Basic components of a monochromatic XPS system.

X-ray Photoelectron Spectroscopy (XPS) is a quantitative spectroscopic technique that measures the empirical formula, chemical state and electronic state of the elements that exist within a material. XPS spectra are obtained by irradiating a material with a beam of X-rays while simultaneously measuring the kinetic energy (KE) and number of electrons that escape from the top 1 to 10 nm of the material being analyzed. XPS requires ultra-high vacuum (UHV) conditions.

XPS is a surface chemical analysis technique that can be used to analyze the chemistry of the surface of a material in its "as received" state, or after some treatment such as: fracturing, cutting or scraping in air or UHV to expose the bulk chemistry, ion beam etching to clean off some of the surface contamination, exposure to heat to study the changes due to heating, exposure to reactive gases or solutions, exposure to ion beam implant, exposure to UV light, for example.

• XPS is also known as ESCA, an abbreviation for Electron Spectroscopy for Chemical Analysis.

• XPS detects all elements with an atomic number (Z) between those of lithium (Z=3) and lawrencium (Z=103). This limitation means that it cannot detect hydrogen (Z=1) or helium (Z=2).

• Detection limits for most of the elements are in the parts-per-thousand (PPTh) range.

• XPS is routinely used to analyze inorganic compounds, metal alloys, semiconductors, polymers, pure elements, catalysts, glasses, ceramics, paints, papers, inks, woods, plant parts, make-up, teeth, bones, human implants, bio-materials, viscous oils, glues, ion modified materials and many others.

Wide Scan Survey Spectrum for all elements.

High Resolution Spectrum for Si(2p) signal.

XPS is used to measure:

• elemental composition of the surface (1–10 nm usually)
• empirical formula of pure materials
• elements that contaminate a surface
• chemical or electronic state of each element in the surface
• uniformity of elemental composition across the top the surface (aka, line profiling or mapping)
• uniformity of elemental composition as a function of ion beam etching (aka, depth profiling)

XPS can be performed using either a commercially built XPS system, a privately built XPS system or a Synchrotron-based light source combined with a custom designed electron analyzer. Commercial XPS instruments in the year 2005 used either a highly focused 20 to 200 micrometer beam of monochromatic aluminum K-alpha X-rays or a broad 10-30 mm beam of non-monochromatic (achromatic or polychromatic) magnesium X-rays. A few, special design XPS instruments can analyze volatile liquids or gases, materials at low or high temperatures or materials at roughly 1 torr vacuum, but there are relatively few of these types of XPS systems.

Because the energy of a particular X-ray wavelength equals a known quantity, we can determine the electron binding energy (BE) of each of the emitted electrons by using an equation that is based on the work of Ernest Rutherford (1914):

E_binding = E_photon - E_kinetic - Φ

where E_binding is the energy of the electron emitted from one electron configuration within the atom, E_photon is the energy of the X-ray photons being used, E_kinetic is the kinetic energy of the emitted electron as measured by the instrument and Φ is the work function of the spectrometer.

History of XPS

In 1887, Heinrich Rudolf Hertz discovered the photoelectric effect. Twenty years later, in 1907, P.D. Innes experimented with a Rontgen tube, Helmholtz coils, a magnetic field hemisphere (electron energy analyzer) and photographic plates to record broad bands of emitted electrons as a function of velocity, in effect recording the first XPS spectrum. Other researchers, Moseley, Rawlinson and Robinson, independently performed various experiments trying to sort out the details in the broad bands. Due to the wars, research on XPS came to a halt. After WWII, Kai Siegbahn and his group in Sweden developed several significant improvements in the equipment and in 1954 recorded the first high energy resolution XPS spectrum of cleaved sodium chloride (NaCl) revealing the potential of XPS. A few years later in 1967, Siegbahn published a comprehensive study on XPS bringing instant recognition of the utility of XPS. In cooperation with Siegbahn, Hewlett-Packard in the USA produced the first commercial monochromatic XPS instrument in 1969. Siegbahn received the Nobel Prize in 1981 to acknowledge his extensive efforts to develop XPS into a useful analytical tool.

Physics of XPS

Rough diagram showing physics of photon induced electron emission (photoemission) in XPS.

A typical XPS spectrum is a plot of the number of electrons detected (Y-axis, abscissa) versus the binding energy of the electrons detected (X-axis, ordinate). Each element produces a characteristic set of XPS peaks at characteristic binding energy values that directly identify each element that exist in or on the surface of the material being analyzed. These characteristic peaks correspond to the electron configuration of the electrons within the atoms, e.g., 1s, 2s, 2p, 3s, etc. The number of detected electrons in each of the characteristic peaks is directly related to the amount of element within the area (volume) irradiated. To generate atomic percentage values, each raw XPS signal must be corrected by dividing its signal intensity (number of electrons detected) by a "relative sensitivity factor" (RSF) and normalized over all of the elements detected.

To count the number of electrons at each KE value, with the minimum of error, XPS must be performed under ultra-high vacuum (UHV) conditions because electron counting detectors in XPS instruments are typically one meter away from the material irradiated with X-rays.

It is important to note that XPS detects only those electrons that have actually escaped into the vacuum of the instrument. The photo-emitted electrons that have escaped into the vacuum of the instrument are those that originated from within the top 10 to 12 nm of the material. All of the deeper photo-emitted electrons, which were generated as the X-rays penetrated 1–5 micrometers of the material, are either recaptured or trapped in various excited states within the material. For most applications, it is, in effect, a non-destructive technique that measures the surface chemistry of any material.

