Critical particle volume fraction

It is usually easier experimentally to specify the critical particle volume fraction rather than the critical particle concentration. Vincent, who has pioneered this approach, usually denotes the critical coagulation particle volume fraction by CF4>. The in Vincent s pseudo-acronym denotes the particle volume fraction whereas the F denotes flocculation, rather than the term coagulation which has been preferred in this text to signify the origin of the attraction. 

The data was obtained at the shear strain of200% and frequency 2 s . There is a clear critical particle volume fraction in this suspension system, about 38 vol%. When the particle volume fraction exceeds this critical value, a sharp increment of the rheological property is observed. Percolation theory was used to explain this phenomenon, which will be discussed in detail in later on. Note that Figure 46 is consistent with Figure 47, and Figure 46 may only show the low particle volume fraction portion of Figure 47. 

The function describing the particle crack for the critical particle volume fraction and for the SiC particle radius greater than critical is presented (Fig. 26). 

Figure 3. Critical flocculation temperature (T) versus log (particle volume fraction ) for the two Si02 g PDMS dispersions in bromocyclohexane O, S15/PDMS5 x, S15/ PDHS3.

Figure 4. Critical flocculation solvent composition toluene + n-hexane (v = volume fraction of toluene), versus log (particle volume fraction, (J>) for various SiC -g-PS systems at 24 1°C V, S12/PS13c 0, S12/PS13a ...

Figure 3.10. Phase diagrams of attractive monodisperse dispersions. Uc is the contact pair potential and (j) is the particle volume fraction. For udk T = 0, the only accessible one-phase transition is the hard sphere transition. If Uc/hgT 0, two distinct scenarios are possible according to the value of the ratio (range of the pair potential over particle radius). For < 0.3 (a), only fluid-solid equilibrium is predicted. For % > 0.3 (b), in addition to fluid-solid equilibrium, a fluid-fluid (liquid-gas) coexistence is predicted with a critical point (C) and a triple point (T).

It is interesting to note that the rate of formation of a doublet is smaller than the rate of dissociation for sufficiently small particles. The difference between the rates of dissociation and formation decreases as the particle size increases, eventually becoming equal at a critical particle radius R. For particles smaller than 27, the doublets are unstable and they reach a dynamic equilibrium with the single particles. For particles greater than 27, however, the doublet is stable and, therefore, its concentration increases with time. The variation of the critical radius 27 with the particle volume fraction (f>p is plotted in Fig. 2.27 is found to be a weak function of 4>P, decreasing from 44 to 18 A as dip increases from 10 to 10 4. 

Fig. 2. The variation of the critical particle radius with the particle volume fraction for particles of unit density, in air at 1 atm and 298 K, and for a Hamaker constant of I0-12 erg.

For dilute suspensions that are colloidally imstable, there is a particle volume fraction where a bridging network is formed. This network is characterized by a probability of filled sites,p. The critical probability. 

The previous expression returns the value of the droplet radius corresponding to the maximum of the monomer chemical potential curve (f ), and says that, for a given amount of costabilizer in the particle (Uj,)>the droplet radius must be smaller than in order to be locally stable. The same equation can be expressed in terms of the critical costabilizer volume fraction, (p, as follows ... 

Flocculated Systems. The viscoelastic responses of flocculated systems are strongly dependent on the suspension structure. The suspension starts to show an elastic response at a critical solid volume fraction of 0ct = 0.05 — 0.07, at which the particles form a continuous three-dimensional network (211-213). The magnitude of the elastic response for flocculated suspensions above 0ct depends on several parameters, such as the suspension structure, interparticle attraction forces and particle size, and shape and volume fraction. Buscall et al. (10) found that the volume fraction dependence of the storage modulus follows a power-law behavior. 

Figure C2.6.10. Phase diagram of colloid-polymer mixtures polymer coil volume fraction jiri n vs particle volume fraction ( ). (a) Narrow attractions, 5/a = 0.1. Only a fluid-crystal transition is present. Tie lines indicate coexisting phases, (b) Longer range attractions, 5/a = 0.4. Gas, liquid and crystal phases (G, L and C) are present, as well as a critical point (CP). The three-phase triangle is shaded (reproduced with permission from [99]. Copyright 1992 EDP Sciences).

S.4. Critical flocculation particle volume fraction Theoretical considerations... 

More recently, the existence of a critical coagulation particle volume fraction has been established by Vincent et al. (1980) for polystyrene latices stabilized by poly(oxyethylene) terminally grafted to the particle surface and by Cowell and Vincent (1982a) for physically attached poly(oxyethylene). Bridget (1979) has found that a parallel situation pertains with silica particles sterically stabilized by terminally anchored polystyrene in ethyl benzene. 

D. Critical coagulation particle volume fraction not observed to-date observed... 

We note further from Fig. 13.12 that fully 40% of the small-particle size distribution falls below the critical particle-size estimate of Dc = 0.93 pm, reducing the effective fraction of the particle volume fraction from 0.22 to 0.13. This results in a substantial upper yield stress of 24 MPa, as Fig. 13.11(c) shows. However, once an effective craze network has developed the actual craze-flow stress of this blend drops to 20.5 MPa, in keeping with expectations. 

Both logarithmic plots of Izod impact strength versus excess volume fraction over the critical stress volume fraction determined by the critical ligament distance gives the slope of 0.45, which is comparable with the critical exponent of 0.44 calculated for monodisperse particles by Bug and coworkers [12]. 

The dominant mechanism of deformation depends mainly on the type and properties of the matrix polymer, but can vary also with the test temperature, the strain rate, and the morphology, shape, and size of the modifier particles (Bucknall 1977, 1997, 2000 Michler 2005 Michler and Balta-Calleja 2012 Michler and Starke 1996). Properties of the matrix determine not only the type of the local yield zones but also the critical parameters for toughening. In amorphous polymers with the dominant formation of crazes, the particle diameter, D, is of primary importance, while in some other amorphous and in semicrystalline polymers with the dominant formation of dilatational shear bands or intense shear yielding, the interparticle distance ID, i.e., the thickness of the matrix ligaments between particles, seems to be also an important parameter influencing the efficiency of toughening. This parameter can be adjusted by various combinations of modifier particle volume fraction and particle size. 

Most percolation studies rely on Monte Carlo simulations, whose properties of interest are critical cluster size, bond densities (i.e. number of bonds that lead to network per site), particle volume in the matrix, particle orientation and network geometry at the threshold. In most cases simulations treat the percolation in a statistical manner which means that the particles are randomly distributed in the matrix and network pathways are formed simply by increasing the particle volume fraction in the composite. Although such an approach has merit and provides valuable insight on the network formation, it is far away from reality, especially when polymers are concerned. Addition of particles in a polymer matrix is mainly performed via solution or melt mixing, which means that both particles and polymer chains are in motion and interact with each other. More advanced theoretical approaches do take into consideration the thermodynamic interactiOTis between the composite constituents (particle-particle, particle-polymer and... 

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