How fast do drugs crystallize from their own melts, and why? Crystal growth rates of 31 undercooled pharmaceutical liquids

Background and motivation

Many new drug candidates are so poorly soluble in water that they are delivered in amorphous form, which has improved the therapeutic effect of several molecules (Kwong 2011, ref 1; Bollag 2010, ref 2). The price is physical instability: the glass must stay non-crystalline for the life of the product, typically 2–3 years. Nobody can wait that long for every candidate, so the authors argue that formulators need a fundamental understanding of crystallization tendency, one that separates compounds at high risk from those that can safely be delivered as glasses (Introduction).

The authors' group had already ranked about 50 organic compounds, mostly drugs, by a simple test (Baird 2010, ref 3). A sample is melted, quenched and then reheated in a DSC. Class I compounds crystallize during cooling (poor glass-forming ability, GFA); class II compounds form a glass on cooling but crystallize on reheating (good GFA, poor glass stability, GS); class III compounds crystallize in neither step (good GFA and GS). Most of the 31 drugs in the present paper were classified in that study. The test is quick, but its result depends on cooling and heating rates and on how nucleation and growth each depend on temperature. Both rates rise and then fall as a melt is cooled ("bell-shaped" curves), and the growth maximum lies at a higher temperature than the nucleation maximum (Okui 1990, ref 4). A class II compound is thought to form nuclei on cooling in a range where growth is slow, and to crystallize only when reheating brings it into the range where growth is fast. A class III compound may grow and nucleate slowly, or it may grow fast but nucleate so poorly that no crystals appear. The test cannot tell these cases apart, and so it cannot say which process controls devitrification.

Polymers complicate the picture. They are thought to stabilize amorphous drugs by lowering molecular mobility, especially when they interact specifically with the drug (Kestur 2010, ref 5; Van den Mooter 2001, ref 6). The polymer must be miscible with the drug, and the amount needed for a given shelf life should depend on how fast the drug itself crystallizes. Whether a polymer acts on nucleation or on growth matters: in acetaminophen (APAP), 15% w/w HPMCAS inhibited nucleation strongly with little effect on growth, whereas poly(acrylic acid) increased nucleation but substantially slowed growth, and both turned APAP from class II into class III (Trasi 2012, ref 7).

Nucleation is hard to study because nuclei are too small to observe; crystal growth of organic compounds has been studied far more (refs 8–16). Naito and Miura (ref 17) proposed that the maximum crystal growth velocity (MCV) of organic compounds depends simply on the summed enthalpies of all phase transitions between the glass transition temperature $T_g$ and the melting temperature $T_m$ and on the number of non-hydrogen atoms; in their set the MCV varied by about five decades.

The paper sets itself three goals: to measure the range of crystal growth rates in a group of pharmaceutically relevant compounds; to test how well these rates correlate with molecular, thermodynamic and kinetic properties; and to find how far a polymer can change the growth rate of compounds that grow at very different intrinsic rates. To do this the authors measured growth rates of 31 drugs as a function of temperature, and the effect of molecularly dispersed polyvinylpyrrolidone (PVP) on a subset.

Background concepts

Undercooled liquid and normalized temperature. A melt cooled below $T_m$ without crystallizing is an undercooled (supercooled) liquid; below $T_g$ it becomes a glass. The extent of undercooling is $\Delta T=T_m-T$. Because the 31 drugs have different $T_g$ and $T_m$, their growth rates can only be compared on a normalized temperature scale. The paper uses two: $T/T_g$, which assumes that all compounds have the same molecular mobility at $T_g$, and $\Delta T$, which compares them at equal undercooling and hence roughly equal thermodynamic driving force (Results, Figure 1).

Why growth rate has a maximum. To grow a crystal, molecules must move to the crystal surface and then stay attached. Just below $T_m$ the liquid is mobile but the driving force for crystallization is small; at lower temperatures the driving force grows, roughly as $\Delta G\propto T_m-T$, while the viscosity rises and mobility falls. The product of the two factors passes through a maximum, the maximum crystal velocity (MCV), at a temperature called $T_{\mathrm{c}\mathrm{r}\mathrm{y}\mathrm{s}\mathrm{t},\mathrm{m}\mathrm{a}\mathrm{x}}$ (Figure 5). Below $T_{\mathrm{c}\mathrm{r}\mathrm{y}\mathrm{s}\mathrm{t},\mathrm{m}\mathrm{a}\mathrm{x}}$ growth is dominated by viscosity; above it, by thermodynamics. Earlier work found that the MCV tends to occur at about 0.94 $T_m$ (Naito 1994, ref 26).

