7682 | So Ma e , 2021, 17, 7682–7696 This jou nal is © The Royal Socie y o Chemis y 2021
Ci e his: So Ma e , 2021,
17, 7682
Ac i e bina y swi ching o so colloids: s abili y
and s uc u al p ope ies
Michael Bley,
a
Joachim Dzubiella*
ab
and A u o Moncho-Jo da
´*
cd
We employ eac i e dynamical densi y unc ional heo y (R-DDFT) and eac i e B ownian dynamics
(R-BD) simula ions o s udy he non-equilib ium s uc u e and phase beha io o an ac i e dispe sion o
so Gaussian colloids wi h bina y in e ac ion swi ching, i.e., we conside a one-componen colloidal
sys em in which e e y pa icle can indi idually swi ch s ochas ically be ween wo in e ac ion s a es
(he e, sizes ‘big’ and ‘small’) a p ede ined a es. We conside he in luence o swi ching ac i i y on he
inhomogeneous densi y p o iles o he colloids con ined by a ious ex e nal po en ials, as well as on
hei pai s uc u e and phase beha io in bulk solu ions. Fo he la e , we ex end he R-DDFT me hod
o inco po a e he Pe cus es -pa icle ou e. Ou esul s demons a e ha swi ching ac i i y s ongly
modi ies he s eady-s a e densi y p o iles and s uc u al (pai ) co ela ions. In pa icula , he swi ching
a e in e pola es om a nea -equilib ium bina y colloidal mix u e o wo s a es a e y low a es o a
non-equilib ium, ‘one-s a e liquid’ a e y high a es cha ac e ized by one, a e age in e ac ion size. The
la e limi can be desc ibed by an equi alen effec i e one-componen (EOC) equilib ium sys em, o
which he exac analy ical exp ession o he effec i e pai po en ial is a diffusion-weigh ed
supe posi ion o he ac i e sys ems’ pai po en ials. This leads o he in e es ing ac ha unde ce ain
condi ions an in e ac ing swi ching sys em can beha e like a non-in e ac ing (ideal) gas in he limi o
high swi ching a es. Mo eo e , o colloids ha a e uns able (i.e., demix) nea equilib ium, we
demons a e ha phase sepa a ion and mic o-clus e ing in bo h con inemen and bulk can be
dynamically con olled by he swi ching a e, and anish o high a es. All R-DDFT esul s a e in
excellen ag eemen wi h ou R-BD simula ions.
1 In oduc ion
Ac i e ma e sys ems a e usually de ined as collec ions o
pa icles con aining in e nal deg ees o eedom wi h he abili y
o ake in and dissipa e ene gy and, in he p ocess, execu e
sys ema ic mo emen . Examples o ac i e so ma e sys ems
a e sel -p opelled nanopa icles,
1,2
ac i e B ownian pa icles,
3–6
ac i e con ac ile biopolyme s such as myosin II mo o s ac ing
on ac in ilamen s inside he cy oskele on o li ing cells,
7,8
o
biological sys ems such as bac e ia.
9
These non-equilib ium
sys ems ha e d awn he a en ion o he so ma e scien i ic
communi y in he ecen yea s due o he e y ich dynamic and
phase beha io .
10
By con inually consuming ene gy, hey
ci cum en he laws o equilib ium he modynamics, leading
o s eady s a es ha depend on kine ic pa ame e s.
11
Biological ac i i y, in pa icula media ed h ough uel-d i en
changes o molecula p ope ies and con o ma ions, has been
made esponsible o liquid–liquid phase sepa a ion and
condensa ion in cells, wi h la ge implica ions o physiology and
disease.
12,13
Li ing cells con ain dis inc sub-compa men s o
acili a e spa io empo al egula ion o biochemical eac ions
whe e ansien mic os uc u ing is key o unc ion. Recen ly,
no el sys ems ha e been designed o achie e p og ammable
ansien con o ma ional s a es ueled by chemical signals wi h
a con olled li e ime.
14–16
All hese applica ions can be included
in a mo e ambi ious p ojec o disco e ing sup amolecula
sys ems wi h non-equilib ium ansien mo phologies o
designing u u e ac i e, adap i e and au onomous ma e ials.
17
The mic oscopic o igins and ea u es o non-equilib ium
s uc u ing, howe e , a e no well unde s ood. Theo e ical
amewo ks o in e ac ing eac ion-di usion sys ems ha e been
linked so a only o mic os uc u ing dynamics o non-ac i e
sys ems d i en by chemical eac ions
18–21
o i us in ec ions.
22,23
In his wo k, in con as o he well-s udied mo ile
ac i i y,
11,24
we ocus on a diffe en kind o ac i e sys em
a
Physikalisches Ins i u , Albe -Ludwigs-Uni e si a
¨ F eibu g,
He mann-He de S aße 3, D-79104 F eibu g, Ge many.
E-mail: joachim[email p o ec ed]g.de
b
Resea ch G oup o Simula ions o Ene gy Ma e ials, Helmhol z-Zen um Be lin
u
¨ Ma e ialien und Ene gie, D-14109 Be lin, Ge many. E-mail: [email p o ec ed]
c
Depa amen o de Fı
´sica Aplicada, Uni e sidad de G anada,
Campus Fuen enue a S/N, 18071 G anada, Spain
d
Ca los I Ins i u e o Theo e ical and Compu a ional Physics, Facul ad de Ciencias,
Uni e sidad de G anada, Campus Fuen enue a S/N, 18071 G anada, Spain
Recei ed 6 h May 2021,
Accep ed 23 d July 2021
DOI: 10.1039/d1sm00670c
sc.li/so -ma e -jou nal
So Ma e
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o med by colloids in which each indi idual pa icle can
ac i ely swi ch be ween wo diffe en s a es a some speci ic
kine ic a e. These s a es, o example, can diffe in he
pa icle con o ma ions and hus ha e a diffe en in e ac ion
size. Such a sys em may ep esen a good model o
mimicking he beha io o so ac i e hyd ogels o esicles
swi ching (o ‘b ea hing’) be ween wo s a es
14,25–27
o espon-
si e, con o ma ionally swi ching biopolyme s.
12,28,29
Recen
de elopmen s ha e p o ided also he oppo uni y o c ea e so
mic omachines wi h p og ammable mo phology.
30
We no e ha such a sys em could also be iewed as a bina y
mix u e o wo diffe en colloidal ypes, whe e each species
swi ches in o he o he a some speci ic kine ic a e. Rega ding
he expe imen al ealiza ion, howe e , his would in ol e mass
ans e and e e sible chemical eac ions be ween he colloidal
pa icles o om some ese oi . He e, we ha e in mind au onomous
so pa icles, like hyd ogels, esicles, o a i icial cells, e c.,
which a e in e nally ueled and hus can ac i ely change size o
shape indi idually. Fo anishing swi ching a es, such a sys em
is in equilib ium, and a wo-componen colloidal mix u e o
wo diffe en species A and B ep esen s he same sys em.
