Therally Activated Asyetric Structural Recovery in a Soft Glassy Nano-Clay Suspension Tanay P. Dhavale, Shweta Jatav and Yogesh M Joshi* Departent of Cheical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016. INDIA. * E-Mail: joshi@iitk.ac.in Abstract In this work we study structural recovery of a soft glassy Laponite suspension by onitoring teporal evolution of elastic odulus under isotheral conditions as well as following step teperature jups. Interestingly, evolution behavior under isotheral conditions indicates the rate, and not the path of structural recovery, to be dependent on teperature. The experients carried out under teperature jup conditions however trace a different path of structural recovery, which shows strong dependence on teperature and the direction of change. Further investigation of the syste suggests that this behavior can be attributed to restricted obility of counterions associated with Laponite particle at the tie of teperature change, which do not allow counterion concentration to reach equilibriu value associated with the changed teperature. Interestingly this effect is observed to be coparable with other glassy olecular and soft aterials, which while evolve in a self-siilar fashion under isotheral conditions, show asyetric behavior upon teperature change. 1
I. Introduction Glassy aterials, such as olecular and soft (colloidal) glasses, are out of therodynaic equilibriu owing to constrained obility of the trapped constituting entities. In olecular glasses kinetically arrested state is achieved by rapidly decreasing the teperature, while in soft glasses the sae is obtained by rapidly increasing the concentration of constituents or by altering the inter-particle interactions. 1 Under quiescent conditions, these aterials undergo structural organization as a function of tie. This process is known as physical aging or structural recovery. 2-4 During this process, the olecular glasses typically undergo specific volue relaxation (or densification) as a function of tie. Aging or structural recovery in soft glasses, on the other hand, can be represented erely as progressive reduction in free energy as a function of tie, as there is no generic structural variable associated with this process. 5 In any glassy aterial, teperature is an iportant variable that affects the tiescale of structural organization, and its effect on olecular glasses has been studied in great details. 6 In this work we study how physical aging in soft glassy nano-clay (Laponite RD) suspension is affected by change in teperature, and the siilarity it shares with structural recovery in olecular and other colloidal glasses. In order to have physical aging in the aterials arrested in therodynaically out of equilibriu state, the trapped entities ust possess sufficient kinetic (theral) energy to carry out structural organization. The extree case of total absence of theral energy in the trapped entities is randoly packed stationary granular edia, wherein the aging is copletely absent, and the syste reains arrested in the high energy state forever. Enhanceent in theral energy caused by increase in teperature is observed to expedite the aging process in the olecular glasses 6-8 as well as in the soft glassy aterials. 9-12 However, depending upon the precise nature of structural rearrangeent present during the physical aging and the nature of low energy structures, additional effects triggered by the change in teperature ay also 2
influence this process. One of the standard ways to assess influence of other teperature dependent variables on structural recovery is to induce step-up and step-down teperature jups and follow the evolution of the affected variable as a function of tie. For exaples, classical experients by Kovacs 7 deonstrated that specific volue relaxation of polyeric glasses upon stepup and step-down change in teperature follows asyetric paths. This behavior is explained by copetition between filled volue dilation/shrinkage and free volue equilibration upon teperature change. The asyetry arises because filled volue dilation is restricted by free volue change in the stepup jup, but filled volue shrinkage is not restricted by free volue change in the step-down jup. 13 Recently McKenna and coworkers 10 carried out siilar experients but on a soft colloidal glass coposed of theroresponsive particles of poly(n-isopropylacrylaide). These particles show rapid swelling/shrinkage dynaics and the volue of the sae can be tuned by changing the teperature (diaeter of particle decreases by increasing teperature). Interestingly this syste also showed qualitative features of asyetry upon step-up and down jups in the volue fraction of particles triggered by change in teperature. In this work we study how the structural recovery behavior of aqueous suspension of Laponite RD, a odel soft glassy aterial, is affected by step-up and step-down teperature jups. Priary particle of synthetic clay Laponite RD has a disk-like shape with diaeter 25±2.5 n and thickness 1 n. 14 This aterial is also known to have very low size polydispersity. 15 A disk of Laponite constitutes three layers wherein two tetrahedral silica layers sandwich an octahedral agnesia layer. Isoorphic substitution of agnesiu by lithiu induces scarcity of positive charge such that face of Laponite particle acquires peranent negative charge. 16 In powder for Laponite particles are present in stacks with sodiu ion residing in the interlayer gallery. After dispersing in water, clay layers swell and owing to osotic pressure gradient, sodiu ions dissociate in water thereby exposing negatively charged faces. 17 At ph 10, edge 3
