1、Simulating Effects of Fiber Crimp, Flocculation, Density and Orientation on Structure Statistics of Stochastic Fiber NetworksJ. Scharcanski , C.T.J. Dodson+ and R. T.Clarke Institute of InformaticsInstitute for Hydraulic ResearchFederal University of Rio Grande do Sul, Porto Alegre, Brasil 91509-900
2、+Department of MathematicsUniversity of Manchester Institute of Science and Technology, Manchester, U.K. M6O 1QDABSTRACTThe Neyman-Scott process is adapted to the problem of simulating the statistical properties of stratified stochastic fibrous materials. The simulations suggest a relationship betwe
3、en mean number of fibers per zone and mean voids, independent of the nature of the stochastic fibrous structure: the characteristic shape of the transfer function curve persists whether the structure contains crimped fibers or not, if it is random or flocculated, isotropic or anisotropic; this could
4、 be an important universal effect. The mean and standard deviation turn out to be positively correlated for some fiber network parameters, such as mean voids, fiber density, and mean number of fiber bounds (i.e. fiber contacts). Also, our simulations suggest that fiber crimp has a higher impact on i
5、sotropic structures. As crimp is increased, isotropic structures tend to present smaller mean voids, higher mean number of fibers per zone, and higher total number of bonds per fiber, than anisotropic structures.1. INTRODUCTIONThe relationship between fiber properties and void space in stochastic fi
6、ber network structures is complex, and has defied several attempts over the years to be accurately modeled in the case of nonwoven fabrics. For example, it is known that considering a given fabric density and structure, smaller pores, better barrier properties and higher flexibility can be obtained
7、utilizing smaller fibers (Kim et al., 2000). However, fiber crimp and fiber orientation also can affect the fabric pore size distribution, and at the same time it influences the number of fiber-to-fiber contacts and hence mechanical properties. Certainly, fiber-to-fiber contacts and fiber orientatio
8、n play a crucial role in determining the physical behavior of nonwoven fabrics. These properties affect the fluid transportation pattern within the fabric, its mechanical properties, surface appearance and hand (Gong et al., 1996). Besides, fiber crimp may affect local fiber density, as we will disc
9、uss later in this work. Therefore, fiber network properties are interrelated in a complex way. Such phenomena is usually difficult to measure experimentally, or it might be costly to acquire sufficient experimental data. Under these circumstances, computer simulation may help clarify such intricate
10、behavior, as we show later.Recently, Kim and Pourdeyhimi (Kim et al., 2000) discovered a relationship that exists in random fiber networks between the properties of fiber crimp and orientation. However, real fiber networks are flocculated (i.e. clumped) and not random (Deng et al.,1994) (Dodson et a
11、l.,1997), and their results need to be extended to this case. Also, it is consensual that network properties such as the density of fiber-to-fiber contacts are intrinsic to fiber networks (Deng et al.,1994), however this property is difficult to measure experimentally, and it has been neglected in p
12、revious works reported in the literature. In this article, we approach these relevant issues.Both meltblown and spunbonded fabrics are stratified stochastic (i.e. multiplanar) fibrous materials, with little or no order or orientation through the thickness. Published work on modeling and simulation o
13、f stratified stochastic porous media tends to use a one-dimensional structure representation (Scharcanski et al., 1998), or a random aggregation process involving extended geometric objects (e.g. disks) (Deng et al.,1994) to limit computation costs. Recently, some analytical models of pore size dist
14、ributions in multiplanar stochastic porous media have been proposed (Dodson et al.,1997). We consider pores as two-dimensional entities (Kim et al., 2000).This work adapts a stochastic model, the Neyman-Scott (NS) process (Neyman and Scott, 1958, 1972), to the problem of modeling the statistical pro
15、perties of stratified materials composed of fibers. Although widely studied (Cox and Isham, 1980), we are not aware that the NS has been used in this context. We use NS as a device for simulating the distribution of fibers in a single or multi-planar network forming a network of inter-connecting por
16、es, thereby deducing properties of the interconnecting pore network from the simulated structures; but as explained above it can also be adapted to the modeling of pores directly if their distributions of size and shape are given. This approach is dealt with in a separate paper.2. SPATIAL FIBER DIST
17、RIBUTION IN STRATIFIED STOCHASTIC FIBER NETWORKS Several attempts have been made to model different aspects of the spatial fiber distribution and the void structure in stratified fiber networks. Most models have focussed on specific issues such as the void size distribution (Dodson et al.,2000; Dods
