of the Mechanics and ofcee.northwestern.edu/people/bazant/PDFs/Papers/598.pdfH.T. Nguyen, S....

Journal of the Mechanics and Physics of Solids 127 (2019) 111–124

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Journal of the Mechanics and Physics of Solids

journal homepage: www.elsevier.com/locate/jmps

Sorption isotherm restricted by multilayer hindered

adsorption and its relation to nanopore size distribution

Hoang Thai Nguyen a , Saeed Rahimi-Aghdam b , Zden ̌ek P. Bažant c , ∗

a Theoretical and Applied Mechanics, Northwestern University, United States b Civil & Environmental Engineering, Northwestern University, United States c Civil and Mechanical Engineering and Materials Science, Northwestern University, 2145 Sheridan Road, CEE/A135, Evanston, IL 60208,

United States

a r t i c l e i n f o

Article history:

Received 7 January 2019

Revised 28 February 2019

Accepted 2 March 2019

Available online 5 March 2019

Keywords:

Porous solids

BET theory

GAB model

Free adsorption

Hindered adsorption

Adsorption surface restriction

Evaporation and condensation

Statistical analysis

Polylogarithm

Jonquière functions

Interlayer water

Poromechanics

a b s t r a c t

Hindered adsorbed layers completely filling the nanopores cause significant deviations

from the classical BET isotherms for multimolecular adsorption of vapor in porous solids.

Since the point of transition from free to hindered adsorption moves into wider nanopores

as vapor pressure increases, the surface area exposed to vapor is decreased by an area re-

duction factor that decreases with increasing adsorbed volume, and thus also with increas-

ing vapor pressure (or humidity). The area reduction factor does not affect the rates of the

local process of direct adsorption or condensation of individual vapor or gas molecules,

but it imposes a lateral constraint on the total area and volume of the free portion of the

adsorption layer that is in direct contact with vapor. A reasonable assumption for the de-

pendence of the area reduction factor on the number of molecular layers is a selfsimilar

function, i.e., a power law. This leads to a sorption isotherm expressed in terms of poly-

logarithms (aka Jonquière functions). The power-law exponent is a property that serves as

an additional data fitting parameter, which is related to the pore size distribution. Com-

pared to BET isotherm with the same initial slope, the proposed isotherm reduces the

growth of the BET isotherm at low and intermediate humidity and the deviation increases

with the exponent. The fitting of isotherm data is reduced to either a series of linear re-

gressions or the minimization of a quadratic expression with respect to one parameter

only. It is shown how to use the optimum fit to calculate the size (or width) distribution

of nanopores < 6 nm. Comparisons with several published isotherms and pore size data

measured on hardened cement pastes show that the present theory gives excellent fits.

Finally, the semi-empirical GAB adsorption model is considered, but its additional param-

eters are not adopted because they weaken the physical foundation and are not constants

as they need to be varied empirically with temperature and, for cements, with the degree

of hydration.

© 2019 Elsevier Ltd. All rights reserved.

1. Introduction and basic concepts

Adsorption of gases or vapors in multimolecular layers in a porous solid is fundamentally described by the classical BET

theory, formulated in 1938 by Brunauer, Emmett and Teller (rumored to stem from a back-of-the envelope calculation of

∗ Corresponding author. E-mail address: [email protected] (Z.P. Bažant).

https://doi.org/10.1016/j.jmps.2019.03.003

0022-5096/© 2019 Elsevier Ltd. All rights reserved.

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112 H.T. Nguyen, S. Rahimi-Aghdam and Z.P. Bažant / Journal of the Mechanics and Physics of Solids 127 (2019) 111–124

Edward Teller during a lunch) ( Brunauer, 1943; Brunauer et al., 1940; 1938 ). One important application of the BET has been

the water desorption and resorption in cement hydrates and concrete, which is here the main application in mind, although

a similar problem arises, e.g., in activated carbon fibers formed by crystallite graphite sheets ( Kaneko et al., 1992 ).

Various useful improvement of the BET theory, particularly its empirical extensions to the capillary range, have been

formulated ( Bažant and Jirásek, 2018; Brunauer et al., 1940; 1969; Freiesleben Hansen, 1985; Halsey, 1948; Hillerborg, 1985;

Künzel, 1995; Lykov and Steffes, 1958; Xi et al., 1994 ). But the effect of varying nanopore sizes on the area exposed to

vapor, important for water in hardened cement paste and concrete, seems to have eluded attention. To take it into account

is the goal of this study, whose basic idea was presented in 2018 in Bažant and Nguyen’s report ( Bazant and Nguyen, 2018 )

( arXiv 1812.11235 ).

The BET theory assumes that the surface area of the molecular adsorbed layers is unrestricted, which requires wide

enough pores. The number of adsorbed water layers increases with the vapor pressure, p , or relative humidity h = p/p sat .For high enough h , there can be up to 5 layers with significant adsorbed mass, which gives the approximate maximum

thickness of 1.335 nm (here p sat = p sat (T ) = saturation vapor pressure, a function of T ). Consequently, in nanopores less than2.67 nm wide (which constitute the major part of pore volume in cement hydrates), and for not too small h , the adsorption

layers on the opposite pore surfaces touch, fill the pores completely and are not in contact with vapor.

