Lamellar modulus

Two approaches have been taken to produce metal-matrix composites (qv) incorporation of fibers into a matrix by mechanical means and in situ preparation of a two-phase fibrous or lamellar material by controlled solidification or heat treatment. The principles of strengthening for alloys prepared by the former technique are well estabUshed (24), primarily because yielding and even fracture of these materials occurs while the reinforcing phase is elastically deformed. Under these conditions both strength and modulus increase linearly with volume fraction of reinforcement. However, the deformation of in situ, ie, eutectic, eutectoid, peritectic, or peritectoid, composites usually involves some plastic deformation of the reinforcing phase, and this presents many complexities in analysis and prediction of properties. 

P. Richetti, P. Kekicheff, P. Barois. Measurement of the layer compressibility modulus of a lamellar mesophase with a surface forces apparatus. J Phys II 5 1129-1154, 1995. 

Similar types of lamellar morphologies were observed for triblock copolymers of diphenylsiloxane and dimethylsiloxane having 40 wt% polydiphenylsiloxane, using electron microscopy, 47-148>. The lamellae thickness was approximately equal to the chain length of the rigid polydiphenylsiloxane blocks. These copolymers showed elastomeric properties comparable to those of conventional silica-reinforced, chemically crosslinked silicone rubbers. Tensile tests yielded an initial modulus of 0.5-1 MPa, tensile strength of 6-7 MPa and ultimate elongation between 400 and 800 %. 

The elastic modulus of composite materials reinforced by discontinuous cylindrical fibers or lamellar shapes is expressed by the Halpin-Tsai equations [106, 230], as shown in the following equation ... 

These observations are valid not only for PE. The changes in the thermomechanical behaviour after annealing of oriented PP having a modulus of elasticity of E = 10-15 GPa and PA (E = 8-10 GPa)187) are similar to those of PE. The thermomechanics of the so-called hardelastic fibres, which are known to have a lamellar structure, is very similar to that of annealed PE and PP 188). 

Fig. Z4 (a) Temperature ramp at a frequency a> = lOrads (strain amplitude A = 2%) for a nearly symmetric PEP-PEE diblock with Mn = 8.1 X 104gmol l, heating from the lamellar phase into the disordered phase. The order-disorder transition occurs at 291 1 °C, the grey band indicates the experimental uncertainty on the ODT (Rosedale and Bates 1990). (b) Dynamic elastic shear modulus as a function of reduced frequency (here aT is the time-temperature superposition shift factor) for a nearly symmetric PEP-PEE diblock with Mn = 5.0 X 1O g mol A Shift factors were determined by concurrently superimposing G and G"for w > and w > " respectively. The filled and open symbols correspond to the ordered and disordered states respectively. The temperature dependence of G (m < oi c) for 96 < T/°C 135 derives from the effects of composition fluctuations in the disordered state (Rosedale and Bates 1990). (c) G vs. G"for a PS-PI diblock with /PS = 0.83 (forming a BCC phase) (O) 110°C (A) 115°C ( ) 120°C (V) 125°C ( ) 130°C (A) 135°C ( ) 140°C ( ) 145°C. The ODT occurs at about 130°C (Han et at. 1995).

The elastic bending modulus Kc for lipid bilayers was found to be of the order of 10 x 10-20 J.6 For the lyotropic lamellar liquid crystals, the additional presence of a cosurfactant at the interface shouldleadto smaller values. Various experimental measurements21 provided values ranging between 0.08 and 5 x 10-20 J. 

In the first part of the paper, a thermodynamic formalism developed earlier10 was used to obtain information about the domains of stability of the lamellar phase. It was shown that, for a set of interaction parameters between layers and bending modulus of the interface, only certain thicknesses are allowed for the water and oil layers. 

Figure 4 The modified stalk mechanism of membrane fusion and inverted phase formation, (a) planar lamellar (La) phase bilayers (b) the stalk intermediate the stalk is cylindrically-symmetrical about the dashed vertical axis (c) the TMC (trans monolayer contact) or hemifusion structure the TMC can rupture to form a fusion pore, referred to as interlamellar attachment, ILA (d) (e) If ILAs accumulate in large numbers, they can rearrange to form Qn phases, (f) For systems close to the La/H phase boundary, TMCs can also aggregate to form H precursors and assemble Into H domains. The balance between Qn and H phase formation Is dictated by the value of the Gaussian curvature elastic modulus of the bIlayer (reproduced from (25) with permission of the Biophysical Society) The stalk in (b) is structural unit of the rhombohedral phase (b ) electron density distribution for the stalk fragment of the rhombohedral phase, along with a cartoon of a stalk with two lipid monolayers merged to form a hourglass structure (reproduced from (26) with permission of the Biophysical Society).

Figure 13.18 Storage shear modulus G for lamellar samples of PEP-PEE 50,000) at 40°C. The...

In contrast to lamellar microdomains, where removal of defects and alignment of domains can, in principle, restore terminal behavior to G and G", even after alignment, G for three-dimensionally ordered spherical domains remains that of a soft solid, albeit with a much lower modulus than that of the unoriented material (Koppi et al. 1994). 

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