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The Importance of Online Viscometer in the Age of Multi-Angle Light Scattering
Stepan Podzimek1,2,3 1Wyatt Technology Europe GmbH
2SYNPO, Pardubice, Czech Republic 3University of Pardubice, Republic
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Outline
• What is (and what is not) the intrinsic viscosity
• Applications of online viscometer
– Universal calibration
– Mark-Houwink equation
– Mark-Houwink plot
– Flory-Fox equation
– Hydrodynamic radius
• Online viscometer for the structural studies
• Online viscometer for fluorescent polymers
• Final remarks
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Intrinsic Viscosity
• Intrinsic viscosity is not viscosity!!!
• Viscosity is the proportionality constant in the Newton´s law for fluids
• Newton´s law for fluids
𝜏 = 𝜂𝑑𝑣
𝑑𝑦
– τ = shear stress (N/m2)
– η = viscosity (Pa.s)
– dv/dy = velocity gradient (s-1)
Velocity profile in liquid placed between moving and stationary plate.
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Temperature Dependence of Viscosity
RT
E
e
viscosity E activation energy of flow T temperature R universal gas constant
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Temperature Dependence of Intrinsic Viscosity
From T. S. Rushing and R. D. Hester, J. Appl. Polym. Sci., 89, 2831 (2003).
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Intrinsic Viscosity
• Intrinsic Viscosity
– Fundamental property characterizing polymer chain conformation and thermodynamic quality of the solvent = volume fraction of polymer in solution c = concentration in mg/mL
• Specific viscosity
η = viscosity of polymer solution η0 = viscosity of pure solvent t = time needed for solution or solvent to flow from one mark to the other
lim0 c
sp
c
5.2sp
10
0
0
0
relativespt
tt
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Classical Determination of Intrinsic Viscosity
• Determination of intrinsic viscosity of polystyrene NIST 706 in THF, 25 °C.
• Data obtained by Ubbelohde capillary viscometer.
0.0 2.0x10-3
4.0x10-3
6.0x10-3
8.0x10-3
1.0x10-2
80
85
90
95
100
105
110
115
sp/c
(m
L/g
)
c (g/mL)
Equation y = a + b*x
Weight No Weighting
Residual Sum of Squares
4.00547
Adj. R-Square 0.98755
Value Standard Error
BIntercept 85.3 0.8
Slope 2764.3 138.6
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Online Determination of Intrinsic Viscosity
IPDP
PIP
Psp
2
4
c
sp
c
0lim
P imbalance pressure across the bridge
IP pressure drop from inlet to outlet
sp specific viscosity
14 16 18 20 22 24
10
100
Volume (mL)
Intr
insic
Vis
co
sity (
mL
/g)
400
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Applications of Intrinsic Viscosity in Polymer Science
• Molar mass from Mark-Houwink equation
• Hydrodynamic radius from [η] and molar mass
• Root mean square (RMS) radius (radius of gyration) from [η] and molar mass
• SEC with universal calibration for the determination of molar mass distribution
• Polymer structure from Mark-Houwink plot: log [] vs. log M
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Mark-Houwink Equation
Traditional way of molar mass determination
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Mark-Houwink Equation
• K, a = constants of Mark-Houwink equation for given polymer, solvent and temperature; M = molar mass
• Traditional method of the determination of molar mass
• Viscosity average (Mv) close to weight-average (Mw)
– NIST 706, [η] = 85.3 mL/g: Mv = 244,000 g/mol; Mw ≈ 270,000 g/mol
– K = 0.0117 mL/g, a = 0.717 (THF, 25 °C, M. Kolinsky and J. Janca, J. Polym. Sci., Polym. Chem. Ed., 12, 1181 (1974).
aKM
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Molar Mass from Mark-Houwink Equation
Molar mass versus elution volume plots of linear polystyrene by MALS and calculated from Mark-Houwink equation.
cK
RM
*
0
16 18 20 22 24
104
105
106
Volume (mL)
Mo
lar
Ma
ss (
g/m
ol)
MALS vs. Mark-Houwink
a
KM
1
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Hydrodynamic Radius
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Hydrodynamic Radius
Vh = hydrodynamic volume M = molar mass NA = Avogadro´s number
• Hydrodynamic radius (Rh)
– Rh is radius of hydrodynamically equivalent sphere, i.e., a hypothetical sphere having the product of intrinsic viscosity and molar mass the same as the polymer molecule.
