Gallium Isotope Effect of Ga-Si Complex Solutions in Water: Theoretical Study Based on Density Functional Theory

Abstract

A Ga isotope is a new proxy for different geochemical processes such as a weathering process, solution process, etc. Si (Si(OH)₄) is ubiquitous in natural water bodies. However, studies on the Ga isotope effect about a Ga³⁺ aqueous solution reacting with Si (Si(OH)₄) are lacking. In this study, the Ga isotope effect of this process will be studied using a theoretical calculation method based on first principles. The results show that the heavy Ga (⁷¹Ga) isotope enrichment ability of different Ga-Si complex solutions is different. The 1000lnβ (‰) sequence of different Ga-Si complex solutions is (OH)₃GaOSi(OH)₃.(H₂O)₃₀ ≈ (OH)₃(H₂O)₂GaOSi(OH)₃.(H₂O)₃₀ > (OH)₂(H₂O)₃GaOSi(OH)₃.(H₂O)₃₀ > (H₂O)₅GaOSi(OH)₃.(H₂O)₃₀ > (OH)(H₂O)₄GaOSi(OH)₃.(H₂O)₃₀. The results show that there are two different reaction mechanisms when a Ga³⁺ aqueous solution reacts with Si-bearing (Si(OH)₄) water; that is, six-coordination Ga-Si complexes and four-coordination Ga-Si complexes are formed at low pH (acidic) and high pH (alkaline), respectively. Compared with a Ga-Si complex aqueous solution under acidic conditions, Ga-Si aqueous solutions under alkaline conditions preferentially enriched the heavy Ga isotope (⁷¹Ga). The Ga isotope fractionation factors (α) between Ga-Si complex solutions and Ga³⁺-bearing aqueous solutions are all negative, which indicates that light Ga (⁶⁹Ga) isotopes preferentially enter the structure of Ga-Si complexes during the formation of Ga-Si complex solutions. At 50 °C, the Ga isotope fractionation factors (1000lnα) of five systems ((H₂O)₅GaOSi(OH)₃.(H₂O)₃₀ vs. [Ga(H₂O)₆]³⁺_(aq), (OH)(H₂O)₄GaOSi(OH)₃.(H₂O)₃₀ vs. [Ga(H₂O)₆]³⁺_(aq), (OH)₃GaOSi(OH)₃.(H₂O)₃₀ vs. [Ga(OH)₃](aq), (OH)₃(H₂O)₂GaOSi(OH)₃.(H₂O)₃₀ vs. [Ga(OH)₃]_(aq), and (OH)₂(H₂O)₃GaOSi(OH)₃.(H₂O)₃₀ vs. [Ga(OH)₃]_(aq)) involved in this study are −0.12, −0.22, −0.07, −0.09, and −0.16 (‰), respectively. Excitedly, Si can affect the enrichment ability of the heavy Ga isotope (⁷¹Ga) in Ga-bearing complex aqueous solutions. This means that when Si is present in aqueous solutions, the enrichment capacity of the heavy Ga isotope (⁷¹Ga) of aqueous solutions will be effectively reduced. Ga in sediments is mainly derived from soluble Ga in the form of adsorbed (Fe, Mn) oxides/hydroxides, and the Ga isotope composition in sediments is heavier than that in basalt. The formation process of Ga-Si complex aqueous solutions influences the Ga isotope fractionation effect and also contributes to the composition of Ga isotopes in sediments. These key Ga isotope fractionation parameters obtained in this study will provide theoretical support for better explaining the reaction mechanism of Ga³⁺ complexes and Si-bearing (Si(OH)₄) water bodies in solution processes and Ga isotope geochemical cycles.

1. Introduction

Gallium (Ga) is located in group IIIA of the periodic table, the same group as Aluminum (Al), which is the most abundant metallic element in the Earth’s crust. Al is involved in many geochemical processes in Earth science. Unfortunately, Al has no stable isotope. The geochemical process of Al can be studied by using the Ga isotope and relevant research work has appeared in this respect [1]. Based on the absolute isotopic composition of Ga, the atomic mass of Ga is 69.72/69.74 [2,3]. Ga has two stable isotopes, ⁶⁹Ga and ⁷¹Ga, with relative abundances of 60.1 and 39.9%, respectively [4,5]. The relative mass difference between the two isotopes of Ga is very large.

Now, Ga is a new proxy indicating various geological processes. It is only in the past decade that the study of Ga isotopes has become more and more focused on. Early studies on Ga mostly stayed in the field of obtaining some basic physicochemical properties such as atomic mass, isotope abundance, and commercial applications of Ga isotopes [2,3,4,5,6,7,8]. With the development of isotope analysis accuracy and improvement in methods, the Ga isotope has been applied more and more in geology, astrochemistry, environmental science, and other fields [1,6,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]. Like Al, Ga is not particularly mobile during weathering [1,28]. Połedniok studied Ga-bearing speciations in soil by a sequential extraction procedure developed by Tessier et al. [28,29]. It was found that the concentration of Ga in soil was mainly related to the regional and total gallium concentration. The enrichment in Ga in some areas is mainly due to the extreme migration of Ga [30]. It is worth mentioning that Ga is a moderately volatile element and has important applications in astrochemistry, such as studying the early formation and evolution of celestial bodies [11]. Some progress has also been made in studying the behavior of the Ga isotope in the evaporation process. In recent years, research using Ga isotopes to study the genesis of ore deposits, the weathering of surface rocks, and the process of soil formation has also emerged [12,13,15,16]. It is found that compared with basalts, the sediments are more enriched with the heavy Ga isotope (⁷¹Ga) [13]. The main reason for this phenomenon is that the Ga in the sediment is mainly from the soluble Ga in the aqueous solution; that is, the solution process has an important effect on the Ga isotopic composition of the sediment. These works are of great significance to the study of the Ga isotope geochemical cycle. Ga is a kind of trace element in the Earth’s crust and it rarely forms independent minerals. In most cases, Ga could enter the lattice of other minerals as a doped element. That is why Ga was discovered so late. The formation of many Ga-bearing minerals is closely related to the sedimentary process. Therefore, the sedimentary environment is closely related to the Ga isotope composition of minerals. Among them, the solution/hydrothermal solution plays an important role in the formation of minerals. This is also an important reason for the urgent completion of this work.