of Laponite particle has weak positive charge. 18 Overall three types of interactions are present aong the Laponite particles in the aqueous edia naely, 5 repulsion aong the faces, attraction between the edge and the face, and van der Waals attraction aong the particles. These interactions are responsible for the soft glassy nature of the Laponite suspension and influence its phase behavior and structural recovery. II. Experiental Procedure We procured Laponite RD fro Southern Clay Products Inc. White powder of Laponite was dried at 120 C for 4 hours before ixing with ultrapure water having ph 10. A detailed procedure to prepare Laponite suspension has been described elsewhere. 5 Freshly prepared unfiltered 2.8 weight % Laponite suspension was preserved in air sealed polypropylene bottle for 3 onths before using the sae. The rheological experients were perfored using MCR 501 rheoeter (Couette geoetry: outer diaeter 5.4 with gap of 0.2 ). In a typical experient saple was shear elted using oscillatory strain ( γ 0 =7000 and frequency 0.1 Hz for 20 in). Subsequently oscillatory shear stress (σ =1 Pa, frequency 0.1 Hz) was applied to onitor evolution of viscoelastic behavior in the isotheral as well as in the teperature step change experients. We verified that application oscillatory flow field with stress agnitude of 1 Pa does not affect the aging process. We also easured ionic conductivity of Laponite suspension saples using Eutech Cyberscan CON 6000 conductivity eter with 4 cell electrode (range 0-500 S and teperature range 0-70 C). We easured conductivity at constant teperatures fro 1 to 40 C as well as in teperatures step change experients. We carried out the conductivity experients in the rheoeter cell itself so as to identically atch the teperature change profiles in the rheology experients. 4
III. Results and Discussion G' [Pa] 900 600 300 (a) 150 10 3 10 4 t [s] 800 600 (b) G'/G 400 200 τ, G 1 0.1 10 3 / T (1/K) 3.2 3.3 3.4 3.5 3.6 τ G 100 10 3 10 4 10 5 t/τ Figure 1. (a) Evolution of G as a function of tie for different teperatures (diaonds 40 C, down triangles 30 C, up triangles 20 C, circles 10 C and squares 1 C. The figure (b) shows superposition of the evolution obtained at different teperatures. The inset shows variation of shift factors as a function of inverse of teperature required to obtain the superposition. 5
In figure 1 we plot evolution of elastic odulus as a function of tie elapsed since stopping the shear elting (aging tie) at five teperatures for Laponite suspension. It can be seen that the evolution of elastic odulus shifts to lower aging ties for experients carried out at higher teperatures. The self siilar curvature of G allows foration of superposition ainly by horizontal shifting. In a glassy aterial, constituents of the sae are kinetically arrested in physical cages fored by neighbors, which allow only a restricted access to its phase space causing ergodicity breaking. In the process of physical aging aterial explores its phase space and progressively attains lower values of free energy as a function of tie. If we represent individual physical cages as energy wells, lowering of free energy as a function of tie is equivalent to increase in well depth. If E is average energy well depth, then through scaling arguents elastic odulus of a aterial can be written as: G 3, where b is characteristic length. 19 Furtherore, the tiescale Eb associated with the process of aging, also known as icroscopic tiescale ( τ ), deterines the rate at which average energy well depth E increases: E = E( t τ ). Owing to faster theral otions, τ is expected to decrease with increase in teperature. Increase in G can be therefore considered as an indication of decrease in free energy, 5 and tie dependence of the sae can be given by: ( ) GT ( ) gtτ ( ) G Tt =. (1), If we introduce G = G GT ( ) and x= t τ, equation (1) can be siply written as: G = gx ( ). (1a) The superposition shown in figure 1b essentially represents function for of equation (1a) with G and τ as vertical and horizontal shift factors respectively. We arbitrarily set ( ) R GT =1 Pa and τ ( T ) R =1 s, where T R is reference teperature ( T R =1 ) as shown in the inset of figure (1b). τ is 6
