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Sep 26, 2013 - These crystallization experiments demonstrate that the fundamental materials property, the velocity of the phase boundary of the crysta...
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Experimental Determination of the Crystallization Phase-Boundary Velocity in the Halozeotype CZX‑1 Eric D. Dill, Amanda A. Josey, Jacob C.W. Folmer, Feier Hou, and James D. Martin* North Carolina State University, Department of Chemistry, Raleigh, North Carolina 27695-8204, United States S Supporting Information *

ABSTRACT: Isothermal crystallization experiments were performed on the halozeotype CZX-1 with 2D temperatureand time-resolved synchrotron X-ray diffraction (TtXRD) and differential scanning calorimetry (DSC). These crystallization experiments demonstrate that the fundamental materials property, the velocity of the phase boundary of the crystallization front, v pb , can be recovered from the Kolmogorov Johnson and Mehl and Avrami (KJMA) model of phase-boundary controlled reactions by introducing the sample volume into the KJMA rate expression. An additional corrective term is required if the sample volume of the crystallization measurement is anisotropic. The concurrent disappearance of the melt and appearance of the crystalline phase demonstrate that no intermediates exist in the crystallization pathway. The velocity of the phase boundary approaches 0 as the glass transition (Tg ≈ 30 °C) is approached and at about 10° below melting point (Tm = 173 °C). The velocity of the phase boundary reaches a maximum of 30 μm s−1 at 135 °C. Single or near-single crystals are grown under conditions where the vpb is much greater than the rate of nucleation. KEYWORDS: crystallization kinetics, Avrami, DSC kinetics, XRD kinetics, phase-boundary velocity, time-resolved diffraction, synchrotron

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INTRODUCTION A fundamental understanding of the mechanism(s) by which homogeneous liquids and amorphous systems crystallize is critical to diverse areas of science and technology ranging from environmental science to advanced information technologies. Despite this importance, an understanding of crystallization mechanisms is limited. Seminal works from 1876 to 1953 remain the basis for much current understanding of nucleation and crystal growth.1−4 In 1876, Gibbs determined the energy required to generate a nucleus and derived the equilibrium form of a crystal based on minimization of the surface energy.5,6 This work was applied to describe the nucleation phenomenon in the vapor-to-liquid and vapor-to-crystal transformations in the works of Volmer and Weber,7 Farkas,8 and Becker and Döring;9 the collection of which is frequently referred to as Classical Nucleation Theory (CNT). Turnbull, Fisher, and Holloman extended CNT into condensed systems,10−12 which, along with the Kolmogorov13−Johnson and Mehl14−Avrami15−17 (KJMA) model, forms the basis for the modern understanding of crystallization kinetics in condensed systems. The KJMA crystalline-growth model is used to extract kinetic parameters for isothermal crystallization processes from supercooled melts and superheated glasses. It describes crystallization as a phase-boundary controlled process and recognizes that the dimensionality of the growth process impacts the observed rate of bulk crystallization. The fraction of material crystallized, α, per time is directly proportional to the progress of the phase boundary for a 1D growth process and © 2013 American Chemical Society

increases with the square or cube of the progress of the phase boundary for 2D and 3D growth, respectively. The rates thus appear to accelerate until growth impinges on domain boundaries and termination ensues, resulting in a sigmoidal shape to the observed rate curves. The KJMA model is commonly expressed as α(t ) = 1 − exp{−[k(t − t0)]n }

(1)

with rate constant k, nucleation time t0, and dimensionality n. These kinetic parameters, particularly the dimensionality, are often used to infer mechanistic detail13−18 in spite of the growing body of literature demonstrating that they are empirical,3,19−22 significantly affected by experimental factors,23−26 and do not necessarily reflect the true reaction mechanism.27−30 Given the broad use of the KJMA model, much effort has been expended to verify the models’ assumptions31−35 and to demonstrate situations in which the model does not hold.36,37 Additionally, numerous extensions to the KJMA model have been investigated.2,38−40 When identified, attempts to correct experimentally dependent kinetic parameters often result in nonphysical values such as rate constants extrapolated to zero sample thickness25 or zero mass.41 To obtain mechanistic information from kinetic parameters, there is a need to remove the empiricism and establish physical Received: August 14, 2013 Published: September 26, 2013 3932

