Example: Uniaxial Deformation. With Axi-symmetric sample cross-section dl l 0 l x. α x since α x α y α z = 1 Rewriting ΔS α ) explicitly in terms of α

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1 Eample: Uniaial Deformation y α With Ai-symmetric sample cross-section l dl l 0 l, d Deform along, α = α = l0 l0 = α, α y = α z = Poisson contraction in lateral directions α since α α y α z = Rewriting ΔS α ) eplicitly in terms of α F ( i k ΔS( αi ) = α α α = T l ΔS(α ) ( ) T,P = kt l α z + 3 α

2 Uniaial Deformation cont d F = kt α l 0 α l 0 α = kt α At small etensions, the stress behavior is Hookean (F ~ const α ) Stress-Etension Relationship σ A l = V 0 0 F ( α ) A zkt = α for z subchains in volume V A 0l0 α z = # of subchains per unit volume = N links V = total 0 Usually the entropic stress of an elastomeric network is written in terms of M where M = avg. molecular weight of subchain between -links mass/vol molesofcrosslinks/volume = M = ρn AkT σ α = ( ) α M α ρ N /N A uniaial

3 Young s Modulus of a Rubber E dσ ρrt lim = + α 3 dα M α E = 3ρRT/M Notice that Young s Modulus of a rubber is :. Directly proportional to temperature. Indirectly proportional to M Can measure modulus of crosslinked rubber to derive M In an analogous fashion, the entanglement network of a melt, gives rise to entropic restoring elastic force provided the time scale of the measurement is sufficiently short so the chains do not slip out of their entanglements. In this case, we can measure modulus of non-crosslinked rubber melt to derive M e!

4 Stress-Uniaial Etension Ratio Behavior of Elastomers σ(α) tensile Image removed due to copyright restrictions. Please see Fig. 5 in Treloar, L. R. G. Stress- Strain Data for Vulcanised Rubber Under Various Types of Deformation. Transactions of the Faraday Society 40 (944): compressive Uniaial Stretching Stress / MN m Theoretical? Small deformations Figure by MIT OCW. High tensile elongations

5 Rubber Demos Tg for Happy Ball and Unhappy Ball Deformation of Rubber to High Strains Heating a Stretched Rubber Band How stretchy is a gel?

6 Gel A Highly Swollen Network Several ways to form a gel: Start with a polymer melt and produce a 3D network (via chemical or physical crosslinks) and then swell it with a solvent Eample: styrene divinyl benzene Start with a concentrated polymer solution and induce network formation, for eample, crosslink the polymer by: Radiation- (UV, electron beam) (covalent bonds) Chemical- (e.g. divinyl benzene & PS) (covalent bonds) Physical Associations -(noncovalent bonds induced by lowering the temperature or adding a nonsolvent) Q: describe some types of physical (ie noncovalent) crosslinks

7 Flory-Rehner Treatment of Gels Assumes elastic effect and miing effect are linearly additive ΔG = ΔG mi + ΔG elastic Polymer-solvent Thermodynamics, ΔGmi Flory Huggins Theory (χ,,, φ, φ ) solvent mi Rubber Elasticity Theory ΔG elastic deform

8 At equilibrium, Flory-Rehner Theory ΔG ΔG mi = ΔG mi + ΔG elastic = ΔG elastic. Thermodynamically good solvent swells rubber favorable χ interaction favorable ΔS mi. BUT Entropy elasticity of network (ΔS elastic ) eerts retractive force to to oppose swelling Swollen Rubber Crosslinked Rubber Swell the network, add n moles of solvent volume, V o subchain, r0 subchain volume, V subchain, r subchain

9 Thermo of Swollen Network cont d α = linear swelling ratio = ' φ = volume fraction polymer = V V 0 N = # of subchains in the network = Isotropic swelling α = α = α = α y z ρv 0 N M s A 3 V0α s = V V φ 0 = V so α = 3 s φ chemical potential, μ 0 μ = μ i ΔG = ni tot T,P,n j chemical potential difference for solvent in gel vs pure solvent μ μ 0 = ΔG n mi T,P,n ΔG + α elastic s α n T,P, n s

