Modeling ion and fluid transport in unsaturated cement systems for isothermal conditions

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Modeling ion and fluid transport in unsaturated cement systems for

isothermal conditions

Samson, E.; Marchand, J.; Snyder, K. A.; Beaudoin, J. J.

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Modeling ion and fluid transport in unsaturated

cement systems in isothermal conditions

Samson, E.; Marchand, J.; Snyder, K.A.;

Beaudoin, J.J.

NRCC-45332

A version of this document is published in / Une version de ce document se trouve dans:

Cement and Concrete Research, v. 35, no. 1, January 2005, pp. 141-153,

Doi:

10.1016/j.cemconres.2004.07.016

ement systems for isothermal onditions E. Samson 1 2 J. Mar hand 1 2 K.A. Snyder 3 J.J. Beaudoin 4 1

CRIB  Départementde génie ivil

Université Laval, Ste-Foy (Q ), Canada, G1K 7P4

2

SIMCO Te hnologiesin .

1400, boul. du Par Te hnologique, Québe (Q ),Canada, G1P4R7

3

Buildingand Fire Resear h Laboratory

National Institute of Standards and Te hnology, Gaithersburg,MD 20899, USA

4

Materials Laboratory -Institute for Resear h in Constru tion

National Resear h Coun il,Ottawa (Ontario), Canada, K1A 0R6

June 23, 2004

Abstra t

Ades riptionofioni transportinunsaturatedporousmaterialsduetogradients

intheele tro- hemi al potentialandthe moisture ontent isdevelopedbyaveraging

the relevant mi ros opi transportequations overa representative volume element.

The omplete set of equations onsist of a time-dependent equations for both the

on entration of ioni spe ies within the pore solution and the moisture ontent

within the porespa e. The ele trostati intera tions areassumed to o ur

instan-taneously and the resulting ele tri al potential satises Poisson's equation. Using

the homogenizationte hnique,moisture transportdueto both theliquidand vapor



Corresponding author. Current address at SIMCO Te hnologies in . Tel.: 418.656.1003, Fax:

ture ontent is found. The nal transport equations ontain transport oe ients

that an be unambiguously related to experimental quantities. The approa h has

the advantage of making the distin tion between mi ros opi and bulk quantities

expli it. KEYWORDS: Diusion (C) Transportproperties (C) Degradation(C) Modeling (E) Homogenization 1 Introdu tion

Overthe past de ade, agreatdeal ofeorthas been spe i allydevoted tothe

investiga-tion ofiontransportme hanismsinunsaturated ementsystems. The topi isimportant

sin e, in many ases, on rete stru tures exposed to ioni solutions are also frequently

subje ted towettinganddrying y les. The oupledtransportofmoistureandions often

tendstoa eleratephysi aland hemi aldegradationme hanismsandredu ethe servi e

life of the material[1, 2,3℄.

Reports re ently published on the subje t have largely ontributed to larify some

fundamental aspe ts of ion transport me hanisms in unsaturated on rete. Many

inves-tigations have also emphasized the intri ate nature of these phenomena. If most of the

di ulties related to the des ription of transport pro esses in on rete are linked to the

intrinsi omplexity of the material, it appears that part of them also lies with the fa t

that authorshave usedmany dierentapproa hes tostudy thesepro esses. Forinstan e,

the denition of the state variables used todes ribe the various transportpro esses tend

to vary signi antly from one study to another. This is most unfortunate sin e the la k

transport me hanisms are des ribed using a well established mathemati al pro edure,

the homogenization te hnique. The te hnique has been re ently used to investigate the

diusion of ions in saturated systems [4℄. A ording to this approa h, the transport

equationsare rst writtenatthe pore s ale. They are then averaged overthe s aleof the

material. Themainadvantageof thehomogenizationte hniqueliesinthe leardenition

of the state variables.

The paper rst addresses the pro ess of moisturetransport inanunsaturated porous

material. Forthe ompletely oupledtransportofionsinanunsaturatedmedia,dynami al

equations are required to express the moisture ontent as a fun tion of time. This is

a hieved by averagingmi ros opi alequationsforbothliquidand watervaportransport.

The mathemati al development yieldsRi hards' equation, and the moisture ontent and

the transport oe ients are well-dened.

The se ondpartof thepaperisdevotedtothe oupledtransportofionsand moisture

in the system. Here, the eld quantity is the on entration of the ions within the pore

solution. The homogenization te hnique is applied to a mi ros opi equation for both

diusive and onve tive transport. Whilediusive equationsalready exist,reformulating

the bulk equations using homogenization ensures that the transport oe ients are well

dened (pore spa e versus mi ros opi quantities) and an, therefore, be unambiguously

relatedto experimental quantities.

2 Water transport in unsaturated porous materials

The rst obje tive isto develop anequation to hara terize the mass transport of water

inanunsaturatedporousmaterial. Ri hards[5℄wasamongthe rstauthorstostudy the

me hanisms of water transport in unsaturated porous solids. In 1931, he proposed the

followingequation todes ribe the ow of water under apillary su tion:



t

apillary potential.

This relationship, known as Ri hards' equation, was later modied to express the

transportof masssolelyasa fun tionofthe gradientinwater ontent. Thismodi ation

is based on the assumption that the apillary potential is a dierentiable fun tion of

the moisture ontent :

=f() (2)

This allows to write:

grad =

d

d

grad (3)

Substituting (3)into (1), one nds:

 t div(D  grad) (4) whereD 

=K(d =d)isthenonlinearwaterdiusivity oe ient. Equation(4)iswidely

used to model the evolution of water ontent in a porous material kept in isothermal

onditions. Equation (4)is alsoknown as Ri hards' equation.

