## SUMMARY

We investigate the rheological properties of microliter quantities of the
spinning material extracted *ex vivo* from the major ampullate gland of
a *Nephila clavipes* spider using two new micro-rheometric devices. A
sliding plate micro-rheometer is employed to measure the steady-state shear
viscosity of ∼1 μl samples of silk dope from individual biological
specimens. The steady shear viscosity of the spinning solution is found to be
highly shear-thinning, with a power-law index consistent with values expected
for liquid crystalline solutions. Calculations show that the viscosity of the
fluid decreases 10-fold as it flows through the narrow spinning canals of the
spider. By contrast, measurements in a microcapillary extensional rheometer
show that the transient extensional viscosity (i.e. the viscoelastic
resistance to stretching) of the spinning fluid increases more than 100-fold
during the spinning process. Quantifying the properties of native spinning
solutions provides new guidance for adjusting the spinning processes of
synthetic or genetically engineered silks to match those of the spider.

## Introduction

Over the past decade, numerous studies have shown that spider dragline silk has a number of unique material properties (Becker et al., 2003; Gosline et al., 1999; Shao and Vollrath, 2002), yet how exactly the spider processes its fiber remains unclear. Recent experiments with recombinant spider silk (Lazaris et al., 2002) and other studies performed with silkworms (Shao and Vollrath, 2002) show that careful control of the processing conditions for fiber spinning is key to obtaining superior mechanical properties in spun silks. Forced silking at different rates results in modified kinematics in the spinning canal and substantial changes in the mechanical properties of the resulting silk fibers (Perez-Rigueiro et al., 2005). A detailed microscopic study of spider silk spinning involving whole spider silk glands suggests that at least two physical processes are important (Knight and Vollrath, 1999): (1) shear-thinning of the silk solution in the ducts that deliver the fluid towards the spinneret (this is, in turn, related to the liquid crystallinity of the protein solution) and (2) pronounced elongation of the long protein chains during the spinning process results in a highly aligned microstructure after the silk dries.

To further understand this complex flow process it is essential to elucidate the rheological properties of the initial liquid spinning material, commonly referred to as `spinning dope' (Vollrath and Knight, 2001), that is stored in the spinning glands of the spider (Chen et al., 2002). Although the spinning dope is a concentrated aqueous solution containing 25-30 wt% protein, all rheological experiments to date have been performed with diluted solutions (typically <5 wt% of protein) (Chen et al., 2002).

Recently, processing experiments have been performed with reconstituted
silk solutions obtained from the silkworm *Bombyx mori*
(Jin and Kaplan, 2003).
Micellar solutions with ∼8 wt% silk were reconstituted using dialysis.
However, in order to produce spinnable fibers, a high-molecular-mass linear
polymer (polyethylene oxide with molecular mass of 0.9 ×10^{6} g
mol^{-1}) was added to the reconstituted solutions. This additional
component augments the `spinnability' of the dope by increasing the
extensional or `tensile' viscosity of the fluid and prevents capillary
break-up of the fluid jet. The resulting spun fibers exhibited morphological
features, such as increased birefringence and alignment, that are similar to
the native silk fiber (Magoshi et al.,
1994). The birefringence properties of raw spider silk have also
been observed and investigated by Knight and Vollrath
(Knight and Vollrath, 1999).
These experiments suggest that native silkworm and spider silk solutions
possess significant non-Newtonian fluid properties
(Chen et al., 2002;
Terry et al., 2004), however
more insight would be gained through direct rheological characterization of
the native, concentrated spider silk dope. For clarity, we provide in the
Appendix a glossary of some of the most important rheological concepts that
are essential for understanding the properties of this complex protein
solution.

Recently published results (Terry et
al., 2004) of bulk shear rheometry on larger scale samples of
silkworm silk solutions indicate strong viscoelastic properties for silk dope
with a zero-shear-rate viscosity on the order of 10^{3} Pa.s and a
critical rate of the onset of shear thinning on the order of 1 s^{-1}.
Phase separation and unstable flow conditions above this critical rate were
interpreted as indication for a shear-induced beta-sheet formation.

