Journal of Colloid and Interface Science 444 (2015) 132–140
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Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Temperature dependence of aging kinetics of hectorite clay suspensions Ruiwen Shu, Weixiang Sun ⇑, Xinxing Liu, Zhen Tong ⇑ Research Institute of Materials Science and State Key Laboratory of Luminescent Materials and Devices, South China University of Technology, Guangzhou 510640, China
g r a p h i c a l a b s t r a c t Increased potential barrier
Synthetic hectorite clay suspension
Increased collision probability
Na+ Na+ Na+ Na+ Na+ Na+
T (°C) 10 15 20 25 30 35 40 45
bTG', bTG'' (Pa)
10
101
100
1.0
0.5
101
102
103
0.0 10
104
tw/aT (s)
a r t i c l e
2 days
4 days
Aging time-temperature superposition
10-1
Model Exp.
1.5
aT
3.5wt%, 4 days 2
20
T (°C)
30
40
i n f o
a b s t r a c t
Article history: Received 28 September 2014 Accepted 20 December 2014 Available online 31 December 2014
The aging of salt-free hectorite suspensions with different concentrations (cL = 2.9, 3.2 and 3.5 wt%) stored for 2 days or 4 days was studied by rheology at different temperatures. The evolution of storage and loss moduli G0 and G00 during aging followed aging time–temperature superposition. The temperature dependence of the shift factor aT, which reflected the aging kinetics, was interpreted by the reaction-limited colloidal aggregation (RLCA) mechanism with counterion condensation in calculating the double-layer interaction of the charged clay particles. Temperature dependence of the plateau modulus and yield stress of the suspension aged for 800 s was modeled with the soft glassy rheology (SGR) theory. The estimated noise temperature x indicated that the sample aged at higher temperature corresponded to a deeper quench in the nonergodic state. Under larger amplitude of oscillatory shear, the suspension exhibited a strain rate-frequency superposition (SRFS). The shearing eliminated the effects of aging and heating. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Hectorite clay suspension Aging Yielding Counterion condensation
1. Introduction Aging behavior is ubiquitous among systems with very slow dynamics that are far from equilibrium [1,2]. The dynamics of these systems is so slow that it cannot completely relax within the experimental time window t. Instead, the dynamics continuously slows down with the waiting-time tw, i.e., the time elapses since the quench of the system. The system relaxation time s l grows at a similar rate with tw as s t w and l is close to 1. This means that the observer can hardly wait for the system to be fully equilibrated. Aging and slow dynamics have been experimentally observed in soft glassy materials, such as gels [3], emulsions [4], microgels [5], ⇑ Corresponding authors. Fax: +86 20 87110273. E-mail (Z. Tong).
addresses:
[email protected]
(W. Sun),
http://dx.doi.org/10.1016/j.jcis.2014.12.073 0021-9797/Ó 2014 Elsevier Inc. All rights reserved.
[email protected]
and even biological systems [6], Although the physics and chemistry of these systems differ dramatically among each other, their dynamics shares similar aging process as mentioned above. Specifically, time-aging time superposition are commonly observed from these systems, where the measured relaxation function at different waiting-times tw can be scaled into a master curve if the time axis is reduced to t/s [7]. One of the popular theoretical interpretations of the aging phenomenon is the trap model [8]. In this model, the system is described as a particle evolving on a complex energy landscape with broad distribution of trap depths. The trap model is adopted into the soft glassy rheology (SGR) model [9,10] to account for the linear and nonlinear rheology and the aging effect of complex fluids. However, the exact relationship between the model parameters with the material elements in the real space is still unsolved [11]. Other models for aging and slow dynamic behavior of glassy materials include the extension of mode-coupling theory (MCT)
R. Shu et al. / Journal of Colloid and Interface Science 444 (2015) 132–140
[12], the non-equilibrium self-consistent generalized Langevin equation (NE-SCGLE) theory [13], and the generalized fluctuation–dissipation relationship [14]. Despite these successful efforts, a full understanding of the microscopic mechanism that governs the common aging behavior remains a significant challenge. Among many model systems used for experimental study of aging dynamics, the synthetic hectorite clay suspension is one of the most investigated [15], which is consist of charged disc-like nanoparticles suspended in water. Depending on the clay concentration and ionic strength, the clay suspension can age into several phases, such as fractal gels, [16] repulsive (Wigner) glasses [17], and attractive glasses [18]. Besides time-aging time superposition [19], the clay suspension exhibits some other scaling relationships during aging, which can be categorized into two classes. The first one is the time-X superposition with X being temperature [20] or adsorbed polymer concentration [21]. The change in these conditions uniformly shifts the relaxation to longer or shorter time without changing the spectrum. The second class is the aging time-X superposition, X being temperature [22,23], salt concentration [24], or the concentration of adsorbed polymer [25]. These superposition phenomena indicate that the aging kinetics of the suspension follows a universal route. Changes in condition X shift the system to earlier or later stage (younger or older age) on the kinetic route. The synergy of the two classes of scaling properties has recently been shown for a hectorite-poly(ethylene glycol) system [21], suggesting the existence of a correlation between the dynamics and kinetics beneath the apparent aging phenomena. It is now wondered what microscopic mechanism governs the phenomenological universality. In the present paper, we will show that the aging kinetics of the hectorite suspension at different temperatures can be understood by the interparticle interaction within the theory of reaction-limited colloidal aggregation (RLCA), and the effect of temperature on the slow dynamics of the aged suspension can be modeled by the soft glassy rheology (SGR) model after mapping the model parameter with experimental results.
2. Experimental section 2.1. Sample preparation Synthetic hectorite clay of gel-forming grade LAPONITEÒXLG (Rockwood Ltd.) was used after dried overnight against anhydrate CaCl2 in vacuum at room temperature. Aqueous suspension of clay concentration cL was prepared by slowly mixing the dried clay with deionized water (18.2 MX cm) under stirring for 15 min and then ultrasonic homogenization for another 15 min until a transparent suspension was obtained. Suspensions of cL = 2.9, 3.2, and 3.5 wt% prepared in this way were stored for 2 days before conductivity, pH, and rheology measurements, which were denoted as L2.9-2d, L3.2-2d, and L3.5-2d, respectively. Another sample of the 3.5 wt% suspension stored for 4 days before measurements was denoted as L3.5-4d. All samples were sealed in vials with parafilm during the storage.
2.2. Conductivity and pH measurements Suspension conductivity was measured using a conductivity meter (DDS-307, Shanghai REX Instrument Factory) with a platinum black electrode (DJS-1c). The pH value of suspensions was detected using a PHB-3 pH tester (Shanghai Sanxin Instrumentation, Inc.). Samples were sealed with parafilm during measurements to avoid CO2 dissolution.
133
2.3. Rheology measurements Rheology measurements were carried out on a stress-controlled rheometer (AR-G2, TA Instruments) using a cone-and-plate fixture with diameter of 40 mm and cone angle of 1°. A thin layer of silicon oil was laid on the rim of the fixture to prevent water evaporation and CO2 dissolution. Before every measurement, the sample was pre-sheared at a shear rate of 3000 s1 for 100 s to achieve a reproducible initial state. Then, the aging of the clay suspension was immediately monitored by applying a sinusoidal strain c(t) = c0sinxt with a constant strain amplitude c0 = 0.5% and frequency x = 6.28 rad/s. The storage and loss moduli G0 and G00 were recorded as a function of the evolution time tw. The start of the aging (tw = 0) was defined as the end of the pre-shearing procedure. The dynamic frequency sweep, dynamic strain sweep, and steady state viscosity measurement were performed on samples after the same pre-shearing and a delay time of 800 s. The delay allowed the sample to reach a repeatable gelled state for the measurements.
