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Suspension Flocculation by Polyelectrolytes: Experimental Verification of a Developed Mathematical Model
JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.
191, 273–276 (1997)
CS974930
Suspension Flocculation by Polyelectrolytes: Experimental Verification of a Developed Mathematical Model Ad. A. Berlin,* ,1 I. M. Solomentseva,† and V. N. Kislenko* *Lviv Polytechnic State University, 12 Bandera Street, Lviv 290646, Ukraine; and †Institute of Colloid and Water Chemistry of NAS, 42 Vernadsky Avenue, Kiev 252680, Ukraine Received July 8, 1996; accepted April 18, 1997
Using experimental data of AgI sol and poly( 2-methyl-5-vinylpyridine ) chloride, we show that there is good agreement between the developed mathematical model of suspension flocculation by polyelectrolytes and the experimental data. The rate constants of bridging and nonbridging aggregation of particles and of floc breakup, as well as their ratios, have been determined. It has been shown that the role of the nonbridging mechanism, as compared with that of the bridging one, diminishes when molecular mass of polyelectrolyte increases from 1.4 1 10 5 to 9.3 1 10 5 . q 1997
of high molecular weight flocculants (4). In contrast, the neutralizing mechanism of flocculation prevails when a strongly charged low-molecular-weight cationic polyelectrolyte, Percol 1697, is being used (4). Elucidation of the roles of both of these mechanisms, i.e., bridging and nonbridging, and also of the role of molecular mass of polyelectrolyte in the flocculation process is the objective of this paper. THEORY AND DISCUSSION
Academic Press
Key Words: AgI sol; poly(methylvinylpyridine); flocculation; mathematical simulation.
INTRODUCTION
Usually industrial and domestic waste, as well as industrial suspensions, contains particles having a negative charge. Therefore cationic polymers are widely used for their flocculation. Therefore, some researchers assume that in many cases a simple charge neutralizing mechanism can explain most of the observed behavior of such colloid systems (1). Really, many authors note that optimum flocculation proceeds when the particle charge has been reduced to around zero (1–3). In such cases some authors (1) note that the molecular mass of the cationic flocculant should not be an important factor, and flocculation efficiency should depend on the charge density of the polyelectrolyte. On the other hand, it has been noted (2, 3) that the bridging mechanism is the most probable cause of flocculation with polyelectrolytes. The effects of the nonbridging mechanism, i.e., the decrease of charge and of a potential of particles, plays only a minor role in a stability loss (3). Therefore, at the higher molecular mass of polyelectrolyte, the lower polymer concentration is necessary to reach the maximum drop in the total particle concentration (2). Investigation on kaolin suspensions flocculation by cationic polyelectrolytes showed that bridge formation between particles is the predominant mechanism of the process at use 1
To whom correspondence should be addressed.
A mathematical model of suspension flocculation by polyelectrolytes (5) takes account of bridging and nonbridging aggregation of suspension particles, as well as breakup of formed flocs under the action of hydrodynamic forces: 0 dN/dt Å £a / £n 0 £b .
Here N is the total particle concentration in a suspension, £a and £n are the rates of bridging and nonbridging aggregation of particles, respectively, £b is the rate of floc break-up, and t is time. Therefore, £ a Å Ka P c N 2 ,
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[2] 2
2
£n Å Kn (1 0 Pc )exp( 0 dj )N ,
[3]
£b Å lKb N[1 0 exp(1 0 N0 /N)],
[4]
where Ka , Kn , and Kb are the rate constants of bridging and nonbridging aggregations of particles, also of floc breakup, 1 is the number of particles forming during floc breakup, Pc Å 2U(1 0 U ) is a probability of successful particle collisions leading to aggregate formation by bridging mechanism, U Å c/cs is the fraction of the particle surface covered by polyelectrolyte, c is the polymer concentration in a suspension, cs is the polymer concentration corresponding to U Å 1, d is a constant, j is the value of j-potential of particles, and N0 Å N(t Å 0) is the initial number of particles per unit volume of a suspension. Experiments show that the rate of suspension flocculation is decreasing with time, and finally the kinetic curves pla-
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TABLE 1 Parameters of PMVP Macromolecules in MEK Solution Designation
[h] (dl/g)
M 1 1005
»R2…1/2 (nm)
PMVP-1 PMVP-4 PMVP-9
0.37 0.74 1.15
1.4 4.4 9.3
29 53 72
F Å Pc exp(dj2 )/(1 0 Pc ). FIG. 1. Kinetic curves of flocculation of AgI sol by PMVP-9 at polymer concentration 0.25 (1) and 2.5 mg/liter (2). Experimental data are marked with points; curves are calculated according to Eq. [15].
