7 Key excerpts on "Diffusion in Materials"

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  eBook - ePub

  Modern Physical Metallurgy

  • R. E. Smallman, A.H.W. Ngan(Authors)
  • 2013(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  Chapter 7

  Diffusion

  Diffusion is an important process by which point defects and solute atoms can migrate inside solid materials at elevated temperatures. Diffusion is governed by Fick’s two laws, the first relating diffusional speed to concentration gradient as driving force and the second on conservation of mass. Mathematical solution to Fick’s laws depends on configurations, such as thin film versus surface. Microscopically, diffusion involves atomic jumps and is governed by activation energies of vacancy formation and migration. Diffusion is important in industrial processes, such as carburization, and material behaviour, such as creep. Enhanced diffusion may take place along short-circuit paths, including dislocation cores and grain boundaries.

  Keywords

  Diffusion; Fick’s laws; steady-state diffusion; non-steady-state diffusion; error function; carburization; thin-film diffusion; surface diffusion; short-circuited diffusion

  7.1 Introduction

  Everyone has some experience with diffusion from an early age. Brewing tea, mixing colours and washing a bleeding cut in water sees the spreading of ‘colour’ molecules by random exchanges with water molecules. They move from regions where they are concentrated to dilute regions. This movement down a concentration gradient is the basis of diffusion. Such diffusion behaviour is not unique to liquids but occurs in solids as well by thermally activated movement of solute atoms in the solvent lattice.

  7.2 Diffusion laws

  A knowledge of diffusion theory is essential in understanding many areas of physical metallurgy, particularly at elevated temperatures. A few examples include such commercially important processes as annealing, heat treatment, the age hardening of alloys, sintering, surface hardening, oxidation and creep. Apart from the specialized diffusion processes, such as grain boundary diffusion and diffusion down dislocation channels, a distinction is frequently drawn between diffusion in pure metals, homogeneous alloys and inhomogeneous alloys. In a pure material self-diffusion can be observed by using radioactive tracer atoms. In a homogeneous alloy diffusion of each component can also be measured by a tracer method, but in an inhomogeneous alloy, diffusion can be determined by chemical analysis merely from the broadening of the interface between the two metals as a function of time. Inhomogeneous alloys are common in metallurgical practice (e.g. cored solid solutions), and in such cases diffusion always occurs in such a way as to produce a macroscopic flow of solute atoms down the concentration gradient. Thus, if a bar of an alloy, along which there is a concentration gradient (Figure 7.1

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  eBook - ePub

  Physical Metallurgy

  Metals, Alloys, Phase Transformations

  • Vadim M. Schastlivtsev, Vitaly I. Zel'dovich(Authors)
  • 2022(Publication Date)
  • De Gruyter

    (Publisher)

  Chapter 4  Diffusion in metals

  Diffusion in a system is usually called the process of the spontaneous motion of atoms for equalizing their concentrations. However, this definition is not accurate. A driving force for the diffusion transfer is the difference in chemical potentials at different points of the system rather than the difference in concentrations. As an example, uphill diffusion (diffusion in a stress field) refers to a process of increasing the concentration difference. Another example is given in Section 4.8, when describing the Darken experiment. Despite this, in many cases, the description of diffusion processes may leave aside the concept of chemical potential.

  Diffusion processes play an important role in physical metallurgy. Firstly, diffusion takes place in many high-temperature phase transformations and structural changes such as decomposition of supersaturated solid solutions, coagulation, recrystallization, homogenization, creep, oxidation, chemical heat treatment, and sintering. Secondly, diffusion is the main tool for studying crystal structure defects, for example, point defects and grain boundaries.

  4.1  Phenomenological theory of diffusion: Fick’s laws

  A diffusion flux is described by the same differential equation as a heat flux in the case of thermal conductivity. In one case, this is the transfer of heat; in the other case, it is the transfer of atoms of a dissolved substance. The one-dimensional heat equation has the form

  d Q = − λ S   ( ∂ T / ∂ x )   d t ,

  where dQ is the heat flux through the area S over a period of time dt, λ is the coefficient of thermal conductivity, and

  ∂ T

  /

  ∂ x

  is the temperature gradient. By analogy, the diffusion equation appears as follows:

  d q = − D S   ( ∂ c / ∂ x ) d t ,

  where dq is the substance flux through the area S over a period of time dt, D is the diffusion coefficient, and

  ∂ c

  /

  ∂ x

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  eBook - ePub

  Developing Solid Oral Dosage Forms

  Pharmaceutical Theory and Practice

  • Yihong Qiu, Yisheng Chen, Geoff G.Z. Zhang, Lawrence Yu, Rao V. Mantri(Authors)
  • 2016(Publication Date)
  • Academic Press

    (Publisher)

  Brownian motion. Diffusion is the process by which atoms, molecules, or small particles are transported from a region of higher concentration to a region of lower concentration due to random motion. In the absence of convection, Brownian motion or diffusion is a primary transport mechanism for small particles (<0.1 µm) or molecules when the transport distance is small (ie, < a few millimeters). Over larger distances, diffusion is a slow process for material transfer, and other factors like convection are important to obtain significant transport of material in a reasonable time period (ie, <1 day). In dissolution testing, diffusion and convection are combined processes that generate dissolved drug in solution, in which convection is the dominating mechanism.

