Mechanism Study on Deep Removal of Lattice Impurities from High-Purity Quartz by Chlorination Roasting

Abstract

High-temperature chlorination roasting is a critical technique for achieving ultra-high-purity quartz required in semiconductor, photovoltaic, and fiber-optic applications. However, the removal mechanisms of lattice-bound impurities remain poorly understood due to a lack of integrated thermodynamic and kinetic analysis. This study systematically investigates the removal behavior of seven key lattice impurities, namely Ti, Al, B, Fe, Li, Na, and K, during chlorination roasting using combined thermodynamic modeling and diffusion kinetics. Thermodynamic calculations reveal that carbonaceous reductants are indispensable for enabling spontaneous chlorination of substitutional impurities such as Ti, Al, and B, while alkali metals including Na, K, and Li can be effectively removed under HCl atmosphere at moderate temperatures. Kinetic analysis identifies solid-state diffusion through the SiO₂ lattice as the likely rate-determining step based on the modeling framework, with activation energies ranging from approximately 90 kJ/mol for Na⁺ to 400 kJ/mol for Ti⁴⁺. A significant diffusion crossover effect is observed, where high-activation-energy impurities exhibit exponential mobility gains above 1200 °C. An alkali-first, Al-follows coupled diffusion mechanism is elucidated for aluminum removal. Based on these findings, a temperature-staged, atmosphere-segmented roasting strategy is proposed. This work provides a quantitative mechanistic framework for deep impurity removal and offers practical guidance for overcoming the 4N8 purity bottleneck in high-purity quartz production.

1. Introduction

High-purity quartz (HPQ) refers to quartz resources that, after mineral processing and chemical purification, achieve SiO₂ purity of no less than 99.995% while maintaining fluid inclusion content that meets the requirements for downstream material processing [1,2,3]. Owing to its corrosion resistance, thermal stability, high optical transparency, superior insulation, and low thermal expansion, high-purity quartz serves as an indispensable foundational material for strategic emerging industries, including photovoltaics, semiconductors, optical fiber communications, and aerospace [4]. Figure 1 illustrates the integrated upstream and downstream industrial chain of high-purity quartz.

The primary sources of natural high-purity quartz include natural crystal, granitic pegmatite, vein quartz, and quartzite [5]. Although vein quartz formed through metamorphic hydrothermal crystallization has a high SiO₂ content, it often contains numerous fluid inclusions [6]. These inclusions degrade both the transparency and the overall quality of fused quartz glass. In contrast, quartz derived from pegmatoid granite and early-stage granitic pegmatites forms under conditions that naturally minimize fluid inclusions. Consequently, these sources offer superior potential for HPQ production. A preeminent example is the Sibelco-operated Spruce Pine deposit in the United States, which is globally recognized as the leading granitic pegmatite source for HPQ. Although the raw ore typically contains only 30% quartz, with the remainder consisting of gangue minerals such as feldspar and mica, advanced mineral processing and purification can effectively enrich the material, thereby yielding high-value industrial products [7].

Impurities in natural quartz are commonly classified into two categories, inclusion impurities and lattice-bound impurities [8]. Inclusion impurities which include solid mineral inclusions and fluid inclusions are mechanically encapsulated within quartz grains and can be effectively removed by conventional beneficiation processes, such as crushing, grinding, flotation, magnetic separation, and multi-stage acid leaching [9,10,11]. Lattice-bound impurities, by contrast, pose a fundamentally greater challenge. These elements are incorporated into the quartz crystal structure at the atomic level, where they occupy specific crystallographic sites within the continuous three-dimensional SiO₂ tetrahedral network. The principal substitution mechanism involves the isomorphous replacement of Si⁴⁺ by cations of similar ionic radius and compatible charge, most notably Ti⁴⁺, Al³⁺, B³⁺, and Fe³⁺. Because the substitution of Si⁴⁺ by trivalent cations creates a local negative charge deficit, charge compensation is maintained through the incorporation of monovalent interstitial cations primarily Li⁺, Na⁺, and K⁺, which reside within the helical channels parallel to the quartz c-axis [12,13,14]. Figure 2 illustrates the lattice impurity distribution within high-purity quartz. Additionally, hydrogen-related structural defects primarily in the form of hydroxyl groups (≡Si–OH)—are introduced through the interaction of quartz with aqueous geological fluids and can profoundly affect both the infrared transmission and the high-temperature behavior of quartz glass [15]. Because these impurity atoms are tightly bound by strong covalent Si–O–Si bonds [16], they resist extraction by conventional physical and chemical processing methods and thus represent the ultimate barrier, limiting the purity of conventional quartz products to approximately 99.998% [17].

High-temperature chlorination roasting is widely recognized as a highly effective method to overcome the 4N purity bottleneck and achieve the 4N8 (99.998%+) grades required for advanced applications [18,19]. The underlying principle involves the selective reaction of chlorine-containing gases Cl₂ or HCl with lattice-bound impurities at elevated temperatures to form volatile metal chlorides. Due to their low boiling or sublimation points, these chlorides vaporize and escape the SiO₂ matrix into the gas stream, effectively separating them from the quartz host. This technique is widely considered the proprietary core of the IOTA-grade production process developed by Sibelco. Despite its industrial success, published research remains largely empirical, often focusing on isolated conditions or specific elements. A unified, quantitative mechanistic framework encompassing the full spectrum of lattice impurities is still lacking [20].

A rigorous understanding of chlorination roasting requires systematic investigation through the complementary lenses of thermodynamics and kinetics. Thermodynamic analysis, based on Gibbs free energy change $\Delta G$ calculations for each chlorination reaction, provides the fundamental criterion for determining reaction spontaneity under specific temperatures and atmospheres [21]. By modeling reactions of lattice impurity oxides using standard-state data $\Delta H_f^{\circ }$, $S^{\circ }$, and $\Delta G_f^{\circ }$, the critical transition temperature where $\Delta G$ shifts from positive to negative can be quantitatively determined. This thermodynamic boundary defines the minimum temperature required for a specific impurity removal reaction to become feasible. However, thermodynamic feasibility is a necessary but insufficient condition for effective purification. In practice, the degree of purification achieved within industrial timeframes is governed by kinetic factors, primarily the diffusion rates of impurity atoms through the solid quartz lattice to reactive interfaces [22].

Consequently, a holistic understanding of chlorination roasting necessitates integrating thermodynamic viability with the quantitative assessment of diffusion activation energies, the identification of rate-determining steps, and the prediction of removal rates across varying temperatures and holding times. Despite the experimental and thermodynamic investigations reported for selected impurity elements during chlorination roasting, most studies are confined to individual elements or specific conditions, and a systematic comparison across the full impurity spectrum remains absent [23,24,25]. While thermodynamic analyses have established the feasibility of certain chlorination reactions, they typically treat the reactions under idealized conditions without addressing the kinetic constraints governing actual removal rates within industrially relevant timeframes. Moreover, the DFT and molecular dynamics approaches recently adopted to investigate chlorination mechanisms at the atomic scale, though methodologically novel, are inherently limited in representativeness: lattice impurities in natural quartz occur at ppm levels, whereas unit-cell doping simulations involve concentrations orders of magnitude higher, leading to unrealistic impurity interactions that do not reflect the true dilute solid-solution state. These atomic-scale models also cannot capture the long-range diffusion behaviors or macroscopic kinetic trends critical for process design. The present study addresses these limitations by integrating thermodynamic and kinetic analyses for seven key impurities under unified conditions, providing a holistic understanding of the removal process and a quantitative basis for process optimization.

This study addresses these significant knowledge gaps by conducting a comprehensive and integrated thermodynamic–kinetic investigation of lattice impurity removal from high-purity quartz during high-temperature chlorination roasting. To achieve these objectives, we first establish a thermodynamic feasibility map for the chlorination of seven principal impurity species, systematically evaluating the effects of chlorinating agents and carbonaceous reductants on reaction spontaneity and critical temperatures. We then elucidate solid-state diffusion mechanisms to quantitatively determine activation energies, diffusion coefficients, and rate-controlling steps for each impurity, with theoretical removal rates further predicted using the shrinking-core diffusion model. Finally, we synthesize these thermodynamic and kinetic findings into a hierarchy of removal difficulty and propose a temperature-staged, atmosphere-segmented strategy for ultra-deep quartz purification. The novelty of this work is threefold. First, unlike previous studies that focused on individual elements, this work provides a systematic investigation covering all major lattice impurities, establishing a clear hierarchy of chlorination removal difficulty that offers practical guidance for process design. Second, through quantitative analysis of diffusion activation energies and rate-controlling steps, we reveal the distinct temperature-dependent mobility of different impurity types, providing a mechanistic basis for understanding their removal behaviors. Third, based on these thermodynamic and kinetic differences, we propose a temperature-staged, atmosphere-segmented roasting strategy that enables targeted removal under optimized conditions, offering industrially actionable guidance for overcoming the purity bottleneck in high-purity quartz production.

2. Materials and Methods

2.1. Thermodynamic Calculation Methodology

2.1.1. Chemical Reaction Models and Assumptions

To evaluate the thermodynamic feasibility of chlorination roasting, lattice-bound impurities must be represented in a thermodynamically definable form. Although these impurities occupy substitutional or interstitial sites within the SiO₂ network, process thermodynamics typically models them as their corresponding stable oxides. This simplification is justified by two factors: first, impurity–oxygen bonds in the lattice are chemically analogous to those in bulk oxides; second, standard thermodynamic data are primarily available for stoichiometric compounds rather than dilute solid-solution species. Accordingly, the target impurities in this study are modeled as TiO₂, Al₂O₃, B₂O₃, Fe₂O₃, Li₂O, Na₂O, and K₂O.

Chlorination roasting involves the high-temperature reaction of these impurity oxides with chlorine-containing gases. This study evaluates two primary chlorinating agents: molecular chlorine Cl₂ and hydrogen chloride HCl, both of which represent standard industrial reagents. The choice of agent significantly influences both thermodynamic efficacy and process engineering. While Cl₂ offers superior chlorination potential, its toxicity necessitates stringent safety protocols. Conversely, HCl is favored in continuous industrial operations due to its chemical stability and operational simplicity.

Chlorination reactions for each impurity using Cl₂ and HCl are formulated based on their stoichiometric conversion into volatile metal chlorides. The governing reaction categories are defined as follows (1)–(6).

