12682
Langmuir 2007, 23, 12682-12686
Shear Modulus of Sintered ‘House of Cards’-Like Assemblies of Crystals H. M. Schaink* Food Physics Laboratory, Wageningen UniVersity, P.O. Box 8129, NL-6700 EV Wageningen, The Netherlands
K. F. van Malssen UnileVer Research and DeVelopment Vlaardingen, P.O. Box 114, NL-3130 AC Vlaardingen, The Netherlands ReceiVed June 27, 2007 A cell model of a ‘house of cards’-like assembly of crystals is used for the study of the evolution of the shear modulus during sintering. The crystals are assumed to have a lozenge shape. The cell model takes different crystal-crystal contacts into account. The force needed to separate two sintered crystals is calculated using the minimum surface area (MSA) approximation. By varying the thickness, long axis, and short axis of the crystals, it is possible to make space-filing configurations which have a nonzero shear modulus at crystal volume fraction that can be as low as φ ) 0.03. This is significantly lower than the volume fractions φ > 0.52 that are found in studies where the MSA approximation is applied to assemblies of spherical particles. It is found that sintering may cause a nonlinear volume fraction dependence of the shear modulus, which depends on the shape of the crystals, the type of crystal-crystal contacts, and the character of the crystal assembly. The calculated shear modulus is analyzed using the phenomenological expression (φ - φ0)β, where φ0 represents the volume fraction at the start of sintering. The exponent β is found to vary between 1 and 2. The interpretation of the shear modulus using a fractal model is also discussed.
Introduction In many oil-based semisolid pharmaceutical, personal care, and food products, a network of fat crystals is responsible for the hardness. Examples can be found in products like ointments, margarine, peanut butter, and cream. For these products it is of interest to know how the strength of the material depends on the solid fat content. In the earliest theories about the rheology of fat crystal networks, it was predicted that the elastic modulus, G′, was proportional to the volume fraction, φ, of the fat crystals.1,2 Later it was recognized that G′ depends more strongly on φ:3-5
G′ ∝ φR
(1)
with R > 1. In all these descriptions it was assumed that the Van der Waals interaction between the fat crystals is the only relevant factor for the strength of the network. However, it has been observed that fat crystals may merge to become one solid object. This phenomenon is called sintering. The effect of sintered fat crystals on the strength of the network has attracted only limited attention.6-8 This is in sharp contrast with the situation in the literature of ceramic materials, where many articles have been published on the Young modulus and the shear modulus of sintered polycrystalline materials.9-14 One of the models used for the phenomenological description of sintered polycrystalline materials is11-14
G ∝ (φ - φ0)β
(2)
where φ0 is the volume fraction at the start of the sintering process * To whom correspondence should be addressed. E-mail: h.schaink@ orange.nl. (1) van den Tempel, M. J. Colloid Sci. 1961, 16, 284. (2) Kamphuis, H.; Jongschaap, R. J. J. Colloid Polym. Sci. 1985, 263, 1008. (3) van den Tempel, M. J. Colloid Interface Sci. 1979, 71, 18. (4) Vreeker, R.; Hoekstra, L. L.; den Boer, D. C.; Agterof, W. G. M. Colloids Surf. 1992, 65, 185. (5) Marangoni, A.G. Fat Crystal Networks; Marcel Dekker: New York, 2005. (6) Johansson, D.; Bergenståhl, B. J. Am. Oil Chem. Soc. 1995, 72, 911. (7) Walstra, P.; van Vliet, T. and Kloek, W., In Food Colloids, Proteins, Lipids, Polysaccharides; Dickinson, E., Bergenstahl B., Eds; Royal Society of Chemistry: Cambridge, 1997; pp 168-181. (8) Ojijo, N. K. O.; Kesselman, E.; Shuster, V.; Eichler, S.; Eger, S.; Neeman, I.; Shimoni, E. Food. Res. Int. 2004, 37, 385.