These bina y sys ems can also be uns able, i.e.,demixin
equilib ium, i pa icle ypes o s a es, A and B, a e incompa ible,
as well known o colloidal mix u es.
31
Fo in ini e a es we
will demons a e ha ou ac i ely swi ching sys em can be
mapped o an effec i e one-componen equilib ium sys em wi h
equi alen s uc u e.
In ou pape we conside in pa icula so colloids in e ac ing
h ough Gaussian pai po en ials wi h bina y swi ching be ween
wo sizes ( e e ed as ‘big’ (b) and ‘small’ (s) om now on). These
Gaussian pai po en ials ep esen a gene ic model o polyme s
and so colloidal hyd ogels
32–34
and cells.
35
To in es iga e
s uc u al ea u es o such an ac i e colloidal dispe sion, we make
use o a eac i e dynamical densi y unc ional heo y (R-DDFT)
p e iously used in simila p oblems
18–23,36
and sol e i in he non-
equilib ium s eady-s a e. To check o he quali y o he R-DDFT,
which makes mean- ield assump ions o spa io empo al
co ela ions, we complemen ou s udy wi h eac i e B ownian
dynamics (R-BD) compu e simula ions. These me hods allows
us o in es iga e he e ec s o ac i e swi ching on he inhomo-
geneous densi y p o iles o mix u es con ined in a ious ex e nal
po en ials, such as slab geome y and inside an sphe ical ca i y.
To s udy he phase beha io and s uc u e also o bulk sys ems,
we gene alize he R-DDFT o inco po a e he Pe cus es -pa icle
ou e.
31
We also conside bina y sys ems which a e uns able
and demix in equilib ium and in es iga e how hey espond o
swi ching in e ac ions.
We demons a e in ou wo k ha he swi ching a e has
d as ic effec s on colloidal s uc u e and phase beha io .
In pa icula , we show ha he a e in e pola es be ween a
sys em compa able o he co esponding equilib ium bina y
mix u e a low a es and a non-equilib ium effec i e ’one-s a e’
liquid o la ge a es, s ongly affec ing he s uc u e and
s abili y in bulk and con inemen . Impo an ly, we show ha
sufficien ly as swi ching impedes he phase sepa a ion o an
(in equilib ium) uns able luid, allowing he con ol o he
deg ee o demixing and local mic os uc u ing by uning he
ac i i y a e. The sys em also demons a es a high deg ee o
e sa ili y, as he in e ac ion pa ame e s can be chosen o
ob ain, o ins ance, a pu ely ideal effec i e sys em in he limi
o as swi ching a es, e en hough pa icle in e ac ions a e
non-negligible. The esul s a e in excellen ag eemen wi h he
eac i e BD compu e simula ions, which u he suppo all
ou da a and he high quali y o he R-DDFT app oach. Hence,
ou wo k desc ibes how ac i e sys ems o swi ching pa icles
modi y he inhomogeneous p ope ies in compa ison o non-
ac i e sys ems, and how his may be exploi ed o con ol he
s uc u e and phase beha io .
2 Theo y
We in es iga e an ac i e sys em o med by colloids in bina y
s a es, i.e., a colloid has ei he a big (b) o a small (s) size.
Pa icles o s a e b ha e he abili y o spon aneously con e
in o s a e s a some ixed a e k
bs
(uni s o ime
1
), and simila ly
pa icles o s a e s swi ch in o pa icles o s a e b a a e k
sb
,
b
Ð
kbs
ksb
s. This kind o swi ching bina y mix u es offe s he
possibili y o s udy he ole played by he swi ching ac i i y on
he non-equilib ium p ope ies.
Wi h he pu pose o cha ac e izing he effec o he
ac i i y on all hese p ope ies, we make use o a well-known
model sys em o med by a bina y mix u e o so
Gaussian colloids, de ined by he ollowing pai in e ac ion
po en ials
37
bu
ij
=e
ij
e

2
/s
ij
2
wi h i,j= s,b, (1)
whe e is he in e pa icle dis ance, b=1/k
B
T(k
B
is he
Bol zmann cons an and T he absolu e empe a u e), e
ij
40
deno es he s eng h o he i–jpai in e ac ions, and s
ij
ep e-
sen s he ange (we will deno e s
bb
and s
ss
by s
b
and s
s
espec i ely, o simpli y no a ion). These so pai po en ials
emain ini e o any in e pa icle dis ance, so pa icles can
in e pene a e each o he .
2.1 S a iona y composi ions o swi ching pa icles
Le i s conside he ime e olu ion o he pa icle concen a ions
o a homogeneous sys em (bulk) in which s a es b and s swi ch
one in o each o he , b "s. We conside he bina y sys em
o med by N
b
pa icles o s a e b and N
s
pa icles o s a e s
con ained wi hin a olume V,a ixed empe a u eT.Thebulk
numbe densi ies o species o each s a e a e gi en by
i
=N
i
/V
(i= b,s). We deno e he o al numbe densi y by
T
=
b
+
s
,and
he mola ac ions by x
s
=
s
/
T
and x
b
=1x
s
. The i s -o de
diffe en ial equa ions ha ule he swi ching kine ics can be
w i en as
d b
d ¼ksb skbs b;d s
d ¼kbs bksb s:(2)
In each case, he i s e m o he igh hand ep esen s he
p oduc ion, whe eas he second one he disappea ance o his
componen . These coupled diffe en ial equa ions a e analy ically
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sol able, s a ing om he ini ial condi ions
b
( =0)=
b0
and
s
( =0)=
s0
, leading o he ollowing ime dependen concen-
a ions
bð Þ¼ 1
Kð b0kbs  s0ksbÞeK þksb T0
½
sð Þ¼ 1
Kð s0ksb  b0kbsÞeK þkbs T0
½;
8
>
>
<
>
>
:
(3)
whe e Kk
bs
+k
sb
and
T0
=
b0
+
s0
. Du ing he whole p ocess,
he o al numbe o pa icles in he sys em is p ese ed, i.e.
b
( )+
s
( )=
b0
+
s0
=
T
. The composi ion x
i
a ies exponen ially wi h
ime a he same a e o inally each he s a iona y s a e, in which
heconcen a ionsbecomecons an .The inals eady-s a e
composi ions a e gi en by x
b
( -N)=k
sb
/(k
bs
+k
sb
)and
x
s
( -N)=k
bs
/(k
bs
+k
sb
). In pa icula , i he ini ial
concen a ions a e adjus ed o ul ill he condi ion k
bs
/k
sb
=
s0
/
b0
=x
s
/x
b
, hen he sys em is o iginally s a iona y a =0and he
in eg a ion o he kine ic equa ions (eqn (2)) leads o a ime-
independen solu ion, in which he composi ions a e cons an :
x
b
( )=x
b0
and x
s
( )=x
s0
.
