High Pressure Investigations

of Liquid Crystals

– a Challenge to Experiment and Theory

Stanisław Urban

M. Smoluchowski Institute of Physics

Jagiellonian University

Krakow, Poland

27th Janik’s Friends Meeting

Zakopane, 10–17 July, 2011

Outline

Introduction

Phase transitions

Entropy separation at the clearing line

Molecular dynamics in different phases

Thermodynamic scaling of dynamical quantities

Summary

Remarks on the profits and difficulties of investigation

under elevated pressures

External pressure applied to a molecular system influences its physical

properties considerably. This is due to relatively weak intermolecular interactions.

Pressure (P) and temperature (T) are equivalent thermodynamic variables, but

they act on the molecular systems differently: T influences mainly excitations of the

rotational and vibrational energy levels whereas P changes mostly the

intermolecular distances.

In liquid crystals (LC) constituted of strongly anisotropic molecules the

interaction potential consists of a distant dependent and orientation dependent parts.

This causes that pressurization may change the phase diagrams, the molecular

ordering within a given LC phase as well as practically all physical properties of a

substance, quite easily. Moreover, a relatively low hydrostatic pressures (up to, say,

hundreds MPa) are sufficient to involve considerable changes of the properties of a

LC system.

The main advantage of the pressure studies of a molecular system arises from

the fact that a given property can be analysed at the isobaric, isothermal and

isochoric conditions which yields much more information than the usual ambient

pressure studies in function of temperature.

Especially important is a possibility of determining the parameters

characterising the interaction potentials in the LC phases.

Measurements under pressure - main difficulties

- Problems connected with the separation „sample - pressurizing medium”

- Measurements can hardly be automated

- Relatively big amounts of samples needed

- Frequent demaging of aparatus parts

- Measurements are time consuming

- Poor theoretical background

Albert Würflinger – Ruhr University Bochum

Mike Roland – Naval Research Laboratory, Washington

Strategy of hp investigations

Selection of samples

Determination of the T(P) phase diagram

Determination of the equation of state V(P,T)

Determination of the dynamical properties of molecules in particular

LC phases

Common analysis of the results with the aim to obtain information about

fundamental properties of the system studied (eg. interaction potential);

the theoretical models are needed.

Substances studied

Two-ring core

Ph-Ph

Cy-Ph

Cy-Cy

Dio-Ph

Pyr-Ph

Tail

Alkyl

Alkoxy

Strong polar group

CN

NCS

Variety of phase sequences

0 50 100 150 200 25040

60

80

100

120

140

Isotropic

Sm A

Crystal I

N

Crystal II

Tc = 77,75 + 0,237xp - 4.64x10

-5xp

2

TS-?

= 95,71 + 0,098xp + 1.58x10-4xp

2

Tm = 59,31 + 0,270xp - 2,49x10

-4xp

2

Tcr = 42.45 + 0,235xp - 2,39x10

-4xp

2

10DBT

T [oC]

P [MPa]

0 50 100 150 200 250320

340

360

380

400

H9C4 OC6H13

CN CN

C O

O

Cr

Sm A

Is

CNCN

T [K]

P [MPa]

S.Urban, A. Würflinger, Z. Naturforsch. 56a, 489 (2001)

A. Würflinger, S.Urban, PCCP 3, 3727 (2001)

0 50 100 150 20040

60

80

100

120O

O

H2n+1Cn NCS

Crystal II

Crystal I

Isotropic

Sm A

8DBT

T [oC]

P [MPa]

0 50 100 150 200300

320

340

360

380

400

(344K, 66MPa)

Cr

SA

N

Is

10CB

T / K

p / MPa

0 50 100 150 200 250300

320

340

360

380

400

11CB

T / K

Cr

SA

Is

N

(382K, 182MPa)

p / MPa

S. Urban, et al., Liq. Cryst., 30, 313-318 (2003).

0 50 100 150 20040

60

80

100

120

H11C5O C C NCS

F

TOLF5

Cr - 52.0o C - N - 65.4oC - Is

TNI

64.9 + 0.310xp -2.58x10-4

p2

Tm 52.4 + 0.263xp -2.59x10

-4p

2

Tfr 49.2 + 0.227xp -1.08x10

-4p

2

Cr

N

IsT / oC

P / MPa

0 20 40 60 80 100 120 140 160 180 20040

50

60

70

80

90

100

110 95oC

107MPa

9BT

Cr

E

I

T/ 0C

P / MPa

A?

