example, R might be the region between two concentric spheres. .... calculate the diffusion current through the entire sphere, we can let SSEL be the entire ...
L. Edmonds 7/30/2016
A Method for Calculating Diffusion Currents through Boundary Surfaces Abstract The diffusion equation describes a variety of physical systems and one of the most frequently cited examples is the heat equation. The rate of heat flow through a boundary surface is an example of a diffusion current that is considered in this report. There are several methods for calculating a diffusion current through a boundary surface. One method is numerical, using a finite difference method or a finite element method. A second method utilizes Green’s functions. A third method, which is the topic of this report, has some resemblance to the Green’s function method but the required function is much easier to calculate than a Green’s function.
I. Introduction The equation considered here that describes some function f is a diffusion equation of the form given by f ( x , t ) f ( x , t ) 2 (1.1) D f ( x, t ) g ( x, t ) t where the spatial coordinates in three dimensions (lower dimensions can be treated as special cases of three dimensions) are represented by the vector x, t is the time coordinate, the deloperator refers to the spatial coordinates, D is a positive constant, 1/τ is a nonnegative constant, and g is typically (not necessarily) a known function. One common application of (1.1) is thermal conduction in some material. In this application, f is the temperature at a given location and time, D is the thermal conductivity of the material divided by the specific heat and density, 1/τ is typically zero, and g is often zero but more generally is proportional to the thermal generation rate density produced by some mechanism (e.g., exposure to microwaves) that heats the interior of the material. Another application is the diffusion of minority charge carriers in a semiconductor. In this application, f is the excess (total minus equilibrium) minority carrier density, D is the diffusion coefficient for minority carriers, τ is the carrier lifetime (approximated as a constant in this application), and g is the carrier generation rate density produced by some mechanism (e.g., exposure to any kind of ionizing radiation) that liberates charge carriers in the interior of the material. The quantity to be solved is explained as follows. We require that (1.1) applies to the interior points of some three-dimensional spatial region denoted R, and whatever time interval that is relevant to the application of interest. We assume that the region R has a closed boundary. For example, R might be the region contained inside of some closed surface. For another example, R might be the region between two concentric spheres. In this case the boundary of R is the union of the two spheres (one sphere is part of the boundary and the other sphere is the other part). For a third example, R might be the region outside of some sphere. For this case the given sphere is part of the boundary while an imagined sphere with an infinite radius is the other part of the boundary. In general, if any portion of the boundary of R is insulated, we require the function f to satisfy reflective boundary conditions there (the normal component of the gradient is zero) and we let SREF denote the reflective portion of the boundary surface. The remainder of 1
the boundary surface is called the active portion and is denoted SACT. Boundary values for f are assumed to be given on this portion of the boundary surface. The diffusion current through a user-selected portion of SACT is the quantity to be calculated. This selected portion of SACT is denoted SSEL. The quantity to be calculated is the diffusion current denoted I(t) and defined by
I (t ) D
S SEL
f ( x , t ) dS
(1.2)
where the unit normal vector in the surface integral is an outer normal and a negative sign was included to produce a positive current when f has the property that it increases when x moves from the boundary towards the interior of R (e.g., this convention regards heat flow as positive when the flow is out of the material contained in R). Several methods are available for calculating the diffusion current. One method is to solve (1.1) numerically using a finite difference method or a finite element method. A problem with this approach is that each example choice of boundary values (or initial conditions or generation rate density) requires a separate calculation. Also, numerical solutions provide only limited physical insight because each solved example is a special case. A second method utilizes a Green’s function. An advantage of this method is that the Green’s function is calculated only once for each geometry (where geometry refers to the geometry of R together with the specification as to how the boundary is partitioned into SREF and SSEL). Having done that, this same Green’s function can accommodate any boundary values, initial conditions, and generation rate density. A disadvantage of this method is that analytical expressions for the Green’s function are available only for a few special geometries while numerical solutions are encumbered by the fact that the Green’s function depends on two sets of coordinates (a set of observation coordinates and a set of source coordinates). Furthermore, any symmetry that might be present in the geometry does not produce symmetry in the Green’s function because of the second set of coordinates. A third method is considered in this report. The method considered in this report has some resemblance to the Green’s function method but instead of a Green’s function we work with a function Ω that is defined and discussed in detail Sections III through V. An advantage that this method shares with the Green’s function method is that Ω has to be calculated only once for each geometry. Having done that, this same Ω can accommodate (via an integration) any boundary values, initial conditions, and generation rate density for the calculation of the diffusion current. A disadvantage that this method shares with the Green’s function method is that analytical expressions for Ω are available only for a few special geometries. Other geometries will require Ω to be evaluated via a numerical solution of the boundary-value problem that defines Ω. However, a numerical evaluation of Ω is much easier than a numerical evaluation of a Green’s function because Ω depends only on one set of coordinates. Also, because Ω depends only on one set of coordinates, any symmetry present in the geometry (if there is any) will also be present in the Ω function, leading to simplifications in the calculations.
II. Assumptions and Notation for the Geometry The three-dimensional spatial region R is a closed set in the sense that it includes all boundary points. Let R’ denote the set of interior points of R so R’ is an open set in which spatial derivatives are defined. As explained in the Introduction, the boundary of R is partitioned into 2
two subsets. One is SREF where reflective boundary conditions are imposed, and the other is SACT where boundary values of f are given. The latter set is also partitioned into two subsets. One is SSEL which is the selected portion of SACT through which the diffusion current is to be calculated. The remainder of SACT will be denoted SREM. The set of boundary points SREF might be a connected set or it might be the union of multiple connected sets that are disconnected from each other. The same options are also allowed for SSEL and SREM. It is also possible for some of these sets to be empty. For example, if R is the region contained within some sphere and the goal is to calculate the diffusion current through the entire sphere, we can let SSEL be the entire sphere with SREF and SREM both empty sets. The selected set SSEL is not completely arbitrary because in order to avoid a singularity that cannot be integrated we require that SSEL and SREM be separated from each other. An example of an allowed geometry is that in which R is the region contained between two concentric spheres of different radii with SSEL selected to be one of the spheres and SREM is the other sphere. However, if the boundary of R is a single connected closed surface, it is necessary for SSEL and SREM to be separated by some portion of SREF. For example, suppose R is the region contained within some sphere and the quantity of interest is the diffusion current through the upper hemisphere when there is not enough symmetry in the boundary values (and/or initial conditions and/or generation rate density) to simply divide the diffusion current through the entire sphere by two. Selecting SSEL to be the upper hemisphere with SREM the lower hemisphere is not allowed but a variation of this arrangement that is allowed is one in which the two hemispheres are separated by a narrow band along the equator where reflective boundary conditions apply. If the diffusion current through the upper hemisphere for this arrangement has a limiting value (it might or might not depending on the continuity conditions, or lack of, satisfied by the boundary values of f) as the width of this band shrinks to zero, this limit is the quantity of interest. If this limit does not exist (e.g., if f = 1 on one hemisphere and f = 0 on the other, which will produce an infinite diffusion current out of one hemisphere and an infinite diffusion current into the other), the quantity of interest is not defined. As previously stated, the normal unit vector in surface integrals, such as the right side of (1.2), is directed away from the interior of R because this is the usual convention when using the divergence theorem. For example, if R is the region contained between two concentric spheres of different radii, the normal unit vector points towards the center for the inner sphere and away from the center for the outer sphere.
