Time-resolved structural evolution during the collapse of responsive hydrogels: The microgel-to-particle transition

The collapse kinetics of microgels is determined experimentally and by mesoscale computer simulations.


section S1.4. Dynamic Light Scattering (DLS)
The thermo-responsive behavior of the PNIPAM microgels at different H 2 O/MeOH compositions was studied by means of DLS at a scattering angle of 30° and wave length of 633 nm. Figure S1 depicts the hydrodynamic radius (computed via the Stokes-Einstein equation using the proper value of the solvent viscosity) of the microgel at different solvent compositions at 10 and 21 °C.
The microgel is swollen in the pure solvents and a minimum size is reached at x MeOH = 0.2. section S1.5. SAXS & SLS measurement in equilibrium states Figure S2 shows the normalized scattering curves obtained by SLS and SAXS for microgel solutions in the initial swollen state (i.e. in pure H 2 O and in pure MeOH at 10 °C) as well as in the collapsed state at x MeOH = 0.20. The solid lines display the fit to the experimental data obtained by the fuzzy spheres model described by Stieger et. al. (37).
The oscillations are correctly reproducing the data, but there are some additional effects concerning the depth and height of minima and maxima that are not accounted for. We speculate that with the extended q-range we are more sensitive to the resolution function, polydispersity and possible deviations from spherical symmetry. However, these influences are not important because the TR-SAXS data can be fitted without including additional effects. The fit results are summarized in table S1.
The scattered intensities of collapsed PNIPAM in x MeOH = 0.20 obtained from static measurements at the beamline ID02 are displayed in fig. S3. The solid line is the fit to the experimental data obtained by the fuzzy sphere model. In table S1 summarizes the fit results.
The scattering curve exhibit several form factor minima, which are shifted to higher q values in comparison to the position of the minima of PNIPAM in pure H 2 O or MeOH fig. S2, indicating particles of smaller sizes and with a narrow size distribution. The scattering intensity decays with q -4 in the high q regime indicating a sharp particle surface similar to compact spheres. The fuzziness of the particle surface described by σ surf is very small for PNIPAM in x MeOH = 0.20 (table S1) indicating that the surface chains are collapsed and shows that the microgel has the structure of a sphere with homogeneous density profile. This small σ surf value is in good agreement with those reported by Stieger et al. (37), when PNIPAM microgel in H 2 O are in the shrunken state at high temperatures (39 °C). On the other hand, σ surf values of PNIPAM dispersed in pure solvents are larger due to the fuzzy character of swollen microgels. For better characterization (see Stieger et al. (37)) is defined here as the radius with half of the excess electron density compared to the core. The total microgel radius R T in x MeOH = 0.20 is less than half of its size in the pure solvents. section S1.6. Stopped-flow experiments with TR-SAXS SAXS experiments were collected at the High Brilliance beamline "ID02 -Time-Resolved Ultra Small-Angle X-Ray Scattering" at the European Synchrotron Radiation Facility (ESRF) in Grenoble, France, using 12.4 keV photons (corresponding to a wavelength of λ = 0.1 nm). The Xray scattering intensity was recorded using a FReLoN (Fast-Readout Low-Noise) Kodak CCD detector. A detector binning of 2 x 2 was used. The sample-detector distance was set to 8 m covering a q-range of approximately 0.01 -0.2 nm −1 .
Static SAXS measurements were performed in a flow-through capillary cell of 1.88 mm diameter to obtain the scattering patterns of the initial and final state of our microgels at 10 °C. The samples were thermally stabilized at 10 °C. The scattering patterns obtained for the microgels at the initial and final state as well the background solutions (H 2 O, MeOH and x MeOH = 0.20) were averaged over 20 frames to improve statistics.
Time-resolved scattering patterns were collected using a Bio-Logic SFM-400 stopped-flow device with four syringes (48). Samples were stored in a fridge at a temperature of ∼ 8 °C. The microgel solutions in pure solvent (either H 2 O or MeOH) were injected from syringe 4, the cononsolvent from the third one and water from the first and second syringes to clean the stopped-flow cell between shots and to calibrate the scattering intensity to absolute scale. The reservoir for the four syringes and stopped-flow cell were kept at 10 °C by circulating water around the reservoir and observation chamber. The observation chamber consisted of the stoppedflow cell (quartz capillary with wall thickness of approximately 10 µm and diameter 1.35 mm) enclosed in an aluminium holder. The volume mixing ratios were calculated based on the initial and the final solvent composition of the PNIPAM dispersion. The final concentration of the PNIPAM dispersion in x MeOH = 0.20 was set to 0.40 wt%. The stopped-flow cell was homogeneously filled by injecting a total volume of ∼ 350 µL of the mixed solution. The total flow rates for the jump from pure water and pure methanol were 