and β is a phenomenological constant. This expression is similar to that describing percolated networks.14,15 For the theoretical description of the strength of sintered polycrystalline materials, often so-called minimum solid area (MSA) models are used.9-13 In these models the polycrystalline material is often modeled as a set of overlapping spheres placed on a regular lattice. The minimum solid area is the area of the smallest surface that has to be created to separate two sintered spheres. It is assumed that the force needed to separate these spheres is proportional to the minimum solid area. The results of these MSA models are fitted to either eq 2 or an exponential model. The usual MSA models predict that sintering starts at volume fractions around 0.5. This is due to the crystalline packing of the spherical particles. In a system with edible oil structured with a mixture of stearic acid + stearyl alcohol16,17 and a monoglyceride + water system,18 the lowest concentration (w/w) at which a space filling network can be made is ∼0.03. This is due to the platelike or needlelike shape of the crystals. Assemblies of platelets occur also in nonfood applications.19,20 Although the possibility of sintering has been mentioned for these types of systems,6-8 no model describing sintering at these low concentrations has been proposed. This work describes a MSA model for study of the evolution of the shear modulus during the liquid-phase sintering21 of a ‘house of cards’-like crystal network. Similar to the MSA work (9) Knudsen, F. P. J. Am. Ceram. Soc. 1959, 42, 376. (10) Bert, C. W. J. Mater. Sci. 1985, 20, 2220. (11) Phani, K. K.; Niyogi, S. K. J. Mater. Sci. 1987, 22, 257. (12) Rice, R. W. J. Mater. Sci. 1996, 31, 102. (13) Rice, R. W. J. Mater. Sci. 2005, 40, 983. (14) Kova´cˇik, J. J. Mater. Sci. Lett. 2001, 20, 1953. (15) Stauffer, D. Phys. Rep. 1979, 54, 1. (16) Schaink, H. M.; van Malssen, K. F.; Morgado-Alves, S.; Kalnin, D.; van der Linden, E. Food Res. Int. 2007, 40, 1185. (17) Gandolfo, F. G.; Bot, A.; Flo¨ter, E. J. Am. Oil Chem. Soc. 2004, 81, 1. (18) Sein, A.; Verheij, J. A.; Agterof, W. G. M. J. Colloid Interface Sci. 2002, 249, 412. (19) Schwenzer, B.; Gomm, J. R.; Morse, D. E. Langmuir 2006, 22, 9829 and Supporting Information. (20) Sinha Ray, S.; Okamoto, K.; Yamada, K.; Okamoto M. Nanoletters 2002, 2, 423. (21) German, R. M. Sintering theory and practice; John Wiley and Sons: New York.
10.1021/la701914m CCC: $37.00 © 2007 American Chemical Society Published on Web 11/03/2007
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Langmuir, Vol. 23, No. 25, 2007 12683
Figure 2. Example of the formation of sintered bonds (indicated by thick black lines).
Figure 1. Unit cell used here, as viewed from different points of view; for clarity, only the lowest three layers are shown for the projection on the xy plane.
materials,9-13
on globular polycrystalline it assumes that the crystals form a regular lattice. Therefore, the fractal dimension22 of the crystal network is always equal to 3. The thickness and the shape of the crystals can be varied, and this makes it possible to form a sintered network at any low crystal concentration. Using this model it is possible to study the influence of the degree of sintering on the shear modulus of the crystal network.
