Elasticity of Spider dragline Silks Viewed as Nematics: Yielding Induced by Isotropic-Nematic Phase Transition

Spider dragline silk shows well-known outstanding mechanical properties. However, its sigmoidal shape of the measured stress-strain curves (i.e. the yield) can not be described by classical polymer theories and recent hierarchical chain model. To solve the long lasting problem, we generalized the Maier-Saupe theory of nematics to construct an elastic model for the polypeptide chain network of the dragline silk. The comprehensive agreement between theory and experiments on the stress-strain curve strongly indicates the dragline silks to belong to liquid crystal elastomers. Especially, the remarkable yielding elasticity of the silk is understood for the first time as the force-induced isotropic-nematic phase transition of the chain network. Our theory also predicts a drop of the stress in supercontracted dragline silk, an early found effect of humidity on the mechanical property in many silks.

Spider dragline silks (SDSs), the main structural web silk regarded as the "spider's lifeline", exhibit fascinating mechanical properties, such as a tactful combination of high tensile strength and high extensibility [1], thus showing a remarkably sigmoidal shape of the measured stress-strain curves [2]. Several experimental studies have been carried out to determine the supra-molecular structure organization of the SDS [3,4,5,6,7] and tried to produce mimic silks with similar properties [8]. It is now widely accepted that SDSs are semicrystalline polymers with β-sheet nanocrystals embedded in amorphous region, which is a polypeptide chain network [9,10,11]; see Fig. 1(a). However, the deformation mechanism, which is essential for understanding the SDS's extraodinary mechanical properties and mimicking the silk, is still in intense debate [3,11,12,13,14].
On the theoretical side, to understand the exceptional mechanical properties of SDS is of longstanding interest, and many models have been proposed [4,9,11,12] and some insights attained [4,9,11,12]. For example, the model by Termonia [9] treated the SDS as a hydrogen-bonded amorphous region embedded with stiff crystals as cross-links. In the interfacial region, an extremely high modulus is required to get the dragline's overall behavior on deformation. While, in the model of Porter and Vallrath [11,12], parameters linking to chemical compositions and morphological order were used to interpret thermo-mechanical properties. But some parameters such as ordered/disordered fractions are difficult to be obtained from experiments. A recent model [4] connecting deformations on macroscopic and molecular length scales still did not consider the change of the orientation of nanocomposites during deformation. Especially, as pointed out by Vehoff at al. recently [13], basic polymer theories such as the freely jointed chain, the freely rotating chain and the worm-like chain, as well as a hierarchical chain model of spider capture silk [15] can not reproduce the sigmoidal shape or even the steep initial regime of the spider dragline silk [ Fig. 2(a)] [13]. In one word, a unified description for SDS as a model biomaterial still seems to be lacking.
Quite a few works [2, 3,7,8,16,17,18] have pointed out that spider silk is liquid crystalline material and liquid crystal (LC) phase plays a vital role in both its spinning process and mechanical properties. In the spinning process, the liquid crystalline 'spinning dope' helps spider to control the folding and crystallization of the main protein constituents at benign condition (close to ambient temperatures and pressures using water as solvent) [2, 16,17]. The liquid crystalline phase also plays an important role in the solid silk's properties [18]. For instance, several works have found out that the orientation of nanocomposites can affect SDS's mechanical properties significantly [3,7,14]. A recent experiment also suggested the existence of conformational transition and the liquid crystalline state of regenerated silk fibroin in water [8]. Therefore, to present an analytically tractable LC model of the SDS that can catch the main physical factors is a current challenge to theorists. In this work, we generalized the Maier-Saupe theory [19] of nematic LC to construct an elastic model for the polypeptide chain network of the SDS. We show that on deformation the SDS undergoes significant changes with orientation of the chain network increased and the dimension of the silk along force direction elongated. The comprehensive agreement between theory and experiments on the stress-strain curve strongly indicates the SDSs to belong to LC elastomers, described as a new class of matter recently [20]. Especially, the remarkable yielding elasticity of the SDS is understood for the first time as the force-induced isotropic-nematic phase transition of the chain network and the self-consistently obtained yield point agrees with experimental data well. The present theory also predicts a drop of the stress in supercontracted SDS, an early found effect of humidity on the mechanical properties in many silks [11,13,21].
We take the polypeptide chain network in the amorphous region of the SDS as a molecular LC field with each chain section corresponding to a mesogenic molecule; see Fig. 1. Because the SDS's high extensibility results primarily from the disordered region [4,5,6], and many experiments showed that the deformation of the crystals is at least a factor of 10 smaller than that of the bulk [3] and that the orientation of the β-sheets is almost unchanged (usually very high) under stress [14,22], we can neglect the deformations and rotations of the β-sheet crystals in current work.
Following the LC continuum theory in the absence of forces, the potential of a mesogenic molecule takes the Maier-Saupe interaction form [19] where θ is the angle between the long axis of the molecule and the silk axis (the z-axis), which is also the direction ofn [ Fig. 1 (b)], a is the strength of the mean field, and S is the orientation order parameter of the LC, defined as the average of second Legendre We notice that the Maier-Saupe potential has been used by Pincus and de Gennes in investigating LC phase transition in a polypeptide system [24]. When a uniform force field f along z-axis is applied, the potential of a molecule is written as where l denotes the length of the mesogenic molecule.
