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1Cellular Polymers, Vol. 33, No. 1, 2014

Understanding the Effects of High Aspect Ratio Fibers on the Mechanical Reinforcement of Soybean Based Polyurethane Foam


Sadakat Hussain, Li-Chi Chang, and M.T. Kortschot*

Advanced Materials Group, Department of Chemical Engineering & Applied Chemistry, University of Toronto, 200 College St. Toronto, Ontario, Canada M5S 3E5

Received: 19 August 2013, Accepted: 7 October 2013

IntROductIOn

In recent years, there has been a push to replace petroleum-based products, and the market for biologically sourced polymers has been growing rapidly. Traditionally, polyurethane foams (PUF) have been made by curing petroleum-based polyol with isocyanate through addition polymerization. A foaming agent such as water is introduced to react with the surplus isocyanate to release gas that creates the foamed structure in the polyurethane. In this study, the polyol used to produce PUF was sourced from the oil found in soya beans (soyol). Soya beans are naturally occurring, renewable and provide a viable alternative to conventional polyols produced from crude oil. With the depletion of oil reserves and the rising cost of petroleum, an alternative natural source for polyols for use in PUF production is desirable. The compressive modulus of PUF can be adjusted during processing, leading to a variety of potential applications. Flexible foams are used in upholstery cushioning and shock absorbers. A common use for rigid PUF is in forming the core material of Structural Insulated Panels (SIPS), which are stiff sandwich panels that can be used as walls in both residential and commercial buildings. SIPS are one of the fastest growing products for use in home building in the United States, where their use doubled between 1997 and 2006 [1].

In this study, we examine the potential to use soyol based PUF in SIPS, motivated by their environmentally friendly, lightweight and cost-effective

*Corresponding Author, [email protected]

2 Cellular Polymers, Vol. 33, No. 1, 2014

Sadakat Hussain, Li-Chi Chang, and M.T. Kortschot

properties. In order for these materials to be effective as the core material for a SIP, they must have sufficient stiffness to resist buckling and localized skin wrinkling effects. For these reasons, current standards [1] suggest that the PUF core must have excellent stiffness under compression. One technique to improve the stiffness of polymers without significantly increasing the weight or cost has been to introduce fibers as reinforcement. In this study, we examine the use of glass fibers in PUF to improve the compressive and shear moduli.

Numerous attempts have been made to reinforce polymeric foams with fibers [2-10]. Gu et al. [2-3] (using a PUF composition identical to that used here) discovered that the compressive strength of PUF can increase by a factor of two when wood fibers are introduced. Dourdanni and Kortschot [4] determined that increasing wood fiber content enhanced the tensile modulus and impact strength of microcellular polystyrene foam composites. They observed an increase in impact strength by a factor of three when 20 weight percent of cherry hardwood fiber was added. Three additional studies reported increases in mechanical properties when adding fibers to PUF. Bledzki et al. [5], observed increases in shear modulus and impact strength with an increase in natural fiber content. Cotgreave et al. [6], observed an increase in tensile modulus and strength with the increase of glass fiber content. Finally, Siegmann et al. [7], found increases in tensile modulus, and compressive modulus with the introduction of glass fiber and powder to PUF.

Although, many experimental studies have shown the benefits of introducing fibers into foams, very little understanding of the fundamental fiber reinforcement mechanisms has been provided. A foam matrix is unique because it can have an extremely low shear modulus, and this means that traditional fibre reinforcement theories may not apply. The main objective of this study was to understand and model the reinforcement of PUF with glass fibers under compression loading. By understanding the effect of fiber aspect ratio and fiber weight fraction it should be possible to optimize the structure, and ultimately, to provide guidance for manufacturing bio-sourced reinforced foams with commercially interesting properties.

AnAlytIcAl MOdels fOR fIbeR ReInfORced fOAMs

Gibson-Ashby Theory for Unreinforced Foams

The theories developed by Gibson and Ashby [11] are the most widely accepted structural models for foam deformation. For PUF compression deformation, their model separates the deformation into three regions: Elastic-Linear, Elastic-



Non-Linear and Densification. For our study, the compressive modulus is the critical parameter, and hence we are concerned with the Elastic-Linear Regime.