2.2 Reac i e dynamical densi y unc ional heo y (R-DDFT)
2.2.1 Basic o malism. We deno e u
ex
i
( )(i= b,s) as he
ex e nal po en ials ac ing on he big and small colloids a
posi ion . These po en ials can be caused by applied ex e nal
o ces (such as elec os a ic o g a i a ional ields) o simply
ep esen he effec o con ining walls. In he absence o
swi ching ac i i y, he ime-dependen densi y p o iles o he
colloids affec ed by hese ex e nal ields, {
i
( , )}, can be p e-
dic ed by he dynamical densi y unc ional heo y (DDFT),
which ep esen s an adap a ion o he classical equilib ium
densi y unc ional heo y o luids o B ownian pa icles o
non-equilib ium condi ions.
38,39
Acco ding o DDFT, he ime
e olu ion o he ensemble a e age densi y p o iles obeys he
ollowing con inui y equa ion
40,41
@ ið ; Þ
@ ¼ Jiwi h i¼b;s;(4)
whe e he ne luxes a e gi en by
J
i
=D
i
[
i
( , )+
i
( , )b (u
ex
i
( )+m
ex
i
( , ))], (5)
wi h i=b,s.He e,D
i
ep esen s he diffusion cons an o
componen i(which is assumed o be independen on he
speci ic loca ion o he pa icles), and mex
ið ; Þ¼dFex½ ið ; Þg
d ið ; Þ.
F
ex
[{
i
( , )}] is he equilib ium excess ee ene gy unc ional wi h
he equilib ium densi y p o iles eplaced by he non-equilib ium
ones
i
( , ).
The equilib ium and non-equilib ium p ope ies o so
Gaussian pa icles desc ibed by eqn (1) a e well ep esen ed by
a weakly co ela ed mean- ield luid o e a su p isingly wide
densi y and empe a u e ange, being mo e accu a e by inc easing
pa icle densi ies.
33
Themean- ield eeene gy unc ional o
colloidal mix u es o wo s a es, i=b,s, eadsas:
Fex½ ið Þg ¼ 1
2X
i;j¼b;sðð ið Þ jð 0Þuij ðj  0jÞd d 0:(6)
Using eqn (6) we ind he mean- ield non-equilib ium excess
chemical po en ial
mex
ið ; Þ¼X
j¼b;sð jð 0; Þuijðj  0jÞd 0:(7)
S a ing om a non-equilib ium ini ial s a e, he ime-
dependen densi y p o iles, {
i
( , )}, e ol e owa ds he inal
equilib ium dis ibu ion, {
eq
i
( )}, in which he diffusi e luxes
o bo h species become ze o a any poin o he space, J
i
=0.
Howe e , he DDFT me hod desc ibed be o e is es ic ed o
he case o non-ac i e sys ems. The ques ion ha na u ally
a ises a his poin is how he DDFT o mula ion can be
gene alized o ac i e sys ems, in which each pa icle swi ches
be ween wo s a es b and s a some p ede ined a es. In his
case, he ime e olu ion o he pa icle concen a ions is no
only caused by he diffusi e luxes {J
i
}, as hey do no accoun
o he p oduc ion and disappea ance o each componen due
o he swi ching ac i i y, p o ided by eqn (2). This p ocess
occu s locally, so he con e sion a e o big colloids in o small
ones and ice e sa only depends on he local concen a ions o
bo h species. The e o e, he classical DDFT amewo k has o
be ex ended o conside his new effec . This can be achie ed by
including in o he con inui y equa ion (eqn (4)) new e ms o
accoun o he p oduc ion and disappea ance o pa icles due
o ac i e swi ching, which ollow he kine ic equa ions (eqn (2)).
The esul ing heo e ical amewo k, called eac i e dynamical
densi y unc ional heo y (R-DDFT), has been success ully
applied o simila p oblems.
18–21,36
Acco ding o R-DDFT, he
ime e olu ion o he densi y p o iles o big and small so
colloids is gi en by he ollowing se o di e en ial equa ions:
@ bð ; Þ
@ ¼ Jbþksb sð ; Þkbs bð ; Þ
@ sð ; Þ
@ ¼ Jsþkbs bð ; Þksb sð ; Þ
8
>
>
>
<
>
>
>
:
;(8)
Eqn (8) oge he wi h eqn (5) and (7) ep esen a closed se o
equa ions o p edic he ime e olu ion o he densi y p o iles
o a mix u e o swi ching Gaussian pa icles.
I he ex e nal po en ials u
ex
i
( ) a e ime independen , hen
he sys em e ol es in ime un il a s eady s a e is e en ually
eached. Howe e , his inal (ac i i y-p esen ) s a e does no
imply ha he ne luxes a e ze o any mo e. Ins ead, he s eady-
s a e densi y p o iles (q
i
/q = 0) in he p esence o swi ching
ac i i y a e he solu ion o
Jb¼ksb sð Þkbs bð Þ
Js¼kbs bð Þksb sð Þ
(;(9)
so he ne diffusi e luxes a e balanced by he p oduc ion
and disappea ance o pa icles due o he swi ching ac i i y.
Since J
i
a0, his inal s eady egime eached a -Nis
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no an equilib ium s a e. In pa icula , he densi y p o iles
o any o he mac oscopic p ope ies as he p essu e will
be dependen on he dynamic p ope ies, namely D
s
,D
b
,k
bs
and k
sb
.
2.2.2 Swi ching ac i i y in s a iona y si ua ions. In he
ollowing we will assume ha he bulk concen a ions (o he
o al numbe o pa icles o s a e b and s o con ined sys ems)
sa is y he s a iona i y condi ion
bulk
s
bulk
b
¼xs
xb
¼kbs
ksb
:(10)
The e o e, he bulk p ope ies (and so he composi ion o he
mix u e o s a es) will emain cons an , e en hough he inhomo-
geneous p ope ies induced by he p esence o ex e nal po en ials,
con ining walls o by phase sepa a ion will s ill be affec ed by he
swi ching ac i i y. As eqn (10) in e connec s bo h kine ic a e
cons an s, we can use only one o hem, o ins ance k
bs
, o
cha ac e ize he swi ching a e. I can be w i en as he in e se
o he cha ac e is ic big- o-small con e sion ime, k
bs
=1/ .We
can compa e wi h he ypical diffusion ime o small pa icles,
0
=s
s2
/D
s
. We he e o e con enien ly de ine he swi ching
ac i i y as
a¼ 0
¼kbs ss
2
Ds
:(11)
Fo a{1, he b "scon e sion a eissoslow ha he ime
e olu ion o he densi y p o ilesisdomina edby hediffusion.In
his case, any swi ching e en a some speci ic loca ion is apidly
compensa ed by diffusi e luxes o pa icles ha balance he effec
o he ac i i y. In he limi a-0 he classical (equilib ium) DFT o
a ue bina y mix u e is eco e ed. In his pa icula case, he
densi y p o iles con e ge o he equilib ium dis ibu ion o -
N.Fo ac1 he exchange a e is so la ge ha he diffusion is
no as enough o compensa e i s effec s, so swi ching e en s
domina e o e diffusion. Thus, in he inal s eady-s a e
he pa icle s a es b and s can no be dis inguished anymo e
because hey do no ha e enough ime o diffuse and eo ganize
acco ding o he applied ex e nal po en ials. Consequen ly, in
e e y poin o he space we ha e ha bo h s eady-s a e densi y
p o iles sha e he same shape. We elabo a e on his in he nex
subsec ion.
