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Transport properties of fractal aggregates calculated by permeability
Colloids and Surfaces A: Physicochem. Eng. Aspects 215 (2003) 173 /179 www.elsevier.com/locate/colsurfa
Transport properties of fractal aggregates calculated by permeability Lech Gmachowski Institute of Physical Chemistry, Polish Academy of Sciences, 01-224 Warsaw, Poland Received 20 June 2002; accepted 26 September 2002
Abstract The Debye /Brinkman concept of treating polymer coils as uniformly permeable spheres is utilized for embryonic aggregates composed of less than a dozen or so primary particles. These can be the direct models for bioparticles or may be equivalent to multi-particle aggregates and macromolecules. The method is based on the calculation of permeability of the system contained inside the sphere circumscribed on the aggregate, which makes it possible to estimate the transport properties such as the translational and rotational diffusion coefficients and the intrinsic viscosity of bead models. With this method it is possible to describe properly the translational diffusion coefficient for the embryonic aggregates analyzed as compared to that deduced from the experimental values of sedimentation velocity. The results obtained for rotational diffusion coefficient and intrinsic viscosity of oligomers are close to that calculated by the Kirkwood /Riseman method using modified Oseen tensor with volume correction. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Oligomers; Permeability; Intrinsic viscosity; Hydrodynamic radius; Diffusion coefficient
1. Introduction Transport properties of fractal aggregates and polymer coils, which are termed as solution or hydrodynamic properties in the field of polymer solutions, are of great interest in colloid and polymer science, biophysics and engineering. Macromolecular coils can be represented by fractal aggregates composed of blobs equivalent to nonporous solid particles [1]. The hydrodynamic properties of oligomeric proteins consisting of several subunits with a polygonal or polyhedral
E-mail address: [email protected] (L. Gmachowski).
geometry can be represented by bead models composed of spherical elements, introduced by Bloomfield et al. [2,3]. Bead models are useful when simple geometric models, such as spheres, ellipsoids or cylinders are inadequate due to specific way in which the subunits are arranged. The method proposed by Kirkwood and Riseman [4], based on the Oseen hydrodynamic interaction tensor between beads, makes it possible to calculate such properties as sedimentation coefficient, translational and rotational diffusion coefficients, as well as intrinsic viscosity. Based on that originally proposed, the method of the determination of hydrodynamic properties is improved to take into account the finite volume of beads, as
0927-7757/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 7 5 7 ( 0 2 ) 0 0 4 4 0 - 5
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Nomenclature a primary particle radius (m) D fractal dimension (-) fR rotational friction coefficient (J) fT translational friction coefficient (kg s 1) g acceleration of gravity (m s 2) i number of blobs or primary particles in an aggregate (-) k internal permeability of aggregate (m2) kB Bolzmann constant (J K 1) m aggregate mass (kg) r hydrodynamic radius of an aggregate (m) R radius of aggregate (radius of circumscribed sphere) (m) ua Stokes free settling velocity of primary particle (m s 1) ur free settling velocity of an aggregate (m s 1) f volume fraction of impermeable blobs or primary particles in an aggregate (-) h0 fluid viscosity (kg m 1 s 1) [h ] intrinsic viscosity (m3 kg 1) rf fluid density (kg m 3) rs solid density (kg m 3) s reciprocal square root of dimensionless internal permeability of aggregate (-) n viscosity increment (-)
described by Carrasco and Garcı´a de la Torre [5,6]. Nevertheless it is currently broadly accepted approach. Approximately at the same time the porous sphere model was introduced by Debye [7] and Brinkman [8] and developed by Brinkman [9] in which a macromolecular coil was modeled as a sphere of uniform permeability. The possibility to use the permeability concept to describe the sedimentation of macromolecules in solution was verified [10] down to the correlation length being of the order of the size of solvent molecule. The progress of the permeable sphere model was limited, however, owing to the difficulties to calculate the permeability of a macromolecule due to its non-uniform structure, as discussed by van Saarloos [11]. The permeability of an aggregate containing not too many particles, however, is possible to calculate much easier than for a multiparticle fractal aggregate, representing a macromolecule. Previously a method was presented [12] for calculation of the intrinsic viscosity of bead models, based on the fractal dimension depen-
dence of hydrodynamic radius. This dependence is valid for aggregates of any number of beads, but of spherical character. This paper is devoted to the description of a method for the calculation of the translational and rotational diffusion coefficients and the intrinsic viscosity for aggregates of any mass and shape. This is based on the determination of the permeability of the system contained inside the sphere circumscribed on the aggregate of sufficiently reduced number of constituents, which is a model of a given multi-particle aggregate, macromolecule or bioparticle.