3. Results and discussion 3.1. Aging time–temperature superposition Fig. 1a shows the evolution of the storage and loss moduli G0 and G00 during aging of the sample L3.5-4d at different temperatures. During the rheology measurement, this sample exhibits a fluid-to-solid transition as indicated by the transition from G0 < G00 to G00 < G0 . The crossover G0 = G00 appears at earlier time (smaller tw) as the temperature increases, meaning that the sample ages faster at higher temperature. When the curves in Fig. 1a are shifted to superimpose the crossover of G0 = G00 , all the curves collapse into a master curve as depicted in Fig. 1b. The curve of Tref = 10 °C is taken as the reference. The horizontal shift factor aT is defined as tc(T)/tc(10 °C), where tc is the time at which G0 = G00 . aT decreases with increasing temperature as shown in the inset of Fig. 1b, indicating that the rate of aging increases with increasing temperature. The vertical shift factor bT varies slightly around 1, meaning that the vertical shift is negligible for optimizing the superposition. The samples L2.9-2d, L3.2-2d and L3.5-2d stored for 2 days exhibit the same superposition as that of L3.5-4d. The horizontal shift factor is plotted in Fig. 2. Unlike the case of L3.5-4d, aT of the samples stored for 2 days shows a maximum in the range of temperatures studied. Upon dispersed in pure water, the sodium cation Na+ between the layers of the clay is released into water, leaving the corresponding number of negative charges on the surface of the clay particle. Without addition of NaOH, there should be a small amount of positive charges on the edge of the particles due to the protonation of OH groups, resulting in the pH of the suspension usually higher than 7 [15]. Fig. 3 depicts that, for sample L3.5-4d, pH generally lies above 9.5 and decreases with increasing temperature. The ion product constant of water increases from 0.292 1014 at 10 °C to 4.195 1014 at 45 °C. Consequently, the OH concentration is very low around 105 to 104 M and also increases accompanying this temperature increase. The number of rim charges per clay particle estimated from [OH] is around 1e 3e, where e is the elementary charge. The results of the other samples are similar with the number of rim charges not higher than 10e (data not shown here). The suspension conductivity rc was measured for L2.9-2d, L3.22d, L3.4-2d and L3.5-4d as a function of temperature and demonstrated in Fig. 4a. For all the samples, the conductivity increases with increasing temperature. The conductivity is contributed by
R. Shu et al. / Journal of Colloid and Interface Science 444 (2015) 132–140
G', G'' (Pa)
102 101 G''
10
0
10-1
γ0 = 0.5%, ω = 6.28 rad/s
101
102
103
t w (s)
T (°C) 10 15 20 25 30 35 40 45
(b) L3.5-4d, Tref = 10°C
102 101 Shift factors
T (°C) 10 15 20 25 30 35 40 45
G'
(a) L3.5-4d
bTG', bTG'' (Pa)
134
0
10
10-1
1.0
bT
0.5
aT
0.0
20
T (°C)
101
102
103
40
104
t w /aT (s)
Fig. 1. (a) Evolution of storage modulus G0 (solid symbols) and loss modulus G00 (open symbols) with waiting-time tw at the indicated temperatures for the sample L3.5-4d; (b) master curve by shifting the curves in (a) with horizontal and vertical shift factors aT and bT, respectively. Inset: the temperature dependence of the shift factors.
L2.9-2d L3.2-2d L3.5-2d L3.5-4d
2.0
aT
1.5
1.0
0.5
0.0 10
20
30
40
T (°C) Fig. 2. Symbols: the horizontal shift factor aT, and lines: the calculated aT with the RLCA model for the indicated samples, see text for details.
1.2
L3.5-4d
1.0
pH
0.8
0.6 9.5
[OH-] (104 M)
10.0
0.4 10
20
30
40
50
T (°C) Fig. 3. Temperature dependence of pH (solid circles) and the concentration of OH anion (open squares) of the sample L3.5-4d.
Na+ and OH- ions as rc = F(lOH[OH] + lNa[Na+]), where F is the Faraday constant, lOH and lNa are the mobility of OH and Na+ ions, respectively. The [Na+] value is estimated from the data of rc and [OH] and plotted against temperature in Fig. 4b. [Na+] decreases with increasing temperature for all the samples. Note that the mobility of the ions also depends significantly on temperature (data taken from reference [26], see the inset of Fig. 4b). The negative charge on the surface of each clay particle estimated from [Na+] is around 600e 900e, much higher than that of the rim charge, which is of the order of 1e. Therefore, the clay particles can be approximately regarded as solely negatively charged.
The temperature dependence of the scaling factor aT is interpreted with the reaction-limited colloidal aggregation (RLCA) model. As the clay concentration (cL = 2.9–3.5 wt%) used in the present study lies within the ‘‘glass’’ region of the reported phase diagram (taking into account the released Na+ cation) [27], the reason why the RLCA mechanism is applicable is explained as follows. The phase diagram of the clay suspensions with different clay and salts concentrations have been controversial for decades [16,17,2732]. The most recent summary of the phase diagram [27] showed that the salt-free clay suspension at concentrations higher than 2.0 wt% belonged to the repulsive glass or Wigner glass state. In the present study, however, the clay volume fraction in the suspension at the highest cL = 3.5 wt% is only 0.0138 (the clay density is 2.53 g cm3). Even taking into account the Debye length (3.40–8.21 nm, see the following calculation), the effective volume fraction is 0.0857, much lower than the typical volume fraction of 0.58 for the glass transition in hard sphere colloidal systems. This excludes the possibility of the repulsive or Wigner glass. Moreover, for the suspensions within this range of clay concentration and ionic strength, appearance of the nonergodic state is favored by adding salts [18,27], i.e., by increasing attractive interaction. This also excludes the possibility of any repulsion-driven nonergodic state, though the SAXS results implied no heterogeneous structure formed in this range of cL [18] and the dynamic light scattering suggested caged dynamics in analogy to the glass [33]. Some researchers, therefore, assigned the clay suspension within this range (cL > 2.0 wt%) to an attractive glass, i.e., the nonergodic state of concentrated colloid driven by attraction interaction [34,35]. Whatever the exact classification, the aging process of the clay suspension in the present study is driven by attractive interaction, because it is promoted by adding salt. Moreover, the electrostatic interaction of the double-layer between clay particles acts as a long-distance repulsive barrier. In the presence of repulsive barrier, a nonzero bond life time between associative particles emerges, and the aggregation behaves an irreversible nature [36]. Particularly, the system dynamics shows the time-barrier height superposition [37]. Indeed, we have previously found that the aging of the clay suspensions modified by adsorption of poly(ethylene glycol) (PEG) can be described by the RLCA model [25], and the system manifests the time-PEG concentration superposition, where the PEG concentration directly controls the barrier height of the interaction between clay particles [21]. Based on the present experimental results and above discussion, we argue that the RLCA model should be the aging mechanism of the clay suspensions here. The scaling factor aT is inversely proportional to the rate con1 stant k of the aggregation [38], i.e., aT / ðk=kref Þ , where kref is the rate constant of aggregation at the reference temperature Tref. If the interaction between two particles is purely attractive
135
(a)
L2.9-2d L3.2-2d L3.5-2d L3.5-4d
0.026
(b)
0.024
OH4
2 Na+ 0
1.2
[Na+] (M)
σc (mS cm-1)
1.4
μ (10-7 m2s-1V-1)
R. Shu et al. / Journal of Colloid and Interface Science 444 (2015) 132–140
1.0
0
50
100
T (°C)
0.022 0.020 0.018
0.8 0.016 10
20
30
10
40
20
30
40
T (°C)
T (°C)
Fig. 4. Temperature dependence of conductivity (a) and Na+ concentration (b) for the indicated samples. Inset: temperature dependence of the mobility of OH and Na+ ions.
and the aggregation is irreversible, the aging kinetics satisfies the diffusion-limited colloidal aggregation (DLCA) with rate constant kD = 4kBT/(3g), where g is the viscosity of water, and kB is the Boltzmann constant [39]. When there is a repulsive barrier (e.g., electrostatic double-layer interaction) between two particles, the rate of aggregation is slowed down by a ratio W = kD/k, where W is called the stability ratio, and k is the rate constant of the RLCA [39]. Fuchs [40] related W to the interaction potential U(h) between two particles by
W ¼ 2r
Z
1
ðh þ 2rÞ
2
exp
0
UðhÞ dh kB T
ð1Þ
where h is the distance between the surface of two particles of radius r. Then, one can calculate the rate constant k from the knowledge of U(h). Within the present study, the scaling factor aT of Fig. 1b is related to W as
aT /
k kref
1 ¼
T ref gW T gref W ref
ð2Þ
where Wref and gref are the stability ratio and water viscosity at Tref, respectively. In order to calculate W according to Eq. (1), the effective radius of the clay particles was evaluated. The clay particle is disc-like with an average diameter of 30 nm and thickness of 1 nm [41]. Assuming that the rotational motion of the particles occurs at a timescale shorter than the translational motion, the clay particles can be considered as an equivalent sphere with the radius equal to the hydrodynamic radius of the particle. For a disc-like particle with radius r0 and thickness d, the hydrodynamic radius r = r0/arctan(2r0/d) [42], so r 9.8 nm in the present study. Within the equivalent sphere assumption, the interaction potential U(h) between two clay particles is the sum of the van der Waals attraction UvdW and the electrostatic double-layer repulsion Udl [43]. The van der Waals attractive energy between two spheres of radius r is
"
U vdW ðhÞ ¼
A 1 1 xðx þ 2Þ þ þ 2 ln 2 xðx þ 2Þ ðx þ 1Þ2 ðx þ 1Þ2
# ð3Þ
where x = h/(2r), and A = 1.06 1020 J is the Hamaker constant for the clay, which was calculated by the Lifschitz theory [44]. The use of the Hamaker constant for spheres may lead to overestimation of the van der Waals force since the volume of a plate is much smaller than the approximate sphere with the equivalent hydrodynamic radius. However, the spherical approximation was acceptable in the previous studies [45-47]. The feasibility of higher Hamaker constant for the disc-like clay seems to come from the need to account for the extra attractive forces, such as the hydration energy and
swelling energy of the clay [48-50]. The value of Hamaker constant used here has been shown successful in calculating the coagulation kinetics and yield stress of the clay suspensions [25,44]. To evaluate the double-layer interaction Udl, the clay particles is considered as charged colloids immersed in salt-free water with only the neutralizing number of the counterions (Na+ cation in the present case). In the case of low surface potential, the Debye–Hückel approximation [51] can be applied to the Poisson– Boltzmann equation, and Udl(h) is given by the Debye-Hückel or Yukawa form [52]
U dl ðhÞ ¼ 4pe0 er r 2 w2s
expðjhÞ h þ 2r
ð4Þ
with the surface potential
ws ¼
Q 4pe0 er r
ð5Þ
and the Debye lengths j1 when neglecting the trace amount of [OH]
j¼
1 1000e2 NA Naþ 2 e0 er kB T
ð6Þ
where Q = 1000e[Na+]NA//(pr20d) is the surface charge per particle, / is the volume fraction of charged particles, NA is the Avogadro’s constant, e0 and er are the vacuum permittivity and the relative permittivity of water, respectively. We fitted the experimental [Na+] data in Fig. 4b with a power law [Na+] = aTb and used the fitted results in the calculation. The calculated j1 value ranges from 3.40 nm to 8.21 nm. The surface potential becomes higher when the charge on the particle surface is higher. There exists an upper limit for the surface charge density, above which the counterion condensation occurs [53]. Ohshima [54] studied this case and derived the condition for the counterion condensation as
Q
ze 4pe0 er r kB T
P ln
1 /
ð7Þ
where z is the valance of the counterion. Thus, the Q limit for the counterion condensation is 57e 62e for the present case, while the lowest charge on the surface of each clay particle is 577e 866e at 40 °C, much higher than the criterion value. Therefore, the counterion condensation is taken into account. The double-layer interaction between two particles is still described by the Yukawa form, provided that ws is replaced by an effective surface potential ws,eff [55], which is the boundary value set by Eq. (7).