to the MSA approximation, this force constant, K, is proportional to the area of contact, A. Using this force, it is possible to calculate the unit cell contribution to the total stress σcell:23
σ c cell )
φc )
2abc (a + c)3
(
∆a 3 - φshared a
)
∑i bu ibr i〉 ) K ∑i 〈ubbu 〉i
(
with
〈ux2〉 〈uxuy〉 〈uxuz〉 〈u bb u 〉i ) 〈uxuy〉 〈uy2〉 〈uyuz〉 〈uxuz〉 〈uyuz〉 〈uz2〉
)
(7a)
(7b)
i
In eq 7 the brackets denote a time average. The cross coefficients are generally not equal to zero. In principle, they can be calculated using a bead-spring simulation.23,24 Since the force constant K is proportional to A, it follows that
〈σ c cell〉 ∝ A
(8)
The elastic modulus is by definition equal to5,23
(4)
Here φshared is the fraction of the volume that is occupied simultaneously by two or more crystals. The elastic modulus is calculated by mapping the crystal network on a bead-spring network,23 where the sintered regions act as rigid springs and the beads belong to the undeformable crystals. The force between two sintered central crystals is given by Hooke’s law:23
bi B F i ) Ku
〈σ c cell〉 ) K〈
(3)
In the next step the crystals are allowed to grow on such a manner that the axis ratios a/b and a/c remain constant. As a result of the crystal growth, areas of contact are formed between neighboring crystals. This is illustrated in Figure 2. The increment of the length of the long axis of a lozenge is 2∆a ) 2(anew a), where 2anew is the length of the long axis after growth. The crystal volume fraction of this sintered crystal network is equal to
φ ) φc 1 +
(6)
The summation is over all the sintered bonds in the unit cell. The vector b ri describes the separation of the beads belonging to different crystals. It can be written as the sum of a vector B Ri that describes the equilibrium position and the vector b ui that describes the shift away from equilibrium, i.e., b ri ) B Ri + b ui. After taking a time average, it is possible to write the cell contribution to the total stress as
Theory A ‘house of cards’-like assembly of crystals in a saturated solution is modeled using a cell model, which mimics an infinite system. The crystals are chosen to be lozenge shaped. The long axis of a single crystal has a length of 2a, the short axis has a length 2b e 2a. The thickness of the crystals is equal to 2c < 2b. The volume of a single crystal is equal to 4abc. Using these crystals, a cross-linked network is made of crystals that share a line of contact. The unit cell used here is a cube with sides of length 4(a + c), as can be seen in Figure 1. The volume occupied by the crystals in the unit cell corresponds to that of 32 lozenges, so the threshold volume fraction at which the crystals just touch each other is equal to
∑i BFibr i
(5)
Here b u is a vector that describes the displacement of the beads with respect to their equilibrium position. The geometry of the chosen unit cell ensures that the force constant is a scalar. Due (22) Mandelbrot, B. B. The fractal geometry of nature: Freeman: New York, 1982.
G ≡ lim γf0
σxy ncell〈σ c cell,xy〉 ≈ lim γf0 ∝A γ γ
(9)
where σxy represents the xy component of the total stress tensor and γ is the strain. The total number of unit cells in the systems is given by ncell. In the cell geometry considered here, there are only a limited number of types of sintered contacts possible. At very low degree of sintering, only those crystals will sinter that are in contact at the threshold volume fraction φ0. At higher degrees of sintering (i.e., larger values of φ), sintering may occur between crystals that are not nearest neighbors. Then more terms have to be accounted for in eq 6. The types of sintered contacts that are possible in this cell model are shown in Figure 2. Taking the different types of sintering, as shown in Figure 3a-d, into account, it is possible to write
G ) lim γf0
ncell γ
∑
j)a,b,c,d
Nj
[κAj
∑i 〈uxuy〉j,i]
(10)
Here use has been made of the MSA definition of the force constant Kj ) κAj. In eq 10, the first summation runs over all (23) Byron Bird, R.; Hassager, O.; Armstrong, R.C.; Curtiss, C.F. Dynamics in Polymeric Liquids, Volume 2: Kinetic theory; John Wiley: New York, 1977. (24) Denneman, A. I. M.; Jongschaap, R. J. J.; Mellema, J. J. Eng. Math. 1998, 34, 75.
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12684 Langmuir, Vol. 23, No. 25, 2007
Figure 3. Schematic representation of the types of sintered contacts that are possible in the cell model; initially, only contacts of type a are formed; at higher degrees of sintering also the other contacts are formed; see also Figure 1.
the types sintered bonds, while the second summation is over all the Nj sintered bonds of type j. If the simplifying approximation is made that all are identical and constant, then eq 10 reduces to
G∝
∑
Ajψj
(11)
Figure 4. Shear modulus, G, for sintered spheres in cubic packing as function of the volume fraction, calculated using MSA.