While Ri hards' equation is ommonly a epted among s ientists, its use over the

past de ades has ontributed to some onfusion on how to des ribe moisture transport

me hanismsinunsaturatedporousmaterials. Ri hardsoriginallywrotethe equationwith

the water ontentexpressed in ubi entimetersof waterpergram ofdrymaterial. Over

the years, some authors have preferred to dene water ontent in kilograms of moisture

perkilogramofdrymaterial[6℄orinkilogramsof waterper ubi meterofmaterial[7,8℄.

However, mostauthorshavetraditionally hosentoexpressthe variablein ubi meterof

water per ubi meterofmaterial[9,10,11℄. Toaddtothe onfusion,manyauthorstend

to dene the moisture ontent as the sum of liquid water and vapor while some others

only onsider the liquidphase.

2.1 General onsiderations

In an attempt to larify these on epts, Ri hards' equation will be derived using the

deformations) kept under isothermal onditions (i.e. the transport of water is solely due

to apillary su tion). Other assumptions willariseduring the development of the model.

The mathemati alrules oftheaveragingte hnique an befound intextbooks [12,13℄.

Onlythe basi denitions willbeexposed in the followingparagraphs. More information

onthete hnique analsobefoundinreferen e[4℄. Thete hniqueisoutlinedherebe ause

it isat the ore of development of allthe transportequations.

As previously mentioned, the homogenization te hnique starts with a onservation

and a transport equation at the mi ros opi level (i.e., at the s ale of the pore). These

equations are then integrated over a Representative ElementaryVolume (REV), su h as

the one depi ted in Figure 1. The size of the volume depends onthe intrinsi properties

of the material. For instan e, for on rete and mortar mixtures, the size of the REV

depends on the maximum diameter of the aggregate parti les. For the hydrated ement

paste, the REV is typi allya few ubi entimeters.

The totalvolumeof theREVisgiven byV

o

. Thevolumeo upied bythe liquidphase

isdesignatedby V

L

o

. The volumetri fra tionofliquid

L

isthe ratioofthe liquidvolume

tothe total volume:

 L = V L o V o (5)

The gaseous phase o upies a volume V

G

o

. It is a mixture of air and water vapor. It is

assumed that both air and the water vapor ll the whole gaseous phase volume. As for

the liquid phase, the volumetri fra tion of gas 

G

is the ratio of the gas volume to the

total volume:  G = V G o V o (6)

In the remainder of the text, the supers ripts L and G will designate the liquid and

gaseous phases, respe tively. Furthermore, the supers ript V will represent the water

vapor phase withinthe totalgaseous phase.

Let a

denote the amountperunit volume of some extensive quantity A inthe phase

 , eithersolid,liquidorgas. Con entration ormass density an serveasexamplesfora

 .

a  = 1 V o Z V  o a  dV (7)

The volumetri intrinsi phase average is dened as:

a   = 1 V  o Z V  o a  dV (8)

The two values are relatedby the following relationship:

a  =  a   (9)

2.2 Transport of liquid water

The ontinuity equation forliquid water is given by [14, 15℄:

 L t +div( L v L )=0 (10) where L

isthemassofliquidwaterperunitvolumeofliquidphase,andv

L

isthevelo ity

of water. The bulk equationis obtained by averagingEquation (10) over the REV:

1 V o Z V L o   L t +div( L v L )  dV =0 (11)

This integral an be divided intwo parts:

1 V o Z V L o  L t dV + 1 V o Z V L o div( L v L )dV =0 (12)

Usingthe denition of the volumetri phaseaverage (equation 7),one an write:

 L t +div( L v L )=0 (13)

The average of the time derivative gives [12, 13℄:

( L  L L ) t =  L t + 1 V o Z S LG o  L u
n LG dS+ 1 V o Z S LS o  L u
n LS dS (14) where S LG o

is the surfa eof the liquid/gas interfa e, S

LS

o

isthe surfa e of the liquid/solid

interfa e, u is the velo ity of the interfa e, n

LG

LS

liquidphaseatthe liquid/solidinterfa e. Sin eitisassumed thatthedeformations ofthe

solid matrix ould be negle ted, the last integral on the right-hand side of equation (14)

an be dropped, whi h leaves:

( L  L L ) t =  L t + 1 V o Z S LG o  L u
n LG dS (15)

The average of the divergen e in equation(13) is given by [12, 13℄:

div( L v L )=div  L ( L v L L )
+ 1 V o Z S LG o  L v L
n LG dS+ 1 V o Z S LS o  L v L
n LS dS (16)

At the solid/liquid interfa e, it is assumed that the liquid velo ity is zero (the so- alled

no-slip ondition of uid me hani s [16℄). Hen e, the last integral onthe right-hand side

of equation (16) an be negle ted, whi h leaves:

div( L v L )=div  L ( L v L L )
+ 1 V o Z S LG o  L v L
n LG dS (17)

Substituting equations(15) and (17) in equation (13), one nds:

( L  L L ) t +div  L ( L v L L )
+ 1 V o Z S LG o  L (v L u)
n LG dS =0 (18)

A ording to Whitaker [15℄, the integral in equation (18) orresponds to the rate of

vaporizationperunitvolumeoftheliquidphaseattheliquid/gasinterfa e,andisdenoted

by m._ Also,the average value 

L L

orresponds tothe density of the liquid

L

, whi h an

be assumed onstant. Equation (18) an thus besimplied:

 L  L t + L div( L v L L )+m_ =0 (19)

The next step onsists of determining the average value of the liquid velo ity. The

starting pointis the Dar y equation [17℄:

v L = K  (gradP + L g) (20) The quantity v L

is the bulkvelo ity of the liquid, K isthe permeability of the material,

 is the vis osity of the uid, P is the pressure on the liquid, and g is gravitational

in materialshavingvery smallpores the apillary for es are dominant: v L = k   L gradp (21) Thequantityp

isthe apillarypressureand k isthepermeabilityof theliquid-lledpore

spa e.

