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Journal of Statistical Physics

, Volume 177, Issue 5, pp 936–959 | Cite as

Anomalous Stretching Dynamics of Tagged Monomer of Branched Polymer in Layered Random Flows

  • Neha
  • Divya Katyal
  • Rama KantEmail author
Article
  • 65 Downloads

Abstract

We extend our formalism to theoretically understand the stretch dynamics of single tagged monomer of flexible branched polymer with arbitrary topology in the presence of layered random flows. We derived an expression for the average squared displacement of the m-th bead of an arbitrary polymer structure with respect to its center of mass. The main focus of our study is the dynamics of star and dendrimer, where the effect of the topology, viz. the number and length of branches for stars and the number of generations and spacer lengths for dendrimers, is analyzed. We predict an increase in stretching as a function of time which finally reaches to the steady-state (plateau) value. In addition, we analyze the influence of variation in external flow which is characterized through its flow exponent (\(\alpha \)), root mean square velocity (\(V_0\)) and flow strength (\(W_{\alpha }\)). The composite measure of flow strength \(W_{\alpha }\) depends upon \(\alpha \), \(V_0\) and persistence length of flow. The magnitude of the maximum stretch (plateau region) increases with increase in flow exponent. The anomalous stretch behavior of star polymer with increasing flow strength shows the non-uniform stretch behavior of polymer.

Keywords

Polymer stretching dynamics Anomalous diffusion Layered random flow Generalized Gaussian structures Average square displacement 

Notes

Acknowledgements

Neha acknowledges UGC for providing JRF and SRF fellowship. Rama Kant acknowledges DST SERB (EMR/2016/07779) for financial assistance.

