# Effects of Concentration of Pore Solution on Stability of Ion-Absorbed Rare Earth Ore Aggregate

Authors: Jie Wang; Guanshi Wang; Bengen Hong
Journal: Advances in Civil Engineering
 (2021)

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## Abstract

The continuous change in solution concentration in ore pores during in situ mineral leaching influences the stability of ore aggregate. In this study, influences of the concentration of ammonium sulfate ((NH4)2SO4) solution on the interaction forces between ore particles were calculated. On this basis, the mechanism by which (NH4)2SO4 solution concentration influences the stability of ore aggregate was analyzed. Furthermore, an empirical formula for estimating the critical (NH4)2SO4 solution concentration for aggregation and dispersion of ore body aggregates with different grain composition was proposed. Some major conclusions were drawn. First, for ore bodies with an initial particle size of less than 0.075 mm, the interaction force between particles was net attraction, with the distance range of this force increasing as the concentration of (NH4)2SO4 solution increased from ≤0.001 to 0.16 mol·L−1, aggregation of ore particles occurring within this distance range. Secondly, for ore bodies with initial particle size of less than 0.075 mm, the interaction force between particles was net attraction, but with the distance range of this force decreasing when the (NH4)2SO4 solution concentration increased from 0.16 to 0.28 mol·L−1, dispersion of ore particles occurring beyond this distance range. Thirdly, for ore bodies with particle sizes of less than 0.038, 0.075 and 0.1 mm, the cation exchange capacity (CEC) was 9.13, 8.96, and 8.8 cmol·kg−1, respectively, and the critical (NH4)2SO4 solution concentration affecting the aggregation and dispersion of ore bodies was 0.12, 0.16, and 0.20 mol·L−1, respectively.


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## Body

## 1. Introduction

Currently, researchers believe that failure of soil aggregate is due to three processes: expansion, explosion, and scattering [1, 2]. For a dry soil aggregate, soil particles are controlled by net attraction when the distance between soil particles is less than 1.5 nm [3, 4]. In this case, soil aggregate will generally not expand and scatter. When the drying soil aggregate is immersed into water, the distance between two adjacent soil particles increases quickly to 1.5 nm as a result of hydraulic repulsive force. This process is called the expansion process of soil aggregates [5]. Since the hydraulic repulsive force is a short-range force, its sphere of effective action is only 1.5 nm and soil aggregates do not scatter beyond this distance. When the distance between two adjacent soil particles is greater than 1.5 nm, electrostatic repulsion becomes the dominant force and it leads to a continuous increase in distance between soil particles, up to 30–60 nm. Small soil particles are released continuously. This process is called explosion and scattering of soil aggregates [6]. In these ways, Van der Waals forces, electrostatic repulsive force, and hydraulic repulsive force can significantly influence the interaction of soil particles in a water system, thus affecting the stability of soil aggregates [7–15].For ion-absorbed rare earth ores, in situ mineral leaching is the preferred technique in China [16]. In this technique, ammonium sulfate solution is injected into the ore body through injecting holes, where it undergoes exchange reactions with rare earth ions. Rare earth ions enter the solution and form part of the mother liquor, which is then discharged from the collection device. Therefore, the injection of ammonium sulfate solution changes the concentrations of ions in solution in ore pores and ion content on ore particle surfaces, thus changing electrostatic repulsive forces between ore particles. In other words, the interaction forces between ore particles are changed and the stability of the ore aggregate is affected. It affects the cohesion of ore body and further affects the strength of ore body and the stability of slope.In this study, influences of (NH4)2SO4 solution concentration on the thickness of the sliding layer in double-electrode layers on ore particle surfaces, and interaction forces of ore particles, were calculated under mixed electrolyte conditions. On this basis, the influencing mechanism of (NH4)2SO4 solution concentration in ore pores on the stability of ore aggregate was determined.


## 2. Model Building

### 2.1. Calculation of the Interaction Forces among Ore Particles

Interaction forces between ore particles consist of the sum of electrostatic repulsion (pR), hydration repulsion (ph), and Van der Waals forces (pvdw), according to the following equation:(1)p=pR+pvdw+ph.


### 2.2. Van der Waals Forces between Ore Particles

There is always mutual attraction between ore particles because there is always Van der Waals attraction between atoms or molecules at close range. Van der Waals forces between particles are dependent on the distance between particles and are expressed according to the following equation [17]:(2)pvdw=−A0.6π10d−3,where A is the Hamaker constant, with values of A for soil or clay in vacuum and in solution being 12 × 10−20 J and 7.8 × 10−20 J, respectively. d is the distance between two adjacent particles.


### 2.3. Hydration Repulsion among Ore Particles

The hydration repulsion force is significantly stronger than the electrostatic repulsion force or Van der Waals force, but it acts over distances of only 1.5 nm. Thus, the hydration repulsion is a short-range acting force [18]. Based on experimental results, Leng [19] proposed that the hydration repulsion force (in atm) between two particles is proportional to the exponent of the distance between the two particles:(3)ph=3.33×104e−5.76×109d.


### 2.4. Electrostatic Repulsion between Ore Particles

#### 2.4.1. Electrostatic Repulsion between Particles

Two ore particles were selected as the research objects. These were conceptualized as two plate-like viscous particles with uniform charge distribution on their surfaces. The surfaces of the plates were treated as planes that extend to infinity in they and z directions; this assumption excludes fringe effects. It was assumed that the two particles equilibrate when the planes that represent the two particles are immersed in an electrolyte solution with a concentration of c0 at a mutual distance of d. At equilibrium, the mutual repulsion of the double electric layers (pd) and the external pressure (p0) exerted by the two planes are equal (Figure 1).Figure 1
Interaction between the double electric layers associated with a pair of particles assumed to behave like parallel plates.The pressure on a unit volume within this space in thex direction is expressed as follows:(4)Fx=dpdx.Unit volumes within this space are also subject to an electrostatic force, defined as the product of the charge density and the electric field intensity according to the following equation:(5)Fe=ρdψdx.The condition for equilibrium between the two particles is expressed as follows:(6)dpdx+ρdψdx=0.Equation (6) can be rearranged to give(7)dp=−ρdψ.The density is defined by equation (8), which can also be expressed as equation (9):(8)ρ=∑iziFci0exp−ziFψRT,where F is the Faraday constant (F = 96490 C·mol−1), R is the universal gas constant (R = 8.314 J·mol−1 K−1), T is the absolute temperature (K), ci is the ion concentration (mol·L−1), the subscript 0 refers to the bulk solution, zi is the ionic valence, and ψ(x) is the potential at x m from the particle surface (V).So,(9)dp=−∑iziFci0exp−ziFψRTdψ.Therefore,pR can be calculated using the following equation:(10)pR=pd−p0=∫0ψd/2−∑iziFci0exp−ziFψRTdψ=∑iRTci0exp−ziFψd/2RT−1.If the pressure is measured in atmospheres (1 atm = 101.325 kPa), then equation (10) becomes(11)pR=RT101∑ici0exp−ziFψd/2RT−1.The potential at the center of the gap between the two particles,ψ(d/2), is expressed as follows [20]:(12)π21+122e2ZiFψd/2/RT+382e4ZiFψd/2/RT−arcsineZiFψ0−ZiFψd/2/2RT=14d8πF2zi2ci0εRTe−ZiFψd/2/2RT.


#### 2.4.2. Calculation of Potential on Ore Particle Surface

According to equation (12), it is necessary to calculate the potential on the surface of the ore particles (ψ0) when calculating ψ(d/2). The method for calculating ψ0 is explained below.When the double layer in only thex direction is considered, and adsorption equilibrium is reached, the formula used to calculate the average ion concentration (c1) in diffusive double electric layers can be defined as follows [21]:(13)c1=N∞V=1V∫0VcxdV=κ∫01/κcxdx,whereN∞ is the total area of adsorbed ions in the diffusion double layer, V is the total area of the diffusion double layer, 1/κ is the thickness of the double layer, and c(x) is the concentration of adsorbed ions from the surface of the solid particle to the distance x. The expression for c(x) is given by(14)cx=∑ici0exp−ziFψRT.Alternatively,c1 can be calculated from(15)c1=CECS1/κ=CECκS,where CEC is the cation exchange capacity (cmol·kg−1) and S is the specific surface area of particles (m2·g−1).Ultimately, equations (13) and (15) can be combined using a minimum error method to calculate the potential on the surface of ore particles (ψ0).


