Skript zur Vorlesung “Einführung in die makromolekulare Chemie

Skript zur Vorlesung
“Einführung in die makromolekulare Chemie – Physikalische
Chemie der Polymeren”
Prof. Dr. Manfred Schmidt, PD Dr. Wolfgang Schärtl, HD Dr. Michael Maskos
Universität Mainz, Institut für Physikalische Chemie
Version 2.1, 23.04.08
Inhalt
1.
2.
3.
4.
5.
6.
7.
Kettenstatistik
Thermodynamics
Molekulargewichtsverteilungen, Mittelwerte
GPC
Lichtstreuung
MALDI-TOF MS
Feld-Fluss Fraktionierung
Einige der wesentlichen Unterschiede der Polymere im Vergleich zu niedermolekuleren
Verbindungen sind auf ihrer Konformation, sowie ihre Molekulargewichtsverteilung
zurückzuführen. Beide Aspekte werden im Folgenden anhand von Modellen und Beispielen
betrachtet.
1. Kettenstatistik
1.1. Gauss-Kette, Kuhn-Kette
Eine Polymerkette besteht aus einer Vielzahl von gleichen Segmenten (Monomere), die
vereinfacht über das Modell der Gauss-Kette beschrieben werden kann:
l
G
lj
G
li
i
G
R
G
R
Modell einer Gauss-Kette (freie Drehbarkeit)
Für den sogenannten Fadenendenabstand bzw. für das entsprechende, für eine statistische
Beschreibung relevantere, mittlere Fadenabstandsquadrat ergibt sich dann:
G N G
R = ∑ li
i =1
GG N N GG
R 2 = RR = ∑ ∑ l i l j
i
j
Mittelwert über sehr viele Konformationen (Gleichgewicht):
N N
GG
R 2 = ∑∑ li l j ;
i
GG
li l j = l 2 [− cosθ
]
i− j
j
Statistische Verteilung der Bindungswinkel:
cosθ = 0
⇒
GG
li l j = 0 , i ≠ j
R 2 = N ⋅ l 2 , weil für N Terme i = j
(wird auch als “Irrflug”-Modell bezeichnet)
Das Kuhn- Modell
Reale Polymere können nicht vollständig durch das Gauss-Kettenmodell beschrieben werden.
Allerdings kann das Modell bei Verzicht auf die Detailstruktur als Vereinfachung dienen, was
von Kuhn entsprechend beschrieben wurde durch:
lk = Kuhnlänge
Nk = Zahl der Kuhnsegmente
R 2 = l k ⋅ N k ≠ l 2 ⋅ N , wenn θ und φ nicht statistisch.
2
Aber: l k ⋅ N k = l ⋅ N = L
Konturlänge
Die unterschiedlichen Modelle lassen sich zueinander ins Verhältnis setzen, um z.B.
Informationen bzgl. der realistischeren, dafür aber physikalisch schwerer deutbaren Modelle
gegenüber den physikalischeren, aber schwerer beschreibbaren Modellen zu erhalten. Hier für
dient z.B. das
Charakteristisches Verhältnis
R2
N ⋅l2
=
lk
l k2 ⋅ N k l k
= ≡ C∞
N ⋅l2
l
l
R
2
2
= lk ⋅ N k
G
R
R2 ≥ l 2 ⋅ N
Im Folgenden einige Beispiele, sowie die statistische Herleitung des Gauss-Modells:
Fragestellung
Mit welcher Wahrscheinlichkeit befinden sich zwei Segmente eines Makromoleküls im
Abstand R?
Ideales Knäuel:
3
3 R2
⎛ 3 ⎞ 2
W(R) = ⎜
exp(
−
)
2 ⎟
2 nl2
⎝ 2πnl ⎠
Gauss
2.Moment der Verteilung liefert <R2> (vgl. Übungen).
∞
R2 =
∫ W ( R) ⋅ R
2
4πR 2 dR
= nl 2
0
∞
∫W ( R) ⋅ 4πR dR
2
0
Verknüpfung mit der Kontourlänge eines Polymers:
Gauss-Modell
Polymer mit Polymerisationsgrad N:
R2 = N ⋅ l2 ⇒
L = N ⋅l
Kuhn- Modell
2
R 2 = lk ⋅ N k = lk ⋅ L
L = lk ⋅ N k
C∞ =
R2
N ⋅l
2
=
lk ⋅ L lk
=
l⋅L
l
in der Literatur:
Zahl der Bindungen
unterschiedlich
Zahl der Monomere
Ausgehend von der oben beschriebenen Gaussverteilung lässt sich die entropische
Rückstellkraft eine Polymerkette beschreiben (vgl. Übungen):
Entropie:
S = k B ln Ω mit Ω > 1, Anzahl der Konfigurationen; Ω = A W(R)
Mit der normierten Gauss-Verteilung:
3
⎛
⎞ 2
R2 − R2
3
⎟ exp ⎜ −
⎟
⎜ 2
R2
⎠
⎝
3 R2
3
R2 ) − k 2 + k
2 R
2
⎛
3
W(R) = ⎜
⎜ 2π R 2
⎝
S(R) = S(
⎞
⎟
⎟
⎠
R2
3
ΔS ( R) = k (1 − 2 )
2
R
Freie Energie:
ΔG = ΔH − TΔS ;
Sei ΔH ≠ f (R)
3
3
R2
ΔG(R) = ΔH − kT + kT ⋅ 2
2
2
R
Rückstellkraft eines elongierten Knäuels:
f=
∂ΔG 3kT
= 2 ⋅ R für R << L = N ⋅ l
∂R
R
Weitere Anwendung von W(R):
∞
1
R
=
∫W (R) ⋅ R
−1
4 π R 2 dR
0
∞
∫ W ( R ) ⋅ 4π R
2
dR
0
Diffusion:
D~
1
R
Höhere Momente R 4 , R 6 …
Im Folgenden ein Beispiel, um die entropischen Rückstellkräfte mittels Rasterkraftmikroskop
zu messen:
1.2. Das reale Valenzwinkel-Modell
GG
Mit li l j = l 2 [− cosθ
]
i− j
−l cos θ
G
l i +1
−l cosθ
G
li
G
li+ 2
θ
l 2 ( − cosθ )
θ
2
GG
li li +1 = l 2 (− cosθ )
G G
li +1li + 2 = l 2 (− cosθ )
GG
li li + 2 = l 2 (− cosθ ) 2
+1
90
-1
x
l
x = l ⋅ cos(180 − θ ) = −l cos(θ )
cos(180 − θ ) =
x
l
x = l ⋅ cos(180 − θ ) = −l cos(θ )
cos(180 − θ ) =
180
270
360
Allgemein:
R2
⎛
⎜
⎜
⎜
=⎜
⎜
⎜
⎜
⎝
GG
l1l1
GG
l 2 l1
GG
l3l1
GG
l1l 2
GG
l2l2
GG
l3 l 2
GG
l1l3
GG
l 2 l3
GG
l3 l3
...
G G
l N l1
...
G G
l N l2
...
G G
l N l3
...
...
...
...
GG
l1l N
GG
l2l N
GG
l3 l N
⎞
⎟
⎟
⎟
⎟
⎟
... ⎟
G G
lN lN ⎟
⎠
GG
GG
GG
GG
R 2 = Nl 2 + 2( N − 1) li li ±1 + 2( N − 2) li li ± 2 + 2( N − 3) li li ±3 + ... + +2 li li ±( N −1)
∟1) Irrflugkette 2)Valenzwinkelkette
GG
li li ± n = l 2 [− cosθ
N =∞
Man benutzt:
∑k
i
]
n
=
i =o
1
1− k
0 ≤ k ≤1
genau: s. unten und s.Flory S.16/17
N =∞
k
∑o ik i = (1 − k )2
0 ≤ k ≤1
und erhält für N → ∞:
(1 − cosθ )
R 2 = Nl 2
(1 + cosθ )
Beispiel Polymethylen : θ = 109.5° ⇒
cosθ = −
1
3
R 2 = 2 Nl 2
R2
2
= C∞ = 2 ; C∞exp = 6.7!! (Modell immer noch nicht in der Lage, die
Nl
experimentellen Daten zu beschreiben).
Die Valenzwinkelkette ist im Vergleich zur Gauss-Kette etwas aufgeweitet für θ > 90° da
cosθ < 0 .
Etwas detaillierter:
R
2
⎛ l11
⎜
⎜ ...
= ⎜ ...
⎜
⎜ ...
⎜l
⎝ 1N
...
...
...
l 22
l33
l 44
...
...
N −1
...
l N1 ⎞
⎟
... ⎟
... ⎟
⎟
... ⎟
l NN ⎟⎠
R 2 = Nl 2 + 2l 2 ∑ ( N − k )α k ;
k =1
α ≡ − cosθ
N −1
N −1
k =1
k =1
= Nl 2 + 2l 2 N ∑α k − 2l 2 ∑ kα k
α −α N
Es gilt: ∑ α =
1−α
k =1
N −1
α (1 − α N ) N α N
k
α
−
k
=
∑
(1 − α )2 (1 − α )
k =1
N −1
k
R 2 = Nl 2 + 2l 2 N
⎧α (1 − α N ) Nα N ⎫
α −α N
−
− 2l 2 ⎨
⎬
2
1−α
1−α ⎭
⎩ (1 − α )
α (1 − α N )
= Nl + 2l N
− 2l
1−α
(1 − α )2
N
2α ⎞
2⎛
2 α (1 − α )
= Nl ⎜1 +
⎟ − 2l
(1 − α )2
⎝ 1−α ⎠
N
2 ⎛ 1 − α + 2α ⎞
2 α (1 − α )
= Nl ⎜
⎟ − 2l
(1 − α )2
⎝ 1−α ⎠
N
2 ⎛1+ α ⎞
2 α (1 − α )
= Nl ⎜
⎟ − 2l
(1 − α )2
⎝1−α ⎠
2
α
2
2
N → ∞: 2. Term vernachlässigbar
R 2 = Nl 2
(1 − cosθ )
(1 + cosθ )
Weitere Verfeinerung des Modells:
Berücksichtigung des Raumwinkels (Drehwinkel um den Bindungswinkel) φ :
lim R 2 = Nl 2
N →∞
(1 − cos θ ) (1 + cos φ )
(1 + cos θ ) (1 − cos φ )
für cos φ << 1
cos φ = 0
frei drehbar (freely rotating chain)
cos φ = 1
all-trans
Berücksichtigung von gauche-trans Zustandsenergien (ungekoppelt) führt zu:
C∞ ≈ 3.2 für Polymethylen
Bezug zum charakteristischen Verhältnis:
lim C n ≡ C∞
n →∞
C∞ = lim
N →∞
R2
charakteristisches Verhältnis!
Nl 2
Charakteristisches Verhältnis
R2
C∞ =
Nl 2
( CH )
2
N
C ∞ = 6 .8
exp.
C∞ = 2
θ = 109 °
C ∞ = 3.7 θ = 109° , φ aus gauche/ trans Pot.Minima
π
cos φ =
∫e
0
−V ( φ ) / kT
π
∫e
cos φ
−V ( φ ) / kT
dφ
dφ
0
Unterschied zur Wirklichkeit:
Potentialbarriere Vg hängt ab von der Nachbarkonfiguration, d.h. Vgg ≠ Vtg etc.
