Moisture Diffusion in Cementitious Materials Adsorption Isotherms Yunping Xi, Zden~k P. Ba~ant, and Hamlin M. Jennings Department of Civil Engineering, Northwestern University, Evanston, Illinois

This article describes an improvement on a previous model proposed by Ba~ant and Najjar, in which moisture diffusivity and moisture capacity are treated as separate parameters. These parameters are evaluated from independent test results, and are shown to depend on the water:cement ratio, curing time, temperature, and cement type. The moisture capacity is obtained as the slope of the adsorption isotherm. A mathematical model is developed and is shown to pre- dict experimental adsorption isotherms of Portland cement paste very well. In the present form, the model is not applicable to high temperatures. ADVANCED CEMENT BASED MATERIALS 1994, 1, 248--257 KEY WORDS: Adsorption, Concrete, Hardened cement paste, Moisture diffusion, Moisture effects, Permeability, Porosity

he properties of cementitious materials de- pend strongly on the moisture content. There- fore, knowledge of the moisture distribution

within concrete structures at various times is of con- siderable practical importance. This is especially true if time-dependent phenomena such as creep, shrinkage, fire resistance, and durability are analyzed. The migra- tion of moisture within concrete is more complex then in most other porous media because a very wide range of pore sizes is present in cement paste, and the pore structure changes with age. An accurate and general theoretical model for cementitious materials does not yet exist.

A very general model is that proposed by Ba~ant and Najjar [1]. It is used widely for diffusion analysis and, especially in computer programs, for predicting diffu- sion through nuclear containment and pressure ves- sels. This model involves an S-shaped curve that de- scribes the dependence of the diffusion coefficient on the relative humidity within the pores in concrete, and, in a later extension by Ba~.ant and Thonguthai [2]

Address correspondence to: Zden6k P. Ba2ant, W.P. Murphy Professor of Civil Engineering and Materials Science, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208.

and Ba~.ant and Wittmann [3], also on the temperature and age of concrete.

However, the effects of composition, particularly the water:cement ratio, are not considered. The slope of the desorption isotherm, called moisture capacity, is assumed to be constant, although actually it is not con- stant. Furthermore, this model was calibrated only by limited test data that were available in 1970, which are not as reliable as more recent data.

Generalizing Ba~ant and Najjar's model [1], Sakata [4] obtained a similar expression which incorporates the effects of the water:cement ratio, in addition to age of concrete. However, he did not give general expres- sions for these parameters. Nor did he relate the ob- served phenomena to implied diffusion mechanisms, although this is essential in order to understand the time-dependent deformations of concrete.

The purpose of this article is to improve the model of Ba~ant and Najjar [1]. First the proper form of the dif- fusion equation will be formulated and its two coeffi- cients, moisture capacity and diffusivity, will be de- scribed separately. The moisture capacity is defined as the derivative of the moisture content with respect to the relative humidity in a pore. Thus, the relationship between the moisture content and the relative humid- ity in a pore at constant temperature, called the ad- sorption isotherm, needs to be established, and then the moisture capacity follows. The first part of this study will deal with the adsorption isotherm and the second part [5] with the moisture capacity and diffu- sivity. A semiempirical expression for the adsorption isotherm will be developed by parametric analysis of the available adsorption test data. The effect of tem- perature will be included automatically, and the effects of water:cement ratio, age, and type of cement will also be taken into account.

Diffusion Equation There are three different types of approaches to the diffusion problem: (1) a simple formulation based upon Fick's law, or Darcy's law [1], (2) computer sim-

© Elsevier Science Inc. Received August 5, 1993 ISSN 1065-7355/94t57.00 Accepted December 22, 1993

Moisture Diffusion in Cementitious Materials Adsorption Isotherms Yunping Xi, Zdenek P. Bazant, and Hamlin M. Jennings Department of Civil Engineering, Northwestern University, Evanston, Illinois

This article describes an improvement on a previous model proposed by Baiant and Najjar, in which moisture diffusivity and moisture capacity are treated as separate parameters. These parameters are evaluated from independent test results, and are shown to depend on the water:cement ratio, curing time, temperature, and cement type. The moisture capacity is obtained as the slope of the adsorption isotherm. A mathematical model is developed and is shown to pre­dict experimental adsorption isotherms of Portland cement paste very well. In the present form, the model is not applicable to high temperatures. ADVANCED CEMENT BASED MATERIALS 1994, I, 248--257 KEY WORDS: Adsorption, Concrete, Hardened cement paste, Moisture diffusion, Moisture effects, Permeability, Porosity

The properties of cementitious materials de­pend strongly on the moisture content. There­fore, knowledge of the moisture distribution

within concrete structures at various times is of con­siderable practical importance. This is especially true if time-dependent phenomena such as creep, shrinkage, fire resistance, and durability are analyzed. The migra­tion of moisture within concrete is more complex then in most other porous media because a very wide range of pore sizes is present in cement paste, and the pore structure changes with age. An accurate and general theoretical model for cementitious materials does not yet exist.

A very general model is that proposed by Bazant and Najjar [1]. It is used widely for diffusion analysis and, especially in computer programs, for predicting diffu­sion through nuclear containment and pressure ves­sels. This model involves an S-shaped curve that de­scribes the dependence of the diffusion coefficient on the relative humidity within the pores in concrete, and, in a later extension by Bazant and Thonguthai [2]

Address correspondence to: Zdenek P. BaZant, W.P. Murphy Professor of Civil Engineering and Materials Science, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208.

© Elsevier Science Inc. ISSN 1065-7355/94/$7.00

and Bazant and Wittmann [3], also on the temperature and age of concrete.

However, the effects of composition, particularly the water:cement ratio, are not considered. The slope of the desorption isotherm, called moisture capacity, is assumed to be constant, although actually it is not con­stant. Furthermore, this model was calibrated only by limited test data that were available in 1970, which are not as reliable as more recent data.

Generalizing Bazant and Najjar's model [1], Sakata [4] obtained a similar expression which incorporates the effects of the water:cement ratio, in addition to age of concrete. However, he did not give general expres­sions for these parameters. Nor did he relate the ob­served phenomena to implied diffusion mechanisms, although this is essential in order to understand the time-dependent deformations of concrete.

The purpose of this article is to improve the model of Bazant and Najjar [1]. First the proper form of the dif­fusion equation will be formulated and its two coeffi­cients, moisture capacity and diffusivity, will be de­scribed separately. The moisture capacity is defined as the derivative of the moisture content with respect to the relative humidity in a pore. Thus, the relationship between the moisture content and the relative humid­ity in a pore at constant temperature, called the ad­sorption isotherm, needs to be established, and then the moisture capacity follows. The first part of this study will deal with the adsorption isotherm and the second part [5] with the moisture capacity and diffu­sivity. A semiempirical expression for the adsorption isotherm will be developed by parametric analysis of the available adsorption test data. The effect of tem­perature will be included automatically, and the effects of water:cement ratio, age, and type of cement will also be taken into account.

Diffusion Equation There are three different types of approaches to the diffusion problem: (1) a simple formulation based upon Fick's law, or Darcy's law [1], (2) computer sim-

Received August 5, 1993 Accepted December 22, 1993

Advn Cem Bas Mat Y. Xi et al. 249 1994;1:248-257

ulation of flow through a random particle system [6-9], or (3) a formulation based directly on the basic physical laws, such as the kinetic theory of ideal gases and the conservation of mass and energy [10,11].

It is well established that, in cement paste, moisture moves from regions where it is plentiful to regions where it is scarce. This suggests that the moisture flux is proportional to the gradient of some variable which measures moisture content. However, the moisture flux may be expressed in two different ways. Either

l = - D w grad(We) (1)

with the mass balance equation

aw a(We+ W.) at at

- div(J) = div(Dw grad We)

(2)

o r

J = - D h grad H

with

(3)

OW OW OH - - div(]) = div(Dh grad H). (4)

at 3H at

H represents pore relative humidity, t is time, W is the total water content (for unit volume of material), We is the evaporable water content, and W, is the nonevap- orable, or chemically bound, water content. Equations 1 and 2 express the moisture flux in terms of the gra- dient of the water content. Equations 3 and 4 express the flux in terms of the gradient of the pore relative humidity (for isothermal conditions only). In eqs I to 4, D w and D h have different physical meanings: D w rep- resents moisture diffusivity, and D h represents perme- ability, or humidity diffusivity [12].

In this s tudy we use the formulation in terms of pore relative humidity, H, because the use of H appears to be more practical, for two reasons: (1) When the changes of evaporable water content due to hydration of cement are taken into account, one finds that, for usual water:cement ratios, the drop in H due to self- dessiccation caused by hydration (as in sealed speci- mens) is rather small, typically H > 0.97 [13]. In fact, it can be neglected even if hydration has not yet termi- nated. On the other hand, OWn/Ot(in eq 2) never has a negligible value unless hydration has ceased. (2) When generalization to variable temperature is considered, grad H can still be considered as a driving force of diffusion, but not grad W or grad W e [2].

In eq 4, two coefficients must be determined: mois-

ture capacity, OW/OH, and diffusivity, D h. Both coeffi- cients depend on H, which causes the nonlinearity of eq 4. Independent test results will be used to evaluate the two coefficients. The moisture capacity, represent- ing the derivative of the equilibrium adsorption iso- therm, can be evaluated from test results. Therefore, the adsorption isotherm will be analyzed first, and then the diffusivity will be evaluated from drying tests [51.

Prediction Formula for Adsorption Isotherm BET Model The best known isotherm model is the famous Brun- auer-Emmett-Teller (BET) model [14], derived from statistical thermodynamics of adsorption. But contrary to early assumptions, the range of validity to the BET equation for cement and concrete does not normally cover the relative pressures (humidity) from 0.05 to between 0.30 and 0.50; rather, it often covers only the range from 0.01 to 0.1 [15-17]. A host of attempts have been made to modify the BET equation in order to obtain better agreement with experimental isotherm data in the multilayer region. Some of the modified models are: the BDDT model [18]; the FHH model [19]; Hillerborg's formula [20]; and the BSB model [21], which is also called the three-parameter BET model, since it is a generalization of the BET model. The BSB model will be used in this study. It is applicable in the relative pressure range from 0.05 to 1.0, and reads:

W = C k V m H

(1 - kH)[1 + (C - 1) k H ] '

( E l - E l ) C = exp R--T

(5)

where H = P/Ps, Ps is the pressure at saturation; C and V m are two constants used in the BET model (V m = monolayer capacity); k is the third constant such that k < 1; E1 is the total heat of adsorption per mole of va- por; El is the latent heat of condensation per mole; R is the gas constant; T is absolute temperature; and W is the quantity of vapor adsorbed at pressure p in grams of water per gram of cement paste.

The adsorption of water in hardened Portland ce- ment paste is influenced by many parameters. In gen- eral, those that affect the adsorption isotherm include any parameter that contributes to the hydration pro- cess of Portland cement and hence to the constitution of the pore structure and the pore-size distribution. Some are as follows: the original water:cement ratio, w/c; type of cement, that is, the chemical composition

Advn Cern Bas Mat 1994;1 :248--257

ulation of flow through a random particle system [6-9], or (3) a formulation based directly on the basic physical laws, such as the kinetic theory of ideal gases and the conservation of mass and energy [10,11].

It is well established that, in cement paste, moisture moves from regions where it is plentiful to regions where it is scarce. This suggests that the moisture flux is proportional to the gradient of some variable which measures moisture content. However, the moisture flux may be expressed in two different ways. Either

J = - Dw grad(We) (1)

with the mass balance equation

aw a(We + Wn) at = at = -div(J) = div(Dw grad We)

(2)

or

J = -Dh grad H (3)

with

aw awaH at = aH at = -div(J) = div(Dh grad H). (4)

H represents pore relative humidity, t is time, W is the total water content (for unit volume of material), We is the evaporable water content, and W n is the nonevap­orable, or chemically bound, water content. Equations 1 and 2 express the moisture flux in terms of the gra­dient of the water content. Equations 3 and 4 express the flux in terms of the gradient of the pore relative humidity (for isothermal conditions only). In eqs 1 to 4, Dw and Dh have different physical meanings: Dw rep­resents moisture diffusivity, and Dh represents perme­ability, or humidity diffusivity [12].

In this study we use the formulation in terms of pore relative humidity, H, because the use of H appears to be more practical, for two reasons: (1) When the changes of evaporable water content due to hydration of cement are taken into account, one finds that, for usual water:cement ratios, the drop in H due to self­dessiccation caused by hydration (as in sealed speci­mens) is rather small, typically H > 0.97 [13]. In fact, it can be neglected even if hydration has not yet termi­nated. On the other hand, aw nlat(in eq 2) never has a negligible value unless hydration has ceased. (2) When generalization to variable temperature is considered, grad H can still be considered as a driving force of diffusion, but not grad W or grad We [2].

In eq 4, two coefficients must be determined: mois-

Y. Xi et al. 249

ture capacity, aW/aH, and diffusivity, Dh • Both coeffi­cients depend on H, which causes the nonlinearity of eq 4. Independent test results will be used to evaluate the two coefficients. The moisture capacity, represent­ing the derivative of the equilibrium adsorption iso­therm, can be evaluated from test results. Therefore, the adsorption isotherm will be analyzed first, and then the diffusivity will be evaluated from drying tests [5].

Prediction Formula for Adsorption Isotherm BET Model The best known isotherm model is the famous Brun­auer-Emmett-Teller (BET) model [14], derived from statistical thermodynamics of adsorption. But contrary to early assumptions, the range of validity to the BET equation for cement and concrete does not normally cover the relative pressures (humidity) from 0.05 to between 0.30 and 0.50; rather, it often covers only the range from 0.01 to 0.1 [15-17]. A host of attempts have been made to modify the BET equation in order to obtain better agreement with experimental isotherm data in the multilayer region. Some of the modified models are: the BDDT model [18]; the FHH model [19]; Hillerborg's formula [20]; and the BSB model [21], which is also called the three-parameter BET model, since it is a generalization of the BET model. The BSB model will be used in this study. It is applicable in the relative pressure range from 0.05 to 1.0, and reads:

W = (1 - kH)[l + (C - 1) kH] ,

(E1 - El)

C = exp RT (5)

where H = pips, Ps is the pressure at saturation; C and V m are two constants used in the BET model (V m =

monolayer capacity); k is the third constant such that k < 1; E1 is the total heat of adsorption per mole of va­por; E/ is the latent heat of condensation per mole; R is the gas constant; T is absolute temperature; and W is the quantity of vapor adsorbed at pressure p in grams of water per gram of cement paste.

The adsorption of water in hardened Portland ce­ment paste is influenced by many parameters. In gen­eral, those that affect the adsorption isotherm include any parameter that contributes to the hydration pro­cess of Portland cement and hence to the constitution of the pore structure and the pore-size distribution. Some are as follows: the original water:cement ratio, wlc; type of cement, that is, the chemical composition

250 Moisture Diffusion in Cementitious Materials Advn Cem Bas Mat 1994; 1:248-257

of the cement; curing time; temperature; curing method; carbonation (for structures with thin cross- sections); added ingredients such as accelerators, wa- ter reducers, retarders, superplasticizers, air entrain- ing admixtures, or antifreezing admixtures; slag- cement [22]; sand-cement ratio, s/c; and gravel-cement ratio, g/c [23].

In this study, only the effects of the original water: cement ratio, w/c; age, t; temperature, T; and cement type are established in an empirical relationship with the amount of adsorption.

Monolayer Capacity The monolayer capacity, V m, is defined as the mass of adsorbate required to cover the adsorbent with a single molecular layer. Presently, there are several experi- mental methods in use to determine V m [24,25]. The following empirical expression for V m fits the observed test data:

Vm = V(t , w/c, C t, T) = Wt ( t )Vwc(W/C)Wc t ( c t )WT(Z ) (6)

where t is the age of specimen in days, and c t indicates the type of cement. The functions in eq 6 may be char- acterized as follows.

1. Vt ( t ) - - -Figure I shows the effect of the curing time on the isotherms. The relationship between V m and t is then obtained as shown in Figure 2. Figures 1 and 2 show V m to increase as time increases. V m, which de- pends upon the extent of hydration, approaches a

.K

0.15

0.12

0.09

0.06

0.03

0.00

T , , 2 ~ K . / / / /

/ / /

o.o oh o:4 o:6 o.~ 1,o

x " pip s

FIGURE 1. Effect of age t o at the start of drying on the ad- sorption isotherms.

