Steady State Morphologies of Paracetamol Crystal from Different

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Steady State Morphologies of Paracetamol Crystal at Different Solvents Jinjin Li, and Michael F. Doherty Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.6b01510 • Publication Date (Web): 12 Dec 2016 Downloaded from http://pubs.acs.org on December 17, 2016

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Crystal Growth & Design

Steady State Morphologies of Paracetamol Crystal at Different Solvents Jinjin Li∗,† and Michael F. Doherty‡ †Key laboratory for Thin Film and Microfabrication of Ministry of Education, Department of Micro/Nano-electronics, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai 200240, P.R.China. ‡Department of Chemical Engineering, University of California, Santa Barbara, California 93106-5080, USA E-mail: [email protected] Abstract Nowadays, mastering crystal growth from different point of view is becoming an essential technique, with the continuous development of pharmaceutical industries and scientific researches. Paracetamol (acetaminophen), a medication used to treat pain and fever, is often at the steady state growth shape but has a complex crystal habit. Due to the non-centrosymmetric structure, paracetamol has multiple types of kink sites on each edge of each face, which brings more difficulties in growth predicting for crystal morphology designers. Therefore, the accurate analysis of paracetamol crystal structure and the ability to design its crystal shapes at as many solvents as possible are of significant meanings and interest. Here, a general mechanistic spiral growth model is introduced to predict the steady state morphologies of paracetamol crystal grown from 30 different solvents. An upgraded kink rate expression and a kink density expression are proposed to describe how the growth units associated with the kink, step, edge, etc., which enable to capture the shape variation of paracetamol in different solvents

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with reasonable accuracy. The proposed algorithm can be applied to the morphology prediction of many kinds of organic molecules.

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1

Introduction

Steady state morphologies of crystals with different crystal habits (i.e., blocks, needles, plates, rods) possess different physical and chemical characteristics, which make great contributions in pharmaceutical manufactures and other practical applications. 1–4 For example, due to the high-aspect-ratios, needle-shaped crystals are usually unaccepted for pharmaceutical industries and related applications because these shapes will influence the downstream processing such as filtering, drying, blending, tabletting etc. 5,6 However, sometimes the needle shape is favorable for the active layered morphologies to increase the electronic and optical properties of pharmaceutical devices. 7,8 Therefore, mastering crystal morphology using designed operating environments and synthesizing crystal with a particular habit are very important for researchers in crystallography. In this context, a mechanistic model which can predict and visualize crystal growth under different solvents is promising to bring the design of crystal growth and the related products into the next generation. 1 Paracetamol(also named as acetaminophen) is commonly used to treat many conditions such as fever, headache, cold, muscle aches, backache, arthritis, toothaches, etc. 9 Paracetamol is available in many retail outlets as tablets/capsules and as liquid medicine. However, experts have pointed out that inappropriate taking of paracetamol will case fever with nausea, stomach pain, dark urine, jaundice etc. 10 For example, people more often take paracetamol during the winter to combat cold and flu. This is an unsafe and ineffective way when the incorrect amount is taken. 11,12 In order to understand pharmaceutical’s crystal habits, a deep research is necessary to figure out the crystal morphology variations in different environment, using an accurate and mechanistic method. 13–15 During the stage of process design, the influence of polymorphy, solvents and other additional impurities is always the most difficult portion to be fixed due to regulatory and safety requirements, which results in a challenge in the scientifical and systematical engineering of pharmaceutical crystal shape. With this need in mind, in the present paper, we investigate the steady state morphologies of paracetamol crystal grown in 30 different solvents, such 3



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as benzene, water, pentane, toluene, acetic acid, amyl acetate, propanol, tetrahydrofuran, etc., using a mechanistic spiral growth (MSG) model, where an upgraded kink rate expression and a kink density expression that enable crystal morphology prediction for organic molecules were used. In MSG model, the impacts of solute, solvent, and their interactions are considered adequately, as well as the the presence of external species to predict the crystal morphology of paracetamol crystal grown from different solvents. External species such as impurities, counterions and additives are all considered in the solution. The proposed scheme, investigating paracetamol crystal in more than 30 different solvents, could provide an advanced theoretical framework to guide experiments explore optimum functionalities of paracetamol medicine and its relevant products. The earliest approach to predict the steady-state crystal morphology is the BFDH model, which is a non-mechanistic method and was proposed by Bravais, Friedel, Donnay and Harker 16–18 through the well-known Frank-Chernov condition. 19,20 According to the easy implementation and acceptable advising, BFDH model found its place during the 20 century although the energetics and surface chemistry effects have not been considered. In 1980, Hartman and Bennema developed the attachment energy (AE) model, 21 which enhance the accuracy to predict the crystal growth rates of crystal faces by taking into account the solid-state interactions. AE model, considering the attachment energies of slices, is generally an improvement to the BFDH model, but still fails to consider the impacts of additives, supersaturation, solvent or other external species. 1,22 The low-ability to capture the impacts of growth environments resulted in the low-accurate prediction of crystal morphology in BFDH model and AE model. Recently, various modified AE models have been published that attempt to overcome these limitations and predict the crystal morphology with higher accuracy. 1,3–5,23,24 To describe the real crystal morphologies with practical conditions, a desirable crystal growth model should take a full consideration to the anisotropic growth rates of the crystallographic faces and the surface energies. In the present paper, the morphology of paracetamol

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crystal was predicted by mechanistic spiral growth(MSG) model, which is a high-fidelity morphological theory and the impacts of temperature, concentration, supersaturation and other imposed growth conditions have been taken into account adequately. The periodic bond chains, containing the information of solid-state interactions, are used to determine the final crystal shape and assist to calculate the relative growth rate of each F-face. The new kink rate expression and the kink density expression are included to analyze the kink behaviors during the crystal growth, which result in the prediction of shape evolution during the cycles of crystal growth or dissolution. MSG model, a high accuracy crystal growth theory, has been tested in many organic molecules to capture the shape variation of crystals within different growth environments. 23–29

