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Extension of Einstein's viscosity equation to that for concentrated dispersions of solutes and particles
JOURNAL OF BIOSCIENCE AND BIOENGINEERING Vol. 102, No. 6, 524–528. 2006 DOI: 10.1263/jbb.102.524
© 2006, The Society for Biotechnology, Japan
Extension of Einstein’s Viscosity Equation to That for Concentrated Dispersions of Solutes and Particles Kiyoshi Toda1 and Hisamoto Furuse1* Institute of Molecular and Cellular Biosciences, The University of Tokyo, 1-1-1 Yayoi, Bunkyo-ku, Tokyo 113-0032, Japan1 Received 27 July 2006/Accepted 5 September 2006
A viscosity equation for concentrated solutions or suspensions is derived as an extension of Einstein’s hydrodynamic viscosity theory for dilute dispersions of spherical particles. The derivation of the equation is based on the calculation of dissipation of mechanical energy into heat in the dispersion, subtracting the energy dissipation in the portion of solutes or particles. The viscosity equation derived thus was well fitted to the viscosity-concentration relationship of the concentrated aqueous solutions of glucose and sucrose. For the suspensions of bakers’ yeast, the concentration dependency of viscosity was expressed well with some modification for the flow pattern around suspended particles. It is suggested that these viscosity equations can be widely applied to both diluted and concentrated dispersions of various solutes and particles. [Key words: viscosity equation, concentrated dispersion, dissipation energy, bakers’ yeast]
having a nearly spherical shape. The viscosity equation was finely applicable to the viscosity of the aqueous solutions of glucose and sucrose. On the other hand, the suspension viscosity of the yeast with a large particle size on the order of micrometers deviated, in the region of high concentration, which was considerably higher than that obtained using the equation. The viscosity equation derived by modifying the flow pattern around suspended particles enabled the simulation of the tendency in the measurement of suspension viscosity.
In a study carried out 100 years ago, Einstein has derived the well known viscosity equation on the basis of a hydrodynamic theory for dilute dispersions of spherical particles (1). Although there have been many trials that simulate the concentration dependencies of viscosity in solutions and suspensions, satisfactory discussions that may elucidate the mechanism underlying this dependency using an extension of the theory have not been offered up to the present, even for the concentrated dispersion of a simple structure of monodispersed spherical particles (2–4). In our previous study, we extended the theoretical treatment to satisfy the viscosity behavior of concentrated dispersion (5). The applicability of the theoretical treatment and derived viscosity equation to actual dispersions of solutes and particles is confirmed in this study with more detail procedures and discussions. The viscosity theory of dilute dispersion is based on the calculation of energy dissipated by suspended spherical particles in liquids. The energy dissipation in the treatment is assumed to be caused even in the portion occupied by the particles. This is permissible only in the case of dilute dispersion. For concentrated dispersions, dissipation energy calculated is overestimated, because dissipation actually cannot occur in the portion occupied by particles. This excess energy must be subtracted from the total dissipation energy in dilute dispersion obtained through a theoretical procedure. The derived viscosity equation for concentrated dispersion was examined with regard to the concentration dependencies of viscosity in the aqueous solutions of sucrose and glucose and in the suspensions of particles of bakers’ yeast
MATERIALS AND METHODS Viscosity theory and experiment The theoretical treatment in this study is based on Einstein’s viscosity theory for dilute dispersions of small spheres (1). The viscosity equation of the dilute dispersion has been derived as ηr = 1 + 2.5φ
(1)
where, ηr is the relative viscosity given as the viscosity ratio of the dispersion to a liquid of dispersant, and φ is the volume fraction of the suspended solutes or particles assumed to be spherical. The derived viscosity equations of concentrated dispersions were examined using the aqueous solutions of glucose and sucrose, and of the suspensions of bakers’ yeast obtained by cultivation, which were compared with the data for spherical particles of polystylene and others cited in the literature (9). Viscosity in the present experiment was measured using a capillary viscometer of the Ostwald type at 20°C. Measurement of volume fractions of solute The volume fractions of glucose and sucrose molecules in concentrated solutions were obtained by multiplying the volume fractions in dilute solutions by dilution ratios. The values of the dilute solutions were derived inversely from the measurement of viscosity using Eq. 1 when the result is φ <0.05. This method of obtaining the volume of the solute molecules in water gives the hydrodynamic molecular
* Corresponding author. e-mail: [email protected] phone/fax: +81-(0)467-52-6267 524
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volume of water molecules attaching to the solute surfaces, on assuming the applicability of Eq. 1 to the dilute solutions. Measurement of volume fractions of yeast The volume fractions of the yeast in concentrated suspensions were obtained by measuring the concentrations of polymers that were diluted after mixing the aqueous solution of a polymer with the suspension composed of yeast and water (7, 8). For the aqueous solution in the present experiment, a polymer of polyvinyl-pyrrolidone was dissolved in water (5). Using the solution of a polymer in this method to obtain yeast volume enables to minimize the osmotic pressure effect derived from dissolved molecules on yeast cells. In this study, the procedure of derivation of Eq. 1 is first surveyed to understand its theoretical background. Flow around one spherical particle First in the derivation of Eq. 1, the flow pattern of the liquid around a single spherical particle is determined by solving the hydrodynamic equation of Navier–Stokes. With the solution derived from the equation for the flow we examine the increase in dissipation energy caused by the presence of the particle in the liquid. As a premise, the liquid of a dispersant with a viscosity η0 is incompressible and the small inertial terms concerning the liquid and the particle are disregarded in the treatment using the hydrodynamic equation. The velocity of the flow in a small domain, in the absence of the particle, is expressed as a linear form, in which u0, v0, and w0 denote the components of the velocity. For an appropriate choice of the coordinates, the velocity components at a point (x, y, z), which is some distance apart from a point (x0, y0, z0), are expressed by u0 = α(x − x0) = αξ v0 = β(y − y0) = βη w0 = γ(z − z0) = γζ