2. Model The diffusion coefficient of an isolated aggregate, macromolecule or bioparticle, either translational or rotational, can be calculated by the Einstein relation [13] DT
kB T fT
or
DR
kB T fR
(1)
L. Gmachowski / Colloids and Surfaces A: Physicochem. Eng. Aspects 215 (2003) 173 /179
where kB is the Boltzmann constant and T is the absolute temperature. The translational friction coefficient of a particle can be written as the product of that for an impermeable particle of the same radius R and the normalized hydrodynamic radius r/R , representing the reduction of coefficient due to the internal permeability of the particle fT 6ph0 R
r R
(2)
The hydrodynamic radius r of a permeable particle is the radius of an impermeable sphere of the same mass having the same dynamic properties. Brinkman [9] combined the Stokes equation of motion with the Darcy law to formulate the equation of motion of a fluid in permeable media. He obtained the following formula for a sphere of uniform permeability, which makes it possible to calculate the translational friction coefficient tanh s 1 r s (3) 3 tanh s R 1 1 2s2 s pffiffiffi where s R= k is the reciprocal square root of dimensionless internal permeability of sphere model. Similarly, the expression for the rotational friction coefficient 3 3 r fR 8ph0 R (4) R with 1=3 3 3 1 R s2 stanh s r
(5)
was derived by Felderhof and Deutch [14]. The intrinsic viscosity of dissolved macromolecules and bioparticles or suspended aggregates of mass m is given by the following formula 10 R3 r 3 (6) [h] p 3 m R with
r R
175
3 3 1 2 s stanh s 10 3 3 1 1 s2 s2 stanh s
1=3
(7)
as derived by Brinkman [9]. The knowledge of the value of reciprocal square root of dimensionless internal permeability s of a given particle makes it possible to determine all the transport properties considered. The permeability of a multi-particle fractal aggregate, also if it represents a macromolecule, is not possible to be calculated straightforwardly by standard formulas, which are applicable for systems with uniform distribution of porosity, because of considerable radial variation of porosity due to fractal structure of macromolecules. If an aggregate has a self-similar structure, the fractal dimension D can be determined by covering the object with the sets of spheres of increasing size, as shown by Feder [15]. The number of spheres will decrease as a negative power of their size. In this way fractals may be produced of lower and lower numbers of constituent clusters (blobs), contained in covering spheres. Since the number of clusters diminishes as a negative power of its diameter, the corresponding volume fraction f of clusters in an aggregate is the following function of the number of clusters f8i1 3=D
(8)
The lower value of the number of clusters, the higher value of f and hence the narrower the range of its variability due to the radial dependence. This is the reason why the standard permeability models, applicable for systems with uniform distribution of porosity, are more reliable when applied to aggregates containing not many particles than to multi-particle fractal aggregates. The above-described method makes it possible to reduce a multi-particle aggregate to a structure containing not too many constituents, for which the permeability can be calculated straightforwardly. Replacing clusters by equivalent impermeable spheres, the aggregate of reduced number of constituents can be represented as composed by primary particles. The calculation of permeability
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L. Gmachowski / Colloids and Surfaces A: Physicochem. Eng. Aspects 215 (2003) 173 /179
of fractal aggregates of reduced number of constituents, was reported by Li and Logan [16] and Woodfield and Bickert [17]. The s-parameter calculated, being the reciprocal square root of the dimensionless permeability, should be the same as for the original aggregate due to self-similarity [17]. Its constancy was demonstrated for the macromolecules of different mass of a given polymer dissolved in a given solvent [1]. This parameter, calculated for an aggregate containing not too many constituents, should thus determine the normalized hydrodynamic radius valid for any aggregate of the same fractal dimension. The following formula [18] is employed for further calculations, previously verified for uniform permeable systems over a very wide porosity interval sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 9 25f4=3 s f 1 (9) a 2 3(1 f)3
particle
where the volume fraction f, of i beads of radius a forming aggregate, contained inside the circumscribed sphere of radius R , is calculated as
The values of normalized hydrodynamic radius used to determine the translational friction coefficient were calculated for some embryonic aggregates by Eqs. (3), (9) and (10). The results are presented in Table 1. For the same structures the values of normalized hydrodynamic radius were determined by Eq. (15) on the basis of sedimentation data of (ur/ua)1/2, measured by Sto¨ber et al. [19] and Sto¨ber and Flachsbart [20]. The results are compared in Fig. 1. A good agreement can be observed, which may be regarded as experimental verification of the permeability method for calculation of the translational friction coefficient for embryonic aggregates. The values of the rotational friction coefficient of an oligomer normalized by that of monomer 3 3 fR (i) R r (16) fR (1) a R
i f 3 R
(10)
a The normalized hydrodynamic radius can be determined employing Eqs. (9) and (10) and one of Eq. (3), Eq. (5) or Eq. (7). For the translational friction coefficient the normalized hydrodynamic radius can be deduced from the sedimentation velocity of a model considered. The sedimentation velocity ur of an individual aggregate can be determined by equating the gravitational force allowing for the buoyancy of the surrounding fluid with the opposing hydrodynamic force, as given by the Stokes law 4 3 pa (rs rf )gi 6ph0 rur 3
(11)
with ur
2 a3 i (rs rf )g 9h0 r
(12)
Relating to the Stokes velocity of primary
ua
2 9h0
(rs rf )ga2
(13)
one gets the following equation ur ua
i
(14)
(r=a)
which can be rearranged to determine the normalized hydrodynamic radius r R
i (R=a)(ur =ua )
(15)
This value can be compared to that obtained by Eqs. (3), (9) and (10).
