Building and Environment 37 (2002) 733–739www.elsevier.com/locate/buildenv

In-life prediction of hygrometric behaviour of buildings materials: anapplication of fractal geometry to the determination of adsorption

and suction properties

Alessandro Stazi ∗, Marco D’Orazio, Enrico QuagliariniUniversity of Ancona, Building Institute, Via Brecce Bianche, 60100 Ancona, Italy

Received 23 August 1999; received in revised form 22 May 2001; accepted 25 May 2001

Abstract

In order to describe the hygrometric behaviour of a porous material such as mortar, it is quite common to resort to the di3usion processtheory [1]. The equations that are obtained depend on the parameters that linearise the dependence on the gradient of the potential Padopted. Such parameters are not constants, but they greatly depend on the hygroscopic content of humidity inside the material. Thehygroscopic content u inside the material depends on the relative humidity of the environment � in a non-linear way. In short, we indicatethe mass 7ux by m: m=f(u) × grad P;with u= u(�).

Current research obtains the constitutive link between u and � by <tting experimental data and there is no theoretical model whichcan interpret the curves obtained. This paper shows the results of research that, on the basis of fractal geometry, has worked out amathematical model in order to express the existing link between the water content inside a porous material and the relative humidity ofthe environment at a given temperature. It shows that the knowledge of the fractal dimension of the pores’ space in a porous medium isenough to work out the suction and adsorption curves characteristic of the medium. c© 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Adsorption and suction curves; Fractal geometry; Building materials

1. Introduction

As it is well known, the thermal properties and the dura-tion of the porous materials used in civil construction (suchas mortars, plasters, bricks) are the function of their hygro-metric properties, i.e. of the way in which they accumulateand release water according to the manner the environmenta3ects them. The variation of such properties and the dam-age that may be caused are so considerable that several stud-ies have been carried out for the last few years in order tohave a deeper understanding of:

• those properties for di3erent materials;• their relationship with the duration of a construction and• the applicability of calculation methods (based on these

properties) aimed at predicting how the construction willbehave during one’s life.

∗ Corresponding author.E-mail address: edilizia@popcsi.unian.it (A. Stazi).

Although research has been making signi<cant progress inthis <eld [2–4], one of the main problems yet unresolvedconcerns the applicability of prediction models to some en-vironmental situations, as they treat separately the physicalprocesses involved such as absorption, condensation, suc-tion and evaporation. The separate treatment of the variousphysical processes is due to the di3erent characterisationprocedures of the materials owing to empirical relationshipsvalid in segments, which makes it very diAcult to predict thebehaviour of porous materials during events (for instancerain) when the di3erent physical phenomena overlap.

A clear example is that of the problems arising with theuse of the methods based on the adsorption and suctioncurves, which, although are to describe with continuity theinertial properties of the system at the passage of the wa-ter, owing to the fact that they are derived from di3erentexperimental procedures give rise to uncertainties on thebehaviour of the material in the segment where condensa-tion gives way to capillary suction. As a matter of fact, theoverlapping of the curves does not make it possible to have

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734 A. Stazi et al. / Building and Environment 37 (2002) 733–739

Nomenclature

u water content (kg=kg)� relative humidity of the environmentr radius of the poresd fractal dimension of the poresD fractal dimension of the water inside the pores�H2O density of water (1000 kg=m3)�vw water 7ux (kg=m2s)�vv vapour 7ux (kg=m2s) H2O surface tension of the water equal to 72 ×

10−3 N=m

Hg mercury surface tension (480×10−3 N=m)�Hg mercury contact angle (141:3◦)Rv universal vapour constant (462 j=kg=K)T absolute temperature assumed as equal to

298 Kp mercury intrusion pressure (Pa)s water suction pressure (Pa)Vpores volume of pores inside the medium (m3)VH2O volume of water adsorpted and suctioned

by the medium (m3)

one single and continuous characterisation in the “linking”interval (956�6 98%).

Our research was carried out on the above-mentionedassumptions and it exploited the concepts of fractal ge-ometry in order to determine the constitutive relationshipbetween the water content in the porous materials and rel-ative humidity [10,13,14,16]. In particular, it demonstratedhow, by representing the complexity of the porous structureof the materials in fractal terms through the fractal dimen-sion, there is a power law useful to describe the water con-tent of the material as a function of relative humidity. Theimportance of the results we have obtained lies in the possi-bility to free oneself of the di3erent empirical laws obtainedfrom the di3erent experimental techniques as, once the frac-tal dimension of the porous medium has been determined, itis possible to describe the hygroscopic behaviour of the ma-terial with continuity in the whole range of relative humidity.

2. Phases, materials and methods

The following research was carried out in three phases:

1. Characterisation of the materials under examination in or-der to determine the porous structure and the hygrometricproperties of its relationship with the suction pressure.

2. Application of the concepts of the fractal geometry andanalytical construction of the adsorption and suctioncurves through “D” (fractal dimension characteristic ofthe material in question).