E 0.05 >

0 . 0 8

0.07

0.06

=

0 . 0 4

o

0 . 0 3

0 0 2

= w/¢ = 0.3 Type N cement T - 2 9 4 K

o w/c = 0.3 Type II cement T== 2 9 4 " K

0 160 260 360 400

t in days

FIGURE 2. Dependence of monolayer capacity on concrete age t.

maximum at a curing time of approximately 6 months. Therefore, after 6 months of curing, the effect of age upon the adsorption isotherm of concrete may be ex- cluded without any significant loss of accuracy. Vt(t ) is therefore assumed to be of the form a + bit. Upon comparison with test data, Vt(t ) behaves according to the following empirical equation

0.22 Vt(t) = 0.068 - T (7)

which is valid for t I> 5 days. However, upon a check of test data gathered when t < 5 days [26,27], the V m values corresponding to the range 0.25 ~< t ~< 4 (in days) are almost equal to the value corresponding to t = 5 days. This indicates that after a certain surface area gets established in cement gel after the initial set, this area remains almost constant during five subsequent days, regardless of the curing period. Hence, for t ~< 5 days, Vt can be treated as constant; V t = 0.024.

2. V c t - - T h e results of some of the adsorption tests are shown for various cements in Figure 3. For these data, Powers and Brownyard [28] did not classify the cement types for the specimens, but only the com- pound compositions. Upon comparison with the com- positions of some commercial U.S. cements [29], the cements were classified into approximately four types.

In spite of extensive studies [22,28], the mechanism by which different compound compositions affect the shape of the adsorption isotherm is not clear. There- fore, we can only give empirical values obtained by fitting the data for different types of cements, in order to describe the effects of compound composition on the adsorption curves:

250 Moisture Diffusion in Cernentitious Materials

of the cement; curing time; temperature; curing method; carbonation (for structures with thin cross­sections); added ingredients such as accelerators, wa­ter reducers, retarders, superplasticizers, air entrain­ing admixtures, or antifreezing admixtures; slag­cement [22]; sand-cement ratio, sic; and gravel-cement ratio, glc [23].

In this study, only the effects of the original water: cement ratio, w/c; age, t; temperature, T; and cement type are established in an empirical relationship with the amount of adsorption.

Monolayer Capacity The monolayer capacity, V m' is defined as the mass of adsorbate required to cover the adsorbent with a single molecular layer. Presently, there are several experi­mental methods in use to determine V m [24,25]. The following empirical expression for V m fits the observed test data:

where t is the age of specimen in days, and Ct indicates the type of cement. The functions in eq 6 may be char­acterized as follows.

1. Vt(t )-Figure 1 shows the effect of the curing time on the isotherms. The relationship between V m and t is then obtained as shown in Figure 2. Figures 1 and 2 show V m to increase as time increases. V m' which de­pends upon the extent of hydration, approaches a

0.15

w/c - 0.3

0.12 Type IV cement . T-284 K

0.09

I •• 0.06 ~

0.03

0.00 0.0 0.2 0.4 0.6 0.8 1.0

X - pIp • FIGURE 1. Effect of age to at the start of drying on the ad­sorption isotherms.

0.08

0.07

0.06

E 0.05

> --0.04

G---<>

0.03

0.02 0 100 200

t in days

Advn Cern Bas Mat 1994; 1 :248-257

w/c - 0.3 Type IV cement T-294 K

w/c - 0.3 Type II cement T-294'K

300 400

FIGURE 2, Dependence of monolayer capacity on concrete age t.

maximum at a curing time of approximately 6 months. Therefore, after 6 months of curing, the effect of age upon the adsorption isotherm of concrete may be ex­cluded without any significant loss of accuracy. Vt(t) is therefore assumed to be of the form a + bit. Upon comparison with test data, Vt(t) behaves according to the following empirical equation

0.22 Vt(t) = 0.068 - -t- (7)

which is valid for t ;,: 5 days. However, upon a check of test data gathered when t < 5 days [26,27], the V m

values corresponding to the range 0.25 .;; t .;; 4 (in days) are almost equal to the value corresponding to t = 5 days. This indicates that after a certain surface area gets established in cement gel after the initial set, this area remains almost constant during five subsequent days, regardless of the curing period. Hence, for t .;; 5 days, V t can be treated as constant; Vt = 0.024.

2. Vet-The results of some of the adsorption tests are shown for various cements in Figure 3. For these data, Powers and Brownyard [28] did not classify the cement types for the specimens, but only the com­pound compositions. Upon comparison with the com­positions of some commercial U.S. cements [29], the cements were classified into approximately four types.

In spite of extensive studies [22,28], the mechanism by which different compound compositions affect the shape of the adsorption isotherm is not clear. There­fore, we can only give empirical values obtained by fitting the data for different types of cements, in order to describe the effects of compound composition on the adsorption curves:

Advn Cem Bas Mat Y. Xi et al. 251 1994;1:248-257

.¢

0.18

0,15

0.12

0.09

0.06

0 .03

wl¢ - 0.3 t , , 28 dlyl L

0.00 o.o o.'2 o~ o.~ o.~ lO

x - p/p=

FIGURE 3. Effect of t he t y p e of c e m e n t o n t he a d s o r p t i o n i s o t h e r m s .

type 1: Vct = 0.9; type 2: V~t = 1; type 3: Vet

= 0.85; type 4: Vct= 0.6.

3. Vwc~Figure 4 shows that V m is linearly related to the water:cement ratio, w/c, within the range 0.3 to 0.6. This relationship means that a low w/c corresponds to a low porosity and a low surface area. Therefore, V m and w/c are assumed to be linked by a linear relation obtained from the test data:

w Vwc = 0.85 + 0.45-- .

c

such as 0.75, no indication exists that this linear rela- tion remains true. In low porosity pastes, it is in fact impossible for V m to decrease with a very low w/c, and a limiting value may exist. Therefore, as an approxi- marion, if the w/c falls below 0.3, Vw¢ should be taken as the value corresponding to w/c = 0.3, and if the w/c is above 0.6, Vwc should be taken as the value corre- sponding to w/c = 0.6.

Parameter C Following the substitution, C O = (E 1 - El)/R, C in eq 5 becomes

C = exp ( - ~ ) , C0 = T ln(C) (10)

where E1 - E~ is the net heat of adsorption, deter- mined from heat of immersion experiments. As shown in Table 1, C varies from approximately 10 to 50. If Vm is held constant, the influence of temperature on the shapes of adsorption curves can be predicted from the

(8) present model, and one finds that near room temper- ature the adsorption curve changes with temperature only slightly. This means that the isotherms can be considered insensitive to changes in C within a rather wide range of values. Thus, it seems reasonable to forgo further analysis of C and simply assume that Co is constant. By eq 10, the expected value of Co can be obtained from test data; Co = 855 (coefficient of vari- ation = 0.098). However, for high temperatures these simplifications are not possible and further phenom-

(9) ena need to be taken into consideration [2].

However, for a low w/c such as 0.15, or a high w/c

0.09

0.08

E 0.07

0.06 t - 180 diys Type It cement T - 294"K

0.05 0.2 o.'~ 0/4 o.'5 0.6

w/¢

FIGURE 4. Dependence of monolayer capacity on the water: cement ratio.

Parameter k Parameter k of the BSB model results from the assump- tion that the number of adsorbed layers is finite, pos- sibly even a small number [21]. To determine k, first the expression for the number of adsorbed layers at the saturation state, n, must be found, and n is then con- verted into parameter k, similar to the expression for vm:

n = N(t,w/c, ct, T) = Nt(t) Nwc(W/C) Nct(Ct) NT(T). (11)

The influences of t and w/c on the parameter n are shown in Figures 5 and 6, respectively. The empirical equations for N t and Nwc obtained by data fitting and the specific values of Net for different types of cements are as follows:

15 Nt(t) = 2.5 + -~- (12)

Advn Cern Bas Mat 1994;1 :248-257

0.16

0.15

0.12

I •• 0.09

~

0.06

0.03

0.00 0.0

w/c - 0.3 t - 28 days T-294'K

0.2 0.4 0.6

x - pip •

0.8 1.0

FIGURE 3. Effect of the type of cement on the adsorption isotherms.

type 1: Vet = 0.9; type 2: Vet = 1; type 3: Vet

= 0.85; type 4: Vet = 0.6. (8)

3. V we-Figure 4 shows that V m is linearly related to the water:cement ratio, w/c, within the range 0.3 to 0.6. This relationship means that a low wlc corresponds to a low porosity and a low surface area. Therefore, V m

and wlc are assumed to be linked by a linear relation obtained from the test data:

w Vwc = 0.85 + 0.45 - .

c (9)

However, for a low w/c such as 0.15, or a high w/c

0.09 -r----------------,

0.08 • • • • • E > 0.07

• 0.06 t - 180 days

Type II cement T-294'K

0.05 +----r-------r-----.-----l 0.2 0.3 0.4 0.5 0.6

w/c

FIGURE 4, Dependence of monolayer capacity on the water: cement ratio.

Y. Xi et al. 251

such as 0.75, no indication exists that this linear rela­tion remains true. In low porosity pastes, it is in fact impossible for V m to decrease with a very low w/c, and a limiting value may exist. Therefore, as an approxi­mation, if the wlc falls below 0.3, V we should be taken as the value corresponding to wlc = 0.3, and if the w/c is above 0.6, V we should be taken as the value corre­sponding to w/c = 0.6.

Parameter C Following the substitution, Co = (E1 - E/)/R, C in eq 5 becomes

C = exp (~) , Co = T In(C) (10)

where E1 - E/ is the net heat of adsorption, deter­mined from heat of immersion experiments. As shown in Table 1, C varies from approximately 10 to 50. If V m

is held constant, the influence of temperature on the shapes of adsorption curves can be predicted from the present model, and one finds that near room temper­ature the adsorption curve changes with temperature only slightly. This means that the isotherms can be considered insensitive to changes in C within a rather wide range of values. Thus, it seems reasonable to forgo further analysis of C and simply assume that Co is constant. By eq 10, the expected value of Co can be obtained from test data; Co = 855 (coefficient of vari­ation = 0.098). However, for high temperatures these simplifications are not possible and further phenom­ena need to be taken into consideration [2].

Parameter k Parameter k of the BSB model results from the assump­tion that the number of adsorbed layers is finite, pos­sibly even a small number [21]. To determine k, first the expression for the number of adsorbed layers at the saturation state, n, must be found, and n is then con­verted into parameter k, similar to the expression for Vm:

n = N(t,wlc, Ct' T) = Nt(t) Nwe(wlc) Nct(ct) Nr(T). (11)

The influences of t and wlc on the parameter n are shown in Figures 5 and 6, respectively. The empirical equations for Nt and Nwe obtained by data fitting and the specific values of Net for different types of cements are as follows:

15 Nt(t) = 2.5 + t (12)

252 Moisture Diffusion in Cementitious Materials Advn Cem Bas Mat 1994;1:248-257

TABLE 1. Test information and opt imized parameters

V~ C k n w/c t T Type

0.0234 15.14 0.8661 7.39 0.309 7.0 294.11 4 0.0302 17.94 0.8282 5.75 0.309 14.0 294.11 4 0.0364 27.16 0.8008 4.79 0.309 28.0 294.11 4 0.0462 25.37 0.7563 4.05 0.309 56.0 294.11 4 0.0525 21.99 0.7021 3.29 0.309 90.0 294.11 4 0.0644 15.31 0.6253 2.570 0.309 180.0 294.11 4 0.0619 23.19 0.6299 2.640 0.309 365.0 294.11 4 0.0371 23.84 0.7989 4.92 0.316 7.0 294.11 2 0.0529 14.32 0.6946 3.18 0.316 14.0 294.11 2 0.0609 10.42 0.6655 2.85 0.316 28.0 294.11 2 0.0580 33.86 0.6685 2.97 0.316 56.0 294.11 2 0.0696 15.84 0.5947 2.37 0.316 90.0 294.11 2 0.0680 13.91 0.6012 2.930 0.316 180.0 294.11 2 0.0512 18.77 0.7467 3.88 0.334 28.0 294.11 3 0.0368 22.61 0.8206 5.52 0.318 28.0 294.11 2 0.0319 19.30 0.8189 5.46 0.324 28.0 294.11 4 0.0465 19.70 0.7808 4.5 0.334 28.0 294.11 1 0.0562 15.96 0.6984 3.23 0.328 28.0 294.11 1 0.0558 20.62 0.7244 3.56 0.334 90.0 294.11 3 0.0530 16.90 0.7227 3.53 0.318 90.0 294.11 2 0.0447 22.24 0.7631 4.16 0.324 90.0 294.11 4 0.0556 19.53 0.7422 3.81 0.334 90.0 294.11 1 0.0669 12.43 0.6343 2.61 0.328 90.0 294.11 1 0.0609 10.42 0.6655 2.85 0.316 28.0 294.11 2 0.0657 9.637 0.7122 3.330 0.433 28.0 294.11 2 0.0632 11.61 0.7805 4.450 0.57 28.0 294.11 2 0.0494 34.60 0.7511 3.980 0.316 28.0 294.11 2 0.0492 34.19 0.7975 4.900 0.432 28.0 294.11 2 0.0481 45.45 0.8363 6.080 0.582 28.0 294.11 2 0.0680 13.91 0.6012 2.390 0.316 180.0 294.11 2 0.0701 13.51 0.7103 3.350 0.433 180.0 294.11 2 0.0751 17.05 0.7553 4.01 0.57 180.0 294.11 2 0.0590 22.54 0.6531 2.820 0.316 180.0 294.11 2 0.0678 15.11 0.7158 3.430 0.432 180.0 294.11 2 0.0690 13.77 0.7670 4.200 0.582 180.0 294.11 2

0.0613 9.753 0.6920 3.100 0.400 730.0 298.00 1 0.0642 18.60 0.7230 3.540 0.450 2555.0 298.00 1 0.0437 47.68 0.8820 8.450 0.700 730.0 298.00 1

0.0149 33.30 0.8740 7.940 0.200 1.000 298.00 2 0.0235 49.98 0.7380 3.790 0.200 3.000 298.00 2

0.0106 7.920 0.9230 12.99 0.200 0.250 308.00 1 0.0209 66.26 0.8070 5.160 0.200 1.000 308.00 1 0.0208 79.87 0.8240 5.680 0.200 3.000 308.00 1 0.0334 17.41 0.7380 3.740 0.200 7.000 308.00 1 0.0393 29.88 0.7190 3.510 0.200 28.00 308.00 1 0.0370 57.76 0.7270 3.640 0.200 90.00 308.00 1

a n d

w Nwc(W/C) = 0.33 + 2 . 2 - (13)

C

p r o v i d e d t ha t t > 5 d a y s ; o t h e r w i s e , Nt(t ) = 5.5. Fu r - t h e r m o r e ,

t y p e 1: Net = 1.1; t y p e 2: Nct= 1; t y p e 3: Nct = 1.15; t y p e 4: Nct= 1.5. (14)

Equa t ion 12 s h o w s that , a t s a tu ra t ion state , the aver - age radi i of the p o r e s accessible to w a t e r dec rea se w i t h p r o g r e s s i n g h y d r a t i o n . This h a s b e e n e x p e r i m e n t a l l y con f i rmed b y o t h e r inves t iga to r s [26]. Equa t ion 12 a lso g ives s o m e idea a b o u t the p o r e s ize at s a tu r a t i on state.