9BT

CrE

100 MPa

N

0 20 40 60 80 100 120 140 160

30

40

50

60

70

80

90

100

110

10BTN?

A?

CrI

E

p [MPa]

T [oC]

55MPa

78oC

101MPa

91oC

I

E A

N

S.Urban, J. Czub, A. Würflinger

Phase Trans. 79, 331 (2006).

Y. Maeda, S.Urban,

Phase Trans. 83, 467 (2010).

88 92 96 100 104

20

24

20

24

20

24

20

24

20

24

20

24

T [oC]

148MPa

U[V]

138MPa

130MPa

122MPa

114MPa

106MPa10BT

70 80 90 10010

15

20

25

30

35

10

15

20

25

30

Cr

IsE

128.6MPa

T / oC

A?

100.2MPa

ECr

Is

9BT

0 20 40 60 80 100 120 140 160 180 20030

40

50

60

70

80

90

100

110

120

Sm E"

H13C6 C6H13

Sm E'

Cr

Sm ESm Bcr

Is

C6-C6

T [oC]

P [MPa]

S.Urban, M. Massalska-Arodz, A. Würflinger, K. Czuprynski,

Z. Naturforsch. 57a, 641 (2002)

• A common feature of smectogenic members is an appearance of splitting of the clearing lines at elevated pressures.

• With increasing the pressure at first the long-range molecular arrangement within the layer is canceled, and then the layer structure is canceled leading to the nematic or isotropic phase.

• The length of the alkyl tail(s) play important role in that effects.

nCB

nPCH

nCCH

nCHBT

nBT

The role of the length of terminal group(s) and a flexibility of the core

should be explained.

The slope of the clearing line

0,2

0,3

0,4

0,5

1011

12

nCB AI

Slo

pe

[K

/MP

a]

nBT EI

6C

HB

T N

I 8765

4 nPCH NI

5C

CH

NI

5P

CH

NI

5C

B N

I

3

6P

CH

NI

6O

CB

NI

6C

B N

I

87

65

4 14

n

The mean-field theories predict that

TNI (P)VΓNI (P) = const

Γ – material constant that reflects the volume dependence of the orientational

contribution to the internal energy, without a priori assumption about the

nature of anisotropic intermolecular potential; larger Γ implies stronger steric

repulsion relative to the attractive interactions.

Equation of state V(p,T) must be known!

)(log

)(log

PVd

PTd

NI

NI

The slope of the clearing line

vs. interaction potential

])()[()( 63

0rr

r

-0,015 -0,010 -0,005 0,000

2,50

2,52

2,54

2,56

2,58

log

( T

c /

K)

log(Vc [ml/g])

=5.03 0.02

log ( )

log ( )

c

c

d T P

d V P

6CHBT

-0.025 -0.020 -0.015 -0.010 -0.005 0.0000.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

6OPB8

log(VNI

/V0)

log

(TN

I/T0)

= 2.51 ± 0.2

-0,06 -0,04 -0,02 0,000,00

0,05

0,10

0,15

0,20

E - I

5BT = 2.2

6BT = 2.4

7BT = 2.2

5CB = 5.3

6CB = 6.3

7CB = 4.7

ln (V/Vo)

ln (

T/T

o)

N - I

nCB

nBT

N

N

O

The transition “LC phase – isotropic liquid” is a (usually weakly) first order phase

transition and is characterised by the entropy ΔStr constituting of two contributions:

a constant volume (or configurational) part ΔSconf and

a dilational part ΔSdil

ΔStr = ΔSconf + ΔSdil

PVT data enable us to separate both contributions:

ΔStr = ΔSconf + (∂S/∂V)TΔVtr

= ΔSconf + (∂P/∂T)VΔVtr.

Combining this expression with the Clausius-Clapeyron equation,

(∂P/∂T)tr = ΔStr/ΔVtr

one gets

ΔSconf = [(∂P/∂T)tr – (∂P/∂T)V ]ΔVtr

or ΔSconf /ΔStr = [(∂P/∂T)tr – (∂P/∂T)V ]/(∂P/∂T)tr.