  The diffusion process is driven by entropy change (ie, ΔS > 0 for diffusion). The process is an example of the second law of thermodynamics; no enthalpy is required in the process (ie, no heat or work is required, but heat can affect the rate of diffusion). Diffusion is irreversible because it can occur in only one direction—from high to low concentrations.

  The exact movement of any individual molecule (or small number of molecules) is completely random. The overall behavior (ie, macroscopic) of a large population of molecules (>106 ) is quite predictable. Diffusion is a process of mass transport that involves the movement of one molecular species through another (ie, a drug in a gel or solution). It occurs by random molecular motion from one point to another, and it can occurs in gas, liquid, or solid states. Diffusion in liquid, gel, or polymeric systems is of primary interest in the pharmaceutical sciences.

  The subject of diffusion has been well studied, and a variety of models and their mathematical expressions related to pharmaceutical applications are available in the literature.

  1 –6

  The first part of this chapter provides a review of the fundamental theory and the basic equations of diffusion to facilitate the further application of diffusion theory in pharmaceutical sciences.

  9.1.2 Basic Equations of Diffusion

  The two basic equations describing the diffusion process are Fick’s first and second laws.

  Fick’s first law : The rate of transfer (dQ/dt ) of a diffusing substance through the unit area (A ) in one dimension (x ) is proportional to the concentration gradient,

  ∂ C / ∂ x

  , as described in Eq. (9.1)

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  eBook - ePub

  An Introduction to Aspects of Thermodynamics and Kinetics Relevant to Materials Science

  • Eugene Machlin(Author)
  • 2010(Publication Date)
  • Elsevier Science

    (Publisher)

  CHAPTER VIII

  Diffusion

  Publisher Summary

  Most of the kinetic phenomena in solids involve diffusion, or a unit step very similar to that operating in diffusion. This chapter provides a study of kinetic phenomena with a study of the fundamentals involved in diffusion in various materials. Further illustrates a phenomenological treatment of linear processes that is based on the thermodynamic theory of irreversible processes. This general approach is made specific by application to diffusion in a binary alloy. Diffusion in ionic crystals and semiconductors essentially follows on the basis of the relations developed for metals and the thermodynamics of defects in these materials. The chapter discusses diffusion along high diffusivity regions, such as grain boundaries. Description of computer assisted strategies for extending the basic concepts to problems involving multicomponent diffusion in technological applications is reviewed. Diffusion of polymer strands is also presented briefly.

  Introduction

  Most kinetic phenomena in solids involve diffusion, or a unit step very similar to that operating in diffusion. It is appropriate therefore to start our study of kinetic phenomena with a study of the fundamentals involved in diffusion in various materials. We cannot provide a complete description of diffusion because, as with nearly every other chapter in this book, such a complete description requires a book in itself. This chapter begins with a phenomenological treatment of linear processes that is based on the thermodynamic theory of irreversible processes. This very general approach is made specific by application to diffusion in a binary alloy. Diffusion in ionic crystals and semiconductors follows on the basis of the relations developed for metals and the thermodynamics of defects in these materials. Then diffusion along high diffusivity regions, such as grain boundaries, is discussed. Finally, the current computer assisted strategies for extending the basic concepts described in this chapter to problems involving multicomponent diffusion in technological applications are discussed. Also, diffusion of polymer strands is treated briefly. Diffusion in liquids is not considered in this chapter.

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  eBook - ePub

  Physical Properties of Concrete and Concrete Constituents

  • Jean-Pierre Ollivier, Jean-Michel Toorenti, Myriam Carcasses(Authors)
  • 2012(Publication Date)
  • Wiley-ISTE

    (Publisher)

  Chapter 6The Fundamentals of Diffusion      

  6.1. The basics of diffusion

  Diffusion is a process that occurs at the atomic level and which can lead, under certain conditions, to the transport of matter on a macroscopic scale. It occurs in many situations, and this chapter is particularly concerned with the diffusion of the particles of a solute dispersed in a fluid (solvent), and with gaseous diffusion, which will be introduced in the latter part of the chapter.

  In fluids, the particles of a solute are subject to Brownian movement under the effect of thermal agitation. This movement was described in 1827 by the botanist R. Brown, who observed the displacement of pollen grains in water. It results from the successive impacts that a particle undergoes with the molecules of the fluid in which it is located. This displacement of particles by a succession of jumps occurring at random in different directions is the phenomenon of diffusion.

  Before considering the macroscopic transport of matter by diffusion, we will define the diffusion coefficient in the following section by considering a microscopic approach to this phenomenon.

  6.1.1. Microscopic approach to diffusion

  Figure 6.1 illustrates a random progression due to Brownian movement.

  Figure 6.1.

