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Proc. R. Soc. A (2009) 465, 1145–1171
doi:10.1098/rspa.2008.0234
Published online 6 January 2009
Predicting crystal growth by spiral motion
B Y R YAN C. S NYDER
AND
M ICHAEL F. D OHERTY *
Department of Chemical Engineering, University of California Santa Barbara,
Santa Barbara, CA 93106, USA
We present a systematic modelling methodology using the spiral growth mechanism of
Burton, Cabrera and Frank to predict the steady-state shape of organic molecular
crystals grown from solution. This methodology has been developed to eliminate the
need for special modifications for each new crystal system studied. Therefore, the
mechanisms and choices for spiral shapes, edges and evolution are mathematically
determined as governed by the underlying solid-state chemistry and physics. The power
of the approach is demonstrated for several crystal systems: naphthalene grown from
both ethanol and cyclohexane; anthracene grown from 2-propanol; and glycine grown
from water. The predicted crystal shapes are in good agreement with experiment.
Keywords: crystal growth; spiral growth; crystal shape
1. Introduction
It has been nearly 60 years since Burton et al. (1951) first published their
landmark work, predicting the growth of crystals via a spiral mechanism
emanating from a screw dislocation. Since that time, many detailed experiments
have observed this mechanism in action for a wide range of crystallizing
materials (Yip & Ward 1996; Paloczi et al. 1998; Vekilov & Alexander 2000;
Sours et al. 2005; Ranguelov et al. 2006). As a result, the Burton, Cabrera and
Frank (BCF) spiral growth mechanism is recognized as the most important
modelling tool in the hands of crystal growers. The model itself has undergone
improvements over the years; nonetheless, its implementation remains
essentially an art owing to the large number of intuitive decisions that must
be made, in particular for faceted crystals. In this paper, we report a systematic
procedure for implementing the BCF spiral growth mechanism to predict
crystal morphology.
Prediction of crystal shape continues to be an important topic of scientific
interest owing to the impact that crystal shapes have on the properties of
materials, including dyes, catalysts, semiconductors and pharmaceuticals.
Crystal shape affects both the performance and processing of the material.
Different crystal faces can have different chemical properties such as surface
reactivity or hydrophobicity, and, since the crystal shape determines the relative
* Author for correspondence ([email protected]).
Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2008.0234 or via
http://journals.royalsociety.org.
Received 5 June 2008
Accepted 28 November 2008
1145
This journal is q 2009 The Royal Society
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1146
R. C. Snyder and M. F. Doherty
area of each face, it also determines the magnitude of these properties.
Additionally, bulk dissolution rate and mechanical properties are dependent on
crystal shape. The crystal shape can also impact downstream processes such as
filtering, washing or drying, as well as slurry viscosity and bulk density. Because
the desired shape is often specific to the industry and the application for each
particular crystal system, a general methodology for predicting shape as a
function of the important design parameters is desired.
The first models that were used to predict crystal growth shapes can be traced
to the Bravais–Friedel–Donnay–Harker model (Bravais 1886; Fridel 1907;
Donnay & Harker 1937), where the only input for crystal shape prediction is
the interplanar spacing. Later, Hartman & Perdok (1955) developed the
attachment energy model to explicitly include the energetic interactions required
to attach a new slice of molecules to a surface. In the past decade, the stateof-the-art methods in modelling have progressed to include the actual
mechanisms for growth. Simulation techniques include the Monte Carlo
algorithm MONTY (Boerrigter et al. 2004) and the kinetic Monte Carlo studies
of Piana & Gale (2005) and Gilmer (Zepeda-Ruiza et al. 2006). Additionally,
a class of mechanistic kinetic models, based upon the methods of Burton et al.
(1951) and Chernov (1984), was developed by Winn & Doherty (1998). These
kinetic models incorporate the effect of solvent, as well as the anisotropic
energetic interactions within crystals (Heng et al. 2006), and have the potential
to be implemented rapidly, making them uniquely suited to both product and
process development. Despite these advances in first-principles modelling, there
are still significant challenges to overcome before they will be widely used.
In this paper, we report the next-generation spiral growth model for faceted
crystals. The model significantly reduces the need for special modifications to be
made for each crystal system, thus unifying the theory. Unification is achieved by
strengthening the connection between spiral motion and the underlying solidstate chemistry. The model has been automated, which enables future
consideration of more complicated systems such as complex pharmaceutical
molecules, co-crystals and the effect of mixed solvents and additives.
New methodologies are reported in several areas. The model specifically
accounts for crystal geometry (distance of growth propagation, detailed edge and
kink area calculations and step heights) along with the physical basis for its
inclusion. The inclusion of important intermolecular interactions is based solely
upon energetics rather than distances, improving on previous formulations
(Winn & Doherty 1998). Also, the selection of step edges to include in the spiral
growth rate calculation is mathematically determined. The critical lengths of
edges on spirals are now determined using a model based on the open geometry of
a spiral rather than the classical Gibbs–Thomson analysis of the equivalent
closed two-dimensional shape. Not only is each of these new methodologies
incorporated into the model and algorithm in order to reduce the number of ad
hoc decisions, but is also applied uniformly across each crystal system, thus
providing for systematic predictions. Despite this added modelling fidelity, the
full calculation can still be performed rapidly (tens of seconds to a few minutes)
using an automated MATLAB program on a desktop PC with a single processor
that takes crystallographic and molecular interaction input data and generates
the resulting steady-state crystal shape, as well as all intermediate data.
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Predicting crystal growth by spiral motion
1147
This paper first describes the modelling theory with an emphasis on new aspects
of spiral dynamics. Then, two example systems, which have not previously been
studied with these kinetic models, are reported and compared with the
experiment: naphthalene grown from cyclohexane and ethanol, and anthracene
grown from 2-propanol. Finally, we perform a calculation for a-glycine grown from
water. This calculation was previously performed (Bisker-Leib & Doherty 2003)
with many of the solid-state chemistry decisions argued intuitively rather than
systematically determined.
2. Growth spirals
For crystals growing by the steady flow of steps across surfaces, the mathematical
expression for the rate of growth normal to a crystal surface (hkl) is
ðv Þ h
Gh k l Z i h k l hk l ;
ð2:1Þ
ðyi Þhk l
where vi is the perpendicular step velocity across the surface in the ith edge
direction; h is the step height; and yi is the interstep distance. Each of these
quantities is face dependent; however, the subscript hkl will be dropped for the ease
of presentation. Prediction of the growth rate of a face growing by the spiral
mechanism thus requires the independent determination of each of these quantities
for each crystal face. One alternative expression for equation (2.1), which is often
convenient for interpreting experiments (Sours et al. 2005), is
G Z v i ri ;
ð2:2Þ
where ri is the slope of the spiral hill averaged over many steps. It should be noted
that the normal growth rate of a given face is independent of the edge direction
selected in equations (2.1) or (2.2). (While vi and yi are dependent on the edge, their
ratio is not; thus, the growth rate is the same for each value of i.) Equation (2.1) can
be reformulated as
h
GZ ;
ð2:3Þ
t
where t is the characteristic rotation time of the spiral. This characteristic time
corresponds to the time between consecutive passes of a step across a point on the
face. The characteristic spiral rotation time can be calculated by the time that it
takes for each edge to appear in one full first turn of the spiral. Beyond the first
turn, the time it takes for yet another pass of a step to cross the same point on the
face is determined solely by the creation of another first turn. Thus, the rotation
time is only relevant during the first turn. In some complex systems, consecutive
layers of a crystal can have alternating properties, including rotation time. In these
cases, it is necessary to evaluate a number of turns corresponding to one full
grouping of the growth pattern. From a modelling perspective, equation (2.3) is the
most important and convenient formulation.
