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Aggregate restructuring by internal aggregation
Colloids and Surfaces A: Physicochem. Eng. Aspects 352 (2009) 70–73
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Colloids and Surfaces A: Physicochemical and Engineering Aspects journal homepage: www.elsevier.com/locate/colsurfa
Aggregate restructuring by internal aggregation Lech Gmachowski a,b,∗ a b
Warsaw University of Technology, Institute of Chemistry, 09-400 Płock, Poland Institute of Physical Chemistry, Polish Academy of Sciences, 01-224 Warsaw, Poland
a r t i c l e
i n f o
Article history: Received 24 November 2008 Received in revised form 28 September 2009 Accepted 28 September 2009 Available online 6 October 2009 Keywords: Internal aggregation Aggregate restructuring Primary aggregate growth Aggregation act Limiting fractal dimension
a b s t r a c t The aggregation of primary aggregates in a multilevel structure aggregate changes the structure of the whole object. A model is presented of cluster–cluster aggregation, in which the growing aggregate has no self-similar structure but restructures to get a limiting fractal dimension, equal to that of primary aggregates. This object serves as a model for larger aggregate containing many two-cluster primary aggregates formed at the same stage of restructuring. The model well describes the evolution of thermal blob size in polymer coils with the solvent quality. The process of sintering of aggregates into compact spherical particles has also a good representation by the model. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Restructuring of the fractal aggregates can occur by internal rearrangement of monomers or by internal aggregation of monomers. In the first case higher values of fractal dimension are observed experimentally [1], than those predicted by the model. The elevation of the fractal dimension is possible as a result of competition between aggregation and simultaneous fragmentation of aggregates [2]. This also can be caused by heating or shear, relaxing the aggregate structure. The particles get the possibility to arrange into a state of lower Gibbs free energy. Because of attracting forces between primary particles, the restructuring causes closer packing of particles and a higher fractal dimension of restructured aggregate. Weber and Friedlander [3] present a schematic diagram of the restructuring process of a single aggregate. The second mechanism can lead either to coalescence of monomers, for example the sintering of the aggregates of the SiO2 particles formed in the laminar diffusion flame into compact spherical particles [4] or to form small aggregates inside a large one. A two-level structure appears as a result of internal aggregation without coalescence. The fractal dimension of small aggregates can differ from that on larger length scales. Such superaggregates were produced in laminar diffusion flames of heavily sooting fuels [5].
∗ Correspondence at: Institute of Physical Chemistry, Polish Academy of Sciences, 01-224 Warsaw, Poland. E-mail address: [email protected]. 0927-7757/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2009.09.052
Aggregates of one structure composed of smaller aggregates of different morphology are observed in aggregated solid–liquid systems [6]. Hydrodynamic behavior of macromolecules in solution can be described using the model of fractal aggregate with mixed statistics [7], as an alternative to chain of Kuhn statistical segments. The size of thermal blobs, being the primary aggregates of effective solid monomers, changes with the thermodynamic quality of the solvent. The crossover from poor solvent to good solvent can be described by the thermal blob growing up to the theta temperature and shrinking again. The cluster–cluster aggregation model seems to be most suitable to describe the process of internal aggregation which causes the primary aggregate growth. At each stage of aggregate restructuring the number I of primary aggregates is reduced twice as a result of I/2 cluster–cluster aggregation acts duplicating the mass of a single primary aggregate.
2. Model Aggregate with mixed statistics is an object which has different fractal dimensions on different length scales. Such aggregate having two-level structure is presented in Fig. 1. Smaller objects are primary fractal aggregates of solid monomers of radius a, whose structure is characterized by the hydrodynamic radius rT , which is the radius of an impermeable sphere of the same mass having the same dynamic properties, and the fractal dimension DT . The fractal dimension D on larger scales results from the spatial arrangement of primary aggregates.