observed to increase with 1 T ( τ can be seen to be following Arrhenius dependence, however the experientally explored range of 1/T is very liited). The vertical shifting is necessary as increase in teperature is observed to decrease odulus in addition to its effect on τ. Vertical shift factor GT ( ) was always closer to unity as shown in the inset. G' [Pa] 700 600 500 400 600 400 (a) (b) 40 C 40 to 30 C 40 to 20 C 40 to 10 C 40 to 1 C 200 1 C 1 to 10 C 1 to 20 C 1 to 30 C 1 to 40 C 0 5000 10000 15000 t [s] Figure 2. Evolution of G upon step jup in teperature at 1800 s. The top figure (a) describes the behavior for step-down jup fro 40 C to entioned lower teperatures, while the botto figure (b) describes step-up jup fro 1 C to entioned higher teperatures. Lines passing through the data represent the predictions of equations (2) and (3) that use the isotheral aging data. In figure 2, we plot effect of step-up and step-down teperature jup on evolution of G as a function of tie. In both the cases teperature was changed at 1800 s. We have oitted the G data associated with the transient in teperature (For coplete raw data refer to the Appendix). Figure 2(a) shows that greater the decrease in teperature is, lower is the rate of evolution of G. Alternatively, figure 2(b) shows that larger the increase in teperature is, faster is evolution of G. This scenario is described ore clearly in figure 3, where evolution of G at 1, 20 and 40 C is shown in addition to that of associated 7
with 1 to 20 C and 40 to 20 C step change. It can be seen than both the step change data approach 20 C evolution curve. In figures 2 and 3, iediately after the step increase in teperature, G can be seen to be decreasing below the isotheral aging curve of previous teperature. This is due to inverse dependence of G on teperature. However, since aging rate is greater at high teperature it soon crosses the isotheral aging curve of previous teperature. Equivalently G is expected to increase iediately after decrease in teperature; however this behavior has been asked by transient in teperature, for which we have not reported the data (Refer to Appendix for ore details). G' [Pa] 700 600 500 400 300 1 C 20 C 40 C 40 to 20 C 1 to 20 C 200 500 1000 3000 10000 t [s] Figure 3. Evolution of G at 1, 20 and 40 C and after step-down jup (40 to 20 C) and step-up jup (1 to 20 C) at 1800 s. The lines passing through the data represent predictions using isotheral evolutions and equations (2) and (3). An assuption that G follows equation (1) suggests that change in teperature does not change the path of aging, but only the rate at which aging takes place. If such scenario exists, knowledge of evolution of G at one teperature, along with the dependences of τ and G on T, can be used to predict not just evolution at any other teperature but also the evolution 8
associated with a step change. Let us assue that step change in teperature fro T 1 to T 2 was introduced at tie t 1. However, on G verses x ( G GT ( ) verses t τ ) scale the aterial will continue to age on the sae path, except with a different rate beyond tie t 1. Therefore the state of the aterial undergoing aging at teperature T 1 for tie t 1 is sae as at tie t 2 if aging would have been carried out at teperature T 2 since the beginning ( x= t1 τ 1 = t2 τ 2, where τ i is associated with teperature T i ). Therefore, t 2 is given by: t ( τ τ ) = t. (2) 2 2 1 1 The value of G at tie t after the teperature step jup ( t > t ) can then 1 siply be represented by: G = gx ( + x), (3) 1 2 where x 1 t 1 τ 1 = and ( ) be written in ters of equations (2) and (3) to yield: x2 = t t1 τ 2. Evolution of G after the step jup can then ( ( )) ( ) ( ) τ associated with the changed teperature using ( ) G = G T2, t t1 t2 = G T2 g t t1 t2 τ 2 for t t1. (3a) Above analysis suggests that, the knowledge of evolution of G under isotheral conditions and shift factors at different teperatures should facilitate prediction of evolution after the step change through equation (3). We therefore use data presented in figure 1, to predict the evolution of G after the step change in teperature in figures 2 and 3. The prediction has been described using thick lines. We have used t 1 =1920s for the step-up data and 2200s for the step-down data, which is different fro actual value of t 1 =1800 s owing to finite tie required to coplete the step change. It can be seen that equations (2) and (3) significantly over predict the evolution of G subsequent to the step-up teperature change. On the other hand predictions of step- 9
down teperature change associated with 40 to 20, 10 and 1 cross each other and do not follow the curvatures of the respective experiental data. Interestingly prediction of 40 to 30 coes closest to the experiental data. The above discussion suggests that assuption of equation (1) results in an excellent superposition (figure 1); but its logical extension, which leads to equation (3), fails to predict the evolution subsequent to the teperature change. Consequently, this result indicates that on G GT ( ) verses t τ scale the aterial will not continue to age on the sae path upon the change in teperature. This further suggests possibility of physicocheical changes to the Laponite suspension upon teperature change which ay be irreversible over the experiental tiescales. Very recently it has been reported that ionic conductivity of Laponite suspension increases with increase in teperature. 