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Method

Paracetamol crystal has a formula of C8 H9 N O2 and a space group of P 21 /a. There are two different hydrogen bonds in the paracetamol structure of to form a pleated sheet. One bond has a length of 2.663(3)˚ A, where OH donates to O=C. Another bond has a length of 2.934(3)˚ A, where OH accepts from H-N. 30 There are four molecules per unit cell in paracetamol, with cell parameters as: a = 12.92˚ A, b = 9.40˚ A, c = 7.10˚ A, α = γ = 90◦ and β = 115.9◦ , measured by Haisa et al. 30 at room temperature. In the present paper, the MSG model will be used to implement the crystal shape prediction of paracetamol in 30 different solvents. For a crystal, the perpendicular growth rate of each face can be described as Gi , which is used to predict the actual crystal shape. In a constant growth environment and a constant Gi , the Frank-Chernov condition 19,20 is used to predict the steady state morphology of a crystal, where the crystal’s final faces have no relationship with its original seed shape. 31,32

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The Frank-Chernov condition is given as: G1 G2 Gi = =···= H1 H2 Hi

(1)

where Hi is the perpendicular distance of face i from the crystal center, as shown in Fig.1. Therefore, the real crystal shapes can be predicted by the determining of perpendicular growth rate Gi of each crystallographic face, which can be expressed as:

Gi =

hi νi hi = y τi

(2)

where νi is the step velocity, y is the inter-step distance. y/νi equals to the spiral rotation time τi . 33 All these parameters can be found in Fig.5(a). The interplanar spacing hi is given as: 34

r hhkl =

1 rhkl

rhkl = (1 − cos2 α − cos2 β − cos2 γ + 2 cos α cos β cos γ)−1 2

2

(3)

(4)

2

h k l 2 2 sin α + sin β + sin2 γ a2 b2 c2 2kl 2lh 2kh + (cos β cos γ − cos α) + (cos α cos γ − cos β + (cos α cos β − cos γ))] bc ac ab [

where h, k, l are Miller indexes for the face with lattice parameters of a, b, c and α, β, γ. For a face with N -sided polygonal spiral, the rotation time can be given as: 33

τi =

N X lc,i+1 sin(αi,i+1 )

νi

i=1

(5)

where on edge i the step velocity is νi ; the angle between edge i and i + 1 is αi,i+1 ; the critical length of edge i + 1 is described as lc,i+1 . The step velocity of the edge i is given by

νi = ap,i ui ρi 6


(6)

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where ui , ρi and ap,i are the kink rate, the kink density and the inter-row distance on the ith edge, respectively (as shown in Fig.4(b)).

2.1

Kink Rate Calculation

To obtain the growth rate of a face, we should calculate the step velocity for each edge in eq.(6), where the kink rate ui and the kink density ρi must be calculated separately. The kink rate represents the difference of the attachment/detachment rate of solute molecules into/from the kink sites. Non-centrosymmetric molecules, such as paracetamol, usually have complicated kink rate expressions where the different types of kink sites, the attachment/detachment rates as well as the kink sites’ rearrangements should be considered. For the kink rate ui , the general expression is given as: − (j + )n − Πnk=1 jk,i Pn ui = n Pn − − − + n−r k=1 (jk jk+1 , · · ·, jk+r−2 )i ] r=1 [(j )

(7)

where n refers to the number of different growth units, the subscripts k and i correspond to the kink and the edge (k and i have the same representations in the following sections). The attachment rate and detachment rate from a kink site can be described using the transition state theory, 35 which is shown in Fig.2. The q in horizontal axis is a reaction coordinate, which is the distance away from the kink site. The product state A and the reactant state B are two different states, which correspond to the detached growth unit solvated in solution and the growth unit attached onto the kink site, respectively. 36 The state between A and B is the transition state, where the growth unit is solvated in solution partially and bonded to neighbor kink sites partially. In this case, the attachment rate (from state A to state B in Fig.2), depending on the mole fraction of solute molecules in the adsorption layer, is identical for each kink site. While the detachment (from state B to state A in Fig.2), different for each kink site, is dependent with the kink detachment work. The

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− attachment rate j + and the detachment rate jk,i can be expressed as:

∆U ) kB T ∆U + ∆Wk = ν0 exp(− ) kB T

j + = Sν0 xeq exp(−

(8)

− jk,i

(9)

where S is the supersaturation, which equals to C/Ceq ; xeq is the mole fraction of solute molecules in solution at equilibrium, which is given as xeq = exp[−∆Wk /(kB T )]; when removing the partially solvated growth unit from the kink site k to a completely solvated state, the work required is ∆Wk . 36 In vapor, when detaching a growth unit from a crystal, ∆Wk is the energy summation of all the breaking of PBC bonds; while in solution, ∆Wk not only includes the PBCs’ breaking, but also involves the energies of surface solvation. ν0 in eqs.(8) and(9) is the frequency factor for attachment and detachment attempts, which relies on the solute, solvent, temperature and other conditions. 37 The energy barrier ∆U , corresponding to the solvation shell breaking around the growth units, is the same on all crystal faces. For solution crystallization, this quantity can be estimated as the enthalpy of desolvation. The frequency ν0 and the exponential exp(−(∆U/kB T )) in eqs.(8) and (9), assumed to be the same everywhere on the crystal surface, can be canceled out in the calculation of relative growth rates and therefore have no effect on the crystal morphology prediction. Fig.3 shows the detachment process (from (a) to (b)) and the attachment process (from (b) to (a)). From (a) to (b), the growth unit 3 was detached from the edge to the solution and results in a new kink site on the top layer. There are three species involved in: the growth unit, which will be detached from the edge; the growth unit that prepare to form the next kink site; and the solvent molecule which make the both these species solvated. Therefore, calculation of ∆Wk in eq.(9) involves the computation of kink site energies of the two successive growth units along the edge.