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155 °C, but the crystallization heat flow was distinguishable from the instrumental quench response until the target isotherm was below 145 °C. The time scale of the raw heat-flow data was adjusted such that the quench initiation for each experiment occurred at t = 0 min. Isothermal crystallization experiments were repeated 3−5 times for each sample, at each isotherm, for a total of 49 isothermal DSC experiments. The instrumental quench response (IQR), which must be subtracted from the raw signal to obtain the crystallization heat flow, is primarily dependent upon the sample mass, but instrumental response as a function of Tiso is also observed. For each sample, the IQR of an isotherm for which crystallization was well separated in time from the quench (e.g., Tiso ≥ 160 °C) was fit with a combination of two Gaussian and one Lorentzian functions. These Gaussian and Lorentzian parameters were then used as the starting parameters to fit the data using SOLVER46 to obtain the IQR for subsequent reactions of the same sample at different isotherms. Isothermal Crystallization: Temperature- and Time-Resolved X-ray Diffraction (TtXRD). Synchrotron diffraction data were obtained on beamlines 11-ID-B (90 KeV, λ = 0.13702 Å, collimated beam 1.0 × 1.0 mm) at the Advanced Photon Source (APS), Argonne National Laboratory, and X6b (19.1 KeV, λ = 0.646 Å, collimated beam 0.3 × 0.3 mm) at the National Synchrotron Light Source (NSLS), Brookhaven National Laboratory. Data were collected in a Debye−Scherer geometry at sampling rates of 0.05 to 8 Hz, with each diffractogram hereafter referred to as a frame, using 2048 × 2048 GE Silicon (APS)47 or 2084 × 2084 SMART CCD detectors (NSLS). Samples were sealed into fused silica capillaries (Charles Supper Co., Natick, MA) and affixed to a single-axis goniometer head with epoxy. During crystallization experiments, the samples were oscillated 10° in synchronization with the duration of X-ray exposure to illuminate a larger region of reciprocal space and minimize thermal gradients in the sample. The wavelength and detector alignment were calibrated to LaB6 or CeO2 standards using fit2d48 to correct all experimental data. A melted and recrystallized ingot of CZX-1 in a fused silica capillary was centered in the synchrotron beam. The sample was heated to 230 °C, noting the temperature at which the sample melted, which when compared to the CZX-1 melting point (Tm) of 173 °C provided an internal temperature calibrant. To ensure melt isotropy, the sample was held at 230 °C, well above Tm, for 5 min. Data collection was initiated for at least 1−3 frames prior to quenching to a Tiso of between 40 and 160 °C. Diffraction data was recorded at the isotherm until crystallization was complete. When a sample did not nucleate within 2 h, the reaction was aborted because of limited synchrotron time, which was the case for Tiso >155 °C. The melt-quench-crystallization cycle was then repeated. Data from a total of 55 crystallization experiments are reported here. Additional data were collected for samples whose diffraction patterns were found to exhibit an excess of ZnCl2. These are excluded from the kinetic analysis to remove as many extraneous effects as possible. No sample degradation was observed after multiple (>15) cycles, indicating that CZX-1 is stable with respect to decomposition under both rapid temperature shifts and multiple hours of high-energy X-ray exposure. Temperature control of the crystallization reaction was afforded by a pneumatically switchable manifold that directs airflow through one of two 3/4′′ in-line air heaters (Omega), schematically depicted in Supporting Information Figure SI-1. One furnace was set to the temperature of the high-temperature melt isotherm (230 °C) and the other to the temperature of the desired Tiso (40−155 °C). The temperature of each heating element was controlled using a Eurotherm 91p temperature controller to a precision of ±0.5 °C. Using the phase transitions of elemental sulfur and the melting temperature of CZX-1 as calibrants, a temperature accuracy of ±2.5 °C was obtained; the relatively large range is a result of sample environment conditions including temperature gradients, variations in air flow, and sample/ thermocouple placement within the air stream. The temperature of the air stream stabilized to the quenched isotherm at the controlling thermocouple T3 (Figure SI-1) within 15 s after the air streams were switched, affording rapid quenching to Tiso, irrespective of the melt and quench temperatures.

significance. It is the goal of this work, and that of a corresponding simulation manuscript,42 to decipher the physical significance of kinetic parameters. Starting with eq 1, we find that with sample specific corrections the transformation kinetics of our experimental system can be described exclusively with physically significant parameters. In this article, investigations into the mechanism of meltcrystal growth of the halozeotype CZX-1, [HNMe 3 ][CuZn5Cl12], by differential scanning calorimetery (DSC) and temperature- and time-resolved synchrotron X-ray diffraction (TtXRD) are described. CZX-1 crystallizes in the cubic space group I4̅3m with a = 10.5887(3) Å, and is isostructural to sodalite. CZX-1 is a glass former and congruently melts.43 The β-cage structure of crystalline CZX-1 was shown to persist into the glassy and liquid states,44 indicating a strong structural similarity between the crystalline, glassy, and liquid phases. As such, it is reasonable to suppose that crystallization follows an A → B type mechanism, which, along with the system’s cubic symmetry, simplifies crystallization studies. As described in the initial part of this article, the use of multiple techniques to investigate the rate of crystallization reveals that depending on the technique employed the KJMA rate constant varied by about 5× at common isotherms. To understand this method dependence, crystallization simulations were performed and are described in a corresponding manuscript.42 In the simulations, all information with respect to the crystallization process is precisely defined or known (nucleation location, orientation, time, and frequency; crystallite growth geometry, time dependent shape and size; and velocity of the phase boundary). Fitting the crystallization simulations to the KJMA model and comparing the fitted kinetic parameters with the defined simulation parameters established a sample-volume and sample-container shape correction. As described in the latter part of this article, application of the volume and shape correction removes the experimental dependence and yields the intrinsic, material-specific rate parameter, the velocity of the phase boundary, vpb. These demonstrate that sample volume and shape are the major contributors to the observed method dependence of the KJMA kinetic parameters in our experimental system.

