Molecular Modeling on Supersaturation-Dependent Growth Habit of 1

Article pubs.acs.org/crystal

Molecular Modeling on Supersaturation-Dependent Growth Habit of 1,1-Diamino-2,2-dinitroethylene Hong-Min Shim,† Hyoun-Soo Kim,‡ and Kee-Kahb Koo*,† †

Department of Chemical and Biomolecular Engineering, Sogang University, Seoul 121-742, Korea Agency for Defense Development, Daejeon 305-600, Korea

‡

S Supporting Information *

ABSTRACT: In the cooling crystallization of 1,1-diamino-2,2dinitroethylene (DADNE), it was found that the aspect ratio of crystals decreases as the cooling rate of the solution increases. To reveal the effect of supersaturation on the growth shape of DADNE, molecular modeling was carried out by the step energy calculation and kinetic Monte Carlo (KMC) simulation. The rodlike shape of DADNE with basal {111}̅ faces was accurately predicted by the step energy calculation, resulting in remarkable agreement with the experiments. The reason behind the slowest growth rate of the {111̅} faces was found to originate from the high energy barrier in the formation of a 2D nucleus on the crystal face. Furthermore, it was shown that the aspect ratio of DADNE decreases by the lowered free energy of 2D nucleation at high supersaturation, in which the distinctive characteristics on the anisotropic growth behavior of DADNE are blurred. The KMC simulation results also provided an understanding of the growth kinetics of growth units on each crystal face: the {111̅} face shows a lower sticking fraction, which means that the {111̅} face offers the surface topology where growth units are difficult to incorporate into the lattice sites. However, as the supersaturation increases, the crystal faces start to be strongly roughened, and the aspect ratio becomes reduced.

1. INTRODUCTION

In the present work, the rodlike DADNE crystals were obtained by cooling crystallization. The growth shape of DADNE was found to be strongly affected by cooling rate, which is a significant factor that determines the supersaturation ratio. However, there is still an absence of mechanistic studies on the crystal growth of DADNE. To control the growth shape of DADNE, the mechanistic approaches such as spiral growth model and 2D nucleation model are required. Therefore, in this article, we aim at simulating the growth shape of DADNE by the mechanistic model; thus, the effect of supersaturation on DADNE will be revealed. Until recently, Doherty’s group has been paving the way in the growth shape prediction of organic materials, where the crystal shapes were accurately predicted by the spiral growth model and 2D nucleation model.16−21 The mechanistic approach developed by Lovette and Doherty allows for predicting the supersaturation-dependent growth shape and proposes the method for estimating critically sized nuclei bounded by the periodic bond chains (PBCs) that are very important when calculating the free-energy barrier of 2D nucleation.20 Meekes’ group extensively studied the supersaturation-dependent growth habit of organic materials such as β-triacylglycerol, paracetamol,

For the last two decades, 1,1-diamino-2,2-dinitroethylene (DADNE) has been referred to as a promising high-energy material available for replacing hexahydro-1,3,5-trinitro-1,3,5triazine (RDX) because of its quite low sensitivity. At the atomic level, the DADNE molecules are packed by 2D wave-shaped layers where the π-stacked interactions with hydrogen bonds extensively exist. This leads to a buffer against external mechanical stimuli.1 A great deal of effort into DADNE has been devoted to the fields of synthesis,2,3 polymorphism,4,5 thermal analysis,6,7 and crystallization8−14 until recent years. In particular, crystallization has attracted a lot of attention because it determines the crystal size distribution and crystal shape that have a great impact on the performance and shock sensitivity of energetic materials15 as well as the efficiency of filtering and washing during downstream processes. Lochert disclosed the hexagonal shape of DADNE crystals obtained by recrystallization from HCl.10 Trzciński et al. reported the effect of cooling rate on the characteristics of DADNE crystals.11 Moreover, Daniel et al. produced DADNE crystals with the sharp block or plate in the presence of ultrasounds,12 and Mandal et al. proposed the methodology for spherical DADNE by using a micellar nanoreactor.13 Recently, Gao et al. prepared nanosized cubic DADNE via an ultrasonic-spray-assisted electrostatic adsorption method.14 © XXXX American Chemical Society

Received: December 25, 2014 Revised: February 6, 2015

A

DOI: 10.1021/cg5018714 Cryst. Growth Des. XXXX, XXX, XXX−XXX

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Crystal Growth & Design aspartame, naphthalene, and β-phthalocyanines by the connected net analysis and kinetic Monte Carlo (KMC) simulations.22−26 Accordingly, those mechanistic approaches facilitate a rather accurate prediction of crystal shape and offer a good understanding of crystal growth as a function of supersaturation ratio. In this article, the procedure of growth shape prediction of DADNE is addressed in detail. First, the system amenable to description on the intermolecular interactions between growth units is determined by comparing the experimental sublimation enthalpy with the calculated lattice energy. All of the bond energies are scaled down based on the dissolution enthalpy because DADNE is crystallized from a dimethylacetamide (DMA)/water solution. Second, the methods of step energy calculation and KMC simulations are specifically tackled. Lastly, the predicted morphologies are compared with the experimental data.