Dies führte zum Modell von Flory: Rotational Isomeric State (RIS)
Zusammenfassung
Statistische Mechanik von Kettenmolekülen
Kettendimension beschreibbar durch:
G
Bindungslänge: l , Bindungsvektor: l
Valenzwinkel: θ
Konformationswinkel: φ
Zahl der Kettenglieder: N
Merke: all-trans Konformation
φ = 0 , denn: 1, 2, 3, 4: coplanar
φ = 180 , denn: 1, 2, 3, 4: sc, eclipsed
R 2 = nl 2
(1 − cos θ ) (1 + cos φ )
(1 + cos θ ) (1 − cos φ )
1) cosθ = 0
cos φ = 0 : Freely Jointed Chain (Irrflug)
2) cosθ ≠ 0
cos φ = 0 : Freely Rotating Chain (Valenzwinkelkette)
3) cosθ ≠ 0
cos φ ≠ 0 : Valenzwinkelkette mit behinderter Rotation
aber: cosφ = 1 ⇒
φ aus Zustandssumme
R 2 = ∞ wegen Grenzwert n → ∞
exakte Lösung für endliche n möglich
U
g+
π
cosφ =
∫e
0
−U (φ ) / kT
π
∫e
cosφ
t
g-
dφ
U, weiter oben verwendet: V
−U (φ ) / kT
dφ
0
trotzdem: U gg ≠ U fg Effekte übernächster Nachbarn
Knäue-Konformation in 2-D?
Flexibilitäten
E
gauche
trans
∆E
gauche
∆E
φ
-180
ΔE < kT :
ΔE >> kT :
-120
-60
0
60
dynamisch flexible Knäuel
starres Knäuel (“eingefroren”)
Konsequenz für die Materialeigenschaft?
“Entropieelastizität”
120
180
Wiederholung und weitere Modelle:
freie Bindungswinkel, freie Rotation
R2
0
= nl 2
feste Bindungswinkel, freie Rotation (Valenzwinkelmodel, freie Drehbarkeit)
R2
0
= nl 2
1 − cosθ
1 + cosθ
feste Bindungswinkel, behinderte Rotation
⎛ 1 − cos θ ⎞⎛ 1 + cos φ ⎞
R 2 = nl2 ⎜
⎜ 1 + cos θ ⎟⎜
⎟⎜ 1 − cos φ ⎟⎟
0
⎝
⎠⎝
⎠
cos φ = 0 ; frei drehbar
cos φ = 1 ; all-trans Kette (Grenzwertproblem lk, lp → ∞ Stäbchen!!)
Maximale Länge: LMax = ( N − 1)l ≈ Nl
⎛θ ⎞
Eff. Konturlänge: L ≅ N ⋅ l ⋅ sin ⎜ ⎟ = N e be
⎝2⎠
θ
Helix
L
Beschreibung der Kettensteifheit:
(1) Kuhnlänge lk
R2
N k l k2 l k
C∞ = 2 =
=
nl
nl 2
l
(2) RIS
(1 − cos θ ) (1 + cos φ )
C∞ =
(1 + cos θ ) (1 − cos φ )
(Konturlänge)
(3) Wormlike chain
⎛ L⎞
R( S ) R′( S ′) = exp⎜ − ⎟
⎜ l ⎟
⎝ p⎠
l
lp ≡ k
2
absolute Kettensteifheit: lk, lp
L L
relative Kettensteifheit: ,
lk l p
Experimentell z.B. in der Viskosität beobachtet:
[η]
kein Potenzgesetz!!!
+0.5
+2
1
L/ lk
100
1.3. Konformation “Realer Makromoleküle”
1) Schmelzen
Dichte:
M
4π 3
ρ=
; VK =
RK ; RK ~ R 2
N L ⋅ VK
3
3M
M
~
= N −1 2
ρ=
3
3
2
N L ⋅ 4π ⋅ RK
R2
12
Dichtefluktuationen gering ⇒ “Mean-field”
d.h. chemische Umgebung ist überall gleich
⇓
“Ungestörte Dimensionen” = ideales Verhalten ( bis auf Eigenvolumen)
2) Verdünnte Lösungen
große Konz.Fluktuationen ⇒ “Now-mean-field”
⇓
WW beeinflussen Konformation ⇒ nicht ideales Verhalten
2. Thermodynamics
1) Solvent quality
ΔG m = ΔH m − TΔS m Gibbs free energy of mixing
⎛ ∂Gim ⎞
⎟⎟
μi ≡ ⎜⎜
n
∂
⎝ i ⎠T , p , ni≠ j
∑ n d (μ ) = 0
i
i
chemical potential
Gibbs- Duhem
i
μ i = μ i0 RT ln ai = μ i0 + RT ln( xi γ i )
μ i0 = chem. pot. of pure component
ai = activity
xi = mole fraction
γ i = activity coefficient
Δμ i = μ i − μ i0 = RT ln xi + RT ln γ i = Δμ iid + Δμ iex
excess chem. pot.
ideal solution:
Δμ i = Δμ iid
ΔH m = 0 ; ΔS m = ΔS m (ideal ) = − R ∑ xi ln xi
Δμ i = RT ln ( xi )
i
athermal solution:
Δμ i ≠ Δμ iid
ΔH m = 0 ; ΔS m ≠ ΔS m (ideal )
regular solution:
Δμ i ≠ Δμ iid
ΔH m ≠ 0 ; ΔS m = ΔS m (ideal ) = R ln (xi )
irregular solution:
Δμ i ≠ Δμ iid
ΔH m ≠ 0 ; ΔS m ≠ ΔS m (ideal )
Special case of irregular solutions:
Pseudo-ideal or “Theta” solution
Δμ i = Δμ iid , but
ΔH m ≠ 0 ; ΔS m ≠ ΔS m (ideal )
ΔH m = T (ΔS m − ΔS m (ideal )) can only be true for a single temp. T = θ
Relation to Second Virial Coefficient A2
⎛ 1
N U ⎞
Δμ1* = RTV10 c2 ⎜⎜
+ A 2 c2 ⎟⎟
⎝ M 2 2M 2 ⎠
(*)
only binary interactions!
V10 : partial molar volumen of solvent
c2 : concentration of polymer (solute)
M 2 : molar mass of polymer (solute)
U : excluded volume of polymer (solute)
⎛ − Δμ1* ⎞
⎜⎜
⎟⎟ = Π
osm. pressure
0
⎝ V1 ⎠T ,n
⎛ 1
⎛ 1
⎞
N U ⎞
− Δμ1*
+ Ac2 ...⎟⎟
= Π = RTc 2 ⎜⎜
+ A 2 c2 ⎟⎟ = RTc 2 ⎜⎜
0
V1
⎝ M2
⎠
⎝ M 2 2M 2 ⎠
Thermodynamik
Schmelze → verdünnte Lösung
ideale Lösung: ΔH m = 0 ; ΔS = ΔS id
bei Polymeren nie erfüllt: Eigenvolumen, Packung Lsgm, WW Lsgm.
Deshalb: Immer pseudoideales Verhalten
ΔH = T ΔS
⇒ T = θ = Theta − Temperatur
analog Boyle-Temp. bei id. Gas
T = θ : Schlechtes Lösungsmittel
(W + W22 )
ΔH ~ ΔW12 = W12 − 11
≈0
2
T >> θ : Gutes Lösungsmittel
ΔW12 >> θ
⇒ Viele Polymer/ Lsgm. Kontakte
1)
Polymerknäuel muss expandieren, um Segmentkonz. niedrig zu halten
2)
Entropieelastizität verursacht Rückstellkraft
1)
Einfluß auf Verteilungsfunktion
ω (R)
R
2)
Einfluß auf R 2
Flory → Skalenargumente
R2
ΔG (R )
~ 2
KT
R
2
0
⎛N ⎞
+ ΔW12 ⎜ 3 ⎟ R 3
⎝R ⎠
2
⎛N⎞
2
⎜ 3 ⎟ :Zahl der Kontaktpaare ~ ρ k
R
⎝ ⎠
R 3 : Knäuelvolumen
R2
N2
~ 2 + ΔW12 3
lk N
R
∂ΔG (R ) R N 2
~ − 4 =0
N R
∂R
5
3
R ~N
R ~ N 3 5 ~ N 0.6
(Flory-Limit)
Ensemble-Mittelwert:
R 2 = α 2 R 2 ~ N 0.6
0
α : Expansionskoeffizient α ~ N 0.1 ∞ gutes Lsgm.
Molmassenabhängigkeiten
Ideal flexible Knäuel, T = θ
R2 ~ N ~ M
Knäuel in gutem Lsgm T >> θ
R 2 ~ M 1, 2
Starre Stäbchen
R 2 ~ L2 ~ M 2
Kugeln
R2 ≡ d 2 ~ M 2 3
Osmose
Membran:
∆p = π
c2 = 0
c2 = c
π
van´t Hoff: lim
c2 →0
=
c2
RT
ideale Lösung
M2
Frage: Welche Molmasse misst man?
Verteilungen der Molmassen bei Polymeren: Exkurs
Welcher Mittelwert bei Osmose?
c
m
c
Π = ∑ Π i ; c = ∑ ci ; Π i = i RT = RTni ; ni = i ~ i
Mi Mi
Mi
M = RT
=
∑m
m
∑M
i
i
∑c
∑Π
=
i
=
∑c
c
∑M
i
; ci ~ mi
i
i
∑n M
∑n
i
i
i
≡ Mn
i
i
Reale Lösungen:
⎧1
⎫
Π
= RT ⎨ + A2 c + A3c 2 + ... ⎬ Virialentwicklung
c
⎭
⎩M
A2 , A3 : Virialkoeffizienten
T = θ : A2 = 0 A3 ≠ 0
A2 beschreibt das “Ausgeschlossene Volumen”
für Kugeln, Stäbchen:
- Eigenvolumen
für Knäuel:
- intramol. Anteil
- intermol. Anteil
“Durchdringung von 2 Knäueln”
Thermodynamik, θ-Zustand
Ideale Lösung:
μi = μ 0i + RT ln xi
Reiner Zustand (Schmelze)
G0 = ∑ ni μ 0i
Mischung/ Lösung:
Gm = ∑ ni μi
ΔG id = Gm − Go = RT ∑ ni ln xi
ΔH mid = o (athermisch)
ΔS mid = − R ∑ ni ln xi
Chemisches Potential der Mischung:
Δμ i = μ i − μ i0 = RT ln ( x1 ) = RT ln (1 − x2 )
x1 >> x2 , d.h. x2 sehr klein und V10 = V 20
3
x22 x 2
−
− ...
2
3
⎛ m2 ⎞
⎜
⎟
n2
n2 ⎜⎝ M 2 ⎟⎠
x2 =
≈
≈
(n1 + n2 ) n1 ⎛⎜ V1 ⎞⎟
⎜V 0 ⎟
⎝ 1 ⎠
ln (1 − x2 ) = − x2 −
V ⎛ Volumen ⎞
m2
und n1 = 10 = ⎜
⎟
M2
V1 ⎝ Molvolumen ⎠
m
c ⋅V 0
x2 = 2 1 ; c2 = 2
M2
V1
mit n2 =
⎡ 1 ⎛ V0 ⎞
⎤
⎛ V 02 ⎞ 2
1
⎟c2 + ⎜ 1 ⎟c + ...⎥
Δμ 1 = − RTV c ⎢
+⎜
2
⎜ 3M 3 ⎟ 2
⎢⎣ M 2 ⎜⎝ 2M 2 ⎟⎠
⎥⎦
2 ⎠
⎝
c2 → 0 :
c
Δμ 1id = − RTV10 2
M2
0
1 2
id
Osmotischer Druck Π = −
Π = RT
Δμ 1
V10
c2
van´t Hoff
M2
Ideale Lösung:
⎧ 1
⎫
Π id
+ 2 A2id c2 + 3 A3id c 22 + ... ⎬
= RTc2 ⎨
c2
⎭
⎩M2
V0
A = 1 2 ;
2M 2
id
2
(
)
2
V0
A = 1 3
3M 2
id
3
(
)
Reale Lösungen:
Berücksichtigung von Aktivitätskoeffizienten, usw.
