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Barycentric Method of Determining the Physical Parameters of a Single-Phase Particle in Liquid Disperse Systems
Journal of Colloid and Interface Science 256, 477–479 (2002) doi:10.1006/jcis.2002.8531
NOTE Barycentric Method of Determining the Physical Parameters of a Single-Phase Particle in Liquid Disperse Systems The present paper deals with the sedimentation of a liquid monodisperse system to define the physical parameters—size, volume, and density—of a single particle of the disperse phase in this system. The proposed physical and mathematical model is based on the use of the barycentric method by which the displacement of the mass center of the liquid disperse system is measured when it has reached the boundary, maximum sedimentation. Spherical form of the particles of the disperse phase is assumed as well as the most compact packing of the spheres in a state of maximum sedimentation at a constant temperature. An example of magnetic suspension of cobalt particles and water is discussed. C 2002 Elsevier Science (USA) Key Words: sedimentation; liquid disperse system; single-particle parameters; barycentric method.
and transparent material. Attached to it is an axis for suspension and other subsidiary elements as shown in Fig. 1, which ensure the measurement of the mass center of the LDS in time. The cuvette is made so that the mass center of the empty cuvette coincides with the geometrical center (Fig. 1). This point O lies on the axis of symmetry ξ of the cavity. The axis of suspension of the cuvette C, not marked on Fig. 1, lies perpendicular to the drawing and is fixed to the point of the surface area of the cylindrical cuvette C (2–4). At this point there is a thin flat mirror that reflects the laser beam. The cuvette C is filled entirely with the LDS and placed in the vertical position. In the course of time, as a result of sedimentation, the mass center of the LDS moves along the axis ξ at a distance of x(t) from point O. For the purpose of measuring, the cuvette C is suspended freely to the base plane of the device. In different moments of time the axis of the cuvette bends at a different angle to the plane. This changes the angle of incidence, respectively, the angle of reflection of the laser beam on the mirror (2–4). The relation between the displacement x(t) of the mass center MC of LDS in the cuvette C and the length of the curve n(t) drawn by the reflected beam on the cylindrical screen of the device at a given moment of time is (2–4)
INTRODUCTION The interaction between the solid particles of the disperse phase and the liquid disperse phase in a liquid disperse system (LDS) causes changes in the physical characteristics of the particles in terms of size, volume, and density. As is noted in (1), for the sake of practice, it is important to know the density of a single particle from the disperse phase. But in the available literature there are no concrete methods for defining the physical parameters of the single particle of the disperse phase of a LDS. The present article proposes the barycentric method for defining the size, volume, and density of a single spherical particle from the disperse phase of a LDS of a given monodisperse system. The density of the disperse medium can also be determined. The proposed barycentric method is based on the displacement of the mass center of the LDS in the state of boundary sedimentation from the geometric center of the vessel in which the system is placed (2–4). In this way one may take into account the behavior of all particles of the disperse phase due to sedimentation. As has been pointed out in (2, 3), the barycentric method is nonperturbative because there is no external impact on it apart from gravitation: the LDS is not submitted to irradiation for measurement, no measuring elements are introduced, and no electric current is passed through it. Apart from that, the proposed method is universal because it can be applied for the study of any kinds of LDSs: transparent, opaque, regardless of coloring. The temperature and atmospheric pressure, however, need to be constant.
EXPERIMENTAL METHOD The proposed barycentric method for defining the physical parameters of a single particle of the disperse phase of the LDS is carried out with a barycentric device described in detail in (2–4). This device has been patented (4) and implemented with the financial support of the University of Plovdiv with Contract F-22 1997/99. Its main body is a cylindrical cuvette C with inside height H and inside radius R (Fig. 1). The cuvette is manufactured from hard, homogenous,
x(t) = (R + a)tg
n(t) , 2L
[1]
where a is the thickness of the wall of the cuvette C and L is the distance from the mirror to the screen. In this case with the barycentric device and a formula, we are able to define the location of the mass center MC of the LDS in each moment of time.