3. Results and discussion
obtained by Garcı´a de la Torre [21] and Garcı´a de la Torre and Carrasco [22] using the method of modified Oseen tensor with the volume correction, were utilized to deduce the values of normalized hydrodynamic radius used to determine the rotational friction coefficient. That was performed for tetrahedron tetramer, octahedron hexamer and cube octamer, indicated in Table 1. The results
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177
Table 1 Values of normalized hydrodynamic radius used to determine the translational and rotational friction coefficients, and values of viscosity increment, all calculated by the permeability method for different embryonic aggregates Oligomeric structure
R /a
s
(r/R)Trans
Linear octamer Linear heptamer Linear hexamer Linear pentamer Linear tetramer Linear trimer Hexagon hexamer Dimer Pentagon pentamer Bipyramid pentamer Square tetramer Octahedron hexamer Triangle trimer Tetrahedron tetramer Pyramid pentamer Compact aggregate of i /9 Trigonal prism hexamer Cube octamer Compact aggregate of i /10 Compact aggregate of i /11 Compact aggregate of i /12 Compact aggregate of i /13
8 7 6 5 4 3 3 2 1/1/sin 368 pffiffiffiffiffiffiffi ffi /12 2=3 pffiffiffi / /1 p2ffiffiffi/ /1 3p / ffiffiffi /12= 3/ffi pffiffiffiffiffiffiffi /1 3=2 pffiffiffi / /1 2/ 3 pffiffiffiffiffiffiffiffi /1 pffiffi7=3 ffi / /1 3/ 3 3 3 3
2.157 2.173 2.201 2.254 2.373 2.711 5.519 4.301 5.928 6.453 6.255 7.465 6.067 8.705 9.071 10.06 10.32 12.40 12.17 14.17 17.74 21.13
0.4662 0.4694 0.4748 0.4849 0.5067 0.5617
(r/R)Rot
13.72
0.8858
6.853 4.700 5.280 3.712 4.901 4.836 4.556 4.959 4.220 4.381
0.9196
4.640 4.716
0.7226
0.8671 0.8077 0.8699 0.8756 0.8888 0.8918
n
0.9094 0.9261 0.9394 0.9504
Aggregates are ranged according to increasing solid volume fraction in model sphere.
Fig. 1. Comparison of normalized hydrodynamic radius used to calculate the translational friction coefficient determined from steady-state falling speed data of Sto¨ber et al. [19] and Sto¨ber and Flachsbart [20] with that calculated using Eqs. (3), (9) and (10), performed for different embryonic aggregates designated in Table 1.
are compared in Fig. 2 to the results calculated with Eqs. (5), (9) and (10). The agreement is good, although the hydrodynamic parameters calculated
Fig. 2. Values of normalized hydrodynamic radius used to determine the rotational friction coefficient, calculated for three bead models designated in Table 1 by the following methods: (m) the permeability method employing Eqs. (5), (9), (10) and (16); (k) using modified Oseen tensor and volume correction, as reported by Garcı´a de la Torre [21] and Garcı´a de la Torre and Carrasco [22].