III. Definition of Ω Analysis of diffusion currents through SSEL is postponed until after a tool that will be used in that analysis has been discussed. This tool is a function denoted Ω. The definition of Ω is given in this section while the next two sections discuss some of its properties. After that is done we can then begin the analysis of diffusion currents through SSEL. Using the notation explained in Sections I and II, the function Ω(x,t) is defined by the following boundary-value problem:
( x, t ) ( x, t ) D ( x, t ) t 2
3
for
x R' , t 0
(3.1)
( x, t ) 1 for x S SEL , t 0 ( x , t ) 0
x SREM , t 0
(3.2b)
for x SREF , t 0
(3.2c)
for
( x, t ) nˆ 0
(3.2a)
( x,0) 0
for x R
(3.2d)
where the n-hat in (3.2c) is the unit normal vector to the boundary surface. This reflective boundary condition applies for t > 0 as required in (3.2c), and also at t = 0 as implied by (3.2d). Note that for any x in SSEL, Ω(x,t) regarded as a function of t is defined and bounded for all t ≥ 0, but it is discontinuous at t = 0. However, for any t ≥ 0, Ω(x,t) regarded as a function of x is continuous in the closed set R. Continuity in x when t = 0 is implied by (3.2d) while continuity in both x and t when t > 0 is implied by (3.1) through (3.2c) together with the requirement (explained in Section II) that there be some separation between SSEL and SREM.
IV. Preliminary Discussion of Singularities Define Ω-dot by
( x , t ) ( x , t ) t
for
x R' , t 0
(4.1)
so (3.1) can be written as
( x , t ) D ( x , t ) ( x , t ) 2
for
x R' , t 0 .
(4.2)
Note that if x is any point on SSEL, we have Ω(x,0) = 0 but Ω(x,t) = 1 for any t > 0. This discontinuity in t of the boundary value produces a singularity in the time derivative, i.e., in Ωdot, at the boundary. This singularity has implications regarding the spatial dependence of Ω-dot when t is close to zero and x is moved towards SSEL. This spatial dependence will first be discussed on a conceptual level, and then a more rigorous analysis will make the conclusions more precise. For a conceptual discussion, consider a time point that is greater than zero but very close to zero. There will be a neighborhood of SSEL, a boundary layer, having the property that Ω is nearly zero outside this boundary layer but changes rapidly from a value of 1 to nearly zero as x moves from SSEL through the boundary layer. As t approaches zero, the boundary layer shrinks to an infinitely thin film covering SSEL, but this boundary layer is still in the open set R’ for any positive t and will be encountered by volume integrals that integrate over the open set R’. A change in Ω from 1 to nearly zero over a thin spatial layer implies that the spatial derivative of Ω, with respect to a coordinate that measures distance from the boundary, becomes a Dirac delta function as the boundary layer shrinks to zero thickness. A second derivative with respect to this coordinate produces a spatial derivative of this δ-function. We can therefore anticipate, from (4.2), that Ω-dot becomes a spatial derivative of a δ-function. Because this singularity is 4
encountered by volume integrals that integrate over the open set R’, we can anticipate that an integration over R’ of a function multiplied by Ω-dot will produce a surface integral of a spatial derivative of that function when taking the limit as t → 0. To make the above statements more rigorous and precise, first note that for any x in the open set R’ and for any t > 0, Ω is sufficiently well behaved in some sufficiently small spatial neighborhood (small enough to not reach the boundary of R) of x so that the limit as t → 0 can be commuted with the spatial derivatives on the left side of (4.2) so that this term becomes zero in the limit. The second term on the right side of (4.2) also goes to zero so we conclude from (4.2) that lim ( x , t ) 0
for any
t 0
x R'
(4.3)
where the “+” superscript indicates that t approaches zero from above. Note that (4.3) is consistent with a statement in the previous paragraph; that the boundary layer shrinks to an infinitely thin film covering SSEL in this limit. However, a casual inspection of (4.3) might give the impression that (4.3) is inconsistent with another earlier statement; that, even in the limit, the boundary layer is encountered by spatial integrations over R’. This apparent paradox is resolved by making the proper distinction between the limit of an integral and the integral of a limit. For a specific illustration, let F(x) be any function that is continuous on R, has first and second derivatives that are continuous and bounded on R’, satisfies reflective boundary conditions on SREF, and has the value of zero everywhere on SACT. Note that Green’s theorem gives D ( x , t ) F ( x ) dS D F ( x ) ( x , t ) dS
D ( x , t ) 2 F ( x ) d 3 x D F ( x ) 2 ( x , t ) d 3 x R'
R'
for t 0
where the surface integrals are over the closed boundary of R. Including boundary conditions for F and Ω on the left side of the above, and using (4.2) to make a substitution on the right side gives
D
S SEL
( x , t ) 3 2 3 F ( x ) dS D ( x , t ) F ( x ) d x F ( x ) ( x , t ) d x R' R'
for t 0
Note that the left side does not depend on t and is therefore equal to the limit of the right side as t → 0+. When taking this limit we note that Ω, like Ω-dot, goes to zero at each point in R’, but, unlike Ω-dot, Ω is bounded on R’ × (0,∞). Therefore, when taking the limit, the integrals containing Ω become zero and the result is