From the definition of the order parameter S, we get a self-consistency equation with α = f l/k B T . The solution of the above equation may not be unique, in order to obtain physically sound solution we still need the requirement of minimization of the free energy given by where Z is the partition function Z = 1 −1 e −U (cos θ)/k B T d cos θ, and the second term at the right-hand side corrects for the double counting arising from the mean field method [20].
We calculate the orientation function S numerically at temperature T * = T /T ni and force f and show the results in Fig. 2(b). Here T ni = a/(4.541k B ) is the isotropic-nematic transition temperature in the absence of forces [19]. We see that, at temperatures below T ni the molecules have spontaneous nematic order, and the force does not induce further order significantly. While for the molecules initially in paranematic states, the applied force field will induce a first-order phase transition, which means S jumps discontinuously to a higher value at a certain critical force f C (T * ). At even higher temperatures, nematic field is weaker and the effect of the force is less dramatic. Interestingly, the α − S curves at different temperatures are qualitatively similar to the stress-orientation curves given by a much more complex nematic elastomer theory [25] (Fig. 5 in Ref. [25]).
To compare with the mechanical experiments of the SDS, we give the expressions for stress and strain in our theoretical framework. Apparently, the stress σ of a bulk is σ ≡ F/A = Nf : is the length of the bulk along z-axis when the force field f is applied and we can take it as L(f ) = l| cos θ| , and L 0 = L(f = 0) = l/2. Then the strain ε is ε = 2 | cos θ| − 1. We show ε versus σ at different temperatures in Fig. 2(c). We see that at temperatures below T ni , the strain grows smoothly with the stress. While for temperatures just above T ni , the strain grows with the stress in almost a linear way under small forces, and then a jump in the strain occurs at the critical force f C (T * ), after which the strain increases smoothly with the stress again. At even higher temperatures, the jump is replaced by a smooth increase in strain, but there is a plateau in a certain range of force.
In our model, the reduced temperature T * is an essential parameter, and we need to choose a proper value for it in order to predict the stress-strain curve of the SDS. Experiments showed that the solution from which the SDS was drawn was in liquid crystalline state at ambient temperature and pressure [2,8,16,17], while the orientation in the amorphous region of solid silk was very low [14]. Thus, we assign the isotropic-nematic transition temperature T ni of the cross-linked chain network slightly lower than the room temperature T r . Namely, the SDS is in paranematic state at ambient temperature and T * is just above 1. The σ − ε curves in the paranematic states in our model indeed exhibit main features of the stress-strain relation of the SDS: there is a linear increase in stress with strain at small values, and then at a certain strain and afterwards, the material becomes softer with lower Young's modulus [18,26]. We then reveal the beginning of the isotropic-nematic phase transition as the yield point. We see that the curves with T * = 1.01 and 1.02 agree well with the measured curve in the beginning linear region and the yield point. But because the actual deformation process of the silk is more complicated and additional factors may be involved, such as the viscoelasticity, defects and poly-domain effect [20],the curve with T * = 1.1 agrees better with the overall stress-strain measurement topologically. We calculate the yield point by choosing curves with T * = 1.01 and 1.02. We get the yield strain ε y ≈ 0.04, the yield stress σ y = αNk B T /l ≈ 8.4MPa, and the Young's modulus at the linear region E ≡ σ y /ε y ≈ 210MPa, given α ∼ 0.2, N/l ∼ 10nm −3 , and k B T r ∼ 4.1pNnm. These results agree with experimental data [11,26] [Fig. 2(d)] satisfactorily.
We notice that our results agree much better with the mechanical properties of the silks with low spinning speed. That is because the spinning speed can induce a low orientation in the amorphous region which makes the silk more stiff. Since this additional order in the amorphous region is not taken into account in the current work, the silks in our model are generally a bit softer than the silks with high spinning speed. Another thing needs pointing out is that we predict there is a phase transition at the yield point which is supported by a few experiments. For instance, in the polarized FTIR spectroscopy experiment by Papadopoulos et al., the orientation of some components in the amorphous region increased by 0.3 when the strain reached 24%. Besides, we would like to discuss about the isotropicnematic transition temperature T ni of the cross-linked chain network. If we choose T * = 1.1 in our calculation, the transition temperature T ni ≈ 270K, which is reasonable.
In addition to describing the stress-strain relation of the SDS, our simple theory can also qualitatively account for the drop of the stress in the wet SDS, i.e. the supercontracted SDS. We take L 0 and R 0 as the initial length and radius of the silk, and L and R as those under stress. Under the assumption of volume conservation we have πR 2 0 L 0 = πR 2 L, so R/R 0 = 1/(1 + ε). The free energy of the bulk can be written as where U is the internal energy of the bulk, f ext is the external force on the bulk and γ is the surface energy coefficient. Minimizing F with respect to ε, we get When the silk is immersed in water, the surface energy coefficient γ increases, so with the same stress σ we get a bigger strain. Thus our theory can predict the softening of supercontracted silk, an effect observed in many experiments [11,13,21,28,29].
In conclusion, we investigate the mechanical properties of the SDS from a point of view of the LC continuum theory. We found out that the deformation process is a force-induced  The z-axis is along the silk axis. (b). The coordinate system of the nematics.n is the director of the nematics,û is the director of the mesogenic molecule, and θ is the angle between the long axis of the molecule and the silk axis z.