Wiederholung:
Special case of irregular solutions:
Pseudo-ideal or “Theta” solution
Δμ i = Δμ iid , but
ΔH m ≠ 0 ; ΔS m ≠ ΔS m (ideal )
ΔH m = (ΔS m − ΔS m (ideal )) can only be true for a single temp. T = θ
Relation to Second Virial Coefficient A2
⎛ 1
N U ⎞
Δμ1* = RTV10 c2 ⎜⎜
+ A 2 c2 ⎟⎟
⎝ M 2 2M 2 ⎠
(*)
only binary interactions!
V10 : partial molar volumen of solvent
c2 : concentration of polymer (solute)
M 2 : molar mass of polymer (solute)
U : excluded volume of polymer (solute)
⎛ − Δμ1* ⎞
⎜⎜
⎟⎟ = Π
osm. pressure
0
⎝ V1 ⎠T ,n
For real solutions: Excluded volume N A ⋅ U statt V10 :
⎞
⎛ 1
⎛ 1
N U ⎞
− Δμ1*
+ A2 c2 ...⎟⎟
= Π = RTc2 ⎜⎜
+ A 2 c2 ⎟⎟ = RTc 2 ⎜⎜
0
V1
⎠
⎝ M2
⎝ M 2 2M 2 ⎠
N AU
A2 =
2 M 22
Excluded volume U:
a) spheres
R
4π
(2 R )3 = 32π R3
3
3
1
N A 4πR 3
= v2 =
3M 2
ρ2
8M 2v2
U=
NA
U=
4v2
for spheres, for M 2 → ∞
M2
not included:
A2 =
A2 ⇒ 0
three particle excluded volume!!
b) rods
more complex: result
1
2 LM 2v2
U = πdL2 =
dN A
2
Lv2
⇒ for constant d : A2 ≠ f (L ) ≠ f (n2 )
A2 =
dM 2
much more complex
c) flexible coils
Flory and others
⎛ θ⎞
U ~ M 22 ⋅ Ψ⎜1 − ⎟
⎝ T⎠
Ψ : Interpenetration function
T = θ : U = 0 ⇒ A2 = 0 why ideal or unperturbed solution?
- conformation unperturbed by solvent effects! (intramolecular effect):
R2 = α 2 R2 ;
α =1
0
R
2
= kM
1
- U = 0 : No finite volume of the chain (Phantom chain)
Elements infinitely small
pseudo-ideal: Analogy to Boyle- temp. for gases!!
Attraction and repulsion compensate at Tg
- three particle interactions not zero, A3 > 0 even if A2 = 0 (comes from derivation of eq. (*))
Dampfdruckosmose
Lösungsmittel
Lösung
T1
Dampf: T = const.
Lösungsmittel
Π ~ ΔT
∆U ~ ∆T
T2
3. Molekulargewichtsverteilungen, Mittelwerte
Mittelwertsbildungen für ein einzelnes Molekül, definiert durch P Wiederholungseinheiten!
Reale Polymerprobe → Viele Makromoleküle mit unterschiedlicher Molmasse ⇒
Molmassenverteilung
Schulz-Zimm Verteilung
(m = 1: Polykond. + Polyaddition + Radikal., Disprop.
m = 2 : Radikal., Rekomb.)
(
)
−
β
M
⇒ s. Abbildung
W (M ) = β m M m exp
m!
m
β=
m = “Kopplungsparameter”
Mn
Poisson-Verteilung (Anionische Polym.)
Mv M −1 exp(− v )
(
)
W M =
⇒ s. Abbildung
(M − 1)!(v + 1)
v = Mn −1
W (M ) ist Massenverteilung
H (M ) ist die Häufigkeitsverteilung: H ( M ) ≡ W(M)
M
M
Molmassen-Mittelwerte:
∞
Mn
∑n M
=
∑n
i
⇒
i
i
∫ H (M )MdM ∫W (M )dM
=
H (M )dM
∫ H (M )dM ∫
0
∞
0
∑ n M ⇒ ∫W (M )MdM
=
M
∑n M
∫W (M )dM
∫W (M )M dM
M
⇒
∫W (M )MdM
M
∑n M ∑n
=
Polydispersität:
M
(∑ n M )
∑m M
=
∑m
∑m M
=
∑m M
i
w
i
i
z
i
2
i
i
i
i
2
i
i
i
2
w
2
i
i
i
2
n
i
i
M
Uneinheitlichkeit: w − 1 ≡ U
Mn
Schulz-Zimm Verteilung
Mw
1
= 1+
Mn
m
Poisson
Mw
M
1
1
= 1 + − 2 ; Pn = n
Mn
Pn Pn
M0
Meßgrößen sind Mittelwerte über die gesamten Proben, d.h. über die gesamte
Molmassenverteilung:
ni R 2
∑
2
i
=
R
n
∑ ni
∑m R = ∑n M R
=
∑m
∑n M
∑m M R
=
∑m M
2
R
i
2
w
i
i
i
i
i
2
i
i
2
R2
i
z
ersetze R
i
i
2
i
i
i
gegen Molmasse M i ⇒ Molmassenmittelwerte!
Zahlen- vs Gewichtsverteilung
H (M)
M
W ( M ) ≡ H(M)
M
M
M
0.20
Poisson
P -1
P: ni(Pi)=exp(1-Pn)(Pn-1) /Γ(Pi)
i
SF
SF
P
P
wi(Pi)=ni(Pi)*Pi/Pn
2
Pw/Pn=1+1/Pn-(1/Pn) =1.09
0.15
ni bzw wi
Pn = 10
0.10
ζ+1
ζ
ζ-1
SZ: ni(Pi)=ζ /(Pn*Γ(ζ+1))*(Pi *exp(-ζPi/Pn))
wi(Pi)=ni(Pi)*Pi/Pn
0.05
Schulz-Flory (Schulz-Zimm mit Pw/Pn=2;
ζ=Pn/(Pw-Pn)=1)
0.00
0
10
20
30
Pi
40
50
Gewichts- bzw. Massenverteilung
(auch Konzentration)
Häufigkeits- bzw. Zahlenverteilung
300
1000000
250
800000
200
mi
ni
600000
150
400000
100
200000
50
0
2000
2500
3000
3500
4000
4500
5000
0
2000
5500
2500
3000
3500
4000
4500
5000
5500
Mi
Mi
Verteilung des Häufigkeitsanteils
(Molenbruch)
Verteilung des Massenanteils
(Gewichtsbruch)
0.07
0.06
0.05
0.05
0.04
0.04
xi
wi
0.06
0.03
0.03
0.02
0.02
0.01
0.01
0.00
2000
2500
3000
3500
4000
Mi
4500
5000
5500
0.00
2000
2500
3000
3500
4000
Mi
4500
5000
5500
xi =
ni
∑ ni
wi =
mi
nM
nM
= i i = i i
∑ mi ∑ ni M i ∑ ni M i
∑n
∑n
i
= xi
i
Mi
Mn
xi und ni, sowie mi und wi vom Verlauf prinzipiell identisch,
ni und mi sowie xi und wi allerdings nicht!
Vorsicht bei Umrechnung von z.B. xi in wi: Mn berücksichtigen!
Wenn nicht, ergibt sich andere (falsche) Verteilung (s.u., rechts)!
70000
0.06
60000
0.05
50000
40000
xi mi
wi
0.04
0.03
30000
0.02
20000
0.01
10000
0.00
2000
2500
3000
3500
4000
Mi
4500
5000
5500
0
2000
2500
3000
3500
4000
Mi
4500
5000
5500
4. Gelpermeationschromatographie (GPC)
Das Trennprinzip:
Kalibrierung mit engverteilten Standards (häufig Polystyrol):
Mi
Elution volume Ve
Ci
Elution volume Ve
Universelle Kalibrierung: Benoit et al., 1967:
Grundlage der universellen Kalibrierung ist die Kuhn Mark Houwink (KMH)-Beziehung:
[η ] = KM a
η
,[η]: Staudinger-Index bzw. Grenzviskosität für c gegen 0, K, aη: KMH
Koeffizient bzw. Exponent. Sie gilt für alle Objekte, bei denen die Molmasse mit der
Dimension skaliert, also für alle Objekte mit fraktaler Dimension (s.o., M ∝ R f ). Zudem
d
trennt die GPC Partikel nach ihrem hydrodynamischen Volumen, dieses skaliert wiederum
entsprechend der KMH-Beziehung mit der Molmasse. Universelle Kalibrierung:
Ve = A − B log ([η ]k M k ) , [η ]k = K k M k
log ([η ]1 M 1 ) = log ([η ]2 M 2 ) ,
log M 2 =
aη,k
, d.h.: bei Ve ,1 = Ve,2
(
⎛ K ⎞ a +1
log ⎜ 1 ⎟ + η ,1
log M 1 .
aη ,2 + 1 ⎝ K 2 ⎠ aη ,2 + 1
1
a
log K1M 1 η ,1
+1
) = log ( K M
2
aη ,2 +1
2
)
5. Lightscattering to Determine Structure and Dynamics of Macromolecules in Solution
(PD Dr. Wolfgang Schärtl)
Introduction:
The phenomenon of light scattering has first been described theoretically by Lord Rayleigh in the
19th century: Rayleigh discovered that our sky looks blue due to the fact that the short
wavelengths “blue part” of the visible spectrum of the sun light is scattered much stronger by the
gas molecules of our atmosphere than the longer wavelengths “red part”. Nowadays, light
scattering has become a very important analytical tool to determine molar mass, size and shape of
nanoparticles in solution. This experiment tries to illustrate the potential of laser light scattering
using selected examples.
I. Theoretical Background:
(A)
Static Light Scattering
I.1
Rayleigh scattering of small particles
Scattering from gases:
Matter scatters electromagnetic waves (light, X-ray) due to the induction of an oscillating electric
JG
dipol which serves as a source for the scattered light wave. This oscillating dipole moment m ( t )
depends on polarizability α (= measure for the potential of creating an induced dipole moment
JG
within a given particle/molecule) and electric field vector E of the incident radiation as:
JG
JG
JG
GG
m ( t ) = α E ( t ) , E ( t ) = E0 exp i ω t − k x
((
))
(1)
with ω = 2πν = 2π c
λ
G
the frequency of light of wavelength λ, and k = 2π
λ the wave vector.
In Eq.(1) we assumed linearly and vertically polarised light propagating in x-direction (s.figure 1).
JG
E
JG
m
JJG
Es
Fig.1
JG
As seen in the figure, the induced dipolmoment within the scattering particle ( m ( t ) ) acts like an
antenna. It emits an electromagnetic wave (the scattered light) isotropically in all directions of the
scattering plane perpendicular to the oscillation direction of, as indicated by the concentric
circles.
