Crossover behavior in the dependence of the …V CHILEAN SYMPOSIUM ON THE CHEMISTRY AND PHYSICAL CHEMISTRY OF POLYMERS G. C. Berry Department of Chemistry Carnegie Mellon University

Carnegie Mellon University 1December 27, 2000

Crossover behavior in the dependence ofthe viscosity on concentration and

molecular weight for semiflexible polymers

CCCC HHHH IIII PPPP OOOO LLLL 2222 0000 0000 0000

V CHILEAN SYMPOSIUM ON THE CHEMISTRY AND PHYSICAL CHEMISTRY OF POLYMERS

G. C. Berry

Department of Chemistry

Carnegie Mellon University

Reprint manuscripts available on requeste-mail: [email protected]

Acknowledgments:

Partial Support:National Science Foundation (GCB)


Concentration Ranges:

Several regimes of viscoelastic behavior are related to the meanseparation Λ of molecular centers relative to the root-mean-square radius of gyration RG:

Λ = (M/cNΑ)1/3

• Infinite dilution (Λ >> RG), describing the limiting behavior ofη~ 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 > RG), defined loosely as the range ofconcentrations for which (ηsp – 1)/c[η] begins to increasewith increasing concentration, but is small enough that ηsp

may be represented by a virial expansion in c[η];

• Moderately concentrated solutions (Λ < 2.5RG), for which thedensity of chains is large enough that certainthermodynamic and hydrodynamic interactions becomeprogressively screened with increasing concentration,vitiating the use of a virial expansion to represent ηsp;intermolecular entanglement effects may develop,depending on the molecular weight;

• Concentrated solutions or bulk (Λ << RG), so that certainthermodynamic and hydrodynamic interactions are fullyscreened, and intermolecular entanglement effects maydevelop, depending on the molecular weight.


Dimensionless reduced viscosity η~:

η~ = η/ηLOC(c) = 1 + c[η](c)

• [η](c) reduces to the intrinsic viscosity [η] at infinite dilution.Expressions for [η](c) will be considered for semiflexiblechains in the following;

• ηLOC(c) is a "Local viscosity", tending to the solvent viscosity

ηsolvent at infinite dilution and to the "viscosity" ηrepeat of arepeat unit for undiluted polymer. We will return to adiscussion of ηLOC

(c) in the following.


Molecular Parameters:

• L: contour length (ML = M/L)

• RG: radius of gyration (root-mean-square)

• RΗ: hydrodynamic radius (RΗ = Ξ/6πηsolvent)

• α: expansion factor

• γΗ: diameter to length ratio of hydrodynamic unit

[η] = [η]FDKηRΗ/γΗL

[η]FD = πNΑRG2 γΗ/ML


Thermodynamic Interactions:

For the wormlike model for a semiflexible chain:

RG ≈ { (âLα2/3)−1 + (L2/12)−1}−1/2

α ≈ 1 + z + kα(z/2)2 ν − 1/2 ; ν ≈ 3/5

z = a1A(â/L)z/(2ν − 1)

z = (3dΤ/16â)(3L/πâ)1/2 ≈ 0.18(dΤ/â)(L/â)1/2

Hydrodynamic Interaction:

KηRΗ ≈ {[(10/3)(RΗ)ND]−2 + [(RΗ)FD]−2}−1/2

where

(RΗ)ND/L = {2·31/2/9}(â/L)1/2α; (RΗ)FD/L = f(L/l, γΗ)

f(L/l, γΗ) ≈ ζ red/{1 + 2κζ redln(3L/2dΗ)}

ζ red = γΗζl/6πηLOC(c) dΗ ∝ γΗ


The Intrinsic Viscosity:

ln (L/â)

ln([η

]M /

â )

L2

α = 1

α > 1


The Infinite Dilution Limit (Λ/RG >> 1):

Λ

• [η](c) ⇒ [η]

• ηLOC(c) ≈ ηsolvent.