Growth models and the Jackson equation. Three classical models describe growth from the melt: normal (continuous) growth on a molecularly rough interface with many equivalent growth sites; screw-dislocation growth, in which a smooth interface grows only at steps supplied by screw dislocations; and two-dimensional (surface) nucleation growth, in which new layers start as two-dimensional nuclei on the face (refs 20–22). Whatever the mechanism, every model has a term for molecular mobility, because molecules must diffuse across the interface to a growth site, and a term for the driving force, which sets the probability that they stay attached rather than return to the liquid. The paper uses the form given by Jackson (ref 23):

Here $U$ is the growth rate, $\lambda$ the average diffusion jump distance, $\eta$ the viscosity, $k_B$ the Boltzmann constant, $T$ the temperature, $\Delta H$ and $\Delta S$ the enthalpy and entropy differences between liquid and crystal, and $f$ the fraction of interface sites that are active growth sites (eq 1). In plain terms: growth speed is mobility (the term in $1/\eta$) times the net probability that an arriving molecule sticks (the bracket, zero at $T_m$ and growing with undercooling) times the fraction of molecules that arrive in the right orientation (the entropy term). The paper's analysis follows these three factors in turn: driving force, viscosity, entropy.

Viscosity as a measure of mobility. Using $\eta$ in eq 1 assumes that the Stokes–Einstein relation holds, so that viscosity describes the diffusion of molecules across the crystal–liquid interface. The authors call this "a reasonable assumption" in the temperature range of their measurements, but note that near $T_g$ diffusion decouples from viscosity, so growth is no longer proportional to the measured viscosity (Rössler 1996, ref 24), and that some crystals then switch from diffusion-controlled to diffusionless growth (Sun 2008, ref 12).

Driving force from the melting entropy. The free energy difference between melt and crystal is approximated as

where $\Delta S_m$ is the entropy difference between melt and crystal at the melting temperature (Discussion, after eq 2). In plain terms: each kelvin of undercooling adds a driving force of about $\Delta S_m$ per mole, so a drug with a large melting entropy gains driving force faster on cooling. (Note by the AI writer: the $\Delta S_m$ column of Table 1 equals the heat of fusion divided by $T_m$ to within about 2% for every drug, so it is calculated rather than measured separately.)

Fragility. A fragile liquid is one whose viscosity rises steeply as it is cooled toward $T_g$; a strong liquid changes more gradually. Two drugs with the same viscosity at $T_m$ can therefore have very different viscosities at the temperature where they grow fastest. The paper invokes fragility, taken from ref 25, to explain some outliers but does not measure it.

Surface versus bulk growth. Crystals often grow faster at a free surface than in the bulk, especially at low temperatures where the material does not flow (Zhu 2008, ref 13), which is thought to reflect higher molecular mobility at the surface near or below $T_g$ (Zhu 2011, ref 19). Whether a compound that grows fast in the bulk also grows fast at the surface had not been examined systematically.

Methods in detail

The study has four parts: a growth-rate data set for 31 drugs measured with one method; a screen of the properties that separate fast from slow growers; an analysis that removes the driving force and viscosity from the growth rate to test the entropy term of eq 1; and a test of how much a polymer, PVP, slows drugs with very different growth rates. The flow below summarizes them.

Contribution 1: one growth-rate data set for 31 drugs

① Problem. To compare growth across many compounds, every rate must come from the same method, must be a pure growth rate with no contribution from nucleation, and must refer to a known crystal form.

