Dynamical critical index

Figure 6 Product of riD versus the index of polymerisation of the PDMS, for N P (full circles) and for N=P (open squares). From this plot, a critical index of polymerisation for a departure from Rouse - like dynamics Ne = 500 can be estimated.

A good quality fit to experimental data can be achieved by assuming polydisperse dynamical fractal clusters with a fractal dimension df = 2.5 and a polydispersity index T = 2.2. From the fits we can deduce the short-range correlation length < o the radius of the cluster elements / (. We obtain ( o = 1.3 0.1 nm and / i = 5 1 nm. When none of the parameters is fixed, the best fit to the data leads to df = 2.5 0.1 and r = 2.3 0.1. These values are in good agreement with the theoretical estimates to within experimental error and lead to critical index values very close to the universal ones. The two other fitted parameters are ( o = 1.2 0.1 nm and / i = 6 1 nm. 

One can try to locate a critical polymerisation index above which the data are no longer compatible with a Rouse-like dynamics, Ng = 500, lager than the Ng= 100 value determined from the diffusion measurements in a frozen matrix. This is an illustration of the fact that the two processes. Rouse motion and entangled motion are in competition the slowest process is the one which is indeed observed.When the matrix chains are mobile, the entangled dynamics becomes more rapid than pure reptation, and the Rouse motion can dominate the dynamics for larger molecular weights than when the matrix chains are immobile. 

Critical points. The critical points (or limit points) of a dynamical system are the points of M for which X mc) = 0. A critical point is either an a or an co limit of a trajectory. The subset of points of M by which are built trajectories having mc as co limit is called the stable manifold of my, the unstable manifold of mc is the set for which mc is an limit. The dimension of the unstable manifold is the index of the critical point. The set of the critical points of a dynamical system satisfies the Poincare-Hopf formula ... 

CCT, critical cracking thickness Boltzmann constant (1.381x10 local permeability [m ] fracture resistance [N m ] average permeability in/of compact [m ] particle shape factor compact thickness [m] initial particle number concentration [m refractive index of particle material refractive index of dispersion material number density of ion i dimensionless number dimensionless number Stokes number Peclet number capillary pressure [N-m ] dynamic pressure [N m ] local liquid pressure in the compact [N-m local solid pressure in the compact [N-m ] superficial fluid velocity [m-s q gas constant [J K ] centre to centre distance [m]... 

To observe the dynamics of small structures such as the growth of single actin filaments in solution or on functionalized beads, a specialized TIRF microscope is needed. Critical for TIRF imaging are furthermore a sensitive CCD Camera, high-quality objectives with a high numeric aperture (NA > 1.4), immersion oil (e.g., Leica with a refractive index of 1.518) as well as suitable lasers and filter sets. These days, complete systems for TIRF microscopy can be purchased from manufacturers such as Olympus, Nikon or Leica. 

We briefly set up a dynamic model of an entangled polymer solution in the semidilute regime, (f) > = and critical volume fraction and N being the polymerization index. In terms of the polymer volume fraction (f> and the conformation tensor W = Wij, the free energy is given by [10]... 

The value of the powers, superscripts s and /, in Eqs. (6) and (7) are 1.3 and 1.8, respectively, by replacing p with the polymer concentration. There are many careful measurements reported on the modulus near the gel point. Tokita and Hikichi [187] measured the modulus of dilute agarose aqueous solution as a function of temperature and determined the value of / to be 1.9 and the critical temperature, = 80°C. The concentration dependence of the agarose aqueous solution at 20°C was also measured and obtained as = 0.0137 g/lOOmL, /=1.93. Gauthier-Manuel et al. [188] measured the dynamic viscoelasticity if = 10 Hz) of silica particle suspension as a function of reaction time and obtained / = 2. The experimental value is closer to the index 1.9 of the 3D percolation theory than is / = 3 of the Flory-Stockmayer... 

The index of a critical point m of a gradient dynamical system is the number of positive eigenvalues of the matrix of the second derivatives of the potential function at m. In this case, a critical point is said to be hyperbolic if none of the eigenvalues are zero. In the case of a gradient dynamical system, the index of a critical point is the number of positive eigenvalues of the matrix of the second derivatives of its potential function at me. A critical point is said hyperbolic if none of the eigenvalues are zero. 

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