Submicrometric Picture of Plaster Hydration: Optimization of the

Article pubs.acs.org/JPCC

Submicrometric Picture of Plaster Hydration: Optimization of the Addition of Gypsum Needles G. Morgado,†,‡ M. Collet,†,‡ Rémi Lespiat,§ Hélène Rétot,§ and Annie Lemarchand*,†,‡ †

Laboratoire de Physique Théorique de la Matière Condensée (LPTMC, UMR 7600), UPMC Université Paris 06, Sorbonne Universités, 4 place Jussieu, case courrier 121, 75252 Paris Cedex 05, France ‡ CNRS (LPTMC, UMR 7600), 3 rue Michel-Ange, 75016 Paris, France § Saint Gobain CREE, 550 Rue Alphonse Jauffret, 84300 Cavaillon, France ABSTRACT: Dissolution of plaster grains and precipitation of gypsum needles are described on a mesoscopic scale in view of the optimization of reaction kinetics when gypsum needles are used as additives. We update a kinetic Monte Carlo algorithm and introduce efficient geometrical tests to check the contacts between monocrystals. The simulations successfully reproduce the acceleration of the reaction in the presence of added gypsum needles. We develop an analytical model which neglects steric hindrance during needle growth and determine the optimum initial length of added needles leading to the shortest induction time. The analytical prediction remarkably agrees with the simulation results. The simulation tool is valuable to predict the dynamics at the end of the reaction, when the interactions between cumbersome needles prevail.

1. INTRODUCTION Despite the accumulation of empirical knowledge, optimizing the manufacture of plasterboard remains a challenge for the building industry. Plasterboard is an abusive term for gypsum board. The hydration of plaster, or calcium sulfate hemihydrate, leads to the precipitation of gypsum, or calcium sulfate dihydrate, according to the reaction: CaSO4 ,

1 3 H 2O + H 2O → CaSO4 , 2H 2O 2 2

modeling of plaster hydration on an intermediate, submicrometric scale and to examine how the tools of statistical physics and stochastic processes can help in understanding the behavior of a large number of interwoven growing needles of gypsum crystal.20,22,27,43 The effect of a well-known setting retarder, citric acid, has been predicted by our simulation model and confirmed by dynamic Raman spectroscopy.27 In this paper, we focus on the addition of micrometric gypsum needles, known to be a setting accelerator, for a water-to-plaster ratio consistent with the manufacture of plasterboards.16,18 The paper is organized as follows. The main steps of the simulation algorithm are given in section 2. Special attention is devoted to the updated procedure used to check the contacts between solids, valid for as small as desired confined gypsum needles and plaster grains. The reaction kinetics deduced from the simulations is studied in section 3 and compared to an analytical approach which neglects steric hindrance during needle growth. The results are discussed in view of the optimization of reaction kinetics when gypsum needles are used as additives. Section 4 contains the conclusion.

(1)

With the hope to play on reaction speed and to improve the mechanical properties of the resulting material at low cost, the addition of diverse substances to the initial mixture of water and plaster has been tested1−17 without leading to the emergence of a predictive model. New additives continue to be often randomly chosen, and the study of their effects requires new but repetitive experiments. Many experimental techniques are available to study the dynamics of the reaction.18 Electrical conductivity2,19,20 and acoustic emission13,21,22 can be used at high water-to-plaster ratios. Calorimetry23,24 and dynamic Raman spectroscopy25−27 can be used to follow the extent of the reaction even at low water-to-plaster ratios. In addition, the structure of the material can be characterized using nuclear magnetic resonance (NMR)28 and cryogenic transmission electron microscopy.29,30 Some relations between the morphology and the mechanical properties have been established.16,17,31−36 However, the interpretation of the results mainly focuses on two remote scales, either the macroscopic scale19,37−40 or the atomic scale.41,42 We recently proposed to address the © 2017 American Chemical Society

2. SIMULATION ALGORITHM The simulation is based on a kinetic Monte Carlo algorithm22 aiming at simulating stochastic trajectories in non equilibrium conditions. Kinetic Monte Carlo methods should not be confused with Monte Carlo simulations consisting in sampling Received: January 17, 2017 Revised: February 17, 2017 Published: February 21, 2017 5657

