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# What is fick's second law of diffusion? Description

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## What is fick's second law of diffusion?

Chapter 5: Atom and Ion Movements in Materials (Continued)

 Composition Profile (Fick’s Second Law):

∙ Describes the dynamic diffusion of atoms. Used for non-steady state  systems.

∂ c

∂ t=∂∂ x(D∂ c

∂ x)

∙ The solution to this past equation depends on the boundary conditions for  a particular situation. One of the solutions is:

cs−c x

cs−co=erf ?(x

2√ Dt)

## Why is it called the error function?

cs = Constant concentration of the diffusing atoms at the surface of the material. co = Initial uniform concentration of the diffusing atoms in the material. cx = Concentration of the diffusing atom at location x below the surface after time t.

∙ This previous equation assumes a one dimensional model (atoms move in  the x-direction).

∙ The erf function is called the error function and it can be evaluated from  Table 5-3 or Figure 5-19. The mathematical function for this function is: erf ( x )=2√ᴨ∫0xexp(−y2) dy

## How do you increase the grain size of a metal?

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∙ These solutions to Fick’s second law allow us to determine the  concentration of diffusing species as a function of time (t) and distance  (x).

 Diffusion and Materials Processing:

∙ Melting and Casting:

- One of the most used processes to process metal, alloys, plastics,  and glasses that mainly consists in the melting and casting of

materials into alloys.

- Diffusion is an important factor in the solidification of metals. For  example, difference in diffusion of dopants in both molten and solid We also discuss several other topics like What is the definition of a classification pathway?

forms must be taken into account during the growth of

semiconductors’ single crystals.

∙ Sintering:

- High temperature treatment that causes particles to join, which  gradually reduces the volume of pore space between them.

- Frequent step in the manufacturing of ceramic components. Also,  very important in the production of metallic parts by powder  metallurgy (process in which metal powders are pressed and  sintered into monolithic components).

- Liquid phase sintering is the process where a small amount of liquid forms and assists densification.

∙ Grain Growth:

- Movement of grain boundaries that allows larger grains to grow at  the expense of smaller grains.

- Used to lower overall energy in the material by reducing the area of the grain boundary (driving force).

- In normal grain growth, the average grain size increases steadily  and the width of the grain size distribution is not severely affected.  On the other hand, disproportionate grain growth tends to occur  when we are dealing with an abnormal situation.

∙ Diffusion Bonding:

- Method used to join materials that occurs in three main steps. - The first step consists in the increase of temperature and pressure  in order to force the two surfaces together, which causes the  flattening of surface, fragmentation of impurities, and production of  high atom to atom contact area.  If you want to learn more check out What is the meaning of active transport?

- The second step is the atom diffusion along grain boundaries to the  remaining voids and the posterior condensation and reduction of  any voids’ size at the interface. This step occurs very quickly  because of the fast grain boundary diffusion; however, grain growth eventually isolates the remaining voids from the grain boundaries.

- The third step volume diffusion must occur in order to eliminate the  voids. This diffusion process is often used for joining reactive  metals, dissimilar metals and materials, and ceramics.  Don't forget about the age old question of Who uses accounting information?

Chapter 6 : Mechanical Properties: Part One

 Terminology for Mechanical Properties:

∙ Stress: Force acting per unit area over which the force is applied. The  main types of stress are: tension, compression, and shear. It is typically  expressed in psi or Pa.

∙ Strain: Change in dimension per unit length. It has dimension and it is  usually expressed in in/in or cm/cm.

 The Tensile Test: Use of Stress-Strain Diagram

∙ Tensile test measures the resistance of a material to a static or slowly  applied force. A strain gage is used to measure the amount that the  specimen stretches between the gage marks when the force is applied. ∙ Engineering Stress:

S=FAo

∙ Engineering Strain:

e=∆ L

Lo

, where

Ao = Original cross-sectional area of the specimen before the test begins Lo = Original distance between the gage marks

∆L = Change in length after force

 Properties Obtained from Tensile Test:

∙ Yield Strength: Don't forget about the age old question of What is meiosis summary?

- As stress is applied to a material, the material initially shows elastic  deformation; however, as this applied stress increases, the material is  eventually going to yield both elastic and plastic deformations.

- Elastic deformations are reversible whereas plastic deformations are  permanent. The critical stress value required to start plastic

deformations is known as the elastic limit.

- The transition from elastic deformation to plastic flow is abrupt (yield  point phenomenon). As plastic deformation begin, the stress value  drops from the upper yield point and then it oscillates around an  average value called lower yield point (See Figure 6-8).