The volume of spinning dope that can be harvested from a single major
ampullate gland of *N. clavipes* is approximately 5-10 μl. Besides
the minute quantity available, the dope is viscous
(Willcox et al., 1996) and
tends to dry with time, making it difficult to obtain reliable viscometric
data. Many of these difficulties can be overcome by using micro-rheometry. The
majority of micro-rheometric techniques available for the characterization of
complex biofluids rely on Brownian forcing of microscopic tracer beads. The
rheological properties of the surrounding fluid matrix are obtained from the
time-correlated displacement of the bead *via* a deconvolution process
(Mukhopadhyay and Granick,
2001; Solomon and Lu,
2001). Such techniques are inherently limited to studies of linear
viscoelastic properties of the test fluid at small shearing strains. By
contrast, the silk spinning process involves large strains and both shearing
and extensional kinematic components
(Knight and Vollrath, 1999).
These large deformations influence the development of non-equilibrium texture
morphologies in the spun silk (Vollrath
and Knight, 2001) that lead to observational phenomena such as
super-contraction (Vollrath et al.,
1996) and shape-memory effects
(Emile et al., 2006). In order
to address these issues, two new micro-rheometric instruments have been
constructed: a flexure-based micro-rheometer
(Clasen et al., 2006;
Clasen and McKinley, 2004;
Gudlavalleti et al., 2005) for
steady and oscillatory shearing measurements and a capillary break-up
micro-rheometer for extensional rheometry
(Bazilevsky et al., 1990;
McKinley and Tripathi,
2000).

Here, we describe the application of these instruments in measuring the
rheological properties of dragline silk solutions extracted from the major
ampullate gland of *Nephila* spiders.

## Materials and methods

Both of the experimental devices utilized in the present study have been
specifically designed to accommodate the small quantities (∼1 μl) of
fluid available from a single major ampullate gland of the *Nephila
clavipes* L. spider, as shown in Fig.
1. Thus, these micro-rheometric devices enable *ex vivo*
testing of the native spinning dope from which the spider spins dragline and
web frame fibers (Gosline et al.,
1999; Vollrath and Knight,
2001).

Dissections were performed using a standard dissecting microscope, and the ampullate glands were extracted and stored under distilled water for less than 5 min whilst being transferred to the micro-rheometers for testing. Each sample was only utilized once due to progressive evaporation of the aqueous phase to the environment.

The flexure-based micro-rheometer generates a plane Couette shearing flow
between two plates that are aligned using white light interferometry and
separated by a precisely controlled gap of 1-150 μm
(Fig. 2a). The plates consist
of cylindrical optical flats that are then diamond-machined to provide the
required rectangular test surface area. The shear stress exerted on the sample
(ranging from 2 to 10^{4} Pa) is calculated from the deflection of the
upper flexure as the lower one is actuated. The imposed shear rate
(), defined as the ratio of the
actuated plate velocity (*v*) and the inter-plate gap (*h*),
, can be varied over
the range
.
Further details of the instrumentation are provided elsewhere
(Clasen et al., 2006;
Clasen and McKinley,
2004).

Recently, it has been noted that the converging flow in the duct of the
major ampullate gland has significant extensional kinematics that can lead to
pronounced molecular extension and orientation
(Knight and Vollrath, 1999;
Vollrath and Knight, 2001). If
macroscopic volumes of the entangled polymer solution are available, then the
extensional viscosity of a viscous liquid can now be measured reliably using
filament stretching rheometry
(Bhattacharjee et al., 2002).
However, given the limited amount of raw dope available, this method becomes
impractical. In the present work we have developed a microscale capillary
break-up extensional rheometer to measure the transient extensional viscosityη
_{e} (McKinley and Tripathi,
2000). The experimental procedure involves placing a drop of the
spinning dope in-between two cylindrical endplates of radius
*R*_{p}=1.5 mm. The plates are then pulled apart to a distance
of 5 mm in order to impose a step axial strain on the sample and form a liquid
thread (Fig. 3). The instrument
can be placed in an environmental chamber to control test temperature and
relative humidity if desired; however the present tests were all performed in
a standard laboratory environment (*T*=22°C; relative humidity
25-35%). The viscoelastic fluid column that is formed by the step extensional
strain subsequently thins under the action of capillarity, while viscous and
elastic forces tend to impede the necking process. The time rate of change of
the liquid filament is monitored using a laser micrometer with a resolution of±
10 μm (Omron model ZL4A; Omron, Schaumburg, IL, USA). Gravitational
drainage is not important for such small samples, and the filament evolves in
a self-similar manner. It has been shown that the apparent extensional
viscosity function can be deduced from the self-similar thinning and pinch-off
dynamics of the viscous fluid thread
(McKinley and Tripathi, 2000)
*via* the expression:
(1)

