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Crystallization kinetics of colloidal binary mixtures with depletion attraction† Anna Kozina,*ab Pedro D´ıaz-Leyva,acd Thomas Palberge and Eckhard Bartschac In this work the crystallization kinetics of colloidal binary mixtures with attractive interaction potential (Asakura–Oosawa) has been addressed. Parameters such as fraction of crystals, linear crystal dimension and crystal packing have been quantified in order to understand how the crystal formation is driven in terms of the depth of the attractive potential and the composition of the binary mixture (described by the number ratio). It was found that inside the eutectic triangle, crystallization is mainly governed by nucleation and the crystal packing is close to the close-packing of hard spheres. Moving out from the eutectic triangle towards small component results in the crystallization of small spheres. Enrichment of the eutectic mixture with large component results in the crystallization of both large and small spheres,

Received 2nd October 2014 Accepted 7th October 2014

however, the kinetics are completely different from those of the eutectic composition. Crosslinked polystyrene microgels with nearly hard sphere interactions were used as model systems. Attraction was

DOI: 10.1039/c4sm02193b

introduced by addition of linear polystyrene. The time evolution of crystallization has been followed by

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static light scattering.

I.

Introduction

Hard sphere uids and crystals are well recognized models in Statistical Physics and So Matter Physics for several reasons. First, they are conveniently studied by analytical theory and computer simulation including important effects like hydrodynamics, polydispersity and potential soness. For example, for size monodisperse hard spheres the positions of freezing and melting are xed solely by the volume fraction f.1 The values bounding the coexistence regime between the uid and face centred cubic crystalline phase are ff ¼ 0.492 and fm ¼ 0.545.2 Allowing for polydispersity, the phase boundaries are shied, the coexistence region is widened, and additional crystal phases appear.3–6 Also, potential soness affects the phase boundary location.7 Second, there are experimental realizations of nearly hard colloidal spheres available.8–11 Suitable choice of the suspending organic solvent assures a good match of solvent and sphere refractive index and mass density.

a

Institut f¨ ur Makromolekulare Chemie, Albert-Ludwigs-Universit¨ at Freiburg, Stefan-Meier-Str. 31, 79104 Freiburg, Germany. E-mail: [email protected]

Instituto de Qu´ımica, Universidad Nacional Aut´onoma de M´exico, Ciudad Universitaria, 04510 M´exico D. F., Mexico

b

c Institut f¨ ur Physikalische Chemie, Albert-Ludwigs-Universit¨ at Freiburg, Albertstr. 21, 79104 Freiburg, Germany

Departamento de F´ısica, Universidad Aut´onoma Metropolitana Unidad Iztapalapa, Av. San Rafael Atlixco 186, 09340 M´exico D. F., Mexico

d

e

Johannes Gutenberg-Universit¨ at Mainz, Institut f¨ ur Physik, Staudingerweg 7, 55128 Mainz, Germany † Electronic supplementary 10.1039/c4sm02193b

information

(ESI)

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available.

See

DOI:

The resulting systems are well accessible by optical methods like microscopy12 and light scattering13–15 up to largest volume fractions. Combining both approaches in a fruitful collaborative effort, important aspects like phase behaviour, glass transition and crystallization kinetics of one component colloidal spheres have been extensively studied.16–27 More recently, also binary mixtures came into focus. These show a rich phase behaviour depending on the size ratio a ¼ RS/ RL (where R denotes the particle radius and the suffixes S and L refer to the small and large component, respectively), on the molar fraction of small spheres, xS, and overall volume fraction f. Early theoretical work on hard spheres suggested the formation of zero miscibility eutectic phase diagrams. Later work revealed a low, but nite miscibility leading to the formation of compounds at certain size ratios and compositions with peritectic and other types of more complicated phase behaviour.28,29 Spindle type phase diagrams are expected for a > 0.94; azeotropes should occur for 0.87 < a < 0.94 and eutectic phase behaviour seems restricted to a small range of 0.75 < a < 0.87.30–33 Further, it is expected for binary HS mixtures, that the pathway to crystallization will change or become complicated due to interfering effects of phase separation and compositional ordering.30 Experimentally, either stoichiometric compounds or amorphous states have been observed.34–39 This is in contrast to theoretical expectation but also to the common observation of simple substitutional alloys in binary charged sphere mixtures.40–43 So far, spindle type, azeotropic or eutectic phase behaviour have not been reported for pure hard spheres. In fact, also in our own recent work we failed to observe eutectic