Elastic-Linear:

Foams are made up of a series of cells. Gibson and Ashby developed their model by assuming that the foam had a unit cell with a simple cubic structure similar to the cell shown in Figure 1.

Figure 1. Simple cubic structure denotes foam cell structure. Figure displays before and after load is applied to the cell structure scenarios

As load is applied vertically for compression, the cell struts at the top and bottom bend leading to deflection. This bending is completely elastic and linear. The bending deflection is related to the applied load through standard beam theory as shown in Equation (1):



δ∝

FL3

EI (1)

where d = deflection, F = applied load, L = length of strut, E = modulus of material and I = second moment of inertia of a strut.

The force – F can be related to the global stress, F ∝ sL2 and the deflection to

the global strain e ∝

δ

L , allowing Equation (1) to be redefined to the following:

ε∝

σL4

EI (2)

Since E*=σ

ε , where E* is the effective modulus of the foam and I ∝ t4, where

t is defined as the thickness of the struts, Equation (2) can be redefined as

follows (where C is a constant of proportionality):

E *

E=

Ct4

L4

(3)

The constant of proportionality was determined by Gibson and Ashby to be approximately 1. The modulus of the foam can be approximated and is directly related to the ratio between the length and thickness of the cells that compose the foam. Furthermore, the ratio of thickness to length of the cell strut in Equation (3) can be further simplified to a ratio of density as shown in the following equation, which is the main relationship of the Gibson-Ashby Theory:

E *

E=

ρ *

ρ

2

(4)

Here, p* is the apparent density of the foam and r is the density of the solid material that composes the foam. Both density values only include the polymeric solid material present in the cell structure in foams. Equation (4) shows that the compressive modulus of foams is independent of cell size and dependent on the apparent density or porosity of the foam. Therefore, two different foam samples may have different cell sizes but would share the same compressive modulus if both samples had the same apparent foam density and porosity. For simplification in this study, Equation (4) can be re-written as:



E *E

=ρ *ρ

2

=M

foam/ V

foam

MPu

/ VPu

2

=V

Pu

Vfoam

2

= Vs( )

2

(5)

Vs is the volume fraction of solid polyurethane in the overall volume occupied by the foam. Mfoam and Vfoam represent the mass and volume of the entire foam sample including solid polyurethane and air. MPu and VPu represent the mass and volume of only the polyurethane in the foam. The mass of the foam will equal the mass of the solid polyurethane since it is the only material that composes the cell structure.

Rule of Mixtures for Fiber Composites

The rule-of-mixtures (ROM) is the simplest micro-mechanical model for fiber reinforced composites. It also represents the ideal structural scenario for fibers to reinforce a matrix. The model assumes that the fibers span the entire length of the specimen and are oriented in the direction of loading. In this model, the strain of the fibers and matrix would equal that of the entire specimen in the loading direction.

ε

m= ε

f= ε

c (6)

The total stress on the composite is the summation of the stress on the matrix and the fiber proportioned to their volume fractions as shown in Equation (7):

σ

c= V

mσ

m+ V

fσ

f (7)

By substituting Hooke’s lawσ

ε, into Equation (7), and the assumption from

Equation (6), we are left with Equation (8) which is the ROM.

E

c= V

mE

m+ V

fE

f (8)

Therefore, it is evident from Equation (8) that the ROM predicts the modulus of the composite to be based on the moduli of the fiber and the matrix and their respective volume fractions. This model represents an ideal upper bound case where any well-bonded fiber reinforced composite modulus predicted cannot exceed the values predicted by the ROM.

The inverse rule-of-mixtures (IROM) is another simple micro-mechanical model for fiber reinforced composites. It represents the worst structural scenario for



fibers reinforcing a matrix where any well-bonded fiber reinforced composite modulus predicted cannot be less than the values predicted by the IROM. The model assumes that the fibers are oriented perpendicular to the direction of loading In this model, the stress of the fibers, matrix and the whole composite would be the same as shown in Equation (9).

σ

m= σ

f= σ

c (9)

As well, it is assumed that the total deflection for the composite is the summation of the deflections in the matrix and the fibers leading to Equation (10).

ε

c= V

mε

m+ V

fε

f (10)

By substituting Hooke’s lawσ

ε, into Equation (10), and the assumption from

Equation (9), we are left with Equation (11) which is the IROM.

