Polymer A polymer (; Greek '' poly-'', "many" + ''-mer'', "part") is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic a ...

crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...

s have different properties than simple atomic crystals. They possess high

density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...

and long range order. They do not possess

isotropy Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...

, and therefore are anisotropic in nature, which means they show

anisotropy Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...

and limited conformation space. However, just as atomic crystals have lattices, polymer crystals also exhibit a periodic structure called a lattice, which describes the repetition of the

unit cell In geometry, biology, mineralogy and solid state physics, a unit cell is a repeating unit formed by the vectors spanning the points of a lattice. Despite its suggestive name, the unit cell (unlike a unit vector, for example) does not necessaril ...

s in the space. The simulation of polymer crystals is complex and not taken from only one state but from

solid Solid is one of the State of matter#Four fundamental states, four fundamental states of matter (the others being liquid, gas, and Plasma (physics), plasma). The molecules in a solid are closely packed together and contain the least amount o ...

-state and

fluid In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...

-state physics as well. Polymer crystals have unit cells that consist of tens of atoms, while the molecules themselves comprise 10⁴ To 10⁶ atoms.

Computational methods

There are two methods for the study of polymer crystals: 1) optimization methods and 2) sampling methods. Optimization methods have some advantages over the sampling method, such as the localization of crystals in phase space. Sampling methods generally cannot localize the crystals, and thus there is no need of the assumptions of localization. Optimization methods include molecular mechanics and lattice dynamics and sampling methods include the

Monte Carlo method Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determi ...

and

molecular dynamics Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the ...

. A brief discussion regarding the methods are as follows: # Optimization method: In this method, we use the optimization technique and optimize the polymer crystal. For this we consider an ideal case where the crystal is free of disorder (this is assumption). Now we have to express the relevant part of the energy surface which can be approximated by using Taylor series expansion to an arbitrary accuracy in small displacements about the local minimum energy structure. Here in optimization method, we introduce the

wave vector In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...

and frequency of

oscillation Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...

term because optimization involves the localization of crystals. We find elastic stiffness moduli here with which modeling is done. # Sampling method: There is no localization of crystals and this sampling method also remove the restriction of approximation of the lattice. There are many disadvantages of this method: a) The Monte Carlo method and molecular dynamics method must use very small polymer crystals for the simulation. These method simulate, approximate the polymers of the order of 10³ to 10⁴ with current generation computers. This is the boundary condition and atoms outside of the simulation box have to be in phase with the atom inside the box. b) Due to heavy computational burden, simple interatomic models are more prevalent in Monte Carlo method and molecular dynamics. There is a variety of methods for studying polymer crystals by molecular simulation. It is especially important in polymer crystals to be cognizant of the limitations imposed by either the assumptions on which a method is based or the robustness of the simulation method.

References

{{reflist Polymers

Computational methods

See also

References