Here, σ is the surface tension of the liquid, *R*(*t*)
is the midpoint radius of the thread measured with the laser micrometer, and
the numerical prefactor is derived from a slender-body lubrication theory for
a viscous incompressible Newtonian fluid in order to account for deviations
from a purely cylindrical geometry in the vicinity of the endplates
(McKinley and Tripathi,
2000).

Due to the high viscosity of the silk and the evaporation of the aqueous
solvent, it was not possible to directly measure the surface tension of the
silk, and furthermore we are not aware of any published data on this topic.
However, we may estimate the range of values to be
30×10^{-3}≤σ≤60×10^{-3} N
m^{-1}. The upper bound is consistent with measured values for other
aqueous polymer solutions (Adamson and
Gast, 1997; Christanti and
Walker, 2001; Cooper-White et
al., 2002). The presence of any additional surfactant components
in the silk dope may lower this number to values closer to those of
non-water-soluble hydrocarbon-based polymers that are typically of the order
of 30×10^{-3} N m^{-1}.

## Results

### Shear viscosity of ex vivo silk solutions

The sliding plate micro-rheometer was used to measure the
shear-rate-dependence of the steady shear viscosityη
() for a ∼1 μl
blob of spinning dope extracted from the major ampullate gland. The dope was
sheared between two 25 mm^{2} optical plates with the gap set to 25μ
m (Fig. 2a), and we further
assume a no-slip boundary condition between the dope and the optical plates.
In the limit of zero shear rate, the data in
Fig. 2b show that the viscosity
of the spinning dope is η_{0}=3500 Pa.s (or
3.5×10^{6} times the viscosity of water). However, under
stronger deformation rates, the dope viscosity drops significantly with
increasing shear rate, i.e. the dope has a shear-thinning viscosity
(Fig. 2b). This effect is
characteristic of concentrated polymer solutions due to the loss of molecular
entanglements and can be described by molecular theories or by
phenomenological constitutive models such as the Carreau-Yasuda equation
(Bird et al., 1987a;
Yasuda et al., 1981):
(2)

where λ is a measure of the relaxation time of the viscoelastic
fluid (its inverse is the critical shear rate that marks the onset of shear
thinning), *n* is the power-law exponent characterizing the
shear-thinning regime observed at high shear rates, and the coefficient
*a* describes the rate of transition between the zero-shear-rate region
and the power-law region.

Nonlinear regression of these parameters to our data yields values ofλ
=0.40 s, *a*=0.68 and *n*=0.18, which are characteristic
for a strongly shear-thinning fluid
(Yasuda et al., 1981). We are
not aware of any other published data on the *ex vivo* rheology of
native *Nephila* dope with which we can compare these values. A
comparison to rheological measurements for *B. mori* dope, determined
with the same experimental setup and shown in
Fig. 2b, gives similar
viscoelastic properties for the silk dopes obtained from the silkworm and the
spider. The constitutive parameters obtained with each sample are tabulated in
Table 1. The shear-thinning
behavior measured in our silkworm dope is consistent with recent experiments
performed with a commercial rheometer
(Terry et al., 2004), in which
the zero-shear-rate viscosity measured was reported to be `approximately 2
kPa.s' (we measured 5±1 kPa.s). Their measurements also showed that the
critical shear-rate above which shear-thinning occurs is of the order 0.5
s^{-1} (we determined 1.7 s^{-1}). No reports of standard
error or sample-to-sample variability were reported in this earlier study; the
relatively small differences between the two sets of measurements may be due
to biological variability as well as to handling of the dope sample.

