Julien Avenet
email: julien.avenet@univ-nantes.fr

Thomas A Cender
email: tcender@udel.edu

Steven Le Corre
email: steven.le-corre@univ-nantes.fr

Jean-Luc Bailleul
email: jean-luc.bailleul@univ-nantes.fr

Arthur Levy
email: arthur.levy@univ-nantes.fr

Experimental correlation of rheological relaxation and interface healing times in welding thermoplastic PEKK composites

Keywords: A. Thermoplastic resin, B. Adhesion, D. Rheology, E

The evolution of polymer welding time to join the interface of thermoplastic Polyetherketoneketone (PEKK) matrix composites was characterized. Using a creep-recovery test protocol to analyse the evolution of rheological behavior, it was shown that for temperatures above the melting point (T > 360 o C), the polymer visco-elastic relaxation time increases exponentially with time-indicating polymer crosslinking or branching. To confirm that relaxation time is proportional to interface healing time, the rheological results were compared to the fracture energy release rate (G 1c ) of samples prepared on a custom bench which is capable of isothermally weld composite samples for controlled time intervals.

The welding time corresponds directly to the relaxation time. Furthermore, the thermal history that the polymer experienced results in cross linking and influences the welding time. The implication is that temperature exposure from prior processing steps (IR oven pre-heating or autoclave consolidation) can limit the ability to weld components in a predictable way.

Introduction

Composite aerospace structures are often composed of large monolithic components because composites are uniquely vulnerable in the vicinity of fasteners [START_REF] Yan | Experimental study on clamping effects on the tensile strength of composite plates with a bolt-filled hole[END_REF][START_REF] Ageorges | Advances in fusion bonding techniques for joining thermoplastic matrix composites: A review[END_REF]. Where possible, bonding without fasteners is a more desirable joining approach. Thermoplastics matrix composites offer the ability to be joined by welding, which has several advantages in terms of materials and manufacturing processes. Due to the lack of chemical crosslinks, thermoplastic polymers can be melted and remolded. In the melt state, thermoplastics molecular chains can relax and slide out of their entanglements. During welding, the process of melted thermoplastic molecular chains diffusing across the interface (to form new entanglements) is known as healing. Aromatic thermoplastics, such as those of the PAEK family, are good candidates for many high performance engineering applications. In particular, carbon fiber reinforced PEKK could be used in many aerospace structures because of its high stiffness and strength [START_REF] Sun | Modeling of elastic-plastic behavior of ldf TM and continuous fiber reinforced as-4/pekk composites[END_REF], high glass transition temperature, and high chemical resistance in the crystalline state [START_REF] Attwood | Synthesis and properties of polyaryletherketones[END_REF][START_REF] Ho | Crystal morphology and phase identifications in poly(aryl etherketone)s and their copolymers. 1. polymorphism in pekk[END_REF].

These same properties which make PAEK polymers high performance, also contribute to it's difficultly in rapid welding processes. The process of thermoplastic welding of composite parts was outlined well by Lee & Springer [START_REF] Woo | A Model of the Manufacturing Process of Thermoplastic Matrix Composites[END_REF].

For their early generation of APC-2 PEEK polymer composites, they reported times to reach full weld to be well over an hour. Such long processes times are completely incompatible with rapid welding processes, such as resistance or induction welding, or even the automated fiber placement process. However, such processes are commonly used with PEEK and PEKK matrix composites with mixed or good result. In this article, we outline a methodology to characterize the weldability of thermoplastic PEKK composites and demonstrate that polymer degradation (resulting from thermal history) will lengthen the time required-thus limit the ability-to weld a polymer interface.

To weld two thermoplastic composite components, the interface must be heated to a temperature near or above the 'melting-point' while the interfaces are in contact [START_REF] Ageorges | Advances in fusion bonding techniques for joining thermoplastic matrix composites: A review[END_REF]. The mechanism for heating varies [START_REF] Yousefpour | Fusion Bonding/Welding of Thermoplastic Composites[END_REF]: hot gas, infrared, friction, electrical resistance, induction-the goal is to localize the application of heat to the joint interface, as to avoid melting and deconsolidating the bulk part.

Since surfaces are never perfectly smooth, the problem of achieving intimate contact involves the deformation of the interface under pressure [START_REF] Woo | A Model of the Manufacturing Process of Thermoplastic Matrix Composites[END_REF][START_REF] Levy | Modeling of the heating phenomena in ultrasonic welding of thermoplastic composites with flat energy directors[END_REF]. At elevated temperature, the polymer is in a viscous state, so by applying pressure, roughness at the interface will be filled by squeeze flow of the polymer. Without sufficient pressure and temperature, achieving full intimate contact could be a rate limiting step. Continuous welding processes such as tape placement, ultrasonic or induction welding, apply high pressure locally to ensure a full intimate contact is achieved quickly (within the process time scale) [START_REF] Butler | An analysis of mechanisms governing fusion bonding of thermoplastic composites[END_REF].

Once intimate contact is achieved, healing often appears as the rate limiting phenomenon in the case of fast or continuous welding processes [START_REF] Butler | An analysis of mechanisms governing fusion bonding of thermoplastic composites[END_REF]. The healing of a thermoplastic interface occurs as chain ends diffuse across the boundary and entangle with the adjacent polymer [START_REF] Wool | A theory of crack healing in polymers[END_REF][START_REF] Hwa | A Theory of Healing at a Polymer-Polymer Interface Young[END_REF][START_REF] Wool | Welding of Polymer Interfaces[END_REF]. The mechanical strength of the interface, σ, increases with time in contact, t, until the interface achieves the strength of the fully healed composite interface σ ∞ . This ratio is defined as the degree of healing, D h .

D h = σ w σ ∞ = G 1c,w G 1c,∞ 1/2 = t w t w,∞ 1/4 (1)
Here, w, denotes that this is a welding process, and t w,∞ is the welding time (t w ) required to complete the welding process, such that the interface strength (σ w ) reaches the strength of the bulk polymer (σ ∞ ). Alternatively, D h can be expressed in terms of the weld fracture energy release rate, G 1c,w , relative to that of the bulk polymer, G 1c,∞ . Eq. (1) has been verified many times for different polymer of varying molecular weight [START_REF] Wool | A theory of crack healing in polymers[END_REF][START_REF] Hwa | A Theory of Healing at a Polymer-Polymer Interface Young[END_REF][START_REF] Wool | Welding of Polymer Interfaces[END_REF]. In isothermal conditions, the welding time, t w,∞ , of a monodisperse linear polymer of given molecular weight may be modelled by an Arrhenius law:

t w,∞ = A exp E a RT (2)
From Eq.( 1) the welding time is classically considered to be identical to the reptation time of the polymer, t R = t w,∞ [START_REF] Wool | A theory of crack healing in polymers[END_REF][START_REF] Bastien | A non-isothermal healing model for strength and toughness of fusion bonded joints of amorphous thermoplastics[END_REF]. The reptation time is intimately linked to the viscoelastic nature of the polymer: it depends on the chemistry, molecular structure, molecular weight, and temperature. The most reliable method to characterize t R is a stress-relaxation test. This can be performed on a displacement controlled rheometer. With the polymer melt situated between the plates (cone or parallel) and in an equilibrium relaxed state, a known shear strain is abruptly applied. Locked into the fix shear state, the shear stress in the polymer decays over time as the polymer relaxes. From this test the relaxation time, λ R , is determined from the Maxwell model for a viscoelastic fluid,

