Glassy states in attractive micellar systems

Physica A 339 (2004) 92 – 100

www.elsevier.com/locate/physa

Glassy states in attractive micellar systems F. Mallamacea;∗ , M. Broccioa , A. Faraonea , W.R. Chenb , S.-H. Chenb a Dipartimento

di Fisica, Istituto Nazionale per la Fisica della Materia, Universita’ di Messina, Vill. S. Agata C.P. 55, I-98199 Messina, Italy b Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, USA

Abstract Recent mode coupling theory (MCT) calculations show that in attractive colloids one may observe a new type of glass originating from clustering e/ects, as a result of the attractive interaction. This happens in addition to the known glass-forming mechanism due to cage e/ects in the hard sphere system. MCT also indicates that, within a certain volume fraction range, varying the external control parameter, the e/ective temperature, makes the glass-to-liquid-to-glass re-entrance and the glass-to-glass transitions possible. Here we present experimental evidence and details on this complex phase behavior in a three-block copolymer micellar system. c 2004 Elsevier B.V. All rights reserved.  PACS: 82.70.Dd; 61.12.Ex; 64.70.Pf Keywords: Structural arrest; Glass; Micelles

1. Introduction Soft materials are of large interest in science and technology [1]. Under appropriate conditions (density and load) they can evolve to a jammed status [2] where the system particle is caught in a small region of phase space with no possibility of escape. In these conditions the material resembles a solid because it is driven into a structural arrest (SA) status. Recently, by considering colloids with attractive interactions, a “general” phase diagram for jamming transition has been proposed and the SA can be driven by 

The research at MIT is supported by a grant from Materials Science Division of US DOE. The research in Italy is supported by the PRIN2002-MIUR project. ∗

Corresponding author. Tel.: +39-090-6765016; fax: +39-090-395004. E-mail address: [email protected] (F. Mallamace).

c 2004 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter
doi:10.1016/j.physa.2004.03.048

Hexagonal

Liquid L G L

2.5

Repulsive Glass

T*

2.0 Liquid

1.5 1.0

A3

0.5 0.0 0.47

(a)

Attractive Glass

0.48

0.49

0.50

0.51

Hard Sphere

0.52

volume fraction

0.53

0.54

0.55

90 80 70 60 50 40 30 20 10

44.0 wt.%

φ=0.479

φ=0.518

42.2 °C T*=0.82 N=106 52.0 °C T*=0.934 N=127

29.0 °C T*=0.95 N=59 51.0 °C T*=1.14 N=99

φ=0.531 90 48.0 wt.% 23.0 °C T*=1.18 80 N=47 70 51.0 °C 60 T*=1.20 N=91 50 40 30 20 10 0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18

q

93

Effective Temperature (T*)

KGT line low temperature liquid high temperature liquid equilibrium liquid-to-crystal phase boundary

-1

110 100 90 80 70 60 50 40 30 20 10 3.0

100 36.0 wt.% 90 80 70 60 50 40 30 20 10

I(q) (cm )

temperature (oC)

F. Mallamace et al. / Physica A 339 (2004) 92 – 100

1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6

35.00 wt% (φ = 0.479)

1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6

36.00 wt% (φ = 0.479)

1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6

44.00 wt% (φ = 0.518)

1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5

48.00 wt% (φ = 0.518) 290 300 310 320 330 340 350

(b)

Temperature (K)

Fig. 1. (a) (bottom panel) The MCT phase diagram for an adhesive hard sphere system ( = 0:03). Phase diagram of the L64=D2 O system (top panel). (b) The theoretical Its of SANS intensity distributions at di/erent c and T and the variations of the Itted e/ective temperature T ∗ , as a function of T .

