23 2 2.1 10 5 6 N/m 2 2.1.1 f x x L dl U 1 du = T ds pdv + fdl (2.1)
24 2 dv = 0 dl ( ) U f = T L p,t ( ) S L p,t (2.2) 2 ( ) ( ) S f = L T p,t p,l (2.3) ( ) U f = L p,t + T ( ) f T p,l (2.4) 1 f e ( U/ L) p,t 2 f S T ( f/ T ) p,l 2.1 x L T 2.12.4 T 0 C 1 AB 20 C L 0 L/L 0
2.1 25 2.2 A B 2.2 2.4 ( ) [ ] f e ln f ln(f/t ) f = 1 = T ln T T p,l p,l (2.5) 2 < r 2 > 0 f e f = T d ln < r2 > 0 dt (2.6) 2.1 T = 298 K PE f e /f = 0.42 -CH 2 -CH 2 -CH 2 - PDMS f e /f = 0.20 -Si-O-Si-O-
26 2 2.1 T = 298 K f e /f d ln < r 2 > 0 /dt 10 3 (K 1 ) 0.18 0.60-1,4-0.13 0.44 0.20 0.67-0.06-0.20-0.42-1.41 0.26 0.87 2.1.2 2.3 L 20 C L 0 L 2.3 L L T=20 C L 0 L/L 0 L/L 0 1 % ( f/ T ) p,t 36% thermoelastic inversion
2.1 27 2.1.3 1805 J.Gough 1859 J.P.Joule 2.4 2.4 500% 10 a20% b
28 2 S ds = ( ) S dt + T p,l ( ) S dl (2.7) L p,t 1 C p T ( S/ T ) p,l 2 2.3 ds = C ( ) p f T dt dl (2.8) T p,l ds = 0 (dt ) S = T ( ) f dl (2.9) C p T p,l 2.2 2.2.1 2.5 P 2.5 1 topological neighbor
2.2 29 spatial neighbor < r 2 > 1/2 0 Γ Γ ( ) 4π 3 < r2 > 3/2 µ 0 V (2.10) µ V Γ = 25 100 < r 2 > 0 µ 1 Γ 1/ µ 2.2.2 f r 0 r 2.6 L x f x λ x 2.6yz λ y = λ z ( ) 3/2 3 P 0 (r 0 ) = 2π < r 2 exp ( 3r 0 2 ) > 0 2 < r 2 (2.11) > 0 r 0 < r 2 > 0 = na 2 n W.Kuhn
30 2 r 0 (λ x, λ y, λ z ) 1936 r 0 r = ˆλ r 0 (2.12) ˆλ λ x 0 0 ˆλ 0 λ y 0 (2.13) 0 0 λ z 2.12 ν r 0 r 0 + dr 0 νp 0 (r 0 )dr 0 r r + dr νp 0 (r 0 )dr 0 = νp (r)dr (2.14) P (r) r r 0 2.12 r φ(r) = 3k BT 2 < r 2 > 0 r 2 (2.15) F (ˆλ) = φ(r)νp (r)dr (2.16) 2.152.12 F (ˆλ) = 3νk BT 2 < r 2 (ˆλ r 0 ) 2 P 0 (r 0 )dr 0 > 0 = νk BT 2 < r 2 (λ 2 x + λ 2 y + λ 2 > z) < r 2 > 0 (2.17) 0 λ x = λ y = λ z = 1 F (ˆλ) F (ˆλ) F (1) F (ˆλ) = ν 2 k BT (λ 2 x + λ 2 y + λ 2 z 3) (2.18)
2.2 31 λ x = λλ y = λ z = 1/ λ f f = ( F/ (λl)) T f = νk BT (λ 1λ ) L 2 (2.19) L 2 σ σ = νk BT L 3 (λ 1λ 2 ) (2.20) L 2 /λ τ τ = νk BT L 3 ( λ 2 1 ) λ (2.21) ν/l 3 ρ M ν/l 3 = ρn A /M E ( ) σ E = λ = ρrt ( λ + 2 ) λ M λ (2.22) λ = 1 E = 3ρk BT M (2.23) T = 300 KM = 10 4 ρ = 1 g/cm 3 ν/l 3 = 10 4 mol/cm 3 E = 7.4 10 6 dyne/cm 2 9 10 11 dyne/cm 3 10 2.7
32 2 f λ S L.R.G.Treloar 1954 2.6.1 S 1 2.7 2.7 σ/(λ 1/λ 2 ) λ 1 λ 1 C 1 C 2 σ = 2C 1 ( λ 1 λ 2 ) + 2C 2 ( 1 1 λ 3 ) (2.24) Mooney Rivlin C 2 C 2
2.2 33 2.2.3 ν 1946 M M c ρ ν = ρ/m c 2ρ/M ν eff = ρ ( 1 2M ) c M c M (2.25) 2.8 τ = ρrt M c ( 1 2M ) ( c λ 2 1 ) M λ (2.26) 1 2M c /M 2.8
34 2 2.9 J.Scanlan L.C.Case 1960 (i, k) i k (i, k) µ ik i 3 1 2.9 2 2.9 1 i i 3 ν eff = 1 2 2k k=2 i=3 iµ ik (2.27) µ ik??