λ R = η 0 G N 0 (3)
where η 0 is the zero shear viscosity (of the newtonian plateau) and G N 0 is the plateau shear modulus [START_REF] Smail | Physically meaningful discrete relaxation spectrum: Rheological behaviour of monodisperse polymer melts[END_REF]. This type of test, as long as it stays within the linear viscoelastic regime, can be very characteristic of the polymer reptation time. In the initial sheared state, polymer chains must disentangle from their strained configuration and gradually reach a state of no stress, as they complete a full reptation process. As such, we can infer that the relaxation time is equivalent to the polymer reptation time [START_REF] Bastien | A non-isothermal healing model for strength and toughness of fusion bonded joints of amorphous thermoplastics[END_REF][START_REF] Smail | Physically meaningful discrete relaxation spectrum: Rheological behaviour of monodisperse polymer melts[END_REF].

Complex fluids like polymer melts are often not well characterized by a single relaxation time. The visco-elastic response can be fitted with a series of relaxation modes

G(t) = G i exp (-t/λ i ) (4)
where G i and λ i are a discrete set of moduli and relaxation times which capture the rheological response. If the set of discrete parameter pairs is small (4 or fewer), a physical meaning can more easily be assigned to each. Conversely, with an infinite set, a continuous relaxation spectrum can be used to transform all linear visco-elastic functions between each other (relaxation modulus, creep-compliance, complex modulus/compliance) [START_REF] Ferry | Viscolelastic Properties of Polymers[END_REF]. This allows a variety of rheological protocols to capture the same behavior on different time scales. From the discrete spectrum, the longest relaxation time is said to be equivalent to the interface healing time [START_REF] Regnier | Modeling of Thermoplastic Welding[END_REF].

In order to directly assess the degree of healing in a welded joint, samples must be welded under controlled conditions (pressure, temperature, and time)

and then tested mechanically to observe the interface strength. Lee and Springer [START_REF] Woo | A Model of the Manufacturing Process of Thermoplastic Matrix Composites[END_REF] proposed to test samples bonded under a press under mode I tensile loading.

Their study was focused on autoclave processing of laminates, rather than rapid welding processes. Lamethe et al. [START_REF] Lamethe | Polymer dynamics applied to PEEK matrix composite welding[END_REF] proposed a methodology to measure the fracture energy of samples welded under controlled nonisothermal conditions. In a previous work, Avenet et al. [START_REF] Avenet | Adhesion of high performance thermoplastic composites: Development of a bench and procedure for kinetics identification[END_REF], developed an apparatus (TACOMA bench)

for welding full double cantilever beam specimens isothermally with residence times as short as 5 seconds by employing rapid cooling. The methodology was very effective at determining the welding time in Eq. ( 2) through a series of tests (varying temperature and time) fit to Eq. 1.

Historically, welding times for commercially available aerospace grad PEEK and PEKK polymers have been shown be on the order of 10 3 s [START_REF] Woo | A Model of the Manufacturing Process of Thermoplastic Matrix Composites[END_REF][START_REF] Yang | Healing of thermoplastic polymers at an interface under nonisothermal conditions[END_REF][START_REF] Yang | Nonisothermal Healing and Interlaminar Bond Strength Evolution During Thermoplastic Matrix Composites Processing[END_REF] ; however, some currently available PAEK polymers are better tuned for welding processes with residence times of the order of a few seconds. Choupin et al.

[21] demonstrated that polymer degradation is a major challenge when processing PEKK (or PEEK) composites: above the melt temperature (340 o C) the polymer backbone can break and initiate crosslinking; thus, limiting the ability to crystallize and increasing the polymer viscosity. An exponential increase in viscosity over time and temperature (in an Nitrogen atmosphere) showed that crosslinkining was a temperature driven kinetic process [START_REF] Choupin | Mechanical performances of PEKK thermoplastic composites linked to their processing parameters[END_REF]. They confirmed an increase in molecular weight with Gel Permeation Chromatography (GPC).

After presenting the PEKK 7002 material used in this study, section 3 presents the procedure to identify the longest relaxation time of the polymer.

In section 4 the mechanical welding time is characterized with the existing TACOMA welding bench and a double cantilever beam test. Finally, the corre-lation between these two approaches is discussed in the last section. Thermal history is also shown to impact the weldability.

Materials

In this study, we characterize PEKK 7002 produced by Arkema. The 7000 series designates that the polymer has a ratio of 70:30 of T:I (Terephthaloyl to Isophthaloyl) and the final digit specifies the melt volume flow index. The PEKK 7002 has a rather low index and thus a high viscosity and high mechanical properties. On the contrary, its processability is limited.

Rheological samples

For rheological characterization, flat discs of diameter 25 mm and thickness 2 mm were manufactured from neat PEKK pellets. Pellets were placed inside of a mold at 350 o C for 5 minutes (until appearing amorphous) and then compression molded at 20 bars. Subsequently, while under pressure, the mold was cooled with water; however, the samples were able to crystallize. The crystallinity was not important at this stage because once in the rheometer, the samples were certain to melt. The importance of this process is that the cycle time was short, so that degradation would be minimized, and samples had no air bubbles when placed in the rheometer.

Adhesion samples

Carbon fiber composite plates were manufactured with a [0, 90] 3s manual layup using 194 g /m 2 carbon-PEKK prepreg from Toho Tenax. On one of the surface, a 50 µm thick film of neat PEKK 7002 polymer was applied to ease the welding process. The plates were then consolidated in an autoclave according to the cycle shown in Fig. 1 where the sample was kept above 350 • C for 90 minutes.

Note that a typical autoclave cycle for thermoplastic composite consolidation include a 30 minute dwell at the peak temperature; however, in this study the cycle was adapted from a standard thermoset procedure and has a longer dwell than the one recommended by the manufacturer.

The 3.2 mm thick composites plates were cut using a Protomax waterjet cutting machine to the sample specification of 25 mm by 125 mm (cf Figure 7)

where the surface ply orientation is along the length direction. A countersunk hole was then machined on the matrix-rich surface, 10 mm from one end, to insert a homemade loading pin used for gripping the sample during the mechanical test [START_REF] Avenet | Adhesion of high performance thermoplastic composites: Development of a bench and procedure for kinetics identification[END_REF]. After a manual cleaning of the surfaces with acetone, the samples were placed into a vacuum chamber at 180 • C for 20 hours to remove the residual water contained into the samples and limit deconsolidation during the heating phase of the welding process.

The subsequent controlled welding process is described later, in section 4.1.

Thermal history effect

Concurrently, in order to quantify the effect of thermal degradation on the welding ability of the composite samples, aging tests were conducted on the already autoclave-consolidated composite plates. To do that, an autoclave composite plate was vacuum-bagged in an oven and heated up to 380 • C with an isothermal step of 2 hours. Vacuum bagging was chosen to prevent deconsolidation during the thermal cycle and temperature was recorded with a thermocouple placed on the top of the plate during the whole aging process (Fig. 1). The aged composite plates were then cut and dried according to the experimental protocol previously described, before welding stage.

