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On measuring wettability in infiltration processing

Veronique Michauda,* and Andreas Mortensenb

aLaboratory for Polymer and Composite Technology (LTC), Ecole Polytechnique Fe de rale de Lausanne (EPFL),

Station 12, CH-1015 Lausanne, Switzerland bLaboratory for Mechanical Metallurgy, Ecole Polytechnique Fe de rale de Lausanne (EPFL),

Station 12, CH-1015 Lausanne, Switzerland

Received 3 January 2007; revised 31 January 2007; accepted 1 February 2007Available online 26 February 2007

Capillary pressures encountered in composite processing are often evaluated by measuring infiltration rates as a function of applied pressure. Such data are generally interpreted assuming slug-flow. Using the Brooks–Corey correlations we relax thisassumption, to indicate possible pitfalls of the slug-flow approach and to show how such data can nonetheless be used to derivemeaningful capillary parameters.Ó 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Infiltration; Composites; Capillary phenomena; Wetting; Threshold pressure

Many composite materials are produced by infiltra-tion. This process is largely governed by capillarity,which acts to drive or oppose motion of the infiltratingfluid into the porous solid preform to be infiltrated.

Quantifying capillary forces, by analysis or measure-ment, is of obvious importance in understanding theprocess.

In the absence of interfacial reactions (which areimportant in some systems but complicate the problemimmensely), the relevant thermodynamic parameter isthe work of immersion W i [1,2]. According to Young’sequation [3]:

W i ¼ rlv cosðhÞ ¼ rsv À rsl ð1Þwhere rlv is the surface tension of the liquid infiltrant, hir its wetting angle on a flat solid substrate, and rsv andrsl are the solid/atmosphere and solid/liquid interfacialenergy, respectively.

Both rlv and h, and hence W i, are measurable directlyusing the sessile drop technique [3]; however, this tech-nique is often not usable for systems of relevance tocomposite processing. Reinforcement materials are gen-erally not available as flat and large substrates. Also,wetting in infiltration is dynamic, which can influenceh [3–5]. Direct methods are therefore often used to

measure capillary forces in infiltration; these come intwo classes.

The first relies on the slug-flow assumption [1,2,4,6,7].In slug-flow, infiltration takes place with a fully satu-

rated infiltration front, across which there is a singlepressure difference, DP c, caused by curved menisci of the liquid surface – as with a liquid in a straight capillarytube.

The second approach is based on methods that weredeveloped in soil science and reservoir engineering.Here, capillary forces are quantified, not with a singlepressure difference but with curves plotting the capillarypressure vs. the fraction of filled void space (or ‘‘satura-tion’’), called drainage or imbibition curves, respec-tively, when the infiltrating fluid does not wet, or wets,the solid [1,2,8–13]. This approach is more complexand also somewhat more cumbersome experimentally,hence it is more rarely adopted in the study of compositeprocessing. However, it is fundamentally more correct.

The point of this note is to examine the formerapproach in light of theory underlying the second.

Consider the first method. It rests on Darcy’s lawwritten for fully saturated flow, which states that therate of flow of a Newtonian and incompressible fluidthrough a solid at sufficiently low Reynolds number(typical of infiltration processing) is proportional tothe local gradient of pressure P within the fluid:

vo ¼ À K

gr P ð2Þ

1359-6462/$ - see front matterÓ

2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.doi:10.1016/j.scriptamat.2007.02.002

* Corresponding author. Tel.: +41 21 693 49 23; fax: +41 21 693 5880; e-mail: [email protected]

Scripta Materialia 56 (2007) 859–862

www.actamat-journals.com

mailto:[email protected]

mailto:[email protected]

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where the fluid ‘‘superficial velocity’’ vo is the volume of fluid passing through a unit surface cut across the por-ous medium per unit time, K is the permeability of theporous medium (in the most general case a tensor andof units m2) and g is the viscosity of the fluid (in Pa s).

