The influence of diol addition on water crystallization kinetics in mesopores
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Abstract
Solidification of liquid occupying material’s pores is one of the main reasons of its skeleton deterioration. So far, it has been usually assumed that confined water undergoes phase transitions more rapidly than temperature changes, which indicates that the kinetic effects are negligible. However, this assumption is often infringed. The main object of this paper is to analyse the crystallization of commonly applied antifreeze, i.e. ethylene as well as propylene glycol, in water solutions contained in voids of mesoporous silica gel with average pore size equal to 11 nm. In case of both solutions, two concentrations are analysed, 5% and 10%. Additionally, the analysis is conducted also for deionized water. The experimental research has been conducted by means of differential scanning calorimetry with multiple cooling rate program applied. The activation energy is estimated according to the differential Friedman method. In order to pick the most appropriate kinetic model to analysed phenomenon, two approaches are applied: the one introduced by Málek as well as the one described by PerezMaqueda et al., the socalled modelfitting method. The former indicates the Šesták–Berggren model, whereas the latter points to nth reaction order model. Both equations demonstrate high accordance with the experimental data. The Šesták–Berggren equation is an empirical formula, which does not provide any explanation about antifreeze solidification. However, the nth reaction order model belongs to the group of geometrical extension/contraction models.
Keywords
Kinetics Water freezing Ethylene glycol solution Propylene glycol solution Mesopores Nucleation Silica gelIntroduction
The inpore crystallization is concerned as the major cause of material’s microstructure damage, which is why it is a frequent subject of scientific research [7, 8, 9, 10, 11]. In vast majority of papers authors assume that temperature changes so slow that considered system remains in thermodynamic equilibrium [12, 13]. However, this condition might be sometimes violated, e.g. during fire in cold climate. In such a situation, when the turbulent heat flow occurs, the kinetic effects cannot be neglected. Phase transitions of inpore liquid in case of nonequilibrium conditions were previously investigated by Setzer [14, 15]. Bronfenbrener et al. introduced a simple model describing mass and energy transport during water phase transition concerning the kinetic effects [16, 17]. Their approach is based on the characteristic time parameter, which remains constant and is not influenced by material microstructure.
So far, a lot of research, which concerns homogenous nucleation of bulk water, has been conducted [18, 19]. However, such crystallization may occur only in sterile, laboratory conditions in case of deionized liquid. In previous papers, we analysed the kinetics of heterogenous crystallization of water confined in mesopores of silica gel [20, 21, 22]. It has been confirmed that it is a thermally activated process. It has been also proved that the empirical Šesták–Berggren model reflects the experimental data in the best way. The aim of presented research is to investigate the phase transition of inpore liquid during rapid temperature changes. This time in addition to deionized water, we intend to analyse the phase transition kinetics of ethylene and propylene glycol in water solution, which are commonly applied as antifreeze. The solidification kinetics of such substances has not been described yet. The mesoporous silica gel with dominant pore diameter of 11 nm is used as a saturated material. The main purpose of this paper is to investigate the mechanism, according to which a solidification of coolant solution happens. To achieve this goal, we determine the parameters of reaction rate function, i.e. the activation energy, the preexponential factor and the kinetic model, for analysed antifreeze solutions. As possible kinetic equations describing the antifreeze solutions, we analysed the following models: nucleation and nuclei growth model, geometrical contraction model, diffusion model and orderbased model. To estimate the kinetic mechanism, we applied the algorithm proposed by Málek [23], which is recommended by the International Confederation for Thermal Analysis and Calorimetry (ICTAC) [24]. In this case, the activation energy is estimated on the basis of Friedman method [25]. As the alternative procedure for kinetic model determination, the linear modelfitting method described by PerezMaqueda et al. is applied [26, 27]. In this paper, we affirm that both of those approaches are suitable ways of finding the kinetic equation describing diol solution crystallization in silica gel mesopores. The experimental research is conducted by means of differential scanning calorimetry (DSC), which is a multipurpose thermoanalytical tool applied in nearly every branch of science [28, 29]. The nonisothermal program is with the following cooling rates employed: 2.5 °C min^{−1}, 5.0 °C min^{−1}, 7.5 °C min^{−1} and 10.0 °C min^{−1}. Additionally, the slowest rate 0.5 °C min^{−1} is introduced in order to conduct calibration procedure. Moreover, the nitrogen absorption/desorption analysis is conducted in order to examine the possible influence of cyclic water freezing from the solution on the silica gel microstructure.
Experimental and theoretical methods
Material properties
Silica gel is chosen as the subject matter of the experimental investigation. Silicon dioxide is a synthetically obtained substance, which forms porous structure and can absorb liquids up to 40% of its own mass. This granular, vitreous material is produced from sodium silicate. The connected hydrophilic pores comprise an extensive surface area, which attracts water by adsorption and capillary condensation. Hence, the main application of silica gels is desiccation and local humidity maintenance. Moreover, the material is commonly used as stationary phase in chromatography and active phase supports in heterogeneous catalysis. The silica gel applied in the experimental research is supplied by SigmaAldrich Co. The producer declares structure properties of the material as follows:

Dominant pore diameter: 15 nm,

Particle size: 250–500 μm,

Pore volume: 1.15 cm^{3} g^{−1},

Surface area: 300 m^{2} g^{−1}.
Textural parameters of silica gel determined by the N_{2} adsorption–desorption
Sample  Treatment  Specific surface area SSA/m^{2} g^{−1}  Total pore volume/cm^{3} g^{−1}  Average pore diameter/nm 

SG1  Virgin sample  287  1.08  11.2 
SG1  Sample saturated with 10% ethylene glycol solution and subjected to 10 freezing cycles  292  1.09  11.3 
The shapes of the obtained isotherms indicate that they represent the IV type according to IUPAC classification. The pore size distributions are unimodal, which proves that investigated material is characterized by pores of one dominant size. In the recorded isotherms, one can notice the capillary condensation step, which corresponds to nitrogen capillary condensation and an evaporation step, which implies nitrogen evaporation from pores. Those observations are indicative of regular material microstructure composed of pores of uniform diameters. The results obtained for both kinds of samples are equal, which indicates that the influence of diol solution freezing on the silica gel microstructure is negligible. Hence, one can assume that the results of kinetic analysis are not affected by the alteration of material microstructure.
Thermodynamic properties of employed solutions [38]
Enthalpy of fusion, \(\Delta H_{\text{f}}\)/J g^{−1}  

