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van-der-Waals interaction between two fractal aggregates
Advanced Powder Technology 22 (2011) 220–225
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Original Research Paper
van-der-Waals interaction between two fractal aggregates Frank Babick a,⇑, Karin Schießl b, Michael Stintz a a b
Institut für Verfahrenstechnik und Umwelttechnik, TU Dresden, 01062 Dresden, Germany Institut für Strömungsmechanik, TU Dresden, 01062 Dresden, Germany
a r t i c l e
i n f o
Article history: Received 30 July 2010 Received in revised form 15 November 2010 Accepted 29 November 2010 Available online 13 December 2010 Keywords: Fractal aggregate van-der-Waals interaction DLCA aggregate Lifshitz–Hamaker function
a b s t r a c t The van-der-Waals forces between particles are of fundamental importance for agglomeration processes or their structuring in suspensions. The paper describes a calculation scheme for the corresponding energy potential when the interacting objects are aggregates of colloidal primary particles. This scheme is further employed to examine the impact of aggregate structure on the van-der-Waals interaction. Calculations were therefore conducted for deterministic aggregates with tuneable fractal dimension and for DLCA aggregates of varying size. It appears that the van-der-Waals interaction is governed by the closest pair(s) of primary particles. That results in an impact of the aggregate structure, when the interaction energy is plotted versus the aggregate centre distance. For DLCA aggregates this relationship can be described by a size-independent, but material specific mastercurve. Ó 2010 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved.
1. Motivation Aggregation of fine particles is a fundamental process when synthesising or processing particulate material. The structure of the resulting aggregates strongly depends on factors like particle concentration, flow condition and interparticle forces. Consequently there exists a huge variety of potential particle configuration within an aggregate. One way to cope with this variety is the concept of fractality which is frequently employed for the description of aggregate structure [1–5]. Fractal-like aggregates have two characteristic properties. At first, their relative aggregate mass, i.e. the number of primary particles within the aggregate – the aggregation number N, scales with their relative size via a power law:
N ¼ kZ
DZ xZ xP
ð1Þ
Secondly, they possess a self-similar structure (in a statistic sense), which is reflected in the pair correlation function:
g / r Dg 3
ð2Þ
The exponent DZ in Eq. (1) is called fractal dimension. It depends on the particle property Z, on which the aggregate size xZ is based. Commonly the diameter of gyration or the hydrodynamic diameter is used. The diameter of gyration xg is proportional to the rmsvalue of the interparticle distance rkl in the aggregate (x2g ¼ 2hr 2kl i). ⇑ Corresponding author. E-mail address: [email protected] (F. Babick).
The structure of an aggregate determines its physical properties (diffusivity, sedimentation, scattering) as well as its interaction with other particles or aggregates. The latter is of particular relevance for colloidal aggregates formed by nanosized primary particles. It affects the aggregation step itself, the flocculation or gelling tendency of aggregate suspensions or their rheological behaviour. The total interaction between two particles or aggregates results from the superposition of attractive and repulsive forces. The main attractive component is the van-der-Waals interaction. Hitherto, there is no analytical approach to calculate the van-der-Waals interaction between two (fractal) aggregates. Approximate expressions, which are applicable to certain fractal dimensions or to either very small or very large distances, are not sufficiently investigated with regard to their validity. This paper addresses the calculation of the van-der-Waals interaction energy between fractal aggregates and the impact of the aggregate’s structure on the results. Particular attention is given to DLCA-like aggregates (DLCA = diffusion-limited cluster–cluster aggregation), which are typical for pyrogenic powders like fumed silica or carbon black [1,2,6]. These powders consist of aggregates in the submicron range made up of nanosized primary particles and are widely used in liquid dispersions (e.g. as thickening or polishing agent). 2. van-der Waals interaction 2.1. Interaction between spheres van-der-Waals interactions are interactions between atoms and molecules with fixed or induced polarisation, which result in a
0921-8831/$ - see front matter Ó 2010 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved. doi:10.1016/j.apt.2010.11.014
F. Babick et al. / Advanced Powder Technology 22 (2011) 220–225
221
Nomenclature Ak3l(h)
Ak3l,0 Ak3l,s Dg DZ d g h hcr
Lifshitz–Hamaker function for the interaction between substances k and l, which are being separated by medium 3 (J) non-retarded Hamaker constant (J) static contribution to Ak3l(h) (J) fractal dimension with respect to the diameter of gyration () fractal dimension with respect to equivalent diameter xZ, Eq. (1) () centre distance of the aggregates (m) pair correlation function (m3) minimum surface distance (m) characteristic distance for retardation (m)
strong attractive force for short distances. Particles obey such an attractive force as well. Hamaker [7] calculated this macroscopic interaction energy by assuming an undisturbed contribution from each elementary pair of atoms/molecules. For two spheres of diameter x1 and x2 he obtained:
A132 x1 x2 x1 x2 þ 12 h ðx1 þ x2 þ hÞ ðx1 þ hÞ ðx2 þ hÞ x1 x2 þ 2 ln 1 ðx1 þ hÞ ðx2 þ hÞ
V v dW ¼
ð3Þ
where h is the minimum surface distance and A132 is a coefficient, which depends on the materials of the particles (1 and 2) and on the medium 3 that separates them. Even though this approach neglects that many-body-aspect of the problem, it gives a good estimate for the dependency on the geometrical parameters. A more elaborated calculation scheme was developed by Lifshitz, which is based on the dielectric response of the involved substances to an electromagnetic field (e.g. [8]). This theory takes account of the decreasing contribution of the induced dipoles to the interaction energy with growing distance between the macro-bodies (retardation effect). That means, that the Hamaker coefficient A132 becomes a function of the interparticle gap h, the Lifshitz–Hamaker function, which can be approximated by [9,10]:
A132 ðhÞ ¼ A132;s þ ðA132;0 A132;s Þ ð1 þ ðh=hcr Þ3=2 Þ2=3
ð4Þ
where A132,0 is the non-retarded Hamaker constant (vanishing gap) and A132,s reflects the static contribution of permanent dipoles [10]. They can be computed from the dielectric function, which is known for several materials [11,12]. The parameter hcr represents a characteristic distance of the retardation and is obtained by fitting the exact Lifshitz–Hamaker function [11]. Strictly speaking, Eq. (4) is valid for two half-spaces only. In the case of two spherical particles there is a complex interrelation between geometry parameter and material properties. The exact analytical solution of the Lifshitz approach was presented by Langbein ([13], as cited in [14]). However, the necessary computations are rather laborious and converge slowly. For that reason, it seems appropriate to approximate the exact analytical solution by combining the Lifshitz–Hamaker function of two half-spaces (Eq. (4)) with the Hamaker geometry function for two spherical particles (Eq. (3)). Pailthorpe and Russel [15] and Arunachalam et al. [16] compared this approximation with the exact solution by Langbein for colloidal polystyrene spheres in an aqueous solution and for aerosol molecule clusters of tetrachloromethane (CCl4), respectively. In both cases the approximate approach underestimates
kZ N rkl xg xp xV VvdW
u j q
fractal prefactor with respect to equivalent diameter xZ, Eq. (1) () aggregation number () centre-to-centre distance between primary particles k and l (m) diameter of gyration (m) diameter of the primary particles (m) diameter of the volume equivalent sphere (m) interaction energy (J) aggregate packing density () Debye–Hückel parameter (m1) radial density function (m3)
the van-der-Waals interaction energy by approx. 10–20%. Pailthorpe and Russel conclude that this deviation is comparable with the uncertainty due to inaccurate dielectric values. The static contribution (of permanent dipoles) to the van-derWaals interaction A132,s is reduced, when the separating medium 3 contains mobile ions. In contrast, the dispersion contribution A132,d = A132,0 A132,s keeps unaffected, because the fluctuations of electrons are much faster than the Brownian motion of ions. Mahanty and Ninham ([17], as cited in [18]) describe the screening for jh 1 as follows:
