concrete micro structure - porosity and permeability
TRANSCRIPT
_, SHRP-C-628
Concrete MicrostructurePorosity and Permeability
D.M. RoyP.W. Brown
D. SkiB.E. Scheetz
W. May
Materials Research LaboratoryThe Pennsylvania State University
University Park, Pennsylvania
v
Strategic Highway Research ProgramNational Research Council
Washington, DC 1993
SHRP-C-628Contract C-201
Program Manager: Don M. HarriottProject Manager: lnam JawedProduction Editor: Marsha Barrett
Program Area Secretary: Ann Saccomano
April 1993
key words:
cement pasteconcrete
permeabilitypore structurepore size distribution
porosimetryporositysurface area
Strategic Highway Research ProgramNational Academy of Sciences2101 Constitution Avenue N.W.
Washington, DC 20418
(202) 334-3774
The publication of this report does not necessarily indicate approval or endorsement of the fmdings, opinions,conclusions, or recommendations either inferred or specifically expressed herein by the National Academy ofSciences, the United States Government, or the American Association of State Highway and TransportationOfficials or its member states.
© 1993 National Academy of Sciences
350/NAP/493
Acknowledgments
The research described herein was supported by the Strategic Highway ResearchProgram (SHRP). SHRP is a unit of the National Research Council that was authorizedby section 128 of the Surface Transportation and Uniform Relocation Assistance Actof 1987.
iii
Contents
Acknowledgments .................................................. iii
Abstract .......................................................... vii
Executive Summary .................................................. 3
A Model for the Distribution of Pore Sizes in Cement Paste .................... 7
Lognormal Simulation of Pore Evolution During Cement and Mortar Hardening ... 19
Concrete Microstructure and Its Relationship to Pore Structure, Permeability,and General Durability ............................................ 25
Porosity/Permeability Relationship ...................................... 43
Relationship Between Permeability, Porosity, Diffusion and Microstructureof Cement Pastes, Mortar, and Concrete at Different Temperatures ........... 76
/4
V
Abstract
A model has been developed that lays the foundation for relating porosity topermeability. This is based on knowledge gained from previous work as well asexperimental and theoretical input from the present program. A linear combination oflognormal distributions may be used to define the pore structure. This report containsfive papers relevant to this topic.
EXECUTIVE SUMMARY
The objective of the current study was to develop an accurate mathematical descriptor for
pore size distribution to relate to permeability and use for permeability prediction. Various
properties of cement-based materials are affected not only by total porosity but also by the size
distribution of the porosity present. In order to model the relationship between pore size
distribution and properties, a suitable mathematical descriptor for pore size distribution must
first be found. Mercury intrusion porosimetry (MIP) is a commonly used method of determining
pore size distributions for the range of pore sizes which significantly affect properties such as
permeability. A suitable mathematical descriptor for pore size distributions determined by MIP
should be one that not only fits the experimental data but, more importantly, provides the basis
for the physical interpretation of pore structure.
Diamond and Dolch (Ref. 1, Paper 1, this supplemental report) showed that the pore size
distributions in cement pastes determined by MIP could be described as lognormal distributions.
The present work in which higher intrusion pressures have been applied, suggests that a single
lognormal distribution is inadequate to describe the smaller pores. A mixture of two lognormal
distributions, or a compound Iognormal distribution was suggested to model the distribution of
pore size determined by MIP in both cement pastes and mortars. However, closer examination of
the data used in our model shows that very coarse pores, pores - > 1 ]an, were rarely included. Pores
in this size range affect properties, such as strength and permeability, and should not be ignored.
In the present study, pores as large as 7 _n are included. It appears that a mixture of three
Iognormal distributions can more closely fit the distribution of pore sizes over the range of sizes
extending 4 orders of magnitude from about I nm to about 10 _rn.
A variate X (0 < x < _) is Iognormally distributed if Y = IogX is normally distributed with mean
and standard deviation ¢_.The probability density function is,
P(X) = (2r_s2)'1/2X" lexp[-0.5{flog(X) - _)/¢_}2] [3]
The range of X is 0 < X < _. _ is defined as the location parameter, _ as the shape parameter.
Some characteristics of a Iognormal distribution are:
mean = exp(_ + 0.5o 2)
median = exp(_)
mode = exp_ - o2)
, coefficient of variation = exp(a 2) - i
variance = mean2[exp(¢_ 2) - I]
., Cement hydration may be regarded as a process of subdivision of void space, or interstices
between anhydrous particles. At the early stages of hydration, the void spaces between anhydrous
particles are large. With proceeding hydration, hydration products bridge the particles and divide
3
the large voids between particles into smaller pores. This phenomenology suggests the lognormal
model of pore size distribution in cementitious materials to have a physical basis.
A mixture of two or more Iognormal distributions is defined by a compound density function:
=zfiP , zfi=l
where fi is the weighting factor of the ith Iognormal sub-distribution P(x, _ti, _i), _i and c i location
parameter and shape parameter of the i th sub-distribution respectively. If an attribute of a system
can be described by two or more Iognormal distributions, it is very likely that two or more
phenomena are occurring in that system. In the case of pore size distributions in cementitious
materials, this may indicate that different origins and formation mechanisms exist for pores in
different size ranges.
A statistical method has been developed to detect if there is a mixture of two lognormal
distributions and to iteratively estimate the parameters in the compound distribution. A similar
approach can be applied to a mixture of more than two lognormal distributions.
The pore size distribution used to demonstrate this method is obtained from a set of
cumulative probability data, P(x) vs. x as obtained by MIP. x is the pore diameter, P(x) is the
cumulative percentage of pore volume with respect to the total pore volume. The graphical
approach to obtain first degree estimates of parameters of a mixture of two or more lognormal
distributions first requires the transformation of x to log(x). Next, the quantfles of N(0,1)
corresponding to P(x) are determined by consulting a standard normal distribution tabulation [e.g.
see ref. 14]. If the cumulative percentage is 50%, the corresponding quantile is 0; if the cumulative
percentage is 84%, the corresponding quantile is 1.0; if the cumulative percentage is 16%, the
corresponding quantfle is the negative of the quantile for (100%-16%L or -1.0; etc. If a plot of log(x)
vs. the quantJJes is linear, x obeys a single Iognormal distribution. The final step in the analysis
is to determine the weighting factors, by locating the intersections between neighboring linear
segments. The goodness of fit can also be checked by comparing selected characteristics of
distribution. The median is often used as an important characteristic of a distribution, and it is
readily determined from the experimental data. Because the median = 5".(fi)(mediani), where
median i is the median of the ith sub-distribution, which is exp(_ti), the overall median can be
estimated from _. Another important characteristic is the inflection point on a plot of pore
volume vs pore radius. The inflection point is easily approximated on an experimental curve. The
inflection point coexists with the intersection between the first two linear segments. The
corresponding x value of the inflection point between the first and second sub-distributions is:
exp [(G2_I - aI112_ / (G2 "¢_1)]
In order to check the generality of fitting pore size distribution data to a compound
lognormal distribution, data from different sources have been examined. Ordinary cement paste,
4
blended cement paste, and mortar hydrated for various lengths of time have been examined. The
results demonstrate that it is reasonable to fit pore size distribution in cementitious materials to a
compound Iognormal distribution. Further, the graphical method used is adequate to provide
initial estimates of parameters in the compound distribution.
Pore size distributions in real materials must exhibit upper and lower bounds. This is the
physical basis underlying the model developed by Diamond and Dolch. The lower limit was
assumed to have little influence on the distribution curve and was neglected in the reduced
diameter expression. It is questionable whether this assumption is valid when small pores are
involved, i.e., when x approaches x1. If the lower limit is included, their model is equivalent to a
four-parameter Iognormal distribution, in which the variate X is confined to the range L < x < U,
and X = (X - L) / (U - X) is L(_ ¢). An attempt to desc_be pore sizes ranging from about 1 nm to about
10 fan by a single four-parameter lognormal distribution was not successful. Because the
simulated compound distribution is fitted to the pore size distribution determined by MIP, it
assumes 100% cumulative probability at the smallest pores intruded by mercury. In other words, it
assumes that the probability that the pore diameters are smaller than those intruded by mercury
is zero. This, of course, is not the real lower limit of pore size. If the real lower limit of pore size
and the percentage of the MIP porosity are both known, the pore size distribution over the range of
pore sizes determined by MIP can also be simulated by using the method we have developed.
The use of the distributions to describe porosity should have physical significance. The first
sub-distribution of the three may be regarded as describing the size distribution of coarse pores.
Pore sizes may extend to include voids. The third sub-distrlbution may be regarded as describing
the size distribution of fine pores. Pore sizes may extend to gel pores. The middle one represents
capillary pores.
It is the pores that belong to the sub-dlstribution representing the finest porosity that are
created by the hydration process. Because the majority of this porosity exists in the hydration
products that are forming, it is these pores that control the kinetics of hydration. Altematlvely,
from the viewpoint of permeabillty-pore structure or fracture mechanics-pore structure
relationships, the majority of porosity in this range is not important. With respect to
permeability, it is well recognized that only pores having diameters greater than some value
significantly contribute to permeability. We have demonstrated that an inflection point can be
calculated using the compound Iognormal model and based on our analysis.
Another characteristic that may be important to permeability is the mean square pore
diameter, or the second moment of pore diameter distribution. If one combines the classic Darcy's
• law and Poiseuflle's law, one can relate the permeability coefficient k to the mean square pore
diameter k = e <D2> / 32 where _ is the porosity and <D2> the mean square pore-diameter. This is
a classic model describing permeability of porous media, assuming that pores axe tubes which are
not interconnected. We have observed that a characteristic pore dimension and a tortuosity factor
are two indispensable variables in all sensible permeability models. If the pore dt_meter
distribution can be modeled by a compound lognormal distribution, the mean square pore
diameter can be readily determined using the equation: <D2> = Z fi exp[2 (iti + _i2)]. The inflection
point, or the mean square pore diameter, can be described in terms of a distribution expressed as
relative pore volumes, as relative pore numbers, or as relative pore surface areas. Depending on
the physical or mechanical property influenced by the distribution in porosity, it may be
beneficial to consider the pore structure in terms of pore surface areas or pore numbers. For
example, the Kozeny-Carman relationship relates permeability to porosity: K ~ e3/(1 - e)2/$2
where e is the total porosity and S is the total surface area of the pores.
This equation requires a value for the porosity expressed in terms of pore surface area. Thus,
multiplicative property of the Iognormal distribution, which allows the interconversion between
volumes and surface areas, anows, in turn, MIP pore volume data to be expressed in appropriate
terms. This attribute, coupled with the ability to deconvolute porosity data, suggests that a basis
has been identified which may allow a more fundamental understanding of relationships between
the behavior of cementitious materials and their pore structures.
In conclusion, a refined model for describing pore size distribution has been developed. This
in turn has been integrated into a model for the prediction of permeability. Finally, an
experimental apparatus to rapidly determine the permeabflities of specimens has been developed.
The predictions from the model show reasonable agreement with experimental values when used
to calculate permeability.
6
A Model for the Distribution of Pore Sizes In Cement Paste
O. Shi, P. W. Brown and S. Kurtz, The Materials Research Laboratory, The PennsylvaniaState• University,UniversityPark,PA 16802
ABSTRACT
The pore size distributionin cement paste overthe range of pore sizes interrogatedby highpressuremercury intrusionporosimetrymay be describedby a mixtureof two lognormaldistributions.The compounddistributiono( poresizesmay be givenas:
)2 (l-f) [(log x-P.2) 2p(x) = f exp- [(log X'lJ.1 exp - 2 ]_r1 xV_-_'_ 20 2 ]+ a2x'V_'x 2(:12
wherep(x) is the probabilitydensityfunctionof poresol sizex, f and (14) are theweightsof
sub-distributions,g.1andP2 are thelocationparametersofsub-distributions,and 01 and(_2are the
shape parametersof sub-distributions.These two sub-distributionsmay representthe largerand
smallercapillaryporesrespectively.The changesinthesub-distribuUonsandthe compound
distributionas functionsol curingage andwater-to-cementratio"are discussed.
INTRODUCTION
The abilitytowedicate the macro-propertiesandbehaviorof porousmaterialsingeneraland
thoseof cement-basedmaterialsinparticulardependsonthe abilityto mathematicallydefinethe pore
structuresinthese materials.Inspiteof varietyol limitations,mercuryintrusionporosimetry(MIP) is e
methodcommonlyusedfordeterminingpore sizedistributionsof a broadrangeof pore sizes. It is
believedthatthe pore size range amenableto MIPanalysisisthat whichaffectsmanymacro-
properties. It is of interesttodevelopa modelforthedistributionof pore sizesincement pastes
determinedby MIP.
Previousinvestigationshave shownthatthe poresizedistributionsincement pastes
determinedby MIP couldbe describedby iognormaldistributionprovidedthepore diametersare
transformedto reduceddiametersby the equation:
Xr= (x- Xl)(Xu-Xl)/ (Xu- x)
where xr is the reduceddiameter,xu andxI are theupperand lower limitsofthe pore sizerange. The
lowerlimitwas assumedto havelittleinfkJenceonthedistributioncurve andwas neglectedinthe
" reduced diameterexpression[1]. However. furtherwork.inwhichhigherintrusionpressureshave
been applied,suggeststhata singleIognormaldistributionmay be inadequateto includethe smaller
pores. Additionally,the assumptionthat the lower limitisnegligiblemaybe invalidwhensmallerpores
are involved.
Basedon the presentlyavailabledata, it appearsIhatthe pore sizedistributionincement paste
7
overthe rangeofporesizesdeterminedbyhighpmssuraMIPmaybedescril:xKIby a mixtureof two
iognormaldistnl:utions.Thesub-distributionsmayrepresentthelargerandsmiler _ pores,
respectively.Thevalidityof assumingtwolog-normallydistributedporesizescanbetestedbya robust
statisticalmethod[2]. A programbasedonthismethodwasdevelopedtodetectwhethera poresize
_strroutionis a mbdureoftwoiognormaldistributions.If it isa mixtureoftwolognonnaldistributions,
threeparametersassociatedwitheachofthetwosub-distributionsareestimated.Thethree
parametersforeachd_ion are:
theweightsofthelargerandsmallerporesizedistdoutions,f and(1.q;
thelocationparametemofthelargerandsmallerporesizedistnbutions,ILI andI_2;
theshapeparametersofthelargerandsmallerporesizedistributions,01 and02;
Thesizedistd0utionofporesincementpastesdeterminedby MIPcanthenbedescribedby a
compounddistributionfromthesesixparameters.
Thechangesoftwosub-distnbutions,withcuringageandwater-to-cementratio,aswellas that
of thecompoundsizedistnbutionareconsideredinthispaperasa preliminaryefforttounderstandthe
physicalmeaningsofthesetwosulPdistritx#tions.
It is a commonphysicaloccurrencethatphenomenaina systemaredescribedbythesumo(two
lognormaldistnbutions.Thegeneralphysicalinterpretationisthattwoindependentphenomenacan
governthatoccurrence.Inthepresentinstance,thissuggeststhattheoriginsand themechanisms
controllingthesizesoflargerandsmallercapiilar/poresmaybedifferent.Otherimp,cationsofthis
modelmayincludethepossibilityofdescrC_ngthewholerangeofporesizesfromtheporesize
distnbutlonsdeterminedbydifferenttachniclues.
FUNDAMENTALSOFTHELOGNORMALDISTRIBUTION
Thenormaldistributionprobabilitydensityfunction,n(.u,o),is:
"X'=, I ex "(x'p')2'Pt '7 22a
where p(x)istheprobabilitythaithe randomvariable,x, willhavea valuex,p is themeananda isthe
standarddeviation.Themean,14,mayalsobereferredtoas thelocationparameterandthestandard
deviation,0, mayalsobereferredtoastheshapeparameter.
A variatex is iognormallydistributedifx - logy wherey isa positivevariate,(0<y < ,,), whichis
normallydistributedwithmean I_ andstandarddeviationa. We thendefinethatx is Iognormally
distributedandthe probabilitydensityfunction,I(_,a),is:
1 x-1_)2lox_/_z exp'[(l°g 2 °
p(x)=
20
wherex istherandomvariable,theporediameterinthiscase.Therangeofxis 0 <x <--. p,isdefined
asthelocationparameter,a is definedastheshapeparameter.Themean,medianandmodeareas
follows:
8
mean = exp(p.+ 0.502)
median,exp(p.)
mode = exp(l_-02")
Figure 1 Isthe comparisonbetweennormaldistributk)nn(O,O.S)and the Iognormaldistribution
I(O,O.S)[3]. Thenormaldistributionis symmetricaboutthe meanwhilethe IogrlormaldistdbuJionis
positivelyskewed.The normaldistributionadsesfroma theoryol elemanlaryerrorscombinedby a
ad_tive process,while the Iognormatdistributionarisesfrom a theoryof elementaryerrorscombinedby
a muItipUcativeprocess.The differencebetweenIognonnatand normaldistributionsas descriptorof
processis similarto thatbetween the geometricandthe arithmeticmeansas measureof Ioca_n.
Thederivationof the multiplicativemodeladsss fromviewingaposiUvevadate as a measureofa
discreterandomprocess.Atthe jth step,thechangein thevariateis a randompmpoflionof thevalue
of variateat the(j-1)th step. Thus,
xj- xj.1 = 8jxj.1
where xj isthe value of variateat the jth step,xj.1 thevalue at the (j-1)thstep, and8] the smallrandomproportionalgrowth/degradationratethattakesthe processfromthe (j-1)thstep tothe jthstep.
0.8 I i I
Q6
0.4
0-4 -2 0 z 2 4 6 8
<w--Z
. 004OLiJW:S =S=S
Figure 1. A comparison between a normal distribution,n(O,0.5), and a IognormaJdistirbution,I(0,O.5).The mode,median, and mean of the IognormaldistnbulJonare shown.
g
It followsthat
xj = [['[(1 + 8i)]xo
Iog(xj)= Iog(xO)+ T..Jog(1+ 8i) = Iog(xo)+ T._
where i = 0,1 ........ j. xo is the initialvalue of the vadate.Applyingthecentrallimit theoryto the sum ofp
the smallrandomquantities,8i, resultsin log(x)havingan approximatenormaldistribution,i.e. x hasa
Iognormal distribution. Many physical and chemical processes, such as corrosion, diffusion,
pulvedzation,crushing,andcracking[4.7] maybe approximatedby _ lognormalmodel.