DOI: 10.1021/acs.jpcc.7b00482 J. Phys. Chem. C 2017, 121, 5657−5666

Article

The Journal of Physical Chemistry C

growth is limited by the impaired diffusion of the reactants in the liquid phase obstructed by already well-grown needles. Above the critical surface Sc, needle growth is assumed to occur with constant probability p. Actual needle growth is accepted if the faces are not blocked by steric hindrance, i.e., by the contact with other solids, such as plaster grains and other needles. In previous works,20,22,43 the contact between the solids, spheres and parallelepipeds, was defined using the discretization of the simulation box into cubic pixels of side length 1 μm. A pixel was considered occupied when it was intersecting a sphere or a parallelepiped. Hence, two solids were said to be in contact as soon as a pixel of one solid was adjacent to a pixel of the other solid. This definition had the drawback that the nonoverlapping constraint was preventing from putting several objects into the same pixel. In the present work, we wish to simulate the impact of many gypsum needles used as additives, including a very large number of close needles of length of the order of 0.15 μm that may be located in a same pixel of side length 1 μm. Rather than decreasing the size of the pixel, which would always introduce a cutoff, we implement a geometrical test to check the contact between a needle and another solid. Hence, the updated algorithm can be used for any initial grain size distribution and any additive size. Such a test would considerably increase the computation time if it would be necessary to check the contact of a needle with all the other solids of the simulation box. The efficiency of the algorithm is preserved by keeping the discretization of the simulation box to first identify the solids in the neighborhood of the needle of interest before applying them geometrical tests. A needle is traced by its eight vertices, a grain is characterized by its center and its diameter. The contact between a needle and a grain is assessed by computing the distance between each face of the needle and the center of the grain and checking if it is smaller than the radius of the grain. To this goal, each face of the needle is discretized using a square mesh of side length Δx. The contact between two needles N1 and N2 is checked as follows. Consider the needle N1 and a point Mf on the mesh of a given discretized face f. According to the numbering of the vertices of the needle N2 shown in Figure 1, we first determine if the point Mf of the needle N1 is inside the two planes associated with the opposite faces (A 1 ,A 2 ,A 3 ,A 4 ) and (A5,A6,A7,A8) of the needle N2 by computing the scalar products sf15 = A1Mf · A1A5 and sf51 = A5Mf·A1A5 and testing if the product sf15sf51 is negative. Then, the same procedure is repeated for the two other couples of opposite faces of the second needle. The face f of the needle N1 is said to be in contact with the needle N2 if the three products sf15sf51, sf12sf21 and sf14sf41 are found negative for at least one point Mf of the face f. In the absence of contact, a face is said to be free. The procedure is repeated for each face of the needle N1. If the six faces of the needle are free or if none of the pairs of opposite faces is blocked, purely homothetic growth is allowed and a same volume w2 Δl0 of gypsum is deposited on each free face, where Δl0 is constant. The rectangular faces receive a thin layer of thickness Δl0/α0 and the square faces, a layer of thickness Δl0, so that the aspect ratio is strictly conserved. Even in the case of blocked opposite faces, the growth of a free face is accepted if the aspect ratio α of the needle is in the allowed interval.5,19 Needles N1 of length smaller than the mesh size Δx are considered as points by the test. For larger values of Δx, the procedure is faster but less accurate. We choose a mesh size Δx = 2 μm to tolerate a certain overlapping between needles and

the phase space by minimizing a potential at equilibrium. In the case of plaster hydration described on a submicrometric scale, an initial configuration of plaster grains with germination sites and possibly gypsum needles in suspension is generated. Then, random processes of gypsum needle growth and plaster grain dissolution are sequentially performed. The main hypothesis on which the model relies is that fast plaster dissolution and homogenization of the liquid phase adiabatically follow the growth process of gypsum crystals. According to experimental variation of conductivity during plaster hydration, it is well admitted that dissolution of plaster is very fast: In less than a minute, the apparent solubility of hemihydrate is achieved in conditions for which the overall reaction takes more than 10 min.19,20 Hence, gypsum precipitation is supposed to be the slowest step, which accordingly imposes the speed of plaster hydration. The hemihydrate grains are modeled by spheres of variable diameter d, that are randomly spread in a cubic box of side L with periodic boundary conditions. The experimental grain size distribution is used to build a distribution of reference and to assign a physical length scale to the simulation unit of length. The grains are placed in decreasing order of diameters with a nonoverlapping constraint. Each plaster paste is characterized by a given water-to-plaster mass ratio, mw/mp, where mw and mp are the masses of water and plaster, respectively. We admit that nuclei preexist on the surface of the hemihydrate grains when the reaction of hydration begins. At the beginning of the simulation, these nuclei are randomly placed on the grains with a surface density g. The nuclei are supposed to already have the shape of embryonic gypsum needles. They are rectangular parallelepipeds of length l0, of the order of a few nanometers, and with a square base of side l0/α0, where α0 is the initial aspect ratio. Moreover, micrometric gypsum needles may be added to water at a given gypsum-to-plaster mass ratio mg/mp, where mg is the mass of gypsum needles initially introduced. The gypsum needles introduced in the bulk are parallelepipeds of length l0′ with the same aspect ratio α0 as the nuclei on the plaster grains. The direction of the needles are randomly chosen. The total number and direction of needles remain unchanged during simulation. At any time, the suspension is supposed to be supersaturated with respect to dihydrate.19 During the simulation time step Δt, the growth of each needle is considered. Local equilibrium, which would imply that, at each time step, a needle reaches the shape that minimizes its surface energy, is not strictly obeyed. A needle is supposed to keep a parallelepipedic shape but with a variable aspect ratio α = l/w in a given interval, where l and w are needle length and width, respectively. The kinetic Monte Carlo procedure requires the definition of the rates of particle growth and shrinking. Two different needle growth regimes are considered, first, associated with a surfacedependent growth probability and, then, a constant growth probability. An initial surface-dependent growth is accepted with probability pS/Sc, until needle total surface S reaches a critical value Sc. This rule ensures the initial autocatalytic growth of the needles, necessary to reproduce the induction time observed during plaster hydration: Initially small germs grow slowly but faster and faster as their surface increases. Such a rule would lead to explosive, unbounded growth probabilities. However, as growth probability increases, needle growth becomes controlled by the arrival time of the calcium and sulfate ions in the neighborhood of the needle. The critical surface Sc is empirically introduced as the surface above which 5658