A132 ðhÞ ¼ A132;s 2jh e2jh þ A132;d ðhÞ
ð5Þ
where j is the Debye–Hückel parameter. 2.2. Interaction between aggregates In the case of the interacting aggregates two limiting situations may be distinguished: At large aggregate separation the two aggregates act like volume equivalent spheres, whereas at short distances only the van-der-Waals interaction of the closest pair of primary particles should be relevant. However, for intermediate distances, the aggregate structure cannot be neglected. In order to study how structure influences the van-der-Waals interaction a simplified calculation scheme will be employed in this paper. That is because analytical expressions of Lifshitz’s macroscopic approach exist only for few regular geometries (e.g. interacting half-spaces [8,19] or spheres [20]). Furthermore, it is helpful to remind, that fractal aggregates are usually not closely packed and that therefore the mean interparticle distance is considerably larger than the primary particle size (in particular for fractal dimensions below 2.5). It thus seems fairly reasonable to assume, that the van-der-Waals interaction between a pair of primary particles is little affected by the presence of the other primary particles (i.e. the many-body aspect can be ignored). Consequently the van-der-Waals interaction energy between two aggregates can be calculated by summation of the interaction between all pairs of primary particles from aggregate A with those of aggregate B:
V v dW;AB ¼
XX
V v dW;ij
ð6Þ
The pair-contribution VvdW,i j is approximated in this paper with Eq. (3) by accounting for the retardation of the Hamaker coefficient at large distance (Eq. (4)). Arunachalam et al. [16] examined the performance of such discrete pairwise sum (i.e. Eq. (6)) for three spherical molecule clusters (CCl4, N = 55, £ 2.82 nm) with two of them in contact. They showed that the deviation from a Lifshitz approach, which accounts for the interaction between the electron fluctuations of
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the single molecules is very small. The contribution of the intermolecular coupling to the total interaction energy was in the order of 1%. Similarly Kim et al. [25] compared several van-der-Waals calculation schemes for a pair of nanoclusters (2340 molecules, £ 5.88 nm) with the coupled dipole method (CDM), which is claimed to be exact in the non-retarding region. For the discrete pairwise sum they observed significant difference only for very small gaps (approx. 15% for 1 nm), while for growing distances the discrete pairwise sum converges relatively fast with the CDM and Langbein solution. Based on these findings we conclude that Eq. (6) is a useful approximation of the total van-der-Waals interaction energy between two arbitrary clusters of particles. However a comprehensive discussion on its validity, in particular for densely packed aggregates, is still open to discussion. The discrete pairwise approach has already been employed by Naumann and Bunz [21] for fractal-like aggregates. In a conference proceeding they present an analytical expression for the interaction energy of two aggregates with a fractal dimensions of 2. For two identical aggregates of geometric size x their solution becomes:
V v dW ðdÞ ¼
u2 27
A131
2 3x 2
d
16x2 2
4d x2
þ
x2
u¼
2
d x2
ð7Þ
rmax,1
rmax,2 d
Fig. 1. Linear arrangement of aggregates for calculating the impact of the minimum surface distance, i.e. the distance between the closest pair of primary particles.
Table 1 Parameters of the Lifshitz–Hamaker function acc. to Eq. (4), for symmetrical, aqueous systems, based on data given in [11]. Material system
A131,0 (J)
A131,s (J)
hcr (nm)
Silica–water–silica Titania–water–titania
0.455 1020 5.35 1020
0.29 1020 0011 1020
9 23
1–3. The fractal prefactor kg was adjusted in a way that follows the correlation between both parameters for DLCA aggregates:
kg ¼ 18:085 e1:4846Dg ð8Þ
However, their paper only briefly sketches the derivation and does not fully explain the assumptions underlying their calculations. Essentially, they assume a spherical shape of the aggregates and a fractal law for the radial mass density:
qðrÞ / rDg 3
center 2
h
where the packing density u of the aggregate is given by:
N x3p x3
center 1
ð9Þ
These are simplifications, because isometric proportions are usually found only for Dg P 2.5 and because aggregates formed by sub-clusters do not behave according to Eq. (9). Additionally, the expression (7) can be only obtained, when the retardation of the dispersive interactions is ignored. Last but not least, Eq. (7) provides the orientation averaged interaction energy. It cannot be applied to a specific spatial configuration like two touching primary particles. Nevertheless, the solution of Naumann and Bunz may serve as useful analytical expression when performing fast calculations of the van-der-Waals interaction. 3. Calculation The calculation of the van-der-Waals interaction energy has been performed for two kinds of virtual aggregates. The first ones are DLCA aggregates, which are created by an algorithm described by Meakin [22,23]. The average fractal dimension of these aggregates is 1.8 [23], but since the algorithm is stochastic a certain variation in the aggregate structure is observed. The second type of virtual aggregates employed in this study is deterministic aggregates with a defined fractality. These aggregates obey the fractal growth condition (Eq. (1)) and show a fractal relationship for the radial density function. They were generated with a particle-cluster-aggregation algorithm that allows for tuning the fractal dimension Dg and the fractal prefactor kg (cf. Eq. (1)). The two parameters unambiguously define the configuration of particles within the aggregate. Hence, structural averaging as in the case of stochastic aggregates can be avoided. Details of the algorithm are presented elsewhere [5]. The deterministic aggregates were used for studying the effect of fractal dimension Dg on the aggregate interaction in the range
ð10Þ
Additionally, the two limiting cases of a linear chain (Dg = 1) and a hexagonal close-packed (hcp) aggregate of spheres (Dg = 3) were created. All calculations were done on MatlabÒ based on the equations given in Section 2. The impact of aggregate structure was investigated for two situations. First, the aggregates are aligned in such a way that their centres of gravity and their outermost primary particles lay on a straight line with the latter forming the closest pair of primary particles (Fig. 1). This configuration allowed for a controlled variation of the minimum surface distance h. Secondly, both aggregates are arbitrarily aligned in space for a given distance of the aggregate centres. In this case the orientation of the aggregates was randomly varied 100 times and the average values were considered. For DLCA aggregates this was repeated for 55 different aggregate pairs (arbitrarily chosen from a set of 100 aggregates). In order to avoid overlap of primary particles the centre distance d was restricted to values larger than the sum of the maximum radial extensions of the two aggregates. Two different materials were assumed for the calculation: amorphous silica and titania (rutile). The parameters for calculating the Lifshitz–Hamaker function A132(h) were taken from Bergström [11]. In Table 1 the thus determined non-retarded Hamaker constant, the static contribution to the Lifshitz–Hamaker function and the critical surface distance according to Eq. (4) are summarised. 4. Results and discussion 4.1. Impact of the aggregate structure This section is to look on the dependency of the interaction energy from the aggregate structure. The interaction energy was calculated for pairs of deterministic aggregates (cf. Section 2.2) with aggregation number N = 200 and primary particles of 20 nm. Fig. 2 shows the results for silica aggregates in a linear arrangement as a function of the minimum surface gap (cf. Fig. 1). The graph contains additionally curves, which are the solutions for a pair of primary particles:
V v dW ðhÞ ¼
A132 ðhÞ xp 24 h
ð11Þ
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F. Babick et al. / Advanced Powder Technology 22 (2011) 220–225
1E-02
1E+01 J -20
1E-01
- interaction energy, 10
point masses
1E-03 chain Dg = 1 Dg = 1.5 Dg = 2 Dg = 2.5 Dg = 3 hcp
1E-05 1E-07 1E-09 1E-11 1E-13 1
primary particles
10 100 1000 surface distance, nm
10000
Fig. 2. Value of the interaction energy vs. surface distance (linear arrangement, cf. Fig. 1) for deterministic aggregates (N = 200, xp = 20 nm) including an ideal chain and a hexagonal closed packing (hcp); asymptotes for a pair of primary particles, a pair of volume equivalent spheres and a pair of point masses are shown as well; material properties for silica (Table 1).