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behaviour in a nominally eutectic mixture of size ratio a ¼ 0.79 and f ¼ 0.567.44 Upon increasing the volume fraction we instead observed vitrication. This observation is in line with earlier work35–37 and has also been exploited to obtain a model system for studies of the hard sphere kinetic glass transition.45 To circumvent vitrication we added small amounts of linear non-adsorbing polymer. In general, added polymer acts as a depletion agent and induces an attractive term in the potential of mean force.46–48 Attraction has a number of interesting effects.23,49,50 It shis the freezing pressure,2 alters the phase behaviour,49,51–53 changes crystallization pathways54 and mechanisms55–57 as well as the kinetics of nucleation growth and ripening.58–60 It further enables the formation of an attractive glass.61 For the eutectic system mentioned above, it should have the effect of increasing the system mobility62–65 and enhancing the tendency to crystallize66 and to fractionate.67 In fact, by adding the free polymer, we were for the rst time able to identify the formation of a eutectic phase diagram with complete miscibility gap for a depletion attractive hard sphere system. The present work goes beyond that study in several ways. First, we have extended the data base considering more samples at different polymer concentration, composition and/or overall volume fraction as well as comparing light scattering to microscopy data. Second, we give a thorough account of the crystallization kinetics in terms of the crystallinity (i.e. the fraction of crystallized material), crystal size and volume fraction within crystallites. This is done separately for small and large spheres. It shows qualitative differences between the two sphere species and in comparison between samples of different compositions and different polymer content. Third, we attempt a kinetic interpretation following lines suggested in literature. In general, all crystals observed in our attractive samples remain much smaller than in the repulsive HS case and further are considerably compressed to volume fractions close to the close packing limit. We nd small conversion exponents m # 1 and extremely small crystal growth velocities of a few microns per day. The overall crystallinity does not exceed 0.6. This all is compatible with a conversion controlled mainly by nucleation events as is expected for attractive systems with large polydispersity. We argue that the nucleation events of both species are controlled by composition uctuations and fractionation rather than by density and/or structural uctuations as seen in single component, low polydispersity HS systems. On the other side, our data also show a strong inuence of the polymer content governing the strength of the attraction. We observe that crystallization proceeds fastest within a small range of polymer concentrations. We argue that for larger attractive strength the overall particle mobility becomes very small, while for lower attractive strengths the nucleation barriers are too large for a rapid conversion. Finally, our systematic study also reveals some unexpected details which are beyond our present understanding. These are shortly discussed, but their explanation will afford further experiments and theoretical assistance. The remainder of the paper is organized as follows. We rst present the experimental details in Section II, including sample characterization (II.A), light scattering experiment (II.B) and

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data interpretation (II.C). Our results are presented in Section III followed by a detailed discussion in Section IV. We close with some short conclusions.

II. A.

Experiment Sample description

The particles used in this study are internally crosslinked polystyrene microgel spheres (crosslink density ¼ 1 : 50) synthesized by the method of emulsion polymerization.68,69 The standard synthesis includes the polymerization of polystyrene, using styrene (Merck Schuchardt OHG, Germany) as the monomer and 1,3-di-iso-propenylbenzene (Fluka Chemie AG, Switzerland) as a crosslinker, and careful purication to remove uncrosslinked polymer. The particles are then suspended in a ‘good’ nearly iso-refractive organic solvent, 2-ethylnaphthalene (Sigma Aldrich Chemie GmbH, Germany; at T ¼ 20  C, n2en ¼ 1.594 and 1.59 < nps < 1.60) which is also density matched for polystyrene (r2en ¼ 0.992 g cm3; rps ¼ 1.05 g cm3). Since the particles swell in the organic solvent, their size was obtained by combining the information about the phase behaviour and interaction potential.45 We presume that the interactions may be reasonably described by the inverse power potential U(r) f 1/rk with k ¼ 40  2. As described elsewhere,63 the found interaction potential can be considered sufficiently hard to approximate that of the repulsive hard-sphere system. Power exponent k was obtained from oscillatory rheological measurements70 and combined with the width of the experimentally observed crystal coexistence region. That allowed us to dene thermodynamic volume fraction f and degree of swelling Q ¼ f/fdry with fdry volume fraction of the microgels in a ‘bad’ solvent (water). Then, the radii of the swollen spheres were obtained as pffiffiffiffi R ¼ Rwater 3 Q. The binary mixture was prepared by combining large particles L (Rh ¼ 193  3 nm) with smaller particles S (Rh ¼ 152  4 nm). The size ratio a ¼ RS/RL is 0.79. To introduce attraction, a certain amount of linear polystyrene (Rg ¼ 13.1 nm, Mw ¼ 133 kg mol1; PSS GmbH, Germany) was added to repulsive dispersions. The attractive potential is the Asakura–Oosawa potential46–48 described by 8 r # 2Rh ; < þN 2Rh \r # 2Rh þ 2Rg ; (1) Udep ðrÞ ¼ rN;pol kB TVOv : r . 2Rh þ 2Rg : 0 The attractive strength varies with the amount of added linear polymer in the system. The interaction potential is proportional to the thermal energy kBT, to the interaction range d ¼ Rg/Rh (where Rh is the colloidal particle radius), to the ‘overlapping volume’ VOv between spherical regions of thickness Rg surrounding the colloidal particles, and to the polymer number concentration rN,pol. For convenience, instead of the numerical value of rN,pol we express the polymer contribution in terms of the relative concentration cP/c*P. Here c*P ¼ 23.5 g l1 is the overlap concentration of linear polystyrene in 2-ethylnaphthalene calculated as 3Mw/(4pNARg3), where NA is the Avogadro constant. The interaction range d was chosen to provide a short-range attraction with dS ¼ 0.086 and dL ¼ 0.0679.