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EXPERIMENTAL METHODS

General Methods and Materials. All manipulations were performed under an inert N2 atmosphere in a glovebox or using vacuum lines. ZnCl2 and HNMe3Cl were purchased from Aldrich and purified via double sublimation prior to use. CZX-144 and CuCl45 were prepared according to previously reported procedures. The purity of all starting materials was confirmed by powder X-ray diffraction (XRD, INEL CPS-120) and differential scanning calorimetery (DSC, TA Instruments Q100). Isothermal Crystallization: DSC. For the DSC isothermal crystallization experiments, three sample masses of CZX-1 were used: 12.2, 20.2, and 26.5 mg. All samples were sealed in high-pressure stainless steel DSC pans with gold foil seals (PerkinElmer product no. B0182901). The samples were melt-crystal cycled five times between 40 and 230 °C at a constant rate of 5 °C min−1 before isothermal crystallization experiments were performed to remove heterogeneous nucleation sites and to ensure uniform thermal contact between the pan and sample. The samples were again heated to 230 °C, held at the melt isotherm for 5 min, and quenched to a crystallization isotherm (Tiso) of between 145 and 160 °C at a maximum instrumental cooling rate of ∼40 °C min−1. Heat-flow data were measured at the isotherm until crystallization was complete, typically between 1 and 100 min. Crystallization began before Tiso was reached for temperatures below 3933

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Sample Volume and Density. To obtain the material-specific vpb from the KJMA rate constant k it is necessary to obtain an accurate estimate of the sample volume.42 In the synchrotron experiments for which the sample is contained within a sealed capillary, an ingot longer than the diameter of the synchrotron beam was utilized. However, the observed signal results only from crystal growth within the irradiated volume, equivalent to the volume of the intersecting rectangular prism of the synchrotron beam and the capillary. Two sizes of capillaries were used, 0.7 and 0.5 mm outer diameter (o.d.) with 0.01 mm wall thickness. The four XRD sample volumes are presented in Table 1.

Table 1. Experimental Sample Volumes (in cm3) Used in 2D TtXRD and DSC Experiments simulation geometry42

sample volume (cm3)

X1 X2 X3 X4

3.8 × 10−5 5.84 × 10−5 1.81 × 10−4 3.63 × 10−4

D1

5.02 × 10−3 8.31 × 10−3 1.09 × 10−2

D2

synchrotron source

capillary diameter (cm)

NSLS 0.05 NSLS 0.07 APS 0.05 APS 0.07 sample mass (mg) 12.2 20.2 26.5

Figure 1. DSC heat-flow curves resulting from a quench of a 20.2 mg CZX-1 sample to isotherms between 145 and 160 °C, as labeled. Five repetitions are shown for crystallization at each isotherm. The instrumental quench response is shown as the hashed region. The inset is plotted with a compressed time scale to highlight the slow crystallization at the 160 °C isotherm.

to the fraction transformed, α. The α versus t data were then fit to eq 1 over the range 0 ≤ α ≤ 0.5 to extract KJMA parameters. For a given isotherm, the sample-to-sample variation in the observed rate of crystal growth is less than a factor of 2, albeit with the greatest being faster rates for the smaller sample and slower rates for the larger sample. The most significant variation in crystal-growth rates is the marked deceleration with increasing Tiso. Such behavior is clearly non-Arrhenius, as has commonly been observed in the literature for crystallization isotherms proximate to the melting point. The amount of heat evolved at a given isotherm exhibited no change over more than 20 cycles, indicating no significant sample degradation. No crystallization is observed for Tiso ≥ 163 °C even after 48 h, which is notably 11 °C below the melting point of CZX-1. Although the focus of this article is on the rate of crystal growth, it is interesting to note that Figures 1, SI-2, and SI-3 also demonstrate significant variation in the time to nucleation. There appears to be both a temperature and sample-size dependence on the time to initial nucleation (t0). In general, t0 appears to increase with increasing temperature and decrease with the smaller sample size. The higher temperature isotherms for the 26.5 mg sample exhibit the greatest irregularity with the t0 for Tiso = 157 °C varying from 8 to 110 min. It is further important to note that t0 does not appear to be determined by the time that the quench is initiated (tq) or the time that the isothermal temperature is achieved (tiso). Isothermal Crystallization: 2D TtXRD. A representative series of 2D TtXRD images collected at the APS for the isothermal crystallization of a sample of CZX-1 at Tiso = 135 °C is given in Figure 2. In the experiment shown, the first evidence of crystallization is observed approximately 604 s after quenching, with crystal growth occurring over the next 30 s. The area detector utilized for the TtXRD experiments is invaluable to visualize and differentiate nucleation and crystal growth. The final crystalline diffraction pattern shown in Figure 2 provides clear evidence that in this reaction the bulk of crystallization is accounted for by a single crystallite, with a few satellite reflections from subsequent nucleation. Such single (or near-single) crystal growth occurs under conditions where nucleation is slow with respect to the rate of crystal growth. By contrast, at both lower and higher temperatures when the rate