Practically, to measure DP c one produces an experi-mental set-up such that infiltration of a homogeneous

and non-deforming preform takes place along a singledirection parallel to a principal direction (Ox) of K . Inslug-flow continuity dictates that vo be constant every-where along the infiltrated preform. When the totalpressure differential DP T driving the motion of the fluidis kept constant, then the position of the infiltrationfront, at x = L, is:

L2 ¼ 2 Kt

gð1 À V sÞ ðD P T À D P cÞ ð3Þ

where x = 0 is the preform entrance, DP c is the capillarypressure, counted positive when it opposes infiltration,and V s is the volume fraction of solid phase in the pre-form. Plotting L2/t (or, when the front position is notdynamically tracked, L2 for a fixed infiltration time t)vs. DP T then yields a straight line that intersects theabscissa axis at DP T = DP c.

Measurements of capillary pressure drop valuesconducted in this manner have been published by manyauthors, for both polymer and metal composite systems[4,6,14–30]; an extensive review of the subject is given inRefs. [20,31]. Methods vary, but reproducible valuesthat obey expectations are often obtained (e.g. DP c is in-versely proportional to the average preform pore diam-eter) [20,31]. The method, however, assumes slug-flowwhile in many such experiments there is clear evidencethat flow is not fully saturated (e.g. [23,27]).

We now consider the same experiment, namely theinfiltration along a single direction (x) of a non-deform-ing preform driven by a constant pressure, DP T, butassume unsaturated flow. Infiltration thus proceedsgradually, over a range of pressures described by thedrainage–imbibition curve [1,2,10–13,32]. Mass conser-vation dictates:

ovo

o x¼ À oV l

ot ð4Þ

where V l is the local volume fraction liquid in thepreform. The permeability K is now a function of V l,and V l itself varies between 0 and (1 À V s) as the localpressure in the liquid, P , increases.

As is well known, for unidirectional infiltrationdriven by a constant pressure the problem can be solvedusing the Boltzman transformation [8,33,34]. We define:

u ¼ x ffiffit

p ð5Þ

and, after substitution of Eqs. (4) and (2) into Eq. (3),the governing equation becomes:

u ¼d À2 K

gd P

dV l

dV ldu

h idV l

ð6Þ

to be solved with boundary conditions:

u ¼ 0 ði:e: x ¼ 0 and t > 0Þ; V l ¼ V lðD P TÞ ð7Þ

and

u ¼ ufront; V l ¼ 0 ð8Þwhere V l(DP T) is the fraction liquid in the preform forP = DP T and ufront is the value of u at the tip of theliquid front advancing into the preform. Solving theproblem requires knowledge of the two functions

K (V l) and V l(P ); these are known in the form of semi-empirical correlations. We use hereafter the correlationsof Brooks and Corey, which are well established in soilscience, and have been successfully confronted withexperimental data [32,35–37], including in compositematerial processing [8,9,38–40].

When h > (p/2), which is generally the case in com-posite processing, the Brooks and Corey correlationreads:

S l ¼ V l

1 À V s¼ 1 À P b

P

k

ð9Þ

and

K ¼ K satS 2l 1 À ð1 À S lÞð2þkÞ

k

h ið10Þ

Here, the liquid saturation S l depends on P via (i) the‘‘bubbling pressure’’ P b, which is the first pressure atwhich the liquid penetrates the preform, and (ii) a ‘‘poresize distribution index’’ k that measures the spread ineffective pore diameter within the preform (the greaterthe spread, the smaller is k). P b is inversely proportionalto the average pore diameter, all else being constant. K satis the permeability of the fully saturated preform.

Substituting Eqs. (9) and (10) in Eq. (6) and integrat-ing once subject to Eq. (8) yields:

dudS l

Z S l0

uð sÞd s ¼ À2 K sat P bð1 À V sÞgð1 À S lÞ

Àðkþ1Þk

k

Â S 2l ½1 À ð1 À S lÞð2þkÞ=k� ð11ÞThis non-linear integro-differential equation is solved

numerically for the function u(S l) subject to Eq. (7)using MathematicaTM (Wolfram Research Inc., Cham-paign, IL). The infiltration front position, ufront

p t, is

predicted; this, of course, is the measured quantity Lin slug-flow infiltration experiments (Eq. (3)).