10 ΔG mi and Flory Huggins n moles of solvent [n + n ]N A = N 0 n moles of polymer recall equivalently ΔG m N 0 φ φ = kt χφφ + lnφ + lnφ ΔG mi = RT [ n lnφ + n lnφ + nχ φ ] some math Δ n G mi = RT[lnφ + φ ( ) + χ φ ] Since for a network (one megamolecule) ΔG n mi = RT[lnφ + φ + χ φ ]

11 ΔG elastic and Rubber Elasticity ΔG elastic = TΔS elastic new term ΔS elastic = k [ (α + α y + α z ) 3 ] ln V V 0 new term is an additional entropy increase per subchain, due to increased volume available on swelling, of k V ln V 0 for N subchains ΔS elastic = k N 3α s 3 ln V V 0

12 ΔG elastic cont d ΔG Δ elastic G = n α s elastic α s n Δ To calculate G elastic α s α s n = NkT 3 α 3α 3 ln sv s α s V0 = NkT 6α s 3 α (3α s s ) 0 = NkT 6α s 3 α s α 3 s = V = V + n v 0 α s = + n v or V 0 V V 0 0 /3 where is the molar volume of the solvent v α s = n 3 + n v V 0 /3 v V 0 = 3α s v V o

13 ΔG elastic cont d collecting terms Now epress in terms of ΔG elastic n = ΔG elastic α s = NkT 6α s 3 α s n α s φ and M ΔG elastic n Collecting the terms we ve worked out, 3α s v V 0 = NkT v φ 3 φ = RTρv φ 3 V 0 φ M μ μ 0 = RT[lnφ + φ + χ φ ] + RT ρv φ 3 φ M At equilibrium, μ = 0 μ ln( φ ) + φ + χ φ = ρv φ 3 φ M

14 Determination of χ, M from Swelling Eperiments I. Generally ρ, v are known φ measured from equilibrium swelling ratio V V 0 If M is known from elastic modulus of dry rubber then, χ available from single swelling eperiment II. ρ, v, φ, φ known as above π Assume, χ known from c measurements for polymersolvent solution. Then, M z available from single swelling eperiment The Most Relaed Gel: Swell polymer in a θ solvent so that the chains overlap and then crosslink. In this case, swollen chains have their unperturbed dimensions. M will be large since there are fewer subchains per unit volume in the dilute and swollen state (c ~ c * in order to form network). The maimum etension of this type of gel can be HUGE!!

15 Practice Problems to Try () Calculate M from known swelling ratio and V = 4 V =.0g/cm 3 v = 40 cm 3 /mole 0 = Š 0. ANS:400 g/mole () Suppose M =5,000 g/mole and = 0.5, =.0g/cm 3, v = 40 cm 3 /mole What is the % solvent in the network for equilibrium swelling? End of material that will be covered by the st eam.

16 Muthukumar, M., Ober, C.K. and Thomas, E.L., "Competing Interactions and Levels of Ordering in Self-Organizing Materials," Science, 77, 5-37 (997). Self Organization Homogeneous state Order Disorder Structurally ordered state Order Structurally ordered state Competing interactions: Enthalpy (H) vs. Entropy (S) Free energy landscape: entropic frustration, multiple pathways Order forming processes - (Macro)Phase separation - Microphase separation - Mesophase formation - Adsorption/compleation - Crystallization Selection of symmetries and characteristic lengths - Chemical affinities (long range correlations) - Interfacial tension

17 Competing Interactions and Levels of Ordering in Self-Organizing (Soft) Materials Materials liquid crystals block copolymers hydrogen bonded complees nanocrystals Structural order over many length scales atomic molecular increasing size scale mesogens domains grains Outcome: Precise shapes, structures and functions