The volume of the DSC samples was determined knowing the pan diameter of d = 0.50 cm and with the assumption of a 90° contact angle between CZX-1 and the sample pan. With such a cylindrical approximation, the sample height is approximated by eq 2.

h=

(m/ρ) π(d /2)2

(2)

Sample density was determined on the basis of the crystallographic density. The crystallographic thermal expansion was computed by a least-squares fit of the lattice constant for six independent TtXRD measurements, detailed in Table 2. The intercept and slope represent

Table 2. CZX-1 Lattice Constant Extrapolated to 0 °C and Thermal Expansion as Determined from Variable Temperature TtRXRD heating rate (°C min−1)

Tstart (°C)

Tfinish (°C)

lattice constant (Å)

thermal expansion (Å K−1) × 104

5 5 10 10 10 20

35 100 31 120 125 107

181 168 170 166 168 167

10.5474(3) 10.5447(10) 10.5590(3) 10.564(3) 10.552(2) 10.543(2)

4.12(2) 3.99(8) 3.75(3) 4.3(2) 5.09(15) 4.47(15)

the lattice constant at 0 °C and the thermal expansion, respectively. The average crystallographic density of CZX-1 (at 273 K) is 2.478(1) g cm−3, with an average thermal expansion resulting in a density decrease of 2.96(7) × 10−4 g cm−3 K−1. Averaged DSC sample volumes (Table 1) are computed using the density at Tiso.

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RESULTS AND DISCUSSION Isothermal Crystallization: DSC. Measurement of liquidto-crystal transformations via DSC ideally exhibits two wellseparated signals corresponding to the instrumental quench response (IQR) and the heat evolved from crystallization. As shown in Figure 1 for the 20.2 mg CZX-1 samples, these are well separated for Tiso ≥ 155 °C. Similar plots for the 12.2 and 26.5 mg samples are given in Supporting Information Figures SI-2 and SI-3, respectively. The IQR was subtracted, and the heat flow was integrated as a function of time and normalized 3934

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Figure 2. TtXRD diffraction frames from the Tiso = 135 °C crystallization of CZX-1. (Top left) Supercooled liquid 610 s after quenching to the isotherm, (top right) fully crystallized sample, and (bottom) time series (0.5 Hz) of diffraction frames.

of growth is slowed with respect to the rate of nucleation, polycrystalline samples are observed. The final frame from each of the TtXRD crystallization experiments in this study is given in Supporting Information Figures SI-4−SI-7. Although no consistent trend has yet been established with respect to nucleation, single crystal growth was most frequently observed for the largest X4 sample size for isotherms between 120 and 150 °C. Approximately 10 or fewer crystallites are observed at Tiso > 90 °C, whereas tens to hundreds of crystallites are observed for Tiso < 90 °C, generally increasing as the isotherm temperature is decreased. Evaluation of the diffraction frames for polycrystalline samples as a function of time clearly demonstrates that nucleation proceeds in a continuous fashion, albeit with subsequent nucleation occurring at a much faster rate than the initial nucleation. Direct analysis of individual crystallite data by 2D TtXRD across a time series is currently under development. For this article, the 2D diffraction patterns are azimuthally averaged to create 1D diffraction patterns and analyzed by singular value decomposition (SVD). The 1D patterns are aggregated into an m × n data matrix A, where m is the number of reciprocal lattice vectors in each pattern and n is the number of diffraction patterns collected. These time-dependent crystallization data are deconvoluted into two sets of orthonormal basis functions according to eq 3. A = UΣVT

Figure 3. SVD analysis of a TTXRD crystallization reaction quenched from a 230 °C melt to Tiso = 133 °C. (a) Basis vectors u1 and u2 and (b) corresponding time traces v1 and v2.

the crystalline phase. These basis vectors describe approximately 80% of the variance of the original A matrix. The third component (not shown) represents 1.2% of the data set and corresponds to a slight shift in peak positions during the transformation, and the remaining 18.8% of the data set is spread across 263 additional basis vectors that have no physical interpretation (i.e., noise). The first two basis vectors of the resultant V matrix, time traces v1 and v2, demonstrate u1 and u2 correspond to simultaneous changes that occur between 65 and 100 s, Figure 3b. Given that all columns of U and V are orthogonal, physical meaning can be ascribed to the basis vectors u1 and u2 by taking the linear combination of basis vectors u1 and u2, producing modified basis vectors u1′ and u2′. Using the rotation matrix shown in eq 4 ⎡ cos(θ) sin(θ ) ⎤ ⎥ [ u1′ u 2 ′] = [ u1 u 2 ]⎢ ⎢⎣−sin(θ ) cos(θ )⎥⎦