lim
t 0 R'
F ( x ) ( x , t ) d 3 x D
S SEL
5
F ( x ) dS
(special case)
(4.4)
which is labelled a special case because of the boundary conditions assumed for F. This result gives a more precise meaning to an earlier statement; that an integration over R’ of a function multiplied by Ω-dot will produce a surface integral of a spatial derivative of that function when taking the limit as t → 0+. Also, an important conclusion from (4.3) and (4.4) is that we cannot indiscriminately move a t → 0+ limit from outside an integral to inside the integral if the integral contains Ω-dot. We now consider a slightly more general case by relaxing the requirement that F be zero on SACT. All other conditions assumed for F still apply. Specifically, F is continuous on R, has first and second derivatives that are continuous and bounded on R’, and satisfies reflective boundary conditions on SREF. Following the same steps that produced (4.4) for this more general case gives
lim F ( x ) ( x , t ) d 3 x D F ( x ) ( x , t ) dS D F ( x ) dS . S SEL t 0 R'
(4.5)
Note that the gradient of Ω on SREM is well behaved because only SSEL has a boundary layer. This gradient goes to zero as t → 0+ so an alternate form of (4.5) is
lim F ( x ) ( x , t ) d 3 x D F ( x ) ( x , t ) dS D F ( x ) dS . S SEL S SEL t 0 R'
(4.6)
An interesting result is obtained from the particular choice of F given by F(x) = 1 for each x in R, which gives lim ( x , t ) d 3 x D ( x , t ) dS 0 . (4.7) S SEL t 0 R' Note that the two integrals in the curly bracket in (4.7) do not individually have finite limits. This is seen by noting that the gradient of Ω on SSEL increases (in magnitude) without bound as t → 0+. This implies that the first term must also become singular because the difference between the terms has a finite limit. Similarly, the two integrals in the curly bracket in (4.6) do not individually have finite limits, if F (x) ≠ 0 on SSEL, although the singularities subtract out to give the curly bracket a finite limit.
V. Integrability of a Singularity It was pointed out in the above discussion that the two integrals inside the curly bracket in (4.7) do not individually have finite limits as t → 0+. However, an important conclusion that can be derived from (4.7) is that these singularities are integrable in t in the sense of an improper Riemann integral. The time-integral of the second integral in the curly bracket is defined as an improper Riemann integral by
t ( x , ) d S d lim ( x , ) d S d 0 S SEL 0 S SEL t
6
(5.1)
providing that the limit exists. To show that the limit does exist, first note that a maximum principle applies to boundary-value problems of the type given by (3.1) and (3.2), and for the specific example of (3.1) and (3.2) this principle concludes that Ω(x,t) ≤ 1 for any (x,t) in R × [0,∞). Therefore, as the point x is moved in a neighborhood of SSEL, the direction of increasing Ω is in the direction of the outer normal vector. This implies that the surface integral in (5.1) is positive, which implies that the outside integral on the right side of (5.1) increases as δ is made closer to zero. This implies that existence of the limit will be established if it can be shown that the outer integral on the right side of (5.1), regarded as a function of δ, is bounded above. To establish this boundedness, select an ԑ > 0. We conclude from (4.7) that there is a δ0(ԑ) > 0 such that 1 ( x , ) d S ( x, ) d 3 x SSEL D R'
if
0 0 ( ) .
(5.2)
Let δ < δ0(ԑ) and note that
( x , ) d S d S SEL t 0 ( ) ( x , ) d S d ( x , ) d S d 0 ( ) S SEL S SEL t
and using (5.2) with the above gives
( x , ) d S d S SEL t 0 ( ) 1 3 ( x , ) d S d ( x , ) d x d 0 ( ) S SEL D R' t 1 0 ( ) ( x , ) d S d ( ) ( x , ) d 3 x d . 0 0 ( ) S SEL D R' t
Interchanging the order of integration in the last term on the far right while using (4.1) gives
( x , ) d S d S SEL t 1 3 ( x , ) d S d 0 ( ) R' x , 0 ( ) x , d x 0 ( ) S SEL D t 1 ( x , ) d S d ( ) x , 0 ( ) d 3 x . 0 0 ( ) S SEL R' D t
7
The far-right bound applies to any δ > 0 satisfying δ < δ0(ԑ), which establishes boundedness of the integral on the far left and verifies the assertion that
( x , ) d S d 0 S SEL t
lim
exists .
(5.3)
Notation that is more applicable to a later section is obtained by changing integration variables to get
t ( x , ) d S d ( x , t ) d S d S SEL 0 S SEL t
(5.4)
and we conclude from (5.3) and (5.4) that
lim
t
0 0
S ( x , t ) dS d SEL
exists .
(5.5)
Notation used to denote this limit is
t ( x , t ) d S d lim ( x , t ) d S d . 0 S SEL S SEL 0 0 t
(5.6)
It is evident from (5.1), (5.4), and (5.6) that
t ( x , t ) d S d ( x , ) d S d . 0 S SEL 0 S SEL t
(5.7)
A corollary to (5.5) is as follows. Let f(x,t) be any function that is continuous on SSEL×(0,∞) and also has the property that for each T > 0, f(x,t) is bounded on SSEL×(0,T]. It is not necessary for f(x,t) to have a limit as t → 0+. An example of an allowed function that does not have such a limit is f(x,t) = sin(1/t) for each x on SSEL. However, if f(x,t) does have such a limit, and satisfies the continuity condition, f(x,t) will also satisfy the boundedness condition. The above boundedness condition implies that for any t > 0 there exists a bound B(t) > 0 such that
f ( x, ) B(t )
for all x S SEL and all (0, t ] .
Combining this with the fact that the surface integral on SSEL of the gradient of Ω is positive gives
S SEL f ( x, ) ( x, t ) dS
B(t )
S SEL
8
( x , t ) dS
for all (0, t ) .