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For the binary mixtures used in this work, taking into account molar fractions of each component in a binary mixture, the average range of attraction is hdi ¼ 0.08  0.005. Polydispersity qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi. index s ¼ hR2 i  hRi2 hRi for each component is about 0.08,

Table 1 Binary mixtures designation for attractive (A) samples: overall suspension volume fraction f, volume fractions of each component in a binary mixture fL,S, particle number ratio NS/NL, molar fraction of S spheres xS, and relative concentration of free polystyrene cP/c*P. For * comparison, relative reservoir concentration cfree P /cP is also given

which was determined by comparison of scattered intensity from the dilute solution with the theoretical form factor curve for a certain polydispersity. Polydispersity index s of the binary mixtures is about 0.1  0.002. Aer preparation, each sample was homogenized by tumbling over a few days and then le undisturbed for the whole time of measurements. The phenomena in the systems of our interest are sensitive to the two main factors: the number ratio between the small and large particles (or in other words, composition) NS/NL and the attraction strength given by cP/c*P. Thus, we varied these two parameters xing the overall volume fraction to f ¼ 0.567  0.006, which was rather close to the glass transition fg ¼ 0.573  0.002 obtained using the predictions of the mode coupling theory71,72 as it is described elsewhere.45 The samples with different compositions in terms of partial volume fractions and attraction strengths are mapped on the phase diagram in Fig. 1 (see gure caption for explanation). The studied samples are also listed in Table 1 where the composition parameters of the binary mixtures such as overall dispersion volume fraction f, volume fractions of each

Sample

f

fL

fS

NS/NL

xSa

cP/c*P

b * cfree /cP P

A1 A2 A2a A3 E6a E6 A4a A4 A5

0.567 0.553 0.573 0.568 0.568 0.568 0.575 0.565 0.568

0.567 0.408 0.290 0.245 0.245 0.245 0.212 0.096 0

0 0.145 0.283 0.323 0.323 0.323 0.363 0.469 0.568

— 0.87 2.41 3.25 3.25 3.25 4.22 11.97 —

0 0.47 0.71 0.77 0.77 0.77 0.81 0.92 1.00

0.53 0.53 0.53 0.53 0.51 0.44 0.53 0.53 0.53

1.82 1.84 1.86 1.82 1.75 1.51 1.88 1.80 1.82

a Calculated as xs ¼ rs/(rS + rL) where ri ¼ 3fi/4pRh,i3 is the particle number density. b Calculated as in ref. 75.

component in a binary mixture fL,S, particle number ratio NS/NL, molar fraction of S spheres xS, and relative concentration * of free polystyrene cP/c*P are shown. The parameter cfree P /cP has been calculated as done in ref. 75. Sample names are chosen in such a way that they correspond to the same names in the previous paper.44 The samples that were not reported previously

Fig. 1 Three dimensional phase diagram representing compositions of the samples as a function of partial volume fractions fL and fS and attraction strength given by cP/c*P. The horizontal plane marked by the blue line is the theoretical phase diagram for binary mixtures of similar size ratio suggested by Bartlett et al.29,73 Blue phase transition lines separate different coexistence regions. The theory predicts the existence of the eutectic region where L crystals + S crystals + fluid are expected to occur. Binary mixtures with purely repulsive interactions but different compositions (equivalent to A-samples) are given by blue circles. Vertical semi-diagonal black plane represents the phase diagram for binary mixtures with the composition of sample A3 when the total volume fraction (left lower corner to almost right upper corner) and attraction strength (from the lower horizontal plane up) are increased. In this plane black solid line shows the gel/attractive glass transition verified for the systems used in this work, dotted line represents kinetic transitions obtained in the previous work on similar systems.74 The intercept of the black dotted line with the semi-diagonal is the repulsive glass transition volume fraction fg obtained experimentally using the scaling laws of the MCT. The samples with the same eutectic composition but different attraction strength A3, E6a and E6 are given by black circles; they lie in the vertical plane. The samples with the same attraction as A3 but different compositions are given by red circles. Their projections to the horizontal and vertical planes are guided by the dashed grey lines. RG and AG denote ‘repulsive glass’ and ‘attractive glass’, respectively.

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have the letter ‘a’ at the end of the name. Samples A1 and A5 are added to the table for completeness, although the kinetics for these samples are not reported here, since the crystallization process was too fast to be tracked as compared to the rest of the samples.

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B.

Light scattering

The crystal structure and crystallization kinetics were studied by time resolved SLS. The spectrometer (modied Soca, SLS Systemtechnik, Germany consists of a cylindrical thermostatted (T ¼ 20  0.1  C.) sample holder lled with toluene (refractive index of toluene nD20 ¼ 1.523  0.002, close to that of quartz nD20 ¼ 1.550  0.002). A cylindrical quartz cell (Hellma 540.110 with inner diameter of 8 mm) containing the dispersion was mounted inside the sample holder and illuminated by 1 mW solid-state laser (wavelength l ¼ 405 nm; Z-LASER Optoelektronik, Germany). The scattered intensity I(q) was detected over a range of angles from q ¼ 25 to 145 with a resolution of 1 . With the laser beam perpendicular to the axis of the rotation of the goniometer arm with the avalanche photodetector mounted, the magnitude of the scattering vector is q ¼ (4pn/l)sin(q/ 2), where n is the refractive index of the solvent and l the laser wavelength. The sample was placed in the spectrometer immediately aer shear melting by tumbling dening waiting time tW ¼ 0. The experiment was repeated with increasing waiting time tW providing data sets for I(q,tW).

C.