L. Gmachowski / Colloids and Surfaces A: Physicochem. Eng. Aspects 352 (2009) 70–73
Nomenclature a D Dlim DT Dw Di Dj i iT I j K K M Mm
MT r rT R R1 Rc RT
primary particle (monomer) radius (m) aggregate fractal dimension limiting value of aggregate fractal dimension primary aggregate fractal dimension particle trajectory fractal dimension fractal dimension of aggregate containing i monomers fractal dimension of aggregate containing j monomers number of primary particles in an aggregate number of monomers in primary aggregate number of primary aggregates in an aggregate number of primary particles in an aggregate Mark–Houwink–Sakurada constant (m3 kg−1 ) Mark–Houwink–Sakurada constant in theta solvent (m3 kg−1 ) mass of a macromolecule (u) mass of non-porous monomer of fractal aggregates representing an individual macromolecule in a theta solvent and a thermal blob (u) thermal blob mass (u) dynamic radius of aggregate (m) dynamic radius of primary aggregate (m) radius of aggregate (m) radius of smallest object similar geometrically to aggregate (m) collision radius (m) radius of primary aggregate (m)
1 · RTDT = iT · R1DT
(2)
where iT is the number of the constituent particles in a primary aggregate. This means that the fractal concept is valid for very small aggregates. Computer simulations [9] show that the fractal dimension is held from very early stages of cluster–cluster aggregation, even for dimers. After rearrangement one obtains iT =
R DT T
(3)
R1
This is a mass–radius relation without prefactor, since it relates the sizes of similar objects. The exponent DT is the fractal dimension of the primary aggregate. Using the concept of the hydrodynamic radius of aggregate rT , which is proportional to the radius of primary aggregate RT and converges to the primary particle radius a for the number of constituent particles equal to unity [10,11], one gets iT =
r DT T
a
=
R DT T
R1
=
r DT R DT T
T
RT
a
(4)
The normalized hydrodynamic radius, appearing in the last formula for the primary aggregate is a general function [12] of the fractal dimension, describing mass–radius relation prefactors for both the primary aggregate
1.56 − 1.728 −
DT 2
2
− 0.228
and the aggregate
The analysis of the structure of a fractal aggregate can be easily performed. The number of spheres needed to cover the primary aggregate increases with the reduction of the size of covering spheres. This number times an adequately chosen power of their radius N() · DT does not depend on the radius of covering spheres and expresses a quantity proportional to the Hausdorff measure [8]. MH ∝ N() · DT
The procedure with the Hausdorff measure can be performed in the radius range from the radius RT of the sphere circumscribed on the primary aggregate down to the radius R1 of the smallest covering sphere, containing one primary particle of radius a, enclosing the object which can be considered as similar geometrically to the aggregate. The constancy of Hausdorff measure, expressed by Eq. (1) implies that
rT = RT
Greek letter solute density (kg m−3 ) s
71
(1)
r = R
D 2
2
− 0.228
(5)
The regarded formula was confirmed by obtaining close values of fractal dimension of aggregates by a method connecting the aggregate structure with its settling behavior and by the light scattering technique [13]. The structural characterization of fractal aggregates was performed [14–18] using the formula (5). Similarly, the same analysis can be performed in the range from the size of primary aggregate up to the size of the whole object to show that the aggregate is self-similar in this interval, regardless of the number of primary aggregates. Moreover, if one imagines that each microaggregate is rearranged to achieve the structure of the whole aggregate, the self-similarity will be extended down to the monomer. In this way one obtains an aggregate of fractal objects similar to the aggregate with the aggregation number characteristic for primary aggregate. Then one can choose a single rearranged microaggregate to represent the whole aggregate structure. The mutual partial interpenetration of aggregating clusters is described using the concept of the collision radii of the constituents. The collision radius of an aggregate is taken to be its contribution to the radius of a new aggregate. The radius of a formed aggregate is thus the sum of the collision radii of joining clusters [11] R = Rci + Rcj =
Fig. 1. Graphical illustration of a two-level structure aggregate with mixed statistics.