RESULTS AND DISCUSSION Let us consider the sedimentation of the LDS that fills the entire cuvette C at constant temperature. In the initial moment t = 0 the LDS is homogenous and its MC coincides with the geometrical center, point O, of its cavity. Let the LDS be monodisperse with a concentration of the particles, c0 (number of particles per unit of volume). In the model under consideration we assume that the particles of the disperse phase are uniform in size and are spherical. In the “dry state,” before the disperse medium has been added, each particle has a certain volume and density. However, in the LDS, each particle correlates with the disperse medium. The properties of the substances which compose the phase and the disperse medium under given external conditions (temperature and pressure) determine how much liquid will diffuse in each particle and what solvent shell will be formed around its surface. The total mass M of the LDS in the closed cuvette, however, remains constant. With ρp , Vp , and rp we designate the effective density, volume, and radius, respectively, of a single particle of the disperse phase, found in a specific disperse medium under specific external conditions. When t > 0 in the conditions of a static homogenous gravitational field with normal sedimentation, the phase particle of the LDS under consideration moves toward the bottom of the cuvette. The LDS is no longer homogenous. The LDS density in the upper part of C decreases and in its bottom part a sediment is formed. The MC of the LDS in the vertically placed cuvette C moves along the 477
0021-9797/02 $35.00
C 2002 Elsevier Science (USA)
All rights reserved.
478
NOTE the phase only is 74.04% of the volume of the sediment, e.g., Vf = π R 2 h. The remaining 25.96% of the volume of the sediment is the volume Vfl of the space between the spherical phase particles filled with liquid. Therefore, the mass Mfl of the liquid in the sediment is Mfl = (1 − k)π R 2 hρl .
[4]
On the other hand, the volume of the space between the spherical particles in the sediment is a subtraction of the volume of the sediment and the volume of all particles in the phase. Therefore, Mfl = (π R 2 h − N Vp ),
[5]
where N = C0 π R 2 H is the number of all particles of the disperse phase. From the equation of the right-hand sides of [4] and [5] we define the effective volume Vp and, consequently, the effective radius rp of the single-phase particle found in the LDS: Vp =
k h ; rp = C0 H
3kh 4C0 H π
1/3 .
[6]
The mass Mf of the disperse phase may be expressed as the difference between the Ms of the sediment and the Mfl of the liquid in the space between the particles of the sediment, e.g., Mf = Ms − Mfl . On the other hand, Mf can result from the multiplication of the Vf and the effective density of the phase. From the equation of these two expressions we define ρp :
FIG. 1. Cylindrical cuvette of a barycentric device (C, cuvette; R, inside radius of cuvette; H , inside height of cuvette; ξ , axis of symmetry of cuvette, O, geometrical center of cuvette; h, height of sediment; Ob , mass center of liquid monodisperse system in boundary, maximum sedimentation).
axis x toward the bottom of C. After an interval of time (theoretically t → ∞), the LDS reaches the boundary, maximum sedimentation. In this state we suppose that all spherical phase particles in the sediment are arranged densely by each other without deformation. The space between these particles is filled with a disperse medium. Apart from that, between the sediment and the pure disperse medium above it, there is a sharp dividing line that is at a height h from the bottom of the cuvette C (Fig. 1). We designate with xb = |OOb | the boundary, maximum displacement of the MC of the LDS from the geometrical center, point O, of the cavity of the cuvette C. As shown in (2) regarding the displacement xb of the MC of the LDS related to point O, we obtain xb =
1 [Ms H − Mh], 2M
[2]
where M is the total mass of the LDS in the cuvette C, Ms the sediment mass, and h the height of the sediment. We shall note again that xb is defined experimentally with the help of the barycentric device (2–4). The mass Ml of the pure disperse medium above the sediment can be found by multiplying the volume of the space occupied by it with the density Pl of the disperse medium, namely, Ml = π R 2 (H − h)ρl .
