Spatial structure of wood composites in relation to processing and ...

Wood Science and Technology 28 (1994) 229-239 9 Springer-Verlag 1994

Spatial structure of wood composites in relation to processing and performance characteristics Part 3. Modelling the formation of multi-layered random flake mats C. Dai, R R. Steiner

229

Summary. The structure of a randomly-formed flake-type wood composite mat is further defined and characterized. A model for the prediction of horizontal mass density distribution in a random flake mat is presented through application of two-dimensional random field theory. This model predicts the small-scale mass density variance and the spatial correlation of flake coverage. The predictions agree well with experimental results and computer simulations. Significance and implications of the model development towards practical manufacturing applications are discussed. In addition, equations for the calculation of general structural properties such as overall mat thickness, between-flake void volume content and maximum potential inter-flake bonded areas are presented.

Introduction In the preceding paper in this series (Dai and Steiner, z994) a mathematical model for describing the formation of a layer of a randomly-formed flake network was presented together with a computer program which simulated the network configuration. The model which partially characterized the structure of a random flake mat in terms of the distribution of flake centres, flake areal coverage, free flake length and void size in a constituent layer, will be further developed in this report. A randomly-formed flake mat has a more or less layered structure since all flakes essentially lie parallel to the horizontal plane of the mat. Such structural feature best manifests itself in a two, rather than three dimensional geometric model. The approach of using a two-dimensional model to describe the mat structure has the following advantages: 2) Structure of the geometric model becomes more mathematically tractable. A complete three-dimensional description of voids inside the mat, for instance, would encounter almost insurmountable mathematical difficulties;

Received 13 August 1992

Chunping Dai', Paul R. Steiner Department of Wood Science Universityof British Columbia Vancouver, B.C. V6T 1Z4 Canada Present address

Wood Science and TechnologyCentre, Universityof New Brunswick,R. R. lo, Fredericton NB, Canada E 3 B 6 H 6 1

Financial support for this work from NSERC/Canadathrough a strategic research grant is gratefully acknowledged.

2) A constituent layer can be prepared in laboratory and the flake arrangement can be measured. This makes it possible to appreciate the nature of the problems and to experimentally test the mathematical model. For this descriptive model a multi-layered random flake mat is defined herein as a summation of a series of two-dimensional randomly-arranged flake layers. The volume fraction and size distribution of voids and the horizontal mass density distribution are two fundamental aspects of the mat formation. The size distribution of voids in one layer, defined as the uncovered areas, was previously described mathematically and tested through a simulation program (Dai and Steiner, 1994). In the following presentation, the non-uniform distribution of horizontal mass density in a flake mat will be emphasized. 230

Model Horizontal mass density distribution The deposition of flake elements to form a random flake mat is characterized by a Poisson process. This allows a complete probabilistic description of point-to-point flake mass density in a two-dimensional mat field. Consider a mat contains N~ number of flakes of A length, w width and 7- thickness over A mat area. The Poisson distribution for describing the probability pi(i) of flake coverage i at an arbitrarily chosen point, is given by: p~(i) -

e-nffi~ i!

(1)

where fif is the mean flake coverage: fif = E(i) = NfA~/A. A characteristic feature of a Poisson distribution is that the variance Var(i) is equal to the mean E(i), or, Var(i) = E(i)

--

fig

-

NfAa; A

(2)

Assuming solid flake density Df, the mean mass density over mat area, E(MD), is determined by dividing the total flake mass by the total mat area, i.e. NfAa;w Df E(MD) -- ~ -- 7-DfE(i)

(3)

Accordingly, the mass density at an arbitrarily chosen point, MD, can be defined as: MD = 7-Dfi

(4)

which is called the point mass density. Thus, the distribution of MD can be obtained from the Eq. (1) through transformation, i.e. MD e nfnf ~

q(MD)--

(5)

(M~f),

According to Equ. (2), the point mass density variance, Var(MD), can be obtained through: Var

= E

= fif

(6)

or:

Var(MD) = T2D~fif ---- 7DfE(MD)

(7)

This point variance is a property of a particular "region" which is different in size from any other regions, i.e., a dimensionless point, and it is the maximum possible variance of mass density (Corte, 1982). The point variance depends on flake thickness and is independent of flake length and width. Although the point mass density distribution can provide in detail the local point-to-point variation occurring on a high level microscale in the mat field, it is usually more useful to model the distribution of local averages. This is because point mass density is practically not observable and details of the microstructure of the wood composites may affect behaviour on the macroscale only through their effect on local averages. In modelling the variation of local average mass density, the whole mat area is viewed as a two-dimensional random field. Thus, based on the known point variance the derivation of the local average variance model can be carried out through the application of theory of random fields (Vanmarcke, 1983). According to this theory, if the point is extended to a finite region, say a rectangle with side length of sx and sy, the variance of the regional mass average, Var(MDa), is expressed as (Vanmarcke, 1983): Var(MDa) ----"7(Sx, sy)Var(MD)

