustrates the gas-handling system which we used to control the amount of hexane inside the sample.. 1. gas. 2. 4. tank (gas). T2. metering valve. liq vase. T3. P3. 3. void. 5. 6. 7. P2. void. T1 8. Vycor sample. cell. trap. Figure 2.1: Initial set-up 15. P1.

(26) CHAPTER 2. EXPERIMENT. This set-up is divided into three distinct parts : a vase containing liquid hexane at equilibrium with its saturated vapor pressure (Psat ), a tank solely meant to contain vapor hexane and used to transfer fluid to the third part consisting of the cell with the sample. The fluid inside the vase, when opened to the tank, can expand to raise the gas pressure up to P ≤ Psat . To prevent the formation of any liquid drops of hexane, special care is taken not to reach Psat inside the tank. Valve 1 is then closed and the gas in the tank is allowed to equilibrate with the sample space by opening valves 5,7 and 8. Since hexane wets Vycor, a lower pressure than Psat is sufficient to fill the sample with liquid, as explained in chapter 1.1. According to Page et al. the Vycor sample fills at a pressure of around 0.6Psat . These volumes can be evacuated by an external scroll pump through valve 6 and a nitrogen trap. Initially I thought about using Polytetrafluoroethylene (PTFE) components in combination with glass since they resist hexane and there is no known desorption of hexane in these materials. After testing several tubes, valves and volumes it became clear that these parts would not be of sufficient quality to withstand the procedures of the experiment. I discovered that most of these elements leaked; they are therefore not appropriate for a precise physical experiment at low pressure and could not be used. As mentioned in the introduction, I wrote my thesis in a group which consists of researchers who regularly work with helium at low temperatures and who therefore need perfectly leak-free systems. I therefore already had access to high-quality, stainless steel components that could resist hexane and would not leak, as well as researchers with the corresponding know-how and experience in that area. All components used in this experiment have been tested with a helium mass spectrometer and no leak higher than 10−8 mbar·L s could be detected. They were checked once before being added to the set-up and several times afterwards to verify that there was still no leak. Despite the advantages of the metal components, for practical and safety reasons, I still wanted to use some glass components in my set-up to be able to see the hexane level in the parts where it is liquid. In the following I will explain the components one by one.. 2.1.1. Vase. To fill the system, the liquid hexane is transferred from a septum bottle with a syringe into the vase (the vase is shown in figure 2.2-a). The vase is used to inject hexane gas into the system by opening valve 1. By cooling the vase with liquid nitrogen, it is also possible to pump out efficiently hexane back into the vase. This is why quartz was used since it can withstand very high thermal shocks. The quartz tube which was cut and closed at one end with a blow-torch and inserted inside a sliding-seal built out from a "raccord trois pièces" (standard CNRS link) on the other end. The seal between the tube and the metallic part is provided by a viton O-ring: by squeezing the seal in the threaded fitting, it is nestled on the tube which seals the tube (as shown on the trap in figure 2.2-b). The sliding seal is a very convenient invention since it makes the transition between glass and metal possible. In this way, it is also possible to adjust the position of the tube. It only works at pressures below 1bar; higher pressures can eject the tube out of the construction. This limit is never exceeded in my experiment with hexane since the pressures vary between 0mbar and 200mbar at the most. Yannick Dupuis. 16.

(27) 2.1. EXPERIMENTAL SET-UP 1. Figure 2.2: a.) The vase with liquid hexane at the bottom; pyrex glass can be filled with liquid nitrogen b.) The trap: the link between glass and metal is opened. The hexane is degassed by successive freezing and melting while pumping (eight cycles) as described in [5] to purify it from the air. The content is pumped with a scroll pump and a turbomolecular pump through the valves 1, 5, 6 and the nitrogen trap. The trap greatly reduces the quantity of hexane passing through the scroll.. 2.1.2. Trap. To start off I used a normal cryogenic trap with activated charcoal. Unfortunately the frozen hexane stopped the gas flow in the trap. I had to think of a different way to trap the out-flowing hexane due to this recurring problem. Hexane condensed inside the charcoal appeared very difficult to evaporate. I then used a similar design for the trap as for the vase but with a much wider tube (diameter 5cm). While the vase has only one gas exit, the trap has two (figure 2.2-b). One comes from the system and has a long tube which reaches roughly the bottom of the trap and the other starts at the top of the trap and goes to the pump. A metal link is brazed on the trap which holds it on the three rods of a Dewar bottle. The vacuum flask is filled with liquid nitrogen to trap the hexane. This works because the melting point of hexane is significantly higher than the boiling point of nitrogen. Only if a large amount of oxygen becomes trapped, can the pressure in the trap eventually exceed the 1bar limit. Therefore no air is pumped through the trap. When the vase is decoupled (valve closed) from the system, the system can be vacuum pumped either through the trap or without it.. 2.1.3. Tank. To fill the tank with hexane gas, the necessary amount of purified hexane can escape from the vase to the tank (valve 5 closed). The tank is a closed cylinder made out of stainless steel holding about 0.5l. 17. Yannick Dupuis.

