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where P is the adsorptive pressure, <math>P_0</math> is the saturation pressure, <math>\theta_m</math> is the surface area covered by a at least one layer, Q is the heat of adsorption, and <math>C\propto\exp(Q/kT)</math>. Here we allow multiple layers to form, so <math>\theta</math> is the total surface coverage counting each layer separately. For example, if half of the adsorbent was covered by a single layer we would have <math>\theta=0.5</math>, if the entire adsorbent was covered by 3 layers, we would have <math>\theta=3</math>, and so on. | where P is the adsorptive pressure, <math>P_0</math> is the saturation pressure, <math>\theta_m</math> is the surface area covered by a at least one layer, Q is the heat of adsorption, and <math>C\propto\exp(Q/kT)</math>. Here we allow multiple layers to form, so <math>\theta</math> is the total surface coverage counting each layer separately. For example, if half of the adsorbent was covered by a single layer we would have <math>\theta=0.5</math>, if the entire adsorbent was covered by 3 layers, we would have <math>\theta=3</math>, and so on. | ||
− | The BET isotherm is a much more powerful tool because it can yield two desirable quantities: the head of adsorption, and the surface area of the adsorbent. | + | The BET isotherm is a much more powerful tool because it can yield two desirable quantities: the head of adsorption, and the surface area of the adsorbent. if we define |
<center><math>Y = \frac{P}{P_0-P}\times\frac{1}{\theta}, \quad X = \frac{P}{P_0} \quad \Rightarrow \quad Y = \frac{1}{C\theta_m}+\frac{C-1}{C\theta_m}X </math></center> | <center><math>Y = \frac{P}{P_0-P}\times\frac{1}{\theta}, \quad X = \frac{P}{P_0} \quad \Rightarrow \quad Y = \frac{1}{C\theta_m}+\frac{C-1}{C\theta_m}X </math></center> | ||
− | + | we have a line whose slope and intercept yield <math>\theta_m</math> and Q. Since <math>\theta</math> is proportional to the total gas adsorbed, which can be determined from pressure and flow measurements, this is a common experimental method to measure the surface area, which is sometimes called the BET area. Adsorbent materials can have BET areas of several thousands of square meters per gram. [Ke] | |
In addition to this isotherm, there is a large selection of advanced methods for finding pore sizes. See [Ke 1.4] for more details. | In addition to this isotherm, there is a large selection of advanced methods for finding pore sizes. See [Ke 1.4] for more details. |
Revision as of 00:32, 11 January 2009
Nick Hutzler
I am a second year Physics graduate student in the Doyle Lab. I am working on an experiment to measure the electron electric dipole moment using a cryogenic molecule beam.
Contents
Final Project: Applications of Gas Adsorption
Introduction
Adsorption is the physical process where molecules (or atoms, though we shall use the word "molecule" to include those as well) in a fluid phase become bound ("stick") to the surface of another solid or liquid. Adsorption is a very broad term that can include a gas or liquid adsorbing onto a liquid or solid through electrostatic attractions, chemical bonds, or some combination of both. In this discussion we will focus primarily on the adsorption of gases onto solid.
Before proceeding, let's introduce some terminology. In adsorption, the fluid-phase adsorptive molecules sticks onto the adsorbent and become adsorbate, or admolecules. When adsorbate unsticks from the adsorbent, whether spontaneously or induced, it is called desorption.
Brief Explanation of Adsorption
The fundamental reason why adsorption occurs is because surfaces have energy. The surface at the interface of a solid or liquid necessarily has some energy density associated with it that is higher than the energy density of bulk material. This is an empirical fact, because materials (both solid and liquid) tend to sacrifice surface area for bulk.
In liquids surface energy arises from the fact that in the bulk, molecular (van der Waals) attractions between some molecule and all its neighbors balance out, resulting in a net force of 0. However, a molecule at a surface will feel a net attraction inwards, which gives rise to a surface energy. This force pulling the surface inward is balanced by the internal pressure of the liquid.
In solids, surface energy arises from disruption of the solid's lattice structure.
Types of Adsorption
Adsorption can be classified by whether the mechanism is "physical" (physisorption) or "chemical" (chemisorption) in nature. Because we will be focusing on physisorption, we then discuss several different adsorption mechanisms that would be classified as physical. The lists presented in this section are adapted from [Ke].