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2.2

Kink Density Calculation

To show the kink density calculation, Fig.4(a) shows the paracetamol structure on [010] edge of {001} face. There are 4 different growth units on face {001}, locating at two different layers. One layer contains growth units 1 and 2, the other has growth units 3 and 4 (Fig.4(a)). Each growth unit contains 2 kink sites based on different orientations, i.e., growth units 1 and 4 contain kink sites B¦ (right orientation) and B? (left orientation), while growth units 2 and 3 contain kink sites A¦ (right orientation) and A? (left orientation). The first step for paracetamol morphology prediction is the calculation of intermolecular interactions, which was performed by GAFF. 38,39 GAUSSIAN 03 40 was used to calculate the partial atomic charges on paracetamol atoms comprising the unit cell, based on the restrained electrostatic potential(RESP) model. The different atom-atom interactions getting condensed into a single intermolecular interaction resulted in different bond energies, as shown in Fig.4(a). For example, the black, blue, orange and green bonds have the energies of −4.45kcal/mol, −0.5kcal/mol, −2.72kcal/mol and −2.43kcal/mol, respectively. All the bond energies leading to the paracetamol has a calculated lattice energy of −28.2kcal/mol, which matches the experimental lattice energy of −28.18kcal/mol 41 very well. With all the bond energies, the kink site A? has a energy of −(4.45 + 2.72 + 2 × 0.5) = −8.17kcal/mol, the kink site B¦ has a energy of −(4.45+2.72) = −7.17kcal/mol, the kink site B? has a energy of −(2.43+2.72) = −5.15kcal/mol, and the kink site A¦ has a energy of −(2.43+2.72+2×0.5) = −6.15kcal/mol. To form a new layer, there are several possible rearrangements that can form 4 different kinks, which resulted in the calculation of kink density on that edge. Fig.4(a) shows one rearrangement which forms A? and B¦ in a new layer by taking out A? and B¦ from a lower layer. For this thermal rearrangement, the energy expense to form one kink is: ε1 = −(4.45 + 4.45)/4 = −2.225kcal/mol. However, we can also take out A? and B¦ from a lower layer and form B? and A¦ on a new layer (ε2 = −(4.45 + 2.43)/4 = −1.72kcal/mol); or take out B? and A¦ from a lower layer and form A? and B¦ on a new layer (ε3 = −(4.45+2.43)/4 = 9



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−1.72kcal/mol); or take out B? and A¦ from a lower layer and form B? and A¦ on a new layer (ε4 = −(2.43 + 2.43)/4 = −1.215kcal/mol). So there are a total of 4 rearrangements to take out one pair of growth units from a lower layer and to form a new pair of growth units on a new layer. The corresponding paracetamol molecules on face {0,0,1} are shown on Fig.4(b), where the green molecule, purple molecule, yellow molecule and white molecule represent the growth units 1, 2, 3 and 4, respectively. The overall kink density for a edge is defined as the sum of all the individual kink densities on that edge, while the individual kink density can be calculated as a weighted sum of the transformations which can produce that type of kink. 1,42 Based on Boltzmann distribution, the kink density expression for [010] edge on (001) face of paracetamol is given by eq.(10) according to the four different kink sites transformations.

ρ=

2.3

exp(− |εkT1 | ) + exp(− |εkT2 | ) + exp(− |εkT3 | ) + exp(− |εkT4 | ) 1 + exp(− |εkT1 | ) + exp(− |εkT2 | ) + exp(− |εkT3 | ) + exp(− |εkT4 | )

(10)

Solvent Impacts

The choice of solvent is a critical design factor during the crystal growth, which will impact the crystal morphology directly. In eq.(9), the work of adhesion ∆Wk only relates to the solid phase in vapor, but refers to the crystalline and solution phases in solution. So different solvent during crystallization gives different values of ∆Wk due to solvation of the crystal surface, which modifies the crystal morphology by impacting the kink densities and the kink rates. 1 In solution, the crystal needs some work to from a disturbance along the edge, which can be given as:

φk,i = −Etot,i /2 + γs si − γk,i si

(11)

where Etot,i , the solid portion, is the total strength of intermolecular interactions that are broken to form a disturbance along edge i; γs , the solvent portion, is the surface energy of

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the solvent; si is the surface area exposed at a kink site along edge i, which is shown in Fig.3; γk,i is the work of adhesion between the crystalline and solution phases. Generalized Amber Force Field (GAFF) 38,39 is used to calculate the atom-atom interactions, which is the basis to obtain the intermolecular interactions. In GAFF, the interaction between atom i and atom j is given by:

Gij = −

Aij Bij qi qi + 12 + 6 rij rij Drij

(12)

where the first term r−6 is the dispersive portion and the last term r−1 refers to the columbic interaction. Assuming the main component in columbic interaction is the hydrogen bond, the work of adhesion in eq.(11) can be expressed in the following two equations, according to the H-bond absence or presence in crystal-solvent interactions: 43 p Without H-bond : γk,i = γc + γs − 2 γc γsd p p p With H-bond : γk,i = γc + γs − 2 γcd γsd − 2( γc+ γs− + γc− γs+ )

(13) (14)

where the subscripts “s” and “c” refer to solvent and crystal, the superscript “d” is the dispersive portion when calculating the intermolecular interactions using GAFF. The superscripts “+” and “−” defined as accepting and donating characteristics, which is the subdivision of hydrogen bond component in the total PBC interactions. The ratio of γ + /γ − closing to unity is a reasonable assumption. 43 If we refer the dispersive energy and the columbic energy in eq.(12) as “Ed = Aij /(rij )6 ” and “Eco = (qi qi )/(Drij )”, the solid contributions γc and γcd can be given as:

γc = Etot,i /si , γcd = (

Ed × Etot,i )/si . Eco + Ed

(15) (16)

To represent cohesive energy densities, the intermolecular interactions between solvent 11



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molecules can be described as solubility parameters. 5 The overall solubility parameter δ can be written as: δ2 =

∆Hvap − RT Vm

(17)

where Vm is the molar volume, and (∆Hvap −RT ) is the enthalpy difference between the condensed phase and isolated molecules. 43–48 Hansen split the overall solubility parameter into three sub-parameters, based on three components of surface energies, 47 which are dispersive portion (δd ), polarization portion (δp ), and hydrogen bonding portion (δh ). To assign the three solubility sub-parameters, solubility parameters should be converted into interfacial energies. Eq.(18) is the relationship between experimental value of the surface energy (γsexp ) and the solubility parameter (δ). γsexp = ²(

Vm 1/3 2 ) δ , NA

(18)

where the empirical constant ² is to scale the surface energy to match the experimental value. In this case, the interfacial energy can be obtained by the empirical constant ² and the three components of solubility parameters. Based on three classes of solvents, Beerbower 49 developed correlations using three subparameters. 5,43 For alcohols, the surface energy and the empirical constant ² are given as: γs = 0.0715Vm1/3 (δd2 + δp2 + 0.06δh2 ), ²1 =

0.0715V

γsexp 1/3 (δ 2 + δ 2 p d

+ 0.06δh2 )

(19)

.

(20)

For acids, phenols, amines and water, the surface energy are given as: γs = 0.0715Vm1/3 (δd2 + 2δp2 + 0.0481δh2 ), ²2 =

γsexp 1/3

0.0715Vm (δd2 + 2δp2 + 0.0481δh2 ) 12


.

(21) (22)

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For all other solvents, the surface energy are given as: γs = 0.0715Vm1/3 (δd2 + 0.632δp2 + 0.632δh2 ), ²3 =

γsexp 1/3

0.0715Vm (δd2 + 0.632δp2 + 0.632δh2 )

,

(23) (24)

Here, Vm is the molar volume of the solvent, with the units of cm3 /mol. In this case, for an alcohol, the predicted hydrogen bonding portion can be written as: γsh = 0.0715²1 Vm1/3 0.06δh2 .

(25)

And γsd and γsp for an alcohol can be given by γsd = 0.0715²1 Vm1/3 δd2 ,

(26)

γsp = 0.0715²1 Vm1/3 δp2 .

(27)

The detailed information of how the solubility parameters and surface energies obtained in solution, please refers to our previous paper. 43

3

Results and Discussion

Periodic Bond Chain (PBC), proposed by Hartman and Perdokc, 50,51 is to describe the solidstate interactions and the crystal growth kinetics. Along a lattice direction, PBCs describe the strongest intermolecular interactions between growth units. During the crystallization process, PBCs are formed between the growth units and exclude any interactions for intra growth units. Only the faces that have two or more PBCs can be the F-faces, where the spiral rings are grown layer by layer. According to the PBC theory, PBCs must satisfy certain rules as shown below: 27 (1) PBCs are directions of strong intermolecular interactions that form uninterrupted

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chains throughout the crystal; (2) PBC vectors are parallel to edges or steps in layered crystal growth; (3) No common interaction is shared between two PBCs in the same plane; (4) PBCs should have stoichiometric compositions; (5) No dipole moment perpendicular to the PBC direction. Fig.5(a) shows the related geometric parameters in MSG model, where the step edge makes forward propagation with the step velocity ν; h is the interplanar spacing; y is the inter-step distance; aP,i refers to the kink size, and Ghkl is the growth rate for the face with Miller indexes (h,k,l ). Crystal grows by the processes of kink site attachment and detachment via thermal roughening, 1 which results in the same kink site interaction on each edge and each face in centrosymmetric crystals and multiple different kink site energies in non-centrosymmetric crystals. In MSG model, a tiered array of steps can be produced by the moving of steps across the surface, which makes the related faces grown layer by layer with a growth rate G. The algorithm of morphology prediction using MSG model is shown in Fig.5(b), where the F-faces of a crystal should be selected as the research objects in the beginning according to the PBC theory. And then, the growth rate for each F-face that determine the crystal morphology depends on the spiral’s rotation time, which was given by P τi = N i=1 [lc,i+1 sin(αi,i+1 )/νi ] (eq.(5)). Step velocity, the only variable to obtain the rotation time of a spiral, can be further expressed as functions of kink density ρi and kink rate ui , which have different expressions in different solutions. In the end, the values of kink density and kink rate for each edge and F-face can be give by the kink attachment rate (eq.(8)) and the kink detachment rate (eq.(9)) via the calculation of detachment work ∆Wk . Using the intermolecular bond energies and the theory of periodic bond chains, the identified F-faces of paracetamol are {110}, {011}, {200}, {20¯1}, {001} and {020}. However, not all of these F-faces will appear in the final morphology of paracetamol due to the different growth rates of that faces and the inclusion of solvents. For example, some F-faces with relative large growth rates will disappear during the crystal growth and the solid-solvent