Accordingto central limittheory, if log(x)obeys a normal distributionn(IJ.,a),then[Iog(x)-p.]/a
obeysa standard normaldistribution,n(0,1). Thus, ifx1 obeys lognormaldistributionI(_,a), x2 obeys
standard normaldistribution n(0,t), and if the cumulativeprobabilityp{x1 < x1,q} = p{x2 _;X2,q},wethen have:
x2,q = [log(x1,q) - _]/o
In statistics,x1,q and x2,q are defined the quantileof orderq of I(p,a) andquantileof order q of
n(0,t) respectively. Thismay be rewrittenas:
log(x1,q) = aX2,q+ I_so that the locusof Ix2,q, log(x1,q]is a straightline. Inotherwords,it x1,qand X?,qare quantilesof the
same order q of I(p.,a)and n(0,t), thenthe locusof (x2,q, x1,q) is a straiglll lineon a semi-logscale.
Suppose now that the cumulativeprobabilityp(x2 < X2,q},insteadof the quantilex2,q, is ruledon an
axis with linear scale; this is the basis of logarithmicprobabilitypaper. If plotting the cumulative
probabilityP{x < x} againstx on the logarithmicprobabilitypaper results in a straightline, we then
reasonablypredict that the variatex obeys the lognormaldistribution. This is one of the mostuseful
propertiesof the lognormaldistribution[3].
MIXTURE OF TWO LOGNORMALDISTRIBUTIONS
The mixtureof twolognormaldistributionis definedby a compounddensityfunction:
= f [(log X-lZ1)2 (l-f) " x'g2) 2p(x exp - ] + exp - [(log ]_r1 x_/2zz 2 a 12 _r2 x _ 2(/22
where f is the w6!_ht of the first sub-distributionI(I.Zl,al), (l-f) is the weigh of the second
sub-distributionI(p2,a2) Pl andP2 are the locationparametersof two sub-distributions,repectively,
and <71and a 2 are the shape parameters of two sub-distributions,respectively. For the first
sub-distribution,the statisticalcharacteristicsare: ,,
mean 1 = exp(_.l + 0.5 o'12)
median 1 = exp(_)
model = exp(141 - a12)
For the second sub-lognormaldistribution,the counterparts are:
10
For the secondsub-lognormaldistribution,the counterpartsare:
mean 2 = exp(_2 + 0.5 (;2 2)
median 2 ,, exp(i.z2)
. mode 2 = exp(p.2 - <_22)
A statisticalmethodhas been presentedto detectthe mixtureof two lognormatdistributions,
andto estimatethe valuesol f, P'I, P'2,(xl and(_2[2,7,8}.
A microprocessor-basedcomputer program has been developed at the MaterialsResearch
Laboratory,PennState University,to implementthismethod. Withinthe context of the presenttopic,
this programfirstplotsthe experimentalcumlativepore size distributioncurveon the log-probability
scale. If the curve appearsS-shaped, it is likelythat the distributionis a mixtureof two Iognormal
distributions. The program then selects a point, which may be but is not constrainedto be the
inflectionpoint,as the startingpoint in the estimationof the value of l, thecorrespondingcumulative
probability.Thispointis also usedto separatethe wholedataset intotwosubsets. Foreach subsetol
data, the programdoes regressionanalysis on porediametersvs. quantilevaluesof n(0,1). The slope
and interceptof the regressionline are the valuesol p.and_. Fromthe estimatesoff, (t-f), P'I, P2, al
and(72.the compounddistributionand the cumulativeprobabilitiesare calculated. Then the program
comparestheexperimentalcumulativeprobabilityand the calculatedcumulativeprobability,andallows
theuse ol iterativeprocedureuntil satisfactoryresultsareobtained.
EXPERIMENTALDATA
Foursetsof MIP dataolotaineclfromthe[9,10]literaturehavebeenanalyzed. Sets1 and 2 are
froma cementpastehydratedat a water-to-cementratioof 0.4 foroneand28 days, respectb/ely.The
pore sizesrangelrom2.9 nm to 700 nm. Sets3 and 4 arefrompasteshydratedfor38 daysat
water-to-cementratios0.4 and0.35, respectively.The rangeof pore sizesforsets3 and 4 is 1.9 nm to
140 nm. Comparisonof calculateddistributionswillbe madebetweenset 1and set2 to showthe effect
of age onpore structure,and between set3 and set4 to showeffectolwater-to-cementratioon pore
structure.Forsimplicity,set1willbe referredto as:1-daydata,set2: 1-monthdata,set 3:0.4 w/c data,
and set4:0.35 w/¢data.
RESULTS
Inthefollowingfigures,"1"denotes the firstsub.distnbutionwhichis the size distributionof the
largerpores,and "2"denotesthe secor_dsub.distribution,the sizedistributionof smallerpores.
Figure2a andb provideexamplesshowingthe estimationOfvaluesof f, 14anda fromthe MIP data. The
values ofp,and (; arethe slopesand interceptsof regressionlineswhichwereplottedon a natural
Ioganthmic-normalquantilescale,notsimply theslopesandinterceptsof thestraightlines shownon
thatfigure. Aninterestingpointis that all the valuesoff are near0.9. Thispointwill be discussed
further.
Rgure3 comparesthe experimentalcumulativeprobabilitiesand thecalculatedcumulative
probabilities.SSE is the sumof square of errors,givenby the equation:
11
e.._=- ,. Experimental Data**.m-I ---- Regressionsf- 0.85
>, **.=,- ol = 2.518 I_l = -0.647:---- o2= 0.882 lz2= -0.131.Q0 90.Om- 1
.¢3O tL I(3. t
50.01- I '
__o I ,,
lO.Om- t%
• %
b0.10m- %
0 01s ....
1 10 100
Size (nm)
eo.om- Experimental Data..u- --- Regressions
f -- 0.90>, ...a=- _1 5.354 I_1= -1.055•-- _z - 1.855 p2 = -0.237.43 X
._0 90.Q=- X
Q. X _
_ 10.01-
0.1(_- a +0.01l-- i
10 100
Size (nm) +
Figure2. Thepa_em f.rL.ando olXainedfroma proba_lilyplolforpastehydratedfor(a)onedayatwit ,, 0.4;Co)38daysatwit, 0.35.
12
N.t- N.IIIB=
W." t I:'xperimentol 04to N.m. _ ,, [xperimentol 04ta
._ .e.i. --- (stimoted__0ota w,-. _ --- [stimated 04ta
•W.m _.B°
0 re.D- ,m.D.
tO.m- IOAI,U 1,0_- t._.
• ¢ ••_e.lm_ iLim._ ,,eom oow •
....... ,i siz.(n.,)'= si_;.(_) "IQ.NB- N._-
w.'.,- \ (xperimentol 0oto w.m- I (xporimentol 0at0\ --- (,timot.d Ooto I --- (,timotod 0_o
_w.m- _ Ill.m.
\N.IB- IO.0l-
_ W.mo _,ll-
SO.I° . _ tO.m-
Lz_ b • • o t_ d- • - • ,•O.Ota- . .... e.otl. . ....... m
,0 $iZ0 (rim),ol $iII (nm) ,m
Figure 3. A comparison of the exl=edmentai and calculated cumulative pmbabiUties based on the useot two lognonnal distributions to model the pore size distribution (a) one day at w/c ,, 0.4; Co)28 claysatw/c - 0.4; (c) 38 days at wlc ,, 0.4; (d) 38 claysat w/c= 0.35.
SSE = I;(px - pc)2
where Px and PCare the expenrnental cumulative probabilities and calculated cumulative probabilities,
respectively. SSE can inclicato the goodness of fit.
Another way to indicato tho goodness of fit is by plotting the expodrnentat probabirdias against
the calculated wobabiities, which is actually the graphical express_n of SSE. Figure 4, which shows
the comparison for the MIP curve for 38 days at 0.4 w/c. is a typicalexample of this.
FigurO 5 shows tho caJculated SUb.diStributions, as well as the calculated compound
distributions, for the 4 data sets. It is worth noting that it is difficult to express the different weights of
_- sub-distributions graphically. Thereforo, we modified the calculated compound distribution to a mixture
of two sub- distributions with equal weights as follows:
loo(_-)-(pl.logf) 2 1 _g(_)-(.2-_(1-f)) 2i _ if
p(x). 1 exp-[. ] + exp-( " ]
13
1.00 -
0.80 -
0i...
N
0.60 - /:/,,,
E ...../..
0 0.4.0
..,_;CL
×0.20- /
0.00 / I I I I I
0.00 0.20 0.40 0.60 0.80 1.00
Est. Cure. Prob.
Rgure4. An exampleof the goodnessof fit shownthe samplehydrated38 days atw/c. 0.4.
Rgure5 is thegraphicalexpressionof the modifiedcompounddistdbulionS.
Rgure 6a showsthe effectof increasinghydrationtime onthe calculatedcompounddistribution.The
shiftof meansand mediansof sub.distributionsis obvious,as are the twocrossoverpointswherethe
sub-distril:uUonsintersect.
Figure61)showsthe effeclof water-to-cementratio onthe calculatedcompounddistribution.
The shiftsinthe mean, medianand crossoverpointare alsoeasilyobserved.
DISCUSSIONAND IMPUCATIONS
ASmaybe seen fromRgure2, thef values(weightingfactors)am near0.9 forthe distribution ._
ol the largerpore sizes. This suggeststhat, ifthe smallerporesareignored,the pore sizesmayfit a
singleIognormaldistribution. As mentionedearlier,previousinvestigationshave shownthatthe pore
14
O.OJ• &IS •
"_, lkN,u Compounddistribution Compounddistribution:_ --- Sub-distribution8 --- Sub-distributions
ta.
o.m .... -........ o._ _" > .........'° sizsC,m)i_' ...... " s;'_o(,,,,) 'i_.
0.0,1 L_I 1
:_ .ou Compounddistribution _L_S Compound distribution¢
_o.ol ------ Sub-distributions _ Sub-distributionsLI,.
m ._LI@c ¢w •
0 0
i J °oOOI .- . ._ : ................. G, e.BI .................10 100 tom t° 101
Size (nm) Size (nm)
Figure5. The probabirdydensityfunctionsof the calculatedcompounddlstnl=utlonando4the twosubdistributionsbasedonthe iognormalmodel(a) oneday atwlc = 0.4; 28 daysatw/c = 0.4; (c)38daysat wlc = 0_4;(cl)38 claysat w/c= 0.35.
sizedistributionincementpastecouldbe desatbed as a singleiognormaldistributionprovidedthat the
pore diametersaretransformedtoreduceddiameters[1]. By examiningthe MIPdata usedinthat
previousinvestigation(Fig.6, andTable II inreference1), whichwas obtainedforpastessimilarto
thoseanalyzedin thispaper(w/c ,, 0.4, age = 28-61 days), it was observedthatthe smallestpore
diameterused tofita singleiognormaldlstdbuttonis 10 rim. It seemsthatinon_r to includesmaller
pores, it iSnecessan/tOuSea mixlureof two Iognormaldistributions.FromFigureS,it wasfoundthat
the (_ametersofthe larger poresrange the orderof magnitude10 to 100 nm,whilethoseof thesmeller
pores are lessthen 10 rim. Resultsalsosuggestthat the f valuedoesnotchangeappmdablywithage
orwater-to-camerara_o,inspiteof the shiftin the crossoverpoint. Thus,0.9 maybe an appropriatef
value forthe la_er pores.
Allrealpanicleorporesizedistribulionsmusthave some maximumandm_imumsize. This
was the physicalinterpretationassociatedwiththe use of reduceddiametersandw_ththe introduction
of the upperand lowerlimitson theparticleor poresize range[1,5,6]. Rgurs5 showsthatthe
maximumsizedoesexistwherethecorrepondtngdensity functionapproacheszero, thoug_no
maximumsizeis explk_tlydefined.On the otherhand, basedonthe cuwentlyavailalMeMIPdata, it is
impossJl_eto locatethe real minumumsize. It mightnotbe validtoassume thatthe minimumsizeis _
negligibleif smallerpores, e.g.thosewithdiametersless than10 nm,are included.Forthis mason
Figure5 leaves thesmallerporedistributioncurvesopen at the lowerend.
15
O.OSO
day old a_. ---- month old-- 0.040
t_ Age: 1 day 28 day_> 0.030
•_ Mean 1 386.64 76.17c Mean 2 6.58 4.21q3
ca 0.020 Median 1 211.37 57.81>, Median 2 6.39 4.17
Crossover 11.52 6.18
.120.010 x
.Q I x0 "_L %
0.000100.0 4OO.0
Size (nm)
0.200I
c .... 0.35o _.m
" lo 0.150c W/C: 0.40 0.35:3la.
>, Mean 1 17.46 15.28"_o.1oo Mean 2 2.75 2.44e. Median 1 13.93 12.40
Median 2 2.44 2.41>, Crossover "3.61 3.15
"--0050--• _""_,mlll
r,l0r's0l,_
13. 0.0000.0 20.0 40.0
Size (nm)q:
Rgure 6. (a) The vafla_ons in the cak:ulated I_mba_fW density functions for Ihe pore size disffibutions(a)betweenI and28daysatw/c- 0.4;(b)asaluncliono_w/cratioataconstantageo138days.
16
It an a_bute of a systemcan be describedbytwo Iognormaldstdbutions,itis veryikely thattwo
phenomenaare __-__-rdnginthatsystem. Inthe presentinstance,this suggeststhat the odgtnsand
the formationmechanlsmeoflargerandmiler capilla_ poresmaybe differenLFigure6a showsthat
the largerporesizedistdbutionshiftsto the leftmarkedlybetween1 and28 days. The meanand
meo_anof the largerpore sizedistributiondecreasefrom387 nmand 211 nmto 76 rimand 58 nm,2
respectively. On the otherhand,the decreasesinthe meanandmedianofsmallerpore size
_stnl:utionare relativelysmallThe shapeof thecurverepresentingthe_ poresalsochangesa
greatdealsugge_lng thatmanyof the largerporesdecreasesignificantlyinsize. AlternaJvely,many
of the smallerporesare alreadypresent alteronlyoneday of hydration.Theobviousspeculationisthat
the smallerporesresulttramhydra_onwhilethe largeronesare formedir_i_ly as intersticesbetween
anhydrousparticles.
Figure6b showsthat the sub-distdbutionsobserved at 38 daysof hydra_onexhibitmedian
pore sizesrelatedto theirwater-to-cementratios,0.4 and 0.35. ComparisOnw_ththechanges
observedin Rgurs 6a, suggeststhat the water-to-cemantratiodeterminesthe initialsub-distributions,
whilethe age determinestheevolutionof the sub-distrtbutlons.However.more samplesneed to be
examinedto inveetigatethe pos___h4eexistence of the (lfferentoriginsandfonna_n mechanismsof
pores incement pastes.
Porosityin cememrangesfrommillimetersto nanometersincementandmanydifferent
techniqueshavebeen usedto determinethe distributionsof pore sizesinthese ranges[11,12].
Inevitably,these measureddlstdbulk)neoverlap. If the uncertaintiesassociatedw#h these
overlappingregionscouldbe solved, it may be possible to describetheentirerangeof pore sizesin
cement paste.Theentiredistritx_ionof pore sizesincement pastesfrommiimetersto nanometers
mightbe a mixtureof multiplelognormaldlstdbutions,eachsub-dlstdbulionmpresentlngthe poresize
distdbutlondata determinedby one spodflctechnique,andreflectingthe spod_ odginandformation
mechanismof poresin thatcorrespondingsizerange.
CONCLUSIONS
1. A mixtureof two iogrlormaldistdbutioneappearsto deschbeporesize (IMdlxjlion detmned
by highpmesureMIPanalysisand suggestsdifferentoriginsandfonnalionmechanismsof
poresincement pastel.
2. The diametersof the larger poreersnge the orderof magnitude10 to 100 nm, wNle thoseof
the smallerporesare less than 10 nm. The sub<istrilxntonofla_11erporeehas a 90% wei_t
in the _ sizedistntxltion.
3. The water-to-cementratioseemsto determinetheinitialsub-distribution,whilethe aging
processatfeds the evolutionof pore structures.
4. Additionalsamples,includingthoseof blendedcements,needto be exanVned,to furmm
eluddete the existenceof the differentoriginsand formationmechanismsof poresin cement
pastel
17
ACKNOWLEDGEMENT& DISCLAIMER
The msearcJ1describedhereinwas sul:fx)rtedbythe StrategicHighwayResearchProgram
(SHRP). SHRP is a unitof theNationalResearchCouncilthatwas authorizedby section128of the
Sudace TransportationandUniformRelocationAssistanceAct of 1987.
Thispaper representsthe viewsof the author(s)only,not necessarilyreflectiveof the viewsof
the NationalResearchCouncil,theviewsof SHRP, orSHRP's sponsors.The resultsreported hereare
not necessarilyinagreementwiththe resultsofotherSHRP researchactivities.Theyare reportedto
stimulatereview anddiscussionwithinthe researchcommunity.
REFERENCES
1. S. DiamondandW. Dolch,J. Colloid& InterfaceSd., _ 234-244 (1972).
2. E. Fowlkes,J. Am. Stat. Assoc.,74. 561-575(1979).
3. J. AJtchisonand JJ_.C.Brown,The LoonormalDistribution.CambridgeUniv.Press,
Ca_ (1957).4. P. Tobiasand D. Trindade,_ Van NostrandReinholdCo., New York, 1986.
5. R. Irani,J. Phys.Chem.,63. 1603 (1959).
6. R. Iraniand A. Callas,ParticleSize:Measurement.Intemmtation.andAootication,Wiley,New
York (1963).
