Bingyu Yu(於冰宇) Yuanchenxi Gao(高袁晨曦) Bin Zheng(鄭斌) Fanlong Meng(孟凡龍)Yu Fang(方羽) Fangfu Ye(葉方富) and Zhongcan Ouyang(歐陽(yáng)鐘燦)

1CAS Key Laboratory of Theoretical Physics,Institute of Theoretical Physics,Chinese Academy of Sciences,Beijing 100190,China

2School of Physical Sciences,University of Chinese Academy of Sciences,Beijing 100049,China

3Beijing 101 Middle School,Beijing 100091,China

4Wenzhou Institute,University of Chinese Academy of Sciences,Wenzhou 325000,China

5Beijing National Laboratory for Condensed Matter Physics,Institute of Physics,Chinese Academy of Sciences,Beijing 100190,China

6Oujiang Laboratory(Zhejiang Laboratory for Regenerative Medicine,Vision and Brain Health),Wenzhou 325000,China

Keywords: auxetic response,liquid crystal elastomers,phenomenological model

1. Introduction

Auxetic response is usually associated with materials with negative Poisson’s ratios. The origin lies in the change of microscopic porous geometry of such materials when subjected to an external stress. Due to their abnormal mechanic properties, materials exhibiting auxeticity possess many potential applications in various fields, such as sports equipment, architecture, aerospace, and biomedical materials.[1–4]Recently, Mistryet al.reported that liquid crystal elastomers in certain circumstances can also exhibit auxetic response,viz.,stretching along a direction perpendicular to the orientation of nematic order causes an increase(rather than decrease)of the elastomer thickness (see Fig. 1).[5]Liquid crystal elastomers(LCEs) are a new class of materials, which combine the orientation properties of liquid crystals with the elastic properties of elastomers.[6]Due to coupling of these two properties,such materials exhibit many novel properties, including soft or semi-soft elasticity.[7–9]In this article, we develop a simple theoretical model to explain the observed auxetic response of LCEs, and further determine by theoretical calculation the critical condition required for appearance of auxetic response.

2. Model and results

To describe the elastic properties of LCEs, a powerful theory, the so-called “neoclassical” theory, has been developed by Warner and Terentjev, which generalizes the microscopic theory for the rubber elasticity to include anisotropic orientational properties of liquid crystals.[6]Alternatively,one can develop, based on symmetry argument, phenomenological theories to describe LCEs. In both types of theories, two essential physical quantities are the orientational order tensor of liquid crystals,Q, and the deformation tensor of the elastomer,Λ= dR/dR0,whereRandR0are,respectively,the positions of a mass point after and before a deformation.

where the symbol Tr represents the trace of a matrix,the coefficientr′is a parameter related to temperature and shear modulus of elastomers, andwandvare, respectively, the coefficients for the third- and fourth-order terms. The coefficienth′represents the strength of frozen-in anisotropy introduced during the cross-linking process of preparing LCE samples,where we have assumed the anisotropy direction aligns along thez-direction,which is also the initial alignment direction of the nematic. Following the rescaling approach in Ref. [10],we can further simplify the above energy expression so as to reduce the parameter number. We thus obtain the following free energy expression:

withr=(v/w2)r′andh=v2/w3h′. Here,fanduhave been rescaled byv3/w4andv/w,respectively,from their counterparts in Eq. (1). Note that the above free energy expression now has only two independent parametersrandh. In the above theory, we have assumed the strain tensoruto be traceless, i.e.,uxx+uyy+uzz= 0. This traceless constraint is a good approximation of the incompressibility requirement,DetΛ=1, for small deformations. When considering large deformations, this traceless constraint does not quantitatively satisfy but still captures main characteristics of the incompressibility requirement. We therefore adopt this traceless constraint throughout the following parts to simplify calculations. A detailed theory following the exact incompressibility requirement will be studied in near future.

We first need to determine the equilibrium state of the free energy given in Eq. (2). Due to the presence of the linear term and the fact that the parameterrcan be negative,theu=0 state is not clearly the equilibrium state. To determine the equilibrium state in the absence of external stress,we can assume thatutakes a diagonal form,with

The value ofu0can then be determined accordingly for givenrandh.

We now proceed to investigate the elastic response of LCEs subjected to an external stress. Note that the new reference state is now the state withu=u0,rather than theu=0 state. When the sample is stretched along thex-direction,uxxgradually increases fromu0, and the other elements of the strain tensoruvary accordingly.As widely studied previously,a stress applied perpendicular to the anisotropy direction usually leads to the rotation of the nematic orientation, and such rotation,together with the boundary constraints,yields a stripe pattern with the nematic orientation alternating from stripe to stripe.[6]In other words,stretching along thex-direction usually causes the occurrence of nonzero off-diagonal elements ofu. However,in some circumstances,a perpendicular stress leads to no rotation of the nematic orientation,instead results in appearance of biaxial order.[12,13]In such cases,we can still assume thatutakes a diagonal form, but withuyybeing not equal touxx. The free energy densityfcan thus be written as

The value ofuyycan then be obtained by solving the above equation numerically. Figure 2 shows howuyyvaries withuxxfor variousrath= 1. For larger,uyydecreases withuxxmonotonically;for smallr,whenuxxincreases,uyyshows non-monotonic behavior,firstly decreasing,then rising up,and finally decreasing again. Such non-monotonic behavior corresponds exactly to LCEs’auxetic response reported recently by Mistryet al.[5]

Fig. 2. Dependence of uyy on uxx for various r at h=1. Note that the three curves start from different uxx because the value change of r results in the change of u0 as well.

After confirming the presence of the auxetic response,we proceed to investigate the exact condition under which the auxetic response may occur. As illustrated in Fig.2, for smallr,the curve ofuyy(uxx) has two extreme points, and the auxetic response starts from the local minimum point and ends at the local maximum point. As the relation betweenuyyanduxxis given by Eq. (3), the derivative?uyy/?uxxcan then be easily calculated,and the positions of the two extreme points can be obtained by solving?uyy/?uxx=,which is equivalent to where the-and+signs are for the local minimum and maximum points,respectively. Whenr=3/8,as shown by the red curve in Fig. 2, the two extreme points on theuyy(uxx) curve merge into one point,at which?2uyy/?2uxx=0. We can thus easily conclude that the condition required for the auxetic response to occur isr ＜rc(=3/8).

3. Discussion and conclusion

In conclusion,we have developed a minimal model to describe the auxetic response of liquid crystal elastomers, and obtained the condition required for the auxetic response to occur.Our theoretical result agrees well with the previous experimental results that liquid crystal elastomers are more likely to exhibit auxetic response at low temperatures.[12]Although being simplified,our model does capture the main feature that the auxetic response is related to the appearance of biaxiality of nematic alignment.[12]However, by assuming the offdiagonal elements of the strain tensor to be zero, the current work does not explain why a stress applied perpendicularly to the initial nematic orientation induces a modification of nematic order rather than a rotation of nematic orientation. Further investigation will be carried out in near future to explore the mechanisms inhibiting the rotation of nematic orientation.

Acknowledgements

We thank Guangle Du and Boyi Wang for helpful discussions. Y.G.and Y.F.thank the support of Beijing 101 Middle School.

Project supported by the National Natural Science Foundation of China(Grant No.22193032).