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73-524: VPI may refer to: Vapor phase infiltration, a synonym for sequential infiltration synthesis Velopharyngeal insufficiency , a medical term Verilog Procedural Interface , in computer programming Vertical Politics Institute , a conservative political action committee Vietnam Petroleum Institute Virginia Polytechnic Institute and State University (a.k.a. Virginia Tech) Virtual path identifier , in computer networking Visegrad Patent Institute ,

146-418: A body with no net movement of matter. An example of a process where both bulk motion and diffusion occur is human breathing. First, there is a "bulk flow" process. The lungs are located in the thoracic cavity , which expands as the first step in external respiration. This expansion leads to an increase in volume of the alveoli in the lungs, which causes a decrease in pressure in the alveoli. This creates

219-489: A formalism similar to Fourier's law for heat conduction (1822) and Ohm's law for electric current (1827). Robert Boyle demonstrated diffusion in solids in the 17th century by penetration of zinc into a copper coin. Nevertheless, diffusion in solids was not systematically studied until the second part of the 19th century. William Chandler Roberts-Austen , the well-known British metallurgist and former assistant of Thomas Graham studied systematically solid state diffusion on

292-530: A liquid medium and just large enough to be visible under an optical microscope exhibit a rapid and continually irregular motion of particles known as Brownian movement. The theory of the Brownian motion and the atomistic backgrounds of diffusion were developed by Albert Einstein . The concept of diffusion is typically applied to any subject matter involving random walks in ensembles of individuals. In chemistry and materials science , diffusion also refers to

365-414: A patent organization created by the four Visegrad countries Vocational Preference Inventory VPI Industries , a manufacturer of phonographs VPIphotonics, photonic simulation tool Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title VPI . If an internal link led you here, you may wish to change the link to point directly to

438-524: A physical and atomistic one, by considering the random walk of the diffusing particles . In the phenomenological approach, diffusion is the movement of a substance from a region of high concentration to a region of low concentration without bulk motion . According to Fick's laws, the diffusion flux is proportional to the negative gradient of concentrations. It goes from regions of higher concentration to regions of lower concentration. Sometime later, various generalizations of Fick's laws were developed in

511-441: A pressure gradient between the air outside the body at relatively high pressure and the alveoli at relatively low pressure. The air moves down the pressure gradient through the airways of the lungs and into the alveoli until the pressure of the air and that in the alveoli are equal, that is, the movement of air by bulk flow stops once there is no longer a pressure gradient. Second, there is a "diffusion" process. The air arriving in

584-435: A range of technology sectors. One of the main applications of SIS is in the enhancement of etch resistance in lithographic photoresist , such as those used in photolithography , microfabrication , and nanolithography . This method involves the sequential deposition of inorganic materials within a patterned resist's micro/nanostructures. By carefully controlling the infiltration of these materials, SIS can precisely engineer

657-497: A region of lower concentration to a region of higher concentration, as in spinodal decomposition . Diffusion is a stochastic process due to the inherent randomness of the diffusing entity and can be used to model many real-life stochastic scenarios. Therefore, diffusion and the corresponding mathematical models are used in several fields beyond physics, such as statistics , probability theory , information theory , neural networks , finance , and marketing . The concept of diffusion

730-406: A wide range of materials, some materials are not suitable for SIS. The relatively slow diffusion of SIS precursor vapors through polymers can make SIS over macroscopic distances time-consuming. For instance, the infiltration of millimeter distances into a polymer may require precursor exposure times of several hours. For comparison, ALD of thin films on dense surfaces that do not involve diffusion into

803-562: Is In case the diffusion coefficient is independent of x {\displaystyle x} , Fick's second law can be simplified to where Δ {\displaystyle \Delta } is the Laplace operator , Fick's law describes diffusion of an admixture in a medium. The concentration of this admixture should be small and the gradient of this concentration should be also small. The driving force of diffusion in Fick's law

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876-479: Is where ( J , ν ) {\displaystyle (\mathbf {J} ,{\boldsymbol {\nu }})} is the inner product and o ( ⋯ ) {\displaystyle o(\cdots )} is the little-o notation . If we use the notation of vector area Δ S = ν Δ S {\displaystyle \Delta \mathbf {S} ={\boldsymbol {\nu }}\,\Delta S} then The dimension of