Ethylene glycol  
Pure substance  \(160.46\) 
5% solution  \(0.95 \cdot [334.1 + 2.119(T  T_{0} )  0.00783(T  T_{0} )^{2} ] + 0.05 \cdot 160.46\) 
10% solution  \(0.90 \cdot [334.1 + 2.119(T  T_{0} )  0.00783(T  T_{0} )^{2} ] + 0.10 \cdot 160.46\) 
Propylene glycol  
Pure substance  \(107.50\) 
5% solution  \(0.95 \cdot [334.1 + 2.119(T  T_{0} )  0.00783(T  T_{0} )^{2} ] + 0.05 \cdot 107.50\) 
10% solution  \(0.90 \cdot [334.1 + 2.119(T  T_{0} )  0.00783(T  T_{0} )^{2} ] + 0.10 \cdot 107.50\) 
Experimental procedure
Applying linear temperature program does not enable user to differentiate the dependence of temperature and reacted fraction from the reaction rate [24, 26]. Hence, the experimental procedure is designed as multitemperature program. The applied program consists of four cooling cycles with the temperature rate equalling 2.5 °C min^{−1}, 5.0 °C min^{−1}, 7.5 °C min^{−1} and 10.0 °C min^{−1}. An auxiliary freezing cycle before each of such main temperature cycle is designed. It consists of cooling sample down to − 40 °C and heating it up to − 0.1 °C with a rate 1.0 °C min^{−1}. The reason of introducing such preliminary cycle is to ensure the presence of ice nuclei in the confined water and moreover to enable to differentiate between the signal corresponding to transitions of excess water and the one related to confined water. The kinetic analysis is conducted on the basis of DSC cooling scans. Then, two equal portions of dry material are put into two crucibles. Additionally, one of the prepared samples is oversaturated with distilled water or diol solution, respectively. The crucible with dried material is placed in the reference side of the DSC furnace whereas the second one in the sample side. Some excessive amount of water or glycol solution is added during saturation. As mentioned before, the signal corresponding to this additional liquid is eliminated by auxiliary cooling cycles, which is why only signal connected to poreconfined liquid freezing is considered in further kinetic analysis. The experimental research is repeated for all of prepared glycol solutions as well as for deionized water. The kinetic analysis is conducted on data representing mean values of three DSC programs performed for particular samples. The temperatures recorded during cooling cycles are determined in accordance with the onset temperature of excess water melting. It is reproducible within ± 0.10 °C for all scans.
Theoretical background
Activation energy
Activation energy \(E_{\upalpha}\) corresponds to the slopes of relation between \(\ln \left( {\frac{{{\text{d}}\alpha }}{{{\text{d}}t}}} \right)_{{\upalpha,\upbeta}}\) and the reciprocal temperature \(1 /T_{{\upalpha,\upbeta}}\), which are analysed for each extent of conversion value, \(\alpha\), and index \(\beta\) denotes various thermal cycles.
Kinetic model determination
Model  Symbol  \(f(\alpha )\) 

Johnson–Mehl–Avrami  JMA(p)  \(p(1  \alpha )[  \ln (1  \alpha )]^{1  1/\rm{p}}\) 
Reaction order  RO(n)  \((1  \alpha )^{\rm{n}}\) 
2D reaction  R2  \(2(1  \alpha )^{1/2}\) 
3D reaction  R3  \(3(1  \alpha )^{2/3}\) 
2D diffusion  D2  \(1/[  \ln (1  \alpha )]\) 
Jander equation  D3  \(1.5(1  \alpha )^{2/3} /[1  (1  \alpha )^{2/3} ]\) 
Ginstling–Brounshtein  D4  \(1.5[(1  \alpha )^{  1/3}  1]\) 
Šesták–Berggren  SB(m,n)  \(\alpha^{\rm{m}} (1  \alpha )^{\rm{n}}\) 
Results and discussion
T_{onset}/°C experimental value  T/°C literature value  

5% Ethylene glycol solution  − 3.67  − 1.80 
10% Ethylene glycol solution  − 6.84  − 3.60 
5% Propylene glycol solution  − 3.10  − 1.50 
10% Propylene glycol solution  − 4.27  − 3.00 
The inconsistency between both values springs from the fact that the literature value has been measured by means of entirely different method, which is introduced by American Society for Testing and Materials [46]. In this case, the temperature of analysed coolant has been monitored by means of thermocouples and the output data have been plotted as the function of time. Moreover, as the freezing point, the authors considered the maximum temperature recorded during temperature increase related to the solidification of overcooled liquid—see Fig. 4.
The ice content is determined in grams of ice per gram of dry material. It can be seen that the amount of developed ice is almost two times lower in case of more concentrated diol solutions. In general, the amount of arisen ice is independent from cooling rate \(\beta\) for all analysed solutions. The ice content equals about 0.636 ± 0.003 g g^{−1} and 0.337 ± 0.002 g g^{−1} for 5% and 10% ethylene glycol solution, respectively, and 0.484 ± 0.007 g g^{−1}, 0.228 ± 0.002 g g^{−1} for 5% and 10% propylene glycol solution, respectively. In case of deionized water, the amount of developed ice is equal to 0.916 ± 0.002 g g^{−1}. Many studies have proved that during inpore water freezing there always remains a nonfreezable liquid layer on the pore wall, whose thickness is about 0.5–2.0 nm [3, 4]. Significantly lower ice content in diol solutions in comparison with deionized water can be caused by considerable expansion of such a layer.
The average values of activation energy
Activation energy E_{a}/kJ mol^{−1}  