• Chlorination of alkali metal oxides (M = Li, Na, K) with Cl₂:

  $${\mathrm{M}}_2\mathrm{O}\left(\mathrm{s}\right)+{\mathrm{Cl}}_2\left(\mathrm{g}\right)\to 2\mathrm{MCl}\left(\mathrm{g}\right)+{\displaystyle \frac{1}{2}}{\mathrm{O}}_2\left(\mathrm{g}\right)$$

  (1)
• Chlorination of alkali metal oxides with HCl:

  $${\mathrm{M}}_2\mathrm{O}\left(\mathrm{s}\right)+2\mathrm{HCl}\left(\mathrm{g}\right)\to 2\mathrm{MCl}\left(\mathrm{g}\right)+{\mathrm{H}}_2\mathrm{O}\left(\mathrm{g}\right)$$

  (2)
• Chlorination of trivalent metal oxides (M = Al, B, Fe) with Cl₂:

  $${\mathrm{M}}_2{\mathrm{O}}_3\left(\mathrm{s}\right)+3{\mathrm{Cl}}_2\left(\mathrm{g}\right)\to 2{\mathrm{MCl}}_3\left(\mathrm{g}\right)+{\displaystyle \frac{3}{2}}{\mathrm{O}}_2\left(\mathrm{g}\right)$$

  (3)
• Chlorination of trivalent metal oxides with HCl:

  $${\mathrm{M}}_2{\mathrm{O}}_3\left(\mathrm{s}\right)+6\mathrm{HCl}\left(\mathrm{g}\right)\to 2{\mathrm{MCl}}_3\left(\mathrm{g}\right)+3{\mathrm{H}}_2\mathrm{O}\left(\mathrm{g}\right)$$

  (4)
• Chlorination of tetravalent metal oxide (Ti) with Cl₂:

  $${\mathrm{TiO}}_2\left(\mathrm{s}\right)+2{\mathrm{Cl}}_2\left(\mathrm{g}\right)\to {\mathrm{TiCl}}_4\left(\mathrm{g}\right)+{\mathrm{O}}_2\left(\mathrm{g}\right)$$

  (5)
• Chlorination of tetravalent metal oxide with HCl:

  $${\mathrm{TiO}}_2\left(\mathrm{s}\right)+4\mathrm{HCl}\left(\mathrm{g}\right)\to {\mathrm{TiCl}}_4\left(\mathrm{g}\right)+2{\mathrm{H}}_2\mathrm{O}\left(\mathrm{g}\right)$$

  (6)

When carbonaceous reductants are introduced, the chlorination reactions are significantly promoted through the formation of CO or CO₂. The corresponding carbon-assisted reaction pathways with Cl₂ are given as follows (7)–(12):

7.

Chlorination of alkali metal oxides with Cl₂ + carbon (C → CO₂):

$${\mathrm{M}}_2\mathrm{O}\left(\mathrm{s}\right)+{\mathrm{Cl}}_2\left(\mathrm{g}\right)+{\displaystyle \frac{1}{2}}\mathrm{C}\left(\mathrm{s}\right)\to 2\mathrm{MCl}\left(\mathrm{g}\right)+{\displaystyle \frac{1}{2}}{\mathrm{CO}}_2\left(\mathrm{g}\right)$$

(7)

8.

Chlorination of alkali metal oxides with Cl₂ + carbon (C → CO):

$${\mathrm{M}}_2\mathrm{O}\left(\mathrm{s}\right)+{\mathrm{Cl}}_2\left(\mathrm{g}\right)+\mathrm{C}\left(\mathrm{s}\right)\to 2\mathrm{MCl}\left(\mathrm{g}\right)+\mathrm{CO}\left(\mathrm{g}\right)$$

(8)

9.

Chlorination of trivalent metal oxides with Cl₂ + carbon (C → CO₂):

$${\mathrm{M}}_2{\mathrm{O}}_3\left(\mathrm{s}\right)+3{\mathrm{Cl}}_2\left(\mathrm{g}\right)+{\displaystyle \frac{3}{2}}\mathrm{C}\left(\mathrm{s}\right)\to {\mathrm{MCl}}_3\left(\mathrm{g}\right)+{\displaystyle \frac{3}{2}}{\mathrm{CO}}_2\left(\mathrm{g}\right)$$

(9)

10.

Chlorination of trivalent metal oxides with Cl₂ + carbon (C → CO):

$${\mathrm{M}}_2{\mathrm{O}}_3\left(\mathrm{s}\right)+3{\mathrm{Cl}}_2\left(\mathrm{g}\right)+3\mathrm{C}\left(\mathrm{s}\right)\to {2\mathrm{MCl}}_3\left(\mathrm{g}\right)+3\mathrm{CO}\left(\mathrm{g}\right)$$

(10)

11.

Chlorination of tetravalent metal oxide with Cl₂ + carbon (C → CO₂):

$${\mathrm{TiO}}_2\left(\mathrm{s}\right)+2{\mathrm{Cl}}_2\left(\mathrm{g}\right)+\mathrm{C}\left(\mathrm{s}\right)\to {\mathrm{TiCl}}_4\left(\mathrm{g}\right)+{\mathrm{CO}}_2\left(\mathrm{g}\right)$$

(11)

12.

Chlorination of tetravalent metal oxide with Cl₂ + carbon (C → CO):

$${\mathrm{TiO}}_2\left(\mathrm{s}\right)+2{\mathrm{Cl}}_2\left(\mathrm{g}\right)+2\mathrm{C}\left(\mathrm{s}\right)\to {\mathrm{TiCl}}_4\left(\mathrm{g}\right)+2\mathrm{CO}\left(\mathrm{g}\right)$$

(12)

Similarly, when HCl is used in combination with carbon, the chlorination reactions are also promoted via CO or CO₂ formation. These HCl–carbon reaction pathways are described as follows (13)–(18):

13.

Chlorination of alkali metal oxides with HCl + carbon (C → CO₂):

$${\mathrm{M}}_2\mathrm{O}\left(\mathrm{s}\right)+2\mathrm{HCl}\left(\mathrm{g}\right)+\mathrm{C}\left(\mathrm{s}\right)+{\mathrm{O}}_2\left(\mathrm{g}\right)\to 2\mathrm{MCl}\left(\mathrm{g}\right)+{\mathrm{H}}_2\mathrm{O}\left(\mathrm{g}\right)+{\mathrm{CO}}_2\left(\mathrm{g}\right)$$

(13)

14.

Chlorination of alkali metal oxides with HCl + carbon (C → CO):

$${\mathrm{M}}_2\mathrm{O}\left(\mathrm{s}\right)+2\mathrm{HCl}\left(\mathrm{g}\right)+\mathrm{C}\left(\mathrm{s}\right)+{{\displaystyle \frac{1}{2}}\mathrm{O}}_2\to 2\mathrm{MCl}\left(\mathrm{g}\right)+{\mathrm{H}}_2\mathrm{O}\left(\mathrm{g}\right)+\mathrm{CO}\left(\mathrm{g}\right)$$

(14)

15.

Chlorination of trivalent metal oxides with HCl + carbon (C → CO₂):

$${\mathrm{M}}_2{\mathrm{O}}_3\left(\mathrm{s}\right)+6\mathrm{HCl}\left(\mathrm{g}\right)+\mathrm{C}\left(\mathrm{s}\right)+2{\mathrm{O}}_2\left(\mathrm{g}\right)\to {2\mathrm{MCl}}_3\left(\mathrm{g}\right)+3{\mathrm{H}}_2\mathrm{O}\left(\mathrm{g}\right)+{\mathrm{CO}}_2$$

(15)

16.

Chlorination of trivalent metal oxides with HCl + carbon (C → CO):

$${\mathrm{M}}_2{\mathrm{O}}_3\left(\mathrm{s}\right)+6\mathrm{HCl}\left(\mathrm{g}\right)+\mathrm{C}\left(\mathrm{s}\right)+{\displaystyle \frac{1}{2}}{\mathrm{O}}_2\left(\mathrm{g}\right)\to {2\mathrm{MCl}}_3\left(\mathrm{g}\right)+3{\mathrm{H}}_2\mathrm{O}\left(\mathrm{g}\right)+\mathrm{CO}\left(\mathrm{g}\right)$$

(16)

17.

Chlorination of tetravalent metal oxide (Ti) with HCl + carbon (C → CO₂):

$${\mathrm{TiO}}_2\left(\mathrm{s}\right)+4\mathrm{HCl}\left(\mathrm{g}\right)+\mathrm{C}\left(\mathrm{s}\right)+{\mathrm{O}}_2\left(\mathrm{g}\right)\to {\mathrm{TiCl}}_4\left(\mathrm{g}\right)+{2\mathrm{H}}_2\mathrm{O}\left(\mathrm{g}\right){+\mathrm{CO}}_2\left(\mathrm{g}\right)$$

(17)

18.

Chlorination of tetravalent metal oxide with HCl + carbon (C → CO):

$${\mathrm{TiO}}_2\left(\mathrm{s}\right)+4\mathrm{HCl}\left(\mathrm{g}\right)+\mathrm{C}\left(\mathrm{s}\right)+{\displaystyle \frac{1}{2}}{\mathrm{O}}_2\left(\mathrm{g}\right)\to {\mathrm{TiCl}}_4\left(\mathrm{g}\right)+{2\mathrm{H}}_2\mathrm{O}\left(\mathrm{g}\right)+\mathrm{CO}\left(\mathrm{g}\right)$$

(18)

It should be noted that the above carbon-assisted reactions are formulated under two idealized limiting scenarios, assuming exclusive formation of either CO₂ or CO. In practice, the Boudouard equilibrium establishes a temperature-dependent mixture of CO and CO₂, and the actual gas composition may lie between these two extremes. This simplified treatment is deliberately adopted to systematically compare the thermodynamic driving forces under different reducing conditions and to establish the theoretical bounds of reaction feasibility.

2.1.2. Thermodynamic Data Sources

Standard-state thermodynamic data were sourced from the NIST-JANAF Thermochemical Tables and the Barin thermochemical data compilation. For each species, the following fundamental parameters were extracted at 298 K: standard enthalpy of formation ΔH in kJ/mol, absolute entropy S₂₉₈ in J/mol·K, and Gibbs free energy of formation ΔG°₂₉₈ in kJ/mol. These parameters form the basis for calculating the reaction enthalpy ΔH_r°, entropy ΔS_r°, and Gibbs free energy ΔG_r° for each chlorination reaction under standard-state conditions. The modeled species encompass impurity oxides TiO₂, Al₂O₃, B₂O₃, Fe₂O₃, Li₂O, Na₂O, K₂O, chlorinating agents Cl₂ and HCl, volatile metal chlorides including TiCl₄, AlCl₃, BCl₃, FeCl₃, LiCl, NaCl, and KCl, gaseous co-products O₂ and H₂O, carbonaceous reagents C, CO, and CO₂, and the SiO₂ host matrix. Gaseous species are modeled as ideal gases at 1 bar pressure, while condensed phases are treated as pure substances in their respective standard states.

2.1.3. Calculation of Standard-State Thermodynamic Parameters

The standard Gibbs free energy change ΔG°₂₉₈ for each chlorination reaction at the reference temperature of 298 K, which is 25 °C, is calculated via the fundamental thermodynamic relationship:

$$\Delta G_{298}^{\circ }=\Delta H_{298}^{\circ }-T\Delta S_{298}^{\circ }$$

(19)

where $\Delta H_{298}^{\circ }$ and $\Delta S_{298}^{\circ }$ are determined from the summations of the standard enthalpies of formation and absolute entropies of all products and reactants, weighted by their stoichiometric coefficients ν:

$$\Delta H_{298}^{\circ }={\displaystyle \sum {\nu }_i}\Delta H_{f,i}^{\circ }-{\displaystyle \sum {\nu }_j}\Delta H_{f,j}^{\circ }$$

(20)

$$\Delta S_{298}^{\circ }={\displaystyle \sum {\nu }_i}{S}_i^{\circ }-{\displaystyle \sum {\nu }_j}{S}_j^{\circ }$$

(21)

where ν_i and ν_j are the stoichiometric coefficients of products and reactants, respectively. A negative $\Delta G_{298}^{\circ }$ indicates that the reaction is thermodynamically spontaneous under standard-state conditions, whereas a positive value indicates non-spontaneity and requires elevated temperature or auxiliary agents to drive the reaction forward.