a pair of volume equivalent spheres:
A132 ðhÞ V v dW ðhÞ ¼ 12
x2V 2
d x2V
þ
x 2 V
d
! x 2 V þ 2 ln 1 d
ð12Þ
and a pair of two point masses
A132 ðdÞ xV 6 V v dW ðh ¼ dÞ ¼ 36 d
ð13Þ
For small surface distances (h xP) all curves converge with the solution for a pair of primary particles, whereas at very large distances (h xagg) all curves merge with the solution of volume equivalent spheres and point mass arrangement. The impact of structure is most pronounced for moderate distances (xp < h < xagg). In general, the interaction energy increases when the aggregate porosity decreases, i.e. the more pairs of primary particles are spaced in a relatively closed distance (with regard to the aggregate surface distance). However, for medium fractal dimensions (1.5 6 Dg 6 2.5) the curves are fairly close and do not exhibit a characteristic dependency on Dg. This is probably due to the negative correlation between the fractal prefactor and the fractal dimension, which is specifically defined for DLCA aggregates in this study (Eq. (10)), but can be observed for any kind of aggregation mechanism [24]. The fractal curves deviate considerably from that of the chain pair and that of the hexagonal close-packed (hcp) aggregates. At moderate surface distances the interaction of the latter is up to 10 times stronger than that of the former. The results imply that the van-der-Waals interaction energy of two aggregates in close proximity (h 6 0.5xp) can be approximated by the interaction energy of the closest pair of primary particles. This is relevant for highly concentrated suspensions or when computing the total interaction energy of two colliding particles. For moderate distances the interaction energy is clearly affected by the aggregate structure and not longer dominated by the closest pair of primary particles. In that region it is therefore more appropriate to study the interaction for a random alignment of the aggregates rather than to keep them in the linear arrangement as before (cf. Section 3). Fig. 3 depicts the corresponding interaction curves as a function of the aggregate centre distance d. For a given centre distance the average surface distance is considerably larger than for a linear arrangement. Consequently, the primary particle approximation
1E-04 1E-06 1E-08 1E-10
chain Dg = 1 Dg = 1.5 Dg = 2 Dg = 2.5 Dg = 3 hcp
1E-12 100
asymptode
1000 center distance, nm
10000
Fig. 3. Value of the orientation averaged interaction energy vs. centre distance for deterministic aggregates with (N = 200, xp = 20 nm) including an ideal chain and a hexagonal close packing (hcp); the asymptote represents a pair of equivalent point masses; material properties for silica (Table 1).
normalised interaction energy
- interaction energy, 10
-20
J
equiv. spheres
19 chain Dg = 1 Dg = 1.5 Dg = 2 Dg = 2.5 Dg = 3 hcp
16 13 10 7 4 1 2.5
3 3.5 4 4.5 normalised center distance
5
Fig. 4. Normalised presentation of Fig. 3; interaction energy is normalised with the asymptote (13) and the distance with the radius of gyration of the aggregates; error bars for Dg = 2 indicate standard deviation due to the variation in the aggregate orientation.
(Eq. (11)) is not applicable and the interaction values are lower than in the case of Fig. 2. For large centre distances all curves fall on the point mass asymptote. The data presentation in Fig. 3 lacks comparability because the minimum centre distance is defined by the aggregate size and therefore depends on the aggregate structure. For that reason the data in Fig. 3 are normalised by scaling the interaction energies of aggregates with those of corresponding point masses (Eq. (13)) and by relating the centre distance to an appropriate aggregate size. Here, the radius of gyration has been chosen (cf. Section 1). The transformed graphs are shown in Fig. 4. Additionally, an indication is given for Dg = 2, to which degree the aggregate orientation affects the interaction energy. Regarding this impact the variation among the different aggregate structures is not very dramatic (with the exception of the chain-like aggregates). 4.2. Impact of retardation All previous examples have been calculated with the material data of amorphous silica. For this substance the non-retarded Hamaker constant A131,0 and the static contribution A131,s are in the same order of magnitude. That means, that the retardation of the dispersion interactions is not very pronounced for silica. This is evidently different for titania (Table 1), where the static contribution
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F. Babick et al. / Advanced Powder Technology 22 (2011) 220–225
1E-03 - interactin energy, 10
-20
J
silica, Dg = 2 1E-04
silica, Naumann titania, Dg = 2
1E-05
titania, Naumann
4.3. Results for DLCA aggregates
1E-06 1E-07 1E-08 400
600 800 center distance, nm
1000
normalised interaction energy
Fig. 5. Interaction energy of deterministic aggregates with Dg = 2 (N = 200, xp = 20 nm) calculated by discrete pairwise summation and using the Lifshitz– Hamaker function (cf. Section 2.2) and by the integral solution of Naumann (Eq. (7)).
a
3 N=45 2.6
N=100 N=200
2.2
N=500 1.8
N=1000
1.4
normalised interaction energy
In contrast to the deterministic aggregates it is not possible to adjust the fractal structure for DLCA aggregates. Due to their stochastic formation there exists certain variability in the particle configuration and in the fractal parameter. For that reason the computation of the van-der-Waals interaction energy has been conducted for different pairs of fractal aggregates randomly aligned in space (cf. Section 3). Fig. 6 shows the resulting interaction curves for silica and titania aggregates of varying size (N). The presentation of data follows the normalisation rules introduced in Section 4.1. In both diagrams the curves of the differently sized aggregates are very close. The differences are much smaller than the variation due to changing orientation (100 times) and replacing the aggregate pairs (55). It is very likely that the existing deviations are mainly due to the relatively small set of employed DLCA aggregates. Anyway, the diagrams in Fig. 6 reveal that the radius (or diameter) of gyration is an appropriate length scale for modelling the interaction energy for this type of aggregate. It facilitates the generation of a mastercurve for the average interaction energy for a given material. The impact of the material is related to its specific retardation of the dispersion interaction. Once such a mastercurve has been established, it can be used for investigating properties of the whole suspension like the internal structure or its stability. 4.4. Remarks on the practical relevance
1 4
b
perfectly fits with our data in the case of silica, but it shows significant deviations in the case of titania (Fig. 5). We therefore believe that accounting for retardation already in the pair interaction of primary particles is necessary when dealing with aggregates of nanosized particles (i.e. when xp is in the order of magnitude of the critical distance hcr of the Lifshitz–Hamaker function).
