How a few percent of PVP slow felodipine crystal growth in the melt: chain length, amount and temperature

Background and motivation

Many new drug candidates are hydrophobic, and their low solubility and slow dissolution can limit oral bioavailability (refs 1, 2). An amorphous form dissolves faster than the crystal (refs 3, 4), but amorphous solids tend to crystallize over pharmaceutically relevant times, which cancels the benefit. Dispersing the drug molecularly in a polymeric carrier is an established way to prevent or at least delay this crystallization (refs 5–7), and several explanations for how polymers stabilize amorphous solids had been put forward (refs 8–13) C1.

A striking observation from the years before this paper was that some polymers inhibit crystallization of amorphous drugs even at small concentrations, below 10 wt% (Konno and Taylor 2006, ref 14; Crowley and Zografi 2003, ref 15; Ishida et al. 2007, ref 16; Korhonen et al. 2008, ref 17). How a few percent of polymer can do this was unexplained. Four ideas were in circulation: polymer accumulating at the interface between the growing crystal and the amorphous matrix (ref 14), a local antiplasticization effect (ref 15), a general loss of molecular mobility (ref 17), and specific drug–polymer interactions such as hydrogen bonding (Matsumoto and Zografi 1999, ref 18) C2. The authors argue that knowing which of these is at work would help in designing better crystallization-inhibiting polymers.

They chose felodipine, an antihypertensive dihydropyridine, as the model drug and poly(vinyl pyrrolidone) (PVP) as the inhibitor. Felodipine crystals grow relatively slowly compared with amorphous nifedipine or indomethacin (refs 16, 19), so growth can be followed in a convenient window of 70–110 °C C3. PVP was used in three grades of very different chain length: K-12, K-29/32 and K-90, with weight-average molecular weights given as 2500, 45 000 and 1 250 000 g/mol in the introduction and as supplier ranges of 2–3 × 10³, 4–5 × 10⁴ and 1–1.5 × 10⁶ g/mol in the abstract and methods M2·M3·M4. The literature on chain length was mixed: in one-component polymers longer chains crystallize less readily (refs 32, 33), several studies found that higher-molecular-weight additives suppress crystallization of a small molecule more (refs 11, 34, 35), but Matsumoto and Zografi found the molecular weight of PVP to matter little for indomethacin (ref 18) C31.

The paper therefore asks three experimental questions, how chain length, polymer amount and temperature each change the growth rate of felodipine crystals from the melt, and one interpretive question: is the slowing caused by a change in the thermodynamic driving force for crystallization, or is it kinetic? The introduction announces the four answers: growth slows dramatically on cooling with and without PVP; longer chains inhibit more; more polymer inhibits more; and the inhibition is "most likely kinetic in origin" C28.

Background concepts

Crystal growth from a supercooled melt. When a drug melt is cooled below its melting temperature $T_m$ without crystallizing, it becomes a supercooled liquid (above the glass transition temperature $T_g$) and, further down, a glass. Once a crystal nucleus exists, it grows by adding molecules at its surface. The growth rate measured here is the linear rate at which the crystal diameter increases, in m/s. The supercooling $\Delta T=T_m-T$ measures how far below the melting point the melt is held. Two effects compete as the temperature falls: the driving force for crystallization grows with $\Delta T$, but the melt becomes more viscous and molecules reach the crystal surface more slowly. Growth rate therefore usually passes through a maximum: at small supercooling it is limited by the driving force, at large supercooling by transport. Felodipine melts at 141.5 °C and has a $T_g$ of 43.3 °C (Table 1) M1.

Spherulites and polymorphs. Felodipine crystallizing in a thin film between two cover glasses forms spherulites: round, polycrystalline aggregates that grow outward from a centre, here as two-dimensional disks. Felodipine has at least two crystal forms. Form I is the stable form that was supplied; form II is metastable and melts at a lower temperature. Raman spectroscopy tells them apart: form II has bands at 1648 cm⁻¹ (C=C stretch) and 1503 cm⁻¹ (aliphatic bending), form I at 1643 and 1483 cm⁻¹ (Figure S2) R2.