several order parameters, in particular it may depend on an e/ective temperature (T ∗ ). The addressing of these new conceptual problems related to attractive systems represents today one of the most challenging subjects of statistical physics. New and known approaches have been used, and in particular the MCT gave important suggestions [3–5], which were explored by di/erent experimental techniques [6–9]. Contrarily to the MCT results on hard spheres (HS) where the kinetic glass transition (KGT) is explained in terms of cage e/ects [10], with volume fraction ( ) as the only control parameter, for attractive hard sphere systems (AHS) a new and rich physical scenario arises, in which clustering processes take place. The corresponding phase diagram (due to the competition between cage and clustering) presents ergodic to non-ergodic transitions having and T as control parameters [4,5]; its properties are: (i) two kinetic glass transition lines, one corresponding to the HS glass at high composition and the other one extending to much lower concentrations. The former is attributed (with the usual mechanism due to excluded volume) to the repulsive part (Repulsive glass (RG)) of the interaction, the latter (temperature-dependent) to the attractive part of the potential (Attractive glass (AG)); (ii) a cusp-like singularity with a glass–liquid–glass reentrant behavior; (iii) a AG–RG transition line, starting where the AG branch crosses the RG one, ending in a point of type A3 [10]. A new phenomenon also arises: at the AG–RG transition the density decays are characterized by a logarithmic time dependence. MCT describes this AHS phase behavior in terms of a

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square-well potential of depth −u and fractional attractive well width  =(R−R )=R (R diameter of a repulsive hard core and R diameter of the particle). Therefore, for a given , besides , a second external control parameter, the e/ective temperature T ∗ = kB T=u, is introduced (Fig. 1, bottom panel, shows the MCT calculated AHS phase diagram for  = 0:03). Several experimental studies have conIrmed some of these MCT predictions [8,6,9]. Here we show that, in a micellar solution having all the AHS characteristics [11,9], the combination of small-angle neutron scattering (SANS) and photon correlation spectroscopy (PCS) techniques can successfully conIrm, in a quantitative way, all the above cited MCT Indings and map out the complete SA phase boundaries.

2. Experiments The investigated system is an aqueous solution of a non-ionic triblock copolymer made of polyethylene oxide (PEO) and polypropylene oxide (PPO), whose chemical structure is [PEO13 −PPO30 −PEO13 ] (L64-Pluronic). These molecules are T -dependent surfactants in water. The intermicellar attraction is mainly due to the fact that at higher temperatures water becomes progressively a poor solvent to both PEO and PPO chains. Its high concentration phase diagram has an ordered liquid crystalline phase (hexagonal). Fig. 1a (top panel) details the equilibrium phase boundary (liquid-to-crystal) between the disordered micellar phase and the ordered hexagonal phase (in particular the liquid to the metastable crystal (dotted line)). Solid symbols represent L(liquid)– G(glass)–L(liquid) transition boundaries. 2.1. SANS For a micellar system the scattered intensity is [9]:
2  M I (q) = cN ; bi − w p P(q)S(q)

(1)

i

M where P(q) is the normalized intra-particle structure factor,  c the polymer concentration, N the aggregation number of polymers in a micelle, i bi the sum of the coherent scattering lengths of polymer atoms, w the scattering length density of D2 0 molecules and p the molecular volume of the polymer. For a suspension of hydrated spherical micelles P(q) is calculated using a proper model [11], where the micelle has a compact spherical hydrophobic core (radius a = R =2), consisting of all the PPO segments with a polymer volume fraction p = 1 (i.e., a dry core) and a di/use corona region (PEO segments and solvent) [9]. The S(q) was obtained using the same square-well potential deIned in terms of T ∗ ,  and R of the MCT approach for AHS colloids. Then, we have solved the Ornestein–Zernike equation in the Percus–Yevick approximation obtaining a formula that gives S(q). The main point in this calculation for I (q) was that an absolute SANS intensity distribution can be Itted uniquely with four parameters: N , ,  and T ∗ [9].