Physisorption vs. Chemisorption
- Physisorption is classified by weakly bound adsorbate, usually by van der Waals or London dispersion forces. An important feature of this type of adsorption is that it is reversible; specifically, desorption can be induced by raising the temperature of the adsorbent, or by decreasing the pressure of the adsorptive. The adsorptive does not suffer any type of chemical change when being adsorbed.
- Chemisorption is classified by strongly bound adsorbate, usually as a result of chemical bonds. Because chemical reactions occur, this process is typically irreversible, and the characteristics of the mechanism are strongly dependent on the species involved. Therefore, we will not focus on this type of adsorption.
Types of Physisorption
- Monolayer/multilayer adsorbates: When the adsorbent is uniformly covered with sites on which the adsorptive can stick, the adsorbate will generally be a roughly uniformly distributed layer. When the adsorption sites are homogeneous and the adsorptive pressure is much smaller than the saturation pressure, this layer is typically one molecule thick. When the adsorptive pressure approaches the saturation pressure, there are typically multiple layers of adsorbate. The admolecules in monolayer often form a lattice gas, where there is typically one admolecule per site but they can hop from site-to-site (see for example [Mo])
- Pore fluids: An adsorbent can have holes or troughs ("pores") that can become filled with adsorbate, and under the right conditions can become a gas or liquid. In fact, some very interesting things can happen because of the complicated thermodynamics of these systems; for example, liquid water a few molecular layers thick has been observed at the surface of certain porous solids... at 77K! [Ro]
- Steric Adsorbates: Adsorbents can have site that can sterically attract certain atomic groups. This mechanism plays is important, for example, in the adsorption of large biomolecules onto activated carbon.
- Ionic Adsorbates: Adsorbent surfaces can be covered with ions that can exchange with ionic parts of a molecule, binding the molecule to the adsorbate.
- Quantum Adsorbates: Adsorption can occur due to purely quantum mechanical effects. One example is when an adsorbent has pores that are comparable to the de Broglie wavelength of the adsorptive, allowing them to enter the pores.
In real life, adsorbents typically have many of the above features.
Adsorption Thermodynamics
Basics
In this section we present some basic quantities and concepts useful for understanding adsorption.
Vapor Pressure
The energies of molecules in a condensed phase are not constant, but instead have some distribution. The higher energy molecules can overcome the energy that binds them to the condensed phase (the enthalpy of vaporization), which leads to evaporation. Therefore, in equilibrium there will be a non-zero pressure in the gas phase in contact with the condensed phase; this pressure is called the vapor pressure. This explains condensation, or why a gas tends to form condensed phases on a surface if the gas phase pressure is higher than the vapor pressure.
Clausius - Clapeyron Relationship
Our discussion will focus on adsorption, a process which typically yields equilibrium pressures below the vapor pressure. However, the thermodynamics of condensation are still important, so we present one of the central relations: the Clausius-Clapeyron equation. The coexistence curve between two phases on a P-T diagram has slope given by
where <math>\Delta H</math> is the enthalpy change of the transition, and <math>\Delta V</math> is the volume change. Notice that this tells us something we already know: dP/dT>0, so the lower the temperature, the more gas condenses.
Enthalpy
At equilibrium, the phase changes between gas and condensed phases do not change pressure, so the energy required to desorb or adsorb a single molecule is equal to the enthalpy change between the phase. This is why the heat or energy of vaporization/adsorption is used interchangeably with the enthalpy of vaporization/adsorption.
Condensation vs. Adsorption
There is a distinction between condensation and adsorption which we should point out. In principle, any gas put in contact with any surface will result in some of the gas phase leaving and sticking to the surface. This is condensation, and is a thermodynamic result of the vapor pressure curve. However, real surfaces can have other features (see "Types of Adsorption") that enhance the amount of gas sticking to the surface, resulting in an equilibrium gas pressure lower than the vapor pressure. Specifically, this enhancement is a binding energy larger than the heat of vaporization. For heavier gases this energy is typically larger by factor of 2-3, but for the light gases Hydrogen and Helium, this factor can be as high as 10 and 30 (resp) for certain porous surfaces! [Re]
Frenkel Equation
The Frenkel Equation is given by [Re]
where <math>Q</math> (kcal/mol) is the heat of vaporization (in the case of condensation) or heat of desorption (in the case of adsorption), R is the gas constant, T is the temperature (K), and <math>\tau_0</math> is some constant. The heat of desorption is dependent on the adsorbent.