Glass transition and antiplasticization. $T_g$ marks where an amorphous solid turns from a glass into a supercooled liquid; molecular mobility rises steeply above it. A polymer with a higher $T_g$ than the drug usually raises the $T_g$ of a miscible blend; this "antiplasticization" lowers mobility at a given temperature and was one proposed reason why polymers slow crystallization (refs 9, 36) C33.

Melting point depression as a test of the driving force. A miscible additive lowers the chemical potential of the drug in the liquid. That makes crystallization less favourable at any temperature, and the most direct sign of it is a lower melting temperature of the drug crystals in the presence of the additive (Marsac et al. 2010, ref 30) C25.

Hydrogen bonding seen by infrared. Felodipine has an NH group that donates a hydrogen bond. In pure amorphous felodipine it bonds to the carbonyl of a neighbouring felodipine molecule and absorbs at 3341 cm⁻¹. A shift to lower wavenumber and a broader band mean stronger or more numerous hydrogen bonds (Tang et al. 2002, ref 22) C12·R9. PVP, with its pyrrolidone carbonyl, can accept such bonds, and earlier work showed that PVP and felodipine are miscible and hydrogen-bonded over the whole composition range up to the felodipine melting point, more so at higher PVP content, and less strongly at higher temperature (refs 14, 37) C36.

Zero-shear viscosity. The viscosity of a melt at vanishing shear rate is its zero-shear viscosity. A Newtonian liquid has the same viscosity at every shear rate; a polymer solution often becomes less viscous when sheared faster (shear thinning). At a fixed mass fraction, longer polymer chains raise viscosity more, because each coil pervades a larger volume; the authors add that favourable interaction with the drug swells the coils beyond their ideal (Gaussian) size, which makes this effect "expected to be strong" (ref 24) C16.

Polymer at a growing crystal front. If a crystal grows into a melt that contains a dissolved polymer and does not take the polymer into its lattice, the polymer is pushed ahead of the front and must diffuse away. Two diffusion coefficients appear: the tracer diffusivity $D_t$, how fast a single polymer coil moves through the drug, and the mutual diffusivity $D_m$, how fast the mixture evens out a concentration gradient. They coincide at infinite dilution. The fraction of polymer that the crystal pushes out rather than engulfs is called ε in this paper. In blends of crystalline and noncrystalline polymers, a large fraction of the amorphous polymer is expelled from the crystals (Wellscheid et al. 1996, ref 42) C39.

Methods in detail

The study has one central measurement, the growth rate of felodipine crystals in films, and a set of side measurements, each aimed at one candidate explanation. Two models then interpret the data: a classical crystal-growth equation and a diffusion model for the polymer at the growing front.

Samples and growth-rate measurement

PVP was dried under vacuum for 2 days. A 10% PVP dispersion of each grade in felodipine was made by rotary evaporation from a 1:1 (by weight) ethanol:dichloromethane solution and dried in a vacuum oven for 48 h. Lower PVP levels, 0.5, 0.75, 1.5, 3 and 4.5% w/w, were made by blending this dispersion with crystalline felodipine in a cryogenic mill, because rotary evaporation below 5 wt% polymer partly crystallized the felodipine and could give nonhomogeneous mixtures P1. For each film, 2–4 mg of mixture was melted on a cover glass, covered with a second glass and quenched to −5 °C (50 °C below the felodipine $T_g$), keeping the relative humidity below 15%. The films were checked to be amorphous under crossed polarizers and stored over P₂O₅ (0% RH) at 40 °C, where crystal nuclei formed P2. The storage time at 40 °C and the melting temperature used to make the films are not stated.

The nuclei were then grown at 70, 80, 90, 95, 100, 105 and 110 °C on a hot stage calibrated with the melting points of naphthalene and adipic acid, and photographed at regular intervals. Under every condition the crystals between the slides were 5.5 ± 2 µm thick, so the authors compared growth through the increase of crystal diameter; the diameter rose linearly with time in every run, over a measurement time scale of 0–60 h, and the slope of a linear fit was taken as the growth rate P3·R19·C5. In pure felodipine some form I crystals nucleated near the cover-glass edges; these were excluded. The number of crystals or films per growth rate is not given; the PVP data in Figures 4, 7 and 8 carry error bars of unstated meaning.

