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Miscibility

Miscibility refers to the ability of two or more substances, typically liquids, to mix completely in all proportions at a given temperature and pressure, forming a homogeneous single-phase solution without separation into distinct layers.^[1] This property contrasts with immiscibility, where substances do not blend fully and instead form separate phases, such as oil and water.^[2] Miscibility is a fundamental concept in physical chemistry, underpinning the behavior of solutions and influencing processes ranging from everyday mixtures to industrial applications.^[1] The thermodynamic basis for miscibility lies in the Gibbs free energy of mixing, \Delta G_{mix} = \Delta H_{mix} - T\Delta S_{mix}, which must be negative for spontaneous mixing to occur.^[3] For ideal mixtures, the entropy of mixing (\Delta S_{mix}) is always positive due to increased disorder, but the enthalpy of mixing (\Delta H_{mix}) depends on intermolecular interactions.^[4] Key examples of miscible liquids include water and ethanol, where hydrogen bonding allows complete dissolution, and dichloromethane and hexane, facilitated by dipole-induced dipole interactions.^[1] In contrast, nonpolar octane and polar water are immiscible because their differing intermolecular forces—London dispersion versus hydrogen bonding—do not favor mixing.^[1] Factors influencing miscibility primarily revolve around the principle of "like dissolves like," where substances with similar polarity and intermolecular forces mix readily.^[1] Polarity is a dominant factor: polar solvents like water mix well with polar solutes, while nonpolar solvents such as hexane prefer nonpolar partners.^[5] Temperature often enhances miscibility by increasing molecular motion and weakening intermolecular attractions, though effects vary by system; for instance, higher temperatures can induce phase separation in some polymer blends via upper critical solution temperature (UCST) phenomena.^[6] Pressure plays a role in gas-liquid or high-pressure liquid systems, such as determining the minimum miscibility pressure in enhanced oil recovery processes.^[2] In practical contexts, miscibility is essential for solvent selection in chemical reactions, extraction processes in organic synthesis, and formulation of pharmaceuticals and cosmetics, where homogeneous mixtures ensure efficacy and stability.^[1] It also impacts engineering fields like petroleum recovery, where miscible gas flooding—using CO₂ at pressures of 2200–3200 psi—improves oil displacement efficiency by 15% or more in pore volume.^[2] Understanding miscibility aids in predicting phase behavior in complex systems, such as polymer blends and refrigerant-lubricant mixtures, optimizing material properties and process design.^[2]

Fundamentals

Definition

Miscibility refers to the property of two or more substances, typically liquids, to mix completely and form a homogeneous solution in all proportions at a given temperature and pressure.^[7] This phenomenon arises when the substances can intermingle at the molecular level without phase separation, resulting in a uniform mixture.^[8] Complete miscibility occurs when the substances are soluble in each other across the entire composition range, whereas partial miscibility is limited to certain ratios, often exhibiting phase separation beyond those limits, such as below an upper consolute temperature or above a lower consolute temperature.^[9] The observation of miscibility dates back to 19th-century chemistry, where early studies of solution behavior contributed to the foundational understanding of liquid interactions.^[10] For instance, ethanol and water are miscible due to their ability to form hydrogen bonds, allowing polar molecules of ethanol to interact favorably with water's polar structure.^[8] In contrast, oil and water are immiscible because oil's nonpolar hydrocarbon chains lack the necessary interactions with water's polar molecules, leading to phase separation driven by differing molecular affinities.^[8] Although miscibility is most commonly associated with liquid systems, the concept extends to other phases: gases are generally completely miscible unless chemical reactions intervene, solids exhibit miscibility through solid solutions like in alloys, and supercritical fluids demonstrate enhanced miscibility due to their unique solvent properties.^[11] This property ultimately depends on thermodynamic favorability, where the free energy of mixing determines whether a homogeneous phase is stable.^[12]

Thermodynamic Basis

Miscibility in binary systems is determined by the thermodynamics of mixing, specifically the condition that the Gibbs free energy of mixing, \Delta G_{\text{mix}}, must be negative for the process to be spontaneous at constant temperature and pressure. This criterion arises from the second law of thermodynamics, where the system evolves toward a state of minimum free energy, favoring the formation of a homogeneous mixture over phase separation.^[4] The Gibbs free energy of mixing is expressed by the fundamental equation

\Delta G_{\text{mix}} = \Delta H_{\text{mix}} - T \Delta S_{\text{mix}}

where \Delta H_{\text{mix}} is the enthalpy of mixing, \Delta S_{\text{mix}} is the entropy of mixing, and T is the absolute temperature. Miscibility occurs when \Delta G_{\text{mix}} < 0, which typically results from a favorable balance between the enthalpic and entropic contributions, with the entropic term often promoting mixing due to increased disorder. In cases where \Delta G_{\text{mix}} > 0, the components remain immiscible, leading to phase separation.^[4] The enthalpic term \Delta H_{\text{mix}} is influenced by intermolecular forces, such as hydrogen bonding, dipole-dipole interactions, and van der Waals forces, which govern the interactions between like molecules (A-A, B-B) versus unlike molecules (A-B) in the mixture. If A-B interactions are stronger than the average of A-A and B-B, \Delta H_{\text{mix}} is negative, enhancing miscibility; conversely, weaker A-B interactions lead to positive \Delta H_{\text{mix}}, opposing mixing. These forces determine the overall favorability of the mixed state relative to the pure components.^[13] Solutions are classified as ideal or non-ideal based on their deviation from this thermodynamic behavior. Ideal solutions assume zero enthalpy of mixing (\Delta H_{\text{mix}} = 0) and obey Raoult's law, where the partial vapor pressure of each component i is P_i = x_i P_i^\circ, with x_i as the mole fraction and P_i^\circ as the pure component vapor pressure; this implies random mixing driven purely by entropy. Non-ideal solutions exhibit \Delta H_{\text{mix}} \neq 0, leading to positive or negative deviations from Raoult's law.^[14] In partially miscible systems, where complete mixing does not occur across all compositions, phase diagrams illustrate the thermodynamic boundaries through binodal curves. These curves delineate the region of single-phase stability from two-phase coexistence; inside the binodal curve (two-phase region), the homogeneous mixture is thermodynamically unstable and separates into two phases in equilibrium, as the free energy of this state exceeds that of the coexisting phases determined by the common tangent construction on the Gibbs free energy-composition curve, while outside it, miscibility prevails. The shape of the binodal reflects temperature-dependent changes in \Delta G_{\text{mix}}, often showing an upper or lower consolute temperature where miscibility gaps close.^[15]^[16]

