J. Chem. Phys. 96, 7696 (1992); h ps://doi.o g/10.1063/1.462370 96, 7696
© 1992 Ame ican Ins i u e o Physics.
Collec i e exci a ions in liquid me hanol:
A compa ison o molecula , la ice-
dynamics, and neu on-sca e ing esul s
Ci e as: J. Chem. Phys. 96, 7696 (1992); h ps://doi.o g/10.1063/1.462370
Submi ed: 03 July 1991 . Accep ed: 14 Feb ua y 1992 . Published Online: 31 Augus 1998
J. Alonso, F. J. Be mejo, M. Ga cía-He nández, J. L. Ma ínez, W. S. Howells, and A. C iado
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Collec i e exci a ions
in
liquid me hanol: A compa ison
o
molecula , la ice-
dynamics, and neu on-sca e ing esul s
J. Alonsoa)
Ru he o d Apple on Labo a o y, Chil on, Didco , Oxon
OXII
OQX,
Uni ed Kingdom
F.
J. Be mejo
Ins i u o de Es uc u a de
la
Ma e ia, Se ano 123, E-28006 Mad id, Spain
M.
Ga c a-He nandeza)
Ru he o d Apple on Labo a o y: Chil on, Didco , Oxon
OXII
OQX,
Uni ed Kingdom
J.
L.
Ma inez
Ins i u
Laue
Lange in, 156X, F-38042 G enoble Cedex, F ance
W.S.
Howells
Ru he o d Apple on Labo a o y, Chil on, Didco , Oxon
OXII
OQx,
Uni ed Kingdom
A.
C iado
Depa amen o de
F'isica
Ma e ia Condensada, Uni e sidad de Se illa, Apa ado 1065, E-4080 Se illa,
Spain
(Recei ed 3 July 1991; accep ed
14
Feb ua y 1992)
The collec i e dynamics
o
liquid me hanol-d4 is s udied by means
o
molecula -dynamics
simula ion.
The
model po en ial
is
alida ed by means
o
la ice ene gy calcula ions and shows
a e y good ag eemen wi h he expe imen ally ob ained c ys al s uc u e. Cen e -o -mass
densi y
and
momen um luc ua ions a e in es iga ed in he
(Q,w)
egion which
is
also
accessible
o
inelas ic neu on-sca e ing
(INS)
echniques. A simple iscoelas ic model
p e iously used o he analysis
o
INS
da a
is es ed agains he dynamic s uc u e ac o
compu ed om he simula ion. A di ec compa ison wi h he INS esul s hemsel es is also
~ade
and
quali a i e ag eemen is ound. Also, a en a i e assignmen
o
he peaks appea ing
m he
cu en -cu en
co ela ions
is
made on he basis
o
la ice-dynamics calcula ions o he
polyc ys alline low- empe a u e a phase.
I.
INTRODUCTION
The collec i e dynamical esponse
o
s ongly associa -
ed liquids has a ac ed a subs an ial e o om heo e ical,
compu e simula ion,
and
expe imen al app oaches (see, o
ins ance, Re . 1). Con a y o wha was obse ed in molecu-
la liquids composed
o
pa icles in e ac ing ia nea ly pu e
Lenna d-Jones po en ials such as liquid
ni ogen/
he p es-
ence
o
s ong in e molecula co ela ions induced by he
hyd ogen bond was expec ed o ende hese sys ems amena-
ble o expe imen al measu emen s using inelas ic neu on
sca e ing. As a
ma e
o
ac , he possibili y
o
obse ing
sho -wa eleng h collec i e exci a ions in he kinema ic
ange accessible o he mal neu ons was poin ed ou some
ime ago by Impey, Madden, and McDonald3 on he basis
o
molecula -dynamics
(MD)
simula ion esul s o liquid wa-
e . A subsequen expe imen on inelas ic neu on sca e ing
(INS)4 opened
up
a deba e conce ning he na u e
o
he
obse ed exci a ion.
On
he o he hand, he in e p e a ion
o
he expe imen al esul s was
pu
in o ques ion by he simula-
ion esul s
o
Wojcik
and
Clemen i,5
bu
a numbe
o
pape s
on a simpli ied e sion
o
he p ojec ion ope a o echnique,
as well as compu e simula ions, seemed o ein o ce he cu -
en in e p e a ion
o
he neu on-sca e ing esul s.6
A s udy
o
he cohe en inelas ic esponse
o
liquid
a)
Pennanen
add ess: Ins i u o de
Es uc u ade
1a
Ma e ia, Se ano 119, E-
28006 Mad id, Spain.
me hanol was pe o med la e 7
and
p o ided addi ional
suppo o he hesis which assigned
an
acous ic na u e
o
he obse ed modes, in opposi ion o hose pos ula ing a
" as -kine ic-mode" na u e
o
he exci a ion.
The
easons o
he choice
o
his ma e ial we e i s ela i e simplici y com-
pa ed o wa e
and
he ac ha i s low mel ing poin enabled
he
s udy
o
he collec i e e ec s wi h a e y small con ibu-
ion om single-pa icle ( ansla ion
and
o a ion) modes.8
The
main di icul y a ising om
he
analysis
o
INS
spec a was he need o de elop a simple analy ic model use-
ul o
da a
analysis pu poses which encompasses mos
o
he
ele an collec i e dynamical pa ame e s. A ela i ely sim-
ple exp ession was ound7 which, based upon a iscoelas ic
ea men ,9 was able o ep oduce he mos p ominen spec-
al ea u es, and he e o e was used
o
in e he dispe sion
beha io om he obse ed inelas ic in ensi ies.
The
p esen s udy has se e al aims. Fi s
o
all, i
is
in-
ended o ob ain p ecise in o ma ion ega ding he collec i e
dynamics
o
his sys em
(a
p elimina y accoun
o
mos ly
single-pa icle esul s has al eady been gi en
10)
which is
suppossed o be simple han wa e since mos
o
he hyd o-
gen-bond ne wo k will be cons i u ed by long winding
chains
10
ins ead
o
he complica ed a angemen s ound in
he
liquid and solid phases
o
wa e .
