U Göschel

W Lutz

N C Davidson

The Influence of a Polymeric Nucleating Additive on the Crystallisation in Glass Fibre Reinforced Polyamide 6 Composites

Keywords: A Polymer-matrix composites (PMCs), B Interface, B Microstructure, B Nucleation, E Injection moulding

Short glass fibre reinforced polyamide 6 composites compounded with a polymeric nucleating additive based on polyamide 2.2 of different content (0, 0.1 and 0.2 wt-%) and processed by injection moulding have been studied to characterise the additive effect on the crystallisation kinetics and morphology. Differential scanning calorimetrical and optical hot stage experiments have shown a significant increase in the overall crystallisation rate as well as the spherulitic growth nucleating additive. Additionally, the presence of glass fibres is proven to enhance crystallisation. Atomic force microscopy confirms the ACCEPTED MANUSCRIPT 2 existence of a transcrystalline layer in the vicinity of the glass fibres. The nucleating additives have significantly decreased the spherulitic size from about 15 to 20 µm towards less than 5 µm, which leads to an improved fibre-to-matrix contact. Those morphological changes are seen as the reason for a strong increase in the Martens heat deflection temperature from 145 to 164 °C characterising an improved thermal stability.

Introduction

The crystallisation behaviour of glass fibre reinforced thermoplastic composites is affected by the materials chosen (polymeric matrix, glass fibre, sizing), the use of nucleating additives as well as the thermal and mechanical conditions during processing and the practical use. During solidification from the melt, the existence of both a gradient in orientation (matrix and fibre) and temperature (core and surface) has a large impact on the structural formation (rate and morphology) and resulting product properties. The complex nature of those parameters can be demonstrated by the presence of differential shrinkage and residual stresses leading to the effect of warpage, which is characteristic for fibre reinforced thermoplastic composites. The structural development during solidification is known to be described as primary crystallisation consisting of nucleation and crystal growth leading to a defined degree of crystallinity depending on the crystallisation conditions applied. Moreover, the long-term process of completion of crystallisation is widely considered as secondary crystallisation to reach the final morphology and resulting properties.

With respect to Alfonso and Ziabicki [START_REF] Alfonso | Memory effects in isothermal crystallization, II. Isotactical polypropylene[END_REF], the primary crystallisation includes the growth of predetermined nuclei as well as sporadic thermal nucleation (N th ) followed by growth. The predetermined nuclei (N 0 ) defined as already be present at the start of the crystallisation consist of heterogeneous (N het ) and athermal homogeneous nuclei (N ath ). Consequently, the total number of growing nuclei (N tot ) at any time (t) can be expressed [START_REF] Alfonso | Memory effects in isothermal crystallization, II. Isotactical polypropylene[END_REF] by N tot (t) = N het + N ath + t 0 th N dt [START_REF] Alfonso | Memory effects in isothermal crystallization, II. Isotactical polypropylene[END_REF] The heterogeneous nuclei are mainly related to the use of nucleating additives.

Catalysts, pigments or impurities are also considered as heterogeneous nuclei, which may enhance crystallisation too. The athermal nuclei strongly depend on the thermal and deformation history of the melt related to the formation of crystal fragments and chain clusters [START_REF] Alfonso | Memory effects in isothermal crystallization, II. Isotactical polypropylene[END_REF]. Those memory effects can be eliminated by appropriate annealing conditions in the melt. However, the required high temperature and annealing time have to be carefully determined.

In the case of PA 6, the effects from degradation starting already at low melt temperatures exceeding 260 °C as well as hydrogen bonding are to be considered.

The use of nucleating additives is essential in industrial processing to control the structure formation and to shorten the cycle time. According to Titzschkau and Beck [START_REF] Tizschkau | Processing auxiliaries for polyamides, nucleationlubrication -chain length modification[END_REF], inorganic compounds (minerals such as talc or kaolin) and organic compounds (e.g. salts of mono-or polycarbonic acids such as Castearate, Ca-montanate, Li-montanate, Na-benzoate, di-benzylidene sorbitols or salts of phosphorous esters) are used as nucleating additives for PA 6.

Furthermore, oligomers of the caprolactam and polymers such as PA 2.2 [START_REF] Tizschkau | Processing auxiliaries for polyamides, nucleationlubrication -chain length modification[END_REF][START_REF] Mudra | Comparative study of efficiency of nucleating agents in PA-6[END_REF] or PA 4.6 [START_REF] Tizschkau | Processing auxiliaries for polyamides, nucleationlubrication -chain length modification[END_REF][START_REF] Brink | Process for the production of polyamide moulded parts with improved crystallisation behaviour[END_REF] are known to enhance nucleation.

Due to low costs and acceptable nucleation efficiency of talc and kaolin [START_REF] Gurato | Influence of nucleating agents on the crystallization of 6-polyamide[END_REF] those materials have been played a major role in industrial processing of PA 6 over years. However, the required homogeneous dispersion of small particle sizes to achieve the anticipated fine crystalline structure remains complex. In the case of organic nucleating compounds well-distributed and small crystalline entities will be achieved during cooling of a completely molten PA 6 [START_REF] Tizschkau | Processing auxiliaries for polyamides, nucleationlubrication -chain length modification[END_REF].

Recently, the nucleating efficiency of the polymeric additive Brueggolen® P22, based on polyamide 2.2, as compounded during injection moulding of the PA 6

homopolymer Ultramid B3 has been proven in preliminary investigations [START_REF] Fracassi | Crystal growth in nucleated polyamide PA 6, 18[END_REF] and by WAXD synchrotron studies at the ANKA Synchrotron Radiation Facility in Karlsruhe [START_REF] Göschel | Crystallisation kinetics in nucleated polyamides, project PX 03-003[END_REF].

In the present study, the main focus will be given to the effect of nucleating additive Brueggolen® P22 content on the primary crystallisation of glass fibre reinforced polyamide (PA 6-GF30) of a given fibre geometry and sizing, processed by injection moulding at optimised conditions for melt temperature, holding pressure and mould temperature.

Experimental

Material

A commercial 30 wt-% short glass fibre reinforced polyamide (PA 6-GF30)

grade Bergamid B70G30 with a fibre diameter of 10 to 13 µm and a fibre length of 3.2 to 4.5 mm in the granule was used. Prior to processing, 0.2 wt-% Ca stearate and the polymeric nucleating additive Brueggolen® P22 (denoted as P22 in the following) at different content of 0, 0.1 and 0.2 wt-% was added to the PA 6 in a powdered form during compounding. The PA 2.2, which is the chemical basis for the P22, is the condensation product of oxalic acid derivatives with ethylene diamine (Figure 1). The polymerisation degree of the used nucleating additive is as low as about 10 [START_REF] Tizschkau | Processing auxiliaries for polyamides, nucleationlubrication -chain length modification[END_REF]. Due to only two CH 2sequences in the monomeric unit of PA 2.2 compared to five in PA 6 an extraordinary large number of hydrogen bonds between the NH and the CO groups can be established, which leads to a high thermal stability of PA 2.2 with no melting up to its decomposition temperature of about 400 °C.

Tensile bars (150 x 10 x 4 mm 3 ) with respect to DIN EN ISO 527 as well as rectangular plates (61 x 61 x 2 mm 3 ) according to DIN EN ISO 294-4 were processed using an injection moulding machine DEMAG ergotec 60/370-120.

