Math 2.0

Statistical mechanics, structure of matter Definition

Reptation

Builds on Ideal chainDiffusion process

Definition

Consider a linear polymer chain of NN segments in a dense entangled system, such as a melt of long chains. The reptation model describes its motion by the following assumptions.

  1. The other chains confine the chain to a tube that follows its contour. The centre line of the tube is an ideal chain of contour length LL proportional to NN.
  2. The chain moves only along the tube, as a one-dimensional diffusion with curvilinear diffusion coefficient Dc=D1/ND_c = D_1 / N, where D1D_1 is a constant independent of NN.
  3. The ends of the chain explore new directions at random, so the part of the tube that the chain has left is forgotten.

The reptation time, or disengagement time, is the time τd\tau_d that the chain needs to leave its initial tube completely. Up to a constant factor it is τd≈L2/Dc\tau_d \approx L^2 / D_c.

Remarks

From clauses 1 and 2, τd∝N2⋅N=N3\tau_d \propto N^2 \cdot N = N^3. In time τd\tau_d the centre of mass moves by about the coil size RR, and R2∝NR^2 \propto N by clause 1, so the self-diffusion coefficient scales as D≈R2/τd∝N/N3=N−2D \approx R^2 / \tau_d \propto N / N^3 = N^{-2}. Example: in a polymer melt above the entanglement length, measured self-diffusion coefficients scale close to N−2N^{-2}, as the model predicts. Non-example: short unentangled chains do not move in a tube, so clause 1 fails; their dynamics is described instead by the Rouse model, with D∝N−1D \propto N^{-1}. The model predicts a melt viscosity proportional to N3N^3, while experiments give an exponent close to 3.43.4; tube length fluctuations and constraint release account for most of the difference.

Reading path

To understand this page from the ground up, read these 72 pages in order. The foundations come first.

  1. Complex numbers object
  2. Construction of the reals from Cauchy sequences concept
  3. Diffusion process concept
  4. Element definition
  5. Integers object
  6. Normal distribution concept
  7. Ordered field definition
  8. Proposition concept
  9. Random experiment concept
  10. Random walk concept
  11. Rational numbers object
  12. Real numbers object
  13. Euler's formula statement
  14. Set definition
  15. Ring of integers statement
  16. Disjunction definition
  17. Implication definition
  18. Predicate definition
  19. Ordered pair definition
  20. Union definition
  21. Sample space definition
  22. Existential quantifier definition
  23. Universal quantifier definition
  24. The rationals form a field statement
  25. Cartesian product definition
  26. Subset definition
  27. Uniqueness quantifier definition
  28. The rationals form an ordered field statement
  29. Binary relation definition
  30. Power set definition
  31. Universal set definition
  32. Order on the reals definition
  33. Function definition
  34. Complement definition
  35. Interval definition
  36. Injective function definition
  37. Indicator function definition
  38. Preimage of a set definition
  39. Sigma-algebra definition
  40. Countable set definition
  41. Event definition
  42. Borel sigma-algebra definition
  43. Probability axioms concept
  44. Probability measure definition
  45. Probability distribution definition
  46. Dirac measure definition
  47. Probability space definition
  48. Conditional probability definition
  49. Random variable definition
  50. Cumulative distribution function definition
  51. Discrete random variable definition
  52. Law of a random variable definition
  53. Independence of two events definition
  54. Random variable with a density definition
  55. Convergence in distribution definition
  56. Probability mass function definition
  57. Expected value definition
  58. Identically distributed random variables definition
  59. Mutual independence of events definition
  60. Probability density function definition
  61. Law of the unconscious statistician statement
  62. Independent sigma-algebras definition
  63. Density of the normal distribution definition
  64. Characteristic function definition
  65. Independence of random variables definition
  66. Normal distribution definition
  67. The characteristic function determines the distribution statement
  68. Independent and identically distributed random variables definition
  69. Lévy's continuity theorem statement
  70. Central limit theorem statement
  71. Ideal chain example
  72. Reptation definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Reptation
About this page
TagVMW2ZNRFTH
URIm2://VMW2ZNRFTH
Kinddefinition
WritingW0 stub
MSC82D60, 82C70
DefinesReptation
UsesIdeal chain, Diffusion process
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.