# Effects of polydispersity on the glass transition dynamics of aqueous
suspensions of soft spherical colloidal particles

Sanjay Kumar Behera, Debasish Saha, Paramesh Gadige and Ranjini Bandyopadhyay
Soft Condensed Matter Group, Raman Research Institute, C. V. Raman Avenue,
Sadashivanagar, Bangalore 560 080, INDIA

###### Abstract

Thermoresponsive poly(N-isopropylacrylamide) (PNIPAM) particles of a nearly
constant swelling ratio and with polydispersity indices (PDIs) varying over a
wide range ($7.4$% - $48.9$%) are synthesized to study the effects of
polydispersity on the dynamics of suspensions of soft PNIPAM colloidal
particles. The PNIPAM particles are characterized using dynamic light
scattering (DLS) and scanning electron microscopy (SEM). The zero shear
viscosity ($\eta_{0}$) data of these colloidal suspensions, estimated from
rheometric experiments as a function of the effective volume fraction
$\phi_{eff}$ of the suspensions, increases with increase in $\phi_{eff}$ and
shows a dramatic increase at $\phi_{eff}=\phi_{0}$. The data for $\eta_{0}$ as
a function of $\phi_{eff}$ fits well to the Vogel-Fulcher-Tammann (VFT)
equation. It is observed that increasing PDIs results in increasingly fragile
supercooled liquid-like behavior, with the parameter $\phi_{0}$, extracted
from the fits to the VFT equation, shifting towards higher $\phi_{eff}$. The
observed increase in fragility is attributed to the prevalence of dynamical
heterogeneities (DHs) in these polydisperse suspensions, while the
simultaneous shift in $\phi_{0}$ is ascribed to the decoupling of the dynamics
of the smallest and largest particles. Finally, it is observed that the
intrinsic nonlinearity of these suspensions, estimated at the third harmonic
near $\phi_{0}$ in Fourier transform oscillatory rheological experiments,
increases with increase in PDIs. Our results are in agreement with theoretical
predictions and simulation results for polydisperse hard sphere colloidal
glasses and clearly demonstrate that jammed suspensions of polydisperse
colloidal particles can be effectively fluidized with increasing PDIs.
Suspensions of these particles are therefore excellent candidates for detailed
experimental studies of the effects of polydispersity on the dynamics of glass
formation.

###### pacs:

Valid PACS appear here

##  I Introduction

Thermoresponsive poly(N-isopropylacrylamide) (PNIPAM) hydrogel suspensions
undergo a reversible volume phase transition above the lowest critical soluble
temperature (LCST) of ${\approx}$ $32$ oC in water Heskins_J_Macromol_Sci_1968
; Hirokawa_JCP_1984 . This property of PNIPAM hydrogels provides an
opportunity to tune the particle size, volume fraction ($\phi$) and inter-
particle interactions in suspension by changing the temperature, thus enabling
the study of the complex phase behaviours of soft colloidal suspensions
Wu_Macromolecules_2003 ; Appel_Softmatter_2016 . Furthermore, suspensions of
PNIPAM colloidal particles have been in focus for their applications in
controlled and self regulated drug delivery L. Dong_J_controlled_release_1991
, bioconjugation Hoffman_clinical_chemistry , biosensors
Zhang_Sensors_actuators_Chemical , pollution control Morris_JCIS_1997 , as
viscosity modifiers of liquids and semisolids and in enhanced oil recovery
Sun_softmatter_2011  and chemical separation.

The zero-shear viscosity $\eta_{0}$ of a suspension of hard sphere colloids
characterized by a significant polydispersity index (width of the size
distribution/mean size $\approx$ 10%) increases significantly when its volume
fraction $\phi$ increases above $\phi\approx 0.53$, with the system entering a
colloidal glassy state at $\phi_{g}\approx 0.58$ Pusey_PRL_1987 ;
Week_Science_2000 . The dynamics of the colloidal particles freeze at
$\phi_{g}$ due to the kinetic constraints imposed on their motion. In order to
understand the dramatic slowdown of the dynamics and the formation of
disordered glassy phases, the use of colloidal suspensions having a
significant polydispersity index (PDI $\approx 10$%) is essential to prevent
the system from entering a crystalline phase Schaertl_statistical-physics_1994
. The increase in viscosity of fragile glass-forming molecular liquids with
decreasing temperature (T) is well described by the Vogel-Fulcher-Tammann
(VFT) law Angell_JNCS_1991 . For colloidal glasses, the inverse of the volume
fraction ($1$/$\phi$) of the colloidal suspension plays the role of
temperature (T) Brambilla_PRL_2009 ; Pusey_Nature_1986 ; Mattsson_Nature_2009
. To use the VFT law for colloidal glasses, therefore, T is replaced with
$1$/$\phi$. The modified VFT equation,
$\eta=\eta_{0}\exp\Big{(}\frac{D\phi}{\phi_{0}-\phi}\Big{)}$, explains the
rise in the viscosity of a colloidal suspension with $\phi$ and its dramatic
increase at $\phi=\phi_{0}$ Brambilla_PRL_2009 ; Mattsson_Nature_2009 . Here,
$1/D$ is the fragility and accounts for the deviation of the viscosity from an
Arrhenius dependence on $\phi$ as the sample approaches the glassy state. In
the modified VFT equation, $\eta_{0}=\eta$ ($\phi=0$). A super-Arrhenius
dependence of viscosity on the appropriate control parameter (1/T or $\phi$)
is an ubiquitous characteristic of a fragile glass former Angell_Science_1995
.

It is known that the increase in width of the particle size distribution in a
mixture of fine powders (fly-ash and silica powder) increases the packing
efficiency due to the decrease in the void fraction of the tapped powder beds
Suzuki_2001 . Experiments and a self consistent theory have shown that
polydispersity affects the linear viscoelasticity of concentrated polymer
solutions and melts Colby_JCP_1988 . In triblock copolymer Pluronic F127
suspensions, it is shown that the addition of an anionic surfactant, sodium
dodecyl sulfate (SDS), breaks the large spherical micelles into smaller,
anisotropic polydisperse micelles Basak_EPJE_2011 . This speeds up the
dynamics of the constituents of these mixtures and results in more efficient
packing configurations, with the particle volume fraction for random close
packing exceeding the value for suspensions of monodisperse hard spheres.

The transition between different arrested states of concentrated binary
colloidal mixtures characterized by large size asymmetries has been well-
documented in the literature Yunker_PRL_2010 ; Hendricks_PRE_2015 ;
Sentjabrskaja_softmatter_2013 . In these experiments, a significant
polydispersity value, which is essential to form colloidal glasses, is
maintained by mixing the colloidal particles of two different sizes in a
certain ratio.

