Dynamic P ope ies
o Magne ic Colloidal Pa icles and Holes
Ma ía del Ca men Miguel López
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Uni e si a
de
Ba celona
Facul a
de
Física
Dynamic
p ope ies
o
magne ic
colloidal
pa icles
and
holes
Memo ia
p esen ada
pe
Ma ía
del
Ca men
Miguel
López
pe
op a
al
G au
de
Doc o
en
Ciencies
Físiques.
Ba celona,
8
de
se emb e
de
1995.
Uni e si a
de
Ba celona
Facul a
de
Física
Dynamic
p ope ies
o
magne ic
colloidal
pa icles
and
holes
Ce i ico
que
la
p esen
esi
doc o al
ha
es a
eali zada
so a
la
meya
di ecció
Ba celona,
8
de
se emb e
de
1995.
D .
J.
M.
Rubí
Capace i,
Ca ed á ic
de
Física
de
la
Ma e ia
Condensada
de
la
Uni e si a
de
Ba celona
P opiedades
dinámicas
de
pa ículas
y
huecos
magné icos
coloidales
In oducción
Las
pa ículas
magné icas
de
pequeño
amaño
son
de
g an
impo ancia
en
mul i ud
de
á eas
de
la ciencia
y
la
ecnología.
No
obs an e,
hay
escasos
ejemplos
de
sis emas
o mados
po
es as
pa ículas
suspendidas
en
un
medio
líquido
de
mane a
que
las
in e acciones
magné icas
desempeñen
un
papel
p incipal,
an o
en
su
compo amien o
dinámico
como
en
la
o mación
de
es uc u as.
En e
es os
pocos
casos
igu an
los
e o luidos
y
las
pa ículas
de
lá ex
dispe sas
en
un
e o uido
(huecos
magné icos).
Un
luido
magné ico
o
e o luido
es
una
suspensión
coloidal
es able
de
pa ículas
mono
dominio
de
un
ma e ial
magné ico
en
un
líquido
común,
como
agua
u
o o
compues o
o gánico.
Inicialmen e
ue on
concebidos
como
la
p ime a
mues a
de
un
líquido
a i icial
en
el
que
las
p opiedades
magné icas
y
eológicas
se
iesen
inc emen
adas
has a
se
compa ables
a
las
de
un
sólido
e omagné ico,
y
se
manu ac u a on
debido
al
hecho
de
que
la
exis encia
de
mono ases
e omagné icas
líquidas
eales
e a
bas an e
imp obable. Aunque
la
es
abilidad
de
la
suspensión
es á
asegu ada
esencialmen e
po
el
mo imien o
b owniano
de
las
pa ículas,
usualmen e
és as
se
ecub en
de
una
capa
de
su ac an e
a
in
de
inhibi
su
coalescencia
a
dis ancias
co as.
Los
e o uidos
ienen
muchas
aplicaciones
ecnológicas,
incluyéndose
en e
ellas
los
p ecin os
de
exclusión
pa a
las
disque e as
de
o denado , amo iguado es
en
al
a oces,
imp eso as,
ins umen os
pa a
an e encia
de
masa
y
calo ,
lub ican es,
e c.
Po
ejemplo,
esul a
complicado
man ene
un
lub ican e
den o
de
la
zona
de
con ac o
de
las
pa e
mó iles
de
un
de e minado mecanismo
ya
que
las
ue zas
cen í ugas
o
la
geome ía
ienden
a
expulsa lo.
Sin
emba go,
un
campo
magné ico
puede
ayuda
a
e ene
un
lub ican e
magné ico
en
la
zona
de
icción
donde
es
más
necesa io.
Más
aún,
la
e icacia
de
un
lub ican e
depende
p incipalmen e
de
su
iscosidad
y
de
su
capacidad
pa a
o ma
una
capa
adso bida.
Los
lub ican es
magné icos
son
más
e ec
i os
que
los
comunes
incluso
en
ausencia
de
campos
magné icos
po que
las
pa ículas
magné icas
que
lo
componen
se
en
a aídas
po
las
supe icies
de
ace o
y
aumen an
así
an o
la
iscosidad
como
el
g oso
de
es a capa
lími e.
Los
e o luidos
encuen
an
ambién
aplicaciones
en
ecología
y
medicina.
Po
ejemplo,
si en
pa a elimina
esiduos
oleosos
de
los
e idos
indus iales.
Los
luidos
magné icos
con
un
hid oca 
bu o
como
base
se
pueden
disol e
en
los
p oduc os
oleosos
pa a
o ma
un
e o luido
más
diluido.
Un
lujo
de
agua
que
con iene
es a
suspensión
ci cula
a
a és
de
un
sepa ado
magné ico
donde
un
campo
magné ico
que a ía
ue emen e
en
el
espacio
sepa a
los
esiduos
magné icos
del
agua.
Pe o
son
los
múl iples
y
cuali a i amen e
di e sos
e ec os
p oceden es
de
la
in e acción
en e
e o luidos
y
campos
magné icos
los
que
han
abie o
la
posibilidad
de
nue as
y
p ome edo as
aplicaciones ecnológicas.
Además
de
sus
p opiedades
magné icas,
óp icas
y
eológicas,
que
discu i emos
a
lo
la go
de
es a
monog a ía,
los
e o luidos
exhiben
o os
enómenos
poco
comunes
de
g an
in e és
cien í ico,
como
las
ines abiliades
supe iciales
ycu iosas
es uc u as
de
no
equilib io.
Hace
algunos
años
se
descub ió
que
las
es e as
mono
dispe sas
de
polies i eno,
dis
pe sas
en
un
e o luido,
cons i uyen
un
modelo
con enien e
pa a
el
es udio
de
a ios
ipos
de
enómenos
que
mues an
ansiciones
o den-deso den.
La
base
de
ello
es
que
las
es e as
desplazan
al
e o luido
y
se
compo an
como
huecos
magné icos
con
un
momen o
magné ico
e ec i o
igual
al
momen o
o al
del
e o luido
desplazado.
Las
es e as
(en e
1
y
100¡, m)
son
mucho
mayo es
que
las
pa ículas
magné icas
del
e 
e luido
(100.4)
y,
po
an o,
se
compo an
como
si
se
hallasen
en
un
medio
magné ico
uni o me.
En
gene al,
las
mic oes e as
mono
dispe sas
p esen an
una
g an
a iedad
de
apli
caciones
indus iales,
cien í icas
y
médicas,
debido
a
la
singula idad
de
sus
muchas
p opiedades
como
el
posee
una
dis ibución
de
amaños
ex emadamen e
uni o me,
una
o ma
es é ica
casi
pe ec a,
con
un
diáme o
de e minado
con
g an
p ecisión,
la
g an
di e sidad
de
ipos
de
pa ículas
(sólidas,
po osas,
magné icas,
e c.)
y
la
posibilidad
de
manu ac u a
sus
p opiedades
supe iciales.
Las
mic oes e as
de
poli
es i eno
con
inclusiones
magné icas
se
han
empleado
sa is ac o iamen e
pa a
sepa a
di e sos
ma e iales
o gánicos
(células,
i us,
micelas,
e c.)
El
p incipio
de
es a
apli
cación
se
basa
en
una
in e acción
selec i a
en e
los
an ígenos
de
la memb ana
de
las
células
y
los
an icue pos
si uados
sob e
las
es e as.
Las
es e as
con
las
células
adhe i
das
pueden
elimina se
pos e io men e
po
medio
de
un
campo
magné ico.
Es
posible
limpia así
la
médula
espinal
de
pacien es
a ec ados
de
neu oblas oma
o
de
cie as
o mas
de
leucemia,
limi ando
los
a amien os
con
adio e apia
y/o
quimio e apia,
que
puede
a ec a
pe judicialmen e
al
es o
de
células
sanas
del
o ganismo.
Median e
es a
écnica,
es
posible
limpia
has a
una
décima
pa e
de
la
médula
de
los
pacien es,

an es
de
ecu i
a
o as
e apias
más adicales.
Las
pa ículas
magné icas
se
pueden
usa
ambién
como
ma cado es
si
se
las
ecub e
con
cie os
componen es
químcicos
o
con
an icue pos.
Du an e
la
mig ación
de
las
pa ículas
dispe sas,
és as
se
adhie en
a
blancos
especí icos
y
acili an
su
localización.
O a
p ome edo a aplicación
es
su
uso
como
po ado es;
se
si úan
cie os
p oduc os
químicos
sob e
la
supe icie
de
las
pa ículas
y
se
anspo a
a
és as
a
los
luga es
donde
las
medicinas
se án
e ec i as.
Los
bioquímicos
es án
alo ando
ac ualmen e
las
po encialidades
de
es e
mé odo
pa a
a a
el
cánce .
Conclusiones
A
lo
la go
de
es a
monog a ía
nos
hemos
ocupado
del
es udio
de
sis emas
luidos,
an o
con
monodominios
magné icos
como
con
dos
ipos
dis in os
de
pa ículas,
magné icas
y
no
magné icas,
en
dispe sión
en
un
líquido
new oniano
en
si uaciones
ue a
del
equi
lib io.
El
compo amien o de
es os
sis emas
se
e
in luenciado
en
g an
medida
po
la
p esencia
de
un
campo
magné ico
ex e no,
lo
que
da
luga
a
nue os
enómenos
que
han
sido
el
undamen o
de
muchas
aplicaciones
p ác icas.
Sin
emba go,
es a
in luencia
depende
de
los
di e en es
p ocesos
de
elajación
que
ienen
luga
den o
las
pa ículas,
con
espec o
a
su
ejes
c is alinos,
así
como
ue a
de
ellas,
con
espec o
al
luido
po a
do .
Hemos
desc i o
cuáles
son
es os
p ocesos
y
ob enido la
dependencia
con
espec o
de
los
pa áme os
que
desc iben
las
pa ículas
y
el
luido,
de
algunos
coe icienes
que
ca ac e izan
las
p opiedades
eológicas,
magé icas
y
óp icas
de
la
suspensión
coloidal.
Pa a
se
más
p ecisos,
en
la
p ime a
pa e
del
Capí ulo
1
hemos
analizado
la
dinámica
de
una
es e a
e omagné ica
en
la
que
el
momen o
magné ico
se
encuen a
ijado
ígidamen e
a
su
cue po,
así
como
ambién
las
iscosidades
de
una
suspensión
diluida
de
es e
ipo
de
pa ículas.
La
base
eó ica
undamen al
es
la
ecuación
de
Na ie -S okes,
en
la
que
se
ha
incluído
una
ue za
inducida
que
ep esen a
las
pe 
u baciones
in oducidas
en
la
dinámica
del
luido
po
el
mo imien o
de
la
pa ícula,
y
una
uen e
de
uido
ipo
Lange in,
p oceden e
de
las
luc uaciones
de
los
campos
hid odinámicos.
Es a
ecuación
da
cuen a
del
acoplamien o
de
las
dinámicas
del
lu
ido
y
la
pa ícula.
Una
expansión
mul ipola
de
las
can idades
que
apa ecen
en
la
solución
o mal
de
la
ecuación
de
Na ie -S okes
nos
pe mi e
calcula
las
exp esiones
pa a
la
ue za
y
el
momen o
de
las
ue zas
eje cidos
sob e
la
pa ícula,
los
cuales
con
ienen con ibuciones
alea o ias
cuyas
p opiedades
es adís icas
es án
dic adas
po
la
eo ía
de
luc uaciones
hid odinámicas.
Hemos
empleado
dos
mé odos
pa a
ob ene
las
iscosidades
de
cizalladu a
y
o acional.
Uno
de
ellos
es
de e minis a,
y
se
basa
en
la
ó mula
de
Ki kwood
pa a
el
enso
de
p esiones
iscosas,
el
cual
a
su
ez
es á
elacionado
con
el
mul ipolo
de
segundo
o den
de
la
ue za
inducida.
El
o o
mé odo
se
basa
en
la
eo ía
de
espues a
lineal,
que
da
los
coe icien es
de
anspo e
en
é minos
de
unciones
de
co elación
dependien es
del
iempo.
La
p esencia
del
campo
magné ico
p o oca
la
apa ición
de
con ibuciones
an isimé icas
al
enso
de
p esiones
y
da
luga
a
la
iscosidad
o acional
que
apa ece
como
un
nue o
coe icien e
de
anspo e.
Es a
pa e
an isimé ica
del
enso
de
p esiones
p o iene
del
balance
que
se
es ablece
en e
el
momen o
de
las
ue zas
eje cidas
po
el
campo
magné ico
sob e
un
dipolo
y
el
momen o
de
las
ue zas
hid odinámicas.
Como
consecuencia,
el
momen o
angula
de
las
pa ículas
puede
di e i
del
alo
de
la
o icidad
del
luido
en
el
pun o
que
aquella
ocupa.
Es e
o malismo
es
bas an e
gene al
y
se
emplea
en
la
segunda
pa e
del
Capí ulo
pa a
calcula
la
dependencia
de
la
iscosidad
o acional
con
espec o de
la
ene gía
magné ica
de
aniso opía
del
ma e ial.
Conside ando
o o
lími e
especí ico,
en
el
cual
los
momen os
magné icos
ya
han
elajado
y
apun an
en
la
di ección
del
campo,
ob enemos que
la
iscosidad
aumen a
al
aumen a
el
pa áme o
de
aniso opía,
alcanzando
un
lími e
de
sa u ación.
Hemos
compa ado
nues os
esul
ados
con
los
de
o os
au o es
y
con
los
da os
expe imen ales
disponibles
pa a
dipolos
ígidos.
Los
esul ados
p oceden es
de
o a
solución
ap oximada
de
la
ecuación
es a
ciona ia
de
Smoluchowski
sob ees iman
los
nues os,
mien as que
los
basados
en
ecuaciones
de
elajación
enomenológicas
p opues as
pa a
el
momen o
angula
in e no
de
la
suspensión
son
muy
p óximos
a
los
que
hemos
ob enido
noso os,
y
ep oducen
bas an e
bien
los
da os
expe imen ales.
Siguiendo
un
p ocedimien o
análogo,
en
el
Capí ulo
11
hemos
p esen ado
un
o 
malismo
gene al
pa a
es udia
la
dinámica
de
elajación
de
pa ículas
e omagné icas,
con
el
p opósi o p incipal
de
p opo ciona
exp esiones
explíci as
an o
pa a
la
iscosi
dad
como
pa a
algunos
iempos
de
elajación
que
ca ac e izan
di e en es
p opiedades
del
ma e ial
(bi e ingencia,
suscep ibilidad
magné ica,
...
).
Los esul ados
cub en
odo
el
ango
de
si uaciones
expe imen ales
posibles.
Hemos
ob enido
la
ecuación
de
Smoluchowski
que
desc ibe
la
e olución
de
la dis ibución
de
p obabilidad
de
los
g a
dos
de
libe ad
ele an es
de
las
pa ículas
y
que
nos
pe mi e
ob ene
una
je a quía
de
ecuaciones
dinámicas
pa a
las
di e en es
unciones
de
co elación.
Es a
je a quía
se
puede
ce a
u ilizando
las
conocidas
ap oximaciones
de
desacoplo
ap opiadas.
A
pa i
de
las
ecuaciones
dinámicas
pa a
las
co elaciones,
uno
puede
encon a
exp e
siones
pa a
los
iempos
de
elajación
ca ac e ís icos
y
és os
cons i uyen
el
pun o
de
pa ida
pa a
de e mina
los coe icien es
de
anspo e
po
medio
de
las
ó mulas
de
G een-Kubo.
En
pa icula ,
hemos
is o
que
la
iscosidad
o acional
alcanza
de
nue o
un
lími e
de
sa u ación,
pe o
aho a
depende
los
dos
pa áme os
que
compa an
la
en
e gía
magné ica
y
la
ene gía
de
aniso opía
con
la
ene gía
é mica.
Pa a
con as a
la
alidez
de
nues o
o malismo,
hemos
compa ado
nues os
esul ados
pa a
el
iempo
de
elajación
de
las
pa ículas
con
los
expe imen os
de
bi e ingencia
lle ados
a
cabo
con
dos
ipos
de
ma e iales
e omagné icos.
En
ambos
casos,
nues os
esul ados
concue dan
muy
bien
con
los
da os
expe imen ales.
También
hemos
p opo cionado
una
exp esión
gene al
pa a
la
suscep ibilidad
magné ica
compleja
del
luido
magné ico
bajo
la
acción
simul ánea
de
un
campo
magné ico
pola izan e
y
de
un
pequeño
campo
al e no,
pe pendicula es
en e
sí,
lo
que
ab e
posibilidades
pa a
nue as
medidas
ex
pe imen ales.
En
el
Capí ulo
111
hemos
es udiado
la
dinámica
de
pa ículas
no
magné icas,
o
hue
cos
magné icos,
suspendidos
en
un
e o luido
en
p esencia
de
un
campo
magné ico
o a i o.
Calculamos
las
ue zas
hid odinámicas
y
el
momen o
de
las
ue zas
eje 
cidas sob e
el
hueco,
de
donde
podemos
iden i ica
los
enso es
de
icción
asla
cional
y o acional.
Es as
can idades
dependen
de
la
acción
olúmica
de
pa ículas
magné icas
y
del
campo
magné ico,
el
cual
o o ga
al
sis ema
un
ca ác e
anisó opo.
El
conocimien o
de
la
ue za
y
del
momen o
de
las
ue zas
que
ac úan
sob e
el
hueco
nos
pe mi e
es udia
las
dinámicas
de
aslación
y
de
o ación
de
las
pa ículas
cuando
aplicamos
un
campo
magné ico
o a i o.
Nos
hemos
concen ado
pa icula men e
en
el
caso
en
que
el
e o luido
es á
en
eposo
y
el
hueco
puede
gi a
bajo
la
in luen
cia
del
campo
magné ico.
Bajo
es as
condiciones,
hemos
p obado
que
la
elocidad
angula
del
hueco
es
p opo cional
a
la
ecuencia
del
campo,
pe o
que
iene
una
di
ección
opues a.
El
coe icien e
de
p opo cionalidad
es
lineal
con
la
acción olúmica
de
las
pa ículas
de
e o luido,
y
depende
de
una
unción
de
la
in ensidad
del
campo
magné ico
que
mues a
un
compo amien o
de
sa u ación.
Hemos
compa ado
nue
s os
esul ados
con
expe imen os
ealizados
con
pa ículas
de
polies i eno
dispe sas
en
di e en es
e o luidos.
En
el
ango
de
ecuencias
conside adas,
hemos
ep oducido
los
esul ados
expe imen ales
conce nien es
a
la elocidad
de
o ación
del hueco
como
unción
del
campo
magné ico
ex e no.
También
hemos
ealizado
un
es udio
p elimina
de
las
in e acciones
hid odinámicas
en e
los
huecos
en
el
e o luido.
Esencialmen e,
hemos
ob enido
exp esiones
pa a
los
enso es
de
Oseen
y
Ro ne-P age
de
un
e 
o luido
como
el
p ime
paso
en
el
es udio
del
impo an e
papel
que
las
in e acciones
hid odinámicas
pueden
desempeña
en
las
p opiedades
ísicas
de
es os
ma e iales
a
bajas
concen aciones.
Pa a
mos a
la
es uc u a
de
es os
enso es
y
su
in luencia
en
la
dinámica
de
los
huecos
magné icos,
hemos
dado
ambién
exp esiones
pa a
las
alocidad
de
una
pa eja
de
pa ículas
deposi ándose
bajo
la
acción
de
la
g a edad
en
el
seno
de
un
e o luido.
Como
espe ábamos,
pa a
di e en es
con igu aciones
iniciales,
la
elocidad
depende
de
los
pa áme os
que
ca ac e izan
el
e o luido.
Resul a
ú il
señala
que,
aunque
hemos
ealizado
el
es udio
pa a
un
e o luido
compues o
po
dipolos
ígidos,
pa a
el
que
la
ene gía
de
aniso opía
es
mucho
mayo
que
la
ene gía
asociada
con
la
in e acción
de
los
momen os
magné icos
con
el
campo
ex e no,
se
puee
hace
una
análisis
simila
en
una
si uación
más
gene al,
en
la
que
es as
dos
ene gías
de
las
pa ículas
magné icas
p esen en
alo es
a bi a ios.
El
Capí ulo
IV
se
p esen a
como
un
es udio
p elimina
de
los
enómenos
de
ag e
gación
que
ienen
luga
en
sis emas
de
pa ículas
magné icas
en
suspensión,
así
como
de
las
es uc u as
esul an es.
Nues o
p ime
p opósi o
ha
sido
elucida
la
in luencia
de
las
in e acciones
hid odinámicas
en
la ciné ica
del
p oceso
de
ag egación.
Hemos
ex endido
la
eo ía
clásica
de
la
coagulación
de
Smoluchowski
pa a
da
cuen a
de
la
p esencia
de
es as
in e acciones,
impo an es
cuando
se
aspasa
el
égimen
de
con
cen aciones
diluidas.
Tales
in e acciones
ac úan
an es
de
que
las
di e en es
pa ículas
lleguen
a
la
es e a
de
in luencia
de
una
pa ícula
dada.
Hemos
ob enido
las
ecuaciones
ciné icas
pa a
el
p oceso
de
ag egación
y,
a
pa i
de
ellas,
hemos
analizado
la
o 
mación
de
ag egados.
Nues a
p incipal
conclusión
es
que la
p esencia
de
in e acciones
hid odinámicas
alen iza
el
p oceso
de
ag egación.
Como
un
segundo
p oblema,
hemos
es udiado
la
dinámica
de
una
cadena
de
pa ículas
magné icas
bajo
la
in luencia
de
una
lujo
ex e no
elongacional.
En
pa icula ,
hemos
calculado
las
con ibuciones
de
la
cadena
al
enso
de
p esiones
del
sis ema
a
pa i
de
la
ecuación
eológica
p opues a
po
K ame s.
De
es a
can idad
hemos
ob enido la
co ección
a
la
iscosidad
debida
a
la
p esencia
de
in e acciones
dipola es.
Es os
esul ados
p elimina es
cons i uyen
el
obje o
de
u u os
abajos.
Di e en es
líneas
de
in es igación
pueden
su gi
de
los
con enidos
de
es a
esis.
En
e
ellas,
podemos
ci a
la
necesidad
de
inco po a
en
el
análisis
e ec os
ine ciales
y
de
la
polidispe sidad
de
las
pa ículas.
Como
comen amos
b e emen e
en
la
in oducción
de
la
esis,
los
e ec os
ine ciales
da ían
luga
a
un
égimen
oscila o io
adicional
en
los
p ocesos
de
elajación
pa a
alo es
muy
al os
del
campo
magné ico.
Sin
emba go,
esul an
se
esenciales
pa a
la
desc ipción
de
la
dinámica o acional
de
las
pa ículas
a
ecuencias
muy
al as.
Con
espec o
a
la
polidispe sidad,
debido
a
que
los
di e 
en es
p ocesos
de
elajación
dependen
de
o ma
dis in a
del
olumen
de
las
pa ículas,
sus
con ibuciones
a
las
magni udes
calculadas
no
son
las
mismas
en
un
medio
po
lidispe so.
Po
ejemplo,
el
espec o
de
los
iempos
de
elajación
puede
ensancha se.
O a
posible
línea
de
in es igación
consis i ía
en
i
más
allá
de
la
espues a
lineal
del
sis ema
en
condiciones
alejadas
del
equilib io.
1.
The
[e omagne ic
pa icles
3
Figu e
.1:
Scanning
ele
on
mic og aph
o
magnelic
nic
osphe es
wi h
an i-H2
mon
oclonal
an iboclies
a .ached
lo
liepa ocy e.
1
The
e omagne ic
pa icles
We
will
dis ega cl
o
.he
momen
he
liquid
in
which
lie
pa icles
a e
dispe sed
and
concen a e
in
he
desc
ip .ion
o
.he
ine
e omagne ic
pa
ic
les.
Fe omagne .ic
pa
.icles
in
a
e o luid
a e
monoclomain
and
possess
almos
con
s
an
magne .ic
momen s.
I
is
well
known
ha
e omagne .ic
c ys als
consis
o
e
gions
whe e
he
magne iza .ion
is
o ien ecl
clilTe en ly.
This
egions
a e
called
magne ic
domains. The
shape
and
size
o
hese
magne ic
do iains
in
he modyna nic
equilib
ium
a e
de e mined
by
he
condi ion
ha
.he
o al
ee
ene gy
o
lie
ma e ial
should
be
a
minimum.
Mo eo e ,
he
magne ic
do iains
a e
sepa a ed
om
each
o he
by
domain
walls,
i.e.,
a
ansi ion
laye
in
which he
dis . ibu ion
o
he
magne iza ion
o ien a ion
is
nonuni o m
bu
changes
con inuously.
F enkel
and
Do man
[7]
p e
dic ed
ha
by
educing
he
size
o
a
e omagne ic
sa nple,
one
would
each
a
poin
a
which
domain
bounda ies
would
no
longe
be
ene ge ically
a o able
so
ha
he
whole
sample
would
become
a
single
do nain.
On
one
hancl,
e e y
domain
wall
means
an
addi ional
a noun
o
su ace
ension
ene gy
which
limi s
he
numbe
o
possible
do nains.
On
he
o he
h
and
,
he
ene gy
associalecl
o
.he
nagne ic
ield
su ounding
he
sample
a o s
hei
o na ion.
Based
on
his
simple
a gumenl
o
compe i ion
one
can
es ima e
he
c i ica!
size
(de)
below
which
he
sa nple
can
be
conside ed
a
mag
ne ic monodomain.
In
he
es i
n
a .ion
one
should
dis inguish
be ween
s ong
and
weak
magne oc ys alline aniso opy
be
ause
i
also
hine!e s
lie
c
ea
ion
o
a
domain
wall