② How. A 3–5 mg portion of drug powder was placed between two coverslips, heated above its melting point and quenched to room temperature on an aluminum block. The film was then seeded at its edge with the starting crystalline material, so that the polymorph of interest grew, and heated until growth had advanced into the bulk. On a Linkam THMS 600 hot stage under a Nikon Eclipse E600 polarizing microscope, the advancing crystal front was photographed at regular intervals; the slope of front position against time gave the radial growth rate (Experimental Methods). Each drug was measured at several temperatures between about 30 and 90 °C above its $T_g$; faster growers could also be measured at lower temperatures. For surface growth the top coverslip was removed. Surface rates could only be measured close to $T_g$, and where the front was irregular the authors converted the total crystallized area into an equivalent radius with ImageJ, as in Rumondor 2010 (ref 18). Melting point, heat of fusion and $T_g$ were measured by DSC (TA Instruments Q2000, temperature calibrated with indium and tin, enthalpy with indium, dry nitrogen purge at 50 mL/min). The paper reports no replicate numbers or uncertainties for the growth rates, and no DSC heating rate.

The 31 drugs are listed in Table 1. They came from Sigma-Aldrich, Hawkins, Spectrum Chemicals, Attix Pharmaceuticals and Euroasia Chemicals; PVP K-12 (Mw printed as "2−3000 g/mol") came from BASF. Note by the AI writer: the Materials paragraph lists nimesulide twice and does not list salsalate, which appears in Table 1 and Figures 1 and 2, so its source is not given.

③ Why it works. Seeding removes nucleation from the experiment: the front starts from added crystals, so what is measured is growth alone, and isothermal measurement gives one rate per temperature. Normalizing temperature by $T_g$ or by $T_m$ then allows compounds with very different transition temperatures to be placed on one plot (Figure 1).

④ Difference from earlier work. The cooling-and-reheating test of ref 3 sees only whether crystals appear, which mixes nucleation and growth. Here growth is measured directly, for a set large enough to look for general trends.

Contribution 2: ranking by maximum growth rate and screening the properties

① Problem. With 31 compounds on different temperature scales, the authors needed one number per drug to rank them, and a way to see which properties go with fast growth.

② How. Table 1 lists, for each drug, the MCV and eleven properties: molecular weight, $T_m$, $T_g$, number of atoms, heat of fusion $\Delta H_f$, melting entropy $\Delta S_m$, the free energy difference $\Delta G$ at $T_{\mathrm{c}\mathrm{r}\mathrm{y}\mathrm{s}\mathrm{t},\mathrm{m}\mathrm{a}\mathrm{x}}$, $T_m-T_g$, crystal density, number of rotatable bonds, and melt viscosity at $T_m$ (taken from ref 25, available for 17 drugs). The drugs were split by MCV into group 1 (10–100 µm/s), group 2 (1–10 µm/s) and group 3 (0.01–1 µm/s). For each property, the authors then plotted (Figure 6)

In plain terms: how far the average of one group lies from the average of all 31 drugs, measured in standard deviations of the whole set. A bar near +1 means that the group sits about one standard deviation above the overall mean for that property.

Note by the AI writer (not discussed in the paper): three things about Table 1 are not explained. How the MCV of each drug was obtained is not stated, and for several drugs it does not match the plotted data (see Experimental analysis). $T_{\mathrm{c}\mathrm{r}\mathrm{y}\mathrm{s}\mathrm{t},\mathrm{m}\mathrm{a}\mathrm{x}}$ is not listed, nor is the method used for "$\Delta G$ at $T_{\mathrm{c}\mathrm{r}\mathrm{y}\mathrm{s}\mathrm{t},\mathrm{m}\mathrm{a}\mathrm{x}}$"; if $\Delta G=\Delta S_m(T_m-T_{\mathrm{c}\mathrm{r}\mathrm{y}\mathrm{s}\mathrm{t},\mathrm{m}\mathrm{a}\mathrm{x}})$, the tabulated values imply growth maxima between about 0.90 and 0.99 $T_m$. And the sources of molecular weight, atom count, crystal density and rotatable bonds are not given. The bars of Figure 6 can, however, be recomputed from Table 1 with the sample standard deviation, to within about 0.06. R1·R8·E5·X3·X4

④ Difference from earlier work. Naito and Miura (ref 17) related the MCV to two quantities; here eleven candidate properties are screened side by side, which lets viscosity and $T_m-T_g$ emerge as the strongest separators.