η~ = η/ηsolvent = 1 + c[η]


With decreasing Λ/RG (increasing c) the effects of screening ofthermodynamic and hydrodynamic interactions becomeimportant, and are here expressed by the relation:

[η](c) = [η]FD(c) K η

(c) RΗ(c) /γΗL

[η]FD(c) = πNΑ(RG

(c) )2γΗ/ML

By analogy to the behavior at infinite dilution, K η(c) RΗ

(c) isrepresented by the expression:

K η(c) RΗ

(c) ≈ {[(10/3)QND(Λ/RG)(RΗ)NDα (c)/α]−2 + [QFD(Λ/RG)(RΗ)FD]−2}−1/2

where both QND(Λ/RG) and QFD(Λ/RG) increase from unity withdecreasing Λ/RG (increasing c).

e.g., at infinite dilution:

KηRΗ ≈ {[(10/3)(RΗ)ND]−2 + [(RΗ)FD]−2 }−1/2


Dilute Solutions:

Λ

• ηLOC(c) ≈ ηsolvent.

• RG(c) ≈ RG

• K η(c)RΗ

(c) increases with decreasing Λ/RG (increasing c)

K η(c) RΗ

(c) ≈ {[(10/3)(RΗ)NDQND(Λ/RG)]−2 + [QFD(Λ/RG)(RΗ)FD]−2 }−1/2

On expanding QND and QFD in a Taylor series with respect to c[η]:

K η(c) RΗ

(c) /KηRΗ = 1 + k'c[η] + k"(c[η])2 + …

≈ exp{k'c[η]} ≈ (1 + c[η])k'

Thus, for dilute solutions:

η/η solvent = 1 + c[η] + k'(c[η])2 + k"(c[η])3 + …


Moderately concentrated solutions:

Λ

• The distribution of molecular centers is liquid-like

• ηLOC(c) ≈ ηsolvent(1 + bϕ) ≈ ηsolventexp(bϕ) for small bϕ

• RG(c) decreases toward RG with decreasing Λ/RG

(α (c) decreases toward unity)

• K η(c) RΗ

(c) /γΗL increases toward unity with decreasing Λ/RG

• [η](c) increases from [η] toward [η]FD(c) :

[η]FD(c) = πNΑ(RG

(c) )2γΗ/ML

(In the absence of chain entanglements)


α (c) ≈ MAX{1; α(1 + [7(RG/Λ)3]2)−1/16}

Rearranging the expression for K η(c) RΗ

(c) :

K η(c) RΗ

(c) /L ≈ γΗQFD{1 + (9QFD/20QNDα (c))2(3L/â)}−1/2

Empirically, for moderately concentrated solutions:

K η(c) RΗ

(c) /L ≈ γΗ(c/ρ)β; γΗ = dΗ/l; 0 ≤ β ≤ k'

Approximate relation (no chain entanglements):

[η](c) ≈ [{[η](1 + c[η])k'}2 + {γΗ(c/ρ)β[η]FD(c)}2]1/2


-0.8 -0.6 -0.4 0 0.2 0.4-0.2

log(R /Λ)G

log

([η]

/ [η

])(c

)α

(c)

0

0.2

0.4

0.6

0.8

1

2

1 EntanglementInteractions

Scaled screening of

IntramolecularInteractions

VirialExpansion


Chain entanglements act to increase K η(c) RΗ

(c) :

[η](c) ≈ [{[η](1 + c[η])k'}2 + {γΗ(c/ρ)β[η]FD(c) E(X/X

c)}2]1/2

X = c[η]FD(c)

Xc = constant ≈ 100; empirical for many systems

E(y) ≈ {1 + [y2m(y)]2}1/2

m(y) ≈ {1 + µy-1/2}3

m(∞) = 1; m(y) ≈ y0.4 for y < 100

E(y) ≈ {1 + y4.8}1/2

η = ηLOC(c) {1 + c[η](c)}


Note:

The scaling of the screening of the thermodynamic andhydrodynamic interactions present in dilute solutions may eachbe scaled with the reduced variable

(RG/Λ)3 = cNΑRG3 /M = c/c*

where c* = M/NΑRG3 .