In materials with multiscale nanoporosity, such as concrete, the maximum thickness of nanopores filled by adsorbed

water molecules decreases with decreasing h ( Fig. 1 b and c). When h exceeds the value corresponding to complete filling,

the adsorption layer is hindered from developing its full thickness, and such a hindered adsorbed water layer (sometimes

less fittingly also called the “interlayer water”) develops a high transverse pressure, called the disjoining pressure ( Bazant

and Bažant, 2012; Bažant, 1970; 1972a; 1972b; Bažant and Bazant, 2012; Deryagin, 1940; Powers, 1966 ).

The mathematical derivation of the BET theory ( Brunauer, 1943; Brunauer et al., 1938 ) is valid only for free, rather than

hindered, adsorbed layers ( Bažant and Jirásek, 2018 ). This derivation is valid only when the surface, of area A ′ , of the mul-timolecular adsorption layer in contact with the vapor is equal to the solid’s adsorbent area A and is independent of the

number of layers. However, in materials such as cement hydrates, there are nanopores of highly variable width ( Fig. 1 b) as

well as nanopores of uniform but very different widths ( Fig. 1 c). In either case, the vapor exposed area, A ′ , in cement hy-drates and some other nanoporous materials must decrease significantly as h increases. In the limit case of sufficiently fast

wetting or drying, in which the filling of the thin nanopores cannot change significantly, as stipulated at the outset (case

1), the decrease of A ′ may be described by an area reduction factor, βn , which depends on the number, n , of multimolecularlayers and reduces the full area A of the bare (or dry) internal pore surface per unit volume of the porous material ( Fig. 1 a),

i.e.

A ′ = βn A (n = 0 , 1 , 2 , 3 , . . . , β0 = 1) (1)

Here βn is a decreasing sequence; see Fig. 1 (in cement paste, A ≈ 500 m/cm 3 , which implies the average nanopore widthto be about 0.7 nm ( Le et al., 2011 )).

As for the hindered adsorption layers, which fill the nanopores completely (and thus are not thicker than 2.67 nm, the

effective diameter of one water molecule), we may assume that the migration of adsorbate (water) molecules along these

layers ( Fig. 1 b and c) is so slow that it does not intervene appreciably with the rates of adsorption, evaporation and con-

densation on the surface that is in contact with vapor.

The decrease of the surface area of multimolecular adsorption layers with increasing number n of the layers may be

schematically represented as shown in Fig. 1 a. Each bold horizontal line in the shaded n th layer marks a site where a

water molecule is either condensing or evaporating(the arrangement of molecules is, of course, only a mean idealization

of a constantly varying random arrangement of molecules). The first complete layer occupies area A of the bare adsorbent

surface. For increasing n , we imagine the dashed lateral constraint to reduce the area of each molecular layer from A to

A ′ = βn A, where βn ( n = 1 , 2 , 3 , . . . ) is a monotonically decreasing sequence, such that β0 = 1 and βn > 0 for all n . A certain restriction on the surface area exposed to vapor was formulated in 1940 in Brunauer et al. (1940) . They mod-

ified the BET theory for the case of two-sided adsorption layers on opposite parallel walls of a planar nanopore of fixed

width. The interference of the opposite two-sided adsorption layers causes that the area exposed to vapor in the nanopore

decreases as the adsorbate volume increases, and vanishes in the limit of full nanopore. A decrease of the area exposed

to vapor also features in the present formulation, but that is where the similarity ends. Here, most of the increase of the

exposed area with the multilayer thickness has a different source, explained by Fig. 2b and c in Brunauer et al. (1940) . In

the 1940 model, this increase is achieved not by growth of the adsorbate volume fraction in filled nanopores of varying

width, but by a densifying random filling of nanopore of constant width (see the pore between two parallel plates in Fig.

2b, or Fig. 4 in Brunauer et al., 1940 ).

The main problem with Brunauer et al. (1940) is the disregard of the vastly increased resistance to the movement

of adsorbate molecules along the nanopore, which occurs by surface diffusion. Unlike here, the statistical analysis in

Brunauer et al. (1940) implies nanopores of uniform width, which is approximately true for some materials (e.g., charcoal or

crystalline dolomite rocks) but is far from true for the cement hydrates. As another difference, in Brunauer et al. (1940) it

is considered that when the molecular layers growing from opposite surfaces of a planar nanopore touch, their heat of liq-

uefaction in the last layer increases, while this increase does not come into play here. The calculated isotherm, given by

equations (E) and 16 in Brunauer et al. (1940) , is much more complicated than what is obtained here. The aim of this 1940

http://arxiv.org/abs/1510.07818v1

H.T. Nguyen, S. Rahimi-Aghdam and Z.P. Bažant / Journal of the Mechanics and Physics of Solids 127 (2019) 111–124 113

Fig. 1. (a) Molecule movements during condensation and evaporation in the n th layer (shaded) balancing each other; (b) Wedge nanopore containing the

equivalent of one or two molecular adsorption layers, showing change of surface area exposed to vapor (unlike (a), randomness of molecular positions not

shown here; (c) Varying of vapor exposed surface when the surface coverage of pores of uniform width changes from one to two molecules (we disregard

the fact that the vapor in (c) is not a normal vapor because its movement along the nanopore is obstructed by surface forces).

theory was to describe the isotherms of types IV and V, as defined in Fig. 1 of Brunauer et al. (1940) and Brunauer (1943) ,

while here we aim at the isotherms of types I, II and III (although a simple adjustment could also fit types IV and V). Since

the 1940 theory ( Brunauer et al., 1940 ) is based on statistics of adsorption in a planar nanopore of constant width, it cannot

be extended to the capillary range, while the present theory can.