Table 1. Crystal Graph of DADNE

2. COMPUTATIONAL METHODS Figure 1a illustrates the crystal structure of DADNE with space group P21/n, Z = 4, a = 6.9396 Å, b = 6.6374 Å, c = 11.3406 Å, and

3. STEP ENERGY CALCULATION The step energy is defined as the released energy or required work when a 2D nucleus is generated on top of the surface. When 2D nucleation occurs, the step up and step down are simultaneously formed; hence, two terms cannot be tackled separately. Therefore, the step energy in one direction was determined as half the energy cost between the flat surface and the surface with a nucleus formed.33 According to the methodology proposed by Deji et al.,34 the specific step energy, εst,i, is defined as Est, i εst, i = lst, i (2)

label

bond direction

number of bonds

length (Å)

Ebond (kcal/mol)

Escaled bond (kcal/mol)

a b c d e f g

[01̅0] [110̅ ] [010̅ ] [000] [100] [11̅ ̅0] [001]

2 1 1 2 1 2 2

4.20 4.90 5.67 6.70 5.99 8.51 6.84

−2.51 −3.39 −3.23 −3.37 −2.50 −0.62 −0.85

−0.65 −0.88 −0.84 −0.87 −0.65 −0.16 −0.22

The F (flat) faces of DADNE that contain two or more PBCs in one layer are shown in Figure 2.

where Est,i is the net formation energy of a single step along edge i and lst,i is length of edge i along the step. To estimate the 2D island shape of a nucleus expected to appear, the relationship of the perpendicular distance from a center within the crystal to face i, H̅ c,i, and edge energy, γedge , derived by Lovette and Doherty was i employed, represented as20 Figure 1. (a) Crystal structure and (b) crystal graph of DADNE. Different colored lines represent different bonds (pink, a; blue, b; mint, ; green, d; purple, e; orange, f; and red, g).

H̅c, i =

ΔHdiss = E bond |E latt|

(3)

where vm is the molecular volume, σ is the supersaturation defined by σ = (XA/X0A) − 1, XA is the concentration of solute molecules in solution, X0A is the equilibrium concentration, k is the Boltzmann’s constant, and T is the absolute temperature. γedge is determined by the relationship γedge = εst,i/dhkl, where dhkl i i is the interplanar distance of the (hkl) face. The critical length of i-th edge can be represented as

β = 90.611°. In the present work, partial atomic charges were obtained by VAMP (Materials Studio, version 7.0) with the MNDO/C semiempirical method.28 Dispersive energy was calculated by using the COMPASS force field.29 Through the geometrical optimization of a unit cell (a = 6.6952 Å, b = 6.5549 Å, c = 11.6263 Å, and β = 91.366°), the calculated lattice energy was −23.8 kcal/mol, which agrees well with an experimental sublimation enthalpy of 25.9 kcal/mol.30 All of the bond energies between DADNE molecules were calculated by MORPHOLOGY (Materials Studio, version 7.0) and visualized by the crystal graph as shown in Figure 1b. Assuming equivalent wetting, all values are rescaled by the dissolution enthalpy in DMA/water solution with a mass ratio of 70:30 by eq 1.31 The dissolution enthalpy was estimated as 6.2 kcal/mol from our solubility data (Figure S1) by using the ideal solution theory.32 Table 1 summarizes the bond energies of DADNE, where the lattice energy is defined as the summation over all bonds (Elatt = 2ϕa + ϕb + ϕc + 2ϕd + ϕe + 2ϕf + 2ϕg), and the scaled value is represented as 27

scaled E bond

edge νm ⎛ γi ⎞ ⎟ ⎜⎜ σ ⎝ kT ⎠⎟

lc, i =

H̅c, i + 1 − H̅c, i cos(αi , i + 1) sin(αi , i + 1)

+

H̅c, i − 1 − H̅c, i cos(αi − 1, i) sin(αi − 1, i) (4)

where αi,i+1 and αi‑1,i are the angles between edges i and i + 1 and between edges i − 1 and i, respectively. As a consequence, the average specific step energy on the (hkl) face, εs̅ t,hkl, was defined as n

εst,̅ hkl =

∑i = 1 εst, ilc, i n

∑i = 1 lc, i

(5)

For a face growing by a 2D nucleation model, the growth rate, Rhkl, is given by35 (1)

* ⎞ ⎡ Δμ ⎤5/6 ⎛ ΔGhkl ⎟ R hkl ∝ βst, hkl ⎢ exp⎜ − ⎥ ⎣ kT ⎦ ⎝ 3kT ⎠

where Ebond, Elatt, and ΔHdiss are scaled bond energy, bond energy, lattice energy, and dissolution enthalpy, respectively. Escaled bond ,

B

(6)


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Crystal Growth & Design

Figure 2. Bonding networks for the (a) {101}, (b) {101̅}, (c) {111̅}, (d) {011}, (e) {110}, (f) {002}_1, and (g) {002}_2 faces of DADNE.