teau ( see Fig. 1 ) . In the equilibrium state at the plateau region d N / dt Å 0. Then Eq. [ 5 ] can be obtained from Eqs. [1] – [ 4 ] :
[10]
One can find the value for Ka /(lKb ) from a plot of the linear relationship between G and F according to Eq. [8]. Then the ratio of the rate constant Kn /Ka can be calculated to estimate the role of nonbridging and bridging aggregations of particles. Integration of Eq. [1] at the initial condition N(t Å 0) Å N0 , taking Eqs. [2] – [4] into account leads to the linear expression
N/[1 0 exp(1 0 N0 /N)] Y Å 0Kat.
Å (lKb /Ka )/[Pc / (Kn /Ka )(1 0 Pc )exp( 0 dj2 )]. [5]
[11]
Here
Then j Å j0 / c/(A / Bc),
[6]
YÅ
*
N
dN/{F1N 2 0 (lKb /Ka )N[1 0 exp(1 0 N0 /N)]},
N0
where j0 Å j at c Å 0, and A and B are the constants (5). As seen from Eqs. [3] and [5], the larger the value of d the more sensitive the aggregation process is to the change of j-potential. Also, the larger the ratio of Kn /Ka the higher the role of nonbridging aggregation of particles is at suspension flocculation by polyelectrolyte. The determination of the values of d and of lKb /Kn can be carried out as follows: at c § cs , Pc Å 0. In this case, Eq. [5] can be transformed into linear relationship: 2
ln{N/[1 0 exp(1 0 N0 /N)]} Å ln(lKb /Kn ) / dj .
[7]
[12] 2
F1 Å Pc / (Kn /Ka )(1 0 Pc )exp( 0 dj ).
[13]
That enables us to determine the value of the rate constant Ka from the tangent of the slope angle of a straight line when plotted Y vs t according to Eq. [11]. EXPERIMENTAL
Monodisperse AgI sol was prepared by gradual addition of 0.1 M AgNO3 into 0.1 M KI with dilution of the sol
A plot of the linear relationship [7] between ln{N/[1 0 exp(1 0 N0 /N)]} and j 2 gives the possibility of finding the values for d and lKb /Kn from the slope of a straight line and the intercept on the ordinate axis. Then it is straightforward to determine the value of Kn / Ka from Eq. [ 5 ] . A little manipulation yields a linear expression, G Å [Ka /(lKb )]F,
[8]
where G Å [1 0 exp(1 0 N0 /N)]/[N(1 0 Pc )exp( 0 dj2 )] 0 Kn /(lKb )
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FIG. 2. Relationship between the total particle concentration in AgI sol and concentration of PMVP-1 (1), PMVP-4 (2), and PMVP-9 (3). Experimental data are marked with points; curves are calculated according to Eq. [5].