5 6 7 8 normalised center distance
9
5 N=45 N=100
4
N=200 N=500
3
N=1000 2
1 4
5 6 7 8 normalised center distance
9
Fig. 6. Averaged normalised interaction energy vs. normalised centre distance for randomly arranged DLCA aggregates of varying aggregation number N (xp = 20 nm); error bars for N = 200 represent variation due to changing orientation and aggregate configuration; (a) result for silica, (b) result for titania (rutile) (cf. Table 1).
to the van-der-Waals interaction is less than 1%. As a consequence, the interactions energy decays more rapidly with aggregate distance than any approach, which neglects the retardation (e.g. CDM [25]). The Naumann solution (Eq. (7)) for example is based on a constant value for A131 when integrating the pair interaction over the aggregate volume. Using the non-retarded Hamaker constant definitely leads to wrong solution. Alternatively the constant A131 in (Eq. (7)) may be replaced by the Lifshitz–Hamaker function A131(h). With this approximation the Naumann solution almost
The results of the previous sections cover only a limited number of aggregate configuration, structure and size and have been restricted to primary particles of 20 nm. It is therefore necessary to discuss the general outcome. For that purpose we will return to Eq. (3) first, which was used to calculate the interaction energy between all pairs of primary particles. This energy does not primarily depend on an absolute length scale but on the ratio h/xp. However, due to the retardation of the dispersion interaction there is a certain dependency on the distance h. Anyway, retardation reduces the strength of interaction but does not change qualitatively the impact of size and structure. Consequently, we can derive general rules from our results. For a linear alignment according to Fig. 1 it was shown, that the aggregate structure affects the interaction only at moderate distances, whereas for very short surfaces distances (h 6 0.5xp) the interaction is completely determined by the closest pair of primary particles. At very large distances (h xagg) the configuration of particles is unimportant and the aggregate mass becomes the decisive parameter. The transition from the primary particle impact to the structure impact at h = 0.5xp corresponds with an interaction energy of 0.0057A132 according to Eq. (3). That are very small values; even for the relatively strong interacting system of titania particles in water (A132,0 = 5.35 1020 J) this is not more than 0.075 kT at room temperature. Hence, the primary particle approximation can be applied to all surface distances but those, where the vander-Waals energy has no physical relevance, i.e. where it is exceeded by the thermal energy. Regarding two approaching DLCA aggregates it is quite conceivable that there are only a few (may be just one) pairs of primary particles that really contribute to the van-der-Waals interaction
F. Babick et al. / Advanced Powder Technology 22 (2011) 220–225
energy of the closest pair(s) of primary particles. Strictly speaking, this approximation is applicable only for surfaces distances within the radius of the radius of the primary particles (h 6 0.5xp), but for larger distances van-der-Waals interactions are negligible. Nonetheless, since the morphology and outer radial density of an aggregate affects the surface distance between two aggregates with given centre distance, there is an apparent impact of the aggregates’ structure on their van-der-Waals interaction. In contrast to simple approximations, the proposed calculation scheme provides a possibility to account for that. Full benefit will certainly be gained when this van-der-Waals calculation is combined with the calculation of other types of interaction.
N×VvdW,PP /VvdW,chain
1 0.8 0.6 N=2
0.4
N=10 0.2
N=20 N=100
0 1
225
10 100 1000 surface distance, nm
10000 References
Fig. 7. Interaction energy vs. surface distance for two linear aggregates that are aligned in parallel; the sum over all (N) facing pairs of primary particles is related to the interaction energy of the aggregates.
between the aggregates. That means, that the attraction between such aggregates is almost independent of their size (for a given xp). The opposite is true for two linear chain-aggregates that are aligned in parallel. In that case the interaction energy grows with the number of opposing pairs of primary particles. For small surface distances (h 6 0.5xp) the dependency is almost linear (Fig. 7). Finally, after having derived that the van-der-Waals interaction energy between two fractal aggregates can be traced back to the pair-interaction of a few primary particles at close proximity, we have to ask, if there is a need for a complete calculation scheme as introduced in this paper and if there is any structural impact? In our opinion both questions can be affirmed. The calculation scheme is required for looking at arbitrary aggregate configurations and alignments as well as for deriving average relationships between the centre distance and the interaction energy. Moreover, such relationships are clearly affected by the morphology and radial density distributions, i.e. by the structure, of the aggregates (cf. Fig. 3). Mastercurves as being indicated by Fig. 6 cannot be (easily) obtained analytically but require appropriate numerical tools. 5. Conclusions This paper has presented a straightforward algorithm for computing the van-der-Waals interactions between aggregates of colloidal particles. It is based on a pairwise summation of interaction energies and employs the Lifshitz–Hamaker function for half-spaces. The simplifications behind this approach (like the complete ignorance of the many-body aspect) have been discussed based on published data of more elaborate calculation schemes. It appears that the model uncertainty is most probably not larger than the uncertainty in the material parameters. The calculation scheme was further applied to deterministic and DLCA aggregates. The former allow for tuning the aggregate structure via fractal parameters, while the latter show a certain variation in the particle configuration, but posses an average fractal dimension of 1.8. DLCA aggregates were chosen as an adequate model for pyrogenic powders, which have a high economic relevance. It could be shown, that the van-der-Waals interaction energy can be fairly well approximated by computing the interaction
[1] D.W. Schaefer, A.J. Hurd, Growth and structure of combustions aerosols, Aerosol Sci. Technol. 12 (1990) 876–890. [2] C.M. Sorensen, Light scattering by fractal aggregates: a review, Aerosol Sci. Technol. 35 (2001) 648–687. [3] P. Sandkühler, M. Lattuada, H. Wu, J. Sefcik, M. Morbidelli, Further insights into the universality of colloidal aggregation, Adv. Colloid Interface Sci. 113 (2005) 65–83. [4] J. Schlomach, Feststoffbildung bei technischen Fällprozessen – Untersuchungen zur industriellen Fällung von Siliziumdoxid und Calciumcarbonat, Dissertation, Universität Karlsruhe (TH), 2006. [5] U. Kätzel, R. Bedrich, M. Stintz, R. Ketzmerick, T. Gottschalk-Gaudig, H. Barthel, Dynamic light scattering for the characterization of polydisperse fractal systems: I. Simulation of the diffusional behavior, Part. Part. Syst. Charact. 25 (1) (2008) 9–18. [6] U. Kätzel, M. Vorbau, M. Stintz, T. Gottschalk-Gaudig, H. Barthel, Dynamic light scattering for the characterization of polydisperse fractal systems: II. Relation between structure and DLS results, Part. Part. Syst. Charact. 25 (1) (2008) 19– 30. [7] H.C. Hamaker, The London – van der Waals attraction between spherical particles, Physica IV 10 (1937) 1058–1072. [8] J. Lyklema, Fundamentals of Interface and Colloid Science, vol. 1: Fundamentals, Academic Press, 1991. [9] W.R. Bowen, F. Jenner, The calculation of dispersion forces for engineering applications, Adv. Colloid Interface Sci. 56 (1995) 201–243. [10] P. Viravathana, D.W.M. Marr, Optical trapping of titania/silica core–shell colloidal particles, J. Colloid Interface Sci. 221 (2000) 301–307. [11] L. Bergström, Hamaker constants of inorganic materials, Adv. Colloid Interface Sci. 70 (1997) 125–169. [12] A.V. Nguyen, Improved approximation of water dielectric permittivity for calculation of Hamaker constants, J. Colloid Interface Sci. 229 (2000) 648–651. [13] D. Langbein, Theory of van der Waals Attraction, Springer, Verlag, 1974. [14] S.N. Thennadil, L.H. Garcia-Rubio, Approximations for calculating van der Waals interaction energy between spherical particles – a comparison, J. Colloid Interface Sci. 243 (2001) 136–142. [15] B.A. Pailthorpe, W.B. Russel, The retarded van der Waals interaction between spheres, J. Colloid Interface Sci. 89 (1982) 563–566. [16] V. Arunachalam, W.H. Marlow, J.X. Lu, Development of a picture of the van der Waals interaction energy between clusters of nanometer-range particles, Phys. Rev. E 58 (1998) 3451–3457. [17] J. Mahanty, B.W. Ninham, Dispersion Forces, Academic Press, 1976. [18] J.N. Israelachvili, Intermolecular and Surface Forces, Academic Press, 1992. [19] E.M. Lifshitz, The theory of molecular attractive forces between solids, Sov. Phys. JETP 2 (1) (1956) 73–83. [20] D. Langbein, Non-retarded dispersion energy between macroscopic spheres, J. Phys. Chem. Solids 32 (1971) 1657–1667. [21] K.-H. Naumann, H. Bunz, van der Waals interactions between fractal particles, J. Aerosol Sci. 24 (1993) S181–S182. [22] P. Meakin, Formation of fractal clusters and networks by irreversible diffusion limited aggregation, Phys. Rev. Lett. 51 (1983) 1119–1122. [23] P. Meakin, Effects of cluster trajectories on cluster–cluster aggregation: a comparison of linear and Brownian trajectories in two- and three-dimensional simulations, Phys. Rev. A 29 (2) (1984) 997–999. [24] C.M. Sorensen, G.C. Roberts, The prefactor of fractal aggregates, J. Colloid Interface Sci. 186 (1997) 447–452. [25] H.-Y. Kim, J.O. Sofo, D. Velegol, M.W. Cole, A.A. Lucas, van der Waals dispersion forces between dielectric nanoclusters, Langmuir 23 (2007) 1735–1740.
