Anomalous elasticity of nematic and critically soft elastomers

Olaf Stenull; T. C. Lubensky

doi:10.1103/PhysRevE.69.021807

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Anomalous elasticity of nematic and critically soft elastomers

Olaf Stenull and T. C. Lubensky

Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

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Phys. Rev. E 69, 021807 – Published 27 February, 2004

DOI: https://doi.org/10.1103/PhysRevE.69.021807

Abstract

Uniaxial elastomers are characterized by five elastic constants. If their elastic modulus $C_{5}$ describing the energy of shear strains in planes containing the anisotropy axis vanishes, they are said to be soft. In spatial dimensions d less than or equal to 3, soft elastomers exhibit anomalous elasticity with certain length-scale-dependent bending moduli that diverge and shear moduli that vanish at large length scales. Using renormalized field theory at $d = 3$ and to first order in $ɛ = 3 - d,$ we calculate critical exponents and other properties characterizing the anomalous elasticity of two soft systems: (i) nematic elastomers in which softness is a manifestation of a Goldstone mode induced by the spontaneous symmetry breaking associated with a transition from an isotropic state to a nematic state, and (ii) a particular version of what we call a critically soft elastomer in which $C_{5} = 0$ corresponds to a critical point terminating the stability regime of a uniaxial elastomer with $C_{5} > 0.$

References (33)

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P. G. de Gennes, in Liquid Crystals of One and Two-Dimensional Order, edited by W. Helfrich and G. Heppke (Springer, New York, 1980), p. 231.
E. M. Terentjev, J. Phys.: Condens. Matter 11, R239 (1999).
P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon, Oxford, 1993).
S. Chandrasekhar, Liquid Crystals (Cambridge University Press, Cambridge, 1992).
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M. Warner and E. M. Terentjev, Prog. Polym. Sci. 21, 853 (1996).
T. C. Lubensky, R. Mukhopadhyay, L. Radzihovsky, and X. Xing, Phys. Rev. E 66, 011702 (2002).
J. Küpfer and H. Finkelmann, Macromol. Chem. Phys. 195, 1353 (1994).
H. Finkelmann, E. Nishikawa, G. G. Pereira, and M. Warner, Phys. Rev. Lett. 87, 015501 (2001).
H. Finkelmann and H. Wermter, Polym. Mater. Sci. Eng. 82, 319 (2000).
M. Hébert, R. Kant, and P.-G. de Gennes, J. Phys. I 7, 909 (1997).
D. L. T. III, P. Keller, J. Naciri, R. Pink, H. Jeon, D. Shenoy, and B. Ratna, Macromolecules 34, 5868 (2001).
L. Golubović and T. C. Lubensky, Phys. Rev. Lett. 63, 1082 (1989).
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G. C. Verwey, M. Warner, and E. M. Terentjev, J. Phys. II 6, 1273 (1996).
M. Warner, J. Mech. Phys. Solids 47, 1355 (1999).
G. Grinstein and R. A. Pelcovits, Phys. Rev. Lett. 47, 856 (1981); Phys. Rev. A 26, 915 (1982).
See, e.g., D. J. Amit, Field Theory, the Renormalization Group, and Critical Phenomena (World Scientific, Singapore, 1984); J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Clarendon, Oxford, 1989).
O. Stenull and T. C. Lubensky, Europhys. Lett. 61, 776 (2003); ibid.see also X. Xing and L. Radzihovsky, 61, 769 (2003).
X. Xing and L. Radzihovsky, Phys. Rev. Lett. 90, 168301 (2003).
L. D. Landau and E. M. Lifshitz, Theory of Elasticity, 3rd ed. (Pergamon, New York, 1986).
P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge University Press, Cambridge, 1995).
Here and in the following, we adopt Einstein’s summation convention.
Provided, of course, there are no aligning fields.
For convenience, we set the Boltzmann constant equal to 1.
The $|_{0}$ indicates that bare quantities are kept fixed while taking the derivatives.
In our notation, these shear moduli are given by $μ_{L} = (ω {+ f + 2 / d}_{⊥} - 2 g) / 4$ and $μ = 1.$
X. Xing, R. Mukhopadhyay, T. C. Lubensky, and R. Radzihovsky, Phys. Rev. E 68, 021108 (2003)..