Rheological time identification

To characterize the rheological behavior of PEKK in the molten state, a

Thermofischer HAAKE MARS III rheometer was used. The temperature controlled chamber (CTC) configuration was used to maintain the sample isother- To prepare the test, the rheometer was calibrated at room temperature.

Without the sample present, the rheometer was preheated to the test temperature. Once the chamber temperature had stabilized, the 25 mm samples was placed on the 35 mm plate and the chamber was closed. The gap was adjusted to 1 mm, where the polymer would wet to the edges of the plates.

Oscillatory tests

It is known that PEKK polymers are not stable above their melting temperature. This is characterized by the increase of viscosity over time [START_REF] Choupin | Macromolecular modifications of poly(etherketoneketone) (pekk) copolymer at the melting state[END_REF] that could be due to crosslinking of the polymer macromolecules.

The effect of temperature on the rheological properties of the polymer was assessed by means of oscillatory rheological tests at constant temperature at a frequency of 1 rad/s with 1% strain. The test was repeated for 4 temperatures: 360, 380, 400 and 420 o C. The temperatures were chosen to be higher than the melting temperature of the materials and are characteristic of most of the industrial processes. Autoclave manufacturing for example is characterized by long residence times at high temperature which could induce macromolecular changes.

Figure 2 shows an important increase in the storage modulus at high temperature which could be attributed to a decrease in chain mobility and confirms crosslinking of the materials. This crosslinking seems also to be faster at higher temperatures validating that this is a thermo-dependent phenomenon as already demonstrated for PEKK [START_REF] Choupin | Macromolecular modifications of poly(etherketoneketone) (pekk) copolymer at the melting state[END_REF] and PEEK [START_REF] Courvoisier | Multi-scale and multitechnique analysis of the thermal degradation of poly (ether ether ketone)[END_REF]. As both the storage and loss moduli increase with dwell time, the intersection between G and G" takes place at lower frequencies and the relaxation times are consequently higher. The experimental procedure starts with a frequency sweep, with 1% strain.

Creep recovery tests

Beginning at 100 rad/s, the frequency is stepped down logarthimically by 5 frequencies per decade and ending at 0.1 rad/s. The end point 0.1 rad/s was chosen for consideration of the acquisition time: each additional frequency takes exponentially longer to complete. In total, the frequency sweep takes a total of 12 min to complete. At the end of the frequency sweep, the material is allowed to recover for 3 min to ensure there is little residual stress from the previous oscillatory test. Once the samples nearly stops deforming ( γ <0.001 s -1 ), the creep-recovery test is initiated. In the creep-recovery test, the applied stress must be set such that the observed shear rate is low enough to ensure the material is within the linear elastic regime, but not too low for the acquisition of the rheometer. Initial tests showed that an applied stress of 5 Pa to 30 Pa is adequate, depending on the state of crosslinking. Additionally, the duration of the applied stress, t = t 2 , must be sufficiently long to ensure the material is deformed past its elastic limit, to the point where the deformation is only viscous (t > t 1 ). This time corresponds roughly to the relaxation time of the material. Initially the deformation time can be short (300 s), but as the material crosslinks, longer deformation time is necessary, (1800 s). After the creep test, the material is allowed at least twice as long (at zero stress) to recover from residual stress caused by the previous oscillatory test. At the end of the creeprecovery test, the sequence is repeated, beginning again with the frequency sweep.

! " # $%"& ' ( # ! ) 1 η 0 t 2 J 0 e t 2 η 0 t 1
The creep compliance

J(t) = γ(t) σ 0 = J e (t) + t η 0 (5)
is the sum of the elastic deformation (recoverable) and the viscous deformation (non-recoverable). In Figure 3 an annotated creep-recovery curve shows how material constants (η 0 and J 0 e ) are measured. After some time (t = t 1 ), the shear rate becomes constant and the elastic deformation reaches a plateau which defines the compliance limit, J 0 e . The compliance limit, J 0 e , was taken from the recovery period since a long recovery period was required in the test protocol to ensure the polymer was in a quiescent state before the next iteration.

A Fourier transform is able to convert the creep compliance from the time domain to a frequency domain and to obtain the storage compliance J [START_REF] Ferry | Viscolelastic Properties of Polymers[END_REF] and the loss compliance J

J (ω) = J 0 e -ω ∞ 0 J 0 e -J(t) + t η 0 sin (ωt) dt (6)
1 ωη 0 + ω ∞ 0 J 0 e -J(t) + t η 0 cos (ωt) dt (7)
Practically, the integration was performed over the duration of the creep period of the test (0 ≤ t ≤ t 2 ). The storage and loss compliance can be simply converted to their respective moduli by the relations:

G = J J 2 + J 2 (8)
and

G = J J 2 + J 2 (9)
Figure 4 shows the evolution of the storage and loss moduli over frequency.

The lines correspond to the data acquired with the classical oscillatory rheology whereas the dotted lines are constructed from the Fourier transform of the creep recovery tests in order to achieve lower frequencies. The Figure shows that with time increasing, the G and G curves are shifting to the higher values indicating a network change attributed to crosslinking.

In the limit of very low frequencies G = J 0 e η 2 0 ω 2 and G = ωη 0 . Therefore, a good approximation for the longest relaxation time would be at the intersection G = G . Using the Cox-Merz rule that ω ≈ γ ≈ 1/λ, we can see that

λ R = J 0 e η 0 (10)
Thus, the longest relaxation times were finally determined from the creep recovery tests and were calculated as the product of the elastic compliance limit and the newtonian plateau viscosity (Eq.10). The procedure was repeated over time at constant temperature to observe the changes in relaxation times induced by thermal history.

Mechanical time identification

In order to correlate the rheological times characterized in the previous section with characteristic mechanical time related to the healing phenomenon, a welding campaign is conducted. Each repetition shows a longer relaxation time.

Figure 5: Principle of the experimental TACOMA setup, designed for welding under isothermal conditions. Welding process is stopped thanks to a fast water cooling.

Welding procedure

Welding experiments were carried out on an experimental welding bench (TACOMA), presented in a previous article [START_REF] Avenet | Adhesion of high performance thermoplastic composites: Development of a bench and procedure for kinetics identification[END_REF] and shown in Fig. 5, under constant isothermal conditions. This experimental welding bench was designed to enable quenching of the composite sample and thus to perform stopped welding experiments. Thermocouples were embedded 1 mm from the sample surface in the copper mold to control and monitor the temperature and the heating was performed thanks to heating cartridges. Thermal homogeneity is ensured since all the samples are heated and pressed together in a confined mold. Furthermore, two channels were machined in each platen near the sample surface for water cooling of the setup. The pressure was applied by means of a pneumatic piston attached in series to the moving platen and the contact force was recorded by a 10 kN load cell.