Substitution of eq 15 in eq 14 and collecting the terms of the same order of (φ - φc)/φc yields expressions for the pre-factors Rn. The first three pre-factors are
j)a,b,c,d
R1 )
where ψj represents the fraction of bonds of type j in the unit cell. It depends on the degree of sintering. So far the influence of the continuous phase on the stress has been neglected. When the crystals are dispersed in a viscous liquid phase, the total stress tensor of the system can be approximated by the Kramers expression:23
σ c ) -ηsz γ + ncellσ c cell + nkBTIc
R2 ) -
R3 )
(12)
Here ηs and z γ represent the viscosity of the fluid phase and the rate of deformation tensor. In eq 12 the first term on the righthand side describes the contribution of the fluid motion, the second term describes the contribution due to the crystal network, and the last term describes the contribution due to the motion of the crystals in the fluid. The shear modulus, G, can be calculated using the definition given in eq 9.
φshared )
3Ncbc
∆a ∆a 1 + )( ) a a 4a(a + c)(
2
(13)
Here Nc stands for the number of contacts of type 3a. For the unit cell shown in Figure 1, Nc ) 48. Combination of eqs 13, 4, and 3 yields
() (
)( )
3Nc(a + c)2 ∆a 2 φ - φc ∆a )3 + 3+ φc a a 8a2
(
1-
)( )
3Nc(a + c)2 ∆a a 8a2
3
(14)
This expression can be solved analytically. However, such an approach yields a lengthy, complicated equation which does not clarify the dependence of ∆a on φ. An approximate, more illustrative, expression can be derived using the definition:
∆a a
)
∑n
( )
Rn
φ - φc φc
n
(15)
(16a)
Nc(a + c) 1 19 8a2
Nc(a + c) 1 1+ 27 8a2
) )
2
(16b)
2
(16c)
For ∆aec, the contact area, A, for a single contact is given by
A)
4cb(a + c) ∆a ∆a 1+ a a a
(
)( )
(17)
Finally, an expression for the shear modulus, G, is obtained by combining eqs 15-17, and 9
Results and Discussion a. The Limit ∆a e c. An impression of the volume fraction dependence of the shear modulus predicted by this model can be obtained by considering the limit ∆a e c. In this case, only sintered connections are formed between crystals that touch each other at the threshold volume fraction, φ0. For ∆a e c, an analytic expression can be obtained for the contact area, A. The derivation starts with the expression for φshared:
( (
1 3
G∝
[( )
( ) ( ) ] (( ) )
φ - φc 2 1 φ - φc + (R12 + R2) + 3 φc φc φ - φc 3 φ - φc +O (2R1R2 + R3) φc φc
4
(18)
When the crystals are very thin, ∆a < c , a, it can be seen from eqs 14 and 18 that the shear modulus G depends linearly on φ - φo. This volume fraction dependence was also found in the earliest descriptions of the shear modulus of fat crystal networks.1,2 For more thick, cubelike crystals, c ≈ a, a deviation from the linear behavior is to be expected on the basis of eqs 14-18. Nonlinear behavior is also found for spherical particles in the MSA approaches,9-13 as is illustrated in Figure 4 for spheres in cubic packing. Here the threshold volume fraction is φ0 ) 0.52. In the next section it is shown how G depends on φ - φ0 for ∆a > c. b. Calculation for Larger Values of ∆a. One of the motivations for performing this work is the observation that ‘house of cards’-like assemblies of crystals have a threshold volume fraction, φ0, that is significantly lower than 0.5. For the crystals arranged in the unit cell given in Figure 1, the threshold volume fraction is given by eq 3. By varying the long axis 2a, the short axis 2b, and the thickness 2c, it is possible to get any small value for φ0. In Figure 5 some examples are shown for the volume fraction dependence of the shear modulus predicted by this model. These values are obtained by defining ∆a and calculating the corresponding contact areas, A, and shared volume fraction, φshared (see eq 4), and hence the total volume fraction, φ, using analytic geometrical methods. These calculations are performed up to an increment of ∆a ) min(a , a(b - c)/(2c)).