Equation (21) isbasedonthe assumptions that gravity ee ts are negligibleand that

the pressure is uniform throughout the liquid and gaseous phases. It should also be

emphasized that the validity of the equation alsorests on the hypothesis that the liquid

phaseis ontinuous. The latterassumptionwillbefurther dis ussed inthelastse tionof

this report.

The bulk velo ity of the liquidv

L

an be related to its intrinsi average ounterpart

through: v L = L v L L (22)

Substituting equations(21) and (22) intoequation (19) gives:

 L  L t  L div  K   L gradp  +m_ =0 (23) Sin e p =f( L

)[15℄, the hain rule allows to write:

gradp =  dp d L  grad L (24)

The substitution of equation (24) inequation (23) gives:

 L  L t  L div  K   L dp d L grad L  +m_ =0 (25) Let D L = K   L dp d L (26)

Equation(25)isnowexpressedasafun tionofasingleeldquantity

L

togivea omplete

des ription of liquidwater transport:

 L  L t  L div(D L grad L )+m_ =0 (27)

L

to awater diusion oe ient. However, it should be emphasized that the movement of

liquid water onsidered in this se tion arises by apillary su tion. It is not, per se, a

diusive phenomenon.

With the denition of D

L

given in equation (26), ombined with equation (24), the

velo ity of the liquidphase (equation 21) an now be writtenas:

v L = D L grad L (28)

2.3 Transport of water vapor

The treatment of the gas transport phenomenon is more ompli ated sin e two phases

have to be onsidered: air and water vapor. However, the problem an be simplied

by onsidering the following assumptions. As mentioned in the previous se tion, the

development of equation (21) rests on the hypothesis that pressure is uniform over the

gaseous phase. This impliesthat there is nobulk movement of airin the gaseousphase.

Consequently, there will be no onve tive transport of water vapor within the material

pore stru ture. Still, there an be movement of mole ules in the gaseous phase as a

resultof their thermalagitation. The otherassumption isthat gravity doesnot haveany

signi ant ee t onthe behavior of the water vapor.

The ontinuityequationforwatervapor omponentofagaseousphaseisthefollowing

[15℄:  V t +div( V v V )=0 (29) The quantity  V

is the mass of water vapor per unit volume of gaseous phase, and v

V

is the velo ity of water vapor. The water vapor will be in movement as a result of its

thermal agitation. It is therefore a diusive pro ess. A ording toDaian [10℄, the water

vapor uxis given as:

 V v V = Dgrad V (30)

 V t div(Dgrad V )=0 (31)

The bulk equation is al ulated from the integration of equation (31) overthe REV:

1 V o Z V G o   V t div(Dgrad V )  dV =0 (32)

This integral an be divided intwo parts,whi h yields:

1 V o Z V G o  V t dV 1 V o Z V G o div(Dgrad V )dV =0 (33)

A ording to the denition of the volumetri phase average (equation 7), equation (33)

an be written as:  V t div(Dgrad V )=0 (34)

The average of the time derivative isgiven by:

( G  V G ) t =  V t + 1 V o Z S GL o  V u
n GL dS+ 1 V o Z S GS o  V u
n GS dS (35) where S GL o

is the surfa e of the liquid/gas interfa e, S

GS

o

is the surfa e of the gas/solid

interfa e, u is the velo ity of the interfa e, n

GL

is a unit ve tor pointing outward the

gaseous phase atthe liquid/gasinterfa e, and n

GS

is a unit ve tor pointing outward the

gaseousphase atthe gas/solidinterfa e. Sin e itisassumed that the deformationsof the

solid matrix are negligible, the last integral on the right-hand side of equation (35) an

be negle ted, whi h leaves:

( G  V G ) t =  V t + 1 V o Z S GL o  V u
n GL dS (36)

The average of the divergen e gives:

div(Dgrad V )=div  G (Dgrad V G )
+ 1 V o Z S GS o Dgrad V
n GS dS+ 1 V o Z S GL o Dgrad V
n GL dS (37)

The rst integral on the right-hand side of equation (37) is negle ted sin e there is no

ex hangeof watervaporbetweenthe solidand thegaseousphases. A ordingly, equation

(37) an besimpliedas:

div(Dgrad V )=div  G (Dgrad V G )
+ 1 V o Z S GL o Dgrad V
n GL dS (38)

as: div(Dgrad V )=div( G Dgrad V G )+ 1 V o Z S GL o Dgrad V
n GL dS (39)

The average of the gradient is given by [12, 13℄:

grad V G = G grad V G + 1 V o Z S GS o Æ x(grad V
n GS )dS+ 1 V o Z S GL o Æ x(grad V
n GL )dS (40)

The quantity  isreferred toby Ba hmatand Bear[12℄ as the tortuosityof the material.