Components of an XPS system

The main components of an XPS system include: a source of X-rays, an ultra-high vacuum (UHV) stainless steel chamber with UHV pumps, an electron collection lens, an electron energy analyzer, mu-metal magnetic field shielding, an electron detector system, a moderate vacuum sample introduction chamber, sample mounts, a sample stage and a set of stage manipulators.

Monochromatic aluminum K-alpha X-rays are normally produced by diffracting and focusing a beam of non-monochromatic X-rays off of a thin disc of natural, crystalline quartz with a <1010> lattice. The resulting wavelength is 8.3386 angstroms (0.83386 nm) which corresponds to a photon energy of 1486.7 eV. The energy width of the monochromated X-rays is 0.16 eV, but the common electron energy analyzer (spectrometer) produces an ultimate energy resolution on the order of 0.25 eV which, in effect, is the ultimate energy resolution of most commercial systems. When working under everyday conditions, the typical high energy resolution (FWHM) is usually 0.4-0.6 eV.

Non-monochromatic magnesium X-rays have a wavelength of 9.89 angstroms (0.989 nm) which corresponds to a photon energy of 1253 eV. The energy width of the non-monochromated X-ray is roughly 0.70 eV, which, in effect is the ultimate energy resolution of a system using non-monochromatic X-rays. Non-monochromatic X-ray sources do not difract out the other nearby X-ray energies and also allow the full range of high energy Bremsstrahlung X-rays (1–12 keV) to reach the surface. The typical ultimate high energy resolution (FWHM) for this source is 0.9–1.0 eV, which includes with the spectrometer-induced broadening, pass-energy settings and the peak-width of the non-monochromatic magnesium X-ray source.

Uses and capabilities

Fundamental XPS Data from pure elements, oxides and compounds.

XPS is routinely used to determine:

• what elements and the quantity of those elements that are present within ~10 nm of the sample surface
• what contamination, if any, exists in the surface or the bulk of the sample
• empirical formula of a material that is free of excessive surface contamination
• the chemical state identification of one or more of the elements in the sample
• the binding energy (BE) of one or more electronic states
• the thickness of one or more thin layers (1–8 nm) of different materials within the top 10 nm of the surface
• the density of electronic states

Capabilities of advanced systems

• measure uniformity of elemental composition across the top the surface (aka, line profiling or mapping)
• measure uniformity of elemental composition as a function of depth by ion beam etching (aka, depth profiling)
• measure uniformity of elemental composition as a function of depth by tilting the sample (aka, angle resolved XPS)

Industries that use XPS

Adhesion, Agriculture, Automotive, Battery, Beverage, Biotech, Canning, Catalyst, Ceramic, Chemical, Computer, Cosmetic, Electronics, Environmental, Fabrics, Food, Fuel cells, Geology, Glass, Laser, Lighting, Lubrication, Magnetic memory, Mineralogy, Mining, Nuclear, Packaging, Paper and wood, Plating, Polymer and plastic, Printing, Recording, Semiconductor, Steel, Textiles, Thin-film coating, Welding

Routine limits of XPS

Quantitative accuracy:

• Quantitative accuracy depends on several parameters such as: S/N, peak intensity, accuracy of relative sensitivity factors, correction for electron transmission function, surface volume homogenity, correction for energy dependency of electron mean free path, and degree of sample degradation due to analysis.
• Under optimum conditions, the quantitative accuracy of the atom % values calculated from the Major XPS Peaks is 90-95% of the atom % values of each major peak. If a high level quality control protocol is used, the accuracy can be further improved.
• Under routine work conditions, where the surface is a mixture of contamination and expected material, the accuracy ranges from 80-90% of the value reported in atom % values.
• The quantitative accuracy for the weaker XPS signals, that have peak intensities 10-20% of the strongest signal, are 60-80% of the true value.

Analysis times

• 1–10 minutes for a survey scan that measures the amount of all elements, 1–10 minutes for high energy resolution scans that reveal chemical state differences, 1–4 hours for a depth profile that measures 4–5 elements as a function of etched depth (usual final depth is 1,000 nm)

Detection limits

• 0.1–1.0 atom % (1 atom% = 1 part per thousand (PPTh) = 1,000 PPM). Ultimate detection limit for most elements is approximately 100 ppm, which requires 8–16 hours.)

Analysis area limits

• Analysis area depends on instrument design. The minimum analysis area ranges from 10 to 200 micrometres. Largest size for a monochromatic beam of X-rays is 1–5 mm. Non-monochromatic beams are 10–50 mm in diameter.

Sample size limits

• Older instruments accept samples: 1x1 to 3x3 cm. Very recent systems can accept full 300 mm wafers and samples that are 30x30 cm.

Degradation during analysis

• Depends on the sensitivity of the material to the wavelength of X-rays used, the total dose of the X-rays, the temperature of the surface and the level of the vacuum. Metals, alloys, ceramics and most glasses are not measurably degraded by either non-monochromatic or monochromatic X-rays. Some, but not all, polymers, catalysts, certain highly oxygenated compounds, various inorganic compounds and fine organics are degraded by either monochromatic or non-monochromatic X-ray sources.
• Non-monochromatic X-ray sources produce a significant amount of high energy Bremsstrahlung X-rays (1–15 keV of energy) which directly degrade the surface chemistry of various materials. Non-monochromatic X-ray sources also produce a significant amount of heat (100 to 200 °C) because the anode that produces the X-rays is typically only 1to 5 cm away from the sample. This level of heat, when combined with the Bremsstrahlung X-rays, acts synergistically to increase the amount and rate of degradation for certain materials. Monochromatic X-ray sources, because they are far away (50–100 cm) from the sample, do not produce any heat effects. Monochromatic X-ray sources are monochromatic because the quartz monochromator system diffracted the Bremsstrahlung X-rays out of the X-ray beam which means the sample only sees one X-ray energy, for example: 1.486 keV if aluminum K-alpha X-rays are used.
• Because the vacuum removes various gases (eg O₂, CO) and liquids (eg water, alcohol, solvents) that were initially trapped within or on the surface of the sample, the chemistry and morphology of the surface will continue to change until the surface achieves a steady state. This type of degradation is sometimes difficult to detect.

