Introduction

Polymorphism is the ability of atoms or molecules to assemble in several different crystalline structures. It is a common phenomenon that has been used to elucidate structure–property relationships in solid-state organic chemistry for decades [1, 2]. Most molecular systems may exhibit polymorphism—and it has been postulated that it is a matter of the amount of resources used to investigate the system that determines the number of polymorphs discovered [3].

Generally, we consider the ability to predict and control the structure of crystals of paramount importance in the production of new solid-state materials. An example is the production of active pharmaceutical ingredients for solid-state dosage forms, where it is vital to produce crystal forms with dissolution and solubility properties that ensure sufficient bioavailability of the compound. It is also important to obtain a solid form that is easy to manufacture and that is stable during transportation and storage. These properties all depend on the crystal structure. The sudden appearance of new crystal forms may be disastrous for the production of pharmaceuticals, as was the case for the HIV drug ritonavir [4]. Last, there are legal issues related to the patentability of new crystal forms [5].

The strategy of most pharmaceutical companies is to screen for polymorphism using a range of crystallization strategies. The strategies involve screening different solvents, temperatures, pH values, etc. The outcome is typically evaluated using powder X-ray crystallography. These experimental screening techniques are tedious and costly, and there is no guarantee that all possible polymorphs have been determined. There is evidently a demand for complementary computational tools that can predict the most likely structures and suggest a strategy to obtain them.

A computational strategy to predict crystal structures has been pursued for decades. Within the last years, several successful approaches have been presented as part of the crystal structure prediction blind tests [6]. These approaches all search for the global minimum on the potential energy hyper-surface of crystal coordinates. In spite of several successful predictions of crystal structures, it remains difficult to assess the relative energies of polymorphic systems [7]. A plausible explanation for these problems is that the potential energy differences between polymorphic forms are so small (1–2 kJ/mol) that the determining factors for polymorph selection must be explained in terms of differences in free energy, including the vibrational entropy [8,9,10,11]. Another explanation is that the structures used are not sufficiently accurate [12]. Kinetic factors may also play an important role, so that the observed structure may be determined by the nucleation and growth of crystals, rather than to the thermodynamic stability of the final crystalline state [1, 2]. In this contribution, we investigate the role of nucleation.

In the early stages of crystal formation, the formation of pre-nuclear clusters, the nucleation stage, and the subsequent growth of crystals, solvent effects, and temperature can influence the growth rates of specific crystal surfaces [13,14,15,16], and heterogeneous nucleation events involving the crystallization vessel or other surfaces can mediate a preference for nucleation of a specific crystal form. Selection of crystal forms, and modification of crystal habits have been demonstrated by epitaxial growth on Langmuir–Blodgett films by Leiserowitz and coworkers [13]. As demonstrated by Matzger and coworkers [17], it is also possible to influence the polymorph selection by using polymer heteronuclei. Polymer-induced heteronucleation (PIHn) has been proposed and tested as a method for discovering new solid forms. The method has been demonstrated for small molecules [17], supramolecular complexes [18], and proteins [19]. In spite of the success of these investigations for scanning for polymorphism, and for promoting new polymorphic forms, little is understood about the underlying mechanism, hampering progress in the design of experimental protocols that can be used to explore the landscape of polymorphic crystal forms.

Because of the largely amorphous character of the solid polymers, epitaxial growth is not a likely explanation for the polymer’s ability to induce different polymorphs. In the work presented here, we investigate whether a heterogeneous solvation model based on electrostatic interactions between the surface and the crystal nuclei can provide new insights into the mechanism of polymorph selection by heterogeneous nucleation. The model system used is acetaminophen (ACM, Fig. 1), a common drug, which crystallizes in a monoclinic and an orthorhombic form. The monoclinic form is known to be the thermodynamically stable form at ambient conditions [20]. Acetaminophen is a commonly used model system in the study of polymorphic molecular crystals. The polymer-induced nucleation of the two forms has previously been studied [17]. Most recently, Lopéz-Mejías and coworkers have investigated the heteronucleation by a combined experimental and computational approach, where polymer-covered surfaces were used to facilitate nucleation. The preferred orientation of acetaminophen crystals on these surfaces were determined using powder X-ray crystallography, and the results were compared to force-field based docking calculations of polymers on crystal surfaces [21].

Fig. 1
figure 1

The conformations of acetaminophen with labeled atoms

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We wish to investigate how acetaminophen interacts with a polymer particle in order to understand how the polymer particle facilitates nucleation. Our aim is to study this for the molecule–polymer particle system in vacuum and in solution. Furthermore, we consider different conformations and orientations of acetaminophen approaching a polymer particle in order to investigate preferential nucleation to the polymer particle.

