Assessment of suitability of some chosen functions for describing of sorption isotherms in building materials

of the sorption isotherms for each material and temperature tested. Abstract This paper presents results of tests and studies conducted on six common building materials, used for con-structing and finishing of external walls. These included: ceramic brick, silicate brick, autoclaved aerated concrete, cement mortar, cement–lime mortar and cement mortar modified with polypropylene fibers. Each of these materials is distinguished by the other structure of porousness, affecting

Greek symbols α, α m Constants ϕ Air relative humidity (%) ρ c The density of the adsorbate (g/cm 3 ) υ The volume of vapour adsorbed Θ The level of the mono-layer coverage of the adsorbent surface

Introduction
Dampness of building materials is a factor recognized as critical. This can be attributed to the destructive impact of moisture to durability of building materials (e.g. Jenisch [1], Rahn and Bonk [2]) and such their technical parameters, as: strength (e.g. Unčik et al. [3]), frost-resistance (e.g. Garbalińska and Wygocka [4], Wygocka [5]), thermal conductivity (e.g. Siwińska [6], Siwińska and Garbalińska [7]) and others. In order to define the quantitative relationships between the balance moisture contained in the material and the moisture-related conditions existing in the ambient air, sorption isotherms are determined, the course of which depends not only on the material porous structure, but on the temperature as well. For many years, in many scientific centers, research works have been conducted on moisture sorption processes in building materials. In general, the sorption research consists of determining the adsorbate mass, based on precise measurements of the sample mass before and after the experiment, i.e. after the mass got stabilized. The way of determining the sorption characteristics with the standard method is described in the norm PN-EN ISO 12571:2002 [8]. Its drawbacks and application of a new method (APM Augenblicksprofilmethode, IPM Instantaneous Profile Method), which shortens the time, were presented by Plagge et al. [9][10][11], Scheffler et al. [12]. On the other hand, Markova et al. [13], Wadsö and Wadsö [14] presented a novelty measuring method, using the so called microcalorimeter, which permits thermodynamic description of the process of sorption. A dynamic vapour sorption (DVS) system was used to determine sorption isotherms in McGregor et al. [15,16] and Wu et al. [17].
Sorption of the mortars, which inter alia were the subject of this paper, was studied also by Anderberg and Wadsö [18], Espinosa and Franke [19], Janz [20], Johannesson [21]. Another team of Garbalińska et al. [22] focused on cement mortars with a different w/c ratio, tested at five relative humidity ranges and in the desorption process. A dynamic determination of sorption isotherm also of cement based materials was performed in Tada and Watanabe [23]. A frequent object of tests and studies was autoclaved aerated concrete, tested, for instance, in Jerman et al. [24]. Those studies of sorption were conducted on autoclaved aerated concrete with different bulk density. However in Koronthalyova [25], there were presented results of the research concerning moisture storage capacity and microstructure of ceramic brick and autoclaved aerated concrete.
The tests referred to materials coming from different manufacturers. On the other hand, Jiřičková and Černý [26] presented test results for moisture parameters of materials based on mineral wool. For the tested materials, they determined a sorption isotherm at 20 °C, at eight levels of sorption dampness. An assessment of moisture qualities of hemp as insulation material was performed in Valovirta and Vinha [27]. In their studies, Valovirta and Vinha focused on hemp in the loose form and three types of insulation in the form of mats. For those materials, sorption isotherms were drawn at five levels of sorption dampness. Moisture-related qualities of many popular building and insulating materials (37 materials) were also assessed by Kumaran [28] in his work. Among many experimentally determined parameters, Kumaran determined sorption isotherms, as well. In the case of sorption measurements, the samples were sized 40 × 40 × 20 mm (8 pieces for a measurement at ϕ = 100 %, T = 22 °C) and 40 × 40 × 6 mm (3 pieces for measurement at ϕ = 88.1; 71.5; 0.6 %, T = 23 °C). An assessment of the impact of adding some glass fibers to cement mortar to its moisture-wise parameters was presented in Podȇbradská et al. [29]. Team of Podȇbradská tested three mortars, different with their composition, content of the fibers and applied additives.