1.56 − 1.728 −
Rci ri
ri +
Rcj rj
rj
(6)
The collision radius normalized by the hydrodynamic radius is dependent on the mechanism of aggregation defined by the character of the lines followed by the colliding particles and aggregates. Straight-line trajectory is characterized by the fractal dimensionality Dw = 1, whereas for Brownian one Dw = 2. It is constant for
72
L. Gmachowski / Colloids and Surfaces A: Physicochem. Eng. Aspects 352 (2009) 70–73
remains unchanged during the process of aggregate restructuring. The model has experimental support reported below. 3. Modeling aggregate restructuring by internal aggregation Fractal aggregate model of macromolecules [7] makes it possible to describe the polymer coils as fractal aggregates with mixed statistics. Two-level structure of the model aggregate is presented in Fig. 1. Inside the thermal blob random walk statistics apply (DT = 2). The fractal dimension D on larger scales results from the spatial arrangement of blobs. The mass of thermal blobs depends on the solvent quality. Starting from poor solvent the blobs grow up to the theta temperature and then shrink again. This fact entitles to test the proposed model with the polymer solution data as first. The intrinsic viscosity, being a hydrodynamic property of dissolved macromolecules, may be represented by the Mark–Houwink–Sakurada equation, which can be written for any solvent quality in the form
Fig. 2. Graphical representation of a cluster–cluster aggregate.
a given mechanism of aggregation, independently of aggregation sequence. Hence R=
Rc (r + rj ) r i
(7)
Employing the mass–radius relation (Eq. (4)) for the resulting aggregate one gets i+j =
D Rc + Rc D i j
r R
(D)
a
=
r D R D r + r D i j c R
r
a
(8)
Then utilizing the mass–radius relation for the joining clusters one obtains i+j =
r
D R D
R
c
(D)
r
(i1/Di + j1/Dj )
D
(9)
If the approaching aggregates are identical (Fig. 2), the sequence cluster–cluster (CCA) occurs described by the formula 2i =
r R
D R D c
(D)
r
or i1−D/Di = 2D−1
r R
(2i1/Di )
D
D R D c
(D)
r
(r/R)(D) (r/R)(Dlim )
(16)
If D = 2, the Mark–Houwink–Sakurada equation describes the intrinsic viscosity for a theta solvent [] = K M 1/2
(17)
The intrinsic viscosity of a given polymer in a solvent crosses over to the theta result at a molecular mass, which is the thermal blob mass [19]. Viscometric expansion factor ˛ can be expressed as follows K 3/D−3/2 [] = M K []
˛3 ≡
(18)
Putting ˛3 = 1 one gets the thermal blob mass
K K
1/(3/D−3/2)
(19)
The fractal aggregate model of a polymer dissolved in a theta solvent [7] makes it possible to describe the intrinsic viscosity as (11)
(12)
Then if two aggregates of fractal dimension Di , each containing iT monomers, join to form a new aggregate, its fractal dimension D is described by the formula
a = 3/D − 1
(10)
If, however, the growing aggregate has no self-similar structure and restructures to get a limiting fractal dimension Dlim , the normalized collision radius is not constant and the last equation should be modified to r r = 21−1/D · (Dlim ) (13) Rc R
iT1−D/Di =
(15)
where
MT =
If the structure of growing aggregate is unchanged (D = Di ), one gets the expression for the constant value of normalized collision radius r r = 21−1/D · (D) Rc R
[] = K M a
D
(14)
According to the model presented, the obtained equation can be used to model the structure evolution of aggregate restructuring by internal aggregation. The limiting value of the fractal dimension is simply the primary aggregate fractal dimension, which
[] =
5 2s
M Mm
1/2
= K M 1/2
(20)
The mass of non-porous monomer of fractal aggregates representing an individual macromolecule in a theta solvent and a thermal blob can be thus determined as Mm =
2.5 2 s K
(21)
Eqs. (19) and (21) make it possible to define the aggregation number of the thermal blob aggregate model, which is described as iT =
MT Mm
(22)
According to the above equations, this value depends on the polymer solvent quality, represented by the fractal dimension. The analysis performed utilizes the values of intrinsic viscosity collected in Ref. [19]. Using Eqs. (14) and (5) for Dlim = 2, the dependences iT (D) have been calculated using CCA simulation, starting from both good and poor solvent regions. The aggregates growing by consecutive CCA events restructured to get a limiting fractal dimension Dlim in an advanced stage of the process. A two-cluster aggregate was
L. Gmachowski / Colloids and Surfaces A: Physicochem. Eng. Aspects 352 (2009) 70–73
73
values of fractal dimension, resulting from experimental connection between the fractal dimension of aggregates and the size of primary particles, are depicted in Fig. 4. 4. Discussion and conclusions
Fig. 3. Fractal dimension dependences of the aggregation number of thermal blobs depicted for several polymers [20]: () poly(oxyethylene); () poly(2vinylpyridyne); () poly(styrene); () poly(␣-methylstyrene). Experimental results are compared to the model curve.