by the method of modified Oseen tensor with the volume correction rise faster with the compactness of the sphere model. Nevertheless this confirms the reliability of the permeability method of the
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178
rotational friction coefficient calculation for oligomers. The intrinsic viscosity may be expressed as the product of a molecular shape parameter known as the viscosity increment n and the specific volume of primary particles [h] n
(4=3)pa3 i m
Comparing to Eq. (6) one gets 2:5 r 3 R 3 n i R a
(17)
(18)
The values of viscosity increment were calculated for some oligomers by Eqs. (7), (9), (10) and (18). The results are presented in Table 1. They are compared in Fig. 3 with the results obtained by the method of modified Oseen tensor with the volume correction as reported by Garcı´a de la Torre [21], Garcı´a de la Torre and Carrasco [22] and Harding [23]. The agreement is good, except for the linear string hexamer. The s -value for this oligomer (equal to 2.201) lies in the range of s B/3, for which the normalized hydrodynamic radius calculated by Eq. (7) for intrinsic viscosity becomes considerably lower than that determined by Eq. (3) for translational friction coefficient [12]. If one used, however, Eq. (3) to calculate the normalized hydrodynamic radius for this oligomer and then put the result (equal to 0.4748) into Eq. (18), the
Fig. 3. Values of viscosity increment for different bead models designated in Table 1, calculated by the following methods: (m) the permeability method employing Eqs. (7), (9), (10) and (18); (k) using modified Oseen tensor and volume correction, as reported by Garcı´a de la Torre [21], Garcı´a de la Torre and Carrasco [22] and Harding [23].
obtained value of the viscosity increment would be of 9.63, a value almost identical to that obtained using modified Oseen tensor and volume correction. For higher values of s -parameter Eqs. (3) and (7), describing the values of normalized hydrodynamic radius used to calculate the translational friction coefficient and the intrinsic viscosity, respectively, give results, which are practically indistinguishable [12]. Both arguments suggest that Eq. (3) is adequate to calculate the viscosity increment for oligomers. So the permeability method of the intrinsic viscosity calculation for oligomers, employing Eqs. (3), (9), (10) and (18), also seems to be reliable. The proposed method of the determination of transport properties of embryonic aggregates is based on the calculation of permeability of the system contained inside the sphere circumscribed on the bead model. This treatment offers the advantage of its simplicity. Permeability is a quantity, which except for the volume fraction of beads inside the sphere, is sensitive to the size of beads. Therefore it may serve as a reference to results obtained by different methods, especially when the bead models are very compact. For such structures the values of hydrodynamic parameters are overestimated [22] when obtained using modified Oseen tensor and volume correction. This tendency is also visible in Figs. 2 and 3 for the values of solid volume fraction in model sphere greater than 0.3. It suggests that the proposed method gives the values of hydrodynamic parameters more reliable than the method using modified Oseen tensor and volume correction. It has been shown that the method of permeability gives the reliable values of transport coefficients when applied to the embryonic aggregates, which can represent multi-particle fractal aggregates and macromolecular coils. To estimate the degree to which the number of constituents should be reduced, let us compare the value of normalized hydrodynamic radius, calculated for D /2 from the following relation [24] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r D 2 1:56 1:728 0:228 (19) R 2 with those determined for different aggregation
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179
References
Fig. 4. Aggregation number dependence of the normalized hydrodynamic radius (solid line) calculated by permeability method employing Eqs. (3), (9), (10) and (20) compared to the value resulting from the fractal dimension by Eq. (19).
numbers by Eqs. (3), (9) and (10), where the normalized aggregate radii were determined by the mass-radius relation [24]: R i1=D sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a D 2 1:56 1:728 0:228 2
(20)
The results are presented in Fig. 4. As expected, in the range of aggregate containing not many primary particles the calculated values of normalized hydrodynamic radius are close to the value resulting from the fractal dimension. The permeability method seems to be reliable up to the number of constituents of about a hundred.
