Temperature-composition Phase Diagram Research Articles

Superconductivity and magnetic and transport properties of single-crystalline CaK(Fe1−xCrx)4As4

Members of the CaK(Fe$_{1-x}$Cr$_{x}$)$_{4}$As$_{4}$ series have been synthesized by high-temperature solution growth in single crystalline form and characterized by X-ray diffraction, elemental analysis, magnetic and transport measurements. The effects of Cr substitution on the superconducting and magnetic ground states of CaKFe$_4$As$_4$ ($T_c$ = 35 K) have been studied. These measurements show that the superconducting transition temperature decreases monotonically and is finally suppressed below 1.8 K as $x$ is increased from 0 to 0.038. The magnetic transition temperature increases in a roughly linear manner as Cr substitution increases. A temperature-composition (\textit{T}-\textit{x}) phase diagram is constructed, revealing a half-dome of superconductivity with the magnetic transition temperature, $T^*$, appearing near 22~K for $x$ $\sim$ 0.017 and rising slowly up to 60~K for $x$ $\sim$ 0.077. The $T$-$x$ phase diagrams for CaK(Fe$_{1-x}$$T$$_{x}$)$_4$As$_4$ for $T$ = Cr and Mn are essentially the same despite the nominally different band filling; this is in marked contrast to $T$ = Co and Ni series for which the $T$-$x$ diagrams scale by a factor of two, consistent with the different changes in band filling Co and Ni would produce when replacing Fe. Superconductivity of CaK(Fe$_{1-x}$Cr$_{x}$)$_{4}$As$_{4}$ is also studied as a function of magnetic field. A clear change in $H^\prime_{c2}$($T$)/$T_c$, where $H^\prime_{c2}$($T$) is d$H_{c2}$($T$)/d$T$, at $x$ $\sim$ 0.012 is observed and probably is related to change of the Fermi surface due to magnetic order. Coherence length and the London penetration depths are also calculated based on $H_{c1}$ and $H_{c2}$ data. Coherence lengths as the function of $x$ also shows changes near $x$ = 0.012, again consistent with Fermi surfaces changes associated with the magnetic ordering seen for higher $x$-values.

Phase and properties prediction of Al–Ag binary system using thermo-calc

The temperature phase diagram stability and physical properties of a binary Silver–Aluminium (Al–Ag) alloy system have been calculated using Thermo-Calc 2021b software at a temperature range of 250–1300 K. Six single-solid phases other than the liquid phase, elemental Ag (fcc_Ag), elemental Al (fcc_Al), hcp_A3, cub_A13, and bcc_Ag were obtained from the equilibrium calculation. Ten regions of the double phase were obtained and evaluated. Three intermediate phases were also obtained from the phase diagram with five invariant reactions, namely one eutectic reaction (at 840.00 K), two peritectic reactions (at 1051.22 K and 1003.18 K), and two peritectoid reactions (at 730.58 K and 881.40 K) were equally found. Sixteen regions other than elemental Ag and Al at 0, and 100, respectively, were obtained. Maximum solid solubility obtained was 5.60 Al mass percent at 730. 58 K for Al in Ag and 58.17 Ag mass percent at 840.00 K for Ag in Al. From the temperature–composition phase diagram, the melting point for pure silver and aluminium were equally inferred as 1234.96 K and 933.50 K, respectively, which agrees with experimental standards. These results play a significant role in the processing of Al–Ag alloy for different industrial-related applications.Graphical abstract

Correlation between multi-factor phase diagrams and complex electrocaloric behaviors in PNZST antiferroelectric ceramic system

Ferroelectric (FE) phase transition with a large polarization change benefits to generate large electrocaloric (EC) effect for solid-sate and zero-carbon cooling application. However, most EC studies only focus on the single-physical factor associated phase transition. Herein, we initiated a comprehensive discussion on phase transition in Pb_0.99Nb_0.02[(Zr_0.6Sn_0.4)_1−<i>x</i>Ti_<i>x</i>]_0.98O₃ (PNZST100<i>x</i>) antiferroelectric (AFE) ceramic system under the joint action of multi-physical factors, including composition, temperature, and electric field. Due to low energy barrier and enhanced zero-field entropy, the multi-phase coexistence point (<i>x</i> = 0.12) in the composition–temperature phase diagram yields a large positive EC peak of maximum temperature change (Δ<i>T</i>_max) = 2.44 K (at 40 kV/cm). Moreover, the electric field–temperature phase diagrams for four representative ceramics provide a more explicit guidance for EC evolution behavior. Besides the positive EC peaks near various phase transition temperatures, giant positive EC effects are also brought out by the electric field-induced phase transition from tetragonal AFE (AFE_T) to low-temperature rhombohedral FE (FE_R), which is reflected by a positive-slope boundary in the electric field–temperature phase diagram, while significant negative EC responses are generated by the phase transition from AFE_T to high-temperature multi-cell cubic paraelectric (PE_MCC) with a negative-slope phase boundary. This work emphasizes the importance of phase diagram covering multi-physical factors for high-performance EC material design.

