Measurement and modelling of the extent of fibre contact in paper - - PowerPoint PPT Presentation

Measurement and modelling of the extent of fibre contact in paper

Warren Batchelor†, Jihong He†‡ and Bill Sampson*

†Australian Pulp and Paper Institute, Monash University, Melbourne, Australia. ‡Now with Amcor Research and Technology Centre, Melbourne, Australia. * School of Materials, University of Manchester, UK.

• Fibre-fibre contacts
• Parameters
• number of fibre-fibre contacts
• free fibre length (fibre segment length)
• Distributions of fibre segment lengths
• Critical factor in mechanical and transport properties

Introduction

An embedded sample ready for examination in confocal microscope A sample on the stage of a confocal microscope

Measurement technique

Sheet cross-sections

Rej-P0 Rej-PH

A B 1 3 2 4

Fibre of interest Cross-section image before (A) and after (B) thresholding and

• binarisation. Fibres 2 and 3 in (B) make two full contacts, fibre 1

makes a partial contact, and fibre 4 is not in contact with the fibre of interest.

Fibre contacts measurement

Depth= 8μm Depth= 10μm Fibre 1 Depth= 4μm Depth= 0μm Fibre 1 Fibre 2 Fibre1 Fibre 2 Fibre 2 Fibre 2 Fibre 1

Cross-sections scanned at different depths

The second image (10µm from the top image) X1 Z2 X2 Z1

α

The first image (top) fibre

Y Z X

Fibre orientation measurement

• Measure fibre cross-section at surface, 10µm down
• Use centre of mass to determine angle to surface
• Correct shape to true cross-section
• Fit bounding box around irregular fibre shape
• Calculate f

Fill factor measurement

Dh Dw Fibre wall area w h D

D f area wall

=

Sheets measured Never dried radiata pine kraft pulp

Fractionation Cutting wet handsheets Accepts (Acc) Rejects (Rej) SL0 SL1 SL2

No pressing: P0 Medium Pressing: PM High pressing: PH Medium Pressing only

Sample

c

N

*

(no./mm) Full Contact (%) Partial contact (%) Free fibre length* (µm) Fibre width* * (µm) Fibre height* * (µm) Fill Factor AccP0 13.0± 1.5 24 76 73.8± 7.7 31.6± 1.3 13.7± 0.7 0.43± 0.018 AccPM 20.8± 2.0 35 65 45.4± 5.1 34.3± 1.4 11.9± 0.6 0.45± 0.016 AccPH 27.7± 2.1 47 53 35.7± 3.0 36.6± 1.5 9.7± 0.4 0.51± 0.018 RejP0 12.9± 4.8 18 82 82.6± 12 29.5± 1.2 15.8± 0.8 0.46± 0.016 RejPM 19.5± 2.0 34 66 50.3± 5.5 32.7± 1.3 14.0± 0.7 0.49± 0.015 RejPH 28.8± 2.6 44 56 35.8± 4.1 34.9± 1.3 11.2± 0.4 0.54± 0.016 SL0 23.4± 2.4 48 52 42.1± 4.2 31.0± 1.2 11.2± 0.5 0.55± 0.016 SL1 22.2± 1.9 53 47 45.6± 5.2 31.8± 1.2 12.3± 0.6 0.52± 0.018 SL2 22.5± 2.1 50 50 45.4± 5.0 33.2± 1.3 10.2± 0.4 0.55± 0.016 * ± is 95% confidence interval

Experimental results

Frequency distributions of free fibre length of samples

• f the accepts

5 10 15 20 25 30 35 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 Free fibre length (μm) Frequency (%) AccP0-measured AccPM-m easured AccPH-measured AccP0-fit b = 82.1, c = 1.62 AccPM-fit b = 48.6, c = 1.55 AccPH-fit b =42.2, c = 1.53

) ) / ( exp( ) / ( ) (

1 c c

b g b g b c g f

− =

−

b is the scale parameter c is the shape parameter

Theory: FCA and number of contacts

• Fractional contact area, Φ is the

structural analogue of RBA and represents its upper limit.

• The expected number of contacts per

fibre is, to a first approximation

• So for fibres of length λ and width ω,

contact a

• f

area expected fibre per area contact total expected

=

n contact a

• f

area expected 2

Φ = ω λ

n

Theory: FCA and number of contacts

θ ω

so the expected area of a contact is

( )

2 sin

2 2

ω π θ ω =

Area of a contact is

( )

θ ω

sin

2

( )

π θ

2 sin

=

and

• Fractional contact area, Φ is the

structural analogue of RBA and represents its upper limit.

