Reaction kinetics of basaltic elements in cementitious matrices: theoretical considerations

Basalt fibers, the frequently mentioned alternative to those made of steel, possess very good mechanical properties and temperature resistance. The alkaline environment of cement matrix makes it vulnerable due to partial fiber decomposition by the effects of OH- ions. This paper aims at computational modelling of such reactions in order to approximate the course of degradation or to predict it lately. The isothermal reaction models are discussed to reveal their strong/weak points by means of fundamental reaction mechanisms analysis. The shape factor and diffusion-based deceleration of the reactions are mentioned as the most significant ones in that respect. The model accuracy is quantified based on fitting the modelling outputs to reference experimental data. The effect of discussion was found to be the most significant factor as the model fitting reached the lowest RMSE (0.0047). Further application of a diffusion model is therefore recommended. The geometrical models need to have reaction rate reduction explicitly incorporated in the reaction constant, otherwise inapplicable data is produced (RMSE = 0.0193).


Introduction
Representing a promising alternative to glass or steel fibers, those made of basalt have gained an increased attention as they are compatible with various matrices while possessing very good mechanical properties, high temperature resistance or chemical resistance.Such properties make these fibers possible to be applied in various environments including the aggressive ones.Unlike steel, however, the alkaline nature of cement matrix might go along with some negatives resulting in partial fibers decomposition by the effects of OH-ions that break the siloxane bonds while releasing silicate into the solution [1,2].Since these processes represent the essence of the fibers corrosion, the understanding and possibility of an accurate estimation of the dissolution rate is of a high importance as it can help to control a cementitious phases formation, to assess the durability or to predict changes of related material properties including the change rates.
The determination of the overall reaction/dissolution kinetics is a complex problem which can be effectively handled by a computational modelling approach that has proved to be indispensable in numbers of scientific disciplines.To achieve precise and reliable results, however, it is important to use an appropriate mathematical model being completed with suitable experimental methods for the corrosion processes monitoring to create a complex methodology.Since there are various models described in the literature, having different demands on input data or computing power [3,4] while producing outputs of various accuracy, this paper summarizes and evaluates the available models and techniques, pointing out their advantages/disadvantage in the light of potential application in basaltic elements reactions when exposed to alkaline environment.

Principles of basalt fibers corrosion in alkaline environment
As mentioned previously, the basalt dissolution is the main phenomenon standing behind the fibers corrosion processes.The basalt fibers, a product of fiberized basalt rock, have composition very similar to Ca-Mg-Al-Si glass, being typical with disordered metastable structures.The polymeric network is composed, among others, of oxygen and Si ions, where the oxygen has bridging/non-bridging attribute depending on the element that it is bonded to.The higher the non-bridging oxygen share in the polymeric network, the easier it can be broken by means of depolymerization [1].The rate of dissolution and the related kinetics depends on various factors.Temperature, basalt fibers composition and surface area, presence of aqueous elements (Ca, Mg, Fe, Al or Si in particular) or acidity/alkalinity of the environment belong among the most important ones.When pH reaches values higher than 9, which is typical for concrete or other cement-based materials, the dissolution of basalt take place due to silicate network hydrolysis [2].

Experimental techniques for monitoring of basalt fibers corrosion processes
The corrosion processes or the progress of chemical reactions can be quantified based on the mass loss observation.The basic data set is usually obtained by initiating the reactions of basalt (in form of fibers or powder) that is added to the NaOH solution.There are some specifics that must be respected, elimination of CO2 effects on the dissolution process in particular, which is ensured by performing the experiments in N2 protective atmosphere and de-aerating of NaOH solution.The mass loss is usually recorded as a function of time.In order to get valuable input data for the modelling procedure, it is highly recommended to arrange such a set of measurements that will be able to provide the mass loss time functions at different liquid/solid ratios of the reactants, specific surface of the basaltic elements or molarity of the solution.As showed by Ramaswamy et al. [2], it is important perform a basalt fiber characterization to get knowledge about the precise mineralogical and/or phase composition, both qualitative and quantitative, and specific surface of the fibers which enables to calculate the mass loss for particular constituents of basalt (mainly for Si, Ca, Al, Fe or Mg).
The mass loss time functions, the key for the subsequent reaction kinetics description, are usually completed with other data to consider or eliminate secondary processes that might occur during the dissolution process such as precipitation, subsequent reactions of dissolved elements, changes of their saturation level or changes in the chemical affinity.Scanning electron microscopy (SEM) is therefore used for the analysis of the precipitation products, being completed by the surface analysis of the fibers (scanning transmission electron microscopy) together with results of chemical composition provided by X-ray diffraction (XRD).