From this equation we can see why lowering the temperature of the adsorbent yields lower equilibrium pressures: the admolecules will spend more time on the adsorbent after each collision. Thus whether we use an ensemble, thermodynamic picture with the Clausius-Clapeyron relationship, or the single-molecule picture with the Frenkel equation, we see that lower adsorbent temperatures will lead to lower equilibrium gas phase pressure.
Equilibrium
Adsorption is a rate-equation governed process, so "true" equilibrium (in a continuum picture) would only occur at <math>t=\infty</math>. However, for real-life industrial applications, [Ke] defines a technical equilibrium as follows: an adsorbtion process is at equilibrium if <math>\Delta m/m < \varepsilon</math>, where <math>\Delta m</math> is the mass change in the total mass <math>m</math> over some time <math>\Delta t</math>, and <math>\varepsilon</math> is some parameter that depends on the application. Typical values are <math>\varepsilon=10^{-5}</math> and <math>\Delta t = </math> 30 minutes.
Industrial processes are often cyclical, and involve adsorption/desorption cycles on some time scale <math>t_c</math>. The distance from equilibrium of a process can then be characterized by a Deborah number:
For <math>De\approx 0</math> the system is at equilibrium.
Adsorption Isotherms
One very important tool for studying adsorption is an adsorption isotherm, a curve of surface coverage <math>\theta</math> versus pressure (or some related variables) at a constant temperature. An application of these models is calculating the internal surface area of the adsorbent.
Langmuir Isotherm
A simple yet powerful model was developed by Langmuir. This model assumes that the adsorptive forms a monolayer on the adsorbent, and that adsorbent covered with a monolayer is passivated (i.e. will no longer adsorb). A quick derivation can be worked out by considering the equilibrium coefficient K of the adsorption reaction <math>M+S\leftrightharpoons MS</math> between molecules M and adsorption sites S:
If <math>\theta</math> is the fraction of adsorbent that is covered by a monolayer, then <math>[S]\propto 1-\theta</math> and <math>[MS]\propto\theta</math>. In the gas phase, the concentration is proportional to the pressure P, so
where b is the constant given in [Ho] as
here T is the temperature in K, Q is the activation energy in eV, k is the Boltzmann constant, N is the number of sites, <math>\sigma_0</math> is the adsorption site area, M is the adsorptive molecular weight, and <math>\tau_0</math> is the adsorption time.
The Langmuir Isotherm is too much of a simplification to find much practical use. Instead, it is more common to use the BET isotherm, which allows for multiple layers.
BET Isotherm
The BET isotherm is given by [Ho]:
where P is the adsorptive pressure, <math>P_0</math> is the saturation pressure, <math>\theta_m</math> is the surface area covered by a at least one layer, Q is the heat of adsorption, and <math>C\propto\exp(Q/kT)</math>. Here we allow multiple layers to form, so <math>\theta</math> is the total surface coverage counting each layer separately. For example, if half of the adsorbent was covered by a single layer we would have <math>\theta=0.5</math>, if the entire adsorbent was covered by 3 layers, we would have <math>\theta=3</math>, and so on.
The BET isotherm is a much more powerful tool because it can yield two desirable quantities: the head of adsorption, and the surface area of the adsorbent. if we define
we have a line whose slope and intercept yield <math>\theta_m</math> and Q. Since <math>\theta</math> is proportional to the total gas adsorbed, which can be determined from pressure and flow measurements, this is a common experimental method to measure the surface area, which is sometimes called the BET area. Adsorbent materials can have BET areas of several thousands of square meters per gram. [Ke]
In addition to this isotherm, there is a large selection of advanced methods for finding pore sizes. See [Ke 1.4] for more details.
Other Isotherms
While the Langmuir and BET isotherms may be the most common, there are a few more that appear every now and then [We,Da]. In each of the following, <math>C</math> represents some constant, but not necessarily the same constant.
- Henry's Law: <math>\theta=C P</math>
- Freundlich Isotherm: <math>\theta=CP^m</math>, where <math>m\leq 1</math>
- DR Isotherm: <math>\log\theta=\log\theta_m-C(\log(P_V/P))^2</math>, where <math>P_V</math> is the vapor pressure of the liquid form of the adsorptive gas. This isotherm is especially useful for cryopumping, as it was designed with microporous solids in mind.