– It can be used as an alternative size parameter to RMS radius.
– Viscometry is often more sensitive than a golden standard dynamic light scattering.
3
1
5.2 4
3
A
hN
MR
A
hN
MV
5.2
3
3
4RVsphere
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Hydrodynamic Radius: DLS versus Viscometry
Conformation plots of polystyrene based on hydrodynamic radius determined by online viscometer and dynamic light scattering.
105
106
10
Hyd
rod
yn
am
ic R
ad
ius (
nm
) V
isco
sity /
DL
S
Molar Mass (g/mol)
3
80
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Flory-Fox Equation
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Flory-Fox: RMS Radius by Viscometry … or [η] by MALS
• T. G. Fox and P. J. Flory, J. Am. Chem. Soc., 73, 1904 (1951)
• O. B. Ptitsyn and Yu. E. Eizner, Sov. Phys. Tech. Phys., 4, 1020 (1960)
3
12
)86.263.21(1086.2 221
a
3
1
6
1
MR
= Flory-Fox constant
a = exponent of Mark-Houwink equation
[η] in dL/g, R in cm
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RMS Radius by MALS vs. Viscometry
20 40 60 80 100
20
40
60
80
100R
MS
Ra
diu
s (
nm
) b
y F
lory
-Fo
x
RMS Radius (nm) by MALS
Intercept = 1.1
Slope = 0.95
Correlation of RMS radius by Flory-Fox with MALS (Polystyrene, THF, 25 °C).
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Intrinsic Viscosity by MALS
From S. Podzimek, J. Appl. Polym. Sci., 54, 91 (1994). PMMA, THF, 25 °C, a = 0.72.
𝜂 =632 𝑅3Φ
𝑀
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Universal Calibration
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Universal Calibration Z. Grubisic, P. Rempp, H. Benoit, Polymer Letters, 5, 753 (1967)
... log bVaM
[] intrinsic viscosity M molar mass
20 22 24 26
106
107
108
109
[]M
Elution Volume (5 mL counts, THF)
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Universal Calibration
• Conventional calibration (left) versus universal calibration (right).
• Once the universal calibration is established typically by narrow standards, the molar mass of polymer is calculated from the intrinsic viscosity determined by viscometer and universal calibration curve.
14 16 18 20 22 24
4
5
6
7
8
9
log
([
]M)
Volume (mL)
Linear PS
Branched PIBMA
PBZMA
Branched PBZMA
Branched PS
Phenoxy resin
14 16 18 20 22 2410
3
104
105
106
107
Mo
lar
Ma
ss (
g/m
ol)
Volume (mL)
Linear PS
Branched PIBMA
PBZMA
Branched PBZMA
Branched PS
Phenoxy resin
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Universal Calibration vs. MALS: Band Broadening in SEC
Ideal and experimental broadened chromatogram and polydispersity within elution volume slice.
