Stability Analysis of Three Dimensional Colloidal Domains

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Stability Analysis of 3D Colloidal Domains J Phys Chem B Vol 109 No 45 2005 21343. temperature colloidal particles aggregate into domains through with respect to shape fluctuations for spherical cylindrical and. a self assembly process and F rb becomes a periodical function lamellar domains. in space At low temperatures domains and dispersing medium. are separated by sharp boundaries across which F r. b jumps from III Spherical Domains, a nonzero value to zero Assuming that colloidal particles are. As shown in our previous paper 20 spherical domains are. uniformly distributed inside domains the density function is. energetically preferred in the low density and weak attraction. reduced to, limits In this section we analyze the stability of spherical. F inside domains,2 outside domains, domains and study the shape transformation driven by spherical. harmonic shape fluctuations, We first introduce a small fluctuation R for a reference. where F2 is set to be zero in this paper for convenience The spherical domain with radius R Following Deutch and Low s. neglect of density fluctuations in eq 2 is acceptable at low approach 27 we expand the shape fluctuation in successive orders. temperatures since shape fluctuations are dominant as discussed. in the Introduction R R0 R1 6, In the mean field framework the short range attraction.
contributes to bulk adhesion and surface free energy The bulk where R0 R1 The same expansion will be applied. adhesion is expressed as FA dr b F r,b where fA r, b is for the stability analysis of cylindrical domains in the next. the adhesive free energy density Consistent with the simplifica section According to this shape fluctuation the domain volume. b is approximated by its average, tion for the density function fA r formally changes from Vd 0 4 R3 3 to. value fA nc 2 where nc is the average contact number per. particle Since the bulk adhesion in a unit system volume 1. FA V F0V0nc 2 is a shape independent constant FA does Vd I d R R 3. not contribute to the shape transformation and will be neglected. For a single domain with surface area Sd its surface free R R2 R3. energy is given by FS ISd dr b rb Sd Although the Vd 0 R3 I d O 7. surface tension is a shape dependent function in general we. apply the zeroth order approximation and consider as a. where I d 0 sin d 02 d is the integral over the solid. constant As an implicit function of temperature the surface. angle The volume of a stable domain remains the same. tension decreases as temperature increases, with respect to fluctuations i e Vd Vd 0 On the basis of. The integrated contribution of the long range Yukawa repul. this constraint the leading order fluctuation satisfies. sion to free energy can be simplified to,I d R0 0 and the next order satisfies R1. N Nd Nd R20 R Using spherical harmonics Ylm we expand. Y m n 3 R0 as,where rij r, rj and Nd is the total number of domains In R0 R0 l clmYlm 8.
the above equation F 1 Y m is the intradomain repulsive free l 1 l 1 m l. energy for domain m,where cl m 1 mc lm is required since R0 is real. r 2 4a Next we calculate the variation of the free energy F with. respect to the shape fluctuation R In the low density limit. the free energy of a colloidal domain is approximated as a sum. whereas F 2, Y m n is the interdomain repulsive free energy of the surface free energy FS and intradomain repulsion F 1 Y. between two domains m and n With a constant surface tension the variation of FS only. depends on the change of surface area Sd Similar to eq 7 the. r 2 4b surface area difference is truncated to quadratic terms. In the low density limit the distance between a pair of domains Sd I d R R. is much larger than the characteristic domain size so that each. domain can be treated as an isolated system and the collective x R R 2 B R 2 R2. contribution of F 2 Y m n is ignored in the lowest order I d B R0 2 R20 9. approximation The resulting effective free energy for a single. domain is written as, where B is the gradient operator for the solid angle. F FS F 1 5 e sin 1 e and e r e e are unit vectors in. spherical coordinates The relation R1 R20 R is, In a more rigorous manner the sum on the right hand side of applied to derive eq 9 Following the expansion in eq 8 and. eq 5 should be treated as energy instead of free energy Since the spherical harmonic differential equation fYlm. the entropic effects can be partially included by the temperature l l 1 Ylm we obtain the surface free energy variation. dependent surface tension and the configurational randomness. of domains is omitted for the case of isolated domains we l. consider eq 5 as a good approximation for the free energy In. the remainder of this paper we will calculate the variation of F. FS Sd l l 1 1 clm 2,21344 J Phys Chem B Vol 109 No 45 2005 Wu and Cao.
