developments and trends in light scattering on ... · ispac- 2005 carnegie mellon university 1...

ISPAC- 2005Carnegie Mellon University

1

Developments and Trends in

Light Scattering on Macromolecular Solutions

Guy C. Berry

Carnegie Mellon UniversityPittsburgh, PA

www.chem.cmu.edu/berry

Carnegie Mellon

Mellon College of Science

Department of Chemistry

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© 2005 Universal Press Syndicate

© 2005 uclick, L.L.C.

Much More Close to Home

Copyright

Published on June 13, 2005

6/13/05 6:08 AMUntitled

Page 1 of 2http://www.uclick.com/client/smc/cl/

Berry, G.C., Total intensity light scattering from solutions of macromolecules, in SoftMatter: Scattering, Manipulation & Imaging, R. Pecora and R. Borsali, Editors.2005, In Press.

Berry, G.C., Light Scattering, Classical: Size and Size Distribution Classification, inEncyclopedia of Analytical Chemistry, R.A. Meyers, Editor. 2000, John Wiley &Sons Ltd: New York. p. 5413-48.

Berry, G.C. and P.M. Cotts, Static and dynamic light scattering, in Experimental methodsin polymer characterization, R.A. Pethrick and J.V. Dawkins, Editors. 1999, JohnWiley & Sons Ltd.: Sussix, UK. p. 81-108.

Berry, G.C., Static and dynamic light scattering on moderately concentrated solutions:Isotropic solutions of flexible and rodlike chains and nematic solutions of rodlikechains. Adv. Polym. Sci., 1994. 114(Polymer Analysis and Characterization): p.233-90.

Berry, G.C., Light scattering, in Encyclopedia of Polymer Science and Engineering, H.F.Mark, et al., Editors. 1987, John Wiley & Sons, Inc.: New York. p. 721-94.

Casassa, E.F. and G.C. Berry, Light scattering from solutions of macromolecules. Tech.Methods Polym. Eval., 1975. 4, Pt. 1: p. 161-286.

www.chem.cmu.edu/berry


3


4

STATIC LIGHT SCATTERING

Rayleigh ratio from a solution with solute concentration c

RSi(q,c) = r2ISi(ϑ)/VobsIINC

S and i designate the polarization state of the electric vectors of the

Scattered and Incident light, respectively, relative to the scattering plane.

• k0 vectoral wave number along the incident beam

• k vectoral wave number along the scattered beam

• q is the vector difference between the and scattered light, respectively:

q = k0 – k

• |k0| = |k| = (4!n/λ) for static scattering, with λ the wavelength in vacuum

• The modulus |q| of the scattering vector q becomes

q = (4!/λ)sin(ϑ/2)


5

RHv(q,c) and RVv(q,c) are given by (Using Isolution – Isolvent):

RHv(q,c) = Raniso(q,c)

RVv(q,c) = Riso(q,c) + (4/3)Raniso(q,c) + Rcross(q,c)

• Riso(q,c): Isotropic component

• Raniso(q,c): Anisotropic component

NOTE: When considering the scattering from a solute with

isotropic scattering elements, RVv(q,c) = Riso(q,c), and

the subscript will be suppressed


6

Isotropic scatterers:

R(q,c) = KopcM S(q,c) ; S(q,c) = P(q,c)F(q,c)

(Note: S(q,c) is sometimes used to denote a different function than that defined here.)

Two expressions are commonly employed to represent F(q,c):

F(q,c) = 1 – cB(c)P(q,c)Q(q,c); F(q,c)-1 = 1 + cΓ(c)P(q,c)H(q,c)

Γ(c) = B(c)/[1 – cB(c)] ; H(q,c) = Q(q,c) [1 – cB(c)]

[1 – cB(c)P(q,c)Q(q,c)]

KopcMR(q,c) =

1S(q,c) =

1P(q,c) + cΓ(c)H(q,c)


7

For a monodisperse solute (only), F(0,c) is related to the equilibrium osmotic

modulus KOS:

F(0,c) = (M/cRT) KOS

KOS = c ∂ Π/∂c

where Π is the osmotic pressure.

With this expression, for a monodisperse solute,

cΓ(c) =MRT∂ Π∂c – 1

Note: This expression is often misapplied for a heterodisperse solute.


8

DYNAMIC LIGHT SCATTERING

g(2)(τ; q,c) = 1 + fC |g(1)(τ; q,c)|2

g(1)(τ; q,c) ≈ µΣ rµ(q,c)exp[–τ γµ (q,c)]; µΣ

rµ = 1

ln[g(2)(τ; q,c) – 1]1/2 =ln[fC]1/2 – K(1)(q,c)τ + 12! K

(2)(q,c)τ2 + …

limq=0

K(1)(q,c)/q2 = DM(c)

aLS(c) = kT

6!ηDM(c) ; limc=0 aLS(c) = RH; Hydrodynamic Radius


9

Static scattering extrapolated to infinite dilution:

[R(q,c)/c]0 = K'n2s(∂n/∂c)2

wMLSPLS(q,0)