Ec=

EfE

m

VmE

f+ V

fE

m

(11)

Shear Lag Theory

Shear-lag theory is a commonly accepted micro-mechanical model to describe the fiber reinforcement of short-fiber composites. The theory was developed by Cox [12], modified for fiber packing considerations by Piggott [13] and further analyzed and refined by Nairn [14]. The theory assumes that the ends of an embedded short fibre debond and are unloaded, and that the matrix transfers the applied load to the fiber through interfacial shear stress. The fiber is stretched with shear stress being a maximum at the ends of the fiber and tensile stress being a maximum at the centre of the fiber. The transfer of applied load from the matrix to the fiber is assumed to occur with no slippage and no yielding of either the matrix or the fiber. The result is a simple rule of mixtures equation, but with a modified fibre contribution, to account for the fact that the fibre is not uniformly strained. Equation (12) shows the classic shear-lag model as defined in Piggott [13]:

E1=E

f1−

tanhnL

f

2

nLf

2

Vf+E

mV

m

(12)

Where Lf is fiber length and n is a correction factor.



Two correction factors are used in this study. One is developed by Cox and is shown in Equation (13) and the other was developed by Nairn and is shown in Equation (14). Nairn’s correction factor is derived from the exact elasticity equations for axisymmetric stress states in axial and radial directions for transversely isotropic materials.

n =1

r

2Em

Ef

1+ vm( )Ln

Pf

Vf

1/2

(13)

where vm – Poisson’s ratio of matrix, and Pf – packing factor assumed to be (2π/√3)

n =2

r2EfE

m

EfV

f+E

mV

m

Vm

4Gf

+1

2Gm

1

Vm

ln1

Vf

−1−

Vm

2

1/2

(14)

where V – Volume fraction, r – fiber radius, G – shear modulus, E – elastic modulus.

Shear-lag theory assumes that the fibers are well dispersed in the matrix and are not touching or interacting with each other. The correction factors are computed based on an assumption that that the fiber is a cylinder embedded in a cylinder of matrix whose outer surface is experiencing the global strain of the composite. The following section highlights a model that assumes that fibers are interacting with each other.

Geometric and Mechanical Percolation

Percolation Theory predicts that fibers embedded in a matrix will form a continuous network at a critical volume fraction known as the percolation threshold, which is dependent on fiber aspect ratio [15]. Figure 2 highlights the formation of a percolating network of fiber as the volume fraction increases.

Traditionally, the theory was applied to explain the increase of electrical conductivity when low volume fractions of highly conductive metal fibers were embedded in electrically insulating polymers. Initially, when the fibers are not



touching as shown in Figure 2a, the conductivity was similar to that of the insulating matrix. However, as a network formed at a critical volume fraction or “percolation threshold” as shown in Figure 2c, the conductivity increased by orders of magnitude. Figure 2d, represents a large volume fraction in comparison to Figure 2c, but this composite is below the percolation threshold, which depends on aspect ratio in a well defined way. [15]

“Mechanical percolation” is a concept that is used to explain the increase in stiffness of a soft matrix when low volume fractions of a high aspect ratio rigid fiber are added. The theory assumes that fibers in close contact deform each other, which results in the increase of stiffness in an otherwise soft matrix. Vanier et al. observed an increase of two orders in the shear modulus of cellulose whisker reinforced latex with only 1 volume % cellulose whisker content [16-17]. They attributed the significant increase in shear modulus to mechanical percolation, since the geometric percolation threshold for the cellulose whiskers they used would be approximately 1 volume % cellulose

Figure 2. (a) low volume fraction, no percolation. (b) - some fiber interaction. (c) - high volume fraction, percolated network of fibers formed (d) - low fiber aspect ratio with high volume fraction, no percolation



whisker content. However, they only observed the mechanical percolation improvements above the glass transition temperature of latex where the shear modulus of the latex dropped from 1 GPa to 0.1 MPa. This is relevant here, as the stiffness of soyol based PUF matrix we produce is comparable to that of latex above the glass transition temperature.

The models discussed above will be compared to experimental results for fiber reinforced PUF samples. The fiber volume fraction and fiber length were varied to determine which theoretical approach is most appropriate for modeling fiber-reinforced foam behavior.

expeRIMentAl pROceduRes And set-up

Fiber Reinforced Foam Sample Development

The PUF was produced by reacting soyol with isocyanate and distilled water. For our purposes, we have to introduce fibers into the procedure to create fiber-reinforced PUF.