#### 2.4.3. Calculation ofψ(x) on Ore Particle Surface

When calculating the surface potentialψ0 of ore particles, it is necessary to define the surface potential ψ(x). In order to calculate these two parameters, the diffusion double layer structure on the particle surface needs to be described mathematically. At present, the Guy–Chapman model is the most accepted method to achieve this, and it describes the double layer structure by the Poisson–Boltzmann equation. The nonlinear Poisson–Boltzmann equation for electrolyte solutions is expressed [22]:(16)dydx=8πF2εRT∑ici0exp−ziy−1,where ε is the dielectric constant. The variable y is defined by y = Fψ(x)/RT, where ψ(x) is the potential at x m from the particle surface (V).When ammonium sulfate solution is injected into rare earth ores, ammonium ions in the solution exchange with rare earth ions on the surface of the ore particles. The solution surrounding the ore particles becomes a mixed electrolyte system. For these solutions, equation (16) becomes(17)dydx=8πF2εRTcCe−zCy−1+cDe−zDy−1+cEe−zEy−1+cHe−zHy−1,where zC, zD, zE, and zH are the valences of the ammonium ions, the sulfate ions corresponding to ammonium ions, the rare earth ions, and the sulfate ions corresponding to rare earth ions, respectively; cC, cD, cE, and cH are the concentrations of the ammonium ions, the sulfate ions corresponding to ammonium ions, the rare earth ions, and the sulfate ions corresponding to rare earth ions, respectively, in mol·L−1. For (NH4)2SO4, zC = 1, zD = −2, and cD = 0.5 cC. For (RE)2(SO4)3, zE = 3, zH = −2, and cH = 1.5 cE. With these substitutions, equation (17) becomes (18)dydx=8πF2εRTcCe−y−1+0.5cCe2y−1+cEe−3y−1+1.5cEe2y−1.Settingy = y0 at x = 0 and y = y(x) at x = x, and rearranging, equation (18) becomes(19)∫y0yx1cCe−y−1+0.5cCe2y−1+cEe−3y−1+1.5cEe2y−1dy=∫0x8πF2εRTdx.Given thaty = Fψ(x)/RT, equation (19) can be substituted to produce the following equation for ψ(x):(20)xs=1κ1ln9cE+3cC−6cE+2cC1−eFζ/RT+6cE+2cC1−eFζ/RT−9cE+3cC2−1−eFζ/RT23cE+cC21−eFζ/RTλ1.Here,λ1 and κ1 are defined by equations (21) and (22), respectively.(21)xs=1κ1ln9cE+3cC−6cE+2cC1−eFζ/RT+6cE+2cC1−eFζ/RT−9cE+3cC2−1−eFζ/RT23cE+cC21−eFζ/RTλ1,(22)xs=1κ1ln9cE+3cC−6cE+2cC1−eFζ/RT+6cE+2cC1−eFζ/RT−9cE+3cC2−1−eFζ/RT23cE+cC21−eFζ/RTλ1.


### 2.5. Thickness of the Sliding Layer in Electric Double Layer

For rare earth ore bodies, when the concentration of (NH4)2SO4 solution exceeds a certain level, ion exchange between the ammonium ions in solution and the rare earth ions on the surface of the ore particles occurs. This changes the ionic composition of the bulk solution around the ore particles and their surface potential, thus changing the thickness of the electric double layer on the surface of the particles. At present, the electric double layer thickness on the surface of particles can be approximately calculated by Debye parameter 1/κ, but this parameter only reflects the influence of the change in solution concentration and does not reflect the influence of the change in surface potential of particles caused by ion exchange. Ding proposed that the sliding layer thickness could be used to describe the electric double layer on the surface of particles [23]. Therefore, this section describes the method for calculating the sliding layer thickness in the double layer on the particle surface, which considers the influence of solution concentration and particle surface potential on the double layer thickness on the particle surface.Equation (20) can be rearranged to obtain the following equation, an expression for the distance to the particle surface (x):(23)xs=1κ1ln9cE+3cC−6cE+2cC1−eFζ/RT+6cE+2cC1−eFζ/RT−9cE+3cC2−1−eFζ/RT23cE+cC21−eFζ/RTλ1.In electrolyte systems,ψ(x) in equation (20) can be equated with the value of the zeta potential (ζ), so that x can be expressed in terms of the thickness of the sliding layer (xs), as shown in the following equation:(24)xs=1κ1ln9cE+3cC−6cE+2cC1−eFζ/RT+6cE+2cC1−eFζ/RT−9cE+3cC2−1−eFζ/RT23cE+cC21−eFζ/RTλ1.


## 2.1. Calculation of the Interaction Forces among Ore Particles

Interaction forces between ore particles consist of the sum of electrostatic repulsion (pR), hydration repulsion (ph), and Van der Waals forces (pvdw), according to the following equation:(1)p=pR+pvdw+ph.


## 2.2. Van der Waals Forces between Ore Particles

There is always mutual attraction between ore particles because there is always Van der Waals attraction between atoms or molecules at close range. Van der Waals forces between particles are dependent on the distance between particles and are expressed according to the following equation [17]:(2)pvdw=−A0.6π10d−3,where A is the Hamaker constant, with values of A for soil or clay in vacuum and in solution being 12 × 10−20 J and 7.8 × 10−20 J, respectively. d is the distance between two adjacent particles.


## 2.3. Hydration Repulsion among Ore Particles

The hydration repulsion force is significantly stronger than the electrostatic repulsion force or Van der Waals force, but it acts over distances of only 1.5 nm. Thus, the hydration repulsion is a short-range acting force [18]. Based on experimental results, Leng [19] proposed that the hydration repulsion force (in atm) between two particles is proportional to the exponent of the distance between the two particles:(3)ph=3.33×104e−5.76×109d.


## 2.4. Electrostatic Repulsion between Ore Particles

### 2.4.1. Electrostatic Repulsion between Particles

Two ore particles were selected as the research objects. These were conceptualized as two plate-like viscous particles with uniform charge distribution on their surfaces. The surfaces of the plates were treated as planes that extend to infinity in they and z directions; this assumption excludes fringe effects. It was assumed that the two particles equilibrate when the planes that represent the two particles are immersed in an electrolyte solution with a concentration of c0 at a mutual distance of d. At equilibrium, the mutual repulsion of the double electric layers (pd) and the external pressure (p0) exerted by the two planes are equal (Figure 1).Figure 1
Interaction between the double electric layers associated with a pair of particles assumed to behave like parallel plates.The pressure on a unit volume within this space in thex direction is expressed as follows:(4)Fx=dpdx.Unit volumes within this space are also subject to an electrostatic force, defined as the product of the charge density and the electric field intensity according to the following equation:(5)Fe=ρdψdx.The condition for equilibrium between the two particles is expressed as follows:(6)dpdx+ρdψdx=0.Equation (6) can be rearranged to give(7)dp=−ρdψ.The density is defined by equation (8), which can also be expressed as equation (9):(8)ρ=∑iziFci0exp−ziFψRT,where F is the Faraday constant (F = 96490 C·mol−1), R is the universal gas constant (R = 8.314 J·mol−1 K−1), T is the absolute temperature (K), ci is the ion concentration (mol·L−1), the subscript 0 refers to the bulk solution, zi is the ionic valence, and ψ(x) is the potential at x m from the particle surface (V).So,(9)dp=−∑iziFci0exp−ziFψRTdψ.Therefore,pR can be calculated using the following equation:(10)pR=pd−p0=∫0ψd/2−∑iziFci0exp−ziFψRTdψ=∑iRTci0exp−ziFψd/2RT−1.If the pressure is measured in atmospheres (1 atm = 101.325 kPa), then equation (10) becomes(11)pR=RT101∑ici0exp−ziFψd/2RT−1.The potential at the center of the gap between the two particles,ψ(d/2), is expressed as follows [20]:(12)π21+122e2ZiFψd/2/RT+382e4ZiFψd/2/RT−arcsineZiFψ0−ZiFψd/2/2RT=14d8πF2zi2ci0εRTe−ZiFψd/2/2RT.


### 2.4.2. Calculation of Potential on Ore Particle Surface

According to equation (12), it is necessary to calculate the potential on the surface of the ore particles (ψ0) when calculating ψ(d/2). The method for calculating ψ0 is explained below.When the double layer in only thex direction is considered, and adsorption equilibrium is reached, the formula used to calculate the average ion concentration (c1) in diffusive double electric layers can be defined as follows [21]:(13)c1=N∞V=1V∫0VcxdV=κ∫01/κcxdx,whereN∞ is the total area of adsorbed ions in the diffusion double layer, V is the total area of the diffusion double layer, 1/κ is the thickness of the double layer, and c(x) is the concentration of adsorbed ions from the surface of the solid particle to the distance x. The expression for c(x) is given by(14)cx=∑ici0exp−ziFψRT.Alternatively,c1 can be calculated from(15)c1=CECS1/κ=CECκS,where CEC is the cation exchange capacity (cmol·kg−1) and S is the specific surface area of particles (m2·g−1).Ultimately, equations (13) and (15) can be combined using a minimum error method to calculate the potential on the surface of ore particles (ψ0).