4
INTRODUCTION
a o ing
he
in
c ease
o
he
c i ical
monodomain
size.
Bu ,
e en
o
he
less
a o ed
cases
o
weak
magne oc ys alline aniso opy,
he
es ima ions
gi e
de
""
300..1
which
is
well
aboye
he
mean
size
o
he
e omagne ic
pa icles
used
in
e o luids
whose
ypical
diame e
does
no
usually
exceed
150..1.
Thus,
e en
allowing
o
a
possible
polydispe si y,
he
colloidal
dispe sed
pa icles
a e
de ini ely
monodomain.
Al hough
we
a e
in e es ed
in
enhancing
o
ins ance
he
magne ic p ope ies
o
he
magne ic
liquid,
and
hese
a e
clea Iy
de e mined
by
he
suspended
pa icles,
he e
is
a
es ic ion
in
hei
size
o
essen ially
a oid
agg ega ion
phenomena
which
a e
mainly
due
o
dipola
magne ic
ene gy.
I we
de ine
a
dimensionless
pa ame e
A
=
m2
jd3kBT
compa ing
dipola
magne ic
and
he mal
ene gies,
whe e
m
and
d
a e
he
magne ic
momen
s eng h
and
he
diame e
o
one
pa icle,
he
s abili y
condi ion
imposes
ha
A
::;
1
which
can
only
be
achie ed
i
he
pa icle
diame e
is
small
enough
( o
a
e omagne ic
ma e ial
whose
sa u a ion
magne iza ion
M.
:::
500G
his
condi ion
holds
i
d
::;
100..1).
An
in e media e
solu ion
o
he
p oblem
can
be
hough
consis ing
in
educing
he
linea
size
o
he
pa icles
bu
keeping
he
same
olume
ac ion
(¡ J
=
n
Vm),
whe e
n
is
he
pa icle
numbe
densi y
and
Vm
he
olume
o
one
monodomain.
Howe e ,
his
s a egy
also
mani es s
so ne
p oblems.
By
pu suing
his
me hod
one
a i es
a
highly
dispe sed
colloidal
suspension
bu
no
longe
magne ic.
I
is
clea
ha
he
elec on
spins
esponsible
o
he
magne ic
o de ing
a e
a ec ed
by
he
p esence
o
he
bounda y
o
he
olume
in
which
hey
a e
enclosed.
The
absence
o
pa ne s
o
exchange
in e ac ion
mani es
i sel
on
he
o ma ion
o
a
hin
demagne ized laye
whose
hickness
can
be
compa able
o
ha
o he
whole
pa icle.
This
ac
is
e i ied
by
expe imen s
[8,
9].
Mo eo e ,
he e exis s
an
al e na i e
poin
o
iew
which
explains
his
demagne ized
laye s
by
chemical
modi ica ion
o
he
pa icles
su aces
due,
o
ins ance,
o
oxida ion
o
o
he
in e ac ion
wi h
he
su ac an
which
is
usually
added
o
s abilize colloidal
suspensions.
This
ques ion
has
been
he
subjec
o
many
s udies
[10,
11,
12]
and
i
emains
open.
Fo
ou
pu poses
in
his
hesis,
his
p oblem
i
is
no
c ucial
because
we
will
always
assume
ha
m
=
M.
Vm
wi h
Vm
he
olume
o
he
magne ic
co e,
which
due
o
he
aboye
conside a ions
can
be
less
han
he
geome ic
olume.
When
looking
a
he
pa icles,
ano he
aspec
ha
should
be
kep
in
mind
is
ha
e en
hough
i
migh
seem
ha
a
monodomain
magne ic
pa icle
beha es
like
a
pe manen
magne ,
his
is
no
always
he
case.
The
o ien a ion
o
he
poles
in
a
magne is
ixed
by
he
magne ic
aniso opy
ene gy.
A
oom
empe a u e,
he
dimensionless
pa ame e
(J'
=
Ka
VmjkBT,
compa ing
aniso opy
and
he mal
ene gies
1.
The
e omagne íc
pa ícles 5
wi h
Ka
he
e ec i e
aniso opy
cons an ,
o
a
monodomain
pa icle
is
no
la ge
(d",
100A,
Ka
'"
104-105
J
1m3).
Consequen ly,
he
p obabili y
o he mal o a ional
luc ua ions
o
he
magne ic
momen
inside
he
pa icle
becomes
impo an .
A
u
::;
1,
he
magne ic
momen
a e
no
p ac ically
a ec ed
by
he
po en ial
ba ie
and
i s
mo ion
is
simila
o
he
B ownian
o a ion
o
a
colloidal
pa icle
in
a
iscous
liquido
Néel
was
he
i s
o
indica e
he
possibili y
o
luc ua ional
emagne iza ion
o
small
pa icles,
ha
is
why
his
p ocess
is
usually
e e ed
o
as
he
N
éel
elaxa ion.
The
cha ac e is ic
ime
o
his
spon aneous
eo ien a ion,
Tq
depends
on
he
pa ame e
U.
Fo
ime
in e als
sho e
han
T
he
pa icle
beha es
like
a
pe manen
magne ,
whe eas
o
measu ing
imes
g ea e
han
T
he
pa icle's
magne ic
momen
can
be
conside ed
ze o.
These
di e ences
can
be
co obo a ed,
o
ins an
ce,
by
means
o
Mossbaue
spec oscopy.
The
alignmen
ends
o
be
dis up ed
by
he mal
agi a ion,
and
o
cou se,
beyond
he
Cu ie
empe a u e
he
domain
possesses
no
magne iza ion
any
longe .
In
he
abo e
discussion
we
ha e
conside ed
he
elaxa ion
o
he
magne ic
momen
in
he
absence
o
an
ez e nal
magne ic
ield.
Bu ,
i
a
magne ic
ield
is
applied
he
ene gy
o
a
magne ically
uniaxial
pa icle
is
he
ollowing
u
=
-m'
H
-
Ka
Vm(71·
kl,
(1.1
)
whe e
m
=
mR
is
he
magne ic
momen
o
he
pa icle,
H
is
he
ex e nal
magne ic
ield,
and
71
is
he
uni
ec o
along
he
di ec ion
o
he
axis
o
easy
magne iza ion,
o
he
aniso opy
axis.
The
e ec i e
cons an
Ka
con ains,
in
gene al,
con ibu ions
coming
om
he
c ys alline
aniso opy
o
he
pa icle
as
well
as
he
shape
aniso opy.
F om
his
exp ession
i
will
be
easy
o
see
ha
he
dynamics
o
he
wo
deg ees
o
eedom
o
a
pa icle
in
a
liquid,
R
and
71,
a e
coupled.
I
is
also
wo h
men ioning
a
his
poin
ha
he e
will
be
essen ially
wo
ele an
dimensionless
pa ame e s
in
he
analysis.
These
pa ame e s
a e
he
abo e
in oduced
u
=
Ka
VmlkBT
and
he
pa ame e
p.
=
mH
IkBT,
compa ing
he
magne ic
ene gy
o
in e ac ion
wi h
he
magne ic
ield
and
he nal
ene gy.
The
de i a i e
o
his
magne ic
ene gy
wi h
espec
o
m
de e mines
he
alue
and
o ien a ion
o
he
e ec i e
magne ic
ield:
H-
-
e :
-H-
2KaVmA(A
RA)
eJJ
-
--a
-
-
+
n n
.
,
m m
(1.2)
which
includes
he
ex e nal
ield
H
and
he
aniso opy
ield
Ha
di ec ed
along
he
aniso opy
axis.
In
he
absence
o
he
ex e nal
ield,
he
magne ic
momen
is jus
unde he
ac ion
o
he
aniso opy
ield
and
he e
a e
wo
equi alen
equilib ium
6
INTRODUCTION
o ien a ions
R
=
71
and
R
=
-71
be ween
which
he
elaxa ion
can
ake
place.
Now,
he
equilib ium
condi ion
is
gi en
by
he
absence
o
magne ic
o ques
ac ing
upon
he
pa icle,
so
ha
in
he
absence
o
he mal
luc ua ions
he
magne ic
momen
is
pa allel
o
Hel .
When
his
is
he
case
he e
a e
wo
di e en
o ien a ional
elaxa ion
p ocesses
o
he
magne ic
momen
o
he
pa icle
ela i e
o
i s
c ys allog aphic
axes.
The
in insec
mo ion
o
he
magne ic
momen
consis
o
a
egula
p ecession
a ound
he
e ec i e
ield
and
o
chao ic
eo ien a ions
due
o
he mal
luc ua ions.
The
egula
mo ion
ela i e
o
he
c ys alline
axes
o
he
pa icle
is
desc ibed
by
he
classical
Landau-Gilbe
equa ion,
which
we
in oduce
in
Chap e
JI.
This
equa ion
ep esen s
he
p ecession
o
he
magne ic
momen
wi h
he
La mo
equency
WL
as
well
as
he
decay
o
his
mo ion
due
o
collisions,
magne oelas ic
in e ac ion,
...
Associa ed
o
his
decay
he e
is
a
cha ac e is ic
ime
To
=
CiWL.,
whe e
Ci
is
a
dimensionless
damping
cons an .
Ano he
cha ac e is ic
ime
TD
=
(2Dm)-1
is
connec ed
wi h
he
o a ional
di usion
o
he
magne ic
momen
inside
he
pa icle
Dm
=
kBTh,
whe e,
as
we
will
see
in
Chap e
JI,
h
plays
he
ole
o
a
o a ional
mobili y
o
he
magne ic
momen o
2
Mo ion
o
he
magne ic
pa icle
in
he
liquid
Besides
he
in e nal
mo ion
ela i e
o
he
pa icle
body,
he
magne ic
momen
also
unde goes
an
ex e nal
o a ional
di usion
as
a
consequence
o
he
mo ion
o
he
mag
ne ic
pa icles
in
he
liquid
hey
a e
suspended.
The
e omagne ic
pa icles
suspended
in
a
nonmagne ic
luid
expe ience
he
ac ion
o
he
ca ie
liquid
h ough
iscous
ie
ion.
We
will
now
conside
he
o a ional
B ownian
mo ion
o
a
colloidal
pa icle.
I s
de e minis ic mo ion
is
desc ibed
by
he
equa ion
o
he
o a ional
dynamics
o
a
solid
body
suspended
in
a
iscous
liquid
(2.1
)
whe e
1
is
he
momen
o
ine ia
o
he
pa icle,
i
i s
angula
eloci y,
�
he
o aional
ic ion
coe icien
o
he
pa icle
in
a
iscous
liquid,
and
he
ex e nal
o ques
ac ing
on
he
pa icle.
Jn
he
absence
o
ex e nal
o ques,
he
o a ional
mo ion
o
he
pa icle
decays
wi
h
he
cha ac e is ic
ime
TI
=
1
/
� .
Fo
a
sphe ical
pa icle
in
he
S okes
app oxima ion
�
=
81 110a3,
whe e
110
is
he
iscosi y
o
he
liquid
and
a
is
he
adius
o
he
pa icle.
3.
The
o a ional
di usion
equa ion
7
I
we
conside
¡
=
1O-2ps
and
d
--
100Á,
i
ollows
ha
TI
--
lO-lIS.
This
alue
is
so
small
ha
in
all
cases
o
p ac ical
in e es
he
ine ial
e m
in
Eq.
(2.1)
may
be
neglec ed
in
compa ison
wi h
he
iscous
one.
Indeed,
his
e m
may
be
ele an
only
o
cha ac e is ic
equencies
o ex e nal
exci a ion
�
100GHz.
Compa ed
o
he
cha ac e is ic
imes
o
hyd odynamic
p ocesses
TI
is
almos
ze o,
Besides
TI
he e
exis s
a
much
la ge
cha ac e is ic
ime
o
he
o a ional
mo ion
o
he
pa icle.
This
ime
is
de e mined
by
he
o a ional
B ownian
mo ion
o
he
axis
n
ep esen ing
he
pa icle.
This
B ownian ime
is
gi en
by
TB
=
(2D
)-1,
which
aking
in o
accoun
he
S okes-Eins ein
ela ion
i
can
be
ew i en
as
TB
=
. /2kBT.
Fo
he
same
alues
gi en
aboye
a
oom
empe a u e,
TB
-
10-6.
Du ing
his
ime,
he
pa icle
unde
he
in luence
o
he mal
luc ua ions
o a es
by
a
ini e
angle.
The
highe
he
iscosi y
o he
ca ie
liquid
he slowe
he
o a ion.
Con e sely,
he
highe
he
iscosi y
he
bigge
he
ime
TI.
Ne e heless,
TB
and
TI
will
be
compa able
jus
o
770
-
1O-4ps,
and
such
a
iscosi y
would
jus
co espond
o
ae osols
o
c yogenic
liquids.
In
eal
e o luids,
depending
on
he
pa icle
olume
and
on
he
magne ic
aniso opy
cons an
he ela ion among
he
di e en
cha ac e is ic
imes,
and
o
he
pa ame e s
J1.
and
(7
may
be
a bi a y.
Thus
he
elaxa ion
ime
o
he
magne iza ion
will
be
in
gene al
a
combina ion
o
he
di e en
imes
al eady
in oduced.
The
limi ing
case
o
an
in ini ely
s ong
coupling
o
(7
�
1,
i.e.
when
he
magne ic
momen
o
he
pa icle
is
igidly
coupled
o
he
easy
axis
o
magne iza ion,
is
known
as
he
igid
dipole
modelo
Despi e
i s
simplici y,
his
app oxima ion
is
widely
used
in
he
heo y
o
magne ic
luids,
and
allows
one
o
explain
a
wide
ange
o
magne ic
and
hyd odynamic
phenomena
expe imen ally
obse ed.
3
The
o a ional
di usion
equa ion
As
a
single
e omagne ic
pa icle
expe iences
bo h
a
sys ema ic
damping
and
andom
he mal luc ua ions
o
i s
magne iza ion
and
i sel ,
he
s udy
o
i s
dynamics
can
be
pe o med
by
ollowing
wo
di e en
me hods:
Lange i 's
app oach
o
he
heo y
o
B ownian
mo ion,
and
by
B own's
in ui i e
me hod
[9]
which
is
an
adap a ion
o
he
a gumen s
de eloped
by
Eins ein
in
1905,
o
in o he
wo ds,
by
Fokke -Planck
o
Smoluchowski
di usion
equa ions
o
he
p obabili y
densi y.
In
o de
o
accom
plish
his,
B own
ollowed
he
p ocedu e
o
Wang
and
Uhlenbeck
[14]
oge he
wi h
he S a ono ich
de ini ion
o
he
de i a i e
o
a
s ochas ic
a iable
[15].
He
also
p oposed
an
al e na i e
and
simple
app oach
o
w i ing
down
he
Fokke -Planck
o
8
INTRODUCTION
Smoluchowski
equa ion
using
a
con inui y
equa ion
a gumen
as
Eins ein
did
in
his
ea men
abou
he
ansla ional
B ownian
mo emen .
Al hough
B own's
app oach
was
de eloped
o
a
e omagne ic
pa icle
in
a
solid
ma ix,
his
o malism
can
be
u he
de eloped
o
desc ibe
he
dynamic
beha io
o
suspensions
o
hese
pa icles
in
luids.
Shliomis
and
co-wo ke s
[10]
i s
ob ained
his
equa ion
o
a
suspension
o
igid
dipoles,
and
hen
o
he
gene al
case
wi h
a bi a y
alues
o
he
a io
¡.JIu,
hey
also
deduced
he
app op ia e
Smoluchowski
equa ion
om
a
model
simila
o
he
i ine an
oscilla o
model,
[12]-[14].
The
no malized
s a iona y
solu ion
o
he
Smoluchowski
equa ion
enables
one
o
ob ain
any
equilib ium
o ien a ional
cha ac e is ics
o
an
assembly
o
magne ic
pa 
icles.
Mo eo e ,
in
mos
o
he
cases
one
has
o
deal
wi h
he
momen s
o
he
dis ibu ion
unc ion.
Pa icula ly
in e es ing
among
hese
momen s
is
he
i s
one,
which
de e mines
he
equilib ium
magne iza ion.
To
ind
ou
he
solu ion
o
he
kine ic
equa ion
o o a ional
di usion
di e en
echniques
ha e
been
p oposed,
especially
o
he
igid
dipole
model.
Fo
ins ance,
o
an
assembly
o
igid
dipoles
unde
cons an
ex e nal
condi ions
any
de ia ion
o
he
dis ibu ion
unc ion
om
i s
equilib ium
alue
may
be
expanded
in o
a
se ies
o
no mal
modes,
each
o
which
decays
acco ding
o
a
simple
exponen ial
law. The
elaxa ion
ime
spec um
o
he
dis ibu ion
unc ion
comes
om
he
eigen alues
o
he
esul ing
equa ion.
In
ac ,
he
li e ime o
any
depa u e
om
equilib ium is
de e mined
by
he
elaxa ion
ime
o
he
mos
long-li ing
modes.
The
e ec i e
ield
me hod
and
he
decoupling
app oxima ions
cons i u e
o he
app oachs
o
he
p oblem.
F om
he
di usion
equa ion
one
can
easily
ob ain
an
in ini e
se
o
coupled
equa
ions
o
he
momen s
o
he
dis ibu ion.
In
o de
o
sol e
he
se ,
one
can
unca e
i
somewhe e.
Bu
,
i
one
wan s
o
s udy
he
ield
dependence
o
he
elaxa ion
p ocess,
he
numbe
o
calcula ions
and
he
di icul y
in
unde s anding
he
esul s
ob ained
g ows
d as ically
wi h
he
ex e nal ield.
To
a oid
hese
incon eniences
ela ed
o
he
nume ical
solu ion,
i
would
help
o
ha e
a
non i ial
scheme
o
closu e
o
he
momen
equa ions
ha
is
capable
o
gi ing
a
compac
analy ical
desc ip ion
o
he
o ien a ional
p ocesses
in
a
e o luid.
Such
a
closu e
p esc ip ion
will
be
be e
he
smalle
he
numbe
o
equa ions.
Ideally,
o
a
suspension
o
igid
dipoles
one
should
ha e
only
one
equa ion,
since
only
he
i s
momen
has
a
di ec
physical
meaning.
In
such
a
way
one
would
be
able
o
gi e
a
co ec
desc ip ion
o
he
a e aged
dynamics
o
he
magne .ic
pa icles
o
a
wide
ange
o
alues
o
he
ield,
aniso opy,
empe a u e,
e c.
The
i s
idea
mee ing
all
hese
equi emen s
o
a
suspension
o
igid
dipoles
is
ihe
e ec i e
jield
me hod
[10],
conside ed
by
Leon o i ch
in
his
book
[21].
Thus,
in

3.
The
o a ionaJ
di usion
equa ion
9
he
nonequilib ium
s a e
one
may
conside
any
a bi a y
alue
o
he
magne íza íon
as
an
equilib ium
one
in
a
ce aín
e ec i e
ield.
Du ing
he
elaxa íon
p ocess,
he
e ec i e
ield
ends
o
he
ue
ield,
so
ha
he
magne íza ion
elaxes
ia
a
sequence
o
quasi-equilib iu n
s a es.
Fu he mo e,
one
should
ep esen
he
nonequilíb ium
dis ibu ion
unc ion
in
he
same
o m
as
he
s a iona y
dis ibu ion
wi h
he
e ec
i e
ield.
This
echnique
also
yields
an
exp ession
o
he
ield
dependence
o
he
elaxa ion
imes
o
a
suspension
o
igid
dipoles
unde
nons a iona y
condí íons
bu
in
a
quiescen
luid.
In
his
monog aph
we
p opose
ano he
al e na i e
and
simple
p ocedu e
o
sol e
he
se
o
in ini e
and
coupled equa ions
o
he
nomen s
o
he
dis ibu ion
unc íon
which
appea s
o
p o íde
good
esul s
no
only
o
he
igid
dipole
model
bu
in
any
a bi a y
si ua íon
includíng
a
mo ing
suspension
o
he
case
o
ini e
aniso opy
ene gy.
The
me hod
is
based
on
he
decoupling
app oximalion
o
so ne
o
he
quan i ies
in ol ed
in
he
analysis,
o
be
p ecise,
o
he
quan i ies
which
anish
a
equilib ium,
when
a e aged,
and
hose
which
a e
di e en
om
ze o.
Thís
app oxima ion
may
be
jus i ied
om
he
ac
ha
in
equilib ium
bo h
quan i ies
a e
no
co ela ed,
so,
in
non-equilib iu n
condi ions
bu
in
a
linea
egime,
o
ins an
ce
when
he e
is
small
bu
non- anishing
alue
o
he
o ici y
o
he
luid
low,
we
will
assume
ha
hese
componen s
emain
unco ela ed.
In
addi ion,
we
ensu e
he
main
cha ac e is ics
o
he
decoupled
quan i ies
such
ha
a e
pe o ming
he
decouplings
hey
a e
s íll
p opo ional
o
he
o ici y,
o
he
in a iance
unde
e lec ions
o
he
easy
axis
o
magne iza íon, ñ,
and
so
on.
We
eally
expec
ha
he
decouplings
a e
mo e
accu a e
o
qui e
small
and
high
alues
o
he
ex e nal
ield,
because
in
he
o me
case
híghe
o de
momen s
will
be
neglígíble
and
in
he
la e
he
dynamícs
o
he
pa icles
ís
mainly
de e mined
by
he ield
a he
han
being
in luenced
by
B ownian
mo ion.
The e
a e
many
possible
ways
o
ca ying
ou
he
decouplings.
Such
app oxi na ions
a e
b oadly
used
in
he
con ex
o
s ochas ic
p ocesses,
and
pa icula ly,
we
will
see
h oughou
he
hesis
he
mos
con enien
decouplings
o
be
pe o med
in
ou
sys em.
In
his
sys em
he
s abili y
o
he
s a iona y
dis ibu ion
in
a
cons an
ex e nal
ield
appea s
as
a
na u al
ac .
Fu he mo e,
om
hese
app oxima ions
one
usu
ally
ob ain
a
single
cha ac e is ic
ime
o
he
elaxa ion
p ocess.
In
he
absence
o
an
ex e nal ield
he
elaxa ion
o
he
ini ial
diso de ed
dis ibu ion
akes
place
by
means
o
ee
o ien a ional
di usion
which
is
a
mono onic
p ocess
wi h
jus
one
cha 
ac e is ic
ime.
In
he
s ong
ield
limi
he
dynamics
o
he
magne ic
momen s
is
mainly
de e minis ic,
and
hey
app oach
he
ield
di ec ion
mono onically
acco ding
o
an
exponen ial
law
wi h
a
cha ac e is ic
ime.
Thus,
al hough
he
ex e nal
ield
10
INTRODUCTION
wilI
educe
he
elaxa ion
ime,
i
does
no
change
he cha ac e
o
he
decay
p ocess.
In
p incipIe,
he
o mal
cause
o
his
beha io
is
ha
we
a e
neglec ing
he
ine ial
e
m
due
o
hei
smalIness.
Keeping
his
e m
would
lead
o
an
oscilla o y
egime
o
he
elaxa ion
p ocess
o
eno mous
and
almos
un eachable
alues
o
he
ex e nal
ield.
Howe e ,
ine ial
e ec s
a e
essen ial
i
one
conside s
he
high- equency
be
ha io
o
o a ing
objec s
o
molecula
size.
He e
he
ea men
becomes
much
mo e
complica ed.
A
e iew
o
he
p oblem
and
pa icula
esul a
is
gi en
by
Co ey
el
al.
[20].
4
Mac oscopic
hyd odynamic
heo y
o
magne ic
luids
Ha ing
cha ac e ized
he
di e en
elaxa ion
mechanism
s udied,
we
p oceed
o
ana
lyze
hei
implica ions
in
he
mac oscopic
beha io
o
he
sys em.
Fe o luids
can
be
ea ed
as
a
homogeneous
one-componen
monophase
luids.
This
app oxima ion
implies
ha
he
p ocesses
ha
should
be
conside ed
a e
such
ha
hei
cha ac e is ic
dimensions
a e
much
la ge
han
he
size
o
he
cons i uen
magne ic pa icles.
Thus,
he
magne iza ion
o he
sys em
is
assumed
o
be
dis ibu ed
h oughou
any
elemen a y
luid
olume.
Es ablishing
he
equa ions
o
mo ion
and
hea
ans e
in
a
e o uid
cons i u e
one
o
he
mos
impo an
p oblems
o
s udy
in
hese
sys ems.
Mo eo e ,
hei
peculia
p ope ies
mani es
hemsel es
mos
e iden ly
in
hei
hyd odynamic
beha io .
The
se
o
hyd odynamic
equa ions
o
an
uncha ged,
magne izable,
and
elec
icalIy
nonpola izable
luid
is
based
on
he
balance
equa ions
o
mass,
linea
and
angula
momen um,
and
hea
ans e
o
a
homogeneous
monophase
medium.
As
ega ds
he
momen um
balance
equa ion
one
should
ake
in o
accoun
he
MaxwelI
enso ,
which in
his
case
is
an isymme ic
unless
he
magne iza ion
densi y
ec o
and
he
ield
s eng h
a e
pa alIel
(due
o
he
ini e
alue
o
he
elaxa ion
ime
o
he
ans e se
componen
o
he
magne iza ion,
his
condi ion
does
no
hold
unde
he
in luence
o
hyd odynamic
o ques
ac ing
upon
he
pa icles
ei he
in
magne ic
ields
ha
change
hei
di ec ion
o
in
a
luid
ha ing
been
se
in
mo ion).
The
asymme y
o
he
s ess
enso
demands
a
gene aliza ion
o
classical
hyd odynamics
in
he
sense
ha
one
should
ake
in o
accoun
he
o a ional
deg ees
o
eedom
o
he
pa icles.
Thus
one
ough
o
conside
ha
he
angula
momen um
densi y
is
made
o
wo
pa s,
he
o bi al
associa ed
wi h
he
ansla ional mo ion
o
he
suspended
pa icles
and
4.
Mac oscopic
hyd odynamic
heo y
o
magne ic
luids
11
molecules
o
he
sol en ,
and
he
spin
momen um
caused
by
he
o a ion
o
he
pa 
icles.
Fo
simple
luids
his
spin
is
ze o
and
he
symme y
o
he
s ess
enso
is
a
necessa y
and
su icien
condi ion
o
he
conse a ion
o
he
angula
momen um.
The
si ua ion
is
di e en
i
he e
exis s
in e na!
o a ion
because
he
s ess
enso
should
no
be
symme ic
any
longe .
The
di e en
luid
elemen s
mo e
wi h
di e en
eloc
i ies
ela i e
o
he
angula
eloci y
o
he
liquid
suspending
hem
gi ing
ise
o
an
in e nal
ic ion.
In
he
absence
o
ex e nal
o ques
ac ing
on
he
pa icles,
he
di e 
ence
be ween
he
a e aged
angula
eloci y
o
he
pa icles
and
he
local
alue
o
he
o ici y
o
he
luid
low
disappea s
as
as
as
ine ial
e ec s
do.
On
he
o he
hand,
his
di e ence
esul s
in
an
i e e sible
p ocess
(wi h
dissipa ion
o
kine ic
ene gy)
o
ans e
o
angula
momen um
be ween
solid
and
liquid
phases
o
he
suspension
due
o
he
iscosi y
o
he
sol en o
Rela ed
o
he
p esence
o
an isymme ic
s esses
in
a
e o luid
he e
appea s
a
new
anspo
coe icien :
he
o a ional
iscosi y,
ha
in
gene al
con ibu es
o
he
e ec i e
iscosi y
o
he
suspension.
This
con ibu ion
ends
o
an
asymp o ic
limi
a
high
ield
s eng h,
i
is
maximum
when
he
magne ic
ield
and
he
o ici y
a e
pe pendicula
and
is
ze o
when
bo h
ec o s
a e
pa allel
since
in
his
case,
i
we
neglec
ine ial
e ec s,
he
angula
eloci y
o
he
pa icle
is
equal
o
he
o ici y
o
he
luid
a
he
poin
occupied
by
he
pa icle
and
only
he
con ibu ion
due
o
symme ic
s esses
will
emain.
In
o he
wo ds,
he
e o luid
mo ion
modi ies
he
alue
o
i s
magne iza ion,
so
i
we
conside
a
magne ic
luid
wi h
nonze o
o ici y
uni o m
shea
(Coue e
low)
in
a
cons an
magne ic
ield
pe pendicula
o
he
plane
o
shea ,
he
ield
ends
o
align
he
pa icle
magne ic
momen s
and,
he e o e,
he
pa icles
hemsel es.
Mo eo e ,
he
o ici y
low
simul aneously
a emp s
o
o a e
he
pa icles
des oying
he
magne iza ion
c ea ed
by
he
ield.
By
means
o
he
Smoluchowski
equa ion
o
he
o a ional
mo ion
o
he
pa icles
and
a e
conside ing
all
hei
possible
mechanism
o
o ien a ion,
we
will
be
able
o
p o ide
a
solu ion
o
he
mac oscopic
equa ion
o
he
magne iza ion
in
a
mo ing
luid.
This
magne iza ion,
in
u n,
en e s
he
equa ions
o
hyd odynamic
mo ion
o
a
magne ic
luid
and
hus
a ec s
he
s a e
o
mo ion
o
he
luid.
The
o a ional
iscosi y
comes
qui e
easily
om
his
analysis
using
a
heologica!
equa ion
o
s a e
o
he
an isymme ic
pa
o
he
p essu e
enso
o
by
using
he
co esponding
G een-Kubo
o mula.
To
co ec
he
hyd odynamics
o
luc ua ions,
one
mus
add
o
he
o al
luid
p essu e
enso
a
he mal
luc ua ing
p essu e
enso
whose
a e age,
bu
no
squa e
a e age,
anishes.
This
luc ua ion
e lec s
he
ac
ha
he
i s
p inciples
om
which
he
comple e
mac oscopic
heo ies
mus
ollow,
a e
ime
e e sible.
AIso
he
12
INTRODUCTION
luc ua ion
a e
seen
o
be
ela ed
o
he
dissipa ion h ough
he
luc ua ion-dissipa ion
ela ion,
showing
indeed
ha
a
dissipa i e
heo y
is
no
comple e
un il
luc ua ions
a e
included.
In
pa icula ,
i
one
conside
he
quiescen
equilib ium
s a es
o
a
magne ic
luid
in
a
nonuni o m
s a iona y
magne ic
ield,
one
obse es
ha
such
a
ield
is
no
able
o
induce
he
e o luid
mo ion,
he
magne ic
p essu e
is
balanced
by
he
hyd os a ic
p essu e.
This
ac
p o ides
he
basis
o
nume ous
p ac ical applica ions
[2].
Once
we
ha e
cha ac e ized
he
mac oscopic
beha io
o
a
e o luid,
gi ing
ex
p essions
o
he
iscosi ies,
as
well
as
o
he
a e age
o
i s
magne iza ion,
one
can
go
u he
and
analyze
he
dynamics
o
a
nonmagne ic
pa icle
(magne ic
hole)
suspended
in
a
e o luid.
Recen ly,
an
inc easing
in e es
in
he
s udy
o
he
dynamic
p ope ies
o
his
magne ic
holes
has
a isen
[5,
3,
4].
Al hough
he
pa icles
a e
no
magne ic,
when
hey
a e
suspended
in
a
ca ie
e o luid
hey
acqui e
an
induced
magne .ic
momen
equal
o
he
magne ic
momen
o
he
e o luid
olume
hey
displace.
The
in e ac ion
o
hese
induced
magne ic
momen s
o
he
holes
causes
a
numbe
o
pecu
lia
phenomena,
such
as
he
o de -diso de
ansi ion
in
magne ic
hole
la ices
and
he
non-linea
phenomena
obse ed
in
assemblies
o
holes
[7].
Fu he mo e,
knowledge
o
he
dynamics
o
such
pa icles
may
cons i u e
a
way o
cha ac e izing
he
anspo
p ope ies
o
he
e o luid.
Fo
example,
he
ic ion
coe icien
o
he
pa icle
gi es
us
in o ma ion
abou
he
iscosi y
o
he
ca ie
luid.
The
o a ional
dynamics
o
he
pa icle
is
s ongly
in luenced
by
he
p esence
o
a
o a ing
magne ic
ield.
We
ha e
ound
ha
he
hole
o a es
in
he
opposi e
di ec ion
o
ha
o
he
ield
a
low
and
mode a e
equencies
o
he
ield
o a ion,
and
his esul
is
co obo a ed
by
ecen
expe imen s,
ob aining
qui e
a
good
ag eemen
in
he
equency
ange
we
a e
conside ing.
As
we
ha e
commen ed
p e iously,
o
high
equencies
one
ough
o
accoun
o
ine ial
e ec s,
and
o
he
e omagne ic
esonance
o
he
e o luid,
which
is
he
La mo
p ecession
o
he
magne ic
momen
inside
he
pa icle
exci ed
by
an
al e na ing magne ic
ield.
The
sys em
holes- e o luid
can
be
modeled
as
a
suspension
o
pa icles
(holes)
m
a
ca ie
luid
( e o luid)
because
di e en
leng h
scales
exis
o
he
e o luid
and
he
holes.
Consequen ly,
he
e o luid
can
be
iewed
as
a
con inuous
medium
h ough
which
he
holes
may
mo e.
The
dynamics
o
he
e o luid
is
go e ned,
a
he
con inuum
le el,
by
he
a o emen ioned
gene alized
equa ions.
BIBLIOGRAPHY
19
[33]
E.
Lemai e,
Y.
G asselli
and
G.
Bossis,
J.
Phys.
JI
F ance
2,
359
(1992);
J.
Phys.
JI F ance
4,
253
(1994).
[34]
D.J.
Klingenbe g,
C.F.
Zukoski,
and
J.C.
Hill,
J.
Appl.
Phys.
73,4644
(1993).
[35]
S.
Miyazima,
P.
Meakin,
and F.
Family,
Phys.
Re .
A
36,
1421
(1987).
[36]
J.J.M.
Janssen,
J.J.M.
Bal ussen,
A.P.
an
Gelde ,
and
J.A.A.J.
Pe enboom,
J.
Phys.
D
23,
1447
(1990).
[37]
M.E.
an
Leeuwen
and
B.
Smi ,
Phys.
Re .
Le .
71,3991
(1993).
[38]
R.
Zhang
and
M.
Widom,
Phys.
Re . E
49,
R3591
(1994).
[39]
G.S.
Rushb ooke,
G.
S ell,
and
J.S.
Roye,
Mol.
Phys.
26,1199
(1973).
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J.J.
Weis
and
D.
Le esque,
Phys.
Re .
Le .
71,
2729
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[41]
M.J.
S e ens
and
G.S.
G es ,
Phys.
Re .
Le .
72,3686
(1994).

