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ARTICLES Modeling Gypsum Crystallization on a Submicrometric Scale G. Dumazer,† V. Narayan,† A. Smith,‡ and A. Lemarchand*,† CNRS, Laboratoire de Physique The´orique de la Matie`re Condense´e, UniVersite´ Pierre et Marie Curie-Paris 6, UMR 7600, 4 place Jussieu, case courrier 121, 75252 Paris Cedex 05, France, and Groupe d’Etude des Mate´riaux He´te´roge`nes, Ecole Nationale Supe´rieure de Ce´ramique Industrielle, E. A. 3178 47-73, aVenue Albert Thomas, 87065 Limoges, France ReceiVed: July 8, 2008; ReVised Manuscript ReceiVed: September 24, 2008
Gypsum needle crystallization from a suspension of calcium sulfate hemihydrate grains in an aqueous solution is studied on a mesoscopic scale. We build a simulation model, which assumes that gypsum formation is limited by heterogeneous nucleation and precipitation with an initial autocatalytic surface-controlled needle growth and a final diffusion-controlled growth. The model introduces a minimal number of parameters whose effect on growth dynamics and gypsum morphology is analyzed. We find that the increase of nucleus number per hemihydrate grain decreases induction time and needle length and increases needle entanglement, i.e., improves the mechanical properties of the material. The simulation results correctly reproduce experiments with and without treatment of the reactants. The simulation model can be used to predict the behavior of the reactive system in conditions that are not compatible with experimental observations. 1. Introduction Plaster is a powder of calcium sulfate hemihydrate (CaSO4 · 1/2H2O), which is obtained by calcining gypsum or calcium sulfate dihydrate (CaSO4 · 2H2O). Reciprocally, when hemihydrate grains are mixed with water, entangled gypsum needles crystallize. The recyclable character of gypsum and its excellent insulating properties has given rise to renewed interest in this environmental material. The literature offers many experimental results describing the effects of various additives on plaster-setting dynamics and the obtained material morphology.1-11 However, the relation between crystallographic structure and dynamical properties is not straightforward and growth rates cannot be easily predicted on the basis of microscopic observations.5 The existing models rely mainly on differential equations.2,12-14 They fit some experimental curves related to macroscopic dynamics that have been obtained for a given additive, but they can hardly predict the effect of a new additive. Our goal is to build a minimal model on a mesoscopic scale, in order to predict the effect of a limited number of crucial parameters on dynamics and morphology. First, we identify the limiting processes which control plaster setting on the submicrometric scale. Then, we build a simulation model on the basis of well-identified hypotheses. The comparison of the simulation results with experiments should enable us to reject the hypotheses or to validate the model. Such an approach was already chosen to perform a submicrometric description of silicon carbide ceramics synthesis.15,16 The paper is organized as follows. In section 2, we present the hypotheses on which our dissolution-precipitation model relies and describe the * Corresponding author. Tel: +33 (0)1 44 27 72 90. Fax: +33 (0)1 44 27 72 87. E-mail:
[email protected]. † Universite´ Pierre et Marie Curie-Paris 6. ‡ Ecole Nationale Supe´rieure de Ce´ramique Industrielle.