From the above condition, together with (5.5), we conclude via the comparison test for improper integrals that t lim f ( x , ) ( x , t ) d S exists . (5.8) d S SEL 0 0 Notation used to denote this limit is
t f ( x , ) ( x , t ) d S d lim f ( x , ) ( x , t ) d S d . (5.9) 0 S SEL 0 0 S SEL t
VI. Sufficient Requirements of the Function f A typical practical application of (1.1) is one in which f is defined by this equation (with the generation rate density g a known function) together with initial conditions and boundary conditions. Given that the initial conditions, boundary conditions, and generation rate density satisfy some continuity (or at least boundedness) conditions, f will also satisfy some continuity conditions when t > 0. However, the theory in the next section can also be applied to a more general context. The function f might have originated from some other physical analysis, not necessarily related to diffusion, but it still satisfies the diffusion equation (1.1) because (1.1) is used to define the function g. In this more general context, continuity conditions satisfied by f are not automatically implied so they must be explicitly imposed in order to insure that the theory in the next section will be valid. The conditions listed in this section are sufficient for the analysis in the next section to be valid when all integrals listed there are interpreted as Riemann integrals or improper Riemann integrals (limits of Riemann integrals). They are not all necessary conditions, and Section VIII will relax some of these conditions, but the easiest way to progress to Section VIII is by starting with the conditions listed in this section, which are: (a) The function f, as a function of both x and t, is continuous on R×(0,∞). (b) For each t > 0, f(x,t), 2 f(x,t), and ∂f(x,t)/∂t as functions of x are continuous and bounded on R’. (c) For each t > 0, f(x,t) as a function of x satisfies reflective boundary conditions on SREF. (d) For each T > 0, f(x,t) is bounded on SSEL×(0,T]. Note that this will be satisfied if (a) above is satisfied and the limit of f(x,t) as t → 0+ exists for each x on SSEL. (e) For each x in R’, the limit of f(x,t) as t → 0+ exists. Furthermore, this limit, denoted fI(x), is continuous and bounded on R’. The initial function fI(x) defined by (e) in the open set R’ is defined as a limit. It is not required that the quantity f(x,0) be defined. If f(x,0) is defined but defined in such a way so that this quantity is not equal to the t → 0+ limit of f(x,t), then f(x,0) is irrelevant to the analysis in the next section because the relevant quantity is the t → 0+ limit of f(x,t). It should be noted that the above requirements do not require that boundary conditions join continuously with initial conditions. In other words, if xS is a point on the boundary of R, the limit as t → 0+ of f(xS,t) is not required to be equal to the limit as x → xS (along a path in R’) of f(x,0). In fact, Ω(x,t) is an example of a function that satisfies the above requirements and, for this example, the former limit is 1 while the latter limit is 0 when xS is a point on SSEL. However, 9
if boundary conditions do not join continuously with initial conditions for f, we can expect singularities in derivatives of f similar to the singularities associated with Ω that were discussed in Section IV. It should also be noted that Condition (b) together with (1.1) implies that g(x,t), as a function of x, is continuous and bounded on R’ for any t > 0. An important special case that is excluded by this requirement is that in which the generation is produced by a point source, i.e., g is a Dirac delta function, because an integral of a Dirac delta function is not a Riemann integral. This case will be treated later in Section VIII.
VII. Diffusion Current and Time-Integrated Current The goal is to solve for the diffusion current I(t) defined by (1.2) where “solve” means to express the quantity in terms of integrals containing Ω together with boundary values for f, initial conditions for f, and the generation rate density function g. The time-integrated current is denoted Q(t) and defined by
t t Q (t ) I ( ) d D f ( x , ) dS d 0 0 S SEL
(7.1)
which gives
I (t )
d Q (t ) . dt
(7.2)
The most convenient way to solve for I(t) is to first solve for Q(t) and then use (7.2). Recall that f satisfies
f ( x, t ) f ( x, t ) D f ( x, t ) g ( x, t ) t 2
x R' , t 0 .
for
(7.3)
The analysis begins by selecting two time points ξ and t satisfying 0 < ξ < t. Note that (3.2) and (4.2) imply that Ω(x,t‒ ξ) satisfies
( x , t ) D ( x , t ) ( x , t ) 2
for
x R' , t
(7.4)
( x , t ) 1
for
x S1 , t
(7.5a)
( x, t ) 0
for
x S0 , t
(7.5b)
( x , t ) nˆ 0
( x, t ) 0
for
x S ref , t
for x R, , t .
We also have, by the chain rule, that
10
(7.5c) (7.5d)
( x , t ) ( x , t ) ( x , t ) t
for
x R' , t .
(7.6)
For any fixed ξ and t satisfying 0 < ξ < t, the two functions f(x,ξ) and Ω(x,t‒ ξ) are sufficiently well-behaved functions of x (recall the conditions listed in Section VI) so that Green’s theorem can be used to get D ( x , t ) f ( x , ) dS D f ( x , ) ( x , t ) dS
D ( x , t ) 2 f ( x , ) d 3 x D R'
R'
f ( x , ) 2 ( x , t ) d 3 x
for t 0
where the surface integrals are over the entire closed boundary of R. Using the reflective boundary conditions satisfied by both f and Ω together with the boundary values for Ω in the surface integrals, and using (1.2) to denote one of the surface integrals gives
I ( ) D
R'
f ( x , ) 2 ( x , t ) d 3 x D ( x , t ) 2 f ( x , ) d 3 x R' D f ( x , ) ( x , t ) dS S ACT
for t 0
The next step uses (7.3) and (7.4) to make substitutions inside each of the two volume integrals. This produces two terms containing 1/τ that subtract out and the result is
f ( x, ) 3 I ( ) f ( x , ) ( x , t ) ( x , t ) d x ( x , t ) g ( x , ) d 3 x R' R' D f ( x , ) ( x , t ) dS for t 0 S ACT
We now use (7.6) to write the above square bracket as the derivative of a product. The result is
I ( )
d f ( x , ) ( x , t ) d 3 x ( x , t ) g ( x , ) d 3 x R' d R' D f ( x , ) ( x , t ) dS S ACT
for t 0
(7.7)
The current I(ξ) is not yet considered to be solved because the first integral on the right side of (7.7) contains f(x,ξ), while a solved quantity is one that is expressed in terms of integrals containing only the initial function fI(x), the function g(x,ξ), and boundary values of f. We can construct a solved quantity by integrating (7.7) with respect to ξ but here it is necessary to pay attention to singularities. We can postpone encountering any singularities until later in the analysis by selecting a δ satisfying 0 < δ < t. When ξ is confined to the interval δ < ξ < t ‒ δ, all
11
relevant functions are sufficiently well-behaved functions of both x and ξ so that the elementary rules of integration can be used with (7.7) to produce t
I ( ) d
R'
f ( x , ) ( x , t ) d 3 x
f ( x , t ) ( x , ) d 3 x
R'
t g ( x , ) ( x , t ) d 3 x d D f ( x , ) ( x , t ) d S d R' S ACT for t 0 .
