Comprehensive Approach to Calculate Oxygen Diffusivity of Cementitious Materials Considering Carbonation
International Journal of Concrete Structures and Materials
Comprehensive Approach to Calculate Oxygen Diffusivity of Cementitious Materials Considering Carbonation
In-Seok Yoon 0
0 Department of Construction Info. Engineering, Induk University , Choansanro 12, Nowon-Gu, Seoul 01878 , Republic of Korea
Oxygen diffusion would not be important for healthy concrete. Once concrete is carbonated, however, oxygen diffusion is very crucial to lead reinforcement corrosion. Therefore, oxygen is always a potential threat of reinforcement corrosion in concrete and oxygen diffusivity is a material parameter for service life determination and durability designing of the concrete. One of the simple approaches is to express the oxygen diffusivity of concrete by a multi-factor function, however, the influences of various factors on the oxygen diffusivity are still ambiguous. The purpose of this study is to establish a simple approach to calculate the oxygen diffusivity of (non)carbonated cementitious materials, which should be defined based on engineering and scientific knowledge of cement and concrete materials. A lot of parameters affecting the oxygen diffusivity, such as the diffusivity in air, tortuosity, micro-structural properties of hardened cement paste, volumetric fraction of aggregate, are taken into consideration for cementitious materials. For carbonated cementitious materials, reduced porosity due to carbonation is considered for the oxygen diffusivity.
service life; oxygen diffusivity; carbonation; porosity; micro-structure
The durability of concrete is closely related to
microstructural properties of the cover concrete because the
harmful substances penetrate through the concrete cover
. The main mechanism of deterioration to
concrete structures is reinforcement corrosion due to harmful
substances such as oxygen, water, oxygen ions, and sulfate
ions. Many researches have been focused on the chloride
diffusivity of cementitious materials
it is very rare to examine the oxygen diffusivity of
In general, gas transport through porous media mainly
occurs by molecular diffusion and/or advection through the
(Rowe 1987; Collin and Rasmuson 1990; Yanful
. For concrete, oxygen gas could not be a threatening
of reinforcement corrosion because pH value of pore
solutiuon is around 13.5. There is a lot more calcium hydroxide
in the concrete pores than can be dissolved in pore water.
This can help maintain the pH at its usual level of around 12
or 13 as the carbonation reaction occurs. However,
eventually all the locally available calcium hydroxide reacts,
precipitating the calcium carbonated and allowing the pH to fall
to a level where passive layer breaks down at the surface of
reinforcement in concrete. Once concrete is carbonated,
however, reinforcement can not be protected by passive
layer and reinforcement corrosion should be possible. This
means that oxygen gas can be regarded as potential
threatening to the concrete durability.
In addition, it is known that the oxygen diffusivity varies
with time due to ongoing hydration of cement that might
influence significantly the prediction of long term durability
performance of concrete
(Patel et al. 2016)
oxygen diffusivity is considered as an important parameter for
the determination of the concrete durability, its magnitude
have been considered as a temporary constant or determined
from regression analysis obtained from experimental results.
However, the results are only trustful for the experiment
duration and not reasonable for forcasting lont term
performance. It is not easy to predict the diffusivity of concrete
realistically because the diffusivity of concrete is influenced
by many factors such as the interfacial zone property
between aggregate particles and bulk cement paste as well as
the micro-structural properties of the cement paste itself.
Meanwhile, a lot of models for the diffusivity of concrete
have been developed with various approaches: from
empirical solution based on the experimental results to
comprehensive multi-scale model with high resolution digital
(Garboczi et al. 1998: Luciano et al. 1999)
. One of
the practical and realistic solutions is to express as a simple
formulation with multi-factors regarded as an individual
influencing effect. However, the effects of the influencing
factors on the oxygen diffusivity of concrete were
ambiguous in previous studies
(Xi et al. 1999: Saetta et al. 1993)
Furthermore, the majority of these researches didn’t deal
with this issue in combination with carbonation of concrete,
although carbonation affects the oxygen diffusivity of
concrete significantly. Since most of the in situ concrete
structures are carbonated, it is necessary to deal with the gaseous
diffusivity of carbonated cementitious materials. The oxygen
diffusivity should be based on the pore structure properties
of cementitious materials.
