In-drop capillary spooling of spider capture thread inspires highly extensible fibres

1.6 MB·English

by  Hervé Elettro

In-drop capillary spooling of spider capture thread inspires highly extensible fibres
Herv´e Elettro,∗ S´ebastien Neukirch, and Arnaud Antkowiak
Sorbonne Universit´es, UPMC Univ Paris 06, CNRS,
UMR 7190 Institut Jean Le Rond d’Alembert, F-75005 Paris, France.
Fritz Vollrath
Oxford Silk Group, Zoology Department, University of Oxford, UK
(Dated: January 6, 2015)
Spiders’websandgossamerthreadsareoftenparadedasparadigmsforlightweightstructuresand
outstandingpolymers. Probablythemostintriguingofallspidersilksisthearaneidcapturethread,
covered with tiny glycoprotein glue droplets. Even if compressed, this thread remains surprisingly
taut –a propertyshared withpure liquidfilms – allowing both threadand web tobein aconstant
5
state of tension. Vollrath and Edmonds proposed that the glue droplets would act as small wind-
1
lassesandberesponsibleforthetension,butotherexplanationshavealsobeensuggested,involving
0
for example the macromolecular properties of the flagelliform silk core filaments. Here we show
2
thatthenanolitregluedropletsofthecapturethreadindeedinducebucklingandcoilingofthecore

n filaments: microscopicin-vivoobservationsrevealthattheslackfibreisspooledintoandwithinthe
a droplets. Wemodelwindlassactivationasastructuralphasetransition, andshowthatfibrespool-
J
ing essentially results from the interplay between elasticity and capillarity. This is demonstrated

5 by reproducing artificially the mechanism on a synthetic polyurethane thread/silicone oil droplet
system. Fibre size is the key in natural and artificial setups which both require micrometer-sized

] fibres to function. The spools and coils inside the drops are further shown to directly affect the
t
f mechanicalresponseofthethread,evidencingthecentralroleplayedbygeometryinspidersilkme-
o
chanics. Besidesheddinglightonaraneidcapturethreadfunctionality,wearguethattheproperties
s
. of this biological system provide novel insights for bioinspired synthetic actuators.
t
a
m
Theorb-websofaraneidspidersarehighlyeffectiveand or indeed in biological applications such as the wetting
-
d efficienttrapsthatareabletofullycushiontheimpactof ofmammalianhairs[10], doesnotcontainanyreportsof
n an insect within a second and then manage to keep the such drop-induced ’windlass’ events.
o prey well ensnared in a wet-sticky capture spiral until Ourresultsfullysupportthewindlasstheoryandiden-
c the spider can reach it [1]. The exceptional extensional tify the fundamental mechanism to be dependent on the
[
propertiesofthespiralthreadsrelyonthemacromolecu- interplaybetweenelasticityandcapillarity. Moreover,we

1 lar architecture of the silk fibroins [2, 3] while the ability demonstrateempiricallyaswellastheoreticallythatthis
v to contract, self-tense and adapt slack length with virtu- mechanism is generic, i.e. not requiring silk proteins in
2 ally no hysteresis has been attributed to a surprisingly either the filaments nor the droplets as fundamental ma-
6
complex mechanical micro-mechanism [4]. Here we are terial components. This not only elucidates the physics
9
setting out to test this micro-windlass hypothesis. To of the system but also opens the way for the design of
0
dateitisbasedmostlyonindirectsnap-shotobservations novel bio-inspired synthetic actuators using the windlass
0
. ofvariousamountsofcoiledthreadsinsidecapturespiral array concept.
