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Liquid-glass re-entrant behavior in a charge-stabilized colloidal dispersion
Journal of Non-Crystalline Solids 307–310 (2002) 812–817 www.elsevier.com/locate/jnoncrysol
Liquid-glass re-entrant behavior in a charge-stabilized colloidal dispersion G.F. Wang *, S.K. Lai Complex Liquids Laboratory, Department of Physics, National Central University, Chung-li 320, Taiwan, ROC
Abstract Using the static structure factor SðqÞ calculated in the rescaled mean spherical approximation and in conjunction with the idealized mode-coupling theory, we determine the loci of the liquid-glass transition phase boundary for a saltfree suspension of charged colloids prepared in different counterions concentrations. For the simplest deionised chargestabilized colloidal liquid, our calculations demonstrate the possibility of observing, in restrictive regions of the phase diagram, the liquid ¢ glass ¢ liquid ¢ glass re-entrant transition. The latter disappears at a lower concentration of counterions and, when this arises, only the glass ¢ liquid ¢ glass is observed. We study this counterion-concentrationdependent re-entrant phenomenon by analyzing the non-ergodic Debye–Waller factor, SðqÞ and their corresponding spatial counterparts. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction A charged stabilized colloidal system is experimentally observed [1,2] to undergo crystallization at a lower volume fraction (g K 0:15) of macroions. This phenomenon can be understood theoretically by the density functional theory [3–5]. In the context of disordered phase, the Ornstein– Zernike (OZ) equation supplemented by the mean spherical approximation (MSA) closure has proved to be a competent tool if the evaluation of the equilibrium fluid structure is further augmented by the notion of size rescaling on charged macroparticles. Hansen and Hayter [6] applied this rescaling idea to study the SðqÞ of micelle solutions
*
Corresponding author. E-mail address: [email protected] (G.F. Wang).
at very low g(103 ). By modeling the interactions between colloidal particles in the so-called Derjaguin–Landau–Verwey–Overbeek (DLVO) approximation [7] for which an analytical MSA SðqÞ solution [8] exists, they found the calculated SðqÞ agreeing excellently with measured data. For an aqueous dilute solution of charge-stabilized colloids, this rescaled MSA (RMSA) has thus made an impressive achievement in interpreting the SðqÞ of charged colloids. Comparatively fewer studies have reported applying the technique to more concentrated charged colloidal dispersions (g > 0:15). A study on these latter systems is both important and worthwhile since it affords an understanding of the liquid-glass phase transformation whose occurrence was observed to fall into the range of g > 0:2 [2,9]. In this work we use the RMSA SðqÞ in conjunction with the idealized mode-coupling theory
0022-3093/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 2 ) 0 1 5 2 4 - 7
G.F. Wang, S.K. Lai / Journal of Non-Crystalline Solids 307–310 (2002) 812–817
(MCT [10]) to determine the liquid-glass transition boundaries under different screening conditions. An interesting result in our calculations is the observation of the transition from the liquid ¢ glass ¢ liquid ¢ glass (LGLG) re-entrant phenomenon in a restrictive region of the concentration of counterions to the GLG behavior at a lower concentration. A similar re-entrant phenomenon has recently been predicted by Wilke and Bosse [11] for an ionic glass.
