Reptation

Last updated

A peculiarity of thermal motion of very long linear macromolecules in entangled polymer melts or concentrated polymer solutions is reptation. [1] Derived from the word reptile, reptation suggests the movement of entangled polymer chains as being analogous to snakes slithering through one another. [2] 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. [3] [4] Sir Sam Edwards and Masao Doi later refined reptation theory. [5] [6] Similar phenomena also occur in proteins. [7]

Contents

Two closely related concepts are reptons and entanglement. A repton is a mobile point residing in the cells of a lattice, connected by bonds. [8] [9] Entanglement means the topological restriction of molecular motion by other chains. [10]

Theory and mechanism

Reptation theory describes the effect of polymer chain entanglements on the relationship between molecular mass and chain relaxation time. The theory predicts that, in entangled systems, the relaxation time τ is proportional to the cube of molecular mass, M: τ ∝ M 3. 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 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. 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 squared 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 ∝ M3).

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. Note that the polymer’s zero-shear viscosity gives an approximation of the actual observed dependency, τ ∝ M3.4; [11] this relaxation time has nothing to do with the reptation relaxation time.

Models

The blob model, explaining the entanglement of long polymer chains. Reptation- blob model.svg
The blob model, explaining the entanglement of long polymer chains.
The tube model, explaining the basically one-dimensional mobility of long polymer chains. Reptation Theory.svg
The tube model, explaining the basically one-dimensional mobility of long polymer chains.

Entangled polymers are characterized with effective internal scale, commonly known as the length of macromolecule between adjacent entanglements.

Entanglements with other polymer chains restrict polymer chain motion to a thin virtual tube passing through the restrictions. [12] 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, [13] the polymer chain is made up of Kuhn lengths of individual length . The chain is assumed to form blobs between each entanglement, containing 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 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 the coefficient of friction on a particular polymer chain, is the Boltzmann constant, and is the absolute temperature.

The linear macromolecules reptate if the length of macromolecule is bigger than the critical entanglement molecular weight . is 1.4 to 3.5 times . [14] There is no reptation motion for polymers with , so that the point 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 and , correspondingly. [15] [16] 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.

See also

Related Research Articles

<span class="mw-page-title-main">Polymer</span> Substance composed of macromolecules with repeating structural units

A polymer is a substance or material that consists of very large molecules, or macromolecules, that are constituted by many repeating subunits derived from one or more species of monomers. Due to their broad spectrum of properties, both synthetic and natural polymers play essential and ubiquitous roles in everyday life. Polymers range from familiar synthetic plastics such as polystyrene to natural biopolymers such as DNA and proteins that are fundamental to biological structure and function. Polymers, both natural and synthetic, are created via polymerization of many small molecules, known as monomers. Their consequently large molecular mass, relative to small molecule compounds, produces unique physical properties including toughness, high elasticity, viscoelasticity, and a tendency to form amorphous and semicrystalline structures rather than crystals.

<span class="mw-page-title-main">Gel</span> Highly viscous liquid exhibiting a kind of semi-solid behavior

A gel is a semi-solid that can have properties ranging from soft and weak to hard and tough. Gels are defined as a substantially dilute cross-linked system, which exhibits no flow when in the steady state, although the liquid phase may still diffuse through this system.

<span class="mw-page-title-main">Strouhal number</span> Dimensionless number describing oscillating flow mechanisms

In dimensional analysis, the Strouhal number is a dimensionless number describing oscillating flow mechanisms. The parameter is named after Vincenc Strouhal, a Czech physicist who experimented in 1878 with wires experiencing vortex shedding and singing in the wind. The Strouhal number is an integral part of the fundamentals of fluid mechanics.

<span class="mw-page-title-main">Polymer physics</span>

Polymer physics is the field of physics that studies polymers, their fluctuations, mechanical properties, as well as the kinetics of reactions involving degradation of polymers and polymerisation of monomers.

The nuclear Overhauser effect (NOE) is the transfer of nuclear spin polarization from one population of spin-active nuclei to another via cross-relaxation. A phenomenological definition of the NOE in nuclear magnetic resonance spectroscopy (NMR) is the change in the integrated intensity of one NMR resonance that occurs when another is saturated by irradiation with an RF field. The change in resonance intensity of a nucleus is a consequence of the nucleus being close in space to those directly affected by the RF perturbation.

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 both shear flow and strain linearly with time when a stress is applied. Elastic materials strain when stretched and immediately return to their original state once the stress is removed.

Fluorescence correlation spectroscopy (FCS) is a statistical analysis, via time correlation, of stationary fluctuations of the fluorescence intensity. Its theoretical underpinning originated from L. Onsager's regression hypothesis. The analysis provides kinetic parameters of the physical processes underlying the fluctuations. One of the interesting applications of this is an analysis of the concentration fluctuations of fluorescent particles (molecules) in solution. In this application, the fluorescence emitted from a very tiny space in solution containing a small number of fluorescent particles (molecules) is observed. The fluorescence intensity is fluctuating due to Brownian motion of the particles. In other words, the number of the particles in the sub-space defined by the optical system is randomly changing around the average number. The analysis gives the average number of fluorescent particles and average diffusion time, when the particle is passing through the space. Eventually, both the concentration and size of the particle (molecule) are determined. Both parameters are important in biochemical research, biophysics, and chemistry.