The colloidal glass transition has also been studied for colloidal suspensions
comprising particles of significant size polydispersity without mixing batches
of monodisperse colloidal particles. The colloidal glass transition of
moderately polydisperse hard sphere colloids is reported to smear out with
increase in polydispersity Brambilla_PRL_2009 . The presence of heterogeneous
dynamics among the small particles, together with the decoupling of the
dynamics of the smallest and largest particles in jammed suspensions of
polydisperse hard sphere colloids, shifts $\phi_{g}$ to higher values
Zaccarelli_softmatter_2015 . Recently, Sollich et al. have reported that the
phase boundaries of fluid, fluid-solid and solid phases of hard sphere
colloids move to higher $\phi$ with increase in polydispersity
Sollich_softmatter_2011 . Even though there are several simulation results and
theoretical models that investigate the effects of polydispersity on the
random close packing volume fraction $\phi_{rcp}$ of hard sphere colloids
Schaertl_statistical-physics_1994 ; Farr_JCP_2009 , these results are
difficult to verify experimentally because of the difficulty in systematically
controlling the polydispersities of particles synthesized in the laboratory.

Particle polydispersity is an extremely crucial parameter in determining the
fragilities and critical volume fractions of soft glassy systems. Local
structural correlations in supercooled liquids, which have been shown in
simulations to vary with polydispersity, can profoundly influence the glass
transition Abraham_PRE_2008 . The number of cooperatively coupled molecules
($N_{corr}$), or the size of DHs, can be obtained by analysing the response at
the third harmonic to the applied sinusoidal perturbation Tarzia_JCP_2010 .
The sizes of the DHs, or alternatively, the numbers of molecules or particles
showing correlated movement ($N_{corr}$), are expected to grow as the glass
transition is approached Thibierge_PRL_2010 . The intrinsic nonlinearity of
the polydisperse colloidal system at the third harmonic also grows as the
glass transition is approached Seyboldt_softmatter_2016 . A detailed study of
the approach of the suspension to a non-ergodic phase in the presence of
varying degrees of polydispersity is therefore important to obtain a clear
understanding of the jamming transition. Both the stiffnesses and the
polydispersities of soft particles are expected to affect the particle
dynamics in suspensions near the colloidal glassy state. Since there is a
conspicuous absence of detailed and systematic experimental studies of the
dependence of the colloidal glass transition on the size polydispersity of the
colloidal particles in suspension, we investigate the glass formation
phenomenon in aqueous suspensions of soft PNIPAM particles by changing the
particle polydispersity in a controlled manner, while keeping the crosslinker
concentration constant to ensure approximately constant particle stiffnesses.

In the present study, PNIPAM particles characterized by a wide range of PDI
values (7.4%-48.9%) are successfully synthesized by using one-pot (OP) and
semi-batch (SB) methods Still_JCIS_2013 ; McPhee_J_Colloid_Interface_Sci_1993
. In OP method, PNIPAM particles of lower polydispersities are synthesized by
varying the sodium dodecyl sulfate (SDS) concentration, while in SB method,
size polydispersity of the particles is controlled by controlling the feeding
rates of the monomer, co-monomer and crosslinker solutions into the reaction
vessel. These simple methods are used to control the polydispersity of soft
colloidal suspensions and to provide an easy alternative for synthesizing
highly polydisperse particles without mixing batches of monodisperse PNIPAM
particles. Dynamic light scattering (DLS) and scanning electron microscopy
(SEM) are performed to study the thermoresponsivity and the size distributions
of these particles.

The dynamic flow behaviors of the suspensions of soft PNIPAM colloids of
varying polydispersities and with increasing particle volume fractions are
studied in rheological experiments. Since these particles are soft and
deformable, we quantify their packing in aqueous suspensions in terms of an
effective volume fraction $\phi_{eff}$. Zero shear viscosity $\eta_{0}$
values, which are extracted from these measurements as a function of
$\phi_{eff}$, are observed to increase with increase in $\phi_{eff}$ and shows
a dramatic increase at $\phi_{eff}=\phi_{0}$. The $\eta_{0}$ vs. $\phi_{eff}$
data are analysed using the VFT law to calculate $D$ and $\phi_{0}$ values for
suspensions of different $\phi_{eff}$. Furthermore, the intrinsic nonlinearity
of these suspensions at the third harmonic is measured from the stress
response to oscillatory strains for several colloidal suspensions comprising
particles over a range of PDIs. Our results, which can be explained in terms
of numerical and analytical studies of polydisperse hard sphere colloidal
systems Schaertl_statistical-physics_1994 ; Farr_JCP_2009 ; Yang_PRE_1996 ;
Zaccarelli_softmatter_2015 , demonstrate that the particle packing behaviour,
and therefore the glass transition of colloidal suspensions, can be controlled
effectively by controlling the polydispersities of the particles in
suspension.

##  II Materials and methods

###  II.1 Synthesis of polydisperse PNIPAM particles by the one-pot (OP)
method

PNIPAM particles of lower polydispersities were prepared by the classical OP
method. All the chemicals were purchased from Sigma-Aldrich. In a typical
experiment, $7.0$ g N-isopropylacrylamide (NIPAM) ($\geq 99\%$), $0.7$ g
N,N’-methylenbisacrylamide (MBA) ($\geq 99.5\%$) and $0.03$ g SDS were
dissolved in $470$ mL of Milli-Q water (Millipore Corp.) in a three-necked
round-bottomed (RB) flask fitted with a reflux condenser, a temperature sensor
and a $N_{2}$ inlet/outlet. The RB flask was placed on a magnetic stirrer with
heating. The solution was stirred at 600 rpm and purged with $N_{2}$ gas for
$30$ min to remove oxygen dissolved in water. The temperature of the heat bath
was raised to $70$∘C. The free radical polymerization reaction was initiated
by the addition of $0.28$ g of potassium persulfate (KPS) ($99.99\%$)
dissolved in $30$ mL of Milli-Q water in the preheated bath at $70$∘C. The
reaction was allowed to proceed for $4$ hours with a constant stirring speed
using a magnetic stirrer. After $4$ hours, the reaction was stopped and the
colloidal suspension was cooled down rapidly in an ice bath. The summary of
the latex recipes is shown in table T$1$ of Supporting Information. PNIPAM
particles of different sizes ($0.16$ $\mu$$m$ \- $0.88$ $\mu$$m$) and PDIs
($7.4$% - $16.2$%) were synthesized by varying the concentration of SDS in the
RB flask. A detailed discussion on how changing the concentration of SDS
changes the PDIs of the PNIPAM particles synthesized in this work can be found
in page 2 of the Supporting Information.