By contrast, the behavior following screening of these, and thedevelopment of entanglements scales with

X = c[η]FD(c) = πNΑ(RG

(c) )2γΗ/ML

No single reduced concentration may be used to scale thereduced viscosity over the entire concentration range of interest.


-2 -1 0 1 0 1 2 32

log(c [η] /100)FD,Θ

0

2

4

6

8

0

2

4

6

8

10

log

(ρ[η

] )(c

)E = 1

c/ρ = 0.01 c/ρ = 0.1

c/ρ = 0.01 c/ρ = 0.1

d = 0Td = 0T

d = 2.78 nmTd = 2.78 nmT

ρ[η]

• â = 1 nm and ML = 400 nm−1


-2 -1 0 1 2

log(c [η] /100)FD,Θ

0

2

4

6

8

0

2

4

6

8

10

3

log

{(η

– 1

)/c

[η

]

}

1+β

FD

,Θ

~

• â = 1 nm; ML = 400 nm-1

• c/ρ is 0.01, 0.03, 0.1, 0.3 and 1pip up, right, down, left and absent, respectively

• log(L/nm) increasing from 2 to 5 in increments of 0.5

• dΤ/nm equal to 0 (lower) and 2.78 (upper)


0

5

10

15

0 10 20 30 40

0 20 40 60 80 100

0

2

4

0

5

10

0 10 20 30 40

c[η]

ln{(

η –

1)/c

[η]}

~

• c/ρ is 0.001, 0.003, 0.01, 0.03 and 0.01log(L/nm) increases from 2 to 5 in increments of 0.5

• â = 1 nm; ML = 400 nm-1; dΤ = 2.78 nmâ = 1 nm; ML = 400 nm-1; dΤ = 0â = 1000 nm; ML = 400 nm-1; dΤ = 2.78 nm


Sodium hyaluronate

An acidic polysaccharide with a disaccharide repeat unit:

NH

CO

CH3

HO

O

H

H

H

H

H O

CH OH2

O

H

HH

OH

OH

H

COONa

OH

Recovered from animal connective tissue, synovial and vitreousfluids, and some bacteria.

For dilute solutions in 0.01 M NaCl,

RG/nm ≈ 2.2 (L/nm)0.5 from light scattering

For L >> â: RG2 = âL/3 (without excluded volume)

∴ â ≈ 3×2.22 = 14.5 nm

[η]/mL·g−1 ≈ 3.05 (L/nm)0.82

∂ln[η]/∂lnT ≈ -1.8


c[η]

Ln

(ηsp

/c[η

])

0

2

4

6

8

10

12

14

0 50 100 150 200 250

0 20 40

2

4

6

Sodium hyaluronate in aqueous 0.10 M NaCl at 25°C.

10−6Mw = 2.22, 2.00, 1.30, 1.00, 0.80, 0.35, and 0.30(unfilled squares, circles, triangles and diamonds and the

filled circles, squares and triangles)


1000/(T/K)

-0.4

-0.2

0.0

0.2

0.4

0.6

3.00 3.25 3.50

-0.4

-0.2

0.0

0.2

0.4

0.6

3.00 3.25 3.50

0.1

1.0

1.45

1.9

Lo

g(η

/⟨

η

⟩)

re

lre

l

• ⟨η rel⟩ is the avg. for the temperature interval (10 to 60°C)

• The nominal value of ln(c[η]) is given for each panel.

• Very unusual behavior--normally ∂ln ηrel/∂T−1 wouldincrease monotonically with increasing c.


0

1 2 3-1 0

4

2

0

Log (c[η])

Lo

g (η

/c

[η])

sp

6

The dashed line has slope 3.15:

η/η solvent ≈ 1 + k'c[η] + c2(c[η])3.15


1 2 3 4 5 6

4

2

0

-2

Log (cMw)

A +

Lo

g(η

/c

M

w)

sp1+

β

• β = 0 (lower) or 0.5 (upper)

• solid lines and dashed lines for higher cMw have slopes2.4 and 2, resp.

