Polymer volume fraction function

Fig. 29. Double logarithmic representation of the entanglement distance in polyethylene at 509 K as a function of the polymer volume fraction >. (Reprinted with permission from [60]. Copyright 1993 American Chemical Society, Washington)...

Fig. 51 Phase diagram for PS-PI diblock copolymer (Mn = 33 kg/mol, 31vol% PS) as function of temperature, T, and polymer volume fraction, cp, for solutions in dioctyl ph-thalate (DOP), di-n-butyl phthalate (DBP), diethyl phthalate (DEP) and M-tetradecane (C14). ( ) ODT (o) OOT ( ) dilute solution critical micelle temperature, cmt. Subscript 1 identifies phase as normal (PS chains reside in minor domains) subscript 2 indicates inverted phases (PS chains located in major domains). Phase boundaries are drawn as guide to eye, except for DOP in which OOT and ODT phase boundaries (solid lines) show previously determined scaling of PS-PI interaction parameter (xodt

For comparison, a telechelic sulfonated polystyrene with a functionality f = 1.95 was prepared. In cyclohexane the material forms a gel independent of the concentration. At high concentrations the sample swells. When lower concentrations were prepared, separation to a gel and sol phase was observed. Thus, dilution in cyclohexane does not result in dissolution of the gel even at elevated temperatures. Given the high equilibrium constant determined for the association of the mono functional sample, the amount of polymer in the sol phase can be neglected. Hence, the volume fraction of polymer in the gel phase can be calculated from the volume ratio of the sol and gel phases and the total polymer concentration. The plot in Figure 9 shows that the polymer volume fraction in the gel is constant over a wide range of concentrations. 

Fig. 4.27 SAXS intensity as a function of wavevector for a PS-P1 diblock (Mw = 60 kg mol-1, 17wt% PS) (points) in dibutyl phthalate with a polymer volume fraction

Fig. 4.34 SANS peak position, q, rescaled by 0(U2S as a function of polymer volume fraction in mixtures of normal and deuterated toluene and toluene at room temperature (Mayes et al. 1994) (a) dPS-PMMA diblock (AL, = 301 kg mol1, /mMA = 0.43) (b) PMMA-c/PS-PMMA triblock (AL — 93 kg mor /pMMA - 0.5). The solid lines indicate the best fit scaling relation q

Fig. 10. The root-mean-square layer thickness tms as a function of the square root of chain length r for four values of polymer volume fraction 0 . Hexagonal lattice,...

Fig. 8 Polymer volume fraction (j) = ho//t,w, where ho is the thickness of the polymer film after spin-coating and hsw is the thickness of a swollen film, measured by in situ spectroscopic ellip-sometry as a function of (a) the relative solvent vapor pressure for thin films of homopolymers PS, PB, and SBS block copolymer. Reprinted from [49], with permission. Copyright 2004 American Institute of Physics, (b) Polymer volume fraction as a function of the swelling time for PS- >-P2VP (SV) block copolymer and for homopolymers PS and P2VP at p/po = 1.0 [118]. The equilibrium degree of swelling indicates that toluene is a selective solvent for the PS block, and that SV block copolymer shows asymmetric swelling under toluene vapor. Reproduced by permission of The Royal Society of Chemistry (RSC) [118]...

Fig. 29 Characteristic wave number q as a function of polymer volume fraction for the systems of protein-like copolymers with L = 63 and random-block copolymers with different block lengths. The domain spacing is defined as r = lir/q. Adapted from [153]...

Figure 15.11 Slopes P=A/(X2-X 1) as a function of the polymer volume fraction at swelling equilibrium in cyclohexane < >e in a series of end-linked PDMS networks with various lengths between junctions (A, A Mn = 25000 g.mol 1 B, B Mn = 10500 g.mol1 C Mn = 3100 g.mol"1) and polymer concentration during crosslinking in toluene (A, B, C vc = 0.7 A , B vc =1)...

Fig.1. Dry thickness h0 of various PDMS layers irreversibly adsorbed on the silica surface of a silicon wafer, as a function of the scaling variable Nm 0m. N is the polymerisation index of the surface anchored chains and(])0 is the polymer volume fraction in the incubation bath. The triangles correspond to 4>0=1, i.e., to the maximum surface density of anchored chains for the particular molecular weight. The range of molecular weights used is 29.6 kg mol-1

The d- values obtained as a function of the polymer volume fraction v for the PEO-added system are shown in Figure 11.6. The values are remarkably similar to the corresponding curve for the PVME-added system shown in Figure 11.3, giving rise... 

This fact can be demonstrated as follows. Let us determine the value of the well-known Flory parameter x, which corresponds to the 6 point (i.e. to the point of inversion of the second virial coefficient of the solution of rods) in the Flory theory of Ref.9). This can be done by expanding the chemical potential of the solvent in the isotropic phase (Eq. (16) of Ref.9 ) into powers of the polymer volume fraction in the solution, and by equating the coefficient at the quadratic term of this expansion to zero this procedure gives Xe = 1/2 independently of p. On the other hand, it is well known26,27) that the value of x decreases with increasing p and that X < 1 at p > 1. The contradiction obtained shows that the expressions for the thermodynamic functions used in Ref.9) are not always correct... 