Fig. 4. Evolution of the characteristics of the SiO2 particles formed in the laminar diffusion flame [4]. Experimental results () are compared to the model curve.
obtained at each stage, which is supposed to be similar to a larger object of any size being under restructuring by internal aggregation. The obtained results, presented as lines in Fig. 3, are compared to the fractal dimension dependences of the aggregation number of thermal blobs calculated for several polymers based on the experimental data [20] reported over a wide interval of solvent quality. A similar analysis has been performed with sintering of the aggregates of the SiO2 particles formed in the laminar diffusion flame [4] into compact spherical particles. For this case Dlim = 3. The obtained line is compared in Fig. 4 to the results of experiment, in which the fractal dimension of aggregates and the size of primary particles were measured as a function of the height above burner. Solid primary particles were treated as compact (DT = 3) aggregates of solid particles existing at the beginning of the sintering process. The calculation with the use of Eqs. (14) and (5) started at iT = 2 and the corresponding fractal dimension of aggregates, equal 1.7. The increasing aggregate number was the third power of the primary particle size related to its initial value. The corresponding
The presented model describes the change in fractal dimension of an aggregate undergoing restructuring by internal aggregation of primary aggregates. Utilizing the self-similarity concept, a rearranged primary aggregate represents the aggregate structure. At each stage of restructuring the number of monomers in primary aggregates duplicates, according to cluster–cluster aggregation acts. Repeating the cluster–cluster aggregation many times, one obtains the aggregate of fractal dimension close to that of primary aggregates. Although the polymer coils are not typical colloidal aggregates, their restructuring under the change of solvent quality is especially spectacular since it can be observed both the growth and shrinkage of thermal blobs when the polymer solvent is heated or cooled. The proposed model well estimates the growth of thermal blobs, being the primary aggregates of DT = 2, for the fractal dimension of polymer coil both increasing up to 2 and decreasing down to 2. Sintering of fractal aggregates is described by a simpler form of the model in which the growing solid monomers (DT = 3) replace the primary aggregates, and hence the growth may cause only the increment of the aggregate fractal dimension. In both cases the model well simulates the restructuring of aggregates by internal aggregation. The presented model is thus applicable to describe of aggregate restructuring by internal aggregation with the formation of both the fractal growing microaggregates and the growing solid monomers. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
H.D. Jang, S.K. Friedlander, Aerosol Sci. Technol. 29 (1998) 81. L. Gmachowski, Colloids Surfaces A: Physicochem. Eng. Aspects 201 (2002) 41. A.P. Weber, S.K. Friedlander, J. Aerosol Sci. 28 (1997) 179. A. Camenzind, H. Schulz, A. Teleki, G. Beaucage, T. Narayanan, S.E. Pratsinis, Eur. J. Inorg. Chem. 6 (2008) 911. W. Kim, C.M. Sorensen, D. Fry, A. Chakrabarti, J. Aerosol Sci. 37 (2006) 386. L. Gmachowski, Colloids Surfaces A: Physicochem. Eng. Aspects 295 (2007) 34. L. Gmachowski, Polymer 48 (2007) 4316. F. Hausdorff, Math. Ann. 79 (1919) 157. M. Kolb, R. Botet, R. Jullien, Phys. Rev. Lett. 51 (1983) 1123. M.M. Takayasu, F. Galembeck, J. Colloid Interface Sci. 202 (1998) 84. L. Gmachowski, Colloids Surfaces A: Physicochem. Eng. Aspects 170 (2000) 209. L. Gmachowski, Colloids Surfaces A: Physicochem. Eng. Aspects 274 (2006) 223. P. Tang, J. Greenwood, J.A. Raper, J. Colloid Interface Sci. 247 (2002) 210. C. Lee, T.A. Kramer, Adv. Colloid Interface Sci. 112 (2004) 49. F. Gruy, P. Cugniet, J. Colloid Interface Sci. 272 (2004) 465. M. Lattuada, H. Wu, M. Morbidelli, Chem. Eng. Sci. 59 (2004) 4401. Y. Mu, H.-Q. Yu, Water Res. 40 (2006) 3596. Z. Ranjbar, S. Rastegar, Colloids Surfaces A: Physicochem. Eng. Aspects 290 (2006) 186. M. Rubinstein, R.H. Colby, Polymer Physics, Oxford University Press, Oxford, 2006, p. 317. M. Kurata, Y. Tsunashima, Viscosity-molecular weight relationships and unperturbed dimensions of linear chain molecules, in: J. Brandrup, E.H. Immergut, E.A. Grulke (Eds.), Polymer Handbook, Wiley-Interscience, New Jersey, 1999.
