History-dependent rheology of a surfactant hexagonal phase
M. L. Sushko,* J. M. Seddon, and R. H. TemplerDepartment of Chemistry, Imperial College, Exhibition Road, London SW7 2AY, United Kingdom
~Received 16 August 2001; revised manuscript received 7 November 2001; published 11 February 2002!
The time-dependent response of a surfactant hexagonal phase of a sodium dodecyl sulphate/pentanol/cyclohexane/brine system to stepped strain is investigated. The dynamics of the system is found to be governedby strain- and noise-induced yielding of the domains of the system. The effects of the applied strain magnitudeand the ionic strength of the brine on the character of the transitions experienced by the system are reported.
Many soft materials exhibit history-dependent rheoloTwo different types of history-dependent responses hbeen defined: rheological aging, and a ‘‘transient’’ behavof a power-law fluid. The recent theory of the rheologicbehavior of soft glassy materials, the soft glassy rheolo~SGR! model, describes both cases. Aging~of the step strainresponse! is defined in the SGR model@1,2# as the propertythat a significant part of the stress relaxation takes placetime scales that grow with the age of the system. Contrarthe case of aging systems, in ‘‘transient’’ behavior all signcant relaxation processes can essentially be observed onite time scales. However, in ‘‘transient’’ systems the timscales of the relaxation after the step strain also depenthe age of the system at the time of strain application. Ththese systems have a short-term memory. Transient behhas been found in dense emulsions@3#, foams@4#, and poly-domain defect textures in ordered mesophases@5#, whoserheology is governed by rearrangements of domains.
The aging phenomenon of pastes@6#, flocculated disper-sions @7#, colloidal glasses@8#, and some other systemswell described experimentally and a good agreement wthe theory has been reported. However, the histodependent rheology of ‘‘transient’’ systems seems nothave been directly investigated. In this paper we will dscribe the history-dependent response upon shear of a sutant hexagonal phase. First we give a brief overview ofexperimental data on the rheology of lyotropic hexagophases.
Surfactant hexagonal phases have been found to exence transitions to some new stable states of director ortation under steady shear or constant stress@9#. Small anglelight scattering studies revealed correlated orientational fltuations perpendicular to the flow direction at low cretimes or at low shear rates, but parallel to the flow directat long creep times or high shear rates@10#. The correlationof orientational fluctuations, perpendicular to the flow diretion, was attributed to log rolling of the cylindrical micellepredicted theoretically as a possible state of orientationpolymeric liquid crystals under shear@11#. However, rheo-nuclear magnetic resonance investigations revealed thaonly stable state of the director orientation under shear
*Corresponding author.
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an orientation parallel to the flow@12#. Thus the correlationof orientational fluctuations perpendicular to the flow diretion could not be attributed to the log rolling of the micellebut was suggested to reflect an undulation of the direcwith a period of several micrometers@13#. It was found thatonce the system underwent a transition to some new stathe director orientation, it remained in this state even acessation of shear@14–16#.
An important feature of the rheological behavior of a heagonal phase is the independence of the director orientaon the shear rate in steady shear experiments. In this regthe tilt angleq of the director, initially aligned along thevelocity gradient was found to be given byg5arctanq,where g is the strain, indicating that the hexagonal phabehaves as a deformable solid@12#.
Recently a class of highly swollen surfactant hexagophases composed of a mixture of sodium dodecyl sulphpentanol, brine, and cyclohexane has been discoveredthis has allowed us to investigate the rheology of the hagonal phase in more detail. The advantage of the systethat the radius of the oil swollen cylinders,R, can be con-tinuously changed by an order of magnitude, while the intcylinder distance remains constant, by varying the oil contand the ionic strength of the polar medium@17#. This leads toa dramatic change in the elasticity of the system@18#. Themacroscopic elastic modulus was found to scale as 1/R3. Theresults of creep tests reported on these soft systems gsome evidence in favor of the SGR model for systems wtransient behavior, revealing, however, that the rheologresponse of the system is more complicated than thatscribed in the model@18#. It was found experimentally that along creep times (t>100 s, t being the stress-on time! thestrain increases as a power law witht
g5zta. ~1!