- Toth Isotherm: <math>\theta = \frac{\theta_m b P}{[1+(bP)^t]^{1/t}}</math>
All of the isotherms we have discussed here have their own regimes where they work, though none of them are universal.
Molecular Description of Adsorption
While surface energies may be able to explain adsorption on thermodynamic scales, it can be explained on the molecular level by considering the van der Waals interaction between the gas molecule and the solid adsorbent. Say that a gas molecule interacts with a molecule of the adsorbent via the 6-12 Lennard-Jones potential
which has an energy minimum of <math>-\epsilon</math> at <math>2^{1/6}r_0</math>. If the adsorbent is a semi-infinite slab of density N, then the interaction energy as a function of z, the normal distance between the molecule and the surface, is [Re]
On the other hand, if the adsorbent is a film (or, more realistically, a crystalline solid having inter-layer spacing much larger than the inter-atomic spacing, for example in graphite), the interaction potential is
It is not always clear whether to use the 3-9 or the 4-10 potential. For example, if one used the above model to experimentally calculate the binding energy E_{b}, the models give different values:
As shown in the table below, the calculated binding energy is model dependent.
We can see that the differences between the models is not that large, with agreement at the roughly 5% level. However, in each case the adsorption binding energy is vastly larger than the typical van der Waals binding energy:
This is due to the enhancement by a factor of N, which is large (on the order of Avogadro's Number) for condensed phases.
This molecular description can be more useful when the gas is at low pressures, and the molecular regime picture is more accurate than the viscous regime picture.
Application: Cryopumping
Introduction
Cryopumping refers to the condensation or adsorption of gas onto a cooled surface. Cryopumping has a molecular explanation that does not rely on surface energies: when a molecule collides with a sufficiently low-temperature surface, the molecule can lose enough of its energy so that it can become bound by weak van der Waals/dispersion forces with the surface. Because cryopumping generally occurs (unless something is not going well) in low-pressure regimes, it is useful to consider the adsorptive gas in the molecular picture. The adsorption binding energies of the molecules arises from the minimum in the van der Waals interaction with the surface. Some values of this adsorption binding energy <math>\varepsilon</math>for various gases on activated carbon is found in [Day]:
Gas | <math>\quad\varepsilon</math> (kJ/mol) |
---|---|
He | 0.5 |
H_{2} | 1.5 |
Ne | 3 |
Ar | 10 |
Kr | 15 |
A typical, commercially available cryopump consists of metal panels covered with a porous adsorbent (coconut charcoal is very common) that is cooled by either a cryorefrigerator or by liquid cryogens.
Properties
Basic Design
A typical cryopump is shown in the figure below:
The sorption panel is covered with a cryosorbent and cooled, typically to (or below) the LHe boiling point around 4K. The LN and LHe shields provide protection from blackbody radiation, and provide additional surface area for cryocondensation. The chevron baffles allow gas to pas through to be cryosorbed, but block blackbody radiation
Regeneration
One drawback of cryopumping is that the adsorbent material must be regenerated. Every material has some finite adsorption capacity, so periodically the adsorbed gas must be desorbed. This can be achieved in a controlled manner by several methods, but a common one is to actively warm the sorption panel.
Pumping Speed and Sticking Coefficient
The flux of molecules impinging upon a cryopump of area A is equal to 1/4 of A times the mean Maxwell-Boltzmann velocity: [Day]
where M is the molar mass of the molecule. However, the probability that a given molecule sticks to the cold surface is not unity; to discuss pumping speed we therefore need to introduce the sticking coefficient <math>\beta</math>, the probability that a molecule sticks to the surface upon collision. The pumping speed of a cryopump panel of area A is therefore
For <math>\beta</math> not too small, this means that a cryopump panel 10 cm on a side can pump hundreds of liters of helium gas per second!