Although great progress has been made in the experimental study of Ga isotope compositions for different samples (terrestrial samples and extraterrestrial samples), the mechanism of the difference in Ga isotopic composition of different samples is still not well understood. Therefore, there is an urgent need to explain the mechanism of Ga isotope fractionation from the molecular level. Fortunately, since the birth of two outstanding works in the field of the theoretical calculation of geochemistry [31,32], the use of theoretical calculation methods to obtain the key parameter in the field of geochemistry (equilibrium stable isotope fractionation factor, α) has become a routine method and has achieved fruitful research results [33,34,35,36,37,38]. The emergence of these theoretical works has strongly promoted the rapid development of the subject of geochemistry.

In this research, the Ga isotope effect between a Si-bearing aqueous solution and Ga³⁺-bearing aqueous solutions is studied based on theoretical calculation. When Si (Si(OH)₄) is contained in an aqueous solution, the aqua-complexes/hydroxides (such as [Ga(H₂O)₆]³⁺_(aq) and [Ga(OH)₃]_(aq)) formed by Ga³⁺ and H₂O/OH under different pH will react with Si(OH)₄ to form various Ga-Si complexes. Potential Ga-Si complexes are (OH)₃GaOSi(OH)₃, (OH)₃(H₂O)₂GaOSi(OH)₃, (OH)₂(H₂O)₃GaOSi(OH)₃, (OH)(H₂O)₄GaOSi(OH)₃, and (H₂O)₅GaOSi(OH)₃ [39]. The speciation forms of these species can be determined using the ab initio method based on first principles, and the Ga isotope effects of these processes can be obtained using the Urey model [31,32]. In the simulation of an aqueous environment (solvation effect), the theoretical method used is the “water-droplet” method. This theoretical method has been proven to be a feasible and reasonable technical means by many researchers [26,36,38,40]. These key Ga isotope fractionation parameters obtained in this study will reveal the aqueous environment of Ga-Si complex solutions and the Ga isotope fractionation mechanism from the molecular level. Further, the database of Ga isotope geochemical cycle research will be enriched.

2. Methods

In the field of theoretical computational geochemistry, there have been two landmark research contributions, namely the work of Urey, Bigeleisen, and Mayer [31,32]. The important significance of their works is that the important parameter of isotope geochemistry (α) is derived and deduced from the basic theory of chemical thermodynamics (partition function). Since then, the theoretical calculation of geochemistry has become another important method to obtain the equilibrium stable isotope fractionation factor in addition to the experimental method and has achieved fruitful research results [26,33,34,36,37,38,41,42,43]. The theoretical basis of this research is based on the results of these two works [31,32]. In the process of obtaining the equilibrium stable isotope fractionation factor (α), the most important intermediate parameter is the reduced partition function ratio (RPFR, also known as β). For the research system involved in this study, take the Ga isotope exchange reaction between (H₂O)₅GaOSi(OH)₃.(H₂O)₃₀ and [Ga(H₂O)₆]³⁺_(aq) as an example:

$${\left(H_{2}O\right)}_5{}^{69}GaOSi{\left(OH\right)}_3.{\left(H_{2}O\right)}_{30}+{\left[{}^{71}Ga{(H_{2}O)}_6\right]}_{(aq)}^{3+}={\left(H_{2}O\right)}_5{}^{71}GaOSi{\left(OH\right)}_3.{\left(H_{2}O\right)}_{30}+{\left[{}^{69}Ga{(H_{2}O)}_6\right]}_{(aq)}^{3+}$$

(1)

Relative to the Ga isotope exchange reaction shown in Formula (1), the mathematical expression of its isotope fractionation parameter (β) is as follows:

$$\mathrm{RPFR}=\mathrm{\beta }=\left(\frac{{\mathrm{s}}^{71}}{{\mathrm{s}}^{69}}\right){\displaystyle \prod _{\mathrm{i}}^{3\mathrm{n}-6}\frac{{{\mathrm{u}}_{\mathrm{i}}}^{69}}{{}^{71}\mathrm{u}_{\mathrm{i}}}\frac{\exp \left(-{{\mathrm{u}}_{\mathrm{i}}}^{69}/2\right)}{\exp \left(-{{\mathrm{u}}_{\mathrm{i}}}^{71}/2\right)}\frac{1-\exp \left(-{{\mathrm{u}}_{\mathrm{i}}}^{71}\right)}{1-\exp \left(-{{\mathrm{u}}_{\mathrm{i}}}^{69}\right)}}$$

(2)

In Formula (2), 69 and 71 represent the two stable isotopes of Ga, namely ⁶⁹Ga and ⁷¹Ga, respectively. “s” represents the symmetry number of the system (molecule or molecular cluster). For the Ga-Si complex aqueous solution involved in this study, the molecular symmetry number does not change before and after the Ga isotope exchange reaction. When obtaining the isotope fractionation parameter (β) according to the theory of statistical thermodynamics, the largest contribution to β comes from the vibration frequency and the contribution of molecular translation and rotation terms can be ignored. Therefore, when calculating the β value of a Ga-Si complex aqueous solution, there is only one unknown, that is, simple harmonic vibration frequency. “u_i” is a function of simple harmonic vibration frequency, and its theoretical expression is as follows:

$${\mathrm{u}}_{\mathrm{i}}=\frac{\mathrm{h}{\mathrm{\nu }}_{\mathrm{i}}}{{\mathrm{k}}_{\mathrm{b}}\mathrm{T}}$$