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needles and their orientation are fixed. Needles and grains are immobile. After a succession of growth and dissolution steps, the simulation ends either when the total amount of hemihydrate has dissolved or when all the needles are blocked by steric hindrance. Table 1 lists the parameters introduced in Table 1. Parameters Characterizing the Plaster Grains and Values Used in the Simulationsa plaster grains ρp = 2.32 × 103 kg·m−3 Mp = 0.145 kg

density molar mass initially

mw/mp = 0.8 D0= 10 ± 5 μm

water-to-plaster mass ratio grain diameter at time t

D N

grain diameter number of grains during Δt radius decrease

Figure 1. Illustration of the procedure followed to determine if the face f of the needle N1 is in contact with needle N2. For any point Mf on the mesh of the face f, the scalar products A1Mf·A1A5 = A1Mf15 × A1A5 and A5Mf·A1A5 = A5Mf15 × A1A5 have the same sign: we conclude that the face f of the needle N1 is outside the region limited by the two planes (A1, A2, A3, A4) and (A5, A6, A7, A8) associated with the corresponding opposite faces of the needle N2.

a

the model and gives the values used in the simulations for the plaster grains and Table 2 gives the corresponding information for the two types of needles, placed on plaster grains or in suspension.

reproduce the experimentally observed mortise-and-tenon joints between large gypsum needles.44,45 On the simulation time scale, the growth process is irreversible and the needles never shrink. After needle growth, grain shrinking is systematically performed. During time t and t + Δt, the increase ΔVg(t) = Vg(t + Δt) − Vg(t) of the total volume of gypsum or dihydrate is equal to the sum of the volumes precipitated during Δt on the needles for which growth has been accepted. Consequently, the number of moles

Table 2. Parameters Characterizing Gypsum Needles on Plaster Grains and in Suspension with the Values Used in the Simulationsa gypsum needles on plaster grains density molar mass

ρg

of dihydrate that precipitated is equal to ng (t ) = ΔVg(t ) M

g

where ρg = 2.32 × 103 kg·m−3 is the density of dihydrate and Mg = 0.1722 kg is the molar mass of dihydrate. A same number of moles of plaster or hemihydrate has to be dissolved. Consequently, the total volume of plaster or hemihydrate to be dissolved is ρg M p ΔVp(t ) = ΔVg(t ) ρp Mg (2)

length aspect ratio critical surface number length width aspect ratio surface growth probability

where ρp = 2.63 × 103 kg·m−3 and Mp = 0.1452 kg are the density and molar mass of hemihydrate, respectively. We assume that all the hemihydrate grains loose an external shell of the same width ΔR, so that

volume increase per free face

N

ΔVp(t ) = ΔRπ ∑ Dk 2 k=1

ΔR

The side of the simulation box is fixed at L = 23 μm.

a

(3)

where Dk is the diameter of the hemihydrate grain k and where the summation is performed over all the N grains existing at time t. To summarize, the algorithm consists of a sequence of two steps: First, growth of each needle is considered with the appropriate probability and performed if steric hindrance and aspect ratio conservation permit it. Second, shrinking of each grain is completed, according to conservation of matter and the hypothesis of supersaturation of the suspension. The number of

ρg = 2.63 × 103 kg·m−3 Mg = 0.172 kg initially surface density: g = 3.5 μm−2 l0 = 0.1 μm α0 = 12 Sc = 0.2 μm2 n at time t l w α = l/w = 12 ± 7 S pS/Sc if S < Sc p = 0.1 if S ≥ Sc during Δt w2Δl0, Δl0 = 0.03 μm

gypsum needles in the water ρg = 2.63 × 103 kg·m−3 Mg = 0.172 kg gypsum-to-plaster mass ratio: mg/mp = 0.0005 variable l′0 α0′ = α0 S′c = Sc n′ l′ w′ α′ = l′/w′ = 12 ± 7 S′ pS′/Sc if S′ < Sc p = 0.1 if S′ ≥ Sc w′2Δl0′ , Δl0′ = Δl0

The side of the simulation box is fixed at L = 23 μm.