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2.2. The scaled intensity In disordered systems the scattered intensity distribution is characterized by a single −1 peak located at qmax or equivalently by a single length scale:  = qmax (i.e., the mean inter-particle distance). In addition, the SANS intensity distribution of a two-phase system (the dispersed phase (in our case micelles) and the solvent) is given by a 3D Fourier transform of the Debye correlation function (r) which must be of the form (r=). Therefore, the dimensionless scaled intensity distribution can be put into a unique function of the scaled scattering wave-vector y = q=qmax , as:  ∞ 3 2 qmax I (q)=!  = d x 4#x2 j0 (xy)(x) ; (2) 0

2

2

∞

where !  = (1=2# ) 0 q2 I (q) dq is the so-called invariant and x = r= = qmax r. Thus by plotting the scaled intensity as a function of y, all the intensity distributions at di/erent T within a single phase region should collapse into a single master curve. In this way, the distinct local structures associated with di/erent phases occurring in di/erent T ranges can be identiIed [13]. 2.3. PCS PCS measures the intermediate scattering functions (ISF), deIned through the dynamical structure factor, S(q; t), as f(q; t) = S(q; t)=S(q; 0) [12]. On approaching the SA the measured fq (t)s show a progressive slowing down of their decay, up to the SA point where they become Pat. The MCT predicts the existence of a critical temperature Tc where the ergodic to nonergodic transition takes place and fq (t) tends to the Inite plateau (Debye–Waller factor (DWF)) fq (t → ∞; )=fqc ¿ 0. The separation parameter is ( ≈ (Tc − T )=Tc and two main density relaxation regimes, with di/erent time scales, are proposed: the ) and * ones. In the ) relaxation fq (t) depends singularly on the time and the control parameter by time-scaling laws of the exponents +, a, b and the exponent parameter , (determined by S(q)) [10]. 3. Results and discussion 3.1. SANS data Fig. 1b gives part of the Its of SANS data, in absolute intensity scale, at di/erent c and T . Symbols are the experimental data and lines are the Its. All the SANS spectra can be well Itted, by taking into account the e/ects of the resolution and the incoherent background, with a choice of  = 0:03 (±0:005) and give unique values of the four parameters, N , ,  and T ∗ . Results give also a unique function (independent from T ) that relates linearly and c [9]. The glass phase spectra are partially resolution limited, thus we cannot perform a precise Itting. Fig. 1b also reports the variations of T ∗ , as a function of T for some values of . It can be clearly seen that T ∗ increases with T . There are no data shown in the glass region, which appear as gaps in the plot.

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Fig. 1a (lower panel) shows a comparison of the phase diagram, in the T ∗ − plane, of the L64 − D2 O micellar solution with that determined by MCT for  = 0:03. The phase points shown in the upper panel have been mapped into the corresponding symbols in the lower panel using results of SANS analyses. These results conIrm the existence of the attractive branch in the predicted SA boundary and the re-entrant L-to-AG-to-L transition. 3.2. Scaling plots Fig. 2a shows SANS intensites and their scaling plots for = 0:522, 0:532 and 0:538, respectively, at di/erent T (291–340 K). Looking at them there are already clues showing the SA, in fact the spectra can be categorized into two groups: a sharp one and a broad one. Taking into account that the observed I (q) single peak rePects the position and the height of the S(q) Irst di/raction peak, the situation clariIes using the scaling approach (Fig. 2b). The position and the height of the Irst S(q) peak rePect the mean inter-particle separation and the degree of local order surrounding a typical particle, respectively. Thus, these quantities can be used to visualize the ordering in the amorphous state. Therefore, a sudden sharpening at a given T signals the onset of the liquid-to-amorphous solid transition. As one can see from the SANS intensities at

= 0:522, there are two degrees of disorder which depend on T . While the narrow peak, resolution limited, represents the glassy state, the broader one represents the liquid state, with a broader distribution of the interparticle distance. This Igure indicates that the system shows a re-entrant L-to-AG-to-L transition as T increases. The sharpness of the scaling peaks, which are resolution limited, indicates that the nearest-neighbor distance in the glassy state (from 306 to 321 K) is more uniform than that in the liquid state (228–303, 325–343 K). The situation is very di/erent at = 0:532. As T rises, the system experiences a re-entrant L-to-AG-to-L transition and when the T increases to 340 K, the system Q −1 . From the corresponding is driven into another glassy state peaked at q = 0:082 A scaling plots one can tell di/erences between these two glasses. While the narrowest peak (340 K) is resolution limited, the slightly broader peak (298–322 K) is also nearly resolution limited, but lower in the scaled intensities. Since the local structure is rePected in scaling plots, we conclude that the degree of disorder is di/erent for these two amorphous states. It can be interpreted that, on varying T , the system shows a L-to-AG-to-L-to-RG transition. According to the MCT results for AHS colloids a glass-to-glass transition is predicted [5]. Although the transitions between di/erent amorphous glassy states are not uncommon in pure substances, there is no detailed investigation on this in a colloidal-like system so far (except for some recent reports [14]). From SANS intensities at = 0:538, one can see that the much broader peak (liquid state) disappears and temperature variations trigger the transition between the two amorphous solid states with di/erent degree of disorder. By increasing T , the variation of the peak heights of the scaling plots given in the inset show a re-entrant RG-to-AG-to-RG transition. The most important prediction of MCT for AHS system is the existence of the end point of this glass-to-glass transition line, the A3 singularity.