Mathematical models-requirements and expectations
There is a number of models or modelling methods which can be used to describe reaction kinetics of basalt fibers in cementitious matrices.While CFD models (computer fluid dynamics) represent the most complex tool for solution of this problem, the simplest way consists in application of the shrinking particle model or shrinking core model.Depending on the complexity, the models can be enhanced by accounting for the pore structure or by involving diffusion mechanisms or reactions inside the porous body.Even if the CFD modelling can provide very detailed data about solid-liquid reaction processes, obtaining the results is often balanced by excessive demands on computing time and power or such computing abilities at all.The compromise between excessive demands and reliable results has been therefore found in single particle models which allows fast computations that are very well correlated so that they can provide reliable data.
The nature of the problem solved decides what kind of a model is suitable to be used.The main objective of the ongoing research is to quantify or estimate the corrosion progress of the basalt fibers exposed to effects of alkali environment.This is expected to be done by means of determination of corrosion layer depth that is formed in the course of reacting time.The solid substance, shape of elements or non-fluid attribute of the systems allows some simplifications to be adopted.This basically excludes CFD models that are too excessive, so suitable for applications in more complex problem solutions [3][4].
Assuming diffusion-controlled reaction mechanisms and cylindrical symmetry of the fibers, the problem can be transformed and solved in a two-dimensional space, where the fiber represents a solid particle being surrounded by the cement matrix.Even if two solids are mentioned, a solid/liquid model can be applied as the cement matrix can be considered as a continuous medium [5].Such models are simpler than CFD but still quite complex than those for description of gas reactions as they can involve chemical interactions, heat and mass transfer or reaction kinetics.The latter is probably the most important factor to understand the whole problem.Its determination or approximation based on experimental data is therefore the main objective of the computational modelling techniques.

Problem solution at ideal conditions
The rate of reactions at isothermal conditions can be described as where A is the pre-exponential factor, Ea is the activation energy, R is the gas constant, T is temperature and f(α) is the reaction model.Ranging between 0 and 1, α expresses a volumetric ratio of material already reacted.The integral form is then written as There is not a mandatory technique to determine the α values, rather called the conversion fraction, so any technique capable to describe the course of reaction can be applied, if the data are converted to this parameter.Gravimetry is one of the most frequently used technique, which express α as a relative mass change on the course between the initial and final values of a sample.However, the geometrical interpretation would be more convenient as it makes possible to describe the corrosion progress by means of the corrosion depth expression.Referring to the identical type, these models are classified as "geometrical contraction" or "shrinking particle".This paper refers to the shrinking particle model to avoid readers misleading by using a duplicate terminology.The basic assumptions of the model are shown in figure 1, exploiting cylindrical symmetry that is more apt for fibers approximation.
Figure 1.Basic principles of a shrinking particle model.
No matter what particle shape is considered in the shrinking particle model, the equation (3) can be applied, describing a relation between the original (r0) and actual radius r of a fiber at time t fiber, which is reduced by the ongoing reactions with the rate constant k.Analogously, the corrosion depth x at time t determines the radius r of the unreacted fiber core.Once the conversion fraction α is rewritten to the volumetric form by adding bulk density ρ to the generally known gravimetric formula, we obtain where l denotes the sample length.Substituting equation (3) into equation ( 5) and rewriting the reaction rate k as k0 = k/r0, the shrinking cylinder model (integral form) is obtained in the final form.: Equation ( 6) represents a simple form of the model based on an assumption of a non-porous particle, so the reactions occur only at the surface.When the reaction interface proceeds towards the center of the particle, the unreacted residue is getting smaller which is understood as the shrinking/contracting process that named this category.The overall size of the particle, however, can remain unchanged [6] when the reaction products are not moved away anyhow, so a visible shrinking is not always observed.The model equation.( 6) reveals that the reaction products, including their effect, are not further involved in the running processes.From the practical point of view, such cases may occur when the products layer is quickly removed, e. g. when solid particles are dissolved in liquids, or such a layer is not form at all, which is typical for gas releasing reactions such as combustion.Corrosion of basalt fibers in cement matrix does not belong among these cases, which has been showed e. g. by Tang et al. [7] or Förster et al. [8] by capturing the products layer on the reacted fibers using scanning electron microscopes.This layer then affects the course of reactions by two main mechanisms.An increase of diffusion resistance of the layer leading to slowing down the reactants supply to the reaction interface represents the first one.The second mechanism is based on a suppression of reaction kinetics due to accumulation of reaction products which increases their concentration.The solid nature (even if treated like a liquid in the model as it is homogeneous and continuous [9]) of the cement matrix exhibits very slow mass transfer which indicates the effects of both phenomena, diffusion and concentration, cannot be neglected.

