research article a modified method to calculate critical ... · : interparticle interaction energy...

Research ArticleA Modified Method to Calculate Critical CoagulationConcentration Based on DLVO Theory

Zhijun Zhang,1,2 Liang Zhao,1 Yanan Li,1 and Mo Chu1

1School of Chemical and Environmental Engineering, China University of Mining and Technology, Beijing 100083, China2Key Laboratory of Coal Processing and Efficient Utilization of Ministry of Education, China University of Mining & Technology,Xuzhou 221008, China

Correspondence should be addressed to Zhijun Zhang; [email protected]

Received 21 August 2015; Accepted 19 October 2015

Academic Editor: George S. Dulikravich

Copyright © 2015 Zhijun Zhang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

1. Introduction

A most important property of a colloidal system is itsstability defined by the tendency of the particles to aggregate.Aggregation of colloidal particles is a result of colloidalforces and was described independently by Derjaguin andLandau and Verwey and Overbeek in their DLVO theory.The DLVO theory involves two major terms: the Van derWaals force/energy based on interparticle distance and theelectrostatic double layer force/energy aroused by overlap-ping of electrical double layer [1–3]. The total interactionforce/energy in theDLVO theory is the summation of the twoforces/energies:

𝑉𝑇= 𝑉𝐴+ 𝑉𝑅, (1)

where 𝑉𝑇here stands for total interaction energy, 𝑉

𝐴stands

for Van der Waals energy which is usually attractive, and 𝑉𝑅

stands for the electrostatic double layer energy.As shown in Figure 1, an energy profile based on the

DLVO theory clarifies interparticle interactions.

For two semi-infinite parallel plate particles, the Van derWaals energy is shown as

𝑉𝐴= −𝐴

12𝜋ℎ2, (2)

where 𝐴 is the Hamaker constant of the system and ℎ is thedistance of closest approach between the particles.

And the electrostatic double layer energy is shown as

𝑉𝑅=64000𝑁

𝐴𝐶𝑘𝑇

𝜅𝛾2

0exp (−𝜅ℎ) , (3)

where𝑁𝐴is Avogadro’s number, 𝐶 is the bulk concentration

of electrolyte, 𝑘 and 𝑇 are the Boltzmann constant and theabsolute temperature, respectively, ℎ is the shortest sepa-ration distance, 𝛾

0is defined as 𝛾

0= (exp(𝑍𝑒𝜓

0/2𝑘𝑇) −

1)/(exp(𝑍𝑒𝜓0/2𝑘𝑇) + 1), and 𝜅 is the inverse Debye length

[4], which is defined by

𝜅 = (1000𝑁

𝐴𝑒2

𝜀𝑘𝑇∑𝑍2

𝑖𝐶𝑖)

1/2

, (4)

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 317483, 5 pageshttp://dx.doi.org/10.1155/2015/317483

2 Mathematical Problems in Engineering

1E − 17

8E − 18

6E − 18

4E − 18

2E − 18

0

−2E − 18

−4E − 18

−6E − 18

−8E − 18

−1E − 17

0

1E−08

2E−08

3E−08

4E−08

5E−08

VR

VT

VA

V(J

)

h (m)

Figure 1: Interparticle interaction energy profile based on DLVOtheory.

where 𝑒 is the elementary charge, 𝜀 is the permittivity of thedispersion medium (𝜀

𝑟𝜀0), and 𝑍

𝑖and 𝐶

𝑖are valency and

concentration of 𝑖th ionic species, respectively.The electrostatic double layer force/energy depended

on the bulk concentration of electrolyte, and the criticalcoagulation concentration (CCC) is one of the most signif-icant characteristics of a colloidal dispersion. It is definedas the minimum concentration of counterions to inducecoagulation of colloidal particles, and it is often applied toassess the status of a colloidal dispersion [5–9].

In the traditional case, it is based on the interactionenergy to calculate CCC. In this paper, a modified calculationmethod is proposed to calculate CCC based on interactionforce, the critical condition of this calculation method isstricter, and the calculation results are more accurate.

2. Traditional Calculation Method of CCC byInteraction Energy

The traditional calculation method of CCC is based on theinteraction energy of two colloidal particles.

For two semi-infinite parallel plate particles, the totalinteraction energy is shown as [10–12]

𝑉𝑇= 𝑉𝑅+ 𝑉𝐴=64000𝑁

𝐴𝐶𝑘𝑇

𝜅𝛾2

0exp (−𝜅ℎ) − 𝐴

12𝜋ℎ2. (5)

As shown in Figure 2, the condition of critical coagulationconcentration based on total interaction energy is 𝑉max = 0.Consider

𝑉𝑇= 0,

𝑑𝑉𝑇

𝑑ℎ= 0.

(6)

Equation (6) can be expressed as

𝑉𝑇=64000𝑁

𝐴𝐶𝑘𝑇

𝜅𝛾2


12𝜋ℎ2= 0,

𝑑𝑉𝑇

𝑑ℎ= −𝜅64000𝑁

𝐴𝐶𝑘𝑇

𝜅𝛾2

0exp (−𝜅ℎ) + 𝐴

6𝜋ℎ3= 0.

