The bird feather vane can be separated into two parts by pulling the barbs apart. The original state can be re-established easily by lightly stroking through the feather. Hooklets responsible for holding vane barbs together are not damaged by multiple zipping and unzipping cycles. Because numerous microhooks keep the integrity of the feather, their properties are of great interest for understanding mechanics of the entire feather structure. This study was undertaken to estimate the separation force of single hooklets and their arrays using force measurement of an unzipping feather vane. The hooklets usually separate in some number synchronously (20 on average) with the highest observed separation force of 1.74 mN (average force 0.27 mN), whereas the single hooklet separation force was 14 μN. A simple numerical model was suggested for a better understanding of zipping and unzipping behaviour in feathers. The model demonstrates features similar to those observed in experiments.
The feather of modern birds is an epidermal outgrowth having complex structure, which evolved since the Late Jurassic . The feather was inherited by birds from a common ancestor of birds and crocodilians, because it demonstrates the presence of a homologous trait in both animals groups . The theropod dinosaurs, closest related to birds, have pinnate feathers, while the feathers of theropods that are distantly related to birds are simple filaments or bunches of filaments of varying lengths and diameters. These hollow filaments or protofeathers are similar to structures seen in feather development: a hollow cylindrical sheath arises first in feather ontogeny from the collar of the feather follicle before the barb ridges .
The main function of feathers in birds and their ancestors is to support flight. However, discovery of non-flying Late Cretaceous feathered dinosaurs suggests that display function played a key role in the early evolution of pinnate feathers . The coloration of feathers is believed to have primarily evolved in response to sexual selection, as it is shown for modern birds . Thermoregulation or waterproofing could have been the original primary functions at least in smaller dinosaurs [5,6]. This is supported by the fact that feather density in smaller birds is higher than in larger birds, because smaller birds lose more heat owing to the relatively larger surface area in proportion to their body volume .
The feathers of modern birds are a waterproof, breathable, lightweight construction combining thermal isolation, rigidity and flexibility [8,9]. These properties are partly related to the feather's ability to hold its parts together temporarily by an elaborate pattern of microhooks (hooklets). Such feather interlocking devices are well known from zoology textbooks . The feather vane (plane part) can be separated into two parts by pulling neighbouring barbs apart. The original state can be re-established easily by lightly stroking through the feather.
The vane of a typical contour feather consists of a number of side branches called barbs, aligned parallel to each other and at an angle to the shaft. The barbs carry, on either side, a similar array of second-level side branches called barbules. In most birds, the row of barbules on the side of the barb towards the tip of the feather bears very fine, backward facing hooklets, as one can see on scanning electron microscopy (SEM) images (figure 1a,b). These barbules are called hook barbules. The barbules on the other side of the barb are sharply curved and have only very short spine-like outgrowths (bow barbules). Hook barbules become automatically anchored on the bow barbules, owing to the preferred orientation of the hooklets and micro-irregularities on both types of barbules.
The separation of the barbules can be caused by some external force. This protects the feathers from real damage and provides the easy repair of a microrupture in the feather coverage. Although the structure and function of the system are well known in general, the rupture forces have not been rigorously measured. Because numerous feather microhooks keep the integrity of the feather, their collective properties are of great interest for understanding rupture/recovery mechanics of the entire feather. Behaviour similar to the collective behaviour of microhooks at feather rupture is observed in many different processes, such as, for example, in the motion of the main plasticity carriers  or in Barkhausen noise in ferromagnetic materials .
This study was performed in order to obtain information about the rupture force dynamics affected by the cooperative hook's behaviour from force measurements and from a numerical model of an unzipping feather vane.
2. Material and methods
Pennaceous feathers of the swan (Cygnus olor) were used for the force measurements (figure 2). The feather was clamped at its vane in a position such that barbs were oriented parallel to the ground. The tip of the next free barbule was clamped to the force sensor (load cell force transducer, FORT 10, World Precision Instruments, Sarasota, FL, USA). The force sensor was attached to a motorized micromanipulator (DC3001R, World Precision Instruments) similar to . Vertical displacement of the transducer using a micromanipulator caused separation of two neighbouring barbs, figure 2b, pulling. Separation was performed with a speed of 200 µm s−1. Afterward, the force sensor was moved to its initial position, figure 2b, healing. Ten such measurements were performed on five individual feathers at the force sampling frequency of 1 kHz. The video of the barbule separation process in pennaceous feathers of swans and a rectrice of a pigeon (Columba palumbus) was recorded with 10 000 frames per second (50 µs exposition) using a high-speed camera (Photron Fastcam SA1) mounted on an inverted microscope Axio Observer.A1 (Carl Zeiss MicroImaging GmbH, Jena, Germany).