Side measurements

Thermal analysis. $T_g$ was taken as the onset on a second DSC heating at 10 °C/min, after heating 20 °C above $T_g$ and cooling 50 °C below it P5. Melting point depression was tested only for pure felodipine and felodipine cryomilled with 3% PVP K-12, heating at 1 °C/min and taking the extrapolated offset of the melting endotherm; with K-29/32 and K-90 this was impossible because their $T_g$ (163.9 and 174.9 °C) lies above the drug's melting point P6·M3·M4.

Infrared spectroscopy. Films of felodipine with 20 wt% PVP of each grade were spin-coated from 1:1 ethanol:dichloromethane onto ZnS windows, dried at 90 °C for several minutes and measured at 25 °C in a nitrogen-purged spectrometer (128 scans). The 20 wt% level was chosen for sensitivity to drug–polymer interactions P7.

Rheometry. Steady-shear viscosity was measured between 25-mm parallel plates (1-mm gap). Each powder sample was melted 5 °C above the felodipine melting point, presheared at 10 s⁻¹ for 5 min, cooled to the test temperature and sheared at 1 s⁻¹ until the reading was steady. Measurements at 110, 105, 100, 95 and 90 °C each took less than 20 min, against roughly 4 h for nucleation at these temperatures (ref 21), so crystallites were absent, as optical inspection showed; below 90 °C the melts crystallized during the run and those data were dropped P4·R10·R24·C14. A shear-rate sweep at 100 °C (Figure S1) checked whether the 3 wt% melts were Newtonian.

Raman and X-ray diffraction. Raman spectra (783-nm laser, 50× objective) identified the polymorph of crystals grown under every combination of temperature, grade and concentration P8. X-ray diffraction (Cu Kα, 5–35° 2θ) was run on films of pure felodipine and of 0.5 and 3 wt% PVP K-29/32 crystallized completely on cover glasses P9. The crystallization temperature of these films is given as 110 °C in the text and 105 °C in the caption of Figure S4 X6.

The crystal-growth equation

To interpret the temperature dependence, the authors start from the textbook expression for the rate of growth of a crystal face (Hiemenz and Lodge, ref 27) C22:

In plain terms: the growth rate $v$ is a mobility term, the inverse of the melt viscosity $\eta$, times the probability $\exp (-\Delta G^{*}/k_{B}T)$ that a new layer of crystal can nucleate on the surface; $k$ is a constant, $\Delta G^{*}$ the free energy barrier for a new monolayer, $k_B$ the Boltzmann constant and $T$ the temperature (Eq. 2) E2. For small molecules the barrier is approximately

In plain terms: the barrier grows with the square of the crystal–melt interfacial free energy $\gamma$ and with the molecular size $b$, and shrinks as the bulk free energy of crystallization per unit volume, $\Delta G_v$ (a negative quantity), becomes more negative (Eq. 3) E3. Since $\Delta G_v=0$ at $T_m$, the entropy of crystallization is $\Delta S_v=\Delta H_v/T_m$, and for a supercooling $\Delta T$ the paper writes

In plain terms: the driving force for crystallization grows in proportion to how far below the melting point the melt is (Eq. 4) E4. Combining the three gives Eq. 5 E5:

In plain terms: if viscosity is the right measure of mobility, then the product $v\eta$ depends on temperature only through the nucleation term, so a plot of $\ln (v\eta )$ against $1/(T\Delta T)$ should be a straight line whose slope is proportional to ${\gamma }^2/\Delta H_v$ C24. Note from the card (extractor note X1): as printed, Eq. 4 and Eq. 5 carry opposite signs; Eq. 5 follows from Eqs. 2 and 3 if $\Delta H_v$ is the negative enthalpy of crystallization, in which case Eq. 4 should read $\Delta G_v=\Delta H_v\Delta T/T_m$. The slope argument is not affected X1.

The empirical log-linear equation

From the concentration data (Figures 8 and 9, discussed below) the authors extract an empirical equation for the growth rate GR at temperature $T$ (in °C) and PVP concentration $x$ (wt%) E1:

In plain terms: at each temperature, every added percent of PVP multiplies the growth rate by the same factor; that factor, set by the slope $aT+b$, moves linearly with temperature. ${\mathrm{G}\mathrm{R}}_0$ is the growth rate without polymer, and for PVP K-29/32 the constants are $a=0.0096$ and $b=-1.494$ E1·R15.