Contribution 3: removing driving force and viscosity to test the entropy term

① Problem. Viscosity and driving force differ between drugs and with temperature. To see what else controls growth, both must be taken out of the measured rate.

② How. First the measured rate $U$ is divided by the driving-force factor of eq 1, giving the "kinetic" growth rate (eq 2):

with $\Delta G\approx \Delta S_m\Delta T$ and $R$ the gas constant. In plain terms: close to $T_m$ the driving force is small and the bracket is far below 1, so dividing by it raises the rate to what it would be if every arriving molecule stayed. $U_{\mathrm{k}\mathrm{i}\mathrm{n}}$ was plotted against the melt viscosity on log–log axes for nine drugs with viscosity–temperature data (Figure 8). Next, following Ediger and co-workers (ref 28), viscosity was removed by multiplying $U_{\mathrm{k}\mathrm{i}\mathrm{n}}$ by ${\tau }_{\alpha }/a$, where

is the structural relaxation time with $C$ and $G$ taken as constants of 4 and 2, and $a$ is the molecular diameter, obtained from the crystal conformation in the Cambridge Structural Database. In plain terms: $a/{\tau }_{\alpha }$ is the speed at which a molecule could move one diameter in one relaxation time, so $U_{\mathrm{k}\mathrm{i}\mathrm{n}}{\tau }_{\alpha }/a$ is the growth rate in units of that molecular speed; what remains should depend on the entropy term of eq 1. This quantity was plotted against $\Delta S_m/R$ (Figure 9). Note by the AI writer: the units of $C$ and $G$ are not given and the values of $a$ are not printed, so the absolute values in Figure 9 cannot be reproduced. E2·E4·X11

④ Difference from earlier work. Ediger and co-workers applied this normalization across inorganic and organic compounds; here it is applied to drugs.

Contribution 4: a polymer test at equal molar ratio

① Problem. If a polymer slowed every drug by the same absolute amount, fast and slow growers would need similar amounts of polymer; if it slowed them by the same factor, fast growers would need more. The authors asked whether PVP is relatively less effective for a fast grower than for a slow one.

② How. Five drugs with quite different bulk growth rates but all with good hydrogen bond donor groups were chosen: acetaminophen, flutamide, nilutamide, aceclofenac and indomethacin (IMC). Because PVP is a hydrogen bond acceptor, the authors expected all five to interact with it similarly. Each drug was cryomilled with 10 molar % PVP K-12 in a Spex 6750 freezer mill, melted between coverslips, nucleated with its starting crystal form and measured as above (Figure 3). The $T_g$ of every mixture was only about 2–3 °C above that of the parent drug. P3·R7·C9

Note by the AI writer: the paper does not say whether "mol%" counts polymer chains or vinylpyrrolidone repeat units. On either basis the mass fraction of PVP differs between the five drugs, whose molecular weights range from 151.1 (acetaminophen) to 357.7 g/mol (indomethacin), so the comparison is at equal molar, not equal mass, ratio. X7

Experimental analysis

Growth rates span four to five orders of magnitude

Figure 1 plots the bulk growth rates of all 31 drugs, from about 4 × 10⁻⁵ to about 100 µm/s, against $T/T_g$ (about 1.03 to 1.36) and against undercooling (about 15 to 120 K). R2 Compared at equal $T/T_g$, the fastest and slowest drugs differ by about 4 orders of magnitude; compared at equal undercooling, by about 5. The ranking also changes with the scale: on the $T/T_g$ scale griseofulvin grows fastest, closely followed by nilutamide, celecoxib and tolbutamide, whereas on the undercooling scale acetaminophen and pimozide are fastest. R3·C7 Ranked by MCV (Table 1 and Table A below), nimesulide is fastest at 216 µm/s and clotrimazole slowest at 0.02 µm/s, a difference of about 4 orders of magnitude. R1·C23 The abstract and the conclusions speak of "about 5" and "up to 5" orders; that figure refers to the undercooling scale. X14

Table A. The 31 drugs in order of maximum crystal velocity, with three of the Table 1 properties (from Table 1; class from the symbol colours of Figure 1: I/II red, III blue, read by the AI writer; — = no viscosity given). R1·M1–M31