3. Experimental veri<cation of the results.

The <rst phase was of an experimental nature and made itpossible to obtain the characterisation data of the materialsnecessary to validate the theoretical constitutive link. Theanalysis was carried out on a porous material commonlyused in the building industry such as mortar, in particular:

• cement mortar (UNI M3): 1 part of cement, 1 part ofhydraulic lime and 5 parts of sand. It is characterised

by a porosimetry distribution tending towards the smallpores (Fig. 1).

The following are the laboratory methods for characteri-sation:

• porosimetry (with Porosimeter 2000 Series, Carlo ErbaInstruments) aimed at the determination of the pores’ dis-tribution and the suction properties (Normal 4=80);

• capillary absorption (Normal 11=85);• water permeability (Normal 44=93; ASTM 1990=C);• measurement of the angle of contact (Normal 33=89).

The second phase was of an analytical nature and madeit possible to evaluate the applicability of the concepts offractal geometry to the materials being tested, thus allowingus to determine the shape of the existing constitutive linkbetween the relative humidity of the environment and thewater content in the porous material.

In particular, the possibility of describing the exchangeprocesses of humidity in a porous medium through the fol-lowing expressions was determined, on the basis of [2,3]:

�vw = − �Dw × (@u=@�) × grad � (liquid phase); (1)

�vv = − �Dv × (@u=@�) × grad � (vapour phase); (2)

where Dw and Dv represent the di3usivity to water, inthe liquid and vapour states, respectively, of the material(m2=s); @u=@� is the slope of the absorption or suctioncurve, according to humidity. We also tried to demonstratethat u= u(�) can be determined on the basis of considera-tions of a merely geometric nature.

For the determination of the fractal geometry “D”, refer-ence was made to [6–8], using the mercury porosimetry.

Assuming that the pores’ space inside a porous materialcan be described by means of fractal modelling (Vpores ˙ rd

instead of Vpores ˙ r3) and that water, on its way in, keepsthis peculiarity (VH2O ˙ rD), the following was the fractalmodel proposed as a function of the pores’ radius:

u=Ar3−D: (3)

A. Stazi et al. / Building and Environment 37 (2002) 733–739 735

Fig. 1. Mercury penetration curve as a function of applied pressure.

Table 1Conditioning environments

Environment �

MgCl2 0.33NaBr 0.58BaCl2 0.80

As a function of relative humidity, bearing in mind the linkbetween r and � (Kelvin equation, [5]):

u=B(ln�)−(3−D); (4)

where A and B are constants.The third phase made it possible to compare the adsorp-

tion and suction curve analytically worked out and the ex-perimental results were obtained by weight analyses at thecontrolled relative humidity.

Three mortar samples were made weighing about 2–3 gand were put into three dryers at controlled humidity, afterbeing stove dried at a temperature of 60◦C, as per the char-acteristics shown in Table 1.

3. Results and discussion

3.1. Characterisation

The analysis carried out by means of the mercuryporosimeter made it possible to evaluate a few physicalcharacteristics of the structure of the material under ex-amination such as total porosity, the pores’ average radiusand volume according to the pressure applied. Mortar wascharacterised by

• total porosity of 26%;• average radius of the pores of 2:46 × 10−7 m and

• volume of the pores as a function of the intrusion pressureof mercury, as indicated in Fig. 1.

The analysis of the distribution of the pores as a functionof their radius (Washburn equation, [5]) shows that it is avariously dispersed unimodal distribution, so that the aver-age radius worked out as a representation of the commonestradius inside the piece tested, which can be assumed to bethe real average radius.

The capillary absorption coeAcient (CA), expressed inkg=cm2 s0:5, was equal to

CA = 0:0066:

Water permeability k, expressed in kg=ms Pa, was equalto

k = 1:42 × 10−9:

Concerning the angle of contact, the quick absorption of thedrop by the material made the evaluation impossible. Thatis why it was assumed to be 0.

3.2. Determination of the fractal dimension

The determination of the fractal dimension was carriedout by the space of the data, or rather in the space of the dataon the basis of the work performed by Professor B.H. Kaye[6–8] and suggesting that the data derived from mercuryporosimetry may be interpreted as a function of the fractaldimension of the object under examination.

We obtained the relationship between the volume of themercury that crept into the porous medium and the pressureapplied to it to push it in (Fig. 2) and the volume of waterthat, with the same radius Fig. 2, would have crept in as afunction of the suction pressure exercised, on the basis ofthe model put forward by Young–Laplace [5].

736 A. Stazi et al. / Building and Environment 37 (2002) 733–739

Fig. 2. Mercury penetration curve as a function of the pressure applied in the logarithmic scale.

Fig. 3. Water penetration curve as a function of the pressure applied.