The scattered light wave emitted by the oscillating dipole is given as:
JG
JJG
GG
⎛ ∂2 m ⎞ 1
−4π 2ν 2α E0
exp
ϖ
−
(2)
Es ( t ) = ⎜⎜ 2 ⎟⎟ 2 =
i
t
k
r
rc 2
⎝ ∂t ⎠ rc
Note that in Eqs.(1) and (2) the complex exponential describes regular oscillations of the electric
field vector both in time and space! These equations therefore also can be called “wave
equations”.
JG JG ∗ JG 2
In a light scattering experiment, the scattered intensity I s = E s E s = E s is detected. For
((
))
very small scattering particles (“point scatterers”) irregularly distributed over the scattering
volume (e.g. gas molecules), it is called Rayleigh scattering and given as:
I s 1 16π 4 2
(3)
=
α N
I0 r 2 λ 4
with I0 the intensity of the incident beam and r the distance between sample and detector. As
seen in Eq.(3) and has been stated already above, this scattering is isotropic and not depending
on observation angle.
I=
Scattering from Solutions:
For pure liquids, the scattering is caused by random density fluctuations within the liquid which
are caused by thermal motion of the molecules. For solutions, on the other hand, the total
scattering intensity depends both on these density fluctuations found also in the pure solvent, as
well as on concentration fluctuations of the dissolved particles within the solution. In the
following, the minor contribution of the pure solvent (density fluctuations !) will be neglected:
within this approximation, the scattering intensity depends only on the scattering power of the
dissolved particles b and on the concentration fluctuations, the later given by the concentration c
– dependence of the osmotic pressure π of our binary solution, as given in Eq. (4):
c
I s ∼ b 2 kT
(4)
( ∂π )T
∂c
According to van´t Hoff :
∂π kT
∂π
1
for ideal solutions,
= kT ( + 2 A2c + ...) for real solutions
=
∂c
M
∂c M
(M = molar mass of dissolved particles, A2 = 2nd virial coefficient)
Note that the expression for real solutions given in brackets is actually a series expansion in c!
The scattering intensity of an ideal solution is just depending on scattering power,
concentration and molar mass of the solute, and given as:
I s ∼ b 2 cM
(5)
The scattering power b2 here depends on the difference in polarizability of solute and solvens
Δα, which itself depends on the respective refractive indices as:
2
2
2
2
n − nD ,0
ε − ε 0 nD − nD ,0
Δα = α − α 0 =
=
= D
(6)
4π N
4π N
4π N
V
V
V
with nD the refractive index of the solute, nD,0 the refractive index of the solvent and N
particle number density.
V
the
Next, we define the so-called Rayleigh ratio R, which is the scattering intensity normalized by the
scattering geometry and therefore neither depending on scattering volume V nor on the distance
sample-detector r:
r 2 4π 2
cM
2 ∂n
R = ( I S − I LM ) = 4 nD ,0 ( D )2
[m-1]
(7)
V
∂
c
N
λ0
L
with the refractive index increment
( ∂nD ) ∼
nD − nD ,0
∂c
(8)
c
To determine this absolute scattering intensity R in the experimental praxis, a so-called scattering
standard, that is a pure solvent (typically toluene) with known absolute scattering intensity ISt,abs, is
used. The numerical value of this absolute scattering intensity of the standard ( in [cm-1]) is found
in textbooks. By measuring the actual scattering intensity of the standard ISt using a given
experimental setup as well as the scattering intensity of the solvent ILM and of the solution Is, the
absolute scattering intensity of the solute sample of interest can be calculated according to:
R=
I Lsg − I LM
I St ,abs
I St
(9)
Comparing Eqs.(5) and (7), we deduce that the scattering power of one individual solute
particle b, also called contrast factor K, is given as:
4π 2
2 ∂n
b = 4 nD ,0 ( D )2 = K
∂c
λ0 N L
2
(10)
[cm2 g-2 Mol ]
For real solutions of small (size < 10 nm) scattering particles, one finally (see also Eq.(4)
obtains:
Kc 1
=
+ 2 A2c + ...
R M
1.2.
(11)
Static Structure Factor and Pair Distribution Function:
For semidilute binary solutions, not just simple concentration fluctuations as considered in
chapter 1.1., but more defined interferences of scattered light originating from neighboring
scattering solute particles have to be taken into account. To approach this problem theoretically,
we first introduce the definition of the scattering vector in order to describe the interference of
two scattering centers B and D with distance r, as sketched in Fig.2:
JJG
A
B
k0
a
α
G
r
JJG
k0
β θ
b C G
k
Fig.2
D
We consider only two beams emitted from the two scattering centers at positions B and D into
one (arbitrarily) selected observation direction, defined by the observation or scattering angle
θ. k 0 and k are the wave vectors of the incident and of the scattered light beam, respectively,
2π
with k 0 = k =
. The difference in distance traveled by the two beams than is given as
λ
Δx = a + b = AB + BC
leading to a phase shift
ϕ=
2π ( a + b)
λ
(12)
(13)
This traveled distance can be expressed as function of r, using a= r cos a and b= r cos b:
ϕ=
2π ( r cos α + r cos β )
λ
(14)
On the other hand, the traveled distance can be expressed via the scattering vector
G G G
G
q = k − k 0 and the distance vector r regarding the following scalar products:
GK
k 0 r = k 0 r cos α
GK
k r = kr cos(180 − β ) = − kr cos β
=>
GK G
G G K
GG
k 0 r − k r = −( k − k 0 ) r = − q r
GK G
2π
k 0 r − k r = k 0 r cos α + kr cos β =
( r cos α + r cos β ) = ϕ
(15)
(16)
λ
Thus, the phase shift is given as
ϕ = − qr
(17)
,that is, the scalar product of the scattering wave vector and the distance vector. (Note: q has the
dimension of an inverse length while r is the distance. The product yields the dimensionless
phase shift.)
Importantly, as seen from Fig.2 the value of the scattering vector q is given as:
q = k −k0 =
4π sin θ
λ
(18)
In solution, the wave length of the incident light changes as given by the refractive index of the
solvent nD. Therefore, the scattering vector in this case is given as:
q = k −k0 =
4π nD sin θ
λ
(19)
Accordingly, the scattered electric field strength is given by:
GG
GG
{
}
{ }
or
(20)
E s ( x , t ) = b e iωt −iq r = b e iωt e −iq r
ω
with b (see above) the contrast factor. The factor ei t corresponding to the time-oscillations of
the electric field vector does not contribute to the average measured scattering intensity, and
therefore can be ignored.
Please note that here the interference was derived for two scattering centers in general.
The two scattering centers could be two particles of a multi-particle crystal or within a
semidilute solution (intermolecular interference, structure factor (see below)), or it could be
two volume elements within one scattering particle (intramolecular interference, form factor
(see below))
GG
E s ( x , t ) = b cos(ωt − q r )
•
•
•
•
•
•
•
•
r
•
•
•
•
r
Fig.3
Before we continue, let us illustrate the important meaning of the scattering vector q in more
detail: simply spoken, the scattering vector can be regarded as an “inverse magnification glass“:
we have shown that the scattering vector has the dimension of an inverse length. The meaning of
this is indeed deeper then “just being a dimension”. The scattering vector acts as sort of an
“inverse magnification glass”: It was shown that the scattered intensity is determined via the
phase shift by the product rq . This means the same change in intensity can be caused by a large
structure with a large characteristic distance r in connection with a small q value, or alternatively
for a small structure, for small r, in connection with a large value of q. On the other hand, in
order to “see” certain structures, for small structures we have to use a large q and for large
structures we have to use a small q. This is where the expression “inverse magnification glass”
results from.
From the magnitude of q as calculated above (see Eq.(18), it is obvious how we can
adjust the value of q in experimental practice: the larger the scattering angle θ and the smaller
the wavelength λ, the larger gets q. In a scattering experiment, one usually chooses a given
wavelength that fits the order of magnitude in size of the scattering particles of interested.
Keeping this wavelength fixed, one investigates the scattering intensity at different scattering
angles θ in order to probe different length scales within the range of interest.
The following figure shows the meaning of q as “inverse magnification glass” and
correlates the different scattering methods (corresponding to different values of λ!) with
approximate size scales.
many point-like
particles
single coils with
persistence length l
many coils
with size d
long thin rod
internal
structure
increasing q images smaller structures
light scattering
SAXS and SANS
Fig.4
Many Scattering Centers
Within the scattering volume of our semidilute solution, there are obviously more than two
scattering centers, scattered waves of which interfere and contribute to the resulting scattering
intensity. In this case, one can calculate the scattering intensity by summing up over all
scattered electric field amplitudes.
The scattering from one scattering center can be expressed as:
⎧
Es
GG ⎫
⎨ −iq r ⎬
i⎭
( x , t ) = b e {iωt }e ⎩
(21)
with b the contrast factor and r the position of scattering particle i within the scattering volume.
i
For N scattering centers, the electric field strength is therefore given by the sum
N
E s (q ) = ∑ b i e
⎧ GG ⎫
⎨ −iq r ⎬
i⎭
⎩
(22)
i =1
and the scattering intensity is
N
I ( q ) = E s ( q ) E * s ( q ) = ∑ bi e
i =1
{−iqG rGi }
N
∑
j =1
bi e
{+iqG rGj }
=
N
N
i =1
j =1
∑ ∑
bi b j e
⎧
G
⎪
⎨ −i q
⎪⎩
(rGi − rGj )
⎫
⎪
⎬
⎪⎭
(23)
Following this procedure, in principle we can calculate the scattering intensity from any
accumulation of scattering centers, e.g. a certain crystal structure or a particle with a certain
shape. We just have to think about what it means to “sum over all scattering centers”, i.e. how
to describe e.g. the particle shape mathematically. We will do this later when we deal with the
form factor.
We have seen so far that the scattered intensity of binary solutions is based on pairwise
interference of scattered light originating from the scattering particles within the scattering
volume. Finally, let us consider the correlation between scattered intensity and this particle
location within the sample by introducing a new important function, the so-called pair
G
distribution G (r ) . First, we divide the scattering volume into very small volume segments, each
()
()
with an individual refractive index difference ΔnD r = nD r − nD ,0 . r is the position vector of
()
the volume segment, nD r its local refractive index which depends both on type and number of
()
the scattering particles within. If all particles are identical, ΔnD r is directly proportional to the
()
N
r (see fig.5).
V
scattering volume V containing a total
of N scattering particles
number density or concentration fluctuation Δ
Δ
()
N
r
V
r
Δ
()
N
0
V
Fig. 5
G
The pair distribution function G (r ) is defined as the space correlation function of the local
density fluctuations (for correlation functions, see also section on “Dynamic Light
Scattering”!):
G
N G N G
N G N G G
G (r ) =< Δ (0) Δ (r ) >= ∫ Δ (0) Δ (r )d r
V
V
V
V
V
(24)
G
G
On the other hand, G (r ) and the detected scattered intensity I (q) are Fourier pairs, that is
they can be transformed into each other by Fourier transformation, respectively:
G
G (r ) = 1
GG
(2π )
3
G
G
∫ exp(iqr )I (q)d q
V −1
G
GG G G
I (q ) = b 2V ∫ exp(−iqr )G (r )d r
(25)
V
Finally, it should be noted that Eqs. (23) and (25) show two different expressions for the same
quantity, namely the scattered intensity obtained by summing up pairwise the interferences of
scattered light emitted from all pairs within the scattering volume of our sample.
1.3.