We begin by assuming that the velocity of a step is governed by
9
v Z 0; l % l c ; =
and
ð2:4Þ
;
v Z v N; l O l c ;
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1148
R. C. Snyder and M. F. Doherty
which was first proposed by Kaischew and later by Voronkov (Kaischew &
Budevski 1967; Voronkov 1973). This profile indicates that, when a spiral edge is
shorter than its critical length, it will not grow at all, and once it is above its
critical length, it will grow with a constant step velocity. The concept of a critical
length for growth has been experimentally validated for calcite (Teng et al. 1998)
and lysozyme (Chernov et al. 2005). These results also demonstrate that
equation (2.4) is a first good approximation for the velocity profile. Voronkov’s
conditions in equation (2.4) are not reversible; if an edge begins to shrink in
length, its perpendicular velocity will continue to be vN as the edge length passes
below lc and smoothly shrinks to zero.
(a ) Spiral edge determination and first turn dynamics
For a convex spiral, the rotation time is given by
tZ
N
X
l c;iC1 sinðai;iC1 Þ
iZ1
vi
;
ð2:5Þ
where ai,iC1 is the angle between edges i and iC1; l c,iC1 is the critical length of
edge iC1; and N is the number of relevant spiral edges. (N will be some or all of
the total set of M edges on a given face. A method for determining which edges in
M to include in N follows.) We define a numbering scheme in which the first edge
to grow is labelled 1 and the last is N. Therefore, side i exposes side iC1. This
numbering scheme is used for the spirals rotating in either a clockwise or
anticlockwise direction. The rotation direction is defined by the advancement of
the outward normal of edge iC1 relative to that of edge i. Equations (2.3) and
(2.5) are used to predict the growth rate of a crystal face. Thus, rather than
directly predicting the step velocity and interstep distance for a single-step
direction, as required for equation (2.1), we predict the critical length and the
step velocity for each edge of the spiral. (The interstep distance is then given by
yiZvit.) Additionally, we must determine the correct number of edges to include
in the set N. Once N is known and the crystallography is given, the angles
between these spiral edges, ai,iC1, are fixed and easily determined. As shown by
equation (2.3) and equation (2.5), the growth rate of a face is determined by the
characteristic time for the first turn of the spiral. Thus, we first focus attention
on the initial spiral turn, its dynamics and which edges of the spiral are
important for the calculation of t. Additionally, the model for the first turn
extends to incorporate spirals away from the dislocation and leads to interesting
insight into their behaviour.
The shape of the spiral is determined by the relationship between the critical
lengths and the velocities of each spiral edge, as well as the angles between them.
Given the perpendicular step velocity of each spiral edge and the angles between
them, the corresponding tangential velocity (the rate at which the edge lengths
increase or decrease) of each spiral edge is also known. The tangential velocity of
a spiral edge i is given by
v ti Z
Proc. R. Soc. A (2009)
v iC1 Kv i cosðai;iC1 Þ v iK1 Kv i cosðaiK1;i Þ
C
;
sinðai;iC1 Þ
sinðaiK1;i Þ
ð2:6Þ
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Predicting crystal growth by spiral motion
(a)
(b)
1149
(c)
not appeared
Figure 1. The evolution of a spiral showing each of the three stages of length change for edge i.
(a) Edge i is lengthening owing to the motion of edge iK1, but edge i has yet to move itself.
(b) Edge i and iK1 are both moving, edge iC1 is lengthening, but has yet to begin moving. (c) All
three edges are moving. The lateral velocity of edge i is different during each of the three different
stages. The filled circle represents the dislocation.
where ai,j is the angle between edges i and j; v ti is the tangential velocity of edge i;
and vi is the perpendicular step velocity of edge i. This is in direct analogy with
the representation of the edge of a two-dimensional crystal, as was first published
by Kozlovskii (1957). While a spiral is not a closed geometric object, in contrast
to a two-dimensional crystal, the calculation is only a function of the edge of
interest, along with each adjacent edge. Thus, the edge under consideration does
not have to be part of a closed object for this equation to be valid.
Since the step velocity profile is assumed to be governed by equation (2.4), each
edge of the spiral undergoes three different stages of tangential growth. Assuming
that an edge (iK1) already exists and its length is greater than its critical length
(liK1Ol c,iK1), these three stages (for edge i) correspond to (i) when edge i begins to
appear and its length is less than its critical length (viZ0) and edge iC1 has yet to
appear (viC1Z0) (figure 1a), (ii) when edge i is larger than its critical length and
edge iC1 has appeared but is less than its critical length (viC1Z0) (figure 1b), and
(iii) when edge i and iC1 are both larger than their critical lengths (figure 1c).
During the first stage, edge i is less than its critical length l c,i. Thus, the only
non-zero step velocity in equation (2.6) is viK1 (edge iC1 has yet to appear) and
the tangential velocity reduces to
v iK1
v t1;i Z
:
ð2:7Þ
sin aiK1;i
Once edge i is larger than its critical length, it begins to move, exposing edge
iC1, which is initially shorter than its critical length l c,iC1; thus viC1Z0. The
tangential velocity expression for edge i is then given by
Kv i cosðai;iC1 Þ v iK1 Kv i cosðaiK1;i Þ
v t2;i Z
C
:
ð2:8Þ
sinðai;iC1 Þ
sinðaiK1;i Þ
Finally, once edge iC1 begins to move, edge i obtains its final tangential velocity
given by equation (2.6) ðv t3;i Þ. During the second stage, the new edge iC1 is being
exposed and has a step velocity of zero. Thus, if the step velocity of edge i is
large, then edge i could have a negative tangential velocity during this second
phase (i.e. it shrinks in length), while it must have had a positive tangential
velocity during the first phase (and either a positive or negative tangential
velocity in the third stage). If step i moves too fast, it is even possible for edge i
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1150
R. C. Snyder and M. F. Doherty
to disappear during stage (ii) before edge iC1 has begun to move. If this is the
case, the spiral rotation time will not depend on edge i, and it should not be
included in the calculation of the spiral time in equation (2.5).
In order to mathematically determine whether spiral edge i disappears before
edge iC1 begins to move, a comparison is made between the time it would take
for edge i to disappear ðtdis Z ðKl c;i =v t2;i ÞÞ and the time it takes for edge iC1 to
reach its critical length ðtcrit Z ðl c;iC1 =v t1;iC1 ÞÞ. In some cases, t dis!0, implying
edge i would never disappear owing to the geometry of the spiral; thus, edge i
should be included in the set of N edges for calculating t. An example where
t dis!0 is a square spiral where v t2;i O 0 since the geometry (see equation (2.8)
with all angles equal to 908) does not allow for the tangential velocity of any edge
to ever be negative. In other cases, edge i may eventually disappear (t disO0). For
such cases, if the time for edge i to disappear is less than the time it takes for edge
iC1 to reach its critical length (t dis!tcrit), then edge i should not be included in
the spiral time calculation, otherwise it should be included. By equating these
two times and solving for the velocity of edge i, a velocity for edge inclusion can
be determined. This inclusion velocity for edge i of the spiral is given by
Kv iK1
v incl
Z l sinða Þ
:
ð2:9Þ
i
c;i
iK1;i
Kcosða
ÞKcotða
Þsinða
Þ
iK1;i
i;iC1
iK1;i
l c;iC1 sinðai;iC1 Þ
Thus, when the step velocity of edge i, vi, is greater than or equal to its inclusion
velocity, v incl
i , (and t disO0), it will disappear during phase 2 of its growth and
should not be included in the spiral rotation time calculation. Whereas if the step
velocity of edge i is less than this inclusion velocity (or if t dis!0), it should be
included in the spiral rotation time calculation. Thus, the number of edges N to
include in the spiral rotation time equation is equal to the number of edges that
pass the inclusion velocity test of equation (2.9), and the crystal growth rate
prediction reduces to a prediction of critical lengths and velocities for each spiral
edge on each crystal face.
