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Confocal microscopy of fluid argon under pressure

ProQuest Dissertations and Theses, 2009
Dissertation
Author: Gabriel Joseph Hanna
Abstract:
Confocal microscopy is a technique used in mainly in the life sciences for producing three-dimensional images of cellular structures. We have adapted the technique to measure volumes and refractive indices of fluids in a diamond anvil cell. While high-precision techniques, such as X-ray diffraction and neutron scattering, exist for measuring lattice volumes of solids, the measurement of fluid volumes is much more difficult. This new technique will allow for quick, inexpensive, and non-destructive measurements of the equation of state of fluids at high pressure. In addition, we have explored the use of carbon dioxide as a probe of the structure of fluid argon and nitrogen, using Fourier transform infrared spectroscopy. New experimental results presented here include: procedures for measuring volume and refractive index with the confocal microscope; the equation of state and refractive index of argon and water along the 300 K isotherm up to about 5 GPa; the dependence on density and pressure of the asymmetric stretching mode of carbon dioxide dissolved in argon and nitrogen at 300 K; and the IR absorption frequency of Ge:O as a function of pressure at 10 K.

Contents List of Figures x List of Tables xiii 1 Introduction 1 1.1 History of argon................................1 1.1.1 Discovery...............................1 1.1.2 Argon and the periodic table.....................2 1.2 Characteristics................................6 1.2.1 Physical and chemical properties..................6 1.2.2 In theory and experiment......................7 1.3 Applications..................................9 1.4 Subject of and motivation for this work...................10 1.5 Bibliography..................................11 2 Interatomic interactions 17 2.1 The argon atom................................18 2.2 Two-body interactions............................23 2.2.1 Van der Waals attraction.......................23 2.2.2 Exchange repulsion..........................26 vi

2.3 Many-body interactions...........................30 2.4 Bibliography..................................33 3 Fluid argon 36 3.1 Classical or quantum mechanics?......................36 3.2 The partition function............................38 3.3 The virial equation of state.........................40 3.4 The pair distribution function........................44 3.5 The hard-sphere model............................45 3.6 Numerical and experimental results.....................46 3.7 Bibliography..................................48 4 Experimental techniques 53 4.1 Diamond anvil cells..............................53 4.1.1 Alignment...............................54 4.1.2 Gasket preparation..........................57 4.1.3 Loading at ambient temperature...................59 4.1.4 Cryogenic loading...........................59 4.1.5 Pressure measurement........................61 4.1.6 Temperature control.........................63 4.2 Confocal microscopy.............................65 4.2.1 Reflected intensity profile......................67 4.2.2 Gaussian beam model........................69 4.2.3 Matrix optics.............................73 4.2.4 Thin lens equations and reflection..................78 4.2.5 Reflected intensity as a function of focus position.........80 4.2.6 Measuring volume:experimental procedure.............84 vii

4.2.7 Optical thickness and area......................87 4.2.8 Refractive index and calibration...................88 4.2.9 Precision of measurements......................89 4.3 Fourier transform infrared spectroscopy...................90 4.3.1 The interferogram...........................90 4.3.2 Using the FTIR spectrometer....................94 4.4 Bibliography..................................96 5 Experimental results 99 5.1 Confocal data.................................99 5.1.1 Equation of state for argon at 300 K................99 5.1.2 Refractive index and Clausius-Mossotti relation..........105 5.1.3 Discussion - confocal data......................107 5.2 FTIR data...................................107 5.2.1 Discussion - FTIR data........................109 5.3 Bibliography..................................117 6 Conclusion 119 A Heitler-London derivation of exchange repulsion 122 A.1 Bibliography..................................129 B Reflected intensity profile:ray optics 130 C Solving the paraxial Helmholtz equation 134 C.1 Bibliography..................................137 D Derivation of the Clausius-Mossotti relation 138 D.1 Bibliography..................................139 viii

E Confocal data for water at 300 K 140 E.1 Bibliography..................................142 F X-ray diffraction experiments 144 F.1 Bibliography..................................144 G IR absorption of Ge:O 146 G.1 Bibliography..................................146 ix