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interaction will also make the crystal grow out of the final morphology. The steady state morphologies of paracetamol are shown in Fig.6-9, where the various shapes are predicted from 30 different solvents at a supersaturation of 4%. The supersaturation of 4% equals to S (in eq.8) and this is realistic for all solvents. The main reason that shapes change with supersaturation is because certain families of faces enter different growth regimes (e.g., spiral growth, 2D nucleation and growth or rough growth). 1 The spiral growth occurs at low supersaturation, while the 2D nucleation and growth is more favorable at higher supersaturation. When faces enter these regimes, their growth accelerates faster with supersaturation and so upon further increasing supersaturation we get a changing morphology (faces growing out, extending aspect ratio etc). In the present paper, the paracetamol morphology was predicted by the spiral growth model for all faces in spiral regime, so the change of supersaturation has no effect on the results. In table 1, the solvents are divided into 6 classes according to their different functional groups. Fig.6-9 show that paracetamol grows into similar shapes within each class of solvents. Fig.10 compares predicted paracetamol morphologies against with experimentally observed shapes grown in vapor and water, 52 indicating excellent agreement. Table 1: Solvent classification according to different functional groups of solvents.

class 1

class 2

Solvents’ type hydrogen bond donating solvents, including hydroxyl group hydrogen bond accepting solvents

class 4

polar solvents without hydroxyl group chlorocarbon solvents

class 5

hydrocarbon solvents

class 6

water

class 3

List of solvents methanol, ethanol,propanol, formic acid, 1-butanol, 2-butanol propylene glycol, ethylene glycol, acetic acid methyl acetate, n-butyl acetate, acetone methyl isobutyl ketone, tetrahydrofuran ethyl acetate, methyl ethyl ketone, nitrobenzene dimethylsulfoxide, dimethylformamide, acetonitrile chlorobenzene n-heptane, toluene, pentane, xylene methyl cyclohexane, benzene, cyclohexane water

Fig.11 shows the aspect ratio of paracetamol crystal as a function of solvent dispersive 15



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Page 16 of 36

surface energy(γsd ) with different class of solvents. This implies that different solvents with different functional groups will have different impact on the solvent surface energy during the crystal growth. The aspect ratio is defined as the ratio of the maximum distance to ¯ k¯¯l). The blue dots in Fig.11, the minimum distance between opposite faces (hkl and h corresponding to the class 2 of solvents, have the cubic-like crystal shape and lie in the bottom of aspect ration figure. While the orange dots, corresponding to the classes 4,5,6 of solvent, have the rod-like crystal shape and lie in the top area of Fig.11. Therefore, large aspect ratio generally predicts elongated shape of a crystal, while a small aspect ratio predicts cubic-like shape. Fig.11 shows that solvent’s functional groups, governing the solvent dispersive energy, strongly determine the crystal morphology within various solvents. In Fig.11, the dots in the lower dispersive energies correspond to paracetamol crystals with lower aspect ratios (< 3), which grown from the solvents of class 2 (hydrogen bond accepting solvents). The higher aspect ratios of paracetamol crystal (∼ 5) correspond to the higher solvent dispersive energies, which belong to the solvent classes 4,5,6. We further present the shape diagram (SD) of paracetamol grown in 30 different solvents as shown in Fig.12. Different colors in morphology map represent the crystal with different faces. Some faces will appear or disappear by moving the loci from one color to another different color. According to the Frank-Chernov condition, 19,20 SD in the space of the relative growth rates is a representation of the accessible habits obtainable from a given subset of crystal faces. 53 If a crystal have more than 3 F faces, SD can be applied in a two-dimensional space defined by the picked growth rate ratios. For example, the sublimation growth shape of paracetamol in Fig.9 shows that there are four F faces, they are {110}, {011}, {20¯1} and {001}. Fig.12 was generated by picking three of these F faces (the face {011} does not appear in the morphology), where the horizontal axis is lg(R{001} /R{110} ) and the vertical axis is lg(R{20¯1} /R{110} ). Paracetamol grown in different solvents lies in different areas of SD, which results in different growth rate ratios of R{001} and R{20¯1} as shown in Fig.12. The boundaries of these

16


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habit domains represent the loci of points in the parameter space in which a new set of faces appears or disappears. By moving any of these points to other areas, the growth rate ratios of (R{001} /R{110} ) and {R{20¯1} /R{110} ) make the change, which results in a different crystal shape with a different solvent. Although the crystal faces are maintained in any given domain of the diagram, the aspect ratio of crystals within each domain may change. For the detailed information regarding to the SD, please refer to the published paper by Matteo et al. 53

4

Conclusion

Crystal growth design is a highly interested topic for pharmaceutical industries and scientists who use crystalline systems to manufacture products and perform researches. By taken into account the impacts of temperature, supersaturation, concentration, and other imposed growth conditions, we have presented a prediction of steady state morphologies of paracetamol crystal grown from 30 different solvents, according to the mechanistic spiral growth(MSG) model. Based on this high-fidelity morphological theory, the predicted paracetamol morphologies match their experimental shapes very well in vapor and water growth, which demonstrated the accuracy of our theory. The aspect ratios and the shape diagram show the crystal habits of paracetamol crystal within different solvents. The extension of the proposed method to other pharmaceutical molecules provides a promising platform for further drug manufacturing developments.

5

Acknowledgement

We are grateful for the financial support provided by Rhodia, Merck, Eli Lilly, Pfizer, Bristol-Meyers-Squibb, Novartis, and the National Natural Science Foundation of China (No.51672176).