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Langmuir, Vol. 23, No. 25, 2007 12685
Figure 6. Log-log plot of the volume fraction dependence of the total shear modulus presented in Figure 5d (thick line); the weak link predictions for df ) 3 (vertical line) and df ) 2.65 are shown as thin straight lines. Table 1. β, as Defined by eq 2, for the Crystal Shapes Studied in Figures 4 and 5
Figure 5. Shear modulus for the crystal network shown in Figure 1 for different crystal shapes as a function of the crystal volume fraction; the dimensions of crystals are given on the left side of each figure; the upper thick curves are calculated using eq 11; the lower curves are the contributions of the different sintered contacts.
The value ∆a < a(b - c)/(2c) is chosen to avoid unrealistic values of φ due to an overestimation of φshared (due to the contacts of type 3a). Figure 5a-c illustrates the volume fraction dependence of G for three different crystals for which φ0 ) 0.03. The shape of the crystals in Figure 5a-c varies from needle-like to platelets. It can be seen that there is a significant influence of the shape of the crystals on the volume fraction dependence of G, although all have the same critical volume fraction. Figure 5a corresponds to the case of thin needle-like crystals. Initially, only sintered bonds are formed of the type illustrated in Figure 3a. At φ = 0.194, the formation of bonds of type 3b starts. The total shear modulus, G, which takes the contribution of both types of sintered bonds into account, is calculated using eq 11. It is found that G depends almost linearly on the volume fraction. More platelet-like crystals have a larger surface area per unit cell. This enhances the ability to form more sintered bonds, as can be seen in Figure 5b and c. Therefore, there are more sintered connects formed in comparison to Figure 5a. This happens at lower volume fractions. For instance, the volume fractions at which new sintered bonds are formed in Figure 5b are, respectively, φ ) 0.030 (type 3a), 0.076 (type 3b), 0.187 (type 3c), and 0.197 (type 3d). It can be seen in Figure 5b that the formation of new sintered bonds has the effect that the total shear modulus deviates from simple linear behavior. In Figure 5c the crystals have almost the shape of a square. For this reason, the separate contributions all start at more or less the same volume fraction. In this case, the thickness c is very small. For this reason, the separate contributions have a nonlinear behavior, as is discussed in the previous section. In Figure 5d the shear modulus is shown for a more bulky crystal, with a ) 7, b ) 6.5, c ) 3. The order of appearance of the different types of bonds in Figure 5d is different from Figure 5b. In Figure 5d new sintered bonds are formed at volume fractions φ ) 0.27 (type 3a), 0.32 (type 3d), 0.49 (type 3c), and 0.69 (type 3b). In this case the separate contributions of the sintered bonds are clearly nonlinear. This is agreement with the prediction that is made on basis of eqs 14-18. In order to get an impression of the nonlinearity of the volume fraction dependence of G, the curves of Figures 4 and 5 are fitted to eq 2. The results of these fits are given in Table 1. The values of the exponent β, shown in Table 1, should be taken as an indication, since the type ordering of the crystals is also a factor
type of crystal
β
spheres (on a cubic lattice)
1.50
lozenge (in the unit cell given in Figure 1) a
b
c
7.000 7.000 7.000 7.000
1.568 3.521 6.445 6.500
0.600 0.230 0.120 3.000
1.05 1.25 1.43 1.67
Table 2. β for the Separate Contributions of the Different Contacts to the Elastic Modulus for the Crystal with (a/b/c ) 7:6.5:3), and the Volume Fraction, Ot, at Which These Contacts Start to Contribute to the Elastic Modulus type of contact
φt
β
Figure 3a Figure 3c Figure 3d
0.27 0.32 0.49
1.31 1.29 2.32
that contributes to β. Here only one type of ordering is considered (see Figure 1). Another factor that influences the value of β is the range of volume fractions that is considered in the fit. It can be seen in Table 1 that the crystal shape has a significant influence on the value of β, which is found to vary between 1 and 2. The relative large value for the bulky crystal (a/b/c ) 7:6.5:3) must be attributed to the connections the of type shown in Figure 3d. This can be seen in Table 2, where the values of β are presented for the separate contributions to the elastic modulus of this bulky crystal. The minimum surface area for the connections of type 3d are in this model the only connections that are not perpendicular to the direction of crystal growth. The consequence is apparently that the minimum surface area of these connections increases more rapidly than that of the other connections.