Figure 3 highlights the procedure used to develop the foam samples.

Figure 3. Diagram of foam sample preparation

The method outlined in Figure 3 is a free-rising method. Glass fibers were added into the soyol and hand stirred for 20 minutes to disperse the fibers in the mixture. The amount of glass fiber added was dependent on the weight fraction desired (0, 5, 10, 15 or 20 wt% of glass fiber). Then catalysts, surfactants, and blowing agents were added and mixed for 20 minutes. Finally, isocyanate was added and stirred briefly for 20 seconds. The mixture was then poured into an open mold where the foam was allowed to rise freely.



The foam was kept in room temperature for 7 days to properly cure. Table 1 highlights the materials used and the specifications. Extensive tomography of samples made in this way showed no evidence of directionality in the cell dimensions, and thus the foams are treated as isotropic materials.

Table 1. Materials used in foam productionMaterial type specific type supplier/details Weight (g)

Surfactant Pyrrolidone, Polysiloxane

Air Products 1.2, 2.0

Catalyst Tertiary amine, Ethanolamine

Sigma Aldrich and Air Products

1.81.2

Blowing agent Distilled Water Available in Laboratory 4.0

Glass fiber Average length: 0.470 mm. Average

Diameter 16 microns. Silane treated

Donated by Fibertec Inc.

0, 5, 10, 15, and 20 wt%

Iso-cyanate Polymeric diphenylmethane

diisocyanate (PMDI)

Specific gravity of 1.23 g/mL at 25°C, NCO

content of 31.2%, and functionality of 2.7 was supplied by Huntsman

NCO index: 100

Soybean based polyol

Soyol® 2102 Hydroxyl number of 63 mg KOH/g was

received from Urethane Soy Systems (Volga, South Dakota, USA)

60

Compression Tests

Compression tests were performed on five specimens for each amount of glass fiber loading according to ASTM - D 1621-10 [18] to determine the compressive modulus. Foams samples were cut into 5 cm x 5 cm x 3 cm specimens. The foams were compressed in a Sintech 20 using a 1000 lb load cell. The foams were compressed to 15% deformation at a speed of 2.5 mm/sec.

chARActeRIzAtIOn study thROugh x-RAy tOMOgRAphy

In order to visualize the structure of the fiber reinforced foams, X-ray tomography images of the fiber-reinforced samples with various weight percentage fiber were taken. Scanning was performed using a SkyScan 1172 with a 10 megapixel



camera at an accelerating voltage of 45 k and a current density of 171. The size of the samples analyzed was approximately 15 mm x 4 mm x 2 mm (height x length x width). The maximum magnification was used, giving an image resolution of approximately 2.0 microns.

Even at the low fiber content used here, the high fiber aspect ratio produces a system near the percolation threshold. X-ray tomographs show that the fibers are actually not isolated and are in close-contact (see Figures 4 and 5). Figure 6 shows that the fibers are longer than the cell diameters and do span multiple cells.

Figure 4. 20 wt% (0.2 volume %) 0.470 mm glass fiber embedded in foam. Densely packed fibers demonstrate that fibers come in close contact and are not isolated. Note: depth of field of image is 2.0 mm. Only fibers shown as PUF was removed with threshold imaging techniques

Figure 5. 20 wt% (0.2 volume %) 0.470 mm glass fiber embedded in foam. Densely packed fibers demonstrate that fibers come in close contact and are not isolated. Note: depth of field of image is 2.0 mm. Only fibers shown as PUF was removed with threshold imaging techniques



It is evident from Figures 4 and 5 that the glass fiber length has been degraded somewhat during mixing, but that even so, the fibers are indeed close enough together to be interacting with another. This leads to possible bending of the fibers. As well, it is evident from Figure 6 that the fibers span multiple cells making it possible to assume that the fibers are longer than the cells and are not solely confined to the cell struts. Since the fibers are much longer than the cell dimensions we will assume for simplicity that the fibers are embedded in a low modulus homogenous matrix and ignore the specifics of the cell structure. The compressive modulus of this homogenous polymeric matrix is determined by Gibson-Ashby Theory as discussed earlier.

fInIte eleMent MOdelIng

In order to further understand and predict physically the deformation of fiber reinforced PUF, we developed a Finite Element Model (FEM). The software used was ABAQUS 6.1.2 CAE. Previous studies [17, 19] have found success in applying FEM to predict mechanical properties of composite materials. The physical experimental results and simulation results were compared to deduce if there were any correlations.