2.3 35 ( i', k' ) ( i, k ) 2.10 i 3i 3 2.2.4 T 0 λ 0 = 1 + ɛ 0 (ɛ 0 1) σ = 3νk B T ɛ 0 T ɛ = ɛ 0 β 3 (T T 0) (2.28) ( β 1 V ) V T p { σ 3νk B T ɛ 0 β } 3 (T T 0) (2.29) ɛ 0 ( ) { σ = 3νk B ɛ 0 β } T ɛ 0 3 (2T T 0) (2.30) ɛ 0 = β 3 (2T T 0) (2.31) T 0 = 293 KT = 343 Kβ = 6.6 10 4 /K ɛ 0 = 0.086 8.6% 2.3
36 2 H.M.JamesE.Guth 1943 (1) (2) r < ( r) 2 >= 2 f < r2 > 0 (2.32) f f = 4 (3) ( F ) ph = ξ 2 k BT (λ 2 x + λ 2 y + λ 2 z 3) (2.33) ξ f ξ = ν(1 2/f) f = 4 ξ = ν/2 σ τ σ ˆλ 2.11 τ
2.3 37 τ τ σ τ 2.3.1 l l l f l 2.12 A f 2.12 A l µ l = f(f 1) l 1 σ τ ν l = f{1 + (f 1) + + (f 1) l 1 } = f[(f 1) l 1]/(f 2) τ A τ ( F ) micro = R l (f) ν l 2 k BT (λ 2 x + λ 2 y + λ 2 z 3) (2.34) R l (f) R l (f) µ l 1 ν l = (f 2)(f 1) l 1 (f 1) l 1 (2.35) l = 2 R 2 (f) = f 1 f (2.36)
38 2 J.Rehner 1943 l lim R l(f) = f 2 l f 1 (2.37) R (4) = 2/3R (3) = 1/2 σ σ τ µ l (τ, τ) ν l µ l µ l /ν l = (f 2)/(f 1) 1 µ l /ν l = 1/(f 1) R (σ, τ) (f 1)/f R = f 2 f 1 = f 1 f f 2 f 1 }{{} (σ,τ) + f 2 f 1 f 1 }{{} (τ,τ) (2.38) (τ, τ) R (f 2)/f ( F ) ph = ν 2 ( 1 2 ) k B T (λ 2 x + λ 2 y + λ 2 z 3 ) (2.39) f ν(1 2/f) ξ ( F ) ph = ξ 2 k BT (λ 2 x + λ 2 y + λ 2 z 3 ) (2.40) 2.3.2 2.17 F (ˆλ) = 3 νk BT 2 < r 2 0 >(< r2 > < r 2 0 >) (2.41) r r 0 < >
2.4 39 r = ˆλ r 0 r = r + r (2.42) r r < r 2 >=< r 2 > + < r 2 > (2.43) < r 2 > = ( 1 2 ) < (ˆλ r 0 ) 2 > (2.44a) f < r 2 > = 2 f < r2 0 > (2.44b) { ( < r 2 >= 1 2 ) } λ 2 x + λ 2 y + λ 3 z + 2 < r0 2 > (2.45) f 3 f ( F ) ph = ξ 2 k BT (λ 2 x + λ 2 y + λ 2 z 3 ) (2.46) ξ = (1 2/f)ν 2.4 ν ξ = ν(1 2/f) 3 L 0 V 0 = L 0 V dry φ c V dry /V 0 L (i) V i = (L (i) ) 3 φ V dry /V i
