A model of the formation and mechanics of a colloidal membrane is obtained within the framework of the hydrodynamic theory of cross diﬀusion. Coupled equations describing advection, diﬀusion and cross-diﬀusion of spherical tracer particles within a colloidal suspension are solved numerically. A dynamic membrane is formed by advecting the colloidal particles toward a porous support; as the particles build up against the support they form a concentrated boundary layer; at suﬃciently high concentrations this layer behaves like a ﬁltration membrane. A tracer pulse is advected toward the steady-state colloidal membrane; the tracer particles are ﬁltered by the colloidal particles via excluded volume and hydrodynamic interactions, quantiﬁed by the colloidal reﬂection coeﬃcient. The ﬁltered tracer particles form a secondary concentrated boundary layer within the membrane. Decreasing the size of the tracer particles allows them to penetrate further into the colloidal membrane, and ﬁnally to pass through below a critical size threshold. 1 Introduction Ultraﬁltration is a membrane separation process used extensively in industry to purify water, process pharmaceuticals and concentrate milk and juices [1]. Ultraﬁltration also occurs in many physiological and natural settings [2,3]. In ultraﬁltration systems a colloidal suspension is advected toward a membrane or porous support that is impermeable to the colloidal particles, but through which the pore ﬂuid and molecular impurities can pass (ﬁgure 1). During the ﬁltration process the colloidal particles build up adjacent to the membrane forming a concentration polarization layer. The large colloid osmotic pressure in this layer can lead to detrimental eﬀects such as membrane fouling and ﬂux decline [4]. On the other hand if the colloidal particles exhibit membrane properties, the concentration polarization layer itself becomes a secondary dynamic ﬁltration membrane that can be used to remove tracer particles from the pore ﬂuid [5 –7]. Recent experimental studies demonstrate the excellent potential of dynamic colloidal membranes for water puriﬁcation applications [5,8]. The membranes form naturally, are relatively inexpensive to produce and can be easily removed and regenerated, mitigating the eﬀects of membrane fouling. A signiﬁcant advantage of such systems is that expensive ultraﬁltration membranes are not required, since the secondary dynamic membrane performs the ﬁltration. Inthisworkamodelofacolloidalmembraneisdevelopedwithinthecontextofthehydrodynamic theory of diﬀusion and cross-diﬀusion in suspensions [9,10]. In Section 2 the diﬀusive ﬂuxes and mass conservation equations are introduced and put in dimensionless form. In Section 3 the equations are solved for the case of a one-dimensional ﬁltration system with two diﬀerent sets of boundary conditions, corresponding to self-forming and pre-deposited dynamic membranes. Section 4 contains a discussion of the results and conclusions. 2 Mass conservation in binary suspensions Conservation of mass in two-component suspensions can be written as @n1 @t+urn1=rJ1and@n2 @t+urn2=rJ2; (1a,b) For God so loved the world, that He gave His only begotten Son, that whosoever believeth in Him should not perish, but have everlasting life. For God sent not His Son into the world to condemn the world; but that the world through Him might be saved. John 3:16-17 1 2 (a) (b)x U U Figure 1: Schematic of a self-forming dynamic colloidal membrane [5,7]. (a) A suspension of initial volume fraction 0 1of colloidal particles (grey) and 0 2of tracer particles (red) is placed adjacent