H€ uckel screening length in which a2i ¼ 4pLB qi Zi2 , LB and Zi being the Bjerrum length and the charge of a colloid or a small ion, respectively. Carrying out an inverse Fourier transform (see Ref. [13] for mathematical details) to Eq. (2), we obtain 2 2 ceff 00 ðrÞ ¼ LB Z0 X
ekD r ekD r ¼ r0 c r r
r > r0 ; ð3Þ
where the coupling parameter j hj j j i cosh þU sinh X ¼ cosh 2 2 2 2
2. Theory In this section we introduce the effective onecomponent model previously developed by Belloni [12] and shown recently by Lai et al. [13,14] to be an adequate model for studying the structure of concentrated charged colloids. In addition, we summarize the idealized MCT for describing the latter system. 2.1. Mean spherical approximation Consider the multicomponent coupled OZ equations given by X Z hij ðrÞ ¼ cij ðrÞ þ q‘ hi‘ r r0 c‘j ðr0 Þdr0 ; ð1Þ ‘¼0
where species i, j and ‘ are that i; j; ‘ ¼ 0 for macroions, i; j; ‘ ¼ 1 for counterions, and i; j; ‘ ¼ 2; 3; . . . for other small ions. Here q‘ is the number density for species ‘, cij ðrÞ is the direct correlation function and hij ðrÞ ¼ gij ðrÞ 1 is the total correlation function which is defined in terms of the pair correlation function gij ðrÞ. Adelman [15] has shown that Eq. (1) can be contracted to an effective direct correlation function ceff ðrÞ whose Fourier transform can be cast [12,13] to read 2 X s s ^ ^ c^eff ðqÞ ¼ c ðqÞ þ ðqÞ c 00 00 0i h
813
i¼1
a0 þ
P
i¼1
ai c^s0iðqÞ
q2 þ kD2
i2 ;
ð2Þ
where superscript s stands for the short range P component, kD1 ¼ 1=ð i¼1 a2i Þ1=2 is the Debye
ð4Þ depends not only on j ¼ kD r0 but also on g through U ¼ ð8f=j3 2t=jÞ in which f ¼ 3g= ð1 gÞ, t ¼ ðKr þ 2fÞ=½2ð1 þ fÞ þ Kr and ðK2r j2 Þ½2ð1 þ fÞ þ Kr 2 ¼ 96LB gðZ02 =r0 Þ which, given Z02 =r0 , j and g, is solved iteratively for Kr (and hence X). Note that X ! expðj=2Þ=ð1 þ j=2Þ in the limit of q0 ! 0, a linearized DLVO result. This implies that the c^eff 00 ðqÞ (and hence the SðqÞ ¼ 1= ½1 q0 c^eff 00 ðqÞ Þ above is appropriate for the description of a suspension of charged colloids at any finite concentration. 2.2. Mode-coupling theory We turn to introduce the MCT appropriate for a suspension of charge-stabilized colloids. The scattering function F ðr; tÞ whose Fourier–Laplace transformed F^ðq; zÞ can be shown to be related to ^ ðq; zÞ by the memory function M F^ðq; zÞ 1 R^ðq; zÞ ¼ 2 ^ ðq; zÞ=SðqÞ SðqÞ z þ q D0 =SðqÞ M ð5Þ where D0 is the Stokes–Einstein diffusion coefficient of a single particle. All the dynamical characteristics are embodied in the memory function ^ ðq; zÞ. It can be inferred from the works of SzaM mel and L€ owen [16] and others [17,18] that there exists an idealized liquid-glass transition at a dynamical transition point which is determined by the following non-linear vitrification equation
814
G.F. Wang, S.K. Lai / Journal of Non-Crystalline Solids 307–310 (2002) 812–817
f ðqÞ SðqÞ ^ ¼ C 1 f ðqÞ D0 q2
ðq; t ¼ 1Þ:
ð6Þ
Here the solution fc ðqÞ ¼ Rðq; t ! 1Þ 6¼ 0 is the glassy Debye–Waller factor, a non-ergodic state, whereas f ðqÞ ¼ 0 is the ergodic state. By making a two-mode approximation [17–19], the above can be put in a form suitable for numerical work Z 1 ^ ðq; tÞ ¼ q0 D0 C dx x 8qp2 0 2 Z jxþqj x y 2 eff c^00 ðxÞ c^eff dy y ðyÞ 00 2q jxqj 2 q eff þ c^eff ðxÞ þ c ðyÞ 00 2 00 SðxÞSðyÞRðx; tÞRðy; tÞ:
ð7Þ
Given SðqÞ, Eqs. (6) and (7) constitute two nonlinear coupled equations which we will solve iteratively for the loci of the liquid-glass transition (for technical details the interested readers should consult Lai et al. [20] applied for cases of atomic liquids).