40 2 i f 3 i V f 2 L (0) L (i) L i f 1 V 0 V i 2.13 L x, L y, L z ˆλ λ x L x L 0 (2.47) V i = V = L x L y L z α x L x L (i) = λ α ( V0 V ) 1/3 λ x (2.48) x f ( ) 1/3 V λ x = α V 0 λ y = λ z = 1 ( ) 1/3 (2.49) V α V 0 f f = ( F/ λ x ) T f = Fk BT L 0 ( ( ) ) ( ) ( ) 2/3 V 1 FkB T V λ V 0 λ 2 = (α 1α ) L (i) V 2 0 (2.50) F ν ξ A= (L (i) ) 2 /α τ x τ x = Fk ( ) 2/3 ( BT V α 2 1 ) V α V 0 (2.51) reduced stress [ f ] [ f ] fφ 1/3 A dry (α α 2 )
2.4 41 ( FkB T [ f ] = V dry α ) φ c 2/3 (2.52) 2.14 2.14 α α Mooney Rivlin 2.15 α 2C 1 φ 2C 2 C 2 λ α 2.50 V 0 V V = V 0 (1 + β T ) λ = 1 + ɛ (ɛ 1) f = Fk j BT 1 + ɛ 1 + β T ff = 3Fk BT (ɛ β T ) (2.53) L 0 (1 + ɛ) 2 L 0 3 ɛ f «f = 3Fk BT {ɛ β T L 0 2 (2T T 0)} (2.54) ɛ
42 2 2.15 2.31 x, y τ x, τ y α x, α y λ x = α x (V/V 0 ) 1/3 λ y = α y (V/V 0 ) 1/3 λ z = (1/ α x α y )(V/V 0 ) 1/3 ««2/3 FkB T V τ x = 2 αx 2 1 «V V 0 αxα 2 y ««2 2/3 FkB T V τ y = 2 α 2 y 1 «(2.55) V V 0 α 2 xαy 2 α x = α y α 2.16 ««2/3 FkB T V τ = 2 α 2 1α «V V 4 0
2.5 43 y f y x α x αα y = 1 ««2/3 FkB T V τ x = 2 α 2 1α «V V «2 0 «2/3 FkB T V τ y = 2 1 1α «(2.56) V 2 V 0 2.16 2 2.5 V 0 L 3 0 = nνa 3 2.17n ν a x f V N 0 V = (N 0 + nν)a 3 V 0 /V φ x λ x yz λ y λ z q V/V 0 = λ x λ y λ z q = 1/φ λ x λ λ y = λ z = 1/ λφ F mix F el F = F mix + F el F mix = V a 3 k BT {(1 φ) ln(1 φ) + χφ(1 φ)} (2.57)
44 2 2.17 x f 0 F el = ν 2 k BT ( λ 2 + 2 ) λφ 3 µ ln φ (2.58) ln φ µ f = F/ (λl 0 ) fl 0 νk B T = λ 1 λ 2 φ (2.59) t φ = 1/λ 3 µ 0 = 0 F ln(1 φ) + φ + χφ 2 + 1 ( 1 n λ µ ) 2 φ = 0 (2.60) λ φ q 10 2.60 φ ( ) 1 2 χ φ 2 = 1 ( 1 n λ µ ) 2 φ (2.61)
2.5 45 µφ/2 q = nψτλ (2.62) χ 1/2 χ = ψττ τ > 0 t = 0 q = λ 3 λ q = (nψτ) 3/5 (2.63) 2.18 2.592.61φ λ ln{nψτ} 2.18 t σ = ln λ y ln λ x (2.64) σ = 1 4 ( 1 ln[nψτ] ) ln λ (2.65)
46 2 *1 0 < σ < 1/2 x yz 2.6 (1) (2) 2.6.1 a f Q 1 (f, T ) n Q 1 (f, T ) λ 0 (t) n (2.66) t fa/k B T λ 0 (t) = sinh t/t (2.67) *1 σ