### Extensional rheology

The data in Fig. 3b show
that at small strains the extensional viscosity η_{e} is three
times larger than the zero-shear-rate viscosity measured with the shearing
micro-rheometer. This observation is consistent with the classical results of
Trouton for a Newtonian liquid (Trouton,
1906). However, at large strains, the necking dynamics are greatly
retarded as the filament simultaneously strain-hardens and undergoes mass
transfer to the surroundings (i.e. evaporative drying). This strain-hardening
stabilizes the spinline and leads to the formation of axially uniform
filaments (Olagunju, 1999).
The apparent extensional viscosity therefore diverges and the thinning fluid
thread ultimately dries to become a solid filament with a fixed finite radius.
In contrast to an actual dragline filament, which is spun under a constant
force corresponding to the weight of a spider
(Gosline et al., 1999), in our
capillary break-up device there is no externally imposed tension. The final
thread radius is measured to be *R*_{f}≈20 μm. The solid
red line in Fig. 3a corresponds
to a one-dimensional model of this drying process, which is discussed in
detail below.

## Discussion

The strong shear-rate dependence of the silk viscosity shown in
Fig. 2 is of considerable
importance during extrusion. During a typical spinning process
(Shao and Vollrath, 2002;
Vollrath and Knight, 2001),
the *Nephila* spider draws out a 4 μm-diameter thread at a speed of
20 mm s^{-1} corresponding to a flow rate of
*Q̇*=0.25 nl s^{-1}. We
approximate the geometry of the long converging spinning canal (or
S-duct*)* shown in Fig.
1b as a truncated cone of length *L*=20 mm and with maximum
and minimum diameters of *D*=200 μm and *d*=4 μm,
respectively. Although little is known about the actual kinematic boundary
conditions at the wall of the spinneret, we assume that there is no-slip
between the fluid dope and the wall, as in previous studies
(Vollrath and Knight, 2001).
For the given geometry and flow rate, the pressure drop associated with steady
flow of a viscous shear-thinning dope through the canal can be estimated from
hydrodynamic lubrication theory [Bird et al.
(Bird et al., 1987a), eq.
4.2-10], which leads to the following expression for the relative pressure
drop for steady flow of a power-law liquid through a linearly tapered tube
compared with that expected for a Newtonian fluid:
(3)

Here, λ and *n* are obtained from the Carreau-Yasuda model
(Eqn 2). This relation is only valid in the shear-thinning regime when the
shear rate is larger than the critical shear rate (1/λ) and the
viscosity is thus well approximated by
.
For our truncated cone geometry, the minimum value of the wall shear rate is
,
which justifies the use of the power law fluid and Eqn 2 and 3 as
approximations. The pressure drop required for the shear-thinning silk dope is
a factor of 500 lower than that associated with a corresponding viscous
Newtonian fluid. Thus, shear-thinning of the liquid crystalline solution
reduces the absolute value of the pressure drop in the spinning canal required
to sustain flow rates of the order of nanoliters per second.

The pressure drop Δ*P*_{silk} necessary to push the
silk dope through the spinneret at the typical flow rate of
*Q̇*=0.25 nl s^{-1} is
approximately 4 ×10^{7} Pa. The corresponding energy dissipation
rate Δ*P*_{silk}
*Q̇*∼10 μW (due to viscous flow of
the silk dope) is comparable to the release rate of potential energy,
*MgV*_{spin}∼20 μW, for a spider descending on a
dragline, where *M* is the mass of the spider (∼0.1 g), *g*
is the gravitational constant, and *V*_{spin}=20 mm
s^{-1} is a typical silking speed. By contrast, if the silk dope did
not exhibit this pronounced shear-thinning behavior (which is associated with
its liquid crystallinity), the viscous dissipation rate required to sustain
the corresponding flow rate of a Newtonian fluid would be 5000 μW and hence
would significantly exceed the potential energy release rate. Additional
sources of energy input would have to be provided by the spider or,
conversely, a much lower natural spinning speed would be selected.