(28) CHAPTER 2. EXPERIMENT. 2.1.4. Tubing. The tubing between the different components is made out of stainless steel, copper or brass. Four different types of valves have been used and the higher-quality valves were installed at vital points in the system.. 2.1.5. Cell. The cell (figure 2.3) is filled with hexane gas by connecting the tank to the cell, directly or in steps as explained in chapter 2.1.7. The cell is made out of a copper cylinder which is closed by two sapphire optical windows (diameter 1 inch) using indium rings. The optical axis of the sapphire is perpendicular to the plane of the window (checked with two crossed linear polarizers). To connect the cell to the gas handling system and the pressures gauges, two thin capillaries (CuNi 0.6-0.8 mm) were soldered on the copper cell. The whole is held by a supporting copper construction which is mounted on an optical rod.. Figure 2.3: Pictures of the cell a.) Cell in copper construction back-illuminated; b.) View of the copper cell where the upper sapphire window has been removed. Both, the thin capillary and the sapphire windows have been chosen to enable the set-up to be used at ambient as well as low temperatures. The sapphire windows resist low temperatures and the thin capillaries reduce the dead volume (discussed in chapter 2.1.6). The reason for using thin capillaries is because the dead volume of the capillaries is particularly important when set up in a cryostat. Both ends would be extended to about 1.5m to reach from the coolest part in the cryostat, where the cell is located, to the outside. Since the temperature between these two areas cannot be measured exactly, the amount of fluid in the capillary can only be estimated.. 2.1.6. Volumes. Prior to the experiments, I had to calibrate all volumes of interest so to measure the mass uptake of the sample volumetrically. The volumes were therefore measured using successive gas expansions starting from a known volume with air. The result is presented in figure 2.4. Yannick Dupuis. 18.

(29) 2.1. EXPERIMENTAL SET-UP 1. V3=(107.6±0.2)cm3 3 1 V2=(7.2±0.2)cm. gas. 2. P3. 3. Vrés=(571.3±0.1)cm3. 4. tank (gaz). metering valve. liq vase. void. 5. 6 void. 7 V4=(15.2±0.3)cm3. P2. 8. P1 sample. V5=(13.4±0.1)cm3 cell Vcell=(15.4±0.4)cm3. trap. Figure 2.4: Volumes initial experimental set-up. Dead Volumes The volumes around the Vycor sample in the cell are dead volumes. At ambient temperature they have little importance but at low temperature they are filled with dense gas and they can amount to a large part of the total fluid in the cell. Fluctuation in temperature then has a substantial effect, which is a problem. To be able to use the same set-up in a cryostat, the existing dead volumes are chosen at spots where they can easily be removed. Volumetric Measurement As explained in chapter 1.1.1, I measure the amount of hexane in the sample volumetrically. I will elaborate this method hereafter. The objective is to fill and drain the Vycor sample in the cell. Therefore the volumes in front of the cell-volume have to be filled up to a higher/lower pressure than the pressure in the cell so as to transfer hexane to and from the cell. Then the valves between the volumes are opened to transfer the gas. To reach a certain pressure in the cell, a variety of combination of volumes can be used at any pressure. E.g. to lower the pressure in the cell considerably, one would need to connect a large volume at low pressure to the cell (for example V3 +V2 +V4 +V5 connected to Vcell ). The smallest step possible in this set-up (without using the metering valve) is to connect V2 to V4 + V5 + Vcell , whereas the initial pressure in V2 can still be varied. In the filling procedure the tank is initially filled with the vase up to a certain pressure. The fairly large tank serves to fill the volumes afterwards or to transfer hexane directly to the cell through the metering valve. For each step, the initial and final pressure in the volume which serves the hexane transfer must be known to deduce the amount of gas transferred in that step. By computing sequentially the number of moles in the cell and subtracting the content in the dead volumes around the sample, I can calculate the amount of fluid in the Vycor. The resulting pressure after relaxation around the sample is then noted. The content of hexane in the Vycor sample as a function of that pressure results in an isotherm. 19. Yannick Dupuis.