In spherical coordinates the intradomain repulsion is explicitly. written as,I d 1 I d 2 dr1 dr2r21r22uY r12 11,where the distance between two vectors b r1 and b. r2 is given by,r12 rb1 b r2 r21 r22 2r1r2 cos 1 2 and is the. angle between these two vectors cos cos 1 cos 2, sin 1 sin 2 cos 1 2 After a straightforward but tedious. derivation the quadratic truncation of the intradomain repulsion. variation is given by,I d 1 I d 2 R0 1 R0 2 uY Re r1 Re r2. I d 1 I d 2 0 dr1r21 R20 2 e r2,r 1 Re r2 12,r is br re r r 1.
where the gradient operator for vector b B, in spherical coordinates We apply the spherical harmonic. expansion and simplify eq 12 to,clm 2 gl R R g1 R R. where gl r1 r2 is defined using the lth order Legendre. polynomial Pl cos as, Figure 1 Examples of spherical harmonic shape fluctuations with. gl r1 r2 A 1 0 d sin Pl cos uY r12 14 R0 cYl0 Three figures on the left side correspond to l 2 and. the other three on the right side correspond to l 3 For figures in the. The details of this derivation are shown in Appendix A The same row their values of c are the same and shown on the left. free energy variation for a spherical domain is the sum of eqs. As discussed in section I the sign of the free energy variation. determines the stable F 0 and unstable F 0 regimes. with respect to a given shape fluctuation Since the free energy. variation for a spherical domain is diagonalized in the basis of. Ylm we focus on spherical harmonic shape fluctuations. Several examples are shown in Figure 1 With respect to the. lth order mode R0 l m l l,clmYlm the critically,stable radius Rl is determined by. R l l 1 1 R4l gl Rl Rl g1 Rl Rl 0 15, where the effective attraction repulsion ratio R F21A 4.
is introduced in ref 20 A direct result of eq 15 is that spherical. domains are always stable with respect to the first order shape. fluctuation mode l 1 Here we recall the equilibrium Figure 2 The stability curves for spherical domains With respect to. spherical radius Req determined by20 the l g2 th order spherical harmonic fluctuation mode spheres are. stable below the corresponding stability curve whereas they are unstable. 1 R R2eq 3 above this curve, 2R3eq 5R2eq 6Req 3 e 2Req 0 16 2 ratios between critical and equilibrium radii are plotted as. functions of R for the first three important fluctuation modes. For conciseness the notation of equilibrium is introduced With respect to the lth order shape fluctuation spherical domains. loosely throughout this paper to represent the optimal state for are stable below the curve of Rl R Req R whereas they are. a given shape instead of the state with the global minimum free unstable above this curve Since the value of Rl is at least twice. energy density In addition the symbol Req is introduced since the equilibrium radius spherical domains in equilibrium are. the original one Rm in ref 20 will be used as the mth order always stable with respect to shape fluctuations in the quadratic. critical radius for cylindrical domains in this paper In Figure approximation In the weak attraction limit R f 0 the critical. Stability Analysis of 3D Colloidal Domains J Phys Chem B Vol 109 No 45 2005 21345. radius Rl is an increasing function of l so that the second order In cylindrical coordinates e r e e z vectors are expressed as. shape fluctuation l 2 is the most important mode Asymp b. r br ze z r e r ze z and the explicit form of intradomain. totically the ratio between critical and equilibrium radii is given repulsion for a cylindrical domain is. L 2 dz1 L 2 dz2 dbr 1 db,Rl 2 2l 1 l l 1,Rf0 Req 5 l 1 uY x z1 z2 2 br 1 b. With the increase of attraction strength the critical radius Rl F21AL dbr 1 dbr 2K0 b. r 2 for L 1, quickly increases Above a critical point Rl l2 l 2 21. 2 l2 l 1 the lth order radius approaches infinity and. the associated free energy variation is always positive For To derive the equation above an integral identity 0 dz. R R 1 2 all the spherical domains can resist shape exp xz2 t2 xz2 t2 K0 t is applied where Km t is the. fluctuations in the quadratic approximation Although other mth order modified Bessel function of the second kind With. shapes such as cylinders become more energetically favorable respect to R we obtain the quadratic truncation for the. than spheres for R 0 788 as shown in Figure 4 of ref 20 the intradomain repulsion variation. transformation from isolated spheres to cylinders cannot be. spontaneously achieved by shape fluctuations, IV Cylindrical Domains F21ALR2 I d 1 I d 2 R0 1 R0 2 K0 Re r 1 Re r 2. In comparison to spheres and lamellae cylindrical domains F21ALR I d 2 R0 2 I d 1 0Rr 1 dr 1. are the most stable state for 0 788 R 0 846 in the low r 2K0 r 1e r 1 Re r 2 22. density limit 20 Shape fluctuations for cylinders can be separated. into two categories about the radius and along the azimuthal. axis In this section we will investigate shape fluctuations about To expand F 1. Y in the Fourier basis we introduce the addition, the radius The azimuthal shape fluctuations will be discussed theorem for Bessel functions34.
