Statistical models of elasticity in main chain and smectic liquid ...

More documents

Recommendations

Info

112 CHAPTER 4. THE ELASTICITY OF SMECTIC-A ELASTOMERSdeformation is tanγ = −λ xz . The following deformation matrix is then obtained⎛ ⎞α 0 0λ = ⎝ 10αβ0 ⎠ (4.137)λ 0 βwhereα = √ λ 2 xx −λ 2 ; β = λ xxλ zz√λ 2 xx −λ 2 ; λ = λ xxλ xz√1+λ 2 xz. (4.138)The free energy is then of the same form as Eq. (4.133) and we have thedecomposition of the mode.A decomposition can be performed on the imposed λ zz deformation. Thestarting point is again Eq. (4.135), but this time setting tanγ = −λ xz . Theresulting matrix is then compared to that of imposed λ zz so that the identificationλ = λ zz√1+λ 2 xz can be made. This decomposition gives a geometricreason for the threshold observed when λ zz is imposed. Suppose that theelastomer deforms with only the λ zz component. The free energy density isthenf zz = 1 2 B(λ−1)2 . (4.139)Alternatively the sample could deform by a shear λ xz , which leaves the layerspacing unchanged, and then rotate to accomplish the same λ zz value. In thiscase the free energy density isf xz = 1 2 µrλ2 xz = 1 2 µr(λ2 −1) . (4.140)Comparing these two energy densities for small ǫ where λ = 1+ǫ, it is clearthat that f zz ∼ 1 2 Bǫ2 < f xz ∼ µrǫ, provided that ǫ < 2µr/B. The latterenergy is first order rather than second order in the strain and explains whyit is so costly and unphysical (it is second order finally where it intercedesafter λ cr ). This decomposition also shows that an imposed λ zz is equivalentto an imposed λ xz deformation plus a rotation and a fixed stretch along thelayer normal such that d/d 0 = λ cr . The decomposition explains why themodulus of the sample after the threshold is the same as that for an imposedλ xz deformation.4.4 Comparison with experimentThe model outlined in §4.2 can be compared with numerous pieces of experimentaldata. Here the Poisson ratios, the elastic moduli and the x-rayscattering are considered.
4.4. COMPARISON WITH EXPERIMENT 1134.4.1 Experimentally measured elastic propertiesFirstly consider the Poisson ratios. When stretched in plane, the model consideredhere produces, for large b, Poisson ratios (0,1) which are observedexperimentally in [63]. When stretched parallel to the layer normal the observedPoisson ratios are (1/2,1/2). The model is consistent with this beforethe threshold occurs, and layer rotation begins. After threshold the modelpredicts Poisson ratios of (1,0) which are not the same as those observed experimentally.This is because the sample, which is cylindrically symmetricabout the stretch axis, maintains its cylindrical symmetry by forming microstructureconsisting of many small domains. This point is crucial in theanalysis of the x-ray data.In the experimental study of Nishikawa and Finkelmann [66] the elasticmoduli of a smectic elastomer were measured. Thesample used in their experimentswas found to have an anisotropy of r ≈ 1.6 from swelling anisotropymeasurements. The threshold strain was found to be ǫ c ≈ 3% with a correspondingstress of σ N ≈ 1.12 × 10 5 Pa. The modulus before thresholdwas E before = 3.2 × 10 6 Pa, and the modulus after threshold was E after =1.1 × 10 5 Pa. Although they comment that after threshold the modulus issimilar to the in-plane modulus they do not give a value for the in-plane modulus.According to the theory outlined above from the reported moduli weexpect ǫ c ≈ E afterE before≈ 3.4% which is extremely close to their reported thresholdstrain given that the transition is not totally sharp when observed experimentally.From the