1 THE WAY TO SOMEWHERE Sub-topics 1 Diffusion Diffusion processes in industry
2 RATE PROCESSES IN SOLIDS At any temperature different from absolute zero all atoms, irrespective of their state of aggregation (gaseous, liquid or solid), are constantly in motion diffusion Diffusion refers to the net flux of any species, such as ions, atoms, electrons, holes, and molecules. Flux = (conductivity) x (driving force) In the case of atomic or molecular diffusion, the conductivity is referred to as the diffusivity or the diffusion constant D The driving force for many types of diffusion is the existence of a concentration gradient. this diffusion constant (D) reflects the mobility of the diffusing species in the given environment The term gradient describes the variation of a given property as a function of distance 2
3 NON-SPONTANEOUS PROCESSES 1. For process to be started, the atoms should have sufficient energy to overcome an activation energy barrier. Q=activation energy 2. State 1 Involves reduction in energy Or Energy change is negative State 2 3
4 DIFFUSION As T of the system is increased, more and more molecules will attain the activation energy level. In statistical mechanics, Maxwell Boltzmann statistics describes the distribution of material particles over various energy states and probabilities to find a particle in definite state: Probability ~ exp(- E/RT) R Boltzmanconstant = 1.38 x10-23 J/atom K 4
5 WHAT IS DIFFUSION? 5
6 SELF-DIFFUSION C C A D B t A B D 6
7 VACANCY OR SUBSTITUTIONAL DIFFUSION Why, in general, is the activation energy for self diffusion higher for materials of high melting point? Atoms move into the vacancies places. More vacancies are created at higher temperature. Diffusion rate is higher at high temperatures. 7
8 ATOMIC DIFFUSION IN SOLIDS Diffusion is a process by which a matter is transported through another matter. Examples: 9 Movement of smoke particles in air : Very fast. 9 Movement of dye in water : Relatively slow. 9 Solid state reactions : Very restricted movement due to bonding. Diffusion processes may be divided into two types: (a) steady state and (b) non-steady state. 8
9 HOW FAST DOES DIFFUSION OCCUR? Cu flux Ni flux Concentration of Cu [kg/m 3 ] Concentration of Ni [kg/m 3 ] flux in x-dir. [kg/m 2 -s] Position, x Concentration Profile, C(x): [kg/m 3 ] J x = D dc dx The rate at which atoms, ions, particles or other species diffuse in a material Diffusion coefficient [m 2 /s] concentration gradient [kg/m 4 ] The flux is defined as the number of atoms passing through a plane of unit area per unit time 9
10 STEADY-STATE DIFFUSION o Diffusion is a time dependent process and the rate of mass transfer is the diffusion flux (J). o In a steady-state condition the concentration gradient is constant. Steady state diffusion takes place at a constant rate - that is, once the process starts the number of atoms crossing a given interface (the flux) is constant with time. This means that throughout the system dc/dx= constant and dc/dt= 0. Fick s First law: Net flow of atoms Per unit area per Unit time = J atoms/cm 2 s The diffusive flux is proportional to the existing concentration gradient. 10
11 STEADY STATE DIFFUSION PROCESS A practical example of steady-state diffusion the purification of hydrogen gas. One side of a thin sheet of palladium metal is exposed to the impure gas composed of hydrogen and other gaseous species such as nitrogen, oxygen, and water vapor. The hydrogen selectively diffuses through the sheet to the opposite side, which is maintained at a constant and lower hydrogen pressure. 11
12 PROBLEM - CONCENTRATION GRADIENT Problem: A plate of iron is exposed to a carburizing (carbon-rich) atmosphere on one side and a decarburizing (carbon-deficient) atmosphere on the other side at 700C. If a condition of steady state is achieved, calculate the diffusion flux of carbon through the plate if the concentrations of carbon at positions of 5 and 10 mm beneath the carburizing surface are 1.2 and 0.8 kg/m 3, respectively. 12 Assume a diffusion coefficient of 3 x m2/s at this temperature.
