Transcript – STRUCTURE FACTORS FOR Chapter 32 PARTICULATE SYSTEMS 32:1. THE ORNSTEIN-ZERNIKE EQUATION

Chapter 32 – STRUCTURE FACTORS FOR
PARTICULATE SYSTEMS
32:1. THE ORNSTEIN-ZERNIKE EQUATION
32:2. THE PERCUS-YEVICK APPROXIMATION
32:3. THE MEAN SPHERICAL APPROXIMATION
32:1. THE ORNSTEIN-ZERNIKE EQUATION


  

The OZ equation: h ( r )  c( r )  N  d r ' c( r  r ' )h ( r ' )
c(r)
Take Fourier transform: H(Q)  C(Q)  N C(Q) H(Q)
Inter-particle structure factor: S I (Q)  1  N H(Q) 
1
1 - NC(Q)
Hard Sphere
Interaction potentials:
U(r)
Screened Coulomb
Square Well
r
h(r)
32:2. THE PERCUS-YEVICK APPROXIMATION
PY closure relation:

 w ( r ) 

c(r )  g (r ) 1  exp  
k
T
B



Hard Sphere
U(r)
Hard-sphere interaction potential: w (r )  0 for r  D
r
w (r )   for r  D
PY solution of the OZ equation: c(r )  0 for r  D
r  r3
c(r)  1  62  1 3 for r  D
D 2 D
Parameters:
Result:
2

1  2 .
1 
1  4
2 
 1   2 
2
1  4

 QD2 cosQD  2QDsin QD  2 cosQD  2 
  sin QD  QD cosQD
NC(Q)  241 

6



2
QD3
QD4




 
 2  QD4 cosQD  4QD3 sin QD  12QD2 cosQD  24QDsin QD  24 cosQD  24  

 

6
2 
QD


RESULT
Inter-particle structure factor: S I (Q) 
Scattering cross section:
1
1  NC(Q)
d(Q)
  2 VP P(Q)S (Q)
I
d
Percus Yevick Model
1.6
1.4
Calculated
1.2
I
S (Q)
1
0.8
0.6
0.4
0.2
0
0
5
10
QR
15
FORM FACTOR P(Q) AND STRUCTURE FACTOR SI(Q)
Percus Yevick Model
Form Factor and Structure Factor
1.2
P(Q)
P(Q)*S (Q)
1
I
0.8
0.6
0.4
0.2
0
0
1
2
3
QR
4
5
32:3. THE MEAN SPHERICAL APPROXIMATION
Screened Coulomb interaction potential:
exp  (r  D) 
U(r )  0D 2 02
for r  D
r
MSA closure relation: c(r )  U(r ) for r  D
h (r )  1 for r  D
U(r)
Screened Coulomb
r
MSA solution of the OZ equation:
1
C sinh( kx ) Fcosh( kx )  1
c(r )  A  Bx  Ax 3 

for x  1
2
x
x
exp( kx )
c( r )   
for x  1
x
Take Fourier transform… to obtain tedious result… not reproduced here
RESULT
Inter-particle structure factor: S I (Q) 
Scattering cross section:
1
1  NC(Q)
d(Q)
  2 VP P(Q)S (Q)
I
d
D = 40 Å, z = 20,  = 0.01, T = 25 C
o
m
1.4
1
I
Structure Factor S (Q)
1.2
0.8
Calculated
0.6
0.4
0.2
0
0
0.05
0.1
0.15
-1
Scattering Variable Q (Å )
0.2
COMMENTS
-- The OZ equation along with one of the closure relations
(the PY or the MSA for charged systems) is used to calculate the
structure factor for scattering particles.
-- Structure factors are needed to mode the cross section for
concentrated systems.