This shear-thinning property of the silk dope may also act in conjunction
with other proposed mechanisms that facilitate the spinning of the thread,
such as a shear-induced transition to a liquid crystalline phase
(Vollrath and Knight, 2001),
localised slip of the polymer solution on the tube wall
(Migler et al., 1993), or a
subtle form of lubrication, such as a watery surfactant layer
(Vollrath and Knight, 2001) or
an analogue to the sericin coat surrounding fibroin fibers spun by *B.
mori* (Kaplan et al.,
1994).

In contrast to the observations of shear-thinning, the measurements of the transient extensional rheology in the micro-capillary break-up extensional rheometer (or μCABER) show that in an elongational flow the material's resistance to stretching increases with elapsed time (and imposed strain). The importance of this strain-hardening phenomena for the spinning of dragline silk appears to have been first noted by Ferguson and Walters (Ferguson and Walters, 1988) and prevents the capillary break-up of an elongating viscoelastic fluid filament (Olagunju, 1999).

In addition to being sheared, the proteins in the spinning dope are also
stretched due to the elongational flow experienced in the converging duct and
the subsequent spinline. An extensional flow of this type is characterized by
the deformation rate and the total Hencky strain accumulated, which can be
defined in the present problem as ϵ=2ln(*D*/*d*) ≈8
(Bird et al., 1987a). This
large value of the extensional strain suggests that the spidroin molecules are
being considerably extended (Perkins et
al., 1997). This extension thus plays a key role in the molecular
alignment necessary for the exceptional mechanical properties of the spun
fiber. The characteristic strain rate for this elongational flow is given by:
(4)

This rate of stretching can be compared with the liquid relaxation time
*via* the Deborah number [see Appendix as well as Bird et al.
(Bird et al., 1987a)], defined
as , which provides a
dimensionless measure of the importance of viscoelastic properties. The
computed value of *De* ≈0.5 indicates that viscoelastic effects
should result in modest strain hardening of the dope (i.e. an increase in the
resistance to stretching with increasing strain)
(Bird et al., 1987a). This
strain-hardening effect is due to chain-stretching of the entangled spidroin
macromolecules, and the presence of this additional elastic stress can be
evaluated from the extensional viscosity of the liquid.

The time evolution in the neck radius that is depicted in
Fig. 3a is driven by the
capillary pressure and resisted by the viscoelastic stresses in the elongating
fluid thread. The necking rate is further modulated by evaporation of solvent
(water) from the thread. This evaporation rate becomes larger as time proceeds
due to the increasing surface area-to-volume ratio. The loss of water also
results in an increase in the fluid viscosity and a further slow-down in the
rate of necking. A simple model that captures the essential physics of this
filament thinning/drying process through a time-dependent viscosity function
is given by Tripathi et al. (Tripathi et
al., 2000). In this analysis, a `lumped parameter' model is
developed that describes the rate of mass transfer in terms of a single
dimensionless group referred to as a processability parameter, *P*.
This parameter is defined as the ratio of the two relevant time scales in the
problem: the time scale for capillary thinning and the time scale for
diffusion of water through the viscous protein dope to the free surface. The
characteristic time for capillary thinning is *t*_{cap}∼η
_{0}*R*_{0}/σ (for a viscous fluid) and the
time scale for diffusion [which in our case limits water removal from the
thread (Kojic et al., 2004)]
is *t*_{diff}∼
*R*_{0}^{2}/*D*_{w}, where
*R*_{0} and *D*_{w} are the initial thread
radius and the diffusivity of water through the dope, respectively. We have
recently reported a value of *D*_{w} =2×10^{-5}
mm^{2} s^{-1} for the diffusivity of water through the
*Nephila* spinning dope (Kojic et
al., 2004). Using this value, along with an initial radius of
*R*_{0}=78 μm (see Fig.
3a) and the expected range 30×10^{-3}≤σ≤
60×10^{-3} N m^{-1} for the surface tension, we obtain
the following estimate of the processability parameter:
(5)

The analysis of Tripathi et al.
(Tripathi et al., 2000)
utilizes the parameter *P* to yield a time-varying fluid viscosity
given by:
(6)

Combining this time-dependent viscosity with Eqn 1 results in an integro-differential equation for calculating the evolution of the radius of the thinning thread.