The processing conditions based on a parameter study by Fracassi [START_REF] Fracassi | Improvement of warpage of glass fibre-reinforced polyamide (PA 6-GF) composites by means of well-defined nucleation[END_REF] are given in Table 1.

Methods

Fibre Orientation

The measurements were performed on tensile bars and rectangular plates as well. From the tensile bars an area perpendicular to the processing direction was used (Figure 2). The measurement was performed on about 16,000 fibres and exhibits a statistical reliability. The fibre orientation was measured from polished sample sections applying an image analysis system designed in house. The normalised first moments for a two dimensional image are used to find the centre of the object while information on size, orientation and ellipticity of the object were derived from the second moments, equ. (2), see Davidson [START_REF] Davidson | Image analysis for fibre orientation measurement[END_REF]. By using a calibrated xy translation stage and overlapping image frames according to Davidson et al. [START_REF] Davidson | Large area high resolution Image analysis of composite materials[END_REF], the fibre orientation with respect to position in the sample could be accurately determined.

( )

yy xx xy M M M 2 2 tan - = φ ( ) ( ) [ ] 2 xy 2 yy xx yy xx 2 M 4 M M 2 M M 2 a + - + + = ( ) ( ) [ ]

Thermal Analysis

Differential scanning calorimetry (DSC) was performed by a Mettler DSC 821 operating with the software STAR V6.10. The sample mass was about 7 mg.

Subsequent to a heating run from -50 to 260 °C at 10 K/min and an annealing at 260 °C for two minutes a cooling run to room temperature at 10 K/min followed by a second heating run to 260 °C at 10 K/min was performed. Nitrogen atmosphere at a flow rate of 80 ml/min was chosen to avoid degradation.

Isothermal Crystallisation Studies

Crystallisation kinetics was investigated under isothermal conditions using differential scanning calorimetry (DSC) and hot stage polarised light microscopy (HSM) characterising the crystallisation rate from the exothermal peak and spherulitic growth, respectively. In both cases a temperature profile according to Figure 3 was applied, see Moritz [START_REF] Moritz | Crystallisation kinetics on glass fibre reinforced polyamide materials[END_REF]. Starting from 23 °C the temperature was raised to the melting stage (T m ) at 265 °C using a rate of 50 K/min. The melt was annealed for three minutes in order to eliminate aggregation and crystal clusters originated from thermal and mechanical loading history, which may substantially affect crystallisation. Thereafter, rapid cooling at 80 K/min to the given temperature for isothermal crystallisation (T c ) ranging from 201 to 211 °C was used.

Differential Scanning Calorimetry (DSC)

For the DSC experiments (Mettler DSC 821) a sample mass of about 5 mg was chosen. The time-dependent development of the degree of crystallinity χ(t) was determined from the heat flow dQ/dt of the exothermal crystallisation peak by means of equ. (3) assuming that the primary crystallisation is completed in the time range from t 0 to t leading to χ(t) ranging from 0 to 100 %. According to equ.

(3), χ(t) can also be expressed by the increase of the crystallisation enthalpy

c c t t t H ) t ( H dt dt dQ dt dt dQ ) t ( 0 0 ∆ ∆ = = χ ∞ (3)
In order to describe quantitatively the crystallisation process under isothermal conditions, where both nucleation and crystal growth are assumed to be timeindependent, the Avrami equation can be used, which reads for heterogeneous nucleation as

χ(t) = 1 -exp (-k•t n ) (4)
Here, the Avrami parameter k = f (T C ) relates to the overall crystallisation rate and (n) to the nucleation mechanism, which includes information on the growth dimensionality and the type of nucleation. A linearisation of equ. ( 4) leads to equ. [START_REF] Gurato | Influence of nucleating agents on the crystallization of 6-polyamide[END_REF], which enables a simple determination of (k) and (n) from the intersection with the ordinate and the slope in the lower log t range, respectively, see e.g. Chrisholm and Zimmer [START_REF] Chisholm | Isothermal crystallization kinetics of commercially important polyalkylene terephthalates[END_REF] and Hiemenz [START_REF] Hiemenz | Polymer Chemistry[END_REF].

log [-ln (1-χ)] = log k + n log t (5)
The Avrami equation is the most applied tool to describe the phenomenology of the crystal development. However, any insight into the molecular processes related to nucleation and crystal growth cannot be given [START_REF] Azzurri | Melt crystallization and polymorphic transformation in isotactic poly (1-butene) based materials[END_REF]. For simplicity, the Avrami equation supposes a radial growth of crystals at constant rate without any interaction with neighbouring crystals. Moreover, it requires a complete phase transformation without a change in volume, a uniform nucleation, a constant density and shape of the growing nuclei. At least, it excludes the secondary crystallisation, see Verhoven [START_REF] Verhoven | Crystallization of polyethylene terephthalathe in injection moulding[END_REF].

The crystallisation rate can be described by the half width t 1/2 , which is defined as the time to reach χ(t) = 0.5 of the overall crystallisation, equ. ( 6):

n 1 2 / 1 k 2 ln t = (6)
The Avrami parameters k and n in equ. ( 6) correspond to those in equ. [START_REF] Gurato | Influence of nucleating agents on the crystallization of 6-polyamide[END_REF]. The value χ(t) = 0.5 can be obtained from the experimental s-shaped curve degree of crystallinity (χ) depending on the crystallisation time (t).

Furthermore, the time-to-peak (t p ), where the overall crystallisation rate reaches a maximum can be determined from the minimum of the exothermal crystallisation peak of the isothermal DSC traces. With respect to Alfonso and

Ziabicki [START_REF] Alfonso | Memory effects in isothermal crystallization, II. Isotactical polypropylene[END_REF], the time-to-peak is related to the number of predetermined nuclei (N 0 ), equ. [START_REF] Göschel | Crystallisation kinetics in nucleated polyamides, project PX 03-003[END_REF].

t p = const. N 0 -1/3 (7)
At the start of crystallisation, the predetermined number of nuclei includes the heterogeneous nuclei (N het ) as well as the athermal nuclei (N ath ), compare with equ. [START_REF] Alfonso | Memory effects in isothermal crystallization, II. Isotactical polypropylene[END_REF].

Hot Stage Polarised Light Microscopy (HSM)

A Leitz DM LB microscope from Leica was used together with a Linkam TMS94 heating cell. The sample with a thickness of 40 to 50 µm was placed between a polariser-analyser set-up in transmission mode [START_REF] Moritz | Crystallisation kinetics on glass fibre reinforced polyamide materials[END_REF]. To avoid thermal decomposition a flow of nitrogen gas was applied. For statistical reasons each measurement was repeated three times at the same conditions. During isothermal crystallisation at the given crystallisation temperatures (T c ) in the chosen range from 201 to 211 °C, the transmitted light intensity I (t) increases with the spherulitic growth. In the molten stage at I (t = 0) the transmitted intensity is zero and reaches a plateau at t = ∞, where the primary crystallisation is completed. From the intensity values the relative degree of crystallinity χ rel (t) can be calculated according to equ. [START_REF] Fracassi | Improvement of warpage of glass fibre-reinforced polyamide (PA 6-GF) composites by means of well-defined nucleation[END_REF].