LS denotes an averaging over heterodispersity appropriate for scattering; K' = 4!2/NAλ40,

MLS = [R(0,c)/c]0/K'n2

s(∂n/∂c)2w

In the RGD limit for a solute:

MLS = ~ψ-2Σν

C wνM

-1ν{Σ

j

nν

~ψj,νmj,ν}2

For identical isotropic scattering elements:Single solvent component

MLS = Σν

C wνMν = Mw

Multiple solvent components

MLS =(∂n/∂c)2

Π

(∂n/∂c)2w Mw


10

Two, optically distinct scattering elements, single solvent component:

MLS = (1 + 2µ1Y + µ2Y2)Mw

Y =~ψA – ~ψB

~ψ ≈

nA – nB

wAnA + (1 – wA)nB

µn = Σν

C

wνMνΔwnν/Mw; n = 1,2

Δwν = wAν – wA = wB – wBν

Special cases for which Δwν = 0, so that µ1 = µ2 = 0:

• random and alternating copolymers

• regular block copolymers or stratified particles with uniform structures

among the molecules


11

(Benoit)


12

Depolarized scattering in the RGD regime:

Identical cylindrically symmetric polarizabilities for all chain elements:

[Raniso(0,c)/c]0

K'n2s(∂n/∂c)2

w= (3/5)Mwδ

2LS

δ2LS = M-1

wΣν

C

wνMνδ2ν ; δ

2ν =

δ20

nν Σ

j

nν Σ

k

nν (3/2)[〈cos2 βij〉 – 1]

• βij is the angle between the major axes of scattering elements i and j

• δ0 = α|| – α⊥

α|| + 2α⊥

For an optically anisotropic homogeneous solute:

MLS,Hv = (3/5)δ2LSMw ; MLS,Vv = {1 + (4/5)δ2

LS}Mw


13

Wormlike chain model:

• persistence length â

• contour length L

• mass per unit length ML = M/L

δ2ν/δ

20 = (2Zν/3){1 – (Zν/3)[1 – exp(–3Z-1

ν )]}; Zν = â/Lν.

rod: Mwδ2LS = δ

20Mw

coil: Mwδ2LS = δ

20m0 << δ

20Mw


14

Scattering beyond the Rayleigh-Gans-Debye (RGD) regime:

• RGD approximation that the electric field giving rise to the dipole radiation of the scattered

light is that of the incident radiation propagating in the medium is the same as that acting on

the solvent is usually valid for "threadlike" molecules;

• RGD approximation may fail for large particles, particularly if the particle refractive index is

very different from the suspending medium.

For nonabsorbing, optically homogeneous particles, two modifications are required to

compute MLS:

• ~ψ ⇒ hsph(ñ)~ψ; hsph(ñ) = 3(ñ + 1)/2(ñ2 + 2); ñ = nsolute/nmedium

• A function m(ñ,λ,Mν) specific for each particle shape is required:

MLS = Σν

C

wνMν[m(ñ,λ,Mν)]2


15

Special case, Mie scattering for spherical particles:

MLS = Σν

C

wνMν[msph(ñ,~αν)]2;

~αν = 2!Rν/λ

Fraunhofer scattering regime for monodisperse spheres (~α > 4 and ñ > 1.4),

MLS ≈ M{3/2~α2(ñ – 1)hsph(ñ)}2


16

Scattering at infinite dilution and "small" scattering angle in the RGD

regime:

[R(q,c)/c]0 = K'n2s(∂n/∂c)2

wMLSPLS(q,0)

(Optically isotropic scattering elements)

PLS(q,0) = 1 – (1/3)q2R2G,LS + …

R2G,LS =

Σν

C

wνM-1ν Σ

j

nν Σ

k

nν ~ψj,ν

~ψk,ν mj,νmk,ν〈|rjk|2ν〉

2Σν

C

wνM-1ν [Σ

j

nν~ψj,νmj,ν]

2


17

Identical scattering elements:

R2G,LS =

1Mw

Σν

C

wνMν R2G,ν; R

2G,ν =

12n2

νΣj

nν Σ

k

nν 〈|rjk|

2ν〉 ≈ (R2

G/Mε)Mεν

Similarly, for the Hydrodynamic Radius:

RH,LS = Mw/ Σν

C

wνMν R-1H,ν ; RH,ν ≈ (RH/Mµ)Mµ

ν

Note: The "power-law" approximations do not apply to the wormlikechain


18

Mean-square radius of gyration for some modelsModel Length scales R2

G

Random-flight linear coil("infinitely thin")

L = contour lengthâ = persistence length

âL/3

Persistent (wormlike) chain(a)

("infinitely thin")

L = contour lengthâ = persistence length

(âL/3)S(â/L)

≈ [(âL/3)-1 + (L2/12)-1]-1

Rod ("infinitely thin") L = length L2/12

Disk ("infinitely thin") R = radius R2/2

Cylinder L = lengthR = radius

L2/12 + R2/2

Sphere R = radius 3R2/5

Spherical shell R = radius (outer)Δ = shell thickness (3R2/5)