Feather material was collected from different sources (a laboratory collection) and stored in sealed plastic bags at room temperature prior to use. The numerical simulations were performed using Matlab v. 7.10 (The MathWorks, Natick, NA, USA).
3.1. Experimental results
High-speed video recording of the feather rupture process demonstrated that because barbules are oriented at some angle to barbs, the barbules were bent near their bases during feather stretching; compare left upper (free) and right lower (fastened) corners (figure 1c). Normally, during the feather stretching, the bending of the linked barbules increases, and at a certain point, the linked barbules uncouple from one another (figure 1c–e). The uncoupling process is very quick and takes just less than 0.1–0.3 ms. Previously deformed barbules return to their initial positions. During the feather stretching (rupture), figure 2a, the individual barbule separation events are seen as force drops, figure 2b. When the pulling force decreases to zero, the barbs return to their initial positions guided by the elastic forces, figure 2b, and anchoring between hooks and bow barbules recovers (healing process). The hysteresis of a force–distance curve at pulling/healing, figure 2b, occurs because barbule bending takes place at pulling but not at healing.
Force and stretching distance increments between uncoupling events (force drops) could be determined from the force–distance curves (figure 2b). The force-to-separation distance ratio remains constant (1.85 N m−1; figure 3a). This means that typically many barbules are involved in the separation process. The distribution of the hooklet connection strength should also be important. Because root mean square of the noise level was 1.5 µN, we were able to detect events of the separation of single hooklets. The distribution of force increments shows that events of single hooklet separation approximately correspond to a force of about 14 µN (see electronic supplementary material, figure S2). The force measurements demonstrate that a few hooklet separation events are most frequently observed (figure 3b). In the experiment, not single hooklets, but their clusters mainly separated from the counterpart barb at once. Average force increment, resulting from the separation of multiple hooklets, was 0.27 mN (s.d. = 0.27, n = 126). The highest force increment was 1.74 mN. Assuming that the force increments are proportional to the number of hooklets holding barbs together before the next uncoupling event takes place, the number of simultaneously separating hooklets can be estimated from the measured pulling force and from the force necessary for single hooklet separation. So, while pulling the row of barbules, 1–40 hooklets (mean = 19.4, s.d. = 19.4, n = 126) were separated at once.
On average, 0.79 hooklets (s.d. = 0.73, n = 158, up to 3.5 hooklets) were separated at once at a pulling distance of 100 µm. The histogram of force increments was fitted with a power law and an exponential function. The exponential fit has lower value of χ2 (figure 3b).
Further, a dynamical mathematical model is introduced, which qualitatively describes zipping/unzipping behaviour in a feather. Using this model, the dynamics of the number of hooklets uncoupling during a feather stretching and statistical properties of the rupture forces were analysed.
3.2. Mathematical model
Our discrete numerical model describes the dynamics of two initially parallel aligned elastic barbs. The conceptual structure of the model is shown in figure 4, and the model parameters and their description are summarized in table 1. Each barb was constructed of 100 elastic segments, each having a length of dR = 60 µm, figure 4b, according to the measurements on SEM images (figure 1a,b). The barbs were provided with longitudinal () and transverse () stiffness, Transverse stiffness tends to hold the angle between the neighbouring segments close to 180°.
As in a real feather, the barbs in the model carry barbules. We constructed the barbules in a manner similar to that in the real barbs. Each of the barbules is elastically attached to a barb. In the model, each barbule was constructed of 10 segments. According to the measurements on SEM images the segment's length (dr) was taken the same as for the barbs, dr = dR. In feathers, the barbules are oriented at some angle to the barbs. The barbules in the model were oriented at β01 = 30° (bow barbules, upper side) and β02 = 45° (hook barbules, lower side), figure 4b. The barbules and barbs have different thickness and therefore different values of their elastic constants, Based on the SEM images, we assume that the barbules have approximately 10 times lower transverse and the same longitudinal elastic constants compared with the barbs: Such approximation allows emphasizing the bending degree of freedom and retaining other geometrical parameters about constant.
As in the experiment, the top right tip of the barb, figure 4a, was moved with constant velocity, V = 50 µm s−1, until the tip reached the final displacement of 2.1 mm. After this, the barb was released and returned spontaneously to its initial position. Low separation velocity compared with velocity of internal motions (for both real and numerical experiments) considerably reduces inertia effects.
A deformation of barbs produces elastic forces proportional to the barb stiffness, figure 4c. The forces are described by the following equations: 3.1where is a position vector of the middle of the segment (the node) j; k = j ± 1. Longitudinal force, , according to this equation, tends to keep a distance between the nodes close to the equilibrium length of the segment dR, and it is linear at small displacements (see electronic supplementary material, figure S1). Transverse force, , is directly proportional to the lateral deflection and tends to keep close to the mean value between its nearest neighbours, The transverse force in the present form is easy to realize, but it is not a purely bending force, because it may include a longitudinal component.