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References (1) Li, J.; Tilbury, C.J.; Kim, S.H.; Doherty, M.F. A design aid for crystal growth engineering. Prog. Mater. Sci. 2016, 82, 1-38. (2) Schmidt, C.; Ulrich, J. Morphology prediction of crystals grown in the presence of impurities and solvents-an evaluation of the state of the art. J. Cryst. Growth 2012, 353, 168-73. (3) Dandekar, P.; Doherty, M.F. Imaging crystallization. Science, 2014, 344, 705-706. (4) Dandekar, P.; Kuvadia, Z.B.; Doherty, M.F. Engineering crystal morphology. Annu. Rev. Mater. Res. 2013, 43, 359-386. (5) Lovette, M.A.; Doherty, M.F. Needle-shaped crystals: Causality and solvent selection guidance based on periodic bond chains. Cryst. Growth Des. 2013, 13, 3341-3352. (6) Panina, N.; Van de Ven, R.; Janssen, F.; Meekes, H.; Vlieg, E.; Deroover, G. Study of the needle-like morphologies of two β-Phthalocyanines. Cryst. Growth. Des 2009, 9,840-7. (7) Freitas, F.; Sarmento, V.; Santilli, C.; Pulcinelli, S. Controlling the growth of zirconia needles precursor from a liquid crystal template. Colloid Surface A 2010, 353,77-82. (8) Giri, G.; Park, S.; Vosgueritchian, M.; Shulaker, M.M.; Bao, Z. High-mobility, aligned crystalline domains of TIPS-pentacene with metastable polymorphs through lateral confinement of crystal. Adv. Mater. 2014, 26,487-93. (9) Pacifici, G.M.; Allegaert, K. Clinical pharmacology of paracetamol in neonates: a review. Curr. Ther. Res. Clin. 2015, 77, 24-30. (10) McNicol, E.D.; Ferguson, M.C.; Haroutounian, S.; Carr, D.B.; Schumann, R. Single dose intravenous paracetamol or intravenous propacetamol for postoperative pain. 2016, 5, CD007126. 18


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(11) Roberts, E.; Nunes, V.D.; Buckner, S.; Latchem, S.; Constanti, M.; Miller, P.; Doherty, M.; Zhang, W.Y.; Birrell, F.; Porcheret, M. Paracetamol: not as safe as we thought? A systematic literature review of observational studies. Ann. Rheum. Dis. 2016, 75, 552-559. (12) Ralapanawa, U.; Jayawickreme, K.P.; Ekanayake, E.M.M.; Dissanayake, A.M.S.D.M. A study on paracetamol cardiotoxicity. BMC Pharmacol. Toxicol. 2016, 17, 30. (13) Perez-Alzate, D.; Blanca-Lopez, N.; Dona, I.; Agundez, J.A.; Garcia-Martin, E.; Cornejo-Garcia, J.A.; Perkins, J.R.; Blanca, M.; Canto, G. Asthma and rhinitis induced by selective immediate reactions to paracetamol and non-steroidal anti-inflammatory drugs in aspirin tolerant subjects. Front. Pharmacol. 2016, 7, 215. (14) Ganesan, V.; Jayaraman, A. Theory and simulation studies of effective interactions, phase behavior and morphology in polymer nanocomposites. Soft Matter 2014, 10, 13-38. (15) Aksimentiev, A.; Fialkowski, M.; Holyst, R. Morphology of surfaces in mesoscopic polymers, surfactants, electrons, or reaction-diffusion systems: Methods, simulations, and measurements. Adv. Chem. Phys. 2002, 121, 141-239. ´ (16) Bravais, A. Etudes crystallographic. Paris: Gauthier Villars; 1866. (17) Friedel, M. Etudes sur la loi de Bravais, Bulletin de la Societe Francaise. Mineralogique 1907,30,326-455. (18) Donnay, J.; Harker, D. A new law of crystal morphology extending the law of Bravais. Am. Mineral. 1937, 22, 446-67. (19) Frank, F.C. On the kinematic theory of crystal growth and dissolution processes. In: Growth and perfection of crystals. New York: John Wiley; 1958, p.411-9.

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(20) Chernov, A. The kinetics of the growth forms of crystals. Sov. Phys. Cryst. 1963, 7, 728-30. (21) Hartman, P.; Bennema, P. The attachment energy as a habit controlling factor: I. Theoretical considerations. J. Cryst. Growth 1980, 49,145-56. (22) Coombes, D.S.; Catlow, R.A.; Gale, J.D.; Rohl, A.L.; Price, S.L. Calculation of attachment energies and relative volume growth rates as an aid to polymorph prediction. Cryst. Growth Des. 2005, 5, 879-85. (23) Kuvadia, Z.B.; Doherty, M.F. Effect of structurally similar additives on crystal habit of organic molecular crystals at low supersaturation. Cryst. Growth Des. 2013, 13, 1412-1428. (24) Kuvadia, Z.B.; Doherty, M.F. Reformulating multidimensional population balances for predicting crystal size and shape. AIChE J. 2013, 59, 3468-3474. (25) Joswiak, M.N.; Duff, N.; Doherty, M.F.; Peters, B. Size-dependent surface free energy and tolman-corrected droplet nucleation of TIP4P/2005 water. J. Phys. Chem. Lett. 2013, 4, 4267-4272. (26) Vetter, T.; Burcham, C.L.; Doherty, M.F. Attainable regions in crystallization processes: their construction and the influence of parameter uncertainty. Comput. Aid. Chem. Eng. 2014, 34, 465-470. (27) Dandekar, P.; Doherty, M.F. A mechanistic growth model for inorganic crystals: solidstate interactions. AIChE 2014, 60, 3707-3719. (28) Dandekar, P.; Doherty, M.F. A mechanistic growth model for inorganic crystals: growth mechanism. AIChE 2014, 60, 3720-3731. (29) Kim, K.J.; Doherty, M.F. Crystallization of selective polymorph using relationship between supersaturation and solubility. AIChE 2015, 61, 1372-1379. 20