Therefore, on the assumption of a Young–Laplace be-haviour

p= − 2 Hg × cos �Hg=r; (5)

s= − 2 H2O=r: (6)

With the pores having equal radius, the result is

s= H2O × p=( Hg × cos �Hg): (7)

Thus, it was possible to determine the characteristics of thecurves shown in Figs. 3 and 4.

The characteristic fractal dimension in that space of datawas obtained by plotting on logarithmic scale axes (Fig. 5).As can be seen, the classes of pores of radius between 10−8

and 10−7 m show a substantially linear dependence on thevolume of water crept in (as a matter of fact, the breakagein the logarithmic linearity of mercury penetration may beexplained by the fact that the pores often break when the

intrusion pressure is increased, or that the macro cavities arepresent). Such linear dependence can be expressed by aninterpolation line that shows that it is a fractal dimension Dequal to

D= logVH2O=log r; (8)

where D is the angular coeAcient of the interpolation line.Thus, we have the following equation, whose symbols are

quite clear:

logVH2O = 0:4735 × log r − 3:7886; (9)

that is to say

VH2O =A× r0:4735 (10)

with A= 1=103:7886 i.e. the volume of water <lling the poreshas a fractionary dependence on the size of the pores.

A. Stazi et al. / Building and Environment 37 (2002) 733–739 737

Fig. 4. Water penetration curve as a function of the pores’ radius.

Fig. 5. Water penetration curve as a function of the pores’ radius in the logarithmic scale.

The rearrangement of the data obtained by mercuryporosimetry, see Eq. (3), led to a fractal dimension forthe volume of water crept in inside the pores, that is D=2:5265.

On the supposition that the porous material analysed wasa fractal object characterised by a porous structure that canbe expressed in terms of roughness and therefore of fractaldimension, a proportionality law between the volume of thewater crept in (VH2O) and the pores’ radius (r) with exponent3 − D (fractionalised) was determined: VH2O ˙ r3−D.

By expressing relationship (4) as a function of the relativehumidity by Kelvin equation, we obtained

u=B(ln�)D−3 =B(ln�)−0:4735; (11)

where B is a function dependent on various physicalparameters such as water temperature and surface

tension:

B= 10−3:7886�H2O=0:9 × 10−3

×(−2 × H2O=�H2O × RvT )0:4735: (12)

3.3. Comparison between theoretical curves andexperimental data

A comparison was carried out by determining experimen-tal points of the absorption curve, as suction values for rela-tive humidity values greater than 90–95% are obtained frommercury porosimetry itself and are just a di3erent represen-tation of the experimental data.

Fig. 6 shows the values obtained (see red points) af-ter the weight stabilisation of the samples inside the dryersand from the porosimetries. The comparison between thevalues obtained from theoretical considerations and those

738 A. Stazi et al. / Building and Environment 37 (2002) 733–739

Fig. 6. Absorption and suction curve for mortar.

Table 2Comparison with the experimental data for mortar

UR Experimental value Fractal model

0.33 0.010 0.010.57 0.012 0.0130.8 0.020 0.0200.9 0.029 0.0290.98 0.080 0.0641 0.182 0.183

based on experimental values has displayed a considerablecorrespondence for the absorption phase, thus con<rmingthat the hypothesis of a fractal structure was well founded(Table 2).

4. Conclusions

This study illustrates the applicability of fractal geometryin order to determine the absorption and suction propertiesof porous materials.

It has, in particular, determined the constitutive link be-tween the porous structure of a material and its water contenton the basis of fractal geometry, and overlooked the prob-lem of the correlation between transportation of the liquidphase and vapour phase, which was always quite uncertain.

This makes it possible to have a deeper knowledge of thebuilding materials with the aim of using them more properlyaccording to the various environmental conditions, as manyof their technological properties such as resistance to frost,thermal conductivity and speci<c heat largely depend ontheir hygrometric status.

Of course, this single study could not deal with the restof the problems connected with the subject, but further re-search will be aimed at the other aspects. An extension ofthis method of research aimed at the determination of otherproperties such as water or vapour permeability seems to be

feasible, given the good results obtained from the compari-son with the experimental data.

Meanwhile, it would also be advisable to further verifythe modelling proposed on other commonly used materialswith a considerable porous structure. A study on the fractalcharacterisation of the absorption and suction curves forbricks is currently under way.

Appendix Fractals and self-similarity

The fractal is a geometric entity, whose (fractal) dimen-sion “d” is the representative parameter indicating how thegiven entity <lls its space [9,11,12,15]. It can be thought ofas a measure of its “roughness” as the irregularity of theshape. While a line will have dimension 1, plane 2 and cube3, more regular objects such as a sponge or a coastline willhave a dimension included between 1, 2 and 3. The exactvalue of d is determined by the use of a series of samplelengths, surfaces or volumes capable of assessing the length,surface or volume of the object. The number of measures(N ) relative to the “measurement scale” r required in orderto cover the object complies with the following law:

N =K × r−d

with K as a constant, so that d can be determined by com-puting the logarithm of both sides.

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