949-418: Is a net movement of oxygen molecules down the concentration gradient. In astronomy , atomic diffusion is used to model the stellar atmospheres of chemically peculiar stars . Diffusion of the elements is critical in understanding the surface composition of degenerate white dwarf stars and their evolution over time. In the scope of time, diffusion in solids was used long before the theory of diffusion

1022-520: Is a tool for surface modifications to improve biocompatibility , bioactivity , and controlled drug release, making it useful in some biomedical applications. Polymers and bioactive macro-molecules treated with SIS can obtain coatings with developed cell adhesion and reduced bacterial adhesion, as well as provide a medium for the controlled release of therapeutics. Such properties are applicable in biomedicine, such as implantable medical devices, tissue engineering , and drug delivery systems. Modification of

1095-536: Is a vector J {\displaystyle \mathbf {J} } representing the quantity and direction of transfer. Given a small area Δ S {\displaystyle \Delta S} with normal ν {\displaystyle {\boldsymbol {\nu }}} , the transfer of a physical quantity N {\displaystyle N} through the area Δ S {\displaystyle \Delta S} per time Δ t {\displaystyle \Delta t}

1168-452: Is based on the consecutive introduction of different precursors into a polymer, taking advantage of the material's porosity on the molecular scale. This allows the precursors to diffuse into the material and react with specific functional groups located along the polymer backbone or pendant group. Through the selection and combination of the precursors, a rich variety of materials can be synthesized, each of which can endow unique properties to

1241-423: Is called a normal diffusion (or Fickian diffusion); Otherwise, it is called an anomalous diffusion (or non-Fickian diffusion). When talking about the extent of diffusion, two length scales are used in two different scenarios: "Bulk flow" is the movement/flow of an entire body due to a pressure gradient (for example, water coming out of a tap). "Diffusion" is the gradual movement/dispersion of concentration within

1314-437: Is comparable to or smaller than the mean free path of the molecule diffusing through the pore. Under this condition, the collision with the pore walls becomes gradually more likely and the diffusivity is lower. Finally there is configurational diffusion, which happens if the molecules have comparable size to that of the pore. Under this condition, the diffusivity is much lower compared to molecular diffusion and small differences in

1387-499: Is important to note that the advantages and limitations of SIS are continually being explored, addressed, and improved upon as research and development efforts in the field are currently ongoing. SIS allows for precise control over the composition, structure, and properties of materials. The sequential nature of the synthesis process enables the integration of multiple materials and the creation of complex and multi-functional nanostructures. SIS enables atomic-level precision in controlling

1460-436: Is intensity of any local source of this quantity (for example, the rate of a chemical reaction). For the diffusion equation, the no-flux boundary conditions can be formulated as ( J ( x ) , ν ( x ) ) = 0 {\displaystyle (\mathbf {J} (x),{\boldsymbol {\nu }}(x))=0} on the boundary, where ν {\displaystyle {\boldsymbol {\nu }}}

1533-418: Is placed in a reactor with an inert atmosphere (typically an inert gas or vacuum ). The first precursor vapor (e.g., trimethylaluminum, TMA ) is introduced at a sufficiently high vapor pressure and duration such that the precursor molecules diffuse into the substrate. Thus the precursor infiltrates the material and then reacts with the interior functional groups. After a suitable diffusion/reaction time,

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1606-528: Is sometimes referred to as "multiple pulsed vapor-phase infiltration" (MPI), "vapor phase infiltration" (VPI) or "sequential vapor infiltration" (SVI). SIS involves the 3D distribution of functional groups in polymers , while its predecessor, ALD, is associated with the impermeable 2D distribution of reactive sites on solid surfaces. In typical ALD processes, the precursor pulses are much shorter in duration and have lower partial pressure compared to SIS, as they only need to provide sufficient exposure to saturate

1679-412: Is the j {\displaystyle j} th thermodynamic force and L i j {\displaystyle L_{ij}} is Onsager's matrix of kinetic transport coefficients . The thermodynamic forces for the transport processes were introduced by Onsager as the space gradients of the derivatives of the entropy density s {\displaystyle s} (he used

1752-528: Is the antigradient of concentration, − ∇ n {\displaystyle -\nabla n} . In 1931, Lars Onsager included the multicomponent transport processes in the general context of linear non-equilibrium thermodynamics. For multi-component transport, where J i {\displaystyle \mathbf {J} _{i}} is the flux of the i {\displaystyle i} th physical quantity (component), X j {\displaystyle X_{j}}

