Non-linear viscoelastic properties of ordered phases of a poly(ethylene oxide)-poly(propylene oxide) triblock copolymer

Rheol Acta (2008) 47:765–776DOI 10.1007/s00397-008-0293-0

ORIGINAL CONTRIBUTION

Non-linear viscoelastic properties of ordered phasesof a poly(ethylene oxide)-poly(propylene oxide)triblock copolymer

Jean-Pierre Habas · Emmanuel Pavie · Alain Lapp ·Jean Peyrelasse

Received: 29 August 2007 / Accepted: 14 May 2008 / Published online: 24 June 2008© Springer-Verlag 2008

Abstract The rheological properties of a PEO–PPO–PEO copolymer in aqueous solutions have been studiedin the gel-like zone of its phase diagram. This polymer isthe Pluronic® P105 (EO37–PO56–EO37). If the temper-ature or the concentration is high enough, the polymerforms micelles and the micelles can condense into crys-talline phases. This linear copolymer has a complex be-havior since the micelles can form different structures:hexagonal, body centered, and lamellar. But, whateverthe structure, the condensed phases exhibit the rheo-logical characteristics of viscoelastic fluids. However,these properties are strongly dependent on the valueof the stress applied during the spectromechanical ex-periment. Stress-time relationships are also observedin steady experiments. Indeed, the solutions behaveas yield stress fluids and the apparent yield stress de-pends on the time during which the stress is applied.A generalization of the Eyring’s model allows a gooddescription of this atypical behavior.

Keywords Block copolymers · Polymer solution ·Non linear viscoelasticity · Relaxation time ·Lyotropic liquid crystal

J.-P. Habas · E. Pavie · J. Peyrelasse (B)IPREM UMR 5254, Université de Pau et des Paysde l’Adour, Hélioparc, 2 avenue du président Angot,64053, Pau cedex 09, Francee-mail: [email protected]

A. LappLaboratoire Léon Brillouin, CEA Saclay, 91191,GIF-sur-YVETTE cedex, France

Introduction

There are various kinds of hydrosoluble associativecopolymers. Nevertheless, all consist of at least twosequences from which solubilities in water becomegradually very different, during a rise of temperatureor an increase of the concentration. Because of theiramphiphilic character, these polymers in solution formmicelles. The most studied systems are probably thetriblock poly(ethylene oxide)–poly(propylene oxide)–poly(ethylene oxide) (PEO–PPO–PEO) in aqueoussolutions (Zhou and Chu 1994; Mortensen 1996, 2001a,b; Prud’homme et al. 1996; Liu et al. 1996; Wu et al.1997; Schmidt et al. 1998; Goldmints et al. 1999a, b;Eiser et al. 2000a, b, c; Hyun et al. 2006). But, thephysicochemical behavior of this kind of polymer wasalso investigated in organic solvents (Chu et al. 1994;Alexandridis and Yang 2000a; Guo et al. 2000) andin water/organic solvents mixtures (Wu et al. 1996;Ivanova et al. 2000, 2001a; Alexandridis and Yang2000b; Guo et al. 2001).

In the literature, the micelles are represented inmost cases as hard spheres constituted by a hydro-phobic core surrounded by a corona containing sol-vent in variable proportion. The models proposedto describe the results obtained from SANS exper-iments even assume that the spherical micelles aremade with two or three layers (Mortensen 1996; Liuet al. 1996; Goldmints et al. 1999a, b; Yang et al. 2000;Perreur et al. 2002, 2006). The fitting of the SANSresults provide precise structural data such as the radiusof the micelles, their volume fraction, or their aggrega-tion number.

766 Rheol Acta (2008) 47:765–776

When the temperature or the polymer concentrationof the solution is increased, the micelles undergo repul-sive interactions and consequently can form lyotropicliquid crystalline phases. Depending on the nature andthe concentration of the polymer, very different struc-tures can be encountered. Many studies report that themicelles based on PEO–PPO–PEO copolymers oftencondense according to a cubic lattice (Wu et al. 1997;Mortensen 1997; Ivanova et al. 2001a, b; Hamley 2001;Perreur et al. 2002). However, the same class of poly-mers is able to produce cylindrical micelles hexagonallypacked (Wu et al. 1996; Schmidt et al. 1998; Ivanovaet al. 2001b) but also lamellar structures (Mortensenet al. 1994; Ivanova et al. 2001b). At fixed polymerconcentration, transitions can occur between differ-ent crystalline phases during a temperature increase(Bahadur and Pandya 1992; Jorgensen et al. 1997; Liet al. 1997).

A significant number of studies have been under-taken to characterize the rheological properties ofseveral PEO–PPO–PEO copolymers in aqueous solu-tion. It seems accepted that the solutions of unimersor micelles are Newtonian fluids. But a certain con-troversy exists about the rheological behavior inthe crystalline phase. Many researchers name “gel”the ordered solutions. This is due to the fact that thedivergence of the viscosity induced by the crystalliza-tion of micelles resembles a classical sol–gel transitionand because the crystallization of PEO–PPO–PEO so-lutions has sometimes been described using percola-tion rules (Lobry et al. 1999; Mallamace et al. 1999;Gambadauro et al. 2001; Chen et al. 2001). Dynamicrheological experiments performed by Wanka et al.(1990) but also Brown et al. (1991, 1992) on differentPEO–PPO–PEO copolymers in aqueous solution wereconsistent with the gel approach. Indeed, the experi-mental data showed that the shear complex modulusof the ordered solution was independent of the appliedshear frequency. Other authors showed that the crys-talline solutions of PEO–PPO–PEO copolymers areyield stress fluids (Wanka et al. 1990; Alexandridis andHatton 1995; Prud’homme et al. 1996; Eiser et al. 2000a,b, c; Pozzo et al. 2005). In a research devoted to EO76–PO29–EO76 and EO127–PO48–EO127 copolymers, Eiseret al. (2000c) proposes that the yield stress exists onlyfor fcc structures since he observes the disappearanceof the yield stress when the solutions undergo a tran-sition to bcc structures. This assessment contradictsthe data previously published. For instance, accordingto Prud’homme experiments, Pluronic F127 (EO100–PO65–EO100) crystallizes in a simple cubic structure

in aqueous solution and nevertheless presents a yieldstress during flow experiments (1996).

