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Solutions To The Linear Diffusion Equation

Martin Eden GlicksmanAfina Lupulescu

Rensselaer Polytechnic InstituteTroy, NY, 12180

USA

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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

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Transform Methods

whereK t, p Kernel of the transform

e pt Laplace kernel

t Independent variable (time).

L F t F t K t, p d t

a

b

˜ F p

L F t Laplace transform converts an object function, F(t), to its image function, F(p). ~

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Linear diffusion into a semi-infinite

medium

time

Initial state

Final state

t

t =4

t =16

C0

t =1

t =0

x

t

t =4

t =16

C0

t =1

t =0

x

t

t =4

t =16

C0

t =1

t =0

x

t

t =4

t =16

C0

t =1

t =0

x

t

t =4

t =16

C0

t =1

t =0

x

t

t =4

t =16

C0

t =1

t =0

x

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Boundary Conditions

1) Initial state:C = 0, for x > 0, t = 0.

2) Left-hand boundary: At x = 0, C0 is maintained for all t > 0.

Ct

D 2Cx2

• The diffusion equation is a 2nd-order PDE and requires two boundary or initial conditions to obtain a unique solution.

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Laplace Transform of the Diffusion Equation

object functionC x, t

e ptLaplace kernel

e pt

0

2 Cx2 dt

2

x2 C e pt

0

dt d2 ˜ C d x2

˜ C x, p C x,t e pt d t0

2 Cx 2

Linear Diffusion Equation

˜ C image function

Laplace transformof C(x,t)

˜ C x, p

1D

Ct

0

e pt

0

2 Cx2 dt

Laplace transform ofthe spatial derivative.

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Laplace Transforms

The Laplace transform of Fick’s second law when C(x,0)=0

d2 ˜ C d x2

pD

˜ C 0

1D

e pt

0

Ct

dt

1D

Ce pt0

p Ce pt

0

dt

pD

˜ C

integration by parts

Ce pt0

0 C t 0 0 boundary

condition

1D

e pt

0

Ct

dt

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Laplace Transforms

˜ C 0 C0e pt

0

dt C0

pe pt

0

˜ C 0 C0

p

Transform of the boundary condition:

˜ C C0

pe

pD

xGeneral transform solution:

˜ C x C0

pe

pD

x

The particular transform solution for the image function arises from the negative root, because the positive root leads to non-physical behavior, . ˜ C (x)

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Laplace Transforms

C x, t C0 erfc x2 Dt

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.

erfc z 1 erf z 1 2

e 2

d0

z

The error function, erf (z) , and its complement, erfc (z) , are defined

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Estimation of the Error Function

erf(z) erf( z) 2

z z3

31 !

z5

52 !

z7

73 ! ...

, ( z 1)

• For small arguments:

erf(z) 1 e z 2

z 1

12z2 ...

, (z )

• For very large arguments:

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Estimation of the Error Function

erf(z)

1.12838z, (0 z 0.15) 0.0198 z 1.2911 0.4262z , (0.15 z 1.5)0.8814 0.0584z, (1.5 z 2)1, (2 z)

• Piecewise approximations for restricted ranges of the argument:

erf z 1 10.278393z 0.230389z2

0.000972z3 0.078108z4

4

z 5 10 5

• Rational approximation for positive arguments, z > 0:

Useful for spreadsheet calculations.

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Error Function

-1

-0.5

0

0.5

1

-3 -2 -1 0 1 2 3

Err

or F

unct

ion,

erf

(z)

zz

Erf(

z)

Antisymmetric: erf(z)=-erf(-z)

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Complementary Error Function

0

0.5

1

1.5

2

-3 -2 -1 0 1 2 3

erfc

(z)

zz

Erfc

(z)

Non-antisymmetric: erfc(z) -erfc(-z)

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Linear diffusion into a semi-infinite

medium

time

Initial state

Final state

t

t =4

t =16

C0

t =1

t =0

x

t

t =4

t =16

C0

t =1

t =0

x

t

t =4

t =16

C0

t =1

t =0

x

t

t =4

t =16

C0

t =1

t =0

x

t

t =4

t =16

C0

t =1

t =0

x

t

t =4

t =16

C0

t =1

t =0

x

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Concentration versus the similarity variable

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

Rel

ativ

e co

ncen

tratio

n, C

/C0

Space-time similarity variable, x/2(Dt)1/2

C x,t C0 erfc x2 Dt

Rel

ativ

e C

onc.

C/C

0

Similarity Variable, x/2(Dt)1/2

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Concentration field versus distance

 = 2(Dt)1/2 is the “time tag”

(Note: has the units of distance!)

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10Rela

tive

Conc

entra

tion,

C(x

)/C0

Distance, x [ in units of =1]

0.20.5

1

23

510

20=100

Rel

ativ

e C

onc.

C/C

0

Distance, x [units of =1]

time

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Diffusion Penetration X*

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10Rela

tive

Conc

entra

tion,

C(x

)/C0

Distance, x [ in units of =1]

0.20.5

1

23

510

20=100

X* = K t1/2

Rel

ativ

e C

onc.