Prior to welding experiments, the coupons were covered with polyimide tape to facilitate removal after the welding sequence. First, the two composite samples were placed in the setup, resting on their sides and the matrix-rich surfaces were facing each other but they were not in contact. A 50 mm long non-adhesive aluminum foil was also placed between the coupons as shown in Fig. 7. This allowed the creation of an initiation crack for fracture test. As shown in Fig. 6, the device is heated from room temperature up to a temperature higher than the melting temperature of the polymer matrix (1) until homogenization of the temperature inside the mold and at the interface of the coupons (2). Once the setpoint temperature is reached and homogeneous inside the mold, contact is initiated between the coupons with the use of the pneumatic piston to weld the samples during a defined contact time t c (3). Welding process is rapidly stopped thanks to the fast cooling of the setup with water circulation (4) and finally the pressure is released (5).

Double cantilever beam fracture test

Mechanical quality of the welded interfaces was assessed by mode I double cantilever beam fracture test according to ASTM standard. Samples were mounted on a 100 kN Zwick Roell tensile machine thanks to the loading pins and homemade gripping system. Tests were conducted in quasi-static condition at a fixed displacement rate of 1 mm.min -1 . During the test, the displacement and force were continuously recorded. The welded samples were not cut after welding so that the dimensions of the Double Cantilever Beam samples are also 25 mm by 125 mm.

The load-displacement curves from the double cantilever beam tests were characterized by a strong unstable crack propagation [START_REF] Avenet | Adhesion of high performance thermoplastic composites: Development of a bench and procedure for kinetics identification[END_REF] probably caused by a non-uniformity of the matrix-rich layer at the interface [START_REF] Sacchetti | Effect of resin-rich bond line thickness and fibre migration on the toughness of unidirectional Carbon/PEEK joints[END_REF]. Interlaminar fracture energy release rate

G 1c = 3P δF 2ba 0 (11)
was then calculated at the initiation of the crack, corresponding to the first drop of load. In the above, P is the peak load, δ the displacement, b the width of the sample, a 0 the initial crack length which is the length of the non-adhesive inserted aluminum foil and F is a correction factor accounting for large displacement effects, i.e. shortening of the moment arm and tilting of the loading pins. It has previously been demonstrated that this value is representative of the overall adhesion of the assembly [START_REF] Avenet | Adhesion of high performance thermoplastic composites: Development of a bench and procedure for kinetics identification[END_REF].

Analysis

The welding time was determined experimentally for three isothermal temperatures (355, 370 and 380 • C) by means of welding tests performed using the TACOMA setup for different residence times. Adhesion energy was determined from load-displacement in DCB fracture tests with Eq. 11. Degree of healing at the crack initiation was evaluated via Eq. 1, where G 1c is evaluated for each sample of corresponding welding time. G1c, ∞ was determined to be 2.3 kJ.m -2 from a stable maximum plateau where additional welding time results in no improvement in interface strength-corresponding to the fracture energy of the fully healed interface. This high value compared to other high performance composites but it has been related to the amorphous state of the interface [START_REF] Avenet | Adhesion of high performance thermoplastic composites: Development of a bench and procedure for kinetics identification[END_REF].

Degree of healing was then plotted against the fourth root of contact time for the three isothermal temperatures (see Figure 8). In this case, the first data points were shown to be not associated to pure healing mechanism but also to the establishment of intimate contact. Therefore, these points were discarded and the welding times were directly taken from the slope of the linear fit. They are given in Table 1. Finally they were plotted against the inverse of temperature to identify the corresponding Arrhenius relationship given in Eq. ( 2), where the parameters E a = 55,5 kJ.mol -1 and A = 3,01.10 -3 s are respectively the activation energy and the identified pre-exponential factor.

Discussion

Rheological test as a mean to quantify adhesion

From the rheological characterization it was possible to predict the welding time of the composite based on its thermal history. The relaxation times are plotted for each isothermal temperatures against dwell time in Fig. 9. It shows that for a fresh polymer matrix, i.e. for t = 0, the relaxation time is of about 1 second. This corresponds to the data obtained in the literature [START_REF] Lamethe | Polymer dynamics applied to PEEK matrix composite welding[END_REF][START_REF] Regnier | Modeling of Thermoplastic Welding[END_REF]. As preprocessing time increases, the relaxation time increases consequently and can even reach values of a few thousand seconds as it was announced in [START_REF] Woo | A Model of the Manufacturing Process of Thermoplastic Matrix Composites[END_REF][START_REF] Yang | Healing of thermoplastic polymers at an interface under nonisothermal conditions[END_REF][START_REF] Yang | Nonisothermal Healing and Interlaminar Bond Strength Evolution During Thermoplastic Matrix Composites Processing[END_REF].

Adhesion measurements on the composite at different temperatures allowed the identification of the healing kinetics and the associated welding time. It should be recalled that the composite plate has originally been processed at 380 • C for one hour (90 minutes above 350 • C) in autoclave before welding experiments. Although this is not the cycle recommended by the manufacturer, the adhesion data is characteristic of these aging conditions. Concurrently, the rheological protocol aimed at replicating the pre-processing of the composite and characterizing the corresponding relaxation time. Thus, for a dwell temperature of 380 • C and dwell time of 3600 seconds the value of the welding time for this composite plate is of a hundred of seconds.

In parallel, the adhesion measurements from the welding experiments showed that the time for a full healing of the interface is also of a hundred seconds depending on the welding temperature (see Table 1). This comparison between rheological and adhesion characterization finally showed a good correlation for a given process condition which is the autoclave pre-processing dwell in this case and proved that rheological characterization of the polymer matrix can thereby be a mean to quantify adhesion time in the case of composite material.

Such procedure is much more rapid since it only takes about one day to gather the data from rheological measurements using a commercial rheometer.

On the contrary, adhesion measurements are characterization campaigns that may last several weeks to perform. Moreover, the welding tests and the associated mechanical characterization require specific apparatus. Finally, adhesion measurements require a significant number of samples, obtained from preconsolidated laminates for the identification of the healing kinetics; when rheological characterization only necessitates polymer pellets which is thus much less expensive.

Effect of Thermal History

To confirm the effect of thermal history, welding tests at 380 • C were performed on the composite plates aged for 2 hours at 380 • C as described in section 2.3 using the TACOMA set-up according to the experimental protocol described in section 4.1. The force-displacement curves from the corresponding DCB tests are shown in Fig. 10 and compared with the force-displacement curves from the reference autoclaved composite plate. First, the initial slopes are the same for the two plates indicating that the elastic performances at room temperature of the composite coupons after thermal aging did not change. Then, micrograph analysis of the aged plate also showed no apparent defects. These validate that no deconsolidation occurred during the aging process. Thereby, the observed changes are solely related to the quality of the welded joint.

For the aged composite plates, only a few adhesion tests were successful and led to few data points. Indeed, such thermal aging was shown to be the lower limit for adhesion development and for most of the samples no adhesion could be obtained even for very long contact time. This is most probably related to a high degree of thermal degradation and thus higher crosslinking of the material.

Considering that the original composite plates were consolidated in autoclave for about 90 minutes at high temperature, the total dwell time of the aged plates is then of about 3.5 hours. From the experimental evolution of the relaxation time of the matrix previously presented, at such dwell times, the relaxation time and therefore the welding time is expected to be extremely long, of about one hour.