2. Adsorption under lateral constraint of expanding hindered adsorption layers

Adsorption is a random process in which water molecules constantly enter the adsorption layer, stay there for a certain

time called the lingering time (of the order of nanosecond de Boer, 1953 ) and then exit into the vapor. In equilibrium, the

number of water molecules within the layer is at any time macroscopically exactly the same. Although the arrangement of


molecules at any moment varies and looks rugged as shown in Fig. 1 c, one can define a precise effective thickness r ef as the

volume occupied by all the water molecules per unit base area.

Let us now follow as closely as possible the original derivation of the BET theory ( Brunauer, 1943; Brunauer et al., 1938 ),

although some vital differences are necessary. With a focus on hydrated cement, let us consider that the adsorbate is water,

although it could be some other substances. Let A be the total base area of the adsorbent base, corresponding to the internal

pore surface in a porous materials, and let αn be the exposed coverage density in the n th adsorption layer, i.e., the areafraction of the adsorbed (or water) molecules that are exposed to vapor in the n th layer ( Fig. 1 a).

Consider now a small area d A of the adsorbent base. The statistical expectation of the rate of condensation or adsorption

density of the water vapor molecules on top of the (n − 1) th layer is q c d A = a n αn −1 p d A where p is the vapor pressureand a n is a constant (to be determined later). The statistical expectation of the rate of evaporation density of the adsorbed

water molecules from the n th layer (i.e., on top of the (n − 1) th layer) is q e d A = b n αn e −Q/RT d A where b n is another constant(to be determined later); T = absolute temperature, R = gas constant, and Q = heat of condensation. For the first layer,n = 1 , Q = Q a = heat of adsorption to the solid (which is the heat dissipated upon severance of van der Waals bonds atthe adsorbent surface as water molecules escape from the surface). For the second and higher layers, we follow the BET

theory by assuming Q = Q l = Q L , where Q L is the heat of liquefaction of the adsorbate. Nevertheless, to facilitate possiblelater modification, we keep separate notation for the heat of condensation, Q l , in the adsorbed layer.

Thermodynamic equilibrium requires that q c = q e , and so for n = 1 : a 1 α0 p = b 1 α1 e −Q a /RT (2) for n = 2 , 3 , 4 , . . . : a n αn −1 p = b n αn e −Q l /RT (3)

This means that, for each layer, the statistical expectation q c of the rate of adsorption or condensation density must be equal

to the statistical expectation q e of the rate of evaporation density.

Note that the area reduction factors, βn , must not appear here because these equations give densities, per unit area ofthe base layer, and are independent of the lateral spread of that layer defined by these factors. However, the area reductions

factors, which represent lateral constraints on the layer areas ( Fig. 2 c), must intervene in the global constraints, which are:

A = ∞ ∑

n =0 βn (αn A ) (4)

v = r 1 ∞ ∑

n =0 nβn (αn A ) (5)

Here r 1 = effective thickness of monomolecular adsorbed layer (representing the volume, v m , of that layer per unit areaof adsorbent surface), and v = total volume of adsorbate over area A . Eq. (4) means that the sum or the top areas of therandom columns of n water molecules, for all possible column heights n ( Fig. 2 c), must be equal to the adsorbent base

area A . Eq. (5) sums the volumes of all the columns of water molecules of various random heights n (the volume and

area per water molecule are omitted from these equations because they later cancel out). The ratio r e f = v /A defines theeffective thickness of the adsorption layer. Taking the dimensionless ratio v / Ar 1 and setting Ar 1 = v m = volume of a fullmonomolecular layer, we obtain the overall lateral constraint:

v Ar 1

= v v m

= ∑ ∞

n =0 nβn αn ∑ ∞ n =0 βn αn

(6)

which represents the key difference from the BET theory ( Brunauer et al., 1938 ) and from its extension to two-sided ad-

sorption ( Brunauer et al., 1940 ).

Assuming that the van der Waals forces of solid surface do not reach beyond the first molecular layer, and noting that

a molecule entering the layers beyond the first essentially undergo liquefaction, we may consider the ratios b n / αn to beconstant for n > 1, i.e.,

b 2 a 2

= b 3 a 3

= b 4 a 4

= · · · = g (7)

where g is a constant. Let us now define two variables:

y = a 1 b 1

p e Q a /RT (8)

h = p g

e Q l /RT (9)

With these notations,

α1 = yα0 (10)

αn = hαn −1 = h 2 αn −2 = h 3 αn −3 = h n −1 α1 = yh n −1 α0 = c T h n α0 (11)


Fig. 2. The adsorption of vapor onto solid wall (left) and corresponding shapes of lateral constraint (right): (a) Example of random arrangement of adsorbed

molecules considered in the BET theory; the vapor-exposed surface area is equal to the adsorbent base area regardless of the number n of layers, which

is equivalent to the lateral constraint shown in (d); (b) interference of two-sided adsorption in nanopores with opposite parallel planar walls, considered

in Brunauer et al. (1940) , which is equivalent to the lateral constraint shown in (e); (c) array of adsorbed molecules with the lateral constraint of exposed

surface due to hindered adsorption, and the corresponding shape of the lateral constraint shown in (e); figure (a) is adapted from Brunauer et al. (1938) ,

and figure (b) from Brunauer et al. (1940) .

in which c T = y h

= a 1 b 1

g e �Q/RT , �Q = Q a − Q l (12)

where always Q a > Q l . To determine the physical meaning of h , we need to introduce the boundary condition for the satu-

ration vapor pressure, p = p sat . We assume it to occur in the limit for v → ∞ . This corresponds to h = 1 and shows that themeaning of notation h is the relative humidity of the vapor, i.e.,

h = p p sat

(13)

We now assume that the boundary condition p = p sat that occurs for v → ∞ is the same as in the BET model. Note that, fortheoretical consistency, our use of the limit v → ∞ requires βn as a function of n to allow v tending to ∞ rather than beingbounded. All the functions βn to be considered here are of that kind.


