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

More documents

Recommendations

Info

98 CHAPTER 4. THE ELASTICITY OF SMECTIC-A ELASTOMERSImposed λ zzIn the firstcase illustrated in Fig. 4.9 a), the following form of the deformationtensor will be considered⎛ ⎞λ xx 0 0λ = ⎝ 0 λ yy 0 ⎠. (4.67)λ zx 0 λThe shear component λ zx is allowed to relax because, when the layers areconstrained to move with the matrix, it allows rotation away from the stretchaxis. The constraints of constant volume and constant layer spacing can beexpressed respectively as1 = λ xx λ yy λ (4.68)1 = λ 2 yy(λ 2 xx +λ 2 zx). (4.69)The director can be calculated from Eq. (4.57) using the tensor λ given inEq. (4.67). The resulting director orientation can be written asn = (−λ yy λ zx ,0,λ xx λ yy ). (4.70)Notethatthisisnormalisedbecauseoftheconstraintofconstantlayerspacing,Eq. (4.69). Substituting these expressions into the free energy density resultsin {f = 1 2 µ 1+λ 2 xxλ 2 + 1λ 2 +r( λ 2 −1 )} . (4.71)xx λ2This expression can be minimized w.r.t. λ xx and has a minimum when λ xx =1λ. The resulting relaxed deformation gradient tensor and director are givenby⎛ 1 ⎞λ0 0 ( √ )λ = ⎝ 0 1 0 ⎠√; n = − λ 2 −1λ,0, 1 λ(4.72)λ 2 −10 λλNote that this deformation is exactly that required to preserve the areas ofthe layers, as can be seen by looking at the projected area and comparing itto n·z. Using this deformation and director, the free energy is given byf = 1 2 µ{ 3+r(λ 2 −1) } . (4.73)This expression for the free energy density is pathological in that it doesnot have zero slope at λ = 1. However at this point the material cannot becompressed along the layer normal due to the infinite value of B (we can neverachieve λ < 1). The free energy thus does have a minimum at λ = 1 althoughit is not smooth there. This is illustrated in Fig. 4.10 together with diagramsof the deformed layers at the different deformation values. Also note that thetransition to the sheared state is not possible in compression because the layerspacing in the z direction can only be increased by a λ zx shear.
4.3. FINITE DEFORMATION EXAMPLES 99654F / µ32100.8 1 1.2 1.4 1.6 1.8 2λ zzFigure 4.10: The figure shows the free energy density of asmecticelastomer withB → ∞asaλ zz deformation isapplied,together withillustrations of thedeformedstate atλ zz = 1and1.3.Imposed λ xxThis deformation is illustrated in Fig. 4.9 b). For a stretch in the plane thefollowing deformation tensor is used⎛ ⎞λ 0 λ xzλ = ⎝ 0 λ yy 0 ⎠. (4.74)λ zx 0 λ zzThe director in this case is given byn = (−λ yy λ zx ,0,λ yy λ) (4.75)Note that to move the director in the xz plane the component λ zx must beincluded. A λ xz alone is not enough. The constraints of constant volume andlayer spacing are given by1 = λλ yy λ zz −λ yy λ zx λ xz (4.76)1 = λ 2 yy (λ2 zx +λ2 ). (4.77)The free energy is given by the following expressionf = 1 2 µ 1 ( (λ2λ 2 zx λ2yy +rλ 2zx +λ2 zz +λ 2 zx +λ 2 )xz +2(r −1)λzz λ zx λ xz λ+ ( λ 2 yy +λ2 zz +2λ2 zx +rλ2 xz)λ 2 +λ 4) . (4.78)The two constraints can be used to eliminate two of the variables. The choiceof which variables to eliminate must be make with care so that the resultingdeformation matrix still incorporates the constraints. The variables λ yy andλ zz are eliminated here.
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 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: 4.3. FINITE DEFORMATION EXAMPLES 97
Page 111 and 112: 4.3. FINITE DEFORMATION EXAMPLES 10
Page 123 and 124: 4.4. COMPARISON WITH EXPERIMENT 113
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?