List of Figures 1.1 Phase diagram for argon...........................7 1.2 Face-centered cubic structure........................8 2.1 Coordinate system for van der Waals potential calculation.........24 2.2 Exchange repulsion energy..........................29 2.3 Argon two-body potential..........................31 2.4 Axilrod-Teller potential............................33 3.1 Lennard-Jones potential and Mayer f-function...............40 3.2 Pair distributions for fluid and solid.....................44 3.3 Fluid argon equation of state........................47 4.1 Diamonds in a DAC.............................54 4.2 Interference fringes and alignment......................55 4.3 Allen screws and dial indicator........................57 4.4 Drilling gaskets................................58 4.5 Cryogenic loading...............................60 4.6 Schematic of confocal microscope......................65 4.7 Sections of confocal image of DAC.....................68 4.8 Reflected intensity profile...........................69 x

4.9 Reflections from interfaces..........................70 4.10 Gaussian beam................................72 4.11 Paraxial ray propagation...........................73 4.12 Paraxial ray refraction............................75 4.13 Gaussian beam reflection...........................78 4.14 Coordinate system for microscope......................80 4.15 Reflections from multiple interfaces.....................82 4.16 Optical thickness and secondary reflections.................83 4.17 Taylor expansion of reflected intensity profile................84 4.18 Zeiss LSM 510 Meta confocal microscope..................85 4.19 Measuring area................................86 4.20 Reflected intensity as a function of incident laser power..........87 4.21 Refractive index calibration.........................89 4.22 Schematic of interferometer.........................91 4.23 Simple interferogram.............................93 4.24 Bomem DA8 FTIR spectrometer.......................94 4.25 Fabry-Perot interference...........................95 5.1 Argon sample area as a function of pressure................100 5.2 Argon refractive index as a function of pressure..............101 5.3 Argon optical thickness and absolute volume as function of pressure...102 5.4 Measured equation of state for argon....................104 5.5 Clausius-Mossotti relation for argon.....................105 5.6 Carbon dioxide in nitrogen at ambient temperature............108 5.7 Absorption spectrum of DAC loaded with Ar and CO 2 ..........109 5.8 Gaussian calculations for carbon dioxide in argon.............110 5.9 Carbon dioxide in argon at ambient temperature..............111 xi

5.10 Widths of IR peaks of carbon dioxide as a function of argon and nitrogen densities.....................................112 5.11 Carbon dioxide in nitrogen (density)....................113 5.12 Frequency of carbon dioxide as a function of density assuming LJ inter- action for argon and oxygen.........................115 A.1 Coordinate system for calculating exchange energy.............123 A.2 Exchange repulsion energy..........................128 B.1 Coordinate system for reflected intensity profile...............131 B.2 Ray diagram of pinhole and pinhole image..................132 B.3 Ray optics vs Gaussian beam........................133 C.1 Gaussian beam................................136 E.1 Equation of state for water at 300 K.....................141 E.2 Index of refraction as a function of density for water at 300 K.......141 F.1 Solid equation of state from X-ray diffraction................145 G.1 IR absorption of Ge:O............................147 xii

List of Tables 1.1 Periodic table of 1870.............................3 2.1 Atomic units.................................18 2.2 Number of n-tuple interactions for system of N atoms...........32 3.1 “Quantumness” of noble gases........................37 4.1 Sample alignment and preindentation record................56 5.1 Fitting parameters for pressure as a function of refractive index......103 5.2 Polarizability of argon.............................106 5.3 Fit parameters for the model given in equation 5.14.............114 E.1 Linear fit of refractive index as a function of density for water at 300 K..142 xiii

Chapter 1 Introduction 1.1 History of argon 1.1.1 Discovery In 1894,Lord Rayleigh noticed that nitrogen produced from chemical reactions seemed to be less massive than nitrogen gas extracted from the atmosphere.The difference was small,less than one-half of one percent,but the precision of the experiments implied that the difference was significant,and unexplained [1]. Impurities in atmospheric nitrogen were suspected,but all known elements were eliminated by one test or another,and in 1895 Lord Rayleigh and William Ramsay [2] presented to the Royal Society their evidence of a new atmospheric component.They isolated it from nitrogen by several methods,and found that it would react with no known substance.They called it argon,which is Greek for “it does no work.” In recreating the experiments of Henry Cavendish in 1785,on what was then called “dephlogisticated air,” Rayleigh and Ramsay showed that argon had demonstrably been isolated even then.But chemistry was not far enough advanced in 1785 for Cavendish to realize what he was seeing. 1