F. Mallamace et al. / Physica A 339 (2004) 92 – 100

97

100 80

φ=0.522

288 K 294 K 303 K 306 K 315 K 321 K 325 K 334 K 343 K

60 40

120 100 80 60 40

20

20

0

60

0 291 K 295 K 298 K 312 K 322 K 329 K 333 K 340 K

φ=0.532

I(q)

50 40 30 20 10 0 40

I(q)qmax3/.<η2>

70

288 K 294 K 303 K 306 K 315 K 321 K 325 K 334 K 343 K RF

140

295 K 300 K 313 K 315 K 320 K 325 K 330 K 333 K 336 K 343 K

φ=0.538

30 20 10

291 K 295 K 298 K 312 K 322 K 329 K 333 K 340 K RF

140 120 100 80 60 40 20 0

295 K 300 K 313 K 315 K 320 K 325 K 330 K 333 K 336 K 343 K RF

140 120 100 80 60 40 20

0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

0 0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

q/qmax

q

(a)

160 140 120 100 80 60 40 20

45% wt (φ = 0.522)

48.5% wt (φ = 0.532)

51% wt (φ = 0.536)

52.5% wt (φ = 0.538)

54.4% wt (φ = 0.544)

54.6% wt (φ = 0.546)

140

Is(q)

120 100 80 60 40 20

140 120 100 80 60 40 20

290 300 310 320 330 340

(b)

290 300 310 320 330 340 350

Temperature (K)

Fig. 2. (a) A series of SANS spectra and their associated scaling plots for three di/erent concentrations at a series of T in the range 291–340 K. (b) The bottom panel gives the peak height of the scaling plots as a function of T .

Beyond this point the two glass became identical (for  = 0:03, is (A3 ) = 0:544). To clarify this point, the scaling plots peak heights (IS (q)) vs. T are given in Fig. 2b. The transition temperatures between the amorphous states can be visualized clearly from them. In addition, for = 0:544 all the scaling peaks have the identical height

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F(q,t)

1. 0 0. 9 0. 8 0. 7 0. 6 0. 5 0. 4 0. 3 0. 2 0. 1 0. 9 0. 8 0. 7 0. 6 0. 5 0. 4 0. 3 0. 2 0. 1 0. 9 0. 8 0. 7 0. 6 0. 5 0. 4 0. 3 0. 2 0. 1 0. 0 10 0

(a) φ = 0.525

(b) φ = 0.535

296 K 300 K 307 K 311 K 325 K

296 K 298 K 300 K 320 K

(c) φ = 0.538

(d) φ = 0.542

298 K 300 K 305 K 310 K 313 K 315 K 318 K 325 K

296 K 298 K 300 K 311 K 315 K 318 K 320 K

(e) φ = 0.544

(f) φ = 0.546 295 298 300 306 308 315 320

295 K 298 K 300 K 303 K 308 K 315 K 320 K

10 1

10 2

10 3

10 4

10 5

10 0

K K K K K K K

10 1

10 2

10 3

10 4

10 5

10 5

Time (µs) Fig. 3. The ISF at di/erent and T . (a) and (b) report the ISFs at = 0:525 and 0:535 respectively, where the L-to-AG transition is predicted. The logarithmic relaxation at intermediate times is highlighted by a straight line. Upon increasing to 0:538 and 0:542, (c) and (d), respectively, all the ISFs can be grouped into two distinct sets of curves having two di/erent values of DWF, one at 0:5 (AG) and the other at 0:4 (RG). ISFs in (e) and (f) indicate that at the A3 point and beyond, the DWFs become identical (fq ∼ 0:46).