The corresponding initiation values of the interlaminar fracture energy are then plotted in Figure 10 against the square root of contact time for both the original and the aged composite plates. For the same welding temperature and contact time, the interlaminar fracture energy value for the aged composite is an order of magnitude lower than the reference composite plate. The thermal aging has affected the weldability of the material by hindering the chain diffusion at the interface. It is believed that the macromolecular network has branched and that it takes longer for the chains to move and cross the interface. As a result, the pre-processing will affect the welding process. For example, in the case of our composite material, the plates were consolidated in autoclave at 380 • C for at least 1 hour. Both rheological and mechanical approaches give an estimate of a hundred seconds for the welding time, but more severe thermal history hinder adhesion development. Continuous welding processes such as automated fibre placement which make use of fresher composite substrates like prepreg tapes are expected to gain full mechanical adhesion for much shorter processing time. But the thermal history experienced by high performance thermoplastic composite coupons could then be a major limiting step in the development of adhesion.

Conclusions

In this work, the welding kinetics of a high performance thermoplastic composite was characterized. The study is twofold. First a rheological procedure gave the longest relaxation time of the polymer. A creep recovery method enabled to overcome the polymer degradation issue. Then a mechanical study of coupons welded using the existing TACOMA bench gave the mechanical healing times. Mechanical tests were also performed with an over-aged composite plate.

The main contributions of the work are the correlations between the matrix rheological times and the composite mechanical bonding times. Additionally, PEKK 7002 was shown to evolve with time at high temperature. This is attributed to crosslinking and impacts the relaxation time. This effect is consistently observed on the healing time identified with mechanical tests. Thus, polymer degradation during the pre-processing step, such as autoclave cycle for instance, influences the subsequent weldability of the structures. As with every polymer processing scheme, the thermal history should be managed in order to minimize polymer degradation. The specimens used in this study reported welding times on the order of 100 seconds but the autoclave cycle included a 90 minute dwell period, which is longer than recommend. It is shown here that a shorter autoclave cycle will produce more weldable samples.

Figure 1 :

 1 Figure 1: Pre-processing cycles used to produce composite laminates. The reference plate is manufactured with a five hour autoclave cycle which includes a 90 min dwell at 330 • C followed by a 80 min dwell at 380 • C. Samples selected to undergo further thermal degradation were post-processed in and oven vacuum-bag for 2hrs at 380 o C.

Figure 2 :

 2 Figure 2: Evolution of the storage modulus and loss modulus with dwell time under a frequency of 1 rad/s with 1% strain at constant temperatures.

Figure 3 :

 3 Figure 3: Creep compliance versus time obtained with the creep recovery test for 100 Pa at 380 • C. The close-up shows that the viscous regime is reached quickly in this case.

Figure 4 :

 4 Figure 4: Storage and loss moduli versus frequency for PEKK 7002 in N 2 gas at 350 o C. The high frequency range (short time response) is captured with an oscillatory frequency sweep, whereas the low frequency range (long time response) is measured by a creep-Recovery method and converted to frequency space with a Fourier transform. Iterating the protocol sequence shows how the visco-elastic response changes as the polymer degrades over time.

Figure 6 :

 6 Figure 6: Experimental welding procedure. The heating (1) and temperature stabilization (2) occur before the samples are put in contact. The welding time is tcontact (3) and quenching occurs in phase (4).

Figure 7 :

 7 Figure 7: Double Cantilever Beam (DCB) fracture test set-up. a 0 is the initiation crack length.

Figure 8 :

 8 Figure 8: Degree of healing at the crack initiation versus the fourth root of contact time for three isothermal temperatures. Welding time can directly be obtained from the slope of the linear fit [18].

Figure 9 :

 9 Figure 9: Evolution of the longest relaxation time of PEKK 7002 versus dwell time. Relaxation time increases with material evolution. This indicates that the pre-processing of the material will subsequently affect the welding process.

Figure 9

 9 Figure 9 presents, the evolution of relaxation time at 385 • C. This corresponds to the data acquired after in-situ crosslinking of the polymer matrix in the rheometer at this temperature of 385 • C and over a wide range of dwell times. If one considers the processing conditions previously recalled for the PEKK-Carbon composite plate of this study, it is possible to estimate the analogous relaxation time and the welding time.

4 )Figure 10 :

 410 Figure 10: Comparison between the reference autoclaved plate and the aged plate at 380 • C for 2 hours. Coupons were welded at 380 • C with different contact times. (a) Force-displacement curves and (b) degree of healing. Even at processing temperature, the additionally dwell drastically reduces the welding potential of the carbon PEKK system

Table 1 :

 1 Mechanical welding times characterized at different isothermal temperatures

Acknowledgements

The authors would like to acknowledge the funding they received from the PERFORM program managed by IRT Jules Verne (French Institute for Research and Technology in Advanced Manufacturing Technologies for Composite, Metallic and Hybrid Structures). The authors wish to associate the industrial partners of the PERFORM project: Airbus, Stelia Aerospace, Faurecia, Naval Group and Daher. Part of this study was also funded by the SIDEFFECT project managed by IRT Jules Verne. The industrial and academic partners of this project are respectively Stelia Aerospace, Airbus, Europe Technologies, Hutchinson, Cetim, Aviacomp, and IREENA Université de Nantes.

The authors would like to thank Professor Erwan Verron for his fruitful discussion about fracture mechanics. Juliana Amirdine also helped with the rheology analysis. Nicolas Boyard provided useful help and insights on the aging effect. Finally, the bench design and manufacturing would not have been possible without the help of Nicolas Lefevre and Jérôme Delmas.


M Spagnuolo

T Theoharis

R Veltkamp

L Mavridis

V Venkatraman

D W Ritchie

N Morikawa

R Andonov

A Cornu

N Malod-Dognin

J Nicolas

M Temerinac-Ott

M Reisert

H Burkhardt

A Axenopoulos

P Daras

Eurographics Workshop on 3D Object Retrieval (2010) I. Pratikakis

Keywords: (according to ACM CCS): I.3.5 [Computer Graphics]: Curve, surface, solid, and object representations-Geometric algorithms, languages, and systems

This paper presents the results of the SHREC'10 Protein Models Classification Track. The aim of this track is to evaluate how well 3D shape recognition algorithms can classify protein structures according to the CATH [CSL * 08] superfamily classification. Five groups participated in this track, using a total of six methods, and for each method a set of ranked predictions was submitted for each classification task. The evaluation of each method is based on the nearest neighbour and area under the curve(AUC) metrics.

Introduction

The specific shapes of protein molecules are central to their biological function. Conventional approaches to compare and classify proteins usually work with their amino acid sequences (e.g. BLAST [AGM * 90] and FASTA [START_REF] Lipman | Rapid and sensitive protein similarity searches[END_REF]). However, in Nature, the 3D structures of proteins are often more conserved than their sequences. Hence, structural alignments can provide significant insights about protein function and can help classify protein families into functional super-families [START_REF] Holm | Dali: a network tool for protein structure comparison[END_REF].