(7)

0 5 10 15 20V(J

)

h (nm)

3E − 18

2E − 18

1E − 18

0E + 00

−1E − 18

−2E − 18

−3E − 18

VT

Figure 2: Conditions of critical coagulation concentration based ontotal interaction energy.

For the first equation of (7) multiplied by −2/ℎ, (7) leadsto

𝑉𝑇= −2

ℎ

64000𝑁𝐴𝐶𝑘𝑇

𝜅𝛾2


6𝜋ℎ3= 0,

𝑑𝑉𝑇

𝑑ℎ= −𝜅64000𝑁

𝐴𝐶𝑘𝑇

𝜅𝛾2


6𝜋ℎ3= 0.

(8)

Solving for ℎ and 𝜅, (8) gives

𝜅ℎ = 2. (9)

For symmetrical type electrolyte, 𝜅 is defined as

𝜅 = (2000𝑍

2𝑒2𝑁𝐴𝐶

𝜀𝑘𝑇)

1/2

. (10)

Substituting (10) into (5) leads to

𝑉𝑇= 64000 ⋅ (

𝜀𝑘3𝑇3𝑁𝐴𝐶

2000𝑍2𝑒2)

1/2

𝛾2

0exp (−𝜅ℎ)

−2000𝑍

2𝑒2𝑁𝐴𝐶𝐴

12𝜋𝜀𝑘𝑇⋅1

(𝜅ℎ)2= 0.

(11)

Substituting (9) into (11) and solving the equation, theCCC can be obtained

𝐶CCC =159648𝜀

3𝑘5𝑇5𝛾4

0

1000𝑁𝐴𝑒6𝐴2𝑍6

. (12)

As we have known 𝑘 = 1.3805 × 10−23 J/K, 𝑒 = 1.602 ×10−19 C, 𝑁

𝐴= 6.023 × 10

23mol−1, and 𝜀 = 𝜀0𝜀𝑟= 8.854 ×

10−12×78.5 = 6.95×10

−10 F/m. Substituting these parametersinto (12) leads to

𝐶CCC = 3.525 × 10−51𝑇5𝛾4

0

𝐴2𝑍6. (13)

Mathematical Problems in Engineering 3

0

0

5 10 15 20

5 10 15 20

Forc

e (N

)

Force curve

Energy curve

V(J

)

h (nm)

h (nm)

3E − 09

2E − 09

1E − 09

0E + 00

−1E − 09

−2E − 09

−3E − 09

−1E − 18

−2E − 18

−3E − 18

3E − 18

2E − 18

1E − 18

0E + 00

Figure 3: Relationship of interaction energy curve and interactionforce curve.

3. Modified Calculation Method of CCC byInteraction Force

The relationship of interaction energy and interaction forceof two colloidal particles is

𝑉 = −∫

𝐷

+∞

𝐹 (ℎ) 𝑑 (ℎ) (14)

or

𝐹 = −𝑑𝑉

𝑑ℎ. (15)

The relationship of interaction energy curve and interac-tion force curve is shown in Figure 3. In the process of twocolloidal particles approaching, when the interaction energyreaches the maximum value, the interaction force is zero, andthe interaction force is greater than zero when the interactionenergy is increased; on the contrary, the interaction force isless than zero when the interaction energy is decreased [13–15].

For two semi-infinite parallel plate particles, the totalinteraction force is shown as

𝐹𝑇= 𝐹𝑅+ 𝐹𝐴= 64000𝑁

𝐴𝐶𝑘𝑇𝛾

2


6𝜋ℎ3. (16)

From Figure 4, we can see that the condition of criticalcoagulation concentration based on total interaction force is𝐹max = 0:

𝐹𝑇= 0,

𝑑𝐹𝑇

𝑑ℎ= 0.

(17)

0 5 10 15 20

FT

Forc

e (N

)

h (nm)

1E − 09

5E − 10

0E + 00

−5E − 10

−1E − 09

Figure 4: Conditions of critical coagulation concentration based ontotal interaction force.

Equation (17) can be expressed as

𝐹𝑇= 64000𝑁


2


6𝜋ℎ3= 0,

𝑑𝐹𝑇

𝑑ℎ= −𝜅 ⋅ 64000𝑁


2


2𝜋ℎ4= 0.

(18)

For the first equation of (18)multiplied by −3/ℎ, (18) leadsto

𝐹𝑇= −3

ℎ⋅ 64000𝑁


2


2𝜋ℎ4= 0

𝑑𝐹𝑇

𝑑ℎ= −𝜅 ⋅ 64000𝑁


2


2𝜋ℎ4= 0.