Data analysis

For the studied systems the natural tendency is to crystallize with formation of many randomly oriented crystallites. Thus, a large number of crystallites in the scattering volume should provide a reasonable estimate of the diffraction pattern. For the uid dispersion of colloidal particles as well as for the polycrystalline samples, time resolved scattering intensity I(q,tW) can be written76 I(q,tW) f P(q)S(q,tW),

(2)

here q is the scattering vector, P(q) is the single-particle form factor and S(q,tW) is the time resolved structure factor. Form factors were determined by measuring I(q) of dilute dispersions (for which S(q) ¼ 1). Time resolved structure factors were obtained by measuring I(q,tW) for the concentrated dispersions, dividing by the appropriate form factors and normalizing by the concentration ratio as Sðq; tW Þ ¼

Iconc: ðq; tW ÞCdilute : PðqÞCconc:

(3)

Stacking faults along with scattering from the coexisting uid make it difficult to accurately isolate the Bragg-reection, especially on the early stage of crystallization. To correct for this contributions to S(q,tW) we used the procedure suggested by Harland and van Megen77 and adopted by other authors.78,79 The scattering from the crystals Sc(q,tW) was calculated as

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Sc(q,tW) ¼ S(q,tW)  b(tW)S(q,tW ¼ 0),

(4)

here b(tW) is a scaling factor chosen so that there are no negative values for Sc(q,tW) between the scattering vectors q1 and q2 limiting a Bragg-reection; S(q,tW ¼ 0) is the structure factor of the metastable uid and is assumed to include the diffuse scattering due to random stacking. From Sc(q,tW) of the Bragg-peaks the following quantities were obtained.77 First, the area under the main reection is proportional to the amount of sample converted to the crystalline phase – crystallinity X(tW). Since our system was composed of two species of particles, each of which might crystallize, we considered separately the fraction of each component in the sample converted from uid into crystal XL,S(tW). For each component we normalized the amount of crystals (the volume taken by the crystals L or S) to the overall volume occupied by the particles in the sample, given that the total volume of the crystalline phase in the binary mixture is [1  b(tW)]. Thus, in our case XL(tW) indicates the fraction of crystals L in the sample. In this case, the sum XL(tW) + XS(tW) gives the total crystallinity of the sample. Crystal fractions are then given by ð q2 Sc ðq; tW Þdq; (5) XL ðtW Þ ¼ cL q1

XS ðtW Þ ¼ cS

ð q2 q1

Sc ðq; tW Þdq:

(6)

here cL and cS are the normalization constants chosen so that XL(tW) + XS(tW) ¼ [1  b(tW)] and XL(tW)/XS(tW) ¼ AL/AS with A being the area under the (111) Bragg-reections; q1 and q2 are the scattering vectors limiting the (111) peaks for each component. By normalization in this way we make the following assumptions: (i) we equate the total crystallinity of the sample to [1  b(tW)] by this underestimating the amount of ordered phase, since the scattering due to random stacking is also subtracted from the scattered intensity; (ii) there is no composition difference between uid and solid phases; (iii) we dene XL,S(tW) as the molar fraction of particles localized in the crystalline phase assuming that all the particles scatter identically. Thus, the determined XL,S(tW) values should not be considered as absolute measures of crystallinity but rather as giving a rough estimate and the order of magnitude. Nevertheless, this allows elucidating the time dependence of crystal nucleation and growth. Second, the average linear crystal dimension Dlin(tW) is shown in units of particle diameters 2R (where R ¼ Reff is the effective radius of the particle) and is given by77 Dlin ðtW Þ ¼

pK ; DqðtW ÞReff

(7)

where K ¼ 1.155 is the Scherrer constant for a cubic shape crystal and Dq(tW) is the width at half maximum of qmax(tW). Although the particles used in this work may be well approximated as hard-spheres, there is a discrepancy between the hydrodynamic radius Rh and the effective radius Reff reecting the interaction distance. We attribute this discrepancy to the slight ‘soness’ of the spheres on their surfaces resulting from

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the solvent penetration, and the ‘fuzzy layer’ on the surface of the spheres due the short polymer chains providing steric stabilization. Third, from the location of the maximum qmax(tW) the volume fraction of a fcc crystal fc(tW) can be calculated as77

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3 2  fc ðtW Þ ¼ pffiffiffi qmax ðtW ÞReff : 2 9 3p

(8)

To obtain Reff we applied the procedure used by Martin et al.78 We assume that in the repulsive system composed only of large or small spheres at the volume fraction of uid-crystal coexistence ff < f < fm there is a thermodynamic equilibrium between the phases aer a certain waiting time. Thus, the expected fc(tW) may be equalled to fm of the one-component dispersion, which was also conrmed by the experiment.77 Therefore, we studied the phase behaviour of one-component repulsive system containing only L or S spheres and xed freezing ff and melting fm volume fractions for each component. Because our particles are slightly soer than hard spheres, we rescale determined ff to ff ¼ 0.49 of ideal hard spheres. Then, aer rescaling we have for large spheres fLm ¼ 0.531 and small spheres fSm ¼ 0.528. We measured S(q) of the repulsive systems made of each component and using qmax of the rst Bragg-reection we obtained effective radii from eqn (7), RLeff ¼ 170.5  2 nm and RSeff ¼ 126.2  3 nm. The error bars for all the data were calculated as follows: the data were adjacently averaged using two points and then a deviation from average for each point was calculated. Only representative error bars are shown.