Parts a, c, and e of Figure 2 give initial configurations of plaster grains, and parts b, d, and f of Figure 2 give the corresponding final configurations of gypsum needles generated by the simulation algorithm. The final configuration (Figure 2b) obtained without additives shows a typical urchin-like structure,30 due to the favorable growth of the radial needles which encounter less needles of the same grain than tangent needles. A similar final configuration (Figure 2d) is obtained for 5659


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Figure 3 is a zoom in on the configuration given in Figure 2d which illustrates the ability of the simulation to reproduce mortise-and-tenon joints between large gypsum needles.

Figure 3. Example of mortise-and-tenon joint: Zoom-in on the final configuration of gypsum needles generated by the simulation algorithm in Figure 2d for an initial gypsum-to-plaster mass ratio mg/mp = 0.0005 and 40 added gypsum needles of initial length l0′ = 2 μm.

3. RESULTS Adding gypsum germs to the powder of plaster is a well-known trick for accelerating the hydration reaction.28,46 Our aim is to examine if the simulations are able to reproduce this phenomenon. 3.1. Simulations of Plaster Hydration Dynamics with Added Gypsum Germs. The simulation results give access to reaction kinetics and, in particular, to the evolution of the extent of the reaction over time. The extent of the reaction, ξ, with 0 ≤ ξ ≤ 1, is defined as the ratio of the volume of gypsum that has precipitated at the considered time tξ and the final volume of gypsum,

Mg ρp

M pρg

Vp(t0), obtained when the reaction is

complete: ξ=

Figure 2. Initial (a, c, e) and final (b, d, f) configurations generated using the simulation algorithm for the parameter values given in Tables 1 and 2. The simulation box of side length L = 23 μm initially contains one plaster grain of diameter D = 5 μm, four grains of diameter D = 10 μm, and one grain of diameter D = 15 μm. (a, b) Without added gypsum in suspension, (c, d) with 40 added gypsum needles of initial length l0′ = 2 μm, (e, f) with 1483 added gypsum needles of initial length l0′ = 0.6 μm, so that the initial gypsum-toplaster mass ratio is kept constant and equal to mg/mp = 0.0005 in parts c−f. See Tables 1 and 2 for the values of the other parameters.

Vg(tξ) − Vg(t0) M pρg Vp(t0)

Mg ρp

(4)

where we take into account the initial volume Vg(t0) of the gypsum needles on the plaster grains or in suspension, that are present before the beginning of the reaction. Figure 4 gives the evolution of the extent of the reaction ξ deduced from the simulations without additives and for three values, l′0 = 0.14, 0.6, 2 μm of the initial length l′0 of the added gypsum needles and a constant gypsum-to-plaster mass ratio mg/mp. Decreasing the initial length l0′ exponentially increases the number of needles. The configurations obtained by the simulations are given in Figure 2 for added gypsum needles of initial length l′0 = 0.6 μm and l′0 = 2 μm. We have omitted the very messy, final configuration obtained for more than 105 added needles of initial length l0′ = 0.14 μm. As shown in Figure 4, the extents of the reaction similarly behave without added needles and for only 40 added needles of initial length l′0=2 μm. Adding gypsum germs has a double effect on dynamics, it reduces the induction time at the beginning of the reaction and it also shortens the end of the reaction. In agreement with experimental observations,28,46 the simulation results show that adding gypsum germs in suspension speeds up the reaction of plaster hydration. For l0′ = 0.14 μm, gypsum is distributed over a

added needles of initial length l′0 = 2 μm. Indeed, at constant gypsum-to-plaster mass ratio, mg/mp, only a few initially long needles are added. The final configuration (Figure 2f) obtained for added needles of initial length l′0 = 0.6 μm looks very different: The simulation box is more homogeneously filled but, in average, the needles in suspension reach a longer final length than the needles that were initially on the grains and started with l0 = 0.1 μm. Because of the early autocatalytic growth, initially longer needles grow faster and have the tendency to drain a larger part of the available matter. 5660


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Figure 4. Extent of the reaction ξ versus time t/Δt deduced from the simulations without added gypsum in suspension (solid line), with added gypsum needles of initial length l′0 = 2 μm (long-dashed line), l′0 = 0.6 μm (short-dashed line), and l0′ = 0.14 μm (dotted line). The values of the other parameters are given in Tables 1 and 2