7. D. Hosmer,CommunicationsinStar., .1,,217-227(1973).
8. V. Hasselblad,Technometrics,_ 431-444 (1966).
9. D. Shl, M.S. Thesis,PurdueUniversity(1984).
10. A. Kumar,Ph.D. Thes_s,Penn. State University(1987).
11. R. Feldman,thisProceedings.
12. R. Gerhaml, thisProceedings.
18
LOGNORMAL SIMULATION OF PORE EVOLUTION DURINGCEMENT AND MORTAR HARDENING
Dexiang Shi, Weiping Ma and Paul W. Brown
Materials Rese_h LaboratoryThe Pennsylvania StateUniversity
University Park,PA 16802
ABSTRACT
A model to describe the pore sizes in cement paste and mortar, as determined by hi.gh pressuremercury intrusion porosimetry, has been developed. The model describes porosRy using acompound lognormal distribution. For given material under a given set of curing conditions, theweighing factors and shape parameters of two sub-distributions in the lognormal model may beconsidered as constants, while the location parametersmay be related to curing time and therelationship can be quantified. Therefore, it is possible to predict both the pore size distribution incement and mortarat any age as well as the evolution in pore size during curing.
INTRODUCTION
Cementidous materials are being used in the immobilization of radioactive waste. Conventionalconcrete structures are being used as engineered barriers. Low level liquid wastes are beingimmobilized in cement-based grouts. While these applications differ from those involving normalconcrete structures,the porosities of cernentitions materialsare important in determining the abilityof a structure to meet its functional requirements, virtuallyregardless of the specific applications. Itis the distribution of porosity throughout cement matrix and at cement-aggregate interfacial zonesthat, in concert with environmental variables, determines the macroscopic rate of transport ofspecies.
The development of an adequate description of thepore size distributions in cements has recentlybeen accomplished [1]. Our work has shown that the pore size distributions in cement pastes overthe size range determined by the high pressure mercury intrusion porosimetry (MIP) may bedescribed by a mixture of two lognormal distributions [1]. The present paperextends the model tocementitious systems containing aggregate.
Many physical and chemical processes may be approximatedby the lognormal model [2-5]. It is acommon physical occurrence that a system can often described by a mixture of two lognormaldistributions, which .govern two different phenomena occurring in that system [6]. Cementhydration may be reganted as a process of subdivision of void space, or interstices betweenanhydrous particles [1,5]. In cement paste and mortar, the origins and the mechanisms controllingthe sizes of largerand smaller capillary pores may be different, suggesting the Iognormal model ofporesizedistributionincementpasteandmortartohaveaphysicalbasis.Previousinvestigationsofporeevolutionincementpastehaveshownthatthedistributionshiftstothesmallerporesizewithincreasingage,andthatthemediandiameterandthresholddiameterofporesdecreaseswithincreaseage{7].Thispaperwillshow that,fora givensetofcuringconditions,theweighingfactors,fand (I-0,andshapeparameters.Ol ando2,ofsub-distributionsmay be consideredconstant, while the location parameters, ot and o2, of sub-distributions may be related to curingtime. Thus, it becomes possible to predict the evolution in pore size during the curing of cementand mortar and to predict pore size distributions of cemem and mortar at any age.
BACKGROUND OF LOGNORMAL DISTRIBUTION
A variate X is lognormally distributed with the location parameter tt and the shape parameter0', if• X = logY where Y is a positive variate, which is normally distributed with mean tt and standasd
deviation o. A compound lognormal distribution of pore sizes can be expressed as follows:
19
where p(x) is the probability density function for pores of size x, f and (I - f) are the weighingfactors of sub-distributions" l.tl and _t2are the location parameters for sub-distributions" o! and 02are the shape parameters for sub-distributions. Based on a robust statistical method [8], a programhas been developed to detect the existence of a mixture of two lognormal distributions. Forcementitious materials, the two sub-distributions may represent the larger and smaller capillarypores, respectively [I].
EXPERIMENTAL
Cement pastes were prepared having water-to-cement ratios (w/c) of 0.3 and 0.5. These werecured for I, 3, 7, 14 and 28 days and for I, 3 and 7 days, respectively. Pore size measurementswere carried out at these ages using mercury intrusion porosimelry (MIP). The details of MIP canbe found in reference [6]. Mortar samples were prepared at a water-to-cement ratio of 0.47. Thecement-to-sand ratio used was 0.623. Saturated surface dry (SSD) sand was used. Samples weretested after curing for 1, 3, 7 and 14 day_. Triplicate MIP measurements were performed for eachsample.
RESULTS AND DISCUSSION
A Comnound Lolnormal Distributionof Pore Sizes in Mortar
Replicate measurementsof pore size dismbutions in mortarshowed excellent agreement. Figure Ishowsthe variauonamong thr_measunmmmtsofthecumulativeporesizedistributionsina mortarsample cured for 3 days.
To evaluate the parametersof the compound lognormal diswibution, log x is plotted versus (log x -p)/o • log x is defined the quantile of the mixture distribution and (log x - p)/o the quantile ofstandard normal distribution [8]. Figure 2 shows Q-Q plots for mortar samples cured for 1, 3, 7and 14 days. The slopes of the two roughly linear segments are initial estimates of Ol and 02, theintercepts of the two segments are initial estimates of $tl and _2, respectively. The initial estimatefor f is the deflection point. These estimates are then iterated. Mathematically, the iterationcan startusing any one of these five estimates. They may result in different sets of estimated parameters,though they may generate the estimated distributions with the same goodness of fiLTherefore, it isnecessary to establish the physical meanings and the reasonable ranges for the five parameters sothat the estimation is not only judged by the goodness of fit, but also by the predetermined,physical meanings and ranges, The meaning of the weighing factors, f and (l-f), is evident. It isnecessary to establish the direct characteristics associated with the shape parameters and thelocation panmctas.
Close examination of Figure 2 shows the linear segments at each end of curves to be almostparallel to each other. This suggests that the values for the shape parameters, Ol and 02 in thelognormal distributions of pore sizes are the same. Figure 3 shows the cumulative probabilitydistributions for the mortars cured for 1, 3, 7 and 14 days. These curves exhibit the same shapebehavior, which can be characterized by the slopes of the linear segments in Figure 2. Thus, theshape parameters in the compound lognormal distributions may be considered inde .pendent ofcuring time. Itfollowsthatonly the location pamn_ters alone may be related to curing un_ whilethe shape parameters are related to other factors than curing time. Table 1 shows the estimatedparameters. Figures 4a-d compare the experiment_y obtained pore size distributions and thosecalculated using the estimated paramem's. The mean of sum of squaredenms between the two are0.001, 0.001,0.001 and0.002, respectively, indicating very good fits.
2O
"_ _ Do,/
X,,v,
"6
o.
t)_ameter (rim)'Q° -°_o -,_o -,:mQuantile¢60of ,.6ON(0.1)_6o_b ,.6o
Figure 1. Reproducibility of porosity measure- Figure 2. Q-Q plots formortarspecimensmeritson mortarspecimens cun_ for 3 days. cured for l, 3, 7, and 14 days.
Estimates of parameters in log-normalmodelforporesizedisn'ibufion n
o--o ,,,1 3.67 1.05-1.005 -0.16 0.87 _o3 3.10 1.03 -I.0 -0.I6 0.85 _ T40ays_7 2.80 0.96-I.0 -0.15 0.85 _,°14 2.43 0.85-0.98 -0.14 0.84
8c_, ....... ;b ...............
Diometer (nm_°
Figure3. Cumulativeprobabilitydiscibudonsformo_ cm'edfor 1, 3. 7, and 14days.
Prediction of Pore Evolution Durine Cement and MortarHardenine
From above analysis, it was found that p! and P.2may be related to curing time. while the f, atand o2 may be assumed to remain constant. Figure 5a shows that Pl for mortars cured for 1, 3and 14 days (can be related to cta'ingtime as in days, t).
P.l = 3.67403 t('°'ls63'tg) [2]
Figu_ 5b shows that g2 for moru_ cured for I, 3 and I4 days (can be related to curing time):
tt2 = 1.07456 exp (-0.016661t) [3]
It must be noted that equations 2 and 3 fit the limited, presently available data. Unlike in the use ofIogm,,mal model to describe pore size distribution, there is no physical basis for these equations.
The 7-day dam were not used to obtain the above expressions. Rather, seven day values werepredicted from the above age-p relationships. The predicted values are Pl = 2.71 and P2 = 0.96,assuming f = 0.85, Ol = -1.0 and 02 = -0.15. Figure 6 shows the comparison between the
21
experimentally obtained pore size dislribudon and that using the predicted parameters. The mean ofsum of.squared em3_ between the two is 0.0011 indicating a very good prediction.
• 1 Day
._ _ %,,, ---Estimated_'_ --- Estimated
0 O "
"_o0
Diameter (,nm) Diameter (nm)
7 Days 14 Days
-,% ___ :__ ,'_ ___ Estimated_,_. Experimental "_ Experimental
_,_o0
o! tO I_ o! I0 IWDiameter (nm) Diameter (nm)
Rg,_ 4. Compa.,'iso. between expedmcm,_llyob_dned CRg,_ 3) andcaJcu_:d ix)n: sizedisu'ibudons usmg d¢ iognormai model.
t_. II.IIB--• _.m- Experimental
--- Predicted
(t ** -.156349) * 3.67403 _ M_e=-
110.(11-
n
O_ ItLQI-
- • t) • .0745| •
Ot_
l,o s.6oAge (day) '°'_° ,s.bo o.0.- . ..... i_lom) . ..Size
Figure 5. Variations in I.tl and 1_2with curing. Figure 6. Comparison between experimentallyobtained and calculateddistribuuons.
22
Figure7 showsthecumulativeprobabilitydistributionsforcementpastesfabricatedatawater-to-cementratio0.3andcuredforI,3,7,14 and28daysandthosefor0.5w/cratiocementpastescuredforI,3 and7 days.The shapesofthecurvesinthefiguresappeartodependsu'onglyonwater-to-cementratiobutverylittleon age.Figures8 showsthecorrespondingQ-Q plots.Atagivenwater-to-cementratio,thelinearsegmentsateachendappearparalleltoeachother.Taken
' together,Figures7 and 8 suggestthatitshouldalsobe reasonabletoassumeconstantshapeparametersforcementpastesatdifferentages,giventhesamecompositionandcuringconditions.
"'-..,,/c=o.sO _ I _'k_. _ • X
.O114 "4& t',_,
.Io.O.3 ,,,,°'.I ','-; ,/e
3 -- ",,,,
ol ll_iorneter (n Quontile of N(O,1)
Figure 7. Cumulativepore sizedistributionsfor Figure8. Q-Q plotsfor cementpastespecimenscementpastes:w/c = 0.3 at l, 3, 7, 14,28 days; havingpore sizedismbutionsshowninw/c = 0.5 at 1, 3, 7 days. Figure7.
Comparing Figures 7 and 8, the 0.3 w/c ratio cement pastesshowlitde change in pore sizedistribution with curing time. Comparedto pore sizedismbutionsfor0.5 w/c ratio cementpastes,there is significantly less large pore space available for size refinement. These data serve toreinforce the idea that the water-to.cement ratio determines the pore structure during the settingprocess and tha_it is from this pore structure that the pore evolution occurs [1].
Because the cumulative distributions of the 0.5 w/c ratio paste show noticeable shifts whilemaintaining the same shape, the method described above was used to predict the pore sizediswibution in 3 day old cement paste. It was determined that f= 0.91, Ol - -1.1 and o2 - -0.24.Figure 9 shows the age-stl and age-st2 relationships. From these relationships, the values predictedfor Stl and $*2are 4.19 and 1.15, respectively. Using the above values for the five parameters, thepredicted compound lognormal distribution of pore sizes in 3 day old, 0.5 w/c ratio cement pastewas obtained and shown in Figure 10. In this instance the mean of sumof squared errors betweenthe experimental and the estimated data is 0.0015.
SUMMARY
Based on presently available data, interpolation has been used to obtain It.values. In order topredict the long term changes in pore s_ucture, it will be necessary to obtain sufficient pore sizedisn-ibudonsdata to allow exlxapolation. In addition, the relationshipsbetween w/c ratio and the oand St values, and those between curing conditions and o and St values have not been fullyelucidated. Furtheracsessment of the effects of w/c ratio andcuringconditions on the parametersin the compound Iognormal model are needed. However, in spite of these limitations, the presentwork has shown that it is reasonable to assume constant values foro and f for both cement pastesand mortars to establishment of simple relationships between curing time and Stvalues. Givensufficient data,it should be possible to predict pore evolution duringcement and mortarhardening,and to predictthe pore size distributionof cement and mortarat any age.
23
ImJllS,-
(t ,* -.0335962) .4.35424 0tern- * Experlmentol_l .. --- Predicted
1tLm*•, (t ** .0712061) * 1.24833 E "_
, ::1_. 0 I._1- *
O.lm-0.01e-
3, AQe (day) '_ize (rim)
Figure 9. Variations in Pl and £t2with curing. Figure I0. Comparison between experimentallyobtainedandcalculateddistributions.
CONCLUSIONS
I. A compound lognormaldistribution model can be used to describe the pore size distributionin themortarandprovidesthebasisforthepredictionof theporesizedistributionatany age.
2. It is reasonable to treatweighing factors and shapeparameters in compound lognormaIporesizedistributionsforcementandmortaras constants.
3. Thelocationparametersinthemodel,_ canberelatedtocuringtimeforcementandmortar.
REFERENCES
I. D.Sift, P.Brown andS.Kurtz,MRS Meeting,Boston,1988,(inpress).2. I. AJtchison and I. A. C. Brown, The Lognormal Distribution, CambridgeUniv. Press,
Cambridge(1957).3. R. kani, J. Phys.Chem.,63. 1603 (1959).4. R. Irani and A. CaLlas,Panicles Size: Measurement. Internretation.andApnlicafion_ Wiley,
New York (1963)., 5. S. Diamond and W. Dolch, J. Colloid and Interface Sci., 38. 234-244 (1972).
6. S. Kunz, S. LevinsonandD. Shi, J.Am. Ccram:Soc., (in press).7. D. W'mslow, J. Mat., 5, 56*.585 .(1970).8. E. Fowikes, J. Am. Star.Assoc., 74. 561-575 (1979).
ACKNOWLEDGEMENT AND DISCLAIMER
The research describedherein was suppo,nedby the StrategicHighway ResearchProgram(SHRP). SHRP is a unitof the National Research Council that was authorized by secdon128 ofthe Surface Tnmspo_tion and Uniform Relocation Assistance Act of 1987.
This paperrepresents theviews of the author(s) only, not necessarily reflective of the views ofthe National Research Council, the views of SHRP, or SHRFs sponsors.The results reportedhere are not necessarilyin agreement with the resultsof other SHRP researchactivities. They arerepottedto stimulate reviewanddiscussionwithin the research community.
c
24
CONCRETE MICROSTRUCTURE AND ITS RELATIONSHIPS TO PORESTRUCTURE, PERMF.ABILITY, AND GENERAL DURABILITY
D.M. ROY, D. SHI, B.E. SCI-IF_ETZ,AND P.W. BROWN
Introduction
The durability of concrete is frequently associated with the transportof dissolved species. Such transport may be considered in terms ofpermeability. It is well recognized that transport occurs through acontinuous network of pores, which exist in the cementitious matrix ofconcrete, as well as through the porosity which exists in the interracialregions with aggregate. Unfortunately, however, the relationships betweenconcrete permeability and the pore structures through which transportoccurs are, at best, qualitative. It is the objective of this paper to describework leading to rapid and accurate measurement of concrete permeabilityand the development of models to describe permeability of concrete interms of pore structure.
Poised Perm¢obility
Permeability measurements have historically been made with wateror various gases including oxygen, nitrogen, argon and air. Darcyian flowhas been used as the theoretical basis for the description of the observedflow conditions. This mathematical treatment required information on thesample dimension (cross-sectional area and permeation length), flowproperties of the fluid, a measurement of the flow rates and an establishedpressure gradient across the test specimen. The vast majority ofexperiments have been conducted at low AP in accordance_ with Darcy'sderivation and usually with ambient pressure being the lower bound onAP. Experimental designs of this type are limited to values of permeabilitythat are typically to the nano-darcy range. In using apparatus typical thistype, water which has permeated through the test specimen is collected onan LVDT, the spring constant of which has been matched to the mass ofwater that would be collected in a reasonable laboratory experiment [Gotoand Roy (1981)]. The resulting displacement of the LVDT by thepermeated fluid is monitored and an equilibrium flow rate is determined.The permeability is calculated from this flow rate. One limitation to thisparticular experimental design is the evaporation of the permeated waterfrom the container which is affixed to the LVDT. A second limitation is
that the lower limit of operation for this apparatus is estimated to be 10-8darcy.
25
To develop porosity-permeability models, measurements onspecimens having low permeabilities need to be carried out. To addressthis need, the pulsed permeability approach was implemented.
Transient decays in pressure have been used in the pressure-pulsepermeability cell to successfully measure hydraulic properties of lowpermeability materials such as cored samples of rocks and sandstones(Brace et al. [1968], Hsieh et al. [1981], Neuzil et al. [1981]). To date, therehas been little work devoted to the application of this technique tocementitious materials (Hooton and Wakeley [1989]), which generally havehigher compressibilities and permeabilities. In the transient pressurepulse method, a jacketed sample is confined between two pressurizedreservoirs which contain the penetrating fluid. The pores of the sampleare also filled with the fluid. A confining pressure higher than thereservoir pressures is maintained on the jacket. When the experimentbegins, the pressure in one of the reservoirs is suddenly changed to ahigher or lower value and the resultant pressure changes on the high orlow pressure side are measured as a function of time.
The optimal way of collecting data is to simultaneously measure thepressure change in both reservoirs. Pommersheim and Scheetz [1989]have recently discussed the advantages of this technique. Hooton andWakeley [1989] have emphasized the sensitivity of test results toenvironmental variables when measuring water permeabilities of concrete.
Figure 1 presents a schematic diagram showing the experimentalconfiguration for the transient pressure pulse test and Figure 2 a detailed,exploded drawing of the cell design. Before the test begins the entiresystem is equilibrated to a constant pressure P0. Then the pressuredownstream reservoir (Pd) is suddenly deccreased to P1, while theupstream reservoir (Pu) remains at P0. Pu will decrease and Pd will rise asfluid is transferred between reservoirs. A constant confining pressure (Pc)is kept outside the sample. By maintaining this pressure at a levelconsiderably higher than P1, leaks are prevented. However, Pe is limitedto avoid creep or micro-cracking.