Further spectromechanical analyses undertaken ina wide frequency range demonstrated that the gelname was incorrect (Alexandridis and Hatton 1995;Jorgensen et al. 1997; Eiser et al. 2000c; Perreuret al. 2001; Gambadauro et al. 2001; Habas et al.2004a, b). Indeed, these researches clearly showed thatcrystalline PEO–PPO–PEO phases behave like entan-gled polymers because G′ and G′′ curves intersect atan angular frequency ωC. At high frequency (ω >ωC),the solution behaves as a viscoelastic solid since thestorage modulus G′ is much higher than the loss mod-ulus G′′. But, for lowest frequencies (ω < ωC), theviscous character of the solution becomes predomi-nant. The critical angular frequency ωC that definesthe limit between the previous domains also permitsthe estimation of the “terminal relaxation time” τ

of the solution by the relationship τ = 1/ωC. Note thatmany associative hydrosoluble copolymers behave likeentangled polymers: PEO–PPO–PEO (Alexandridisand Hatton 1995; Eiser et al. 2000c; Gambadauroet al. 2001; Jorgensen et al. 1997), PEO–PBO [wherePBO means poly(butylene oxide); Hamley et al. 1998],hydrophobic ethoxylated urethane (Annable et al.1993; Pham et al. 1999), lecithin (Shchipunov andHoffmann 1999).

In a recent paper, we studied the viscoelastic proper-ties of a four-branched PEO–PPO–PEO copolymer inaqueous solution (Habas et al. 2004b). Our resultsshowed without ambiguity that in the bcc orderedphase, the system is neither a gel, neither a yield stressfluid. The rheological behavior is highly non-linear sinceit is not possible to find a linear viscoelastic region.Furthermore, the terminal relaxation time τ is stronglydependent on the value of the stress used in the dyna-mic mechanical test and the apparent yield stressstrongly depends on the time during which the stressis applied.

The aim of this paper is to explore the possiblegeneralization of this peculiar viscoelastic behavior toother PEO–PPO–PEO copolymers in aqueous solu-tion. As the previous results were obtained with a four-branched polymer, the study of a linear copolymer(Pluronic P105) will make it possible to characterizethe influence of the polymer morphology. In addition,Pluronic P105 is able to crystallize into varied struc-tured phases: hexagonal, bcc or lamellar (Habas et al.2004a). Thus, it represents an ideal candidate to exam-ine the stress-dependence of the rheological propertiesfor each structured phase.

Rheol Acta (2008) 47:765–776 767

Experimental section

Samples

In this paper, we study the Pluronic® 105 (P105)copolymer. It is a linear copolymer (EO)37(PO)56

(EO)37 with a molecular weight of 6,500 g mol−1. It wasobtained from BASF and was used as received withoutany further purification. Aqueous solutions were pre-pared by slow stirring at low temperature (T = 5◦C)where water is a good solvent for both PEO and PPOblocks. We used bi-distilled water for all rheologicalexperiments and D2O for the small-angle neutron scat-tering studies. The polymer concentration in solutionwas expressed in weight percentages and is the samefor all measurements (p = 30% w/w).

Small-angle neutron scattering

SANS experiments were performed at the laboratoireLéon Brillouin, CEA de Saclay (France) on the PAXYspectrometer. The neutron wavelength was λ = 6 Å,and the effective distance between sample and de-tector was 3.1 m. The data collection is done in atwo-dimensional detector that counts the number ofscattered neutrons during a given time. A 2D scat-tering pattern is obtained and the one-dimensionalI(q) data set is generated by circular integration ofthe corresponding two-dimensional patterns. Detailsfor static experiments (data corrections, references)have already been given elsewhere (Perreur et al.2001). For the measurements performed under shear,we used a thermostated Couette cell made of quartz.In this geometry, the external cylinder was mobile(� = 47.24 mm) around a vertical axis, whilst the in-ternal cylinder (� = 43.43 mm) was fixed. The heightof the inner cylinder was 55 mm. The rotation speedcould vary from 0 to 800 rpm that allowed a shear-rate varying between 0 and 955 s−1. The cell was po-sitioned so that the neutron beam (horizontal) wasperpendicular to the axis of rotation of the cell whichis vertical.

Rheometry

Dynamic mechanical measurements were performedwith a Rheometric Scientific DSR stress-controlledrheometer equipped with a Couette geometry (the di-ameter and the length of the rotating inner bob are,respectively, 29.5 and 44.3 mm while the gap widthis 1.25 mm). The oscillatory frequency can be varied

from 10−4 to 100 rad s−1 while the applied stress canvary between 0.02 and 320 Pa. In order to avoid anypre-shear of the sample, the following procedure wasrespected for all rheological experiments: The solutionwas inserted in the pre-cooled cup so that the samplewas liquid and after introducing the rotor, it was recov-ered by a thin layer of silicone oil with low viscosityto prevent evaporation. Then, it was heated to thedesired temperature. Stress sweep experiments wereperformed with the same apparatus in the same stressrange. For each value, the stress was applied during ameasurement time tm and the DSR measures the meanshear rate. Various stress sweep analysis were carriedout with different tm varying between 1 to 5,500 s. Dy-namic viscosimetric measurements were carried out ona strain-imposed rheometer (ARES from RheometricScientific). The instrument was used in steady mode.As previously, a Couette geometry was used, but in thiscase, the mobile part is the cup (cup diameter: 34 mm,gap width: 1 mm, inner height: 33.4 mm).