In this paper a simple model depicting the time evolution
of the oxygen diffusivity of (non)carbonated concrete is
presented, based on the previous research
purpose of this study is to develop a mathematical model for
the estimation of diffusivity of concrete taking into
consideration various factors and carbonation process. The model
is realstic but, yet simple enough for practical application in
durability design. This formulation is believed to make it
possible to take into consideration durability design of
concrete structures in connection with the time evolution of
2. Material Modeling of Oxygen Diffusivity for Cementitious Materials
2.1 Oxygen Diffusivity
Xi et al. (1999)
Saetta et al. (1993)
and reasonable way to compute the chloride diffusivity of
concrete. However, the solution didn’t clearly depict the
effect of some important factors such as the temperature and
its effect on the viscosity. These have been only depicted
using the temperature equation of Arrhenius type in most of
In this study, the previous study of author
was used to depict new analytical approach with a
multifactor function as illustrated in Fig. 1. A functional
multifactors were composed of chloride diffusivity in bulk fluid,
F(Do(S,T)), pore structural properties, F(Stmicro), tortuosity,
F(s2T), hindrance effect, F(H). In what follows a description
of these factors is given.
DClðcpÞ ¼ FðDoðS; T ÞÞ FðStmicroÞ F s2T
2.1.1 Oxygen Diffusivity in Air
The flow of oxygen can be ignored for full saturated
concrete, however, oxygen gas flows for dried or partial
saturated concrete. The flow media of oxygen can be the
bulk pore as the gases can flow through air space.
Diffusion theory for gases is based on the kinetic
molecular theory of gases
(Bird at al. 1960)
. The diffusivity D is a
function of both temperature and pressure. Diffusion
increases with increasing temperature (as molecules move
more rapidly), and decreases with increasing pressure (which
packs more molecules in a given volume, making it harder
for them to move). These temperature and pressure effects
are illustrated by Eq. (2), which applies to the diffusivity
between any two components of a mixture (called a binary
DABTjpj ¼ DABTipi pj
where DAB : diffusivity for the binary pair at A, B,
XD/T : collision integral for molecular diffusion, which is a
function of kT/EAB,
k: Boltzman constant = 1.38 9 1016 ergs/ok,
EAB : energy of molecular interaction(ergs),
i,j : reference and modeled conditions, respectively.
For binary pairs of oxygen with nitrogen, carbon dioxide,
and water, and in the temperature range from 0 C to 80 C,
kT/EAB ranges from about 1.3 to 3.5. Using tabulated values
for kT/EAB for each of the gases in the mixture, kT/EAB was
calculated for each binary pair according the square root
The collision integral X can be approximated from
tables relating it to kT/EAB. For the kT/EAB values of interest
X ranges from about 1.3 to 0.9 (decreasing as kT/EAB
increases). For this series of calculations a 5th order
polynomial was fit to the kT/EAB—X. From the basis of diffusion
in air, oxygen diffusivity should be equivalent to 0.176 cm2/
s in the air pore of cementitious materials.
2.1.2 Pore Structural Properties in Cement Paste
Porosity and pore size distribution function is necessary to
consider the effect of micro-structural properties of
cementitious materials on oxygen diffusion.
Maekawa et al.(1999
suggested porosity distribution function as;
dVp ¼ Br expð BrÞd ln r
Vp : fractional pore volume of the distribution up to pore
B : sole porosity distribution parameter, which is a peak
point of porosity distribution of the cement paste on a
From Eq. (4), total porosity of cement paste, u, can be
Br expð BrÞ d ln r
If ln r substitutes for x, r is replaced with exp(x). Then,
Eq. (5) can be rewritten as;
Vp ¼ B
expðxÞ expf B expðxÞg dx
If the total porosity Vp is known, sole porosity distribution
parameter B can be calculated from Eq. (6). The total
porosity Vp can be calculated by HYMOSTRUC
, a numerical simulation program for the
hydration of cementitious materials. The result is shown in
2.1.3 Pore Blocking due to Moisture
The gaseous diffusivity depends largely on the degree of
saturation of the material. According to the research of Houst
and Wittmann, diffusivity had function of the relative
humidity in which the porous samples of aerated concrete
had been equilibrated. Since gaseous substance can not
penetrate through pore filled with moisture, saturation is
very important to estimate gaseous diffusivity.