The e x p r e s s i o n for k is eas i ly o b t a i n e d f r o m the ex- p r e s s i o n s for n a n d C. F r o m eq 5,

W C k H (15) n

V--~ = (1 - kH)[1 + (C - 1)kH]

252 Moisture Diffusion in Cementitious Materials

TABLE 1. Test information and optimized parameters

Vm C

0.0234 15.14 0.0302 17.94 0.0364 27.16 0.0462 25.37 0.0525 21.99 0.0644 15.31 0.0619 23.19 0.0371 23.84 0.0529 14.32 0.0609 10.42 0.0580 33.86 0.0696 15.84 0.0680 13.91 0.0512 18.77 0.0368 22.61 0.0319 19.30 0.0465 19.70 0.0562 15.96 0.0558 20.62 0.0530 16.90 0.0447 22.24 0.0556 19.53 0.0669 12.43 0.0609 10.42 0.0657 9.637 0.0632 11.61 0.0494 34.60 0.0492 34.19 0.0481 45.45 0.0680 13.91 0.0701 13.51 0.0751 17.05 0.0590 22.54 0.0678 15.11 0.0690 13.77

0.0613 9.753 0.0642 18.60 0.0437 47.68

0.0149 33.30 0.0235 49.98

0.0106 7.920 0.0209 66.26 0.0208 79.87 0.0334 17.41 0.0393 29.88 0.0370 57.76

and

w Nwc(w/c) = 0.33 + 2.2 -c

k n

0.8661 7.39 0.8282 5.75 0.8008 4.79 0.7563 4.05 0.7021 3.29 0.6253 2.570 0.6299 2.640 0.7989 4.92 0.6946 3.18 0.6655 2.85 0.6685 2.97 0.5947 2.37 0.6012 2.930 0.7467 3.88 0.8206 5.52 0.8189 5.46 0.7808 4.5 0.6984 3.23 0.7244 3.56 0.7227 3.53 0.7631 4.16 0.7422 3.81 0.6343 2.61 0.6655 2.85 0.7122 3.330 0.7805 4.450 0.7511 3.980 0.7975 4.900 0.8363 6.080 0.6012 2.390 0.7103 3.350 0.7553 4.01 0.6531 2.820 0.7158 3.430 0.7670 4.200

0.6920 3.100 0.7230 3.540 0.8820 8.450

0.8740 7.940 0.7380 3.790

0.9230 12.99 0.8070 5.160 0.8240 5.680 0.7380 3.740 0.7190 3.510 0.7270 3.640

(13)

provided that t > 5 days; otherwise, Nt(t) = 5.5. Fur­thermore,

type 1: Net = 1.1; type 2: Net = 1; type 3: Net = 1.15; type 4: Net = 1.5. (14)

Advn Cem Bas Mat 1994;1 :248-257

wlc t T Type

0.309 7.0 294.11 4 0.309 14.0 294.11 4 0.309 28.0 294.11 4 0.309 56.0 294.11 4 0.309 90.0 294.11 4 0.309 180.0 294.11 4 0.309 365.0 294.11 4 0.316 7.0 294.11 2 0.316 14.0 294.11 2 0.316 28.0 294.11 2 0.316 56.0 294.11 2 0.316 90.0 294.11 2 0.316 180.0 294.11 2 0.334 28.0 294.11 3 0.318 28.0 294.11 2 0.324 28.0 294.11 4 0.334 28.0 294.11 1 0.328 28.0 294.11 1 0.334 90.0 294.11 3 0.318 90.0 294.11 2 0.324 90.0 294.11 4 0.334 90.0 294.11 1 0.328 90.0 294.11 1 0.316 28.0 294.11 2 0.433 28.0 294.11 2 0.57 28.0 294.11 2 0.316 28.0 294.11 2 0.432 28.0 294.11 2 0.582 28.0 294.11 2 0.316 180.0 294.11 2 0.433 180.0 294.11 2 0.57 180.0 294.11 2 0.316 180.0 294.11 2 0.432 180.0 294.11 2 0.582 180.0 294.11 2

0.400 730.0 298.00 1 0.450 2555.0 298.00 1 0.700 730.0 298.00 1

0.200 1.000 298.00 2 0.200 3.000 298.00 2

0.200 0.250 308.00 1 0.200 1.000 308.00 1 0.200 3.000 308.00 1 0.200 7.000 308.00 1 0.200 28.00 308.00 1 0.200 90.00 308.00 1

Equation 12 shows that, at saturation state, the aver­age radii of the pores accessible to water decrease with progressing hydration. This has been experimentally confirmed by other investigators [26]. Equation 12 also gives some idea about the pore size at saturation state.

The expression for k is easily obtained from the ex­pressions for nand C. From eq 5,

W CkH n = V m = (1 - kH)[1 + (C - l)kH]

(15)

Advn Cem Bas Mat Y. Xi et al. 253 1994;1:248-257

C

8

:tl 5-

4 -

3-

2-

1 0 ~6o 260

o o w/o, , ,0 .3 Tgpe N 0ement T . , 204 K

- - ~ c - 0 . 3 "type. cement

T - 2 9 4 " K

~D

360 400

t in d a y s

FIGURE 5. D e p e n d e n c e of p a r a m e t e r n o n concre te age t.

Therefore, H = 1.0:

W C k n - V m - (1 - k)[1 + (C - 1)k]" (16)

Then, from eq 16, an expression for k can be deter- mined:

k = (17) 2(1 - C)

4-.5

4 .0

3 .5

3 .0

2.5

• t ,, 180 clays

Type II o o r v ~

• T " 294"K

2.0 o.2s o.~s o.~,s o.~s

wlc

FIGURE 6. D e p e n d e n c e of p a r a m e t e r n o n t he w a t e r : c e m e n t ratio.

If the positive sign in eq 17 is used, k = 1/(1 - C). Upon comparison of the shapes of the experimental isotherms in Figures 1 and 3 with BDDT classifications [17,18], the isotherm curves for cement paste corre- spond to the t ype 2 isotherm. Hence, the values of C will be grater than 2.0 [15]. This is true for the test data in Table 1, which range form 10 to 50. Therefore, if the positive sign in eq 20 were used, k would be negative. However, the BSB model requires that k be within the range 0 < k < 1 [21]. For this reason, the negative sign must be used in eq 20. The expression for k then be- comes

k = (18) ( C - 1)

The condition 0 < k < 1 is equivalent to the condi- tion 0 < 1 - (l/n) < 1; therefore, n > 1. At saturation state, the number of molecular layers covering a pore surface will always be greater than 1.

Influence of Temperature Given one known isotherm, other unknown isotherms can be determined if the relationship of C, V,,, and k (or n) to temperature is studied. C is exponentially de- pendent on temperature. In eq 10, co is related to the heat of adsorption. Although Co has been assumed to be constant, the reality is more complex. Actually, the values of C display no definite trend [17], nor do those of Co, as seen in Table 1. Perhaps C o can be assumed to change only slightly with temperature [14], but this needs to be examined more carefully through further research. For the first approximation, C o may perhaps be taken as independent of temperature.

The variations of Vm with changes in temperature are due only to the thermal expansion or contraction of the adsorbed layer. The volumetric changes of the ad- sorbate, water, are very small at room temperature. Therefore, V T may be taken as 1 in eq 6. For the same reason, n varies only slightly as well. In fact, empirical evidence shows that for moderate changes in temper- ature, the variation of n is negligible [15]. Therefore, in eq 11, N r may also be taken as 1. k varies with tem- perature according to eq 18, since C is temperature dependent. For temperatures much higher than room temperature, and especially above 100°C, further phe- nomena come into play and a more complex model, based on the thermodynamic properties of water, is required (see ref 2).

Advn Cern Bas Mat 1994;1 :248-257

8~------------------------------~

7 ~ w/c-O.3 TyperoJ~

6

5

c: 4

3

2

o 100 200

t In days

T-284 K

-- w/c-O.3 Type II cement T-284'K

300 400

FIGURE 5. Dependence of parameter n on concrete age t.

Therefore, H = 1.0:

W Ck n = V m = (1 - k)[1 + (C - l)k] . (16)

Then, from eq 16, an expression for k can be deter­mined:

k = -------------

4.5

4.0

3.5

c:: 3.0

2.5

2.0 0.25

2(1 - C)

•

•

0.35

w/c

• •

0.45 0.55

(17)

• •

FIGURE 6, Dependence of parameter n on the water:cement ratio.

Y. Xi et al. 253

If the positive sign in eq 17 is used, k = 1/(1 - C). Upon comparison of the shapes of the experimental isotherms in Figures 1 and 3 with BOOT classifications [17,18], the isotherm curves for cement paste corre­spond to the type 2 isotherm. Hence, the values of C will be grater than 2.0 [15]. This is true for the test data in Table 1, which range form 10 to 50. Therefore, if the positive sign in eq 20 were used, k would be negative. However, the BSB model requires that k be within the range 0 < k < 1 [21]. For this reason, the negative sign must be used in eq 20. The expression for k then be­comes

k==------ (18) (C - 1)

The condition 0 < k < 1 is equivalent to the condi­tion 0 < 1 - (lin) < 1; therefore, n > 1. At saturation state, the number of molecular layers covering a pore surface will always be greater than 1.

Influence of Temperature

Given one known isotherm, other unknown isotherms can be determined if the relationship of C, V m' and k (or n) to temperature is studied. C is exponentially de­pendent on temperature. In eq 10, Co is related to the heat of adsorption. Although Co has been assumed to be constant, the reality is more complex. Actually, the values of C display no definite trend [17], nor do those of Co, as seen in Table 1. Perhaps Co can be assumed to change only slightly with temperature [14], but this needs to be examined more carefully through further research. For the first approximation, Co may perhaps be taken as independent of temperature.

The variations of V m with changes in temperature are due only to the thermal expansion or contraction of the adsorbed layer. The volumetric changes of the ad­sorbate, water, are very small at room temperature. Therefore, V T may be taken as 1 in eq 6. For the same reason, n varies only slightly as well. In fact, empirical evidence shows that for moderate changes in temper­ature, the variation of n is negligible [15]. Therefore, in eq 11, NT may also be taken as 1. k varies with tem­perature according to eq 18, since C is temperature dependent. For temperatures much higher than room temperature, and especially above 100°C, further phe­nomena come into play and a more complex model, based on the thermodynamic properties of water, is required (see ref 2).

254 Moisture Diffusion in Cementitious Materials Advn Cem Bas Mat 1994;1:248-257

Proposed Model and Its Comparison with Test Data The m o d e l a n d the f inal f o r m u l a s for V m, C, a n d k are

as fo l lows:

V m C k H W = (1 - kH)[1 + (C - 1)kH] (19)

a n d

( 022, .. w) t > 5 days , 0.3 < w/c < 0.7

(20)

b u t for t ~ 5 days , set t = 5 days ; for w/c ~ 0.3, set w/c = 0.3; for w/c --> 0.7, set w/c = 0.7; Vet is g i v e n by eq 8;

t is age of s p e c i m e n in days ; a n d

TABLE 2. Optimized parameters and statistics

Vm C k n Ss~, Sc Sk S,,

(21)

• 0217 18.3032 .8496 7.0325 .9275 1.2089 .9809 .9516 • 0310 18.3032 .8045 5.4096 1.0274 1.0202 .9713 .9408 .0357 18.3032 .7700 4.5982 .9805 .6739 .9615 .9600 • 0380 18.3032 .7477 4.1925 .8230 .7215 .9886 1.0352 • 0389 18.3032 .7381 4.0392 .7410 .8323 1.0513 1.2277 • 0396 18.3032 .7297 3.9130 .6153 1.1955 1.1669 1.5226 .0400 18.3032 .7252 3.8490 .6461 .7893 1.1513 1.4580 • 0363 18.3032 .7778 4.7599 .9781 .7678 .9735 .9675 • 0519 18.3032 .7111 3.6614 .9807 1.2782 1.0238 1.1514 • 0597 18.3032 .6601 3.1122 .9799 1.7565 .9919 1.0920 • 0636 18.3032 .6272 2.8376 1.0961 .5406 .9383 .9554 • 0650 18.3032 .6131 2.7339 .9345 1.1555 1.0309 1.1535 .0663 18.3032 .6006 2.6484 .9744 1.3158 .9990 .9039 .0511 18.3032 .7154 3.7173 .9988 .9751 .9581 .9581 .0597 18.3032 .6616 3.1256 1.6230 .8095 .8062 .5662 • 0359 18.3032 .7772 4.7485 1.1265 .9484 .9491 .8697 .0541 18.3032 .7025 3.5557 1.1644 .9291 .8997 .7901 • 0540 18.3032 .6988 3.5116 .9608 1.1468 1.0005 1.0872 .0557 18.3032 .6761 3.2654 .9989 .8876 .9333 .9172 .0651 18.3032 .6147 2.7456 1.2284 1.0830 .8506 .7778 .0392 18.3032 .7464 4.1712 .8762 .8230 .9781 1.0027 • 0590 18.3032 .6613 3.1234 1.0615 .9372 .8910 .8198 • 0589 18.3032 .6571 3.0847 .8798 1.4725 1.0359 1.1819 .0597 18.3032 .6601 3.1122 .9799 1.7565 .9919 1.0920 .0628 18.3032 .7283 3.8936 .9565 1.8993 1.0226 1.1693 • 0665 18.3032 .7800 4.8086 1.0530 1.5765 .9994 1.0806 • 0597 18.3032 .6601 3.1122 1.2080 .5290 .8789 .7820 .0628 18.3032 .7279 3.8869 1.2767 .5353 .9127 .7933 • 0669 18.3032 .7836 4.8887 1.3903 .4027 .9370 .8041 • 0663 18.3032 .6006 2.6484 .9744 1.3158 .9990 1.1081 • 0698 18.3032 .6808 3.3134 .9953 1.3548 .9584 .9891 .0739 18.3032 .7415 4.0920 .9839 1.0735 .9817 1.0204 • 0663 18.3032 .6006 2.6484 1.1230 .8120 .9196 .9392 • 0697 18.3032 .6802 3.3077 1.0287 1.2113 .9503 .9643 .0743 18.3032 .7457 4.1602 1.0761 1.3292 .9723 .9905

• 0628 17.6216 .6840 3.3548 1.0238 1.8068 .9884 1.0822 .0643 17.6216 .7086 3.6385 1.0020 .9474 .9801 1.0278 .0682 17.6216 .7683 4.5748 1.5616 .3696 .8710 .5414

• 0226 17.6216 .7497 4.2350 1.5141 .5292 .8577 .5334 .0226 17.6216 .7497 4.2350 .9600 .3526 1.0158 1.1174

• 0203 16.0543 .7711 4.6585 1.9155 2.0271 .8354 .3586 • 0203 16.0543 .7711 4.6585 .9715 .2423 .9555 .9028 • 0203 16.0543 .7711 4.6585 .9762 .2010 .9358 .8202 • 0309 16.0543 .7288 3.9325 .9263 .9221 .9876 1.0515 .0509 16.0543 .5852 2.5713 1.2947 .5373 .8140 .7325 .0555 16.0543 .5279 2.2587 1.4989 .2779 .7261 .6205

254 Moisture Diffusion in Cementitious Materials Advn Cem Bas Mat 1994;1 :248-257

Proposed Model and Its Comparison ( 0.22)( w)

with Test Data v m = 0.068 - -t- 0.85 + 0.45 ~ Vet,

(20)

The model and the final formulas for V m' C, and k are t > 5 days, 0.3 < w!c < 0.7

as follows: but for t ,,;;; 5 days, set t = 5 days; for wlc ,,;;; 0.3, set w/c = 0.3; for wlc ;:;,; 0.7, set wlc = 0.7; Vet is given by eq 8;

VmCkH (19)

t is age of specimen in days; and

W = (1 - kH)[1 + (C - 1)kH] C = exp (;), Co = 855 (21)

and

TABLE 2. Optimized parameters and statistics

Vm C k n SSm Sc Sk Sn

.0217 18.3032 .8496 7.0325 .9275 1.2089 .9809 .9516

.0310 18.3032 .8045 5.4096 1.0274 1.0202 .9713 .9408

.0357 18.3032 ,7700 4.5982 .9805 .6739 .9615 .9600

.0380 18.3032 .7477 4.1925 .8230 .7215 .9886 1.0352

.0389 18.3032 .7381 4.0392 .7410 .8323 1.0513 1.2277

.0396 18.3032 .7297 3.9130 .6153 1.1955 1.1669 1.5226

.0400 18.3032 .7252 3.8490 .6461 .7893 1.1513 1.4580

.0363 18.3032 .7778 4.7599 .9781 .7678 .9735 .9675

.0519 18.3032 .7111 3.6614 .9807 1.2782 1.0238 1.1514

.0597 18.3032 .6601 3.1122 .9799 1.7565 .9919 1.0920

.0636 18.3032 .6272 2.8376 1.0961 .5406 .9383 .9554

.0650 18.3032 .6131 2.7339 .9345 1.1555 1.0309 1.1535

.0663 18,3032 .6006 2.6484 .9744 1.3158 .9990 .9039

.0511 18.3032 .7154 3.7173 .9988 .9751 .9581 .9581

.0597 18.3032 .6616 3.1256 1.6230 .8095 ,8062 .5662

.0359 18.3032 .7772 4.7485 1.1265 .9484 .9491 .8697

.0541 18.3032 .7025 3.5557 1.1644 .9291 .8997 .7901

.0540 18.3032 .6988 3.5116 .9608 1.1468 1.0005 1.0872

.0557 18.3032 .6761 3.2654 .9989 .8876 .9333 .9172

.0651 18.3032 .6147 2.7456 1.2284 1.0830 .8506 .7778

.0392 18.3032 .7464 4.1712 .8762 .8230 .9781 1.0027

.0590 18.3032 .6613 3.1234 1.0615 .9372 .8910 .8198

.0589 18.3032 .6571 3.0847 .8798 1.4725 1.0359 1.1819

.0597 18.3032 .6601 3.1122 .9799 1.7565 .9919 1.0920

.0628 18.3032 .7283 3.8936 .9565 1.8993 1.0226 1.1693

.0665 18.3032 .7800 4.8086 1.0530 1.5765 .9994 1.0806

.0597 18.3032 .6601 3.1122 1.2080 .5290 .8789 .7820

.0628 18.3032 .7279 3.8869 1.2767 .5353 .9127 .7933

.0669 18.3032 .7836 4.8887 1.3903 .4027 .9370 .8041

.0663 18.3032 .6006 2.6484 .9744 1.3158 .9990 1.1081

.0698 18.3032 .6808 3.3134 .9953 1.3548 .9584 .9891

.0739 18.3032 .7415 4.0920 .9839 1.0735 .9817 1.0204

.0663 18.3032 .6006 2.6484 1.1230 .8120 .9196 .9392

.0697 18.3032 .6802 3.3077 1.0287 1.2113 .9503 .9643

.0743 18.3032 .7457 4.1602 1.0761 1.3292 .9723 .9905

.0628 17.6216 .6840 3.3548 1.0238 1.8068 .9884 1.0822

.0643 17.6216 .7086 3.6385 1.0020 .9474 .9801 1.0278

.0682 17.6216 .7683 4.5748 1.5616 .3696 .8710 .5414

.0226 17.6216 .7497 4.2350 1.5141 .5292 .8577 .5334

.0226 17.6216 .7497 4.2350 .9600 .3526 1.0158 1.1174

.0203 16.0543 .7711 4.6585 1.9155 2.0271 .8354 .3586

.0203 16.0543 .7711 4.6585 .9715 .2423 .9555 .9028

.0203 16.0543 .7711 4.6585 .9762 .2010 .9358 .8202

.0309 16.0543 .7288 3.9325 .9263 .9221 .9876 1.0515

.0509 16.0543 .5852 2.5713 1.2947 .5373 .8140 .7325

.0555 16.0543 .5279 2.2587 1.4989 .2779 .7261 .6205

Advn Cem Bas Mat Y. Xi et al. 255 1994;1:248-257

TABLE 3. Statistics

M = 37 M = 46

Myra 1.019 1.072 M c 1.076 0.984 M k 0.974 0.957 M. 1.004 0.952 Dv. 0.016 0.398 D c 0.120 0.717 D k 0.007 0.241 D. 0.024 0.348

M = 37: T h e test data of the last n i n e r o w s in T a b l e 1 (w/c < 0 .3 , a n d w/c > 0 .6) are n o t i n c l u d e d . M = 46: A l l of the test data in Table 1 are i n c l u d e d .

