Relationship between air diffusivity and permeability coefficients of cementitious materials

In this study, the relationship between air diffusivity and permeability in cementitious materials was investigated. First, we derived an equation to correlate air diffusivity and permeability in a straight circular tube. Then, we reviewed existing studies that measured both air diffusivity and permeability and compared reported data and calculated values to verify the applicability of the derived equation to cementitious materials. Although a correction factor was not used, the two sets of data showed good agreement quantitatively. This indicates that the derived equation can be applied to cementitious materials including concrete, and measured air diffusivity can be converted to permeability of concrete and vice versa using the derived equation.


Introduction
The evaluation of concrete durability is becoming more important for rational design and maintenance.
Carbon dioxide and oxygen are important deterioration factors of concrete structures as they cause carbonation of concrete and corrosion of reinforcement [1]. Therefore, the appropriate evaluation of resistance against air penetration should be performed to estimate the durability of concrete structures. In general, air diffusivity test [2,3] or air permeability test [4][5][6] are conducted to evaluate air penetration in concrete. The driving force of the former is the concentration gradient, whereas that of the latter is the pressure gradient. The condition in a diffusivity test is closer to the real condition of oxygen and carbon dioxide penetration in concrete. However, the experimental setup in a diffusivity test is complicated because the pressure on the two flat surfaces of the sample plate should be kept the same and the concentration of air should to be monitored during the test. On the other hand, in air permeability test, the volume of air penetrating through concrete due to pressure gradient is determined by measuring the volume of air penetration or air pressure, and this test can be conducted using a relatively simple setup.
Lately, devices for in-situ investigation of air permeability have been developed [7][8][9][10]; however, it is unclear if the actual penetration of oxygen or carbon dioxide due to diffusion can be evaluated using the air permeability test. Correlation between air diffusivity and permeability coefficients has been reported [11][12][13][14], but a method to convert air permeability to diffusivity has not yet been established. Once the relationship between air diffusivity and permeability is established, the penetration of carbon dioxide and oxygen can be estimated from air permeability measured with a simple experimental setup or even nondestructive testing.
In this study, a straight circular tube was used to theoretically investigate the relationship between diffusion and air permeability coefficients. Then, studies that measured both the diffusion and permeability coefficients of concrete, mortar and paste were reviewed and a comparison of reported and calculated values was carried out to confirm if the obtained relationship is applicable to actual cementitious materials with complicated pore structure.
where λ is the mean free path (m) and L s is the space size (m). λ can be calculated by the following equation. 2 2 Pd where k B is the Boltzmann constant (= 1.3807 × 10 -23 N·m/K), T is the temperature (K), P is the pressure (Pa), and d is the molecular diameter (m). In general, a flow with K n < 0.01 is considered to be viscous flow, that with K n > 1 is considered to be molecular flow, and that with 0.01 ≤ K n ≤ 1 is considered to be transient flow [15,16]. When molecular flow is dominant, the diffusion coefficient is expressed as follows [17]: where C m is the conductance in molecular flow (m 3 /s), l is the distance between two points (m), A is the inner cross-sectional area of a circular tube (m 2 ), β is the coefficient of surface roughness, H is the tube perimeter (m), s is a constant (less than 1; 1s indicates the fraction of a specular reflected molecule),  is the root mean velocity of a gas molecule (m/s), and τ is the mean sojourn time of molecules absorbed on a tube surface. The second term in the denominator of Eq. 3 can be ignored because air consists of mostly nitrogen, which has a very short τ of 10 -12 s, and β and s in concrete have not been established quantitatively. The effect of this term was estimated by Sakai and Kishi [18] and was not large when D m > 10 -5 m 2 /s. In a straight circular tube, C m is expressed as follows: where r is the tube radius (m). The air permeability coefficient when molecular flow is dominant is expressed as follows [19]: where μ is the viscosity of air (= 2.0 × 10 -5 Pa·s). By combining Eqs. 3-5, the relationship between air diffusion and permeability coefficients when molecular flow is dominant is obtained as follows: On the other hand, in a large space where viscous flow is dominant, the diffusion coefficient is equal to that in balk. In this case, the diffusion coefficient is expressed as follows: where D 0 is the diffusion coefficient in balk (m 2 /s). The diffusion coefficient considering molecular flow, viscous flow and transient flow is expressed as follows: Combining Eqs. 6-8, D can be expressed as follows:

Data from existing studies
Related studies on air diffusivity and permeability were reviewed as listed air diffusivity and permeability were not the same in the studies reviewed; however, the reported values were adopted as they were because the original data were not available and hence, it is impossible to calculate the air diffusivity and permeability using the same approach. In the studies in [12,21], data of various moisture contents were presented. Fly ash and granulated blast furnace slag were used in [11] and [14], respectively. The samples in [22] were immersed in different aqueous solutions before drying, whereas the samples in [20,23] were conditioned at various humidity and temperature values.   [11] Mortar CH4 Air Air Flow rate cm 2 N / A [12] Concrete O2 N2 Air Flow rate cm 4 /(s·N) 200 [13] Concrete O2 N2 Air Pressure m 2 3 N2 O2 Concentration m 2 20-110 [14] Paste CO2 CO2 Air Pressure m 2 150-250 [20] Concrete O2 N2 O2 Pressure m/s 100 → Lower [21] Concrete O2 N2 Air Flow rate cm 4 /(s·N) 200 [22] Mortar O2 N2 O2 Pressure m/s 100 → 50 [23] Mortar is dominant when r = 62 nm. In logarithmic scale, the midpoint of these radii is 620 nm and this radius is the boundary between viscous flow and molecular flow. Therefore, when the measured air permeability is 50×10 -16 m 2 , the representative pore radius in terms of air penetration is 620 nm. Sakai, Nakamura [24] proposed a relationship between air permeability and the representative pore radius as follows:

Results and discussion
According to Eq. 10, when k = 50×10 -16 m 2 , r is 325 nm, which is close to 620 nm in logarithmic scale.
This result further validates Eq. 10.
The agreement in Fig. 3 validates the conversion of air permeability to diffusion coefficient using Eq. 9.
As introduced earlier, devices that can evaluate the air permeability of concrete in a non-destructive manner are presently available and we can now obtain the diffusion coefficient of concrete on site using such devices and Eq. 9. Furthermore, Fig. 3 indicates that we do not need to evaluate both air diffusivity and permeability because one of these can be obtained by conversion from the other one. The results obtained in this research will contribute to rational evaluation of the durability of concrete structures.  In this study, the relationship between air diffusivity and permeability was investigated using theoretical approach and literature survey. An equation that describes the relationship between air diffusivity and permeability in molecular flow, transition flow and viscous flow was derived. Although a straight circular tube was assumed in the derivation of the equation, the calculated values showed good agreement quantitatively with experimental data. This indicates that air diffusion and permeability are governed by the same factor, possibly the pore structure, and air diffusion can be converted to permeability coefficients and vice versa, using the equation derived in this paper. The studies reviewed in this paper already contain data for concrete, mortar and cement paste of various mix designs prepared under various conditions;

Conclusion
however, further tests on samples prepared at extreme conditions are required to determine the limitation of the equation derived in this paper.