Page 1 B: DIFFUSION & MECHANICAL PROPERTIES B1: Diffusion. B2: Deformation in Materials. B3: Fracture, Fatigue and Creep. B4: Strengthening Mechanisms. Page 2 B: DIFFUSION & MECHANICAL PROPERTIES B1: Diffusion. B2: Deformation in Materials. B3: Fracture, Fatigue and Creep. B4: Strengthening Mechanisms. B1: DIFFUSION B1.1 Introduction In a sample having non-uniform composition of certain kind of atoms, a concentration gradient is set in, which leads to migration of atoms from region of their higher concentration to the region of lower concentration. This phenomenon is referred to as Diffusion and continues till atomic distribution becomes homogeneous throughout. The diffusion process plays an important role in the field of metallurgy and fabrication of extrinsic semiconductors. The technology of controlled diffusion plays a sensitive role in accomplishment of desired devices. The metals, in their pure form, are soft and do not possess large tensile strength. Their properties can be modified by the process of alloying in a controlled manner. The desired properties of alloys are achieved by proper choice of solute (atoms to be diffused) and their amount to be diffused. This is accomplished through sophisticated techniques and instruments. Further the dynamical processes of bio-systems utilize the phenomenon of diffusion in wide variety of ways which include the role of blood circulation leading to nourishment of various body organs and also waste disposal from them. The control of heart beat through hormonal diffusion and digestion of food through enzymatic action are some of the examples. Further the role of diffusion in pharmaceutical applications is undisputed through the drug therapy curing a specific portion of body. B1.2 Basics of Diffusion The process of diffusion involves two components: (i) solvent is the medium in which atoms of similar or different kinds are to be diffused and (ii) solute are the atoms of one or more kinds which are to be diffused into the environment of solvent. The solute atoms migrate and distribute themselves in solvent in process of which they may become interstitial impurity (occupying interstitial spaces in solvent lattice) or substitution impurity (substituting the solvent atom from its regular position in lattice). The process of diffusion can be classified into two types which are (a) Self-diffusion in which solute and solvent atoms are identical. For example diffusion of copper in copper (b) Inter-diffusion in which the solute atoms are different from the solvent atoms as in the case steel where solvent is iron whereas solute atoms are those of carbon and many other elements. The phenomenon of diffusion of solute atoms into the solvent environment may occur through the following mechanisms: (a) Vacancy mechanism where the solute atoms lie in the vacancy and moves to other available vacancy. Atoms can move from one site to Prof. J.K.Goswamy’s Notes on Materials Science: Properties of Materials. Page 150 Page 3 B: DIFFUSION & MECHANICAL PROPERTIES B1: Diffusion. B2: Deformation in Materials. B3: Fracture, Fatigue and Creep. B4: Strengthening Mechanisms. B1: DIFFUSION B1.1 Introduction In a sample having non-uniform composition of certain kind of atoms, a concentration gradient is set in, which leads to migration of atoms from region of their higher concentration to the region of lower concentration. This phenomenon is referred to as Diffusion and continues till atomic distribution becomes homogeneous throughout. The diffusion process plays an important role in the field of metallurgy and fabrication of extrinsic semiconductors. The technology of controlled diffusion plays a sensitive role in accomplishment of desired devices. The metals, in their pure form, are soft and do not possess large tensile strength. Their properties can be modified by the process of alloying in a controlled manner. The desired properties of alloys are achieved by proper choice of solute (atoms to be diffused) and their amount to be diffused. This is accomplished through sophisticated techniques and instruments. Further the dynamical processes of bio-systems utilize the phenomenon of diffusion in wide variety of ways which include the role of blood circulation leading to nourishment of various body organs and also waste disposal from them. The control of heart beat through hormonal diffusion and digestion of food through enzymatic action are some of the examples. Further the role of diffusion in pharmaceutical applications is undisputed through the drug therapy curing a specific portion of body. B1.2 Basics of Diffusion The process of diffusion involves two components: (i) solvent is the medium in which atoms of similar or different kinds are to be diffused and (ii) solute are the atoms of one or more kinds which are to be diffused into the environment of solvent. The solute atoms migrate and distribute themselves in solvent in process of which they may become interstitial impurity (occupying interstitial