1 – [1 – (Δ/R)5]

1 – [1 – (Δ/R)3]

Spherical shell ("infinitely thin") R = radius (outer) R

Spheroid 2R1 = unique axis2R2 = transverse axis R2

1

2 + (R2/R1)

2

5

a) S(Z) = 1 – 3Z + 6Z2 –6Z3[1 – exp(–Z-1)] ≈ (1 + 4Z)-1; Z = â/L


19

R2G,LS and RH,LS for some power-law approximations

R2G,LS RH,LS

Exact Relation(a) (1/Mw)Σν

C wνMν R

2G,ν Mw/Σ

ν

C wνMν R

-1H,ν

Approximation for(b)

RG ∝ RH ∝ Mε/2

(R2G/Mε) M ε+1

(ε+1)/Mw (RH/Mε/2) Mw/M 1 − ε(1 − ε)

Random-flight coil(c) ;

ε = 1

(R2G/M) Mz (RH/M1/2) Mw/M 1/2

(1/2) ≈

(RH/M1/2) M1/2w (Mw/Mn)

0.10

Rodlike chain (thin)(c) ;

ε ≈ 2

(R2G/M2) MzMz+1 (RH/M) Mw

Sphere(c) ; ε = 2/3 (R2G/M2/3) M 5/3

(5/3)/Mw ≈

(R2G/M2/3) M2/3

z (Mw/Mz)0.10

(RH/M1/3) Mw/M 2/3(2/3) ≈

(RH/M1/3) M1/3w (Mw/Mn)

0.10

M µ(µ) = Σ

ν

C

wνMµν ; M(µ) is Mn, Mw, (MwMz)

1/2 and (MwMzMz+1)1/3 for µ = –1, 1, 2 and 3


20

Wormlike Chain (not a power-law):

R2G,LS = (Lzâ/3)SLS(â/Lz)

SLS(Zz) = 1 – 3Zz + 6Z2zh + 2h + 1 –6Z3

z (h + 2)2

h(h + 1)2{1 – [1 + (Z(zh + 2))-1]-h}

≈ [1 + 4Zz(h + 2)/(h + 3)]-1 = [1 + 4â/Lz+1]-1

Branched flexible chain polymers:

For regular star- and comb-shaped branched random-flight chains:

R2G = gR2

G,LIN ; g ≈ λbr + gstar(1 – λbr)7/3

• f is the number of branches• λbr is the fraction of the mass in the backbone, i.e., λbr = 0 for a star-shaped

molecule, and λbr = 1 for a linear chain.

Theoretical expressions for g are available for a variety of structures, includingrandomly branched chains.

The combination of SEC with light scattering measurements of M and R2G for

the eluent is particularly useful, especially if a model is available for g for the

anticipated branched chains (e.g., randomly branched "fractions"):


22

Randomly branched poly(methylmethacrylate), PMMA.

(a) linear (•), branched polymers (°)

(b) The ratio using the power-law (°)

and polynomial (•) extrapolations


23

Optically diverse scattering elements

R2G,LS =

Σν

C

Σj

nν Σ

k

nν ~ψj,ν

~ψk,ν mj,νmk,ν〈|rjk|2ν〉

2[Σν

C

Σj

nν~ψj,νmj,ν]

2

Special case:

• two scattering elements, A and B

• weight fraction wA = 1 – wB

= nAmA/(nAmA + nBmB)

• ~ψ = wA~ψA + wB

~ψB

• ~wA = 1 – ~wB = wA~ψA/

~ψ ;~wA and ~wB many be +, –, or 0.

R2G,LS = ~wAR2

G,A + (1 – ~wA)R2G,B + ~wA(1 – ~wA)Δ2

AB

Δ2AB is the mean-square separation of the c.g.'s of the A & B constellations.

R2G,geo = wAR2

G,A + (1 – wA)R2G,B + wA(1 – wA)Δ2

AB


24

Example:

A stratified sphere: outer diameter RB and inner core diameter RA ( ΔAB = 0)

R2G,LS = (3/5)

~wAR2A +(1 – ~wA)

R5B – R5

A

R3B – R3

A

For a thin shell enclosing the solvent, such that ~wA = 0:

R2G,LS = (3/5)R2

B

1 – [1 – (Δshell/RB)5]

1 – [1 – (Δshell/RB)3] ; Δshell = RB – RA

The expression for the ratio of the volume of the shell to its surface area in terms of

experimentally determined parameters may be solved to give

Δshell ≈ (v2Mw/4!NAR2G,LS){1 + (1.3)β2 + 0.06β3 + …}-1

where β = v2M/4!NA(R2G)3/2 << 1


25

Optically anisotropic scattering elements

Homopolymer comprising cylindrically symmetric scattering elements:

[RSi(q,c)/c]0 = K'n2s(∂n/∂c)2

wMLS,SiPLS,Si(q,0)

PLS,Hv(q,0) = 1 – (3/7)R2G,LS,Hvq

2 + …

PLS,Vv(q,0) = 1 – (1/3)R2G,LS,Vvq

2 + …

For the persistent coil model,

R2G,LS,Hv =

Σν

C

wνMνδ2νf

23,νR

2G,ν

Σν

C

wνMνδ2ν

; R2G,LS,Vv =

Σν

C

wνMν(1 + 4δ2ν/5)J(δν)R

2G,ν

Σν

C

wνMν(1 + 4δ2ν/5)

J(δν) =1 – (4/5)f1,νδν + (4/7)(f2,νδν)

2

1 + (4/5)δ2ν


26

The functions fi, appearing in the reciprocal scattering factors for anisotropicchains as a function of the contour length L divided by the persistence length â:––––, f1;–– - ––, f2; - - -, f3; and – – –, f4 .