Furthermore, according to Eq. (9) , (p sat /g) e Q l /RT = 1 or

g = p sat (T ) e Q l /RT (14) So we may conclude that

c T = c 0 e Q l /RT (15) where c 0 is an empirical calibration parameter. A reasonable range for c T is recommended in Thommes et al. (2015) ; which

is 0 ≤ c T ≤ 150. Substitution of Eqs. (10) and (11) into Eq. (6) now furnishes for the sorption isotherm the result:

v v m

= θ (h, T ) = c T ∑ ∞

n =1 βn n h n

1 + c T ∑ ∞

n =1 βn h n (16)

For no hindered adsorption, i.e. βn = 1 for all n , this formula reduces to the BET isotherm as a special case.

3. Isotherms obtained for various area reduction factors

Because of their self-similarity, power functions of n appear to be a suitable choice for βn . Besides, they allow evaluatingthe infinite sums analytically.

1. The simplest choice, satisfying the condition that β0 = 1, is

βn = 1 1 + n (17)

for which we can introduce U and X functions as:

θ (h, T ) = c T U(h ) 1 + c T X (h )

(18)

with the definitions

X (h ) = ∞ ∑

n =1 βn h

n = ∞ ∑

n =1

h n

1 + n = −h + ln (1 − h )

h (19)

and U(h ) = ∞ ∑

n =1 βn nh

n = ∞ ∑

n =1

nh n

1 + n = h + ln (1 − h ) − h ln (1 − h )

h (1 − h ) (20)

Eq. (16) then yields, for the hindered isotherm, the final expression ( Fig. 3 a):

θ (h, T ) = c T h + ln H − h ln H H[(1 − c T ) h − c T ln H]

, H = 1 − h (21)

Fig. 3 a compares the plot of this hindered isotherm with the BET isotherm. For the hindered isotherm, c T = 28 (whichis the typical value for normal hardened cement paste, according to Bažant and Jirásek, 2018 ). Note that, similar to BET,

lim h → 1 θ = ∞ , which is erroneous, because of ignoring capillarity, and that θ is too high in the capillary range, i.e.,roughly for h > 0.90. This is, of course, inevitable since the capillarity is governed by Kelvin–Laplace equation and should

not be modeled as adsorption (in contrast to the GAB theory discussed later).

For the 5 th molecular layer, the area reduction factor due to hindered adsorption is β5 = 1/16. It is by luck if thisagrees with experiments (or MD simulations), because Eq. (17) for βn has no fitting parameter to optimize the fit ofexperimental adsorption data.

2. For data fitting, it helps to have a free fitting parameter. A simple way to introduce a fitting parameter, s , into a self-

similar function is to set:

βn = 1 (1 + n ) s (22)

Varying s , one has a continuous transition to the BET theory, which is attained for s → 0. For arbitrary s , the infinite sumsin Eq. (16) can be expressed in terms of a special function, the polylogarithm, aka Jonquière’s function, which is denoted

by Li s ( h ) (and is defined only for | h | < 1). We have ( Fig. 3 a):

X (h ) = ∞ ∑

n =1

h n

(1 + n ) s = ∞ ∑

n =0

h n

(1 + n ) s − 1 = Li s (h )

h − 1 (23)

and U(h ) = ∞ ∑

n =1

nh n

(1 + n ) s = Li s −1 (h ) − Li s (h )

h (24)


Fig. 3. Adsorption Isotherms corresponding with Eqs. (25) , (32), (40) , and (44) ( c T = 28 ) of the present hindered adsorption theory compared with BET prediction.

Hence θ (h, T ) = c T Li s −1 (h ) − Li s (h ) h + c T [ Li s (h ) − h ]

(25)

A common feature of the foregoing functions (and some functions below) is that the reduced area vanishes for n → ∞ .Physically this is not objectionable since the adsorption and hindered adsorption are negligible for n > 5. But it may be

useful to introduce a second parameter γ which gives a finite reduced area for n → ∞ . 3. A simpler way to introduce a free fitting parameter, γ , is to set:

βn = γ + 1 − γ1 + n (26)

which reduces to the BET theory when γ = 1 . Evaluating the infinite sums in Eqs. (19) and (20) , we obtain:

X (h ) = γ h 1 − h − (1 − γ )

h + ln (1 − h ) h

(27)

U(h ) = γ h (1 − h ) 2 − (1 − γ )

h ln (1 − h ) − h − ln (1 − h ) h (1 − h ) (28)

4. For more flexibility, parameter γ may be introduced into Eq. (22) , which already has one free parameter, s :

βn = γ + 1 − γ(1 + n ) s , n = 0 , 1 , 2 , 3 , 4 , . . . (29)


X (h ) = ∞ ∑

n =1

[ γ + 1 − γ

(1 + n ) s ]

h n = γ h 1 − h + (1 − γ )

[Li s (h )

h − 1

](30)

and U(h ) = ∞ ∑

n =1

[ γ + 1 − γ

(1 + n ) s ]

nh n = γ h (1 − h ) 2 + (1 − γ )

Li s −1 (h ) − Li s (h ) h

(31)

