SIMULACIJE TEKO"CIH KRISTALOV Z GAY-BERNE-OVIM POTENCIALOM










R. LUKAC
Gimnazija Murska Sobota
SSolsko naselje 12
9000 Murska Sobota

in

Computational Physics Group
Institute of Experimental Physics
University of Vienna, Strudlhofgasse 4
A-1090 Vienna

E-mail: Renato@s-gms.ms.edus.si



POVZETEK

Gay-Bernov potencial je zelo koristen kompromis med kvantitativnim opisom intermolekularnega potenciala tekocekristalnih molekul in efektivnim izkorißcanjem moci sodobnih racunalnikov.

Raziskali smo meßanico molekul dolßine 3.0 in 1.5. Velik niz podatkov pridobljen z obseßnimi racunalnißkimi simulacijami smo analizirali tako z vizualizacijskimi kot statisticnimi metodami. Osredotocili smo se na takoimenovane 'sendvic' strukture zgrajene iz izmenjujocih se plasti daljßih molekul v smekticni fazi in krajßih molekul, ki so ostale v izotropni fazi tudi pri visokem pritisku.

POTENCIAL IN METODA


Za simulacijo interakcije dveh tekocekristalnih molekul smo uporabili Gay-Berne-ov potencial [1]:

$$U(\hat{\vec{u}_1},\hat{\vec{u}_2},\vec{r})=4\varepsilon(\hat{\vec... ...igma(\hat{\vec{u}_1},\hat{\vec{u}_2},\hat{\vec{r}}) + \sigma_0}\biggr\}}^{12} -$$

$${\biggl\{{\sigma_0 \over r - \sigma(\hat{\vec{u}_1},\hat{\vec{u}_2},\hat{\vec{r}}) + \sigma_0}\biggr\}}^6\Biggr] ,$$

kjer je potencialna jama dolocena z $\varepsilon(\hat{\vec{u}_1},\hat{\vec{u}_2},\hat{\vec{r}})$ in $\sigma(\hat{\vec{u}_1},\hat{\vec{u}_2},\hat{\vec{r}})$. Odvisna je od relativne orientacije molekul ( $\hat{\vec{u}_1},\hat{\vec{u}_2}$) glede na intermolekularni vektor ($\hat{\vec{r}}$). Pomembna sta parametra anizotropije:
$\chi'={\Bigl\{1- ({\varepsilon_e \over \varepsilon_s})^{1 \over \mu} \Bigr\} \over \Bigl\{1+({\varepsilon_e \over \varepsilon_s})^{1 \over \mu}\Bigr\}}$, $\chi={\Bigl\{({\sigma_e \over \sigma_s})^2-1\Bigr\} \over \Bigl\{({\sigma_e \over \sigma_s})^2+1\Bigr\}}$.

Uporabili smo standardno GB parametrizacijo [2]:
$\mu=2, \nu=1,{\varepsilon_e \over \varepsilon_s}={1 \over 5}$, ${\sigma _e \over \sigma _s}=$3.
Meßanico molekul dolßine 3 in 1.5 smo simulirali z Lorentz-Berthelot-ovimi pravili, tako da smo za skupino krajßih molekul uporabili dolßino: ${\sigma _e \over \sigma _s}=$1.5.

Produkcija je bila izvedena z (NVT) Monte Carlo simulacijo N=1000 molekul v kubicnem prostoru s periodicnimi robnimi pogoji in konvecijo najblißjih sosedov [3,4]. Predstavljeni so samo rezultati za ekvimolarno meßanico (500 molekul dolßine 3 in 500 molekul dolßine 1.5).

REZULTATI:


Ureditveni parameter $P_2$ smo dolocili iz tenzorja Q [5]:

$$Q_{\alpha,\beta}={{1}\over{N}} {\sum_{i=1}^{N}{{3u_\alpha u_\beta - \delta_{\alpha,\beta}}\over{2}}}$$

kot najvecjo lastno vrednost in direktor $\vec n$ kot pripadajoci lastni vektor. $P_2$ smo izracunali za celotni sistem, za referencne delce $P_2$(1) ( ${\sigma _e \over \sigma _s}=$3) in za dodane krajße $P_2$(2) ( ${\sigma _e \over \sigma _s}=$1.5 ).

