ISPAC- 2005 Carnegie Mellon University 1 Developments and Trends in Light Scattering on Macromolecular Solutions Guy C. Berry Carnegie Mellon University Pittsburgh, PA www.chem.cmu.edu/berry Carnegie Mellon Mellon College of Science Department of Chemistry
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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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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
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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)
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RHv(q,c) and RVv(q,c) are given by (Using Isolution – Isolvent):
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.
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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
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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.
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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|〉
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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
)
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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
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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):
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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
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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)
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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
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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
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Optically diverse scattering elements
Limiting to two scattering elements, A and B, with ~wA = 1 – ~wB = wA~ψA/
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
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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.
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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
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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):
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R(q,c) = µΣRµ(q,c)
≈ K µΣ
Mc
P(q,c) + 2A2Mc µ,c
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Thanks for your patience and attention!
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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.
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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.