2.6 47 R = f (nk BT ln λ 0 ) (2.68) na l R/na l = t ln λ 0(t) (2.69) t t = ψ(l) φ(l) = R 0 = nk B T f dr = nk B T t 0 l 0 t d l d t dt = nk BT ψ(l)d l [ t l t 0 ] ld t (2.70) φ(l) = nk B T [ψ(l)l ln λ 0 (ψ(l))] (2.71) R P 0 (R) = Ce βφ(l) = Ce ng(l) P 0 (l) (2.72) g(l) g(l) ln λ 0 (ψ(l)) + ψ(l)l (2.73) C C = 1 0 4πl 2 d le ng(l) (2.74) φ(r) F (ˆλ) = ν [φ(ˆλ R 0 ) φ(r 0 )]P 0 (R 0 )dr 0 (2.75) ˆλ R 0 P 0 (R 0 ) ν l R/na F (ˆλ) νk B T = n d l[g(ˆλ l) g(l)]p 0 (l) (2.76)
48 2 λ x = λλ y = λ z = 1/ λ ˆλ l = [λ 2 x 2 +1/λ(y 2 +z 2 )] 1/2 η(λ, θ)l η η(λ, θ) [( λ 2 + 1 ) cos 2 θ 1 1/2 (2.77) λ λ] L 0 f fl 0 = F (ˆλ)/ λ fl 0 νk B T = n 1 0 2πl 3 d l 1 0 d cos θ g( ˆλ l ) ζ(λ, θ) η(λ, θ) ψ(ηl)p 0(l) (2.78) g( ˆλ l ) λ = λ g(ηl) = g (ηl) dη dλ l = ψ(ηl) ζ η l (2.79) ζ(λ, θ) (2λ + 1λ 2 ) cos 2 θ 1 λ 2 (2.80) 2.19 λ 0 = e t2 /6 ln λ 0 = t 2 /6λ = t/3 t ψ(l) = 3 l g(l) = 3 l 2 (3 l) 2 /6 = 3 l 2 /2 ψ(ηl)ζ/η = 3 lζ R 1 d cos θζ(λ, θ) = 0 2(λ 1/λ 2 )/3 fl 0 /νk B T = λ 1/λ 2 λ 0 = (sinh t)/tln λ 0 = ln(sinh t/t)l = L(t) = coth t 1/t ψ(l) = L 1 (l) fl 0 νk B T = 2nπ 1 0 d l l 3 P 0 (l) 1 0 d cos θ ψ (ηl) ζ η (2.81) L.R.G.Treloar cos θ θ = 0x π/2y, z 1954 x, y, z θ = 0 ζ/η = 2θ = π/2 ζ/η = 1/λ 3/2 fl 0 νk B T = n 1 0 [ ψ(λl) 1 ( )] l λ ψ P 3/2 0 (l)4πl 3 d l (2.82) λ
2.6 49 l R 0 = n a l = l 0 n a/na = 1/ n [ ( fl 0 νk B T = n1/2 L 1 3 λ n 1/2 ) 1 ( )] 1 L 1 λ3/2 λ 1/2 n 1/2 (2.83) S 2.6.2 2.19 ζ 1 n nζ l R/nζat fa/k B T λ 0 (t) = σ + n exp ( ζ6 ) t2 (2.84) l = ζ t ln λ 0(t) = t 3 1 (2.85) 1 + σe ζ 6 t2 σ/u σ t = ψ(l) 2.78
50 2 2.6.3 λ 0 ( )} λ {λ uk v2 λ e αt uk ln 2αt λ e αt = v(1 v)uk sinh αt αt k exp( t 2 /6) α b/a u v λ 0 (t) t = ψ(l) 2.78