Soil is the basis for life and soil science is regarded as the final frontier; however, as compared to chemistry, physics, biology, and other disciplines, soil science undergoes an obviously slower development and remains almost stagnant in the past few decades, mainly due to two reasons: (1) wrong and outdated perceptions for a large portion of soil researchers; (2) complexity of soil systems that are difficult to characterize by current experimental techniques. Computer simulations have unique advantages to handle complex systems while currently, its role during soil researches is far from being recognized. In this chapter, several examples are given with respect to application of density functional theory (DFT) calculations to soil science, focusing on the adsorption of uranyl ion and SO2 onto mineral surfaces and reaction mechanisms to form acid rain. In this way, insightful clues at the atomic level are provided for the adsorption, interaction, and reactions regarding soil systems. We believe that computer simulations including DFT are the right key to unravel the complicated processes occurring in soils. More efforts of computer simulations are anticipated for soil science with aim to decipher the experimental results and probe the uncharted principles that may result in a revolutionary in the near future.
- soil science
- computer simulations
- density functional theory
- interfacial adsorption
- reaction mechanism
- complex systems
According to Natural Resources Conservation Service (NRCS), soil is defined as a natural body comprised of solids (mainly minerals and organic matters), liquids, and gases that occurs at the intermediate surface of the Earth, occupies space and is characterized by one or both of the following properties: horizons and layers, which are distinguishable from the initial materials as a result of addition, loss, transfer, and transformation of energy and matter or the ability to support rooted plants in natural circumstances. Soil constitutes the basis for life and bridges the biosphere, atmosphere, hydrosphere, and geosphere. Despite these facts, apparently less attention has been given to soil science than to other disciplines such as physics, chemistry, and biology. As said by Gardner (the past president of Soil Science Society of America) , “not a few people mistakenly perceive that everything worth knowing about soils has already been understood, and all we need to do is merely to apply that knowledge properly. Even knowledgeable scientists assume that principles and theories developed from other disciplines can be applied to the researches of soil science in a straightforward way, requiring little imagination or creativity. In their opinions, soil science is just one of expressions for the applied physics, chemistry, or biology.”
The situation of soil science research is alarming. The core concepts of current soil science textbooks remain almost unchanged as compared to those of half a century ago. Obsolete or even incorrect standpoints are a commonplace . Fortunately, a few researchers have recognized such a crisis. On the other hand, because of the complexity of systems and co-function of multiple factors, it seems challenging for experimental techniques to
The relationship between system size and computational accuracy for representative theoretical levels is shown in Figure 1. For huge systems (100,000 or even more atoms), classical methods (Monte Carlo and Molecular Dynamics) seems to be the good choice, although the computational accuracy is relatively low; In contrast,
Let us look back to
where a, b, and c are empirical parameters.
There is a five-rung Jacob’s ladder of common density functional approximations, as local spin-density approximation (LSDA), generalized-gradient approximation (GGA), meta-GGA, hyper-GGA, and random phase approximation (PRA-like functionals), see more details in . As far as we know, the most widely density functionals are at GGA (e.g., B3LYP, PBE) and meta-GGA (e.g., M06 L, M06-2X) levels.
Computer simulations, as discussed above, are particularly useful to handle the complex soil systems, and two applications were illustrated in this chapter. Section 2 summarized DFT calculated results of uranyl adsorption onto mineral surfaces and Section 3 elaborately discussed DFT calculated results of SO2 (one of the main precursors for acid rain) adsorption and formation mechanisms of acid rain. In the end, concluding remarks were provided and some suggestions for DFT applications to the future soil researches were presented as well. A more critical role of computer simulations should have played in soil science, and this chapter aims to arouse the attention of general soil researchers regarding to the applications of computer simulations. More efforts in this regard are greatly beneficial to decipher the experimental results and probe the uncharted principles that will result in a revolutionary for soil science in the near future.
2. Uranyl adsorption onto mineral surfaces
Release of radionuclides into the environment seriously threatens the ecosystem and human health, and adsorption of radionuclides onto mineral surfaces significantly affects their migration and transport into the environment . Accordingly, knowledge about the interaction between radionuclides and minerals is essential for the long-term risk assessment of radioactive waste repositories. Uranium is usually present in the uranyl (UO22+) form [14, 15] and DFT calculations have been conducted to understand the adsorption of UO22+ onto mineral surfaces [16, 17, 18, 19, 20, 21, 22, 23, 24, 25].