[3]
The mass Ms is a sum of the Mf of the phase and the Mfl of the small quantity of disperse medium filling the space between the spherical particles of the phase in the sediment. According to (5), the volume of densely packed spheres that touch without deforming in the three-dimensional space is 74.04% or k = 0.7404 of the volume of the space occupied by them. Consequently, the volume Vf of
ρp =
Ms − Mfl . kπ R 2 h
[7]
From [1] and [7], we obtain the relation between ρp and the maximum displacement xb of the MC of the LDS in a state of boundary sedimentation: ρp =
M 2xb (H − kh) 1+ . 2 πR H kh(H − h)
[8]
The density ρl of the liquid can be expressed in an analogous way. This is done by expressing Ml from formula [2] and equating it with [3]. After transformation we obtain ρl =
M 2xb 1 − . π R2 H H −h
[9]
Considering formulae [8] and [9], we can easily find the mass Mf of the phase in the LDS, namely, Mf =
2xb (H − kh) M hk + . H H −h
[10]
The mass Mls of the disperse medium above the sediment becomes Mls = Ml + Mfl =
M 2xb (H − kh) 1 − . H H −h
[11]
The maximum relative error module for ρp is evaluated as follows: ρp M R H h xb . ≤ + + + + ρp M R H h xb
[12]
A computer simulation was carried out using the described barycentric device. We assumed the dimensions of the cuvette to be H = 12 × 10−2 m and
479
NOTE R = 8 × 10−3 m. In the absence of a magnetic field, at T = 20◦ C, we studied the sedimentation of spherical cobalt particles in water. It was assumed that the volume concentration of these particles was assumed to be 0.025, e.g., C0 = 9.32 × 1022 m−3 . The mass of the whole LDS was M = 28.786 × 10−3 kg, the height of the sediment was h = 5 × 10−3 m, and xb = 8.974 × 10−3 m. Through formulae [6] and [8] we calculated that the effective radius of a single cobalt particle from the LDS is rp = 4.3 × 10−9 m, and the effective density is ρp = 8.4 × 103 kg/m3 . It has been pointed out in (6) that the density of a single cobalt particle outside the liquid is ρp0 = 8.8 × 103 kg/m3 , and the thickness of the hydrate shell around it may reach 2.0 × 10−9 m. On the basis of the data presented above about M, R, H , h, and xb and the absolute errors M = 10−6 kg and R = H = h = xb = 10−4 m, we calculate the module of maximum relative error as follows: ρp ≤ 4.4%. ρp
ACKNOWLEDGMENT The authors are grateful to Dr. V. Dimitrov for his valuable suggestions and discussion.
REFERENCES 1. Hodakov, G. S., and Udkin, U. P., “Sedimentationii analyse.” Chemistry, Moscow, 1981. 2. Krustev, G. A., Christozov, D. D., and Machev, K. Sv., Travaux Scientifiques, Universite de Plovdiv 28, 4 (1990). 3. Christozov, D. D., Z . Lebensm. Unters. Forsch. A 206, 303 (1998). 4. Christozov, D. D., and Krustev, G. A., Device for sedimentary analysis of liquid disperse systems, Patent No. 26040, Reg. No. 37476, Rep. Bulgaria, 1977. 5. Hales, T. C., Nature 395, 435 (1998). 6. Fertman, V. E., in “Magnitniye zhidkosti” (Magnetic fluids), p. 135. Vysshaya shkola, Moscow, 1988.
SUMMARY On the basis of these observations, it is evident that the experimental determination of the maximum displacement of xb of the mass center of the LDS in a state of boundary sedimentation, together with the knowledge of the k coefficient, allow the theoretical determination of the effective density ρp of a spherical phase particle in a LDS, all the more that the particle need not be separated from the liquid monodisperse system or be under external impact for the sake of measuring. With all that, the described physical and mathematical model in the case of the sedimentation of a monodisperse system leads to the determination of the values of the density ρl of the disperse medium, the mass of the phase Mf , and the mass of the disperse medium of the LDS Ml . The barycentric measuring of the displacement of the mass center of the LDS in the state of boundary sedimentation and the proposed formulae allow the determination of the changes of the physical parameters of a single particle of the disperse phase of a LDS caused by the interaction of the particle with the disperse medium. The changes lead to other values, the speed of sedimentation and the sedimentary stability of LDS.
Diana Dakova∗,1 Dimo Christozov† Maryana Beleva† ∗ Department of Theoretical Physics University of Plovdiv Plovdiv, Bulgaria †Department of Physics Higher Institute of Food and Flavour Industries Plovdiv, Bulgaria Received April 10, 2001; accepted June 14, 2002
1 To whom correspondence should be addressed: Department of Theoretical Physics, University of Plovdiv, 24, Tzar Assen Str., 4000 Plovdiv, Bulgaria. E-mail: [email protected].