The segments of barbules are connected by the forces described in the same way as the forces between segments of the barbs (equation (3.1)), figure 4c,d: 3.2where is a position vector of the barbule's node j; k = j ± 1.
Additionally, a barbule–barb joint rotational stiffness, B, was introduced, which tends to keep the joint angles, βn, constant. The rotational force, figure 4e, acting on adjacent segments of the barb and barbules is linearly proportional to the difference β0n − βn: 3.3where index n = 1,2 numerates the barbs. At B > 0, the force in equation (3.3) tends to keep the angle as close to the equilibrium value as possible. As a result, the system dynamically maintains a shape similar to the one depicted in figure 4a.
The main feature of the feather is that bow and hook barbules stay connected during pulling until some critical force/displacement is achieved in the connection between hooklets and short spines. To simulate this, in the model, we assume that if a node of a hook barbule is close enough to nodes of the bow barbules, then it can form a ‘bond’ (connection between hooklet and spines, figure 4a,b,e). For the sake of simplicity, in an initial state, all possible bonds appear at a certain distance between the barbule segments.
The bonds can be characterized by the vectors , where indices n and m numerate the contacting nodes on both subsystems (hooklets and spines on bow barbules) and are position vectors of the nodes. When one end of a barb starts to move vertically, it pulls barbule segments from their initial position to a new one. This deformation causes an interaction force , figure 4e, which tends to return the distance between bound barbules to equilibrium. For small deviations, linearly increases with the distortion: 3.4where kint is a stiffness of the bond, is the equilibrium bond vector. At large deformations, the interaction force also becomes strong, and the hooklets detach. In the framework of this model, it means that if deformation exceeds some threshold value rth: 3.5then the bond breaks. In a real system, the threshold depends on the spatial realization of the particular contact and is different for different interconnections. In the model, it could be realized by a random distribution of the thresholds: 3.6where is a constant, is a uniformly distributed random number. For further simulations, we take The force-to-separation distance ratio was found to be in good agreement with experimental data for such a relation between and kint (see Discussion). Figure 4a reproduces the described model system in some intermediate configuration.
A single hooklet can form only one bond; therefore, all the bonds can be numerated. The total interaction force between two barbs was defined as a sum of these forces: 3.7where i as an index, N is the number of bonds. This is a force which has to be applied to the pulled edge of the upper barb to let it move with a constant velocity.
The motion of barb/barbule segments was calculated according to a standard equation of a particle motion in a viscous medium: 3.8where m is a mass of segment (m = 1), γ is a coefficient damping oscillations in a system (γ = 1.1), Σf is a sum of all the forces acting on the segment. The integration time step was taken 0.02 s.
The further the upper barb moves from its initial position the greater is the number of broken bonds. Owing to the barbule's elasticity, clearly visible in figure 4a (as well as in the electronic supplementary material, movie S1), at some intervals, the system can gradually deform to some extent without breaking any bond. The longer such intervals last, the larger is the force F, and the higher the probability of simultaneous separation of many hooklets at once or generation of an avalanche of the consequent bond-breaking becomes (figure 1c–e and the electronic supplementary material, movie S1). Such collective effect makes the feather unzipping process nontrivial.
To describe the collective behaviour of the bonds quantitatively, we introduce a specific measure. The difference between total number of connected bonds at the beginning N(t) and at the end N(t + τ) of a time interval [t,t + τ]. Being normalized to the length of the interval τ, this value coincides with a time derivative of function N(t) averaged over the corresponding time interval: 3.9This value is a convenient measure to characterize a rate of bond-breaking and -restoring. The time interval chosen was τ = 0.5 s. During this time, a typical avalanche of bond-breaking develops.
3.3. Results of numerical simulations
Typical results of numerical simulations are summarized in figure 5. Figure 5a–c presents the dependences of N, averaged derivative and force on the displacement of the pulled end of the barb, respectively. The dashed line crossing all these subplots separates an initial transient process, during which no bond-breaking occurs, from the subsequent stationary regime of bond rupture. For convenience, details of the model behaviour, shown in figure 5a–c, are given in figure 5d–f in smaller time intervals. Dashed-dotted lines here mark pronounced correlations between force drops after long periods of its continual growth and avalanches of bond-breaking, clearly recognizable by the minimums of
The upper barb gradually returns to its equilibrium position when the external pulling force disappears. At some stage of the recovery process (healing), the broken bonds start to be restored. A bond appears again when the distance between a hooklet and its correct contact position became less than rth. The thin line in figure 6 represents evolution of N during a typical unzipping–zipping cycle. First, the unzipping takes place until displacement reaches a terminal distance, indicated by the vertical black line. Then, the healing takes place until the displacement reaches the initial position. One can see that the healing ends up with a number of contacts smaller than it was in the initial state. This is clearly seen from the inset in figure 6 (thin line). The recovery is not complete, because not all hooklets find the correct contact position to restore the bonds, even if the barbs and barbules have returned to their original configuration.