References

  1. 1.
    Chan, E.Y., et al.: DNA mapping using microfluidic stretching and single-molecule detection of fluorescent site-specific tags. Genome Res. 14, 1137–1146 (2004)Google Scholar
  2. 2.
    Schwartz, D.C., et al.: Ordered restrictions maps of Saccharomyces cerevisiae chromosomes constructed by optical mapping. Science 262, 110–114 (1993)ADSGoogle Scholar
  3. 3.
    Dhingra, J.K., et al.: A single-molecule barcoding system using nanoslits for DNA analysis. Proc. Natl. Acad. Sci. 104(8), 2673–2678 (2007)ADSGoogle Scholar
  4. 4.
    Doyle, P.S., et al.: Self-assembled magnetic matrices for DNA separation chips. Science 295, 2237 (2002)Google Scholar
  5. 5.
    Han, J., Craighead, H.G.: Separation of long DNA molecules in a microfabricated entropic trap array. Science 288, 1026–1029 (2000)ADSGoogle Scholar
  6. 6.
    Graham, M.D.: Fluid dynamics of dissolved polymer molecules in a confined geometries. Annu. Rev. Fluid Mech. 43, 273–298 (2011)ADSMathSciNetzbMATHGoogle Scholar
  7. 7.
    Smith, S.B., Cui, Y., Bustamante, C.: Overstretching B-DNA: The elastic response of individual double-stranded and single-stranded DNA molecules. Science 271, 795–799 (1996)ADSGoogle Scholar
  8. 8.
    Ferree, S., Blanch, H.W.: Electrokinetic stretching ofa tethered DNA. Biophys. J. 85, 2539–2546 (2003)ADSGoogle Scholar
  9. 9.
    Perkins, T.T., Smith, D.E., Larson, R.G., Chu, S.: Stretching of a single tethered polymer in a uniform flow. Science 268, 83–87 (1995)ADSGoogle Scholar
  10. 10.
    Lumley, J.: Drag reduction by additives. Annu. Rev. Fluid. Mech. 1, 367–384 (1969)ADSGoogle Scholar
  11. 11.
    Lumley, J.: On the solution of equations describing small scale deformation. Symp. Math. 9, 315–334 (1972)Google Scholar
  12. 12.
    Morikawa, K., Yanagida, M.: Visualization of individual DNA molecules in solution by light microscopy: DAPI staining method. J. Biochem. 89, 693–696 (1981)Google Scholar
  13. 13.
    Biswas, P., Kant, R., Blumen, A.: Stretch dynamics of flexible dendritic polymers in solution. J. Chem. Phys. 114, 2430–2441 (2001)ADSGoogle Scholar
  14. 14.
    Kant, R., Biswas, P., Blumen, A.: Hydrodynamic effects on the extensions of stars and dendrimers in external fields. Macromol. Theory Simul. 9, 608–620 (2000)Google Scholar
  15. 15.
    Biswas, P., Kant, R., Blumen, A.: Polymer dynamics and topology: extension of stars and dendrimer in external fields. Macromol. Theory Simul. 9, 56–67 (2000)Google Scholar
  16. 16.
    Roovers, J., Comanita, B.: Dendrimers and dendrimer-polymer hybrides. Adv. Polym. Sci. 142, 179–228 (1999)Google Scholar
  17. 17.
    Ganazzoli, F., LaFerla, R., Raffaini, G.: Intramolecular dynamics of dendrimer under excluded-volume conditions. Macromolecules 34, 4222–4228 (2001)ADSGoogle Scholar
  18. 18.
    Cai, C., Chen, Z.Y.: Rouse dynamics of a dendrimer model in the \(\theta \) condition. Macromolecules 30, 5104–5117 (1997)ADSGoogle Scholar
  19. 19.
    LaFerla, R.: Conformations and dynamics of dedrimers and cascade macromolecules. J. Chem. Phys. 106, 688–700 (1997)ADSGoogle Scholar
  20. 20.
    Satmarel, C., von Ferber, C., Blumen, A.: Spacers role in the dynamics of hyperbranched polymers. J. Chem. Phys. 124, 174905-1–17490-10 (2006)ADSGoogle Scholar
  21. 21.
    Blumen, A., von Ferber, C., Jurjiu, A., Koslowski, T.: Generalized Vicsek fractals: regular hyperbranched polymers. Macromolecules 37, 638–650 (2004)ADSGoogle Scholar
  22. 22.
    Jurjiu, A., Koslowski, T., von Ferber, C., Blumen, A.: Dynamics and scaling of polymer networks: Vicsek fractal and hydrodynamics interactions. Chem. Phys. 294, 187–199 (2003)Google Scholar
  23. 23.
    Gurtovenko, A.A., Blumen, A.: Relaxation of disordered polymer network: regular lattice made up of small-world Rouse networks. J. Chem. Phys. 115, 4924–4929 (2001)ADSGoogle Scholar
  24. 24.
    Perkins, T.T., Smith, D.E., Chu, S.: Single polymer dynamics in a elongational flow. Science 276, 2016–2029 (1997)Google Scholar
  25. 25.
    Smith, D.E., Chu, S.: Response of flexible polymers to a sudden elongational flow. Science 281, 1335–1340 (1998)ADSGoogle Scholar
