Active binary switching of soft colloids: stability and structural properties

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 PAPER Open Access A icle. Published on 29 July 2021. Downloaded on 9/2/2021 12:26:49 PM. This a icle is licensed unde a C ea i e Commons A ibu ion-NonComme cial 3.0 Unpo ed Licence. View A icle Online View Jou nal | View Issue This jou nal is © The Royal Socie y o Chemis y 2021 So Ma e , 2021, 17, 7682–7696 | 7683 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 Pape So Ma e Open Access A icle. Published on 29 July 2021. Downloaded on 9/2/2021 12:26:49 PM. This a icle is licensed unde a C ea i e Commons A ibu ion-NonComme cial 3.0 Unpo ed Licence. View A icle Online 7684 | So Ma e , 2021, 17, 7682–7696 This jou nal is © The Royal Socie y o Chemis y 2021 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 So Ma e Pape Open Access A icle. Published on 29 July 2021. Downloaded on 9/2/2021 12:26:49 PM. This a icle is licensed unde a C ea i e Commons A ibu ion-NonComme cial 3.0 Unpo ed Licence. View A icle Online This jou nal is © The Royal Socie y o Chemis y 2021 So Ma e , 2021, 17, 7682–7696 | 7685 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 Pape So Ma e Open Access A icle. Published on 29 July 2021. Downloaded on 9/2/2021 12:26:49 PM. This a icle is licensed unde a C ea i e Commons A ibu ion-NonComme cial 3.0 Unpo ed Licence. View A icle Online 7686 | So Ma e , 2021, 17, 7682–7696 This jou nal is © The Royal Socie y o Chemis y 2021 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 So Ma e Pape Open Access A icle. Published on 29 July 2021. Downloaded on 9/2/2021 12:26:49 PM. This a icle is licensed unde a C ea i e Commons A ibu ion-NonComme cial 3.0 Unpo ed Licence. View A icle Online This jou nal is © The Royal Socie y o Chemis y 2021 So Ma e , 2021, 17, 7682–7696 | 7687 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. Pape So Ma e Open Access A icle. Published on 29 July 2021. Downloaded on 9/2/2021 12:26:49 PM. This a icle is licensed unde a C ea i e Commons A ibu ion-NonComme cial 3.0 Unpo ed Licence. View A icle Online 7688 | So Ma e , 2021, 17, 7682–7696 This jou nal is © The Royal Socie y o Chemis y 2021 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. So Ma e Pape Open Access A icle. Published on 29 July 2021. Downloaded on 9/2/2021 12:26:49 PM. This a icle is licensed unde a C ea i e Commons A ibu ion-NonComme cial 3.0 Unpo ed Licence. View A icle Online This jou nal is © The Royal Socie y o Chemis y 2021 So Ma e , 2021, 17, 7682–7696 | 7689 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. Pape So Ma e Open Access A icle. Published on 29 July 2021. Downloaded on 9/2/2021 12:26:49 PM. This a icle is licensed unde a C ea i e Commons A ibu ion-NonComme cial 3.0 Unpo ed Licence. View A icle Online 7690 | So Ma e , 2021, 17, 7682–7696 This jou nal is © The Royal Socie y o Chemis y 2021 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. So Ma e Pape Open Access A icle. Published on 29 July 2021. Downloaded on 9/2/2021 12:26:49 PM. This a icle is licensed unde a C ea i e Commons A ibu ion-NonComme cial 3.0 Unpo ed Licence. View A icle Online