∙ Tensile Strength: Stress obtained at the highest applied force, which can  be located in the engineering stress-strain curve as the maximum stress.  ∙ Elastic Properties: Modulus of elasticity (Young’s modulus) ,E, is the slope  of the stress-strain curve in the elastic region. The relationship between  stress and strain is known as Hooke’s Law:

E=Se

, where

S = Stress

E = Strain

∙ Tensile Toughness: Energy absorbed by a material prior to fracture, which  is sometimes measured as the area under the true stress-strain curve

(work fracture). Engineers usually equate tensile toughness to the area  under the stress-strain curve since it is easier to measure stress-strain.  ∙ Ductility: Ability of a material to be permanently deformed without  breaking when a forced is applied. There are two common measures of  ductility: Don't forget about the age old question of Does photosynthesis occur in autotrophs or heterotrophs?

- Percent Elongation: Quantifies the permanent plastic deformation at  failure by measuring the distance between gage marks on the

specimen before and after the test.

Elongation=lf−lo

lox 100

lf = Distance between gage marks after the specimen breaks

lo = Distance between gage marks before breaking

- Percent Reduction in Area: Measure the percent change in the cross sectional area at the point of fracture before and after the test. It  describes the amount of thinning undergone by the specimen during  the test.

Reduction∈Area=Ao−Af

Aox 100

Af = Final cross-sectional area at the fracture surface

Ao = Initial cross-sectional area at the fracture surface

∙ Toughness:

- Measure of the ability of a material to absorb energy without fracture.  It is used to describe the combination properties of strength and  ductility, and it has the units of J/m3 or MPa.

- High toughness means that there is high yield strength and ductility.   True Stress and True Strain:

∙ True Stress:

σ=FA

∙ True Strain:

dl

lo l

¿¿

l=ln ?¿ l

ε=∫ lo

¿

A = Instantaneous area

F = Force applied

l = Instantaneous sample length

lo = Initial length

 The Bend Test for Brittle Materials:

∙ Many brittle materials may crack when placed in the tensile testing  machine, so the bend test is used. This test consists in the application of  a load at three points causing bending, so a tensile force acts on the  material opposite the midpoint.

∙ Flexural Strength: Also called modulus of rupture, it describes the  material’s strength

Flexural strength for three point bend test: σbend=3FL

2w h2

L = Distance between the two outer points

w = Width of the specimen

h = Height of the specimen

∙ Flexural Modulus: Modulus of elasticity in bending

Ebend=L3F

4wh3δ

δ = Deflection of the beam

∙ The test can also be conducted using the four-point bend test, which  maximum stress is given by:

σbend=3F L1

4 wh2

L1 = L/4

NOTE: Check Figure 6-7 for the complete engineering stress-strain curve. Chapter 7: Mechanical Properties: Part Two

 Fracture Mechanics:

∙ Discipline concerned with the behavior of materials containing cracks or  other small flaws> this latter term refers to features such as small pores,  inclusions, or microcracks.

∙ Fracture Toughness: Measures the ability of a material containing a flaw  to resist an applied load. The typical test is performed by applying a  tensile stress to a specimen prepared with a flaw of known size and  geometry.

K= fσ√πa

K = Stress intensity factor

f = Geometry factor for the specimen and flaw

σ = Applied stress

a = Flaw size

NOTE: When infinite width is assumed, f ~ 1

∙ The value of K that causes the flaw to grow and cause failure can be  determined by performing a test on a specimen with a known flaw size.  This critical stress intensity factor is known as the fracture toughness  (Figure 7-2):

Kc=K required for a crack ¿ propagate

∙ Fracture toughness depends on the thickness of the sample ( As  thickness increases, Kc decreases to a constant value). This constant is  known as the plane strain fracture toughness K1c, which is the one that is usually reported as the property of a material. This latter term doesn’t  depend on the thickness of the sample.

 Importance of Fracture Mechanics:

∙ Brittle Fracture: Any small crack or imperfection limits the capability of a  ceramic to resist a tensile stress. This occurs when a crack, usually

called Griffith flaw, concentrates and magnifies the applied stress (Figure 7-4):

σactual ? 2σ√a/r

σ = Tensile stress

r = Very thin cracks

a = Long cracks

NOTE: If σactual exceeds the yield strength, the crack is going to grow and eventually  cause failure, even though the nominal applied stress σ is small.

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