Our model is based on the use of energy functionals for dielectric media and generally the basic idea is to divide a large system into two or more subsystems depending on the actual system. The subsystem of principal interest, acetaminophen, will be described by a quantum mechanical approach, whereas the other subsystems are described by classical electrostatics [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62]. We determine the interactions between the molecular subsystem and the polymer particle by the induced polarization in the polymer particle and the electric field due to the charge distribution of the molecular subsystem [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62]. The molecular subsystem and classical subsystems are coupled by an effective interaction operator and thereby one obtains a modified quantum mechanical equation for calculating the interaction energy of a molecular subsystem interacting with the polymer particle in vacuum or in solution. We consider the molecular subsystem, acetaminophen, to interact with the polymer particle and the surrounding solvent and we utilize a heterogeneous solvation model [63,64,65] for calculating the interaction energies of the different conformations and orientations of acetaminophen with the heterogeneous environment.

Theory

The heterogeneous solvation model for the polymer and molecular system in vacuum

Heterogeneous and homogeneous solvent effects and the interactions between particles and molecules have been treated in a number of models and the general trend in these models is to divide the system into a molecular and a surrounding part, and treat the two subsystems with different levels of theory. This could be quantum mechanics/molecular mechanics models or a heterogeneous solvation model where the molecule interacts with a particle represented by a dielectric medium. The later model is the approach taken in the following.

We consider the heterogeneous solvation model containing a molecule and a polymer particle subsystem and the outer solvent is neglected. The polymer particle is represented by a semi-infinite dielectric medium, and the coordinate system is defined so z = 0 is the interface between the two subsystem. Further, the coordinate system is restrained by the dielectric medium being located at z < 0 and the molecule at z > 0. The molecule is represented by quantum mechanically determined partial charges located at the atomic positions. The molecular partial charges are determined prior to this, by either the scheme proposed by Cioslowski [67,68,69] or the CHelpG [70] method. Since the interface is at z = 0 the z component of the point charge qi at the position r→i = (xi, yi, zi) is restricted to zi > 0.

The interaction between the charges and the dielectric medium is modeled by image charges in the dielectric medium. The image charges and their positions are given by

$$\begin{array}{@{}rcl@{}} q_{i,s}=-\frac{\epsilon-1}{\epsilon+ 1}q_{i} \\ \text{at}~ \vec{r}_{i,s}=(x_{i},y_{i},-z_{i}) \end{array} $$
(1)

where the subscript s refers to an image charge.

The induced potential from N image charges at an arbitrary point in the molecular subsystem, r→ is given by

$$\begin{array}{@{}rcl@{}} U_{pol}(\vec{r})=\sum\limits_{i = 1}^{N}U_{pol}^{(i)}(\vec{r})=\sum\limits_{i = 1}^{N}\frac{q_{i,s}}{|\vec{r}-\vec{r}_{i,s}|} \end{array} $$
(2)

The energy resulting from the interaction between the molecular charge distribution, ρm, and the dielectric medium is given by

$$\begin{array}{@{}rcl@{}} E_{pol}=\frac{1}{2}\int d\vec{r}\rho_{m}(\vec{r})U_{pol}(\vec{r}) \end{array} $$
(3)

Since the distribution of the molecular charges is discrete, the integral can be substituted with a summation:

$$\begin{array}{@{}rcl@{}} E_{pol}=\frac{1}{2}\sum\limits_{k = 1}^{N}q_{k}U_{pol}(\vec{r}_{k}) \end{array} $$
(4)

It is important to note that the potential in Eq. 2 is the potential arising from the interaction between one point in the molecular subsystem with all the image charges in the dielectric medium, i.e., a sum over all the image charges. On the other hand, (4) is a sum over all the charges and potentials in the molecular subsystem.

The heterogeneous solvation model for the polymer and molecular system in solution

The dielectric continuum model above neglects that the molecule in many cases is surrounded by a medium other than vacuum. The inclusion of a surrounding solvent in the heterogeneous solvation model is done by including in addition to the dielectric continuum at z < 0 a second dielectric medium placed at z > 0 and x2 + y2 + z2 > a with a being the radius of a half-sphere cavity containing the molecule. The first medium is the solid particle and modeled by the dielectric constant 𝜖1, while the second dielectric medium is the solvent with the dielectric constant 𝜖2. In this model, each charge in the molecular system gives rise to three new image charges; an image charge directly induced in the solid continuum qs1, an image charge in the solvent continuum qs2, and an image charge induced in the solid qs3 by the induced charge in the solvent qs2. Thus, the charge qi at ri = (xi, yi, zi) in the molecular system induces the following image charges