1
droplets laid down on a microscope slide [4] combined Figure 1 outlines the macroscopic observation of the
0
with the interpretation of mechanical behaviour. The system. The natural set-up is fully self-assembling, with
5
1 lack of direct, dynamic observation of the phenomenon the spider coating the core fibres of the capture spiral
: has given rise to criticism refuting the conclusions of an with a thin layer of hygroscopic compounds [11]. This
v
active mechanism driving the system (see [5] with reply layerthenswellsbytakingupwaterfromtheatmosphere
i
X [6] and summary [7]). Much of the criticism was dis- and go through a Plateau-Rayleigh instability to form a
r missed by empirical studies [6] but a fundamental ques- string of interconnected silk droplets sitting astride the
a tion must be taken seriously: after all, the physics of core silk fibres [12]. This natural micron-sized windlass
such a mechanism would require the watery coating of array system ensures that the capture thread is always
the thread to be able to (i) exert forces sufficiently pow- under tension – independent of any extension or relax-
erfultorapidlytensionsubstantiallengthsofthefilament ation. Asthecapturethreadisstretched, lengthsofcore
whichbearshundredsofnanolitredroplets,and(ii) hold filament are simply strained or, depending on past ac-
the composite system in place without sagging against tion, pulled out of a droplet. It appears that the surface
gravity. Actually,thevastliteratureonliquidcoatingon tension of the droplet is strong enough to pull-in the silk
fibres,whetherintextile[8]orglassfilaments[9]products filament, which buckles and then coils up inside the liq-
uid.
Figure 2 shows force-elongation measurements per-
formed on a single capture thread, accompanied by mi-
∗Electronicaddress: [email protected] croscopicobservationsofcorefibrecoilingwithintheglue
2
I T > T
P
II T T
P
x
1 mm
III T < T
P
Figure 1: A liquid-like fibre. Whether stretched or relaxed, the typical capture silk thread of an araneid orb spider (here
Nephila edulis) remains taut. Force monitoring reveals a threshold tension T above which the fibre behaves like a spring (I).
P
As the force is decreased a force plateau T (cid:39)T is reached, along which the fibre adopts a wide range of lengths, just as soap
P
films do (II). At lower forces, T <T , the fibre is totally contracted (III). See also Supplementary Videos for full cycle.
P
droplets. This data reveals the self-organisation of the T (µN)
core filaments into coils within the droplet, and deter-
7
mines the link between the shape of the filament and I
the mechanical response of the thread. Indeed the ob- 6
served mechanical response encompasses three different
stages,eachcharacterizedbyatypicalforce-displacement 5 STRAIGHT
relation and a specific filament shape within the drop. 4 COILED II
At very low tension T exerted on the fibre, the cen-
tral glue droplet, resulting from the successive merging TP
of tiny drops, has ‘ingested’ a large amount of spare fi- BUCKLED
2
bre and presents a roughly linear relation between force
and extension, see region III. As a plateau force T (cid:39) T 1 III
P
is reached, the filament starts uncoiling and exiting the
0
drop,allowingthiscompositesystemtoeffortlesslyreach 0 2 4 6 8 10 12
very large extensions, see region II. Interestingly this x (mm)
mechanical response, associated with unspooling, bears
little resemblance with the typical response of a fibre, Figure 2: Shape-induced functionalization. Quasi-static
and is much more reminiscent of the mechanics of soap forcemeasurementsonspidercapturethreadscombinedwith
microscopicobservationsdemonstratethattheslackcorefila-
films [13]. Past the threshold where all the coils have
mentisspooledintothedroplets(typically250-300µmwide)
been straightened, the capture thread recovers a classic
alongtheforceplateauT =T . ForlargerforcesT >T ,the
drop-on-(straight) fibre appearance along with a linear P P
fibre straightens and a spring-like behaviour is shown. The
stretched spring regime response, see region I. This tri-
characteristicJ-shapeforthisforce-extensioncurvecanbeat-
modal response of the capture thread is not unlike the tributed to a shape-induced functionalization of the fibre by
typical response of a Shape Memory Alloy (SMA) fibre, the glue droplets (see also Supplementary Video).