Scattering of Interacting Particles
The pair distribution function G(r) and its Fourier transform I(q) depend on the particle
interaction pair potential u(r). For illustration, let us consider two small hard spheres. R << q-1,
and G(r) therefore is given as:
u (r )
r < 2 R : u (r ) = ∞ → G (r ) = 0
G (r ) = exp(−
)
with
(26)
kT
r ≥ 2 R : u (r ) = 0 → G (r ) = 1
Note that the exponential is a Boltzmann term, expressing the probability of finding a certain
interparticle distance r in case of a competition between thermal energy kT (“chaos”) and
interaction u(r) (“order”).
For the hard sphere system, G(r) is a step function a shown in fig.6:
G(r)
1-G(r)
-1
1-
2R
r
2R
r
Fig.
6
In general, the scattered intensity I(q) in dependence of u(r) is given for the hard sphere system
and an isotropic sample as:
I (q)
N
N
= [1 −
b2 V
V
∞
u (r ) sin(qr )
N
N
2
∫0 (1 − exp(− kT )) qr 4π r dr ] = V [1 − V
2R
∫
0
sin(qr )
4π r 2 dr ]
qr
(27)
Next, we consider G(r) and I(q) for more concentrated solutions, that is increased particle
interactions. At a given particle concentration, we can increase the effective particle volume
fraction by exchanging the short-range hard sphere repulsion by a long-range repulsive potential,
e.g. the Lennard-Jones-potential
u (r ) = 4ε[( r
which has been sketched in figure 7:
1.3d
)−12 − ( r 1.3d )−6 ]
(28)
u(r)
0
r
Fig. 7
d=2R
G(r)
In this case G(r) shows well-pronounced correlations (and oscillations) even for larger
interparticle distances, as sketched below:
1
2R
r
Fig.8
If the scattering particles are larger in size (10 nm < radius < 1 µm), the scattered intensity
I(q) contains both interparticular interferences (structure factor = Fourier transform of G(r) =
S(q)) and intraparticular interferences (particle form factor = P(q)):
N N Z
Z
G G
G
I (q) = b 2 ∑∑∑∑ < exp(−iq(r pi − r qj ) >t
(29)
p =1 q =1 i =1 j =1
Consider N identical particles within the scattering volume, each consisting of Z scattering
centers. The meaning of the distance vector rpi - rqj is illustrated below:
interparticle distance: rpi - rqj
particle p, scattering center i
rpi
rqj
particle q, scattering center j
Fig. 9
Separating intraparticular (p = q, formfactor) and interparticular (p ≠ q) interferences, we get:
I (q)
1.4.
N
2 =
b
V
2
GG
GG
⎛N⎞ Z Z
< exp(−iqr i1 j1 ) > + ⎜ ⎟ ∑ ∑ < exp(−iqr i1 j2 ) >
∑∑
⎝ V ⎠ i1 =1 j2 =1
i1 =1 j1 =1
Z
Z
(30)
The Zimm-Equation
For very dilute solutions the interparticle scattering distribution or structure factor S(q)= 1, and
the measured scattered intensity only contains the particle form factor P(q), which for isotropic
particles is given as:
Z
Z
Z
Z
GG
sin(qrij )
>r
P (q ) = 1 2 ∑∑ < exp(−iqr ij ) > Gr = 1 2 ∑∑ <
(31)
Z i =1 j =1
Z i =1 j =1
qrij
Series expansion yields:
Z
Z
2
(32)
(1 − 1 q 2 rij + ...)
6
Z 2 ∑∑
i =1 j =1
Next, we need the so-called center-of-mass coordinate system. For this purpose, the origin of
the coordinate system is transferred to the particle’s center of mass, as shown in the figure:
P(q) = 1
i
0=S
j
Next, we assume a homogeneous particle density, i.e.. ρ(ri) = ρ. Therefore
Fig. 10
Z
G
∑s
i
=0
and
< s 2 >= 1
i =1
G2
Z
s
Z∑
i
(33)
≠0
i =1
2
with si the position vector of scattering center i of the particle. Note that < s 2 >=< Rg > ≠ 0.
With the distance vector r ij = s j − s i we get:
1
G
2Z
2
∑∑ < r
2
ij
>= 1
2Z
2
∑∑ (s
i
2
G G
2
+ s j − 2( s i s j ))
(34)
G G
s i s j = 0 for i ≠ j , and we get for the form factor P(q):
P(q) = 1 − 1 < s 2 > q 2 + ...
3
(35)
Note that this is a series expansion in q where higher order terms are not shown explicitly (see
below). Thermodynamic fluctuation theory shows that the absolute scattering intensity
depends on P(q) as:
(36)
= 1
+ 2 A2 c + ...
R
MP(q)
This is a series expansion in c. For comparatively dilute solutions, higher order terms in c can
be neglected, however.
Kc
Inserting P(q) from eq. (42) we get the very important Zimm-equation:
Kc
1 + 1 < s2 > q2
3
=
+ 2 A2 c ≈ 1 (1 + 1 < s 2 > q 2 ) + 2 A2 c
R M (1 − 1 < s 2 > 2 q 4 )
M
3
9
(37)
This equation provides the basis for analyzing the scattered intensity from comparatively small
particles (<s2>q2 << 1, in which case the series expansion (Eq.(35)) is terminated with the
< s 2 > q 2 -term. For light scattering, this size regime is corresponding to: 10 nm < radius < 50
nm)) to determined the molar mass, the radius of gyration <s2>0.5 or the 2nd Virial coefficient A2 ,
the later providing a quantitative measure for the particle-solvent interactions. Note that for
polydisperse samples the Zimm analysis yields the following averages:
(i)
Mass-average of the molecular mass Mw
∑N M M
∑N M
∑N M < s
> =
∑N M
Mw =
i
i
i
i
2
(ii)
Z-average of the squared radius of gyration
<s
2
(38)
i
i
i
i
2
z
i
2
>
(39)
i
How are the quantities Mw, A2 and s2z determined in experimental praxis (Zimm-Plot) ??
As an exercise, try to explain all details of the Zimm-Plot given in figure 11 !!
6,0
5,5
5,0
4,0
3,5
-7
Kc/R / 10 mol/g
4,5
3,0
2,5
2,0
1,5
1,0
0,0
5,0
10,0
2
15,0
10
20,0
-2
(q +kc) / 10 cm
Fig.11
1.5.
Particle Form Factor for “Large” Spheres
As shown above (see Eq.(31)), the form factor P(q) is given as:
P(q) = 1
Z
Z
2
Z
GG
∑∑ < exp(−iqr
i =1 j =1
ij
(40)
) > Gr
For homogeneous spherical particles, one gets:
P (q ) =
9
( qR )
6
( sin ( qR ) − qR cos ( qR ) )
2
(41)
with R the radius of the sphere.
This corresponds to an oscillating function as shown in figure 12. Note that the position of the
first minimum is found at qR = 4.49, which can be used to easily determine the particle size.
0
10
-1
10
P(q)
-2
10
-3
10
-4
10
-5
10
0
2
4
6
qR
8
10
12
Fig.
12
Note also that for polydisperse samples these oscillations are not as well pronounced, as shown
in figure 13:
0
10
-1
10
ΔR/R=0.01
ΔR/R=0.05
ΔR/R=0.10
P(q)
-2
10
-3
10
-4
10
-5
10
0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07
-1
q [nm ]
Fig.13: form factor P(q) for spherical particle with size polydispersity 1, 5 and 10 %.
(B)
Dynamic Light Scattering
So far, we have ignored the fact that particles in solution show thermally incited diffusion, socalled Brownian motion. The origin of this Brownian motion are the random thermal density
fluctuations of the solvent molecules already mentioned above. While we have neglected their
direct influence on the scattering intensity before, they nervertheless push the scattering
particles along and therefore cause a time-dependence of our pair correlation function G(r).
As characteristic for this Brownian motion, also called a “Random walk”, the mean squared
displacement of the scattering particles depends linearly on the time of motion: <ΔR(t)2> =
6Dst, with Ds the selfdiffusion coefficient.
2.1.
Time-Intensity-Autocorrelation Function and Particle Motion
The so-called dynamic structure factor Fs(q,t) contains the complete informations concerning the
particle motion. In analogy to Eqs.(25), Fs(q,t) it is the Fourier transform of the so-called vanHove-autocorrelation function Gs(r,t):
G
G
GG G
Fs (q, t ) = ∫ Gs (r , t ) exp(iqr )d r
(42)
G
N G N G
Gs (r , t ) =< (0, 0) (r , t ) >V ,T
V
V
and
(43)
G
Gs (r , t ) is a measure for the probability to find a given scattering particle at time t and position
G
r , if the same particle previously at time 0 has been located at position 0. Note that not the
G
absolute position vectors but only the relative distance vector r is important. The average < > is
taken both over the whole scattering volume and the total measuring time. For an isotropic
G
diffusive particle motion (= Brownian motion), also called “random walk“, Gs (r , t ) is:
Gs (r , t ) = [2π
3
3
3r (t ) 2
)
2 < ΔR (t ) 2 >
(44)
) = exp ( − Ds q 2t )
(45)
< ΔR (t ) 2 >] 2 exp(−
Fourier transform leads to:
Fs (q, t ) = exp(−q 2 < ΔR 2 (t ) >T t
6
The Stokes-Einstein-equation
Ds =
kT
kT
=
f
6πη RH
(46)
allows to determine the hydrodynamic radius of the scattering particle if sample temperature T
and solvent viscosity η are known and the selfdiffusion coefficient is measured by dynamic light
scattering. This is the underlying principle of important modern analytical apparatus, so-called
particle sizers.
Finally, let us consider the theoretical background of the dynamic light scattering experiment
itself:
2.2.
Theory of the Dynamic Light Scattering Experiment
I(t)
<I(t)I(t+τ)>T
The principle of dynamic light scattering is shown in the following sketch:
τ1
τ2
t
τ
Fig. 14
On the left, the signal detected by the photomultiplier at a given scattering angle is shown. Note
that the particle motion, as mentioned above, causes statistic fluctuations with measurement time
in G(r) and therefore also in I(q). For static light scattering experiments, the average scattered
intensity as indicated by the dotted line is determined. For dynamic light scattering, on the other
hand, the detailed analysis of the fluctuating intensity is important. For this purpose, the
fluctuation pattern is transferred into an intensity correlation function, using the following
scheme as indicated in fig.14 for two different correlation times τ: the time-dependent scattered
intensity is multiplied with itself shifted by a distance τ in time, and these products are averaged
over the total measurement time. This intensity correlation function <I(q,t)I(q,t+τ)>=g2(τ) , which
for diffusing particles should exponentially decay from 2 to 1, is related to our dynamic structure
factor or amplitude correlation function Fs(q,τ) via the so-called Siegert-relation:
Fs (q,τ ) = exp(− Ds q 2τ ) =< Es (q, t ) Es (q, t + τ ) >=
or
Fs (q,τ ) = g1 ( q,τ ) =
< I ( q, t ) I ( q, t + τ ) > − A
A
g 2 ( q, τ ) − A
A
Here, A = <I>2 is the base line of the correlation function.
(47)
(48)
log Fs(q,τ)
Fs(q,τ) (= g1(q,τ))
In experimental practice, Fs (q,τ ) is plotted not only in a lin-lin-scale (leading to the wellknown expontential function) but also in lin-log and log-lin scale as sketched below for a bimodal
sample, that is a sample containing scattering particles with two different sizes:
τ
log τ
Figs.15: log-lin and lin-log plot of Fs (q,τ ) for a bimodal sample.
