3.2 Shear stresses of olloidal dispersions at the glass transition in equilibrium
4.1.2 Coagulation kinetis
As shown by Holthoet al. [244℄ the kinetisof Brownianoagulation of partilesinthe
earlieststage with the transitionfromsingle partiles todoublets an bedesribed by
dN z
where
k ij
is a seond-order oagulation rate onstant, t is the time, andN z
is the totalpartileonentration of
z
-foldaggregates.The partile onentration of singlets
N 1
and doubletsN 2
as a funtion of time an beobtained by solving eq. 4.1 analytially. Assuming
k ij = k 11
and the initial partileonentration
N 0
we obtainN 1 = 2N 0
2 + k 11 N 0 t
(4.2)using eq. 4.1and integrating leads to
N 2 = 4N 0 2 k 11 t
(2 + k 11 N 0 t) 3
(4.3)Developing
N 1
andN 2
forthe short times leads toN 1
IfweonsiderthattheoagulationofspherialpartilesisjustontrolledbyBrownian
dif-fusion, aordingtothe theoryof Smoluhsovski[245, 246℄,the oagulationrateonstant
beomes:
k 11 = 2k s = 8k B T
3η
(4.6)where
k B
is the Boltzmannonstant,T
the temperature, andη
the visosityof the uid.Considering the hydrodynami interations and the interpartiles potential leads to a
more ompliatedexpression [247℄.
Inthe earlystage oftheaggregation,wherethe doubletsare formed,the sattering
inten-sity an beexpressed asthe sum of the sattering intensity
I(q, t)
of the single partilesI 1 (q)
and doubletsI 2 (q)
.I(q, t) = N 1 (t)I 1 (q) + N 2 (t)I 2 (q)
(4.7)Using the expression of the singlets and the doublets of the equations 4.4 and 4.5, the
sattered light intensity results in
the value for the form fator of an aggregate of
z
idential spheres an be alulatedwithin the Rayleigh-Gans-Debye approximation [247, 248℄, onsidering that independent
primarypartileswithin anaggregatewith identialsatterers. The oagulationrate an
be determined fromthe stati intensity with
1
with
a
the radius of the primary partiles.In the dynami light sattering the intensity-weighted average of the diusion oeient
D(q, t)
between single partiles and doublets an be alulated fromthe rst moment ormean deay rate
Γ(q, t)
.| D(q, t) | = | Γ(q, t) |
q 2 = N 1 (t)I 1 (q)D 1 + N 2 (t)I 2 (q)D 2
N 1 (t)I 1 (q) + N 2 (t)I 2 (q)
(4.10)Aording to the Stokes-Einstein equation, the diusion oeient is related to the
hy-drodynamiradius of the partile
D(q, t) = k b T
6πηr h (q, t)
(4.11)If we onsider oagulation at the early stage, where only doublets are formed, and if we
assume that
a = r h (q, t = 0) = r h,1
weobtain1
r h (q, t) = N 1 (t)I 1 (q)/r h,1 + N 2 (t)I 2 (q)/r h,2
N 1 (t)I 1 (q) + N 2 (t)I 2 (q)
(4.12)where
r h,1
andr h,2
are the hydrodynami radii of the singlet and doublet, respetively. Eliminating the number of partiles with eq. 4.4 and 4.5 and after dierentiation, eq.4.13 an bewritten as
CCD
Figure 4.1: Experimental setup for the laser ontrolled oagulation experiment.
1
where the doublethydrodynamiradius is given by
r h,2 ≈ 1.38r h,1
[249℄.Aombinationof bothstatiand dynamilightsatteringfromtheequation4.9and4.13
allows the determination of
k 11
independently onthe form fator.k 11 N 0 = r h,2
Theore-shellpartileswith5
mol.%
rosslinkerdesribed inthehapter2.1were usedinthishapter. Thekinetisofthe oagulationwasinvestigatedby dynamilightsattering
(DLS) using a Peters ALV 5000 light sattering goniometer where the temperature was
ontrolledwith anaurayof
±
0.1o
Cas. The reversibilityof oagulationwasmeasured by DLS with a Zetaziser (Malvern model nanoZS) in bak sattering at 173o