3. Numerical results and discussion In all of the calculations below we have (a) kept the charged colloids in water and maintained the colloidal dispersion at room temperature, (b) considered monovalent counterions as the only source of screening small ions and (c) imposed the condition of charge neutrality, q1 Z1 ¼ q0 Z0 , for
Fig. 1. Phase diagrams of charge-stabilized colloidal dispersions for (a) cc –gc , (b) Z0c –gc , (c) rc0 –gc and (d) rc0 –Z0c . Notations used are: j ¼ 1:5 (solid circles), 2 (open circles), 2.65 (solid diamonds), 3 (solid triangles), 3.8 (open triangles). The region of lowest values gc (xaxis is self-explanatory and y-axis varies from rc0 ¼ 0 in step of 1000 up to 5000) is enlarged in the insert in Fig. 1(c) for j ¼ 1:5 (solid circles), 2 (open circles), 2.65 (full curve), 3 (dashed curve) and 3.8 (open triangle).
G.F. Wang, S.K. Lai / Journal of Non-Crystalline Solids 307–310 (2002) 812–817
the self-consistent calculation of j. As a result, the Z0 and r0 of macroions have to be adjusted [13]. Fig. 1(a) shows the cc gc phase diagram calculated at j ¼ 1:5 and 2. For convenience, we have included in the same figure our previous results of j ¼ 2:65, 3 and 3.8 [21]. We note first of all that an increase in j will lead to a larger ergodic region. This general scenario typical for the hard-core Yukawa potential can be understood physically by analyzing the dependence of c on the Z02 =r0 and X [21]. Fig. 1(b) and (c) show the detailed changes of Z0c gc and rc0 gc respectively and, for completeness, we depict in Fig. 1(d) the relation of rc0 and Zc0 . An important finding of Fig. 1(b)–(d) is the occurrence of a LGLG re-entrant phenomenon for 2 < j < 3:8; the LGLG transition disappears for j < 2, and when this happens, only the GLG transition is observed. To delve into the re-entrant behavior, we con) charged sider a monodisperse (r0 12 606 A colloidal system (see the interception of the horizontal lines in Fig. 1(c) and (d)). Corresponding to this r0 , we display respectively in Fig. 2(a) and (b) the two fc ðqÞ and SðqÞ calculated at j ¼ 1:5 (solid circles). One notices immediately that the fc ðqÞ and SðqÞ show concerted oscillatory structures. To see the role played by the charge of a macroion, we compare these results with a system of neutral hard spheres previously shown to have undergone structural arrested at gc ¼ 0:516 [20]. For this system the purely excluded volume effect has resulted in its fc ðqÞ delineating a rather short-ranged behavior. Thus, for the case of higher concentrated lower charged colloids (gc ¼ 0:5, Z0c ¼ 329e), it will be driven predominantly by the geometric restriction with its fc ðqÞ mainly of a shorter-range order. This is in contrast to the less concentrated (gc ¼ 0:175) higher charged (Z0 ¼ 943e) case which reflects a change in localization mechanism being manifested primarily by the electrostatic effect and secondarily by the hard-core factor. Accordingly, the longer-range order necessary for ensuring the structural arrest of charged colloidal particles should differ drastically from the hard-core effects. This feature has in fact shown up in Fig. 3(a) for the spatial critical amplitude H ðrÞ which is the Fourier transform of
815
Fig. 2. (a) Glassy Debye–Waller form factor fc ðqÞ and (b) static ) structure factor SðqÞ for a monodisperse (r0 12 606 A charge-stabilized colloidal dispersion calculated at j ¼ 1:5 for the cases (gc ¼ 0:5, Z0 ¼ 329e) (dashed curve) and (gc ¼ 0:175, Z0 ¼ 943e) (full curve). The neutral hard-sphere results (dotdashed curves) are included for comparison.
