# A Modified Method to Calculate Critical Coagulation Concentration Based on DLVO Theory

Authors: Zhijun Zhang; Liang Zhao; Yanan Li; Mo Chu
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

The critical coagulation concentration (CCC) is defined as the minimum concentration of counterions to induce coagulation of colloidal particles. A modified calculation method was proposed to calculate CCC. Comparing the modified calculation method of CCC with the traditional calculation method, the critical condition of modified calculation method is stricter than traditional calculation method. The critical condition of modified calculation method is the maximum value of interaction force that is zero, and the critical condition of traditional calculation method is the maximum value of interaction energy that is zero. The calculation result of CCC based on interaction force is greater than the calculation value based on interaction energy. The CCC value of modified calculation method can ensure particles to coagulate definitely.


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## Body

## 1. Introduction

A most important property of a colloidal system is its stability defined by the tendency of the particles to aggregate. Aggregation of colloidal particles is a result of colloidal forces and was described independently by Derjaguin and Landau and Verwey and Overbeek in their DLVO theory. The DLVO theory involves two major terms: the Van der Waals force/energy based on interparticle distance and the electrostatic double layer force/energy aroused by overlapping of electrical double layer [1–3]. The total interaction force/energy in the DLVO theory is the summation of the two forces/energies: (1)
V
T
=
V
A
+
V
R
,where V
T here stands for total interaction energy, V
A stands for Van der Waals energy which is usually attractive, and V
R stands for the electrostatic double layer energy.As shown in Figure1, an energy profile based on the DLVO theory clarifies interparticle interactions.Figure 1
Interparticle interaction energy profile based on DLVO theory.For two semi-infinite parallel plate particles, the Van der Waals energy is shown as(2)
V
A
=
-
A
12
π
h
2
,where A is the Hamaker constant of the system and h is the distance of closest approach between the particles.And the electrostatic double layer energy is shown as(3)
V
R
=
64000
N
A
C
k
T
κ
γ
0
2
exp
⁡
-
κ
h
,where N
A is Avogadro’s number, C is the bulk concentration of electrolyte, k and T are the Boltzmann constant and the absolute temperature, respectively, h is the shortest separation distance, γ
0 is defined as γ
0
=
(
e
x
p
⁡
(
Z
e
ψ
0
/
2
k
T
)
-
1
)
/
(
e
x
p
⁡
(
Z
e
ψ
0
/
2
k
T
)
+
1
), and κ is the inverse Debye length [4], which is defined by(4)
κ
=
1000
N
A
e
2
ε
k
T
∑
Z
i
2
C
i
1
/
2
,where e is the elementary charge, ε is the permittivity of the dispersion medium (ε
r
ε
0), and Z
i and C
i are valency and concentration of ith ionic species, respectively.The electrostatic double layer force/energy depended on the bulk concentration of electrolyte, and the critical coagulation concentration (CCC) is one of the most significant characteristics of a colloidal dispersion. It is defined as the minimum concentration of counterions to induce coagulation of colloidal particles, and it is often applied to assess the status of a colloidal dispersion [5–9].In the traditional case, it is based on the interaction energy to calculate CCC. In this paper, a modified calculation method is proposed to calculate CCC based on interaction force, the critical condition of this calculation method is stricter, and the calculation results are more accurate.