Structural transformations and phase transitions in hexagonal La-containing lyomesophases

Understanding and identification of phase equilibrium in self-assembling systems like lyotropic liquid crystals (LLCs) plays a key role in their further practical applications. In this study the phase transitions in LLCs based on С12ЕО10 nonionic surfactant, lanthanum nitrate hexahydrate, and water are reported. Polarized optical microscopy, X-ray powder diffraction, and fluorimetry were used to find correlations between system composition and structural changes. A new approach to identify a reverse hexagonal mesophase by the analysis of concentration dependencies of mesophase structural parameters was proposed. Data from these experiments were further compared with pyrene fluorescence studies to confirm the phase inversion. Based on complex analyses of experimental data an inversion of hexagonal mesophases via intermediate lamellar phase in the С12ЕО10(LaIII)/H2O system was found at the concentration ratio of 85/15 wt.%. The interpreted temperature-composition phase diagram was plotted. The regions of existence of direct hexagonal, intermediate mesophases and reverse hexagonal were found in the С12ЕО10(LaIII)/H2O system.

Unexpected Slow Kinetics of Poly(Methacrylic Acid) Phase Separation in the Semi-Dilute Regime.

Poly (methacrylic acid) (PMAA) solutions are known to exhibit a lower critical solution temperature (LCST). A temperature-composition phase diagram of PMAA has been constructed by standard cloud point determination through transmittance measurements, and also by studying the steady states reached under phase separation. This allows us to reconstruct the binodal curve describing the phase behavior of PMAA for both low and high concentration regimes, and to determine accurately the LCST temperature. In a second step, the structures formed following a temperature jump above the cloud point and their evolution in time have been investigated at the nanoscale using small angle neutron scattering (SANS). This approach shows that the formation of phase-separated nanostructures is a slow process, requiring more than 12 h. The formed structures are then shown to depend on the amplitude of the temperature jump above the cloud point. An original mechanism of phase separation is identified in the semi-dilute regime. The growth of micrometric-size droplets with an inner structure displaying the rheological properties of a gel leads to the formation of a percolating network which hinders the influence of gravity. Such a result can explain the slow kinetics of the PMAA LCST transition.

Light-tunable three-phase coexistence in mixed halide perovskites

Mixed iodine-bromine perovskites used in solar cells undergo below a critical temperature an intrinsic demixing into phases with different iodine-bromine compositions. In addition, under illumination they show nucleation of an iodine-rich phase. We predict from thermodynamic considerations that in mixed iodine-bromine perovskites like MAPb(I$_{1-x}$Br$_x$)$_3$ the interplay of these effects can lead to coexistence of a bromine-rich, iodine-rich, and nearly iodine-pure nucleated phase. This three-phase coexistence occurs in a region in the composition-temperature phase diagram near the critical point for intrinsic demixing. We investigate the hysteresis in the evolution of this coexistence when temperature or illumination intensity are cycled. Depending on the particular way the coexistence is established, nearly iodine-pure nuclei should form either in the iodine-rich phase only or both in the bromine-rich and iodine-rich phases. Experimental verification of this fundamentally novel type of light-tunable three-phase coexistence should be possible by a combination of absorption and photoluminescence experiments.