Con eptually,itisthe ratio ofthe ma ros opi system lengthtothe shortest path length

through the pore (liquidor gas) spa e. As su h, it is a quantity that stri tly equalto or

less than one. The parameter

Æ

x is dened as

Æ

x = x x

o

, where x is a position ve tor

withinthe REV andx

o

isthe positionve tor ofthe enter ofthe REV. Therst integral

ontheright-handsideofequation(40)involvesthesolid/gasinterfa e. Ex eptforthevery

low water ontent onditions, there will be no dire t onta t between these two phases

be ause water willbe adsorbed onthe surfa e of the solid. A ordingly, the integral an

be negle ted. It is assumed that the term (grad

V

n

GL

) in the se ond integral on the

right-hand side of equation (40) varies very slightly over the surfa e S

GL

o

. Under this

assumption, itleavesan integralof a position ve tor times a s alarover a losed surfa e,

whi hgiveszero. Equation (40) is thussimpliedas:

grad V G = G grad V G (41)

Repla ingequations (36), (39) and (41) into equation(34) gives:

( G  V G ) t div( G D G grad V G )+ 1 V o Z S GL o (Dgrad V  V u)
n GL dS =0 (42)

Substituting equation(30) in equation (42),one nds:

( G  V G ) t div( G D G grad V G ) 1 V o Z S GL o  V (v V u)
n GL dS =0 (43)

Withaker[15℄ showed that the integral inequation (43) has the same value as the one in

equation (18). It represents the rate of ondensationper unit volume of the water vapor

phase atthe liquid/gasinterfa e. Therefore, equation (43) an bewritten as:

( G  V G ) t div( G D G grad V G ) m_ =0 (44)

L V

f(

L

) [10℄. Applying the hain rule, it gives:

( G  V G ) t div   G D G d V G d L grad L  _ m=0 (45)

The quantities pre eding the gradient within the parenthesis an be lumped together to

forma single vapor diusion oe ient:

D V = G D G (46)

Equation (45) an be writtensu in tly:

( G  V G ) t div  D V d V G d L grad L  _ m=0 (47)

2.4 Total moisture transport

Inthe previousse tions, the transportequationsforthe liquidand thevaporphaseswere

onsideredseparately(equations27and47). Inordertoget a omplete des riptionofthe

transport, both equationsshould beadded together:

 t ( L  L + G  V G ) div   L D L + d V G d L D V  grad L  =0 (48)

Asthedensityofwatervaporhasamu hlowervaluethanthe oneofliquidwater(

V G   L ) and  G  L

,equation (48) an be simpliedas:

 L  L t div   L D L + d V G d L D V  grad L  =0 (49) Let D  =  L D L + d V G d L D V  L (50)

Substituting equation(50) into(49) gives:

 L t div(D  grad L )=0 (51)

This isRi hards' equation. As an beseen, the equationfully des ribes the movementof

both vapor and liquid water on the basis of a single variable 

L



tothe vaportransportanda se ondonerelatedtothe transportofthe liquidphase. The

demonstration also indi ates that the variable  of the original equation (4) stands for

the volumetri liquidwater ontent, whi h isexpressed in ubi meter of water per ubi

meter of material.

2.5 Determination of the moisture transport properties of

hy-drated ement systems

The des ription of moisturetransportme hanismson the basis of equation (51) requires

the determination of the fun tion D



. An interesting dis ussion of the variation of this

fun tion with the water ontent of the material has re ently been published by Martys

[19℄. The author learly emphasizes the non-linear hara ter of this fun tion.

Measurementsmadeonasand olumn[20℄showthatwhenthehumidityinthemedium

ishigherthan4%(by weight),the ontributionofthe vaporphasetothe overall moisture

transfer is negligible. In that ase, one an assume that D



=D

L .

Over the years, numerous experimental te hniques have been used to determine the

moisturetransportproperties ofhydrated ement systems. Athorough dis ussion of this

subje tis beyond the s ope ofthis paper. Comprehensive riti alreviews ofthis problem

an be found in referen es [21, 22℄.

3 Ioni transport in unsaturated porous materials

Several mathemati al models have been developed to predi t the movement of ions in

ement-basedmaterials. Mostoftheseapproa hesare single-ionmodels, onsideringonly

hlorideanditsdetrimentalee tonthedurabilityofthematerial. Mostofthetime,su h

models onsider the transport of ions under the ee t of diusion and adve tion (uid

ow). Also onsidered is the ee t of the hemi al rea tions involving the onsidered

However, these models oversimplify some basi physi al phenomena. For instan e,

the ele tri al oupling between the ions [18℄ and its ee t on their movements is often

overlooked. This is parti ularlytrue for ement-based materials sin e they ontain

on- entrated porous solution. The ele tri al oupling between the ions for on entrated

solutions was re ently put in eviden e in two papers by Snyder [26, 27℄ that report on

diusion experiments through non rea tive erami frits. Multiioni models taking into

a ountele tri al ouplingwere re entlypublishedbyMasielal. [28℄andTru etal. [29℄.

Unfortunately,asitwasthe ase withRi hards'equation,thereisala kof agreement

with regard to the denition and the use of some parameters in these models. For

ex-ample, the diusion oe ient is sometimes alled the intrinsi diusion oe ient, the

apparent diusion oe ient, or the ee tive diusion oe ient. On e again, the

aver-aging pro edure is used to generate an ioni transport model. The method will larify

some of the basi on epts behind the modeling of ioni transport. Su h a work was

previously published [4℄ but was applied only to non-rea tive saturated materials. The

modelpresented inthe following se tionsis more general.

3.1 Transport of ions in the liquid phase

Thetransportmodelisbasedonthe observationthat thetransportof ionsonlyo urs in

the liquidphase. Hen e, noequation has tobe developed forthe solid orgaseousphases.

The onservation equation foran ioni spe ies i inthe liquidphase atpore s ale isgiven

by the following mi ros opi equation [13℄:

 i t +div(j i )+r i =0 (52) The quantity i is the on entration, j i

is the ioni ux and r

i

is a sour e/sink term

a ountingforthehomogeneous hemi alrea tions[39℄betweenionsinsolution. Thebulk

equationis obtained fromaveragingthis equationover the REV,followingthe pro edure

that lead toequations (13) and (34):

 i t +div(j i )+r i =0 (53)

( L i L ) t =  i t + 1 V o Z S LS o i u
n LS dS+ 1 V o Z S LG o i u
n LG dS (54)

The rst integral on the right-hand side of equation (54) ontains a term that a ounts

for the velo ity of the solid/liquidinterfa e. Whilethis interfa e may possibly move as a

resultofsomedissolution/pre ipitation hemi alrea tions,itwilldosoveryslowly. It an

thus be negle ted. The other integral involvesthe movement of the liquid/gas interfa e.