to a membrane or porous support. The membrane is impermeable to the colloidal particles, but permeable to the tracer particles. (b) A ﬂow u=U^iis imposed, consolidating the colloidal particles against the support; the colloidal particles form a dynamic membrane that ﬁlters the tracer particles from the pore ﬂuid. whereniis the concentration (number of particles per unit volume) of component i,uis the volume-average velocity of the mixture and Ji=ni(uiu)is the diﬀusive ﬂux of irelative to u. In systems experiencing strong cross-diﬀusion eﬀects the ﬂuxes take the form J1=D11rn1D12rn2and J2=D21rn1D22rn2; (2a,b) whereD11andD22are the main Fickian diﬀusion coeﬃcients of the colloidal particles and tracer particles, and D12andD21are cross-diﬀusion coeﬃcients. Expressions for the Dijas functions of particle concentration have been obtained previously for hard-sphere suspensions [9 –11] and for polymer solutions [12]. Here the hard-sphere expressions are used, as summarized in the Appendix. 2.1 Governing equations Figure 1ashows a schematic of the system to be studied. At x= 0is a porous support or mesh that is permeable to the ﬂuid and tracer particles, but impermeable to the colloidal particles. Adjacent to the support is a colloidal suspension of initial concentration n0 1containing tracer particles at n0 2. Att= 0a volume ﬂow u=U^iis imposed and the colloidal particles are advected toward the porous support, forming a concentration polarization layer (ﬁgure 1 b). The boundary conditions atx= 0are that u1= 0andu2=U^i. In terms of the ﬂuxes Jithis gives J1=n1U^iand J2= 0 (x= 0): (3a,b) For the one-dimensional system in ﬁgure 1, conservation of mass equations (1) take the form @n1 @tU@n1 @x=@J1 @xand@n2 @tU@n2 @x=@J2 @x(x>0);(4a,b) and the ﬂuxes (2) become J1=n1(u1+U) =D11@n1 @xD12@n2 @x; (5) 3 and J2=n2(u2+U) =D21@n1 @xD22@n2 @x: (6) It will be convenient numerically to make the equations dimensionless. This can be accomplished by introducing the dimensionless lengths ^x=x=Land times ^t=tU=L ;dimensionless velocities ^ui=ui=U;volume fractions i=nivi;wherevi=4 3R3 iis the particle volume and Rithe particle radius; dimensionless ﬂuxes ^Ji=viJi=U; and dimensionless diﬀusivities ^Dij=Dij=D0 i;where D0 i=kBT=(60Ri)is the Stokes-Einstein diﬀusivity of a particle and 0is the viscosity of the pore ﬂuid. In terms of these quantities equations (4)–(6) take the dimensionless form @ 1 @^t=@ 1 @^x@^J1 @^x(^x>0); (7) @ 2 @^t=@ 2 @^x@^J2 @^x(^x>0); (8) where ^J1=1 Pe ^D11@ 1 @^x+3^D12@ 2 @^x ; (9) and ^J2=1 Pe 3^D21@ 1 @^x+^D22@ 2 @^x : (10) In equations (9) and (10) Pe=UL=D0 1is the Peclet number and =R2=R1=D0 1=D0 2is the particle size ratio. The boundary conditions (3) become ^J1= 1and ^J2= 0 (^x= 0): (11a,b) The far-ﬁeld boundary conditions for the system in ﬁgure 1 are 1! 0 1and 2! 0 2 (^x!1 ); (12a,b) where 0 i=n0 iviis the initial volume fraction of component iin the suspension. 3 Results 3.1 Self-forming dynamic membrane Figure2showstheresultsofasimulationofaself-formingdynamicmembrane, inwhichasuspension with initial concentrations 0 1= 0:1and 0 2= 0:001, size ratio= 0:7and Peclet number Pe= 20, is advected toward the porous support. The equations have been solved using the MatLab partial diﬀerential equation solver pdepe (Matlab ﬁles included as Supplementary Material). At early times ( ^t= 0:05and^t= 0:1), the colloid volume fraction increases rapidly against the support at ^x= 0(ﬁgure 2a), but the tracer particle concentration is relatively unchanged from it’s initial value 0 2(ﬁgure 2b). The colloidal suspension is still relatively porous at this stage