The resulting sorption isotherm, shown in Fig. 3 b, is:

θ (h, T ) = c T γ h 2 + (1 − γ ) H 2 [ Li s −1 (h ) − Li s (h )]

hH 2 + c T H[ γ h 2 + (1 − γ ) H(Li s (h ) − h )] (32)

5. Another way to introduce a free parameter βn is to consider the first term, β1 ( ≤ 1) as a free parameter;

βn = β1 n

, n = 1 , 2 , 3 , 4 , . . . (33)

The summed series and the θ function then are ( Fig. 3 c):

X (h ) = ∞ ∑

n =1

β1 h n

n = −β1 ln (1 − h ) (34)

U(h ) = ∞ ∑

n =1

β1 nh n

n = β1 h

1 − h (35)

Hence θ (h, T ) = c T β1 h (1 − h ) ( 1 − c T β1 ln (1 − h ) )

(36)

6. A further alternative is to add another fitting parameters s together with β1 , which would be:

βn = β1 n s

, n = 1 , 2 , 3 , 4 , . . . (37)

For β1 → β0 = 1 and s → 0, one can have again a continuous transition to the BET theory ( Fig. 3 c).

X (h ) = ∞ ∑

n =1

β1 h n

n s = β1 Li s (h ) (38)

and U(h ) = ∞ ∑

n =1

β1 nh n

n s = β1 Li s −1 (h ) (39)

Leading to θ (h, T ) = c T β1 Li s −1 (h ) 1 + c T β1 Li s (h )

(40)

7. If the experimental (or MD) data are abundant enough to optimize more fitting parameters, β1 , β2 , and s , one mayconsider that

βn = β2 (n − 1) s , n = 2 , 3 , 4 , 5 , . . . (41)

For s = 1 :

X (h ) = β1 h + ∞ ∑

n =2

β2 h n

n − 1 = β1 h − β2 h ln (1 − h ) (42)

and U(h ) = β1 h + ∞ ∑

n =2

β2 nh n

n − 1 = β1 h + β2 h + h ln (1 − h ) − ln (1 − h )

1 − h (43)

Therefore, the sorption isotherm can be obtained as ( Fig. 3 d):

θ (h, T ) = c T β1 hH + β2 (h + h ln H − ln H) (1 − h )[1 + c T (β1 h − β2 h ln H)]

, H = 1 − h (44)

Function βn in the exponential form would also allow an easy solution but is not considered here since it is not selfsimilar.


Comment on long-term sorption with surface diffusion into nanopores

The adsorbed water in partially filled nanopores less than about 5 nm wide cannot interact with the vapor directly.

Rather, the interaction requires diffusion along the nanopore under the influence of surface forces, which can be a very slow

process. But, with a long enough delay, the diffusion would eventually come to a standstill. This would alter the sorption

isotherm. Such long-term isotherm should also be analytically tractable, but is beyond the scope of this study.

4. Parameter identification from measured sorption isotherm

Unlike the BET isotherm, none of foregoing isotherms is amenable to linear regression. Nevertheless, the isotherm param-

eters can be identified from sorption data almost instantly, by optimum fitting of the measured isotherm with a powerful

nonlinear optimization algorithms such a the Levenberg–Marquardt and using the BET isotherm as the initial estimate. The

BET isotherm is a special case for s = 0 (or γ = 1 ). The convergence of optimization is better if the equation for θ is put in the form of linear regression:

Y = BX + C (45)even though the coefficients B and C are known only if one parameter is fixed. To bring the general equation of Eq. (18) to

a quasi-linear regression form, we may set:

B = 1 v m

, C = 1 c T v m

, Y = U ( h, γ ) v

(46)

while X is already defined by Eq. (19) .

The different equations for θ ( h, T ) can be written in terms of various quasi-linear forms. For example, to bring Eq. (25) tothis quasi-linear form, we may set:

B = 1 c T v m

, C = − 1 v m

, (47)

X = h + c T [ Li s −1 ( h ) − h ] Li s −1 ( h ) − Li s ( h )

, Y = 1 v

(48)

The optimization algorithm should then be applied to minimize the square:

� = (Y − BX − C) 2 ⇒ min (49)Alternatively, we can iterate linear regressions. We choose a sequence of discrete values or γ or s , e.g., γ =

0 , 0 . 02 , 0 . 04 , 0 . 06 , . . . 1 . For each of them, and for each h for which data were taken, we calculate the data points ( X, Y )

and run linear regression. This yields B as the slope of the regression line and C as the intercept. For each regression case,

we then calculate the �value and search the case that yields the smallest �. Finally, from the optimum B and C , we calculate

the optimum v m and c T .

Another way is to begin with the BET isotherm. First we apply the BET regression to identify the values of v m , c T (and k T ,

if applicable). This gives Q a and Q l values, which can give a good fit only for the range of low enough h . Next, a decreasing

function βn is chosen and its relevant parameters are fitted to match the remaining part of the sorption isotherm.

5. Pore size distribution and its relation to exposed area reduction function

At this point, function βn can be identified by optimizing the fit of sorption isotherm. However, does it have anythingto do with the nano-scale pore structure, as proposed at the beginning? It does indeed. As illustrated in Fig. 1 , the cover-

age of pore surface by the adsorbed water is restricted by the pore structure, and thus the surface area on which further

water molecules can get adsorbed is getting reduced, as reflected in function βn . The corresponding maximum thickness ofadsorbed water layer that can be accommodated on each of two opposite pore walls is nr 1 .