  The Brownian movement of a particle. It shows random forward movement under the influence ofsuccessive impacts [TOU 05]

  6.1.1.1. The case of fluids

  Imagine the displacement of a particle via the random movement described in Figure 6.1 , and consider the approach proposed by Tourrenc [TOU 05].

  A solute particle starting from position A reaches position B after N impacts, including the first impact at position A. The trajectory between two collisions at points

  Pn−1

  and

  Pn

  is rectilinear, and can be characterized by the vector .

  Its length, l , and its orientation at each point, P , have the same laws of probability (homogeneity hypothesis), and the orientations of are equiprobable (isotropy hypotheses).

  Let us take . The hypotheses of homogeneity and isotropy imply that the average value of is zero, and that it can be written as:

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  eBook - ePub

  Introduction to Microfabrication

  • Sami Franssila(Author)
  • 2010(Publication Date)
  • Wiley

    (Publisher)

  14 Diffusion

  The power of silicon microelectronics technology stems from the ability to tailor dopant concentrations over eight orders of magnitude by introducing suitable n- or p-type dopants into the silicon. The upper limit is set by the solid solubility of the dopants (about 1021 cm−3 ), the lower limit (about 1012 cm−3 ) by impurities which result from the silicon crystal growth. This enables a wealth of microstructures and devices, witnessed by the multiplicity of diode, transistor, thyristor and other semiconductor device designs. In silicon IC technology dopant diffusion is such a key step that the country of origin of semiconductor devices is defined as the country where diffusions were made.

  Dopants can be introduced into silicon by five different methods:

  • during crystal growth
  • by neutron transmutation doping (NTD)
  • during epitaxy
  • by diffusion
  • by ion implantation.

  The first two techniques are applied to whole ingots, and epitaxy results in a uniformly doped layer all over the wafer. Diffusion and ion implantation are techniques to locally vary the dopant concentration, and they are the topics of this and the following chapter.

  14.1 Diffusion Process

  Diffusion is the movement of atoms along concentration gradients. Atoms from high-concentration areas move to areas of lower concentration (and if we wait long enough, there will be no concentration gradients). In microtechnology diffusion is a technique to introduce and drive boron, phosphorus and other dopant atoms into the silicon lattice.

  Thermal diffusion is a high-temperature process: diffusion temperatures for the common dopants are in the range of 900–1200 °C. Diffusion furnaces are identical to oxidation furnaces, and diffusion is a batch process where long process times are compensated by a huge loads, 100 or even 200 wafers, in a batch.

  Thermal diffusion can be done from the gas phase. In gas phase doping the wafers are put in a furnace and a suitable doping gas, POCl3 for phosphorus doping, or BBr3 for boron doping, is introduced. The wafers are exposed to dopant atom vapors and doped (Figure 14.1 ). The alternative technique is diffusion from doped thin films. For example, boron-doped polysilicon, phosphorus-doped silica glass (PSG) or doped spin-on glass is deposited on the wafer, which is then put into a furnace. Dopants from the doped film diffuse into the silicon. The junction depth (x j ) is the depth where diffused dopant concentration equals substrate dopant concentration. Solid solubilities of dopants in silicon are shown in Figure 14.2

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  eBook - ePub

  Advances in Heat Transfer Unit Operations

  Baking and Freezing in Bread Making

  • Georgina Calderon-Dominguez, Gustavo F. Gutierrez-Lopez, Keshavan Niranjan, Georgina Calderon-Dominguez, Gustavo F. Gutierrez-Lopez, Keshavan Niranjan(Authors)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  entropy. The change in entropy is equal to the heat transfer divided by the temperature:

  Δ S = Δ Q T ,

  (2.3)

  where S is entropy, Q is heat, and T is temperature.

  2.1.2  MOLECULAR DIFFUSION AND MASS TRANSPORT

  Molecular diffusion refers to the diffusive transport of species (movement of molecules) due to the concentration gradients in a mixture. The diffusion of molecules occurs in the direction required to balance the concentration gradient.

  Diffusion is caused by a random molecular motion being the consequence of thermally induced agitation of molecules, which finally tends to complete the homogenization of the mixture. The very low rate of diffusion is due to molecular collisions, which occur at a rate of millions per second per cubic centimeter producing an extremely strong hindering of the movement of molecules.

  Therefore, diffusion occurs more intensively at high temperatures (high-molecular average velocities) and at low pressures (lower concentration of molecules, fewer collisions). The molar mass of the molecules also influences the rate of diffusion as light molecules move more rapidly than the heavy ones.

  Mass transport is a macroscopic process in which the portions of fluid are moved over much larger distances than in the diffusion process, carrying the transferred component from regions of high concentrations to low concentrations. This process is generated by agitation or by currents and eddies of the turbulent flow. However, the mass transfer between the newly adjacent currents of fluids proceeds by means of diffusion, which mixes the portions of fluids.

  In general, all the process of heat, mass, and momentum transfer are explained by the general transport equation:

  Process   transport   velocity =

  driving   force

  resistance

  .

  This equation expresses the need of a driving force that is opposite to a resistance to transfer a property. Mathematically, the general equation of molecular transport of a property is expressed as

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