The tests, which were described in the literature, performed in various centers, are the sign of a wide range of problems discussed in relation to sorption of moisture in building materials. To high extent they concern the influence of different factors on the scale of the equilibrium dampness. They also give evidence of a diversification of research techniques applied.
The experimentally determined equilibrium sorption moisture values are used for determining sorption isotherms, which are then attempted to describe in mathematical categories. One has to note that the literature provides a lot of empirical equations of adsorption isotherms, developed for particular purposes and using various concepts and also some new mathematical techniques, as, for instance, neuron networks or genetic codes (Keller and Staudt [30]). Nevertheless, there is no universal sorption isotherm equation available, which could describe satisfactorily the course of this curve for all building materials, in the entire relative humidity. Thus, the analysis get down to verification of equations available in the ever-growing literature of the subject (Chen and Chen [31], Kumar [32], Kumar and Sivanesan [33,34], Keller and Staudt [30], Lowell et al. [35], Lagorosse et al. [36], Osanyintola and Simonson [37], Perry and Green [38], Rouquerol et al. [39], Yuh-Shan [40], Pavlik et al. [41], Furmaniak [42]), while the authors achieve various degrees of adjustment of their theoretical curves to the courses observed in the experiments.
In the extensive literature of physical chemistry, two classifications of sorption isotherms dominate, namely the Brunauer's classification (Perry and Green [38], Rouquerol et al. [39], Brunauer et al. [43]) and the one of IUPAC (Keller and Staudt [30]; Rouquerol et al. [39], Lowell et al. [35]). Materials of different internal structures present different kinetics of the adsorption process and different balance dependencies. Therefore it is difficult to find one ideal function, which could describe sorption isotherms for all materials in a satisfactory manner. A number of frequently used empiric description of adsorption isotherms, referring to non-organic, organic and biological materials of the adsorbent, can be found in Keller and Staudt [30], where references to sources providing a broad review of isotherms are given.
In this paper, suitability of 24 mathematical models, describing dependence of sorption dampness on relative humidity, are analyzed and assessed. In many cases in study, equations were insufficient to give a full explanation or description of processes taking place during the sorption process. Namely, we come across Henry's dependence at very low pressures. With this equation, at a very low pressure of the gas, adsorption value is proportional to concentration or pressure in the volumetric phase. On the other hand, in Langmuir's equation, within the scope of low pressers in the gaseous phase, adsorption is proportional to steam pressure and in this range of pressures Langmuir's equation gets transformed into Henry's equation [30]. As far as Langmuir's isotherm is concerned, the value of adsorption grows initially proportionally to pressure, and after some time this growth ceases gradually, and at sufficiently high pressures of the gas adsorption achieves a constant value. At that time, the adsorbent surface becomes saturated with a one-molecule layer of adsorbate. Langmuir's isotherm describes chemi-sorption cases quite well. In case of physical adsorption, it reflects the reality well only for small degrees of coverage, failing at higher pressure values. The course of Langmuir's isotherm features a linear section within the scope of low pressures and a proximity to a horizontal line for higher pressures. On its whole length the isotherm is convex. The isotherm is usable the most for describing adsorption in micro-porous materials [30]. Langmuir's isotherm equation is frequently an initial equation for a number of more detailed studies, as in many cases it is not sufficient for getting a full picture of what is really happening. On the other hand, Freundlich's equation does not always deliver a satisfactory set of values calculated with those experimental ones, especially for higher pressures. According to this equation, quantity of the adsorbed substance may increase with no limits, along with the growth of the pressure, while, in fact, the phenomenon of saturation occurs. The Freundlich's isotherm equation describes adsorption on heterogenic surfaces and experimental data for small concentrations pretty well. Meanwhile, erroneous results may be achieved with Dubinin and Raduszkiewicz's equation-particularly, where a microporous adsorbent is heterogenic [35].