spaces in solvent lattice) or substitution impurity (substituting the solvent atom from its regular position in lattice). The process of diffusion can be classified into two types which are (a) Self-diffusion in which solute and solvent atoms are identical. For example diffusion of copper in copper (b) Inter-diffusion in which the solute atoms are different from the solvent atoms as in the case steel where solvent is iron whereas solute atoms are those of carbon and many other elements. The phenomenon of diffusion of solute atoms into the solvent environment may occur through the following mechanisms: (a) Vacancy mechanism where the solute atoms lie in the vacancy and moves to other available vacancy. Atoms can move from one site to Prof. J.K.Goswamy’s Notes on Materials Science: Properties of Materials. Page 150 another if there are vacancies and the atom has sufficient energy to overcome a local activation energy barrier. The activation energy for diffusion is the sum of energy required to form a vacancy and further cause its motion. (b) Interstitial Mechanism is applicable in case of small atoms like hydrogen, helium, carbon, nitrogen diffuse through the interstitial spaces in a crystal. The activation energy for diffusion is the energy required for these atoms to squeeze through the small voids between the host lattice atoms. B1.3 Macroscopic Model of Diffusion B1.3.1 Steady State Diffusion: Fick’s I Law The process of diffusion is time dependent which implies that one needs to study the quantity of solute material that gets transported in solvent is a function of time. This time dependent mass transfer is often represented by a quantity called diffusion flux (J) which is defined as the solute mass transferred in unit time through a unit cross- sectional area placed perpendicular to the direction of flow. The diffusion flux can be mathematically represented as: dt dM A J 1 = The units of J are kg/m 2 S or atoms/m 2 s. If the diffusion flux does not change with time, a steady state condition exists. Common example of such a process is diffusion of atoms of gas through a metal plate for which the concentration of diffusing species (see figure B1.1(a) ) on both sides of plates is kept constant. If concentration of diffusing gas atoms is plotted as a function of position or depth, then the curve is referred to as concentration profile, which is shown for the steady state case under discussion in figure B1.1(b) as: Figure B1.1: (a) Steady state diffusion across a thin plate (b) a linear concentration profile for the given diffusion situation. Mathematically, the concentration gradient is expressed as: dx dC gradient ion Concentrat = The kinetics of steady state diffusion process is described by Fick’s first law which states that the diffusion current (J) is directly proportional to the concentration gradient ? ? ? ? ? ? ? ? x c . Hence we express this mathematically as: ) 1 . 1 (B x c D J ? ? - = Prof. J.K.Goswamy’s Notes on Materials Science: Properties of Materials. Page 151 Page 4 B: DIFFUSION & MECHANICAL PROPERTIES B1: Diffusion. B2: Deformation in Materials. B3: Fracture, Fatigue and Creep. B4: Strengthening Mechanisms. B1: DIFFUSION B1.1 Introduction In a sample having non-uniform composition of certain kind of atoms, a concentration gradient is set in, which leads to migration of atoms from region of their higher concentration to the region of lower concentration. This phenomenon is referred to as Diffusion and continues till atomic distribution becomes homogeneous throughout. The diffusion process plays an important role in the field of metallurgy and fabrication of extrinsic semiconductors. The technology of controlled diffusion plays a sensitive role in accomplishment of desired devices. The metals, in their pure form, are soft and do not possess large tensile strength. Their properties can be modified by the process of alloying in a controlled manner. The desired properties of alloys are achieved by proper choice of solute (atoms to be diffused) and their amount to be diffused. This is accomplished through sophisticated techniques and instruments. Further the dynamical processes of bio-systems utilize the phenomenon of diffusion in wide variety of ways which include the role of blood circulation leading to nourishment of various body organs and also waste disposal from them. The control of heart beat through hormonal diffusion and digestion of food through enzymatic action are some of the examples. Further the role of diffusion in pharmaceutical applications is undisputed through the drug therapy curing a specific portion of body. B1.2 Basics of Diffusion The process of diffusion involves two components: (i) solvent is the medium in which atoms of similar or different kinds are to be diffused and (ii) solute are the atoms of one or more kinds which are to be diffused into the environment of solvent. The solute atoms migrate and distribute themselves in solvent in process of which they may become interstitial