Reptation Theory
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A peculiarity of
thermal motion A thermal column (or thermal) is a rising mass of buoyant air, a convective current in the atmosphere, that transfers heat energy vertically. Thermals are created by the uneven heating of Earth's surface from solar radiation, and are an example ...
of very long linear
macromolecules A macromolecule is a very large molecule important to biophysical processes, such as a protein or nucleic acid. It is composed of thousands of covalently bonded atoms. Many macromolecules are polymers of smaller molecules called monomers. The ...
in ''entangled''
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 ...
melts or concentrated polymer solutions is reptation. Derived from the word
reptile Reptiles, as most commonly defined are the animals in the class Reptilia ( ), a paraphyletic grouping comprising all sauropsids except birds. Living reptiles comprise turtles, crocodilians, squamates (lizards and snakes) and rhynchocephalians ( ...
, reptation suggests the movement of entangled polymer chains as being analogous to
snake Snakes are elongated, Limbless vertebrate, limbless, carnivore, carnivorous reptiles of the suborder Serpentes . Like all other Squamata, squamates, snakes are ectothermic, amniote vertebrates covered in overlapping Scale (zoology), scales. Ma ...
s slithering through one another.
Pierre-Gilles de Gennes Pierre-Gilles de Gennes (; 24 October 1932 – 18 May 2007) was a French physicist and the Nobel Prize laureate in physics in 1991. Education and early life He was born in Paris, France, and was home-schooled to the age of 12. By the age of ...
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. Sir Sam Edwards and
Masao Doi (born 29 March 1948) is a Professor Emeritus at Nagoya University and The University of Tokyo. He is a Fellow of the Toyota Physical and Chemical Research Institute. In 1978 and 1979 he wrote a series of papers with Sir Sam Edwards expanding on ...
later refined reptation theory. Similar phenomena also occur in proteins. Two closely related concepts are reptons and entanglement. A repton is a mobile point residing in the cells of a lattice, connected by bonds. Entanglement means the topological restriction of molecular motion by other chains.


Theory and mechanism

Reptation theory describes the effect of
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 ...
chain entanglements on the relationship between
molecular mass The molecular mass (''m'') is the mass of a given molecule: it is measured in daltons (Da or u). Different molecules of the same compound may have different molecular masses because they contain different isotopes of an element. The related quanti ...
and chain
relaxation time In the physical sciences, relaxation usually means the return of a perturbed system into equilibrium. Each relaxation process can be categorized by a relaxation time τ. The simplest theoretical description of relaxation as function of time ' ...
. The theory predicts that, in entangled systems, the relaxation time is proportional to the cube of molecular mass, : . The prediction of the theory can be arrived at by a relatively simple argument. First, each polymer chain is envisioned as occupying a tube of length , through which it may move with snake-like motion (creating new sections of tube as it moves). Furthermore, if we consider a time scale comparable to , we may focus on the overall, global motion of the chain. Thus, we define the tube mobility as :, where is the
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
of the chain when it is pulled by a
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
, . will be
inversely proportional In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constan ...
to the
degree of polymerization The degree of polymerization, or DP, is the number of monomeric units in a macromolecule or polymer or oligomer molecule. For a homopolymer, there is only one type of monomeric unit and the ''number-average'' degree of polymerization is given by ...
(and thus also inversely proportional to chain weight). The
diffusivity Diffusivity is a rate of diffusion, a measure of the rate at which particles or heat or fluids can spread. It is measured differently for different mediums. Diffusivity may refer to: *Thermal diffusivity, diffusivity of heat *Diffusivity of mass: ...
of the chain through the tube may then be written as :. By then recalling that in 1-dimension the
mean squared displacement In statistical mechanics, the mean squared displacement (MSD, also mean square displacement, average squared displacement, or mean square fluctuation) is a measure of the deviation of the position of a particle with respect to a reference positio ...
due to
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
is given by :, we obtain :. The time necessary for a polymer chain to displace the length of its original tube is then :. By noting that this time is comparable to the relaxation time, we establish that . Since the length of the tube is proportional to the degree of polymerization, and μtube is inversely proportional to the degree of polymerization, we observe that (and so ). From the preceding analysis, we see that molecular mass has a very strong effect on relaxation time in entangled polymer systems. Indeed, this is significantly different from the untangled case, where relaxation time is observed to be proportional to molecular mass. This strong effect can be understood by recognizing that, as chain length increases, the number of tangles present will dramatically increase. These tangles serve to reduce chain mobility. The corresponding increase in relaxation time can result in
viscoelastic In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly wi ...
behavior, which is often observed in polymer melts. Note that the polymer’s zero-shear
viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inte ...
gives an approximation of the actual observed dependency, ; this relaxation time has nothing to do with the reptation relaxation time.