Structural Evolution and Enhanced Piezoelectric Activity in Novel Lead-Free BaTiO3-Ca(Sn1/2Zr1/2)O3 Solid Solutions

In this study, a series of solid solutions of (1−x)BaTiO3-xCa(Sn1/2Zr1/2)O3 (abbreviated as (1−x)BT-xCSZ, x = 0.00–0.15) ceramics have been prepared by the conventional solid-state reaction method to search for high performance lead-free piezoelectric materials. The structural evolution, microstructure, and piezoelectric properties are investigated. X-ray diffraction (XRD) results indicate that the phase symmetry strongly depends on the CSZ content. A tetragonal phase is well-maintained in the compositions of 0 ≤ x ≤ 0.03, and coexistence of tetragonal and cubic phases is obtained in the range of x = 0.06–0.09, beyond which a pure cubic phase becomes stable. More importantly, a significantly enhanced piezoelectric coefficient of d33 = 388 ± 9 pC/N is attained in the composition of x = 0.06 in the MPB region, where a tetragonal ferroelectric phase and an ergodic relaxor phase with average cubic symmetry coexist. Based on the analysis of crystal structure and dielectric properties, a temperature-composition phase diagram consisting of four phase regions is established. This study indicates that the lead-free BT-CSZ binary system has great potential for use in electromechanical transducer applications.

Superconductivity and phase diagrams of CaK(Fe1−xMnx)4As4 single crystals

Members of the CaK(Fe$_{1-x}$Mn$_{x}$)$_{4}$As$_{4}$ series have been synthesized in single crystalline form and characterized by elemental analysis, thermodynamic and transport measurements. These measurements show that the superconducting transition temperature decreases monotonically and is finally suppressed below 1.8 K. For $x$-values greater than 0.016, signatures of a magnetic transition can be detected in both thermodynamic and transport measurements in which kink-like features allow for the determination of the transition temperature, $T^*$, that increases as Mn substitution increases. A temperature-composition ($T$-$x$) phase diagram is constructed, revealing a half-dome of superconductivity with the magnetic transition temperature, $T^*$, appearing near 26 K for $x$ $\sim$ 0.017 and rising slowly up to 33 K for $x$ $\sim$ 0.036. Specific heat data are used to track the jump in specific heat at $T_c$; The CaK(Fe$_{1-x}$Mn$_x$)$_4$As$_4$ data does not follow the scaling of $\Delta$$C_{p}$ with $T_{c}^3$ as many of the other Fe-based superconducting systems do. Elastoresistivity coefficients, $2m_{66}$ and $m_{11}-m_{12}$, as a function of temperature are also measured. $2m_{66}$ and $m_{11}-m_{12}$ are qualitatively similar to CaK(Fe$_{1-x}$Ni$_x$)$_4$As$_4$. This may indicate that the magnetic order in Mn substituted system may be still the same as CaK(Fe$_{1-x}$Ni$_x$)$_4$As$_4$. A clear change in $H^\prime_{c2}$($T$)/$T_c$, where $H^\prime_{c2}$($T$) is d$H_{c2}$($T$)/d$T$, at $x$ $\sim$ 0.015 is observed and probably is related to change of the Fermi surface due to magnetic order. Coherence lengths and the London penetration depths are also calculated based on $H_{c1}$ and $H_{c2}$ data. Coherence lengths as the function of $x$ also shows the changes near $x$ = 0.015.

Structural and dynamical aspects of extremely swollen lyotropic phases

In this contribution, the composition-temperature phase diagram of a swollen microemulsion system was investigated using high resolution small-angle X-ray scattering to explore the morphological landscape over the selected region of interest. As the oil content was increased, the morphology transformed from a bicontinuous microemulsion, XD < 0.6, to a disordered emulsified-microemulsion, XD > 0.6, consisting of a dense packing of microemulsion droplets, and a hexagonal phase at XD = 0.6. During the heating from room temperature up to the highest temperature, T = 304 K, each composition transitioned into the disordered emulsified-microemulsion morphology, however, while cooling more complex structural changes were observed. Interestingly, for each composition the same sequence of phase transitions was observed during the cooling process, first there was a transition to the emulsified hexagonal phase and then to the bicontinuous microemulsion, likely due to further expulsion of oil, resulting in a composition that would favour a bicontinuous morphology. The dynamics of the microemulsion droplets were probed using X-ray photon correlation spectroscopy, which revealed two relaxation modes, typically found in dense colloidal systems below their glass transition. These results demonstrate the importance of a combined structural and dynamical approach in the studies of dense microemulsion systems, relevant for many industrial applications.