Influencing Factors

Enthalpy Contributions

The enthalpy of mixing, denoted as ΔH_mix, represents the change in enthalpy when two or more components are combined to form a homogeneous mixture at constant temperature and pressure; it is calculated as the difference between the enthalpy of the mixture and the sum of the enthalpies of the pure components.^[17] A negative ΔH_mix indicates exothermic mixing, which provides a thermodynamic driving force that favors miscibility by releasing energy through attractive intermolecular interactions.^[18] Conversely, a positive ΔH_mix signifies endothermic mixing, which opposes miscibility by requiring energy input to overcome repulsive or mismatched interactions between unlike molecules.^[18] In regular solution theory, the enthalpy of mixing is modeled as arising solely from pairwise molecular interactions, assuming random mixing and no volume change upon mixing. The equation for ΔH_mix in this framework is given by

\Delta H_\text{mix} = B \phi_1 \phi_2 V

where B is the interaction parameter (often B = (\delta_1 - \delta_2)^2, with \delta_1 and \delta_2 being the Hildebrand solubility parameters of the components), \phi_1 and \phi_2 are the volume fractions of the two components, and V is the total volume of the mixture.^[19] This formulation highlights how differences in cohesive energy densities between components lead to enthalpic contributions that can either promote or hinder complete miscibility, with positive B values indicating limited solubility.^[19] Key factors influencing ΔH_mix include the matching of molecular polarities, encapsulated in the "like dissolves like" principle, which posits that miscibility is enhanced when solute-solvent interactions are energetically comparable to self-interactions within each pure component.^[12] For instance, attempts to mix nonpolar solvents like hydrocarbons with polar ones like water encounter significant enthalpic barriers, resulting in positive ΔH_mix due to the unfavorable disruption of hydrogen bonds in water and weak van der Waals forces in the hydrocarbon.^[12] Unlike entropy effects, which vary strongly with temperature through the T\Delta S term in the Gibbs free energy, ΔH_mix is typically considered independent of temperature in regular solution approximations, allowing it to dominate miscibility behavior across moderate temperature ranges.^[20] In surfactant systems, the critical micelle concentration (CMC) reflects enthalpic stability by marking the point where amphiphilic molecules aggregate to minimize unfavorable hydrophobic interactions with the aqueous environment, effectively enhancing local miscibility within micellar structures.^[21] The enthalpic component of micellization free energy, derived from tail-solvent repulsion, determines the sharpness of the transition and the overall solubility limit before phase separation occurs.^[21]

Entropy Contributions

The entropy of mixing, \Delta S_{\text{mix}}, primarily arises from configurational contributions, reflecting the increased number of possible molecular arrangements when distinct components intermix, thereby enhancing the system's disorder.^[22] This entropic gain drives miscibility in scenarios where enthalpic interactions are neutral or unfavorable, as the greater randomness of the mixed state outweighs the ordered pure components.^[22] For ideal binary solutions, the entropy of mixing is given by the expression

\Delta S_{\text{mix}} = -R (x_1 \ln x_1 + x_2 \ln x_2),

where R is the gas constant and x_1, x_2 are the mole fractions of the components, assuming random mixing without volume changes or specific interactions.^[22] This formula, derived from statistical mechanics, quantifies the positive \Delta S_{\text{mix}} that favors spontaneous mixing at constant temperature and pressure.^[22] In polymer systems, the Flory-Huggins theory modifies this ideal entropy to account for the asymmetry in molecular sizes and chain stiffness, using volume fractions instead of mole fractions for the configurational term; the interaction parameter [\chi](/page/Chi) incorporates entropic penalties arising from reduced conformational freedom in the mixed state.^[23] These adjustments highlight how polymeric entropy contributions can limit miscibility compared to small-molecule solvents, emphasizing the role of molecular architecture in entropic effects.^[23] Temperature plays a pivotal role in amplifying entropic favorability, as the -T \Delta S_{\text{mix}} term in the Gibbs free energy becomes more negative with rising T, often promoting miscibility at elevated temperatures even if lower-temperature mixing is enthalpically hindered.^[24] In non-ideal mixtures, excess entropy arises from deviations such as volume contraction or expansion upon mixing, which alter the available configurational space beyond ideal random distribution; positive excess entropy enhances miscibility by further increasing disorder, while negative values impose restrictions.^[25] These non-ideal entropic effects, often linked to specific intermolecular associations, provide critical insights into phase behavior in complex systems.^[26]