• Transition gives âγΗ ≈ 9-12 nm; close to measured â


log c[ η]

-1.00 0.00 1.00 2.00

0.1 1 10 100

log(c/gL )-1

0

2

6

10

4

8

10

0

2

6

4

8

W/3

00

• The slow increase is consistent with the knowndependence of [η] on T (∂ln[η]/∂lnT ≈ -1.8); this reflectsâ decreasing with increasing T

• The extremum is unexpected, and may reflect somedecrease in the temperature dependence of â throughintermolecular effects; no theoretical treatment available.


More on the Local Viscosity:

Postulate: The dependence of ηLOC(c) on composition is

similar to that of the viscosity ηMIX of mixtures ofsmall molecules on composition.

In many treatments of ηMIX it is assumed that

ηMIX = Aexp[Γ(T, {x}, …)]

where {x} is the set of mole fractions of the components.

With small molecule components at temperatures well abovethe Tg of any of the components, it sometimes assumed that

Γ(T, {x}, …) ≈ ∑µ

xµΓµ + ∑µ≠α

∆Γµα

For example, then if all of the ∆Γµα = 0:

ln(ηMIX) ≈ ∑µ

xµln(ηµ)

Arrhenius (1887) utilized a similar expression with xµ replacedby the volume fraction ϕµ of component µ.


In several treatments, RTΓµ is taken to be an activation freeenergy for flow, and is approximated as the "ideal" freeenergy of mixing, and the RT∆Γ are the non-ideal "residualterms in the free energy of mixing. Thus for a binary mixture:

ln(ηMIX) ≈ (1–x2)ln(η1) + x2ln(η2) + ∆Γ12(x2,T,…)

e.g., with ∆Γ12(x2) = x2(1 – x2)γ12(T,…) a simple approximation,so that positive or negative curvature then results in plots ofln(ηMIX) vs x2 through the choice of γ12.

A hybrid expression has been utilized for mixtures with atleast one component with a Tg in the range of T of theexperiment:

Γ(T, {x}, …) ≈ ∑µ

xµΓµ + ∑µ≠α

∆Γµα + Ψ(T – Tg,…)

In which case, for a binary mixture

ln(ηMIX) ≈ (1–x2){ln(η1) – Ψ1(T – Tg,1,…)}

+ x2{ln(η2) – Ψ2(T – Tg,2,…)}

+ ∆Γ12(x2,T,…) + ΨMIX(T – Tg,…)


With the Vogel relation for Ψ(T – Tg,…):

Ψ(T – Tg,…) = K/(T – Tg + ∆)

In the WLF approximation, K and ∆ are "universal" constants:K ≈ 2300 K, ∆ ≈ 57.5 K.

There are very few data available to assess this expressionfor mixtures of small molecules.

Three examples will be discussed:

♦ An example for poly(vinyl acetate) with η as a function ofM at fixed ϕ, thereby fixed Tg (except for possible effectsat low M) and fixed ηLOC

(c)

♦ An example for solutions of trehalose, a disaccharide witha relatively high Tg

♦ An example for polystyrene at a fixed M, as a function of Tand ϕ


Poly(vinyl acetate): Cetyl alcohol & diethyl phthalate:

3 4 5 6

log( ϕM )w

0

2

4

-2

log

(η/P

a·s)

1

3

5

-1

1.0

0.75

0.50

0.25

0.125


Aqueous solutions of Trehalose (Tg ≈ 120°C):

For this system,

Tg =x2Tg;2 + k(1 – x2)Tg;1

x2 + k(1 – x2)

where k is a system-dependent (essentially empirical)constant, sometimes related to the difference in thevolumetric thermal expansion of the two components.

Two examples of possible correlations will be discussed:

♦ An example in which it is assumed that η/K(x2) shouldscale with T – Tg(x2), where K(x2) is some function of themole fraction of trehalose, to be determined from the data.

♦ An example in which it is assumed that η should scalewith T – T0(x2), where T0(x2) is a parameter to bedetermined from the data.


