Lennard-Jones interaction polymers

Figure 7(c). Normalized steady state radial distribution of difference stress contributed by Lennard-Jones interactions, for a simple liquid Nb = 0 (solid line) and polymer melts with Nb — 1,3 and 11 (dashed lines). Reduced density p — 0.9 for all cases. The results for the polymer melts are indistinguishable to within the statistical scatter of the simulations. 

One model which has been extensively used to model polymers in the continuum is the bead-spring model. In this model a polymer chain consists of Nbeads (mers) connected by a spring. The easiest way to include excluded volume interactions is to represent the beads as spheres centered at each connection point on the chain. The spheres can either be hard or soft. For soft spheres, a Lennard-Jones interaction is often used, where the interaction between monomers is... 

Baumgartner, A. Statics and dynamics of the freely jointed polymer chain with Lennard-Jones interaction. J. Chem. Phys. 1980, 72, 871-9. 

Figure 4. Demixing temperatures for Lennard-Jones polymer in Lennard-Jones solvent at P, = 2 as a function of polymer concentration for 16-mers (circles) and 64-mers (triangles). Temperature and pressure units are relative to the critical solvent properties. The Lennard-Jones interaction potential energy is cut at 2.5a and shifted [80].

Brown (24,29) proposed a model of homogeneous yielding based on the Mie (Lennard-Jones) interaction potential. According to Brown, there are three types of molecular motions in an amorphous polymer up to shear 5deld strain ... 

An example of a comparison by Honnell et al. of PRISM theory with molecular dynamics simulations are shown in Figure 3. Details of the model are given elsewhere. Briefly, a meltlike density was studied for /V = 50-150 unit chains. The linear polymers were modelled as freely jointed beads with a purely repulsive, shifted Lennard-Jones interaction between all segment pairs. The corresponding chain aspect ratio is F-1.4. PRISM theory with the PY closure (plus a standard correction... 

Fig. 9.6 Radius of gyration (Rq)/(R j (lower curve) and average squared end-to-end distance (upper curve) versus / for a good solvent. ) is the mean-squared radius of gyration for a single polymer chain. The solid circles are data for polystyrene and polyisoprene from Ref 180. The crosses are from the off-lattice MC simulations of Freire et al. for N = 49 or 55. The other symbols are from MD simulations for monomers interaeting with a purely repulsive Lennard-Jones interaction, eq. (9.3), at T=l.2e/ke for A=100(o) and at r=4.0e/fc s for = 2.5(7 for A = 50(A) and 100 ( ). The solid line has slope of 0.41. (From Ref 117.)...

Fig. 9.9 Form factor P q) versus qu for four-star polymers in a good solvent with/=10-50 for JV = 50 simulated by MD for a purely repulsive Lennard-Jones interaction between nonbonded monomers from Ref. 96. Also shown are the results for two linear polymers with 50 and 100 monomers. The data have been offset for clarity.

Fig. 9.14 Projection of a typical configuration of two branched polymers both with/= 40 arms and N =50 monomers per arm for a good solvent. The results are at T = 4.0e/ke for monomers interactions with a Lennard-Jones interaction truncated at = 2.5central core, while (b) is a bottlebrush made of 40 arms which are tethered to a flexible backbone. 

The strength, Ewaii, is parameterized by the Hamaker constant. A, which is proportional to the energy parameter, e, of the Leimard-Jones potential. Typically, in a coarse-grained model, the Lennard-Jones interactions are cut-off at a finite distance, cf. Eq. (1.9). Therefore, the strength, AA, is the difference between the Hamaker constant of the interactions inside the polymer liquid and that between polymer and solid. 

Atomistically detailed models account for all atoms. The force field contains additive contributions specified in tenns of bond lengtlis, bond angles, torsional angles and possible crosstenns. It also includes non-bonded contributions as tire sum of van der Waals interactions, often described by Lennard-Jones potentials, and Coulomb interactions. Atomistic simulations are successfully used to predict tire transport properties of small molecules in glassy polymers, to calculate elastic moduli and to study plastic defonnation and local motion in quasi-static simulations [fy7, ( ]. The atomistic models are also useful to interiDret scattering data [fyl] and NMR measurements [70] in tenns of local order. 

We have studied, by MD, pure water [22] and electrolyte solutions [23] in cylindrical model pores with pore diameters ranging from 0.8 to more than 4nm. In the nonpolar model pores the surface is a smooth cylinder, which interacts only weakly with water molecules and ions by a Lennard-Jones potential the polar pore surface contains additional point charges, which model the polar groups in functionalized polymer membranes. 

Concluding this section, one should mention also the method of molecular dynamics (MD) in which one employs again a bead-spring model [33,70,71] of a polymer chain where each monomer is coupled to a heat bath. Monomers which are connected along the backbone of a chain interact via Eq. (8) whereas non-bonded monomers are assumed usually to exert Lennard-Jones forces on each other. Then the time evolution of the system is obtained by integrating numerically the equation of motion for each monomer i... 

In an early attempt to model the dynamics of the chromatin fiber, Ehrlich and Langowski [96] assumed a chain geometry similar to the one used later by Katritch et al. [89] nucleosomes were approximated as spherical beads and the linker DNA as a segmented flexible polymer with Debye-Huckel electrostatics. The interaction between nucleosomes was a steep repulsive Lennard-Jones type potential attractive interactions were not included. 

We now present results from molecular dynamics simulations in which all the chain monomers are coupled to a heat bath. The chains interact via the repiflsive portion of a shifted Lennard-Jones potential with a Lennard-Jones diameter a, which corresponds to a good solvent situation. For the bond potential between adjacent polymer segments we take a FENE (nonhnear bond) potential which gives an average nearest-neighbor monomer-monomer separation of typically a 0.97cr. In the simulation box with a volume LxL kLz there are 50 (if not stated otherwise) chains each of which consists of N -i-1... 

In the absence of specific penetrant/polymer interactions, solubility of the penetrant is determined mainly by its chemical namre and depends on condensability, which is represented by boiling temperamre (Tb), critical temperature (Ter), or Lennard-Jones constant (s/fe) [7,8]. It is known that in the hydrocarbon series the increase in condensability is accompanied by a parallel increase in the size of molecules (Table 9.1 [9-17]). It is therefore not surprising that in both glassy and rubbery polymers correlations of hydrocarbon solubility in the polymers with condensability and sizes of hydrocarbon molecules are observed (Figures 9.1 through 9.3). 

Fig. 12. Equation of state (a) and phase diagram (b) of a bead-spring polymer model. Monomers interact via a truncated and shifted Lennard-Jones potential as in Fig. 6 and neighboring monomers along a molecule are bonded together via a finitely extensible non-linear elastic potential of the form iJpENE(r) = — 15e(iJo/

The solubility parameter approach is a thermodynamically consistent theory and it has some links with other theories such as the van der Waals internal pressure concept, the Lennard-Jones pair potentials between molecules, and entropy of mixing concepts of the lattice theories. The solubility parameter concept has found wide use in industry for nonpolar solvents (i.e. solvent selection for polymer solutions and extraction processes) as well as in academic endeavor (thermodynamics of solutions), but it is unsuccessful for solutions where polar and especially hydrogen-bonding interactions are operating. 

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