where βst,hkl is the step kinetic coefficient, which is assumed to be constant and independent of face orientation, and ΔG*hkl is the excess free energy of the 2D nucleus defined as16 * = ΔGhkl

edge 2 π (γhkl ) vmdhkl

Δμ

(7)

γedge hkl

where is the edge energy and Δμ is the driving force for crystallization, which is defined as Δμ a = ln kT aeq

(8)

where a is the activity of solute molecules in a solution, which can be replaced by the mole fraction XA of solute molecules for the crystallization from an ideal solution. If the ratio XA/X0A is sufficiently close to unity, then the driving force can be written as a supersaturation, Δμ/kT = σ. Therefore, the growth rate represented by eq 6 can be written as ⎛ πv (ε )2 ⎞ m st, ̅ i ⎟ R hkl ∝ βst, hkl σ 5/6exp⎜⎜ − ⎟ 3 kTd ⎝ hkl Δμ ⎠

(9)

3.1. {101} Faces. As shown in Figure 2a, the {101} face in a layer comprises four different bonds: a, b, e, and g. First, the PBCs that are expected to form a stable edge on a crystal face should be determined in order to calculate the step energy. Once a straight PBC consisting of only bond g is assumed to form an edge along the [101]̅ direction, two PBC candidates, a and a−e−a−b, remain along the [010] and [121̅] directions, respectively. According to Hartmann’s approach, two chains cannot coexist to form one edge of a nucleus. Therefore, a rather strong PBC that leads to a stable edge was determined by comparing the average bond energy between them. As a result, PBC [010] has a value of −0.65 kcal/mol, whereas the PBC [121̅] results in −0.71 kcal/mol. Therefore, PBCs [101̅] and [121̅] are considered on the {101} face. The step energy was calculated by the projection of a bonding structure to the direction of PBCs. In Figure 3a, two steps that are energetically different each other can be formed along PBC

Figure 3. (a) Steps on the {101} face projected in the [101]̅ (up) and [121]̅ (down) directions. (b) Critical lengths of the [101̅] (left) and [121]̅ (right) steps.

[101̅]. The energy costs for steps A and B are 4a + 2b + 2e + 2c + 4f and 4a + 2b + 2e − 2c − 4f, respectively; hence, the step energies C


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Crystal Growth & Design Table 2. Details of Nuclei on the {101} Face step

direction

no.

length (Å)

lst,i (Å)

energy cost

Est,i (kcal/mol)

εst,i (kcal/mol Å)

A B C

[101]̅ [101̅] [121]̅

2 2 4

13.55 13.55 18.86

6.78 6.78 4.72

4a + 2b + 2e + 2c + 4f 4a + 2b + 2e − 2c − 4f 4a + 2b + 2e + 8g

2.00 0.84 0.93

0.29 0.12 0.19

Figure 5. (a) Steps on the {111̅} face projected in the [101] (up) and [011] (down) directions. (b) Critical lengths of the [101] (left) and [011] (right) steps.

[1̅21] exist on the {101} face. Hence, it was assumed that a sixsided nucleus appeared on the face. Figure 3b presents the critical length of an edge that is evaluated by eqs 3 and 4, where the edge with relatively large step energy tends to decrease its length. 3.2. {101̅} Faces. In Figure 2b, the {101̅} face in a layer involves the bonds a, c, and d, where a and d PBCs exist along the [101] and [010] directions, respectively. As presented in Figure 4a, two different types of a step can be formed along the [101] direction because the bonds normal to the crystal surface vary depending on the nucleation sites. It is similar to the case of PBC [101̅] on the {101} face. Given in Table 3, the energy cost for step A is 4a + 2c + 2b + 2e − 4f, whereas step B results in 4a + 2c + 4f − 2b − 2e. Step B, with a rather smaller energy cost, was accounted for in the calculation of growth rate. However, along the [010] direction, only one kind of a step exists, and the step energy is 1.30 kcal/mol. In Figure 4b, the critical length of an edge along the [010] direction was found to be considerably smaller than that along the [101] direction because the step in

Figure 4. (a) Steps on the {101̅} face projected in the [101] (up) and [010] (down) directions. (b) Critical lengths of the [101] (left) and [010] (right) steps.

per mole for A and B are 2.00 and 0.84 kcal/mol, respectively. Nucleus A generates the surface on which the bonds 2c + 4f newly appear compared to the initial flat surface, whereas nucleus B makes the surface on which the bonds 2c + 4f disappear in opposite. Along the [121]̅ direction, there is one kind of step where the step energy is 0.93 kcal/mol. The number of growth units and length that are taken into account in the calculation are summarized in Table 2. To calculate the average specific step energy, the critical size of a nucleus that is expected to appear in the 2D nucleation should be calculated. Because nucleation tends to minimize the energy cost during nucleation, the step energy with the rather smaller value is selected. PBCs, [101̅], [1̅01], [121̅], [1̅21], [12̅1̅], and Table 3. Details of Nuclei on the {101̅} Face step

direction

no.