On
he
o he
hand,
he
p esen exe cise will se e o es he alidi y
o
he app oxi-
ma ion used
o
analyze he neu on
da a
by pe o ming a
simila analysis
o
he simula ed dynamical s uc u e ac-
o s. Finally, i will y o shed some ligh in o
he
o igin
o
7696 J. Che n. Phys. 96 (10), 15 May 1992 0021-9606/92/107696-14$06.00 @ 1992 Ame ican Ins i u e
o
PhySics
Alonso
e
a/.:
Collec i e exci a ions
in
liquid
me hanol
7697
collec i e modes o he han he no mal sound whose phys-
ical o igin is s ill an open con o e sy.4,5
The ou line
o
he pape is as ollows. Some ele an
o mulas conce ning densi y and pa icle cu en luc u-
a ions a e e iewed in he nex sec ion.
In
Sec.
III
we
sum-
ma ize he main cha ac e is ics
o
he molecula -dynamics
simula ions as well as he way in which he in e molecula
po en ial has been alida ed, and
we
in oduce some esul s
on mos ly s a ic p ope ies. Sec ion IV is de o ed o he
s udy
o
he spec al ea u es
o
he au oco ela ion unc-
ions
o
densi y and pa icle cu en luc ua ions. A com-
pa ison wi h he ele an unc ions ob ained om a la ice-
dynamical model o he polyc ys al will also be made and
he o igin
o
high- equency modes which a e shown o be
common o he liquid and polyc ys alline phases will be
discussed. The modeling
o
S(
Q,liJ) and he compa ison wi h
inelas ic neu on-sca e ing esul s a e he subjec
o
Sec.
V.
Finally,
ou
conclusions and summa y a e p esen ed in he
las sec ion. De ails abou he la ice-dynamics calcula ions
a e gi en in he Appendix.
II. THEORETICAL BACKGROUND
Dynamic p ocesses in molecula liquids ha e a g ea e
complexi y han in mona omic liquids due o he exis ence
o
addi ional o a ional and ib a ional deg ees
o
eedom. As
a i s app oxima ion,
i
is cus oma y o analyze he dynam-
ics
o
molecula cen e s
o
mass in e ms
o
he o mulas
de eloped o mona omic liquids. I
1,12
In his s udy
we
will be mainly in e es ed in he collec-
i e beha io as cha ac e ized by he luc ua ions in he den-
si y and pa icle cu en s. As usual, his will be done h ough
hei ime-au oco ela ion unc ions
(ACFs),
F(Q, )
=J...
<PQ( +s)p_Q(s»,
N
J/(Q, )
=
~
([q'jQ( +s)]
[q'LQ(s)]>,
J/(Q, )
=J...
([b'jQ( +s)][b'LQ(s)]>,
(1)
N
known as he in e media e sca e ing unc ion and co ela-
ion unc ions
o
he longi udinal and ans e se compo-
nen s
o
he pa icle cu en , espec i ely.
In
he p eceding
equa ions N
is
he numbe
o
molecules; he a e age
is
aken
o e ini ial condi ions s and wa e ec o s Q
o
magni ude
Q;
q and b a e uni ec o s pa allel and pe pendicula , espec-
i ely, o he wa e ec o Q; inally, he spa ial Fou ie com-
ponen s
o
he mic oscopic numbe densi y and pa icle cu -
en a e de ined
by
N
pQ( )
= L exp[
-iQ·Ra( )],
a=1
N
jQ
(I)
= L
Va
( )
exp[ -iQ·Ra
( )],
a=l
(2)
whe e Ra
( )
and
Va
( )
a e he posi ion and eloci y ec o s
o
he cen e
o
mass
o
molecule a
a
ime
.
The powe spec um
o
he in e media e sca e ing
unc ion is known as he dynamic s uc u e ac o and is
gi en by
I
l~
S(Q,liJ) = -
d
F(Q, )
cos(liJ ) ,
1T
0 (3)
wi h simila exp essions o he spec a
o
he cu en au o-
co ela ion unc ions (CACFs)
J[(Q, )
and J
(Q, ).
F om heo e ical and compu a ional s andpoin s i
is
ad an ageous o s udy he sho - ime beha io
o
he in e -
media e sca e ing unc ion, which p o ides in o ma ion
abou he equency momen s
o
he dynamic s uc u e ac-
o . Expanding
F(
Q, )
in a Taylo se ies,
we
ob ain
F(Q, )
=
Scm
(Q)
I -
liJ~
-+
liJ~liJ7
- -
'"
,
(
2
4 )
2!
4!
(4)
whe e
(5)
and liJi a e he educed second and ou h equency mo-
men s
o
S(Q,liJ), espec i ely. The e a e analogous exp es-
sions o he longi udinal and ans e se componen s
o
he
pa icle CACF; and hei educed second momen s, liJ7(Q)
and liJ;
(Q),
a e ela ed o he Q-dependen elas ic cons an s
el1
(Q)
and
G~
(Q),
espec i ely.
11
F om he de ini ions
o
Eqs.
(I)
and
(2)
and exploi ing
he s a iona y p ope y
o
ime ACFs i ollows ha
F(
Q, )
and J[ (
Q, )
a e no independen .
3
In ac , hei powe spec-
a a e ela ed h ough
II
liJ
2S(Q,liJ) = Q2J[(Q,liJ).
(6)
III. COMPUTATIONAL DETAILS AND STATIC
PROPERTIES
A. The
po en ial
model
A subs an ial numbe
o
a emp s ha e been egis e ed
whe e model po en ials we e de eloped in o de o ep o-
duce he mal and s uc u al p ope ies
o
he liquid phase as
well
as
single-pa icle ime-co ela ions unc ions and he
dynamics
o
bond b eaking and o ming in he hyd ogen-
bond
(HB)
ne wo k.
14
-
18
Howe e , o he ex en
o
ou
knowledge no se ious a emp o in es iga e he collec i e
dynamics in me hanol om a molecula -dynamics poin
o
iew has been made so a .