(3)

In Formula (3), h, k_b, and T are known quantities, namely Planck’s constant, Boltzmann’s constant, and temperature in Kelvin. “v_i” is the simple harmonic vibration frequencies of the system.

$$1000\ln \beta =a{x}^3+b{x}^2+cx+d$$

(4)

In Formula (4), a, b, c, and d represent the coefficient terms of variables of different powers, respectively. x is a function of temperature and the mathematical expression is

$$x=\raisebox{1ex}{${10}^6$}\!\left/ \!\raisebox{-1ex}{$T^2$}\right.$$

(5)

In Formula (5), T is the absolute temperature (temperature in Kelvin).

The equilibrium stable isotope fractionation factor (1000lnα) is widely used in the field of geochemistry. The theoretical calculation formula (taking system (H₂O)₅GaOSi(OH)₃.(H₂O)₃₀ vs. [Ga(H₂O)₆]³⁺_(aq) as an example) is shown as follows:

$$\begin{array}{l}1000\ln {\alpha }_{{\left(H_{2}O\right)}_{5}GaOSi{\left(OH\right)}_3.{\left(H_{2}O\right)}_{30}vs {\left[Ga{\left(H_{2}O\right)}_6\right]}_{(aq)}^{3+}}\\ =1000\ln \frac{{\beta }_{{\left(H_{2}O\right)}_{5}GaOSi{\left(OH\right)}_3.{\left(H_{2}O\right)}_{30}}}{{\beta }_{{\left[Ga{\left(H_{2}O\right)}_6\right]}_{(aq)}^{3+}}}\\ =1000\ln {\beta }_{{\left(H_{2}O\right)}_{5}GaOSi{\left(OH\right)}_3.{\left(H_{2}O\right)}_{30}}-1000\ln {\beta }_{{\left[Ga{\left(H_{2}O\right)}_6\right]}_{(aq)}^{3+}}\end{array}$$

(6)

After obtaining the simple harmonic vibration frequencies of the system, the equilibrium stable isotope fractionation factor can be obtained by combining Formulas (2), (3), and (6). Therefore, how to accurately obtain the harmonic vibrational frequency of the study system becomes the key to accurately calculate the equilibrium stable isotope fractionation factor (α). When obtaining the equilibrium stable isotope fractionation factor by theoretical calculation, there is only one independent variable, that is, temperature.

By Formula (6), the equilibrium stable isotope fractionation factor calculated theoretically (1000lnα) is associated with the isotope fractionation value obtained experimentally (Δ). The method for obtaining equilibrium stable isotope fractionation factors (Δ) between different substances experimentally is shown in Formula (7), i.e.,

$${\Delta }_{A-B}={\delta }_A-{\delta }_B$$

(7)

In Formula (7), A and B represent two different substances or two different phases of the same substance. δ represents the isotope composition of different substances (phases).

In this study, the theoretical calculation software package of Gaussian 16 with the theoretical basis set B3LYP/6-311+G(d, p) is used to obtain the harmonic vibrational frequencies of the research systems [44,45,46]. The version of Gaussian 16 used in this study is Revision B.01. The B3LYP method has been used to study the isotope effects of various systems by previous researchers [33,36,45,47]. Among them, Li et al. (2009) used the B3LYP/6-311+G(d, p) theoretical level to systematically study the isotope fractionation factors of Ge (the adjacent element of Ga) and its geological applications [36]. The work of Bruin et al. (1999) recommends a theoretical basis of 6-311+G(d, p) for the study of the geometry and electronic structure of Cu-bearing complexes [47]. Ga is located in the fourth period of the periodic table, which is the same period as Cu and Ge. It is reasonable to use B3LYP/6-311+G(d, p) for the structural optimization and harmonic vibrational frequency calculations of Ga-Si complex aqueous solutions. The “water-droplet” method is used to deal with the solvation effect of the Ga-Si complex aqueous solution. This method has been widely used to study solvation effects of different stable isotope systems [36,38,48,49].

3. Results

3.1. Structural Characteristics of Ga-Si Complexes after Optimization

The information on bond length and bond angle of the optimized Ga-Si complex aqueous solution is given in

Table 1. The information of bond length and bond angle for the inner layer coordination structure of different Ga-Si complex aqueous solutions after optimization. The theoretical method of B3LYP/6-311+G(d, p) is used to calculate harmonic vibrational frequencies.

	Parameters	Bond Length (Å)					Bond Angle (°)
Species		Ga-Si	Bridge O		Non-Bridge O		∠Ga-O-Si
			Ga-O	Si-O	Ga-OH	Ga-OH2
(OH)3GaOSi(OH)3		3.17	1.87	1.62	1.86		130.60
(OH)3(H2O)2GaOSi(OH)3		3.15	1.86	1.60	1.87		130.43
(OH)2(H2O)3GaOSi(OH)3		3.03	1.85	1.62	1.84	2.19	121.02
(OH)(H2O)4GaOSi(OH)3		3.17	1.85	1.61	1.81	2.13	132.49
(H2O)5GaOSi(OH)3		3.09	1.82	1.63		2.05	127.46

. Figure 1 shows the inner layer coordination space structure of the five Ga-Si complex aqueous solutions. Figure 2 shows the spatial structure of different Ga-Si complex aqueous solutions (with 30 water molecules added to the inner layer). The bond length, bond angle, and spatial structure of Ga-Si complexes with different ligand types are different. As the ligands of Ga-Si complexes change from H₂O to OH, the spatial structures of Ga-Si complexes also change from hexa-coordinate (octahedral) to four-coordinate (tetrahedral). Previous experimental findings through the high-resolution X-ray absorption fine structure (XAFS) also gave the same conclusion [39]. Therefore, the theoretical calculation results are in good agreement with the experimental results. Moreover, with the increase in OH ligands in Ga-Si complexes, the Ga-O bond length of bridge-oxygen in the structure gradually increased from 1.82 Å ((H₂O)₅GaOSi(OH)₃) to 1.87 Å ((OH)₃GaOSi(OH)₃) (Table 1). The concentration of OH ligands is mainly affected by the pH of a water body. Therefore, the pH of water is an important factor that determines the type of Ga-Si complexes and affects the equilibrium Ga stable isotope fractionation effect.

