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Hard-Needle Elastomer in One Spatial Dimension

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Abstract

We perform exact statistical mechanics calculations for a system of elongated objects (hard needles) that are restricted to translate along a line and rotate within a plane, and that interact via both excluded-volume steric repulsion and harmonic elastic forces between neighbors. This system represents a one-dimensional model of a liquid crystal elastomer, and has a zero-tension critical point that we describe using the transfer-matrix method. In the absence of elastic interactions, we build on previous results by Kantor and Kardar, and find that the nematic order parameter Q decays linearly with tension \(\boldsymbol{\sigma}\). In the presence of elastic interactions, the system exhibits a standard universal scaling form, with \(\boldsymbol{Q / \vert \sigma \vert}\) being a function of the rescaled elastic energy constant \(\boldsymbol{k / \vert \sigma \vert ^\Delta}\), where \(\boldsymbol \Delta\) is a critical exponent equal to 2 for this model. At zero tension, simple scaling arguments lead to the asymptotic behavior \(\boldsymbol{Q \sim k^{1/\Delta }}\), which does not depend on the equilibrium distance of the springs in this model.

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Notes

  1. We used \(\ell ^{-N}\) in the spatial measure so that Y is dimensionless. In classical statistical mechanics, it is usual to consider the measure in phase space \(d \mu = h^{-N} \prod _i dx_i d p_i\), with the inclusion of Planck’s constant h, where \(p_i\) is the momentum of particle i. This ensures a dimensionless partition function and agreement with the classical limit of an analogous quantum system. In our case, we could combine \(\ell\) with h so that the contribution from the momentum variables is also dimensionless.

  2. Note that Eq. (24) is valid for bulk particles. For boundary particles, \(< X_{k_i} > = (\sum _{\beta =1}^M \nu _{\beta \, \alpha ^*}X_{\beta })/(\sum _{\beta =1}^M \nu _{\beta \, \alpha ^*})\).

  3. The average spacing s can be calculated from a derivative of the free energy as \(s/\ell = \partial (\beta f) / \partial \,\tau\). For the hard-needle elastomer, this calculation results in \(s/\ell = -(1/\lambda _{\alpha ^*}) \partial \lambda _{\alpha ^*} / \partial \, \tau + a - \tau / \Lambda\). The first term yields the expected scaling behavior, whereas the last term provides important corrections when \(\Lambda \sim \tau ^2\).

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Acknowledgements

DBL thanks the financial support through FAPESP grants # 2021/14285-3 and # 2022/09615-7. AP is grateful to ICTP-SAIFR for a 2-month visiting grant.

Funding

This work was partially supported by FAPESP, through grants # 2021/14285-3 and # 2022/09615-7.

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Authors and Affiliations

  1. ICTP South American Institute for Fundamental Research, Rua Dr. Bento Teobaldo Ferraz, São Paulo, 01140-070, SP, Brazil

    Danilo B. Liarte

  2. Institute of Theoretical Physics, São Paulo State University, Rua Dr. Bento Teobaldo Ferraz, São Paulo, 01140-070, SP, Brazil

    Danilo B. Liarte

  3. CNR–Istituto dei Sistemi Complessi, Dipartimento di Fisica, Università Sapienza, Piazzale Aldo Moro, Rome, 00185, Italy

    Alberto Petri

  4. Institute of Physics, University of São Paulo, Rua do Matão, São Paulo, 05508-090, SP, Brazil

    Silvio R. Salinas

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Correspondence to Danilo B. Liarte.

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Liarte, D.B., Petri, A. & Salinas, S.R. Hard-Needle Elastomer in One Spatial Dimension. Braz J Phys 53, 73 (2023). https://doi.org/10.1007/s13538-023-01289-7

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