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TABLE 2 Parameters of Flocculation of AgI Sol by Poly(methylvinylpyridine) Quantity
PMVP-1
PMVP-4
PMVP-9
cs (mg/liter) Uz d 1 104 (mV02) 1013 1 Kn/(1 Kb) (liters) 1012 1 Ka/(1 Kb) (liters) Kn/Ka
8.2 0.29 5.8 { 1.6 5.5 { 2.0 5.1 { 1.3 0.11 { 0.07
4.2 0.48 4.1 { 1.8 4.1 { 0.8 3.7 { 1.4 0.11 { 0.06
2.2 0.73 0.042 { 0.019 3.7 { 1.1 4.5 { 1.3 0.08 { 0.05
formed one-fifth of previous value. Salt excess was removed by dialysis for 1 week. The sol obtained had pH 5.9 and conductivity of 3 1 10 05 V 01 cm01 . The average radius of particles was 80 nm (by electron microscopy). Poly(2-methyl-5-vinylpyridine) chloride fractionated (PMVP) was used as polyelectrolyte (Table 1). Molecular mass of PMVP was measured by viscometry in methyl ethyl ketone (MEK) (6). Root mean square radius of particles was calculated according to the Flory–Fox equation (7). The flocculation technique was carried out as follows. PMVP solution of a given concentration was added into the constant volume of AgI sol. The system was stirred and then after a certain time interval, diluted to 151 th to 201 th of the previous value with KCl solution of the same conductivity. The count of the particle number per unit volume was carried out in a cell of a flow ultramicroscope described in Ref. (8). RESULTS AND DISCUSSION
The relationship between the number of particles per unit volume of AgI sol and PMVP concentration is shown in Fig. 2. The values of cs found according to Ref. (5) from inflection points of curves N(c) decrease with increasing molecular mass of polyelectrolyte (Table 2). Therefore the higher molecular mass of PMVP the lower polymer concentration is necessary to reach U Å 1. Let cz denote the PMVP concentration where j-potential
FIG. 3. Effect of j-potential of particles in AgI sol on total particle concentration according to Eq. [7] at flocculation by PMVP-1 (1) and PMVP-4 (2).
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FIG. 4. Plot of G vs F according to Eq. [8] at flocculation of AgI sol by PMVP-4 (1) and PMVP-9 (2).
of particles in AgI sol goes to zero. The value of cz can be calculated from formula [6] for j Å 0. Then one can determine Uz Å cz /cs . As seen in Table 2, the value of Uz increases with rising molecular mass of polyelectrolyte. Consequently, j-potential vanishes at the higher particle surface coverage with polymer for high-molecular-weight PMVP. As seen in Fig. 2, the values of N at c Å cs , i.e., U Å 1, are lower than initial total particle concentration, N0 . This illustrates the fact that both bridging and nonbridging aggregations of particles occur at flocculation of AgI sol under the action of PMVP (5). Therefore a mathematical model [1] – [4] and equations following from it can be used for description of this flocculation process. Figure 3 illustrates relationship between total particle concentration in sol and j-potential of particles. The experimental data lie on straight lines when plotted according to Eq. [7]. Values of d and of the rate constant ratio Kn /(lKb ) calculated from the slope of the straight lines and from the intercepts on the ordinate axis are represented in Table 2. The value of d increases when molecular mass of PMVP diminishes; i.e., the sensitivity of the aggregation process to the change of j-potential rises. The increasing value of Kn / (lKb ) points to the fact that the contribution of nonbridging aggregation into flocculation process increases. Consequently, the scale of nonbridging aggregation of particles
FIG. 5. Change of Y with time at flocculation of AgI sol by PMVP-9.