(∼ 140), indicating two glasses with the same degree of local order. At the same it can be observed that all the scaled intensities collapse into a single master curve, i.e., all states are identical. Increasing further to 0:546 the situation remains the same as the A3 volume fraction one. This is a compelling proof that MCT predictions are accurate. 3.3. ISF PCS data can give further conIrmations, from the dynamical point of view, of the proposed SANS results, and in particular on the novel singularity at the A3 point. In Fig. 3 the ISFs measured at six di/erent volume fractions ( = 0:525, 0.535, 0.538, 0.542, 0,544 and 0:546) are shown as a function of temperature. Figs. 3a and b report the ISFs measured at = 0:525 and 0:535, respectively, where the L-to-AG transition is predicted, vs. T . In the liquid state fqc values are zero, whereas in the attractive glass state fqc is about 0:5. The SA is thus characterized by a discontinuous change in fqc . The observed occurrence of a region of logarithmic relaxation at intermediate

F. Mallamace et al. / Physica A 339 (2004) 92 – 100

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time preceding the plateau region for the system in the non-ergodic state just before the transition, is highlighted by a straight line in the log-lin plot. Upon increasing

to 0.538 and 0.542 Figs. 3c and d, respectively, we see that all the ISFs can be grouped into two distinct sets of curves having two di/erent values of the DWF, one of 0:5 (AG) and the other of 0:4 (RG). According to MCT, there is the possibility to observe a AG-to-RG transition by varying T ∗ . Because −u is T -dependent and increases on heating, T ∗ actually decreases as T rises, making the transition from the RG region to the AG region possible. By comparing the long-time limit of the ISF’s with the MCT predictions, we can identify the two di/erent types of glasses by their DWFs. The reason for observing two di/erent DWF values can be interpreted as the di/erent degree of localization of density Puctuations in the two glasses. These Igures, combined with Fig. 2 give Irm evidence of the repulsive glass (fqR ∼ 0:4)-to-attractive glass (fqA ∼ 0:5) transition. ISFs in Figs. 3e and f indicate that at the A3 point and beyond, the DWFs of the two glasses states become identical (fq ∼ 0:46). These Igures share the same features. It hints at critical point-like characteristics of the A3 point. This can be considered as a deInitive proof of the existence of the A3 point singularity in the phase diagram occurring exactly at the volume fraction predicted by MCT. It is however worth noting that at the A3 point and beyond, the intermediate time relaxations (the ) relaxation region) of the two glassy states are clearly di/erent, in spite of the fact that the long-time relaxations become identical. This is the Irst experimental Inding of this interesting point that was perhaps not predicted by MCT [14].

4. Conclusions We have used SANS and PCS to verify that L64 − D2 0 micellar system have the complex phase diagram and follows the overall structural arrest transition behavior predicted by MCT for attractive colloids. We show experimentally the existence of a L-to-AG-to-L re-entrant transition, a RG-to-AG transition and an A3 singularity in the MCT phase diagram [14]. Our SANS experiments further show that, while the local structures of the AG and the RG glasses are in general di/erent, they become identical at the A3 singularity. However, PCS results indicate that the two glasses relaxations are di/erent in the intermediate temporal region. The main result of this report is that the use of the SANS method makes possible to pinpoint the exact volume fraction where the cusp singularity and the A3 point are located [14] and map out the whole structural arrest transition boundaries in the studied systems. From the SANS experiments, a signiIcant di/erence in the local structure factor before and after the SA is detected at various temperatures for all the volume fractions. The reason behind this experimental fact still remains unclear. It is our conjecture that the reported SANS data is rePecting the aging e/ects of the sample on the time scale of our measurements. Another important suggestion regards the criticality of the A3 point: it is intriguing to speculate the extent to which one can draw the analogy between this singularity and the ordinary equilibrium critical point.

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