SðqÞhc ðqÞ [20]. It is interesting to see the lower-g higher charged colloidal H ðrÞ [10] revealing a ‘softening’ in the local collective excitation in the process of b-relaxation. Consistent behavior is also observed for the pair correlation function gðrÞ depicted in Fig. 3(b) where the gðrÞ of the gc ¼ 0:175 (gc ¼ 0:5) charged colloids shifts outwards (inwards) in agreement with the overall change of fc ðqÞ in Fig. 2(a). At this point, we should emphasize that in approaching the lowest values g, the macroion charge Z0 needs increased to strong enough (Coulombic) coupling to ensure structural arrest of charged colloidal particles. In view of this
816
G.F. Wang, S.K. Lai / Journal of Non-Crystalline Solids 307–310 (2002) 812–817
we determine the parametric phase diagram for a deionized charge-stabilized dispersion. In the restrictive region 2 < j 6 3:8 of the phase diagram r0 g or Z0 g, we observe the LGLG re-entrant transformation. The LGLG re-entrant phenomenon disappears for j < 2 and is replaced instead by the GLG transition. The fc ðqÞ and SðqÞ and their spatial counterparts, H ðrÞ and gðrÞ, were analyzed to illustrate their correlations with the reentrant behavior. It is conjectured that the disappearance of the LGLG transformation for j < 2 is due to the ineffectiveness of Coulombic coupling alone in driving the system into an ergodic–nonergodic transition at a much lower g. We note in passing that the calculation presented here is quite general for the method, with slight and straightforward modification, could be used to investigate the phase diagrams of a more realistic chargestabilized colloidal solution such as a system of monodisperse polystyrene charged spheres in the presence of electrolytes [22].
Acknowledgements
Fig. 3. (a) Critical spatial amplitude H ðrÞ and ðbÞ pair correlation function gðrÞ for the same charge-stabilized colloids as in Fig. 2. Refer to Fig. 2 for notations.
we conjecture 1 that the disappearance of the LGLG phenomenon for j < 2 is indicative of the ineffectiveness of Coulombic coupling alone in driving charged colloids into an ergodic–nonergodic transition at a much lower g. 4. Conclusion Using a realistic hard-core Yukawa potential to construct SðqÞ and applying the idealized MCT,
1
We are currently looking into the possibility of understanding it analytically following along the line of Ref. [11] in the lowest g region.
This work has been supported in part by the National Science Council (NSC89-2112-M-008064). S.K.L would like to thank Professor W.K. Liu for hosting his visit to the University of Waterloo during which period this paper is written to fruition. We acknowledge continual support from the National Central University.
References [1] Y. Monovoukas, A.P. Gast, J. Colloid Interface Sci. 128 (1989) 533. [2] E. Sirota, H.D. Ou-Yang, S.K. Sinha, P.M. Chaikin, Phys. Rev. Lett. 62 (1989) 1524. [3] S. Sengupta, A.K. Sood, Phys. Rev. A 44 (1991) 1233. [4] P. Salgi, R. Rajagopalan, Langmuir 7 (1991) 1383. [5] N. Choudhury, S.K. Ghosh, Phys. Rev. E 51 (1995) 4503. [6] J.P. Hansen, J.B. Hayter, Molec. Phys. 46 (1982) 651. [7] E.J. Verwey, J.G. Overbeek, Theory of the Stability of Lyophobic Colloids, Elsevier, Amsterdam, 1948. [8] J.B. Hayter, J. Penfold, Molec. Phys. 42 (1981) 109. [9] W. H€artle, H. Versmold, X.J. Zhang-Heider, J. Chem. Phys. 102 (1995) 6613.
