AbstractA nonwoven fibrous filter media is modeled as a planar sto-

chastic array of straight lines defining multiple polygons. Thecumulative distribution and mean of the hydraulic diameterof these polygons is determined, and related theoretically tothe “mean flow” pore diameter commonly measured by com-mercial partial flow testers. This model is tested against dataobtained for a range of wetlaid glass microfiber handsheetsand electrospun polymeric nanofiber webs. The results indi-cate that the simple two-layer model is inadequate. However,a statistically powerful correlation between “mean flow” porediameter, total fiber length per unit area, and fiber diameter, isdemonstrated to hold.

IntroductionThe general concept of modeling a nonwoven fibrous web

as a planar stochastic array of straight lines is at least fortyyears old. Citations begin with Corte and Lloyd [1] in 1966,and Piekaar and Clarenburg [2] in 1967, and continue withJohnston [3,4] in 1983 and 1998. Deng and Dodson [5] gave acomprehensive discussion of the stochastic geometry of ran-dom fibrous networks in 1994, and most recently, the area wasreviewed by Sampson [6] in 2001. All of these models dealtwith random networks of fibers of finite length.

In all of these references there is theoretical discussion ofthe pore size and the pore size distribution, but no actual mea-surement of pore size. The measurement of pore size distrib-ution by through flow testing also has a considerable history,from its beginnings as a bubble point evaluation or max poretest, through the manual partial flow tester, to the discussionof the Coulter automated liquid porosimeter [7] by Batchu, tothe current instruments offered by PMI [8], Xonics [9], andTopas [10].

The aim of this work was to revisit the simplest model of anonwoven fibrous web using a planar stochastic array of

straight lines, assuming that the fibers are effectively infinite.Then, by using image analysis software on the resultinggraphical images, I determine the simplest and most appro-priate distribution for the resulting hydraulic pore diameterdistribution. This was then integrated to provided a directrelationship between the “mean flow” pore diameter and themean hydraulic pore diameter, and finally this model wascompared to actual measurements of “mean flow” pore diam-eters measured on a range of wet laid glass microfiber handsheets and electrospun polymeric nanofiber webs.

The Geometric Model I begin by modeling the web as a random collection of lines.

Imagine a small circular region of the web, as seen through amicroscope. Each fiber traversing the region of radius R isrepresented as a line connecting two points whose polar coor-dinates are {R, θ1} and {R, θ2}. Thus N fibers are generated byrandomly selecting N pairs of angles, all in the range from 0to 2π. The Cartesian coordinates of these entry and exit pointsare then given by:

(eq. 1)

These equations were set up pairwise in two columns in anExcel spread sheet. Each fiber was given by a pair of rows,with an empty row in between. A graph was then createdwhich was just the image of the N fibers. For example, Figure1 is such an image for 400 fibers.

This graphical image was copied and pasted through animage viewing shareware program from which it was export-ed as a TIF file. The TIF file was finally imported into animage analysis shareware program named ImageJ, which isfreely available from the National Institute of Health. Thissoftware automatically measures and tabulates the area, Ai,

On the ‘Mean Flow’ Pore SizeDistribution of Microfiber andNanofiber WebsBy Dr. Norman Lifshutz, Senior Research Fellow, Hollingsworth & Vose Co.

ORIGINAL PAPER/PEER-REVIEWED

18 INJ Spring 2005

and the perimeter, Pi, of each of the polygonal regions gener-ated by the intersection of all the fibers. From these two mea-surements we calculate the “hydraulic diameter” from equa-tion 2.

(eq. 2)

We choose the hydraulic diameter because it arises natural-ly in the hydrodynamics of flow through a conduit and wenote that for a circle it reduces to the diameter and for a squareit reduces to the length of a side.

The resulting cumulative probability distribution can thenbe easily determined and displayed. Figure 2 shows thecumulative hydraulic diameter of the polygons that resultfrom 50, 100, 200 and 400 lines. This cumulative distributionP>(D) is just the fraction of the polygons with hydraulic diam-eter greater than D.