The diffusion model for PVP at the growing front

The crystals grew at a constant rate, which the authors found surprising if PVP were pushed out by the growing crystal and accumulated at its surface C38·C39. To test this they solved the diffusion of PVP around a disk-shaped crystal. The starting point is Fick's second law, printed as (Eq. 6) E6

In plain terms: the local PVP concentration $c$ at radial distance $r$ and time $t$ changes as polymer diffuses down its concentration gradient, with mutual diffusivity $D_m$. The operator for Fick's second law is the Laplacian ${\nabla }^2$; the printed $\nabla$ is taken here as a misprint X2. With $c(r,0)=c_{\infty }$ and $c(\infty ,t)=c_{\infty }$, Frank's similarity solution in cylindrical coordinates (Frank 1950, ref 43) is (Eqs. 7 and 8) E7·E8

In plain terms: ahead of the crystal the polymer concentration decays from a peak at the crystal surface back to the bulk value $c_{\infty }$, with a shape given by the exponential integral $E_1$; $A(t)$ sets the height of the peak. The amount of polymer in the peak must equal what the crystal has pushed out (Eq. 9) E9:

In plain terms: the excess polymer outside the crystal of radius $R(t)$ equals the fraction ε of the polymer that was in the volume now occupied by the crystal; volume change on crystallization is neglected. This fixes $A(t)$ (Eq. 10) E10:

In plain terms: the more polymer is expelled and the larger the crystal, the higher the peak; faster diffusion spreads the excess over a wider zone and lowers it. The front moves as $R(t)=vt$ with the measured growth rate $v$. The authors set $D_m$ equal to the tracer diffusivity at infinite dilution, $D_t$, which is exact only as $c\to 0$, and estimate $D_t$ from the Stokes–Einstein equation (Eq. 11) E11·C40:

In plain terms: a coil of hydrodynamic radius $R_h$ diffuses more slowly in a more viscous solvent; ${\eta }_s$ is the viscosity of pure felodipine. The radius comes from the Kirkwood–Riseman relation for a Gaussian chain, $R_h=0.665R_g$ (ref 44), and the radius of gyration from the characteristic ratio, printed as $R_g=(C_{n}nl/6{)}^{1/2}$, with $n$ the number of backbone C–C bonds, $l=1.5$ Å the C–C bond length, $C_n=5$ for K-12 and 10 for K-29/32 and K-90, and weight-average degrees of polymerization of 22, 360 and 11260 (ref 45) E12·E13·M2·M3·M4. For 3 wt% K-29/32 at 100 °C ($c_{\infty }=0.03$) the inputs were $D_t=4.98\times {10}^{-16}$ m²/s and $v=2.37\times {10}^{-8}$ m/s, with ε = 1, 0.1 and 0.01 R23·P10.

Note by the AI writer (from the card's extractor notes, not discussed in the paper): three details of this model do not match as printed. The argument of $E_1$ appears with a minus sign in Eqs. 7 and 10, but the published curves (Figures 11 and 12) are reproduced only with the positive argument $r^2/(4D_{m}t)$ X2. The Stokes–Einstein value cannot be reproduced with the stated solvent viscosity: pure felodipine at 100 °C (about 44 Pa·s in Figure 5) gives about 1.8 × 10⁻¹⁵ m²/s, 3.6 times the printed value, which is matched only with the viscosity of the 3 wt% K-29/32 mixture (about 156 Pa·s); the $R_g$ formula is also printed without the square on $l$ X4. And the growth rate is defined from the crystal diameter, while the model uses it as the speed of the radius X5. The figure captions also give the unit of $D_t$ as m/s² X3.

Experimental analysis

All growth is form II, and PVP does not change the crystal shape

Raman spectra showed that crystals grown in the bulk of the films were the metastable form II under every combination of temperature, grade and concentration; only in pure felodipine did some form I crystals appear near the cover-glass edges, and these were excluded. On the hot stage form II melted below form I, consistent with the Raman assignment, although the melting temperature of form II is not given R2·R3·C7. All growth rates in the paper are therefore form II growth rates. Pure felodipine grew as spherulites at all seven temperatures, with faceted, laterally straight crystal–melt interfaces below 100 °C and more zigzag interfaces above (Figure 1) R4. With 3 wt% of any grade at 90 °C (Figure 2), and with 0–4.5 wt% K-29/32 at 80 °C (Figure 6), the crystals remained two-dimensional spherulites with a smooth, distinct interface; the authors saw no considerable change in macroscopic shape at other temperatures either R5·R12·C8.