G.F. Wang, S.K. Lai / Journal of Non-Crystalline Solids 307–310 (2002) 812–817 [10] W. G€ otze, in: J.P. Hansen, D. Levesque, J. Zinn-Justin (Eds.), Liquids, Freezing and the Glass Transition, NorthHolland, Amsterdam, 1991, p. 287. [11] S.D. Wilke, J. Bosse, Phys. Rev. E 59 (1999) 1968. [12] L. Belloni, J. Chem. Phys. 85 (1986) 519. [13] S.K. Lai, G.F. Wang, Phys. Rev. E 58 (1998) 3072. [14] S.K. Lai, J.L. Wang, G.F. Wang, J. Chem. Phys. 110 (1999) 7433. [15] S.A. Adelman, Chem. Phys. Lett. 38 (1976) 567. [16] G. Szamel, H. L€ owen, Phys. Rev. A 44 (1991) 8215. [17] P. Baur, G. N€ agele, R. Klein, Phys. Rev. E 53 (1996) 6224; P. Baur, G. N€ agele, R. Klein, Physica A 245 (1997) 297.
817
[18] U. Bengtzelius, W. G€ otze, A. Sj€ olander, J. Phys. C 17 (1984) 5915. [19] L. Sj€ ogren, Phys. Rev. A 22 (1980) 2883. [20] S.K. Lai, H.C. Chen, J. Phys.: Condens. Matter 5 (1993) 4325; S.K. Lai, H.C. Chen, J. Phys.: Condens. Matter 7 (1995) 1499; S.K. Lai, S.Y. Chang, Phys. Rev. B 51 (1995) R12869. [21] S.K. Lai, G.F. Wang, W.P. Peng, in: F. Tokuyama, H.E. Stanley (Eds.), The 3rd Tohwa University International Conference on Statistical Physics, vol. CP 519, AIP, 2000, p. 99. [22] G.F. Wang, S.K. Lai, Phys. Rev. Lett. 82 (1999) 3645.
Liquid-glass re-entrant behavior in a charge-stabilized colloidal dispersion G.F. Wang *, S.K. Lai Complex Liquids Laboratory, Department of Physics, National Central University, Chung-li 320, Taiwan, ROC
Abstract Using the static structure factor SðqÞ calculated in the rescaled mean spherical approximation and in conjunction with the idealized mode-coupling theory, we determine the loci of the liquid-glass transition phase boundary for a saltfree suspension of charged colloids prepared in different counterions concentrations. For the simplest deionised chargestabilized colloidal liquid, our calculations demonstrate the possibility of observing, in restrictive regions of the phase diagram, the liquid ¢ glass ¢ liquid ¢ glass re-entrant transition. The latter disappears at a lower concentration of counterions and, when this arises, only the glass ¢ liquid ¢ glass is observed. We study this counterion-concentrationdependent re-entrant phenomenon by analyzing the non-ergodic Debye–Waller factor, SðqÞ and their corresponding spatial counterparts. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction A charged stabilized colloidal system is experimentally observed [1,2] to undergo crystallization at a lower volume fraction (g K 0:15) of macroions. This phenomenon can be understood theoretically by the density functional theory [3–5]. In the context of disordered phase, the Ornstein– Zernike (OZ) equation supplemented by the mean spherical approximation (MSA) closure has proved to be a competent tool if the evaluation of the equilibrium fluid structure is further augmented by the notion of size rescaling on charged macroparticles. Hansen and Hayter [6] applied this rescaling idea to study the SðqÞ of micelle solutions
*
Corresponding author. E-mail address: [email protected] (G.F. Wang).
at very low g(103 ). By modeling the interactions between colloidal particles in the so-called Derjaguin–Landau–Verwey–Overbeek (DLVO) approximation [7] for which an analytical MSA SðqÞ solution [8] exists, they found the calculated SðqÞ agreeing excellently with measured data. For an aqueous dilute solution of charge-stabilized colloids, this rescaled MSA (RMSA) has thus made an impressive achievement in interpreting the SðqÞ of charged colloids. Comparatively fewer studies have reported applying the technique to more concentrated charged colloidal dispersions (g > 0:15). A study on these latter systems is both important and worthwhile since it affords an understanding of the liquid-glass phase transformation whose occurrence was observed to fall into the range of g > 0:2 [2,9]. In this work we use the RMSA SðqÞ in conjunction with the idealized mode-coupling theory
0022-3093/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 2 ) 0 1 5 2 4 - 7
G.F. Wang, S.K. Lai / Journal of Non-Crystalline Solids 307–310 (2002) 812–817
(MCT [10]) to determine the liquid-glass transition boundaries under different screening conditions. An interesting result in our calculations is the observation of the transition from the liquid ¢ glass ¢ liquid ¢ glass (LGLG) re-entrant phenomenon in a restrictive region of the concentration of counterions to the GLG behavior at a lower concentration. A similar re-entrant phenomenon has recently been predicted by Wilke and Bosse [11] for an ionic glass.