Owing to the decrease in δ/δο with increasing L/â, the influence of thedepolarized components decreases rapidly for L/â >1, and one can simply put allfi ≈ 1 with negligible error in the analysis of data.


27

Upper:[Kc/RVv(0,c)]1/2 for cis-PBO.

Middle:Kc/RHv(0,c) for cis-PBO.

Lower:[Kc/RVv(0,c)]1/2 for ab-PBO.

310

Kc/

RHv

(ϑ)

10 K

c/R

Vv(ϑ)

5

1

2

3

4

0 0.2 0.4 0.6 0.8 11

2

3

4

sin ( ϑ/2)2

10 K

c/R

Vv(ϑ)

5

0


28

Scattering beyond the RGD regime

R2G,LS =

Σν

C

wνMνy(ñ,λ,Mν)[m(ñ,λ,Mν)]2R

2G,RGD,ν

Σν

C

wνMν[m(ñ,λ,Mν)]2

Mie scattering theory:

R2G,LS = (3/5)

Σν

C

wνMν[msph(ñ,~αν)]2ysph(ñ,~αν)R

2ν

Σν

C

wνMν[msph(ñ,~αν)]2

Evaluation of an average R from R2G,LS requires

an iterative process.


29

Scattering at infinite dilution and arbitrary q

PLS(q,0) =

Σν

C

wνM-1ν Σ

j

nν Σ

k

nν ~ψj,ν

~ψk,ν mj,νmk,ν〈[sin(q|rjk|ν)]/q|rjk|ν〉

Σν

C

wνM-1ν [Σ

j

nν

~ψj,νmj,ν]2

Identical scattering elements

PLS(q,0) = 1

Mw Σν

C

wνMν Pν(q,0); Pν(q,0) = 1n2ν Σ

j

nν Σ

k

nν 〈[sin(q|rjk|)/q|rjk|〉


30

Particle scattering functions for some optically isotropic models

Model R2G P(q,0)

Random-flight linear coil

âL/3 u = âLq2/3 pc(u) = (2/u2)[u - 1 + exp(-u)]

Disk ("infinitely thin") R2/2 y = Rq (2y2)[1 - J1(2y)/y]

Sphere 3R2/5 y = Rq (9/y6)[sin(y) – ycos(y)]2

Shell ("infinitely thin") R y = Rq [sin(y)/y]2

Rod ("infinitely thin") L2/12 x = Lq p1(x) = (2/x2)[xSi(x) – 1 + cos(x)]

Monodisperse Random-Flight model:

P(q,0)-1 = 1 + 13u + 136u2 – 1

540u3 + O(u4)

P(q,0)-1/2 = 1 + 16u – 0×u2 – 11080u3 + O(u4

)


31


32

Chain length dispersion:

Linear random-flight chain heterodisperse in M:

PLS(q,0) = (2/rMwq2){1 – (1/rMnq2)[1 – MnΣ

ν

C

wνM-1νexp(–rMνq

2)]}

PLS(q,0)-1 = 1 + rMzq2/3; For the most-probable distribution of M

Linear rodlike chain with a Schulz-Zimm distribution in M:

PLS(q,0) =2

(1 + h)ξ

arctan(ξ) + Σj = 1

h - 1

1

h – j – 1h (1 + ξ2)(j - h)/2sin[(h - j)arctan(ξ)]

where ξ = qMw/ML(1 + h).

PLS(q,0) =2ML

qMw arctan

qMw

2ML; For a most-probable distribution of M


33


34

g(1)(τ; q,c) ≈ µΣ rµ(q,c)exp[–τ γµ (q,c)]; µΣ

rµ = 1

γµ(q,c) =kTq2

6!ηâµ(c) ; lim

c=0 aµ(c) = RH,µ; Hydrodynamic Radius

Inverse Laplace transform of g(1)(t; q,c):


35

Behavior at large R2Gq2

Random-flight chain:

lim u >> 1

P(q,0)-1 = C + u/2 + O(u-1)

lim u >> 1

uP(q,0) ≈ 2

lim u >> 1

[Κοπc/R(q,c)]0 = (1/2)[M-1 + (R2G/M)q2 + …]

where C = 1/2 for a linear chain. Note: ∂[Κopc/R(q,c)]0/∂q2 = â/2ML

Rodlike chain:

lim u >> 1

P(q,0)-1 = C + Lq/! + O(q-1)

lim u >> 1

uP(q,0) ≈ !Lq/12

where C = 2/!2 and u = R2Gq2 = L2q2/12. Note: ∂[Κopc/R(q,c)]0/∂q = L/!M = 1/!ML


36

Kratky-Porod wormlike chain model

Three ranges of behavior in q2P(q,0) vs q:

I. Wormlike chain behavior for R2Gq2 < 1

II. Flexible chain like asymptote (q2P(q,0) ∝ 2) for 1/RG < q < 1/â

III. Rodlike asymptotic behavior (q2P(q,0) ∝ q) for âq > 1

Approximate models to mimic this behavior may be fitted by a Padé relation:

P(q,0) ≈

PRF(q,0)m +

1 – exp(–(âq)2)

1 + Lq2/!m

1/m

; m ≈ 5-7

PRF(q,0) for the random-flight chain, with âL/3 replaced by R2G = (âL/3)S(â/L)


37

Kratky-Porod plot

The intersection of the

extrapolated lines for regions

II and III occurs for

âq* ≈ (6/π)S(â/L)-1

≈ (6/!)(1 + 4â/L)

------------------------------------------------

Holtzer Plot

A maximum in qP(q,0) vs q

that marks the transition from

regions I to II occurs for

RGq** = 1.466

L/â = 640, 160, 80, 40, 20, 10, 5 top to bottom


38

In neutron scattering, the

effect of the chain element

diameter may be seen in

region III, and modeled by

replacing multiplying !/Lq

by Psection(q,0),

Psection(q,0) ≈

2J1(Rcq)

Rcq2

≈ exp[–(Rcq)2/4]

Two different wormlike micelles


39

Optically diverse scattering elements

Limiting to two scattering elements, A and B, with ~wA = 1 – ~wB = wA~ψA/

~ψ:

PLS(q,0) =~wAPA(q,0) + (1 – ~wA)PB(q,0) + ~wA(1 – ~wA)PAB(q,0)

Pν(q,0) =

1n2νΣj

nν Σ

k

nν 〈[sin(q|rjk|ν)]/q|rjk|ν〉 ; ν = A, B

PAB(q,0) =1

nAnBΣj

nA

Σk

nB

〈[sin(q|rjk|)]/q|rjk|〉 – [PA(q,0) + PA(q,0)];

PAB(q,0) ≈ (ΔABq)2/3 + …

• Note the similarity in form to the expression the for R2G,LS.

• Has been applied to mixtures of chemically diverse polymers.

Example:

Stratified sphere (2-layers):

PLS(q,0) =

~wA[PA(q,0)]1/2 + (1 – ~wA)R3

B[PB(q,0)]1/2 – R3A[PA(q,0)]1/2

R3B – R3

A

2

where PA(q,0) and PB(q,0) are the functions for spheres with radii RA and RB,

respectively.

• Reduces to a solvent-filled shell with thickness RB – RA if ~wA =0

• If ~ψ = 0, this form may be recast to give [R(q,c)/c]0 that exhibits min and

max similar to those for charged spheres, discussed later, but for an entirely

different reason.


41

Optically anisotropic scattering elements

[RSi(q,c)/c]0 = K'n2s(∂n/∂c)2

wMLS,SiPLS,Si(q,0)

For rodlike chains monodisperse in M, with x = Lq:

(1 + 4δ2/5)PVv(q,0) = p1(x) + δ2{(4/5)p3(x) – (2 – δ-1)m1(x) +

(9/8)m2(x)+ m3(x)}

PHv(q,0) = p3(x) + (5/8)sin2(ϑ/2)m2 (x)

The functions of x are known trigonometric functions:

p1(x) = (2/x2)[xSi(x) – 1 + cos(x)]

p2(x) = (6/x3)[x – sin(x)]

p3(x) = (10/x5)[x3 + 3xcos(x) – 3sin(x)]

m1 = p1 – p2

m2 = 3p1 – p2 –p3

m3 = p3 – p2


42

Scattering Beyond the RGD Regime

• Numerical methods are available to compute the scattering for a number of particleshapes, beyond the scope here.

• PVv(q,0) may be calculated for homogeneous spheres using the Mie theory as

functions of ~α and ñ.


43

Fraunhofer Limit:

For both ñ >> 1 and a phase shift magnitude ~α|ñ – 1| > 10, the angular dependence is

independent of ñ, and is the same for absorbing and nonabsorbing particles.