Models

Entangled polymers are characterized with effective internal scale, commonly known as ''the length of macromolecule between adjacent entanglements'' M_. Entanglements with other polymer chains restrict polymer chain motion to a thin virtual ''tube'' passing through the restrictions. Without breaking polymer chains to allow the restricted chain to pass through it, the chain must be pulled or flow through the restrictions. The mechanism for movement of the chain through these restrictions is called reptation. In the blob model, the polymer chain is made up of n
Kuhn length The Kuhn length is a theoretical treatment, developed by Hans Kuhn, in which a real polymer chain is considered as a collection of N Kuhn segments each with a Kuhn length b. Each Kuhn segment can be thought of as if they are freely jointed with ...
s of individual length l. The chain is assumed to form blobs between each entanglement, containing n_ Kuhn length segments in each. The mathematics of random walks can show that the average end-to-end distance of a section of a polymer chain, made up of n_ Kuhn lengths is d=l \sqrt. Therefore if there are n total Kuhn lengths, and A blobs on a particular chain: :A= \dfrac The total end-to-end length of the restricted chain L is then: :L=Ad = \dfrac = \dfrac This is the average length a polymer molecule must diffuse to escape from its particular tube, and so the characteristic time for this to happen can be calculated using diffusive equations. A classical derivation gives the reptation time t: :t=\dfrac where \mu is the coefficient of friction on a particular polymer chain, k is Boltzmann's constant, and T is the absolute temperature. The linear macromolecules reptate if the length of macromolecule M is bigger than the critical entanglement molecular weight M_. M_ is 1.4 to 3.5 times M_. There is no reptation motion for polymers with M, so that the point M_ is a point of dynamic phase transition. Due to the reptation motion the coefficient of self-diffusion and conformational relaxation times of macromolecules depend on the length of macromolecule as M^ and M^3, correspondingly. The conditions of existence of reptation in the thermal motion of macromolecules of complex architecture (macromolecules in the form of branch, star, comb and others) have not been established yet. The dynamics of shorter chains or of long chains at short times is usually described by the
Rouse model The Rouse model is frequently used in polymer physics. The Rouse model describes the conformational dynamics of an ideal chain. In this model, the single chain diffusion is represented by Brownian motion of beads connected by harmonic springs. The ...
.


See also

* Important publications in polymer physics *
Polymer characterization Polymer characterization is the analytical branch of polymer science. The discipline is concerned with the characterization of polymeric materials on a variety of levels. The characterization typically has as a goal to improve the performance of t ...
*
Polymer physics Polymer physics is the field of physics that studies polymers, their fluctuations, mechanical properties, as well as the kinetics of reactions involving degradation and polymerisation of polymers and monomers respectively.P. Flory, ''Principles of ...
*
Protein dynamics Proteins are generally thought to adopt unique structures determined by their amino acid sequences. However, proteins are not strictly static objects, but rather populate ensembles of (sometimes similar) conformations. Transitions between these stat ...
* Soft matter


References

{{reflist, 2 Polymer physics Materials science Polymers 1971 introductions