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BERLIN, SOLOMENTSEVA, AND KISLENKO
in AgI sol rises when molecular mass of polyelectrolytes decreases. Then the opportunity arises to test the linear relationship [8]. As seen in Fig. 4, experimental data for the AgI sol– PMVP system are situated on straight lines when plotted according to Eq. [8]. The slope of the straight line gives the value of Ka /(lKb ) (Table 2). Then one can determine the value of the ratio Kn /Ka taking account of the Ka /(lKb ) value found earlier (see Table 2). These values and the size of d show that the role of nonbridging aggregation as compared with bridging aggregation of particles in AgI sol is less for high-molecular-weight polyelectrolyte (PMVP-9). The values found for the ratios of the rate constant and for d (Table 2) allow calculation relationship between the total particle concentration, N, and PMVP concentration in AgI sol using Eq. [5]. Computer calculations were executed by means of ZEROIN algorithm (9) using expansion of the left side of Eq. [5] as an initial approximation. The curves in Fig. 2 illustrate the satisfactory agreement of calculated curves with experimental data and about adequacy of the suggested mathematical model [5] to the experiment on flocculation of AgI sol by PMVP. It is worth noting that the minimums of the plot of N vs c/cs do not coincide with the values of Uz for PMVP-1 and PMVP-9 (Table 2). The kinetic data presented in Fig. 1 give an opportunity to determine the values of the individual rate constants using Eqs. [11] – [13]. Numerical integration of Eq. [12] was performed by using an adaptive recursive Simpson’s rule. Figure 5 presents a plot of the linear relationship between N and time according to Eq. [11]. The slope of the straight line yields the value Ka Å (2.4 { 0.3) 1 10 013 liters/min. Using the values of the ratios of the rate constants found earlier (see Table 2), one can determine the values of Kn Å (2.0 { 1.1) 1 10 014 liters/min and of lKb Å (5.4 { 2.4) 1 10 02 min 01 for flocculation of AgI sol by PMVP-9. The values found for the rate constants enable us to calculate the change of the number of primary particles in a floc N0 /N in time. For this purpose, Eqs. [1] – [3] can be rewritten as d(N0 /N)/dt Å Ka {N0 F1 0 (lKb /Ka )(N0 /N) 1 [1 0 exp(1 0 N0 /N)]}.
[14]
Integration of Eq. [14] at the initial condition N0 /N(t Å 0) Å 1 gives t Å Y1 /Ka ,
[15]
where Y1 Å
*
N0/N
d(N0 /N)/ {N0 F1 0 (lKb /Ka )(N0 /N)
1
1 [1 0 exp(1 0 N0 /N)]}.
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FIG. 6. Relationship between £a , liter 01 min 01 (1), £n , liter 01 min 01 (2), and £a / £n (3) and PMVP-9 concentration in AgI sol.
Numerical integration of Eq. [16] was performed by using an adaptive recursive Simpson’s rule. Equation [15] defines the relationship between the number of primary particles in a floc, N0 /N, and time. Curves in Fig. 1 demonstrate a satisfactory agreement between the kinetic curves calculated in such a way and experimental data on flocculation of AgI sol by PMVP-9. The relationships of the rates of bridging and nonbridging aggregation of AgI particles from PMVP-9 concentration can be calculated by Eqs. [2] and [3]. Figure 6 demonstrates these functions and the ratio £a / £n . It is interesting that the curve for £a has two maximums not coinciding with the minimum of the curve N vs c/cs (see Fig. 2). At the same time, the curve for £n has the minimum and the curve of £a / £n has the maximum coinciding with the minimum of the curve N(c/cs ) as discussed in Ref. (5). Consequently, the agreement between the experimental data on AgI sol aggregation by PMVP and the mathematical model of suspension flocculation by polyelectrolytes supports correctness of the suggested model.
REFERENCES 1. Gregory, J., Flocculation. Prog. Filtr. Sep. 4, 55 (1986). 2. Baran, A. A., Tusupbayev, I. K., Solomentseva, I. M., Deryagin, R. M., and Musabekov, K. B., Colloid J. (USSR) 42, 11 (1980). 3. Solomentseva, I. M., Tusupbayev, I. K., Baran, A. A., and Musabekov, K. B., Ukr. Chem. J. (USSR) 46, 928 (1980). 4. Baran, S., and Gregory, J., Coll. J. (Russia) 58, 13 (1996). 5. Berlin, Ad. A., Kislenko, V. N., and Solomentseva, I. M., Colloid J. (Russia) (in press). 6. Jarlugio, C., Crescentini, L., Mula, A., and Jechele, J. B., Macromolec. Chem. 97, 97 (1966). 7. Flory, P. J., ‘‘Mechanics of Chain Molecules.’’ Interscience, New York–London–Sydney–Toronto, 1969. 8. Kudryavtseva, N. M., and Deryagin, B. V., Colloid J. (USSR) 25, 739 (1963). 9. Forsythe, G. E., Malcolm, M. A., and Moler, C. B., ‘‘Computer Methods for Mathematical Computations,’’ C7632. Prentice-Hall, Englwood Cliffs, NJ, 1977.
