Free Volume of Polymers using Positron Annihilation Lifetime
Spectroscopy (PALS)
M. Qasim Shaikh
Martin-Luther-Universität
Halle-Wittenberg
Fachbereich Physik
Structure of the free volume
• free volume due to structural, static or dynamic, disorder
• important for several macroscopic properties of these materials,
• viscosity, molecular transport, structural relaxation, and physical aging
Schematic representaion of a single Poly (Propylene) microstructure (X=76)
(Simulation, Theodoru et al. 1985)
Experimental Ways to determine Free Volume in Polymers
• Positron Annihilation Lifetime Spectroscopy (PALS)– Detection of subnanometric local free volumes (holes):
Size distribution (mean hole volume <vh> and mean dispersion σh)
• Pressure-Volume-Temperature-Experiments (PVT)Analysis by Simha-Somcynsky lattice-hole model EOS
– Fraction of vacancies h, specific hole free. Vf = hV, and occupied volumes, Vocc = (1-h)V
• Correlation of PALS and PVT– allows estimation of PALS hole density Vf = Nh´<vh>
→All parameters of the structure of hole free volume can be obtained from PALS and PVT
Basics of PALS
•Thermalization
•Diffusion
•Annihilation
Basic Principles and Theories of Positron formation in Polymers
Ps Formation in Polymers
Ps
-
--
-
----
---
e+β+ source
Pick-Off Annihilation
Ps
= 2 - 4 ns
τ = 1 ns
τ
Ps
0.5 nm
Ps localization in a hole of the (excess) free volume
Ps localization in interstitial free volume gives the packing co-efficient ´C´ of the crystals
Theory of Tau-Eldrup (TE) Model
Sources of Positron
The height of the potential is infinity, and δr – empirical parameter δr = 0.166 nm
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+
++
−
=−
rrrSin
rrr
ns
h
h
h
hpickoffPso
δπ
πδ
τ2
211
5.0
0 1 2 3 4 5 6 70123456789
10
o-Ps
life
time τ p
o (ns
)hole radius rh (Å)
Tao-Eldrup Standard Model
threshold
mean hole volume
vh(τ3) = (4/3)πrh3(τ3)
.......
Typical PALS Spectrum
Sources of Positron2
3
1
n(t) = I1 exp(-t/τ1) + I2 exp(-t/τ2)+ I3 exp(-t/τ3)
p-Ps
τ1 = 0,125 ns
Free Annihilation
τ2 = 0,4 ns
o-Ps (Pick-off annihilation)
τ3 › 0,5 ns
PC
The Positron Lifetime Measurement
Sources of Positron
Typical Lifetimes in Holes of:
Sources of PositronSources of Positron
Analysis of COC and PC by PALS
Cyclic Olefin Copolymer (COC) Poly Carbonate (PC)
V and Vocc vs T (K) [PVT]
300 350 400 450 500 550 6000.85
0.90
0.95
1.00
1.05
1.10
Tg(P)
Voc
c (cm
3 /g)
V
(cm
3 /g)
T (K)
COC 0.14080
120160200
0.1
200
V
Vocc
Tg(0)
300 350 400 450 500 550 600
0.75
0.80
0.85
0.90
0.95
Tg(P)
Voc
c (cm
3 /g)
V (c
m3 /g
)
T (K)
PC
Vocc
V0.14080
120160200
0.1
200
Tg(0)
The specific total, V, (black open symbols), and occupied, Vocc = (1 – h)V(blue symbols) volume as a function of temperature T and as selection of isobars (P in MPa) for COC and PC.
Vf vs T(K)
The specific hole free volume Vf = hV as a function of temperature T and as selection of isobars (P in MPa) for COC and PC.
250 300 350 400 450 500 550 6000.000.020.040.060.080.100.120.140.160.18
Tg(P)
Tg(0)
Vf (c
m3 /g
)
T (K)
COC
T0
Vf0.14080
120160200
250 300 350 400 450 500 550 6000.000.020.040.060.080.100.120.140.16
Tg(P)
Tg(0)
Vf (c
m3 /g
)
T (K)
T0
PCVf
0.14080
120160200
τ3 and σ3 vs T (K) [PALS]
The mean, τ3, and the mean dispersion, σ3, of o-Ps lifetimes as a function of temperature T for densified at 200 Map (blue), gas-exposed (read) and untreated (black) COC and PC.
300 350 400 450 5001.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
σ3
σ3 (n
s)
τ3 (n
s)
T (K)
τ3
Tg
untreated gas-exposeddensified
COC
300 350 400 450 5001.41.61.82.02.22.42.62.83.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Tk
untreated gas-exposeddensified
σ 3 (ns)
τ3 (n
s)
T (K)
PC
Tg
τ3
σ3
Ps yield vs T(K)
The Ps yield P = I1 + I3 with I1/I3 = 1/3 as a function of temperature T for densified at 200 Map (blue), gas-exposed (read) and untreated (black) COC and PC.
300 350 400 450 50020
25
30
35
40
P =
I 1 + I 3 (%
)
T (K)
COC
300 350 400 450 50030
35
40
45
P =
I 1 + I 3 (%
)
T (K)
PC
Vh and σh vs T(K)
The mean, <vh>, and the mean dispersion, σh, of the hole volume as a function of temperature T for densified at 200 MPa (blue), gas-exposed (red) and untreated (black) COC and PC.
300 350 400 450 50040
60
80
100
120
140
160
20
304050
607080
90
σ h (Å3 )
<vh>
(Å3 )
T (K)
σh
<vh>
COC
Tg
300 350 400 450 500
60
80
100
120
140
160
180
20
30
40
50
60
70
80
90
σ h (Å3 )
<vh>
(Å3 )
T (K)
PC
Tg
<vh>
σh
Vf vs T(K)
The specific free volume Vf = <vh>Nh’ from PALS at 10-5 Pa as a function of temperature for untreated (black filled circles), densified at 200 MPa (blue squares), and gas-exposed (read diamonds) COC and PC. The black empty circles show 0.1 MPa isobars from PVT experiments for the untreated polymers, Vf = hV.
300 350 400 450 5000.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
Vf =
<v h>N
h' (cm
3 /g)
T (K)
COC
300 350 400 450 5000.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
V f = <
v h>Nh' (
cm3 /g
)
T (K)
PC
Thanks for your time and patience!