(19)

Solving for ℎ and 𝜅, (19) gives

𝜅ℎ = 3. (20)


𝐹𝑇= 64000𝑁


2

0exp (−𝜅ℎ)

− (2000𝑍

2𝑒2𝑁𝐴𝐶

𝜀𝑘𝑇)

3/2

⋅𝐴

6𝜋 (𝜅ℎ)3= 0.

(21)


𝐶 ⋅ 64000𝑁𝐴𝑘𝑇𝛾2

0exp (−3)

= 𝐶3/2⋅ (2000𝑍

2𝑒2𝑁𝐴

𝜀𝑘𝑇)

3/2

⋅𝐴

6𝜋 (3)3.

(22)

Solving for 𝐶 from (22), the CCC can be obtained

𝐶CCC =328724𝜀

3𝑘5𝑇5𝛾4

0

1000𝑁𝐴𝑒6𝐴2𝑍6

. (23)

Substituting those parameters into (23) leads to

𝐶CCC = 7.258 × 10−51𝑇5𝛾4

0

𝐴2𝑍6. (24)

4 Mathematical Problems in Engineering

0 5 10 15 20

0 5 10 15 20

Repulsive force

Attractive force

Energy curve

Force curve

A B

Forc

e (N

) V

(J)

h (nm)

h (nm)

3E − 09

2E − 09

1E − 09

0E + 00

−1E − 09

−2E − 09

−3E − 09

3E − 18

2E − 18

1E − 18

0E + 00

−1E − 18

−3E − 18

−2E − 18

Figure 5: Relationship of interaction energy curve and interactionforce curve when maximum value of interaction energy is zero.

4. Comparison of Two Calculation Methods

Comparing the modified calculation method of CCC withthe traditional calculation method, the critical conditionof modified calculation method is stricter than traditionalcalculation method. When maximum value of interactionenergy is zero, the relationship of interaction energy curveand interaction force curve is shown in Figure 5. We can seethat the interaction force is zero when the maximum valueof interaction energy is zero. We also now know that theinteraction energy is less than zero in the region of A to B,but the interaction force is greater than zero in this region. Sothere is a repulsive forcewhen two colloidal particles’ distanceis in this region, and the colloidal particles cannot coagulate.In conclusion, the traditional calculation method of CCC isnot accurate based on maximum value of interaction energythat is zero. And the strict calculation condition of CCC isthat maximum value of interaction force is zero.

In addition, due to Brownian motion or stirring effect inthe real colloidal solution, colloidal particles are moving, socoagulation can occur even if the energy barrier or force isnot strictly less than zero.

The calculation results of CCC with two calculationmethods are shown in Table 1. The coefficients of two resultsare different, the coefficient of calculation result of CCCbasedon interaction force is 7.258 × 10−51, and the other one is 3.525× 10−51. Therefore, the calculation result of CCC based oninteraction force is greater than the calculation value basedon interaction energy.TheCCC value ofmodified calculationmethod can ensure particles to coagulate definitely.

Montmorillonite is main inorganic mineral in coal andoil shale. Since the addition of Ca2+ to montmorillonite

Table 1: Calculation results of critical coagulation concentrationwith two calculation methods.

Calculation method Interaction energy Interaction forceCalculation condition 𝑉max = 0 𝐹max = 0

CCC, mol/L 3.525 × 10−51𝑇5𝛾4

0

𝐴2𝑍67.258 × 10

−51𝑇5𝛾4

0

𝐴2𝑍6

Table 2: CCC of Ca2+ for montmorillonite by different calculationand experimental method.

Calculation or experimental method CCC, mmol/LInteraction energy (traditional) 0.34Interaction force (modified) 0.48Microscope observation [17] 0.45

suspensions results in coagulation [16], the results of CCC ofCa2+ for montmorillonite with calculation and experimentalmethods are shown in Table 2. The experimental result ofCCC of Ca2+ for montmorillonite is 0.45mmol/L, and thecalculation results of CCC of Ca2+ for montmorillonite basedon interaction energy and force are 0.34 and 0.48mmol/L,respectively. The experimental result of CCC is close to thecalculation result based on interaction force, so the CCCvalue of modified calculation method is more accurate thantraditional calculation method.

5. Conclusions

A modified calculation method was proposed to calculateCCC based on interaction force, the critical condition ofmodified calculation method is the maximum value ofinteraction force that is zero, and the critical condition oftraditional calculation method is the maximum value ofinteraction energy that is zero. There is a repulsive forcebetween two colloidal particles when interaction energy isless than zero in the specific region, and the colloidal parti-cles cannot coagulate. Comparing the modified calculationmethod of CCC with the traditional calculation method,the critical condition of modified calculation method isstricter than traditional calculation method. The calculationresult of CCC based on interaction force is greater thanthe calculation value based on interaction energy. The CCCvalue of modified calculation method can ensure particles tocoagulate definitely.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The authors would like to acknowledge the financial supportfrom National Key Basic Research Program of China (Grantno. 2014CB744301) and Beijing Natural Science Foundation(Grant no. 3154037).

Mathematical Problems in Engineering 5

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