$$\begin{array}{@{}rcl@{}} q_{i,s1}=-\frac{\epsilon_{1}-1}{\epsilon_{2}+ 1}q_{i} &~\text{at}~&\vec{r}_{i,s1}=(x_{i},y_{i},-z_{i}) \end{array} $$
(5)
$$\begin{array}{@{}rcl@{}} q_{i,s2}=-\frac{a}{|\vec{r}_{i}|}\frac{\epsilon_{2}-1}{\epsilon_{2}+ 1}q_{i} &~\text{at}~&\vec{r}_{i,s2}=\frac{a^{2}}{|\vec{r_{i}}|^{2}}(x_{i},y_{i},z_{i}) \end{array} $$
(6)
$$\begin{array}{@{}rcl@{}} q_{i,s3}=-\frac{\epsilon_{1}-1}{\epsilon_{2}+ 1}q_{i,s2} &~\text{at}~&\vec{r}_{i,s3}=\frac{a^{2}}{|\vec{r_{i}}|^{2}}(x_{i},y_{i},-z_{i}) \end{array} $$
(7)

The induced potential from the 3N charges at an arbitrary point r→ in the half-sphere cavity is given by

$$\begin{array}{@{}rcl@{}} U_{pol,solv}(\vec{r})=\sum\limits_{i = 1}^{N}U_{pol}^{(i)}(\vec{r})=\sum\limits_{i = 1}^{N}\sum\limits_{j = 1}^{3}\frac{q_{i,sj}}{|\vec{r}-\vec{r}_{i,sj}|} \end{array} $$
(8)

and we find that the energy of the interaction between molecule and solvent, and solid particle is obtained as

$$\begin{array}{@{}rcl@{}} E_{pol,solv}=\frac{1}{2}\int d\vec{r}\rho_{m}(\vec{r})U_{pol,solv}(\vec{r}) \end{array} $$
(9)

and since the molecule is represented by a set of discrete quantum mechanically determined charges, the integral is substituted by a summation

$$\begin{array}{@{}rcl@{}} E_{pol,solv}=\frac{1}{2}\sum\limits_{k = 1}^{N}q_{k}U_{pol,solv}(\vec{r}_{k}) \end{array} $$
(10)

and the radius of the half spherical cavity is determined by the atom where the distance to origin plus the van der Waal radius of that given atom is largest (Fig. 2).

Fig. 2
figure 2

Slabs of the monoclinic and orthorhombic forms of acetaminophen with the most energetically favorable orientation towards the polymer surface

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Results and discussion

In this study, we consider three conformations of acetaminophen, Fig. 1, approaching a polymer surface. In addition, each conformer is approaching the surface in six different orientations (Figs. 6 and 11).

The molecular structures were determined by geometry optimization at the CAM-B3LYP/aug-cc-pVTZ level of theory. The partial molecular charges were calculated from the geometries using the CHelpG procedure implemented in the Gaussian electronic structure program [66] with different basis set (see Table 1). The partial charges were converged when using the aug-cc-pVQZ basis set, and the partial charges were used in the heterogeneous solvation model for calculating the interaction energy. The polymer particle was represented as a dielectric medium with a dielectric constant, 𝜖1, set to 10 and the solvent used here is water and represented as a dielectric medium with the dielectric constant 𝜖2 = 78

Table 1 The partial charges of acetaminophen in the flat and scorpion conformations, derived using the CHelpG method and CAM-B3LYP and different basis sets
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In Figs. 4, 7, and 9, the interaction energies between the polymer particle and acetaminophen in the flat and scorpion conformations are plotted for all orientations. It must be emphasized that the distance axis is not equivalent to the z component of the position vector. The distance axis is zero when the van der Waal radius of the molecule equals the distance from the closest atom to the surface. It should be mentioned that the van der Waal radius is not the same in all directions, that is, the closest atom at different orientations is not necessarily at the same distance from the surface. To avoid mixing, the z component of the position and the distance from the van der Waal radius, the later is renamed z∗. The van der Waal radii used for the respective atoms are, rw(C) = 1.70Å, rw(H) = 1.20Å, rw(O) = 1.50Å, and rw(N) = 1.55Å.

The interaction energies of the solvated molecule are shown as a function of the distance z∗ (Fig. 3).

Fig. 3
figure 3

Energies of the af conformation as a function of the distance to the surface, using different combinations of dielectric constants for the solvent and surface

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The polymer and molecular system in vacuum

The flat conformation, af

In Figs. 4 and 5, the interaction energies between the polymer and acetaminophen in the flat conformation are plotted for all six orientations in standard and logarithmic scale. The af5 orientation has the lowest energy closely followed by the af2 orientation. This is due to the large negative charge on the nitrogen and oxygen atoms in the amide group, the oxygen atom in hydroxy group, and the corresponding induced positive charge in the dielectric medium. The reason for the af5 orientation to have slightly lower energy than the af2 orientation is that the positively charged carbon atom in the amide group is further away from the positive charge induced by the oxygen of the same group. In the af2 orientation, all the atoms in the amide group are close to the surface and thereby the induced charges. Since the central carbon in the amide group, C(7), is positively charged, all the surrounding negatively charged atoms, N(18), O(20), and C(8), are repelled by the induced negative charge and vice versa. This repulsion is larger than the gain in partial charges of the atoms being close to the surface.