algorithm governing gypsum needle growth. In section 3, we give the simulation results and compare plaster-setting dynamics with experimental results that we obtained in dilute conditions with and without a confidential treatment of the reactants. The results on final gypsum morphology are discussed in section 4. Section 5 is devoted to the conclusion. 2. A Dissolution-Precipitation Model Gypsum formation from β-hemihydrate grains in an aqueous solution is known to obey a dissolution-precipitation mechanism.1,2 At room temperature, β-hemihydrate solubility is more than 3 times larger than dihydrate solubility.2 It is commonly accepted that hemihydrate dissolution is fast compared to the other processes taking place in the medium.2 We admit that precipitation is the limiting process. Acceleration of precipitation in the presence of a small amount of finely ground gypsum seeds1 reveals that dihydrate crystallization results from heterogeneous nucleation. Gypsum needles9,11,17 grow from nuclei which are located on the surface of the hemihydrate grains. Our model does not describe the early formation of these nuclei, and we admit that they preexist in the form of embryonic needles. Gypsum precipitation is characterized by an induction period followed by a sudden acceleration. Such a behavior is typically observed in thermal explosion, due to the exponential dependence of reaction rate on temperature.18-20 Explosion may also occur in autocatalytic reactions, when the reaction rate increases with the amount of product previously formed.21-23 In order to reproduce such an explosive phenomenon, we choose an autocatalytic growth law for the embryonic needles with a growth rate which is proportional to their surface. This choice is motivated by the following experimental results: the order of magnitude of the apparent activation energy (Ea ) 40 kJ/ mol at T ) 20-40 °C) is typical of a surface-controlled reaction and the growth rate of dihydrate is independent of the stirring
10.1021/jp806028v CCC: $40.75 2009 American Chemical Society Published on Web 01/06/2009
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Figure 1. Details of a configuration for G ) 1 nucleus per grain (a) at the beginning of the simulation (after 50 time steps) and (c) at the end of the simulation (after 616 time steps). Idem for G ) 10 nuclei per grain (b) at the beginning of the simulation (after 50 time steps) and (d) at the end of the simulation (after 351 time steps). The selected areas in (b) and (d) are magnified in Figure 2. Physical value of cell size: ∆x ) ∆y ) 50 nm. Optical microscopy image of the obtained material (e) for nontreated reactants and (f) for treated reactants. The experiments are performed in quasi two-dimensional conditions, between two plate glasses 20 µm distant, in dilute conditions (H/W ) 0.15), for β-hemihydrate grains of mean radius R ) 5 µm at an initial temperature of 20 °C.
rate at the beginning of the reaction.8 As the reaction proceeds, the reaction speed is determined by the stirring rate, which points to a reaction controlled by diffusion in the liquid phase.8 Whereas the medium is globally out of equilibrium during the reaction, each needle is supposed to reach partial and local equilibrium at any time. Dissolution and precipitation of dihydrate in the vicinity of a crystal face is supposed to be fast and reversible with respect to diffusion of the reactant in the liquid phase, so that each needle fits to the equilibrium shape, whereas the set of needles has globally an irreversible evolution. According to the Wulff theorem, equilibrium crystal growth is ruled by minimization of the surface free energy.24,25 Such a crystal is faceted and the growth rate of each face is proportional to its specific surface energy:25 a needle keeps its shape and grows homothetically. In other words, growth under thermodynamic control leads to a constant needle aspect ratio, l/w, where l and w are respectively needle length and needle width. Local thermodynamic regime is all the more reached since the global reaction is slow. This condition is satisfied in the dilute
conditions considered here, i.e., for a small hemihydrate/water volume ratio. If we admit the existence of a small liquid layer between the two solid surfaces, the contact with another solid does not modify the surface tension of a crystallographic face. Consequently, surface free energy and needle shape remain unchanged in the presence of steric hindrance. In particular, needle growth is stopped as soon as two opposite sides are blocked, since any further growth without overlap would modify needle aspect ratio and violate Wulff theorem. We perform two-dimensional (2D) simulations of the formation of gypsum on a mesoscopic scale in dilute conditions. Hemihydrate grains are modeled by disks of radius R. They are randomly spread without overlap in a square box with periodic boundary conditions as shown in Figure 1a,b. G nuclei are randomly placed without overlap on the perimeter of each grain. These embryonic needles are rectangles of width w0 and length l0 ) Rw0, where R is the aspect ratio. They are turned in randomly chosen directions, which definitely impose needle directions as shown in Figures 1a-d and 2a,b. The value of G
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Figure 3. Different types of growth cases during time step ∆t. White rectangles correspond to needle state at time t. Gray parts correspond to dihydrate which precipitates during t and t + ∆t. Black rectangles mimic growth blocking due to steric hindrance. A thin liquid layer is supposed to remain between the solids so that needle specific surface energy is unchanged in the case of contact. (a) Free growth, (b) one blocked side, (c) two perpendicular blocked sides.