The mass of argon was determined by Rayleigh and Ramsay to be,in modern units, about 40 u,and to compose about 1 % of the atmosphere.William Crookes [2,3] measured its visible emission spectrum and determined that it corresponded with no known element.Rayleigh and Ramsay also determined,from the velocity of sound in argon,the ratio of specific heats (C p /C v );they found this ratio to be about 5/3, indicating a monatomic gas [2]. At the request of Rayleigh and Ramsay,Olszewski [2,4] successfully liquefied and solidified argon.Among other things,he measured the critical point to be at (in modern units) 5 MPa,152 K;current accepted values are 4.86 MPa,150.7 K [5–9].Hartley [10] showed that the argon spectrum was present in the spectrum of air,and Newall [11] showed that the argon spectrumhad been measured previously in 1894,though of course not identified as argon at that time.Further,MacDonald and Kellas [12],at the request of Rayleigh and Ramsay,showed that argon,though found along with nitrogen in the atmosphere,was not present along with nitrogen in the tissues of plants and animals. This ruled out argon as possibly being a form of nitrogen. The evidence,then,pointed to a newly discovered element,abundant in the envi- ronment,which did not engage in any known chemical reaction. 1.1.2 Argon and the periodic table Argon presented a serious challenge to the periodic table [13–15]. The periodic table of 1895 was not,like that of today,based on quantum mechanics. It was a list of elements arranged by atomic mass.The chemical properties of elements were thought to be periodic with respect to atomic mass,though what atomic mass had to do with chemical properties was unknown.Furthermore,it was not even yet accepted by all chemists whether such things as atoms really existed,though almost all chemists accepted the atomic model as useful for understanding chemical reactions 2

Groups: I II III IV V VI VII VIII Periods: (transition) 0 H Series: Li Be B C N O F 1 1 Na Mg Al Si P S Cl 2 K Ca – Ti V Cr Mn Fe Co Ni Cu 2 3 Cu Zn – – As Se Br 4 Rb Sr Y?Zr Nb Mo – Ru Rh Pd Ag 3 5 Ag Cd In Sn Sb Te I 6 Cs Ba – Ce – – – – – – – 4 7 – – – – – – – 8 – – – – Ta W – Os Ir Pt Au 5 9 Au Hg Ti Pb Bi – – 10 – – – Th – U – – – – – Table 1.1:Mendeleev’s periodic table of 1870 [16].Atomic mass increases from left to right and from top to bottom.“–” indicates empty slots for elements yet to be discovered.“Y?” refers to the doubtful location of yttrium.Note that the last element in Group VIII is the first element of Group I in the next period. [13–15]. The most successful periodic table was Mendeleev’s,but his was only the latest in a long line of systems of classification of elements,and his table was published nearly contemporaneously with those of five other authors.While Mendeleev’s table had held up well over twenty years,the periodic system had no theoretical justification [13–17]. A periodic table of Mendeleev’s that may well be the one referred to by Rayleigh and Ramsay [2] is shown in Table 1.1. Mendeleev’s Group VIII contained the “transition elements;” these were the ele- ments of multiple valency that were thought to have chemical properties “intermedi- ate” with respect to Group VII and Group I.The last-listed element in Group VIII of a period is the first-listed element in Group I of the next period [16]. Mendeleev used his table to correct erroneously measured atomic masses of known elements,and to predict chemical properties and atomic masses of those yet to be discovered.In a fewcases atomic masses stubbornly refused to obey the periodic law;for 3