( )

) ( ) ( ) ( ) ( 0 t I t I 0 t I t I t rel = - ∞ = = - = (8)
Consequently, the experimental curve I = f (t) provides the half-width (t 1/2 ), which corresponds to χ rel = 50 %, compare equ. (6).

Light Microscopy (LM)

Polarised light microscopy studies in transmission mode were performed on thin microtom slices with a thickness of about 100 µm by a Leitz DM RXP microscope from Leica. The cuts were performed with a microtom Polycut E from Reichert Jung at room temperature.

Scanning Electron Microscopy (SEM)

For investigation of the morphology with SEM, tensile bars were prepared via cryo fracture. The samples were cracked under nitrogen atmosphere, both parallel as well as normal to the processing direction (Figure 2) and then gold sputtered. Further samples resulted from a tensile experiment at room temperature according to DIN EN ISO 527. The cross-section of the samples was investigated by means of a Jeol JSM 6300 F microscope operating at an acceleration voltage of 10 kV and an angle of 30°.

Atomic Force Microscopy (AFM)

Pulsed Force Mode (PFM)

The PFM is an external module for AFM from WITec GmbH, Germany. Using this method it is possible to map images of topography, adhesion and stiffness simultaneously as discussed by Krotil et al. [START_REF] Krotil | Pulsed force mode: a new method for the investigation of surface properties[END_REF]. The PFM applies a sinusoidal modulation on the vertical piezo of the AFM. At each data point a complete force-to-distance cycle was performed providing the stiffness from the slope of the curve. The lowest force of the curve characterises the maximum adhesive force (or pull-off force). The PFM is an intermittent contact mode and has the main advantages of low contact time, low lateral shear forces acting between probe and sample and the possibility to control the forces applied on the sample which enables the imaging of soft samples such as polymers [START_REF] Grandy | Microthermal characterization of segmented polyurethane elastomers and a polystyrene-poly (methyl methacrylate) polymer blend using variable-temperature pulsed force mode atomic force microscopy[END_REF].

Measuring Conditions and Sample Preparation

The atomic force microscopic data were obtained on an Explorer from TopoMetrix. The experiments were performed using the pulsed force mode in ambient air at 500 kHz. A rectangular silicon cantilever (2.82 x 25 x 225 µm³) with a free resonance frequency of 73 kHz and a stiffness of 2.1 N/m from Nanosensors was used. The radius of curvature of the tip was approximately 10 nm. Samples of each material for AFM were cut perpendicular and parallel to the processing direction according to Figure 3. Thereafter, the samples were embedded in epoxy resin and subsequently polished by abrasive paper ranging from FEPA norm 320 to 4000 (grain size 46 to 5 µm) followed by a diamond polishing paste with a grain size of 1 µm and ultrasonic bath cleaning.

Martens Heat Deflection Temperature (HDT)

The Martens heat deflection temperature (HDT) was determined on sections of tensile bars according to DIN 53462 applying a bending stress during heating at a slow rate of 50 K/hour starting from room temperature. The HDT corresponds to the temperature at a given tensile bar deflection of about 0.2 % according to DIN 53462.

Results

Fibre Orientation

The measurements were carried out on tensile bars as well as rectangular plates.

Tensile Bar

The PA 6-GF30 samples of different additive content (0, 0.1 and 0.2 wt-%) have shown a preferential alignment of the fibres parallel to the processing direction, expressed by a 11 (Table 2). The parameter a 11 ranges between 0.677 and 0.700 and reveals a small increase with increasing nucleating additive content. The fibre orientation in the plane 1-2, perpendicular to the processing direction is much lower. The corresponding parameter varies between 0.188 and 0.201.

With respect to the limited thickness of 4 mm, the fibre orientation along the thickness (a 33 ) is very small and ranges between 0.111 and 0.123. It can be concluded, that the effect of the nucleating additive content (0, 0.1 and 0.2 wt -%) on the fibre orientation applying identical processing conditions can almost be neglected.

Rectangular Plate

By the example of the 0.1 wt-% nucleated PA 6-GF30, the a 22 component of the second order orientation tensor is represented in the plane 2-3, which corresponds the plate area of 61 x 2 mm 2 . (Figure 5). The higher the value of a 22 , the more the fibres are oriented perpendicular to the processing direction.

Consequently, in the core region the fibres have a preferential orientation perpendicular to the processing direction (along the 2-axis) with a 22 close to 1.

Approaching the surface region, the a 22 value decreases down to about 0.2, which goes along with a preferential fibre orientation parallel to the processing direction. The a 33 component has been found to be negligible. Thus the a 11 component is approximately the negative of the above plot. All three samples (0, 0.1 and 0.2 wt-% nucleated) have shown almost the same fibre orientation distribution with a classic 'core' region of comparable size. Consequently, any differences in the thermo-mechanical behaviour will not be related to the fibre orientation.

Thermal Analysis

The melting peak temperature as determined from the first temperature run of DSC measurements is about 222 °C and does not change with the P22 additive content (Table 3). The corresponding melting enthalpy and thus the degree of crystallinity calculated from the measured melting enthalpy divided by the melting enthalpy of ideal crystals ∆H crystal = 188.4 J/g according to Wunderlich

[19] seems to be somewhat higher for nucleated compared to the nonnucleated samples. The degree of crystallinity of 32 % and 32.5 % for a P22 content of 0 and 0.1 as well as 0.2 wt-%, respectively (Table 3) is in the normal range of PA 6 GF30 materials. The additive effect is seen more pronounced in the onset of crystallisation leading to 192.4, 194.1 and 194.6 °C for a P22 content of 0, 0.1 and 0.2 wt-%, respectively (Table 3) by applying an identical cooling rate of 10 K/min. Those findings are in good agreement with isothermal DSC results in chapter 3.3.1.

Isothermal Crystallisation

Two different methods namely the differential scanning calorimetry (DSC) and the hot stage polarised light microscopy (HSM) were applied. Despite of large differences in the underlaying phenomena that are the heat flow (DSC) and spherulitic growth (HSM), both methods are complementary in describing the phenomenon of the primary crystallisation.

Differential Scanning Calorimetry (DSC)

The time-to-peak (t p ) as determined from the exothermal crystallisation peak during isothermal crystallisation according to equ. ( 7) is related to the number of predetermined nuclei (N 0 ). At the crystallisation temperature of 205 °C, the value t p decreases from 8.2 min (0 wt-%) over 6.1 min (0.1 wt-%) to 5.9 min (0.2 wt-%) with increasing nucleating additive P22 content (Figure 6). Such a decrease in t p documents an increase in the number of predetermined nuclei, which consequently leads to an increase in the overall crystallisation rate. As is seen in Figure 6, the width of the exothermal crystallisation peak decreases with a decrease in t p .

At all temperatures of isothermal crystallisation in the range from 201 to 211 °C, the 0.1 wt-% nucleated sample crystallises much faster than the non-nucleated one (Figure 7), which underlines the high nucleation efficiency of the nucleating additive P22.

Furthermore, the comparison between the curves (a) and (b) in Figure 6 reveals that the presence of glass fibres enhances crystallisation. This is in agreement with results on glass fibre reinforced polyalkylene terephthalates by Chisholm and Zimmer [START_REF] Chisholm | Isothermal crystallization kinetics of commercially important polyalkylene terephthalates[END_REF].

