This document summarizes simulations and theory on the phase behavior of hard rods that are restricted to movement within a two-dimensional substrate plane. Both Monte Carlo simulations and density functional theory predict a continuous isotropic-nematic phase transition as rod density increases. The transition remains continuous even when an attractive substrate potential is introduced. Future work will extend the model to multilayers and investigate rod deposition dynamics.

1. Phase Behavior of Restricted System of Rods
Theory and Simulations
Mohit Dixit∗1
, Martin Oettel2
, Miriam D. Klopotek2
, Tanja Schilling1
1
Theory of Soft Condensed Matter, University of Luxembourg, Luxembourg
2
Institut f¨ur Angewandte Physik, Universitat T¨ubingen, Germany
*mohit.dixit@uni.lu
Introduction
•In literature, bulk phase behavior of hard rods (spherocylinders) has
been determined using simulations [2,3] and density-functional the-
ory [4] predicting Isotropic-Nematic (I-N) transition to be of 1st
order.
•In bulk, anisotropic particles like rods already posses a rich phase
behavior like Nematic, Smectic, Crystalline, Liquid.
•May become even more complex when rods are near a substrate.
•Relevant for morphology studies of films in organic semiconductor
devices like Solar Cells and Transistors.
Simulations
Hard rods (spherocylinders) with their mid points fixed on the
substrate plane.
•Monte Carlo Simulations of hard rods. Centers are fixed on a plane
while the orientation is free for 3 dimensions.
•Measure nematic order parameter Qnem of the system defined as:
Qnem = P2(cosθ) =
3
2
cosθ −
1
2
(1)
where θ is the angle between rod and director.
•Isotropic phase ( Qnem = 0 ) and Nematic phase ( Qnem = 1 ).
Theory Collaborators from T¨ubingen
•Considering hard sperocylinders with length L and diameter D
whose centers are fixed on a plane.
•Applying Classical Density Functional Theory.
•Nematic order parameter Qnem in terms of Legendre coefficients
Bij can derived as [5]:
Qnem ≈ −ρ0B10 (2)
•The numerical evaluation of the Legendre coefficients gives:
Qnem ≈ 0.45 LDρ0 (L/D → ∞) . (3)
•which shows Qnem ∝ ρ0 i.e. Continuous Isotropic-Nematic phase
transition.
Results
•Observe a Continuous Isotropic-Nematic phase transition.
0.0001 0.001 0.01 0.1
ρ0
D
2
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
Qnem
N = 500
N = 6400
0.45 ρ0
LD
(a)
0.0001 0.01 1
ρ0
LD
1e-05
0.0001
0.001
0.01
0.1
1
Qnem
L/D = 10, N= 10000
L/D = 10, N= 6400
L/D = 5, N = 10000
L/D = 5, N = 6400
L/D = 3, N = 10000
(b)
•With Attractive Substrate.
0 0.1 0.2 0.3 0.4 0.5 0.6
ρ0
D
2
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Qnem
no substrate pot.
ε = 0.2
ε = 1
ε = 2
ε = 5
(a)
0 0.1 0.2 0.3 0.4 0.5 0.6
ρ0
D
2
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Qnem
no substrate pot.
ε = 0.2
ε = 0.5
ε = 1
(b)
Conclusions
•Observe Continuous Isotropic-Nematic phase transition in re-
stricted system of rods.
•Transition remains Continuous even with the introduction of At-
tractive Substrate.
Forthcoming Research
•Extending Monolayer model to Multilayer.
•Investigating Dynamics via Rod Deposition onto substrate.
References
[1] L. Onsager, Ann. New York Acad. Sci. 51, 627 (1949)
[2] J.A.C. Veerman and D. Frenkel, Phys. Rev. A 41, 3237 (1990).
[3] P. Bolhuis and D. Frenkel, J. Chem. Phys. 106, 666 (1997).
[4] M. Schmidt and H. Lowen, Phys. Rev. E 55, 7228 (1997).
[5] ”Monolayers of hard rods on planar substrates: I. Equilibrium” ; M. Oettel, M.
Klopotek, M. Dixit, E. Empting, T. Schilling, and H. Hansen–Goos ; J. Chem.
Phys. 145, 074902 (2016)
Acknowledgements
• DFG/FNR INTER project ”Thin Film Growth”
• Discussions with Frank Schreiber.