Weight root mean square

Synthetic, nonionic polymers generally elute with little or no adsorption on TSK-PW columns. Characterization of these polymers has been demonstrated successfully using four types of on-line detectors. These include differential refractive index (DRI), differential viscometry (DV), FALLS, and MALLS detection (4-8). Absolute molecular weight, root mean square (RMS) radius of gyration, conformational coefficients, and intrinsic viscosity distributions have... 

Table IV lists comparative SD and / values for fittings of all of the reactions of Table II and III with each of the gr scales derived in this paper. The comparison includes figures for fittings with F and R values of Swain and Lupton (S L) and fitting with the Hammett equation. We believe the results given in Table IV provide a clear confirmation of the uniqueness and limited generality of the o/2(ba) scale. Very consistently, the fit achieved by the or (BA) scale is shown in Table IV to be superior by significant factors to that achieved by any of the other scales or by the simple op treatment. This fact is clearly reflected in the overall / values and the similarly weighted root-mean-square / values, F = y/lfnif lN, sum taken over all reactions. The value of F is. 067 for the basis sets of Table II (compare with overall / of. 058). The comparable F values are. 140 for Or, . 088 for S L, and. 155 for 0(p) with the data differences as explained in Table IV. For all sets of Table IV, the corresponding figures are. 073 for o (ba). -143 for a%, . 097 for S L, and. 209 for 0(py...

The parameters calculated for the acids are given in Table 3 and those for the bases in Table 4. The weighted root-mean-square deviation between the experimental enthalpies and those calculated from the parameters in Tables 3 and 4 using Eq. (13) is about 0.016 corresponding to a deviation of about 0.2 kcal/mole for a heat of 8 kcal/mole. The excellent agreement between the experimental enthalpies of adduct formation and the calculated enthalpies for aU of the interactions are reported in the literature (40). 

A search for the optimal polynomials was made by minimizing a weighted root-mean-square error (RMSE) around the mean of absolute errors of the expansion of In b u) ... 

Another characteristic of a polymer surface is the surface structure and topography. With amorphous polymers it is possible to prepare very smooth and flat surfaces (see Sect. 2.4). One example is the PMIM-picture shown in Fig. 7a where the root-mean-square roughness is better than 0.8 ran. Similar values are obtained from XR-measurements of polymer surfaces [44, 61, 62], Those values compare quite well with observed roughnesses of low molecular weight materials. Thus for instance, the roughness of a water surface is determined by XR to 0.32 nm... 

Equation (10) directs attention to a number of important characteristics of the molecular expansion factor a. In the first place, it is predicted to increase slowly with molecular weight (assuming t/ i(1 — 0/T) >0) and without limit even when the molecular weight becomes very large. Thus, the root-mean-square end-to-end distance of the molecule should increase more rapidly than in proportion to the square root of the molecular weight. This follows from the theory of random chain configuration according to which the unperturbed root-mean-square end- o-end distance is proportional to (Chap. X), whereas /r = ay/rl. 

For a statistical coil, the product of polymer intrinsic viscosity and molecular weight is directly proportional to the cube of the root-mean-square radius of gyration RG 77137... 

Figure 9 Scaling relationship between root-mean-square radius and molecular weight for polystyrene using GPC/MALLS. (From Wyatt, P. Hicks, D. L., Jackson, C., and Wyatt, G. K., Am. Lab., 20, 108,1988. With permission.)...

Since the masses of the molecules are proportional to their molecular weights and the average velocity of the molecules is a measure of the rate of effusion or diffusion, all we have to do to this equation to get Graham s law is to take its square root. (The square root of v2 is not quite equal to the average velocity, but is a quantity called the root mean square velocity. See Problem 12.18.)... 

Effect of PVA Molecular Weight on Adsorbed Layer Thickness. Figure 4 shows the variation of reduced viscosity with volume fraction for the bare and PVA-covered 190nm-size PS latex particles. For the bare particles, nre(j/ is independent of and the value of the Einstein coefficient is ca. 3.0. For the covered particles, rired/ t increases linearly with tp. Table IV gives the adsorbed layer thicknesses calculated from the differences in the intercepts for the bare and covered particles and determined by photon correlation spectroscopy, as well as the root-mean-square radii of gyration of the free polymer coil in solution. The agreement of the adsorbed layer thicknesses determined by two independent methods is remarkable. The increase in adsorbed layer thickness follows the same dependence on molecular weight as the adsorption density, i.e., for the fully hydrolyzed PVA s and... 

Figure 2. Root-mean-square thickness in the plateau region of polystyrene, adsorbed on chrome ferrotype as a function of the square root of the molecular weight, in cyclohexane, 36.4°C.

Fig. 21. Ratio between the correct molecular weight (M) and that calculated from dissymmetry (Md), as a function of the root mean square end-to-end distance calculated from dissymmetry (< h2 >y2) for different values of e 30 ...

It should be mentioned that another validation technique, called leverage correction [1], is available in some software packages. This method, unlike cross validation, does not involve splitting of the calibration data into model and test sets, but is simply an altered calculation of the RMSEE fit error of a model. This alteration involves the weighting of the contribution of the root mean square error from each calibration... 

FIGURE 3.15 Standard plot of the log of the mean radium of gyration versus log molecular weight for differently shaped macromolecules. Essentially, for a sphere the radius is proportional to the root-mean-square (RMS) radius, and with a slope in the logrg versus log M of 1/3 for rod-shaped polymers, length is proportional to RMS radius and M with a slope of 1 and for random coils the end-to-end distance is proportional to the RMS radius and with a slope of about 0.5-0.6. 

Fig. 10 Root-mean-square-error (RMSE) plots for the deconvolutions of Fig. 9 (a) no relaxation, (b) Jansson-type relaxation, (c) clipping, (d) Gaussian weighting.

Stromberg et al.68 applied ellipsometry to the adsorption of polystyrene samples with a narrow M-distribution on a chrome plate from cyclohexane solutions at 35 °C (the theta condition for polystyrene) and found that t was approximately proportional to the square root of molecular weight, M up to 1.8 x 106. These values of t were compared with the root-mean-square end-to-end distances calculated for the chains which are attached at one end to either a reflecting or an absorbing wall. They fell between the computed values for these two walls, though somewhat closer to the values for the absorbing wall. 

The model was also validated by full cross-validation. Four PCs were necessary to explain the most variation in the spectra (99.9%), which best described the molecular weight. The root mean square error of prediction... 

In static LLS [43], the angular dependence of the excess absolute time-averaged scattering intensity, known as the Rayleigh ratio Rw(< )> is measured. For a dilute solution measured at a relatively small angle (6), Rw(q) can be related to the weight average molar mass (Mw), the root mean square... 

Fig.6 Temperature dependence of z-average root-mean square radius of gyration ((Pg>) and average hydrodynamic radius ( Rh)) of copolymers NIPAM-co-VP/60/5 and NIPAM-co-VP/30/5 in water, where the weight average molar masses are 2.9 x 106 and 4.2 x 106 g/mol, respectively [56]...

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