Three features of Table 1 need care. The printed group 1 range is 10–100 µm/s, yet nimesulide (216) and flutamide (104) are placed in group 1. X2 Celecoxib's $T_m-T_g$ is printed as 125, although its $T_m$ (436 K) and $T_g$ (328 K) differ by 108 K; Figure 7 plots the printed 125. X1 And, as noted above, how the MCVs were obtained is not stated. Note by the AI writer: several MCVs lie far above the highest growth rate plotted for the same drug, for example nimesulide (216 µm/s against about 8.6 µm/s in Figure 2b), salicin (95.5 against about 4.4) and griseofulvin (66.6 against about 15); others lie below plotted rates, for example felodipine (0.86 against about 1.2 µm/s in Figures 1 and 2b) and indomethacin (0.47 against about 1.25 µm/s at 120 °C in Figure 3). A value below a measured rate cannot be the maximum of that data set, so the MCVs should be read as values of unstated origin. The grouping of drugs near the group limits depends on them. X3

Good glass formers can be fast growers

In Figure 1 each drug is coloured by its class in the cooling-and-reheating test: red for class I and II, blue for class III. "Interestingly", the authors write, several class III drugs, the good glass formers, had very high growth rates; "in fact", class III growth rates span the entire range, with both some of the fastest and some of the slowest growers. Drugs with poor glass stability (class II or I-b) had moderate to fast growth rates, in the upper half of the plot. C14 Counted from Table A, the 19 class III drugs include 7 in group 1 (among them nimesulide, the fastest), 4 in group 2 and all 8 of group 3; the 12 class I/II drugs split 6 and 6 between groups 1 and 2, with MCVs from 1.8 to 104 µm/s, and none is a slow grower. R1

The authors read the fast-growing class III drugs as good glass formers only because they "clearly" have a low tendency to form nuclei under the conditions of the test. C15 This matters for risk. Processing can leave residual or process-induced nuclei, and then such a drug "may have a high probability" of crystallizing over the shelf life, because its growth rate is high; a class III drug with an inherently low growth rate will crystallize much more slowly even when nuclei are present. These statements assume that relative growth rates carry over to storage temperatures. C16 On this basis the authors propose an extended classification that combines nucleation behaviour with growth rate (Figure 4), summarized in Table B.

Table B. The extended classification proposed in Figure 4 and the Discussion. Nucleation is inferred from the cooling-and-reheating behaviour of ref 3; it was not measured. C17·C18

Note by the AI writer: the class labels of the two schemes overlap and are easy to confuse. Acetaminophen is "class II" in the older scheme (Introduction) but class IB in Figure 4; nimesulide, nilutamide and celecoxib are class III in the older scheme but class II in the new one; cinnarizine and droperidol are red (class I or II) in Figure 1 but class III in the new scheme. Always state which scheme a class label refers to. X10

Surface and bulk growth rank the drugs differently

Figure 2 compares surface growth near $T_g$ (panel a, $T/T_g$ about 0.98 to 1.14) with bulk growth of the same 14 drugs (panel b, about 1.07 to 1.33). Surface rates ranged from about 0.00005 µm/s (loratadine) to about 1 µm/s (nimesulide). R4·R5 The order "does not necessarily match": felodipine grew faster at the surface than carbamazepine and nilutamide, yet in the bulk it was slower than nilutamide at all temperatures and slower than carbamazepine at higher temperatures. On the other hand loratadine, the slowest surface grower, was also slow in the bulk, and the faster surface growers, piroxicam and acetaminophen, were also fast in the bulk. C8 Note by the AI writer: because the surface and bulk data cover different $T/T_g$ windows, most of this comparison is between different temperatures. X16

Slow growers have viscous melts and a small Tm − Tg

Figure 6 shows how far the average of each group lies from the overall average for each property. Table C gives the values read from the figure.