and the tracer particles are able to pass through into the ﬁltrate. However, as the colloid is further consolidated (^t= 0:15), the volume fraction at ^x= 0approaches close packing and the tracer particles can no longer easily ﬁt through the pore space. At longer times ( ^t&0:2) the colloidal membrane almost completely blocks the tracer particles, and a large tracer polarization layer forms within the membrane. The inset to ﬁgure 2b shows the time evolution of the tracer concentration entering the ﬁltrate at ^x= 0. Once the dynamic membrane has formed ( ^t&0:1), the tracer concentration entering the ﬁltrate drops rapidly, approaching zero at longer times. 4 Sheet1 Page 100.050.10.150.20.250.00E+001.00E-012.00E-013.00E-014.00E-015.00E-016.00E-017.00E-01 Column E Column F Column G Column H Column I 00.10.20.30.40.50.60.7 1 0 0.05 0.1 0.15 0.2 0.25 ^x^t= 0:0 ^t= 0:05 ^t= 0:1 ^t= 0:15 ^t= 0:2(a) Sheet1 Page 100.050.10.150.20.250.00E+005.00E-041.00E-031.50E-032.00E-032.50E-033.00E-03 Column K Column L Column M Column N Column O 00.51.01.52.02.53.0103 2 0 0.05 0.1 0.15 0.2 0.25 ^x^t= 0:0 ^t= 0:05 ^t= 0:1 ^t= 0:15 ^t= 0:2(b) Sheet1 Page 100.050.10.150.200.00020.00040.00060.00080.001 00.20.40.60.81.0103 2 0 0.05 0.1 0.15 0.2 ^t Figure 2: Simulation of a self-forming dynamic membrane, in which a binary suspension with = 0:7and initial concentrations 0 1= 0:1and 0 2= 0:001is advected toward a porous support at ^x= 0. The inset to (b) shows the tracer concentration at ^x= 0as a function of time. 3.2 Pre-deposited colloidal membrane In the next simulation, with results shown in ﬁgure 3, a layer of colloidal particles of initial concentration 0 1and widthLis ﬁrst advected toward the porous support, in the absence of the tracer particles, in order to form a pre-deposited dynamic membrane [7]. In this case 2= 0and equations (7)–(10) reduce to the classic ultraﬁltration equations studied by Russel et al.[13,14]: @ 1 @^t=@ 1 @^x+1 Pe@ @^x ^D11@ 1 @^x (^x>0); (13) with boundary conditions Pe 1=^D11@ 1 @^x(^x= 0;1): (14) Figure 3 shows the solution at various times approaching steady state for the case 0= 0:1and Pe= 20:Similarly to Davis and Russel [13], the concentration at ^x= 0quickly approaches the close-packing value p= 0:64;and a dynamic membrane with diﬀuse boundary layer forms; at ^t'10the system has reached steady state. The steady-state porosity n= 1 1of the dynamic membrane is lowest at ^x= 0and increases signiﬁcantly within the diﬀuse boundary layer. As will be seen in the next section, the varying porosity of the dynamic membrane allows a tracer pulse to penetrate some distance within the membrane before being blocked by the reduction in pore size. 5 Sheet1 Page 1 00.10.20.30.40.50.60.70.80.910.00E+001.00E-012.00E-013.00E-014.00E-015.00E-016.00E-017.00E-01 Column BE Column BF Column BG Column BH Column BI Column BJ Column BK 00.10.20.30.40.50.60.7 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ^x^t= 0 ^t= 0:01 ^t= 0:1 ^t= 0:2 ^t= 0:5 ^t= 1:0 ^t= 10 Figure 3: Simulation of the development of a pre-deposited colloidal membrane by advecting a layer of colloidal particles with initial volume fraction 0 1= 0:1against a porous support at ^x= 0. At^t= 10the dynamic membrane has reached a steady-state conﬁguration. 