5 Ionic conductivity of aqueous Laponite suspension originates fro NaOH used to aintain ph 10 of water and the sodiu counterions (Na + ) associated with Laponite particles. Ionic conductivity of ultrapure water having ph 10 (aintained by adding NaOH) is around 20 µs/c. Therefore increase in conductivity beyond this is due to dissociation of the Na + counterions fro Laponite particles. In order to study counterion (Na + ) dissociation behavior of Laponite suspension upon teperature change, we easured ionic conductivity of the sae under isotheral conditions as well as upon teperature step change. The behavior of conductivity variation is shown in figure 4. Typically the easured values of conductivity of the suspension are over 40 ties higher than the value associated with water having ph 10. Therefore ajor contribution to ionic conductivity is fro counterions. It should be noted that before easuring the ionic conductivity, suspension saples were echanically rejuvenated at the respective teperatures. The conductivity reported in figure 4 is under quiescent conditions and easured as a function of tie after the echanical rejuvenation was stopped. It can be seen fro figure 4 that conductivity of 10
suspension kept under isotheral conditions reains constant over the duration of experient, but indeed increases with increase in teperature. We also plot conductivity upon step change in teperature in the sae plots without disturbing the aterial as is the case with rheology experients. It can be seen that upon teperature step-up and step-down change, conductivity respectively increases and decreases subsequent to the change, but does not reach the sae value associated with the isotheral easureents of the changed teperature. In addition, for saller changes in teperature the difference of isotheral conductivity value and that of upon teperature change is sall, but increases with increase in the agnitude of teperature change. σ [S/c] 1.02 0.96 0.90 1.02 0.96 0.90 (a) (b) 0 5000 10000 t [s] Figure 4. Ionic conductivity of Laponite suspension is plotted as a function of tie under isotheral conditions (open sybols) and following the teperature change conditions (filled sybols). The teperature change was carried out at 1800 s. In both the figures open sybols fro botto to top: 1 C, 10 C, 20 C, 30 C, and 40 C. Filled 11
sybols in figure (a) represent teperature down jup: black squares 40-1 C, red circles 40-10 C, blue up triangles 40-20 C, green down triangles 40-30 C. Filled sybols in figure (b) represent teperature up jup: red circles 1-10 C, blue up triangles 1-20 C, green down triangles 1-30 C, and agenta left triangles 1-40 C. Usually subsequent to the teperature change ionic conductivity (or the concentration of counterions) is expected to equilibrate with respect to the changed teperature. In case of increase in teperature ore dissociation of counterions and their diffusion away fro the particle surface is expected. On the other hand with decrease in teperature part of the counterions are expected to recobine with the faces of Laponite particle. Therefore, the plausible reason behind the difference in counterion concentration under isotheral conditions and that of after teperature change ay be the soft solid like consistency associated with Laponite suspension at the tie of teperature change. In the conductivity experients we ensure theral equilibriu before the echanical rejuvenation is perfored. During the echanical rejuvenation the aterial is in liquid state and has low viscosity, consequently the obility of the counterions is high and equilibriu value of ionic conductivity associated with the corresponding teperature is achieved. However, at the tie of teperature change the aterial is not in the liquid state but in an apparent solid state ( G is uch above 200 Pa and significantly greater than G ) and owing to which, reduction in obility of the counterions ay take place. Such reduction in the obility of counterions leads to restricted diffusion of counterions hindering achieveent of equilibration of conductivity in the teperature step change experients. Consequently, in teperature step-down experients conductivity reains at high value even at reduced teperature, while in step-up experients conductivity reains at low value though the teperature is high. We believe that this phenoenon is responsible for asyetry in the rheological behavior in the up and down teperature jup experients. 12
Concentration of counterions is known to profoundly influence interactions aong the Laponite particles as it directly affects the Debye screening length (1/κ ) as well as the surface potential ( Φ 0 ) associated with the faces of Laponite particles. Typically dependence of Debye screening length on concentration of Na + ions is given by: is given by: ( n n ) Φ0 ~ Na 0, 5 where 0 1/ κ ~ n Na 0.5, 20 while that of surface potential n is nuber density of Na + ions due to the sources other than Laponite (such as NaOH). Shahin and Joshi 5 recently analyzed interactions aong the Laponite particles using DLVO theory and observed that enhanceent in conductivity owing to counterions increases the height of repulsive energy barrier, however decreases the width of the sae when particles approach each other in a face-to-face fashion. Therefore, due to difference in conductivity associated with isotheral and teperature change experients, equation (2) and in turn equation (3) cannot be applied to the sae. Fro the coparison of the