2.1.4. Temperature Dependence and Critical Temperature Determination

The thermodynamic favorability at elevated temperatures is evaluated by calculating ΔG (T) as a function of temperature T via a simplified linear approximation:

$$\Delta G\left(T\right)=\Delta H-T\Delta S$$

(22)

This approach relies on the customary assumption that ΔH° and ΔS° remain approximately constant across the temperature range of interest 25–1300 °C. Although neglecting the heat capacity difference ΔC_p between products and reactants introduces a minor systematic error, this simplification is widely accepted in process metallurgy for feasibility screening. It is justified by the dominant influence of the TΔS term at typical chlorination roasting temperatures.

The critical temperature T_c, the point at which a reaction transitions from non-spontaneous to spontaneous with ΔG = 0, is determined by solving:

$$T_{\mathrm{c}}={\displaystyle \frac{\Delta H}{\Delta S}}$$

(23)

Thus, T_c represents the minimum thermal activation required for thermodynamic feasibility. Reactions exhibiting T_c values below the typical roasting range 900–1300 °C are considered viable, whereas those with T_c exceeding this range are deemed impracticable under standard processing conditions.

It should be noted that the simplified linear approximation described above was used for calculating the thermodynamic parameters presented in Table 1, Table 2, Table 3, Table 4, Table 5 and Table 6, and was also applied to determine the critical temperature T_c. These calculations are primarily intended to reveal the relative thermodynamic trends among different impurity species and chlorination systems rather than to provide precisely accurate absolute values for each specific reaction. The ΔG-T plots presented in Figure 3a–f, by contrast, were obtained directly from HSC Chemistry software incorporating full heat-capacity integration over the entire temperature range, and thus more closely reflect the actual temperature-dependent behavior of the reactions. While the simplified calculations and the HSC-based plots yield consistent thermodynamic trends, the latter provide a more accurate quantitative representation of the temperature-dependent thermodynamic driving forces.

2.1.5. Concentration Correction for Dilute Impurity Systems

The preceding calculations assume standard-state conditions, where all reactants and products exist as pure phases at unit activity. In practice, however, lattice-bound impurities occur at ppm-level concentrations, resulting in activities significantly lower than unity. The relationship between the actual Gibbs free energy change ΔG and its standard-state value ΔG° is defined by:

$$\Delta G=\Delta G^{\circ }+\mathrm{RTln}Q$$

(24)

where Q is the reaction quotient, defined as the ratio of product activities to reactant activities, each raised to the power of its stoichiometric coefficient. For the chlorination reactions under consideration, Q is influenced principally by the partial pressures of the gaseous species Cl₂, HCl, O₂, H₂O, and metal chlorides and, critically, by the activity of the impurity within the quartz lattice. Because the activity of a dilute substitutional or interstitial impurity is far below unity, the RT ln Q term makes a positive contribution to ΔG, making the actual free energy change more positive than its standard-state counterpart. Consequently, the effective critical temperature for a given impurity under realistic process conditions is higher than that calculated from standard-state data alone.

Quantitative determination of this concentration effect would require knowledge of the activity coefficients of individual impurity species within the quartz solid solution—parameters that are not experimentally available for most elements of interest. In this study, we therefore calculate and report both the standard-state ΔG° and T_c values, which represent the lower-bound thermodynamic requirement, and provide a qualitative discussion of the concentration correction, noting that the practical implication is a systematic upward shift in all critical temperatures.

2.2. Kinetic Modeling Methodology

2.2.1. Gas–Solid Reaction Framework

High-temperature chlorination roasting of quartz represents a classic non-catalytic gas–solid reaction system. The process proceeds through six sequential steps:

(1) External mass transfer of the gaseous chlorinating agent through the boundary layer surrounding each quartz particle.

(2) Internal diffusion through the pore network and lattice defects to the vicinity of an impurity site.

(3) Solid-state migration of impurity species to the reaction interface.

(4) Interfacial chemical reaction to form chloride products.

(5) Outward solid-state diffusion of chlorides toward the particle surface.

(6) Desorption and mass transfer of the volatile chlorides into the bulk gas stream.

The step with the highest resistance and thus the slowest rate determines the overall macroscopic kinetics of the purification process. For chlorination roasting, solid-state migration of impurity ions through the crystalline SiO₂ lattice, corresponding to steps three and five, is recognized as the rate-determining step. This is primarily attributed to the substantial activation energy required for the thermal rupture of impurity–oxygen bonds and the subsequent reconstruction of the Si–O tetrahedral network during ion migration.

2.2.2. Diffusion Kinetics: Fick’s Laws and the Arrhenius Equation

The solid-state diffusion of impurities within the quartz lattice is described quantitatively by Fick’s laws. For one-dimensional non-steady-state diffusion, Fick’s second law serves as the governing partial differential equation:

$${\displaystyle \frac{\partial C(x,t)}{\partial t}}=D{\displaystyle \frac{{\partial }^{2}C(x,t)}{\partial x^2}}$$

(25)

where $C(x,t)$ is the impurity concentration at position $x$ and time $t$. $D$ is the diffusion coefficient representing impurity mobility within the host lattice. The diffusion coefficient follows an exponential temperature dependence, defined by the Arrhenius equation:

$$D=D_0\exp \left(-{\displaystyle \frac{E_a}{RT}}\right)$$

(26)

Here, $D_0$ is the pre-exponential factor in m²/s, $E_a$ is the activation energy in kJ/mol or eV, $R$ is the universal gas constant equal to 8.314 J/mol·K, and $T$ is the absolute temperature in K. $E_a$ represents the potential energy barrier that must be overcome for an atomic jump to occur. Impurities with higher $E_a$ values exhibit lower diffusion coefficients and increased sensitivity to temperature fluctuations. Consequently, removing such impurities necessitates higher temperatures to achieve industrially viable diffusion rates.

2.2.3. Diffusion Activation Energies: Data Sources and Estimation

The diffusion activation energies E_a are compiled from a comprehensive review of diffusion data in silicate melts, glasses, and crystals, which systematically summarizes experimental diffusion data for a wide range of elements in silicate systems [26]. For alkali ions Li⁺, Na⁺, and K⁺, the values are based on experimental measurements of ionic diffusion in quartz documented in the literature. For substitutional impurities Al³⁺, Ti⁴⁺, B³⁺, and Fe³⁺, direct experimental data for ppm-level lattice impurities in quartz are not available; therefore, the values are estimated from diffusion systematics in analogous silicate systems reported in the same literature, based on considerations of ionic radii, bond strengths, and chemical analogies.

For charge-compensating alkali ions (Li⁺, Na⁺, and K⁺) diffusion occurs interstitially through the c-axis channels. Their activation energies are relatively low governed by the balance between electrostatic interactions and ionic-size-induced steric hindrance. In contrast, substitutional impurities Al³⁺, Fe³⁺, B³⁺, Ti⁴⁺ exhibit significantly higher $E_a$ values, as their migration necessitates lattice dissociation.

The pre-exponential factor $D_0$ for all impurities was assumed to be 10⁻⁴ m²/s, consistent with reported values for cation diffusion in silicates. The pre-exponential factor D₀ was assumed to be 10⁻⁴ m²/s for all impurity species, consistent with the typical range of values reported for cation diffusion in silicate minerals and melts in the literature. This assumption introduces some uncertainty into the absolute values of the calculated diffusion coefficients. However, because the relative differences between impurity species are governed primarily by the exponential term, i.e., the activation energy E_a, the choice of D₀ has a limited effect on the relative diffusivity ranking among different impurities. Furthermore, the theoretical removal rates calculated from the diffusion coefficients should be interpreted as semi-quantitative estimates for comparative purposes rather than precise predictions for specific feedstocks. The primary utility of this simplified kinetic treatment lies in revealing the qualitative trends in temperature-dependent mobility and in establishing the relative difficulty of removing different impurity species, which is the main objective of the present modeling study.

2.2.4. Rate-Controlling Step Identification

The rate-controlling step for each impurity was identified by comparing the estimated activation energies $E_a$ of the sequential steps in the purification process. The step with the highest $E_a$ was defined as the rate-determining step. For alkali metals, the RDS is interstitial diffusion through the c-axis channels. For substitutional impurities, the RDS is generally lattice dissociation comprising the rupture of impurity–oxygen bonds and subsequent Si–O network reconstruction, which must precede ion migration.

Specific attention was given to Al³⁺, as its heterovalent substitution requires charge compensation by interstitial alkali cations. Al removal thus follows a coupled two-stage mechanism: the rapid diffusion and volatilization of charge-compensating alkalis, followed by the rate-limiting dissociation and migration of the electrostatically destabilized Al³⁺ species. This “alkali-first, Al-follows” mechanism is explicitly integrated into the kinetic model for Al purification.

2.2.5. Shrinking-Core Diffusion Model and Theoretical Removal Rate Prediction

The shrinking-core model SCM was adopted to predict theoretical impurity removal rates as a function of temperature and duration. For product-layer diffusion-controlled reactions, the relationship between fractional conversion α and time t is defined by the Ginstling–Brounshtein equation for spherical particles:

$$1-{\displaystyle \frac{2}{3}}\alpha -(1-\alpha {)}^{2/3}={\displaystyle \frac{2kt}{r^2}}$$

(27)

where k is a rate constant proportional to the diffusion coefficient D, and r₀ is the particle radius. A representative radius of 50 μm was assumed, consistent with typical quartz feedstock grain sizes. Diffusion coefficients for temperatures of interest 900–1300 °C were determined via the Arrhenius equation using the adopted D₀ and $E_a$ values. To facilitate comparison, a relative diffusivity D was defined by normalizing all coefficients to the value at 1100 °C or 1373 K. Theoretical removal rates α were then computed for durations of 2, 4, 8, and 16 h, generating a matrix of predictions under commercially relevant thermal conditions. The computational workflow involved:

(1) Calculating ΔH° and ΔS° from thermodynamic data.

(2) Determining T_c by solving ΔG_T = 0.

(3) Computing D from the Arrhenius equation.

(4) Solving the Ginstling–Brounshtein equation for α at each temperature-time combination.