Figure 2. Magnification of the selected area in (a) Figure 1b and (b) Figure 1d.
imposes the maximum number of needles that will grow around each grain. Indeed, we do not create new nucleation sites during the simulation, since their formation would require a longer time than the precipitation of dihydrate on the existing nuclei. Moreover, even if one would consider some new nuclei that would appear on the surface of the shrunk hemihydrate grains, their growth would be difficult due to the presence of the already big needles that started to grow earlier. Note that G gives the upper limit of the number of needles per grain, since it is possible that some nuclei never grow due to steric hindrance. At any time, the solution is supposed to be supersaturated with respect to dihydrate. Time t is discretized. During the time step ∆t, the growth of each needle is considered. Growth is accepted with a given probability. In the initial autocatalytic growth phase, an embryonic needle of width w and length l grows with sizedependent probability p0(w + l), which is proportional to its perimeter. The coefficient p0 characterizes the surface state of dihydrate crystal. Then, the needle grows with constant probability p, which is controlled by diffusion in the liquid phase. The transition between the two growth regimes occurs when the needle perimeter reaches the critical size wc + lc ) p/p0. If we consider the set of needles in a given configuration, it means that some needles grow whereas the others remain unchanged. According to our assumption, precipitation is the limiting process. During ∆t, dihydrate precipitation and needle growth are adiabatically followed by hemihydrate dissolution and grain shrinking as follows. At each time step, the needles, for which free growth occurs with probability p0(w + l) or p, increase their width by 2∆w and their length by ∆l ) 2R∆w, as shown in Figure 3a. To check if a needle is in contact with another solid, space is discretized into square cells of side ∆x. We consider that two solids are in contact if the distance between them is less than or equal to ∆x. The face of a needle in contact with another solid is blocked by steric hindrance and does not grow. The different types of blocking which do not prevent growth are given in Figure 3b,c. If only one face is blocked, the growth rate of the needle is divided by 2. For example, if a short side is blocked, a slice of thickness ∆l is added on the opposite face and a slice of thickness ∆w/2 is added on each
long face. If two orthogonal faces are blocked, a slice of thickness ∆w is added on the free long face and a slice of thickness ∆l is added on the free short face. In all other cases, growth is not accepted. In particular, a needle with two blocked opposite sides is not allowed to grow. These growing rules ensure that needle aspect ratio, l/w, remains constant and fixed at R, as required by Wulff theorem. After the growth step, we consider dissolution. For the sake of simplicity, we neglect the difference between the molar volumes of hemihydrate and dihydrate: we dissolve a total volume of hemihydrate equal to the total volume of dihydrate which precipitated during the growth step. The total volume which precipitated determines the width ∆R(t) of the external crown which is lost by each hemihydrate grain. In summary, the simulation consists in a succession of needle growth and grain shrinking steps. The simulation ends either when the total amount of hemihydrate has dissolved or when all the needles are blocked by steric hindrance. Note that the grains and the needles do not move during reaction. The skeleton of entangled needles rigidifies the structure. Our aim is to analyze the influence of the different parameters of the model on the simulation results, i.e., on the dynamics of the reaction and on the morphology of the resulting material. The simulation results will be compared with time-resolved experiments that we carried out in dilute conditions for treated and nontreated reactants. We perform a series of simulations for variable number G of nuclei per grain in the range 1 e G e 30, and variable rate constants p0 and p in the range 2 × 10-4 e p0 e 10-1 and 0.1 e p e 1. The other parameters are fixed at the following values, which agree with the experimental conditions: volume ratio of hemihydrate to water H/W ) 0.15, needle aspect ratio R ) 20, hemihydrate grain radius R ) 5 µm ) 100∆x, which imposes spatial cell size ∆x ) 50 nm. For each initial configuration, we generate a set of 600 embryonic needles of size arbitrarily determined by the initial width w0 ) 0.1∆x. We choose to simulate the same number of embryonic needles for each set of parameters, in order to be able to compare results that are associated with the same statistical error. Since all the simulations are performed at fixed density (fixed H/W) and fixed hemihydrate grain radius R, variable nucleation site number G leads to (i) a variable simulation box surface that scales as [(R2)/(H/W)][1/G] and (ii) a variable number of simulated hemihydrate grains that scales as 1/G. In these conditions, the number of needles per unit surface increases as the nucleation site number G increases. As p0 or p varies, we keep constant both simulation box surface and hemihydrate grain number. Needle width increase is fixed at ∆w ) 0.15∆x. For each set (G, p0, p), we perform averages over 10 realizations
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Figure 4. (a) Experimental conductivity curves of a suspension obtained from the dissolution of 10 g of β-hemihydrate into 200 mL of deionized water (volume ratio H/W = 0.15), without treatment (solid squares), with a weak treatment (empty triangles), with a stronger treatment (solid triangles). Initial temperature 20 °C, WTW conductivity meter LF538. (b) Averaged time evolution of hemihydrate percentage deduced from simulations, %HH, for G ) 1 (solid line), G ) 10 (dotted line), G ) 25 (dashed line) nuclei per grain. Dissolution is not complete for G ) 25.