Another theory, the BET theory, hardly ever describes the isotherms for the entire relative pressure range [35]. The BET model fails for adsorbents with a porous structure because of capillary condensation. The BET isotherm is correct within the p/p s range (relative partial steam pressure) up to 0.5. For pressures close to saturated steam pressure values the compliance is worse, where for p/ p s = 1 the dependence of the adsorbed gas amount on p/ p s departs from the one described with the BET equation. The BET equation envisages a too low adsorption under a low pressure and a too high adsorption under a high pressure. According to [30,35], the BET equation can be used for 0.05 < p/p s < 0.35, and sometimes 0.5. For p/p s > 0,5 the BET equation loses its suitability because of the occurrence of physical adsorption and capillary condensation. It is not recommended to use the BET equation for adsorbents with very narrow pores. The Harkins-Jury's equation ensures a larger range of coherence with experiments than the BET equation does. One of modifications of the BET equation is the Hüttig's adsorption isotherm. Another formula, the Hansen's equation, describes the course of sorption isotherms within the humidity range from 20 to 98 %. According to information available in the literature, for conditions with low pressure and high temperature values, physical adsorption of gas molecules decreases and the isotherms are a straight-line function as the Henry's or Freundlich's ones. For high pressures and low temperatures physical adsorption of gas molecules grows and we may expect isotherms similar to those of the Langmuir's. On the other hand, at high pressures and low temperatures physical adsorption leads to condensation in the pores and isotherms typical of the BET can be expected.
Many models of sorption isotherms, including the ones mentioned above, were tested on building materials.
A popular cellular concrete, was subject to studies presented in [44]. Its authors tested that material for seven temperature values (5,10,15,20,25,30,35 °C) on small (30 × 30 × 10 cm) and large (40 × 40 × 20 cm) samples. For getting a description of the sorption isotherms obtained from the nine tested models, the Oswin equation proved to be the best. The same material was studied by the authors of [45], as well, but this time for one temperature of 25 °C and eight humidity levels. Seven models were analyzed. Again, two sample sizes were tested (small ones: 3 × 3 × 1× cm and large ones: 4 × 4 × 2× cm). As a result of analyzing many equations, the authors chose their own new model as the best.
According to the data available in the literature, in case of many building materials, the above mentioned equations do not reflect the form of the isotherms with a sufficient precision for the entire scope of relative humidities. Hence, referring to the whole range of pressures, from p/ p s = 0 to p/p s = 1, we may say that for such a type of materials it is impossible to identify a simple isotherm, which would make it possible to describe all the phenomena of physical adsorption, taking place where gas is in touch with the surface of porous solid bodies. This is the effect of the energetic heterogeneity on the adsorption surfaces, an intense diversification of the shapes of the pores, their sizes, mutual interconnections and various qualities of adsorbed molecules. Nevertheless, the above mentioned isotherm equations find their practical application in experiments or industrial processes [30]. When describing experimental data, many scientists use those models, as e.g. [36,40], and obtain high coherence. Some reach for linearized forms of the equations to estimate the isotherm parameters, e.g. Kumar and Sivanesan [32][33][34].
Having that in mind, the authors found it recommendable to make an attempt to describe sorption isotherms for some selected popular building materials, using the approximating functions that are proposed in the literature. The authors focused, however, only on those, for which their suitability in engineering has been proven. Bearing in mind studies presented in the literature, the authors decided to conduct an extensive experiment on moisture sorption and to analyse a mathematical description of the obtained experimental data. In this paper, suitability of 24 mathematical models describing dependence of sorption dampness on relative humidity, were analyzed and assessed. As the tested materials were used: ceramic brick, silicate brick, autoclaved aerated concrete and three different cementbased mortars. The results and the obtained dependencies of the dampness on the relative humidity w(ϕ), determined for each of the materials concerned at three temperature levels, are presented below.