Let N cylindrical rods be situated in volume V, their concentration being c = N/V. The polymer volume fraction in the solution is then - jrpcd V4. Let us introduce the orientational distribution function for the rods f(u) cf(u)df2 is the number of rods per unit volume, which have the orientations within the small spatial angle dQ around the unit vector u. It is dear that in the isotropic state f(u) = const = l/4a. In the liquid-crystalline state the function f(u) has two maxima along the anisotropy axis. 

As already noted, the measured nonlinear shear relaxation modulus, for linear molecules with little polydispersity, is in excellent agreement with the Doi-Edwards model at long times. However, for melts or concentrated solutions of very high molecular weight (e.g., 10 for polystyrene, where 0 is the polymer volume fraction), the measuredfiamping function, h(y), is drastically lower than the Doi-Edwards prediction (Einaga et al. 1971 Vrentas and Graessley 1982 Larson etal. 1988 Morrison and Larson 1992). This anomalous... 

Figure 11.19 Splay constant K as a function of molecular aspect ratio L/d at polymer volume fractions of 0.16 and 0.20, obtained from light-scattering data, taking = 0.6 x 10 dynes, in solutions of PBG in a mixed solvent composed of 18% diox-ane and 82% dichloromethane. An aspect ratio of 100 corresponds to a molecular weight of 210,000. (From Lee and Meyer 1990, by permission of Taylor Francis.)...

Figure 3.18 Polymer volume fraction (V2) at swelling equilibrium for poly(methyl trifluoropropylsiloxane) (PMTFPS) networks as a function of the temperature in selected solvents (20). (O, ) Butyl acetate ( , ) tetrahydrofuran (A, A) ethyl acetate (O, ) -chlorobutane. Empty and filled symbols represent values of V2 obtained by increasing and decreasing the temperature, respectively. (From Ref. 20.)...

Analysis of the scattering data yielded information about the structure of the polymer layer at the interface and, in particular, allowed determination of the polymer volume fraction profile as a function of distance from the oil-water interface. The profiles obtained were consistent with the hydrophobic poly (propylene oxide) blocks bound to the surface of the droplet while the hydrophilic poly(ethylene oxide) blocks were largely present as tails. The authors also determined the effect of salt on the conformation of the adsorbed polymer layer. 

Monomer diffusion inside a particle is a function of the polymer volume fraction. 

FIG. 6.1 Structural information obtained from SANS for an 8 wt% EO37PO58EO37 solution at 60°C plotted as a function of the cosolvent (glycerol, formamide, or ethanol) content in the mixed solvent. First row micellar association number ( association) second row radii of core and core + corona (Wcore and / micelle) third row polymer volume fraction of core and corona (acore and acorona). (From Alexandridis, P. and Yang, L., Macromolecules, 33, 5574, 2000.)... 

Fig. 9 The acoustic contrast easily saturates. The figure shows a sketch of the contrast function (integrands in Eqs. 78 and 79) as a function of the polymer volume fraction of an adsorbed polymer film. It is assumed that both the shear modulus Gf and the dielectric constant Sf are roughly proportional to the polymer concentration. However, Gf increases much more strongly than gf. If, for instance, a swollen polymer film contains 50% water, this will not appreciably decrease the apparent acoustic thickness because the modulus of the film is still much larger than the modulus of water and (Gf - Gijq)/Gf remains about unity. This is different in optics because the contrast is roughly proportional to the concentration. As a consequence, the apparent optical film thickness is proportional to the product of concentration and thickness, which is the adsorbed amoimt. In acoustics, the apparent thickness is close to the geometric thickness. Trapped water appears as part of the film in acoustics...

Paul et al. (25) observed that for polymer volume fractions less than 0.8, the functional dependence of the diffusion coefficients on the polymer volume fraction was, generally, in accordance with Equation 40. Muhr and Blanshard (26) provide additional supporting data on different polymers than those reported by Paul et al, Roucls and Ekerdt (27) measured the diffusion of cyclic hydrocarbons in benzene-swollen polystyrene beads their diffusion coefficients satisfy the general form of Equation 40. The effective dlffuslvltles of organic substrates in crossllnked polystyrene reported by Marconi and Ford (17) also follow trends predicted in Equation 40. In the absence of experimental data, it appears that Equation 40 provides a reasonable, and the simplest, means to estimate D for use in detailed modeling or in estimation methods such as Equation 38. Equation 40 was used by Dooley et al. (11) in their study of substrate diffusion and reaction in a macroreticular sulfonic acid resin which involved vapor phase reactants. 

Methods for calculating cloud-point and spinodat curves from Eq. (36) have been discussed in the literature [57,61, 79-88]. The interaction parameter, x, may be empirically expressed as a function of temperature, polymer volume fraction and number-average degree of polymerization [87]. 

The simulations yielded the positions of the fronts in the problem as a function of time (Fig. 11) as well as the polymer volume fraction profiles as a function of both time and position (Fig. 12). An increase in the external... 

The simulations yielded the polymer volume fraction profile as a function of time, both in the rubbery and in the diffusion boundary layer, as shown in Figs. 15 and 16. The variation of the positions of the various interfaces in the problem as a function of time is shown in Fig. 17. The effect of molecular weight... 

Fig. 15. Polymer concentration profile in the slab, expressed as the polymer volume fraction, Uj, as a function of dimensionless position with 0 indicating the center of the slab. Theoretical predictions have been adapted from the workofDevottaet al. [48], using the following parameters u,., = 0.15 Bd = 1.5 i)f = 0.3 x = 0.35 and KU/Do = 10 ...

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