0.20

0.15

0,20

.,~0.10

0.05

0.00 O0

Powers et al. (1947) w/¢ = 0.300 Type N cement T = 294"K t - 28 days o o

o

0.'2 0.'4 0.'6 0.'8 1.0

k =

(, C - 1

and

(22)

n = (2"5 + 1 5 ~ (0"33 + 2"2 w) N c t ' t e j \

t > 5 days, 0.3 < w/c < 0.7 (23)

but for t ~ 5 days, set t = 5 days; for w/c ~ 0.3, set w/c =

0.3; for w/c >t 0.7, set w/c = 0.7; Xct is given by eq 14.

wle - 0.316 Powers et al. (1947) Type II cement T - 2 9 4 "K _/91 0 . 1 5

.1~ 0.10

0.05

0 . 0 0 o.o o.'2 o.'4 o.'e 0;8 ,.o

x = p , / p x = p , / p

0.20

0.15

~ . 1 0

0.05

0.00 0.0

Powers et al. (1947) w/c = 0.316 d Type II cement / T=294 "K /

o'2 0'4 0.'6 o'8 1.o

0 . 2 0

o

wlc = 0.334 Powers et al. 1 7 Type I cement

0 . 1 5 T = 284"K t = 28 days

" ~ 0.10 E

0.05 ~.~

0 . O 0 o.o o.~ 0;4 o.% o.~ 1.o

x = p~/p x = p~/p

FIGURE 7. Comparison of predicted curves with isotherms measured for various concretes.

Advn Cem Bas Mat Y. Xi et al. 255 1994;1 :248-257

TABLE 3. Statistics (1-~)C-1 M= 37 M= 46

k= (22) Mv 1.019 1.072 C-1 M

m 1.076 0.984 c

and Mk 0.974 0.957 Mn 1.004 0.952

n = (2.5 + ~~) (0.33 + 2.2~) Net, Ov 0.016 0.398 Oem 0.120 0.717

(23) Ok 0.007 0.241 On 0.024 0.348 t > 5 days, 0.3 < w/c < 0.7

M = 37: The test data of the last nine rows in Table 1 (wle < 0.3, and wle > 0.6) are not included.

but for t ,,;:; 5 days, set t = 5 days; for w/c ,,;:; 0.3, set w/c = 0.3; for w/c ~ 0.7, set w/c = 0.7; Net is given by eq 14. M = 46: All of the test data in Table 1 are included.

0.20 .,.-----------------,

0.15

.g> C)

.50.10

~

0.05

0.00 0.0

0.20

0.15

.g> C)

.510

~

0.05

0

0.00 0.0

w/c == 0.309 Type IV cement T-294 'K t - 28 days

o

0.2 0.4

X=

w/c - 0.316 Type II ~ment T-294 K t - 28 days

0

0

0.2 0.4

X=

Powers at al. (1947)

o

o o

0.6 0.8 1.0

~ /p S

Powers at al. (1947)

'Q

0.6 0.8 1.0

~ /p S

.g

0.20 .,.----------------~

0.15

w/c - 0.316 Type II cement T-294 ·K

t - 180 days

Powers at al. (1947)

.5 0.10

~

.g>

0.05 o

0.00 0.0 0.2 0.4 0.6 0.8 1.0

X= ~ /p S

0.20 .,.------------------,

0.15

w/c = 0.334 Type I cement T=294·K t == 28 days

o

Powers at al. (1947)

o

C) 0.10

.5 ~

0.05 o

0.00 +-----,-----,r----~--__,_----I 0.0 0.2 0.4 0.6

x = ~ /p s

0.8 1.0

FIGURE 7. Comparison of predicted curves with isotherms measured for various concretes.

256 Moisture Diffusion in Cementitious Materials Advn Cem Bas Mat 1994;1:248-257

0.30

0 ) 0 . 2 0

.c_ '0.10

0.25 I w/¢ == 0.45 Odleretal. (1072) o'

0.20 I T - 2 9 8 " K /

~ 0.15

C 0.10

0.05

0.00

w/c = 0.582 Powers et al. (1947) Type II ce. ment /

000/o 0.'2 0.4 0.'6 0.18 1.0 °-°°o. o o.o o.'2 o.'4 o.'6 o.~ ~.o

x = ~ / p x = ~ / p

w/¢ = 0.2 Mikhail et al. (1975)

T = 3 0 8 K Type I cement /

0.25] 0]0. ,5 w/c = 0.432 Powers et al. (1947)

0.20 -1 Type II cement / I / T= 'K °/ I ~ 0 15 0.10 • ~ t = 7 days O) r.. .l:::

0.10

o 'iii 0"00070 0.'2 0/4 0.'6 0.~ ,.0 0.0 0.'2 0:4 0.'6 0.~

x = I ~ / P x = ~ / p

1.0

FIGURE 8. Further comparison of predicted curves with isotherms measured for various concretes.

Table 2 shows the calculated values of Vm, C, n, and k and the ratios of the calculated values to the test data Svm, Sc, S,, and S k, respectively, measured by Powers and Brownyard [28], Hagymassy et al. [27], and Mikhail and Abo-EI-Enein [26]. Table 3 shows the ex- pected values Mvm, M c, M, , and M k of these ratios and the coefficients of variation Dvm, De, D n, and D k. It is evident that the calculated results agree with the test data quite well (see the second column of Table 3, M = 37). Therefore, the present empirical equation can be used to estimate the parameters V m, C, and n. Based on these parameters, approximate information about the pore structure, such as the surface area and the pore volume, can be obtained [30]. For the cases of

w/c > 0.6 and w/c < 0.3, the present formula is valid also, but the deviation from the real values increases (see the third column of Table 3, M = 46).

Figures 7 and 8 show comparisons of the calculated curves of the present empirical formulas with the mea- sured curves. It is obvious that for different curing times, different w/c, and various cement types, these curves agree closely. This confirms that the chosen governing parameters, t, w/c, cement type and T, rep- resent the major factors affecting the adsorption iso- therms. Most importantly, this also confirms that the present semiempirical equation can represent the ad- sorption isotherm over the complete pressure range (P/Ps = 0 to 1).

256 Moisture Diffusion in Cernentitious Materials

0.30

0 0 .20

a, c:::

3: 0.10

w/c - 0.582 Type II cement T-294 'K t - 28 days

o o

o

Powers et a!. (1947)

o o

o

o o

o

o

0.00 +-----r---.----,------r----i 0.0 0.2 0.4 0.6 0.8 1.0

x = ~/p

0.25 -.------------------..,

0.20

~0.15 o .S 3: 0.10

0.05

w/c - 0.432 Type II cement T-294'K t"" 180 days

Powers et al. (1947) 0

o

0.00 +----.---~--_,._---,__----l 0.0 0.2 0.4 0.6 0.8 1.0

x = ~/p

0.25

w/c - 0.45 0.20 T-298 K

Type I cement

~0.15 t - 2555 days

.S 3: 0.10

0.05 0

Advn Cern Bas Mat 1994;1 :248-257

Odler et al. (1972) 0

0

0

0

0.00 ..... ----r----.------,r----~--__l 0.0 0.2 0.4 0.6 0.8 1.0

x = ~/p

0.15 -r------------------,

w/c - 0.2 Mikhail et a!. (1975)

T-3os'K

0.10 Type I cement 0

~ t - 7 days o o

c::: o

3: 0.05

o

0.00 +----,-----,---,__--,------1 0.0 0.2 0.4 0.6 0.8 1.0

x = ~/p FIGURE 8. Further comparison of predicted curves with isotherms measured for various concretes.

Table 2 shows the calculated values of V m' C, n, and k and the ratios of the calculated values to the test data Sv , Sc' Sn' and Sk' respectively, measured by Powers a;d Brownyard [28], Hagymassy et al. [27], and Mikhail and Abo-El-Enein [26]. Table 3 shows the ex­pected values Mvm, Mc' Mw and Mk of these ratios and the coefficients of variation Dv , Dc, Dn, and Dk . It is evident that the calculated res{ilts agree with the test data quite well (see the second column of Table 3, M = 37). Therefore, the present empirical equation can be used to estimate the parameters V m' C, and n. Based on these parameters, approximate information about the pore structure, such as the surface area and the pore volume, can be obtained [30]. For the cases of

wlc > 0.6 and wlc < 0.3, the present formula is valid also, but the deviation from the real values increases (see the third column of Table 3, M = 46).

Figures 7 and 8 show comparisons of the calculated curves of the present empirical formulas with the mea­sured curves. It is obvious that for different curing times, different wlc, and various cement types, these curves agree closely. This confirms that the chosen governing parameters, t, wlc, cement type and T, rep­resent the major factors affecting the adsorption iso­therms. Most importantly, this also confirms that the present semiempirical equation can represent the ad­sorption isotherm over the complete pressure range (pips = 0 to 1).

Advn Cem Bas Mat Y. Xi et al. 257 1994;1:248-257

Conclusions

1. The drying process of concrete can be described by the diffusion equation governing the pore rel- ative humidity. The moisture capacity and diffu- sivity in the diffusion equation should be treated as two separate coefficients to be evaluated by independent test results. Both the moisture ca- pacity and diffusivity depend on the pore struc- ture of concrete, and the pore structure depends on the basic material parameters, such as the wa- ter:cement ratio, curing time, temperature, and type of cement.

2. The BSB model is chosen as a prediction model for the adsorption isotherm. Some empirical for- mulas are established for determining the three parameters of this model based on the available adsorption test data. The calculated results show that present empirical formulas predict the pa- rameters quite accurately and therefore predict the adsorption isotherms for various Portland ce- ment pastes very well.

3. The parameters V m and n of the adsorption iso- therm, predicted by the present formulas, can be used to obtain approximate information about the surface area, pore volume, and pore-size dis- tribution on the basis of a given w/c, curing time, cement type, and temperature. This is useful for cases where the adsorption test data are unavail- able.

4. Among the numerous parameters influencing the adsorpt ion isotherm of Portland cement paste, the major parameters are the curing time, water:cement ratio, cement type, and tempera- ture. The results show that: (1) the temperature does not have much influence on the isotherm in the range of room temperatures; (2) after 6 months of curing, the parameters V m and n, which represent the monolayer capacity and the number of adsorbed layers at saturation, are al- most independent of the curing time; and (3) w/c has a linear relationship with V m and n, and it seems that at w/c = 0.6, V m and n reach their maximum values.

Acknowledgments The theoretical part of the present research was supported under NSF grant 0830-350-C802 to Northwestern University, and the test data analysis was supported by NSF Science & Technology Center for Advanced Cement-Based Materials at Northwestern University.

R e f e r e n c e s 1. Ba~ant, Z.P.; Najjar, L.J. Mat~riaux et Constructions 1972,

5, 3-20.

2. Ba~ant, Z.P.; Thonguthai, W. J. Eng. Mech. Div., Proceed- ings, ASCE 1978, 104, 1059-1079.

3. Ba~ant, Z.P.; Wittmann, F.H., Eds. Mathematical Models for Creep and Shrinkage of Concrete; John Wiley & Sons: New York, 1982; pp 163-256.

4. Sakata, K. Cement Concrete Res. 1983, 13, 216--224. 5. Xi, Y.; Ba~ant, Z.P.; Molina, L.; Jennings, H.M.J. Adv.

Cement Based Mater. 1994, 1,258-266. 6. Kamp, C.L.; Roelfstra, P.E.; Wittmann, F.H. Proc. Int.

Conf. on Struc. Mech. In Reactor Tech. (SMiRT), Lausanne, 1987, H, 157-166.

7. Garboczi, E.J. Cement Concrete Res. 1990, 20, 591-601. 8. Garboczi, E.J.; Bentz, D.P. In Scientific Basis for Nuclear

Waste Management XIII, MRS Symposium Series Proc.; Oversby, V.M.; Brown, P.M., Eds.; Materials Research Society: Pittsburgh, PA, 1990.

9. Schwartz, L.M.; Banavar, J.R. Phys. Rev. B 1989, 39, 11965-11970.

10. Huang, C.L.D.; Siang, H.H.; Best, C.H. Int. J. Heat Mass Transfer 1979, 22, 257-266.

11. Chou, W.T-H.; Whitaker, S. Proc. of the Third International Drying Symposium 1982, 1, 135-148.

12. Jonasson, J.-E. Proc. Int. Conf. on Struc. Mech. in Reactor Tech. (SMiRT), Brussels, 1985, H5/11,235-242.

13. Copeland, L.E.; Bragg, R.H. Proc. Am. Soc. for Testing Materials 1955, 204 (PCA Bulletin 52).

14. Brunauer, S.; Emmett, P.H.; Teller, E. J. Am. Chem. Soc. 1938, 60, 309-319.

15. Gregg, S.J.; Sing, K.W.S. Adsorption, Surface Area and Po- rosity; Academic Press, Inc.: London, 1982.

16. Mikhail, R.S. Microstructure and Thermal Analysis of Solid Surfaces, John Wiley & Sons: New York, 1983.

17. Brunauer, S. The Adsorption of Gases and Vapors, Princeton University Press: Princeton, NJ, 1943.

18. Brunauer, S.; Deming, L.S.; Deming, W.E.; Teller, E. J. Am. Chem. Soc. 1940, 62, 1723-1732.

19. Halsey, G. J. Chem. Phys. 1948, 16, 931-937. 20. Hillerborg, A. Cement Concrete Res. 1985, 15, 809-816. 21. Brunauer, S.; Skalny, J.; Bodor, E.E.J. Colloid Interface

Sci. 1969, 30, 546-552. 22. Mikhail, R.S.; Abo-E1-Enein, S.A.; Gabr, N.A.J. Appl.

Chem. Biotechnol. 1975, 25, 835-847. 23. Gleysteen, L.F.; Kalousek, G.L.J. Am. Conc. Inst. 1955,

26, 437-446. 24. Young, D.M.; Crowell, A.D. Physical Adsorption of Gases;

Butterworths: Washington, 1962. 25. Adamson, A.W. Physical Chemistry of Surfaces, 4th ed;

John Wiley & Sons: New York, 1982. 26. Mikhail, R.S.; Abo-EI-Enein, S.A. Cement Concrete Res.

1972, 2, 401-414. 27. Hagymassy, J., Jr.; Odler, I.; Yudenfreund, M.; Skalny,

J.; Brunauer, S. J. Colloid Interface Sci. 1972, 38, 20-34; 265-276.

28. Powers, T.C.; Brownyard, T.L. Proc. Am. Conc. Inst. 1946-1947, 43, 101, 149, 469, 549, 845, 933.

29. Ramachandran, V.S. Concrete Admixtures Handbook: Prop- erties, Science, and Technology; Noyes Publications, 1984.

30. Brunauer, S.; Mikhail, R.S.; Bodor, E.E.J. Colloid Interface Sci. 1967, 24, 451-463.

Advn Cern Bas Mat 1994;1 :248-257

Conclusions

1. The drying process of concrete can be described by the diffusion equation governing the pore rel­ative humidity. The moisture capacity and diffu­sivity in the diffusion equation should be treated as two separate coefficients to be evaluated by independent test results. Both the moisture ca­pacity and diffusivity depend on the pore struc­ture of concrete, and the pore structure depends on the basic material parameters, such as the wa­ter:cement ratio, curing time, temperature, and type of cement.