3.2.1 Colloidal ﬁltration of a Gaussian tracer pulse In ﬁgure 4 a Gaussian pulse of tracer particles of the form 2= 0 2exp[a(^x^x0)2)]is introduced at^x=^x0and advected toward the steady-state dynamic membrane obtained in ﬁgure 3, which is used as an initial condition for 1(^x). Figure 4a shows results for the case 0 2= 0:001; a= 2000 and^x0= 0:4, withPe= 20and= 2. At early times the tracer plume experiences traditional advection-diﬀusion eﬀects, with the plume spreading diﬀusively as it moves towards the membrane. As the tracer particles enter the membrane, however, the restricted pore space begins to hinder their motion. Once the tracer particles reach a critical depth within the membrane they are unable to advect further, and they form a concentrated tracer boundary layer. Physically, the tracer particles are blocked by the membrane because the pore size of the membrane eventually becomes smaller than the tracer particle size, as discussed further in Section 4. As a comparison and to illustrate the eﬀects of cross-diﬀusion on the system behaviour, ﬁgure 4b shows results under identical conditions, but with the cross-diﬀusion coeﬃcients set to zero, D12=D21= 0. In this case, the tracer particles are able to pass through the membrane. The tracer diﬀusivity D22still approaches zero at a critical depth within the dynamic membrane and the tracer particles no longer experience Brownian diﬀusion. This can be seen in the proﬁles for ^t>0:2, where the width of the tracer plume remains ﬁxed as it moves through the membrane. While diﬀusion ceases, the plume is still able to advect through the membrane as a travelling wave, and eventually passes through the system and into the ﬁltrate ( ^t= 0:5). Figure 5 shows the eﬀect of changing the size of the tracer particles on the ^t= 0:5proﬁle in ﬁgure 4a. The solid curves show the tracer concentration proﬁle at ^t= 0:5for several values of the particle size ratio =R2=R1. As the size ratio decreases the tracer particles are able to reach further into the dynamic membrane, until for .0:4they are able to pass through into the ﬁltrate. 4 Discussion The membrane eﬀects in ﬁgures 2–5 are caused by the cross-diﬀusion D21term in the tracer ﬂux equation (1 b). In the ultraﬁltration system in ﬁgure 2, rn1<0whilern2>0at^x= 0, and the two terms on the right hand side of (1 b) are of opposite sign. The main D22diﬀusion term acts to relax the tracer concentration gradient and move the particles through the membrane in the ^x direction. The cross-diﬀusion term is driving them in the opposite direction, against their own concentration gradient (uphill diﬀusion) [11]. When the colloid volume fraction 1approaches close-packing, hindered diﬀusion causes the tracer Fickian diﬀusion coeﬃcient D22to reduce to zero (ﬁgure 7d). The cross-diﬀusion coeﬃcient D21, however, becomes large in this limit (ﬁgure 7 c) and uphill diﬀusion dominates, prohibiting the tracer particles from passing through the membrane. 6 Sheet1 Page 1 00.10.20.30.40.50.60.70.80.910.00E+005.00E-041.00E-031.50E-032.00E-03 Column AA Column AB Column AC Column AD Column AE Column AF Column AG 00.51.01.52.0103 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ^x^t= 0 ^t= 0:01 ^t= 0:1 ^t= 0:2 ^t= 0:3^t= 0:4 ^t= 0:5(a) Sheet1 Page 1 00.10.20.30.40.50.60.70.80.910.00E+005.00E-041.00E-031.50E-032.00E-03 Column AM Column AN Column AO Column AP Column AQ Column AR Column AS 00.51.01.52.0103 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ^x^t= 0 ^t= 0:01 ^t= 0:1 ^t= 0:2 ^t= 0:3^t= 0:4 ^t= 0:5(b) Figure 4: Simulation of the advection of a tracer pulse toward a dynamic colloidal membrane, for the case Pe= 20and= 2. In (a) cross-diﬀusion eﬀects stop the tracer pulse from moving through the membrane. In (b) the cross-diﬀusion coeﬃcients are set to zero and the tracer is able to pass through the system into the ﬁltrate. The dotted curve shows the steady-state position of the dynamic membrane obtained in ﬁgure 