2.2.3 The effec i e one-componen (EOC) equilib ium
sys em desc ibes ac1. The mic os uc u e, i.e., all he
s eady-s a e adial dis ibu ion unc ions, con e ges o he
same unc ion o he in e pa icle sepa a ion, g
ij
( )-g
eff
( ),
o ac1. The e o e, we can de ine an effec i e pai po en ial
lim
a-N
u
ij
( )=u
eff
( ) o an equi alen equilib ium sys em ha
desc ibes he non-equilib ium ac i e sys em in he limi ac1.
Fo he same eason, he ex e nal po en ial ac ing on
bo h species may also be exp essed by an effec i e po en ial
lim
a-N
u
ex
i
( )=u
ex
eff
( ). I can indeed be shown ma hema ically
(see Appendix) om he R-DDFT equa ions ha he effec i e
ex e nal po en ial can be w i en as an a e age o he indi idual
ex e nal po en ials
36
uex
e ð Þ¼Dbxbuex
bð ÞþDsxsuex
sð Þ
DbxbþDsxs
:(12)
Simila ly, he effec i e in e pa icle pa icle pai po en ial
akes he ollowing analy ical o m
ue ð Þ¼Dbxb
2ubbð ÞþðDbþDsÞxbxsubsð ÞþDsxs
2ussð Þ
DbxbþDsxs
:
(13)
These a gumen s show ha he mic os uc u e in he limi
a-Nac ually co esponds o he one o an effec i e one-
componen sys em (EOC) in equilib ium.
I is impo an o emphasize he e some impo an poin s.
Fi s , eqn (12) and (13) o he effec i e ex e nal and in e pa icle
in e ac ions o he EOC equilib ium luid a e only co ec ly
de ined in he limi o e y as ac i i y a es, ac1. I we a e
no in his limi , he non-equilib ium sys em canno be mapped
on o an non-ac i e effec i e one-componen luid. Second, u
eff
( )
depends on he assump ions made o he excess ee ene gy in
he DFT. The analy ical o m p o ided by eqn (13) ep esen s
a pa icula esul o he bina y mean- ield luid. Diffe en
p esc ip ions o F
ex
will lead o diffe en exp essions o he
effec i e pai po en ial. Finally, u
ex
eff
( )andu
eff
( ) depend on he
dynamic p ope ies o he sys em. I we change he diffusion
coefficien s, he effec i e pai po en ial will be diffe en oo, and
so he s eady-s a e densi y p o iles. Consequen ly, p ope ies
such as he bulk p essu e o he luid, ob ained ia in eg a ion
o he comp essibili y equa ion, will depend on he pa icle
diffusi i ies, which is a clea signa u e ha he ac i e sys em
is no in equilib ium o aa0, e en hough i eaches he
s eady-s a e o -N.
2.3 R-DDFT o Pe cus’ es pa icle ou e
Fo a bina y mix u e o big and small colloids in e ac ing wi h
pai po en ials ha only depend on in e pa icle dis ance, , he
mic os uc u e is ully de e mined by he big–big, big–small
and small–small ime-dependen adial dis ibu ion unc ions
g
bb
( , ), g
bs
( , ) and g
ss
( , ) o he homogeneous bulk suspension
(wi hou con ining ex e nal po en ial).
In o de o calcula e g
ij
( ), we need o gene alize he R-DDFT
amewo k. Fo his pu pose, we i s make use o he Pe cus
es pa icle ou e.
42,43
Wi hin his me hod, in p inciple one
should sol e he R-DDFT equa ions in he p esence o a es
pa icle o s a e iloca ed a he o igin = 0 ha ac s as an
ex e nal po en ial o he mix u e. The densi y p o iles o
pa icle o s a e ja ound his cen al pa icle,
ij
( , ), no mal-
ized by he co esponding bulk densi y
bulk
j
, p o ide he adial
dis ibu ion unc ion, g
ij
( , )=
ij
( , )/
bulk
j
. Howe e , his
p ocedu e s ill lacks a e y impo an ea u e o he eal ac i e
swi ching mix u e: he ex e nal po en ial exe ed by he cen al
pa icle is no ixed, bu i luc ua es due o he swi ching o he
cen al colloid.
To in oduce he swi ching o he ex e nal po en ial, he
wo-s a e R-DDFT amewo k mus be ex ended o a ou -s a e
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R-DDFT ha also akes he ansi ion p obabili y be ween he
wo possible s a es o he luc ua ing cen al pa icle in o
accoun . We deno e
p
ij
( , ) as he p obabili y densi y o inding
a pa icle o s a e jloca ed a a dis ance om a cen al pa icle
o s a e ia ime , and de ine he he ec o
p
( , )as
p
( , )=(
p
bb
( , ),
p
bs
( , ),
p
sb
( , ),
p
ss
( , )). (14)
Analogously, we can de ine he ec o including he ou
cu en s
J=(J
bb
,J
bs
,J
sb
,J
ss
). (15)
In he con ex o he mean- ield app oxima ion, he diffusi e
cu en s a e gi en by
Jij ¼Dj p
ijð ; Þ
Dj p
ijð ; Þb uijð Þþ X
k¼b;sð ikð 0Þukjðj  0jÞd 0
"#
:
(16)
The ime e olu ion o he p obabili y densi ies
p
ij
( , ) obeys
he ollowing se o ou coupled Fokke –Planck equa ions,
which include pa icle in e ac ions (modeled h ough he
mean- ield app oach)
44–46
@
@ pð ; Þ ¼   JþW pð ; Þ:(17)
The ansi ion a e ma ix W akes in o accoun no only he
swi ching be ween bo h kind o colloids (b and s) dis ibu ed
a ound he cen al po en ial, bu also he swi ching o he
luc ua ing ex e nal po en ial in i sel . I is gi en by
W¼
2kbs ksb ksb 0
kbs ðkbs þksbÞ0ksb
kbs 0ðkbs þksbÞksb
0kbs kbs 2ksb
0
B
B
B
B
B
B
@
1
C
C
C
C
C
C
A
(18)
Wsa is ies he equi ed p ope ies o any ansi ion ma ix,
namely Wij 0iajand P
i
Wij ¼08j.