length (Å)

lst,i (Å)

energy cost

Est,i (kcal/mol)


A B C

[101] [101] [010]

2 2 2

13.28 13.28 6.56

6.64 6.64 3.28

4a + 2c + 2b + 2e − 4f 4a + 2c + 4f − 2b − 2e 2c + 4d

1.68 0.46 1.30

0.25 0.07 0.39

D


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Crystal Growth & Design Table 4. Details of Nuclei on the {111̅} Face step

direction

no.

length (Å)

lst,i (Å)

energy cost

Est,i (kcal/mol)


A B C

[101] [101] [011]

2 2 4

13.28 13.28 18.86

6.64 6.64 4.72

2b + 2e 4a + 2c 8d + 2c

0.77 1.07 1.08

0.11 0.16 0.23

Figure 6. (a) Steps on the {011} face projected in the [100], [11̅ 1]̅ , and [11̅ 1̅ ] directions in series. (b) Critical lengths of the [100], [11̅ 1]̅ , and [11̅ 1̅ ] steps in series.

Table 5. Details of Nuclei on the {011} Face step

direction

no.

length (Å)

lst,i (Å)

energy cost

Est,i (kcal/mol)


A B C

[100] [11̅ 1]̅ [1̅1̅1]

1 4 4

3.35 14.81 15.06

3.35 3.70 3.77

b+c−e 4b + 4f + 4g 4c + 4d + 4f

0.53 0.63 0.93

0.08 0.17 0.24

lst,i (Å)

energy cost

Est,i (kcal/mol)


c−b−e 4f + 4g + e

−0.34 0.27

−0.03 0.07

2f

0.16

0.02

2e

0.65

0.09

Table 6. Details of Nuclei on the {110} and {002} Faces step

direction

no.

length (Å)

A B

[001] [11̅1]

1 4

11.46 14.47

A

[01̅0]

1

6.61

A

[100]

1

6.69

{110} Face 11.46 3.62 {200}_1 Face 6.61 {200}_2 Face 6.69

3.4. {011} Faces. For the {011} face (Figure 2d), three PBCs, a−f, c−d, and b−g, exist along the [100], [1̅11̅], and [1̅1̅1] directions, respectively. In Figure 6a, the values of the average step energy on the {011} face were calculated (Table 5). Figure 6b demonstrates the critical length with respect to σ and T, where each step results in a very similar value. For the {011} face, a sixsided nucleus on a surface was considered because of the six PBCs along the [100], [10̅ 0], [11̅ 1]̅ , [111̅ ], [1̅1̅1], and [111]̅ directions. 3.5. {110} and {002} Faces. For the {110} face (Figure 2e), the stable edge was initially determined as PBC [11̅1], a−c, with an average bond energy of 0.75 kcal/mol, and the other edge becomes PBC [001], d−g, with an average bond energy of 0.55 kcal/mol. PBC [11̅0] is assumed not to form a stable edge because it shares the bonds a and c with the edge [111̅ ]. In Table 6, the negative step energy of nucleus A means that nucleation along the [001] direction does not require an additional energy cost. Hence, the {110} face easily forms a lot of 2D nuclei (Figure 7). The PBCs on the {002} face were found to be much

the [010] direction with a larger energy cost tends to be minimized. 3.3. {111̅} Faces. In Figure 2c, three PBCs, c−e, d, and a−b− a−e, were found to exist along the [11̅0], [101], and [011] directions, respectively. However, two PBCs, [11̅0] and [011], share bond e in common, where the values of the average bond energy are 0.75 and 0.71 kcal/mol, respectively. In this work, PBC [011] that passes through all lattice sites on the {111}̅ face was considered because PBC [110̅ ] skipping the lattice sites is a mechanistically unacceptable case. Figure 5a presents the step on the {111̅} face. The two energetically different steps can be generated along the [101] direction. In Table 4, step A and B need the energy cost of 2b + 2c and 4a + 2c, respectively, where step A with a smaller energy cost is favorably generated. On the basis of the step energies, the critical lengths for the two steps were estimated as shown in Figure 5b. On top of the {111̅} face, a four-sided nucleus consisting of four PBCs along the [11̅0], [011], [101], and [01̅1̅] edges was assumed to be generated. E


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Figure 7. (a) Steps on the {110} face projected in the [001] (up) and [111̅ ] (down) directions.

obscure, where PBCs could not be clearly defined. For the {002}_1 face, once PBC [010] consisting of bond a is selected, PBC [100] cannot be allowed because it shares the same bond of a. For the {200}_2 face, two PBCs cannot be simultaneously generated because of the coexistence of bond b. These faces behave like S (step) faces where only one type of PBC exists in a layer; hence, the energy cost for generating the step is limited to only one direction. In general, S faces grow faster than F faces, and then those faces tend to disappear. Therefore, the {110} and {002} faces were not considered in growth shape prediction by the step energy calculation.