## 2. Traditional Calculation Method of CCC by Interaction Energy

The traditional calculation method of CCC is based on the interaction energy of two colloidal particles.For two semi-infinite parallel plate particles, the total interaction energy is shown as [10–12](5)
V
T
=
V
R
+
V
A
=
64000
N
A
C
k
T
κ
γ
0
2
exp
⁡
-
κ
h
-
A
12
π
h
2
.As shown in Figure2, the condition of critical coagulation concentration based on total interaction energy is V
m
a
x
=
0. Consider(6)
V
T
=
0
,
d
V
T
d
h
=
0
.Figure 2
Conditions of critical coagulation concentration based on total interaction energy.Equation (6) can be expressed as(7)
V
T
=
64000
N
A
C
k
T
κ
γ
0
2
exp
⁡
-
κ
h
-
A
12
π
h
2
=
0
,
d
V
T
d
h
=
-
κ
64000
N
A
C
k
T
κ
γ
0
2
exp
⁡
-
κ
h
+
A
6
π
h
3
=
0
.For the first equation of (7) multiplied by -
2
/
h, (7) leads to(8)
V
T
=
-
2
h
64000
N
A
C
k
T
κ
γ
0
2
exp
⁡
-
κ
h
+
A
6
π
h
3
=
0
,
d
V
T
d
h
=
-
κ
64000
N
A
C
k
T
κ
γ
0
2
exp
⁡
-
κ
h
+
A
6
π
h
3
=
0
.Solving forh and κ, (8) gives(9)
κ
h
=
2
.For symmetrical type electrolyte,κ is defined as(10)
κ
=
2000
Z
2
e
2
N
A
C
ε
k
T
1
/
2
.Substituting (10) into (5) leads to(11)
V
T
=
64000
·
ε
k
3
T
3
N
A
C
2000
Z
2
e
2
1
/
2
γ
0
2
exp
⁡
-
κ
h
-
2000
Z
2
e
2
N
A
C
A
12
π
ε
k
T
·
1
κ
h
2
=
0
.Substituting (9) into (11) and solving the equation, the CCC can be obtained (12)
C
CC
C
=
159648
ε
3
k
5
T
5
γ
0
4
1000
N
A
e
6
A
2
Z
6
.As we have knownk
=
1.3805
×
1
0
-
23 J/K, e
=
1.602
×
1
0
-
19 C, N
A
=
6.023
×
1
0
23 mol−1, and ε
=
ε
0
ε
r
=
8.854
×
1
0
-
12
×
78.5
=
6.95
×
1
0
-
10 F/m. Substituting these parameters into (12) leads to(13)
C
CCC
=
3.525
×
1
0
-
51
T
5
γ
0
4
A
2
Z
6
.


## 3. Modified Calculation Method of CCC by Interaction Force

The relationship of interaction energy and interaction force of two colloidal particles is(14)
V
=
-
∫
+
∞
D
F
h
d
hor(15)
F
=
-
d
V
d
h
.The relationship of interaction energy curve and interaction force curve is shown in Figure3. In the process of two colloidal particles approaching, when the interaction energy reaches the maximum value, the interaction force is zero, and the interaction force is greater than zero when the interaction energy is increased; on the contrary, the interaction force is less than zero when the interaction energy is decreased [13–15].Figure 3
Relationship of interaction energy curve and interaction force curve.For two semi-infinite parallel plate particles, the total interaction force is shown as(16)
F
T
=
F
R
+
F
A
=
64000
N
A
C
k
T
γ
0
2
exp
⁡
-
κ
h
-
A
6
π
h
3
.From Figure4, we can see that the condition of critical coagulation concentration based on total interaction force is F
m
a
x
=
0:(17)
F
T
=
0
,
d
F
T
d
h
=
0
.Figure 4
Conditions of critical coagulation concentration based on total interaction force.Equation (17) can be expressed as(18)
F
T
=
64000
N
A
C
k
T
γ
0
2
exp
⁡
-
κ
h
-
A
6
π
h
3
=
0
,
d
F
T
d
h
=
-
κ
·
64000
N
A
C
k
T
γ
0
2
exp
⁡
-
κ
h
+
A
2
π
h
4
=
0
.For the first equation of (18) multiplied by -
3
/
h, (18) leads to(19)
F
T
=
-
3
h
·
64000
N
A
C
k
T
γ
0
2
exp
⁡
-
κ
h
+
A
2
π
h
4
=
0
d
F
T
d
h
=
-
κ
·
64000
N
A
C
k
T
γ
0
2
exp
⁡
-
κ
h
+
A
2
π
h
4
=
0
.Solving forh and κ, (19) gives(20)
κ
h
=
3
.Substituting (10) into (16) leads to(21)
F
T
=
64000
N
A
C
k
T
γ
0
2
exp
⁡
-
κ
h
-
2000
Z
2
e
2
N
A
C
ε
k
T
3
/
2
·
A
6
π
κ
h
3
=
0
.Substituting (20) into (21) leads to (22)
C
·
64000
N
A
k
T
γ
0
2
exp
⁡
-
3
=
C
3
/
2
·
2000
Z
2
e
2
N
A
ε
k
T
3
/
2
·
A
6
π
3
3
.Solving forC from (22), the CCC can be obtained(23)
C
CCC
=
328724
ε
3
k
5
T
5
γ
0
4
1000
N
A
e
6
A
2
Z
6
.Substituting those parameters into (23) leads to(24)
C
CCC
=
7.258
×
1
0
-
51
T
5
γ
0
4
A
2
Z
6
.