The shear rate varied between 10−4 and 103 s−1, andthe temperature was controlled with an accuracy of±0.1◦C.

Results

Thermomecanical measurements

Figure 1 shows the effect of the temperature on thecomplex shear modulus of P105 in aqueous solution(p = 30%). At the lowest temperatures (zone 1), thesystem is a viscoelastic liquid, since the loss modulusG′′ is significantly higher than the storage modulus G′.The first decrease of the loss modulus with tempera-ture is characteristic of a solution made of individualmolecules (unimers) as described elsewhere in the caseof branched copolymers (Perreur et al. 2001, 2006).The consecutive increase of G′′ is likely due to themicelles’ formation as suggested by Lau et al. (2004)and Pozzo et al. (2005). For T = 18◦C, the complexshear modulus increases with several orders of mag-nitude and the elastic character becomes predominant(G′ > G′′). This transition resembles a sol–gel tran-sition, but by analogy with already published results(Habas et al. 2004b), it can be related to the micelle’scrystallization. A second transition is clearly observedespecially on the curve G′′(T) between 50 and 55◦Cand defines the lower limit of zone 3. An ultimatedecrease of both moduli appears at T = 70◦C. Notethat in this zone 4, the system remains more elasticthan viscous. On the contrary of the results reported by

768 Rheol Acta (2008) 47:765–776

-3

-2

-1

0

1

2

3

4

5

0 20 40 60 80

Temperature (°C)

log 1

0 G

', lo

g 10

G"

(Pa)

2 31 4

Fig. 1 Thermomechanical analysis of the solution (ω = 1 rads−1): G′ ( ) G′′ ( ). Zone 1: unimers and/or micelles. Zone 2:micelles condensed in hexagonal phase. Zone 3: bcc phase. Zone4: lamellar phase

Park and Char (2002) on a similar copolymer (P103),no turbidity change is detected between these differentzones.

Small-angle neutron scattering measurements

Small-angle neutron scattering experiments were car-ried out at various temperatures from zones 1 to 4.The objective was to identify the exact nature of thecrystalline phases and detect possible transitions be-tween organized structures. It is important to remem-ber that the 2D patterns obtained on solutions at restare isotropic in all cases.

At low temperature in zone 1, the scattered intensityremains weak and independent of q. The solutionsbehave as homogeneous mixtures and this result con-firms that in this zone the polymer is not micellized.For temperature higher than the critical micellizationtemperature, a peak appears on the I(q) spectra. Thispeak is due to significant inter-micellar correlations andits intensity increases when the temperature increases(i.e. when the volume fraction of micelles increases).

In zone 2, the main peak at q1 is followed by asmall secondary peak at q2 but the exact position ofthis peak is difficult to determine exactly. In zone 3,only the main peak is highlighted and in zone 4 a smallpeak at q2 = 2q1 shows that the structure is probably

lamellar. These results were already published and willnot be shown here (Perreur et al. 2001; Habas et al.2004a). The appearance of secondary Bragg peaks in-dicates the presence of crystalline structures but sincethe diffraction patterns are isotropic the samples arepowder like and the polycrystals have all the possibleorientations in space. When a solution in the crystallinestate is sheared, spots are observed to appear on the2D pattern. These spots become increasingly more in-tense when the shear rate γ̇ is increased. This result isgeneral and was observed with all the samples studiedin this work. At low shear rate, the ring patterns ofthe unsheared sample coexist with broad peaks havingbanana shape. When increasing shear rate, the peaksbecome sharper, and the isotropic part of the scatteredintensity disappears. So the transition was graduallyfrom a powder like sample to a sample with completealignment but the crystalline structure is the same one.When the sample is completely aligned, one does notobserve any further modification of the diffraction pat-tern up to the maximum shear rates accessible withour experimental set-up (955 s−1). In all cases, it wasalso noted that, a pre-oriented sample does not presentat rest any disorientation process even after a longperiod of time (t > 10 h). As shown on Fig. 2, the spotsare located on vertical layers and concentric circles.We previously gave a rigorous interpretation of thediffraction patterns of condensed solutions subjected toa shear rate (Perreur et al. 2002; Habas et al. 2004b).Under flow, the polycrystals are oriented with one oftheir direction in the direction of the flow, but have allthe possible orientations around this direction.

Figure 2a is obtained in zone 2. A shear rate of100 s−1 is high enough to obtain a maximum orien-tation. The diffraction pattern is characteristic of ahexagonal structure. The system form rodlike micelleswith hexagonal packing. As observed with compara-ble systems (Holmqvist et al. 1997), the spot patterncan be completely explained by an alignment perpen-dicular to the flow direction. Figure 2b was registeredin zone 3 and shows that the spots are separated byangles of 55◦ and 70◦, which is characteristic of a bodycentered cubic lattice whose crystals are directed withthe row [1, 1, 1] in the direction of the flow (Perreuret al. 2002). The shear rate necessary to obtain theorientation maximum (about 600 s−1) is higher than inthe preceding case.

In zone 4, the pattern (Fig. 2c) is symmetric withspots located at q1 and 2q1. This shows that the struc-ture is lamellar with an orientation parallel to the flow.It is necessary to notice that in this case, the orientationis easy. A shear rate of 30 s−1 is sufficient to obtain amaximum orientation.

� �

Rheol Acta (2008) 47:765–776 769

Fig. 2 2D diffraction patternsfor the solution under flow.a hexagonal in zone 2 of theFig. 1; b bcc in zone 3;c lamellar in zone 4

a b c

Dynamic mechanical analyses

Figure 3 shows the spectromechanical properties ofa 30% solution at different temperatures when thestructure of the ordered phase is a hexagonal crystallinelattice. These experiments were performed with thesame value of the applied stress (σ = 320 Pa). Inthe high frequency range, G′ is higher than G′′ andboth moduli depend little on the frequency (solid likebehavior). But, for low frequencies and at the lowesttemperatures, one reaches the flow zone where G′′ ∝ ω.