Tested materials
Six commonly used building materials, used for erection and finishing of external walls, were tested. The performed measurements concerned three types of brick and three mortars based on cement, with the following contents in 1 dm 3 : • cement mortar CM: cement 490, water 270, sand 1519 g, • modified mortar MM: cement 490, water 270, sand 1519 g, polypropylene fibers with length of 3 mm 0.9 g, • cement-lime mortar CLM: cement 194, lime 194, water 336, sand 1164 g.
In the brick group, the common, full ceramic brick CB, autoclaved aerated concrete AAC and silicate brick SB were tested. The data on the materials tested are collected in Table 1. Their specific gravity and structural parameters were determined by means of the Pore Master 60 mercury porosimeter. The parameters given in Table 1 were determined considering specific capacity of the measuring instrument available. Mercury porosimeter, used in the measurements, enables to assess data for the pores of diameters from ≈950 to ≈0.003 μm. The method applied is the most common technique used to measure structural parameters of porous building materials, also those which were the subject of the research performed. The textural parameters obtained by Mercury Intrusion Porosimetry will be supplemented in the future by additional data from measurements performed with the application of scanning electron microscopy. The SEM analysis it will make it possible to perform both an assessment of the chemical composition using energy dispersive spectroscopy analyzer, but also to identify better the microstructural details of the tested materials. It will enable a more complete analysis of the influence of microstructural parameters on moisture sorption processes.

Moisture sorption measurements
In order to propose a function dependence, which could provide a satisfactory description of the relationship between the material moisture content w and the air relative humidity ϕ, it was necessary to conduct tests aimed at determining the values equilibrium sorption dampness at different ambient conditions. The moisture sorption studies were performed according to norm PN-EN ISO 12571:2002 [8]. They were conducted at three temperatures and at six relative humidity levels: Maintenance of the temperature at the desired level was ensured by means of a chamber thermostat, while stability of the relative humidity was obtained by means of saturated water solutions of appropriate salts: LiCl, MgCl 2 , Mg(NO 3 ) 2 , NaCl, KCl, K 2 SO 4 . Use of the mentioned saturated salt solutions in the Raoult's law. The saturated conditions were ensured by the presence of crystals of salt at the bottom of the vessel, whereas only the molecules of the solvent, i.e. water reached the space above the solution and provided the appropriate air relative humidity. Table 2 shows the values of the air relative humidities above the saturated solutions in the state of equilibrium at three temperature levels pursuant to PN-EN ISO 12571:2002 [8] standard including the uncertainty interval for the each temperature. The samples of the materials tested were situated on a grille above the saturated solutions and there was no contact between them.
The selected materials were cut into 1 cm thick samples, insulated on the side surfaces and dried up to constant mass in a dried adjusted to the temperature of 105 °C. Next, the samples were put in tight containers with the particular solutions and the containers were moved into a climatic chamber with a stable temperature. In the thermostatic chamber, in each air-tight container, three samples of each material were stored.
The sorption measurements came down to recording the changes the mass of the samples. At the commencement of the research (t = 0 h), the samples were completely dried. The process of absorption of humidity in each climate (at ϕ > 0 %) manifested by the mass increase. The intervals among each particular weighing, initially being of 6, 8, 12 h, extended gradually to 7 days. The measurements were performed until the moisture equilibrium was achieved in all the samples kept in the given thermal/humidity conditions. The tests were continued for 6 months (temperatures 5 and 20 °C) and 5 months (35 °C) for the wall materials, and for 9 months (20 and 35 °C) and 10 months (5 °C) for the mortars.
The graphs, showing the kinetics of the sorption process at the temperature of 20 °C for three mortars (CM, MM, CLM) can be found in the article [7]. Below are shown the graphs (Figs. 1, 2, 3) for the remaining materials (CB, SB, AAC) obtained at the temperature of 20 °C for six levels of humidity ϕ. The entire set of the graphs for all the temperatures and materials is included in the study [6].