t
(7.8)
Now consider the limit as δ → 0+ of the first integral on the right side of (7.8). An important distinction between this volume integral and volume integrals containing Ω-dot is that the integrand is bounded in the former integral. The conditions listed in Section VI are sufficient to conclude, via Arzela’s dominated convergence theorem, that the limit can be moved from outside the integral to inside the integral. But the δ → 0+ limit of f(x,δ) is the definition of fI(x) so
0 R' lim
f ( x , ) ( x , t ) d 3 x
R'
f I ( x ) ( x, t ) d 3 x .
(7.9a)
Similar considerations apply to the second integral on the right side of (7.8) but now we use (3.2d) to obtain
0 R' lim
f ( x , t ) ( x , ) d 3 x 0 .
(7.9b)
The δ → 0+ limit of the third integral on the right side of (7.8) exists because the volume integral is an integration of a function that is continuous and bounded in R’, so the improper Riemann integral is also a proper Riemann integral. This gives
lim
t
0
R'
t g ( x , ) ( x , t ) d 3 x d
0
R'
g ( x , ) ( x , t ) d 3 x d .
(7.9c)
The last integral on the right side of (7.8) is the one in which the existence of the δ → 0+ limit is less obvious because of the singularity of the gradient of Ω(x,t ‒ ξ) when x is on SSEL and ξ = t. However, the conditions listed in Section VI are sufficient for (5.8) to apply and we conclude from (5.8) that this singularity is integrable as an improper Riemann integral. The result is
lim
t
0
f ( x , ) ( x , t ) d S S d ACT f ( x , ) ( x , t ) d S d 0 S ACT t
where the integral on the right is an improper (if not a proper) Riemann integral.
12
(7.9d)
The final result for the time-integrated current Q(t) is obtained by combining (7.1), with the integral interpreted as an improper (if it is not a proper) Riemann integral, with (7.8) and (7.9) to get f I ( x ) ( x , t ) d 3 x
Q (t )
R'
t
0
t g ( x , ) ( x , t ) d 3 x d D f ( x , ) ( x , t ) dS d . R' 0 S ACT
(7.10)
The final result for the current I(t) is obtained by combining (7.2) with (7.10) to get
I (t )
d f I ( x ) ( x , t ) d 3 x d t R' t
0
t g ( x , ) ( x , t ) d 3 x d D f ( x , ) ( x , t ) dS d . R' 0 S ACT
(7.11)
What might seem to be the natural next step after listing (7.11) is to move the time derivative inside of the square bracket and use the sum rule for derivatives together with the chain rule to obtain terms with the derivative inside an integral, which will be integrals containing Ω-dot. The resulting expression will be called the expanded derivative. A motivation for not expanding the derivative becomes clear after making a distinction between avoidable singularities and an unavoidable singularity. If the boundary conditions for f on SSEL do not join continuously with the initial conditions (the meaning of this statement was explained in Section VI), the absolute value of I(t) will increase without bound as t → 0+. This is called an unavoidable singularity. However, the expanded derivative will contain terms that individually have singularities even if there is no unavoidable singularity, i.e., even if the boundary conditions join continuously with initial conditions. If there is no unavoidable singularity, the individual singularities in the expanded derivative will subtract out. For example, the two terms in the curly bracket in (4.7) individually become singular in the limit but the singularities subtract out in the sense that the combination of terms shown there has a finite limit. Singularities that subtract out are called avoidable because they can be avoided. The easiest way to avoid them in (7.11) is by not expanding the derivative, i.e., by leaving the time derivative on the left side of the square bracket as is shown in (7.11).
VIII. Discontinuities and Point Sources A. Discontinuities The conditions listed in Section VI require that fI(x) be continuous and bounded on R’. This requirement can be relaxed by replacing it with a more lenient requirement that will include most functions, including most discontinuous functions, that are likely to be encountered in scientific or engineering applications. The relaxed requirements are: (i) The function fI(x) is bounded in R’ and Riemann integrable on R’.
13
(ii) There exists a sequence of continuous (in R’) functions, denoted f1(x), f2(x), …, that are uniformly bounded in the sense that there is a constant M > 0 such that |fn(x)| ≤ M for all n and all x in R’, and the sequence converges pointwise to fI(x). (Uniform convergence is impossible if fI(x) is discontinuous but uniform convergence is not required.) When fI(x) satisfies the above conditions, Q(t) is defined here to be the n → ∞ limit of Qn(t) where Qn(t) is the time-integrated current produced when fI(x) is replaced by fn(x) and is given by Qn ( t )
R'
f n ( x ) ( x , t ) d 3 x t
0
t g ( x , ) ( x , t ) d 3 x d D f ( x , ) ( x , t ) dS d . R' 0 S ACT
When taking the n → ∞ limit to obtain Q(t), Arzela’s dominated convergence theorem allows us to commute the limit across the integral and the result is identical to (7.10). Therefore (7.10) can be used without modification when fI(x) is discontinuous as long as it satisfies conditions (i) and (ii). A similar analysis concludes that (7.10) can be used without modification when g(x,t) is discontinuous as long as it satisfies: (I) The function g(x,ξ) is bounded in R’×[0,t] and Riemann integrable on R’×[0,t]. (II) There exists a sequence of continuous (in R’×[0,t]) functions, denoted g1(x,ξ), g2(x,ξ), …, that are uniformly bounded in the sense that there is a constant M > 0 such that |gn(x,ξ)| ≤ M for all n and all (x,ξ) in R’×[0,t], and the sequence converges pointwise to g(x,ξ).