The same figure also shows that the crossover point(G′ = G′′) at the angular frequency ωC, shifts to lowerfrequencies when the temperature is increased. At the

-4 -3 -2 -1 0 1 2

log10 ω (rad s-1)

log 1

0 G

', lo

g 10

G"

(a.u

)

39°C

26°C

21°C

Fig. 3 Effect of the temperature on the spectromechanical re-sponse of the solution in the hexagonal phase. σ = 320 Pa. G′( ) G′′ ( ). For clarity, the different series have been shiftedalong the vertical axis

highest temperature (T > 39◦C), the crossover point iseven out of the accessible frequency range. This evo-lution indicates that the terminal relaxation time (τ =1/ωC) of the ordered solution strongly increases withthe temperature. At a fixed temperature, the relaxationtime is also highly dependent on the stress as shown inFigs. 4 and 5 at two temperatures in the hexagonal zone(T = 23 and 26◦C). In both cases, one notes that thecrossover between the curves G′ and G′′ strongly movestowards the high frequencies as the stress increases(i.e., the relaxation time decreases). This stress-timedependency is much more important at 23◦C, i.e., whenthe temperature is close to the liquid–crystal transitiontemperature. Nevertheless, it is to note that the plateau

-4 -3 -2 -1 0 1 2

log10 ω (rad s-1)

log 1

0 G

', lo

g 10

G"

(a. u

.)

100 Pa

130 Pa

200 Pa

290 Pa

Fig. 4 Effect of the applied stress value on the spectromechanicalresponse of the solution in the hexagonal phase. T = 23◦C. G′( ) G′′ ( ). For clarity, the different series have been shiftedalong the vertical axis

� � � �

770 Rheol Acta (2008) 47:765–776

-4 -3 -2 -1 0 1 2

log10 ω (rad s-1)

log 1

0 G

' et l

og10

G"

(a. u

.)

200 Pa

320 Pa

100 Pa

25 Pa

Fig. 5 Effect of the applied stress value on the spectromechanicalresponse of the solution in the hexagonal phase. T = 26◦C. G′( ) G′′ ( ). For clarity, the different series have been shiftedalong the vertical axis

modulus remains independent on the applied stress(the curves are arbitrarily shifted along the vertical axisfor clarity).

Two series of spectromechanical measurements werealso conducted in the body-centered cubic phase (T =54◦C) and in the lamellar phase (T = 72◦C). The exper-imental data respectively represented in Figs. 6 and 7show that the terminal relaxation time decreases whenthe applied stress increases. The global spectromechan-ical behavior of a bcc condensed solution is very similarto that observed in the hexagonal phase. Consequently,it is impossible to differentiate a hexagonal structurefrom a bcc ordered phase using only spectromechanicalmeasurements. But, the lamellar phase presents quitedistinct viscoelastic properties: its plateau modulus ismuch smaller (≈ 102 to 103 Pa) than that observed withthe bcc or hexagonal phases (≈ 4 104 Pa). Moreover,the terminal relaxation time is significantly shorter andthen the flow zone is reached for smaller values of theapplied stress.

Steady experiments

We reminded in the introduction part, that some au-thors show that condensed copolymer solutions canbehave as yield stress fluids. In order to investigate thepossible existence of a yield stress in P105 structuredsolutions, stress sweep measurements were performed

-4 -3 -2 -1 0 1 2

log10 ω (rad s-1)

log 1

0 G

', lo

g 10

G"

(a. u

.)

175 Pa

320 Pa

Fig. 6 Effect of the applied stress value on the spectromechanicalresponse of the solution in the bcc phase. T = 54◦C. G′ ( ) G′′( ). For clarity, the different series have been shifted along thevertical axis

in steady mode. The following procedure was respectedfor each measurement. The sample is initially cooledin order to be in the liquid state. Then, it is heated

-4 -3 -2 -1 0 1 2

log10 ω (rad s-1)

log 1

0 G

', lo

g 10

G"

(a. u

.)

25 Pa

50 Pa

7 Pa

Fig. 7 Effect of the applied stress value on the spectromechanicalresponse of the solution in the lamellar phase T = 72◦C. G′ ( )G′′ ( ). For clarity, the different series have been shifted alongthe vertical axis

�

�

� �

�

�

Rheol Acta (2008) 47:765–776 771

until the test temperature where the solution presentsa structured phase. For each value of the applied stress,the shear rate is measured after a time tm that remainsconstant along each stress sweep analysis. Various mea-surements were undertaken at the same temperaturebut for different values of tm ranging from 2 to 2,400 s.Figure 8 shows the results obtained for the solutions ofP105 in the hexagonal phase (T = 26◦C). In a doublelogarithmic scale, the first point on each curve gives theyield stress value σc. Indeed, it is the first point where·γ is different of zero. One can easily notice that the σc

value depends on the time during which the stress isapplied. As a consequence, the critical stress σc shouldbe called “apparent yield stress”. It is important to notethat the flow did not change the equilibrium structureof the sample since these experiments are immediatelyreproducible. Figure 9 represents the variations of theapparent yield stress upon the measurement time tm. Itis clear that σc tends toward very low values when tmincreases and it is even likely that σc vanishes when tmtends toward infinity. Similar results were obtained withthe bcc and lamellar phases.

The knowledge of the stress variations with the shear

rate·γ makes it possible to calculate the apparent vis-

cosity for each measurement time tm with the relation:η = σ

·γ

. Figure 10 shows the results obtained for a 30%

solution in the hexagonal phase (T = 26◦C). The struc-

1

1.5

2

2.5

3

-7 -6 -5 -4 -3 -2 -1

log10 γ (s-1)

log 1

0σ

(Pa

s)

2s

10s

60s

600s

2400s

.