7.04 mL s -1 and 8.54 mL s -1 respectively. PNIPAM dispersion in pure solvent was mixed turbulently with the cononsolvent and injected for at least 50 ms continuously into the flow path of the stopped-flow device corresponding to a steady state condition. X-ray data acquisition was triggered directly at 50 ms and later according to the time steps. The sample age of the mixed solution during this steady state was mainly determined by the necessary time to flow from the mixer inside the stopped-flow device to the point of X-ray exposure. The transfer time from the last mixer to the capillary cell was estimated to be approximately 3.00 ms. The effective exposure time for each frame was 1.5 ms, i.e., each scattering curve was integrated over 1.5 ms. Consequently, the first scattering curve describes the kinetics at a time of 5 ms. The minimum detector readout time during the experiment was 320 ms (2 x 2 binning), hence a stroboscopic data strategy was applied to access points in time well before 320 ms. This was achieved by varying the dead time before the first Xray exposure, i.e. a delay time of 40 ms, 50 ms, 55 ms, 60 ms, 70 ms etc. was introduced. Each acquisition was repeated at least three times to improve the statistics and to verify the reproducibility. Subsequently, all curves were averaged for each point in time.
The time axis for the measurement data was calculated with the following equation where is the dead time of the stopped-flow device, is the onset time of the frame, the exposure time or lifetime of the beam and the mixing time for the purge to reach steady state.
Measurements of the size of the microgels before and after passing the mixer showed no difference, thus the microgels are not destroyed when passing the mixer. section S1.7. Form factor model comparison For the analysis of the diffraction patterns obtained through time-resolved SAXS (TR-SAXS), a form factor model is necessary. There are various form factor models which differ in complexity available in literature. For microgels with a constant excess electron density profile which smoothly drops at the edge (36), a fuzzy sphere model is applied. For microgels with a more complex density profile a core-shell fuzzy sphere model (37) is used. This core-shell fuzzy sphere model allows for two sections with different excess electron densities and smooth transitions. For the fit of our TR-SAXS measurement data we tried the models of different complexity for the different states during the microgel collapse. Figure S4 shows a comparison of the form factor models fitted to the TR-SAXS measurement at 5 ms.
section S1.8. Modelling of the Scattering Curves Scattering data are an intensity distribution I(q) as a function of momentum transfer q = (4π/λ)sinθ, where 2θ is the scattering angle. For data on absolute scale, it is usual to express the scattering intensity distribution in terms of the differential scattering cross section . It can for a very dilute dispersion of monodisperse and spherical particles without concentration effects be expressed as where n is the number density of microgel particles, ∆ρ SAXS is the excess scattering length density between polymer and solvent, V pol is the 'dry' volume of the polymer in a particle and P(q) is the normalized scattering form factor (P(q=0) = 1) of the particle.
The number density of microgel particles n can be calculated from the mass fraction of microgel in the sample c as follows where is apparent specific density of the polymer and ρ solvent is the density of the solvent.
For SAXS the excess scattering length density is proportional to the electron density difference of the polymer , and of the solvent , .
The electron density (electrons per unit of volume) of the polymer can be calculated from its apparent specific density as follows and similarly for the solvent , . = . 7 0 : where Z i and M i denote, respectively, the number of electrons and molecular weight of species i, N A is Avogadro's number, and ϕ MeOH is the weight fraction of methanol in the solvent.
The size polydispersity of particle has been considered in the modelling and fitting of the data. It is described by a normalized Gaussian number distribution as with 〈 〉 describing the average particle radius and σ poly denoting the relative particle size polydispersity.
Thus, the differential scattering cross section is expressed as For the SAXS data of the large particle investigated in the present work, there are significantly instrumental smearing effects and these have to be taken into account. It is done by introducing a resolution function which describes the probability distribution of the actual scattering vectors q for a given nominal scattering vector 〈q〉. The parameters of the resolution function can be estimated from the geometry of the setup, the wavelength distribution and the detector resolution. The instrument resolution for the beamline ID02 is approximately described by a Gaussian function as follows where the width of the instrument smearing is σ =0.0015nm -1 .
The model intensity is thus described by (S10) Different form factor functions P(q,R) have been used to fit the scattering curves, depending on in which state is the PNIPAM microgel, as will be described below.