Fig. 4
figure 4

Energy plots of the six orientations of the flat conformations of acetaminophen

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Fig. 5
figure 5

Energy plot of the six orientations of the flat conformation of acetaminophen in a logarithmic scale

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The af1 orientation behaves overall very similar to that of the af2 orientation, which is expected due to the similarity in orientation. The difference between af1 and af2 is the angle between the benzene moiety and the surface; in af1 they are parallel, while the benzene moiety is tilted in af2 so the line connecting the lower hydrogen atom in the methyl group and the hydroxy group are parallel to the surface. At long distances, the interaction energies of the two orientations are almost identical. When the molecules are within a distance of 1.5Å, (z∗ = 1.5), the energy of the af2 orientation drops significantly compared to af1, the reason being that the hydroxy oxygen, O(19), is closer to the surface in this orientation.

The af1 orientation has a slightly lower energy than the af3 orientation at z∗ = 0, but as the distance is increased, the energy difference increases as well, to around z∗ = 2, from where it decreases again. In the logarithmic plot it is clearly observed that the af3 orientation is significantly less stable than the rest of the orientations at distances longer than 2Å. In the af3 orientation, the hydroxy group is the driving force in the increased interaction energy. Since the amide group at all times is approximately 4Å further away from the surface, the interaction energy contribution from this part of the molecule plays a less significant role. At short distances, however, the proximity of the hydroxy group and dielectric medium results in lower energies than af6 and af4. The proximity of the hydroxy group has, in the af6 orientation, been sacrificed to get the amide group closer to the surface, but since the oxygen in this group is pointing away, the resulting interacting energy at zero distance is lowered.

The af6 and af4 orientations have remarkably higher interaction energies than the rest of the orientations. This is due to a strong induction of a negative charge, by C(7), in the proximity of the negatively charged C(8) and N(18). In all the other cases, the amide oxygen, O(20), and/or the hydroxy oxygen, O(19), have been closer to the surface, and thus contributed to a large interaction. In af6, these effects are strongly decreased. Though somewhat counter intuitive, the lowest interaction energy is observed in the case of orientation af4. One might suspect this orientation to yield a lower interaction energy due to the amide pointing down towards the dielectric medium, but this is obviously wrong. The reason for the smaller energy contribution may be found in the increased distance from the surface to the charged nitrogen and oxygen in the amide group. Furthermore, the positively charged carbon, C(7), is closest to the surface and thereby inducing a negative charge close to the negatively charged carbon in the methyl group.

The ac conformation

The only difference between the two flat conformations af and ac is the direction of the hydroxy bond compared to the amide group. In af, the N–H bond is in the opposite direction of the O–H bond, while in ac the bonds are in the same direction. In Table 1 the charges of ac with cc-pVQZ and aug-cc-pVQZ basis sets are listed. The atomic charges of af and ac are overall the same, but the ac conformation has slightly larger charges and significantly larger dipole moment.

The investigated orientations of the ac conformation are the same as those for af and named in the same manner (see Fig. 6). Because of the high degree of similarity between the conformations, the orientations of af and ac are expected to behave similarly. The difference in dipole moment does, however, indicate there might be some changes. Indeed, the relative stability and the minimum energy have been changed (Fig. 7): the lowest energy is now obtained by orientation 2, which gains about –0.03 eV, while orientation 5, which has the lowest energy in af, is elevated by 0.01 eV.

Fig. 6
figure 6

The orientations of the flat conformations

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Fig. 7
figure 7

Energy plots of the six orientations of the flat ac conformations of acetaminophen

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The two orientations parallel or almost parallel to the surface, 1 and 2, are stabilized by the rotation of the O–H bond. The reason for this is that the negative induced charge from the hydroxy hydrogen is further away from O(20). This can be seen as an energy lowering of O(20) and H(13) for orientation 1 and 2 in ac conformation compared to af (Fig. 12). One might expect this to be a minor change, but the energy contribution from these atoms is decreased; the difference is about 0.065 eV, on the same scale as the overall energy (Fig. 8).

Fig. 8
figure 8

Energy plot of the six orientations of the flat ac conformation of acetaminophen in a logarithmic scale

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The sideways orientations, 5 and 6, are raised or lowered in energy depending on whether H(13) is pointing down towards the surface or away from it. The energy is lower in the conformation where the hydrogen is near the surface, though, ac6 have gained 0.005 eV compared to af6, while ac5 loses around 0.01 eV.