(8)

= 7(Sx, sy)TDfE(MD) 7(sx, sy) is defined as the variance function of mass density MD, which measures the reduction of the point variance Var(MD) under regional averaging. It is a dimensionless function of value range: 0 ~ "y(s,, Sy) _< 1. The upper limit occurs when the region shrinks to a point. The reduction of variance of regional mass average is merely due to the spatial correlation of the flake coverage between points within the region. A flake of finite area content covered at one point can always contribute to the coverage of its neighbouring points. The degree of the coverage contribution from one point to another depends on the distance the two points separate. It decreases with an increase in the distance and becomes nil as the distance exceeds the maximum dimension of the flake. This feature is measured by the correlation coefficient (Vanmarcke, 1983) of the flake coverage. Assuming stationary and isotropic properties, the variance function 7(Sx, sy) is given by (Dodson, 1971):

7(sx, Sy) =

/

~(r; A, :v)b(r; sx, sy)dr

(9)

0 where ~(r; A, ~v) is the correlation coefficient of coverage between pairs of points which are a distance r apart. It depends only on the distance r, flake length ), and width •. b(r; sx, sy) is the probability density function for the distance r between two points arbitrarily chosen in the rectangle, and depends only on the shape and size of the sample region. The equations for the determination of~/(r; A, o;) and b(r; s~, sy) are given in the Appendix. It should be noted that the exact solution to the integral (Eq. 9) can not be reached and the approximation may be obtained through numerical methods on a computer. Figure 1 shows a typical result of the correlation coefficient ~(r; A, ~v) and the probability density function b(r; Sx, st). The distribution of mass density of a finite zone can be obtained by fitting the mean density E(MD) and its variance Var(MDa) with a normal distribution. The normal distribution of the mass density is assumed here as a result of a direct application of the Central Limit Theorem (Olkin et al., 198o). According to this theorem, the normal distribution can provide a good approximation for the probabilistic characterization of a Poisson associated distribution as the mean of the distribution increases. This mean is represented here by the average flake coverage (number of layers).

231

1.2

0.1

1

0.08

b(r; s•

p-

o~

~E 0.8 0.06 %0.6 0 0 t-.O_

232

>, (,9 c-

;

'

0.04

nO

0.02

o~ ..Q

0.4 0.2

0

s

O_

00

20

40

60

80

1800

distance r (mm) Fig. 1. The correlation coefficient 7/(r; A, co) and probability density b(r, sx, sy) as functions of distance r between pairs of points in a 20 x 20 sq. mm zone (A = 83.8mm and co = 0.31 mm)

General structural characteristics Because of the random distribution of flake coverage, the apparent thickness of a mat varies locally from point to point. In addition, flakes are not always flat, therefore, measurements of overall mat thickness could be quite arbitrary. However, to quantitatively describe such macrostructural properties as mat density, relative void volume and in particular to evaluate the compression strain of a mat during pressing, an appropriate definition of the initial mat thickness is necessary. The overall mat thickness Tmo is defined here as the height of mat points where the probability of flake coverage about 0.001. This permits an analytical approach to estimating Tmo, namely, Tmo e

nffifr ~ 0.001

(10)

The average mat thickness Tmo is obtained by multiplying the mean flake coverage fif by flake thickness ~- and equals the thickness Tmm of the flake mat where the packing is maximized without inducing any voids between flakes. It can be shown as follows: Tmo z ~fq- -- NfAa;'r _ (W/Of_......~)_ Tmm (11) A A where W is the total flake weight, equal to the product of total flake volume NfAwT and its density Df. The relative between-flake void volume of a flake mat, RVbfo, is defined as the ratio of the air volume between the flakes and total flake volume, or: RVbfo --

TmoA -- "FmoA "rmo TmoA -- 1 - Tm~

(12)

The relationship between number i of flakes stacked on top of one another and its probability Pi (i) has been given by Eq. (1). This relationship can lead to the calculation of inter-flake bonded area, a parameter of interest for predicting the development of internal bond strength of wood composites during the hot-pressing. A potential bonded area results wherever an overlap can occur between a pair of flakes. As shown in Fig. 2, there are (i - 1) interfaces or poten-

potential bonded a (PBA)

r

e

a

~

[ (i-1) PBA

~

i

Fig. 2. Schematic arrangement of a stack of flakes in a mat

i flakes

tial bonded areas (PBA) in an i-flake stack. The maximum potential bonded area (PBAmax) in a random flake mat, resulting from complete mat compression, is determined by: O-