Examples in Chemical Systems

Organic Liquids

Miscibility among organic liquids is primarily governed by the similarity in their molecular structures and intermolecular forces, allowing for complete mixing in various proportions. For instance, benzene and toluene, both nonpolar aromatic hydrocarbons, are fully miscible at room temperature due to their comparable van der Waals interactions and molecular sizes.^[27] Similarly, acetone, a polar aprotic solvent, and ethanol, a polar protic solvent, exhibit complete miscibility, facilitated by dipole-dipole interactions and hydrogen bonding between the carbonyl group of acetone and the hydroxyl group of ethanol.^[27] These examples illustrate how structural similarities, such as the presence of aromatic rings in nonpolar pairs or polar functional groups in protic/aprotic pairs, promote homogeneous solutions without phase separation. Key factors influencing miscibility in organic liquids include the nature of functional groups and quantitative measures like Hildebrand solubility parameters, which estimate cohesive energy densities. Hydrocarbon chains, being nonpolar, enhance miscibility among similar apolar molecules but reduce compatibility with polar organics; conversely, functional groups like hydroxyl (-OH) or carbonyl (C=O) increase polarity, favoring interactions with other polar species through hydrogen bonding or dipole moments. The Hildebrand parameter (δ) provides a predictive tool: liquids with δ values differing by less than about 2 MPa^{1/2} are typically miscible, as seen in benzene (δ = 18.5 MPa^{1/2}) and toluene (δ = 18.2 MPa^{1/2}), where the close values reflect their nonpolar cohesion. For polar pairs like acetone (δ = 19.9 MPa^{1/2}) and ethanol (δ = 26.5 MPa^{1/2}), the larger difference is offset by specific interactions, underscoring Hildebrand's utility primarily for nonpolar systems.^[28] Partial miscibility occurs in certain organic pairs where temperature modulates solubility, leading to upper critical solution temperature (UCST) or lower critical solution temperature (LCST) behaviors. In systems exhibiting UCST, the liquids are miscible above a critical temperature but form two phases below it due to dominant unfavorable enthalpy at lower temperatures. A representative example is the aniline-hexane system, which displays an UCST of approximately 60 °C, resulting in partial miscibility at ambient conditions where specific compositions separate into coexisting layers.^[29] Such temperature-dependent phase behavior is less common in simple organic liquids than in polymer solutions but highlights the role of molecular interactions in limiting complete mixing. The impact of chain length on miscibility is evident in homologous series like alkanes, where increasing carbon chain length progressively reduces solubility in polar solvents such as water. Short-chain alkanes like methane or ethane show marginal solubility due to minimal hydrophobic surface area, but as chain length grows (e.g., from pentane to hexadecane), the nonpolar hydrocarbon dominates, rendering them essentially immiscible and forming distinct layers. This trend arises from the increasing dominance of London dispersion forces over any potential polar interactions, with solubility decreasing exponentially with carbon number. In organic chemistry extraction processes, miscibility patterns of organic liquids are essential for designing efficient separations, as selecting solvents with appropriate polarity ensures the target solute partitions favorably between phases while avoiding unwanted mixing. For example, nonpolar organics like toluene are chosen for extracting nonpolar compounds from aqueous media due to their immiscibility with water yet miscibility with similar solutes, enabling clean phase splits and high recovery yields. This principle underpins techniques like liquid-liquid extraction, where solvent compatibility directly impacts purification efficiency in synthesizing pharmaceuticals and fine chemicals.

Inorganic Solutions

Inorganic solutions primarily involve the mixing of ionic compounds or polar inorganic liquids with solvents like water, where miscibility is driven by strong ion-dipole interactions and solvation effects. Many common salts, such as sodium chloride (NaCl), exhibit high miscibility in water, dissolving to form homogeneous solutions up to saturation limits around 36 g/100 mL at 25°C, due to the exothermic lattice energy release being offset by favorable hydration energies. Similarly, concentrated acids like sulfuric acid (H₂SO₄) and hydrochloric acid (HCl) are completely miscible with water in all proportions, releasing significant heat upon mixing as a result of strong hydrogen bonding and dissociation.^[30]^[31]^[32] Immiscibility arises in systems lacking compatible polarities, such as liquid mercury with water, where mercury's metallic bonding and low polarity prevent uniform mixing, forming distinct layers with negligible solubility (approximately 60 μg/L at 25°C).^[33] Inorganic salts like NaCl are also immiscible in nonpolar solvents such as ethanol or hexane, as the absence of polar interactions hinders ion solvation, leading to precipitation or phase separation. This contrast highlights the role of polarity matching in determining miscibility outcomes. Central to aqueous miscibility of ions is the formation of hydration shells, where water molecules orient their oxygen atoms toward cations and hydrogen atoms toward anions, stabilizing the dissolved state through electrostatic attractions. The solvation energy, often on the order of -400 to -1000 kJ/mol for common ions like Na⁺ and Cl⁻, overcomes the lattice energy of the solid, enabling dissolution; for instance, in NaCl(aq), the first hydration shell typically coordinates 6 water molecules around each ion. These shells reduce ion mobility and influence solution properties, with partial dehydration occurring at interfaces.^[34]^[35] In concentrated inorganic solutions, such as brines, miscibility limits lead to phase behaviors like eutectic mixtures, where specific salt-water compositions exhibit the lowest melting point (e.g., -21.1°C for NaCl-H₂O), allowing separation of ice and concentrated salt via processes like eutectic freeze crystallization. Beyond saturation, further addition of salts can induce phase separation into a concentrated aqueous phase and solid precipitates, as seen in multi-component brines where ion pairing and reduced water activity promote immiscibility.^[36]^[37] Gas-liquid miscibility in inorganic systems, particularly non-reactive gases in water, is limited and described by Henry's law, which states that the solubility concentration C is proportional to the partial pressure P of the gas above the liquid: C = k_H \cdot P, where k_H is the Henry's law constant. For oxygen (O₂) in water at 25°C and 1 atm, solubility is low at approximately 8 mg/L, reflecting weak van der Waals interactions and reflecting the upper limit for complete miscibility under ambient conditions.^[38]