From the linearisation of the Avrami equation, see eq. ( 5), two regions of different slope are obtained (Figure 8). In accordance to ref. [START_REF] Azzurri | Melt crystallization and polymorphic transformation in isotactic poly (1-butene) based materials[END_REF], the linear range with a strong increase in the slope can be related to the primary crystallisation, followed by a slower crystallisation at larger times denoted as secondary crystallisation. The kinetics of crystallisation is seen by the intersection of the curve in the linear part with the ordinate or by the corresponding shift along the time axis with increasing crystallisation temperature. Both, the non-nucleated and the 0.1 wt-% nucleated PA 6-GF30

show a significant enhancement of crystal growth with decreasing crystallisation temperature in the range from 201 to 209 °C (Figure 8). The slope at lower crystallisation times (t), which characterises the nucleation mechanism, changes only slightly with the crystallisation temperature.

Hot Stage Polarised Light Microscopy (HSM)

Applying the HSM method, the spherulitic growth is described by an increase in the transmitted polarised light with increasing crystallisation time at a given crystallisation temperature from 201 to 211 °C during isothermal conditions, as is shown in Figure 9 for the non-nucleated and the 0.1 wt-% nucleated PA 6-GF30, respectively [START_REF] Moritz | Crystallisation kinetics on glass fibre reinforced polyamide materials[END_REF].

Starting in the melt, the transmitted light intensity after passing the polariseranalyser set-up at t = 0 is zero. Spherulitic growth changes the polarisation plane and leads to an increase in the transmitted light intensity reaching a plateau, where the primary crystallisation is completed (at t = ∞). The resulting intensity-time curves are s-shaped (Figure 9).

Light microscopy (LM)

Light microscopical studies show a significant decrease in the size of the spherulites (diameter of less than 5 µm) together with an improved distribution of the spherulites for the 0.1 and 0.2 wt-% nucleated in comparison to the nonnucleated (diameter 15 to 20 µm) PA 6-GF30 samples (Figures 10 and11).

Similar results have been described on the corresponding non-reinforced PA 6 [START_REF] Moritz | Crystallisation kinetics on glass fibre reinforced polyamide materials[END_REF].

Scanning Electron Microscopy (SEM)

For the discussion of the fibre matrix adhesion three representative SEM images out of a large variety of images will be used. Chosen are the nonnucleated samples prepared via tensile experiments at room temperature (Figure 12a) and at cryo conditions (Figure 12b) in comparison to a 0.2 wt-% nucleated sample prepared via tensile experiments at room temperature (Figure 12c). The SEM images reveal that in some areas of the fracture surface the polymeric matrix adheres well to the glass fibre. In other areas, the fibre surface is smooth and does not show matrix residuals, which suggests a weak fibre matrix adhesion. With respect to fibre-matrix adhesion there are no significant differences, which arise from the temperature of fracture as well as the use of nucleating additives.

Atomic Force Microscopy (AFM)

Samples cut perpendicular to the processing direction have been investigated by means of AFM using the pulsed force mode (PFM). The cross section exhibits a highly oriented skin layer and a spherulithic core region, which is in agreement to the fibre orientation measurements in Figure 5.

Topography

The topography image in Figure 13 on 0.1 wt-% nucleated PA 6-GF30 shows a morphology in the vicinity of the glass fibre, called interphase, which is different to the spherulitic structure in the polymeric matrix according to Kroh [START_REF] Kroh | Investigation of the structure of glass fibre reinforced polyamide using atomic force microscopy[END_REF]. Such an interphase is also found non-nucleated and the 0.2 wt-% nucleated PA 6-GF30. The thickness is in the range between 1.4 and 2.9 µm showing no definite correlation with the nucleating additive content. According to Kocker [START_REF] Kocker | A new test method for the determination of the fibre/matrix adhesion of fibre reinforced thermoplastic resins[END_REF] it is known that the interphase layer thickness has a direct influence on the shear strength in single fibre pullout tests and therefore, the thermo-mechanical composite properties.

Adhesion

Applying the PFM mode, the adhesion image shows the highest contrast. That originates from the minimal force of the force-to-distance curve leading to a very clear signal, which contains additional information compared to that of the topography image. The adhesion image in Figure 14 on 0.2 wt-% nucleated PA 6-GF30 confirms the existence of an interphase already seen in the topography image in Figure 13. Additionally, the adhesion image suggests that the interphase consists of a transcrystalline layer besides a sizing layer [START_REF] Kroh | Investigation of the structure of glass fibre reinforced polyamide using atomic force microscopy[END_REF]. With respect to Harel and Maron [START_REF] Harel | On crystalline interfaces in composite materials[END_REF], the phenomenon of transcrystallisation can take place if the nuclei density at the fibre surfaces exceeds that in the polymer matrix.

Martens Heat Deflection Temperature

The non-nucleated PA 6-GF30 has shown an averaged heat deflection temperature according to Martens of 145 °C, represented in Figure 15. Both nucleated samples exhibit significant higher heat deflection temperatures with 155 °C and 164 °C for the 0.1 and 0.2 wt-% nucleated samples, respectively [START_REF] Kroh | Investigation of the structure of glass fibre reinforced polyamide using atomic force microscopy[END_REF]. It is assumed that the increase in the heat deflection temperature with nucleating additive content is due to the strong decrease in the spherulite size and improvement in the spherulite distribution, which may support the fibre-tomatrix contact in the vicinity of the fibres.

Conclusions

A new type of nucleating additive Brueggolen® P22, based on the polymeric short chain polyamide (PA 2.2), has been studied with respect to the effect of the additive content (0, 0.1 and 0. Tables Table 1: Parameters chosen in the injection moulding processing. Cycle time (sec) 25.9 22.5

Parameters

  -major axis length b = semi-minor axis length θ = out of plane angle

  time (t) divided by the corresponding crystallisation enthalpy of ideal crystals ( 0 c H ∆ ), see Azzuri[START_REF] Azzurri | Melt crystallization and polymorphic transformation in isotactic poly (1-butene) based materials[END_REF].

  0

  The crystallisation temperatures were chosen to reach the maximum transmitted light intensity within the shorttime range up to 20 minutes. With increasing crystallisation temperature (T c ) the curves shifts towards longer crystallisation times and the slope in the linear part decreases. The shift of the intensity-time curves can be quantified by an increase in the half-width (t 1/2 ) of the relative degree of crystallinity (χ rel ) applying equ.[START_REF] Fracassi | Improvement of warpage of glass fibre-reinforced polyamide (PA 6-GF) composites by means of well-defined nucleation[END_REF]. At T c = 211 °C, the half-width is about 7 min for the nucleated PA 6-GF30 in comparison with about 11 min for the non-nucleated one showing a strong increase in the primary crystallisation rate due to the use of the P22 nucleating additive.