10 12 14 16 18
0.0
0.1
0.2
0.3
0.4
Re
sp
on
se
Elution Volume
13.6 13.8 14.0 14.2 14.4
0.0
0.1
0.2
0.3oooo
oooo
oooo
oooo
oooo
oooo
oooo
oooo
oooo
oooo
oooo
oooo
oooo
oooo
oooo
oooo
oooo
oooo
oooo
oooo
oooo
oooo
oooo
oooo
oooooooo
oooo
oooo
oooo
Response
Elution Volume
14/2
oooo
oooo
oooo
oooo
oooo
oooo
oooo
oooo
oooo
oooo
oooo
oooo
oooo
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Universal Calibration vs. MALS
Polydisperse slice
– Slice polydispersity is always larger in case of branched polymers compared to
linear polymers
– Molar masses are not Mi
– MALS measures weight-average Mw,i
– SEC-VIS-UC measures number-average Mn,i (Balke, S. T., Mourey, T. H., and
Harrison, C. A. J. Appl. Polym. Sci. 1994, 51, 2087)
– Molar mass versus elution volume plots from MALS and UC should be parallel
and the difference should indicate the slice polydispersity
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Universal Calibration vs. MALS: Number-Average Mn
103
104
105
106
103
104
105
106
103
104
105
106
0
20
40
60
80
100
120
140
Mn(g
/mol)
, U
C
Mn(g/mol), MALS
Mn(%
), U
C v
s M
AL
S
Mn(g/mol), MALS
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Universal Calibration vs. MALS: Weight-Average Mw
103
104
105
106
107
103
104
105
106
107
103
104
105
106
107
60
80
100
120
140
Mw (
g/m
ol)
, U
C
Mw (g/mol), MALS
Mw (
%),
UC
vs
MA
LS
Mw (g/mol), MALS
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Universal Calibration vs. MALS: Molar Mass Plots
10 12 14 1610
3
104
105
106
Elution Volume (mL)
Mo
lar
Ma
ss (
g/m
ol)
10 12 14 1610
2
103
104
105
106
107
Elution Volume (mL)
Mola
r M
ass (
g/m
ol)
Molar mass versus elution volume plots from MALS and universal calibration for linear polystyrene (left) and branched polystyrene (right).
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Universal Calibration vs. MALS: Molar Mass Distribution
Cumulative molar mass distributions from MALS and universal calibration for linear polystyrene (left) and branched polystyrene (right).
104
105
106
0.0
0.2
0.4
0.6
0.8
1.0
Cu
mu
lative
We
igh
t F
ractio
n
Molar Mass (g/mol)
Mn = 110,000 g/mol
Mw = 270,000 g/mol
Mn = 73,000 g/mol
Mw = 244,000 g/mol
Mn (MO) = 137,000 g/mol
104
105
106
107
0.0
0.2
0.4
0.6
0.8
1.0
Cu
mu
lative
We
igh
t F
ractio
n
Molar Mass (g/mol)
Mn = 240,000 g/mol
Mw = 938,000 g/mol
Mn = 35,000 g/mol
Mw = 548,000 g/mol
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Universal Calibration vs. MALS: Molar Mass Plots
Molar mass versus elution volume plots from MALS and universal calibration for fluorescent polymer (lignin). RI chromatogram is shown here.
12 14 16 18
102
103
104
105
106
Elution Volume (mL)
Mola
r M
ass (
g/m
ol)
Mn = 850,000
Mn = 1800
Mw = 1,650,000
Mn = 13,000
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Mark-Houwink Plot Characterization of Polymer Structure
log─log [η] versus molar mass
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Molecular Structure from Mark-Houwink Plot
Exponent of Mark-Houwink equation
– Linear macromolecules in thermodynamically good solvents: a ≈ 0.7
– Linear macromolecules in thermodynamically poor solvents: a ≈ 0.5
– Oligomers: a ≈ 0.5
– Hard spheres: a ≈ 0
– Extended chains: a ≈ 0.8 to ≈1.5
– Linear polymers have linear MH plots
– Curved plots indicates branching
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Chain Stiffness from VIS/MALS Chromatograms
SEC- MALS-VIS-RI chromatograms of PS (left) and cellulose tricarbanilate (right) in THF.
9 10 11 12 13 14 15 16
0.0
0.5
1.0
Re
lative
Sca
le
Volume (mL)
MALS
VIS
RI
9 10 11 12 13 14 15 16
0.0
0.5
1.0
MALS
VIS
RI
Re
lative
Sca
leVolume (mL)
a < 1 a > 1
cS
McS
McS
RI
a
VIS
MALS
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High-Molar-Mass Highly Branched Fractions
• Chromatograms from MALS, viscometer and infrared detector for NIST 1476 branched PE. • Mark-Houwink exponent a → 0 makes response of viscometer proportional to concentration.