The FEM assumes that a block of fiber-reinforced foam is placed in between two rigid bodies. The fiber content varies from (0 wt% up to 20 wt%) fiber. The foam block is deformed 10 percent in compression by the rigid bodies as shown in Figure 7 to determine the compressive modulus of the foams.

In the model, we assumed the foam was a homogenous very low modulus matrix, ignoring the cell structure for simplicity. For modeling purposes, the compressive modulus of the foam matrix was based on the stress-strain curves

Figure 6. 20 wt% (0.2 volume %) 0.470 mm glass fiber embedded in foam. Intact Fibers span multiple cells. Note: depth of field of image is 2.0 mm



from 0 wt% fiber foam samples, adjusted for any changes in density (induced by cell nucleation) at the fiber surfaces. This adjustment is further discussed below, and was made using basic Gibson-Ashby theory, as expressed by Equation (5). The Poisson’s ratio was assumed to be 0 (information also used for micro-mechanical models).

It was assumed that the fibers used in the model were uniformly sized cylinders, 16 microns in diameter and 470 microns long. The modulus and density values for material properties were based on those of glass (Elastic Modulus = 80 GPa, Density = 2550 Kg/m3, Poisson’s ratio = 0.3, information also used for micro-mechanical models). Fibers were designated as beams in the ABAQUS model. Beam elements are one-dimensional, are straight initially, and are allowed to bend and elongate during deformation. They were randomly placed and oriented inside the foam block structure by a Python script written to set up the model in ABAQUS. The code first randomly selects an X, Y, Z coordinate for one end of the fiber to be placed. It then randomly selects three angles respective to the three axes to orient the fiber in 3d space. It repeats the process until the desired weight fraction of fibers is obtained. The fibers are treated as embedded region objects in ABAQUS, which allows them to share nodes with the foam block. This mimics how a fiber would be encased by the foam matrix in a real setting.

Figure 7. Compression of foam samples in FEM simulation



Results And dIscussIOn

Ideally, we would like to create a sample set containing various fiber loadings in a consistent foam matrix. Unfortunately, however, the fibers can act as cell nucleating sites, which increases the porosity of the foam and decreases the apparent foam density. Based on Equation (5), the compressive modulus of the foam matrix would also decrease. Figure 8 shows the decrease in volume fraction of solid PU as fiber weight percentage increases and the expected modulus of the foam matrix, calculated using Equation (5). The foam modulus corresponding with actual measured density was then used in all micromechanical modeling.

Figure 8. Volume fraction of Solid PU decreasing as fiber weight percentage increases. Corresponding foam Modulus expected through Gibson-Ashby Theory. 0.470 mm fibers samples

Compression tests were performed on the samples of varying fiber wt% (0, 5, 10, 15, 20). Figure 9 shows the compressive modulus for the samples.

Figure 9 shows a drop in compressive modulus of the composite samples from 0 wt% fiber to 10 wt% which correlates with the drop expected in foam modulus as shown in Figure 8. Although the foam matrix modulus remains low at 15 and 20% fiber content, the composite modulus increased, showing that significant fiber reinforcement is obtained.

In order to understand the changes in compressive modulus when 470 microns long glass fibers were added, we compared the experimental results to the accepted micro-mechanical models for fiber- reinforced polymers. In Figure 9,



the ROM provides the upper possible bound modulus where the fibers are assumed to be long, oriented in the direction of loading and span the entire length of the foam. The IROM provides the lower bound of values where the fibers are assumed to be perpendicular to the loading. Essentially, the IROM curve matches the expected foam modulus curve in Figure 8 where it is assumed that the fibers will have an insignificant effect on the modulus. The experimental results curve in Figure 9 is higher than the IROM curve suggesting that fibers are contributing to improving the foam modulus.