,,,,j

--

2-layer

ii

~ 0.2 -, ', -Q ;~'AA / O

a_~

5-lager

"~i' ' (

lO-.ayer

L ~ ', ',k,"''~"

o. I~,, ~, ,~,

', ~'

.... .., 0

",

~,

?; oo 10

;e

/

20-layer

/

,,0o%/ ... 20

,,z%

30-layer

-,%

9~176 . . . . . _'t- . . . . . ,30

40

number of flakes Fig. 5. Distribution o f flake coverage over mat area as a function of number of

flake layers. Total flake number = 80x No. of layers with other conditions as in Fig. 3

50

235

a 4o

3O 'f~A i &

m

2O

A

9~x,

AA

Ai

A

&

&&

,

&

' A

A

tI

E

,~

&

,,

.

&

A n

~

,.

&

'

i

r

10

~ b

;o

r

2'o 30

io

i

i

50

x-value

7'o 80

r

1oo.o

(mm)

Fig. 6 a, b. Illustration of the non-uniform formation of a random flake mat. The data is based on a computer simulation with input parameters: total flake No = 1600 and other conditions as in Fig. 3. a Three-dimensional version; b One-dimensional profile scanned along y = 72 mm

layer mats is demonstrated in Fig. 5. As layers are added, the mean and variance of coverage increase equally in value - a unique feature of the Poisson process. The non-uniformity of the flake coverage distribution of a randomly formed mat is illustrated in Fig. 6. The local flake coverage appears to vary in such irregular manners that it is spatially correlated with an invariant mean and variance. The structures of this type are mathematically described as homogeneous two-dimensional random fields (eg. Vanmarcke, 1983), therefore, the theory of random fields is applicable here.

Mass density distribution Figure 7 shows the distribution of mass density for sampling zone with size 1 • 1 and 5 x 5 m m 2, in terms of mathematically predicted, computer simulated and experimentally measured results. The formation of the hand-formed mat is close to that of computer simulated mat, and both are nearly normally distributed with a slightly positive skewness, which stems from the Poisson distribution of the point mass density (Corte, 1982). Such skewness becomes

. Experiment I

4

~'~

a

D Simulation

g3

"o

_O

o

Q.

1

236 O O

o.a

0.4 0.6 0.8 mass density (g/cm2)

1

1.2

Fig. 7 a, b. Distribution of mass density of flake mats as experimentally measured, computer simulated and mathematically predicted; a Zone size: 1 • 1 mm2 and total flake No = 800 and, b zone size: 5 • 5 mm2 and total flake No = 1 600. (other conditions as in

more noticeable as the zone size and the total layer number decrease and less so as they increase (compare (a) and (b) in Fig. 7). Nevertheless, the most characteristic parameter for describing non-uniformity of the mat formation is numerically expressed as the variance of the mass density distribution, which is independent of the shape of the distribution. In Table 1 the size-dependent mass density variances are compared between experimental, simulated and predicted data. Again a close correlation is found between the three procedures. The results also show that the mass density variance is a monotonic decreasing function of the sampling zone size, namely, the larger the zone size, the less the mass density variance. The slightly higher values for simulation compared to that of mathematical prediction is due to the digitization process from real numbers in practice to integers in simulation. This difference will decrease as the digitization resolution is increased. It is important to note that the variance of the hand-formed mat is slightly greater than that of both simulated and model mats. This means less uniformity and, therefore, less randomness of the hand-formed mat structure. This seems to be generally true for most machine-formed flake mats in wood composite manufacturing processes due to the likelihood of such non-random effects as flocculation and preferential orientation of flakes in the machine direction. As such, knowledge about random flake mat formation can establish a standard of maximum uniformity of flake mat structures for practical production. Achieving this formation would minimize the detrimental effect of non-uniformity in products. The maximum uniformity or the minimum mass density variation depends on flake geometry, i.e., flake length A, width w and thickness r. With the development of the present model, this relationship is rigorously known. Such information about mat formation is without doubt of great importance for evaluating and controlling the quality of flake-type wood composites during the manufacturing processes. Table 1. Comparison of mass density variances (flake size: 83.8 mm by 9.31 mm by 0.8, No. 800; mat: 250 mm by 250 mm; flake density: 0.4 g/cm3) Zone size (mm2)

Experiment

Simulation ((g/cm2)2)