1825-421: Is the change in the value of a quantity; for example, concentration, pressure , or temperature with the change in another variable, usually distance . A change in concentration over a distance is called a concentration gradient , a change in pressure over a distance is called a pressure gradient , and a change in temperature over a distance is called a temperature gradient . The word diffusion derives from

1898-434: Is the normal to the boundary at point x {\displaystyle x} . Fick's first law: The diffusion flux, J {\displaystyle \mathbf {J} } , is proportional to the negative gradient of spatial concentration, n ( x , t ) {\displaystyle n(x,t)} : where D is the diffusion coefficient . The corresponding diffusion equation (Fick's second law)

1971-449: Is then purged or evacuated to complete the first SIS cycle. The second precursor may also create new functional groups for reaction with the first precursor for subsequent SIS cycles. Sequential infiltration steps can then be repeated using the same or different precursor species until the desired modifications are achieved. When the desired infiltrations are achieved, the modified material can undergo further post-treatment steps to enhance

2044-458: Is universally recognized that atomic defects are necessary to mediate diffusion in crystals. Henry Eyring , with co-authors, applied his theory of absolute reaction rates to Frenkel's quasichemical model of diffusion. The analogy between reaction kinetics and diffusion leads to various nonlinear versions of Fick's law. Each model of diffusion expresses the diffusion flux with the use of concentrations, densities and their derivatives. Flux

2117-460: Is widely used in many fields, including physics ( particle diffusion ), chemistry , biology , sociology , economics , statistics , data science , and finance (diffusion of people, ideas, data and price values). The central idea of diffusion, however, is common to all of these: a substance or collection undergoing diffusion spreads out from a point or location at which there is a higher concentration of that substance or collection. A gradient

2190-457: The i {\displaystyle i} th component. The corresponding driving forces are the space vectors where T is the absolute temperature and μ i {\displaystyle \mu _{i}} is the chemical potential of the i {\displaystyle i} th component. It should be stressed that the separate diffusion equations describe the mixing or mass transport without bulk motion. Therefore,

2263-495: The Latin word, diffundere , which means "to spread out". A distinguishing feature of diffusion is that it depends on particle random walk , and results in mixing or mass transport without requiring directed bulk motion. Bulk motion, or bulk flow, is the characteristic of advection . The term convection is used to describe the combination of both transport phenomena . If a diffusion process can be described by Fick's laws , it

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2336-472: The kinetic coefficients L i j {\displaystyle L_{ij}} should be symmetric ( Onsager reciprocal relations ) and positive definite ( for the entropy growth ). The transport equations are Here, all the indexes i , j , k = 0, 1, 2, ... are related to the internal energy (0) and various components. The expression in the square brackets is the matrix D i k {\displaystyle D_{ik}} of

2409-710: The photoresist in the extreme ultraviolet range to improve EUV lithography . SIS has applications in the field of surface coatings, particularly in the development of coatings with specific functional properties. With the sequential infiltration of different precursors into the material, SIS allows for the creation of coatings with enhanced properties and performance such as durability, corrosion resistance, oleophilicity / lipophilicity , anti-reflection, and/or improved adhesion to substrates. Such an application of SIS can be used for protective coatings for metals, anti-fouling coatings for biomedical applications, and coatings for optical and electronic devices. In this application,

2482-568: The PMMA-enriched microphase subdomains. Subsequent removal of the PS-b-PMMA by using oxygen plasma or by annealing in air will convert the combined organic and inorganic mesoscale pattern into a purely inorganic Al 2 O 3 pattern that shares the mesoscale structure of the block copolymer but is more chemically and thermally robust. The capability of SIS to modify polymeric materials in a controlled manner has resulted in utility across

2555-648: The SIS precursors are selected to react with just one of the BCP components but not with the second component, then the inorganic material will only nucleate and grow in that component. For instance, TMA will react with the PMMA side chains of PS-b-PMMA but not with the PS side chains. Consequently, SIS using TMA and H 2 O as precursor vapors to infiltrate a PS-b-PMMA microphase-separated substrate will form Al 2 O 3 specifically in

2628-401: The alveoli has a higher concentration of oxygen than the "stale" air in the alveoli. The increase in oxygen concentration creates a concentration gradient for oxygen between the air in the alveoli and the blood in the capillaries that surround the alveoli. Oxygen then moves by diffusion, down the concentration gradient, into the blood. The other consequence of the air arriving in alveoli is that