Fo
his kind
o
compu a ion e y long
MD
uns a e
needed in o de o p o ide accep able s a is ics; he e o e,
lexible molecula models become imp ac ical since hey
usually equi e ime s eps
a
leas I o de
o
magni ude
smalle han he igid models. Among he la e ones, wo
ha e been a o ed by mos esea che s.
14,15
As he eliabili y
o
bo h models is
compa able,15,19
he choice be ween hem
becomes a ma e
o
p e e ence.
Fo
his wo k
we
ha e cho-
sen he model p oposed by Haughney, Fe a io, and Mc-
Donald
ls
which is a si e-si e model, each si e-si e in e ac-
ion being a Lenna d-Jones plus a Coulombic e m.
B.
La ice
ene gy
minimiza ion
The adop ed model o he in e molecula in e ac ions
has been also applied o he low- empe a u e c ys al phase
20
o
me hanol-d4. The eqUilib ium c ys al s uc u e co e-
sponding o his po en ial model has been ob ained as he
minimum-ene gy con igu a ion.
Fo
his pu pose, a New-
J.
Chem.
Phys.,
Vol.
96,
No.1
0,
15
May
1992
7698 Alonso
e
al.: Collec i e exci a ions
in
liquid me hanol
on-Raphson minimiza ion p ocess wi h he p og am
21
WMIN
has been ca ied
ou
s a ing
a
he expe imen al c ys-
al s uc u e, using he cell pa ame e s and molecula o a-
ions and ansla ions as a iables. The Ewald me hod
22
has
been used o deal wi h he Coulombic sums, and he cu o
dis ance
o
12
A has been adop ed o he Lenna d-Jones
in e ac ions.
The changes occu ing in he p ocess by he la ice pa-
ame e s
a,
b,
and
ca e
2.9%,3.8%,
and 1.6%, espec i ely,
whe eas he maximum shi in he a omic coo dina es is 0.3
A.
This o de
o
disc epancy be ween expe imen al and cal-
cula ed c ys al s uc u es is usual in c ys al packing calcula-
ions using
a om-a om
po en ial models; he e o e,
we
can
assume ha he p oposed model ep oduces ai ly well he
expe imen al c ys al s uc u e.
C. The
MD
algo i hm
Simula ions
o
a 256
CD
3
OD
molecule sys em we e ca -
ied ou using he compu e p og am desc ibed in Re .
10,
which was w i en ollowing he hin s gi en by Allen and
Tildesley.23 The p og am compu es he ajec o y
o
he sys-
em subjec ed o cubic pe iodic bounda y condi ions using
New onian classical mechanics (NVE-P ensemble). The
molecules we e ea ed as igid bodies composed
o
six mass
poin s (modeling he a oms in he molecule) and h ee in e -
ac ion si es loca ed
a
he oxygen, ca bon, and hyd oxylic
deu e ium posi ions.
The
Ca esian equa ions
o
mo ion
we e in eg a ed using he eloci y e sion
o
he Ve le algo-
i hm
23
wi h a ime s ep
o
10
-
14
s,
and he
RA
TILE
algo-
i hm
24
was used o implemen he holonomic cons ain s
equi ed o keep all in amolecula dis ances ixed. The mo-
lecula geome y
and
pa ame e s de ining he in e ac ion
po en ial we e hose p oposed by Haughney, Fe a io, and
McDonald.
15
Only wo mino modi ica ions on he po en-
ial model we e in oduced. Fi s , he masses
o
he hyd o-
gen a oms ha e been subs i u ed by ha
o
deu e ium in
o de o simula e he ully deu e a ed compound (which
is
he one s udied by neu on sca e ing); and, second, he
long- ange in e ac ions we e unca ed by using a swi ch
unc ion
23
,25
o
um
o smoo hly all he in e ac ions be-
ween pai s
o
molecules whose cen e s-o -mass sepa a ion
was g ea e han he cu o dis ance.
Following he in oduc ion
o
he swi ch unc ion he
in e ac ion ene gy be ween wo molecules, a and/3, becomes
Uap( ;a,
jp ) =
S(R
~p)U~p( ;a, jp),
(7)
whe e
RaP
is
he dis ance be ween cen e s
o
mass o bo h
molecules, and
Sex)
is he unique i h-o de polynomial
sa is ying
{I o
x<R
L
Sex)
= 2
o o
x>R
u,
(8)
and ha ing con inuous i s and second de i a i es
a
he end
poin s
o
he in e a1.
25
In
he p esen calcula ions
we
ook
RL =
12.25
A and
Ru
= 12.65 A o bo h uns. No e ha
due o he la ge cu o adius and he elec oneu ali y
o
me hanol molecules he leading e m in he unca ed po en-
ial co esponds o he dipole-dipole in e ac ion and be-
ha es as
R;;/.
The
unc ion
U~( ;a, jp)
is
he po en ial
unc ion e e ed o as model
HI
by Haughney, Fe a io,
and McDonald
I 5 and has he o m
o
a sum
o
si e-si e in e -
ac ions.
D.
Accu acy
o
he esul s
The
use
o
pe iodic bounda y condi ions imposes some
es ic ions upon he leng h and ime scales
o
he phenome-
na
ha
can be s udied wi h con en ional molecula dynam-
ics.
F om
a dynamical poin
o
iew he quan i y o in e es
is
he ecu ence ime (de ined as 'T ee = LboJc, whe e c
is
he
eloci y
o
sound)
,26
which
is
abou 1.4 ps o he simula-
ions epo ed he e.
