Observation spheres

An as yet unresolved discrepancy between these two studies should be pointed out. Keller et al. (1970) observed cylinders for a material containing 25% polystyrene even after annealing on the other hand, McIntyre and Campos-Lopez observed spheres for a material containing a higher concentration of polystyrene, 38.5 %. In view of the work of Meier (1969) and others, spheres should have been obtained at the lower concentration, and cylinders at the higher. (See Section 4.7.) It is possible that the inversion of phase shape may be related to the fact that the specimens studied by Keller et al. (1970) were extruded rather than cast certainly the shear forces present in extrusion can themselves affect particle shape (VanOene, 1972) (see Section 9.6). 

We begin by describing the physical mechanism underlying polymorphism. We shall rediscover a key idea introduced in Chap. 4, viz. the notion of optimum or spontaneous curvature cq of an amphiphilic film. It will provide a qualitative understanding for the classical sequence of forms observed sphere, cylinder and bilayer. 

Originally, only the classical set of phases was observed, spheres, cylinders and lamellae. With time and increasing refinement of both theory and experiment other configurations, stable and metastable, were predicted and observed. The ordered bicontinuous double diamond was observed in a star block copolymer (Thomas et al. 1986) and the gyroid in a diblock copolymer (Hajduk et al. 1994). A recent theoretical calculation (Matsen 2012) is shown at left in the figure below, and compared with experimental observations complied from different sources (Matsen 2002) (Figs. 1.12 and 1.13). 

To determine experimentally the function in Eq. (5.64), first-year engineers at Cornell observed spheres of various materials (steel, brass, aluminum, glass, lucite, nylon, teflon, and polypropylene) falling through various fluids (water, vegetable oil. 

After some time the anodie and cathodie areas beeome separated as a barrier of corrosion products is formed in between them. This wall stabilizes the pH differences that oeeur due to the separation of the anodic and cathodic reactions. Wall and cap together form the observed sphere... 

STM and AFM profiles distort the shape of a particle because the side of the tip rides up on the particle. This effect can be corrected for. Consider, say, a spherical gold particle on a smooth surface. The sphere may be truncated, that is, the center may be a distance q above the surface, where q < r, the radius of the sphere. Assume the tip to be a cone of cone angle a. The observed profile in the vertical plane containing the center of the sphere will be a rounded hump of base width 2d and height h. Calculate q and r for the case where a - 32° and d and h are 275 nm and 300 nm, respectively. Note Chapter XVI, Ref. 133a. Can you show how to obtain the relevent equation ... 

Monte Carlo computer simulations of spheres sectioned into a disc [104, 105] show tliat steric interactions alone can produce a nematic phase of discotic molecules. Columnar phases are also observed [104, 105]. 

In the previous section, non-equilibrium behaviour was discussed, which is observed for particles with a deep minimum in the particle interactions at contact. In this final section, some examples of equilibrium phase behaviour in concentrated colloidal suspensions will be presented. Here we are concerned with purely repulsive particles (hard or soft spheres), or with particles with attractions of moderate strength and range (colloid-polymer and colloid-colloid mixtures). Although we shall focus mainly on equilibrium aspects, a few comments will be made about the associated kinetics as well [69, 70]. 

Experimentally, tire hard-sphere phase transition was observed using non-aqueous polymer lattices [79, 80]. Samples are prepared, brought into the fluid state by tumbling and tlien left to stand. Depending on particle size and concentration, colloidal crystals tlien fonn on a time scale from minutes to days. Experimentally, tliere is always some uncertainty in the actual volume fraction. Often tire concentrations are tlierefore rescaled so freezing occurs at ( )p = 0.49. The widtli of tire coexistence region agrees well witli simulations [Jd, 80]. 

The fonnation of colloidal crystals requires particles tliat are fairly monodisperse—experimentally, hard sphere crystals are only observed to fonn in samples witli a polydispersity below about 0.08 [69]. Using computer... 

We will focus on one experimental study here. Monovoukas and Cast studied polystyrene particles witli a = 61 nm in potassium chloride solutions [86]. They obtained a very good agreement between tlieir observations and tire predicted Yukawa phase diagram (see figure C2.6.9). In order to make tire comparison tliey rescaled the particle charges according to Alexander et al [43] (see also [82]). At high electrolyte concentrations, tire particle interactions tend to hard-sphere behaviour (see section C2.6.4) and tire phase transition shifts to volume fractions around 0.5 [88]. 

As shown in section C2.6.6.2, hard-sphere suspensions already show a rich phase behaviour. This is even more the case when binary mixtures of hard spheres are considered. First, we will mention tire case of moderate size ratios, around 0.6. At low concentrations tliese fonn a mixed fluid phase. On increasing tire overall concentration of mixtures, however, binary crystals of type AB2 and AB were observed (where A represents tire larger spheres), in addition to pure A or B crystals [105, 106]. An example of an AB2 stmcture is shown in figure C2.6.11. Computer simulations confinned tire tliennodynamic stability of tire stmctures tliat were observed [107, 1081. 

All that can be concluded from the data given in the preceding example is that the particle is not an unsolvated sphere. However, when an appropriate display of contours is examined for f/fo (e.g.. Ref. 2), the latter is found to be consistent with an unsolvated particle of axial ratio about 4 1 or with a spherical particle hydrated to the extent of about 0.48 g water (g polymer). Of course, there are a number of combinations of these variables which are also possible, and some additional experimental data—such as the intrinsic viscosity—are needed to select that combination which is consistent with all experimental observations. 

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