ws;eff ¼
Q eff 4pe0 er r
¼
kB T 1 ln ze /
ð8Þ
R. Shu et al. / Journal of Colloid and Interface Science 444 (2015) 132–140
The effective charge per particle Qeff, which is about 60e in the present case, is also close to the reported values [56,57]. The total interaction potential U(h) = UvdW(h) + Udl(h) can then be calculated using Eqs. 3, 4 and 8, as well as [Na+] = aTb from Fig. 4b. The stability ratio W is evaluated by numerical integral of Eq. (1), and the scaling factor aT is calculated with Eq. (2). The calculated aT is plotted in Fig. 2 compared with the experimental data. The agreement is qualitatively well, especially the appearance of the maxima in the aT vs. T curves. The acceleration effect of heating on the aggregation of spherical colloids in an electrolyte solution is rarely reported [58,59]. In those reports, the interaction potential was not affected very much by the increase of temperature, and the heating effect on the aggregation kinetics was attributed to the increase in the stick probability of the particles or the clusters and the decrease in the water viscosity (see Eqs. (1) and (2)). However, for the salt-free clay suspension, the counterion concentration [Na+] varies with temperature [20,23,60]. Fig. 4b illustrates that [Na+] in the present work decreases with increasing temperature. The total interaction potential U(h) calculated from Eqs. 3, 4 and 8, and the normalized potential U/kBT with the thermal energy kBT are presented in Fig. 5. For easy comparison, the maxima of the potential Umax and the normalized maximum potential Umax/kBT are also plotted against T in Fig. 5. The maximum potential Umax increases monotonically with temperature for all samples (Fig. 5c), but the normalized maximum potential Umax/kBT depends non-monotonically on T for the samples L2.8-2d, L3.2-2d and L3.52d, and decreases with T for the sample L3.5-4d (Fig. 5d). The above results suggest that the increase in the potential barrier does not catch up with the increase in the kinetic energy, which enhances the collision probability of the clay particles. This U/kBT characteristics is preserved in the subsequent result of W through Eq. (1), and finally in aT through Eq. (2). This means that the aggregation kinetics of our clay suspensions upon heating is affected by two competitive factors: (1) increased potential barrier by the increase
(a) L3.5-4d
T/°C 10 15 20 25 30 35 40
Umax
U (10-19J)
3.5
in ws,eff (Eq. (8)), and (2) increased collision possibility of the particles by increasing kinetic energy. For the samples L2.8-2d, L3.2-2d and L3.5-2d, these two effects compete comparatively, resulting in a maximum in aT; and for the sample L3.5-4d, the latter predominates, resulting in a decrease in aT. As known from Fig. 2, the calculated curves do not capture the clay concentration dependence of aT for the samples stored for 2 days. The observed aT maximum becomes smaller and appears at lower temperature with increasing clay concentration cL. However, the calculated trend is just the opposite. aT is comprehensively determined by multiple factors, including water viscosity, Na+ concentration, degree of the counterion condensation, interaction potential between two clay particles, and thermal motion of the clay particles, which are all considered in the model in the present study. It is therefore difficult to point out the exact origin for the above discrepancy between the observation and model. The discrepancy may also come from the spherical approximation for the plate-like clay particles. The RLCA model applied in this study is a simplified version of the population balance equations (PBEs) theory originally proposed by Smoluchowski [61] and developed by later researchers, which predicted the master curve of average radius of the growing clusters during colloidal aggregation (see [62] and references therein). However, the PBEs are applicable for initial aggregation only. There were also several experimental reports of master curves for the modulus growth measured during the post-gel stage [63-66]. Why the post-gel aging stage also follows the same master curve as the initial aggregation stage remains an open question. Previously, we interpreted the aging kinetics and master curve of the hectorite clay-NaCl-poly(ethylene glycol) suspension with the RLCA model based on the classical DLVO potential without consideration of the counterion condensation, because the added NaCl concentration was much higher than that of the counterion [25]. In the present study, in contrast, no salt was added, so the effect of counterions was not negligible.
(b) L3.5-4d
Umax/kBT
85
U/kBT
136
T/°C 10 15 20 25 30 35 40
80
3.0 0
1
0
1
h (nm)
h (nm) 95
(c)
(d)
4.0
L2.9-2d
Umax/kBT
Umax (10-19J)
L2.9-2d
3.8 L3.2-2d
3.6
L3.2-2d
90
L3.5-4d
3.4
L3.5-2d
85 L3.5-2d
L3.5-4d
3.2 10
20
30
T (°C)
40
10
20
30
40
T (°C)
Fig. 5. The total interaction potential U (a) and the reduced value U/kBT (b) for L3.5-4d at indicated temperatures; the maximum of the total potential Umax (c) and the reduced value Umax/kBT (d) varying with temperature for the indicated samples.
137
R. Shu et al. / Journal of Colloid and Interface Science 444 (2015) 132–140
3.2. Yielding and shear melting behavior
r ¼ ry þ K c_ n
ð9Þ
where ry is the yield stress, K and n are fitting parameter. The estimated yield stress in the inset of Fig. 7 increases with increasing temperature. The temperature dependence of the plateau modulus and yield stress for the sample L3.5-4d at tw = 800 s reflects the effect of ageing rate on the viscoelasticity of the suspension. The synergistic behavior of the aging and viscoelasticity of a glassy material can be described by the soft glassy rheology (SGR) model [9], which attributes the slow dynamics to the energetic trap and relies on the external force to yield the material by activation process. A dimensionless noise temperature x is introduced to parameterize the depth of quenching trap or glassiness of the system. If x > 1, the system is ergodic and equilibrated within finite time; when x < 1, the SGR model predicts the glassy state that exhibits nonequilibrium aging behavior. Prediction of the SGR model was tested by both colloidal glasses [69] and gels [70]. However, the original SGR model is in the dimensionless form [10], and quantitative comparison of its prediction to the experimental results is still rare [5,69,70]. Particularly, the waiting-time dependent dynamic modulus is given by the SGR model in a dimensionless form (for the case x < 1 and ~t 1) as
~ x; ~tÞ ¼ 1 1 ðix~tÞx1 Gð CðxÞ
ð10Þ
To apply Eq. (10) to the experimental data, the reduced complex ~ and reduced waiting time ~t are defined respectively by modulus G
20
102
10
20
30
T (°C)
40
50
T (°C) 10 15 20 25 30 35 40
101 10-2
10-1
100
101
102
103
104
γ (s ) -1
Fig. 7. Steady flow stress (symbols) of the sample L3.5-4d after aged for tw = 800 s and the fitted curve by Eq. (9) (lines) at the indicated temperatures. Inset: yield stress ry (squares) and calculated curve by Eq. (12) (line) varying with temperature.