Comparing the experimentally obtained strain time depdence having the form of Andrade’s law for primary creepmetals@19#, with the general equation for creep
g5sJ~ t !, ~2!
and by substituting the creep complianceJ(t) with the ex-pressions given in the SGR model
M. L. SUSHKO, J. M. SEDDON, AND R. H. TEMPLER PHYSICAL REVIEW E65 031501
~1,x,2!: J~ t2tw!}~ t2tw!x21 for transient regime ,~4!
one can see that Andrade’s law~1! for a polycrystalline sys-tem corresponds to expression~4! given for the ‘‘transient’’regime in the SGR model with the exponenta equal tox21 @note, (t2tw) in the SGR model is the measuring timi.e., time elapsed from the beginning of the measurementtw is the time of stress application, the waiting time#. Theexponenta has been reported to be independent of thetory of the sample for a given applied stress so long apolycrystalline texture was maintained in agreement with~4! showing no waiting time dependence of the creep copliance. However, the exponent was found to depend onimposed stress whilex is taken to be a constant in the SGmodel. According to the experimental results@18#, the expo-nenta increases with the applied stress from 0.62 ats51 Paup to 0.9 ats59 Pa. This corresponds to an increase ox5a11 with applied stress from 1.62 up to 1.9 in the pocrystalline regime. Further increase of the applied strleads to monodomain formation, and, consequently, to arheological behavior of the system. It was found thatlarge stress, the system reaches a steady state witha51. Thesame steady state, characterized by a linear variation ogwith time, was immediately reached, if the initially polycrytalline system was presheared by the application of a sciently large stress@18#. These results are in agreement wthe theory, which gives the expression for the creep comance in the case ofx.2 different to that for lowerx, with thecompliance becoming linearly dependent on the measutime @see Eq.~3!#.
Thus, this surfactant system seems to be very suitablea detailed investigation of history-dependent rheology intransient regime.
II. EXPERIMENTAL PROCEDURE
We investigated the quaternary system of sodium dodesulfate~SDS! ~BDH Laboratory Suppliers, England, 99% prity!, pentanol, brine, and cyclohexane. Systems with 0.50.4M brine were used. In both systems the brine/SDS weratio was equal to 2.5 while the cyclohexane and pentacontent was 65 and 4 wt. % of the total weight of the mture, respectively, for the system with 0.5M brine, and 61.5%and 4.7% for the system with 0.4M brine. At the above com-positions the systems are reported to display swollen hexnal phases at room temperature@17#, and we confirmed thisby polarizing microscopy.
The rheological measurements were performed in a Cette geometry on a strain-controlled Bohlin VOR rheome~BRS VOR 7:9, Sweden!. The Couette cell consisted offixed inner cylinder of 7 mm radius and a rotating oucylinder, with a 0.7 mm sample gap.
In the relaxation tests, a straing0 was applied and thetime dependence of the relaxation modulusG(t)5s(t)/g0was measured@heres(t) is the stress#. A series of experi-ments with applied strains of 20%, 40%, 60%, 80%, a100% were performed. Each of the series of measuremconsisted of 19 relaxation experiments lasting 300 s e
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with a strain rise time of 0.1 s~Fig. 1!. Each successiverelaxation measurement in the series started immediatelyter the previous one was finished. In other words the waittime tw , i.e., the time elapsed from the beginning of the fimeasurement in the series, had an increment of 300 s fmeasurement to measurement. All relaxation experimewere performed at 2560.2 °C.
To avoid any preshearing of the sample, it was coodown to12 °C once loaded in the measuring system~at thistemperature the sample is in an isotropic state! and thenheated up to 25 °C and left at this temperature for 4 h ormore before the measurements were performed. This prdure gave reproducible results. All experiments werepeated at least three times.
III. THEORY: THE SOFT GLASSY RHEOLOGY MODEL
In the SGR model@1# the macroscopic sample is regardas a combination of mesoscopic elements. Each elemeassigned a local strainl, and a corresponding stresskl (k isan elastic constant!, which describes deformations awafrom some local position of unstressed equilibrium relatto neighboring elements. The local strain of an elemensupposed to follow the imposed strain until it reachesyield valuel y . At this point the element rearranges to a neconfiguration, where it is less deformed. Thus, yielding pvides a mechanism of stress relaxation, while between yevents the material behaves as an elastic solid of springstantk.
Yielding in the SGR model is regarded not as a purstrain-induced phenomenon, but as an ‘‘activated’’ procesmesoscopic element strained by an amountl has a certainprobability of yielding in a unit time interval. This rate it21, where the characteristic yield time for the element wa yield energyE5 1
2 kly2 is
t5t0 expF S E21
2kl2D Y xG , ~5!
wheret0 is the ‘‘attempt’’ time andx is an activation factor,regarded as an effective ‘‘noise’’ temperature or, alterntively, as the typical energy available for the activated p
FIG. 1. Time dependence of strain applied to SDS/pentacyclohexane/brine systems. The values forg0 are given in the text.
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HISTORY-DEPENDENT RHEOLOGY OF A SURFACTANT . . . PHYSICAL REVIEW E 65 031501
cess. Forx,1 the theoretical flow curve, i.e., a macroscopstress responses(g) to a steady shear rateg, has a macro-scopic yield stress:s(g→0)5sy.0. This behavior is char-acteristic of a glass phase. If such a system is strained besy , it ages, while ifs.sy , the system achieves a steastate and aging no longer occurs. In the so-called ‘‘transieregime, 1,x,2 aging is absent and the behavior of tsystem is dominated by yielding of the mesoscopic eleme
There are two possibilities of yielding for the elements.the element was strained up to the yield point, 1/2kl2'E, itexperiences a ‘‘strain-induced’’ yield event. On the othhand, an element with an energy much below the yield pocan yield through activation dynamics onceE21/2kl2'x ~a‘‘noise-induced’’ yield event!.