CHAPTERI
DYNAMICS
OF
FERROMAGNETIC
PARTICLES
IN
SUSPENSION:
LIMIT
CASES
In
his
chap e
we
s udy
he
dynamics
o
a
e omagne ic
pa icle
and
compu e
he
anspo
coe icien s
o
a
dilu e
suspension
cons i u ed
by
hese
pa icles
unde
he
ac ion
o
a
cons an
magne ic
ield,
We
will
de elop
a
o malism
which
makes
i
possible
o
cons uc
a
gene al
scheme
o
analyze
he
dynamics
o
he
sys e n.
In
pa 
icula ,
we
ca y
ou
an
explici
calcula ion
o
he
iscosi ies
o
a
dilu e
suspension
o
sphe ical
pa icles.
The
iscosi ies
a e
essen ially
compu ed
by
means
o
G een-Kubo
o mulas
in
he
linea
esponse
heo y
amewo k.
Mo eo e ,
he
Smoluchowski
equa
ion
o
he
o ien a ional
deg ees
o
eedom
o
he
pa icles
enable
us
o
ob ain
he
a e aged
alues
and
he
co ela ion unc ions
in ol ed
in
he
calcula ions.
Depend
ing
on
he
a io
Ji/u,
compa ing
bo h
magne ic
and
aniso opy
ene gies,
wo
simple
expe imen ally
eachable
egimes
show
up,
and
hei
analysis
cons i u es
he
scope
o
his
chap e .
The
i s
pa
o
i
deals
wi h
he
limi
u
�
Ji,
ha
will
be
e e ed
o
as
suspension
o
igid
dipoles.
In
he
second
pa ,
we
ha e
analyzed
he
implica ions
o
he
sphe e's
aniso opy
ene gy
on
i s
dynamics
a
high
magne ic
ield
(Ji
�
u).
The
o a ional
iscosi y
has also
been
calcula ed
as
a
unc ion
o
he
aniso opy
pa ame e .
Fo
a
gi en
geome y,
his
magni ude
is
esponsible
o
he
in
c ease
o
he e ec i e
iscosi y
o
he
sys em
due
o
he
p esence
o
an
ex e nal
magne ic
ield.
Ou
esul s
a e
compa ed
wi h
ha
ob ained
by
o he au ho s
based
on
di e en
app oaches.
20
Pa
1
RIGID
DIPOLES
21
22
CHAPTER
l.
LIMIT
CASES
1
In oduc ion
T anspo
phenomena
in
colloidal
suspensions
and
polyme
solu ions
ha e
played
a
p ominen
ole
in
he
s udy
o
hese
sys ems
due
o
hei
po en ial
applica ions
o
di e en
a e
as
o
physics,
physico-chemis y
and
biophysics.
In
dealing
wi h
anspo
phenomena,
one
is
mainly
con
ce ned
wi h
wo
le els
o
desc ip ion.
In
he
mac oscopic
le el,
one
conside s
he
sys em
as
a
whole
and
es ablishes
e olu ion
equa ions
o
he
ele an
quan i ies
in
he
amewo k
o
heo ies
o
con inuum
media
and
non
equilib ium
he modynamics.
In
he
o he
le el,
one
dis inguishes
he
suspended
objec s
om
he
ca ie
luid
and
uses
Lange in
o
Smoluchowski
desc ip ions
which
should
be
complemen ed
wi h
he
knowledge
o
he
dynamics
o
he
indi idual
objec s,
i.e.,
we
ha e
o
know he
o ce
and
he
o que
exe ed
by
he
luid
on
he
pa .icle
mo ing h ough
i .
Fu he mo e,
one
o
he
main
p oblems
abou
anspo
phenomena
in
di e en
sys ems
is
he
calcula ion
o
anspo
coe icien s,
which
a e
cha ac e is ic
o
he
esponse
o
he
sys em
o
ex e nal
o ces.
This
ask
has
been
acco nplished
ex en
si ely
o
pa icles
o
di e en
shape
and
he e
exis
many
well- ounded
esul s
in
he
li e a u e
[1, 2].
In
his
pa ,
we
will
deal
wi h
suspensions
o
dipola
pa icles
co nposed
o
a
ca ie
luid
and
pa icles
ha ing
dipola
momen s
igidly
a ached.
In
pa icula
we
will
discuss
he
case
o
e omagne ic
pa icles.
Thus,
one
may
conside
ha
he
magne ic
momen s
a e
only
o ien ed
by
he
magne ic
ield,
he
o a ional
B ownian
mo ion
o
he
e omagne ic
pa icles
and
by
he
ex e nal
low.
This
las
mechanism
depends
on
he
shape
o
he
pa icles.
As
an
example,
when
he
pa icles
a e
sphe es
hey
a e
o ien ed
by
he
o ici y
o
he
low,
whe eas
elonga ed
pa icles
a e
o ien ed
by
he
elonga ional
low
as
well.
The
possibili y
o
he
o ien a ion
o
he
pa icles
by
he
magne ic
ield
is
esponsible
o
he
peculia
beha io
o
he
iscosi ies
o
e o luids.
Pa icula ly,
i
has
been
shown
ha
in
he
case
o
a
suspension
o
e omagne ic
pa icles,
he e
appea s
a
new
anspo
coe icien :
he
o a ional
iscosi y,
ela ed
o
he
p esence
o
an isymme ic
s esses
[3].
The
o a ional
iscosi y
o
he
suspension
was
calcula ed
by
Shliomis
[4]
using
a
con inuum
medium
app oach
in which
he
di e ence
be ween he
o ici y
o
he
luid
and
he
a e aged
angula
eloci y
o
he
pa icles
(spin)
leads
o
he
in oduc ion
o
an
in e nal
angula
momen um
o
he
luid
elemen s.
Fo
his
eason,
i
beco
mes
necessa y
o
o mula e
a
new
balance
equa ion
o
his
quan i y,
which
is
coupled
wi h
he
momen um
balance
equa ion.
Ou
pu pose
in his
pa
o
he
hesis
is
o
p esen
a
uni ied
o malism
able
o
1.
In oduc ion
n=R.
n=R
H
•
n=R
e:=R
e-n=R
•
23
Figu e
1.1:
In
suspension
o
igid
dipoles
he
magne ic
momen s
a e
igidly
coupled
o
he
easy
axis
o
magne iza ion
o
he
pa icles
and
elax
oge he .
desc ibe
he
dynamics
o
he
pa icles
and
o
compu e
he
anspo
coe icien s.
He e
we
p esen
wo
e sions:
one
whe e
he
iscosi ies
a e
calcula ed
om
a
heological
equa ion
o
s a e,
gi ing
he
exp ession
o
he
p essu e
enso ,
and
he
o he
based
on
he
linea
esponse
heo y,
which
assumes
he
exis ence
o
luc ua ions
in
he
l
uid
and
equi es
he
knowledge
o
he
luc ua ing
dynamics,
To
his
end,
we
ha e
o ganized
his
pa
o
he
chap e
in
he
ollowing
way.
In
Sec ion
2,
we
analyze
he
dynamics
o
a
e omagne ic
sphe e
by
using
an
induced
o ce
in
he
Na ie -S okes
equa ion,
accoun ing
o
he
pe u ba ion
caused
by
he
sphe e
due
o
i s mo ion.
We
ind he
exp ession
o
he
o ce
and
o que
exe ed
on
he
pa icle
which
spli
up
in o
sys ema ic
and
andom
con ibu ions,
and
discuss
he
o igin
o
an isymme ic
s esses.
Likewise,
in
Sec ion
3
he
exp essions
o
he
shea
and
o a ional
iscosi ies
a e
ob ained
using
a
heological
equa ion
o
s a e
o
he
con ibu ion
o
he
p essu e
enso
due
o
he
p esence
o
pa icles
in
suspension.
We
in oduce
he
Smoluchowski
equa ion
and
we
ge
he
exp ession
o
he
a e age
o
he
magne ic
momen
o
ob ain
he
o a ional
iscosi y.
Sec ion
4
is
de o ed
o
he
calcula ion
o
he iscosi ies
by
using
G een-Kubo
o mulas.
Fo
his
pu pose,
we
need
o
know
he
luc ua ion
dynamics
which
ollows
om
luc ua ing
hyd odynamics
[5],
and
om
he
analysis
o
he
co esponding
Smoluchowski
equa ion
o
he
o ien a ion
o
he
pa icles
[6]-[8].
Finally,
in
he
las
sec ion
we
summa ize
ou
main
esul s.
24
CHAPTER
1.
LIMIT
CASES
2
Dynamics
o
a
e omagne ic
sphe e
Le
us
conside
a
dilu e
suspension
o
e omagne ic
sphe ical
pa icles
o
adius
a
imme sed in
a
nonpola
incomp essible sol en ,
unde
he
in luence
o
an
ex e nal
magne ic
ield.
Ou
s a ing
poin
o
analyzing
he
s a iona y
mo ion
o
he
pa icle
will
be
he
linea ized
s a iona y
Na ie -S okes
equa ion
(2.1
)
in
which
p( ,
)
is
he
p essu e
ield,
1/0
he
iscosi y
o
he
sol en
and
e ,
)
is
he
eloci y
ield.
Owing
o
he
incomp essible
na u e
o
he
ca ie
luid
he
eloci y
ield
also
sa is ies
V'.
=
O.
(2.2)
In
Eq,
(2.1)
we
ha e
assumed
ha
he
pe u ba ion
caused
by
he
mo íon
o
he
pa 
icle
may
be
aken
in o
accoun
by
ín oducíng
an
induced
o ce
ield
¡i( , )
[10,11].
Fu he mo e,
we
ha e
conside ed
he
possíbilí y
o
luc ua ions
in
he
luid
by
means
o
he
Lange in-Iike
luc ua ing
sou ce -V'.
IlR,
coming
om
he
decomposi ion
o
he
iscous
p essu e
enso
in
i s
sys ema ic
and
andom
IlR
pa s
[5].
To
be
consís en ,
he
induced
o ce
should
be
de ined
in
such
a
way
ha
s ick
bounda y
condi ions
a
he su ace
o
he
pa icle,
e ,
)
=
ü( )
+
i( )
x
aii,
o
I
-
Rcm( )1
=
a,
(2.3)
a e
sa is ied.
In
his
equa ion
ü
and
ñ
a e
he
ansla ional
and
o a ional
eloci ies
o
he
pa icle,
espec i ely,
Rcm( )
is
he
posi ion
o
he
cen e
o
mass
o
he
sphe e,
and
n
==
(
-
Rcm( »/W
-
Rcm( )l.
Since
ine ial
and
memo y
e ec s
ela i e
o
he
mo ion
o
he
pa icle
a e
neglec ed
in
he
s a iona y
case,
he
induced
o ce
is
a
su ace
o ce
and
can
be
exp essed
as
[10,
11]
(2.4)
whe e
1(n)
is
an
induced
o ce
densi y
pe
uní
a ea.
The
eloci y
ield
gi en
by
Eq.
(2.1)
ís
he e o e
alíd
ín
he
whole
space,
e en
ínside
he
sphe es.
To
compu e
he
mobili y
enso
we
need,
i s
o
all,
o
know
he
o mal
solu ion
o
he
elocí y
ield.
Thís
solu ion
ollows
om
(2.1)
by
Fou íe
ans o ming
in
.
In
ac ,
a e
elímina íon
o
he
p essu e,
by
applying
he
ans e sal
p ojec o
(1
-
kk),
wí h
k
==
{,
one
ob ains

2.
Dynamics
oE
a
Ee omagne ic
sphe e
25
(2.5)
whe e
we
ha e
used
he
incomp essibili y
condi ion
which,
in Fou ie
space,
eads
k
.
=
O
and
we
ha e
in oduced
he
p opaga o
-
1
""
T(/c)
=
-/c2
(1
-
H),
'lo
(2.6)
wi h
1
as
he
uni
ma ix
and
Vo
he
unpe u bed
eloci y
ield
in
he
absence
o
he
pa icle.
The
p opaga o
T(k)
is
he
Fou ie
ans o m
o
he
Oseen
enso
gi en
by
T( j
=
_1_(1
+
),
811'770
whe e
==
.
This
o mal
solu ion
can
be
ew i en
in
eal
space
in
he
o m
(2.7)
( j
=
o( j
+
J
di
'T(i
-
i')
.
i(i')
+
R( j,
whe e
he
andom
eloci y
ield
is
gi en
by
he
in e se
Fou ie ans o m
o
(2.8)
(2.9)
The
a e age
and
co ela ion
o
his
quan i y
hen
ollows
om
he
s ochas ic
p op
e ies
o
he
ando n
pa
o
he
iscous
p essu e
enso .
Acco ding
o
luc ua ing
hy
d odynamics,
llR
in oduces
a
Gaussian
whi e
noise
s ochas ic
p ocess
o
ze o
mean
and
luc ua ion-dissipa ion
heo em
[5]
R
-
R
-,
3
-
-,
,
(
Ilij(k,
)IlI:1(k
, »)
=
2kBT
'1o :.ijl:l
(211')
8(k
+
k)
8(
-
),
(2.10)
whe e
we
ha e
de ined
:.ijl:1
=
8ik8j1
+
8il8jk
-
�8ij8kl.
The
o mal
solu ion
(2.8)
gi es
he
eloci y
ield
a
any
poin . The e o e,
o
a
gi en
poin
a
he
su ace
o
he
sphe e,
in
iew
o
he
s ick
bounda y
condi ion,
one
has
(2.11)
whe e
we
ha e
in oduced
he
esponse
unc ion
p(n,
n')
p(n,
n')
=
(2a:)3
J
i
exp
(iak.
(n
-
n'»
T(k).
(2.12)
26
CHAPTER
1.
LIMIT
CASES
To
p oceed
u he ,
we
will
expand
he
eloci ies
and
he
induced
o ce
o
(2.11)
in
mul ipoles
[10,
11].
Fo
an
unspeci ied
quan i y
�(a71)
o
ha
equa ion,
one
has
�(
A)
=
�
(21
+
1)!!
Al.
!li'+1
an
L..J
I!
n
0
.
'=0
Be e
n'
is
an
i educible
enso
o
ank
1,
i.e.
he
enso
o
ank
1
aceless
and
symme ic
in
any
pai
o
i s
indexes,
cons uc ed
wi h
he
ec o
ñ.
The
symbol
0
s ands
o
ull
con ac ion
and
!li'+1
is
he
mul ipole
o
o de
1
+
1,
which
is
a
enso
(2.13)
o
ank
1
+
1.
These
quan i ies
a e
gi en
by
!li'+1
=
_!_
J
dñ
n'�(ñ).
411'
Inse ing
hese
exp essions
in o
he
o mal
solu ion
(2.11),
one
ge s
(2.14)
(l+I)
=
�
(2/'
+
1)!!
1'(/+1,1'+1)
0
p"+I)
+
(l+I)
+
(l+I)
L..J
I'!
o
R'
1'=0
which
ela es
he
mul ipoles
o
he
eloci y
and
o
he
induced
o ce
densi y.
In
his
(2.15
)
exp ession
he
ma ix
elemen s
1'(1+1,1'+1)
ha e
been
de ined
as
1'(1+1,1'+1)
=
(417 )
J
dñ
J
dñ'
n'
1'(71,
ñ')
ñ/'.
(2.16)
The
ep esen a ion
o
he
mobili y
ke nel
in
e ms
o
i educible
mul ipoles
is
diagonal
in
he
indexes
1
and
I'
[10,
11].
As
an
example,
we
ha e
I'C1,1)
=
.?:!:.I.
37]0
Acco ding
o
equa ion
(2.15)
o
1
=
O,
we
ge
(2.17)
(2.18)
The
mul ipole
(l)
can
be
compu ed
h ough
i s
de ini ion
ou lined
in
equa ion
(2.14).
One
has
(l)
=
_!_Jdñ
(ü+ñ
x
añ)
=
ü.
47
(2.19)
In
he
same
way,
by
assuming
ha
o(
')
=
�.
(Rcm
+añ)
a
he
su ace
o
he
pa icle,
we
ob ain
�l)
=
_!_
J
dñ
�.
(Rcm
+
añ)
=
�
.
Rcm.
411'
(2.20)
2.
Dynamics
o
a
e omagne ic
sphe e
27
Since
p(ii.)
is
he
o ce
pe
uni
a ea
exe ed
by
he
pa icle
on
he
luid,
he
d ag
o ce
exe ed
by
he
luid
on
he
pa icle
is
gi en
by
(2.21)
acco ding
o
Eq.
(2.14).
Joining
he
esul s
gi en
h ough
Eq.
(2.19)-(2.21)
and
using
Eq.
(2.18)
and
he
exp ession
o
he
na ix
ele nen
gi en
in
Eq.
(2.17),
we
ge
he
hyd odyna nic
o ce
-H
--
::(l)
F
=-67 7]oa(u- 3·Rem- R),
(2.22)
o n
which
one
iden i ies
he
ic ion
coe icien
67 7]oa
co esponding
o
he
in e se
o
he
nobili
y,
as
well
as
he
B ownian o ce
FB
=
-67 7]oa ¡i).
We
can
also
p oceed
by
conside ing
he
case
1
=
1
in
Eq.
(2.12).
One
nay
show
ha
his
equa ion
gi es
ise
o
V(2)
=
31l(2,2)
0
:¡::{2)
+
�2)
+
�),
whe e
he
nobili y
1l(2,2)
ollows
o n
Eq.
(2.12)
and
is
gi en
by
(2.23)
1l(2,2)
=
_a_S
+
�A.
157]0
97]0
In
his
exp ession
S
and
A
ha e
been
de ined
as
he
iso opic
enso s
(2.24
)
(2.25)
and
1
Aijkl
=
"2
(6il6jk
-
6ik6jl),
sy n ne ic
and
an isy n ne ic
in
any
pai
o
i s
indexes.
The
nobili y
(2.24),
oge he
wi h
he
esul s
(2.26)
V(2)
=
_!_
J
dñ
il
[11
+
ñ
x
aii.]
=
�
E
.
ñ
47
3
(2.27)
and
(2.28)
ob ained
by
neans
o
Eq.
(2.14),
wi h
E
being
he
Le i-Ci i a
enso
and
T
s anding
o
he
ansposed
na ix,
can
be
used
in
(2.23).
One
inally
a i es
a
he
exp ession
28
CHAPTER
l.
LIMIT
CASES
a
-
T
(2)
a
1
1
.-1'2)
-(E·n-�
)-VR
=-(-S+-A)0. '
.
3
'70
5 3
(2.29)
The induced
o ce
mul ipole
:¡:(2)
may
be
ob ained
om his
equa ion.
Mul iplying
bo h
sides
o
Eq.
(2.29)
by
S,
one
has
(2.30)
whe e,
o
an
a bi a y
enso
o second
ank
T,
T(')
s ands
o
i s
symme ic
aceless
pa
T��)
=
�(T."
+
T"
-
�TLL6
..
)
'}
-
2
'}
}'
3
.""
'}
.
On
he
o he
hand,
mul iplying
bo h
sides
o
Eq.
(2.29)
by
A,
one
ob ains
(2.31)
:¡:(2a)
=
'7o(E'
ñ
+
�(a)
_
3'70
�a),
a
whe e, again
o
an
a bi a y
enso o
second
ank,
T(a)
is
de ined
as
(2.32)
(2.33)
Mo eo e ,
om
Eq.
(2.29)
we
can
ob ain
he
equa ion
o
he
hyd odynamic
o que
since,
as
in
Eq.
(2.21),
we
ha e
(2.34
)
so
ha
only
he
an isymme ic
pa
o
:¡:(2)
is
in ol ed.
Making
use
o
he
ac
ha
E
:
E
=
-21,
in
Eq.
(2.29),
we
a i e
a
(2.35)
whe e
we
ha e
de ined
he
o ici y
Wo
==
�E
:
�(a),
and
one
can
iden i y
he
B ownian
o que
TB
=
-127 '7oa2E:
�).
As
ollows
om
(2.22)
and
(2.35),
he
exp essions
o
he
o ce
and
o que
spli
up
in o
sys ema ic
and
andom
con ibu ions.
The
la e
o igina es
om
he
p esence
o
luc ua ions
in
he
luid
and
hei
s a is ical
p ope ies
ollow
om
luc ua ing
hyd odynamics
acco ding
o
Eq.
(2.9)
and
(2.10).
Once
we
ha e
ob ained
he
hyd odynamic
o ce
and
o que,
we
can
p oceed
o
ob ain
he
equa ions
o
he
ansla ional
and
o a ional
mo ion
o
he
pa ícle.
Since
e omagne ic
pa icles
a e
e y
small
( adius
abou
100
Á)
ine ial
e ec s
can
be
3.
Shea
and
o a ional
iscosi ies
o
he
e o luid
35
3.3
Ro a ional
iscosi y
and
he
ex e nal
magne ic
ield
The
quan i y
be ween
b ake s
in
Eq.
(3.5)
can
be
ans o med
by
using
he
equa ion
o
he
o a ional
mo ion
o
a
e omagne ic
pa icle,
which
ollows
om
he
o al
angula
momen um
equa ion
dñ
-1
d ñ
-
_
_
-
I-;¡¡+"Yo
T =-{ (O-wo)+mxH
(3.3.1)
whe e
1
is
i s
momen
o
ine ia,
and
"Yo
he
elec on
gy omagne ic
ac o .
We
will
neglec
he
e m
coming
om
he
angula
momen um
o
he elec ons
de e mining
he
magne ic
momen
o
he
pa icles.
Mo eo e ,
i
he
pa icle
is
small
enough,
as
occu s
o
e o luids,
ine ial
e ec s
can
also be
neglec ed.
The e o e
he
hyd odynamical
and
magne ic
o ques
balance
each
o he
ou
and
Eq.
(3.5)
ans o ms
in o
(3.3.2)
In
iew
o
(3.3.2),
we
conclude
ha
he
an isymme ic
pa
o
he
p essu e
enso
depends
on
he
ela i e
o ien a ion
o
he
magne ic
momen
and
he
ield.
To
compu e
he
co esponding
anspo
coe icien
namely,
he
o a ional
iscosi y,
T¡ ,
we
ha e
o
a e age
his
pa
o e
all
possible
o ien a ions
o
k
We
hen
ob ain
(3.3.3)
whe e
he
a e age
on
he
igh -hand
side
ollows
om
he
exp ession
o
he
i s
momen
o
R
compu ed
by
he
Smoluchowski
equa ion
and
gi en
by
(3.2.6)
along
wi h
(3.2.8).
We
inally
ob ain
(3.3.4)
This
equa ion
yields
he
o a ional
iscosi y
3
�- anh�
T¡
=
"2T¡0l/J
�
+
anh
s
'
This
las
quan i y
beha es
o
�
«:
1
as
(3.3.5)
1
2
T¡
==
¡T¡ol/J�
and
is
hen
quad a ic
in
he
ield.
Fo
�
�
1,
one
has
(3.3.6)
(3.3.7)

36
CHAPTER
l.
LIMIT
CASES
1.0
�--------�--------�----------�--------�
0.8
---
__
-
0.6
0.4
0.2
0.0
�
�
_'
L-
�
O
5
10
15
20
Jl
Figu e
1.2:
The
quan i y
¡ / ¡.
e sus
he
pa ame e
p.
The
solid
line
co esponds
o
ou
esul
gi en
in
Eq.
(3.3.5)
and
wi h
he
esul
coming
om
a
phenomenological
heo y
p oposed
in
Re .[4],
whe eas
he
dashed
line
is
ob ained
om
he
solu ion
o
he
Smoluchowski
equa ion
gi en
in
Re .[6].
The
do s
ep esen
expe imen al
da a
om
Re .
[13].
4.
Calcula íon
o
he
iscosi ies
using
G een-Kubo
o mulas
37
Consequen ly,
he
o a ional
iscosi y
ends
owa ds
he
asymp o ic
limi
�1104>.
In
Fig.
1.2
we
ha e
plo ed
he
educed
o a ional
iscosi y
11 /11.,
wi h
11.
==
�1104>,
as
a
unc ion
o
he
pa ame e
¡J.
The
p esence
o
he
o a ional
iscosi y
leads
o
he
in oduc ion
o
an
e ec i e
iscosi y
o
he
e o luid.
Fo
he
case
o
aplana
Coue e
low
whe e
he
applied
magne ic
ield
is
choosen
pe pendicula
o
he
o ici y,
his
quan i y
gi es
he
inc ease
o
he
iscosi y
o
he
suspensión
due
o
he
ac ion
o
he
ex e nal
ield.
These
esul a
ag ee
wi h
he
ones
ob ained in
Re .
[4]
by
means
o
a
phenomenological
heo y
and
wi h
he
expe imen s
pe o med
by
Mc
Tague
[17].
In
ou
o me
calcula ion
o
he
i s
momen
o
R,
we
ha e
neglec ed
a
e m
o
second
o de
in
he
o ici y.
This
app oxima ion
can
be
emo ed
by
compu ing
he
dis ibu ion
unc ion
up
o
second
o de
in
his
quan i y
a
low
magne ic
ield.
The
esul ing
a e ages
can
be
gi en,
o
example,
when
he
o ici y
is
pa allel
o
.
Fo
he
a e age
o
he
y-componen ,
one
has
(3.3.8)
which
is
in
good
ag eemen
wi h
he
co esponding
esul
ob ained
in
Re .
[15]
by
means
o
B ownian
dynamics. No e,
howe e ,
ha
since
he
e omagne ic
pa icles
a e
e y
small,
second
o de
co ec ions
in
he
o ici y
a e
only
impo an
a
e y
high
shea
a es.
4
Calcula ion
o
he
iscosi ies
using
G een-Kubo
o mulas
Ou
pu pose
in
his
sec ion
is
o
p esen
an
al e na i e
de i a ion
o
he
shea
and
o
a ional
iscosi ies.
These
quan i ies
wiJl
be
compu ed
om
he
co esponding
G een
Kubo
o mulas
a ising
om
linea
esponse
heo y,
which
ha e
been
widely
used
o
calcula e
anspo
coe icien s.
The
con ibu ion
o
he
pa icles
o
he
shea
iscosi y,
in oduced
in
Eq.
(3.4),
is
hen
gi en
by
(4.1
)
In
o de
o
compu e
he
co ela ion
o
he
symme ic
pa
o
he
p essu e
enso ,
we
will
use
he
ela ionship
be ween
he
p essu e
enso
and
he
second
o de
mul ipole
o
he
induced
o ce,
gi en
h ough
eqs.
(3.2)
and
(2.30).
P o ided
ha
ou
e e ence
s a e
should
be
in
equilib ium
condi ions
( 3
=
O),
his
ela ionship
is
he
oJlowing
38
CHAPTER I.
LIMIT
CASES
TI
(.)
(
)
-
20
2
(2.)
(
)
p,:Cz
-
-
1 a
7]oV
R,:c'
,
(4.2)
which
used
in
(4.1),
yields
(4.3)
The
co ela ion
in
his
exp ession
can
be
compu ed
om
luc ua ing
hyd odynam
ics.
In
ac ,
om
(2.9)
and
(2.10)
we
ge
In
eal
space
his
exp ession
is
gi en
by
(4.5)
Using
now
he
mul ipole
expansion,
o
poin s
a
he
su ace
o
he
pa icle,
we
can
ob ain
he
co esponding
co ela ion
o
he
second
mul ipole
o
he
andom
eloci y.
One
a i es
a
(4.6)
whe e
he
ma ix
ele nen
1-'(2,2)
has
been
gi en
in
(2.24).
Fo
he
symme ic
pa
o
his
co ela ion,
one
has
(4.7)
Employing
his
exp ession
in
(3.3)
and
pe o ming
he
in eg al,
we
hen
come
o
he
esul
(4.8)
which
co esponds
o
he
con ibu ion
o
he
iscosi y
o
he
suspension
owing
o
he
p esence
o
pa icles.
The
o a ional
iscosi y
can
be
compu ed
by
neans
o
he
same
p ocedu e.
The
G een-Kubo
o mula
is
now
T/
=
Vk1BT
100
d (TI��lz( )TI��lz(O))
o
in
e ms o
he
co esponding
axial
ec o s
(4.9)
4.
Calcula ion
o
he
íscosi es
using
G een-Kubo
o mulas
39
n-
=
Vk�T
100
d (n��J( )n��J(o»).
Making
use
o
equa ions
(3.2)
and
(2.32)
and
he
de ini ion
(3.1.3),
we
ge
(4.10)
(4.11)
whe e
we
ha e
employed
again
he
equi emen
3
=
O,
o
he
e e ence
s a e.
The co ela ion
in
Eq.
(4.10)
hen
ollows
om
his
las
exp ession.
One
has
(II��J( )n��J(o»)
=
(4'11"a3)2 ¡�([Oy( )
+
23a
(
:
�")( »Yl
[01/(0)
+
23a
(
:
�")(O»!ll)
(4.12)
o
al e na i ely
2
(n��J( )n��J(O»)
=
(41Ta3)2��
(m
x
H)y( )(m
x
H}y(O»),
(4.13)
wi h
�
=
81Ta3 ¡o
being
he
o a ion ic ion
coe icien .
To
ob ain
his
equali y
we
ha e
used
he
equa ion
o
o a ional
mo ion
o
he
pa icle
(4.14)
whe e
he
exp ession
o
he
hyd odynamic
o que
is
gi en
in
(2.35)
and
M
is
he
magne ic
o que.
In
(4.14)
we
ha e
neglec ed
ine ial
and
gy omagne ic
e ec s,
hus
he
o ques
balance
each
o he
ou .
This
ac
p o ides
he
equa ion
o
de i e
(4.13)
om
(4.12).
The
o a ional
iscosi y
can
be
inally
exp essed
as
3
2
oo
,,
TJ
=
2¡ JTJoD Jl
Jo
d (R.,( )R.,(O»),
(4.15)
whe e
Jl
is
he
Lange in
pa ame e
n;;�,
compa ing
magne ic
and
he mal
ene gies,
and
we
ha e
aken
he
magne ic
ield
poin ing
owa ds
he
z-di ec ion.
To
pe o m
he
in eg al
in
(4.15)
we
ha e
o
know
he
co ela ion
unc ion
o
he
x-componen
o
he
ec o
Él.
The
e olu ion
equa ion
o
his co ela ion
comes
om
he
Smoluchowski
equa ion
(4.16)
alid
in
he
absence
o
o ici y.
In
ac ,
s a ing
om
(4.16)
one
may
de i e
he
e olu ion
equa ion
o
he
co ela ion
40
CHAPTER
l.
LlMlT
CASES
d(R(�/l(O»)
=
D {
-2(R( )R(0»)
+
¡d!
(R(O)}
-
IJ(R( )R(O)R( »)
.
Ji},
(
4.17)
which
o
he
pe pendicula
componen
yields
(4.18)
To
sal e
his
equa ion
we
will
in oduce
a
decoupling
app oxima ion
as
in
he
p e ious
sec ion.
This
app oxima ion
consis s
o
decoupling
he
pe pendicula
and
pa allel
componen s
o
he
ec o
R,
due
o
he
di e en
na u e
o
he
dynamics
o
bo h
componen s,
as
we
al eady
poin ed
ou ,
and
can
be
o mula ed
in
he
ollowing
way:
(4.19)
In
addi ion,
in
he
linea
egime,
we
may
app oxima e
in
(4.19)
(RII( »)
by
i s
equi
lib ium
alue
(RII(O»)eq
=
C(IJ).
This
equilib ium
a e age
is
compu ed
wi h
he
equilib ium
dis ibu ion
unc ion
IJ
A A
/!eq
=
.
h
exp(IJR·
H).
411'SIn IJ
Consequen ly,
Eq.(
4.18)
ans o ms
in o
he
elaxa ion
equa ion
(
4.20)
(4.21)
om
which
we
may
iden i y
he
elaxa ion
ime
TJ.
(4.22)
Thus
we
expec
ha
he
co ela ion
decays
exponen ially
acco ding
o
(4.23)
The
o a ional
iscosi y
hen
ollows
om
he
G een-Kubo
o mula
(4.15)
oge he
wi h
(4.23).
Pe o ming
he
in eg al,
one
a i es
a
(4.24)
In
iew
o
he
esul
(R.:(0)2)eq
=
�,
which
can
also
be
ob ained
by
compu ing
he
equilib ium
a e age
wi h
he
equilib ium
dis ibu ion
unc ion
(4.20),
one
ob ains