The vertical orientations, 3 and 4, have approximately the same interaction energy in the ac and af conformation. In the upright position, only the atoms near the surface contribute significantly to the energy. Since the difference between the two conformations are a 180∘ rotation of the hydroxy group, the energy is not affected in these orientations. When only the benzene ring and amide group or hydroxy group is considered, the two flat conformations are identical.

The scorpion conformation

The scorpion conformation of acetaminophen has been investigated in six orientations as well, and the interaction energies can be seen in Figs. 9 and 10. The greatest interaction is observed for the orientations as6 and af1, with as6 having the lowest interaction energy. It is expected due to the negatively charged oxygen and nitrogen in the amide group, O(20) and N(18), that the two conformations have the lowest energies. Further, this explanation also justifies why the as6 orientation has the lowest interaction energy (Figs. 11 and 12).

Fig. 9
figure 9

Energy plots of the six orientations of the scorpion conformation of acetaminophen

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Fig. 10
figure 10

Energy plots of the six orientations of the scorpion conformation of acetaminophen in a logarithmic scale

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Fig. 11
figure 11

The orientations of the scorpion conformation

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Fig. 12
figure 12

The energy contribution from the hydroxy group for ac1 and af1

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At z∗ = 0, the as3 orientation has the third lowest energy, which is due to the attraction between the amide nitrogen, N(18), and the hydroxy oxygen, O(19), and the corresponding induced charges. However, around z∗ = 1 the as4 orientation has lower interaction energy, and this is true for the longer distances as well. The orientation as3 has lower interaction energy than as4 at short distances because of the short distance between the charged atoms, H(14), N(18), O(19), and O(20), to the surface.

In Figs. 13 and 14, the atomic contributions related to the overall interaction energy of the atoms in the amide and the hydrogen atoms in the benzene moiety are presented. For the as3 orientation, the energy contribution related to N(18) is increased and the energy related to H(14) is decreased as the molecule approaches the surface, which implies a great negative charge being induced by the hydrogen. This effect is rather short ranged, and around 1Å the effect is hardly seen. A similar but longer-ranged behavior is observed for the energy contributions related to N(18) in the as4 orientation. The difference is that the repulsion arises from the negative charges mainly induced by H(14) and C(7), and at longer distances smeared out charges are better than localized.

Fig. 13
figure 13

The atomic contribution to the overall interaction energy of the amide group

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Fig. 14
figure 14

The atomic contribution to the overall interaction energy of the benzene hydrogen atoms

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The effect of the delocalized induced negative charges is evident internally in the molecule as well; the atoms further away from the amide group have lower energy in the as4 orientation than in the as3 orientation. In both orientations, the neighboring hydrogen atoms in the benzene moiety, H(10) and H(11), experience an attraction from the induced charges, but this time the energy contributions are lowest for the as4 orientation. At first, it may come as a surprise that H(11), which is pointing away from the surface in as4, has a lower energy in as4 than in as3, where it is closest to the surface. The reason is the same as before; the induced smeared out charges. H(9), in the other end of the benzene moiety, experiences an even greater attraction, which is due to the charges induced by H(13) in the hydroxy group. Since the oxygen is further away from the surface in the as4 orientation than in as3, the corresponding induced charge is lower, and the effect on the atoms is lowered as well. The energy of the last hydrogen, H(12), in the benzene moiety in the as4 orientation decreases as the molecule gets closer to the surface. At longer distances the same is true for H(12) in as3, but when the distance is shorter than 1Å, the energy increases. This is due to the effect of the charge induced by O(19). Along with the smeared out positive charges H(9) and H(10) is closer to the surface in the as4 orientation and thereby induces greater charge and can feel the other induced charges better. At longer distances, it is mainly the hydrogen atoms in the benzene moiety, which are responsible for as4 having lower energy than as3 (Table 2).

Table 2 The energy contributions from the different parts of the molecule having the scorpion conformation with the as3, as4, and as5 orientations
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The lowest interactions at z∗ = 0 are observed for the as2 and as5 orientations. The first has the disadvantage of the methyl group blocking the amide and hydroxy group approaching the surface. As a result, the interaction is remarkably lower than that observed for the rest of the orientations. The behavior of the as5 orientation is somewhat more difficult to explain; one might expect a molecule in this orientation to behave similar to a molecule in the as4 orientation. This is clearly not the case; for one thing there is a substantial energy difference at z∗ = 0 (Table 2), and though decreasing, this difference seems to exist at longer distances as well. The nitrogen and oxygen atoms in the amide group are flipped up in the as5 orientation compared to the as4 orientation, and the increased distance accounts for some of the energy difference. This flip also shortens the distance of the amide hydrogen, H(14), and the surface, which induces a negative charge in the dielectric continuum. In Figs. 15, 16 and 17, the atomic contributions to the overall interaction energy of the acetamide and hydroxy subgroup, and of the hydrogens on the benzene moiety, are shown for the as4 and as5 orientation. Though having the lowest interaction energy, the as4 orientation has the highest positive energy contribution from a single atom; the nitrogen in the amide group, N(18).