~t ¼ t w =s0 ~ x; ~tÞ ¼ G ðx; ~tÞ=Gp Gð where s0 is a microscopic time and Gp the plateau modulus of the suspension [5,71]. s0 is related to the local yielding of a mesoscopic element with characteristic yielding time s
s ¼ s0 exp
1 2 E kl =x 2
ð11Þ
where E is the yield elastic energy of the mesoscopic element, l is the local strain applied on the mesoscopic element, and k is an elastic constant. Since the mesoscopic element in the SGR model is not related to any realistic structure, but merely defined as a region ‘‘large enough for a local strain l to be defined, but small enough for this to be approximately uniform within the region’’ [9], the choice of the s0 value is somewhat arbitrary. Previous reports mostly applied Eq. (10) to the master curve of time-aging time superposition and equated the experimental axis xtw to the x~t in Eq. (10) [5,69,71], setting implicitly s0 = 1.0 s. Yin and Solomon [70] chose s0 more carefully and found it to be of the order of 0.01 s. In the present study, we found that s0 = 0.1 s was most appropriate. The value of x at different temperatures (Fig. 8) was obtained by fitting the results of Fig. 6 with Eq. (10) and tw = 800 s (~t = 8000) and using G0 p in the inset of Fig. 6 as Gp. The x values for all samples are smaller than 1, which means the sample is already in the non-equilibrium glassy state. x decreases with
L3.5-4d, tw = 800 s, γ0 = 0.5%
G' 10
40
2
800
400 0
20
40
60
T (°C)
0.9
0.9
L3.5-4d x
T (°C) 10 15 20 25 30 35 40 45 50
0.8 0.7
0.8
0.0
0.5
1.0
aT = W / W ref
x
3
G' p (Pa)
G', G'' (Pa)
10
L3.5-4d
σ (Pa)
At tw = 800 s after the pre-shear, the structure evolution was slow enough to measure the viscoelasticity of the aged samples at the strain amplitude c0 = 0.5%. Fig. 6 presents the frequency dependence of the storage and loss moduli G0 and G00 for the sample L3.5-4d at different temperatures after aged for 800 s. At these temperatures, G0 is more than one order of magnitude higher than G00 and nearly independent of frequency, exhibiting the characteristics of a soft solid. The plateau modulus G0p (G0 at x = 0.1 rad/s) shown in the inset of Fig. 6 increases with increasing temperature. Fig. 7 displays the steady flow curves of the sample L3.5-4d at different temperatures after aged for 800 s. At all the temperatures, the stress r reaches a finite value at the shear rate limit c_ ? 0, which is the characteristics of the yield stress fluid [67]. The data are fitted with the Herschel–Bulkley model for the yield stress fluid [68]
σy (Pa)
60
0.7
G'' 10
1
10
-1
10
0
10
1
10
2
ω (rad/s) Fig. 6. Frequency dependence of storage modulus G0 (solid symbols) and loss modulus G00 (open symbols) for sample L3.5-4d aged at tw = 800 s and indicated temperatures. Inset: temperature dependence of the plateau modulus G0p .
0.6 10
20
30
40
50
T (°C) Fig. 8. Tempetaure dependence of the noise temperature x for the sample L3.5-4d. Inset: dependence of x on the shift factor aT.
R. Shu et al. / Journal of Colloid and Interface Science 444 (2015) 132–140
increasing temperature, indicating that the dynamics of the sample at tw = 800 s is trapped more deeply at higher temperature. The yield stress ry is predicted by the SGR model as [72]
R
r~ y ¼
ry 01 dll½ZðlÞx ¼ R1 x r0 dl½ZðlÞ 0
ð12Þ
with
ZðlÞ ¼
1 c_
Z
l
02
0
dl expðl =2xÞ
ð13Þ
0
γ0 = γ0ω (s-1) L3.5-4d, T = 25°C
0.01 0.05 0.1 0.5 1 2 5 10
G'1, G''1 (Pa)
10 2
10 1
10 0
10 -1 10-2
10-1
100
101
102
ω (rad/s) Fig. 9. Real and imagine parts of the first harmonic G01 (solid symbols) and G001 (open symbols) from frequency sweep at constant shear rate amplitude c_ 0 for the sample L3.5-4d at 25 °C.
(a)
-1 G'1 γ0 = γ0ω (s ) 0.01 0.05 0.1 0.5 1 2 5 G''1 10
L3.5-4d, T = 15°C
G1' , G1'' (Pa)
102
101
0
10
r0 is a scaling factor introduced to normalize the observed ry. Using x in Fig. 8 and r0 = 101.5 Pa, the calculated curve expresses the observed trend in the inset of Fig. 7. The noise temperature x in the SGR model is a signature of the quench depth for a glassy system below the glass or nonergodic state transition temperature [9,10]. Its relationship with temperature is interesting experimentally [69] and theoretically [73,74]. In the present study, the temperature dependence of x is opposite to the jamming phase diagram for attractive particles proposed by Trappe, et al. [75], and is also opposite to the previously reported results for attractive particles from experiments [75-77] and simulations [78-81]. Particularly, in the scenario of the jamming phase diagram, either decreasing temperature T or increasing the attractive potential Uattr promotes the formation of nonergodic (jammed) state [75], whereas the case of the present study is the contrary as shown in Fig. 8. This discrepancy seems to originate from the aging of the clay, which is not considered in the attractive jamming phase diagram. If a jammed system has been completely evolved, raising temperature only facilitates the thermal activation of the jammed particles and is unfavored for the jamming. For the clay suspensions here, the system is undergoing aggregation and far from completely evolved. Since the aggregation follows the RLCA, which is an irreversible aggregation mechanism, raising temperature mainly leads to a higher collision probability of the particles, hence promoting the formation of nonergodic state. To this end, the nongergodicity of the clay suspension (characterized by the noise temperature x) was related to the aggregation kinetics (characterized by stability ratio W). The inset of Fig. 8 shows the dependence of x on the shift factor of aging aT, which equals to the reduced stability ratio W/Wref. The more stable (reluctant to aggregate) the clay suspension, the less nonergodic of the system. At even larger strain or strain rate, the aged sample may be driven back to the earlier state or even fluidized. This behavior is
102
101 G''1
100
10-1
10-1 10-3
10-2
10-1
100
10-3
101
(c)
10-1
100
101
T (°C) 15 25 35
(d)
G'1
L3.5-4d, T = 35°C
10-2
τ (γ0)ω (rad/s)
τ (γ0)ω (rad/s)
103
G'1
(b) L3.5-4d, T = 25°C
G'1, G''1 (Pa)
138
100
10-1 101
τ (γ0)
G1' , G''1 (Pa)
102
G''1
10-2
0
10
-1
10
10-1 10-3
10-2
10-1
τ (γ0)ω (rad/s)
100
101
-3
10-2
10-1
100
101
γ0 (s ) -1
Fig. 10. Strain rate-frequency superposition (SRFS) master curves at constant c_ 0 for the sample L3.5-4d at indicated temperatures: (a) 15 °C, (b) 25 °C, (c) 35 °C, and the c_ 0 dependence of the horizontal shift factor s (c_ 0 ).
R. Shu et al. / Journal of Colloid and Interface Science 444 (2015) 132–140
common among soft glassy materials and is often called ‘‘shear melting’’ or ‘‘rejuvenation’’ [82-86]. Under a large amplitude oscillatory shear c(t) = c0sinxt with the shear amplitude c0 higher than the linear viscoelastic limit, the sample response becomes nonlinear with multiple higher harmonic components [87]. In this case, the measured G0 and G00 should be denoted differently as G01 and G001 because they become the first harmonic real and imagine moduli, respectively. Fig. 9 shows the curves of frequency dependent G01 and G001 at different shear rate amplitude c_ 0 (c_ 0 ¼ c0 x) at 25 °C for the sample L3.5-4d. As the shear rate amplitude increases, the G01 and G001 curves shift horizontally to higher frequency without apparent change in the shape. Also the fluidic regime (G01 < G001 ) extends with increasing c_ 0 . Wyss et al. [88] showed that for many colloid systems the frequency dependence of G01 and G001 can be scaled into the master curve when plotted against the reduced frequency sðc_ 0 Þx [64]. This was called the strain-rate frequency superposition (SRFS). The dependence of the scaling factor s on c_ 0 characterizes quantitatively how much an aged glassy material is ‘‘molten’’ by the oscillatory shear. Fig. 10a–c shows the SRFS master curves of the sample L3.5-4d at 15 °C, 25 °C, and 35 °C. Fig. 10d presents the c_ 0 dependence of the scaling factor s at three temperatures. At these temperatures, G01 and G001 can be scaled into master curves, except the high frequency portion of G001 . Wyss et al. attributed this to local viscous contribution that was shear independent [88]. The scaling a factor follows the relationship of s / c_ 0 with a 1. The difference of sðc_ 0 Þ at different temperatures is negligible. For an aging colloidal suspension, the shear field with strain rate c_ sets a limit of relaxation time scale of the sample. If the sample is aged to a state that the structure relaxation time is longer than the reciprocal of the shear rate c_ 1 , the relaxation is largely reset by c_ a with a 1 [88,89]. In other words, the structure and slow dynamics of the aged suspension were rejuvenated by shearing. The temperature effect on the aging kinetics and the aged structure is suppressed by the shearing field.
References
4. Conclusion
[36]
The aging process of the salt-free suspensions of hectorite clay at different temperatures exhibited an aging time–temperature superposition. The temperature dependence of the aging kinetics was modeled by the RLCA mechanism with counterion condensation for the double-layer interaction potential of charged clay particles. The variation of aging kinetics with increasing temperature was interpreted by the competition of increased repulsive interaction and increased collision probability of the particles. The temperature dependence of plateau modulus and yield stress of the suspensions aged for 800 s was described by the SGR model. The obtained noise temperature x decreased with increasing temperature, which indicated that the sample aged at higher temperature was equivalent to a deeper quench. Finally, under larger amplitude of shearing, the sample exhibited strain rate-frequency superposition (SRFS). The strain rate amplitude dependence of the shift factor showed negligible temperature dependence, which implied that shearing eliminated the effect of aging as well as its temperature dependence.
[37]
Acknowledgments The authors owe a great deal of thanks to Prof. Hiroyuki Ohshima for the valuable discussion about the counterion condensation in the double-layer interaction. The financial support from the Natural Science Foundation of China (21204023) and the National Basic Research Program of China (973 Program, 2012CB821504) is gratefully acknowledged.