2. The BSB model is chosen as a prediction model for the adsorption isotherm. Some empirical for­mulas are established for determining the three parameters of this model based on the available adsorption test data. The calculated results show that present empirical formulas predict the pa­rameters quite accurately and therefore predict the adsorption isotherms for various Portland ce­ment pastes very well.

3. The parameters V m and n of the adsorption iso­therm, predicted by the present formulas, can be used to obtain approximate information about the surface area, pore volume, and pore-size dis­tribution on the basis of a given wle, curing time, cement type, and temperature. This is useful for cases where the adsorption test data are unavail­able.

4. Among the numerous parameters influencing the adsorption isotherm of Portland cement paste, the major parameters are the curing time, water:cement ratio, cement type, and tempera­ture. The results show that: (1) the temperature does not have much influence on the isotherm in the range of room temperatures; (2) after 6 months of curing, the parameters V m and n, which represent the monolayer capacity and the number of adsorbed layers at saturation, are al­most independent of the curing time; and (3) wle has a linear relationship with V m and n, and it seems that at wle = 0.6, V m and n reach their maximum values.

Acknowledgments

The theoretical part of the present research was supported under NSF grant 0830-350-CS02 to Northwestern University, and the test data analysis was supported by NSF Science & Technology Center for Advanced Cement-Based Materials at Northwestern University.

References 1. Bazant, Z.P.; Najjar, L.J. Materiaux et Constructions 1972,

5, :>-20.

Y. Xi et al. 257

2. Bazant, Z.P.; Thonguthai, W. ]. Eng. Mech. Div., Proceed­ings, ASCE 1978, 104, 1059-1079.

3. Bazant, Z.P.; Wittmann, F.H., Eds. Mathematical Models for Creep and Shrinkage of Concrete; John Wiley & Sons: New York, 1982; pp 16:>-256.

4. Sakata, K. Cement Concrete Res. 1983, 13, 21&-224. 5. Xi, Y.; Bazant, Z.P.; Molina, L.; Jennings, H.M. ]. Adv.

Cement Based Mater. 1994, 1, 258-266. 6. Kamp, e.L.; Roelfstra, P.E.; Wittmann, F.H. Proc. Int.

Conf. on Struc. Mech. In Reactor Tech. (SMiRT), Lausanne, 1987, H, 157-166.

7. Garboczi, E.J. Cement Concrete Res. 1990,20,591-601. 8. Garboczi, E.J.; Bentz, D.P. In Scientific Basis for Nuclear

Waste Management XIII, MRS Symposium Series Proc.; Oversby, V.M.; Brown, P.M., Eds.; Materials Research Society: Pittsburgh, P A, 1990.

9. Schwartz, L.M.; Banavar, J.R. Phys. Rev. B 1989, 39, 11965--11970.

10. Huang, e.L.D.; Siang, H.H.; Best, e.H. Int. ]. Heat Mass Transfer 1979, 22, 257-266.

11. Chou, W.T-H.; Whitaker, S. Proc. of the Third International Drying Symposium 1982, 1, 135--148.

12. Jonasson, J.-E. Proc. Int. Conf. on Struc. Mech. in Reactor Tech. (SMiRT), Brussels, 1985, H5/11, 235--242.

13. Copeland, L.E.; Bragg, R.H. Proc. Am. Soc. for Testing Materials 1955, 204 (PCA Bulletin 52).

14. Brunauer, S.; Emmett, P.H.; Teller, E. ]. Am. Chem. Soc. 1938, 60, 309-319.

15. Gregg, S.J.; Sing, K.W.S. Adsorption, Surface Area and Po­rosity; Academic Press, Inc.: London, 1982.

16. Mikhail, R.S. Microstructure and Thermal Analysis of Solid Surfaces, John Wiley & Sons: New York, 1983.

17. Brunauer, S. The Adsorption of Gases and Vapors, Princeton University Press: Princeton, NJ, 1943.

18. Brunauer, S.; Deming, L.S.; Deming, W.E.; Teller, E. ]. Am. Chem. Soc. 1940, 62, 172:>-1732.

19. Halsey, G. ]. Chem. Phys. 1948, 16, 931-937. 20. Hillerborg, A. Cement Concrete Res. 1985, 15, 809-816. 21. Brunauer, S.; Skalny, J.; Bodor, E.E. ]. Colloid Interface

Sci. 1969, 30, 54&-552. 22. Mikhail, R.S.; Abo-El-Enein, S.A; Gabr, N.A J. App/.

Chem. Biotechno/. 1975, 25, 835--847. 23. Gleysteen, L.F.; Kalousek, G.L. ]. Am. Conc. Inst. 1955,

26, 437-446. 24. Young, D.M.; Crowell, AD. Physical Adsorption of Gases;

Butterworths: Washington, 1962. 25. Adamson, AW. Physical Chemistry of Surfaces, 4th ed;

John Wiley & Sons: New York, 1982. 26. Mikhail, R.S.; Abo-El-Enein, S.A. Cement Concrete Res.

1972, 2, 401-414. 27. Hagymassy, J., Jr.; adler, I.; Yudenfreund, M.; Skalny,

J.; Brunauer, S. ]. Colloid Interface Sci. 1972, 38, 2(}-34; 265--276.

28. Powers, T.e.; Brownyard, T.L. Proc. Am. Conc. Inst. 1946--1947, 43, 101, 149, 469, 549, 845, 933.

29. Ramachandran, V.S. Concrete Admixtures Handbook: Prop­erties, Science, and Technology; Noyes Publications, 1984.

30. Brunauer, 5.; Mikhail, R.S.; Bodor, E.E. ]. Colloid Interface Sci. 1967, 24, 451-463.

Moisture Diffusion in Cementitious Materials Moisture Capacity and Diffusivity Yunping Xi, Zdeng~k P. Ba~ant, Larissa Molina, and Hamlin M. Jennings Department of Civil Engineering, Northwestern University, Evanston, Illinois

Based on a model by Ba~ant and Najjar, and using a new model for adsorption isotherms, moisture capacity and diffusivity of concrete are analyzed. The moisture capacity, obtained as a derivative of the adsorption isotherm, first drops as the humidity increases from zero, then levels off as a constant, and finally again increases when the humidity approaches saturation, regardless of the age, cement type, temperature, and water:cement ratio. The well-known diffusion mechanisms, including the ordinary diffusion, Knudsen diffusion, and surface diffusion, are analyzed and the diffusion in concrete is treated as a combination of these mechanisms. An improved formula for the dependence of diffusivity on pore humidity is proposed. The improved model for moisture diffusion is found to give satisfactory diffusion profiles and long-term drying predictions. The model is suited for incorporation into finite element programs for shrinkage and creep effects in concrete structures. ADVANCED CEMENT BASED MATEmALS 1994, 1, 258--266 KEY WORDS: Adsorption, Concrete, Hardened cement paste, Moisture diffusion, Moisture effects, Permeability, Po- rosity

~ oisture diffusion is very important for the ~long-term performance of cementitious mate- i rials. The moisture diffusion can be described

by diffusion equations and solved by various numeri- cal methods, provided that the coefficients are known. However, even though the diffusion-related coeffi- cients have long been studied in the research on trans- port behaviors of the materials, they still remain an unsolved problem, although many different models have been proposed [1-5].

The major difficulty in establishing reliable diffusion parameters is that diffusion of moisture inside cemen- titious materials is basically controlled by the micro- structure of the material, and especially by the pore- size distribution. The microstructure is changing with age as well as with relative humidity in the pores. Therefore, all of the parameters, such as the water/ cement ratio, type of cement, and curing time, which affect the formation of the microstructure of cementi- tious materials, have significant effects on diffusion

© Elsevier Science Inc. ISSN 1065-7355/94/$7.00

parameters. However, few models proposed in the lit- erature have taken enough parameters. Most recently, Daian [2,3] proposed a model for mortar in which the pore-size distribution is analyzed and the relationship between the diffusivity and the diffusion mechanisms is explored. But this model was not calibrated with available test results and did not take into account all the relevant influencing parameters. In the context of freeze-thaw analysis, Ba~ant et al. [6] developed a model for the isotherms based on the filling of capillary meniscus and the pore-size distribution. This model, however, does not cover gel pores.

For the two reasons mentioned in the preceding ar- ticle [7], the basic idea from Ba~ant and Najjar [1] is followed in this article. An expression for moisture dif- fusion is formulated in terms of the relative humidity. Some other models which use the moisture content as the basic variable have also been formulated [8-10]. However, they have some limitations [1,7]. In any case, a formulation of drying in terms of the relative humidity, rather than moisture content, is preferable.

Xi et al. [7] present a diffusion equation that has been formulated in terms of two separate parameters, the moisture capacity and the diffusivity, both of which control the diffusion process. This is a natural result of abandoning the approximation that assumes the slope of adsorption isotherm to be constant. Both of the parameters are considered to be functions of the relative humidity in the pores, which makes the prob- lem nonlinear. Because the moisture capacity is the derivative of the adsorption isotherm, the adsorption isotherm must be studied first [7]. A semiempirical for- mula for the adsorption isotherm has been established based on the available test results for the water adsorp- tion isotherm. The effects of water:cement ratio, tem- perature, type of cement, and curing time have been taken into account.

The main purposes of this article are, first, to analyze how moisture capacity is related to the parameters: water:cement ratio, curing time, temperature, and type of cement, and second, to assess qualitatively the diffusivities corresponding to various diffusion mech-

Moisture Diffusion in Cementitious Materials Moisture Capacity and Diffusivity Yunping Xi, Zdenek P. Bazant, Larissa Molina, and Hamlin M. Jennings Department of Civil Engineering, Northwestern University, Evanston, Illinois

Based on a model by Baiant and Najjar, and using a new model for adsorption isotherms, moisture capacity and diffusivity of concrete are analyzed. The moisture capacity, obtained as a derivative of the adsorption isotherm, first drops as the humidity increases from zero, then levels off as a constant, and finally again increases when the humidity approaches saturation, regardless of the age, cement type, temperature, and water:cement ratio. The well-known diffusion mechanisms, including the ordinary diffusion, Knudsen diffusion, and surface diffusion, are analyzed and the diffusion in concrete is treated as a combination of these mechanisms. An improved formula for the dependence of diffusivity on pore humidity is proposed. The improved model for moisture diffusion is found to give satisfactory diffusion profiles and long-term drying predictions. The model is suited for incorporation into finite element programs for shrinkage and creep effects in concrete structures. ADVANCED CEMENT

BASED MATERIALS 1994, 1, 258-266 KEY WORDS: Adsorption, Concrete, Hardened cement paste, Moisture diffusion, Moisture effects, Permeability, Po­rosity

II oisture diffusion is very important for the long-term performance of cementitious mate­rials. The moisture diffusion can be described

by diffusion equations and solved by various numeri­cal methods, provided that the coefficients are known. However, even though the diffusion-related coeffi­cients have long been studied in the research on trans­port behaviors of the materials, they still remain an unsolved problem, although many different models have been proposed [1-5].

The major difficulty in establishing reliable diffusion parameters is that diffusion of moisture inside cemen­titious materials is basically controlled by the micro­structure of the material, and especially by the pore­size distribution. The microstructure is changing with age as well as with relative humidity in the pores. Therefore, all of the parameters, such as the water/ cement ratio, type of cement, and curing time, which affect the formation of the microstructure of cementi­tious materials, have significant effects on diffusion

© Elsevier Science Inc. ISSN 1065-7355/94/$7.00

parameters. However, few models proposed in the lit­erature have taken enough parameters. Most recently, Daian [2,3] proposed a model for mortar in which the pore-size distribution is analyzed and the relationship between the diffusivity and the diffusion mechanisms is explored. But this model was not calibrated with available test results and did not take into account all the relevant influencing parameters. In the context of freeze-thaw analysis, Bazant et al. [6] developed a model for the isotherms based on the filling of capillary meniscus and the pore-size distribution. This model, however, does not cover gel pores.

For the two reasons mentioned in the preceding ar­ticle [7], the basic idea from Bazant and Najjar [1] is followed in this article. An expression for moisture dif­fusion is formulated in terms of the relative humidity. Some other models which use the moisture content as the basic variable have also been formulated [8-10]. However, they have some limitations [1,7]. In any case, a formulation of drying in terms of the relative humidity, rather than moisture content, is preferable.

Xi et al. [7] present a diffusion equation that has been formulated in terms of two separate parameters, the moisture capacity and the diffusivity, both of which control the diffusion process. This is a natural result of abandoning the approximation that assumes the slope of adsorption isotherm to be constant. Both of the parameters are considered to be functions of the relative humidity in the pores, which makes the prob­lem nonlinear. Because the moisture capacity is the derivative of the adsorption isotherm, the adsorption isotherm must be studied first [7]. A semiempirical for­mula for the adsorption isotherm has been established based on the available test results for the water adsorp­tion isotherm. The effects of water:cement ratio, tem­perature, type of cement, and curing time have been taken into account.

The main purposes of this article are, first, to analyze how moisture capacity is related to the parameters: water:cement ratio, curing time, temperature, and type of cement, and second, to assess qualitatively the diffusivities corresponding to various diffusion mech-

Advn Cem Bas Mat Moisture Diffusion in Cementitious Materials 259 1994;1:258-266

0.4

T , , 298°K t - 2 S c k ~

O.3 Type I oement

8

. ................. *1 o.1

0.0 o.o 0.'4 0:6 0'8 1.o

Relat ive H u m i d i t y

FIGURE 1. Effect of water-cement ratio on sorption iso- therms.

anisms and the influences of the water:cement ratio. Finally, an empirical formula for diffusivity is pro- posed and calibrated by recent data on diffusion. The resulting general expression can be employed in prac- tice.

M o i s t u r e C a p a c i t y

The diffusion equat ion for cementit ious materials reads [7]:

0W OH 0---H 0t - div(Dh grad H) (1)

where the moisture capacity, represented by OW/aH, can be obtained from the equilibrium adsorption iso- therm (derivative of eq 19 in ref 7):

0W

aH

CkVm + Wk[1 + (C - 1)kill - Wk(1 - kH) ( C - 1)

(1 - kH) [1 + (C - 1)k/-/] (2)

Figures 1 to 3 illustrate the influences of the water- cement ratio and age on the isotherms and moisture capacity curves. The effect of age is analogous to the effect of the degree of hydration, which was analyzed by Jonasson [4].

From Figures 2 and 3 it is evident that moisture ca- pacity is not a constant. First it drops, then it becomes constant, and finally it increases. The physical mean- ing of such a variation in moisture capacity may be explained by comparing Figure 1 with Figure 2. The first turn point in Figure 2 corresponds to point A in

Figure 1. At this point, the adsorbent reaches its mono- layer capacity, above which the moisture capacity does not decrease steeply with increasing H. The second turn point in Figure 2 corresponds to point B in Figure 1. This is the initial point of capillary condensation. For the usual practical range (from 50 to 100%), one can see that the moisture capacity increases.

Equation 1 can be rearranged as follows:

OH OH at - OW div(Dh grad H). (3)

The curves of OH/aW, which is the reciprocal of mois- ture capacity, are shown in Figure 4 for various values of w/c. Figure 5 shows the effects of different ages upon aH/OW. From Figures 4 and 5 one can also see clearly that aH/aW is not a constant. This is one of the im- provements over the previous model of Ba~.ant and Najjar [1].

Note that the present adsorpt ion isotherm was based upon test data for cement paste. To apply the adsorption isotherm to concrete, a few additional ef- fects need to be considered, such as the aggregate con- tent. However, few test data for concrete are found in the literature [11].

Another point requiring explanation is why we have limited our attention to the adsorption isotherms even though, strictly speaking, desorption isotherms should be used for drying processes, and adsorption iso- therms for wetting processes. Actually, the exact val- ues of the adsorption and desorption curves are not as important as the shapes of the curves if we study one- way drying or one-way wetting only. In fact, upon comparison of experimental results, it is apparent that

1.6

wl¢ ,, 0.6 T , , ;18 °K

1.2 t - ~ d a t m t

~ 0.8 "l~l~ I oelnlnt

0.0 , , , , 0.0 0.2 0.4 0.6 0.8 1.0

Re la t i ve H u m i d i t y

FIGURE 2. Effect of water:cement ratio on moisture capacity.

Advn Cern Bas Mat 1994;1 :258-266

0.4 -r-----------------,

0.3

~

i 8 0.2

i 0.1

0.0 -f----,------,,----,-----,------i 0.0 0.2 0.4 0.6 0.8 1.0

Relative Humidity

FIGURE 1. Effect of water-cement ratio on sorption iso­therms.

anisms and the influences of the water:cement ratio. Finally, an empirical formula for diffusivity is pro­posed and calibrated by recent data on diffusion. The resulting general expression can be employed in prac­tice.