3. The ﬁltration mechanism of the colloidal membrane can be further illustrated by considering the advection of a single tracer particle within the steady-state membrane. In this case u1= 0; and for a single tracer particle @ 2=@x= 0. The ﬂux equations (5) and (6) become n1U=D11@n1 @x(15) and n2(u2+U) =D21@n1 @x: (16) As the tracer moves into the membrane the pore size decreases and the tortuosity factor approaches 0 while the reﬂection coeﬃcient approaches 1 (see Appendix); the cross-diﬀusion factor`= (+ 1)= also approaches 1 in this limit, so that the tracer cross-diﬀusion coeﬃcient (18c) becomesD21=D0 2K^1( 2=2 1) =D11n2=n1:Substituting this into (16) and solving for the tracer velocity u2then gives u2=UD21 n2@n1 @x=U+Un1D21 n2D11= 0: (17) Therefore, the tracer particle is trapped and unable to move within the membrane when = 0and = 1[10]. 7 Sheet1 Page 100.10.20.30.40.50.60.70.80.910.00E+005.00E-041.00E-031.50E-032.00E-03 Column AG Column AX Column AW Column AV Column AJ 00.51.01.52.0103 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ^x= 2 = 1 = 0:6 = 0:4 Figure 5: Illustration of the eﬀect of changing the particle size ratio =R2=R1on the tracer proﬁles within the dynamic membrane. The blue curve ( = 2) shows the ﬁnal proﬁle obtained in ﬁgure 4a ( ^t= 0:5). Asdecreases (red, green and yellow curves) the tracer particles are able to reach further into the membrane; for .0:4the tracer particles are no longer completely blocked and can pass through into the ﬁltrate. Quantitatively, the model results depend on the expressions for the cross-diﬀusion coeﬃcients summarized in the Appendix. Expressions for D11andD22have been compared with experiment for the hard-sphere system [10,11], and similar equations for all four Dijare in good agreement with experiments on salt cross diﬀusion in polyethylene glycol solutions [12]. Perhaps the most uncertain aspect of the model is the expression for the colloidal reﬂection coeﬃcient . In this work an expression (22 b) forobtained previously [10] for hard-sphere suspensions has been used. An alternative expression for relevant to porous media was also obtained in [10] and may be more appropriate for dynamic membranes as it ensures the tracer particles are blocked by the membrane whenever the particle size is equal to or greater than the physical pore size. However, either expression gives qualitatively similar results ( u2!0as !0and!1). Expressions forthat also depend on particle charge and shape in cylindrical pores have been obtained by Anderson and Quinn [15] and Bhalla and Deen [16], which have a qualitatively similar behaviour to the expression used here ( !0as the pore size increases while !1as the pore size approaches the tracer particle size). 5 Conclusions A model of a dynamic colloidal membrane has been developed within the framework of the hydrodynamic theory of cross diﬀusion. The model simulates two experimental conﬁgurations: a self-forming membrane in which the colloidal particles are part of the feed solution; and a pre-deposited membrane with subsequent ﬁltration of a tracer plume. The ﬁltration eﬀect of the colloidal membrane is caused by strong cross-diﬀusion eﬀects, which tend to push the tracer particles in a direction opposite to their own concentration gradient (uphill diﬀusion). The model accounts for the eﬀects of hydrodynamic and excluded volume interactions between the colloidal membrane and tracer particles, and simulates the eﬀect of varying the tracer particle size. For suﬃciently small particles, the tracers can pass through the pores of the membrane; once the tracer radius approaches the pore size, however, the tracer advection is hindered, and the particles instead form a concentrated boundary layer within the membrane. The generic hard-sphere model developed here should provide a basic framework within which to study dynamic membrane eﬀects in more complex systems. 