[1] L. Gmachowski, Polym. J. 22 (1990) 771. [2] V.A. Bloomfield, W.O. Dalton, K.E. van Holde, Biopolymers 5 (1967) 135. [3] V.A. Bloomfield, W.O. Dalton, K.E. van Holde, Biopolymers 5 (1967) 149. [4] J.G. Kirkwood, J. Riseman, J. Chem. Phys. 16 (1948) 565. [5] B. Carrasco, J. Garcı´a de la Torre, Biophys. J. 75 (1999) 3044. [6] B. Carrasco, J. Garcı´a de la Torre, J. Chem. Phys. 111 (1999) 4817. [7] P. Debye, Phys. Rev. 71 (1947) 486. [8] H.C. Brinkman, Physica 13 (1947) 447. [9] H.C. Brinkman, Proc. R. Dutch Acad. Sci. 50 (1947) 618, 821. [10] L. Gmachowski, Polym. J. 29 (1997) 870. [11] W. van Saarloos, Physica 147A (1987) 280. [12] L. Gmachowski, Eur. Biophys. J. 30 (2001) 453. [13] A. Einstein, Investigations on the Theory of the Brownian Movement, Dover Publications, Dover, 1956, pp. 1 /119. [14] B.U. Felderhof, J.M. Deutch, J. Chem. Phys. 62 (1975) 2391. [15] J. Feder, Fractals, Plenum Press, New York, 1988, pp. 14 / 15. [16] X.-Y. Li, B.E. Logan, Wat. Res. 34 (2001) 3373. [17] D. Woodfield, G. Bickert, Wat. Res. 35 (2001) 3801. [18] L. Gmachowski, J. Chem. Eng. Japan 29 (1996) 897. [19] W. Sto¨ber, A. Berner, R. Blaschke, J. Colloid Interface Sci. 29 (1969) 710. [20] W. Sto¨ber, H. Flachsbart, Environ. Sci. Technol. 3 (1969) 1280. [21] J. Garcı´a de la Torre, in: S.E. Harding, A.J. Rowe (Eds.), Dynamic Properties of Biomolecular Assemblies, The Royal Society of Chemistry, Cambridge, 1989, pp. 3 /31. [22] J. Garcı´a de la Torre, B. Carrasco, Eur. Biophys. J. 27 (1998) 549. [23] S.E. Harding, Prog. Biophys. Mol. Biol. 68 (1997) 207. [24] L. Gmachowski, Colloids Surf. A: Physicochem. Eng. Aspects 211 (2000) 197.
Transport properties of fractal aggregates calculated by permeability Lech Gmachowski Institute of Physical Chemistry, Polish Academy of Sciences, 01-224 Warsaw, Poland Received 20 June 2002; accepted 26 September 2002
Abstract The Debye /Brinkman concept of treating polymer coils as uniformly permeable spheres is utilized for embryonic aggregates composed of less than a dozen or so primary particles. These can be the direct models for bioparticles or may be equivalent to multi-particle aggregates and macromolecules. The method is based on the calculation of permeability of the system contained inside the sphere circumscribed on the aggregate, which makes it possible to estimate the transport properties such as the translational and rotational diffusion coefficients and the intrinsic viscosity of bead models. With this method it is possible to describe properly the translational diffusion coefficient for the embryonic aggregates analyzed as compared to that deduced from the experimental values of sedimentation velocity. The results obtained for rotational diffusion coefficient and intrinsic viscosity of oligomers are close to that calculated by the Kirkwood /Riseman method using modified Oseen tensor with volume correction. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Oligomers; Permeability; Intrinsic viscosity; Hydrodynamic radius; Diffusion coefficient
1. Introduction Transport properties of fractal aggregates and polymer coils, which are termed as solution or hydrodynamic properties in the field of polymer solutions, are of great interest in colloid and polymer science, biophysics and engineering. Macromolecular coils can be represented by fractal aggregates composed of blobs equivalent to nonporous solid particles [1]. The hydrodynamic properties of oligomeric proteins consisting of several subunits with a polygonal or polyhedral
E-mail address: [email protected] (L. Gmachowski).
geometry can be represented by bead models composed of spherical elements, introduced by Bloomfield et al. [2,3]. Bead models are useful when simple geometric models, such as spheres, ellipsoids or cylinders are inadequate due to specific way in which the subunits are arranged. The method proposed by Kirkwood and Riseman [4], based on the Oseen hydrodynamic interaction tensor between beads, makes it possible to calculate such properties as sedimentation coefficient, translational and rotational diffusion coefficients, as well as intrinsic viscosity. Based on that originally proposed, the method of the determination of hydrodynamic properties is improved to take into account the finite volume of beads, as
0927-7757/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 7 5 7 ( 0 2 ) 0 0 4 4 0 - 5
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L. Gmachowski / Colloids and Surfaces A: Physicochem. Eng. Aspects 215 (2003) 173 /179
Nomenclature a primary particle radius (m) D fractal dimension (-) fR rotational friction coefficient (J) fT translational friction coefficient (kg s 1) g acceleration of gravity (m s 2) i number of blobs or primary particles in an aggregate (-) k internal permeability of aggregate (m2) kB Bolzmann constant (J K 1) m aggregate mass (kg) r hydrodynamic radius of an aggregate (m) R radius of aggregate (radius of circumscribed sphere) (m) ua Stokes free settling velocity of primary particle (m s 1) ur free settling velocity of an aggregate (m s 1) f volume fraction of impermeable blobs or primary particles in an aggregate (-) h0 fluid viscosity (kg m 1 s 1) [h ] intrinsic viscosity (m3 kg 1) rf fluid density (kg m 3) rs solid density (kg m 3) s reciprocal square root of dimensionless internal permeability of aggregate (-) n viscosity increment (-)
described by Carrasco and Garcı´a de la Torre [5,6]. Nevertheless it is currently broadly accepted approach. Approximately at the same time the porous sphere model was introduced by Debye [7] and Brinkman [8] and developed by Brinkman [9] in which a macromolecular coil was modeled as a sphere of uniform permeability. The possibility to use the permeability concept to describe the sedimentation of macromolecules in solution was verified [10] down to the correlation length being of the order of the size of solvent molecule. The progress of the permeable sphere model was limited, however, owing to the difficulties to calculate the permeability of a macromolecule due to its non-uniform structure, as discussed by van Saarloos [11]. The permeability of an aggregate containing not too many particles, however, is possible to calculate much easier than for a multiparticle fractal aggregate, representing a macromolecule. Previously a method was presented [12] for calculation of the intrinsic viscosity of bead models, based on the fractal dimension depen-
dence of hydrodynamic radius. This dependence is valid for aggregates of any number of beads, but of spherical character. This paper is devoted to the description of a method for the calculation of the translational and rotational diffusion coefficients and the intrinsic viscosity for aggregates of any mass and shape. This is based on the determination of the permeability of the system contained inside the sphere circumscribed on the aggregate of sufficiently reduced number of constituents, which is a model of a given multi-particle aggregate, macromolecule or bioparticle.