Approximation of the solution towards real-world conditions
If diffusion is the dominant controlling mechanism, the reaction rate depends on the product layer formed as it slows down and reduces the amount of reactants entering the system.The reaction rate decreases proportionally to the thickness of this layer as the reactant A has to be transferred through the product layer AB to reach the element B and react, as shown in figure 2. The mass of A entering the system through the unit area of the A-AB interface can be written as where MX is the molecular weight of the constituent A, D is the diffusion coefficient, ρ denotes the bulk density of the layer (AB), l is product layer thickness and x a position inside being described by its distance from the A-AB interface, C is the concentration of A in AB layer.If the linear concentration gradient is assumed, the third term on the right-hand side of the equation ( 7) is equal to -(CABB -CAAB)/l and the equation ( 8) is obtained, Basalt fiber (B) Reaction products (AB) in which the substitution of the term 2D[MAB(CABB-CAAB)]/MBρ by the constant k gives 22 Obeying the parabolic law, the equation ( 9) represents the one-dimensional diffusion model, in which the product layer thickness and the conversion fraction are directly proportional.The 1D model assumes an infinite plane, which corresponds principally to a sphere with the infinite radius.To allow a solution of real fiber-related problems using this model, the shape factor must be respected and implemented.It requires the same modification principles to be followed as in case of the shrinking cylindrical model derived within equations ( 3)- (6).In this way, Jander 2D diffusion model (equation ( 10)) is obtained [10].This model is often questioned due to the flat plane assumption which it is based on.The commonly preferred alternative, that is physically more appropriate, has been derived using the Ginstling-Brounshtein approach [10] which is based on the solution of the Fick's first law for radial diffusion in cylinder-like elements (equation ( 11)).
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Comparison of model outputs
As previously discussed, the outputs of individual models may vary depending on their fundamental principles, complexity and precision at the derivation phase, reflection of reaction-participating processes or number and nature of simplifications adopted.The final decision about the models' accuracy, however, must be done based on model outputs confrontation with real (experimental) data, which is the main principle of validation procedures.This paper provides rather a theoretical discussion focused on modeling mechanisms and principles to analyze their effects on results with respect to specifics of fiber-related problems.Comparison of model fitting results to reference data are then shown in the end, being supposed to provide information for the decision about models' appropriateness.
A closer look to the shrinking particle model reveals, that the shape effect is not the part of the mathematical derivation.To be considered, it must be included in the reaction rate constant to reflect the reaction area reduction.From this point of view, k is not the constant anymore but a time dependent parameter expressed based on g(α)-t 1/2 plot.Comparison of the model outputs with/without the shape effect being assumed is given in figure 3.This model shortage, or rather the model deriving principles, can be further exploited in this paper, as it makes it quite universal as the reaction rate constant can incorporate other factors as well, the diffusion effect in particular.Such a phenomenon can be then easily added and compared with the effect of the shape factor when k is evaluated based on log g(a)-log t plot.These results are shown in figure 3.
The results indicate that both, the shape factor and diffusion, must be considered when basalt fiber corrosion is predicted/approximated using mathematical modelling as the plots of SPM model with or without the shape factor gave less accurate results.Moreover, these outputs were achieved when the initial point αt=0 = 0 was excluded at the k parameter expression (see figure 3).Even after that, the RMSE values yielded, 0.0127 and 0.0193, respectively, are higher than that obtained using diffusionincorporated model (0.0047).

Conclusions
The results presented in this paper showed, that chemical reaction leading to basalt fibers degradation in alkaline environment are driven by diffusion, which cannot be neglected in the reaction course approximation using mathematical modelling approaches.Diffusion models should be therefore used, these incorporating the shape factor are recommended the most.Here, the effect of diffusion or the shape factor are considered directly due to the mathematical nature of the model.The simple models (not derived from the diffusion law), mostly those based on the particle geometry, must incorporate the deceleration factors explicitly.The separate modification of the reaction constant is the best way, but more demanding processing of experimental data is required while the outputs remain still less accurate.

Figure 2 .
Figure 2. Basic principles of a diffusion model.

Figure 3 .
Figure 3. Evaluation of shape factor and diffusion effect on modelling outputs, shrinking particle model (SPM) used as the reference.Upper left: theoretical model outputs, upper right: evaluation of reaction rate (constant) based on different α-t plots to incorporate individual reaction decelerating principles.Bottom: comparison of model outputs.