## 4. Comparison of Two Calculation Methods

Comparing the modified calculation method of CCC with the traditional calculation method, the critical condition of modified calculation method is stricter than traditional calculation method. When maximum value of interaction energy is zero, the relationship of interaction energy curve and interaction force curve is shown in Figure5. We can see that the interaction force is zero when the maximum value of interaction energy is zero. We also now know that the interaction energy is less than zero in the region of A to B, but the interaction force is greater than zero in this region. So there is a repulsive force when two colloidal particles’ distance is in this region, and the colloidal particles cannot coagulate. In conclusion, the traditional calculation method of CCC is not accurate based on maximum value of interaction energy that is zero. And the strict calculation condition of CCC is that maximum value of interaction force is zero.Figure 5
Relationship of interaction energy curve and interaction force curve when maximum value of interaction energy is zero.In addition, due to Brownian motion or stirring effect in the real colloidal solution, colloidal particles are moving, so coagulation can occur even if the energy barrier or force is not strictly less than zero.The calculation results of CCC with two calculation methods are shown in Table1. The coefficients of two results are different, the coefficient of calculation result of CCC based on interaction force is 7.258 × 10−51, and the other one is 3.525 × 10−51. Therefore, the calculation result of CCC based on interaction force is greater than the calculation value based on interaction energy. The CCC value of modified calculation method can ensure particles to coagulate definitely.Table 1
Calculation results of critical coagulation concentration with two calculation methods.
Calculation method
Interaction energy
Interaction force
Calculation condition
V
max
⁡
=
0
F
max
⁡
=
0
CCC, mol/L
3.525
×
1
0
-
51
T
5
γ
0
4
A
2
Z
6
7.258
×
1
0
-
51
T
5
γ
0
4
A
2
Z
6Montmorillonite is main inorganic mineral in coal and oil shale. Since the addition of Ca2+ to montmorillonite suspensions results in coagulation [16], the results of CCC of Ca2+ for montmorillonite with calculation and experimental methods are shown in Table 2. The experimental result of CCC of Ca2+ for montmorillonite is 0.45 mmol/L, and the calculation results of CCC of Ca2+ for montmorillonite based on interaction energy and force are 0.34 and 0.48 mmol/L, respectively. The experimental result of CCC is close to the calculation result based on interaction force, so the CCC value of modified calculation method is more accurate than traditional calculation method.Table 2
CCC of Ca2+ for montmorillonite by different calculation and experimental method.
Calculation or experimental method
CCC, mmol/L
Interaction energy (traditional)
0.34
Interaction force (modified)
0.48
Microscope observation [17]
0.45


## 5. Conclusions

A modified calculation method was proposed to calculate CCC based on interaction force, the critical condition of modified calculation method is the maximum value of interaction force that is zero, and the critical condition of traditional calculation method is the maximum value of interaction energy that is zero. There is a repulsive force between two colloidal particles when interaction energy is less than zero in the specific region, and the colloidal particles cannot coagulate. Comparing the modified calculation method of CCC with the traditional calculation method, the critical condition of modified calculation method is stricter than traditional calculation method. The calculation result of CCC based on interaction force is greater than the calculation value based on interaction energy. The CCC value of modified calculation method can ensure particles to coagulate definitely.



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*Source: 317483-2015-11-02.xml*