Figure 2 demonstrates the differences between two spirals, each with six edges
considered, but one which contains two edges that do not persist beyond the
growth of the following edge. Figure 2a corresponds to the resulting spiral for the
edge velocities (in arbitrary units) v1Zv4Z2.0, v2Zv5Z1.0 and v3Zv6Z1.5, with
critical lengths l c,1Zl c,4Z2.0, l c,2Zl c,5Z3.0 and l c,3Zl c,6Z4.0 and angles a1,2Z
a4,5Z508, a2,3Za5,6Z608 and a3,4Za6,1Z708. Figure 2b corresponds to the
resulting spiral for v1Zv4Z2.0, v2Zv5Z1.5 and v 3Zv 6Z1.0, with critical lengths
l c,1Zl c,4Z2.0, l c,2Zl c,5Z4.0 and l c,3Zl c,6Z3.0 and angles a1,2Za4,5Z708, a2,3Z
a5,6Z608 and a3,4Za6,1Z508. For the spiral in figure 2a, each of the edges of the
spiral passes the test to determine whether it should be included in the
calculation of t. However, for the spiral in figure 2b, the velocities of edges 1 and
4 are greater than those of the inclusion velocities (and t disO0); thus, these two
edges are not included in the calculation of t for that spiral. The spiral rotation
time for the first spiral is then given by equation (2.5) with NZ6 (tx11.7),
whereas that for the second spiral is given by
l c;3 sinða2;3 Þ l c;5 sinða3;5 Þ l c;6 sinða5;6 Þ l c;2 sinða6;2 Þ
tZ
C
C
C
ð2:10Þ
v2
v3
v5
v6
(tx10.4). The smaller value of the rotation time leads to a larger value of the
growth rate, Gi, for the same face.
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Predicting crystal growth by spiral motion
(a)
1151
(b)
1
1
2
6
5
2
6
3
5
4
3
4
Figure 2. (a,b) Two clockwise spirals are presented with identical edge orientations, but different
velocities and critical lengths for each edge. Note that two symmetry related edges in (b) have
disappeared during their second stage of growth and are thus not included in the rotation time for
that spiral.
(b ) Spiral shape evolution
Based upon equations (2.3) and (2.5), the growth rate of a crystal face is
determined purely by the dynamics of the first turn. Once the first turn is
completed, the interstep distance in each edge direction between one rotation
and the next is fixed. Also, the step velocity is assumed to be constant when an
edge is longer than its critical length and the height of a step is constant.
According to equation (2.1), the face growth rate does not depend on anything
beyond the first turn, since each of the terms in the equation is fixed.
Nonetheless, the shape of the spiral evolves as it moves away from the dislocation
at its centre, and eventually achieves a steady-state spiral shape. Spiral edges can
also disappear during this evolution process after they reach their third stage of
growth (where edge iC1 has also begun to grow), but it is rather unlikely that
new edges would appear without a change in the system (e.g. a change in
temperature or the addition of a surface active impurity into the solution).
Nevertheless, even the disappearance of spiral edges in (and beyond) their third
stage of growth will not affect the normal growth rate of a face.
Since the tangential velocity of a spiral edge is determined by the same
expression as the tangential velocity of a two-dimensional crystal, the critical
step velocity that determines whether a spiral edge will disappear at some
distance away from the centre of the spiral (beyond stage (ii) of its growth) is
also given by the same expression. This critical step velocity for the
disappearance of a spiral edge beyond its second stage of growth is given by
v crit
Z
i
v iC1 sinðaiK1;i Þ C v iK1 sinðai;iC1 Þ
;
sinðaiK1;i C ai;iC1 Þ
ð2:11Þ
which is identical to the expression derived by Szurgot & Prywer (1991) for the
two-dimensional crystals. As a spiral evolves in shape and extends outwards
from the dislocation, a steady-state spiral shape will emerge by analogy to
the steady-state growth shapes seen for bulk crystal shapes (in both two and
three dimensions).
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R. C. Snyder and M. F. Doherty
The simplest expression for a spiral shape evolution model is based on
following the length of each edge in time,
dli
ð2:12Þ
Z v ti ; i Z 1; .; M ;
dt
where M is the number of spiral edges that could exist in the first turn and li and
v ti are the length and tangential velocity of edge i, respectively. The tangential
velocity is determined by equations (2.6)–(2.8). This spiral shape evolution
model tracks the shape of one turn of a spiral from its initial formation as it
evolves outwards away from the dislocation. Thus, in order to generate the full
spiral from the set of li, equation (2.12) must be solved for many turns of the
spiral and they must be placed end to end with the corresponding angle between
each of the spiral sides. Nonetheless, it is only necessary to follow the dynamics of
one turn (i.e. one revolution) of edges as they evolve away from the dislocation
because each additional turn will follow identical dynamics while lagging in time
by an integer multiple of the rotation time, t. This spiral evolution model has no
apparent steady state; however, by non-dimensionalizing the variables, steadystate features of the model are revealed. Gadewar & Doherty (2004) derived a
similar set of equations for the two-dimensional crystals growing in time. The
characteristic quantities used to define dimensionless variables for our system
need to be chosen appropriately for a growing spiral. The characteristic length
P is
chosen to be the perimeter of the turn of the spiral being considered, LZ M
iZ1 li
(the total length of all of the edges in one set M). In order to monitor the spiral
evolution, a set of edges M is tracked as they move away from the dislocation. As
the spiral moves away from the dislocation, L continually increases. The
characteristic velocity is defined
Pas thet sum of all of the tangential velocities of all
of the edges in that turn, V Z M
iZ1 v i . Thus, the dimensionless length is defined
by xiZli/L and the dimensionless velocity is defined by ui Z v ti =V .
Using these dimensionless variables, the dynamic model is reformulated as
follows:
dxi
V
Z ðui Kxi Þ; i Z 1; .; M ;
ð2:13Þ
L
dt
and
dL
ZV:
dt
ð2:14Þ
A dimensionless warped time can then can be defined, dxZ ðV =LÞdt, and
equation (2.13) can be rewritten in fully dimensionless form,
dxi
Z ui Kxi ;
dx
i Z 1; .; M :
ð2:15Þ
The steady state of this system corresponds to the condition when each of the
states of the model, xi, is at a value that does not change with time. The steady
state occurs when
ui;ss Z xi;ss ;
i Z 1; .; M :
ð2:16Þ
When the step velocities (vi) are constant in time, then ðv ti Þ, and hence ui are also
constant beyond stage (ii) of each edge’s growth. Under these constant relative
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Predicting crystal growth by spiral motion
0.3
0.2
0.1
0
2
4
6
8
rotation number
10
12
14
Figure 3. The relative length of each of the three characteristic edges (solid line, edge 1; dot-dashed
line, edge 2; dotted line, edge 3) of the spiral in figure 2a as they grow away from the dislocation.
Each relative length approaches a steady state, as predicted by the model for spiral shape evolution.
velocity conditions, the model is a linear system of ordinary differential
equations, with all the eigenvalues equal to K1. It follows that the steadystate spiral shape (which corresponds to the shape of the spiral far away from the
dislocation) is unique and stable.
The steady-state nature of the spiral shape is demonstrated for the spiral shown
in figure 2a. Figure 3 shows the dimensionless length, xi, of each of the three
characteristic spiral edges (1–3), from their initial appearance at the dislocation and
as they move outward away from the spiral. In the initial stages of the first rotation,
the tangential velocities transition through their three stages of growth; thus, the
curves are not smooth throughout that process. Far away from the dislocation, the
dimensionless length approaches the steady state (u1,ssZx1,ssx0.027; u2,ssZ
x2,ssx0.188; u3,ssZx3,ssx0.285). This highlights the stable steady state that all
spirals characteristically evolve towards as they emanate from their dislocations.
