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

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1 Mathematical Problems in Engineering Volume 215, rticle ID , 5 pages Research rticle Modified Method to Calculate Critical Coagulation Concentration Based on DLVO Theory Zhijun Zhang, 1,2 Liang Zhao, 1 Yanan Li, 1 andmochu 1 1 School of Chemical and Environmental Engineering, China University of Mining and Technology, Beijing 183, China 2 Key Laboratory of Coal Processing and Efficient Utilization of Ministry of Education, China University of Mining & Technology, Xuzhou 2218, China Correspondence should be addressed to Zhijun Zhang; zzjun1984@126.com Received 21 ugust 215; ccepted 19 October 215 cademic Editor: George S. Dulikravich Copyright 215 Zhijun Zhang et al. This is an open access article distributed under the Creative Commons ttribution 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 of colloidal particles. 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. 1. Introduction most important property of a colloidal system is its stability defined by the tendency of the particles to aggregate. ggregation 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: V T =V +V R, (1) where V T here stands for total interaction energy, V stands forvanderwaalsenergywhichisusuallyattractive,andv R stands for the electrostatic double layer energy. s shown in Figure 1, an energy profile based on the DLVO theory clarifies interparticle interactions. For two semi-infinite parallel plate particles, the Van der Waalsenergyisshownas V = 12πh 2, (2) where is the Hamaker constant of the system and h is the distance of closest approach between the particles. nd the electrostatic double layer energy is shown as V R = 64N CkT exp ( h), (3) where N is vogadro 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 is defined as γ = (exp(zeψ /2kT) 1)/(exp(Zeψ /2kT) + 1), and is the inverse Debye length [4], which is defined by =( 1N e 2 εkt 1/2 Z 2 i C i), (4)

2 2 Mathematical Problems in Engineering 1E 17 3E 18 8E 18 V R 6E 18 4E 18 V T 1E 18 V (J) 4E 18 6E 18 8E 18 V 1E 8 2E 8 h (m) 3E 8 4E 8 5E 8 V (J) E E 18 V T 1E 17 Figure 1: Interparticle interaction energy profile based on DLVO theory. where e is the elementary charge, ε is the permittivity of the dispersion medium (ε r ε ), 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 [1 12] V T =V R +V = 64N CkT γ 2 exp ( h) 12πh 2. (5) sshowninfigure2,theconditionofcriticalcoagulation concentration based on total interaction energy is V max =. Consider Equation (6) can be expressed as V T =, dv (6) T dh =. V T = 64N CkT γ 2 exp ( h) 12πh 2 =, dv T dh = 64N CkT exp ( h) + 6πh 3 =. (7) 3E 18 Figure 2: Conditions of critical coagulation concentration based on total interaction energy. to For the first equation of (7) multiplied by 2/h, (7) leads 64N CkT exp ( h) + 6πh 3 =, V T = 2 h dv T dh = 64N CkT Solving for h and,(8)gives exp ( h) + 6πh 3 =. (8) h = 2. (9) For symmetrical type electrolyte, is defined as =( 2Z2 e 2 1/2 N C ). (1) εkt Substituting (1) into (5) leads to V T = 64 ( εk3 T 3 1/2 N C 2Z 2 e 2 ) exp ( h) 2Z2 e 2 N C 12πεkT 1 (h) 2 =. (11) Substituting (9) into (11) and solving the equation, the CCC can be obtained C CCC = ε3 k 5 T 5 γ 4 1N e 6 2 Z 6. (12) s we have known k = J/K, e = C, N = mol 1,andε=ε ε r = = F/m. Substituting these parameters into (12) leads to C CCC = T 5 γ 4 2 Z 6. (13)