An alternative description of the material elements usedthe SGR model is the description of a particle moving inlandscape of quadratic potential wells of depthE. The bot-tom of each potential well corresponds to the unstrainstate. Yielding is associated with the hopping of the partito the bottom of a neighboring potential well.
The constitutive equations for the SGR model for a stem prepared at time zero in an initial state of zero stressstrain and subjected to a time-dependent macroscopic sg(t)(g(t)50 for t<0) are@2#
s~ t !5g~ t !G0„Z~ t,0!…1E0
t
@g~ t !
2g~ t8!#Y~ t8!Gp„Z~ t,t8!…dt8, ~6!
15G0„Z~ t,0!…1E0
t
Y~ t8!Gp„Z~ t,t8!…dt8, ~7!
with functionsZ(t,t8), Gp(Z), andG0(Z) defined as
Z~ t,t8!5Et8
t
exp~@g~ t9!2g~ t8!#2/2x!dt9, ~8!
Gp~Z!5E0
`
r~E!exp~2Ze2E/x!dE, ~9!
G0~Z!5E0
`
P0~E!exp~2Ze2E/x!dE. ~10!
In these equationsY(t8) is the average yield rate at timet8,P0(E), andr(E) are correspondingly the probability distrbution for the yield energies in the initial state att50 and inthe consequent states,t.0. As shown in@1# the predictionsof the SGR model are independent of the details of sampreparation andP0(E) could be taken equal tor(E).
The same choice of units and of distribution of tyield energies as in @1# have been made:r(E)5(1/xg)exp(2E/xg) where the mean yield energyxg5^E& istaken to be equal to unity. Consequently, with this choiceunits r(E)5exp(2E).
An alternative form of the constitutive equations that wwill use to calculate the relaxation modulus could be otained by substitution of Eq.~7! into Eq. ~6! @1# as
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s~ t !5g~ t !2E0
t
g~ t8!Y~ t8!Gp„Z~ t,t8!…dt8. ~11!
For the case described in our experimental procedwhen the strain of the same amplitude is applied to the stem after equal time intervalsDt, strain can be expressed a
g~ t !5g0u~ t !1g0u~ t2Dt !1g0u~ t22Dt !1•••
1g0u„t2~n21!Dt…
5 (j 50
n21
~ j 11!g0u~ t2 j Dt ! ~12!
with the functionu(t)51 for t.0 and zero elsewhere. Substitution of theg(t) function into Eq.~8! gives for thenthmeasurement @ t.(n21)Dt#, n.1 and (j 21)Dt,t8, j Dt, j 51, . . . ,n21,
Zj~ t,t8!5a (n2 j )2~ t2tw!1DtS j 1 (
i 50
n2 j -1
a i 2D 2t8,
~13!
wherea5exp(g02/2x).
The relaxation modulus of thenth step could be foundfrom the constitutive Eq.~11!:
G~ t2tw ,tw ,g0!
512E0
tFg~ t8!
ng0GY~ t8!Gp„Z~ t,t8!…dt8
512E0
twFg~ t8!
ng0GY~ t8!Gp„Z~ t,t8!…dt8
2Etw5(n21)Dt
t
Y~ t8!Gp„Z~ t,t8!…dt8
512 (j 51
n21j
nE( j 21)Dt
j Dt
Y~ t8!Gp„Zj 21~ t,t8!…dt8
2E(n21)Dt
t
Y~ t8!Gp„Zn21~ t,t8!…dt8. ~14!
To find an analytical expression for the relaxation modlus, the hopping rate has to be calculated. The expressionthe hopping rate could be derived following the logic of tyielding rate calculation in the linear case@1#. By substitut-ing Zj (t,t8) into the second constitutive Eq.~7! and takingthe Laplace transform of it, we derive the expression foryielding rate forx.1 as
Yj5x21
xa (n2 j )2
1x21
xG~x!a (n2 j )2Fa (n2 j )2
~ t2tw!
1DtS j 1 (i 50
n2 j 21
a i 2D G12x
1~x21!G~x!a12xt2x
1O~ t2(12x),t122x, . . . !, ~15!
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M. L. SUSHKO, J. M. SEDDON, AND R. H. TEMPLER PHYSICAL REVIEW E65 031501
whereG is the gamma function.Substituting this into Eq.~14! and performing the integra
tion we have forx.1 andt2tw!tw
G~ t2tw ,Dt,n,g0!} (j 50
n21j
na (n2 j )2
G~x!S a (n2 j )2~ t2tw!