Table C. (Group average − overall average) / overall standard deviation for each Table 1 property (read from Figure 6, approximate). R8

The authors read the departures of molecular size, $T_m$, melting entropy, $\Delta G$, $T_m-T_g$ and viscosity as "suggesting" that these are important factors. Viscosity "stands out": group 3 has a much higher than average melt viscosity. $T_m-T_g$ also varies considerably, and crystal density "seems to vary greatly" between groups 1 and 2. C24 The abstract condenses this into: fast growers had smaller molecular weights, higher melting temperatures, lower melt entropies, lower melt viscosities and higher crystal densities. C25 Note by the AI writer: for density this holds only for group 1 against group 2; group 3 sits at the average, so density does not fall steadily with growth rate. The viscosity bars, the largest in the figure, rest on the 17 drugs with a known melt viscosity, only 5 of the 13 in group 1. X8·X9

Figure 7 looks at the two strongest factors together. For the 17 drugs with a melt viscosity, viscosity at $T_m$ falls as $T_m-T_g$ rises, with the printed fit

where $y$ is the viscosity at $T_m$ in Pa·s and $x$ is $T_m-T_g$ in K. In plain terms: drugs whose melting point lies far above their glass transition tend to have fluid melts, and drugs with a small gap tend to have viscous ones. The group 3 drugs sit at $T_m-T_g$ of 85–117 with viscosities of 0.0703–0.191 Pa·s, the group 1 drugs at 121–146 with 0.0136–0.0513 Pa·s. R9 The authors call it a "somewhat linear trend" with considerable scatter, especially at high viscosity. C26 (Note by the AI writer: the fit reproduces to four digits from the Table 1 values, including the printed 125 for celecoxib. E6)

Their explanation: "If it is assumed" that all melts have the same viscosity at $T_g$, then a drug with a low melting point relative to its $T_g$ will have a low MCV. Because the MCV lies near 0.94 $T_m$, a drug with a viscous melt at $T_m$ and a short range to traverse down to $T_g$ is still viscous at the temperature of its growth maximum. The authors add that fragility "also needs to be taken into account", and cite Angell's finding that the lowest-melting xylene isomer is the best glass former (ref 27). C27·C28 Three group 2 drugs, tolazamide, aceclofenac and bifonazole, lie among the group 1 drugs in Figure 7 ($T_m-T_g$ of 154, 143 and 134; viscosities of 0.0066, 0.0143 and 0.0101 Pa·s). "The most likely reason" aceclofenac and bifonazole are not fast growers, the authors write, is that their viscosity rises more steeply with cooling ("they appear to be more fragile", ref 25), while tolazamide "could be" a slower than expected grower because of its complex molecular structure. Neither fragility nor structural complexity was measured; tolazamide's 21 atoms and 5 rotatable bonds are close to the averages of the set. C29·X18

Driving force and viscosity are not enough

Figure 8 plots $U_{\mathrm{k}\mathrm{i}\mathrm{n}}$ against melt viscosity for nine drugs on log–log axes. Within each drug the corrected rate falls steeply as viscosity rises: for example from about 66 to about 2.2 µm/s for nilutamide as $\log \eta$ rises from about −0.2 to about 2.8, and from about 0.049 to about 0.00046 µm/s for miconazole between about 0.5 and 3.1. Across drugs, however, the rates at similar viscosity differ widely: near $\log \eta \approx 1$ (about 10 Pa·s), nilutamide grows at about 20 µm/s and nimesulide at about 3.3 µm/s, against about 0.1 µm/s for aceclofenac and bifonazole and about 0.02 µm/s for miconazole (all read from Figure 8, approximate). R10 The authors see two sets of drugs whose growth rates differ by around 2–3 orders of magnitude at similar viscosity and driving force, and conclude that "Clearly" other parameters contribute. C30 Note by the AI writer: the upper set includes felodipine, a group 3 drug by MCV, so these two sets are not the MCV groups. X15

Melting entropy accounts for part of the rest

The remaining factor in eq 1 is the entropy term. Following Jackson, a molecule must have the correct orientation to join the crystal, and only a fraction of the molecules in the melt, reflected by the entropy difference, have it. C31 Figure 9 plots $\log (U_{\mathrm{k}\mathrm{i}\mathrm{n}}{\tau }_{\alpha }/a)$ against $\Delta S_m/R$ for the same nine drugs, with the printed fit