Temperature: a growth maximum near 105 °C

Figure 3 shows the raw data for pure felodipine: crystal diameter rose linearly with time at 90, 95, 100 and 105 °C, for example from about 119 to about 407 µm between about 600 and 4800 s at 90 °C R6. The slopes, together with three more temperatures, give the growth-rate curve of Figure 4 (Table A). Pure felodipine grows fastest near 105 °C, which the authors place at "around 105 ± 5 °C", and slows on cooling; the text describes the drop from 110 to 70 °C as "approximately 2 orders of magnitude" C9. Read from Figure 4, the change is from about 1.5 × 10⁻⁷ to about 3.5 × 10⁻⁹ m/s, a factor of about 43, or about 1.6 orders of magnitude R7·X9.

Table A. Linear growth rate of felodipine form II crystals (m/s), read from Figure 4 (approximate, log axis), and the fold reductions stated in the text (pp.3588–3589) R7·R8.

Molecular weight: more inhibition with longer chains, at a shrinking rate

Three weight percent of any grade lowered the growth rate at every temperature, and the shape of the curve changed: with PVP the rate still rose from 105 to 110 °C, where pure felodipine had already passed its maximum (Table A) R7. The inhibition was much weaker at high temperature. At 110 °C, K-12 and K-29/32 each cut the rate by a factor of 3 and K-90 by 6; at 70 °C the factors were 11, 15 and 21 R8. The authors put it as the inhibiting effect "appears to be smaller at high temperatures" and, in the abstract, as "much greater at lower temperatures" C10. At a given concentration the longest chains inhibited most C11.

The gain from longer chains is small next to the effect of adding the polymer at all, which the authors stress: it is "not nearly as large as the impact of simply adding the polymer" C29. At 70 °C and 3 wt%, K-29/32 was about 1.55 times as effective as K-12, while K-90 was only 1.4 times as effective as K-29/32, although the jump in chain length between those two grades is larger R8·M2·M3·M4. From these two ratios the authors "suspect" that the molecular-weight dependence of crystallization inhibition "asymptotically disappears in the large molecular weight limit", which would reconcile the earlier studies that did and did not see a molecular-weight effect; their own wording is that the results "appear to hint at this possibility" C32·C31. At 110 °C the pattern is the reverse: K-12 and K-29/32 gave the same 3-fold reduction and the step to K-90 doubled it X13.

Concentration: a log-linear law with a temperature-dependent slope

With PVP K-29/32 the growth rate fell with every step in concentration, 0, 0.5, 0.75, 1.5, 3.0 and 4.5 wt%, at every temperature (Figure 7), and the concentration effect grew monotonically on cooling R13·C18. At 110 °C, 1.5 and 4.5 wt% slowed growth by factors of 2 and 6; at 70 °C by 3 and 35 R13. Table B gives the curves at the two end temperatures.

Table B. Growth rate (m/s) at three temperatures versus PVP K-29/32 concentration, read from Figure 7 (approximate). The text gives 2- and 6-fold reductions at 110 °C and 3- and 35-fold reductions at 70 °C for 1.5 and 4.5 wt% R13.

Plotted against concentration on a log scale (Figure 8), each temperature series from 70 to 105 °C falls on a straight line, and the slopes of these lines, plotted against temperature (Figure 9), lie on a straight line too: about −0.82 per wt% at 70 °C, −0.73 at 80, −0.64 at 90, −0.58 at 95, −0.54 at 100 and −0.49 at 105 °C R14·R15. The fitted line, $aT+b$ with $a=0.0096$ and $b=-1.494$, reproduces these values (−0.822 at 70 °C, −0.486 at 105 °C) and turns the data into Eq. 1 E1. The authors call both correlations "rather strong" C19. Two details limit the reach of this law: the 110 °C series, where pure felodipine is already past its growth maximum, is not included in Figures 8 and 9, and no regression statistics are printed; at 70 and 80 °C several points lie visibly off the lines X14.