It is clear that this cumulative probability distribution canbe effectively represented by a two-parameter exponentialdistribution, where p(D) is the corresponding differential dis-tribution. The fact that a small but finite lower bound on D isrequired is an artifact created by the finite pixel size involvedin the digital image and its analysis. For an infinitely smallpixel size we can use:

(eq. 3)

Of course, these are the number distributions, referring to

the counted number of pores of each size. However, we areinterested in the mean flow pore, which is that size which willallow half of the full airflow through the web. Most peoplewould use the Hagen Poisuelle law to argue that flow througha pore should vary as D4. However, a pore in this planarmodel is not a capillary tube, and so might best be consideredan orifice. For orifices, one usually assumes that the flow ratevaries as D2. Thus, for the orifice model the cumulative flowdistribution, F>(D), would be given by:

On the other hand, for the capillary model the cumulativeflow distribution would be given by:

By definition, the mean flow pore diameter, DMF, is thatvalue for which the flow is reduced by half in a partial flowtest instrument. It is that value of D for which F>(D) = 1/2, sothat we must satisfy one of two equations, depending onwhich model we choose.

19 INJ Spring 2005

Figure 1400 FIBERS WITH RANDOM STARTING AND ENDING POINTS

(eq. 4)

(eq. 5)

The most accessible average diameter is the mean hydraulicdiameter, DH, which is four times the area of all pores dividedby the total perimeter of all pores.

(eq. 7)

Thus, depending on which model we choose we have a rela-tionship between the mean hydraulic pore diameter and themean flow pore diameter:

Real NonwovensLet us now turn the discussion to real fibrous nonwovens.

Let us begin by assuming that there is a web of grammage G,made up of fibers having a diameter df and material density ρ.The specific length or length per unit mass of a fiber is just Λf=4/πdf

2ρ, so that Λ, the total length of fiber per unit area of themat, is given by:

(eq. 10)

As in our model, the fibers dividethe mat into polygonal areas, and wecan calculate PP, the total polygonalperimeter per unit area, by simplydoubling the length of fiber per unitarea, since each segment of fiber isbounding two polygons, one on eachside of the fiber. On the other hand,AP, the polygonal area per unit areaof the mat, is not 1, but rather wemust subtract the coverage or pro-jected area of the fibers. Of coursethis only has meaning for sheets withcoverage less than 1:

Thus the mean hydraulic pore diameter for a low coveragesheet is given by the equation:

(Eq. 14)

Partial Flow TestingIn partial flow testing a sample is saturated with a fluid of

known surface tension, γ, and low vapor pressure. Pressure isapplied to one side of the saturated web, forcing liquid tomigrate to the other side. As the pressure is increased a pointis reached where a first bubble can escape from the largestpore on the down stream side of the saturated web, and as thepressure is increased bubbles can escape from progressivelysmaller and smaller pores. Knowing the surface tension wecan immediately relate the pore size, DP, from which bubblesare escaping, to the pressure, P, that is being applied, using theequation DP = 4γ/P. The flow rate of gas is measured as afunction of the applied pressure, and the “mean flow” porediameter is the pore diameter corresponding to the pressure atwhich the gas flow is half what it would be for a dry sheet.The question now arises, from the model we developed aboveand real "mean flow" pore data, can we calculate the effectivebasis length that actually determines the pore size distribu-tion? In other words, combining equations 8 and 14 givesallows us to define the effective basis length, ΛEFF.

20 INJ Spring 2005

Figure 2CUMULATIVE DIAMETER DISTRIBUTION

(eq. 6)

(eq. 13)

(eq. 15)

(eq. 8)

To test this approach we prepared wet laid hand sheets witha range of Evanite glass microfiber grades, and PPG choppedDE glass, having a range of basis weights. The specific surface

area of these samples were measured by BET analysis using aMicromeritics Gemini 2370 instrument and the surface areaequivalent fiber diameter was then calculated. An additional

21 INJ Spring 2005

Table 1HAND SHEET DATA

set of samples was prepared by electrospinning a range ofpolyvinyl alcohol webs, having a range of basis weights. Thefiber diameters of these sheets were determined from SEMimages, measuring 100 fibers and calculating the surface areaequivalent average diameter for consistency with the BETmeasurements. The solidity of the glass samples was deter-mined by measuring the liquid retention of the sheet withGalden Fluid. Mean flow pore measurements on all the sam-ples were carried out using an Automated Capillary FlowPorometer from Porous Materials Inc with Galden Fluid. Thecomplete data set is shown in Table 1.