}
(2)
From the assumption of the incompressibility of the liquid, the following condition is realized. α+β+γ=0
(3)
In the presence of a spherical particle of the radius r, the center of which is placed at the point (x0, y0, z0), the velocity components around the particle are expressed as u = αξ + u′ v = βη + v′ w = γζ + w′
}
(4)
where u′, v′, and w′ are the additional terms of the velocity due to the presence of the particle. It follows from the contact of molecules that the velocity of liquid molecules agrees with that of the particle on its surface. In addition, from the symmetry of the flow, the spherical particle stands still. These conditions provide a boundary condition which is u= v=w= 0 on the surface ρ = ξ 2 + η 2 + ζ 2 = r, where ρ denotes the distance from the point (x0, y0, z0). Furthermore, the flow in a position far from the particle approaches the flow obtained using Eq. 2, that is, u′= v′= w′= 0 when ρ =∞. The hydrodynamic equation of Navier–Stokes is given by ∂p ∂p ∂p (5) --------- = η0∆u, --------- = η0 ∆v, -------- = η0 ∆w ∂ξ ∂η ∂ζ where p is the hydrostatic pressure, and the operator ∆ is expressed as 2 2 2 ∂ - --------∂ ∂ ∆ = ---------+ ---------+ 2 2 2 ∂ξ ∂η ∂ζ Einstein obtained the solution of the hydrodynamic equation under the boundary conditions.
525
5 r3 u = αξ − ------ ------5- ξ(αξ 2 + βη2 + γζ 2) 2 ρ 5- -----r 5- ξ(αξ 2 + βη2 + γζ 2) − -----r 5- αξ + ----7 2 ρ ρ5
(6)
The expressions for v and w are given by analogy. Moreover, for the pressure p,
{
}
2 1 2 1 2 1 ∂ ---∂ ---∂ --- ρ ρ ρ 5 3 - + β --------------- + γ --------------- +C p = – ------η0r α --------------2 2 2 3 ∂ξ ∂η ∂ζ
(7)
where C is a constant of the pressure. Calculation of dissipation energy The energy transformed into heat per unit time within the region of the spherical volume V of the radius R, which is indefinitely larger than the particle radius r, is calculated by the above solution. The energy Q to be calculated is given by the integration extended over the surface of the sphere. That is,
∫
Q = (Unu + Vnv + Wnw)ds
(8)
where Un, Vn, and Wn are the components of the pressure exerted on the surface of the sphere. The first two terms on the right-hand side of Eq. 6 are used in the following calculation, because the other terms vanish for ρ=R assumed for a very large size compared with r. That is, 5- -----r3- ξ(αξ 2 + βη2 + γζ 2) u = αξ − ----(9) 2 ρ5 The analogous omission is carried out similarly for v and w, and for p, α ξ 2 + βη2 + γζ 2 p = −5η0r3 ---------------------------------------+C (10) ρ5 The dissipation energy given by Eq. 8, consumed in the volume V, is derived by calculation using Eqs. 9 and 10. This was reported in the literature (1), as 1- φ ) (11) Q = 2Λ2η0V(1 + ----2 1 where Λ2 =α2 +β 2 +γ2, φ1 =4/3πr3/V, and V =4π/3R3. The single-sphere Extension to system of many particles case in the preceding study is extended to the dispersion composed of a liquid and many particles placed at points (xi, yi, zi), where i denotes the number of particles. It is considered from the marked distance dependency obtained from Eqs. 9 and 10 that the additional effect of the particles on energy dissipation mainly occurs around the particle standing still with the center placed at (x0, y0, z0). Furthermore, the small effects on Eq. 11 due to the flow around the dispersed particles are disregarded, because the particles, except the particle at (x0, y0, z0), are move with the stream. The expression given by Eq. 11 is realized for each center of the other particles with an appropriate choice of coordinates. The number of the dispersed particles in the volume V is assumed to be so large that the dispersion can be regarded as a liquid of a smooth structure consisting of many particles. First, we determine the total dissipation energy in the volume V in the presence of dispersed particles having a number n in unit volume, on summing the above results in the single-particle case. The dissipation energy in the volume V, in which the volume fraction of the particles is φ1 ×V ×n, is therefore given by 1 Qn = 2Λ2η0V(1 + ------ φ) 2
(12)
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where the suffix n in Qn denotes the case with the concentration n. On the other hand, we regard the dispersion including many particles as a liquid of a smooth structure with the viscosity η*. The energy consumed in the volume V of the liquid under the same flow is calculated using the results in the preceding treatment. This energy is equivalent to the above dissipation energy obtained by summing the results from the particles. Equating both dissipation energies consumed in the volume V, we have the viscosity equation of Eq. 1 for the relative viscosity ηr =η*/η0. The procedure and condition in deriving the equation are briefly examined in the following paragraph. For the dilute dispersion of many particles, the velocity components u, v, and w of the flow affected by the presence of the particles at a point (x, y, z) are expressed as a linear superimposition of the effects of the particles. That is, on omitting the terms of small values from u, v, and w, u = αx − ∑ ui
(13)
ξ i (αξ i 2 + βη i 2 + γζ i 2 ) 5- ------r 3- -------------------------------------------------ui = ----2 ρi 2 ρi3
(14)
Here,
where ξi =x−xi, ηi =y− yi, ζi = z−zi, and ρi = ξ i + η i + ζ i . The expressions for v and w are given by analogy. Derivation of viscosity equation for dilute dispersion of small spheres When we consider the dispersion as a liquid of a smooth structure, the energy Qn* consumed in the volume V is Qn* = 2Λ*2η*V
(15)
where η* is the dispersion viscosity and Λ* =α* +β* +γ*2, in which α* is given by 2
2
2
ui xi ∂ui ∂u- = α + --------α* = -------(16) ∑ ∂xi-x = 0= α + n R---------ρi - ds ∂x x = 0 In the expression, the sum is replaced by the integral under the condition of consistently irregular distribution of dispersing particles. The last term of Eq. 16 is according to the Gauss’s divergence law in the vector analysis. After integrating the term inserting ui for x= y=z =0 in Eq. 14, the equation becomes α*= α(1 −φ). Similarly, β* and γ* are given by β* =β(1 − φ) and γ*= γ(1 −φ). With the above values for α*, β*, and γ*, Eq. 15 is expressed as
∫
Qn* = 2Λ2η*V(1 − φ)2
(17)
Equating Qn of Eq. 12 with Qn* of Eq. 17, we have Eq. 1 for the relative viscosity ηr at the first order-term of φ, as a viscosity equation of dilute dispersion.