NOTE Barycentric Method of Determining the Physical Parameters of a Single-Phase Particle in Liquid Disperse Systems The present paper deals with the sedimentation of a liquid monodisperse system to define the physical parameters—size, volume, and density—of a single particle of the disperse phase in this system. The proposed physical and mathematical model is based on the use of the barycentric method by which the displacement of the mass center of the liquid disperse system is measured when it has reached the boundary, maximum sedimentation. Spherical form of the particles of the disperse phase is assumed as well as the most compact packing of the spheres in a state of maximum sedimentation at a constant temperature. An example of magnetic suspension of cobalt particles and water is discussed. C 2002 Elsevier Science (USA) Key Words: sedimentation; liquid disperse system; single-particle parameters; barycentric method.
and transparent material. Attached to it is an axis for suspension and other subsidiary elements as shown in Fig. 1, which ensure the measurement of the mass center of the LDS in time. The cuvette is made so that the mass center of the empty cuvette coincides with the geometrical center (Fig. 1). This point O lies on the axis of symmetry ξ of the cavity. The axis of suspension of the cuvette C, not marked on Fig. 1, lies perpendicular to the drawing and is fixed to the point of the surface area of the cylindrical cuvette C (2–4). At this point there is a thin flat mirror that reflects the laser beam. The cuvette C is filled entirely with the LDS and placed in the vertical position. In the course of time, as a result of sedimentation, the mass center of the LDS moves along the axis ξ at a distance of x(t) from point O. For the purpose of measuring, the cuvette C is suspended freely to the base plane of the device. In different moments of time the axis of the cuvette bends at a different angle to the plane. This changes the angle of incidence, respectively, the angle of reflection of the laser beam on the mirror (2–4). The relation between the displacement x(t) of the mass center MC of LDS in the cuvette C and the length of the curve n(t) drawn by the reflected beam on the cylindrical screen of the device at a given moment of time is (2–4)
INTRODUCTION The interaction between the solid particles of the disperse phase and the liquid disperse phase in a liquid disperse system (LDS) causes changes in the physical characteristics of the particles in terms of size, volume, and density. As is noted in (1), for the sake of practice, it is important to know the density of a single particle from the disperse phase. But in the available literature there are no concrete methods for defining the physical parameters of the single particle of the disperse phase of a LDS. The present article proposes the barycentric method for defining the size, volume, and density of a single spherical particle from the disperse phase of a LDS of a given monodisperse system. The density of the disperse medium can also be determined. The proposed barycentric method is based on the displacement of the mass center of the LDS in the state of boundary sedimentation from the geometric center of the vessel in which the system is placed (2–4). In this way one may take into account the behavior of all particles of the disperse phase due to sedimentation. As has been pointed out in (2, 3), the barycentric method is nonperturbative because there is no external impact on it apart from gravitation: the LDS is not submitted to irradiation for measurement, no measuring elements are introduced, and no electric current is passed through it. Apart from that, the proposed method is universal because it can be applied for the study of any kinds of LDSs: transparent, opaque, regardless of coloring. The temperature and atmospheric pressure, however, need to be constant.
EXPERIMENTAL METHOD The proposed barycentric method for defining the physical parameters of a single particle of the disperse phase of the LDS is carried out with a barycentric device described in detail in (2–4). This device has been patented (4) and implemented with the financial support of the University of Plovdiv with Contract F-22 1997/99. Its main body is a cylindrical cuvette C with inside height H and inside radius R (Fig. 1). The cuvette is manufactured from hard, homogenous,
x(t) = (R + a)tg
n(t) , 2L
[1]
where a is the thickness of the wall of the cuvette C and L is the distance from the mirror to the screen. In this case with the barycentric device and a formula, we are able to define the location of the mass center MC of the LDS in each moment of time.