4. KMC SIMULATION KMC simulations on crystal growth of DADNE were carried out on the basis of the solid-to-solid (STS) model. In this model, growth units are randomly attached to the surface forming at least one bond, and the configurations of them are neglected.36 According to Gilmer’s approach,37 the attachment rate of a growth unit is represented as +

K =

⎛ Δμ ⎞ ⎟ K 0+exp⎜ ⎝ kT ⎠

Figure 8. Flowchart of the KMC simulation. (10)

boundary conditions were imposed to reduce the edge effects. At first, all possible detachment rates that depend on ΔW were prepared by random insertion of a growth unit on each surface, and then T and Δμ were specified. The algorithm used is based on the n-fold way proposed by Bortz et al.,38 where a number n of events is classified with their probability. Therefore, this method needs all possible ways of events in advance, but it is very useful for sampling without loss of attempts compared to the Metropolis scheme. For a given configuration, the rate of an event j is defined by ρjrj, where ρj and rj are the density and the rate of an attachment or detachment event j, respectively. It simultaneously conciders a majority of situations in which insertion and annihilation of a growth unit competitively occur on a crystal face. The event j is selected by employing the KMC algorithm, which determines the event of interest from the cumulative rates by generating a random number, and insertion and annihilation trials continue to run until the simulation step elapses. The growth rate of a crystal face is correlated to the sticking fraction defined as

K+0

where is the frequency of attachment trials. The detachment rate of a growth unit, which is dependent on the specific lattice site i where a growth unit exists, can be expressed as ⎛ ΔWi ⎞ ⎟ K i− = K 0−exp⎜ − ⎝ kT ⎠

(11)

where ΔW is the energy required to remove a molecule from a crystal surface. At equilibrium, the attachment and detachment rates are equal. Therefore, the attachment rate at Δμ = 0 is K+0 , which becomes K−0 exp(−ΔW/(kT)), where K−0 is the frequency of detachment trials. The average ΔW is half the lattice energy, which can be substituted by the dissolution enthalpy. ⎛ E ⎞ K 0+ = K 0−exp⎜ − latt ⎟ ⎝ 2kT ⎠

(12)

Figure 8 presents the flowchart of the KMC simulations designed for the 2D nucleation. In the present work, the periodic F


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Figure 9. Microscope and scanning electron microscope (upper and lower panels, respectively) images of DADNE crystals obtained from DMA/water solution with the cooling rates of (a) 0.1, (b) 0.3, (c) 0.5, and (d) 0.7 °C/min, respectively.

R hkl ∝ Shkl =

Natt − Ndet Natt

because of the rapidly expanding metastable zone width. Hence, the main reason behind the change of growth shape would be from the difference in degree of supersaturation acting on crystals in a solution. First, the typical approaches to the growth shape prediction, such as the Bravais−Friedel−Donnay−Harker (BFDH) model39 and attachment energy (AE) model,40 were carried out. Figure 11 shows the predicted growth shape of DADNE by the BFDH model, where the {111}̅ faces with a smaller interplanar distance are not generated. However, the {101}, {101}̅ , {011}, and {002} faces appear extensively because of a large interplanar distance (Table 7). The AE model predicts a growth shape where the {111}̅ faces are newly generated and the surface area of the {002} faces tends to decrease. According to the AE model, the face growth rates are proportionally correlated to the magnitude of attachment energy; hence, the {101}̅ , {011}, {111}̅ , and {101} faces are dominantly generated rather than the {110} and {002} faces. Unfortunately, they could not predict the growth shape of DADNE that is dominantly faceted by the {111}̅ faces. The crystal faces are grown by the movement of monolayer steps, where solute molecules spread out over a face. At low supersaturation, the source of a step is provided by dislocations on a crystal face, and then the spiral motion of monolayers mainly occurs. In that case, the kink energy and kink rate are considered as key factors influencing the growth shape, but they are independent of supersaturation except for the kink rate when the growth units are noncentrosymmetric. Nonetheless, the kink rate cannot affect the relative growth rate of F faces; hence, the change of growth shape with supersaturation is rarely observed.19,41 However, in the present work, the growth shape of DADNE was found to be changed by supersaturation, and this implies that DADNE grown in a DMA/water solution cannot be described by the spiral growth model. At high supersaturation, the source of a step is mainly supported by 2D nucleation on a crystal face because the chemical potential is sufficiently large enough to form 2D nuclei. In that case, the energy involved in the formation of a step on a crystal face becomes an important factor affecting the facial growth rates. * < 3RT, as is the case when γedge is small, Furthermore, if ΔGhkl

(13)

where Natt is the number of attached growth units on the crystal face and Ndet is the number of detached growth units from the crystal face.