This model for spiral shape is general, in that it can also be used for more complex
velocity profiles to determine the growth rate of the crystal face.
(c ) Spiral rotation direction
For an organic molecular system, the exposed chemical moieties along each edge
of a given face are often different from one another. Thus, the edges have different
velocities and critical lengths. Since the spiral direction will dictate which edge
velocity is paired with the corresponding next edge’s critical length, the growth rate
resulting from the clockwise and anticlockwise spiral rotation directions can be
different (see equations (2.3) and (2.5)). Additionally, different rotation directions
could result in the disappearance of different edges. While there could be some
preference to the direction of the spiral based upon the relative number of
dislocations in each of the rotation directions, the actual growth rate of the face
only depends on the most active spiral direction (Frank 1949). This is because the
most active, or fastest growing, spiral direction will overtake the slower growth
spiral direction. Thus, the characteristic spiral time of each spiral direction is
calculated and the smaller t, corresponding to the more active spiral (faster growth
rate), will determine the normal growth rate of that face.
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R. C. Snyder and M. F. Doherty
(a)
(b)
1
1
2
6
5
2
3
4
3
6
5
4
Figure 4. Two spirals are presented with each corresponding edge direction having the same step
velocities and critical lengths. The edges are taken in opposite order such that the spirals are
rotated in different directions: (a) clockwise and (b) anticlockwise. Since the spiral in (b) has a
smaller rotation time (faster growth rate), it is the dominant spiral on that face and is used to
determine the face’s normal growth rate.
Figure 4 shows the spiral shape for each of the two rotation directions for a
growth spiral. For this case, consider three independent edges (h, j and k) and thus
six possible spiral sides. They have velocities and critical lengths (in arbitrary
units) of vhZ2.0, vkZ1.0 and vjZ1.5, and l c,hZ2.0, l c,kZ3.0 and l c,jZ2.5, as well as
angles ah,kZak,hZ458, ak,jZaj,kZ658 and aj,hZah,jZ708. In both the clockwise and
anticlockwise cases, we begin with side 1 as h. Thus, the clockwise spiral of figure 4a
has its sides 1 and 4 as h, 2 and 5 as k, and 3 and 6 as j, giving velocities v1Zv4Z2.0,
v2Zv5Z1.0 and v3Zv6Z1.5, with critical lengths l c,1Zl c,4Z2.0, l c,2Zl c,5Z3.0 and
l c,3Zl c,6Z2.5 and angles a1,2Za4,5Z458, a2,3Za5,6Z658 and a3,4Za6,1Z708. The
anticlockwise spiral of figure 4b has its sides 1 and 4 as h, 2 and 5 as j and 3 and 6
as k, giving velocities v1Zv4Z2.0, v2Zv5Z1.5 and v3Zv6Z1.0, with critical lengths
l c,1Zl c,4Z2.0, l c,2Zl c,5Z2.5 and l c,3Zl c,6Z3.0 and angles a1,2Za4,5Z708, a2,3Z
a5,6Z658 and a3,4Za6,1Z458. In this case, the dominant growth spiral (thus the one
that determines the face’s normal growth rate) is the anticlockwise spiral in
figure 4b, since its rotation time (tanticlockx8.80) is less than that of the clockwise
spiral in figure 4a (tclockx9.16).
3. Predicting faces, edges, edge height, critical length and step velocity
In order to implement equations (2.3) and (2.5) to determine the crystal growth
rates, we now need to predict which faces will be flat (F) on the crystal, as well as
the edge directions and their critical lengths, step velocities and heights for each
of the F crystal faces.
(a ) Face and edge determination
The number of strong periodic bond chains (PBCs) present on a particular
crystal face helps to determine the mechanism by which that crystal face will
grow. On faces with zero PBCs (kinked or K faces), or one PBC (stepped or S
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Predicting crystal growth by spiral motion
1155
faces), the face tends to grow by the incorporation of molecules throughout the
crystal surface. On faces where there are multiple PBCs, so termed F faces,
growth occurs via a layer-by-layer mechanism where molecules incorporate along
specific edges on the crystal surface, which are present owing to (i) spirals
emanating from screw dislocations terminating at the surface or (ii) the
formation of the two-dimensional nuclei on the surface. Since molecules
incorporate throughout the surface on S and K faces, the growth rates of these
faces are fast relative to the growth of F faces and are often limited by transport
processes. Thus, we assume that all of the S and K faces grow much (10–1000
times) faster than the F faces (Lovette et al. 2008). Normally, the steady-state
growth shape of a crystal will be bounded by slow-growing F faces. However, in
some cases, particular directions may have no F faces to bound that direction
and needle- or plate-like crystals will result.
At low levels of supersaturation, the spiral growth mechanism is dominant
(Ohara & Reid 1973). In order to implement a spiral growth prediction, it is
necessary to determine the edges that form the sides of the spiral. We assume
that each of the PBCs forms a potential edge of the spiral and is included in the
set M. These edges are relatively slow growing since the number of kink sites
along them is far fewer than along any of the other directions on the face. In
other words, edges running in any other direction on the face are expected
to move with a velocity faster than their inclusion velocity (vincl) and do not need
to be included in the growth rate prediction.
(b ) Spiral edge height
The spiral edge height is determined by the distance between layers of growth.
This corresponds to the smallest magnitude of the dislocation’s Burgers vector
that can provide for layerwise growth of the crystal. While some larger
dislocations have been shown to exist on some crystal faces (Vekilov et al. 1992),
each face is expected to be dominated by the smallest size Burgers vectors since
the energy of the dislocation increases quadratically with the magnitude of the
Burgers vector (Ewjbj2). Additionally, if larger sized dislocations are present
(say twice the height of the smallest Burgers vector) when the steps flowing from
them intersect with those from the smallest Burgers vector, those steps of twice
the height will be reduced to half the size of the steps of the smallest height.
The height corresponding to the smallest magnitude Burgers vector is
determined by first selecting a slice of the plane with a thickness of one
interplanar spacing and determining whether it contains two or more PBCs. It is
possible that the actual height of the step will be a fraction of the interplanar
spacing. Therefore, the slice is then divided into half (or thirds depending on the
symmetry). If each of the new slices contains growth units and is an F plane, then
the division is performed again. This division continues until the newly divided
slices do not each contain a set of growth units that form an F plane. The step
height is then given by the thickness of the slice that corresponds to the last
division, where each of the divided slices contained growth units that formed an
F plane. In some complex systems, consecutive growth layers of a crystal face can
have alternating properties, including rotation time. In these cases, it is
necessary to evaluate the number of turns corresponding to one full grouping of
the growth pattern. Additionally, as mentioned previously, Burgers vectors of
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R. C. Snyder and M. F. Doherty
magnitude greater than one have been seen. These situations can lead to
potential step splitting or step doubling (Land 1997). The explicit modelling
details of systems with these properties are not considered here and will be the
subject of a future paper.
(c ) Spiral edge critical length
Predicting the critical length of a spiral edge requires the determination of the
length below which the edge has a velocity of zero and above which the edge
moves at a positive normal velocity (see equation (2.4)). While the growth of a
spiral edge is a kinetic process, the determination of this critical length is often
treated with thermodynamic theories. Kinetic theories for the determination of a
critical length exist (Voronkov 1970); however, the physical properties necessary
to implement them are not easily accessible or quantitatively predictable. On the
other hand, the physical properties required for the thermodynamic models are
more readily and rapidly attainable. Thus, thermodynamic methods are used
here to estimate critical lengths.