The same might happen in a real feather. However, such a stalemate can be effectively overcome by an action simulating the behaviour of a bird, which actually helps to restore the hooklet contacts by periodic motions of its beak (preening) or producing feather vibrations during flight. To simulate these in the model, we applied periodic force along the horizontal x-direction to the internal rows of the nodes, which start to oscillate. The period of the oscillations was optimized. It was adjusted to a time close to that typical for the usual zipping–unzipping avalanche of the bonds. Thick line in figure 6 presents an unzipping–zipping loop of the number of contacts, N, in the presence of the periodic force applied at the end of the healing process (the barbs were close to the initial position, but many of the bonds were not yet restored). It is clearly seen from the plot that this line deviates from the thin line, starting from the moment when the periodic force was applied. After this moment, the curve in the presence of the forced oscillations goes higher than that without (thin line), and, finally, it ends with the number of contacts which practically coincides with the initial number for the original unperturbed system.
Figure 7 shows an obvious correlation between lateral oscillations of the force and rate of bond recovery. Dashed-dotted lines mark the correspondence between the force maximums (associated with the maximal displacement of the elastic fibres) and stimulated avalanches of the bond recoveries, quantitatively represented by the positive bursts of the derivative .
The zigzag shape of the distance–force dependence during the separation of interlocking hooklets on hook barbules from the spine-like outgrowths on bow barbules is important for mechanical stability of covering feathers under external forces caused by air microturbulence in flight. The variation of the rupture force from 0.014 to 1.7 mN demonstrates that a feather can withstand not some average pulling force (0.27 mN), but a much higher one (1.7 mN as a maximum in our case). Such high rupture force may be observed because of the cooperative effect arising from the inhomogeneous hooklet binding strength distribution. If all the bonds are equally strong, then the constant average rupture force is measured . A strong correlation between the force increment (ΔFp) and the pulling distance (Δδ) required for a single separation event, figure 3a, means that the feather during the stationary phase of pulling possesses almost constant stiffness. Our model demonstrated the same correlation (figure 8a). The ratio between and kint was varied in the model to achieve the correlation between ΔFp and Δδ close to that observed in experiment. Absolute value of was also varied to achieve a shape of the force increment (ΔFp) histogram in the model similar to the shape of the experimental histogram (the number of bins was also varied).
Typically, the histogram of jump (avalanche) sizes in the motion of the main plasticity carriers  or in Barkhausen noise of ferromagnetic materials  follows a power law distribution. However, in our model and in experiments with real feathers, the best fit of the histograms of the force increment distribution ΔFp is an exponential function (figures 3b and 8b). This indicates an independence of a separation event from the previous one. This conclusion is supported by a low correlation coefficient for the consecutive force increments (R2 = −0.087).
An interesting feature of such an interlocking mechanism is that microstructures are not damaged after separation, but can resist multiple zipping–unzipping cycles. The origin of hooklets permitted the origin of compacted closed and aerodynamic efficient vanes in large asymmetric contour feathers [15,16]. The hooklets are grasping devices holding together barbules and barbs, which are built from keratin with Young's modulus of about 2.5 GPa . Microhooks responsible for keeping mechanical stability of the wing structure during flight are known from insects, such as representatives of Heteroptera and Hymenoptera. Here, hooks provide functional diptery. Forewing microhook arrays holding the hindwings of the bug Coreus marginatus in flight can be separated from the hindwing at a force of 0.8 mN. The array consists of numerous (more than 20) 40–60 µm long hooks . The number of wing hooks holding fore- and hindwings together correlates with the flight distance for bees .
Because in the plumage of birds the feathers have different morphology, it would be interesting to compare how the rupture force differs in different types of feathers and how the force is related to the stiffness of the vane, and to a particular function of the feather. Interspecific comparison might provide some insights into the microstructural adaptations to feather specializations.
The numerical model presented here differs from similar models of artificial systems such as Velcro models in geometry  and in the introduction of bonding strength distribution compared with . Our model demonstrates the features of real feathers observed in mechanical experiments. It demonstrated the same shape of the force–distance curves and system healing when an external force disappeared. Moreover, the model demonstrated a more effective system healing when a periodic force, modelled birds preening, was applied to the feather vane.
This work was supported by the German Science Foundation (DFG) Initiative ‘Bionik’ (DFG grant no. GO995/7-1 to SG).
Victoria Kastner (Max Planck Institute for Intelligent Systems, Stuttgart, Germany) helped with the linguistic corrections of the manuscript.
- Received October 26, 2013.
- Accepted November 26, 2013.
- © 2013 The Author(s) Published by the Royal Society. All rights reserved.