What are the advantages/disadvantages of the two representations, respectively ??
Finally, it should be noted that only in case of very dilute samples the selfdiffusion
coefficient, and therefore the hydrodynamic radius of the scattering particles, can be determined
by dynamic light scattering.
Polydisperse samples:
For polydisperse samples with size distribution P(R) we also get a distribution function of the
corresponding selfdiffusion coefficients P(Ds). There Fs (q,τ ) is not simply a monoexponential
decay, but a superposition of several exponential functions:
∞
Fs (q,τ ) = ∫ P( Ds ) exp ( −q 2 Dsτ )dDs
(49)
0
One practicable way of analyzing these signals is the so-called „cumulant analysis“, which in
practice is a series expansion of Fs(q,τ) and therefore only is valid for small size polydispersities <
20 %:
1
1
ln Fs (q,τ ) = −κ1τ + κ 2τ 2 − κ 3τ 3 + ...
(50)
2!
3!
The first cumulant κ1 = Ds q ² yields the average diffusion coefficient Ds and therefore an
2
average hydrodynamic radius R H , the second cumulant κ2 = ( Ds2 − Ds )q 4 provides a
quantitative measure for the polydispersity of diffusion coefficients σD which is given as:
σD =
Ds2 − Ds
Ds
2
=
κ2
κ12
(51)
Note that determination of the actual size polydispersity is far more complicated and depends on
the size distribution function of the sample, for example Gaussian or rectangular. Corresponding
expressions for the diffusion coefficient polydispersity
κ2
and the corresponding size
κ12
polydispersities are shown in figure 16:
Fig. 16
Finally, it should be noted that for polydisperse samples the average selfdiffusion coefficient
determined from the correlation function Fs(q,τ), e.g. by cumulant analysis, is q-dependent.
Therefore, it is also called apparent diffusion coefficient Dapp(q). The “true” average diffusion
coefficient, which by the way is a z-average, is determined by interpolation of the apparent
diffusion coefficient towards q = 0. For small particles 10 nm < radius < 120 nm, this
interpolation is given as:
(
Dapp ( q ) = Ds , z 1 + K Rg
2
q2
)
(52)
Importantly, the constant K not only depends on sample polydispersity but also on the particle
topology (sphere, cylinder etc.). Plotting Dapp(q) vs. q2 in this case leads to a linear increase, the
intercept with the q = 0 axis yielding the z-average diffusion coefficient Ds,z and therefore an
inverse z-average hydrodynamic radius RH,z-1. The fact that the apparent diffusion coefficient due
to polydispersity effects increases with increasing scattering vector q is simple to understand: the
scattered intensity which determines the relative contribution of a respective particle size to the
correlation function depends both on particle concentration and particle form factor. Consider
for illustration the particle form factors of 3 different particle sizes as shown in figure 17.
q1
q2
0
10
-1
10
P(q)
-2
10
-3
R = 100 nm
R = 120 nm
R = 140 nm
10
-4
10
-5
10
0,00
0,01
0,02
-1
q [nm ]
0,03
Fig. 17
Obviously, P ( q ) of the largest particles decays first with increasing q (due to destructive
intraparticular interferences). This causes a loss in contribution of larger particles to the scattered
intensity and therefore decreases their contribution to the average correlation function with
increasing q, leading to the fact that the apparent diffusion coefficient becomes larger (“faster”)
with increasing q.
How does Dapp(q) vs. q2 look like for large polydisperse particles 200 nm < <R> < 500
nm ??
6. MALDI-TOF Mass Spectrometry
(HD Dr. Michael Maskos)
Matrix Assisted Laser Desorption/Ionization Time-of-Flight Mass Spectrometry
1
Ekin = U ⋅ z ⋅ e = mv 2 , U: applied electrical field, z: number of charges, m: mass, v : average
2
velocity
v = at , a : acceleration, t: time
U
a 2
Force: F = m ⋅ a = ⋅ z ⋅ e , d = ⋅ t acc
2
d
t acc = d ⋅
tdrift =
2m
z
U ⋅e
m
distance L
z
= =L
velocity v
2 ⋅U ⋅ e
t = tacc + tdrift =
const. m
z
U
real (dead times etc.):
m
= a ⋅ t meas + b
z
Resolution:
⎧
⎫
2
2
2
⎪⎪⎛
d m ⎞ ⎛ ⎛ ΔE ⎞ ⎞ ⎪⎪
U Δt I ,D ⎞ ⎛
⎟ + ⎜ cTA
= ⎨⎜⎜ c I ,D
Δv ⎟ + ⎜ c E ⎜
⎟⎟ ⎬
L U ⎟⎠ ⎜⎝ ⎝ U ⎠ ⎟⎠ ⎪
L ⎟⎠ ⎜⎝
⎪⎝m
⎪
⎪⎩
C
⎭
A
B
12
⎛ Δt ⎞
⎜ ⎟
⎝ t ⎠ linear
A: distribution of ionization times, detector response time Δt I , D → given by exp. setup
B: velocity distribution due to desorption process Δv (“turn-around-time”)
C: initial speed and place of ionization → energy distribution ΔE independent of m and L
Reflectron: energy refocussing leads to c ≈ 0
dominant:
m
A for
≤ 3000amu
z
m
B for
≥ 3000amu
z
MultiChannel Plate (MCP)
Reflectron
7. Field-Flow-Fractionation FFF
(HD Dr. Michael Maskos)
Theory of FFF:
Flow density J x = − D
dc ( x )
+ Uc ( x ) , D : diffusion coefficient, U : field induced velocity
dx
from the wall
to the wall
at equilibrium: J x = 0
⎡
U⎤
concentration profile: c( x ) = c0 exp ⎢− x ⎥ , c0 : conc. at the wall, x : coordinate in channel
D⎦
⎣
height direction
D
⎛ x⎞
⇒ c( x ) = c0 exp⎜ − ⎟ , l: characteristic average of distance
U
⎝ l⎠
(depending on sample and field)
F
kT
D=
(Stokes – Einstein), U =
, F : force, f : friction coefficient
f
f
kT
=> l =
, typically 1 < l < 10 μ m
F
effective layer thickness: l =
Normal Retention Mode (“Brownian”)
L
, L : channel length, v : particle velocity ( v of an exponential
v
distribution in a parabolic flow profile)
Retention time t R =
Parabolic flow profile:
Retention (as in Chromatography) described by dimensionless retention parameter λ
l
kT
, w : channel height (typically 75-300 μm)
λ= =
w Fw
average zone velocity v in z-direction v =
Retention ratio R =
c(x )v( x )
c( x )
(averaging over channel height)
v
, v : average v of sample component, v ( x ) : average v of solvent
v( x )
w
Integral expression: R =
∫ c( x)v( x)dx
0
w
w
0
0
∫ c( x)dx ∫ v( x)dx
Flow profile of an isoviscous liquid between two parallel plates (Hagen-Poiseuille):
v( x) =
Δp
x( w − x) , η : viscosity of solvent
2ηL
Δpw 2
=> average velocity v( x) : v( x) =
12ηL
⎛
⎞
⎛ 1 ⎞
Integration yields: R = 6λ ⎜⎜ coth⎜ ⎟ − 2λ ⎟⎟
⎝ 2λ ⎠
⎝
⎠
if w >> l (requirement for efficient operation): λ -> 0, thus lim (
)
- Term → 1 and so
limit λ → 0 lim R = 6λ
λ →0
Alternative description:
V
t
R = 0 = 0 , V0 : dead volume (channel volume), VR : retention volume, t 0 : dead-(void-) time
VR t R
t R : retention time
→
tR w F w
= =
t 0 6l 6kT
→ tR ~ F
⇒ seperation if ΔF is high enough (typically: 10-16 N)
size selectivity: S d =
d log t R
d log R
Flow FFF
viscous force on particle due to cross flow: F = f U =
kT U
= 6πη R U
D
→ tR ~ f , D −1 , R , thus S d ≅ 1 (GPC typically S d ~ 0.2 )
Band broadening: average plate height H
H = H neg + H long + ∑ H i , H neg : velocity gradient → mass transfer (non-equilibrium), H long :
longitudinal diffusion: can often be neglected, H i : instrumental effects: can often also be
neglected
Theory of separation systems with non-uniform flow profile leads to:
L
χw 2
H neg =
v , v : average Carrier-velocity = 0 , χ = f ( λ )
t
D
3
for small λ (λ ≈ 0.06) : χ ≅ 24λ
L
Dt 0
= 2
H w 24λ3
typical example of channel parameters yields:
H of 0.18 mm ⇒ N ≅ 1550 possible
plate height N =
The FFF-Family
Sedimentation FFF:
4π 3
R Δρ G , m´ : effective mass (mass – mass of flowtation), V p :
3
particle volume, R: radius, G: gravitational acceleration
t R ~ m´,V p , T , Δρ , parameter G
F = m´G = V p Δρ G =
→ Sd ≅ 3
→ run time problem with samples of totally different size
→ programmed field (simple equations no longer valid!)