$Method$ | SDS conc./Flow rate |  $<$$d_{25^{\circ}C}$$>$ |  $\sigma$($\mu$$m$) | PDI | $\alpha$
---|---|---|---|---|---
OP |  $0.3$ g/L |  $0.16$ $\mu$$m$ | $0.0119$ | 7.4% | $1.72\pm 0.01$
OP |  $0.06$ g/L |  $0.40$ $\mu$$m$ | $0.0464$ | 11.6% | $1.71\pm 0.01$
OP |  $0.01$ g/L |  $0.47$ $\mu$$m$ | $0.0635$ | 13.5% | $1.69\pm 0.01$
OP |  $0.005$ g/L |  $0.88$ $\mu$$m$ | $0.1424$ | 16.2% | $1.68\pm 0.01$
SB |  $1.5$ mL/min |  $1.24$ $\mu$$m$ | $0.1900$ | 15.3% | $1.81\pm 0.02$
SB |  $1.0$ mL/min |  $1.73$ $\mu$$m$ | $0.5000$ | 28.9% | $1.87\pm 0.01$
SB |  $0.7$ mL/min |  $2.38$ $\mu$$m$ | $0.8690$ | 36.5% | $1.84\pm 0.01$
SB |  $0.5$ mL/min |  $2.78$ $\mu$$m$ | $1.3595$ | 48.9% | $1.83\pm 0.02$
Table 1: Mean hydrodynamic diameters ($<$$d_{H}$$>$), standard deviation of
particle sizes ($\sigma$) at $25$∘C, polydispersity indices (PDI) and swelling
ratios ($\alpha$) measured from dynamic light scattering experiments at
$\theta$=$90$∘ for suspensions of PNIPAM particles synthesized by varying the
SDS concentration in the one-pot (OP) method and by varying the flow rate of
the reaction ingredients in the semi-batch (SB) method.

###  II.2 Synthesis of polydisperse PNIPAM particles by the semi-batch (SB)
method

PNIPAM particles of higher polydispersities were prepared in an SB method.
$7.0$ g of NIPAM, $0.46$ g of MBA and $34.4$ g of $2$-aminoethylmethacylate
hydrochloride (AEMA) were dissolved in $200$ mL of water in a three-necked
round bottom (RB) flask fitted with a reflux condenser, a temperature sensor
and a $N_{2}$ inlet/outlet. Homogeneously crosslinked PNIPAM particles of
different sizes ($1.24$ $\mu$$m$ \- $2.78$ $\mu$$m$) and PDIs ($15.3$% -
$48.9$%) were synthesized by varying the feeding rates of monomer, crosslinker
and AEMA solutions. The three-necked RB flask was placed on a magnetic stirrer
with heating. The solution was stirred at $600$ rpm for $20$ min and also
purged with nitrogen for $20$ min. $180$ mL of this solution was taken out in
a syringe and $120$ mL of water was added to the $20$ mL solution remaining in
the flask. The temperature of the heat bath was raised to $80^{\circ}$C and
the bath was purged with $N_{2}$ gas for $20$ min. The free radical
precipitation reaction was initiated by the addition of $41.6$ mg ammonium
persulfate (APS) dissolved in $8$ mL of water. Primary nucleation sites
started to form after about $5$ min, following which the solution present in
the syringe was fed into the reaction vessel at a constant flow rate (between
$0.5$ and $1.5$ ml/min) using a syringe pump (Chemyx Inc.). The reaction was
stopped $5$ min after all the ingredients were added. The latices were cooled
down rapidly in an ice bath. The preparation of the latex recipes is
summarized in table T$1$ of Supporting Information. PNIPAM particles of
different sizes ($1.24$ $\mu$$m$ \- $2.78$ $\mu$$m$) and PDIs ($15.3$% -
$48.9$%) were synthesized by varying the flow rates of the reaction
ingredients into the RB flask. A detailed discussion on the effects of flow
rates of the reaction ingredients on the PDIs of the PNIPAM particles
synthesized here can be found in page 2 of the Supporting Information.

Latices obtained from SB and OP methods were purified by four successive
centrifugations, decantations and redispersions in Milli-Q water to remove
SDS, the remaining monomers, oligomers and impurities. Centrifugation was
performed at a speed of $11,000$ rpm for $60$ min. Dried polymers were
collected by evaporating water from both the supernatant and residue. Fine
PNIPAM powders were prepared by crushing and grinding the dried polymers with
a mortar and pestle.

###  II.3 Dynamic Light Scattering:

The sizes and size distributions of the PNIPAM suspensions and the
thermosensitive responses of these suspensions were determined using dynamic
light scattering (DLS) experiments. DLS experiments were performed using a
Brookhaven Instruments Corporation (BIC) BI-200SM spectrometer. The details of
the setup are given elsewhere Debasish_YMJ_Ranjini_Soft_Matter_2014 . A
temperature controlled water bath (Polyscience Digital) attached to the DLS
setup was used to maintain the temperature of the suspension. A Brookhaven
Instruments BI-9000AT digital autocorrelator was used with the DLS setup to
generate the intensity autocorrelation function $g^{(2)}(q,t)$ of the
scattered light. Here,
$g^{(2)}(q,t)=\frac{<I(q,0)I(q,t)>}{<I(q,0)>^{2}}=1+A|g^{(1)}(q,t)|^{2}$
Bern_Pecora , where $q$, $I(q,t)$, $g^{(1)}(q,t)$ and $A$ are the scattering
wave vector, the intensity at a particular $q$ and a delay time $t$, the
normalized electric field autocorrelation function and the coherence factor
respectively. $q$ is related to the scattering angle $\theta$ by the equation
$q=(4\pi n/\lambda)\sin(\theta/2)$, where $n$ and $\lambda$ are the refractive
index of the medium and the wavelength of the laser respectively. For a
suspension of monodisperse particles, the normalized electric field
autocorrelation function has an exponential decay:
$g^{(1)}(q,t)=[\exp(-t/\tau)]$. Here, $\tau$ is the relaxation time of the
particle which is directly related to its diffusion coefficient $D_{0}$ by
$1/(\tau q^{2})=D_{0}$. The hydrodynamic radius $R_{h}$ of the particle can be
calculated using the Stokes-Einstein relation $R_{h}=\frac{k_{B}T}{6\pi\eta
D_{0}}$ Bern_Pecora ; Einstein_1905 . Here $k_{B}$, $T$ and $\eta$ are the
Boltzmann constant, the absolute temperature and the viscosity of the solvent
respectively.