[c1] H. Finkelmann, H. J. Koch, and G. Rehage, Makromol. Chem., Rapid Commun. 2, 317 (1981).

[c2] P. G. de Gennes, in Liquid Crystals of One and Two-Dimensional Order, edited by W. Helfrich and G. Heppke (Springer, New York, 1980), p. 231.

[c3] E. M. Terentjev, J. Phys.: Condens. Matter 11, R239 (1999).

[c4] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon, Oxford, 1993).

[c5] S. Chandrasekhar, Liquid Crystals (Cambridge University Press, Cambridge, 1992).

[c6] S. T. Kim and H. Finkelmann, Macromol. Rapid Commun. 22, 429 (2001).

[c7] R. Zentel, Angew. Chem. Adv. Mater. 101, 1437 (1989).

[c8] C. N. Bowman and C. A. Guymon, MRS Bull. 15 (1997).

[c9] Liquid Crystals in Complex Geometries, edited by G. P. Crawford and S. Zumer (Taylor and Francis, London, 1996).

[c10] M. Warner and E. M. Terentjev, Prog. Polym. Sci. 21, 853 (1996).

[c11] T. C. Lubensky, R. Mukhopadhyay, L. Radzihovsky, and X. Xing, Phys. Rev. E 66, 011702 (2002).

[c12] J. Küpfer and H. Finkelmann, Macromol. Chem. Phys. 195, 1353 (1994).

[c13] H. Finkelmann, E. Nishikawa, G. G. Pereira, and M. Warner, Phys. Rev. Lett. 87, 015501 (2001).

[c14] H. Finkelmann and H. Wermter, Polym. Mater. Sci. Eng. 82, 319 (2000).

[c15] M. Hébert, R. Kant, and P.-G. de Gennes, J. Phys. I 7, 909 (1997).

[c16] D. L. T. III, P. Keller, J. Naciri, R. Pink, H. Jeon, D. Shenoy, and B. Ratna, Macromolecules 34, 5868 (2001).

[c17] L. Golubović and T. C. Lubensky, Phys. Rev. Lett. 63, 1082 (1989).

[c18] Peter D. Olmsted, J. Phys. II 4, 2215 (1994).

[c19] H. Finkelmann, I. Kundler, E. M. Terentjev, and M. Warner, J. Phys. II 7, 1059 (1997).

[c20] G. C. Verwey, M. Warner, and E. M. Terentjev, J. Phys. II 6, 1273 (1996).

[c21] M. Warner, J. Mech. Phys. Solids 47, 1355 (1999).

[c22] G. Grinstein and R. A. Pelcovits, Phys. Rev. Lett. 47, 856 (1981); Phys. Rev. A 26, 915 (1982).

[c23] See, e.g., D. J. Amit, Field Theory, the Renormalization Group, and Critical Phenomena (World Scientific, Singapore, 1984); J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Clarendon, Oxford, 1989).

[c24] O. Stenull and T. C. Lubensky, Europhys. Lett. 61, 776 (2003); ibid.see also X. Xing and L. Radzihovsky, 61, 769 (2003).

[c25] X. Xing and L. Radzihovsky, Phys. Rev. Lett. 90, 168301 (2003).

[c26] L. D. Landau and E. M. Lifshitz, Theory of Elasticity, 3rd ed. (Pergamon, New York, 1986).

[c27] P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge University Press, Cambridge, 1995).

[c28] Here and in the following, we adopt Einstein’s summation convention.

[c29] Provided, of course, there are no aligning fields.

[c30] For convenience, we set the Boltzmann constant equal to 1.

[c31] The $|_{0}$ indicates that bare quantities are kept fixed while taking the derivatives.

[c32] In our notation, these shear moduli are given by $μ_{L} = (ω {+ f + 2 / d}_{⊥} - 2 g) / 4$ and $μ = 1.$

[c33] X. Xing, R. Mukhopadhyay, T. C. Lubensky, and R. Radzihovsky, Phys. Rev. E 68, 021108 (2003)..

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Anomalous elasticity of nematic and critically soft elastomers

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