predictions of equation (3) and the experiental data shown in figure 2, it is plausible to conclude that decrease in concentration of Na + ions causes increase in G. This is ore apparent for the coparison of step-down teperature change experients (for step-up teperature change experients this effect is observable over a sall duration of tie). In figure 2, prediction subsequent to step-down teperature jup shows higher increase in G for greater decrease in T as odel considers isotheral data for which concentration of Na + ions substantially decreases with decrease in T. As tie passes, owing to higher rate of aging at higher T, the predictions cross each other. Qualitatively siilar behavior is also observed for step-up teperature jup predictions with cross over occurring soon after the teperature change. In reality, however, decrease in teperature does not decrease concentration of Na + ions to the level of isotheral values. Consequently, increase in G in step-down experients and decrease in G in step-up experients is significantly saller than that predicted by the odel. 13
Interestingly the conductivity after the teperature step down jup fro 40 C to 30 C is very siilar to that of isotheral conductivity associated with 30 C as shown in figure 4 (a). As a result, we can apply equations (2) and (3) to the G evolution data. Consequently as shown in figures 2, the prediction of the behavior upon teperature change for 40 C to 30 C coes closest to that of experiental data. In soft glassy aterials the interparticle interactions are priarily responsible for their inability to achieve equilibriu. 1 Therefore for those soft glassy aterials, where the constituents of the sae bear dissociable counterions and charged surfaces, hindered obility of the ions cannot be ruled out. For such systes aging behavior upon step teperature change can be siilar to that of observed in the present work. The present results therefore suggests apparent siilarity with the observation of asyetric structural recovery seen in the in the polyeric glasses as reported by Kovacs 7 and for thero-sensitive soft glassy paste as observed by McKenna and coworkers. 10 In all these cases asyetry can be attributed to other nonlinear physical effects induced due to change in teperature. However these other nonlinear effects originate fro kinetically arrested nature of the aterial. In the first case it is asyetric dissociation of counterions upon change in teperature while in the latter cases it is asyetry associated with filled volue expansion/contraction triggered by the teperature change. IV. Conclusion In this work, we onitor evolution of G of a shear rejuvenated soft glassy Laponite suspension under isotheral as well as step teperature change experients. It is typically observed that structural recovery described by increase in G is faster at greater teperatures, and speeds up or slows down respectively for the step-up and step-down changes in teperature. Superposition of isotheral evolution curves of G at different teperatures 14
supports an assuption that only the rate of structural recovery and not the path depends on teperature. However, when teperature change was carried out in the solid state, this assuption fails to predict the evolution of G following step change fro the isotheral data suggesting teperature change induces such effects that are irreversible over the observation tiescales. We also easure the ionic conductivity of the Laponite suspension which indicates concentration of dissociated counterions. For experients carried out at constant teperatures, conductivity is observed to be higher at greater teperatures. However during the teperature jup experients, owing to restricted obility of the ions at the tie of jup when the aterial is in solid state, change in conductivity does not reach the isotheral conductivity values associated with the changed teperature. We feel that this nonlinearity associated with conductivity change causes rheological response to be asyetric. Acknowledgeent: We would like to thank Departent of Atoic Energy Science Research Council (DAE-SRC) for financial support. We also thank an anonyous reviewer for thoughtful coents that led to substantially iproved analysis of the experiental data. 15
Appendix G' [kpa] 0.6 0.5 0.4 0.3 0.2 1 C 1-10 C 1-20 C 1-30 C 1-40 C T 80 40 0 T [ C] G' [kpa] 0.1 0.7 0.6 0.5 0.4 0.3 40 C 40-1 C 40-10 C 40-20 C 40-30 C Region of transient (a) 10 3 10 4 t [s] T Region of transient (b) 10 3 10 4 t [s] 80 40 0 T [ C] Figure S1. Raw data of G evolution upon teperature step up (a), and step down (b) jup for Laponite suspension. The corresponding change in teperature for one case each is also plotted in the respective figures. It can be seen that tie required to overcoe transient in teperature decrease ( 1000 s) is higher than that required for teperature increase ( 300 s). During such transients, since properties of the aterial change rapidly over a duration of a single cycle to onitor G, response ceases to be haronic. This induces errors in the estiation of G. Therefore we have copletely oitted the data associated with the transient in the anuscript. 16
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