Applications in Materials

Metal Alloys

In metal alloys, miscibility refers to the ability of two or more metallic elements to form homogeneous solid solutions, where solute atoms substitute for or occupy positions within the solvent's crystal lattice without phase separation. Complete miscibility occurs when elements are fully soluble across all compositions in the solid state, as exemplified by the copper-nickel (Cu-Ni) system. In Cu-Ni alloys, both elements share a face-centered cubic (FCC) crystal structure and have similar atomic radii (approximately 128 pm for Cu and 125 pm for Ni), electronegativities (1.90 for Cu and 1.91 for Ni), and valences, enabling the formation of a single α-phase solid solution over the entire concentration range from pure Cu to pure Ni.^[39]^[40] This complete solid solubility is thermodynamically favored due to minimal strain energy and favorable mixing enthalpy, resulting in continuous solidus and liquidus lines in the phase diagram without intermediate phases.^[41] Partial miscibility arises when solubility is limited, governed by the Hume-Rothery rules, which predict the extent of solid solution formation based on atomic compatibility. These rules stipulate that for substantial substitutional solubility, the solute and solvent must exhibit: (1) atomic size difference no greater than 15% to minimize lattice strain; (2) identical crystal structures (e.g., both FCC); (3) similar electronegativities to ensure comparable electron affinities and prevent compound formation; and (4) the same valence electron count to maintain electronic balance.^[42]^[43] Violations of these criteria, such as significant size mismatch or electronegativity differences, restrict solubility to narrow composition ranges, leading to multiphase structures. For instance, in systems like Al-Cu, limited solubility (up to about 5.7 wt% Cu in Al at 548°C) follows these rules, promoting precipitation hardening rather than full solid solutions.^[42] Phase diagrams illustrate miscibility behaviors in alloys, particularly through eutectic and peritectic reactions in partially immiscible systems. In the lead-tin (Pb-Sn) system, the metals exhibit negligible mutual solid solubility (less than 0.1 wt% at room temperature) due to differences in atomic size (Pb: 175 pm, Sn: 158 pm in white tin form) and crystal structures (Pb: FCC, Sn: body-centered tetragonal), resulting in a classic eutectic diagram. The eutectic point occurs at 61.9 wt% Sn and 183°C, where the liquid decomposes into nearly pure Pb-rich α and Sn-rich β phases, facilitating low-melting solders but limiting homogeneous alloying.^[44] Peritectic points, as seen in systems like Fe-Ni, mark boundaries where limited miscibility leads to reactions forming new phases, influencing casting and heat treatment processes.^[45] Solid-state miscibility in alloys is enabled by atomic diffusion mechanisms that allow solute atoms to redistribute within the lattice. Vacancy diffusion predominates in substitutional alloys, where atoms exchange positions with lattice vacancies, requiring thermal activation to create and migrate vacancies (activation energy typically 1-3 eV).^[46] This mechanism homogenizes compositions during annealing, as seen in Cu-Ni, by countering concentration gradients through Fickian diffusion. Interstitial diffusion, faster due to lower activation barriers (often <1 eV), occurs when small solute atoms (e.g., carbon in iron) move through octahedral or tetrahedral voids without displacing host atoms, enhancing miscibility in systems like austenitic steels.^[47] Both processes are exponentially temperature-dependent, with interstitial rates exceeding vacancy rates by factors of 10^4-10^6 at homologous temperatures, enabling equilibrium phase formation.^[48] A key application of solid solution miscibility is strengthening in metallurgy, where solute atoms distort the lattice to impede dislocation motion, increasing yield strength without forming second phases. In solid solution strengthening, misfit strains from atomic size differences (e.g., 10-20% mismatch) create elastic barriers, as described by Fleischer's model, which predicts critical resolved shear stress proportional to solute concentration and misfit parameter.^[49] Labusch's statistical theory extends this for concentrated solutions, emphasizing collective solute-dislocation interactions and explaining observed concentration exponents near 2/3 in FCC alloys like Cu-Zn.^[50] This mechanism is widely applied in alloys such as austenitic stainless steels (e.g., Ni and Cr in Fe) and aluminum alloys (e.g., Mg in Al), enhancing tensile strength by 20-50% while maintaining ductility for structural components in aerospace and automotive industries.^[51]

Polymers and Solids

Miscibility in polymer systems is relatively rare due to the entropic penalties associated with mixing long, rigid chains, but certain blends achieve compatibility through favorable specific interactions. A well-known example is the polystyrene (PS) and poly(vinyl methyl ether) (PVME) blend, which remains miscible across a wide composition range owing to intermolecular interactions such as hydrogen bonding between the phenyl groups of PS and the ether oxygen in PVME.^[52] These interactions lower the enthalpic contribution to the free energy of mixing, enabling a single-phase morphology confirmed by techniques like differential scanning calorimetry showing a composition-dependent glass transition temperature. Such miscible blends exhibit enhanced mechanical properties compared to their individual components, making them useful in applications like adhesives and coatings. The thermodynamic criterion for miscibility in polymer blends is often evaluated using the Flory-Huggins interaction parameter χ, where values of χ < 0.5 indicate miscibility for high-molecular-weight polymers by favoring a negative free energy of mixing.^[53] This parameter, derived from lattice models, quantifies the balance between enthalpic repulsion and entropic mixing in solid polymer matrices. In contrast, most polymer pairs are immiscible because χ > 0.5, leading to phase separation into distinct domains that can coarsen over time. For instance, poly(vinyl chloride) (PVC) blends with nitrile butadiene rubber (NBR) demonstrate this immiscibility, resulting in a phase-separated microstructure with co-continuous domains that influence toughness and barrier properties.^[54] In non-polymeric solid systems, miscibility is more readily achieved through atomic substitution or doping, particularly in semiconductors where complete solid solutions form. Silicon-germanium (Si-Ge) alloys exemplify this, exhibiting full miscibility across all compositions due to their similar lattice parameters and bonding characteristics, allowing uniform incorporation of Ge into the Si lattice.^[55] This solid-solid miscibility enables bandgap engineering for devices like high-speed transistors. Factors such as molecular weight in polymers narrow the miscibility window—higher weights increase the critical χ threshold—while processing conditions like elevated temperatures and prolonged mixing times promote better homogeneity by reducing viscosity and enhancing diffusion.^[56] In PVC/NBR blends, for example, optimizing shear rate and temperature during extrusion minimizes domain size, improving blend performance despite underlying immiscibility.^[57]