C(x

)/C0

Rel

ativ

e C

onc.

C/C

0

Distance, x [units of =1]

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Penetration versus square-root of time

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

X*(0.8)X*(0.6)X*(0.4)X*(0.9)

Pen

etra

tion

Dis

tanc

e, X

* (C/C

0)

Time tag, 2(Dt)1/2

Pen

etra

tion

Dis

tanc

e, X

*(C

/C0)

0.40.6

0.8

0.9

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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

Tim

e

0 x

M

t = 0

-

t

t =1

t =10

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0 x

M

t = 0

-

t

t =1

t =10

0 x

M

t = 0

-

t

t =1

t =10

0 x

M

t = 0

-

t

t =1

t =10

0 x

M

t = 0

-

t

t =1

t =10

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. Ti

me

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Instantaneous planar diffusion source

pD

˜ C x, p d2 ˜ C x, p

d x2 C x, 0

D

C x,t

dx M

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

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Instantaneous planar source

pD

˜ C x, p d2 ˜ C x, p

d x2 0

˜ C Ae p D x Be p D x

Reduction of the Laplace transform:

The general solution for which is:

˜ C Ae p Dx , x 0, B 0

˜ C Be p D x, x 0 , A 0

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Instantaneous planar source

or e pt

0

C x,t dx dt0

M20

e ptdt

or Bp

De

pD

x

0

M2 p

and so B M

2 pD

C x,t 0

dx M2

Mass constraint for the field:

L C x,t d x

0

L M

2

Laplace transform the mass constraint:

˜ C x, p d x0

M2p

The integral constraint for the image function is:

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Instantaneous planar source solution

˜ C x, p M

2 De

xD

p

p

Laplace transforms table shows, L-1 e a p

p

1 t

e a24t

The transform solution

where a = x / (D)1/2.

L-1 ˜ C C x,t = M2 D

L-1 p 12e a p

Inverting the transform solution

C x,t = M

2 Dte

x2

4 Dt .

Diffusion solution

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0

0.5

1

1.5

2

-3 -2 -1 0 1 2 3

C(x

)/M [

leng

th]-1

x [distance]

.075

.05

Dt=0

0.025

0.25

13

C (x

) M [l

engt

h] -1

x [distance]

Normalized plot of the planar source solution

time

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Conservation of mass for a planar source

x

dx 1C x,t M

(x), thus

C x,t M

d x

1

1

e u2

du

1Gauss’s integral

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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.

C x,t = M

2 Dte

x2

4 Dt .C x, t Mthin film

Dte x 2

4 Dt

where Mthin-film represents the instantaneous thin-film source “strength.”

2Mthin-film=M

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Procedure for Analysis of Thin-Film Data

C x,t Mthin film

Dte x 2

4 DtTake logs of both sides of

ln C x, t lnMthin film

Dt

x 2

4Dt

A plot of lnC versus x2 yields a slope=-1/4Dt

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Thin-Film Experiment

0

1 105

2 105

3 105

4 105

5 105

0 10 20 30 40 50

Counts 100s

Cou

nts

in 1

00 s

Distance, [microns]

Geiger counter data after microtoning 25 slices from the thin-film specimen.

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Log Concentration versus x2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.5 1 1.5 2 2.5 3 3.5 4

log e R

adio

activ

ty, l

nA*

x2, [cm2 25104]

Slope=-1/4Dt

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Exercise1. 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.

C(x,t)d x

M

2 Dte

x 2

4 Dt d x

The mass conservation integral is given by:

C(x, t)d x

2 M2 Dt

e

x2

4Dt d x0

Symmetry of diffusion flow allows:

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Exercise

Introduce the variable substitution u= x /2 (Dt )1/2 and obtain:

C(x,t)d x

MDt

e u2

2 Dt d u0

Simplification yields:

C(x, t)d x

M 2

e u2

du0

The diffusant mass is conserved according to:

C(x,t)d x

M erf M

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Exercise2a) 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=10-11 cm2/sec, M=25 g/cm2, and the sources are both located 1m to either side of the origin.

2c) Plot the concentration at the plane x=0 against time.

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Exercise

2a) C x, t M

2 Dte x 1 2 4 Dt e x1 2 4 Dt

0

0.5

1

1.5

2

-4 -3 -2 -1 0 1 2 3 4

C(x

,t)/M

x

0.5

0.05

2

0.025

Dt=0

0.1

0.25

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Exercise

C 0,t;ˆ x MDt

e ˆ x 2 4 Dt

2b)For two sources of strength M the concentration is:

C 0,t;ˆ x t

MDt

e ˆ x 24 Dt ˆ x 2

4Dt 2 12t

Differentiate with respect to t :

Dt ˆ x 2

2

The concentration reaches its maximum at t *:

Cmax 0,t M

ˆ x e / 2 0.484 M

ˆ x

The maximum concentration reached at x = 0

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Exercise

2c)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 1 2 3 4 5 6 7

C(0

,t) [g

/cm

3 ]

Time [103 sec]

D=10-11 cm/sM=5.10-4 g/cm2

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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.