### 2.4.3. Calculation ofψ(x) on Ore Particle Surface

When calculating the surface potentialψ0 of ore particles, it is necessary to define the surface potential ψ(x). In order to calculate these two parameters, the diffusion double layer structure on the particle surface needs to be described mathematically. At present, the Guy–Chapman model is the most accepted method to achieve this, and it describes the double layer structure by the Poisson–Boltzmann equation. The nonlinear Poisson–Boltzmann equation for electrolyte solutions is expressed [22]:(16)dydx=8πF2εRT∑ici0exp−ziy−1,where ε is the dielectric constant. The variable y is defined by y = Fψ(x)/RT, where ψ(x) is the potential at x m from the particle surface (V).When ammonium sulfate solution is injected into rare earth ores, ammonium ions in the solution exchange with rare earth ions on the surface of the ore particles. The solution surrounding the ore particles becomes a mixed electrolyte system. For these solutions, equation (16) becomes(17)dydx=8πF2εRTcCe−zCy−1+cDe−zDy−1+cEe−zEy−1+cHe−zHy−1,where zC, zD, zE, and zH are the valences of the ammonium ions, the sulfate ions corresponding to ammonium ions, the rare earth ions, and the sulfate ions corresponding to rare earth ions, respectively; cC, cD, cE, and cH are the concentrations of the ammonium ions, the sulfate ions corresponding to ammonium ions, the rare earth ions, and the sulfate ions corresponding to rare earth ions, respectively, in mol·L−1. For (NH4)2SO4, zC = 1, zD = −2, and cD = 0.5 cC. For (RE)2(SO4)3, zE = 3, zH = −2, and cH = 1.5 cE. With these substitutions, equation (17) becomes (18)dydx=8πF2εRTcCe−y−1+0.5cCe2y−1+cEe−3y−1+1.5cEe2y−1.Settingy = y0 at x = 0 and y = y(x) at x = x, and rearranging, equation (18) becomes(19)∫y0yx1cCe−y−1+0.5cCe2y−1+cEe−3y−1+1.5cEe2y−1dy=∫0x8πF2εRTdx.Given thaty = Fψ(x)/RT, equation (19) can be substituted to produce the following equation for ψ(x):(20)xs=1κ1ln9cE+3cC−6cE+2cC1−eFζ/RT+6cE+2cC1−eFζ/RT−9cE+3cC2−1−eFζ/RT23cE+cC21−eFζ/RTλ1.Here,λ1 and κ1 are defined by equations (21) and (22), respectively.(21)xs=1κ1ln9cE+3cC−6cE+2cC1−eFζ/RT+6cE+2cC1−eFζ/RT−9cE+3cC2−1−eFζ/RT23cE+cC21−eFζ/RTλ1,(22)xs=1κ1ln9cE+3cC−6cE+2cC1−eFζ/RT+6cE+2cC1−eFζ/RT−9cE+3cC2−1−eFζ/RT23cE+cC21−eFζ/RTλ1.


## 2.4.1. Electrostatic Repulsion between Particles

Two ore particles were selected as the research objects. These were conceptualized as two plate-like viscous particles with uniform charge distribution on their surfaces. The surfaces of the plates were treated as planes that extend to infinity in they and z directions; this assumption excludes fringe effects. It was assumed that the two particles equilibrate when the planes that represent the two particles are immersed in an electrolyte solution with a concentration of c0 at a mutual distance of d. At equilibrium, the mutual repulsion of the double electric layers (pd) and the external pressure (p0) exerted by the two planes are equal (Figure 1).Figure 1
Interaction between the double electric layers associated with a pair of particles assumed to behave like parallel plates.The pressure on a unit volume within this space in thex direction is expressed as follows:(4)Fx=dpdx.Unit volumes within this space are also subject to an electrostatic force, defined as the product of the charge density and the electric field intensity according to the following equation:(5)Fe=ρdψdx.The condition for equilibrium between the two particles is expressed as follows:(6)dpdx+ρdψdx=0.Equation (6) can be rearranged to give(7)dp=−ρdψ.The density is defined by equation (8), which can also be expressed as equation (9):(8)ρ=∑iziFci0exp−ziFψRT,where F is the Faraday constant (F = 96490 C·mol−1), R is the universal gas constant (R = 8.314 J·mol−1 K−1), T is the absolute temperature (K), ci is the ion concentration (mol·L−1), the subscript 0 refers to the bulk solution, zi is the ionic valence, and ψ(x) is the potential at x m from the particle surface (V).So,(9)dp=−∑iziFci0exp−ziFψRTdψ.Therefore,pR can be calculated using the following equation:(10)pR=pd−p0=∫0ψd/2−∑iziFci0exp−ziFψRTdψ=∑iRTci0exp−ziFψd/2RT−1.If the pressure is measured in atmospheres (1 atm = 101.325 kPa), then equation (10) becomes(11)pR=RT101∑ici0exp−ziFψd/2RT−1.The potential at the center of the gap between the two particles,ψ(d/2), is expressed as follows [20]:(12)π21+122e2ZiFψd/2/RT+382e4ZiFψd/2/RT−arcsineZiFψ0−ZiFψd/2/2RT=14d8πF2zi2ci0εRTe−ZiFψd/2/2RT.


## 2.4.2. Calculation of Potential on Ore Particle Surface

According to equation (12), it is necessary to calculate the potential on the surface of the ore particles (ψ0) when calculating ψ(d/2). The method for calculating ψ0 is explained below.When the double layer in only thex direction is considered, and adsorption equilibrium is reached, the formula used to calculate the average ion concentration (c1) in diffusive double electric layers can be defined as follows [21]:(13)c1=N∞V=1V∫0VcxdV=κ∫01/κcxdx,whereN∞ is the total area of adsorbed ions in the diffusion double layer, V is the total area of the diffusion double layer, 1/κ is the thickness of the double layer, and c(x) is the concentration of adsorbed ions from the surface of the solid particle to the distance x. The expression for c(x) is given by(14)cx=∑ici0exp−ziFψRT.Alternatively,c1 can be calculated from(15)c1=CECS1/κ=CECκS,where CEC is the cation exchange capacity (cmol·kg−1) and S is the specific surface area of particles (m2·g−1).Ultimately, equations (13) and (15) can be combined using a minimum error method to calculate the potential on the surface of ore particles (ψ0).


## 2.4.3. Calculation ofψ(x) on Ore Particle Surface

When calculating the surface potentialψ0 of ore particles, it is necessary to define the surface potential ψ(x). In order to calculate these two parameters, the diffusion double layer structure on the particle surface needs to be described mathematically. At present, the Guy–Chapman model is the most accepted method to achieve this, and it describes the double layer structure by the Poisson–Boltzmann equation. The nonlinear Poisson–Boltzmann equation for electrolyte solutions is expressed [22]:(16)dydx=8πF2εRT∑ici0exp−ziy−1,where ε is the dielectric constant. The variable y is defined by y = Fψ(x)/RT, where ψ(x) is the potential at x m from the particle surface (V).When ammonium sulfate solution is injected into rare earth ores, ammonium ions in the solution exchange with rare earth ions on the surface of the ore particles. The solution surrounding the ore particles becomes a mixed electrolyte system. For these solutions, equation (16) becomes(17)dydx=8πF2εRTcCe−zCy−1+cDe−zDy−1+cEe−zEy−1+cHe−zHy−1,where zC, zD, zE, and zH are the valences of the ammonium ions, the sulfate ions corresponding to ammonium ions, the rare earth ions, and the sulfate ions corresponding to rare earth ions, respectively; cC, cD, cE, and cH are the concentrations of the ammonium ions, the sulfate ions corresponding to ammonium ions, the rare earth ions, and the sulfate ions corresponding to rare earth ions, respectively, in mol·L−1. For (NH4)2SO4, zC = 1, zD = −2, and cD = 0.5 cC. For (RE)2(SO4)3, zE = 3, zH = −2, and cH = 1.5 cE. With these substitutions, equation (17) becomes (18)dydx=8πF2εRTcCe−y−1+0.5cCe2y−1+cEe−3y−1+1.5cEe2y−1.Settingy = y0 at x = 0 and y = y(x) at x = x, and rearranging, equation (18) becomes(19)∫y0yx1cCe−y−1+0.5cCe2y−1+cEe−3y−1+1.5cEe2y−1dy=∫0x8πF2εRTdx.Given thaty = Fψ(x)/RT, equation (19) can be substituted to produce the following equation for ψ(x):(20)xs=1κ1ln9cE+3cC−6cE+2cC1−eFζ/RT+6cE+2cC1−eFζ/RT−9cE+3cC2−1−eFζ/RT23cE+cC21−eFζ/RTλ1.Here,λ1 and κ1 are defined by equations (21) and (22), respectively.(21)xs=1κ1ln9cE+3cC−6cE+2cC1−eFζ/RT+6cE+2cC1−eFζ/RT−9cE+3cC2−1−eFζ/RT23cE+cC21−eFζ/RTλ1,(22)xs=1κ1ln9cE+3cC−6cE+2cC1−eFζ/RT+6cE+2cC1−eFζ/RT−9cE+3cC2−1−eFζ/RT23cE+cC21−eFζ/RTλ1.


