Shell spring constant

Abstract In this paper we report on AFM force spectroscopy measurements on hollow polymeric spheres of colloidal dimensions made from polyelectrolyte multilayers of polyal-lylamine and polystyrenesulfonate in water. We find that the shells show a linear force-deformation characteristic for deformations of the order of the shell wall thickness. This experimental outcome is discussed in terms of analytical results of continuum mechanics, in particular the scaling behaviour of the shell spring constant with wall thickness, shell radius and speed of the deformation is analysed. The experimental results agree well with the predictions of Reissner for thin shells and allow... 

In the solution of Reissner (Eq. 1) the shell spring constant sheii should scale oc/ 2 and scaling relation, we have prepared shells with different wall thickness and different radius. Figure 3 shows a plot of sheii multiplied with the radius versus the square of the shell wall thickness, so that the slope of the plot reflects only the fixed material properties. 

Fig. 3 Shell spring constant multiplied with the shell radius, which makes the data from shells of different radius collapse onto a master curve, and plotted against the square of the shell thickness. Black triangles shells with 9.6 micrometer radius, grey squares shells with 7.85 micrometer radius. Full lines correspond to linear regressions performed on each data set separately. In both cases the predicted scaling of the shell spring constant proportional to h2 is found in the experiment...

This allows us to derive the elastic properties of the capsules from the measured shell spring constants. The most precise way to do so is to use the combination of the data in Fig. 3, since the slope of the rescaled data contains the Young s modulus of the material and is an average over all measurements. However, since the measurements were carried out in water and not in dried state, like the thickness measurements, the thickness has to be modified to account for the swelling of PAH/PSS. Kiigler et al. showed that the thickness of the multilayers depends on the degree of humidity [27]. According to their study, PAH/PSS multilayers swell by 15% between multilayers dried under ambient conditions and immersed into water. Similar results were found by Losche et al. [28], who... 

Fig. 5 Dependency of the measured shell spring constant on speed the speed of the AFM probe deforming the surface is varied over 3 orders of magnitude which results, within the accuracy of the measurement, in no change of the elastic constants...

Ocore + 0.848 Ocore- -)shel Core-shell spring constant k = 74.92 eV A2... 

Figure 2 In the shell model, a core charge z, + q, is attached by a harmonic spring with spring constant kj to a shell charge - ij,. For a neutral atom, Zi = 0. The center of mass is at or near the core charge, but the short-range interactions are centered on the shell charge. (Not drawn to scale the displacement dj between the charges is much smaller than the atomic radius.)...

Equations [28] and [29] correspond directly to Eqs. [7] and [6], but for the case of dipoles with finite extent. In that sense, models based on point dipoles can be seen as idealized versions of the shell model, in the limit of infinitely small dipoles. That is, the magnitude of the charges qi and spring constants ki approach infinity in such a way as to keep the atomic polarizabilities a, constant. Indeed, in that limit, the displacements will approach zero in the shell model, and the two models will be entirely equivalent. 

We recall from Chapter 1 that for ionic materials, ionic polarizability can be taken into account using the shell model of Dick and Overhauser (1958), which treats each ion as a core and shell, coupled by a harmonic spring. The ion charge is divided between the core and shell such that the sum of their charges is the total ion charge. The free ion polarizability, a, is related to the shell charge, Y, and spring constant, k, by ... 

Ro is determined by fitting to the experimental lattice parameter. Shell charges and spring constants were derived for the binary oxides by fitting to dielectric and elastic constants. All the binary potentials and shell parameters were transferred mchanged for the ternary and quaternary cuprates. 

Frame structure model with a spring constant for axial deformation and bending of C-C bond using analytical method 0.61-0.48 A two dimensional continuum-shell model which is composed of the discrete molecular structures linked by the carbon-carbon bonds... 

Polarization of ions can be included in one of two ways. The natural approach is to use point ion polarizabilities, which has been successfully explored by Wilson and Madden (1996). An alternative, which has been used for many decades, is the so-called shell model (Dick and Overhauser 1958) as illustrated schematically in Figure 1. This is a simple mechanical model, in which an ion is represented by two particles-a core and a shell-where the core can be regarded as the representing the nucleus and inner electrons, while the shell represents the valence electrons. As such, all the mass is assigned to the core, while the total ion charge (qt = qc + qs) is split between both of the species. The core and shell interact by a harmonic spring constant, Kcs, but are Coulombically screened from each other. The polarizability is then given by ... 

The most popular model which takes into account both the ionic and electronic polarizabilities is the shell model of DICK and OVERHAUSER [4.12]. It is assumed that each ion consists of a spherical electronic shell which is isotropically coupled to its rigid ion-core by a spring. To begin with we consider a free ion which is polarized by a static field E. The spring constant is k, the displacement of the shell relative to its core is v and the charge of the shell is ye (Fig.4.7). In equilibrium, the electrostatric force yeE is equal to the elastic force kv yeE = kv. The induced dipole moment is d = yev = aE from which we obtain the free ion polarizability... 

Fiq.4.8a,b The coupling of two neighbouring ions in the simple shell model, a) Ions in equilibrium positions and b) ions in displaced positions, ki, k2 and S are the spring constants of the model ... 

Dynamic models for ionic lattices recognize explicitly the force constants between ions and their polarization. In shell models, the ions are represented as a shell and a core, coupled by a spring (see Refs. 57-59), and parameters are evaluated by matching bulk elastic and dielectric properties. Application of these models to the surface region has allowed calculation of surface vibrational modes [60] and LEED patterns [61-63] (see Section VIII-2). 

Electron distribution is affected by the presence of external electric fields and the polarisation interaction that arises in the presence of other atoms. One of the most common models to take into accoimt polarisation is the shell model. In the shell model the total charge of the atom is distributed between the nucleus and a mass-less spherical shell. This shell is connected to the nucleus by a spring of a given force constant. 

Shanahan, T. M., Pigati, J. S., Dettman, D. L. Quade, j. 2005. Isotopic variabiUty in the aragonite shells of freshwater gastropods living in springs with nearly constant temperature and isotopic composition. Geochimica et Cosmochimica Acta, 69, 3949—3966. 

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