Fig. 8 Influence of the measurement time tm on the determina-tion of the apparent yield stress of the solution in the hexagonalphase (T = 26◦C)

0

40

80

120

160

0 1 2 3 4

log10 tm (s)

σ c (

Pa)

Fig. 9 Apparent yield stress σc as a function of the measurementtime tm for the solution in the hexagonal phase (T = 26◦C)

tured solution presents a rheothinning behavior. Forlow shear rate, the curves present a Newtonian plateauwith viscosity η0. The η0 value increases with the time ofmeasurement tm. For the highest values of

·γ , all curves

are superimposed showing that after a short transitionzone, the viscosity of the solution does not depend onthe measurement time tm. Similar results are obtainedin the cubic zone. These results demonstrate that evenif the solution is able to flow for very low values ofthe stress, the viscosity of the system remains very high(> 108 Pa s). For this reason, the solution at rest lookslike a gel.

3.5

4.5

5.5

6.5

7.5

8.5

-7 -6 -5 -4 -3 -2 -1

log10 γ (s-1)

log 1

0η

(Pa

s)

2s

10s

60s

600s

2400s

.

Fig. 10 Apparent viscosity as a function of the shear rate for thesolution in the hexagonal phase (T = 26◦C)

772 Rheol Acta (2008) 47:765–776

The rheothinning properties of solutions in the crys-talline phases can be described with Carreau’s model:

η( ·γ)

= ηo

1 +(τo

·γ)n (1)

where ηo is the zero-shear viscosity, τ o, the average re-laxation time of the solution and n the pseudo-plasticityindex. The values of these different parameters, ob-tained for all measurement times tm in the hexagonalzone are presented in Table 1. It is easy to check that,in log-log scale, the viscosity and average relaxationtime are linear functions of the measurement time. Thisshows that for very long measurement time, the viscos-ity and relaxation time tend towards infinite values.

Several research groups have already described therheothinning properties of structured PEO–PPO–PEOsolutions (Prud’homme et al. 1996; Jorgensen et al.1997; Pham et al. 1999; Eiser et al. 2000c; Guo et al.2001). In these studies, the decrease of the viscositywith the shear gradient is often allotted to the pro-gressive orientation of the polycrystals in the flow di-rection. In our case, this interpretation seems to beincorrect. We performed a first shear sweep in order toorient the polycrystals and observed that the viscositydecrease following the Carreau model. However, therecovery of the rheological properties was quite fast(lower than 2 mn) since the result is immediately repro-ducible. On the contrary, the SANS patterns showedthat after several hours at rest, a pre-oriented sampledoes not present any disorientation mechanism. As aconsequence, the particular rheological properties ofordered solutions are independent of the orientationof the polycrystals. Similar conclusion was obtained byLinemann et al. (1995) in the study of aqueous solutionsof a branched nonionic surfactant. This surfactant wasable to condense into a cubic phase where the samplebehaved like an elastic solid for shearing frequencieshigher than 5 10−3 rad s−1. But in the same region of itsphase diagram, the sample was able to flow when a highshear rate was applied. As in our experiments, the flow

Table 1 Zero-shear viscosity ηo, average relaxation time τo andpseudo-plasticity index n of the Carreau model

Measurement time tm (s) ηo (Pa s) τo (s) n

2,400 2.12 108 1.00 106 0.95600 6.58 107 2.88 105 0.9660 9.37 106 4.63 104 0.9310 2.88 106 1.27 104 0.942 8.75 105 3.36 103 0.95

did not change the structure since the same equilibriummodulus was obtained before and immediately after acreep experiment.

Discussion

Several authors have investigated the dynamic of mi-cellar systems by various experimental techniques suchas nuclear magnetic resonance (Almgren et al. 1995;Cau and Lacelle 1996), dynamic light scattering (Brownet al. 1992; Malmsten and Lindman 1992; Waton et al.2001) or SANS (Huibers et al. 1999). If the existence ofrelaxation phenomena is indisputable, their structuralorigin remains subject of controversy. Indeed, a verylarge range of characteristic times is observed from1 μs to 1 h. Different theoretical approaches were evenproposed to relate these relaxation times to molecu-lar mechanisms. In the simplest model, the relaxationprocess is due to the life time of a micelle (Shchipunovand Hoffmann 1999; Candau and Oda 2001). Othertheories are based on micelles’ diffusion. Pham et al.(1999) propose a process that is similar to lacunarmechanism where micellar units can move in free vol-umes produced by density fluctuations. According toAnnable et al. (1993), the micellar diffusion can bedescribed by the transient network rules. At least, inrheological studies dealing with associative copolymers,the terminal relaxation time of the solution is associatedto the disentanglement of neighboring micellar coronas(Prud’homme et al. 1996; Annable et al. 1996). Never-theless, it is important to note that none of the hereabove reported mechanisms is able to take into accountthe influence of an applied stress. More generally, ifthe nonlinearity of the dynamic rheological propertiesof structured micellar systems was already observed inthe literature, it was rarely interpreted (Jorgensen et al.1997; Pople et al. 1997; Hamley et al. 1998, 2000; Lobryet al. 1999; Daniel et al. 2001).

First of all, the decrease of the relaxation time of theordered solution with the applied stress cannot be dueto the breaking of the structure. Indeed, the rheologicalexperiments are immediately reproducible and are fullyreversible.

In a previous paper dealing with the description ofa branched PEO–PPO–PEO (T908), we demonstratedthat this copolymer was able to condense in a bccphase and that its viscoelastic properties were stronglydependent on the applied stress (Habas et al. 2004b).In the same work, we proposed to describe the stress-time dependency of the rheological behavior by usingan approach derived from the Eyring’s theory. In this

Rheol Acta (2008) 47:765–776 773

model, the terminal relaxation time τ is characteristicof the diffusion of a micelle that jumps in a vacancy ofthe crystalline structure. It can be expressed by:

τ =h

kT exp( Ea

kT

)

exp(

σSb2kT

) − exp(−σSb

2kT

) (2)

where h and k are, respectively, the Planck andBoltzmann constants, T the temperature and Ea theactivation energy. S is the surface of a micelle trappedin a potential gap of width b produced by surround-ing interacting micelles. The application of an externalstress σ to the ordered solution induces the micellediffusion and consequently, the displacement of thewhole crystalline structure because of the persistenceof the micellar interactions. As the surface of a micellecan be determined from SANS experiments, the onlyadjustable parameters of the model are the interactionenergy Ea and the diffusion length b .