1Dt (i 50
n2 j 21
a i 2D 12x
1G~x!a22x~ t2tw!12x. ~16!
The first term in Eq.~16! reflects the influence of all previous steps on the relaxation in thenth step. While the second term describes the relaxation as if there were no prevsteps and a single large amplitude strain was applied tosystem at timetw5(n21)Dt. In our experiments the residual stress is subtracted froms(t2tw) at the beginning ofeach measurement. Consequently, the relaxation moduluthe (n11)th step will be expressed as
G~ t2tw ,Dt,n11,g0!
}(j 50
nj
n11a (n112 j )2
G~x!S a (n112 j )2~ t2tw!
1Dt(i 50
n2 j
a i 2D 12x
2n
n11G~ t5nDt,Dt,n,g0!
1G~x!a22x~ t2tw!12x ~17!
or
G~ t2tw ,Dt,n,g0!}G~x!a22x~ t2tw!12x1d ~18!
with
d5G~x!F (j 50
nj
n11a (n112 j )2S a (n112 j )2
~ t2nDt !
1Dt(i 50
n2 j
a i 2D 12xG2G~x!~Dt !12xF (j 50
n21j
n11a (n2 j )2
3S (i 50
n2 j -1
a i 2D 12x
1n
n11a22xG . ~19!
An estimation ofudu gives
udu,n22xan2~Dt !12x/2. ~20!
Noting that Eq.~16! was derived in the short-time limit, i.et2tw!tw , and taking into account that 12x,0 we have(t2tw)12x@tw
12x . Consequently, substituting for the (n11)th steptw by nDt and using the relation~20! we findthat
~ t2tw!12x@2udu/~nan2!. ~21!
According to this relation the second term in Eq.~18! issmall in comparison with the first term provided that t
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number of stepsn and the amplitude of the applied strain anot too high. In our case the system was subjected to strof amplitude in the range of 0.03–0.15~20–100 %! and thenumber of steps in strain was 19. Figure 2 illustrateseffect of the second term in Eq.~18! on the relaxation modu-lus. The termd becomes comparable with the main term fthe highest applied strain~100%! andn>14 only. In all othercases the approximation of the relaxation modulus withmain term in Eq.~18! is legitimate.
IV. RESULTS AND DISCUSSION
A. 20% strain
It can be seen from Fig. 3 that all the relaxation curvexcept those where full relaxation is achieved, i.e., whereGdrops to zero, are linear in a log-log plot. Thus, in these cathe relaxation modulus is a power function of the measurtime G(t2tw ,tw)}(t2tw)b with a negative powerb. Ex-amples of power-law fits of the relaxation curves are giventhe figure. The deviation of the experimental data fromfits was found not to exceed 5%.
Comparison of the experimental results with theoretipredictions shows that the experimental curves could bescribed by the function
G~ t2tw ,tw!}~ t2tw!12x ~22!
for the casex.1. Thus, in the following, the experimentaresults will be discussed in terms of the SGR model andterms of the ‘‘noise temperature’’x associated with the probability of yielding rather than in terms ofb. In terms of themodel, stress relaxation proceeds through yield events,through the rearrangement of mesoscopic elements ofsystem into new configurations with zero stress. Conquently, the deviation from a power-law dependence ofrelaxation modulus observed at some waiting times indicayielding of most of the elements of the system. For thecurves the slope of the first linear region following the ho
FIG. 2. Dependence ofd on the number of steps in strain fox51.2, t2tw55, Dt5300, andg050 ~curve 1!, g050.05 ~curve2!, g050.1 ~curve 3!, andg050.15 ~curve 4!. The main term andthe resulting relaxation modulus are shown forg050.1 on curves 5and 6, respectively.
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HISTORY-DEPENDENT RHEOLOGY OF A SURFACTANT . . . PHYSICAL REVIEW E 65 031501
zontal region,t2tw,0.1 s~not shown!, in log-log plots wastaken to calculatex ~Fig. 3, dashed lines!.
The slope of the relaxation modulus varies from measument to measurement, i.e., thex value is a function of thewaiting time. During the first two measurements the valuethe noise temperature is high~Fig. 4! and the system experiences yielding, presumably associated with the orientaof domains with hexagonal symmetry in the shear field. Tx value decreases monotonically until it reaches a consvalue. The decrease of the noise temperature and, coquently, the decreasing probability of yielding can be undstood in the following way. After the yield energy is on aerage achieved, mesoscopic elements are able to hop tbottom of neighboring potential wells with lower energy ththe current one. The higher the number of yielded elemethe lower the average energy of the system. The lower aage energy leads to a decrease in the probability of noinduced yield events, i.e., to a decrease in thex value, asobserved experimentally. Consequently, the noise tempture decreases until most of the mesoscopic elementsyielded and some new state with lower energy is reache
It can be seen in Fig. 3~for tw<3000 s) that the systemdoes not fully relax during all measurements, except thetwo. This means that the energy of the system does not reto the value characteristic of the unstrained state, but iscumulating in the system during the subsequent measments. Consequently, the average energy of the mesosc
FIG. 3. Relaxation modulus of the SDS/pentancyclohexane/0.5M brine system subjected to 20% strain vs the msuring time. The waiting time valuestw at the beginning of each tesare shown in the figure. For the sake of clarity not all of the expmental curves are shown. The dashed lines show fits to thewith a power law functionC(t2tw)12x.