The decays of the autocorrelation functions estimated for the samples studied
here can be fitted to stretched exponential functions,
$C(t)=\frac{g^{(2)}(q,t)-1}{A}=\big{[}\exp(-t/\tau)^{\beta}\big{]}^{2}$. Here,
$A$ is used to normalize the autocorrelation data to $1$ at $t=0$. The
stretching exponent $\beta$ is used to calculate the mean and the second
moment of the relaxation time spectrum of the particles in suspension using
the relations $<$$\tau$$>$$=$$(\frac{\tau}{\beta})\Gamma(1/\beta)$ and
$<$$\tau^{2}$$>$$=$ $(\frac{\tau^{2}}{\beta})\Gamma(2/\beta)$ respectively
Lindsey_JCP_1980 , where $\Gamma$ is the Euler Gamma function. The mean
hydrodynamic diameter $<$$d_{H}$$>$ and the standard deviation or width of the
particle size distribution $\sigma$ were obtained by using
$<$$d_{H}$$>$$=\frac{k_{B}T<\tau>q^{2}}{3\pi\eta}$ and
$\sigma=\frac{k_{B}T\sqrt{<\tau^{2}>-<\tau>^{2}}q^{2}}{3\pi\eta}$ respectively
Segre_PRL_1996 .

For the DLS measurements reported here, PNIPAM suspensions of a very dilute
concentration ($c=0.01$ wt%) were prepared by adding dried PNIPAM powder in
Milli-Q water. $5.0$ mL of each suspension was filled in a glass cuvette and
the cuvette was placed in the sample holder of the DLS setup attached to a
temperature controller. The swelling and shrinkage behaviors of PNIPAM
particles were characterized at various temperatures at $3$∘C intervals in the
range $20$∘C - $50$∘C during cooling and heating. The swelling ratio $\alpha$
of the PNIPAM particles, calculated by measuring the mean hydrodynamic
diameter $<$$d_{H}$$>$ of the particle in the completely swollen state
($<$$d_{20^{\circ}C}$$>$) and in the completely shrunken state
($<$$d_{45^{\circ}C}$$>$), is defined as
$\alpha$$=$$<$$d_{20^{\circ}C}$$>$$/$$<$$d_{45^{\circ}C}$$>$ (table $1$).

The normalized electric field autocorrelation function $g^{(1)}(q,t)$ is
related to the line width distribution $G(\Gamma)$ by the relation
$g^{(1)}(q,t)=$$<$$E(t,q)E^{*}(0,q)$$>$$=\int_{0}^{\infty}G(\Gamma)\exp(-\Gamma
t)d\Gamma$ Koppel_JCP_1972 ; Xia_langmuir_2004 . Here, $\Gamma$ is the decay
rate which is related to $D_{0}$ by the relation $\Gamma=D_{0}q^{2}$. The
particle size distribution of each PNIPAM particle suspension is obtained from
the inverse Laplace transform of $g^{(1)}(q,t)$ Koppel_JCP_1972 ;
Xia_langmuir_2004 . The values of mean relaxation time $<$$\tau$$>$ and mean
hydrodynamic diameter $<$$d_{H}$$>$, obtained by fitting the intensity
autocorrelation data of a PNIPAM suspension synthesized in the OP method at an
SDS concentration of 0.06 g/liter (figure S$1$(a) of Supporting Information),
and in the SB method at a flow rate of $0.7$ mL/min of the flow rates of the
reaction ingredients(figure S$1$(b) of Supporting Information), are seen to
decrease with the increase in temperature (table T$2$ of Supporting
Information). The size distribution of PNIPAM particles is quantified by a
dimensionless polydispersity index (PDI) which is defined as the ratio of the
width of the particle size ($\sigma$) and the mean hydrodynamic diameter
($<$$d_{H}$$>$) of the particle.

###  II.4 Scanning Electron Microscopy:

PNIPAM particles were directly visualized using a scanning electron microscope
(SEM, GEMINI column, ZEISS, Germany). An ITO substrate was cleaned with
acetone and a PNIPAM suspension of concentration $0.01$ wt% was added on the
ITO coated side of the substrate using a pipette. The sample placed on the ITO
substrate was dried overnight at $25$∘C and was then loaded on the SEM stage
for imaging. The electron beam interacts with the atoms of the sample and the
backscattered secondary electrons are used to produce the surface images of
the samples. There were some particle doublets and triplets that were visible
in the images. These were not considered in the size analysis. The average
particle sizes and size distributions were estimated from approximately
$1,000-1500$ particles.

###  II.5 Rheology:

Rheological measurements were performed with a stress controlled Anton Paar
MCR $501$ rheometer. For these measurements, PNIPAM suspensions of very high
concentrations were prepared by adding dried PNIPAM powder in Milli-Q water.
The suspension was then stirred for $24$ hours and sonicated for $45$ minutes.
PNIPAM suspensions of different concentrations were prepared by diluting
suspensions of the highest PNIPAM particle concentration with Milli-Q water. A
concentric cylinder geometry (CC-17) of measuring bob radius $r_{i}=8.331$ mm,
measuring cup radius $r_{e}=9.041$ mm, cone angle $\alpha=120^{\circ}$ and gap
length $L=24.990$ mm was used for dilute samples. For concentrated samples, a
cone-plate geometry (CP-25) of cone radius $r_{c}=12.491$ mm, cone angle
$\alpha=0.979^{\circ}$ and measuring gap $d=0.048$ mm was used. The inner
surfaces of concentric cylinder and cone-plate geometries are made of smooth
stainless steel. A solvent trap oil was used to minimize solvent evaporation.
The temperature of the samples were maintained constant at $25$∘C using a
Peltier unit and a water circulation system for counter cooling. For each
experiment, a sample volume of $4.7$ mL was loaded in the concentric cylinder
geometry, while a sample volume of $0.07$ mL was loaded in the cone-plate
geometry. Measurements with both roughened (sandblasted) and smooth geometries
produced comparable results and verify the absence of wall slip in the smooth
geometries used in this work (figure S$2$ of Supporting Information).

Steady state flow experiments of PNIPAM suspensions were performed at $25$∘C.
The viscosity $\eta$ of each suspension was measured by varying the shear rate
$\dot{\gamma}$ between $0.001$ and $4000$ $s^{-1}$. The zero-shear viscosity
$\eta_{0}$ of each suspension was calculated by fitting the viscosity $\eta$
versus shear rate $\dot{\gamma}$ curve (figure S$3$ of Supporting Information)
with the Cross model Cross_JCS_1965 :

$\frac{\eta-\eta_{\infty}}{\eta_{0}-\eta_{\infty}}=\frac{1}{1+(k\dot{\gamma})^{m}}$
(1)

where $\eta_{0}$ and $\eta_{\infty}$ correspond to the viscosity plateaus at
very low and very high shear rates respectively, $k$ is a time constant
related to the relaxation time of the polymer in solution and $m$ is a
dimensionless exponent. The relative viscosity $\eta_{rel}$ of a PNIPAM
suspension is defined as the ratio of its zero-shear viscosity $\eta_{0}$ and
the viscosity of water $\eta_{s}$: $\eta_{rel}=\eta_{0}/\eta_{s}$.