## 2.5. Thickness of the Sliding Layer in Electric Double Layer

For rare earth ore bodies, when the concentration of (NH4)2SO4 solution exceeds a certain level, ion exchange between the ammonium ions in solution and the rare earth ions on the surface of the ore particles occurs. This changes the ionic composition of the bulk solution around the ore particles and their surface potential, thus changing the thickness of the electric double layer on the surface of the particles. At present, the electric double layer thickness on the surface of particles can be approximately calculated by Debye parameter 1/κ, but this parameter only reflects the influence of the change in solution concentration and does not reflect the influence of the change in surface potential of particles caused by ion exchange. Ding proposed that the sliding layer thickness could be used to describe the electric double layer on the surface of particles [23]. Therefore, this section describes the method for calculating the sliding layer thickness in the double layer on the particle surface, which considers the influence of solution concentration and particle surface potential on the double layer thickness on the particle surface.Equation (20) can be rearranged to obtain the following equation, an expression for the distance to the particle surface (x):(23)xs=1κ1ln9cE+3cC−6cE+2cC1−eFζ/RT+6cE+2cC1−eFζ/RT−9cE+3cC2−1−eFζ/RT23cE+cC21−eFζ/RTλ1.In electrolyte systems,ψ(x) in equation (20) can be equated with the value of the zeta potential (ζ), so that x can be expressed in terms of the thickness of the sliding layer (xs), as shown in the following equation:(24)xs=1κ1ln9cE+3cC−6cE+2cC1−eFζ/RT+6cE+2cC1−eFζ/RT−9cE+3cC2−1−eFζ/RT23cE+cC21−eFζ/RTλ1.


## 3. Materials and Methods

### 3.1. Sample Preparation

Ionic clay rare earth ore was used as the test material in this study. The ionic clay was ground and screened using a 0.075 mm sieve. In order to analyze the effect of interaction between ore particles on the measured shear strength of the ore material, the total specific surface area and cation exchange capacity (CEC) of the samples were measured to enable the potential on the ore particle surfaces and the interaction forces between ore particles to be calculated. The total specific surface area was measured using a fully automated specific surface and pore diameter analyzer (Bates 3H-2000PS) utilizing a N2-B.E.T method. The CEC was measured using an ammonium acetate exchange method.


### 3.2. Mineral Leaching Experiment Method

Raw ores, which were screened using a 0.075 mm sieve, were collected and (NH4)2SO4 solution (0, 0.04, 0.08, 0.12, 0.16, 0.20, 0.24, and 0.28 mol·L−1) was added. After thorough mixing, a controlled moisture content of 21.8% was reached throughout the sample. A cutting ring was used to form samples with a compaction density of 1.71 g·cm−3, diameter 6.18 cm, and height 2 cm. The ore samples were saturated in the cutting ring using the extraction saturation method. The saturation solution used the corresponding concentration of (NH4)2SO4 solution and then the cutting ring was put in a 2 L beaker for mineral leaching at a solid to liquid ratio of 1 : 5. To enable enough time for adequate water-soil chemical reaction of the reshaped ore samples, and to prevent acid erosion of the ore cement by (NH4)2SO4 solution, the reaction time was set to 1 day.


### 3.3. Ore Gradation Experiment Method

For raw ore with initial particle sizes of less than 0.075 mm, wet samples were tested in a laser particle analyzer (BT2002, China) after leaching in (NH4)2SO4 solution. Each sample analysis was repeated at least five times.


### 3.4. Zeta Potential Test Method on Ore Particle Surfaces

A set of 0.5 g·L−1 mineral particle suspensions were prepared using four different concentrations of ammonium sulfate solution (0, 0.04, 0.08, 0.12, 0.16, 0.20, 0.24, and 0.28 mol·L−1) and tested in a microelectrophoresis apparatus (JS94H2, China) without pH adjustment. The temperature was maintained at room temperature (25°C) during the test and each sample was tested at least five times. The error range on the measurements was ±2 mv.


## 3.1. Sample Preparation

Ionic clay rare earth ore was used as the test material in this study. The ionic clay was ground and screened using a 0.075 mm sieve. In order to analyze the effect of interaction between ore particles on the measured shear strength of the ore material, the total specific surface area and cation exchange capacity (CEC) of the samples were measured to enable the potential on the ore particle surfaces and the interaction forces between ore particles to be calculated. The total specific surface area was measured using a fully automated specific surface and pore diameter analyzer (Bates 3H-2000PS) utilizing a N2-B.E.T method. The CEC was measured using an ammonium acetate exchange method.


## 3.2. Mineral Leaching Experiment Method

Raw ores, which were screened using a 0.075 mm sieve, were collected and (NH4)2SO4 solution (0, 0.04, 0.08, 0.12, 0.16, 0.20, 0.24, and 0.28 mol·L−1) was added. After thorough mixing, a controlled moisture content of 21.8% was reached throughout the sample. A cutting ring was used to form samples with a compaction density of 1.71 g·cm−3, diameter 6.18 cm, and height 2 cm. The ore samples were saturated in the cutting ring using the extraction saturation method. The saturation solution used the corresponding concentration of (NH4)2SO4 solution and then the cutting ring was put in a 2 L beaker for mineral leaching at a solid to liquid ratio of 1 : 5. To enable enough time for adequate water-soil chemical reaction of the reshaped ore samples, and to prevent acid erosion of the ore cement by (NH4)2SO4 solution, the reaction time was set to 1 day.


## 3.3. Ore Gradation Experiment Method

For raw ore with initial particle sizes of less than 0.075 mm, wet samples were tested in a laser particle analyzer (BT2002, China) after leaching in (NH4)2SO4 solution. Each sample analysis was repeated at least five times.


## 3.4. Zeta Potential Test Method on Ore Particle Surfaces

A set of 0.5 g·L−1 mineral particle suspensions were prepared using four different concentrations of ammonium sulfate solution (0, 0.04, 0.08, 0.12, 0.16, 0.20, 0.24, and 0.28 mol·L−1) and tested in a microelectrophoresis apparatus (JS94H2, China) without pH adjustment. The temperature was maintained at room temperature (25°C) during the test and each sample was tested at least five times. The error range on the measurements was ±2 mv.


## 4. Results and Discussion

### 4.1. Variation Law of Interaction Force between Ore Particles

Changes in solution concentration result in changes to the particle surface potential, the zeta potential, and the types of ions contained in the electric double layer. The thickness of double-electrode layers on ore particle surfaces and interaction forces between ore particles also change accordingly. To calculate the thickness of the sliding layer in the double-electrode layers on a particle surface and the interaction force between particles, ammonium ion concentration, rare earth ion (RE) concentration, zeta potential on an ore particle surface, CEC of ore sample, and specific surface area in solution with different concentrations of (NH4)2SO4 solution after mineral leaching were determined. Main indexes of ore samples with initial particle size of less than 0.075 mm are shown in Table 1.Table 1
Main indexes of ore samples with initial particle size of less than 0.075 mm.
Sample no.Ammonium ion concentration (mol·L−1)RE concentration (mol·L−1)Zeta potential on particle surface (mv)Specific surface area of particles (m2·g−1)BeforeAfter1≤0.00100−42.2523.8720.080.0783.6 × 10−4−34.2019.9930.160.1544.5 × 10−4−31.1018.1440.240.2244.3 × 10−4−27.4518.5750.320.3084.2 × 10−4−25.2517.5760.400.3834.4 × 10−4−15.5018.5470.480.464.3 × 10−4−10.4017.5480.560.5313.6 × 10−4−7.6018.65According to a previously reported theoretical calculation model, the concentration of bulk solution cannot be zero when calculating the interaction force between ore particles. Therefore, it is hypothesized that when the concentration of bulk solution is lower than a certain value, the zeta potential on ore particle surfaces does not change, and the electric double layer on the surface of ore body particles is not affected. In this study, (NH4)2SO4 solution was used as the pore solution. The concentration of bulk solution was calculated to be approximately 2 × 10−4 mol·L−1 from equations (20)–(22) according to the zeta potentials when the concentration of (NH4)2SO4 solution was 0 mol·L−1.Next, parameters were introduced into the ore particle surface potential calculations. The variation in particle surface potential when the (NH4)2SO4 solution concentration increased from 0 to 0.28 mol·L−1 was obtained. As can be seen from Figure 2, particle surface potential decreased from –372 to –126 mv when the (NH4)2SO4 solution concentration increased from 0 to 0.28 mol·L−1. This was mainly due to the negatively charged particles with one double-electrode layer absorbed on the particle surfaces, resulting in a relatively low initial concentration of ions in the double-electrode layer. With increasing (NH4)2SO4 concentration in the ore pores, the concentration difference drove the movement of ammonium ions in the pore solution towards the double-electrode layer adsorbed onto the soil particle surfaces and then gradually through the double-electrode layer. In addition to ion exchange between some ammonium ions and rare earth ions, some ammonium ions were finally adsorbed on the ore particle surfaces and neutralized negative charges on the ore particles. This decreased the charge of ore particles and the soil particle surface potentials. This research conclusion was consistent with Liu’s [24].Figure 2
Relationship between (NH4)2SO4 solution concentration and particle surface potential.


#### 4.1.1. Calculation of Thickness of Sliding Layer on Ore Particle Surface

Given the zeta potential on the particle surfaces and the calculated surface potential, the thickness of the sliding layer in the double-electrode layer on the particle surfaces could be calculated using equation (24). The results are shown in Figure 3. The thickness of the sliding layer decreased from 43.50 to 3.00 nm, when the (NH4)2SO4 solution concentration increased from 0 to 0.16 mol·L−1, and from 3.00 to 3.47 nm, when the (NH4)2SO4 solution concentration increased from 0.16 to 0.28 mol·L−1.Figure 3
Variation of sliding layer thickness on ore particle surfaces.