Spheres:

lim R/λ >> 1

PVv(q,0) = {2J1(~α sin(ϑ))/~α sin(ϑ)}2

lim R/λ >> 1

[RVv(q,c)/c]0 ∝Σ νnν~α

6ν{2J1(~αν sin(ϑ))/~αν sin(ϑ)}2

Cylinders of length Lcyl and radius R, with Lcyl >> R >> λ,

lim R/λ >> 1

PVv(q,0) = sin2(~αϑ))/(~αϑ)3 Note: No dependence on L

Extrema pattern similar for the functions for spheres and cylinders


44

Data on Hollow Spherical Shells


45

Scattering from a dilute solution

R(q,c) = KopcM S(q,c)

S(q,c) = P(q,c)F(q,c)

F(q,c) = 1 – cB(c)P(q,c)Q(q,c); F(q,c)-1 = 1 + cΓ(c)P(q,c)H(q,c)

Γ(c) = B(c)/[1 – cB(c)]

H(q,c) = Q(q,c) [1 – cB(c)]

[1 – cB(c)P(q,c)Q(q,c)]

B(c) = M~B(c)

~BLS(c)QLS(q,c) =Σν

C

Σµ

C

wνwµ~ψν

~ψµ MνMµPν(q,c)Pµ(q,c)~Bνµ(c)Qνµ(q,c)

[MLSPLS(q,c)]2


46

Scattering in the RGD regime at zero scattering angle:

~BLS(c) = MLS-2 Σ

ν

C

Σµ

C

wνwµ~ψν

~ψµ MνMµ~Bνµ(c)

For a solute with optically identical scattering elements,

~BLS(c) = M-2w Σ

ν

C

Σµ

C

wνwµ MνMµ~Bνµ(c)

For a dilute solution, ~BLS(c) may be expanded in a Taylor series and used to

compute ΓLS(c):

ΓLS(c) = 2A2,LSMw + 3A3,LSMwc + …


47

KopcR(q,c) = M-1

w{1 + cΓLS(c)}

= M-1w{1 + 2A2,LSMwc + 3A3,LSMwc2 + …}

By comparison

aLS(c) = RH,LS{1 – [(2 – k2)(A2M/[η]) – k1][η]c + …}

ΠRTc = M-1

n{1 + A2,ΠMnc + A3,ΠMnc2 + …}

A2,LS = A2,Π, etc., only for a monodisperse solute


48

Special case:

Solution with A2 >> 0: Put A3,LSM = ϒ3(A2,LSMw)2

KopcR(q,c) = M-1

w{1 + 2A2,LSMwc + 3ϒ3(A2,LSMwc)2 + …}

Kopc

R(q,c) 1/2

= M-1/2w {1 + A2,LSMwc + [(3ϒ3 – 1)/2](A2,LSMwc)2 + …}

Monodisperse spheres interacting through a hard-core potential:

ϒ3 = 5/8; (3ϒ3 – 1)/2 = 7/16 ≈ 0.44


49

Representative behavior for a flexible chain in a 'very good solvent'


50

Heterodisperse solute, identical optically isotropic scattering elements

A2,LS = M-2w Σ

ν

C

Σµ

C

wνwµ MνMµA2,νµ ≈ kA2M-γ

W ΩLS

By comparison, ,

ΠMn

RTc = 1 + A2,ΠMnc +…

A2,Π = Σν

C

Σµ

C

wνwµ A2,νµ ≈ kA2M-γ

n ΩΠ

A2,LS and A2,Π may be correlated with Mw and Mn, resp., if the Ω are close to

unity


51

γ = 1/5 (Good solvent).

ΩLS ∝ A2,LS/M-γW (curves 2 and 3) and ΩΠ ∝ A2,Π/M-γ

n (curves 1 and 4)

1 and 2: MνMµA2,νµ = kA2[M(2-γ)/3

ν + M(2-γ)/3µ ]3/2

3 and 4: A2,νµ = (A2,ννA2,µµ)1/2


52

Optically diverse, isotropic scattering elements

A2,LS = M-2w Σ

ν

C

Σµ

C

wνwµ ~ψν

~ψµ MνMµA2,νµ

• Some work to study the "cross-terms" as a function of composition for

mixtures of homopolymers, but theory is lacking.

Optically identical, anisotropic scattering elements

KcRVv(0,c) =

1

M(1 + 4δ2/5) {1 + 2

1 – δ2/10

1 + 4δ2/5 A2M c + …}

KcRHv(0,c) =

53Mδ2 {1 – A2M c/4 + …}

with the latter limited to rodlike chains.

Note the weak dependence of Kc/RHv(0,c) on c.


53

Upper:[Kc/RVv(0,c)]1/2 for solutions of cis-PBO.Middle:Kc/RHv(0,c) for solutions of cis-PBO.Lower:[Kc/RVv(0,c)]1/2 ab-PBO.

Upper:[Kc/RVv(ϑ,c)]0 for solutions of cis-PBO.Middle:[Kc/RHv(ϑ,c)]0 for solutions of cis-PBO.Lower:[Kc/RVv(ϑ,c)]0 for solutions of ab-PBO.