(30) CHAPTER 2. EXPERIMENT. 2.1.7. Hexane Transfer. To measure an isotherm, we measure the pressure at a constant temperature in the system while transferring matter. There are several ways to add and remove matter from the system. Temperature Controlled Box One method is to control the temperature of a box which is filled with gas and connected to the system. A change in temperature results in an expansion or contraction of the gas and changes the pressure in the system. The temperature can be controlled very precisely (±0.01K) and a computer program can be used to regulate the temperature. This is the method used by our team in the experiments with helium. Since the existing apparatus for this method works at the temperature of liquid nitrogen (made for experiments with helium) it could not be used for an experiment with hexane (the temperature of liquid nitrogen is below the triple point of hexane). I therefore needed to develop another method. Dosage In this scenario a reference volume is filled up to a known pressure and opened towards the cell volume. The pressure then stabilizes and the transferred matter can be calculated. Unfortunately it can take a very long time to get a stable value for every step. Also the sudden strong change of pressure creates perturbation. This is a method used in my experiment. Small Flow Rate Another way to transfer matter is by having a small flow. It creates less perturbation than with large doses and, if well calibrated, the system stays close to an equilibrium state. This method then gives many points on the isotherm, in contrast to the last method. We could not fill our cell using a constant flow since the corresponding apparatus did not guarantee a consistent flow. Nevertheless, this method is used in my experiment with a metering valve. Knowing the volumes and pressures on both sides of the valve serves to calculate the matter transfer.. 2.2. Characterization of the Vycor Sample. For this experiment it was possible to choose between Vycor discs with thicknesses of 2 or 4 mm. For the first measurements, a thick disc was used (these results are not published in this thesis), which was then replaced by a thin disc in order to reduce the number of moles necessary to fill the sample. All Vycor discs are delivered unpolished which makes them unsuitable for optical measurements. Therefore, they need to be reworked. Our samples were polished at the "pôle Cristaux Massifs" of the Institut Néel. A pressure of 300 cmg 2 is exerted on the sample during the lapping-process. The Vycor disc, which is tightened in a ring held by screws, is then polished in a three-step process with gradually finer abrasive (9µm, 6µm - 20min and 3µm - 30min) for several hours. In the end and after each step the sample is cleaned. Yannick Dupuis. 20.

(31) 2.2. CHARACTERIZATION OF THE VYCOR SAMPLE. Figure 2.5: Picture of Vycor sample 1 Two thin discs were used, which I call sample 1 and sample 2. Sample 1 was cleaned with ethanol and sample 2 with H2 O. The results presented in this thesis originate from sample 2.. 2.2.1. Porosity. The two samples were used to calculate the porosity. The manufacturer indicates it to be approximately 28%[10]. It is possible to check this data by weighing the Vycor-disc (apparent mass mapp ) and calculating its volume V geometrically. With the porosity φ (0 ≤ φ ≤ 1) the relation between the apparent density %app of the Vycor disc and its bulk density %bulk is mapp mbulk = (1 − φ) = (1 − φ) %bulk V V %app ⇔ (1 − φ) = %bulk %app =. For the first sample we get the apparent density %app by measuring the weight mapp = (0.525 ± 0.001)g and the volume V = (0.356 ± 0.003) cm3 of the Vycor disc (diameter d = (16.00 ± 0.02) mm, height h = (1.77 ± 0.01) mm): %app = (1.48 ± 0.01). g cm3. With the data from table 3.1 for %bulk we obtain the porosity φ = 32% ± 1% For sample 2 (m = (0.433 ± 0.0005)g, d = (15.8 ± 0.1)mm, h = (1.45 ± 0.01)mm) we get φ = 30% ± 1%. In the production of Vycor glass, the porosity is not controlled very precisely on the scale of the sample. It is more important to have a narrow pore size distribution. Therefore, it is not surprising that these results for the porosity do not quite correspond to the indication of φ = 28% in [10]. Consequently, it was useful to check the porosity of our sample before using it. If we consider that the Vycor samples were not clean, we would assume some of the pores to be filled with water vapor. Therefore, the actual porosity would be equal or higher to the one calculated in this example. This does not explain the discrepancy between the values and we must assume that the porosity of our samples are slightly higher than the 21. Yannick Dupuis.

(32) CHAPTER 2. EXPERIMENT. value given by the manufacturer. The value of porosity obtained for my samples corresponds to the larger porosity found in other publications: Page et al. φ = 30.9% [19], Grüner φ = 30%, φ = 31.5% [8].. 2.2.2. Number of Mole - Sample 2. From the porosity, the amount of fluid required to fill the sample can be estimated. We measure this quantity in moles. The void space Vvoid in the sample is Vvoid = V · φ for the sample 2 this gives us a void volume of Vvoid,2 = (0.086 ± 0.006) cm3 If the fluid is able to penetrate every last pore and use all the available space, it will use the entire void space. We then obtain the maximal number of mole nVycor,liq,max = Vvoid ·. %liq Mmol. assuming that that %liq = const (incompressibility) for pressures in the bulk and for pressures in the porous environment. For the adsorption of hexane (chapter 1.3) in sample 2, this gives us the following number of mole nVycor,2 = (6.6 ± 0.4) · 10−4 mol. 2.3. (2.1). Attempted Isotherms. My Bachelor Thesis was intended to cover the analysis of isotherms with hexane in Vycor not only with optical but also with acoustic measurement techniques. Furthermore we aimed at producing isotherms at different temperatures, and maybe comparing the results to other fluids. In the end it took much longer than expected to solve problems which occurred, especially with the use of hexane. Most of the time had to be spent searching for a way to make a decent isotherm, which was an essential objective of my thesis and the first step to be able to understand the effects taking place in the sample. Therefore the actual analysis section had to be shortened to cover only optical measurements. In the following section I will explain the challenges with hexane that I have come across during my Bachelor Thesis while measuring my first isotherms. I will point out the issues and the solutions I thought of to explain them. Figure 2.6 shows the first entire isotherm that I measured. This was the first hysteresis cycle I measured until the end and I was glad that it seems to close again at a lower nVycor pressure. nVycor,full corresponds to the condensed fraction, while nVycor,full refers to the result obtained in equation 2.1. Yannick Dupuis. 22.