see Figure 3. And indeed in both systems the plateau
regimecorrespondstoaphasetransition: fromaustenite
to martensite in SMA, and from coils-in-drop to straight
fibre in capture silk. extended phase – where the thread runs straight outside
To elucidate the conditions for the windlass action, we the drop. The extended phase is modelled as a spring
examine this phase transition further. We start by con- with stiffness k = πEh2/(cid:96), where (cid:96) stands for the rest
e
sidering a model capture thread composed of a soft fi- length, E the Young’s modulus of the fibre and h its ra-
bre capable of being bent or stretched and subjected to dius. The strain energy of the extended phase is then
an external tension T, supporting a wetting liquid drop 1k (x − (cid:96))2, where x is the phase’s extension. The
2 e e e
standingastride[14]. Thoughthin,silkfilamentsarenot coiled phase typically describes the system at low ap-
expected to be significantly excited with thermal fluc- plied tension T. In the limit T = 0 the coiled phase
tuations: the persistence length of such filaments is in- extensionisx =D whereD isthediameterofthe,then
c
deedestimatedtobekilometer-sized[15],tobecompared spherical, liquid drop. As tension starts rising, the liq-
withafewtensofnanometersforDNA.Disregardingen- uiddropdeformsintoanelongatedshape. Modellingthe
tropic elasticity effects, we describe windlass activation deformation as ellipsoidal yields the following spring-like
as a phase transition between a coiled phase – where the relation: T = 4πγ(x −D), where γ is the liquid surface
5 c
fibre is entirely packed within the liquid drop – and an tension. The strain energy of the coiled phase is then
3
E = 1k (x −D)2 wherethespringstiffnessoftheliquid TD2/4 2EI
c 2 c c
drop directly arises from surface tension k = 4πγ. As 2.0 SHAPE MEMORY ALLOY BEHAVIOUR
c 5 T
typicalforphasetransitionproblems,thereisanenerget-
k
ical cost per unit length associated with the transforma- c
k
tion from the coiled phase to the extended phase, which 1.5 T0 eX / L
we note (cid:15)0. This energy originates from both the un- 0D / L 1
coilingofthefibreandtheenergydifferencebetweenwet
anddrystates. Theuncoilingprocessresultsinanelastic
1.0
energy gain −1EIκ2 per unit length, where the curva-
2
ture κ = 2/D and I = πh4. The latent energy per unit
4
lengthalsoembodiesthedifferenceinsurfaceenergies: in
0.5
THEORETICAL
the coiled phase the fibre is surrounded by liquid, hence PREDICTION
bears a surface energy 2πhγ per unit length whereas in
sl x /
the extended phase the air surrounding the fibre yields
0
an energy 2πhγ per unit length. Here γ and γ de- 0 0.2 0.4 0.6 0.8 1.0 1.2
sv sl sv
note solid-liquid and solid-air interface energies respec-
tively. It is to be noted that a surface energy constrast Figure 3: Windlass as a structural phase transition.
between the coiled and extended phase persists even if Numericalsimulationsofaslenderfibresupportingadropsit-
a thin liquid film sheathes the core filament, for the en- tingastrideunravelasubcriticalbucklingtransitionwherethe
ergy of a filament coated with a thin liquid layer differs slack fibre is suddenly localised within the droplet. Continu-
from that of a filament immersed in liquid [13]. Upon ation of equilibria, with FγD2/(4π2EI)=0.5 and L=20D,
reveal the fine details in the micro-mechanical response, re-
using Young-Dupr´e wetting relation γ −γ = γcosθ,
sv sl sulting in inhomogeneities in the Maxwell plateau. The ap-
theenergeticalcostforphasetransformationisobtained:
proximate plateau value T D2/(4π2EI) (cid:39) 0.45, given by
(cid:15) = 2πhγcosθ− 1πEh4/D2, where θ is the liquid con- P
0 2 equation (2), is drawn for comparison. Several equilibrium
tact angle on the fibre. From this expression we readily
configurationsillustratethecoilingofthefiberanditsstrong
obtainconditionsforwindlassactivation. Indeed,forthe packing within the droplet. The mechanical response is typ-
coiled phase to be stable at small extensions x, (cid:15)0 has to ical of systems exhibiting first-order phase transitions (such
be positive. This condition can be recast into a condi- as shape-memory alloys - SMA), and the numerical results
tion for the radius, where it becomes apparent that only are confirmed by an analytical model involving a wet/coiled
extremely thin fibres can deploy a windlass: (analog to the austenite phase in SMA) and a dry/stretched
(analog to the martensite phase in SMA) phase (see inset).
