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

Lin-ying Cui<EMAIL_ADDRESS>Department of physics, Tsinghua
University, Beijing, 100084, China  Fei Liu  Center for Advanced Study,
Tsinghua University, Beijing, 100084, China  Zhong-can Ou-Yang  Center for
Advanced Study, Tsinghua University, Beijing, 100084, China  Institute of
Theoretical Physics, The Chinese Academy of Sciences, P.O.Box 2735 Beijing
100190, China

###### Abstract

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.

###### pacs:

81.05.Lg, 81.40.Jj, 82.35.Pq, 61.30.Dk

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 gosline ,
thus showing a remarkably sigmoidal shape of the measured stress-strain curves
Vollrath2 . Several experimental studies have been carried out to determine
the supra-molecular structure organization of the SDS Anja ; Hayashi ;
Oroudjev ; Simmons ; Krasnov  and tried to produce mimic silks with similar
properties Li . It is now widely accepted that SDSs are semicrystalline
polymers with $\beta$-sheet nanocrystals embedded in amorphous region, which
is a polypeptide chain network Termonia ; Vollrath1 ; Lefevre ; 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 Anja ; Porter ; Vollrath1 ; Vehoff ; Papa .

On the theoretical side, to understand the exceptional mechanical properties
of SDS is of longstanding interest, and many models have been proposed
Termonia ; Porter ; Vollrath1 ; Krasnov  and some insights attained Termonia ;
Porter ; Vollrath1 ; Krasnov . For example, the model by Termonia Termonia
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 Porter ; Vollrath1 , 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
Krasnov  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 Vehoff ,
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 haijun  can not reproduce the sigmoidal shape or even the steep
initial regime of the spider dragline silk [ Fig. 2(a)] Vehoff . In one word,
a unified description for SDS as a model biomaterial still seems to be
lacking.

Quite a few works Vollrath2 ; Knight ; Anja ; Simmons ; Li ; Kerkam ; willcox
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) Vollrath2 ; Kerkam ; willcox . The
liquid crystalline phase also plays an important role in the solid silk’s
properties Knight . For instance, several works have found out that the
orientation of nanocomposites can affect SDS’s mechanical properties
significantly Anja ; Papa ; Simmons . A recent experiment also suggested the
existence of conformational transition and the liquid crystalline state of
regenerated silk fibroin in water Li . 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 Maier  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 warner . 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 Bell ; Vehoff ; Vollrath1 .

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 Krasnov ; Oroudjev ; Hayashi , and many experiments
showed that the deformation of the crystals is at least a factor of 10 smaller
than that of the bulk Anja  and that the orientation of the $\beta$-sheets is
almost unchanged (usually very high) under stress Philip ; Papa , we can
neglect the deformations and rotations of the $\beta$-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 Maier

$\displaystyle V(\cos\theta)=-aS(\frac{3}{2}\cos^{2}\theta-\frac{1}{2}),$ (1)

where $\theta$ is the angle between the long axis of the molecule and the silk
axis (the z-axis), which is also the direction of $\hat{\bf n}$ [ 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 polynomial
gennes

$\displaystyle
S=\left\langle\frac{3}{2}\cos^{2}\theta-\frac{1}{2}\right\rangle.$ (2)

We notice that the Maier-Saupe potential has been used by Pincus and de Gennes
in investigating LC phase transition in a polypeptide system Pincus . When a
uniform force field $\bf f$ along z-axis is applied, the potential of a
molecule is written as

$\displaystyle U(\cos\theta)=V-fl\cos\theta,$ (3)

where $l$ denotes the length of the mesogenic molecule.

From the definition of the order parameter $S$, we get a self-consistency
equation

$\displaystyle\begin{array}[]{lll}S&=&\int_{-1}^{1}(\frac{3}{2}\cos^{2}\theta-\frac{1}{2})\exp(\frac{3aS}{2{\rm
k_{B}T}}\cos^{2}\theta+\alpha\cos\theta)d\cos\theta\\\
&&\left/\int_{-1}^{1}\exp(\frac{3aS}{2{\rm
k_{B}T}}\cos^{2}\theta+\alpha\cos\theta)d\cos\theta\right.,\end{array}$ (6)

with $\alpha=fl/{\rm k_{B}}T$. 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

$\displaystyle F_{MS}=-{\rm k_{B}}T\ln Z+\frac{1}{2}aS^{2},$ (7)

where $Z$ is the partition function $Z=\int_{-1}^{1}e^{-U(\cos\theta)/{\rm
k_{B}}T}d\cos\theta$, and the second term at the right-hand side corrects for
the double counting arising from the mean field method warner .