In
o de o ensu e
ha
no measu able
a i ac s we e in oduced due o ecu ence e ec s, he
F(
Q, ) o he lowes Q alue analyzed (0.25 A -
I)
was ex-
amined in de ail and no spu ious ea u es ha could be a -
ibu ed o ecu ence e ec s we e ound o imes smalle
han 4 ps. Besides, co ela ion unc ions compu ed om
i-
ni e ime-leng h
MD
ajec o ies a e subjec o nume ical
and s a is ical
e o s;27,23
he e o e, he measu able
ACF
is
gi en by
MD( )
= ( )
+
€( ),
whe e ( )
is
he ue
ACF
and
€( )
ep esen s he e o e m. Taking his in o accoun ,
we
see ha he spec um
o
a
MD
ACF
is gi en by
1
L""
MD(W)
= -
d MD( )W( )
cos(w )
1T
0
=
[ ew)
+
€(w)]
®
W(w),
(9)
whe e
w( )
is
some window unc ion and
W(w)
i s Fou ie
ans o m. The e o e, he a ainable spec um
is
a con olu-
ion
o
he ue spec um wi h he window unc ion plus a
noise e m. Besides, due o he possible exis ence
o
ecu -
ence e ec s as men ioned ea lie , he window should decay
o ze o o imes
o
he o de
o
'T ee o a oid he con amina-
ion
o
he spec a by ecu ence e ec s. Howe e , his wo -
sens he esolu ion in he equency domain and a low
Q,
whe e he spec al ea u es o in e es a e na ow and loca ed
a
low equency, i seems ha he e
is
no choice bu o use a
wide window hough i could esul in
an
inc eased spec al
noise le el.
The in e media e sca e ing unc ion and pa icle cu -
en au oco ela ion unc ions ha e been compu ed using
he as Fou ie ans o m
(FFT)
me hod
23
o bo h he -
modynamic s a es, and o many alues
o
Q spanning om
0.25 o 5 A -
I.
In
all cases an a e aging o e he whole se
o
allowed wa e ec o s (compa ible wi h
he
pe iodic bound-
a y condi ions) was pe o med.
In
o de o ob ain an es ima ion
o
he quali y
o
ou
esul s he 200 K un was subdi ided in o eigh sub uns
o 20
ps leng h. Au oco ela ion unc ions we e compu ed o
hese sub uns and la e a e aged and used o es ima e he
s anda d de ia ion
o
he mean.
Fo
Q = 0.25 A
-I
( ha
wi h only h ee independen wa e ec o s could be consid-
e ed a un a o able case) he e o in
F(Q, )
was a ound
10% du ing he i s 5 ps. Fou ie ans o ming he eigh
sub uns sepa a ely
we
ound he
e o
in
S(
Q,w
)/S(
Q) o be
again 10% and cons an in he equency ange s udied.
As
he numbe
o
allowed wa e ec o s inc eases wi h Q he
e o s diminish acco dingly down o 3 % o
Q>
2 A -
I.
The
as decay
o
cu en au oco ela ions unc ions wi h ime
J. Che n. Phys., Vol. 96, No. 10, 15 May 1992
Alonso e
al.:
Collec i e exci a ions
in
liquid
me hanol
7699
enables us o employ he
e o
analysis
o
Zwanzig and
Ailawadi,27 inding a maximum
e o
o
± 0.05 o he no -
malized CACFs
a
low Q which educes o ± 0.02 as Q
inc eases.
E.
Some esul s on s a ic p ope ies
Two
o
he h ee he modynamic s a es s udied in Re .
10
a e eanalyzed in his wo k, wi h special emphasis
on
he
collec i e, ime-au oco ela ion unc ions.
The
a e age al-
ues
o
simple he modynamic p ope ies we e compu ed
along wi h he ajec o ies, and a e shown oge he wi h den-
si ies
and
un leng hs in Table
I.
The
un leng hs we e chosen
long enough so ha
we
can a e age o e he longes pe iod
luc ua ions
o
he he modynamics p ope ies obse ed in
he sys em.
The
s a ic s uc u e ac o as well as he pa ial pai -
co ela ion unc ions ha e been analyzed in a p e ious pa-
pe . 10
The
educed second
and
ou h momen s
o
S(
Q,{i)
,
{i)o,
and
(i)[o
ha e been p esen ed in Fig. 1 oge he wi h hei
espec i e ideal-gas limi s.
Ou
esul s a e e y simila o
hose
o
Wojcik and Clemen i o wa e S hough he e-
quencies in ol ed a e somewha smalle o me hanol.
The
second momen shows a s ong dependence on he s a ic
s uc u e ac o Q < 2.5 A -I as expec ed om Eq.
(5),
dis-
playing a p onounced dip
a
Qp
[i.e., he posi ion
o
he main
maximum
o
Scm
(Q)
],
and emaining close o i s ideal alue
om he e on.
The
ou h momen does no app oach i s
ideal limi so quickly hough i oscilla es a ound i s sel - al-
ue om ela i ely low
Q.
Besides, he dimensionless quo ien
o
(i)/
and
i s single-pa icle coun e pa , namely
(i) 'l (Q)
=
(n~
+ 3Q2kB
TIM)
112,
(10)
exhibi s li le
o
no dependence on he he modynamic s a e,
n~
being he "Eins ein equency" (see Table
I).
On
he
o he hand, he oscilla ions a ound he sel - alue decay e y
slowly, being s ill no iceable a
Q~
5 A -I hough hey lose
ampli ude o
Q>
2.5
A-I,
which demons a es
ha
collec-
i e e ec s a e impo an down o leng h scales
o
jus
a
ew
angs oms (i.e., nea es -neighbo dis ances).
Resul s o he second equency momen
o
he spec a
o
he ans e se CACFs also show independence
o
he
he modynamic s a e
(a
leas o he wo s a es s udied
he e) when exp essed in dimensionless o m:
{i)
(Q)/{i),;,,' ( Q), whe e
(i)~el (Q)
=
(n~
+
Q2kBTIM)
112.
(11)
Collec i e e ec s seem o be impo an only in he e-
gion Q < I A -
I;
o la ge momen um ans e s he
CACF
beha es mos ly like i s single-pa icle coun e pa .
The
in i-
ni e- equency mac oscopic shea modulus,
Goo'
can be es i-
ma ed om he low-Q alues
o
(i)
(Q).
App oxima ing
Goo
by
Goo
(Q = 0.25 A -
I)
we ge alues (Table
I)
ha
a e
abou hal hose ound o liquid wa e .s
IV. DENSITY AND CURRENT FLUCTUATIONS
A. Densi y luc ua ions
The
dynamic s uc u e ac o
S(
Q,{i)
has been plo ed
in Figs. 2 and 3 as a unc ion
o
he angula equency
(i)
o
he smalles accessible alues
o
he momen um ans e .
