--- trunk/xDissertation/Conclusion.tex	2008/01/30 16:01:02	3336
+++ trunk/xDissertation/Conclusion.tex	2008/03/05 23:34:04	3360
@@ -1 +1,64 @@
 \chapter{\label{chap:conclusion}CONCLUSION}
+
+This dissertation has shown the efforts to the understanding of the
+structural properties and phase behavior of lipid membranes. In
+Ch.~\ref{chap:mc}, we present a simple model for dipolar elastic
+membranes that gives lattice-bound point dipoles complete
+orientational freedom as well as translational freedom along one
+coordinate (out of the plane of the membrane). There is an additional
+harmonic term which binds each of the dipoles to the six nearest
+neighbors on either triangular or distorted lattices.  The
+translational freedom of the dipoles allows triangular lattices to
+find states that break out of the normal orientational disorder of
+frustrated configurations and which are stabilized by long-range
+anti-ferroelectric ordering.  In order to break out of the frustrated
+states, the dipolar membranes form corrugated or ``rippled'' phases
+that make the lattices effectively non-triangular.  We observe three
+common features of the corrugated dipolar membranes: 1) the corrugated
+phases develop easily when hosted on triangular lattices, 2) the wave
+vectors for the surface ripples are always found to be perpendicular
+to the dipole director axis, and 3) on triangular lattices, the dipole
+director axis is found to be parallel to any of the three equivalent
+lattice directions.
+
+Ch.~\ref{chap:md} we developed a more realistic model for lipid
+molecules compared to the simple point dipole one. To further address
+the dynamics properties of the ripple phase, the simulation method is
+switched to molecular dynamics.  Symmetric and asymmetric ripple
+phases have been observed to form in the simulations. The lipid model
+consists of an dipolar head group and an ellipsoidal tail.  Within the
+limits of this model, an explanation for generalized membrane
+curvature is a simple mismatch in the size of the heads with the width
+of the molecular bodies.  The persistence of a {\it bilayer} structure
+requires strong attractive forces between the head groups.  One
+feature of this model is that an energetically favorable orientational
+ordering of the dipoles can be achieved by out-of-plane membrane
+corrugation.  The corrugation of the surface stabilizes the long range
+orientational ordering for the dipoles in the head groups which then
+adopt a bulk anti-ferroelectric state. The structural properties of
+the ripple phase we observed in the dynamics simulations are
+consistant to that we observed in the Monte Carlo simuations of the
+simple point dipole model.
+
+To extend our simulations of lipid membranes to larger system and
+longer time scale, an algorithm is developed in Ch.~\ref{chap:ld} for
+carrying out Langevin dynamics simulations on complex rigid bodies by
+incorporating the hydrodynamic resistance tensors for arbitrary shapes
+into an advanced symplectic integration scheme.  The integrator gives
+quantitative agreement with both analytic and approximate hydrodynamic
+theories for a number of model rigid bodies, and works well at
+reproducing the solute dynamical properties (diffusion constants, and
+orientational relaxation times) obtained from explicitly-solvated
+simulations. A $9$ times larger simulation of the lipid bilayer are
+carried out for the comparison with the molecular dynamics simulations
+in Ch.~\ref{chap:md}, the results show the structural stability of the
+ripple phase.
+
+The structural properties and the formation mechanism for the ripple
+phase of lipid membranes are elucidated in this dissertation. However,
+the importance of the ripple phase in the experimental view is still a
+mystery, hopefully, this work can contribute some flame to the
+lighting of the experimental field. Further insights of the phase
+behavior of the lipid membranes can be obtained by applying a atomic
+or more detailed molecular model with information of the fatty chains
+of the lipid molecules.