47
2.4 Bounda y condi ions and nume ical de ails
To sol e he R-DDFT (eqn (8)), h ee bounda y condi ions a e
equi ed. Fi s , he ini ial densi y p o iles ( o = 0) ha e o be
speci ied:
i
( , =0)=
i0
( ). The second and hi d condi ion
in ol e he knowledge o he numbe densi ies in wo diffe en
loca ions. Fo bina y ac i e mix u es con ined be ween wo
plana walls sepa a ed by a dis ance L, hese condi ions impose
ze o diffusi e luxes a bo h con ining walls: J
i
(z=0, )=J
i
(z=L, )=0.
Fo mix u es con ined inside a sphe ical ca i y o adius R,we
ge J
i
( =0, ) = 0 due o sphe ical symme y a he cen e o he
ca i y and J
i
( =R, ) = 0 (closed con ining sphe ical wall).
Finally, o in eg a e he ou -s a e R-DDFT (eqn (17)), we also
ha e he condi ion J
ij
( =0, ) = 0 (sphe ical symme y), whe eas
he hi d condi ion imposes ixed p obabili y densi ies a away
om he cen al pa icle -N, gi en by
p
bb
(N, )=k
sb2
/(k
bs
+
k
sb
)
2
,
p
ss
(N, )=k
bs2
/(k
bs
+k
sb
)
2
and
p
bs
(N, )=
p
sb
(N, )=k
bs
k
sb
/
(k
bs
+k
sb
)
2
. In his way, he composi ion o he mix u e in he
bulk is p ese ed.
In o de o nume ically in eg a e he R-DDFT and he
s ochas ic R-DDFT equa ions, we use a spa ial g id o Dz=
D =0.01s
s
. The ime s ep has been ixed o D =10
5
0
,whichis
small enough o a oid he appea ance o nume ical ins abili ies
(D oDz
2
/(2D
s
)). Fo he case o he calcula ion o he adial
dis ibu ion unc ions, he in eg a ion ex ends o e a dis ance
la ge ha
max
=80s
s
o a oid ini e-size effec s, using
Fas Fou ie T ans o ms o e alua e he con olu ion in eg als
appea ing in eqn (16).
3 Reac i e B ownian dynamics (R-BD)
compu e simula ions
Complemen a y o he R-DDFT, all sys ems ha e been simula ed
using a eac i e B ownian dynamics (R-BD) algo i hm, which
includes ac i e swi ching be ween he wo diffe en s a es o
colloids, b and s. The o e damped Lange in equa ion o mo ion
o a pa icle iw i es
x
i
:
i
= U(
i
)+ ( ), (19)
whe e :
i
and
i
deno e eloci y and posi ion o he i- h pa icle,
he d ag coefficien x
i
and he diffusion coefficien D
i
a e ela ed
as D
i
=k
B
T/x
i
,and ( ) is he andom o ce ec o . The compo-
nen s o he andom o ce ec o ul ill he p ope ies hR
a
( )i=0
and hR
a
( )R
b
( 0)i=2x
i2
D
i
d
ab
d(  0) wi h aand bdeno ing he
spa ial dimensions, and dand d
ab
as Di ac and K onecke del a
unc ions, espec i ely. The i s e m on he igh hand side in
eqn (19) yields he o ce ac ing on he i- h pa icle
Fi¼ Uð iÞ¼ uex
ið iÞX
N
jai
uijð ijÞ;(20)
which consis s o he con ibu ion h ough he posi ion-
dependen ex e nal ield and he pai wise in e ac ions o all
o he neighbo s ound a dis ances
ij
. The posi ions o all N
pa icles a e upda ed using he Eule –Ma uyama p opaga ion
scheme,
48
which w i es
ið þD Þ¼ ið ÞþD
xi
Fiþffiffiffiffiffiffiffiffiffiffiffiffiffi
2DiD
p i;(21)
whe e D is he in eg a ion ime-s ep, which is 10
3
0
o sys ems
M3 and M4 (Table 1), and 10
4
0
o he sys ems M1 and M2,
espec i ely, and
i
is a ec o consis ing o andom alues
ollowing a s anda d no mal dis ibu ion.
Table 1 Main pa ame e s desc ibing he pa icle in e ac ions and con-
cen a ions o ou diffe en bina y Gaussian sys ems. Sys ems M1 o M3
a e s able mix u es in equilib ium, whe eas sys em M4 is loca ed inside he
uns able egion o he phase diag am in equilib ium
Sys em e
bb
e
ss
e
bs
s
b
/s
s
s
bs
/s
s
T
s
s3
x
s
D
s
/D
b
M1 2.0 2.0 2.0 2.0 1.5 0.191 0.5 2.0
M2 2.0 2.0 2.0 1.0 1.0 0.76 0.5 1.0
M3 2.0 2.0 1.0 1.504 1.277 2.4 0.75 1.504
M4 2.0 2.0 1.888 1.504 1.277 2.4 0.75 1.504
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The possibili y o ac i e swi ching be ween he pa icle s a es
is checked a e e e y in eg a ion s ep, and he p obabili ies o
swi ching a e de ined as
p
bs
=1e
k
bs
D
,p
sb
=1e
k
sb
D
(22)
whe e k
bs
=a/
0
and k
sb
=x
b
a/(x
s
s
s
). In he ac i e swi ching
R-BD amewo k, p ope ies o he i- h pa icle a e swi ched, i
he andom a ia e ollowing a uni o m dis ibu ion be ween
ze o and one p
i
( ) is below p
bs
i he pa icle is o s a e b o is
below p
sb
i he pa icle is o s a e s.
These R-BD simula ions o up o 8100 swi ching pa icles
ha e been conduc ed using an own code o p oduc ion imes
up o 25
B
and he pa ame e s p esen ed in Table 1. The box
sizes ha e been a ied o he bulk simula ions a he gi en
densi ies o educe ini e size effec s. Fo he cubic and sphe ical
sys em, cu -off dis ances o he pai wise in e ac ions ha e been
se o alues close o hal o he espec i e box size anging om
5.9 o 7.4s
s
o cubic box dimensions wi h edge leng hs 12 and
15s
s
, espec i ely. Fo he bulk simula ions o he sys ems M3
and M4, pe iodic bounda y condi ions (pbc) ha e been applied
on he cubic simula ion cells. In he simula ions o he mix u es
M1 and M2 con ined be ween wo walls, he pbc ha e been
main ained in he wo dimensions pe pendicula o he
sepa a ion ec o o he walls wi h box sizes l
x
=l
y
=10s
s
a a
sepa a ion dis ance l
z
=2.5s
s
,usingacu -offdis anceo 3.5s
s
.The
sphe ical ca i y has adius 6.0s
s
(same o hepo en ialcu -off).