example,no matter how many times they were measured,the atomic mass of tellurium came out greater than that of iodine.But their respective chemical properties required that tellurium come before iodine in the table [13–15]. While there was room for more elements in the table–it was this very feature that made Mendeleev’s table so widely accepted among chemists–there was no place for argon,unless some major finding of chemistry or physics were wrong.The properties which made argon so difficult to fit into the periodic system were these:it reacted with no known chemical,it had an atomic mass of 40 u,and it was a monatomic gas. There was no reason to suppose that such a thing as an inactive element should exist.As suggested by Rayleigh and Ramsay [2],argon’s chemical inactivity might be only apparent,and the conditions required for it to react might yet be discovered.Until they were discovered it would be impossible to definitively assign argon its place in the table,or to definitively say that the table must be wrong. None of Mendeleev’s groups were characterized by chemical inactivity,but Rayleigh and Ramsay [2] suggested that argon might go in Group VIII,after chlorine,as a transition to potassium.Unfortunately the atomic mass of argon was the same as that of calcium.It seemed too ad hoc to just assume that argon could be that far out of order with respect to atomic mass,or that the periodic table would put argon between two elements with properties so dissimiliar to its own. Another possibility,suggested by Rayleigh and Ramsay [2],was that “argon” could be a mixture of two gases;“argon” of atomic mass 37 u,and “krypton” of atomic mass 82 u.If the mixture consisted of 93 % argon and 7 % krypton,this would explain the measured atomic mass of 40 u.Argon and krypton would both be inert,and argon could go in Group VIII between chlorine and postassium while krypton could go in Group VIII between bromine and rubidium.However,it seemed that such a relatively large fraction of krypton should have been detected when the argon-krypton mixture was 4

liquefied,and the diffusion experiments of Rayleigh and Ramsay should have isolated a measurable quantity of krypton. Other than mercury vapor,no monatomic gases were known in 1895.If argon were a diatomic molecule,then its atomic mass would be,at 20 u,just right to go in Group VIII as a transition from fluorine to sodium.Alternatively,argon might just be a compound of nitrogen;as Mendeleev suggested [13–15],it might be N 3 .In both cases the kinetic theory of gases must be wrong,and atoms must not really exist. Several other possibilities were suggested by other researchers,but all were unsatis- factory enough that none was accepted.It seemed,in 1895,that chemists must either abandon the periodic table,abandon the atomic theory and the kinetic theory of gases, or just ignore argon altogether [13–15].As Lord Rayleigh said, The facts were too much for us;and all that we can do now is to apologise for ourselves and for the gas.[13] All that could be definitively said about argon was that it was a newly discovered element which didn’t seem to make any sense in relation to the others.Furthermore,it was quite abundant and had been “under everyone’s nose” at least as far back as 1785. The work previously done on argon enabled Ramsay,in the next few years,to isolate helium,neon,krypton,and xenon (from Greek for “sun,” “new,” “secret,” and “strange,” respectively).These had the same chemical inactivity as argon,but also had atomic masses that made sense in the periodic table,and were eventually regarded as a group in their own right,either coming before Group I or after Group VII. The periodic table was vindicated–although it had not predicted a new class of elements characterized by chemical inactivity,it was able to accomodate them;and the new class of elements,in general,obeyed the periodic law.Argon was considered just one more exception to the general rule of increasing atomic mass [13–15]. 5

Within a few decades,the periodic table was put on a solid theoretical foundation. Largely due to the experimental work of Moseley on X-rays and K-shell electrons [15], the role of electrons in chemistry was worked out according to quantum mechanical theories.The elements of the periodic table were then arranged by atomic number, rather than atomic mass,which eliminated the exceptions.Argon at 18 fits perfectly between chlorine at 17 and potassium at 19. Lord Rayleigh and WilliamRamsay received the Nobel Prizes for Physics and Chem- istry (respectively) in 1904 for their discovery of argon and the other noble gases. 1.2 Characteristics 1.2.1 Physical and chemical properties The atomic number of argon is 18.It has three stable isotopes: 36 Ar, 38 Ar, 40 Ar.The first two are more abundant in the universe at large.However,on Earth 40 Ar,which is produced by the decay of 40 K,makes up 99.6 % of argon and is found in rocks,the atmosphere (1 % by volume),and dissolved in the oceans [5–9]. At ambient pressure,argon liquefies at 87.3 K and solidifies at 83.8 K [5–9].At ambient temperature argon solidifies at 1.35 GPa [20].Its triple point is at 83.81 K, 68.95 kPa.Its critical point is at 150.7 K,4.86 MPa [5–9].Figure 1.1 shows the phase diagram up to 10 GPa and 800 K. Solid argon is face-centered cubic (fcc).Figure 1.2 shows the structure.Around 30 GPa hexagonal close-packed (hcp) structures begin to form.The fcc and hcp phases are predicted to coexist up to around 300 GPa,at which point the solid should be hcp. Body-centered cubic phases and metallization are predicted at much higher pressures but have not so far been observed [21–24]. Our experimental work was done at ambient temperature.For the lowest pressures 6