Figure captions Figure 1 :

 captions1 Figure captions

Figure 2 :

 2 Figure 2: Samples taken from a tensile bar (DIN EN ISO 527) for fibre

Figure 3 :

 3 Figure 3: Temperature profile for the isothermal crystallisation studies at

Figure 4 :

 4 Figure 4: Coordinate system for the rectangular plate.

Figure 5 :

 5 Figure 5: Fibre orientation distribution in the plane 2-3 of a 0.1 wt-% nucleated

Figure 6 :

 6 Figure 6: DSC traces of isothermal crystallisation of (a) non-nucleated and non-

Figure 7 :

 7 Figure 7: Time-to-peak temperature (t p ) from DSC exothermal crystallisation peak of (a) non-nucleated and (b) 0.1 wt-% nucleated PA 6-GF 30 at different

Figure 8 :

 8 Figure 8: Linearised Avrami plot from DSC measurements of the non-nucleated

Figure 9 :

 9 Figure 9: Transmitted light intensity vs. crystallisation time from HSM

Figure 10 :

 10 Figure 10: LM image, non-nucleated PA 6-GF30 at T c = 209 °C.

Figure 11 :

 11 Figure 11: LM image, 0.1 wt-% nucleated PA 6-GF30 at T c = 209 °C.

Figure 12a :

 12a Figure 12a: SEM image of non-nucleated PA 6-GF30 prepared via fracture at

Figure 12b :

 12b Figure 12b: SEM image of non-nucleated PA 6-GF30 prepared via fracture at

Figure 12c :

 12c Figure 12c: SEM image of 0.2 wt-% nucleated PA 6-GF30 prepared via tensile

Figure 13 :

 13 Figure 13: Left: AFM (PFM) image of 0.1 wt-% nucleated PA 6-GF30,

Figure 14 :

 14 Figure 14: a) AFM (PFM) image of 0.2 wt-% nucleated PA 6-GF30, adhesion,

Figure 15 :

 15 Figure 15: Heat deflection temperature of non-nucleated as well as 0.1 and

Figure 2 :Figure 3 :Figure 4 :Figure 5 :Figure 6 :

 23456 Figure 2: Samples taken from a tensile bar for fibre orientation, SEM and AFM

Figure 7 : 1 Figure 8 :Figure 9 :

 7189 Figure 7: Time-to-peak temperature (t p ) from DSC exothermal crystallisation peak of (a) non-nucleated and (b) 0.1 wt-% nucleated PA 6-GF 30 at different

Figure 10 :

 10 Figure 10: LM image, non-nucleated PA

Figure 11 :

 11 Figure 11: LM image, 0.1 wt-% nucleated

Figure 12a :

 12a Figure 12a: SEM image of nonnucleated PA 6-GF30 prepared via fracture at

Figure 12b :

 12b Figure 12b: SEM image of nonnucleated PA 6-GF30 prepared via fracture at

Figure 12c :

 12c Figure 12c: SEM image of 0.2 wt-% nucleated PA 6-GF30 prepared via

.

Figure 13 :Figure 14 :

 1314 Figure 13: Left: AFM (PFM) image of 0.1 wt-% nucleated PA 6-GF30,

Figure 15 :

 15 Figure 15: Heat deflection temperature of non-nucleated (0.0 wt-%) as well as

Table 2 :

 2 Fibre orientation in PA 6-GF30 tensile bar samples of different P22 nucleating additive content.

	Additive content	a 11	a 22	a 33
	(wt-%)
	0	0.677	0.201	0.123
	0.1	0.692	0.193	0.116
	0.2	0.700	0.188	0.111

Table 3 :

 3 Melting and crystallisation parametres of PA 6-GF30 tensile bar samples of different P22 nucleating additive content from DSC measurements.

	Additive	Melting peak	Melting	Degree of	Onset of
	content	temperature	enthalpy	crystallinity	crystallisation
	(wt-%)	T m	∆H	χ	T c, onset
		(°C)	(J/g)	(%)	(°C)
	0	221.9	42.2	32.0	192.4
	0.1	221.9	42.9	32.5	194.1
	0.2	222.4	42.7	32.4	194.6

Acknowledgement

The authors would like to thank K. Titzschkau and P. Beck (BrüggemannChemical) for the materials and processing. Financial support is acknowledged from the "Max Buchner Stiftung" in the frame of the project 2459 ("Verbesserte Wärmeformbeständigkeit thermoplastischer Mehrphasenwerkstoffe durch kontrolliertes Kristallwachstum"). Furthermore, the authors thank the co-workers at the Institute for Polymer Technology (IKT), Germany S. Osterloh, U. Müller and U. Fritz and the former students M. Kroh and S. Moritz for their substantial technical support.

heat deflection temperature is found to increase significantly with nucleating additive content, probably supported by an improved fibre-to-matrix contact in relation to the spherulite size and distribution.


D Trias

R García

J Costa

N Blanco

J E Hurtado

Quality control of CFRP by means of digital image processing and statistical point pattern analysis

Keywords: fiber-reinforced composites, optical microscopy, digital image processing, statistical properties, quality control

Although fiber-reinforced composite materials have often been considered as periodic materials in theoretical models, the distribution of fibers is random in real materials. This random distribution of fibers is closely related to their transverse failure behavior. This paper proposes the use of statistical functions which describe random point patterns as a quantification of the dispersion of the transverse failure properties of several carbon fibre reinforced polymers (CFRP). It is shown that the analysis of the K function is the most meaningful for this purpose.

Introduction

At the microscopical level, the main morphological characteristics of long fibre reinforced polymers are heterogeneity and anisotropy. In spite of this, composite materials have classically been modeled by means of periodical unit cells, that is, without taking into account neither the heterogeneity nor the geometrical disorder of fibers. The periodicity hypothesis leads to simplifications which make possible the application of homogenization methods [START_REF] Suquet | Homogenization Techniques for Composite Media, chapter Elements of homogenization for inelastic solid mechanics[END_REF][START_REF] Sanchez-Palencia | Homogenization Techniques for Composite Media, chapter Boundary Layers and Edge Effects in Composites[END_REF][START_REF] Sanchez-Hubert | Introduction aux méthodes asymptotiques et à l'homogénéisation[END_REF], it provides good estimations for the elastic properties [START_REF] Sun | Prediction of composite properties from a representative volume element[END_REF], and it can also be employed with good results in non-linear two-scale methods [START_REF] Michel | Effective properties of composite materials with periodic microstructure: a computational approach[END_REF][START_REF] Li | General unit cells for micromechanical analyses of unidirectional composites[END_REF][START_REF] Huyse | Random field modeling of elastic properties using homogenization[END_REF][START_REF] Car | Numerical simulation of composite materials: two procedures[END_REF][START_REF] Feyel | A multilevel finite element method (FE 2 ) to describe the response of highly non-linear structures using generalized continua[END_REF]. Also, in computational mechanics, the periodicity assumption leads to lower computational costs whereas other approaches may be computationally unaffordable.

However, a simple optical microscope observation reveals that long fiber reinforced composites (i.e. carbon or glass fiber-reinforced thermoset matrices) are far from being ordered materials since the fiber is randomly distributed through the matrix, sometimes showing areas with fiber clusters and resin pockets. These heterogeneities lead to local stress values in the matrix which are higher than those obtained assuming a periodical distribution and, consequently, they are more likely to produce damage, matrix cracking, or to cause degradation phenomena [START_REF] Trias | Random models versus periodic models for fibre reinforced composites[END_REF]. For this reason, the local damage in a transverse section of the composite (that is, matrix cracking and matrix-fiber debonding) is expected to depend strongly on the random distribution of the reinforcement.