(3)

(4)

where θ is the angle over which the original basis vectors are rotated. A rotation of θ = 205° was found to minimize the difference between u1′ and the experimental liquid diffraction pattern (Figure 4a), and correspondingly the difference between u2′ and the crystalline pattern (Figure 4b). Basis vectors v1 and v2 must also be rotated over the angle of θ = 205°, producing modified basis vectors v1′ and v2′. Because u1′ and u2′ represent the time-independent experimental liquid and crystalline diffraction patterns, v1′ and v2′ correspond to the time-dependent loss of liquid scattering and appearance of crystalline diffraction, respectively. The normalized basis vectors v1′ and v2′ along with their fit to eq 1 are shown in Figure 4c. Note specifically that the loss of the liquid scattering

T

where U and V are the left singular vectors (LSV) and right singular vectors (RSV) corresponding to time-independent diffraction patterns and their time-dependence, respectively, and Σ is composed of singular values (i.e., weighting factors) that describe the contribution of the corresponding singular vectors to the data and are sorted in decreasing order.49 Detailed descriptions of the general theory behind SVD and applications are available.49−53 An example of this SVD analysis of a crystallization reaction is shown in Figure 3 for a reaction quenched from T = 230 °C to Tiso = 133 °C. The first two basis vectors of the resultant U matrix, u1 and u2 in Figure 3a, contain elements of the broad amorphous scattering from the liquid phase and the sharp diffraction from 3935

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Figure 5. Rotated basis vectors v1″ and v2″ (rotation of v1 and v2 by θ = 6°) to maximize variance in v1″ and minimize variance in v2″. The variance, v1″, is fit to the KJMA model (solid line).

reactions conducted in the high-temperature region within about 40° of the melting point exhibited a slowing in the rate of crystallization with increasing Tiso. However, for isotherms above Tg but below 135 °C, the rate of crystallization is observed to increase with increasing Tiso. Notably, the single or near-single crystal pattern observed in the final frame for a range of higher-temperature experiments (Figures SI-4−SI-7) demonstrates that the slowed rate of crystallization in the higher-temperature region does not correspond to a slowed rate of nucleation as suggested by Classical Nucleation Theory.7−12 The observed variation in crystallization rates is observed both for experiments, resulting in single- and polycrystalline products. Details of the crystallization rate as a function of temperature are given below in the presentation of measured rate constants. KJMA Parameter Correlations. To obtain significant mechanistic information from rate measurements, a minimum requirement is that any physical parameters be intrinsic to the material. However, substantial variation between all KJMA parameters extracted from the DSC and TtXRD experiments was observed, even for experiments conducted at common isotherms. Nucleation is presumed to be a random event, but n and k are expected to be material, not technique dependent. The KJMA exponent n is classically understood to include the dimensionality of growth, λ = 1, 2, or 3, as well as the probability of nucleation β such that n = λ + β. β is considered to be 0 for fixed nucleation and 1 for continuous nucleation. The many single or near-single crystal reactions observed (Figures SI-4−SI-7) should result in β ≈ 0, yet values of n as low as 2 and well above 4 were obtained. Similarly, widely variant values of k were observed for reactions measured at a common isotherm but with different techniques. Initial insight into the KJMA parameter problem is obtained by understanding the extent to which the KJMA parameters t0, n, and k are correlated.42 Consider, for example, the TtXRD isothermal crystallization experiment shown in Figure 6 where the sample reached the Tiso = 139 °C at 0 s. Fixing n = 3, a reasonable assumption given the cubic structure of CZX-1 and

Figure 4. Modified basis vectors u1′ and u2′, from rotating basis functions u1 and u2 by θ = 205°, which represent the experimental liquid and crystalline diffraction patterns in (a) and (b), respectively. (c) Corresponding rotated basis vectors v1′ and v2′ demonstrate the simultaneous disappearance of liquid diffraction (red) and the appearance of crystalline diffraction (blue).

and the emergence of the crystalline scattering cross at 50%, providing strong support for a mechanistic conclusion that CZX-1 crystallization proceeds via an A → B type mechanism (i.e., with no intermediates). An A → B crystallization mechanism indicates that the crystalline phase can only grow at the expense of the liquid phase. As a result, the time-dependence of crystallization is spread across v1′ and v2′, which serves to decrease the signal-tonoise (S/N) ratio in each. However, taking further advantage of the orthogonality of the U and V matrices, it is possible to transfer essentially all of the time-dependence of crystallization into one of the basis vectors by rotating v1′ and v2′ (or v1 and v2) to produce v1″ and v2″. A rotation angle of 6° from the original basis vectors v1 and v2 achieves this, as shown in Figure 5. The corresponding u1″ and u2″ (not shown), which contain a mixture of both liquid and crystalline scattering, are physically meaningless. However, rotating all time-dependence into one singular vector maximizes the S/N for these TtXRD kinetic experiments. The superior quenching capability in the TtXRD experiments provides access to a greater range of temperatures over which the crystallization rate can be measured than was accessible by DSC. As was observed in the DSC measurements, 3936