Determination and Prediction

Experimental Methods

Experimental methods for assessing miscibility involve direct observation and measurement of physical properties in laboratory settings to determine whether substances form a single homogeneous phase or separate into distinct layers. Visual and turbidity-based techniques, such as cloud point determination, are commonly employed for partially miscible liquid systems. In this approach, a mixture is heated or cooled while monitoring for the onset of opacity, indicating phase separation; the temperature at which turbidity appears defines the cloud point, providing insight into the boundaries of the miscible region.^[58] This method is particularly useful for binary liquid mixtures where temperature influences solubility limits, allowing construction of phase diagrams through repeated measurements at varying compositions.^[59] Refractometry offers a quantitative means to evaluate miscibility by tracking changes in the refractive index of mixtures. For fully miscible liquids, the refractive index typically varies linearly or follows a predictable curve with composition, while deviations signal specific interactions or limited solubility.^[60] Instruments like Abbe refractometers measure these indices precisely, enabling calculation of excess properties that confirm homogeneity across the composition range.^[61] Similarly, densitometry assesses volume changes upon mixing by measuring mixture densities and deriving excess molar volumes. Positive or negative excess volumes indicate expansive or contractive behavior, respectively, which correlates with miscibility; for example, ideal miscible pairs show near-zero excess volumes.^[62] Thermal analysis via differential scanning calorimetry (DSC) assesses miscibility, particularly in polymer systems, by examining glass transition temperatures (Tg). A single Tg indicates a homogeneous miscible blend, while multiple Tgs suggest phase separation in partially miscible systems. In cases of partial miscibility, DSC traces may show peaks corresponding to phase transitions, validating miscibility limits against thermodynamic predictions.^[63] Spectroscopic techniques, including nuclear magnetic resonance (NMR) and infrared (IR) spectroscopy, further distinguish homogeneous from heterogeneous states. NMR assesses molecular environments through chemical shift uniformity and line widths, where broadening suggests phase separation in immiscible mixtures. IR spectroscopy detects intermolecular interactions via vibrational band shifts; in miscible liquids, consistent spectra across compositions confirm a single phase, while splitting indicates heterogeneity.^[64] Historical methods from the early 20th century, such as shake-flask tests, laid the foundation for miscibility assessment. These involve vigorously agitating equal volumes of two liquids in a flask, allowing settling, and visually inspecting for layer formation; a single layer denotes miscibility, while separation indicates immiscibility. Though qualitative, these tests remain a simple benchmark for initial screening before advanced quantification.

Computational Models

Computational models provide a powerful framework for predicting miscibility in mixtures by simulating molecular interactions and thermodynamic properties at the atomic or group level, enabling predictions without conducting physical experiments. These approaches range from classical simulations to quantum mechanical calculations, often relying on parameters derived from experimental data or first principles to estimate mixing energies, activity coefficients, and phase behaviors. Such models are particularly valuable for screening large chemical spaces in industries like pharmaceuticals and materials design, where empirical testing is resource-intensive.^[65] Molecular dynamics (MD) simulations model miscibility by computing the free energy of mixing through trajectories of molecular motions under defined force fields, which approximate intermolecular potentials to capture van der Waals, electrostatic, and hydrogen-bonding interactions. Force fields such as OPLS-AA or CHARMM are commonly employed to parameterize these potentials, allowing calculation of solubility parameters or interfacial tensions that indicate miscibility limits, as demonstrated in predictions of minimum miscibility pressure for CO₂-hydrocarbon systems where MD-derived interfacial tension decreases to zero at the miscibility point. For instance, in polymer-drug formulations, MD has been used to predict miscibility by evaluating the Flory-Huggins interaction parameter χ from simulated radial distribution functions, achieving agreement with experimental solubility limits within 10-20% for small organic molecules. These simulations typically run on the nanosecond timescale for systems of thousands of atoms, providing insights into entropy-driven mixing in liquids.^[66]^[67]^[65] The Conductor-like Screening Model for Real Solvents (COSMO-RS) predicts miscibility in liquid mixtures by combining quantum mechanical surface charge densities (σ-profiles) with statistical thermodynamics to compute activity coefficients and solubility parameters, particularly effective for polar and hydrogen-bonding systems. Developed from COSMO solvation theory, it estimates the chemical potential μ_i(T,x) for components in mixtures, identifying miscibility gaps when the Gibbs free energy of mixing becomes positive, as seen in water-ether binaries where COSMO-RS accurately reproduces liquid-liquid equilibrium compositions with average deviations of 5-15% in phase fractions. For organic liquids, COSMO-RS derives Hansen solubility parameters from σ-moments, enabling predictions of mutual solubilities; for example, it forecasts the upper critical solution temperature in alcohol-hydrocarbon blends by integrating dispersion, electrostatic, and hydrogen-bonding contributions to the excess free energy. This model excels in a priori predictions without fitted parameters beyond universal σ-potentials, making it suitable for screening solvents in formulation design.^[68]^[69]^[70] The UNIversal Functional Activity Coefficient (UNIFAC) group contribution method forecasts miscibility in organic mixtures by decomposing molecules into functional groups and estimating activity coefficients γ_i from group-group interaction parameters, which determine phase stability via the Gibbs excess free energy G^E = RT Σ x_i ln γ_i. Seminal work established UNIFAC parameters from regression of vapor-liquid equilibrium data, allowing prediction of liquid-liquid immiscibility when γ_i exceeds critical values leading to binodal curves; for binary organics like alcohols and alkanes, it predicts miscibility limits with root-mean-square errors in composition below 0.05 mole fraction. In applications to partially miscible systems, modified UNIFAC variants incorporate temperature-dependent interactions to model azeotrope formation and solubility edges, as in ether-water mixtures where group contributions for -OH and -CH₂- accurately delineate the two-phase region. This semi-empirical approach is computationally efficient for multicomponent predictions, relying on a database of over 100 group interactions validated across thousands of systems.^[71]^[72]^[73] Ab initio calculations, grounded in quantum mechanics, predict miscibility in small systems by directly computing interaction energies from the Schrödinger equation without empirical parameters, often using density functional theory (DFT) or coupled-cluster methods to evaluate dimer binding energies that inform lattice models for phase separation. For binary mixtures like hydrogen-helium solids, ab initio calculations yield positive mixing enthalpies, revealing immiscibility due to weak van der Waals forces. In molecular systems, supermolecular approaches compute pairwise interaction energies E_int = E_AB - E_A - E_B with basis set superposition error corrections, enabling derivation of Flory-Huggins χ parameters for predicting upper consolute temperatures in small organics with errors under 5 K. These methods are limited to clusters of 10-50 atoms due to O(N^4) scaling but provide benchmark data for force fields in larger simulations.^[74]^[75] Despite their strengths, computational models face limitations in accuracy for complex polymers and metals, where MD force fields often underestimate long-range correlations in entangled chains, leading to 20-50% errors in χ for high-molecular-weight systems, and ab initio methods are infeasible beyond minimal clusters due to exponential cost. UNIFAC and COSMO-RS struggle with ionic or metallic bonding, overpredicting miscibility gaps in polymer blends by ignoring conformational entropy, with deviations up to 30% in critical compositions for polydisperse materials. Recent post-2020 advances in machine learning address these by training neural networks on MD-generated datasets to predict polymer-solvent miscibility, achieving R² > 0.9 for χ in over 1,000 pairs via multitask graph convolutional models that incorporate molecular fingerprints and Flory-Huggins theory. As of 2025, advanced XGBoost models have enhanced predictions of minimum miscibility pressure for impure CO₂ in enhanced oil recovery, demonstrating high reliability.^[76]^[77]^[68]^[78] Hybrid ML frameworks integrate quantum descriptors with empirical data, enhancing predictions for metallic alloys by emulating DFT energies, reducing computational time by orders of magnitude while maintaining sub-10% error in solubility limits.