#### 4.1.2. Calculation of Electrostatic Repulsive Force between Ore Particles

With the particle surface potentials now known, variation in the electrostatic repulsion force between particles at different (NH4)2SO4 solution concentrations could be calculated using equation (10). Results are shown in Figure 4.Figure 4
Relationship between electrostatic repulsive force and distance between two particles. (a) ≤001∼0.16 mol/L (NH4)2SO4 solution. (b) 0.16∼0.28 mol/L (NH4)2SO4 solution.
(a)(b)It could be seen from Figure4 that the electrostatic repulsive force between two fixed ore particles was negatively correlated to the (NH4)2SO4 solution concentration between 0.04 and 0.16 mol·L−1 but was positively correlated from 0.16 to 0.28 mol·L−1. For the electrostatic repulsive force between two fixed ore particles, the main influencing factors included solution concentration and thickness of double electric layer on particle surface [25]. Thus, the variation of electrostatic repulsive force in Figure 4 could be analyzed as follows.First, when the (NH4)2SO4 solution concentration was lower than 0.16 mol·L−1, there was a negative correlation with the electrostatic repulsive force between ore particles. The existence of an electrostatic repulsive force between particles means there must be an overlap of the double electric layer on the two particle surfaces. Given that there is a certain number of ions in the double-electrode layer, this number of ions increases with expansion of the overlapping area and the electrostatic repulsive force between particles also increases (shadowed areas in Figure 5(a)). According to calculations, the thickness of the sliding layer on the ore particle surface decreased from 43.50 to 3.00 nm when the (NH4)2SO4 solution concentration increased from 0 to 0.16 mol·L−1; in other words, although the number of ions in the double electric layer on the particle surfaces increased with increasing (NH4)2SO4 solution concentration, the thickness of the double-electrode layer decreased and the overlapping region of the double electric layer between two particles became narrower (Figure 5(b)). The net result was a decrease in electrostatic repulsive force between ore particles.Figure 5
Diagram of variations in double electric layer on particle surfaces at different concentrations of (NH4)2SO4 solution. (a) 0.04 mol·L−1. (b) 0.16 mol·L−1. (c) 0.28 mol·L−1. Notes: yellow solid dots are soil particles, red solid dots are cations, and black circles show the double-electrode layer adsorbed onto the particle surface.
(a)(b)(c)Secondly, when the (NH4)2SO4 solution concentration was greater than 0.16 mol·L−1, the electrostatic repulsive force between particles was positively correlated with concentration. As the (NH4)2SO4 solution concentration increased from 0.16 to 0.28 mol·L−1, the thickness of the sliding layer increased from 3.00 to 3.47 nm. In other words, the thickness of the double electric layer on the particle surfaces increased with increasing (NH4)2SO4 solution concentration and the overlapping area expanded. Furthermore, the number of ions in the double electric layer on the particle surfaces increased accordingly. Therefore as shown in Figures 5(b) and 5(c), the net result was an increase in the electrostatic repulsive force between ore particles.


#### 4.1.3. Calculation of Net Force between Ore Particles

When the surface potential is known, the variation in net force between ore particles at different (NH4)2SO4 solution concentrations can be calculated using equation (1). Results are shown in Figure 6.Figure 6
The relationship between net force and particle distance.Figure6 shows that when the (NH4)2SO4 solution concentration was ≤0.001 mol·L−1, the distance between particles ranged from 1.55 to 1.75 nm; when the (NH4)2SO4 solution concentration was 0.04 mol·L−1, the distance between particles ranged from 1.55 to 2.70 nm; when the (NH4)2SO4 solution concentration was 0.08 mol·L−1, the distance between particles ranged from 1.55 to 2.80 nm; when the (NH4)2SO4 solution concentration was 0.12 mol·L−1, the distance between particles ranged from 1.55 to 2.90 nm; when the (NH4)2SO4 solution concentration was 0.16 mol·L−1, the distance between particles ranged from 1.55 to 3.05 nm; when (NH4)2SO4 solution concentration was 0.20 mol·L−1, the distance between particles ranged from 1.55 to 3.00 nm; when the (NH4)2SO4 solution concentration was 0.24 mol·L−1, the distance between particles ranged from 1.55 to 3.00 nm; and when the (NH4)2SO4 solution concentration was 0.28 mol·L−1, the distance between particles ranged from 1.55 to 2.90 nm.Additionally, Figure6 shows that when the (NH4)2SO4 solution concentration ranged from 0.04 to 0.28 mol·L−1, the net force between ore particles transformed from net repulsion to net attraction when the distance between two particles increased; this net attraction reached a maximum at approximately 1.55 nm. When the (NH4)2SO4 solution concentration ranged from 0.04 to 0.16 mol·L−1, the maximum net attraction was positively correlated with the (NH4)2SO4 solution concentration. In contrast, the maximum net attraction was negatively correlated with the (NH4)2SO4 solution concentration from 0.16 to 0.28 mol·L−1 (Figure 7).Figure 7
Relationship between net force and (NH4)2SO4 solution concentration.


### 4.2. Effects of (NH4)2SO4 Solution Concentration on the Stability of Ore Aggregate

#### 4.2.1. Determining the Critical (NH4)2SO4 Solution Concentration at Particle Size Less Than 0.075 mm

Particle grading curves of ore bodies at different (NH4)2SO4 solution concentrations were determined. A shown in Figure 8(a), the content of fine particles (<50 μm) decreased while the content of larger particles (>50 μm) increased when the (NH4)2SO4 solution concentration increased gradually from 0 to 0.16 mol·L−1, indicating the clustering of ore particles. Figure 8(b) reveals the opposite phenomenon when the (NH4)2SO4 solution concentration increased from 0.16 to 0.28 mol·L−1, indicating the scattering of ore particles. Therefore, for ore bodies with a particle size of less than 0.075 mm, the critical (NH4)2SO4 solution concentration affecting aggregation and dispersion of ore particles was 0.16 mol·L−1.Figure 8
Grading of ore particles. (a) ≤0.001–0.16 mol·L−1 (NH4)2SO4 solution. (b) 0.16–0.28 mol·L−1 (NH4)2SO4 solution.
(a)(b)Changes in the interaction forces between ore particles determine the clustering or scattering behavior of particles. Therefore, the influences of (NH4)2SO4 solution concentration on clustering of ore particles were explained on the basis of the interaction forces between ore particles described in Section 3.1, with further details provided below.First, when the (NH4)2SO4 solution concentration was lower than the critical concentration, ore particles clustered gradually with increasing (NH4)2SO4 solution concentration. According to the calculated interaction force, the interaction force between particles became net attraction when the (NH4)2SO4 solution concentration was ≤0.001 mol·L−1 and the distance between ore particles was within 1.55–1.75 nm, resulting in ore particles in this distance range clustering together, while ore particles beyond this distance range did not. Similarly, the interaction force between particles was net attraction when the (NH4)2SO4 solution concentration was 0.04, 0.08, 0.12, and 0.16 mol·L−1 and the distance between ore particles was 1.55–2.70, 1.55–2.80, 1.55–2.90, and 1.55–3.05 nm, respectively. This indicated that ore particles clustered into aggregates in these three distance ranges. Therefore, the distance range of net attraction increased when the (NH4)2SO4 solution concentration increased from ≤0.001 to 0.16 mol·L−1 and the ore particles within this distance gradually agglomerated into new aggregates, as shown in Figure 8(a) [26, 27].Secondly, when the (NH4)2SO4 solution concentration was higher than the critical concentration, ore particles scattered with increasing (NH4)2SO4 solution concentration. According to calculations, the interaction force between particles was net attraction at four (NH4)2SO4 solution concentrations (0.16, 0.20, 0.24, and 0.28 mol·L−1) and four corresponding distance ranges (1.55–3.5, 1.55–3.00, 1.55–3.00, and 1.55–2.90 nm). In other words, ore particles clustered into aggregates in these distance ranges. Therefore, the distance range of net attraction decreased when (NH4)2SO4 solution concentration increased from 0.16 to 0.28 mol·L−1 and ore particles beyond this distance were gradually dispersed, as shown in Figure 8(b) [26, 27].


#### 4.2.2. Effects of CEC on Critical (NH4)2SO4 Solution Concentration

Based on the above analysis, for ore bodies with particle sizes of less than 0.075 mm, the critical (NH4)2SO4 solution concentration affecting aggregation and dispersion of ore particles was determined and the relevant mechanism elucidated. Ore grading after mineral leaching for different particle sizes (0.038 and 0.1 mm) and different (NH4)2SO4 solution concentrations was performed according to the experimental method described in Section 2.1. The results are shown in Figures 9 and 10.Figure 9
Grading of ore particles smaller than 0.038 mm.Figure 10
Grading of ore particles smaller than 0.1 mm.Figures9 and 10 reveal that the critical (NH4)2SO4 solution concentration was 0.20 mol·L−1 for particles smaller than 0.038 mm and 0.12 mol·L−1 for particles smaller than 0.1 mm. According to calculations of the interaction force between ore particles, solution concentration was the main influencing factor. Therefore, changes in the critical (NH4)2SO4 solution concentration arising from different grading levels was related to changes in CEC, thus influencing changes to the concentration of solution surrounding ore particles [28, 29]. Therefore, the relationship between CEC and critical concentration of (NH4)2SO4 solution was further analyzed (Figure 11).Figure 11
Relationship between critical concentration and CEC.It can be seen from Figure11 that there was an exponential relationship between critical (NH4)2SO4 solution concentration and CEC of ore particles: ccri = a + b exp(CEC/c), where ccri is the critical concentration of (NH4)2SO4 solution, CEC is the cation exchange capacity on the ore particle surface, and a, b, and c are fitting parameters (a = 0.058, b = 5.18 × 10−11, and c = 0.42). The critical (NH4)2SO4 solution concentration can be estimated when the CEC is fixed; on this basis, critical concentrations of (NH4)2SO4 solution at different particle compositions could be determined.