5.
Conclusions
41
3
Il-
anh
Il
TJ
=
"2
TJo4J
Il +
anh
Il
'
(4.25)
which,
as
we
expec ed,
also
coincides
wi h
he
co esponding
exp ession
ob ained
by
Shliomis
using
a
con inuum
heo y
[4]
and
wi h
he
esul
o
[7].
5
Conclusions
In
his
pa
we
ha e
analyzed
he
dynamics
o
a
e omagne ic
sphe ical
pa icle,
as
well
as
he iscosi ies
o
a
dilu e
suspension
cons i u ed
by
hese
pa icles.
Ou
heo e ical
amewo k
is
based
on
he
Na ie -S okes
equa ion
in
which
an
induced
o ce, esul ing
om
he
pe u ba ion
in oduced
in
he
dynamics
o
he
luid
by
he
pa icle,
and
a
s ochas ic
Lange in
sou ce,
coming
om
he
luc ua ions
o
he
hyd odynamical
ields,
ha e
been
included.
This
equa ion
accoun s
o
he
coupled
dynamics
o
he
luid
and
he
pa icle.
A
mul ipola
expansion
o
he
quan i ies
appea ing
in
he
o mal
solu ion
o
he
Na ie -S okes-Lange in
equa ion
leads
o
he
exp essions
o
he
o ce
and
o que
exe ed
on
he
pa icle
which
con ain
andom
con ibu ions
whose
s a is ical
p ope ies
a e
dic a ed
by
luc ua ing
hyd odynamics
[5].
We
use
wo
me hods
o
ob ain
he
shea
and
o a ional
iscosi ies. One
is
de e 
minis ic
and
is
based
on
he
Ki kwood
o mula
o
he
iscous
p essu e
enso
which
is
shown
o
be
ela ed
o
he
second
o de
mul ipole
o
he
induced
o ce.
The
o he
is
based
on
he
linea
esponse
heo y
gi ing
he
anspo
coe icien s
in
e ms
o
ime-depe iden
co ela ion
unc ions.
The
p esence
o
he ield
is
esponsible
o
he
appea ance
o
an
an isymme ic
con ibu ion
o
he
p essu e
enso
and
in oduces
he
o a ional
iscosi y
as
a
new
anspo
coe icien .
The
exis ence
o
an isymme ic
s esses
comes
om
he
ac
ha
he
o que
exe ed
by
he
magne ic
ield
on
a
dipole
and
he
hyd odynamic
o que
balance
each
o he
ou .
Consequen ly,
he
angula
e
loci y
o
he
pa icle
may
di e
om
he
alue
o
he
o ici y
o
he
luid
a
he
poin
i
occupies.
We
ha e
s udied
he e
a
simple
case
in which
he
anspo
coe icien s
ha e
al eady
been
calcula ed
by
o he
au ho s
by
means
o
di e en
me hods.
Ou
main
pu pose
has been
o
de elop
a
a he
gene al
o malism
ha
can
be
use ul
in
he
s udy
o
anspo
phenomena
in
e o luids
in
di e en
and
mo e
in ica e
si ua ions,
which
cons i u e
he
subjec
o
so ne
o
he
ollowing
pa s
o
he
monog aph.
Fu he mo e,
he
me hods
de eloped
he e
can
also be
applied
o
s udy
he
dependence
o
he
iscos
i y
on
equency
[16],
o
o
highe
concen a ions,
in
which
case
dipola
in e ac ions
42
CHAPTER
J.
LJMJT
CASES
play
an
essen ial
ole.
Wo k
is
being
done
in
he
las
case
in
o de
o
ind
u he
heo e ical
esul s.
Pa
II
80FT
MAGNETIC
MATERIAL8
IN
HIGH
MAGNETIC
FIELD8
43
44
CHAPTER
l.
LIMIT
CASES
n
__
n
n
H
Figu e
1.3:
Fo
so
magne ic
ma e ials
in
su icien ly
high
magne ic
ields he
magne ic
momen s
o ien
hemsel es
in
he
di ec ion
o
he
ield
e y
quickly.
Then
i
akes
place
he
mechanical
o a ion
o
he
pa icles
o
he
s a iona y
o ien a ion.
6
In oduc ion
In
his
pa
o
he
chap e
we
apply
he
o malism
de eloped
in
he
p e ious
pa
o
he
case
in
which
a e
pe u bing
he
sys em,
he
elaxa ion
owa ds
he
magne ic
ield
akes
place
in
wo
s eps:
i s
a
quick
elaxa ion
o
he
magne ic
momen ,
hen
a
mechanical
o a ion
o
he
pa icle
owa ds
he
equilib ium
o ien a ion,
wi h
he
magne ic
momen
also
pa allel
o
he
easy
axis
o
magne iza ion
[19].
This
si ua ion
occu s
o
high
magne ic
ields
and
he
ene gy
o
he
magne ic
pa icles
educes
o
he
ene gy
o
aniso opy.
Expe imen ally,
i
is
ound
ha
he
magne iza ion
ends
o
lie
along
ce ain
c ys allog aphic
axes;
his
e ec
is
known
as
c ys alline
aniso opy.
I
is
addi ional
o
he
di ec ional
e ec s
ha
occu
when
he
samples'
shape
lacks
sphe ical
o
cubic
symme y.
The
exis ence
o
c ys alline
aniso opy
may
be
de non
s a ed
by
he
magne iza ion
cu es.
I
is
clea
ha
much
smalle
ields
a e
equi ed
o
magne ize
he
c ys als
o
sa u a ion
along
ce ain
di ec ions han
along
o he s.
The
c ys allog aphic
axes
along
which
he
magne iza ion
ends
o
lie
a e
called
easy
di ec ions;
he
axes
along
which
i
is
mos
di ícul
o
p oduce
sa u a ion
a e
called
ha d
di ec ions.
Wi h
his
goal
in
mind,
his
pa
has
been
dis ibu ed
as
ollows.
Sec ion
7
is
in ended
as
a
sho
e iew
o
he
o a ional
dynamics
which
was
p e iously
analyzed.
9.
Conclusions
51
Re .
[18]
using
a
phenomenological
app oach,
based
on
he
o mula ion
o
a
elaxa ion
equa ion
o
he
in e nal
angula
momen um
o
he
sys em
in
a
con inuum
desc ip ion.
Howe e ,
we
ha e
obse ed
again
so ne
disc epancies
in
ou
esul s
wi h
espec
o
he
ones
ob ained
by
hose
au ho s
om
an
app oxima ed
solu ion
o
he
Smoluchowski
equa ion
[18],
especially
o
in e media e
alues
o
he
aniso opy
pa ame e .
9
Conclusions
By
means
o
he
o malism
de eloped
in
he
p e ious
pa ,
we
ha e
p esen ed
in his
pa
a
calcula ion
o
he
o a ional
iscosi y
based
on
he
G een-Kubo
o mula
o
his
anspo
coe icien .
Ou
esul s
show
ha
he
iscosi y
inc eases
when
inc eas
ing
he
aniso opy
pa ame e
and
eaches
a
sa u a ion
limi o
We
ha e
compa ed
ou
cu e
wi h
he
ones
ob ained
in
Re .[18].
Thei
esul
coming
om
a
solu ion
o
he
s a iona y
Smoluchowski
equa ion
o e es ima es
ou s,
whe eas
he
one
based
on
a
phenomenological
elaxa ion
equa ion hey
p opose
o
he
in e nal
angula
momen
um
o
he
suspension,
is
close
o
ou s,
I
would
be
in e es ing
o
ga he
expe imen al
da a
in
o de
o
decide
which
o
he h ee
app oaches
is
he
mos
con incing.
We
ha e
also
compu ed
he
o a ional
iscosi y
by
means
o
he
heological
equa
ion
o
s a e
p oposed
by
Ki kwood,
gi ing
he
p essu e
enso
in
e ms
o
he
hy
d odynamic
o ce
exe ed
by
he
luid
on
he
magne ic
pa ide.
The
a e age
o
he
p essu e
enso
has
been
calcula ed
om
he
s a iona y
solu ion
o
he
Smoluchowski
equa ion
(9.1
)
which
now
con ains
a
con ibu ion
due
o
he
o ici y
o
he
low.
Following
he
s eps
indica ed
in
[7]
we
ge
he
same
esul
(8.22).
In
he
same
way,
we
could
sol e
he
mo e
gene al
case
in
which
bo h,
he
elaxa ion
o
he
magne ic
momen
owa ds
he
ield
and
owa ds he easy
axis
o
magne iza ion,
ake
place.
So ne
ela ed
heo e ical
esul s
ha e
been
ob ained
in
[1]
and
i
will
be
he
subjec
o
he
ollowing
chap e .

Bibliog aphy
[1]
J.
Happel
and
H.
B enne ,
Low
Reynolds
Numbe
Hyd odynamics (Kluwe
Aca
demic
Publishe s,
Do d ech ,
The
Ne he lands,
1991).
[2]
H.L.
F isch
and
R.
Simha,
in:
Rheology,
ol.Il,
F.
R.
Ei ich,
ed.
(Acad.
P ess,
New
Yo k,
1956).
[3]
S.R.
de
G oo
and
P.
Mazu ,
Non-Equilib ium
The modynamics
(Do e ,New
Yo k,1984).
[4]
M.
1.
Shliomis,
So .
Phys.
JETP
34
(1972)
1291.
[5]
L.
Landau
and
E.
M.
Li shi z,
S a is ical
Physics,
Pa
2
(Pe gamon
P ess,
Ox
o d,
1981).
[6]
M. A.
Ma senyuk,
Yu.
L.
Raikhe
and
M.
1.
Shliomis,
So .
Phys.
JETP
38
(1974)
413.
[7]
J.M.
Rubí
and
M.C.
Miguel,
Physica
A
194
(1993)
209.
[8]
J.
M.
Rubí,
C.
Salueña,
and
A.
Pé ez-Mad id,
in:
Complex
Fluids,
L.
Ga ido,
ed.,
Lec u e
No es
in
Physies,
ol.
415
(Sp inge -Ve lag,
Be lin).
[9]
M.C.
Miguel,
J.
Bone
A alos,
A.
Pé ez-Mad id
and
J.M.
Rubí,
Physiea
A
193
359
(1993).
[10]
P.
Mazu
and
D.
Bedeaux,
Physica
76
(1974)
235.
[11]
P.
Mazu
and
W.
Van
Saa loos,
Physiea
115
A
(1982)
21.
[12]
H.
B enne ,
J.
Colloid.
In e ace
Sci.
1
(1970)
141.
[13]
J.
P.
Me
Tague,
J.
Chem.
Phys.
51
(1969)
133.
52
BIBLIOGRAPHY
53
[14]
C.
Salueña,
A.
Pé ez-Mad id
and
J.
M.
Rubí,
J.
Colloid.
In e ace
Sci. 164,
(1994)
263.
[15]
A.
O.
Cebe s,
Magni . gid odin.4
(1984)
17
(Magne ohyd odynamics,1984).
[16J
C.
Salueña
and
J.
M.
Rubí,
J.
Chem.
Phys.
102
(1995)
3812.
[17]
J.-C.
Bac i,
K.
Dje i,
S.
Ne eu
and
R.
Pe zynski,
J.
Mag.
Mag.
Ma e .
123
(1993)
67.
[18]
Yu.
L.
Raikhe
and
M.
1.
Shliomis,
SOy.
Phys.
J.
Appl.
Mech.
Tech.
Phys.
15,
(1974)
470.
[19]
M.I.
Shliomis
and V.1.
S epano ,
J.
Magn.
Magn.
Ma e .
122
(1993)
196.
CHAPTER
II
GENERAL
DYNAMICS
In
he
p e ious chap e
we
ha e
analyzed
he
dynamics
o
a
e omagne ic
pa icle
and
he
anspo
coe icien s
o
a
dilu e
suspension
cons i u ed
by
hem
in
wo
lim
i ing
cases,
namely,
a
suspension
o
igid
dipoles
and
a
suspension
o
a
so
magne ic
ma e ial
unde
he
in luence
o
a
high magne ic
ield.
The
main
goal
o
his
chap e
is
o
p o ide
a
gene al
heo y
co e ing
he
whole
ange
o
expe imen al
si ua ions.
Indeed,
di e en
measu emen s
ca ied
ou
o
mag
ne ic
liquids
show
ha
he e
is
ini e
coupling
be ween
he
o ien a ion
o
he
mag
ne ic
momen
o
a
e omagne ic
monodomain
and
he
o ien a ion
o
he
pa icle
i sel
(cha ac e ized
by
he
o ien a ion
o
i s
c ys alline
axes).
Because
o
his
coupling,
he
elaxa ion
o
he
magne ic
momen s
akes
place
in
wo
di e en
ways
ha
p oceed
simul aneously:
o a ion
wi hin
he
pa icle
and
oge he
wi h
he
pa icle
wi h
e
spec
o
he
ca ie
liquido
Bo h
p ocesses
a e
o
o a ional
di usion
ype.
Thus,
o
all
p ac ical
pu poses,
he
Smoluchowski
equa ion,
desc ibing
he
e olu ion
o
he
p obabili y
densi y
o
he
di ec ions
o
bo h
he
magne ic
momen
and
he
axis
o
easy
magne iza ion
o
he
pa icles,
has
been used.
In
pa icula ,
we
compu e
he
o a ional
iscosi y
om
a
G een-Kubo
o mula
and
gi e
an
exp ession
o
di e en
elaxa ion
imes.
These
cha ac e is ic
imes
come
om
he
dynamic
equa ions
o
he
co ela ion unc ions
which,
in
he
linea
esponse
heo y
amewo k,
a e
in ol ed
in
he
calcula ion
o
so ne
o
he
ma e ial's
physical
p ope ies
we
a e
in e es ed in
(op ical,
magne ic,
...
).
Ou
esul s
ag ee
qui e
well
wi h
expe imen s pe o med
wi h
di e en
samples
o
e omagne ic
pa icles,
which
pe mi
o
dis inguish
he
di e en
elaxa ion
egimes
occu ing
when
he
size
and
he
na u e
o
he
magne ic
ma e ial
o
he
g ains
a e
eely
modi ied.
54
1.
In oduc ion
55
1
In oduc ion
Sys ems
o
single-domain e omagne ic pa icles
imme sed
in
a
solid
o
liquid
phase
exhibi
a
numbe
o
in e es ing
elaxa ion
phenomena
which
ha e
been
he
subjec
ma e
o
many
expe imen al
and
heo e ical
analyses
[1]-[4].
These
phenomena
a e
essen ial
in
he
s udy
o
he
dynamics
o
hese
pa icles,
and,
pa icula ly,
ha e
a
c1ea
in luence
when
de e mining
he
e ec i e
iscosi y,
he
dynamic
bi e ingence,
and
he
magne ic
suscep ibili y.
One
o
he
main
peculia i ies
o
hese
sys ems
is
ha
hei
p ope ies
a e
g ea ly
in luenced
by
he
p esence
o
an
ex e nal
magne ic
ield.
I
is
p ecisely
his
ac
which
has
been
he
basis
o
many
p ac ical applica ions
[1].
The
o a ional
dynamics
o
a
e omagne ic
pa icle
embedded
in
a
liquid
phase
is
he
esul
o
he
compe i ion
o
h ee
o ien a ional
mechanisms
ela ed
o
he
ex e 
nal
ield,
he
axis
o
easy
magne iza ion,
and
o a ional
B ownian
mo ion.
Tha
is,
whe eas
he
magne ic
momen
o
he
e omagne ic
pa icle
elaxes
owa ds
he
di ec
ion
o
he
magne ic
ield,
he
axis
o
easy
magne iza ion
ends
o
be
aligned
wi h
he
magne ic
momen ,
hus
gi ing
ise
o
di e en
coupled
elaxa ion
phenomena.
Un
il
ecen ly,
he
mos
equen
case
ha
has
been
s udied
in
he
li e a u e
deals
wi h
igid-dipoles
[4]-[6],
o
which he
aniso opy
ene gy
is
dominan
due
o
he
la ge
alue
o
he
aniso opy
cons an ,
and
because
he
adius
o
he
pa icle
usually
exceeds
a
c i ical
alue.
Wha
is
mo e,
when
looking
o
he
elaxa ion
phenomena
desc ibed
by
a
igid
dipole,
one
dis ega ds
he
p ecessional
mo ion
o
he
magne ic
momen ,
and,
consequen ly,
he
associa ed
dissipa ion.
Unde
hese
condi ions,
we
canno
alk
abou
he
elaxa ion
o
he
axis
o
easy
magne iza ion
owa ds he
magne ic
momen
any
longe ,
ins ead
bo h
ec o s
elax
oge he .
Howe e ,
he e
a e
ma e ials
o
which
he
aniso opy
ene gy
may
be
compa able
o
he
ene gy
associa ed
wi h
he
in e ac ion
wi h
he
magne ic
ield,
o
e en
smalle .
The e o e,
a
gene al
heo y
en
compassing
such
a
wide
a ie y
o
si ua ions
and
accoun ing
o
expe imen al
esul s
should
be
de eloped.
The
p esence
o
di e en
elaxa ion
mechanisms
has
implica
ions
in
he
o m
o
he
e ec i e
iscosi y
o
he
sys em,
which
exhibi s
signi ican
co ec ions
when
compa ed
o
he
iscosi y
o
a
suspension
o
non-magne ic
pa icles
o
he
same
shape.
Ano he
poin
o
in e es
is
he
appea ance o
elaxa ion
imes
which
a e
usually
in ol ed
in
he
cha ac e iza ion
o
ce ain
physical
p ope ies,
and
which
a e
sui able
o
being
measu ed
by
means
o
di e en
expe imen al
echniques.
The
pu pose
o
his
chap e
is
o
p esen
a
heo y
capable
o
gi ing
exp essions
o
he
ele an
anspo
coe icien s
o
he
sys em
and
o
he
co esponding
cha ac
e is ic
elaxa ion
imes
de e mining
o
ins ance,
he
e ec i e
iscosi y,
he
dynamic
56
CHAPTER
Il.
GENERAL
DYNAMICS
n
n
H
Figu e
11.1:
The
magne ic
momen s
a e
o ien ed
along
an
in e media e
di ec ion
be
ween
ha
o
he
magne ic
ield
and
he
easy
axis
o
magne iza ion
when
bo h
he
aniso opy
ene gy
and
he
magne ic
ene gy
o
in e ac ion
wi h
he ield
a e
compa a
ble.
The
elaxa ion
o
hese
wo
deg ees
o
eedom
is
coupled.
bi e ingence,
and
he
magne ic
suscep ibili y
o
he
suspension.
We
will
ocus
on
he
gene al
si ua ion
in
which
he
magne ic
and
aniso opy
ene gies
o
he
pa icles
may
ha e
a bi a y
alues.
The
o malism
we
ha e
de eloped
is
based
on
he
linea
esponse
heo y
whe e
he
co ela ion
dynamics
comes
om
a
Smoluchowski
equa
ion.
As
we
will
show
in
one
o
he
sec ions,
ou
esul
o
he
elaxa ion
ime
o
he
o a ion
o
he
pa icle
is
compa ed
o
expe imen al
da a
and
ag ees
qui e
well
wi h
bi e ingence
expe imen s.
We
ha e
dis ibu ed
he
chap e
in
he
ollowing
way:
in
Sec ion
2,
we
es ablish
basic
equa ions
desc ibing
he
dynamics
o
he
deg ees
o
eedom.
O
pa icula
in e es
is
he Smoluchowski
equa ion
o
he
p obabili y
densi y
which
is
gi en
in
a
gene al
case
o
unspeci ied
alues
o
he
magne ic
and
aniso opy
ene gies.
The e
a e
di e en
ways
o
de i ing
such
equa ion.
Al hough
we
will
no
go
h ough
his
ques ion,
he
equa ion
we
p opose
can
be
compa ed
o
ano he
one
ob ained
p e iously
om
a
di e en
heo e ical
me hod.
In
Sec ion
3,
we
deal
wi h
he
calcula ion
o
he
o a ional
iscosi y
using
a
G een-Kubo
equa ion
p oposed
om
he
linea esponse
heo y.
This me hod
leads
o
an
exp ession
o
his
anspo
coe icien
which
is
s udied
in
pa icula
si ua ions
o in e es .
Sec ion
4
is
de o ed
o
he
calcula ion
o he
elaxa ion
imes
o
he
pa icles
and
he
ans e sal
componen
o
he
magne iza ion

2.
Coupled
dynamícs oE
he
deg ees
oi
E eedom
57
when
conside ing
he
di e en
o ien a ional
mechanisms.
We
ha e
compa ed
ou
esul s
o
expe imen s
done
o
wo
samples
o
e y
common
e omagne ic
pa icles
o
which
he
size
and
he
na u e
o he
magne ic
ma e ial
clea ly
es ablish
di e en
alues
o
he
magne ic
ene gy
and
he ene gy
o
aniso opy,
ob aining
a
good
ag eemen
in
bo h
si ua ions.
Finally,
in
he
las
sec ion
we
summa ize
ou
main
esul s.
2
Coupled dynamics
o
he
deg ees
o
eedom
The
ene gy
o
a
sphe ical
single-domain
e omagne ic
pa icle
unde
he
ac ion
o
an
ex e nal
magne ic
ield
is
he
sum
o
wo
con ibu ions.
These
con ibu ions
o igi
na e
om
he
ex e nally
imposed
magne ic
ield
and
he
p esence
o
an
axis
o
easy
magne iza ion
( o
uniaxial
c ys als).
I s
exp ession
is
gi en
by
(2.1
)
whe e
ñi
=
mR
is
he
magne ic
momen
o
he
pa icles,
H
is
he
ex e nal
magne ic
ield,
Ka
is
he
i s
aniso opy
cons an
(assumed
posi i e),
Vm
is
he
magne ic
olume
o
one
o
hese
sphe es,
and n
is
he
uni
ec o
along
he
di ec ion
o
he
axis
o
easy
magne iza ion
o
.uniaxial
magne ic c ys als.
I
is
clea
om
Eq.
(2.1)
ha
in
he
gene al
case
whe e
bo h
con ibu ions
may
ake
a bi a y
alues,
he
elaxa ion
mechanisms
o
he
deg ees
o
eedom,
R
and
ñ,
o
he
e omagne ic
sphe es
in
suspension
a e
coupled.
The
de e minis ic
dynamics
o
R
is
go e ned
by
he
Landau-Gilbe
equa ion
[7],
p oposed
o
s udy
he
elaxa ion
o
he
magne ic
momen s
o
magne ic
pa icles
e n
bedded
in
a
solid
ma ix
dR
'Yo
{)U
•
dR
.
-
=
---_
x
R-
0/-
X
R.
d
m
{)R
d
(2.2)
F om
his
equa ion,
one
may
iden i y
he
wo
mechanisms
esponsible
o
he
a ia
ion
o
k
he
e ec i e
ield
HeJl
ex:
-�,
which
causes
a
La mo
p ecessional
mo ion
o
R,
and
he
mean
ield,
H
d
ex:
-
4 ,
which
in oduces
a
damping
due
o
he
col
lisions
o
he
elec ons
de e nining
he
magne ic
momen
o
he
domain
in
a
me al,
o
in
a
semiconduc o ,
due
o
magne oelas ic
in e ac ions.
In
Eq.
(2.2),
'Yo
is
he
gy omagne ic
a io
o
an
elec on,
and
he
quan i y
O/
plays
he
ole
o
a
da nping
coe icien .
The Landau-Gilbe
equa ion
can
be
ew i en
such
ha
58
CHAPTER
n.
GENERAL
DYNAMICS
dEl
_
au
_
_
-
=
-hR
X
-_
x
R
+
WL
X
R
d
aR
'
(2.3)
wi h
WL
=
gDe"
as
he
La mo
equency
o
he
p ecessional
mo ion,
and
whe e
9
==
1'0(1
+
0'2)-1
and
h
==
�(1
+
0'2)-1
[2].
This
equa ion
is
alid
in
a
s a iona y
ame
o
e e ence.
I
he
e omagne ic
pa icle
is
o a ing
i sel
wi h
he
angula
eloci y
ñ,
we
mus
modi y
Eq.(2.3)
by
adding
on
i s
igh
hand
si
de
he
co esponding
con ibu ion
coming
om
he
o a ion.
One
hen
has
dH
(
_
au)
_
_
_
di
=
-h
R
x
aH
x
R
+
(WL
+
n)
x
R.
(2.4)
Fu he mo e,
he
dynamics
o
ñ
ollows
om
he
kinema ic
ela ion
dñ
-
di
=
n
x
ñ,
(2.5)
This
exp ession
can
be
ew i en
as
dñ
[_
1
-
-]
_
-d
=
Wo
+
-m
x
H
x
n,
�
wi h
�
=
811'7]oa3
being
he
o a ional
ic ion
coe icien
o
he
pa icles;
7]0
is
he
(2.6)
iscosi y
o
he
ca ie
luid,
a
is
he
hyd odynamic
adius
o
he
pa icles,
and
Wo
he
o ici y
o
he
ca ie
luid.
One
a i es
a
his
exp ession
a e
using
he
de e minis ic
pa
o
he
balance
equa ion
o
he
o al
angula
momen um
(2.7)
p o ided
ha
we
neglec
he
e
m
accoun ing
o
he
ine ial
e ec s and
he
e
m
com
ing
om
he
angula
momen um
o
he
elec ons
de e mining
he
magne ic
mo nen
o
he
pa icles.
He e
B
is
he
B ownian
o que
ac ing
on
he
pa icle
[8].
The e olu ion
o he
p obabili y
densi y,
1/;(¡, ),
wi h
"Y
==
(H,ñ),
is
go e ned
by
he
Smoluchowski
equa ion.
When
ñ
=
O,
which
co esponds
o
he
case
o
pa icles
embedded
in
a
solid
ma ix,
he
Smoluchowski
equa ion
was
deduced
by
B own
[9]
om
he
Landau-Gilbe
equa ion.
In
hese
las
condi ions,
and in
o de
o
p ese e
he
o al
angula
momen um
conse a ion,
he
small
magne ic
pa icle
should be
embedded
in
a
la ge
igid
solid
ma ix.
O he wise,
he e
will
appea
an
elas ic
wis
in
he
ma ix,
and
one
should
ake
in o
accoun
he
co esponding
elas ic
o que
in
he
equa ion
o
he
o al
angula
momen um
o
he
pa icle.
The
manne
in
which
B own
exp essed
his
in ui i e
me hod
o
de i ing
he
Smoluchowski
equa ion
was
o
conside
2.
Coupled
dynamics
oi
he
deg ees
oi
E eedom
59
he
e ec
o
he mal luc ua ions
on
he
p obabili y
densi y.
B own
sugges ed
ha
he mal
agi a ion
causes
P
o
become
mo e
uni o m
so
ha ,
in
an
equa ion
desc ibing
i s
ime
e olu ion,
he mal
agi a ion
gi es
ise
o
a
di usion
e m
in
P.
Shliomis
and
co-wo ke s
[10]
ob ained
his
equa ion
o
a
suspension
o
igid
dipoles.
I
we
de ine
he
dimensionless
pa ame e s
/J
=
�l
and
(7
=
K,,"aF,
compa ing
magne ic
and
aniso opy
ene gies
o
he mal
ene gy,
espec i ely,
his
las
si ua ion
co esponds
o
he
limi
(7
�
/J.
Rhaike
and
Shliomis
[11]
also
p oposed
he
Smoluchowski
equa ion
o
he
opposi e
limi
(7
<€::
/J,
in
which he
dipoles
a e
apidly
o ien ed
owa ds
he
ield
di ec ion.
As
ega ds
he
gene al
case
o
a bi a y
alues
o
he
a io
/JI
(7,
Shliomis
and
co-wo ke s
also
deduced
he
app op ia e
Smoluchowski
equa ion
om
a
model
simila
o
he
i ine an
oscilla o
model,
[12]-[14].
In
such
a
gene al
si ua ion,
he
Smoluchowski
equa ion
can
al
so
be
ob ained
om
he
con inui y
equa ion
in
he
space
spanned
by
he
deg ees
o
eedom
"'{
==
(R,
n)
J Pb, )
a
«
)
(
).)
a
=-8"'(.
J"'{,
+ P"'{, "'{,
whe e
�
==
d"'{ld
and
he
cu en
Jb, )
is
gi en
by
he
Fick's
law
(2.8)
Jb,
)
=
-D
.
a Pb,
),
8"'(
(2.9)
wi h
D
being
a
di usion
ma ix.
Combining
(2.8)
and
(2.9)
we
hen
a i e
a
a P
a a
.
8
=
8"'(
.
(D
.
8"'(
P
-
P"'{).
(2.10)
The
di usion
ma ix
is
ela ed
o
he
mobili y
ma ix
b
h ough
he
Eins ein
ela ion,
D
=
kBTb.
A e
inse ing
eqs.
(2.4)
and
(2.6)
in o
(2.10),
we
can
ew i e
Eq.
(2.10)
in
he
o n
(2.11)
whe e
we
ha e
a
con ibu ion
coming
om
a
non-po en ial
cu en
-Ynon-po ,
and
he
mobili ies
b
a e
ound
o
be
( o
mo e
de ails
see
Appendix
A)
-
;
(1
-
nn),
1
'-
-
(h
+
{ )(1
-
RR),
TI,'
"
bRn
==
{
[(n
.
R)1
-
nR].
(2.12)
60
CHAPTER
II.
GENERAL
DYNAMICS
He e
he
symbol
T
s ands
o
ansposi ion.
These
exp essions
can
be
employed
in
Eq.(2.11).
A e
so ne
ma hema ical ans o ma ions
(see
de ailed
calcula ions in
Appendix
A),
one
hen
a i es
a
he
Smoluchowski
equa ion
N
&
=
---
-
U
--
D (nR
+
nñ)·
[ P(nR
+
nñ)
kBT
+
(nR
+
nñ) P]
- -
U
-
-
DmnR·
{,pnR
kBT
+
nR P}
-
nR·
(WL P)
('RR
+
ññ)
.
(wo P),
(2.13)
+
h
-ñ='
8
d--R'
8
·1
kT·
w
e e
"-ñ
n
x
8ñ
an
n
R
=
x
7iR
a e
o a iona
ope a o s,
D;
==
T
IS
he
B ownian
o a ional
di usion
coe icien ,
and
Dm
==
kBTh
can
be
in e p e ed
as
he
di usion
coe icien
o
he
magne ic
momen
inside
he
pa icles.
These
di usion
coe icien s
a e
ela ed
o
wo
elaxa ion
imes
in ol ed
in
he
Smoluchowski
equa ion,
namely
TD
=
(2Dm)-1
ela ed
o
he
chao ic
eo ien a ions
o
ni
inside
he
pa icle
due
o
he mal
luc ua ions,
and
he
B ownian
ime
TB
=
(2D
)-1.
The
Smoluchowski
equa ion
(2.13)
ag ees
wi h
he
co esponding
one
ob ained
in
Re .
[12]
by
using
a
model
simila
o
he
i ine an
oscilla o
model
and
will
be used
in
ou
subsequen
analysis.
3
G een-Kubo
o mula o
he
o a ional
iscosi y
In
his
sec ion,
we
we
will
ocus
on
he
de e mina ion
o
he
o a ional
iscosi y
om
he
co esponding
G een-Kubo
o mula.
This
o mula
gi es
his
anspo
coe icien
in
e ms
o
he
co ela ion
unc ion
o
he
axial
ec o ,
ñ�a),
ela ed
o
he
an isy n
me ic
pa
o
he
con ibu ion
o
he
pa icles
o
he
p essu e
enso
[8],
(3.1
)
whe e
V
is
he
olume
o
he
sys em.
In
Re .
[8],
W!
ob ained
a
ela ionship
be ween
he
pa icle
con ibu ion
o
he
p essu e
enso ,
ñp,
and
he
ex e nal
o que
expe ienced by
he
pa icle
du ing
i s
mo ion.
As
a
esul ,
i
we
conside
he
magne ic
ield
poin ing
owa ds
he
z-di ec ion,
i
is
ound
ha
he
o a ional
iscosi y
can
be
inalIy
ew i en
as
3
['''''
T}
=
"2
PT}o
i.
1-'2
Jo
d (R ( )R (O»),
wi h
P
=
4/3;a'
being
he
olume
ac ion
o
pa icles.
(3.2)
3.
G een-Kubo
o mula
{o
he
o a ional
iscosi y
67
1.0
�----...,.-----�-----�----�
0.8
DjD,=1
11=100
0.6
0.4
0.2
0.0
�----�----�------._----�
O
5
10
15
20
o
Figu e
11.3:
Ro a ional
iscosi y
e sus
he
pa ame e
O"
o
di e en
alues
o
he
a io
Dm/
D,.
and
o
he
pa ame e
¡J.