Because of the randomness of pore structure, the overall thickness of adsorbed layers grows not in integer steps but

continuously. So we replace n with the continuous variable

ρ = r r 1

= n c r r c

, n c = r c r 1

(50)

where n c is the maximum effective number of monomolecular adsorbed layers exposed to vapor that can be held on a

solid surface of the adsorbent; approximately, n c = 5 and r c is the corresponding thickness; r 1 is the effective thicknessof a monomolecular layer, about 0.267 nm for water (in our calculations we summed up to ∞ because it was simpler andequivalent since the layers beyond the fifth have negligible densities ( < 10 −3 of liquid water). For water in cement paste, n cequal to 5 is reached approximately at humidity 79% according to βn in Eq. (22) .

Referring to Fig. 4 a, and considering adsorbent solid surface A = 1, we see that the volume of adsorbed layers of effectivethickness ≤ r is the sum of the darkly and lightly shaded volumes, and is equal to ∫ ρ

0 β( ̄ρ) r 1 d ρ where β( ̄ρ) = βn when ρ̄


Fig. 4. Prediction of pore size distribution from fitted sorption isotherm of results based on available data from Baroghel-Bouny (2007) on hardened

portland cement paste (PC I): (a) probability density function, (b) cumulative density function (the sub-figures show the same in semi-log scale).

is an integer. The lightly shaded area represents an unrestricted volume of vapor-exposed layers not thicker than r , and is

equal to n βn r 1 or, in a continuous form, to ρ β( ρ) r 1 . This volume must be subtracted from the total volume of the sum oflightly and darkly shaded volumes, to obtain the darkly shaded volume, ϕ( ρ), which represents the volume of all restrictedadsorbed layers up to thickness r and corresponds to pore width w = 2 r since a filled pore consists of two mutually touchinglayers adsorbed on opposite pore walls. To normalize the cumulative volume, ϕ( ρ) should be divided by the total volume ofwater when the entire pore is filled ( h → 1). This causes cancellation of r 1 , the effective thickness of monomolecular layer.So, the cumulative distribution of the half-width of nanopores and its density are:

ϕ(ρ) = 1 C

(∫ ρ0

β( ̄ρ) d ̄ρ − ρβ(ρ) )

, C = ∫ ∞

0

β( ̄ρ) d ̄ρ (51)

ϕ ′ (ρ) = −ρC

d β(ρ)

d ρ(52)

The cumulative distribution of the total pore width (or size) w and the distribution density then are:

ψ(w ) = ϕ (

wn c

2 r c

)= n c

2 Cr c

∫ w 0

β

(w̄ n c

2 r c

)d ̄w − wn c

2 Cr c β(

wn c

2 ρc

)(53)

ψ ′ (w ) = d ψ(w ) d w

= −w C

(n c

2 r c

)2 d βd ρ

(54)

where r c = 1.335 nm as mentioned earlier. The volume fraction of pores whose width is between w and w + d w is ψ ′ ( w )d w .In particular, if the experimental sorption isotherm is optimally fitted by Eq. (22) with parameter s , then the cumulative

distribution of total pore width (or size) w and its density are:

ψ(w ) = n c 2 Cr c

∫ w 0

1

( 1 + w̄ n c / 2 r c ) s d ̄w − wn c

2 Cr c

1

( 1 + wn c / 2 r c ) s (55)

= ( 1 + wn c / 2 r c ) s − swn c / 2 r c − 1

( 1 + wn c / 2 r c ) s (56)

ψ ′ (w ) = w (s − 1) (

n c

2 r c

)2 s (1 + wn c / 2 r c ) s +1 (57)

6. Comment on empirical GAB model with an extra parameter k T

It may be noted that, in the literature, there are disagreements regarding Q l . The main one is the GAB model of Guggen-

heim, Anderson and de Boer ( Anderson, 1946; de Boer et al., 1968; Guggenheim, 1966 ) (popular mainly in food science),

which was later formulated by Brunauer et al. (1969) as a modification of BET. Both GAB and BET consider Q l to be the

same for all the molecular layers except the first (indeed, the decay of the van der Waals forces with distance is so fast

that a significant effect beyond the first layer is virtually impossible). However, the GAB model assumes Q l to be appreciably

smaller than Q L (typically 20% smaller).

The significant difference of Q l from Q L in the GAB model is motivated by the desire to fit the entire sorption isotherms,

including the capillary range even if the capillarity is not modeled. This difference cannot be justified by the van der Waals

forces, because they decay too fast and have virtually no effect beyond the first molecular layer. Especially, these forces


cannot affect all the layers beyond the first in the same way. Furthermore, Q l being different from Q L would imply the bond

of two contacting adsorbate molecules in adjacent adsorbed layers to be significantly weaker than when the same molecules

are in contact in a liquid, but this has apparently never been justified physically and is doubtful.

At low h , there is relatively sharp boundary between adsorption multi-layer, and generally beyond the 5 th molecular

layer the adsorbed volume is negligible. But for h very close to saturation, as all the vapor is approaching liquefaction,

the vapor boundary becomes fuzzy and disappears. In deriving the isotherm, same as in the derivation of BET theory, we

imposed the boundary condition p = p sat of h = 1 for v → ∞ . In the semi-empirical GAB model, by contrast, the volume, v , at saturation is considered to be finite and the saturation

condition, p = p sat , is assumed to occur at some humidity h = k T , which represents an extra fitting parameter, necessarilyvarying with temperature. Furthermore, as argued in de Boer et al. (1968) , the heats of condensation of water into bulk

liquid and into an adsorption layer are considered to be different (though not by much, or else the ratios a n / b n and a L / b Lwould lead to a physical nonsense). By assuming volume v at p sat to be finite, one conveniently introduces an extra free

parameter to be determined empirically by fitting sorption data for high humidity at which capillarity matters. This, of

course, disregards the fact that the surface adsorption is not a correct model for the capillary range.