RESULTS AND DISCUSSION The volume fraction of suspended spherical particles and dispersant viscosity only functioned as the viscosity factors in the above-derived equation of the dilute dispersion, although the derivation of the equation required the calculation concerning the complicated flow caused by the suspended spherical particles (cf. Eqs. 13 and 14). Furthermore, the fundamental characteristics that depends on the distance of the flow pattern used in the calculation will not be associated noticeably with the shape of the dispersing particles, because the velocity components of the flow might be expressed by the linear superimposition of solutions derived from the hydrodynamic equation for the flow around the particles, irrespective of the particle shape. It is surmised from the concept that the viscosity of dispersion depends mainly on the size or the void volume of suspended par-
ticles, irrespective of the shape of particles, and apparent spherical models for actual solutes or particles can be realized at any concentration in the treatment. The viscosity equation derived above for dilute dispersion is extended to an equation applicable to the dispersion at a high concentration. The derived viscosity equation of the concentrated dispersion is applied to the aqueous solutions of glucose and sucrose and to a suspension of bakers’ yeast, which is nearly spherical having a radius of about 5 µ. Derivation of viscosity equation for concentrated dispersions The flow pattern derived previously, which is expressed by Eq. 9 for the single-sphere case, is assumed to be applicable even for concentrated dispersions including many particles, because the influence of the other particles on the flow is small owing to the canceling of the effects of the particles with each other from the symmetrical property of the flow (6). Thus, it is suggested that the dissipation energy in concentrated dispersion is given as extended forms of Eqs. 12 and 17 for the dilute case. In the previous treatment of the dissipation energy in the dilute case, the portion of the volume occupied by the particles has been taken as a part of the liquid. Because energy dissipation is actually cannot occur at each particle portion, the excess energy calculated should be subtracted from Eqs. 12 and 17, particularly for high concentrations. When the volume occupied by dispersed particles are regarded as a part of the liquid, the dissipation energy in the volume V is 2Λ2η0V as determined using Eq. 12. The volume occupied by the particles is Vφ. Then, the energy in the volume is approximately 2Λ2η0Vφ, where the small addition due to the flow around the particles to the value is neglected. We subtract this value from Qn of Eq. 12, which results in 1 Qc = 2Λ2η0V(1 − ------- φ) (18) 2 where the suffix c in Qc indicates the correction in energy calculation. On the other hand, in the case when we regard the dispersion as a liquid of a smooth structure of the viscosity η*, the dissipation energy in the volume V is given by Eq. 17. It is conceded as an extension of the dilute case that, on considering the above-mentioned symmetrical property of the flow, the velocity components are expressed with the forms of Eqs. 13 and 14. The flow is given by the sum of the effects of the particles, and is used in Eq. 16 which is required for the calculation of Eq. 15. However, in the treatment of concentrated dispersion it must be considered that the calculation of Eq. 16 provides no realistic value and what disappears at each particle portion. That is, energy dissipation cannot actually occur at the particle portion, similarly to the previous calculation. Subtracting the excess energy 2Λ*2η*Vφ in the volume Vφ from Eq. 15, we refine the value Qn* of Eq. 17 to Qc* = 2Λ2(1 − φ)3η*V
(19)
Equating Qc of Eq. 18 with Qc* of Eq. 19, we have a viscosity equation of the concentrated dispersion, which is, for the relative viscosity ηr,
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1 – 0.5φ ηr = ---------------------(20) ( 1 – φ )3 This equation includes Eq. 1 at the first order approximation of φ. Viscosity equation for dispersion of large particles In the case when the dispersed particles are large such as yeast of about 5 µ radius in suspension in the present case, Eqs. 13 and 14 expressing the flow affected by the presence of particles in concentrated dispersion could not necessarily be adopted for the flow. Equation 13 is expressed as, with Eq. 14 modified with a correction factor (κ) that may depend on the size and concentration of the particles, 2 2 2 5 r3 ξi (αξ i + βηi + γζ i ) u = α x − ∑ κ ----(21) - -----------------------------------------------------2 ρi2 ρi3 The expressions for v and w are given by analogy. When we merge the value κ into the radius r in Eq. 21, the above modification on the velocity components means that the effect of the perturbed flow caused by the presence of large particles reflects the apparent increase in particle size, which is κ1/3r for κ > 1. The previous dissipation energy given by Qc of Eq. 18 is replaced by
FIG. 1. Concentration dependencies of viscosity in aqueous solutions of glucose (circles) and sucrose (triangles). The ordinate and abscissa represent the relative viscosity of the solutions and the volume fractions of the solutes. The solid curve expresses the calculations using Eq. 20.