Note by the AI writer (not discussed in the paper): in Eq. 1 the factor per added wt% is $\exp (aT+b)$, about 0.44 at 70 °C and about 0.61 at 105 °C. Applied to 4.5 wt%, Eq. 1 predicts a 40-fold reduction at 70 °C where 35-fold was measured, and, extrapolated to 110 °C, 7.2-fold where 6-fold was measured X14.

The side tests: nothing obvious changes

Glass transition. Table 1 gives an onset $T_g$ of 43.3 °C for felodipine and 43.6, 43.4, 43.5 and 43.6 °C with 0.5, 1.5, 3.0 and 4.5 wt% K-29/32, 44.1 °C with 3 wt% K-12 and 44.0 °C with 3 wt% K-90, with standard deviations within ±1 °C, although the polymers themselves have $T_g$ values of 102.9, 163.9 and 174.9 °C R1·M1·M2·M3·M4. There was "no detectable difference", as earlier reported for other polymer/drug dispersions (refs 15, 16) C6.

Melting point. With 3 wt% K-12, felodipine melted at 140.7 °C against 141.5 °C without polymer, which the authors read as an unaffected melting point, consistent with their earlier report (ref 30) R1·C26.

Hydrogen bonding. In amorphous films with 20 wt% PVP, the felodipine NH band moved from 3341 to 3337 cm⁻¹ and broadened (Figure S3), which the authors read as "stronger and/or more numerous interactions" between drug and polymer than between drug molecules. The shift and broadening were the same for all three grades, so the felodipine–PVP interaction does not depend on molecular weight in this range R9·C13.

Crystal lattice. X-ray patterns of crystals from pure felodipine and from 0.5 and 3 wt% K-29/32 showed "no discernible difference", which the authors take to suggest that PVP does not co-crystallize and the crystal domain is pure felodipine R18·C17. Each pattern in Figure S4 shows three reflections, near 7.6, 15.0 and 22.5° 2θ, with lower peaks in the PVP samples R18.

Viscosity rises steeply with chain length but does not track inhibition

The zero-shear viscosity of the supercooled melts (Figure 5) depends strongly on the PVP grade (Table C). Relative to pure felodipine, 3 wt% K-12 changed it only slightly, K-29/32 raised it by a factor of 3–4 and K-90 by 10–20 R10·C15. At 1 s⁻¹ the 3% K-12 melt was Newtonian, while the K-29/32 and K-90 melts depended slightly on shear rate; for K-90 the authors give a scaling exponent of 0.28. In the 100 °C sweep of Figure S1, the 3% K-90 melt has a plateau of about 760 Pa·s at low shear and falls to about 450 Pa·s at 1 s⁻¹ and about 180 Pa·s at 10 s⁻¹ R10·R11.

Table C. Zero-shear (or near zero-shear) viscosity in Pa·s, read from Figure 5 (approximate, log axis) R10.

Two observations led the authors to doubt that bulk viscosity measures the mobility of the felodipine molecules. First, the molecular-weight trends disagree: K-90 raised viscosity 10–20-fold over pure felodipine, yet its inhibition was only modestly larger than that of K-12, whose viscosity effect was small; so "there is no simple correlation" between the two C29. Second, when the growth rates of pure felodipine and of 3% K-12 were multiplied by viscosity and plotted as $\ln (v\eta )$ against $1/(T\Delta T)$ (Figure 10, 90–110 °C; K-29/32 and K-90 left out as non-Newtonian), the K-12 line was less steep. Under Eq. 5 a less steep line would mean a larger driving force with PVP, and so a faster viscosity-corrected growth; instead the K-12 values lie below those of pure felodipine at every temperature, from about 5.2 × 10⁻⁶ against 2.0 × 10⁻⁵ m·Pa at 90 °C to about 5.9 × 10⁻⁷ against 1.1 × 10⁻⁶ m·Pa at 110 °C R16·C24. The authors conclude that viscosity "cannot be used as a precise measure" of felodipine mobility C24. The points in Figure 10 are not labelled by temperature; the temperatures given here were assigned from the abscissa X18.