where $y=\log (U_{\mathrm{k}\mathrm{i}\mathrm{n}}{\tau }_{\alpha }/a)$ and $x=\Delta S_m/R$. In plain terms: after driving force and viscosity are taken out, drugs with a larger melting entropy grow more slowly, by about 0.6 log units per unit of $\Delta S_m/R$. Nilutamide (about −5.1 at $\Delta S_m/R$ of about 8.7) is highest; miconazole (about −8.2 at about 10.9) is lowest; cinnarizine, with the largest entropy (about 12.5), lies at about −7.9 (read from Figure 9, approximate). R11·E7 The authors find that the compounds "follow the expected trend but with considerable scatter", similar to the scatter Ediger and co-workers found, and conclude that entropy "certainly plays a role" but that further factors, which "could include" molecular interactions in the melt and at the interface, are not accounted for. C32 Note by the AI writer: the text calls 0.61 a "correlation coefficient" while the figure labels it $R^2$, and the text calls the x-variable "configurational entropy" and "melt entropy" while the axis and caption use the melting entropy $\Delta S_m$; the plotted x-values agree with $\Delta S_m/R$ from Table 1. X12

PVP slows fast and slow growers by a similar factor

Figure 3 shows the growth rates of the five drugs with and without 10 mol% PVP. Table D gives the readings at the lowest and highest temperature of each drug.

Table D. Growth rates with and without PVP at the lowest and highest measured temperature of each drug (read from Figure 3, log axis, approximate). R6

At 70 °C neat flutamide grows about 170 times faster than neat aceclofenac, yet PVP cuts both by similar factors. The authors conclude that the initial growth rate "did not appear" to influence the extent of growth-rate reduction, since the polymer reduced growth for all five drugs "by around the same extent"; in the Discussion they put the factor at "around 5−10". C10 Note by the AI writer: over all temperatures the readings give reductions of about 9–14-fold for flutamide, 3–6-fold for acetaminophen, 4–8-fold for aceclofenac, 5–8-fold for indomethacin and about 5-fold for nilutamide at 80–100 °C, but about 50-fold for nilutamide at 70 °C. No replicates are given. X6 From the equal factor the authors infer that faster growers "will require a significantly larger amount of polymer" to stabilize their amorphous form (Results: "This observation suggests"). C11

The chain from these findings to the authors' risk argument can be drawn as follows.

Discussion

Where the results apply. All growth rates were measured above $T_g$. The authors therefore see them as "directly pertinent" to crystal growth during high-temperature processing such as melt extrusion: post-extrusion cooling is usually slow, so the product may spend enough time in the undercooled range for nucleation and growth to occur. C34 Amorphous products are stored at lower temperatures, and the authors accept that growth-rate trends "cannot readily be extrapolated" from high to low temperatures. They nevertheless judge it "likely" that drugs that grow fast in the undercooled liquid, relative to other drugs, also grow relatively fast in the glass, "in particular for compounds where faster growth is driven by small melt entropy values". C35·C36 Near $T_g$, where products are made and stored, the Stokes–Einstein assumption behind eq 1 weakens and some crystals switch to diffusionless growth. C21

Polymers. Specific drug–polymer interactions such as hydrogen bonding are thought to govern how strongly a polymer inhibits growth (Kestur 2010, ref 5; Taylor and Zografi 1997, ref 29), and above a certain concentration a high-$T_g$ polymer also raises the viscosity of the mixture. C33 For drugs with an OH or NH donor available to interact, "our data suggest" that, for a given polymer, the relative reduction in growth rate is very similar between drugs, whatever their starting rate. Hence, "if it is desirable to reduce the growth rate to a given value", fast growers need more polymer than slow ones. C12·C11 At the level tested, PVP raised the $T_g$ of each mixture by only about 2–3 °C. R7

What the paper explains and what it leaves open. In the authors' words, differences in melt viscosity (molecular mobility) and configurational entropy "could explain much of the observed variation in growth rates", while "additional factors that have not yet been elucidated clearly play a role". C38 The residual scatter in Figures 8 and 9, up to 2–3 orders of magnitude at similar viscosity before the entropy correction, is where those factors would have to act. C30·C32

