Diffusion Reaction Interaction for a Pair of Spheres - ETD ...

ate constant λ 50 . The figures 2.2 – 2.4 are produced from the rapidly
1 =
convergent equation (2.55), they are within 1% of the true value using only the
unity and the first 30, 15, and 8 terms from the infinite sum, respectively.
One intuitive statement is well illustrated in these plots i.e., for a large
very reactive sink γ → ∞ , λ 0 with the two spheres in contact
1 =
d a + a ) = 1,
the probability tends to unity, perfect trapping. The converse is
( 1 2
also true for γ = 0 , λ → ∞ with the two spheres at infinite dilution
1
d a + a ) → ∞ , the sink is a perfect reflector ( P = 0)
. These two statements
( 1 2
require that the probability of any real interactive sink-source pair be bounded
below by 0 and above by 1, a tighter lower limit is fixed by P which contains the
−1
correction factor ( 1+
λ1)
.
As the center to center distance approaches infinity the equation (2.31)
should reduce to equation (2.72) Pm . In fact, P and Pm
are within 0.1% for
figure 2.2 whereγ
= 10 for d a + a ) > 1.
01,
Figure 2.3 where γ = 1 if
( 1 2
d a + a ) > 1.
5 , and Figure 2.4 whereγ = 0.
10 and d a + a ) > 1.
9 . When
( 1 2
m
( 1 2
the two spheres are within these ranges the multipole correction plays a
significant role and must be considered. Outside of this range all three curves are
−1
harmonic relations in d1
where ( 1+
λ1)
controls the probability such that
decreasing
λ 1 (fast reaction, diffusion-controlled) increases the P value.
All of the curves in figure 2.2, 2.3 and the reaction-limited ( λ 5)
curves
in figure 2.4 the maximum occurs at the y-axis where d a + a ) = 1 and for all
39
( 1 2
1 ≥