It is similarto the rst integralin equation (14). While itwas used in the mathemati al

development of the moisture transport, it is assumed that it has only a small ee t on

the ioni transport, and an thus be negle ted,simplifyingequation (54) to:

( L i L ) t =  i t (55)

The average of the divergen e is given by:

div(j i )=div( L j i L )+ 1 V o Z S LS o j i
n LS dS+ 1 V o Z S LG o j i
n LG dS (56)

Thelastintegralontheright-handsideofequation(56)a ountsfortheioni ux rossing

the liquid/gaseous interfa e. The value of this ux is zero sin e ions do not go into the

gaseous phase. The other integral, related to the ux of ions a ross the solid/liquid

interfa e, will be used to model the various hemi al rea tions involving those phases.

A ordingly,equation (56) an beredu ed to:

div(j i )=div( L j i L )+ 1 V o Z S LS o j i
n LS dS (57)

Substitutingequations (55)and (57) inequation(53) and averagingthe term r

i by  L r i L

(see equation 9),one nds:

( L i L ) t +div( L j i L )+ L r i L + 1 V o Z S LS o j i
n LS dS =0 (58)

The next step onsists of writing the proper ux expression at the mi ros opi level

(ions in bulk ele trolyte) and averaging it over the REV. Due to the harged nature

of ions, this expression has to onsider the ele tri al oupling between ioni parti les.

hemi al a tivity. Finally, the movement of the uid itself will have an impa t on the

movement of ions. All these physi al phenomena an be taken into a ount through the

extended Nernst-Plan k modeltowhi his added anadve tion term [18℄:

j i = D  i grad i D  i z i F R T i grad D  i i grad(ln i )+ i v L (59) The parameter D  i

is the self-diusion oe ient [30℄ of spe ies i in diluted, free water

onditions,

i

is the hemi al a tivity oe ient, is the ele tri al potential, z

i

is the

valen e number of the ion, F is the Faraday onstant, R is the ideal gas onstant and

T is the absolute temperature. The terms on the right-hand side of equation (59) are

asso iated with diusion, ele tri al oupling between the ions, hemi al a tivity ee ts

and water transport, respe tively.

TheintegrationoftheuxovertheREV,similartothepro edurefollowedinequations

(9)and (11) to(13), leads to:

 L j i L = D  i  L grad i L D  i z i F R T  L i grad L D  i  L i grad(ln i ) L + L i v L L (60)

The next steps onsistin averagingthe various gradientsand variablesin equation (60).

The average of the on entration gradient is given by [12, 13℄:

grad i L = L grad i L + 1 V o Z S LS o Æ x(grad i
n LS )dS+ 1 V o Z S LG o Æ x(grad i
n LG )dS (61) The quantity  L

is the tortuosity of the aqueous phase. It is a purely geometri al fa tor

a ounting for the omplexity of the paths the ions must travel through in liquidspa e.

It is a fun tion of the water ontent 

L

, sin e it is related tothe volume of liquid in the

pore spa e, and itsvalue is less than one [12℄. The parameter

Æ

xwas rst en ountered in

equation (40).

To evaluatethe surfa e integralsin equation(61), onehas torefer tothe doublelayer

theory [31℄. Figure 2 shows the ross-se tion of a pore and the s hemati shape of the

on entration prole along its radius. The solid bears an ele tri al surfa e harge 

solid .

It is neutralizedby harges of the opposite sign in two dierent zones, the Stern and the

diuse layers, bearing respe tively 

stern

and 

di

 solid = stern + di (62)

Theexternallimitofthe Sternlayer, alledthe outerHelmholtzplaneortheshearplane,

separates the solid fromthe aqueous phase, inwhi h ioni diusion o ur. The aqueous

phase is divided in the diuse layer and the free water zone, where ions do not feel

the ee t of the solid/liquid interfa e. A re ent study by Revil [32℄ showed that ioni

transportmay o ur inthe Stern layer. Butit wasalsomentioned inthe paperthat this

phenomenon isnegligiblewithrespe ttotransportin the bulk porewhenthe pore solution

ishighly on entrated,asitisthe ase in ement-basedmaterials. Consequently,onlythe

ioni transport in the aqueous phase is onsidered in this paper. Finally, the des ription

of the ross-se tion of the pore is omplete by onsidering a gaseous phase atthe enter

of the pore, whenthe latter isnot saturated [33℄.

It is assumed that the on entration proles at the liquid/gas interfa e is at (see

gure 2). Consequently, the se ond integral in equation (61) is negligible sin e there

is no on entration gradient along the radius at the liquid/gas interfa e. The situation

is dierent for the rst integral be ause of the on entration gradient along the radius

at the solid/liquid interfa e aused by the ele tri harge at the surfa e of the solid.

Simple double layer al ulations made with the Gouy-Chapman model [31℄ are shown

on Figure 3. They emphasize that in reasing the ioni strength of a solution in the

vi inity of a harged surfa e de reases dramati ally the thi kness of the double layer.