68
CHAPTER
Il.
GENERAL
DYNAMICS
espec i ely,
which
also
coincide
wi h
he
esul s
gi en
in
Re .[12].
E en
in
his
case,
wi h
¡
�
1,
i
u
akes
mode a e
o
small
alues,
he
o a ional
iscosi y
does
no
each
i s
sa u a ion
alue
any
longe .
In
pa icula ,
i
u
-
O,
his sa u a ion
alue
depends
on
he
a io
Dm/D
as
we
ha e
poin ed
ou
in
Eq.
(3.22).
Tha
is,
he
dissipa ion
no
only
depends
on
he
sol en
iscosi y,
bu
also
on
he
damping
cons an
O'
and
he
gy omagne ic
ac o
10.
Addi ionally,
he
o a ional
iscosi y
inc eases
wi h
u
un il
i
eaches
i s
sa u a ion
alue
when
u
-
oo.
The beha io
o
'7
/
'7.
as
a
unc ion
o
u,
and
o
di e en
alues
o
¡
is
depic ed
in
Fig.
11.3,
om
which
we
can
co obo a e
he
main
ea u es
o
ou
p e ious
analysis.
The
a io
Dm/
D;
has been
ob ained
a e
conside ing
he
ollowing
alues
o
he
in ol ed
quan i ies:
O'
'"
10-2,
10
'"
107e-1
s-1,
M.
'"
103e,
'70
'"
1O-2c.!J;
and
he
ac
ha
he
magne ic
olume
is
almos
he
same
as
he
hyd odynamic
olume
o
he
pa icles,
wi h
a
=
10-6cm.
An
al e na i e
p ocedu e
used
o
calcula e
he
iscosi y
in ol es
a
heological
equa ion
o
s a e
o
he
p essu e
enso .
In
his
way,
i
is
also
possible
o
ob ain
exac ly
he
same
exp ession
o
he
o a ional
iscosi y
(3.16),
bu
now
in
he
p esence
o
a
linea
o ici y
ield
in
s a iona y
condi ions.
4
Relaxa ion
imes.
Compa ison
wi h
expe imen s
4.1
T ansien
bi e ingence
in
c ossed
ields
The
elaxa ion
o
he
op ical
bi e ingence
induced
by magne ic
g ains
dispe sed
m
he
analyzed
medium,
is
among
he
simples
expe imen al
ools
a ailable
o
heologi
cal
s udies
o
iscoelas ic
solu ions.
A
iscosime e
based
on
he
de e mina ion
o
he
elaxa ion
o
small
magne ic
pa icles
in
suspension
in
he s udied
medium
u ns
he
iscosi y
de e mina ion
in o
an
op ical
bi e ingence
measu emen
wi h
many
ad an
ages:
i)
i
is
a
nondes uc i e
me hod,
ii)
we
jus
need
o
add
a
e y
small
amoun
o
pa icles
(<p
'"
10-4),
iii)
i
does
no
need
any
mechanical
sys em,
i )
he
use
o
a
lase
beam
o
he
bi e ingence
measu emen
enables
us
he
de e mina ion
o
he
iscosi y
in
he
olume
o
a
ew
mm3.
In
addi ion,
as
he
size
o
he
magne ic
p obe
is
o
he
o de
o
100A,
he
measu ed
iscosi y
is
a
local
quan i y.
The
main
limi a ion
o
he
usual
ansien
bi e ingence
de ices
is
he
polydispe si y
o
he
magne ic
pa icles.
Usually,
a
log-no mal
dis ibu ion
o
sphe ical
g ain
diame e s,
d,
as
he
one
gi en
h ough
he
exp ession
4.
Relaxa ion
imes.
Compa ison
wi h
expe imen e
69
P(d)
=
�CTd
exp
(
-
2�2
In2
(�)
)
,
(4.1.1)
whe e
do
and
CT
a e
he
mean
diame e
and
he
a iance
o
he
dis ibu ion,
espec
i ely,
is
sui able
o
desc ibe
he
samples.
Thus,
in
o de
o
ge
an
exponen ial
elax
a ion
wi h
only
one
cha ac e is ic
ime
a
sample
whose
alue
o
CT
is
abou
0.3
o
less
mus
be
used.
Each
pa icle
is
i s
cha ac e ized
h ough
a
s a ic
bi e ingence
measu emen
as
a
unc ion
o
he
magne ic
ield.
This
is
due
o
he
aniso opy
o
he
elec ic
suscep ibili y
enso
ela ing
he
pola izabili y
o
he
sys em
o
he
inciden
elec ic
ield.
The main
causes
o
he
aniso opy
a e
he
in e nal
op ical aniso opy
o
he
magne ic
ma e ial
(c ys alline
aniso opy)
and
he
shape
aniso opy
o
he
pa icles.
Conce ning
he
dynamic
beha io
o
he
bi e ingence,
di e en
expe imen s
can
be
ca ied
ou :
In
a
liquid,
i
he
magne ic
ield
is
swi ched
o
ab up ly,
pa icles
he mally
elax
owa ds
andom
di ec ions. Thus
looking
a
he
bi e ingence
elaxa ion
is
compa ing
iscous
o
he mal
ene gies.
I
a
squa e
pulse
o
magne ic
ield
H
is
applied
o
he
e o luid
solu ion,
magne ic
pa icles
i s
end
o
align
along
he
ield
leading
o
bi e ingence
6n( ), and,
as
he
ield
is
swi ched
o ,
hey
he nally
elax
o
andom
di ec ions.
Bi e ingence
dec eases
exponen ially
acco ding
o
Pe in's
law
[18)
8n( )
=
8n(d,
H)exp- /T(d)
(4.1.2)
whe e
he
cha ac e is ic
ime
is
T(d)
=
(6D
)-1,
wi h
D;
he
o a ional
di usion
coe icien .
In
he
quan i y
8n( ),
he
polydispe si y
o
he
samples
modi ies
bo h
8n(d,
H)
and
T(d).
Owing
o
his
ac ,
.6.n( )
is
no
a
simple
exponen ial
unc ion
o
ime.
Bu ,
in
o de
o
cha ac e ize
Lln( ),
one
can
use
he
sho es
ime
deduced
om
he
ini ial
slope
o
he
unc ion
In(.6.n( ))
e sus
.
In
a
iscoelas ic
medium,
looking
a
he
bi e ingence
esponse
o
an
al e na ing
magne ic
ield
allows
o
each
he
iscosi y
and
he
elas ic
modulus
o
he
medium
a
high
equencies.
I
a
la ge
magne ic
ield
is
supe imposed,
he e
is
a
magne ic
es o ing
o ce
and
iscous
ene gy
is
compa ed
o
ha .
Ins ead
o
elaxing
a
andom,
magne ic
momen s
elax
owa ds
he
di ec ion
o
he
supe imposed
ield.
In
his
case,
he
elaxa ion
ime
associa ed
wi h
he
o ien a ion
mechanism
o
he
pa icles,
TR,
has
been
measu ed
ecen ly
by
Bac i
e
al.
[19,
20).
This
quan i y
comes
om
he
elaxa ion
ime
o
he
ligh
in ensi y
collec ed
in
a
pho ocell
a e
c ossing
he
sample
in
he
p esence
70
CHAPTER
TI.
GENERAL
DYNAMICS
o
an
ex e nal
magne ic
ield,
and
when
applying
addi ional
pulses
o
magne ic
ield
o
pe u b
he
sample.
These
expe imen s
pe mi
us
o
dis inguish
he
di e en
elaxa ion
egimes
occu ing
when
he
size
and
he
na u e
o
he
magne ic
ma e ial
o
he
pa icles
a e
modi ied.
These
egimes
a e
de e mined
by
he
pa ame e s
1-'
and
u,
compa ing
magne ic
and
aniso opy
ene gies
wi h
he mal
ene gy,
espec i ely.
In
he
expe imen s,
i
was
obse ed
ha
e o luid
pa icles,
p e en ed
om
mo ing
by
being
quenched
in
a
igh
gel
ne wo k,
do
no
exhibi
bi e ingence al hough
hey
s ill
show
magne iza ion.
Consequen ly,
he
bi e ingence
o
he
solu ion
is
closely
ela ed
o
a
mechanical
alignmen
o
he
pa icles
along
he
equilib ium
o ien a ion.
In
Re s.
[8,
15,
21],
wo
opposi e
limi s
we e
conside ed,
one
whe e
u
�
1-'
( igid
dipole
app oxima ion)
and
ano he
o
which
u
«:
1-'.
Fo he
o ne
limi ,
he
e
laxa ion ime
o
he
pe pendicula
componen
o
he
magne iza ion
is
ound
o
be
[8,
15]
(
4.1.3)
which,
in
he
case
when
1-'
-+
00,
ends
o
-DI
=
1!..a.
In
he
la e
case,
elaxa ion
IJ
IJ
occu s
in
wo
s eps,
i s
a
quick
elaxa ion
o
R
owa ds
il,
hen
a
mechanical
o a ion
o
he
pa icle
o
he
equilib ium
o ien a ion
wi h
he
easy
axis
o
magne iza ion
pa allel
o
R
and
ñ.
Unde
hese
condi ions,
he
cha ac e is ic
elaxa ion
ime
could
be
ob ained
om
he
equa ion
o
mo ion
o
wha e e
componen
o
he
co ela íon
unc ion
(ñl.nz)( )(nl.nz)(O)}
[21].
Pe o ming
he
co espondíng
decouplings
in
i s
e olu ion
equa ion,
we
ob ain
he
elaxa ion
ime
(
)-1
(
1
)-1
(
1
)-1
T
=
2D
u
+
Q(u)
=
TB
U
+
Q(u)
,
(4.1.4)
which
o
u
-+
00,
ends
o
2D1
=
zu
,
-o
u
In
he
p esen
analysis,
we
conside
he
less
s ingen
case
o
which
he
a io
1-'/
u
may
ake
a bi a y
alues.
Consequen ly,
bo h
pa a ne e s
1-'
and
u,
a e
expec ed
o
de e mine
he
elaxa ion
ime
associa ed
wi h
he
o a ional
elaxa ion
o
he
pa icles.
As
o
he
limi
u
«:
1-'
discussed
p e iously,
he
app op ia e
quan i y
o
desc ibing
his
mechanical
elaxa ional
mo ion
is
again
a
componen
o
he co ela ion
unc ion
(nl.nz)( )(ñl.nz)(O)}.
F om
he
Smoluchowski
equa ion,
we
ob ain i s
dynamic
equa
ion
�
d(n nz)( l nz)(O))
=
I-'(n (n·
R))( )(n nz)(O))
-1-'(n;R )( )(n nz)(O)}
-1-'(n nzRz)( )(n nz)(O)}
-
6(n",nz)( )(n nz)(0)}.
(4.1.5)
4.
Relaxa ion
imes.
Compa ison
wi h
expe imen s
71
P oceeding
along
he
same
lines
as
in
he
p e ious
sec ion,
we
a i e
a
a
closed
se
o
h ee
di e en ial
equa ions
o
he
co ela ion
unc ions
(Rz( )(nznz)(O»),
«n
...
(ñ.
R))( )(nznz)(O)),
and
«nznz)( )(nznz)(O»).
I
is
wo hwhile
poin ing
ou
ha
he
quan i ies
a
ime
appea ing
in
he
h ee
independen
co ela ion
unc ions
a e
he
same
as
in
he
p e ious
sec ion. Fo
he
sake
o
simplici y,
we
wiIl
in oduce
he
ec o
Q
=
(Q
1,
Q2,
Q3),
w
hose
com
ponen s
a e
he
Laplace
ans o ms
o
(Rz
( )(
nz
n
z
)(
O»),
{(nz(ñ.R»( )(nznz)(O»),
and
{(nznz)( )(nznz)(O)},
espec i ely;
and
he
ec o
QO
=
(Q¡,
Q:í,
Q3)
ep esen ing
he
ini ial
alues
o
hese
co ela ion
unc ions.
These
ini ial
alues
can
be
calcula ed
wi h
he
equilib ium
p obabili y
densi y
a
=
O.
The
sys em
o
di e en ial
equa ions
can
be
w i en
in
ma ix
no a ion
as
(4.1.6)
wi h
Á
he
coe icien s
ma ix
(3.13).
We
a e
pa icula ly
in e es ed in
he elaxa ion
dynamics
o
«n nz)( )(nznz)(O»).
F om
Eq.
(4.1.6),
we
ob ain i s
Laplace
ans o m
om
which
we
can
iden i y
he
elaxa ion
ime
we
a e
in e es ed
in
-
D-1
l'
Q3(S)
TR
-
Im-Qo
.-0
3
(
4.1.8)
In
igu e
H.4,
we
ha e
ep esen ed
he
elaxa ion
ime
TR
e sus
jJ
o
di e en
e omagne ic
samples,
bu ,
in
o de
o
compa e
wi h
expe imen al
da a
om
Re .
[19],
we
ha e
also
ep esen ed
he
elaxa ion
ime
TR
e sus
H-1
in
igu e
11.5.
The
da a
co espond
o
wo
samples
o
magne ic
pa icles
o
he
same
mean
size
bu
made
o
di e en
magne ic
ma e ials,
namely
CoFe204
and
I
-
Fe203.
The
Co
e i e
sample
has
an
aniso opy
cons an
K
=
2
.
105�,
and
he
sa u a ion
alue
o
he
magne iza ion
is
M. ::::
250�A.
Fo
he
maghemi e,
he
aniso opy
cons an
is
K
=
4
.
103�
and
M.
::::
270�A.
Wi h
hese
alues,
he
Co- e i e
sample
can
be
conside ed
as
a
igid
dipole
(O"
�
jJ).
On
he
o he
hand,
he
maghemi e pa icles
a e
such
ha
jJ
�
0".
Rega ding
he
alues
o
he
aniso opy
cons an ,
0",
and
he
a io
m/kBT
=
jJoM.
Vm/kBT
,
we
ha e
aken
0""'"
15,
m/kBT,...,
2.8·
1O-4:¡
o
he
maghemi e
and
0""'"
565,
m/kBT,...,
1.8·
1O-4:¡
o
he
Co- e i e.
Fo
bo h
samples,
he
alues
o
he
emaining
quan i ies
a e:
TB
,...,
4.5ms
and
Dm/
D;
,...,
1.
As
i
was
obse ed
in
he
expe imen s
o
he Co- e i e
sample,
TR
ends
o
ze o
when
H
-
oo.
Bo h
R
and
ñ,
quickly
elax
owa ds
he
ield
di ec ion
due
o
he
72
CHAPTER
II.
GENERAL
DYNAMICS
1.5
l'
(ms)
1.0
R
Figu e
H.4:
Relaxa ion
ime
o
he
pa icles
as
a
unc ion
o
J.l.
o
he
Co- e i e
and
maghemi e
samples.

4.
ReJaxa ion
imes.
Compa ison
wi h
expe imen e
73
igidi y
o
he
dipoles.
Fo
he
maghemi e
sample
TR
ends
o
a
ixed,
non- anishing
alue
(
.....
0.3),
which
can
also
be
ob ained
om
Eq.
(4.1.4).
Unde
hese
pa icula
condi ions,
he
magne ic
momen s
apidly
elax
owa ds he
ield
di ec ion,
bu
due
o
he
mode a e
alue
o
(T,
he
elaxa ion
o
he
easy
axis
o
magne iza ion,
ñ,
o
in
o he
wo ds,
he mechanical
elaxa ion
o
he
pa icles,
akes
place
in
a
ini e
pe iod
o
ime.
Conce ning
he
beha io
o
TR
when
H
-
O,
we
obse e
ha
TR
-
6b.
=
Z
independen ly
o
he
alue
o
he
pa ame e
(T.
This ime
co esponds
o
he
well
known
cha ac e is ic
elaxa ion
ime
o
he
co ela ions
o
he
componen s
o
he
second
o de
enso
( in
-
)
o
a
pu ely
di usi e
p ocess
[27].
In
igu e
11.5,
we
ha e
also
ep esen ed
he
ex apola ion
o
he
elaxa ion
ime
coming
om
a he
simple
conside a ions
made
in
Re .
[22]
o
he
igid
dipole
limi
unde he
ac ion
o
a
e y
la ge
ex e nal
magne ic
ield.
Ou
esul s
ag ee
wi h
he
asymp o ic
beha io
in
i s
alidi y
ange,
bu
a
he
same
ime,
hey
show
he
de ia ions
a
in e media e
and
low
magne ic
ield.
These
simple
a gumen s
can
also
be
p oposed
o
he
opposi e
case
p
�
(T
ep oducing
he
asy np o ic
alue
TR
.....
0.3ms
o he
maghemi e
sample,
bu
hey
a e
no
able
o
explain
he
p-dependence
o
he
elaxa ion
ime
o his
ma e ial.
4.2 T ans e se
complex
suscep ibili y
In
his
sec ion,
we
a e
in e es ed
in
he
linea
esponse
o
he
dispe sion
o
an
ae
magne ic
ield
o
small
ampli ude.
F om
an
expe imen al
poin
o
iew,
i
is
much
easie
o
obse e
di e en
e ec s
such
as
he
sa u a ion
o
he
magne iza ion
o
he
equency
dependen
ce
o
i s
elaxa ion
o
a
e o luid
han o
a
pola
dielec ic
luid,
because
bo h
he
ield
s eng h
and
he
equencies
equi ed
a e
much
lowe .
Mo eo e ,
as
we
will
see
below,
in
a
magne ic
luid
he
a ia ion
o
he
complex
suscep ibili y
wi h
equency
depends
on
he
elaxa ion
ime
o
he
magne ic
momen o
In
he
absence
o
an
ex e nal
de
ield,
Raikhe
and
Shliomis
[23]
de i ed
exp es
sions
o
he
complex
ae
suscep ibili y,
X(w)
=
X'(w)
-
iX"(w),
o
a
single
domain
uniaxial
pa icle. They
calcula ed
bo h
he
pa allel
and
pe pendicula
suscep ibili ies
wi h
espec
o
he easy
axis
o
magne iza ion.
Fo
ha
pa icula
si ua ion,
hey
showed
ha
he
equency
dependen
ce
o
he
suscep ibili y
was
a
he
sa ne
ime
a
unc ion
o
he
pa ame e
(T,
compa ing aniso opy
ene gy
o
he
he mal
ene gy.
The
applied
ac
ield
o ien a es bo h
he
magne ic
momen s
and
he
pa icle
axes,
bu
in
a
linea
app oxima ion,
i.e.
o
small
alues
o
he
applied
ac
ield,
hey
neglec ed
74
CHAPTER
JI.
GENERAL
DYNAMICS
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
0.8
0.4
0.2
0.0
...._
...._
.....__
...._
.........
--'
0.0 0.5
1.0
1.5
H-l(lO·sm/A)
2.0
2.5
Figu e
11.5:
Relaxa ion
ime
o
he
pa icles
as
a
unc ion
o
H-1
o
he
Co- e i e
and
maghemi e
samples.
Expe imen al
da a
o
Re .
[19]
co espond
o
he
do s.
The
dashed
line
co esponds
o
he
limi
J.l
�
1
o
he
Co- e i e,
Re .[22].
4.
Relaxa ion
imes.
Compa ison
wi h
expe imen s
75
he
ield-induced
pa iele
o ien a ion.
Thus,
o
a
weak
measu ing
ield,
he
pa í
ele
aniso opy
axes
we e
jus
o ien ed in
a
andom
ashion
due
o
B ownian
mo ion.
Mo eo e ,
in
Re .
[4]
he
au ho s ob ained
he
exp essions
o
he
longi udinal
and
ans e se
componen s
o
he
complex
dielec ic
suscep ibili y
enso
o
a
sys em
o
nonin e ac ing
pola
molecules
unde he
simul aneous
ac ion
o
a
cons an
ex e nal
elec ic ield
and
a
small
ae
elec ic
ield.
This
si ua ion
would
be
equi alen
o
he
p e iously
men ioned
igid
dipole
app oxima ion.
In
his
case,
he
suscep ibili ies
depend
on
he
bias
ield.
Fo
expe imen al
measu es
in
a
liquid,
i
is
much
simple
o
de e mine
he
pa allel
and
pe pendicula
suscep ibili ies
wi h
espec
o
a
ixed
di ec ion like
ha
o
he
de
ield
han
wi h
espec
o
he
axis
o
easy
magne iza ion
o
he
pa ieles.
Thus,
in
his
subsec ion,
we
ob ain
he
exp ession
o
he
pe pendic
ula
suscep ibili y
wi h
espec
o
he
bias
ield
ii,
i.e.
he
pe u bing
ex e nal
ield
j{'
.1
j{,
when
no
only
he
magne ic
momen s
bu
also
he
pa ieles
hemsel es
a e
o ien ed
by
he
ex e nal
ield,
such
as
we
ha e
desc ibed
h oughou
he
chap e .
We
ob ain
ha
he
suscep ibili y
depends
on
he
bias
ield
h ough
he
pa ame e
/-l.
F om
linea
esponse
heo y,
he
decay
o
he
magne iza ion
pe pendicula
com
ponen
unde
he
in luence
o
a
cons an
ield
H,
ano he
small
cons an
ex e nal
ield
H'
(H'
.1
H
and
such
ha
mH'JkBT
<
1)
ha ing
been
swi ched
o
a
ime
=
O,
is
(MJ.( ))
-
(MJ.(O))
=
Xl.H'CJ.( ),
(4.2.1)
whe e
(4.2.2)
is
he
pe pendicula
componen
o
he
magne iza ion,
2
xl.
=
�:T
({Ri(O))
-
(RJ.(0))2)
is
he
pe pendicula
componen
o
he
s a ic
magne ic
suscep ibili y,
and
(4.2.3)
(
4.2.4)
is
he
au oco ela ion
unc ion
o
any
pe pendicula
componen
o
he
magne iza ion.
The
co esponding
complex
magne ic
suscep ibili y
XJ.(w)
is
(4.2.5)
76
CHAPTER
JI.
GENERAL
DYNAMICS
In
he
limi
o low
equencies,
Eq.
(4.2.5)
may
be
w i en
as
X.L(w)
�
x�(1-
iWTol),
(4.2.6)
whe e
(4.2.7)
is
he
elaxa ion
ime.
Mo eo e ,
Col
(s)
is
he
Laplace
ans o m
o he
au oco ela ion
unc ion.
Eq.
(4.2.6)
can
be
w i en
down,
up
o
he
same
o de
o
accu acy,
in
he
o m
o
he
Debye
equa ion
(4.2.8)
This
is
he
o a ional
di usion
limi
whe e
he
beha io
o
Col( )
and,
consequen ly,
o
(Mol
( ))
-
(Mol(O))
may
be
app oxima ed
by
he
exponen ial
(4.2.9)
The
elaxa ion
ime
ollows
om
(4.2.7)
oge he
wi h
(4.2.4),
and
(3.8-3.14).
In
ac ,
when
he
pe u bing
ex e nal
ield
is
poin ing
owa ds he
e",
axis,
H'
=
H'
É""
and
he
cons an
pola izing
magne ic
ield
coincides
wi h
he
ez
axis,
H
=
H
«.,
ou
au oco ela ion
unc ion
educes
o
C
( )
=
(R",( )R",(O))
'"
(R;(O))
(4.2.10)
Consequen ly,
D-1
l'
R1(s)
Tol
=
1m--;:;o
.
•
-0
"'1
In
Fig.
11.6,
we
ha e
ep esen ed
Tol
e sus
JJ
o
he
di e en
alues
o
he
pa ame e
(4.2.11)
(J'
co esponding
o
di e en
a ailable
ma e ials.
The
Debye
spec a
(single
elaxa ion
ime
app oxima ion)
is
gi en
by
(
4.2.12)
whe e
we
ha e
de ined
XO
=
nm2/3kBT
as
he
s a ic
alue
o
he
suscep ibili y
in
he
absence
o
he
cons an
ield.
In
Figs.
11.7
and
11.8
we
plo
he
eal
X�
(w)
and
imagina y
x1
(w)
pa s
o
he
ans e se
componen
o
he
no malized
complex
suscep ibili y
o
di e en
alues
o
APPENDIX
A
83
Analogously,
he
po en ial
e ms
(A3),
(A6),
and
(A8)
can
be
w i en
in
he
o m
a
(
(,
OU)
,)
a
(
"OU)
h-,·
'1/;
R
x
-,
x
R
=
-,
'l/;h(1
-
RR)
.
-.
oR oR oR oR
'
(A9)
1
a
(-
-
,)
a
(1
,,0U
1,
'OU)
--,'
'I/;(R l
+
Rñ)U
x
R
=
-,'
'1/;-(1
-
RR)·
-,
+
'I/;-{(R·
ñ)1
-
Rñ}·
-,
,
{ oR
oR
{
oR
{
on
(AlO)
_!_�.
('I/;(1l l
+
1lñ)U
x
ñ)
=
�.
('I/;_!_(1
-
ññ)
.
o�
+
'I/;_!_
{(R-
ñ)1
_
nR}
.
o�)
,
{
on
on
{ on {
a
R
(All)
whe e
he
unde lined
ac o s
may
be
iden i ied
o
he
mobili y
ma ices
de ined
in
Eq,
(2.12).
Mo eo e ,
he
di usion
ma ix
is
gi en
by
D
=
kBTb,
so
ha
we
can
also
ew i e
he
di usi e
ac o s
in
e ms
o
he
o a ional
ope a o s
(A13)
a
(
0'1/;
)
a
"'
,0'1/;
--
_.
DR
.-
=D
-·{(R·n)I-Rn}·-=DR·
·R··I.
oR
n
oil
oR
oñ
R
n
'P,
(A14)
:n
.
(DnR.
!�)
=
D Rñ
·1l l'l/;·
(A15)
Finally,
subs i u ing
eqs.
(A2)-(A8)
and
(A12)-(A15)
in
Eq.
(A
1),
we
a i e
a
equa ion
(2.13).