2701-413: The cell (against the concentration gradient). Because there are more oxygen molecules outside the cell, the probability that oxygen molecules will enter the cell is higher than the probability that oxygen molecules will leave the cell. Therefore, the "net" movement of oxygen molecules (the difference between the number of molecules either entering or leaving the cell) is into the cell. In other words, there

2774-411: The chemical composition and density of the resist, thus enhancing its resistance to common etching processes. This enables technologists to achieve finer feature patterns and increased durability in microfabrication, ultimately advancing the capabilities of semiconductor manufacturing and nanotechnology applications. Another recent application for SIS in lithography is to enhance the optical absorption of

2847-433: The coefficient of diffusion for CO 2 in the air. The error rate is less than 5%. In 1855, Adolf Fick , the 26-year-old anatomy demonstrator from Zürich, proposed his law of diffusion . He used Graham's research, stating his goal as "the development of a fundamental law, for the operation of diffusion in a single element of space". He asserted a deep analogy between diffusion and conduction of heat or electricity, creating

2920-410: The concentration of carbon dioxide in the alveoli decreases. This creates a concentration gradient for carbon dioxide to diffuse from the blood into the alveoli, as fresh air has a very low concentration of carbon dioxide compared to the blood in the body. Third, there is another "bulk flow" process. The pumping action of the heart then transports the blood around the body. As the left ventricle of

2993-630: The deposition of precursor materials. This high level of precision allows creating nanostructures with uniform dimensions, well-defined interfaces, and tailored properties. SIS is a versatile fabrication technique amenable to a diverse range of combinations of polymer chemistries and precursor species. By selecting specific precursor materials, researchers can tune the properties of the fabricated materials, which include but are not limited to electrical conductivity, optical properties, and catalytic activity. This empowers various applications in electronics, photonics, energy devices, separations, and more. One of

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3066-595: The diffusion ( i , k  > 0), thermodiffusion ( i  > 0, k  = 0 or k  > 0, i  = 0) and thermal conductivity ( i = k = 0 ) coefficients. Under isothermal conditions T  = constant. The relevant thermodynamic potential is the free energy (or the free entropy ). The thermodynamic driving forces for the isothermal diffusion are antigradients of chemical potentials, − ( 1 / T ) ∇ μ j {\displaystyle -(1/T)\,\nabla \mu _{j}} , and

3139-519: The diffusion and reaction of the SIS precursors below the polymer surface facilitate a bulk-like transformation such that the effective thickness of the surface coating (e.g., several microns) is much larger than the film thickness that would result using the same number of atomic layer deposition (ALD) cycles on a conventional, dense substrate (e.g., a few nanometers). SIS, with its precise control over material properties, can be used to develop sensors and actuators. The functional layers created through

3212-463: The diffusion and reactions of ALD precursors into polymer films were considered challenges to address rather than opportunities. However, potential benefits of these phenomena were demonstrated by Knez and coworkers in a 2009 report describing the increased toughness of spider silk following vapor-phase infiltration. Sequential infiltration synthesis was developed by Argonne National Laboratory scientists Jeffrey Elam and Seth Darling in 2010 with

3285-397: The diffusion flux is [flux] = [quantity]/([time]·[area]). The diffusing physical quantity N {\displaystyle N} may be the number of particles, mass, energy, electric charge, or any other scalar extensive quantity . For its density, n {\displaystyle n} , the diffusion equation has the form where W {\displaystyle W}

3358-428: The example of gold in lead in 1896. : "... My long connection with Graham's researches made it almost a duty to attempt to extend his work on liquid diffusion to metals." In 1858, Rudolf Clausius introduced the concept of the mean free path . In the same year, James Clerk Maxwell developed the first atomistic theory of transport processes in gases. The modern atomistic theory of diffusion and Brownian motion

3431-443: The frame of thermodynamics and non-equilibrium thermodynamics . From the atomistic point of view , diffusion is considered as a result of the random walk of the diffusing particles. In molecular diffusion , the moving molecules in a gas, liquid, or solid are self-propelled by kinetic energy. Random walk of small particles in suspension in a fluid was discovered in 1827 by Robert Brown , who found that minute particle suspended in

3504-421: The heart contracts, the volume decreases, which increases the pressure in the ventricle. This creates a pressure gradient between the heart and the capillaries, and blood moves through blood vessels by bulk flow down the pressure gradient. There are two ways to introduce the notion of diffusion : either a phenomenological approach starting with Fick's laws of diffusion and their mathematical consequences, or