with a
strit ontrol of the temperature set by Peltier elements at
±
0.1o
C. Cryogeni eletronmirosopy was performedas desribed insetion 2.1.2.
Sanning fore mirosopy (SFM) experiments were arried on at room temperature a
Dimension3100 mirosope(Veeo InstrumentsIn.). The SFMwas operatedintapping
modeusingsiliontipswithaspringonstantofira40
Nm −1
andaresonanefrequenyrangingfrom200to300
Hz
. Thesanratewasvariedbetween 0.5and1.0Hz
tooptimizethe image quality. The samples have been prepared as followed: a drop 30
µL
0.1wt%
latexsolutionhasbeendepositedonsiliiumwaferand driedat60
o C
. Thesame sampleswere then examined ina Zeiss Axioteh variopolarized optialmirosope inreetion.
The experimentalsetup for the laser ontrolled self-assemblyonsists onan inverted
mi-rosope with a laser port (Olympus IX-71) (see g. 4.1). Two mirrors mounted on
magnet losed loop galvano sanners are situated in a onjugate onfoal plane with a
A) B)
Figure4.2: A)Hydrodynami radius of the ore-shell latex versustemperature, asdetermined by
dynamilightsattering. Withoutsalt (hollowsymbols)thePNIPAMnetworkshrinks
upon heating and remains stable even at high temperature. With addition of salt
(5.10
−2
M KCl)the partiles oagulate above the Low Critial Solution Temperature
(LCST)around 33
o
C.B) Reversible oagulation ofa 2.5.10
−3
wt.
%
dispersionmea-sured by dynami light sattering withthe Zetasizeras funtion ofthe time hanging
the temperature above andbeyond the LCST.
sanning point in the sample. The onjugate planes are formed with the help of two
lensesteleentri systemand anobjetive. Thelatterserves bothforfousingof thelaser
beam and for observation of the sample. The laser (Coherent Verdi V-5,
λ = 532 nm
)an be foused down to below 1
µm
by the objetive (Olympus LCACHN 40xPH) andtypialpowervaluesare between
0.1
and 100mW
. The ellismounted horizontallyin a temperature-ontrolledxyz-stage. Ithas agap thikness of50µm
(Hellma106-QS).Thesample is illuminated by a white light soure (halogen lamp) with Köhler illumination
[250℄. The interferene lter in front of the amera avoids damage to the CCD hip by
foussedlaseroruoresent light[251℄. The pituresare takenwithaCCDamera(PCO
pixely).
4.1.4 Results and Disussion
Reversible oagulation and stability
The struture and the swelling of the partiles used in this setion have been arefully
investigated in hapter 2. The volume transition within the shell an easily be studied
by dynami lightsattering (DLS) asdesribed inthe setion 2.1.4. Figure4.2A) shows
the dependene of the hydrodynamiradius
R H
determinedbyDLS onthe temperature.R H
dereasesgraduallywith temperatureuntilasharpvolumetransitionfromswollen to unswollen states takes plae, reahing a nal ollapsed size at a transition temperaturearound 38
o C
. Without addition of salt this proess is thermoreversible without any hysteresis.Addition of 5.10
−2 mol.L −1
KCl leads to a slight shrinking of the partiles. Thisphe-nomenonhas beenalreadyinvestigated inreentstudies[72℄. The additionofsaltsreens
the residual eletrostati interation of the partiles as shown by eletrophoreti
mobil-ity measurements [72℄. Hene, at higher temperatures the dispersions beome unstable
and aggregate [72℄. Aggregation takes plae between 32 and 33
o C
, whihis lose to theLCST value determined from the dynami light sattering analysis at 32.2
o C
(seese-tion 2.1.4). This assertsthat the system interats solelythrough steri interationbelow
the LCST, whih makes it suitable as a model dispersion as demonstrate in the setion
rystallization (see hapter 2.3) and in the investigation of the dynamis in equilibrium