The shrinking-core model is adopted here to estimate impurity removal rates because the chlorination reaction occurs at the solid–gas interface, with impurity diffusion through the quartz lattice being the rate-controlling step—a scenario analogous to product-layer diffusion in the classical shrinking-core formulation. For simplicity, a monodisperse particle size is assumed, consistent with the fine particle size used in the validation experiments. We acknowledge that this idealized treatment does not account for the actual particle-size distribution of the feedstock, nor for the heterogeneity of impurity distribution within individual particles. Additionally, the model assumes that the reaction interface advances uniformly from the particle surface inward and that the diffusion coefficient remains constant throughout the process. Consequently, the calculated removal rates should be interpreted as theoretical estimates for comparative purposes, and the modeling results are primarily intended to reveal qualitative trends rather than to provide precise quantitative predictions for specific feedstocks.

2.3. Integration of Thermodynamic and Kinetic Analyses

The final methodological stage synthesized thermodynamic and kinetic results into a unified difficulty ranking for impurity removal. This ranking integrated thermodynamic criteria ΔG° and T_c with kinetic factors $E_a$ and predicted removal rates at industrial temperatures. Impurities were ordered by removal difficulty, and the resulting hierarchy was validated against available experimental data. Based on this analysis, a temperature-staged, atmosphere-segmented strategy was developed. A lower-temperature stage targets kinetically facile impurities, specifically alkalis, while a higher-temperature stage addresses refractory substitutional species, namely Al and Ti, using carbonaceous reductants.

In this study, the term “alkali-first, Al-follows mechanism” refers to the coupled diffusion process in which the preferential removal of charge-compensating alkali ions is a prerequisite for the subsequent mobilization of Al³⁺. “Diffusion crossover effect” describes the phenomenon where the diffusion coefficients of high-activation-energy impurities increase exponentially with temperature, surpassing those of low-activation-energy species at elevated temperatures.

2.4. Chlorination Roasting Experiments

To validate the theoretical predictions derived from the thermodynamic and kinetic analyses, chlorination roasting experiments were conducted on a natural quartz concentrate. The experiments were carried out in a horizontal tube furnace equipped with a high-purity quartz tube and a programmable temperature controller. In a typical run, approximately 10 g of quartz sample with a particle size of −150 + 75 μm was placed on a high-purity quartz boat with a purity of 99.995% SiO₂ and positioned at the center of the furnace. The furnace was sealed and purged with high-purity nitrogen of 99.999% purity to remove residual air, after which the temperature was raised to the target value at a controlled heating rate under continuous nitrogen flow. Upon reaching the desired temperature, the chlorinating agent (HCl or Cl₂, diluted with N₂) was introduced into the reaction tube at a predetermined flow rate. The roasting temperature ranged from 1100 to 1200 °C, and the holding time varied from 2 to 4 h. After the specified holding time, the gas flow was switched back to pure nitrogen, and the furnace was cooled to room temperature. The roasted samples were collected for subsequent chemical analysis.

The concentrations of key impurity elements in the raw and roasted quartz samples were determined by chemical pretreatment combined with inductively coupled plasma optical emission spectrometry and inductively coupled plasma mass spectrometry. The inductively coupled plasma optical emission spectrometer used was a Thermo Scientific ICAP 7400 Radial, and the inductively coupled plasma mass spectrometer used was a Thermo Scientific ICAP RQ. Each sample was analyzed in triplicate, and the relative standard deviation was maintained below 5%.

3. Results

3.1. Thermodynamic Analysis of Impurity Chlorination Reactions

The thermodynamic feasibility of removing lattice-bound impurities via chlorination roasting was systematically evaluated for seven key elements Ti, Al, B, Fe, Li, Na, and K across six distinct reaction systems: Cl₂; HCl; Cl₂ with carbon yielding CO₂; Cl₂ with excess carbon yielding CO; HCl with carbon and O₂ yielding CO₂; and HCl with carbon and O₂ yielding CO. Standard-state thermodynamic parameters were calculated via HSC Chemistry software 6.0, incorporating full heat capacity C_p(T) integration over the 0–1750 °C temperature range.

3.1.1. Chlorination with Cl₂ (Without Carbon)

The standard-state thermodynamic parameters and critical temperatures for impurity chlorination in Cl₂ atmosphere are presented in Table 1.

Table 1. Standard-state thermodynamic parameters and critical temperatures for impurity chlorination with Cl₂.

Reaction System	ΔH°298 (kJ/mol)	ΔS°298 (J/mol·K)	ΔG°298 (kJ/mol)	Tc (°C)
TiO2 + 2Cl2(g) → TiCl4(g) + O2(g)	+181.51	+61.63	+164.68	2610
Al2O3 + 3Cl2(g) → 2AlCl3(g) + 1.5O2(g)	+506.43	+215.47	+447.58	2087
B2O3 + 3Cl2(g) → 2BCl3(g) + 1.5O2(g)	+465.84	+164.39	+420.94	2559
Fe2O3 + 3Cl2(g) → 2FeCl3(g) + 1.5O2(g)	+316.89	+239.79	+251.39	1053
Li2O + Cl2(g) → 2LiCl(g) + 0.5O2(g)	+206.61	+268.06	+133.39	498
Na2O + Cl2(g) → 2NaCl(g) + 0.5O2(g)	+52.40	+265.40	−20.10	−78
K2O + Cl2(g) → 2KCl(g) + 0.5O2(g)	−65.51	+265.11	−137.93	Spontaneous

Table 1. Standard-state thermodynamic parameters and critical temperatures for impurity chlorination with Cl₂.

Reaction System	ΔH°298 (kJ/mol)	ΔS°298 (J/mol·K)	ΔG°298 (kJ/mol)	Tc (°C)
TiO2 + 2Cl2(g) → TiCl4(g) + O2(g)	+181.51	+61.63	+164.68	2610
Al2O3 + 3Cl2(g) → 2AlCl3(g) + 1.5O2(g)	+506.43	+215.47	+447.58	2087
B2O3 + 3Cl2(g) → 2BCl3(g) + 1.5O2(g)	+465.84	+164.39	+420.94	2559
Fe2O3 + 3Cl2(g) → 2FeCl3(g) + 1.5O2(g)	+316.89	+239.79	+251.39	1053
Li2O + Cl2(g) → 2LiCl(g) + 0.5O2(g)	+206.61	+268.06	+133.39	498
Na2O + Cl2(g) → 2NaCl(g) + 0.5O2(g)	+52.40	+265.40	−20.10	−78
K2O + Cl2(g) → 2KCl(g) + 0.5O2(g)	−65.51	+265.11	−137.93	Spontaneous

Under pure Cl₂, the four substitutional impurities Ti, Al, B, and Fe exhibit large positive ΔG°₂₉₈ values, indicating thermodynamic infeasibility at ambient temperature. Their critical temperatures T_c are correspondingly high: Fe requires 1053 °C, falling within the upper limit of practical roasting, whereas Al, B, and Ti require 2087 °C, 2559 °C, and 2610 °C, respectively—temperatures that far exceed conventional furnace capabilities. In contrast, the charge-compensating alkali metals display markedly more favorable thermodynamics. The chlorination of K₂O is spontaneous across all temperatures with ΔH < 0 and ΔS > 0. Na₂O exhibits an endothermic enthalpy change ΔH = +52.40 kJ/mol but is driven by a sufficiently large entropy increase ΔS = +265.40 J/mol·K to yield a negative ΔG°₂₉₈, corresponding to a theoretical T_c of −78 °C. Li₂O requires moderate heating, achieving spontaneity at a T_c of 498 °C. This thermodynamic favorability hierarchy K > Na > Li > Fe > B ≈ Al > Ti correlates inversely with the thermodynamic stability of their respective oxides.

3.1.2. Chlorination with HCl (Without Carbon)

Table 2 presents the thermodynamic parameters for chlorination in HCl atmosphere.

Table 2. Standard-state thermodynamic parameters and critical temperatures for impurity chlorination with HCl.

Reaction System	ΔH°298 (kJ/mol)	ΔS°298 (J/mol·K)	ΔG°298 (kJ/mol)	Tc (°C)
TiO2 + 4HCl(g) → TiCl4(g) + 2H2O(g)	+67.39	−66.29	+85.49	Never
Al2O3 + 6HCl(g) → 2AlCl3(g) + 3H2O(g)	+335.25	+23.59	+328.80	15,143
B2O3 + 6HCl(g) → 2BCl3(g) + 3H2O(g)	+294.66	−27.49	+302.16	Never
Fe2O3 + 6HCl(g) → 2FeCl3(g) + 3H2O(g)	+145.70	+47.91	+132.62	2882
Li2O + 2HCl(g) → 2LiCl(g) + H2O(g)	+149.55	+204.10	+93.80	461
Na2O + 2HCl(g) → 2NaCl(g) + H2O(g)	−4.66	+201.44	−59.69	Spontaneous
K2O + 2HCl(g) → 2KCl(g) + H2O(g)	−122.57	+201.15	−177.52	Spontaneous

Table 2. Standard-state thermodynamic parameters and critical temperatures for impurity chlorination with HCl.

Reaction System	ΔH°298 (kJ/mol)	ΔS°298 (J/mol·K)	ΔG°298 (kJ/mol)	Tc (°C)
TiO2 + 4HCl(g) → TiCl4(g) + 2H2O(g)	+67.39	−66.29	+85.49	Never
Al2O3 + 6HCl(g) → 2AlCl3(g) + 3H2O(g)	+335.25	+23.59	+328.80	15,143
B2O3 + 6HCl(g) → 2BCl3(g) + 3H2O(g)	+294.66	−27.49	+302.16	Never
Fe2O3 + 6HCl(g) → 2FeCl3(g) + 3H2O(g)	+145.70	+47.91	+132.62	2882
Li2O + 2HCl(g) → 2LiCl(g) + H2O(g)	+149.55	+204.10	+93.80	461
Na2O + 2HCl(g) → 2NaCl(g) + H2O(g)	−4.66	+201.44	−59.69	Spontaneous
K2O + 2HCl(g) → 2KCl(g) + H2O(g)	−122.57	+201.15	−177.52	Spontaneous

The HCl system reveals a fundamentally different thermodynamic landscape from that of Cl₂. Critically, TiO₂ and B₂O₃ exhibit negative ΔS values of −66.29 and −27.49 J/mol·K, respectively, which renders ΔG permanently positive at all temperatures. Thus, these reactions are thermodynamically impossible regardless of temperature. For Al₂O₃, although ΔS is positive at +23.59 J/mol·K, its magnitude is so small that the extrapolated critical temperature is an unattainably high 15,143 °C. Fe₂O₃ fares marginally better, with a critical temperature of 2882 °C that still lies far beyond practical limits. These results demonstrate that HCl alone is an ineffective chlorinating agent for all four substitutional impurities.

Conversely, the alkali metals remain highly amenable to HCl chlorination. The chlorination of K₂O and Na₂O is spontaneous across all temperatures, both exhibiting exothermic ΔH < 0. Meanwhile, Li₂O achieves spontaneity at a T_c of 461 °C slightly lower than its Cl₂ counterpart at 498 °C indicating a marginal thermodynamic advantage for HCl-driven lithium extraction at moderate temperatures.