Contents lists available at ScienceDirect
Advanced Powder Technology journal homepage: www.elsevier.com/locate/apt
Original Research Paper
van-der-Waals interaction between two fractal aggregates Frank Babick a,⇑, Karin Schießl b, Michael Stintz a a b
Institut für Verfahrenstechnik und Umwelttechnik, TU Dresden, 01062 Dresden, Germany Institut für Strömungsmechanik, TU Dresden, 01062 Dresden, Germany
a r t i c l e
i n f o
Article history: Received 30 July 2010 Received in revised form 15 November 2010 Accepted 29 November 2010 Available online 13 December 2010 Keywords: Fractal aggregate van-der-Waals interaction DLCA aggregate Lifshitz–Hamaker function
a b s t r a c t The van-der-Waals forces between particles are of fundamental importance for agglomeration processes or their structuring in suspensions. The paper describes a calculation scheme for the corresponding energy potential when the interacting objects are aggregates of colloidal primary particles. This scheme is further employed to examine the impact of aggregate structure on the van-der-Waals interaction. Calculations were therefore conducted for deterministic aggregates with tuneable fractal dimension and for DLCA aggregates of varying size. It appears that the van-der-Waals interaction is governed by the closest pair(s) of primary particles. That results in an impact of the aggregate structure, when the interaction energy is plotted versus the aggregate centre distance. For DLCA aggregates this relationship can be described by a size-independent, but material specific mastercurve. Ó 2010 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved.
1. Motivation Aggregation of fine particles is a fundamental process when synthesising or processing particulate material. The structure of the resulting aggregates strongly depends on factors like particle concentration, flow condition and interparticle forces. Consequently there exists a huge variety of potential particle configuration within an aggregate. One way to cope with this variety is the concept of fractality which is frequently employed for the description of aggregate structure [1–5]. Fractal-like aggregates have two characteristic properties. At first, their relative aggregate mass, i.e. the number of primary particles within the aggregate – the aggregation number N, scales with their relative size via a power law:
N ¼ kZ
DZ xZ xP
ð1Þ
Secondly, they possess a self-similar structure (in a statistic sense), which is reflected in the pair correlation function:
g / r Dg 3
ð2Þ
The exponent DZ in Eq. (1) is called fractal dimension. It depends on the particle property Z, on which the aggregate size xZ is based. Commonly the diameter of gyration or the hydrodynamic diameter is used. The diameter of gyration xg is proportional to the rmsvalue of the interparticle distance rkl in the aggregate (x2g ¼ 2hr 2kl i). ⇑ Corresponding author. E-mail address: [email protected] (F. Babick).
The structure of an aggregate determines its physical properties (diffusivity, sedimentation, scattering) as well as its interaction with other particles or aggregates. The latter is of particular relevance for colloidal aggregates formed by nanosized primary particles. It affects the aggregation step itself, the flocculation or gelling tendency of aggregate suspensions or their rheological behaviour. The total interaction between two particles or aggregates results from the superposition of attractive and repulsive forces. The main attractive component is the van-der-Waals interaction. Hitherto, there is no analytical approach to calculate the van-der-Waals interaction between two (fractal) aggregates. Approximate expressions, which are applicable to certain fractal dimensions or to either very small or very large distances, are not sufficiently investigated with regard to their validity. This paper addresses the calculation of the van-der-Waals interaction energy between fractal aggregates and the impact of the aggregate’s structure on the results. Particular attention is given to DLCA-like aggregates (DLCA = diffusion-limited cluster–cluster aggregation), which are typical for pyrogenic powders like fumed silica or carbon black [1,2,6]. These powders consist of aggregates in the submicron range made up of nanosized primary particles and are widely used in liquid dispersions (e.g. as thickening or polishing agent). 2. van-der Waals interaction 2.1. Interaction between spheres van-der-Waals interactions are interactions between atoms and molecules with fixed or induced polarisation, which result in a
0921-8831/$ - see front matter Ó 2010 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved. doi:10.1016/j.apt.2010.11.014
F. Babick et al. / Advanced Powder Technology 22 (2011) 220–225
221
Nomenclature Ak3l(h)
Ak3l,0 Ak3l,s Dg DZ d g h hcr
Lifshitz–Hamaker function for the interaction between substances k and l, which are being separated by medium 3 (J) non-retarded Hamaker constant (J) static contribution to Ak3l(h) (J) fractal dimension with respect to the diameter of gyration () fractal dimension with respect to equivalent diameter xZ, Eq. (1) () centre distance of the aggregates (m) pair correlation function (m3) minimum surface distance (m) characteristic distance for retardation (m)
strong attractive force for short distances. Particles obey such an attractive force as well. Hamaker [7] calculated this macroscopic interaction energy by assuming an undisturbed contribution from each elementary pair of atoms/molecules. For two spheres of diameter x1 and x2 he obtained:
A132 x1 x2 x1 x2 þ 12 h ðx1 þ x2 þ hÞ ðx1 þ hÞ ðx2 þ hÞ x1 x2 þ 2 ln 1 ðx1 þ hÞ ðx2 þ hÞ
V v dW ¼
ð3Þ
where h is the minimum surface distance and A132 is a coefficient, which depends on the materials of the particles (1 and 2) and on the medium 3 that separates them. Even though this approach neglects that many-body-aspect of the problem, it gives a good estimate for the dependency on the geometrical parameters. A more elaborated calculation scheme was developed by Lifshitz, which is based on the dielectric response of the involved substances to an electromagnetic field (e.g. [8]). This theory takes account of the decreasing contribution of the induced dipoles to the interaction energy with growing distance between the macro-bodies (retardation effect). That means, that the Hamaker coefficient A132 becomes a function of the interparticle gap h, the Lifshitz–Hamaker function, which can be approximated by [9,10]:
A132 ðhÞ ¼ A132;s þ ðA132;0 A132;s Þ ð1 þ ðh=hcr Þ3=2 Þ2=3
ð4Þ
where A132,0 is the non-retarded Hamaker constant (vanishing gap) and A132,s reflects the static contribution of permanent dipoles [10]. They can be computed from the dielectric function, which is known for several materials [11,12]. The parameter hcr represents a characteristic distance of the retardation and is obtained by fitting the exact Lifshitz–Hamaker function [11]. Strictly speaking, Eq. (4) is valid for two half-spaces only. In the case of two spherical particles there is a complex interrelation between geometry parameter and material properties. The exact analytical solution of the Lifshitz approach was presented by Langbein ([13], as cited in [14]). However, the necessary computations are rather laborious and converge slowly. For that reason, it seems appropriate to approximate the exact analytical solution by combining the Lifshitz–Hamaker function of two half-spaces (Eq. (4)) with the Hamaker geometry function for two spherical particles (Eq. (3)). Pailthorpe and Russel [15] and Arunachalam et al. [16] compared this approximation with the exact solution by Langbein for colloidal polystyrene spheres in an aqueous solution and for aerosol molecule clusters of tetrachloromethane (CCl4), respectively. In both cases the approximate approach underestimates
kZ N rkl xg xp xV VvdW
u j q
fractal prefactor with respect to equivalent diameter xZ, Eq. (1) () aggregation number () centre-to-centre distance between primary particles k and l (m) diameter of gyration (m) diameter of the primary particles (m) diameter of the volume equivalent sphere (m) interaction energy (J) aggregate packing density () Debye–Hückel parameter (m1) radial density function (m3)
the van-der-Waals interaction energy by approx. 10–20%. Pailthorpe and Russel conclude that this deviation is comparable with the uncertainty due to inaccurate dielectric values. The static contribution (of permanent dipoles) to the van-derWaals interaction A132,s is reduced, when the separating medium 3 contains mobile ions. In contrast, the dispersion contribution A132,d = A132,0 A132,s keeps unaffected, because the fluctuations of electrons are much faster than the Brownian motion of ions. Mahanty and Ninham ([17], as cited in [18]) describe the screening for jh 1 as follows:
A132 ðhÞ ¼ A132;s 2jh e2jh þ A132;d ðhÞ
ð5Þ
where j is the Debye–Hückel parameter. 2.2. Interaction between aggregates In the case of the interacting aggregates two limiting situations may be distinguished: At large aggregate separation the two aggregates act like volume equivalent spheres, whereas at short distances only the van-der-Waals interaction of the closest pair of primary particles should be relevant. However, for intermediate distances, the aggregate structure cannot be neglected. In order to study how structure influences the van-der-Waals interaction a simplified calculation scheme will be employed in this paper. That is because analytical expressions of Lifshitz’s macroscopic approach exist only for few regular geometries (e.g. interacting half-spaces [8,19] or spheres [20]). Furthermore, it is helpful to remind, that fractal aggregates are usually not closely packed and that therefore the mean interparticle distance is considerably larger than the primary particle size (in particular for fractal dimensions below 2.5). It thus seems fairly reasonable to assume, that the van-der-Waals interaction between a pair of primary particles is little affected by the presence of the other primary particles (i.e. the many-body aspect can be ignored). Consequently the van-der-Waals interaction energy between two aggregates can be calculated by summation of the interaction between all pairs of primary particles from aggregate A with those of aggregate B:
V v dW;AB ¼
XX
V v dW;ij
ð6Þ
The pair-contribution VvdW,i j is approximated in this paper with Eq. (3) by accounting for the retardation of the Hamaker coefficient at large distance (Eq. (4)). Arunachalam et al. [16] examined the performance of such discrete pairwise sum (i.e. Eq. (6)) for three spherical molecule clusters (CCl4, N = 55, £ 2.82 nm) with two of them in contact. They showed that the deviation from a Lifshitz approach, which accounts for the interaction between the electron fluctuations of
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F. Babick et al. / Advanced Powder Technology 22 (2011) 220–225
the single molecules is very small. The contribution of the intermolecular coupling to the total interaction energy was in the order of 1%. Similarly Kim et al. [25] compared several van-der-Waals calculation schemes for a pair of nanoclusters (2340 molecules, £ 5.88 nm) with the coupled dipole method (CDM), which is claimed to be exact in the non-retarding region. For the discrete pairwise sum they observed significant difference only for very small gaps (approx. 15% for 1 nm), while for growing distances the discrete pairwise sum converges relatively fast with the CDM and Langbein solution. Based on these findings we conclude that Eq. (6) is a useful approximation of the total van-der-Waals interaction energy between two arbitrary clusters of particles. However a comprehensive discussion on its validity, in particular for densely packed aggregates, is still open to discussion. The discrete pairwise approach has already been employed by Naumann and Bunz [21] for fractal-like aggregates. In a conference proceeding they present an analytical expression for the interaction energy of two aggregates with a fractal dimensions of 2. For two identical aggregates of geometric size x their solution becomes:
V v dW ðdÞ ¼
u2 27
A131
2 3x 2
d
16x2 2
4d x2
þ
x2
u¼
2
d x2
ð7Þ
rmax,1
rmax,2 d
Fig. 1. Linear arrangement of aggregates for calculating the impact of the minimum surface distance, i.e. the distance between the closest pair of primary particles.