14 16 18 20 22 24
0.0
0.2
0.4
0.6
0.8
1.0R
ela
tive
Sca
le
Volume (mL)
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Molecular Structure from Mark-Houwink Plot
• Mark-Houwink plots of epoxy resin, linear polystyrene, linear poly(methyl methacrylate, linear poly(benzyl methacrylate), linear poly(iBuPOSSMA) and star-branched poly(isobutyl methacrylate).
103
104
105
106
107
[]
(mL
/g)
Molar Mass (g/mol)
a ≈ 0.71
a ≈ 0.69
a ≈ 0.50
a ≈ 0.70
a ≈ 0.69
a ≈ 0
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Molecular Structure from Mark-Houwink Plot
Mark-Houwink plots of alginate, xanthan, pullulan, and dextran in aqueous NaNO3.
104
105
106
10
100
1000
slope = 0.62
slope = 1.49
[]
(mL
/g)
Molar Mass (g/mol)
slope = 1.09
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Branching from Mark-Houwink Plot
Mark-Houwink plots of linear polystyrene and branched polystyrene prepared by copolymerization of styrene with various amounts of divinylbenzene. The degree of branching growths in the order of blue, magenta and green.
105
106
10
100
Intr
insic
Vis
co
sity (
mL
/g)
Molar Mass (g/mol)
400
a = 0.7
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Branching from Mark-Houwink Plot
Mark-Houwink plots of linear PLGA 50/50 and branched PLGA containing 0.5 %; 3 % and 5 % dipentaerythritol.
103
104
105
10
20
[]
(mL
/g)
Molar Mass (g/mol)
2
a ≈ 0.56
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Micro-Viscometer
Advanced Polymer Chromatography, APC
μ-SEC
μ-DAWN, UT-rEX, μ-ViscoStar
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SEC-MALS versus APC-μMALS: Molar Mass Distribution
104
105
106
0.0
0.2
0.4
0.6
0.8
1.0C
um
ula
tive W
eig
ht F
raction
Molar Mass (g/mol)
APC @ 0.5 mL/min
APC @ 0.9 mL/min
SEC @ 1 mL/min
Cumulative molar mass distribution of NIST 706 polystyrene.
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APC-μMALS- μVIS: Molar Mass and Intrinsic Viscosity Averages
Set-up Mn
(103 g/mol) Mw
(103 g/mol) Mz
(103 g/mol) [η]
(mL/g)
SEC-MALS-VIS* 1 mL/min
110 ± 1 269 ± 1 424 ± 4 87.8 ± 0.1
SEC-μMALS-μVIS** 0.9 mL/min
127 ± 7 259 ± 3 410 ± 3 89.4 ± 0.5
SEC-μMALS-μVIS** 0.5 mL/min
116 ± 4 263 ± 1 426 ± 2 91.1 ± 0.3
*Agilent 1100 HPLC; PLgel Mixed-C 300 × 7.5 mm 5 μm columns; HELEOS, T-rEX, ViscoStar III
**Waters Acquity APC; 900 Å, 4.6 × 150 mm, 2.5 µm; 450 Å, 4.6 × 75 mm, 2.5 µm; 125 Å, 4.6 × 150 mm, 2.5 µm; and 45 Å, 4.6 × 150 mm, 1.7 µm; μDAWN, UT-rEX, μViscoStar
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SEC-MALS versus APC-μMALS: Mark-Houwink Plot
104
105
106
10
100
APC @ 0.5 mL/min
SEC @ 1 mL/min
Intr
insic
Vis
cosity (
mL/g
)
Molar Mass (g/mol)
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Summary
• Universal calibration is valid, but less accurate and robust than MALS.
• UC for highly branched polymers (Mn) and fluorescent polymers.
• The major application of online viscometer is in the area of structural studies.
• Mark-Houwink plot provides information about polymer chain conformation and branching.
• Intrinsic viscosity allows calculation of hydrodynamic radius with higher sensitivity than DLS.
• Wyatt Technology ViscoStar® III is a successor of the previous two generations of the instrument with several advanced features.
– Bridge balance autotune
– Faster pressure sensors (less peak broadening)
• Wyatt Technology μ-ViscoStar is compatible with APC