Perron et al.  examined UO22+ adsorption onto rutile(110) surface, the most stable face of natural rutile reported by Jones and Hockey [26, 27]. UO22+ forms a bidentate inner sphere complex with three H2O molecules to fill its first hydration shell, see Figure 2. There exist two types of surface-O atoms, as Ot (terminal-O) and Ob (bridging-O) that are singly and doubly coordinated, respectively. As a result, a total of three adsorption structures are produced: UO22+ coordinated to two Ob atoms (bb), two Ot atoms (tt) and both of Ob and Ot atoms (bt). The bb mode is the most preferred and has a lower energy than the bt and tt modes by 5.0 and 13.6 kJ/mol, respectively.
In addition to the [UO2(H2O)n]2+ adsorption complexes, other uranyl species containing anionic ligands such as OH− and CO32− also play a critical role in the environmental circumstances. Pan and co-workers  investigated the effects of different ligands (H2O, OH−, CO32−) on the adsorption on rutile(110) surface. Note that only the most stable adsorption mode (bb) has been taken into account therein. The uranyl ion (UO22+) interacts more strongly with anionic ligands (OH− and CO32−) than H2O in the first coordination sphere. The bond lengths of the U6+ center and two Ob atoms are respectively calculated at 2.248 and 2.333 Å for [UO2(H2O)3]2+, 2.808 and 2.930 Å for [UO2(OH)3]−, 2.852 and 2.960 Å for [UO2(CO3)2]2−, suggesting that the adsorption strengths of the uranyl ion with rutile(110) surface are significantly impaired in the latter two cases. Adsorption of these uranyl complexes on partially hydrated surfaces were then studied, which exhibit stronger interactions and are attributed to the formation of H-bonds between H2O molecules and uranyl ion (UO22+) that strengthen the adsorption processes.
Kaolinite is the major constituent of sedimentary clay rocks, and adsorption of uranyl species onto its (001) surface [20, 21] and (010) edge surface [22, 23] have been theoretically studied. The adsorption of uranyl species on tetrahedral SiO4 surfaces is thermodynamically unfavorable, with the adsorption energies of inner- and outer-sphere complexes being, respectively, 239 and 206 kJ/mol ; in contrast, the adsorption at octahedral AlO6 surfaces is preferred due to the presence of upright (perpendicular to the surface) and lying (parallel to the surface) OH groups. The most stable configuration corresponds to an inner-sphere monodentate complex, where UO22+ bonds directly to the lying surface-OH group and has an adsorption energy of −155 kJ/mol. Martorell et al.  continued to study the adsorption of uranyl ion on the bare and solvated octahedral AlO6 surfaces of kaolinite, where UO22+ forms two direct bonds with the surface-O atoms of deprotonated OH groups. Two different adsorption structures are assigned according to the coordination of surface-O atoms: two surface-O atoms connected to one and two Al atoms are designated to be AlOO (short-bridge site) and AlO-AlO (long-bridge site), respectively. The formation of adsorption complex at the short-bridge site (AlOO) requires an energy of 195 kJ/mol and is obviously less than at the long-bridge site (AlO-AlO, 261 kJ/mol). Thus, the uranyl ion prefers to adsorption at the short-bridge site; furthermore, similar trends for uranyl adsorption remain when including solvent effects by adding a monolayer of water molecules.
Gibbsite is a primary mineral form of aluminum hydroxide, and adsorption of the uranyl ion onto its (001) surface [18, 24] and edge surface  have been investigated, similar to the situation of kaolinite discussed above. Figure 3a depicts three potential adsorption sites for uranyl adsorption: Sites I and III hold almost the same O-O distances (about 2.7 Å), while site II shows an apparently elongated O-O distance (about 3.4 Å). Site I represents the lowest-energy adsorption structure (Figure 3), and the adsorption structures of sites II and III are less stable with relative energies being 22.2 and 46.3 kJ/mol, respectively. Three H-bonds constructed between surface-OH groups and UO22+ center as shown in Figure 3 stabilize the interactions between uranyl ion and gibbsite surfaces.