Temperature Dependence of Sticking Coefficient
The sticking coefficient depends on both the gas temperature and the surface temperature. The gas temperature dependence can be worked out by a simple argument [Daw]. For a Maxwell-Boltzmann gas at temperature T_{g} there is some critical energy E_{c} above which a gas molecule will not stick. The fraction of molecules with energy above this critical energy is
However, the fraction of unstuck molecules is also equal to 1-<math>\beta</math> in equilibrium, so
Therefore, the gas temperature dependence can be written as
The dependence on the surface temperature is much more complicated, since it depends on the specific physics of the gas-surface interaction. One calculation of the temperature dependence is presented by Asnin et al [As], but assumes a linear adsorption isotherm, valid only at very small surface coverage:
where E_{a} is the energy of adsorption, and <math>\Delta S</math> is the entropy change on adsorption. Without making restrictive assumptions, only empirical formulas for the sticking coefficient exist (to my knowledge). For example, Gurevich et al [Gu] calculated the sticking coefficient of Helium on activated charcoal by using measured adsorption isotherms, and found the following complicated curves:
Capture Coefficient vs. Sticking Coefficient
There are two types of coefficients that evaluate cryopump effectiveness: sticking and capture. Sticking coefficient is discussed above, and in fact most literature does not differentiate between the two (for example, the above table of capture coefficients should really be "sticking coefficients" given our terminology). The capture coeffient c is the probability that a molecule entering the entire cryopump apparatus will become adsorbed. For example, instead of colliding with the cold cryosorbent, the molecule could reflect off the radiation shield back into the region to be pumped. If the overall transmission probability w is the probability that a molecule entering the apparatus will impinge upon the cryosorbent, then
Water-like molecules such as H_{2}O, CO_{2}, and hydrocarbons are typically condensed on the first/radiation (typically around 80K) stage, and have typical values of w and <math>\alpha</math> near 1 for a commercial cryopump. Air-like gases such as N_{2}, O_{2}, CO, and Ar are typically condensed in the second stage (between 4K and 20K) and typically have w=0.2, <math>\beta\approx 1</math>. Light gases such as He, H_{2}, and Ne must be cryosorbed, and typically have w=0.2 and <math>\beta<1</math> that is very dependent on the species and temperature.
Cryosorbent Material
- Zeolites: Zeolites are aluminosilicates with very regular crystal structures. They are commonly honeycomb structures with length scales on the order of a few angstroms. This porosity gives them a huge pumping ability, which is why zeolites have so many industrial applications. However, zeolites tend to be hydrophilic, which limits their use as a cryosorbent: to regnerate the zeolite, temperatures of around 300 Celsius (too hot for vacuum seals) are needed to desorb the water [Day].
- Porous Metals: Metals such as copper and aluminum can be prepared in such a manner as to give them a large porosity. These porous metals have been tested and do indeed work as cryosorbers [An]. These materials have an advantage over activated charcoal because they can be formed into monolithic shapes, as opposed to attaching little bits (which can crumble and break off) to a substrate. However, these materials have not yet demonstrated the necessary adsorption capacity to compete with charcoal.
- Activated Charcoal: Activated charcoals are widely regarded as the best overall cryosorbent material. They have huge internal surface area and porosity, are readily available and inexpensive, are robust against thermal cycling, and can be completely regenerated by raising the temperature to the moderate temperature of around 125 Celcius [Day]. Most commercial cryopumps use activated charcoal.
Pumping Capabilities
Activated charcoal is an excellent cryosorbent material. The graph below shows pumping speed data for several different kinds of charcoal:
As we can see, a 10 cm x 10 cm panel covered with activated charcoal can easily pump at a speed of 800 liters/s! To compare, a turbomolecular pump with a pumping speed of a few hundred liters/s would cost around $10,000. A small panel of a charcoal cryopump can be made for under $100, and will be completely clean, require no electricity, and make no noise.
Optimization Variables
The study of optimizing the pumping characteristics of activated charcoal is a non-trivial one, and many groups have studied it. Though I will not go into details here, I will mention some major contributing factors.
- Bonding Method: One must somehow anchor the charcoal to a cold surface to make the charcoal itself cold. It turns out that the pumping characteristics of the charcoal are heavily dependent on the characteristics of the bond, and in fact some bonding methods will result in no pumping whatsoever!
- Charcoal Type: Not all activated charcoals are created equal. It turns out that activated charcoal made from coconut is on average superior to those made from coal, wood, petroleum byproducts, or bone.
- Contamination: The charcoal will adsorb contaminants, like water vapor or vacuum pump oil. These can seriously hinder pumping capabilities, which is why charcoal should be heated ("baked out") before use, which forces desorption of these contaminants.