B. Point Sources Now consider a point source so g(x,t) is a Dirac delta function. This requires a different kind of analysis because useful integrations of Dirac delta functions are not Riemann integrations. Analysis of a point source can be done with Riemann integrals by taking suitable limits and that is the approach used here. A point source can be distributed over time or it can be impulsive. An impulsive point source can be thought of as a generation rate density that is nonzero over a vanishingly small neighborhood of some point in R’ and is “turned on” over a vanishingly small time interval but has enough intensity to produce a time-integrated current that is not vanishingly small. A point source that is distributed over time is similarly defined except that the “turned on” time is not vanishingly small and the intensity has a finite time integral. The current produced by any collection of point sources added to any generation rate density satisfying conditions (I) and (II) above is the sum of the currents from the individual contributions so it is sufficiently general to consider a single point source by itself. Also, contributions to the current from boundary conditions and initial conditions for f are additive (as seen in (7.10)) so it is sufficiently general to consider the current produced by a point source when this current is defined to be the current produced by only the point source, i.e., it excludes the contributions from boundary conditions and initial conditions and includes only the second integral on the right side of (7.10). We first consider an impulsive point source located at some point x0 in the open set R’ and is turned on at some time t0 > 0. Because x0 is a point in an open set and t0 > 0, there exists a δ1 and δ2 satisfying
14
if x x0 1 then x R' , if t t0 2 then t 0 . For each positive integer n we define a neighborhood of the point (x0,t0), a set of points (x,t) and denoted Nn, by
N n ( x , t ) | x x0 1 and t t0 2 n n
(8.1)
so that each neighborhood is a subset of R’×(0,t), and with the neighborhoods becoming smaller with larger n. This sequence of neighborhoods is used to define a sequence of functions, denoted h1(x,t), h2(x,t), …, to be any sequence of functions satisfying
hn ( x, t ) is continuous and bounded in R' (0, ) hn ( x, t ) 0 if ( x, t ) N n , hn ( x, t ) 0 if ( x, t ) N n
0 R' hn ( x, ) d
3
x d 1 .
(8.2a) (8.2b) (8.2c)
From this sequence of functions we obtain another sequence of functions, denoted g1(x,t), g2(x,t), …, and given by
gn ( x, t ) A hn ( x, t ) (impulsive point source)
(8.3)
where A is any real number and is interpreted to be the strength of the source. The significance of A is seen by combining (8.2c) with (8.3) to get
0 R'
g n ( x , ) d 3 x d A (impulsive point source) .
(8.4)
A finite representation of an impulsive point source of strength A is gn(x,t), which becomes a better representation for larger values of n. It is nonzero over a very small (when n is very large) neighborhood of x0, and over a very small (when n is very large) time interval. However, the amplitude of this pulse-like function, while finite, is large enough to compensate for the smallness of the neighborhood and time interval as needed to satisfy (8.4). The timeintegrated current from an impulsive point source of strength A is defined here to be the n → ∞ limit of Qn(t) where Qn(t) is what the second integral on the right side of (7.10) becomes when g(x,ξ) is replaced by gn(x,ξ), so Qn(t) is given by
Qn (t )
t 0
R'
g n ( x , ) ( x , t ) d 3 x d .
15
(8.5)
If t satisfies 0 ≤ t ≤ t0 ‒ δ2/n, the integration does not include any points where gn(x,ξ) differs from zero so Qn(t) = 0. If t satisfies t > t0 ‒ δ2/n, the integration includes at least some points where gn(x,ξ) differs from zero. The fact that gn(x,ξ) does not change sign (it has the same sign as A where it differs from zero), and differs from zero only in the neighborhood Nn, allows us to conclude from the mean value theorem for integrals that t Qn (t ) ( x ' , t t ' )
0
R'
g n ( x , ) d 3 x d
for some (x’,t’) in Nn. If t satisfies t > t0 + δ2/n, the integration includes all points where gn(x,ξ) differs from zero so the integral on the right is equal to A. The conclusion is
2 0 if t t0 n Qn ( t ) A ( x ' , t t ' ) for some ( x ' , t ' ) N n
if t t0
2
.
n
If we take the limit as n → ∞, the neighborhood Nn shrinks to the point (x0,t0) and the result is
0 if 0 t t0 Q (t ) A ( x0 , t t0 ) if t t0
(impulsive point source)
(8.6)
where Q(t0) is defined to be zero in (8.6) so that Q(t) will be continuous. If t approaches t0 either from above or from below, Q(t) → 0. We next consider a point source that is distributed over time and is located at some point x0 in the open set R’. Using steps similar to those used for the impulsive point source, we define a sequence of neighborhoods by
M n x | x x0 n where δ > 0 is selected to be small enough so that each neighborhood is contained in R’. From this sequence of neighborhoods we define a sequence of functions, denoted k1(x), k2(x), …, to be any sequence of functions satisfying
kn ( x ) is continuous and bounded in R' kn ( x ) 0 if x Mn , kn ( x ) 0 if x Mn
R' k n ( x ) d
16
3
x 1.
From this sequence of functions we obtain another sequence of functions, denoted g1(x,t), g2(x,t), …, and given by
gn ( x, t ) B(t ) kn ( x ) (point source distributed over time) where B(t) is sectionally continuous and bounded on (0,∞) but otherwise is an arbitrary function. The significance of B(t) is seen by combining the above equations to get
R'
g n ( x , t ) d 3 x B(t ) (point source distribute d over time) .
(8.7)
The remainder of the analysis is similar to the analysis used for an impulsive point source and the result is t Q(t ) B( ) ( x0 , t ) d
(point source distribute d over time) .
0
(8.8)
IX. A Practical Consideration Eq. (7.10) is not useful for calculating Q(t) unless we are able to calculate Ω and its gradient on boundary surfaces. If the geometry is anything other than one of a few special cases, the most likely approach is to use modern commercial software to obtain a numerical solution to (3.1) and (3.2). Having done that for the special boundary conditions and initial conditions satisfied by Ω, we can then accommodate arbitrary boundary conditions, initial conditions, and generation rate density satisfied by f via (7.10). Also, a numerical solution is as easy to obtain with as without the inclusion of the term Ω/τ on the right side of (3.1). There are a few special geometries for which Ω can be expressed as an infinite series derived by using the method of separation of variables. This method is almost as easy to use when the term Ω/τ is present on the right side of (3.1) as it would be without this term present. However, the examples that require the least amount of effort are those examples for which the solution for Ω can be copied from some textbook. Unfortunately from the point of view of (7.10), most textbook examples do not include the term Ω/τ on the right side of (3.1). The goal here is to calculate Q(t) from another function, denoted Ѱ, that is easier to find in the literature and is defined by
( x , t ) D ( x , t ) t 2
( x, t ) 1 ( x , t ) 0
for
x R' , t 0
(9.1)
for x S SEL , t 0
(9.2a)
x SREM , t 0
(9.2b)
for x SREF , t 0
(9.2c)
for
( x , t ) nˆ 0
( x,0) 0 17
for x R .