The high energy contribution related to the nitrogen atom and the low energy contributions related to the H(14) and C(7) atoms suggest a strong induced negative charge in the dielectric medium. In the other end of the molecule, a similar event is observed; because a significant negative charge is induced by H(13), the energy of the H(13) is lowered and the energy O(19) is raised compared to the as5 orientation. Further, the induced positive image charges by H(13) and H(14) lowers the energy of H(10) and H(9) more in as4 than in as5 (see Fig. 17).

For all the conformations, the major difference in energy arises from the amide, where as5 have the lowest energy due to H(14) being flipped up, and the hydrogens of the benzene moiety, where as4 has the lowest energy due to the H(14) and H(13) being near the surface (Table 2). The impact of hydrogen on the interaction energy is surprising because of the low partial charges, but nevertheless it is this interaction which gives the as4 orientation its stability.

Fig. 15
figure 15

Acetamide group

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Fig. 16
figure 16

Hydroxy group

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Fig. 17
figure 17

Benzene hydrogen atoms

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Comparison of the molecular conformation

In line with the previous discussion of the orientations, the lowest interaction energy is achieved with the scorpion conformation and the as6 orientation, because this geometry allows the nitrogen and oxygen atoms to be close to the surface, and the positive amide carbon to be furthest away. This results in the most stable interaction of the scorpion conformation, the energy being a factor of two greater than that of the flat af conformation and 0.04 eV greater than the flat ac conformation.

Though the scorpion conformation can obtain the lowest interaction energy, the energy of the molecule is 0.11 eV higher than the energy of the flat conformation. Since the interaction energy difference of the most favorable orientation in the scorpion and the flat conformations is 0.04 eV, the conversion energy loss can be halved. This suggests that the molecular structure can be altered during aggregation on a polymer particle represented by a dielectric medium.

The two flat conformations have some differences as well. The major difference is the most stable orientation is changed from orientation 5 to 2 going from the af to the ac conformation.

The polymer and molecular system in solution

The flat af conformation

The interaction energies of the different orientations and polymer particle along with the surrounding solvent are shown in Fig. 22. The relative stability at z∗ = 0 is changed compared to that in vacuum, since the solvent interacts strongly with the molecular charge distribution. The larger dielectric constant of the solvent leads to larger induced image charges. A clear indication of this, is the af6 orientation which was the second least stable orientation in vacuum, but the most stable when the solvent is taken into account. This is due to the amide oxygen, O(20), and the hydroxy group being so close to the solvent surface, and thus inducing larger image charges than in the solid. This is evident by comparison to the same orientation in vacuum, and the af5 orientation, which is the af6 orientation rotated 180∘. The bump on the af6 curve is caused by a shift in the atom determining the radius, at z∗ < 1.7 H(12) is determining, while at z∗ > 1.7 H(7) determines.

Though the opposite oriented conformation compared to af6, the af5 orientation has the second lowest energy near the solid surface. The orientations af5 and af6 have the advantage over the other orientations of having the highly charged atoms pointing either towards the solvent surface or the polymer particle surface. When the distance to the particle is increased, the af6 orientation remains the most stable orientation, while af5 is overtaken by af1 and af4 before z∗ = 1.2. When the distance to the surface is increased, the radius of the cavity is increased as well, resulting in the amide and hydroxy group being further way from the solvent. At longer distances, the larger charged oxygen atom’s contribution is limited by the increased radius, which raises the energy. All the orientations, except af3, have the amide oxygen, the hydroxy group, or both near the solvent surface, which accounts for the increased stability at longer distances.

The orientations parallel to the solid surface, af1 and af2, are the third and fourth stable orientations near the surface of the polymer particle. They both have similar behavior when the distance is increased, which is expected due to the similarity in orientation. The difference in energy may, to some extent, be assigned to the difference in the cavity radius. Further, and more importantly, the hydroxy hydrogen, H(13), in af2 does not gain any energy, opposite to the one in af1 (Fig. 18). The reason may be that this hydrogen experiences the positive image charge induced by O(19) in the solvent, while the hydrogen itself is not inducing a large image charge, due to the increased distance to the surface. The enhanced atom–solvent distance for hydrogen compared to oxygen is due to the slightly tilted molecule.