Materials routinely analyzed by XPS

Inorganic compounds, metal alloys, semiconductors, polymers, pure elements, catalysts, glasses, ceramics, paints, papers, inks, woods, plant parts, make-up, teeth, bones, human implants, bio-materials, viscous oils, glues, ion modified materials

Organic chemicals are not routinely analyzed by XPS because they are readily degraded by the either the energy of the X-rays or the heat from non-monochromatic X-ray sources.

Related methods

• UPS, Ultra-violet photoelectron spectroscopy (aka PES)
• PES, Photo-electron spectroscopy (aka UPS)
• ZEKE, Zero Electron Kinetic Energy spectroscopy
• AES, Auger electron spectroscopy (AES)

Literature references

• Handbooks of Monochromatic XPS Spectra, Volumes 1-5, B. Vincent Crist, published by XPS International, LLC, 2005, Mountain View, CA, USA
• Surface Analysis by Auger and X-ray Photoelectron Spectroscopy, ed. John T. Grant and David Briggs, published by IM Publications, 2003, West Chester, UK
• Practical Surface Analysis by Auger and X-ray Photoelectron Spectroscopy, ed. Martin P. Seah and David Briggs, published by Wiley & Sons, 1983, Chichester, UK ISBN 0-471-26279-X
• Practical Surface Analysis by Auger and X-ray Photoelectron Spectroscopy, 2nd edition, ed. Martin P. Seah and David Briggs, published by Wiley & Sons, 1992, Chichester, UK
• Surface Chemical Analysis -- Vocabulary, ISO 18115 : 2001, International Standards Organization, TC/201, Switzerland, [1]
• Handbook of X-ray Photoelectron Spectroscopy, C.D.Wagner, W.M.Riggs, L.E.Davis, J.F.Moulder, and G.E.Mullenberg, published by Perkin-Elmer Corp., 1979, Eden Prairie, Minn, USA
• Handbook of X-ray Photoelectron Spectroscopy, J.F.Moulder, W.F.Stickle, P.E.Sobol, and K.D.Bomben, published by Perkin-Elmer Corp., 1992, Eden Prairie, Minn, USA

See also

• Photoemission spectroscopy
• List of materials analysis methods

External links

• The Science of Spectroscopy - supported by NASA. Spectroscopy education wiki and films - introduction to light, its uses in NASA, space science, astronomy, medicine & health, environmental research, and consumer products.

Instrument suppliers

• JEOL, Japan [2]
• Omicron, Germany [3]
• Shimadzu-Kratos, USA [4]
• Specs, Germany [5]
• Thermo Electron, USA [6]
• Ulvac-Phi, Japan [7]

Reference data sources

• XPS International, USA [8]
• NIST, US Government, USA [9]
• LaSurface, France [10]

Service labs - companies

• Materials Analysis Services, USA [11]
• Evans Analytical Group, USA [12]
• Ulvac-PHI, USA [13]
• RBD Enterprises, USA [14]

Fick's laws of diffusion describe diffusion, and define the diffusion coefficient D.

History

Fick's laws of diffusion were derived by Adolf Fick in the year 1855.

Fick's first law

Fick's first law is used in steady state diffusion, i.e., when the concentration within the diffusion volume does not change with respect to time (J_in=J_out).

Where

• J is the diffusion flux in dimensions of [(amount of substance) length^-2 time^-1], [mol m^-2 s^-1]
• D is the diffusion coefficient or diffusivity in dimensions of [length² time^-1], [m² s^-1]
• ɸ is the concentration in dimensions of [(amount of substance) length^-3], [mol m^-3]
• x is the position [length], [m]

Fick's second law

Fick's second law is used in non-steady or continually changing state diffusion, i.e., when the concentration within the diffusion volume changes with respect to time.

Where

• ɸ is the concentration in dimensions of [(amount of substance) length^-3], [mol m^-3]
• t is time [s]
• D is the diffusion coefficient in dimensions of [length² time^-1], [m² s^-1]
• x is the position [length], [m]

It can be derived from the First Fick's law and the mass balance:

Assuming the diffusion coefficient D to be a constant we can exchange the orders of the differentiating and multiplying on the constant:

and, thus, receive the form of the Fick's equations as was stated above.

For the case of 3-dimensional diffusion the Second Fick's Law looks like:

,

where is the usual del operator.

Finally if the diffusion coefficient is not a constant, but depends upon the coordinate and/or concentration, the Second Fick's Law looks like:

Applicability

Equations based on Fick's law have been commonly used to model transport processes in foods, neurons, biopolymers, pharmaceuticals, porous soils, semiconductor doping process, etc. A large amount of experimental research in polymer science and food science has shown that a more general approach is required to describe transport of components in materials undergoing glass transition. In the vicinity of glass transition the flow behavior becomes "non-Fickian". See also non-diagonal coupled transport processes (Onsager relationship).