3.2. Ga Isotope Fractionation Parameters (1000lnβ) of Different Ga-Si Complex Aqueous Solutions

In this study, the structures of five different Ga-Si complex aqueous solutions were optimized and their simple vibration frequencies were calculated. Based on the simple harmonic vibration frequencies of these structures, their Ga isotope fractionation parameters (1000lnβ) could be obtained. When calculating the parameters for each species, it is necessary to optimize the structure of a series of molecular clusters with different numbers of water molecules and calculate their harmonic vibration frequencies. Taking (OH)₂(H₂O)₃GaOSi(OH)₃ as an example, it was defined as the inner coordination structure. When simulating the solvation effect, water molecules are added to the periphery of (OH)₂(H₂O)₃GaOSi(OH)₃ in batches in groups of 6H₂O until the (OH)₂(H₂O)₃GaOSi(OH)₃.(H₂O)₃₀ structure is obtained. (OH)₂(H₂O)₃GaOSi(OH)₃.(H₂O)₃₀ is selected as the structure of the (OH)₂(H₂O)₃GaOSi(OH)₃ complex aqueous solution. This is because when the number of water molecules exceeds 30, its Ga isotope fractionation parameter (1000lnβ) will no longer change (Figure 3). Therefore, when calculating the relevant parameters (such as 1000lnβ) for each species, five structures with different numbers of water molecules (6, 12, 18, 24, and 30 water molecules) should be calculated. To ensure the accuracy and rationality of the calculation results, the structure of each molecular cluster is calculated four times in parallel, labeled A, B, C, and D. In the study of these five Ga-Si complex aqueous solutions, the structure optimization and simple harmonic vibration frequency calculation of more than 100 molecular clusters will be needed.

Table 2. The energies of structural optimization for Ga-Si complex aqueous solutions with different numbers of water molecules. n is the number of water molecules. The theoretical method of B3LYP/6-311+G(d, p) is used to calculate harmonic vibrational frequencies.

	Energy	E(RB3LYP) (Hartree)
Species
n		0		6	12	18	24	30
(OH)2(H2O)3GaOSi(OH)3.(H2O)n		−2898.64	A	−3357.50	−3816.36	−4275.22	−4734.09	−5192.95
			B	−3357.50	−3816.37	−4275.23	−4734.10	−5192.96
			C	−3357.51	−3816.37	−4275.23	−4734.08	−5192.95
			D	−3357.39	−3816.35	−4275.19	−4734.05	−5192.95
(OH)3(H2O)2GaOSi(OH)3.(H2O)n		−2898.13	A	−3357.00	−3815.86	−4274.72	−4733.59	−5192.44
			B	−3357.01	−3815.87	−4274.71	−4733.57	−5192.44
			C	−3357.00	−3815.87	−4274.73	−4733.58	−5192.47
			D	−3356.99	−3815.86	−4274.73	−4733.60	−5192.47
(OH)3GaOSi(OH)3.(H2O)n		−2745.18	A	−3204.05	−3662.91	−4121.77	−4580.64	−5039.49
			B	−3204.04	−3662.92	−4121.77	−4580.62	−5039.48
			C	−3204.04	−3662.90	−4121.77	−4580.64	−5039.47
			D	−3204.05	−3662.91	−4121.77	−4580.62	−5039.51
(H2O)5GaOSi(OH)3.(H2O)n		−2899.24	A	−3358.19	−3817.09	−4275.96	−4734.82	−5193.70
			B	−3358.19	−3817.08	−4275.94	−4734.82	−5193.67
			C	−3358.19	−3817.08	−4275.96	−4734.82	−5193.66
			D	−3358.18	−3817.08	−4275.95	−4734.81	−5193.67
(OH)(H2O)4GaOSi(OH)3.(H2O)n		−2899.01	A	−3357.90	−3816.76	−4275.63	−4734.48	−5193.34
			B	−3357.89	−3816.76	−4275.62	−4734.49	−5193.35
			C	−3357.89	−3816.75	−4275.62	−4734.49	−5193.35
			D	−3357.90	−3816.77	−4275.62	−4734.49	−5193.37

shows the energies of the optimized structures of all Ga-Si complex clusters. As can be seen from Table 2, the maximum energy difference of all molecular clusters is only 0.02 Hartree (four parallel computations of the same structure). Hence, from the point of view of energy difference, the results of these theoretical calculations are reasonable. Ga isotope fractionation parameters (1000lnβ) of five Ga-Si complex aqueous solutions are listed in detail in

Table 3. Isotope fractionation parameters (1000lnβ) of different Ga-Si complex aqueous solutions at temperatures 0, 25, 50, 100, 150, 200, 300, and 500 °C.