1Qcm = 2Λ2η0V(1 + -----κφ − φ) (22) 2 where the suffix m in Qcm indicates the value calculated with the modified flow of Eq. 21. On the other hand, Qc* of Eq. 19 for concentrated dispersion regarded as a smooth structure is replaced by Qcm* = 2Λ2(1 − κφ)2(1 − φ)η*V
(23)
Equating Qcm of Eq. 22 with Qcm* of Eq. 23, we have a modified viscosity equation for the concentrated dispersion of large particles as, for the relative viscosity, 1 + 0.5κφ – φ ηr = ---------------------------------------(24) (1 – κφ )2 (1 – φ ) As suggested from Eq. 21, this equation shifts to Eq. 20 at κ= 1 in which the size effect caused by the dispersed particles on the flow is neglected. The applicability of the derived equations to measurements of viscosity for actual solutions and suspensions was examined with regard to the viscosity-concentration relationship. Comparison with measurements of concentrated aqueous solutions of glucose and sucrose Figure 1 shows the concentration dependencies of the relative viscosity in the aqueous solutions of glucose and sucrose. The ordinate and abscissa represent respectively the relative viscosity and the volume fraction of the solute molecules. In Fig. 1, the open symbols denote the measurements of the glucose (circles) and sucrose (triangles) solutions, respectively. The concentration dependencies of the relative viscosity in the solutions of glucose and sucrose show almost the same tendency. The solid line expresses the calculations using Eq. 20. The viscosity equation derived by the theoretical treatment in the case of concentrated dispersion gave a good agreement with the tendency in the measurements.
FIG. 2. Concentration dependencies of viscosity in the suspensions of yeast (open) and spherical particles (filled). The ordinate and abscissa represent the relative viscosity of the suspensions and the volume fractions of the yeast or the particles. The solid and dotted curves respectively express the calculations using Eqs. 20 and 24 in which κ= 1+ 0.6φ.
Comparison with measurements of concentrated suspensions of yeast Figure 2 shows the concentration dependencies of the relative viscosity in the yeast suspensions. The ordinate and abscissa represent the relative viscosity, which is the viscosity ratio of suspensions to water, and the volume fraction of the suspended yeast, respectively. In Fig. 2, the open circles denote the measurements of the yeast suspensions. The filled small circles quoted from a reference denote the data of the suspensions of spherical particles with sizes on the order of micrometers, such as polystylene and glass beads. These data were reduced by Thomas with minimizing secondary effects such as non-Newtonian
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behavior, inertial forces, and instrument wall effects (9). The concentration dependencies of the relative viscosity in the suspensions of yeast and spherical particles showed similar tendencies. However, the tendency deviated considerably in upper regions of the gradients in the glucose and sucrose solutions, that is, in the high-concentration region except the concentration of φ < 0.05. The measurement trend in the suspensions of the yeast and the spherical particles was simulated using the modified viscosity equation derived theoretically. In Fig. 2, the dotted line expresses the calculations using Eq. 24, in which κ was given by κ = 1 + 0.6φ
(25)
The viscosity-concentration dependence in the suspensions of yeast and spherical particles, whose sizes were on the order of micrometers, was expressed well up to the highconcentration region using the modified viscosity of Eq. 24 with the correction factor of Eq. 25, although the correlation was unnecessary for the solution viscosity of glucose and sucrose molecules. The equation expressing the factor depended on the concentration given by the volume fraction φ of the suspended large particles. The viscosity equations derived from the several modifications simulated the concentration dependencies in the dispersions of various solutes and particles. It is confirmed from the good applicability that the derivation process carried out to extend the viscosity equation of the dilute disper-
sion of small spheres to that useful for the concentrated dispersions of actual solutes and particles is reasonable. REFERENCES 1. Einstein, A.: Eine neue Bestimmung der Moleküldimensionen. Annalen der Physik, 19, 289-306 (1906). Correction, ibid., 34, 591–592 (1911). A new determination of molecular dimensions, p. 36–54. In Fürth, R. (ed.) and Cowper, A. D. (tr.), Investigation on the theory of the Brownian movement. Dover Publications, USA (1956). 2. Rutgers, R.: Relative viscosity and concentration. Rheologica Acta, 2, 305–348 (1962). 3. Batchelor, G. K.: Transport properties of two-phase materials with random structure. Annu. Rev. Fluid Mech., 6, 227–255 (1974). 4. Jeffrey, D. J. and Acrivos, A.: The rheological properties of suspensions of rigid particles. AIChE J., 22, 417–432 (1976). 5. Furuse, H. and Aiba, S.: Viscosity of concentrated suspension of spherical particles. Kagaku Kogaku Ronbunshu, 6, 109–111 (1980). 6. Pokrovski, V. N.: Refinement of the results of the theory of suspension viscosity. Soviet Physics JETP, 28, 339–340 (1969). 7. Conway, E. J. and Downey, M.: An outer metabolic region of the yeast cell. Biochem. J., 47, 347–355 (1950). 8. Toda, K. and Aiba, S.: Measurement of microbial cell density. Hakkokogaku, 44, 431–436 (1966). 9. Thomas, D. G.: A note on the viscosity of Newtonian suspensions of uniform spherical particles. J. Colloid Sci., 20, 267–277 (1965).