Currently, the most widely used protein structure classification systems are CATH [CSL * 08] and SCOP [START_REF] Murzin | Scop: a structural classification of proteins database for the investigation of sequences and structures[END_REF], both of which are curated by human experts. In CATH, the classification is initially performed using the SSAP [START_REF] Orengo | Ssap: sequential alignment program for protein structure comparison[END_REF] structural alignment tool, whereas SCOP relies more on visual inspection by the curators. However, with the rapid growth of the three-dimensional (3D) protein structures in the Protein Data Bank (PDB [BWF * 00]), it would be desirable to be able to assemble and update structural classifications in a more automated way.

Task

The task of this track is to classify protein structures according to their CATH superfamilies. Five groups (listed in the order in which they registered) participated in this track, and each group was intially provided with a data set of 1000 proteins, selected by the track organisers, with which they could train or prepare their algorithms. All information about the nature of the proteins and their primary amino acid sequence information was masked to prevent the participants from using such knowledge in conventional protein sequence analysis software. Five days before the deadline for the track, 50 further protein structures (Figure 1) were made available to be used as queries against the initial set. The participants were asked to rank the initial dataset in order of similarity to each of the query proteins. Thus each group was asked to submit 50 ranked lists for each similarity method used.

Data

Using CATH version 3.3, the track organisers assembled a dataset of 1000 protein structures from 100 CATH superfamilies, where each superfamily consisted of at least 10 structures, and where each structure contained at least 50 amino acids. From 50 of the superfamilies, one additional member was selected at random to serve as a query structure. The protein file names and protein sequence information were masked to try to prevent the participants from using conventional protein sequence matching techniques. Hence, the supplied data files included only the x, y and z coordinates and radii for the atoms within each protein

Evaluation

For the evaluation, the participants were asked to provide a ranked list for each of the query proteins against the 1000 protein dataset. Using these ranked lists, the perfomance of each method was measured in two different ways.

• Nearest neighbour: If the first protein of each ranked list was found to be a member of the same CATH superfamily as the query, this was counted as a correct prediction. The overall percentage of correct predictions was calculated over the 50 queries submitted by each group. • ROC plot (Receiver Operating Characteristic [START_REF] Egan | Signal detection theory and roc analysis[END_REF]):

By construction, each ranked list contained 10 true positives (TPs) and 990 true negatives (TNs). In order to measure the overall ability of each method to distinguish the TPs from the TNs, the list was traversed sequentially and the rate of TPs (TPR) against the rate of FPs (FPR) was plotted. The area under the curve (AUC) of each ROC plot was calculated to give a single numerical performance measure. A perfect prediction would consist of a list of 10 TPs followed by 990 TNs, giving an AUC of 1.0.

Methods

Brief decriptions of the methods are provided in this section.

Spherical Polar Fourier Shape Density Functions (SPF) by L. Mavridis and D.W. Ritchie

In the SPF approach, protein shapes are represented as 3D density functions expressed as expansions of orthonormal basis functions: where N is the order of the expansion, R nl (r) are Laguerre-Gaussian radial functions, y lm (ϑ, ϕ) are spherical harmonics, and a nlm are the expansion coefficients which are calculated numerically as described previously [START_REF] Kemp | Protein docking using spherical polar fourier correlations[END_REF]. Figure 2 shows the SPF representations of a pair of similar nitrogenase domains at several expansion orders. For this track, we used expansions to order N = 25 for all calculations. In order to superpose a pair of protein structures we calculate a rotation-dependent Carbo-like similarity score S ROT using:

ρ(r) = N ∑ n=1 n-1 ∑ l=0 l ∑ m=-l a nlm R nl (r)y lm (ϑ, ϕ) (1)
S ROT = N ∑ nlm a nlm b nlm [ N ∑ nlm a 2 nlm ] 1 2 [ N ∑ nlm b 2 nlm ] 1 2
(2)

c The Eurographics Association 2010.

Conceptually, one protein is held fixed and a sixdimensional (6D) rotational/translation search over positions of the second protein is performed. However, in practice it is more efficient to implement the search using one translational and five Euler angle rotational coordinates [START_REF] Kemp | Protein docking using spherical polar fourier correlations[END_REF].

3DZernike by V. Venkatraman

3D Zernike descriptors [NK04, LLL * 08], an extension of spherical harmonics, have been used for molecular shape retrieval and more recently for protein-protein docking [START_REF] Venkatraman | Protein-protein docking using region-based 3d zernike descriptors[END_REF]. A key point in favour of this representation is that of rotational invariance while allowing for a compact shape representation to an arbitrary expansion order. Mathematical and implementation details can be found in the papers by Novotni and Klein [START_REF] Novotni | Shape retrieval using 3d zernike descriptors[END_REF] and Mak et al. [START_REF] Mak | An extension of spherical harmonics to region-based rotationally invariant descriptors for molecular shape description and comparison[END_REF]. For the current protein classification task, the following procedure was used: Surface Generation Molecular surfaces for the proteins were generated using the MSMS software [START_REF] Sanner | An extension of spherical harmonics to region-based rotationally invariant descriptors for molecular shape description and comparison[END_REF]. Binary Voxelization The program binvox [Min] was used to produce a binary voxel grid (voxel dimension set to 128). Zernike moments Software provided by Novotni and Klein [Nov] was used to calculate Zernike moments upto an expansion order N = 20. Each protein is thus represented by a vector of 121 coefficients. Similarity Measurement Two protein shapes A and B represented by their respective Zernike moments were com-

pared using the Euclidean metric d = 121 ∑ i=1 (A 2 i -B 2 i ).

GENOCRIPT / D2 encoding by N. Morikawa

We performed the retrieval of the dataset of 1000 protein structures for structurally similar proteins of 50 query structures in the following three steps. First, the "CA" (or α-Carbon atom) traces of the proteins were extracted from the supplied data files by considering the pattern of atom radii. Next, the D2 codes of the 1000 protein structures were computed by program "ProteinEncoder" and saved in a ".code" file (392 KB): target_SHREC2010.code. Also computed were the D2 codes of the 50 query structures: query_structure.code. Then, retrieval of the dataset was carried out with program "ComSubstruct," which computes the length of the longest common subsequence of two D2 codes. For example, the top 100 D2 code-similar fragments are obtained by typing the following command: "ComSubstruct -l -o1 -s -w1.1 -b100 query_structure.code target_SHREC2010.code." Because more than one fragment may correspond to a protein, the top-most fragment was chosen for each protein to obtain the ranked list of protein names. See below for more detail. The programs ProteinEncode and ComSubstruct are available from http://www. genocript.com.

Extraction of the CA trace of a protein

To identify the main-chain fragments of N-CA-C atoms, the supplied data files were examined for the atom radius pattern of 1.70-2.00-1.74. Only the CA atoms of the N-CA-C fragments are considered in our method.