	Temperature (°C)	0	25	50	100	150	200	300	500
Species
(OH)2(H2O)3GaOSi(OH)3		7.77	6.65	5.75	4.41	3.49	2.82	1.95	1.09
(OH)2(H2O)3GaOSi(OH)3.(H2O)6		7.76	6.63	5.73	4.39	3.47	2.80	1.94	1.08
(OH)2(H2O)3GaOSi(OH)3.(H2O)12		8.08	6.90	5.96	4.57	3.61	2.92	2.02	1.13
(OH)2(H2O)3GaOSi(OH)3.(H2O)18		8.11	6.93	5.99	4.59	3.62	2.93	2.03	1.13
(OH)2(H2O)3GaOSi(OH)3.(H2O)24		8.15	6.96	6.02	4.61	3.64	2.95	2.04	1.14
(OH)2(H2O)3GaOSi(OH)3.(H2O)30		8.24	7.05	6.09	4.67	3.69	2.98	2.07	1.15
(OH)3(H2O)2GaOSi(OH)3		8.43	7.20	6.22	4.77	3.76	3.04	2.11	1.17
(OH)3(H2O)2GaOSi(OH)3.(H2O)6		8.37	7.15	6.17	4.73	3.74	3.02	2.09	1.17
(OH)3(H2O)2GaOSi(OH)3.(H2O)12		8.32	7.11	6.14	4.71	3.72	3.01	2.08	1.16
(OH)3(H2O)2GaOSi(OH)3.(H2O)18		8.27	7.06	6.10	4.68	3.69	2.99	2.07	1.15
(OH)3(H2O)2GaOSi(OH)3.(H2O)24		8.33	7.12	6.14	4.71	3.72	3.01	2.08	1.16
(OH)3(H2O)2GaOSi(OH)3.(H2O)30		8.35	7.14	6.16	4.72	3.73	3.02	2.09	1.16
(OH)3GaOSi(OH)3		8.41	7.19	6.21	4.76	3.76	3.04	2.10	1.17
(OH)3GaOSi(OH)3.(H2O)6		8.39	7.17	6.20	4.75	3.75	3.03	2.10	1.17
(OH)3GaOSi(OH)3.(H2O)12		8.36	7.15	6.17	4.73	3.73	3.02	2.09	1.17
(OH)3GaOSi(OH)3.(H2O)18		8.35	7.13	6.16	4.72	3.73	3.01	2.08	1.16
(OH)3GaOSi(OH)3.(H2O)24		8.40	7.17	6.19	4.75	3.75	3.03	2.10	1.17
(OH)3GaOSi(OH)3.(H2O)30		8.38	7.16	6.18	4.73	3.74	3.02	2.09	1.17
(H2O)5GaOSi(OH)3		6.73	5.73	4.94	3.77	2.97	2.40	1.66	0.92
(H2O)5GaOSi(OH)3.(H2O)6		6.76	5.75	4.95	3.78	2.98	2.40	1.66	0.92
(H2O)5GaOSi(OH)3.(H2O)12		6.75	5.74	4.94	3.77	2.97	2.40	1.65	0.92
(H2O)5GaOSi(OH)3.(H2O)18		6.77	5.76	4.96	3.79	2.98	2.41	1.66	0.92
(H2O)5GaOSi(OH)3.(H2O)24		6.75	5.75	4.95	3.78	2.97	2.40	1.66	0.92
(H2O)5GaOSi(OH)3.(H2O)30		6.75	5.75	4.95	3.77	2.97	2.40	1.66	0.92
(OH)(H2O)4GaOSi(OH)3		7.01	5.99	5.17	3.97	3.13	2.54	1.75	0.98
(OH)(H2O)4GaOSi(OH)3.(H2O)6		6.76	5.76	4.97	3.80	3.00	2.42	1.67	0.93
(OH)(H2O)4GaOSi(OH)3.(H2O)12		6.62	5.64	4.86	3.71	2.92	2.36	1.63	0.91
(OH)(H2O)4GaOSi(OH)3.(H2O)18		6.55	5.58	4.81	3.67	2.89	2.33	1.61	0.90
(OH)(H2O)4GaOSi(OH)3.(H2O)24		6.58	5.60	4.82	3.68	2.90	2.34	1.62	0.90
(OH)(H2O)4GaOSi(OH)3.(H2O)30		6.62	5.63	4.85	3.70	2.91	2.35	1.62	0.90

. In the temperature range of 0~500 °C, Ga isotope fractionation parameters (1000lnβ) of these five Ga-Si complex aqueous solutions ((OH)₂(H₂O)₃GaOSi(OH)₃.(H₂O)₃₀, (OH)₃(H₂O)₂GaOSi(OH)₃.(H₂O)₃₀, (OH)₃GaOSi(OH)₃.(H₂O)₃₀, (H₂O)₅GaOSi(OH)₃.(H₂O)₃₀, and (OH)(H₂O)₄GaOSi(OH)₃.(H₂O)₃₀) ranged from 8.24~1.15, 8.35~1.16, 8.38~1.17, 6.75~0.92, and 6.62~0.90 (‰), respectively. In Table 3, the 1000lnβs of Ga-Si aqueous solutions with 6, 12, 16, 24, and 30 water molecules are the arithmetic means of four parallel calculations (A, B, C, and D). The polynomial expansion coefficients of 1000lnβs for different Ga-Si complex aqueous solutions are shown in

Table 4. The constant terms of polynomial expansion for Ga isotope fractionation parameters (1000lnβ) of different Ga-Si complex aqueous solutions.