So, similar to de Boer et al. (1968) , we may consider the state h = k T at p = p sat to account for such a deviation in theheat of condensation, where k T is not a constant but a variable parameter depending on temperature T (and only on T ), in

some unspecified empirical way determined by data fitting (rather than activation energy). To derive the GAB model, the

BET derivation is repeated except that:

h is replaced by h ′ = k T p p sat

= k T h (58)

where k T = p sat g

e Q l /RT (59)

in which k T = 1 while in BET k T = 1. This leads to: c T = c ′ 0 e Q L /RT (60)

where c ′ 0

is another empirical calibration parameter. All of the foregoing solution is valid if h is replaced by k T h (although

this replacement ignores physics). For the special case, k T = 1 (i.e., for no difference between the heats of condensation andliquefaction), the present solution reduces to the BET isotherm.

Parameter k T should be used with caution. Data fitting may give a value k T 1, which is a warning of a false estimationof c T and v m . Brunauer et al. (1969) showed fitting results with k T ≥ 0.79, but the corresponding heat of liquefaction wasoff by > 500 J/mol compared with the heat of condensation in bulk liquid at ( p = p sat ), which is unreasonable. As will beseen in the next section, the GAB model can be deceptively good in fitting, but with parameter values far from physically

realistic.

By virtue of its greater number of free parameters, the GAB model is very flexible in the fitting of diverse adsorption

isotherms. But this comes at the expense of losing a close fit of the initial rise of the sorption isotherm, of having to

recalibrate k T for each different temperature, and of losing connection to pore size distribution.

The fact that the GAB model cannot be related to the pore size distribution, i.e., to Eq. (51) , is an inconvenient limitation.

Introducing function βn would not make sense. Thus, e.g., the effect of the changes of pore structure due to hydrationcannot be predicted with GAB. The problem is that the additional parameters decouple the GAB model from the adsorption

theory and make it semi-empirical.

A fundamental flaw is that the GAB model attempts to distort the adsorption theory so as to fit the high humidity portion

of the isotherm corresponding to large pores, although this portion should properly be modeled by capillarity, rather than

adsorption (this should be the objective of further research). It is for these reasons that we prefer not to make here the

same modifications as in the GAB model.

7. Comparisons with some previously observed isotherms and with GAB model

To illustrate the behavior and usefulness of the present theory, its isotherm predictions are compared to the BET and GAB

isotherms for materials with nanopores small enough to exhibit hindered adsorption. These materials can be the normal

Portland cement paste with a low water-cement ratio or cements of various special types with extra minerals or crystals to

be formed. In these adsorbents, the C-S-H platelets or needles grow and create nanopores in which full adsorption layers

exposed to vapor cannot develop.

By means of the procedure described here, parameter c T , and the monomolecular layer volume v m or mass u m (per gram

of dry specimen) were obtained by fitting the BET theory up to a value of h where the least-square error begins to increase.

After that, the remaining relevant parameters (depending on which function was used) were fitted to match the further

regime of the sorption isotherm data.

To make a comparison to the GAB model, the v m (or u m ) of the BET fit is kept the same (based on the argument of

Brunauer et al., 1969 ), while the parameters c T,GAB and k T (defined in the previous section) are re-fitted to get a broader

match of the isotherms. As will be shown later, to obtain a good fit with GAB, parameters c T,GAB and k T will have to deviate

from their reasonable values.


Fig. 5. Fitting of published experimental results on cement pastes using the BET, the GAB theory, and the present theory; Tests of: (a) Baroghel-

Bouny (2007) on hardened portland cement pastes (PC I) and (b) Wang et al. (2012) on phosphoaluminate cement paste (PAC) and (c) Powers and Brown-

yard (1946) on hardened portland cement paste (PC IV).

In one experiment, Baroghel-Bouny (2007) measured adsorption of hardened cement paste with w/c = 0 . 20 at the equiv-alent age of t e ≈ 30 days. Up to h = 0 . 4 , BET gave a good fit of their data points shown in Fig. 5 a, but overestimated theisotherm for higher h . Both the present the GAB models allow a better fit up to h = 0 . 9 . However, to obtain it with the samevalue u m = 2.317, the GAB model requires that k T = 0 . 798 and c T,GAB = 132.9, which is much greater than the realistic valueof c T = 29.2 ( Bažant and Jirásek (2018) recommended that c T ≤ 50 for normal concrete and Thommes et al. (2015) recom-mended a value ≤ 150 for the usual adsorbate-adsorbent pairs). The present theory, on the contrary, maintains the value ofc T and k T = 1 for all the data. And unlike GAB, the initial isotherm slope is exactly the same as in BET.

In the same series of experiments, Baroghel-Bouny (2007) reported the pore size distribution corresponding to the fore-

going sorption isotherm. The results plotted in Fig. 4 b and c show a comparison of the peak pore radius in the nano-scale

pores.