Figure 3 is a schematic representation of how the upstream anddownstream pressures change with time for two typical classes ofmaterials. The figure is illustrated for the case where the two reservoirvolumes, Vu and Vd, are equal. Nomenclature is provided beneath thefigure. The curves marked A. are the pressure decay and pressure risecurves for "tight rocks" such as granite which typically have lowpermeabilities and compressibilities. Here the response of the two curvesis symmetric around a horizontal line drawn to the final pressure Pfreached by both reservoirs. These data collected for an actual sample arepresented in Figure 4. As discussed by Pommersheim and Scheetz (1989),the analysis of Brace et. al. [1968] predicts that this pressure will liemidway between P0 and P1, while the analysis of Hsieh et. al. [1981] °predicts that Pf will lie more towards the upstream side. The difference isattributable to the fact that Brace and co-workers assumed in their
26
development that the compressive storage of the sample was negligible,whereas Hsieh and co-workers did not. Pf can also be estimated from amass balance knowing the pore volume of the sample and the volumes andinitial pressures in the reservoirs (Trimmer [1981]).
Model Development
Previous mathematical models for pressure pulse testing have beendeveloped by Brace et. al. [1968], Lin [1977], Hsieh et. al. [1981] andPommersheim and Scheetz [1989].
As a model system consider the cylindrical sample depicted in Figure1, having total volume V = AL, where A is the cross-sectional area of thespecimen and L is its length. The sample is confined between the twopressurized reservoirs, the upstream one at Pu and the downstreampressures are P1 and Po, respectively.
The partial differential equation which governs pressure changes asa function of distance and time P(x,t) within the sample is given by:
o_p ,( &p ,_2 fl' # 6P (1)6)(2 + #\6-x-) - k at
where P = pressure within the sample;
g = pore fluid viscosity;
k = sample permeability;
x = axial distance within the sample, measured from the upstreamreservoir;
t = time;
13' = lumped compressibility.
Equation (1) represents how the pressure P(x,t) varies within thesample as a function of position x and time t. It is a non-linear partialdifferential equation subject to one time and two boundary conditions.These are given by:
P(x,0) = P0 initial condition
P (0,t) = Pu(t) boundary conditions
P(L,t) = Pd(t)
27
where L is the sample thickness, and Pu and Pd are the upstream anddownstream pressures, respectively.
Assumptions involved in the derivation of equation (1) include: onedimensional mass transfer with constant transport area, constanttemperature, and constant physical properties (g, k, [_' and porosity).These assumptions are the same as those stated or implied by previousworkers [Brace et. al. (1968), Hsieh et. al. (1981)]. Most of them are likelyto be met in laboratory tests with homogeneous corings. Samplecempressibilities will he most likely to remain constant when the ratio ofthe confining pressure to the initial pressui'e difference, i.e., Pc/ (P1-P0)ishigh [Hooton and Wakeley (1989)].
13' is a lumped compressibility. It depends on the compressibility ofthe fluid, 13, the compressibility of the sample, 13s, the effectivecompressibility of the jacketed sample 13e, and the sample porosity e,according to:
13' = (13e - 13s) + e(13 - [3e) (2)
A similar equation has been presented by Brace et. al. [1968] and Hsieh et.al. [1981]. Of all three quantities be is the one which is least likely to beknown and which is also potentially the largest, especially in experimentalconfigurations where the sample is retained in a flexible rubber or plasticsleeve. In effect this makes 13e, and thus 13', an arbitrary parameter. Neuzilet. al. [1981] found values for this parameter which were several orders ofmagnitude greater than the fluid compressibility.
By introducing the dimensionless distance z = x/L, time 0 = t/T, andpressure P = (P- P0)/AP, equation (1) and its attendant conditionsbecome:
a,AdSP/6z2 + _,oz ,, = 6o (3)
(I) (II) (III)
where AP = P1 P0
with conditions: p(0,q) = Pu(q) reduced boundary conditions
p(1, q) = Pd(q) reduced initial conditions
p(z,0) = 0¢
T = 13'ttL2/k is a characteristic time for the transfer of mass from thehigh pressure to the low pressure side.
28
The terms in equation (3) have been labeled (I), (II) and (III). Term(I) represents the transfer of mass between reservoirs caused by thepressure difference. It is present in all formulations of the problem. Theunsteady state term (III) corresponds to the accumulation of mass withinthe sample. If the observed experimental times t are much greater thanthe characteristic time T, i.e., t >> T, then the accumulation term can beneglected. This is a requirement of a steady state process. However, since
, the pressures at the two ends of the sample are continuously changing,pressures within the system slowly adjust to accommodate these changes,in effect adjusting to each new steady-state. Such behavior is called quasi-static and the system is said to be at quasi-steady state.
Term (II) in equation (3) represents the dynamic response to thecompression of the jacketed sample. This non-linear term will be smallwhen the dimensionless compressibility 13'AP is small. Calculations showthat term (II) would be small if _' is of comparable magnitude to thecompressibility of the fluid, but, as discussed, this is often not the case inpractice.
Application of these equations to data from Figure 4 should result ina strength line when stated at time vs.
Pu - PLk
From the slope of this line the characteristic chance time can bedetermined and subsequently the permeability. Figure 5 represents theprocessed data and the regression fit to the data.Pore Structure
Recent studies by Shi et al. [Shi et al., 1989, 1990a, 1990b] haveshown that the pore size distributions in cementitious materials may bedescribed in terms of a mixture of lognormal distributions. A variate X islognormally distributed if Y = logX where X is a positive variate, (0 < X < **),which is normally distributed with mean _t and standard deviation ¢_ withregard to Y. The probability density function of a lognormal distribution is[Aitchison and Brown, (1957)],
p(x) = (2xa2) -1/2 X-1 exp[-0.5{(log(x) - lX)/o}2]
The range of x is 0 < x < _t. Ix is defined as the location parameter, a isdefined as the shape parameter. The mean, median, mode and variance
, are as follows:mean = exp(_t + 0.502)median = exp(lx)_
" mode = exp(].t - 0 2)variance = mean2[exp(o2)-l]
29
The mixture of two or more lognormal distributions is defined by acompound density function:
P(x) = _: fi p(x, _ti, a i)
Zfi=l
where fi is theweightingfactorof the ithsub-distributionp(x,IAb cri),andIxi and c_i are locationparameterand shape parameterof the ith sub-distribution,respectively.A statisticalmethod was developedto detectifthereis a mixtureof two lognormaldistributionsand iterativelyestimatethe parametersin the compound distribution[Fowlkes,(1979)]. Withoutacomputer program, the iterativeestimationof parametersis extremelydifficult.However, one can use a simplegraphicalmethod to obtainfirstdegree estimatesof parametersfor a compound distributionof two ormore lognormaldistributions.The method is describedas follows.Supposethatone has obtaineda set of cumulativeprobabilitydata,P(X) vs. Xwhere X isthepore sizeand P(X) istherelativepercentageof porevolumewith respectto the totalpore volume. The graphicalapproachto obtainestimates of parameters of a mixture of two or more lognormaldistributionscan be summarizedas follows.
Step 1: Transform X to log(X).
Step 2: Find the quantiles of N(0,1) corresponding to P(X), by consulting thestandard normal distribution tabulation which can be found in anystatistics text book. For example, if the cumulative percentage is 50%, thecorresponding quantile is 0; if the cumulative percentage is 84%, thecorresponding quantile is 1.0; etc.
Step 3: Plot log(X) vs. the quantiles.
If one single straight line is found, one can conclude that X obeys alognormal distribution. If more than a single straight line is observed, it islikely that the porosity can be described by a mixture of lognormaldistributions.
Step 4: Determine the intercepts and slopes of each linear portion. Thesecorrespond to _ti, and ci, i =l,...,n, respectively. From the intersectionsbetween any two neighboring linear segments, one can find the weightingfactors, fi-
Step 5: Determine the quality of the fit by drawing lognormal distributionsusing the parameters obtained and comparing them to the data.
30
An important characteristic of a mixture of lognormal distributions isthe inflection point. This is located on the experimental curve at theintersection between the first two linear segments. The correspondinglog(X) value is:
4
[a2 lXl - al Ix2] / [a2 - all7
Comparing the calculated inflection point with that from the experimentaldata allows a comparison of the model fit.
Pore Structure, Percolation and Permeability
Among the models relating pore structure and permeability,percolation theory-based models are physically the most meaningful.Percolation theory assumes a lattice containing a very large number oflattice points [Stauffer, (1985)]. It is assumed that a point can berandomly occupied or vacant independent of the states of the neighboringsites. However, depending on the situation, additional assumptions can bemade which influence site occupancy. Accordingly, occupied sites may beisolated from each other or connected with neighboring sites to form acluster. When only a small fraction of the total lattice sites are occupied,the probability, p, that a given site will be isolated is high. With increasingsite occupancy, the probability of the formation of a large cluster, whichextends from one side of the lattice to the other, becomes non-zero. Sucha large cluster is called "infinite-path." It percolates through the lattice inthe similar way that fluid percolates through porous materials along thenetwork of "infinite-path" pores. There is a statistically-based criticalpoint (percolation threshold), Pc, below which no 'infinite-path' can form.
It is clear that these concepts are relevant to the transport of liquidsthrough the "open" porosity in concrete. In the determination ofpermeability, only the liquid which flows through the network ofconnected pores is determined. The network of connected pores isequivalent to the "infinite-path" in percolation theory. Therefore, it is thestructure of the infinite network, not the total porosity and its distribution,which controls permeability. It is generally accepted that only pores withdiameter greater than some value effectively contribute to permeability.The observed inflection point-on the cumulative pore size distributiondetermined by mercury porosimetry locates the minimum diameter ofpores which are continuous through all regions of the hydrated cementpaste [Nyame and Illston, (1980); Mehta and Manmohan, (1980); Hughes,(1985); Goto and Roy, (1981)]. It is this observation which provides theconnection between the lognormal model for pore structure, discussed
-. above, percolation theory and permeability.
31
A Preliminary PCrqol_tion-Permeability Model
Most of the percolation models which have been developed havebeen applied to describing flow in sedimentary rock. This work suggeststhat a power law may exist between permeability and a characteristic poredimension, and that a tortuosity factor which may be quantified by usingpercolation concepts. These are important aspects of modeling porestructure-permeability relationship in cement and concrete. "
In percolation theory, the relationship: f = C(p' - pc) n can be used torelate f, the fraction of connected pores in the infinite network, to thecritical porosity, Pc. C and n are constants, Pc the critical porosity belowwhich there is no flow permeating at all, p' the probability that two poresinterconnect, and f is related to tortuosity [Guegeun and Dienes, (1989)]. Ithas been suggested that n = 1.9 or 2 [Fisch and Harris, (1978); Berman, etal, (1986)].
In the following, the three remaining parameters are calculated.Assuming p' to be proportional to porosity p and represented by Pc', wehave f = C'(p- pc') 2, where pc'= Pc/c'. Pc' can then be obtained byextrapolating the porosity-permeability curve to the point where thepermeability equals zero. C' can be obtained by fitting the experimentaldata to the model. From very limited data on permeability and pore sizedistribution, we calculated P.C' = 0.15 and C' = 0.23.
We chose the second moment of the pore diameter distribution,<D2>,as the characteristic pore dimension. The predicted permeability is (thederivation is omitted here):
k = (f/32) p<D2>
The second moment of a lognorrnal distribution is exp(2ix + 2a2).In our compound lognormal model for pore size distribution,
<D2> = Y_fi exp(2 (IX+ a2))
This approach can be demonstrated using the data shown in Figure 6,which are for a fly ash blended cement (fie = 0.536), mixed at a water-to-solids ratio of 0.55 and hydrated for 28 days at 23°C. The value of<D2> wascalculated as 2383 nm2. The porosity and permeability were determinedto be 0.45 by MIP and 3.86 x 10-5 Darcy by flow-through permeability,respectively. The calculated value of the permeability is 1.93 x 10-5 Darcy.Although the prediction is satisfactory in this case, the model must bevalidated by comparison to additional data to obtain reliable values for C',Pc, and to verify its general validity.
32
S_mmary
We have briefly discussed the means for the rapid measurement ofpermeability, the means by which pore size data obtained from mercury
, intrusion porosimetry can be quantified, the basis for using percolationtheory to treat the pore data, and the development of a preliminary modelto describe permeability.
33
References
J. Aitchison and J.A.C. Brown, The Lognormal Distribution, Ch. 2_ CambridgeUniv. Press, Cambridge (1957).
P
D. Berman et al., Conductances of filled two-dimensional networks, Phys.Rev. B, 3_, 4301, 1986.
W.F. Brace, J.B. Walsh and W.J. Frangos, Permeability of granite under highpressure, J. Geophys. Res. 7_ (6), 2225-2236 (1968).
R. Fisch and A. Harris, Critical behavior of random resistor networks nearthe percolation threshold, Phys. Rev. B, 18, 416, 1978.
E. Fowlkes, J. Am. Stat. Assoc., 74, 561-575 (1979).
S. Goto and D. Roy, The effect of w/c ratio and curing temperature on thepermeability of hardened cement paste, Cem. Cone. Res., 11, 575, 1981.
Y. Gueguen and J. Dienes, Transport properties of rocks from statistics andpercolation, Math. Geology, 21, 1, 1989.
J.D. Hooton and J.D. Wakeley, Influence of test conditions on the waterpermeability of concrete in a triaxial cell, Pore Structure andPermeability of Cementitious Materials, 157-64, L.R. Roberts and J.P.Skalny, Eds., MRS (1989).
P.A. Hsieh, J.V. Tracy, C.E. Neuzil, J.D. Bredehoeft and S.E. Silliman, Atransient laboratory method for determining the hydraulic propertiesof 'tight' rocks--I. Theory, Intl. I. Rock Mech. Min. Sci. & Geomech.Abstr. 18, 245-252 (1981).
D. Hughes, Pore structure and permeability of hardened cement paste, Mag.Cone. Res., 37, 227, 1985.
L.B.W. Jolley, Summation of Sorie_, 2nd revised ed., Dover Publications,New York (1961).
P. Mehta, and D. Manmohan, Pore size distribution and permeability ofhardened cement paste, in 7th International Congress on the Chemistryof Cement, 1980, Vol. 3, pp. VII 1-5.
C.E. Neuzil, C. Cooley, S.E. Silliman, J.D. Bredehoeft and P.A. Hsieh, Atransient laboratory method for determining the hydraulic propertiesof 'tight' rocks II. Applieati0n, Intl. J. Rock Mech. Min. Sci. & Geomech.Abstr. 18, 253-258 (1981)
34
B. Nyame, and J. Illston, Capillary pore structure and permeability ofhardened cement paste, in 7th International Congress on the Chemistryof 17ement. 1980, Vol. 3, pp. VI 181-185.
J. Pommersheim and B. Scheetz, "Extension of standard methods formeasuring permeabilities of pressure pulse testing, to be submited forpublication (1989).
D. Shi, P. Brown and S. Kurtz, "A model for the distribution of pore sizes incement paste," 23-34, Pore Structure and Permeability of CementitiousMaterials, L.R. Roberts and J.P. Skalny, Eds., MRS (1989).
D. Shi, W. Ma and P. Brown, "Lognormal simulation of pore evolution duringcement and mortar hardening," 143-48, Scientific Ba_i_ for NuclearWaste Management XIII, V. Oversby and P.W. Brown, Eds., MRS(1990a).
D. Shi, P.W. Brown and W. Ma, "Lognormal Simulation of Pore SizeDistribution in Cementitious Materials," J. Am Ceram Soc, submitted(1990b).
D. Stauffer, Introduction tQ Percolation Theory, Ch. 2, Taylor & Francis,London, 1985.
D. Trimmer, Design criteria for laboratory measurements of lowpermeability rocks, Geophys. Res. Lett. 8 (9), 973-975 (1981).
35
Figure (_aptions
Figure 1. Schematic drawing of the pulsed permeability apparatus.
Figure 2. Exploded drawing of permeability cell.
Figure 3. Theoretical pressure-time behavior for pulsed permeabilityexperiment.
Figure 4. Typical raw data for pulsed permeability experiment.
Figure 5. Typical processed data for pulsed permeability experiment.
Figure 6. Cumulative pore size distribution of FA cement (w/c = 0.55, f/c =0.536, cured for 28 days).
36
37
:! ,i
ei
38
eJnsseJd
39
0
- I0IJJ
03¢0LU
- ,_0 a.0 a:
®X ®
0 0 0 0 0 0 0 0
40
¢D
41
42
POROSITY/PERMEABILITY RELATIONSHIPS
P.W. BrownDepartment of Materials Science and Engineering
. Pennsylvania State University
Dex ShiCivil Engineering DepartmentPennsylvanic State University
J.P. SkalnyW.R. Grace
Iotroduq_ign
The structure of the porosity in concrete strongly influences itsperformance. Specifically, porosity determines the rates at whichaggressive species can enter the mass and cause disruption. Rates ofintrusion are related to the permeability of the concrete. In the mostgeneral way, permeability depends on the total porosity. Moreimportantly, however, permeability depends on way in which thetotal porosity is distributed. Porosity, in turn, is related to theoriginal packing of the cement, mineral admixtures, and theaggregate particles, to the water-to-solids ratio, to the rheology,which is related to the degree of dispersion of the solids originallypresent, and to the conditions of curing.
This contribution considers the nature of porosity in cementand concrete, discusses its measurement and the limitations in theinterpretation of porosity data. Models developed to describeporosity both in cement paste and concrete are reviewed. Withemphasis on permeability models developed for a variety of porousinorganic materials, relationships between pore structure andpermeability are discussed in terms of their applicability to concrete.
The Nature of Porosity in Cement Paste. Mortar and Concrete
From a pragmatic standpoint, porosity of a material is not ofinterest as an end in itself. Rather porosity is of interest because itdirectly influences both mechanical and transport properties of
o cementitious materials. For example, the pore shape-dependentrelationship between porosity and strength, <_, of the form:
<_/o'o = exp-(be) [1]
*Materials Science of Concrete II (in press).