© 2006, The Society for Biotechnology, Japan
Extension of Einstein’s Viscosity Equation to That for Concentrated Dispersions of Solutes and Particles Kiyoshi Toda1 and Hisamoto Furuse1* Institute of Molecular and Cellular Biosciences, The University of Tokyo, 1-1-1 Yayoi, Bunkyo-ku, Tokyo 113-0032, Japan1 Received 27 July 2006/Accepted 5 September 2006
A viscosity equation for concentrated solutions or suspensions is derived as an extension of Einstein’s hydrodynamic viscosity theory for dilute dispersions of spherical particles. The derivation of the equation is based on the calculation of dissipation of mechanical energy into heat in the dispersion, subtracting the energy dissipation in the portion of solutes or particles. The viscosity equation derived thus was well fitted to the viscosity-concentration relationship of the concentrated aqueous solutions of glucose and sucrose. For the suspensions of bakers’ yeast, the concentration dependency of viscosity was expressed well with some modification for the flow pattern around suspended particles. It is suggested that these viscosity equations can be widely applied to both diluted and concentrated dispersions of various solutes and particles. [Key words: viscosity equation, concentrated dispersion, dissipation energy, bakers’ yeast]
having a nearly spherical shape. The viscosity equation was finely applicable to the viscosity of the aqueous solutions of glucose and sucrose. On the other hand, the suspension viscosity of the yeast with a large particle size on the order of micrometers deviated, in the region of high concentration, which was considerably higher than that obtained using the equation. The viscosity equation derived by modifying the flow pattern around suspended particles enabled the simulation of the tendency in the measurement of suspension viscosity.
In a study carried out 100 years ago, Einstein has derived the well known viscosity equation on the basis of a hydrodynamic theory for dilute dispersions of spherical particles (1). Although there have been many trials that simulate the concentration dependencies of viscosity in solutions and suspensions, satisfactory discussions that may elucidate the mechanism underlying this dependency using an extension of the theory have not been offered up to the present, even for the concentrated dispersion of a simple structure of monodispersed spherical particles (2–4). In our previous study, we extended the theoretical treatment to satisfy the viscosity behavior of concentrated dispersion (5). The applicability of the theoretical treatment and derived viscosity equation to actual dispersions of solutes and particles is confirmed in this study with more detail procedures and discussions. The viscosity theory of dilute dispersion is based on the calculation of energy dissipated by suspended spherical particles in liquids. The energy dissipation in the treatment is assumed to be caused even in the portion occupied by the particles. This is permissible only in the case of dilute dispersion. For concentrated dispersions, dissipation energy calculated is overestimated, because dissipation actually cannot occur in the portion occupied by particles. This excess energy must be subtracted from the total dissipation energy in dilute dispersion obtained through a theoretical procedure. The derived viscosity equation for concentrated dispersion was examined with regard to the concentration dependencies of viscosity in the aqueous solutions of sucrose and glucose and in the suspensions of particles of bakers’ yeast
MATERIALS AND METHODS Viscosity theory and experiment The theoretical treatment in this study is based on Einstein’s viscosity theory for dilute dispersions of small spheres (1). The viscosity equation of the dilute dispersion has been derived as ηr = 1 + 2.5φ
(1)
where, ηr is the relative viscosity given as the viscosity ratio of the dispersion to a liquid of dispersant, and φ is the volume fraction of the suspended solutes or particles assumed to be spherical. The derived viscosity equations of concentrated dispersions were examined using the aqueous solutions of glucose and sucrose, and of the suspensions of bakers’ yeast obtained by cultivation, which were compared with the data for spherical particles of polystylene and others cited in the literature (9). Viscosity in the present experiment was measured using a capillary viscometer of the Ostwald type at 20°C. Measurement of volume fractions of solute The volume fractions of glucose and sucrose molecules in concentrated solutions were obtained by multiplying the volume fractions in dilute solutions by dilution ratios. The values of the dilute solutions were derived inversely from the measurement of viscosity using Eq. 1 when the result is φ <0.05. This method of obtaining the volume of the solute molecules in water gives the hydrodynamic molecular
* Corresponding author. e-mail: [email protected] phone/fax: +81-(0)467-52-6267 524
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volume of water molecules attaching to the solute surfaces, on assuming the applicability of Eq. 1 to the dilute solutions. Measurement of volume fractions of yeast The volume fractions of the yeast in concentrated suspensions were obtained by measuring the concentrations of polymers that were diluted after mixing the aqueous solution of a polymer with the suspension composed of yeast and water (7, 8). For the aqueous solution in the present experiment, a polymer of polyvinyl-pyrrolidone was dissolved in water (5). Using the solution of a polymer in this method to obtain yeast volume enables to minimize the osmotic pressure effect derived from dissolved molecules on yeast cells. In this study, the procedure of derivation of Eq. 1 is first surveyed to understand its theoretical background. Flow around one spherical particle First in the derivation of Eq. 1, the flow pattern of the liquid around a single spherical particle is determined by solving the hydrodynamic equation of Navier–Stokes. With the solution derived from the equation for the flow we examine the increase in dissipation energy caused by the presence of the particle in the liquid. As a premise, the liquid of a dispersant with a viscosity η0 is incompressible and the small inertial terms concerning the liquid and the particle are disregarded in the treatment using the hydrodynamic equation. The velocity of the flow in a small domain, in the absence of the particle, is expressed as a linear form, in which u0, v0, and w0 denote the components of the velocity. For an appropriate choice of the coordinates, the velocity components at a point (x, y, z), which is some distance apart from a point (x0, y0, z0), are expressed by u0 = α(x − x0) = αξ v0 = β(y − y0) = βη w0 = γ(z − z0) = γζ
}
(2)
From the assumption of the incompressibility of the liquid, the following condition is realized. α+β+γ=0
(3)
In the presence of a spherical particle of the radius r, the center of which is placed at the point (x0, y0, z0), the velocity components around the particle are expressed as u = αξ + u′ v = βη + v′ w = γζ + w′
}
(4)
where u′, v′, and w′ are the additional terms of the velocity due to the presence of the particle. It follows from the contact of molecules that the velocity of liquid molecules agrees with that of the particle on its surface. In addition, from the symmetry of the flow, the spherical particle stands still. These conditions provide a boundary condition which is u= v=w= 0 on the surface ρ = ξ 2 + η 2 + ζ 2 = r, where ρ denotes the distance from the point (x0, y0, z0). Furthermore, the flow in a position far from the particle approaches the flow obtained using Eq. 2, that is, u′= v′= w′= 0 when ρ =∞. The hydrodynamic equation of Navier–Stokes is given by ∂p ∂p ∂p (5) --------- = η0∆u, --------- = η0 ∆v, -------- = η0 ∆w ∂ξ ∂η ∂ζ where p is the hydrostatic pressure, and the operator ∆ is expressed as 2 2 2 ∂ - --------∂ ∂ ∆ = ---------+ ---------+ 2 2 2 ∂ξ ∂η ∂ζ Einstein obtained the solution of the hydrodynamic equation under the boundary conditions.