of the initial condition, i.e., averages over different positions of the hemihydrate disks and embryonic needles. The order of magnitude of the time step ∆t will be determined thanks to the comparison of the simulation results with the experiments. 3. Growth Dynamics Optimization of induction period and sufficiently slow setting are essential to enable convenient plaster processing. Understanding of plaster-setting dynamics and devising a theoretical predictive tool could be an alternative to trial and error experiments consisting of probing the effects of various additives. We propose to perform systematic analyses of the influence of the parameters of the model on the dynamics of gypsum cristallization and on the morphology of the obtained material. First, we study the effects of different parameters on kinetics of dihydrate formation. Figure 4a gives the time evolution of the experimental conductivity for different intensities of the confidential treatment in dilute conditions. Conductivity is proportional to Ca2+ concentration in solution. At the scale of the measurement, conductivity immediately reaches the value associated with the saturation of the solution with respect to hemihydrate, which corroborates our assumption about the very fast dissolution of hemihydrate. Then, conductivity is nearly constant as long as some solid hemihydrate remains. The sudden decrease of conductivity toward the value corresponding to the saturation of the solution with respect to dihydrate reveals the end of the reaction. Qualitatively, the reaction becomes faster as the experimental treatment intensifies. The simulation results given in Figure 4b show the averaged hemihydrate percentage versus time for different values of the number G of nuclei per grain. In the following, averaged quantities are obtained by ensemble averages over 10 realizations of 600-needle configurations. The autocatalytic growth coefficient is fixed at p0 ) 0.01, and the diffusion-controlled growth probability is p ) 0.8 in
Dumazer et al.
Figure 5. (a) Schematic representation of the different dynamical regimes during autocatalytic evolution of hemihydrate percentage. A fast regime (S2) between two slow regimes (S1 and S3) is identified. (b) Averaged time evolution of hemihydrate percentage deduced from simulations, %HH, for G ) 1 (solid line), two realizations of hemihydrate percentage evolution for different initial configurations of hemihydrate disks and nucleus position for G ) 1 (dashed lines). The induction time, t90, maximum dissolution speed, Vmax, and reaction time, t5, are indicated for one of the two realizations.
