Fluorescence Spectroscopy as a Tool for Investigating the Self-Organized Polyelectrolyte Systems - Self Organized Nanostructures of Amphiphilic Block Copolymers - page 232

4.2 Monte Carlo Simulation of Shell-Forming Chain

Conformations

4.2.1 Models and Simulation Techniques

In our computer studies of the conformational behavior of the shell-forming chains,
we used MC simulations [ 91 , 95 ] on a simple cubic lattice and studied the shell
behavior of a single micelle only. Because we modeled the behavior of shells of
kinetically frozen micelles, we simulated a spherical polymer brush tethered to the
surface of a hydrophobic spherical core. The association number was taken from
the experiment. The size of the core, lattice constant (i.e., the size of the “lattice
Kuhn segment”) and the effective chain length were recalculated from experimental
values on the basis of the coarse graining parameterization [ 95 ] .
The interactions are described by contact energies and the electrostatic energy.
For contact energies we used the common matrix of interaction parameters in
which the “reference interactions”( i.e., those where solvent is involved) are zero.
The optimized parameters in units of kT are

ε S - S = 0, ε PMA - S = 0, ε C - S = 0,

ε PMA - PMA = − 0.8, where S, C and PMA stand

for solvent (i.e., empty lattice site, occupied implicitly by solvent), C core (lattice
point at the surface of the core or an attached hydrophobic pendant group), and the
PMA bead (i.e., the Kuhn lattice segment, irrespectively of the ionization).
The electrostatic interactions (in aqueous solutions of annealed PEs, such as
PMA, where the potentially ionizable groups get charged) are treated indirectly
by solving the spherically symmetrical Poisson-Boltzmann equation (PBE) for the
electrostatic potential

0.27, ε PMA - C = 0.8, and ε C - C =

ϕ ( r ) [ 95 ] :

r 2 d

1
r 2

d
d r

φ ( r )

= − ρ (

r )

r ,

(19)

d r ε ε

0

where ε r are the dielectric vacuum permittivity and the relative (position de-

pendent) permittivity of the dielectric medium, and the charge density

ε 0 and

includes
both the charge of the micelle (in the mean-field approximation) and of all small
ions.
We consider the following components: -COOH, -COO − ,H 3 O + ,OH − ,Na + ,
and Cl − , but not all are independent since they have to fulfill the relations, K w

ρ ( r )

=

= a H 3 O + ·

a A − / a H 3 O + · a COOH ,where K w and K A are the ionization

product of water and the effective dissociation constant describing the dissociation
of carboxylic groups in PMA, respectively. The effective dissociation constant for
PMA in solutions as a function of ionic strength and degree of ionization was mea-
sured by Morcellet et al. [ 151 ] and by Porasso et al. [ 152 ] . Nevertheless, we use
the constant value p K A =

a OH − and K A

4.69 for the monomeric methacrylic acid as a reasonable
first approximation [ 153 ] because most theoretical calculations for PEs employs
p K A for the monomeric unit, which facilitates the comparison with data of other
authors on similar systems. The activities of components a i are calculated using