The diffusion model: expelled PVP would pile up within seconds

Figure 11 shows the predicted PVP profiles for 3 wt% K-29/32 at 100 °C if all polymer is expelled (ε = 1). At 5, 10, 20, 50, 100, 150 and 200 s the front sits at about 0.12, 0.24, 0.47, 1.19, 2.37, 3.56 and 4.74 µm, and the peak PVP weight fraction there rises from about 0.08 to about 0.13, 0.22, 0.48, 0.90, 1.33 and 1.75; within about 0.2–0.3 µm ahead of the front the concentration is back at 0.03 R20. Figure 12 follows the surface value: for ε = 1 it reaches 0.5 within about 52 s; for ε = 0.1 it climbs to about 0.46 after 500 s; even for ε = 0.01, when 99% of the polymer is engulfed, it rises from 0.03 to about 0.07 over 500 s and "remains to be considerably time-dependent" R21. Raising $D_t$ by two orders of magnitude did not change the qualitative result R22·C41.

From the constant growth rate the authors therefore conclude that "it seems reasonable" that all the polymer chains are "encaged in the crystalline domain", which keeps the composition at the front at the bulk value; with the unchanged diffraction pattern, they place PVP "in separate domains within the crystalline phase", with the felodipine crystal structure unaltered C42·C43.

Note by the AI writer (not discussed in the paper): the model peaks above a weight fraction of 1 at 150 and 200 s cannot be real concentrations; they reflect a model with no upper bound and with $D_m=D_t$, an approximation the authors themselves say fails when polymer accumulates strongly X16·C40. The qualitative point, that expelled polymer would accumulate quickly at this growth rate and diffusivity, survives the corrections listed in the methods section; for example, halving the front speed lowers the dimensionless group $v^{2}t/(4D_t)$ four-fold but still leaves a large accumulation X5.

Discussion

The authors' argument: kinetic, not thermodynamic

The discussion works through the candidate explanations one by one and keeps the one that is not ruled out. The chain, with the authors' hedges, runs as follows.

The introduction states the conclusion as "most likely kinetic in origin", the discussion as "unlikely" to be thermodynamic and "perhaps of purely kinetic origin", and the conclusions section as "believed to be kinetic rather than thermodynamic", adding that the model analysis "favors the idea" of reduced mobility through drug–polymer specific interactions C28·C35. The strength of the wording thus varies across the paper, but each version is an inference, not a measurement of mobility.

Where the argument is thin

Note by the AI writer (based on the card's extractor notes): the argument proceeds by elimination, and the eliminations are uneven. The thermodynamic route is ruled out with a single melting point, for K-12 only, whose 0.8 °C drop lies within the stated ±1 °C standard deviation; $\gamma$ was never measured, and the authors' own reading of Figure 10, a less steep line implying a larger driving force with PVP, points the other way before being set aside X11. The hydrogen-bond mechanism rests on infrared spectra recorded at 25 °C with 20 wt% PVP, whereas growth was measured at 70–110 °C with 0.5–4.5 wt%. And the hydrogen-bond signal is identical for the three grades, while the inhibition differs between them, so hydrogen bonding alone cannot account for the molecular-weight effect, as the authors themselves note X12·C30. Felodipine mobility itself was not measured. The PXRD support for an unchanged lattice comes from three reflections that look like one family of planes from films oriented between glass slides X10. The paper's experimental core, the growth-rate data and the log-linear law, is much firmer than its mechanistic interpretation.

Molecular weight

The molecular-weight result places this paper between the conflicting earlier reports: chain length matters, but much less than the presence of the polymer. The authors' suggestion, that the molecular-weight effect fades for long chains, fits the 70 °C ratios (1.55 then 1.4) but is tested with only three grades and is offered as a hint C32. At 110 °C the largest step was from K-29/32 to K-90, not from K-12 to K-29/32 X13.

What it suggests for formulation

The authors' own practical suggestion is that, with Eq. 1, the growth rate at a given polymer level "might be reasonably estimated" for conditions closer to real processes, such as spray drying, from measurements made quickly at high temperature where growth is fast. They add two limits: kinetics change drastically near or below $T_g$, so data from $T\gg T_g$ should not be carried there carelessly, and the extrapolation cannot go above the temperature of maximum growth (refs 25, 26) C20·C21. They also flag that how the polymer is distributed inside the crystalline drug may determine how fast the drug later dissolves, a question left for future work C44.