the series of simulations for variable G. The qualitative evolution of the experimental conductivity compares well with the evolution of hemihydrate percentage deduced from the simulations. The effect of the experimental treatment is recovered by an increase of the simulation parameter G. The comparison between the experiments and the simulation results given in Figure 4 enables to allocate physical values to the simulation time step in dilute conditions. We find that ∆t is of the order of 2 s. The simulation model reproduces the typical S-shape of explosive or autocatalytic phenomena with three steps: an induction period with a slow dynamics S1, a fast regime S2, and a relaxation S3 toward the final state at which reaction ends. Figure 5a gives an illustration of these three different regimes. We introduce three quantities to characterize these three steps, as shown in Figure 5b. The averaged induction time, t90, at which 90% of hemihydrate remains, characterizes the induction period S1. The maximum dissolution speed Vmax is defined as the maximum slope of the curve which gives the percentage of hemihydrate versus time associated with a given realization of the dynamics. The value of Vmax characterizes the fast intermediate regime S2. Note that the averaged value Vmax is not the slope of the averaged hemihydrate percentage evolution given in Figure 5a but is obtained after collecting the maximum dissolution speed for 10 realizations and computing the mean value. The two definitions do not lead to the same results, due to the high sensitivity of the dynamics to initial conditions and fluctuations, as shown in Figure 5b for G ) 1: the curve which gives the evolution of the averaged hemihydrate percentage is much smoother in the leading edge than the two individual realizations. The reaction time t5, at which 5% of hemihydrate remains, characterizes the time for which reaction can be considered complete. The value of t5 depends on the duration
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Figure 7. Averaged number-distribution function P(l) of the needle length l (in units of ∆x ) 50 nm) at the end of the simulation for G ) 1 (dotted line) and G ) 15 (solid line).
Figure 6. (a) Averaged time t90 at which 90% of hemihydrate remains versus number of nuclei per grain G. (b) Averaged maximum dissolution speed Vmax versus G. (c) Averaged time t5 at which 5% of hemihydrate remains versus G. The solid triangles correspond to incomplete dissolution for at least one of the 10 realizations.
of the different steps S1, S2, and possibly S3 if the end of the reaction is very slow. As the number G of nuclei per grain increases, two phenomena compete: due to the larger number of needles per grain, both reactive surface and steric hindrance increase. As shown in Figure 6a, the induction time t90 decreases for increasing G. As G varies, the duration of regime S1 is controlled by the increase of the reactive surface per grain, i.e., the surface of already precipitated dihydrate, on which further precipitation will occur. If the induction time t90 is chosen as a criterion to characterize the reaction speed, we conclude that the reaction is faster for an increasing number of nuclei per grain. On the contrary, if we consider the maximum dissolution speed Vmax, we deduce from Figure 6b and from the decrease of Vmax as G increases that the reaction is slower as the number of nuclei per grain increases. As shown in Figure 5b for G ) 1, the reaction stops suddenly due to the disappearance of a solid phase (the hemihydrate) in the medium before steric hindrance begins and the maximum dissolution speed Vmax is reached at the very end of the reaction. As G increases, the fast regime S2 slows down, which reveals that the dynamical behavior of the fast regime versus G is not controlled by reactive surface but by steric hindrance, i.e., needle entanglement. We will come back later on the behavior of Vmax, when examining the morphology of the obtained material. The competition between the two opposite phenomena leads to a minimum for t5 versus G as observed in Figure 6c. For small G (G < 10), the reaction time t5 decreases as G increases: the behavior of t5 is mainly imposed by the increase of reactive surface during the slow regime S1 and t5 behaves as the induction time t90. At large G (G g 10), for which needle entanglement becomes limiting, t5 increases as G increases: the behavior of t5 is strongly influenced by the slowing down of the fast regime S2 due to steric hindrance. Moreover, the simulation results reveal the existence of a transition for which reaction stops slowly before hemihydrate has been entirely dissolved. The final nonvanishing value of