This theoretical approach was powerful enough todescribe the stress-time dependency of T908 orderedsolutions. We have applied the same formalism to thestructured solutions of P105. Figure 11 shows the resultsobtained for the solution at T = 26◦C where the mi-celles are ordered according to a hexagonal structure.In these experimental conditions, the radius of thecylindrical micelle determined from SANS experimentswas close to Rm = 40 Å. One can note that the modelallows a quite satisfactory prediction of the experi-mental data τ = f (σ). The optimization of the modelparameters provides the diffusion length b = 4350 Å

2

3

4

5

0 50 100 150 200 250 300 350

σ (Pa)

log 1

0τ

(s)

Fig. 11 Terminal relaxation time at T = 26◦C in the hexagonalphase as a function of the stress. The full line corresponds to Eq. 2

and the activation energy Ea = 1.63 × 10−19 J. Thisvalue is comparable with that previously determinedfor branched PEO–PPO–PEO (Habas et al. 2004b).

If the experimental data presented in Fig. 11 seemto be fitted by a linear regression, the model predictsthe divergence of the terminal relaxation time of thestructured phase at low value of the applied stress.This information is relevant because structured PEO–PPO–PEO solutions are, at rest, gel-like phases. Onthe contrary, at high σ values, the micelle diffusion isquicker and the characteristic relaxation time shorter.

The same phenomenon is also responsible for thestress-time dependency observed during the rheolog-ical characterization of structured P105 solution insteady mode. The apparent yield stress of the solution isa direct consequence of the evolution of the relaxationtime with the applied stress. The sample’s flow will bedetected whenever the measurement time will be largerthan the relaxation time. This explains why the yieldstress is all the more small as measurement time is large.

The diffusion mechanism described above is suit-able for micellar structures. As regards P105 solu-tion, it should only be convenient for hexagonal andbcc phases. Nevertheless, stress-time relationships werealso observed for P105 ordered solution in the lamellarphase. Once more, the same model is able to fit theevolution of the terminal relaxation time of the solutionin this phase as shown in Fig. 12 at T = 72◦C. In thislatter case, the layer-sliding mechanism remains pos-sible but the diffusing object is more likely a molec-ular cluster. The activation energy obtained from the

Fig. 12 Evolution of the terminal relaxation time at T = 72◦Cwith the applied stress in the lamellar phase. The full line corre-sponds to Eq. 2

774 Rheol Acta (2008) 47:765–776

optimization of the model parameters is slightly smaller(Ea = 1.58 × 10−19 J) than in the hexagonal phase. Thismeans that the interaction energy between moleculesis lower and agrees with the reduction of the shearmodulus in the lamellar phase in comparison with thehexagonal and bcc phases.

Conclusion

The viscoelastic properties of aqueous Pluronic 105ordered solutions were investigated using steady-stateshear and oscillatory shear experiments. Whatever thenature of the phase studied, the spectromechanical be-havior of the system resembles to that usually observedwith entangled polymer. If the lamellar phase seems topresent a lower value of the plateau modulus, there isno important difference between the hexagonal and thebcc arrangements. Nevertheless, in all cases, the termi-nal relaxation time of the structured solution is stronglydependent on the value of the stress imposed during thefrequency analysis. Similar stress-time dependency wasalso observed in stress sweep experiments. Indeed, theordered solution presents an apparent yield stress, thevalue of which is directly related to the measurementtime. All these results were described and explainedusing a model derived from Eyring’s theory. In the caseof bcc or hexagonal phase, this approach is based onthe diffusion of a micelle from an energy gap producedby neighboring and interacting units. In the lamellarphase, the diffusing object is rather represented by amolecule cluster. This relaxation model agrees withthe fast recovery of the rheological properties of theordered solution. Moreover, it does not contradict theresults of SANS under flow since the proposed mech-anism does not induce any macroscopic disorientationof the crystalline structure. All the results reported inthis article with linear P105 block copolymer have alsobeen observed in a previous paper with branched PEO–PPO–PEO. In other words, the non-linearity of therheological behavior of this copolymer class seems tobe independent of the material morphology and of thecrystalline structure. This finding reveals that the differ-ent mechanism described and their interpretation arelikely applicable to other PEO–PPO–PEO in aqueoussolutions.

References

Alexandridis P, Hatton TA (1995) Poly(ethylene oxide)-poly(propylene oxide)-poly(ethylene oxide) block copolymerssurfactants in aqueous solutions and at interfaces: thermo-

dynamics, structure, dynamics and modeling. Colloids SurfA 96:1–46

Alexandridis P, Yang L (2000a) Micellization of polyoxyalky-lene block copolymers in formamide. Macromolecules 33:3382–3391

Alexandridis P, Yang L (2000b) SANS investigation of poly-ether block copolymer micelle structure in mixed solvents ofwater and formamide, ethanol or glycerol. Macromolecules33:5574–5587

Almgren M, Brown W, Hvidt S (1995) Self-aggregation andphase-behavior of poly(ethylene oxide) poly(propylene ox-ide) poly(ethylene oxide) block-copolymers in aqueous-solution. Colloid and Polym Sci 273:2–15

Annable T, Buscall R, Ettelaie R, Whittlestone D (1993) Therheology of solutions of associating polymers: comparisonof experimental behavior with transient network theory.J Rheol 37:695–726

Annable T, Buscall R, Ettelaie R (1996) Network formationand its consequences for the physical behavior of associatingpolymers in solution. Colloids Surf A 112:97–116

