Analytical and Computer Methods to Evaluate Mechanical Properties of the Metamaterials Based on Various Models of Polymeric Chains

Part of the Advanced Structured Materials book series (STRUCTMAT, volume 59)


The formation of polymer coating on a solid substrate is investigated by means of computer simulation (Monte-Carlo method). The sticking coefficient depending on different factors affecting the adhesion of monomer units is calculated. Mechanical properties are stimulated on the base of the hybrid discrete-continuous model, which describes the system consisting of flexible substrate and polymer coating. At different temperatures and intermolecular interactions constants, the dependencies of Young modulus on the deformation degree are calculated. Ferroelectric properties of the polymer coating depending on frequency and amplitude of external electric field, temperature and interchain interactions are investigated.


Metamaterials Ferroelectric polymer Monte-Carlo method Stockmayer potential Hybrid discrete-continual model Tension and bending deformations Hysteresis 



The work is performed within the framework of the project “Methods of microstructural nonlinear analysis, wave dynamics and mechanics of composites for research and design of modern metamaterials and elements of structures made on its base” (grant №15-19-10008 of by the Russian Science Foundation).


  1. 1.
    Bhushan, B., Jung, Y.: Natural and biomimetic artificial surfaces for superhydrophobicity, self-cleaning, low adhesion, and drag reduction. Prog. Mater Sci. 56, 1–108 (2011)CrossRefGoogle Scholar
  2. 2.
    Bico, J., Thiele, U., Quere, D.: Wetting of textured surfaces. Colloid Surf. A Physicochem. Eng. Asp. 206(1–3), 41–46 (2002)CrossRefGoogle Scholar
  3. 3.
    Boinovich, L.B., Emelyanenko, A.M.: Russ. Chem. Rev. 77(7), 583 (2008)CrossRefGoogle Scholar
  4. 4.
    Callies, M., Quere, D.: On water repellency. Soft Matter 1(1), 55–61 (2005)CrossRefGoogle Scholar
  5. 5.
    Kushch, V.I., Chernobai, V.S., Mishuris, G.S.: Longitudinal shear of a composite with elliptic nanofibers: local stresses and effective stiffness. Int. J. Eng. Sci. 84, 79–94 (2014)CrossRefGoogle Scholar
  6. 6.
    Zheng, L., Wu, X., Lou, Z., et al.: Superhydrophobicity from microstructured surface. Chin. Sci. Bull. 49, 1779–1787 (2004)CrossRefGoogle Scholar
  7. 7.
    Lisichkina G.V.: Chemistry of grafted of the surface compounds. FIZMATLIT, Moscow (2003)Google Scholar
  8. 8.
    Shirtcliffe, N., McHale, G., Atherton, S., et al.: An introduction to superhydrophobicity. Adv. Colloid Interface Sci. 161, 124–138 (2010)CrossRefGoogle Scholar
  9. 9.
    Zhang, X., Shi, F., Niu, J., et al.: Superhydrophobic surfaces: from structural control to functional application. J. Mater. Chem. 18, 621–633 (2008)CrossRefGoogle Scholar
  10. 10.
    Ageev, A.A., Aksenova, I.V., Volkov, V.A., Eleev, A.F.: About fluorine-containing surface-active substances capable to modify polymer fabrics’ fibers. Fluorine Notes 4(83), 5–6 (2012)Google Scholar
  11. 11.
    Rigoberto, C.A., et al.: Polymer Brushes: Synthesis, Characterization, Applications. WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim (2004)Google Scholar
  12. 12.
    Birstein, T.M.: The polymer brushes. Soros Educ. J. 5, 42–47 (1999)Google Scholar
  13. 13.
    Mamonova, M.V., Prudnikov, V.V., Prudnikova, V.V., Prudnikova, I.A.: The theoretical and experimental methods in surface physics, Omsk (2009)Google Scholar
  14. 14.