Appendix
B
Decoupling
app oxima ions
In
Sec ion
3,
we
ha e
in oduced
decouplings
o
so ne
co ela ion
unc ions
appea ing
in
he
e olu ion
equa ions
o
he
co ela ions.
The
pu pose
o
his
appendix
is
o
gi e
mo e
de ails
abou
he
p ocedu e
ollowed
o
ca y
ou
such
decouplings.
In
Eq.
(3.4),
he
app oxima ed quan i ies
a e
(R
n,
)( )R (O))
'"
(R (
)R (O))
en,
)eq
=
.c(¡¡
)(R ( )R (O)),
(Bl)
and
whe e
linea iza ions
in
ime
ha e
been
pe o med.
The
app oxima ion
(Bl)
was
al
eady
discussed
in
he
ex
(see
Eq.
(3.5)
and
commen s
below).
Fo
he
co ela ion
(B2),
we
decouple
he
quan i ies
R
and
(ñ·
R)2
because
hey
a e
no
coupled
in
equilib ium
condi ions,
(R (ñ·
k)2)eq
=
O,
(R )eq
=
O,
and
(n·
R)2)eq
#
O.
Thus,
in
a
si ua ion
no
a
o n
equilib ium,
we
will
assume
ha
bo h
quan i ies
emain
also
decoupled.
By
simila
a gumen s,
in
Eq.
(3.6)
we
ha e
al
so
pe o med
he
ollowing
app oxi
ma ions
(R (nz(n·
R)))( )R (O))"""
(R",( )R (O))«nz(n·
R)))eq
=
.c(¡¡)Q(u)(R ( )R (O)),
(B3)
(n (Rz(n·
R)))( )R (O))
'"
«n (n·
R))( )R (O))(Rz)eq
=
.c(¡¡)(n (n·
R))( )R (O)),
(B4)
and
«n (n
.
R)(l
-
(n·
R)2))( )R (0))
.....,
(n (n
.
R))( )R (O))(l
_
(�.
�)4)eq).
(B5)
«n
.
R)2)eq
In
Eq.
(3.7),
we
ha e
used
84
APPENDIX
B
85
(n�R.,)( )R.,(O))
-
(R.,( )R.,(O))(n;)eq
=
CC(JJ)
(1
-
3Q(u))
+
Q(u))(R ( )R (O)),
JJ
(B6)
(n nzRz)( )R.,(O))
-
(n nz)( )R (O))(Rz)eq
=
C(JJ)(n.,nz)( )R (O))o
(B7)
No ice
ha
he
decoupling
in
he
co ela ion
(n.,(ñ
o
R)3)( )R (O))
o
Eq.
(B5)
is
(n (ñ
o
R))( )R (O))
¡�:::�:l::
and
no
(n (ñ
o
R))( )R (O))(ñ
o
R)2)eq,
which
leads
o
di e gencies
o
he
o a ional
iscosi y
a
s nall
alues
o
JJo
As
ega ds
his
ac ,
i
is
wo hwhile
o
emphasize
ha
hese
app oxima ions
a e
mo e
accu a e
o
mode a e
and
highe
alues
o pa ame e s
JJ
and
a
.
This
ype
o unca ion
was
al eady
p oposed
by
S a ono ich
in
he
con ex
o
s ochas ic
p ocesses,
Bibliog aphy
[1]
P oceedings
01
he
Six h
In e na ional
Con e ence
on
Magne ic
Fluids,
edi ed
by
V.
Cabuil,
J.-C.
Bac i,
and
R.
Pe zynski
(No h-Holland,
A ns e da n,
1993).
[2]
W.T.
Co ey,
P.J.
C egg
and
Yu.P.
Kal nyko ,
On
he
Theo y
o
Debye
and
Neel
Relaxa ion
o
Single
Domain
Fe omagne ic
Pa iicles,
edi ed
by
1.
P igogine
and
S.A.
Rice,
Ad .
in
Che n.
Phys.,
Vol.
83,
(Wiley
In e science,
New
Yo k,
1992),
p.263.
[3]
P.C.
Fannin
and
S.W.
Cha les,
J.
Phys.
D:
Appl.
Phys.
22,
187
(1989);
J.
Phys.
D:
Appl.
Phys.
24,
76
(1991).
[4]
J.T.
Wald on,
Yu.P.
Kal nyko
and
W.T.
Co ey,
Phys.
Re .
E
49,3976
(1994).
[5]
Yu.
L.
Raikhe
and
M.
I.
Shlio nis,
Relaxa ion
phenomena
in
Condensed
M
al e ,
edi ed
by
Willia n
Co ey,
Ad .
in
Che n.
Phys.,
Vol.
87
(Wiley
In e science,
New
Yo k,
1994)
p.
595.
[6]
V.G.
Bash o oy,
B.M.
Be ko sky
and
A.
N.
Vislo ich,
In oduc ion
o
The mo
mechanics
o
Magne ic
Fluids
(Sp inge -Ve lag,
Be lin,
1988).
[7]
T.
L.
Gilbe ,
Phys.
Re .
100,
1243
(1955).
[8]
M.C.
Miguel,
J.
Bone
A
alos
,
A.
Pé ez-Mad id
and
J.M.
Rubí,
Physica
A
193
359
(1993).
[9]
W.
F.
B own,
Phys.
Re .
130,
1667
(1963).
[10]
M.
A.
Ma senyuk,
Yu.
L.
Raikhe
and
M.
I.
Shlio nis,
SOy.
Phys.
JETP
38,413
(1974).
[11]
Yu.
L.
Raikhe
and
M.
J.
Shlio nis,
SOy.
Phys.
J.
Appl.
Mech. Tech.
Phys,
15,
470
(1974).
86
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[12]
M.I.
Shliomis
and
V.I.
S epano ,
J.
Magn.
Magn.
Ma e .
122
196
(1993).
[13]
J.R.
Calde wood
and
W.T.
Co ey,
P oc.
Roy.
Soco
A
356
269
(1977).
[14]
N.G.
an
Kampen,
S ochas ic
P ocesses
in
Physics
and
Chemis y,
(No h
Holland,
Ams e dam,
1992).
[15]
M.1.
Shliomis,
SOy.
Phys.
JETP 34
1291
(1972).
[16]
M.
Doi
and
S.F.
Edwa ds,
The
Theo y
01
Po/yme
Dynamics,
(Cla endon
P ess,
Ox o d,
1986).
[17]
J.P.
McTague,
J.
Chem.
Phys.
51133
(1969).
[18]
F.
Pe in,
J.
Phys.
Radium
533
(1934).
[19]
J.-C.
Bac i,
K.
Dje i,
S.
Ne eu and
R.
Pe zynski,
J.
Magn. Magn.
Ma e .
123
67
(1993).
[20]
J.-C.
Bac i,
J.
Dumas,
D.
Go se,
R.
Pe zynski
and
D.
Salin,
J.
Physique
Le .
46
L-1l99
(1985).
[21]
M.C.
Miguel,
J.M.
Rubí
and
A.
Pé ez-Mad id,
Physica
A
20324
(1994).
[22]
J.-C.
Bac i
and R.
Pe zynski,P oceedings
01
he
XII
Silges
Con e ence,
edi ed
by
L.
Ga ido
(Sp inge -
Ve lag,
Be lin,
1993).
[23]
Yu.L.
Raikhe
and
M.I.
Shliomis,
SOy.
Phys.
JETP
(Engl.
ansl.)
40
526
(1974).
[24]
P.C.
Fannin,
B.K.P.
Scai e
and
S.W.
Cha les,
J.
Magn.
Magn.
Ma e .
122
159
(1993).
[25]
M.1.
Shliomis
and
V.I.
S epano ,
J.
Magn.
Magn.
Ma e .
122
176
(1993);
Relax
a ion
phenomena
in
Condensed
Mai e ,
edi ed
by
William
Co ey,
Ad .
in
Chem.
Phys.,
Vol.
87
(Wiley In e science,
New
Yo k,
1994)
p.I.
CHAPTER
III
DYNAMICS
OF
MAGNETIC
HOLES
DISPERSED
IN
A
FERROFLUID
Once
we
ha e
cha ac e ized
he
mac oscopic
beha io
o
a
e o luid,
gi ing
exp essions
o
he
iscosi ies
and,
co espondingly,
o
bo h
he
symme ic
and
an isymme ic
pa s
o
he
p essu e
enso ,
as
well
as
o
he
a e age
o
i s
magne iza ion,
in
his
chap e
we
analyze
he
dynamics
o
a
nonmagne ic
pa icle
(magne ic
hole)
suspended
in
a
e o luid.
The
o a ional
dynamics
o
he
pa icle
is
s ongly
in luenced
by
he
p esence
o
a
o a ing
magne ic
ield.
As
a
esul ,
we
ha e
ound
ha
he
hole
o a es
in
he
opposi e
di ec ion
o
ha
o
he
ield.
Ou
analysis
is
alid
a
low
and
mode a e
equencies
o
he
ield
o a ion
and
may
be
compa ed
o
ecen
expe imen s
ob aining
qui e
a
good
ag eemen
in
he
equency
ange
we
a e
conside ing,
in
which
a
linea
ela ionship
be ween
bo h
he
o a ion
equency
o
he
ield
and
he
angula
eloci y
o
he
pa icle
is
ound.
In
addi ion,
he
dependence
o
he
equency
o
he
pa icle
on
he
magne ic
ield
s eng h
is
also
compa ed
o
he
expe imen s.
Fo
a
sligh ly
g ea e
concen a ion
o
holes,
we
also
de e mine
he
Ro ne-P age
and
Oseen
equi alen
enso s,
as
he
i s
s eps
in
he
cha ac e iza ion
o
hyd odynamic
in e ac ions
be ween
he
nonmagne ic
pa icles
in
he
e o luid.
88

1.
In oduc ion
89
1
In oduc ion
Recen ly,
an
inc easing
in e es
in
he
s udy
o
he
dynamic
p ope ies
o
he
so
called
magne ic
holes,
which
a e
colloidal
nonmagne ic
pa icles
dispe sed
in
a
ca ie
magne ic
luid,
has
a isen
[1]-[4].
Al hough
he
pa icles
a e
no
magne ic,
when
hey
a e
suspended
in
a
ca ie
e o luid
hey
acqui e
an
induced
magne ic
momen
equal
o
he
magne ic
momen
o
he
e o luid
olume
hey
displace.
The
in e ac ion
o
hese
induced
magne ic
momen s
o
he
holes
causes
a
numbe
o
peculia
phenomena,
such
as
he
o de -diso de
ansi ion
in
magne ic
hole
la ices
[5,
6]
and
he
non
linea
phenomena
obse ed
in
assemblies
o
holes
[7].
Fu he mo e,
knowledge
o
he
dynamics
o
such
pa icles
may
cons i u e
a
way
o
cha ac e izing
he
anspo
p ope ies
o
he
e o luid.
Fo
example,
he
ic ion
coe icien
o
he
pa icle
gi es
us
in o ma ion
abou
he
iscosi y
o
he
ca ie
luid.
The
sys em
holes- e o luid
can
be
modeled
as
a
suspension
o
pa icles
(holes)
in
a
ca ie
luid
( e o luid).
This
simpli ica ion
can
be
accomplished
when
di e en
leng h
scales
exis
o
he
e o luid
and
he
holes.
Consequen ly,
he
e o luid
can
be
iewed
as
a
con inuous
medium
h ough
which
he
holes
may
mo e.
The
dynamics
o
he
e o luid
is
go e ned,
a
he con inuum
le el, by
a
gene alized
Na ie -S okes
equa ion.
The
o a ions
o
he
e omagne ic
pa icles
lead
o
he
p esence
o
an
an isymme ic
pa
o
he
p essu e
enso
gi ing
ise
o
he
appea ance
o
a
new
anspo
coe icien :
he
o a ional
iscosi y
[8,
9].
This
coe icien
en e s
he
exp ession
o
he e ec i e
iscosi y
o
he
suspension
[10].
The
pu pose
o
his
chap e
is
o
analyze
he
ansla ional
and
o a ional
dynamics
o
a
magne ic
hole
imme sed
in
a
e o luid
unde
he
in luence
o
a
o a ing
magne ic
ield.
In
pa icula ,
we
a e
in e es ed in
explaining
he
phenomenon
o
he
o a ion
o
he
hole
in
he
opposi e
di ec ion
o
ha
o
he
o a ing
ield,
as
has
been
obse ed
in
ecen
expe imen s.
As
we
will
see,
he
exp ession
o
he
angula
eloci y
o
he
hole
comes
om
i s
co esponding
o a ional
equa ion
o
mo ion
in ol ing
he
hyd odynamic
o que
exe ed
by
he
luid
on
he
pa icle,
which
is
calcula ed
in
a
nons a iona y
si ua ion.
Fu he mo e,
we
in end
o
cha ac e ize
he
hyd odynamic
in e ac ions
be ween
he
non-magne ic
pa icles
in
a
e o luid
by
gi ing
exp essions
o
he
co espond
ing
Oseen
and
Ro ne-P age
equi alen
enso s.
In
pa icula ,
he
Oseen
enso
o
a
simple
luid
is
well known
o
being
he
simples
app oxima ion
when
compu ing
hyd odynamic
in e ac ions,
i.e.
when
he
suspension
is
dilu e
enough
ha
he
pa 
icles
a e
loca ed
a
a
ela i e
la ge
dis an
ce
om
each
o he .
This
means
ha
he
90
CHAPTER
III.
DYNAMICS
OF
MAGNETIC
HOLES
suspension
o
non-magne ic
pa icles
unde
conside a ion
is
dilu e
enough
o
a oid
agg ega ion
phenomena
among
he
holes,
bu
a
he
same
ime,
he e
is
a
su icien
numbe
o
pa icles
ha
makes
i
necessa y
o
ake
in o
accoun
hyd odynamic
in
e ac ions
be ween
pai s
o
sphe es.
As
an
example
o
unde s and
he
in luence
o
hyd odynamic
in e ac ions
on
he
dynamics
o
he
magne ic
holes,
we
will
also
s udy
hei
sedimen a ion
in
he
p esence
o
an
ex e nal
magne ic
ield.
Wi h
his
goal
in
mind,
we
ha e
dis ibu ed
he
chap e
in
he
ollowing
way.
In
Sec ion
2,
we
o mula e he
basic
equa ions
o
he
whole
sys em
in
he
con inuum
app oxima ion.
Unde
his
app oxima ion,
he
magne ic
hole
may
be
iewed
as
a
mac oscopic
pa icle
mo ing
h ough
a
con inuum
medium
wi h
in e nal
deg ees
o
eedom.
In
Sec ion
3,
we
analyze
he
mo ion
o
he hole
un il
ob aining
explici
exp essions
o
he
o ce
and
o que
exe ed
on
he
pa icle.
F om
hese
equa ions,
we
hen
de i e
he
ansla ional
and
o a ional
ic ion
enso s.
Mo eo e ,
om
he
analysis
o
he
o a ional
mo ion,
we
in e
a
linea
law
ela ing
he
angula
eloci y
o
he
hole
and
he
o a ion
equency
o
he
ield.
These
esul s
a e
compa ed
o
ex
pe imen s
in
Sec ion
4.
In
Sec ion
5
we
s udy
he
hyd odynamic
in e ac ions
be ween
pai s
o
magne ic
holes.
We de i e he
exp essions
o
he
Oseen
and
Ro ne-P age
equi alen
enso s
and
we
s udy
he
sedimen a ion
o
wo
holes
in
he
p esence
o
an
ex e nal
magne ic
ield
o
di e en
con igu a ions
in
o de
o
show
he
in luence
o
such
in e ac ion.
Finally,
in
he
las
sec ion,
we
summa ize
ou
main
esul s.
2
Basic
equa ions
in
he
con inuum
app oxima ion
Le
us
conside
a
dilu e
suspension
o sphe ical
magne ic
holes
o adius
a
in
a
e o luid
unde
he
in luence
o
an
ex e nal
magne ic
ield,
which
o a es
wi h
he
cons an
equency
WO'
In
he
con inuum
app oach,
he
e o luid
(unde s ood
as
a
suspension
o
small
magne ic
pa icles
in
a
non-pola
sol en )
is
assumed
o
be
a
con inuum
medium
wi h
a
new
hyd odynamic
ield: he
spin
o
mean
angula
eloci y
o
he
olume
elemen s
o
he
con inuum
[8].
This
app oxima ion
is
jus i ied
unde
he
mild
equi emen
ha
he
size
o
he
holes
( ipically
abou
10-3
-1O-4cm)
be
much
la ge
han
ha
o
he
e omagne ic
pa icles
(l0-6cm).
The
magne ic
ield
o a ion
equencies
a e
aken
low
enough
o
conside
he
qua
sis a iona y
limi
in
which
he
de i a i es
o
he
hyd odynamic
ields,
al hough
ime
dependen ,
may
be
neglec ed.
Unde
his
app oxima ion,
he
equa ions
o
mo ion
a e
(2.1)
2.
Basie
equa ions
in
he
eon inuum
app oxima ion 91
0=
2ña
+
M
x
¡¡
+
T',
(2.2)
whe e
p( , ),
( ,
),
and
ñ(a)( ,
)
==
-1/2E
:
II(a)( ,
)
a e
he
p essu e,
eloci y,
and
he
axial
ec o
ela ed
o
he
an isymme ic
pa
o
he
p essu e
enso ,
II(a),
e
spec i ely.
Mo eo e ,
E
is
he
Le i-Ci i a
enso ,
M( ,
)
is
he a e age
magne iza ion
densi y,
and
¡¡
=
H
¡
is
he
ex e nal
magne ic
ield.
The
appea ance
o
an isym
me ic
s esses
in
he
e o luid
comes
om
he
di e ence
be ween
he
mean
angula
eloci y
and
he
o ici y
o
he
low,
1/2"V
x
.
In
addi ion,
we
ha e
in oduced
he
induced
o ce
and
o que
densi ies,
i( ,
)
and
:¡:i( ,
),
which
o igina e
om
he
pe u ba ion
caused
by
he
mo ion
o
he
holeo
The
shea
iscosi y
'1,
acco ding
o
Eins ein's
law
is
5
T/
=
T/o(l
+
'24».
(2.3)
This
exp ession
holds
up
o
linea
o de
in
he
olume
ac ion
o
magne ic
pa icles
4>
=
411V
IV,
whe e
b
is
he
adius
o
one
e omagne ic
monodomain,
and
V
is
he
olume
occupied
by
he
sys em.
We
ha e
also
de ined
T/o
as
he
iscosi y
o
he
ca 
ie
luid.
Fu he mo e,
hese
equa ions
a e
complemen ed
wi h
he
incomp essibili y
condi ion
"V
.
=
O
.
The
induced
o ce
and
o que
densi y
ields,
i( ,
)
and
:¡:i( ,
)
a e
in oduced
so
ha
he
equa ions
o
mo ion
o
he
luid,
gi en
by
eqs.
(2.1)
and
(2.2),
a e
also
alid
o
he
poin s
inside
he
sphe e
[11,
12].
This
conside a ion
imposes
he
equi emen
i( ,
)
=
T'( ,
)
=
O,
o
¡
-
Rcm( )1
>
a.
(2.4)
Addi ionalIy,
hey
mus
be
chosen
in
such
a
way
o
he
eloci y
and
he
p essu e
ields
o
sa is y
( ,
)
=
i( )
+
ñ( )
x
añ,
o
¡
-
Rcm( )1
=
a,
(2.5)
p( , )
=
O,
o
¡
-
Rcm( )1
<
a,
(2.6)
and
he
magne iza ion
o
he
hole
M( ,
)
=
O,
o
¡
-
Rcm( )1
�
a.
(2.7)
In
hese
equa ions,
we
ha e
in oduced
i
and
ñ
as
he
ansla ional
and
o a ional
eloci ies
o
he
pa icle,
espec i ely;
Rcm( )
as
he
posi ion
o
he
cen e o
mass
o
he
sphe e,
and
ñ
==
(
-
Rcm( ))/¡
-
Rcm( )l.
92
CHAPTER
IIl.
DYNAMICS
OF
MAGNETIC
HaLES
A e
conside ing
he
condi ions
(2.4)
-
(2.7),
and
o
he
quasis a iona y
case,
(ü( )
+
ñ( )
x
añ)
::=
O,
we
ha e
i( , )
=
(ñ, )8(Ji-
Rcm( )l-
a),
(2.8)
and
Í(i,
)
=
Í( )8(a
-
Ji
-
Rcm( )l),
(2.9)
whe e
8
is
he
Hea iside
s ep
unc ion.
We
will
now
p oceed
o
de i e
a
o mal
solu ion
in
e ms
o
he
induced
o ce
and
o que
densi ies
wi h
he
pu pose
o
ob aining
exp essions
o
he
hyd odynamic
o ce
and
o que
on
he
nonmagne ic
pa icle
as
a
unc ion
o i s
eloci y
and
he
eloci y
ields
in
he
absence
o
he
sphe e.
The
p ocedu e
we
use
does
no
equi e
explici
knowledge
o
ei he
i(i,
),
?( ,
)
o
he
eal
eloci y
ield
(i,
).
Equa ions
(2.1)
and
(2.2)
can
be
combined
o
elimina e
he
e m
p opo ional
o
he
an isymme ic
axial
ec o ,
ñ-,
in
Eq.
(2.1).
We
hen
ob ain
(2.10)
whe e
we
ha e
de ined
ji
as
he
combina ion
o
he
induced
o ce
and
o que
ields
(2.11)
To
sol e
o
he
eloci y
ield
om
Eq.
(2.10)
,
we
need
o
know
he
exp ession
o
he
magne iza ion
densi y.
The
p ocedu e
used
o
ob ain
his
quan i y
was
in oduced
in
Re .
[13]
o
he
case
o
a
cons an
magne ic
ield,
and
was
based
upon
he
solu ion
o
he
co esponding
Smoluchowski
equa ion.
In
appendix
A,
we
p esen
an
ex ension
o
he
me hod
o
o a ing
magne ic
ields.
Ou
soIu ion
is
(2.12)
whe e
n
is
he
numbe
densi y
o
dipoles,
(R)
is
he
a e aged
o ien a ion
ec o
o
each
magne ic
momen ,
p.
=
mH
/kBT
is
he
Lange in
pa ame e ,
compa ing
magne ic
and
he nal
ene gies,
wi h
m
he
magne ic
momen
s eng h,
assumed
cons an ,
kB
he
Bol zmann
's
cons an ,
and
D;
=
k T
is
he
o a ionaI
di usion
coe icien ;
wi h
�,.
=
81 7Job3
he
o a ionaI ic ion
coe icien
o
a
e omagne ic
pa icle.
Addi ionally,
C(p.)
=
co h
p.
-
1/
u,
is
he
Lange in
unc ion
and
F(p.)
==
2+c;i ¡J)
is
in oduced
and
ob ained
in
appendix
A.
4.
Compa ison
wi h
expe imen s
99
This
exp ession
is
also
simila
o
he
co esponding
one
ob ained
by
Selle s
and
B en
ne
o
g a i a ional
dipoles.
I
is
wo h
men ioning
ha ,
o
he
o a ional
mo ion,
he
ime
dependen
ce
o
he
magne ic
ield
gi es
ise
o
an
ex a
e m
in
he
hyd odynamic
o que
p opo ional
o
dd1
x
H
o
he
quasis a iona y
analysis.
As
we
will
see,
o
he
ield
o a ion
equencies
unde
conside a ion,
his
e m
is
esponsible
o
he
coun e - o a ion
o
he
hole
in
espec
o
he
ield
o a ion,
as
obse ed
by
Helgesen
and
Skjel o p
[2]
and
by
Popplewell
ei
al.
[15].
The
magne ic
hole
o a ional
equa ion
o
no ion
is
gi en
by
(3.2.16)
whe e
1
is
he
momen
o
ine ia
o
he
pa ícle.
Again,
since
he
ine ial
e
m
is
negligible,
he
esul
is
ha
H
=
O.
Fo
a
e o iuid
ini ially
a
es ,
whe e
í
=
O,
and
up
o
he
i s
o de
in
he
olume
ac ion,
q¡,
om
Eq.
(3.2.14)
we
ob ain
-
dH
-
2( )
=
q¡JjF(Jj)(
d
x
H).
(3.2.17)
In
pa icula ,
i
we
conside
an
ex e nal
magne ic
ield
o a ing
in
he
XY
plane
wi h
angula equency
Wa,
H
( )
=
cos
wa e
-í-sin
wa e",
he
magne ic
hole
angula
eloci y
educes
o
(3.2.18)
F om
his
exp ession,
we
hen
conclude
ha
he di ec ion
o
he
o a ion
o
he
sphe es
is
opposi e
he
ield
o a ion.
Addi ionally,
we
ob ain
a
linea
ela ionship
be ween
he
ield
equency
and
he
o a ion
equency
o
he
sphe es
o
he
ange
o
ield
equencies
we
a e
conside ing
in
ou
analysis,
ha
is,
o
equencies
which
enable
us
o
pe o m
a
quasis a iona y
ea men .
Mo eo e ,
unde
hese
condi ions,
ñ
is
independen
o
he
size
o
he
sphe es,
i
only
depends
on
he
olume
ac ion
o
magne ic
pa icles
and
on
he
Lange in
pa ame e .
4
Compa ison
wi h
expe imen s
The
o a ion
o magne ic
holes
induced
by
a
o a ing
magne ic
ield
has
been
obse ed
in
ecen
expe imen s
[2,
15].
The
expe imen al
se up
consis s
o
a
hin
laye
o
e o iuid,
con ined
be ween
wo
glass
pla es,
in
which
sphe ical
pa icles
o
polys y ene
a e
dispe sed.
When
applying
a
magne ic
ield
o a ing
in
he
plane
o
he
pla es,

100
CHAPTER
III.
DYNAMICS
OF
MAGNETIC
HOLES
i
was
obse ed
ha
he
pa icles
o a e
in
he
opposi e
di ec ion.
The
expe imen s
we e
pe o med
o
ke osene-based
[2]
and
wa e -based
e o luids
[2,
15].
The
o a ion
equency
o
he
hole,
ñ,
o
he
ke osene-based
e o luid,
which
showed
good
signs
o
homogenei y,
was
measu ed
o
di e en
samples
con aining
polys y ene
sphe es
o
a ious
sizes.
They
saw
ha
ñ
was
independen
o
he
size
o
he
sphe es,
o
he
size
ange
hey
ook
in o
conside a ion.
Fu he mo e,
in
he
in e media e
equency
ange,
hey
obse ed
a
linea
ela ionship
be ween
he
angula
eloci y
o
he
sphe e
and
he
angula
eloci y
o
he
magne ic
ield.
In
he
case
o
a
wa e -based
e o luid,
i
has
been
cla i ied
ha
i
shows
a
weak
endency
o
sedimen a ion
and
agg ega ion
phenomena
unde
he
ac ion
o
he ield
[2].
In
he
same
e e ence,
he
di e en
esponse
o
he
magne ic
hole
when
suspended
in
bo h
ypes
o
e o luids
was
also
emphasized.
A
his
poin ,
i
should
be
no ed
ha
he e
is
no
o al
ag eemen
be ween
he
esul s
o
[2]
and
[15]
conce ning
such
a
e o luid. In
pa icula ,
Popplewell
and cowo ke s
conclude
ha
pa icles
o
abou
2a ==
70J.lm
o a e
opposi e
he
di ec ion
o
he
ield,
o
any
alue
o
"-'o,
whe eas
smalle
pa icles,
2a
==
20J.lm,
o a e
in
he
ield
di ec ion
a
ield
equencies
below
70H
z
and
in
he
opposi e
di ec ion
a
highe
equencies.
Howe e ,
Helgesen
and
Skjel o p
we e
no
able
o
ep oduce
hese
esul s.
Addi ionalIy,
in
bo h
pape s
he e
a e
al
so
expe imen al
da a
ela ing
he
holes
o a ion
equency
and
he
ex e nal
ield
s eng h
a
in e media e
alues
o
"-'o'
In
ega ds
o
he
coupling
be ween
ansla ional
and
o a ional
mo ion,
he
ex
pe imen al
obse a ions
co obo a e
ou
esul s.
This
can
easily
be
concluded
a e
inspec ion
o
he
equa ions
o
he
o ce
and
o que
gi en
byeqs.
(3.1.7)
and
(3.2.14)
.
Mo eo e ,
ou
esul
gi en
h ough
Eq.
(3.2.18)
is
in
comple e
ag eemen
wi h
he
obse a ions
ca ied
ou
o
he
ke osene-based
e o uid
in
he
ange
o
in e media e
ield
equencies,
which,
as
we
ha e
s a ed
be o e,
enables
us
o
pe o m
a
quasis a
iona y
s udy
o
he
sys em.
This
can
be
seen
in
Fig.
1Il.l
o
a
e o uid
wi h
he
same
cha ac e is ic
as
ha
conside ed
in
Re .
[2],
ha
is,
o
olume
ac ion
o
mag
ne ic
pa icles
P
==
0.1
and
sa u a ion
magne iza ion
o
200G.
On
he
o he
hand,
as
a
as
he
heo y
is
conce ned,
we
can
also
indica e
ha
we
ha e
ob ained
a
o a ion
equency
which
does
no
depend
on
he
iscosi y
o
he
sol en
(3.2.18)
.
Despi e
he
possibili y
o
he
inhomogenei ies
in
wa e -based
e o luids,
men ioned
in
Re .
[2],
he
dependence
o
n
on
he
ield
s eng h
seems
o
be
ep esen ed
by
a
sa u a ion
cu e
as
he
one
co esponding
o
he
quan i y
J.lF(J.l)
=
J.l-
anhJ.l
J.l
+
anh
u
(4.1
)
4.
Compa íson
wí h
expe ímen s
101
0.010
0.001
10
100
H
(Hz)
1000
Figu e
111.1:
Ro a ion
equency
o a
magne ic
hole
in
a
ke osene-based
e o luid
laye
e sus
he
equency
H
o
he
o a ing
magne ic
ield
o
H
=
140e.
The
di ec ion
o
o a ion
o
he
sphe es
is
opposi e
o
ha
o
he
ield.
102
CHAPTER
IIl.
DYNAMICS
OF
MAGNETIC
HOLES
0.4
s
(Hz)
0.3
0.6
0.5
0.2
0.1
0.0
...___
'--
__
----''-
__
--'
---I.
--'
O
20
40
60 80
100
H
(G)
Figu e 111.2:
Va ia ion
o
pa icle
o a ion
wi h
he
ield
s eng h.
Ro a ion
coun e
o
he
ield
o a ion.
(a
=
35¡..¿m,
lB
=
60Hz)
5.
Hyd odynamic
in e ac ions
be ween
pai s
o
magne ic
hoJes
103
in ol ed
in
ou
Eq.
(3.2.18).
In
Fig.
11I.2,
we
ha e
ep esen ed
he
equency
as
a
unc ion
o
he
imposed
ield
o
a
e o luid
wi h
he
same
magne iza ion
as
ha
conside ed
by
Popplewell
and
cowo ke s.
Finally,
i
is
wo h
men ioning
ha
in
acco dance
wi h
Re .
[2],
we
canno
ep oduce
he
dependence
o
he
equency
on
he
size
o
he
pa icles
as
obse ed
in
Re .
[15].
5
Hyd odynamic
in e ac ions
be ween
pai s
o
mag
ne ic
holes
Hyd odynamic
in e ac ions
among di e en
objec s
mo ing
in
a
iscous
luid
ha e
been
ex ensi ely
s udied
because
o
hei
e ec s
on
he
dynamics
and
consequen ly
on
ce ain
p ope ies
o
he
whole
dispe sion
[17, 18].
The
i s
s udies
o
hese
in e 
ac ions
we e
unde aken
by
Smoluchowski
using
he
so-called
me hods
o
e lec ion.
Bu ,
due
o
hei
complexi y,
hey
ha e
only
been
applied
o
he
case
o
a
pai
o
pa icles.
On
he
o he
hand,
he
me hod
o
induced
o ces,
i s
in oduced
by
Mazu
[19]
and
gene alized
in
Re .
[12]
o
a
sys em
o
N
sphe es
mo ing
in
a
simple
liquid,
pe mi s
o
calcula e
he
mobili y
enso s
o
any
desi ed
o de
o
app oxima ion
as
an
expansion
in
he
in e se
dis an
ce
be ween
he
pa icles.
In
his
sec ion,
we
i s
de elop
he
la e
echnique
o
s udy
he
hyd odynamic
in e ac ions
be ween
pai s
o
sphe ical
nonmagne ic
pa icles
mo ing
in
an
aniso opic
magne ic
liquido
5.1
Oseen and
Ro ne-P age
enso s
o he
e o uid
We
will
now
conside
a
suspension
o
N
sphe ical
magne ic
holes
o
adii
ai,
(i
=
1,
...
,
N),
in
an
unbounded
incomp essible
e o luid
unde
he
in luence
o
an
ex e nal
magne ic
ield,
which
may
o a e
wi h
cons an
angula
eloci y
Wo.
The
suspension
unde
conside a ion
is
dilu e
enough
o
a oid
agg ega ion
due
o
he
in e ac ion
o
he
induced
magne ic
momen s
o
he
holes.
On
he
o he
hand,
i
con ains
a
su icien
numbe
o
pa icles
so
ha
hyd odynamic
in e ac ions
be ween
pai s
o
sphe es
mus
be
conside ed.
As
in
he
p e ious
sec ions,
he
e o luid
is
assumed
o
be
a
con inuum
medium
based
on
he
ac
ha
he
size
o
he
holes
is
much
la ge
han
ha
o
he
e omagne ic pa icles
cons i u ing
such
a
e o luid.
Ou
s a ing
poin
o
analyzing
he
quasis a iona y
mo ion
o
he
pa icles
will
be
he
linea ized
equa ions
o
conse a ion
o
he
linea
and
angula
momen a,
which
now
ead
104
CHAPTER
IJI.
DYNAMICS
OF
MAGNETIC
HOLES
N
0=
- 7p+
¡ 72V
-
7
x
ü(a)
+
L
j"d,
j=1
(5.1.1
)
N
0=
2ña
+
M
x
.i1
+
L
T1nd,
i=1
(5.1.2)
oge he
wi h
0=
7
.
V,
(5.1.3)
whe e
we
ha e
in oduced
a
se
o
induced
o ces
and
o ques
densi ies,
j"d( ,
)
and
T1nd( ,
),
(j
=
1,
...
,
N)
which
o igina e
om
he
pe u ba ion
caused
by
he
mo ion
o
he
holes.
As
we
ha e
explained
be o e,
he
ex ension
o
he
luid
eloci y
ield
inside
he
pa icles
imposes
he
equi emen
j"d( ,
)
=
Tjnd( ,
)
=
O,
o
¡
-
Rj( )1
>
ai
and
(j
=
1,
...
,
N).
Ri( )
gi es
he
posi ion
o
he
sphe e
cen e
a
ime
.
Fu he mo e,
he se
o
equa ions
including
induced
o ces
and
o ques
mus
be
comple ely
equi alen
wi h
he
o iginal
bounda y
alue
p oblem
e ,
)
=
uj( )
+
nj( )
x
(
-
Rj( »,
o
¡
-
Rj( )1
�
aj,
(5.1.4)
i.e.,
s ick
bounda y
condi ions
a
he
su aces
o
he
sphe es.
Simila ly,
o
he
hyd o
s a ic
p essu e
we
impose
he
condi ion
p( ,
)
=
O,
o
¡
-
Ri( )1
<
ai'
(5.1.5
)
and
o
he
magne iza ion
(5.1.6)
whe e
we
ha e
in oduced
iij
and
ñi
as
he
ansla ional
and
o a ional
eloci ies
o
he
pa icle
i.
espec i ely.
Consequen ly,
conside ing
hese
condi ions
and
o
he
quasis a iona y
case,
all
he
induced
o ce
and
o que
densi ies
mus
be
o
he
same
o m
as
in
Sec ion
2.
I we
wan
o
conside
a
o a ing
magne ic
ield,
i s
angula
equency
should
be
low
enough
o
conside
he
quasis a iona y
limi
in
which
he
de i a i es
o
he
hyd odynamic
ields,
al hough
ime
dependen ,
may
be
neglec ed.
In
appendix
A
we
ob ain
he
exp ession
o
he
magne iza ion
densi y.
Subs i u ing
his
esul
in
Eq.
(5.1.2),
and
combining
eqs.
(5.1.1)
and
(5.1.2)
we
can
w i e
he
ollowing
equa ion