Entropy separation at the clearing point

0

40

80

120

160

200

0 20 40 60 80 100 120 140

Crystal Nematic Isotropic

6CHBT

1.0150.995

0.980cm3/g0.950cm

3/g

T [oC]

P [

MP

a]

0 20 40 60 80 100 120 140

0,88

0,90

0,92

0,94

0,96

0,98

1,00

1,02

1,04

10MPa20

30 40

50 60

7080 90

100120 140

160

Vs

[cm

3/g

]

T [oC]

6CHBT - Isobars

ambient pressure

180MPa

0 50 100 150 200 2500,0

0,2

0,4

0,6

0,8

1,0

6CB N - I

6DBT A - I

6CHBT N - I

S

conf /S

tr

P [MPa]

6BT E - I

H13C6

NCS

ΔSconf /ΔStr = [(∂P/∂T)tr – (∂P/∂T)V ]/(∂P/∂T)tr.

Molecular dynamics in LC phases

Typical dielectric absorption spectra

103

104

105

106

107

0,00

0,05

0,10

0,15

0,20

0,25

0,30

6OPB8

T = 328 K

10MPa

18

25

33

40

63

70

93

116

130

145

"

f / Hz

104

105

106

107

0,0

0,1

0,2

0,3

0,4

a)

p = constant = 1 atm

62.9oC N

60.5

58.1

54.6 SA

51.0

47.3

43.7 SC

37.5

32.6 "

f / Hz

N

N

O

T = constant

τ|| = 1/(2πfmax), where fmax corresponds to ε”max

J. Czub, et al. Z. Naturforsch., 58a, 333 (2003)

10-5

10-4

10-3

10-2

10-1

100

101

102

103

10-3

10-2

10-1

100

10-3

10-2

10-1

100

"/" m

ax

f/fmax

T=64.0C

6BT

Debye

"/" m

ax

P=0.1MPa

Representative dielectric loss curves

for 6BT

C.M.Roland, et al. J. Chem. Phys., 128, 224506 (2008) 2,9 3,0 3,1 3,2 3,3 3,4

0,0

0,1

0,2

'

"Sm A

38.60C

6OPB8

105

106

107

0

1

2

3

4

5

6

307 K

309.7

312.2

315.2

f [Hz]

"

7PCH

1 atm

a

105

106

107

0

1

2

3

4

5

6b

20 MPa

60

100

120

7PCH

336 K

"

f [Hz]

2,7 2,8 2,9 3,0 3,1 3,2 3,310

-8

10-7

266 cm3/mol

1000/T [K-1]

1 atm

60 MPa

260 cm3/mol

|| [s]

7PCH

a)

0 50 100 150 200

255 260 265 2700

10

20

30

40

50

60

70

b)

#H

U

P [MPa]

U

,

H [

kJ/m

ol]

Vm [cm

3/mol]

7PCH

VT

pTUH

TRU

TRH

pRTV

V

V

p

T

###

1#

1#

#

)(

)/ln(

)/ln(

)/ln(

0 40 80 120 160 200 240

10-8

10-7

1.2410-7s 1.5810

-7s

384K

P / MPa

1.0710-8s

312K

7PCH

|| [s]

#H/#U 2

Thermodynamic scaling of the dynamical quantities

It was demonstrated by Roland and co-workers that the structural

relaxation times and viscosity measured as function of P, T and V can

be rescaled to one master curve with one adjusting parameter using

the simple exponential form

τ = A exp(B/TVγ) (A,B,C,D – constants)

η = C exp(D/TVγ)

The same was done by me for ~20 LC phases.

2,8 2,9 3,0 3,1 3,2 3,3 3,4-15,5

-15,0

-14,5

-14,0

-13,5

-13,0

-12,5

-12,0

1000/(TV)

6OPB8

c)

ln ( [s])

= 2.7 ± 0.1

N

N

O

Cr – SmC – SmA – N – I

S. Urban, C.M. Roland, J. Czub, K. Skrzypek, J. Chem. Phys, 127, 094901 (2007)

0 20 40 60 80 100 120 140 160 180 200-15,5

-15,0

-14,5

-14,0

-13,5

-13,0

-12,5

-12,0

2,85 2,90 2,95 3,00 3,05 3,10 3,15 3,20-15,5

-15,0

-14,5

-14,0

-13,5

-13,0

-12,5

-12,0

ln

p / MPa

313.3K

323.0K

332.2K

342.2K

346.2K

1000/T / K-1

ln

0.97

0.98

0.99

1.00

1.01 cm3/g

160 MPa

120

80

40 Δ #U /Δ # H ~53/78 ~0.68 SmA

~46/68 ~0.68 SmC

2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.210

-8

10-7

10-6

10-5

10-4

6CHBT N

=5.07CB N

=3.3

7PCH N

=3.9

6OPB8 SA,C

=2.7

8BT SE

=4.1

T=const

p=const

V=const

1000/(TV)