Temperature dependence of the diffusion coefficient

The diffusion coefficient at different temperatures is often found to be well predicted by

Where:

• D is the diffusion coefficient
• D₀ is the maximum diffusion coefficient (at infinite temperature)
• E_A is the activation energy for diffusion in dimensions of [energy (amount of substance)^-1]
• T is the temperature in units of [absolute temperature] (kelvins or degrees Rankine)
• R is the gas constant in dimensions of [energy temperature^-1 (amount of substance)^-1]

Typically, a compound's diffusion coefficient is 10,000x greater in air than in water. Carbon dioxide in air has a diffusion coefficient of 16 mm²/s, and in water, its coefficient is 0.0016 mm²/s [1].

Biological perspective

The first law gives rise to the formula

It states that the rate of diffusion of a gas across a membrane is

• Constant for a given gas at a given temperature by an experimentally determined factor, K
• Proportional to the surface area over which diffusion is taking place, A
• Proportional to the difference in partial pressures of the gas across the membrane, P₂ - P₁
• Inversely proportional to the distance over which diffusion must take place, or in other words the thickness of the membrane, D.

Fick's first law is also important in radiation transfer equations. However, in this context it becomes inaccurate when the diffusion constant is low and the radiation becomes limited by the speed of light rather than by the resistance of the material the radiation is flowing through. In this situation, one can use a flux limiter.

The exchange rate of a gas across a fluid membrane can be determined by using this law together with Graham's law.

Bernoulli's equation redirects here; see Bernoulli differential equation for an unrelated topic in ordinary differential equations.

Bernoulli's principle states that in fluid flow, an increase in velocity occurs simultaneously with decrease in pressure. This principle is a simplification of Bernoulli's equation which states that the sum of all forms of energy in a fluid flowing along an enclosed path (a streamline) is the same at any two points in that path. It is named after the Dutch/Swiss mathematician/scientist Daniel Bernoulli, though it was previously understood by Leonhard Euler and others. In a fluid flow with no viscosity, and therefore one in which a pressure difference is the only accelerating force, it is equivalent to Newton's laws of motion. It is important to note that the only cause of the change in fluid velocity is the difference in pressures either side of it. It is very common for the Bernoulli effect to be quoted as if it states that a change in velocity causes a change in pressure. The Bernoulli principle does not make this statement.

Examples used to demonstrate the effect

Lift

One common way of understanding how an airfoil develops lift relies upon the pressure differential above and below a wing. In this model the pressures can be calculated by finding the velocities around the wing and using Bernoulli's equation. However, this explanation often uses false information, such as the incorrect assumption that the two parcels of air which separate at the leading edge of a wing must meet again at the trailing edge, and the assumption that it is the difference in air speed that causes the changes in pressure.

Venturis

A common model to demonstrate the Bernoulli effect is a convergent, divergent nozzle also called a venturi. This is simply a large diameter tube feeding into a smaller diameter tube and then further feeding into another larger tube. Venturis are easier to understand when considering a gas rather than a liquid, but the functions for either are much the same. In order for any gas flow to occur it is essential that the exit pressure is lower than the entry pressure for this system. This pressure difference causes the fluid to accelerate from the intake larger tube into the smaller tube. The stored spring energy available to the fluid because of the pressure difference results in the fluid not only expanding as it goes from higher to lower pressure, but effectively overshooting in its expansion as a result of the mass of the gas particles and compressibility of the gas, springing apart beyond the point where all the forces would be balanced. Before the fluid can spring back, there is more fluid behind it, also at this lower pressure. This first sample of fluid then has no pressure difference either side of it to cause it to spring back. This part of the fluid then remains at a lower pressure until it merges with the slower fluid in the exit tube. The pressure in the exit tube will be higher than that in the smaller middle tube, and so the fluid moving from the smaller to larger tube is slowed down by this pressure difference.

Venturi effect and carburetors

Bernoulli's principle can be used to analyze the venturi effect that is used in carburetors and elsewhere. In a carburetor, air is passed through a Venturi tube to increase its speed and by the mechanisms explained above, decrease its pressure. The low pressure air is routed over a tube leading to a fuel bowl. The low pressure sucks the fuel into the airflow so that the combined fuel and air can be sent to the engine. The pressure reduction is proportional to the rate of air flow squared, so that more fuel is sucked in as the air flow increases, and the fuel:air mixture remains roughly the same proportion over a wide range of speeds. The pressure reduction effect can be observed by blowing over the top end of a straw with the bottom of the straw in a container of water; the water level will rise in the straw as the flow over the top of the straw increases in speed.

Bernoulli equations

There are typically two different formulations of the equations; one applies to incompressible flow and the other applies to compressible flow.

Incompressible flow

The original form, for incompressible flow in a uniform gravitational field (such as on Earth), is:

v = fluid velocity along the streamline

g = acceleration due to gravity on Earth

h = height from an arbitrary point in the direction of gravity

p = pressure along the streamline

ρ = fluid density

These assumptions must be met for the equation to apply:

• Inviscid flow − viscosity (internal friction) = 0
• Steady flow
• Incompressible flow − ρ = constant along a streamline. Density may vary from streamline to streamline, however.
• Generally, the equation applies along a streamline. For constant-density potential flow, it applies throughout the entire flow field.

The decrease in pressure simultaneous with an increase in velocity, as predicted by the equation, is often called Bernoulli's principle.