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Contents lists available at ScienceDirect
Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Temperature dependence of aging kinetics of hectorite clay suspensions Ruiwen Shu, Weixiang Sun ⇑, Xinxing Liu, Zhen Tong ⇑ Research Institute of Materials Science and State Key Laboratory of Luminescent Materials and Devices, South China University of Technology, Guangzhou 510640, China
g r a p h i c a l a b s t r a c t Increased potential barrier
Synthetic hectorite clay suspension
Increased collision probability
Na+ Na+ Na+ Na+ Na+ Na+
T (°C) 10 15 20 25 30 35 40 45
bTG', bTG'' (Pa)
10
101
100
1.0
0.5
101
102
103
0.0 10
104
tw/aT (s)
a r t i c l e
2 days
4 days
Aging time-temperature superposition
10-1
Model Exp.
1.5
aT
3.5wt%, 4 days 2
20
T (°C)
30
40
i n f o
a b s t r a c t
Article history: Received 28 September 2014 Accepted 20 December 2014 Available online 31 December 2014
The aging of salt-free hectorite suspensions with different concentrations (cL = 2.9, 3.2 and 3.5 wt%) stored for 2 days or 4 days was studied by rheology at different temperatures. The evolution of storage and loss moduli G0 and G00 during aging followed aging time–temperature superposition. The temperature dependence of the shift factor aT, which reflected the aging kinetics, was interpreted by the reaction-limited colloidal aggregation (RLCA) mechanism with counterion condensation in calculating the double-layer interaction of the charged clay particles. Temperature dependence of the plateau modulus and yield stress of the suspension aged for 800 s was modeled with the soft glassy rheology (SGR) theory. The estimated noise temperature x indicated that the sample aged at higher temperature corresponded to a deeper quench in the nonergodic state. Under larger amplitude of oscillatory shear, the suspension exhibited a strain rate-frequency superposition (SRFS). The shearing eliminated the effects of aging and heating. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Hectorite clay suspension Aging Yielding Counterion condensation
1. Introduction Aging behavior is ubiquitous among systems with very slow dynamics that are far from equilibrium [1,2]. The dynamics of these systems is so slow that it cannot completely relax within the experimental time window t. Instead, the dynamics continuously slows down with the waiting-time tw, i.e., the time elapses since the quench of the system. The system relaxation time s l grows at a similar rate with tw as s t w and l is close to 1. This means that the observer can hardly wait for the system to be fully equilibrated. Aging and slow dynamics have been experimentally observed in soft glassy materials, such as gels [3], emulsions [4], microgels [5], ⇑ Corresponding authors. Fax: +86 20 87110273. E-mail (Z. Tong).
addresses:
[email protected]
(W. Sun),
http://dx.doi.org/10.1016/j.jcis.2014.12.073 0021-9797/Ó 2014 Elsevier Inc. All rights reserved.
[email protected]
and even biological systems [6], Although the physics and chemistry of these systems differ dramatically among each other, their dynamics shares similar aging process as mentioned above. Specifically, time-aging time superposition are commonly observed from these systems, where the measured relaxation function at different waiting-times tw can be scaled into a master curve if the time axis is reduced to t/s [7]. One of the popular theoretical interpretations of the aging phenomenon is the trap model [8]. In this model, the system is described as a particle evolving on a complex energy landscape with broad distribution of trap depths. The trap model is adopted into the soft glassy rheology (SGR) model [9,10] to account for the linear and nonlinear rheology and the aging effect of complex fluids. However, the exact relationship between the model parameters with the material elements in the real space is still unsolved [11]. Other models for aging and slow dynamic behavior of glassy materials include the extension of mode-coupling theory (MCT)
R. Shu et al. / Journal of Colloid and Interface Science 444 (2015) 132–140
[12], the non-equilibrium self-consistent generalized Langevin equation (NE-SCGLE) theory [13], and the generalized fluctuation–dissipation relationship [14]. Despite these successful efforts, a full understanding of the microscopic mechanism that governs the common aging behavior remains a significant challenge. Among many model systems used for experimental study of aging dynamics, the synthetic hectorite clay suspension is one of the most investigated [15], which is consist of charged disc-like nanoparticles suspended in water. Depending on the clay concentration and ionic strength, the clay suspension can age into several phases, such as fractal gels, [16] repulsive (Wigner) glasses [17], and attractive glasses [18]. Besides time-aging time superposition [19], the clay suspension exhibits some other scaling relationships during aging, which can be categorized into two classes. The first one is the time-X superposition with X being temperature [20] or adsorbed polymer concentration [21]. The change in these conditions uniformly shifts the relaxation to longer or shorter time without changing the spectrum. The second class is the aging time-X superposition, X being temperature [22,23], salt concentration [24], or the concentration of adsorbed polymer [25]. These superposition phenomena indicate that the aging kinetics of the suspension follows a universal route. Changes in condition X shift the system to earlier or later stage (younger or older age) on the kinetic route. The synergy of the two classes of scaling properties has recently been shown for a hectorite-poly(ethylene glycol) system [21], suggesting the existence of a correlation between the dynamics and kinetics beneath the apparent aging phenomena. It is now wondered what microscopic mechanism governs the phenomenological universality. In the present paper, we will show that the aging kinetics of the hectorite suspension at different temperatures can be understood by the interparticle interaction within the theory of reaction-limited colloidal aggregation (RLCA), and the effect of temperature on the slow dynamics of the aged suspension can be modeled by the soft glassy rheology (SGR) model after mapping the model parameter with experimental results.
2. Experimental section 2.1. Sample preparation Synthetic hectorite clay of gel-forming grade LAPONITEÒXLG (Rockwood Ltd.) was used after dried overnight against anhydrate CaCl2 in vacuum at room temperature. Aqueous suspension of clay concentration cL was prepared by slowly mixing the dried clay with deionized water (18.2 MX cm) under stirring for 15 min and then ultrasonic homogenization for another 15 min until a transparent suspension was obtained. Suspensions of cL = 2.9, 3.2, and 3.5 wt% prepared in this way were stored for 2 days before conductivity, pH, and rheology measurements, which were denoted as L2.9-2d, L3.2-2d, and L3.5-2d, respectively. Another sample of the 3.5 wt% suspension stored for 4 days before measurements was denoted as L3.5-4d. All samples were sealed in vials with parafilm during the storage.
2.2. Conductivity and pH measurements Suspension conductivity was measured using a conductivity meter (DDS-307, Shanghai REX Instrument Factory) with a platinum black electrode (DJS-1c). The pH value of suspensions was detected using a PHB-3 pH tester (Shanghai Sanxin Instrumentation, Inc.). Samples were sealed with parafilm during measurements to avoid CO2 dissolution.
133
2.3. Rheology measurements Rheology measurements were carried out on a stress-controlled rheometer (AR-G2, TA Instruments) using a cone-and-plate fixture with diameter of 40 mm and cone angle of 1°. A thin layer of silicon oil was laid on the rim of the fixture to prevent water evaporation and CO2 dissolution. Before every measurement, the sample was pre-sheared at a shear rate of 3000 s1 for 100 s to achieve a reproducible initial state. Then, the aging of the clay suspension was immediately monitored by applying a sinusoidal strain c(t) = c0sinxt with a constant strain amplitude c0 = 0.5% and frequency x = 6.28 rad/s. The storage and loss moduli G0 and G00 were recorded as a function of the evolution time tw. The start of the aging (tw = 0) was defined as the end of the pre-shearing procedure. The dynamic frequency sweep, dynamic strain sweep, and steady state viscosity measurement were performed on samples after the same pre-shearing and a delay time of 800 s. The delay allowed the sample to reach a repeatable gelled state for the measurements.