In another set of experiments, Wang et al. (2012) studied a compacted pore structure of phosphoaluminate cement (PAC)

paste with a different water-cement ratio, w/c = 0 . 32 . Under scanning electron microscope (SEM) image, the formation ofcotton-shaped gels was observed, resulting in relatively small pore sizes and a complex pore structure. As a result, Fig. 5 b

displayed a significant deviation from the BET isotherm at h > 0.25, and also from the GAB isotherm at h > 0.5. The param-

eters of GAB model ( c T,GAB = 218 . 6 and k T = 0 . 772 ) scale down the entire curve and do not allow for a transition to a porescale that is one or more orders-of-magnitude larger. Hence, a better match can be obtained by fitting these data with the

present isotherm using function βn given by Eq. (22) , with parameters β1 ≈ 1 and r ≈ 1. As can be seen, the fit is better forh up to 0.95.

Important early adsorption data were those of Powers and Brownyard (1946) , shown by data points in Fig. 5 c. Both the

present theory and GAB fit the data very well, except that GAB shows poorer asymptotic behaviors as h approaches 0 or 1

and leads to a significantly different c T , with k T much less than 1 (1 being the value corresponding to BET).

Generally, the existing adsorption data do not suffice to reveal major differences between the present theory and GAB.

It will be necessary to conduct adsorption tests at significantly different tem peratures and, for cements, also at significantly

different degrees of hydration. For example, according to the present theory as well as BET, Eq. (15) , the c T value depends

only on the heat of water condensation, Q l , in the adsorption layer (or the heat of liquefaction of water) and thus must not

depend on the degree of hydration, nor temperature.

The foregoing comparisons prove the versatility of the present theory, with a major improvement over BET and an ap-

preciable improvement over GAB. The present model also allows distinctly better isotherm predictions when h approaches

1, and does so without sacrificing a good fit near h = 0 .

Conclusions

1. Because the area of a multimolecular hindered adsorbed layer increases with the relative humidity of vapor, the area

of the free (or unhindered) adsorption layers exposed to pore vapor decreases with the thickness of the multimolecular

adsorbed layer, and thus also with the increasing humidity of vapor in the pores.

2. The basic idea in extending the BET isotherm to nano-porous solids with hindered adsorption is to treat the transi-

tions from free to hindered adsorption as lateral constraints imposing an area reduction factor that decreases from one

molecular adsorption layer to the next.

3. The key point in the derivation of the isotherms is that the area reduction factors apply only to the overall volume and

area of the exposed free adsorbed layers but not to the local rates (or statistical expectations) of water evaporation,

liquefaction and adsorption on the solid adsorbent surface.


1

1

4. Like in BET, the isotherm is obtained as the ratio of the sums of two infinite series.

5. Considering the area reduction factor to be inversely proportional to the layer number leads to a simple analytical for-

mula for the sorption isotherm. A general inverse power-law dependence of this factor with exponent s on the layer

number is also analytically tractable, and yields to an isotherm expressed in terms of the polylogarithm (aka Jonquière)

functions. For the same initial slope, the resulting isotherms deviate from the BET isotherm downward. The deviation

grows with increasing exponent s .

6. The inverse power law exponent, s , is an additional empirical parameter providing flexibility in test data fitting. An

additive constant or exclusion of the fist layer from the power law can provide additional fitting parameters.

7. If the area reduction function is calibrated by optimum fitting of sorption test data, one can calculate the size (or width)

distribution of the nanopores less than about 6 nm wide. Vice versa, from the nanopore size distribution, one can predict

the isotherm for not too high humidities.

8. Comparisons with some published isotherms observed experimentally on various cement pastes indicate that the present

theory leads to significantly closer fits than the BET theory. It also fits these isotherms slightly better than the popular

GAB model and matches the initital slope of the BET isotherm.

9. Isotherm shapes significantly improved over the BET are also achieved with the semi-empirical GAB adsorption model,

which gains higher flexibility in data fitting by loosening the physical basis of BET—particularly by (1) arbitrary decou-

pling the heat of condensation in adsorbed layer from the heat of liquefaction, (2) redefining the state of saturation, (3)

replacing model constants with variable parameters of arbitrary temperature dependence, and (4) precluding any math-

ematical relation to nanopore size distribution. The empirical adjustments in GAB lend it more flexibility in the fitting

of isotherm data, though only for one temperature, and also improve extrapolation to high humidities even though the

capillarity is not modeled. However, the present model works better, despite loosening no physics.

0. To reveal the significant differences between GAB and the present theory, the adsorption tests would have to be con-

ducted at significantly different temperatures and, for cements, also at significantly different degrees of hydration.

1. The present analysis might need adjustment for extremely slow long-term drying or wetting which would allow signifi-

cant mass exchange by surface diffusion along almost full nanopores.

Conflict of interest

The authors declare no conflict of interest.

Acknowledgments

Partial financial support from the Department of Energy through Los Alamos National Laboratory grant number 47076

to Northwestern University is gratefully acknowledged. Preliminary research relevant for concrete was supported by the U.S.

Department of Transportation through Grant 20778 from the Infrastructure Technology Institute of Northwestern University,

and from the NSF under grant CMMI-1129449 .

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Sorption isotherm restricted by multilayer hindered adsorption and its relation to nanopore size distribution1 Introduction and basic concepts2 Adsorption under lateral constraint of expanding hindered adsorption layers3 Isotherms obtained for various area reduction factorsComment on long-term sorption with surface diffusion into nanopores

4 Parameter identification from measured sorption isotherm5 Pore size distribution and its relation to exposed area reduction function6 Comment on empirical GAB model with an extra parameter kT7 Comparisons with some previously observed isotherms and with GAB modelConclusionsConflict of interestAcknowledgmentsReferences

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