4 Resul s and discussion
We explo e wo diffe en condi ions o con inemen (i.e.,
be ween wo pa allel plana walls and inside a sphe ical
compa men ) as well as bulk solu ions (wi hou ex e nal
po en ials). S a ing om an ini ial non-equilib ium con igu a ion,
he pa icle densi ies
i
( , ) e ol e in ime un il a inal s eady-
s a e egime is eached. This inal s a e depends on he swi ching
ac i i y a e, a. We use ou mean- ield R-DDFT o de e mine he
inhomogeneous s eady-s a e densi y p o iles o con ined colloids,
i
( ), and he ou -s a e R-DDFT o calcula e he s eady-s a e adial
dis ibu ion unc ions o he bulk suspension, g
ij
( ). In all cases,
heo e ical esul s a e compa ed wi h he R-BD simula ions.
We co e ac i i ies anging om a=0(non-ac i eequilib ium
s a e) o a= 1000 ( as swi ching a e egime), which yields esul s
compa able o a-N.
Table 1 shows he in e ac ion pa ame e s, numbe densi ies
and s a e composi ions o ou diffe en ac i e sys ems o
Gaussian colloids (sys ems M1 o M4) ha will be ma e o
in es iga ion. Sys ems M1 o M3 co espond o s able mix u es
( hey do no phase sepa a e in equilib ium), whe eas sys em M4
is loca ed inside he uns able egion o he phase diag am, so i
unde goes luid- luid demixing.
49
In all cases, he kine ic a e
cons an s ha e been chosen o ul ill he condi ion gi en by
eqn (10) o p ese e he composi ion o he mix u e o s a es.
The pa icle diffusi i ies a e in all cases assumed o ollow he
Eins ein ela ion, D
i
=k
B
T/(3pZs
i
), so D
b
=(s
s
/s
b
)D
s
.
In addi ion o he ac i e sys ems, we also explo e he effec i e
one-componen (EOC) in equilib ium, which is equi alen o he
ac i e sys em in he limi a-Nwhe e all ex e nal po en ials
and all pa icle–pa icle pai in e ac ions con e ge o he same
e ec i e po en ials, gi en by eqn (12) and by eqn (13), espec i ely.
4.1 Inhomogeneous p ope ies and mic os uc u e o s able
mix u es
4.1.1 Symme ic sli -po e con inemen . We i s in es iga e
he non-equilib ium s eady-s a e densi y p o iles o sys em M1
con ined in o a na ow sli po e. The o al a e age densi y and
composi ion o his sys em a e
T
s
s3
= 0.191 and x
s
=x
b
= 0.5.
We emphasize ha
T
is a ixed quan i y because he sli
ep esen s a closed sys em. The dis ance be ween bo h plana
walls is L= 2.5s
s
. In o de o con ine he colloids, he same
ex e nal auxilia y po en ial is applied o bo h componen s:
bu
ex
b
(z)=bu
ex
s
(z) = 10(e
50z/s
s
+e
50(Lz)/s
s
) (23)
o 0 ozoL, and u
ex
b
(z)=u
ex
s
(z)=N o zo0o z4L.
The heo e ical densi y p o iles a e ob ained s a ing a ime
= 0 om he equilib ium p o iles o he non-ac i e sys em (a=
0), and hen u ning on he ac i i y a 40. The local densi ies
e ol e apidly a sho imes and inally con e ge o he inal
s eady s a e, in which he effec o he ac i i y is exac ly
balanced by he diffusi e cu en s. Black solid and ed dashed
lines in Fig. 1 show he s eady-s a e p o iles o big and small
colloids o a= 0, 10, and 1000. As obse ed, ac i i y modi ies
he densi y p o iles nea each wall, e en hough he a e age
concen a ions o each componen emain unal e ed. The
adso p ion peak o big pa icles dec eases whe eas he concen-
a ion peak o small pa icles close o he wall shows an
inc ease coupled wi h a educ ion o he deple ion egion.
In addi ion o he heo e ical calcula ions, we pe o med
R-BD simula ions in he same condi ions o es ima e he
eliabili y o ou mean- ield R-DDFT me hod. Fig. 2(a) depic s
a ep esen a i e snapsho o sys em M1 con ined be weeen wo
plana walls o a= 0. The nea ly quan i a i e ag eemen
be ween he R-DDFT heo e ical p edic ions and he R-BD
simula ion da a ob ained in he s eady-s a e egime (depic ed
as hollow symbols in Fig. 1) indica es ha he eac i e mean-
ield DDFT is able o cap u e he non-equilib ium s uc u e o
he ac i e mix u e o s a es o he so colloids o any ac i i y
a e a. I should be ema ked ha he mean- ield app oach
used in ou heo e ical model (eqn (6)) compa es e y well o
he simula ion da a, e en o he small alues o he pa icle
numbe densi ies o sys em M1.
Fo e y la ge ac i i ies, he sys em inally ends o a s eady-
s a e in which he densi y p o iles o big and small pa icles
con e ge o each o he . This ac can be clea ly obse ed in
Fig. 1(c) whe e he densi y p o iles o a= 1000 a e shown.
As he ac i i y a e inc eases, pa icle swi ching en o ces a
dec ease o he concen a ion o big colloids and an inc ease
o small ones close o he walls. Fo a= 1000, bo h p o iles
con e ge o a common o m, hus indica ing ha he bina y
sys em beha es as an effec i e one-componen sys em (EOC) in
he limi o la ge swi ching ac i i y.
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This esul can be a ionalized as ollows: he swi ching
ac i i y in oduces in he sys em a delocaliza ion o he pa icle
s a es (big and small), in he sense ha a colloid o s a e b in
ce ain loca ion is suddenly con e ed in o a pa icle o s a e s.
I a{1 he swi ching e en s a e a e and he diffusi e cu en s
a e s ill able o p ese e he dis inc ion be ween bo h s a es.
Howe e , o ac1 many swi ching effec s occu du ing he
diffusi e ime
0
, so pa icles do no mo e as enough o
ea ange hei loca ion by diffusion. Consequen ly, hey end
o expe ience he same a e age in e pa icle effec i e in e -
ac ion (u
ij
( )-u
eff
( ) o a-N). In o de o con i m his,
we include in plo Fig. 1(c) he co esponding p edic ion om
he equilib ium EOC, whe e he colloids a e in e ac ing h ough
an a e age ex e nal and pa icle–pa icle pai po en ials gi en by
eqn (12) and (13), espec i ely (blue lines o heo y and blue
symbols o R-BD simula ions). The ag eemen be ween he
densi y p o iles ob ained o a= 1000 and he EOC con i ms
ha he ac i e sys em indeed con e ges o he EOC model in he
limi o la ge swi ching a es.
