Crossover behavior in the dependence of the viscosity on concentration and molecular weight for semiflexible polymers

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1 Crossover behavior in the dependence of the viscosity on concentration and molecular weight for semiflexible polymers CCC HI III PPPP OO LL V CHILEAN SYMPOSIUM ON THE CHEMISTRY AND PHYSICAL CHEMISTRY OF POLYMERS G. C. Berry Department of Chemistry Carnegie Mellon University Reprint manuscripts available on request gcberry@andrew.cmu.edu Acknowledgments: Partial Support: National Science Foundation (GCB) Carnegie Mellon University 1

2 Concentration Ranges: Several regimes of viscoelastic behavior are related to the mean separation Λ of molecular centers relative to the root-meansquare radius of gyration R G : Λ = (M/cN Α ) 1/3 Infinite dilution (Λ >> R G ), describing the limiting behavior of η~ as c[η] tends to zero, such that ~ η 1 is equal to c[η] (except possibly for charged chains under some conditions); Dilute solutions (Λ > R G ), defined loosely as the range of concentrations for which (η sp 1)/c[η] begins to increase with increasing concentration, but is small enough that η sp may be represented by a virial expansion in c[η]; Moderately concentrated solutions (Λ <.5R G ), for which the density of chains is large enough that certain thermodynamic and hydrodynamic interactions become progressively screened with increasing concentration, vitiating the use of a virial expansion to represent η sp ; intermolecular entanglement effects may develop, depending on the molecular weight; Concentrated solutions or bulk (Λ << R G ), so that certain thermodynamic and hydrodynamic interactions are fully screened, and intermolecular entanglement effects may develop, depending on the molecular weight. Carnegie Mellon University

3 Dimensionless reduced viscosity η ~ : η~ = (c) η/ηloc = 1 + c[η] (c) [η] (c) reduces to the intrinsic viscosity [η] at infinite dilution. Expressions for [η] (c) will be considered for semiflexible chains in the following; (c) η LOC is a "Local viscosity", tending to the solvent viscosity η solvent at infinite dilution and to the "viscosity" η repeat of a repeat unit for undiluted polymer. We will return to a (c) discussion of η LOC in the following. Carnegie Mellon University 3

4 Molecular Parameters: L: contour length (M L = M/L) R G : radius of gyration (root-mean-square) R Η : hydrodynamic radius (R Η = Ξ/6πη solvent ) α: expansion factor γ Η : diameter to length ratio of hydrodynamic unit [η] =[η] FD K η R Η /γ Η L [η] FD = πn Α R G γ Η /M L Carnegie Mellon University 4

5 Thermodynamic Interactions: For the wormlike model for a semiflexible chain: R G { (âlα /3) 1 + (L /1) 1 } 1/ α 1 + ẑ + k α (ẑ/) ν 1/ ; ν 3/5 Hydrodynamic Interaction: ẑ =a 1 A(â/L)z/(ν 1) z = (3d Τ /16â)(3L/πâ) 1/.18(d Τ /â)(l/â) 1/ K η R Η {[(1/3)(R Η ) ND ] + [(R Η ) FD ] } 1/ where (R Η ) ND /L = { 3 1/ /9}(â/L) 1/ α; (R Η ) FD /L = f(l/l, γ Η ) f(l/l, γ Η ) ζ red /{1 + κζ red ln(3l/d Η )} ζ red (c) = γ Η ζ l /6πη LOC d Η γ Η Carnegie Mellon University 5

6 The Intrinsic Viscosity: α > 1 ln([η]m /â ) L α = 1 ln (L/â) Carnegie Mellon University 6

7 The Infinite Dilution Limit (Λ/R G >> 1): Λ [η] (c) [η] (c) η LOC η solvent. η~ = η/ηsolvent = 1 + c[η] Carnegie Mellon University 7