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stronger than the noise inherent to the measurement, further exacerbating the determination of t0. Several features of this CZX-1 system present a strong case for the KJMA exponent to be fixed at 3, indicative of 3D crystal growth with little or no contribution from continuous nucleation. The cubic space group of CZX-1 (I4̅3m) suggests that the system should exhibit isotropic growth in three dimensions, unless constrained into an anisotropic sample geometry.42 As shown in Figure 7, crystals of CZX-1 with a

Figure 6. TtXRD data for a crystallization reaction at Tiso = 139 °C (black circles). Data are fit to the KJMA model with n = 3 (red) or with t0 fixed to 0 (blue), 30 (cyan), or 60 s (yellow). Insets are an expansion of the onset and termination portions of the transformation.

Figure 7. Single crystal of CZX-1 (2 × 2 × 2 mm3) demonstrating the cubic morphology of crystal growth.

the observed single crystal growth, yields a very good fit to the data when t0 = 91 s and k = 2.28(6) × 10−2 s−1. However, as shown in Table 3 and Figure 6, the first half of the

cubic morphology have been observed to grow from a sample sealed within a fused silica tube that was quenched from the melt to a glass and then held at room temperature (23−25 °C) for several weeks. Furthermore, several of the 2D TtXRD experiments, Figures 2 and SI-4−SI-7, demonstrate single crystal growth, and nearly all of those with Tiso greater than 90 °C isotherms exhibit fewer than 10 crystallites. For these systems, a continuous nucleation probability term β is likely at or near 0. Thus, to mitigate correlation effects, all XRD experimental data in were fit with n fixed at 3. Isothermal Crystallization Rate. Crystallization rate constants determined by fitting isothermal DSC and TtXRD crystallization data to eq 1 with n fixed at 3 are presented in Figure 8a. As described above, the distribution of rate constants as a function of temperature clearly demonstrate that the rate of crystallization increases as the isothermal reaction temperature increases from above the Tg to some maximum Tiso, after which the rate decreases with increasing reaction temperature. However, on more careful inspection, it becomes apparent that the rate constants are distributed with respect to the sample size used in the respective experiments. At common isotherms, the rate constants extracted from TtXRD data collected at the NSLS synchrotron are distinctly greater than those obtained for data collected at the APS synchrotron, which are significantly greater than those obtained from DSC experiments. Given that condensed phase crystallization is a phaseboundary controlled process (i.e., atomic rearrangement is required at the phase boundary between the melt and crystal), it is implausible that the physical and chemical processes controlling the reaction have changed when measured with different volumes. That the volume of a sample must be considered with respect to the rate at which the sample crystallizes is intuitive. Starting from an assumption that the velocity of the crystallization phase boundary should be a material-specific and constant value, albeit temperature dependent, a large sample will take longer to crystallize than a small sample. As the sample is transformed, the crystal volume must grow at a rate proportional to the velocity of the phase boundary, vpb, raised to the power of dimensionality of crystal growth. Conversely, in the simplest system of a cube growing in

Table 3. KJMA Parameters for Fits of the Data Given in Figure 6 Demonstrating Strong Correlation between Parameters k

n

t0

Fisher’s Z

0.0228(6) 0.0134(6) 0.00976(3) 0.007561(15)

3 6.39(15) 9.6(2) 12.9(3)

91(8) 60 30 0

2.60 2.76 2.76 2.70

transformation is equally fit when t0 is fixed to earlier time points and n is allowed to vary. When t0 is set to 0 s, the time at which Tiso was achieved, the data are well fit by n = 12.9(3) and k = 7.56(1) × 10−3 s−1. These data demonstrate the strong, positive correlation between k and t0 and the negative correlation between both of these values and n. Although each of the four cases considered numerically fit the KJMA model with statistical significance, three produce physically unreasonable values for t0 and n. The above analysis of parameter correlations demonstrates that care must be taken before ascribing physical significance to systems described with large KJMA dimensionalities. It is likely that the cases in which large KJMA exponents are reported are in fact ones in which t0 was not accurately determined. Correlation problems,54 such as those noted above, can be avoided with either prior knowledge, or by independent determination of t0 or n. In the CZX-1 crystallization experiments, it is difficult to predetermine t0 because nucleation has been observed to occur both as quickly as a few seconds after the system has reached Tiso and as slow as hours after quenching to the growth isotherm. Nevertheless, it is clear that t0 is not equivalent to the time at which the system reaches the quench isotherm. In fact, one can only determine an upper bound to t0 because nucleation must have occurred prior to the observation of the first experimental signal. Assuming a nucleation event is the result of organization of a few hundred unit cells out of ∼1017 unit cells in a TtXRD sample, it is unlikely that any signal resulting from nucleation will be 3937