Magnetic phase diagram of the solid solution LaMn2(Ge1−xSix)2 (0 ≤ x ≤ 1) unraveled by powder neutron diffraction

The structural and magnetic properties of the ThCr2Si2-type solid solution LaMn2(Ge1−xSix)2 (x = 0.0 to 1.0) have been investigated employing a combination of X-ray diffraction, magnetization and neutron diffraction measurements, which allowed establishing a magnetic composition-temperature phase diagram. Substitution of Ge by Si leads to a compression of the unit cell, which affects the magnetic exchange interactions. In particular, the magnetic structure of LaMn2(Ge1−xSix)2 is strongly affected by the unit cell parameter c, which is related to the distance between adjacent Mn layers. Commensurate antiferromagnetic layers and a canted ferromagnetic structure dominate the Si-rich part of the solid solution, whilst an incommensurate antiferromagnetic flat spiral and a conical magnetic structure are observed in the Si-poor part.

Temperature-dependent enthalpy and entropy stabilization of solid solution phases in non-equiatomic CoCrFeNiTi high entropy alloys: computational phase diagrams and thermodynamics

Research interest in multi-principal element high entropy alloys (HEAs) has increased drastically since the field was first formally introduced in 2004. Since then, HEAs have become important candidate materials for many key applications. However, despite the progress made in this field, there remains much ambiguity surrounding HEA phase stabilities. To that end, the calculation of phase diagrams (CALPHAD) method was used to construct extensive temperature-composition phase diagrams of the CoCrFeNi x Ti2−x , Co x CrFeNiTi2−x , CoCrFe x NiTi2−x , and CoCr x FeNiTi2−x HEA systems. Due to its potentially favorable properties, the current work was focused on the single face-centered cubic (fcc) solid solution phase and an extensive thermodynamic analysis was carried out to examine the underlying thermodynamic factors of its stabilization. The mixing enthalpies and entropies of the alloys in the studied systems were calculated, where it was found that the single fcc solid solution phase can be either enthalpy- or entropy-stabilized depending on the temperature. The deviation of these quantities from the ideal solid solution thermodynamic behavior was considered, and it was found that close to and within the single fcc solid solution regions, the deviation is smallest in all systems. Furthermore, a preliminary exploration of the impact of interstitial nonmetals such as C, N, and O showed noticeable alteration of the phase equilibria of the studied systems. This work emphasizes the importance of exploring non-equiatomic compositions of HEAs as well as the necessity of a comprehensive thermodynamic analysis to understand HEAs phase stabilities.

Adding a New Dimension to the Amorphous Solid Dispersion Phase Diagram: Studying Dissolution Kinetics of Crystalline Drugs in a Polymer Matrix Using Temperature Dependent XRPD and DSC

Hot-melt-extrusion (HME) is an enabling technology used for poorly soluble active pharmaceutical ingredients (APIs) to increase the bioavailability by embedding the drug in a water soluble and often amorphous carrier such as a polymer. Knowledge of the most critical factors impacting the dissolution rate of crystalline API in the polymer during manufacturing will provide useful insight for process improvement. In this study, crystalline APIs (Acetaminophen, APAP and Indomethacin, IMC) were analyzed in a polymeric matrix (Copovidone, PVP-VA64) via X-Ray Powder Diffraction (XRPD) and Differential Scanning Calorimetry (DSC) to follow the dissolution process under various conditions in a down-scaled static laboratory system. The combination of in-situ XRPD measurements and a kinetic model based on DSC data proved to be a suitable tool to investigate the dissolution process and can be applied to various APIs and polymers to avoid residual crystallinity and thermal degradation. Thus, the temperature-composition phase diagram in a thermodynamic equilibrium is augmented by the kinetic component as new dimension. The obtained findings set the foundation for investigating the dissolution kinetics and enable the transition from a static to a dynamic system.

Equilibrium phase diagrams of isostructural and heterostructural two-dimensional alloys from first principles

SummaryAlloying is a successful strategy for tuning the phases and properties of two-dimensional (2D) transition metal dichalcogenides (TMDCs). To accelerate the synthesis of TMDC alloys, we present a method for generating temperature-composition equilibrium phase diagrams by combining first-principles total-energy calculations with thermodynamic solution models. This method is applied to three representative 2D TMDC alloys: an isostructural alloy, MoS2(1-x)Te2x, and two heterostructural alloys, Mo1-xWxTe2 and WS2(1-x)Te2x. Using density-functional theory and special quasi-random structures, we show that the mixing enthalpy of these binary alloys can be reliably represented using a sub-regular solution model fitted to the total energies of relatively few compositions. The cubic sub-regular solution model captures 3-body effects that are important in TMDC alloys. By comparing phase diagrams generated with this method to those calculated with previous methods, we demonstrate that this method can be used to rapidly design phase diagrams of TMDC alloys and related 2D materials.