One method, which has been used by Chernov (1984) and Winn & Doherty
(1998) to determine the critical length of a spiral edge, is to predict the size of a
two-dimensional critical nucleus using a classical Gibbs–Thomson approach. The
critical length of a spiral edge is then assumed to be equal to the critical length
for that same edge on an equivalent hypothetical two-dimensional nucleus. For
this calculation, the critical length is determined by performing a free energy
balance on the system and asking whether adding another molecule to that
nucleus would increase or decrease the free energy of the system.
Rather than assuming that the critical length of a two-dimensional nucleus
corresponds to the critical length on a spiral edge, we propose to determine the
critical length using a free energy balance on a single spiral edge. This is similar
to the approach proposed by DeYoreo and co-workers (Thomas et al. 2004). The
critical length of edge i is then determined by asking whether adding a new row
of molecules to this edge at its current instantaneous length either increases or
decreases the free energy of the system (figure 5). If adding a row of molecules of
length li increases the free energy of the system (DGO0), the edge is assumed to
be unable to grow, while if adding a row of molecules of length li decreases the
free energy of the system (DG!0), the edge is assumed to be able to grow. Thus,
the critical length is the length at which the change in free energy caused by
adding a new row of molecules to the system is zero.
Consider the case where edge iK1 is growing and continually lengthening edge
i. To determine the critical length of edge i, consider the change in total Gibbs
free energy (DG) when a new row of molecules is added to edge i of the spiral
(figure 5), given by
X
DG ZKnDm C
gj DAj ;
ð3:1Þ
j
where n is the number of molecules that were added in the new row; Dm
(dimensions of energy per molecule) is the difference between the chemical
potential of the molecules in solution and in the crystal; gj (dimensions of energy
per area) is the specific surface energy of each new area being exposed; and Aj is
the size of each new area being exposed. We assume that the surface energy along
the length of edge i after the new row i is added is energetically identical to the
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Predicting crystal growth by spiral motion
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Figure 5. A schematic of the variables used in the edge velocity and critical length calculations.
The critical length of edge i is determined by calculating when the row of molecules in the area
enclosed by the dashed line can be added to the newly appearing edge, resulting in a net decrease in
the free energy of the system.
surface energy along edge i before the new row is added. For this to be true for
complex molecular systems (e.g. non-centrosymmetric molecules), multiple rows
of molecules may need to be added. Thus, ðhgedge
a e;i Þ cancels from the
i
calculation, and equation (3.1) can be rewritten as
hap;i li
edge
DG ZK
Dm C hðgedge
ð3:2Þ
iK1 a e;iK1 C giC1 a e;iC1 Þ;
Vm =NA
where h is the height of the spiral edge; ap,i is the distance the edge propagates
with the addition of a row on edge i; li is the length of edge i; Vm/NA is the
molecular volume; gedge
iK1 is the edge energy (energy per area) of edge iK1; and ae,i
is the distance between molecules along edge i (figure 5). This can then be
simplified to
l
edge
DG ZK i Dm C ðfedge
ð3:3Þ
iK1 C fiC1 Þ;
a e;i
where fedge
is the edge energy (energy per molecule) of edge i (fedge
Z gedge
Aedge
;
i
i
i
i
edge
Ai Z ae;i h). A characteristic plot for the free energy change for the addition of a
new row is shown in figure 6. The critical length where DGZ0 is given by
edge
ðfedge
iK1 C fiC1 Þa e;i
:
ð3:4Þ
Dm
Note that the critical length of edge i depends on the edge energy of the adjacent
edges of the spiral (iK1 and iC1). The distance between each molecule on edge i
(a e,i) is a simple function of the geometry. The chemical potential driving force
(Dm) is assumed to be isotropic, thus is equal on each edge of each face (variables
that are isotropic have the same value on each edge of each face and do not affect
the relative rates necessary for shape prediction since they cancel as common
factors). Critical lengths are expected to be in the range from 10 to 1000 nm.
The only terms that remain to be predicted to determine the critical lengths are the
edge energies per molecule, fedge
, for each edge of the spiral.
i
l c;i Z
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R. C. Snyder and M. F. Doherty
c,i
Figure 6. A plot of the change in Gibbs free energy against the length of the spiral edge on which a
row of molecules is being added. The critical length for this edge is given by the location where the
function is zero.
(d ) Spiral edge velocity
A constant competition exists between the attachment of molecules to the
crystal and the detachment of molecules from the crystal. The most important
area of molecular incorporation is along step edges. When the rate of
attachment is greater than the rate of detachment, a crystal edge (step) will
grow; however, when the rate of detachment is greater than the rate of
attachment, the step will dissolve. When the rate of attachment is equal to the
rate of detachment, the edge is in equilibrium. This attachment or detachment
takes place as a four-step process. For attachment, the solute first diffuses from
the bulk solution through a boundary layer to the crystal surface. Next, the
solute molecule adsorbs onto the surface, then diffuses across the surface and
finally is incorporated into the lattice. The detachment process is identical, but
occurs in reverse. For organic molecular crystals grown in solution, the
molecules often diffuse from the bulk directly to the locations on the surface
where they incorporate (i.e. surface diffusion is a fast process). Additionally, for
solution growth, the growth rate is often limited by the incorporation of
molecules into the lattice (Chernov 1984). Thus, we use a model which
assumes that lattice incorporation is the rate-limiting step. The most
important site for the incorporation of molecules into the crystal is the kink
site. In this paper, we focus on the growth velocity of crystals where a kink site
can be defined classically as a ‘half-crystal position’ (Kossel 1927; Stranski
1928). The half-crystal position is defined as a location where exactly half the
bonds available to that site are exposed to the solution, and the other half are
interacting with the solid. This can only be valid for the case where the bond
network surrounding each molecule is centrosymmetric. Determination of the
step velocity for more complex bond network systems will be the subject of a
future paper.
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Predicting crystal growth by spiral motion
1159
An expression for the step velocity of the ith edge where classical kinks along
step edges are the primary sites of incorporation and disincorporation from the
crystal, and where kink integration is rate limiting is given by
a DU
e
v i Z ap;i n
exp K
ð3:5Þ
Vm ðC KCsat Þ;
kT
d i
where ap,i is the distance of propagation (figure 5) of edge i and n is the frequency
of attachment and detachment attempts (Markov 2003). The average spacing
between kinks is d and the molecular spacing along the step is a e; thus, the
quantity a e =d is the probability of finding a kink at any location along the step
edge. The concentration of the solute in the solution is C, the saturation
concentration is Csat and Vm is the molar volume of the solute. Finally, DU is the
energy barrier for the incorporation of molecules into a kink site. For solution
crystallization, this quantity can be estimated as the enthalpy of desolvation.
The frequency (n), the difference in concentration between the bulk and
saturation (CKCsat) and the exponential ðexpðKðDU =kTÞÞÞ terms are each
assumed to be isotropic, and thus all cancel in the calculation of relative growth
rates and therefore have no effect on crystal shape. (For complex systems, the
half-crystal position does not exist; therefore, these isotropic assumptions are not
necessarily all valid. For such systems, the step velocity calculation may require
a more specific face-dependent determination of these variables.) The kink
density, or the probability of finding a kink site, was first derived by Burton et al.