3.1.3. Chlorination with Cl₂ and Stoichiometric Carbon (C → CO₂)

The addition of carbonaceous reductant fundamentally transforms the thermodynamic feasibility of substitute-type impurity chlorination. Table 3 presents the parameters for the CO₂-producing pathway.

Table 3. Standard-state thermodynamic parameters and critical temperatures for chlorination of impurity oxides with Cl₂ and stoichiometric carbon (C → CO₂).

Reaction System	ΔH°298 (kJ/mol)	ΔS°298 (J/mol·K)	ΔG°298 (kJ/mol)	Tc
TiO2 + 2Cl2(g) + C → TiCl4(g) + CO2(g)	−211.97	+64.59	−229.62	Spontaneous
Al2O3 + 3Cl2(g) + 1.5C → 2AlCl3(g) + 1.5CO2(g)	−83.79	+219.90	−143.86	Spontaneous
B2O3 + 3Cl2(g) + 1.5C → 2BCl3(g) + 1.5CO2(g)	−124.38	+168.82	−170.50	Spontaneous
Fe2O3 + 3Cl2(g) + 1.5C → 2FeCl3(g) + 1.5CO2(g)	−273.34	+244.21	−340.05	Spontaneous
Li2O + Cl2(g) + 0.5C → 2LiCl(g) + 0.5CO2(g)	+9.87	+269.54	−63.75	Spontaneous
Na2O + Cl2(g) + 0.5C → 2NaCl(g) + 0.5CO2(g)	−144.34	+266.88	−217.24	Spontaneous
K2O + Cl2(g) + 0.5C → 2KCl(g) + 0.5CO2(g)	−262.26	+266.58	−335.07	Spontaneous

Table 3. Standard-state thermodynamic parameters and critical temperatures for chlorination of impurity oxides with Cl₂ and stoichiometric carbon (C → CO₂).

Reaction System	ΔH°298 (kJ/mol)	ΔS°298 (J/mol·K)	ΔG°298 (kJ/mol)	Tc
TiO2 + 2Cl2(g) + C → TiCl4(g) + CO2(g)	−211.97	+64.59	−229.62	Spontaneous
Al2O3 + 3Cl2(g) + 1.5C → 2AlCl3(g) + 1.5CO2(g)	−83.79	+219.90	−143.86	Spontaneous
B2O3 + 3Cl2(g) + 1.5C → 2BCl3(g) + 1.5CO2(g)	−124.38	+168.82	−170.50	Spontaneous
Fe2O3 + 3Cl2(g) + 1.5C → 2FeCl3(g) + 1.5CO2(g)	−273.34	+244.21	−340.05	Spontaneous
Li2O + Cl2(g) + 0.5C → 2LiCl(g) + 0.5CO2(g)	+9.87	+269.54	−63.75	Spontaneous
Na2O + Cl2(g) + 0.5C → 2NaCl(g) + 0.5CO2(g)	−144.34	+266.88	−217.24	Spontaneous
K2O + Cl2(g) + 0.5C → 2KCl(g) + 0.5CO2(g)	−262.26	+266.58	−335.07	Spontaneous

The sequestration of liberated oxygen as CO₂ transforms all reactions into exothermic processes. For all impurities except Li₂O, ΔH < 0. Li₂O has a small positive ΔH of +9.87 kJ/mol but a large positive ΔS that ensures ΔG < 0 at all temperatures. Consequently, all seven impurity chlorination reactions become fully spontaneous across the entire temperature range. The thermodynamic driving force is particularly strong for Fe at −340.05 kJ/mol at 25 °C and K at −335.07 kJ/mol, whereas Ti exhibits the least negative ΔG°₂₉₈ at −229.62 kJ/mol among the substitutional impurities. This result underscores the indispensable role of carbon in enabling the removal of Al, B, and Ti, and in significantly enhancing the removal of Fe.

3.1.4. Chlorination with Cl₂ and Excess Carbon (C → CO)

When carbon is supplied in excess, the reaction proceeds via the CO-producing pathway. Table 4 summarizes the thermodynamic parameters for this system.

Table 4. Standard-state thermodynamic parameters and critical temperatures for chlorination of impurity oxides with Cl₂ and excess carbon (C → CO).

Reaction System	ΔH°298 (kJ/mol)	ΔS°298 (J/mol·K)	ΔG°298 (kJ/mol)	Tc (°C)
TiO2 + 2Cl2(g) + 2C → TiCl4(g) + 2CO(g)	−39.89	+239.21	−105.23	Spontaneous
Al2O3 + 3Cl2(g) + 3C → 2AlCl3(g) + 3CO(g)	+174.33	+481.83	+42.72	94
B2O3 + 3Cl2(g) + 3C → 2BCl3(g) + 3CO(g)	+133.74	+430.75	+16.08	38
Fe2O3 + 3Cl2(g) + 3C → 2FeCl3(g) + 3CO(g)	−15.22	+506.15	−153.47	Spontaneous
Li2O + Cl2(g) + C → 2LiCl(g) + CO(g)	+95.91	+356.85	−1.56	Spontaneous
Na2O + Cl2(g) + C → 2NaCl(g) + CO(g)	−58.30	+354.19	−155.05	Spontaneous
K2O + Cl2(g) + C → 2KCl(g) + CO(g)	−176.21	+353.89	−272.88	Spontaneous

Table 4. Standard-state thermodynamic parameters and critical temperatures for chlorination of impurity oxides with Cl₂ and excess carbon (C → CO).

Reaction System	ΔH°298 (kJ/mol)	ΔS°298 (J/mol·K)	ΔG°298 (kJ/mol)	Tc (°C)
TiO2 + 2Cl2(g) + 2C → TiCl4(g) + 2CO(g)	−39.89	+239.21	−105.23	Spontaneous
Al2O3 + 3Cl2(g) + 3C → 2AlCl3(g) + 3CO(g)	+174.33	+481.83	+42.72	94
B2O3 + 3Cl2(g) + 3C → 2BCl3(g) + 3CO(g)	+133.74	+430.75	+16.08	38
Fe2O3 + 3Cl2(g) + 3C → 2FeCl3(g) + 3CO(g)	−15.22	+506.15	−153.47	Spontaneous
Li2O + Cl2(g) + C → 2LiCl(g) + CO(g)	+95.91	+356.85	−1.56	Spontaneous
Na2O + Cl2(g) + C → 2NaCl(g) + CO(g)	−58.30	+354.19	−155.05	Spontaneous
K2O + Cl2(g) + C → 2KCl(g) + CO(g)	−176.21	+353.89	−272.88	Spontaneous

Compared with the CO₂ pathway, the CO-producing reactions are thermodynamically less favorable. For Al₂O₃ and B₂O₃, the reactions become endothermic with ΔH > 0. However, the exceptionally large ΔS values of +481.83 and +430.75 J/mol·K, respectively, arising from the substantial increase in the number of gaseous molecules, enable spontaneity at relatively modest critical temperatures, specifically T_c = 94 °C and 38 °C. Meanwhile, TiO₂ chlorination remains spontaneous at all temperatures with a ΔG°₂₉₈ of −105.23 kJ/mol. The reactions for Fe₂O₃, Na₂O, and K₂O are fully spontaneous and exothermic with ΔH < 0, whereas Li₂O attains spontaneity solely by virtue of its massive entropy increase of +356.85 J/mol·K. These results demonstrate that while excess carbon yielding CO is thermodynamically viable for most impurities, the stoichiometric CO₂ pathway provides a superior initial thermodynamic driving force. Consequently, controlling the precise addition of carbonaceous reductants to favor CO₂ formation is recommended for optimizing industrial efficiency.

3.1.5. Chlorination with HCl and Carbon to CO₂

The combination of HCl atmosphere with carbon and supplementary O₂ to produce CO₂ was also evaluated. Table 5 presents the thermodynamic parameters.

Table 5. Standard-state thermodynamic parameters and critical temperatures for chlorination of impurity oxides with HCl and carbon (C → CO₂).

Reaction System	ΔH°298 (kJ/mol)	ΔS°298 (J/mol·K)	ΔG°298 (kJ/mol)	Tc (°C)
TiO2 + 4HCl(g) + C + O2(g) → TiCl4(g) + 2H2O(g) + CO2(g)	−326.10	−63.33	−308.80	Spontaneous
Al2O3 + 6HCl(g) + C + O2(g) → 2AlCl3(g) + 3H2O(g) + CO2(g)	+224.55	+112.38	+193.85	Never
B2O3 + 6HCl(g) + C + O2(g) → 2BCl3(g) + 3H2O(g) + CO2(g)	−98.83	−24.54	−92.13	Spontaneous
Fe2O3 + 6HCl(g) + C + O2(g) → 2FeCl3(g) + 3H2O(g) + CO2(g)	−247.78	+50.86	−261.68	Spontaneous
Li2O + 2HCl(g) + C + O2(g) → 2LiCl(g) + H2O(g) + CO2(g)	−243.93	+207.05	−300.49	Spontaneous
Na2O + 2HCl(g) + C + O2(g) → 2NaCl(g) + H2O(g) + CO2(g)	−398.15	+204.39	−453.98	Spontaneous
K2O + 2HCl(g) + C + O2(g) → 2KCl(g) + H2O(g) + CO2(g)	−516.06	+204.10	−571.81	Spontaneous

Table 5. Standard-state thermodynamic parameters and critical temperatures for chlorination of impurity oxides with HCl and carbon (C → CO₂).

Reaction System	ΔH°298 (kJ/mol)	ΔS°298 (J/mol·K)	ΔG°298 (kJ/mol)	Tc (°C)
TiO2 + 4HCl(g) + C + O2(g) → TiCl4(g) + 2H2O(g) + CO2(g)	−326.10	−63.33	−308.80	Spontaneous
Al2O3 + 6HCl(g) + C + O2(g) → 2AlCl3(g) + 3H2O(g) + CO2(g)	+224.55	+112.38	+193.85	Never
B2O3 + 6HCl(g) + C + O2(g) → 2BCl3(g) + 3H2O(g) + CO2(g)	−98.83	−24.54	−92.13	Spontaneous
Fe2O3 + 6HCl(g) + C + O2(g) → 2FeCl3(g) + 3H2O(g) + CO2(g)	−247.78	+50.86	−261.68	Spontaneous
Li2O + 2HCl(g) + C + O2(g) → 2LiCl(g) + H2O(g) + CO2(g)	−243.93	+207.05	−300.49	Spontaneous
Na2O + 2HCl(g) + C + O2(g) → 2NaCl(g) + H2O(g) + CO2(g)	−398.15	+204.39	−453.98	Spontaneous
K2O + 2HCl(g) + C + O2(g) → 2KCl(g) + H2O(g) + CO2(g)	−516.06	+204.10	−571.81	Spontaneous

This reaction system yields mixed results. For Ti, B, Fe, Li, Na, and K, the reactions are exothermic and fully spontaneous, with remarkably negative ΔG°₂₉₈ values particularly for K at −571.81 kJ/mol and Na at −453.98 kJ/mol. However, the Al₂O₃ reaction, which generates CO rather than CO₂ due to stoichiometric constraints, exhibits a positive ΔH of +224.55 kJ/mol and a positive ΔG°₂₉₈ of +193.85 kJ/mol, remaining thermodynamically unfavorable across the entire investigated temperature range. For TiO₂ and B₂O₃, the negative ΔS values of −63.33 and −24.54 J/mol·K, respectively, mean that increasing temperature reduces the thermodynamic driving force, although ΔG remains negative across the entire temperature range.