The PNIPAM particles are soft and start deforming in the presence of
neighbouring particles at volume fractions above the random close packing
volume fraction of undeformed monodisperse spheres ($\phi_{RCP}=0.64$)
Pusey_Nature_1986 ; Mattsson_Nature_2009 . Since the volume fraction $\phi$
does not account for the particle deformations of these soft particles, we
consider a modified parameter called the effective volume fraction
$\phi_{eff}$. $\phi_{eff}$ of a PNIPAM suspension was obtained from the
relation $\phi_{eff}=nV_{d}$, where $n$ is the number of particles per unit
volume and $V_{d}=\pi$$<$$d_{H}$$>$${}^{3}/6$ is the volume of an undeformed
particle of mean hydrodynamic diameter $<$$d_{H}$$>$ in a very dilute
suspension Mattsson_Nature_2009 . The above relation can be written in terms
of the polymer concentration of the microgel packing $c$ as
$\phi_{eff}=c/c_{p}$ Lorenzo_macromolecules_2013 , where $c_{p}=m_{p}/V_{d}$
is the polymer concentration inside each particle in the swollen state and
$m_{p}$ is the molecular weight of each particle. The Batchelor’s equation
relates $\eta_{rel}$ of the suspension with its $\phi_{eff}$ as follows:
Batchelor_JFM_1997 ; Brady_J.Rheology_1995 :

$\eta_{rel}=1+2.5\phi_{eff}+5.9\phi_{eff}^{2}$ (2)

The polymer mass concentration of the suspension can be converted into the
effective volume fraction by substituting $\phi_{eff}=c/c_{p}$ in equation 2:

$\eta_{rel}=1+2.5(c/c_{p})+5.9(c/c_{p})^{2}$ (3)

The Batchelor’s law is valid for only dilute and relatively monodisperse hard
sphere suspensions Batchelor_JFM_1997 .

In our experiments, only dilute concentrations ($\phi_{eff}$ $<$ 0.2) of
polydisperse PNIPAM suspensions are used to rule out inter-particle
interactions Poon_colloidal_suspension . Our assumption is validated by the
excellent fits of the data to equation (3) (figure S4 of Supporting
Information). The values of $c_{p}$, calculated by fitting equation (3) to the
experimental data displayed in figure S$4$ of the Supporting Information, are
tabulated in table T$3$ for PNIPAM particles of different mean hydrodynamic
diameters $<$$d_{H}$$>$ and polydispersity indices (PDIs). The fitted values
of the polymer concentration inside each particle, $c_{p}$, is independent of
the polymer mass concentration of the suspension $c$ and depends only on the
particle preparation protocol. $\phi_{eff}=c/c_{p}$ is therefore used to
determine the value of the effective volume fraction $\phi_{eff}$ at higher
concentrations for the same batch of particles.

Medium amplitude oscillatory shear (MAOS) tests were performed for suspensions
of PNIPAM particles of lower PDIs synthesized by the OP method using the
CP-$25$ geometry. PNIPAM suspensions of different volume fractions near
critical volume fractions $\phi_{c}$ were prepared using several different
batches of particles. The Fourier coefficients of the elastic and viscous
moduli at the third harmonic, $G^{\prime}_{3}$($\omega$,$\gamma_{0}$) and
$G^{\prime\prime}_{3}$($\omega$,$\gamma_{0}$), and at the fundamental
harmonic, $G^{\prime}_{1}$($\omega$,$\gamma_{0}$) and
$G^{\prime\prime}_{1}$($\omega$,$\gamma_{0}$), were measured by varying the
strain amplitude $\gamma_{0}$ between 0.1% to 50% at a fixed angular frequency
$\omega$. The intrinsic moduli [$G^{\prime}_{3}$($\omega$)] and
[$G^{\prime\prime}_{3}$($\omega$)] at a very low strain amplitude were
determined by fitting the strain amplitude dependence of the Fourier
coefficients of the elastic and viscous moduli at the third harmonic,
$G^{\prime}_{3}$($\omega$,$\gamma_{0}$) and
$G^{\prime\prime}_{3}$($\omega$,$\gamma_{0}$), with quadratic functions,
$G^{\prime}_{3}=[G^{\prime}_{3}(\omega)]\gamma_{0}^{2}$ and
$G^{\prime\prime}_{3}=[G^{\prime\prime}_{3}(\omega)]\gamma_{0}^{2}$ (figure
S$5$ of Supporting Information) Seyboldt_softmatter_2016 . The frequency
dependent equilibrium linear elastic and viscous moduli
$G^{\prime}_{eq}(\omega)$ and $G^{\prime\prime}_{eq}(\omega)$ were determined
by fitting the strain amplitude dependence of the fundamental Fourier
coefficients of the elastic and viscous moduli,
$G^{\prime}_{1}$($\omega$,$\gamma_{0}$) and
$G^{\prime\prime}_{1}$($\omega$,$\gamma_{0}$), to the equations
$G^{\prime}_{1}=G^{\prime}_{eq}(\omega)+[G^{\prime}_{1}(\omega)]\gamma_{0}^{2}$
and
$G^{\prime\prime}_{1}=G^{\prime\prime}_{eq}(\omega)+[G^{\prime\prime}_{1}(\omega)]\gamma_{0}^{2}$
(figure S$6$ of Supporting Information). The intrinsic nonlinearity
$Q_{0}(\omega)$ was obtained from the relation
$Q_{0}(\omega)=\frac{|[G_{3}(\omega)]|}{|G_{eq}(\omega)|}$
Seyboldt_softmatter_2016 . Here,
$|[G_{3}(\omega)]|=[[G^{\prime}_{3}(\omega)]^{2}+[G^{\prime\prime}_{3}(\omega)]^{2}]^{1/2}$
and
$|G_{eq}(\omega)|=[[G^{\prime}_{eq}(\omega)]^{2}+[G^{\prime\prime}_{eq}(\omega)]^{2}]^{1/2}$
are the magnitudes of the intrinsic moduli at the third harmonic and the
frequency dependent equilibrium linear complex modulus respectively.

We observe some difference in the quality of the $G^{\prime}_{3}$ and
$G^{\prime\prime}_{3}$ data acquired from suspensions of PNIPAM particles
synthesized by OP and SB methods at very low $\gamma_{0}$ (figure S$7$ of
Supporting Information). It is difficult to determine the intrinsic parameters
$[G^{\prime}_{3}(\omega)]$ and $[G^{\prime\prime}_{3}(\omega)]$ for the highly
polydisperse PNIPAM suspensions synthesized by the SB method due to the
presence of large fluctuations in the $G^{\prime}_{3}$ and
$G^{\prime\prime}_{3}$ values measured at very low $\gamma_{0}$ (figure
S$7$(b) of Supporting Information). These observed fluctuations arise from the
inhomogeneities present in these highly polydisperse suspensions. Since the
data from the suspensions of PNIPAM particles synthesized by the OP method are
less noisy (figure S$7$(a) of Supporting Information), we therefore use their
aqueous suspensions to study the nonlinear Fourier transform rheology at the
first higher harmonic.