Sin e ementitiousmaterialsbear ahighly hargedsolution,the doublelayerextend over

a very small region. Outside this region, the on entration prole is unae ted by the

surfa e harge. Followingthis,theterm

R Æ x(grad i
n LS )dS inequation(61)isnegle ted, leaving: grad i L = L grad i L (63)

Then one needs to average the term in equation (60) on erned with the ele tri al

oupling between the ions. A ording to the pro edure for averaging a produ t [12, 13℄,

one an write:

i grad L = i L grad L + Æ i Æ grad L (64)

Æ i = i i L (65)

Itisassumedthatthedeviationsleadtosmallterms,whi hallowstonegle tthedeviation

produ t inequation (64): i grad L = i L grad L (66)

Followingthe same pro edure astheone used forthe on entrationgradient,the average

of the potentialgradientgives:

grad L = L grad L + 1 V o Z S LS o Æ x(grad
n LS )dS+ 1 V o Z S LG o Æ x(grad
n LG )dS (67)

Figure 2 shows a potential prole a ross the se tion of a pore. A ording to the double

layer models [31℄, it has a shape similar to the on entration prole, i.e. it is disturbed

near the solid/liquid interfa e but tends to a at prole toward the enter of the pore.

And like the on entration proles shown on Figure 3, in reasing the ioni strength of

the solution redu es the thi kness of the area where the gradient of potentialis dierent

from zero. A ordingly, the integrals are negle ted, assuming again that the ele tri al

phenomenaneartheinterfa edonotae t ioni movement. Equation(67)thussimplies

to: grad L = L grad L (68)

Substituting equation(68) into(66) gives:

i grad L = i L  L grad L (69)

The same approa h is used to average the hemi al a tivity term in equation (60).

The same assumptions on erning the deviations, as well as those on erning the ee t

of the ele tri alphenomena at the solid/liquidinterfa e lead to:

i grad(ln i ) L = i L  L grad(ln i L ) (70)

It is assumed that the term ln

i L

orresponds to the hemi al a tivity oe ients

al- ulated with the average on entrations

i L . For simpli ity, ln i L is approximated by ln i L : i grad(ln i ) L = i L  L grad(ln i L ) (71)

i v L = i L v L + Æ i Æ v L (72)

The term in equation(72) ontaining the deviations is alled the dispersive ux [13, 34℄.

It is shown inthe previousreferen es that it an be writtenunder aFi kian form:

Æ i Æ v L = D disp grad i L (73) where D disp

is alled the oe ient of adve tive dispersion, and is due to ngering, not

diusion. Consequently,thisterm anbeaddedtotheioni diusiontermthatwouldthen

exhibitanew diusion oe ientbeing thesum ofthe lassi alone plusthe oe ientof

adve tive dispersion. When the uid is in movement under the ee t of a water ontent

gradient, as des ribed in the pre eding se tion, the velo ity is relatively weak. In that

ase, the dispersion term an be negle ted [35℄, leadingto:

i v L = i L v L (74)

Substituting equations (63), (69), (71), and (74) in equation (60), gives the average

ux expression:  L j i L = D  i  L  L grad i L D  i z i F R T  L  L i L grad L D  i  L  L i L grad(ln i L )+ L i L v L (75)

The diusion oe ient atthe ma ros opi levelD

i isdened as: D i = L D  i (76)

To simplifythe expression, let:

C i  i L (77)  L (78)

Substituting equations(76) to (78) inequation (75) gives:

 L j i L = D i  L gradC i D i z i F R T  L C i grad D i  L C i grad(ln i L )+ L C i v L (79)

the ma ros opi ioni transportequation: ( L C i ) t div  D i  L gradC i + D i z i F R T  L C i grad +D i  L C i grad(ln i L )  L C i v L  + L r i L + 1 V o Z S LS o j i
n LS dS =0 (80)

In order to simplify this equation, the integral must be expressed in a manner that is

more friendly to a further numeri al analysis. The term (j

i

n

LS

) gives the amount of

ions rossingthe solid/aqueous phaseinterfa e, asaresult of dissolution/pre ipitationor

ionex hangerea tions. Itispossible toexpress itdierentlyby performingthe averaging

operationontheionsinthesolidphase[13℄. The onservationequationatthemi ros opi

s ale for the ions in solid phase is:

 is t +div(j is )=0 (81)

wherethesubs ript sdesignatesthe solidphase. Contrarytoequation(52),itisassumed

thatno hemi alrea tionso urwithinthe solidphase,sin e allpre ipitation/dissolution

phenomenaaretakingpla eatthesolid/aqueousphaseinterfa e. Averagingequation(81)

over the REV leads to:

( s C is ) t +div( s j is s )+ 1 V o Z S SL o j is
n SL dS =0 (82) where s

isthe volumetri fra tion ofsolid phase andn

SL

isanoutward (tothe S-phase)

unit ve tor on the solid/aqueous phase interfa e (designated as S

SL

). The integral in

equation (82) has the same value as the one in equation (80) but with an opposite sign

sin e the ions oming out of the aqueous phase are being bound by the solid phase.

Furthermore, the ux j

is

within the solid is zero sin e there is noioni movement in this

phase. This allows one to write:

( s C is ) t = 1 V o Z S SL o j is
n SL dS = 1 V o Z S LS o j i
n LS dS (83)

Substituting equation(83) in equation (80) gives:

( L C i ) t div  D i  L gradC i + D i z i F R T  L C i grad +D i  L C i grad(ln i L )  L C i v L  + L r i L + ( s C is ) t =0 (84)

Thisisthegeneralexpressionfortheioni transport inporousmaterialsunderisothermal

To model the transport of ions under the inuen e of apillary su tion, it would seem

straightforward to substitute equation (28) in (84). However, the development of the

water transport equations was made for the ase of pure water in a porous material.