III.

melting two main peaks corresponding to (111) fcc-reection of each component appear. Since the appearance of the peaks occurs simultaneously, this indicates that the crystallization of both components takes place at the same time. The S(111) peak is more prominent in the beginning, however, as the crystals made of large spheres grow, two more reections L(220) and L(311) appear on the 6th day of measurement corresponding to crystallization of large spheres. Fig. 3 shows the fraction of particles of each component converted into crystal X(tW) by the waiting time tW normalized according to eqn (5) and (6). Samples A6 and A7 showed the kinetics very similar to A3 and, to avoid plot overloading, are presented in the ESI.† There are two main regions in each graph. The initial stage of conversion is attributed to the nucleation and growth and usually has the fastest rate. The following region is the slower rate conversion that is attributed to coarsening.77,80,81 The crossover time separating these two regimes for A3 tcross is also shown. Samples A3, A4a, A2a as well as very similar to A2a samples E6a and E6 (see ESI†) show simultaneous crystallization of L and S particles, conrming their eutectic nature. However, at long waiting times the total fraction of crystals slightly increases for A4a and decreases for A2a as compared to A3. Sample A4 showed only the crystals made of small spheres, which is expected, since the sample is in the region of S + uid coexistence. Only the main (111) reection of the crystallites made of S

Results

The evolution of the structure during crystallization of sample A3 is represented in Fig. 2. On the second day aer shear

Structure factor Sc(q) for sample A3 from bottom to top showing the appearance and evolution of the crystal structure. The Bragg-reflections are indexed on the fcc structure with L for large and S for small spheres. Fig. 2

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Fig. 3 Fraction of crystals X(tW) made of L or S spheres in the overall volume of the samples vs. waiting time tW, the upper plot for small and the lower plot for large spheres. The symbols for each sample are given in the legend as well as the molar fraction xS. Crossover time tcross shown for A3 separates two conversion stages.

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component was detected and no peaks of crystals L appeared over 30 days of waiting. Since the crystallinity of the sample A4 is only due to the crystals S, we did not normalize X(tW) to the total amount of crystals in the binary mixture. Thus, here X(tW) is the total crystallinity of the sample given by [1  b(tW)]. We could observe that A4 crystallized rather fast and we detected only the last stage of rapid conversion when the nucleation is almost complete. It took the system less than one day to form about 20% of crystals out of the whole volume. That was much faster as compared to the samples in other regions of the phase diagram. Sample A2 showed the most curious behaviour out of all the samples. Since A2 is located outside the eutectic triangle, by analogy with A4, one would expect that only large spheres should crystallize. Indeed, they crystallize and form rather large percentage of crystals as it is seen from Fig. 3. However, surprisingly, small spheres also crystallize, although their crystallization is not expected. Moreover, small spheres do not crystallize in the same way as in the other samples. They rst form a very small fraction of crystals that persist during the rst 7–10 days and then the crystal fraction increases dramatically up to 40 days of measurement, while the fraction of L crystals stays already constant. This is a very important experimental nding. Possible origins of such behaviour will be discussed below. The crystallinity of each component in samples E6a and E6 (see ESI†) practically follows the kinetics of sample A2a, however, the total crystallinity at long waiting times for these two samples is slightly higher than that for A2a. The data are summarized in Table 2 where all the main parameters of crystallization kinetics are shown. Fig. 4 shows the logarithm of the fraction of crystals formed by each component log10[X(tW)] vs. the logarithm of waiting time log10(tW) for A3 and A2 as the most representative samples. The stage of rapid conversion (before tcross) may be described by the power-laws X(tW)  tWm. Power-law exponents m are obtained from the slopes of the straight lines of the log10[X(tW)] vs. log10(tW) in Fig. 4. While sample A3 shows simultaneous crystallization for both species, the crystallization kinetics of S in A5 up to coarsening may be divided into two regions: slow and fast conversion indicated by the two exponents mi and mS, respectively. The exponents m for the rest of the samples are shown in Table 2. The crystallization rate reected in m does not change much from A3 to A4a, though for all the three samples mS is a bit higher than mL, which implies faster crystallization of S spheres. For A2a the crystallization rate is the highest among the rst

Table 2

Crystallization parameters for studied samples

Sample

XL

XSa

mL

mS

Dlin,L

Dlin,S

fc,L

fc,S

A2 A2a A3 A4a A4 E6a E6

0.334 0.107 0.129 0.162 0 0.117 0.119

0.190 0.246 0.295 0.367 0.386 0.286 0.293

1.03 0.73 0.52 0.61 — 0.71 0.75

0.88b 1.05 0.67 0.74 — 1.23 1.18

7.7 7.7 9.0 7.6 — 17.0a 17.7a

10.1 9.1 8.0 8.2 8.7 17.6a 17.3a

0.733 0.718 0.718 0.718 — 0.696 0.686a

0.724 0.724 0.724 0.724 0.724 0.689 0.705a

a

At the longest tW. b mi  0.1.