Figure 5. Top: Total volume of gypsum Vg (black solid line), volume of gypsum Vg′ (red dashed line) that precipitates on added germs, and (blue dotted line) that precipitates on the volume of gypsum Vgrains g nuclei on the surface of the plaster grains versus time t/Δt. Bottom: Mean needle length ⟨ltot⟩ (black solid line), mean length ⟨l′⟩ (red dashed line) of the initially added seeds, and mean length ⟨l⟩ of the needles initially located on grain surface versus time t/Δt. The results are deduced from simulations with 1483 gypsum needles of initial length l′0 = 0.6 μm initially added to the suspension and 7220 needles of initial length l0 = 0.1 μm on the surface of the plaster grains. The values of the other parameters are given in Tables 1 and 2

very large number of needles, which are consequently very small and do not interfere with each other: As shown in Figure 4, the reaction remains fast until the very end and abruptly stops. Without or with a few added needles such as for l′0 = 0.6 μm, the total amount of gypsum precipitates on a small number of needles. The resulting needles are thus expected to be long and cumbersome. The evolution of mean needle lengths and the volumes of gypsum that precipitate on the nuclei on the surface of the plaster grains and on added seeds are given in Figure 5. During the evolution of the different volumes, the inflection point is sooner reached and the induction time is accordingly smaller for the volume precipitated on the added germs with initial length l′0 = 0.6 μm than for the volume precipitated on the nuclei on the plaster grains, with initial length l0 = 0.1 μm. Clearly, the initially longer added germs rapidly reach the critical length lc and enter into a linear regime of length increase. However, their final growth is slowed down, due to their bigger length which induces a larger number of contacts and blocking by steric hindrance. The properties of the end of the reaction shown in Figures 4 and 5 are thus related to the interaction between monocrystals and to the influence of steric hindrance. This result highlights the interest in studying dissolution−precipitation on a mesoscopic scale in order to capture the macroscopic dynamics of the phenomenon. The decrease of the induction time when gypsum germs are added and l0′ is decreased is related to the increase of the surface on which gypsum may precipitate. Hence, the ability of the simulations to reproduce the acceleration of the beginning of the reaction confirms the main hypotheses of the model. Gypsum precipitation imposes the global reaction speed and the increase of gypsum crystal surface on which further precipitation can occur speeds up the reaction. This observation is in line with the slowing down of the reaction in the presence of citric acid which leads to the formation of more compact needles.13,18,27 Needles with a smaller aspect ratio α have a smaller surface and, consequently, a smaller growth probability, which leads to a slower reaction. The initial length l0′ of the needles used as additives can be considered as a control parameter that the experimentalist can

choose to optimize the dynamics of the reaction. Intuitively, we expect a non trivial behavior as the initial needle length varies for a constant quantity of gypsum introduced, i.e., a constant initial gypsum-to-plaster mass ratio mg/mp. For freely growing needles in bulk, the reaction rate is the product of two terms: the growth probability and the volume of gypsum that precipitates during the time step. The volume of gypsum deposited per needle and time step varies as (l0′ )2 and the number of needles as 1/(l0′ )3 at constant mg/mp, so that the total volume of gypsum precipitated per unit time scales as 1/l′0. Two different regimes are obtained depending on whether the initial length l0′ is smaller or larger than the critical length lc. For l0′ < lc, the early growth probability is pS/Sc ≃ (l0′ )2, whereas, for l′0 ≥ lc the growth probability p is constant and independent of l′0. For added needles initially smaller than the critical length lc ≃ 0.76 μm, increasing the length l0′ at constant mg/mp has two opposite effects on the reaction rate: the growth probability increases but the volume of precipitated gypsum decreases. These two competing effects may result in a maximum for the reaction rate. In order to investigate this issue, we consider two characteristic times, the induction time before the sudden increase of the extent of the reaction and the time for completion of the reaction. Defining tξ as the time at which the extent of the reaction reaches the value ξ, we evaluate the induction time by t0.1 for which ξ = 0.1. In this framework, the time for completion is denoted by t1. Figures 6 and 7 give the variation of the induction time t0.1 and the time for completion t1 for different values of the initial length l0′ of added gypsum germs, respectively. For each value of l′0, five simulations are performed for different seeds of the random number generator. The error bars indicate the resulting standard deviation. The two characteristic times t0.1 and t1 attain a minimum value, 5661


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derive analytical relationships between the time tξ at which the extent of the reaction reaches the value ξ and the initial length l0′ of the needles in suspension. The grains are assumed to have the same initial diameter D0 and the needles are supposed to grow freely, i.e., without steric hindrance, so that they keep the same aspect ratio α0. In these conditions, the length l of a needle on a plaster grain and the length l′ of a needle in the suspension obey the following differential equations: pS Δl0 dl =2 , for S < Sc dt Sc Δt =2p

Δl0 , for S ≥ Sc Δt

pS′ Δl0 dl′ =2 , for S′ < Sc dt Sc Δt Figure 6. Induction time: Time t0.1/Δt necessary for the extent of the reaction to reach ξ = 0.1 versus initial length l0′ of added gypsum needles. Key: solid blue line, analytical results; dashed red line, approximate analytical results obtained by neglecting the germs on the plaster grains; green semidisk, analytical results obtained by considering only germs on the plaster grains; solid squares, simulations results for the parameter values given in Tables 1 and 2. The error bars corresponds to ±σ, where σ is the standard deviation.