(33) 2.3. ATTEMPTED ISOTHERMS nVycor nVycor,full 0.20. ò. æ. æææ. ò. 0.15 0.10. ò. 0.05. ò. ò. æ æ æ. æ æ. æ æ. ò ò. ò ò. ò ò. ò ò. ò. æ æ. æ. æ æ. 20. 40. 60. 80. 100. 120. 140. P in mbar. Figure 2.6: 3rd attempt to measure an isotherm: adsorption in blue circles, desorption in red triangles. The first obvious problem was that the fraction of liquid filled in the Vycor did not correspond to the total number of moles expected to be condensed in the sample but was smaller by a factor of 5. Several factors could have led to that result. I had a doubt about the precise volume calibration; this could have been partly a consequence of the dead volumes of the valves. Also, I filled and emptied the cell by dosage as explained in chapter 2.1.7. The considered expansion volume V2 = 7.2cm3 (volume which was connected to the cell) was rather small, and thus comparable to the volume which gets added and deducted (estimation: 1cm3 ) when some of the valves are opened and closed. These uncertainties could explain why the curve ends at a lower level because I measured the adsorbed fluid volumetrically. Since I did not have a pressure gauge in that volume, I could only rely on P3 in the larger volume V1 . I also had a doubt about the metering valve which tended to leak. To seal it again, it was sufficient to tighten a screw on the valve. Additionally, the curve only resembles a normal hysteresis cycle of a sorption isotherm. In comparison to [19], the kink is not very sharp and the shape of both curves is very linear. This could be explained by the same factors as set out above. Furthermore I suspected a fluctuation or non-linearity of the pressure gauge. Even though the sensors had been tested before with air, I was not sure whether they would resist hexane or whether hexane might influence their behavior. At the same time, I had not tested them for dependence on temperature.. While testing some ideas, I measured a quicker isotherm which is shown in 2.7. 23. Yannick Dupuis.

(34) CHAPTER 2. EXPERIMENT nVycor nVycor,full æ. æ. æ. æ. 0.1. ò. æ. 50. 100. 150. P in mbar. -0.1 ò. -0.2. -0.3. ò. Figure 2.7: 4th attempt to measure an isotherm: adsorption in blue circles, desorption in red triangles In this isotherm I used the volume V = V2 + V1 to transfer the hexane. This isotherm shows that a very different problem arises: it appears that I withdraw more matter from the cell than I inject. The cell was practically empty in the beginning and could not possibly be more empty at a pressure of 75mbar. Therefore there must have been a measuring problem. My first assumption was that I might have condensed some hexane in the volumes that I used to fill the cell. It is possible that hexane condenses on the metal if it is a few degrees colder than the rest of the system (297K) because I filled the volumes up to a pressure of ∼ 170mbar, which corresponds to Psat at about 294K. Another suspicion was that hexane got adsorbed in nano-cavities of membranes in the valves, which I suspected to be porous or in the rubber seals (made out of viton) even though the compatibility with hexane is supposed to be excellent. This could have been the case in the new volume. On adsorption we have a pressure drop in these volumes from a higher pressure, at which there might have been some liquid, to a lower pressure, where the liquid could have evaporated. This would explain a lower change in pressure, and I would have falsely concluded that less matter had been adsorbed by the Vycor. That would explain why I obtained a result indicating that the sample was only partly filled. At the same time, it explains the desorption part of the hysteresis. That curve even crosses the adsorption and goes into the negative. This is normal if one believes in the idea of liquid in the volume during the adsorption process. This liquid would not exist during desorption because the pressure is never very high in the considered volumes which are used for the matter transfer. Therefore, it would appear as though more hexane had been taken out of the sample. Nevertheless, I would expect the curve to drop from +16% to −84% rather than +16% to −30%, if the volumes were correctly calibrated. To confirm this theory, I tested the system with hexane in different situations, never exceeding 150mbar to be sure not to condensate any hexane in the bulk volumes. However, the problems persisted. I suspected the seals and membranes were the cause and decided to make a new volume to test my theory. That volume had only one seal made of metal and a perfectly airtight metering valve. No relaxation could be seen in that volume. The idea in the next step was to insert a rubber seal and a valve-membrane successively to see whether they would adsorb fluid or not. Yannick Dupuis. 24.