Concluding Remarks In this work a cell model is used to study the influence of sintering to the shear modulus of a ‘house of cards’-like assembly of crystals. By varying the shape of the crystals, it is possible to create a start configuration of the sintering process, with a volume fraction, φ0, that is significantly lower than the value of φ0 of an assembly of spherical particles, which is used often for the study of ceramic materials. Sintering may cause the shear modulus to depend nonlinearly on the crystal volume fraction. When the volume fraction dependence of the calculated shear moduli, presented in Figure 5, is fitted to (φ - φ0)β, the exponent β is found to vary between 1 and 2. Needle-like crystals tend to have smaller value for β than platelet-like crystals. The values of β fall in the range 0.72.3 which is reported in experimental studies of sintering (ref 14, and references therein). The fit equation (φ - φ0)β is identical to the equation that appears in the percolation theory. However,
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12686 Langmuir, Vol. 23, No. 25, 2007
Table B1. Number of Contacts Per Unit Cell, the Contact Criterion for ∆a, the Minimum Surface Area, A, and the Volume Shared for One Pair of Crystals for the Different Types of Crystal-Crystal Contacts Defined in Figure 3 type of no. of contact contacts criterion ∆a > 3a 3b 3c 3d
48 48 24 16 16 16 8 16
8
0 c c (a - b + 2c)/ (1 + b/a) a(a - b)/(b + c) a(a - b + c)/b a(a - b + c)/b (abc + a3 - ab2)/ (ab + ac + b2) a(a - b + c)/b
A
Vshared
4(c + ∆c)(1 + c/a)∆a tan θ
2(c + ∆c)(1 + c/a)2(∆a)2 tan θ 2(c + ∆c)2[2∆a + ∆c - c] tan θ 4(c + ∆c)(∆a - c)2 tan θ (∆a + ∆b + b - a - 2c)2(c + ∆c) sin 2θ
2(∆a + ∆b + b - a - 2c)(c + ∆c) sin 2θ 4(c + ∆c)(∆b + ∆c + b - a) tan((π - 2θ)/2)
2(c + ∆c)(∆b + ∆c + b - a)2 tan((π - 2θ)/2) 2(c + ∆c)2(2(b + ∆b - a) - c + ∆c) tan((π - 2θ)/2) 4(c + ∆c)(b + ∆b - a - c)2 tan((π - 2θ)/2) min (2Vshared,1,Vshared,2) with
min (2A1,A2) with
A1 ) [tan2((π - 2θ)/2) + 1]1/2(b + ∆b - c - a2/b + ∆a + a∆c/b)2 Vshared,1 ) 1/3 (b + ∆b - c - a2/b + ∆a + a∆c/b)3 A2 ) 4 tan ((π - 2θ)/2)(c + ∆c)(∆b + ∆c + b - a) Vshared,2 ) 2(c + ∆c)(∆b + ∆c + b - a)2 tan((π - 2θ)/2) 4(c + ∆c)(b + ∆b - a - c)2 tan((π - 2θ)/2)
for percolation phenomena, the values of the exponent β are typically larger than 2.25 This is probably because the start configuration in the unit cell is not a percolation cluster. The used cell model is based on 32 crystals. However, using the rules that are applied to construct this cell, it is possible to accommodate up to 48 crystals in the unit cell. A similar calculation with a 48 crystal unit cell has the effect that the values of ψi in eq 11 are slightly different. This indicates that the shear modulus depends also on the ordering of the crystals. Therefore, it can be concluded that there are three factors that influence the shear modulus of a sintered crystal network: i) the ordering of the crystals ii) the geometry of the crystal-crystal contacts (Table 2) iii) the shape of the crystals. (Figures 4 and 5) In the fractal description of (fat)-crystal networks, only the first of these three factors is considered. Analysis of the log-log plot of Figure 5d, as given in Figure 6, shows that there is an apparent scaling behavior for 0.58 < φ < 0.78. The exponent is found to be equal to R ) 2.9 (see also eq 1). The fractal description makes a distinction of a weak link regime and a strong link regime. For these regimes, the exponent R is given by26