H€ uckel screening length in which a2i ¼ 4pLB qi Zi2 , LB and Zi being the Bjerrum length and the charge of a colloid or a small ion, respectively. Carrying out an inverse Fourier transform (see Ref. [13] for mathematical details) to Eq. (2), we obtain 2 2 ceff 00 ðrÞ ¼ LB Z0 X
ekD r ekD r ¼ r0 c r r
r > r0 ; ð3Þ
where the coupling parameter j hj j j i cosh þU sinh X ¼ cosh 2 2 2 2
2. Theory In this section we introduce the effective onecomponent model previously developed by Belloni [12] and shown recently by Lai et al. [13,14] to be an adequate model for studying the structure of concentrated charged colloids. In addition, we summarize the idealized MCT for describing the latter system. 2.1. Mean spherical approximation Consider the multicomponent coupled OZ equations given by X Z hij ðrÞ ¼ cij ðrÞ þ q‘ hi‘ r r0 c‘j ðr0 Þdr0 ; ð1Þ ‘¼0
where species i, j and ‘ are that i; j; ‘ ¼ 0 for macroions, i; j; ‘ ¼ 1 for counterions, and i; j; ‘ ¼ 2; 3; . . . for other small ions. Here q‘ is the number density for species ‘, cij ðrÞ is the direct correlation function and hij ðrÞ ¼ gij ðrÞ 1 is the total correlation function which is defined in terms of the pair correlation function gij ðrÞ. Adelman [15] has shown that Eq. (1) can be contracted to an effective direct correlation function ceff ðrÞ whose Fourier transform can be cast [12,13] to read 2 X s s ^ ^ c^eff ðqÞ ¼ c ðqÞ þ ðqÞ c 00 00 0i h
813
i¼1
a0 þ
P
i¼1
ai c^s0iðqÞ
q2 þ kD2
i2 ;
ð2Þ
where superscript s stands for the short range P component, kD1 ¼ 1=ð i¼1 a2i Þ1=2 is the Debye
ð4Þ depends not only on j ¼ kD r0 but also on g through U ¼ ð8f=j3 2t=jÞ in which f ¼ 3g= ð1 gÞ, t ¼ ðKr þ 2fÞ=½2ð1 þ fÞ þ Kr and ðK2r j2 Þ½2ð1 þ fÞ þ Kr 2 ¼ 96LB gðZ02 =r0 Þ which, given Z02 =r0 , j and g, is solved iteratively for Kr (and hence X). Note that X ! expðj=2Þ=ð1 þ j=2Þ in the limit of q0 ! 0, a linearized DLVO result. This implies that the c^eff 00 ðqÞ (and hence the SðqÞ ¼ 1= ½1 q0 c^eff 00 ðqÞ Þ above is appropriate for the description of a suspension of charged colloids at any finite concentration. 2.2. Mode-coupling theory We turn to introduce the MCT appropriate for a suspension of charge-stabilized colloids. The scattering function F ðr; tÞ whose Fourier–Laplace transformed F^ðq; zÞ can be shown to be related to ^ ðq; zÞ by the memory function M F^ðq; zÞ 1 R^ðq; zÞ ¼ 2 ^ ðq; zÞ=SðqÞ SðqÞ z þ q D0 =SðqÞ M ð5Þ where D0 is the Stokes–Einstein diffusion coefficient of a single particle. All the dynamical characteristics are embodied in the memory function ^ ðq; zÞ. It can be inferred from the works of SzaM mel and L€ owen [16] and others [17,18] that there exists an idealized liquid-glass transition at a dynamical transition point which is determined by the following non-linear vitrification equation
814
G.F. Wang, S.K. Lai / Journal of Non-Crystalline Solids 307–310 (2002) 812–817
f ðqÞ SðqÞ ^ ¼ C 1 f ðqÞ D0 q2
ðq; t ¼ 1Þ:
ð6Þ
Here the solution fc ðqÞ ¼ Rðq; t ! 1Þ 6¼ 0 is the glassy Debye–Waller factor, a non-ergodic state, whereas f ðqÞ ¼ 0 is the ergodic state. By making a two-mode approximation [17–19], the above can be put in a form suitable for numerical work Z 1 ^ ðq; tÞ ¼ q0 D0 C dx x 8qp2 0 2 Z jxþqj x y 2 eff c^00 ðxÞ c^eff dy y ðyÞ 00 2q jxqj 2 q eff þ c^eff ðxÞ þ c ðyÞ 00 2 00 SðxÞSðyÞRðx; tÞRðy; tÞ:
ð7Þ