	Parameter	a	b	c	d
Species
(OH)2(H2O)3GaOSi(OH)3.(H2O)30		7.0429 × 10−5	−7.1015 × 10−3	6.9692 × 10−1	6.2006 × 10−3
(OH)3(H2O)2GaOSi(OH)3.(H2O)30		1.1109 × 10−4	−7.8732 × 10−3	7.0885 × 10−1	−1.6438 × 10−3
(OH)3GaOSi(OH)3.(H2O)30		6.1993 × 10−5	−6.5505 × 10−3	7.0076 × 10−1	1.5884 × 10−2
(H2O)5GaOSi(OH)3.(H2O)30		4.2861 × 10−5	−4.3055 × 10−3	5.5300 × 10−1	1.0022 × 10−2
(OH)(H2O)4GaOSi(OH)3.(H2O)30		6.9455 × 10−5	−4.8448 × 10−3	5.4640 × 10−1	−7.5287 × 10−4

. The Ga isotope fractionation parameters (1000lnβ) of these Ga-Si complex aqueous solutions as a function of temperature are shown in Figure 4. According to Formula (1), that is, the polynomial expansion of 1000lnβ, the value of 1000lnβ at any temperature can be easily obtained (in the temperature range of 0~500 °C).

3.3. Ga Isotope Fractionation Factors (1000lnα) between Different Ga-Si Complex Aqueous Solutions and Ga³⁺-Bearing Aqua-Complex/Hydroxide Solutions

Si is a very widely distributed non-metallic element in nature and its content in the Earth’s crust is very considerable. Si is found in water, mainly in the form of Si(OH)₄. Ga³⁺-bearing aqueous solutions at different pH will react with Si(OH)₄ to form various Ga-Si complexes. The core content of this study is to study the Ga isotope fractionation effect of this chemical reaction process. Under different temperature conditions, the equilibrium Ga stable isotope fractionation factors (1000lnα) between different Ga-Si complex aqueous solutions and Ga³⁺-bearing aqueous solutions are shown in

Table 5. The equilibrium stable Ga isotope fractionation factors (1000lnα) between different Ga-Si complex aqueous solutions and Ga³⁺ free ion aqueous solutions at different temperatures. “aq” stands for aqueous solution. The data of Ga isotope fractionation parameters (1000lnβ) for Ga³⁺ free ion aqueous solutions used are the data of the previously published article of our project [26].

	Temperature (°C)	0	25	50	100	150	200	300	500
Species
(H2O)5GaOSi(OH)3.(H2O)30 vs. [Ga(H2O)6]3+(aq)		−0.16	−0.14	−0.12	−0.1	−0.08	−0.06	−0.04	−0.03
(OH)(H2O)4GaOSi(OH)3.(H2O)30 vs. [Ga(H2O)6]3+(aq)		−0.29	−0.26	−0.22	−0.17	−0.14	−0.11	−0.08	−0.05
(OH)3GaOSi(OH)3.(H2O)30 vs. [Ga(OH)3](aq)		−0.06	−0.06	−0.07	−0.07	−0.05	−0.05	−0.04	−0.02
(OH)3(H2O)2GaOSi(OH)3.(H2O)30 vs. [Ga(OH)3](aq)		−0.09	−0.08	−0.09	−0.08	−0.06	−0.05	−0.04	−0.03
(OH)2(H2O)3GaOSi(OH)3.(H2O)30 vs. [Ga(OH)3](aq)		−0.2	−0.17	−0.16	−0.13	−0.1	−0.09	−0.06	−0.04

. The variations of equilibrium Ga stable isotope fractionation factors (1000lnα) between different Ga-Si complex aqueous solutions and Ga³⁺-bearing aqueous solutions with temperature are shown in Figure 5.

4. Discussion

4.1. Influence of Ga-Si Complex Structures on the Ga Isotope Fractionation Effect

It has become a consensus in the field of isotope geochemistry that the types of ligands, the spatial structures of the complexes, and the coordination bonds (chemical bonds) can significantly affect the isotope fractionation effect. The morphology and structural information of Ga adsorbed on calcite, magnesite, and δ-MnO₂ surfaces were systematically studied [50]. Persson et al. (2006) used quantitative adsorption experiments, extended X-ray absorption fine structure (EXAFS) spectroscopy, and surface complexation modeling to investigate the adsorption of Ga(III) at the water–α-FeOOH (goethite) interface [51]. Previous studies have shown that under acidic conditions, the Ga-O bond length of the first Ga coordination shell is 1.81 ± 0.01 Å [39]. In this study, the optimization result for Ga-O bond length in the first Ga coordination shell of (H₂O)₅GaOSi(OH)₃ (acidic conditions) is 1.82 Å, which is in good agreement with the results of the Si-bearing aqueous solution analyzed by EXAFS [39,52] (Table 1). Under alkaline conditions, previous studies have found that Ga-Si bond lengths are 3.12–3.20 Å [39] and 3.11–3.14 Å [52], respectively. The results of this study show that Ga-Si bond lengths vary from 3.15 to 3.17 Å ((OH)₃GaOSi(OH)₃ and (OH)₃(H₂O)₂GaOSi(OH)₃) (Table 1). Under acidic conditions, previous studies showed that the Ga-Si bond length varied in the range of 3.20 ± 0.03 Å [39], and the Ga-Si bond length of (OH)(H₂O)₄GaOSi(OH)₃ in this study was 3.17 Å. Previous research results showed that the variation range of Ga-O-Si bond angles was 125°–135° [39], and the results of this study showed that the variation range of Ga-O-Si bond angles was 121°–131°. When the content of Ga and Si in the water body is low, it mainly exists in the form of monomer complexes, that is, ${\mathrm{Ga}\left(\mathrm{OH}\right)}_{\mathrm{n}}{{(\mathrm{H}}_2\mathrm{O})}_{6-\mathrm{n}}^{3-\mathrm{n}}$ (n = 0–3) and $\mathrm{Si}{\left(\mathrm{OH}\right)}_4$. At low pH, the coordination number of Ga in Ga-Si complexes is 6, and at high pH, the coordination number of Ga-Si complexes is 4 [39]. The results of this study show that the coordination number of the inner layer of (H₂O)₅GaOSi(OH)₃.(H₂O)₃₀ and (OH)(H₂O)₄GaOSi(OH)₃.(H₂O)₃₀ is 6 (6Ga-O bonds, octahedral structure). The inner layer structures of (OH)₂(H₂O)₃GaOSi(OH)₃.(H₂O)₃₀, (OH)₃(H₂O)₂GaOSi(OH)₃.(H₂O)₃₀, and (OH)₃GaOSi(OH)₃.(H₂O)₃₀ have a coordination number of 4 (4Ga-O bonds, tetrahedral structure). The theoretical results show that the theoretical optimizations of Ga-Si complex aqueous solutions are in good agreement with the experimental results [39,52].