hemihydrate percentage obtained for G ) 25 in Figure 4b gives an example of this behavior. As shown in Figure 6, the critical value of nuclei per grain is Gc ) 18 for the choice of the other parameter values. Whereas a variation of parameter G modifies both reactive surface and steric hindrance, the autocatalytic growth coefficient, p0, only plays on the surface-controlled reaction rate at the beginning of the reaction for given steric hindrance. Similarly, probability p determines the growth rate of the diffusioncontrolled regime, which controls the last part of the reaction. We performed a series of simulations at fixed G for variable p0 or variable p. The induction time t90 and the time t5 decreases as p0 or p increases. As expected, the reaction is faster as the growth rate constant p0 or p increases. The maximum dissolution speed Vmax is a decreasing function of p0 and an increasing function of p (vide infra). With regard to dynamics, we conclude that the effect of the experimental treatment can be modeled by an increase of the number G of nuclei per grain. Moreover, the simulation results given in Figure 6a predict that further increase of G beyond G = 10 does not sensitively decrease the induction time. 4. Morphology In this section, we discuss how final product morphology varies with the parameters of the model. The morphology of the obtained gypsum determines the mechanical properties of the material. Intuitively, better resistance to deformation is obtained for finely entangled needles. Figure 1e,f gives experimental results that are obtained in dilute conditions, with and without treatment. These figures show optical microscopy images of a thin layer of solution where crystal growth is observed in quasi two-dimensional conditions that compare well with the 2D simulations. The experimental parameters are the following. The weight ratio of water to hemihydrate is fixed to 2, which leads to a volume ratio of hemihydrate to water close to the value H/W ) 0.15 that is chosen in the simulations. The mean value of the hemihydrate grain radius is equal to 5 µm. Final configurations that are deduced from the simulations for a small and a larger number G of nuclei per hemihydrate grain are given in Figure 1c,d. Qualitatively, experimental treatment effect as well as G increase leads to an increase of gypsum needle number per unit volume. Figure 7 gives the averaged number-distribution functions P(l) of needle length for a small
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Figure 8. (a) Averaged mean needle length 〈l 〉, and (b) averaged standard deviation σ(l) according to the number-distribution function of the needle length versus number of nuclei per grain G. The solid triangles correspond to incomplete dissolution for at least one of the 10 realizations. The length unit is the cell size ∆x ) 50 nm.
and a larger number G of nuclei. The periodic modulation that appears in Figure 7 corresponds to the mean distance between two hemihydrate grains. It is an artifact due to the rigorously identical values of the hemihydrate grain radius that we consider in the simulations. The modulation would disappear if we would introduce a radius distribution function. An increase of the number of nuclei per grain from G ) 1 to G ) 10 decreases the mean needle length and the standard deviation of the distribution function. Figure 8 shows the variation of averaged mean needle length 〈l 〉 and averaged standard deviation σ(l) versus G. Let us recall that averaged value refers to ensemble averaged over 10 realizations of the initial configuration whereas mean length, 〈l〉, and standard deviation, σ(l), refers to the distribution function P(l) associated with a given initial condition. We give in Figure 8a the variation of 〈l 〉 versus G. The averaged mean needle length is a monotonous decreasing function of G: for a larger number of nuclei per hemihydrate grain, the same total number of needles shares a smaller amount of dihydrate: the needles are consequently smaller. With a spatial cell size ∆x fixed at 50 nm, an autocatalytic growth coefficient p0 ) 0.01, and a diffusioncontrolled growth probability p ) 0.8, these results lead to an averaged mean needle length which decreases from 〈l〉 ) 413∆x = 20 µm to 〈l 〉 ) 207∆x = 10 µm as G increases from 3 to 10. Such values are in good agreement with the experimental results deduced from the microscopic images that are given in Figures 1e,f for treated and non treated reactants. The increase of G at constant p0 and p reproduces correctly the effect of the treatment on morphology. Moreover, simulation predicts that a further increase of the number of nuclei per grain would not sensitively decrease the mean needle length. As shown in Figure 8b, the standard deviation σ(l) of needle length distribution decreases as the number G of nuclei per grain increases. For small values of G, the dispersion of needle length is large. Actually, a needle which has the chance to grow at the beginning of the simulation, i.e., which is randomly chosen for a growth step according to the probability p0(w + l), will have more chance to grow during the next step, since its perimeter