A
200 K he wo lowes -Q spec a (0.25 and 0.35 A -
I)
exhibi
dis inc i e B illouin side peaks
a
equencies
(i)
B
o
0.44 and
0.56 X
10
13
adls,
espec i ely. Then he peak becomes o e -
damped as he momen um ans e is inc eased, hough i
is
s ill no iceable as a weak shoulde in he spec a o momen-
um ans e s below 0.6 A -
I.
Ano he in e es ing ea u e
o
S(
Q,{i)
is
he inc ease
o
he in ensi y in he equency e-
gion a ound
2X
1013
ad/s,
which a Q = 0.86 A
-I
mani-
es s i sel clea ly as a shoulde in he spec um.
The
p es-
ence
o
his shoulde sugges s he exis ence
o
a second
exci a ion in he liquid coexis ing wi h he lowe - equency
mode.
A
300 K he o e all si ua ion
is
e y simila , al-
hough he B illouin peaks al eady appea as o e damped
a
he lowes alue
o
he momen um ans e accessible o
ou
MD
expe imen (see Fig.
3).
F om
Fig. 2 i is clea
ha
he peak maxima whene e
isible show a no iceable dispe sion
(a
leas up o 0.4
A-I).
TABLE
I.
A e age alues
o
simple he modynamic p ope ies o simula ed liquid me hanol·d4. Run
numbe s a e as
in
Re .
10.
Quan i y Uni s Run 1 Run 3
Densi y
kglm
3 943.0 885.0
L"'
n nm 2.5333 2.5875
Elapsed ime
ps
177.00 100.10
Tempe a u e" K 202.3 ± 0.3 297.7 ±
1.0
Uln e
b
kl/mol
-39.88 -34.79
P essu e"
MPa
52.0 105.3
n.3
ps
-2
463.7 402.2
Scm
(0) 0.029 ± 0.009 0.038 ± 0.003
XT
10-
'0
Pa-'
6.5 ±
1.9
6.3 ± 0.5
cT
mls
1290 ± 190 1340
±60
CII
(Q
= 0.25
A.
-') 10·
J/m
3 15.77 13.16
G~
(Q=0.25A.··')
10·
J/m
3 6.72 5.96
"The
s anda d de ia ion
o
he a e age empe a u e has been es ima ed by aking in o accoun he s a is ical
ine iciency
o
he da a (Re . 23).
"No long· ange co ec ions
ha e
been
applied.
J.
Che n.
Phys.,
Vol.
96,
No.1
0,
15
May
1992

7700 Alonso
e
al. : Collec i e exci a ions in liquid me hanol
2.5-
..--...
n
2.0-
---
'"Cl
o
I-<
M
1.5-
.-
0
,........
'--"
---
1.0-
:3
0.5-
c»
*
*
*
•
wz(Q)
•
.....
.......
4>+'
~
.•.••
4>
#,
+
~
...
..-"
...
++
..
, +
..
,
.'
......
.
.......
-........ e
.........
.
.....
-
...
.
::::::::::~:::::::::~
...........
.
.'
*
• .. *
* • • • * •
-
....
~
o
0.5
1.0 1.5
2.0
2.5
3.0
3.5
4.0
4.5
Q / A-I
FIG.
1.
"Dispe sion ela ions" o he educed second
and
ou h momen s
o he
dynamic s uc u e ac o ,
S(
Q,(()
a
200 K.
Thei
ideal-gas (high-Q) limi s,
which a e gi en by
Q(kBT
1M)
1/2
and
Q(3k
BT
1M)
112,
espec i ely, a e shown as s aigh do ed lines.
The eloci y
o
p opaga ion o his exci a ion can be es i-
ma ed om
he
equency
o
he side maxima
o
S(
Q,w),
W B
(Q),
acco ding 0
28
(12)
which in he hyd odynamic limi educes o he adiaba ic
sound eloci y. Subs i u ing he abo e-men ioned equen-
cies, we ob ain sound eloci ies
o
1760 and 1600
m/s
o
Q = 0.25
and
0.35 A-I, espec i ely.
Ou
simula ion esul s
seem o ag ee easonably well wi h ecen measu emen s
o
he hype sonic eloci y.29 In pa icula , he adiaba ic sound
eloci ies epo ed a e 1475 and 1080
m/s
o liquid me ha-
nol-d4
a
200
and
300
K,
espec i ely.
The
highe alue ob-
ained om
ou
200 K simula ion can be in e p e ed as a
posi i e sound dispe sion in he simula ed sys em.
B.
Longi udinal cu en luc ua ions
In o de
o
in es iga e ho oughly he dispe sion
o
he
B illouin mode
and
o cla i y whe he a second mode exis s,
we ha e unde aken he calcula ion
o
he spec um o longi-
udinal cu en luc ua ions which is ela ed o
S(
Q,w)
h ough Eq.
(6).
Some well-known bene i s
o
s udying he
spec a
o
he
cu en
ACF
a e
(i)
J1
(Q, )
is oscilla o y
and
decays as e
han
F(
Q, ); he e o e, he spec a a e less a -
ec ed by unca ion
o
windowing e ec s;
(ii)
he e
is
no
cen al peak
and
consequen ly no o e lapping wi h he
quasielas ic componen ;
(iii)
he spec a always exhibi ,
a
leas , a pai
o
peaks (S okes
and
an i-S okes);
and
(i ) he
high- equency
pa
o
he spec a is enhanced, allowing he
s udy
o
weak high- equency exci a ions.
Cons an -Q sec ions
o
he spec um
o
he longi udinal
cu en luc ua ions a e shown in Fig.
4.
A
i s sigh , he
mos s iking ea u e
is
undoub edly he appea ance
o
a sec-
ond peak
a
equencies a ound 2.25 X
1013
ad/s,
which is
he
mos in ense in he in e al 0.70 < Q < 1.25 A-I.
The
Q
dependence
o
he equency
o
hese wo maxima has been
plo ed in Fig.
5.