4.2. The 1000lnβs of Different Ga-Si Complex Aqueous Solutions

In the field of theoretical geochemical calculation, the β value is a key intermediate parameter for the calculation of the isotope fractionation factor (α), which is a parameter of geochemical concern. It has no theoretical and practical significance to study the absolute data size of β. A more valuable operation is to compare the size of βs between different species. The general rule is that under the same environmental conditions, the greater the β value, the stronger the ability of the species to enrich heavy isotopes. The 1000lnβ (‰) changes of these five Ga-Si complex aqueous solutions are as follows: (OH)₃GaOSi(OH)₃.(H₂O)₃₀ ≈ (OH)₃(H₂O)₂GaOSi(OH)₃.(H₂O)₃₀ > (OH)₂(H₂O)₃GaOSi(OH)₃.(H₂O)₃₀ > (H₂O)₅GaOSi(OH)₃.(H₂O)₃₀ > (OH)(H₂O)₄GaOSi(OH)₃.(H₂O)₃₀. As can be seen from Figure 3, at 25 °C, the 1000lnβs of these five Ga-Si complex aqueous solutions are divided into two relatively large ranges, that is, 5.6~6.0 (‰) ((H₂O)₅GaOSi(OH)₃.(H₂O)₃₀ and (OH)(H₂O)₄GaOSi(OH)₃.(H₂O)₃₀) and 6.6~7.2 (‰) ((OH)₃GaOSi(OH)₃.(H₂O)₃₀, (OH)₃(H₂O)₂GaOSi(OH)₃.(H₂O)₃₀, and (OH)₂(H₂O)₃GaOSi(OH)₃.(H₂O)₃₀). This also shows that there are two different types of chemical reaction mechanisms when Si(OH)₄ reacts with the hydrated/hydroxyl complexes of Ga³⁺. When the coordination number of the hydrated/hydroxyl complexes of Ga³⁺ is 4 and 6, there are two different reaction mechanisms. The 1000lnβs of (OH)₃GaOSi(OH)₃.(H₂O)₃₀ and (OH)₃(H₂O)₂GaOSi(OH)₃.(H₂O)₃₀ complex aqueous solutions were not significantly different (7.16 and 7.14‰ at 25 °C, respectively). The reason for this phenomenon is that the coordination structure of these two Ga-Si complex aqueous solutions is the same. This can be seen from the structures of (OH)₃GaOSi(OH)₃ and (OH)₃(H₂O)₂GaOSi(OH) in Figure 1 and Figure 2. In the (OH)₃(H₂O)₂GaOSi(OH)₃ complex, the two H₂O do not enter the inner coordination structure of (OH)₃GaOSi(OH)₃; that is, Ga exists in the form of four-coordination. At 25 °C, the 1000lnβs of species (OH)₃(H₂O)₂GaOSi(OH)₃ were 7.19, 7.17, 7.15, 7.13, 7.17, and 7.16 (‰) for clusters with different water molecule numbers (0, 6, 12, 18, 24, and 30), respectively. The difference between these 1000lnβs is very small. This is because previous studies have found that the isotope fractionation effect is a local effect, and the largest contribution to isotope fractionation is H₂O near the inner layer structure of Ga-Si complexes. The contribution of the external H₂O to the isotope fractionation effect decreases with increasing distance [53]. Therefore, when choosing the optimal model of Ga-Si complex aqueous solutions, the structures with 30H₂O are reasonable. Other Ga-Si complex aqueous solutions have the same rule (see Figure 3).

4.3. Equilibrium Ga Stable Isotope Fractionation Factors of Ga-Si Complexes in Natural Waters (1000lnα, ‰)

The latest research has been able to use the Ga to trace ocean water [24]. However, there are no reports on the use of the Ga isotope as a technical means to trace water bodies. This section will discuss equilibrium Ga stable isotope fractionation factors of Ga-Si complexes in natural waters in depth. When discussing the equilibrium Ga stable isotope fractionation factors (1000lnα) of Ga-Si complex aqueous solutions, the corresponding non-Si aqueous solution species are [Ga(H₂O)₆]³⁺_(aq) and [Ga(OH)₃]_(aq), respectively. The reasons for this operation will be explained in the second half of this section. The data of the isotope fractionation parameter (1000lnβ) for these two substances used in this study are derived from the previous work of our research group [26].