D2 encoding of local protein structures

We used a discrete differential geometrical technique called "D2 encoding" to analyse local protein structures, where the conformation of all five-CA fragments (i.e. fragments of five CA atoms) of a protein are encoded using a five-tetrahedron sequence [START_REF] Morikawa | Discrete differential geometry of tetrahedrons and encoding of local protein structure[END_REF]. First, the conformation of each five-CA fragment is represented by a folded sequence of five tetrahedrons. Next, the corresponding (0,1)-valued sequence of length five, which are denoted as a base-32 number, are assigned to the center CA atom of the fragment. Then, we obtain a description of the conformation of a protein by arranging base-32 numbers in the order that the corresponding CA atoms appear in the CA trace. The base-32 number sequence is called the D2 code of a protein.

Dataset search by ComSubStruct

One of the simplest measures of sequence similarity is the length of the longest common subsequence (LCS). We used the length of the LCSs of two D2 codes to quantify the differences between two protein backbone conformations. The width of compare window was set to the product of "1.1" and the length of the shorter sequence using the "-w" option. The width of slide step was then the product of 0.1 and the length of the shorter sequence.

Sorting structures

Protein names are ranked based on the length of LCS. The length of a protein sequence is used for tie-break purposes (the shorter, the better). The similarity scores are obtained by dividing the LCS-length by the protein-length. More precisely, the maximum value is (protein-length -4) / proteinlength.

Contact Map Overlap maximization by R.

Andonov, A. Cornu, N. Malod-Dognin and J.Nicolas

Principle

This approach compares protein structures based on common inter-atomic contacts. Formally, the contact map of a protein is a graph, CM = (V, E), with vertices V associated to the amino-acids of the protein and contact edges E associated to close amino acids (Euclidean distance between CA atoms, which are known to form the backbone of the protein) smaller than a given threshold. The similarity between two proteins is then determined by the maximum overlap of their contact maps (equivalent to their maximum Number c The Eurographics Association 2010.

of Common Contacts (NCC)). Finding this number and the associated alignment between the amino-acids of both proteins, known as Contact Map Overlap maximization (CMO), is an NP-hard problem [START_REF] Goldman | Algorithmic aspects of protein structure similarity[END_REF] and has been extensively studied in the bioinformatics and computer science communities [CCI * 04, XS07].

The A_purva solver

To classify the queries in the context of SHREC_10 we used the solver A_purva which has been recently proposed in [START_REF] Andonov R | An efficient lagrangian relaxation for the contact map overlap problem[END_REF]. A detailed description can be found in [START_REF] Malod-Dognin N | Protein Structure Comparison: From Contact Map Overlap Maximisation to Distance-based Alignment Search Tool[END_REF].

A_purva is able to solve CMO in an exact manner in the framework of a classical branch and bound approach (B&B) where upper (UB) and lower (LB) bounds are generated by Lagrangian relaxation. When an instance is optimally solved, we have the relation LB = NCC = UB. Otherwise UB > LB and the so called relative gap U B-LB U B gives an idea of the precision of the results. This property was very useful in the context of SHREC_10 where, because of the time limitation, we were forced to limit the search process on the root of the B&B only.

A_purva was launched without branch and bound, with a limit of 10 000 subgradient descent iterations (i.e. about 20sec per instance). For most query instances (with less than 700 CA atoms) a limit of 2 000 iterations and 4 sec gave the same results.

Extraction of the backbone and generation of the contact map

In order to adapt A_purva to SHREC_10 conditions where only the coordinates of the atoms have been provided, without identifying their names, we proceeded as follows. Interesting atoms have been filtered on the basis of stable distances that could correspond to the protein backbone (in a PDB file, consecutive atoms N, CA, C and O exhibit N-CA, CA-C and C-O bonds with relatively fixed distances of 1.45Å, 1.53Å and 1.24Å, respectively). Note that we did not use atom radii for this purpose. Globally, the procedure tends to filter all CA and a few other carbon atoms in each protein that we consider as CA in the rest of the treatment.

The contact maps were generated with a distance threshold of 7.5Å between two CA atoms, excluding natural contacts between consecutive amino-acids.

Scoring scheme

Based on the obtained values, two scoring functions were tested in order to detect the similarity between a query Q and each protein P of CATH superfamilies. The first one was first proposed in [START_REF] Xie | A reduction-based exact algorithm for the contact map overlap problem[END_REF]:

SIM(Q, P) = 2 × LB |E Q | + |E P | . (3)
Once results known, this default score appeared to be the best one for the classification task. The nearest neighbour score with it reaches 88%.

The second index used the confidence in the results of A_purva, C = LB U B , and was finally retained for the contest. The score is given by:

Cscore(Q, P) = C ×UB + (1 -C) × LB |E Q | × (1 + abs(|EP|-|EQ|) max(|EP|,|EQ|) ) (4)
A final step used the knowledge of superfamily labels: the mean rank of the three best scores for each superfamily was computed. It allowed classifying all proteins of a same superfamily together if they got a good rank.

All Contact Maps computations were done on the Ouestgenopole bioinformatics platform http://genouest.org.

Group Integration for Protein Structure Description by M. Temerinac-Ott, M. Reisert and H. Burkhardt

Group Integration (GI) is a powerful tool for describing three dimensional structures [START_REF] Burkhardt | Invariant features in pattern recognition -fundamentals and applications[END_REF]. The main idea is to average the representatives of a transformation group (e.g. Euclidean) in order to obtain group invariant descriptors, which can be compared in order to determine similarities. Group integration can be extended by Spherical Harmonics [START_REF] Reisert | Invariant features for 3d-data based on group integration using directional information and spherical harmonic expansion[END_REF] in order to obtain more robust descriptors. The details of our method are explained in [START_REF] Temerinac | Invariant features for searching in protein fold databases[END_REF].

Modelling Protein Shape

Proteins can be described by the position of the atoms of the protein and their order in the amino acid sequence. In order to apply group integration to proteins, the proteins are modelled as superpositions of Gaussian distributions centered at the positions of the atoms.

In the SHREC'07 protein track [START_REF] Temerinac M | Shrec 2007: 3d shape retrieval contest, protein retrieval track[END_REF], only CA atoms were used, whereas here we now use all atoms to compute GI features. However, we did not use the provided atom radius data.

Classifying Proteins based on GI features

The result of group integration is a multidimensional histogram H α,β,γ,∆,µ,ł with 2048 bins. Through concatenation of the histogram dimension, we obtain one feature vector for each protein. The similarity measure s(x, y) between two feature vectors is obtained using the χ 2 2 distance.

3D protein classification using the Spherical trace transform by A. Axenopoulos and P. Daras

Our 3D shape-based approach is presented for the efficient search, retrieval, and classification of protein molecules. The method relies on the geometric 3D structure of the proteins, which is produced from the corresponding PDB files. After proper positioning of the 3D structures, in terms of translation and scaling, the Spherical Trace Transform is applied c The Eurographics Association 2010.

to them so as to produce geometry-based descriptor vectors, which are completely rotation invariant and perfectly describe their 3D shape.

Preprocessing

Since the exact 3D position and radius of the protein's atoms is known from the available PDB file, the protein can be represented as a set of spheres. Then, the Solvent Excluded Surface is computed using the MSMS algorithm [START_REF] Sanner | Fast and robust computation of molecular surfaces[END_REF]. The protein is now represented as a triangulated mesh which provides a sufficient approximation of the protein's 3D shape. As a next step, a voxelization process, similar to the one presented in [DZA * 06] takes place. More specifically, the 3D mesh is placed into a bounding cube, which is partitioned in equal cube shaped voxels. Voxels that lie inside the 3D model or on the surface are assigned non-zero values.