5.
Hyd odynamic
in e ac ions
be ween
pai s
oi
magne ic
holes
105
N
0=
-
7p
+
(7]
+
7]
) 72V'
+
7]
7
X
h(Ji
.
7
X
V)
+
'L
. jnd,
i=l
whe e
7]
is
he
o a ional
iscosi y,
and
whe e
we
ha e
de ined
: d
as
he
combina ion
o
he
induced
o ce
and
o que
ield
densi ies
(5.1.7)
(5.1.8)
In
o de
o
sal e
o mally
he
equa ion
o
mo ion
o
he
luid
(5.1.7),
we
in oduce
Fou ie
ans o ms
o
he
eloci y
ield
V'.
F om
now
on,
we
wíll
omi
he
explici
dependence
on
ime
o
he
di e en
ields.
We
also
de ine
he
Fou ie
ans o m
o
he
induced
o ce
densi y
:
in
a
e e ence
ame
in which
sphe e
j
is
a
es
a
he
o igin
(5.1.9)
A e
Fou ie
ans o ming
his
equa ion
and
applying
he ans e sal
p ojec o
(1
-
H:),
wi h
k
==
k/k,
one
a i es
a
he
o mal
solu ion
o
he
ans e sal
componen
o
he
eloci y
ield
N
(k)
=
T(k,
)
.
'L
exp(
-ik
.
Ri
): jnd(k),
i=l
whe e,
as
in
Sec ion
2,
we
ha e
in oduced
he
p opaga o
(2.16).
As
we
ha e
indica ed
be o e,
he
sphe es
a e
allowed
o
mo e
wi h
a bi a y
e
loci y
h ough
he
luid,
which
may
i sel
be
in
a bi a y
s a iona y
non-uni o m
un-
(5.1.10)
pe u bed
low.
Fo
he
sake
o
simplici y,
we
will
conside
ha
he
unpe u bed
luid
is
a
es ,
and
we
wíll
s udy
he
hyd odynamic
in e ac ions
which
a e
se
up
be ween
he
sphe es
when
hey
mo e.
In
pa icula
ou
main
goal
is
o
calcula e
he
ansla
ional
mobili y
up
o
a
gi en
o de
o
app oxima ion
in
a
se ies
expansion
in
powe s
o
in e se
dis ances
be ween
he
sphe es,
when
only hyd odynamic
in e ac ions
be ween
wo
sphe es
con ibu e.
The
o mal solu ion
can
be
ew i en
in
eal
space
N
( , )=
'Lj
d 'T( - ,).:Fjnd( ').
i=l
In
pa icula ,
o
hose
poin s
on
he
su ace
o
he
sphe e
i,
and
a e
aking
in o
(5.1.11)
conside a ion he
p ope ies
o
he
gene alized
induced
o ce,
we
ha e
106
CHAPTER JII.
DYNAMICS
OF
MAGNETIC
HOLES
-
N
J
-
ñ·
X
¡!nd
ii(Ri
+a¡ñ¡)
=
�
dñj¡.J.(ñ¡,ñj)' [/ d(ñj)
-
)
2
J
],
3=1
whe e
we
ha e
in oduced
he
esponse
unc ion
(5.1.12)
(5.1.13)
wi h
he
de ini ion
R;j
=
R;
-
Rj.
Fo
he
pu pose
o
e alua ing
hyd odynamic
in e ac ions,
i
is
con enien
o
in oduce i educible
induced
o ce
mul ipoles,
de ined
in
e ms
o
he su ace induced
o ces
acco ding
o
1
J
A
=ind
.....{1+1)=_
dñ,
A�[/,!nd(A.)_n¡XT¡
]
:
i
471'
nI
nI
I
nI
2'
(5.1.14)
whe e
ñ:
is
an
i educible
enso o
ank
1,
aceless
and
symme ic
in any
pai
o
i s
indices.
The
expansion
in
e ms
o
i educible
o ce
mul ipoles,
which
is
w i en
in
a
coo dina e
ee
way,
is
equi alen
o
an
expansion
in
sphe ical
ha monics,
o
which
i
can
be
educed
i
pola
coo dina es
a e
used.
We
also
in oduce
i educible
su ace
momen s
o
he
luid
eloci y
ield,
and
we
ob ain
a
ela ionship
be ween
he
induced
o ce
mul ipoles
and
he
su ace
momen s
o
he
luid
eloci y
ield
h ough
a
hie a chy
o
equa ions.
I
is
his
hie a chy
which
will
hen
enable
us
o
ob ain
exp essions
o
he
mobili y
enso s
which ela e
he
o ces
and
o ques
on
he
sphe es
o
hei
lineal
and
angula
eloci ies.
The
gene al
ela ionship
be ween
he
(1
+
1)
o de
mul ipoles
o
he
quan i ies
de ined
o e
he
su ace
o
he
sphe e
is
ound
o
be,
(1+1)
_
�
�
(2/'
+
1)!!
(1+1,1'+1)
�I'+l)
V¡
-
L... L...
1"
¡.J.¡j
8
j
,
j=ll'=O
.
(5.1.15)
which
in ol es
he
enso s
(5.1.16)
As
we
al eady
poin ed
ou
in
Sec ion
3,
he
i s
o de
i educible
nul ipole
will
allow
us
o
de e mine
he
ansla ional
mobili y.
Thus,
ou
s a ing
poin
will
be
he
ela ionship
be ween
eloci y
and
o ce
mul ipoles
which
comes
om
Eq.
(5.1.15)
o
1
=
O
(1)
_
�
�
(2/'
+
1)!!
(1,/'+1)
�I'+l)
V¡
-
�
L...
I'!
¡.J.¡j
8
j
,
)=1/'=0
(5.1.17)
5.
Hyd odynamíc
ín e ac íons
be ween
paí s
o
magne íc
hoJes
107
whe e
he
mul ipole
P)
is
P)
=
411 '
J
dn¡
(ü¡
+
ñ¡
x
a¡n¡)
=
Ü¡.
(5.1.18)
Mo eo e ,
using
he
ac
ha
he
d ag
o ce
exe ed
by
he
luid
on
he
pa icle
j
(w i en
in
e ms
o
he
induced
o ce)
is
(5.1.19)
i
ollows
ha
�1)
=
_!_
Jdn.( lnd
_
nj
x
J)
=
_
FjH
.
J
41 '
J
J
2
41 'a�
J
(5.1.20)
In
Sec ion
3,
we
al eady
ga e
he
exac
o m
o
his
ela ionship
o
he
case
i
=
j.
We
ob ained,
5 3
..
-61 '7]oa¡{[1
+
24>
+
¡4>¡. F(¡. )](l
-
H
H)
5 3
..
+
[1
+
24>
+
24>¡. F(¡. )]H
H}
.
Ü¡,
(5.1.21)
Now,
we
willlook
a
he
case
i
j.
In
his
case,
he
enso s
¡. g+I,II+1)
a e
no
longe
diagonal
in
hei
uppe
indices,
so
ha
he e
is
a
coupling
be ween
di e en
mul ipoles.
Bu ,
i
we
se
l
=
O,
we
will
es i-
ou sel es
o
he
lowes alue
o
l'
(1'
=
O),
because
beyond
ha ,
we
will
ob ain
con ibu ions
p opo ional
o
(al
R)P
wi h
p
2::
4
[12],
negligible
o
he
concen a ion
o
holes
we
a e
conside ing.
The
co esponding
ma ix
is
�U,l)
=
(;!2)
J
dk
J
dk
exp(ik.
R;j)
h-1(k).
(1
-
kk)jo(ka¡)jo(kaj).
(5.1.22)
whe e
jo(x) is
he
sphe ical
Bessel
unc ion
o
ze o
o de .
In
appendix
e
we
ob ain
ha
(1,1
)
�¡j
aJ
5 3
A•
27]oR;j
([1-"24>
-
"24>¡. F(¡. )](l
+
R;jR;j)
3
AA
A
A
A
A
+
24>¡. F(¡. )[1
-
H
H
-
(R;j
x
H)(R;j
x
H)]}
aJ(a¡+aJ)
5
3
l·
A
27]oR�j
{[1-"24>
-
"24>¡. F(¡. )](-31
-
R;jR;j)
3
1
'A
A
A
A A
"24>¡. F(¡. )[3(1
-
H
H)
-
in;
x
H)(R;j
x
H)]),
(5.1.23)
+
108
CHAPTER
lII.
DYNAMICS
OF
MAGNETIC
HaLES
om
which he
ansla ional
mobili y
enso
comes
up
s aigh o wa dly,
a e
aking
in o
accoun
eqs.
(5.1.17)
-
(5.1.20)
RP
1
53
AA
1'''
=
{[1-
- /¡
-
- /¡pF(p)](1
+
R;.R;.)
IJ
81 7]0
R;j
22
J
J
3
AA
AA A A
+
2 /¡pF(p)[1
-
H
H
-
(R;j
x
H)(R;j
x
H)]}
(a;
+
aJ)
5
3
1
A'
+
3
([l-
- /¡
-
-lPpF(p)](-1
-
R;jR;j)
81 7]oRij
2
2
3
3
1
A A
•
••
•
2 /¡pF(p)[3(1
-
H
H)
-
(R;j
x
H)(R;j
x
H)]),
(5.1.24)
This
mobili y
enso
ep esen s
he
eloci y
o
a
sphe e
i
wi h
espec
o
he
luid
eloci y
due
o
he
mo ion
o
he
pa icle
i.
and
pe
uni
o
o ce
exe ed
on
sphe e
i.
I
is
alid
up
o
he
i s
o de
in
he
olume
ac ion
o
magne ic
pa icles,
/¡,
and
up
o
hi d
o de
in
he
a io
(al
R;j),
o
any
alue
o
he
Lange in
pa ame e
p.
F om his
exp ession,
we
can
also
ob ain
he
co esponding
Oseen
enso
o
he
e o luid.
The
Oseen
enso
is
he
i s
app oxima ion
when
compu ing
hyd odynamic
e ec s.
Ne e heless,
i
is
a
alid
app oxima ion
when
he
pa icles
a e
a
apa
enough,
and
we
can
assume
ha
pa icle
j
gene a es
he
same
pe u ba ion
as
would
be
p oduced
by
a
poin
o ce
si ua ed
a
i s
cen e .
Thus,
he
Oseen
enso
is
ob ained
a e
neglec ing
he
hi d
powe
o
(al
R;j),
o
I'ij
1
{[1-
� /¡
-
� /¡pF(p)](1
+
Ri·Ri'}
81 7]0
R;j
2
2
J
}
3
.
+
2 /¡pF(p)[1
-
H
H
-
(R;j
X
H)(Rij
x
Hm,
(5.1.25)
Ob iously,
i
we
se
he
olume
ac ion
o
magne ic
pa icles
equal
o
ze o,
/¡
=
O,
his
Oseen
enso
educes
o
he
well-known
exp ession
1
. .
I'P)·
=
8
R;
(1
+
R;jR;j),
1 7]0
'j
which
is
he
p opaga o
o
he
pe u ba ions
in
a
simple
luid.
(5.1.26)
5.2
Sedimen a ion
o
wo
magne ic
holes
in
a
e o luid
To
illus a e
he in luence
o
hyd odynamic
in e ac ions
on
he
suspension
o
magne ic
holes,
we
will
s udy
he
sedimen a ion
in
he
p esence
o
an
ex e nal
magne ic
ield.
Appendix
B
Explici
de i a ion
o
Eq.
(5.1.24)
S a ing
om
Eq.
(5.1.22)
we
can
make
an
expansion
such
ha
I-'g,l)
=
(;!2)
J
dk
J
dk
exp(ik.
Hij)
h-1(k).
(1
-
kk)
(1
_
(a?
;
aJ)
k2
+
O(k4»)
.
(B2)
In
a
e e ence
ame
in
which
he
ez-axis
is
pa allel
o
he
uni a y
ec o
�i
(H
=
Rii�j),
we
may
w i e
dk
=
d�ijd<Pii,
whe e
�ij
=
Rij
.
k
is
he
cosine
o
he
pola
angle
be ween
�j
and
k,
and
�ij
is
he
azimu al
angle.
(1,1)
I-'ij
(2�2)
{11
d�ij
121(
d�ij
100
dk
exp(ikR;i�ii)
h-1(k).
(1
-
kk)
(1-
(a?;
aJ)
e
+
O(k4»)
.
(B3)
Making
use
o
he
iden i y
1
¡1
loo
.
¡1
dq
-
dxxP
dyyqe,: :y
=
iq
dxxP-8(x)
=
8pqp!(-1)P,
211"
-1
-00
-1
dxq
(B4)
i
hen
ollows
ha
he
e ms
o
o de
k4
and
highe
o
he
expansion
in
he
in eg and
o
Eq.
(B3)
gi e
anishing
con ibu ions
upon
in eg a ion,
since
any
elemen
o
he
enso
h-1(k).
(1
-
kk)
will
be,
a e
in eg a ion
o e
�ij,
a
polynomial
in
�ij
whose
highes
o de
is
p opo ional
o
( i
[12],
(B5)
whe e
1
1
(
5
3
)
el
=
--
'"
-
1
-
-�
-
-�¡.JF(¡.J)
,
71
+
¡
710
2 2
(B6)
and
115

116
APPENDIX
B
'I
1
3
)
C2
=
....,
--l/JJJF(JJ
.
'1(
'1
+
'I )
'10
2
Thus,
we
can
ew i e
Eq.
(B3)
in
he
ollowing
o m
(B7)
2
JI
1211"
(
2
+
2)
�2
)
(1,1)
aj
-1
'"
ai
aj
u
lJij
=(21T)R.;j
_ld{¡j
o
dl/Jijh
(k)·(I-kk)
6(�ij)+
6R j
{)2�ij6(�¡j)
.
(B8)
Fo
he
sake
o
simplici y,
we
de ine
(B9)
wi h
GU,l)
=
(;!)
¡11
d{ij
1211"
dl/Jij
(
Cl
(1
-
H)
+
C2(k
x
il)(k
x
il»)
6(�ij),
and
(BlO)
2(
2
+
2)
JI
1211"
J::l2
(1,1)
aj
ai
aj
(".
"
')
u
Hij
=
(121T)
-1
d�ij
o
d�ij
C1(1
-
kk)
+
C2(k
x
H)(k
x
H)
{)2�ij
6(�ij).
(B11)
A e
pe o ming
he
in eg als
appea ing
in
eqs.
(B10)
and
(B11),
hey
educe
o
2
d;],l)
=
a�
(Cl(l
+
R;Jl-ij)
+
C2[1
-
il
il
-
(R;j
x
il)(R;j
x
il)l)
,
(B12)
The
way
o
sol ing
he
in eg al
in
Eq.
(B10)
is
he
ollowing
whe e
we
ha e
in oduced
he
cons an s
(91,
...
,94)
which
can
be
ound
by
con ac ing
he
enso s
in
bo h
si
des
o
he
equa ion,
i s ly
wi h
he
iden i y
ma ix,
1,
and
sec
ondly
wi h
he
enso
R;j
R;j.
In
doing
so,
and
a e
calcula ing
he
simple
scala
in e
g als
esul ing
om
hese
con ac ions,
we
a i e
a
a
de e mined
sys em
o
equa ions
o
he
aboye
in oduced
cons an s.
The solu ions
a e
91
=
92
=
93
=
-94
=
aJ
/2.
APPENDIX
B
117
The
in eg al
in
Eq.
(B11)
can
be
ca ied
ou
ollowing
exac ly
he
same
p ocedu e.
Finally,
once
we
ha e
subs i u ed
he
cons an s
Cl
and
C2
gi en
h ough
eqs.
(B6)
and
(B7),
Eq.
(B9)
u ns
ou
o
be
Eq.
(5.1.23).
Bibliog aphy
[1]
H.S.
Selle s
and H.
B enne ,
PCH
PhysicoChemical
Hyd odynamics
11,
(1989)
455.
[2]
G.
Helgesen
and
A.T.
Skjel o p,
J.
Magn.
Magn.
Ma .
97,
(1991)
25.
[3]
G.
Helgesen
and
A.T.
Skjel o p,
J.
Appl.
Phys.
69
(12),
(1991)
8277.
[4]
P.
Da ies,
J.
Popplewell,
G.
Ma in,
A.
B adbu y
and
R.W.
Chan ell,
J.
Phys.
D,
Appl.
Phys.
19,
(1986)
469.
[5]
A.T.
Skjel o p,
Phys.
Re .
Le .
51,
(1983)
2306;
J.
Magn. Magn.
Ma .
65,
(1987)
195;
J.
Appl.
Phys.
57
(1),
(1985)
3285.
[6]
G.
Helgesen
and
A.T.
Skjel o p,
Physica
A
170,
(1991)
488;
Physica
A
176,
(1991)
37.
[7]
G.
Helgesen,
P.
Pie anski
and
A.T.
Skjel o p,
Phys.
Re .
Le .
64,
(1990)
1425;
Phys.
Re .
A
42,
(1990)
7271.
[8]
H.
B enne ,
J.
Colloid
In e ace
Sci.
1,
(1970)
141;
Ann.
Re .
Fluid
Mech.
2,
(1970)
137;
In .
J.
Engng.
Sci. 22,
(1984)
645.
[9]
V.G.
Bash o oy,
B.M.
Be ko sky
and
A.
N.
Vislo ich,
In oduc ion
io
The mo
mechanics
01
Magne ic
Fluids
(Sp inge -Ve lag,
Be lin,
1988).
[10]
J.
P.
Mc
Tague,
J. Chem.
Phys.
51,
(1969)
133.
[11]
P.
Mazu
and
D.
Bedeaux,
Physica
76,
(1974)
235.
[12]
P.
Mazu
and
W.
Van
Saa loos,
Physica
115
A,
(1982)
21.
[13]
J.M.
Rubí
and
M.C.
Miguel,
Physica
A
194,
(1993)
209.
118
BIBLIOGRAPHY
119
[14]
M.C.
Miguel,
J.
Bone
A ales,
A.
Pé ez-Mad id
and
J.M.
Rubí,
Physica
A
193,
(1993)
359.
[15]
J.
Popplewell,
R.E.
Rosensweig
and
R.J.
Johns on,
IEEE
T ans.
Magn.
26,
(1990)
1852.
[16]
M.
A.
Ma senyuk,
Yu.
L.
Raikhe
and
M.
1.
Shliomis,
SOy.
Phys.
JETP
38,
(1974)
413.
[17]
W.B.
Russel,
n.A.
Sa ille,
W.R.
Schowal e ,
Col/oidal
Dispe sions,
(Camb idge
Uni e si y
P ess,
Camb idge,
1989).
[18]
J.
Happel
and
H.
B enne ,
Low
Reynolds
Numbe
Hyd odynamics (Kluwe ,
The
Ne he lands,
1991)
[19]
P.
Mazu ,
Physica
HOA,
(1982)
128.
CHAPTERIV
AGGREGATION
PHENOMENA
This
chap e
is
in ended
o
be
an
in oduc o y
analysis
o
he
agg ega ion
phenomena
ha
a ise
o
ins ance,
as
he
size
o
he
pa icles
in
a
e o uid
inc eases
o
a
highe
concen a ions.
In
such
condi ions
he
dipole-dipole
i e ac ion
be ween
he
pa icles
is
enhanced and
B ownian
mo ion
is
no
longe
able
o
s abilize
he
suspension.
Mo eo e ,
in
he
p esence
o
a
high
magne ic
ield
he
pa icles
show
up
a
endency
o
o m
chains.
We
s udy
he
kine ics
o
he
o ma ion
o
he
agg ega es
by
means
o
he
Smoluchowski
heo y
o
coagula ion
in
colloids
accoun ing
o
hyd odynamic
in e ac ions.
These
in e ac ions
become
ele an
o
he
concen a ion
o
pa icles
which
gi e
ise
o
hese
phenomena
and
slow
down
he
agg ega ion
p ocess.
In
addi ion,
he
heology
o
he
chains
ha
a e
usually
obse ed
in
sys ems
wi h
dipola
in e ac ions
is
s udied
o
a
a he
simpli ied
si ua ion
in
o de
o
elucida e
he
e ec s
o
he
dipola
magne ic
in e ac ions
in
he
con ibu ion
o
he
chains
o
he p essu e
enso
o
he
suspension.
120