  26. 26.
    Schroeder, C.M., Babcock, H.P., Shaqfeh, E.S.G., Chu, S.: Observation of polymer conformation hysteresic in extensional flow. Science 301, 1515–1519 (2003)ADSGoogle Scholar
  27. 27.
    Smith, D.E., Babcock, H.P., Chu, S.: Single-polymer dynamics in steady shear flow. Science 283, 1724–1727 (1999)ADSGoogle Scholar
  28. 28.
    LeDuc, P., Haber, C., Boa, G., Wirtz, D.: Dynamics of individual flexible polymers in a shear flow. Nature 399, 564–566 (1999)ADSGoogle Scholar
  29. 29.
    Gerashchenko, S., Steinberg, V.: Statistics of tumbling of a single polymer molecule in shear flow. Phys. Rev. Lett. 96, 038304 (2006)ADSGoogle Scholar
  30. 30.
    Babcock, H.P., Teixeira, R.E., Hur, J.S., Shaqfeh, E.S.G., Chu, S.: Visualization of molecular fluctuations near the critical point of the coil-stretch transition in polymer elongation. Macromolecules 36, 4544–4548 (2003)ADSGoogle Scholar
  31. 31.
    Rzehak, R., Kromen, W., Kawakatsu, T., Zimmermann, W.: Deformations of a tethered polymer in uniform flow. Europhys. J. E 2, 3–30 (2000)Google Scholar
  32. 32.
    Brochard Wyart, F.: Deformations of one tethered chain in strong flows. Europhys. Lett. 23, 105–111 (1993)ADSGoogle Scholar
  33. 33.
    Huang, C.-C., Winkler, R.G., Sutmann, G., Gompper, G.: Semidilute polymer solutions at equilibrium and under shear flow. Macromolecules 43, 10107–10116 (2010)ADSGoogle Scholar
  34. 34.
    Oshanin, G., Blumen, A.: Rouse chain dynamics in layered random flows. Phys. Rev. E. 49, 4185–4191 (1994)ADSGoogle Scholar
  35. 35.
    Oshanin, G., Blumen, A.: Dynamics and conormational properties of Rouse polymers in random layered flows. Macromol. Theory Simul. 4, 87–109 (1995)Google Scholar
  36. 36.
    Groisman, A., Steinberg, V.: Stretching of polymers in a random three-dimensional flow. Phys. Rev. Lett. 86, 934–937 (2001)ADSGoogle Scholar
  37. 37.
    Gerashchenko, S., Chevallard, C., Steinberg, V.: Single-polymer: coil stretch transition in a random flow. Europhys. Lett. 71, 221–227 (2005)ADSGoogle Scholar
  38. 38.
    Martins Afonso, M., Vincenzi, D.: Nonlinear elastic polymers in random flow. J. Fluid Mech. 540, 99–108 (2005)ADSzbMATHGoogle Scholar
  39. 39.
    Celani, A., Musacchio, S., Vincenzi, D.: Polymer transport in random flow. J. Stat. Phys. 118, 531–554 (2005)ADSMathSciNetzbMATHGoogle Scholar
  40. 40.
    Le Doussal, P.: Diffusion in layered random flows, polymers, electron in random potential, and spin depolarization in random fields. J. Stat. Phys. 69, 917–954 (1992)ADSMathSciNetzbMATHGoogle Scholar
  41. 41.
    Zumofen, G., Klafter, J., Blumen, A.: Trapping asects in enhanced diffusion. J. Stat. Phys. 65, 991–1013 (1991)ADSGoogle Scholar
  42. 42.
    Katyal, D., Kant, R.: Dynamics of generalized Gaussian polymeric structures in random layered flows. Phys. Rev. E 91(1–13), 042602 (2015)ADSGoogle Scholar
  43. 43.
    Ripoll, M., Winkler, R.G., Gompper, G.: Star polymers in shear flow. Phys. Rev. Lett. 96, 188302-01–188302-04 (2006)ADSGoogle Scholar
  44. 44.
    Singh, S.P., Chatterji, A., Gompper, G., Winkler, R.G.: Dynamics and rheological properties of ultrasoft colloids under shear flow. Macromolecules 46, 8026–8036 (2013)ADSGoogle Scholar
  45. 45.
    Lyulin, S.V., Darinskii, A.A., Lyulin, A.V., Michels, M.A.J.: Computer simulations of the dynamics of neutral and charged dendrimers. Macromolecules 37, 4676–4685 (2004)ADSGoogle Scholar
  46. 46.
    Nikoubashman, A., Likos, C.N.: Branched polymers under shear. Macromolecules 43, 1610–1620 (2010)ADSGoogle Scholar
  47. 47.
    De Gennes, P.G.: Coil-stretch transition of dilute flexible polymers under ultrahigh velocity gradients. J. Chem. Phys. 60, 5030–5042 (1974)ADSGoogle Scholar
  48. 48.
    Doyle, P.S., Shaqfeh, E.S., Gast, A.P.: Dynamic simulation of freely draining flexible polymers in steady linear flows. J. Fluid Mech. 334, 251–291 (1997)ADSzbMATHGoogle Scholar
  49. 49.
    Garoisman, A., Steinberg, V.: Elastic turbulence in a polymer solution flow. Nature 405, 53–55 (2000)ADSGoogle Scholar
  50. 50.
    Groisman, A., Steinberg, V.: Elastic turbulence in a curvilinear flows of polymer solutions. New J. Phys. 6, 29 (2004)ADSGoogle Scholar