• The expected number of contacts per

fibre is, to a first approximation

• So for fibres of length λ and width ω,

contact a

• f

area expected fibre per area contact total expected

=

n contact a

• f

area expected 2

Φ = ω λ

n

Theory: FCA and number of contacts

θ ω

so the expected area of a contact is

( )

2 sin

2 2

ω π θ ω =

Area of a contact is

( )

θ ω

sin

2

( )

π θ

2 sin

=

and

• Fractional contact area, Φ is the

structural analogue of RBA and represents its upper limit.

• The expected number of contacts per

fibre is, to a first approximation

• So for fibres of length λ and width ω,

contact a

• f

area expected fibre per area contact total expected

=

n contact a

• f

area expected 2

Φ = ω λ

n

Φ = ω λ π

4 n

Theory: FCA – 2D networks 1963

1 for 1 1

D 2

≤ − − = Φ

−

c c e

c

• Kallmes et al. considered the statistics
• f networks where less than 1% is

covered by more than two fibres and obtained

( )

! c e c c P

c c

−

=

Theory: FCA – 2D networks, 1963

1 for 1 1

D 2

≤ − − = Φ

−

c c e

c

• Kallmes et al. considered the statistics
• f networks where less than 1% is

covered by more than two fibres and obtained

• Limitation is that Kallmes’ theory

gives Φ in terms of coverage only yet we know that we can influence fibre contact independently of coverage through density.

0.2 0.4 0.6 0.8 1 5 10 Mean coverage

Φ

1

= φ

3 4

= φ

2 3

= φ

5 8

= φ

( ) ( )

c c c 1 2

− = φ

( ) ( )

) / 1 log( ; 1

1 D 2

ε φ = = Φ

∑

∞ =

c c P c c c

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + − + − = Φ K

75 ) log( 16 ) log( 9 ) log( 2 2 1 ) log(

3 2 D 2

ε ε ε ε ε

Theory: FCA – 2D networks, 2003

Theory: Multiplanar networks, 2006

• Probability of contact between

adjacent layers is

( )

2

1

ε −

• Fraction of fibre surface available

for additional contact between layers is

( )

D 2

1

Φ −

• So total FCA for network of infinite

coverage is

( ) ( ) ( )( )

D 2 D 2 2 D 2

1 2 1 1 1

Φ − − − = Φ − − + Φ = Φ∞ ε ε ε

Theory: Multiplanar networks, 2006

• Probability of contact between

adjacent layers is

( )

2

1

ε −

• Fraction of fibre surface available

for additional contact between layers is

( )

D 2

1

Φ −

• So total FCA for network of infinite

coverage is

( ) ( ) ( )( )

D 2 D 2 2 D 2

1 2 1 1 1

Φ − − − = Φ − − + Φ = Φ∞ ε ε ε

• FCA for network of finite coverage

is approximately

∞

Φ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = Φ

c 1 1

• Recall

Φ = ω λ π

4 n

• So

∞ ∞

Φ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = Φ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ω β δ ω λ π ω λ π

1 4 1 1 4 c n

Validation of theory

( )( )

D 2

1 2 1

Φ − − − = Φ∞ ε ε ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + − + − = Φ K

75 ) log( 16 ) log( 9 ) log( 2 2 1 ) log(

3 2 D 2

ε ε ε ε ε Dh Dw

cell sheet

1

ρ ρ ε

f

− =

2

partial full equiv

n n n

+ =

∞

Φ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ω β δ ω λ π

1 4 n

Validation of theory

10 20 30 40 50 60 70 80 20 40 60 80 n from model n equiv from experiment This study Elias (Tappi, 1967)

Conclusions

• FCA can be expressed as a function of porosity and coverage only.
• Number of equivalent contacts per fibre is proportional to

fibre length and FCA and inversely proportional to fibre width.

• The fill factor seems to provide an appropriate weighting for

apparent density permitting calculation of an accessible porosity.

• Agreement between theory and experiment is good.

Acknowledgements

• Richard Markowski, for his assistance in the fractionation

experiment.

• Funding received from the Australian Research Council

(ARC) is greatly appreciated.