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elements at the beginning of the relaxation measurementhe seriesE(t5tw), tw>600 s, is an increasing function othe waiting time. It can be supposed thatE(t5tw) will in-crease monotonically until it becomes close enough toyield energy for the mesoscopic elements to hop to the btom of a neighboring potential well. This transition is oserved at a waiting time of 3250 s at the end of the 1measurement~Fig. 3, upper graph, the thick curve!. Accord-ing to the SGR model, yielding of a mesoscopic elemenassociated with its rearrangement into a new equilibriconfiguration with zero local strain. However, as shownFig. 2, the relaxation modulus, i.e., the average shear stin the system (G}s), does not decrease to zero when yieing occurs during the 11th measurement, but becomesorder of magnitude lower than that before yielding. Conquently it is likely that only some mesoscopic elements hayielded attw53250 s. If so, the average stress of the syst
s51
N (i 51
N
s i51
N (i 51
n
s i11
N (i 51
N2n
051
N (i 51
n
s i , ~23!
where N is the number of the mesoscopic elements insystem,n is the number of elements that have not yielde
-
i-ta FIG. 4. Waiting time dependence of~a! noise temperaturex and~b! relaxation modulus G(t2tw5200 s, tw) for theSDS/pentanol/cyclohexane/0.5M brine system subjected to 20%strain. The dashed line on the lower figure shows the funct150(t2tw)12x(tw) for t2tw5200 s.
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M. L. SUSHKO, J. M. SEDDON, AND R. H. TEMPLER PHYSICAL REVIEW E65 031501
and s i is the stress of the elementi. Thus the higher thenumber of yielded elements the lower the resulting macscopic stress of the system.
It can be supposed that two different states with differcharacteristic energies, presumably states with differententations of the domains with hexagonal symmetry, coein the system after this yielding. The system remains in ttransient state during the seven consequent measurem~waiting time interval 3250–5300 s! until the system experi-ences another yielding at the end of the 18th measuremThis time the stress in the system drops essentially to zindicating that most of the mesoscopic elements have rranged, adopting a new state.
The system in the transient state between the two yieldevents is very unstable and experiences a number of wyields before it adopts the new state of orientation. It is likethat the ratio of the number of mesoscopic elements instates of orientation is changed via yielding. It shouldnoted that theG value, i.e., the average stress of the mescopic elements, in the transient state is low~Fig. 3!. Conse-quently the only way for yielding in this case is througactivation dynamics, associated with interelement intertions. On the other hand attw53250 s and 5300 s the aveage energy of the system presumably comes close enouthe yield energy (G value is high! to cause strain-induceyield events to become more probable. The value ofnoise temperature is equal to unity for these two measments supporting the supposition that the transitions havstrain-induced character.
According to Eq.~18! the relaxation modulus, approxmated by the first term, should be independent of the waitime. However, as shown in Fig. 4~a! strong dependence ofGon the waiting time appears to be observed. This dependcannot arise from the termd in Eq. ~18!, which depends onthe number of steps in strain, i.e., it is waiting time depedent. First of all this term is small compared to the main tein Eq. ~18! and it has a waiting time dependence quite dferent from the experimental one~Figs. 2 and 4!. The otherpossibility is that the waiting time dependence ofG is asso-ciated with the waiting time dependence of the noise teperature. As noted in@2#, this is supposed to be a minimaextension to the SGR model, however the analysis ofSGR with x evolving in time was not yet reported. On thbase of our experimental data we propose to incorporawaiting time dependence ofx into Eq.~18! not allowingx toevolve with measuring time. Taking this into account a retively good fit of the experimental curve was obtained wthe function
G~ t2tw ,tw!}~ t2tw!12x(tw) ~24!
~Fig. 4!. Thus, according to this empirical relation,G is in-dependent of the waiting time when the noise temperaturconstant and decreases or increases withtw according to thewaiting time dependence of thex value.
We propose a qualitative interpretation of the waiting timdependence of the relaxation modulus. First we assumsituation where the probability of yield events is low, i.ewhen the energies of the mesoscopic elements are wel
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low the yield energy and thex value is low. In this case theaverage stress of the system will have the same incremethe beginning of each measurement and theG value will beindependent of the waiting time@the residual stress is subtracted from thes(t2tw) at the beginning of each measurment#.