(35) 2.4. EXPERIMENTAL SET-UP 2. In the meantime I pursued another idea. Hexane is known to dissolve grease and I had most recently discovered a small layer of grease in V1 , which made me suspicious. In my set-up I used very little vacuum-grease on the rubber seals to seal the volumes correctly. In my opinion, this small amount of vacuum-grease could not have affected the measurements to that extent. I had not cleaned all components in the set-up before using them. Most parts were made for this set-up only and could hardly be contaminated with grease, although the exact origin of the tank was not clear. To see the effect grease has on hexane, I put just a fingertip of grease in different volumes, emptied them and filled them with hexane. The drop in pressure when the hexane dissolves in the grease and the increase of pressure when the hexane leaves again at low pressure is remarkable, as shown in figure 2.8. Considering this effect, I did not pursue the idea of porous membranes or seals.. Figure 2.8: The pressure drop due to hexane dissolving in a fingertip of vacuum-grease in two different volumes; the different colors have no deep meaning. Given this result, I believed it would be impossible to use this experimental set-up for further isotherms. Grease is needed on most of the rubber seals in order to make them completely air-tight; it is used in several valves in that set-up and I would have had to clean all components to ensure that they are grease-free. I therefore decided that I would make another set-up without any rubber seals between the tank and the cell and with high quality valves which are absolutely leak-free without any grease. With this new information I could create an experimental set-up which is perfectly suitable for this experiment.. 2.4. Experimental Set-up 2. This second experimental set-up was built with the experience gained from the problems in the previous set-up. The right valves and links were chosen as to have a leak-free system with absolutely no grease. I cleaned all components in three different ultrasonic baths, once with soap, once in clear water and finally in deionised water. The cell, the trap and the vase stayed the same, while all other components were exchanged. The size of the volumes was chosen in order to have a decent resolution but also to facilitate the execution of the experiment. The set-up with the corresponding volumes is shown in figure 2.9. 25. Yannick Dupuis.

(36) CHAPTER 2. EXPERIMENT Vcap=(29.0±0.1)cm3 Vcell=(3.9±0.5)cm3. gas. P2 T3. 1. liq vase. tank (gas). T1 Vycor sample. T2 2. metering valve. Vmicro=(2.5±0.5)cm3. Vtank=(605±1)cm. P1. cell. 3. void. trap. Figure 2.9: Set-up 2 The part on the left side of valve 1 is only used to fill or empty the tank, while everything on the right side of that valve is used to condensate and evaporate hexane in the cell. A new metering valve has been installed, which is leak-free and which can be used to slowly fill and empty the cell as explained in 2.1.7.. 2.5. Final Isotherms. With this new set-up I expected to be able to produce isotherms without the severe problems encountered with the previous set-up. The result of the first entire isotherm with the new set-up (isotherm 6) is presented in figure 2.10-a. As a comparison, the isotherm obtained by Page et al. is shown in image b.). a.). b.). Figure 2.10: a.) Isotherm 6 with the new experimental set-up, T = (296.9 ± 0.2)K; b.) Isotherm by Page et al. in [19] Isotherm 6 has been measured at a temperature around 296.9K, which corresponds to a saturated vapor pressure of Psat = 193mbar. The two isotherms in figure 2.10 are similar, the hysteresis loop is clearly visible with a sharp kink on desorption and the adsorption and desorption curve join for low (me: ∼ 0.5Psat , Page: ∼ 0.4Psat) and high pressures (me: ∼ 0.8Psat, Page:∼ 0.7Psat ). There appears to be a shift between the two isotherms, which can be explained by a slightly different pore-size distribution. For pressures around 0mbar and Psat , the isotherm of Page et al. reveals more details than isotherm 6. For pressures around Psat I did not want to approach the saturated vapor pressure, in order not to risk the condensation of liquid in the bulk volumes. Therefore I do not have many Yannick Dupuis. 26.

(37) 2.5. FINAL ISOTHERMS. measuring points on the highest level of the curve. I suspect the sample to be partially filled when starting the experiment. To entirely evaporate all the liquid in the pores and to evacuate the gas, one needs to heat and pump the sample. Before I start an isotherm measurement the sample has been pumped during several hours (or several days). However, in our set-up it is not easy to heat the sample at the same time because fragile equipment is all around the cell. If the cell was not empty at the beginning, this could also explain the fact that the sample seems only to be filled up to 84% for high pressures. The incomplete drainage of the sample can be seen when comparing the two adsorption curves. Isotherm 6 looks similar to an exponential growth curve whereas the isotherm of Page shows a strong ascent for very low pressures and a rather linear curve after that. At 0.3Psat the sample used by Page is filled up to 30% (estimation) while my sample appears to be 13% filled. The moles of fluid which were still in the sample at the start of the measurements are then missing in the calculation. They account for the missing 16% to fill the sample. The error bars are calculated with the propagation of uncertainty of the pressure gauges, the temperature and the volumes. This bar can be reduced by replacing the pressure gauges and thermometers by more precise sensors and by calibrating the volumes (in particular Vmicro ) more precisely. All in all, isotherm 6 is a satisfying curve which shows the essential effects and helps me to situate myself in the adsorption and desorption process. Another isotherm (isotherm 7) was then traced to demonstrate the reproductibility of the curve and to evaluate whether it is worth measuring the curve at a slower pace. There are two reasons why I wanted to try that. I wanted to see whether it makes a difference to change the pressure in the cell abruptly and to make large pressure-steps or a slow relaxation of gas through the metering valve while frequently stopping. I had also noticed that the relaxation after some gas expansions (dosage) can take several hours to reach a constant value (explanation 2.7). It is sometimes hard to estimate whether the relaxation process is over or still happening. I therefore lowered the speed significantly: The measurements for isotherm 6 took 12 hours, while isotherm 7 was measured over 8 days. The result is shown in figure 2.11. Both isotherms are calculated with the Van der Waals equation and the number of moles transferred to fill and drain the cell is corrected by the temperature fluctuation (see chapter 2.6.3). Isotherm 6 was measured at (296.9 ± 0.2)K and isotherm 7 at T = (297.5 ± 0.5)K.. Figure 2.11: Isotherm 6 (duration: 12h) and isotherm 7 (duration: 8 days) 27. Yannick Dupuis.