(9.2d)
For example, suppose SSEL is a complete sphere and R’ is the region contained within the sphere. The boundary value problem for Ѱ is then the same boundary value problem that is solved in order to calculate how long it takes to cook a turkey when the turkey geometry is approximated by a sphere. The solution for this Ѱ contains an infinite series and is cumbersome to write but it can be found in the literature. Another example is given more attention here because the solution is simple to write. This applies to the case in which SSEL is a complete sphere, R’ is the region outside the sphere, and Ѱ(x,t) approaches zero as the distance from x to the center of the sphere approaches infinity. The solution for this example is
rR R ( x , t ) erfc 4D t r
(example)
( r R ) 2 rˆ R ( x , t ) ( x , t ) exp r 4 D t Dt
(example)
where R is the radius of the sphere, r is the distance from the center of the sphere to the point x, r-hat is the unit vector directed away from the center of the sphere, and erfc is the complimentary error function defined by
erfc(z)
2
2
z
e
d .
The above equations for Ѱ also apply on the sphere and inside the sphere but (9.1) is not satisfied at the center of the sphere because Ѱ has a singularity at the center. There still remains the task of calculating Q(t) when Ѱ is known instead of Ω. The equation for Q(t) expressed in terms of Ѱ is a little more cumbersome than (7.10) when 1/τ ≠ 0 but that is the penalty for using Ѱ instead of Ω. The equation is derived by defining the function h by
h( x , t ) e t / f ( x , t ) .
(9.3)
It is easy to show from (9.3) and (7.3) that h satisfies
h( x , t ) D h( x , t ) g ( x, t ) et / t 2
for
x R' , t 0 .
(9.4)
We now define q(t) to be what Q(t) becomes when f is replaced by h, and we define i(t) to be what I(t) becomes when f is replaced by h, so the defining equations are 18
i (t ) D
S SEL
h( x , t ) dS ,
t
q(t ) i ( ) d 0
(9.5)
which gives
i (t )
d q( t ) . dt
(9.6)
We also have, from (1.2), (9.3) and (9.5), that
i (t ) e t / I (t ) .
(9.7)
The equation (9.4) governing h is the same as (7.3) governing f except for the et/τ coefficient in the g term and there is no 1/τ term on the right side of (9.4), so the appropriate weighting function Ω becomes Ѱ when calculating diffusion currents for h. The equation for q(t) is obtained from (7.10) by changing notation with Q replaced by q, f replaced by h (which is et/τf), g replaced by et/τg and Ω replaced by Ѱ. The result is q( t )
R'
f I ( x ) ( x , t ) d 3 x
0 R' e t
t /
t g ( x , ) ( x , t ) d 3 x d D e t / f ( x , ) ( x , t ) dS d . (9.8) 0 S ACT
After calculating q(t) via (9.8), we can then calculate i(t) from (9.6) and then I(t) from (9.7) to get d q( t ) I ( t ) e t / i (t ) e t / . (9.9) dt
X. K+1 Terminals This section considers the case in which SACT is a collection of terminals, where a terminal is a portion of the boundary of R where f(x,t) has the same value for each x on that portion of the boundary. The terminals are denoted S0, S1, …, SK. The boundary values of f are denoted by
f ( x, t ) 0 f ( x, t ) fi (t )
for x S 0 , t 0
for x Si , t 0 when i {1,2,...,K}
(10.1a) (10.1b)
where each fi(t) is Riemann integrable over whatever time interval that is relevant to an application of interest but is otherwise an arbitrary function of t. The terminal S0 is included in the notation because this is convenient for some special applications (explained below) but if this is not convenient for the application of interest (e.g., if there is no terminal satisfying (10.1a)), 19
this terminal can be removed from the equations in this section by letting the set of points S0 be the empty set. For each i in the set {0, 1, 2,…,K}, the set of boundary points Si might be a connected set or it might be the union of multiple connected sets that are disconnected from each other. However, each terminal is required to be separated from all other terminals in the same way that SSEL and SREM are required to be separated from each other as explained in Section II. Notation that includes the terminal S0 is convenient for some applications and an example is the steady-state (large-t limit) diffusion equation when 1/τ = 0, the generation rate density is zero, and boundary values are constant in time so that a steady-state solution is well defined. For this case, the diffusion equation reduces to Laplace’s equation. The sum of the diffusion currents summed over all terminals (which is the current density integrated over all of SACT) is zero for this example. The boundary-value problem governing the diffusion currents through the terminals for this example is mathematically the same as the boundary value problem governing the electrostatic charges on a collection of K+1 conductors when the electrostatic potentials are given. Therefore we can use the vocabulary from the electrostatics problem to describe this example. Using the electrostatics vocabulary, the sum of the charges on all conductors is zero (if the problem is an exterior problem, it might be necessary for one of the conductors to be an imagined sphere of infinite radius to satisfy this condition). Also, the charge on each conductor can be expressed as a linear combination of the potentials of the conductors. The coefficients to the potentials are called coefficients of electrostatic induction and are elements of a square matrix. A difficulty occurs when we attempt to invert this matrix to express the potentials in terms of the charges (using the vocabulary of the example diffusion problem considered here, a difficulty occurs when we attempt to solve for the terminal values of f when the terminal currents are given). The reason is that the charges on the conductors are not all independent (because the sum of the charges on all conductors is zero), and the result is that the coefficient of induction matrix does not have an inverse as a (K+1)×(K+1) matrix. A trick used in the electrostatics problem to obtain an invertible matrix is to select one of the conductors, S0, and use it to define the reference potential. This is equivalent to assigning zero potential to S0, with the potentials of all other conductors interpreted as potential differences with respect to the potential of S0. The charges on the K conductors S1, …, SK can be assigned arbitrarily and independently (this assignment implies the charge on S0) and the coefficient of induction matrix can be inverted as a K×K matrix to calculate the potentials of all other conductors. Using the vocabulary of the example diffusion problem considered here, notation that includes a terminal S0 is a convenient way to solve for the terminal values of f when the terminal currents are given. However, as already pointed out, if this notation is not convenient for a particular application of interest, the terminal S0 can be removed from the equations that will be derived in this section by letting S0 be an empty set. A single equation can be used to calculate the time-integrated diffusion currents through all terminals if we define a set of Ω-functions denoted Ω0(x,t), Ω1(x,t), …, ΩK(x,t). The definition of each Ω-function is the same as (3.1) together with (3.2) except for a slight change in notation. With the understanding that i and j are each arbitrary elements of the set {0, 1, …, K}, the definition is written as
i ( x , t ) i ( x , t ) D i ( x , t ) t 2
20
for
x R' , t 0
(10.2)
i ( x, t ) 1 i ( x, t ) 0
for
i ( x , t ) nˆ 0
i ( x,0) 0
for x Si , t 0
(10.3a)
x S j when j i, t 0
(10.3b)
for
x SREF , t 0
for x R .