Rational Development of a Carrier-Free Dry Powder Inhalation Formulation for Respiratory Viral Infections via Quality by Design: A Drug-Drug Cocrystal of Favipiravir and Theophylline.

Formulating pharmaceutical cocrystals as inhalable dosage forms represents a unique niche in effective management of respiratory infections. Favipiravir, a broad-spectrum antiviral drug with potential pharmacological activity against SARS-CoV-2, exhibits a low aqueous solubility. An ultra-high oral dose is essential, causing low patient compliance. This study reports a Quality-by-Design (QbD)-guided development of a carrier-free inhalable dry powder formulation containing a 1:1 favipiravir–theophylline (FAV-THP) cocrystal via spray drying, which may provide an alternative treatment strategy for individuals with concomitant influenza infections and chronic obstructive pulmonary disease/asthma. The cocrystal formation was confirmed by single crystal X-ray diffraction, powder X-ray diffraction, and the construction of a temperature–composition phase diagram. A three-factor, two-level, full factorial design was employed to produce the optimized formulation and study the impact of critical processing parameters on the resulting median mass aerodynamic diameter (MMAD), fine particle fraction (FPF), and crystallinity of the spray-dried FAV-THP cocrystal. In general, a lower solute concentration and feed pump rate resulted in a smaller MMAD with a higher FPF. The optimized formulation (F1) demonstrated an MMAD of 2.93 μm and an FPF of 79.3%, suitable for deep lung delivery with no in vitro cytotoxicity observed in A549 cells.

Exploration of theoretical and practical evaluation on Kollidon®SR matrix mediated amorphous filament extrusion of norfloxacin by melt extrusion

The selection of an appropriate polymer for the preparation of amorphous solid dispersions via hot-melt extrusion (HME) deals with the study of various solid-state properties of drugs and polymers. Hence, before going to HME, it is necessary to have appropriate knowledge of drug-polymer miscibility, drug-polymer interactions, and Gibb's free energy of mixing for the respective drug-polymer system. In the present work, we evaluated the miscibility and interaction parameters between norfloxacin (NOR) and Kollidon®SR Powder (KSR-P). The solubility parameter approach, based on theoretical calculations, was performed as a preliminary step for the prediction of miscibility between NOR and KSR-P. A theoretical and experimental approach was used as a validation tool using Flory-Huggin's interaction parameter value determined using the melting point depression method. The effect of NOR loading on the interaction parameter was systemically predicted to construct a thermodynamic phase diagram and Gibb's free energy of mixing for the NOR and KSR-P binary systems for the preparation of amorphous filament extrudate. A temperature-composition phase diagram was used to predict the processing temperature at the respective NOR loading to obtain a homogenous amorphous extrudate. Further, the obtained extrudate was evaluated through dimension studies, NOR assay, DSC, XRD, ATR analysis, and release behaviour.

Peculiarities of Phase Formation in Mn-Based Na SuperIonic Conductor (NaSICon) Systems: The Case of Na1+2x Mn x Ti2-x (PO4)3 (0.0 ≤ x ≤ 1.5).

NAtrium SuperIonic CONductor (NASICON) structured phosphate framework compounds are attracting a great deal of interest as suitable electrode materials for “rocking chair” type batteries. Manganese-based electrode materials are among the most favored due to their superior stability, resource non-criticality, and high electrode potentials. Although a large share of research was devoted to Mn-based oxides for Li- and Na-ion batteries, the understanding of thermodynamics and phase formation in Mn-rich polyanions is still generally lacking. In this study, we investigate a bifunctional Na-ion battery electrode system based on NASICON-structured Na1+2xMnxTi2–x(PO4)3 (0.0 ≤ x ≤ 1.5). In order to analyze the thermodynamic and phase formation properties, we construct a composition–temperature phase diagram using a computational sampling by density functional theory, cluster expansion, and semi-grand canonical Monte Carlo methods. The results indicate finite thermodynamic limits of possible Mn concentrations in this system, which are primarily determined by the phase separation into stoichiometric Na3MnTi(PO4)3 (x = 1.0) and NaTi2(PO4)3 for x < 1.0 or NaMnPO4 for x > 1.0. The theoretical predictions are corroborated by experiments obtained using X-ray diffraction and Raman spectroscopy on solid-state and sol–gel prepared samples. The results confirm that this system does not show a solid solution type behavior but phase-separates into thermodynamically more stable sodium ordered monoclinic α-Na3MnTi(PO4)3 (space group C2) and other phases. In addition to sodium ordering, the anti-bonding character of the Mn–O bond as compared to Ti–O is suggested as another important factor governing the stability of Mn-based NASICONs. We believe that these results will not only clarify some important questions regarding the thermodynamic properties of NASICON frameworks but will also be helpful for a more general understanding of polyanionic systems.

