06.07.2013 Views

Dual Fronts in a Phase Field Model Karl Glasner* and Robert ...

Dual Fronts in a Phase Field Model Karl Glasner* and Robert ...

Dual Fronts in a Phase Field Model Karl Glasner* and Robert ...

SHOW MORE
SHOW LESS

Create successful ePaper yourself

Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.

¢¡¤£¦¥¨§¤© £ £§¥ ¥<br />

¤<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

¦<br />

¨¤<br />

<br />

<br />

<br />

<br />

<br />

¦<br />

£¢¥¤§¦©¨£¢§§§<br />

¡<br />

<br />

¡¦¦<br />

£¢§§¤¡ £¢ §<br />

<br />

£¨§§


£¢¦<br />

¡<br />

¥¤§¦ ¨¢<br />

<br />

<br />

©<br />

<br />

<br />

¦<br />

<br />

¨©¤<br />

<br />

<br />

<br />

<br />

¨¢<br />

<br />

<br />

<br />

<br />

<br />

<br />

¥¦<br />

<br />

<br />

<br />

<br />

<br />

<br />

¤<br />

<br />

¥<br />

<br />

<br />

¤ ¦<br />

¤¨<br />

<br />

<br />

¤ <br />

<br />

<br />

¤¤<br />

<br />

<br />

¤¨<br />

<br />

<br />

<br />

<br />

©<br />

<br />

<br />

<br />

<br />

¦<br />

<br />

¨ <br />

<br />

<br />

<br />

¢¤<br />

¦¦


¦<br />

<br />

¦ <br />

¤<br />

<br />

<br />

<br />

<br />

¦<br />

§¦§¨©<br />

¡£¢¥¤ <br />

<br />

¦¦<br />

¨<br />

¦<br />

§¦<br />

¦<br />

<br />

<br />

<br />

¤<br />

<br />

© <br />

<br />

© <br />

<br />

<br />

<br />

©<br />

<br />

¤ <br />

<br />

© ¨ ¦<br />

¢<br />

<br />

<br />

<br />

¥ <br />

<br />

<br />

<br />

¦¦<br />

<br />

£ <br />

©


¢<br />

<br />

<br />

¤ ¤ ¦ <br />

<br />

<br />

¢ £ ¢ ¡ <br />

¥¤ ¤ ¢<br />

¢ <br />

<br />

¦ §¦¤<br />

<br />

<br />

<br />

¨ <br />

¨<br />

<br />

<br />

© © © <br />

<br />

<br />

<br />

<br />

¨<br />

<br />

<br />

¤<br />

<br />

<br />

<br />

<br />

¢<br />

¤ <br />

<br />

<br />

§<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

§<br />

£ <br />

<br />

§£ <br />

<br />

<br />

<br />

<br />

¤ ¤ § <br />

£<br />

<br />

§<br />

¤<br />

<br />

<br />

¦¦ §¤<br />

<br />

¢ £<br />

£


§¦¦<br />

<br />

<br />

¦¦<br />

<br />

<br />

<br />

<br />

¡ ¡ £¢<br />

<br />

¤¢¤¨¨<br />

<br />

¡ ¦¥ <br />

§¢ <br />

¨¢ ¢ § <br />

<br />

<br />

<br />

¡ <br />

¤ ¤<br />

<br />

<br />

<br />

<br />

¡ ©¢ ¡ <br />

<br />

¢¢ <br />

¢<br />

¤¢ <br />

<br />

<br />

<br />

<br />

© <br />

<br />

¤ ¢ ¨ <br />

¥<br />

¢¡ <br />

¢<br />

¦ <br />

<br />

¦¤¦¦<br />

¦


1<br />

0<br />

−1<br />

λ = 10<br />

0 5 10 15 20<br />

¦<br />

<br />

