Supporting information: The numerical study of the adsorption of
flexible polyelectrolytes with the annealed charge distribution onto
an oppositely charged sphere by the self-consistent field theory
Chaohui Tong
Department of Physics, Ningbo University, Ningbo, 315211, China
Derivation of the SCF equations
In this supplementary material, we present a detailed derivation of the
self-consistent field equations for an open system of an incompressible weak-base
type polyelectrolyte solution by using the grand canonical ensemble. The derivation is
similar to that by Witte, et al for weak-base type polyelectrolyte brushes, which used
the (semi)grand canonical emsemble.1 It is noted that the grand canonical ensemble
was also employed to derive the SCF equations for neutral polymer blends by
others.2,3
Here, we consider a weak-base type polyelectrolyte aqueous solution in contact
with a PE bulk solution maintained at a constant temperature T , a constant chemical
potential  for each species, and a fixed pH and ionic strength. The PE segments
participate in the following acid-base equilibrium:
Kb

 PH   OH 
P  H 2O 

where P denotes a monomer segment. The base equilibrium constant in the bulk
solution is defined as K b   PH   OH    P  , where the square bracket stands for
the basis of a molar concentration. For a given system configuration, the above
1
chemical equilibrium leads to the coexistence of nC charged segments and
nU  n p N p  nC uncharged segments in the solution, where n p and N p denote the total
number of polyelectrolyte chains in the system and the number of monomer segments
per chain, respectively.
In a grand canonical ensemble, the grand partition function   exp   G  with
G the grand potential having the following expression





1 1 1 1 1


exp   p n p   m nm  C nC  U nU 
np
nm
V
m


p ! nC ! nU ! nm ! V
1
n
m n p  0 nC  0 nU  0 nm  0
np
nm
nC
nU
k 1
k 1
k 1
k 1
  DRk  s    Drmk   DrCk   DrUk  D exp    H 


    j ˆ j  r   1   fˆ  r  ˆ p  r   gˆ  r  ˆ p  r   ˆ p  r  
r
 j
 r
nC
np
   dr ˆ e  r     
k 1 i 1
Np
0
nU
np
ds  rCk  Ri  s     ds  rUk  Ri  s  
0
Np
k 1 i 1
(A1)
where the subscripts j , m represent all the species and the small molecular species
(the solvent, mobile ions) in the system, respectively, V is the volume of the system,
nm denotes the number of small molecules of species type m . In the above equation,
  1 kBT  with k B the Boltzmann constant,  j represents the chemical potential of
each species in the bulk. Rk  s  and rmk , respectively, denote the position vectors of
the s th segments of the k th polymer chain and the k th small molecule of type m .
rCk and rUk represent the position vectors of the k th charged and uncharged segments,
respectively. The intrinsic volume of each species j is  j , and its microscopic
density operator ˆ j is defined as follows
np
ˆ p  r     ds  r  Rk  s  
0
k 1
Np
(A2)
2
nm
ˆ m  r      r  rmk 
(A3)
k
The microscopic polymer charge and non-charge fraction operators are defined as
nC
fˆ  r      r  rCk 
k 1
nU
gˆ  r      r  rUk 
k 1
np

k 1
np

k 1
Np
0
Np
0
ds  r  Rk  s  
(A4)
ds r  Rk  s  
(A5)
The charge density operator ˆ e  r  is
ˆ e  r   fˆ  r  ˆ p  r    zion ˆion  r 
(A6)
ion
where zion denotes the charge valence of each mobile ion species. The Hamiltonian
H is defined as
np
H  
k 1
3  Rk  s  

  drVint ˆ p  r   ps ˆ s  r  
2b 2  s  
2
Np
0
1

 dr   e  r  ˆ  r   2   r    r 
e
2

 

(A7)

  dr  U0 gˆ  r  ˆ p  r   C0 fˆ  r  ˆ p  r    m0 ˆ m  r  

m

where b represents the monomer size, Vint is a characteristic interaction volume for
monomer segments and the solvent molecules, and  ps is the Flory-Huggins type
interaction parameter,   r  is the electric potential,   r  is the electric permittivity,
e denotes the elementary charge, i0 denotes the standard chemical potential of
species i . In equation (A1), the first delta function is for the enforcement of the
incompressibility condition for the system, the second delta function ensures the
proper normalization of the polymer charge and non-charge fractions, the third delta
function is to maintain the charge neutrality of the system, and the final two delta
functions constrain the charged and non-charged monomer segments to lie on the
3
polymer chains.
Introducing the following identities to the grand partition function allows the
replacement of the microscopic density operators ˆ j  r  , fˆ  r  and ĝ  r  by the
field variables  j  r  , f  r  and
g  r  , which in essence are the ensemble
averages of the corresponding density operators.
  D  r      r   ˆ  r   1
(A8)
 Df  r   f  r  ˆ  r   fˆ  r  ˆ  r   1
(A9)
 Dg  r   g  r  ˆ  r   gˆ  r  ˆ  r   1
(A10)
j
j
j
j
p
p
p
p
Using the Fourier representation of the delta-function for the   functions in the
above equations and the first three ones in equation (A1), after a standard procedure,
we obtain the following grand partition function