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191, 273–276 (1997)
CS974930
Suspension Flocculation by Polyelectrolytes: Experimental Verification of a Developed Mathematical Model Ad. A. Berlin,* ,1 I. M. Solomentseva,† and V. N. Kislenko* *Lviv Polytechnic State University, 12 Bandera Street, Lviv 290646, Ukraine; and †Institute of Colloid and Water Chemistry of NAS, 42 Vernadsky Avenue, Kiev 252680, Ukraine Received July 8, 1996; accepted April 18, 1997
Using experimental data of AgI sol and poly( 2-methyl-5-vinylpyridine ) chloride, we show that there is good agreement between the developed mathematical model of suspension flocculation by polyelectrolytes and the experimental data. The rate constants of bridging and nonbridging aggregation of particles and of floc breakup, as well as their ratios, have been determined. It has been shown that the role of the nonbridging mechanism, as compared with that of the bridging one, diminishes when molecular mass of polyelectrolyte increases from 1.4 1 10 5 to 9.3 1 10 5 . q 1997
of high molecular weight flocculants (4). In contrast, the neutralizing mechanism of flocculation prevails when a strongly charged low-molecular-weight cationic polyelectrolyte, Percol 1697, is being used (4). Elucidation of the roles of both of these mechanisms, i.e., bridging and nonbridging, and also of the role of molecular mass of polyelectrolyte in the flocculation process is the objective of this paper. THEORY AND DISCUSSION
Academic Press
Key Words: AgI sol; poly(methylvinylpyridine); flocculation; mathematical simulation.
INTRODUCTION
Usually industrial and domestic waste, as well as industrial suspensions, contains particles having a negative charge. Therefore cationic polymers are widely used for their flocculation. Therefore, some researchers assume that in many cases a simple charge neutralizing mechanism can explain most of the observed behavior of such colloid systems (1). Really, many authors note that optimum flocculation proceeds when the particle charge has been reduced to around zero (1–3). In such cases some authors (1) note that the molecular mass of the cationic flocculant should not be an important factor, and flocculation efficiency should depend on the charge density of the polyelectrolyte. On the other hand, it has been noted (2, 3) that the bridging mechanism is the most probable cause of flocculation with polyelectrolytes. The effects of the nonbridging mechanism, i.e., the decrease of charge and of a potential of particles, plays only a minor role in a stability loss (3). Therefore, at the higher molecular mass of polyelectrolyte, the lower polymer concentration is necessary to reach the maximum drop in the total particle concentration (2). Investigation on kaolin suspensions flocculation by cationic polyelectrolytes showed that bridge formation between particles is the predominant mechanism of the process at use 1
To whom correspondence should be addressed.
A mathematical model of suspension flocculation by polyelectrolytes (5) takes account of bridging and nonbridging aggregation of suspension particles, as well as breakup of formed flocs under the action of hydrodynamic forces: 0 dN/dt Å £a / £n 0 £b .
Here N is the total particle concentration in a suspension, £a and £n are the rates of bridging and nonbridging aggregation of particles, respectively, £b is the rate of floc break-up, and t is time. Therefore, £ a Å Ka P c N 2 ,
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[2] 2
2
£n Å Kn (1 0 Pc )exp( 0 dj )N ,
[3]
£b Å lKb N[1 0 exp(1 0 N0 /N)],
[4]
where Ka , Kn , and Kb are the rate constants of bridging and nonbridging aggregations of particles, also of floc breakup, 1 is the number of particles forming during floc breakup, Pc Å 2U(1 0 U ) is a probability of successful particle collisions leading to aggregate formation by bridging mechanism, U Å c/cs is the fraction of the particle surface covered by polyelectrolyte, c is the polymer concentration in a suspension, cs is the polymer concentration corresponding to U Å 1, d is a constant, j is the value of j-potential of particles, and N0 Å N(t Å 0) is the initial number of particles per unit volume of a suspension. Experiments show that the rate of suspension flocculation is decreasing with time, and finally the kinetic curves pla-
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TABLE 1 Parameters of PMVP Macromolecules in MEK Solution Designation
[h] (dl/g)
M 1 1005
»R2…1/2 (nm)
PMVP-1 PMVP-4 PMVP-9
0.37 0.74 1.15
1.4 4.4 9.3
29 53 72
F Å Pc exp(dj2 )/(1 0 Pc ). FIG. 1. Kinetic curves of flocculation of AgI sol by PMVP-9 at polymer concentration 0.25 (1) and 2.5 mg/liter (2). Experimental data are marked with points; curves are calculated according to Eq. [15].