18、on et al., 1997), adjacent inter-fiber distances (Dodson et al., 1994), or the spatial distribution of fiber density (Dodson et al., 1994). Usually, such models are used to predict particular structural properties (e.g. void size distribution), given the information known a priori about other, relat
19、ed, properties (e.g. deviation from randomness of the spatial distribution of fiber density). Fiber networks result from the stochastic spatial deposition of fibers, and the fiber deposition process is well described by a homogeneous Poisson process (Dodson et al., 1994) when fibers are positioned c
20、ompletely at random. However, complete randomness is the ideal, and in practice fiber networks deviate significantly from a Poisson process (Dodson et al., 1997); in industrial fibrous materials, mean fiber density often varies spatially and fibers form clusters. The NS process used in our work is a
21、 spatial clustering process describing the spatial occurrences of fiber groupings (i.e. fiber clusters) (Cressie, 1993). Such clusters are generated by a spatial point process having parameters that control the deposition of fibers within each cluster, as described below. As mentioned above, some as
22、pects of the fiber and void structure have already been reported elsewhere. Consequently, it is our concern that our simulation model generate structures that are compatible with those found in practice, whilst any new model that is proposed must retain those features of earlier models that are know
23、n to be consistent with reality, whilst improving, where possible, on those aspects where existing models are less satisfactory. Statistical properties of the NS process, originally proposed as a model of galaxy clusters in cosmology, have been described by Cox and Isham (1980) and Cressie (1993) am
24、ong others. It can be used either (a) to model voids of given shape and size distributions occurring in different layers of a multi-layer structure, so as to allow for correlation between void position and size in different layers, or (b) as in the application described below, to model fiber positio
25、n and orientation. For this second application, the structure of the NS process is as follows.1. A set of N parent events (i.e. cluster centroids) is realized from a Poisson process (in its most general form, not necessarily homogeneous) with mean (i.e. density) fibre_density. 2. Each parent produce
26、s a random number k of offspring, according to a discrete probability distribution p(K=k). Each offspring corresponds to a fiber. A simple form for p(K=k) is the Poisson distribution, either truncated (K0) or untruncated (K 0); other discrete distributions are also possible.3. The locations of the k
27、 offspring, relative to their parent, are the points (xi, yi), i=1.k, where the coordinates xi, yi are realizations of random variables having a bivariate, continuous probability distribution f(x, y). Relative to the parent event, each pair of coordinates defines the midpoint of a fiber. A particula
28、rly simple model is obtained by taking f(x, y) to be bivariate Normal with zero correlation, with deviations along the axes x and y given by x and y respectively; a further simplification takes x = y , giving circular symmetry of fiber mid-points about the parent event. With x and y unequal, fiber m
29、id-points are distributed anisotropically.4. Each offspring is a fiber of length L, where L is a random variable with continuous distribution defined on the interval 0, ). In our simulations, the distribution of L was taken as the lognormal probability distribution with parameters l , l2.5. The orie
30、ntation of each fiber is a realization of a random variable with distribution defined over the interval 0, ). In its simplest form, the distribution of is uniform, p()d = d / , for which fiber distribution is isotropic; the uniform distribution is a special case (a=1, b=1) of the beta distribution(0
31、 1 or if a1 and b1, b1.6. The final realization is composed of the superposition of offspring only.In our model, at each layer, the incidence of parent events within a zone is considered to be a Poisson process such that its probability density is (Stoyan et al. 1995)(Dodson et al.,2000):p(N=n) = (
32、)n exp (-S) / n! n =0, 1where n is the number of parent events in a zone of area S and is the mean number of parent events per unit area. We use a multi-planar model with layers of fibers. In this work, we assume that layers are independent (although as noted above, the NS process can be used to mod
33、el the correlation between pore positions in different layers). Given a simulated multi-layer void structure, structural properties can be evaluated, such as the spatial density distribution of matter and voids, and properties of the network of communicating pores.2.1 MODELLING FIBER CRIMP One of th