and in ow (see hapter 3.2). In the dilute regime the reversibility of the aggregation
proess has been investigated by dynami light sattering (see g. 4.2b)). For this
pur-pose the Zetasizer was used, where the temperature ontrolled by Peltier elements an
be rapidly adjusted. In this experiment the system is rst maintained 10
min
at 25o C
,then heated rst at 32
o C
, and after at dierent temperatures from 33o C
to 35o C
. Theooling proess was thenreorded rst at32
o C
andthen at25o C
. Whereasthe system isstableat25and32
o C
itaggregatesat33o C
. Theaggregationspeedwasfoundtoinreasewith inreasing temperature and the hydrodynami radius varies between 150 and 400
nm
within 10min
. Cooled down at 32o C
the hydrodynami radius sharply dereased toreahed the same value as beforethe aggregationourred, whihwas alsoheked at25
o C
. In opposition to ore-shell system with linear PNIPAM, whih usually aggregatesin a non reversible way [237℄, the aggregation was found to be totally reversible in this
experiment. Above 32
o C
the partiles shrink whih is aompanied by a diminution ofsteriinterations. Thislastonesarenotabletoompensatethe vanderWaalsattrative
interations whih results in the aggregation of the system. The dense ross-linked shell
around the polystyrene partiles prevents the interpenetration of the networks. Cooling
down indues the reswelling of the partile and the onset of the strong osmoti pressure
bringingthe system to separate again. The response of the system is fastdue to the size
ofthe mirogel. Moreoverwhen theaggregation proess ismostly limitedtothediusion
of the partilesin the fastmode regime [244℄ the dissolution ofthe aggregateswas found
tobemuh faster forthe investigated onentrations.
The kinetis of the aggregation proess has been investigated by DLS asfuntion of the
temperature for dierent onentrations in presene of 5.10
−2 molL −1
KCl. Thetreat-ment was performed following the method proposed by Holthof et al. [244℄. Fig. 4.3
presents the evolution of the normalized hydrodynami radius for the dierent
temper-atures measured at 90
o
for a 1.25.10
−3 wt%
solution. The same experiment has beenrepeated for dierent onentrations ranging between 2.5.10
−3
and 2.5.10
−5 wt%
. Theexperiment wasperformed asfollow: 2.3
mL
ofthe latexsolutionwasequilibrated atthe wished temperaturein the DLS for 20min
. Then 0.2mL
of a 0.625molL −1
KClmain-tainedatthesametemperaturewere quikly addedtosetthe saltonentrationat5.10
−2
molL −1
. After homogenization the measurement was started. The ountrate and the hydrodynami radius resulting from the seond umulant analysis were then monitoredas funtion of the time at a sattering angle of 90
o
. The hydrodynami radius was rst
foundtodereaseafterthe additionofthesaltand thentoinrease followingthe
aggrega-tion proess. The intensity was rst onstant and then inreased. After few minutes the
intensitydereased. This an beattributed totheformfator of theaggregates following
the aggregation proess [244, 247, 248℄. Only the dynami light sattering experiments
have been treated in the following study onsidering the diulty to analyze the stati
lightsatteringexperiments. Theinitialhydrodynamiradius
R H,0
wasonsideredatthe1
Figure4.3: Normalized hydrodynami radius as a funtion of time measured by dynami light
sattering at 90
o
for a 1.3.10
−3 wt%
solution at dierent temperatures. The time has been resaled to the beginning of the oagulation proess. The full lines indiatethe initialslope onsidered inthe alulationof the oagulationrate onstant (see eq.