(10.3c) (10.3d)
Note that if S0 is an empty set, then Ω0(x,t) = 0 for all (x,t) in R×[0,∞). For any i in the set {0, 1, …, K}, the time-integrated diffusion current through the terminal Si can be calculated by letting SSEL = Si, so Ωi becomes the function satisfying (3.1) and (3.2). Using (7.10) with this choice for SSEL gives Qi (t )
R'
t
0
f I ( x ) i ( x , t ) d 3 x
t g ( x , ) i ( x , t ) d 3 x d D f ( x , ) i ( x , t ) dS d R' 0 S ACT
(10.4)
where t
Qi (t ) I i ( ) d , 0
I i (t ) D f ( x , t ) dS . Si
(10.5)
The surface integral on SACT is the sum of the surface integrals summed over all terminals and this fact together with (10.1) allows us to write (10.4) as Qi (t )
R'
f I ( x ) i ( x , t ) d 3 x t
0
K t g ( x , ) i ( x , t ) d 3 x d f j ( ) i , j (t ) d R' 0
(10.6)
j 1
where the notation was shortened by defining γi,j by
i, j (t ) D i ( x, t ) dS . Sj
(10.7)
The γi,j coefficients satisfy a reciprocity relation that can be derived from the following considerations. We are not allowed to let f = Ω0 in (10.6) because the derivation of (10.6) used (10.1a) which is incompatible with (10.3a) when i = 0 and S0 is not an empty set. However, (10.4) did not use (10.1) so (10.4) applies to an arbitrary f as long as g is selected to satisfy (7.3). When using (10.4) we are allowed to let f = Ωj (so the g satisfying (7.3) is zero) for any j in the set {0, 1, …, K}. Doing so while using (10.5) and (10.7) gives 21
t
t
0 j,i ( ) d 0 i, j (t ) d . A change in the integration variable on the right gives t
t
0 j,i ( ) d 0 i, j ( ) d which applies to arbitrary t ≥ 0 so we conclude that
j,i (t ) i, j (t )
(10.8a)
for any i and j in the set {0, 1, …, K}. The above equation can also be written, via (10.7), as
S j
i ( x , t ) dS j ( x , t ) dS .
(10.8b)
Si
Note that if S0 is an empty set, we get
0,i (t ) i,0 (t ) 0 if S 0 is an empty set .
(10.9)
Another set of constraints on the γi,j coefficients is obtained by using (10.4) with f given by f(x,t) = e‒ t/τ (so the g satisfying (7.3) is zero).1 The gradient of f is zero so Qi(t) = 0 for each i in the set {0, 1, …, K}. Making these substitutions in (10.4) while using (10.7) gives K
0 e / i, j (t ) d R' i ( x, t ) d 3 x . t
j 0
An equivalent result is obtained by changing the integration variable on the left side to write the equation as K
0 e / i, j ( ) d et / R' i ( x, t ) d 3 x . t
(10.10a)
j 0
Differentiating with respect to t and dividing out a common factor gives K
i, j (t ) R' i ( x, t ) d 3 x d t R' i ( x, t ) d 3 x 1
d
when t 0
(10.10b)
j 0
1
The result derived from this choice for f will be (10.10). An alternate derivation that will ultimately lead to (10.10b) starts with f = 1, which satisfies (7.3) when g = 1/τ.
22
which is defined when t > 0 and applies for each i in the set {0, 1, …, K}. An interesting special case is that in which 1/τ = 0 and we are interested in the large-t limit of γi,j. In this limit the square bracket in (10.10b) approaches a constant so the right side of (10.10b) becomes zero. We conclude that the matrix [γi,j] is not invertible as a (K+1)×(K+1) matrix for this case because the right side of (10.10b) being zero for each i in the set {0, 1, …, K} is a statement that the matrix [γi,j] maps a column matrix having all elements equal to 1 (which is not a null column matrix) into a null column matrix. The signs of the γi,j coefficients can be determined by noting that the values of Ω i are between 0 and 1. The maximum value (which is 1) is on the terminal Si. Therefore, if the spatial coordinate x is moved away from this terminal towards the interior of R, Ωi either remains constant (a possibility for some special cases) or decreases. The gradient of Ωi evaluated on the terminal Si is directed outward from the interior of R. Similar considerations applied to the terminal Sj when j ≠ i, where Ωi has its minimum value, conclude that the gradient of Ωi evaluated on the terminal Sj is directed into the interior of R. These conclusions combined with (10.7) give
i,i (t ) 0,
i, j (t ) 0 when j i
(10.11)
which applies for each i and j in the set {0, 1, …, K}. The primary result of this section is (10.6) which gives the time-integrated current through each terminal S0, S1, …, SK when the γi,j coefficients are calculated from (10.7). It might be helpful to know that these coefficients satisfy (10.8) through (10.11). The instantaneous diffusion current through each terminal is calculated from
I i (t )
d Qi (t ) . dt
23
(10.12)