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within a factor of about 2.67. It is also shown that the observed rate of crystallization is strongly dependent upon the location and orientation of the growing crystal within the sample volume. Random nucleation location and orientation results in an intrinsic variation in the measured rate constants of about a factor of 2. The simulations also demonstrate that the relative anisotropy of the sample volume in which the crystal grows can impact the measured rate. The KJMA model with the volume correction is appropriate for reasonably isotropic sample volumes, but outside of this range the impingement of crystal growth will begin earlier as the sample geometry approaches the limiting cases of 2D (disc) and 1D (capillary). Such progressively earlier termination blocks one (disc) or two (capillary) available growth dimensions, producing a lower apparent dimensionality, n′, a lower apparent k′, and therefore a lower apparent phaseboundary velocity, vpb′. To obtain the most accurate vpb, it is preferable to conduct experiments in sample geometries that are as isotropic as possible. The XRD geometries reported in this work are sufficiently isotropic (0.4 ≤ d/h ≤ 0.7) that no anisotropy correction is required. However, the experimental DSC geometries result in aspect ratios of 19.5, 11.8, and 9.0 for the 12.2, 20.2, and 26.5 mg samples, respectively, under the cylindrical approximation. As demonstrated by the series of constant volume simulations,42 a decreasing apparent dimensionality n′ and apparent rate constant k′ are observed with increasing sample anisotropy. However, this effect can be removed with an empirical expression whose numerical terms result from a fit to Figure 6a in ref 42 shown in eq 6.

Figure 8. (a) KJMA rate constants as a function of temperature extracted from isothermal DSC (red) and TTXRD (blue/cyan) experiments. Samples X1 and X2 were collected at the NSLS and X3 and X4 were collected at the APS synchrotron sources, respectively. (b) Phase-boundary velocities as a function of temperature.

⎛ ⎞ d ⎛d⎞ ac = exp⎜ −0.29 ln⎜ ⎟ + 0.24⎟ , > 2.67 ⎝h⎠ ⎝ ⎠ h

an unrestricted volume and not impacted by termination effects, vpb should be directly proportional to the cube root of the transformed volume. The normalization of the crystallization transformation to α in the KJMA model implicitly exchanges the volume dependence of the experimental signal for a volume-dependent rate constant k. We thus propose to reintroduce the sample volume into the KJMA model according to eq 5 to produce the chemically meaningful phase-boundary velocity vpb ∝ k 3 V

(6)

The function to describe the variation of the dimensionality and rate constant with respect to the aspect ratio of the sample must be scaled to the relative number of crystallites in that sample (thus, a function of nucleation rate, growth rate, and sample size) because for any given sample aspect ratio, additional nuclei further subdivide the sample, causing each crystallite to be less anisotropic. To account for the experimentally observed sample-container shape effect, we simulated the relative volume of each DSC sample (1.5 × 106 to 3.2 × 106 volume elements) with otherwise identical simulation parameters.42 The DSC simulations were fit using the KJMA model with t0 fixed at the known simulated tnuc to obtain an apparent dimensionality, n′, which was then averaged over all simulations performed. The aspect ratio corresponding to that fit value of n′ in the simulation shown in Figure 6a of ref 42 was then used to fit the experimental data for each of the DSC sample geometries. Specifically, the modified exponents used to correct the DSC data are n′ = 2.5, 2.62, and 2.7 for the 12.2, 20.2, and 26.5 mg DSC samples, respectively, corresponding to apparent aspect ratios of 5.17, 4.75, and 3.95. As applied to the DSC data reported in Figure 8, the sample anisotropy correction results in an increase from the values obtained for vpb when n was fixed to 3 of 75, 54, and 35% for the 12.2, 20.2, and 26.5 mg samples, respectively. The sample volume and anisotropy corrections, eqs 5 and 6, along with a term to describe the intrinsic geometry of crystal growth, g (g = 1 for cubic growth), are combined to give eq 7.