4.13).
beginningofthe aggregation. The initialslopeofthe normalizedhydrodynami radiusor
intensity was used toalulate the oagulation rate onstant
k 11
from the dynami lightsattering experiment using the equation 4.13. This method onsiders the form fator
and the hydrodynami radius of a doublet of hard spheres [244, 247, 248℄. The
assump-tion does not neessary fully hold inthe ase of the omposite mirogelwhihare rather
ompressible atleast below 40
o C
.The dierent results are presented in the Fig. 4.4 in form of a stability ratio
W
whihorrespondstotheratiobetweendiusionlimitedoagulationrateonstant
2k s
aordingto the Smoluhsovski theory (see equation 4.6) and the experimental oagulation rate
onstant
k 11
.W
beomesnearlyonstantafter34o C
with anaveragevalueof3.8.10−18 ±
0.4
m 2 s −1
. Holthof et al. investigated the inuene of dierent salts on the oagulation rate measurement of sulfonatestabilizedpolystyrene partiles[244℄. They reporta valueof 4.3.10
−18 ±
0.4m 2 s −1
for the fast oagulation rate onstant using KCl as eletrolyte[244℄. Our experiment is in reasonable agreement with the ones reported partiularly
onsidering the omplexity of our system and the experimental error mostly related to
the sensitivityof
k 11
tothe preseneof traeamountsofimpurities[244℄. Asaonlusiontheoagulationoftheompositepartilesanbeontrolledbythetemperatureandbythe
salt onentration. Steri interations ontribute mostly to the stability of the system.
The fast oagulation rate onstant is reahed in the viinity of the end of the volume
transitionin the polymeri shell after34
o C
.In g. 4.5 the hydrodynami o size is plotted as funtion of the time in a log-log
representation for the longer times. Four sets of data are presented for a 1.3.10
−3 wt%
solution at 32.8
o C
, 33o C
, 36o C
and 40o C
. For aggregates showing fratal strutures,with fratal dimension
d f
, the radiusR
of the os inreases with the timet
aordingto a power law [252℄,
R ∝ t z
wherez = 1/d f
. The fratal dimension an be alulatedfrom the slope. For temperature higher than 32.8
o C
the slope beomes onstant and amean value of
d f = 2.05 ± 0.03
is obtained. This value is signiantly greater than thevalue of 1.7-1.8, whih is normally expeted for partiles undergoing diusion-ontrolled
10 0 10 1 10 2 10 3
32 34 36 38 40 42
1.25.10 -3 wt % 2.5.10 -3 wt % 2.5.10 -4 wt % 2.5.10 -5 wt %
T [°C]
W = 2 k s /k 1 1
Figure4.4: Stabilityratio(
W = 2k s /k 11
)measured atdierent temperatures andonentrations.The experimentalaggregationrate onstant
k 11
has beendetermined bydynami lightsattering fromtheequation 4.13whereas
2k s
isobtained fromthe expressionderivedby Smoluhsovski (see eq. 4.6) .
2.00 2.25 2.50 2.75 3.00
2.2 2.4 2.6 2.8 3.0 3.2
logt [s]
lo g R [n m ]
Figure4.5: Log-log plotsof average o radius (symbols)as funtionof timefor the longer times
for a 1.3.10
−3 wt%
solutionat 32.8o C
(irles), 33o C
(squares), 36o C
(downtrian-gles) and 40
o C
(uptriangles).irreversible aggregation. The same observation was done by Rasmusson et al. [236℄ on
the temperature ontrolled oulation of mirogelspartiles with sodium hloride. The
authors reported a fratal dimension of about 2.0 from the same kind of analysis below
the LCST. They interpreted this value as anindiation of a weak reversible oulation
model, whih was supported by their estimation of the depth of the minimum in the
interpartilepair potential. Nevertheless forsimilarPNIPAMbasedmirogelsRouthand
Vinent reported that the
d f
dereases from 2 around the CTF to 1.75 at temperature greatlyhigherthantheCFTandthantheLCST[253℄. Itisnot theasefortheore-shellpartiles were the fratal dimension remains equal to approximately 2.0 even at higher
temperatures. This hints to the higher stability of the ore-shell system under these
experimental onditions. The reversibility of the oagulation presented in g. 4.2 B)
onrms this assumption.