[s]

4

8

12

4

8

12

1,02 1,04 1,06 1,08 1,10 1,12

4

8

12

333 K

343 K

353 K

363 K

373 K[

mP

a]

5PCH

2

7PCHIsotropic

3PCH

Vm [cm

3/g]

1,6 1,8 2,0 2,2 2,4 2,6

0,2

0,4

0,6

0,8

1

1000/TV

Isotropic

7PCH

*= 4.0

5PCH

*= 3.9

3PCH

* = 3.9

2,2 2,4 2,6 2,8 3,0 3,2

0,5

1

1,5

1000/TV

5PCH

*= 4.0

*

Nematic

3PCH

*= 4.2

7PCH

*= 3.8

1,6 1,8 2,0 2,2 2,4 2,6

5

10

15

20

333 K

343 K

353 K

363 K

373 K

[

mP

a]

1000/TV

3PCH

= 4.6

5PCH

= 4.57PCH

= 4.4

Isotropic

2,2 2,4 2,6 2,8 3,0 3,25

10

15

20

25

30

35

1000/TV

[

mP

a]

3PCH

= 4.8

5PCH

= 4.3

323 K

333 K

343 K

353 K

363 K

7PCH

= 4.1

Nematic

0,96 0,98 1,00 1,02 1,04 1,06 1,085

10

15

20

10

20

30

10

15

202530

3PCH

Vm [cm

3/g]

5PCH

CN

H15C7

7PCH

[

mP

a] 323 K

333 K

343 K

353 K

363 K

Nematic

2,2 2,3 2,4 2,5 2,6 2,7 2,810

-8

10-7

10-6

[s]

*

1000/TV

7PCH

= 3.9

* = 4.1

2.4 2.5 2.6 2.7 2.8 2.9 3.0

10-8

10-7

10-6

* = 3.8

= 3.6

1000/TV

5PCH

[s]

*

τ* = υ -1/3(kBT/m)1/2τ ~ Vm -1/3T 1/2 τ

* = υ 2/3(kBTm)-1/2 ~ Vm 2/3T -1/2

D. Fragiadakis and C.M. Roland, J. Chem. Phys.2011, 134, 044504.

S. Urban, Liq. Cryst., accepted

The question arises whether the three material constants, , τ and , can simply be compared.

The parameter characterises the thermodynamic phase transition

whereas the two others concern two different aspects of the dynamic

behaviour of molecules within the nematic phase. In fact, there is no a

theoretical basis for discussion a connection of the thermodynamic

conditions associated with the nematic – isotropic transition (or other

transitions) and the time scale of the flip-flop molecular motions in the

nematic phase.

The question arises whether the three material constants, , τ and , can simply be compared.

The parameter characterises the thermodynamic phase transition

whereas the two others concern two different aspects of the dynamic

behaviour of molecules within the nematic phase. In fact, there is no a

theoretical basis for discussion a connection of the thermodynamic

conditions associated with the nematic – isotropic transition (or other

transitions) and the time scale of the flip-flop molecular motions in the

nematic phase.

However, one can speculate about that taking into consideration some

experimental facts and well known relationships:

(i) and τ values were found to be identical or close for many LCs

hitherto studied.

Substance Nematic phase Isotropic phase

τ τ* * * 3PCH - - 4.8 4.2 5.0 4.6 3.9 5.0 5PCH 3.6 3.8 4.3 4.0 4.1 4.5 3.9 4.1 7PCH 3.9 4.1 4.1 3.8 3.8 4.4 4.0 3.8

Accuracy 0.15

The data determined from the calorimetric DTA, PVT and dielectric spectroscopy studies of liquid crystalline substances in

different phases. The activation parameters ΔH, ΔU and ΔV exhibit some changes within a given phase and were taken in a

mid part of the phase.