13 CONCENTRATION GRADIENT The concentration gradient shows how the composition of the material varies with distance: c is the difference in concentration over the distance x atoms/m 2 s Kg/m2 s Kg/m 3 13
14 PROBLEM - SEMICONDUCTOR DOPING One way to manufacture transistors, which amplify electrical signals, is to diffuse impurity atoms into a semiconductor material such as silicon (Si). Problem: Suppose a silicon wafer 0.1 cm thick, which originally contains one phosphorus atom for every 10 million Si atoms, is treated so that there are 400 P- atoms for every 10 million Si atoms. Calculate the concentration gradient (a) in atomic percent/cm and (b) in atoms/(cm 3 x cm) The lattice parameter of silicon is A. 14
15 DIFFUSIVITY Diffusivity depends on 9 Type of diffusion : Whether the diffusion is interstitial or substitutional. 9 Temperature: As the temperature increases diffusivity increases. 9 Type of crystal structure: BCC crystal has lower APF than FCC and hence has higher diffusivity. 9 Type of crystal imperfection: More open structures (grain boundaries) increases diffusion. 9 The concentration of diffusing species: Higher concentrations of diffusing solute atoms will affect diffusivity. 15
16 DIFFUSION COEFFICIENT 16
17 EFFECT OF TEMPERATURE ON DIFFUSION A large activation energy results in a relatively small diffusion coefficient. Temperature has a most profound influence on the coefficients and diffusion rates. When the temperature increases, the diffusion coefficient D increases and, therefore, the flux of atoms increases as well. At higher temperatures, the thermal energy supplied to the diffusing atoms permits the atoms to overcome the activation energy barrier and more easily move to new sites in the atomic arrangements. 17 At low temperatures often below about 0.4 times the absolute melting temperature diffusion is very slow and may not be significant.
18 IMPURITY DIFFUSION INTO SILICON WAFER the activation energies in ionic materials are high and the rates of diffusion are low SiO 2 Impurities are made to diffuse into silicon wafer to change its electrical characteristics. Used in integrated circuits. Silicon wafer is exposed to vapour of impurity at 1100C in a quartz tube furnace. The concentration of impurity at any point depends on depth and time of exposure. Doping Silicon with P 1. Deposit P rich layers on surface. silicon 2. Heat it. 3. Result: Doped semiconductor regions. silicon 18
19 DESIGN PROBLEM: INTEGRATED CIRCUIT INTERCONNECTS Top layers that serve as the wire for this device (interconnect). Diffusion-layer doped silicon that have been coated with an interlayer dielectric. What material can be used for interconnects? 19
20 DESIGN OF AN IRON MEMBRANE A cylinder 3 cm in diameter and 10 cm long contains a gas that includes 0.5 x10 20 N atoms per cm 3 and 0.5 x10 20 H atoms per cm 3 on one side of an iron membrane. Gas is continuously introduced to the pipe to assure a constant concentration of nitrogen and hydrogen. The gas on the other side of the membrane includes a constant N atoms per cm3 and 1 x H atoms per cm 3. The entire system is to operate at 700 C (at this T iron has the BCC structure). Design an iron membrane that will allow no more than 1% of the nitrogen to be lost through the membrane each hour, while allowing 90% of the hydrogen to pass through the membrane per hour. 20
21 TYPES OF DIFFUSION In volume diffusion, the atoms move through the crystal from one regular or interstitial site to another. Because of the surrounding atoms, the activation energy is large and the rate of diffusion is relatively slow. Atoms can also diffuse along boundaries, interfaces, and surfaces in the material. Atoms diffuse easily by grain boundary diffusion, because the atom packing is disordered and less dense in the grain boundaries. Because atoms can more easily squeeze their way through the grain boundary, the activation energy is low. Surface diffusion is easier still because there is even less constraint on the diffusing atoms at the surface. 21
22 TUNGSTEN -THORIUM DIFFUSION COUPLE Consider a diffusion couple between pure tungsten and a tungsten alloy containing 1 at% thorium. After several minutes of exposure at 2000 C, a transition zone of 0.01 cm thickness is established. What is the flux of thorium atoms at this time if diffusion is due to (a) volume diffusion, (b) grain boundary diffusion, and (c) surface diffusion? The lattice parameter of BCC tungsten is Å. 22