Given SðqÞ, Eqs. (6) and (7) constitute two nonlinear coupled equations which we will solve iteratively for the loci of the liquid-glass transition (for technical details the interested readers should consult Lai et al. [20] applied for cases of atomic liquids).
3. Numerical results and discussion In all of the calculations below we have (a) kept the charged colloids in water and maintained the colloidal dispersion at room temperature, (b) considered monovalent counterions as the only source of screening small ions and (c) imposed the condition of charge neutrality, q1 Z1 ¼ q0 Z0 , for
Fig. 1. Phase diagrams of charge-stabilized colloidal dispersions for (a) cc –gc , (b) Z0c –gc , (c) rc0 –gc and (d) rc0 –Z0c . Notations used are: j ¼ 1:5 (solid circles), 2 (open circles), 2.65 (solid diamonds), 3 (solid triangles), 3.8 (open triangles). The region of lowest values gc (xaxis is self-explanatory and y-axis varies from rc0 ¼ 0 in step of 1000 up to 5000) is enlarged in the insert in Fig. 1(c) for j ¼ 1:5 (solid circles), 2 (open circles), 2.65 (full curve), 3 (dashed curve) and 3.8 (open triangle).
G.F. Wang, S.K. Lai / Journal of Non-Crystalline Solids 307–310 (2002) 812–817
the self-consistent calculation of j. As a result, the Z0 and r0 of macroions have to be adjusted [13]. Fig. 1(a) shows the cc gc phase diagram calculated at j ¼ 1:5 and 2. For convenience, we have included in the same figure our previous results of j ¼ 2:65, 3 and 3.8 [21]. We note first of all that an increase in j will lead to a larger ergodic region. This general scenario typical for the hard-core Yukawa potential can be understood physically by analyzing the dependence of c on the Z02 =r0 and X [21]. Fig. 1(b) and (c) show the detailed changes of Z0c gc and rc0 gc respectively and, for completeness, we depict in Fig. 1(d) the relation of rc0 and Zc0 . An important finding of Fig. 1(b)–(d) is the occurrence of a LGLG re-entrant phenomenon for 2 < j < 3:8; the LGLG transition disappears for j < 2, and when this happens, only the GLG transition is observed. To delve into the re-entrant behavior, we con) charged sider a monodisperse (r0 12 606 A colloidal system (see the interception of the horizontal lines in Fig. 1(c) and (d)). Corresponding to this r0 , we display respectively in Fig. 2(a) and (b) the two fc ðqÞ and SðqÞ calculated at j ¼ 1:5 (solid circles). One notices immediately that the fc ðqÞ and SðqÞ show concerted oscillatory structures. To see the role played by the charge of a macroion, we compare these results with a system of neutral hard spheres previously shown to have undergone structural arrested at gc ¼ 0:516 [20]. For this system the purely excluded volume effect has resulted in its fc ðqÞ delineating a rather short-ranged behavior. Thus, for the case of higher concentrated lower charged colloids (gc ¼ 0:5, Z0c ¼ 329e), it will be driven predominantly by the geometric restriction with its fc ðqÞ mainly of a shorter-range order. This is in contrast to the less concentrated (gc ¼ 0:175) higher charged (Z0 ¼ 943e) case which reflects a change in localization mechanism being manifested primarily by the electrostatic effect and secondarily by the hard-core factor. Accordingly, the longer-range order necessary for ensuring the structural arrest of charged colloidal particles should differ drastically from the hard-core effects. This feature has in fact shown up in Fig. 3(a) for the spatial critical amplitude H ðrÞ which is the Fourier transform of