(cid:18)4γcosθ(cid:19)1/3
h< D2/3 (1)
E
Noting the total energy Ee of the extended phase as ing (x,T) diagram, shown in Fig. 3, reveals inhomo-
Ee = 21ke (xe − (cid:96))2 + (cid:15)0(cid:96), we write the global en- geneities in the Maxwell line. These inhomogeneities
ergy when part of the fibre is in the coiled phase, and are due to fine details in the micro-mechanical response
part in the extended phase. Denoting the extended (spooling and unspooling events) of the system. Setting
phase fraction ρ, the global energy E(xe,xc,ρ) reads: Fγ =2πhγcosθ,weplotinFig.3thehorizontallinecor-
E = (1−ρ)Ec(xc)+ρEe(xe). Minimizing this energy respondingtoequation(2),andseethatitcompareswell
we find that the fibre can be entirely in the coiled phase with numerical computations.
(ρ = 0; filament fully packed in the drop) with tension
Our theoretical results show that, provided capillarity
T =k (x−D), or entirely in the extended phase (ρ=1)
c prevails over filament elasticity, elastocapillary spooling
with tension T =k (x−(cid:96)). A third interesting possibil-
e can be initiated on any flexible fibre, and that there is
ity consists in a mixture of phases 0 < ρ < 1. In this
no need for any rearrangement in the molecular archi-
latter case, part of the filament is packed in the drop
tecture. We experimentally demonstrated this material
but the outer part is taut, consistent with our observa-
independence by triggering windlass on a soft Thermo-
tions. InthelimitwhereD (cid:28)(cid:96)and(cid:15) (cid:28)k ,k ,wehave
0 e c plastic PolyUrethane (TPU) micron-sized filament. As
x (cid:39)D+(cid:15) /k , x (cid:39)(cid:96)+(cid:15) /k and T (cid:39)T (cid:39)(cid:15) , so that
c 0 c e 0 e P 0 soon as a single silicone oil droplet was deposited on the
the plateau tension is
sagging TPU filament, the fibre straightened and a ten-
1 h4 sion force on the fibre of typically 1 µN immediately de-
TP (cid:39)2πhγcosθ− 2πED2 (2) veloped. We performed tensile tests and observed that
the force curve, see Fig. 4, exhibited the plateau regime
We performed detailed numerical computations of equi- with T ∼ 1µN until the filament was totally straight
P
librium of an extensible and flexible elastic fibre [16, 17]. and the spring regime started. Snapshots of the filament
Thefilament, heldatbothextremitieswithimposeddis- within the oil droplet in the plateau and spring regimes
tance x, was subjected to attracting meniscus forces F are shown in Fig. 4 and confirm the link between fila-
γ
at entrance and exit of a confining sphere. The load- ment shape and the mechanical response of the system.
4
erly tinkers by coopting simple laws of physics [23].
T (µN)
100
Materials
80
Capture silk samples. Our Nephila edulis spider
60 waskeptina80×80×30cmvivarium,consistingofwood
STRAIGHT
panels, PMMA windows and artificial plants. The spi-
derwaskeptathighhumidity(above70%)andcomfort-
40 DRY
abletemperature(above22◦C)witha12/12hrday/night
WET schedule. Thespiderwasfedcricketsandfliesthreetimes
20
aweek. Sectionsofwebwerecarefullyexcisedusingasol-
COILED
deringironfortransferwithinarigidframe. Tovisualise
thefibrerunningthrougheachdroplet,thehumiditywas
2 4 6 8 setto100%rHfor15minutes,thensteppeddownto50%
x (mm)
before observation.
Artificial samples. PolyUrethane (TPU, Elastollan
Figure 4: A synthetic windlass mechanism. Combined 1185A from BASF(cid:13)R) granules were deposited on a hot
quasi-staticmechanicalmeasurementsandpolarizer-analyzer plate at 230◦C. After melting, we used a tweezer to pick
observations of silicone oil droplet sitting astride thin TPU
upasmalldropletwhichwasthenstretchedquicklywhile
fibre (h=2.3±0.15µm, E =17±3MPa, and wetting angle
at the same time being released into ambient room tem-
θ=36±7◦fortheleftinset)demonstratethatthekeyfeatures
perature. This resulted in the creation of micron-sized,
ofthespidercapturethreadcanbeobtainedwithanartificial
metre-long, soft filaments. The filament was then de-
system. The natural mechanical response of TPU, shown in
dashed line, is significantly altered by the addition of an oil posited on the measuring setup as outlined below. A
droplet (wet length 380µm for the left inset), resulting in a droplet of silicone oil (Rhodorsil(cid:13)R 47V1000) was then
highlyextensiblesystem(here+9000%breakingstrain). The deposited by gently touching and brushing the filament
resulting J-shape curve appears to be associated with elasto- with a drop hanging from a pipette.