Contents lists available at ScienceDirect
Colloids and Surfaces A: Physicochemical and Engineering Aspects journal homepage: www.elsevier.com/locate/colsurfa
Aggregate restructuring by internal aggregation Lech Gmachowski a,b,∗ a b
Warsaw University of Technology, Institute of Chemistry, 09-400 Płock, Poland Institute of Physical Chemistry, Polish Academy of Sciences, 01-224 Warsaw, Poland
a r t i c l e
i n f o
Article history: Received 24 November 2008 Received in revised form 28 September 2009 Accepted 28 September 2009 Available online 6 October 2009 Keywords: Internal aggregation Aggregate restructuring Primary aggregate growth Aggregation act Limiting fractal dimension
a b s t r a c t The aggregation of primary aggregates in a multilevel structure aggregate changes the structure of the whole object. A model is presented of cluster–cluster aggregation, in which the growing aggregate has no self-similar structure but restructures to get a limiting fractal dimension, equal to that of primary aggregates. This object serves as a model for larger aggregate containing many two-cluster primary aggregates formed at the same stage of restructuring. The model well describes the evolution of thermal blob size in polymer coils with the solvent quality. The process of sintering of aggregates into compact spherical particles has also a good representation by the model. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Restructuring of the fractal aggregates can occur by internal rearrangement of monomers or by internal aggregation of monomers. In the first case higher values of fractal dimension are observed experimentally [1], than those predicted by the model. The elevation of the fractal dimension is possible as a result of competition between aggregation and simultaneous fragmentation of aggregates [2]. This also can be caused by heating or shear, relaxing the aggregate structure. The particles get the possibility to arrange into a state of lower Gibbs free energy. Because of attracting forces between primary particles, the restructuring causes closer packing of particles and a higher fractal dimension of restructured aggregate. Weber and Friedlander [3] present a schematic diagram of the restructuring process of a single aggregate. The second mechanism can lead either to coalescence of monomers, for example the sintering of the aggregates of the SiO2 particles formed in the laminar diffusion flame into compact spherical particles [4] or to form small aggregates inside a large one. A two-level structure appears as a result of internal aggregation without coalescence. The fractal dimension of small aggregates can differ from that on larger length scales. Such superaggregates were produced in laminar diffusion flames of heavily sooting fuels [5].
∗ Correspondence at: Institute of Physical Chemistry, Polish Academy of Sciences, 01-224 Warsaw, Poland. E-mail address: [email protected]. 0927-7757/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2009.09.052
Aggregates of one structure composed of smaller aggregates of different morphology are observed in aggregated solid–liquid systems [6]. Hydrodynamic behavior of macromolecules in solution can be described using the model of fractal aggregate with mixed statistics [7], as an alternative to chain of Kuhn statistical segments. The size of thermal blobs, being the primary aggregates of effective solid monomers, changes with the thermodynamic quality of the solvent. The crossover from poor solvent to good solvent can be described by the thermal blob growing up to the theta temperature and shrinking again. The cluster–cluster aggregation model seems to be most suitable to describe the process of internal aggregation which causes the primary aggregate growth. At each stage of aggregate restructuring the number I of primary aggregates is reduced twice as a result of I/2 cluster–cluster aggregation acts duplicating the mass of a single primary aggregate.