18 Strategic Design for Materials with Multiple Length Scales Synthetic design strategy - Intramolecular shapes and interaction sites (molecular docking, etc) - Control multistep processing to achieve long range order Interactions - sequential Reduction of disorder (S ) - simultaneous - synergistic Strengthening of intra- and (H ) - antagonistic inter-molecular interactions Structural design strategy - organize starting from initially homogeneous state - organize from largest to smallest length scale (induce a global pattern, followed by sequential development of finer details) Selection of growth directions - applied bias field(s) - substrate patterning Prior-formed structures impose boundary conditions - commensuration of emergent and prior length scales - compatibility of structures across interfaces

19 Principles of Self Organization: Microphase Separation Block Copolymers The min - ma principle: Minimize interfacial area Maimize chain conformational entropy Result: Morphology highly coupled to molecular characteristics Morphology serves as a sort of molecular probe Homogeneous Disordered State Ordered State Gas of junctions T < T ODT Junctions on Surfaces IMDS Figure by MIT OCW.

20 Microdomain Morphologies and Symmetries - Diblock Copolymers IMDS Spheres Cylinders 0-% -33% Double Gyroid Double Diamond 33-37% Lamellae 37-50% "A" block Increasing volume fraction of minority phase polymer "B" block Junction point Figure by MIT OCW.

21 Hierarchical Structure & Length Scales [ CH CH CH CH] n Polybutadiene [ CH CH] n Polystyrene -0.5 nm loop -0 nm bridge -40 nm Figure by MIT OCW.

22 Computing the characteristic length scale: Equilibrium Domain Spacing Min-Ma Principle G = Free Energy per Chain N = # of segments = N A + N B a = Step Size aa ~ a B λ = Domain Periodicity Σ = Interfacial Area/Chain kt γ AB = Interfacial Energy = a segmental volume ~ a 3 χ 6 AB Sharp Interface λ, γ AB χ AB z kt = Segment - Segment Interaction Parameter = ε [ ε + ε ] AB AA BB Strong Segregation Limit Nχ very large (high MW and positive χ), => pure A domains & pure B microdomains

23 Characteristic Period (Lamellae) State Melt State Microdomains ΔG =ΔH TΔS IMDS = γ AB Nχ AB φ A φ B kt + 3 λ / kt Na ( ) Note: Na 3 = λ Σ Enthalpic Term Entropic Spring Term ΔG(λ) = kt a χ AB 6 Na 3 ( λ /) N χ AB φ A φ B kt + 3 λ / kt Na ΔG(λ) = α λ const + βλ const ( ) ΔG λ = 0 α = + βλ λ 0

24 Free Energy of Lamellae con t Thus, the optimum period of the lamellae repeat unit is : λ opt = α 3 an /3 χ /6 β Important Result: Domain dimensions scale as λ ~ / 3 N Chains in microdomains are therefore stretched compared to the homogeneous melt state ΔG(λ opt ) =.ktn 3 χ AB 3 3 kt

25 Order-Disorder Transition (ODT) Estimating the Order-Disorder Transition:.kTN /3 χ /3 Nχ G LAM G Disordered ABφ A φ B kt since both terms >> kt 3 For a 50/50 volume fraction, φ A φb = / 4 so.n / 3 χ / 3 The critical Nχ is just (Nχ) = Nχ / 4 c = (4.8) 3/ ~ 0.5 N χ < 0.5 N χ > 0.5 Homogeneous, Mied Melt Lamellar Microdomains Original Order-Disorder Diblock Phase Diagram computed by L. Leibler, Macromolecules, 980

26 Diblock Copolymer Morphology Diagram B B B B Spheres Cylinders Double Gyroid Lamellae Im3m p6mm Ia3d pm Figure by MIT OCW. strong segregation limit Nχ >> S C L C' S` χ N 40 G CPS CPS` weak segregation Disordered limit Nχ ~ f A Figure by MIT OCW.