23 NONSTEADY-STATE DIFFUSION Concentration of solute atoms at any point in metal changes with time in this case. Change of concentration of solute atoms with change in time in different planes Fick s Second Law Ficks second law: Rate of compositional change is equal to diffusivity times the rate of change of concentration gradient. 23
24 NON-STEADY STATE DIFFUSION Copper diffuses into a bar of aluminum. Surface conc., Cs of Cu atoms bar pre-existing conc., Co of copper atoms Cs C(x,t) Co to t 1 t 2 t 3 position, x 24 15
25 SOLUTION One practically important solution is for a semi-infinite solid in which the surface concentration is held constant. Frequently, the source of the diffusing species is a gas phase, the partial pressure of which is maintained at a constant value. Furthermore, the following assumptions are made: 1.Before diffusion, any of the diffusing solute atoms in the solid are uniformly distributed with concentration of C x at the surface is zero and increases with distance into the solid. 3. The time is taken to be zero before the diffusion process begins. 25
26 FICK S SECOND LAW SOLUTION 26
27 TABULATION OF ERROR FUNCTION VALUES Z 27
28 CARBURIZING Diffusing carbon atoms Low carbon Steel part Carbon Gradients In Carburized metals Result: --hard to deform: C atoms "lock" planes from shearing. --hard to crack: C atoms put the surface in compression. 28
29 INDUSTRIAL APPLICATIONS OF DIFFUSION CASE HARDENING Sliding and rotating parts needs to have hard surfaces. These parts are usually machined with low carbon steel as they are easy to machine. Their surface is then hardened by carburizing: Steel parts are placed at elevated temperature (927C) in an atmosphere of hydrocarbon gas (CH4). Carbon diffuses into iron surface and fills interstitial space to make it harder. Photograph of a steel gear that has been case hardened. The outer surface layer was hardened by a hightemperature heat treatment during 29 which carbon from the surrounding atmosphere diffused into the surface.
30 EFFECT OF TEMPERATURE ON DIFFUSION- EXAMPLE If diffusivity at two temperatures are determined, two equations can be solved for Q and D0 30
31 TIME COMPUTATION 31
32 DIFFUSIVITY DATA FOR SOME METALS 32
33 PROBLEM: TIME FOR DIFFUSION Carburizing: the steel piece is exposed, at an elevated temperature, to an atmosphere rich in a hydrocarbon gas, such as methane (CH4 ). Consider an alloy that initially has a uniform carbon concentration of 0.25 wt% and is to be treated at 950 C. If the concentration of carbon at the surface is suddenly brought to and maintained at 1.20 wt%, how long will it take to achieve a carbon content of 0.80 wt% at a position 0.5 mm below the surface? The diffusion coefficient for carbon in iron at this temperature is 1.6 x m 2 /s. 33
34 DESIGN PROBLEM The wear resistance of a steel gear is to be improved by hardening its surface. This is to be accomplished by increasing the carbon content within an outer surface layer as a result of carbon diffusion into the steel; the carbon is to be supplied from an external carbon-rich gaseous atmosphere at an elevated and constant temperature. The initial carbon content of the steel is 0.20 wt%, whereas the surface concentration is to be maintained at 1.00 wt%. In order for this treatment to be effective, a carbon content of 0.60 wt% must be established at a position 0.75 mm below the surface. Specify an appropriate heat treatment in terms of temperature and time for temperatures between 900C and 1050 C. Use data in Table for the diffusion of carbon in γ-iron. The gas const = 8.31 J/mol K 34
35 DESIGN OF A MORE ECONOMICAL HEAT TREATMENT 10 h are required to successfully carburize a batch of 500 steel gears at 900 C, where the iron has the FCC structure. We find that it costs $1000 per hour to operate the carburizing furnace at 900 C and $1500 per hour to operate the furnace at 1000 C. Is it economical to increase the carburizing temperature to 1000 C? What other factors must be considered? 35
36 SUMMARY: STRUCTURE & DIFFUSION Diffusion FASTER for... open crystal structures lower melting T materials materials w/secondary bonding smaller diffusing atoms Diffusion SLOWER for... close-packed structures higher melting T materials materials w/covalent bonding larger diffusing atoms lower density materials higher density materials 36 20
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