The results of this study show that the general trend of the equilibrium stable Ga isotope fractionation effect between different Ga-Si complex aqueous solutions and non-Si aqueous solution species ([Ga(H₂O)₆]³⁺(aq) and [Ga(OH)₃](aq)) is that Ga-Si complex aqueous solutions will lose heavy Ga (⁷¹Ga) isotopes compared with species in aqueous solutions without Si (Table 4 and Figure 4). In the temperature range of 0~500 °C, the equilibrium stable Ga isotope fractionation factors (1000lnαs) of these five systems ((H₂O)₅GaOSi(OH)₃.(H₂O)₃₀ vs. [Ga(H₂O)₆]³⁺_(aq), (OH)(H₂O)₄GaOSi(OH)₃.(H₂O)₃₀ vs. [Ga(H₂O)₆]³⁺_(aq), (OH)₃GaOSi(OH)₃.(H₂O)₃₀ vs. [Ga(OH)₃](aq), (OH)₃(H₂O)₂GaOSi(OH)₃.(H₂O)₃₀ vs. [Ga(OH)₃]_(aq), and (OH)₂(H₂O)₃GaOSi(OH)₃.(H₂O)₃₀ vs. [Ga(OH)₃]_(aq)) are as follows: −0.16~−0.03, −0.29~−0.05, −0.06~−0.02, −0.09~−0.03, and −0.2~−0.04 (‰) (Table 4). At 50 °C, the equilibrium Ga stable isotope fractionation factors of these five systems are −0.12, −0.22, −0.07, −0.09, and −0.16 (‰), respectively. At this temperature, the equilibrium Ga stable isotope fractionation effect is very obvious. The equilibrium Ga stable isotope fractionation factors of these systems are all negative, which indicates that light Ga (⁶⁹Ga) isotopes preferentially enter the structure of Ga-Si complexes during the formation of Ga-Si complexes. The fundamental reason for this is that ⁶⁹Ga has a smaller atomic mass than ⁷¹Ga.

Previous studies have found that when Ga is adsorbed to the mineral surface, the Ga isotope fractionation factors between the solid phase and the solution phase (Δ_{solid-solution}) can reach up to −1.27‰ (calcite) and −0.89‰ (goesite) [14]. This also shows that when adsorption occurs, the light Ga isotope (⁶⁹Ga) is preferentially adsorbed to the mineral surface. In a natural water body, the distribution of Si is very wide. Various Ga-Si complex aqueous solutions formed by the reaction between Ga³⁺-bearing and Si-bearing solutions will affect the Ga isotope effect in different processes such as adsorption. Previous studies on the Ga isotope effect in different Ga-bearing deposits have found that the difference in Ga isotope composition in different Ga-bearing deposits is relatively large. The Ga isotope composition of claystone with kaolinite as its main mineral component is −0.22‰ to +0.19‰, and that for bauxite with diaspore as its main mineral component is −0.60‰ to 0.15‰. The Ga isotope composition of dolostone from the Loushanguan group is −0.28‰ to −0.18‰ [12]. This difference in Ga isotope composition may be closely related to their sedimentary environment (species morphology, pH, etc.). The Ga isotope fractionation effect in the formation of Ga-Si complex aqueous solutions covered in this study can also indicate that the solution process has a significant influence on the Ga isotope composition of the deposit.

There is relatively little work involving Ga³⁺ aqueous species [1]. Similar to Al³⁺, Ga³⁺ is also subject to hydrolysis in aqueous solutions, which is why the species of Ga³⁺ in aqueous solutions are mainly hydration/hydroxide complexes. Previous studies have also found that the species morphology of Ga³⁺ in an aqueous solution is a function of pH and temperature [54]. It was also indicated that the hydrolysis reaction of Ga³⁺ mainly occurred at low pH, and the morphology of Ga³⁺-bearing species was strongly influenced by Ga(OH)₄⁻ (25 °C). In natural fluid systems, Ga(OH)₄⁻ is the dominant Ga³⁺-bearing species. At constant pH, the hydrolysis of Ga³⁺ is strongly affected by temperature [1]. As a typical dispersive element, researchers mostly study Ga with elements that have similar properties (such as Al³⁺, Fe³⁺, etc.). The main indexes involved are the element differentiation, migration ability, hydrolysis reaction, and so on. These important studies can reveal the geochemical behavior of Ga³⁺ in natural fluid systems. Unfortunately, the reality is that there are few studies on the Ga isotope fractionation effect in solution systems. This also limits the geochemical application of Ga as a new proxy. In this study, the Ga isotope effects of five Ga-Si complex systems were studied to fill this gap to some extent. Based on the previous work of our research group, the theoretical calculation of the equilibrium stable Ga isotope fractionation effect in a Si-bearing solution environment should be helpful to better understand the geochemical behavior of Ga³⁺ in solution processes and the geochemical cycle of the Ga isotope [26].

5. Conclusions

In this study, Ga geochemical parameters (isotope fractionation effect, α) of Si-bearing solution processes were systematically studied using the ab initio method based on first principles. Some preliminary results were obtained: (1) Species morphology, pH, and temperature are the key factors affecting the Ga isotope effect. (2) During the formation of Ga-Si complexes, Ga³⁺ mainly enters the Ga-Si complex structures in the form of four and six coordination structures; that is, the reactions between Ga³⁺-bearing aqueous solutions and Si(OH)₄ have two different mechanisms, and are significantly affected by the pH value. (3) Similar to the existing Ga isotope effect involved in the Ga³⁺ adsorption process, the light Ga isotope (⁶⁹Ga) will preferentially react when forming Ga-Si complexes. This research investigated the effect of Si (Si(OH)₄) on the isotope fractionation of Ga-bearing complex aqueous solution from the molecular level. As a new technology, there is limited research about the Ga isotope effect in different geochemical processes, such as weathering, fluid, adsorption, and biological processes. By far, the in-depth study of the Ga isotope effect in the interaction between Ga and Si in a solution is temporarily lacking. The acquisition of these key Ga isotope fractionation parameters will fill the gap in the study of the Ga isotope fractionation effect in the solution processes that contain Si. At the same time, the geochemical behavior and geochemical cycle of Ga can be better revealed. With the rise in Ga isotope research, more and more studies involving the Ga isotope effect will appear. Shortly, the application of the Ga isotope as a geochemical proxy will be more and more mature and extensive.