3 Mathematical Problems in Engineering 3 V (J) Force (N) 3E 18 1E 18 E + 1E 18 3E 18 3E 9 2E 9 1E 9 E + 1E 9 2E 9 Energy curve Force curve Force (N) 1E 9 5E 1 E F T 5E 1 1E 9 Figure 4: Conditions of critical coagulation concentration based on total interaction force. Equation(17)canbeexpressedas 3E 9 Figure 3: Relationship of interaction energy curve and interaction force curve. 3. Modified Calculation Method of CCC by Interaction Force The relationship of interaction energy and interaction force of two colloidal particles is or D V= F (h) d (h) (14) + F= dv dh. (15) The relationship of interaction energy curve and interaction force curve is shown in Figure 3. 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]. For two semi-infinite parallel plate particles, the total interaction force is shown as F T =F R +F = 64N CkT exp ( h) 6πh 3. (16) From Figure 4, we can see that the condition of critical coagulation concentration based on total interaction force is F max =: F T =, df (17) T dh =. to F T = 64N CkT exp ( h) 6πh 3 =, df T dh = 64N CkT exp ( h) + 2πh 4 =. (18) For the first equation of (18) multiplied by 3/h,(18)leads F T = 3 h 64N CkT exp ( h) + 2πh 4 = df T dh = 64N CkT exp ( h) + 2πh 4 =. Solving for h and,(19)gives Substituting (1) into (16) leads to F T = 64N CkT exp ( h) (19) h = 3. (2) ( 2Z2 e 2 3/2 N C ) εkt 6π (h) 3 =. Substituting (2) into (21) leads to C 64N kt exp ( 3) =C 3/2 ( 2Z2 e 2 3/2 N ) εkt 6π (3) 3. Solving for C from(22),theccccanbeobtained (21) (22) C CCC = ε3 k 5 T 5 γ 4 1N e 6 2 Z 6. (23) Substituting those parameters into (23) leads to C CCC = T 5 γ 4 2 Z 6. (24)

4 4 Mathematical Problems in Engineering Force (N) V (J) 3E 18 1E 18 Energy curve E E 18 3E 18 3E 9 2E 9 1E 9 Force curve Repulsive force E E 9 ttractive force 2E 9 3E 9 Figure 5: Relationship of interaction energy curve and interaction force curve when maximum value of interaction energy is zero. 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 Figure 5. We can see that the interaction force is zero when the maximum value of interaction energy is zero. We also now know that the interactionenergyislessthanzerointheregionoftob, 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. nd the strict calculation condition of CCC is that maximum value of interaction force 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 Table 1. The coefficients of two results are different, the coefficient of calculation result of CCC based on interaction force is , and the other one is 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 methodcanensureparticlestocoagulatedefinitely. Montmorillonite is main inorganic mineral in coal and oil shale. Since the addition of Ca 2+ to montmorillonite B Table 1: Calculation results of critical coagulation concentration with two calculation methods. Calculation method Interaction energy Interaction force Calculation condition V max = F max = CCC, mol/l T5 γ 4 2 Z T5 γ 4 2 Z 6 Table 2: CCC of Ca 2+ for montmorillonite by different calculation and experimental method. Calculation or experimental method CCC, mmol/l Interaction energy (traditional).34 Interaction force (modified).48 Microscope observation [17].45 suspensions results in coagulation [16], the results of CCC of Ca 2+ for montmorillonite with calculation and experimental methods are shown in Table 2. The experimental result of CCC of Ca 2+ for montmorillonite is.45 mmol/l, and the calculation results of CCC of Ca 2+ for montmorillonite based on interaction energy and force are.34 and.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. 5. Conclusions 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. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. cknowledgments The authors would like to acknowledge the financial support from National Key Basic Research Program of China (Grant no. 214CB74431) and Beijing Natural Science Foundation (Grant no ).