limits:
- Maximum load/pressure on the seals: max 105 g
- Lower limit of separation: approx. d ~ 50 nm, Polymer only > 104 – 107 g/mol
Thermal FFF:
Temperature gradient:
100 K
dT
K
~ 10 4
→ ΔT ≈
( ≅ w)
100μm
dx
cm
thermal diffusion of polymers → cold wall
D dT
F = kT T
, DT : thermal diffusion coefficient
D dx
⎛ tR ⎛ DT ⎞
⎞
b
with D ≅ AM − b , b ≈ 0.6 ⎟
⎜ ≅⎜
⎟ ΔTM
⎝ t0 ⎝ A ⎠
⎠
Problem: DT often unknown
DT
≅ Soret coefficient
D
with known D determination of DT
D : sensitive to polymer dimension
DT : sensitive to chemical composition
Electrical FFF:
So far not established, because problmes with elektrolyte gases
⎡⎣ F ~ q ( effective charge ) ⎤⎦
F = kT
μE
D
, μ : elektrical mobility, E : applied electrical field
General FFF Theory
Velocity vectors
Parabolic Flow Profile
FFF-channel
no flow, only field
Field
Field
only flow, no field
concentration
flow and field
General FFF Theory
te
M1 =
∫ tc(t )dt
ta
te
R=
∫ c(t )dt
ta
υ zone
R=
υ
w
υ zone =
M 1,u
M 1,r
tu
R=
tr
c ( x )υ ( x )
c ( x)
1
υ = ∫ υ ( x )dx
1-x/w
w0
w
1
c ( x )υ ( x ) = ∫ c ( x )υ ( x )dx
w0
w
1
c ( x ) = ∫ c ( x )dx
w0
⎡ x ⎛ x ⎞2 ⎤
υ ( x ) =6 υ ⎢ − ⎜ ⎟ ⎥
⎢⎣ w ⎝ w ⎠ ⎥⎦
v(x)/<v>
FFF Theory: Force Field
dc ( x )
J x = U xc ( x ) − D
dx
Jx = 0
⎛ xU x ⎞
Solution: c ( x ) = c0 exp ⎜
⎟
D
⎝
⎠
F
Forces: U x =
f
dc ( x )
U xc ( x ) = D
dx
⎛ x Ux ⎞
Convention: c ( x ) = c0 exp ⎜ −
⎟
D
⎝
⎠
Fritction: f = kT
D
⎛ xF ⎞
c ( x ) = c0 exp ⎜ −
⎟
kT
⎝
⎠
Flow-FFF:
•
wV c
Ux =
V0
⎛ xwV• ⎞
c
⎜
⎟
c ( x ) = c0 exp −
⎜ DV0 ⎟
⎝
⎠
FFF Theory: Retention Parameter
Average height l:
kT
l=
F
Retention Parameter λ:
l kT
λ= =
w Fw
w
Exact Average
height le:
le =
∫ xc ( x ) dx
0
w
∫ c ( x ) dx
⎛ x⎞
∫0 x exp ⎜⎝ − l ⎟⎠ dx
le = w
⎛ x⎞
∫0 exp ⎜⎝ − l ⎟⎠ dx
w
le
Exact Retention Parameter λe: λe =
w
0
λe = λ +
1
⎛1⎞
1 − exp ⎜ ⎟
⎝λ⎠
⎛ w⎞
exp ⎜ − ⎟ ( l + w ) − l
w
l ⎠
⎝
le =
=l+
⎛ w⎞
⎛ w⎞
exp ⎜ − ⎟ − 1
1 − exp ⎜ ⎟
⎝ l ⎠
⎝l ⎠
Flow-FFF Theory: Retention Parameter
λ=
kTV0
•
f Vq w
f = 3πη d
2
λ=
kTV0
•
3πη V q w2 d
FFF Theory: Retention Parameter
⎛ x ⎞
c ( x ) = c0 exp ⎜ −
⎟
⎝ λw ⎠
c ( x)
⎛ − λ1 ⎞
= −c0 λ ⎜ e − 1⎟
⎝
⎠
1
1
⎛
⎞
−
−
⎛
⎞
2
λ
λ
c ( x )υ ( x ) =6λ c0 υ ⎜ 2λ ⎜ e − 1⎟ + e + 1⎟
⎜
⎟
⎠
⎝ ⎝
⎠
1
2λ
−
1
2λ
⎛ 1 ⎞ e +e
coth ⎜
⎟= 1
1
−
2
λ
⎝
⎠
e 2λ − e 2λ
FFF Theory: Retention Ratio and Retention Parameter
R=
c ( x )υ ( x )
c ( x) υ ( x)
Approximations:
⎛ 1 ⎞
2
12
λ
= 6λ coth ⎜
−
⎟
2
λ
⎝
⎠
1. empirical:
λ=
(
1− 1− 4 R
3
2
2. square: R = 6λ − 12λ ⇒ λ =
4
R = 6λ ⇒ λ = R
3. linear:
6
Upper limits of R und
Approx.
)
R
6 (1 − R )
1
3
λ , with max. λ -
%deviation
Analyt.
limits of R
2%
5%
10%
R
λ
R
λ
R
λ
1
<1
0.64
0.15
0.76
0.20
0.85
0.27
2
<3/4
0.68
0.17
0.72
0.20
0.74
0.22
3
-
0.06
0.01
0.14
0.02
0.27
0.04
Relation between R and λ
1
3 lin
2 quad
1 empir
exact
Appr.
Approximation
A
Appr.
Approximation
B
Appr.
ApproximationGl.
C
ex.Funktion
0.8
λ
0.6
0.4
smaller
distance from
accum. wall
0.2
0
0
0.2
0.4
0.6
0.8
1
R
longer sample retention times
unretained
Band Broadening
diffusional flux
x
υH
υ zone
υL
l
c
concentration profile
z
x
diffusional flux in time td
υH
l
υ zone
υL
l
+z
+z
z
Band Broadening
Elution:
(z direction)
Plate height H:
∂c
∂ 2c
=D 2
∂t
∂z1
H=
σ2
Z
small, monodisperse:
Einstein:
with
σ 2 = 2Dt
υ zone = Z t
z1 = z − υ zonet
second moment of dislocation (in z direction)
first moment of dislocation (in z direction)
2
⎛
z − υ zonet ) ⎞
(
m
exp ⎜ −
c=
⎟
⎜
⎟
4
Dt
4π Dt
⎝
⎠
D = Dz + ∑ Di
∑D
i
= Dn
2 Dz t + 2 Dnt 2 Dz 2 Dn
H=
=
+
υ zone υ zone
Z
H = Hd + Hn
i
Dz : axial diffusion coefficient
(z direction)
H d ==
2 Dz
υ zone
2 Dz
Hd =
R υ
H n ==
2 Dn
υ zone
Band Broadening
Nonequilibrium
c∗ = c0 ( z ) e
c = c∗ (1 + ε )
sample concentration c(x,z,t)
−
x
l
c*: concentration without flow, c0: concentration at accum. wall,
ε: equlilibrium departure term, accounts for the differential displacement
of analyte in axial direction (z direction)
cυ = c∗υ + c∗ευ
average convective flux:
Fick‘s 1. law:
J z = − Dn
Dn = −
(cross-sectional average)
∂ c∗
∂z
c∗ευ
c∗ ( ∂ ln c∗ ∂z )
υ: from Navier-Stokes equation for incompressible
flow between two infinite parallel plates
⎡ x ⎛ x ⎞2 ⎤
υ ( x ) =6 υ ⎢ − ⎜ ⎟ ⎥
⎢⎣ w ⎝ w ⎠ ⎥⎦
Band Broadening
Nonequilibrium
determination of ε:
solving
with
assumption
∂c
∂ ⎛
∂c ⎞ ∂ ⎛
∂c ⎞
= − ⎜ Uc − Dx ⎟ − ⎜υ c − Dz ⎟
∂t
∂x ⎝
∂x ⎠ ∂z ⎝
∂z ⎠
∂c∗
c∗
=−
l
∂x
and
i) axial gradient negligible (analyte cloud spans a much
greater width in axial (z) than in lateral (x) dimension)
ii) near-equilibrium (lateral (x) equilibrium maintained during
axial (z) migration) ∂c
∂ 2 c∗
∂c∗
∂t
∂ ε −1 ∂ε υ − υ zone ∂ ln c
−l
=
2
∂x
∂x
∂z
Dx
⎛ dε ⎞
(no analyte flux
with ⎜
=
0
⎟
across accum. wall)
dx
⎝
⎠ x =0
2
∗
≈ Dz
∂z
2
− υ zone
c∗ε = 0
∂z
(assumpt. 1,
c = c∗
)
Band Broadening
Nonequilibrium
Final solution
Hn =
ψ l 2υ zone
Dx
=
χ w2 υ
Dx
1
2 ⎞
⎛
2
−1 λ
−1 λ
28λ + 1)(1 − e ) − 10λ (1 + e ) − 2 −
(
⎜
⎟
−1 λ
λ
λ
3
4 (1 − e )
⎜
⎟
ψ=
⎛ ⎛
⎞⎟
2 ⎜
⎞ 1
−1 λ
−1 λ
1
1
⎡(1 + e ) − 2λ (1 − e ) ⎤ ⎜ +4 −
⎜ 4λ ⎜1 +
⎟−
− 6⎟⎟
⎣
⎦
−1 λ
−1 λ
⎟⎟
⎜
λ (1 − e ) ⎜ ⎜⎝ λ (1 − e ) ⎟⎠ 3λ
⎝
⎠⎠
⎝
where
and
χ = ψλ 2 R
if λ is small:
limψ = 4; lim χ = 24λ 3
λ →0
λ →0
power series expansion (λ < 0.10; R < 0.48):
ψ = 4 (1 − 6λ + 24λ 3 + 96λ 4 )
χ = 24λ 3 (1 − 8λ + 12λ 2 + 24λ 3 + 48λ 4 )
Resolution and Fractionating Power
Resolution: measure of separation between two zones
∆z: gap between center of gravity of neighboring zones,
σ: standard deviation of the zones
related to retention time:
Fractionating Power F: Resolution
between particles by difference in
diameter or molecular weight
∆z
∆z
Rs =
=
2 (σ 1 − σ 2 ) 4σ
δ tr
Rs =
4σ t
Rs
Fd =
δd d
tr ∂ ln tr
tr
d ∂tr
Fd =
Sd
=
=
∂d 4σ t 4σ t ∂ ln d 4σ t
Rs
Fm =
δM M
d ( log tr )
Sd =
d ( log d )
Selectivity S
M ∂tr
tr ∂ ln tr
tr
Fm =
Sm
=
=
∂M 4σ t 4σ t ∂ ln M 4σ t
d ( log tr )
Sm =
d ( log M )
Resolution and Fractionating Power
Theoretical plate number N
Fd , M
High retention:
⎛ tr ⎞
N =⎜ ⎟
⎝ σt ⎠
N
=
Sd , M
4
2
L
N=
H
random dispersion: N
selective dispersion: S
24λ 3w2 υ
H ≈ Hn =
D
Fd , M = S d , M
LD
384λ 3 w2 υ
differentiating R
1⎞
⎛ R
+ 1 − ⎟ S max
S = 3⎜
2
R⎠
⎝ 36λ
Smax
d ( ln λ )
=
d ( ln d )
d ( log tr ) d ( log λ )
Sd =
d ( log λ ) d ( log d )
analyte-field
interaction
Th-, F-FFF: Smax 0.5 – 0.7
SEC: Smax 0.1 – 0.2
Theory of Asymmetrical Flow FFF (AF-FFF)
Bolt
Inlet
Tubing
Sample
Inlet
Block
Outlet
Tubing
Upper Wall
(Glass/Plexiglass)
Spacer
Membrane
Frit
Crossflow
Outlet
K
b0
bL
L
zk
z
Theory of Asymmetrical Flow FFF (AF-FFF)
Corrections for the flow force field velocity:
width in trapezoidal channel:
⎛ 3x 2 2 x3 ⎞
u ( x ) = − u0 ⎜ 1 − 2 + 3 ⎟
w
w ⎠
⎝
b ( z ) = b0 − z ( b0 − bL ) L
•
average flow velocity:
impact on plate height:
unretained time:
(void time)
υ =
H=
Vin − u0 ( b0 z − z 2 ( b0 − bL ) ( 2 L ) )
w ( b0 − z ( b0 − bL ) L )
χ w2
D
υ
⎛
•
V0 ⎜
Vc
tu = • ln ⎜ 1 + •
Vc ⎜ V out
⎜
⎝
b0 − bL 2
⎛
⎛
⎞ ⎞⎞
⎜ w ⎜ b0 zk − 2 L zk − K ⎟ ⎟ ⎟
⎠ ⎟⎟
⎜1 − ⎝
V0
⎜
⎟⎟
⎜
⎟⎟
⎝
⎠⎠
Theory of Asymmetrical Flow FFF (AF-FFF)
K
b0
bL
L
zk
Parameter
•
Vc
•
z
value
1 ml/min
V out
1 ml/min
zk
2.1 cm
V0
0.68 ml
w
105 µm
b0
2.12 cm
bL
0.47 cm
L
28.6 cm
K
2.25 cm2
tu = 27.6 s
Asymmetrical Flow FFF (AF-FFF) in Toluene
bim.Kugeln
monodisp.Kugeln
0,16
detector
a.u.