On the other hand, because of the growing importance of composite materials in mechanical and structural engineering together with the lack of knowledge about many issues related to their failure, damage and fatigue behavior, there is a demand from the industry for quality control methods. This quality control methods should provide information on the defects within the material produced during manufacturing, the tolerance to these kind of defects, the relation between the material properties and its micro-scale structure. Traditionally, volume fraction is used as a measure of the quality of a laminate and ultrasound devices are normally used to complement this information by detecting voids and bubbles within the matrix. Some researchers have proposed sophisticated and highly technological procedures such as thermal imaging techniques [START_REF] Almond | Thermal imaging of composites[END_REF][START_REF] Blanco | Study of carbon fibres and carboncarbon composites by scanning thermal microscopy[END_REF], optical coherence tomography [START_REF] Dunkers | The application of optical coherence tomography to problems in polymer matrix composites[END_REF], near-infrared spectroscopy [START_REF] Li | Rapid and non-destructive analysis of quality of prepreg cloth by near-infrared spectroscopy[END_REF][START_REF] Lachenal | Advantages of near infrared spectroscopy for the analysis of polymers and composites[END_REF] or X-ray tomography [START_REF] Schilling | X-ray computed microtomography of internal damage in fibre reinforced polymer matrix composites[END_REF] for the inspection of fiber-reinforced composites. Although these techniques are extremely precise, they usually require high technology machinery, sophisticated interpretation techniques and highly specified and qualified personnel. This makes them unusable for most industries.

The widespread use of computers in industry prompted some pioneering work, like that by Berryman [START_REF] Berryman | Measurement of spatial correlation functions using image processing techniques[END_REF], in data acquisition using digital image processing for heterogeneous materials. The quantitative techniques for digital image processing of composites are widely employed in metal matrix composites (MMCs) [START_REF] Karnezis | Characterization of Reinforcement Distribution in Cast Al-Alloy/SiC p Composites[END_REF][START_REF] Huang | Quality control and nondestructive tests in metal matrix composites[END_REF] and some research has applied Fourier transformation to detect the orientation of reinforcement in reinforced concrete [START_REF] Redon | Assessment of fibre orientation in reinforced concrete using Fourier image transform[END_REF].

In fiber-reinforced polymers, much of the work devoted to the geometrical characterization of materials via digital image processing has been focused on braided composites [START_REF] Lian | Quantitative determination of morphological features of triaxially braided composites by the use of machine vision[END_REF]. Summerscales and co-workers computed total perime-ter and total area of inter-tow pore spaces in woven laminates produced by RTM [START_REF] Summerscales | Quantitative microstructural examination of RTM fabrics designed for enhaced flow[END_REF][START_REF] Pearce | The use of automated image analysis for the investigation of fabric architecture on the processing and properties of fibre-reinforced composites produced by RTM[END_REF] and applied Voronoi tessellation and fractal dimensions to quantify the microstructure of woven composites [START_REF] Summerscales | Voronoi cells, fractal dimensions and fibre composites[END_REF]. The full characterization of glass and carbon fiber reinforced composites has also been addressed by means of optical microscopy [START_REF] Davidson | Large-area, high-resolution image analysis of composite materials[END_REF] and digital image processing of micrographies has been employed by Joffe and Mattsson[26].

This work is part of a line of research which tries to bridge stress and strain fields at the macroscale with damage initiation and other microstructural phenomena by considering the random distribution of the fibers within the composite. This approach provides probability distribution functions for the stress and strain components, and is therefore useful for structural reliability purposes. The methodology presented here starts from micrographies and, using image processing techniques together with spatial statistics tools, measures the homogeneity of the distribution of the fiber within the composite. Although the distribution of the fiber within the material is random, it is homogeneous -as will be shown in the next section the statistical homogeneity can be mathematically defined -if the fiber is correctly spread through out the material and, in this way, regions containing matrix pockets are avoided. This homogeneity can be seen as a measure of the quality of the fiber distribution since, as this paper will show, homogeneity in the material leads to lower mechanical property dispersion.

Spatial point patterns

This section summarizes the basics on spatial point patterns focusing on those aspects which are specially relevant for the statistical analysis of the microstructural characteristics of heterogeneous materials. Details on these topics can be found elsewhere [START_REF] Diggle | Statistical analysis of spatial point patterns[END_REF][START_REF] Ohser | Statistical Analysis of Microstructures in Materials Science[END_REF][START_REF] Sobczyk | Stochastic Modeling of Microstructures[END_REF].

A spatial pattern is a set of points which are located irregularly in a domain.

The points' position is governed by some stochastic mechanism. Consider an image of area A of a fiber-reinforced composite material , with the number (N ) and position (x) of the center of the fibers being a random variable. Then, the set of the position of the centers is a spatial point pattern.

The first-order properties of a spatial point pattern can be described by the intensity function, λ(x):

λ(x) = lim |dx|→0 E[N (dx)] |dx| ( 1
)
where E[•] is the mathematical expectance operator. The second-order intensity function, λ 2 (x, y), can be defined as:

λ 2 (x, y) = lim |dx|,|dy|→0 E[N (dx)N (dy)] |dx||dy| (2)
which corresponds to the intensity function at x conditional on knowing that there is a fiber located at y. The scaled function:

g(x, y) = λ(x, y)/λ 2 (3)
is called the radial distribution function.

A usual assumption for the spatial point patterns found in heterogeneous ma-terials is that they are second-order stationary. This assumption implies that their statistical properties are invariant under translation, they have a constant mean, λ((x)) = λ, and second-order properties can be expressed only as a function of the vector r = x -y. Moreover, if a spatial point pattern can be considered isotropic, second order properties only depend on the modulus of vector r, so we can write λ 2 (x, y) = λ 2 (r), g(r) = g(r). The point patterns given by the positions of fiber centres in fiber-reinforced composite materials can be considered isotropic second-order stationary and, consequently, the second order properties analyzed in this work only depend on r.

The usual probabilistic function which is assumed to describe the position of inclusions in a material is the Poisson point field [START_REF] Ohser | Statistical Analysis of Microstructures in Materials Science[END_REF][START_REF] Sobczyk | Stochastic Modeling of Microstructures[END_REF]. This model describes complete spatial randomness (CSR) in the distribution of fibers. That means that the probability of finding N fibers in a subdomain of area A is the same for any chosen subdomain. Consequently, this model assumes that clusters of inclusions (fibers) do not take place.

The probability of finding k fibers in a window W of area A(W ) is given by:

P [N = k] = (λ • A(W )) k k! • e -λ•A(W ) k = 0, 1, . . . ( 4
)
where λ is the fiber density, that is, the number of fibers per unit area.

Nevertheless, the Poisson distribution is physically unattainable due to finite dimension of the inclusions. For this reason, the Poisson distribution is often used for comparison purposes and may serve to distinguish between aggregated and more regular patterns. Therefore, a slight modification in the Poisson point field model is usually taken into account in order to reproduce real situations, in which the inclusions have a finite radius. In this case, the center of two inclusions cannot be closer than their diameters. So this model, which is also known in the literature as Matérn's model [START_REF] Ohser | Statistical Analysis of Microstructures in Materials Science[END_REF][START_REF] Sobczyk | Stochastic Modeling of Microstructures[END_REF], is normally employed to describe the random position of the fibers within the composite. For the cases in which the radius of the fiber r is much smaller than the sample window, it can be considered that the Poisson hard-core model and the Poisson model can be considered equivalent. This is the situation found in the materials described in this paper and, for this reason, only the Poisson distribution will be employed.