2. Model Aggregate with mixed statistics is an object which has different fractal dimensions on different length scales. Such aggregate having two-level structure is presented in Fig. 1. Smaller objects are primary fractal aggregates of solid monomers of radius a, whose structure is characterized by the hydrodynamic radius rT , which is the radius of an impermeable sphere of the same mass having the same dynamic properties, and the fractal dimension DT . The fractal dimension D on larger scales results from the spatial arrangement of primary aggregates.
L. Gmachowski / Colloids and Surfaces A: Physicochem. Eng. Aspects 352 (2009) 70–73
Nomenclature a D Dlim DT Dw Di Dj i iT I j K K M Mm
MT r rT R R1 Rc RT
primary particle (monomer) radius (m) aggregate fractal dimension limiting value of aggregate fractal dimension primary aggregate fractal dimension particle trajectory fractal dimension fractal dimension of aggregate containing i monomers fractal dimension of aggregate containing j monomers number of primary particles in an aggregate number of monomers in primary aggregate number of primary aggregates in an aggregate number of primary particles in an aggregate Mark–Houwink–Sakurada constant (m3 kg−1 ) Mark–Houwink–Sakurada constant in theta solvent (m3 kg−1 ) mass of a macromolecule (u) mass of non-porous monomer of fractal aggregates representing an individual macromolecule in a theta solvent and a thermal blob (u) thermal blob mass (u) dynamic radius of aggregate (m) dynamic radius of primary aggregate (m) radius of aggregate (m) radius of smallest object similar geometrically to aggregate (m) collision radius (m) radius of primary aggregate (m)
1 · RTDT = iT · R1DT
(2)
where iT is the number of the constituent particles in a primary aggregate. This means that the fractal concept is valid for very small aggregates. Computer simulations [9] show that the fractal dimension is held from very early stages of cluster–cluster aggregation, even for dimers. After rearrangement one obtains iT =
R DT T
(3)
R1
This is a mass–radius relation without prefactor, since it relates the sizes of similar objects. The exponent DT is the fractal dimension of the primary aggregate. Using the concept of the hydrodynamic radius of aggregate rT , which is proportional to the radius of primary aggregate RT and converges to the primary particle radius a for the number of constituent particles equal to unity [10,11], one gets iT =
r DT T
a
=
R DT T
R1
=
r DT R DT T
T
RT
a
(4)
The normalized hydrodynamic radius, appearing in the last formula for the primary aggregate is a general function [12] of the fractal dimension, describing mass–radius relation prefactors for both the primary aggregate
1.56 − 1.728 −
DT 2
2
− 0.228
and the aggregate
The analysis of the structure of a fractal aggregate can be easily performed. The number of spheres needed to cover the primary aggregate increases with the reduction of the size of covering spheres. This number times an adequately chosen power of their radius N() · DT does not depend on the radius of covering spheres and expresses a quantity proportional to the Hausdorff measure [8]. MH ∝ N() · DT
The procedure with the Hausdorff measure can be performed in the radius range from the radius RT of the sphere circumscribed on the primary aggregate down to the radius R1 of the smallest covering sphere, containing one primary particle of radius a, enclosing the object which can be considered as similar geometrically to the aggregate. The constancy of Hausdorff measure, expressed by Eq. (1) implies that
rT = RT
Greek letter solute density (kg m−3 ) s
71
(1)
r = R
D 2
2
− 0.228
(5)
The regarded formula was confirmed by obtaining close values of fractal dimension of aggregates by a method connecting the aggregate structure with its settling behavior and by the light scattering technique [13]. The structural characterization of fractal aggregates was performed [14–18] using the formula (5). Similarly, the same analysis can be performed in the range from the size of primary aggregate up to the size of the whole object to show that the aggregate is self-similar in this interval, regardless of the number of primary aggregates. Moreover, if one imagines that each microaggregate is rearranged to achieve the structure of the whole aggregate, the self-similarity will be extended down to the monomer. In this way one obtains an aggregate of fractal objects similar to the aggregate with the aggregation number characteristic for primary aggregate. Then one can choose a single rearranged microaggregate to represent the whole aggregate structure. The mutual partial interpenetration of aggregating clusters is described using the concept of the collision radii of the constituents. The collision radius of an aggregate is taken to be its contribution to the radius of a new aggregate. The radius of a formed aggregate is thus the sum of the collision radii of joining clusters [11] R = Rci + Rcj =
Fig. 1. Graphical illustration of a two-level structure aggregate with mixed statistics.
