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.

Introduction • 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

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

Sheet cross-sections Rej-P H Rej-P 0

Fibre contacts measurement Fibre of A B interest 3 4 2 1 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.

Cross-sections scanned at different depths Fibre 2 Fibre 2 Fibre 1 Fibre1 Depth= 4 μ m Depth= 0 μ m Fibre 2 Fibre 2 Fibre 1 Fibre 1 Depth= 10 μ m Depth= 8 μ m

Fibre orientation measurement Y X 1 X The first image Z fibre (top) 0 Z 1 α The second image (10µm Z 2 from the top image) X 2

Fill factor 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 Fibre wall area wall area = f D h D h D w D w

Sheets measured Never dried radiata pine kraft pulp Fractionation Cutting wet handsheets Accepts Rejects SL 0 SL 1 SL 2 (Acc) (Rej) Medium Pressing only No pressing: P 0 Medium Pressing: P M High pressing: P H

Experimental results Full Partial Free fibre Fibre Fibre Fill Factor * N c length * width * * height * * Sample Contact contact (no./mm) (%) (%) (µm) (µm) (µm) AccP 0 13.0± 1.5 24 76 73.8± 7.7 31.6± 1.3 13.7± 0.7 0.43± 0.018 AccP M 20.8± 2.0 35 65 45.4± 5.1 34.3± 1.4 11.9± 0.6 0.45± 0.016 AccP H 27.7± 2.1 47 53 35.7± 3.0 36.6± 1.5 9.7± 0.4 0.51± 0.018 RejP 0 12.9± 4.8 18 82 82.6± 12 29.5± 1.2 15.8± 0.8 0.46± 0.016 RejP M 19.5± 2.0 34 66 50.3± 5.5 32.7± 1.3 14.0± 0.7 0.49± 0.015 RejP H 28.8± 2.6 44 56 35.8± 4.1 34.9± 1.3 11.2± 0.4 0.54± 0.016 SL 0 23.4± 2.4 48 52 42.1± 4.2 31.0± 1.2 11.2± 0.5 0.55± 0.016 SL 1 22.2± 1.9 53 47 45.6± 5.2 31.8± 1.2 12.3± 0.6 0.52± 0.018 SL 2 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

35 30 AccP0-measured AccPM-m easured AccPH-measured 25 AccP0-fit b = 82.1, c = 1.62 Frequency (%) AccPM-fit b = 48.6, c = 1.55 20 AccPH-fit b =42.2, c = 1.53 15 c − = − c 1 c f ( g ) ( g / b ) exp( ( g / b ) ) b b is the scale parameter 10 c is the shape parameter 5 0 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 Free fibre length ( μ m) Frequency distributions of free fibre length of samples of the accepts

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 expected total contact area per fibre = n expected area of a contact • So for fibres of length λ and width ω , λ ω Φ 2 = n expected area of a contact

Theory: FCA and number of contacts • Fractional contact area, Φ is the structural analogue of RBA and represents its upper limit. ω θ • The expected number of contacts per fibre is, to a first approximation expected total contact area per fibre = ω 2 n expected area of a contact Area of a contact is ( ) θ sin • So for fibres of length λ and width ω , ( ) 2 θ = and sin π λ ω Φ 2 = n expected area of a contact so the expected area of a contact is ω π ω 2 2 = ( ) θ sin 2

Theory: FCA and number of contacts • Fractional contact area, Φ is the structural analogue of RBA and represents its upper limit. ω θ • The expected number of contacts per fibre is, to a first approximation expected total contact area per fibre = ω 2 n expected area of a contact Area of a contact is ( ) θ sin • So for fibres of length λ and width ω , ( ) 2 θ = and sin π λ ω Φ 2 = n expected area of a contact so the expected area of a λ contact is 4 ω π ω = Φ 2 2 n = π ω ( ) θ sin 2

Theory: FCA – 2D networks 1963 • Kallmes et al. considered the statistics of networks where less than 1% is covered by more than two fibres and obtained − − c 1 e Φ = − ≤ 1 for c 1 2 D c − c c ( ) c e = P c c !

Theory: FCA – 2D networks, 1963 1 • Kallmes et al. considered the statistics of networks where less than 1% is 0.8 covered by more than two fibres 0.6 Φ and obtained 0.4 − − c 1 e Φ = − ≤ 1 for c 1 2 D 0.2 c 0 • Limitation is that Kallmes’ theory 0 5 10 gives Φ in terms of coverage only Mean coverage yet we know that we can influence fibre contact independently of coverage through density.

Theory: FCA – 2D networks, 2003 ( ) − 4 8 ( ) 2 c 1 3 φ = φ = φ = φ = φ = 1 c 3 5 c 2 ∞ 1 ( ) ( ) ∑ Φ = φ = ε c P c ; c log( 1 / ) 2 D c c = 1 ⎛ ⎞ ε ε ε 2 3 1 2 log( ) log( ) log( ) ⎜ ⎟ Φ = ε ε − + − + + K log( ) ⎜ ⎟ 2 D ⎝ ⎠ 2 9 16 75

Theory : Multiplanar networks, 2006 • Probability of contact between adjacent layers is ( ) − ε 2 1 • Fraction of fibre surface available for additional contact between layers is ( ) − Φ 1 2 D • So total FCA for network of infinite coverage is ( ) ( ) Φ ∞ = Φ + − ε − Φ 2 1 1 2 D 2 D ( )( ) = − ε − ε − Φ 1 2 1 2 D

Theory: Multiplanar networks, 2006 • Probability of contact between • FCA for network of finite coverage adjacent layers is is approximately ⎛ − ⎞ 1 ∞ ( ) Φ = Φ ⎜ ⎟ 1 − ε 2 1 ⎝ ⎠ c • Recall • Fraction of fibre surface available λ 4 for additional contact between = Φ n π ω layers is ( ) • So − Φ 1 2 D λ ⎛ − ⎞ 4 1 ∞ = Φ ⎜ ⎟ n 1 π ω • So total FCA for network of infinite ⎝ ⎠ c coverage is λ ⎛ − δ ⎞ 4 ∞ = Φ ⎜ ⎟ 1 π ω β ω ( ) ( ) ⎝ ⎠ Φ ∞ = Φ + − ε − Φ 2 1 1 2 D 2 D ( )( ) = − ε − ε − Φ 1 2 1 2 D

Validation of theory λ ⎛ − δ ⎞ 4 ( )( ) ∞ = Φ ⎜ ⎟ Φ ∞ = − ε − ε − Φ n 1 1 2 1 π ω β ω 2 D ⎝ ⎠ ⎛ ⎞ ε ε ε 2 3 1 2 log( ) log( ) log( ) Φ = ε ε ⎜ − + − + + ⎟ K log( ) ⎜ ⎟ 2 D ⎝ ⎠ 2 9 16 75 D h ρ ε = − sheet 1 ρ f cell D w n = + partial n n equiv full 2

Validation of theory 80 This study Elias (Tappi, 1967) 70 60 n equiv from experiment 50 40 30 20 10 0 0 20 40 60 80 n from model

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.