<span class="mw-page-title-main">Dynamic light scattering</span> Technique for determining size distribution of particles

Dynamic light scattering (DLS) is a technique in physics that can be used to determine the size distribution profile of small particles in suspension or polymers in solution. In the scope of DLS, temporal fluctuations are usually analyzed using the intensity or photon autocorrelation function. In the time domain analysis, the autocorrelation function (ACF) usually decays starting from zero delay time, and faster dynamics due to smaller particles lead to faster decorrelation of scattered intensity trace. It has been shown that the intensity ACF is the Fourier transform of the power spectrum, and therefore the DLS measurements can be equally well performed in the spectral domain. DLS can also be used to probe the behavior of complex fluids such as concentrated polymer solutions.

Rubber elasticity refers to the ability of solid rubber to be stretched up to a factor of 10 from its original length, and return to close to its original length upon release. This process can be repeated many times with no apparent degradation to the rubber.

Methods have been devised to modify the yield strength, ductility, and toughness of both crystalline and amorphous materials. These strengthening mechanisms give engineers the ability to tailor the mechanical properties of materials to suit a variety of different applications. For example, the favorable properties of steel result from interstitial incorporation of carbon into the iron lattice. Brass, a binary alloy of copper and zinc, has superior mechanical properties compared to its constituent metals due to solution strengthening. Work hardening has also been used for centuries by blacksmiths to introduce dislocations into materials, increasing their yield strengths.

In the case of systems composed of subsystems, the classification of quantum-entangledstates is richer than in the bipartite case. Indeed, in multipartite entanglement apart from fully separable states and fully entangled states, there also exists the notion of partially separable states.

<span class="mw-page-title-main">Rouse model</span>

The Rouse model is frequently used in polymer physics.

The term file dynamics is the motion of many particles in a narrow channel.

<span class="mw-page-title-main">Michael Cates</span> British physicist (born 1961)

Michael Elmhirst Cates is a British physicist. He is the 19th Lucasian Professor of Mathematics at the University of Cambridge and has held this position since 1 July 2015. He was previously Professor of Natural Philosophy at the University of Edinburgh, and has held a Royal Society Research Professorship since 2007.

Masao Doi 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 1976, he introduced a second quantised formalism for studying reaction-diffusion systems. In 1978 and 1979 he wrote a series of papers with Sir Sam Edwards expanding on the concept of reptation introduced by Pierre-Gilles de Gennes in 1971. In 1996 he authored the textbook Introduction to Polymer Physics.

Hoffman nucleation theory is a theory developed by John D. Hoffman and coworkers in the 1970s and 80s that attempts to describe the crystallization of a polymer in terms of the kinetics and thermodynamics of polymer surface nucleation. The theory introduces a model where a surface of completely crystalline polymer is created and introduces surface energy parameters to describe the process. Hoffman nucleation theory is more of a starting point for polymer crystallization theory and is better known for its fundamental roles in the Hoffman–Weeks lamellar thickening and Lauritzen–Hoffman growth theory.

Vladimir Nikolajevich Pokrovskii is a Russian scientist known for his original contributions to polymer physics and economic theory. He was the founder of the Altai school of dynamics of nonlinear fluids.

Mechanics of gelation describes processes relevant to sol-gel process.

In cosmology, Gurzadyan-Savvidy (GS) relaxation is a theory developed by Vahe Gurzadyan and George Savvidy to explain the relaxation over time of the dynamics of N-body gravitating systems such as star clusters and galaxies. Stellar systems observed in the Universe – globular clusters and elliptical galaxies – reveal their relaxed state reflected in the high degree of regularity of some of their physical characteristics such as surface luminosity, velocity dispersion, geometric shapes, etc. The basic mechanism of relaxation of stellar systems has been considered the 2-body encounters, to lead to the observed fine-grained equilibrium. The coarse-grained phase of evolution of gravitating systems is described by violent relaxation developed by Donald Lynden-Bell. The 2-body mechanism of relaxation is known in plasma physics. The difficulties with description of collective effects in N-body gravitating systems arise due to the long-range character of gravitational interaction, as distinct of plasma where due to two different signs of charges the Debye screening takes place. The 2-body relaxation mechanism e.g. for elliptical galaxies predicts around years i.e. time scales exceeding the age of the Universe. The problem of relaxation and evolution of stellar systems and the role of collective effects are studied by various techniques, see. Among the efficient methods of study of N-body gravitating systems are the numerical simulations, particularly, Sverre Aarseth's N-body codes are widely used.