2. Model The diffusion coefficient of an isolated aggregate, macromolecule or bioparticle, either translational or rotational, can be calculated by the Einstein relation [13] DT
kB T fT
or
DR
kB T fR
(1)
L. Gmachowski / Colloids and Surfaces A: Physicochem. Eng. Aspects 215 (2003) 173 /179
where kB is the Boltzmann constant and T is the absolute temperature. The translational friction coefficient of a particle can be written as the product of that for an impermeable particle of the same radius R and the normalized hydrodynamic radius r/R , representing the reduction of coefficient due to the internal permeability of the particle fT 6ph0 R
r R
(2)
The hydrodynamic radius r of a permeable particle is the radius of an impermeable sphere of the same mass having the same dynamic properties. Brinkman [9] combined the Stokes equation of motion with the Darcy law to formulate the equation of motion of a fluid in permeable media. He obtained the following formula for a sphere of uniform permeability, which makes it possible to calculate the translational friction coefficient tanh s 1 r s (3) 3 tanh s R 1 1 2s2 s pffiffiffi where s R= k is the reciprocal square root of dimensionless internal permeability of sphere model. Similarly, the expression for the rotational friction coefficient 3 3 r fR 8ph0 R (4) R with 1=3 3 3 1 R s2 stanh s r
(5)
was derived by Felderhof and Deutch [14]. The intrinsic viscosity of dissolved macromolecules and bioparticles or suspended aggregates of mass m is given by the following formula 10 R3 r 3 (6) [h] p 3 m R with
r R
175
3 3 1 2 s stanh s 10 3 3 1 1 s2 s2 stanh s
1=3
(7)
as derived by Brinkman [9]. The knowledge of the value of reciprocal square root of dimensionless internal permeability s of a given particle makes it possible to determine all the transport properties considered. The permeability of a multi-particle fractal aggregate, also if it represents a macromolecule, is not possible to be calculated straightforwardly by standard formulas, which are applicable for systems with uniform distribution of porosity, because of considerable radial variation of porosity due to fractal structure of macromolecules. If an aggregate has a self-similar structure, the fractal dimension D can be determined by covering the object with the sets of spheres of increasing size, as shown by Feder [15]. The number of spheres will decrease as a negative power of their size. In this way fractals may be produced of lower and lower numbers of constituent clusters (blobs), contained in covering spheres. Since the number of clusters diminishes as a negative power of its diameter, the corresponding volume fraction f of clusters in an aggregate is the following function of the number of clusters f8i1 3=D
(8)
The lower value of the number of clusters, the higher value of f and hence the narrower the range of its variability due to the radial dependence. This is the reason why the standard permeability models, applicable for systems with uniform distribution of porosity, are more reliable when applied to aggregates containing not many particles than to multi-particle fractal aggregates. The above-described method makes it possible to reduce a multi-particle aggregate to a structure containing not too many constituents, for which the permeability can be calculated straightforwardly. Replacing clusters by equivalent impermeable spheres, the aggregate of reduced number of constituents can be represented as composed by primary particles. The calculation of permeability
176
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of fractal aggregates of reduced number of constituents, was reported by Li and Logan [16] and Woodfield and Bickert [17]. The s-parameter calculated, being the reciprocal square root of the dimensionless permeability, should be the same as for the original aggregate due to self-similarity [17]. Its constancy was demonstrated for the macromolecules of different mass of a given polymer dissolved in a given solvent [1]. This parameter, calculated for an aggregate containing not too many constituents, should thus determine the normalized hydrodynamic radius valid for any aggregate of the same fractal dimension. The following formula [18] is employed for further calculations, previously verified for uniform permeable systems over a very wide porosity interval sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 9 25f4=3 s f 1 (9) a 2 3(1 f)3