¦¤¦<br />

<br />

¦¤¤¦<br />

¢ ¢ ¢¡ ¢ ¨ <br />

<br />

¦¦<br />

¦¤<br />

<br />

¤<br />

¤¤ <br />

<br />

<br />

<br />

¦<br />

<br />

<br />

<br />

¤<br />

¤ <br />

¤ <br />

<br />

¢ ¦¦<br />

<br />

¢<br />

¢


¢¡¤£¦¥¨§©¡¥¡§¥©¦©¥§<br />

¦<br />

©<br />

<br />

<br />

<br />

¦©<br />

<br />

<br />

<br />

¤<br />

<br />

<br />

<br />

¦<br />

<br />

§¦¥¥§ ¡¦¦ ¦¡ ¦ §¦ ¨ ¦ ¨ ¢<br />

¦<br />

<br />

¢¨ ¤ ¦ <br />

§¦¨ ¢¦<br />

<br />

©¦ §¦ ¦ ¦ ¡£¢¦¦ ©¦¤ ¢¥¤ §¦¤ ¦ <br />

§¦¤ ¦¤ ¦¤ ¦¤ ¢¡ ¦¡<br />

<br />

<br />

<br />

¢¡ ¦¢¥¤<br />

© <br />

<br />

<br />

¥ <br />

<br />

<br />

¦<br />

<br />

<br />

¤ <br />

<br />

<br />

©©¡©¥§¥ ¢ <br />

¢¤<br />

<br />

<br />

<br />

<br />

<br />

¢ £ <br />

<br />

<br />

<br />

<br />

<br />

©¢<br />

¨<br />

<br />

¤<br />

<br />

<br />

<br />

¦


Negative entropy s<br />

Negative entropy s<br />

1<br />

0.5<br />

0.4<br />

0.3<br />

0.2<br />

0.1<br />

0<br />

6 0<br />

4<br />

λ =λ =2.72<br />

s m 2<br />

Parameter λ<br />

λ =3.76<br />

s 4<br />

λ =2.72<br />

m<br />

2<br />

Parameter λ<br />

Δ = 0.65 e = −0.15<br />

0<br />

0<br />

−1<br />

Δ = 0.35 e = 0.15<br />

−1<br />

−0.5<br />

0<br />

0.5<br />

1<br />

<strong>Phase</strong> variable φ<br />

−0.5<br />

0<br />

0.5<br />

<strong>Phase</strong> variable φ<br />

¤ ¦ <br />

<br />

¤¡ ¢ ¨¢ ¦ <br />

¡ ¢ ¤¢ ¤<br />

§¦¦<br />

¤ <br />

£¢<br />

¢ ¢ ¢ ¤ <br />

<br />

¡¢<br />

¦ §¦¦<br />

<br />

<br />

<br />

£¢ <br />

<br />

<br />

<br />

1


¤<br />

©<br />

<br />

<br />

<br />

¤<br />

¢ §¦ <br />

<br />

<br />

<br />

§¦¥¥§¡ ¡¦¦ ¦¡ ¦ §¦¨ ¢¦ ¨<br />

© ¡£¢¥¤<br />

¢¨ ¡¦¤ §¦¤ §¦¤ ¦ £¢¦ ¤ ©<br />

¦ ¦¨§ ¥¨ © ¢¨¥© ¤ © ¦¡ ¢<br />

<br />

¤¥¤<br />

<br />

<br />

¤ <br />

<br />

<br />

¦¦<br />

£¢ § <br />

<br />

¢£¢ §£¢ <br />

¤<br />

¨ ¢¤<br />

<br />

<br />

<br />

¦<br />

<br />

<br />

¢ © ¦ ¦<br />

<br />

<br />

<br />

<br />

<br />

¢¦<br />

¦<br />

¦<br />

<br />

<br />

<br />

<br />

¢ <br />

¢<br />

<br />

¤ <br />

<br />

¦<br />

<br />

<br />

<br />

¤


¡ <br />

<br />

<br />

<br />

¤<br />

¤<br />

£¢¦<br />

¤¢<br />

<br />

<br />

<br />

© ¥§¦©¨<br />

<br />

§¦<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

©<br />

<br />

<br />

<br />

¤<br />

<br />

<br />

¨ <br />

<br />

<br />

<br />

<br />

¤ <br />

<br />

¢ <br />

§ <br />

<br />

¤¢ ¢<br />

<br />

©<br />

<br />

¢ <br />

¢¤<br />