Figure 1.1:Phase diagram for argon,showing the melting line to 800 K [18].Inset:the liquid-gas phase boundary [19],encountered at far lower pressures and temperatures than were achieved in this work. we can achieve in a diamond anvil cell (about 0.2 GPa) argon is a supercritical fluid. The highest pressures we achieved in this work were under 10 GPa. 1.2.2 In theory and experiment Only one chemical compound of argon is known,argon fluorohydride (HArF),discovered only in 2000 and stable only under unusual conditions [25].The heavier noble gases, xenon and radon,can be induced to form a few compounds,and helium and neon are not known to form any [26,27]. The noble gases are chemically inactive due to their filled valence shells;reminding us that Argan,in 1673,found that opiumputs people to sleep due to its dormitive virtue [28].A better explanation can be found in the spherical symmetry of their electronic wavefunctions [29–31],as will be detailed later (section 2.1). 7

Figure 1.2:Face-centered cubic structure (fcc).Left:fcc unit cell.The length of the side of the unit cell,L,and the interatomic distance a,are related by L = a √ 2.Right: atom-centered view of fcc structure.The central atom has 12 nearest neighbors,all at the same interatomic distance a.There are 3-fold as well as 4-fold symmetry axes. The interatomic forces between noble gas atoms are isotropic and very weak,and as gases they show nearly ideal behavior.Their deviations fromideality provide important clues to the details of their interatomic interactions [20,32,33]. Noble gas atoms can be treated to high accuracy as interacting through purely atomic potentials (without considering interactions between electrons and nuclei) [20, 32,33].The atoms repel at short distances and attract at longer ones,with a potential minimum between,as will be derived later (section 2.2) from the Schr¨odinger equation and the Pauli exclusion principle.This was first done in the 1920s and 1930s by Lennard- Jones [29,30],London [34],Slater [35],Pauling [36],and others and was an early success for quantum mechanics. 8

For substances interacting by hard-sphere two-body potentials,the equation of state can be worked out exactly and analytically [37].The computers of the 1940s and 1950s could carry out numerical calculations for more complicated potentials such as that of Lennard-Jones,but for many years calculations involving anisotropy or many-body forces were computationally prohibitive [38].In noble gases these effects are very small, so they were the first substances for which chemical and macroscopic propertes could be worked out from first principles. Within the noble gases argon occupies a “sweet spot”–massive enough to be treated classically or semiclassically to good accuracy,unlike heliumand neon;but small enough that its electrons hardly interact with other elements,unlike the electrons of xenon and radon [26,27].Argon has the added virtues of being cheap,abundant,and easy to handle [20]. In summary,argon is a model substance for fundamental atomic and condensed matter physics. 1.3 Applications Argon is interesting in its own right for what it can tell us about fundamental atomic physics,but it also has practical applications. Argon is by far the cheapest and most abundant of the noble gases [20].It is produced by fractional distillation of air,usually as a byproduct of oxygen production. This has the interesting consequence that when demand for steel drops,the price of argon goes up [39]. Due to its relative cheapness,commercial and industrial applications of argon are widespread.Argon is generally used in circumstances when an inert gas is needed and nitrogen will not work.Nitrogen may not be unreactive enough,or nitrogen may be 9

too thermally conductive. Some examples of commercial and industrial applications are:inert atmospheres for welding,glove boxes,electronics manufacture,crystal growth,incandescent and fluorescent lighting,and as a thermal insulator in double-pane windows [20,39–41]. Some examples of scientific applications of argon are:as a carrier gas for mass spectrometry,as a hydrostatic pressure mediumin diamond anvil cells,and as a method of dating rocks and fossils [20,41,42]. 1.4 Subject of and motivation for this work In this work we study the equation of state of fluid argon at 300 K in a diamond anvil cell,and we use the vibrational modes of dissolved carbon dioxide to probe the fluid argon structure. From the experimental data we collect,we hope to draw some conclusions about the interatomic interactions of argon,and the structure of the fluid state.This will be useful to theorists who wish to work fromfirst principles to understand macroscopic and chemical properties.It will also be useful to researchers in the static high pressure field who wish to use argon as a hydrostatic pressure medium and as a pressure standard. Our experimental techniques include:confocal microscopy,used to obtain equation of state data for fluids,which will be developed in this work;and Fourier transform infrared spectroscopy,used to measure the localized modes of carbon dioxide in argon. To our knowledge,this is the first time that confocal microscopy has been used to measure the volume of a fluid under pressure.Fluid volumes are difficult to measure. Argon is a model systemthat provided a good test of this novel experimental technique. There are other techniques of measuring fluid volumes.One is Fabry-Perot interfer- ence,which yields the product of refractive index and sample thickness.Consequently 10