Alternatively, it is possible to apply this theory directly to the present
microcapillary break-up measurements treating *P* as an arbitrary
fitting parameter. The results of using a best-fit value of
*P*=2.715×10^{-2} are shown in
Fig. 3a by the solid red line.
This best fit value of the processability parameter is in good agreement with
the *a priori* estimate given above.

The resistance of the fluid thread to further stretching is characterized by the apparent extensional viscosity (derived from Eqns 1, 6) as presented in Fig. 3b over the entire course of the filament evolution. At large strains, the filament undergoes strain-hardening due to the combined action of molecular elongation and solvent evaporation and ultimately becomes a solid thread with a constant diameter. The extensional viscosity increases by 100-fold during the capillary thinning of the filament radius. This strain-hardening plays an important role in the fiber spinning process by inhibiting capillary thread break-up and stabilizing the spinline.

### Conclusions

In this work, we have used two new micro-rheometric devices that utilize
less than 5 μl of fluid for a test and enable the measurement of the steady
and transient rheological properties of *ex vivo* samples of biopolymer
solutions such as spider and silkworm spinning dope. The devices are able to
impose large deformation rates and large strains that match the range of
deformations experienced *in vivo.* Our measurements show that the
steady shear viscosities
of *N. clavipes* and *B. mori* spinning solutions have very
large zero-shear-rate viscosities but shear-thin dramatically above a critical
deformation rate (see Table 1).
By contrast, in extensional flow, the apparent extensional viscosity of the
spider silk dope increases without limits due to the combined action of
molecular elongation and solvent evaporation.

Orb-weaving spiders use a specialized fiber-spinning process that exploits
the nonlinear rheology of a complex fluid. In the spinning canal of *N.
clavipes*, the shear viscosity of the spinning dope decreases by an order
of magnitude and thus reduces the pressure-drop along the canal, whereas the
extensional viscosity increases by a factor of 100 to stabilise the fluid
thread and inhibit capillary break-up of the spun thread. Tailoring the
rheological properties of artificial spinning dopes containing genetically
modified or reconstituted silks to match the *ex vivo* properties of
the natural dope may prove essential in enabling us to successfully process
novel synthetic materials with mechanical properties comparable to, or better
than, those of natural spider silk.

## Appendix/glossary

In this glossary we provide brief working definitions for some of the most important rheological concepts utilized in this work and provide references to other primary sources for additional reading.

### Constitutive equation

Also often described as a `rheological equation of state'. Such equations relate the tensorial state of stress in a complex fluid to the entire deformation history imposed on it. If the relationship between an imposed shear-rate and the resulting shear stress is nonlinear then the fluid is `non-Newtonian'. Constitutive equations may be constructed empirically or derived from molecular-based kinetic theories (Bird et al., 1987a; Bird et al., 1987b).

