Curves Blending functions

Characteristics of Curves Industrial practice of shape modeling applies only a small number of representations for curves. Polynomial are the preferred class of mathematical functions for the description of curves and surfaces. Basis functions are often called blending functions because they affect the shape of the entire curve (global control) or only several of its segments (local control). They are connected to control vertices or interpolation points. 

If statements provided above are justified, than according to [155], there must always be some threshold concentration and temperature values where the flow mechanism of blended systems changes. This conclusion has been confirmed experimentally the curves of functions - T for the studied compositions have well-shown bend, where viscosity drop rate changes. The processing of these data involving peak Newtonian viscosity ( 7n), obtained by extrapolation to zero shear rate, demonstrates (Figure 3.40), that dependencies Ig n on HT can be approximated by two linear sections - 20-40°C and 50-80°C, that can be the effect of the blend flow mechanism varied near 45 °C. 

Fig. 5. Phase behavior of blends of a styrene—acrylonitrile copolymer containing 19 wt % of acrylonitrile with other SAN copolymers of varying AN content and as a function of the molecular weight of the two copolymers (° ) one-phase mixture ( ) two-phase mixtures as judged by optical clarity. Curve...

Figure 11 Force displacement curves of a-PP, b-nonreac-tive PP-NBR blend, c-reactive blend containing 13 wt% GMA functionalized PP, and d-reactive blend containing 25 wt% GMA functionalized PP. Source Ref. 73.

Figure 13.6 (a) Elongation as a function of wind-up speed for partially oriented yarn, (b-d) Stress-strain curves of fibers of PET blends with 3% copolyester of 1,4-phenyleneterephthalate and p-oxybenzoate (CLOTH) and 3% copolymer of 6-oxy-2-naphthalene and p-oxybenzoate (CO), spun at 3500, 4000 and 4500 m/min (1) PET control (2) 3 % CLOTH (3) 3 % CO the loci of the theoretical extensions of the PET control are shown as dashed curves [17]. From Orientation suppression in fibers spun from melt blends, Brody, H., J. Appl. Polym. Sci., 31, 2753 (1986), copyright (1986 John Wiley Sons, Inc.). Reprinted by permission of John Wiley Sons, Inc. 

Curves resulting from the choice of Bezier functions blend the values of the known boundary potentials to produce interior potential values and have the appropriate smoothness properties desired in the final solution. Further, the 5, have maximum values that distribute evenly through the mesh regions. For instance, for u between 0 and j in Eq. (15.20), the value of 5q,4 is greatest, and all other B variables approach minimum values. Thus, Bq,4 serves to sample that particular range of u values. 

Needless to say, the rheological properties of polymer mixtures are complex and nearly impossible to predict. Figure 4.12 shows the viscosity of a natural rubber (NR)/poly(methyl methacrylate) (PMMA) blend (top curve) as a function of percentage NR [2]. For comparison, the predictions of four common equations are shown. The equations are as follows ... 

An interesting method of fitting was presented with the introduction, some years ago, of the model 310 curve resolver by E. I. du Pont de Nemours and Company. With this equipment, the operator chose between superpositions of Gaussian and Cauchy functions electronically generated and visually superimposed on the data record. The operator had freedom to adjust the component parameters and seek a visual best match to the data. The curve resolver provided an excellent graphic demonstration of the ambiguities that can result when any method is employed to resolve curves, whether the fit is visually based or firmly rooted in rigorous least squares. The operator of the model 310 soon discovered that, when data comprise two closely spaced peaks, acceptable fits can be obtained with more than one choice of parameters. The closer the blended peaks, the wider was the choice of parameters. The part played by noise also became rapidly apparent. The noisy data trace allowed the operator additional freedom of choice, when he considered the error bar that is implicit at each data point. 

Figure 3. Relationship between % of crystallinity and lignin content for blends prepared with the acetylated lignin series. The degree of substitution of hydroxy functionality is given with each curve.

Suzuki et al. reported cloud-point temperatures as a function of pressure and composition in mixtures of poly(ethyl acrylate) and poly(vinylidene fluoride) [9], Their data in terms of p(T) curves at constant composition show that miscibility in the same system may either improve or decline with rising pressure, depending on the blend s composition. Important consequences for blend-processing ensue. A planned two-phase extrusion may easily be jeopardized by the pressure building up in the extruder. Conversely, a homogeneous melt may be turned into a two-phase system when the pressure on the blend increases. 

Figure 9.7-1 Experimental cloud-point curve of the polymer blend Poly(methyl methacrylate)/Poly(styrene-co-acrylonitrile (28%AN)) as a function of pressure.