impurity (occupying interstitial spaces in solvent lattice) or substitution impurity (substituting the solvent atom from its regular position in lattice). The process of diffusion can be classified into two types which are (a) Self-diffusion in which solute and solvent atoms are identical. For example diffusion of copper in copper (b) Inter-diffusion in which the solute atoms are different from the solvent atoms as in the case steel where solvent is iron whereas solute atoms are those of carbon and many other elements. The phenomenon of diffusion of solute atoms into the solvent environment may occur through the following mechanisms: (a) Vacancy mechanism where the solute atoms lie in the vacancy and moves to other available vacancy. Atoms can move from one site to Prof. J.K.Goswamy’s Notes on Materials Science: Properties of Materials. Page 150 another if there are vacancies and the atom has sufficient energy to overcome a local activation energy barrier. The activation energy for diffusion is the sum of energy required to form a vacancy and further cause its motion. (b) Interstitial Mechanism is applicable in case of small atoms like hydrogen, helium, carbon, nitrogen diffuse through the interstitial spaces in a crystal. The activation energy for diffusion is the energy required for these atoms to squeeze through the small voids between the host lattice atoms. B1.3 Macroscopic Model of Diffusion B1.3.1 Steady State Diffusion: Fick’s I Law The process of diffusion is time dependent which implies that one needs to study the quantity of solute material that gets transported in solvent is a function of time. This time dependent mass transfer is often represented by a quantity called diffusion flux (J) which is defined as the solute mass transferred in unit time through a unit cross- sectional area placed perpendicular to the direction of flow. The diffusion flux can be mathematically represented as: dt dM A J 1 = The units of J are kg/m 2 S or atoms/m 2 s. If the diffusion flux does not change with time, a steady state condition exists. Common example of such a process is diffusion of atoms of gas through a metal plate for which the concentration of diffusing species (see figure B1.1(a) ) on both sides of plates is kept constant. If concentration of diffusing gas atoms is plotted as a function of position or depth, then the curve is referred to as concentration profile, which is shown for the steady state case under discussion in figure B1.1(b) as: Figure B1.1: (a) Steady state diffusion across a thin plate (b) a linear concentration profile for the given diffusion situation. Mathematically, the concentration gradient is expressed as: dx dC gradient ion Concentrat = The kinetics of steady state diffusion process is described by Fick’s first law which states that the diffusion current (J) is directly proportional to the concentration gradient ? ? ? ? ? ? ? ? x c . Hence we express this mathematically as: ) 1 . 1 (B x c D J ? ? - = Prof. J.K.Goswamy’s Notes on Materials Science: Properties of Materials. Page 151 The negative sign indicates that solute atoms migrate down the concentration gradient. The constant D is called diffusion coefficient which is characteristic of solute atoms and solvent environment and depends upon the following factors: (i) temperature of solvent medium (ii) mechanism of diffusion (iii) type of crystal structure of solvent, (iv) imperfections in crystal (v) concentration of diffusing species. The Fick’s law can be understood to a greater extent through a kinetic model. Let’s consider a one dimensional flow of solute atoms in normal direction through section S as shown in the figure B1.2 below: Figure B1.2: The diagram showing the one dimensional flow of atoms through the intermediate plane S through their jumping in either of two possible directions. The concentration of solute atoms in adjacent planes are indicated by c 1 and c 2 (c 1 > c 2 ) respectively. The atoms on both the planes jump randomly to right as well as left. The atoms of plane 1 which jump to right and those of plane 2, which jump to left will cross the section S in opposite directions. Since the net flow of solute atoms will be towards right so we can write diffusion current as: ) 2 . 1 ( 2 1 2 1 2 1 B v n v n J - = In the equation (B1.2), n 1 and n 2 represent planar densities of solute atoms at planes 1 and 2 respectively while v represents the jump frequency. The factor of half accounts for equal probability of jumping in one of the two possible directions leading to net rate of flow through section S. The concentrations c 1 and c 2 are defined as number of solute atoms per unit volume at plane 1 and 2 respectively. Hence we can have relation between atomic density and concentration of solute atoms at a plane as n = ca. Using this relation in (B1.2), we get: ) 3 . 1 ( ) ( 2 1 2 1 B c c av J - = If the concentration does not vary rapidly between the planes 1 and 2, then we can write as: ) 4 . 