3577-410: The infiltration and synthesis via SIS allows the creation of materials with tailored properties such as composition , mechanics , stoichiometry , porosity , conductivity , refractive index , and chemical functionality on the nanoscale . SIS has been applied in various fields, including electronics , energy storage , AI and catalysis , due to its ability to control material properties. SIS

3650-489: The intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=VPI&oldid=1184613633 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Sequential infiltration synthesis This synthesis involves metal-organic vapor-phase precursors and co-reactants dissolving and diffusing into polymers , interacting with

3723-430: The kinetic diameter of the molecule cause large differences in diffusivity . Biologists often use the terms "net movement" or "net diffusion" to describe the movement of ions or molecules by diffusion. For example, oxygen can diffuse through cell membranes so long as there is a higher concentration of oxygen outside the cell. However, because the movement of molecules is random, occasionally oxygen molecules move out of

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3796-584: The liquid and solid lead. Yakov Frenkel (sometimes, Jakov/Jacob Frenkel) proposed, and elaborated in 1926, the idea of diffusion in crystals through local defects (vacancies and interstitial atoms). He concluded, the diffusion process in condensed matter is an ensemble of elementary jumps and quasichemical interactions of particles and defects. He introduced several mechanisms of diffusion and found rate constants from experimental data. Sometime later, Carl Wagner and Walter H. Schottky developed Frenkel's ideas about mechanisms of diffusion further. Presently, it

3869-498: The main challenges of SIS is the need to perform the process in an inert environment. Creation of a vacuum and/or introduction of inert gas carries costs that may be prohibitive for applications. A second challenge is the complexity of the diffusion-reaction process. Specifics of the reactor configuration and process parameters can impact the final product material substantially, which can complicate process optimization, reproduction, and scalability. Even though SIS can be applied to

3942-451: The main phenomenon was described by him in 1831–1833: "...gases of different nature, when brought into contact, do not arrange themselves according to their density, the heaviest undermost, and the lighter uppermost, but they spontaneously diffuse, mutually and equally, through each other, and so remain in the intimate state of mixture for any length of time." The measurements of Graham contributed to James Clerk Maxwell deriving, in 1867,

4015-446: The material. The process of SIS involves various key steps, the first of which is materials selection . A suitable substrate material , such as a polymer film, and precursors, typically molecules that can react with the substrate's functional groups, are selected for the infiltration synthesis. The pairing of polymer chemistry and precursor species is vital for acquiring the desired functionalisation and modification. The substrate

4088-401: The matrix of diffusion coefficients is ( i,k  > 0). There is intrinsic arbitrariness in the definition of the thermodynamic forces and kinetic coefficients because they are not measurable separately and only their combinations ∑ j L i j X j {\textstyle \sum _{j}L_{ij}X_{j}} can be measured. For example, in

4161-436: The mechanical properties of proteins is an early example of SIS application. For Spider dragline silk the toughness characteristic was significantly enhanced when metallic impurities Titanium or Aluminium were infiltrated in to the fibres. This fibre doping using SIS techniques attempts to mimic the effect of metallic impurities on silk properties observed in nature. As the advantages and disadvantages of SIS are outlined, it

4234-417: The modified layers' properties, including stability. Post-treatment may include heating, chemical treatment, or oxidation to remove the organic polymer. With SIS it is natural to apply to block copolymer substrates. Block copolymers such as polystyrene -block- poly(methyl methacrylate ), PS-b-PMMA, can spontaneously undergo microphase separation to form a rich variety of periodic mesoscale patterns. If

4307-415: The movement of fluid molecules in porous solids. Different types of diffusion are distinguished in porous solids. Molecular diffusion occurs when the collision with another molecule is more likely than the collision with the pore walls. Under such conditions, the diffusivity is similar to that in a non-confined space and is proportional to the mean free path. Knudsen diffusion occurs when the pore diameter

4380-466: The performance and stability of energy storage and conversion systems. Employing SIS and the correct precursors, the technique can modify the surfaces and interfaces of materials used in batteries , super-capacitors , and fuel cells , enhancing charge transport, electrochemical stability, and energy density. SIS is also being explored for its applications in photovoltaics , in which it can be used to engineer interfaces and increase light absorption. SIS