5. RESULTS AND DISCUSSION Figure 9 presents the DADNE crystals obtained by cooling crystallization, where 15.8 wt % of DADNE in DMA/water solution (mass ratio = 70:30) was cooled down from 80 to 20 °C at cooling rates of 0.1, 0.3, 0.5, and 0.7 °C/min under mixing with a mechanical stirrer at 400 rpm. As shown in Figure 10, the basal

Figure 10. A typical rodlike DADNE crystal.

faces of a DADNE crystal were indexed by measuring the interplanar angles between crystal faces from lots of pictures using an optical microscope. At a slow cooling rate, the crystals grow as a rodlike shape with {111̅} basal faces. However, as cooling rate increases, it was shown that the major axis length of crystals becomes shorter and rather small crystals are obtained. In cooling crystallization, a fast cooling rate usually induces obviously a higher supersaturation than a slow cooling rate G


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Figure 11. Crystal shapes of DADNE predicted by the BFDH (left) and AE (right) models.

Table 7. Interplanar Distance and Attachment Energy of DADNE face

dhkl (Å)

Eatt (kcal/mol)

{101} {101̅} {111}̅ {011} {110} {002}

5.74 5.86 4.37 5.71 4.68 5.81

−11.21 −8.83 −10.02 −9.86 −11.99 −14.71

Figure 13. Results of the KMC simulation for DADNE. The sticking fraction is plotted as a function of Δμ/kT. The simulations were carried out on a crystal face of 40 × 40 unit cells during 10 000 steps.

Figure 14. Crystal shape of DADNE predicted by the KMC simulations at Δμ/kT = (a) 4.7, (b) 4.9, and (c) 5.1.

shape of DADNE at Δμ/kT = 0.1 is in good agreement with that from experiments, as shown in Figures 9a. The aspect ratio of DADNE at Δμ/kT = 0.3 dramatically diminishes, and the {011} faces start to appear. The aspect ratio of DADNE at Δμ/kT = 0.5 becomes almost one-third of that at Δμ/kT = 0.1. The main reason for shape change by a degree of supersaturation is the excess free energy of a 2D nucleus (ΔGhkl * in eq 7). The anisotropic terms, γedge and dhkl, are key factors determining the i rodlike shape of DADNE when the driving force, Δμ, is small. * becomes dominated by these anisotropic terms, Therefore, ΔGhkl which subsequently leads to the anisotropic growth behavior. However, when the driving force, Δμ, is sufficiently large enough to form a lot of 2D nuclei, the effects of γedge and dhkl on G*hkl i become considerably weaker compared with that of Δμ so that it induces the kinetic roughening to a large extent regardless of

Figure 12. Crystal shape of DADNE predicted by the step energy at Δμ/kT = (a) 0.1, (b) 0.3, and (c) 0.5.

then growth by the 2D nucleation mechanism extremely * was found to be occurs.16 In Figure S2, a great part of ΔGhkl smaller than 3RT over all supersaturation. Therefore, the 2D nucleation model is applicable for describing the supersaturationdependent growth shape of DADNE. Figure 12 shows the supersaturation-dependent growth shapes of DADNE predicted by step energy calculation. The rodlike H


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Figure 15. Snap shots of the KMC simulations of faces (a) {101}, (b) {101̅}, (c) {111̅}, (d) {011}, (e) {110}, and (f) {002}, where Δμ/kT = 4.0, 4.6, 4.6, 3.8, 3.6, and 3.6, respectively. A blue slice is located on the third layer of a crystal face from the bottom.

faces. Therefore, a relatively large ΔGhkl * of the {111̅} face allows DADNE crystals to grow with a rodlike shape at a low driving force. Figure 13 represents the sticking fraction as a function of Δμ/kT obtained from the KMC simulation. The {101̅} and {111̅} faces do not grow until Δμ/kT = 4.6, above which they dramatically grow. However, the {101}, {011}, {002}, and {110} faces are grown at a relatively smaller driving force than the {101̅} and {111̅} faces. In section 3.5, it was anticipated that the {002} and {110} faces would grow faster than the other faces because the bonding network of {002} face was similar to that of an S face, and the {110} face provided the negative value of εs̅ t. Interestingly, the KMC simulation also predicts the high sticking fraction of the {002} and {110} faces. This means that both of the faces offer surface topology where growth units are favorably incorporated. The growth shape of DADNE predicted by the sticking fractions of the KMC simulation is presented in Figure 14. The aspect ratio was found to decrease as the driving force increased, but the major axis length of a crystal is rather smaller than those predicted by the step energy calculation. Theoretically, the step energy calculation considers only the energy change when forming a 2D nucleus on a flat face. However, the KMC simulation considers all possible events that depend strongly on the work, ΔW, for detaching growth units; hence, it offers the kinetics of growth units with time. Figure 15 shows the KMC simulation results for each face, where a moderately low supersaturation is employed to avoid extreme kinetic roughening. A manifold of 2D islands was observed on the {101}, {101̅}, {111̅}, and {011} faces. However, the {110} face shows 0D deposition where a number of isolated growth

units reside, and the {002} face produces 1D deposition of growth units along the [010] direction.