525
5 r3 u = αξ − ------ ------5- ξ(αξ 2 + βη2 + γζ 2) 2 ρ 5- -----r 5- ξ(αξ 2 + βη2 + γζ 2) − -----r 5- αξ + ----7 2 ρ ρ5
(6)
The expressions for v and w are given by analogy. Moreover, for the pressure p,
{
}
2 1 2 1 2 1 ∂ ---∂ ---∂ --- ρ ρ ρ 5 3 - + β --------------- + γ --------------- +C p = – ------η0r α --------------2 2 2 3 ∂ξ ∂η ∂ζ
(7)
where C is a constant of the pressure. Calculation of dissipation energy The energy transformed into heat per unit time within the region of the spherical volume V of the radius R, which is indefinitely larger than the particle radius r, is calculated by the above solution. The energy Q to be calculated is given by the integration extended over the surface of the sphere. That is,
∫
Q = (Unu + Vnv + Wnw)ds
(8)
where Un, Vn, and Wn are the components of the pressure exerted on the surface of the sphere. The first two terms on the right-hand side of Eq. 6 are used in the following calculation, because the other terms vanish for ρ=R assumed for a very large size compared with r. That is, 5- -----r3- ξ(αξ 2 + βη2 + γζ 2) u = αξ − ----(9) 2 ρ5 The analogous omission is carried out similarly for v and w, and for p, α ξ 2 + βη2 + γζ 2 p = −5η0r3 ---------------------------------------+C (10) ρ5 The dissipation energy given by Eq. 8, consumed in the volume V, is derived by calculation using Eqs. 9 and 10. This was reported in the literature (1), as 1- φ ) (11) Q = 2Λ2η0V(1 + ----2 1 where Λ2 =α2 +β 2 +γ2, φ1 =4/3πr3/V, and V =4π/3R3. The single-sphere Extension to system of many particles case in the preceding study is extended to the dispersion composed of a liquid and many particles placed at points (xi, yi, zi), where i denotes the number of particles. It is considered from the marked distance dependency obtained from Eqs. 9 and 10 that the additional effect of the particles on energy dissipation mainly occurs around the particle standing still with the center placed at (x0, y0, z0). Furthermore, the small effects on Eq. 11 due to the flow around the dispersed particles are disregarded, because the particles, except the particle at (x0, y0, z0), are move with the stream. The expression given by Eq. 11 is realized for each center of the other particles with an appropriate choice of coordinates. The number of the dispersed particles in the volume V is assumed to be so large that the dispersion can be regarded as a liquid of a smooth structure consisting of many particles. First, we determine the total dissipation energy in the volume V in the presence of dispersed particles having a number n in unit volume, on summing the above results in the single-particle case. The dissipation energy in the volume V, in which the volume fraction of the particles is φ1 ×V ×n, is therefore given by 1 Qn = 2Λ2η0V(1 + ------ φ) 2
(12)
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where the suffix n in Qn denotes the case with the concentration n. On the other hand, we regard the dispersion including many particles as a liquid of a smooth structure with the viscosity η*. The energy consumed in the volume V of the liquid under the same flow is calculated using the results in the preceding treatment. This energy is equivalent to the above dissipation energy obtained by summing the results from the particles. Equating both dissipation energies consumed in the volume V, we have the viscosity equation of Eq. 1 for the relative viscosity ηr =η*/η0. The procedure and condition in deriving the equation are briefly examined in the following paragraph. For the dilute dispersion of many particles, the velocity components u, v, and w of the flow affected by the presence of the particles at a point (x, y, z) are expressed as a linear superimposition of the effects of the particles. That is, on omitting the terms of small values from u, v, and w, u = αx − ∑ ui
(13)
ξ i (αξ i 2 + βη i 2 + γζ i 2 ) 5- ------r 3- -------------------------------------------------ui = ----2 ρi 2 ρi3
(14)
Here,
where ξi =x−xi, ηi =y− yi, ζi = z−zi, and ρi = ξ i + η i + ζ i . The expressions for v and w are given by analogy. Derivation of viscosity equation for dilute dispersion of small spheres When we consider the dispersion as a liquid of a smooth structure, the energy Qn* consumed in the volume V is Qn* = 2Λ*2η*V
(15)
where η* is the dispersion viscosity and Λ* =α* +β* +γ*2, in which α* is given by 2
2
2
ui xi ∂ui ∂u- = α + --------α* = -------(16) ∑ ∂xi-x = 0= α + n R---------ρi - ds ∂x x = 0 In the expression, the sum is replaced by the integral under the condition of consistently irregular distribution of dispersing particles. The last term of Eq. 16 is according to the Gauss’s divergence law in the vector analysis. After integrating the term inserting ui for x= y=z =0 in Eq. 14, the equation becomes α*= α(1 −φ). Similarly, β* and γ* are given by β* =β(1 − φ) and γ*= γ(1 −φ). With the above values for α*, β*, and γ*, Eq. 15 is expressed as
∫
Qn* = 2Λ2η*V(1 − φ)2
(17)
Equating Qn of Eq. 12 with Qn* of Eq. 17, we have Eq. 1 for the relative viscosity ηr at the first order-term of φ, as a viscosity equation of dilute dispersion.