section 1.9. Modelling and fitting of the microgel form factor in the initial and final states
Scattering data for PNIPAM microgels have been successfully described by a fuzzy sphere model (37). The smooth decay of the density profile of the microgel at its periphery arises from an inhomogeneous distribution of cross-linker within the particle. This is modeled in real space as a convolution of the radial scattering length distribution of a compact sphere with a Gaussian to get a function with a gradual drop-off in scattering length density generating the fuzziness of the particle surface, described by the form factor P(Q) where A(q) defines the amplitude of the form factor P(Q). R and σ surf are the two adjustable parameters, representing the radius of the particle where the scattering length density profile decreased to half the core density and the width of the smeared particle surface, respectively. The core of the microgel that exhibit a higher degree of cross-linking density is described by the radial box profile extending to a radius of about R box = R -2σ surf . In dilute solution, the profile approaches zero at R T = R + 2σ surf . Thus, the overall size of the particle is approximated given by R T . A small number of chains reaching outside the particle will contribute only to the hydrodynamics of the particle, and therefore the size obtained by scattering methods is, expected to be smaller than the hydrodynamic radius R H determined by DLS.
To account for polymer-like scattering due internal structure of the microgel layer, a Lorentzian function is added to P(Q) in Eq. S11. The average correlation length in the network is described by ξ and I L (0) is the value of the intensity arising from fluctuations for q → 0. Finally, a constant background I back is added to correct the residual incoherent scattering.
Incorporating all above-mentioned contributions, the model expression for the scattering intensity distribution is given by As described above, the time-resolved series data have a bump at higher q, which was impossible to fit with the simple core-shell model proposed by Berndt, Pedersen and Richtering (36). We ascribe the bump to internal density fluctuations as one has in the core-shell fuzzy sphere model with density fluctuations (63). An empirical term of L &K n A \ A X ⁄ was included with a scale factor and R g taken as a fit parameter.
In this model, the density profile is defined in terms of piecewise parabola as is described in Ref.
(36). The density profile is based on a profile with a constant density in the center in a region up to r = W and a decay of the outer surface, which is determined by σ. The radial density profile ∆ρ(r) of a particle with such a graded surface is expressed by the half-height radius R = W + σ.
Thus, the inner interface is given by R in = W core + σ in ; and the outer interface by R T = W core + 2σ in Size polydispersity of the outer radius R T (Eq. S7), and instrumental smearing (Eq. S9) were included, however the Lorentzian function I L (q) that described the internal polymer scattering was omitted as the q-range and data at high q do not allow inclusion of this term.
An initial model-independent analysis using an approach similar to that described in (65) showed that the profiles in the time-resolved series all have a low density in the core of the particles and a higher density closer to the surface. Therefore, this was the initial guess for the profiles.
The final homogeneous state was first fitted on absolute scale using a contrast factor estimated from the partial density of PNIPAM in water and calculation of the electron density of the 20 mol % of MeOH mixture. This allowed the particle number density to be determined and this was kept fixed in all the fits of the time-resolved series. With this, the actual excess electron densities of the particles in the time-resolved series could be determined in units of electrons per cubic Angstrom.
The model contains a high number of fit parameters and this gave some instabilities of the fits. Therefore, it was decided to fix some of the parameters in the model to reasonable values.
Precipitation polymerization of NIPAM leads to microgels with very narrow size distribution (<10 %) and fixing the relative polydispersity of the size to 5 % give a reasonable smearing of the minima in the intensity expression as a function of q and it is also in agreement with the value determined by dynamic light scattering. The surface smearing (σ out ) was fixed to 3 nm, which is below the resolution limit of the data. We note that usually collapsed states of microgels have sharp surfaces so therefore it is reasonable to keep it fixed at a low value (37). We also experienced that the fits were not very sensitive to the electron density in the core of the particles and the width of the interface (σ in ) between the expanded core and the collapsed outer shell. Therefore, we kept the width of the interface (σ in ) fixed at 40 nm. And the core density was determined by fitting to the data at late stages of the time-resolved measured (at about 2 s when making jumps from pure MeOH and at about 3 s when jumping from pure water) and fixed in the fits of the time series.
In practice, an automated fitting procedure was used in which the results at a given step is used as initial values in the next step. The data were fitted in the direction from late stages to early stage.
The range of the data from 0.01 to 0.12 nm -1 was used, which seems to have good signal to noise ratio.
The model was fitted to the experimental data using a least-square routine minimizing the reduced The mixing ratio was calculated based on the initial and the final solvent composition. Each kinetics measurement corresponds to an average of 5-10 runs to improve statistics and to ensure reproducibility. For all measurements, the times were corrected for t dead and the t pre-trigger .
The turbidity τ is defined as (68) (S13) where I 0 is the intensity of the incident light, I t the transmitted intensity and l the length of the optical path. The increase of temperature inside the SF cuvette upon H 2 O and MeOH mixing was measured experimentally (as will be explained in detailed below) and estimated numerically using Eq. (S14)