Opazovali smo tudi pritisk p

$$p=\rho T + {{1}\over{3 V}} {\sum_{i}\sum_{j > i} { - \vec{r_{ij}^*} * { {\partial U_{ij}} \over {\partial \vec{r^*}}} }} .$$

Pozicijsko strukturo smo opisali s parkorelacijsko funkcijo

$$g(r^*)= {V \over{N^2}} \langle {\sum_{i}\sum_{j \ne i}{\delta(r^*-r_{ij}^*)}} \rangle$$

orientacijsko pa z orientacijsko parkorelacijsko funkcijo

$$g_2(r^*) \propto < cos ( u_i, u_j; r_{ij}=r* ) >.$$

Longitudinalna parkorelacijska funkcija $g_l(r^*)$ je g($r^*$) proicirana na direktor $\vec n$. Podobno je transverzalna parkorelacijska funkcija $g_t(r^*)$ g($r^*$) proicirana pravokotno na direktor $\vec n$. Vse so bile izracunane za celotni sistem in za obe komponenti posebej.

Slika: Potential energy

Slika: Pressure

Slika: Order parameter $P_2(1)$ for molecules with length 3

Slika 4: $g(r^*)$ for a 50% 3/1.5 mixture at T*=1.25: a)1-1 correlations (reference particles, ${\sigma _e \over \sigma _s}=$3); b)2-2 correlations ( ${\sigma _e \over \sigma _s}=$1.5 particles); c)1-2 correlations (3/1.5)

Slika 5: Longitudinal $g(r^*)$ for a 50% 3/1.5 mixture at T*=1.25: a)1-1 correlations (reference particles, ${\sigma _e \over \sigma _s}=$3); b)2-2 correlations ( ${\sigma _e \over \sigma _s}=$1.5 particles); c)1-2 correlations (3/1.5)

Slika 6: Transversal $g(r^*)$ for a 50% 3/1.5 mixture at T*=1.25: a)1-1 correlations (reference particles, ${\sigma _e \over \sigma _s}=$3); b)2-2 correlations ( ${\sigma _e \over \sigma _s}=$1.5 particles); c)1-2 correlations (3/1.5)

Slika 7: $g_2(r^*)$ for a 50% 3/1.5 mixture at T*=1.25: a)1-1 correlations (reference particles, ${\sigma _e \over \sigma _s}=$3); b)2-2 correlations ( ${\sigma _e \over \sigma _s}=$1.5 particles); c)1-2 correlations (3/1.5)

Slika 8: Longitudinal $g_2(r^*)$ for a 50% 3/1.5 mixture at T*=1.25: a)1-1 correlations (reference particles, ${\sigma _e \over \sigma _s}=$3); b)2-2 correlations ( ${\sigma _e \over \sigma _s}=$1.5 particles); c)1-2 correlations (3/1.5)

Slika 9: Transversal $g_2(r^*)$ for a 50% 3/1.5 mixture at T*=1.25: a)1-1 correlations (reference particles, ${\sigma _e \over \sigma _s}=$3); b)2-2 correlations ( ${\sigma _e \over \sigma _s}=$1.5 particles); c)1-2 correlations (3/1.5)

Slika 10: Snapshots of the 50% 3/1.5 mixture at T*=1.25 and $\rho *=$0.57 after the first compression; a) 3/1.5 particles; b) ${\sigma _e \over \sigma _s}=$3 particles; c) ${\sigma _e \over \sigma _s}=$1.5 particles

Slika 11: Snapshots of the 50% 3/1.5 mixture at T*=1.25 and $\rho *=$0.57 after the second compression; a) 3/1.5 particles; b) ${\sigma _e \over \sigma _s}=$3 particles; c) ${\sigma _e \over \sigma _s}=$1.5 particles

DISKUSIJA:

• Termodinamicne vrednosti:

  Pri obeh kompresijah/dekompresijah smo opazili podobni histerezi.
  

• Statisticne vrednosti:

  Pri majhni gostoti (pod 0.42) sta obe komponenti v izotropni fazi. Pri veliki gostoti (nad 0.47) daljße molekule preidejo v smekticno fazo, pri cemer se jasno formirajo plasti, vendar krajße molekule ostanejo kljub plastovitosti v izotropni fazi.
  

• Vizualizacija:

  Direktorja nista obakrat isto usmerjena za obe kompresiji, vendar je stopnja urejenosti za oba primera enaka. Jasno je vidna 'sendvic struktura'.

LITERATURA:

1. J.G.Gay and B.J.Berne,
   J.Chem.Phys., 74, 3316, (1981)
2. G.R.Luckhurst, R.A.Stephens and R.W.Phippen,
   Liq. Crystals, 8, 451, (1990)
3. F.J.Vesely:
   Computational Physics. An Introduction.
   (Plenum Press, London/New York, (1995))
4. M.P.Allen and D.J.Tildesley:
   Computer Simulation of Liquids.
   (Oxford University Press, Oxford, (1990))
5. G.R. Luckhurst and C.A.Veracini:
   The Molecular Dynamics of Liquid Crystals.
   (Kluwer Academic Publishers, Dordrecht, (1994))

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