8 With decreasing Λ/R G (increasing c) the effects of screening of thermodynamic and hydrodynamic interactions become important, and are here expressed by the relation: [η] (c) (c) (c) (c) =[η] FD K η R Η /γ Η L (c) (c) [η] FD =πn Α (R G ) γ Η /M L (c) (c) By analogy to the behavior at infinite dilution, K η R Η is represented by the expression: (c) (c) K η R Η {[(1/3)Q ND (Λ/R G )(R Η ) ND α (c) /α] + [Q FD (Λ/R G )(R Η ) FD ] } 1/ where both Q ND (Λ/R G ) and Q FD (Λ/R G ) increase from unity with decreasing Λ/R G (increasing c). e.g., at infinite dilution: K η R Η {[(1/3)(R Η ) ND ] + [(R Η ) FD ] } 1/ Carnegie Mellon University 8

9 Dilute Solutions: Λ (c) η LOC η solvent. R G (c) R G (c) (c) K η R Η increases with decreasing Λ/R G (increasing c) (c) (c) K η R Η {[(1/3)(R Η ) ND Q ND (Λ/R G )] + [Q FD (Λ/R G )(R Η ) FD ] } 1/ On expanding Q ND and Q FD in a Taylor series with respect to c[η]: (c) (c) K η R Η /K η R Η = 1 + k'c[η] + k"(c[η]) + exp{k'c[η]} (1 + c[η]) k' Thus, for dilute solutions: η/η solvent = 1 + c[η] + k'(c[η]) + k"(c[η]) 3 + Carnegie Mellon University 9

10 Moderately concentrated solutions: Λ The distribution of molecular centers is liquid-like (c) η LOC η solvent (1 + bϕ) η solvent exp(bϕ) for small bϕ R G (c) decreases toward R G with decreasing Λ/R G (α (c) decreases toward unity) (c) (c) K η R Η /γ Η L increases toward unity with decreasing Λ/R G [η] (c) (c) increases from [η] toward [η] FD : (c) (c) [η] FD =πn Α (R G ) γ Η /M L (In the absence of chain entanglements) Carnegie Mellon University 1

11 α (c) MAX{1; α(1 + [7(R G /Λ) 3 ] ) 1/16 } (c) (c) Rearranging the expression for K η R Η : (c) (c) K η R Η /L γ Η Q FD {1 + (9Q FD /Q ND α (c) ) (3L/â)} 1/ Empirically, for moderately concentrated solutions: (c) (c) K η R Η /L γ Η (c/ρ) β ; γ Η =d Η /l; β k' Approximate relation (no chain entanglements): [η] (c) [{[η](1 + c[η]) k' } + {γ Η (c/ρ) β (c) [η] FD } ] 1/ Carnegie Mellon University 11

12 (c) log([η] / [η]) α (c) 1 1 Entanglement Interactions Scaled screening of Intramolecular Interactions. Virial Expansion log(r /Λ) G Carnegie Mellon University 1

13 (c) (c) Chain entanglements act to increase K η R Η : c)} ] 1/ [η] (c) [{[η](1 + c[η]) k' } + {γ Η (c/ρ) β (c) [η] FD E(X /X (c) X =c[η] FD X c = constant 1; empirical for many systems E(y) {1 + [y m(y)] } 1/ m(y) {1 + µy -1/ } 3 m( ) = 1; m(y) y.4 for y < 1 E(y) {1 + y 4.8 } 1/ (c) η = η LOC {1 + c[η] (c) } Carnegie Mellon University 13

14 Note: The scaling of the screening of the thermodynamic and hydrodynamic interactions present in dilute solutions may each be scaled with the reduced variable (R G /Λ) 3 = cn Α R G 3 /M = c/c* where c* = M/N Α R G 3. By contrast, the behavior following screening of these, and the development of entanglements scales with (c) (c) X =c[η] FD = πn Α (R G ) γ Η /M L No single reduced concentration may be used to scale the reduced viscosity over the entire concentration range of interest. Carnegie Mellon University 14

15 1 8 6 c/ρ =.1 c/ρ =.1 d =.78 nm T E = 1 d =.78 nm T 4 (c) log(ρ[η] ) 8 6 c/ρ =.1 c/ρ =.1 d = T ρ[η] d = T log(c [η] /1) FD,Θ â = 1 nm and M L = 4 nm 1 Carnegie Mellon University 15