1 ( 2 1 B x c a c c ? ? - = - Putting (B1.4) in (B1.3), we get: ) 5 . 1 ( 2 1 2 B x c v a J ? ? - = Comparing the equations (B1.1) and (B1.5), the diffusion coefficient D in one dimensional case can be expressed as: S c 1 c 2 a Prof. J.K.Goswamy’s Notes on Materials Science: Properties of Materials. Page 152 Page 5 B: DIFFUSION & MECHANICAL PROPERTIES B1: Diffusion. B2: Deformation in Materials. B3: Fracture, Fatigue and Creep. B4: Strengthening Mechanisms. B1: DIFFUSION B1.1 Introduction In a sample having non-uniform composition of certain kind of atoms, a concentration gradient is set in, which leads to migration of atoms from region of their higher concentration to the region of lower concentration. This phenomenon is referred to as Diffusion and continues till atomic distribution becomes homogeneous throughout. The diffusion process plays an important role in the field of metallurgy and fabrication of extrinsic semiconductors. The technology of controlled diffusion plays a sensitive role in accomplishment of desired devices. The metals, in their pure form, are soft and do not possess large tensile strength. Their properties can be modified by the process of alloying in a controlled manner. The desired properties of alloys are achieved by proper choice of solute (atoms to be diffused) and their amount to be diffused. This is accomplished through sophisticated techniques and instruments. Further the dynamical processes of bio-systems utilize the phenomenon of diffusion in wide variety of ways which include the role of blood circulation leading to nourishment of various body organs and also waste disposal from them. The control of heart beat through hormonal diffusion and digestion of food through enzymatic action are some of the examples. Further the role of diffusion in pharmaceutical applications is undisputed through the drug therapy curing a specific portion of body. B1.2 Basics of Diffusion The process of diffusion involves two components: (i) solvent is the medium in which atoms of similar or different kinds are to be diffused and (ii) solute are the atoms of one or more kinds which are to be diffused into the environment of solvent. The solute atoms migrate and distribute themselves in solvent in process of which they may become interstitial impurity (occupying interstitial spaces in solvent lattice) or substitution impurity (substituting the solvent atom from its regular position in lattice). The process of diffusion can be classified into two types which are (a) Self-diffusion in which solute and solvent atoms are identical. For example diffusion of copper in copper (b) Inter-diffusion in which the solute atoms are different from the solvent atoms as in the case steel where solvent is iron whereas solute atoms are those of carbon and many other elements. The phenomenon of diffusion of solute atoms into the solvent environment may occur through the following mechanisms: (a) Vacancy mechanism where the solute atoms lie in the vacancy and moves to other available vacancy. Atoms can move from one site to Prof. J.K.Goswamy’s Notes on Materials Science: Properties of Materials. Page 150 another if there are vacancies and the atom has sufficient energy to overcome a local activation energy barrier. The activation energy for diffusion is the sum of energy required to form a vacancy and further cause its motion. (b) Interstitial Mechanism is applicable in case of small atoms like hydrogen, helium, carbon, nitrogen diffuse through the interstitial spaces in a crystal. The activation energy for diffusion is the energy required for these atoms to squeeze through the small voids between the host lattice atoms. B1.3 Macroscopic Model of Diffusion B1.3.1 Steady State Diffusion: Fick’s I Law The process of diffusion is time dependent which implies that one needs to study the quantity of solute material that gets transported in solvent is a function of time. This time dependent mass transfer is often represented by a quantity called diffusion flux (J) which is defined as the solute mass transferred in unit time through a unit cross- sectional area placed perpendicular to the direction of flow. The diffusion flux can be mathematically represented as: dt dM A J 1 = The units of J are kg/m 2 S or atoms/m 2 s. If the diffusion flux does not change with time, a steady state condition exists. Common example of such a process is diffusion of atoms of gas through a metal plate for which the concentration of diffusing species (see figure B1.1(a) ) on both sides of plates is kept constant. If concentration of diffusing gas atoms is plotted as a function of position or depth, then the curve is referred to as concentration profile, which is shown for the steady state case under discussion in figure B1.1(b) as: Figure B1.1: (a) Steady state diffusion across a thin plate (b) a linear concentration profile for the given diffusion situation. Mathematically, the concentration gradient is expressed as: dx dC gradient ion Concentrat = The kinetics of steady state diffusion process is described by Fick’s first law which states that the diffusion current (J) is directly proportional to the concentration gradient ? ? ? ? ? ? ? ? x c . Hence we express this mathematically as: ) 1 . 