8 Acknowledgements This work was made possible by the love and merciful kindness of Jesus, Lord of heaven and earth and Saviour of all who believe. Believe on the Lord Jesus Christ and thou shalt be saved, and thy house.Acts 16:31. A Diﬀusion coeﬃcients For tracer diﬀusion in concentrated suspensions the four cross-diﬀusion coeﬃcients can be written in the form D11=D0 1K^1; D 12=D0 1K= ; (18a,b) D21=D0 2(^`K^1 ); D 22=D0 2 ; (18c,d) whereD0 i=kBT=(6Ri0)is the Stokes-Einstein diﬀusivity of particle i,kBis Boltzmann’s constant,Tis temperature, and 0is the viscosity of the suspending ﬂuid [10]. In (18 a)Kis the dimensionless permeability (viscous mobility) of the host-particle matrix and ^1=  1=(kBT) = d( 1Z)=d 1, where 1= (@=@n 1)Tis the derivative of the host-matrix osmotic pressure with respect to concentration; 1=n1v1is the host-particle volume fraction, v1=4 3R3 1is the volume of a host particle and Z( 1)is the host-matrix compressiblity factor. In (18 b)is the reﬂection coeﬃcient of the host matrix accounting for hindrance eﬀects on tracer advection within the pore space, and is the equilibrium partition coeﬃcient of the tracer particles between the pore space and the bulk ﬂuid. In (18 c) is a preferential interaction coeﬃcient accounting for excluded volume eﬀects; and ^`= 2`=(2 1);where`= ( +1)= is the viscous cross-diﬀusion factor between the tracer and host particles, 2=n2v2is the tracer-particle volume fraction and =R2=R1is the ratio of particle sizes. Finally, in (18 d) is the diﬀusive tortuosity factor accounting for the hindrance eﬀect on tracer diﬀusion within the pore space. For spherical particles approximate expressions for the friction and compressibility factors of the host-particle matrix are K= (1 1)6:55andZ=1 +a 1 1 1= p; (19a,b) wherea= 41= pand p= 0:64is the volume fraction at random close packing [11]. In the dilute limit 1 1equations (19) reduce to the exact results K= 16:55 1andZ= 1 + 4 1[17,18]. GivenZ,^1can be obtained as ^1=d( 1Z) d 1=1 + 2a 1a 2 1= p (1 1= p)2: (20) Approximate expressions for the partition coeﬃcient and preferential interaction coeﬃcient in hard-sphere suspensions are = (1 2= p)eb12 1+ 2= pand =n2b12v1= 2b12=3; (21a,b) where=R2=R1is the size ratio of the tracer and host particles, b12= (1+)3is a thermodynamic coupling coeﬃcient, 2=n2v2is the tracer volume fraction and v2=4 3R3 2is the tracer particle volume [10,11]. Constitutive equations for the diﬀusive tortuosity factor and reﬂection coeﬃcient in hard sphere suspensions are =( (1d)a ;if0d1 0 ifd>1and=( 1(1d)a;if0d1 1 ifd>1(22a,b) where d=3 1 cd(1 1)andcd= 3 g (1 g)(1 +2) (1 +): (23a,b) 9 Sheet1 Page 10.00E+002.00E-014.00E-016.00E-01050100150200250 0.00E+002.00E-014.00E-016.00E-0100.20.40.60.81 0.00E+002.00E-014.00E-016.00E-010.00E+002.00E-014.00E-016.00E-018.00E-011.00E+00 00.20.40.60.81.0 0 0.1 0.2 0.3 0.4 0.5 0.6 1(a) 210.60.4 Sheet1 Page 10.00E+002.00E-014.00E-016.00E-0100.20.40.60.81 0.00E+002.00E-014.00E-016.00E-010.00E+002.00E-014.00E-016.00E-018.00E-011.00E+00 00.20.40.60.81.0  0 0.1 0.2 0.3 0.4 0.5 0.6 1(b) 2 1 0.6 0.4 Figure 6: (a) Diﬀusive tortuosity factor and (b) colloidal reﬂection coeﬃcient as functions of 1for the case 2= 0:001and four values of , corresponding to those in ﬁgure 5. Heredis the ratio of the tracer size R2to the locally