3. Acid rain
In 1852, Smith demonstrated the relationship between acid rain and atmospheric pollution in Manchester, and after 20 years (i.e., 1872), he coined the term “acid rain.” Now acid rain has become a popular term and one of the world’s biggest environmental concerns, especially in North America, Europe, and China . Acid rain refers to any form of precipitations with acidic components that fall to the ground from the atmosphere, including rain, snow, flog, hail, or even dust that is acidic. It results mainly from SO2 and NOx emissions to the atmosphere and the further transport by wind and air currents, during when SO2 and NOx react with water, oxygen, and other substances leading to the formation of sulfuric (H2SO4) and nitric (HNO3) acids. The pH of acid rain is approximately 5.6 and since 1940s, researchers began to recognize its strong impacts on the ecosystem and human health so that soils, freshwaters, forests, and buildings will be damaged. With regard to soils, acid rain inhibits the decomposition of organic matter , fixation of nitrogen , elution of calcium (Ca2+), magnesium (Mg2+), potassium (K+), and other nutrients . As a result, soil fertility and microbial activity show an obvious reduction . Geochemical modeling indicated that Ca2+ leaching in marble due to acid rain neutralization approximates 0.158 mmol/L, in contrast to 10.5 mmol/L by dry deposition, and the corresponding Cu2+ losses in bronze are ca. 0.21 and 47.3 mmol/L, respectively .
As aforementioned, SO2 emissions are one of the principal causes of acid rain and also represent a primary source of atmospheric aerosols, which can lead to respiratory diseases, premature deaths, and even climate changes by affecting the properties of clouds and the balance of solar radiation. Therefore, it is of great significance to convert SO2 to other less contaminated compounds, and a number of measures to control SO2 emissions have been proposed. During 1982–1999, SO2 emissions have reduced by approximately 65% in Europe and 40% in the United States, and SO2 emissions in China decline in the late 1990s while again increase after then. DFT calculations provide useful information about the adsorption of SO2 onto mineral surfaces as well as reaction mechanisms that seem difficult to capture by current experimental techniques [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], which are, however, critical to understand the formation of acid rain at the molecular level and to remediate the ecosystem. Clay minerals, such as alumina (Al2O3), iron oxides (FexOy), are good candidates for the adsorption of acid components from acid rain and then convert them into less hazardous compounds. Lo et al.  studied the adsorption of SO2 on clean (100), dehydrated (110), and hydrated (110) surfaces of γ-Al2O3, finding that significant adsorption differences exist for the various surfaces and the calculated adsorption energies (−13 to –85 kcal/mol) are consistent with experimental results.
The γ-Al2O3(100) surface is composed by bridging-O and five-coordinated Al atoms, and a total of five stable configurations are produced for SO2 adsorption (Figure 4). The feeble interaction between S and surface-O atoms results in the physisorption configuration (CM3) with the S-O distance of 2.915 Å, and the corresponding binding energy is very small (−2.0 kcal/mol). The interaction between O@SO2 and Al atoms leads to a chemisorption state named CM4, and the O-Al bond distance and adsorption energy are 2.123 Å and −23.9 kcal/mol, respectively. The other three configurations are also ascribed to chemisorption. In CM5, one O@SO2 atom is coordinated to two Al atoms in the vicinity of an octahedral vacancy, and CM7 and CM8 can be considered to generate from CM5 conversion and recombination. As compared to CM5 and CM7, the adsorption configuration CM8 possesses a superior symmetry, and both O@SO2 atoms participate in the formation of direct bonds with the Al atoms. The adsorption energies of SO2 are calculated to be −39.2, −15.1, and −45.2 kcal/mol, respectively, for CM5, CM7, and CM8. In consequence, three types of sulfite (SO32−) are produced during the adsorption of SO2 onto γ-Al2O3(100) surface. For all adsorption configurations including three with positive adsorption energies (CM1: 1.0 kcal/mol, CM2: 2.7 kcal/mol, and CM6: 21.4 kcal/mol), no direct coupling is detected between S@SO2 and Al atoms. When γ-Al2O3(110) surface is hydrated, five stable adsorption configurations arise that are distinct from dehydrated condition: two physisorption modes (HM1 and HM2) and three chemisorption modes (HM3, HM4, and HM5), see Figure 4. HM1 and HM2 are structurally similar in that their S atoms are coordinated to a surface hydroxyl, while the coordination numbers of their Al atoms are different from each other. HM3 is produced by interaction of O@SO2 atom with five-fold Al sites. HM4 and HM5 contain the sulfite species where the S atom is coordinated with surface-O atoms. The adsorption energies are calculated to be −20.4, −25.3, −31.1, −17.5, and −35.0 kcal/mol, respectively, for HM1, HM2, HM3, HM4, and HM5. In consequence, HM5 with formation of the sulfite species represents the lowest-energy adsorption configuration, which is the same as in dehydrated condition (CM8). Two IR peaks at 1214 and 1349 cm−1 are assigned to the sulfate species, which can be been finely interpreted by DFT calculated results.