    Oura, K., Lifschic, V.G., Sarznin, A.A., Zotov, A.V., Katayama, M.: Introduction to Surface Physics, Moscow, Nauka, p. 496 (2006)Google Scholar
  15. 15.
    Huang, G.Y., Yu, S.W.: Effect of surface piezoelectricity on the electromechanical behavior of a piezoelectric ring. Phys. Status Solid. B 243(4), 22–24 (2006)CrossRefGoogle Scholar
  16. 16.
    Wang, G.F., Feng, X.Q.: Effect of surface stresses on the vibration and buckling of piezoelectric nanowires. EPL 91, 56007 (2010)CrossRefGoogle Scholar
  17. 17.
    Yan, Z., Jiang, L.Y.: Surface effects on the electromechanical coupling and bending behaviors of piezoelectric nanowires. J. Phys. D Appl. Phys. 44, 075404 (2011)CrossRefGoogle Scholar
  18. 18.
    Yan, Z., Jiang, L.Y.: Vibration and buckling analysis of a piezoelectric nanoplate considering surface effects and in-plane constraints. Proc. R. Soc. A 468, 3458–3475 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Zhang, J., Wang, Ch., Adhikari, S.: Surface effect on the buckling of piezoelectric nanofilms. J. Phys. D Appl. Phys. 45, 285301 (2012)CrossRefGoogle Scholar
  20. 20.
    Pan, X.H., Yu, S.W., Feng, X.Q.: A continuum theory of surface piezoelectricity for nanodielectrics. Sci. China: Phys., Mech. Astron. 54(4), 564–573 (2011)Google Scholar
  21. 21.
    Yan, Z., Jiang, L.: Electromechanical response of a curved piezoelectric nanobeam with the consideration of surface effects. J. Phys. D Appl. Phys. 44, 365301 (2011)CrossRefGoogle Scholar
  22. 22.
    Yan, Z., Jiang, L.: Surface effects on the electroelastic responses of a thin piezoelectric plate with nanoscale thickness. J. Phys. D Appl. Phys. 45, 255401 (2012)CrossRefGoogle Scholar
  23. 23.
    Altenbach, H., Eremeyev, V.A., Lebedev, L.P.: On the existence of solution in the linear elasticity with surface stresses. ZAMM 90(3), 231–240 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Altenbach, H., Eremeyev, V.A., Lebedev, L.P.: On the spectrum and stiffness of an elastic body with surface stresses. ZAMM. 91(9), 699–710 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Eremeyev, V.A., Lebedev, L.P.: Existence of weak solutions in elasticity. Math. Mech. Solids 18(2), 204–217 (2013)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Eremeyev, V.A., Lebedev, L.P.: Mathematical study of boundary-value problems within the framework of Steigmann-Ogden model of surface elasticity. Continuum Mech. Thermodyn. 28(1–2), 407–422 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Schiavone, P., Ru, C.Q.: Solvability of boundary value problems in a theory of plane–strain elasticity with boundary reinforcement. Int. J. Eng. Sci. 47, 1331–1338 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Nasedkin, A.V., Eremeyev, V.A.: Spectral properties of piezoelectric bodies with surface effects. In: Surface Effects in Solid Mechanics, pp. 105–121. Springer (2013)Google Scholar
  29. 29.
    Nasedkin, A.V., Eremeyev, V.A.: Harmonic vibrations of nanosized piezoelectric bodies with surface effects. ZAMM 94(10), 878–892 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Nasedkin, A.V., Eremeyev, V.A.: Some models for nanosized magnetoelectric bodies with surface effects. In: Advanced Materials, pp. 373–391. Springer (2016)Google Scholar
  31. 31.
    Binder, К.: Monte-Carlo Methods in Statistical Physics. Mir, Moscow (1982)Google Scholar
  32. 32.
    Binder, К., Heermann, D.W.: Monte-Carlo Simulation in Statistical Physics, 5th edn. Springer Heidelberg, Dordrecht, London, NY (2010)Google Scholar
  33. 33.