Figure 1: (a) The hydrodynamic particle size distribution of PNIPAM particles
measured from DLS experiments for particles synthesized by the OP method at
different SDS concentrations (g/liter) at $25$∘C and at a scattering angle
$\theta$=$90$∘. (b) The hydrodynamic particle size distribution of PNIPAM
particles measured from DLS experiments for particles synthesized by the SB
method at different flow rates (ml/min) of the reaction ingredients at $25$∘C
and at a scattering angle $\theta$=$90$∘.

##  III RESULTS AND DISCUSSIONS

###  III.1 PNIPAM particles characteristics:

The plots of hydrodynamic particle size distributions, estimated from DLS at a
scattering angle $\theta$=$90$∘, for PNIPAM particles synthesized by the one-
pot (OP) and the semi-batch (SB) methods are shown in figure $1$. The sizes
($<$$d_{H}$$>$), swelling ratios ($\alpha$) and polydispersity indices (PDIs),
estimated for all the synthesized PNIPAM particles in aqueous suspensions in
the DLS experiments, are listed in table 1. It is seen from figure 1 and table
$1$ that the mean and the width of the size distribution of the PNIPAM
particles increase with the decrease in concentration of SDS. Using the OP
method, we have synthesized particles of PDIs varying from 7.4% to 16.2%.
Particles of higher polydispersity (PDIs between 15.3% and 48.9%) are
synthesized in the SB method by varying the flow rate of the reaction
ingredients . It is seen from figure 1 and table 1 that the decrease in the
flow rate of the reaction ingredients in the SB method increases the mean and
the width of the size distribution of the PNIPAM particles. The increase in
mean particle sizes and PDIs with decreasing SDS concentration in the OP
method and with decreasing flow rates of the reaction ingredients in the SB
method have been verified by making additional measurements at lower
scattering angles (figure S$8$ of Supporting Information). The range of PDI
values of the soft PNIPAM particles synthesized here varies between 7.4% and
48.9%. This wide range allows us to systematically investigate the effects of
PDI values on the approach of concentrated PNIPAM suspensions towards the
colloidal glass transition.

Figure 2: (a-b) Representative SEM micrographs of dry PNIPAM particles
synthesized by the OP method (a) particle PDI 11.9% at SDS concentration of
$0.06$ g/liter and (b) particle PDI 16.2% at SDS concentration $0.01$ g/liter.
(c-d) Representative SEM micrographs of dry PNIPAM particles, synthesized by
the SB method (c) particle PDI 46.9% at a flow rate of $0.7$ mL/min and (d)
particle PDI 53.8% at a flow rate of $0.5$ mL/min.

Direct visualization of the PNIPAM particles synthesized by the OP and the SB
methods is achieved using SEM imaging. Representative SEM images are displayed
in figure 2. The average sizes and particle size distributions estimated from
SEM images are shown in table T1 and figure S$9$ of the Supporting
Information. These results are in excellent agreement with the DLS results
(table 1 and figure$1$). However, the sizes of the particles measured from SEM
images are found to be less than those obtained by DLS at $45$∘C by a factor
of almost $1.2-2.0$ (table T$1$ of Supporting Information). This is explained
by considering that above the LCST ($45$∘C), the particles in suspension
shrink by expelling water from their interior, but are still not collapsed
completely. The DLS measurements therefore reveal larger values of
$<$$d_{H}$$>$ when compared to the SEM experiments in which the particles are
in the dried state.

The particle stiffness is characterized by measuring the temperature
dependence of the sizes of the PNIPAM particles using DLS. The corresponding
thermoresponsive curves of the PNIPAM particles in suspension are shown in
figure 3. It is seen that the swelling ratio $\alpha$ is approximately
insensitive to the method of synthesis (figure $3$). This is explained by
considering that a fixed concentration of crosslinker MBA is used in both the
OP and SB methods (table T$1$ of Supporting Information). It can therefore be
concluded that the swelling behaviour, and therefore, the CLD inside each
particle is approximately insensitive to changes of PDI.

Figure 3: Temperature dependent swelling ratio
$d_{T^{\circ}C}/d_{45^{\circ}C}$ as a function of temperature $T^{\circ}C$ for
suspensions of PNIPAM particles of different PDIs synthesized by the SB method
at different flow rates (ml/min) of the reaction ingredients, and by the OP
method, synthesized using different concentrations (g/L) of SDS, estimated
from DLS experiments at a scattering angle $\theta$=$90$∘. Figure 4: (a)
Relative viscosity $\eta_{rel}$ vs. effective volume fraction $\phi_{eff}$ of
suspensions of PNIPAM particles synthesized by the OP method at different
concentrations of SDS and characterized by different PDIs: 7.4 $\%$ ($\Box$),
11.6 $\%$ ($\circ$), 13.5 $\%$ ($\triangle$), 16.2 $\%$ ($\nabla$) at
$25^{\circ}$C. (b) $\eta_{rel}$ vs. $\phi_{eff}$ of suspensions of PNIPAM
particles synthesized by the SB method at different flow rates of reaction
ingredients and characterized by different PDIs: 15.3 $\%$ ($\Box$), 28.9 $\%$
($\circ$), 36.5 $\%$ ($\triangle$), 48.9 $\%$ ($\nabla$) at $25^{\circ}$C. The
solid lines are the VFT fits (equation $4$) to the data. Insets show the
fitted values of fragility $1/D$ ($\blacksquare$) and $\phi_{0}$ ($\bullet$)
vs. PDI of the PNIPAM particles in suspension.

###  III.2 Glass formation dynamics of polydisperse PNIPAM suspensions:

The dynamics of glass formation as a function of PDIs in suspensions of soft
PNIPAM particles having PDIs between $7.4\%$ and $48.9\%$ is investigated
using rheology at $25^{\circ}$C. The colloidal glass transition is expected to
occur at a volume fraction $\phi_{g}$ at which the viscosity shows an apparent
divergence. It is well known that for colloidal glasses, the volume fraction
$\phi$ plays a role analogous to the inverse of the temperature $T$, with the
dynamics slowing down with increasing $\phi$ due to the increase in kinetic
constraints Hunter_RPP_2012 ; Megen_PRL_1993 . Since $\phi_{eff}$ is a correct
measure of the concentration of soft PNIPAM particles in aqueous suspensions,
we estimate the relative zero shear viscosity $\eta_{rel}$ as a function of
$\phi_{eff}$. $\eta_{rel}$ vs. $\phi_{eff}$ plots are depicted in figure $4$.
$\eta_{rel}$ values are observed to increase by four decades over the range of
$\phi_{eff}$ explored here.