Goethite (α-FeO(OH)), which can be found in soils and other low-temperature environments, is an iron-bearing hydroxide. Because of the considerable adsorption capacity for organic acids and anions, goethite has also been widely used in environmental remediation and protection . Zubieta et al.  investigated the adsorption of SO2 on partially and fully hydrated (110) surfaces of goethite and obtained eight stable products: six sulfite, one bisulfate, and one sulfate (Figure 5). The six adsorption structures containing sulfite species, created only on two types of partially dehydrated goethite surfaces, are further divided into two monodentric mononuclear (MdMn) and four bidentate (Bd) configurations. In the MdMn configurations (I and II), the S-OFe distances are elongated as compared to the other S-O distances, and they display two symmetrical stretching modes (OSO and OSOFe) centered at ca. 1126 and 976 cm−1. In BdBn configurations (I and II), the two S-OFe distances are approximately 1.62 Å, and although with similar geometries and stretching modes, the vibrational frequencies deviate significantly from those of MdMn configurations and fall at around 672 and 661 cm−1. In BdPn configurations (I and II), one S-OFe distance is optimized at 1.75 Å and lengthened as compared to those of BdPn configurations, while the other S-OFe distance equals 1.55 Å and is obviously contracted. The Bader analyses indicate that all sulfite species carry approximately −1.4 |e| charges.
The bisulfate species is formed when the S atom constructs direct bonds with μ1-OH (Figure 5), which is further stabilized by two H-bonds (1.77 and 1.81 Å). A similar bisulfate species was observed by Liu et al.  during the adsorption of SO2 on γ-Al2O3 surfaces. The Bader analyses indicate that the HSO3 species (bisulfate) carries −0.8 |e| charges. The computational sulfite and bisulfate species have IR spectra that are consistent with experimental results [37, 38]. The sulfate species emerges only when two water molecules are deprived from the (110) surface of goethite. In this way, two O atoms are singly coordinated and one O atom is three-fold coordinated with the Fe ions of goethite(110) surface. The sulfate species is corroborated by one H-bond (1.81 Å) and carries −1.6 |e| charges according to the Bader charge analyses. A stable sulfite structure can be produced under identical conditions, whereas its adsorption energy is obviously less, indicating that the formation of the sulfate species is significantly preferred.
The adsorption of SO2 onto the Cu(100), MgO and carbon surfaces was discussed as well, which may provide insightful clues for resembling processes onto mineral surfaces. It was proposed that SO2 and H2O are co-adsorbed onto Cu(100) surfaces , through the direct coupling of Cu atom with S and O@H2O atoms. The Cu-O distances ascend in the order of co-adsorption of SO2 and H2O < adsorption of only SO2 < < adsorption of only H2O. Accordingly, the interaction between SO2 and Cu(100) surface is stronger than that of H2O, and the co-adsorption of SO2 conduces to the enhanced interaction of H2O with Cu(100) surface. At the same time, the Cu-S distance of the co-adsorption configuration is optimized at 2.385 Å and is shorter than that with only SO2 adsorption. That is, water exhibits a promoting effect for the adsorption of SO2 on Cu(100) surface, as corroborated by the calculated adsorption energies. Eid and collaborators  found that as compared to regular MgO surface, the adsorption capacity of SO2 at MgO(Fs-center) defects is higher, and MgO(Fs-center) corresponds to an enhanced catalytic activity. With regard to pure carbon materials, SO2 is physisorbed and van der Waals (vdW) is the driving force therein . When carbon materials are modified with functional groups such as carboxyl, lactone, or/and phenolic hydroxyl, the adsorption strengths of SO2 are enhanced pronouncedly, especially for the sites at edge surfaces. In addition, these functional groups show little effects on SO2 adsorption, suggesting that the enhanced adsorption is mainly due to regulation of carbon surface properties.