6. CONCLUSIONS In the present experiments, rodlike DADNE crystals with the {111̅} basal faces were obtained from a DMA/water solution, and the aspect ratio was found to decrease as the supersaturation increased. According to the mechanism of crystal growth, the supersaturation-dependent growth shapes are intimately associated with the rate of 2D nucleation. When the driving force is sufficiently large enough to overcome the energy barrier of forming a step on the crystal face, the rate of 2D nucleation determines the relative growth rates of crystal faces. Therefore, the step energy is of utmost importance when predicting the growth shape with respect to supersaturation. In this work, the step energy calculation accurately predicts the growth morphology of DADNE that could not be anticipated by the BFDH and AE models. As a result, it was concluded that the rodlike shape of DADNE can be ascribed to a high-energy barrier of 2D nucleation, ΔG*hkl, on the {111̅} face. Furthermore, the analysis of PBCs on each face makes it possible to filter out the faces that would grow much faster and then eventually disappear. It is surmised that the {002} faces would behave like S faces in which only one PBC could reside. The {110} faces are expected to grow much faster because of the negative step energy, which implies that the additional energy is not required for the 2D nucleation. The KMC simulation shows the same results where {002} and {110} faces start to grow even at a very low driving force. However, a disagreement between the step energy calculation and KMC simulations was found on the {101̅} face. The growth rate estimated by the step energy is much faster than that by the I


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Korea; Asian Crystallization Technology Society: Gyeonggi-do, Korea; p 68. (10) Lochert, I. J. FOX-7−A New Insensitive Explosive; Technical Report DSTO-TR-1238 for DSTO Aeronautical and Maritime Research Laboratory: Fishermans Bend, Victoria, Australia, 2001. (11) Trzciński, W. A.; Cudziło, S.; Chyłek, Z.; Szymańczyk, L. J. Hazard. Mater. 2008, 157, 605−612. (12) Daniel, M. A.; Davies, P. J.; Lochert, I. J. FOX-7 for Insensitive Boosters; Technical Report DSTO-TR-2449 for DSTO Defence Science and Technology Organisation: Edinburgh, South Australia, Australia, 2010. (13) Mandal, A. K.; Thanigaivelan, U.; Pandey, R. K.; Asthana, S.; Khomane, R. B.; Kulkarni, B. D. Org. Process Res. Dev. 2012, 16, 1711− 1716. (14) Gao, B.; Wu, P.; Huang, B.; Wang, J.; Qiao, Z.; Yang, G.; Nie, F. New J. Chem. 2014, 38, 2334−2341. (15) Van der Steen, A. C.; Verbeek, H. J.; Meulenbrugge, J. J. In Proceedings of the 9th International Detonation Symposium on Detonation, August 28−September 1, 1989, Portland, OR; Office of the Chief of Naval Research: Arlington, VA; pp 83−88. (16) Winn, D.; Doherty, M. F. AIChE J. 1998, 44, 2501−2514. (17) Winn, D.; Doherty, M. F. AIChE J. 2000, 46, 1348−1367. (18) Snyder, R. C.; Doherty, M. F. Proc. R. Soc. A 2009, 465, 1145− 1171. (19) Kuvadia, Z. B.; Doherty, M. F. Cryst. Growth Des. 2011, 11, 2780− 2802. (20) Lovette, M. A.; Doherty, M. F. Cryst. Growth Des. 2012, 12, 656− 669. (21) Kim, S. H.; Dandekar, P.; Lovette, M. A.; Doherty, M. F. Cryst. Growth Des. 2014, 14, 2460−2467. (22) Boerrigter, S. X. M.; Hollander, F. F. A.; van de Streek, J.; Bennema, P.; Meekes, H. Cryst. Growth Des. 2002, 2, 51−54. (23) Boerrigter, S. X. M.; Cuppen, H. M.; Ristic, R. I.; Sherwood, J. N.; Bennema, P.; Meekes, H. Cryst. Growth Des. 2002, 2, 357−361. (24) Cuppen, H. M.; van Eerd, A. R. T.; Meekes, H. Cryst. Growth Des. 2004, 4, 989−997. (25) Cuppen, H. M.; Graswinckel, W. S.; Meekes, H. Cryst. Growth Des. 2004, 4, 1351−1357. (26) Panina, N.; van de Ven, R.; Janssen, F. F. B. J.; Meekes, H.; Vlieg, E.; Deroover, G. Cryst. Growth Des. 2009, 9, 840−847. (27) Gilardi, R. CCDC 127539: Experimental Crystal Structure Determination. Cambridge Crystallographic Data Centre: Cambridge, U.K., 1999.https://summary.ccdc.cam.ac.uk/structure-summary?ccdc= 127539. (28) Thiel, W. J. Am. Chem. Soc. 1981, 103, 1413−1420. (29) Sun, H. J. Phys. Chem. B 1998, 102, 7338−7364. (30) Sorescu, D. C.; Boatz, J. A.; Thompson, D. L. J. Phys. Chem. A 2001, 105, 5010−5021. (31) Cuppen, H. M.; Day, G. M.; Verwer, P.; Meekes, H. Cryst. Growth Des. 2004, 4, 1341−1349. (32) Prausnitz, J. M.; Lichtenthaler, R. N.; Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria; Prentice Hall: Englewood Cliffs, NJ, 1986; Chapter 9. (33) Deij, M. A.; Los, J. H.; Meekes, H.; Vlieg, E. J. Appl. Crystallogr. 2006, 39, 563−570. (34) Deij, M. A.; Meekes, H.; Vlieg, E. Cryst. Growth Des. 2007, 7, 1949−1957. (35) van der Eerden, J. P. Crystal Growth Mechanisms. In Fundamentals; Hurle, D. J., Ed.; Handbook of Crystal Growth, Vol. 1; North-Holland Elsevier: Amsterdam, 1993; p 307. (36) Boerrigter, S. X. M.; Josten, G. P. H.; van de Streek, J.; Hollander, F. F. A.; Los, J.; Cuppen, H. M.; Bennema, P.; Meekes, H. J. Phys. Chem. A 2004, 108, 5894−5902. (37) Gilmer, G. H. J. Cryst. Growth 1976, 36, 15−28. (38) Bortz, A. B.; Kalos, M. H.; Lebowitz, J. L. J. Comput. Phys. 1975, 17, 10−18. (39) Donnay, J. D. H.; Harker, D. Am. Mineral. 1937, 22, 446−467. (40) Hartman, P.; Bennema, P. J. Cryst. Growth 1980, 49, 145−156. (41) Shim, H.-M.; Koo, K.-K. Cryst. Growth Des. 2014, 14, 1802−1810.