TR-turbidity of collapse transition kinetics of PNIPAM microgels
for the heat capacity C p definition (70) (S14) with heat capacity C p in J/mol K and excess enthalpy mixing H E in J/mol. The increase of the temperature inside the TC-100/10T cuvette before the measurement and during the mixing of MeOH and H 2 O was performed by using a cable sensor Pt100 installed to a digital thermometer (fig. S14). The change in temperature with time was recorded using ESM-4450 software (from National Instruments Labview TM ) installed to the PC of the stopped-flow device.
The temperature in the thermostat was set to 10 °C. The reservoir chambers for syringe 2 and 3 were filled with MeOH and H 2 O, respectively. Both liquids were let there for 30 min in order to ensure that they have reached the temperature of 10 °C. Afterwards, the liquids were pushed up from the reservoirs to the mixer and to the cuvette. The mixture flowed through the cuvette for more than 40 s.
As displayed in fig. S14 and fig. S16 the temperature inside the cuvette was around 11.5 °C at the time t = 0 s (before mixing). This means the temperature was 1.5 °C higher inside the cuvette than set in the water bath of the thermostat coupled to the stopped-flow device (temperature 10 °C).
When the liquids were pushed up from the syringes to the mixer and to the cuvette, the temperature increased until a maximum temperature was reached and was kept at this value for a few seconds, before it decreased with a slower rate to the initial value. The increase in temperature inside the cuvette was around 11.3 K, when H 2 O and MeOH were mixed to get a final solvent composition of 20 mol % MeOH, which is very close to the theoretical predictions that we have made by assuming an adiabatic process without friction. The same procedure was performed for different H 2 O/MeOH mixtures. The resulting temperature changes inside the cuvette are summarized in fig. S15. section S1.16. Temperature dependent size of the PNIPAM microgel as determined by DLS As mentioned in the previous chapter the enthalpy of mixing can lead to a temperature rise, which could have an effect on the microgel collapse transition. Therefore, we adapted the starting temperature of the system, such that an increase of the temperature inside the stopped-flow cell will not affect significantly the volume change of the microgel. Figure S17 displays the size of the microgel (hydrodynamic radius) as function of temperature in the pure solvents and at x MeOH = 0.2 . The figure shows a temperature increase of ca. 10 K which does not affect the initial size and hardly the final state, thus we can conclude that the microgel collapse is mainly determined by the change of the solvent composition.

section S2. Computer simulation
We apply a hybrid simulation approach, combining Molecular Dynamics Simulations (MD) for microgel particles with Multiparticle Collision Dynamics (MPC) simulations for the embedding fluid.

section 2.1. Multiparticle collision dynamics approach
MPC is a mesoscale hydrodynamic simulation approach for fluids (49,72,73). Thereby, the fluid is presented by 7 point particles of mass p distributed in a cubic simulation box of length q with periodic boundary conditions. The dynamics of the fluid particles proceeds by subsequent streaming a collision steps. In the streaming step, MPC particles move ballistically within the time interval ℎ s ( + ℎ) = s ( ) + ℎt ( ) (S15) where s and t are the position and velocity of particle u. In the collision step, the solvent particles are sorted in cubic collision cells of length v. Subsequently, the relative velocity of a particle, with respect to the center-of-mass velocity of the cell t , is rotated by a fixed angle α around a randomly oriented axis, i.e.