From the measured mean flow pore values and the mea-sured fiber diameter we calculated an effective basis length,LEFF, for each sample, using both the orifice model and thecapillary model. The relationship between effective basislength calculated with the orifice model, the sheet basis lengthand the fiber diameter is shown in Figure 3.

Both the fiber diameter term and the basis length term arevery statistically significant, with p values of 10-8 and 10-14

respectively. The implication of Figure 3 is that the mean flow

pore diameter is determined by a very small fraction of thefiber in the sheet, which I take to be a thin down stream layer,and the exact quantity of fiber in that thin downstream layerof fiber is a function only of the basis length of the web, andthe fiber diameter. Figure 4 shows the same result for theeffective basis length calculated using the capillary model.The only difference is a slightly higher R2 and slightly lower pvalues of 10-11 and 10-14 for fiber diameter and basis lengthrespectively. Thus, the dependence of mean flow pore on avery small fraction of the fiber length in the sheet is not a largefunction of the flow model chosen.

An alternative way of looking at the data would be to seeka simple correlation between the raw “mean flow” pore data,the basis length of the sheet, and the fiber diameter, as shownin Figure 5. It is clear that there is indeed a very strong corre-lation between the “mean flow” pore diameter, the fiber diam-eter, and the basis length of fiber in the sheet, as expressed inthe regression equation shown there.

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Figure 3EFFECTIVE BASIS LENGTH VS BASIS LENGTH AND FIBER DIAMETER

Conclusions:1. A simple stochastic model of a fibrous nonwoven web

leads us to an exponential distribution of the “hydraulicdiameter” of the pores resulting from the intersection of allthe fibers.

2. The “mean flow” pore diameter can then be shown to bea simple multiple of the mean hydraulic diameter. The con-stant is 1.337 or 2.336 depending on whether one uses an ori-fice model or a capillary model.

3. Based on these models and extensive experimental data,the effective length of fiber determining the “mean flow” poresize is one to two orders of magnitude less than the totallength of fiber in a fibrous nonwoven web, suggesting that the“mean flow” pore diameter is determined by a very thindownstream layer of the web. While sample basis lengthranged from less than 1.0 to over 400 mm/m2, the effectivebasis length varied from .04 to over 1.0 mm/m2. In terms ofcoverage, the samples ranged from 2 to 100 m2/m2 while theeffective coverage ranged from .1 to 3 m2/m2. Finally, in termsof thickness the samples ranged from 20 to 3400 fiber diame-ters while the effective thickness varied from 3.5 to 15.5 fiber

diameters.4. The “mean flow” pore diameter of a fibrous nonwoven

web appears to depend only on the diameter of the fiber andthe basis length of the web.

References:1. Corte, H. and E.H. Lloyd, Trans. IIIrd Fund. Res. Symp.

BPBMA, London, 1966, p981-10092. Piekaar, H.W. and L.A. Clarenberg, Chem. Eng. Sci. 1967

22 p1399-1408.3. Johnston, P.R., J. Testing & Evaluation 1983 11(2) p117-

121.4. Johnston, P.R., Filtration & Separation 1998 35(3) p287-

292.5. Deng, M. and C.T.J. Dodson, “Paper: An Engineered

Stochastic Structure” Tappi Press, Atlanta, 19946. Sampson, W.W., “Trans. XIIth Fund. Res. Symp.” Pulp

and Paper Fundamental Research Society, Bury, 2001 p20017. Batchu, H..R., “Proc. 1990 Nonwovens Conference”

TAPPI Press, Atlanta, 1990, p367-3718. Porous Materials Inc. Capillary Flow Porometer,

23 INJ Spring 2005

Figure 4EFFECTIVE BASIS LENGTH VS BASIS LENGTH AND FIBER DIAMETER

http://www.pmiapp.com/products/capillary_flow_porom-eter.html

9. Xonics Capillary Flow Porometer.http://www.xonics.com/Products/XonicsProducts/Porometer3G.html

10. Topas Pore Size Meter. http://www.topas-gmbh.de/poro.htm — INJ

24 INJ Spring 2005

Figure 5PORE DIAMETER VS BASIS LENGTH AND FIBER DIAMETER