Adsorption of SO2 is the first step for the formation of acid rain. According to our preliminary studies, the reaction mechanisms of gas phase and mineral surfaces resemble each other, and hence the gas-phase results are beneficial to understand acid rain formation at mineral surfaces. The reaction of SO2 and H2O produces two isomers that have close electronic energies ; in addition, the two isomers have apparently lower electronic energies than H2SO3. Accordingly, the gaseous SO2 and H2O mixture is likely to exist as the SO2·H2O complex rather than H2SO3. The activation barrier of SO2 reacting with H2O to form H2SO3 is so high (146.7 kJ/mol) that it becomes very difficult to produce the sulfite species (H2SO3) in gas phase. Five years later, Stirling  investigated the hydrolysis of SO2 in aqueous solutions, finding that hydrated SO2 forms the bisulfite anion and hydronium ion after overcoming an energy barrier of about 17 kcal/mol, while the one-step formation of H2SO3 has not been detected. The orientation of water molecules in the hydration shell of SO2 implies a more facile formation of the bisulfite anion rather than H2SO3 , in line with the results of meta-dynamics calculations . When HO2 participates in the reaction, the S atom constructs a new bond with O@H2O atom (S-O: 1.716 Å). The energy barrier reduces considerably and equals 56.6 kJ/mol. HO2 exists widely in the atmosphere and participates in a variety of chemical reactions . In addition to HO2, a number of computational studies explicitly indicated that acidic substances in the atmosphere play similar catalytic effects and reduce significantly the energy barriers for the hydrolysis of SO2 [48, 49, 50]. The effects of H2SO3 and H2O on the hydrolysis of SO2 were studied by Liu et al. , showing that the energy barriers ascend in the order of H2SO3 + SO2·H2O (26.4 kJ/mol) < H2O + SO2·H2O (84.0 kJ/mol) < SO2·H2O (154.6 kJ/mol). The catalytic effect of H2SO3 is obviously more pronounced than that of H2O. As reflected by NBO analyses, the reactant complexes in presence of sulfurous acid have the enhanced second-order stabilization energies as compared to those with addition of only water molecules, which provides strong supports for the pronounced catalytic effect of H2SO3. The kinetic calculations further show that the hydrolysis of SO2 is a nearly autocatalytic reaction.
Alternatively, acid rain can be formed by release of SO2 into the atmosphere and oxidization of SO3, which then reacts with H2O to form sulfuric acid (H2SO4). There are many oxidants that are able to convert SO2 to SO3, and Calvert  have provided the formation mechanism of sulfuric acid (H2SO4)
SO2 + OH → HSO3
HSO3 + O3 → SO3 + HO2
SO3 + H2O → H2SO4
Interaction between SO2 and the OH radical (OH•) has been addressed by
SO2 + O3 → SO3 + O2
SO2 + H2O2 → H2SO4
SO2 and O3 interact mainly through the S and O@O3 atoms, and their reaction causes the formation of S-O bond (1.708 Å) and the rupture of O-O bond in O3. This is one-step process and requires to overcome an energy barrier of 46.5 kJ/mol. The reaction of SO2 with H2O2 proceeds via the OH-abstraction mechanism, and two OH radicals generated from the dissociation of H2O2 are appended to SO2 forming H2SO4. However, the energy barrier of this reaction is 299.5 kJ/mol, which is extremely difficult to proceed at normal conditions. Chen et al.  reported the reaction mechanism between SO2 and HO2 and showed that there exist two types of SO2·HO2 complexes: one is to combine the terminal O@HO2 and S atoms. In this reaction, the O-S distance decreases from 2.966 Å (the initial complex: SO2·HO2) to 1.598 Å while the O-O bond of HO2 is gradually elongated until broken. The reaction of SO2 and H2O is divided into three stages, and the energy barrier of SO3·OH formation is so small (1.7 kJ/mol) that can be considered negligible. The final product of this reaction is HSO4. The other complex is characterized by two types of intermolecular H-bonds that form between the terminal O@HO2 and S atoms and between the H@HO2 and O@SO2 atoms. It is a one-step process and the product for this reaction is HOSO·O2. When a single water is added, it has a combined effect on the overall process, not only accelerating the reaction of generating HSO4 by reducing the activation barrier of the second step from 48.8 to 44.8 kJ/mol but also inhibiting the reaction between SO2 and HO2 to produce HOSO due to blocking the interaction between the H@HO2 and O@SO2 atoms.