The equation is named for Daniel Bernoulli although it was first presented in the above form by Leonhard Euler.

Compressible flow

A second, more general form of Bernoulli's equation may be written for compressible fluids, in which case, following a streamline:

= gravitational potential energy per unit mass, in the case of a uniform gravitational field

= fluid enthalpy per unit mass, which is also often written as (which conflicts with the use of in this article for "height"). Note that where is the fluid thermodynamic energy per unit mass, also known as the specific internal energy or "sie".

The constant on the right hand side is often called the Bernoulli constant and denoted b. For steady inviscid adiabatic flow with no additional sources or sinks of energy, b is constant along any given streamline. More generally, when b may vary along streamlines, it still proves a useful parameter, related to the "head" of the fluid (see below).

When shock waves are present, in a reference frame moving with a shock, many of the parameters in the Bernoulli equation suffer abrupt changes in passing through the shock. The Bernoulli parameter itself, however, remains unaffected. An exception to this rule is radiative shocks, which violate the assumptions leading to the Bernoulli equation, namely the lack of additional sinks or sources of energy.

Derivation

Incompressible fluids

The Bernoulli equation for incompressible fluids can be derived by integrating the Euler equations, or applying the law of conservation of energy in two sections along a streamline, ignoring viscosity, compressibility, and thermal effects.

The simplest derivation is to first ignore gravity and consider constrictions and expansions in pipes that are otherwise straight, as seen in Venturi effect. Let the x axis be directed down the axis of the pipe.

The equation of motion for a parcel of fluid on the axis of the pipe is

In steady flow, v = v(x) so

With ρ constant, the equation of motion can be written as

or

where C is a constant, sometimes referred to as the Bernoulli constant. We deduce that where the speed is large, pressure is low. In the above derivation, no external work-energy principle is invoked. Rather, the work-energy principle is inherently derived by a simple manipulation of the momentum equation. The derivation that follows includes gravity and applies to a curved trajectory, but a work-energy principle must be assumed.

A streamtube of fluid moving to the right. Indicated are pressure, height, velocity, distance (s), and cross-sectional area.

Applying conservation of energy we find that:

the work done by the forces in the fluid + decrease in potential energy = increase in kinetic energy.

The work done by the forces is

The decrease of potential energy is

The increase in kinetic energy is

Putting these together,

or

After dividing by Δt, ρ and A₁v₁ (= rate of fluid flow = A₂v₂ as the fluid is incompressible):

or, as stated in the first paragraph:

Further division by g implies

A free falling mass from a height h (in vacuum), will reach a velocity

or .

The term is called the velocity head.

The hydrostatic pressure or static head is defined as

, or .

The term is also called the pressure head.

A way to see how this relates to conservation of energy directly is to multiply by density and by unit volume (which is allowed since both are constant) yielding:

and

Compressible fluids

The derivation for compressible fluids is similar. Again, the derivation depends upon (1) conservation of mass, and (2) conservation of energy. Conservation of mass implies that in the above figure, in the interval of time Δt, the amount of mass passing through the boundary defined by the area A₁ is equal to the amount of mass passing outwards through the boundary defined by the area A₂:

.

Conservation of energy is applied in a similar manner: It is assumed that the change in energy of the volume of the streamtube bounded by A₁ and A₂ is due entirely to energy entering or leaving through one or the other of these two boundaries. Clearly, in a more complicated situation such as a fluid flow coupled with radiation, such conditions are not met. Nevertheless, assuming this to be the case and assuming the flow is steady so that the net change in the energy is zero,

where ΔE₁ and ΔE₂ are the energy entering through A₁ and leaving through A₂, respectively.

The energy entering through A₁ is the sum of the kinetic energy entering, the energy entering in the form of potential gravitational energy of the fluid, the fluid thermodynamic energy entering, and the energy entering in the form of mechanical work:

A similar expression for ΔE₂ may easily be constructed. So now setting 0 = ΔE₁ - ΔE₂:

which can be rewritten as:

Now, using the previously-obtained result from conservation of mass, this may be simplified to obtain

which is the Bernoulli equation for compressible flow.

External links

• Daniel Bernoulli and the making of the fluid equation the story of what happened.
• Testing Bernoulli: a simple experiment here is an experiment that you can easily do yourself to test Bernoulli's equation. There are also 2 questions and answers.
• Animated Demonstration of Bernoulli's Principle Adjustable animation of Bernoulli's principle with explanation and links.

Millau Viaduct

Official name	Le Viaduc de Millau
Carries	4 lanes of the A75 autoroute
Crosses	valley of the River Tarn
Locale	Millau, France
Design	Cable-Stayed
Longest span	342 metres (1,122 ft)
Total length	2,460 metres (8,071 ft)
Width	32 metres (105 ft)
Clearance below	270 metres (886 ft) at maximum
Opening date	December 14, 2004

The 'Millau Viaduct' (French: le Viaduc de Millau) is a cable-stayed road bridge that spans the valley of the River Tarn near Millau in southern France. Designed by French bridge engineer Michel Virlogeux in collaboration with British architect Norman Foster, it is the tallest vehicular bridge in the world, with one pier's summit at 343 metres (1,125 ft)—slightly higher than the Eiffel Tower and only 38 m (125 ft) shorter than the Empire State Building. It was formally opened on 14 December 2004 and opened to traffic on 16 December 2004.

