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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, and

N z

is the total

partileonentration of

z

-foldaggregates.

The partile onentration of singlets

N 1

and doublets

N 2

as a funtion of time an be

obtained by solving eq. 4.1 analytially. Assuming

k ij = k 11

and the initial partile

onentration

N 0

we obtain

N 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

and

N 2

forthe short times leads to

N 1

IfweonsiderthattheoagulationofspherialpartilesisjustontrolledbyBrownian

dif-fusion, aordingtothe theoryof Smoluhsovski[245, 246℄,the oagulationrateonstant

beomes:

k 11 = 2k s = 8k B T

(4.6)

where

k B

is the Boltzmannonstant,

T

the temperature, and

η

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 partiles

I 1 (q)

and doublets

I 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 alulated

within 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 or

mean 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

weobtain

1

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

and

r 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 usedin

thishapter. Thekinetisofthe oagulationwasinvestigatedby dynamilightsattering

(DLS) using a Peters ALV 5000 light sattering goniometer where the temperature was

ontrolledwith anaurayof

±

0.1

o

Cas. The reversibilityof oagulationwasmeasured by DLS with a Zetaziser (Malvern model nanoZS) in bak sattering at 173

o

with a

strit ontrol of the temperature set by Peltier elements at

±

0.1

o

C. Cryogeni eletron

mirosopy 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

andaresonanefrequeny

rangingfrom200to300

Hz

. Thesanratewasvariedbetween 0.5and1.0

Hz

tooptimize

the image quality. The samples have been prepared as followed: a drop 30

µL

0.1

wt%

latexsolutionhasbeendepositedonsiliiumwaferand driedat60

o C

. Thesame samples

were 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.

%

dispersion

mea-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) and

typialpowervaluesare between

0.1

and 100

mW

. The ellismounted horizontallyin a temperature-ontrolledxyz-stage. Ithas agap thikness of50

µm

(Hellma106-QS).The

sample 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 temperature

around 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. This

phe-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 the

LCST value determined from the dynami light sattering analysis at 32.2

o C

(see

se-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 25

o C

,

then heated rst at 32

o C

, and after at dierent temperatures from 33

o C

to 35

o C

. The

ooling proess was thenreorded rst at32

o C

andthen at25

o C

. Whereasthe system is

stableat25and32

o C

itaggregatesat33

o C

. Theaggregationspeedwasfoundtoinrease

with inreasing temperature and the hydrodynami radius varies between 150 and 400

nm

within 10

min

. Cooled down at 32

o C

the hydrodynami radius sharply dereased toreahed the same value as beforethe aggregationourred, whihwas alsoheked at

25

o C

. In opposition to ore-shell system with linear PNIPAM, whih usually aggregates

in 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 of

steriinterations. 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. The

treat-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 been

repeated for dierent onentrations ranging between 2.5.10

−3

and 2.5.10

−5 wt%

. The

experiment wasperformed asfollow: 2.3

mL

ofthe latexsolutionwasequilibrated atthe wished temperaturein the DLS for 20

min

. Then 0.2

mL

of a 0.625

molL −1

KCl

main-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 monitored

as 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

wasonsideredatthe

1

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 indiate

the 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 light

sattering 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

whih

orrespondstotheratiobetweendiusionlimitedoagulationrateonstant

2k s

aording

to the Smoluhsovski theory (see equation 4.6) and the experimental oagulation rate

onstant

k 11

.

W

beomesnearlyonstantafter34

o 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 value

of 4.3.10

−18 ±

0.4

m 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℄. Asaonlusion

theoagulationoftheompositepartilesanbeontrolledbythetemperatureandbythe

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

, 33

o C

, 36

o C

and 40

o C

. For aggregates showing fratal strutures,

with fratal dimension

d f

, the radius

R

of the os inreases with the time

t

aording

to a power law [252℄,

R ∝ t z

where

z = 1/d f

. The fratal dimension an be alulated

from the slope. For temperature higher than 32.8

o C

the slope beomes onstant and a

mean value of

d f = 2.05 ± 0.03

is obtained. This value is signiantly greater than the

value 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 light

sattering fromtheequation 4.13whereas

2k s

isobtained fromthe expressionderived

by 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 32.8

o C

(irles), 33

o C

(squares), 36

o C

(down

trian-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-shell

partiles 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: 12.1

wt %

(full line), 10.75

wt

%

(dashed line), 8

wt %

(dotted-dashed line) and 6

wt %

(dotted line). B) Diret

observation ofthe 12.1

wt %

solutionatdierenttemperatures. C) Redued visosity

η s

of the dierent onentrations asfuntion ofthe eetivevolume fration

φ 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 PNIPAM

mirogels 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 system

behaves 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 phase

sepa-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 maintained

at dierent temperatures for half an hour. At 10

o C

the solution is in the glassy state

and the solution appears bluish. The orresponding volume fration alulated for this

temperatureis equalto 0.63. At 17

o C

whih orresponds toan eetive volume fration

of 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 liquid

state. The samples at 35 and 40

o C

present the solutionafter the osetof the attrative

interations. The solutionbeomes white and opaque and remainsrelatively stable after

30

min

, whereas the solutionat 45

o 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 1

Hz

have been used whih remains in the

linear visoelasti domain at least for temperatures below 32

o C

. Thus the transitions within the system an be measured without disturbing the system. The measurements

were performed on four onentrations (6, 8, 10.75 and 12.1

wt%

) by inreasing the

A)

B)

C)

D)

Figure4.7: A)C) SFM mirographs and optial mirosopy of a 0.1

wt%

omposite mirogels

solution ontaining 5.10

−2 molL −1

KCl after dropasting and drying at 60

o C

on

siliiumwafer. 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 shearing5

min

at 100

s −1

toremove all the

history 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 25

o C

, the redued

visosity

η /η s

has been plotted as funtion of

φ ef f

. The alulation of

φ ef f

for this

system 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, whih

orresponds 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

a

gelationproessanbeobserved,wheretheomplexvisosityinreasesformorethanfour

deades within approximately 2

o C

. A maximum was observed at 35.3

o C

for 12.1, 10.7

and 8.0

wt%

solutions, whereas the 6

wt%

solution ontinuously inreases. This an be related to a earlier phase separation for less onentrated systems. After this maximum

the visosity dereases again and reahes a plateau. This an beattributed to the phase

separationofthe systemasshownonthephotographing. 4.6B)at45

o C

. Atthispoint

the 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

inthe

oven. 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 time

of irradiation of a 10.75

wt.%

latex solution with a laser power of 1

W

. The

ab-sorbane

A(r)

is plotted as funtion of the radial distane. The dashed line presents

the 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 to

0 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 absorption

A

(in

µm −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 32.2

o C

for 30

min

beforestarting the experiment. Sine the temperatureis measured at the sample holder, it does not represent the exat temperature within the liquid and

needs to be orreted. The temperature oset was determined by slowly heating the

samplewitharateof0.013

o C/min

,therebykeepingthelaseroandobservingtheoverall

optialtransmittane. 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

mW

and foused into the enter of the ell.