3. Results and discussion 3.1. Aging time–temperature superposition Fig. 1a shows the evolution of the storage and loss moduli G0 and G00 during aging of the sample L3.5-4d at different temperatures. During the rheology measurement, this sample exhibits a fluid-to-solid transition as indicated by the transition from G0 < G00 to G00 < G0 . The crossover G0 = G00 appears at earlier time (smaller tw) as the temperature increases, meaning that the sample ages faster at higher temperature. When the curves in Fig. 1a are shifted to superimpose the crossover of G0 = G00 , all the curves collapse into a master curve as depicted in Fig. 1b. The curve of Tref = 10 °C is taken as the reference. The horizontal shift factor aT is defined as tc(T)/tc(10 °C), where tc is the time at which G0 = G00 . aT decreases with increasing temperature as shown in the inset of Fig. 1b, indicating that the rate of aging increases with increasing temperature. The vertical shift factor bT varies slightly around 1, meaning that the vertical shift is negligible for optimizing the superposition. The samples L2.9-2d, L3.2-2d and L3.5-2d stored for 2 days exhibit the same superposition as that of L3.5-4d. The horizontal shift factor is plotted in Fig. 2. Unlike the case of L3.5-4d, aT of the samples stored for 2 days shows a maximum in the range of temperatures studied. Upon dispersed in pure water, the sodium cation Na+ between the layers of the clay is released into water, leaving the corresponding number of negative charges on the surface of the clay particle. Without addition of NaOH, there should be a small amount of positive charges on the edge of the particles due to the protonation of OH groups, resulting in the pH of the suspension usually higher than 7 [15]. Fig. 3 depicts that, for sample L3.5-4d, pH generally lies above 9.5 and decreases with increasing temperature. The ion product constant of water increases from 0.292 1014 at 10 °C to 4.195 1014 at 45 °C. Consequently, the OH concentration is very low around 105 to 104 M and also increases accompanying this temperature increase. The number of rim charges per clay particle estimated from [OH] is around 1e 3e, where e is the elementary charge. The results of the other samples are similar with the number of rim charges not higher than 10e (data not shown here). The suspension conductivity rc was measured for L2.9-2d, L3.22d, L3.4-2d and L3.5-4d as a function of temperature and demonstrated in Fig. 4a. For all the samples, the conductivity increases with increasing temperature. The conductivity is contributed by
R. Shu et al. / Journal of Colloid and Interface Science 444 (2015) 132–140
G', G'' (Pa)
102 101 G''
10
0
10-1
γ0 = 0.5%, ω = 6.28 rad/s
101
102
103
t w (s)
T (°C) 10 15 20 25 30 35 40 45
(b) L3.5-4d, Tref = 10°C
102 101 Shift factors
T (°C) 10 15 20 25 30 35 40 45
G'
(a) L3.5-4d
bTG', bTG'' (Pa)
134
0
10
10-1
1.0
bT
0.5
aT
0.0
20
T (°C)
101
102
103
40
104
t w /aT (s)
Fig. 1. (a) Evolution of storage modulus G0 (solid symbols) and loss modulus G00 (open symbols) with waiting-time tw at the indicated temperatures for the sample L3.5-4d; (b) master curve by shifting the curves in (a) with horizontal and vertical shift factors aT and bT, respectively. Inset: the temperature dependence of the shift factors.
L2.9-2d L3.2-2d L3.5-2d L3.5-4d
2.0
aT
1.5
1.0
0.5
0.0 10
20
30
40
T (°C) Fig. 2. Symbols: the horizontal shift factor aT, and lines: the calculated aT with the RLCA model for the indicated samples, see text for details.
1.2
L3.5-4d
1.0
pH
0.8
0.6 9.5
[OH-] (104 M)
10.0
0.4 10
20
30
40
50
T (°C) Fig. 3. Temperature dependence of pH (solid circles) and the concentration of OH anion (open squares) of the sample L3.5-4d.
Na+ and OH- ions as rc = F(lOH[OH] + lNa[Na+]), where F is the Faraday constant, lOH and lNa are the mobility of OH and Na+ ions, respectively. The [Na+] value is estimated from the data of rc and [OH] and plotted against temperature in Fig. 4b. [Na+] decreases with increasing temperature for all the samples. Note that the mobility of the ions also depends significantly on temperature (data taken from reference [26], see the inset of Fig. 4b). The negative charge on the surface of each clay particle estimated from [Na+] is around 600e 900e, much higher than that of the rim charge, which is of the order of 1e. Therefore, the clay particles can be approximately regarded as solely negatively charged.
The temperature dependence of the scaling factor aT is interpreted with the reaction-limited colloidal aggregation (RLCA) model. As the clay concentration (cL = 2.9–3.5 wt%) used in the present study lies within the ‘‘glass’’ region of the reported phase diagram (taking into account the released Na+ cation) [27], the reason why the RLCA mechanism is applicable is explained as follows. The phase diagram of the clay suspensions with different clay and salts concentrations have been controversial for decades [16,17,2732]. The most recent summary of the phase diagram [27] showed that the salt-free clay suspension at concentrations higher than 2.0 wt% belonged to the repulsive glass or Wigner glass state. In the present study, however, the clay volume fraction in the suspension at the highest cL = 3.5 wt% is only 0.0138 (the clay density is 2.53 g cm3). Even taking into account the Debye length (3.40–8.21 nm, see the following calculation), the effective volume fraction is 0.0857, much lower than the typical volume fraction of 0.58 for the glass transition in hard sphere colloidal systems. This excludes the possibility of the repulsive or Wigner glass. Moreover, for the suspensions within this range of clay concentration and ionic strength, appearance of the nonergodic state is favored by adding salts [18,27], i.e., by increasing attractive interaction. This also excludes the possibility of any repulsion-driven nonergodic state, though the SAXS results implied no heterogeneous structure formed in this range of cL [18] and the dynamic light scattering suggested caged dynamics in analogy to the glass [33]. Some researchers, therefore, assigned the clay suspension within this range (cL > 2.0 wt%) to an attractive glass, i.e., the nonergodic state of concentrated colloid driven by attraction interaction [34,35]. Whatever the exact classification, the aging process of the clay suspension in the present study is driven by attractive interaction, because it is promoted by adding salt. Moreover, the electrostatic interaction of the double-layer between clay particles acts as a long-distance repulsive barrier. In the presence of repulsive barrier, a nonzero bond life time between associative particles emerges, and the aggregation behaves an irreversible nature [36]. Particularly, the system dynamics shows the time-barrier height superposition [37]. Indeed, we have previously found that the aging of the clay suspensions modified by adsorption of poly(ethylene glycol) (PEG) can be described by the RLCA model [25], and the system manifests the time-PEG concentration superposition, where the PEG concentration directly controls the barrier height of the interaction between clay particles [21]. Based on the present experimental results and above discussion, we argue that the RLCA model should be the aging mechanism of the clay suspensions here. The scaling factor aT is inversely proportional to the rate con1 stant k of the aggregation [38], i.e., aT / ðk=kref Þ , where kref is the rate constant of aggregation at the reference temperature Tref. If the interaction between two particles is purely attractive
135
(a)
L2.9-2d L3.2-2d L3.5-2d L3.5-4d
0.026
(b)
0.024
OH4
2 Na+ 0
1.2
[Na+] (M)
σc (mS cm-1)
1.4
μ (10-7 m2s-1V-1)
R. Shu et al. / Journal of Colloid and Interface Science 444 (2015) 132–140
1.0
0
50
100
T (°C)
0.022 0.020 0.018
0.8 0.016 10
20
30
10
40
20
30
40
T (°C)
T (°C)
Fig. 4. Temperature dependence of conductivity (a) and Na+ concentration (b) for the indicated samples. Inset: temperature dependence of the mobility of OH and Na+ ions.
and the aggregation is irreversible, the aging kinetics satisfies the diffusion-limited colloidal aggregation (DLCA) with rate constant kD = 4kBT/(3g), where g is the viscosity of water, and kB is the Boltzmann constant [39]. When there is a repulsive barrier (e.g., electrostatic double-layer interaction) between two particles, the rate of aggregation is slowed down by a ratio W = kD/k, where W is called the stability ratio, and k is the rate constant of the RLCA [39]. Fuchs [40] related W to the interaction potential U(h) between two particles by
W ¼ 2r
Z
1
ðh þ 2rÞ
2
exp
0
UðhÞ dh kB T
ð1Þ
where h is the distance between the surface of two particles of radius r. Then, one can calculate the rate constant k from the knowledge of U(h). Within the present study, the scaling factor aT of Fig. 1b is related to W as
aT /
k kref
1 ¼
T ref gW T gref W ref
ð2Þ
where Wref and gref are the stability ratio and water viscosity at Tref, respectively. In order to calculate W according to Eq. (1), the effective radius of the clay particles was evaluated. The clay particle is disc-like with an average diameter of 30 nm and thickness of 1 nm [41]. Assuming that the rotational motion of the particles occurs at a timescale shorter than the translational motion, the clay particles can be considered as an equivalent sphere with the radius equal to the hydrodynamic radius of the particle. For a disc-like particle with radius r0 and thickness d, the hydrodynamic radius r = r0/arctan(2r0/d) [42], so r 9.8 nm in the present study. Within the equivalent sphere assumption, the interaction potential U(h) between two clay particles is the sum of the van der Waals attraction UvdW and the electrostatic double-layer repulsion Udl [43]. The van der Waals attractive energy between two spheres of radius r is
"
U vdW ðhÞ ¼
A 1 1 xðx þ 2Þ þ þ 2 ln 2 xðx þ 2Þ ðx þ 1Þ2 ðx þ 1Þ2
# ð3Þ
where x = h/(2r), and A = 1.06 1020 J is the Hamaker constant for the clay, which was calculated by the Lifschitz theory [44]. The use of the Hamaker constant for spheres may lead to overestimation of the van der Waals force since the volume of a plate is much smaller than the approximate sphere with the equivalent hydrodynamic radius. However, the spherical approximation was acceptable in the previous studies [45-47]. The feasibility of higher Hamaker constant for the disc-like clay seems to come from the need to account for the extra attractive forces, such as the hydration energy and
swelling energy of the clay [48-50]. The value of Hamaker constant used here has been shown successful in calculating the coagulation kinetics and yield stress of the clay suspensions [25,44]. To evaluate the double-layer interaction Udl, the clay particles is considered as charged colloids immersed in salt-free water with only the neutralizing number of the counterions (Na+ cation in the present case). In the case of low surface potential, the Debye–Hückel approximation [51] can be applied to the Poisson– Boltzmann equation, and Udl(h) is given by the Debye-Hückel or Yukawa form [52]
U dl ðhÞ ¼ 4pe0 er r 2 w2s
expðjhÞ h þ 2r
ð4Þ
with the surface potential
ws ¼
Q 4pe0 er r
ð5Þ
and the Debye lengths j1 when neglecting the trace amount of [OH]
j¼
1 1000e2 NA Naþ 2 e0 er kB T
ð6Þ
where Q = 1000e[Na+]NA//(pr20d) is the surface charge per particle, / is the volume fraction of charged particles, NA is the Avogadro’s constant, e0 and er are the vacuum permittivity and the relative permittivity of water, respectively. We fitted the experimental [Na+] data in Fig. 4b with a power law [Na+] = aTb and used the fitted results in the calculation. The calculated j1 value ranges from 3.40 nm to 8.21 nm. The surface potential becomes higher when the charge on the particle surface is higher. There exists an upper limit for the surface charge density, above which the counterion condensation occurs [53]. Ohshima [54] studied this case and derived the condition for the counterion condensation as
Q
ze 4pe0 er r kB T
P ln
1 /
ð7Þ
where z is the valance of the counterion. Thus, the Q limit for the counterion condensation is 57e 62e for the present case, while the lowest charge on the surface of each clay particle is 577e 866e at 40 °C, much higher than the criterion value. Therefore, the counterion condensation is taken into account. The double-layer interaction between two particles is still described by the Yukawa form, provided that ws is replaced by an effective surface potential ws,eff [55], which is the boundary value set by Eq. (7).