48
is well known [1], where b is a shape-dependent parameter, e theporosity and 0o the ideal strength. This relationship is illustrated inFigure 1 [2].
With respect to durability, the ability of concrete to resistvarious forms of deterioration is often related to its impermeability.Analysis of the sources of porosity and its connectivity canfrequently provide the means to understand the mechanisms bywhich aggressive species can intrude concrete. This is because thepore structure defines the paths along which liquid or vaporpreferentially moves. This, in turn, is of obvious importance withrespect to specific durability considerations including the transportof freezable water and electrolytes, such as chlorides, throughconcrete.
It is widely accepted that permeability is determined bymicrostructure. Microstructure in this context is defined in terms ofpore and crack structures. A large number of models have beendeveloped, particularly for sedimentary rocks, to predict thepermeability from measurements of pore structures and cracks [3-12]. For cementitious materials, it is well recognized that both totalporosity and its distributions determine the permeability [13-16],and that only pores with diameters greater than a specific valuecontribute significantly to permeability [13, 14, 16]. Figure 2, forexample, illustrates the dependence of permeability on the porosity[17]. It has also been observed that the inflection point on thecumulative pore size distribution obtained by mercury porosimetrylocates the minimum diameter of pores which form a continuousnetwork through hydrating cement paste [13, 14, 16].
Large variations in permeability of concrete having nominallysimilar porosities are frequently observed [13-15, 18]. Verbeck evenimplied that the permeability of concrete may not be a fundamentalmaterial property [19]. Alternatively Podvaley and Prozenko [20]claimed that the variations should in most instances be considered anobjective characteristics of the phenomenon, and that the source ofvariation in permeability results from small variations in concretemicrostructure. As a consequence, it is useful to enumerate thevarious types of porosity present in concrete and to establish theirrelative contributions to permeability.
There are a variety of "types" of porosity in concrete. Thesetypes may be classified in terms of their origin or in terms of theiranticipated effect on measurable parameters such as strength orpermeability. Sources of porosity in concrete include:
44
1. gel pores2. smaller capillary pores3. larger capillary pores4. large voids (also included in this category may be intentionally
added voids such as by air entrainment)5. porosity associated with paste-aggregate interracial zones6. microcracks and discontinuities associated with dimensional
instabilities that occur during curing7. porosity in aggregate
The diameter of a stable gel pore is assumed to be about 2 nm[21]. The reason for the selection of this value is based on theassumption that hydration products cannot precipitate in poreshaving diameters smaller than about 2 nm. Because the gel porosityresides in the hydration products that accumulate between the liquidphase and the anhydrous cement grains, gel porosity has a majoreffect on hydration rates but only a minor effect on transportprocesses involving liquids. However, there is at present nojustification for ignoring the other types of pores listed above. Thus,the contribution of each of the remaining types of porosity topermeability must be considered. Unfortunately, deconvolution ofthe relative contributions of each of these sources of porosity topermeability has not been carried out. Therefore, conclusionsreached regarding concrete permeability are frequently based onextrapolation of results obtained for cement pastes. However, it maybe reasonable to subdivide the porosity in concrete into two classes:(1) that in the paste matrix and (2) that associated with theaggregate and paste interface.
The principle source of the matrix porosity contributing topermeability is that associated with residual space between cementgrains which was originally filled with water. The contribution of thissource of porosity may be amenable to assessment by investigationsof cement paste. However, the contribution of the porosityassociated with the interracial zones between paste and aggregateand the microcracks that develop in this interfacial region, whichextend into the paste, to permeability must be assessed bydeterminations carried out directly on aggregate-containingmaterials.
Two types of porosity can be considered to form the network ofcapillary porosity present in cement and concrete: large and smallcapillary porosity. One reason for categorizing porosity in thismanner is related to the influence of chemical and mineraladmixtures on the two types. Capillary porosity is assumed to have amajor effect on transport processes but only a minor effect on
45
hydration rates. The diameters of capillary pores could in theoryrange from very small values to large ones. However, it has beenassumed that the lower diameter limit of capillary porosity is 100nm [14, 16, 22]. It is worthy of note that there is an apparentdiscrepancy between the size of a gel pore (2 nm) and the lowerbound on the size of capillary pore. According to the IUPAC [23]classification of pore sizes, the diameter of a micropore is 2 nm orless while that of a mesopore is between approximately 2 and 50 nm.Mesopores are of a size range for which electrostatic interactionsbetween the pore walls and the liquid would extend over asignificant fraction of the cross-sectional area. A consequence of thismay be that transport processes through pores having diameters inthis range are hindered by electrostatic effects. It is well known thatthe mineral admixtures affect permeability; the basis for this effectcan be understood in terms of the formation of a larger amount ofporosity in the mesopore range. However, this view requires furtherexperimental verification for both portland-pozzolan and pc.tland-slag systems.
Overview of Porosity Measurement and Jntcrpretation of D_ta
Three important types of techniques have been used toexperimentally measure the porosity and/or its distribution inporous materials. These techniques are gas adsorption, mercuryintrusion, and direct observation techniques including serialsectioning and pore casting followed by optical or SEM observation ofthe sections or the casts.
Vapor or Gas Adsorption
This technique involves the adsorption of a gas (includingwater vapor) on the accessible internal surfaces of a specimen.Because a significant degree of uncertainty regarding the inter-relationships among hydration phenomena, pore size distributions,and properties of cement systems remain, studies using thistechnique have been carried out over the past decades to elucidatethem. However, internal surface areas determined by gas adsorptionexperiments are dominated bythe gel porosity, which has only aminor effect on bulk transport.
Mercury Intrusion
A second, important technique to determine porosity ismercury intrusion. In this technique, mercury is forced into poroussamples with increasing pressure. Because the mercury is assumednot to wet the internal sample surfaces (the angle of contact between
46
a drop of mercury on a cement surface representative of the theinternal surface and the surface would be greater than 90°), theintrusion pressure can be related to the diameter of the poreintruded provided the contact angle between the mercury and the
, sample are known or can be assumed. The value assumed for thecontract angle allows the calculation of the diameter of the poreintruded according to the Washburn equation [24]:
P = -2_cosO/r [2]
where P is the pressure needed to intrude a pore of radius r, x is thesurface tension of mercury, • is the angle of contact.
The application of mercury intrusion porosimetry to measurethe distributions of pore sizes in cements was pioneered by Winslowand Diamond [25, 26]. Although there are certain limitations to theuse of mercury intrusion in the analysis of the pore structures ofcements, the technique is very attractive since it measures pores ofthe general size range that appear to control permeability.
Data obtained by mercury intrusion tiorosimetry (MIP) must,however, be interpreted with caution. Pores in the general sizerange or 50 to 100 nm are of a size for which capillary condensationis of importance. The occurrence of capillary condensation is basedon the assumption proposed by Zsigmondi [27] that the equilibriumvapor pressure in a pore of a given diameter can be determined bythe radius of curvature of the meniscus of the liquid in the pore.This leads to the familiar Kelvin equation:
ln(po/p) =-2xM/(SaRT) [3]
where Po is the saturation vapor pressure of the bulk liquid, p is thesaturation vapor pressure of the liquid in a pore of diameter a, x isthe interracial (surface) tension, M is the molecular weight, 8 is thedensity, T is the absolute temperature and R is the ideal gas constant.Rearranging the Kelvin equation gives:
p = po/(e x) [4]
where x is the right-hand term in Eq. [3], and shows that thesaturation vapor pressure in a pore is reduced with respect to that inthe bulk liquid. A consequence of this is that pores in this size rangeremain filled with liquid at relative humidities below 100%.Therefore, attempts to minimize disruption to pore structures bydrying under conditions less aggressive than 105"C, for example,may result in incomplete desiccation.
47
Another important factor complicating interpretation of poresize data for cements is the presence of "ink bottle" pores. Gasadsorption experiments on cements have shown that there issignificant hysteresis between the adsortion branch and thedesorption branch of the curves [see for example 28]. Katz [29] wasprobably the first to point out that sorption hysteresis occurs whensmaller capillary pores do not empty during desorption with theeffect of blocking the emptying of larger capillary pores. Generallysimilar behavior is observed with mercury intrusion experiments;some of the mercury intruded under pressure remains trapped incement specimens when the pressure is removed even though is itassumed that the mercury does not wet the specimen surfaces. Thisis the result of mercury being forced into small-necked pores. Aconsequence of this phenomenon is that large pores with small necksappear as many small pores. Cerbesi [3.0] reported thatapproximately 75% of the pore volume of cement paste interrogatedby MIP is composed of pores uniform in cross section. The remaining25% are "ink bottle" pores. Because bulk transport is primarilyaffected by the minimum pore diameters, this phenomenon mayinfluence the interpretation of MIP data with respect topermeability.
An additional factor that must be considered when porestructures are determined by mercury intrusion porosimetry is thatclosed porosity is difficult to determine unless the pore walls aredamaged by the intrusion process. It is not necessarily undesirablethat closed porosity is not intruded because it is unlikely that closedporosity would significantly contribute to transport processes.Alternatively, the rupture of pore walls would result in regions ofclosed porosity being included in the measurement of porositybecause the membranes of hydration product that isolate theseregions from the capillary porosity network are destroyed duringmercury intrusion [see for example 31].
Another limitation to the interpretation of pore size data isrelated to the uncertainty in the value of the contact angle used inthe Washburn equation. Finally, because specimens are routinelydesiccated before mercury intrusion, mesopores will appear tocontribute more to the pore size distribution of the fine fraction thanis appropriate.
Taken together, these phenomena serve to complicate theinterpretation of experiments using mercury intrusion porosimetrywhich attempt to assess both the volumes and size distributions ofpores that significantly contribute to permeability.
48
Direct Observation of Porosity
There are two types of techniques used to directly observe theporosity in cement paste, mortar, and concretes. The first of these isthe direct observation of large pores on a polished section using
optical microscopy. The observation of air void spacings is oneexample of this. However, the observation of finer pores usuallyinvolves the intrusion of a monomer and its polymerization or theintrusion and hardening of an epoxy. Porosity can then be observed
in polished sections or in thin sections via optical microscopy [32, 33].The observation of porosity with an optical microscope is frequentlyenhanced by the use of ordinary or fluorescent dyes [34]. Thistechnique has also been applied to the observation of microcracks inmortar [35]. The observation of porosity by optical techniques islimited by the resolution of the optical microscope. Typically theseanalyses are carried out at low magnification, 40-400X.
Optical microscopic observations may also be carried onpolished sections from which material is systematically removed bycyclic polishing between observations. Analysis of the pore size datafrom the serial sections can then be used to assemble a montage ofthe three dimensional pore structure. A less tedious variant of the
technique of preparing serial sections is pore casting. Pore castinghas found application in the analysis of the pore systems in rocksusing both optical and electron microscopy [36, 37] but very limitedapplication in cements [38]. In this technique the pore structure isfilled with a plastic material and the inorganic matrix is dissolved toreveal the three-dimensional pore network. The advantage to pore
casting is that it is possible to analyze the pore structure as anetwork.
Electron microscopy has also been used to analyze the porosity
on polished sections. This is accomplished either by directobservation of the pores or by intruding the pores with a compoundcontaining an element readily detected by energy dispersive x-rayanalysis, such as a chlorine-containing polymer, followed by an arealcompositional analysis for regions high in chlorine. Typically, thiscan be done with a resolution of 3-5 v m.
Regardless of the specific technique selected, there are twomajor limitations of direct observation of pore structures. The f'Lrst isthe limitation on resolution when optical microscopy is involved. Thesecond limitation is that regarding the ability to intrude epoxy ormonomer into small pores. However, regardless of these limitations,direct observations of pore casts, polished sections, and thin sections
49
in concert with image processing have developed into a powerfultechnique for pore structure analysis [39-43].
Models for Pore Size Di_ribution_ in Cements
In spite of the variety of limitations, previously discussed,mercury intrusion has come to be regarded as a standard means bywhich the pore structures of cement pastes are examined. Thepressure at which mercury will flow into a pore is related to the sizeof that pore. Thus, by determining the volume of mercury intrudedas a function of intrusion pressure, plots as shown in Figure 3 [25]are obtained. While data of this type are useful in comparingrelative pore size distributions in various cement pastes, furtheranalysis is required to extract descriptors of these pore sizedistributions. Such descriptors can then be used in permeabilityexpressions.
The most common method of analyzing porosity data is todetermine an averaged pore size. This is the most straightforwardapproach. However, from the standpoint of those processes thataffect cement porosity this may not be the best approach. Rather,the nature of the pore size distributions need to be defined in termsthat can be treated quantitatively. This is most readily done bydescribing pore sizes using a distribution function. The general formof a function that may be used to describe a cumulative pore sizedistribution is:
fr °Q
P(R >r) = V(r)dr; P(O) = 1,[5]
where P(R>r) is the probability that a pore will have a radius largerthan r, i.e. the cumulative pore size distribution, V(r)dr is the volumefraction of pore space whose radii are between r and r + dr. Theprobability that a pore will have a radius larger than zero is unity.
Little appears to have been done in terms of developingdistribution functions describing the porosity of cement pastes. Theexception to this is an investigation by Diamond and Dolch [44]. Inthis investigation it was reported that the distribution in pore sizesin a hardened cement paste could be described by a log-normaldistribution function. This is a significant finding because the datawere obtained by mercury intrusion porosimetry. Diamond [45] alsomeasured the variations in pore size distributions with temperatureover a range from 6 to 400C. He observed pore size distributions tobe initially coarser in pastes cured at elevated temperatures but thatthe differences became negligible after about 1 month of curing. In
50
accord with earlier work, Diamond also observed that thedistribution in porosity could be treated as log-normal. Recently, Shiand coworkers [46, 47] have extended the analysis to MIP dataobtained at high intrusion pressures. Using robust statisticaltechniques Shi, et al have observed that is possible to describe thedistributions in pore size in both cement paste and mortar in termsof a linear combination of lognormal distributions. The equations
, used to describe distributions of porosity are of the form:
p(r) = Z fi p(r, _ti, ci), Z fi = 1.0 [6]
where p(r) is the probability density function, fi the weighing factorof the i tla lognormal sub-distribution, i.ti and _i the location parameterand shape parameter in the i th lognormal sub-distribution p(r, I.ti, ai),respectively. As hydration proceeds, the location parameters changesignificantly, while the relative weights of the sub-distributions andthe shape parameter change relatively little. The significance of thisfinding is that it is possible to describe the variation in porositydistribution as a function of water-to-cement ratio and as a functionof curing time in terms of the variation of parameters having directstatistical significance. These parameters can be incorporated, inturn, into pore structure-permeability relationships.
Techniques Used to Measur_ P_l'mcability
A variety of techniques have been developed for permeabilitymeasurement. A number of these are summarized in ref. [48]. Themost common technique is the measurement of the amount of waterthat can be forced through a specimen subjected to a large hydraulichead under conditions which approach steady state. Suchpermeability measurements suffer from several disadvantages. If asample is relatively impermeable, the use of a relatively small lengthis required. This raises the question of representativeness. Thesolubilities of hydrated cement phases will vary in response to thelocal hydrostatic pressure. Thus a pressure gradient through asample can lead to the redistribution of the more soluble phases,Ca(OH) 2 and gypsum, if present. Redistribution of solids can affectthe pore structure and, thus, the permeability. A third disadvantage,which is ubiquitous to essentially all permeability measurements, isthat specimens having low permeabilities are saturated only withgreat difficulty.
A second, relatively common method for measuringpermeability is the rapid chloride permeability method [49]. In thismethod, chloride ions migrate through a specimen under a potentialgradient. The advantage to the method is its relative rapidity.
51
Disadvantages are that there is resistive heating associated with thevoltage drop across the sample and that the chloride can rcac_ withaluminam phases and influence the pore structure.
One approach to eliminating the problems inherent in the moretraditional methods of permeability measurement is to minimize thepressure gradient across the sample. Such a method has beendeveloped to measure the permcabilities of rocks [50]. According tothe method, the pressure gradient across the sample is initially zero.A zero pressure gradient is attained by using two pressurizedreservoirs. In order to make a measurement, the pressure in one ofthe reservoirs is rapidly elevated or lowered and the rate of decay inthe pressure gradient across the sample is measured. The rate ofdecay can then be mathematically related to the permeability [50].While this method offers the advantages of rapidity in measurementat relatively low pressure gradients [51], obtaining fully saturated,low permeability samples remains a problem [52].
Models for the Pcrmeabiliti¢_ of t_¢rnent and (_oncret¢
Consideration of models developed to "describe the permeabilityof porous media will include empirical models, network models,probabilistic models and models based on percolation theory.
Empirical Models
In general, permeability is a measure of the ease with which afluid passes through a porous body [53]. The most commonexpression used to describe the permeability, k, is Darcy's Law:
k = -p.Q/[ A 8g(dh/dz) ] [7]
where Q is the volume of fluid discharged per unit time through thecross-sectional area A, I_ is the viscosity of the fluid, 8 is the densityof the fluid, g is the acceleration of gravity, dh/dz is the hydraulicgradient in the direction of flow, z.
Another simple model is that of Poiseuille [4]. Poiseuille's lawstates that the volume flow through a capillary tube of diameter r is
Q = -xr4/8ttdP/dl
where p. is the viscosity of the fluid and dP/dl is the pressuregradient causing flow along a tube of length I.
52
These simple models Can be combined to develop an expressionfor permeability. Assuming the porosity in a cross-section through amaterial with total porosity p to result from the intersections ofpores having different diameters ri, the term A in Darcy's law can beexpressed as S/p. S is the cross-sectional pore area. In this case, thetotal volume flow can be expressed in both Darcy's law and inPoiseuiUe's law as:
Q = -S/e k/gdP/dl (Darcy's law) [9]
Q = -Z_(ri)4/SgdP/dl (PoiseuiIlie's law) [10]
(The terms g and 8 only result in different units)
Combining both equations results in:
k = e/S_;_z(ri)4/8 = e(_(ri)2(ri)2/8)/S
= e(Y_Si(ri)2/8)/S = e <r2>/8 [11]
where Si is the cross-section area of pores of radius ri, <r2> the meansquared pore radius, or the second moment of the pore radiusdistribution. This model relates permeability to porosity and theaverage pore size by assuming that pores are tubes and do notinterconnect.