Substance Tc

[K]

∂Tc/∂P [K/MPa]

Phase ΔH [kJ/mol]

ΔU [kJ/mol]

ΔV [cm3/mol]

ΔU/ΔH Exper. Eq. (5) Eq. (12)

γ Г 104αP [1/K]

-104ατ [1/K]

-ατ/ αP

5CB 308.3 0.424 N 62 38 59 0.61 0.48 0.54 4.1 5.3 6.8 6.3 0.919

6CB 301.2 0.390 N 62 27 63 0.44 0.54 0.54 4.1 6.3 7.0 8.1 1.15

7CB 314.6 0.370 N 64 30 63 0.47 0.52 0.53 3.3 4.7 8.5 9.2 1.15

8CB 313.8 0.370 N, A 60/40 32/24 60/38 0.53/0.57 0.50 0.50 4.2 4.0 7.7 7.7 1.00

5PCH 328.1 0.440 N 69 39 70 0.57 0.59 0.58 3.5 5.2 6.3 9.0 1.42

7PCH 331.0 0.420 N 70 35 65 0.50 0.51 0.52 3.9 3.3 7.1 7.3 1.03

8PCH 328.3 0.412 N 70 36 60 0.51 0.58 0.56 3.6 3.4 6.7 9.2 1.37

6CHBT 316.7 0.419 N 63 33 65 0.52 0.54 0.52 5.0 5.0 5.8 6.8 1.174

6DBT 350.1 0.259 A 51 20 42 0.40 0.41 0.43 4.0 2.9 9.4 6.6 0.72

6OPB8 339.2 0.234 A, C 79/69 52/46 58/45 0.66/0.66 0.61 0.61 2.7 2.5 7.0 11.1 1.63

5BT 347.1 0.284 E 2.3 2.2

6BT 344.7 0.285 E 3.1 2.4

7BT 345.6 0.241 E 2.3 2.2

8BT 341.1 0.237 E 70 36 59 0.51 0.47 0.48 4.1 2.7 7.4 6.60 0.90

0 20 40 60 80 100 120 140 160

-8.0

-7.5

-7.0

-6.5

-6.0

8CB

7CB

60PB8

6CB

7PCH

60PB8

log

(

/s)

P [MPa]

0 50 100 150 200

0,6

0,8

1

1,2

1,4

1,6 Nematic

Isotropic

7PCH

[m

Pa]

P [MPa]

323 K

333

343

353

363

373

383

TNI

0 40 80 120 160 200 240

10-8

10-7

1.2410-7s 1.5810

-7s

384K

P / MPa

1.0710-8s

312K

7PCH

|| [s]

0 40 80 120 160 200 240300

320

340

360

380 7PCH

Cr

N

P / MPa

T [K]

I

It seems that the constancy of both

dynamical quantities at Tc is well fulfilled.

(ii) The relaxation time τ||(Tc) is independent of pressure and volume. {The same

concerns the structural relaxation times along the characteristic temperature-pressure

lines in glass-formers}.

(iii) The nematic order parameter S is constant along Tc(Pc).

(iv) The relaxation time τ|| is directly related to S via the so-called

retardation factor which in turn depends on the Maier-Saupe strength

parameter :

g = τ|| /τo ~ υ(V) S

Taking the above into consideration one can state

that , τ and should be the same if τ|| (and ) are

constant along the clearing line.

Conclusions

Pressure studies are indispensable for discussing the interaction potentials.

The thermodynamic potential parameter, Γ, measuring the variation of the interaction

energy with volume is associated with the stability limits of the ordered state.

The scaling parameter, γ, reflects the volume dependence of the dynamical quantities

within the whole range of a LC phase.

Equivalence of both parameters affirms the connection between the longitudinal

dynamics and the repulsive part of the interaction potential.

The fact that the thermodynamic conditions associated with the stability limits of the

ordered state bear a direct relationship to the time scale of molecular rotations is

unanticipated by any model and should guide theoretical progress in this class of

materials.

Large values of the scaling parameters (Γ and γ > 3, say) implies that the intermolecular

potential for LCs can be approximated as a repulsive inverse power law with the weaker

attractive forces treated as a spacially-uniform background,

U(r) ~ r -Γ + const

Thank you for

your attention

Predicted hp studies:

• substances with another phases (C, B, N*, …)

• substances forming the glassy state

• stimulation of theoretitions for developing appropriate models

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

0 2 4 6 8

14

13

12

10

11

897

6 5

4

32

U

/H

1

1 5CB N

2 6CB N

3 7CB N

4 8CB N

5 8CB A

6 5PCH N

7 7PCH N

8 8PCH N

9 6CHBT N

10 6DBT A

11 6OPB8 A,C

12 6BT E

13 8BT E

14 5*CB Is

U/H=1/(1+0.18)

The parameter provides a measure of the

relative importance of V as opposed to T.

For strictly activated dynamics, in

which thermal energy dominates the

behavior, = 0, whereas for the hard

sphere limit .

Typical LC phases

Liq

uid

-like p

hase

s C

rysta

l-like p

hase

s