10 0
Figure4.6: A) Complex visosity
η ∗
asfuntion of the temperature (10 min/o
C) for 1 Hzand 1
%
for dierent onentrations with 5.10−2 M
KCL: 12.1wt %
(full line), 10.75wt
%
(dashed line), 8wt %
(dotted-dashed line) and 6wt %
(dotted line). B) Diretobservation ofthe 12.1
wt %
solutionatdierenttemperatures. C) Redued visosityη ∗ /η s
of the dierent onentrations asfuntion ofthe eetivevolume frationφ ef f
for temperatures below 25
o C
.From repulsive to attrative glasses
The inuene of the attrative interations has been then investigated for more
onen-trated solutions varying between 6 and 12
wt%
. The phase diagram of the PNIPAMmirogels has been already intensively studied experimentally [7,2527, 30, 234℄, and is
rather well understood theoretially [27℄. The omposite mirogels were found to follow
the same features. Indeed the rystallization and the glass transition of this system has
been presented in the hapter 2.3 and 3.2. It was found that below 25
o C
the systembehaves like hard spheres. For eetive volume frations below 0.49 the solution are in
the liquid state whih is haraterized by a smaller turbidity. Raising the temperature
above 32
o C
the system beomes white and opaque. After a longer time a phasesepa-ration ours. For less onentrated systems this results immediately in the oagulation
of the system. As an example g. 4.6 B) displays the a 12.1
wt%
solution maintainedat dierent temperatures for half an hour. At 10
o C
the solution is in the glassy stateand the solution appears bluish. The orresponding volume fration alulated for this
temperatureis equalto 0.63. At 17
o C
whih orresponds toan eetive volume frationof 0.54, partiles rearrange into rystals. This is aompanied by the hange of the
solu-tion olor to green due to the Bragg's reetions. At 30
o C
the solution is in the liquidstate. The samples at 35 and 40
o C
present the solutionafter the osetof the attrativeinterations. The solutionbeomes white and opaque and remainsrelatively stable after
30
min
, whereas the solutionat 45o C
learly shows a phase separation.An appropriate method to follow the phase transition of the system is to measure the
omplexvisosity
η ∗
ofthesystemasfuntionofthetemperature(seeg. 4.6A))[25,30℄.A deformation of 1
%
and a frequeny of 1Hz
have been used whih remains in thelinear visoelasti domain at least for temperatures below 32
o C
. Thus the transitions within the system an be measured without disturbing the system. The measurementswere performed on four onentrations (6, 8, 10.75 and 12.1
wt%
) by inreasing theA)
B)
C)
D)
Figure4.7: A)C) SFM mirographs and optial mirosopy of a 0.1
wt%
omposite mirogelssolution ontaining 5.10
−2 molL −1
KCl after dropasting and drying at 60o C
onsiliiumwafer. C)D)SFMmirographs andpolarizedoptialmirosopyofa0.1
wt%
omposite mirogels solutionwithout addition of salt afterdropasting and drying at
60
o C
on siliium wafer.temperatureat a rate of 0.175
o C/min
after shearing5min
at 100s −1
toremove all thehistory of the sample. As shown in g. 4.2 the hydrodynami radius dereases with
inreasing temperatures. This leads to a derease of the eetive volume fration
φ ef f
aompaniedbyaderease of
η ∗
. Forabetter understandingofthis experimentwhenthe system is purely repulsive, whih orresponds to temperatures below 25o C
, the reduedvisosity
η ∗ /η s
has been plotted as funtion ofφ ef f
. The alulation ofφ ef f
for thissystem desribed inthe setion 2.3.3 onsiders the evolutionof the hydrodynamiradius
as funtion of the temperature.
η ∗ /η s
dereases slowly between 0.63 and 0.545, whihorresponds to the rystalline and glassy state (solid state). In the liquid/rystalline
oexistenedomainbetween0.494and0.545thereduedomplexvisositydereasesmuh
faster. For eetivevolume below0.494, orresponding tothe melting of the rystallites,
thesolutionisinthe liquidstateand theredued visosity dereasesslower. After33
o C
agelationproessanbeobserved,wheretheomplexvisosityinreasesformorethanfour
deades within approximately 2
o C
. A maximum was observed at 35.3o C
for 12.1, 10.7and 8.0
wt%
solutions, whereas the 6wt%
solution ontinuously inreases. This an be related to a earlier phase separation for less onentrated systems. After this maximumthe visosity dereases again and reahes a plateau. This an beattributed to the phase
separationofthe systemasshownonthephotographing. 4.6B)at45
o C
. Atthispointthe measurement an just be onsidered qualitatively as the assumption of an isotropi
materialis not respeted anymore.