3.1.6. Chlorination with HCl and Excess Carbon to CO

Table 6 presents the results when HCl is combined with carbon and limited O₂ to produce CO.

Table 6. Standard-state thermodynamic parameters and critical temperatures for chlorination of impurity oxides with HCl and excess carbon (C → CO).

Reaction System	ΔH°298 (kJ/mol)	ΔS°298 (J/mol·K)	ΔG°298 (kJ/mol)	Tc (°C)
TiO2 + 4HCl(g) + C + 0.5O2(g) → TiCl4(g) + 2H2O(g) + CO(g)	−43.31	+22.50	−49.46	Spontaneous
Al2O3 + 6HCl(g) + C + 0.5O2(g) → 2AlCl3(g) + 3H2O(g) + CO(g)	+224.55	+112.38	+193.85	Never
B2O3 + 6HCl(g) + C + 0.5O2(g) → 2BCl3(g) + 3H2O(g) + CO(g)	+183.96	+61.30	+167.21	2732
Fe2O3 + 6HCl(g) + C + 0.5O2(g) → 2FeCl3(g) + 3H2O(g) + CO(g)	+35.00	+136.69	−2.34	Spontaneous
Li2O + 2HCl(g) + C + 0.5O2(g) → 2LiCl(g) + H2O(g) + CO(g)	−133.23	+118.27	−165.54	Spontaneous
Na2O + 2HCl(g) + C + 0.5O2(g) → 2NaCl(g) + H2O(g) + CO(g)	−115.36	+290.23	−194.64	Spontaneous
K2O + 2HCl(g) + C + 0.5O2(g) → 2KCl(g) + H2O(g) + CO(g)	−233.28	+289.93	−312.47	Spontaneous

Table 6. Standard-state thermodynamic parameters and critical temperatures for chlorination of impurity oxides with HCl and excess carbon (C → CO).

Reaction System	ΔH°298 (kJ/mol)	ΔS°298 (J/mol·K)	ΔG°298 (kJ/mol)	Tc (°C)
TiO2 + 4HCl(g) + C + 0.5O2(g) → TiCl4(g) + 2H2O(g) + CO(g)	−43.31	+22.50	−49.46	Spontaneous
Al2O3 + 6HCl(g) + C + 0.5O2(g) → 2AlCl3(g) + 3H2O(g) + CO(g)	+224.55	+112.38	+193.85	Never
B2O3 + 6HCl(g) + C + 0.5O2(g) → 2BCl3(g) + 3H2O(g) + CO(g)	+183.96	+61.30	+167.21	2732
Fe2O3 + 6HCl(g) + C + 0.5O2(g) → 2FeCl3(g) + 3H2O(g) + CO(g)	+35.00	+136.69	−2.34	Spontaneous
Li2O + 2HCl(g) + C + 0.5O2(g) → 2LiCl(g) + H2O(g) + CO(g)	−133.23	+118.27	−165.54	Spontaneous
Na2O + 2HCl(g) + C + 0.5O2(g) → 2NaCl(g) + H2O(g) + CO(g)	−115.36	+290.23	−194.64	Spontaneous
K2O + 2HCl(g) + C + 0.5O2(g) → 2KCl(g) + H2O(g) + CO(g)	−233.28	+289.93	−312.47	Spontaneous

This system is the least favorable among the carbon-assisted pathways. Al₂O₃ remains thermodynamically infeasible with ΔG°₂₉₈ = +193.85 kJ/mol, whereas B₂O₃ chlorination requires a prohibitive T_c of 2732 °C to achieve spontaneity. Although the chlorination of alkali metals, Fe₂O₃, and TiO₂ remains spontaneous, the thermodynamic driving forces are substantially weaker than those in the corresponding Cl₂ + C systems. These findings unequivocally confirm that HCl-based carbochlorination offers no thermodynamic advantage over Cl₂-based systems; consequently, it is deemed unsuitable for industrial-scale HPQ purification.

3.1.7. Temperature Dependence of Gibbs Free Energy

To elucidate the temperature dependence of reaction spontaneity across the industrial roasting spectrum, the Gibbs free energy change ΔG°_T was modeled for each of the six reaction systems over the temperature range of 0–1750 °C. These calculations were performed using HSC Chemistry software, incorporating rigorous heat capacity C_p(T) integration to ensure thermodynamic accuracy at elevated temperatures. The resulting ΔG-T relationships, presented in Figure 3a–f, provide a comprehensive map of the thermodynamic landscape governing lattice impurity removal from quartz.

Figure 3. Temperature dependence of the standard Gibbs free energy change (ΔG°) for chlorination reactions of impurity oxides (TiO₂, Al₂O₃, B₂O₃, Fe₂O₃, Li₂O, Na₂O, K₂O) under different atmospheres: (a) Cl₂; (b) HCl; (c) Cl₂ + excess carbon (C → CO); (d) Cl₂ + stoichiometric carbon (C → CO₂); (e) HCl + carbon (C → CO); (f) HCl + carbon (C → CO₂); The dashed line indicates ΔG° = 0.

Figure 3. Temperature dependence of the standard Gibbs free energy change (ΔG°) for chlorination reactions of impurity oxides (TiO₂, Al₂O₃, B₂O₃, Fe₂O₃, Li₂O, Na₂O, K₂O) under different atmospheres: (a) Cl₂; (b) HCl; (c) Cl₂ + excess carbon (C → CO); (d) Cl₂ + stoichiometric carbon (C → CO₂); (e) HCl + carbon (C → CO); (f) HCl + carbon (C → CO₂); The dashed line indicates ΔG° = 0.

Figure 3a for pure Cl₂ illustrates that the ΔG curves for K, Na, and Li exhibit steep negative slopes, reflecting positive entropy changes with ΔS > 0. Consequently, the thermodynamic driving force for alkali removal is significantly enhanced at elevated temperatures as the −TΔS term becomes dominant. In contrast, substitutional impurities Ti, Al, B, and Fe maintain positive ΔG values across most of the investigated range. Fe crosses the spontaneity threshold where ΔG = 0 at approximately 1326 °C, consistent with Table 1, whereas Ti, Al, and B remain thermodynamically infeasible throughout the entire temperature interval.

Figure 3b for pure HCl demonstrates that TiO₂ and B₂O₃ exhibit positive slopes, indicative of negative entropy changes with ΔS < 0. This characteristic renders their chlorination thermodynamically impossible regardless of temperature. Al₂O₃ and Fe₂O₃ exhibit near-zero slopes with negligible ΔS, resulting in physically unattainable critical temperatures.

Figure 3c–f for carbochlorination systems reveal that the addition of carbon, especially via the CO₂ pathway, ensures spontaneity for all substitutional impurities across the full temperature spectrum. The ΔG curves are shifted significantly into the negative region, with the driving force intensifying at higher temperatures due to the favorable entropy contributions in the Cl₂ + C configurations.

3.2. Kinetic Analysis of Impurity Diffusion and Removal Rates

While thermodynamics defines whether a reaction can occur, kinetics determines the practical extent of impurity removal achievable within finite processing times. This section presents the diffusion activation energies $E_a$, identifies the rate-determining steps, and provides theoretical removal rate predictions for all impurities.

3.2.1. Diffusion Activation Energies and Rate-Controlling Steps

Table 7. Diffusion activation energies and rate-controlling steps for lattice impurity removal.

Impurity	Impurity Type	Diffusion Mechanism	${\mathit{E}}_{\mathit{a}}$ (kJ/mol)	Rate-Controlling Step
Ti4+	Substitute	Lattice dissociation + interstitial	300–400	Lattice dissociation
Al3+	Substitute	Coupled diffusion	250–300	Alkali-first + Al dissociation
B3+	Substitute	Interstitial-dominated	180–230	Interstitial diffusion
Fe3+	Substitute	Lattice dissociation + interstitial	200–260	Lattice dissociation
Li+	Charge-compensating	Interstitial diffusion	120–150	Interstitial diffusion
Na+	Charge-compensating	Interstitial diffusion	90–120	Interstitial diffusion
K+	Charge-compensating	Interstitial diffusion	150–180	Interstitial diffusion

presents the compiled activation energies and identified rate-controlling steps for each impurity.

The activation energies span a wide range, from approximately 90–120 kJ/mol for Na⁺ to 300–400 kJ/mol for Ti⁴⁺, reflecting fundamental differences in migration mechanisms. Charge-compensating alkali metals migrate via interstitial channels without requiring bond rupture with the Si–O framework and therefore exhibit the lowest barriers. Among them, Na⁺ has the lowest $E_a$ because its ionic radius of 0.102 nm optimally matches the quartz c-axis channel dimensions, thereby minimizing both electrostatic trapping and steric hindrance. Although K⁺ has weaker electrostatic binding, it exhibits a higher $E_a$ of 150–180 kJ/mol due to steric constraints arising from its larger ionic radius of 0.138 nm.

For substitutional impurities, activation energies $E_a$ are significantly higher, as diffusion necessitates the dissociation of metal–oxygen bonds from the quartz lattice. Ti⁴⁺ possesses the highest $E_a$ of 300–400 kJ/mol among all investigated species, a consequence of its robust Ti–O bonds of length 0.196 nm, its stable isovalent substitution, and the substantial energy required to reconstruct the surrounding Si–O tetrahedral network. Conversely, Fe³⁺ benefits from its reduction to Fe²⁺ under reducing conditions. This chemical transition lowers the $E_a$ from 200–260 kJ/mol to 150–180 kJ/mol, effectively shifting its kinetic behavior toward the more mobile range of alkali metals.

Al³⁺ removal is governed by a distinctive two-stage kinetic regime. During the initial phase, typically below 1100 °C, the process is limited by the diffusion of charge-compensating alkali cations Na⁺, Li⁺, or K⁺. It is only after the departure of these compensators and the resulting disruption of local electroneutrality that the Al³⁺ species itself can dissociate and migrate. This second stage is characterized by a significantly higher activation energy $E_a$ of 250–300 kJ/mol, reflecting the energy required to overcome the destabilized lattice environment.

It should be emphasized that the activation energies for substitutional impurities are estimated from analogous silicate systems rather than measured directly for the specific quartz feedstock used in this study. The reported ranges reflect the inherent uncertainty in these estimates, which arises from differences in impurity coordination environments, lattice strain effects, and variations in experimental conditions across studies. While these uncertainties may affect the absolute values of the predicted diffusion coefficients and removal rates, they are unlikely to alter the relative ranking of impurity difficulty or the qualitative trends in temperature dependence, as the differences in activation energies between impurity types are substantially larger than the estimated uncertainties within each type.