(1951). Similarly, we assume that the rearrangement of kinks occurs on a time
scale that is faster than that of kink integration; therefore, the kinks are assumed
to be in their most probable distribution and are Boltzmann distributed. Thus,
the expression for the kink density when positive and negative kinks have
different kink energies can be expressed as
kink
kink
exp KðfC
=kTÞ C expðKfK
=kTÞ
ae
Z
;
kink
kink
d
1 C expðKfC =kTÞ C expðKfK =kTÞ
ð3:6Þ
kink
kink
where fC
and fK
are the energies required to form a positive and negative
kink, respectively (energy per molecule). For the case of equal positive and
kink
kink
Z fK
Z fkink Þ, this reduces to the familiar expression
negative kink energies ðfC
kink K1
ae
1
f
Z 1 C exp K
:
2
d
kT
ð3:7Þ
The final form of the step velocity is then given by
v i wap;i
a e
d
i
;
ð3:8Þ
where a e/d is given by equation (3.6). Since ap,i is readily calculated from the crystal
geometry (i.e. crystallography), it only remains to predict the kink energy to
determine the relative step velocities.
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R. C. Snyder and M. F. Doherty
4. Interfacial energies
For crystal growth from solution, the edge and kink energies used in determining
critical lengths and step velocities should be the interfacial energies between the
solid and solvent. Using the above methods, these interfacial energies are the key
variables remaining to be predicted for shape calculations. For this work, we use
a classical method of interfacing two surfaces by using known and calculated data
on the pure solute and solvent to determine the interfacial energies. In order to
account for interfacial interactions in this way, we first estimate the solid and
solvent interactions independently. Then, we use these values to estimate the
interfacial energy between them.
To calculate the solid component of the edge energy, each of the solid broken bond
energies that are exposed at the edge for a single molecule is summed component by
component. In other words, the dispersive components to each of the broken bond
energies exposed are summed together resulting in the solid-phase dispersive edge
energy ðfedge
solu;dis Þ, and the Coulombic components to each of the broken bond energies
are summed together resulting in the solute side Coulombic edge energy ðfedge
solu;coul Þ.
The solute phase component to the kink energy has previously been shown to
correspond to the broken bond energy of the intermolecular interaction of the
molecules parallel to the edge (Winn & Doherty 1998). Thus, the dispersive ðfkink
solu;dis Þ
and Coulombic ðfkink
Þ
components
to
the
solute
phase
kink
energy
are
simply
the
solu;coul
dispersive and Coulombic broken bond energies of that interaction. These quantities
are each in the units of energy/molecule; however, the interfacial calculation
requires the quantities to be on a per area basis. Thus, each of the above components
edge
is then recalculated on a per area basis (i.e. fedge
Z gedge
solu;dis =A
solu;dis ).
In order to estimate the solvent side surface energy, we use a technique based upon
solubility parameters, which has been proposed by Kaelble (1971) and is given by
gsolv;z Z f ðVmsolv Þ1=3 ðdz Þ2 ;
ð4:1Þ
where Vmsolv is the molar volume of the solvent; dz is the solubility parameter
associated with interaction component z (zZCoulombic, h-bond donating, h-bond
accepting, dispersive, etc.); and f is a fractional factor of the solvent internal
energy displayed at an interface. Data for these variables for a variety of common
solvents, in particular for the molar volume and solubility parameters, are
available from a variety of sources (Kaelble 1971; Barton 1975). Sometimes, only a
single solubility parameter for hydrogen bonding is reported and this then needs to
be distributed into both accepting and donating components. This apportionment
is carried out using d2hbond Z 2ddon dacc , where the amount of donating and accepting
fractions are estimated based upon the molecule (i.e. for equally
pffiffiffidonating and
accepting solvents such as water or ethanol ddon Z dacc Z ðdhbond = 2Þ).
Finally, the solute and solvent side components are used to determine an estimate
for the energy required to form an interface between the solute and the solvent. This
is done using a classical method where the interfacial energy between two surfaces
can be estimated as the sum of the cohesive energy of each surface (the energy
required to create the interface of each of the surfaces in a vacuum) minus the work
of adhesion (the energy relieved by putting the two surfaces together). In general,
the interfacial energy between two surfaces (a,b) is thus given by
ga;b Z ga C gb KWad ;
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Predicting crystal growth by spiral motion
1161
where ga and gb are the interfacial energies of surfaces a and b (the cohesive energy)
and Wad is the work of adhesion. Furthermore, the
work of adhesion can be
pffiffiffiffiffiffiffiffiffiffi
estimated using the geometric mean rule Wad Z 2 ga gb (Israelachvili 1992).
This method has been extended by Winn & Doherty (2002) to also focus on
hydrogen bonding.
Here, we will use a generalized version of these equations that includes all of
the existing interactions in both the solute and the solvent. These interactions, in
general, consist of both a dispersive component (due primarily to van der Waals
interactions) and a Coulombic component (which can stem from a variety of
phenomenon such as an ionically charged species, hydrogen bonding components
or electrostatic interactions resulting from pi bonding, particularly in aromatic
compounds). Since the cohesive energy terms are those of the pure component,
all (dispersive and all Coulombic) of the interactions from both the solute and
the solvent need to be included. However, the work of adhesion only should
include the appropriate terms where the solute and the solvent can interact.
For example, for systems where either the solute or the solvent has no
Coulombic nature at all, the work of adhesion only results from the dispersive
terms and we use (note that either or both of the Coulombic components of the
cohesive energy in this expression will be zero)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
gsolu;solv Z gsolu;dis C gsolv;dis C gsolu;coul C gsolv;coul K2 gsolv;dis gsolu;dis :
ð4:3Þ
Alternatively, for a system where significant hydrogen bonding is present, we use
the following:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
gsolu;solv Z gsolu;dis C gsolv;dis C gsolu;coul C gsolv;coul K2 gsolv;dis gsolu;dis
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð4:4Þ
K2 gsolv;don gsolu;acc K2 gsolv;acc gsolu;don ;
where the donating and accepting components to the hydrogen bonding
character are distributed from the total Coulombic energy using the method
described for those solvents in the previous paragraph.
In other systems where substantial pi bonding and aromatic rings are present,
we use the following:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
gsolu;solv Z gsolu;dis C gsolv;dis C gsolu;coul C gsolv;coul K2 gsolv;dis gsolu;dis
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K2 gsolv;don gsolu;coul :
ð4:5Þ
In this case, we have included a term in the work of adhesion to account for the
negatively charged pi bonds interacting with the positively charged hydrogen
bond-donating groups of the solvent. (If a solvent such as benzene were used and
the solute had hydrogen bond-donating groups, the solute and the solvent in the
final term would be reversed.)
In each case, the first four terms represent the work of cohesion and the
remaining terms represent the work of adhesion from each of the components at
the interface. In some cases, for more complex molecular crystals, the Coulombic
character may stem from a combination of components (e.g. a combination of
pi bonds and hydrogen bonding). For these cases, a more detailed account for
each of these interactions may be necessary.
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R. C. Snyder and M. F. Doherty
Table 1. Naphthalene solid-state interactions/PBCs.
bond label
bond direction
Coulombic energy
(kcal molK1)
dispersive energy
(kcal molK1)
total energy
(kcal molK1)
h
h
i
j
j
k
[110]
½110
[010]
[102]
½102
[001]
0.62
0.62
0.94
0.18
0.18
0.09
2.09
2.09
1.47
0.77
0.77
0.65
2.71
2.71
2.51
0.95
0.95
0.74
Table 2. Solvent information: molar volume (Vm); dispersive solubility parameter (dd); hydrogen
bonding solubility parameter (dh); and fractional factor (f ). (Data from Kaelble (1971) for columns
marked (1) and Barton (1975) for columns marked (2).)
solvent
Vm (cc molK1) (1)
dd (cal ccK1)1/2 (2)
dh (cal ccK1)1/2 (2)
f (1)
cyclohexane
ethanol
2-propanol
water
108.7
57.5
76.0
18.0
8.2
7.7
7.2
7.6
0.0
6.7
6.3
14.6
0.159
0.073
0.080
0.110
(a ) Crystal shape determination
Using the calculated relative growth rates outlined above, the steady-state
crystal shape is given by the Frank–Chernov condition (Frank 1958; Chernov
1963) as
R1
R
R
R
Z 2 Z 3 Z/Z N ;
ð4:6Þ
x1
x2
x3
xN
where Ri is the relative growth rate of face i and xi is the relative perpendicular
distance from the centre of the crystal to face i. Additionally, faceted crystal
shape dynamics in both growth and dissolution can be estimated using faceted
shape evolution methodologies (Zhang et al. 2006; Snyder & Doherty 2007).