Sanning Fore Mirosopyand optial mirosopy were performed inorderto imagethe
inueneoftheattrativeinterationsonthestrutureofadensesuspensionsafterdrying
on a siliium wafer at temperature above the LCST (see g. 4.7). To this purpose 0.1
wt%
solutions have been prepared, one without salt and one with 5.10−2 molL −1
KCl.After drop astingonsiliium wafer, the solutionhave been quikly dried at60
o C
intheoven. The salty solutionshows ametastable struture and arather rough surfaearising
fromthe attrative interations. This was onrmed by the optial mirosopeequipped
333s
Figure4.8: Miroaggregation of the ompositemirogelsmaintained at34
o C
after dierent timeof irradiation of a 10.75
wt.%
latex solution with a laser power of 1W
. Theab-sorbane
A(r)
is plotted as funtion of the radial distane. The dashed line presentsthe alulation from the absorbane alulated after the LCST at 35.4
o C
from eq.4.15. The full linespresentthe best tonsidering the adsorptionof theaggregate as
adjustable parameter.
with a polarizer where no iridesene ould be observed. Indeed at high temperatures
the partilesstart to stik together, whih prevents any orderingwith inreasing
onen-tration. On the ontrary without salt the solution is still stable at high temperatures.
The eletrostati interations prevent the partilesto ome intoontat, and allowthem
to orderduring the drying proess. Ordered domainsand a more ompat struture an
then be observed by SFM, asalready shown for similarsystems by Hellweg and al. [59℄.
The ordering an be visualized by polarized optial mirosopy in the form of photoni
rystals.
Laser ontrolled miro-patterning
As shown by Lyonand oworkers, phase transitionan be obtained onthe mirosale by
fousing a green laser on a mirogel solution doped with gold nanopartiles [11, 243℄. A
transition from the glassy to the rystalline state, and from rystal to liquid ould thus
be obtained by loallyheatingthe sample.
We have performed similar experiments, however, with a temperature losely below the
LCST in order to indue a transition from liquid to attrative glass. Due to a slight
absorption of the sample at the laser wavelength of 532
nm
we did not have to use to0 2x10 -2 4x10 -2 6x10 -2
0 1000 2000 3000 4000 5000 0
5
Figure4.9: Evolution of the radius in
µm
(hollow symbols) and of the absorptionA
(inµm −1
)(fullsymbols)oftheaggregateobtainedduringtheirradiationofa10.75
wt%
solution.Dashed line indiatesthe essation of the irradiation.
additionaldye or gold doping. The reason for the optial absorption is still unlear and
theabsorptionspetrumisdiulttomeasurebyonventionalUV-VIS-spetrosopy due
to the strong sattering bakground. Nevertheless, as will be demonstrated below, we
have been able to determine the absorption oeient, at least at the laser wavelength,
fromthe observed sampleheating.
The wholeexperimentwas performedwith aninvertedoptial mirosopeequipped with
a CCD amera and a laser port through whih the beam of a diode-pumped frequeny
doubledsolidstatelaserouldbeoupledinandfousedontothesample. Aellof
d = 50 µm
thikness was employed and the samplewas equilibrated at ameasured temperature of 32.2o C
for 30min
beforestarting the experiment. Sine the temperatureis measured at the sample holder, it does not represent the exat temperature within the liquid andneeds to be orreted. The temperature oset was determined by slowly heating the
samplewitharateof0.013
o C/min
,therebykeepingthelaseroandobservingtheoveralloptialtransmittane. The ollapseofthe PNIPAM shellandtheformationofaggregates
are aompanied by a strong inrease of the turbidity at a temperature of 33.0
o C
(g.4.6). This temperature has been used as an internal referene in order to orret the
temperaturemeasured outside of the sampleell.
The laser power was adjusted to 100
mW
and foused into the enter of the ell.The laser power was adjusted to 100