5 Mathematical Problems in Engineering 5 References [1] J. N. Israelachvili, Intermolecular and Surface Forces, Elsevier, London, UK, 211. [2]L.S.Dorobantu,S.Bhattacharjee,J.M.Foght,andM.R. Gray, nalysis of force interactions between FM tips and hydrophobicbacteriausingdlvotheory, Langmuir, vol.25, no. 12, pp , 29. [3] J. J. Spitzer, consistent alternative to the DLVO theory, dhesion,vol.79,no.1,pp ,23. [4] G. Livadiotis and D. J. Mccomas, Electrostatic shielding in plasmas and the physical meaning of the Debye length, Journal of Plasma Physics,vol.8,no.3,pp ,214. [5] J.-P. Hsu and B.-T. Liu, Critical coagulation concentration of a colloidal suspension at high particle concentrations, The JournalofPhysicalChemistryB,vol.12,no.2,pp ,1998. [6] J.-P. Hsu and B.-T. Liu, Effect of particle size on critical coagulation concentration, Colloid and Interface Science, vol. 198, no. 1, pp , [7]L.Liu,L.Moreno,andI.Neretnieks, novelapproachto determine the critical coagulation concentration of a colloidal dispersion with plate-like particles, Langmuir,vol.25,no.2,pp , 29. [8] S. García-García, S. Wold, and M. Jonsson, Kinetic determination of critical coagulation concentrations for sodium- and calcium-montmorillonite colloids in NaCl and CaCl 2 aqueous solutions, Colloid & Interface Science,vol.315,no.2, pp ,27. [9] J.-D. Hu, Y. Zevi, X.-M. Kou, J. Xiao, X.-J. Wang, and Y. Jin, Effect of dissolved organic matter on the stability of magnetite nanoparticles under different ph and ionic strength conditions, Science of the Total Environment, vol. 48, no. 16, pp , 21. [1]. Saejiew, O. Grunberger, S. runin, F. Favre, D. Tessier, and P. Boivin, Critical coagulation concentration of paddy soil claysinsodium-ferrousironelectrolyte, Soil Science Society of merica Journal, vol. 68, no. 3, pp , 24. [11] M. Jia, H. Li, H. Zhu, R. Tian, and X. Gao, n approach for the critical coagulation concentration estimation of polydisperse colloidal suspensions of soil and humus, JournalofSoils& Sediments,vol.13,no.2,pp ,213. [12] S.. Semyonov, study of electrolytic coagulation of gold sols by the method of localized surface plasmon resonance spectroscopy, Colloid Journal, vol. 75, no. 4, pp , 213. [13] L.Meagher,G.V.Franks,M.L.Gee,andP.J.Scales, Interaction forces between α-alumina fibres in aqueous electrolyte measured with an atomic force microscope, Colloids and Surfaces : Physicochemical and Engineering spects,vol.146,no.1 3,pp , [14] T. Isobe, Y. Nakano, Y. Kameshima, K. Okada, and. Nakajima, FM interaction study of poly(ammonium acrylate) adsorbed on a Si surface in aqueous KCl solution, Chemistry Letters,vol. 39, no. 12, pp , 21. [15] J. Fréchette and T. K. Vanderlick, Double layer forces over large potential ranges as measured in an electrochemical surface forces apparatus, Langmuir,vol.17,no.24,pp ,21. [16] Z. Zhang, J. Liu, L. Feng, Y. Wang, and M. Zhang, Calculation of critical hardness of coal slime water system based on DLVO theory, China University of Mining & Technology,vol. 43,no.1,pp ,214. [17] P.Chheda,D.Grasso,andC.J.vanOss, Impactofozoneon stability of montmorillonite suspensions, Colloid and Interface Science,vol.153,no.1,pp ,1992.

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Module 8: "Stability of Colloids" Lecture 37: "" The Lecture Contains: DLVO Theory. Effect of Concentration. Objectives_template

Module 8: Stability of Colloids Lecture 37: The Lecture Contains: DLVO Theory. Effect of Concentration. Objectives_template The Lecture Contains: DLVO Theory Effect of Concentration file:///e /courses/colloid_interface_science/lecture37/37_1.htm[6/16/2012 1:02:12 PM] Studying the stability of colloids is an important topic