RESULTS AND DISCUSSION The volume fraction of suspended spherical particles and dispersant viscosity only functioned as the viscosity factors in the above-derived equation of the dilute dispersion, although the derivation of the equation required the calculation concerning the complicated flow caused by the suspended spherical particles (cf. Eqs. 13 and 14). Furthermore, the fundamental characteristics that depends on the distance of the flow pattern used in the calculation will not be associated noticeably with the shape of the dispersing particles, because the velocity components of the flow might be expressed by the linear superimposition of solutions derived from the hydrodynamic equation for the flow around the particles, irrespective of the particle shape. It is surmised from the concept that the viscosity of dispersion depends mainly on the size or the void volume of suspended par-
ticles, irrespective of the shape of particles, and apparent spherical models for actual solutes or particles can be realized at any concentration in the treatment. The viscosity equation derived above for dilute dispersion is extended to an equation applicable to the dispersion at a high concentration. The derived viscosity equation of the concentrated dispersion is applied to the aqueous solutions of glucose and sucrose and to a suspension of bakers’ yeast, which is nearly spherical having a radius of about 5 µ. Derivation of viscosity equation for concentrated dispersions The flow pattern derived previously, which is expressed by Eq. 9 for the single-sphere case, is assumed to be applicable even for concentrated dispersions including many particles, because the influence of the other particles on the flow is small owing to the canceling of the effects of the particles with each other from the symmetrical property of the flow (6). Thus, it is suggested that the dissipation energy in concentrated dispersion is given as extended forms of Eqs. 12 and 17 for the dilute case. In the previous treatment of the dissipation energy in the dilute case, the portion of the volume occupied by the particles has been taken as a part of the liquid. Because energy dissipation is actually cannot occur at each particle portion, the excess energy calculated should be subtracted from Eqs. 12 and 17, particularly for high concentrations. When the volume occupied by dispersed particles are regarded as a part of the liquid, the dissipation energy in the volume V is 2Λ2η0V as determined using Eq. 12. The volume occupied by the particles is Vφ. Then, the energy in the volume is approximately 2Λ2η0Vφ, where the small addition due to the flow around the particles to the value is neglected. We subtract this value from Qn of Eq. 12, which results in 1 Qc = 2Λ2η0V(1 − ------- φ) (18) 2 where the suffix c in Qc indicates the correction in energy calculation. On the other hand, in the case when we regard the dispersion as a liquid of a smooth structure of the viscosity η*, the dissipation energy in the volume V is given by Eq. 17. It is conceded as an extension of the dilute case that, on considering the above-mentioned symmetrical property of the flow, the velocity components are expressed with the forms of Eqs. 13 and 14. The flow is given by the sum of the effects of the particles, and is used in Eq. 16 which is required for the calculation of Eq. 15. However, in the treatment of concentrated dispersion it must be considered that the calculation of Eq. 16 provides no realistic value and what disappears at each particle portion. That is, energy dissipation cannot actually occur at the particle portion, similarly to the previous calculation. Subtracting the excess energy 2Λ*2η*Vφ in the volume Vφ from Eq. 15, we refine the value Qn* of Eq. 17 to Qc* = 2Λ2(1 − φ)3η*V
(19)
Equating Qc of Eq. 18 with Qc* of Eq. 19, we have a viscosity equation of the concentrated dispersion, which is, for the relative viscosity ηr,
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1 – 0.5φ ηr = ---------------------(20) ( 1 – φ )3 This equation includes Eq. 1 at the first order approximation of φ. Viscosity equation for dispersion of large particles In the case when the dispersed particles are large such as yeast of about 5 µ radius in suspension in the present case, Eqs. 13 and 14 expressing the flow affected by the presence of particles in concentrated dispersion could not necessarily be adopted for the flow. Equation 13 is expressed as, with Eq. 14 modified with a correction factor (κ) that may depend on the size and concentration of the particles, 2 2 2 5 r3 ξi (αξ i + βηi + γζ i ) u = α x − ∑ κ ----(21) - -----------------------------------------------------2 ρi2 ρi3 The expressions for v and w are given by analogy. When we merge the value κ into the radius r in Eq. 21, the above modification on the velocity components means that the effect of the perturbed flow caused by the presence of large particles reflects the apparent increase in particle size, which is κ1/3r for κ > 1. The previous dissipation energy given by Qc of Eq. 18 is replaced by
FIG. 1. Concentration dependencies of viscosity in aqueous solutions of glucose (circles) and sucrose (triangles). The ordinate and abscissa represent the relative viscosity of the solutions and the volume fractions of the solutes. The solid curve expresses the calculations using Eq. 20.