Dumazer et al. has increased. In the absence of steric hindrance, i.e., for small values of G, such a needle will rapidly reach a large size and consume the reactives before the other needles have time to begin to grow. The dispersion of needle length is therefore large. However, for large values of G, the advantage of the already long needles which began to grow early becomes less important in the presence of steric hindrance which prevents them from reaching an even longer length. At the end of the simulation, steric hindrance favors the final growth of smaller needles which began to grow later and homogenizes needle size. For completeness, we consider the behavior of 〈l 〉 and σ(l) versus autocatalytic growth coefficient p0, which controls the beginning of the reaction, and diffusion-controlled growth probability p, which controls the end of gypsum crystallization. Both the mean length and standard deviation are decreasing functions of p0 and increasing functions of p. The initial autocatalytic surface-controlled growth introduces discrimination into the needle set and sensitively favors the needles which first start to grow, whereas the final diffusion-controlled growth acts more equitably on the needles. For small values of p0, i.e., small initial reactivity, only some needles have a chance to grow and then to attract the main part of the dihydrate. This phenomenon results in large mean lengths and large dispersions. The needle size decreases and homogenizes as the initial reactivity increases. Parameter p mainly plays on the dynamics of the reaction but has little direct effect on the morphology of the resulting material. However, the increase of growth probability p results in an increase of the critical needle size wc + lc ) p/p0 after which the growth becomes diffusion controlled. Consequently, larger values of p lead to a larger range of needle length l0 e l e lc for which autocatalytic growth rules the dynamics. Therefore, increasing p can be compared to decreasing p0. The analysis of the results concerning morphology brings a new insight into the behavior of the averaged maximum dissolution speed. Surprisingly, the maximum speed Vmax of the intermediate step of plaster setting and the mean needle length 〈l〉 behave in the same way. They decrease as G and p0 increase and increase with p. A big needle grows more rapidly than a small one. This is particularly true in the autocatalytic growth step controlled by p0 and still valid, to a smaller extent, in the diffusion-controlled growth step, since the amount of dihydrate which precipitates on a needle during a time step increases with the needle perimeter. This increase of precipitation rate is directly linked to the mean dissolution rate. The nonintuitive decrease of Vmax as G and p0 increase, i.e., as reactive surface and autocatalytic rate constant increase, can be related to the decrease of the mean needle length 〈l 〉 at constant needle number. Similarly, we observe that Vmax increases as the diffusion-controlled rate constant p increases, i.e., as 〈l 〉 increases at constant needle number. In order to characterize the entanglement of the structure and, consequently, the mechanical properties of the material, we determine the averaged mean number 〈C〉 of contacts per needle. As seen in Figure 9, 〈C 〉 increases as the number of nuclei per grain increases. The increase of needle number per unit volume leads to a larger value of the number of contacts, i.e., to a better resistance to deformation. The increase of p0, which decreases needle length at constant needle number, induces a decrease of the number of contacts per needle. Parameter p variation does not sensitively play on the number of contacts per needle. Our experiments reveal that the Young’s modulus, which characterizes elasticity, the flexural strength and the compressive strength of gypsum increase when reactants are treated. More intense
Modeling Gypsum Crystallization
Figure 9. Averaged mean number 〈C 〉 of contacts per needle versus number of nuclei per grain G. The solid triangles correspond to incomplete dissolution for at least one of the 10 realizations.