Most intensively, the sorption process took its course in the initial phase, i.e. during the first 7 days. Afterwards only some slight increments in the sorption dampness were observed, extended, however over many months. Obtained for the equilibrium state, the results were used for determining the sorption dampness w (%), expressed as a percentage of water content, referred to the dry sample mass: For each of the 18 climatic conditions, the stabilized sorption dampness w was determined as an arithmetic mean of the three samples. The test results were shown in Tables 3,  4, and 5.
Among all the materials in study, the highest sorption dampness was recorded for autoclaved aerated concrete at 5 °C and ϕ ≈ 98 %, amounting to 25.25 %. The lowest equilibrium dampness in each of the climates in question was observed in the ceramic brick-it featured a very low sorptivity, not exceeding 2 %.

Sorption isotherms: experimental results
Experimental data permitted to draw up sorption isotherms for the tested materials at the each of tested temperatures. Figure 4 gathers the sorption isotherms, drawn for each of the six materials at the three temperatures. The presented above groupings of obtained curves confirmed the influence of both the various structures of the tested materials, and the temperature on the nature of sorption isotherms. An assessment of the impact of temperature on the character of sorption isotherms was based on the graphs shown on Fig. 4. For the ceramic brick, no noticeable influence of temperature on the obtained isotherms was observed in the entire humidity range. For the other materials, the influence of the temperature on the course of the isotherms of sorption was manifested in a clearer way particularly in the scope of the higher humidities ϕ > 75 %. There, a specific tendency is clearly seen, as the curves determined for 5 °C remain above the other ones. At 35 °C the isotherms generally are distinguished with the lower values of the equilibrium humidities.
In the case of structural materials (CB, SB, AAC) it was observed that the isotherms do not match one another throughout the entire range of the tested humidity, at each of the temperature levels chosen for the tests. The isotherms indicate plainly different internal structures of those materials. The cellular concrete, i.e. the material of the highest porousness and the specific surface, demonstrated the highest values of the sorption humidity in particular climates.
As far as the mortar group is concerned, the situation is quite different. The isotherms of the cement and modified mortars (CM, MM) are very similar at the three temperature values. Generally, the polypropylene fibers that had been added to the modified mortar did not affect the course of the sorption isotherm, while addition of lime to the mortar (CLM) contributed to the fact that the cement-lime mortar isotherm locates itself below the curves determined for the other two mortars.
The achieved sorption isotherms run smoothly from the moment of formation of a single adsorption film, through appearance of other layers, until the beginning of capillary condensation, when the dampness grows dramatically. The observable differences concerning the sorption dampness value in the materials in question and the course of the sorption isotherms are attributed to different specific surface areas of the tested materials and their different porosity structures. Only when relative moisture grew above 90 %, at all the temperature values, i.e. 5, 20 and 35 °C, the isotherms of the ceramic brick disclosed a slight increase in the material dampness. For silicate brick and autoclaved aerated concrete, the obtained isotherms disclose three stages of adsorption clearly. Their first sections are convex. Their points of inflexion and increase of moisture above 80 % are clearly shown. For cement-based materials, the courses of isotherms are more moderate. Similarly to the other materials, sorption isotherms of mortars can be qualified as type II in the Brunauer's and IUPAC classifications.

Estimation of sorption isotherms
The main objective of this study was to test the possibility of describing the experimentally obtained isotherms, applying the sorption isotherm equations proposed in the literature. Several dozens of equations, drawn from the extensive literature of the subject, were used. The authors decided to perform such a big testing scheme in terms of the equations suitability because of the fact that the literature they referred to did not offer any universal model that could help describe experimental data in a satisfactory way in reference to different materials, within the entire relative humidity range and at different temperature levels. Functions used in the analysis are presented in Table 6 (Chen and Chen [31], Keller and Staudt [30], Siwińska [6]).
With statistical methods, compliance of the adsorption isotherm equations (Table 6), as proposed in the literature, with experimental data was determined. For that purpose the STATISTICA programme was used. To estimate the coefficients of the equations, the authors used one of the programme modules, i.e. the non-linear estimation. To assess the quality of the coherence between the equations to the experimental data, correlation coefficient R and the least squares estimator were used (Yuh-Shan [40]). Two estimation methods, the ones of Levenberg-Marquardt's and Gauss-Newton's were applied. The data collected in Table 7 manifest to what extent the models tested are suitable to description of the isotherms of sorption for each of the materials in a given temperature. Lack of specified value of the R parameter manifests lack of such possibility.