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Fig. 4 Logarithm of the fraction of crystals log10[X(tW)] formed by each component vs. the logarithm of waiting time log10(tW), for samples A3 (green) and A2 (grey). Diamonds for large and triangles for small spheres. Power-laws X(tW)  tWm are drawn by the dashed lines of the same colour for exponents m indicated.

three samples. For sample A4 it was difficult to obtain the rate exponent m with a good precision but we expect it to be higher than that for the eutectic samples. In A2 the large spheres crystallize faster as compared to those in A3 and compared to the small spheres. Already aer 3 days the particle transfer from uid to crystal is signicantly slowed down. For the small spheres, the initial relaxation stage has a very low crystallization rate with exponent mi ¼ 0.095, whereas the rate of rapid conversion is signicantly higher with m ¼ 0.88. Samples E6a and E6 showed the highest conversion rate for the small spheres out of all the samples. The crystal growth (in terms of average linear crystal dimensions Dlin(tW)) for the most representative samples is shown in Fig. 5. The size of the crystallites in A3 grows slightly during the whole crystallization process, however, the growth is so small (about 2–3 particle diameters) that we can consider crystal size being virtually constant during the fast crystallization stage. Similar picture is observed for A4a, A2a and A4 with the nal crystal size shown in Table 2. For A2 the data suggest that the size of all the crystallites is about constant starting from the beginning of the solidication process. However, there is a hint of faster growth during the rapid crystallization stage for S crystals even though more precise data would be helpful to interpret them more accurately. Thus, we conclude that the same almost constant crystal size is observed in A2 as it is in A2a. Lower attraction allows crystals to grow almost twice as much as in the deeper quenched samples as it is observed in E6a. Even lower attraction in E6 results in the crystals made of small spheres of the size similar to those in E6a but the crystals made of large spheres grow differently. First, they are very tiny of the order of 4–5 particles during the rst 27 days and then they grow intensively up to 35 days of observation. Thus, it looks like there is an extended stage of crystal growth (up to 40 days) for E6 that is not observed in A3 and E6a.

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Fig. 5 Average linear crystal dimensions Dlin(tW) vs. waiting time tW for the crystallites in samples A3, A2, E6a and E6, the upper plot for small and the lower plot for large spheres. The symbols for each sample are given in the legend.

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colouration but no well dened coloured spots. This indicates that although one nds well-dened Bragg-reections in the scattering pattern of A3, there are no crystals of macroscopic size noticeable by the naked eye in this sample. A rough estimation of the relative proportions of amorphous regions to the crystalline ones by visual inspection of the sample indicated the ratio of 4 : 1. The average tendency with decrease of attraction is quite clear: the crystal size increases with time and reaches average values higher for samples E6a and E6 then for sample A3. Packing fraction of the crystalline phase fc(tW) composed either of L or S is represented by Fig. 7. As it is seen from Fig. 7, the packing of the crystal phase composed of both large and small spheres in A3 is very close to the close-packing fraction for hard spheres fcp ¼ 0.74. This observation is remarkable and unexpected, since in the systems of repulsive hard spheres such a high crystal packing was never observed. Moreover, this packing is seen already on a very early stage of crystallization, practically starting with nucleation, and it stays that high during the whole process. The crystal packing is not affected signicantly by a composition change from A4a to A4. However, moving from the eutectic triangle towards L + uid region represented by A2 results in the appearance of the initial stage of packing where fc slightly grows but eventually reaches almost the same values as in the previous samples. Moreover, in the end, L spheres become even more closely packed in A2 than in A3 – A4a – A2a samples. As we lower attraction, the crystals are allowed to relax like in A6, where the crystal packing stays

Talking about the crystal size it is worth to mention that it is rather small and depends on the quench depth. One could see the difference already with the naked eye. Fig. 6 demonstrates the difference. For example, the crystals in E6a and E6 could be seen by looking at the sample cells under a horizontal microscope without any magnication, whereas the crystallites in A3 could be seen only with the magnication of 2.5. The magnied sample image shows that there is a strong

Fig. 6 Crystallites as seen by horizontal microscope without magnification (upper row) and with 2.5 magnification (lower row). The sample names with magnifications are given below the pictures. Red circles mark the region with crystallites.

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Fig. 7 Crystal packing fraction of the crystalline phase fc(tW) vs. waiting time tW for the crystallites in samples A3, A2, E6a and E6, the upper plot for small and the lower plot for large spheres. The symbols for each sample are given in the legend.

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constant and provides on average the value of fc(tW) ¼ 0.69  0.01. Interestingly, for even weaker attraction in E6 there is a time dependence of crystal packing. S crystals are rather densely packed in the beginning but then they relax as they grow. L crystals, on the contrary, are getting more packed with time reaching the value of fc close to that in E6a.

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IV. Discussion The process of crystallization in a metastable colloidal liquid includes three main stages. First, the crystal nuclei containing a few particles are created. This stage is known as ‘crystal nucleation’, which is usually preceded by the induction period of formation of precursors and subcritical nuclei. When the size of the nuclei is large enough to cross the free energy barrier (critical nucleus), these nuclei start to grow while the new nuclear continue to form. This stage is called ‘rapid conversion’, which is supposed to last until the crystals ll the whole volume. The following it stage is the ‘crystal coarsening’, where the large crystals grow at the expense of small ones and the total crystallinity stays constant. The diffusion is an important issue in particle crystallization. The adaptation of the classical nucleation theory to colloidal systems82 predicts nucleation rate N f Ds with Ds the self diffusion coefficient within the disordered phase. Thus, inside the liquid-crystal coexistence region of HS phase diagram the nucleation rate increases with the increase of f up to melting point fm.77 This is also accompanied by the fast increase of the crystal size. For high colloid volume fractions fm < f < fg, the high crystallization rate is accompanied with the formation of very tiny crystallites as a result of hindered particle diffusion. This is known as ‘nucleation dominated crystallization’. The polydispersity usually brings an additional stage of local fractionation to the crystallization process.78,83 The slight increase of the total crystallinity at long waiting times for A4a and its decrease for A2a as compared to A3 is connected to the change of the particle size distribution. For the original sample A3 it has the form of the bell with the small shoulder on the right due to L spheres. If we add more S spheres to this bell, we make the system slightly less polydisperse. However, adding more L spheres broadens the bell and introduces more disorder. For this reason less polydisperse sample A4a can form more crystals and in more polydisperse A2a less total amount of crystals can be formed. Moving from the eutectic triangle towards the corners of the binary phase diagram changes the tendency completely. Now even less polydisperse sample A4 has lower total crystallinity and more polydisperse A2 results in the highest crystallinity. We connect it with the crystallization of the second component. While in A4 sample L spheres do not crystallize and do not contribute to the overall crystallinity, in A2 sample S spheres do. This indicates that the sample composition is really important for the amount of created crystals. Surprisingly, the lowering of attraction at the same sample composition does not affect the total crystallinity but rather the crystal properties. The crystallization rate reected in m (Table 2) indicates that for all the samples except A2 small spheres crystallize faster than large spheres. For samples A1 and A4a we connect it with

