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!