## 4.1. Variation Law of Interaction Force between Ore Particles

Changes in solution concentration result in changes to the particle surface potential, the zeta potential, and the types of ions contained in the electric double layer. The thickness of double-electrode layers on ore particle surfaces and interaction forces between ore particles also change accordingly. To calculate the thickness of the sliding layer in the double-electrode layers on a particle surface and the interaction force between particles, ammonium ion concentration, rare earth ion (RE) concentration, zeta potential on an ore particle surface, CEC of ore sample, and specific surface area in solution with different concentrations of (NH4)2SO4 solution after mineral leaching were determined. Main indexes of ore samples with initial particle size of less than 0.075 mm are shown in Table 1.Table 1
Main indexes of ore samples with initial particle size of less than 0.075 mm.
Sample no.Ammonium ion concentration (mol·L−1)RE concentration (mol·L−1)Zeta potential on particle surface (mv)Specific surface area of particles (m2·g−1)BeforeAfter1≤0.00100−42.2523.8720.080.0783.6 × 10−4−34.2019.9930.160.1544.5 × 10−4−31.1018.1440.240.2244.3 × 10−4−27.4518.5750.320.3084.2 × 10−4−25.2517.5760.400.3834.4 × 10−4−15.5018.5470.480.464.3 × 10−4−10.4017.5480.560.5313.6 × 10−4−7.6018.65According to a previously reported theoretical calculation model, the concentration of bulk solution cannot be zero when calculating the interaction force between ore particles. Therefore, it is hypothesized that when the concentration of bulk solution is lower than a certain value, the zeta potential on ore particle surfaces does not change, and the electric double layer on the surface of ore body particles is not affected. In this study, (NH4)2SO4 solution was used as the pore solution. The concentration of bulk solution was calculated to be approximately 2 × 10−4 mol·L−1 from equations (20)–(22) according to the zeta potentials when the concentration of (NH4)2SO4 solution was 0 mol·L−1.Next, parameters were introduced into the ore particle surface potential calculations. The variation in particle surface potential when the (NH4)2SO4 solution concentration increased from 0 to 0.28 mol·L−1 was obtained. As can be seen from Figure 2, particle surface potential decreased from –372 to –126 mv when the (NH4)2SO4 solution concentration increased from 0 to 0.28 mol·L−1. This was mainly due to the negatively charged particles with one double-electrode layer absorbed on the particle surfaces, resulting in a relatively low initial concentration of ions in the double-electrode layer. With increasing (NH4)2SO4 concentration in the ore pores, the concentration difference drove the movement of ammonium ions in the pore solution towards the double-electrode layer adsorbed onto the soil particle surfaces and then gradually through the double-electrode layer. In addition to ion exchange between some ammonium ions and rare earth ions, some ammonium ions were finally adsorbed on the ore particle surfaces and neutralized negative charges on the ore particles. This decreased the charge of ore particles and the soil particle surface potentials. This research conclusion was consistent with Liu’s [24].Figure 2
Relationship between (NH4)2SO4 solution concentration and particle surface potential.


### 4.1.1. Calculation of Thickness of Sliding Layer on Ore Particle Surface

Given the zeta potential on the particle surfaces and the calculated surface potential, the thickness of the sliding layer in the double-electrode layer on the particle surfaces could be calculated using equation (24). The results are shown in Figure 3. The thickness of the sliding layer decreased from 43.50 to 3.00 nm, when the (NH4)2SO4 solution concentration increased from 0 to 0.16 mol·L−1, and from 3.00 to 3.47 nm, when the (NH4)2SO4 solution concentration increased from 0.16 to 0.28 mol·L−1.Figure 3
Variation of sliding layer thickness on ore particle surfaces.


### 4.1.2. Calculation of Electrostatic Repulsive Force between Ore Particles

With the particle surface potentials now known, variation in the electrostatic repulsion force between particles at different (NH4)2SO4 solution concentrations could be calculated using equation (10). Results are shown in Figure 4.Figure 4
Relationship between electrostatic repulsive force and distance between two particles. (a) ≤001∼0.16 mol/L (NH4)2SO4 solution. (b) 0.16∼0.28 mol/L (NH4)2SO4 solution.
(a)(b)It could be seen from Figure4 that the electrostatic repulsive force between two fixed ore particles was negatively correlated to the (NH4)2SO4 solution concentration between 0.04 and 0.16 mol·L−1 but was positively correlated from 0.16 to 0.28 mol·L−1. For the electrostatic repulsive force between two fixed ore particles, the main influencing factors included solution concentration and thickness of double electric layer on particle surface [25]. Thus, the variation of electrostatic repulsive force in Figure 4 could be analyzed as follows.First, when the (NH4)2SO4 solution concentration was lower than 0.16 mol·L−1, there was a negative correlation with the electrostatic repulsive force between ore particles. The existence of an electrostatic repulsive force between particles means there must be an overlap of the double electric layer on the two particle surfaces. Given that there is a certain number of ions in the double-electrode layer, this number of ions increases with expansion of the overlapping area and the electrostatic repulsive force between particles also increases (shadowed areas in Figure 5(a)). According to calculations, the thickness of the sliding layer on the ore particle surface decreased from 43.50 to 3.00 nm when the (NH4)2SO4 solution concentration increased from 0 to 0.16 mol·L−1; in other words, although the number of ions in the double electric layer on the particle surfaces increased with increasing (NH4)2SO4 solution concentration, the thickness of the double-electrode layer decreased and the overlapping region of the double electric layer between two particles became narrower (Figure 5(b)). The net result was a decrease in electrostatic repulsive force between ore particles.Figure 5
Diagram of variations in double electric layer on particle surfaces at different concentrations of (NH4)2SO4 solution. (a) 0.04 mol·L−1. (b) 0.16 mol·L−1. (c) 0.28 mol·L−1. Notes: yellow solid dots are soil particles, red solid dots are cations, and black circles show the double-electrode layer adsorbed onto the particle surface.
(a)(b)(c)Secondly, when the (NH4)2SO4 solution concentration was greater than 0.16 mol·L−1, the electrostatic repulsive force between particles was positively correlated with concentration. As the (NH4)2SO4 solution concentration increased from 0.16 to 0.28 mol·L−1, the thickness of the sliding layer increased from 3.00 to 3.47 nm. In other words, the thickness of the double electric layer on the particle surfaces increased with increasing (NH4)2SO4 solution concentration and the overlapping area expanded. Furthermore, the number of ions in the double electric layer on the particle surfaces increased accordingly. Therefore as shown in Figures 5(b) and 5(c), the net result was an increase in the electrostatic repulsive force between ore particles.


### 4.1.3. Calculation of Net Force between Ore Particles

When the surface potential is known, the variation in net force between ore particles at different (NH4)2SO4 solution concentrations can be calculated using equation (1). Results are shown in Figure 6.Figure 6
The relationship between net force and particle distance.Figure6 shows that when the (NH4)2SO4 solution concentration was ≤0.001 mol·L−1, the distance between particles ranged from 1.55 to 1.75 nm; when the (NH4)2SO4 solution concentration was 0.04 mol·L−1, the distance between particles ranged from 1.55 to 2.70 nm; when the (NH4)2SO4 solution concentration was 0.08 mol·L−1, the distance between particles ranged from 1.55 to 2.80 nm; when the (NH4)2SO4 solution concentration was 0.12 mol·L−1, the distance between particles ranged from 1.55 to 2.90 nm; when the (NH4)2SO4 solution concentration was 0.16 mol·L−1, the distance between particles ranged from 1.55 to 3.05 nm; when (NH4)2SO4 solution concentration was 0.20 mol·L−1, the distance between particles ranged from 1.55 to 3.00 nm; when the (NH4)2SO4 solution concentration was 0.24 mol·L−1, the distance between particles ranged from 1.55 to 3.00 nm; and when the (NH4)2SO4 solution concentration was 0.28 mol·L−1, the distance between particles ranged from 1.55 to 2.90 nm.Additionally, Figure6 shows that when the (NH4)2SO4 solution concentration ranged from 0.04 to 0.28 mol·L−1, the net force between ore particles transformed from net repulsion to net attraction when the distance between two particles increased; this net attraction reached a maximum at approximately 1.55 nm. When the (NH4)2SO4 solution concentration ranged from 0.04 to 0.16 mol·L−1, the maximum net attraction was positively correlated with the (NH4)2SO4 solution concentration. In contrast, the maximum net attraction was negatively correlated with the (NH4)2SO4 solution concentration from 0.16 to 0.28 mol·L−1 (Figure 7).Figure 7
Relationship between net force and (NH4)2SO4 solution concentration.