Location

Millau Viaduct's coordinates are 44.077165° N 3.022887° E. Before the bridge was constructed, traffic had to descend into the Tarn River valley and pass along the route nationale N9 near the town of Millau, causing heavy congestion at the beginning and end of the July and August vacation season. The bridge now traverses the Tarn valley above its lowest point, linking the causse du Larzac to the causse rouge, and is inside the perimeter of the Grands Causses regional natural park.

The bridge forms the last link of the A75 (la Méridienne) autoroute, providing a continuous high-speed route south from Paris through Clermont-Ferrand to Béziers. The purpose of the A75 is to increase the speed and reduce the cost of vehicle traffic travelling along this route. Many tourists heading to southern France and Spain follow this route because it is direct and without tolls for the 340 km between Clermont-Ferrand to Béziers, except for the bridge itself.

The Eiffage group operates the viaduct as a toll bridge, with the toll currently set at €4.90 for light automobiles (€6.50 during the peak months of July and August). The bridge was constructed by the Eiffage group, which also built the Eiffel Tower, under a government contract which allows the company to collect tolls for up to 75 years.

Description

Panoramic view of Millau Viaduct from south-east side

The Millau Viaduct consists of an eight-span steel roadway supported by seven concrete piers. The roadway weighs 36,000 tonnes and is 2,460 m long, measuring 32 m wide by 4.2 m deep. The six central spans each measure 342 m with the two outer spans measuring 204 m. The roadway has a slope of 3% descending from south to north, and curves in plan section on a 20 km radius to give drivers better visibility. It carries two lanes of traffic in each direction.

The piers range in height from 77–246 m, and taper in their longitudinal section from 24.5 m at the base to 11 m at the deck. Each pier is composed of 16 framework sections, each section weighing 2,230 tonnes. These sections were assembled on site from pieces of 60 tonnes, 4 m wide and 17 m long, made in factories in Lauterbourg and Fos-sur-Mer by Eiffage. The piers each support 97 m tall pylons. The piers were assembled first, together with some temporary supports, before the decks were slid out across the piers by satellite-guided hydraulic rams that moved the deck 600 mm every 4 minutes.

The viaduct is the tallest vehicular bridge in the world, nearly twice as tall as the previous tallest vehicular bridge in Europe, the Europabrücke in Austria. (The proposed Strait of Messina Bridge in Italy, if constructed, would be taller.)

The Millau Viaduct is the second highest vehicular bridge measured from the roadway elevation. Its deck, at approximately 270 m above the Tarn, is slightly higher than the New River Gorge Bridge in West Virginia in the United States, which is 267 m above the New River. The Royal Gorge Bridge in Colorado, United States has a deck considerably higher than either, at 321 m above the Arkansas River.

Construction

The viaduct under construction, seen from the south in early 2004

Construction began on 10 October 2001 and was intended to take three years, but weather conditions put work on the bridge behind schedule. A revised schedule aimed for the bridge to be opened in January 2005. The viaduct was officially inaugurated by President Chirac on 14 December 2004 to open for traffic on 16 December, several weeks ahead of the revised schedule. The construction of the bridge is depicted in a documentary of the Discovery Channel 'Megastructures' series.

Preliminary studies

In initial studies, four options were examined:

1. bypass Millau to the east, requiring two large bridges over the Tarn and the Dourbie;
2. bypass Millau to the west (12 km longer), requiring four bridges;
3. follow the path of Route Nationale 9, providing good access to Millau but at the cost of technical difficulties and intrusion on the town; and
4. traverse the middle of the valley.

The fourth option was selected by the government on 28 June 1989. It consisted of two possibilities: the high solution, and the low solution, requiring the construction of a 200 m bridge to cross the Tarn, then a viaduct of 2300 m extended by a tunnel on the Larzac side. After long construction studies, the low solution was abandoned because it would have intersected the water table, had negative effects on the town, cost more, and the driving distance would have been longer.

After the choice of the high viaduct's path, five teams of architects and researchers simultaneously worked on a technical solution. The original concept for the bridge was devised by French designer Michel Virlogeux. The architects of the bridge are the British firm Foster and Partners. They worked together with the Dutch engineering firm ARCADIS, responsible for the technical design of the bridge.

Implementation

The nearly completed bridge in September 2004

The bridge deck was constructed on land at the ends of the viaduct and rolled lengthwise from one tower to the next, with eight temporary added towers also in place. The movement was accomplished by a computer-controlled system of pairs of wedges under the deck; the upper and lower wedges of each pair pointed in opposite directions. These were hydraulically operated, and moved repeatedly in the following sequence:

• Lower wedge slides under the upper wedge, raising it to the roadway above and then forcing the upper wedge still higher to lift the roadway.
• Both wedges move together, advancing the roadway a short distance.
• Lower wedge retracts from under the upper wedge, lowering the roadway and then allowing the upper wedge to drop away from the roadway.
• Upper wedge moves backward, placing it into position farther (back) along the roadway, ready to repeat the cycle and advance the roadway again.

The builders

PERI Formwork technology for the construction of the highest bridge pier

Four consortia were in competition for the building contract:

• One led by Dragados (Spanish), with Skanska (Swedish) and Bec (French);
• Société du viaduc de Millau, made up of ASF, Egis, GTM, Bouygues Travaux Publics, SGE, CDC Projets, Tofinso (all French) and Autostrade (Italian); and
• One led by Générale Routière, with Via GTI (French), and Cintra, Necso, Acciona, and Ferrovial Agroman (all Spanish).
• The successful bidders, lead by the Eiffage group, product of the Fougerolles-SEA fusion, the third largest French group in public works, and the sixth largest in Europe.