Another frequently encountered permeability model is theCarmen-Kozeny model. This model is sometimes referred to as the
hydraulic radius model since it assumes pore diameters to be 4 timesthe void volume of the medium divided by the pore surface area. Itrelates the permeability to total porosity and the specific surfacesarea of the pores. The mathematical form of the Carman-Kozenymodel is
k = e3/[ko (Le/L)2(1 - e)2S 2] [12]
where e is the total porosity, ko is the permeability of an infinitelydilute bed, Le is the average path length for flow, L is the path length(Le/L is often called the tortuosity), S is the specific surface area [52].
Archie's law model [55] relates porosity and permeability usinga power law. This model is expressed as
k **eta [13]
53
being the total porosity and m a constant. In the Archie modelfactors as connectivity and tortuosity, which are important topermeability of cementitious materials, are not considered.
Each of the above models relates permeability to the porosityin some fashion. It is of significant interest to relate thepermeabilities of cementitious materials to parameters that controlthe development of microstructure. As a consequence, a variety ofinvestigations have been carried out to elucidate possiblerelationships between pore structures and permeability incementitious materials. Auskern and Horn [56] compared the poresize distributions obtained by mercury intrusion porosimetry forpastes hydrated at water-to-cement ratios of 0.35 and 0.55. After90 days of hydration both the distribution in pore sizes and the totalporosities were similar for the pore diameters of 90 nm and below.This suggests that the variation in permeability with water-to-cement ratios is dominated by the larger porosity. Nyame andIllston [13] reported a linear relationship between the maximumradius of a continuous pore and the permeability of a paste. Mehtaand Manmohan [14] found that the permeability of pastes could becorrelated with the volume of pores having diameters greater than132 rim. Goto and Roy [16] observed that pores in the size rangefrom 75 to 750 nm have a major influence on paste permeabilities.However, it has also been established that the techniques used todetermine the pore structures of cementitious materials may affectthe pore structures themselves. For example, Hughes [15] confirmedthe observations of Feldman [31] and others that mercury intrusionsignificantly damages the pore structure of cement paste. Hughesalso reviewed the work of Marsh, which indicated the lack of anycorrelation between permeability and pore size distribution, anddeveloped a simple tube model to predict permeability.Unfortunately, a generic model having predictive capabilities doesnot seem to have emerged from these investigations and thecapabilities of the models described above appear to be limited. Atbest it appears that, while there not agreement regarding therelationships between porosity and permeability, the weight ofevidence suggests the larger capillary porosity to primarilycontribute to the permeability in cement paste. However, care mustbe taken in generalizing these observations to concrete.
P_rmeability Models
The following sections describe number of classes of modelswhich have not been applied to cementitious systems but that haveproven successful in predicting permeabilities of catalysts, _of
54
geological materials including rocks and soils, and of permeabilityanalogs in electrical systems.
Network Models
The Daxcy-Poiseuille model, the Archio model and the Carman-Kozeny model all attempt to relate permeability to some averageddescriptor of the porosity, or to the total porosity. They do notconsider the structures in pore network. In spite of this limitationthese models can, in some instances, qualitatively predict concretepermeability [see for example 57, 58]. However, the predictionsbased on these models are frequently considered to be inadequate[59]. This recognition occurred in the geosciences in the 1950's andled to the development of a "network model" to describepermeability [60-62]. Since then, network models describing poroussystems have been developed in the areas of geophysics [63],petroleum geology [64, 65], soils [4] and chemical engineering [65].However, none seems to have been specifically developed in the areaof cement and concrete.
Network models axe based on the analogy of Darcy's law forfluid flow to Ohm's law for current flow. Seeburger and Nur [59], forexample, investigated the effects of confining pressure onpermeability and bulk modulus in rocks using a network pore spacemodel as a tool. In their network model, the elements are capillarytubes of given length, cross section and shape. The change ofpermeability with hydrostatic confining pressure is caused by theeffect of the stress field on the shape and resulting flowcharacteristics of each element. In a model developed by Dullien[54], a network consists of many sub=networks of pores that have aspecific smallest entry pore size, Figure 4. Each sub-network istreated as a flow channel. The volume flow in each channel isdetermined by the mean squared diameter of pores in that channel.Total volume flow is the sum of that in all flow channels. DuUien'smodel [3, 54, 70] is written as foliows:
k= ¢<D2>/96 [14]
where <D2> is the mean square pore diameter and e theporosity. Thevalue of the constant in Eq. [14] differs from that in the classical
. Poiseuille model, Eq. [11], by a factor of 3. Dullien interpreted thisdifference as a tortuosity factor. It is doubtful, however, whetherthe tortuosity factor is a constant. A tortuosity factor is more likely avariable depending on microstructure. The mean square porediameter in the model is determined from a bivariate pore diameterdistribution proposed by Dullien [71], as will be discussed as follows.
55
Mercury intrusion porosimetry can only determine pore entrysizes. Quantitative image analysis can determine the "true" poresizes, although only in two dimensions. If the size distributioncurves determined by these methods overlap over the entire rangeof pore sizes, a bivariate pore size distribution can be expressed asp(D, De) where D is true pore diameter and De is pore entry diameter,Figure 5. Dullien developed a simple, graphical procedure toestimate the volume of pores having the true diameter D to D + dDand the entry diameter De to De + dDe. A bivariate distribution wasfound to be suitable to account for both distributions [3, 57, 71].From this bivariate distribution, Dullien claimed that the relationbetween pore entry diameter and pore diameter can be determined.It must be noted, however, that the bivariate distribution cannot beestablished unless two distributions overlap over the whole range ofpore sizes. In less porous materials, such as cement and concrete,usually there are many fine pores, which are not amenable toobservation by image analysis. In addition there are very largepores (voids) which can be difficult to measure by mercuryporosimetry. Thus, overlap in the whole range of porosity may notbe possible. It is also worth noting that the pore size distributiondetermined by image analysis represents the size distribution of theintersections of pores by a plane. Three-dimensional, or the truevolume distribution of pores must be estimated using stereologicalprocedures [72-77].
Probabilistic Models
Probabilistic models were first developed by Childs and Collis-George [67]. Subsequently, models have been developed by Marshall[68] and others [4, 69]. The basis for a probabilistic model is that aporous solid can be divided into two parts by a plane normal to theflow direction and that the two cross-sections can be rejoined. Ifpores are distributed at random and the term f(D)dD is theprobability that a pore has diameter within the range D to (D +dD),then the probability that pores with diameter Di to Di + dDi on cross-section i is connected to pores of size Dj to Dj + dDj on cross-section jis f(Di)f(Dj)dDidDj. If the interconnection between pores of the twocross-sections is assumed to be completely random, the permeabilitycoefficient can be expressed as:
k **e2j'j"D2f(Di)f(Dj)dDidDj [15]
where e is the porosity and D the smaller of Di and Dj. Theproportionality factor is 1/32 if circular capillary tubes are assumed[4]. Juang and Holtz [4] noticed that in reality, the connection
56
between pores of two cross-sections would not be completelyrandom. They added a factor called the "connection function" g(y, Di,D j), where y is the sample length, to control the connectivity andrealized that it accounts for tortuosity of flow. When y_ 0, it
, becomes k = e2<D2>/32, a form of the classical Poiseuille capillarymodel [4]. As Gueguen and Dienes pointed out [6], the tortuosityconcept is related to percolation concept, as will be discussed
Models Based on Percolation Theory
In 1957, Broadbent and Hammersley introduced the term"percolation theory" to describe critical phenomenon, or phasetransition behavior [78]. The basis for percolation theory can beconsidered in terms of the site occupancy of a lattice. A primaryassumption is that every site is either "occupied" or "empty" basedon an entirely random process which is independent of theoccupancy of neighboring sites. The probability that a site isoccupied is p and (I - p) is the probability that a site is empty.Occupied sites are either isolated from each other or adjacent toneighboring sites to form clusters. If the site occupancy is near zero,most occupied sites will be isolated. If, °n the other hand, the siteoccupancy is close to unity, then almost all occupied sites will beconnected to form a large cluster, which extends from one side of thelattice to the other. This large cluster represents an "infinite-path."It percolates through the lattice in the similar way that fluidpercolates through porous materials along the network of connectedpores. Increasing the site occupancy from zero to unity results in acritical point (percolation threshold), Pc, above which an "infinite-path" can form. The occurrence of an infinite path is illustrated inFigure 6 for a simple model of fluid flow through a porous medium.
The percolation described is called "site percolation" [79-83].There is a counterpart called "bond percolation" [79-83]. In thisinstance, each site on the lattice is occupied, and interconnectionsbetween neighboring sites are regarded as bonds. In this instance, pis the probability that a bond is "open" and (l-p) is the probabilitythat a bond is "closed." A cluster is a group of neighboring sitesconnected by open bonds. The percolation threshold, Pc, is thefraction of open bonds, below which an 'infinite-path' cannot form.Fluid flowing through porous materials can be modeled in terms ofits passage through a network of interconnected pores.
Intermediate between site percolation and bond percolationthere is a type of percolation called "site-bond percolation" [79, 80,84]. The lattice sites are no longer all occupied as in bondpercolation. In site-bond percolation, p is the probability that there
57
is a bond between neighboring sites. A cluster is then a group ofneighboring occupied sites connected by bonds. The bondpercolation threshold decreases from unity, when the portion ofoccupied sites equals the site percolation threshold, to the normalbond percolation threshold, when the fraction of occupied sites isunity. This type of percolation appears to most closely represent thenature of fluid flow through porous materials. The fraction ofoccupied sites is equivalent to porosity, the bonds represent thechannels connecting pores, and a cluster represents "infinite-path"pores. The channels may consist of pore entries and cracks.
In the percolation theory, P is defined as the strength ofinfinite network. The strength, P, is the percolation probability thata site belongs to the infinite-path. It is actually the fraction of totalnumber of lattice sites that are on an infinite network [79]. A powerlaw expression may be written:
P ** (p - pc) t [16]
if p approaches Pc from above where p is the concentration ofoccupied site or bonds, and Pc is the critical concentration. Transportproperties are proportional to (p - pc) cx. ct is the transport exponent.However, t usually differs from a [85] because P counts for bothbackbone and dead ends. Dead ends contribute to the mass of theinfinite network but not to transport properties. However, dead endsentrap fluid.
From the standpoint of flow in porous materials, an infinitenetwork consist of two parts: backbones and dead ends. Fluid flowsthrough a porous materials along the backbones and are entrappedin dead ends. Permeability measurement usually only deals with theflow-through fluid; however, entrapped fluid can significantly affectproperties, such as freeze-thaw resistance of concrete. Therefore, itis necessary to consider phenomena associated with flow in bothbackbones and dead ends. Computer simulation can be used todifferentiate dead ends from backbones and to calculate fractions ofbackbone and dead ends with respect to the infinite network andnumerical studies have been made to quantify backbones [86].
Several of the models previously described above can beconsidered in terms of percolation concepts. Both percolation theoryand Archie's law use a power law to describe the criticalphenomenon. Thus, Archie's law may be regarded as a special casein which the percolation threshold is zero [87]. For dense yetpermeable materials, the percolation threshold can approach-a verylow value. For example, rock salt has porosity of only 0.6% but a
58
permeability of 7.3 x I0-6 Darcy [88]). Dullien observed thepermeability of sandstone to mercury to depend on the degree ofpore prefilling by mercury. That is pores of a given size that werepreviously filled with mercury contributed to permeability while
. those that were not filled did not contribute to bulk flow. Dullienconcluded that at a low mercury saturation there must be continuousflow channels in the medium. At low saturation, only the largestentry pores are penetrated by mercury. In other words, these flowchannels consist only of pores with large entries that can be intrudedby the mercury. However, if the saturation is reduced to a very lowlevel, no permeability is recorded [3]. This low saturation must berelated to percolation threshold. Thus, Dullien's model as well asJuang's statistical model can be regarded as transitional modelsbetween non-percolation-based models and percolation-basedmodels.
Gueguen and Dienes [6, 89, 90] used statistics in combinationwith percolation theory to correlate pore and crack structures withpermeability. Their models assume narrow size distributions andcan be expressed as:
k = fE<r>2/32, for pore structure and[17]
k = 2fe<w>2/15, for crack structure,
f is the fraction of connected pores and is proportional to (p - pc) a,<r> is the mean pore radius and <w> is one half the mean crackaperture, and ¢ is the porosity. Unfortunately, the assumption ofnarrow size distribution is not valid for cementitious materials.However, the model explicitly includes f, the fraction of connectedpores and thereby provides a means to quantify the tortuosity factor[6]. Additionally, their model provides insight to crack-permeabilityrelationships; in cementitious materials pores and cracks are notdistinct and the effect of cracks on permeability must be considered.
Katz and Thompson [5, 91, 92] developed a percolation modelusing an approach based on the work carried out by Ambegaokar,Halperin and Langer (AHL) [93] on electron hopping in amorphoussemiconductors. They showed that transport in a random systemwith a broad distribution of conductances is dominated by those
: conductances with magnitudes greater than some characteristic valuego. gc is the largest conductance such that the set of conductances
; {g:g > go) forms an infinite network. Transport in such a systemreduces to a percolation problem with threshold ge- Kirk-patrick [82]and Shante [94] extended these ideas and assigned the value ge to alllocal conductances with values g >-gc and set all conductances with
59
values g < gc to zero. They arrived at a trial solution for the sampleconductance of the form:
g = agc[p(gc)-Pc]a, [18]
where P(gc) denotesthe probabilitythata given conductanceisgreaterthan or equalto gc, and a is a constant.Pc is the criticalconcentration of conductance, below which no infinite cluster canform. The transport exponent, cc is approximately 1.9 [95, 96]. Thismodel is similar to a site-bond percolation model. Katz andThompson [5, 92] applied these concepts in the development of amodel describing the permeabilities of porous rock. They definedthreshold conductance value gc as function of the characteristiclength 1c" gc oo lc. Assuming cylindrical pores, the characteristiclength is the pore diameter. Then the hydraulic conductance of thesample is the function of the length parameter 1:
g(1) - Egc(1)[p(1) - pc]ct [19]
where e is the porosity. As 1 is lowered past the threshold 1c thefunction g(1) also decreases but the power-law term [p(1) pc] 0_increases as more and more pores are included in the infinitenetwork. Mathematically, this means that there is a maximum valuefor 1 that places a lower limit on g(l). Physically, because onlyconductances with characteristic lengths I >_.Ic are included in thecalculation, the sample conductance, g(lmax), must be always lessthan or equal to real sample conductance. In other words, g(lmax)approaches the true conductance as closely as possible. Aftermaximizing, the model is expressed as follows:
k = (1/226)1c a/ao, [20]
where 1c is the threshold characteristic length, which is inflectionpoint on the cumulative pore size distribution determined bymercury porosimetry. _ is the electric conductivity of the rock, andao is the conductivity of the pore solution. They also proposed thatelectrical conductivity can be determined directly from mercuryinjection [91, 92]. The result is a pore structure-permeability modelwhich can predict permeability simply from mercury intrusion dataas follows:
k ** (lmax)2(lmax/lc)ef(lmax), [21]
where f(lmax) is the volume fraction of pores intruded by mercury.lmax is the maxima of 1. The advantage to this percolation-basedmodel is that the use of percolation factors that are difficult to
60
determine, such as those used in Gueguea's model (f, p, and pc), isavoided. The disadvantage is that the model is based on themaximizing a lower bound [97]. As a consequence, the calculatedpermeability is typically less than the measured value. Anotherdisadvantage to this model is that mercury porosimetry may not beable to describe crack structures correctly, although crack structuremay be as important as pore structure in terms of their effects on
' permeability. Garboczi has recently applied Katz and Thompsonmodel to cement and obtained encouraging results [968], howeverpermeability and pore size distribution data have not yet beenobtained on the same samples.
61
tmmaz.
A variety of models have been developed to predictpermeability from pore structure. A common feature of thesemodels is that a power law may exist between permeability andsome characteristic pore dimension. The tortuosity factor has beenincluded explicitly in J'uang's model, Dullien's model (though it isassumed as a constant) and all percolation-based models (in the formof percolation factors).
Each model has its rational base. Statistical models consider allpores, whereas the Katz-Thompson model only considers large pores.Dullien's work provides the the basis for relating entry pore volumeand true pore volume. Gueguen and Dienes have considered crack-permeability relationship and explicitly used f, the fraction ofconnected pores in their-models. Alternatively, Katz and Thompsonavoid calculating f, which is difficult. Rather, they predictpermeability from mercury porosimetry data. Even the traditionalCarmen-Kozeny model has its advantages. This pore structure-permeability model requires as input is a pore size distribution.Although a pore size distribution is more" easily determined thanpermeability per se, it usually assumes shape for pores. TheCarman-Kozeny model uses the specific surface area of pores asinput. This does not require any shape assumption and can,therefore, avoid a source of error.
The determination of permeability is of obvious importance ina variety of disciplines. Investigators in these various disciplineshave developed both methodologies for the determination ofpermeability and models to describe their results relativelyindependently. It has been the objective of this review to brieflydescribe measurement methods and models, along with theiradvantages and limitations, that may have relevance to cementitioussystems.
62
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49. D. Whiting, "Rapid Determination of the Chloride Permeability ofConcrete," FHWA Rept. RD-81/l19, Federal HighwayAdministration, Washington DC (1981).
50. W.F. Brace, J.B. Walsh, and W.J. Frangos, "Permeability of GraniteUnder High Pressure," J. Geophys. Res. 73, 2225-36 (1968).
51. R.D. Hooton and L.D. Wakely, "Influence of Test Conditions onWater Permeability of Concrete in a Triaxial Cell," in poreStructure and Permeability 0f Cementitigu8 Materials. 157-64L.R. Roberts and J.P. Skalny, Eds., MRS (1989).
52. B. Scheetz, private communication (1990).
6(3
53. S. Davis, "Porosity and Permeability of Natural Materials," Ch. 2in Flow Through Porous Media, R.J.M. De Weist, Ed., AcademicPress, NY (19(_9).