Assuming that a reduced viscosity should scale with T – Tg:

-3

-2

-1

0

1

2

3

4

250 300 350 400 450 500

0.0034

0.0045

0.0060

0.0090

0.0109

0.0212

0.0316

0.0432

0.0465

0.0601

0.0749

0.1339

T – Tg(x) + Tg(x=0.075); (K)

log(

η/P

a·s)

+ l

og(∆

/…)

Volume fraction trehalose

0.0 0.10 0.15 0.20

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0.0 0.2 0.4 0.6 0.8 1.0

Mole fraction trehalose

log(

∆/…

)

Assuming that the viscosity should scale with T – T0:

-4

-3

-2

-1

0

1

2

3

4

150 200 250 300 350 400 450 500 550

00.00340.0450.00600.00900.01090.02120.03160.04330.04650.06010.07490.1339

-80

-60

-40

-20

0

20

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

MOLE FRACTION TREHALOSE

T – ∆(x, x = 0.075); (K)

∆(x,

x =

0.0

75)

– (T

g(x)

– T

g(0.

075)

); (

K)

log(

η/P

a·s)


With some systems, it appears that two such expressionsmay be required to approximate Tg for the blend:

Tg = Min

x2Tg;2 + k1(1 – x2)Tg;02

x2 + k1(1 – x2);

x2Tg;∞1 + k2(1 – x2)Tg;1x2 + k2(1 – x2)

introducing additional empirical constants, and where Tg;2 forthe polymer may depend approximately linear in 1/Mn, and.

0 0.2 0.4 0.6 0.8 1

100

50

0

-50

-100

-150

Weight fraction solvent

Tg (

°C)

Polystyrene/toluene

Dilatometry

DTA

Braun and Kovacs (1965)

Polystyrene/tritolyl phospate (TCP)

Dilatometry or DSC DTA

Plazek et al (1970)


0 0.2 0.4 0.6 0.8 1

Weight fraction solvent

Tg

(K

)

Poly(vinyl chloride)

dibutyl phthalate

dicyclohexyl phthalate

Pezzin (1968)

350

250

200

300

150


Polystyrene/styrene (Mw = 2.4 × 105)lo

g(η/

Poi

se)

-2

-1

0

1

2

3

4

150 175 200 25

(T – Tg)/K

T = 20°C

30° 40° 50°

60-67°

-2.5

-2.0

-1.5

-1.0

-0.5

0

0.508

0.456

0.446

0.407

0.326

0.319

0.304

0.263

0.217

0.207

0.181

0.168

0.130

0.094

0.061

0.036

0.022

log(

η LO

C/P

oise

)


300 350 400 450 500-2.5

-2.0

-1.5

-1.0

-0.5

0

(T – Tg)/K

-1.5

-1.0

-0.5

0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.0 0.1 0.2 0.3 0.4 0.5

Weight Fraction Polymer

Log(

K/…

)

log(

η LO

CK

-1/P

oise

)150 175 200 225

log(

η LO

C/P

oise

)

T – ∆(w, w = 0.51); (K)

∆(w

, w=

0.5

1)

– (T

g(w

) –

Tg(0

.51)

); (

K)


-140

-120

-100

-80

-60

-40

-20

0

0 0.1 0.2 0.3 0.4 0.5

0.508

0.456

0.446

0.407

0.326

0.319

0.304

0.263

0.217

0.207

0.181

0.168

0.130

0.094

0.061

0.036

0.022


-2.5

-2.0

-1.5

-1.0

-0.5

0

0.508

0.456

0.446

0.407

0.326

0.319

0.304

0.263

0.217

0.207

0.181

0.168

0.130

0.094

0.061

0.036

0.022

150 175 200 225

(T – Tg)/K

log(

η LO

C/P

oise

)

-1.5

-1.0

-0.5

0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.0 0.1 0.2 0.3 0.4 0.5


Log(

∆/…

)

log(

η LO

C∆-

1 /P

oise

)

Documents

Crossover behavior in the dependence of the …V CHILEAN SYMPOSIUM ON THE CHEMISTRY AND PHYSICAL CHEMISTRY OF POLYMERS G. C. Berry Department of Chemistry Carnegie Mellon University