=2p

Δl0 , for S′ ≥ Sc Δt

(5)

(6)

(7)

(8)

which takes into account that both needle types have a given probability to increase their length by 2Δl0 during the time step Δt, the same critical surface S c and critical length lc = α0

Sc 2(1 + 2α0)

at which the process switches from an

autocatalytic regime to a growth at constant rate. Introducing the parameters

J=

2p Δl0 lc2 Δt

(9)

Δl0 Δt

(10)

K = 2p

we rewrite eqs 5-8) as follows:

Figure 7. Time for completion of the reaction: time t1/Δt necessary for the extent of the reaction to reach ξ = 1 versus initial length l′0 of added gypsum needles. See the caption of Figure 6.

dl(t ) = Jl(t )2 , for l0 ≤ l < lc dt

(11)

= K , for lc ≤ l ≤ l1

(12)

dl′(t ) = Jl′(t )2 , for l′0 ≤ l′ < lc dt

(13)

= K , for lc ≤ l′ ≤ l′1

(14)

where l0 and l1 are the initial length and the final length of a needle on a plaster grain, whereas l′0 and l′1 are the initial length and the final length of a needle in the suspension, respectively. The number of needles on plaster grains per unit volume of plaster initially considered

confirming the existence of an optimal length l′0,min associated with the fastest reaction. Different asymptotic values are reached for vanishing l0′ and large l0′ . At constant gypsum-toplaster mass ratio mg/mp, the limit of large l0′ corresponds to a negligible number of added needles with respect to the preexisting germs on the plaster grains. In these conditions, the asymptotic value of tξ at large l0′ converges toward the time t0ξ obtained without added needles and represented at l0′ = 0 for the sake of clarity in Figures 6 and 7 for ξ = 0.1 and ξ = 1, respectively. On the contrary, in the limit of vanishing l′0, the number of added needles diverges as well as the induction time and, consequently, the time for completion. In the next subsection, we examine how our model can be used to analytically predict the effects of added gypsum needles. 3.2. Analytical Expressions for Induction Time and Time for Completion. We use the following hypotheses to

6g n = Vp(t0) D0

(15)

is fixed by the surface density g of needles and the diameter D0 of the grains. The number of needles in the water per unit volume of plaster obeys

(α0)2 mg ρp n′ = Vp(t0) (l′0 )3 mpρg

(16)

The volume of gypsum Vg(tξ) and the initial volume of plaster, Vp(t0), obey a conservation relation leading to 5662


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Table 3. Analytical Relationships between the Initial Length l0′ of the Needles Added to the Water and the Time tξ Necessary for the Extent of the Reaction to Reach the Value ξa l0′ versus tξ

⎛ l′0 = ⎜1 − ⎝

with A(tξ) = b −

⎛ l′0 = ⎜1 − ⎝

domain of validity

⎞ ⎟/Jt (1 + aξA(tξ))1/3 ⎠ ξ 1

l′0 < lc, l(tξ) < lc, l′(tξ) < lc

cl03 ξ((1 − Jl0 tξ)3 − 1)

⎞ ⎟/Jtξ

1

l0′ < lc, l(tξ) ≥ lc, l′(tξ) < lc

(aξB(tξ))1/3 ⎠

⎛ ⎛ c with B(tξ) = ξ ⎜l03 − ⎜2lc − ⎝ ⎝

l′0 = (2lc + Ktξ ±

lc 2 l0

⎞3⎞ + Ktξ⎟ ⎟ + ⎠⎠

1 aξ

+b l0′ < lc, l(tξ) < lc, l′(tξ) ≥ lc

ΔC )/2C(tξ)1/3

⎛ with C(tξ) = 1 + abξ + ac ⎜1 − ⎝

⎞ ⎟ (1 − Jl0 tξ)3 ⎠ 1

and ΔC = (2lc + Ktξ)2 − 4C(tξ)1/3lc2

l′0 = (2lc + Ktξ ±

l0′ < lc, l(tξ) ≥ lc, l′(tξ) ≥ lc

ΔE )/2E(tξ)1/3

⎛ ⎛ with E(tξ) = 1 + abξ + ac ⎜l03 − ⎜2lc − ⎝ ⎝

2

lc l0

3⎞

⎞ + Ktξ⎟ ⎟ ⎠⎠

and ΔE = (2lc + Ktξ)2 − 4E(tξ)1/3lc2 l0′ = Ktξ/(C(tξ)1/3 − 1) l′0 = Ktξ/(F(tξ)1/3 − 1) with F(tξ) = 1 + abξ + ac(l03 − (lc + K(tξ − tc))3) and tc = a

(

1 l0

−

1 lc

)/J

We have introduced the parameters a =

1 ⎛ l′(tξ) ⎞ ⎟ = cl(tξ) + ⎜ Vp(t0) a ⎝ l′0 ⎠ Vg(tξ)

mpρg mg ρp

Mg ρp Mpρg

, and c =

6g D0α0 2

.