16 1 8 6 ~ 1+β log{(η 1)/c [η] } FD,Θ log(c [η] /1) FD,Θ 3 â = 1 nm; M L = 4 nm -1 c/ρ is.1,.3,.1,.3 and 1 pip up, right, down, left and absent, respectively log(l/nm) increasing from to 5 in increments of.5 d Τ /nm equal to (lower) and.78 (upper) Carnegie Mellon University 16

17 ln{(η ~ 1)/c[η]} c[η] c/ρ is.1,.3,.1,.3 and.1 log(l/nm) increases from to 5 in increments of.5 â = 1 nm; M L = 4 nm -1 ; d Τ =.78 nm â = 1 nm; M L = 4 nm -1 ; d Τ = â = 1 nm; M L = 4 nm -1 ; d Τ =.78 nm Carnegie Mellon University 17

18 Sodium hyaluronate An acidic polysaccharide with a disaccharide repeat unit: CH OH H OH HO H H O H H O H OH H H H O O H NH CO CH 3 COONa Recovered from animal connective tissue, synovial and vitreous fluids, and some bacteria. For dilute solutions in.1 M NaCl, R G /nm. (L/nm).5 from light scattering For L >> â: R G = âl/3 (without excluded volume) â 3. = 14.5 nm [η]/ml g (L/nm).8 ln[η]/ lnt -1.8 Carnegie Mellon University 18

19 14 1 Ln(ηsp/c[η]) c[η] 4 Sodium hyaluronate in aqueous.1 M NaCl at 5 C. 1 6 M w =.,., 1.3, 1.,.8,.35, and.3 (unfilled squares, circles, triangles and diamonds and the filled circles, squares and triangles) Carnegie Mellon University 19

20 Log(η / η ) rel rel /(T/K) η rel is the avg. for the temperature interval (1 to 6 C) The nominal value of ln(c[η]) is given for each panel. Very unusual behavior--normally ln η rel / T 1 would increase monotonically with increasing c. Carnegie Mellon University

21 6 4 Log (η sp /c[η]) Log (c[η]) The dashed line has slope 3.15: η/η solvent 1 + k'c[η] + c (c[η]) 3.15 Carnegie Mellon University 1

22 4 1+β A + Log(η sp /c Mw) Log (cmw) β = (lower) or.5 (upper) solid lines and dashed lines for higher cm w have slopes.4 and, resp. Transition gives âγ Η 9-1 nm; close to measured â Carnegie Mellon University

23 1-1 log(c/gl ) W/ log c[ η] The slow increase is consistent with the known dependence of [η] on T ( ln[η]/ lnt -1.8); this reflects â decreasing with increasing T The extremum is unexpected, and may reflect some decrease in the temperature dependence of â through intermolecular effects; no theoretical treatment available. Carnegie Mellon University 3

24 More on the Local Viscosity: Postulate: (c) The dependence of η LOC on composition is similar to that of the viscosity η MIX of mixtures of small molecules on composition. In many treatments of η MIX it is assumed that η MIX = Aexp[Γ(T, {x}, )] where {x} is the set of mole fractions of the components. With small molecule components at temperatures well above the T g of any of the components, it sometimes assumed that Γ(T, {x}, ) x µ Γ µ + µ µ α Γ µα For example, then if all of the Γ µα = : ln(η MIX ) x µ ln(η µ ) µ Arrhenius (1887) utilized a similar expression with x µ replaced by the volume fraction ϕ µ of component µ. Carnegie Mellon University 4