Moisture Capacity

The diffusion equation for cementitious materials reads [7]:

awaH aH at = div(Dh grad H) (1)

where the moisture capacity, represented by aW/aH, can be obtained from the equilibrium adsorption iso­therm (derivative of eq 19 in ref 7):

CkV m + Wk[l + (C - l)kH] -aw Wk(l - kH) (C - 1) aH (1 - kH) [1 + (C - l)kH]

(2)

Figures 1 to 3 illustrate the influences of the water­cement ratio and age on the isotherms and moisture capacity curves. The effect of age is analogous to the effect of the degree of hydration, which was analyzed by Jonasson [4].

From Figures 2 and 3 it is evident that moisture ca­pacity is not a constant. First it drops, then it becomes constant, and finally it increases. The physical mean­ing of such a variation in moisture capacity may be explained by comparing Figure 1 with Figure 2. The first turn point in Figure 2 corresponds to point A in

Moisture Diffusion in Cernentitious Materials 259

Figure 1. At this point, the adsorbent reaches its mono­layer capacity, above which the moisture capacity does not decrease steeply with increasing H. The second turn point in Figure 2 corresponds to point B in Figure 1. This is the initial point of capillary condensation. For the usual practical range (from 50 to 100%), one can see that the moisture capacity increases.

Equation 1 can be rearranged as follows:

aH aH at = aw div(Dh grad H). (3)

The curves of aHlaW, which is the reciprocal of mois­ture capacity, are shown in Figure 4 for various values of w/c. Figure 5 shows the effects of different ages upon aHlaW. From Figures 4 and 5 one can also see clearly that aHlaW is not a constant. This is one of the im­provements over the previous model of Bazant and Najjar [1].

Note that the present adsorption isotherm was based upon test data for cement paste. To apply the adsorption isotherm to concrete, a few additional ef­fects need to be considered, such as the aggregate con­tent. However, few test data for concrete are found in the literature [11].

Another point requiring explanation is why we have limited our attention to the adsorption isotherms even though, strictly speaking, desorption isotherms should be used for drying processes, and adsorption iso­therms for wetting processes. Actually, the exact val­ues of the adsorption and desorption curves are not as important as the shapes of the curves if we study one­way drying or one-way wetting only. In fact, upon comparison of experimental results, it is apparent that

1.6 -.------------------,

w/c - 0.8 T-.oK

1.2 t - 28 daya Type I cement

.c ~ ! 0.8

0.4 The MCOnd turn point The lrat tum point

0.0 +---~--~--~--"T""_----1 0.0 0.2 0.4 0.6 0.8 1.0

Relative Humidity

FIGURE 2. Effect of water:cement ratio on moisture capacity.

260 Y. Xi et al. Advn Cem Bas Mat 1994;1:258-266

2.0

1,5

"I- 1.0

0.5

0.0 0.0

w/c ~ 0.6 T , , 2 ~ °K j t - 20 days J

t . lo / / /

0.'2 0.'4 0.'6 o.'a 1.0

Relative humidity

FIGURE 3. Effect of age on moisture capacity.

adsorption and desorption curves possess virtually the same shape [12,13]. This can be checked further by classification of the hysteresis loops. According to de Boer's classification [14], the cement paste displays a loop of type B. Another classification recommended in the International Union of Pure and Applied Chemis- try (IUPAC) manual [14] shows that the cement paste displays a loop of type H3. Both classifications indicate that adsorption and desorption have nearly the same shape in the practical range of H. On the other hand, the BSB model (eq 19 in ref 7) is valid for both adsorp- tion and desorption. Additionally, since systematic ex- perimental results on adsorption are much more abun- dant in the literature than those for desorption, the use of the adsorption isotherm is most convenient.

In eq 3, aH/aW can be computed by the adsorption isotherm for a given w/c, t o, type of cement, and tem- perature [7]. Therefore, the only unknown coefficient that remains in eq 3 is the diffusivity.

Diffusivity and Diffusion Mechanisms Diffusivity of concrete depends strongly on the diffu- sion mechanisms. Diffusion mechanisms are influ- enced by the pore structure of concrete. Three distinct transport mechanisms may operate singularly or si- multaneously: molecular diffusion (ordinary diffu- sion), Knudsen diffusion, and surface diffusion. Thus, the total diffusivity is a complex property that often includes contributions from multiple mechanisms [15]. Although each individual mechanism is reasonably understood, it is not always easy to make an accurate prediction of the total diffusivity because it depends strongly on the details of the pore structure.

Figure 6a shows the molecular diffusion process in- side the macropores (capillary pores) of concrete. At a low relative humidity level, the field force of the pore wall captures water molecules to form the first at- tached layer. Other water molecules continue to move ahead and, as the humidity increases, more layers of water molecules cover the pore walls. As a result, the free space available to vapor inside the macropore de- creases. However, the force field of the wall thus weakens and, at the same time, the mean free path of water molecules decreases because the mean free path of water molecules, surrounded by the solid wall with water molecules attached to it, is smaller than the mean free path of water molecules surrounded only by solid walls. These trends affect the resistance to diffu- sion oppositely.

When the pore humidity is high enough, the ad- sorbed water will form a meniscus at a neck (a narrow connection between larger pores). At high humidity, menisci form on both ends of the neck, and the neck is completely filled. At this point, water molecules con- dense at one end of the neck, while at the other end they evaporate, as shown in Figure 6b. Since part of the transport is through gas, this condensation and evaporation process strongly accelerates the diffusion process.

The foregoing diffusion process will dominate whenever the mean free path of the water vapor (which is 800 A at 25°C) is small relative to the diameter of the macropore, which is generally regarded to have a diameter of about 50 nm to 10 ~m [16]. Pores of this size constitute only a small portion of the pores in concrete. Therefore, molecular diffusion or ordinary

1 6

4 -

Tyl~ I ¢m~nt 0 o.o o/2 o.'4 o.'8 o.~ .o

Relative Humidity

FIGURE 4. Effect of water-cement ratio on reciprocal of mois- ture capacity.

260 Y. Xi et al.

2.0 --r---------------------.

w/c - 0.8 T __ oK

1.5 t-28days

:I: 'tJ ~ 1.0

0.5

Type I cement

t - 10 days

t - 28 days

0.2 0.4 0.6

Relative humidity

FIGURE 3. Effect of age on moisture capacity.

0.8 1.0

adsorption and desorption curves possess virtually the same shape [12,13]. This can be checked further by classification of the hysteresis loops. According to de Boer's classification [14], the cement paste displays a loop of type B. Another classification recommended in the International Union of Pure and Applied Chemis­try (IUPAC) manual [14] shows that the cement paste displays a loop of type H3. Both classifications indicate that adsorption and desorption have nearly the same shape in the practical range of H. On the other hand, the BSB model (eq 19 in ref 7) is valid for both adsorp­tion and desorption. Additionally, since systematic ex­perimental results on adsorption are much more abun­dant in the literature than those for desorption, the use of the adsorption isotherm is most convenient.

In eq 3, aHlaW can be computed by the adsorption isotherm for a given w!c, tal type of cement, and tem­perature [7]. Therefore, the only unknown coefficient that remains in eq 3 is the diffusivity.

Diffusivity and Diffusion Mechanisms Diffusivity of concrete depends strongly on the diffu­sion mechanisms. Diffusion mechanisms are influ­enced by the pore structure of concrete. Three distinct transport mechanisms may operate singularly or si­multaneously: molecular diffusion (ordinary diffu­sion), Knudsen diffusion, and surface diffusion. Thus, the total diffusivity is a complex property that often includes contributions from multiple mechanisms [15]. Although each individual mechanism is reasonably understood, it is not always easy to make an accurate prediction of the total diffusivity because it depends strongly on the details of the pore structure.

Advn Cem Bas Mat 1994;1 :258-266

Figure 6a shows the molecular diffusion process in­side the macropores (capillary pores) of concrete. At a low relative humidity level, the field force of the pore wall captures water molecules to form the first at­tached layer. Other water molecules continue to move ahead and, as the humidity increases, more layers of water molecules cover the pore walls. As a result, the free space available to vapor inside the macropore de­creases. However, the force field of the wall thus weakens and, at the same time, the mean free path of water molecules decreases because the mean free path of water molecules, surrounded by the solid wall with water molecules attached to it, is smaller than the mean free path of water molecules surrounded only by solid walls. These trends affect the resistance to diffu­sion oppositely.

When the pore humidity is high enough, the ad­sorbed water will form a meniscus at a neck (a narrow connection between larger pores). At high humidity, menisci form on both ends of the neck, and the neck is completely filled. At this point, water molecules con­dense at one end of the neck, while at the other end they evaporate, as shown in Figure 6b. Since part of the transport is through gas, this condensation and evaporation process strongly accelerates the diffusion process.

The foregoing diffusion process will dominate whenever the mean free path of the water vapor (which is 800 A at 25°C) is small relative to the diameter of the macropore, which is generally regarded to have a diameter of about 50 nm to 10 !-Lm [16]. Pores of this size constitute only a small portion of the pores in concrete. Therefore, molecular diffusion or ordinary

16--r------------------~

12

~ '% ~

~

J: 8 Q~

"0

,..~

w/c - 0.8 ~Q

4 . .,

T-_OK ,..~

t - 28 days ~

Type I cement 0.6'

0 0.0 0.2 0.4 0.6 0.8 1.0

Relative Humidity

FIGURE 4. Effect of water-cement ratio on reciprocal of mois­ture capacity.

Advn Cem 8as Mat Moisture Diffusion in Cementitious Materials 261 1994;1:258-266

20

1 5 -

!1o15 0 0.0

Type I c q m ~

o'2 0.'4 o '8 o ' s l o

Relative humidity

FIGURE 5. Effect of age on reciprocal of moisture capacity.

diffusion occurs in concrete only occasionally. It is not a dominant mechanism.

Mesopores (25 to 500 A) and micropores (<25 A) comprise the largest portion of concrete pores. In these pores, the collisions be tween molecules as well as against pore walls provide the main diffusion resis- tance. In that case, the diffusion is called Knudsen diffusion. Similar phenomena occur at various humid- ity levels, but a difference exists between molecular diffusion and Knudsen diffusion. For Knudsen diffu- sion, the diffusion resistance is related to pore size. Some formulas for Knudsen diffusion in straight cylin- drical pores have been deduced [2,17]. However, for cement matrix, which is an amorphous colloidal mate- rial with randomly oriented pores and variable pore radii, it is necessary to consider the pore connections

oOOO I, O

(a) Low humidity - - adsorption

(b) High humidity - - evaporation and condensation

FIGURE 6. Diffusion mechanisms at different humidity lev- els.

and tortuosity. For smaller pores, the resistance is larger and thus the diffusivity is smaller.

Figure 7 displays the surface diffusion process that occurs in certain mesopores and micropores, such as the pores of parallel walls. The water molecules never escape the force field of the pore surface. The transport involves a thermally activated process with jumps be- tween the adsorption sites. Such a process, represent- ing the surface diffusion, poses greater resistance to transport than Knudsen diffusion for pore sizes typical of concrete. Thus, the surface diffusion is insignificant unless most of the water is adsorbed water. Therefore, surface diffusion is significant in concrete only at very low humidity.

To sum up, water diffusion in concrete occurs by one or more of the three mechanisms described: (1) ordi- nary diffusion, (2) Knudsen diffusion, and (3) surface diffusion. When multiple mechanisms occur, their ef- fects do not necessarily combine in a simple manner. Previous investigators obtained some models for each mechanism individually. Since they based their results upon diffusion in regular cylindrical pores, a tortuosity factor was introduced to correct for pore irregularity. The contributions of each mechanism were then com- bined [2,17,18]. However, as mentioned above, the pore structure changes when the water-cement ratio changes. Furthermore, the formation process of fine micropores depends strongly on time and humidity level. Therefore, it is difficult to derive a general ex- pression for diffusivity. The model we will now present does not treat each mechanism individually, but tries to predict the general combined trend.

Upon comparison of all three diffusion mechanisms, it is clear that they share similarities. At low humidity, the pore volume decreases, the surface force field weakens, and the mean free path decreases. These be- haviors may just offset each other such that the effec- tive diffusivity for all the mechanisms becomes con-

///////////////

/ / / / / / / / / / / / /

Pore wall

FIGURE 7. Mechanism of surface diffusion.

Advn Cem Bas Mat 1994;1 :258-266

20,---------------------------------,

15

5

w/c - 0.8 T-_oK

Type I cenW1t

04-----~------~----_.------._----~ 0.0 0.2 0.4 0.6 0.8 1.0

Relative humidity

FIGURE 5. Effect of age on reciprocal of moisture capacity.

diffusion occurs in concrete only occasionally. It is not a dominant mechanism.

Mesopores (25 to 500 A) and micropores «25 A) comprise the largest portion of concrete pores. In these pores, the collisions between molecules as well as against pore walls provide the main diffusion resis­tance. In that case, the diffusion is called Knudsen diffusion. Similar phenomena occur at various humid­ity levels, but a difference exists between molecular diffusion and Knudsen diffusion. For Knudsen diffu­sion, the diffusion resistance is related to pore size. Some formulas for Knudsen diffusion in straight cylin­drical pores have been deduced [2,17]. However, for cement matrix, which is an amorphous colloidal mate­rial with randomly oriented pores and variable pore radii, it is necessary to consider the pore connections

(a) Low humidity - adsorption

~o

o 0 o 0 ~ o ~rdon

00 0_

_----__ 0 0

(b) High humidity - evaporation and condensation

FIGURE 6. Diffusion mechanisms at different humidity lev­els.

Moisture Diffusion in Cementitious Materials 261

and tortuosity. For smaller pores, the resistance is larger and thus the diffusivity is smaller.

Figure 7 displays the surface diffusion process that occurs in certain mesopores and micropores, such as the pores of parallel walls. The water molecules never escape the force field of the pore surface. The transport involves a thermally activated process with jumps be­tween the adsorption sites. Such a process, represent­ing the surface diffusion, poses greater resistance to transport than Knudsen diffusion for pore sizes typical of concrete. Thus, the surface diffusion is insignificant unless most of the water is adsorbed water. Therefore, surface diffusion is significant in concrete only at very low humidity.

To sum up, water diffusion in concrete occurs by one or more of the three mechanisms described: (1) ordi­nary diffusion, (2) Knudsen diffusion, and (3) surface diffusion. When multiple mechanisms occur, their ef­fects do not necessarily combine in a simple manner. Previous investigators obtained some models for each mechanism individually. Since they based their results upon diffusion in regular cylindrical pores, a tortuosity factor was introduced to correct for pore irregularity. The contributions of each mechanism were then com­bined [2,17,18]. However, as mentioned above, the pore structure changes when the water-cement ratio changes. Furthermore, the formation process of fine micropores depends strongly on time and humidity level. Therefore, it is difficult to derive a general ex­pression for diffusivity. The model we will now present does not treat each mechanism individually, but tries to predict the general combined trend.

Upon comparison of all three diffusion mechanisms, it is clear that they share similarities. At low humidity, the pore volume decreases, the surface force field weakens, and the mean free path decreases. These be­haviors may just offset each other such that the effec­tive diffusivity for all the mechanisms becomes con-

///////////////

Pore wall

FIGURE 7. Mechanism of surface diffusion.

262 Y. Xi et al. Advn Cem Bas Mat 1994 ;1 :258-266

stant at low humidity (see Figure 8). At high humidity, capillary condensation occurs and hence the diffusion resistance lessens. Consequently, the effective diffu- sivity of the system may be assumed to follow the simple empirical curve shown in Figure 8d. The inev- itable empirical aspect of this curve makes it unneces- sary, and in fact unjustified, to distinguish among these mechanisms. The following simple empirical for- mula, which can capture the aforementioned trends, is proposed:

D h = OLh + ~h[1 - 2 - ' o ~ h ( " - ' ) ] . (4)

a) D h 004

0 0 3

002

a h = 0.02

#h = 0.01

"Yh = 2

/

Here, ~h, ~h, and ~/h are coefficients to be calibrated from test data (Figure 9 displays their meanings). ~a represents the lower bound on diffusivity approached at low humidity level. The value of f3h/2 is the diffusiv- ity increment from low humidity level to saturation state; ~/h characterizes the humidity level at which the diffusivity begins to increase.

The coefficients %, [3 a, and ~/h are strongly affected by w/c. The effect of curing time on these coefficients could be regarded as negligible. The curing time, of course, does affect the diffusion process, but this is taken into account by the moisture capacity.

b) D h

0.04

0.03

0.02

cx h = 0.02

#h = 0.03

/ 7 h = 4

0.01 0.01 o o 0/2 0.'4 0/6 o'8 10 00 o'2 o'4 o~ o'a 10

Relative humidity Relative humidity

Ordinary diffusion Knudsen diffusion

c) D h d) D h 0.04 0.04

0.03

002

a h = 0.013

flh = 0.01

7 h = 1 0.03

0.02

c( h = 0.018

flh = 0.025

7 h ==3 / 0.01 , 0.01

00 o'z o'4 06 o~ lo 00 o'2 o'4 o'6 o'a ~ o

Relative humidity Relative humidity

Surface diffusion Total diffusion

FIGURE 8. Diffusivity dependence on humidity for various diffusion mechanisms and for their combined effect.