54

Concentrated and moderately concentrated solutions

at zero scattering angle

KopcMR(q,c) = 1 + cΓ(c)

With this expression, for a monodisperse solute,

cΓ(c) =MRT∂ Π∂c – 1

From Π for monodisperse, optically homogeneous spheres interacting through a

hard-core potential:

cΓ(c) = ϕ 8 – 2ϕ + 4ϕ2 – ϕ3

(1 – ϕ)4 ≈ ϕ8 – 2ϕ (1 – ϕ)4

where the volume fraction ϕ = (4π/3)NAR3c/M for spheres of radius R, andA2Mc = 8ϕ


55

Flory-Huggins theory of concentrated polymer solutions:

ΠV1/RT = –{ln(1 – ϕ) + (1 – 1/r)ϕ + χϕ2}

Γ(c) =v2

2MV1

1

1 - ϕ – 2χ – ϕ∂χ∂ϕ

χ is a constant in FH theory, but it depends on ϕ in experiment


56

Data for various concentrated

polymer solutions over a range of

M:

Empirical fit:

χ = χ1 + χ2ϕ

√1 – ϕ

χ1 = 1/2

Circle, PStyr/CHx;

Square, PDMS/MEK;

Diamond, PIB/Bz


57

Γ(c) =v2

2MV1

1

1 – ϕ – 2χ1 – χ2ϕ

2 + 1 – (1 + n)ϕ(1 – ϕ)1+n


58

Moderately concentrated solutions

Rodlike chains, for c < cmesophase transition

Γ(c) = 2A2M + 3A3Mc

Solutions of poly(benzyl glutamate):


59

A similar truncation appears to hold for flexible chain polymers under Flory

Theta conditions: cΓ(c) ≈ 3A3Mc2

The data are for polystyrene of

various M, under Flory Theta

conditions.

Line has slope 2


60

"Scaling behavior" for moderately concentrated polymer solutions in "good

solvents":

For a monodisperse solute, with ϒ3 = A3Mc/(A2Mc)2:

ΠM/RTc ≈ 1 + A2Mc{1 +(ϒ3/p)A2Mc}p

KcM/R(0,c) ≈ 1 + 2A2Mc{1 + (3ϒ3/4p)(2A2Mc)}p

p = (4 – 3ε)/(3ε – 2) ≈ 1/4 is determined by the stipulations that:

• A2M2/NAR3

G tends to a constant, with R2G ∝ Mε and ε ≈ 6/5, and

• Π should not depend on M for such a system in moderately concentrated

solutions.


61

Data on various polymers in systems with A2 > 0


62

Scattering for arbitrary concentration and q

KopcMR(q,c) =


Dilute to low concentrations

KopcMR(q,c) =

1P(q,0) + cΓ(c)H(q,c)

H(q,c) =2ψ2W2(q)c + {3ψ3W3(q) + 4[P(q,0)W2(q)2 – W3(q)]ψ2

2}c2

2ψ2c + 3ψ3c2 + …

ψj = AjM(M/NAR3G)j-1

are dimensionless virial coefficients

In the "Single-Contact" approximation of Zimm, H(q,c) = 1, and Γ(c) is given

by the virial expansion discussed above, giving rise to the "Zimm plot".


63

Moderately Concentrated and Concentrated Solutions/Suspension

KopcMR(q,c) =


For moderately concentrated spheres interacting through a hard-core potential,

P(q,c) = P(q,0), and a first-order theory, correct to 3-body interactions gives

H(q,c) = P(2q,0)1/2/P(q,0)

To a first-approximation for polymer solutions, one might adopt the same form,

but use P(q,c) for the random-flight chain, calculated with the value of R2G

appropriate for the concentration of interest.

Mean-field theory for concentrated polymer solutions gives H(q,c) ≈ 1


64

Spheres: Hard-core potential

Upper:

Heavy curve is for 3-body

approximation;

Lighter curves are for the

all-interactions approximation;

these are for ϕ = 0.05, 0.1, 0.2,

0.3 and 0.4 (left to right)

Lower:

Curves 1 and 2 are for the 3-

body and full interaction

approximations, resp.

1

2

0.1 0.2 0.3 0.400

10

20

30

ϕ

cΓ(c)

(a)

0 0.5 1.0 1.5 2.0 2.5 3.0 3.50

0.2

0.4

0.6

0.8

H(q,c)

qR

1.0


65

Data on polystyrene at Flory Theta

conditions, c increasing from bottom to

top from [η]c = 0.60 to 6.54

Data on poly(α-methyl styrene in a good

solvent, c increasing from bottom to top.


66

Angular Dependence

b(q,c)2 = R(0,c){∂R-1(q,c)/∂q2}

b(q,c)2 = ξP(q,c)2 + ξH(q,c)2

ξP(q,c)2 = {1/[1 + cΓ(c)]}{∂P-1(q,c)/∂q2}

ξH(q,c)2 = {cΓ(c)/[1 + cΓ(c)]}{∂H(q,c)/∂q2}

For dilute and concentrated polymer solutions, H(q,c) ≈ 1

For moderately concentrated polymer solutions, ∂H(q,c)/∂q2 ≤ 0


67

Solutions of Poly(benzyl glutamate Upper: Polystyrene under Flory Theta

Conditions

Lower: Various polymers in Good Solvents

(PDMS, Psty, PMMA)


68

Behavior for large u (as in neutron scattering):

u >>1: 1/P(q,0) ≈ 1/2[1 + u + O(1/u)]

2KOPcMR(q;c) ≈ 1 + u + 2cΓ(c)H(q;c)

With the assumption that H(q;c) ≈ 1 in this regime, the Ornstein-Zernike form emerges