    Multer-Krumbhaar, H., Burkhardt, T.W., Kroll, D.: A generalized kinetic equation for crystal growth. J. Cryst. Growth. 38 (1977)Google Scholar
  34. 34.
    Wang, J., Huang, Z., Duan, H., Yu, S., Feng, X., Wang, G., Zhang, W., Wang, T.: Surface stress effect in mechanics of nanostructured materials. Acta Mech. Solida Sin. 24, 52–82 (2011)CrossRefGoogle Scholar
  35. 35.
    Bogdanova, YuG: The Adhesion and its Role in the Strength of Polymer Composites. Moscow University Press, Moscow (2010)Google Scholar
  36. 36.
    Pratton, M.: Introduction to Surface Physics, p. 256. Izhevsk, NIC RHD (2000)Google Scholar
  37. 37.
    Roldugin, V.I.: Physical Chemistry of Surface, p. 586. Dolgoprudnyy, Intellect (2008)Google Scholar
  38. 38.
    Vaz, C.A.F., Bland, J.A.C., Lauhoff, G.: Magnetism in ultrathin film structures. Rep. Prog. Phys. 71 (2008)Google Scholar
  39. 39.
    Lipatov, Y.S.: The Colloidal Polymer Physics, p. 344. Kiev, Nauka, Dumka (1984)Google Scholar
  40. 40.
    Berlin, A.A., Basin, V.E.: Fundamentals of Polymer Adhesion, p. 319. AM, Chemistry (1969)Google Scholar
  41. 41.
    Proutorov, E.V., Maksimova, O.G., Maksimov, A.V.: Simulation of the adhesion contact on the interface of polymer-metal. J. Phys: Conf. Ser. 633, 012044 (2015)Google Scholar
  42. 42.
    Allen, M.P., Tildesley, D.J.: Computer Simulation of Liquids. Clarendon, Oxford (1987)zbMATHGoogle Scholar
  43. 43.
    Binder, K., Levek, D., Weiss, J., et al.: Monte-Carlo Methods in Statistical Physics. Springer, Berlin (1979)CrossRefGoogle Scholar
  44. 44.
    Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equations of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1091 (1953)CrossRefGoogle Scholar
  45. 45.
    Yakovlev, A.D.: Chemicals and Coatings Technology, p. 448. SPb., HIMIZDAT (2008)Google Scholar
  46. 46.
    Gerasimov, R.A., Maksimov, A.V., Petrova, T.O., Maksimova, O.G.: Ordering and the relaxation properties of macromolecules in ferroelectric polymer films. Phys. Solid State 54(5), 1002–1004 (2012)CrossRefGoogle Scholar
  47. 47.
    Gerasimov, R.A., Eremeyev, V.A., Petrova, T.O., Egorov, V.I., Maksimova, O.G., Maksimov, A.V.: Computer simulation of the mechanical properties of metamaterials. J. Phys: Conf. Ser. 738(1), 012100 (2016)Google Scholar
  48. 48.
    Gotlib, Y.Y., Maximov, A.V.: A theory of orientational ordering in two-dimemsional multichain polymer systems with dipole interaction. Polym. Sci. A 34(11), 902–907 (1992)Google Scholar
  49. 49.
    Maksimov, A.V., Pavlov, G.M.: The molecular orientational order in surface layers of polymer films. Polym. Sci. A 49(7), 828–836 (2007)CrossRefGoogle Scholar
  50. 50.
    Maksimov, A.V., Gerasimov, R.A.: Anisotropic models of polymer ferroelectrics. Phys. Solid State 51(7), 1365–1369 (2009)CrossRefGoogle Scholar
  51. 51.
    Maksimov, A.V., Gerasimov, R.A., Maksimova, O.G.: Ordering and large-scale relaxation properties of macromolecules in ferroelectric polymer films. Ferroelectrics 432(1), 32–40 (2012)CrossRefGoogle Scholar
  52. 52.