815
Fig. 2. (a) Glassy Debye–Waller form factor fc ðqÞ and (b) static ) structure factor SðqÞ for a monodisperse (r0 12 606 A charge-stabilized colloidal dispersion calculated at j ¼ 1:5 for the cases (gc ¼ 0:5, Z0 ¼ 329e) (dashed curve) and (gc ¼ 0:175, Z0 ¼ 943e) (full curve). The neutral hard-sphere results (dotdashed curves) are included for comparison.
SðqÞhc ðqÞ [20]. It is interesting to see the lower-g higher charged colloidal H ðrÞ [10] revealing a ‘softening’ in the local collective excitation in the process of b-relaxation. Consistent behavior is also observed for the pair correlation function gðrÞ depicted in Fig. 3(b) where the gðrÞ of the gc ¼ 0:175 (gc ¼ 0:5) charged colloids shifts outwards (inwards) in agreement with the overall change of fc ðqÞ in Fig. 2(a). At this point, we should emphasize that in approaching the lowest values g, the macroion charge Z0 needs increased to strong enough (Coulombic) coupling to ensure structural arrest of charged colloidal particles. In view of this
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G.F. Wang, S.K. Lai / Journal of Non-Crystalline Solids 307–310 (2002) 812–817
we determine the parametric phase diagram for a deionized charge-stabilized dispersion. In the restrictive region 2 < j 6 3:8 of the phase diagram r0 g or Z0 g, we observe the LGLG re-entrant transformation. The LGLG re-entrant phenomenon disappears for j < 2 and is replaced instead by the GLG transition. The fc ðqÞ and SðqÞ and their spatial counterparts, H ðrÞ and gðrÞ, were analyzed to illustrate their correlations with the reentrant behavior. It is conjectured that the disappearance of the LGLG transformation for j < 2 is due to the ineffectiveness of Coulombic coupling alone in driving the system into an ergodic–nonergodic transition at a much lower g. We note in passing that the calculation presented here is quite general for the method, with slight and straightforward modification, could be used to investigate the phase diagrams of a more realistic chargestabilized colloidal solution such as a system of monodisperse polystyrene charged spheres in the presence of electrolytes [22].
Acknowledgements
Fig. 3. (a) Critical spatial amplitude H ðrÞ and ðbÞ pair correlation function gðrÞ for the same charge-stabilized colloids as in Fig. 2. Refer to Fig. 2 for notations.
we conjecture 1 that the disappearance of the LGLG phenomenon for j < 2 is indicative of the ineffectiveness of Coulombic coupling alone in driving charged colloids into an ergodic–nonergodic transition at a much lower g. 4. Conclusion Using a realistic hard-core Yukawa potential to construct SðqÞ and applying the idealized MCT,
1
We are currently looking into the possibility of understanding it analytically following along the line of Ref. [11] in the lowest g region.
This work has been supported in part by the National Science Council (NSC89-2112-M-008064). S.K.L would like to thank Professor W.K. Liu for hosting his visit to the University of Waterloo during which period this paper is written to fruition. We acknowledge continual support from the National Central University.
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