Biophotonics_Concepts_to_Applications

GðsÞ¼

hidIðtÞdIðtþsÞ

hiIðtÞ^2

¼

hiIðtÞIðtþsÞ

hiIðtÞ^2

þ 1 ð 9 : 5 Þ

where the pointed brackets signify averaging over all values of time t and where
dI(t)¼I(t)þhI(t)iis the deviation from the mean intensity.
For a two-dimensional sample with N particles the correlation function can be
written as

GðsÞ¼

1

N

1

1 þðÞs=sD

ð 9 : 6 Þ

whereτDdefines thediffusion time(the time it takes a particle to move a specific
distance by diffusion). Examples of a typical correlation function are shown in
Fig.9.11. The shape of G(τ) gives information on molecular diffusion. As Fig.9.11
indicates, for a given liquid viscosity the diffusion time becomes longer for larger
particles, that is, the diffusion is slower and the curve shifts to the right. In addition,
Fig.9.11shows that for an increasing number of particles in the observation vol-
ume the correlation function curve moves downward. Note that atτ= 0 the value of
G(0) is inversely proportional to the average number of particles in the measure-
ment volume. For illustration purposes on the right-hand side of Fig.9.11, two and
four different sized particles are shown to be in the measurement volume.

Example 9.7Suppose curve 2 in Fig.9.11gives the result for a measure-

ment volume that contains two medium sized particles. What do curve 1 and

curve 3 represent in relation to curve 2 for a liquid with the same viscosity?

Solution: Because G 1 (0) = G 2 (0), the measurement volume for curve 1

contains the same average number of particles as for the setup of curve 2, but

the particles for test 1 are larger (so they diffuse slower). Because G 3 (0) is

half the value of G 2 (0), there are twice as many particles in the measurement

volume for G 3 (τ) compared to that for G 2 (τ).

Correlation function G(

τ)

Increasing

number

of molecules

Slower

molecular

diffusion

Time (arbitrary units)

Curve 1

Curve 2

Curve 3

G 1 (0),

G 2 (0)

G 3 (0)

Curve 1

Curve 2

Curve 3

Size and particle number

Fig. 9.11 Examples of the correlation function for different number and sizes of molecules

272 9 Spectroscopic Methodologies