Viscoelasticity, lateral forces

The AFM probe tip interacts via various mechanisms with the sample surface, including simple quasi-static and dynamic normal and lateral forces. These (and other) interactions depend on local physical properties of the sample, on the scale of the probed area. Tribological, viscoelastic, and microhardness mechanical properties can be probed under (more or less) controlled environmental conditions. 

Figure 6 shows the master curves for the PS films with M of 4.9k and 140k drawn by horizontal and vertical shifts of each curve shown in Fig. 5 at the reference temperatures of 267 and 333 K, respectively [26]. The master curves obtained from the dependence of lateral force on the scanning rate were very similar to the lateral force-temperature curves, as shown in Fig. 3. Hence, it seems plausible as a general concept that the scanning rate dependence of the lateral force exhibits a peak in a glass-rubber transition. Also, it is clear that the time-temperature superposition principle, which is characteristic of bulk viscoelastic materials [35], can be applied to the surface relaxation process as well. Assuming that Uj has a functional form of Arrhenius type [36, 37], the apparent activation energy for the aa-relaxati(Mi process, A//, is given by ...

The glass transition of amorphous polymeric films was investigated by SPM imder various conditions. We established, using lateral force measurements, that the pressure exerted by the tip does not have an effect similar to a hydrostatic pressure on the properties of the polymer. Instead, an apparent transition is observed due to the viscoelastic nature of the sanq)le. We confirm the existence of a critical load and scan speed, which need to be determined to obtain accurate glass transition temperature measurements. 

Polymeric materials exhibit viscoelastic phenomena, which must be taken into account in designing the materials applications. For example, rubber in a tire receives stimuli over a wide frequency and temperature range from the road surface. In the case of bulk samples, the frequency and temperature can be converted mutually based on the time-temperature superposition (TTS) principle [72]. However, TTS is a kind of empirical rule and, consequently, an actual measurement method with a wide frequency and temperature range is necessary to precisely predict the properties of practical products. Various AFM-based conventional methods have been proposed to measure viscoelasticity such as lateral force microscopy (LFM) [73-75], force modulation (FM) [76-78], and contact resonance (CR) [79-81]. Even tapping mode can report energy-dissipative phenomena [44,82-84] and further offers loss tangent mapping [85,86]. 

The surface molecular motion of amorphous polymeric solids was measured by lateral force microscopy, scatming viscoelasticity microscopy, and differential X-ray photoelectron spectroscopy. Data are given for PS and styrene-methyl methacrylate copolymers. 35 refs. 

LFM is another widely used AFM technique to laterally probe the polymer surface, sometimes referred to as friction force microscopy (FFM). In this mode of operation, the AFM tip slides aaoss the polymer surface at a range of scanning speeds. The friction and adhesion force with the surface cause the cantilever to twist, and the friction force is measured from the torsion of the sliding cantilever. The frictional behavior of polymeric solids is closely related to their dynamic viscoelastic properties.The friction force depends on both the temperature and the scanning speed. The scanning rate dependence of the lateral force corresponds to the frequency dependence of the loss modulus ". Master curves can be constmcted with measurement at different temperatures and scanning speeds. The results based on this technique reflea the controversies commonly seen in the field of polymer dynamics in thin films and confined geometries. There are multiple observations of polymer surfaces and thin films with either bulk-like behavior or enhanced mobility. " " " ... 

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

The frequency correlation time xm corresponds to the time it takes for a single vibrator to sample all different cavity sizes. The fluctuation-dissipation theorem (144) shows that this time can be found by calculating the time for a vertically excited v = 0 vibrator to reach the minimum in v = 1. This calculation is carried out by assuming that the solvent responds as a viscoelastic continuum to the outward push of the vibrator. At early times, the solvent behaves elastically with a modulus Goo. The push of the vibrator launches sound waves (acoustic phonons) into the solvent, allowing partial expansion of the cavity. This process corresponds to a rapid, inertial solvent motion. At later times, viscous flow of the solvent allows the remaining expansion to occur. The time for this diffusive motion is related to the viscosity rj by Geo and the net force constant at the cavity... 

Hallworth and Carless (1 ) discuss several possibilities for the effect of light liquid paraffin on the stability of emulsions with light petroleum or chlorobenzene as the main components. They seem to prefer an explanation previously advanced by them and several other authors for the effect of fatty alcohol, namely that the increased stability is due to the formation of an interfacial complex between the additive and sodium hexadecyl sulphate. The condenced mixed film will resist coalescence primarily by virtue of its rheological properties. With mixed films of the present type, the importance of the film viscoelasticity lies in its ability to maintain electrical repulsion between approaching droplets by preventing lateral displacement of the adsorbed ions. The effective paraffinic oil has chains at least as long as those of the alkyl sulphate and will be associated by van der Waals forces with the hydrocarbon chain of the alkyl sulphate at the interface. 

Calculate the longitudinal strain of a viscoelastic rod of a material that behaves like (a) a Maxwell model, (b) a Maxwell solid in shear but an elastic solid in bulk and (c) a viscoelastic solid standard in shear but an elastic solid in bulk. The material is constrained in such a way that the lateral dimensions cannot vary when it is under uniform forces of compression at both ends of the rod. 

Free vibration, the motion that persists after the excitation is removed, is governed by Eq. (17.85), in which the applied transverse force has been made zero. Let us assume a solution of the form Uy(x, t) = /(x)exp(io)0 where f x) specifies the lateral displacement and is the angular frequency of the motion. For low loss viscoelastic materials, the free vibrations can be assumed to be quasi-harmonic, and therefore the complex modulus in the equation of motion can be used. The Laplace transform of Eq. (17.85) gives... 

Stress relaxation experiments involve the measurement of the force required to maintain the deformation produced initially by an applied stress as a function of time. Stress relaxation tests are not performed as often as creep tests because many investigators believe they are less readily understood. The latter point is debatable, and it may only be that the practical aspects of creep measurements are simpler. As will be shown later, all the mechanical parameters are in theory interchangeable, and so all such measurements will contribute to the understanding of viscoelastic theory. Whereas stress relaxation measurements are useful in a general study of polymeric behavior, they are particularly useful in the evaluation of antioxidants in polymers, especially elastomers, because measurements on such systems are relatively easy to perform and are sensitive to bond rupture in the network. 

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