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(30) Haisa, M.; Kashino, S.; Kawai, R.; Maeda, H. The Monoclinic Form of pHydroxyacetanilide. Acta Crystallogr. Sect. B: Struct. Crystallogr. Cryst. Chem. 1976, 32, 1283-2085. (31) Zhang, Y.; Doherty, M.F. Simultaneous prediction of crystal shape and size for solution crystallization. AIChE J. 2004, 50, 2101-12. (32) Zhang, Y.; Sizemore, J.P.; Doherty, M.F. Shape evolution of 3-dimensional faceted crystals. AIChE J. 2006, 52, 1906-15. (33) Snyder, R.C.; Doherty, M.F. Predicting crystal growth by spiral motion. P. Roy. Soc. A-Math. Phy. 2009, 465, 1145-71. (34) Giacovazzo, C.; Monaco, H.L.; Artioli, G.; Viterbo, D.; Milanesio, M.; Gilli, G.; Gilli, P.; Zanotti, G.; Catti, M. Fundamentals of crystallography. Third Edition. pp.71 Oxford Univeristy Press Inc., New York 2011. (35) Eyring, H. The activated complex in chemical reactions. J. Chem. Phys. 1935, 3, 107115. (36) Kim, S.H.; Dandekar, P.; Lovette, M.A.; Doherty, M.F. Kink rate model for the general case of organic molecular crystals. Cryst. Growth Des. 2014, 14, 2460-2467. (37) Lovette, M.A.; Doherty, M.F. Predictive modeling of supersaturation-dependent crystal shapes. Cryst. Growth Des. 2012, 12, 656-669. (38) Wang. J.; Wolf, R.M.; Caldwell, J.W.; Kollman, P.A.; Case, D.A. Development and testing of general amber force field. J. Comput. Chem. 2004, 25, 1157-1174. (39) Weiner, P.K.; Kollman, P.A. AMBER: Assisted model building with energy refinement. A general program for modeling molecules and their interactions. J. Comput. Chem. 1981, 2, 287-303.

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(40) Frisch, M. J. Gaussian 03, Revision C.02. 2003. (41) Li, T.; Feng, S. Empirically augmented density functional theory for predicting lattice energies of aspirin, acetaminophen polymorphs, and ibuprofen homochiral and racemic crystals. Pharm. Res. 2006, 23, 2326-2332. (42) Kuvadia, Z.B.; Doherty, M. F. Spiral growth model for faceted crystals of noncentrosymmetric organic molecules grown from solution. Cryst. Growth. Des. 2011, 11, 2780-2802. (43) Tilbury, C.; Daniel, G.; Will, M. Doherty, M.F. Predicting the effect of solvent on the crystal habit of small organic molecules. Cryst. Growth Des. 2016, 16,2590-2604. (44) Hildebrand, J.; Scott, R. The Solubility of Nonelectrolytes, 3rd ed.; Reinhold Publishing Corporation: New York, 1950. (45) Hildebrand, J. H.; Scott, R. L. Regular Solutions; Prentice-Hall: Upper Saddle River, NJ, 1962. (46) Gardon, J. L. Critical review of concepts common to cohesive energy density, surface tension, tensile strength, heat of mixing, interfacial tension, and butt joint strength. J. Colloid Interface Sci. 1977, 59, 582-596. (47) Barton, A. F. M. Solubility parameters. Chem. Rev. 1975, 75, 731-753. (48) Barton, A. F. CRC Handbook of Solubility Parameters and Other Cohesion Parameters; CRC press: Boca Raton, FL, 1991. (49) Beerbower, A. Surface free energy: a new relationship to bulk energies. J. Colloid Interface Sci. 1971, 35, 126-132. (50) Hartman, P.; Perdok, W. G. On the relations between structure and morphology of crystals. I. Acta Crystallogr. 1955, 8, 49-52. 22


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(51) Hartman, P.; Perdok, W. G. On the relations between structure and morphology of crystals. III. Acta Crystallogr. 1955, 8, 525-529. (52) Ristic, R.; Finnie, S.; Sheen, D.; Sherwood, J. Macro- and micromorphology of monoclinic paracetamol grown from pure aqueous solution. J. Phys. Chem. B 2001, 105, 9057-9066. (53) Salvalaglio, M.; Vetter, T.; Mazzotti, M.; Parrinello, M. Controlling and predicting crystal shapes: the case of urea. Angew. Chem. Int. Ed., 2013, 52, 13369-13372.

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G6 G5 G1 G2 = = = = H6 H 5 H1 H2

G1 G6

H1

H6

H2 G5

G2

H5 H4

H3

G3 G4

Figure 1: The Frank-Chernov condition to predict the steady-state growth morphology, assuming there are 6 F faces for a crystal and each face has a growth rate of Gi .

24


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E jk,ij+

Detachment Attachment

ΔU

State B q

State A ΔWk

Figure 2: Schematic energy barrier for attachment and detachment of solute growth units from kink sites, where q is distance of the growth unit from the kink site. The filled red circles represent the growth units which are attaching to/detaching from the kink site and the open black circles represent the unoccupied kink sites. States A and B represent growth units − in solution and incorporated to the crystal, respectively; j + and jki are the corresponding attachment and detachment rates.

25



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(a)

Page 26 of 36

kink site with surface area si 4

3

4

3

aP,i 1

2

1

1

2

2

1

Kink detachment ΔWk

new kink site

(b)

kink rate u i

4

3

4

1

2

1

2

1

2

Step velocity

1

3

Solvated molecule

Figure 3: Illustration of the detachment process, where the growth unit 3 was detached from the edge to the solution and results in a new kink site on the top layer. The inverse process from (b) to (a) is the attachment process.

26


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(a)

4

A

3

B

3

4

2

1

A

B

3

4

4 -4.45kcal/mol

1

2

1

2

2

1

1

2

1

-2.72kcal/mol -0.5kcal/mol

4

3

4

3

3

4

3

4

4 -2.43kcal/mol

1

2

A

1

2 B

2

1

1

B

2

1

A

[010] edge on {001} face of paracetamol

(b)

aP,i

1

2

4

3

Figure 4: (a) Rearrangement for one pair of growth units on edge [010] of {001} face of paracetamol, which forms A? and B¦ in a new layer by taking out A? and B¦ from a lower layer. The new layer contains growth units 1 and 2, while the lower layer contains growth units 3 and 4. The four different kink sites are: A?, B¦, B?, A¦. The bonds with different colors have different bond strengths, which are shown in the right. (b)The corresponding paracetamol molecules to the actual crystal structure, where the molecules are colored by symmetry operation (the green, purple, yellow and white molecules correspond to the growth units 1, 2, 3 and 4, respectively).