treatments of the reactants do not improve further the mechanical properties of the resulting material. In summary, the crucial intermediate stage S2, during which easy processing of the material can be achieved, is characterized by the speed Vmax. During the induction time, the mixture remains fluid and cannot be fashioned. A too large value of Vmax would lead to a nearly instantaneous setting that would prevent from shaping plaster. We show that the surface state of the hemihydrate grains, in particular the nucleus properties, are essential factors that play on Vmax. The increase of the number of nuclei per hemihydrate grain in the simulations leads to the same morphological changes as the effect of the treatment of the reactants in the experiments. 5. Conclusion We have built a simulation model which reproduces the dissolution of hemihydrate grains in water and the crystallization of gypsum needles on a mesoscopic scale in dilute conditions. Stochastic rules are introduced to govern growth dynamics, and typical simulation length scale is 50 nm. The model relies on well-identified hypotheses and introduces a small number of parameters whose effect on growth dynamics and gypsum morphology is studied in detail. Dissolution is supposed to be faster than crystallization. Needles grow from nuclei. Initially, growth dynamics is supposed to be autocatalytic and controlled by the size of the previously precipitated needles. Then, reaction rate is imposed by diffusion in the liquid phase. Such a minimal model has the advantage to be predictive and to enable a systematic characterization of the reaction in wide parameter ranges. We find that the increase of the number G of nuclei per hemihydrate grain leads to a faster reaction, in the sense where it decreases the induction period. It also leads to shorter, more entangled needles, as revealed by the increase of the number of contacts per needle. The effect of parameter p0, which controls
J. Phys. Chem. C, Vol. 113, No. 4, 2009 1195 the reaction rate during the initial autocatalytic growth, has a similar consequence on dynamics but the opposite effect on morphology. Gypsum morphology is nearly independent of probability p, which governs the reaction rate during the diffusion-controlled growth regime. The comparison of the simulation results with experiments in dilute conditions has validated the model. The latter can be used for parameter values that are not easily accessible from an experimental point of view. In particular, we intend to use the model to investigate the dynamical properties at higher densities. We also wish to devise an analytical, stochastic growth model, whose results will be compared to the simulations. Acknowledgment. V.N. acknowledges CNRS for a 3-month position in the laboratory LPTMC (Paris). References and Notes (1) Amathieu, L.; Boistelle, R. J. Cryst. Growth 1986, 79, 169–177. (2) Amathieu, L.; Boistelle, R. J. Cryst. Growth 1988, 88, 183–192. (3) De Vreugd, C. H.; Witkamp, G. J.; van Rosmalen, G. M. J. Cryst. Growth 1994, 144, 70–78. (4) Rinaudo, C.; Lanfranco, A. M.; Boistelle, R. J. Cryst. Growth 1996, 158, 316–321. (5) Bosbach, D.; Junta-Rosso, J. L.; Becker, U.; Hochella Jr., M. F. Geochim. Cosmochim. Acta 1996, 60, 3295–3304. (6) Badens, E.; Veesler, S.; Boistelle, R. J. Cryst. Growth 1999, 198/ 199, 704–709. (7) Boisvert, J.-P.; Domenech, M.; Foissy, A.; Persello, J.; Mutin, J.C. J. Cryst. Growth 2000, 220, 579–591. (8) Brandt, F.; Bosbach, D. J. Cryst. Growth 2001, 233, 837–845. (9) Rashad, M. M.; Mahmoud, M. H. H.; Ibrahim, I. A.; Abdel-Aal, E. A. J. Cryst. Growth 2004, 267, 372–379. (10) Hill, J.-R.; Plank, J. J. Comput. Chem. 2004, 25, 1438–1448. (11) Reynaud, P.; Saadaoui, M.; Meille, S.; Fantozzi, G. Mater. Sci. Eng., A 2006, 442, 500–503. (12) Rosmalen, G. M.; Daudey, P. J.; Marche´e, W. G. J. J. Cryst. Growth 1981, 52, 801–811. (13) Hand, R. J. Cem. Concr. Res. 1994, 24, 885–895. (14) Hernandez, A.; La Rocca, A.; Power, H.; Graupner, U.; Ziegenbalg, G. J. Cryst. Growth 2006, 295, 217–230. (15) Lemarchand, A.; Bonnet, J.-P. J. Eur. Ceram. Soc. 2006, 26, 2389– 2396. (16) Lemarchand, A.; Bonnet, J.-P. J. Phys. Chem. C 2007, 111, 10829– 10835. (17) Rinaudo, C.; Franchini-Angela, M.; Boistelle, R. J. Cryst. Growth 1988, 89, 257–266. (18) Lemarchand, A.; Ben Aim, R. I.; Nicolis, G. Chem. Phys. Lett. 1989, 162, 92–98. (19) Nowakowski, B.; Lemarchand, A. Physica A 2002, 311, 80–96. (20) Nowakowski, B.; Lemarchand, A. Phys. ReV. E 2003, 68, 031105. (21) Schlo¨gl, F. Z. Phys. 1972, 253, 147–161. (22) Antoine, C.; Lemarchand, A. J. Chem. Phys. 2007, 126, 104103. (23) Beretka, J.; van der Touw, J. W. J. Chem. Tech. Biotechnol. 1989, 44, 19–30. (24) Venables, J. A. Introduction to Surface and Thin Film Processes; Cambridge University Press: Cambridge, UK, 2000. (25) Herring, C. Phys. ReV. 1951, 82, 87–93.
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