    Petrova, T.O., Maksimova, O.G., Gerasimov, R.A., Maksimov, A.V.: Application of analytical and numerical methods to simulation of systems with orientation interactions. Phys. Solid State 54(5), 937–939 (2012)CrossRefGoogle Scholar
  53. 53.
    Maksimova, O.G., Maksimov, A.V.: Orientational order in two-dimensional polymer systems as described in terms of the Vaks-Larkin model. Polym. Sci. A 45(9), 1476–1485 (2003)Google Scholar
  54. 54.
    Bakingem, E.: Basic theory of intermolecular forces. Application to the small molecules. In: Pullman, B. (ed.) Intermolecular interactions: from diatomic molecules to biopolymers, pp. 9–98. Moscow, Mir (1981)Google Scholar
  55. 55.
    Askadskii, A.A.: Deformation of Polymers. Khimiya, Moscow (1973)Google Scholar
  56. 56.
    Cui Lian, XuXu, JiXin, Che, Zelong, He, Huijie, Xue, Tianquan, Lv: Properties of phase transformation of ferroelectric thin films with surface layers. J. Mod. Phys. 2, 1037–1040 (2011)CrossRefGoogle Scholar
  57. 57.
    Landau, L.D.: Theoretical Physics. Nauka. VIII, Moscow (1982)Google Scholar
  58. 58.
    Sonin, A.S.: Introduction to the Physics of Liquid Crystals. Nauka, Moscow (1983)Google Scholar
  59. 59.
    Landau, L.D., Lifschic, E.M.: Statistical Physics. Nauka. V, Moscow (1976)Google Scholar
  60. 60.
    Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solid. Struct. 14(6), 431–440 (1978)CrossRefzbMATHGoogle Scholar
  62. 62.
    Duan, H.L., Wang, J., Huang, Z.P., Karihaloo, B.L.: Size-dependent effective elastic constants of solids containing nanoinhomogeneities with interface stress. J. Mech. Phys. Solids 53, 1574–1596 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Duan, H.L., Wang, J., Karihaloo, B.L.: Theory of elasticity at the nanoscale. In: Aref, H., Van der Giessen, E. (eds.) Advances in Applied Mechanics, vol. 42, pp. 1–68. Elsevier, Amsterdam (2008)Google Scholar
  64. 64.
    Duan, H.L., Wang, J., Karihaloo, B.L., Huang, Z.P.: Nanoporous materials can be made stiffer than non-porous counterpartsby surface modification. Acta Mater. 54, 2983–2990 (2006)CrossRefGoogle Scholar
  65. 65.
    Eremeyev, V.A.: On effective properties of materials at the nano- and microscalesconsidering surface effects. Acta Mech. 227, 29–42 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Javili, A., McBride, A., Steinmann, P.: Thermomechanics of solids with lower-dimensional energetics: on the importanceof surface, interface, and curve structures at the nanoscale. A unifying review. Appl. Mech. Rev. 65, 010802–1–31 (2012)Google Scholar
  67. 67.
    Zhu, H.X., Wang, J.X., Karihaloo, B.L.: Effects of surface and initial stresses on the bending stiffness of trilayer plates and nanofilms. J. Mech. Mater. Struct. 4, 589–604 (2009)CrossRefGoogle Scholar
  68. 68.
    Chen, C., Shi, Y., Zhang, Y., Zhu, J., Yan, Y.: Size dependence of Young’s modulus in ZnO nanowires. Phys. Rev. Lett. 96, 075505 (2006)CrossRefGoogle Scholar
  69. 69.
    Cuenot, S., Frétigny, C., Demoustier-Champagne, S., Nysten, B.: Surface tension effect on the mechanical properties of nanomaterials measured by atomic force microscopy. Phys. Rev. B 69, 165410 (2004)CrossRefGoogle Scholar
  70. 70.