(38) CHAPTER 2. EXPERIMENT. We can see that the curves are similar. Three significant differences can be observed: On adsorption isotherm 7 is slightly higher than isotherm 6 until the sample is almost full. At this point the isotherms cross, which means that I did not wait long enough. In that situation I expand some gas into the cell and wait until the pressure drops to a constant value. As a matter of fact I only waited 12 minutes for these steps. We realize that the sample in isotherm 7 fills to a higher percentage than in isotherm 6. This could be due to the fact that I pumped longer on the sample to drain the fluid and I heated it slightly with a heat-gun. On the desorption branch, the kink of the pressure drop is at a higher pressure for isotherm 7. This is most likely the result of a higher temperature of the sample. At that time, the temperature of the cell was about 298K compared to ≤ 297K on isotherm 6. The shift by 1K results in a change of ∆Psat ≈ 9mbar and therefore correlates with the measured shift of the kink by ∆P = 5mbar (situated at ∼ 0.5Psat ). In conclusion the difference for a slow to a fast isotherm-measurement is hardly noticeable. As long as one waits long enough to see the pressure saturate, the measurement is proper.. 2.6. Experimental Apparatus. Some of the apparatus used in this experiment is presented hereafter.. 2.6.1. Pressure Gauge. Three pressure gauges are used in the set-ups: two 0-1bar gauges and one 0-6bar gauge, which was later exchanged with a 0-200mbar gauge to obtain a better resolution. All gauges have a 4-20mA exit and are read by a HP 34970A data aquisition unit. The information is then sent to the computer. The 0-1bar pressure gauges were first calibrated at 0bar and at the ambient pressure, then compared to each other. Since I still had doubts about their linearity, I compared all the pressure gauges to a very precise Digiquartz pressure instrumentation from Paroscientific,Inc. (0.01% accuracy) in a helium system. No major inaccuracy was noticed and it served to recalibrate them. The resolution of the 0-1bar gauge is δP1 = 0.7mbar (steps in pressure signal) while the error of the 0-200mbar gauge is δP2 = 0.3mbar deduced by the pressure shift in time.. 2.6.2. Temperature Sensor. For this experiment, three resistance thermometers were made with 100Ω resistances and four-terminal sensing (with the intent to use the sample setup at low temperatures with a different fluid in a cryostat). They were calibrated at the boiling point of nitrogen and the melting point of water. The thermometers are read by a HP 34970A data aquisition unit. The absolute error is δabs T = 0.53K (derived from the measuring error of the resistance) and the relative error is δrel T = 0.2%. These temperature sensors serve to control the stability of the temperature. Yannick Dupuis. 28.

(39) 2.6. EXPERIMENTAL APPARATUS. 2.6.3. Room Thermalization. To measure isotherms the temperature must be constant and also the volumetric measuring method requires a certain regulation of the temperature. My experiment was set up in a room with an industrial air-conditioning system. The computers were in the adjacent room. To limit temperature fluctuations the heating and cooling mode were enabled to alternate. This mode was appropriate because the experiment took place in the spring time, which means that the outside temperature fluctuated from day to day between 10◦ C and 30◦ C. The air-conditioning regulates around a temperature of 25.5◦ C. The cooling is activated when the temperature exceeds 26◦ C until it is cooled below 25.5◦ C and the heating of the air-conditioning turns on when the temperature falls under 25.3◦ C until it reaches 25.5◦ C. The temperature sensor of the air-conditioning is in the center of the room at a height of about 2.5m, causing the temperature of the experiment to be lower, around 24◦ C. However the air-conditioning does not control the temperature to a satisfying level. Long term temperature fluctuation of 1K and short term fluctuations of 0.3K are possible. In comparison, Page et al. immerses the cell as well as the tank in a temperature-controlled bath, the bath of the cell being stabilized to better than 0.01◦ C around 22.6◦ . a.). b.). Tair. T1. elapsed time in h. elapsed time in h. Figure 2.12: a.) Temperature measured close to the exit of the air-conditioning; b.) Temperature of the cell; the different colors have no deep meaning. Image a.) in figure 2.12 shows the temperature T air measured close to the ceiling not far from the exit of the air-conditioning. The temperature sensor controlling the cycles of the air-conditioning is at a similar position. T air is moving up and down, as the airconditioning heats and cools the room. The average temperature is not constant, at the time 105h for example, it has a drop in temperature. This means that the air-conditioning is not controlling the temperature of the room correctly. In image b.) we see the temperature of the cell T 1 (the temperatures measured on the other components of the system have a similar behavior and value as T 1 ). It fluctuates less over a short period of time because it is not close to the air-conditioning. Nevertheless there are considerable fluctuations. The shortest temperature variations correspond to moments, when I entered the room, opening or closing valves, or working on the optic. We can also see a period of 24h which corresponds to the day-night sequence. During night the temperature drops, while the sun heats during the day. We notice that there is a rise in temperature starting from 120h. This is due to the outside temperature, which rose up to 35◦ C from previously about 20◦ C during daytime. 29. Yannick Dupuis.