Fig. 6.30 Order-disorder transition in a binary homopolymer/diblock blend as a function of volume fraction of homopolymer computed using SCFT (Whitmore and Noolandi 1985a). The symmetric diblock has N = 100. Curves are shown for different values of /3 = NJNC. The temperature scale on the left-hand side was calculated for PS-PB/PB blends.

The heart of the pilot plant study normally involves varying the speed over two or three steps with a given impeller diameter. The analysis is done on a chart, shown in Fig. 36. The process result is plotted on a log-log curve as a function of the power applied by the impeller. This, of course, implies that a quantitative process result is available, such as a process yield, a mass transfer absorption rate, or some other type of quantitative measure. The slope of the line reveals much information about likely controlling factors. A relatively high slope (0.5-0.8) is most likely caused by a controlling gas-liquid mass transfer step. A slope of 0, is usually caused by a chemical reaction, and a further increase of power is not reflected in the process improvement. Point A indicates where blend time has been satisfied, and further reductions of blend time do not improve the process performance. Intermediate slopes on the order of 0.1-0.4, do not indicate exactly which mechanism is the major one. Possibilities are shear rate factors, blend time requirements, or other types of possibilities. 

A calibration curve is prepared, using absorbance versus concentration plot, so that the concentration of the unknown component can be determined. But quantitative analysis for a complex system like vulcanised rubber or a blend of two or three components, is not possible. The use of computers with the FTIR spectrometer, increases the rapid scanning capability, data processing for analysis of chemical or physical structural changes in polymers as a function of time over the entire mid-IR frequency. 

The viscosity functions of homopolymers and compatible polymer blends measured at different temperatures can be shifted together by displacement along a 45° axis to form a single curve (mastercurve) by time-temperature-superposition (see Fig. 3.12). This... 

Fig. 7. Miscibility door for 50/50 blends of PB and SB as a function of copolymer composition. The circles refer to experimentally determined LCSTs and UCSTs. The curve was calculated using the equation-of-state theory discussed in Sect. 2.1. Miscibility occurs to the left of the curve. Inside the dashed area, solution cast films are transparent [39]...

From the pure component data it is possible to calculate the expected a behavior as a function of temperature for a blend of the two polymers using the equation ah = sas + < rar, where the subscripts b, s, and r refer to the blend, polystyrene, and rubber, respectively, and the < s represent the volume fractions of the two components in the blend. The calculated curves (Figure 7) are reasonably smooth and exhibit only the polystyrene Tg. The calculated curve for TR-41-2445 is in good agreement with that found experimentally for the solution-blended material. The only significant difference is that below the polystyrene Tg the calculated values of a are about 0.5 X 10 4 deg1 lower than the experimentally determined data points. This may be attributable to the density differences in the samples, particularly for the blended material where density variations and void formation can occur at the interfaces between the polymer phases. 

Calculation of Master Curves from Mechanical Models. The only way to obtain valid master curves for the thermorheologically complex systems (75/25 and 50/50 blends) is to calculate the moduli of the blends as a function of time, using an appropriate mechanical model. This method requires knowledge of the time and temperature dependence of the mechanical properties of the constituent phases. 

One of the most difficult problems when characterizing copolymers and polymer blends by SEC-viscometry is the accurate determination of the polymer concentration across the SEC elution curve. The concentration detector signal is a function of the chemical drift of the sample under investigation. To overcome this problem, Goldwasser proposed a method where no concentration detector is required for obtaining Mn data [72]. In the usual SEC-viscometry experiment, the determination of the intrinsic viscosity at each slice of the elution curve requires a viscosity and a concentration signal ... 

Curves obtained by plotting the decrease of film thickness after MO min exposure to Op-RIE as a function of weight ratio of GR-950 blended OFPR-800 are shown in Fig.9. The oxygen etching rates of... 

Duff et al. [27] reported a study made by means of DSC and WAXD on SPS/ PPE blends of various compositions, precipitated from ethylbenzene solutions, compression molded at 330 °C for 2 min and then slowly cooled to room temperature. In particular, the WAXD patterns show that in sPS-rich blends (>50 50 wt%) sPS is in a 0 or (3 form, while small amounts of a are present in the 50 50 wt% blend. The kinetics of crystallization and the mechanism of nucleation of sPS were investigated under isothermal and nonisothermal conditions as a function of blend composition and molecular weights of the components. The experimental curves show that the half-time to crystallization, t j2, increases with increasing content and molecular weight of PPE, but is not influenced by the molecular weight of sPS. The crystallization kinetics were... 

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