5. Results
(a ) Naphthalene
The first example system used to demonstrate our crystal shape prediction
methodology is naphthalene. The morphology of naphthalene has been studied
previously using both the attachment energy method (Grimbergen et al. 1998) and
the molecular simulation (Cuppen et al. 2004). Each of these methods was
accurate at reproducing the shape of naphthalene grown from the vapour;
however, they did not account for solvent effects. Experimental results for the
shape of naphthalene grown from multiple solvents (cyclohexane and ethanol) are
also available (Grimbergen et al. 1998). Thus, this system is a good choice for
demonstrating the predictive power of our model, including its ability to predict
solvent effects.
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Predicting crystal growth by spiral motion
(b)
(c)
(001)
(a)
(d)
(001)
(110)
(201)
(110)
(201)
Figure 7. The crystal shape prediction scheme for naphthalene. (a) The experimentally determined
crystal structure is input to the model. Next, the intermolecular interactions in the solid state are
calculated, and the faces that have multiple PBCs resulting from these interactions are
determined. (b) These interactions are shown schematically for each face, where different colours
represent different magnitudes of interactions. (c) The calculation proceeds to calculate velocities
and critical lengths resulting in the corresponding predicted spiral shapes for each face. (d ) These
are then used to predict the relative growth rate of each face leading to the predicted crystal shape.
Table 3. Naphthalene: predicted relative growth rates (R) of crystal faces.
face
bond chains
h (Å)
R (ethanol)
R (cyclohexane)
{001}
f111g
{110}
f201g
{200}
i, h, h
j, h
j, h, k
i, j, j
i, k
7.14
4.65
4.67
4.07
3.38
1.00
6.00
2.02
1.77
4.61
1.00
8.26
2.95
1.54
5.27
Naphthalene crystallizes in the monoclinic space group P21/a with the
following lattice parameters: aZ8.266 Å; bZ5.968 Å; cZ8.669 Å; aZgZ908;
and bZ122.92 (Pawley & Yeats 1969). There are two molecules per unit cell and
they are located at the unit cell positions (0,0,0) and ð1=2; 1=2; 0Þ. The energetic
interactions between the molecules in the solid are calculated using HABIT
(Clydesdale et al. 1996), with the force field of Scheraga (Némethy et al. 1983)
assuming a monomer growth unit. The atomic charges for the HABIT calculation
are obtained using GAUSSIAN (Frisch et al. 2004). The strong intermolecular
interactions in the solid state (those with an energy greater than kT) are given in
table 1. Owing to the crystal and molecular symmetry, each of these interactions
repeats to form one of the six strong bond chains that are used to determine the
growth rate of the crystal faces. (A bond chain with an asterisk identifier is
related to the bond chain of the same letter by symmetry.) Based upon the
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R. C. Snyder and M. F. Doherty
interactions listed in table 1, the families of the F faces of naphthalene are {001},
{110}, f201g
and {200}. Since experimentally obtained crystal shapes are
f111g,
available for naphthalene grown from ethanol and cyclohexane, these two
solvents are chosen for comparison with the predicted morphologies. While no
direct evidence exists for the spiral mechanism on naphthalene, atomic force
microscopy (AFM) results have shown it to be the dominant mechanism for some
cases of growth of a similar molecule, anthracene (Cuppen et al. 2004). Thus, for
each of the F faces, we implement the spiral growth rate calculation. The solvent
data required to implement the shape prediction model is provided in table 2.
For the case of growth in ethanol, the solute–solvent interactions are determined
using equation (4.5) because of the polar ethanol molecule’s interactions with the
directional pi bonding of naphthalene.
Table 3 lists each of the F faces of naphthalene, the PBCs present on that face
and the predicted relative growth rates of each face grown in both ethanol and
cyclohexane. Detailed results for all the intermediate data (kink energies, edge
energies, kink probabilities, etc.) for growth in cyclohexane and ethanol are given
in the electronic supplementary material. Figure 7 shows the progression of the
calculation. We begin with a known crystal structure (figure 7a). Then
the intermolecular interactions in the solid state are calculated, which give rise
to the PBCs in the system (figure 7b). These bond chains then correspond to the
edges used for the predicted spiral shapes (figure 7c). Finally, the growth rates
corresponding to these spirals lead to a crystal shape (figure 7d). Note that the
spiral shapes indeed evolve towards a steady-state growth shape as they move
away from the spiral centre. The crystal shapes based upon the growth rates in
table 3 for growth in ethanol and cyclohexane are shown in figure 8, alongside
the redrawings of the experimental results from Grimbergen et al. (1998).
The predicted results using our spiral growth model are in very good agreement
with the experimental results.
(b ) Anthracene
The morphology of anthracene has also been studied previously using both the
attachment energy method (Docherty & Roberts 1988; Grimbergen et al. 1998) and
the molecular simulation (Cuppen et al. 2004). These methodologies were accurate
at reproducing the shape of anthracene grown from the vapour; however, they did
not account for solvent effects. Experimental results have been previously reported
for the growth of anthracene in an unspecified alcohol (Groth 1919). However,
owing to unreported details associated with these results, we have performed our
own experiments by slow evaporation of anthracene from a 2-propanol solution.
The crystals were grown by preparing a slightly undersaturated solution of
2-propanol at room temperature and adding approximately 20 ml of it to the
bottom of several small Petri dishes. The dishes were then each partially covered to
varying extents and were let stand until all of the solvent was evaporated. The
crystals grown in the mostly covered Petri dishes were well formed, and the results
from these are used for comparison with growth predictions.
Anthracene crystallizes in the monoclinic space group P21/a with the
following lattice parameters: aZ8.562 Å; bZ6.038 Å; cZ11.184 Å; aZgZ908;
and bZ124.70 (Mason 1964). There are two molecules per unit cell and they are
located at the unit cell positions (0,0,0) and ð1=2; 1=2; 0Þ. The interactions
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Predicting crystal growth by spiral motion
(a)
(b)
(c)
(d)
Figure 8. A comparison of the (a,c) predicted and (b,d ) experimental steady-state shapes of
naphthalene grown from (a,b) ethanol and (c,d ) cyclohexane. Experimental images are redrawn
from Grimbergen et al. (1998).
Table 4. Anthracene solid-state interactions/PBCs.
bond label
bond direction
Coulombic energy
(kcal molK1)
dispersive energy
(kcal molK1)
total energy
(kcal molK1)
h
h
i
j
j
k
[110]
½110
[010]
[102]
½102
[001]
1.30
1.30
1.93
0.40
0.40
0.04
3.11
3.11
2.31
0.88
0.88
0.54
4.41
4.41
4.24
1.28
1.28
0.58
between the molecules in the solid were predicted using HABIT (Clydesdale et al.
1996), with the force field of Scheraga (Némethy et al. 1983) assuming a
monomer growth unit. The atomic charges for the HABIT calculation were
obtained using GAUSSIAN (Frisch et al. 2004). The strong interactions in the solid
state (greater than kT) are given in table 4. Again, each of these interactions
repeats to form one of the six strong bond chains that are used to determine the
edges on each of the F faces and hence the growth rate of the crystal faces. The
spiral growth mechanism is again used for growth rate determination of the F
faces, and thus is modelled using the methodology described earlier in this paper.