ws;eff ¼
Q eff 4pe0 er r
¼
kB T 1 ln ze /
ð8Þ
R. Shu et al. / Journal of Colloid and Interface Science 444 (2015) 132–140
The effective charge per particle Qeff, which is about 60e in the present case, is also close to the reported values [56,57]. The total interaction potential U(h) = UvdW(h) + Udl(h) can then be calculated using Eqs. 3, 4 and 8, as well as [Na+] = aTb from Fig. 4b. The stability ratio W is evaluated by numerical integral of Eq. (1), and the scaling factor aT is calculated with Eq. (2). The calculated aT is plotted in Fig. 2 compared with the experimental data. The agreement is qualitatively well, especially the appearance of the maxima in the aT vs. T curves. The acceleration effect of heating on the aggregation of spherical colloids in an electrolyte solution is rarely reported [58,59]. In those reports, the interaction potential was not affected very much by the increase of temperature, and the heating effect on the aggregation kinetics was attributed to the increase in the stick probability of the particles or the clusters and the decrease in the water viscosity (see Eqs. (1) and (2)). However, for the salt-free clay suspension, the counterion concentration [Na+] varies with temperature [20,23,60]. Fig. 4b illustrates that [Na+] in the present work decreases with increasing temperature. The total interaction potential U(h) calculated from Eqs. 3, 4 and 8, and the normalized potential U/kBT with the thermal energy kBT are presented in Fig. 5. For easy comparison, the maxima of the potential Umax and the normalized maximum potential Umax/kBT are also plotted against T in Fig. 5. The maximum potential Umax increases monotonically with temperature for all samples (Fig. 5c), but the normalized maximum potential Umax/kBT depends non-monotonically on T for the samples L2.8-2d, L3.2-2d and L3.52d, and decreases with T for the sample L3.5-4d (Fig. 5d). The above results suggest that the increase in the potential barrier does not catch up with the increase in the kinetic energy, which enhances the collision probability of the clay particles. This U/kBT characteristics is preserved in the subsequent result of W through Eq. (1), and finally in aT through Eq. (2). This means that the aggregation kinetics of our clay suspensions upon heating is affected by two competitive factors: (1) increased potential barrier by the increase
(a) L3.5-4d
T/°C 10 15 20 25 30 35 40
Umax
U (10-19J)
3.5
in ws,eff (Eq. (8)), and (2) increased collision possibility of the particles by increasing kinetic energy. For the samples L2.8-2d, L3.2-2d and L3.5-2d, these two effects compete comparatively, resulting in a maximum in aT; and for the sample L3.5-4d, the latter predominates, resulting in a decrease in aT. As known from Fig. 2, the calculated curves do not capture the clay concentration dependence of aT for the samples stored for 2 days. The observed aT maximum becomes smaller and appears at lower temperature with increasing clay concentration cL. However, the calculated trend is just the opposite. aT is comprehensively determined by multiple factors, including water viscosity, Na+ concentration, degree of the counterion condensation, interaction potential between two clay particles, and thermal motion of the clay particles, which are all considered in the model in the present study. It is therefore difficult to point out the exact origin for the above discrepancy between the observation and model. The discrepancy may also come from the spherical approximation for the plate-like clay particles. The RLCA model applied in this study is a simplified version of the population balance equations (PBEs) theory originally proposed by Smoluchowski [61] and developed by later researchers, which predicted the master curve of average radius of the growing clusters during colloidal aggregation (see [62] and references therein). However, the PBEs are applicable for initial aggregation only. There were also several experimental reports of master curves for the modulus growth measured during the post-gel stage [63-66]. Why the post-gel aging stage also follows the same master curve as the initial aggregation stage remains an open question. Previously, we interpreted the aging kinetics and master curve of the hectorite clay-NaCl-poly(ethylene glycol) suspension with the RLCA model based on the classical DLVO potential without consideration of the counterion condensation, because the added NaCl concentration was much higher than that of the counterion [25]. In the present study, in contrast, no salt was added, so the effect of counterions was not negligible.
(b) L3.5-4d
Umax/kBT
85
U/kBT
136
T/°C 10 15 20 25 30 35 40
80
3.0 0
1
0
1
h (nm)
h (nm) 95
(c)
(d)
4.0
L2.9-2d
Umax/kBT
Umax (10-19J)
L2.9-2d
3.8 L3.2-2d
3.6
L3.2-2d
90
L3.5-4d
3.4
L3.5-2d
85 L3.5-2d
L3.5-4d
3.2 10
20
30
T (°C)
40
10
20
30
40
T (°C)
Fig. 5. The total interaction potential U (a) and the reduced value U/kBT (b) for L3.5-4d at indicated temperatures; the maximum of the total potential Umax (c) and the reduced value Umax/kBT (d) varying with temperature for the indicated samples.
137
R. Shu et al. / Journal of Colloid and Interface Science 444 (2015) 132–140
3.2. Yielding and shear melting behavior
r ¼ ry þ K c_ n
ð9Þ
where ry is the yield stress, K and n are fitting parameter. The estimated yield stress in the inset of Fig. 7 increases with increasing temperature. The temperature dependence of the plateau modulus and yield stress for the sample L3.5-4d at tw = 800 s reflects the effect of ageing rate on the viscoelasticity of the suspension. The synergistic behavior of the aging and viscoelasticity of a glassy material can be described by the soft glassy rheology (SGR) model [9], which attributes the slow dynamics to the energetic trap and relies on the external force to yield the material by activation process. A dimensionless noise temperature x is introduced to parameterize the depth of quenching trap or glassiness of the system. If x > 1, the system is ergodic and equilibrated within finite time; when x < 1, the SGR model predicts the glassy state that exhibits nonequilibrium aging behavior. Prediction of the SGR model was tested by both colloidal glasses [69] and gels [70]. However, the original SGR model is in the dimensionless form [10], and quantitative comparison of its prediction to the experimental results is still rare [5,69,70]. Particularly, the waiting-time dependent dynamic modulus is given by the SGR model in a dimensionless form (for the case x < 1 and ~t 1) as
~ x; ~tÞ ¼ 1 1 ðix~tÞx1 Gð CðxÞ
ð10Þ
To apply Eq. (10) to the experimental data, the reduced complex ~ and reduced waiting time ~t are defined respectively by modulus G
20
102
10
20
30
T (°C)
40
50
T (°C) 10 15 20 25 30 35 40
101 10-2
10-1
100
101
102
103
104
γ (s ) -1
Fig. 7. Steady flow stress (symbols) of the sample L3.5-4d after aged for tw = 800 s and the fitted curve by Eq. (9) (lines) at the indicated temperatures. Inset: yield stress ry (squares) and calculated curve by Eq. (12) (line) varying with temperature.