    Jing, G.Y., Duan, H.L., Sun, X.M., Zhang, Z.S., Xu, J., Li, Y.D., Wang, J.X., Yu, D.P.: Surface effects on elastic properties of silver nanowires: contact atomic-force microscopy. Phys. Rev. B 73, 235409–6 (2006)Google Scholar
  71. 71.
    Kim, C., Ru, C., Schiavone, P.: A clarification of the role of crack-tip conditions in linear elasticity with surface effects. Math. Mech. Solids 18, 59–66 (2013)MathSciNetCrossRefGoogle Scholar
  72. 72.
    Kim, C.I., Schiavone, P., Ru, C.Q.: Effect of surface elasticity on an interface crack in plane deformations. Proc. R. Soc. A 467, 3530–3549 (2011)Google Scholar
  73. 73.
    Mishuris, G.S.: Interface crack and nonideal interface concept (Mode III). Int. J. Fract. 107, 279–296 (2001)CrossRefGoogle Scholar
  74. 74.
    Mishuris, G.S., Kuhn, G.: Asymptotic behaviour of the elastic solution near the tip of a crack situated at a nonidealinterface. ZAMM 81, 811–826 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  75. 75.
    Arroyo, M., Belytschko, T.: An atomistic-based finite deformation membrane for single layer crystalline films. J. Mech. Phys. Solids 50, 1941–1977 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  76. 76.
    Miller, R.E., Shenoy, V.B.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139 (2000)CrossRefGoogle Scholar
  77. 77.
    Sfyris, D., Sfyris, G., Galiotis, C.: Curvature dependent surface energy for a free standing monolayer graphene: someclosed form solutions of the non-linear theory. Int. J. Non-Linear Mech. 67, 186–197 (2014)CrossRefGoogle Scholar
  78. 78.
    Shenoy, V.B.: Atomistic calculations of elastic properties of metallic fcc crystal surfaces. Phys. Rev. B 71, 094104 (2005)CrossRefGoogle Scholar
  79. 79.
    Steigmann, D.J., Ogden, R.W.: Plane deformations of elastic solids with intrinsic boundary elasticity. Proc. R. Soc. A 453, 853–877 (1997)Google Scholar
  80. 80.
    Steigmann, D.J., Ogden, R.W.: Elastic surface–substrate interactions. Proc. R. Soc. A 455, 437–474 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  81. 81.
    Povstenko, Y.: Mathematical modeling of phenomena caused by surface stresses in solids. In: Altenbach, H., Morozov, N.F. (eds.) Surface Effects in Solid Mechanics, pp. 135–153. Springer, Berlin (2013)CrossRefGoogle Scholar
  82. 82.
    Rubin, M., Benveniste, Y.: A Cosserat shell model for interphases in elastic media. J. Mech. Phys. Solids 52, 1023–1052 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  83. 83.
    dell’Isola, F., Steigmann, D., Della Corte, A.: Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response. Appl. Mech. Rev. 67(6), 060804-060804-21 (2016)Google Scholar
  84. 84.
    dell’Isola, F., Giorgio, I., Pawlikowski, M., Rizzi, N.L.: Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium. Proc. Roy. Soc. A 472(2185) (2016). doi:10.1098/rspa.2015.0790Google Scholar
  85. 85.
    Scerrato, D., Giorgio, I., Rizzi, N.L.: Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations. Z. Angew. Math. Phys. 67(3), 1–19 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  86. 86.
    Scerrato, D., Zhurba Eremeeva, I. A., Lekszycki, T., and Rizzi, N. L.: On the effect of shear stiffness on the plane deformation of linear second gradient pantographic sheets, Z. Angew. Math. Mech. (2016). doi:10.1002/zamm.201600066Google Scholar
  87. 87.
    Leonova, T.M., Kastro, R.A.: The dielectric properties of the MIS structures based on aluminum oxide. Materials of the XII-th International Conference “Physics of Dielectrics”, St. Petersburg (2011)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Southern Federal UniversityRostov-on-DonRussian Federation
  2. 2.Cherepovets State UniversityCherepovetsRussian Federation

Personalised recommendations