We calculate the orientation function $S$ numerically at temperature
$T^{*}=T/T_{\rm ni}$ and force $f$ and show the results in Fig. 2(b). Here
$T_{\rm ni}=a/(4.541\rm k_{B})$ is the isotropic-nematic transition
temperature in the absence of forces Maier . We see that, at temperatures
below $T_{\rm ni}$ 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 $f_{\rm C}(T^{*})$. At even higher temperatures,
nematic field is weaker and the effect of the force is less dramatic.
Interestingly, the $\alpha-S$ curves at different temperatures are
qualitatively similar to the stress-orientation curves given by a much more
complex nematic elastomer theory warner2  (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
$\sigma$ of a bulk is $\sigma\equiv F/A=Nf$: $F$ is the force on the surface
of the bulk, $A$ is the area of the surface, and $N$ is the number of
molecules per area. The strain $\varepsilon$ of the bulk is defined as
$\varepsilon=[L(f)-L_{0}]/L_{0}$, where $L(f)$ is the length of the bulk along
z-axis when the force field $\bf f$ is applied and we can take it as
$L(f)=\langle l|\cos\theta|\rangle$, and $L_{0}=L(f=0)=l/2$. Then the strain
$\varepsilon$ is $\varepsilon=2\langle|\cos\theta|\rangle-1$. We show
$\varepsilon$ versus $\sigma$ at different temperatures in Fig. 2(c). We see
that at temperatures below $T_{\rm ni}$, the strain grows smoothly with the
stress. While for temperatures just above $T_{\rm ni}$, 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 $f_{\rm C}(T^{*})$, 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
Vollrath2 ; Kerkam ; willcox ; Li , while the orientation in the amorphous
region of solid silk was very low Papa . Thus, we assign the isotropic-nematic
transition temperature $T_{\rm ni}$ 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
$\sigma-\varepsilon$ 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
Knight ; Du . 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 warner ,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 $\varepsilon_{y}\approx 0.04$,
the yield stress $\sigma_{y}=\alpha N{\rm k_{B}}T/l\approx 8.4{\rm MPa}$, and
the Young’s modulus at the linear region
$E\equiv\sigma_{y}/\varepsilon_{y}\approx 210{\rm MPa}$, given $\alpha\sim
0.2$, $N/l\sim 10{\rm nm}^{-3}$, and ${\rm k_{B}}T_{r}\sim 4.1{\rm pNnm}$.
These results agree with experimental data Du ; Vollrath1  [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 isotropic-nematic transition
temperature $T_{\rm ni}$ of the cross-linked chain network. If we choose
$T^{*}=1.1$ in our calculation, the transition temperature $T_{\rm ni}\approx
270{\rm K}$, 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 $\pi R_{0}^{2}L_{0}=\pi R^{2}L$,
so $R/R_{0}=\sqrt{1/(1+\varepsilon)}$. The free energy of the bulk can be
written as

$\displaystyle\begin{array}[]{lll}F&=&U-f_{ext}(L-L_{0})+2\pi RL\gamma\\\
&=&U-f_{ext}(L-L_{0})+2\pi R_{0}L_{0}\gamma\sqrt{1+\varepsilon},\end{array}$
(10)

where $U$ is the internal energy of the bulk, $f_{ext}$ is the external force
on the bulk and $\gamma$ is the surface energy coefficient. Minimizing $F$
with respect to $\varepsilon$, we get

$\displaystyle\sigma=\frac{f}{\pi R_{0}^{2}}=\frac{1}{\pi
R_{0}^{2}L_{0}}\frac{\partial
U}{\partial\varepsilon}+\frac{\gamma}{R_{0}\sqrt{1+\varepsilon}}.$ (11)

When the silk is immersed in water, the surface energy coefficient $\gamma$
increases, so with the same stress $\sigma$ we get a bigger strain. Thus our
theory can predict the softening of supercontracted silk, an effect observed
in many experiments Bell ; Vehoff ; Vollrath1 ; work2 ; work3 .

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 isotropic-nematic phase transition process.
Remarkably, such a simple model with Maier-Saupe theory is able to reproduce
the stress-strain curve of the SDS, get the yield point, and qualitatively
interpret the drop of the stress in the supercontracted silk.

This work is supported by the National Innovation Research Project for
Undergraduates.

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Figure 1:  (a). A schematic diagram of the structure of the dragline silk. The
bold lines represent the $\beta$-sheet crystals, and the thin lines represent
the polypeptide chains in the amorphous region. The z-axis is along the silk
axis. (b). The coordinate system of the nematics. $\hat{\bf n}$ is the
director of the nematics, $\hat{\bf u}$ is the director of the mesogenic
molecule, and $\theta$ is the angle between the long axis of the molecule and
the silk axis $z$.  Figure 2: (Color online.) (a) Comparison of a typical
measured dragline silk’s stress-strain curve (black solid line) with
theoretical curves evaluated by the freely jointed chain (red dash dot line)
and the hierarchical chain model (olive dash line) [After T. Vehoff et al.].
(b) The orientation order parameter $S$ as a function of $\alpha$($=fl/{\rm
k_{B}}T$). (c) The stress-strain curves at different temperatures
$T^{*}(=T/T_{\rm ni})$. (d) The stress-strain curve of the SDS spinned with
the speed of $1{\rm mms}^{-1}$ [After N. Du et al.].