Note by the AI writer (not discussed in the paper): what this suggests for formulation. First, the two cheapest numbers in Table 1, $T_m$ and $T_g$, already carry information: in this set a small $T_m-T_g$ went with a viscous melt and slow growth, so a large gap between $T_m$ and $T_g$ is a warning sign for a drug intended to be formulated amorphous. Second, a cooling-and-reheating test should be read together with a growth rate: drugs such as nimesulide, nilutamide and celecoxib, which form glasses easily but grow fast, need care in any process that can leave nuclei, such as milling, incomplete melting or slow cooling of an extrudate. Third, the PVP result is relative: cutting a flutamide-like drug tenfold still leaves it much faster than an untreated indomethacin-like drug, so the polymer level should be chosen from the neat-drug growth rate rather than set as a fixed percentage. Finally, the comparison was made at equal molar ratio; on a mass basis the light acetaminophen received relatively more polymer than indomethacin.

Limitations

Limits the authors state. Near $T_g$, diffusion decouples from viscosity and growth is "no longer proportional to the experimentally determined viscosity"; some crystals change to diffusionless growth (Discussion). Growth-rate trends "cannot readily be extrapolated from high to low temperatures". The link between $T_m-T_g$ and the MCV rests on the assumption that all drugs have the same viscosity at $T_g$ ("If it is assumed"), and fragility "also needs to be taken into account". Factors beyond viscosity and entropy "have not yet been elucidated". C21·C35·C27·C38

The AI writer's own judgement (1): the maximum growth velocities. The MCV is the paper's central ranking quantity, but the paper does not say how it was obtained, and Table 1 does not list $T_{\mathrm{c}\mathrm{r}\mathrm{y}\mathrm{s}\mathrm{t},\mathrm{m}\mathrm{a}\mathrm{x}}$. For several drugs the MCV is far above any plotted growth rate (nimesulide 216 against about 8.6 µm/s), and for felodipine and indomethacin it is below plotted rates (0.86 against about 1.2, and 0.47 against about 1.25 µm/s), which a maximum cannot be. The group 1 limit of 10–100 µm/s is broken by nimesulide and flutamide, celecoxib's $T_m-T_g$ conflicts with its $T_m$ and $T_g$, and the conclusions present the 0.94 $T_m$ location of the MCV as an observation although the Discussion cites it from ref 26. The spread of "about four orders of magnitude" is robust to these issues; the exact MCV of a single drug, and the group of drugs near the limits, are not. X1·X2·X3·X4·X5

The AI writer's own judgement (2): sample sizes and uncertainty. No replicate numbers, uncertainties or error bars are given for any growth rate. Melt viscosity, the property that separates the groups most strongly, is taken from the literature and known for only 17 of the 31 drugs, 5 of them in group 1; the entropy analysis rests on 9 drugs, with constants whose units are not given. The PVP test uses one polymer at one molar level in five drugs that all have hydrogen bond donors, with reductions that range from about 3- to 14-fold and one point near 50-fold rather than a single factor. X17·X9·X11·X6·X7

The AI writer's own judgement (3): the risk classification. The extended classification rests on nucleation behaviour inferred from the cooling-and-reheating test; no nucleation rate was measured, and the explanations that carry the risk argument (poor nucleation of fast class III growers, fragility of the Figure 7 outliers, similar hydrogen bonding with PVP) are proposed rather than tested. The surface–bulk comparison mostly compares different temperatures. Overall the paper is solid as a map of how widely growth rates vary and which properties go with them; it is weaker as a quantitative model or as a validated risk scheme. C15·C29·C9·X10·X16

Conclusion

This paper gives a single, comparable data set of crystal growth rates from the undercooled melts of 31 drugs. Maximum growth rates span about four orders of magnitude, from 216 µm/s for nimesulide to 0.02 µm/s for clotrimazole. Slow growers stand out by a much more viscous melt and a smaller gap between $T_m$ and $T_g$; after the driving force and viscosity are removed, the melting entropy accounts for part of what is left, with clear room for other factors. Drugs that pass a cooling-and-reheating test as good glass formers can be among the fastest growers, which leads the authors to an extended classification that combines nucleation tendency with growth rate. And 10 mol% PVP slowed five drugs by broadly similar factors, which the authors read as fast growers needing more polymer. The next steps the paper points to are direct measurements of nucleation, growth rates below $T_g$, and the additional molecular factors behind the remaining scatter.