Foamed polymers calculation

It is easy to see that these models are all based on the same (microstructural) principle, viz. that there is an elementary structural unit that can be described and then used for calculation. Remember that the corresponding unit cell for foamed polymers is the gas-structure element8 10). Microstructural models are a first approximation to a general theory describing the deformation and failure of gas-filled materials. However, this approximation cannot be extended to allow for all macroscopic properties of a syntactic foam to be calculated 166). In fact, the approximation works well only for the elastic moduli, it is satisfactory for strength properties, but deformation... 

This survey deals with the fundamental morphological parameters of foamed polymers including size, shape and number of cells, closeness of cells, cellular structure anisotropy, cell size distribution, surface area etc. The methods of measurement and calculation of these parameters are discussed. Attempts are made to evaluate the effect and the contribution of each of these parameters to the main physical properties of foamed polymers namely apparent density, strength and thermoconductivity. The cellular structure of foamed polymers is considered as a particular case of porous statistical systems. Future trends and tasks in the study of the morphology and cellular structure-properties relations are discussed. 

However, a real foam structure is composed of cells having differing shapes, sizes and volumes. In studying the properties of foamed polymers as well as in developing and elaborating preparative processes, it is necessary to find out cell size, shape and volume distribution. The methods for calculating the respective distribution functions will be discussed in Sect. 9.2, 9.3 here, we only note that the cell size distribution function is a most comprehensive and valuable characteristic of plastic foam structures. 

F g. 24a—c. Model of foamed polymer with isolated cells (a) block dissected by plane X (b) relationship between cell radius r and radius of circles s on section surface (c) calculation of functions f(r) and f(s) under limiting conditions 0 < x < 2x x — 2rx -f a = 0... 

Vice versa, starting from the specific surface of a foamed polymer one can find the geometrical size of the constituent cells. Indeed, in those rare cases where the real cellular foam structure may be represented by sphere of the same size, the diameter of the spheres can be easily calculated from Eq. (58) . In the case of more irregular structures, the diameter given by this formular is a mean surface-volume diameter . 

Cellular Structure Models and Calculation of Mechanical Properties of Foamed Polymers... 

Here is an example of a foamed polymer measured in shear where the activation energy (enthalpy) can be calculated from the Arrhenius equation... 

It is very useful to be able to predict melt temperature in extrusion, particularly in the extrusion of temperature sensitive polymers. Examples are extrusion of crosslinkable polymers, foamed polymers, and polymers that are susceptible to degradation. Unfortunately, the proper calculation of melt temperature is rather involved and requires the use of numerical techniques, the most popular being finite element analysis. 

Where n is the number of cells in the defined area A, and yOp and pf are the density of the unfoamed and foamed polymer, respectively. The foam density was measured by the buoyancy method. The relative foam density was calculated as the ratio of the foamed density to the density of unfoamed sample. 

The samples most commonly tested in compression are foams and rubbers, which experience compressive forces during use. Very often, the polymer foams that experience compression are not readily visible to us, even though they are all around. Polymer foams are widely used in carpet underlay, upholstery, shoe insoles, backpack straps, bicycle helmets, and athletic pads. Solid rubbers are much more visible, including automobile and bicycle tires, gaskets and seals, soft keys on calculators, and shoe soles. 

Research and development technologists at the Dow Chemical Company can characterize materials in a variety of ways. One material property that is especially critical in polymer foaming and processing technology is density. A tool used for measuring the density of a material is called a pycnometer. There are many different manual and automatic types to choose from. For extremely accurate and precise density measurements, an easy-to-use, fully automatic gas displacement pycnometer is utilized. Analyses are commenced with a single keystroke. Once an analysis is initiated, data are collected, calculations performed, and results displayed without further operator intervention. 

An examination of the experimental findings and the calculation model shows that the deformability of a syntactic foam depends mainly on the elastic properties of the polymer matrix, whereas the filler concentration mainly affects its compressibility. In fact, monolithic (unfilled) samples do deform elastically at the start of the compression curve, but when the material is deformed further, the forced elasticity limit is reached (Fig. 21). Thus, the nominal ultimate strength for non-brittle failure is determined by the fact that the forced elastic limit is reached, and not because the adhesive ties have lost their stability (as it is the case with light plastic foams) 8 10). 

Recently, Kinra and Ker 137) published data of the shear modulus of syntactic foams consisting of hollow glass spheres in a poly(methyl methacrylate) matrix. The glass spheres had a mean radius of 45 pm and a wall thickness of 1.2 pm. Reliable values are known for the shear modulus of the polymer G0, the shear modulus of glass Gs, and Poisson s ratio of the polymer G0 = 1120 MPa, Gs = 2800 MPa, and v0 = 0.35. Using these values, the upper curve 1 of Fig. 24 was calculated by Nielsen for the modulus of the foam as a function of the volume fraction of hollow spheres. These calculated values are, however, too high compared with the experimental values reported by Kinra and Ker. 

Fig. 9.52 Separation efficiency of a three-chamber co-rotating disk devolatilizer of 450°F PS melt containing 1500-3000 ppm styrene, fed at 42-lb/h into 0.54-in-wide chambers at 50-torr absolute pressure, as a function of disk speed and with flow rate as a parameter. Broken curves show calculated residence times. [Reprinted by permission from P. S. Mehta, L. N. Valsamis, and Z. Tadmor, Foam Devolatilization in a Multichannel Co-rotating Disk Processor, Polym. Process. Eng., 2, 103-128 (1984).]...

The foam formation during the addition of the stripping agent is not taken into account here, because the distribution of the stripping agent within the polymer, the number of bubbles and their growth, the bubble size, and the resulting increase of the phase interface cannot be calculated. 

Comparing the calculated and experimental stress-strain diagrams for real plastic foams, we will take Eo in Eq. (77) as the elasticity modulus of the polymer base Y and y are foam and polymer base densities (volumetric weights), respectively ... 

Eq. (77) may be valid for other rigid foams as well, since the model underlying the calculation procedure encompasses morphologies of a large enough range of real cellular polymers. 

Certain methods of calculating friction pressures involve the use of pressure charts that are in the form of pressure drop per length of pipe versus the flow rate of the foam. These types of charts incorporate foam quality and tubular geometry but may neglect consideration of one or more of the parameters such as temperature, pressure, foam texture, polymer or surfactant type, and concentration. Each of these parameters is often unspecified but drastically affects the friction pressures of foams. Information derived from such charts should be taken only as a guideline. 

---