values of r and E after we would expect an in plane modulus of4 E afterr≈ 2.75×10 5 Pa. Fig. 4.16 shows a fit of this theory to the elastic dataof [66].The fit to the experimental data is good given the small number of parametersin the model for such a complicated material.4.4.2 X-ray scatteringReference [66] also contains x-ray data for a smectic elastomer as it is beingstretched along the layer normal. It is observed that the x-ray peaks correspondingto the layers rotate as the sample is stretched. There is also a sharpdrop in the intensity of these peaks which was attributed to melting of thesmectic phaseto thenematic or isotropic phasebyNishikawa andFinkelmann.Fig. 4.17 below compares the calculated orientation of the director withthe experiment of [66]. As is clear from the figure the rotation of the smecticlayers is in good agreement with the model discussed here.Theabove theory does not predictthat the smectic phasemelts on stretching.The energy cost to perform this melting can be calculated as follows: theentropy change found in [66] for the smectic-isotropic phase transition was∆S = 2.4 × 10 −2 JK −1 g −1 . Thus the cost for melting at 300K for a samplewith density ρ ∼ 1g cm −3 is T∆Sρ ∼ 7.2 × 10 6 J m −3 . To pay the cost of
Page 3:
iStatistical models of elasticity i
Page 7:
AcknowledgementsDuring the course o
Page 10 and 11:
viiiCONTENTS5 Soft elasticity of sm
Page 12 and 13:
2 CHAPTER 1. INTRODUCTION1.2 Neo-cl
Page 14 and 15:
4 CHAPTER 1. INTRODUCTIONticular in
Page 17 and 18:
Chapter TwoThe elasticity of hairpi
Page 19 and 20:
2.1. INTRODUCTION 9Fig. 2.1. This b
Page 21 and 22:
2.2. MODEL OF HAIRPIN CHAINS 11nFig
Page 23 and 24:
2.2. MODEL OF HAIRPIN CHAINS 13za)
Page 25 and 26:
2.2. MODEL OF HAIRPIN CHAINS 15in
Page 27 and 28:
2.2. MODEL OF HAIRPIN CHAINS 17The
Page 29 and 30:
2.2. MODEL OF HAIRPIN CHAINS 1910.8
Page 31 and 32:
2.2. MODEL OF HAIRPIN CHAINS 21This
Page 33 and 34:
2.2. MODEL OF HAIRPIN CHAINS 23f g
Page 35 and 36:
2.3. NON-AFFINE DEFORMATION MODEL 2
Page 41 and 42:
2.4. HAIRPIN CHAIN NETWORK FREE ENE
Page 43 and 44:
2.4. HAIRPIN CHAIN NETWORK FREE ENE
Page 45 and 46:
2.5. CONCLUSIONS 35to their limit
Page 47 and 48:
2.A. GEOMETRY OF A HAIRPIN AT ZERO
Page 49 and 50:
2.B. UNDULATING CHAIN STATISTICS 39
Page 51 and 52:
2.C. COUNTING CONFIGURATIONS BY IND
Page 53 and 54:
2.C. COUNTING CONFIGURATIONS BY IND
Page 55 and 56:
2.D. INTERPRETATION OF FN 452.D Int
Page 57:
2.E. SPRING CONSTANT OF AN EXTENDED
Page 60 and 61:
50 CHAPTER 3. POLARISATION OF CHIRA
Page 62 and 63:
Page 64 and 65:
Page 66 and 67:
Page 68 and 69:
Page 70 and 71:
Page 72 and 73: 62 CHAPTER 3. POLARISATION OF CHIRA
Page 91 and 92: Chapter FourThe elasticity of smect
Page 93 and 94: 4.1. INTRODUCTION 83wherer 12 is th
Page 95 and 96: 4.1. INTRODUCTION 85HzyxGlobal rota
Page 97 and 98: 4.1. INTRODUCTION 87Figure 4.3: The
Page 99 and 100: 4.1. INTRODUCTION 89−D 2[(v (a)xz
Page 101 and 102: 4.2. MICROSCOPIC, FINITE DEFORMATIO
Page 107 and 108: 4.3. FINITE DEFORMATION EXAMPLES 97
Page 121: 4.3. FINITE DEFORMATION EXAMPLES 11
Page 125 and 126: 4.4. COMPARISON WITH EXPERIMENT 115
Page 127: 4.5. CONCLUSIONS 117formation in th
Page 130 and 131: 120 CHAPTER 5. SOFT ELASTICITY OF S
Page 147 and 148: Bibliography[1] P. J. Flory. Statis
Page 149 and 150: 139[27] J-.C. Meiners and S. R. Qua
Page 151 and 152: 141[58] W. L. McMillan. Measurement
Page 153 and 154: 143[86] L. Golubovic and T. C. Lube
show all

Statistical models of elasticity in main chain and smectic liquid ...

Create successful ePaper yourself

Delete template?

Save as template?