Reptation
Encyclopedia
Reptation is the snake-like thermal motion of very long linear, entangled macromolecules in polymer
melts or concentrated polymer solutions.
Entanglement means the topological restriction of molecular motion by other chains.
The concept of reptation was introduced into polymer physics in 1971
by Pierre-Gilles de Gennes
to explain the dependence of the mobility of a macromolecule on its length. It is used as a mechanism to explain the viscous flow in an amorphous polymer.
Effective cross-links from entanglements with other polymer chains restrict polymer chain motion to a 'tube' within these restrictions. Since polymer chains would have to be broken to allow the restricted chain to pass through them, the mechanism for movement (flow) of this restricted chain is reptation. In the 'blob' model, the polymer chain is made up of
Kuhn length
s of individual length
. The chain is assumed to form tangled 'blobs' between each effective cross-link, containing 
Kuhn length segments in each. The mathematics of random walks can show that the average end-to-end length of a polymer chain, made up of 
Kuhn lengths is 
. Therefore if there are 
total Kuhn lengths, and 
blobs on a particular chain:
The total end-to-end length of the restricted chain
is then:
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
:
where
is cofficient of friction on a particular polymer chain, 
is Boltzmann's constant, and 
is the temperature.
The linear macromolecules reptate, if the length of macromolecule M is bigger than ten times ‘the length of macromolecule between adjacent entanglements’ Me . There is no reptation motion for polymers with M<10 Me, so that the point 10 Me 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−2 and M3, 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
.
chain entanglements on the relationship between molecular mass
and chain relaxation time (or similarly, the polymer’s zero-shear viscosity
). The theory was originally developed in 1971 by Pierre-Gilles de Gennes
, and has since been evolved by Sir Sam Edwards
and Masao Doi. As the name suggests, the theory envisions the movement of entangled polymer chains as analogous to snake
s slithering through one another. The theory predicts that, in entangled systems, the relaxation time τ is proportional to the cube of molecular mass. This is a reasonable approximation of the actual observed relationship, τ~(Molecular Mass)3.4 .
The prediction of the theory is arrived at by a relatively simple argument. First, each polymer chain is envisioned as occupying a tube of length L, 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 τ, we may focus on the overall, global motion of the chain. Thus, we define the tube mobility as μtube=v/f , where v is the velocity
of the chain when it is pulled by a force
f. Note also that μtube will be inversely proportional to the degree of polymerization
(and thus also inversely proportional to chain weight).
The diffusivity of the chain through the tube may then be written as Dtube=kBTμtube. By then recalling that in 1-dimension the mean square displacement due to Brownian motion
is given by s(t)2 = 2Dtubet, we obtain s(t)2=2kBTμtubet. The time necessary for a polymer chain to displace the length of its original tube is then t=L2/(2kBTμtube). By noting that this time is comparable to the relaxation time , we establish that τ~L2/μtube. 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 τ~(DPn)3 (and so τ~(Molecular Mass)3).
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
behavior, which is often observed in polymer melts.
Polymer
A polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...
melts or concentrated polymer solutions.
Entanglement means the topological restriction of molecular motion by other chains.
The concept of reptation was introduced into polymer physics in 1971
by Pierre-Gilles de Gennes
Pierre-Gilles de Gennes
Pierre-Gilles de Gennes was a French physicist and the Nobel Prize laureate in physics in 1991.-Biography:...
to explain the dependence of the mobility of a macromolecule on its length. It is used as a mechanism to explain the viscous flow in an amorphous polymer.
Mechanism
Entangled polymers are characterized with effective internal scale, commonly known as ‘the length of macromolecule between adjacent entanglements’ Me.Effective cross-links from entanglements with other polymer chains restrict polymer chain motion to a 'tube' within these restrictions. Since polymer chains would have to be broken to allow the restricted chain to pass through them, the mechanism for movement (flow) of this restricted chain is reptation. In the 'blob' model, the polymer chain is made up of
Kuhn lengthKuhn length
The Kuhn length is a theoretical treatment, developed by Werner 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 each other...
s of individual length
. The chain is assumed to form tangled 'blobs' between each effective cross-link, containing Kuhn length segments in each. The mathematics of random walks can show that the average end-to-end length of a polymer chain, made up of Kuhn lengths is . Therefore if there are total Kuhn lengths, and blobs on a particular chain:The total end-to-end length of the restricted chain
is then: 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
:where
is cofficient of friction on a particular polymer chain, is Boltzmann's constant, and is the temperature. The linear macromolecules reptate, if the length of macromolecule M is bigger than ten times ‘the length of macromolecule between adjacent entanglements’ Me . There is no reptation motion for polymers with M<10 Me, so that the point 10 Me 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−2 and M3, 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
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...
.
To be merged
Reptation theory describes the effect of polymerPolymer
A polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...
chain entanglements on the relationship between molecular mass
Molecular mass
The molecular mass of a substance is the mass of one molecule of that substance, in unified atomic mass unit u...
and chain relaxation time (or similarly, the polymer’s zero-shear viscosity
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
). The theory was originally developed in 1971 by Pierre-Gilles de Gennes
Pierre-Gilles de Gennes
Pierre-Gilles de Gennes was a French physicist and the Nobel Prize laureate in physics in 1991.-Biography:...
, and has since been evolved by Sir Sam Edwards
Sam Edwards (physicist)
Sir Samuel Frederick Edwards FLSW FRS is a British physicist.-Early life and studies:Sir Samuel was born on 1 February 1928 in Swansea, the son of Richard and Mary Jane Edwards....
and Masao Doi. As the name suggests, the theory envisions the movement of entangled polymer chains as analogous to snake
Snake
Snakes are elongate, legless, carnivorous reptiles of the suborder Serpentes that can be distinguished from legless lizards by their lack of eyelids and external ears. Like all squamates, snakes are ectothermic, amniote vertebrates covered in overlapping scales...
s slithering through one another. The theory predicts that, in entangled systems, the relaxation time τ is proportional to the cube of molecular mass. This is a reasonable approximation of the actual observed relationship, τ~(Molecular Mass)3.4 .
The prediction of the theory is arrived at by a relatively simple argument. First, each polymer chain is envisioned as occupying a tube of length L, 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 τ, we may focus on the overall, global motion of the chain. Thus, we define the tube mobility as μtube=v/f , where v is the velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
of the chain when it is pulled by a force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
f. Note also that μtube will be inversely proportional to the degree of polymerization
Degree of polymerization
The degree of polymerization, or DP, is usually defined as the number of monomeric units in a macromolecule or polymer or oligomer molecule.For a homopolymer, there is only one type of monomeric unit andthe number-average degree of polymerization is given by...
(and thus also inversely proportional to chain weight).
The diffusivity of the chain through the tube may then be written as Dtube=kBTμtube. By then recalling that in 1-dimension the mean square displacement due to Brownian motion
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...
is given by s(t)2 = 2Dtubet, we obtain s(t)2=2kBTμtubet. The time necessary for a polymer chain to displace the length of its original tube is then t=L2/(2kBTμtube). By noting that this time is comparable to the relaxation time , we establish that τ~L2/μtube. 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 τ~(DPn)3 (and so τ~(Molecular Mass)3).
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
Viscoelasticity
Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like honey, resist shear flow and strain linearly with time when a stress is applied. Elastic materials strain instantaneously when stretched and just...
behavior, which is often observed in polymer melts.