t ( + ℎ) = t ( ) + w(x)(t ( ) − t ( ))
(S16) where w(x) is the rotation matrix. For each collision step, a random shift of the entire collision grid is applied in order to ensure Galilean invariance of the system (74).

section 2.2. Molecular dynamics simulations: Microgel model
Microgel particles are represented as a regular network of polymers of length 7 , which are tetrafunctionally crosslinked. The monomers are modeled as pointlike particles, each of mass 9.
They are connected through the harmonic potential where ~ is the spring constant and } is the equilibrum bond length. The interactions between nonbonded points is described by the Lennard-Jones potential (LJ) where ‚ is the quenching depth, F is the dameter of the monomers, ‰ Š is the distance between monomer u and •, ‰ is the cutoff distance and ˆ= 4‚$(F ‰ ⁄ ) %H − (F ‰ ⁄ ) † '. To study the collapse dynamics, microgels are first simulated in good solvent conditions, where ‰ = 2 %/ † F, untill reaching equilibrium and then the cutoff distance is set to ‰ = 2.5F to model the poor solvent condition. The coupling between the MPC fluid and the monomers is achieved in the collision step (49). Newton's equations of motion for the microgel are integrated by the velocity Verlet algorithm with a time step ∆ , which is smaller than the collision-time step ℎ.
The size of the microgel is characterized by its radius of gyration " , which is defined as

Effect of cononsolvent transport
The cononsolvent transport into the microgel is described in a coarse-grained manner. Thereby, an imaginary sphere, centered at the center of mass of the microgel, is assumed around the microgel. Then, the radius of the sphere is reduced with a speed ¦ and all the monomers outside the sphere are marked as attractive (poor solvent condition), while the monomers inside the sphere are treated as being in good solvent condition. The collapse speed of the microgels ¦ " is determined by measuring the slope of the collapse curves (cf. fig. S19(a)) at the time, where the radius of gyration radius meets the condition " ( ) = " (0)/2. Figure S19(b) shows the normalized collapse speed of a microgel versus the normalized transport speed. It should be noted that ¦ m is the intrinsic collapse speed of the microgel for infinitely fast transport of the cononsolvent into the microgel (i.e., the quenching depth, ‚, instantaneously changes for all monomers); the transport depends on the network structure and the quenching depth. Figure S19 suggests that if the transport of cononsolvent particles is faster that the intrinsic collapse speed of the microgel (i.e., ¦ /¦ m > 1), then the initial fast collapse regime becomes independent of the diffusion process. However, if the transport speed of cononsolvent particles is slower that the intrinsic collapse speed of the microgel (i.e. ¦ /¦ m < 1), the microgel collapse speed is defined by the speed of the cononsolvent transport.
section 2.6. Evolution of structure during microgel collapse for low quenching depth The radial distribution function of monomers during the microgel collapse for a low quenching depth is shown in fig. S20(a). The figure, together with snapshots in fig. S20(b), show the structural evolution from early stages of the collapse, where small clusters are formed near the crosslinking sites ( = 100 ), to the formation of a hollow core-shell-like structure (i.e., = 400 ), which, at the end, turns into a collapsed globule ( = 2000 ). To characterize the coreshell-like structure, we determine the total radius § ( ) of the microgel and the thickness of its core ¨ ( ), where these quantities are defined as the maximum and minimum radius at which the radial density distribution assumes half of the value of its maximum. The thickness of the shell is correspondingly defined as ¨ © ( ) = § ( ) − ¨ ( ). The thickness analysis (inset) shows that the structural changes in the core region of the microgel continue while the overall size of the particle does not show a considerable change anymore ( / > 1250). This could be due to problems in estimating the width of the resolution function, polydispersity, and also due to possible minor deviations from spherical symmetry, which are not included in the model.  for the radial monomer density distribution for a microgel with 7 = 20 and ε = 1.5. The inset shows the time T , W/a R T /a evolution of the outer radius ± ² , shell thickness ³´µ ¶·· , and core thickness ³¸² s ¶ , respectively. (b) Snapshots of microgel conformations (thin slice through the center is shown on the right) at various times during the simulation.