References

[1]
7.9: Miscibility - Chemistry LibreTexts
Oct 4, 2022 · Two liquids that appear to mix completely together are said to be miscible. Water and ethanol are one example of a pair of miscible liquids, ...
[2]
Miscibility - an overview | ScienceDirect Topics
Miscibility refers to the ability of two fluids to mix completely in all proportions without separating into different phases.
[3]
Miscibility and Thermodynamics of Mixing of Different Models of ...
The thermodynamic condition that two liquid compounds, A and B, are fully miscible with each other can be formulated as the inequality (42) (1)holds in the ...
[4]
Thermodynamics of Mixing - Chemistry LibreTexts
Jan 29, 2023 · This module will discuss the effect that mixing has on a solution's Gibbs energy, enthalpy, and entropy, with a specific focus on the mixing of two gases.
[5]
¿What is a Miscibility and how does it work? - Area Cooling Solutions
Jul 24, 2023 · Miscibility is the property of two substances to mix completely. Thus, it can also be defined as the ability of two substances to form a homogeneous solution.
[6]
Solubility and Factors Affecting Solubility - Chemistry LibreTexts
Jan 29, 2023 · An increase in pressure and an increase in temperature in this reaction results in greater solubility. An increase in pressure results in more ...
[7]
Illustrated Glossary of Organic Chemistry - Miscible; immiscible
Miscible: Two liquids that combine in any ratio to form a homogeneous solution. Liquids which have little or no mutual solubility are immiscible.
[8]
[PDF] Miscibility of Liquids
The tendency of some liquids to "mix" that is, to form homogeneous mixtures is referred to as their miscibility. Two liquids are miscible if they are each.
[9]
12.3 Types of Solutions and Solubility – Chemistry Fundamentals
The distinction between immiscibility and miscibility is really one of degrees, so that miscible liquids are of infinite mutual solubility, while liquids ...
[10]
How the Theory of Solutions Arose - chemteam.info
Van't Hoff was well launched on the theory of ideal solutions, with its fundamental equation the exact equivalent of the ideal gas law, pv = RT.
[11]
Lesson 8: The Chemistry of Solutions
A few substances are infinitely soluble, or miscible, which means that they can be mixed together in any proportion. Ethyl alcohol and water are an example of ...Missing: definition | Show results with:definition
[12]
A first-principles theory for predicting liquid miscibility and mixture ...
Feb 12, 2021 · The miscibility between two liquids is determined by the mismatch in both the dielectric constant and the molar volume. For two liquids A and B ...
[13]
[PDF] Chapter 5: The Thermodynamic Description of Mixtures - Chemistry
Copper and nickel are completely miscible in all proportions and form true alloys, not merely physical admixtures. The line above which the mixture is a liquid ...
[14]
Intermolecular Forces in Mixtures And Solutions
Jan 29, 2023 · Some forces that interact within pure liquids are also present during mixtures and solutions. Forces such as Cohesive as well as Adhesive forces still apply to ...
[15]
Raoult's Law - Chemistry LibreTexts
Jan 29, 2023 · Ideal solutions satisfy Raoult's Law. · In an ideal solution, it takes exactly the same amount of energy for a solvent molecule to break away ...
[16]
13.2: Phase Diagrams- Binary Systems - Chemistry LibreTexts
Apr 12, 2022 · When two liquids that are partially miscible are combined in certain proportions, phase separation occurs (Sec. 11.1.6). Two liquid phases in ...
[17]
[PDF] Thermodynamics of mixing
Ideal mixtures are always homogeneous as a result of the mixing entropy always being positive. Figure 4.3 shows the mixing free energy of an ideal regular ...
[18]
Enthalpy of Mixing - an overview | ScienceDirect Topics
Enthalpy of mixing (Δ H mix) is defined as the difference between the sum of the enthalpies of the solute and solvent and the enthalpy of the mixture, ...
[19]
[PDF] Chapter 4 Solution Theory
Hence, the enthalpy of mixing at constant T and P tells us how much heat will be released or absorbed when combining chemical species. Before, further ...
[20]
Physics-Based Solubility Prediction for Organic Molecules
... an approximation of the condensed phase intermolecular forces, which can be related to the enthalpy of mixing as. ΔHmix=Vt(δ1−δ2)2ϕ1ϕ2. (22). where V ...
[21]
[PDF] Lecture 3: Models of Solutions
In the regular solution model, the enthalpy of mixing is obtained by counting the different kinds of near neighbour bonds when the atoms are mixed at random; ...
[22]
Surfactant Self-Assembling and Critical Micelle Concentration
May 6, 2020 · Critical micelle concentration (CMC) is the main chemical–physical parameter to be determined for pure surfactants for their ...Missing: miscibility | Show results with:miscibility
[23]
Entropy of Mixing - Chemistry LibreTexts
Apr 1, 2023 · Multiplied by the negative in the front of the equation gives a positive quantity. Equation 1 applies to both ideal solutions and ideal gases.Missing: source | Show results with:source
[24]
Flory-Huggins Theory - an overview | ScienceDirect Topics
Flory–Huggins theory is defined as a mean-field, lattice model theory that explains the change in Gibbs free energy upon mixing two dissimilar polymers, ...
[25]
Entropy of mixing – Knowledge and References - Taylor & Francis
With increasing temperature, the term –TΔSmix becomes more negative and might counteract the positive enthalpy of formation in stabilising a phase with ...
[26]
Thermodynamics of solid solutions - DoITPoMS
The excess entropy is called the entropy of mixing (ΔSmix), and is mainly configurational in origin (i.e. it is associated with the large number of ...
[27]
Thermodynamics of Mixing - UMD Geology - University of Maryland