together with shape fluctuations for lamellae in the next section. because of their spatial similarities, Similar to the approach for spherical domains we expand. Km ar Im ar eim 1 2, the shape fluctuation in the Fourier basis under the constraint. of constant volume For a cylindrical domain with radius R and where Im t is the mth order modified Bessel function of the. azimuth length L 1 a fluctuation R about R formally first kind and r and r denote the larger and smaller of r 1. changes the volume from Vd 0 R2L to and r 2 respectively Substituting eq 23 into eq 22 the. intradomain repulsion variation is given by,Y 2 2 F1ALR. cm 2 Km R Im R K1 R I1 R,Vd 0 R2L I d 18 24, As a sum of eqs 20 and 24 the free energy variation for a. where I d 2 d Following the successive expansion cylindrical domain is diagonalized in the Fourier basis so that. R R0 R1 the constant volume constraint we will concentrate on harmonic shape fluctuations Examples. for stable domains requires that the leading order fluctuation in Figure 3 show that an original cylindrical domain can evolve. satisfies I d R0 0 and the next order satisfies R1 into m subdomains induced by the mth order fluctuation mode. R20 2R The leading order fluctuation can be expanded R0 m cm eim c m e im Since the stability of domains. is determined by the sign of F the mth order critically stable. in the Fourier basis as R0 m 0cm eim where c m c m. radius Rm with respect to R0 m is given by,since R0 is real.
Using shape fluctuations in the expansion form we calculate R. the free energy variation Truncated to quadratic terms the 3. m2 1 Km Rm Im Rm K1 Rm I1 Rm 0 25, change of surface area with respect to R is given by. The equation above shows that cylindrical domains remain. Sd L I d x R R 2 R R 2 R marginally stable with respect to the first order mode m 1. To discuss higher order fluctuation modes we recall the. L equilibrium cylindrical radius Req20,I d R0 2 R20 19. eq 2K1 Req I1 Req 0 26,where the relation R1 R20 2R is applied The dReq. expansion of the surface free energy variation in in the Fourier. basis is thus written as Here we reemphasize that the equilibrium radius in the equation. above is defined only for cylindrical domains without the. consideration of other shapes The results of Rm R Req R are. 20 plotted in Figure 4 as functions of the control parameter R. Similar to fluctuation modes for spheres the mth order shape. 21346 J Phys Chem B Vol 109 No 45 2005 Wu and Cao,V Lamellar Domains. With the lowest spatial symmetry isolated lamellar domains. are energetically favorable in the case of strong attraction. R 0 846 In this section we investigate the stability of. lamellae with respect to shape fluctuations The length height. and width of a lamellar domain are denoted by L1 1. L2 1 and h respectively Two small shape fluctuations. s x y and s x y are introduced on boundaries z 0 and z. h respectively The domain volume changes from Vd 0. dx L 2 dy s x y,L 2 h s x y,Vd 0 dx dy s x y s x y 28.
For convenience we introduce two dimensional vectors b r. xe x ye y in the real space and b,q qxe x qye y in the Fourier. space To satisfy the constant volume constraint for stable. domains the shape fluctuation around z 0 is expanded as. Figure 3 Examples of harmonic shape fluctuations with R0 c r. eiqb br 29, cos m projected in the xy plane Three figures on the left side. correspond to m 2 and the other three on the right side corre. Three dimensional domain patterns can self assemble in a charged colloidal suspension with competing short range attraction and long range Yukawa repulsion Following the investigation of the ground state domain shapes in our previous paper we study the stability of isolated spherical cylindrical and lamellar domains

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