## 4.1.1. Calculation of Thickness of Sliding Layer on Ore Particle Surface

Given the zeta potential on the particle surfaces and the calculated surface potential, the thickness of the sliding layer in the double-electrode layer on the particle surfaces could be calculated using equation (24). The results are shown in Figure 3. The thickness of the sliding layer decreased from 43.50 to 3.00 nm, when the (NH4)2SO4 solution concentration increased from 0 to 0.16 mol·L−1, and from 3.00 to 3.47 nm, when the (NH4)2SO4 solution concentration increased from 0.16 to 0.28 mol·L−1.Figure 3
Variation of sliding layer thickness on ore particle surfaces.


## 4.1.2. Calculation of Electrostatic Repulsive Force between Ore Particles

With the particle surface potentials now known, variation in the electrostatic repulsion force between particles at different (NH4)2SO4 solution concentrations could be calculated using equation (10). Results are shown in Figure 4.Figure 4
Relationship between electrostatic repulsive force and distance between two particles. (a) ≤001∼0.16 mol/L (NH4)2SO4 solution. (b) 0.16∼0.28 mol/L (NH4)2SO4 solution.
(a)(b)It could be seen from Figure4 that the electrostatic repulsive force between two fixed ore particles was negatively correlated to the (NH4)2SO4 solution concentration between 0.04 and 0.16 mol·L−1 but was positively correlated from 0.16 to 0.28 mol·L−1. For the electrostatic repulsive force between two fixed ore particles, the main influencing factors included solution concentration and thickness of double electric layer on particle surface [25]. Thus, the variation of electrostatic repulsive force in Figure 4 could be analyzed as follows.First, when the (NH4)2SO4 solution concentration was lower than 0.16 mol·L−1, there was a negative correlation with the electrostatic repulsive force between ore particles. The existence of an electrostatic repulsive force between particles means there must be an overlap of the double electric layer on the two particle surfaces. Given that there is a certain number of ions in the double-electrode layer, this number of ions increases with expansion of the overlapping area and the electrostatic repulsive force between particles also increases (shadowed areas in Figure 5(a)). According to calculations, the thickness of the sliding layer on the ore particle surface decreased from 43.50 to 3.00 nm when the (NH4)2SO4 solution concentration increased from 0 to 0.16 mol·L−1; in other words, although the number of ions in the double electric layer on the particle surfaces increased with increasing (NH4)2SO4 solution concentration, the thickness of the double-electrode layer decreased and the overlapping region of the double electric layer between two particles became narrower (Figure 5(b)). The net result was a decrease in electrostatic repulsive force between ore particles.Figure 5
Diagram of variations in double electric layer on particle surfaces at different concentrations of (NH4)2SO4 solution. (a) 0.04 mol·L−1. (b) 0.16 mol·L−1. (c) 0.28 mol·L−1. Notes: yellow solid dots are soil particles, red solid dots are cations, and black circles show the double-electrode layer adsorbed onto the particle surface.
(a)(b)(c)Secondly, when the (NH4)2SO4 solution concentration was greater than 0.16 mol·L−1, the electrostatic repulsive force between particles was positively correlated with concentration. As the (NH4)2SO4 solution concentration increased from 0.16 to 0.28 mol·L−1, the thickness of the sliding layer increased from 3.00 to 3.47 nm. In other words, the thickness of the double electric layer on the particle surfaces increased with increasing (NH4)2SO4 solution concentration and the overlapping area expanded. Furthermore, the number of ions in the double electric layer on the particle surfaces increased accordingly. Therefore as shown in Figures 5(b) and 5(c), the net result was an increase in the electrostatic repulsive force between ore particles.


## 4.1.3. Calculation of Net Force between Ore Particles

When the surface potential is known, the variation in net force between ore particles at different (NH4)2SO4 solution concentrations can be calculated using equation (1). Results are shown in Figure 6.Figure 6
The relationship between net force and particle distance.Figure6 shows that when the (NH4)2SO4 solution concentration was ≤0.001 mol·L−1, the distance between particles ranged from 1.55 to 1.75 nm; when the (NH4)2SO4 solution concentration was 0.04 mol·L−1, the distance between particles ranged from 1.55 to 2.70 nm; when the (NH4)2SO4 solution concentration was 0.08 mol·L−1, the distance between particles ranged from 1.55 to 2.80 nm; when the (NH4)2SO4 solution concentration was 0.12 mol·L−1, the distance between particles ranged from 1.55 to 2.90 nm; when the (NH4)2SO4 solution concentration was 0.16 mol·L−1, the distance between particles ranged from 1.55 to 3.05 nm; when (NH4)2SO4 solution concentration was 0.20 mol·L−1, the distance between particles ranged from 1.55 to 3.00 nm; when the (NH4)2SO4 solution concentration was 0.24 mol·L−1, the distance between particles ranged from 1.55 to 3.00 nm; and when the (NH4)2SO4 solution concentration was 0.28 mol·L−1, the distance between particles ranged from 1.55 to 2.90 nm.Additionally, Figure6 shows that when the (NH4)2SO4 solution concentration ranged from 0.04 to 0.28 mol·L−1, the net force between ore particles transformed from net repulsion to net attraction when the distance between two particles increased; this net attraction reached a maximum at approximately 1.55 nm. When the (NH4)2SO4 solution concentration ranged from 0.04 to 0.16 mol·L−1, the maximum net attraction was positively correlated with the (NH4)2SO4 solution concentration. In contrast, the maximum net attraction was negatively correlated with the (NH4)2SO4 solution concentration from 0.16 to 0.28 mol·L−1 (Figure 7).Figure 7
Relationship between net force and (NH4)2SO4 solution concentration.


## 4.2. Effects of (NH4)2SO4 Solution Concentration on the Stability of Ore Aggregate

### 4.2.1. Determining the Critical (NH4)2SO4 Solution Concentration at Particle Size Less Than 0.075 mm

Particle grading curves of ore bodies at different (NH4)2SO4 solution concentrations were determined. A shown in Figure 8(a), the content of fine particles (<50 μm) decreased while the content of larger particles (>50 μm) increased when the (NH4)2SO4 solution concentration increased gradually from 0 to 0.16 mol·L−1, indicating the clustering of ore particles. Figure 8(b) reveals the opposite phenomenon when the (NH4)2SO4 solution concentration increased from 0.16 to 0.28 mol·L−1, indicating the scattering of ore particles. Therefore, for ore bodies with a particle size of less than 0.075 mm, the critical (NH4)2SO4 solution concentration affecting aggregation and dispersion of ore particles was 0.16 mol·L−1.Figure 8
Grading of ore particles. (a) ≤0.001–0.16 mol·L−1 (NH4)2SO4 solution. (b) 0.16–0.28 mol·L−1 (NH4)2SO4 solution.
(a)(b)Changes in the interaction forces between ore particles determine the clustering or scattering behavior of particles. Therefore, the influences of (NH4)2SO4 solution concentration on clustering of ore particles were explained on the basis of the interaction forces between ore particles described in Section 3.1, with further details provided below.First, when the (NH4)2SO4 solution concentration was lower than the critical concentration, ore particles clustered gradually with increasing (NH4)2SO4 solution concentration. According to the calculated interaction force, the interaction force between particles became net attraction when the (NH4)2SO4 solution concentration was ≤0.001 mol·L−1 and the distance between ore particles was within 1.55–1.75 nm, resulting in ore particles in this distance range clustering together, while ore particles beyond this distance range did not. Similarly, the interaction force between particles was net attraction when the (NH4)2SO4 solution concentration was 0.04, 0.08, 0.12, and 0.16 mol·L−1 and the distance between ore particles was 1.55–2.70, 1.55–2.80, 1.55–2.90, and 1.55–3.05 nm, respectively. This indicated that ore particles clustered into aggregates in these three distance ranges. Therefore, the distance range of net attraction increased when the (NH4)2SO4 solution concentration increased from ≤0.001 to 0.16 mol·L−1 and the ore particles within this distance gradually agglomerated into new aggregates, as shown in Figure 8(a) [26, 27].Secondly, when the (NH4)2SO4 solution concentration was higher than the critical concentration, ore particles scattered with increasing (NH4)2SO4 solution concentration. According to calculations, the interaction force between particles was net attraction at four (NH4)2SO4 solution concentrations (0.16, 0.20, 0.24, and 0.28 mol·L−1) and four corresponding distance ranges (1.55–3.5, 1.55–3.00, 1.55–3.00, and 1.55–2.90 nm). In other words, ore particles clustered into aggregates in these distance ranges. Therefore, the distance range of net attraction decreased when (NH4)2SO4 solution concentration increased from 0.16 to 0.28 mol·L−1 and ore particles beyond this distance were gradually dispersed, as shown in Figure 8(b) [26, 27].