The values of $\eta_{rel}$ of suspensions of soft PNIPAM particles in the
arrested state are higher than the $\eta_{rel}$ values reported for hard
sphere colloidal particle suspensions Meeker_PRE_1997 ; Phan_PRE_1996 . The
observed higher values of $\eta_{rel}$ in these suspensions of polydisperse,
deformable PNIPAM particles approaching $\phi_{g}$ are due to the increase in
the number of contacts and the number of neighbours surrounding a given
particle, which result in the enhancement of the particle packing efficiency.
The values of the fragility, $1/D$, and the effective volume fraction
$\phi_{0}$, at which $\eta_{rel}$ increases dramatically, are obtained by
fitting the $\eta_{rel}$ vs. $\phi_{eff}$ data (figure $4$) to the VFT
equation Mattsson_Nature_2009 ; Brambilla_PRL_2009 :

$\eta_{rel}=\exp\Big{(}\frac{D\phi_{eff}}{\phi_{0}-\phi_{eff}}\Big{)}$ (4)

Here, $D$ is the fragility parameter. The fragility ($1/D$) in glass-forming
liquids is defined as the deviation of the viscosity from Arrhenius growth
upon cooling rapidly (for supercooled liquids) or with increasing $\phi$ (for
colloidal suspensions) Angell_Science_1995 ; Tanaka_PRL_2003 .

The values of $1/D$ and $\phi_{0}$, obtained from fits to the VFT law for
suspensions of PNIPAM particles with varying PDIs are reported in table 2. It
is seen that $1/D$ and $\phi_{0}$ increase with increase in PDIs.

Method | PDI | $1/D$ | $\phi_{0}$
---|---|---|---
OP | $7.4\%$ | $0.27\pm 0.01$ | $0.99\pm 0.04$
OP | $11.6\%$ | $0.29\pm 0.01$ | $1.31\pm 0.01$
OP | $13.5\%$ | $0.31\pm 0.01$ | $1.37\pm 0.03$
OP | $16.2\%$ | $0.35\pm 0.02$ | $1.41\pm 0.02$
SB | $15.3\%$ | $0.37\pm 0.05$ | $0.66\pm 0.03$
SB | $28.9\%$ | $0.53\pm 0.04$ | $0.74\pm 0.02$
SB | $36.5\%$ | $0.68\pm 0.03$ | $0.78\pm 0.01$
SB | $48.9\%$ | $0.79\pm 0.02$ | $0.85\pm 0.01$
Table 2: Fragility $1/D$ and $\phi_{0}$ values obtained by fitting the VFT
equation to the relative viscosity $\eta_{rel}$ vs. effective volume fraction
$\phi_{eff}$ data (figure $4$) for suspensions of PNIPAM particles of
different PDIs.

The observed increase of $\phi_{0}$ with increasing PDIs for the PNIPAM
particle suspensions studied here is in agreement with theoretical predictions
and simulation results for close packed hard sphere colloids
Schaertl_statistical-physics_1994 ; Farr_JCP_2009 ; Yang_PRE_1996  and can be
explained by considering that the packing of smaller spherical particles in
between bigger spherical particles becomes increasingly efficient as the
sample PDI increases. It has been shown that suspensions of polydisperse hard
spheres can stay fluidized at volume fractions higher than the random close
packing volume fraction of monodisperse hard spheres
Zaccarelli_softmatter_2015 ; Brambilla_PRL_2009 . This is due to the high
degree of heterogeneous dynamics prevalent among the small particles whose
mobilities are higher than those of the bigger particles in the polydisperse
mixture Zaccarelli_softmatter_2015 . Heterogeneous dynamics, and the resultant
decoupling of the dynamics of the small and big particles in the polydisperse
suspensions, result in the big particles forming an arrested state, with the
smaller particles of higher diffusivities smearing out the glass transition to
a certain degree Zaccarelli_softmatter_2015 . The increased prevalence of
heterogeneous dynamics and the increased packing efficiency of PNIPAM
microparticles with increase in the polydispersity index can therefore explain
the data in figure 4.

It is seen from table 2 that the $\phi_{0}$ values for suspensions of
particles of lower polydispersities, synthesized by the OP method, are higher
than those of the suspensions of the particles synthesized by the SB method
(table 2). This can be attributed to the rigid cores and loose shells of the
particles produced in the OP method Saunders_Langmuir_2004 . The heterogeneous
crosslinking density (CLD) of the PNIPAM particles synthesized by the OP
method produces highly crosslinked rigid cores, and less crosslinked hairy
shells Pelton_Langmuir_1989 . At sufficiently high volume fractions, the core-
shell particles deform up to the core because of the presence of the hairy
shells. This shifts $\phi_{0}$ to higher values. The PNIPAM particles
synthesized by the SB method, in contrast, have a uniform CLD within each
particle Still_JCIS_2013 . This is confirmed by direct visualization (figure
S$10$ of the Supporting Information). The larger polymer densities at the core
of the particles obtained in the OP method create a larger refractive index
mismatch between the particles and the background solvent (water). This leads
to stronger light scattering and more suspension turbidity in suspensions of
these particles when compared to the particles synthesized by the SB method.
At fixed volume fraction above $\phi_{RCP}$, therefore, the particles
synthesized by the SB method deform less when compared to the particles
synthesized by the OP method.

Figure 5: (a) Intrinsic nonlinearity $Q_{0}$ at the third harmonic vs.
$\omega<R_{h}>^{2}/D_{0}$ for suspensions of PNIPAM particles synthesized by
the OP method for different $\phi_{eff}$ near $\phi_{0}$ at a PDI = 11.6% at
$25^{\circ}$C. (b) Intrinsic nonlinearity $Q_{0}(\omega)$ at the third
harmonic vs. $\omega<R_{h}>^{2}/D_{0}$ for suspensions of PNIPAM particles
synthesized by OP method and characterized by different PDIs at a
$\phi_{eff}=1.0$ at $25^{\circ}$C. Inset shows the peak height ($\Box$) and
peak position ($\circ$) of $Q_{0}(\omega<R_{h}>^{2}/D_{0})$ curve vs. PDI of
the PNIPAM particles in suspension.

The fragility of glass forming liquids is ascribed to the presence of slow and
fast moving groups of molecules or particles called dynamical heterogeneities
(DHs). The observed high fragility values of these glassy suspensions of
polydisperse particles can be understood by considering an increase in the
concentration of the DHs He_PRE_1999 ; Kiriushcheva_PRE_2001 . Indeed, the
fragility values increase by almost a factor of three over the range of PDIs
explored in this work (table II and solid squares in the insets of figure
$4$(a) and $4$(b)). The enhancement of heterogeneous dynamics with increasing
PDIs is further supported by the third harmonic nonlinear response data
discussed in the following section.