54. F.A.L. Dullien "Single Phase Flow Through Porous Media andPore Structure," Chem. Eng. J., 10. 1-34 (1975).
55. G.E. Archie, "The electrical resistivity log as an aid indetermining some reservoir characteristics," AIME Trans., 146,54 (1942).
56. A. Auskern, and W. Horn, "Capillary Porosity in HardenedCement Paste," JTEVA 1(1), 74-79 (1973).
57. J. Vuorinin, "Applications of Diffusion Theory to PermeabilityTests on Concrete Part I: Depth of Water Penetration intoConcrete and Coefficient of Permeability," Mag. Contr. Res.37(132), 145-52 (1985).
58. J. Vuorinin, "Applications of Diffusion Theory to PermeabilityTests on Concrete Part II: Pressure-Saturation Test on Concreteand Coefficient of Permeability," Mag. Contr. Res. 37(132), 153-61 (1985).
59. D.A. Seeburger and J. Nur, "A Pore Space Model for RbekPermeability and Bulk Modulus," Geophys. Res. 89(B1), 527-36(1981).
60. I. Fatt, "The Network Model of Porous Media I. CapillaryPressure Characteristics," Pet. Trans AIME 207, 144-59 (1956).
61. I. Fatt, "The Network of Porous Media II. Dynamic Properties ofa Single Size Tube Network," Pet. Trans AIME 207, 160-63(1956).
62. I. Fatt, "The Network of Porous Media III. Dynamic Properties ofNetworks with Tube Radius Distribution," Pet. Trans. AIME 207.164-81 (1956).
63. M. Rink and J.R. Schopper, "Computations of Network Models ofPorous Media," Geophys. Prospect. 15. 262 (1967).
64. R.W. Ostensten, "Microcrack Permeability in Tight GasSandstone," SPEJ, 919-27 (1983).
67
65. J.H. Thomeer, "Air Permeability as a Function of the Three Pore-Network Parameters," JPT, 809-14 (1983).
66. A.C. Payatakes, C. Tein, and R.M. Turian, "A New Model forGranular Porous Media: Part 1. Model Formulation," AIChE J.19(1) 58-67 (1973).
67. E.C. Childs and N. Collis-George, "The permeability of porousmaterials," Proc. Royal Soc., London, 201A, 392 (1950).
68. T.J. Marshall, "A relation between permeability and sizedistribution of pores," J. Soil Sci., 9, 1 (1958).
69. R.J. Millington and J.P. Quirk, "Permeability of porous media,"Nature, 183, 387 (1959).
70. F.A.L. Dullien, "Prediction of tortuosity factors from porestructure data," AIChE. J. 21, 820 (1975).
71. F.A.L. Dullien, "Bivariate pore-size distributions of somesandstones," J. Colloid and Interface Sci., 52, 129 (1975).
72. D. Shi, "Discussion on estimation of 3-D properties from 2-Dmeasurements without shape assumption", in Advances inC_men_ Manufacture and Use, E. Oartner, Ed., EngineeringFoundation, New York (1989).
73. D. Shi, Ph.D. Thesis, Purdue University (1987).
74. D. Shi, "A probabilistic approach to evaluate distribution ofparticles/pores without shape assumption," J. Microscopy,submitted.
75. E.E. Underwood, Quantitative Stereology, 1-48, Addison-Wesley,Reading, Massachusetts (1970).
76. DeHoff, R.T., The statistical background of quantitativemetallography, in Quantitativ_ Microscopy. R.T. DeHoff and F.N.Rhines, Eds., McGraw-Hill, New York (1968).
77. E.E. Underwood, "Surface area and length in volume," inQuantitative Microscopy. Ed. R.T. Dehoff and F.N. Rhines, Eds.,McGraw-Hill, New York (1968).
78. J.M. Ziman, Model_ of Disorder, 370-386, Cambridge UniversityPress, Cambridge (1979).
68
79. D. Stauffer, Introduction to. Percolation Theory, Ch. 2, Taylor &Francis, London (1985).
80. D. Stauffer, Scaling Theory of Percolation Clusters, Ch. 1, North-Holland, Amsterdam (1974).
, 81. R. Zallen, The Physics of Amorphous Solid_, Ch.4, John Wiley,New York (1983).
82. S. Kirkpatrick, in Ill-Condensed Matter, R. Balian, et al. Eds., Ch.5, North-Hollan, New York (1979).
83. R. Zallen, in Fluctuation Phenomena, Ed. Montroll, E.W. andLebowitz, J.L., Ch. 3, Horth-Holland, Amsterdam (1987).
84. D. Stauffer, "Percolation and cluster size distribution," in O nGrowth and Form, H.E. Stanley and N. Ostrowsky, Eds., MartinusNijhoff, Boston, (1986).
85. D. Stauffer, Introduction to Percolation Theory_, Ch. 5, Taylor &Francis, London (1985).
86. S.S.Manna, "Structure of backbone perimeters of percolationclusters," J. Phys. A: Math. Gen. 22, 433 (1989).
87. I. Balberg, "Exclude-volume explanation of Archie's law," Phys.Rev. B, _3, 3618 (1986).
88. S.N. Davis, in Flow through Porous Media, R.J.M. De Wiest, Ed.,v
Ch.2, Academic Press, New York (1969).
89. J.K. Dienes, in Is_o¢_ in Rock Mechanics, R.E. Goodman and andF.E. Heuze, Eds., Ch. 9, Ame. Inst. Mining, Metall. & Pet. Engr.,New York (1982).
90. Y. Gueguen, et al., "Models and time constants for permeabilityevolution," Geophys. Res. Lett., 13, 460 (1986).
91. A.J. Katz and A.H. Thompson, "Prediction of rock electricalconductivity from mercury injection measurements," J. GeophysRes., 92, 599 (1987).
92. A.H. Thompson et al., "The microgeometry and transportproperties of sedimentary rock," Adv. in Phys., 36, 625, (1987_).
69
93. Ambegaokar, V. et al., "Hopping conductivity in disorderedsystems," Phys. Rev. B, 4_, 2612, (1971).
94. V.K.S Shant, "Hopping conduction in quasi-one-dimentionaldisordered compounds," Phys. Rev. B, 2597 (1977).
95. R. Fisch and A.B. Harris, "Critical behavior of random resistornetworks near the percolation threshold," Phys. Rev. B, 1_, 416(1978).
96. D. Berman, et al., "Conductances of filled two-dimensionalnetworks," Phys. Rev. B, 33, 4301, (1986).
97. Le Doussal, "Permeability versus conductivity for porous mediawith wide distribution of pore sizes," P., Phys. Rev. B, 39, 4816(1989).
98. E.J. Garboczi and D.P. Bentz, "Analytical and Numerical Models ofTransport in Porous Cementitious Materials," 675-81 inScientific Ba_i_ for Nuclear Waste Management XIII, V. Oversbyand P.W. Brown, Eds. (1990).
LIST OF FIGURE CAPTIONS
Figure 1. The relationship between capillary porosity and compressivestrength [2].
Figure 2. The dependence of permeability on the capillary porosity [17].
Figure 3. Cumulative pore volumes intruded depending on age for a TypeI portland cement paste hydrated at 24C at a water-to-cement ratio of 0.4[25].
Figure 4. An illustration of one element in a pore network in which poresizes may vary [54].
Figure 5. A comparison of the distribution in the diameters of pore entriesand pore sizes [73].
Figure 6. An illustration showing the formation of an infinite path. Thisillustration represents, in two-dimensions, flow paths across a bulk samplein which a pore structure is defined by the application of percolationtheory. The hydraulic gradient is from left to right. _"
70
28
24 1_
0 I I1O0 80 60 40 20 0
Capillary porosity (%)
71
100 f90
"to 80 ,, ,!
7o" O0 "
50 /40 , /
•_ 20 ,_,
0U
0 10 20 30 40
Capillary Porosity-per cent.
72
I I iI I 1I I Ii I II I I
i II ' I i
T ,---I , 'ii I ' I I ' I
w I
I I II I I
I CAPILLARY ELEMENT I I, I
OF NETWORK
73
1400 " G O PORE ENTRY DIAMETER
_ [ I! DISTRIBUTION (De)m / J_ &. -=. COMPLETE PORE SIZE
1200r. _ DISTRIBUTION (D'
'?f'tt" i "
0 20 40 60 80 100 120
PORE DIAMETER, D OR De (IJn'l)
74
_ ..... _ °_oO o_o o_Od oo ..o _.° o • • ., _o o_o _ .... o._. • _o° • •
, o o oQo o °, • .° • _ °° • _° • • o. ....
_ _!i!___° iJl_i!_!_!_i_i_ii_!ili_iilli!_i_iii_iiiiii_i_iiii_ooo.
_!_.°.._!_._.i_i_._i_._ ._i._!_'_.._._.!i_'i_ i_"_ii_i_ _
• : _i _i_!_ _ _i'°_i_'" _ .! _i!_i°_._.i_.._ o_!.:_i_.i_i_i_
liiii,i i iiiii iiii! t!ii !i i!i iii!i_°_ °o,_ .oo° • Oo. • °o _'._ _'_
75
RELATIONSHIPS BETWEEN PERMEABILITY, POROSITY,
DIFFUSION AND MICROSTRUCTURE OF CEMENT PASTES,
MORTAR, AND CONCRETE AT DIFFERENT TEMPERATURES
DELLA M. ROY, MaterialsResearchLaboratory,The Pennsylvania State University,UniversityPark,Pennsylvania16802.
ABSTRACT
Permeabilitiestowateranddiffusionofionicspeciesincementitiousgrouts,pastesandmortarsareimportantkeystoconcretedurability.InvestigationshavebeenmadeofnumerousmaterialscontainingDonlandandblendedcements,andthosewithfine-grainedfdler,atroomtemperatureandafterprolongedcuring at severalelevatedtemperaturesup to 90"C. Theseconstitutepartof studiesof fundamentalmaterialrelationshipsperformedin ordertoaddressthequestionof long-termdurability. In general,thepermeabilit_esof thematerialshavebeenfound to be low [many <10"8 Darcy (10"13 m.s'l)l after curing for 28 days Orlongerattemperaturesupto60"C. Theresultsobtainedat90"Caresomewhatmorecomplex.In somesetsof studiesof blendedcementpasteswith w/cvarying from 0.30 to 0.60 andcuredattemperaturesupto90"Cthemoreopen-poresu'ucture(at theelevatedtemperatureandhigherw/c) as evident from SEM microstructuralstudiesas well as mercuryporosimetryaregenerallycorrelatedalsowith a higherpermeabilityto liquid. The degreeof bondingandpermeabilityevidentinpasteormortar/rockinterfacialstudiespresentsomewhatmoreconflictingresults.Thebondstrength(tensilemode)hasbeenshowntobeimprovedinsomematerialswithincreasedtemperature.The resultsofpermeabilitystudiesofpaste/rockcouplesshowexampleswithsimilarlowpermeabilities,andsomewithincreasedpermeabilitywithtemperature.
Ionicdiffusionstudiesalsobringimportantbearingtounderstandingtheeffectofporesu'ucture.Thebestinterrelationshipsbetweenchloridediffusionandporestructureappeartorelatediffusionratetomedianporesize.SimilarresultswerefoundwiththeAASHTO"chloridepermeability"test.
INTRODUCTION
The_ ofhaxdenedcementpastetochemicalattackandphysicaldegradzdionis
theresult_l_,ed composition andmicrostructuralfactorswith environmentalexposureconditions.t'-t,:j Farm's conwalling the initial ,,n_'ostructure developmentarethereforestronglyrelatedto eventtutlconcretedurability.t";J
Tim tesismn_ of cementirlonsmaterials to chemical attackandphysicaldegradationisonlyindite_yrelatedtothemechanicalsu'engthof thematerial,i.e.,strongmaterialsdonotalwtysresistattackanddisintegrativeforcesinanyenvironmentinwhichtheyareplaced.Nevertheless_itisu'uethathigherstrengthconcretesgenerallyhavelowpet,meabillties,andtherebywillprovidebenerresistancetochemicalattackandphysicaldegradationthanIow-su'engthmaterials.Thewatercontent(andw/cratio)ofacementpasteisprobablythemostimportantsinglefactorindeterminingthesubsequentporosityandpermeabilityofthe
" hardenedpaste,anddecreasingthewatercontentboth decreasespermeabilityandporosity,andusuallyincreasesthe strength.
Theporosityandporesnmcnn'eofcementpastesexertmajorconn'olontheingressofpotentiallydeleterioussubstances.Theporestructureandmicrostrucmmconstitutemajorrate-controilingfactorsfor potentiallyharmfultransportprocesses, whether thes_s beingtran_ iswater(togiverisetoaficali-siiicaexpansion),sulfateions(causingsulfateattack),chlorideions(todeR._ivatetheprotectivelayeronreinforcingsteel),acids,oroxygenorcarbondioxidettZl.
76
Thefactorscontrollingmicrostructureandporestructurehavenotbeenfullydiscernedfrompastknowledge.E.g..certainsupplementarycementingmaterialsorpozzolanas"promotedevelopmentofafineporestructure_ resultinlowerp_,,e,ability(anddiffusivityofpotentiallydeleteriousspecies)farbeyondthatwhichwouldbepredictedonthebasisofthewater/cementirioussolidsratioalone.[12"14]
Thediffusionofionsandtherateofpermeationoffluidsacrossaporousccmendtiousmatrixare,_uitively.relatedtothevolumeofporesandthesizeandinterconnectivityofthese pores.tOJThe basisfor such a statementlies in the fact thatif these two factorswere notrelatedto thepores in a porousmatrixbutto the solidpart,theobserved rates of diffusionof_;10"i2t_/._c a!_ temperaturein porous cen'_ntitious.materialswould instead resemblethatof aton'ncdiffuslon m polycrystallineceramicsof -I0 "'_m'/sec or lower. Thepermeabi_tyofwaterthroughwatersaturatedcementitiousmaterialshasbeenshowntoberelatedtotheporosityandporesizedistribution.[1.21Inthecaseofionicdiffusioninwater-sann'atedporouscementitiousmaterials,thediffusioncoefficientsformoderately-sized,singly-chargedcationsgenerallyrangebetweenI0_IIandI0"_3m2/sec[3"61whilethediffusioncoefficientsfortheseionsinpurewateris-I0""m'_/sec,tlj
Theobviousquestionthusarisesastowhythediffusionisretardedthroughthematrix.Oneofthefactorscontributingtothedeceaseisthesmallerareathroughwhichtheionsmaymigr_ in a porousmauix viz. the areaof the pores. This areafactormay explain a two- orthn_-foldclane.aseinthediffusioncoefficientofanionacrossaporousrnau'ixascomparedtothatinthepu_ aqueousphase.Itwaspointedoutearlierthattheobservc_lreductionsaregenerallymuchl_g_p',d_pntwo-orthree-fold,inthevicinityofI0_toI0'*times.ConsideringrecentevMuadonsr'.J,'-'Jthep_bable re&sonforsuchalargeretardationarisesfromtheconsu'ictionandtormosityoftheporepaththroughwhichanionmigratesinawater-saturatedporousroan'ix;Theion-wallinteractionleadstoasubstantialdecreaseindiffusion.Recentpapershaveelucidatedthispoint.[I0.III
Th=_ aretwostagesinthepr_atadonofcementitiousmaterialswhichshouldbe_y con=oiledfortheyarecriticalindevelopingtheultimateresistanceofsuchmaterialtoforcesofdegradation:I)thefreshstate,duringmixing,placingandconsolidation;and2)thecuringstage.Beyondthechemistryofthecemenddousmaterials,factorswhichinfluenceearlystage theological propertiescan significantlyaffectthe physical propertiesof thehardenedpasB which conuol their ultimatedurd_ty. Poor mixing, i_,_,_yate di_on,bleedingsegregationandrelatedphenotta_ cangive riseto inhomogencidesinthehat,nedpastemic_=ucmre, which can _ pathwaysforrapidtransportof harmfulspecies.Them am _ _ limiuforwwm reductimwhichpem_ adequatewodmbility.
md sula__ we ccmm_y used to improvefluidity, buttheiruse mayalso gemate effem as yet. pcxx'lyundmmcaL
A combimdon of chemical and physical facu_ areinfluendMinthecuring stage: e.g.,ch_..l_g thechemistry,as by pan_ sulzt_ution of Ign:mlan_ granulatedblastfunug¢ slag
or :_lim fume for cemmt can preventexcessive heat¢voludon which would remit iamicm-cracHng. This is above and beyond theireffect in generat_g a finer pore mucture.The chemical, physical, andmk:mstrucn_ pmperuesof hardenedcement pastes, therefore,arecriticallyinm_elatedwith respectto theireffectson degr_n.
Poro_rv andPore StructureofCement Pastes
• A wide varietyof techniques have been usedfor thed_adon of thepore sizedisuibudon andpore charact_sdcsofpordandcementpastes. Althoughmm_-myintrusionpomsimmry(MIP)hassomelimitations"ithasbeengenerallyrecognizedthatthepore_, w.hjc_.i___n_Asure.s,isrelated to the same factorswhichconlxolpm_ne.ationoffluids 10115. t l'_'°'|W'_ t i ¢• ,q_l" Gas adsorpdon methods describethe finer pore sizes of gel -pon_t_'J butthesearenotbelievedtocontributesignificandytotheu'anSlX_rateof thevariousspecies.The resultsof..MIP._nts report.edherehavebeenobtainedrlwistlhfreeze-d_jsamplestreatedto numm_ damageto thespocu'nensbe_'oremeasm-.,m_t.L'o_
77
Sometypical resultsof mercuryintrusionporosJmea-ystudiesare shown in figures I and2. Plotswere rm_ of va_rlousexperimen_ panmemrs, primarily: a) in_rude,d volume(expressedascm_/cm_ providing a total porosity) asa functionof porev_dius,and b) dV/cLPasa functionof poreradius. Surfaceareameasurementswerealso made. One of the mostusefulparametershasprovedtobe thedV/dPvs._.pgr¢,,_,diusplot.fromwhichthe"critical
' poreradius" or the "maximum continuousporeradius t_)could bedetermined. Since the poreradiusandinmasionpressurearerelatedreciprocally, the functiondV/dP may be usedinsteadof a direct functionof the poreradius. The radius at which the maximum in theplotoccursrepresentsthegroupingof the largest fractionof interconnectedpores,.w.htchin effect,
conn'olsthe n'ansmissivityof the material. The 'median'pore radiust to,t_slis also,similar.The critical pore radiusis related to, but not identical to Diamond's[20| 'thresholdradius, thelaner occurringat a larger radius.