(40) CHAPTER 2. EXPERIMENT. Figure 2.13: Isotherm 7 with and without a correction of the transferred matter by the temperature fluctuation. This temperature fluctuation is not a problem for the volumetric calculation of transferred matter because it can be corrected by the temperature fluctuation (figure 2.13). For measuring an isotherm however, which requires a constant temperature, a shift of 1K is not reasonable. According to the derivative of the saturated vapor pressure curve (figure 1.14) this results in a change of Psat of more than 4%. A shift of temperature can prevent the experimenter from stabilizing and observing a certain condition of the sample. To face this fluctuation problem, we plan to introduce a cooling system for the cell in the future. A water circulation will then control the temperature of the copper construction which holds the cell. Because copper is a good thermal conductor, this may sufficiently reduce the temperature fluctuations and the set-up does not need to be considerably changes.. 2.7. Relaxation Time. When considering a physical system, the relaxation time is the amount of time it takes the system to return from a perturbed state to its state of thermodynamic equilibrium. In my experiment, the perturbed state is when two volumes with different pressures get connected. The equilibrium is reached when the pressure in both volumes is equal. I would like to calculate how much time I have to wait during my experiment until a stable point has been reached. When connecting a volume with a different pressure to the cell, two relaxation times are expected. One due to the thin capillaries leading towards the cell and another corresponding to the relaxation in the Vycor sample itself on adsorption and desorption.. 2.7.1. Relaxation through Capillary. The external factor is the relaxation time on account of the capillaries around the sample. We consider the case of two volumes V1 and V2 filled with gas only which are connected by fine tubes (capillaries) of a length L and a diameter d (figure 2.14). Yannick Dupuis. 30.

(41) 2.7. RELAXATION TIME capillary. volume 2 V2, P2. volume 1 V1, P1. Figure 2.14: Considered system In the initial state the volumes have different pressures P1,in , P2,in . The system is therefore perturbed. Let us consider a higher pressure in volume 1: P1,in = P2,in + ∆P with ∆P > 0. We know that the pressure will be equal in the entire system once the system has reached its final state, the thermodynamic equilibrium: P1,fin = P2,fin C Pfin , whereby P1,in > Pfin > P2,in . We can therefore express the initial pressures as P1,in = Pfin + ∆P1 P2,in = Pfin + ∆P2 with ∆P1 > 0, ∆P2 < 0. (2.2) (2.3). During this process, the amount of gas-substance in the system stays constant. By using the ideal gas law, neglecting the volume of the capillary, and assuming that T 1 ≈ T 2 we deduce that dntot dt V2 ⇔ Ṗ1 = −Ṗ2 V1 0=. ⇔ Pfin = P2,in + ∆P. V1 V1 + V2. and ∆P2 = −. V1 ∆P1 V2. (2.4). The Hagen-Poiseuille equation[27] dP Z = µQ̇V dx L 128L Z= πd4 µ: viscosity of the flowing substance x: distance along the tube. (2.5). describes the pressure drop of flowing fluid in capillaries. It is therefore suitable to describe our situation: a pressure drop in a gas moving through thin tubes [17]. We will use this equation to calculate the outflow of the gas from volume 1 to volume 2 through the capillary. 31. Yannick Dupuis.