I is impo an o emphasize again ha he ac i e sys em in
he limi a-Ncanno be con ound wi h a uly equilib ium
sys em. Indeed, he effec i e po en ials u
eff
( ) and u
ex
eff
( )in
con as o he con en ional ones, include he diffusion
cons an s, which is a signa u e o he non-equilib ium na u e
o he unde lying mix u e. In ac , he componen wi h a la ge
diffusi i y con ibu es mo e o he a e age.
4.1.2 Asymme ic sli -po e con inemen . In o de o s udy a
con ined ac i e sys em in which big and small colloids in e ac
diffe en ly wi h he con ining walls, we sol e again he wo-
s a es R-DDFT diffe en ial equa ions using an asymme ic
ex e nal ield o small and big colloids in such a way ha
componen b is a ac ed o he le wall and epelled om he
igh one, whe eas componen s is epelled om he le and
a ac ed o he igh wall. In pa icula , we selec
buex
bðzÞ¼2e5z=ssþ2e5ðLzÞ=ssþbuauxðzÞ
buex
sðzÞ¼þ2e5z=ss2e5ðLzÞ=ssþbuauxðzÞ
8
<
:
;(24)
whe e u
aux
(z) is an auxilia y sho - ange po en ial included o
a oid pa icle pene a ion inside he walls, gi en by bu
aux
(z)=
10(e
100z/s
s
+e
100(Lz)/s
s
). Fo hese condi ions, he sys em
con ined inside bo h pla es is he one iden i ied in Table 1 as
sys em M2. This sys em has a symme ical composi ion, bo h
componen s ha e he same pa icle size (s
b
=s
s
=s
bs
) and
diffusion cons an s. Pa icles o he same s a e a e epelled,
whe eas pa icles o diffe en s a e a e a ac ed (e
bb
=e
bb
=
e
bs
= 2). This combina ion o pa ame e s ies o ep oduce
he ypical sc eened elec os a ic in e ac ions appea ing in a
bina y mix u e o opposi ely cha ged colloids con ined inside
wo cha ged elec odes.
Fig. 1 S eady-s a e densi y p o iles o big and small so colloids o
sys em M1 con ined wi hin a na ow plana sli (wall sepa a ion L=
2.5s
s
), o ac i i ies (a) a=0,(b)a= 10 and (c) a= 1000. Black solid and
ed dashed lines show R-DDFT p edic ions o
b
(z)and
s
(z), espec i ely.
Black squa es and ed iangles deno e he same p o iles calcula ed om
R-BD simula ions esul s. Blue lines and symbols a e R-DDFT and R-BD
esul s ob ained o he effec i e one-componen sys em (EOC).
Fig. 2 Snapsho s o B ownian dynamics simula ion. (a) Sys em M1
con ined inside a na ow sli o a= 0. (b) Sys em M4 con ined inside a
sphe ical compa men o a= 0 (equilib ium)and a= 100. Blue and ed
sphe es ep esen colloids in he big and small s a es, espec i ely.
(b) Sys em M4 in bulk (box wi h pe iodic bounda y condi ions) o a=0,
10 and 1000.
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Fig. 3 depic s he s eady-s a e densi y p o iles o he ac i e
M2 sys em o a= 0 (equilib ium), 10 and 1000. The acco dance
be ween heo y and simula ion is e y good in all cases.
Diffe ences a e only obse ed a ound he adso p ion peak close
o he a ac i e wall o each componen , whe e he heo y
o e es ima es he local densi y. This disc epancy can be a ibu ed
o he limi a ion o he mean- ield app oach when p edic ing he
s uc u e in egions wi h s ong densi y a ia ions.
Again, he s eady-s a e densi y p o iles in he egime o e y
as swi ching a es con e ge o he ones ob ained o he
co esponding EOC. Fo he pa icula choice o in e ac ion
pa ame e s o sys em M2, he esul ing effec i e pai in e -
ac ion po en ial and ex e nal po en ial a e gi en by u
eff
( )=0
and u
ex
eff
(z) = 0 (excluding he auxilia y sho - ange po en ial ha
p e en s wall pene a ion). In o he wo ds, in he limi o as
swi ching ac i i ies, ac1, pa icle in e ac ions a e dynamically
neu alized leading o a sys em ha effec i ely beha es as an
ideal sys em. This can clea ly app ecia ed in Fig. 3(c), in which
he densi y p o iles o bo h componen s end o a la dis ibu ion
ypical o a non-in e ac ing sys em. I is well known ha ideal
sys ems do no exis in na u e; hey a e an idealiza ion limi o
dilu ed and weakly in e ac ing sys ems. Consequen ly, hese
esul poin s ou a su p ising p ope y o po en ial p ac ical
in e es o ac i e swi ching sys ems: hey can be designed o ully
ep oduce he beha io o a eal ideal sys em, e en hough pa icle
in e ac ions a e non-negligible.
4.1.3 Bulk. The swi ching ac i i y no only has impo an
implica ions on he inhomogeneous densi y p o iles o con ined
sys ems, bu also on he local o de ing o he pa icles a ound
each o he , i.e. he mic os uc u e in bulk. Fo an homogeneous
bulk suspension, he mic os uc u e o he mix u e o s a es
is cha ac e ized by means o he pa ial adial dis ibu ion
unc ions, g
ij
( ), so ha
bulk
j
g
ij
( ) is he numbe densi y o
pa icle o s a e ja dis ance om a cen al colloid o s a e i
loca ed a = 0. Due o he condi ion gi en by eqn (10), bulk
concen a ions a away om he cen al pa icle ( -N)a e
no affec ed by ac i i y. Howe e , he shape o g
ij
( )canbe
signi ican ly modi ied close o he cen al pa icle o wo
easons: (1) swi ching e en s o he colloids a ound he cen al
pa icle; (2) swi ching e en s o he cen al pa icle in i sel ,
which beha es as a luc ua ing ex e nal po en ial o he es o
he mix u e due o he non-equilib ium ac i i y.
To explo e he ole o he swi ching ac i i y on he mic o-
s uc u e, we employed ou ou -s a e R-DDFT and R-BD
simula ions o de e mine g
ij
( ) o inc easing ac i i y a es. We
selec ed he sys em M3 o epulsi e Gaussian colloids shown in
Table 1. The in e ac ion ange o his sys em ep esen s ai ly
well he simula ion da a o mix u es o linea lexible polyme s
wi h a numbe o monome s N
m
=200(big)and100(small),
imme sed in a good sol en .
33
Since e
bs
oe
bb
=e
ss
, he eis
a dec ease o ene gy penal y by placing unlike species as
neighbo s, which in u n a o s mixing be ween bo h
componen s. In ac , mix u e M3 is a s able sys em ha does
no exhibi luid- luid demixing.