2KOPcMR(q;c) ≈ [1 + 2cΓ(c)]{1 + (ξq)2}; ξ2 = R2

g/[1 + 2cΓ(c)]

-1.0

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

-3.0 -2.0 -1.0 0.0 1.0 2.0

log cΓ( c )

log

(ξ/

Rg

)

172K233K600K600K950K2,000K1/√(1+2cG(c))1/√(1+cG(c))


69

Charged Spheres:

R(q,c)KopcM = S(q,c) =

P(q,c)1 + cΓ(c)P(q,c)H(q,c) ≈

1cΓ(c)H(q,c)

Debye screening length κ−1 provides a

measure of the range of the electrostatic

interactions, where

κ−1 = (8πNALBIo)−1/2

where LB = e2/εkT is the Bjerrum length,

with ε the dielectric strength ( LB/nm ≈

57/ε at 25°C, or LB ≈ 0.7 nm for water),

and Io is the ionic strength (Io = Σνi2mi/2,

with νi and mi the molarity and charge of

species i).


70

ln[g(2)(τ; q,c) – 1]1/2 =ln[fC]1/2 – K(1)(q,c)τ + 12! K

(2)(q,c)τ2 + …

limq=0 K(1)(q,c) = DM(c) q2

Data for a variety of Mand solvents

Upper curve:Good solvents

Lower CurveFlory Theta Solventconditions


71

Non Ergodic Behavior, e.g., Many Gels

g(2)t (τ; q,c) = 1 + x(1 – x)f1/2

C |fE(τ,q) – fE(∞,q)| +

x2fC[f2E(τ,q)(τ,q) – f2

E(τ,q)(∞,q)]

Ensemble-averaged normalized concentration fluctuation correlation function:fE(τ,q) = 〈∆c(τ,q)∆c(0,q)〉E/〈∆c2〉E; 1 ≥ fE(τ,q) ≥ 0

Ergodic system, fE(∞,q) = 0; nonergodic medium, 1 ≥ fE(∞,q) ≥ 0

If the gel network with spatial fluctuations in the network junctions:

x=〈n(q)〉E/〈n(q)〉t; fE(∞,q) = exp(–q2〈δ2〉)

〈δ2〉 is the mean square amplitude of the spatial fluctuation of the constraint


72

Intermolecular association in polymer solutions

• Association of small molecule surfactants to form wormlike micelles, usually

an equilibrium process

• Association of linear flexible chain polymers, often forming meta-stable

states, sometimes in the form of quasi-randomly branched structures, and

sometimes as more dense, colloidal particles.


73

Intermolecular association

in micelles

Aqueous solutions of

wormlike micelles of

hexaoxyethylene dodecyl

ether

association-dissociation

equilibria for wormlike

micelles suggest that for a

range of c, Mw might

increase as Mw ∝ c1/2

Dashed lines: Slope –1/2Solid lines: Slope 2


74

Metastable association

The analysis of metastable behavior is sometimes facilitated by an approximate

representation with a few 'pseudo components' (often two or three), each of which

dominates the scattering over a limited range of q, with M, A2 and P(q,c):


75

R(q,c) = µΣRµ(q,c)

≈ K µΣ

Mc

P(q,c) + 2A2Mc µ,c


76

Thanks for your patience and attention!


77

Low concentrations: The third virial coefficient (isotropic elements)

~BLS(c) ≈ MLS-2 Σ

ν

C

Σµ

C ~ψν

~ψµ

wνwµMνMµ~Bνµ

0 + Σκ

C

wκ [~Bνµκ

0 – Mκ~Bνκ

0 ~Bµκ0 ]c

Optically identical scattering elements:

A3,LS ≈Mw-2 Σ

ν

C

Σµ

C

Σκ

C

wνwµwκMνMµA3,νµκ

– (4/3)Mw-3 Σ

ν

C

Σµ

C

Σκ

C

Σσ

C

wνwµwκwσMνMµMκMσ[A2,νκA2,µκ – A2,νκA2,µσ]

where A2,νµ = ~Bνµ0 /2 (as above) and A3,νµκ = ~Bνµκ

0 /3.

• Very little work to explore the terms in this expression.


78

1. Berry, G.C., Total intensity light scattering from solutions of macromolecules, in Soft Matter: Scattering, Manipulation &Imaging, R. Pecora and R. Borsali, Editors. 2005, In Presss.

2. Berry, G.C., Light Scattering, Classical: Size and Size Distribution Classification, in Encyclopedia of Analytical Chemistry,R.A. Meyers, Editor. 2000, John Wiley & Sons Ltd: New York. p. 5413-48.

3. Berry, G.C. and P.M. Cotts, Static and dynamic light scattering, in Experimental methods in polymer characterization, R.A.Pethrick and J.V. Dawkins, Editors. 1999, John Wiley & Sons Ltd.: Sussix, UK. p. 81-108.

4. Casassa, E.F. and G.C. Berry, Light scattering from solutions of macromolecules. Tech. Methods Polym. Eval., 1975. 4,Pt. 1: p. 161-286.

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