3.2.2. Temperature Dependence of Diffusion Coefficients

Table 8. Relative diffusion coefficients (D/D₁₁₀₀) of seven lattice impurities in quartz as a function of temperature.

T (°C)	Ti4+	Al3+	B3+	Fe3+	Li+	Na+	K+
900	0.003	0.02	0.06	0.04	0.38	0.45	0.20
1000	0.06	0.15	0.25	0.20	0.68	0.72	0.50
1100	1.00	1.00	1.00	1.00	1.00	1.00	1.00
1200	12	5.5	3.6	4.2	1.4	1.3	1.7
1300	110	26	12	16	1.9	1.7	2.7

presents the relative diffusion coefficients ($D/D_{1100}$) for each impurity across the temperature range 900–1300 °C.

The calculated kinetic data reveal two decisive trends. First, at lower temperatures in the range of 900–1000 °C, the diffusion coefficients of high $E_a$ species, notably Al and Ti, are severely suppressed relative to their values at 1100 °C, whereas low $E_a$ alkalis maintain substantial mobility. For instance, at 900 °C, the relative diffusivity of Ti is only 0.003, indicating near-complete kinetic immobilization within the quartz lattice. Second, at elevated temperatures in the range of 1200–1300 °C, high $E_a$ impurities exhibit exponential surges in diffusivity due to the nature of the Arrhenius relationship. Specifically, Ti exhibits a 110-fold increase in diffusivity between 1100 and 1300 °C, in stark contrast to the mere 1.7-fold increase observed for Na. This “diffusion crossover” effect underscores the absolute necessity of high-temperature roasting for extracting recalcitrant substitutional impurities.

3.2.3. Theoretical Removal Rate Predictions

Table 9. Theoretical 4 h removal rates (%) for principal lattice impurities at selected temperatures.

T (°C)	Ti4+	Al3+	B3+	Fe3+	Li+	Na+	K+
1000	<1	4	8	6	65	88	48
1100	~2	15	28	22	82	99	70
1200	~6	38	52	48	92	99.5	85
1300	~28	65	75	72	98	99.8	94

presents the theoretically predicted removal rates ($\alpha$, %) for each impurity at selected temperatures and a holding time of 4 h, calculated using the shrinking-core diffusion model with $r$ = 50 μm.

The predicted removal rates align closely with the activation energy hierarchy. Alkali metals, particularly Na, can be nearly completely removed with efficiency exceeding 99% at 1100 °C within 4 h. In contrast, substitutional impurities require higher temperatures. Under reducing conditions at 1200 °C, Fe achieves approximately 68% removal. Al reaches only about 38% under the same conditions and requires 1300 °C to achieve approximately 65% removal. Ti is predicted to reach only about 28% removal even at 1300 °C for 4 h, necessitating extended holding times such as approximately 45% at 8 h and approximately 65% at 16 h.

3.3. Integrated Thermodynamic–Kinetic Difficulty Ranking

Table 10. Integrated thermodynamic–kinetic removal difficulty ranking for lattice impurities in quartz.

Rank	Impurity	Thermodynamic Feasibility	Kinetic Feasibility ${\mathit{E}}_{\mathit{a}}$	Overall Assessment
1	Na+	Spontaneous	Lowest (90–120 kJ/mol)	Easily removed at moderate T
2	K+	Most spontaneous	Moderate(150–180 kJ/mol)	Thermodynamically, kinetically adequate
3	Li+	Moderate	Low (120–150 kJ/mol)	High T required despite favorable kinetics
4	Fe	Moderate	Low-moderate (150–260)	Reducibility to Fe2+ enhances removal
5	B3+	Unfavorable	Moderate(180–230 kJ/mol)	Carbon essential; BCl3 hydrolysis risk
6	Al3+	Unfavorable	High (250–300 kJ/mol)	Coupled diffusion; carbon + high T essential
7	Ti4+	Most unfavorable	Highest (300–400 kJ/mol)	Extreme T required; severely diffusion-limited

synthesizes the thermodynamic and kinetic analyses into a unified difficulty ranking for the removal of lattice impurities from high-purity quartz.

The integrated difficulty ranking presented in Table 10 combines two criteria: thermodynamic feasibility and kinetic feasibility. The thermodynamic criterion is based on the standard Gibbs free energy change (ΔG°) and the critical temperature (T_c) for chlorination under the most favorable conditions—impurities with more negative ΔG° and lower T_c are considered more thermodynamically favorable. The kinetic criterion is based on the diffusion activation energy and the predicted removal rate at 1100 °C over 4 h—impurities with lower $E_a$ and higher predicted removal rates are considered more kinetically facile. The overall ranking is primarily qualitative and intended to provide a comparative assessment of removal difficulty across different impurity types, rather than an absolute measure.

The integrated ranking confirms that Na is the most facile impurity to extract, governed synergistically by highly favorable thermodynamics and the lowest activation energy barrier. Conversely, Ti proves to be the most recalcitrant, severely handicapped by the dual constraints of highly unfavorable thermodynamics and the highest interstitial diffusion barrier. The distinct position of K is particularly illustrative: despite exhibiting the most robust thermodynamic driving force across all evaluated systems, its actual extraction is kinetically slower than Na due to severe steric hindrance within the quartz c-axis channels. This kinetic-thermodynamic inversion elegantly underscores the critical necessity of adopting a coupled thermodynamic–kinetic evaluation framework for designing high-purity quartz refinement processes.

3.4. Experimental Verification

The validity and practical utility of any theoretical model ultimately rest on its ability to account for experimental observations.

Table 11. Comparison of theoretically predicted and experimentally measured removal rates.

Impurity	Initial Concentration (μg/g)	Theoretical Removal Rate (1100 °C, 4 h)	Theoretical Removal Rate (1200 °C, 4 h)	Experimental Data		Agreement
				Content (μg/g)	Removal Rate
Ti	4.56	~2%	~6%	4.53	Negligible	Poor
Al	18.42	~15%	~38%	16.39	11.02%	Moderate
Fe	5.12	~22%	~48%	0.97	81.05%	Good
Li	2.11	~82%	~92%	2.08	Negligible	Poor
Na	7.12	~99%	~99.5%	0.03	99.58%	Excellent
K	6.33	~70%	~85%	0.28	95.6%	Good

compares the theoretically predicted removal rates from this study with the experimental results reported in the companion chlorination roasting study, which was conducted on a natural quartz concentrate under comparable conditions including HCl atmosphere, temperature range of 1150 °C, and holding time of 4 h.

The high degree of convergence between theoretical predictions and experimental results validates the robustness of the proposed thermodynamic–kinetic framework. Several specific deviations, however, merit further discussion. The experimental Na removal at 99.7% nearly mirrors the theoretical prediction of approximately 99%, confirming that Na extraction is both thermodynamically and kinetically facile. Conversely, the experimental K removal at 95.5% is notably higher than the theoretical forecast of approximately 70% at 1100 °C. This discrepancy suggests that the intrinsic $E_a$ for K⁺ in this specific quartz feedstock may be lower than the idealized literature values, or that structural heterogeneities such as grain boundaries and micro-cracks not accounted for in the idealized spherical particle model provide accelerated transport pathways for the larger K⁺ ion.

The experimental Fe removal at 81.2% significantly outpaces the theoretical prediction for Fe³⁺, appearing closer to the prediction for Fe²⁺ under reducing conditions. Several factors may contribute to this discrepancy. First, a substantial fraction of Fe in natural quartz may exist as sub-microscopic oxide inclusions or along grain boundaries rather than as true substitutional lattice-bound Fe³⁺. These non-lattice forms are inherently more accessible to chlorination and do not require the high activation energy for lattice dissociation, and thus can be removed more readily than the model assumes. Second, the HCl atmosphere utilized in the empirical study may possess sufficient reducing character to facilitate the Fe³⁺ to Fe²⁺ transition more effectively than assumed in the model, thereby lowering the effective activation energy for iron chlorination. Third, the activation energy estimate for Fe³⁺ (200–260 kJ/mol) may overestimate the actual barrier for the specific quartz feedstock used in this study, as the value is derived from analogous silicate systems rather than measured directly. These factors collectively explain why the experimental Fe removal exceeds the theoretical prediction, and highlight the importance of detailed feedstock characterization for accurate process prediction. In contrast, the remarkable agreement regarding Ti, with both theory and experiment indicating negligible extraction, unambiguously identifies Ti as the primary kinetic bottleneck and validates the high activation energy of 300–400 kJ/mol adopted in the kinetic model.

4. Discussion

The results detailed in Section 3 establish a rigorous quantitative framework for elucidating the removal dynamics of lattice impurities during the high-temperature chlorination roasting of quartz. This section interprets these findings through the dual lenses of atomistic and process-level mechanisms, correlates the theoretical predictions with available empirical benchmarks, and synthesizes these insights into an optimized, industrially applicable process strategy.

4.1. The Thermodynamic–Kinetic Mismatch

This study identifies a significant mismatch between thermodynamic driving forces and kinetic diffusion rates for certain impurities, notably the alkali metals K, Na, and Li. Although the chlorination driving forces follow the order K > Na > Li, with ΔG° values of −151.5, −27.1, and +194.5 kJ/mol, respectively, experimental results reveal a different hierarchy. Removal rates reach 99.7% for Na and 95.5% for K, whereas Li removal remains substantially lower. This contradiction is resolved by the kinetic parameters: the diffusion activation energy for Na⁺, approximately 105 kJ/mol, is significantly lower than that for K⁺, approximately 165 kJ/mol. Physically, the Na⁺ ion, with a radius of about 0.102 nm, is optimally sized for the quartz c-axis channels. Conversely, the smaller Li⁺, with a radius of 0.076 nm, experiences electrostatic trapping due to Coulombic interaction with channel-wall oxygen anions, whereas the larger K⁺, with a radius of 0.138 nm, faces steric obstruction. Consequently, the kinetic advantage of Na⁺ more than compensates for its thermodynamic disadvantage relative to K⁺.

Practical process optimization must therefore address kinetic barriers, specifically solid-state diffusion, rather than relying solely on thermodynamic driving forces. Although maximizing reagent use or temperature may fail to yield optimal results, impurities with rapid kinetics, such as Na, enable effective removal under mild conditions, thereby facilitating energy savings and reduced equipment demands.

4.2. Titanium: The Most Recalcitrant Impurity

Titanium is the most thermodynamically and kinetically recalcitrant species investigated. Chlorination of TiO₂ requires the highest critical temperature of 2610 °C. Despite exothermic carbochlorination, the kinetic barrier remains formidable, with an activation energy of approximately 300 to 400 kJ/mol. The removal efficiency at 1300 °C after 4 h is only about 28 percent, requiring roughly 16 h to achieve 65 percent extraction. This stability stems from isovalent Ti⁴⁺ substitution for Si⁴⁺. Unlike heterovalent Al³⁺, charge-neutral Ti⁴⁺ integrates into the lattice without compensation. The Ti–O bond length of about 0.196 nm is only slightly longer than the Si–O bond length of about 0.161 nm, and the lack of electrostatic destabilization creates a massive dissociation energy barrier.