However, the Cox and Nairn’s shear lag models fail to predict the improvement of the compressive modulus of the samples correctly. Shear-lag theory models the shear-transfer of load from the matrix to the ends of the fiber. For the glass fibers, the modulus of the fiber is more than one hundred thousand times larger than the modulus of the foam, and so Cox’s shear lag theory suggests that the shear stress transfer is essentially negligible due to the disparity in modulus, and the fibers are essentially unloaded. For this reason, Cox’s theoretical line is virtually identical to the foam modulus line.

Nairn’s shear lag model, which corrects some of the basic assumptions built into the original Cox model, significantly over predicts the experimental results. Nairn compared his model to finite element results for composites where the fiber modulus was only ten to a hundred times larger than the modulus of the matrix. Nairn suggested that shear-lag theory models will not work well when fibers are encased in an infinite matrix volume. For our materials

Figure 9. Compressive modulus at different fiber weight percentage for 0.470 mm fiber length samples in comparison to micro-mechanical models



with 20 wt% fiber (fiber to solids ratio), the fiber volume % (volume of fiber divided by volume of foamed composite) is only 0.3 %, and together with the extremely low matrix modulus, this explains the discrepancy between the model and experimental results.

Since there was an improvement in composite modulus at high fiber loads, even though the foam matrix density and modulus decreased, the fibers must be loaded, in spite of the fact that basic Cox shear lag theory predicts essentially no stress transfer. If the fibers are loaded, but the matrix is not responsible, a more direct fiber-to-fiber load transfer mechanism is supported, suggesting that mechanical percolation could form the basis of an appropriate micro-mechanical model.

FEM simulations of fiber reinforced foams were conducted to understand if it was possible to predict the increase in mechanical properties with the introduction of fibers and to provide insight into what micro-mechanical mechanism was contributing to the increase. As can be seen in Figure 9 the results for the FEM simulations coincided reasonably well with the experimental results. Therefore, it is appropriate to use the FEM simulations to predict the physical deformation of the fiber reinforced foam samples.

The FEM revealed that fibers that are close together bend one another during compression and their strain energy is much higher than would be predicted by shear lag alone. In fact, isolated fibers in the simulation are essentially unstrained, as Cox theory predicts. (See Figures 10, 11.)

Figure 10. Fiber 1 is a close-contact fiber where three other fibers are in close contact to it. Fiber 2 is an isolated fiber



Figure 11. Fiber 1 – close contact fiber has significantly higher strain energy than Fiber 2 – isolated fiber after the simulation completes

Figure 12. Total strain energy and total fiber strain energy at 5% deformation

Furthermore, the strain energy in the fibers was the main contributor to the total strain energy of the composite after 5 percent compressive deformation in the FEM simulation. As well, the fiber strain energy contribution increased at higher weight fractions of glass fiber as shown in Figures 12 and 13.



Based on these results we believe that the fibers begin to mechanically percolate and interact at about 10 wt% and beyond. Figure 14 shows the micro-structure of a 5 wt% fiber reinforced foam sample with the PU struts removed through manipulating the image threshold. This image confirms that at 5 wt%, there is very little contact between fibers, and thus there would be very little mechanical percolation.

The combination of experimental results and FE simulations strongly support the idea that mechanical percolation plays a significant role in the reinforcement of low modulus foams. Because of this, the addition of fibers to the system can be beneficial for volume fractions greater than the percolation threshold.

Figure 13. Fiber strain energy percentage contribution to total system strain energy at 5% deformation

Figure 14. 5 wt% (0.07 volume %) 0.470 mm glass fiber embedded in foam. Fibers are not densely packed. Note: depth of field of image is 1.8 mm. Only fibers shown as PUF was removed with threshold imaging techniques



cOnclusIOns

The introduction of glass fibers to PUF results in an increase of compressive modulus. In particular, samples that had higher than 10 wt% glass fiber content had an improved compressive modulus. The increase in compressive properties could not be explained by traditional micro-mechanics models including shear-lag theory and rule of mixtures. However, using a FEM simulation we were able to predict the increase in compressive properties and determine that fiber-fiber interactions and fiber bending was a contributing factor to the reinforcement of the foam matrix. Hence the concept of mechanical percolation is necessary to explain the reinforcement of the high modulus glass fiber in the low modulus PUF. Finally, we investigated the micro-structure of samples with X-ray tomography in order to understand the relationships between structure and compressive modulus.

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