RESULTS AND DISCUSSION Let us consider the sedimentation of the LDS that fills the entire cuvette C at constant temperature. In the initial moment t = 0 the LDS is homogenous and its MC coincides with the geometrical center, point O, of its cavity. Let the LDS be monodisperse with a concentration of the particles, c0 (number of particles per unit of volume). In the model under consideration we assume that the particles of the disperse phase are uniform in size and are spherical. In the “dry state,” before the disperse medium has been added, each particle has a certain volume and density. However, in the LDS, each particle correlates with the disperse medium. The properties of the substances which compose the phase and the disperse medium under given external conditions (temperature and pressure) determine how much liquid will diffuse in each particle and what solvent shell will be formed around its surface. The total mass M of the LDS in the closed cuvette, however, remains constant. With ρp , Vp , and rp we designate the effective density, volume, and radius, respectively, of a single particle of the disperse phase, found in a specific disperse medium under specific external conditions. When t > 0 in the conditions of a static homogenous gravitational field with normal sedimentation, the phase particle of the LDS under consideration moves toward the bottom of the cuvette. The LDS is no longer homogenous. The LDS density in the upper part of C decreases and in its bottom part a sediment is formed. The MC of the LDS in the vertically placed cuvette C moves along the 477
0021-9797/02 $35.00
C 2002 Elsevier Science (USA)
All rights reserved.
478
NOTE the phase only is 74.04% of the volume of the sediment, e.g., Vf = π R 2 h. The remaining 25.96% of the volume of the sediment is the volume Vfl of the space between the spherical phase particles filled with liquid. Therefore, the mass Mfl of the liquid in the sediment is Mfl = (1 − k)π R 2 hρl .
[4]
On the other hand, the volume of the space between the spherical particles in the sediment is a subtraction of the volume of the sediment and the volume of all particles in the phase. Therefore, Mfl = (π R 2 h − N Vp ),
[5]
where N = C0 π R 2 H is the number of all particles of the disperse phase. From the equation of the right-hand sides of [4] and [5] we define the effective volume Vp and, consequently, the effective radius rp of the single-phase particle found in the LDS: Vp =
k h ; rp = C0 H
3kh 4C0 H π
1/3 .
[6]
The mass Mf of the disperse phase may be expressed as the difference between the Ms of the sediment and the Mfl of the liquid in the space between the particles of the sediment, e.g., Mf = Ms − Mfl . On the other hand, Mf can result from the multiplication of the Vf and the effective density of the phase. From the equation of these two expressions we define ρp :
FIG. 1. Cylindrical cuvette of a barycentric device (C, cuvette; R, inside radius of cuvette; H , inside height of cuvette; ξ , axis of symmetry of cuvette, O, geometrical center of cuvette; h, height of sediment; Ob , mass center of liquid monodisperse system in boundary, maximum sedimentation).
axis x toward the bottom of C. After an interval of time (theoretically t → ∞), the LDS reaches the boundary, maximum sedimentation. In this state we suppose that all spherical phase particles in the sediment are arranged densely by each other without deformation. The space between these particles is filled with a disperse medium. Apart from that, between the sediment and the pure disperse medium above it, there is a sharp dividing line that is at a height h from the bottom of the cuvette C (Fig. 1). We designate with xb = |OOb | the boundary, maximum displacement of the MC of the LDS from the geometrical center, point O, of the cavity of the cuvette C. As shown in (2) regarding the displacement xb of the MC of the LDS related to point O, we obtain xb =
1 [Ms H − Mh], 2M
[2]
where M is the total mass of the LDS in the cuvette C, Ms the sediment mass, and h the height of the sediment. We shall note again that xb is defined experimentally with the help of the barycentric device (2–4). The mass Ml of the pure disperse medium above the sediment can be found by multiplying the volume of the space occupied by it with the density Pl of the disperse medium, namely, Ml = π R 2 (H − h)ρl .