Bahadur P, Pandya K (1992) Aggregation behavior of PluronicP-94 in water. Langmuir 8:2666–2670

Brown W, Schillén K, Almgren M, Hvidt S, Bahadur P(1991) Micelle and gel formation in a poly(ethylene ox-ide)/poly(propylene oxide)/ poly(ethylene oxide) triblockcopolymer in water solution. Dynamic and static light scat-tering and oscillatory shear measurements. J Phys Chem95:1850–1858

Brown W, Schillén K, Hvidt S (1992) Triblock copolymers inaqueous solution studied by static and dynamic light scatter-ing and oscillatory shear measurements. Influence of relativeblock sizes. J Phys Chem 96:6038–6044

Candau SJ, Oda R (2001) Linear viscoelasticity of salt-free worm-like micellar solutions. Colloids Surf A 183:5–14

Cau F, Lacelle S (1996) 1H NMR relaxation studies of the mi-cellization of a poly(ethylene oxide)-poly(propylene oxide)-poly(ethylene oxide) triblock copolymer in aqueous solu-tions. Macromolecules 29:170–178

Chen SH, Liao C, Fratini E, Baglioni P, Mallamace F (2001)Interaction, critical, percolation and kinetic glass transitionsin Pluronic L-64 micellar solutions. Colloids Surf A 183:95–111

Chu B, Wu G, Schneider DK (1994) A scattering study on inter-micellar interactions and structures of polymeric micelles. JPolym Sci Part B: Polym Phys 32:2605–2614

Daniel C, Hamley IW, Wilhelm M, Mingvanish W (2001) Non-linear rheology of a face-centred cubic phase in a diblockcopolymer gel. Rheol Acta 40:39–48

Eiser E, Molino F, Porte G (2000a) Correlation between theviscoelastic properties of a soft crystal and its microstructure.Eur Phys J E 2:39–46

Eiser E, Molino F, Porte G, Diat O (2000b) Nonhomogeneoustextures and banded flow in a soft cubic phase under shear.Phys Rev E 61:6759–6764

Eiser E, Molino F, Porte G, Pithon X (2000c) Flow in micellarcubic crystals. Rheol Acta 39:201–208

Gambadauro P, Venuti V, Liao C (2001) Dynamical propertiesin dense triblock copolymer micellar system. Colloids SurfA 183:133–147

Goldmints I, Yu GY, Booth C, Smith KA, Hatton TA (1999a)Structure of (deuterated PEO)-(PPO)-(deuterated PEO)block copolymer micelles as determinated by small angleneutron scattering. Langmuir 15:1651–1656

Goldmints I, von Gottberg FK, Smith KA, Hatton TA (1999b)Small angle neutron scattering study of PEO-PPO-PEO

Rheol Acta (2008) 47:765–776 775

micelle structure in the unimer-to-micelle transition region.Langmuir 13:3659–3664

Guo C, Liu HZ, Chen JY (2000) A fourier transform infraredstudy on water-induced reverse micelle formation of blockcopoly(oxyethylene-oxypropylene-oxyethylene) in organicsolvent. Colloids Surf A 175:193–202

Guo L, Colby RH, Lin MY, Dado GP (2001) Micellar structurechanges in aqueous mixtures of nonionic surfactants. J Rheol45:1223–1243

Habas JP, Pavie E, Perreur C, Lapp A, Peyrelasse J (2004a)Nanostructure in block copolymer solutions: rheology andsmall-angle neutron scattering. Phys Rev E 70:Art. No.61802

Habas JP, Pavie E, Lapp A, Peyrelasse J (2004b) Understand-ing the complex rheological behavior of PEO-PPO-PEOcopolymers in aqueous solutions. J Rheol 48:1–21

Hamley IW (2000) The effect of shear on ordered block copoly-mer solutions. Curr Opin Colloid Polym Sci 5:342–350

Hamley IW (2001) Amphiphilic diblock copolymer gels: the rela-tionship between structure and rheology. Philos T Roy SocA 359:1017–1044

Hamley IW, Pople JA, Fairclough JPA, Ryan AJ, Booth C, YangYW (1998) Shear-induced orientational transitions in thebody-centered cubic phase of a diblock copolymer. Macro-molecules 31:3906–3911

Holmqvist P, Alexandridis P, Lindman B (1997) Modificationof the microstructure in poloxamer block copolymer-water-“Oil” systems by varying the “Oil” type. Macromolecules30:6788

Huibers PDT, Bromberg LE, Robinson BH, Hatton TA (1999)Reversible gelation in semidilute aqueous solutions of as-sociative polymers: a small-angle neutron scattering study.Macromolecules 32:4889–4894

Hyun K, Nam JG, Wilhellm M, Ahn KH, Lee SJ (2006) Largeamplitude oscillatory shear behavior of PEO-PPO-PEO tri-block copolymer solutions. Rheol Acta 45(3):239–249

Ivanova R, Lindman B, Alexandridis P (2000) Evolution in struc-tural polymorphism of Pluronic F127 poly(ethylene oxide)-poly(propylene oxide) block copolymer in ternary systemswith water and pharmaceutical acceptable organic solvents:From “glycols” to “oils”. Langmuir 16:9058–9069

Ivanova R, Lindman B, Alexandridis P (2001a) Modification ofthe lyotropic liquid crystalline microstructure of amphiphilicblock copolymers in the presence of cosolvents. Adv ColloidInterfac Sci 89–90:351–382

Ivanova R, Alexandridis P, Lindman B (2001b) Interaction ofpoloxamer block copolymers with cosolvents and surfac-tants. Colloids Surf A 183–185:41–53

Jorgensen EB, Hvidt S, Brown W, Schillén K (1997) Effects ofsalts on the micellization and gelation of a triblock copoly-mer studied by rheology and light scattering. Macromole-cules 30:2355–2364

