Summary:  Morphology of Complex Materials

(Philosophically what you should walk away with.)

This quarter we have examined the concept of hierarchy in morphology for complex systems of importance to materials application and research.  We have considered 5 topics, Table 1,  though there are many other examples of hierarchy some of which are mentioned in the special topics section (spider silk, block copolymers, liquid crystalline polymers, bone, hair etc.) 

Table 1.  A possible way to organize what we have covered in this quarter.

The intent of the course was to demonstrate that the hierarchical approach represents a new view of morphology that is particularly useful with complex materials.  This new view is partially based on old ideas but in the context of a unifying framework to view complex materials as a whole it is a new concept.  Structural hierarchy had its basis in structural biology since systems such as nucleic acids and proteins were difficult to describe using a conventional single structural level model such as the unit cell for metals.  In a hierarchical view we divide a morphology into discrete levels of structure such as bricks to build walls to build floors to build buildings to build cities.  Each level of structure is considered independently in terms of thermodynamics, kinetics and the resulting morphology.  The Hoffman equation for lamellar crystalline thickness in polymers is a good example of the application of pseudo-equilibrium thermodynamics to a single structural level in a complex material that displays hierarchy.  The hope of a hierarchical view is that the separate levels and their interaction can be assimilated into an understanding of a material that would be too complicated to consider as a whole.  A similar approach has been applied to modeling (computer calculations) of complex systems such as protein folding where the molecular and atomic details are worked out on a low level of structure while global scaling laws and bulk simplifications are used on larger scales.  In most areas of materials science the hierarchical view is found at the heart of new views and for this reason it is worthy of separate consideration as a topic or even as a unifying theme in the field of morphology.

The application of hierarchy in the static structure of polymers has generated answers to unresolved questions concerning coil expansion in semi-dilute solutions and at temperatures between very good solvent and theta conditions.  The use of a blob hierarchical level allows for continuous structural transitions under these conditions.  Structural renormalization using a hierarchical level such as a blob is a concept of wide applicability and should be considered an independent advancement in thermodynamics similar to Gibb's contribution of the pseudo-equilibrium approach in the late 1800's.  Similar to the wide applicability of pseudo-equilibirum thermodynamic models such as the Gibbs-Thompson equation for nano-particle size, we can expect hierarchical models to make major contributions to advancements in many fields beyond polymer chain scaling. 

Similarly, the application of hierarchy to dynamic models of polymers first by Rouse in the 1930's and 1940's and later by Degennes and Edwards in Reptation theory can be expected to be widely applicable to a wide range of materials including proteins though application of hierarchical dynamic models to protein folding is still a cutting edge field.  We should expect a hierarchical view to be the basis of many future models and polymer scientists, with their exposure to Rouse and reptation theory should be at an advantage to contribute in a wide range of applications of Polymer hierarchical dynamics theory. 

The hierarchical view can be seen to contribute to modeling, dynamics and static morphological views of a wide range of materials and structures.

Rules for Hierarchical Models:

1)  A complex structure is based on a primary level beyond which there is no reason to subdivide the structure, for instance, because the primary level has 0 conformational entropy, represents a chemical repeat unit, or represents a structural limit for the material.

2)  The primary units can be considered independently in terms of dynamics, kinetics and thermodynamics, for instance the Kuhn unit displays a spring constant and a friction factor or the persistence unit displays a length that can be calculated using RISM theory.

3)  The primary units are arranged in a secondary structure in a coordinated and understandable manner using statistics, thermodynamics, kinetics, and/or dynamics.  For example the Ramachandran plot of protein structure is a mechanism to categorized complex relationships of amino acid primary structures in protein secondary structures such as alpha heilices and beta sheets. 

4)  Secondary structures, such as blobs, can be considered independently in terms of dynamics, kinetics and thermodynamics, for instance the Hoffman equation for chain folded crystals, the scaling relationships for concentration blob size, bond angle considerations and hydrogen bonding considerations for alpha-helicies. 

5)  There is a continuum of behavior between primary-secondary-tertiary structure that can allow for calculation of structural and dynamic features such as the Kuhn size, blob size and extent of helical structure.  For many hierarchical structures this continuum is of main importance to the usefulness of the hierarchical approach.  In many cases this continuum concept has not been fully developed, such as for proteins.

6)  Secondary structures compose tertiary structures such as the native state protein, polymer coil, Rouse chain and lamellar crystal.  The tertiary structures can be considered independently in terms of dynamics, kinetics and thermodynamics yet represent a continuum with the secondary structures, that is at the secondary structure size physical properties can be described both by the secondary and tertiary models allowing for solution to certain problems in thermodynamics, kinetics and structure.

7)  In many but not all hierarchical materials a quarternary structure exists that is composed of tertiary structures in either an organized or a statistical manner.  For proteins the quarternary structure is often organized in to a functional unit such as a ribosome in combination with nucleic acids and metal ions.  For polymer crystals the quartenary structure might be a spherulite.  In the polymer melt, the reptation theory really describes the dynamics of an ill described quartenary structure involving interaction between chains through entanglements. 

8)  Quarternary structures are the least understood for hierarchical materials and often rely on weak bonding and display high polydispersity in structure such as in agglomerates of aggregates and in entangled polymer chains. For this reason and simply because larger structures display larger lever arms for applied forces, quarternary structures are the first to be broken apart by mechanical stress and other applications of energy to a hierarchical system.  When agglomerates of carbon black are milled in rubber they break apart quickly on application of shear while aggregates usually remain intact through processing.