The excess vibrational (or in liquids and gases - rotational or translational) entropies may be positive or negative. Highly ordered liquid solutions may be " ...<|control11|><|separator|>
[28]
[PDF] Solvent Miscibility Table
Solvent Miscibility Table acetic acid acetone acetonitrile benzene n -butanol butyl acetate carbon tetrachloride chloroform cyclohexane. 1,2-dichloroethane.
[29]
Surface Tension, Hansen Solubility Parameters, Molar Volume ...
This table highlights the surface tension and Hansen Solubility Parameters of a variety of solvents and other liquids. An analogous table, which also ...
[30]
prepare it once, use it many times. V. liquid-liquid phase transition ...
After the weak complex is formed, the increase in temperature results in complex disruption leading to partial miscibility. ... UCST or LCST or both, are shown.
[31]
[PDF] Solubility of ionic solids in water | Notes on General Chemistry
Oct 18, 2007 · The solubility of ionic solids in water depends on two things: (1) the energy change, DEdissolve, that occurs when the ionic solid goes into ...Missing: miscibility | Show results with:miscibility
[32]
Fate and toxicity of spilled chemicals in groundwater and soil ... - NIH
Sulfuric acid is totally miscible in water. When water is present in the ... The acute toxicity of six accident preparedness substances (sulfuric acid, nitric ...
[33]
Hydrochloric acid | 7647-01-0 - ChemicalBook
Sep 25, 2025 · It is miscible with water and with alcohol. Concentrations of hydrochloric acid are expressed in percent by weight, or may be expressed in ...
[34]
Solvation Structure and Energetics of Single Ions at the Aqueous ...
Similar partial dehydration occurs in the first solvation shell; the probability of water hydrating the ion in the direction of the vapor phase is reduced. ...Missing: miscibility | Show results with:miscibility
[35]
[PDF] Thermodynamics of Solvation of Ions
The thermodynamics of ion solvation involves electrostatic effects, solvent immobilization, electrostriction, dielectric saturation, and a cavity formation in ...
[36]
Eutectic freeze crystallization simultaneous formation and ...
Eutectic freeze crystallization (EFC) separates aqueous inorganic solutions into pure water and pure salt. By operating at the eutectic point, ...Missing: mixtures | Show results with:mixtures
[37]
Phase Equilibria and Phase Separation of the Aqueous Solution ...
In addition, the most important physical and chemical processes occurring in brines are evaporation, concentration, crystallization, precipitation, dissolution, ...
[38]
The solubility of oxygen in water and saline solutions - Frontiers
Apr 13, 2023 · Henry's law constant (KO2(aq)) using the partial pressure of O2 in the gas phase describes the equilibrium condition for dissolution reaction of ...
[39]
complete solubility of components in solid state (solid solution)
Jun 13, 2018 · The copper/nickel alloy system exhibits such perfect miscibility over the entire concentration range (complete solid solution series). In the ...
[40]
Introduction - DoITPoMS
Both metals are completely soluble in each other. The α phase is a substitutional solid solution. This occurs because both the Cu and Ni form face centred cubic ...
[41]
Copper-Nickel Alloys: Properties, Processing, Applications
The two elements are closely related and are completely miscible in both the liquid and solid state. Cu-Ni alloys crystallise over the whole concentration range ...General information on Cu-Ni... · Cu-Ni alloys in EN standards · Properties · Melting
[42]
[PDF] 8 - 1 CHAPTER 8 METALS AND ALLOYS 8.1 Types of Alloying 8.2 ...
8.2 Hume-Rothery Rules for Alloying. It is not possible for every combination of materials to form an alloy. The following rules are for maximum solubility ...Missing: limits | Show results with:limits
[43]
[PDF] CHAPTER 4: IMPERFECTIONS IN SOLIDS
Atomic Sizes: should not differ by more than 14%;. 2. Crystal structures: same for “pure” metals;. 3. Electronegativity: The atoms should have similar ...
[44]
[PDF] Chapter 9: Phase Diagrams
Note that Pb-Sn materials were widely used as solders, since the melting temperature (183°C) of the eutectic is considerably below that of pure Sn (232.0°C) and ...<|separator|>
[45]
Binary Phase Diagrams - Learn Metallurgy
Examples : Pb-Sn, Cu-Ag, Pb-Sb, Cd-Zn, Sn-Bi. phase diagram of Pb-Sn Alloy. The above figure shows a eutectic phase diagram of Pb - Sn alloy. The solid phase ...
[46]
[PDF] chapter 5: diffusion in solids
•Temperature - diffusion rate increases very rapidly with increasing temperature. •Diffusion mechanism - interstitial is usually faster than vacancy.
[47]
[PDF] Chapter 5: Diffusion in Solids
Vacancy Diffusion: • atoms exchange with vacancies. • Both self-diffusion and inter-diffusion occur by this mechanism; for the latter ...
[48]
Diffusion mechanisms [The heart of crystalline materials]
Diffusion occurs by stepwise migration of atoms from lattice site to lattice site across the crystal. It occurs in interstices and vacancies.
[49]
[PDF] Grain Size and Solid Solution Strengthening in Metals - DiVA portal
It is also one of the methods most extensively used in the development of higher strength in commercial alloys. A great deal of research has been published on ...
[50]
[PDF] Solid solution strengthening theories of high-entropy alloys
We present a review of solid solution strengthening models for random concentrated solid solu- tions of which high entropy alloys are an interesting subset.Missing: seminal | Show results with:seminal
[51]
Solid solution strengthening – Knowledge and References
Solid solution strengthening is a process of improving the strength of a pure metal by adding atoms of an alloying element to its crystalline lattice, ...
[52]
Spectroscopy of polystyrene and poly(vinyl methyl ether) blends
... Miscible Polymer Blends. Macromolecules 2003, 36 (21) , 8156-8163 ... Specific Intermolecular Interactions in Polymer Blends. 1986, 138-147. https ...