The values of the R parameters placed in Table 7 show the accuracy of matching provided by a given model.
In Figs. 5, 6, 7, 8, 9 and 10 the authors presented graphs for each of the materials at the three temperature levels, obtained with the application of the best experiment-matching equations, i.e. the ones, for which the values of R were the highest (shown bold in the Table 7). Analysis of the values of the R coefficients placed in Table 7 proves that the solutions were not found for all the 24 adopted models. In such instances, dashes are placed in the table above. Amongst all the analyzed equations, the highest compatibility between the mathematical models and the experimental data for the six materials in question was observed in the equations of Chen and Chen Jovanovič's-for multi-layer adsorption, and Hüttig's. For those models, the highest values of the R correlation coefficient were obtained. Further, some satisfactory results were recorded, using the Dubinin-Serpiński's and D'Arcy-Watt's models; not in all cases, however, solutions were found. For the ceramic brick, the equation of Chen and Chen describes the isotherms satisfactorily for 5 °C, 20 °C and 35 °C. At 20 °C, the D'Arcy-Watt's equation is better; however, it does not provide solution for the above mentioned two temperature levels. Nevertheless, the situation with the silicate brick seems to be quite different, as in this case there are several functions, which yield quite good descriptions of the experimental data for all the temperature levels. These equations include: the Chen and Chena, the Jovanovič's-for multi-layer adsorption, the Dubinin-Serpinski's and the Hüttig's. However, from among them, the highest accuracy is provided by the Chen and Chen models. It is non-disputably the best model also in case of the cellular concrete.
Amongst all the analysed equations, the highest compatibility between the mathematical models and the experimental data for the three mortars was observed also in the equations of Chen and Chen. With this model, the highest values of the R coefficient were recorded in the most of temperatures.

Conclusions
Obtained in non-stationary tests of sorption processes, the equilibrium sorption moisture values for the six materials, all collected at the three temperature and six humidity levels, were used to draw sorption isotherms. Then the isotherms were confronted with the isotherms under the Brunauer's and IUPAC classifications. The obtained isotherms represent by their nature type II isotherm, according to the classification mentioned before. This isotherm type relates to appearance of a multi-molecular adsorption film during physical adsorption. On those curves, it is possible to identify three sections respective to three stages of gas adsorption on a solid body. The first section is typical of isotherm type I, where, at low relative humidities, a monomolecular film of the adsorbed substance gets formed on the surface of the adsorbent. Further on, along with the increasing relative humidity, formation of a multi-molecular film takes place. Above the humidity of ca. 80 %, the process of capillary condensation in the adsorbent mezzopores begins. Obtained results of the research manifest the influence of the structural construction of materials to the quantity of the sorption humidity. Because a big specific surface, porousness and sizes of pores are one of the factors affecting the sorption ability of porous bodies.
The authors undertook to test whether exists a possibility to describe very different courses of the sorption isotherms by means of one universal formula. In order to describe the experimentally obtained sorption isotherms for the porous building materials in study, the functions listed in Table 6 were used. The analysis results, reflecting the suitability of the individual models applied, are presented in Table 7.
Having analysed all the data, it is declared that the Chen's equation reflects relation w = f(ϕ) the best. For this model, a high accordance between the experimental and the theoretical courses was achieved for all the tested materials,  Table 7, the authors may state that the Chen's equation approximates all experimental data in a satisfactory manner and can be used successfully to describe sorption isotherms for such building materials as ceramic brick, silicate brick, autoclaved aerated concrete or cement mortars of the different kinds.

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Hansen Chen and Chen Hill and de Boer       Under planning is applying of the microstructural parameters typical for every material to describe their sorption isotherms.