It can be seen from the above discussions that the SO3 and water reaction is an integral section for the formation of acid rain. The direct reaction of SO3 with one water to produce sulfuric acid (H2SO4) requires to overcome a large energy barrier (28.7 kcal/mol) and seems difficult to occur at normal conditions [55, 56, 57]. When the second water participates, the energy barrier reduces substantially and the reaction becomes almost barrierless, which is probably due to the formation of a stable six-membered cyclic transition state, see Figure 6 . The two water molecules transfer their protons in a concerted manner, and a new S-O bond is formed between the newly generated OH and SO3 fragments. In consequence, multiple reaction pathways may co-exist as illustrated below
SO3 + (H2O)2 → H2SO4 + H2O
SO3·H2O + H2O → H2SO4 + H2O
SO3 + (H2O)2 → H2SO4 + H2O
The energy barriers for the reaction of SO3 and water clusters (H2O)n to form H2SO4·(H2O)n-1 are calculated to be 28.7, 11.1, and 4.6 kcal/mol for n = 1, 2, 3, respectively . With increase of water numbers, more water molecules are available to solvate and stabilize the charged transition state complexes, which further reduce the energy barriers. That is, it is favorable to hydrolyze SO3 to form sulfuric acid in presence of sufficient water vapor. The second molecule acts as a good catalyst that promotes proton transfer from water to SO3, and according to both experimental and theoretical reports, the rate constant of SO3 + 2H2O reaction is approximately 10−15 cm3·molecule−1·s−1 [60, 61, 62]. HO2 , HCOOH [62, 64], H2SO4 , HNO3  can replace the role of the second water molecule and exhibit a similar catalytic effect on the formation of sulfuric acid (H2SO4).
4. Concluding remarks
In contrast to the rapid development of chemistry, physics, and biology and other disciplines, soil science remains almost stagnant in the past few decades, and to best of our knowledge, no breakthroughs have been reported for rather a long time. Despite that, no one can deny the vital significance of soil to our life, and soil science has been widely acknowledged as the final frontier.
The slow progresses for soil science, we think, should be attributed to two reasons: (1) wrong and outdated perceptions. A majority of soil researchers mistakenly believe that all knowledge worth knowing about soils has already been understood and no revolutionary progresses would take place; (2) complex systems. Soils are very structurally complicated and there are multiple factors to co-function, which makes it very difficult to characterize by experimental techniques. Computer simulations have unique advantages to handle complex systems, while currently its role in soil science is far from being recognized. In this chapter, two examples are elaborately discussed with regard to application of DFT calculations in soil science: one focuses on the adsorption of uranyl onto mineral surfaces, and the other involves the adsorption of SO2 onto mineral surfaces and reaction mechanisms to form acid rain. It can be seen from these discussions that DFT calculations are able to provide useful and detailed information about the adsorption, interaction and reactions at the atomic level that greatly promote our understanding about soil science.
With advent of high-performance computing platforms, the same DFT calculation tasks of 10 years ago can now finish within a remarkably shorter time, even if you increase model size (periodic model is also an option), consider solvent effects by adding explicit solvent molecules or/and choose more accurate methods. The methodological developments regarding to DFT calculations have also made remarkable progresses over the recent three decades, and as a result, thermodynamics and reaction barriers can now be predicted with nearly chemical accuracy (≤1 kJ/mol). In consequence, computer simulations including DFT are the right key to unravel the complicated phenomena and processes occurring within soils; e.g., with DFT calculations, the aggregation mechanisms of “real” soils and the driving force therein were unveiled at an atomic level .
In addition to DFT methods, there are a number of other computational methods, such as QM/MM and Molecular Dynamics (MD). The choice of suitable computational methods is strongly recommended. We are pleased to see the birth of the ClayFF force-field  that was developed specially for clay minerals and the capability of the ReaxFF force-field  to handle reaction mechanisms. The 2013 Noble Prize in Chemistry was awarded to the development of “multiscale methods for complex systems,” and now it is time to apply these methods to tackle complex soil systems.
This work was sponsored by the National Natural Science Foundation of China (21473137), the Fourth Excellent Talents Program of Higher Education in Chongqing (2014-03) and the Natural Science Foundation Project of CQ CSTC, China (cstc2017jcyjAX0145).
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