teau ( see Fig. 1 ) . In the equilibrium state at the plateau region d N / dt Å 0. Then Eq. [ 5 ] can be obtained from Eqs. [1] – [ 4 ] :
[10]
One can find the value for Ka /(lKb ) from a plot of the linear relationship between G and F according to Eq. [8]. Then the ratio of the rate constant Kn /Ka can be calculated to estimate the role of nonbridging and bridging aggregations of particles. Integration of Eq. [1] at the initial condition N(t Å 0) Å N0 , taking Eqs. [2] – [4] into account leads to the linear expression
N/[1 0 exp(1 0 N0 /N)] Y Å 0Kat.
Å (lKb /Ka )/[Pc / (Kn /Ka )(1 0 Pc )exp( 0 dj2 )]. [5]
[11]
Here
Then j Å j0 / c/(A / Bc),
[6]
YÅ
*
N
dN/{F1N 2 0 (lKb /Ka )N[1 0 exp(1 0 N0 /N)]},
N0
where j0 Å j at c Å 0, and A and B are the constants (5). As seen from Eqs. [3] and [5], the larger the value of d the more sensitive the aggregation process is to the change of j-potential. Also, the larger the ratio of Kn /Ka the higher the role of nonbridging aggregation of particles is at suspension flocculation by polyelectrolyte. The determination of the values of d and of lKb /Kn can be carried out as follows: at c § cs , Pc Å 0. In this case, Eq. [5] can be transformed into linear relationship: 2
ln{N/[1 0 exp(1 0 N0 /N)]} Å ln(lKb /Kn ) / dj .
[7]
[12] 2
F1 Å Pc / (Kn /Ka )(1 0 Pc )exp( 0 dj ).
[13]
That enables us to determine the value of the rate constant Ka from the tangent of the slope angle of a straight line when plotted Y vs t according to Eq. [11]. EXPERIMENTAL
Monodisperse AgI sol was prepared by gradual addition of 0.1 M AgNO3 into 0.1 M KI with dilution of the sol
A plot of the linear relationship [7] between ln{N/[1 0 exp(1 0 N0 /N)]} and j 2 gives the possibility of finding the values for d and lKb /Kn from the slope of a straight line and the intercept on the ordinate axis. Then it is straightforward to determine the value of Kn / Ka from Eq. [ 5 ] . A little manipulation yields a linear expression, G Å [Ka /(lKb )]F,
[8]
where G Å [1 0 exp(1 0 N0 /N)]/[N(1 0 Pc )exp( 0 dj2 )] 0 Kn /(lKb )
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FIG. 2. Relationship between the total particle concentration in AgI sol and concentration of PMVP-1 (1), PMVP-4 (2), and PMVP-9 (3). Experimental data are marked with points; curves are calculated according to Eq. [5].