¢<br />

<br />

¦¤ ¤ ¦ ¢ ¦¦ ¤<br />

<br />

¢<br />

<br />

© <br />

¤<br />

<br />

©©<br />

<br />

¤ <br />

<br />

¦<br />

<br />

£<br />

¢


¦<br />

<br />

<br />

<br />

<br />

¢ <br />

©©©¦ ¤<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

¤ <br />

¡ ©<br />

<br />

¦<br />

<br />

<br />

<br />

¥§¦£¢ ¥¤ <br />

¤ £<br />

¤<br />

<br />

<br />

<br />

¤ ¦ ¤ ¢ <br />

<br />

<br />

¤<br />

¦<br />

¢ <br />

¤ <br />

¡ £¢ ¡ ¡ £ ¡ ¢<br />

<br />

¦ <br />

<br />

<br />

£¢ <br />

<br />

£¢ <br />

<br />

<br />

¢<br />

<br />

¦


V<br />

2<br />

1.5<br />

1<br />

0.5<br />

0<br />

10 15 20<br />

λ<br />

25 30<br />

<br />

¡<br />

¢ ¢ <br />

<br />

<br />

¢ ¢<br />

¤¦<br />

¢ <br />

<br />

¦<br />

¦<br />

¥§¦¥ ¢ ¤£¦¥ ¨§ <br />

¤ <br />

<br />

<br />

<br />

<br />

<br />

¤<br />

<br />

<br />

¨ <br />

<br />

<br />

<br />

© <br />

<br />

<br />

¢ <br />

<br />

¦ ¢


¦©¨ ¡<br />

©©£¢<br />

<br />

<br />

¥¤<br />

¦ § <br />

¡ <br />

<br />

<br />

<br />

£ §¦ § <br />

§<br />

<br />

¢ <br />

<br />

<br />

<br />

<br />

<br />

<br />

¨<br />

©¨<br />

¦ <br />

£¢ <br />

<br />

¢ <br />

©¨¦<br />

<br />

<br />

<br />

¦£¢ ¥¤©£¢<br />

¥¤ ¢¤<br />

<br />

<br />

<br />

<br />

<br />

¦<br />

<br />

<br />

<br />

¦<br />

<br />

<br />

§ <br />

<br />

£¢<br />

¦ ¢<br />

<br />

<br />

<br />

£¢ £¢ ¤<br />

<br />

¨¦


¦<br />

<br />

¨<br />

¡ ¢ ¢<br />

¨ £ ¢ ¢ <br />

¨ £ ¢ ¢<br />

¦ ¢<br />

<br />

<br />

<br />

¤ ¤ ¤ <br />

¢ <br />

<br />

¤ ¢ <br />

<br />

©<br />

<br />

<br />

¨<br />

<br />

<br />

<br />

§<br />

¦ ¢<br />

<br />

£ ¦§ <br />

<br />

¢¦<br />

£¢<br />

¦ <br />

<br />

<br />

<br />

<br />

<br />

<br />

¢<br />

<br />

<br />

¤ £¢<br />

¦ <br />

<br />

¥ ¢ ¢ ¨ <br />

¦ ¢ ¢ <br />

<br />

<br />

¨<br />

¨ <br />

<br />

¥ ¢ ¢


¦ ¢ ¦ <br />

<br />

¤ ¤ ¤ <br />

¤ ¢ ¢ <br />

<br />

<br />

¨<br />

§ ¢ <br />

<br />

<br />

<br />

<br />

<br />

¢¡ <br />

<br />

<br />

¤ <br />

<br />

¤ ¤ ¤ <br />

<br />

<br />

¤ <br />

¢<br />

<br />

<br />

©<br />

¦¤<br />

¦ ¤ £¢ ¤ <br />

<br />

<br />

<br />

£ <br />

¥<br />

<br />

¤£ ¤¦¥ <br />

¤ § <br />

¢ <br />

¢§ ¢ <br />

¤ <br />

¤ <br />

¤ ¤ ¦ <br />

<br />

<br />

©¨<br />

¤ ¤¦¥ <br />

£¢£<br />

¤ <br />

<br />

<br />

¢ <br />

£¢£ § <br />

¥ £<br />

<br />

¥ §


£ <br />

¢ ¦ <br />

<br />

¡ ¢§<br />

¦<br />