Fig. 18
figure 18

Solvated af1 and af2 hydroxy group

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The last two orientations, af3 and af4, are the least stable near the surface of the polymer particle. As before, the relative stability of the two are reversed as compared to that in vacuum. The least stable orientation, af3, gains very little compared to the other orientations due to the fact that the methyl group is insulating the whole molecule from the solvent by the larger radius. The orientation af4, on the other hand, gains a lot by interacting with the solvent, but very little from the interacting with the polymer particle. This is evident since the energy is lowered by only 0.01 eV when the molecule goes from z∗ = 7Å to 0Å. At all distances, the radius of the cavity is determined by hydroxy group, ensuring a constant energy reduction (Figs. 19, 20, 21, 22, 23 and 24).

Fig. 19
figure 19

Solvated as1 and as6 hydroxy group

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Fig. 20
figure 20

Solvated as2 and as3 hydroxy group

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Fig. 21
figure 21

Solvated as4 and as5 hydroxy group

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Fig. 22
figure 22

Solvated af orientations

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Fig. 23
figure 23

Solvated as orientations

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Fig. 24
figure 24

Solvated Ac

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The ac conformation

The same relative stability of the orientations of the solvated flat conformations is the same, but the ac conformation generally obtains lower energies, Fig. 24.

As in vacuum the vertical orientations, 3 and 4, have approximately the same energy in both flat conformations. The same is true for orientation 2, while orientation 1 is lowered 0.02 eV going from the af to the ac conformation. The main reason is H(13), which goes from contributing with a slightly positive energy in the af conformation, to contributing negative (–0.027 eV) in ac. The two hydrogen atoms on the H(14) side, H(11) and H(12), are lowered in energy, while for the hydrogen atoms on the opposite side it is raised, Fig. 25. This matches the idea that H(13) in ac is close to positive or neutral charges, which are near the surface. Thus H(13) is close to more induced negative charges in ac than in af.

Fig. 25
figure 25

Energy contribution from the benzene hydrogen in ac1 and af1

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It is worth noting that orientation 6 gains around –0.02 eV going from the af conformation to ac, while orientation 5 only gains –0.01 eV. This is due to a lower energy contribution from the hydroxy group and O(20). In ac5 H(13) and H(14) are close to the solvent surface and, thus, they contribute greatly to the energy. The two oxygen atoms do not induce strong charges, but are affected by the charges produced by H(13) and H(14). In ac6 this is reversed and the oxygen atoms are closest to the solvent surface. Due to the greater charge of the oxygen, greater charges are induced in the solvent-medium. The significant difference between ac5 and 6 is O(20), C(7), and H(14), which accounts for the increased stability of orientation 6, Fig. 26.

Fig. 26
figure 26

Energy contribution from O(20), C(7), and H(14) in the amide group in ac5 and ac6

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The scorpion conformation

The interaction energies of the different orientations of the scorpion conformation and polymer particle surrounded by solvent are shown in Fig. 23. In vacuum, when solvated, we expect the opposite to be the case: The amide or hydroxy group is, in the most stable orientations, expected to be near the solvent surface.

When that is said, at the surface of the polymer particle the most stable form is, as in vacuum, the as6 orientation closely followed by the orientations af1, as2, and as5. Overall the orientations af1 and as6 behave similarly as in vacuum, but the energies are lowered due to the interactions with the surrounding solvent. The orientation as6 is lowered somewhat more because of the hydroxy oxygen, O(19), having a lower energy due to the image charge induced by the hydrogen H(13), see Fig. 23. In the af1 orientation the image charge induced by H(13) is close to the atom O(19), while it is further away in the as6 orientation, due to the angle between the position vectors of O(19) and H(13) being larger in the as6 orientation.

The two conformations with energies almost the same as the af1 orientation at z∗ = 0, as2 and as5, have lowered the energy the most. When the distance is increased to around 0.2Å both orientations have lower energy than the rest of the orientations. The as2 orientation has two bumps due to a change in which atom determines the radius of the cavity. The radius is determined by H(13) between z∗ = 0Å to 1.61Å, by O(19) between 1.61Å and 5.32Å, and by H(14) at longer distances than 5.32Å.

The as2 orientation has the lowest energy at z∗ > 0.2 because both oxygen atoms are close to the solvent surface. The as3 orientation, which is the as2 orientation rotated 180∘, decreases little in energy compared to as2, because the radius of the half-sphere is determined by the methyl group. The hydroxy group is therefore further away from the solvent surface, and, consequently, the stabilizing contribution from this group is lower, see Fig. 20.

As observed previously, the relative stability of two related orientations is reversed when going from vacuum to solvation, and this hold for the as4 and as5 orientations as well. A part of the extra stability of the as5 orientation comes from the amide oxygen being slightly closer to the surface of the solvent than in the as4 orientation, see Fig. 21. The other contribution comes from the hydroxy hydrogen inducing a charge in the vicinity of atoms H(9) and H(10).

