© meg/aol ‘02 Solutions To The Linear Diffusion Equation Martin Eden Glicksman Afina Lupulescu Rensselaer Polytechnic Institute Troy, NY, USA

© meg/aol ‘02 Outline Transform methods Linear diffusion into semi-infinite medium –Boundary conditions –Laplace transforms Behavior of the concentration field Instantaneous planar diffusion source in an infinite medium Conservation of mass for a planar source –Error function and its complement –Estimation of erf(x) and erfc(x) Thin-film configuration

© meg/aol ‘02 Transform Methods where Kernel of the transform Laplace kernel Independent variable (time). Laplace transform converts an object function, F(t), to its image function, F(p). ~

© meg/aol ‘02 Linear diffusion into a semi-infinite medium time Initial state Final state

© meg/aol ‘02 Boundary Conditions 1) Initial state: C = 0, for x > 0, t = 0. 2) Left-hand boundary: At x = 0, C 0 is maintained for all t > 0. The diffusion equation is a 2nd-order PDE and requires two boundary or initial conditions to obtain a unique solution.

© meg/aol ‘02 Laplace Transform of the Diffusion Equation object function Laplace kernel Linear Diffusion Equation image function Laplace transform of C(x,t) Laplace transform of the spatial derivative.

© meg/aol ‘02 Laplace Transforms The Laplace transform of Fick’s second law when C(x,0)=0 integration by parts boundary condition

© meg/aol ‘02 Laplace Transforms Transform of the boundary condition: General transform solution: The particular transform solution for the image function arises from the negative root, because the positive root leads to non-physical behavior,.

© meg/aol ‘02 Laplace Transforms The concentration field associated with the image field is found by inverting the transform either by formal means, a look-up table, or using a computer-based mathematics package. The error function, erf (z), and its complement, erfc (z), are defined

© meg/aol ‘02 Estimation of the Error Function For small arguments: For very large arguments:

© meg/aol ‘02 Estimation of the Error Function Piecewise approximations for restricted ranges of the argument: Rational approximation for positive arguments, z > 0: Useful for spreadsheet calculations.

© meg/aol ‘02 Error Function z Erf(z) Antisymmetric: erf(z)=-erf(-z)

© meg/aol ‘02 Complementary Error Function z Erfc(z) Non-antisymmetric: erfc(z)  -erfc(-z)

© meg/aol ‘02 Linear diffusion into a semi-infinite medium time Initial state Final state

© meg/aol ‘02 Concentration versus the similarity variable Relative Conc. C/C 0 Similarity Variable, x/2(Dt) 1/2

© meg/aol ‘02 Concentration field versus distance = 2(Dt) 1/2 is the “time tag” (Note: has the units of distance!) Relative Conc. C/C 0 Distance, x [units of =1] time

© meg/aol ‘02 Diffusion Penetration X * X * = K t 1/2 Relative Conc. C(x)/C 0 Relative Conc. C/C 0 Distance, x [units of =1]

© meg/aol ‘02 Penetration versus square-root of time Penetration Distance, X*(C/C 0 )

© meg/aol ‘02 Instantaneous planar diffusion source in an infinite medium These diffusion problems concern placing a finite amount of diffusant that spreads into the adjacent semi-infinite solid. Initial state Final state Time

© meg/aol ‘02 Instantaneous planar diffusion source in an infinite medium These diffusion problems concern placing a finite amount of diffusant that spreads into the adjacent semi-infinite solid. Time

© meg/aol ‘02 Instantaneous planar diffusion source Application of the Laplace transform to Fick’s second law gives: The diffusion process is subject to the mass constraint for a unit area: t = 0, C (x, 0), for all x  0 Initial condition C (  ∞, t) = 0 Boundary condition

© meg/aol ‘02 Instantaneous planar source Reduction of the Laplace transform: The general solution for which is:

© meg/aol ‘02 Instantaneous planar source or and so Mass constraint for the field: Laplace transform the mass constraint: The integral constraint for the image function is:

© meg/aol ‘02 Instantaneous planar source solution Laplace transforms table shows, The transform solution where a = x / (D) 1/2. Inverting the transform solution Diffusion solution

© meg/aol ‘02 C (x) M [length] -1 x [distance] Normalized plot of the planar source solution time

© meg/aol ‘02 Conservation of mass for a planar source thus Gauss’s integral

© meg/aol ‘02 Thin-film configuration Thin-film diffusion configuration is used in many experimental studies for determining tracer diffusion coefficients. It is mathematically similar to the instantaneous planar source solution. where M thin-film represents the instantaneous thin-film source “strength.” 2M thin-film =M

© meg/aol ‘02 Procedure for Analysis of Thin-Film Data Take logs of both sides of A plot of lnC versus x 2 yields a slope=-1/4Dt

© meg/aol ‘02 Thin-Film Experiment Geiger counter data after microtoning 25 slices from the thin-film specimen.

© meg/aol ‘02 Log Concentration versus x 2 Slope=-1/4Dt

© meg/aol ‘02 Exercise 1. Show by formal integration of the concentration distribution, C(x,t), given by eq.(3.31), that the initial surface mass, M, redistributed by the diffusive flow is conserved at all times, t>0. The mass conservation integral is given by: Symmetry of diffusion flow allows:

© meg/aol ‘02 Exercise Introduce the variable substitution u= x /2 (Dt ) 1/2 and obtain: Simplification yields: The diffusant mass is conserved according to:

© meg/aol ‘02 Exercise 2a) Two instantaneous planar diffusion sources, each of “strength” M, are symmetrically placed about the origin (x=0) at locations =  1, respectively, and released at time t=0. Using the linearity of the diffusion solution develop an expression for the concentration, C(x,t), developed at any arbitrary point in the material at a fixed time t>0. Plot the concentration field as a function of x for several fixed values of the parameter Dt to expose its temporal behavior. 2b) Find the peak concentration at x=0 and determine the time, t *, at which it develops, if D= cm 2 /sec, M=25  g/cm 2, and the sources are both located 1  m to either side of the origin. 2c) Plot the concentration at the plane x=0 against time.

© meg/aol ‘02 Exercise 2a)

© meg/aol ‘02 Exercise 2b) For two sources of strength M the concentration is: Differentiate with respect to t : The concentration reaches its maximum at t *: The maximum concentration reached at x = 0

© meg/aol ‘02 Exercise 2c)

© meg/aol ‘02 Key Points Solutions to the linear diffusion equations require two initial or boundary conditions. Examples of problems with constant composition and constant diffusing mass are demonstrated. Laplace transform methods were employed to obtain the desired solutions. Solutions are in the form of fields, C(r, t). Exposing the behavior of such fields requires careful parametric description and plotting. Similarity variables and time tags are used, because they capture special space- time relationships that hold in diffusion. Diffusion solutions in infinite, or semi-infinite, domains often contain error and complementary error functions. These functions can be “called” as built-in subroutines in standard math packages, like Maple ® or Mathematica ® or programmed for use in spreadsheets. The theory for the classical “thin-film” method of measuring diffusion coefficients was derived using the concept of an instantaneous planar source in linear flow.