1Qcm = 2Λ2η0V(1 + -----κφ − φ) (22) 2 where the suffix m in Qcm indicates the value calculated with the modified flow of Eq. 21. On the other hand, Qc* of Eq. 19 for concentrated dispersion regarded as a smooth structure is replaced by Qcm* = 2Λ2(1 − κφ)2(1 − φ)η*V
(23)
Equating Qcm of Eq. 22 with Qcm* of Eq. 23, we have a modified viscosity equation for the concentrated dispersion of large particles as, for the relative viscosity, 1 + 0.5κφ – φ ηr = ---------------------------------------(24) (1 – κφ )2 (1 – φ ) As suggested from Eq. 21, this equation shifts to Eq. 20 at κ= 1 in which the size effect caused by the dispersed particles on the flow is neglected. The applicability of the derived equations to measurements of viscosity for actual solutions and suspensions was examined with regard to the viscosity-concentration relationship. Comparison with measurements of concentrated aqueous solutions of glucose and sucrose Figure 1 shows the concentration dependencies of the relative viscosity in the aqueous solutions of glucose and sucrose. The ordinate and abscissa represent respectively the relative viscosity and the volume fraction of the solute molecules. In Fig. 1, the open symbols denote the measurements of the glucose (circles) and sucrose (triangles) solutions, respectively. The concentration dependencies of the relative viscosity in the solutions of glucose and sucrose show almost the same tendency. The solid line expresses the calculations using Eq. 20. The viscosity equation derived by the theoretical treatment in the case of concentrated dispersion gave a good agreement with the tendency in the measurements.
FIG. 2. Concentration dependencies of viscosity in the suspensions of yeast (open) and spherical particles (filled). The ordinate and abscissa represent the relative viscosity of the suspensions and the volume fractions of the yeast or the particles. The solid and dotted curves respectively express the calculations using Eqs. 20 and 24 in which κ= 1+ 0.6φ.
Comparison with measurements of concentrated suspensions of yeast Figure 2 shows the concentration dependencies of the relative viscosity in the yeast suspensions. The ordinate and abscissa represent the relative viscosity, which is the viscosity ratio of suspensions to water, and the volume fraction of the suspended yeast, respectively. In Fig. 2, the open circles denote the measurements of the yeast suspensions. The filled small circles quoted from a reference denote the data of the suspensions of spherical particles with sizes on the order of micrometers, such as polystylene and glass beads. These data were reduced by Thomas with minimizing secondary effects such as non-Newtonian
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behavior, inertial forces, and instrument wall effects (9). The concentration dependencies of the relative viscosity in the suspensions of yeast and spherical particles showed similar tendencies. However, the tendency deviated considerably in upper regions of the gradients in the glucose and sucrose solutions, that is, in the high-concentration region except the concentration of φ < 0.05. The measurement trend in the suspensions of the yeast and the spherical particles was simulated using the modified viscosity equation derived theoretically. In Fig. 2, the dotted line expresses the calculations using Eq. 24, in which κ was given by κ = 1 + 0.6φ
(25)
The viscosity-concentration dependence in the suspensions of yeast and spherical particles, whose sizes were on the order of micrometers, was expressed well up to the highconcentration region using the modified viscosity of Eq. 24 with the correction factor of Eq. 25, although the correlation was unnecessary for the solution viscosity of glucose and sucrose molecules. The equation expressing the factor depended on the concentration given by the volume fraction φ of the suspended large particles. The viscosity equations derived from the several modifications simulated the concentration dependencies in the dispersions of various solutes and particles. It is confirmed from the good applicability that the derivation process carried out to extend the viscosity equation of the dilute disper-
sion of small spheres to that useful for the concentrated dispersions of actual solutes and particles is reasonable. REFERENCES 1. Einstein, A.: Eine neue Bestimmung der Moleküldimensionen. Annalen der Physik, 19, 289-306 (1906). Correction, ibid., 34, 591–592 (1911). A new determination of molecular dimensions, p. 36–54. In Fürth, R. (ed.) and Cowper, A. D. (tr.), Investigation on the theory of the Brownian movement. Dover Publications, USA (1956). 2. Rutgers, R.: Relative viscosity and concentration. Rheologica Acta, 2, 305–348 (1962). 3. Batchelor, G. K.: Transport properties of two-phase materials with random structure. Annu. Rev. Fluid Mech., 6, 227–255 (1974). 4. Jeffrey, D. J. and Acrivos, A.: The rheological properties of suspensions of rigid particles. AIChE J., 22, 417–432 (1976). 5. Furuse, H. and Aiba, S.: Viscosity of concentrated suspension of spherical particles. Kagaku Kogaku Ronbunshu, 6, 109–111 (1980). 6. Pokrovski, V. N.: Refinement of the results of the theory of suspension viscosity. Soviet Physics JETP, 28, 339–340 (1969). 7. Conway, E. J. and Downey, M.: An outer metabolic region of the yeast cell. Biochem. J., 47, 347–355 (1950). 8. Toda, K. and Aiba, S.: Measurement of microbial cell density. Hakkokogaku, 44, 431–436 (1966). 9. Thomas, D. G.: A note on the viscosity of Newtonian suspensions of uniform spherical particles. J. Colloid Sci., 20, 267–277 (1965).