When ions are in solution, the vapour pressure above a solution is lower than in pure

water [36℄. This ee t is quantied through Raoult's law. A ordingly, the relationship

p

=f(

L

)should instead bewritten as:

p =f( L ; i ) (85)

sin e the presen e of ions in solution is likely to disturb the equilibrium between the

aqueous and gaseous phases in a pore. To evaluate in what extent the presen e of ions

will ae t the vapor pressure of water, one an use Raoult's law to al ulate the vapor

pressure hange between pure water and a 500 mmol/L NaCl solution with water as

solvent. A ording toRaoult's law [36℄, the vapor pressure hange is given as:

p v =X solute p Æ v (86) where X solute

is the molar fra tion of solute (NaCl) in the solution and p

Æ v

is the vapor

pressure of pure water. At 25

Æ

C , the vapor pressure of bulk water is 3.17 kPa [36℄.

Knowing that inone literof water thereare 56moles:

X solute

=

0.5 moleNaCl

0.5 moleNaCl +56 moles water

=0:009 (87)

This gives a hange in vapor pressure of
p

v

=0:03 kPa, whi h is obviously very weak.

A ording tothe result of this simple al ulation,the ee t ofioni on entration onthe

apillarypressure isnegle ted. Itwasalsonegle tedinthe modelspresented inreferen es

[23, 24,25, 28℄.

Substituting equation (28) inequation (84) gives:

( L C i ) t div  D i  L gradC i + D i z i F R T  L C i grad +D i  L C i grad(ln i L )+C i D L grad L  + L r i L + ( s C is ) t =0 (88)

This equation an be used to model the transport of ions in unsaturated ement-based

expression to al ulate the hemi al a tivity oe ients. These topi s are addressed in

the followingse tions.

3.3 Cal ulation of the potential

The ele tri al potentialin equation (88) arises in the materialto enfor e the

ele troneu-trality ondition. If two spe ies are diusing ina materialwithone of the spe ies having

a greater self-diusion oe ient. To maintain a neutral solution, the potential reated

slows the fastest ions and a elerates the slowest ones.

The mathemati alrelationship thatrelates ele tri alpotentialtoele tri al harges in

a given media is given by Poisson's equation [18℄

r 2 +   =0 (89)

where  is the ele tri al harge density and  is the medium permittivity. The harge

density an be relatedto the ioni on entration through:

=F N X i=1 z i i (90)

where N isthe numberof ioni spe ies. Substituting equation(90) in(89) gives:

r 2 + F  N X i=1 z i i =0 (91)

It may seem awkward tohave anequation from ele trostati s ina model where the ions

aremovingthroughtime. However, sin etheele tromagneti signalismovingmu hmore

rapidly than the ions, Poisson'sequation is perfe tlysuitable.

To use equation(91) in the transport model, ithas tobe averaged overthe REV. As

it was done previously, it is assumed that the boundary ee ts at the liquid/solid and

liquid/gas interfa es are negligible. Following the same average rules as in the previous

se tions, we get the following relationship:

div( L  L grad )+ L F  N X i=1 z i C i =0 (92)

The models to al ulate the hemi al a tivity oe ients are numerous. The rst ones

developed are the Debye-Hü kel and extended Debye-Hü kel models [37℄. From purely

ele trostati onsiderations, they relate the hemi al a tivity oe ients of ioni spe ies

to the ioni strength of a solution. They are valid for ioni strengths up to 10 and 100

mmol/L respe tively.

In ement-based materials, the ioni strength is mu h higher. To suit this parti ular

situation, a hemi al a tivity relationship was developed re ently by Samson et al. [38℄

whi hgivesgoodresults for highly on entrated solutions:

ln i = Az 2 i p I 1+a i B p I + (0:2 4:1710 5 I)Az 2 i I p 1000 (93)

where I is the ioni strength of the solution, and A and B are temperature dependent

parameters. The parameter a

i

inequation (93) varies with the ioni spe ies onsidered.

3.5 Modeling of hemi al rea tions

Two terms appear in equation (84) to a ount for hemi al rea tions. The term r

i L

is

a sink/sour e term that models homogeneous hemi al rea tions [39℄, i.e. rea tions that

solelyinvolvethe aqueousphase, as for instan e:

Ca 2+ (aq) +OH (aq) !CaOH + (aq) (94)

The other term related to hemi al rea tions is (

s C

is

)=t. As mentioned before, it

a ountsforioni ex hangesbetweenthe aqueousandsolidphases. This typeof hemi al

rea tion is alled heterogeneous [39℄. It in ludes dissolution/pre ipitation and surfa e

exhange phenomenons. The formation of portlandite is an example of heterogeneous

rea tion: Ca 2+ (aq) +2OH (aq) !Ca(OH) 2(s) (95)

In most ases, hemi al rea tions are modeled by assuming that they are faster than

tious materials. Under LEA, hemi alrea tions are modeled by algebrai mathemati al

relationships [39℄. Following a paper published in 1989 by Yeh and Tripathi [41℄, the

urrent trendfor solving ioni transportproblems inrea tivematerialsis toseparate the

transport and hemi alrea tion parts. The partial dierential equationsdes ribing ioni

transportare solved withthe nitedieren eorniteelementmethodwhereas aNewton

algorithmisused tosolve the nonlinear algeabri system of equation asso iated with the

hemi al rea tions. Depending on the type of hemi al rea tions involved in a problem,

dierentalgorithms anbeusedtosplittransportand hemistry,asreviewedinreferen es

[42, 43℄.

When the lo al equilibrium assumption is not valid, hemi al rea tions are modeled

with kineti expressions [39℄ involving rea tion rate. This ase arises for problems in

groundwater ioni transport where large pressure head gradients an be at the origin of

high uid velo ity. The topi of kineti ally ontrolled rea tion modeling is dis ussed in

referen es [44, 45, 46℄.