Application: Activated Charcoal
Activated charcoal is not useful only for cryopumping. It is one of the most prevalent adsorbent materials and finds applications in many industries.
Properties of Activated Charcoal
The reason charcoal is such a powerful adsorbent material is because it has a huge internal surface area and porosity. The figure below shows a 2000x magnified image of coconut charcoal, allowing us to see its pore structure.
Activated charcoals have a surface area that is typically 1000 - 1700 m^{2}/g, which works out to a square mile of surface area in just a few pounds of charcoal! This surface area is made available by activation.
In addition, activated charcoal is cheap, non-toxic, easy to handle, and strong.
Production of Activated Charcoal
The production of activated charcoal is an important part of understanding why it has such huge porosity. The base material (coal, wood, nutshells, etc), which must contain a large portion of carbon, is dried and then heated to remove byproducts. However, at this phase the pores of the charcoal are typically filled. The key to unblocking these pores is activation: the material is then heated and exposed to an oxidizer, such as CO_{2}, which burns away the material lodged in the pores.
Applications
There are far too many applications of activated charcoal to give a complete list, so we will mention just a few here.
CIP Gold Extraction
A common technique for mining metals is leach mining, where ore is dissolved in a chemical from which the desired metal is extracted. One such technique is Gold cyanidation, where ore is dissolved in Cyanide and Gold-Cyanide complexes are created. Carbon in Pulp (CIP) is a method to isolate these compounds, which simply involves flowing the dissolved ore through many containers of activated charcoal. The charcoal is then removed from the tanks, where the gold complexes are extracted using high temperatures, or chemicals such as ethanol. The final step is to remove the gold from the complexes, which is usually achieved by electroplating the gold onto cathodes.
Metal Finishing
A common metal finishing technique is electoplating, which can completely cover an object with a thin, uniform layer of metal. The basic technique is to submerge an anode made of the plating metal and a cathode made of the object to be plated into an electrolyte. As current flows through the anode, the metal oxidizes and dissolves into the solution, where the ions are then attracted to the cathode, where they deposit.
Typically the electrolyte solution contains is treated with organic compounds that enhance the features of the plating, however the compounds can (and do) break down yielding undesirable results. Therefore, activated charcoal is often added to the solution because it will adsorb and remove these unwanted byproducts.
Impurity Removal
- Drinking Water Filtration: When combined with zeolites and ion exchange resins, activated charcoal based water filter systems (such as Brita) can remove Chlorine, Benzene, Lead, Copper, Mercury, Cadmium, Zinc, and a large number of other impurities.
- Sewage Treatment: Activated charcoal is often used as a filter for treated sewage. After most of the solid and particulate material is remove through various processes, activated charcoal helps to remove many leftover toxins and impurities.
- Air Treatment: Industrial applications include the removal of volatile organic compounds (VOCs), such as fumes from gasoline, paints, and dry-cleaning chemicals. Special types of charcoal can be used to remove mercury from emissions. Residental applications include dehumidification and odor removal. Charcoal is also found in gas/filter masks.
- Liquor Purification: Flowing vodka or whiskey through activated charcoal can improve its taste, color, and odor.
Medical Application
Activated charcoal is itself not toxic (provided that it is a charcoal designed for consumption), so it can be ingested in the event of poisoning to adsorb the poisons and prevent them from being absorbed into the body. This is an extremely effective method, and is a very common modern treatment for poisoning.
In addition to tested medical applications, charcoal is often used as a home remedy for various digestive problems.
References
[Ke] Keller & Staudt, Gas Adsorption Equilibria
[We] Welch, Capture Pumping Technology
[Ro] Robens, E. "Some Remarks on the Interface Ice/Water." Proceedings VIII Ukrainian-Polish Symposium: Theoretical and Experimental Studies of Interfacial Phenomena and their Technological Applications (2004)
[Mo] Morgenstern, K. and Rieder, K. H. PRL 93 (2004) 056102
[Ho] Hoffman, Singh, and Thomas, Handbook of Vacuum Science and Technology.
[Day] Day, Use of Porous Materials for Cryopumping (link?)
[Daw] Dawson, J. and Haygood, J. Cryopumping, Cryogenics, Vol. 5, No. 2, April 1965.
[As] Asnin, L. et al. Calculation of the sticking coefficient in the case of the linear adsorption isothem, Russian Chemical Bulletin, Vol 52. No. 12, December 2003