The effective pore volume V eff of the gaseous inflow is the
residual pore fraction excluding the degree of saturation (Sr)
in the whole pore and can be expressed as follows
Vpeff ¼ Vp Vg ¼ Vpð1
In this study, the pore system with saturation condition is
shown in Fig. 3. The residual amount of the compounding
water was calculated and reflected from the pore structure
system of the cementitious material. The pore structure
system of the cementitious material is composed of the vapor
and the moisture volume under the blocking condition from
texternal environment. If concrete is exposed to blocking
from external environment so that moisture evaporation is
prevented and the relative humidity is isolated, however, the
amount of pore solution is equivalent to the amount of
capillary water in pore system.
Since mixing water is ongoing consumped by hydration
reaction in pore system, the amount of capillary water
decreases. Thus, the volume of gas in the void per unit
volume of the cement hardened body, Vg, is the amount of
volume other than the capillary water content in the void
existing in the volume Vcp of the cement paste as follows;
qwþqce w=c ðw=c
0:4 aÞ Vcp
Tortuosity is classically defined as a ratio l to le. Here le is
effective path length in the pore and l is the shortest path
(Mota et al. 1998)
. Unlike porosity, the tortuosity
factor can’t be measured directly. This tortuosity factor
depends on packing arrangement, channel shape, media
homogeneity and so on. It is quite difficult to define the
tortuosity as a specific value, however, this study focuses on
Fig. 3 Schematic representation of the border limits for the
pore system with water
(Van Breugel 1991)
the derivation of a simple and approximate expression for
tortuosity of flow path in cement paste.
In this study, toruosity factor was considered based on the
previous study of author
. In principal, the
morphology of cement core is similar to the shape of circle,
however, the shape is transformed into a rectangular type in
order to simplify the fluid flow
(Yu and Li 2004)
Considering the shape function, averaged tortuosity can be thus
pffi1ffiffiffi1ffiVffiffipffi 1 þ 14 þ 12 pffi1ffiffiffiffiffiffiffiffiVffiffiffipffi þ 1
Since the particles in actual cement paste are randomly
distributed, some particles may overlap to each other and
hence it is difficult to express the tortuosity. However, the
presented approach is expected to depict simplified
tortuosity. In the study, oxygen diffusivity is assumed to be s2T.
2.1.5 Effect of Hindered Diffusion
The forth term of Eq. (1), (d) F(H), means hindrance effect
due to narrow pore diameter. As the molecular diameter of
the solute approaches the diameter of the pore, the diffusive
transport of the solute through the solvent is hindered by the
presence of the pore and the pore wall. This is known as a
‘hindered diffusion’. The function of the hindered diffusion
can be expressed as
(Welty et al. 2001)
Two correction factors, f 0ðuÞ and f 00ðuÞ, are theoretically
bounded by 0 and 1. Furthermore, both correction factors are
function of the reduced pore diameter u.
FðH Þ ¼ f 0ðuÞf 00ðuÞ
u ¼ dpore
2.1.6 Effect of Aggregate
Concrete consists of cement paste and aggregate and is a
random composite material in terms of media for fluid
transportation. Based on the micro-structural characteristics
of cement paste, an interactive behavior in compliance with
the presence of aggregates should be taken into accounted in
the calculation of composite diffusivity. Two possible
considerations should be discussed here. First, the aggregate can
reduce the whole diffusivity of concrete. The diffusivity of
aggregate is generally much lower than that of hardened
cement paste. This means that high volume fraction of
aggregates can lead to a reduction of the whole diffusivity in
the unit volume of concrete. However, it should be noticed
that high volume of aggregate has the adverse effect which
can induce increasing the whole diffusivity of concrete due
to ITZ (interfacial transition zone) effect. As more
aggregates are added, the additional, highly porous ITZ regions
force the bulk matrix to be denser in order to conserve the
overall w/c ratio. Thus, the existence of aggregate has
advantage and disadvantage at the same time in terms of the
oxygen diffusivity of concrete.
In this study, EMT (effective medium theory) is used to
convert the diffusivity of cement paste into that of concrete.