~t ¼ t w =s0 ~ x; ~tÞ ¼ G ðx; ~tÞ=Gp Gð where s0 is a microscopic time and Gp the plateau modulus of the suspension [5,71]. s0 is related to the local yielding of a mesoscopic element with characteristic yielding time s
s ¼ s0 exp
1 2 E kl =x 2
ð11Þ
where E is the yield elastic energy of the mesoscopic element, l is the local strain applied on the mesoscopic element, and k is an elastic constant. Since the mesoscopic element in the SGR model is not related to any realistic structure, but merely defined as a region ‘‘large enough for a local strain l to be defined, but small enough for this to be approximately uniform within the region’’ [9], the choice of the s0 value is somewhat arbitrary. Previous reports mostly applied Eq. (10) to the master curve of time-aging time superposition and equated the experimental axis xtw to the x~t in Eq. (10) [5,69,71], setting implicitly s0 = 1.0 s. Yin and Solomon [70] chose s0 more carefully and found it to be of the order of 0.01 s. In the present study, we found that s0 = 0.1 s was most appropriate. The value of x at different temperatures (Fig. 8) was obtained by fitting the results of Fig. 6 with Eq. (10) and tw = 800 s (~t = 8000) and using G0 p in the inset of Fig. 6 as Gp. The x values for all samples are smaller than 1, which means the sample is already in the non-equilibrium glassy state. x decreases with
L3.5-4d, tw = 800 s, γ0 = 0.5%
G' 10
40
2
800
400 0
20
40
60
T (°C)
0.9
0.9
L3.5-4d x
T (°C) 10 15 20 25 30 35 40 45 50
0.8 0.7
0.8
0.0
0.5
1.0
aT = W / W ref
x
3
G' p (Pa)
G', G'' (Pa)
10
L3.5-4d
σ (Pa)
At tw = 800 s after the pre-shear, the structure evolution was slow enough to measure the viscoelasticity of the aged samples at the strain amplitude c0 = 0.5%. Fig. 6 presents the frequency dependence of the storage and loss moduli G0 and G00 for the sample L3.5-4d at different temperatures after aged for 800 s. At these temperatures, G0 is more than one order of magnitude higher than G00 and nearly independent of frequency, exhibiting the characteristics of a soft solid. The plateau modulus G0p (G0 at x = 0.1 rad/s) shown in the inset of Fig. 6 increases with increasing temperature. Fig. 7 displays the steady flow curves of the sample L3.5-4d at different temperatures after aged for 800 s. At all the temperatures, the stress r reaches a finite value at the shear rate limit c_ ? 0, which is the characteristics of the yield stress fluid [67]. The data are fitted with the Herschel–Bulkley model for the yield stress fluid [68]
σy (Pa)
60
0.7
G'' 10
1
10
-1
10
0
10
1
10
2
ω (rad/s) Fig. 6. Frequency dependence of storage modulus G0 (solid symbols) and loss modulus G00 (open symbols) for sample L3.5-4d aged at tw = 800 s and indicated temperatures. Inset: temperature dependence of the plateau modulus G0p .
0.6 10
20
30
40
50
T (°C) Fig. 8. Tempetaure dependence of the noise temperature x for the sample L3.5-4d. Inset: dependence of x on the shift factor aT.
R. Shu et al. / Journal of Colloid and Interface Science 444 (2015) 132–140
increasing temperature, indicating that the dynamics of the sample at tw = 800 s is trapped more deeply at higher temperature. The yield stress ry is predicted by the SGR model as [72]
R
r~ y ¼
ry 01 dll½ZðlÞx ¼ R1 x r0 dl½ZðlÞ 0
ð12Þ
with
ZðlÞ ¼
1 c_
Z
l
02
0
dl expðl =2xÞ
ð13Þ
0
γ0 = γ0ω (s-1) L3.5-4d, T = 25°C
0.01 0.05 0.1 0.5 1 2 5 10
G'1, G''1 (Pa)
10 2
10 1
10 0
10 -1 10-2
10-1
100
101
102
ω (rad/s) Fig. 9. Real and imagine parts of the first harmonic G01 (solid symbols) and G001 (open symbols) from frequency sweep at constant shear rate amplitude c_ 0 for the sample L3.5-4d at 25 °C.
(a)
-1 G'1 γ0 = γ0ω (s ) 0.01 0.05 0.1 0.5 1 2 5 G''1 10
L3.5-4d, T = 15°C
G1' , G1'' (Pa)
102
101
0
10
r0 is a scaling factor introduced to normalize the observed ry. Using x in Fig. 8 and r0 = 101.5 Pa, the calculated curve expresses the observed trend in the inset of Fig. 7. The noise temperature x in the SGR model is a signature of the quench depth for a glassy system below the glass or nonergodic state transition temperature [9,10]. Its relationship with temperature is interesting experimentally [69] and theoretically [73,74]. In the present study, the temperature dependence of x is opposite to the jamming phase diagram for attractive particles proposed by Trappe, et al. [75], and is also opposite to the previously reported results for attractive particles from experiments [75-77] and simulations [78-81]. Particularly, in the scenario of the jamming phase diagram, either decreasing temperature T or increasing the attractive potential Uattr promotes the formation of nonergodic (jammed) state [75], whereas the case of the present study is the contrary as shown in Fig. 8. This discrepancy seems to originate from the aging of the clay, which is not considered in the attractive jamming phase diagram. If a jammed system has been completely evolved, raising temperature only facilitates the thermal activation of the jammed particles and is unfavored for the jamming. For the clay suspensions here, the system is undergoing aggregation and far from completely evolved. Since the aggregation follows the RLCA, which is an irreversible aggregation mechanism, raising temperature mainly leads to a higher collision probability of the particles, hence promoting the formation of nonergodic state. To this end, the nongergodicity of the clay suspension (characterized by the noise temperature x) was related to the aggregation kinetics (characterized by stability ratio W). The inset of Fig. 8 shows the dependence of x on the shift factor of aging aT, which equals to the reduced stability ratio W/Wref. The more stable (reluctant to aggregate) the clay suspension, the less nonergodic of the system. At even larger strain or strain rate, the aged sample may be driven back to the earlier state or even fluidized. This behavior is
102
101 G''1
100
10-1
10-1 10-3
10-2
10-1
100
10-3
101
(c)
10-1
100
101
T (°C) 15 25 35
(d)
G'1
L3.5-4d, T = 35°C
10-2
τ (γ0)ω (rad/s)
τ (γ0)ω (rad/s)
103
G'1
(b) L3.5-4d, T = 25°C
G'1, G''1 (Pa)
138
100
10-1 101
τ (γ0)
G1' , G''1 (Pa)
102
G''1
10-2
0
10
-1
10
10-1 10-3
10-2
10-1
τ (γ0)ω (rad/s)
100
101
-3
10-2
10-1
100
101
γ0 (s ) -1
Fig. 10. Strain rate-frequency superposition (SRFS) master curves at constant c_ 0 for the sample L3.5-4d at indicated temperatures: (a) 15 °C, (b) 25 °C, (c) 35 °C, and the c_ 0 dependence of the horizontal shift factor s (c_ 0 ).
R. Shu et al. / Journal of Colloid and Interface Science 444 (2015) 132–140
common among soft glassy materials and is often called ‘‘shear melting’’ or ‘‘rejuvenation’’ [82-86]. Under a large amplitude oscillatory shear c(t) = c0sinxt with the shear amplitude c0 higher than the linear viscoelastic limit, the sample response becomes nonlinear with multiple higher harmonic components [87]. In this case, the measured G0 and G00 should be denoted differently as G01 and G001 because they become the first harmonic real and imagine moduli, respectively. Fig. 9 shows the curves of frequency dependent G01 and G001 at different shear rate amplitude c_ 0 (c_ 0 ¼ c0 x) at 25 °C for the sample L3.5-4d. As the shear rate amplitude increases, the G01 and G001 curves shift horizontally to higher frequency without apparent change in the shape. Also the fluidic regime (G01 < G001 ) extends with increasing c_ 0 . Wyss et al. [88] showed that for many colloid systems the frequency dependence of G01 and G001 can be scaled into the master curve when plotted against the reduced frequency sðc_ 0 Þx [64]. This was called the strain-rate frequency superposition (SRFS). The dependence of the scaling factor s on c_ 0 characterizes quantitatively how much an aged glassy material is ‘‘molten’’ by the oscillatory shear. Fig. 10a–c shows the SRFS master curves of the sample L3.5-4d at 15 °C, 25 °C, and 35 °C. Fig. 10d presents the c_ 0 dependence of the scaling factor s at three temperatures. At these temperatures, G01 and G001 can be scaled into master curves, except the high frequency portion of G001 . Wyss et al. attributed this to local viscous contribution that was shear independent [88]. The scaling a factor follows the relationship of s / c_ 0 with a 1. The difference of sðc_ 0 Þ at different temperatures is negligible. For an aging colloidal suspension, the shear field with strain rate c_ sets a limit of relaxation time scale of the sample. If the sample is aged to a state that the structure relaxation time is longer than the reciprocal of the shear rate c_ 1 , the relaxation is largely reset by c_ a with a 1 [88,89]. In other words, the structure and slow dynamics of the aged suspension were rejuvenated by shearing. The temperature effect on the aging kinetics and the aged structure is suppressed by the shearing field.
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4. Conclusion
[36]
The aging process of the salt-free suspensions of hectorite clay at different temperatures exhibited an aging time–temperature superposition. The temperature dependence of the aging kinetics was modeled by the RLCA mechanism with counterion condensation for the double-layer interaction potential of charged clay particles. The variation of aging kinetics with increasing temperature was interpreted by the competition of increased repulsive interaction and increased collision probability of the particles. The temperature dependence of plateau modulus and yield stress of the suspensions aged for 800 s was described by the SGR model. The obtained noise temperature x decreased with increasing temperature, which indicated that the sample aged at higher temperature was equivalent to a deeper quench. Finally, under larger amplitude of shearing, the sample exhibited strain rate-frequency superposition (SRFS). The strain rate amplitude dependence of the shift factor showed negligible temperature dependence, which implied that shearing eliminated the effect of aging as well as its temperature dependence.
[37]
Acknowledgments The authors owe a great deal of thanks to Prof. Hiroyuki Ohshima for the valuable discussion about the counterion condensation in the double-layer interaction. The financial support from the Natural Science Foundation of China (21204023) and the National Basic Research Program of China (973 Program, 2012CB821504) is gratefully acknowledged.
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