###  III.3 Study of the intrinsic mechanical nonlinear susceptibility of
polydisperse PNIPAM suspensions:

The nonlinear susceptibilities of glass forming liquids have been studied
recently to evaluate its dependence on the length scales of the dynamical
heterogeneities Brambilla_PRL_2009 ; Thibierge_PRL_2010 ; Ladieu_PRB_2012 ;
Paramesh_Sci-Rep_2017 . Moreover, the nonlinear rheological response of visco-
elastic materials have been studied to understand material rigidity and its
microscopic origin Hyun_macromolecules_2009 . A recent report on soft PNIPAM
core-shell particle suspensions showed the divergence of the mechanical
nonlinear susceptibility at the third harmonic while approaching the glass
transition Seyboldt_softmatter_2016 . In order to understand the heterogeneous
behaviour and fragility of polydisperse PNIPAM suspensions, we record the
mechanical nonlinear susceptibility arising at the third harmonic in the
stress response of concentrated polydisperse PNIPAM suspensions using Fourier
transform oscillatory rheology. To estimate the nonlinearity of these
suspensions, we calculate the intrinsic nonlinearity $Q_{0}$, which is the
ratio of the mechanical susceptibilities at the third harmonic with respect to
the first harmonic at a very low strain amplitude $\gamma_{0}$. $Q_{0}$ is
estimated as a function of angular frequency $\omega$ of the applied strain
for PNIPAM suspensions synthesized by the OP method at several different
$\phi_{eff}$ near $\phi_{0}$ at a fixed PDI value and for several different
PDIs at a fixed $\phi_{eff}$.

In figures $5(a)$ and figure $5(b)$, $Q_{0}$ is plotted as a function of
$\omega$, normalized by the intrinsic relaxation frequency
($D_{0}/<R_{h}>^{2}$) of a single particle, obtained from dilute suspensions
for PNIPAM suspensions at several different $\phi_{eff}$ at a fixed PDI and
for several different PDIs at a fixed $\phi_{eff}$, respectively. $Q_{0}$
exhibits a peak with increase in $\omega<R_{h}>^{2}/D_{0}$ in both plots. It
is seen from figure 5(a) that the peak height increases by half a decade and
the peak position shifts to lower frequencies in PNIPAM suspensions when
$\phi_{eff}$ increases from 0.93 to 1.20 at a fixed particle PDI. The values
of the peak heights and the peak positions at various $\phi_{eff}$ are plotted
in the inset of figure 5(a). When the PDIs of the particles in these
suspensions are increased from $7.4\%$ to $16.2\%$ at a fixed $\phi_{eff}$,
the peak height is observed to increase by a half a decade, while the peak
positions shift to higher frequencies (figure $5(b)$). The values of the peak
positions and the peak heights vs. PDI are plotted in the inset of figure
5(b). The plots of $Q_{0}$, estimated as a function of
$\omega<R_{h}>^{2}/D_{0}$ for PNIPAM suspensions of different $\phi_{eff}$
near $\phi_{0}$ at a PDI = 11.6%, are seen to collapse at high frequencies
(figure $5(a)$). This observation indicates the presence of a single fast
relaxation time in PNIPAM suspensions of different $\phi_{eff}$ values near
$\phi_{0}$ when the particle PDI is kept constant. However, in figure $5(b)$,
the plots corresponding to different PDIs at fixed $\phi_{eff}$ are observed
to have different shapes and therefore cannot be collapsed onto a master
curve. This suggests the presence of different relaxation rates in the PNIPAM
suspensions having different PDIs at a fixed $\phi_{eff}$.

The observed increase in the peak heights with increase in $\phi_{eff}$ at a
fixed PDI (inset of figure 5(a)) and with increase in PDI at a fixed
$\phi_{eff}$ (inset of figure 5(b)) confirm an increase in the intrinsic
nonlinearities of the suspensions when the relevant control parameter
($\phi_{eff}$ or PDI) is changed. This can be interpreted in terms of
increasingly heterogeneous dynamics which is also apparent from the fragility
measurements reported in figure 4. In general, for suspensions characterized
by higher particle PDIs, the dynamics of small particles decouple from the
dynamics of big particles, which leads to a broadening of the relaxation
spectra of the particles. This is expected to increase the heterogeneity in
the dynamics of suspensions of higher particle PDIs.

Furthermore, the shift of the peak position to a higher frequency with
increase in particle PDI indicates a decrease in the structural relaxation
times (inset of figure 5(b)) of the particles in aqueous suspension. This can
be explained by considering that the smaller particles in the highly
polydisperse suspensions still remain free to diffuse even at very high volume
fractions. This results in the emergence of an arrested state at larger
particle concentrations when compared to suspensions having particles of
smaller PDIs. From the nonlinear rheology results shown in figures $5(a)$ and
$5(b)$, we can therefore separate the effects of changes in $\phi_{eff}$ and
PDI on the glass transition of PNIPAM suspensions. Increase in $\phi_{eff}$ at
a fixed PDI accelerates the jamming of PNIPAM particles in suspension, while
increase in PDI at a fixed $\phi_{eff}$ fluidizes the jammed state of PNIPAM
suspensions. Qualitatively, the increase in $Q_{0}$ with increase in particle
PDI indicates an increase in the concentrations and length scales of dynamical
heterogeneities and supports the viscosity data shown in figure 4.

##  IV Conclusions

We have synthesized thermoresponsive PNIPAM particles of a wide range of
polydispersity indices (PDIs) using the one-pot and semi-batch synthesis
methods. The thermoresponsive behaviour and the size distributions of these
particles are characterized by DLS and SEM. A rheological study of the
approach of these suspensions towards the glass transition shows that the soft
and polydisperse PNIPAM particles form highly fragile glasses at volume
fractions well above the random close packing volume fraction $\phi_{rcp}$ of
monodisperse hard sphere colloidal suspensions. The nonlinear stress response
at the third harmonic to applied oscillatory strains is measured to quantify
the intrinsic nonlinearity of these samples. The intrinsic nonlinearity of
these suspensions is monitored by changing $\phi_{eff}$ at a fixed PDI and by
changing particle PDIs at a fixed $\phi_{eff}$. The nonlinearity of the
dynamics of these suspensions is observed to increase with increase in both
$\phi_{eff}$ and PDI. The increase in fragility and $\phi_{0}$ of these
concentrated suspensions with increase in particle PDI is supported by our
nonlinear rheology results. For a jammed packing of polydisperse particles,
the local packing fraction, the number of contacts and the number of
neighbours surrounding a given particle are the parameters that decide the
packing efficiency at the single particle level. All these parameters are
expected to increase with increase in $\phi_{eff}$, PDI and the particle
softness, thereby enhancing the packing efficiency of the soft and deformable
PNIPAM particles. Suspensions of PNIPAM particles are therefore excellent
candidates for studying the dynamics of highly polydisperse particles in soft
glassy suspensions.

##  V Acknowledgements

We thank A. Dhason and K. M. Yatheendran for their help with SEM and cryo-SEM
imaging.

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