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Chapter 2 Interatomic interactions It is extremely difficult to make argon react chemically with anything,and the phase diagram of argon (up to about 100 GPa) is very simple.What is it about argon (and the other noble gases) that is different from all other elements? There are many different types of chemical “bonds”.Ionic bonds form when an electron fromone atomis transferred to another,as in the case of table salt.In covalent bonds electrons are shared between atomic nuclei,as in the cases of hydrogen,oxygen, and nitrogen.There are bonds intermediate in some degree,as covalent and ionic bonding represent extremes on a continuum [1,2].In metallic bonds,some electrons in the metal are not localized to any one atom.There are hydrogen bonds,where a hydrogen atom,covalently bonded to another atom,is made sufficiently non-polar to strongly attract a third atom.This type of bond is present in water and many proteins. Van der Waals bonds are responsible for the solid and liquid phases of the noble gases. They are present to some degree between all atoms and molecules,though nearly always weaker than any other bond that may be present [1–3]. Words mean what we want them to mean–it is a question of which is to be master [4].In this work a “chemical bond” is when an electron from one atom is associated 17

Quantity Atomic unit Name SI equivalent Other equivalent Mass m e electron mass 9.109 38 ×10 −31 kg 1 1823 u Charge e c electron charge 1.602 18 ×10 −19 C — Angular momentum ¯h ¯h 1.054 57 ×10 −34 Js — Length a 0 Bohr radius

4π 0 ¯ h 2 m e e 2 c

0.052 917 7 nm 0.529 177 ˚ A Energy E h hartree

e 2 c 4π 0 a 0

4.359 74 ×10 −18 J 27.2114 eV Table 2.1:Atomic units and their SI equivalents,along with equivalents in other com- monly used units.Bohr radius and hartrees are calculated assuming infinite nuclear mass,which simplifies comparisons between atoms of differing masses. with the nucleus of another.While it is not unknown in the literature to refer to,for example,two helium atoms at low temperature as a “molecule” [5],in this work we do not.Of the various types of bonds only covalent,ionic,and metallic bonds are “chemical bonds” by this definition.The other kinds could be called “atomic bonds”, if it is necessary to name them. We choose this definition so that we can characterize the difference between noble gases and the other elements in terms of their electronic structures,which are deter- mined by the number of protons Z in their nuclei.Using this approach we can show why noble gases are so reluctant to form chemical bonds and what sort of interatomic interactions they do exhibit. In dealing with electrons and nuclei,it is most convenient to work in atomic units, which are defined in table 2.1 [6]. 2.1 The argon atom The chemical behavior of any atomis determined by the structure of its electrons,which in turn is determined by its atomic number [3]. 18

Full document contains 161 pages
Abstract: Confocal microscopy is a technique used in mainly in the life sciences for producing three-dimensional images of cellular structures. We have adapted the technique to measure volumes and refractive indices of fluids in a diamond anvil cell. While high-precision techniques, such as X-ray diffraction and neutron scattering, exist for measuring lattice volumes of solids, the measurement of fluid volumes is much more difficult. This new technique will allow for quick, inexpensive, and non-destructive measurements of the equation of state of fluids at high pressure. In addition, we have explored the use of carbon dioxide as a probe of the structure of fluid argon and nitrogen, using Fourier transform infrared spectroscopy. New experimental results presented here include: procedures for measuring volume and refractive index with the confocal microscope; the equation of state and refractive index of argon and water along the 300 K isotherm up to about 5 GPa; the dependence on density and pressure of the asymmetric stretching mode of carbon dioxide dissolved in argon and nitrogen at 300 K; and the IR absorption frequency of Ge:O as a function of pressure at 10 K.