AFM results for vapour-grown anthracene crystals have been previously reported
(Grimbergen et al. 1998), which showed spiral growth.
Based upon the interactions listed in table 4, the families of F faces of
{110}, f201g
and {200}. The data on 2-propanol
anthracene are {001}, f111g,
required to implement the shape prediction model is provided in table 2. Table 5
lists each of the F faces of anthracene, the molecular interactions corresponding
to each bond chain present on that face and the relative growth rates of that face
grown from 2-propanol. As was the case for naphthalene, the solute–solvent
interactions are estimated using equation (4.5) owing to the pi-bonded rings in
the compound. Detailed results for all of the intermediate data (kink energies,
edge energies, kink probabilities, etc.) can be found in the electronic
supplementary material.
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R. C. Snyder and M. F. Doherty
(a)
(b)
(c)
Figure 9. A comparison of the (a,b) experimental and (c) predicted steady-state growth shapes of
anthracene grown from 2-propanol. The experimental shapes show (a) some dispersion;
nonetheless, a (b) characteristic growth shape is very similar to the (c) predicted growth shape.
Table 5. Anthracene: predicted relative growth rates (R) of crystal faces.
face
bond chains
h (Å)
R (2-propanol)
{001}
f111g
{110}
f201g
{200}
i, h, h
j, h
j, h, k
i, j, j
i, k
9.19
4.89
4.58
4.17
3.52
1.00
10.29
2.99
1.86
5.38
The predicted crystal shape based upon the growth rates in table 5 for growth
from 2-propanol is shown in figure 9, alongside several of the experimentally grown
crystals. The experimental results show some degree of variation owing to the
varied conditions of growth during solvent evaporation. Nonetheless, the predicted
results using our spiral growth model are in good agreement with the typical shape
of the experimentally grown crystals. Moreover, our experimental results are also
similar to the shape reported by growth from alcohol (Groth 1919).
(c ) a-Glycine
Finally, we report the predicted results for a-glycine grown from water. The
morphology of a-glycine has been studied previously using kinetic crystal growth
methods (Bisker-Leib & Doherty 2003), but here we report successful morphology
predictions without the need for expert decisions during implementation of the
model. Experimental crystal morphologies of a-glycine grown in water are available
from multiple literature sources (Boek et al. 1991; Poornachary et al. 2007).
a-Glycine crystallizes in the monoclinic space group P21/n with the following
lattice parameters: aZ5.1054 Å; bZ11.9688 Å; cZ5.4645 Å; aZgZ908; and
bZ111.697 (Jönsson & Kvick 1972). There are four molecules per unit cell, and
the growth unit is taken to be a hydrogen bonded cyclic dimer incorporating the
zwitterionic interactions. The dimer growth unit can be identified from the solidstate interactions between the molecules, and has also been confirmed both by
diffusion (Chang & Myerson 1986) and AFM (Carter et al. 1994). The two
growth units are located at the unit cell positions (0,0,0) and ð1=2; 1=2; 1=2Þ. The
interactions between the growth units in the solid were predicted using HABIT
(Clydesdale et al. 1996), with the force field of Scheraga (Némethy et al. 1983)
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Predicting crystal growth by spiral motion
(a)
(b)
(020)
(110)
200 µm
(011)
Figure 10. A comparison of the (a) predicted shape of a-glycine grown from water and (b) experimental
shape from Poornachary et al. (2007).
Table 6. a-Glycine solid-state interactions/PBCs.
bond label
bond direction
Coulombic energy
(kcal molK1)
dispersive energy
(kcal molK1)
total energy
(kcal molK1)
i
j
k
k
[100]
[001]
½111
½11 1
6.18
8.58
0.43
0.43
4.38
0.29
1.76
1.76
10.56
8.87
2.18
2.18
Table 7. a-Glycine: predicted relative growth rates (R) of crystal faces.
face
bond chains
h (Å)
R (water)
{020}
{011}
{110}
i, j
k,i
k, j
5.98
4.67
4.41
1.00
13.12
1.84
based upon the dimer growth unit. The atomic charges for the HABIT
calculation were obtained using GAUSSIAN (Frisch et al. 2004). The strong
interactions in the solid are given in table 6. They repeat as four bond chains, two
of which have identical interaction energies.
Based upon the strong interactions listed in table 6, the families of the F faces of
a-glycine are {020}, {110} and {011}. a-Glycine is commonly grown from water and
the solvent data required to implement our model is given in table 2. Since the
growth unit consists of the hydrogen bonded cyclic dimer that includes
the zwitterionic effects, all of the hydrogen bonding and zwitterionic capabilities
of the solute are assumed to be satisfied internally to the growth unit (gsolu,accZ0;
gsolu,donZ0). The solute–solvent interactions are calculated using equation (4.4)
(which reduces to equation (4.3) for this case of gsolu,accZ0; gsolu,donZ0). Table 7
lists each of the flat faces of a-glycine, the molecular interactions corresponding to
each bond chain present on that face and the relative growth rates of that face based
on the spiral growth mechanism for growth in water. The detailed results for all of
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R. C. Snyder and M. F. Doherty
the intermediate data (kink energies, edge energies, kink probabilities, etc.) can be
found in the electronic supplementary material. The crystal shapes based upon the
growth rates in table 7 for growth from water are shown in figure 10a, and are very
similar to the experimental results from Poornachary et al. (2007), shown in
figure 10b.
6. Conclusions
We have reported on the next generation of kinetically based crystal growth
models for the spiral growth mechanism. The evolution of spiral shape and the
determination of which edges to include in the spiral shape calculation have been
mathematically determined. This methodology has now advanced to become
completely automated using a MATLAB program, which can take crystallographic,
solvent and molecular interaction information as input data and use the spiral
growth methodology to systematically predict the steady-state crystal shape of
organic molecular solids grown from solution.
While the model has been automated, and it appropriately accounts for
crystallographic orientation, solvent and spiral dynamics, it also has some
remaining limitations that are areas of future research. The determination of
edge velocity has been derived only for systems where classical kinks (half-crystal
positions) are present. Many crystal systems do fall into this category; however,
many complex molecular organic crystals such as pharmaceutical molecules do
not. Thus, a methodology for a more complete description of the edge velocity for
systems without classical kinks is necessary. Additionally, for those more
sophisticated systems, a more complex methodology for edge and face
determination will also be necessary. For this work, the resulting relative crystal
growth rates that determine the steady-state crystal shape are currently
independent of supersaturation. However, it is well known that some crystals
have supersaturation-dependent relative growth rates when grown from solution
(e.g. paracetamol grown from water; Shekunov & Grant (1997)). Methods to
account for supersaturation dependencies on relative growth rate for the spiral
growth model, and methods to appropriately determine the transition from spiral
growth to the two-dimensional nucleation and growth are also desired.
One hallmark of our approach is that it is modularized in nature. When higher
fidelity methods to determine edge velocities, critical lengths, solute–solvent
interactions or any other estimated values in the model are developed, they can
be readily incorporated into the calculation procedure. Furthermore, the model
can even be implemented when some of these values are taken from the
experiments. The key idea is to base the crystal growth rates on the simplest
physically realistic representation of the underlying molecular mechanisms,
thereby linking crystal engineering with crystal chemistry. Using these
strategies, we have successfully developed and implemented a rapid method for
the prediction of crystal shape that serves as a basis for future developments in
crystal shape engineering, including mixed solvents and additives.
We are grateful for the financial support provided by Merck & Co. and the National Science
Foundation (grant no. CBET-0651711).
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