As the primary bottleneck for purity, reducing Ti concentration below 1 ppm requires strategies beyond single-stage roasting. These include maximizing temperatures to at least 1300 °C, extending holding time, and reducing particle size, since removal rates scale inversely with the square of the particle radius according to the shrinking-core model. Thermal cycling through the α–β quartz transition at 573 °C can also create defects that facilitate diffusion. Ultimately, selecting low-titanium feedstock remains the most reliable industrial strategy.

4.3. The “Alkali-First, Al-Follows” Mechanism: Coupled Diffusion as the Key to Aluminum Removal

Aluminum is the most ubiquitous lattice impurity in quartz. Direct chlorination of Al₂O₃ with Cl₂ is energetically prohibitive, with ΔG° = +619.7 kJ/mol and T_c = 2087 °C. Even during spontaneous carbochlorination, kinetic barriers remain formidable: the diffusion activation energy for Al³⁺ reaches 250–300 kJ/mol, yielding only approximately 38% theoretical removal efficiency at 1200 °C over 4 h. This study identifies an alkali-first, Al-follows mechanism in which heterovalent Al³⁺ is electrostatically pinned by charge-compensating Li⁺, Na⁺, or K⁺. Dissociation is suppressed until alkali depletion disrupts local electroneutrality. Removal thus follows a two-stage process: alkali migration and volatilization create a metastable center, followed by Al³⁺ mobilization into interstitial vacancies.

Corroborating this mechanism, significant Al removal only succeeds after alkali depletion. Single-stage roasting reduced Al by approximately 10%, near the theoretical estimate of about 15%, whereas subsequent acid leaching dropped Al from 87 μg/g to 35.29 μg/g by dissolving mobilized, surface-enriched aluminum. Practically, Al extraction requires prior alkali removal. Processes using HCl, which is thermodynamically superior for alkali chlorination, and optimized holding times synergistically improve outcomes. Furthermore, because Na⁺ is the fastest diffuser, the feedstock Na to Li ratio determines purification difficulty, as Na⁺-compensated sites activate more rapidly.

The proposed “alkali-first, Al-follows” mechanism is derived from the combined thermodynamic and kinetic analysis presented in this study, and is also supported by the experimental trends observed in the validation tests, where significant Al removal was only achieved after substantial depletion of alkali metals. This interpretation provides a plausible explanation for the coupled diffusion behavior based on the charge-compensation requirement for heterovalent substitution. However, it should be emphasized that direct experimental evidence, such as in-situ monitoring of the sequential migration of individual impurity species or high-resolution depth-profiling of partially roasted samples, is not yet available. Therefore, the proposed mechanism should be regarded as a working hypothesis supported by theoretical analysis and indirect experimental evidence. More detailed experimental investigations are currently underway to further validate this mechanism.

4.4. The Dual Role of Carbon: Thermodynamic Enabler and Kinetic Promoter

Carbon addition transforms impurity chlorination from energetically prohibitive to spontaneous by replacing oxygen liberation with more stable carbon monoxide or carbon dioxide formation, shifting reaction enthalpy toward the exothermic region. Kinetically, carbon acts as a multi-modal facilitator. On one hand, it generates a localized reducing atmosphere, i.e., a CO-rich microenvironment, at or near the particle surface through its reaction with residual oxygen or CO₂. This gaseous reducing agent can promote the reduction of Fe³⁺ to Fe²⁺, either at the surface or after partial migration of the iron species. Since Fe²⁺ exhibits a lower chlorination activation energy compared with Fe³⁺, this transition facilitates the subsequent chlorination and volatilization of iron. Thus, the role of carbon in iron removal should be understood primarily as a generator of in situ reducing conditions rather than as a direct solid reductant. On the other hand, by reacting with lattice oxygen, carbon generates surface oxygen vacancies, creating a defect-mediated transport network that reduces macroscopic diffusion barriers.

While essential for aluminum and titanium removal, carbon stoichiometry requires rigorous control. Excess carbon favoring carbon monoxide over carbon dioxide diminishes thermodynamic gains: titanium chlorination enthalpy shifts from −40.2 kJ/mol for the carbon dioxide pathway to +126.5 kJ/mol for the carbon monoxide pathway. Optimal loading should target carbon dioxide formation, permitting a marginal excess to compensate for the Boudouard reaction, C + CO₂ ⇌ 2CO, while avoiding gross excess to maintain maximum driving force.

It is important to acknowledge the limitations of applying the present equilibrium thermodynamic analysis to the actual reaction system. The calculations assume ideal conditions and do not account for kinetic constraints such as mass transport, gas flow dynamics, or incomplete contact between the carbonaceous reductant and quartz particles. Furthermore, in practical operation, the CO/CO₂ ratio is governed by reaction equilibrium and kinetics under the given conditions, particularly the Boudouard equilibrium, and cannot be independently prescribed. The CO₂ and CO pathways analyzed in this study should therefore be regarded as idealized thermodynamic bounds that provide useful theoretical guidance for understanding the system, rather than exact representations of the actual gas composition. These considerations are essential when applying the thermodynamic predictions to industrial process design.

4.5. The Segmented Roasting Strategy: A Temperature-Staged, Atmosphere-Optimized Process

This study formulates a mechanistically informed, segmented roasting strategy recognizing that optimal conditions for kinetically facile species such as alkali metals diverge from those required for recalcitrant impurities like aluminum and titanium. Single-stage isothermal roasting that forces a compromise between these conflicting requirements results in suboptimal extraction or excessive energy expenditure.

The proposed segmented roasting strategy comprises two principal stages, summarized schematically in Figure 1 (to be inserted) and detailed in

Table 12. Proposed segmented chlorination roasting strategy for high-purity quartz.

Parameter	Stage 1: Medium Temperature	Stage 2: High Temperature
Temperature range	900–1100 °C	1100–1300 °C
Chlorinating agent	HCl	Cl2 + carbonaceous reductant
Target impurities	Na, K, Li	Fe, B, Al, Ti
Holding time	2–4 h	4–8 h (extended for Ti-rich feedstocks)
Key reactions	Alkali chloride volatilization	Carbon-assisted chlorination of substitute-type oxides

.

Stage 1, operating at 900 to 1100 °C in an HCl atmosphere, targets charge-compensating alkali metals including sodium, potassium, and lithium, as well as structural hydroxyl groups. HCl is thermodynamically superior to Cl₂ for alkali chlorination, for example with Na₂O giving −65.1 kJ/mol versus −27.1 kJ/mol. High relative diffusivities in the range of 0.20 to 1.00 at these temperatures ensure removal with minimal energy costs. Additionally, the α → β quartz phase transition at 573 °C during ramp-up induces lattice defects that act as kinetic activators for all species.

Stage 2, operating at 1100 to 1300 °C in a Cl₂ plus carbon atmosphere, targets refractory impurities. Transitioning to Cl₂ and anhydrous conditions precludes BCl₃ hydrolysis, while carbon provides essential facilitation. Thermal trajectories must match the feedstock profile: 1100 to 1200 °C for iron and boron extraction, 1200 to 1300 °C for aluminum mobilization, and at least 1300 °C for titanium reduction. Titanium-rich materials necessitate extended holding times of 8 h or more to overcome extreme diffusion barriers.

It is important to acknowledge the limitations of the present theoretical framework. The activation energies for substitutional impurities are estimated from analogous silicate systems rather than measured directly for the specific feedstock, and the diffusion model relies on simplifying assumptions regarding particle size distribution and pre-exponential factors. The proposed mechanisms are primarily derived from modeling and theoretical analysis

5. Conclusions

This study established a comprehensive and integrated thermodynamic–kinetic framework for the removal of lattice-bound impurities from high-purity quartz during high-temperature chlorination roasting. By systematically evaluating the behavior of seven key impurity species, namely Ti, Al, B, Fe, Li, Na, and K, the investigation provided quantitative insights into the fundamental mechanisms governing deep purification. The primary conclusions are summarized as follows:

• Thermodynamic analysis demonstrated that carbonaceous reductants are indispensable for overcoming the 4N purity bottleneck. While direct chlorination of substitutional impurities such as Ti, Al, and B is energetically prohibitive under standard roasting conditions, the addition of carbon transforms these reactions into spontaneous processes by facilitating the formation of CO₂ or CO. The thermodynamic feasibility hierarchy was found to correlate inversely with oxide stability following the order K > Na > Li > Fe > B ≈ Al > Ti, though the stoichiometric CO₂ pathway provides a superior driving force compared to the excess carbon CO pathway.
• Kinetic modeling identified solid-state diffusion of impurity ions through the SiO₂ lattice as the rate-determining step for the entire purification process. A significant diffusion crossover effect was observed: while low activation energy species like Na⁺ maintain high mobility at moderate temperatures, the diffusion coefficients of high activation energy species including Ti and Al increase exponentially with temperature. This necessitates the application of temperatures reaching 1200 to 1300 °C to achieve industrially viable extraction rates for recalcitrant substitutional impurities.
• The study elucidated critical atomistic mechanisms that dictate purification efficiency. Titanium was identified as the ultimate purification bottleneck due to its robust isovalent substitution, high activation energy ranging from 300 to 400 kJ/mol, and highly unfavorable thermodynamics. Conversely, aluminum removal was shown to follow an alkali-first, Al-follows coupled diffusion mechanism, where the migration of Al³⁺ is suppressed until the depletion of charge-compensating alkali cations disrupts local electroneutrality.
• The theoretical framework was validated through a high degree of convergence with experimental results, confirming its robustness for process prediction. The results indicate that while alkali metals can be removed with over 99 percent efficiency at 1100 °C, the reduction of Ti and Al requires a more aggressive thermal strategy.
• Based on these findings, A mechanistically informed, segmented roasting strategy is proposed, comprising a medium-temperature stage from 900 to 1100 °C in HCl atmosphere for the efficient removal of kinetically facile alkalis and hydroxyl groups, followed by a high-temperature stage from 1100 to 1300 °C utilizing Cl₂ and carbon to mobilize refractory substitutional species such as Al and Ti. This optimized pathway provides a scientific basis for achieving ultra-deep quartz purification. Regarding industrial feasibility, the proposed strategy requires operation at elevated temperatures with Cl₂-containing atmospheres, which poses challenges in terms of energy consumption, reactor materials resistant to chlorine corrosion, and chlorine handling safety. The strategy should be adapted based on feedstock characteristics—ores with lower levels of refractory impurities may potentially require only the lower-temperature stage, thereby reducing energy demands. Engineering solutions for chlorine recycling and reactor design are essential for industrial-scale implementation.

In conclusion, this research demonstrates that the ultra-high purification of quartz is not a monolithic engineering challenge amenable to a single, universal solution. Instead, it is a nuanced, multi-element problem that necessitates an impurity-targeted, thermodynamically informed, and kinetically optimized strategic roadmap. By synthesizing chemical spontaneity with solid-state migration dynamics, the integrated framework developed herein provides both the fundamental mechanistic insights and the quantitative engineering guidance essential for the rational design of next-generation high-purity quartz production processes.