262 Y. Xi et al.

stant at low humidity (see Figure 8). At high humidity, capillary condensation occurs and hence the diffusion resistance lessens. Consequently, the effective diffu­sivity of the system may be assumed to follow the simple empirical curve shown in Figure 8d. The inev­itable empirical aspect of this curve makes it unneces­sary, and in fact unjustified, to distinguish among these mechanisms. The following simple empirical for­mula, which can capture the aforementioned trends, is proposed:

Dh = ah + J3h[l - 2- IO"Yh(H-l)]. (4)

a)

c)

0.04 ,---------------,

0.03

a h ,. 0.02

Ph = 0.01

'Yh '" 2

0.02 J-----~

0.01 0.0

Dh 0.04

0.03

0.02

0.2 0.4 0.6

Relative humidity

Ordinary diffusion

a h ... 0.013

Ph = 0.01

'Yh .. 1

0.8 1.0

0.01 +----,---r---.,----r----i 0.0 0.2 0.4 0.6

Relative humidity

Surface diffusion

0.8 1.0

b)

d)

Advn Cern Bas Mat 1994;1 :258-266

Here, ah' J3h' and 'Yh are coefficients to be calibrated from test data (Figure 9 displays their meanings). ah represents the lower bound on diffusivity approached at low humidity level. The value of J3h12 is the diffusiv­ity increment from low humidity level to saturation state; 'Yh characterizes the humidity level at which the diffusivity begins to increase.

The coefficients ah' J3h' and 'Yh are strongly affected by w!c. The effect of curing time on these coefficients could be regarded as negligible. The curing time, of course, does affect the diffusion process, but this is taken into account by the moisture capacity.

0.04,..---------------,

0.03

a h = 0.02

Ph '" 0.03

'Yh '" 4

0.02 4--------

0.01 0.0

Dh 0.04

0.2 0.4 0.6 0.8

Relative humidity

Knudsen diffusion

a h = 0.018

Ph ... 0.025

1.0

0.03

'Yh =3 /

0.02 I----~ 0.01 +----,---r---r----,.--~

0.0 0.2 0.4 0.6 0.8 1.0

Relative humidity

Total diffusion

FIGURE 8. Diffusivity dependence on humidity for various diffusion mechanisms and for their combined effect.

Advn Cem Bas Mat Moisture Diffusion in Cementitious Materials 263 1994;1:258-266

0.5

f 12 h

0 . 4

0.3 e- 13

0.2

0 . 1

D h "h + /lh{1-tlXp[-II'~lO 7h(H'l) = 1}

7h = ~

o.0 o o o'2 0.'4 0?6 o.~ ~o

Ph/2

RetatNe humidity

FIGURE 9. Diffusivity dependence on humidity for various values of parameter "/h-

Before calibrating eq 4 with test data, the possible relations between ot h, f3h, ~/h, and w/c must be analyzed. As described earlier, the porosity increases as w/c in- creases. This means that the volume fraction of macro- pores increases also. As a result, the diffusivity at low humidity levels increases with increasing w/c because water molecules migrate much faster in the macro- pores than in the micropores. Thus, ot h generally in- creases with increasing w/c. As for 13 h, it first increases with increasing w/c, for the same reason as does ah. After a certain point, however, f3h decreases with in- creasing w/c because the increase of diffusivity from a low humidity level to the saturation state will gradu- ally weaken with increasing volume fraction of macro- pores. An increase in w/c leads to an increase in ~h. From Figure 9, a larger ~/h corresponds to a higher hu- midity level at which the diffusivity initially begins to increase, and it also corresponds to a higher rate of diffusivity increase. This trend is reasonable because a higher w/c corresponds to a larger volume fraction of macropores, and because the humidity level, or the pressure level, necessary for capillary condensation is higher in a larger pore than in a smaller pore, accord- ing to capillary theory. Knowledge of the foregoing general trends is helpful to choose a correct function to be calibrated by drying test results (Figures 10-12).

TABLE 1. Test information and optimized parameters

No. w/c t o l ~h ~h "~h

66-07-08 0. 657 7 7.5 0. 003 5.170 7.03 66-28-08 0.657 28 7.5 0.003 5.240 7.66 50-03-15 0.50 3 15.0 0.0423 0.432 3.54 59-01-10 0.59 1 10.0 0.020 0.875 8.97 63-03-15 0.63 3 15.0 0.088 0.864 8.58 75-03-15 0.75 3 15.0 0.195 0.002 20.60

0 . 3 0

0 . 1 5

0 . 1 0

0 . 0 ~

0 " 0 0 0 ~ o . ~ o o . ~ o . a o o . a ~ o . ' ~o o . ~ o . - o

wlc

FIGURE 10. Dependence of % on water-cement ratio of con- crete mix.

Calibration by Test Results Two methods are used to identify D h from test results. First, upon application of the Boltzmann transforma- tion [19] to eq 3, the diffusivity in eq 3 can be expressed as follows: Dh(H ) = f ~ u/(2dH/du), dH, where u = y/~/t is the Boltzmann variable, and y is the depth from the surface of drying. This method, however, is valid only when the diffusion equation is linear, that is, when the moisture capacity in eq 3 is constant. The second method consists of trial and error selections of various expressions for Dh, such as eq 4. For each selection, the nonlinear diffusion equation must be solved numeri- cally to determine which selection best fits the experi- mental results. The optimum values of the coefficients in the formula can be found by an optimization algo- rithm that minimizes the sum of squared deviation from test data, which has been used in the present study.

The equation to be solved is the same as eq 3. As- suming that drying occurs only in one dimension, we have the diffusion equation with the boundary and initial conditions

at - OW Ox Dh-~x (5a)

fo r t = 0 , 0 ~ < x ~ I : H = 1 f o r x = 1, t > 0 : H = 0

forx = 0, t > 0 : OH/Ox= O. (5b)

c ~ . a

o . 6

o . 2

FIGURE 11. Dependence of ~ h o n water:cement ratio.

o . e

w/c

Advn Cem Bas Mat 1994;1 :258-266

0.5 ....----------------,

.c o

0.4

0.3

0.2

To. 1

4.

L 0.0 -+-----..---,----,---,-------1 0.0 0.2 0.4 0.6 0.8 1.0

Relative humidity

1

FIGURE 9. Diffusivity dependence on humidity for various values of parameter 'Yh'

Before calibrating eq 4 with test data, the possible relations between Cth' I3h' 'Yh' and wle must be analyzed. As described earlier, the porosity increases as wle in­creases. This means that the volume fraction of macro­pores increases also. As a result, the diffusivity at low humidity levels increases with increasing wle because water molecules migrate much faster in the macro­pores than in the micropores. Thus, Cth generally in­creases with increasing wle. As for I3h' it first increases with increasing wle, for the same reason as does o'h'

After a certain point, however, I3h decreases with in­creasing wle because the increase of diffusivity from a low humidity level to the saturation state will gradu­ally weaken with increasing volume fraction of macro­pores. An increase in wle leads to an increase in 'Yh'

From Figure 9, a larger 'Yh corresponds to a higher hu­midity level at which the diffusivity initially begins to increase, and it also corresponds to a higher rate of diffusivity increase. This trend is reasonable because a higher wle corresponds to a larger volume fraction of macropores, and because the humidity level, or the pressure level, necessary for capillary condensation is higher in a larger pore than in a smaller pore, accord­ing to capillary theory. Knowledge of the foregoing general trends is helpful to choose a correct function to be calibrated by drying test results (Figures 10-12).

TABLE 1. Test information and optimized parameters

No. w!c to Cth I3h 'Yh

66-07-08 0.657 7 7.5 0.003 5.170 7.03 66-28-08 0.657 28 7.5 0.003 5.240 7.66 50-03-15 0.50 3 15.0 0.0423 0.432 3.54 59-01-10 0.59 1 10.0 0.020 0.875 8.97 63-03-15 0.63 3 15.0 0.088 0.864 8.58 75-03-15 0.75 3 15.0 0.195 0.002 20.60

Moisture Diffusion in Cementitious Materials 263

0.20 ....----------------:1.--..., O.1S

lj~ 0.10 • •

• w/c

FIGURE 10. Dependence of Cth on water-cement ratio of con­crete mix.

Calibration by Test Results Two methods are used to identify Dh from test results. First, upon application of the Boltzmann transforma­tion [19] to eq 3, the diffusivity in eq 3 can be expressed as follows: Dh(H) = IiI ul(2dHldu), dH, where u = ytv't is the Boltzmann variable, and y is the depth from the surface of drying. This method, however, is valid only when the diffusion equation is linear, that is, when the moisture capacity in eq 3 is constant. The second method consists of trial and error selections of various expressions for Dh, such as eq 4. For each selection, the nonlinear diffusion equation must be solved numeri­cally to determine which selection best fits the experi­mental results. The optimum values of the coefficients in the formula can be found by an optimization algo­rithm that minimizes the sum of squared deviation from test data, which has been used in the present study.

The equation to be solved is the same as eq 3. As­suming that drying occurs only in one dimension, we have the diffusion equation with the boundary and initial conditions

Itt

for t = 0, 0 ,,;;; x ,,;;; 1: H = 1 for x = I, t > 0: H = 0 for x = 0, t > 0: aHlax = O.

o.~

0.0 O.4e. o ... o. 0

FIGURE 11. Dependence of I3h on water:cement ratio.

(Sa)

(5b)

w/c

264 Y. Xi et al. Advn Cem Bas Mat 1994;1:258-266

2 ~

2 0

1 0

FIGURE 12. Dependence of "Yh on water:cement ratio.

O . 8 O

w/c

This diffusion problem is solved by the Crank- Nicolson finite difference algorithm [20] for both cylin- drical and Cartesian coordinate systems. D h (eq 4) is a function with unknown parameters ~h, ~h, and ~h. Therefore, a nonlinear curve-fitting program is used and the program for the finite difference solution is combined with a nonlinear optimization subroutine in order to calibrate the three parameters with available test data.

An important point is the effect of age. The speci- mens for adsorption test were cured up to t o days, and then all the evaporable water held inside the specimen was evaporated by the vacuum method. After this pro- cess the pore structure is supposed to be the same as that at age t o , since the hydration process ceases when the pore relative humidity drops below approximately H = 0.8 [21]. This means that time t in the equations of adsorption isotherm [7] corresponds roughly to the time when the hydration process terminates, and the termination of the process depends mainly on the pore relative humidity. For real drying specimens, the pore relative humidity in the outer layers drops after a short exposure to the environment, while the inner pores are still under conditions similar to the curing room. Thus, the age t is actually an equivalent age t e, not the real age, and it is significantly different at different loca- tions.

For the numerical solution, an equivalent time incre- ment At e may be assumed to be a function of the rela- tive humidity in the pores as follows [1]:

Ate = At[1 + (7.5--7.5/-/)4] -1 (6)

where At is the real age increment when the specimen is cured in a fog room. Equation 6 satisfies two asymp- totic trends. When H is near saturation, At e = At , and when H approaches 0, At e is almost 0. Through eq 6, the t ransfer from hydra t ion to n o n h y d r a t i o n is achieved in a smooth way, and no sudden jump oc- curs.

To calibrate a general expression for diffusivity (eq 4), large amounts of test data are needed. The experi- ments should be carried out systematically. By keeping all the parameters constant except one, the effect of that parameter on diffusivity can be checked. This needs to be done for the effects of w/c, concrete com- position, temperature, and to. The only test data that serve this purpose are those obtained by Molina [23], which will be used in this study. These tests have var- ious w/c values and almost the same t o as listed in Table 1. They are labeled 50-03-15, 63-03-15, and 75-03-15, where, for example, the first two digits in 50-03-15 mean w/c = 0.5, the next two digits 03 mean t o = 3 days, and the last two digits 15 mean specimen depth 15 cm. Parrott's drying test results [22], labeled 59-01- 10, are also used for parameter calibrations.

The test results used by Ba~ant and Najjar [1], la- beled 66-07-08 and 66-28-08 in Table 1, will not be used here, although a good fit of those test data can be achieved by eq 4 (see Figures 13 and 14). By compari- son of those test data with the data shown in Figures 15 to 18, one finds that the trend at small t - t o (i.e., high relative humidity) is completely different from that in Figures 13 and 14. The old data (Figures 13 and 14) showed a sharp drop of H in both the inner and

l-~rulon 0968)

. ~ ~. o j ~ . w/= - 0.~7

E i . 3 ,

~" o . 8 x - 1 . 9 x - 7 . W

0 I '~ 0 6

o 4 o 2 6 0 4 6 0 6 6 0 8 6 o ~ o ' o o ~ 2 0 o

Time in days

FIGURE 13. Comparison of calculated time dependence of humidity with Hanson's data (]968) for curing period t o = 7 days.

, • " 1 0

E r -

o B ._~

r ~ 0 . 6

Hanson (1968)

x -O .6 X-1 ,g

0 2 . 0 0 , ~ 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0

Time in clays

FIGURE 14. Comparison of calculated time dependence of humidity with Hanson's data [24] for curing period t o = 28 days.

264 Y. Xi et al.

• •

w/c

FIGURE 12. Dependence of 'Yh on water:cement ratio.

This diffusion problem is solved by the Crank­Nicolson finite difference algorithm [20] for both cylin­drical and Cartesian coordinate systems. Dh (eq 4) is a function with unknown parameters Uh' ~h' and 'Yh' Therefore, a nonlinear curve-fitting program is used and the program for the finite difference solution is combined with a nonlinear optimization subroutine in order to calibrate the three parameters with available test data.

An important pOint is the effect of age. The speci­mens for adsorption test were cured up to to days, and then all the evaporable water held inside the specimen was evaporated by the vacuum method. After this pro­cess the pore structure is supposed to be the same as that at age to, since the hydration process ceases when the pore relative humidity drops below approximately H = 0.8 [21]. This means that time t in the equations of adsorption isotherm [7] corresponds roughly to the time when the hydration process terminates, and the termination of the process depends mainly on the pore relative humidity. For real drying specimens, the pore relative humidity in the outer layers drops after a short exposure to the environment, while the inner pores are still under conditions similar to the curing room. Thus, the age t is actually an equivalent age te, not the real age, and it is significantly different at different loca­tions.

~ 1.0

:2 E ::I

J::. Q)

i O.B

16 a: 0.6

1t·0.1

04 0 200 400

x.4.4

600

Henaon (1968)

w/c -0.8157

'.-7_ ,- 3"

BOO 1000 1200

Time in days

FIGURE 13. Comparison of calculated time dependence of humidity with Hanson's data (1968) for curing period to = 7 days.

Advn Cem Bas Mat 1994;1 :258-266

For the numerical solution, an equivalent time incre­ment Me may be assumed to be a function of the rela­tive humidity in the pores as follows [1]:

(6)

where t1t is the real age increment when the specimen is cured in a fog room. Equation 6 satisfies two asymp­totic trends. When H is near saturation, t1te = t1t, and when H approaches 0, t1te is almost O. Through eq 6, the transfer from hydration to nonhydration is achieved in a smooth way, and no sudden jump oc­curs.

To calibrate a general expression for diffusivity (eq 4), large amounts of test data are needed. The experi­ments should be carried out systematically. By keeping all the parameters constant except one, the effect of that parameter on diffusivity can be checked. This needs to be done for the effects of w!c, concrete com­position, temperature, and to. The only test data that serve this purpose are those obtained by Molina [23], which will be used in this study. These tests have var­ious wlc values and almost the same to as listed in Table 1. They are labeled 50-03-15, 63-03-15, and 75-03-15, where, for example, the first two digits in 50-03-15 mean wlc = 0.5, the next two digits 03 mean to = 3 days, and the last two digits 15 mean specimen depth 15 cm. Parrott's drying test results [22], labeled 59-Ol­IO, are also used for parameter calibrations.

The test results used by Bazant and Najjar [I], la­beled 66-07-08 and 66-28-08 in Table I, will not be used here, although a good fit of those test data can be achieved by eq 4 (see Figures 13 and 14). By compari­son of those test data with the data shown in Figures 15 to 18, one finds that the trend at small t - to (i.e., high relative humidity) is completely different from that in Figures 13 and 14. The old data (Figures 13 and 14) showed a sharp drop of H in both the inner and

~ 1.0

"C 'E ::I

J::. Q) 0 B

i ~

0.6

Hanson (1968)

wle -0.857

x·O.8 x-UI

0.4 -±O---=-2rOO=---4"Oc:::0-----=6TOO=---S=-=0...,0---::-, 0-'-0'-:0--1-=-200

Time in days

FIGURE 14. Comparison of calculated time dependence of humidity with Hanson's data [24] for curing period to = 28 days.