The aboveplotscontrastthefinerporesn'uctureofacementpastecon_ninggranulatedblast-furnaceslagwithapurepordandcementpaste(thecriticalporeradiusoftheformerisabout 1/3of the latter). Figure 3 conn'aststhecritical pore radii of a seriesof cementpastesofdifferent w/c curedat different temperatures.The critical pore r_Liusincreasesby a ten-foldfactorwhen increasingthewit ratio of the slagcontainingmaterialsfrom 0.3 to 0.6, while thetotal porosityincreasesby abouta factorof two (2) in similar materials. Theseresultsandsh-'nilarresultsfrom other workclearly indicatethat there is aneffecton the porestructurefarbeyondthat anticipated from the increased porosityalone. Furthermore,the effect differs with.cementcomposition and type. Figure 4 gives comparisonsof mecumulative pore vommesofportland and portland-slag mixtures.
A second r_t_scntationof pore size diswibution dam, the median pore size is the poreradius at which 50% of the pore volume is observed in the pore size range considered. Inmost c_mentitious materials rhe critical po_ radius and the median pore radius have similarvalues. Pml:x_es of different cen'_ntitious compositions may be conu'asted by comparing theporosity, median pore radius or the critical pore radius.
The mercury porosimetrymeasu_ment results for cementpastes,andthose blendedwithflyash,u'4nulatcdslag,silicafumeandEriequa='=(min-u-siDshowsignificantdtt'ferences.Aseriesofsampleswerecuredata tempe_nn_of38"CtheresultsofwhicharegiveninTableI.
Re!atiy.eJy.cgmpar,,blevaluestothoselisted in TableIwereobtainedwithspecimenscuredat27 C.tte, =aj Except for the pure portland cement, and mixtures with fine quartz, a steadydec'tease in medi:_n pore sizeis shownwithprolongedcuringtime.
In general,the additionof mineraladmixturesto portlandcementleadsto a decreaseinpore sizeor an increasein the fraction of porosityin the finer porerangeof 15nm or less.(FigA.lThe porosity may increasethrough this incorporationas in silica fume and fly ashcasesbuttheincreaseinporo=tyis only_hthecluant/.tyo.ffreerpores.The _'¢sultsan=inkeepingwiththosefoundpreviouslyforsimilarsysteras.[lalThe effectoftheincreaseinfinerporecontentindiz_dy leads to an appat_nt increase in the tortuosity or interconnectivit'yof the pores.
When porosityandmean poreradius,determinedby MIP,areexpressedasa functionofwater/cementitioussolidvolurn¢ratio(w/s),aninterestingrelationshipisshown.Figure5givesporositydataforcementitiousmixturesasa functionofw/s(vol).A familyofcurvesisshown,whichisdifferentforeachcementitiousmaterial.However,when themean poreradiusisplotted._ a.functionofw/s,(Figure6),r,he valuesfallon a singlelinearplot.decreasing with dimiaishing wls.[21]
Diffusion and Permeability
As a key to the factorsinfluencing the rateof penetntion of chloride ion intohardenedcemendrlousmaterials,and their consequenteffects, the diffusion of C1" ionsthroughhatdenedcementpasteshasbeenstudiedbya numberof investigators.[4,5] The diffusionofCI" ionsis stronglyinfluenc___ by the typeof cement,and blendedcement pasteswith 30% flvashor 65% slaggave lower diffusion thanportland cement pastes.[5] The durability of "blendedcement,=to highly ooncenlxatedchloride solution(27.5% C.aCI.,4.3.9% MgCI-, .,-1.8% NaC1)is another matter, as was reportedby Feldman.[221He als_ found that bl¢'nds
78
I I I _ ] ] l i J I
0,7CRITICAl. PORE RAOItJ$
aGW/C * 0,40i4 DAYS
-'-- a= 4S'C
OIFIr[RIrNTIAL
"o
O I 1 r I i I I r , [I I I i I I i z I | (
(2J _a
2_Q _25 7'3 SO 40 30 ZS ?.0
POR[ RADIUS (ANGSTROMS ; 140* WETTING ANGLE)
Fig. 1. clV/aP plots of comparable portland cement and slagcement pastes (60:40).
t l I I [ _ t [ I I
CUMULATIVEPORE VOI.UME
020,P" W/C • 0.40
0.1,/ ,: 0AYS CEMIrNT/SLAG..,/
F .?
0 '[,_ I _ I 1 I I i I I I
:) CI[MI[NT --..I
o>0.o-
0.10 _
°-;- / •0 ;..,,,¢¢ r l _ ; I I I I _ t
ZSQ 12S 7_ 60 _0 40 3S _ 2S ZO IS
I=OR¢"RADIUS111
Fig. 2. Cumulative Ix)re volume versus pore radius ofportland cement and slag cement (60:40) pastes.
7g
SLAG/CEMENT
/
_: /
'- ; / / ,_ ,, ....oJO0
/_"..." o z. II /#'=r..' o 4_" // '/"_....' ° _o- t/._
,o. o.,o,_ o..o, o,WIC RATIO
Fig. 3 Critical pore radius (dV/dP maximum) versusw/c ratio of slag cement pastes cured at differenttemperatures.
0,_ l I | /l /(
___ 14 OAYS _L Ze DAYS'_ 0.3
O CEMENT 0"° t2 oz o.....o%,,.,,._-o• o_,,,_,-_ /
o,- o .I/ _ _.._..,__.._t.
SLA(
W 0.04 • r W,'_-- ! 0 o.e
_ f oo.,.
AO.3 •• 002 CEMENT SLAG
(.1
0.01 I I I .¢," t l t I //" )2T 45 60 90 27 45 60 90
:_ CURING TEMPERATURE('_)
Fig. 4 Cumulative pore volume of portland cementcompared with slag cement pastes.
8O
I I I Iz¢
w/l?O.6vCl o ¢IFA (8012_I4 e C/CA (70/.I0)I wls, "¢IFA (60140)
• C/S 1351651 ,,
_, 0.4 ii//' (
Q2 [
0.I I , I , I I'2.0 1.7S LS 1,25 t.O 0.?5
wol'er/solid (voi)
Fig. 5 Porosily as a function of w/s volume ratio;abbre-(Hg porosimetry). C=cement,FA=fly ash, SF=silica fume, S:,granulatedblast-furnace slag.
i i i i
%\
- %,_\ '| -\
1 \,. \
" io- \ -_- \
,%I I I !
2.0 1.75 LS 1.25 1.0waterlsolld ratio (voi)
Fig. 6 Mean pore radiusvs, w/s (vol.) ratio. .,
£
with 35% fly ash or 70% slag had a finer pore size disu-iburion than Type 129nland cement
and a Iowa"Ca(OH) 2 content, which improved their durability. Page et.al.t_J rcponedchlo_'j'_lediff_idvit7fromsodiumchloridewhichfollowedFickslawwhileKumar andRoytOJshowedthecomparisongiveninTable2 fordifferenttypesofcementpastecomparedwithpordandcement(OPC),when CsClwas usedasa d/ffusam.
81
Table1. PorosityaadMed_ Po_SizeforType! PortlandCementsBlendedwtd_VariousAdmixturesandCaredtt 38"C.
porosity (%_ _?edian Pore Size inrn_wls w/s w/s
Compaddm Ap w/s w/s 0.35 0.30 w/s */s 0.35 w/s(by v_.) (De/s) 0.40 0.35 * SP 0.40 0.35 + SP 0,30
TypeI 7 27.0 24.0 21.5 12.5 12.0 9.2Portland 28 23.5 22.0 14.0 15.0Cement 90 24.5 19.0 19.0 12.5
180 , 16.5 12.8 7.5 5.4365 27.0 18.0 17.0 13.6 17.5 14.0 8.8 12.5
35%TyI_ [ PC 7 190 14.5 13.5 37+ 65%BI_ 28 11.5 12.5 2.75 2.65FurnaceSlag 90 10.0 6.5 2.45 2._5
180 8.0 5.7 2.40 2.55365 6.0 4.5 2.50 2.65
90% Type[ PC 7 27.7 8.010%Silica 28 27.5 7.5Fume 90 24.0 20.5 6.2 4.1
180 23.0 ,:.2365 22,0 16.5 8.0 4.2
75%Typel PC 90 20.5 17.0 8.5 5.5+ 12.5%5_un 180 19.5 16.5 8.5 5.0Min-u-sil12.5%10tin1 365 23.5 18.5 10.0 90
70%TypeIPC 7 26.6 25.0 29.0 100+ 30%FlyAsh 28 26.0 21.3 9.2 6.-;
Min-u-sil= finepowderedqua_ (SiO2).
Table 2Chloride Diffusion Coefficients with CsC1 Diffusion
w/c = 0.35
Cemmt Dxl09 cm2S'I& Blended 28 Days 90 Days
OPC 75.1 57.2_ Slag 65 9.62 4.87
Si Fume 2.9 1.12Fee Quartz 25 -- 67.7Fly Ash 30 55.8 -
82
The rcvas¢ of the a-_,.sponof ions orchemical speciesinto a cement pasteis the movementof water through the paste and remov.alof specie_i_,!eaching, which dependson the poresu.ucmm and wat_ permeability or me paste.t',.o._._._'qAs water permeationinvolves theu'anspor¢of species undera pressuregndiem, leachinginvolves the transportof species undera concenwadon gradient,e.g., in the case withvery purewater in contact with cement paste.Solubleions,particularlymonovalentones,havebeenfoundtobereleasedbyadiffusionmechanism,whilethereleaserateofotherionsisgenerallyslower,controlledby acombinationofmechanisms.[12]
P.r,mmlgilx
Numerousexperimentalstudieshavebeenmadeoftheeffectofwater/cementratio,curingtemperatureandotherfactorsonthepgrositv,portsizedistributionandpermeabilityof
• • • 4, "_ • • *
pastesandmortarsofcementmousmatenals.['15.2"1.25-,8lThe permeabdttytowaterwasmeasuredonnumerousspecimensandwascalculatedfromtheflowrateasfollows:
(viscosityofH20)(flowrate)(samplelength)Permeability=
(crosssectionalareaofsample)(pressuredifferenceacrossthesamplel
Inonestudy[4]itwasshownthat.althoughtotalporositiesofcementpastesamplescuredat 60"C are smallerthan in those cured at 27"C,thepore volume larger than 750]kradiusis greater in the 60 C samples and it related to higherpermeabilitiesalso in the latter.[q Plotsof log permeability vs. w/c in Fig. ) show a nearly linearincreasewith w/c.
Commonly it has been the experience that well cured materials with low w/c have lowpermeabilities below the limit of measurement of many,.tzpesof apparatus.[15.21._.-31! Thiswould be expected fromoriginal predictionsof Powerst_'J and others that the "capillary"porostty is essentially "zero"in cement paste materialswith w/c <0.40, and that the remainin,.,porosityis fine gel pores. Inone study of slag-cement pastes[15]water permeabilitymeasurementswhich were made on samples taken directlyafter curing,mildly surfacedried.and epoxied into rings, showed that Rermeabilitiesof all the 0.40 and 0.50 w/c slag cementsamples were below 10"_Darcy [I.tm':](no measurable flowunder the given test conditions)upto60"C.
• Inotherstudiesarelationshipbetweenwaterpem_abilityandmeanporeradiuswasfound.[21]Figure8showstheapproximatelylinearrelationshipfoundbetweenlogwaterpermeabilityandmeanporeradius,forthesamematerialsasinFigs.5 and6,whichincludepastesofdifferentw/cmadeofpureportlandcement,andofblendswithflyash,silicafu,ne.andslag.Finally,aseriesoflinearrelationshipswasalsofoundbetweentherapidchloride"permeability"(coulombschargetransported)andmeanporeradiusfortypeIcementpaste
iO-S-
.IO "4-
._ IO-7 permeabitttyand.wlc
g
"o7 /l T I I I
0.35 0.40 0.45 0.50W/¢
83
IO'W I 1 ! I I
I0"e - I
T. //
/ o
t co_' / Fig. 8 Water per_eability
i _t / vs.mean pore radius8 of pas=es from Fig. 5,o_, .-- • •
m
I I I I IiO 20 30tte_mPorei_Ivs (nml
I..,,L,.... I I I I I I
6000 - "" """-i\\
\\
- \\
\
4000- _\\ Fig. 9 Coulombs charge cransporl:ed- \x (rapid chloride permeabillry)vs. water permeability, pastes
Q of Fig. 5
ZOO0- "_
I e_......._ %0i _-- "-;_._-.._,L II , i I I I 17 I
,o-' ,o_ ,o-",04 to_',o-iio_'water Permeability cm,s"4
andforeachofthreelevelsofflyashsubsrimuon.Withineachmaterial,thechlorideu"ansponrateincreasedlinearlywithmean IX_S,iZ_.Therewerenotenoughdataforslagandsilicafume toestablishclearrelationships,t"JTherelationshipsbetween"rapidchloridepermeability"andwaterpe:meabilityfollowsimilartrends,asshown inFig.9. Surprisingly,the silica fume material exhibited higherchloride u-ansportnot consistent with the low waterpermeability;, however, not enough samples of silica fume (or slag,) were investigated to makevalidcomparisonswiththeotherblendedruaterlab.Additionalstudiesarerequiredtovalidatethese relations for all types of materials.
DiscussionandConclusions
As discussedabove,permeabilitiestowateranddiffusionofionicspeciesincementitiousgrams,pastesandmortarsareimportantkeystoconcretedurability.Thereforeinvestigations have been made of numerous materials containing portland and blendedcements, and those with fine-grained admixtures (fly ash, slag, silica fume), at roomtemperatureandafter prolongedcuringat severalelevatedtemperaturesup to 90"C Theseconstitutepartof studiesof fundamentalmaterialrelationshipsperformedin order toaddressthequestion of long-termdurability. In zeneral,thepermeabilitiesof the materialshavebeen
I '13m I' found to be low [many < 10"8Darcy ( 0" •s" )] after curing for 28 days or longer attemperatures up to 60"C. The results obtained at 90"Care somewhat more complex.
: Porosity, mean poreradiusandporesizedistributionin cementitiousmaterialsareveryimportant micro-structuralcharacteristics;theyarerelated to a seriesof properuesof materials,suchas flexural sta'ength,fracture toughness,durability, and resistivity to ionic diffusion.The work discussedhere revealsthefactthatthe poresmactureof materials in questionisessentiallyaffectedbythe fineness,w/s ratio,chemicalcomposition,andpozzolanicreaction.Chloride ion diffusion under anappliedelectricfield appearsto be affectednotonly by thepore smactu_ of the materials, butalsobythediffusion mechanism.
84
The relationship between porosity, pore structure and ionic diffusion was suggested asfollows:
cc= [Ca rCb]3/e
whereCa andCb areconstants;e isporosity,rismedianporeradius,ctffiratiooftheoreticaltoeffectivediffusion.[6.16]ValuesforCa andCb weregeneratedfortheneatportlandcementoastesand C, = I03 + 2.6andCb ffi-0.41± 0.13forCI"diffusionandC a = 4.6± 2.5andCb= -0.62± 0.0_forCs �diffusxonwereobtainedoverthetemperaturerangeof27 to60 C.
A relationshiphasbeenfoundbetweenmean poreradiusand"rapidchloridepermeability,"aswellaswaterpermeability.Additionalstudiesarerequiredtorefinetheserelationshipsanddetermine their applicability to a variety of ceracnridous materials.
Acknowledgement
The research described herein was supportedby the Su'ategic Highway Research Program(SHRP). SHRP is a unit of the National Research Council that was authorized by section 128 ofthe Surface Trauspormtion and Uniform Relocation Assistance Act of 1987.
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86
Concrete and Structures Advisory Committee
Chairman Liaisons
James J. MurphyNew York State Department of Transportation Theodore 1L Ferragut r
Federal Highway AdministrationVice Chairman
Howard H. Newlon, Jr. Crawford F. Jencks
Virginia Transportation Research Council (retired) Transportation Research Board "
Members Bryant MatherUSAE Waterways Eaperiment Station
Charles J. Arnold
Michigan Department of Transportation Thomas J. Pasko, Jr.Federal Highway Administration
Donald E. Beuerlein
Koss Construction Co. John L. Rice
Federal Aviation Administration
Bernard C. Brown
Iowa Department of Transportation Suneel Vanikar
Federal Highway Administration
Richard D. Gaynor
National Aggregates Association/rNational Ready Mixed 11/19/92Concrete Association
Expert Task GroupRobert J. Girard
Missouri Highway and Transportation Department Amir Hanna
David L. Gress Transportation Research Board
University of New Hampshire Richard H. Howe
Gary Lee Hoffman Pennsylvania Department of Transportation (retired)
Pennsylvania Department of Transportation Stephen Forster
Brian B. Hope Federal Highway Administration
Queens University Rebecca S. McDaniel
Indiana Department of TransportationCarl E. Locke, Jr.
University of Kansas Howard H. Newlon, Jr.
Clellon L. Love.all Virginia Transportation Research Council (retired)
Tennessee Department of Transportation Celik H. Ozyildirim
David G. Manning Virginia Transportation Research Council
Ontario Ministry of Transportation Jan P. Skalny
W.R. Grace and Company (retired)Robert G. Packard
Portland Cement Association A. Haleem Tahir
American Association of State Highway and Transportation
James E. Roberts OfficialsCalifornia Department of Transportation
Lillian Wakeley
John M. Scanlon, Jr. USAE Waterways Experiment StationWiss lanney Elsmer Associates
Charles F. Seholer
Purdue University
Lawrence L. Smith
Florida Department of Transportation
John 1L Strada
Washington Department of Transportation (retired)