1.56 − 1.728 −
Rci ri
ri +
Rcj rj
rj
(6)
The collision radius normalized by the hydrodynamic radius is dependent on the mechanism of aggregation defined by the character of the lines followed by the colliding particles and aggregates. Straight-line trajectory is characterized by the fractal dimensionality Dw = 1, whereas for Brownian one Dw = 2. It is constant for
72
L. Gmachowski / Colloids and Surfaces A: Physicochem. Eng. Aspects 352 (2009) 70–73
remains unchanged during the process of aggregate restructuring. The model has experimental support reported below. 3. Modeling aggregate restructuring by internal aggregation Fractal aggregate model of macromolecules [7] makes it possible to describe the polymer coils as fractal aggregates with mixed statistics. Two-level structure of the model aggregate is presented in Fig. 1. Inside the thermal blob random walk statistics apply (DT = 2). The fractal dimension D on larger scales results from the spatial arrangement of blobs. The mass of thermal blobs depends on the solvent quality. Starting from poor solvent the blobs grow up to the theta temperature and then shrink again. This fact entitles to test the proposed model with the polymer solution data as first. The intrinsic viscosity, being a hydrodynamic property of dissolved macromolecules, may be represented by the Mark–Houwink–Sakurada equation, which can be written for any solvent quality in the form
Fig. 2. Graphical representation of a cluster–cluster aggregate.
a given mechanism of aggregation, independently of aggregation sequence. Hence R=
Rc (r + rj ) r i
(7)
Employing the mass–radius relation (Eq. (4)) for the resulting aggregate one gets i+j =
D Rc + Rc D i j
r R
(D)
a
=
r D R D r + r D i j c R
r
a
(8)
Then utilizing the mass–radius relation for the joining clusters one obtains i+j =
r
D R D
R
c
(D)
r
(i1/Di + j1/Dj )
D
(9)
If the approaching aggregates are identical (Fig. 2), the sequence cluster–cluster (CCA) occurs described by the formula 2i =
r R
D R D c
(D)
r
or i1−D/Di = 2D−1
r R
(2i1/Di )
D
D R D c
(D)
r
(r/R)(D) (r/R)(Dlim )
(16)
If D = 2, the Mark–Houwink–Sakurada equation describes the intrinsic viscosity for a theta solvent [] = K M 1/2
(17)
The intrinsic viscosity of a given polymer in a solvent crosses over to the theta result at a molecular mass, which is the thermal blob mass [19]. Viscometric expansion factor ˛ can be expressed as follows K 3/D−3/2 [] = M K []
˛3 ≡
(18)
Putting ˛3 = 1 one gets the thermal blob mass
K K
1/(3/D−3/2)
(19)
The fractal aggregate model of a polymer dissolved in a theta solvent [7] makes it possible to describe the intrinsic viscosity as (11)
(12)
Then if two aggregates of fractal dimension Di , each containing iT monomers, join to form a new aggregate, its fractal dimension D is described by the formula
a = 3/D − 1
(10)
If, however, the growing aggregate has no self-similar structure and restructures to get a limiting fractal dimension Dlim , the normalized collision radius is not constant and the last equation should be modified to r r = 21−1/D · (Dlim ) (13) Rc R
iT1−D/Di =
(15)
where
MT =
If the structure of growing aggregate is unchanged (D = Di ), one gets the expression for the constant value of normalized collision radius r r = 21−1/D · (D) Rc R
[] = K M a
D
(14)
According to the model presented, the obtained equation can be used to model the structure evolution of aggregate restructuring by internal aggregation. The limiting value of the fractal dimension is simply the primary aggregate fractal dimension, which
[] =
5 2s
M Mm
1/2
= K M 1/2
(20)
The mass of non-porous monomer of fractal aggregates representing an individual macromolecule in a theta solvent and a thermal blob can be thus determined as Mm =
2.5 2 s K
(21)
Eqs. (19) and (21) make it possible to define the aggregation number of the thermal blob aggregate model, which is described as iT =
MT Mm
(22)