Advn Cem Bas Mat Moisture Diffusion in Cementitious Materials 265 1994;1:258-266

e) , • 1 o

" 0

l - o s

e r o 6

Mo l ln l i I t Id. (1891)

x = 12.5

x - O x - 3 , S

x - 2

b) -g E

c) ,_~ ~o -g_ E t - o . 8

"~ o 6

e r

w/c = 0.63 I = 15 cm t d= 3 daya H ~ , - 50 %

5 ' 0 1 (DO

Time in days

Mal ine I t el. (1991)

W/C = 0,75 I = 15 c m

t O= 3 days Harp, 50 %

~ '© 1 (Do

Time in days ~ o

x . 12.5

\ Z, w/c = O.ti

to" 3 daym I - 15¢m

Hen- 50 % M01ir~i t t Id. (1991)

o 5 0 1 o o 1 5 0 2 0 0

Time in days

FIGURE 15. Comparisons of calculated time dependence of humidity with Molina's data [23].

outer layers of the specimens immediately after the exposure, which might not be correct since the inner layer cannot respond that quickly to the change of boundary condition. The recent data (Figures 15 to 16) show the more reasonable trends: the outside layer exhibits a sharp drop of H and the inner layer exhibits a smooth transition from high H to low H. The possible reason for such a difference is that the advanced tech- nique to precisely measure the high relative humidities (80 to 100%) has been developed only during the last 20 years. The parameters o~ h, 13 h, and "/h corresponding to the best fit of the old data are not consistent with the values corresponding to the recent results (see Table 1), due to the different trends. Thus, the old data are excluded.

The optimum formula for oL h, illustrated in Figure 10, is

w ah = 1.05 -- 3.8 -- + 3.56 (7)

c

Upon examination of Figure 10, it is apparent that ah increases as w/c increases. The optimum curves for 15h and ~/h are:

w (wt ~h = --14.4 + 5 0 . 4 - -- 41.8 (8) C

~/h = 31.3 -- 1 3 6 - + 162 (9) C

Figures 11 and 12 i l lustrate eqs 8 and 9. It is apparent that the trends obtained w i th eqs 7 to 9 agree w i th those obtained earlier by analysis of the d i f fus ion mechanisms.

The theoretical curves based upon moisture capacity (eq 2) and diffusivity (eqs 4 and 7 through 9) are shown in Figures 13 to 16 in comparison with the test data. All the theoretical curves fit the test curves satisfactorily and give reasonable long-term predictions as well.

Conclusions

. The moisture capacity, obtained as the derivative of adsorption isotherm, is not a constant. First it drops, then becomes constant, and finally in- creases. This trend is generally true regardless of the age, type of cement, temperature, and water: cement ratio. The physical meaning of the first turn points in moisture capacity, marking the transition from the initial drop to the constant region at increasing humidity, may correspond to the reaching of monolayer capacity in adsorp- tion isotherm; and the meaning of the second turn point, marking the transition from the con- stant region to the final increase, may correspond to the beginning of capillary condensation. For

.'~'_ Parrott (1988) 1 0

i . o

" ~ x - 3.36 r "

o . 9

o . 8 .... ~ x - 1,66

i f ' wi t - 0.sin ..... x . , ! . l 5

o 7 I - 1o¢m x - o.7~ Hen- 10 %

o 6 ° o o 5 o , ' o 0 1 o o ' o o 1 5 o ' . o o 2 o o . o o

Time in days

FIGURE 16. Further comparison of calculated time depen- dence of humidity with Parrott [22].

a)

b)

c)

Advn Cem Bas Mat 1994;1 :258-266

. ~ 1 0

'C 'E :::J ..c: ~

0.8

~ Qi c:

06

Molina II Ii. (1991)

;--..'-'-,~~==-=~-'-=~~----<_~~~--:.!x .12.5

.-8

w/c - 0.83 ~1:_~15~cm:-"----L..-~-----~· t ~ 3 days H..,- 50"

o 4 +----~----___,_---_____:<__:__-----1 a 50 100 150 200

~ 10

'C 'E .E 08

~ ~ 0.6

Qi c:

0.4

w/c - 0.75 t.- 3 day.

Time in days

1-15cm

H..,- 50"

Molina II Ii. (1991)

o 2 +-____ -,-____ -----:< _____ ~-----.J

~ 'C 'E :::J ..c:

.~ ia Qi c:

1.0

0.8

06

04

o

./e - 0.5

to-3~

1-15c:m

Hen- 50"

50 100 150

Time in days

Molina II aI. (1991) o 2 +0-----=5~0----,~0-0---~--------l

Time in days

FIGURE 15. Comparisons of calculated time dependence of humidity with Molina's data [23].

outer layers of the specimens immediately after the exposure, which might not be correct since the inner layer cannot respond that quickly to the change of boundary condition. The recent data (Figures 15 to 16) show the more reasonable trends: the outside layer exhibits a sharp drop of H and the inner layer exhibits a smooth transition from high H to low H. The possible reason for such a difference is that the advanced tech­nique to precisely measure the high relative humidities (80 to 100%) has been developed only during the last 20 years. The parameters (Xh' I3h' and 'Yh corresponding to the best fit of the old data are not consistent with the values corresponding to the recent results (see Table I), due to the different trends. Thus, the old data are excluded.

The optimum formula for (Xh' illustrated in Figure 10, is

Moisture Diffusion in Cementitious Materials 265

W (W)2 (Xh = 1.05 - 3.8 ~ + 3.56 ~ . (7)

Upon examination of Figure 10, it is apparent that (Xh

increases as wle increases. The optimum curves for I3h and 'Yh are:

W (W)2 I3h = -14.4 + 50.4 ~ - 41.8 ~ (8)

W (W)2 'Yh = 31.3 - 136 ~ + 162 ~ . (9)

Figures 11 and 12 illustrate eqs 8 and 9. It is apparent that the trends obtained with eqs 7 to 9 agree with those obtained earlier by analysis of the diffusion mechanisms.

The theoretical curves based upon moisture capacity (eq 2) and diffusivity (eqs 4 and 7 through 9) are shown in Figures 13 to 16 in comparison with the test data. All the theoretical curves fit the test curves satisfactorily and give reasonable long-term predictions as well.

Conclusions

.~ 'C 'E :::J ..c: CD > ~ Qi c:

1. The moisture capacity, obtained as the derivative of adsorption isotherm, is not a constant. First it drops, then becomes constant, and finally in­creases. This trend is generally true regardless of the age, type of cement, temperature, and water: cement ratio. The physical meaning of the first turn points in moisture capacity, marking the transition from the initial drop to the constant region at increasing humidity, may correspond to the reaching of monolayer capacity in adsorp­tion isotherm; and the meaning of the second turn point, marking the transition from the con­stant region to the final increase, may correspond to the beginning of capillary condensation. For

Parrott (1988)

1.0

0.9

0.8 X .1.56

./e·O.58

07 to. 1 day l-l0cm x·O.1$ Hen- 10%

06 000 50.00 100.00 150.00 200.00

Time in days

FIGURE 16. Further comparison of calculated time depen­dence of humidity with Parrott [22].

266 Y. Xi et al. Advn Cem Bas Mat 1994;1:258-266

the usual practical range from 50 to 100%, the moisture capacity keeps increasing with increas- ing relative humidi ty .

2. There are three possible diffusion mechanisms for concrete drying: ordinary diffusion, Knudsen diffusion, and surface diffusion. By individual analysis of each diffusion mechanism, the com- mon features of the mechanisms are found: for low humidi t ies , the diffusivities may become constant, and at high humidities, the diffusivities increase, regardless of the mechanism. There- fore, instead of considering combinations of the diffusivities of various mechanisms, a total diffu- sivity expression that simply reflects these com- mon features is proposed.

3. The improved formula for the dependence of dif- fusivity on pore humidi ty , developed herein, gives satisfactory agreement with test data.

4. The influence of the water:cement ratio on the diffusivity is analyzed. With an increasing water: cement ratio, (1) the diffusivity at low humidi ty level increases; (2) the incremental part of diffu- sivity increases to a certain point, then decreases, and vanishes asymptotically with a very large water:cement ratio; and (3) the humidi ty level at which the diffusivity begins increasing becomes higher.

5. The p r e s e n t m o d e l for m o i s t u r e d i f fu s ion through concrete gives satisfactory diffusion pro- files and correct long-term drying predictions. It is suitable for incorporation in finite element pro- grams for shrinkage and creep effects in concrete structures.

Acknowledgments

The theoretical part of the present research was supported under NSF grant 0830-350-C802 to Northwestern University, and the test data analysis was supported by NSF Science & Technology Center for Advanced Cement-Based Materials at Northwestern University. L. Molina wishes to thank Swedish Cement and Concrete Institute (CBI), Stockholm, for supporting her one-year Visiting Research As- sociate appointment at Northwestern University.

References 1. Ba~ant, Z.P.; Najjar, L.J. Mat~riaux et Constructions 1972,

5, 3-20. 2. Daian, J.-F. Transport in Porous Media 1988, 3, 563-589. 3. Daian, J.-F. Transport in Porous Media 1989, 4, 1-16. 4. Jonasson, J.-E. Proc. of 8th SMIRT 1985, H5/11,235-242. 5. Saetta, A.V.; Scotta, R.V.; Vitaliani, R.V. Materials Jour-

nal, ACI 1993, 90, 441-451. 6. Ba~ant, Z.P.; Chern, J.C.; Rosenberg, A.M.; Gaidis, J.M.

J. Am. Ceram. Soc. 1988, 71,776--783. 7. Xi, Y.; Ba~ant, Z.P.; Jennings, H.M.J. Adv. Cement Based

Mater. 1994, 1,248-257. 8. Mensi, R.; Acker, P.; Attolou, A. Materials and Construc-

tion 1988, 21, 3-12. 9. Pihlajavaara, S.E.; Vaisanen, J. Numerical Solution of Dif-

fusion Equation with Diffusivity Concentration Dependent; Publ. No. 87, State Institute for Technical Research, Hel- sinki.

10. Wittmann, X.; Sadouki, H.; Wittmann, F.H. Proc. of lOth SMIRT 1989, Q, 71-79.

11. Gleysteen, L.F.; Kalousek, G.L.J. Am. Conc. Inst. 1955, 26, 437 ~6.

12. Hagymassy, J., Jr.; Odler, I.; Yudenfreund, M.; Skalny, J.; Brunauer, S. J. Colloid Interface Sci. 1972, 38, 20-34, 265-276.

13. Mikhail, R.S.; Abo-EI-Enein, S.A.; Gabr, N.A.J. Appl. Chem. Biotechnol. 1975, 25, 835-847.

14. Gregg, S.J., Sing, K.S.W. Adsorption, Surface Area and Po- rosity; Academic Press Inc.: London, 1982.

15. Karger, J.; Ruthven, D.M. Diffusion in Zeolites and Other Microporous Solids; John Wiley & Sons, Inc.: New York, 1992.

16. Ba~ant, Z.P., Ed. Mathematical Modeling of Creep and Shrinkage of Concrete; John Wiley and Sons: Chichester, 1988.

17. Ruthven, D.M. Principles of Adsorption and Adsorption Pro- cesses; John Wiley & Sons, Inc.: New York, 1984.

18. Satterfield, C.N. Mass Transfer in Heterogeneous Catalysis, MIT Press, 1970.

19. Sakata, K. Cement Concrete Res. 1983, 13, 216-224. 20. von Rosenberg, D.U. Methods for the Numerical Solution of

Partial Differential Equations; American Elsevier Publish- ing Company, Inc.: New York, 1969.

21. Powers, T.C. Proc. of the Highway Research Board 1947, 27, 178-188 (PCA Bulletin No. 25).

22. Parrott, L.J. Adv. Cement Res. 1988, 1, 164-170. 23. Molina, L. Private communication on test results ob-

tained at Swedish Cement and Concrete Institute (CBI), Stockholm, 1991.

24. Hanson, J.A.J. Am. Conc. Inst. 1968, 65, 535-543 (also: PCA Bull. D141).

266 Y. Xi et al.

the usual practical range from 50 to 100%, the moisture capacity keeps increasing with increas­ing relative humidity.

2. There are three possible diffusion mechanisms for concrete drying: ordinary diffusion, Knudsen diffusion, and surface diffusion. By individual analysis of each diffusion mechanism, the com­mon features of the mechanisms are found: for low humidities, the diffusivities may become constant, and at high humidities, the diffusivities increase, regardless of the mechanism. There­fore, instead of considering combinations of the diffusivities of various mechanisms, a total diffu­sivity expression that simply reflects these com­mon features is proposed.

3. The improved formula for the dependence of dif­fusivity on pore humidity, developed herein, gives satisfactory agreement with test data.

4. The influence of the water:cement ratio on the diffusivity is analyzed. With an increasing water: cement ratio, (1) the diffusivity at low humidity level increases; (2) the incremental part of diffu­sivity increases to a certain point, then decreases, and vanishes asymptotically with a very large water:cement ratio; and (3) the humidity level at which the diffusivity begins increasing becomes higher.

5. The present model for moisture diffusion through concrete gives satisfactory diffusion pro­files and correct long-term drying predictions. It is suitable for incorporation in finite element pro­grams for shrinkage and creep effects in concrete structures.

Acknowledgments

The theoretical part of the present research was supported under NSF grant 0830-350-C802 to Northwestern University, and the test data analysis was supported by NSF Science & Technology Center for Advanced Cement-Based Materials at Northwestern University. L. Molina wishes to thank Swedish Cement and Concrete Institute (CBI), Stockholm, for supporting her one-year Visiting Research As­sociate appointment at Northwestern University.

References

Advn Cern Bas Mat 1994; 1 :258-266

1. Bazant, Z.P.; Najjar, L.J. Materiaux et Constructions 1972, 5, 3--20.

2. Daian, J.-F. Transport in Porous Media 1988, 3, 563--589. 3. Daian, J.-F. Transport in Porous Media 1989, 4, 1-16. 4. Jonasson, J.-E. Proc. of 8th SMIRT 1985, H5111, 235-242. 5. Saetta, A.V.; Scotta, RV.; Vitaliani, RV. Materials Jour­

nal, ACI 1993, 90, 441-451. 6. Bazant, Z.P.; Chern, J.e.; Rosenberg, A.M.; Gaidis, J.M.

J. Am. Ceram. Soc. 1988, 71, 776-783. 7. Xi, Y.; Bazant, Z.P.; Jennings, H.M. J. Adv. Cement Based

Mater. 1994, 1, 248-257. 8. Mensi, R; Acker, P.; Attolou, A. Materials and Construc­

tion 1988, 21, 3--12. 9. Pihlajavaara, S.E.; Vaisanen, J. Numerical Solution of Dif­

fusion Equation with Diffusivity Concentration Dependent; Pub!. No. 87, State Institute for Technical Research, Hel­sinki.

10. Wittmann, X.; Sadouki, H.; Wittmann, F.H. Proc. of 10th SMIRT 1989, Q, 71-79.

11. Gleysteen, L.F.; Kalousek, G.L. J. Am. Cone. Inst. 1955, 26, 437-446.

12. Hagymassy, J., Jr.; Odler, I.; Yudenfreund, M.; Skalny, J.; Brunauer, S. J. Colloid Interface Sci. 1972, 38, 20-34, 265-276.

13. Mikhail, RS.; Abo-El-Enein, S.A.; Gabr, N.A. J. Appl. Chern. Biotechnol. 1975,25, 835-847.

14. Gregg, S.J., Sing, K.S.W. Adsorption, Surface Area and Po­rosity; Academic Press Inc.: London, 1982.

15. Karger, J.; Ruthven, D.M. Diffusion in Zeolites and Other Microporous Solids; John Wiley & Sons, Inc.: New York, 1992.

16. Bazant, Z.P., Ed. Mathematical Modeling of Creep and Shrinkage of Concrete; John Wiley and Sons: Chichester, 1988.

17. Ruthven, D.M. Principles of Adsorption and Adsorption Pro­cesses; John Wiley & Sons, Inc.: New York, 1984.

18. Satterfield, e.N. Mass Transfer in Heterogeneous Catalysis, MIT Press, 1970.

19. Sakata, K. Cement Concrete Res. 1983, 13, 216-224. 20. von Rosenberg, D.U. Methods for the Numerical Solution of

Partial Differential Equations; American Elsevier Publish­ing Company, Inc.: New York, 1969.

21. Powers, T.e. Proc. of the Highway Research Board 1947, 27, 178-188 (PCA Bulletin No. 25).

22. Parrott, L.J. Adv. Cement Res. 1988, 1, 164-170. 23. Molina, L. Private communication on test results ob­

tained at Swedish Cement and Concrete Institute (CBI), Stockholm, 1991.

24. Hanson, J.A. J. Am. Conc. Inst. 1968, 65, 535-543 (also: PCA Bull. D141).