146 4.6  Advanced Biophysical Techniques Using Elastic Light Scattering

particles) to approximate the scattering as emerging from scattering centers that are local

fluctuations δC in the continuum concentration function for molecules in solution. This

fluctuation approach is a common theoretical biophysical tool used to interface continuum

and discrete mathematical regimes (see Chapter 8). Here, δC can be estimated from the

Boltzmann distribution through a fluctuation in chemical potential energy Δμ, which relates

to the osmotic pressure of the solution Π in the sample volume V, through a simple Taylor

expansion of the natural logarithm of (1 − δC), as

(4.24)

∆

Π

µ

δ

δ

=

−

(

) ≈

= −

RT

C

RT C

V

ln 1

where

R is the molar gas constant

T is the absolute temperature

But Π can also be related to C through a virial expansion that is in essence an adaptation of

the ideal gas law but taking into account real effects of interaction forces between molecules

and the solvent (usually an organic solvent such as toluene or benzene). This indicates

(4.25)

Π

C

RT

M

BC

O C

w

≈

+

+

( )



^



^

1

2

where B is the second virial coefficient. Combining Equations 4.23 through 4.25 gives the

Debye equation:

(4.26)

KC

R

M

BC

w

(θ) ^≈

+

1

2

where R is the Rayleigh ratio given by:

(4.27)

R

d I

I

(

(

θ

θ

)=

)

2

0( )

And the factor K depends on the parameters of the SLS instrument and the molecular

solution:

(4.28)

K

n

C

n

P T

=

∂

∂



^



^



^



^

+

,

(

2

1

2

2

4

2

π

λ

θ

cos

)

Therefore, a linear plot of KC/​R versus C has its intercept at 1/​Mw. Thus, Mw can be determined.

For larger molecules in the Mie/​Tyndall scattering regime, the Debye equation can be

modified to introduce a form or shape factor P(Q), where Q is the magnitude of scattering

vector (the change in the photon wave vector upon scattering with matter):

(4.29)

KC

R

M

BC P Q

w

θ( )

≈

+



^



^

( )

1

2

1

The exact formulation of P depends on the 3D shape and extension of the molecule, embodied

in the RG parameter. The most general approximation is called the “Guinier model,” which can

calculate RG from any shape. A specific application of the Guinier model is the Zimm model

that assumes that each molecule in solution is approximated as a long random coil whose