34、e simulation parameters is the degree of fiber crimp k=P/L, which is defined as the ratio between the fiber perimeter P, and the end to end distance, or length L of each fiber. Therefore, fiber crimp k 1 implies that fiber perimeter P is larger than fiber length L, and that the fiber is not straight
35、. Given a step p, fiber crimp is then simulated as if the fiber was folded in n=L/p pieces, and each piece is an isosceles triangle with height a=kp/2, and angle t=cos-1(p/(2a) to the base. 3. SIMULATION OF NEYMAN-SCOTT PROCESSESThe first requirement is to obtain realizations of the spatial distribu
36、tion of parent events with parameter ; for simplicity, we assume that is constant. In a spatial region with area A, the number N of parent events in A has a Poisson distribution p(N=n) = (A)n exp(A)/n! and the n events form an independent random sample from the uniform distribution over A . Ripley (
37、1983) states that care must be taken when choosing the pseudo-random number generator, since although most generators will yield a uniform distribution over the interval (0, 1), many will yield quite regular patterns on the unit square (Cressie, 1993). If A is a rectangle of dimension (0, a1)(0, a2)
38、, Lewis and Shedler (1979) recommend that the x-co-ordinates of events located in (0, )(0, a2) be generated from a Poisson process with intensity .a2, such that the distances between the x-co-ordinates of the parent events are exponentially-distributed with parameter .a2 (i.e., a pseudo-random varia
39、ble u is generated from a uniform distribution over (0,1), and the exponential pseudo-random variable is obtained from the transformation v= - log(u)/( .a2) ). The variables v1, v1+v2, v1+v2+v3,. define the x-coordinates, the process stopping when the sum exceeds the upper x-limit a1 , thus determin
40、ing the value of n, the number of parent events for the simulation run. Having generated the x-coordinates in this way, the y-coordinates, n in number, are obtained by generating uniformly-distributed pseudo-random numbers over the interval (0, a2).Having generated the positions of the parent events
41、, the next step is to generated the k offspring from the discrete distribution pK(k), k=0,1,2,. . An important point to bear in mind when generating the positions of the offspring is that the area of interest A must be embedded within a larger area, such that the offspring from parent events generat
42、ed in this larger area but which lie outside A, may fall within it. Failure to take this precaution may introduce appreciable bias from the edge effect.Generation of fiber lengths from a log-Normal distribution, and of fiber orientations from a beta distribution, present no particular problems.4. EX
43、PERIMENTAL RESULTS AND DISCUSSION To illustrate the performance of our simulation procedure, several simulated samples were generated with different conditions of anisotropy, fiber crimp, fiber clumping, and fiber spatial density. Simulation may help to understand phenomena that would otherwise be c
44、ostly or difficult to measure. From the analysis of these structures we may learn, for example, how to model different aspects of their structure, such as the existing relationship between fiber crimp, fiber bonding and void structure, or the effect of different parameters on the final void structur
45、e.Figures 1 (a), (b), (c) and (d) show some simulated crimp fiber networks. Usually, in papermaking mean fiber length ranges from 1.5 6 mm (Smook, 1994). In our experiments we utilize short fibers, i.e. mean fiber length of 2 mm, and standard deviation of 0.5 mm. The fiber density of the simulated s
46、amples in Figure 1 vary from 15 70 fibers per mm2. The area represented in those images corresponds approximately to 100 mm2. Notice that Figures 1(a) and (b) show random isotropic and anisotropic simulated samples with 20 fibers per mm2, and k=1.35. Figures 1(c) and (d) show the polar plots obtaine
47、d for those samples, using the technique proposed in (Scharcanski et. al., 1996), indicating that the sample shown in Figure 1(a) is in fact nearly isotropic (i.e. e=1.30), while the sample shown in Figure 1(b) is anisotropic (i.e. e=2.33).Both samples were simulated with the same fiber density and
48、fiber length distributions. Nevertheless, we used x=4 mm and y=4 mm to generate the nearly random sample. For image analysis purposes, the simulated images were sub-divided into approximately 200 x 200 square blocks, namely, pixels (i.e. each pixel is 25 x 10 4 mm2), and after that, the fiber densit
49、y (i.e. pixels occupied by fibers) as well as void areas (i.e. pixels not occupied by fibers) were computed. Figures 2 and 3 show logarithmic scale plots of the mean versus the standard deviation of voids and fiber density distributions for a wide range of structures (i.e. random and flocculated, isotropic and anisotropic). The observation of those plots indicate that, in general, mean and standard deviation of voids, as well as of fiber density, are correlated (Dodson et al., 1999). Also, a similar conclusion may be derived about the number of con