Lau BK, Wang QQ, Sun W (2004) Micellization to gelation ofa triblock copolymer in water: Thermoreversibility and scal-ing. J of Polym Sci Part B – Polym Phys 42(10):2014–2025

Li H, Yu GE, Price C, Booth C, Hetch E, Hoffmann H(1997) Concentrated aqueous micellar solutions of diblockcopoly(oxyethylene/oxybutylene) E41B8: a study of phasebehavior. Macromolecules 30:1347–1354

Linemann R, Läuger J, Schmidt G, Kratzat K, Richtering W(1995) Linear and nonlinear rheology of micellar solutionsin the isotropic cubic and hexagonal phase probed by rheo-small-angle light scattering. Rheol Acta 34:440–449

Liu YC, Chen SH, Huang JS (1996) Relationship betweenthe microstructure and rheology of micellar solutions

formed by a triblock copolymer surfactant. Phys Rev E 54:1698–1708

Lobry L, Micali N, Mallamace F, Liao C, Chen SH (1999) Interac-tion and percolation in the L64 triblock copolymer micellarsystem. Phys Rev E 60:7076–7087

Mallamace F, Chen SH, Liu Y, Lobry L, Micali N (1999) Per-colation and viscoelasticity of triblock copolymer micellarsolutions. Physica A: Stat and Theor Phys 266:123–135

Malmsten M, Lindman B (1992) Self-assembly in aqueous blockcopolymer solutions. Macromolecules 25:5440–5445

Mortensen K (1996) Structural studies of aqueous solutionsof PEO-PPO-PEO triblock copolymers, their micellar ag-gregates and mesophases; a small-angle neutron scatteringstudy. J Phys: Condens Matter 8:103–124

Mortensen K (1997) Cubic phase in a connected micellarnetwork of poly(propylene oxide)-poly(ethylene oxide)-poly(propylene oxide) triblock copolymers in water. Macro-molecules 30:503–507

Mortensen K (2001a) PEO-related block copolymer surfactants.Colloids Surf A 183:277–292

Mortensen K (2001b) Structural properties of self-assembledpolymeric aggregates in aqueous solutions. Polym AdvTechnol 12:2–22

Mortensen K, Brown W, Jorgensen E (1994) Lamellarmesophase of poly(ethylene oxide)-poly(propylene oxide)-poly(ethylene oxide) melts and water-swollen mixtures.Macromolecules 27:5654–5666

Park MJ, Char K (2002) Two gel states of a PEO-PPO-PEOtriblock copolymer formed by different mechanisms. Macro-mol Rapid Com 23:688–692

Perreur C, Habas JP, Peyrelasse J, François J, Lapp A (2001)Rheological and small-angle neutron scattering studies ofaqueous solutions of branched PEO-PPO-PEO copolymers.Phys Rev E 63:Art. No.31505

Perreur C, Habas JP, François J, Peyrelasse J, Lapp A (2002)Determination of the structure of the organized phase of theblock copolymer PEO-PPO-PEO in aqueous solution underflow by small-angle neutron scattering. Phys Rev E 65: Art.No. 41802

Perreur C, Habas JP, Lapp A, Peyrelasse J (2006) Salt influenceupon the structure of aqueous solutions of branched PEO-PPO-PEO copolymers. Polymer 47:841–848

Pham QT, Russel W, Thibeault JC, Lau W (1999) Micellarsolutions of associative triblock copolymers: the relation-ship between structure and rheology. Macromolecules 32:5139–5149

Pople JA, Hamley IW, Fairclough JPA, Ryan AJ (1997)Ordered phases in aqueous solutions of diblockoxyethylene/oxybutylene copolymers investigated bysimultaneous small-angle X-ray scattering and rheology.Macromolecules 30:5721–5728

Pozzo DC, Hollabaugh KR, Walker LM (2005) Rheology andphase behavior of copolymer-templated nanocomposite ma-terials. J Rheol 49:759–782

Prud’homme RK, Wu G, Schneider DK (1996) Structureand rheology studies of poly(oxyethylene-oxypropylene-oxyethylene) aqueous solution. Langmuir 12:4651–4659

Schmidt G, Richtering W, Lindner P, Alexandridis P (1998)Shear orientation of a hexagonal lyotropic triblock copoly-mer phase as probed by flow birefringence and small-anglelight and neutron scattering. Macromolecules 31:2293–2298

Shchipunov YA, Hoffmann H (1999) Indicative evidence forcoexistence of long and short polymer-like micelles inlecithin organogel from rheological studies. Langmuir 15:7108–7110

776 Rheol Acta (2008) 47:765–776

Wanka G, Hoffmann H, Ulbricht W (1990) The aggregationbehavior of poly-(oxyethylene)-poly-(oxypropylene)-poly-(oxyethylene)-block-copolymers in aqueous solution. Col-loid Polym Sci 268:101–117

Waton G, MichelsB, ZanaR (2001)Dynamicsofblockcopolymermicelles in aqueous solution. Macromolecules 34:907–910

Wu G, Liu L, Buu VB, Chu B, Schneider DK (1996) SANSand SAXS studies of pluronic L64 in concentrated solution.Physica A 231:73–81

Wu C, Liu T, Chu B, Schneider DK, Graziano V (1997) Char-acterization of the PEO-PPO-PEO triblock copolymer and

its application as a separation medium in capillary elec-trophoresis. Macromolecules 30:4574–4583

Yang L, Alexandridis P, Steytler DC, Kositza MJ, HolzwarthJF (2000) Small-angle neutron scattering investigationof the temperature-dependent aggregation behavior ofthe block copolymer L64 in aqueous solution. Langmuir16:8555–8561

Zhou Z, Chu B (1994) Phase behavior and associationproperties of poly(oxypropylene)-poly(oxyethylene)-poly-(oxypropylene) triblock copolymer in aqueous solution.Macromolecules 27:2025–2033