4453-459: The polymer constituents entirely to derive purely inorganic materials that maintain the structure of the original polymer morphology, including mesoporosity . Although the early research in SIS focused on a small number of inorganic materials such as Al 2 O 3 , TiO 2 , and ZnO, the technology diversified over the next decade and came to include a wide variety of both inorganic materials and organic polymers, as detailed in reviews. SIS

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4526-427: The polymers functional groups via reversible complex formation and/or irreversible chemical reactions , and yielding desired composite materials , which may be nanostructured . The metal-organic precursor (A) and co-react vapor (B) are supplied in an alternating ABAB sequence. Following SIS, the organic phase can be removed thermally or chemically to leave only the inorganic components behind. The precise control over

4599-401: The purpose to synthesize nanoscopic materials starting from block copolymer templates. A patent application was filed in 2011 and the first patent was issued in 2016. SIS involves vapor diffusing into an existing polymer and chemically or physically binding to it. This results in the growth and formation of inorganic structures by selective nucleation throughout the bulk polymer. With SIS,

4672-453: The reactor is purged with inert gas or evacuated to remove reaction byproducts and un-reacted precursors. A second vapor-phase species, often a co-reactant, such as H 2 O , is introduced. Again, the precursor partial pressure and exposure time are selected to allow sufficient time and thermodynamic driving force for diffusion into the polymer and reaction with the functional groups left by the first precursor exposure.  The second precursor

4745-454: The selective infiltration of specific precursors can enhance the sensitivity, selectivity, and response of sensors, which have applications in gas sensing, chemical sensing, biosensing, and environmental monitoring. SIS is also promising when used to engineer actuators with tunable properties, enabling the fabrication of devices on micro- and nanoscales for various applications. SIS has also shown promise in energy devices, especially in improving

4818-451: The shapes of various inorganic materials can be tailored by applying their precursor chemistries to patterned or nano-structured organic polymers, such as block copolymers. SIS was developed to intentionally enable the infusion of inorganic materials such as metal oxides and metals within polymers to yield hybrid materials with enhanced properties. Hybrid materials created via SIS can further be subjected to thermal annealing steps to remove

4891-451: The substrate would require exposure times of <1 s using the same precursors. Diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical potential . It is possible to diffuse "uphill" from

4964-517: The surface chemical groups on the substrate surface. In SIS, the partial pressures and exposure times for the precursor pulse are typically larger compared to ALD to allow for sufficient precursor to infiltrate a 3D volume by dissolution and diffusion . SIS utilizes the diffusive nature of precursor transport within polymers, making the distribution of precursors within the material sensitive to time, pressure , temperature , polymer chemistry , and micro-structure . The diffusion of precursors below

5037-504: The surfaces of polymers during ALD was observed in 2005 by the Steven M. George group when they observed that polymers could uptake trimethylaluminium (TMA) via absorption within their free volume. In this study, the interactions between the ALD precursors and the polymer functional groups were not recognized, and the diffusion of precursors into polymer films was considered a problem. Hence,

5110-407: The term "force" in quotation marks or "driving force"): where n i {\displaystyle n_{i}} are the "thermodynamic coordinates". For the heat and mass transfer one can take n 0 = u {\displaystyle n_{0}=u} (the density of internal energy) and n i {\displaystyle n_{i}} is the concentration of

5183-444: The terms with variation of the total pressure are neglected. It is possible for diffusion of small admixtures and for small gradients. For the linear Onsager equations, we must take the thermodynamic forces in the linear approximation near equilibrium: where the derivatives of s {\displaystyle s} are calculated at equilibrium n ∗ {\displaystyle n^{*}} . The matrix of

5256-500: Was created. For example, Pliny the Elder had previously described the cementation process , which produces steel from the element iron (Fe) through carbon diffusion. Another example is well known for many centuries, the diffusion of colors of stained glass or earthenware and Chinese ceramics . In modern science, the first systematic experimental study of diffusion was performed by Thomas Graham . He studied diffusion in gases, and

5329-554: Was developed by Albert Einstein , Marian Smoluchowski and Jean-Baptiste Perrin . Ludwig Boltzmann , in the development of the atomistic backgrounds of the macroscopic transport processes , introduced the Boltzmann equation , which has served mathematics and physics with a source of transport process ideas and concerns for more than 140 years. In 1920–1921, George de Hevesy measured self-diffusion using radioisotopes . He studied self-diffusion of radioactive isotopes of lead in

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