capillary spooling within the oil droplet, in agreement with Measurementmethods. Filamentsamplesweretrans-
our theoretical description (see also Supplementary Video). ferred to the measuring setup by coiling one end around
the tip of a FemtoTools FT-FS1000 capacitive deflection
forcesensor(50nN-1mNrange)andgluingtheotherend
toaglassslideasbase. Theforcesensorwasmountedon
In addition we show in Fig. 5 that a synthetic TPU fibre a linear micro positioner SmarAct SLC-1730 (repetabil-
displayed the same overall properties of hypercontrac- ity 0.5 µm) and measurements are performed through
tion than capture silk: the liquid drop put the system in a work station by USB connection. All the tests were
tension and there was virtually no sagging, even up to performed in stretching at a speed of 25µm/s, and con-
fivefold length reduction. sideringthecentimetersizeinlengthofthesample, they
Interestingly the addition of the oil droplet allows to can be considered to be quasi-static. The optical setup
turn (and tune) the mechanical response into the J- consisted of a Leica macroscope (VZ85RC) mounted on
shaped curve so typical of many collagenous tissues [18]. a micro-step motor and a 3 megapixels Leica DFC-295
Mimicking the mechanical response of biological tissues camera (400× zoom, 334 nm/pixel picture resolution) or
with synthetic materials certainly offers appealing per- aD800ENikoncamerawith310mmC-mountextension
spectives for e.g. the design of artificial muscles. Actu- rings (937 nm/pixel video resolution and 374 nm/pixel
ally, more than the material properties, this is the shape photo resolution) alternatively. We used a Phlox 50x50
of the strongly curled spools of core filament, reminis- mmbacklight, at60000luxoralternativelyanopticalfi-
cent of packed DNA in bacteriophages [19], that drives brewithLEDlamp(MoritexMHF-M1002)withcircular
themechanicalresponseofboththenaturalandthearti- polarizer. SideviewswereacquiredwithasecondD800E
ficial systems. These typical examples of shape-induced camera,witha70mmextensiontubeanda100mmmacro
functionalization [20] illustrate the fact that geometry Zeiss lense (7,27 microns/pixel video resolution). The
plays a key role in spider silk and TPU/oil droplet ’ex- force sensor was tared to zero with the fibre compressed
treme’ mechanics [21, 22]. slightly more than its slack length, so that it sags, but
In conclusion we have shown that the mechanical re- only minutely, be it for fibres with or without droplet.
sponse of capture silk relies on it being a thin fibre dec- The measurement of the slack length was performed by
orated with drops powering the micro-windlass action. pulling on the filament at one end by a few micrometers
Thus the extraordinary mechanical behaviour of araneid to straighten the fibre.
spider capture silk relies more on the macroscopic com- TheTPUfibrediametermeasurementwasperformedus-
posite structure than on the microscopic material prop- ing Fiji software. A high-resolution picture of the fibre
erties,underliningthatnaturalselectionsometimesclev- is analysed using the following steps : the contrast is
5
a) BARE FIBRE c) FIBRE + SILICONE OIL DROPLET
b) d)
PUSH PUSH PUSH PUSH
Figure 5: Artificial spider capture thread. Left: a thin micrometer-sized TPU filament is held on a caliper. Bringing the
ends together leads the filament to exhibit a characteristic catenary-like sagging shape (note: the outlines serve as guides for
theeye). Right: Thisbehaviourchangesafterdepositionofasmallsiliconeoildropletonthefilament: bringingtheextremities
closer leaves the filament taut while the excess filament enters the central droplet.
enhanced up to the point that 0.4% of the pixels are
saturated,thenthegreyvalueofthepixelsonalineper-
pendicular to the fibre axis is plotted. The typical curve
obtained this way resembles a downward pointing gaus-
sian, thus the diameter of the fibre is extracted as the
full width at half minimum of the peak.