Lines in plo s (a)–(d) o Fig. 4 show he s eady-s a e adial
dis ibu ion unc ions (g
bb
( ), g
ss
( ) and g
ss
( )) ob ained om
ou heo y o ou di e en alues o he swi ching ac i i y a e,
namely a= 0 (equilib ium), 1, 10 and 1000. The co esponding
simula ed adial dis ibu ion unc ions in he s eady-s a e
egime a e shown as hollow symbols in Fig. 4. Excellen
ag eemen is ound be ween simula ions and heo e ical
p edic ions, hus con i ming ha inco po a ing he es -
pa icle ou e o he ou -s a es R-DDFT ep esen s a eliable
me hod o p edic he mic os uc u e o non-equilib ium ac i e
swi ching mix u es. Since he mean- ield app oach in ol ed in
ou model (see eqn (6)) becomes exac in he high densi y limi ,
his ag eemen is expec ed o imp o e e en mo e by inc easing
he bulk o al numbe densi y o he sys em,
T
.
All pai in e ac ions in sys em M3 a e pu ely epulsi e, and
hus he esul ing g
ij
( ) ha e a so co ela ion hole a small
in e pa icle dis ances, ypically obse ed in his kind o so
epulsi e po en ials. The hole is smalle o g
bs
( ) due o he
weake epulsion be ween big and small colloids. This so
co ela ion hole is g adually educed as
T
inc eases, a beha io
Fig. 3 S eady-s a e densi y p o iles o sys em M2 con ined wi hin a
na ow plana sli (wall sepa a ion L=2.5s
s
), o ac i i ies (a) a=0,(b)
a= 10 and (c) a= 1000. An ex e nal asymme ic po en ial is applied o bo h
species. Black solid and dashed lines ep esen heo e ical p edic ions o
b
(z)and
s
(z), espec i ely, ob ained wi h R-DDFT. Black squa es and ed
iangles show he same p o iles oba ined om R-BD simula ions. Blue
lines and symbols a e R-DDFT and R-BD esul s o he EOC.
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ypical o ini e co e po en ials, app oaching he ideal-gas-like
beha io in he high densi y limi . Fo a= 0, he h ee pa ial
adial dis ibu ion unc ions a e e y diffe en om each o he ,
bu inc easing he swi ching ac i i y leads o a p og essi e
app oach o he h ee dis ibu ion unc ions. In he limi o
la ge a, he h ee unc ions con e ge o he same common
p o ile, which in u n ma ches he co esponding g( ) o he
EOC (u
eff
( ) gi en by eqn (13)).
Fig. 4(e) depic s he heo e ical and simula ed numbe –
numbe s a ic s uc u e ac o o sys em M3 wi h a=0,a=
1000 and he EOC. I is de ined as S
NN
(q)=1+
T
h
ˆ
a e
(q), whe e
h
ˆ
a e
(q) is he Fou ie ans o m o he a e age dis ibu ion
unc ion, ha eð Þ¼P
i;j
xixjðgijð Þ1Þ. Fo a= 0, he co ela ion
wells o g
bb
( ) and g
ss
( ) close o = 0 a e deepe han he one o
g
bs
( ), whe eas o a= 1000 he h ee dis ibu ion unc ions
ha e con e ged o he same unc ional o m, lea ing he a e age
almos una ec ed. In con as o his beha io , we will show in
he ollowing sec ion ha S
NN
(q) shows impo an changes wi h
a o a sys em ha phase-sepa a es in equilib ium, so i becomes
an excellen indica o o he onse o luid– luid demixing.
4.2 Phase sepa a ion dynamically con olled by in e ac ion
swi ching
4.2.1 Sphe ical ca i y. We showed ha swi ching ac i i y
plays a e y impo an ole de e mining he inhomogeneous
p ope ies and mic os uc u e. These esul s sugges ha he
inhomogeneous dis ibu ion o a phase-sepa a ed mix u e
should be also deeply affec ed by he ac i i y.
36
The condi ions o s abili y and he phase diag ams o
epulsi e Gaussian mix u es ha e been ex ensi ely s udied
unde he mean- ield app oach.
33,49
We selec a pa icula
mix u e wi h he same in e ac ion pa ame e s han sys em
M3, bu shi ing he big-small in e ac ion o e
bs
= 1.888.
Al hough his change seems o be small, i has a e y ele an
impac o he phase beha io since he sys em now phase
sepa a es in o wo solu ions o diffe en composi ion abo e
he c i ical poin , loca ed a 
Tss
3¼1:647 and x
s= 0.7.
33,49
In o de o ensu e he phase sepa a ion o ou Gaussian
mix u e o s a es, we chose a o al numbe densi y gi en by
T
s
s3
= 2.4 and composi ion x
s
= 0.75 (sys em M4 in Table 1).
This bina y mix u e demixes spon aneously o a= 0, as shown
in he R-BD simula ion snapsho in Fig. 2(c).
To in es iga e he effec o he ac i i y on he phase
coexis ence, we apply he me hod desc ibed by A che ,
40,50
i.e.
we con ine he sys em inside a sphe ical ca i y o adius R
ca
=6s
s
by means o epulsi e sphe ically symme ic ex e nal po en ials
buex
ið Þ¼
Eið =RÞ10 R
1 4R
(i¼b;s(25)
whe e E
b
=E
s
=20andR=5s
s
. The esul ing s eady-s a e densi y
p o iles o a=0a eshowninFig.5(a)(lines o heo e ical
p edic ions ans symbols o R-BD simula ions). We obse e clea
signa u es o phase sepa a ion a equilib ium: big pa icles a e
mos ly adso bed close o he ex e nal wall o he ca i y, whe eas
small ones a e mainly dis ibu ed in he cen al egion o he
ca i y.Thispa icula seg ega iono pa iclesiscausedby he
exis ence o an effec i e a ac ion be ween he wall and he la ge
componen , which a ises because he epulsion induced by he
wall has a longe ange on he scale o he small pa icles, leading
Fig. 4 Plo s (a)–(d) s eady-s a e adial dis ibu ion unc ions g
bb
( ) (black line and squa es), g
ss
( ) ( ed dashed line and ci cles), and g
bs
( ) (g een dashed
line and iangles) o sys em M3 ob ained om sol ing he 4-s a es R-DDFT equa ions (lines) and om R-BD simula ions (symbols) in he bulk solu ion
o ac i i ies a= 0 (equilib ium), a=1,a= 10, and a= 1000. Blue squa es and lines in (d) ep esen he g( ) o he EOC om R-DDFT and R-BD,
espec i ely. Plo (e) numbe –numbe s uc u e ac o S
NN
(q) o a= 0, 10, 1000 and he EOC.
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