According to the above equations, this value depends on the polymer solvent quality, represented by the fractal dimension. The analysis performed utilizes the values of intrinsic viscosity collected in Ref. [19]. Using Eqs. (14) and (5) for Dlim = 2, the dependences iT (D) have been calculated using CCA simulation, starting from both good and poor solvent regions. The aggregates growing by consecutive CCA events restructured to get a limiting fractal dimension Dlim in an advanced stage of the process. A two-cluster aggregate was
L. Gmachowski / Colloids and Surfaces A: Physicochem. Eng. Aspects 352 (2009) 70–73
73
values of fractal dimension, resulting from experimental connection between the fractal dimension of aggregates and the size of primary particles, are depicted in Fig. 4. 4. Discussion and conclusions
Fig. 3. Fractal dimension dependences of the aggregation number of thermal blobs depicted for several polymers [20]: () poly(oxyethylene); () poly(2vinylpyridyne); () poly(styrene); () poly(␣-methylstyrene). Experimental results are compared to the model curve.
Fig. 4. Evolution of the characteristics of the SiO2 particles formed in the laminar diffusion flame [4]. Experimental results () are compared to the model curve.
obtained at each stage, which is supposed to be similar to a larger object of any size being under restructuring by internal aggregation. The obtained results, presented as lines in Fig. 3, are compared to the fractal dimension dependences of the aggregation number of thermal blobs calculated for several polymers based on the experimental data [20] reported over a wide interval of solvent quality. A similar analysis has been performed with sintering of the aggregates of the SiO2 particles formed in the laminar diffusion flame [4] into compact spherical particles. For this case Dlim = 3. The obtained line is compared in Fig. 4 to the results of experiment, in which the fractal dimension of aggregates and the size of primary particles were measured as a function of the height above burner. Solid primary particles were treated as compact (DT = 3) aggregates of solid particles existing at the beginning of the sintering process. The calculation with the use of Eqs. (14) and (5) started at iT = 2 and the corresponding fractal dimension of aggregates, equal 1.7. The increasing aggregate number was the third power of the primary particle size related to its initial value. The corresponding
The presented model describes the change in fractal dimension of an aggregate undergoing restructuring by internal aggregation of primary aggregates. Utilizing the self-similarity concept, a rearranged primary aggregate represents the aggregate structure. At each stage of restructuring the number of monomers in primary aggregates duplicates, according to cluster–cluster aggregation acts. Repeating the cluster–cluster aggregation many times, one obtains the aggregate of fractal dimension close to that of primary aggregates. Although the polymer coils are not typical colloidal aggregates, their restructuring under the change of solvent quality is especially spectacular since it can be observed both the growth and shrinkage of thermal blobs when the polymer solvent is heated or cooled. The proposed model well estimates the growth of thermal blobs, being the primary aggregates of DT = 2, for the fractal dimension of polymer coil both increasing up to 2 and decreasing down to 2. Sintering of fractal aggregates is described by a simpler form of the model in which the growing solid monomers (DT = 3) replace the primary aggregates, and hence the growth may cause only the increment of the aggregate fractal dimension. In both cases the model well simulates the restructuring of aggregates by internal aggregation. The presented model is thus applicable to describe of aggregate restructuring by internal aggregation with the formation of both the fractal growing microaggregates and the growing solid monomers. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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