[53]
Blend Miscibility and the Flory−Huggins Interaction Parameter
The Flory−Huggins interaction parameter, χ, is often used in the literature to describe the binary interactions of polymer blends, yet to what extent does this ...
[54]
Phase-Separated Structure of NBR/PVC Blends with Different ... - NIH
Aug 9, 2023 · A polymer blend is a mixture of chemically different polymers. Dissimilar polymers are only miscible via a favorable specific interaction ...
[55]
Silicon–Germanium: The Legacy Lives On - MDPI
Apr 18, 2022 · Silicon–germanium is a substitutional solid solution alloy with complete miscibility ... Study on diffusion barriers of doping elements in Si-Ge ...
[56]
The Effect of Processing Conditions on the Microstructure of ...
Mar 22, 2024 · This result can be explained when considering that the higher processing temperature, causing a decrease in the polymer viscosity, induces a ...
[57]
Microstructural characterisation and compound formation in rapidly ...
Jul 15, 2024 · Compounds formation is researched in rapidly solidified Si-14.2 at% Ge alloy. Higher Ge-concentration regions are observed as the cooling rate is increased.
[58]
Revising Concepts on Liquid–Liquid Extraction: Data Treatment and ...
Dec 30, 2021 · The cloud-point method is not a rigorous method to obtain the binodal curve, since it depends on the visual perception of the observer to ...
[59]
[PDF] Liquid-liquid equilibria of the phosphoric acid, ester and water ...
The solubility curves were investigated via the cloud point method in an equi- librium glass cell. For this purpose, binary mixtures with specified composition ...
[60]
Physical Properties and Liquid−Liquid Equilibrium of the Ternary ...
The results were used to calculate excess molar volumes and deviations of refractive index over the miscibility composition range for the mixtures ...
[61]
[PDF] VAPOUR EQUILIBRIUM IN A BINARY SYSTEM OF MISCIBLE ...
2 Determine the refractive index of samples No.1-6 (n) using the Abbe's refractometer. Write down the refractive indexes to the Table 1. 3 Calculate the molar ...
[62]
[PDF] Densities and Viscosities of Mixtures of Two Ionic Liquids ... - PATh
Jul 14, 2016 · ABSTRACT: Density and dynamic viscosity data of binary mixtures of ionic liquids (ILs) were determined in this work,.
[63]
[PDF] Characterizing the miscibility of polymer blend systems by Dynamic ...
If the glass transition temperatures are shifting towards each other, but do not fully merge into one single temperature, the components are soluble to a ...<|separator|>
[64]
[PDF] Fourier Transform Infrared and Raman Spectroscopic ...
Our objective was to establish the extent to which preferred composition fluctuations in homogeneous miscibility-gap type solutions influence solidification.
[65]
Molecular simulation as a computational pharmaceutics tool to ... - NIH
In this review we will discuss how computational methods, and in particular classical molecular dynamics simulations, can be used to calculate solubility.
[66]
Molecular dynamics simulation to estimate minimum miscibility ...
In this study, we used molecular dynamics (MD) simulation to estimate the MMP of pure and impure CO 2 with oil by obtaining interfacial tensions (IFTs)
[67]
Molecular Dynamics Prediction Verified by Experimental Evaluation ...
Nov 29, 2021 · The reported MDS method can be utilized to predict a drug's solubility/miscibility in PDL to estimate the level of drug loading.
[68]
Assessment of COSMO-RS for Predicting Liquid–Liquid Equilibrium ...
Jun 12, 2024 · This could be attributed to the fact that COSMO-RS tends to overestimate the miscibility gap (i.e., overestimating the upper critical solution ...
[69]
Prediction of miscibility gaps in water/ether mixtures using COSMO ...
The absolute average error in the predicted ether composition is found to be 85% in the water-rich phase and 17% in the ether-rich phase using COSMO-SAC (2002).
[70]
Using COSMO-RS to Predict Hansen Solubility Parameters
Jul 13, 2022 · Understanding the Top-down Fragmentation of 2D Material in Miscible Liquid Environment Based on Hansen Solubility Parameters Theory.
[71]
Group‐contribution estimation of activity coefficients in nonideal ...
A group-contribution method is presented for the prediction of activity coefficients in nonelectrolyte liquid mixtures.
[72]
Prediction of Minimum Miscibility Pressure Using the UNIFAC Group ...
Feb 18, 2016 · Using the mPR-LCVM-UNIFAC method and the group contribution characterization of undefined fractions, the results show good agreement with those ...
[73]
Flash points of partially miscible aqueous–organic mixtures ...
This work predicts the flash points of partially miscible aqueous–organic mixtures using the flash point prediction model of Liaw et al. (J. Chem.
[74]
Ab initio study of the miscibility for solid hydrogen–helium mixtures ...
Feb 18, 2020 · On the basis of first-principles calculations combined with the structure prediction method, we investigate the miscibility for solid H2–He ...
[75]
Use of ab initio interaction energies for the prediction of phase ...
Ab initio molecular orbital methods have been used for calculations of interaction parameters of the UNIQUAC and NRTL activity coefficient model to predict ...Missing: miscibility | Show results with:miscibility
[76]
Multitask Machine Learning to Predict Polymer–Solvent Miscibility ...
Jul 10, 2023 · We develop a machine-learning framework to achieve highly generalized and robust prediction of Flory–Huggins χ parameters for polymer–solvent solutions.
[77]
Integrating theory with machine learning for predicting polymer ...
This work combines Flory-Huggins theory with ML to improve predictions of polymer solution phase behavior, especially in data-sparse situations, and provide ...Missing: metals post-