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TABLE 2 Parameters of Flocculation of AgI Sol by Poly(methylvinylpyridine) Quantity
PMVP-1
PMVP-4
PMVP-9
cs (mg/liter) Uz d 1 104 (mV02) 1013 1 Kn/(1 Kb) (liters) 1012 1 Ka/(1 Kb) (liters) Kn/Ka
8.2 0.29 5.8 { 1.6 5.5 { 2.0 5.1 { 1.3 0.11 { 0.07
4.2 0.48 4.1 { 1.8 4.1 { 0.8 3.7 { 1.4 0.11 { 0.06
2.2 0.73 0.042 { 0.019 3.7 { 1.1 4.5 { 1.3 0.08 { 0.05
formed one-fifth of previous value. Salt excess was removed by dialysis for 1 week. The sol obtained had pH 5.9 and conductivity of 3 1 10 05 V 01 cm01 . The average radius of particles was 80 nm (by electron microscopy). Poly(2-methyl-5-vinylpyridine) chloride fractionated (PMVP) was used as polyelectrolyte (Table 1). Molecular mass of PMVP was measured by viscometry in methyl ethyl ketone (MEK) (6). Root mean square radius of particles was calculated according to the Flory–Fox equation (7). The flocculation technique was carried out as follows. PMVP solution of a given concentration was added into the constant volume of AgI sol. The system was stirred and then after a certain time interval, diluted to 151 th to 201 th of the previous value with KCl solution of the same conductivity. The count of the particle number per unit volume was carried out in a cell of a flow ultramicroscope described in Ref. (8). RESULTS AND DISCUSSION
The relationship between the number of particles per unit volume of AgI sol and PMVP concentration is shown in Fig. 2. The values of cs found according to Ref. (5) from inflection points of curves N(c) decrease with increasing molecular mass of polyelectrolyte (Table 2). Therefore the higher molecular mass of PMVP the lower polymer concentration is necessary to reach U Å 1. Let cz denote the PMVP concentration where j-potential
FIG. 3. Effect of j-potential of particles in AgI sol on total particle concentration according to Eq. [7] at flocculation by PMVP-1 (1) and PMVP-4 (2).
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FIG. 4. Plot of G vs F according to Eq. [8] at flocculation of AgI sol by PMVP-4 (1) and PMVP-9 (2).
of particles in AgI sol goes to zero. The value of cz can be calculated from formula [6] for j Å 0. Then one can determine Uz Å cz /cs . As seen in Table 2, the value of Uz increases with rising molecular mass of polyelectrolyte. Consequently, j-potential vanishes at the higher particle surface coverage with polymer for high-molecular-weight PMVP. As seen in Fig. 2, the values of N at c Å cs , i.e., U Å 1, are lower than initial total particle concentration, N0 . This illustrates the fact that both bridging and nonbridging aggregations of particles occur at flocculation of AgI sol under the action of PMVP (5). Therefore a mathematical model [1] – [4] and equations following from it can be used for description of this flocculation process. Figure 3 illustrates relationship between total particle concentration in sol and j-potential of particles. The experimental data lie on straight lines when plotted according to Eq. [7]. Values of d and of the rate constant ratio Kn /(lKb ) calculated from the slope of the straight lines and from the intercepts on the ordinate axis are represented in Table 2. The value of d increases when molecular mass of PMVP diminishes; i.e., the sensitivity of the aggregation process to the change of j-potential rises. The increasing value of Kn / (lKb ) points to the fact that the contribution of nonbridging aggregation into flocculation process increases. Consequently, the scale of nonbridging aggregation of particles
FIG. 5. Change of Y with time at flocculation of AgI sol by PMVP-9.