Numerical computations. The windlass system is
modeled as an elastic filament, obeying Kirchhoff equi-
librium equations, in interaction with a sphere. Except
at the two ‘meniscus’ points, the filament is prevented
from touching or crossing the sphere through a soft-
wall barrier potential. The equilibrium of the system
is solved using two-points boundary-value problem tech-
niques (shooting method in Mathematica, and colloca-
tion method using the Fortran - AUTO code).
Supplementary information
The coiled phase has extension x . It is modeled as a
c
linear spring with stiffness k = 4/5πγ, where γ is the
c
liquid surface tension. The tension-extension relation is
then
Figure 6: Two phase model. The windlass system com-
T =4/5πγ(x −D) (3)
c prises two phases. The extended phase consists of the fiber
thatliesoutsidethedrop(s). Thecoiledphaseconsistsofthe
whereDisthediameterofthedropatT =0. Thestrain
the fiber that is coiled inside the drop(s). Each phase has its
energy of the coiled phase is then
own extension, and there is a fraction ρ (resp. 1−ρ) of the
E =1/2k (x −D)2 (4) complete fiber is the extended (resp. coiled) phase.
c c c
The extended phase has extension x . It is modelled
e
as a spring with stiffness k = πh2E/(cid:96) and rest length
e
(cid:96), whereE istheYoung’smodulusofthefiberandhthe Asexplainedinthemaintext,(cid:15)0 istheenergyassociated
radiusofitscircularcross-section. Thetension-extension with the transformation of a unit length of the coiled
relation of the extended phase is then phase into a unit length of the extended phase:
T =πh2E(x /(cid:96)−1) (5) (cid:15) =2πhγcosθ−1/2πEh4/D2 (6)
e 0
6
where θ is the wetting angle of the liquid on the fiber. A second solution is ρ = 1, T = λ = k (x−(cid:96)), and
e
The total energy of the extended phase is then x =x; the system is entirely in the extended phase.
e
Athirdsolutionis0<ρ<1,thatisthesystemisina
E =1/2k (x −(cid:96))2+(cid:15) (cid:96) (7)
e e e 0 mixture of phases. The tension is given by T =E(cid:48)(x )=
c c
E(cid:48)(x )=(E −E )/(x −x ) yielding
When a fraction ρ (respectively 1−ρ) of the system is e e e c e c
in the extended (resp. coiled) phase, the global energy
E(x ,x ,ρ) is given by: (cid:18) λ2 λ2 (cid:19)(cid:46)(cid:18) D(cid:19)
e c T =λ=(cid:15) 1+ − 1− (11)
0 2(cid:15) k 2(cid:15) k (cid:96)
0 c 0 e
E =(1−ρ)E (x )+ρE (x ) (8)
c c e e
which, as D (cid:28)L and (cid:15) (cid:28)k ,k , simplifies to T (cid:39)(cid:15) .
where (cid:96) is the natural length of the fiber. 0 e c 0
We minimize this energy under the fixed-end con-
straint of given global extension x:
x=(1−ρ)xc+ρxe (9) Acknowledgments
Consequently we work with the function L=E−λ[(1−
ρ)x +ρx ] and solve for The present work was supported by ANR grant
c e
ANR-09-JCJC-0022-01, ‘La Ville de Paris - Programme
(∂L/∂x ,∂L/∂x ,∂L/∂ρ)=0 (10) E´mergence’, Royal Society International Exchanges
c e
Scheme 2013/R1 grant IE130506, and the PEPS PTI
The Lagrange multiplier λ is easily identified with the program from CNRS. We thank R´egis Wunenburger for
external force T needed to enforce the constraint (9). discussions and experimental advices on thread visual-
A first solution is ρ = 0, T = λ = k (x−D), and ization within droplets and Christine Rollard for advices
c
x =x; the system is entirely in the coiled phase. in spider housing.
c
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