27



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G hkl (a)

h

kink

y

aP,i

v step

(b)

Growth Rate

Selection of F-faces

( Gi = Step height (Eq.3): h =

hiν i = hi / τi ) y

1 rhkl

Rotation time of a sprial (Eq.5) N

τi = ∑

lc,i +1 sin(α i ,i +1 ) νi

i =1

Step velocity (Eq.6) νi = aP, i ui ρi Kink density (Eq.10) Kink rate (Eq.7)

Kink detachment rate (Eq.9)

Kink attachment rate (Eq.8)

j − k ,i = ν 0 exp((−∆U + ∆Wk ) / k BT )

j + = Sν 0 xeq exp(−∆U / k BT )

Vapor

Solid-solid interaction (by GAFF)

Solid-solid interaction (by GAFF)

Solvent

Solvent-solvent interaction (Eqs.19, 21 and 23) Solid-solvent interaction (Eqs.13 and 14)

Figure 5: (a)Illustration of parameters in MSG model for a crystal structure with a step height h (usually the interplanar spacing), the inter-step distance y and the distance aP,i . The forward velocity (ν) of successive steps gives rise to the normal growth rate of the face Ghkl . (b)Algorithm of morphology prediction in MSG model, using the equations in the present paper.

28


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Solvent class 1

Formic acid

Methanol

1-Butanol

2-Butanol

Ethylene glycol

Propylene glycol

Ethanol

Acetic acid

Propanol

Figure 6: The predicted steady state shapes of paracetamol in class 1 of solvents including formic acid, methanol, ethanol, 1-butanol, 2-butanol, acetic acid, ethylene glycol, propylene glycol and propanol.

29



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Solvent class 2

Tetrahydrofuran

Amyl acetate

Acetone

Methyl acetate

Nitrobenzene

Methyl ethyl ketone

Methyl isobutyl ketone

Ethyl acetate

n-Butyl acetate

Figure 7: The predicted steady state shapes of paracetamol in class 2 of solvents including tetrahydrofuran, amyl acetate, acetone, methyl acetate, nitrobenzene, methyl ethyl ketone, methyl isobutyl ketone, ethyl acetate, and n-butyl acetate.

30


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Solvent class 3

Acetonitrile

Dimethylsulfoxide

Dimethylformamide

Figure 8: The predicted steady state shapes of paracetamol in class 3 of solvents including acetonitrile, dimethylsulfoxide and dimethylformamide.

31



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Solvent classes 4, 5, 6

Vapor

Water

Xylene

Benzene

Methyl cyclohexane

Toluene

Pentane

Chlorobenzene

n-Heptane

Figure 9: The predicted steady state shapes of paracetamol in vapor and classes 4,5,6 of solvents including water, pentane, xylene, benzene, chlorobenzene, methyl cyclohexane, toluene and n-heptane.

32


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(b) Observed shape of paracetamol in Vapor

(a) Predicted shape of paracetamol in Vapor

(d) Observed shape of paracetamol in Water

(c) Predicted shape of paracetamol in Water

Figure 10: A comparision between predicted and observed shapes 52 of paracetamol under sublimation growth ((a) and (b)) and when grown from water ((c) and (d)), indicating good agreement between MSG model and experiments in both cases.

33



6

Aspect ratio

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 34 of 36

5

Class1 of solvents

4

Class 2 of solvents Class 3 of solvents Classes 4,5,6 of solvents

3

2

1 5

10

15

20

25

30

Dispersive surface energy (erg/cm 2 ) Figure 11: Aspect ratios of predicted paracetamol crystal morphology as a function of solvent dispersive surface energy γsd . All the dots are colored by different classes of solvents. Class 1 refers to the hydrogen bond donating solvents, including hydroxyl group; class 2 refers to the hydrogen bond accepting solvents; class 3 represents the polar solvents without hydroxyl group; while classes 4,5,6 are the chlorocarbon solvents, hydrocarbon solvents and water, respectively.

34


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Morphology Map 2

lg(G{2,0,-1} /G{1,1,0} )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Vapor

1

0

Acetic acid -1

-2 -2

-1

0

1

2

lg(G {0,0,1} /G {1,1,0}) Water

Vapor

Methyl ethyl ketone

Butyl acetate

Methyl acetate

Benzene

Acetone

Toluene

Amyl acetate

n-Heptane

Dimethylsulfoxide

Xylene

Dimethylformamide

Chlorobenzene

1-Butanol

Pentane

2-Butanol

Chlorobenzene

Acetic acid

Methyl cyclohexane

Propanol

Propylene glycol

Acetonitrile

Water

Tetrahydrofuran

Ethylene glycol

Methanol

Nitrobenzene

Ethanol

Ethyl acetate

Formic acid

Methanol

Figure 12: Shape diagram (SD) of paracetamol in 30 different solvents. The accessible habit space of paracetamol crystals as a function of the relative growth rates of three relevant crystal faces, {20¯1}, {001} and {110}, is displayed. Different points in SD, having different crystal shapes, correspond to different solvents.

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Title: Steady State Morphologies of Paracetamol Crystal at Different Solvents Authors: Jinjin Li and Michael F. Doherty TOC graphic:

Synopsis: Mastering the crystal habits of paracetamol at different solvents plays an important role to dig the huge potential of its pharmacological functions. Here, the mechanistic spiral growth model and the shape diagram are proposed to predict and explain the shape variations of paracetamol crystal grown from 30 different solvents with reasonable accuracy.


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Steady State Morphologies of Paracetamol Crystal from Different

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