If the energies of most of the mesoscopic elementsclose to the yielding energy, but not close enough fortransition to take place during the current measuremsome of the mesoscopic elements may yield during the strise time of the subsequent measurement when an additistrain is applied to the system. This may change thex valueand lower the average stress of the system. ConsequeG(t2tw50) will be lower than that at the beginning of thprevious measurement. The energy distribution is expecto be bimodal in this case, i.e., one part of the elementshave low or zero energies and the other part will have engies close to the yielding value. The noise temperatureexpected to increase relative to the value in the previmeasurement reflecting the interactions between the yieand unyielded elements. The higher the fraction ofyielded elements the higher the probability of yielding fthe rest of the elements, and the higher thex value.
Finally, we need to interpret our observation that thecrease of the relaxation modulus is accompanied by a sdrop in the noise temperature@Fig. 4, pointsa andb on theG(tw) curve correspond to pointsa andb on thex(tw) curve,respectively#. The low x value suggests that the energy dtribution of the elements during this measurement~we willdenote this measurement by its numbern) is narrow, i.e., theenergies of the elements are close to the average value.ing that thex value is always relatively high in the previoumeasurement~the measurement numbern21), we supposethat the transition from the broad to narrow energy distribtion takes place during the (n21)th measurement. Providethat no structural rearrangements were observed in then21)th measurement, the average energy of the system aend of the (n21)th measurement is close to, but below tyield energy. Consequently, at the beginning of thenth mea-surement the mesoscopic elements of the system receivincrement in strain, i.e., an increment in energy, such thataverage energy of the system may become close enougthe yielding value for a strain-induced transition to taplace. The difference between the situation when the disbution at the end of the (n21)th measurement is broad onarrow is the following. If the system with the broad distrbution of energies of mesoscopic elements is strained,elements with high energies, i.e., with the energies closethe yielding value, can yield thereby acquiring zero strain.was pointed out above, this will lower the average energythe system. On the other hand, if the energy distributionnarrow no yielding will occur during the strain rise time anthe average stress of the system att2tw50 will be higherthan that in the previous case.
Thus the increase inG(t2tw50) is presumably associated with the mechanism of the strain-induced transitiowhile the decrease in this value is due to the noise-induyield events.
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HISTORY-DEPENDENT RHEOLOGY OF A SURFACTANT . . . PHYSICAL REVIEW E 65 031501
B. 40–100 % strain
The profile~variation with tw) and the value of the noistemperature both depend on the applied strain. For the sof relaxation measurements with an applied strain of 40the noise temperature and the probability of yieldinghigh. When 40% strain is applied to the system, the notemperature exhibits a maximum during the second measment in the series~Fig. 4!. An abrupt drop in the noise temperature is observed in the next measurement, presumdue to yielding of most of the mesoscopic elements ofsystem. After this drop in thex value, the profile of the noisetemperature dependence on the waiting time in the reg600<tw<3000 s repeats that for the previous series withapplied strain of 20% in the region between the two trantions, i.e., for 3000<tw<5400 s @inset in Fig. 5~a!#. Thisindicates that when higher strain is applied the first transittakes place during the third measurement in the series insof in the 11th in the previous case. In other words the enerequired for most of the mesoscopic elements to rearrathemselves into a new state is available almost immedia~note that it is a strain-induced transition!. Conversely whenlower strain is applied, the system has to collect the eneduring a number of measurements until it is sufficient fortransition to take place. After the second transition, the stem submitted to 40% strain undergoes a number of strand noise-induced transitions.
FIG. 5. Waiting time dependence of~a! noise temperaturexand ~b! relaxation modulus G(t2tw5200 s,tw) for theSDS/pentanol/cyclohexane/0.5M brine system subjected to 40%strain ~squares! and 60% strain~triangles!. The inset in~a! showsthe waiting time dependence ofx shifted by 2400 s along thetw axisfor the system subjected to 40% strain~solid line! andx(tw) for thesame system subjected to 20% strain~dashed line!.
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Structural data are required to detect the orientation ofdomains in the system in the shear field and they willreported in a future publication. However, according to trheo-small angle light scattering data of Richteringet al.@10#, a surfactant hexagonal phase subjected to consshear stress undergoes transitions from an isotropic distrtion of the domains, to a perpendicular orientation, and ththrough a regime of coexistence of parallel and perpendicorientations to the formation of a monodomain with therector in the flow direction. It can be supposed that in tcase of applied strain, our system undergoes similar trations. Consequently, the first transition may be associawith the orientation of the domains in the perpendicular oentation. The second—with the rearrangement of some ofdomains parallel to the flow direction—and the subsequtransitions may be associated with an increase of the fracof parallel-oriented domains and then with monodomain fmation.