In quantum physics, the "monogamy" of quantum entanglement refers to the fundamental property that it cannot be freely shared between arbitrarily many parties.

References

  1. ↑ Pokrovskii, V. N. (2010). The Mesoscopic Theory of Polymer Dynamics. Springer Series in Chemical Physics. Vol. 95. Bibcode:2010mtpd.book.....P. doi:10.1007/978-90-481-2231-8. ISBN   978-90-481-2230-1.
  2. ↑ Rubinstein, Michael (March 2008). Dynamics of Entangled Polymers. Pierre-Gilles de Gennes Symposium. New Orleans, LA: American Physical Society. Retrieved 6 April 2015.
  3. ↑ De Gennes, P. G. (1983). "Entangled polymers". Physics Today. 36 (6): 33. Bibcode:1983PhT....36f..33D. doi:10.1063/1.2915700. A theory based on the snake-like motion by which chains of monomers move in the melt is enhancing our understanding of rheology, diffusion, polymer-polymer welding, chemical kinetics and biotechnology
  4. ↑ De Gennes, P. G. (1971). "Reptation of a Polymer Chain in the Presence of Fixed Obstacles". The Journal of Chemical Physics. 55 (2): 572. Bibcode:1971JChPh..55..572D. doi:10.1063/1.1675789.
  5. ↑ Samuel Edwards: Boltzmann Medallist 1995, IUPAP Commission on Statistical Physics, archived from the original on 2013-10-17, retrieved 2013-02-20
  6. ↑ Doi, M.; Edwards, S. F. (1978). "Dynamics of concentrated polymer systems. Part 1.?Brownian motion in the equilibrium state". Journal of the Chemical Society, Faraday Transactions 2. 74: 1789–1801. doi:10.1039/f29787401789.
  7. ↑ Bu, Z; Cook, J; Callaway, D. J. (2001). "Dynamic regimes and correlated structural dynamics in native and denatured alpha-lactalbumin". Journal of Molecular Biology. 312 (4): 865–73. doi:10.1006/jmbi.2001.5006. PMID   11575938.
  8. ↑ Barkema, G. T.; Panja, D.; Van Leeuwen, J. M. J. (2011). "Structural modes of a polymer in the repton model". The Journal of Chemical Physics. 134 (15): 154901. arXiv: 1102.1394 . Bibcode:2011JChPh.134o4901B. doi:10.1063/1.3580287. PMID   21513412. S2CID   1979411.
  9. ↑ Rubinstein, M. (1987). "Discretized model of entangled-polymer dynamics". Physical Review Letters. 59 (17): 1946–1949. Bibcode:1987PhRvL..59.1946R. doi:10.1103/PhysRevLett.59.1946. PMID   10035375.
  10. ↑ McLeish, T. C. B. (2002). "Tube theory of entangled polymer dynamics". Advances in Physics. 51 (6): 1379–1527. Bibcode:2002AdPhy..51.1379M. CiteSeerX   10.1.1.629.3682 . doi:10.1080/00018730210153216. S2CID   122657744.
  11. ↑ Berry, G. C.; Fox, T. G. (1968). "The viscosity of polymers and their concentrated solutions". Fortschritte der Hochpolymeren-Forschung. Advances in Polymer Science. Vol. 5/3. Springer Berlin Heidelberg. p. 261. doi:10.1007/BFb0050985. ISBN   978-3-540-04032-3.
  12. ↑ Edwards, S. F. (1967). "The statistical mechanics of polymerized material". Proceedings of the Physical Society. 92 (1): 9–16. Bibcode:1967PPS....92....9E. doi:10.1088/0370-1328/92/1/303.
  13. ↑ Duhamel, J.; Yekta, A.; Winnik, M. A.; Jao, T. C.; Mishra, M. K.; Rubin, I. D. (1993). "A blob model to study polymer chain dynamics in solution". The Journal of Physical Chemistry. 97 (51): 13708. doi:10.1021/j100153a046.
  14. ↑ Fetters, LJ; Lohse, DJ; Colby, RH (2007). "25.3". In Mark, James E (ed.). Chain Dimensions and Entanglement Spacings in "Physical properties of polymers handbook" (2nd ed.). New York: Springer New York. p. 448. ISBN   978-0-387-69002-5.
  15. ↑ Pokrovskii, V. N. (2006). "A justification of the reptation-tube dynamics of a linear macromolecule in the mesoscopic approach". Physica A: Statistical Mechanics and Its Applications. 366: 88–106. Bibcode:2006PhyA..366...88P. doi:10.1016/j.physa.2005.10.028.
  16. ↑ Pokrovskii, V. N. (2008). "Reptation and diffusive modes of motion of linear macromolecules". Journal of Experimental and Theoretical Physics. 106 (3): 604–607. Bibcode:2008JETP..106..604P. doi:10.1134/S1063776108030205. S2CID   121054836.
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.