§ ¢ ¤£¦¥ ¦¥<br />

£ §¦ ¢<br />

¦ <br />

<br />

¨¢<br />

¤ ¦ <br />

¦<br />

¦<br />

<br />

¦<br />

<br />

<br />

<br />

¦<br />

<br />

<br />

¢<br />

£<br />

¦<br />

<br />

<br />

¦ ©<br />

<br />

<br />

¤ <br />

<br />

¦<br />

¤<br />

<br />

<br />

¨<br />

¦


V<br />

0.6<br />

0.5<br />

0.4<br />

0.3<br />

0.2<br />

0.1<br />

0<br />

2.2 2.4 2.6 2.8 3 3.2 3.4<br />

λ<br />

<br />

<br />

<br />

¦<br />

¢ ¢ ¢ ¢ <br />

¦<br />

<br />

¨¨<br />

<br />

<br />

¨<br />

¡<br />

¦<br />

<br />

<br />

<br />

<br />

<br />

<br />

¢ <br />

¦<br />

¢¥<br />

<br />

¦<br />

<br />

¦<br />

¤¤<br />

<br />

©¦ <br />

<br />

<br />

<br />

¡ <br />

<br />

<br />

¦ <br />

<br />

<br />

<br />

¤


¦<br />

<br />

¤ <br />

<br />

<br />

<br />

<br />

<br />

¢¡¤¤£ <br />

¦ © <br />

<br />

© ¢ ¢ © ¡© <br />

<br />

¡ ¢¢<br />

§¦ ¥ <br />

¨© ¨ <br />

<br />

¤<br />

<br />

¤<br />

¤¤<br />

©©<br />

<br />

¨<br />

©<br />

© <br />

¨<br />

¦<br />

©©<br />

<br />

¨©<br />

¦<br />

¢©©©<br />

<br />

§<br />

¨©<br />

<br />

<br />

©©©


© §¢ <br />

¨<br />

©©©©<br />

¡<br />

¦ <br />

¨<br />

£¢¤©<br />

<br />

© ¥¤ §¦ <br />

¨<br />

¢©©<br />

©¨<br />

<br />

¨<br />

©<br />

¢© <br />

¨<br />

©©<br />

£<br />

¦<br />

¨<br />

<br />

¡<br />

©<br />

§ <br />

¨<br />

¢©<br />

© <br />

¨<br />

©©<br />

<br />

<br />

¨<br />

¦ ¢ ¥¤ <br />

<br />

¤¢© <br />

<br />

©<br />

©© <br />

¨<br />

©©©<br />

¢ <br />

¨<br />

<br />

<br />

©©<br />

<br />

© <br />

¨<br />

©©<br />

<br />

© <br />

¨<br />

<br />

¡<br />

§¤<br />

<br />

©©


© ¤ <br />

¨<br />

©©<br />

<br />

¢¡<br />

<br />

© <br />

¨<br />

¤£<br />

©©<br />

<br />

<br />

¨<br />

¨ © <br />

¢¦¥<br />

<br />

¤ ¢ ¤ ©<br />

<br />

<br />

©©©<br />

© ¦ <br />

¨<br />

<br />

©©©©<br />

© §¤<br />

¨<br />

<br />

<br />

©©<br />

<br />

¢© <br />

¨<br />

¦ ¢ <br />

<br />

©©©<br />

© ¡ <br />

¨<br />

¢ <br />

¢©©©©<br />

<br />

¨©<br />

<br />

© ©©<br />

<br />

¤<br />

¨©<br />

<br />

©¨<br />

©©©<br />

<br />

¨¢©<br />

<br />

©©<br />

<br />

<br />

¨<br />

©<br />

<br />

¢


¨<br />

<br />

©©<br />

<br />

© <br />

¨<br />

<br />

©©©©<br />

<br />

© <br />

¨<br />

© ©©<br />

<br />

© § <br />

¨¡<br />

¥<br />

<br />

©©©©

Hooray! Your file is uploaded and ready to be published.

Saved successfully!

Ooh no, something went wrong!