EMT has been developed for the estimation of the overall
conductivity of a multiphase material originally, however,
this can be applied to variety of situations in disordered or
random networks. Recently
Tatlier et al.
have extended the EMT to account for mass transfer
in the presence of possible concentration gradients between
the distinct phases of a composite material. Based on the
EMT model, the effective diffusivity can be expressed as:
1 þ V ð1
d2 þ 21 k1
D1, D2: the diffusivities in the two distinct components in
V1, V2 : the volume fractions of the different components
in the multiphase material and may be replaced by 1-V and
V, respectively (V1 = 1-V, V2 = V),
C1, C2 : the concentrations of the species in the different
k = C1/C2, a measure of the distribution of the diffusing
species between the components in the composite material.
When k is equal to 1, a uniform distribution of the diffusion
species exists in the material.
1 ¼ D1=D2;
d ¼ ð3V
1 þ k1ð2
A significant assumption of EMT is that the neighborhood
of a certain region in a mixture can be treated as a uniform
medium having a conductivity or diffusivity. That is, there
should be exists no correlation between the positions of the
different types of regions
(Tatlier et al. 2004)
. The oxygen
ds : kinetic diameter (& 346 pm for oxygen gas).
dpore : diameter of pore,
The first correlation factor, partition coefficient, is based
on simple geometrical arguments, as;
f 0ðuÞ ¼
ðtd þ dsÞÞ
in which, td means a twice thickness of adsorbed layer. For
the second correlation factor, Renkin equation, which is
reasonable for 0 B u B 0.6, is used.
f 00ðuÞ ¼ 1
2:104u þ 2:09u3
These expressions for the hindrance effects would be
useful to depict delayed diffusion rate which is occurred in
consequence of narrow pore diameter and collision of
oxygen gas with each others.
diffusivity of aggregate is assumed to be 1 9 10-11 cm2/s,
based on the research dealt with quartz
(Elphick et al. 1986)
2.2 Decreased Porosity of Concrete Due to Carbonation
Carbonation of concrete leads to a change of porosity and
this can have significant impact on oxygen diffusivity. In this
study, changed porosity of concrete due to carbonation is
calculated for limited mix proportion of OPC concrete.
had suggested the development of major
constituents of cement paste in order to estimate the
performance of concrete versus time;
2C3S þ 6HrH!;C3SC3S2H3 þ 3CH
2C2S þ 4HrH!;C2SC2S2H3 þ CH
C4AF þ 2CH þ 2CSH2 þ 18HrH;C4AF
2C3A þ CSH2 þ 10HrH!;C3AC4ASH12
C4AF þ 4CH þ 22HrH;C4AF
2C3A þ CH þ 12HrH!;C3AC4AH13
½CaðOH Þ2 ¼ 2 ½C3S oFC3S þ 2 ½C2S oFC2S
þ ½CSH2 o; t t
t* can be defined as;
t ¼ kH;C3Að1
In addition, initial concentration of compound i can be
Molar concentration of major ingredients (i = C3S, C2S,
C4AF, C3A) is calculated based on the reaction rate with
water. A reaction rate of compound i can be expressed as:
i ni 1
rH,i : constant of reaction rate of compound i,
[i] : the current concentration of compound i,
[i]o : the concentration of compound i at initial time.
The constant of reaction rates rH,i can be obtained from the
fraction Fi(t) of compounds i. This also can be expressed by
Eq. (22). The concentration of hydration product with
elapsed time can be expressed by Eqs. (23)–(25).
FiðtÞ ¼ 1
niÞ 1=ð1 niÞ
0 t t
¼ 2 ½C3S oFC3S þ 2 ½C2S oFC2S
½CaðOH Þ2 ¼ 2 ½C3S oFC3S þ 2 ½C2S oFC2S
2½C4AF oFC4AF ;
½i o ¼
w qc a qc
MWi 1 þ c qw þ c qagg
mi : the weight fraction of compound i in the clinker (kg/
mcl : the weight fraction of clinker (kg/m3),
qc: density of cement (kg/m3),
qw: density of water (kg/m3),
qagg: density of aggregate (kg/m3).
The values of the exponents ni and the coefficients kH,i are
listed in Table 1.