3

1 + bξ a

where we have introduced the notations mpρg a= mg ρp

c=

,b=

(blue curve) and the simulation results (solid squares) is excellent in Figure 6 for the induction time t0.1. Indeed, for a small extent of the reaction such as ξ = 0.1, the needles are still small compared to the size of the simulation box and steric hindrance is negligible. In particular, the good agreement obtained for small values of l0′ validates the updated algorithm used to test the contacts, specifically between small solids. A worse agreement between the analytical predictions and the simulation results is obtained in Figure 7 in the case of the time for completion, t1. We observe that the simulation results deviate more and more from the analytical prediction as the initial length l′0 of the added needles increases. At the end of the reaction, the effects of steric hindrance are more important, in particular when matter is distributed over a few, consequently cumbersome, needles. Eqs 11 and 12) can be solved in the simpler case where no gypsum needles are added in bulk and germs on plaster grains obey l0 < lc. We find

3

= cl0 3 +

b=

l0′ ≥ lc, l(tξ) < lc, l′(tξ) ≥ lc l′0 ≥ lc, l(tξ) ≥ lc, l′(tξ) ≥ lc

(17)

(18)

Mg ρp M pρg

(19)

6g D0α0 2

(20)

Different solutions l(tξ) and l′(tξ) are obtained when solving eqs 11-14,17) depending on the initial values of the lengths l0 and l0′ with respect to the critical length lc and on whether l(tξ) and l′(tξ) have reached lc or not. The initial length l0 of the embryonic needles on the plaster grains is always fixed at a value smaller than lc. We look for the dependence of the time tξ, at which the extent of the reaction reaches the value ξ, on the initial length l0′ of the needles added to the suspension. It is easier to obtain the analytical expression of l′0 versus tξ and Table 3 gives the results for all the different cases. We apply the results given in Table 3 to two values of the extent of the reaction, ξ = 0.1 and ξ = 1, in order to compare the analytical predictions and the simulation results for these two typical times. The analytical results are obtained in the limit of free growth whereas the simulations take into account steric hindrance. The agreement between the analytical predictions

tξ0

=

l(tξ0) − lc K

with tc =

(

1 l0

−

1 lc

+ tc , for l(tξ0) ≥ lc

(21)

)/J . The conservation relation is deduced

from eq 17 in the limit a → ∞, where the parameter a is given in eq 18: cl(tξ0)3 = cl0 3 + ξb

(22)

It reads: tξ0 = 5663

(l03 + ξb/c)1/3 − lc + tc K

(23) DOI: 10.1021/acs.jpcc.7b00482 J. Phys. Chem. C 2017, 121, 5657−5666

Article

The Journal of Physical Chemistry C which is valid for ξ = 0.1 and ξ = 1 for the parameter values given in Tables 1, 2. The analytical prediction for the asymptotic behavior reached in the absence of added gypsum needles is represented by a green semidisk at l0′ = 0. Once again, the agreement with the simulations without added gypsum needles is much better in the case of the induction time t0.1 than in the case of the time for completion. 3.3. Optimization of Reaction Dynamics According to the Size of Added Gypsum Germs. In order to guide the choice of experimental conditions, an analytical expression of the length of added gypsum needles associated with the fastest reaction could be valuable. In the limit of small l0′ , the number of added needles diverges and the germs on the plaster grains may be neglected. Table 3 provides an approximate analytical approach to this limit by imposing c = 0 on the parameter c defined in eq 20. Choosing c = 0 amounts to imposing g = 0, i.e., no germs on the surface of the plaster grains. In Figure 7, the relatively good agreement between the approximate analytical approach (red dashed curve) and the simulation results (squares) for large values of l0′ is fortuitous. More relevant is the agreement between the approximate analytical approach and the simulations for both the induction time (Figure 6) and the reaction time (Figure 7) in the neighborhood of the minimum. Consequently, the approximation can be used to derive an analytical expression for the initial length l′0 of added needles associated with the minimum of the curve. Following Table 3 with c = 0 and for l0′ < lc and l′(tξ) > lc, which is the case for ξ = 0.1 and a fortiori for ξ = 1, we have

( t = ξ

1 l ′0

− J

1 lc

) + (l′ (1 + abξ)

1/3

0

then be used to assign a physical value to the critical length lc. The analytical calculations have been performed assuming that the plaster grains have the same initial diameter D0. The results could be improved to take into account the actual grain size distribution used in the experiments. For example, in the case of a Gaussian grain size distribution of mean diameter D0 and standard deviation σ0, only the parameter c defined eq 20 should be replaced by cG =