The
low- equency mode
is
acous ic in na-
u e
and
i
is
ela ed o he peak appea ing in
S(
Q,w)
a
low
Q,
hough i now appea s
a
somewha highe equencies
(see Fig. 5) due o he w2 ac o in Eq.
(6);
i s "dispe sion
ela ion" shows no signi ican quali a i e di e ence om
wha has been ound in simple liquids.
28
Namely, i
is
s ongly a ec ed by s uc u al e ec s
a
low Q [showing a
p onounced dip in he neighbo hood
o
Qp
],
bu beyond 2.5
A-I
kine ic e ec s become p edominan , as
can
be seen
om he compa ison wi h he ideal-gas-limi ing alue
w:::(Q) =
Q(2k
B
T/M)
112.
The
second mode exhibi s a mo e complex beha io .
To
begin wi h, i canno be unambiguously dis inguished as a
peak in he whole Q ange examined in his s udy,
bu
usual-
ly a shoulde can be obse ed
a
equencies a ound 2 X
10
13
ad/s
in all he spec a no showing a peak. Second, because
o
he s ong o e lap be ween he wo modes i
is
di icul o
asce ain whe he his highe - equency mode shows any
measu able dispe sion. In addi ion, we ound in
ou
p e ious
single-pa icle s udy
10 ha he cen e -o -mass eloci y au o-
co ela ion unc ion
(VACF)
has a seconda y peak
a
2.04 X
1013
ad/s
which ag ees app oxima ely wi h he posi-
ion obse ed he e. The in ensi y
o
bo h modes exhibi s im-
po an oscilla ions
a
low Q
ha
a e p og essi ely damped
J.
Chem. Phys., Vol. 96, No.1 0, 15 May 1992
Alonso
e
al.: Collec i e exci a ions
in
liquid me hanol
7701
0.50
A-I
0.55
A-I
0.5
1.0 1.5
2.0
W / (
10
13
ad/s
)
0.61
A-I
0.70
A-I
0.74
A-I
o78A-I
0.82
A-I
0.86
A-I
2 3
w / ( 1013
ad/s
)
FIG.
2.
Cen e s-o -mass dynamic s uc u e ac o o liquid me hanol a
200 K. S( Q, iJ)
is
shown as a unc ion
o
he angula equency,
iJ,
o he
lowes
II
alues
o
he momen um ans e . No e he di e en equency
anges used in bo h ames.
The
o dina e uni s a e (10
12
ad/s)
-1.
M
o
......
x
o
0.5
0.24
A-I
0.34
A-I
0.42
A-I
0.49
A-I
0.54
A-I
1.0 1.5
2.0
w / ( 10
13
ad/s
)
FIG.
3.
Cen e s-o -mass dynamic s uc u e ac o o liquid me hanol a
oom empe a u e. S( Q, iJ) is shown as a unc ion
o
he angula equency,
IJ, o he lowes
i e
alues
o
he momen um ans e . The o dina e uni s
a e
(10
12
ad/s)
-
I.
0'0.15
II
...,
~
...;-
_0.10
~
2:
...;-
0.05
(a)
0.08
0.07
0'0.06
II
do.0
5
...;-
0.04
-
'3
dO.0
3
...;-
0.02
0.01
(b)
0.25
A-I
0.35
A-I
0.43
A-I
0.55
A-I
0 2 3
w / ( 1013
ad/s
)
0.61
A-I
0.70
A-I
0.74
A-I
0.78
A-I
0.86
A-I
°
234
5
W / ( 1013
ad/s
)
FIG.
4.
Cons an -Q sec ions
o
he spec a
o
he no malized longi udinal
cu en au oco ela ion a 200 K as a unc ion
o
he equency. No e he
di e en equency anges used in bo h ames. The o dina e uni s a e
(10
12
ad/s)
-1.
as Q inc eases; howe e , hey oscilla e in opposi e di ec ions.
The
in ensi y
o
he high- equency mode
is
ound o be
oughly p opo ional o
OJ
m
(Q)
while he acous ic mode ol-
lows a
end
p opo ional o i s in e se, which
is
in ag ee-
men wi h i s asymp o ic beha io
a
high Q (i.e., ideal gas).
A new sound eloci y c,
(Q)
can be de ined simila ly
o
cB(Q)
bu
using he equencies om he maxima
o
he
CACF
spec a ins ead
o
hose
o
S(
Q,OJ).
The
high- and
low- equency limi s
o
he sound eloci y a e deno ed by
COO
(Q)
and
CO
(Q),
espec i ely, and a e gi en by
COO
(Q)
= OJ,CQ)/Q,
CO
CQ)
=
YcT(Q),
C
13)
whe e is he quo ien
o
he speci ic hea s and
cT(Q)
=
OJ
o
(Q)/Q
is
a wa e- ec o -dependen gene aliza-
ion
o
he
iso he mal sound eloci y. Finally, in Fig. 6 we
ha e collec ed all
ou
da a
conce ning sound eloci ies. No e
ha
we ha e plo ed
CT
(Q)
ins ead
o
he low- equency
J. Chem. Phys., Vol. 96,
No.
10,15
May 1992
7702 Alonso
e
a/.:
Collec i e exci a ions
in
liquid me hanol
"'
-
2.~
11
2.0-
....
.,.,
'0
.....
1.~
-
0'
1.0-
1
o.~
I~I
}~'! i+ dl~~H
H
i ·
o Q j
A-I
FIG.
5.
F equency
o
he maxima
o
he longi udinal cu en spec a o
me hanol a 200 K as a unc ion
o
he momen um ans e . Open ci cles
ep esen he acous ic mode while as e isks gi e he loca ion
o
he second
exci a ion. The maxima obse ed in
S(
Q,Cll)
a low Q ha e been indica ed as
squa es. Finally, he s aigh do ed line ep esen s he ideal-gas asymp o ic
limi . E o ba s ha e been es ima ed as he
hal
wid h a hal maximum
(HWHM)
o
he window unc ion used o Fou ie ans o ming he cu -
en ACF. To aid compa ison he igu e has been plo ed on he same scale
as Fig.