The work leader is the Compagnie Eiffage du Viaduc de Millau, owner of the government contract. The construction consortium is made up of the Eiffage TP company for the concrete part, the Eiffel company for the steel roadway (Gustave Eiffel built the Garabit viaduct in 1884, a train bridge in the neighboring Cantal département), and the Enerpac company for the roadway's hydraulic supports. The engineering group Setec has authority in the project, with SNCF engineering having partial control.

The formwork technology for the bridge piers came from PERI.

Costs and resources

The bridge's construction cost up to €394 million, with a toll plaza 6 km north of the viaduct costing an additional €20 million. The builders, Eiffage, financed the construction in return for a concession to collect the tolls for 75 years, until 2080. However, if the concession is very profitable, the French government can assume control of the bridge in 2044.

The project required about 127,000 m³ of concrete, 19,000 metric tons of steel for the reinforced concrete, and 5,000 metric tons of pre-stressed concrete for the cables and shrouds. The builder claims that the bridge's lifetime will be at least 120 years.

Statistics

• 2,460 metre: total length of the roadway
• 7: number of piers
• 77 m: height of Pier 7, the shortest
• 343 m: height of Pier 2, the tallest (245 m at the roadway's level)
• 87 m: height of a pylon
• 154: number of shrouds
• 270 m: average height of the roadway
• 4.20 m: thickness of the roadway
• 32.05 m: width of the roadway
• 85,000 m³: total volume of concrete used
• 290,000 tonnes: total weight of the bridge
• 10,000–25,000 vehicles: estimated daily traffic
• €4.90–6.50: typical automobile toll, as of 2005
• 20 km: horizontal radius of curvature of the road deck

Project timeline

• 28 June 1989: governmental approval of the middle route
• 19 October 1991: selection of the high solution, with the viaduct at 2500 m
• 10 January 1995: declaration of utilité publique (public usefulness), as needed to apply eminent domain or compulsory purchase.
• 9 July 1996: choice of the cable-stayed bridge type
• 1998: decision to contract out both construction costs and future tolls to a private enterprise
• 16 October 2001: work begins
• 14 December 2001: laying of the first stone
• January 2002: laying pier foundations
• March 2002: start of work on the pier support C8
• June 2002: support C8 completed, start of work on piers
• July 2002: start of work on the foundations of temporary, height adjustable roadway supports
• August 2002: start of work on pier support C0
• September 2002: assembly of roadway begins
• November 2002: first piers complete
• 25 February–26 February 2003: laying of first pieces of roadway
• November 2003: completion of the last piers (Piers P2 at 221 m and P3 at 245 m are the highest piers in the world.)
• 28 May 2004: the pieces of roadway are several centimetres apart, their juncture to be accomplished within two weeks
• 2nd half of 2004: installation of the pylons and shrouds, removal of the temporary roadway supports
• 14 December 2004: official inauguration
• 16 December 2004: opening of the viaduct, ahead of schedule
• 10 January 2005: initial planned opening date

Gallery

Panoramic view of the Millau Viaduct, as seen from the south. The red temporary supports are still visible in this 29 June 2004 photograph.

Single segment of the construction on the small exhibition under viaduct

Large numbers of people stop to observe the viaduct from the viewing area on the northern side

See also

• List of bridges by length
• Royal Gorge Bridge

External links

Wikimedia Commons has media related to:

Viaduc de Millau

• Viaduc de Millau
• Structurae: Millau Viaduct
• Millau Viaduct website
• France shows off tallest bridge (BBC report of official opening, 14 December, 2004)
• Millau Viaduct - Worlds tallest bridge Stunning Panoramic Pictures
• In Pictures: The Millau Bridge (BBC images of the Millau Viaduct, including (page 2) a map showing the bridge's location)
• Project on Viaduc de Millau From Road Traffic Technology
• France opens world's tallest bridge (MSNBC article on the official opening, 14 December, 2004)
• Eiffage group website
• Le Viaduc de Millau (in French)
• Technical popularisation conference on the Millau Viaduct by head engineer Michel Virlogeux (in French): RealAudio, Windows Media
• Google Maps Satellite photo (under construction)
• Millau, City and Bridge Map, photos, diagrams and text.

View Poster

• Born: December 1770
• Birthplace: Bonn, Germany
• Died: 26 March 1827 (cirrhosis of the liver, plus dropsy)
• Best Known As: The composer of Beethoven's Fifth

Mozart aside, Ludwig van Beethoven is the most famous classical composer of the western world. He is remembered for his powerful and stormy compositions, and for continuing to compose and conduct even after he began to go deaf at age 28. The ominous four-note beginning to his Fifth Symphony -- bom bom bom bommmmm -- is one of the most famous moments in all of music.

Beethoven never married. After his death his friends found letters to a lover he called "Immortal Beloved," whose identity has never been discovered... Beethoven's precise date of birth is unknown; he was baptized on 17 December 1770, and it is presumed he was born on 16 December.