Prediction

lx 1 5• 10x 10 25•

0.0111 0.0092 0.0073 0.0044

0.0104 0.0084 0.0066 0.0034

0.0098 0.0080 0.0061 0.0029

Table2. Some general structural properties (conditions as in Table 1) Properties

Eq. No

Experiment

Simulation

Prediction

Overal mat thickness a Tmo(mm) Ave. mat thickness Tmo(mm) Relative Btwn-flake void vol. RVbfo Max. potential bonded area PBAm~, ( X105mm2)

(10)

16.8 (0.0012)

16.8 (0.0013)

16.8 (0.0009)

(11)

7.960

7.952

7.989

(12)

0.526

0.527

0.524 237

(13)

5.594

5.588

5.616

a Numbers in the brackets are the probability values

General structural characteristics Some macro-structural properties of hand-formed and simulated mats and that of mathematical prediction are compared in Table 2. The prediction results agree well with the experiment and simulation. This generally verifies the derived equations (Eqs. lo-13). It provides a basis for predicting changes of such characteristic parameters as between-flake void volume and inter-flake bonded area as the mat is compressed during hot pressing.

Summary and conclusion The formation of a random flake mat has been further investigated by presenting a descriptive model for the characterization of small-scale mass density distribution of the mat as well as other structural properties. The prediction agrees well with experimental results and computer simulation. This in general verifies the proposed theory and methodology and leads to following conclusions: 1. The formation of an ordinary flake mat can be approximated by a random process, which is essentially characterized by the Poisson distribution of flake centers and flake coverage. 2. In terms of spatial correlation, a multi-layered random flake mat can be modelled as a two-dimensional random field. As such, random variance of the mass density distribution, the most characteristic index of the non-uniformity of mat formation, can be mathematically described by random field theory. 3. The computer simulation program, which at present serves as a tool to test the mathematical prediction, may play a more important role in further modelling processes where the problems become too complicated to be mathematically tractable. The development of a mass density distribution model allows a standard to be established for maximizing mat uniformity, which can be compared with present industrial mat formation processes. The quantitative description of mat structural features such as flake coverage distribution, void volume and its size distribution and inter-flake bonded area, can establish a microstructural approach to modelling the complicated hot-pressing operation in wood composite manufacture.

Appendix Equations for calculating q(r; ~,, o) and b(r; Sx, Sy) Assuming the flake length A and width w and a pair of points r distance apart, the equations for the determination of the correlation coefficient ~(r; A, w) is obtained as follows (Dodson, 1971):

,m(r;A,~) = 1 -- 2 (~ + r 7r w

238

rl2(r; A, a ; ) 2= -

~3(r; A, w)

r2 ) 2Aw

(arcsin(~)a; G

(0 < r < w) -r+

( r2 - 1/ ~ ) J

=_( (w / (~) 2 arcsin -arccos 7r (r2)05 -4- ~ - 1

(r 2 + ~-1

w 2A

A 2w

(w < r < A) -

r2 2Aw

)o.5) (A Sy) is given by (Ghosh, 1951;Dodson, 1971): r) b(r; Sx, Sy)= ( 4~Sx

b(r; Sx, Sy) =

(~ 7rSxSy _ r(Sx + sy) + ~ r 2 )

4r Sxsyarcsin r SxS-T~

(0 < r < sy)

+ sx(r2 -- s2y)0"5-- Sxr --

(sy < r < Sx) (15)

r ) ~( b(r; Sx, Sy)= ( 4Sx--~

sxsy (arcsin ( ~ ) -

arccos ( ~ ) )

+sx(r 2 - S2y)~ + Sy(r2 - sZx)~ _ ~1 ( r2 + sx2 + s2y)) (Sx < r G (sZx+ Sy)2"~ For square zones, Sx = Sy, and the central range for r is non-existent. References Corte, H. 1982" The structure of paper. In: Rance,H. F. (Ed):Handbookof paper science.Vol.z: 175-281.Amsterdam: Elsevier Dai, C.; Steiner,P. R. 1994:Spatial structure of wood compositesin relationto processingand performance characteristics:Part II. Modellingand simulationof randomly-formedflakelayernetwork.WoodSci. and Technol.28:135-146

Dodson, C. T. I. 1971: Spatial variability and the theory of sampling in random fibrous network. I. Roy. Statist. Soc. B. Vol. 33, No. 1:88-94 Ghosh, B. 1951: Random distances within a rectangle and between two rectangles. Calcutta Math. Soc., Vol. 43: 17-24

Olkin, 1.; Gleser, L. J.; Derman, C. 198o: Probability models and applications. N.Y.: Macmillan Publishing Co., Inc. Suchsland, O. 1959: An analysis of the particle board process. Michigan Quarterly Bulletin. Vol. 42, No. 2: 35o372 Vanmarcke, E. 1983: Random fields: Analysis and synthesis. Cambridge: The MIT Press

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