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the higher volume fraction of S as well as with their higher mobility. For A2a the crystallization rate is the highest among the three samples. Apparently, on increase of disorder in the system the mobility (diffusion) of the particles, especially of the small ones, grows. More mobile particles can faster rich each other to form a nucleus, which results in the increase of the total crystallization rate. Recent experiments showed that in polydisperse hard-sphere systems the limited crystallization at early times is governed by the local fractionation process.83 The large number of crystalline precursors appears but the crystal growth is limited resulting into a very low crystallization rate. This picture is very similar to what we observe for crystallization of small spheres in sample A2. Very low conversion rate given by mi suggests that our binary mixture also fractionates prior the stage of rapid conversion and we detect the initial stage when S spheres hardly start forming the nuclei. We presume that all our samples pass this local fractionation stage, though, it cannot be detected easily. We believe that it is connected tightly with the partial volume fraction of S particles fS. If the number of S spheres is sufficiently high (e.g. larger than 0.2), fractionation seems to occur too fast to be detected on a day time scale. However, when fS is low enough (less than 0.2, see Fig. 1), the fractionation stage is rather long (up to 10 days) and is reected in the initial plateau. Larger exponents m for E6a and E6 than for A3 indicate that on weaker attraction the crystallization process is faster although it results in almost the same amount of crystals. The sublinear increase of crystallinity given by m < 1 for S or L spheres in practically all the samples indicates a signicant slowdown of crystallization process. Especially this is true for A3. The fraction of crystals formed by the time tW may be written as X(tW) ¼ Dlin3(tW)Nc(tW)  tWm ¼ tWgtWn,77 where Nc(tW) is the number of average-sized crystals, g is the exponent associated to the average crystal dimension (dened by Dlin3(tW) f tWg) and n the exponent associated to the number of average-sized crystals (dened by Nc(tW) f tWn). The crystal size in A3 stays practically constant during the whole crystallization process together with the increase of crystallinity. This indicates that X(tW)  Nc(tW)  tWn and crystallization is dominated by nucleation. This is also applicable to samples A4a, A2a, A4 and A2. This feature is reminiscent to what was observed in repulsive almost monodisperse hard spheres at high volume fractions of f > 0.55.77 However, in the case of repulsive systems reported in ref. 77 the nucleation was slightly accelerated with n z 1.7 as well as in the case of other experiments on hard spheres reporting m ¼ 4 during the rst crystallization stage.80 A quite recent work on similar single colloid–polymer mixture with low polydispersity of colloids s z 0.06 reported m z 1.33 (ref. 59) together with a hindered crystal growth. However, our case of binary colloid–polymer mixture is the particular one when not only the crystal growth but even the nucleation is extremely slowed down. Why does this happen? There are two main conditions for this. First, a very high polymer concentration close to dynamical arrest. This implies that the particle dynamics are extremely slow, there are more long-lived bonds between the particles, which prevents them from the diffusion over the distances long enough to be included into a crystal.

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Another implication is the saturation of the system with the polymer that brings the issue of considering the system as a ternary mixture. This is because when two particles try to get closer to form a nucleus, they rst need to exclude the polymer from the volume between them to make a contact. The addition of each next particle to the cluster (or pair) requires the further diffusion of the polymer away from the cluster and appearance of the polymer depleted (cluster) and polymer rich (around the cluster) zones. Therefore, the polymer number density becomes the controlling parameter for the crystal precursor formation, nucleation and growth, which was also demonstrated in ref. 59. The second condition is the polydispersity. Not every particle can be included to the cluster having overcome the difficulty made by the polymer. Thus, rst, the system has to locally fractionate. This process slows down the crystallization even further. Of course when we say ‘slowed down’, we compare it with the repulsive or attractive almost monodisperse colloids, because our analogous to A1 repulsive composition did not crystallize at all on a very long time scale.44 In the case of slightly lower attraction (sample E6a), the stage of rapid conversion (up to 10 days) is accompanied by a slight crystal growth (up to 10 days in Fig. 5). It means that the rapid crystallization stage is governed not only by the nucleation but also by the crystal growth. This is the result of slightly less hindered dynamics as compared to A3. However, the dynamics are still slow enough to allow only a few more particles to be included into a crystal, which allows only a small increase of the crystal dimensions. When the attraction lowered further (sample E6), the kinetics also change. The rapid crystallization of S spheres is governed by nucleation and growth, while the rst 27 days of crystallization of L are only due to nucleation. Then, at almost constant crystallinity, there is a burst of crystal growth indicating crystal ripening. A possible origin of this ‘delayed growth’ stems from the difference in particle mobilities. Since S spheres are more mobile just because of their size, it is easier for them to be included into a growing nucleus. This may open some free volume so that L spheres can also rearrange. It is worth to mention that at even lower attraction close to c*P it took the sample several months to crystallize, while at even lower polymer concentration it did not crystallize at all. Thus, there is a certain interval of attraction strength where the crystallization may be observed. It should be large enough to promote crystallization by helping the system to form a stable critical nucleus, and at the same time it should not slow down the particle dynamics too much in order to allow the crystal nucleation and growth. It is known for hard spheres77 that the packing of crystals depends strongly on the total volume fraction f of the metastable uid before crystallization. Thus, colloidal dispersions in the uid-crystal coexistence region should form crystals with an equilibrium packing fc(tW) close to the melting concentration. For f > fm one expects at equilibrium crystals with fc(tW) equal to f. Thus, in this work in the absence of depletion ideally one would expect to obtain maximum crystal packing fc(tW) ¼ 0.568, at least for one of the components. However, we obtained much higher fc(tW) ¼ 0.73, which stays constant in time for the majority of samples. Such a high packing of the crystals is most