particle
where the volume fraction f, of i beads of radius a forming aggregate, contained inside the circumscribed sphere of radius R , is calculated as
The values of normalized hydrodynamic radius used to determine the translational friction coefficient were calculated for some embryonic aggregates by Eqs. (3), (9) and (10). The results are presented in Table 1. For the same structures the values of normalized hydrodynamic radius were determined by Eq. (15) on the basis of sedimentation data of (ur/ua)1/2, measured by Sto¨ber et al. [19] and Sto¨ber and Flachsbart [20]. The results are compared in Fig. 1. A good agreement can be observed, which may be regarded as experimental verification of the permeability method for calculation of the translational friction coefficient for embryonic aggregates. The values of the rotational friction coefficient of an oligomer normalized by that of monomer 3 3 fR (i) R r (16) fR (1) a R
i f 3 R
(10)
a The normalized hydrodynamic radius can be determined employing Eqs. (9) and (10) and one of Eq. (3), Eq. (5) or Eq. (7). For the translational friction coefficient the normalized hydrodynamic radius can be deduced from the sedimentation velocity of a model considered. The sedimentation velocity ur of an individual aggregate can be determined by equating the gravitational force allowing for the buoyancy of the surrounding fluid with the opposing hydrodynamic force, as given by the Stokes law 4 3 pa (rs rf )gi 6ph0 rur 3
(11)
with ur
2 a3 i (rs rf )g 9h0 r
(12)
Relating to the Stokes velocity of primary
ua
2 9h0
(rs rf )ga2
(13)
one gets the following equation ur ua
i
(14)
(r=a)
which can be rearranged to determine the normalized hydrodynamic radius r R
i (R=a)(ur =ua )
(15)
This value can be compared to that obtained by Eqs. (3), (9) and (10).
3. Results and discussion
obtained by Garcı´a de la Torre [21] and Garcı´a de la Torre and Carrasco [22] using the method of modified Oseen tensor with the volume correction, were utilized to deduce the values of normalized hydrodynamic radius used to determine the rotational friction coefficient. That was performed for tetrahedron tetramer, octahedron hexamer and cube octamer, indicated in Table 1. The results
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177
Table 1 Values of normalized hydrodynamic radius used to determine the translational and rotational friction coefficients, and values of viscosity increment, all calculated by the permeability method for different embryonic aggregates Oligomeric structure
R /a
s
(r/R)Trans
Linear octamer Linear heptamer Linear hexamer Linear pentamer Linear tetramer Linear trimer Hexagon hexamer Dimer Pentagon pentamer Bipyramid pentamer Square tetramer Octahedron hexamer Triangle trimer Tetrahedron tetramer Pyramid pentamer Compact aggregate of i /9 Trigonal prism hexamer Cube octamer Compact aggregate of i /10 Compact aggregate of i /11 Compact aggregate of i /12 Compact aggregate of i /13
8 7 6 5 4 3 3 2 1/1/sin 368 pffiffiffiffiffiffiffi ffi /12 2=3 pffiffiffi / /1 p2ffiffiffi/ /1 3p / ffiffiffi /12= 3/ffi pffiffiffiffiffiffiffi /1 3=2 pffiffiffi / /1 2/ 3 pffiffiffiffiffiffiffiffi /1 pffiffi7=3 ffi / /1 3/ 3 3 3 3
2.157 2.173 2.201 2.254 2.373 2.711 5.519 4.301 5.928 6.453 6.255 7.465 6.067 8.705 9.071 10.06 10.32 12.40 12.17 14.17 17.74 21.13
0.4662 0.4694 0.4748 0.4849 0.5067 0.5617
(r/R)Rot
13.72
0.8858
6.853 4.700 5.280 3.712 4.901 4.836 4.556 4.959 4.220 4.381
0.9196
4.640 4.716
0.7226
0.8671 0.8077 0.8699 0.8756 0.8888 0.8918
n
0.9094 0.9261 0.9394 0.9504
Aggregates are ranged according to increasing solid volume fraction in model sphere.