Table 1 Parameters of the Lifshitz–Hamaker function acc. to Eq. (4), for symmetrical, aqueous systems, based on data given in [11]. Material system
A131,0 (J)
A131,s (J)
hcr (nm)
Silica–water–silica Titania–water–titania
0.455 1020 5.35 1020
0.29 1020 0011 1020
9 23
1–3. The fractal prefactor kg was adjusted in a way that follows the correlation between both parameters for DLCA aggregates:
kg ¼ 18:085 e1:4846Dg ð8Þ
However, their paper only briefly sketches the derivation and does not fully explain the assumptions underlying their calculations. Essentially, they assume a spherical shape of the aggregates and a fractal law for the radial mass density:
qðrÞ / rDg 3
center 2
h
where the packing density u of the aggregate is given by:
N x3p x3
center 1
ð9Þ
These are simplifications, because isometric proportions are usually found only for Dg P 2.5 and because aggregates formed by sub-clusters do not behave according to Eq. (9). Additionally, the expression (7) can be only obtained, when the retardation of the dispersive interactions is ignored. Last but not least, Eq. (7) provides the orientation averaged interaction energy. It cannot be applied to a specific spatial configuration like two touching primary particles. Nevertheless, the solution of Naumann and Bunz may serve as useful analytical expression when performing fast calculations of the van-der-Waals interaction. 3. Calculation The calculation of the van-der-Waals interaction energy has been performed for two kinds of virtual aggregates. The first ones are DLCA aggregates, which are created by an algorithm described by Meakin [22,23]. The average fractal dimension of these aggregates is 1.8 [23], but since the algorithm is stochastic a certain variation in the aggregate structure is observed. The second type of virtual aggregates employed in this study is deterministic aggregates with a defined fractality. These aggregates obey the fractal growth condition (Eq. (1)) and show a fractal relationship for the radial density function. They were generated with a particle-cluster-aggregation algorithm that allows for tuning the fractal dimension Dg and the fractal prefactor kg (cf. Eq. (1)). The two parameters unambiguously define the configuration of particles within the aggregate. Hence, structural averaging as in the case of stochastic aggregates can be avoided. Details of the algorithm are presented elsewhere [5]. The deterministic aggregates were used for studying the effect of fractal dimension Dg on the aggregate interaction in the range
ð10Þ
Additionally, the two limiting cases of a linear chain (Dg = 1) and a hexagonal close-packed (hcp) aggregate of spheres (Dg = 3) were created. All calculations were done on MatlabÒ based on the equations given in Section 2. The impact of aggregate structure was investigated for two situations. First, the aggregates are aligned in such a way that their centres of gravity and their outermost primary particles lay on a straight line with the latter forming the closest pair of primary particles (Fig. 1). This configuration allowed for a controlled variation of the minimum surface distance h. Secondly, both aggregates are arbitrarily aligned in space for a given distance of the aggregate centres. In this case the orientation of the aggregates was randomly varied 100 times and the average values were considered. For DLCA aggregates this was repeated for 55 different aggregate pairs (arbitrarily chosen from a set of 100 aggregates). In order to avoid overlap of primary particles the centre distance d was restricted to values larger than the sum of the maximum radial extensions of the two aggregates. Two different materials were assumed for the calculation: amorphous silica and titania (rutile). The parameters for calculating the Lifshitz–Hamaker function A132(h) were taken from Bergström [11]. In Table 1 the thus determined non-retarded Hamaker constant, the static contribution to the Lifshitz–Hamaker function and the critical surface distance according to Eq. (4) are summarised. 4. Results and discussion 4.1. Impact of the aggregate structure This section is to look on the dependency of the interaction energy from the aggregate structure. The interaction energy was calculated for pairs of deterministic aggregates (cf. Section 2.2) with aggregation number N = 200 and primary particles of 20 nm. Fig. 2 shows the results for silica aggregates in a linear arrangement as a function of the minimum surface gap (cf. Fig. 1). The graph contains additionally curves, which are the solutions for a pair of primary particles:
V v dW ðhÞ ¼
A132 ðhÞ xp 24 h
ð11Þ
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F. Babick et al. / Advanced Powder Technology 22 (2011) 220–225
1E-02
1E+01 J -20
1E-01
- interaction energy, 10
point masses
1E-03 chain Dg = 1 Dg = 1.5 Dg = 2 Dg = 2.5 Dg = 3 hcp
1E-05 1E-07 1E-09 1E-11 1E-13 1
primary particles
10 100 1000 surface distance, nm
10000
Fig. 2. Value of the interaction energy vs. surface distance (linear arrangement, cf. Fig. 1) for deterministic aggregates (N = 200, xp = 20 nm) including an ideal chain and a hexagonal closed packing (hcp); asymptotes for a pair of primary particles, a pair of volume equivalent spheres and a pair of point masses are shown as well; material properties for silica (Table 1).
a pair of volume equivalent spheres:
A132 ðhÞ V v dW ðhÞ ¼ 12
x2V 2
d x2V
þ
x 2 V
d
! x 2 V þ 2 ln 1 d
ð12Þ
and a pair of two point masses
A132 ðdÞ xV 6 V v dW ðh ¼ dÞ ¼ 36 d
ð13Þ
For small surface distances (h xP) all curves converge with the solution for a pair of primary particles, whereas at very large distances (h xagg) all curves merge with the solution of volume equivalent spheres and point mass arrangement. The impact of structure is most pronounced for moderate distances (xp < h < xagg). In general, the interaction energy increases when the aggregate porosity decreases, i.e. the more pairs of primary particles are spaced in a relatively closed distance (with regard to the aggregate surface distance). However, for medium fractal dimensions (1.5 6 Dg 6 2.5) the curves are fairly close and do not exhibit a characteristic dependency on Dg. This is probably due to the negative correlation between the fractal prefactor and the fractal dimension, which is specifically defined for DLCA aggregates in this study (Eq. (10)), but can be observed for any kind of aggregation mechanism [24]. The fractal curves deviate considerably from that of the chain pair and that of the hexagonal close-packed (hcp) aggregates. At moderate surface distances the interaction of the latter is up to 10 times stronger than that of the former. The results imply that the van-der-Waals interaction energy of two aggregates in close proximity (h 6 0.5xp) can be approximated by the interaction energy of the closest pair of primary particles. This is relevant for highly concentrated suspensions or when computing the total interaction energy of two colliding particles. For moderate distances the interaction energy is clearly affected by the aggregate structure and not longer dominated by the closest pair of primary particles. In that region it is therefore more appropriate to study the interaction for a random alignment of the aggregates rather than to keep them in the linear arrangement as before (cf. Section 3). Fig. 3 depicts the corresponding interaction curves as a function of the aggregate centre distance d. For a given centre distance the average surface distance is considerably larger than for a linear arrangement. Consequently, the primary particle approximation
1E-04 1E-06 1E-08 1E-10
chain Dg = 1 Dg = 1.5 Dg = 2 Dg = 2.5 Dg = 3 hcp
1E-12 100
asymptode
1000 center distance, nm
10000
Fig. 3. Value of the orientation averaged interaction energy vs. centre distance for deterministic aggregates with (N = 200, xp = 20 nm) including an ideal chain and a hexagonal close packing (hcp); the asymptote represents a pair of equivalent point masses; material properties for silica (Table 1).
normalised interaction energy
- interaction energy, 10
-20
J
equiv. spheres
19 chain Dg = 1 Dg = 1.5 Dg = 2 Dg = 2.5 Dg = 3 hcp
16 13 10 7 4 1 2.5
3 3.5 4 4.5 normalised center distance
5
Fig. 4. Normalised presentation of Fig. 3; interaction energy is normalised with the asymptote (13) and the distance with the radius of gyration of the aggregates; error bars for Dg = 2 indicate standard deviation due to the variation in the aggregate orientation.
(Eq. (11)) is not applicable and the interaction values are lower than in the case of Fig. 2. For large centre distances all curves fall on the point mass asymptote. The data presentation in Fig. 3 lacks comparability because the minimum centre distance is defined by the aggregate size and therefore depends on the aggregate structure. For that reason the data in Fig. 3 are normalised by scaling the interaction energies of aggregates with those of corresponding point masses (Eq. (13)) and by relating the centre distance to an appropriate aggregate size. Here, the radius of gyration has been chosen (cf. Section 1). The transformed graphs are shown in Fig. 4. Additionally, an indication is given for Dg = 2, to which degree the aggregate orientation affects the interaction energy. Regarding this impact the variation among the different aggregate structures is not very dramatic (with the exception of the chain-like aggregates). 4.2. Impact of retardation All previous examples have been calculated with the material data of amorphous silica. For this substance the non-retarded Hamaker constant A131,0 and the static contribution A131,s are in the same order of magnitude. That means, that the retardation of the dispersion interactions is not very pronounced for silica. This is evidently different for titania (Table 1), where the static contribution
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F. Babick et al. / Advanced Powder Technology 22 (2011) 220–225
1E-03 - interactin energy, 10
-20
J
silica, Dg = 2 1E-04
silica, Naumann titania, Dg = 2
1E-05
titania, Naumann
4.3. Results for DLCA aggregates
1E-06 1E-07 1E-08 400
600 800 center distance, nm
1000
normalised interaction energy
Fig. 5. Interaction energy of deterministic aggregates with Dg = 2 (N = 200, xp = 20 nm) calculated by discrete pairwise summation and using the Lifshitz– Hamaker function (cf. Section 2.2) and by the integral solution of Naumann (Eq. (7)).