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likely due to depletion. In hard sphere crystallization the newly formed crystals experience an osmotic pressure due to the surrounding uid. As the crystals grow they create a depletion zone around them with a reduced particle number density. The concomitant reduction of osmotic pressure allows the crystals to relax and expand, thereby decreasing their volume fraction. Thus, one normally observes a decrease of fc(tW) with time. In the case of colloid–polymer mixtures the osmotic pressure is dominated by the contribution of the free polymer which adds onto that from the coexisting uid. Thus, the crystals are compressed to a much higher packing fc(tW) 0.73. Furthermore, the depletion of single particles in the uid surrounding the growing crystal no longer leads to a decrease of the osmotic pressure as the (dominant) contribution from the free polymer remains essentially unchanged. In addition, the much slower growth velocity of the crystals as compared to a purely repulsive monodisperse colloidal system leads to a much weaker particle depletion from the surrounding uid. Therefore, the crystals cannot relax and remain in a highly compressed state even for long observation times. Our observation of the increase in the crystal packing for A2 during the initial crystallization stage is quite remarkable. Especially, it is pronounced for small spheres and coincides with the time when L spheres rapidly nucleate (up to 10 days) and S spheres fractionate. Thus, the increase of fc of S may be explained by taking into account the fractionation process at that early stage. As the nucleus grows, the particles of the same or similar size may be included into it. However, the free polymer excluded from the nuclei S and L would press them stronger because of the increased polymer number density outside crystals. In other words, the nucleus undergoes higher osmotic pressure, since the ratio d ¼ Rg/Rh decreases, and its growth is hindered as discussed above. There is only one way to include more particles into a nucleus without a signicant crystal growth: use the particles as similar in size as possible and pack them more closely. Aer the nucleation of L is complete, the crystal packing of S stays constant with fc(tW) close to fcp(HS) ¼ 0.74. It is very probable that at very short times the same increase of fc occurs for the other samples A3, A4a, A2a and A4, however, the fractionation is too fast to be detected on our measurement time-scale. Obviously, as we lower the attraction, the crystals are allowed to relax as in E6a and E6. Interestingly, L crystal packing increases in E6, which we connect with the same fractionation process like for S spheres in A2. S spheres in E6 are the only ones showing the expected tendency of crystal relaxation in HS, although they do not relax to the initial binary mixture colloid volume fraction but rather reach much higher packing of 0.705. This happens because the crystal growth is allowed: the crystal that grows relaxes, the nucleus that cannot grow neither can relax.

V. Conclusions We presented a detailed study on the phase behaviour and crystallization kinetics of colloid binary mixtures with depletion attraction. We found that the binary mixture with the eutectic composition does crystallize on introduction of depletion

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attraction with formation of fcc crystals made by each, smaller and larger, component. Both components crystallize simultaneously with formation of fcc crystals suggesting the solidication of a truly eutectic composition. The crystallites are found to be much smaller in size and more closely packed than in the case of repulsive systems. The enrichment of the eutectic composition with S results in the formation of only S crystals. The enrichment of the eutectics with L results in formation of both L and S crystals but crystallization of S is signicantly hindered and there is a stage of local fractionation prior the stage of rapid conversion. At deeper quenches the initial stage is governed mainly by the nucleation and the growth is hampered because of the dynamics restriction. For shallow quenches the nucleation is accompanied by the crystal growth resulting in larger crystals but the growth of one component may be delayed. Crystal packing in attractive systems is extremely high as compared to purely repulsive hard spheres. The average values of crystal packing at long waiting times suggest that the crystals become more densely packed on increase of attraction strength.

Acknowledgements Work supported by German Research Council (Deutsche Forschungsgemeinscha) under Grant no. SFB 428 and Marie Curie Research/Training Network on Dynamical Arrest under Contract no. MRTN-CT-2003-504712. AK thanks “Red Tem´ atica de la Materia Condensada Blanda” (CONACyT, Mexico) and IQUNAM for nancial support during completion of this work. Authors also thank Taco Nicolai for important comments.

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