After these assumptions, the pair distribution function g(r) can be defined as the probability of finding an inclusion whose center lies in an infinitesimal circular region of radius dr around the point r, provided that the coordinate system is located at the center of a second inclusion.

Another useful function for the statistical analysis of point patterns is Ripley's K-function, which can be defined as the number of further points expected to lie within a radial distance r of an arbitrary point and divided by the number of points per unit area. Ripley's estimator [START_REF] Ripley | The second-order analysis of stationary point processes[END_REF] seems to be the most appropriate [START_REF] De | Statistical Analysis of Particle Distributions in Composite Materials[END_REF]:

K(r) = A N 2 N k=1 w -1 k I K (r) ( 5
)
where N is the number of points in the observation area A; I K (r) is the number of points lying within the circle of radius r and with center located in the k-th point; and w k is the proportion of the circumference contained within the sampling area A to the whole circumference with radius r. The weight w k can be computed numerically or using analytical expressions given in the literature [START_REF] Diggle | Statistical analysis of spatial point patterns[END_REF].

[Figure 9 about here]

The following relation between g(r) and K(r) can be found [START_REF] Pyrz | Quantitative description of the microstructure of composites. part 1: Morphology of unidirectional composite systems[END_REF]:

g(r) = 1 2πr dK(r) dr (6)
The second-order intensity function of a complete random pattern (CSR or Poisson set), K P (r), in a two dimensional domain is given by [START_REF] Ohser | Statistical Analysis of Microstructures in Materials Science[END_REF][START_REF] Sobczyk | Stochastic Modeling of Microstructures[END_REF]: The second order intensity function, K(r), and the pair distribution function, g(r), are useful to describe long-range interactions between points. Another interesting measure of how inclusions or fibers are distributed within the material is given by the nearest-neighbor distribution, which can be obtained easily as the probability distribution function of the shortest distance to a fiber for each fiber. Analogously, second-or third-nearest-neighbor distributions may be computed. These nearest neighbor functions focus on short-range interactions between points.

K P (r) = πr 2 r > 0 (7)

Materials and techniques

In this work, the analyzed composite materials were UD Laminates containing The elastic properties of HTA fibers and the different matrices can be found in Table 1. Results of tension tests are given in Table 2.

[Table 1 about here]

[Table 2 about here] The next step is the automatic detection of the fibers. In order to achieve this goal, image processing algorithms have been applied to the sequence of acquired images.

Digital image processing techniques

The automatic detection of the fibers should face two main challenges. Firstly, the radiometric properties of the acquired images prevent the fibers from being segmented from the matrix by a simple binarization. This radiometric artifact can be observed in the images as a non-uniform illumination field, brighter at the top-right corner of the image, and darker at the bottom-left corner.

Figure 1 shows a sample image to be processed. Its radiometric properties can be observed by plotting the intensity component of the image as a depth-map, as shown in Figure 2. On the other hand, it is not possible to directly binarize the acquired images, since their histogram is not bimodal and, therefore, no unique threshold exists to correctly separate the pixels belonging to the fibers and from those of the matrix (see Figure 3). The second problem to be solved is the detection of the individual fibers in the segmented image. In view of the fact that a binary image segmenting the fibers presents a large number of connected fibers (e.g. see Figure 3a), it is not possible to directly apply standard image processing techniques -such as Connected Component Labeling (CCL) [START_REF] Chang | A linear-time component-labeling algorithm using contour tracing technique[END_REF]-to label the fibers. On the other hand, we could consider using the a priori knowledge of the circular shape of the fibers. One alternative could be the use of the Hough transform [START_REF] Ballard | Generalizing the Hough transform to detect arbitrary shapes[END_REF]. Unfortunately, some of the fibers present radii that are quite different from the average (e.g. see Figure 4), generating a very large parameter space. The amount of possible radius in the parameter space, together with the lack of knowledge in the number of fibers to detect in every image, advises against the use of the Hough transform for circles.

[Figures 1, 2, 3 and 4 about here]

In this context, the proposed procedure to detect the fibers involves 2 phases:

(i) filtering and radiometric correction, and (ii ) fiber segmentation and localization.

Filtering and radiometric correction

In order to carry out the radiometric correction of the images, the procedure illustrated in Figure 5 is applied.

[Figure 5 about here]

First, the image is divided into buckets. Every bucket has to be small enough to neglect local differences in the radiometric values inside the bucket, but big enough to ensure that a reasonable amount of pixels belonging to the two regions to be segmented (fiber and matrix) fall inside the bucket. Next, a local segmentation of the grey levels detected in every bucket is carried out.

We should look for the two main regions, fiber and matrix, and compute the average value of the matrix grey levels. In some cases, a small third region, which belongs to the dust forming a deposit onto the sample, may appear .

The pixels belonging to this third region can be observed in Figure 2b as the vertical stripes at the lower bound of the depth map. It is expected that the amount of dirt will be very small when compared with the other two regions, and that it will have lower grey values. A fixed threshold is set to remove the pixels that belong to the dust (if any). The remaining pixels are segmented into the two regions (fiber and matrix), using the Otsu algorithm [START_REF] Otsu | A thresholding selection method from gray-level histogram[END_REF]. The local brightness-constancy assumption has proved to be valid for every bucket, enabling a correct segmentation of the matrix region. Then, the average value of the region corresponding to the matrix is computed for every bucket. Once a grid of mean grey level values is obtained, a second order 2D surface is fitted to them.

Finally, the original image is subtracted from the obtained surface. In this way, a corrected image presenting similar values for both fiber and matrix regions is obtained. The good performance of this procedure can be observed comparing the histograms of an original and a processed image (see Figure 6).

[Figure 6 about here]

Fiber segmentation and localization

Once the image has been radiometrically corrected, it presents a perfect bimodal histogram, as illustrated in Figure 6b. Therefore, a binarization can be carried out by choosing the correct threshold using the algorithm proposed by Otsu [START_REF] Otsu | A thresholding selection method from gray-level histogram[END_REF]. Then, a Connected Component Labeling (CCL) algorithm can be applied to the binary image [START_REF] Chang | A linear-time component-labeling algorithm using contour tracing technique[END_REF]. Since the points in a connected component form a candidate region for representing an object, this algorithm finds the connected blobs in the image. After running CCL on the image, the algorithm assigns a unique label to all points in the same component. Our final aim is then to obtain a labeled image where every label belongs to an individual fiber in the image. Unfortunately, if we apply CCL to the binarized image, and then look for the largest connected component, the object illustrated with a bounding rectangle in Figure 7a is obtained. For this reason, further processing is required to segment the individual fibers. One way to "disconnect" objects that are linked together is to apply the erode operator as defined by the mathematical morphology [START_REF] Gonzalez | Digital Image Processing[END_REF]. This operator shrinks the objects by "eroding" their contour, but erosion of a binary image has the problem: the objects that are smaller than the size of the structuring element used for eroding are eliminated. A priori, this should not be a problem, since a small 3 × 3 structuring element is used. However, due to the nature of the images illustrated in this work, several iterations of erosion are required to "disconnect" individual fibers. For this reason, instead of applying the erosion to the whole image, the biggest connected component of the image is segmented from the rest of the image, and the erode operation is applied only to this element.