3.6 Evaluation of the ioni transport properties

Two dierent transport parameters appear in equation (88). There is the diusion

o-e ient D

i

asso iated with the diusion pro ess and the liquid water diusivity D

L to

hara terizethe ee t ofthe uidvelo ity onthe ioni transport. Adis ussion ofD

L was

already given inse tion 2.5.

The diusion oe ient is evaluated with the migration experiment test. It onsists

of a elerating hloride ionswith anapplied externalpotentialthrough adisk of

ement-based materials glued between two ells lled with ioni solutions. The analysis of the

results yieldsthe diusion oe ients. Dierent analysis methodsare found in the

liter-ature. One is based onsteady state measurements of hloride having rossed the sample

[47,48℄. Another[49℄isbasedonmeasuringnon-steady state hlorideprolesbygrinding

the sample after a short exposure. A re ent paper by Samson et al. [50℄ des ribes a

method based on urrent measurements during the migration test. The measurements

All these methods are performed insaturated onditions. As shown in equation(76),

the diusion oe ientD

i

depends, through 

L

,onthe saturation ondition. Nomethod

ould be found in the literature to evaluate this parameter for unsaturated onditions.

However, it is possible that D

i

might not be ae ted by the saturation state of the

material above a given saturation level, the latter being dened as s = 

L

=, where 

is the porosity. Revil [32℄ showed that for shalysand, the diusion of the ions is almost

unae ted for a water saturation above 0.6. We thus infer that for on rete stru tures

exposed tohigh relativehumidityenvironment, the diusion oe ient isindependentof

the water ontent.

In the ux equation (75), the re urring quantity 

L 

L

, whi h ould also be written

 L

s,is analogous toa saturation-dependent formation fa tor for the liquidphase of the

poresystem. ThesaturationsresultsfromtheaveragingovertheREV.Thetortuosity

L

is alsoa fun tion of the saturation and ree ts the onne tedness of the moisturephase.

At a riti al moisture ontent s

, the liquid phase is no longer onne ted, the tortuosity

 L

goes tozero, the transport withinthe liquidphase eases.

The remaining question is the dependen e of the tortuosity 

L

on the saturation.

Althoughnopre isedataexistfor ementitioussystems,thereexistsqualitativedatafrom

whi hinferen es an bemade. Thesedata typi ally expressthe relative total ondu tion

as a fun tionof the saturation s. The relative total ondu tion =

o

is analogous to the

produ t of the saturation and the relative tortuosity:

(s) (s=1) = s  L (s)  L (s =1) = s  L  Lo (96)

Therefore, dividing these results by s will yieldthe relative hange inthe tortuosity.

TheworkofMartys[19℄suggeststhatforapreferentiallywettingliquidbeingdispla ed

by a non-wetting one, the limiting behavior of 

L

near saturation an be approximated

by the diluteee tive mediumtheory result:

s  L  Lo =1 3 2 (1 s)+ 1 2 (1 s) 2 s!1 (97)

ap- L  Lo = s 2 + 1 2 (98)

Therefore, a de rease to 80 % saturation will result in a 10 % hange in the tortuosity.

Given that transport oe ients an routinely hange by orders of magnitude, a 10 %

hange in the tortuosity is relatively quite minor. Sin e this result is only approximate

near saturation, further redu tions insaturation would have a fargreater ee t.

4 Con lusion

Themathemati almodeldevelopedinthispaperisrstsummarized. Formaterialswhere

the water transport o urs as a result of apillary su tion, the water ontent prole an

be al ulatedwith Ri hards'equation(51). The ions willmove inthe materialunderthe

ombinedee tofdiusion(in ludingele tri alanda tivityee ts)andwatermovement

a ording to equation (88). The ele tri al potential, arising from the ele tri al oupling

between the ions in order to maintain a neutral solution, is al ulated with Poisson's

equation (92). The hemi al a tivity oe ients, for the highly harged pore solution

of ement-based materials, an be evaluated with expression (93). Finally, several

refer-en es were given to address the modeling of hemi al rea tions o uring in ementitous

materials.

The useoftheaveragingte hnique learly helpsto larifythe meaningofsome

impor-tantparametersinthe model. A ordingtothiste hnique,thewater ontentinRi hards'

model orrespondstoavolumetri water ontent. Thewaterdiusivitywas learlyshown

to be a ontribution of both liquid water and vapor transport. The diusion oe ient,

the parameter that hara terize the ioni diusion pro ess, isdire tly related tothe

geo-metri alproperties of the materialthrough aparameter alled the tortuosity.

The averaging te hnique proved to be a powerful mathemati al tool to lay the

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Solid

Solid

Solid

Solid

Liquid

Gas

V

o

V

o

L

V

_o

G

Toward the center of the pore

Surface of the material

Stern

layer

Diffuse

layer

Free

water

Solid phase

Aqueous phase

Shear plane

Concentration

Potential

n

_LG

Gaseous phase

n

_LS

grad c

_i

Figure2: Con entrationandpotentialprolea rossaporenear thesolid/liquidinterfa e

0

0.5

1

1.5

2

2.5

3

0

5

10

15

20

25

30

35

40

45

50

Concentration (mmol/L)

Distance from the surface (nm)

c

+

c

−

c(

∞

)

κ

−1

0

20

40

60

80

100

120

140

0

5

10

15

20

25

30

35

40

45

50

Concentration (mmol/L)

Distance from the surface (nm)

c

+

c

−

c(

∞

)

κ

−1

Figure 3: Con entration proles of a 1-1 ele trolyte near a harged surfa e al ulated

with the Gouy-Chapmandoublelayer model. The al ulations were made with asurfa e

potentialof 25 mV. The Debye length

1