The results of this calculation are plotted in Figure 1in adimensional form, defining the infiltration front po-sition as

F ¼ ðufrontÞ2 ð1 À V sÞg2 K sat P b

ð12Þ

while dimensionless applied pressure is defined as

p ¼ P

P bð13Þ

This adimensionalization of pressure and infiltrationvelocity is such that infiltration under slug-flow will givea straight line of slope 1, intersecting the horizontal axisat p = 1. Indeed, in this case DP c = P b, k tends towardsinfinity since the preform structure tends towards one of perfectly uniform pores (e.g. a bundle of straight capil-

laries) and Eq. (3) applies. As seen, as p increases, allcurves gradually become straight lines of slope unity.

860 V. Michaud, A. Mortensen / Scripta Materialia 56 (2007) 859–862

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Also, as k increases, i.e. as the degree of uniformity of pore size and pore geometry within the preform in-creases, the curve becomes increasingly linear.

Suppose now that a series of experiments be con-ducted to measure the capillary pressure according tothe first, slug-flow-based, method described above.According to Eq. (3), data are most conveniently plottedas L2 divided by t (i.e. (ufront)

2), vs. DP T: this is simply adimensional form of Figure 1. Examining Figure 1, asthe degree of pore uniformity decreases, i.e. as k de-creases, much of the data from experiments of this typemay lie within the non-linear portion of the curve; exam-ples of non-linear plots such as those in Figure 1 can befound in many studies (e.g. Ref. [25]). Running astraight line through a limited set of data and extrapo-lating to the horizontal axis will then lead to a measuredapparent capillary pressure drop, DP c, that has no phys-ical meaning. Given the finite experimental error thatcomes in such experiments, this is a pitfall that will easilybe encountered in practice.

If, on the other hand, a sufficiently wide range of ap-plied pressures is explored that the strictly linear portionof the curve is obtained, then taking the intercept of theresulting straight line with the horizontal axis, as illus-trated in Figure 2 for a low value of k, will yield a pres-sure, P ex, which increasingly exceeds P b as k decreases.For realistic values of k, such as 2 [8,9,38–40], the differ-ence between P ex and P b is quite significant (Fig. 1).

Now, although P ex itself has no direct fundamentalsignificance, the fact that it increasingly deviates from

P b as k decreases suggests that the deviation betweenthe two may be used to access the pore size distributionindex, k. Analyzing results from the present calculations,it turns out that P ex can be expressed, over the range of kexplored here, as a simple function of P b and k, namely:

P ex ¼ P bkþ 0:71

k À 1

ð14Þ

as shown in Figure 3. Therefore, provided the range of pressures explored in experiments of the first type ismade sufficiently wide to obtain both (i) a trustworthyvalue for the lowest (or ‘‘bubbling’’) pressure P b atwhich the infiltrant first penetrates the preform and (ii)

the strictly linear, high-pressure portion of the curvesin Figure 1, it is still possible to use the data to arrive

at the two parameters that describe the drainage curveaccording to the Brooks and Corey correlation, namely(i) the bubbling pressure P b, which is measured directly,and (ii) the pore size distribution index k, which is sim-ply computed from Eq. (14) as:

k ¼ P ex þ 0:71 P b P ex À P b

ð15Þ

We also note that, if infiltration of the porous solidpreform by the liquid takes place quasi-statically, theintegral of the drainage curve

Z 1

0

P dS l ¼Z

1

0

P bð1 À S lÞÀ1=k dS l ¼ P b kðk À 1Þ ¼ W iRl

ð16Þis the work of immersion W i, multiplied by the total areaof preform/infiltrant interface created per unit volumeof liquid infiltrated into the preform, Rl. Since Rl

can be measured directly using the BET technique,drainage/imbibition curves can then be used to measuredirectly the work of immersion W i, defined in Eq. (1) – provided again that infiltration takes place quasi-stati-cally. Then, inserting Eq. (16) into Eq. (14), experimentsof the first type, conducted over a sufficiently wide range

of pressures to measure both P b and P ex, can also beused to assess W i as:

Figure 1. Adimensional infiltration front kinetic parameter F as afunction of P /P b, for various values of k.

Figure 2. Adimensional infiltration front kinetic parameter F as afunction of P /P b, for k = 1.57 (observed with Saffil alumina shortfibres), showing extensive non-linearity at lower values of P andillustrating the definition of P ex.

Figure 3. Graphic validation of the correlation given in Eq. (14).

V. Michaud, A. Mortensen / Scripta Materialia 56 (2007) 859–862 861

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