Carbonation usually proceeds in the volume of concrete in
the form of a front, separating a completely carbonated
region from the rest, in which carbonation has not started
yet. In this latter region, the value of decreased porosity due
to carbonation (DVc) is zero, whereas in the former, DVc is
approximately equal to
CaðOHÞ2 DVCH þ ½CSH DVCSH
in which, the concentrations of Ca(OH)2 and C-S–H mean
those at the completion of hydration and DVCH and DVCSH
equal 3.85 9 10-6 m3/mol, 15.39 9 10-6 m3/mol,
Accordingly, the porosity of carbonated concrete can be
Vc ¼ Vp
Vc: porosity of carbonated concrete,
Vp: porosity of non-carbonated concrete, which can be
calculated by HYMOSTRUC, as shown in Fig. 2.
As a result, the porosity of carbonated concrete is
calculated and presented as shown in Fig. 4. For carbonated
concrete, the changed porosity is inputted into the multi
factor function of Eq. (1) again.
for the flow of oxygen gas. The oxygen diffusivity decreased
significantly until 28 days because the micro-structural
desification of the concrete is not yet sufficiently developed.
The trend of the reduction lasted even at long term because
of on-going hydration of cement, however, the reduction rate
was not excessive. For concrete with w/c ratio 0.50, the
oxygen diffusivity of the concrete at 28 days decreased up to
4.5 times. The trend shows a noticeable decrease since then,
however, this continues slightly. Therefore, it is very harmful
for young concrete to be exposed to harmful substances
directly and this requires very careful attention from the
initial stage of construction. Meanwhile, the decrease in
oxygen diffusivity is clear for concrete with high w/c ratio
and this is identical to the experimental result of
et al. (2006)
. This would be because micro-structures of
concrete with high w/c ratio is more developed than that of
concrete with low w/c ratio as time elapsed.
Figure 6 represents the calculation of effective oxygen
diffusivity of concrete which means diffusivity with
effective pore due to moisture. The calculation is based on the
assumption that oxygen gas only flows through effevtive
pore, in which the pores filled with water is excluded in total
pore, to consider the effect of pore blocking due to water.
Like Fig. 5, the oxygen diffusivity of concrete with effective
pore tends to decrease with elapsed time. Two points should
be discussed. Firstly, it can be seen that the effective oxygen
diffusivity of the concrete was considerably reduced than
apparent diffusivity of the concrete. This means that the pore
blocking due to moisture has a very large effect on the
3. Results and Discussion
3.1 Oxygen Diffusivity of Noncarbonated
Figure 5 shows the result of calculation of apparent
oxygen diffusivity of concrete for different w/c ratio. The
calculation is based on assumption that total pore is available
oxygen diffusivity of concrete. Secondly, the value of the
oxygen diffusivity was extremely reduced after 100 days.
This implies that the effect of reducing oxygen diffusivity
due to moisture was obvious in long-term age.
Figure 7 presents the effect of pore blocking due to
moisture on the oxygen diffusivity of concrete. The
influence of moisture was not significant in the early age of
concrete, however, the effective diffusivity of concrete
decreased remarkably with elapsed time and the trend was
evident depending on w/c ratio of concrete. The effective
diffusivity has decreased to less than 1/10 of the apparent
diffusivity, as 1,000 days have passed. It is confirmed,
therefore, that the effect of moisture is very important for
calculating the gaseous diffusivity of cementitious materials
3.2 Oxygen Diffusivity of Carbonated Concrete
For carbonated concrete, (a) oxygen diffusivity in bulk
fluid is constant, however, (b) the function of pore structural
properties, (c) tortuosity and (d) hindrance effect are
changed. That is, the carbonation of concrete leads to reduce
porosity and to more complicated and narrow path. Figure 8
represents calculation result of oxygen diffusivity of
carbonated concrete with apparaent pore. The oxygen
diffusivity of concrete was also greatly reduced by carbonation,
compared with Fig. 7.
The oxygen diffusivity of carbonated concrete is
computed and compared with that of non-carbonated concrete in
Fig. 9. It is clear that the oxygen diffusivity of carbonated
concrete decreases. According to experimental result of
Saeki et al. (2006)
, chloride diffusivity of OPC carbonated
concrete decreased, while that of blended concrete with
flyash increased or decreased, depending the replacement
ratio. For blended concrete with granulated blast furnace
slag, the chloride diffusivity increased. The reason is
because concrete become more porous if concrete with
Fig. 7 Comparison of effective oxygen diffusivity of concrete
(De) and apparent oxygen diffusivity (Da) of concrete.
granulated blast slag is carbonated
(Ngala et al. 1997)
. It is
necessary to construct the theoretical formulation of oxygen
diffusivity for blended concrete.
