6.2.2. Definition of Effective Properties 3064 6.3. Response Properties to Magnetic Fields 3066 6.3.1. Nuclear Shielding 3066 6.3.2. Indirect Spin−Spin Coupling 3067 6.3.3. EPR Parameters 3068 6.4. Properties of Chiral Systems 3069 6.4.1. Electronic Circular Dichroism (ECD) 3069 6.4.2. Optical Rotation (OR) 3069 6.4.3. VCD and VROA 3070 7. Continuum and Discrete Models 3071 7.1. Continuum Methods within MD and MC Simulations 3072

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Quantum mechanical continuum solvation models.

We critically review dissipative particle dynamics (DPD) as a mesoscopic simulation method. We have established useful parameter ranges for simulations, and have made a link between these parameters and χ-parameters in Flory-Huggins-type models. This is possible because the equation of state of the DPD fluid is essentially quadratic in density. This link opens the way to do large scale simulations, effectively describing millions of atoms, by firstly performing simulations of molecular fragments retaining all atomistic details to derive χ-parameters, then secondly using these results as input to a DPD simulation to study the formation of micelles, networks, mesophases and so forth. As an example application, we have calculated the interfacial tension σ between homopolymer melts as a function of χ and N and have found a universal scaling collapse when σ/ρkBTχ0.4 is plotted against χN for N>1. We also discuss the use of DPD to simulate the dynamics of mesoscopic systems, and indicate a possible problem with...

https://www.physics.iitm.ac.in/~iol/lecture_notes/groot-warren.pdf

Dissipative particle dynamics : bridging the gap between atomistic and mesoscopic simulation

We present an extensive molecular‐dynamics simulation for a bead spring model of a melt of linear polymers. The number of monomers N covers the range from N=5 to N=400. Since the entanglement length Ne is found to be approximately 35, our chains cover the crossover from the nonentangled to the entangled regime. The Rouse model provides an excellent description for short chains N<Ne, while the dynamics of the long chains can be described by the reptation model. By mapping the model chains onto chemical species we give estimates of the times and distances of onset of the slowing down in motion due to reptation. Comparison to neutron spin‐echo data confirm our mapping procedure, resolving a discrepancy between various experiments. By considering the primitive chain we are able to directly visualize the confinement to a tube. Analyzing the Rouse mode relaxation allows us to exclude the generalized Rouse models, while the original reptation prediction gives a good description of the data.

Dynamics of entangled linear polymer melts: A molecular‐dynamics simulation

https://www.tcd.ie/Physics/people/Graham.Cross/SF%20Poster%202011/Schuh-ActaMater07-Mechanical%20behaviour%20of%20amorphous%20alloys.pdf

Mechanical behavior of amorphous alloys

Block copolymers are all around us, found in such products as upholstery foam, adhesive tape and asphalt additives. This class of macromolecules is produced by joining two or more chemically distinct polymer blocks, each a linear series of identical monomers, that may be thermodynamically incompatible (like oil and vinegar). Segregation of these blocks on the molecular scale (5–100 nm) can produce astonishingly complex nanostructures, such as the “knitting pattern” shown on the cover of this issue of PHYSICS TODAY. This striking pattern, discovered by Reimund Stadler and his coworkers, reflects a delicate free‐energy minimization that is common to all block copolymer materials.

Block Copolymers—Designer Soft Materials

An inhomogeneous polymer system is analyzed on the basis of a lattice model, in the spirit of the Flory–Huggins approach. Although the theory is developed in some detail, so as to point the way to future applications, only equations for the problem of polymer–polymer interfaces are derived. In the limit of broad interfaces the results of Helfand and Tagami are recovered. For the more general cases numerical results are presented.

Theory of inhomogeneous polymers: Lattice model for polymer–polymer interfaces

Rigid constraints fix the value of some degrees of freedom, such as bond lengths or angles, in the allowed motion of a system. ’’Flexible constraints’’ is a term which describes the limit as the amplitudes of vibration in these coordinates are made to approach zero by infinitely increasing appropriate force constants. The statistical properties of flexibly and rigidly constrained systems are, in general, different. How the difference arises is illustrated in terms of a simple model. A suggestion of Fixman on how to compensate for the difference in terms of an effective potential is rederived for the statics of the model. This paper examines dynamical problems more deeply. The quantum mechanical case is also discussed.

Flexible vs rigid constraints in statistical mechanics

Central concepts of the reptation model of polymer relaxation by disentanglement are the confining tube, the primitive path, and the entanglement net. These are given more precise definitions and are expounded upon here. The entanglement net is regarded as the set of primitive paths of the molecules. The scaling relation connecting distance between entanglements and concentration is derived by a self‐consistency argument. Explicit results on probability distributions characterizing unentangled loops and primitive path lengths are derived. The polymers are modeled as random walks in a regular entanglement net. Results are presented as a function of the spacing between entanglements.

Statistics of the entanglement of polymers: Concentration effects

Time-correlation functions from computer simulations of polymers

Effects of polydispersity on rheological properties of entangled polymers are analyzed. A number of models of tube renewal are discussed and compared with each other and with experiments. A theory incorporating reptation, tube renewal, and fluctuations in the tube length into a general description of stress relaxation is developed. Dynamical moduli were calculated for monodispersed and bidispersed systems and compared with experiments. The product of viscosity and recoverable shear compliance is predicted to be of the order of the longest relaxation time of the higher molecular weight component with a weakdependence on the relative volume fraction of the binary mixture for a large enough concentration of longer molecules.

Eugene Helfand

Papers

Theory of inhomogeneous polymers: Lattice model for polymer–polymer interfaces

Flexible vs rigid constraints in statistical mechanics

Statistics of the entanglement of polymers: Concentration effects

Time-correlation functions from computer simulations of polymers

Theory of Polydispersity Effects on Polymer Rheology. Binary Distribution of Molecular Weights