[3]
The mass Ms is a sum of the Mf of the phase and the Mfl of the small quantity of disperse medium filling the space between the spherical particles of the phase in the sediment. According to (5), the volume of densely packed spheres that touch without deforming in the three-dimensional space is 74.04% or k = 0.7404 of the volume of the space occupied by them. Consequently, the volume Vf of
ρp =
Ms − Mfl . kπ R 2 h
[7]
From [1] and [7], we obtain the relation between ρp and the maximum displacement xb of the MC of the LDS in a state of boundary sedimentation: ρp =
M 2xb (H − kh) 1+ . 2 πR H kh(H − h)
[8]
The density ρl of the liquid can be expressed in an analogous way. This is done by expressing Ml from formula [2] and equating it with [3]. After transformation we obtain ρl =
M 2xb 1 − . π R2 H H −h
[9]
Considering formulae [8] and [9], we can easily find the mass Mf of the phase in the LDS, namely, Mf =
2xb (H − kh) M hk + . H H −h
[10]
The mass Mls of the disperse medium above the sediment becomes Mls = Ml + Mfl =
M 2xb (H − kh) 1 − . H H −h
[11]
The maximum relative error module for ρp is evaluated as follows: ρp M R H h xb . ≤ + + + + ρp M R H h xb
[12]
A computer simulation was carried out using the described barycentric device. We assumed the dimensions of the cuvette to be H = 12 × 10−2 m and
479
NOTE R = 8 × 10−3 m. In the absence of a magnetic field, at T = 20◦ C, we studied the sedimentation of spherical cobalt particles in water. It was assumed that the volume concentration of these particles was assumed to be 0.025, e.g., C0 = 9.32 × 1022 m−3 . The mass of the whole LDS was M = 28.786 × 10−3 kg, the height of the sediment was h = 5 × 10−3 m, and xb = 8.974 × 10−3 m. Through formulae [6] and [8] we calculated that the effective radius of a single cobalt particle from the LDS is rp = 4.3 × 10−9 m, and the effective density is ρp = 8.4 × 103 kg/m3 . It has been pointed out in (6) that the density of a single cobalt particle outside the liquid is ρp0 = 8.8 × 103 kg/m3 , and the thickness of the hydrate shell around it may reach 2.0 × 10−9 m. On the basis of the data presented above about M, R, H , h, and xb and the absolute errors M = 10−6 kg and R = H = h = xb = 10−4 m, we calculate the module of maximum relative error as follows: ρp ≤ 4.4%. ρp
ACKNOWLEDGMENT The authors are grateful to Dr. V. Dimitrov for his valuable suggestions and discussion.
REFERENCES 1. Hodakov, G. S., and Udkin, U. P., “Sedimentationii analyse.” Chemistry, Moscow, 1981. 2. Krustev, G. A., Christozov, D. D., and Machev, K. Sv., Travaux Scientifiques, Universite de Plovdiv 28, 4 (1990). 3. Christozov, D. D., Z . Lebensm. Unters. Forsch. A 206, 303 (1998). 4. Christozov, D. D., and Krustev, G. A., Device for sedimentary analysis of liquid disperse systems, Patent No. 26040, Reg. No. 37476, Rep. Bulgaria, 1977. 5. Hales, T. C., Nature 395, 435 (1998). 6. Fertman, V. E., in “Magnitniye zhidkosti” (Magnetic fluids), p. 135. Vysshaya shkola, Moscow, 1988.
SUMMARY On the basis of these observations, it is evident that the experimental determination of the maximum displacement of xb of the mass center of the LDS in a state of boundary sedimentation, together with the knowledge of the k coefficient, allow the theoretical determination of the effective density ρp of a spherical phase particle in a LDS, all the more that the particle need not be separated from the liquid monodisperse system or be under external impact for the sake of measuring. With all that, the described physical and mathematical model in the case of the sedimentation of a monodisperse system leads to the determination of the values of the density ρl of the disperse medium, the mass of the phase Mf , and the mass of the disperse medium of the LDS Ml . The barycentric measuring of the displacement of the mass center of the LDS in the state of boundary sedimentation and the proposed formulae allow the determination of the changes of the physical parameters of a single particle of the disperse phase of a LDS caused by the interaction of the particle with the disperse medium. The changes lead to other values, the speed of sedimentation and the sedimentary stability of LDS.
Diana Dakova∗,1 Dimo Christozov† Maryana Beleva† ∗ Department of Theoretical Physics University of Plovdiv Plovdiv, Bulgaria †Department of Physics Higher Institute of Food and Flavour Industries Plovdiv, Bulgaria Received April 10, 2001; accepted June 14, 2002
1 To whom correspondence should be addressed: Department of Theoretical Physics, University of Plovdiv, 24, Tzar Assen Str., 4000 Plovdiv, Bulgaria. E-mail: [email protected].