This procedure produces an image where only this connected component is eroded, and the rest of the image remains unmodified (see Figure 7). For each of the materials described in Section 3, 40 images have been analyzed with the techniques just described. As a result, the position of the center of the fibers in each image has been determined.

Volume fraction

The first magnitude which can be easily computed from the acquired data is the fiber content (v f ). This is a key magnitude which governs the composite behavior and from which elastic constants are usually estimated. Table 3 shows the mean (µ), variance (σ 2 ) and coefficient of variation (ρ) computed for each of the three analyzed materials. While HTA5131/RTM6 and HTA5131/977-2 have similar properties, clearly HTA5131/6376 has a higher mean and much lower values of variance and coefficient of variation.

[Table 3 about here]

Second-order functions

K(h) and g(h) (Equations 3 and 5, respectively) can be used to assess if a point pattern describes complete spatial randomness.

For each of the images of all three materials, functions K(h) and g(h) were computed. Let us denote with K(j) i (h) and g(j) i (h) the estimations of functions K(h) and g(h) computed for the i-th image for the j-th material. Then, for each of the materials, we can analyze the statistics of K(r) and g(r):

µ K (j) (r) K(j) (r) = 1 m m j i=1 K(j) i (h) (8)
Var K(j) (r) 1 m(m -1)

m j i=1 { K(j) i (r) -K(j) (r)} (9) ρ K(j) (r) Var K(j) (r) K(j) (r) (10)
µ g (j) (r) g(j) (r) = 1 m m j i=1 g(j) i (r) (11)
Var g(j) (r) 1 m(m -1)

m j i=1 {g (j) i (r) -g(j) (r)} (12)
ρ g(j) (r) Var g(j) (r) for each material together with the plot of the analytical expression for a

g(j) (r) (13)
Poisson set (Equation 7). All four plots are nearly superposed, showing that, on average, all three materials could be considered to contain a CSR pattern for the distribution of fibers.

[Figure 10 about here]

[Figure 11 about here]

A closer observation of the plots of K(j) (r) reveals that the superposition of the plots is more remarkable for r/R < 15, where r is the inter-fiber distance and R the fiber radius. For larger values of r/R the plot for HTA5131/977-2 seems to depart slightly from the CSR plot. The difference with respect a CSR pattern can be analyzed by computing the relative error for each material j:

E K (j) (r) = K(j) (r) -K P (r) K P (r) (14)
According to the plot for this function, given in Figure 12, the HTA5131/6376 material is clearly much closer to the CSR pattern than the others.

[Figure 12 about here]

The plots for the variance and coefficient of variation of K(r), also given in Figure 10. As happened in the results for the volume fraction, given in Table 3, the variance found in HTA5131/6376 is much lower than in the other materials, .

Regarding Figure 11, analogous conclusions can be derived: all three materials seem to have a g(r) function close to that corresponding to a CSR pattern and the variance and coefficient of variation is much lower for the HTA5131/6376 material. The relative difference with respect to a CSR pattern can be computed for g(r) as was done for K(r) in Equation 14. The resulting plots for E g (j) (r) , shown in Figure 11, and the relative error, shown in Figure 13, are not as meaningful as they were for E K (j) (r) .

[Figure 13 about here] However, from the fiber distribution analysis and the statistics of volume frac-tion and from the results of tensile tests shown in Table 2 a clear conclusion can be drawn: those materials showing a large dispersion on tensile test also exhibit a large dispersion on volume fraction and a large dispersion from a CSR distribution. Analysis of the K function reveals this phenomenon more clearly than the analysis of the fiber volume fraction.

Conclusions

The statistical point patterns of fiber positions have been analyzed for three different carbon fiber-reinforced polymers. Data for the statistical analysis has been obtained through digital image processing: (i) filtering and radiometric correction, and (ii ) fiber segmentation and localization. In this way, 40 digital images for three different CFRPs have been analyzed. The volume fraction and the position of the fiber centers has been obtained for each digital image.

Together with a statistical analysis of the volume fraction, the functions K(r) and g(r) which describe the random distribution of fibers within the material have been computed and compared with the respective functions for a Complete Spatial Random (CSR) pattern.

The results of this analysis reveal that the materials showing a distribution of fibers which is more different from a CSR pattern have a larger dispersion in their failure behavior,and the analysis of the K function is the most meaningful.

The tools presented could be easily employed in an industrial environment as quality control measurements, since the small dispersion in failure behavior is a desired property in mechanic engineering materials.

  Although g(r) and K(r) are related, they provide quite different physical information. K(r) can distinguish different patterns and detect regularities, whereas the pair distribution function g(r) describes the occurrence intensity of inter-inclusion distances. In this later function, a local maximum indicates the most frequent distances between points and a local minimum the least frequent ones in the pattern. The pair distribution function, , instead of the two-point probability function, can be used for the statistical description of a composite sample when the material can be considered ergodic and statistically isotropic.

  a nominal fiber volume fraction of 60% (HTA 5131 400TEX 6K TO AERO Tenax). Three different matrices, RTM6 and 6376 from Hexcel and 977-2 epoxy resin from Cytec, were used. The composite with the RTM6 matrix was produced by resin transfer moulding and the others by a standard autoclave technique. All composites were cured at 180 o C for 2 h and post cured at 190 o C for 4 h . From the panel, standard tensile test specimens were cut according to DIN EN ISO 527-4, Type 3, and 45 o GFRP tabs were applied. Finally, for each material, seven specimens were tested in tension in the 90 o direction in a universal tensile machine (Zwick 1475) at a cross head speed of 1 mm/min.

  Next, the biggest connected component of the object is located through CCL, and the segmentation/erosion procedure is repeated recursively until the biggest object is the size of an individual fiber. Finally, CCL provides a unique label for every fiber of the image. Then, computing the gravity center of every connected component provides the localization of the fibers, as shown in Figure 8. Provided that most image processing algorithms involving the use of neighborhood operations (i.e., applying structuring elements, masks, etc.) fail to correctly process the boundaries of the image, the outline of the image is ignored to obtain statistical data. This outline has been set to the radius of a standard fiber. [Figures 7 and 8 about here]

Figures 10 and 11

 11 Figures 10 and 11 show plots for these statistics. Figure 10 (top) shows µ [K(r)]

29 3 30 4 31 5 32 6 34 8 35 9Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .Fig. 6 .Fig. 7 .Fig. 8 .Fig. 9 .

 29303132343512346789 Figures

Fig. 10 .

 10 Fig. 10. Mean(top), variance(middle) and coefficient of variation (bottom) of K(h) for the analyzed materials

Fig. 11 .Fig. 12 .Fig. 13 .

 111213 Fig. 11. Mean(top), variance(middle) and coefficient of variation (bottom) of g(h) for the analyzed materials

Table 2

 2 Ultimate strain statistics for three different CFRPs

7 Acknowledgements The work of D. Trias, J. Costa and N. Blanco has been partially funded by the Spanish Government under research project MAT2003-09768-C03-001. The work of R. Garca has been partially funded by the Spanish Government under research project CTM2004-04205. D. Trias would like to express gratitude to the University of Girona for doctoral research grant BRAE00/02. Digital images and tensile test results were provided for a former collaboration by Bodo Fiedler and Thomas Hobbiebrunken of the composites group the Technical University of Hamburg-Harburg (TUHH).

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List of Tables