(5) −1

where vpb has units of distance × s . A similar relation was previously introduced to describe sample mass effects on kinetics of thermal decomposition reactions.41,55 As shown in Figure 8b, application of eq 5 to the crystallization rate constants obtained from crystallization experiments successfully accounts for the majority of the technique-dependence from the rate constant. To establish the quantitative validity of the empirical relationship proposed in eq 5, an extensive set of crystal growth simulations were performed.42 The simulations allowed precise definition of vpb, nucleation time/nucleation rate, crystallite shape, sample size and shape, nucleus location, and orientation. The simulated data were then fit in the same manner as the experimental data to compare the fit KJMA parameters t0, n, k, and vpb to the defined simulation parameters. The simulations validate eq 5 with the addition of a crystallite geometric shape term, g, when the crystallization is performed in reasonably isotropic sample containers, which are defined by the minimum and maximum sample axes being 3938

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kg 3 V ac

Article

Additionally, DSC experiments measure the entire sample in contrast to TtXRD, which can only probe the portion of the sample that intersects the synchrotron beam. However, using an area detector for the TtXRD experiments affords insight into the number of nucleation events that occurred during the sample’s transformation, which is in contrast to DSC measurements that cannot differentiate between single and polycrystalline transformations. Our sample volume and anisotropy correction to the KJMA model affords an unprecedented measurement of the velocity of the phase boundary from bulk crystallization kinetic measurements. These range from 0.02 μm s−1 at 37 °C near Tg to a maximum of 32 μm s−1 at 135 °C (Tmax) and 1.5 μm s−1 at our highest observed crystallization temperature of about 11 °C below the melting temperature. Given the unit-cell length for CZX-1 is 10.85 Å, the maximal phase-boundary velocity indicates that the crystalline axes increase by approximately 29 unit-cell lengths per millisecond. Finally, we recognize that vpb at temperatures between Tg and Tmax seems to exhibit Arrhenius-type kinetics, although it has been noted in the literature that the theoretical justification for application of the Arrhenius model to reactions in the condensed phase is unclear.19,56 The observation that the rate of crystal growth exhibits a maximum is not unique to this system. To explain this effect in an Arrhenius context, CNT presumes that the reduction in the rate is due to a reduction in nucleation. However, the observation here that the crystallization rate is slowed even for the growth of single crystals cannot be explained by classical models. Additionally, these models cannot account for the observation that crystallization is not observed until ∼10 °C below Tm. Although an energetic model that accounts for the observed crystallization behavior is not yet developed, replacing the empirical kinetic parameters by those with clear intrinsic physical significance is a necessary starting point.

(7)

This correction affords the intrinsic, material-specific kinetic parameter for a condensed phase reaction, the velocity of the phase boundary, to be determined from the simple form of the KJMA expression (eq 1). Application to the experimental data of this work yields the vpb reported in Figure 9, demonstrating complete consistency between the TtXRD and DSC measurements.

Figure 9. Plot of vpb as a function of temperature with the DSC experiments corrected for their anisotropic sample shape.

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CONCLUSIONS Utilization of multiple experimental techniques to measure the rate of crystallization revealed the need for a critical sample volume correction that, when applied to the KJMA model for phase-boundary controlled reactions, provides the physically meaningful and material-specific kinetic parameter, the velocity of the phase boundary, vpb. This is in clear contrast to the method-dependent and therefore not material-specific KJMA rate constant, k.23−26 The set of experimental measurements and corresponding simulations42 demonstrate that to obtain the most accurate kinetic information it is necessary to independently determine at least one of the KJMA parameters, most reasonably the growth dimensionality n, to reduce parameter correlation effects. Furthermore, it is demonstrated that kinetic measurements should be performed in isotropic sample volumes to maximize unrestricted crystallite growth, thereby mitigating sample-container effects. Because nucleation location and orientation can cause the rate of crystallization to vary by about a factor of 2, a large number of kinetic measurements should be made to ensure appropriate statistical sampling.2 Specifically, with respect to the rate of crystallization of the halozeotype CZX-1, the combination of DSC and TtXRD techniques provides access to kinetic measurements over the entire temperature range between Tg and Tm. The DSC measurements provide invaluable access to measurement of the slow crystallization rates at temperatures near the melting point. By contrast, the TtXRD measurements, which clearly differentiate between liquid and crystalline phases, afford a rapid quenching such that kinetic measurements in deeply supercooled melts can be achieved.

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ASSOCIATED CONTENT

S Supporting Information *

Schematic diagram of the forced air furnace used for isothermal crystallization experiments at the synchrotrons located at Argonne and Brookhaven National Labs; heat-flow curves resulting from a quench of 12.2 and 26.5 mg CZX-1 samples to various isotherms; and final frames from the X-1, X-2, X-3, and X-4 XRD experiments. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

This work was supported by NSF via contracts DMR-0305086 and DMR-0705190. Use of the National Synchrotron Light Source, Brookhaven National Laboratory and the Advanced Photon Source, Argonne National Laboratory, Office of Science User Facilities operated for the U.S. Department of Energy (DOE) Office of Science, was supported by the U.S. DOE under Contract No. DE-AC02-98CH10886 and DE-AC0206CH11357, respectively. Beam line scientists Jonathan 3939

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Hanson, Elaine DiMasi, Peter Chupas, and Karina Chapman are gratefully acknowledged.

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NOTE ADDED AFTER ASAP PUBLICATION This article published October 10, 2013 with errors throughout the text. The corrected version published October 22, 2013.

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