0,14
0,1095
0,1090
0,1085
0,1080
0,1075
0,12
0,1070
400
600
0,10
0
200
400
600
tR / s
FFF: General Setup
Inlet
Field
Outlet
x
Spacer
Chann
el
y
z
Accu
mu
lation
wall
Historical Overview
Thermal-
Sedimentation-
Electrical-
Capillary/Flow-
1966 - 1974
1976 Flow-FFF
Concentration-FFF
1977
Gravitational-FFF 1978
1979
Pressure-FFF
Magnetical-FFF
1980
Shear-FFF
1984
1987
Asym. Flow-FFF
1993 - 1995 Frit-Inlet-Outlet-FFF
2D-Sed.-Flow-FFF
1994
Publications on FFF until 1998
90
Utah
Rest of the world
Total
80
Papers / year
70
60
50
40
30
20
10
0
1965
1970
1975
1980
1985
1990
Year
J. Calvin Giddings
1966
1995
2000
FFF: General principle
Basic Mode: „Normal Mode“
(„small“ particles/molecules: diffusion)
Field
Parabolic flow profile
Flow
Flow
vectors
X
Accumulation wall
lY
Y
lX
X
Steric/Hyperlayer Mode
x = 0.2 w
x=0
Accumulation wall
Accumulation wall
0.12
better retained
(exit later)
(small first)
Steric-Hyperlayer Mode (large first)
Normal Mode
less retained
(exit earlier)
0.10
Retention Ratio
x=0
x = 0.2 w
0.08
0.06
0.04
0.02
S-Fl-FFF
0.00
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Particle size, µm
3.5
4.0
4.5
5.0
0.001
0.01
0.1
1.0
10
100
StericHyperlayer
FFF
Fumed Silica
Silica Beads
Cyan Blue Pigment
Carbon Black
Lipoproteins
Viruses
Yeast
Microsomes
Transition Region
Normal
FFF
Chromatographic
Silica
Ground
Limestone
Ground Coal
Glass Beads
Cells
Pollens, Spores
Polystyrene Latexes
Polymers
Proteins
DNS, Vesicles
0.001
0.01
0.1
1.0
10
Hydrodynamic diameter [µm]
100
General Experiment - Fractogram
2
tintr
Vintr
F
3
4
4
5
5
stop
trel
void time
retentation time
flow
relaxation
on channel sample introduction
loop filling
1
V°
t°= F
textra =
tr
Vextra
F
textra
Separation Example
SEC
Thermal-FFF
PMAA 240 000
and
PS 200 000
PS
200 000
PMAA
240 000
inj.
inj.
Void
peak
0
1
2
3
4
0
2
4
6
8 10
Practical Overview over the FFF Methods
FFFTypical application range
technique
Modes
Sd-FFF
Polymers > 106 g/mol, and colloids or
Normal,
particles > 30 nm, useful for particular matter steric,
and biological applications. Applicable to focussing,
adhesion
water and organic solvents. The only
technique operating in all modes of FFF.
Gr-FFF
Particles > 1 µm. Applicable to water and
organic solvents.
Th-FFF
Lipophilic synthetic polymers > 104 g/mol. Normal, steric
Very useful for large shear sensitive
polymers or aggregates. Applicable to water
and organic solvents.
Fl-FFF
Polymers, colloids and particles from 1000 Normal, steric
g/mol or 1 nm to ≈ 50 µm. Most universal of
all FFF techniques. Applicable to water and
organic solvents.
El-FFF
Biopolymers and colloids from 40 nm to > 1 Normal, steric
µm. Applicable to water.
Steric,
adhesion
Band Broadening
A)
A) Correction for zone broadening of a model fractogram. (a)
represents the original curve and the corrected one whereas (b) is
the uncorrected fractogram.
B) Comparison of differential particle size distributions of narrowly
distributed polystyrene latex standards derived by MALLS and FlFFF without correction for zone broadening.
B)
Flow FFF: Membrane Fowling
(b)
scale µm
0
2
4
6
8
SEM micrograph of a polycarbonate Fl-FFF membrane after the fractionation of
a mixture of 121 nm, 265 nm and 497 nm polystyrene standards
Separation for the FFF-Family
FFF technique
λ=
Sedimentation
(Sd)
λ=
RT
⎛
ρ
ω 2 r M ⎜⎜1 −
⎝ ρs
6k T
π
dH3
physico-chemical
parameter
⎞
⎟⎟ w
⎠
ω r w (ρ s − ρ )
2
ρs, M
dH
Thermal
(Th)
λ=
D
DT (dT / dx) w
D, DT
Electrical
(El)
λ=
D
µe E w
D, µe
Flow
(SF, AF)
λ=
D V0
V w 2
D, dH
c
Steric
λ=
dH
2w
Magnetic
(Mg)
λ=
RT
M w χ m H m ∆H m
dH
M, χm
Relation between λ and the physical
solute properties using different FFF
techniques with R = gas constant, ρ =
solvent density, ρs = solute density, ω2r
= centrifugal acceleration, V0 = volume
of the fractionation channel, V c = cross
flowrate, E = electrical field strength,
dT/dx = temperature gradient, M =
molecular mass, dH = hydrodynamic
diameter, DT = thermal diffusion
coefficient, µe = electrophoretic mobility,
χM = molar magnetic susceptibility, Hm =
intensity of magnetic field, ∆Hm =
gradient of the intensity of the magnetic
field.
Sedimentation FFF (Sd-FFF)
t0
1800 rpm
0.22 µm
0.40
0.30
Response
0.27
tr shift
0.50
0.40
0.60
800 rpm
5
0
1800 rpm
15
10
Time (min)
0.40
C
25
20
D
0.50
t0
Response
0.22 µm
0.60
0 rpm
0.27
0.30
0
5
15
10
Time (min)
20
25
Sedimentation FFF (Sd-FFF)
F = meff G = v p ∆ρ G =
meff = m
∆ρ
ρp
π d H3 G ∆ρ
6
vp: particle volume
buoyancy-corrected particle mass
G = ω 2 racc
centrifugal acceleration
2π rpm
ω=
60
rotation angular velocity
kT
kT
6kT
λ=
=
=
meff G w
v p ∆ρ G w
π d H3 ∆ρ G w
Sedimentation FFF (Sd-FFF)
Separation of components of partially aggregated latex by Sd-FFF.
Thermal FFF (Th-FFF)
Thermal FFF (Th-FFF)
DT dT
|F| = kT
D dx
⎡ dc
⎞ dT ⎤
⎛α
Jx = −D⎢ + c⎜ + γ ⎟ ⎥
⎝T
⎠ dx ⎦
⎣ dx
γ is the thermal expansion coefficient and α the thermal diffusion factor α = (DT/D)T
1 dc
⎞ dT
⎛α
= −⎜ + γ ⎟
c dx
⎠ dx
⎝T
⎛− x⎞
c( x ) = c 0 exp⎜
⎟
⎝ l ⎠
with
1 ⎛α
⎞ dT
= ⎜ + γ⎟
l ⎝T
⎠ dx
< c( x) >= c 0 λ (1− exp( −1/ λ ))
-1
⎡ ⎛ DT
D
⎞ dT ⎤
λ = ⎢⎜
+ γ⎟w
≈
⎥
D T ⋅ ∆T
⎠ dx ⎦
⎣⎝ D
γ small compared to α/T,
small dT/dx = ∆T/w
Thermal FFF (Th-FFF)
η is η(T) => η(x)!
⎡
⎢x
∆p ⌠ x
dx −
v ( x ) = − ⎢⎮
⎢
L ⌡ η( x )
⎢0
⎣
d ⎛ dv( x) ⎞
∆p
⎜η
⎟= −
dx ⎝ dx ⎠
L
virial expansion
1
η
⎤
x
x
∫0 η( x ) dx ⌠ 1 ⎥
dx ⎥
⎮
w
1
⌡ η( x ) ⎥
⎥
∫0 η( x ) dx 0
⎦
w
= a0 + a1T + a 2 T 2 + a3 T 3
dT/dx is a function of the thermal conductivity κ, and κ is κ(T)
κ = κc +
dκ
T − Tc )
(
dT
if dκ/dT = constant:
2
x ∆T 2 ⎛ 1 dκ ⎞
∆T +
−1+ 1+
⎟
⎜
w κc dT
w ⎝ κc dT ⎠
T( x) = Tc +
1 dκ
2x 1 dκ
κc dT
difficult to calculate 1/η, approximation (within 0.25% error):
⎛ dT ⎞
x2 ⎛ d2T ⎞
x3 ⎛ d3 T ⎞
⎜⎜ 2 ⎟⎟ +
⎜⎜ 3 ⎟⎟ + .......
T( x) = Tc + x ⎜ ⎟ +
⎝ dx ⎠c 2 ⎝ dx ⎠
3! ⎝ dx ⎠
c
c
Thermal FFF (Th-FFF)
⎛ x ⎞i
v( x) = −
hi ⎜ ⎟
L i=1 ⎝ w ⎠
∆p
< v( x) >= −
R=
R=
∑
∆p
L
1
5
∑
i=1
For R → 0:
5
hi
hi
(i + 1)
λ
5
∑ h(ii +/ h1)1
i=1
5
hi: calculated polynominal coefficient
hi
∑hi (i + 1)
i=1
⎧
⎡5
⎪
1
⎢ hi
⎨
⎪⎩ ( 1− exp( −1/ λ )) ⎢⎣ i = 1
⎫
⎤ 5
j⎥
i⎪
λ + i! hi λ ⎬
⎪⎭
j = 1 ( i − j )! ⎥
⎦ i=1
i −1
∑ ∑
i!
∑
Detector response (A.U.)
Thermal FFF (Th-FFF)
232
30
272
91
330
135
198
0
20
40
60
80
Retention time, tr(min)
426
100
120
Separation of eight polystyrene latex particles in aqueous suspension
by Th-FFF. The numbers above each peak correspond to the particle
diameter in nm.
Thermal FFF (Th-FFF)
80
∆T, °C
60
∆T
9k
40
35 k
575 k
200
k
90 k
1970 k
5480 k
20
0
0
5
10
15
Time, min
20
25
Thermal FFF (Th-FFF)
Micelle with Au in core
Empty micelle
Response [a.u.]
1500
Elugram ∆T = 20.7K
w = 75µm
1000
500
0
0.0
0.5
1.0
1.5
2.0
2.5
VE [ml]
Th-FFF measurements of polystyrene-poly-4-vinylpyridine
(PS123-b-P4VP118) micelles in toluene. The core consists of
poly-4-vinylpyridine which can be used as a nanoreactor for
Au synthesis to generate a significantly different DT of the
core. However, the detected DT is that of polystyrene.
Electrical FFF (El-FFF)
U = µe E
F = f µe E
D
λ=
µe Ew
including ζ-potential
(ζ-potential < ± 25 mV):
µe = ζ ( 2ε 3η ) f (κ D d H )
κD: inverse Debye length
1.0 < f(κD dH) < 1.5
Magnetic FFF (Mg-FFF)
Mχ mHm dHm
F=
dx
RT
λ=
MwχmHm ∆Hm
Literature
M.E. Schimpf, K. Caldwell, J.C. Giddings (eds.), Field-Flow
Fractionation Handbook, Wiley-Interscience, New York 2000
M. Martin, „Theory of Field Flow Fractionation“, Advances in
Chromatography 1998, 39, 1 – 138
H. Cölfen, M. Antonietti, „Field-flow fractionation techniques for
polymer and colloid analysis“, Adv. Polym. Sci. 2000, 150, 67 – 187
M. Maskos, W. Schupp, „Circular Asymmetrical Flow Field-Flow
Fractionation for the Semipreparative Separation of Particles“, Anal.
Chem. 2003, 75, 6105-6108