Collapse of polymer brushes grafted onto planar ... - Wageningen UR

DYNAMIC EXTENSION OF THE SELF-CONSISTENT FIELD THEORY OF
INHOMOGENEOUS POLYMER SYSTEMS
T. Kawakatsu
Department of Physics, Tohoku University,
Aza-Aoba, Aramaki, Aoba-ku, Sendai 980-8578, Japan
email: kawakatu@cmpt.phys.tohoku.ac.jp
ABSTRACT
1. Introduction
Self-consistent field (SCF) theory is one of the useful techniques for studying mesoscopic structures of
inhomogeneous polymer systems, such as phase separating polymer blends, polymer films on solid
surfaces, and polymer brushes (Fleer 1993, Matsen 1996). Combined with a numerical evaluation of the
polymer path integral, i.e. canonical statistical weight of the chain conformation, this theory well reproduces
the equilibrium domain structures and segment density profiles. Despite of such success in
equilibrium systems, extensions to dynamic non-equilibrium systems have not completely understood yet.
In the present study, we propose several techniques to introduce dynamic processes into the SCF theory,
and evaluate their usefulness and efficiency.
2. Dynamic SCF method for weakly non-equilibrium systems
The dynamic extension of the SCF theory has been initiated by Fraaije (Fraaije 1993) by combining
the SCF theory with the Fick’s law of segment diffusion driven by the gradient of the chemical potential.
To obtain the chemical potential, the system is assumed to be in local equilibrium, which is essential for
the use of the path integral formalism. Using this technique, one can calculate the ordering processes in
phase separating polymer blends or block copolymer systems. For example, we can reproduce the phase
diagram of a mixture of long and short block copolymers that shows complex domain structures (Morita
2002). Using the segment interaction parameters and the block ratio (the ratio between the lengths of the
two blocks) as input parameters, the dynamic SCF calculation gives a quantitatively correct phase
diagram and domain structures including metastable transient structures.
The success of these dynamic SCF simulations owes to the weak non-equilibrium nature of the
phenomena, which is crucial to the use of the path integral formalism of the polymer chain conformation.
3. Highly non-equilibrium systems -- Rheological properties --
As in the case of a sheared polymer melt, an external deformation force makes the chain conformation
strongly stretched. In such a situation, the assumption of the local equilibrium becomes unreliable. To
overcome this difficulty, we extend the path integral formalism by introducing the bond orientation tensor,
and combined it with the reptation dynamics (Kawakatsu 2001, Shima 2002). In this formalism, the
polymer path integral Q ( s, r; s′ , r ' ) , i.e. the statistical weight of a subchain between the s -th and s′ -th
segments whose ends are found at r and r ′ , is given by
∂
1
Q( s, r; s′ , r') = ∇∇: { A( s, r) Q( s, r; s′ , r') } −∇⋅{ u( s, r) Q( s, r; s′ , r') } −βV(
r) Q( s, r; s′ , r ')
,
∂s
2
where u( s,
r ) and A( s,
r ) are the average and covariance of the distribution of the s -th bond vector,
and V ( r ) is the self-consistent field. The temporal evolutions of u( s,
r ) and A( s,
r ) are determined by
the reptation theory and the Navier-Stokes equation (Kawakatsu 2001, Shima 2002). A similar treatment
has recently been used by Fredrickson (Fredrickson, 2002).
We applied this method to a sheared two parallel plates on which melt brushes are grafted. Due to the
disentanglement between the two brushes, the shear stress and the density profile of the segments
changes as is shown in Figure 1. Such a behavior is consistent with a microscopic Monte Carlo
simulation using a many chain system.