averaged pore size within the suspension, Rd=cdRp, whereRp= 3 1=[R1(1 1)]is the physical pore size [19]. The quantity cdis a scaling factor accounting for ﬂuctuations in the pore size and g= 0:58is the volume fraction at the hard-sphere glass transition. The scaling factor ensures that !0=in the large tracer limit R2!1;where( 1)is the dynamic viscosity of the host suspension. When d1the tracer particles are trapped in the pores and = 0while= 1(ﬁgure 6). This trapping eﬀect is the physical cause of the ﬁltration behaviour occurring in ﬁgures 3–5. Settingd= 1in (23a) gives c 1= (1 + 3=cd)1as the critical colloid volume fraction at which particle trapping occurs [10]. As decreases (smaller tracer particles), higher volume fractions are required to cause particle trapping [10] and for suﬃciently small tracer particles trapping does not occur, as illustrated in ﬁgure 5. The exponents a andain (22) are a = 1cd 3anda=1cd 3; (24a,b) where 1= 2:5=(:22 +)and1=b12(1 + 3+2) +2=(1 +3). Figure 6 shows the diﬀusive tortuosity factor and reﬂection coeﬃcient as functions of 1 for the case 2= 0:001and four values of corresponding to those in ﬁgure 5. When = 2(blue curves), as 1increases !0while!1. When = 0the tracer particles are unable to diﬀuse within the pore space, while when = 1the tracer particles are not able to advect within the pore space [10]. As decreases, higher values of 1are required to block the tracer particles; this explains the curves in ﬁgure 5, where the tracers are able to advect further into the membrane as decreases. For = 0:4(yellow curves), the reﬂection coeﬃcient is less than one even at the close-packed limit; the tracer particles in ﬁgure 5 in this case (yellow curve) are therefore able to pass completely through the membrane and into the ﬁltrate. Finally, given and, the cross-diﬀusion factor can be obtained as `=(+ 1) and ^`= 2` 2 1: (25) With equations (19)–(25), the diﬀusion coeﬃcients in (18) can be determined as functions of the volume fractions 1and 2and the particle size ratio . Figure 7 shows the Dijas functions of 1for the case = 2and 2= 0:001:The proﬁles are qualitatively similar to those obtained previously for the hard-sphere system [10,18] and for salt diﬀusion in polymer solutions [12,20]. A signiﬁcant aspect of these proﬁles is the fact that the tracer Fickian diﬀusivity D22approaches zero 10 Sheet1 Page 10.00E+002.00E-014.00E-016.00E-0100.511.522.533.544.5 0.00E+002.00E-014.00E-016.00E-01050100150200250 0.00E+002.00E-014.00E-016.00E-0100.511.522.533.544.55 01.02.03.04.0 ^D11 0 0.1 0.2 0.3 0.4 0.5 0.6 1(a) Sheet1 Page 10.00E+002.00E-014.00E-016.00E-0100.511.522.533.544.5 0.00E+002.00E-014.00E-016.00E-01050100150200250 0.00E+002.00E-014.00E-016.00E-0100.20.40.60.81 050100150200250 ^D12 1 0 0.1 0.2 0.3 0.4 0.5 0.6 1(b) Sheet1 Page 10.00E+002.00E-014.00E-016.00E-0100.20.40.60.81 0.00E+002.00E-014.00E-016.00E-01-101234560.00E+002.00E-014.00E-016.00E-0100.511.522.533.544.55 0.00E+002.00E-014.00E-016.00E-0100.20.40.60.81 -1.001.02.03.04.05.06.0 ^D21 2 0 0.1 0.2 0.3 0.4 0.5 0.6 1(c) Sheet1 Page 10.00E+002.00E-014.00E-016.00E-0100.20.40.60.81 0.00E+002.00E-014.00E-016.00E-01-10123456 00.20.40.60.81.0 ^D22 0 0.1 0.2 0.3 0.4 0.5 0.6 1(d) Figure 7: Cross-diﬀusion coeﬃcients ^Dij=Dij=D0 ias functions of volume fraction 1for the case = 2and 2= 0:001. in the close-packed limit 1! p;while the tracer cross-diﬀusion coeﬃcient D21remains positive. Therefore the tracer motion is determined by cross-diﬀusion in this limit. This eﬀect enables the colloidal membrane to stop the advection of the tracer particles, as discussed further in Section 4.