  51. 51.
    Liu, Y., Steinberg, V.: Stretching of polymer in a random flow: effect of a shear rate. Europhys. Lett. 90(1–5), 44005 (2010)ADSGoogle Scholar
  52. 52.
    Liu, Yonggang, Steinberg, Victor: Single polymer dynamics in a random flow. Macromol. Symp. 337, 34–43 (2014)Google Scholar
  53. 53.
    Chertkov, M., Kolokolov, I., Lebedev, V., Turitsyn, K.: Polymer statistics in a random flow with mean shear. J. Fluid Mech. 531, 251–260 (2005)ADSMathSciNetzbMATHGoogle Scholar
  54. 54.
    Turitsyn, K.: Polymer dynamics in chaotic flows with a strong shear component. J. Exp. Theor. Phys. 105, 655–664 (2007)ADSGoogle Scholar
  55. 55.
    Balkovsky, E., Fouxon, A., Lebedev, V.: Turbulent dynamics of polymer solutions. Phys. Rev. Lett. 84, 4765–4768 (2000)ADSGoogle Scholar
  56. 56.
    Chertkov, M.: Polymer stretching by turbulence. Phys. Rev. Lett. 84, 4761–4764 (2000)ADSGoogle Scholar
  57. 57.
    Celani, A., Puliafito, A., Vincenzi, D.: Dynamicsal slowdown of polymers in laminar and random flows. Phy. Rev. Lett. 97(1–4), 118301 (2006)ADSGoogle Scholar
  58. 58.
    Davoudi, J., Schumacher, J.: Stretching of polymers around the kolmogorov scale in a turbulent shear flow. Phys. Fluids 18(1–11), 025103 (2006)ADSGoogle Scholar
  59. 59.
    Rouse, P.E.: A theory of the linear viscoelastic properties of dilute solutions of coiling polymers. J. Chem. Phys. 21, 1273–1280 (1953)ADSGoogle Scholar
  60. 60.
    Zimm, B.H.: Dynamics of polymer molecules in dilute solution: Viscoelasticity flow birefringence and dielectric loss. J. Chem. Phys. 24, 269–278 (1956)ADSMathSciNetGoogle Scholar
  61. 61.
    Kirkwood, J.G., Riseman, J.: The intrinsic viscosities and diffusion constants of flexible macromolecules in solution. J. Chem. Phys. 16, 565–573 (1948)ADSGoogle Scholar
  62. 62.
    Sommer, J.U., Blumen, A.: On the statistics of generalized Gaussian structures: collapse and random external fields. J. Phys. A 28, 6669 (1995)ADSMathSciNetzbMATHGoogle Scholar
  63. 63.
    Schiessel, H.: Unfold dynamics of generalized Gaussian structures. Phys. Rev. E 57, 5775–5781 (1998)ADSGoogle Scholar
  64. 64.
    Friedrich, Chr., Schiessel, H., Blumen, A.: In: Siginer, D.A., DeKee, D., Chhabra, R.P. (eds.) Advances in the Flow and Rheology of Non-Newtonian Fluids, p. 429. Elsevier, Amsterdam (1999)Google Scholar
  65. 65.
    Schiessel, H., Friedrich, Chr., Blumen, A.: In: Hilfer, R. (ed.) Applications of Fractional Calculus in Physics. World Scientific, Singapore (1999)Google Scholar
  66. 66.
    Gurtovenko, A.A., Blumen, A.: Generalized Gaussian structures: model for polymer systems with complex topologies. In: Polymer Analysis Polymer Theory, Advances in polymer Science, vol. 182, pp. 171–282. Springer, Berlin, Heidelberg (2005)Google Scholar
  67. 67.
    Satmarel, C., von Ferber, C., Blumen, A.: Dynamics of end-linked star-polymer structures. J. Chem. Phys. 123(3), 034907 (2005)ADSGoogle Scholar
  68. 68.
    Blumen, A., Jurjiu, A.: Multifractal spectra and the relaxation of model polymer networks. J. Chem. Phys. 116, 2636–2641 (2002)ADSGoogle Scholar
  69. 69.
    Jurjiu, A., Koslowski, Th, Blumen, A.: Dynamicsmof deterministic fractal polymer networks: hydrodynamics interactions and the absence of scaling. J. Chem. Phys. 118, 2398–2404 (2003)ADSGoogle Scholar
  70. 70.
    Blumen, A., Gurtovenko, A.A., Jespersen, S.: Anomalous diffusion and relaxation in macromolecular systems. J. Non-Cryst. Solids 305, 71–80 (2002)ADSGoogle Scholar
  71. 71.
    Chen, Z.Y., Cai, C.: Dynamics of starburst dendrimers. Macromolecules 32, 5423–5434 (1999)ADSGoogle Scholar
  72. 72.
    Matheron, De Marsely: Is transport in porous media always diffusive? A counterexample. Water Resour. Res. 16, 901–917 (1980)ADSGoogle Scholar
  73. 73.
    Katyal, D., Kant, R.: Dynamics of comb-of-comb network polymers in random layered flows. Phys. Rev. E 94, 062503-01–062503-11 (2016)ADSGoogle Scholar
  74. 74.
    Katyal, D., Kant, R.: Semidiluted polymer solutions at equillibrium and under shear flow. Macromol. Theory Simul. 1700009, 1–15 (2017)Google Scholar
