Diffusion continued Lecture 17 Ficks First Law Last
- Slides: 22
Diffusion (continued) Lecture 17
Fick’s First Law Last time we introduced Fick’s First Law Note minus sign. This means diffusion moves down the concentration gradient, i. e. , from high to low concentration.
Deriving Fick’s Law • On a microscopic scale, the mechanism of diffusion is the random motion of atoms. • Consider two adjacent lattice planes in a crystal spaced a distance dx apart. The number of atoms (of interest) at the first plane is n 1 and at the second is n 2. • We assume that atoms can randomly jump to an adjacent plane and that this occurs with an average frequency n (i. e. , 1 jump of distance dx every 1/ν sec) and that a jump in any direction has equal probability. • At the first plane there will be nn 1/6 atoms that jump to the second plane (there are 6 possible jump directions). At the second plane there will be nn 2/6 atoms that jump to first plane. The net flux across area dx 2 from the first plane to the second is then:
Deriving Fick’s Law • We’ll define concentration, c, as the number of atoms/unit volume n/x 3, so in a volume dx 3: • Letting dc = -(c 1 - c 2) and multiplying by dx/dx • Letting then
Continuity Equation • Imagine a volume and a flux of a species into that volume, Jx, at x and a flux out, Jx+dx, at x+dx. • Suppose nx atom/sec at x and nx+dx atoms/sec pass through the plane at x and x+dx, respectively. • The fluxes at the two planes are Jx = nx/dx 2 -sec and Jx+dx = nx+dx/dx 2 -sec. • Conservation of mass dictates that the increase in the number of atoms, dn, within the volume is just what goes in minus what comes out. or: • Over an increment of time dt this will be dn = (Jx - Jx+dx)dt. The concentration change is At fixed x and t
Deriving Fick’s Second Law • Continuity equation relates any change in flux along the gradient to change in concentration with time. • Since: • we have:
Fick’s Second Law • The second law says that the change in concentration with time at some point on a conc. gradient depends on the second derivative of the concentration gradient. • The straight line (∂2 c/∂x 2) is the steady state case. It is also the situation towards which a system will evolve over time.
Solutions to the Second Law • There is no single ‘solution’ to the second law, rathere a variety of solutions that depend on the boundary conditions of a particular situation. • As a first example, consider a infinitesimally thin layer (e. g. , Ir at K-T boundary) of material at x = 0 at t = 0. These, together with mass conservation, are our boundary conditions. • Mass conservation is
Thin Film Solution • Boundary conditions were x = 0 at t = 0 and • Solution, from Crank (1975), fitting these conditions is: • If diffusion can occur only in one direction:
Adjacent Layers • Now suppose with have one layer of finite thickness with concentration co abutting a second with c=0. • Crank’s solution is to superimpose the solutions for an infinite number of thin flims, dξ, and to sum their contributions to the total concentration at Xp. • Solution is:
Adjacent Layers • The solution to: • is • where erf is the error function and has the approximate form: • also available as the ERF function in Excel.
Other examples • Book has several other examples such as diffusion of Ni in a spherical, zoned olivine crystal with an initially homogeneous conc. of 2000 ppm and a conc. fixed at the rim by equilibrium with a magma at 500 ppm.
Diffusion in Multicomponents Systems • So far, we have considered only what is called tracer or self diffusion - diffusion of a species who concentration is so small it has no effect on the system. • By definition, diffusion is a process in which there is no net transport across the boundary of interest (otherwise it is advection). Mass & charge balance require diffusion in one direction to be balanced by diffusion in another. • More generally, we can consider o “Chemical” diffusion o Multicomponent chemical diffusion
Chemical Diffusion • “Chemical” diffusion refers to non-ideal situations where for one reason or another, we cannot treat our diffusing species as an inert particle, i. e. , chemical interactions occur, we replace concentration with the chemical potential gradient. • where L is the chemical or phenomenological coefficient. • We have the additional task of deducing chemical potential from concentration, otherwise, treatment is similar.
Multicomponent Diffusion • This is where the concentration of an species is great enough that its diffusion depends on the diffusion of another species, e. g. , Fe and Mg in ferromagnesian silicates (these elements tend to occupy the same site, so Mg can move in only if Fe moves out, etc). • Di, k is called the interdiffusion coefficient and computed as: o where n’s are the mole fractions. • The total solution is J = -DC (minus sign missing in book) • where J and C are (vertical) vectors of species and D is a matrix.
Multicomponent Chemical • Multicomponent chemical diffusion is a combination of the latter two, where we use interdiffusion coefficients and chemical potential. • Diffusion matrix is: J = -LC • Bruce Watson’s experiments dissolving a quartz sphere in molten basalt show an example where chemical diffusion must be used - note Na and K diffusing up a concentration gradient!
Mechanisms of Diffusion in Solids • Exchange: low probability • Interstitial: pretty much limited to He, etc. • Interstitiality: also low probability • Vacancy: most likely o Diffusion will then depend on both the probability of a jump and the probability of a vacancy. o Vacancies will increase with T: o where k is a rate constant and EH and activation energy for vacancy creation.
Diffusion Rates • Probability of atom making a jump to vacancy is • where ℵ is number of attempts and EB is the activation or barrier energy. • Combining, total diffusion rate will be product of probability of a vacancy plus the probability of a jump: o where m and n are constants • At higher T, thermal vacancies dominate and • Bottom line: like other things in kinetics, we expect diffusion rates to increase exponentially with temperature.
Diffusion Rates • In general, the temperature of the diffusion coefficient is written as: • In log form: • At lower temperature, where permanent vacancies dominate, we might see different behavior because:
Determining Diffusion Coefficients • In practice, one would do a experiments at a series of temperatures with a tracer, determine the profile, solve Fick’s second law for each T. • Plotting ln D vs 1/T yields Do and EB. • Diffusion coefficient is different for each species in each material. • Larger, more highly charged ions diffuse more slowly.
- Fick's second law
- Ficks second law of diffusion
- Ficks laws
- Perfusion limited vs diffusion limited
- Diffusion coefficient formula
- Explain fick's law of diffusion
- Bulk movement
- Ficks law
- Ficks second law
- Newton's first law and second law and third law
- Newton's first law of motion
- Diffusion vs facilitated diffusion
- Relocation diffusion vs expansion diffusion
- Erik ficks
- Ficks lw
- 01:640:244 lecture notes - lecture 15: plat, idah, farad
- Randy pausch the last lecture summary
- Boyle's law charles law avogadro's law
- Charles law constant
- Fick's 2nd law of diffusion
- Mass transfer
- Rate of diffusion formula
- Grahams law