I.
sound eloci y due o
ou
lack
o
eliable in o ma ion on
o he simula ed sys em. The disag eemen be ween C B
(Q),
C I
(Q),
and C T
(Q)
a
low Q
is
a clea symp om
o
he s ong
sound dispe sion [no e
ha
in o de o econcile he alues
o c
B
(Q)
and Co
(Q)
we would ha e o
assume ~2].
A 200
Ke en C B
(Q)
and C I
(Q)
do no coincide, indica ing ha he
B illouin peaks
a e
b oad and o e lap subs an ially wi h he
cen al peak.
On
he o he hand, he low-Q beha io
o
c,
(Q)
expe iences an impo an change a 300 K, eaching a
maximum alue and, possibly, beginning o con e ge
owa ds Co
(Q)
jus
beyond he lowes Q
we
a e able o in es-
iga e in ou simula ion (0.25
A-I).
4.
300 K
3.
200 K
---.
en
3.
-
S
~
2.~
-2.
C;
'(:)1.
1.
o.
o
0.5
1.0 1.5
2.0
0
0.5
1.0 1.5
2.0
QjA-I
Q/A-l
FIG.
6.
Sound eloci ies
cT(Q),
c~
(Q),
c,(Q),
and cB
(Q)
as a unc ion
o
he momen um ans e (see ex o de ini ions). The open ci cles co e-
spond o
c,(Q),
he squa es o
cB(Q),
and he solid lines o
cT(Q)
and
c
~
(Q),
as indica ed.
c.
T ans e se cu en luc ua ions
We ha e unde aken he calcula ion
o
ans e se
CACF
in o de o ge some insigh in o a
pa
o
he collec-
i e dynamics no easily amenable o expe imen a ion. Ex-
pe imen al echniques such as neu on sca e ing do no cou-
ple easily o ans e se modes (unless some kind
o
di use
"umklapp"
p ocess
is
pos ula ed).
In
ac , o da e molecula
dynamics is he mos eliable sou ce
o
in o ma ion abou
ans e se exci a ions in dense luids o he kinema ic ange
o
ou
in e es .
The equency
o
he maximum
o
he ans e se
CACF
spec a has been plo ed in Fig. 7 as a unc ion
o
Q.
A
i
o
he low-Q da a
o
he "dispe sion ela ion" o a s aigh line
gi es an es ima e
o
he eloci y
o
p opaga ion which
is
940 ± 20
mls
a
200
K.
Co esponding spec a and dispe -
sion ela ion a 300 K look quali a i ely he same. Howe e ,
a his empe a u e he e
is
a subs an ial o e lap
o
he
S okes and an i-S okes peaks o he lowes Q which dis o s
o some ex en he dispe sion ela ion; o example, a he
lowes Q
he alueo J,
(Q,liJ
=
0)
is abou
1/2
o
he alue a
he maximum, while a 200 K i is
1/20
o
i .
In
addi ion, he
eloci y
o
p opaga ion has now educed o 750 ±
25
m/s.
The li e ime
o
he exci a ion
30
can be es ima ed om he
ecip ocal
o he
hal wid h
a
hal maximum
o he
peaks. A
200 K we ind li e imes
o
1.06, 0.71, 0.42, and 0.35 ps a
Q = 0.25, 0.35, 0.43, and 0.50 A -
I,
espec i ely, while a
300 K he li e imes a e 0.64,0.49,0.35, and 0.30 ps o simi-
la alues
o
momen um ans e .
An
ex apola ion
o
he
dispe sion ela ion o lowe
Q,
a
300 K, sugges s ha he
peaks
a
± liJ m
(Q)
will coalesce in o a single one o Q <
0.1
A-I,
bu he e is no indica ion
o
ha
happening o any
ini e
Qa
200 K. Undoub edly, he eason why me hanol
seems o be able o suppo eely p opaga ing shea wa es
wi h wa eleng hs as la ge as 50-100 A
is
he exis ence
o
a
e y s able hyd ogen-bond ne wo k, which is i sel connec -
ed o he s ong di ec ionali y
o
he hyd ogen-bond in e ac-
ion.
---.0.
en
-
~
~
o.
o
.....
c
'1
o.
o
0.5
1.0
1.5
2.0
Q /
A-I
FIG.
7.
F equency
o
he maximum
o
he ans e se cu en au oco ela-
ion spec a o me hanol a 200 K as a unc ion
o
he momen um ans e .
E o
ba s as in Fig.
5.
J. Chem. Phys., Vol. 96, No.1 0, 15
May
1992
Alonso
e
al.: Collec i e exci a ions
in
liquid me hanol
7703
Fo
momen um ans e s g ea e
han
1.0
A-I
he
ans e se
CACF
beha es mos ly as a single-pa icle coo di-
na e. Finally, o end he analysis
o
he spec a
o
he ans-
e se cu en we
jus
men ion
ha
a conspicuous seconda y
maxima can be obse ed
a
he
same equency as
he
hi d
( e y weak) maxima in
he
VACFspec a,
10
i.e., 3.57X 1013
ad/s,
in almos all he spec a wi h Q be ween 0.5
and
1.5
A-I, hough i
is
ex emely weak.
In
he low-Q egion
he
spec a con ain some ipples ha p eclude
he
obse a ion
o
weak signals as his seconda y peak.
D.
la ice-dynamics
calcula ions
In
o de
o explo e he physical o igin
o
he modes ap-
pea ing in he simula ion as well as
o
ha e a e e ence
o
compa e wi h, we ha e unde aken a la ice-dynamics
(LD)
calcula ion o a polyc ys al sample ollowing he p ocedu e
desc ibed in he Appendix.
F om
he calcula ed dispe sion ela ions
o
he 24
modes co esponding o o a ions
and
ansla ions
o
he
ou molecules wi hin
he
uni cell, a quan i y di ec ly com-
pa able wi h he
cu en
co ela ion unc ions has been de-
i ed. Se e al plo s co esponding
o
di e en alues
o
he
momen um ans e a e shown in Fig.
8.
