Yet, statistical approaches can yield results and are often pertinent, since large polymers (i.e., polymers with many monomers) are describable efficiently in the thermodynamic limit of infinitely many monomers (although the actual size is clearly finite).
Thermal fluctuations continuously affect the shape of polymers in liquid solutions, and modeling their effect requires the use of principles from statistical mechanics and dynamics.
As a corollary, temperature strongly affects the physical behavior of polymers in solution, causing phase transitions, melts, and so on.
The simplest possible polymer model is presented by the ideal chain, corresponding to a simple random walk.
Experimental approaches for characterizing polymers are also common, using polymer characterization methods, such as size exclusion chromatography, viscometry, dynamic light scattering, and Automatic Continuous Online Monitoring of Polymerization Reactions (ACOMP)[5][6] for determining the chemical, physical, and material properties of polymers.
This assumption is valid for certain polymeric systems, where the positive and negative interactions between the monomer effectively cancel out.
Ideal chain models provide a good starting point for the investigation of more complex systems and are better suited for equations with more parameters.
This causes a reduction in the conformational possibilities of the chain, and leads to a self-avoiding random walk.
In the limit of a very bad solvent the polymer chain merely collapses to form a hard sphere, while in a good solvent the chain swells in order to maximize the number of polymer-fluid contacts.
Therefore, polymer in good solvent has larger size and behaves like a fractal object.
A path of this walk of N steps in three dimensions represents a conformation of a polymer with excluded volume interaction.
Because of the self-avoiding nature of this model, the number of possible conformations is significantly reduced.
Derived from the word reptile, reptation suggests the movement of entangled polymer chains as being analogous to snakes slithering through one another.
[14] Pierre-Gilles de Gennes introduced (and named) the concept of reptation into polymer physics in 1971 to explain the dependence of the mobility of a macromolecule on its length.
Reptation is used as a mechanism to explain viscous flow in an amorphous polymer.
[15][16] Sir Sam Edwards and Masao Doi later refined reptation theory.
[22] The study of long chain polymers has been a source of problems within the realms of statistical mechanics since about the 1950s.
Suppose that the train moves either a distance of +b or −b (b is the same for each step), depending on whether a coin lands heads or tails when flipped.
As a result of the central limit theorem, if N ≫ 1 then we expect a Gaussian distribution for the end-to-end vector.
Recall that according to the principle of equally likely a priori probabilities, the number of microstates, Ω, at some physical value is directly proportional to the probability distribution at that physical value, viz; where c is an arbitrary proportionality constant.
An example of this is a common elastic band, composed of long chain (rubber) polymers.
By stretching the elastic band you are doing work on the system and the band behaves like a conventional spring, except that unlike the case with a metal spring, all of the work done appears immediately as thermal energy, much as in the thermodynamically similar case of compressing an ideal gas in a piston.
In both the ideal gas and the polymer, this is made possible by a material entropy increase from contraction that makes up for the loss of entropy from absorption of the thermal energy, and cooling of the material.