  75. 75.
    Doi, M., Edwards, S.F.: The Theory of Polymer Dynamics. Clarendon Press, Oxford (1986)Google Scholar
  76. 76.
    Eichinger, B.E., Martin, J.E.: Distribution function for Gaussian molecules. II. Reduction of the Kirchhoff matrix for large molecules. J. Chem. Phys. 69, 4595–4599 (1978)ADSGoogle Scholar
  77. 77.
    Kloczkowski, A., Mark, J.E., Frisch, H.L.: The relaxation spectrum for Gaussian networks. Macromolecules 23, 3481–3490 (1990)ADSGoogle Scholar
  78. 78.
    Gaveau, B., Schulman, L.S.: Anomalous diffusion in a random velocity field. J. Stat. Phys. 66, 375–383 (1992)ADSMathSciNetzbMATHGoogle Scholar
  79. 79.
    Abramowitz, M., Stegun, I. A.: Handbook of Mathematical Functions. Dover Publications Inc., New YorkGoogle Scholar
  80. 80.
    Bauer, B.J., Fetters, L.J., Graessley, W.W., Hadjichristidis, N., Quack, G.F.: Chain dimensions in dilute polymer solution: a light-scattering and viscometric study of multiarmed polyisoprene stars in good and \(\theta \) solvents. Macromolecules 22, 2337–2347 (1989)ADSGoogle Scholar
  81. 81.
    Huber, K., Bantle, S., Burchard, W., Fetters, L.: Semidilute solutions of star branched polystyrene: a light and neutron scattering study. Macromolecules 19, 1404–1411 (1986)ADSGoogle Scholar
  82. 82.
    Stivala, S.S., Khorramian, B.A., Patel, A.: Small-angle X-ray scattering from 12-arm polystyrene fraction in MEK. Polymer 27, 517–522 (1986)Google Scholar
  83. 83.
    Jin, S., Jin, K.S., Yoon, J., Heo, K., Kim, J., Kim, K.-W., Ree, M., Higashihara, T., Watanabe, T., Hirao, A.: X-ray scattering studies on molecular structures of star and dendritic polymers. Macromol. Res. 16, 686–694 (2008)Google Scholar
  84. 84.
    Willner, L., Jucknischke, O., Richter, D., Roovers, J., Zhou, L.-L., Toporowski, P.M., Fetters, L.J., Huang, J.S., Lin, M.Y., Hadjichristidis, N.: Structural investigation of star polymers in solution by small-angle neutron scattering. Macromolecules 27, 3821–3829 (1994)ADSGoogle Scholar
  85. 85.
    Dozier, D.W., Huang, J.S., Fetters, L.J.: Colloidal nature of star polymer dilute and semidilute solutions. Macromolecules 24, 2810–2814 (1991)ADSGoogle Scholar
  86. 86.
    Georgiou, T.K., Phylactou, L.A., Patrickios, C.S.: Synthesis, characterization and evaluation as transfection reagents of ampholytic star copolymers: effect of star architecture. Biomacromolecules 7, 3505–3512 (2006)Google Scholar
  87. 87.
    Georgiou, T.K.: Star polymers for gene delivery. Polym. Int. 63, 1130–1133 (2014)Google Scholar
  88. 88.
    Frchet, J.M.J.: Functional polymers and dendrimers: reactivity, molecular architecture and interfacial energy. Science 263, 1710–1715 (1994)ADSGoogle Scholar
  89. 89.
    NewKome, G.R.: Advances in Dendritic Macromolecules. JAI, London (1996)Google Scholar
  90. 90.
    Esfand, R., Tomalia, D.A.: Poly(amidoammine) (PAMAM) dendrimers:from biomimiery to drug delivary and biomedical applications. Drug Discov. Today 6, 427–436 (2001)Google Scholar
  91. 91.
    Svenson, S., Tomalia, D.A.: Dendrimers in biomedical applications-reflections on the field. Adv. Drug Deliv. Rev. 64, 102–115 (2012)Google Scholar
  92. 92.
    Bar-Haim, A., Klafter, J., Kopelman, R.: Dendrimers as controlled artificial energy antennae. J. Am. Chem. Soc. 119, 6197–6198 (1997)Google Scholar
  93. 93.
    Nantalaksakul, A., Reddy, D.R., Bardeen, C.J., Thayumanavan, S.: Light harvesting dendrimers. Photosynth. Res. 87, 133–150 (2006)Google Scholar
  94. 94.
    Biswas, P., Cherayil, B.J.: Radial dimensions of starburst polymers. J. Chem. Phys. 100, 3201–3209 (1994)ADSGoogle Scholar
  95. 95.
    Ganazzoli, F., LaFerla, R.: The unperturbed state of dendrimers. J. Chem. Phys. 113, 9288–9293 (2000)ADSGoogle Scholar
  96. 96.
    Gurtovenko, A.A., Markelov, D.A., Gotlib, Y.Y., Blumen, A.: Dynamics of dendrimer-based polymer networks. J. Chem. Phys. 119, 7579–7590 (2003)ADSGoogle Scholar

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Authors and Affiliations

  1. 1.Department of ChemistryUniversity of DelhiDelhiIndia

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