F om
inspec ion
o
he igu e he ollowing commen s a e in o de :
(a)
The
mani old
o
acous ic peaks appea s as a well-de ined en i y
up
o Q alues
o
abou 0.5
A,
showing no iceable dispe sion;
and
(b)
wo se ies
o
peaks
o
a mixed ( o a ional
and
ans-
la ional) cha ac e a e clea ly appa en om 0.5 A onwa ds,
cen e ed
a
abou 2.2
and
3.0X
1013
ad/s.
Such equencies
can be easily co ela ed wi h p ominen ea u es appea ing
in he calcula ed ib a ional densi y
o
s a es
(DOS)
which
is displayed in Fig.
9.
A close inspec ion
o
he igu e whe e a
compa ison
o
a om-a om
DOS
is
made o he c ys al
and
liquid phases e eals he ollowing ea u es:
•
The
pu ely acous ic modes a e appa en in he polyc ys al
as a Debye con ibu ion (Le., wi h a dependence upon (2)
up
o equencies
o
5 X
10
12
ad/s.
The
i s wo in ense
peaks cen e ed
a ound
lOX
10
12
ad/s
a e
o
mixed ( o a-
ional
and
ansla ional) cha ac e as has been e idenced
om he analysis
o
he mode eigen ec o s.
•
The
polyc ys alline DOS
up
o
w = 15X
10
12
ad/s
be-
comes in
he
liquid phase a b oad, s uc u eless o a ional
con ibu ion cen e ed
a
abou he same equency as in he
solid.
The
sound mode ( ansla ional
o
cen e -o -mass)
con ibu ion appea s in
he
liquid as a b oad dis ibu ion
cen e ed
a
abou 5 X
10
12
ad/s,
which will gi e ise
a
low-Q
alues o a ini e- equency esponse in
he
S(
Q,w).
A no-
iceable amoun
o
ansla ion- o a ion coupling is appa -
en , which e idences he ac
ha
no pu e ansla ional
o
o a ional modes
a e
o be expec ed o occu in a molecula
ma e ial wi hin
he
kinema ic ange accessible
o
MD
o
INS
s udies.
• A se ies
o
in ense peaks cen e ed
a
22, 30, and 38 X
10
12
ad/s
in he polyc ys al appea as a b oad s uc u e be ween
w =
15x
10
12
and
w =
30X
10
12
ad/s
in he liquid in a e-
quency egion showing also a no iceable amoun
o
o a ion-
ansla ion coupling.
The
in ense peak which appea s in
he
solid
a
abou 50 X
10
12
ad/s
also shows
up
in
he
liquid,
al hough he modes con ibu ing o i a e now so ened,
which leads o a educ ion in he peak equency o abou
37 X
10
12
ad/s.
F om
he analysis
o
some
o
he mode eigen ec o s he
componen s
o
he
mode pola iza ions can be ob ained. As
an
example, i was ound
ha
he
wo low- equency peaks
co espond o modes wi h an a e age cha ac e
o
Tx(y,z)
= 0.20 (0.23, 0.36)
and
RX(Y,z)
= 0.74 (0.47, 0.11)
o he low- equency peak, and
Tx(y,z)
= 0.21 (0.22,0.67)
and
Rx(y,z)
= 0.24 (0.62,0.14) o he highe - equency one.
He e
Tx(y,z)
and
Rx(y,z)
deno e
he
a e aged ampli Udes
o
he ansla ional and o a ional molecula deg ees
o
ee-
dom
exp essed in he p incipal axis
o
he ine ia molecula
ame (in inc easing ine ia momen
o de ).
The e o e, i seems clea
ha
he compa ison
o
he cu -
en -cu en
co ela ions as well as he DOS o bo h poly-
c ys al and liquid phases e eals
ha
mos
o
he obse ed
ea u es in he luid phase can be assigned by aking he
LD
esul s as a e e ence.
Fu he
discussion on his will be de-
e ed o he Discussion sec ion.
V.INELASTIC NEUTRON SCATTERING AND
MODELING OF S{Q, o)
Recen ly, Be mejo e
aU
ha e measu ed he inelas ic
neu on-sca e ing spec a
o
liquid me hanol-d4,
I(
Q,w),
which
is
ela ed o he
a om-a om
dynamic s uc u e ac o s
no U no no
I(Q,w)
ex:
I
~S~l (Q,W)
+ I I
bibjSi/Q,w)
,
i=1
41T
i=lj=1
(14)
whe e
na
is
he numbe
o
a oms
pe
molecule,
(J
inc
and
bi
a e he incohe en sca e ing c oss sec ion and he cohe en
sca e ing leng h o neu ons
o
a om ype i, and S
ij
(
Q,w)
is
he
a om-a om
dynamic s uc u e ac o o a oms i and j
( he
supe sc ip sel makes e e ence
o
i s sel -pa ). How-
e e igo ous, he p eceding equa ion would no be e y use-
ul as a model due o i s la ge numbe o ee pa ame e s
and
a simpli ied one based in a well-known iscoelas ic app oxi-
ma ion o simple liquids has been p oposed7
I
model
(
Q,w)
ex:
Scm
(Q)
exp( -p,Q 2)
X
{Rqe
(Q,W) +
P6
(pi -
P6
)7
}
[W7(W2 -PF)] 2 +
(W
2 -P6)2
®
O(w),
(15)
whe e
Scm
(Q)
deno es he s uc u e ac o o molecula
cen e s, he exponen ial is a Debye-Walle e m,
Rqe
(Q,w)
ep esen s he cen al quasi elas ic componen , he second
e m
inside he cu ly b acke s
is
he usual iscoelas ic sca -
e ing law o simple liquids,9
O(w)
is he expe imen al es-
olu ion unc ion; and ® deno es a con olu ion ope a ion.
To
keep he numbe
o
ee pa ame e s o a minimum he Max-
wellian elaxa ion ime 7 is es ima ed using he p esc ip ion
due o Lo esey,9
7
-1=2[(PF-P6)!1 ]1I2.
(16)
As usual,
he
Q dependence
o
he educed momen s om
J. Chem. Phys., Vol. 96, No.1 0, 15
May
1992