4. CONCLUSION In this paper, we present an updated simulation model of plaster hydration and gypsum crystallization.22,27 On the chosen length scale, the simulation reproduces the growth of a large number of interacting gypsum needles, for which statistical physics and stochastic processes offer well-adapted modeling tools. Ignoring the very beginning and the very end of the reaction, we assume that precipitation of gypsum on needles imposes the overall reaction speed and that partial dissolution of plaster grains rapidly adapts to maintain supersaturated conditions. The initial needle growth probability is assumed to be proportional to needle surface. Growth may be forbidden due to nonoverlapping constraints. The procedure followed to check the contacts between solids differs from the one used in our previous works22,27 and is adapted to the simulation of a large number of solids confined in a box. The model is used to describe the effect of adding ground gypsum mineral, known as an accelerator of the reaction. The simulation results satisfactorily reproduce the increase of reaction speed when gypsum needles are added to the initial mixture. The previous comparison between dynamic Raman spectroscopy and simulation results already confirmed the ability of the model to account for the effects of a retarder such as citric acid, known to produce more compact needles.27 It is worth noting that the effects of both an accelerator and a retarder have been predicted by the same model without ad hoc introduction of new hypotheses and fitting parameters. The simulation results confirm the main hypothesis: At the beginning of the reaction, the reaction speed is controlled by the total surface on which gypsum may precipitate, the number of existing germs of gypsum and their aspect ratio. Assuming free growth and neglecting the germs on the plaster grains, we derive an analytical expression for the optimal initial length of added gypsum needles leading to the fastest reaction for a constant quantity of added gypsum. This prediction can be useful to easily select favorable experimental conditions. The agreement between the analytical approach which neglects steric hindrance and the simulation results is better for the induction time than for the time for completion. When the induction time is reached, the gypsum needles are still small whereas long, cumbersome needles interact at the end of the reaction. The discrepancies between the approximate analytical approach and the simulation results point out the failure of a naive interpretation which would assume that the extent of the reaction evolves as if a single monocrystal of gypsum was growing. The results underline that accounting for steric hindrance makes the simulations so significant and suggest the ability of the model to reproduce the mesoscopic structure of the final material. As a perspective, we wish to use the model to predict the mechanical properties of plasterboard. We conclude that all the arguments involving a modification of dissolution to explain the effect of an additive should be

(24)

The time tξ appears as the sum of two functions of l0′ . The first term, associated with the early dynamics given in eq 13, decreases as l′0 increases and the second term, related to eq 14 when the length of the growing needle has overcome lc, increases as l0′ increases. The competition between the two terms leads to the existence of an extremum value. Deriving eq 24 with respect to l′0, we find that tξ reaches a minimum value for: l′0,min = lc(1 + abξ)−1/6

where eqs 9,10) have been used to obtain

(25)

l2c =K/J. Following mpMg

18,19), the product ab is simply given by ab =

mg M p

c , all things being equal. In

particular, eq 25 for the minimum would be unchanged.

− lc)

K

σ0 2 + D0 2 3σ0 2 + D0 2

eqs

. According

to eq 25, the length of added needles associated with the minimum of the induction time only depends on the quantity of added needles through mg/mp and on the critical length lc, which marks the transition between the autocatalytic growth and the growth at constant rate. For the parameter values given in Tables 1, 2 and for ξ = 0.1, we find l0,min ′ =0.305 μm, which remarkably agrees with the simulation results given in Figure 6 and is, as expected, smaller than the critical length lc ≃ 0.76 μm. It is worth noting that the existence of a minimum for the induction time as a function of the initial length of added gypsum germs is a consequence of the main hypothesis of the model, i.e. the autocatalytic character of early needle growth. We suggest that a series of experiments for variable l′0 values could be performed to check the model. The smallest induction time observed, if any, could be determined and the simple expression given in eq 25 could 5664


Article

The Journal of Physical Chemistry C handled with care.18,24,34 Invoking a modification of precipitation, the limiting process, is certainly more relevant. In particular, all the processes which affect the number of nucleation sites, the available surface on which gypsum may precipitate, and the obstacle to needle growth due to steric hindrance should not be underestimated. It is crucial to describe the evolution of a large number of grains and needles to understand plaster hydration. A simulation model which correlates macroscopic dynamics and the submicrometric structure of the material offers rational arguments to guide the experimentalist in the choice of an additive, in view of the optimization of a given property.

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AUTHOR INFORMATION

Corresponding Author

*(A.L.) E-mail: [email protected]. ORCID

Annie Lemarchand: 0000-0001-7122-8043 Notes

The authors declare no competing financial interest.

■

REFERENCES

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Submicrometric Picture of Plaster Hydration: Optimization of the

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