19 Approximate Solutions of Chemical Separation Equations with Diffusion
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G E O R G E H . WEISS Physical Sciences Laboratory, Division of Computer Research and Technology, National Institutes of Health, Bethesda, M d . 20014 M E N A C H E M DISHON Department of Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel Many problems in the analysis of biochemical separation techniques require the solution of a Fickian equation of the form
where є is proportional to the diffusion constant. This paper presents an approximate solution to this equation valid for small є. For many separation systems єis 10 or less, while f ( x ) and g(x) are of the order of 1. Applications are given to velocity centrifugation experiments, with pressure dependent sedimentation, and to pore gradient electrophoresis. -3
"any problems related to biochemical separation systems require • solution to a transport equation of the form (1) in which c is solute concentration, g(x) is the transport term, and ef(x) is the diffusion term. Some examples of systems which fall into this category are: ( 1 ) Pressure dependent systems in velocity sedimentation (1, 2, 3, 4). (2) G e l pore electrophoresis (5,6). (3) Analytical gradient chromatography (7,8). 207 In Polymer Molecular Weight Methods; Ezrin, Myer; Advances in Chemistry; American Chemical Society: Washington, DC, 1973.
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POLYMER
MOLECULAR
WEIGHT METHODS
Undoubtedly other examples can be found. In the study of the systems cited above, a common property is that the ( dimensionless ) parameter c is very small compared with the (dimensionless) terms f(x) and g(x) which can be chosen to be of the order of 1. In all of the examples cited c is 2.5 χ 10" or less. In this paper we summarize a singular perturbation technique for the solution of Equation 1, which is to be solved subject to an initial condition c(x,0), and on the assumption that boundary effects (in x) can be ignored. To see why it is impossible to obtain a solution to Equation 1 by ordinary perturbation theory, consider the simple system Downloaded by UNIV OF MONTANA on January 25, 2016 | http://pubs.acs.org Publication Date: June 1, 1973 | doi: 10.1021/ba-1973-0125.ch019
2
dc οτ
dc dx 2
2
subject to an initial condition c(x,0) = δ ( χ ) , a delta function. If we set c = 0, then the solution is C(X,T) = δ ( χ ) . O n the other hand, the solution to Equation 2 is known to be:
'
c(x T) =
^ e x p (~^)
(3)
that is, spreading occurs because of diffusion. The solution (Equation 3 ) cannot be found by expanding the variables i n Equation 2 i n a power series in c. In particular if we consider a point χ ^ 0, the concentration is identically equal to 0 when e = 0, but it can be much greater than 0 for any nonzero c To derive an approximate solution we start by transforming the space variable χ to one that moves by the convective transport mechanism— i.e., a coordinate that remains fixed with the solute motion when diffusion can be neglected. This coordinate w i l l be denoted by $ and is τ
du
If we define a function H(z) to be the solution to
JI ο
g(u)
=
()
z
5
then a molecule initially at position £o (or x = H ( £ o ) ) w i l l be trans ported to position £ + τ at time τ i n the absence of diffusion. If we also define a new dependent variable ψ(χ,τ) by 0
0
Ψ(ζ, τ) = g(x)c(x, τ)
In Polymer Molecular Weight Methods; Ezrin, Myer; Advances in Chemistry; American Chemical Society: Washington, DC, 1973.
(6)
19.
Chemical Separation Equations
WEISS A N D DISHON
209
then Equation 1 can be transformed to a [ F « + τ) d ( ψ W Ιθ(ξ + τ) θξ \G(£ + τ))]
θψ ck
οξ
m
'
Κ Ί
where
Downloaded by UNIV OF MONTANA on January 25, 2016 | http://pubs.acs.org Publication Date: June 1, 1973 | doi: 10.1021/ba-1973-0125.ch019
F{u)
= f(H(u)),
G(u) = g{H{u))
(8)
Equation 7 is exact, no approximation having been made. At this point we can introduce the perturbation procedure based on the fact that c is small. Let us assume that the initial condition is c(x,0) = c &(x), where δ(χ) is a delta function (—i.e., an initial pulse injection). The principal consequence of a small c is the fact that the bandwidth of the diffusion broadened peak is narrow, going to zero as c vanishes. Thus, we make the approximation that only the region near £ = 0 in the terms F (ξ + Τ) and G ( £ + τ ) is of interest. The resulting value, denoted by ψο(έ,τ), therefore satisfies the equation 0
ck
G (T) θξ 2
2
W
subject to the initial condition ψο(χ,Ο) = c g(x)S(x). This is a diffusion equation with a time-dependent diffusion constant. It can be reduced to a more familiar form by introducing a new dimensionless time Δ ( τ ) by 0
^ = £9ê
du
G (u) 2
(10)
in which case Equation 9 becomes θφο
θψ 2
0
The solution to this equation is straightforward and can be written
*·«' -νϊ3Μ (-5&)) Δ )
Μ ρ
(12)
If we can assume that g(0) = 1 (this can always be done without loss of generality), then the final expression for c(x,r) becomes Co,
In Polymer Molecular Weight Methods; Ezrin, Myer; Advances in Chemistry; American Chemical Society: Washington, DC, 1973.
210
POLYMER
MOLECULAR
WEIGHT METHODS
where £ is given i n Equation 4. Notice that c(x,r) is not generally sym metric in χ because $ is equal to a function of χ that is not necessarily symmetric. The maximum peak concentration is very closely given by C m a x
_
Downloaded by UNIV OF MONTANA on January 25, 2016 | http://pubs.acs.org Publication Date: June 1, 1973 | doi: 10.1021/ba-1973-0125.ch019
Co
1
(14)
0(#(τ))\/4πεΔ(τ)
This relation enables one to determine c (or equivalently, the diffusion constant) experimentally. To put the preceding analysis into a more applied framework, let us consider the peak broadening in pore gradient electrophoresis (5, 6, 9). For this problem, let D and D represent the diffusion coefficient in the gel and in the absence of a gel, and M and M the respective mobilities. For many gels these variables experimentally satisfy Equation 5: 0
0
D/D
= exp ( -
= M/Mo
0
x/L)
(15)
for a linear gel gradient, where χ is distance and L is an experimentally measured parameter. Let V be the voltage gradient. Then the dimensionless parameters τ and c that characterize the Fick equation are τ = MoVt/L,
ε = D /(M VL) 0
0
(16)
In addition we w i l l let ζ = x / L be a dimensionless distance. The trans port equation for this system is
S--ÏK)-S