Fig. 1. Comparison of normalized hydrodynamic radius used to calculate the translational friction coefficient determined from steady-state falling speed data of Sto¨ber et al. [19] and Sto¨ber and Flachsbart [20] with that calculated using Eqs. (3), (9) and (10), performed for different embryonic aggregates designated in Table 1.
are compared in Fig. 2 to the results calculated with Eqs. (5), (9) and (10). The agreement is good, although the hydrodynamic parameters calculated
Fig. 2. Values of normalized hydrodynamic radius used to determine the rotational friction coefficient, calculated for three bead models designated in Table 1 by the following methods: (m) the permeability method employing Eqs. (5), (9), (10) and (16); (k) using modified Oseen tensor and volume correction, as reported by Garcı´a de la Torre [21] and Garcı´a de la Torre and Carrasco [22].
by the method of modified Oseen tensor with the volume correction rise faster with the compactness of the sphere model. Nevertheless this confirms the reliability of the permeability method of the
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178
rotational friction coefficient calculation for oligomers. The intrinsic viscosity may be expressed as the product of a molecular shape parameter known as the viscosity increment n and the specific volume of primary particles [h] n
(4=3)pa3 i m
Comparing to Eq. (6) one gets 2:5 r 3 R 3 n i R a
(17)
(18)
The values of viscosity increment were calculated for some oligomers by Eqs. (7), (9), (10) and (18). The results are presented in Table 1. They are compared in Fig. 3 with the results obtained by the method of modified Oseen tensor with the volume correction as reported by Garcı´a de la Torre [21], Garcı´a de la Torre and Carrasco [22] and Harding [23]. The agreement is good, except for the linear string hexamer. The s -value for this oligomer (equal to 2.201) lies in the range of s B/3, for which the normalized hydrodynamic radius calculated by Eq. (7) for intrinsic viscosity becomes considerably lower than that determined by Eq. (3) for translational friction coefficient [12]. If one used, however, Eq. (3) to calculate the normalized hydrodynamic radius for this oligomer and then put the result (equal to 0.4748) into Eq. (18), the
Fig. 3. Values of viscosity increment for different bead models designated in Table 1, calculated by the following methods: (m) the permeability method employing Eqs. (7), (9), (10) and (18); (k) using modified Oseen tensor and volume correction, as reported by Garcı´a de la Torre [21], Garcı´a de la Torre and Carrasco [22] and Harding [23].
obtained value of the viscosity increment would be of 9.63, a value almost identical to that obtained using modified Oseen tensor and volume correction. For higher values of s -parameter Eqs. (3) and (7), describing the values of normalized hydrodynamic radius used to calculate the translational friction coefficient and the intrinsic viscosity, respectively, give results, which are practically indistinguishable [12]. Both arguments suggest that Eq. (3) is adequate to calculate the viscosity increment for oligomers. So the permeability method of the intrinsic viscosity calculation for oligomers, employing Eqs. (3), (9), (10) and (18), also seems to be reliable. The proposed method of the determination of transport properties of embryonic aggregates is based on the calculation of permeability of the system contained inside the sphere circumscribed on the bead model. This treatment offers the advantage of its simplicity. Permeability is a quantity, which except for the volume fraction of beads inside the sphere, is sensitive to the size of beads. Therefore it may serve as a reference to results obtained by different methods, especially when the bead models are very compact. For such structures the values of hydrodynamic parameters are overestimated [22] when obtained using modified Oseen tensor and volume correction. This tendency is also visible in Figs. 2 and 3 for the values of solid volume fraction in model sphere greater than 0.3. It suggests that the proposed method gives the values of hydrodynamic parameters more reliable than the method using modified Oseen tensor and volume correction. It has been shown that the method of permeability gives the reliable values of transport coefficients when applied to the embryonic aggregates, which can represent multi-particle fractal aggregates and macromolecular coils. To estimate the degree to which the number of constituents should be reduced, let us compare the value of normalized hydrodynamic radius, calculated for D /2 from the following relation [24] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r D 2 1:56 1:728 0:228 (19) R 2 with those determined for different aggregation
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179
References
Fig. 4. Aggregation number dependence of the normalized hydrodynamic radius (solid line) calculated by permeability method employing Eqs. (3), (9), (10) and (20) compared to the value resulting from the fractal dimension by Eq. (19).
numbers by Eqs. (3), (9) and (10), where the normalized aggregate radii were determined by the mass-radius relation [24]: R i1=D sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a D 2 1:56 1:728 0:228 2
(20)
The results are presented in Fig. 4. As expected, in the range of aggregate containing not many primary particles the calculated values of normalized hydrodynamic radius are close to the value resulting from the fractal dimension. The permeability method seems to be reliable up to the number of constituents of about a hundred.
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