Unified theory for light-induced halide segregation in mixed halide perovskites

Mixed halide perovskites that are thermodynamically stable in the dark demix under illumination. This is problematic for their application in solar cells. We present a unified thermodynamic theory for this light-induced halide segregation that is based on a free energy lowering of photocarriers funnelling to a nucleated phase with different halide composition and lower band gap than the parent phase. We apply the theory to a sequence of mixed iodine-bromine perovskites. The spinodals separating metastable and unstable regions in the composition-temperature phase diagrams only slightly change under illumination, while light-induced binodals separating stable and metastable regions appear signalling the nucleation of a low-band gap iodine-rich phase. We find that the threshold photocarrier density for halide segregation is governed by the band gap difference of the parent and iodine-rich phase. Partial replacement of organic cations by cesium reduces this difference and therefore has a stabilizing effect.

Existence of a quadruple point in a binary ferroelectric phase diagram

In experimentally measured temperature-composition ferroelectric phase diagrams of ${\mathrm{BaTiO}}_{3}$-based binary systems, a quadruple point where cubic (C), tetragonal (T), orthorhombic (O), and rhombohedral (R) phases converge has been frequently reported in previous work. More interestingly, the quadruple points are experimentally found to behave as a critical point with large enhancement in properties. However, it has remained a fundamental question as to whether a quadruple point in a binary ferroelectric system defies the thermodynamic phase rule and whether such a point necessarily goes critical. In this study, it is demonstrated by Landau theory that a C-T-O-R quadruple point in a binary ferroelectric system can only exist in the form of a unique type of critical point at which two first-order transition lines and two second-order ones meet, and such critical quadruple points do not defy the thermodynamic phase rule. It is further shown that at such a critical C-T-O-R quadruple point, the system exhibits infinitely large piezoelectric coefficients, which agrees with the high piezoelectricity observed at the C-T-O-R quadruple point in a number of ${\mathrm{BaTiO}}_{3}$-based binary ferroelectric systems and also helps to explain the large piezoelectricity obtained at the morphotropic phase boundaries of these quadruple point based systems.

Incommensurately modulated structures in Pb(Zr1−xSnx)O3 single crystals by x-ray diffraction

Pb(Zr1−xSnx)O3 single crystals were characterized using x-ray diffraction as a function of temperature and composition. The information on the structure of two intermediate phases, situated between antiferroelectric and paraelectric phases in the composition–temperature phase diagram, has been obtained. The lower-temperature intermediate phase is characterized by incommensurate displacive modulations in the Pb sublattice. The higher temperature intermediate phase is characterized by distortions of the oxygen sublattice primarily in the form of anti-phase tilts of the oxygen octahedra. The structured diffuse scattering was also observed in the paraelectric phase along Γ–M and M–R lines pointing to the high-temperature correlated disorder of Pb ions and octahedral rotations, respectively.

Mannitol Crystallization at Sub-Zero Temperatures: Time/Temperature-Resolved Synchrotron X-ray Diffraction Study and the Phase Diagram.

Mannitol, a common pharmaceutical ingredient, exhibits complex polymorphism even in simple binary mannitol/water mixtures, with four crystalline forms observed. In this investigation, time/temperature-resolved synchrotron X-ray diffraction measurements are performed during freezing and thawing of mannitol/water mixtures. Mannitol crystallization depends strongly on the cooling rate and is initiated during cooling, if the cooling rate is lower than the critical cooling rate; otherwise, mannitol remains amorphous during freezing and crystallizes during subsequent heating above -30 °C. A temperature-composition phase diagram is constructed, reflecting eutectic and peritectic points and lower-temperature equilibria involving mannitol hemihydrate, hexagonal ice, and β-mannitol. Comparison of the experimental data with the phase diagram reveals that the mannitol crystallization behavior does not follow the equilibrium but appears to obey the Ostwald crystallization rule. Novel insights on equilibrium and kinetics phase relationships in mannitol/water systems could lead to improved formulations and manufacturing processes for pharmaceuticals and biopharmaceuticals.