   D   D j  D j   D  D  Df  Dg  D C  D U  D exp   G 
 j

(A11)
with the grand potential
 




    j  r   j  r    Vint  p  r   ps  s  r     r    j  j  r   1 


 j

  j

 1

2
 G   dr    r    r    C  r  f  r    U  r  g  r    p  r     r   p  r  
2





  f  r   g  r   1   e  r   f  r   p  r    zion ion  r  

ion




 Q p   Qion  Qs  QC  QU
ion
(A12)
In equation (A11),  j ,  C ,  U ,  ,  ,  are the corresponding conjugate fields
introduced in using the Fourier representation of the delta-function. The single
4
polymer chain and species partition functions are given by

Q p   DR  s  exp  

Np
0


 DR  s  exp   3 2b   R  s  s 
exp   

 drq  r , N 
V
1



 3 2b 2  R  s  s 2   R  s   ds 
p


Np
2
0
2

 ds

(A13)
p
p
Qs 
exp  s   s0 
Qion 
 dr exp   r 
(A14)
s
V
0
exp  ion  ion
 zion  
 dr exp   r 
(A15)
ion
V


QC  exp  C  C0     dr exp  C  r   p  r 


QU  exp  U  U0   dr exp   U  r   p  r 
(A16)
(A17)
In equation (A13), q  r , N p  is the reduced partition function for a single chain and
is determined by the modified diffusion equation (see equation (9) in the main text).
After taking the zero functional derivatives of the grand potential with respect to
the field variables  j ,  j , f , g ,  C ,  U ,  ,  and  , the following SCF
equations are obtained:
 p  r   Vint  ps s  r    p  r     r  1
(A18)
s  r   Vint  ps  p  r   s  r 
(A19)
ion  r   ion  r    ezion  r 
(A20)
p r  
s  r  
exp   p 
V

Np
0
dsq  r , s  q  r , N p  s 
exp  s  s0 
ion  r  
V
exp  s  r  
0
exp  ion  ion
 zion  
V
(A21)
(A22)
exp  ion  r  
5
(A23)
 C  r    e  r     r 
(A24)
U  r     r 
(A25)
f  r   exp  C  C0    exp   C  r  
(A26)
g  r   exp  U  U0  exp   U  r  
(A27)
   r   1
(A28)
j
j
j
f r   g r   1
(A29)
   r    r    ef  r   p  r   e zion ion  r   0
(A30)
ion
The constant pre-factors in equations (A21) – (A23) can be used to define the bulk
b
concentrations  bp ,  sb and  ion
. Taking equation (A21) as an example, we have
 Pb   p  r    
exp   p 
V

Np
0
dsq  r  , s  q  r  , N p  s 
(A31)
so the constant pre-factor can be fixed as
exp   p  V  Pb

Np
0
dsq  r  , s  q  r  , N p  s  .
(A32)
The constant pre-factors in equations (A26) and (A27) can be fixed by using the
values of f and g in the bulk and by noting that in the bulk solution,   0 , and
 is a constant, with the results of exp  C  C0     fb exp   r     and
exp  U  U0   gb exp   r     . Then equations (A26) and (A27) become
f  r   fb exp   r     exp    e  r     r  
(A33)
g  r   gb exp   r     exp    r  
(A34)
Plugging the above two equations into eq. (A29), the variable  can be solved as
  r   ln 1  fb  fb exp  e  r     r   
(A35)
Further simplification of the SCF equations can be made by eliminating the dependent
variables f , g and  in the SCF equations, leading to the following SCF equations
6
 p  r   Vint  ps s  r    p  r   ln 1  fb  fb exp  e  r     r     1 (A36)
s  r   Vint  ps  p  r   s  r 
(A37)
ion  r   ion  r    ezion  r 
(A38)
 p  r    bp  dsq  r , s  q  r , N p  s 
Np
0

Np
0
dsq  r  , s  q  r  , N p  s 
(A39)
 s  r    sb exp s  r     s  r  
(A40)
b
ion  r   ion
exp ion  r     ion  r  
(A41)
   r   1
(A42)
j
j
j
fb exp    e  r  
   r    r    e p  r 
 e zion ion  r   0
1  fb  fb exp    e  r  
ion


(A43)
Please note that the last two constants in equation (A36) can be dropped out since it
has no effect on the equilibrium properties of the PE solution.
Instead of using the number density for each species in the SCF equations, now
we proceed to write down the SCF equations in terms of the volume fractions. For
simplicity and without losing generality, let’s assume that  p  s  0  b3 , mobile
ions are point charges, and invoke the volume fractions  j  r  by using the fact that
 j  r   0  j  r  . By performing the same non-dimensionlization scheme as that in
Ref 4, we finally arrive at the SCF equations in terms of the volume fractions of
concerning species in the system as shown in the main text.
References
(1) 1 K. N. Witte, S. Kim and Y. Y. Won, J. Phys. Chem. B 113, 11076-11084 (2009).
(2) S. M. Wood and Z. G. Wang, J. Chem. Phys. 116, 2289-2300 (2002).
(3) M. W. Matsen, Phys. Rev. Lett. 74, 4225-4228 (1995).
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(4) Q. Wang, T. Taniguchi and G. H. Fredrickson, J. Phys. Chem. B 108, 6733-6744
(2004).
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