Figure 10 illustrates effective oxygen diffusivity of
carbonated concrete. The combined effect of carbonation and
moisture influences oxygen diffusivity, compared with
Fig. 8. The effective oxygen diffusivity of carbonated
concrete was around 25 * 38% lower than that of
Figure 11 presents the effect of pore blocking due to
moisture on the oxygen diffusivity of carbonated concrete.
The influence of carbonation was significant from initial
time and that is similar to Fig. 7. The effect of moisture was
not significant in the early age of concrete and the effective
diffusivity of concrete decreased remarkably with elapsed
time. However, the trend was not severe, compared with
Fig. 10 Effective oxygen diffusivity of carbonated concrete.
Fig. 10. It is confirmed, therefore, that the moisture and
carbonation are significantly related to oxygen diffusivity
and it is very important to calculate the diffusivity of
3.3 Comparion with Previous Researches
For cementitous materials, a lot of studies have been
accomplished on chloride diffusivity, however, study on
oxygen diffusivity is rare. It is worth mentioning that the
CEB 1990 Model Code states that the effective diffusivity of
CO2 and oxygen in concrete is in the range of 0.5 9 10-4 to
6 9 10-4 cm2/s. Gaseous penetration is higly dependent on
the degree of saturation of cementitious materials rather than
luquid penetration. Gaseous penetration decreased rapidly at
the degree of saturation of more than 50%
Houst and Wittmann examined that CO2 diffusivity was
10 times higher than oxygen diffusivity in the same concrete.
Accoding to their study, the diffusivity was constant when
the relative humidity was less than 60%, however, the values
were reduced if the relative humidity exceeds 60%. The
results was coincided with experiment work accompliashed
Martin et al. (2006)
, who measured oxygen diffusivity
In the calculation of this study, the apparent oxygen
diffusivity of matured concrete at 28 days was in the range of
3 9 10-3 to 6.5 9 10-3 cm2/s, which is similar to the
results of Houston and Wittmann. On the other hand, the
effective oxygen diffusivity of the same cured concrete was
calculated to be in the range of 3 9 10-4 to 8 9 10-4 cm2/s,
which was within the limits of the CEB 1990 Model Code. It
has been shown, however, that a 9 year elapsed period has to
be elapsed for the reduction to the limit of 0.5 9 10-4 cm2/s,
which is the limit range of CEB 1990 Model Code.
Therefore, the calculation results of this study are approximately
consistent with the ranges presented in the previous
For the microstructural characteristics of paste, it should
be observed that specific properties of concrete like ITZ
(Interfacial Transition Zone) might also affect the diffusion
of oxygens through the path. The ITZ effect on oxygen
diffusivity is very ambiguous in carbonated concrete. It is
necessary to investigate the effect of chemical
decomposition and changed ITZ of concrete due to carbonation on the
oxygen diffusivity and to calculate changed porosity due to
The oxygen diffusivity is of particular quantitative
parameter for estimating durability performance
quantitatively and for taking into account the effect of oxygen on
calculation of service life of concrete. The purpose of this
study is to establish the oxygen diffusivity for carbonated as
well as noncarbonated concrete. A comprehensive approach
for formulation with the multi-factor function is constructed,
considering many affecting factors. In this study,
apparent/effective oxygen diffusivity of concrete and reduction of
oxygen diffusivity of carbonated concrete were calculated.
For noncarbonated concrete, the oxygen diffusivity
decreased significantly from initial casting time to 28 days
and continued to decrease slightly. For carbonated concrete,
the changed porosity due to carbonation was computed and
considered in the calculation of the oxygen diffusivity. The
porosity of OPC concrete due to carbonation decreased and
this led to the reduction of the oxygen diffusivity. The
effective oxygen diffusivity of carbonated concrete was
around 25 * 38% lower than that of non-carbonated
The proposed formulation is simple, yet realistic which
can be used for estimating quantitative durability
performance and reasonable durability design of concrete
This work was supported by the Korea Research Foundation
Grant funded by the Korean Government
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