Descriptor Extraction

Every 3D object is expressed in terms of a binary volumetric function. In order to achieve translation invariance, the center of mass of the 3D object is calculated and the model is translated so that its center of mass coincides with the coordinate system origin. Scaling invariance is also accomplished, by scaling the object in order to fit inside the unit sphere. Then, a set of concentric spheres is defined. For every sphere, a set of planes which are tangential to the sphere is also defined. Further, the intersection of each plane with the object's volume provides a spline of the object, which can be treated as a 2D image.

Next, 2D rotation invariant functionals, F, are applied to this 2D image, producing a single value. Thus, the result of these functionals when applied to all splines, is a set functions defined on every sphere whose range is the results of the functional. Finally, a rotation invariant transform, T, is applied on these functions, in order to produce rotation invariant descriptors. For the needs of the SHREC, the implemented functionals F are the 2D Krawtchouk moments, and the Polar Fourier Transform, while the T function is the Spherical Fourier Transform.

A more detailed description of the extraction of these descriptors is available in [DZA * 06]. The dimension of descriptor vectors is N Fourier = 1080 for the descriptors based on the Polar-Fourier 2D functional and N Krawtchouk = 1080 for the descriptors based on the Krawtchouk 2D functional.

Matching

Firstly, the descriptors are normalized so that their absolute sum is equal to 1. Then, the Minkowski L1 distance is computed for a pair of descriptor vectors. The L1 distance is a measure of dissimilarity between two descriptor vectors. In order to transform this dissimilarity into a similarity metric, a decreasing sigmoid function was applied so that lowdissimilarity values are closer to 1 and high-dissimilarity values are closer to 0.

Results

In this section, we present the perfomance evaluation results of the track. Each participating group submitted one set of results based on their selected set of parameters. This was a blind experiment and each group could only submit one set of results. Therefore, it was not possible for partipants to tune the parameters of their algorithms.

Nearest neighbour : Table 3 summarizes the retrieval rates for all the methods. There were five cases in which none of the methods found the nearest neighbour. These were: Q12 (1wwjA00), Q30 (1iicA02), Q40 (3c4aA01), Q43 (1nyaA00), and Q48 (3bioA02). In a further seven cases, only one method found the nearest method as the top match. However, there were 11 additional cases in which several methods found the nearest neighbour as the second hit (i.e. 4 for GENOCRIPT, 3 for Group Integration, 3 for 3DBlast and 1 for 3DZernike).

Method

Correct ROC plots : For each of the submitted result lists, a ROC plot and its corresponding AUC was calculated. Figure 4 shows the resulting AUC of all methods for each target. Because early recognition of TPs is at least as important as obtaining a good overall AUC score, we also calculated another set of AUC values which correspond to the first part, up to 10% of the database, of the ROC curves. An aggregate c The Eurographics Association 2010.

ROC plot was also calculated to summarize the overall performance of each method as a single ROC curve, as shown in Figure 5.

1

Conclusions

In this paper, we have presented and compared the perfomance of six algorithms submitted by the five research groups who participated in this track.

Contact Maps and 3D-Blast were conceived specifically to compare proteins structures, and these approaches give the best results, although the Group Integration and GENOCRIPT/D2 approaches also perform very well. The Contact Maps and GENOCRIPT approaches both used a preselection step to try to infer the CA backbone structure of the corresponding proteins from simple geometrical invariants.

Contact Maps compares proteins on the basis of conserved proximities between atoms, where Genocript encodes the CA backbone structure of length N into a 16 valuedsequence of length (N-4). Both 3D-Blast and 3D-Zernike compare shapes globally, but 3D-Blast uses FFT-based rotational comparisons, whereas 3D-Zernike uses a fast scale-and rotation-invariant scoring technique derived from a spherical harmonic plus Zernike polynomial expansion of each protein. The Spherical Trace Transform approach calculates scale-and rotationinvariant descriptors from 2D slices of the protein volumes using polar Fourier transforms. The Group Integration approach constructs and compares group invariant descriptors from the given atomic coordinates of each protein. With the exception of the 3D-Zernike approach, which gave unexpectedly disappointing results, the general shape classification approaches also gave very encouraging predictions when one considers the generic nature of those approaches and the very tight timetable under which this experiment was conducted.

Although in this experiment, some superfamilies may have been easier to identify than others, it is worth noting that no approach can reproduce the classification of the human experts in all cases. This suggests that protein modelc The Eurographics Association 2010.

ing and classification is a difficult task for current 3D shape recognition methods. Therefore adopting a benchmark based on protein shape classification, such as the one presented here, will provide a challenging dataset with which to evaluate new 3D object recognition algorithms.

Figure 2 :

 2 Figure 2: The superposition of a pair of nitrogenase proteins, shown as ribbon cartoons (left), backbone traces (middle), and as 3D SPF density expansions to order N=25 (right). The protein in the top row is from azotobacter vinlandii (PDB code 2MIN). Top row: PDB code 2MIN; middle row PDB code 1MIO; bottom row their superposed orientation. The two proteins have a sequence identity of 43%.

Figure 3 :

 3 Figure 3: 3D representation of a protein with a) spheres and b) Solvent Excluded Surface.

Figure 4 :

 4 Figure 4: Bar chart analyses for each method showing the calculated AUC for each of the 50 query proteins. The upper bar charts show the total AUC, whereas the lower bar charts show the AUCs calculated for the top 10% of the database.

Figure 5 :

 5 Figure 5: The upper figure shows aggregate ROC plots for each method obtained when quering the 1000 protein dataset using the 50 query proteins. The lower figure shows an expanded view of the first 10% of the upper figure to highlight the early recognition behaviour of each method.

Table 1 :

 1 An

		. A simple table
	was also provided which associates each given protein struc-
	ture file with a synthetic superfamily name (e.g. "F001"), as
	shown in Table 1.
	Protein File Protein Family
	P0001.pdb	F001
	P0002.pdb	F001
	...	...
	P0500.pdb	F050
	...	...
	P0999.pdb	F100
	P1000.pdb	F100

extract of the classification file, which was provided with the initial data set.

Table 2 :

 2 Participating groups and methods.

	Method Name	Participants
	3DBlast		L. Mavridis and D.W. Ritchie
	3DZernike		V. Venkatraman
	GENOCRIPT		N. Morikawa
	Contact Maps		R. Andonov, A. Cornu, N.
			Malod-Dognin, and J. Nicolas
	Group Integration	M. Temerinac-Ott, M. Reisert,
			and H. Burkhardt
	Spherical	Trace	A. Axenopoulos and P. Daras
	Transform

Table 3 :

 3 Nearest neighbour results.

		Predictions
	3DBlast	68%
	3DZernike	8%
	GENOCRIPT	56%
	Contact Maps	80%
	Group Integration	52%
	Spherical Trace Transform	0%

c The Eurographics Association 2010.