Published works

Novels by year of composition

The Game-Players of Titan

Dates are for completion of first (and usually only) draft. Publication dates follow. + indicates subsequent significant expansion, * subsequent revision or minor expansion

1950

Gather Yourselves Together (1994)

1952

Voices From the Street (forthcoming 2006)

1953

Vulcan's Hammer (1960+)

Dr. Futurity (1960+)

The Cosmic Puppets (1957*)

1954

Solar Lottery (1955*)

Mary and the Giant (1987*)

The World Jones Made (1956)

1955

Eye in the Sky (1957)

The Man Who Japed (1956)

1956

A Time for George Stavros (ms. lost)

Pilgrim on the Hill (ms. lost)

The Broken Bubble (1988)

1957

Puttering About in a Small Land (1985)

1958

Nicholas and the Higs (ms. lost)

Time Out of Joint (1959)

In Milton Lumky Territory (1985)

1959

Confessions of a Crap Artist (1975)

1960

The Man Whose Teeth Were All Exactly Alike (1982)

Humpty Dumpty in Oakland (1986)

1961

The Man in the High Castle (1962)

1962

We Can Build You (1972)

Martian Time-Slip (1964)

1963

Dr. Bloodmoney, or How We Got Along After the Bomb (1965)

The Game-Players of Titan (1963) (ISBN 0-679-74065-1)

The Simulacra (1964)

The Crack in Space (1966+)

Now Wait for Last Year (1966)

Do Androids Dream of Electric Sheep?

1964

Clans of the Alphane Moon (1964)

The Three Stigmata of Palmer Eldritch (1965)

The Zap Gun (1967)

The Penultimate Truth (1964)

Deus Irae with Roger Zelazny (1976*+)

The Unteleported Man (1966 / 1983+ / 1984*+ as Lies, Inc.)

1965

The Ganymede Takeover with Ray Nelson (1967*)

Counter-Clock World (1967)

1966

Do Androids Dream of Electric Sheep? (1968)

Nick and the Glimmung (for children) (1988)

Ubik (1969)

1968

Galactic Pot-Healer (1969)

A Maze of Death (1970)

1969

Our Friends from Frolix 8 (1970)

1970

Flow My Tears, The Policeman Said (1974*)

1973

A Scanner Darkly (1977*)

1976

Radio Free Albemuth (1985)

1978

VALIS (1981)

The Crack in Space

1980

The Divine Invasion (1981)

1981

The Transmigration of Timothy Archer (1982)

Short stories

The short stories of Philip K. Dick have recently been republished in five omnibus volumes, as follows:

1. The Short Happy Life of the Brown Oxford and Other Stories, ISBN 0-8065-1153-2
2. We Can Remember It for You Wholesale and Other Stories, ISBN 0-8065-1209-1
3. Second Variety and Other Stories, ISBN 0-8065-1226-1
4. The Minority Report and Other Stories, ISBN 0-8065-1276-8
5. The Eye of the Sibyl and Other Stories, ISBN 0-8065-1328-4

1952

Beyond Lies the Wub

The Gun

The Little Movement

The Skull

The Variable Man

1953

The Builder

Colony

The Commuter

The Cookie Lady

The Cosmic Poachers

The Defenders

Expendable

The Eyes Have It

The Great C

The Hanging Stranger

The Impossible Planet

Impostor

The Indefatigable Frog

The Infinities

The King of the Elves – optioned by Disney Animation in June 2006

Martians Come in Clouds

Mr. Spaceship

Out in the Garden

Paycheck

Piper in the Woods

Planet for Transients

The Preserving Machine

Project: Earth

Roog

Second Variety

Some Kinds of Life

The Trouble with Bubbles

The World She Wanted

1954

A World of Talent

The Last of the Masters

Adjustment Team

Beyond the Door

Breakfast at Twilight

The Crawlers

The Crystal Crypt

Exhibit Piece

The Father-thing

The Golden Man

James P. Crow

Jon's World

The Little Black Box

Meddler

Of Withered Apples

A Present for Pat

Prize Ship

Progeny

Prominent Author

Sales Pitch

Shell Game

The Short Happy Life of the Brown Oxford

Small Town

Souvenir

Strange Eden

Survey Team

Time Pawn

Tony and the Beetles

The Turning Wheel

Upon the Dull Earth

1955

Autofac

Captive Market

The Chromium Fence

Foster, You're Dead!

The Hood Maker

Human Is

The Mold of Yancy

Nanny

Psi-man Heal My Child!

Service Call

A Surface Raid

Vulcan's Hammer

War Veteran

1956

A Glass of Darkness

Minority Report

Pay for the Printer

To Serve the Master

1957

Misadjustment

The Unreconstructed M

1958

Null-o

1959

Explorers We

Fair Game

Recall Mechanism

War Game

1963

All We Marsmen

The Days of Perky Pat

If There Were No Benny Cemoli

Stand-by

What'll We Do With Ragland Park?

1964

Cantata 140

A Game of Unchance

Novelty Act

Oh, to be a Blobel!

Orpheus with Clay Feet

Precious Artifact

The Unteleported Man

The War with the Fnools

Waterspider

What the Dead Men Say

1965

Project Plowshare

Retreat Syndrome

1966

Holy Quarrel

We Can Remember It For You Wholesale

Your Appointment Will Be Yesterday

1967

Faith of our Fathers

Return Match

1968

Not By Its Cover

The Story To End All Stories

1969

A. Lincoln, Simulacrum

The Electric Ant

1972

Cadbury, the Beaver Who Lacked

1974

The Different Stages of Love

The Pre-persons

A Little Something For Us Tempunauts

1979

The Exit Door Leads In

1980

I Hope I Shall Arrive Soon - originally titled Frozen Journey

Rautavaara's Case

Chains of Air, Web of Aether

1981

The Alien Mind

1984

Strange Memories Of Death

1987

The Day Mr. Computer Fell Out of Its Tree

The Eye of The Sibyl

Fawn, Look Back

Stability

1988

Goodbye, Vincent

1989

11-17-80

1992

The Name of the Game is Death

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