(42) CHAPTER 2. EXPERIMENT. The conversion of the volumetric flow Q̇V into a change of moles ṅ can be done using the ideal gas law, which, by integrating, brings us to (with 2.5) P21 − P22 = ZµRT ṅ2 = −ZµṖ1 V1 2. (2.6). Using the equations 2.2, 2.3 and 2.4 we get P21 − P22 ≈ 2Pfin. V1 + V2 ∆P1 V2. and therefore with equation 2.6 Ṗ1 Pfin 1 1 =− + ∆P1 Zµ V1 V2. !. which results in the relaxation time Zµ 1 1 τ= + Pfin V1 V2. !−1 (2.7). This result shows us that the relaxation time will increase when we are considering a low resulting pressure Pfin . I tested the theoretical calculations regarding the relaxation time in capillaries. I chose helium gas to relax from volume V1 = (220 ± 1) cm3 to volume V2 = (550 ± 5) cm3 through capillary of the length L = (330 ± 5) cm and the diameter d = 0.1cm. Helium is known to react very little with other elements. We can therefore be sure that there are no side effects, especially since we are working in a very clean system, which has practically not been in contact with anything else than helium. The pressure curve of the gas relaxing in V2 is shown in figure 2.15.. Figure 2.15: Relaxation curve in helium, horizontal axis in time of day, vertical axis pressure in mbar With the resulting pressure of 4990Pa we obtain a relaxation time of τtheo = (85 ± 2) s Yannick Dupuis. 32.

(43) 2.7. RELAXATION TIME. In the following I will compare this calculated result to the experimental result. I expect the curve to behave as t P(t) = 49.9 − exp − τ Subtracting the base line and changing the vertical axis to a logarithmic axis gives us figure 2.16.. Figure 2.16: Semi-log plot The slope of this graph m = −43h−1 gives us τexp = 84s In conclusion, the theoretical and experimental values correspond, which leads me to believe that the approximations made in chapter 2.7.1 are correct for this problem. In this experiment only the capillary leading to the cell is rather thin (d = 0.6mm). With its length of L = 800mm, the viscosity of hexane (table 3.2), and considering the expansion from (V2 + V4 + V5 ) to Vcell , we obtain a relaxation time of τ(Pfin = 100mbar) ≈ 59s τ(Pfin = 40mbar) ≈ 147s This effect was therefore taken into consideration when working on the experiment, because the waiting period was always at least 10 minutes.. 2.7.2. Relaxation in a Vycor Disc. The internal factor is the relaxation time in the Vycor sample itself. The gas surrounding the Vycor sample takes some time to penetrate the Vycor until the system reaches the thermodynamic equilibrium. I would like to approximate this relaxation time in this part.. 33. Yannick Dupuis.

(44) CHAPTER 2. EXPERIMENT. Darcy’s Law [19] describes the flow of a fluid through a porous network: k ~u = − ∇P η ~u: flow velocity k: permeability of the Vycor sample η: viscocity of hexane, table 3.2 With the continuity equation for the flow of hexane, as detailed in [19], we calculate for a filled state (liquid): ∂P − DP ∇2 P = 0 ∂t k with the diffusion coefficient DP = η·κ (compressibility κ). According to the law of diffusion, with the values for κ and η from table 3.2, k from table 3.1, and the thickness of the sample (h = 1.45mm), we obtain a relaxation time of !2 2 h2 τ= = 10.5s π DP. In comparison with the result of the relaxation time due to the capillary (chapter 2.7.1), the relaxation time of the disc is short. Both calculated relaxation times have been taken into consideration during the experimental procedure because the waiting period after each pressure step has been of several minutes.. 2.7.3. Experimental Data. To understand which role the relaxation process plays during the experimental procedure, the pressure curve in the cell during isotherm 6 is shown in figure 2.17. Pcell. elapsed time in h. Figure 2.17: Pressure of the cell The part until h ≈ 19.5 is during adsorption and the rest is during desorption. The sample is filled by dosage in this example. When filling the sample, in each step the pressure in the cell is increased and then the cell is isolated. Now the hexane vapor penetrates the sample and condenses within the pores. As a result, the pressure drops progressively. In this phase, a relaxation curve can be seen Yannick Dupuis. 34.

(45) 2.8. CONCLUSION. on the pressure. Once the pressure is constant, another pressure-step is taken. On desorption the pressure is lowered step-by-step. After each pressure decrease, the sample releases fluid and a relaxation can be observed. As can be seen in figure 2.17, the relaxation times can be much longer than calculated in the previous part (several minutes) and are therefore not explained by the assumptions made in the two previous chapters. These long relaxation times have also been seen by Page et al. (time constant 370s). A more thorough calculation for the relaxation time in the Vycor disc might explain them. It would have to consider the redistribution of the liquid in the pores and the transport of gas which could form liquid.. 2.8. Conclusion. In this chapter, I described the experimental approach I followed to obtain isotherms with hexane in Vycor. I challenged the difficulties set by the use of hexane and managed to obtain reliable and reproducible isotherm curves. This was an essential objective of my Bachelor Thesis. It will be interesting to improve this set-up with a temperature-regulated stage in the future. First, this will allow to obtain a more accurate control on the temperature of the sample. Indeed, the room thermalization is not stable enough for this experiment. Second, this will enable isotherms to be taken for different temperatures and hence to study the transition from the percolation to the cavitation regime.. 35. Yannick Dupuis.

(46) CHAPTER 2. EXPERIMENT. Yannick Dupuis. 36.