Supporting information
Dynamic light scattering: Brief theory and instrumentation
Comprehensive descriptions of scattering theory are available in the literature, e.g. [1]. The normalized intensity correlation function measured in a dynamic light scattering experiment is defined as
wherei, j denote the two detectors and l, m the two intersecting laser beams in 3D configuration,q is themodulus of the scattering vector and the lag time with reference point .For autocorrelation experiment and ; for cross-correlation experiment and . Brackets denote time averaging.Schätzel[2] has shown that the electric field cross-correlation and autocorrelation functions are similar in respect to the single-scattering contributions but higher order scattering is greatly suppressed in the cross-correlation experiment. The difference in the correlation functions of single scattered photons in auto and cross-correlation experiment lies in the maximum amplitude factor , which is 1 for autocorrelation and 0.25 for cross-correlation. The decreased maximum amplitude in 3D-cross-correlation experiment is due to the fact that both detectors can register photons scattered from both intersecting beams and therefore contribute to the baseline of the cross-correlation function. In the turbid samples, where multiple scattering becomes significant, autocorrelation approach fails but cross-correlation scheme produces accurate results.
The intensity autocorrelation functions are converted to electric field autocorrelation functions using the Siegertrelation
where is the amplitude parameter, which depends on the alignment and correlation scheme, and g(1) is the electric field autocorrelation function. The g(1)can be expanded in terms of cumulants
where is the intensity-weighted decay rate of the exponential, and and are expansion factors. The mean diffusion coefficient is obtained from the relationship
where the subscript refers to the value obtained from the second order cumulant fit.
An LS InstrumentsAG supplied goniometer in 3D-configuration with the laser wavelengthof 633 nm was used for the measurements. The two detectors were coupled to a 4 channel ALV 7004 hardware correlator to enable simultaneous recording of auto and cross-correlation data. For our instrument is approximately 0.95 for autocorrelation and 0.11 for 3D-cross-correlation configuration.
Expression for the particle concentration
From the results of scaling law approach for polymer solutions [3] we can expect the PNIPAM chains to collapse into dense globules of constant density. If we assume that all the monomer ends up in particles we then expect to be able to polymerize total amount of collapsed polymer from the given amount of monomer in the batch
Here is the total collapsed volume of all the microgels, is the molar volume of collapsed main monomer, is the collapsed volume of the cross-linker and the fraction of the cross-linker of the total monomer amount . If we assume that the particles have a monomodal and narrow size distribution (so that we can determine the average particle size by DLS reliably), then the number of particles is
where is the mean volume of the collapsed particles and the number of particles.Given that we deal with dispersions, the concentration of the particles is then
We can express the volume of cross-linker units in the polymer by their excess volume , which gives the expression in the form of Eq. 1
/ (1)In the case we don’t lose monomer in side reactions and there are no additional contributions to the volume of the collapsed particles, we would expect the number density of particles to determine the final particle volume.
Particle homogeneity
Reaction kinetics
Final particle volume
As discussed earlier we expect the number of the particles to be the parameter, which determines the final particle volume in accordance to Eq. 1. Combining this expression with the empirical Eq. 7, we arrive at an empirical expression describing the number concentration of the particles. If we choose to work with relative quantities so that the molar volume of the collapsed network is excluded from the terms A and B then this expression is
/ (8)Figure S 7A shows the behavior of Eq. 8 in the case of constant A’-to-B’-ratio, analogous to the synthesis of constant monomer-to-initiator ratio at different temperatures. Figure S 7B shows the corresponding final particle volumes. The number concentration goes through a maximum and then decreases leading to the characteristic deviations from the linear dependence of volume on monomer concentration, discussed in the context of Figure 6B.Figure S 8A and B show the number concentration function and final particle size for variable A’-to-B’-ratio, respectively. The constant B’ term translates to constant intercept in Figure S 8B.
Final particle size data
References
1. Lindner P, Zemb T (2002) Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter. Amsterdam: North Holland Delta Series
2. Schätzel K (1991) Suppression of Multiple Scattering by Photon Cross-correlation Techniques. J Mod Optic 38:1849–1865.
3. Rubinstein M, Colby RH (2003) Polymer Physics. Oxford University Press