a
3 N=45 2.6
N=100 N=200
2.2
N=500 1.8
N=1000
1.4
normalised interaction energy
In contrast to the deterministic aggregates it is not possible to adjust the fractal structure for DLCA aggregates. Due to their stochastic formation there exists certain variability in the particle configuration and in the fractal parameter. For that reason the computation of the van-der-Waals interaction energy has been conducted for different pairs of fractal aggregates randomly aligned in space (cf. Section 3). Fig. 6 shows the resulting interaction curves for silica and titania aggregates of varying size (N). The presentation of data follows the normalisation rules introduced in Section 4.1. In both diagrams the curves of the differently sized aggregates are very close. The differences are much smaller than the variation due to changing orientation (100 times) and replacing the aggregate pairs (55). It is very likely that the existing deviations are mainly due to the relatively small set of employed DLCA aggregates. Anyway, the diagrams in Fig. 6 reveal that the radius (or diameter) of gyration is an appropriate length scale for modelling the interaction energy for this type of aggregate. It facilitates the generation of a mastercurve for the average interaction energy for a given material. The impact of the material is related to its specific retardation of the dispersion interaction. Once such a mastercurve has been established, it can be used for investigating properties of the whole suspension like the internal structure or its stability. 4.4. Remarks on the practical relevance
1 4
b
perfectly fits with our data in the case of silica, but it shows significant deviations in the case of titania (Fig. 5). We therefore believe that accounting for retardation already in the pair interaction of primary particles is necessary when dealing with aggregates of nanosized particles (i.e. when xp is in the order of magnitude of the critical distance hcr of the Lifshitz–Hamaker function).
5 6 7 8 normalised center distance
9
5 N=45 N=100
4
N=200 N=500
3
N=1000 2
1 4
5 6 7 8 normalised center distance
9
Fig. 6. Averaged normalised interaction energy vs. normalised centre distance for randomly arranged DLCA aggregates of varying aggregation number N (xp = 20 nm); error bars for N = 200 represent variation due to changing orientation and aggregate configuration; (a) result for silica, (b) result for titania (rutile) (cf. Table 1).
to the van-der-Waals interaction is less than 1%. As a consequence, the interactions energy decays more rapidly with aggregate distance than any approach, which neglects the retardation (e.g. CDM [25]). The Naumann solution (Eq. (7)) for example is based on a constant value for A131 when integrating the pair interaction over the aggregate volume. Using the non-retarded Hamaker constant definitely leads to wrong solution. Alternatively the constant A131 in (Eq. (7)) may be replaced by the Lifshitz–Hamaker function A131(h). With this approximation the Naumann solution almost
The results of the previous sections cover only a limited number of aggregate configuration, structure and size and have been restricted to primary particles of 20 nm. It is therefore necessary to discuss the general outcome. For that purpose we will return to Eq. (3) first, which was used to calculate the interaction energy between all pairs of primary particles. This energy does not primarily depend on an absolute length scale but on the ratio h/xp. However, due to the retardation of the dispersion interaction there is a certain dependency on the distance h. Anyway, retardation reduces the strength of interaction but does not change qualitatively the impact of size and structure. Consequently, we can derive general rules from our results. For a linear alignment according to Fig. 1 it was shown, that the aggregate structure affects the interaction only at moderate distances, whereas for very short surfaces distances (h 6 0.5xp) the interaction is completely determined by the closest pair of primary particles. At very large distances (h xagg) the configuration of particles is unimportant and the aggregate mass becomes the decisive parameter. The transition from the primary particle impact to the structure impact at h = 0.5xp corresponds with an interaction energy of 0.0057A132 according to Eq. (3). That are very small values; even for the relatively strong interacting system of titania particles in water (A132,0 = 5.35 1020 J) this is not more than 0.075 kT at room temperature. Hence, the primary particle approximation can be applied to all surface distances but those, where the vander-Waals energy has no physical relevance, i.e. where it is exceeded by the thermal energy. Regarding two approaching DLCA aggregates it is quite conceivable that there are only a few (may be just one) pairs of primary particles that really contribute to the van-der-Waals interaction
F. Babick et al. / Advanced Powder Technology 22 (2011) 220–225
energy of the closest pair(s) of primary particles. Strictly speaking, this approximation is applicable only for surfaces distances within the radius of the radius of the primary particles (h 6 0.5xp), but for larger distances van-der-Waals interactions are negligible. Nonetheless, since the morphology and outer radial density of an aggregate affects the surface distance between two aggregates with given centre distance, there is an apparent impact of the aggregates’ structure on their van-der-Waals interaction. In contrast to simple approximations, the proposed calculation scheme provides a possibility to account for that. Full benefit will certainly be gained when this van-der-Waals calculation is combined with the calculation of other types of interaction.
N×VvdW,PP /VvdW,chain
1 0.8 0.6 N=2
0.4
N=10 0.2
N=20 N=100
0 1
225
10 100 1000 surface distance, nm
10000 References
Fig. 7. Interaction energy vs. surface distance for two linear aggregates that are aligned in parallel; the sum over all (N) facing pairs of primary particles is related to the interaction energy of the aggregates.
between the aggregates. That means, that the attraction between such aggregates is almost independent of their size (for a given xp). The opposite is true for two linear chain-aggregates that are aligned in parallel. In that case the interaction energy grows with the number of opposing pairs of primary particles. For small surface distances (h 6 0.5xp) the dependency is almost linear (Fig. 7). Finally, after having derived that the van-der-Waals interaction energy between two fractal aggregates can be traced back to the pair-interaction of a few primary particles at close proximity, we have to ask, if there is a need for a complete calculation scheme as introduced in this paper and if there is any structural impact? In our opinion both questions can be affirmed. The calculation scheme is required for looking at arbitrary aggregate configurations and alignments as well as for deriving average relationships between the centre distance and the interaction energy. Moreover, such relationships are clearly affected by the morphology and radial density distributions, i.e. by the structure, of the aggregates (cf. Fig. 3). Mastercurves as being indicated by Fig. 6 cannot be (easily) obtained analytically but require appropriate numerical tools. 5. Conclusions This paper has presented a straightforward algorithm for computing the van-der-Waals interactions between aggregates of colloidal particles. It is based on a pairwise summation of interaction energies and employs the Lifshitz–Hamaker function for half-spaces. The simplifications behind this approach (like the complete ignorance of the many-body aspect) have been discussed based on published data of more elaborate calculation schemes. It appears that the model uncertainty is most probably not larger than the uncertainty in the material parameters. The calculation scheme was further applied to deterministic and DLCA aggregates. The former allow for tuning the aggregate structure via fractal parameters, while the latter show a certain variation in the particle configuration, but posses an average fractal dimension of 1.8. DLCA aggregates were chosen as an adequate model for pyrogenic powders, which have a high economic relevance. It could be shown, that the van-der-Waals interaction energy can be fairly well approximated by computing the interaction
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