For the 60% strain experiment the averagex value ishigher than that obtained at 40% strain and fluctuatestweenx51.2 and 1.5, indicating that the system is in a trasient state~Fig. 5!. Presumably the external strain is higenough for the system to adopt the same state as in thestrain experiment after the second transition, i.e., the statcoexistence of domains perpendicular and parallel toflow orientations. On the other hand, the 60% applied strmight not be sufficient to induce monodomain formatioHowever, it is likely that a monodomain is formed during thseries of experiments with 80% applied strain~Fig. 6!. In thiscase after yielding during the first measurement in the sethe noise temperature is almost constant and equal tothat is close to thex value for the case of the 60% appliestrain. An abrupt increase in thex value observed attw51500–1800 s is followed by a region of nearly constanoise temperature. The noise temperature is close to unithis region, indicating that the system has reached a stestate with a higher degree of order where the probabilitynoise-induced yield events is very low. It can be suppothat from close to the beginning of the series most ofdomains are aligned in the flow direction. The size of thedomains might increase due to interdomain interactions ua monodomain is formed attw'2000 s.
The profile of the noise temperature dependence onwaiting time in the case of the 100% applied strain is simito that in the previous case~Fig. 6!. The only differencebetween these two cases is that the transition is shifted onwaiting time scale totw5900 s and is narrower than in thprevious case.
The value of the relaxation modulus depends on the wing time, in agreement with the above qualitative picture. Fall cases except the case of the first transition in the syssubjected to 20% strain theG value drops to zero when thtransitions take place, indicating that most of the mesoscoelements have yielded to the unstrained state. The avestress values when 60% strain is applied are the samthose in the cases of 80% and 100% applied strain, bethe transition, indicating that the system is in the same st
However, this is not the case for the strain-induced trsitions. It can be seen from Fig. 5 that the relaxation modu
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M. L. SUSHKO, J. M. SEDDON, AND R. H. TEMPLER PHYSICAL REVIEW E65 031501
obtained in the experiments with 40% applied strain is mthan an order of magnitude lower than that in 20% strexperiment in the region around the first two transitions. Tresult is the manifestation of the speedup of the relaxaprocess in the nonlinear regime. As shown in@2#, a largesingle step straing0 speeds up the relaxation by a factexp(g 0
2/2x). In the single step strain approximation this rsult may be applied to each test in our series. Thus the higthe applied strain the higher the speedup of relaxation. Teffect explains also the shift of the transition times onwaiting time scale to lower waiting times withg0.
C. Effect of composition
Investigations of the relaxation in the SDS/pentancyclohexane/brine system in lower ionic strength brine wa slightly smaller spacing between cylinders@17#, revealedthat the system experiences similar transitions as inhigher ionic strength case. However, the transitions induby low strain, i.e., the transitions presumably associated wthe rearrangement of domains, are shifted on the waitime scale to shorter waiting times. For example, the trations, induced by 20% strain, are shifted by 3000 s towashorter waiting times. However, a decrease in the valuethe applied strain results in a decrease in the transition tshift. A linear dependence of the shift of transition timeapplied strain was observed~Fig. 7!. Thus a 19.3% strainwas found to produce exactly the same effect on the 0.M
FIG. 6. Waiting time dependence of~a! noise temperaturexand ~b! relaxation modulus G(t2tw5200 s, tw) for theSDS/pentanol/cyclohexane/0.5M brine system subjected to 80%strain ~circles! and 100% strain~crosses!.
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system as a 20% strain on the 0.5M system. It should benoted that when 19% strain is applied no transitions excthe first one at a waiting time of 4200 s were observbecause the subsequent transition is expected to occur 15after the first one, i.e., at a waiting time of 5700 s, which wnot experimentally accessible.
Stronger electrostatic interactions in the system withlower ionic strength of brine may be responsible for the oserved shift in the transition times. Provided that in bosystems the weight ratio of brine/SDS is the same,screening of the SDS charge is different: the number ofexcess counterions per SDS charge is 0.35 for the 0.5M sys-tem ~the Debye length is 0.4 nm! and 0.28 for the 0.4Msystem~the Debye length is 0.5 nm!. Consequently, the electrostatic repulsion between the cylinders is higher in the stem with 0.4M brine. This suggests a higher cooperativfor domain rearrangement. The stronger correlation betwthe elements suggests also that less strain is required toduce the same transitions. These considerations areported by the fact that the noise temperature, reflectingdegree of interdomain interactions, is higher for the 0.4Msystem before the transitions take place, while the drop inx value after the transitions is steeper~Fig. 8!. The noisetemperature value oscillates between 1 and 1.27 in the rebetween the transitions, while in the 0.5M system this valueis almost constant and equal to 1.12. This means that wstronger intercylinder, and, consequently, interdomain inactions are present, more domains rearrange themselvesa new state simultaneously, causing a steeper drop inx value.
