Density increment

Here c(r) is the polymer mass concentration at a radial distance r from the axis of rotation, 0p/0c the specific density increment of polymer at constant T and p0, and co the angular velocity. 

In this paper we have used the quantity (1 — vp0) in writing equations for sedimentation equilibrium experiments. Some workers prefer to use the density increment, 1000(dp/dc)Tfn, instead when dealing with solutions containing ionizing macromolecules. This procedure was first advocated by Vrij (44), and its advantages are discussed by Casassa and Eisenberg (39). Nichol and Ogston (13) have used the density increment in their analysis of mixed associations. The subscript p. means that all of the diffusible solutes are at constant chemical potential in the buffer... 

A determination of the molecular weight by small-angle x-ray scattering requires, therefore, knowledge of the scattering intensity as a function of protein concentration and angle of observation, as well as of the electron density increment, ( — ), which replaces the refractive index... 

As in Equation 14 for the density increment, the right side of Equation 13a is equal to (dn/dg2)T,m,ns, i-e., the refractive index increment measured by using a dialyzed macromolecule solution and the dialyzate as reference solvent.)... 

Data used to generate these surfaces were collected at 80°C in density increments of 0.1 g/mL. Other conditions as described in the text... 

It is easier experimentally to determine 5 than to perform an equilibrium sedimentation measurement, but, because we are interested in the molar mass and the density increment, D has to be determined independently. 

The constant depends on the radiation used, as does (dp Jdc ), which is the refractive index increment for light, the electron density increment for X-rays, and the neutron scattering density increment for neutrons. 

In Section IV,E we saw how to obtain experimental data that are related to the molar mass of a particle and the appropriate density increment of the solution. We shall assume that both these parameters have been measured and discuss how they can be interpreted. In certain cases, the mass density increment could be measured directly by weighing calibrated volumes of the solution at dialysis equilibrium. These, however, are difficult experiments to perform in high salt concentrations. 

The thermodynamics approach to macromolecules in solution and their interactions with solvent components have been reviewed by Eisenberg (1990). Briefly, the density increment of a solution containing three components—(1) water, (2) macromolecules, and (3) salt—due to an increase in the concentration of component 2 at constant chemical potential of the other components, is given by... 

Fig. 6. Mass (top and right-hand y axis) and neutron scattering length (bottom and left-hand y axis) density increments for AMDH as functions of the respective solvent densities. From Zaccai et al. (1989), with permission.

Composition Parameters of the hMDH Solution Particles, Calculated from Neutron-Scattering and Mass Density Increments... 

Because mass density increments could not be determined for potassium phosphate solutions, a value of v2 = 0.75 was assumed in order to interpret the neutron scattering density increment. 

The solute contributing to the density increment must be highly soluble in electrophoresis buffer, of a high density increment per gram, electrically neutral, nontoxic to cells, of a low intrinsic viscosity, not, or only slightly, contributing to osmolarity, and easy to remove after electrophoretic separation. 

In both cases, the distance from the light source to the sample plane was 10 cm. The light intensity was modulated using a 30 step transparent wedge of 0.1 optical density increments. Before each exposure, the sample and the mask were thermally equilibrated for 5 min at a designated temperature. 

Samples were exposed to the light through a photographic step tablet with 0.1 optical density Increments. This provided the necessary range of exposure levels at a single exposure time ( ). Specific details on the exposure time and light sources used are given separately in the discussion. 

Fig. 1. Electron density of acetylenedicarboxylic acid dihydrate projected down the short crystal axis. Each contour line represents a density increment of approximately one electron per A, the one-electron line being dotted (from [8]).

The interstitial Br molecules of TMA /g Br2 are found in the larger of these voids in disordered fashion. Presumably the other ests listed in Table 2 are in the same set of cavities but this has not been tested directly. It was suggested that these cavities may contain water molecules even in the supposedly anhydrous a-TMA. However the density increment that would result is within the limits of experimental error, while the original intensity data are no longer available so that this suggestion cannot be subjected to experimental test without making a new set of intensity measurements. 

In the two-dimensional case, the integral in Eq.(2) is straightforward. Therefore, in putting L = 1.5 mm, the density increment corresponding to one fringe shift is given by... 

Why does the free energy density acquire this particular form First, in the curvature term with modulus Ku, we must use the second derivatives because the first derivatives correspond to a pure rotation of all the layers that does not cost energy. The higher derivatives are ignored for small distortions. For the compressibility term, the first derivative (du/dz) is sufficient. Second, both the compressibility and the curvature terms must be squared due to head-to-tail symmetry and parabolic form of the density increment gdisrgo as a function of distortion (Hooke s law). However, the question arises why is only splay modulus taken into account in (8.44) and not the other two Frank moduli K22 and 33. Considering the splay and bend distortions of the SmA phase in Fig. 8.24 we can see that only the splay distortion is allowed because it leaves the interlayer distance and the... 

Take different values of C and, and calculate the values of C2 and the annulus apparent-density increment. 

From the above calculation results, with the particle volume concentration increased, the annulus apparent-density increments increases. Take K = 0.5, C = 2% and C = 3%, the annulus apparent-density increments is too large, which may fracture the formation and cause accidents. Take K = 0.7, Cl = 1%, the annulus apparent-density increments is acceptable. 

We specialize for simplicity in the following to the decomposition of the density and refractive index increments to a three component system in which component 3 is a low molecular weight salt or uncharged component. It is possible to define an equivalent particle with which are asssociated B, and 83=83— gram of component 1 or component 3 respectively per gram of component 2 3 derives from Dorman exclusion of an ionic component 3 in the case of highly charged component 2. The density increment is now restated in the form... 

In multicomponent systems (more than one solvent component), the refractive index increment dn/dc is replaced by an experimentally measurable increment at constant chemical potential ft of solvent components, (jdn/dc2 in analogy to the density increments discussed above. In the limit of low concentrations, C2... 

Density increments (Op/Ocj) , and scattering length density increments (O n/Oc ) have been calculated on the assumption that there are four labile exchangeable protons per base pair for a DNA with 50%... 

Column 2 Hydrodynamic properties radius of gyration, Rq translational diffusion coefBcient, Djo.w persistence length, Oq density increment, Ag/Ac sedimentation coefficient, S2o.wi nd refractive increment, An/Ac. 

Density Increment for Mass Movement Mass Fraction Increment Momentum Increment in the X or R Direction Momentum Increment in Z Direction 1 for Slab, 2 for Cylindrical Geometry Time Increment in Microseconds Density (g/cc)... 

Figure 11.25 The MWD pn p,D)) from MQ-NMR measurements for various radiation dosages. The data was interpreted by using N = 5.7 in Eq. 11.1. Inset Corresponding chain density increment /, -/ j = pjO)/pJD) - 1 showing linear increase as a function of D. Reproduced with permission from [112] Copyright (2011) American Physical Society.

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