Deionized water  − 79.097 
5% Ethylene glycol solution  − 77.085 
10% Ethylene glycol solution  − 104.76 
5% Propylene glycol solution  − 135.655 
10% Propylene glycol solution  − 164.645 
It can be readily noticed that the righthand side of Eq. (21) is a straight line. Furthermore, we maximize the Pearson correlation coefficient between the lefthand side of Eq. (21) and reciprocal temperature by optimizing n and m parameters of Eq. (13). A slope of this line corresponds to value of activation energy, whereas an intercept to preexponential factor and constant c.
Parameters of reduced SB model fitted to different reaction models
Model  \(f(\alpha )\)  Parameters of the equation \(c(1  \alpha )^{n} \alpha^{m}\) 

JMA(2)  \(2(1  \alpha )[  \ln (1  \alpha )]^{1/2}\)  \(2.079(1  \alpha )^{0.806} \alpha^{0.515}\) 
JMA(3)  \(3(1  \alpha )[  \ln (1  \alpha )]^{2/3}\)  \(3.192(1  \alpha )^{0.748} \alpha^{0.693}\) 
F1  \(1  \alpha\)  \(1  \alpha\) 
R2  \(2(1  \alpha )^{1/2}\)  \(2(1  \alpha )^{1/2}\) 
R3  \(3(1  \alpha )^{2/3}\)  \(3(1  \alpha )^{2/3}\) 
D2  \(1 /[  \ln (1  \alpha )]\)  \(0.973(1  \alpha )^{0.425} \alpha^{  1.008}\) 
D3  \(1,5(1  \alpha )^{2/3} /[1  (1  \alpha )^{2/3} ]\)  \(4.431(1  \alpha )^{0.951} \alpha^{  1.004}\) 
Equation (22) is a general expression, which includes both R2 and R3 model for \(n = 0.5\) and \(n = 0.67\), respectively—see Table 6. In case of those models, one assumes that nucleation occurs rapidly on the surface of the crystal. The rate of crystallization is controlled by the reaction interface progress from the centre of the crystal. Different cases of such a model can be derived depending on a shape of emerging crystal [48]. The R2 model corresponds to cylindrical volume whereas the R3 model to spherical/cubic volume. In case of analysed substances, we obtain up to four times larger values of n parameter, which does not provide any physical meaning. However, the experimental data is reflected with precision and accuracy by reaction order model characterized by such parameter values.
Conclusions
 1.
The ice content in diol solutions in comparison with deionized water is significantly lower. This fact may arise from the existence of much thicker nonfreezable layer on pore walls in case of antifreeze solutions than in water. The thickness increases with rising diol solution concentration.
 2.
The value of activation energy obtained by means of Friedman method is negative and varies slightly as a function of temperature for all of analysed substances. This fact indicates that the process rate increases with the decreasing temperature. The activation energy decreases with the increasing concentration of glycols. Furthermore, ethylene and propylene glycol solutions solidifications are the complex processes, in which the apparent activation energy fluctuates, but activation energy values for individual processes are constant. To determine the rate functions, we assume the constant, average value of activation energy for all analysed solutions.
 3.
According to applied isoconversional method, it is settled that the empirical Šesták–Berggren formula reflects the experimental data with the high accuracy. Moreover, the second employed approach, the modelfitting method indicates that the nthorder reaction model with fractal value of n parameter also greatly resembles experimental results. To sum up the nthorder reaction formula can be successfully applied to description of ethylene and propylene glycol solidification. The empirical SB formula might be applied for both water and glycol solutions.
 4.
The nitrogen adsorption–desorption isotherms as well as the pore size distributions obtained by means of nitrogen adsorption analysis for virgin samples as well as for samples subjected to ten freezing cycles are identical. This fact indicated that the silica gel microstructure does not undergo any deterioration caused by cyclic freezing of inpore liquid.
Notes
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