The above considerations relate to the transitions assated with domain rearrangements only, but not to thosesociated with monodomain formation. Contrary to the lostrain case, positive shifts in transition times are obserwhen the 0.4M system was submitted to high strain. Fexample, monodomain formation presumably takes plactw5900 s in the 0.5M system subjected to 100% straiwhile under the same experimental conditions the 0.4M sys-tem experiences such a transition 3300 s later, attw54200 s~Fig. 9!.
As in the case of the low applied strain the shift in ttransition time is equivalent to the shift in the applied stra
FIG. 7. Strain dependence of the transition time shift in tSDS/pentanol/cyclohexane/brine system with 0.4M brine with re-spect to the transition times in SDS/pentanol/cyclohexane/brinetem with 0.5M brine subjected to 20% strain.
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HISTORY-DEPENDENT RHEOLOGY OF A SURFACTANT . . . PHYSICAL REVIEW E 65 031501
value. The waiting time dependence of the relaxation molus for the 0.4M sample with 80% applied strain is similar iprofile to that for the 0.5M system with a 60% applied strai~Fig. 10!. The lowest strain that~presumably! induces mon-odomain formation in the 0.4M system was found to beequal to 83.3%~Fig. 11!, while for the 0.5M system 80%strain induces such a transition in the middle of the se~Fig. 6!. Thus, on the applied strain scale the transitionssociated with monodomain formation are shifted towahigher strain. A linear dependence of the transition time son the applied strain was also observed~Fig. 9!. Again weattempt to interpret these results from the point of viewthe intercylinder, interdomain interactions.
In the case of the low applied strain, stronger interdomrepulsion in the 0.4M system induces stronger element~do-main! coupling, because the rearrangement of a domain mlead to a decrease in the distances between the cylindethis domain and those of the neighboring ones. This induan increase in the electrostatic repulsion between the yiedomain and the neighboring ones and, consequently, leaan increase in the probability of the neighboring domainarranging. Provided that the Debye lengths in 0.5M and
FIG. 8. Noise temperature dependence on waiting time forSDS/pentanol/cyclohexane/brine system with 0.5M brine subjectedto 20% strain~crosses! and with 0.4M brine subjected to 19.3%strain ~squares!.
FIG. 9. Strain dependence of the transition time shift inSDS/pentanol/cyclohexane/brine system with 0.4M brine with re-spect to the transition times in the same system with 0.5M brinesubjected to 100% strain.
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0.4M systems do not differ much, the shift on the applistrain scale required to induce the transitions in both systat the same waiting times is low,Dg50.7%. However,Dgis higher than 20% in the case of the high applied strain,when most of the domains presumably have the same ortation and the straining of the system induces a monodomformation. Monodomain formation implies the annihilatioof domain boundaries due to cylinder fusion. Consequenin this case electrostatic repulsion along the cylinders rathan much weaker intercylinder repulsion plays the mrole. Thus, the higher the repulsion along the cylinder is, i
eFIG. 10. Waiting time dependence of the relaxation modulus
the SDS/pentanol/cyclohexane/brine system with 0.5M brine sub-mitted to 60% strain~triangles! and with 0.4M brine subjected to80% strain~squares!.
FIG. 11. Waiting time dependence of the relaxation modulusthe SDS/pentanol/cyclohexane/0.4M brine system subjected to83.3%, 85%, 90%, and 100% strain.
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M. L. SUSHKO, J. M. SEDDON, AND R. H. TEMPLER PHYSICAL REVIEW E65 031501
the lower the SDS charge screening, the higher the strainshould be applied to induce monodomain formation.
V. CONCLUSION
An expression for the relaxation modulus for the casethen steps of equal strain amplitude and equal length appto a sample in the transient regime has been derived withe framework of the SGR model. The experimental datarelaxation in the SDS/pentanol/cyclohexane/brine systemthis case showed a more complicated behavior of the reation modulus than proposed in the theory. However, alloing the noise temperature to evolve with the waiting time,found a qualitative agreement between the calculated andexperimental data.
Depending on the amplitude of the applied strain and
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number of steps in strain, the SDS/pentanol/cyclohexabrine system was found to experience a number of strand noise-induced transitions. The transitions in the sysare presumably associated with the rearrangement of themains having two-dimensional hexagonal symmetry whthe applied strain is low, and with monodomain formatiwhen a high strain is applied to the system. However, simtaneous rheological and x-ray investigations are requiredprove this supposition, and these experiments are currentprogress.
ACKNOWLEDGMENTS
This work was funded by EPSRC Grant No. GRM50850. We thank Professor P.E. Luckham for the genergift of the Bohlin rheometer.
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