25 In several treatments, RTΓ µ is taken to be an activation free energy for flow, and is approximated as the "ideal" free energy of mixing, and the RT Γ are the non-ideal "residual terms in the free energy of mixing. Thus for a binary mixture: ln(η MIX ) (1 x )ln(η 1 ) + x ln(η ) + Γ 1 (x,t, ) e.g., with Γ 1 (x ) = x (1 x )γ 1 (T, ) a simple approximation, so that positive or negative curvature then results in plots of ln(η MIX ) vs x through the choice of γ 1. A hybrid expression has been utilized for mixtures with at least one component with a T g in the range of T of the experiment: Γ(T, {x}, ) x µ Γ µ + Γ µα + Ψ(T T g, ) µ µ α In which case, for a binary mixture ln(η MIX ) (1 x ){ln(η 1 ) Ψ 1 (T T g,1, )} + x {ln(η ) Ψ (T T g,, )} + Γ 1 (x,t, ) + Ψ MIX (T T g, ) Carnegie Mellon University 5

26 With the Vogel relation for Ψ(T T g, ): Ψ(T T g, ) = K/(T T g + ) In the WLF approximation, K and are "universal" constants: K 3 K, 57.5 K. There are very few data available to assess this expression for mixtures of small molecules. Three examples will be discussed: An example for poly(vinyl acetate) with η as a function of M at fixed ϕ, thereby fixed T g (except for possible effects (c) at low M) and fixed η LOC An example for solutions of trehalose, a disaccharide with a relatively high T g An example for polystyrene at a fixed M, as a function of T and ϕ Carnegie Mellon University 6

27 Poly(vinyl acetate): Cetyl alcohol & diethyl phthalate: log( η/pa s) log( ϕm ) w Carnegie Mellon University 7

28 Aqueous solutions of Trehalose (T g 1 C): For this system, T g = x T g; + k(1 x )T g;1 x + k(1 x ) where k is a system-dependent (essentially empirical) constant, sometimes related to the difference in the volumetric thermal expansion of the two components. Two examples of possible correlations will be discussed: An example in which it is assumed that η/k(x ) should scale with T T g (x ), where K(x ) is some function of the mole fraction of trehalose, to be determined from the data. An example in which it is assumed that η should scale with T T (x ), where T (x ) is a parameter to be determined from the data. Carnegie Mellon University 8

29 log(η/pa s) Assuming that a reduced viscosity should scale with T T g : 4 3 Mole fraction trehalose log(η/pa s) + log( / ) log( / ) Volume fraction trehalose T Tg(x) + Tg(x=.75); (K) Assuming that the viscosity should scale with T T : (x, x =.75) (Tg(x) Tg(.75)); (K) MOLE FRACTION TREHALOSE T (x, x =.75); (K) Carnegie Mellon University 9

30 With some systems, it appears that two such expressions may be required to approximate T g for the blend: T g = Min x T g; + k 1 (1 x )T g; x + k 1 (1 x ) ; x T g; 1 + k (1 x )T g;1 x + k (1 x ) introducing additional empirical constants, and where T g; for the polymer may depend approximately linear in 1/M n, and. 1 Polystyrene/tritolyl phospate (TCP) Dilatometry or DSC 5 DTA Plazek et al (197) Tg ( C) -5-1 Polystyrene/toluene Dilatometry DTA Braun and Kovacs (1965) Weight fraction solvent Carnegie Mellon University 3

31 Poly(vinyl chloride) 35 dibutyl phthalate dicyclohexyl phthalate Pezzin (1968) 3 Tg (K) Weight fraction solvent Carnegie Mellon University 31

32 Polystyrene/styrene (M w = ) log(ηloc/poise) log(η/poise) T = C (T Tg)/K Carnegie Mellon University 3

33 (T Tg)/K log(ηlock -1 /Poise) Log(K/ ) Weight Fraction Polymer -1.5 log(ηloc/poise) (w, w=.51) (Tg(w) Tg(.51)); (K) Weight Fraction Polymer T (w, w =.51); (K) Carnegie Mellon University 33

34 log(ηloc -1/Poise) Log( / ) Weight Fraction Polymer -1.5 log(ηloc/poise) (T Tg)/K Carnegie Mellon University 34

Notes on Schroeder's minor scales: Troubles with bare parameters in modeling macroscopic properties

Notes on Schroeder's minor scales: Troubles with bare parameters in modeling macroscopic properties Guy C. BERRY Department of Chemistry Carnegie Mellon University Pittsburgh, PA, USA "One upon a time,