1 3 - df
(19a)
3+x Strong link: R ) 3 - df
(19b)
Weak link: R )
Interpreting the curve in Figure 6 using the weak link theory yields a fractal dimension of df ) 2.65. In experimental studies of fat crystal networks where the weak link theory is used for the interpretation of the data, the fractal dimension is found to be in the range 2.0-2.9.4,5,27,28 The strong link interpretation of R ) 2.9, using a backbone fractal dimension of 1 < x < 1.3 yields 1.5 < df < 1.6. Similar results for the strong link and the weak link interpretations are obtained for the other crystal assemblies studied in Figure 5. However, for this cell model the fractal dimension is by definition equal to 3. Therefore, these results show that the determination of the fractal structure of a sample using only rheological data is awkward. It is better to study the fractal character of a sample using a scattering technique or an optical technique and use the outcome of these experiments for the interpretation of a rheology experiment.4,5,27,28 A discrepancy between a fractal dimension obtained using a scattering technique or an optical technique on the one hand and rheology on the other could be indicative for a sintered crystal network.
Another indication of sintering could be the frequency dependence of the shear modulus. During the sintering process, new bonds are formed, as is shown in Figure 5. This is likely to affect the frequency spectrum of the crystal network. This can also be studied using computer simulations.
Appendix A In this appendix expressions are given for the volume fraction and the minimum surface area for fused spherical particles of radius R. The distance between the centres of the spheres is equal to r < R. The minimum surface area is equal to
A ) 4π(R2 - r2) ∝ G
(A1)
The volume shared by a pair of spheres is given by29
V(R) )
2π 3 (2R - 3R2r + r3) 3
(A2)
Thus, the volume fraction for a set of N spheres in a cubic arrangement is given by
φ)
Vtot 2πN ) [(2 - 2Nc)R3 + 3NcR2r - Ncr3] (A3) V 3V
where Nc represents the number of nearest neighbors; Nc ) 6 for a cubic packing. Expressions A1 and A2 are used for the calculation of Figure 4.
Appendix B The shape of the crystals is determined by tan θ ) b/a and (long axis, short axis, thickness) ) (2a + 2∆a, 2b + 2∆b, 2c + 2∆c) with ∆a/∆b ) a/b and ∆a/∆c ) a/c. Using standard geometry, it is possible to determine at which values of ∆a which crystal-crystal contacts are important and the corresponding values of the minimum solid area A and the correction of the volume fraction. These expressions are presented in Table B1. In the actual calculation, more ∆a values for crystal-crystal contacts are taken into account. However, these were not found to be relevant for the results presented here. Note Added in Proof. Recently a review on structure formation in oil was presented by Pernetti et al.30 Acknowledgment. H.M.S. thanks Prof. P. Walstra for a discussion about fat crystallisation. LA701914M
(25) van der Linden, E.; Sagis, L. M. C. Langmuir. 2001, 17, 5821. (26) Shih, W.-H.; Shih, W. Y.; Kim, S.-I.; Liu, J.; Aksay, I. A. Phys. ReV. A 1990, 42, 4772. (27) Marangoni, A. G.; Narine, S. S. Food Res. Int. 2002, 35, 957. (28) Narine S. S.; Marangoni A. G. Phys. ReV. E 1999, 60, 6991.
(29) Hill, T.L. Introduction to Statistical Thermodynamics; Dover Publications: New York, 1986. (30) Pernetti, M.; van Malssen, K. F.; Flo¨ter, E.; Bot, A. Curr. Opin. Colloid Interface Sci. 2007, 12, 221.