1 (B x c D J ? ? - = Prof. J.K.Goswamy’s Notes on Materials Science: Properties of Materials. Page 151 The negative sign indicates that solute atoms migrate down the concentration gradient. The constant D is called diffusion coefficient which is characteristic of solute atoms and solvent environment and depends upon the following factors: (i) temperature of solvent medium (ii) mechanism of diffusion (iii) type of crystal structure of solvent, (iv) imperfections in crystal (v) concentration of diffusing species. The Fick’s law can be understood to a greater extent through a kinetic model. Let’s consider a one dimensional flow of solute atoms in normal direction through section S as shown in the figure B1.2 below: Figure B1.2: The diagram showing the one dimensional flow of atoms through the intermediate plane S through their jumping in either of two possible directions. The concentration of solute atoms in adjacent planes are indicated by c 1 and c 2 (c 1 > c 2 ) respectively. The atoms on both the planes jump randomly to right as well as left. The atoms of plane 1 which jump to right and those of plane 2, which jump to left will cross the section S in opposite directions. Since the net flow of solute atoms will be towards right so we can write diffusion current as: ) 2 . 1 ( 2 1 2 1 2 1 B v n v n J - = In the equation (B1.2), n 1 and n 2 represent planar densities of solute atoms at planes 1 and 2 respectively while v represents the jump frequency. The factor of half accounts for equal probability of jumping in one of the two possible directions leading to net rate of flow through section S. The concentrations c 1 and c 2 are defined as number of solute atoms per unit volume at plane 1 and 2 respectively. Hence we can have relation between atomic density and concentration of solute atoms at a plane as n = ca. Using this relation in (B1.2), we get: ) 3 . 1 ( ) ( 2 1 2 1 B c c av J - = If the concentration does not vary rapidly between the planes 1 and 2, then we can write as: ) 4 . 1 ( 2 1 B x c a c c ? ? - = - Putting (B1.4) in (B1.3), we get: ) 5 . 1 ( 2 1 2 B x c v a J ? ? - = Comparing the equations (B1.1) and (B1.5), the diffusion coefficient D in one dimensional case can be expressed as: S c 1 c 2 a Prof. J.K.Goswamy’s Notes on Materials Science: Properties of Materials. Page 152 ) 6 . 1 ( 2 1 2 B va D = If we consider the three dimensional case then the atom can jump in any random direction and its probability of crossing the section S will be 6 1 instead of half and the diffusion coefficient will become: ) 7 . 1 ( 6 1 2 B va D = The values of D can range from 10 -20 –10 -50 m 2 /s which correspond to jumping frequencies ranging from 10 -20 Hz to 1Hz. This indicates that the diffusion is an extremely slow process. B1.3.2 Non-Steady State Diffusion: Fick’s II Law The diffusion of solute atoms in solvent environment obeys the continuity equation given as: ) 8 . 1 ( 0 B x J t c = ? ? + ? ? The above given equation B1.8 can be interpreted as: If we consider a closed solvent volume characterized by certain concentration of solute in it. Then ant change in concentration of solute as a function of time implies inflow (or outflow) of solvent through the considered closed volume. This equation represents generalized form of conservation law. In this particular case, it is indicative conservation of solute mass. Putting equation (B1.1) in (B1.8), we get: ) 9 . 1 ( 2 2 B x c D t c x c D x t c ? ? = ? ? ? ? ? ? ? ? ? ? - ? ? - = ? ? The equation (B1.9) is called the Fick’s Second law. The solution of equation (B1.9) (i.e. concentration as a function of position and time) are possible when meaningful boundary conditions are specified. The Fick’s second law can be applied to the case of one dimensional case flow of solute in semi-infinite medium. The basic assumptions for this case are: (a) Before diffusion, the diffusing solute atoms in the solid are uniformly distributed with the concentration C o . (b) The value of x is zero at surface of semi-infinite medium and increases with distance into the solid. (c) The time is taken to be zero at the instant the process of diffusion begins. Hence the boundary conditions for this diffusion process are: ) 10 . 1 ( 0 0 0 0 0 B x at C C x at C C t For x at C C t For s s ? ? ? ? ? ? 8 = = = = > 8 = = = = The application of boundary conditions given by equation B1.10 in equation B1.9 gives the solution as: ) 11 . 1 ( 2 1 0 0 B Dt x erf C C C C s x ? ? ? ? ? ? ? ? - = - - Here Cx is the concentration of solute atoms at depth x after time t. The ? ? ? ? ? ? ? ? Dt x erf 2 denotes the Gaussian error function. The equation B1.11 Prof. J.K.Goswamy’s Notes on Materials Science: Properties of Materials. Page 153Read More

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