### 4.2.2. Effects of CEC on Critical (NH4)2SO4 Solution Concentration

Based on the above analysis, for ore bodies with particle sizes of less than 0.075 mm, the critical (NH4)2SO4 solution concentration affecting aggregation and dispersion of ore particles was determined and the relevant mechanism elucidated. Ore grading after mineral leaching for different particle sizes (0.038 and 0.1 mm) and different (NH4)2SO4 solution concentrations was performed according to the experimental method described in Section 2.1. The results are shown in Figures 9 and 10.Figure 9
Grading of ore particles smaller than 0.038 mm.Figure 10
Grading of ore particles smaller than 0.1 mm.Figures9 and 10 reveal that the critical (NH4)2SO4 solution concentration was 0.20 mol·L−1 for particles smaller than 0.038 mm and 0.12 mol·L−1 for particles smaller than 0.1 mm. According to calculations of the interaction force between ore particles, solution concentration was the main influencing factor. Therefore, changes in the critical (NH4)2SO4 solution concentration arising from different grading levels was related to changes in CEC, thus influencing changes to the concentration of solution surrounding ore particles [28, 29]. Therefore, the relationship between CEC and critical concentration of (NH4)2SO4 solution was further analyzed (Figure 11).Figure 11
Relationship between critical concentration and CEC.It can be seen from Figure11 that there was an exponential relationship between critical (NH4)2SO4 solution concentration and CEC of ore particles: ccri = a + b exp(CEC/c), where ccri is the critical concentration of (NH4)2SO4 solution, CEC is the cation exchange capacity on the ore particle surface, and a, b, and c are fitting parameters (a = 0.058, b = 5.18 × 10−11, and c = 0.42). The critical (NH4)2SO4 solution concentration can be estimated when the CEC is fixed; on this basis, critical concentrations of (NH4)2SO4 solution at different particle compositions could be determined.


## 4.2.1. Determining the Critical (NH4)2SO4 Solution Concentration at Particle Size Less Than 0.075 mm

Particle grading curves of ore bodies at different (NH4)2SO4 solution concentrations were determined. A shown in Figure 8(a), the content of fine particles (<50 μm) decreased while the content of larger particles (>50 μm) increased when the (NH4)2SO4 solution concentration increased gradually from 0 to 0.16 mol·L−1, indicating the clustering of ore particles. Figure 8(b) reveals the opposite phenomenon when the (NH4)2SO4 solution concentration increased from 0.16 to 0.28 mol·L−1, indicating the scattering of ore particles. Therefore, for ore bodies with a particle size of less than 0.075 mm, the critical (NH4)2SO4 solution concentration affecting aggregation and dispersion of ore particles was 0.16 mol·L−1.Figure 8
Grading of ore particles. (a) ≤0.001–0.16 mol·L−1 (NH4)2SO4 solution. (b) 0.16–0.28 mol·L−1 (NH4)2SO4 solution.
(a)(b)Changes in the interaction forces between ore particles determine the clustering or scattering behavior of particles. Therefore, the influences of (NH4)2SO4 solution concentration on clustering of ore particles were explained on the basis of the interaction forces between ore particles described in Section 3.1, with further details provided below.First, when the (NH4)2SO4 solution concentration was lower than the critical concentration, ore particles clustered gradually with increasing (NH4)2SO4 solution concentration. According to the calculated interaction force, the interaction force between particles became net attraction when the (NH4)2SO4 solution concentration was ≤0.001 mol·L−1 and the distance between ore particles was within 1.55–1.75 nm, resulting in ore particles in this distance range clustering together, while ore particles beyond this distance range did not. Similarly, the interaction force between particles was net attraction when the (NH4)2SO4 solution concentration was 0.04, 0.08, 0.12, and 0.16 mol·L−1 and the distance between ore particles was 1.55–2.70, 1.55–2.80, 1.55–2.90, and 1.55–3.05 nm, respectively. This indicated that ore particles clustered into aggregates in these three distance ranges. Therefore, the distance range of net attraction increased when the (NH4)2SO4 solution concentration increased from ≤0.001 to 0.16 mol·L−1 and the ore particles within this distance gradually agglomerated into new aggregates, as shown in Figure 8(a) [26, 27].Secondly, when the (NH4)2SO4 solution concentration was higher than the critical concentration, ore particles scattered with increasing (NH4)2SO4 solution concentration. According to calculations, the interaction force between particles was net attraction at four (NH4)2SO4 solution concentrations (0.16, 0.20, 0.24, and 0.28 mol·L−1) and four corresponding distance ranges (1.55–3.5, 1.55–3.00, 1.55–3.00, and 1.55–2.90 nm). In other words, ore particles clustered into aggregates in these distance ranges. Therefore, the distance range of net attraction decreased when (NH4)2SO4 solution concentration increased from 0.16 to 0.28 mol·L−1 and ore particles beyond this distance were gradually dispersed, as shown in Figure 8(b) [26, 27].


## 4.2.2. Effects of CEC on Critical (NH4)2SO4 Solution Concentration

Based on the above analysis, for ore bodies with particle sizes of less than 0.075 mm, the critical (NH4)2SO4 solution concentration affecting aggregation and dispersion of ore particles was determined and the relevant mechanism elucidated. Ore grading after mineral leaching for different particle sizes (0.038 and 0.1 mm) and different (NH4)2SO4 solution concentrations was performed according to the experimental method described in Section 2.1. The results are shown in Figures 9 and 10.Figure 9
Grading of ore particles smaller than 0.038 mm.Figure 10
Grading of ore particles smaller than 0.1 mm.Figures9 and 10 reveal that the critical (NH4)2SO4 solution concentration was 0.20 mol·L−1 for particles smaller than 0.038 mm and 0.12 mol·L−1 for particles smaller than 0.1 mm. According to calculations of the interaction force between ore particles, solution concentration was the main influencing factor. Therefore, changes in the critical (NH4)2SO4 solution concentration arising from different grading levels was related to changes in CEC, thus influencing changes to the concentration of solution surrounding ore particles [28, 29]. Therefore, the relationship between CEC and critical concentration of (NH4)2SO4 solution was further analyzed (Figure 11).Figure 11
Relationship between critical concentration and CEC.It can be seen from Figure11 that there was an exponential relationship between critical (NH4)2SO4 solution concentration and CEC of ore particles: ccri = a + b exp(CEC/c), where ccri is the critical concentration of (NH4)2SO4 solution, CEC is the cation exchange capacity on the ore particle surface, and a, b, and c are fitting parameters (a = 0.058, b = 5.18 × 10−11, and c = 0.42). The critical (NH4)2SO4 solution concentration can be estimated when the CEC is fixed; on this basis, critical concentrations of (NH4)2SO4 solution at different particle compositions could be determined.


## 5. Conclusions

For ion-absorbed rare earth ores, the concentration of the injected (NH4)2SO4 solution in in situ mineral leaching is generally no higher than 0.23 mol·L−1. And due to the ion exchange effect, interaction forces between ore particles are changed. This changes the stability of the ore body aggregates, and the particles are aggregated or diffused, which directly affects the cohesive force of the ore body and thus affects the shear strength and stability of the ore body slope. At the same time, the agglomeration or diffusion of ore body particles will directly affect the effective diameter of the pores in the ore body, thus affecting the permeability and directly affecting the mining efficiency. Therefore, the effect of (NH4)2SO4 solution concentration on the interaction force between ore particles was calculated and the mechanism underpinning the effect of (NH4)2SO4 solution concentration on ore aggregate stability was determined. Ultimately, an empirical formula for estimating the critical (NH4)2SO4 solution concentration for aggregation and dispersion of ore body aggregates at different grain compositions was proposed. Some major conclusions could be drawn, which can provide reference for the concentration selection of ammonium sulfate solution in the leaching and mining process of ammonium sulfate solution in different gradation ionic rare earth mines.(1)
For ore bodies with initial particle size smaller than 0.075 mm, when the (NH4)2SO4 solution concentration was ≤0.001 mol·L−1, the interaction force between ore particles was net repulsion resulting in ore particle dispersion. The interaction force between particles was net attraction at four (NH4)2SO4 solution concentrations of 0.04, 0.08, 0.12, and 0.16 mol·L−1 and the four corresponding distance ranges of 1.55–2.70, 1.55–2.80 nm, 1.55–2.90  nm, and 1.55–3.05 nm, resulting in ore particle clustering into aggregates within these distance ranges. Therefore, the net attraction distance range increased when the (NH4)2SO4 solution concentration increased from ≤0.001 to 0.16 mol·L−1, resulting in the aggregation of ore particles within this range.(2)
For ore bodies with initial particle size of less than 0.075 mm, the interaction force of particles was net attraction at four (NH4)2SO4 solution concentrations of 0.16, 0.20, 0.24, and 0.28 mol·L−1 and the four corresponding distance ranges of 1.55–3.05, 1.55–3.00, 1.55–3.00, and 1.55–2.90 nm, with ore particles clustering into aggregates in these distance ranges. Therefore, the net attraction distance range decreased with increasing (NH4)2SO4 solution concentration from 0.16 to 0.28 mol·L−1, with the dispersion of ore particles beyond the range of net attraction.(3)
For ore bodies with particle sizes of less than 0.038, 0.075, and 0.1 mm, the CEC was 9.13, 8.96, and 8.8 cmol·kg−1, respectively. The critical (NH4)2SO4 solution concentration that discriminated between clustering and scatting of ore particles was 0.12, 0.16, and 0.20 mol·L−1, respectively. Therefore, there was an exponential correlation between the critical (NH4)2SO4 solution concentration and CEC.



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