Note by the AI writer (not discussed in the paper): for formulation, three points follow from the data. First, a few weight percent of PVP slowed growth 11–35-fold at 70 °C but only 3–6-fold at 110 °C, so the protection is weakest close to the growth maximum, the regime of hot processing, and strongest in the cooler, more viscous regime. Second, the middle grade K-29/32 delivered most of the molecular-weight benefit, while K-90 added little inhibition but a large rise in melt viscosity. Third, $T_g$ and the hydrogen-bond signal did not distinguish the grades at all; only the growth-rate measurement ranked the polymers, so a $T_g$-based screen would miss this kind of inhibition.

Limitations

Limits the authors state. Extrapolating growth rates from high temperature to temperatures near or below $T_g$ needs care, and extrapolation above the temperature of maximum growth is not possible C21. The diffusion model uses the tracer diffusivity at infinite dilution for the mutual diffusivity, which becomes invalid if the polymer accumulates strongly; the authors state that the conclusion is not affected and support this with a 100-fold change in $D_t$ C40·R22. The 3% K-29/32 and K-90 melts were non-Newtonian, so only K-12 entered the viscosity-scaled analysis (Figure 10) R16. How PVP is arranged inside the crystals is not known C44.

The AI writer's own judgement (from the card's extractor notes and its assessment of how solid the paper is):

• The growth-rate data are direct, consistent across Figures 4, 7 and 8 and with the fold changes in the text; this part of the paper is strong. The number of crystals or replicate films behind each rate and the meaning of the error bars are not given, and diameter–time data are shown only for pure felodipine, so the statement that growth was linear in time with PVP over 0–60 h is reported, not shown X17·X19.
• The log-linear law is fitted for 70–105 °C and PVP K-29/32 only, without regression statistics; it should not be applied outside that window or to other grades without new data X14.
• The kinetic interpretation rests on one melting point within its standard deviation and on an unmeasured interfacial energy; the hydrogen-bond mechanism rests on room-temperature IR that does not vary with molecular weight X11·X12.
• The supercooling in Figure 10 and in the text (71.4–31.4 °C) is computed from the melting point of form I (Table 1: 141.5 °C), whereas the crystals that grew were form II, which melts lower; the true supercooling relative to form II is smaller X15.
• The printed model contains inconsistencies: the sign of the exponential-integral argument, the gradient in place of the Laplacian, a tracer diffusivity that matches the mixture rather than the stated solvent viscosity, a diameter-based growth rate used as a radius speed, and units of m/s² in the captions; predicted fractions above 1 show that the curves are qualitative X2·X4·X5·X3·X16.
• Viscosities at 100 °C differ by about 30% between Figure 5 and Figure S1, and the shear exponent is printed as positive although the viscosity falls with shear rate X8·X7. The equations for the growth model carry a sign inconsistency between Eq. 4 and Eq. 5 X1. The PXRD crystallization temperature is 110 °C in the text and 105 °C in the SI X6.
• One drug, one polymer chemistry, a thin-film geometry and dry conditions: whether the log-linear law and the weak molecular-weight effect hold for other drug–polymer pairs, for nucleation rather than growth, or in the presence of water, is not addressed.

Conclusion

This paper measured, with simple hot-stage microscopy, how three variables control how fast felodipine crystals grow from the supercooled melt. A few weight percent of PVP slowed growth at every temperature from 70 to 110 °C, by factors of 3–6 at 110 °C and 11–21 at 70 °C for 3 wt%, and by 35-fold at 70 °C for 4.5 wt% K-29/32. The logarithm of the growth rate fell linearly with PVP concentration, with a slope that rose linearly with temperature, which the authors condensed into the empirical Eq. 1. Longer chains inhibited more, but with a gain far smaller than that of adding the polymer at all. $T_g$, the one measured $T_m$ and the hydrogen-bond signal did not change in ways that could explain this, and viscosity did not track inhibition. The authors conclude that PVP slows felodipine growth kinetically, "speculate" that drug–polymer hydrogen bonds reduce the mobility of the drug, and infer from a diffusion model that the polymer is engulfed by the growing spherulites rather than piling up at their surface.