1.
In oduc ion
121
1
In oduc ion
Colloidal
dispe sions
play
an
impo an
ole
in
many
na u al
phenomena
as
well
as
in
a ious
indus ial
p ocesses.
The
s abili y
o
he
suspension
agains
agg ega ion
o
he
pa icles
is
o
essen ial
impo an
ce
o
i s
beha io
[1].
As
long
as
one can
neglec
in e pa icle
in e ac ions,
one
does
no
obse e
agg ega ion
phenomena
and
hen
he
main
p oblem
is
o
unde s and
how
he
physical
p ope ies
o
he
luid,
o
example
i s
iscosi y,
a e
modi ied
due
o
he
p esence
o
he
pa icles.
Bu
when
in e ac ions
be ween
pa icles
s a
o
be
impo an ,
he
p eceding
p oblem
becomes
much
mo e
complica ed.
In
his
case,
he
agg ega ion
mechanism
in
a
colloidal
solu ion
is
qui e
complex
because
he
luid
does
keep
in o
an
ac i e
ole.
The
na u e
o
in e -pa icle
in e ac ions
depends
on
he
p ope ies
o
he
luid,
and,
conce ning
i s
d i ing
mo ion,
hyd odynamic
e ec s
canno
be
neglec ed.
The
concep s
o
ha d
and
so
pa icles
a e
commonly
used
in
he
li e a u e
[2].
A
ha d
pa icle
is
a
igid
sphe e
on
which
only
hyd odynamic
o ces
a e
ac ing,
hey
do
no
unde go
o he
in e ac ion
o ces
excep
o
an
in ini ely s ong
epulsion
on
con ac
o
p e en
in e pene a ion.
In
con as ,
a
so
pa icle
is
a
sphe e
subjec
o
in e ac ions,
o he
han
hyd odynamic
ones,
el
a
ini e
dis ances.
In
he
las case,
one
can
hink o
such
sphe es
as
ha ing
a
su ace
bounda y
laye
(sphe e
o
in luence)
whose
hickness
is
de e mined
by
a
cha ac e is ic
in e ac ion
leng h.
Thus,
o
elec o
s a ically
s abilized
dispe sions
his
hickness
can
be
iden i ied
wi h
he
double-Iaye
hickness,
o
an
de
Waals
in e ac ions
wi h
he hickness
whe e
he
in e ac ion
en
e gy
becomes
compa able
o
he
he mal
ene gy,
and
o
s e ically
s abilized
colloids
wi h
he
leng h
o
he
polyme
ails
adso bed
on
he
su ace
o
he
suspended
pa icles.
This
idea
was
in
a
ce ain
way
exploi ed
by
on
Smoluchowski
[3],
who
i s
analized
he
p oblem
o
coagula ion
in
he
absence
o
any
epulsi e
ba ie
( apid
coagula ion).
A
sphe e
wi h
a
su ace
bounda y
laye
can
be
conside ed
so
because
his
laye
can
be
de o med
and
pene a ed
du ing
he
app oach
o
wo
pa icles.
Mo eo e ,
in
he
li e a u e
he
e m
coagula ion
is
applied
o
agg ega ion
ha
is
induced
by
he
an
de
Waals
a ac ion
be ween he
colloidal
pa icles.
On
he
o he
hand,
he
e m
loccu
la ion
is
ese ed
o
polyme -induced
agg ega ion.
Coagula ion
usually
gi es
ise
o
compac
agg ega es
whe eas
loccula ion
equen ly
p oduces
mo e
open
s uc u es.
Pa icula ly,
he
monodomain
magne ic
pa icles
a e
usually
su icien ly
small
ha
B ownian
o ces,
along
wi h
sho - ange
s e ic
epulsion
due
o
ei he
na u al
o
syn
he ic
polyme s,
domina e
in
he
ze o- ield
limi
o
gua an ee
he
s abili y
o
he
pa icles.
Howe e ,
when
a
non
e y
dilu e
suspension
o
hese
pa icles
is
unde
122
CHAPTER
IV.
AGGREGATION
PHENOMENA
he
ac ion
o
an
applied magne ic
ield,
pa icle
agg ega ion
occu s
when
he nag
ne ic
a ac ion
be ween
he
pe manen
momen s
is
s ong
enough
o
ou weigh
he
s abilizing
o ces.
Thus
he
hickness
o
he
bounda y
laye
can
be
iden i ied
wi h
he hickness
o
which
he
magne ic
in e ac ion
ene gy
is
compa able
o
he
he mal
ene gy
a he
han
wi h
he
leng h
o
he
polyme ,
due
o
he
long
ange
cha ac e
o
such
in e ac ion.
In
he
la e
1930's,
Winslow
[4]
obse ed
in e es ing
phenomena
when
dielec ic
pa icles
suspended
in
oil
we e
subjec
o
an
elec ic
ield.
He
saw
he
induced
o ma
ion
o
chains
o
pa icles
aligned
wi h
he
elec ic
ield
and,
e en
o
mo e
p ac ical
impo an
ce,
ha
he
e ec i e
iscosi y
o
he
suspension
could
be
a ied
by
o de s
o
magni ude
by
modi ying
he
applied
elec ic ield.
Howe e ,
s udies
on
elec o he
ological
luids
a e
hinde ed
by
many
p oblems
ela ed
o
su ace
cha ge,
elec o
de
pola iza ion,
adso bed
wa e ,
ield
inhomogenei ies,
e c.
An
analogous
ield-induced
beha io
is
shown
by
magne o heological
luids,
e.g.,
in
a
suspension
o
magne izable
pa amagne ic
pa icles
in
a
nonmagne ic
luid
[5]
o
in
a
suspension
o
nonmagne iz
able
sphe es
in
a
e o luid
(magne ic
holes,
see
p e ious
chap e ).
In
apu e
e o luid
one
obse es
simila
phenomena
bu ,
due
o
ac
ha
he
pa icles
a e
pe manen ly
magne ized,
he e
a e
pola iza ion
o ces
e en
wi hou
an
applied
magne ic
ield.
In
all
hese
cases,
one
has
he
ad an age
ha
hey
a e
no
suscep ible
o
he
aboye
men ioned
p oblems.
Thus,
s udying
he
kine ics
o
he
o ma ion
o
he
agg ega es
and
hei
heology
a e
opics
o
p ac ical impo ance
and
has
p o ided
he
bases
o
heo e ical
and
expe imen al
s udies
[6]-[8]
as
well
as
nume ical
simula ions
[9]-[11].
Ha ing
all
hese
in
mind,
he
s uc u e
o
his
chap e
is
as
ollows,
In
Sec ion
2
we
discuss
he
in luence
o
hyd odynamic
in e ac ion
in
he
kine ics
o
agg ega es
o ma
ion
by
using
he
Smoluchowski's
heo y
o
coagula ion.
We
w i e
he
co esponding
kine ic
di e en ial
equa ions
gi ing
he
ime
e olu ion
o
he
dynamic
clus e
size
dis ibu ion
unc ion
and
sol e
hem
o
a
a he
simpli ied
case.
In
his
sec ion
we
also
discuss
he
scaling
beha io
o
he
clus e
size
dis ibu ion.
Sec ion
3
is
in ended
o
be
an
in oduc o y
analysis
o
he
heology
o
he
chains
which
commonly
appea
when
he
agg ega ion
p ocess
occu s
unde
he ac ion
o
an
ex e nal
ield.
( al a)
The
conclusions
a e
summed
up in
he las
sec ion.
2. Hyd odynamic
in e ac ions
in
he
Smoluchowski
heo y
o
coagula ion
123
2
Hyd odynamic
in e ac ions
in
he
Smoluchowski
heo y
o
coagula ion
The
in es iga ion
o
agg ega ion
p ocesses
by
means
o
compu e
simula ions
ha e
been
enhanced
du ing
he
las
10
yea s
a e
he
pionee ing
wo k
o Wi en
and
Sande
whe e
hey
p oposed
hei
di usion-limi ed
agg ega ion
model
e e ed
o
as
DLA
[12].
This
wo k
s imula ed
he
de elopmen
o
he
di usion-limi ed
clus e -clus e
agg ega ion
model
by
Meakin
[13]
and
Kolb
e
al.
[14],
which,
pa icula ly,
p o ides
a
be e
unde s anding
o
s uc u al
and
kine ic
aspec s
o
colloid
agg ega ion.
The
coagula ion
p ocess
can
also
be
discussed
in
e ms
o he
Smoluchowski's
heo y
which,
as
we
will
see
below,
due
o
he
na u e
o
he
app oxima ion
inhe en
in
i s
de i a ion
is
a
mean- ield
heo y.
Ne e heless,
his
heo y
is
expec ed
o
hold
o
dimensions
g ea e
han
he
uppe
c i ical
dimension
aboye
which
luc ua ions
become
i ele an
and
ha
o
his
coagula ion
p ocesses
is
2.
Smoluchowski
i s
ound
ou
a
e y
in e es ing
applica ion
o
he
heo y
o
B ow
nian
mo ion
in
he
coagula ion
exhibi ed
by
colloidal
pa icles
when
an
elec oly e
is
added
o
he
solu ion.
His
heo y
is
based
on
he
sugges ion
o
Zsigmondy
ha
coag
ula ion
is
a
consequence
o
he
exis en
ce
o
a
sphe e
01
in iuence
o
a
ce ain adius
R
su ounding
each
colloidal
pa icle
such
ha
i s
B ownian
mo ion
emains
una ec ed
unless
ano he
pa icle
en e s
wi hin
i s
sphe e
o
in luence.
When
his
happens
hey
s ick
o
one
ano he
o
o m
a
single
uni .
Fo his
pa icula
p oblem,
he
sphe es
o
in luence
a e
supposed
o
o igina e
in
he
o ma ion
o
elec ic
double
laye s
a ound
each
pa icle.
The
double
pa icle
con inues
mo ing
andomly
so
long
as
i
does
no
come
wi hin
he
sphe es
o
in luence
o
a
single
o
ano he
double
pa icle.
Then
we
will
ha e
he
o ma ion o
a
iple
o
a
quad upole
pa icle,
and
so
on.
This
p o
cess
wilI
e en ualIy
lead
o
he
o al
coagula ion
o
alI
he
colloidal
pa icles
in o
one
clus e .
Recen
eal- ime
expe imen s
pe o med
wi h
suspensions
o
supe pa amagne ic
la ex
pa icles
[15]
co obo a e
Smoluchowski's
hypo heses.
In
his
case,
he
mag
ne ic
ield
induces
chain
o ma ion,
and
one
obse es
wo
agg ega ion
ime
scales.
When
pa icles
a e
a
enough
away
om
each
o he
ha
hei
dipola
in e ac ion
is
weak
ela i e
o
kBT,
B ownian
mo ion
domina es.
When
his
andom
mo ion
hap
pens
o
b ing
wo
pa icles
close
enough
ha
hei
dipola
in e ac ion
is
o
he
same
o de
as
kBT,
a
apid
ansi ion
be ween
andom
and
ballis ic
mo ion
occu s
and
he
pa icles
agg ega e immedia ely. Mo eo e ,
nume ical
simula ions
inco po a ing
induced-dipola
in e ac ion
o ces
and
B ownian
mo ion
also
co obo a e
ha
he e
is
124
CHAPTER
IV.
AGGREGATION
PHENOMENA
a
ci ical
alue
o
he
in e pa icle
sepa a ion
wi hin
which
pa icle
mo ion
ceases o
be
andom.
Thus,
each
pa icle
can
be
hough
o
as
ha ing
a
cap u e
olume
de ined
by
equa ing
dipola
in e ac ion
ene gy
o
he mal
ene gy.
I
one
pa icle
en e s
his
olume,
he
wo
pa icles
unde go
ballis ic
agg ega ion.
On
he
o he
hand,
he
appli
cabili y
o
he
Smoluchowski
equa ion
is
limi ed
o
low
clus e
concen a ions
because
i
assumes
ha
he e
a e
only
bina y
collisions.
We
a e
in e es ed
in
de e mining
he
concen a ions
nI,
n2,
...
,
o
single,
double,
iple,
quad upole,
e c.,
pa icles
a
ime
gi en
ha
a
ime
=
O
he e
a e
no
single
pa icles.
Mo eo e ,
once
we
know
hese
concen a ions
we
can
w i e
he
scaling
laws
which
show
how
hese
quan i ies
beha e
o
long
imes
o
o
la ge
adius
o
he
sphe es
o
in luence.
Bu ,
in
his
sec ion
we
will
in oduce
a
modi ica ion
o
he
simple
heo y
p oposed
by
Smoluchowski
by
conside ing
hyd odynamic
e ec s,
Fo
dis an
ces
g ea e
o
o
he
same
o de
o
magni ude
as
he
in e pa icle
sepa a ion
h eshold
(-
2a
-
7a,
wi h
a
he
adius
o
one
pa icle),
hyd odynamic
in e ac ions
should
be
aken
in o accoun
when
s udying
he
di usi e
mo ion
o
he
pa icles.
The e
a e
p e ious
a emps
o
inco po a e
his
e ec s
in
he
inal
s ages
o
he
app oach
o
wo
non- nagne ic
pa icles
(i.e.
when
jus
sho
ange
in e ac ions
be ween
he
pa icles
a e
conside ed)
when,
because
i
is
di icul
o
he
emaining
ilm
o
Iiquid
o
escape,
he
p ocess
is
clea ly
slowed
down.
Honig
e
al.
[16]
o
ins ance
examined
he
p oblem
de i ing
an
app oxima e
ela ion o
a
modi ied
di usion
coe icien :
D(h)
1
+
2aj3h
------�-
=
--------�-------
D(h
-+
00)
1
+
13aj6a
+
a2
j3h2
(2.1
)
whe e
h
=
-
2a
is
he
dis an
ce
o
closes
app oach.
This
e ec
was
also
es ed
expe imen ally
by
Lich enbel
el
al.
[17]
who
ound
ha
he
a e
o
coagula ion
was
educed
o
less
han
hal
o
he
Smoluchowski
alue
o
hose
colloidal
suspensions.
Howe e ,
when
dealing
wi h
long
ange
in e ac ions,
we
belie e
ha
i
would
be
enough
o
in oduce
hyd odynamic
in e ac ions
ep esen ed
by
he
Oseen
and
Ro ne
P age
enso s
discussed
in
he
p e ious chap e .
This will
enable
us
o
ind
analy ic
exp essions
no
only
o
he
a e
o
coagula ion
bu
also
o
he
di e en
concen a ions
o
clus e s
nI,
n2,
...
2.1
Kine ic
equa ion
o
i e e sible
agg ega ion
Fo
he
ime
in e als
we
a e
in e es ed
in
he
mo ion
o
he
pa icles
is
di usi e,
i.e.,
local
luc ua ions
in
he
concen a ion
due
o
B ownian
mo ion
de e mine
a
global
di usion
om
he
highe
o
he
lowe concen a ed
egions.
We
will
s a
conside ing
2.
Hyd odynamic
in e ac ions
in
he
Smoluchowski
heo y
o
coagula ion
131
-
Fo
la ge
alues
o
R,
i.e.,
o
a
long
ange
in e ac ion,
and
p o ided
ha
3a/2R
<
1
L
will
be
also
la ge
so
ha
we
can
ew i e
Eq.
(2.2.15)
as
ollows
no
(
1
)-(1:+1)
no
(
k
)2
_
.
nI:
=
(noL )2
1
+
noL
::::
k2
no L
e;;;n,
o ,
equi alen ly
(2.3.1)
nI:
ex
k-T
(;'1)
,
whe e
we
ound
ou
he
cha ac e is ic
exponen s
T
=
2
and
=
1.
-
On
he
o he
hand,
o
su icien ly
la ge
imes
(2.3.2)
nI:
=
no
(1
+
_1_)
-(1:+1)
....,
no
(_k_)
2
e--;;!n
(noL )2
noL
-
k2
noL
'
om
which
i
is
possible
o
w i e
he
size
dis ibu ion unc ion
unde
he
ollowing
(2.3.3)
scaling
o n
(2.3.4)
whose
cha ac e is ic
exponen s
a e
T
=
2
and
z
=
1.
Vicsek and
Family
[21]
and
independen ly
Kolb
[22]
in oduced
a
dynamic
scaling
desc ip ion
o
he
clus e
size
dis ibu ion
in
he
clus e -clus e
agg ega ion
model.
Mon e
Ca Io
simula ions
showed
ha
a
dynamic
scaling
o
he
aboye
in oduced
o m
(2.3.5)
ep esen ed
well
he
beha io
o
nl:( ).
He e
(x)
is
a
scaling
unc ion
which
depends
on
he
dimension
and
on
he
clus e
mobili y.
The
exponen
z
has been
measu ed
bo h
expe imen ally
and
by
Mon e
Ca Io
simula ions
o
di usion-limi ed
clus e -clus e
agg ega ion.
The
expe imen al
esul
a e
gene ally
in
ag eemen
wi h
he
simula ions
and,
a
he
same
ime,
he
Mo e
Ca lo
esul s
ag ee
wi h
he
Smoluchowski
app oach
in
d
=
2,3,
bu
disag ee
in
one
dimensiono
Tha
was
consis en
wi h
he
obse a ion
ha
Smoluchowski's
heo y,
wi hou
he
inco po a ion
o
spa ial
luc ua ions,
ails
below
a
c i ical
dimension
de
=
2.
As
an
applica ion
o
he
scaling
laws,
le
us
look
a
he
case
o
magne ic
pa icles
wi h
dipola
in e ac ions,
when
we
neglec
hyd odynamic
e ec s,
i.e.
o
la ge
alues
o
R.
In
his
case,
as
we
al eady
poin ed
ou
in
he
in oduc ion,
we
can
es ima e
he
alue
o
R
in
he
ollowing
way

132
CHAPTER
IV.
AGGREGATION PHENOMENA
(2.3.6)
which
ollows
by
equa ing
bo h
magne ic
and
he nal
ene gies.
Consequen ly,
(2.3.7)
so
ha
he
concen a ion
o
clus e s
o
k
pa icles
scales
wi h
he
pa icle's
nagne ic
momen
(2.3.8)
wi h
cha ac e is ic
exponen s
T
=
2
y
'Y
=
2/3.
Janssen
el
al.
[11]
used
a
nume ical
app oach
o
sol
e
he
basic
equa ion
o
loccula ion,
conside ing
a
cylind ically
sym
me ic
in e ac ion
in
o de
o
model
he
dipola
in e ac ion
in
pa amagne ic
pa icles.
In
pa icula
hey
in es iga ed
he
a e
o
ini ial
loccula ion
J
=
87
DnoR/W,
whe e
in
hei
no a ion
W
s ands
o
he
s abili y
ac o
which
indica es
how
he
in e ac
ions
modi y
he
o al
lux
in
compa ison
wi h
he
case
o
apid
agg ega ion,
in
which
no
epulsi e
o ces
a e
p esen
o
slow
down
he
p ocess.
They
ob ained
ha
his
s abili y
ac o
depends
on
he
dimensionless
pa ame e
cha ac e izing
he
dipola
in e ac ion,
o
in
o he
wo ds
on
he
magne ic
momen
o
he
pa icles,
Wex
m-2/3,
wi h
he
same
cha ac e is ic
exponen .
This
esul
was
also
e i ied
expe imen ally
a
high magne ic
ields.
Mo eo e ,
ecen
expe imen al
in es iga ions
on
he
magne ic- ield-induced
chain
o ma ion
o
supe pa amagne ic
la ex
pa icles
[15]
show
a
powe -law
dependen
ce
on
ime
as
can
also
be
ob ained
by
he
Smoluchowski
equa ion
and
h ee-dimensional
simula ions
o
dipola
pa icles,
Bu
hey
also
ound
ha
he
alue
o
he
exponen
z
ha e
a
weak
in e se
dependen
ce
on
he
pa icle
olume
ac ion
and
in
he
di nen
sionless
cons an
cha ac e izing
he
dipola
in e ac ion
s eng h.
We
plan
o
pu sue
wo k
on
his
expe imen al
obse a ions
in
he
nea
u u e.
3
Chains
o
magne ic
pa icles
in
an
elonga ional
low
Colloidal
pa icles
may
gi e
ise
o
ei he
ixed
o
ee
s uc u e
agg ega es
depending
on
he
na u e
o
he
agg ega ion
p ocesses
and
he
ype
o
in e pa icle
bonds
es ab
lished
[25].
We
will
conside
he
case
in
which
he
sphe es
a e
in
close
con ac
bu
a e
3.
Chains
o
magne ic
pa icles
in
an
elonga ional
low
133
Figu e
IV.1:
Chain
o
magne ic
pa icles
in
an
elonga ional
low.
All
he
magne ic
momen s
owa ds
he
di ec ion
o
he
high
ex e nal
ield
applied.
s ill
able
o
ansla e ela i e
o
one
ano he
unde
he
ac ion
o
an
elonga ional
low.
I
he
pa icles
a e
non-magne ic,
his
in e nal
mo ion
is
esis ed
by
hyd odynamic
lub ica ion
o ces
and, al hough
he
ela i e
mo ion
is
e y
small,
i
has
seen
o
ha e
a
big
e ec
on
he
s ess
ansmission.
Now,
he
pa icles
in e ac
wi h
each
o he
because
o
he
dipola
magne ic
in e ac ion
and
we
wan
o
know he
e ec
o
his
in e ac ion
on
he
in e nal mo ion
and
he
iscosi y
o
such
a
sys em.
We
de ine
he
ollowing
dimensionless
pa ame e s
..
==
m2
j(d3kBT)
and
J.l
=
mH
jkBT,
whe e
m
and
d
a e
he
magne ic
momen
and
he
diame e
o
each
sphe e,
H
is
he
ex e nal
magne ic
ield
s eng h,
kB
is
he
Bol zmann
cons an ,
and
T
he
absolu e
empe a u e.
When
J.l
-
00,
all
he
magne ic dipoles align
pa allel
o
he
ex
e nal
magne ic
ield.
Unde
his
condi ion,
he
ene ge ically
a ou able
a engemen
o
N
g ains
is
a
linea
chain.
As
we
will
see
below,
he
analy ic
s udy
o
hese
pa 
icula
agg ega es
equi es
conside able
app oxima ions,
e en
in
he
limi
case
..
�
1
and
J.l
-
oo.
On
he
o he
hand,
en opy
conside a ions
will
sugges
ha
o he
mo e
complex
s uc u es
may
a ise
when
he
pa icle
densi y
is
oo
la ge
o
..
is
oo
small
[26].
Bu
o
low
densi y
and
la ge
.. ,
he
mos
a ou able
phase
is
a
linea
chain.
Di ec
calcula ions
[26]
show
ha
despi e
he e
a e
long- ange
con ibu ions
o
he ene gy
o
a
chain
con aining
N
pa icles,
he
ampli ude
o
he
ib a ions
and
oscilla ions
o
he
sphe es
a e
almos
en i ely
aken
in o
accoun
by
jus
conside ing
he
e ec s
due
o
i s
and
second
nea es -neighbou
in e ac ions.
The
long
ange
e ec s
s ill
exis ,
bu
hey
a e
domina ed
by
sho
ange
a ac i e
con ibu ions.
As
a
ma e
o
ac ,
he
impo an
egions
co espond
o
ela i e
dis ances
among
he
134
CHAPTER
IV.
AGGREGATION
PHENOMENA
pa icles
"""
d
and
ela i e
de ia ions
(J
"""
O
o
11'.
In
addi ion,
o
ou
magne ic
colloids
he
in e ac ions
o
one
g ain
a e
essen ially
sa u a ed
when
i
has
come
in o
close
con ac
wi h
wo
o he s.
Mo eo e ,
in
he
high
ield
limi ,
he
s a e
o
a
sphe e
will
be
en i ely
desc ibed
by
he
posi ion
o
i s
cen e.
This
will
simpli y
conside ably
he
equa ion
o
con inui y
ha
desc ibes
he
conse a ion
o
sys em
poin s
in
he
con igu a ion
space.
Finally,
we
will
dis ega d
London-
Van
de
Waals
o ces
as
well
as
s e ic
e ec s.
3.1
Con ibu ion
o
he
chains
o
he
p essu e
enso
o
he
sys em
Apa
om
he
con ibu ion
o
he
sol en
o
he
o al
p essu e
enso
o
a
supension,
he e
is
ano he
con ibu ion
coming
om
he
di ec
in e ac ion
o
he
suspended
pa icles
[27,
28].
Thus, ollowing
K ame s'
heo y,
he
con ibu ion
o
a
chain
o
he
p essu e
enso
o
he
whole
sys em
is
1
�
(�
8Vmag
)
N
-
1
IIp
=
-
V
�
qk�
+
-V-kBT1,
k=1
s»
whe e
V
is
he
olume
o
he
sys em,
q¡,
a e
he ela i e
posi ion
ec o s,
Vmag
is
he
dipole-dipole
magne ic
po en ial,
N
is
he
numbe
o
pa icles
o
he
chain,
and
1
is
(3.1.1
)
he
uni a y
ma ix.
In
o de
o
do
he
a e ages
appea ing
on
he
igh
hand
side
o
Eq.
(3.1.1),
we
need
o
sol e
he
co esponding
di usion
equa ion
in
he
con igu a ional
space:
al/;
'"
{8
'"
TT
(
8l/;
8Vmag)
8
�}
-
=
�
-=-.
�""'ij
.
kBT-�
+
-_-l/;
-
-_
.
( 3.
H¡)l/;
8
i
en;
j
8Rj 8Rj
8H¡
(3.1.2)
whe e
i;
is
he
pos ion
ec o
o
he
í-sphe e,
,..,.Tl
is
he
ela i e
ansla ional
nobil
i y,
and
3
is
he
elonga ional
a e
( 3
=
3T)
co esponding
o
he
s a iona y
ho noge
neous
ex e nal
low,
Vo
=
3
.
.
We
a e
also
assuming
ha
he
sphe es
mo e
h ough
he
sol en
wi hou
dis u bing
he
eloci y
ield
( ee
d aining),
so
ha
TT
1
,..,.
..
=
--IÓij.
1)
611'7]a
whe e
we
conside
ha
all
he
sphe es
in
he
chain
ha e
exac ly
he
same
adius
(3.1.3)
a.
Mo eo e ,
we
will
dis ega d
he
o a ional
con ibu ion
o
he
di usion
equa ion
3.
Chains
oi
magne ic
pa icles
in
an
eJonga ionaJ
low
135
because
in
he
high
ield
limi
he
s a e
o
a
pa icle
is
ully
ep esen ed
by
he
posi ion
o
i s
cen e.
Subs i u ing
Eq.
(3.1.3)
in o
Eq.
(3.1.2),
we
a i e
a
o /J
=
kBT
I:
{02_ /J
+
o_
.
(oVmag!kBT
/J)
-
.!.
.
( 3
.
R;) /J}
.
&
61!'1Ja
i
oR?
aH; aH; aH;
In
addi ion,
as
we
a e
conside ing
he
limi
A
::»
1,
J-l
->
00,
we
will
assu ne
ha
(3.1.4
)
he
po en ial
can
be
exp essed
as
a
sum
o
he
nea es
neighbou s
in e ac ion
V.
N-l
mag
"'"'
k
T
=
L....J
�;,i+l
B
;=1
As
we
ha e
p e iously
indica ed,
he
ampli ude
o
he
ib a ions
and
oscilla ions
o
(3.1.5)
he
sphe es
a e
well
ep esen ed
by
such
in e ac ions.
A e
aking
in o
accoun
ha
he
mos
impo an
con ibu ions
come
om
ela i e
dis ances
.....
dand
ela i e
de ia ions
(J
"-'
O,
each
e m
in
he
sum
o
Eq.
(3.1.5)
can
be
w i en
as
ollows
�;,;+1
"-'
-A(2
-
3(J?
-
6�;)
(3.1.6)
whe e
we
ha e
in oduced
he
ela i e
posi ion
ec o
q;
=
Ri+1
-
R;
whose
co npo
nen s
in
sphe ical
coo dina es
a e
q;
==
(q;,
(Ji,
<p).
Mo eo e ,
we
ha e
ca ied
ou
a
de elopmen
a ound
he
maximum
con ibu ion
q;
d(1
+
�;)
wi h
�i
"-'
O
(Ji
O.
(3.1.7)
Now,
we
will
w i e
he
di usion
equa ion
in
e ms
o
he
ela i e
posi ion
ec o s,
q;.
Bu
be o e
ha ,
we
in oduce
he
ans o ma ion
q;
=
I:
Bil:RI:
wi h
s.,
=
6i+1,1:
-
6i,1;,
1:
om
which
one
in e s
he
ela ion
(3.1.8)
a
a
-_
=
I:
Bki-;;=
aH;
k
os»
(3.1.9)
wi h
i
í
=
j
i
í
=
j
±
1
o he wise
(3.1.10)
136
eHAPTER
IV.
AGGREGATION
PHENOMENA
Taking
in o accoun
Eqs.
(3.1.8)-(3.1.10),
he
di usion
equa ion
eads
The
s a iona y
solu ion
o
his
homogeneous
po en ial
low
ield
is
./.
(-
-)
e
{-Vmag
311''1a(a
"''''c
-
-)}
'I',
q1,"
·,QN-1
=
eep
-¡¡-;¡;-
+
k
T
p
:
L...,;L...,;
ij
qi
qj
,
B
B
i
j
(3.1.12)
whe e
e
is
a
no maliza ion
cons an
and
eij
is
he
K ame s'
ma ix
de ined
in
he
ollowing
way:
e
..
_
{
i(N
-
j)/N
i i
s
j
1)
-
j(N
_
i)/N
i
j
�
i
(3.1.13)
Once
we
know
he
s a iona y
solu ion
o
he
di usion
equa ion,
we
will
compu e
he
con ibu ion
o
he chain
o
he
p essu e
enso ,
i.e.,
he
a e age
appea ing
on
he
igh
hand
si
de
o
equa ion
(3.1.1)
( ik
O��ag)
=
J
l
di ¡
( ik
O��ag)
1/J. (Q1,""
V -1)'
s»
1=1
s»
As
we
a e
in e es ed
in
inding
he
New onian
iscosi y
enso ,
we
can
expand
he
exponen ial
ac o
(37 '1a)/(kBT)( 3:
Li
Li
eijq'¡ijj)
in
Eq.
(3.1.12)
up
o
i s
o de
in
3.
I
we we e
also
in e es ed
in
he
i s
non-New onian
con ibu ion
we
would
include
he
second
o de ,
...
A
i s
we
will
use
(3.1.14)
o
i
we
in oduce
he
ansla ional
di usion
coe icien
o
a
single
pa icle
D
=
kBT/(67 '1a)
and
Eq.
(3.1.5)
whe e
e'
is
he
new
no maliza ion
cons an
ha
we
compu e
in
Appendix
A

3.
ehaíns
o
magne ic
pa icJes
in
an
elonga ional
Iow
137
1
C'
(3.1.17)
+
Consequen ly,
Eq.
(3.1.14)
educes
o
(
_
OVmag)
q
1:
----;:;-:;
os»
e'
¡/i
dq¡
exp(
-�I,I+l))
(iJ:
o�,:ag)
�l
Ü
{l
+
2�(�.
��C;;;;q;)}.
(3.1.18)
In
o de
o
e alua e
his
a e age,
we
will
decompose
i
in o
di e en
pa s
(de ailed
calcula ions
ha e
been
collec ed
in
Appendix
B).
Up
o
i s
in
{3
we
ha e
(iJ:
O�'ikag
)
=
kBT
{1
+
�
�
Cil:{3·
z z
+
6�A
(1
-
l�A)
el;k{3·
(1
-
Z Z)}
(3.1.19)
Once
we
ha e
been
able
o
compu e
his
a e age,
he
con ibu ion
o
one
chain
o
he
p essu e
enso
(3.1.1)
In
iew
o
Eq.
(3.1.20),
we
can
conclude
ha
we
ob ain
an
expansion
in
powe s
o
he
in e se
o
he
pa ame e
A,
compa ing
dipola
and
he mal
ene gies,
s a ing
om
he
assymp o ic
alue
which
co esponds
o
he
igid
chain
limi
and
inc easing
o
dec easing
when
A
dec eases,
i.e.
when
he
chain
becomes
mo e
lexible,
depending
upon
he
s uc u e
o
he
low
a e.
I
wo hs
poin ing
ou
ha ,
as
we
a e
no
aking
in o
accoun
he
hyd odynamic
in e ac ion
be ween
he
pa icles,
he
igid
chain
limi
p essu e
enso
g ows
simply
as
he
hi d
powe
o
he
chain
leng h.
This
in e ac ions
a e
he
esponsible
o
he
loga i hmic
e m
ha
should
also
appea
when
dealing
wi h
a
long
s aigh
line
o
sphe es
[27],
bu
which
is
less
impo an
o sho e
chains.
138
CHAPTER
IV.
AGGREGATION PHENOMENA
Mo eo e ,
om
his
exp ession
we
can
also
ob ain
he
con ibu ion
o
he
chain
o
he
iscosi y
enso ,
'Ip
IIp
=
-2'1p
:
13
(3.1.21)
and
(3.1.22)
He e
S
is
a
symme ic
enso
such
ha
Sijkl
=
1/2(6ik6jl
+
6iI6jk).
As
he
ex e nal
magne ic
ield
ha
we
a e
applying
is
such
ha
J1.
--4
00,
he
chain
will
be
always
o ien ed
in
he
di ec ion
o
he
ield
and
smoo hly
ib a es
and
oscilla es
a ound
his
o ien a ion.
In
his
case,
he
symme ies
o
he
luid
low
essen ially
de e mine
he
cha ac e is ics
o
bo h
he
p essu e
and
he
iscosi y
enso .
Thus,
as
he
chain
is
o ien ed
along
he
z-axis,
i
he
low
ield
a e
3
is
diagonal,
he
p essu e
enso
will
also
be
diagonal,
i.e.
i
will
be
a
symme ic
enso .
On
he
o he
hand,
we
will
ha e,
in
gene al,
bo h
a
symme ic
and
an
an isymme ic
con ibu ion
o
a
non-diagonal
low
a eo
Le
us
conside
so ne
pa icula
cases:
i)
Flow
h ough
a
po e
o ien ed
along
he
same
di ec ion
as
he chain.
In
hís
case
he
low
a e
13
has
he
ollowíng
o m:
(3.1.23)
whe e
{3
is
he
low
a e
s eng h,
and
he
sys e n
has
a
o a ional
symme y
a ound
he
z-axis.
The
chain
in
hese
condi ions
is
s e ched
by
he
elonga ional
low
gi ing
ise
o
he
so-called
eloga ional
iscosi y
de ined
as
ollows
_
IIzz
-
1I.u
=
11' lod3
(N2
_
1)
{2N
�
(1
_
_3_)}
2{3
8V
+
3>'
18>.·
(3.1.24)
Fo
a
luid
low
wi h
o a ional
sy nme y
a ound
he
x
(o
y)
axis,
we
will
ha e
wo
di e en
con ibu ions
coming
om
he di e ences
_
1I.u
-
IIzz
=
1 7]od3
(N2
_
1)
{N
�
(1
_
_3_)}
2{3
8V
+
3>'
18>"
(3.1.25)
3.
Chaíns
o
magne ic
pa icles
in
an
elonga ional
low
139
(3.1.26)
Mo eo e ,
in
bo h
cases
we
ha e
a
con ibu ion
coming
om
he
ace
o
he
p essu e
enso ,
which
o
he
low
a e
gi en
by
Eq.
(3.1.23)
eads
(3.1.27)
and
o
he
low
wi h
o a ional
symme y
a ound
he
x
axis
is
T TI
=
1 ¡od3
(N2
_
1)
{N
_
2.
(1
_
�)}
6/3
8V
3>'
18>.·
(3.1.28)
ii)
Plana
elonga ional
low.
This
low
can
be
gene a ed
by
ou
o a ing
in ini e
cylinde s.
I
we
loca e he
cylinde s
such
ha
he
low
a e
is
again
a
diago
nal
ma ix,
he
p essu e
enso
will
be
diagonal
and
symme ic
in
iew
o
Eq.
(3.1.20),
i.e.
o
3=/3(���
)
O O
-1
(3.1.29)
we
will
ind
a
simila
beha io
o
he
sys em
as
he
one
desc ibed
aboye.
On
he
o he hand
i
we
o a e
he
ou
cylinde s
450,
i.e.
o
3=/3(�
�
�)
1
O
-1
(3.1.30)
he
p essu e
enso
has
bo h
a
symme ic
and
an
an isymme ic
pa .
Rela ed
o
hese
pa s,
we
will
ind
no
only
elonga ional
iscosi ies
bu
al
so
a
shea
iscosi y
and
a
o a ional
iscosi y
gi en
by
(.)
_
_
_
_
1 Tlod3
2
_
{
2.
(
_�)}
TI.,z
-
2 ¡/3
-
2
16V
(N
1)
N
+
3>'
1
18>'
/3,
(3.1.31)
(a)
_
_
_
_
1 ¡od3
(N2
_
)
{
_
_!_
(
_�)}
TI.,z
-
2 ¡ /3
-
2
16V
1
N
3>'
1
18>'
/3,
(3.1.32)
whe e
we
iden i y
he
shea and
o a ional
iscosi ies
140
CHAPTER
IV. AGGREGATION
PHENOMENA
(3.1.33)
(3.1.34)
4
Conclusions
This
chap e
is
in ended
as
a
p elimina y s udy
o
he
agg ega ion
phenomena
aking
place
in
sys ems
o
magne ic
pa icles
in
suspension
and
o
he
esul ing
s uc u es.
Ou
i s
poin
has
been
o
elucida e
he
in luence
o
he
hyd odynamic
in e ac
ions
(HI)
in
he
kine ics
o
he
agg ega ion
p ocess.
We
ha e
ex ended
he
classic
Smoluchowski
heo y
o
coagula ion
o
accoun
o
he
p esence
o
HI
ocu ing
when
one
goes
beyond
he
dilu e
egime.
Such
in e ac ions
ac
be o e
he
di e en
pa i
eles
a i e
a
he
sphe e
o
in luence
o
a
gi en
pa icle.
We
ha e
ob ained
he
kine ic
equa ions
o
he
agg ega ion
p ocess
and
om
hem
we
ha e
analyzed
he
clus e
o 
ma ion.
Ou
main
conclusion
is
ha
he
p esence
o
HI
slows
down he
agg ega ion
p ocess.
As
a
second
p oblem
we
ha e
s udied
he
dynamics
o
a
chain
o
magne ic
pa icles
unde
he
in luence
o
an
ex e nal
elonga ional
low.
In
pa icula ,
we
ha e
compu ed
he
con ibu ion
o
he
chain
o
he
p essu e
enso
o
he
sys em
om
he
heological
equa ion
o
s a e
p oposed
by
K ame s.
F om
his
quan i y
we
ha e
ob ained
he
co ec ion
o
he
iscosi ies
due
o
he
p esence
o
dipola
in e ac ions.
These
p elimina y
esul s
will
cons i u e
he
subjec
o
u u e
wo k.