### Shear-thinning viscosity

One of the most common rheological features of complex fluids is a nonlinear relationship between the shear stress () and the shear rate (). For most polymeric systems the steady shear viscosity (defined as the ratio of the measured shear stress to the imposed shear rate at steady state; ) decreases as the deformation rate increases due to increasing flow-alignment of the underlying microstructure. The process is particularly dramatic and leads to a very pronounced decrease in the viscosity and increasing optical anisotropy for liquid crystalline polymers (Burghardt, 1998). This effect is known generically as `shear-thinning' and is demonstrated in Fig. 2. In the limit of low shear rates, the relationship between stress and rate reduces to a simple linear one (i.e. the fluid approaches the limit of a simple Newtonian fluid) and the steady shear viscosity approaches a constant value that is defined as the zero-shear-rate viscosity; lim as . The Carreau-Yasuda model presented in the text is a relatively simple example of a constitutive model that can describe this transition from Newtonian to shear-thinning. The model is derived by considering the rate of creation and destruction of molecular entanglements in a concentrated polymer solution or melt. Numerous other constitutive equations (for example, the Giesekus model or the Phan-Thien-Tanner model) can also capture the general trends shown by our data; however, in many cases, these equations contain additional model parameters that can only be determined from a more extensive range of rheological tests. The texts by Bird et al. (Bird et al., 1987a; Bird et al., 1987b) compare and contrast the relative benefits of these different constitutive models.

### Liquid crystalline solutions

Liquid crystalline solutions are distinguished by the rigidity and local ordering of the constituent molecules (in contrast to the random walk conformation associated with flexible macromolecules). This local molecular ordering can lead to phase transitions as the concentration is increased or the system temperature is reduced. In addition, the coupling between imposed mechanical deformations and molecular ordering leads to optical anisotropy in the solutions that is manifested in effects such as flow-induced birefringence (Burghardt, 1998). Such effects have been measured in protein solutions obtained from silkworms and from spiders (Magoshi et al., 1994; Willcox et al., 1996) and are discussed in detail in the review of Vollrath and Knight (Vollrath and Knight, 2001).

### Extensional viscosity

The extensional viscosity of a fluid is a measure of the resistance to
elongational (stretching) deformations and is defined as a ratio of the
measured tensile stress difference to the imposed rate of stretching. Although
perhaps this concept is initially puzzling to contemplate, some physical
understanding may be attained by recognizing that the extensional viscosity
holds the same relationship to the shear viscosity of a fluid as the Young's
modulus (*E*) does to the shear modulus (*G*) for an elastic
solid. Indeed, for an incompressible Newtonian fluid, the extensional
viscosity is precisely three times the shear viscosity, a result first
obtained by Trouton 100 years ago
(Trouton, 1906), just as the
Young's modulus is three times the shear modulus for an incompressible Hookean
solid. For non-Newtonian fluids such as polymer solutions, the extensional
viscosity is an independent material function that cannot be determined from
the shear viscosity. Typically, the extensional viscosity of a complex fluid
is a function of both the rate of elongation and the total strain imposed and
this governs the `spinnability' of a fluid thread
(Macosko, 1994).

### Strain-hardening

For many polymeric systems it is found that the extensional viscosity increases with the total strain imposed on the system. This is a consequence of the increasing molecular elongation of the flexible polymer chains as the external strain is increased and is referred to as strain-hardening or sometimes `strain-stiffening' (Nguyen and Kausch, 1999). This increase in the extensional viscosity is only to be expected if the rate of deformation imposed on the fluid is sufficiently rapid to exceed local relaxation of the chain back towards equilibrium; this criterion is parameterized by the Deborah number of the flow.

### The Deborah number

The Deborah number provides a dimensionless measure of how important
non-Newtonian effects are expected to be in a given deformation. The Deborah
number represents a ratio of the intrinsic relaxation time of the polymeric
liquid to the characteristic flow time scale (or equivalently the product of
relaxation time with the rate of deformation) of a particular flow process
(McKinley, 2005). For example,
if a polymer chain can relax back to its equilibrium configuration (through
Brownian motion) faster than it is deformed (*De*<1) then the
material will not show strain-hardening in elongation, or shear-thinning in
steady shear flow and will instead flow in the same manner as a viscous
Newtonian fluid.

## ACKNOWLEDGEMENTS

This research was supported by funds from the NASA Biologically-Inspired
Technology Program, the DuPont-MIT Alliance and in part by the U.S. Army
through the Institute for Soldier Nanotechnologies, under Contract
DAAD-19-02-D0002 with the U.S. Army Research Office. Adult female *Nephila
clavipes* spiders were kindly provided by Rachel Rogers of the Miami
MetroZoo.

- © The Company of Biologists Limited 2006