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in AgI sol rises when molecular mass of polyelectrolytes decreases. Then the opportunity arises to test the linear relationship [8]. As seen in Fig. 4, experimental data for the AgI sol– PMVP system are situated on straight lines when plotted according to Eq. [8]. The slope of the straight line gives the value of Ka /(lKb ) (Table 2). Then one can determine the value of the ratio Kn /Ka taking account of the Ka /(lKb ) value found earlier (see Table 2). These values and the size of d show that the role of nonbridging aggregation as compared with bridging aggregation of particles in AgI sol is less for high-molecular-weight polyelectrolyte (PMVP-9). The values found for the ratios of the rate constant and for d (Table 2) allow calculation relationship between the total particle concentration, N, and PMVP concentration in AgI sol using Eq. [5]. Computer calculations were executed by means of ZEROIN algorithm (9) using expansion of the left side of Eq. [5] as an initial approximation. The curves in Fig. 2 illustrate the satisfactory agreement of calculated curves with experimental data and about adequacy of the suggested mathematical model [5] to the experiment on flocculation of AgI sol by PMVP. It is worth noting that the minimums of the plot of N vs c/cs do not coincide with the values of Uz for PMVP-1 and PMVP-9 (Table 2). The kinetic data presented in Fig. 1 give an opportunity to determine the values of the individual rate constants using Eqs. [11] – [13]. Numerical integration of Eq. [12] was performed by using an adaptive recursive Simpson’s rule. Figure 5 presents a plot of the linear relationship between N and time according to Eq. [11]. The slope of the straight line yields the value Ka Å (2.4 { 0.3) 1 10 013 liters/min. Using the values of the ratios of the rate constants found earlier (see Table 2), one can determine the values of Kn Å (2.0 { 1.1) 1 10 014 liters/min and of lKb Å (5.4 { 2.4) 1 10 02 min 01 for flocculation of AgI sol by PMVP-9. The values found for the rate constants enable us to calculate the change of the number of primary particles in a floc N0 /N in time. For this purpose, Eqs. [1] – [3] can be rewritten as d(N0 /N)/dt Å Ka {N0 F1 0 (lKb /Ka )(N0 /N) 1 [1 0 exp(1 0 N0 /N)]}.
[14]
Integration of Eq. [14] at the initial condition N0 /N(t Å 0) Å 1 gives t Å Y1 /Ka ,
[15]
where Y1 Å
*
N0/N
d(N0 /N)/ {N0 F1 0 (lKb /Ka )(N0 /N)
1
1 [1 0 exp(1 0 N0 /N)]}.
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[16]
07-11-97 06:51:55
FIG. 6. Relationship between £a , liter 01 min 01 (1), £n , liter 01 min 01 (2), and £a / £n (3) and PMVP-9 concentration in AgI sol.
Numerical integration of Eq. [16] was performed by using an adaptive recursive Simpson’s rule. Equation [15] defines the relationship between the number of primary particles in a floc, N0 /N, and time. Curves in Fig. 1 demonstrate a satisfactory agreement between the kinetic curves calculated in such a way and experimental data on flocculation of AgI sol by PMVP-9. The relationships of the rates of bridging and nonbridging aggregation of AgI particles from PMVP-9 concentration can be calculated by Eqs. [2] and [3]. Figure 6 demonstrates these functions and the ratio £a / £n . It is interesting that the curve for £a has two maximums not coinciding with the minimum of the curve N vs c/cs (see Fig. 2). At the same time, the curve for £n has the minimum and the curve of £a / £n has the maximum coinciding with the minimum of the curve N(c/cs ) as discussed in Ref. (5). Consequently, the agreement between the experimental data on AgI sol aggregation by PMVP and the mathematical model of suspension flocculation by polyelectrolytes supports correctness of the suggested model.
REFERENCES 1. Gregory, J., Flocculation. Prog. Filtr. Sep. 4, 55 (1986). 2. Baran, A. A., Tusupbayev, I. K., Solomentseva, I. M., Deryagin, R. M., and Musabekov, K. B., Colloid J. (USSR) 42, 11 (1980). 3. Solomentseva, I. M., Tusupbayev, I. K., Baran, A. A., and Musabekov, K. B., Ukr. Chem. J. (USSR) 46, 928 (1980). 4. Baran, S., and Gregory, J., Coll. J. (Russia) 58, 13 (1996). 5. Berlin, Ad. A., Kislenko, V. N., and Solomentseva, I. M., Colloid J. (Russia) (in press). 6. Jarlugio, C., Crescentini, L., Mula, A., and Jechele, J. B., Macromolec. Chem. 97, 97 (1966). 7. Flory, P. J., ‘‘Mechanics of Chain Molecules.’’ Interscience, New York–London–Sydney–Toronto, 1969. 8. Kudryavtseva, N. M., and Deryagin, B. V., Colloid J. (USSR) 25, 739 (1963). 9. Forsythe, G. E., Malcolm, M. A., and Moler, C. B., ‘‘Computer Methods for Mathematical Computations,’’ C7632. Prentice-Hall, Englwood Cliffs, NJ, 1977.
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