KMC simulations. It means that the step energy could not always be directly linked to the kinetics of growth units on crystal faces. The step energy method is profoundly based on classical thermodynamics that consider the energy change when a critically sized nucleus is formed on a crystal face. Fundamentally, it implies that the {101̅} face provides the surface structure on which it is difficult to form a 2D nucleus compared with other faces although the small quantity of step energy is gained. DADNE comprises noncentrosymmetric growth units that do not have a center of symmetry that leads to the generation of complex bonding networks. Basically, the centrosymmetric growth units behave in a Kossel-like fashion;19 therefore, the detachment events of a growth unit are countable by estimating all possible kink and edge sites. However, noncentrosymmetric growth units tend to generate lots of detachment events. Therefore, numerous kink sites that are energetically different from each other exist even in one edge direction,41 which imposes enormous burdens on the KMC simulation. In the present work, we tried to solve this problem by screening possible events with random insertion. It helps to count the number of events as much as possible; hence, it is expected that the sampling based on the solid-to-solid model will become more efficient.

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ASSOCIATED CONTENT

S Supporting Information *

The mole fraction solubility of DADNE in the DMA/water (70/30) and excess free energy of 2D nucleus. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Phone: +82-2-705-8680. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Next-Generation Converged Energy Materials Research Center and by the Hanwha and Agency for Defense Development (UC120019GD; Republic of Korea).

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REFERENCES

(1) Zhang, C.; Wang, X.; Huang, H. J. Am. Chem. Soc. 2008, 130, 8359−8365. (2) Anniyappan, A.; Talawar, M. B.; Gore, G. M.; Venugopalan, S.; Gandhe, B. R. J. Hazard. Mater. 2006, B137, 812−819. (3) Latypov, N. V.; Johansson, M.; Holmgren, E.; Sizova, E. V.; Sizov, V. V.; Bellamy, A. J. Org. Process Res. Dev. 2007, 11, 56−59. (4) Evers, J.; Klapötke, T. M.; Mayer, P.; Oehlinger, G.; Welch, J. Inorg. Chem. 2006, 45, 4996−5007. (5) Crawford, M.-J.; Evers, J.; Göbel, M.; Klapötke, T. M.; Mayer, P.; Oehlinger, G.; Welch, J. M. Propellants, Explos., Pyrotech. 2007, 32, 478− 495. (6) Gao, H.-X.; Zhao, F.-Q.; Hu, R.-Z.; Pan, Q.; Wang, B.-Z.; Yang, X.W.; Gao, Y.; Gao, S.-L.; Shi, Q.-Z. Chin. J. Chem. 2006, 24, 177−181. (7) Xu, K.; Song, J.; Zhao, F.; Ma, H.; Gao, H.; Chang, C.; Ren, Y.; Hu, R. J. Hazard. Mater. 2008, 158, 333−339. (8) Kim, J.-K.; Kim, J.-W.; Park, D.-B.; Kim, H.-S.; Koo, K.-K. In Proceedings of the 18th International Symposium on Industrial Crystallization, September 13−16, 2011, Zurich, Switzerland; European Federation of Chemical Engineering: Paris, France; pp 447−448. http://www.efce.info/home.html. (9) Na, J.-Y.; Kim, J.-K.; Kim, H.-S.; Koo, K.-K. In Proceedings of the Asian Crystallization Technology Symposium, May 23−25, 2012, Seoul, J


Molecular Modeling on Supersaturation-Dependent Growth Habit of 1

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