Comparison of the flat conformations and scorpion conformation

As in vacuum, the energy of the scorpion conformation is lower than the flat conformations when they are solvated. The flat conformations are reduced a bit more than the scorpion conformation, but the energy difference between the most stable orientations of the flat ac and scorpion conformation is still around 0.02 eV.

The interactions between the atom O(20) and the polymer surface is the strongest, and when combined with N(18), the lowest interaction energies are obtained. If they are blocked from getting close to the surface by the rest of the molecule, the surrounding positive lower-charged atoms have a significant role to play. Since the positive charge is distributed between more atoms than the negative charge, the charges are generally lower and the charge distribution more diffuse. The delocalized charges can in some cases interact over longer distances than the atoms O(20) and N(18), and stabilizing an orientation through the, at first sight innocent, hydrogen atoms in the benzene moiety.

Investigating the role of polymer and solvent dielectric constants

Within the presented heterogenous dielectric solvation model, it is straightforward to investigate the role of different solvents and polymers. Polymers have dielectric constants from 1 to 20, while solvents can have dielectric constants from about 2 (cyclohexane) to above 100 (formamide). In Fig. 3 we compare the influence of these extreme situations on the aggregation of the flat (ac) conformation of acetamide.

The two known crystal forms of acetaminophen have significantly different intermolecular packing patterns. The sideways stacking of molecules on the polymer surface (orientations 5 and 6) were the most stable. In both crystal forms, a slab of molecules can be identified showing the sideways stacking pattern, however in the orthorhombic form, this slab consists of only orientation 5, whereas in the case of the monoclinic form, both orientation 5 and the reverse orientation 6 are involved (Fig. 2).

To suppress the formation of the monoclinic polymorph and favor the formation of the orthorhombic form, it is necessary to chose a solvent with a high dielectric constant combined with a surface with a low dielectric constant, e.g., combining water with poly tetrafluoroethylene (TEFLON). As shown in Fig. 3b, the energy gap between the two sideways conformations as5 and as6 is more than 0.05 eV per molecule (about 130 kJ/mol). If instead the monoclinic form is preferred, a combination involving heptane and a high-dielectric solid such as graphite should be used; this will favor both conformations 5 and 6, and thus allow the monoclinic polymorph to nucleate on the surface.

Conclusions

We have investigated the prospects of using a heterogenous dielectric solvation model to study the nucleation of molecular crystals on polymer surfaces. We demonstrate that the orientation of the molecule has a major influence on the interaction energy, and hence the likelihood of the molecule aggregating on the surface. In the case of acetaminophen, the least stable gas-phase conformation (scorpion) is the form having the most favorable interaction with the surface. This conformation is not observed in the known crystal forms, but our model suggests that heterogenous nucleation on a polymer surface can be used to initiate aggregation of such otherwise metastable crystal nuclei by tuning the dielectric constants of solvent and polymer.

It is important to note that the inclusion of the outer solvation changes the relative stability of the orientations and between the two conformations. Neglecting a solvent with a significantly higher dielectric constant will in many cases result in wrong results and predictions. This is pronounced in the flat conformations where the relative stabilities of the orientations 1–2, 3–4, and 5–6 (Fig. 6) are reversed when going from vacuum to solvation.

The model suggests that the aggregation on a polymer surface will allow the nucleation of the orthorhombic form of acetaminophen, in line with experimental evidence. The preference for the nucleation of the orthorhombic polymorph is pronounced when an apolar surface is combined with a polar solvent.

By using the heterogenous dielectric solvation model described here, it is straightforward to change dielectric constants, and hence to study the effect of other solvents and other surfaces. This may inspire the experimentalist in the choice of solvent and polymer for crystallization of polymorphic systems.

Our approach demonstrates a non-epitaxial approach to heterogenous nucleation. It is a computationally inexpensive approach which, once the atomic charges have been derived from ab initio calculations, scales linearly with the size of the molecules, and it is thus possible to study the aggregation of very large molecules.

In our present investigation, we have not included any explicit solvent molecules in the description of the solvent interactions between the heterogeneously solvated molecule and the neighboring solvent molecules. It is clearly, as indicated by one of the reviewer, a very important aspect to consider in future investigations utilizing the presented method. In the present investigation, we have not included van der Waals interactions between the molecule and the polymer but we have in ref. [71] developed a model that includes dispersion interactions between the polymer and the molecule. We have clearly shown that dispersion interactions are important for describing the interactions between the polymer and the molecule [71]. For future investigations, it would be beneficial to combine the two approaches when investigating the interactions between the polymer and molecule. Presently, we are trying to establish procedures for checking how well the model is able to explain and predict experimental results but we are still in the initial phase of utilizing the presented model.