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
𝑔𝑖𝑗 (2) (𝑞, 𝜏) ≝
⟨𝐼𝑖 𝑙 (𝑞, 𝜏0 )𝐼𝑗 𝑚 (𝑞, 𝜏)⟩
⟨𝐼𝑖 𝑙 (𝑞)⟩⟨𝐼𝑗 𝑚 (𝑞)⟩
where i, j denote the two detectors and l, m the two intersecting laser beams in 3D configuration, q is
the modulus of the scattering vector and 𝜏 the lag time with reference point 𝜏0 . For autocorrelation
experiment 𝑖 = 𝑗 and 𝑙 = 𝑚; for cross-correlation experiment 𝑖 ≠ 𝑗 and 𝑙 ≠ 𝑚. 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 𝛽,
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 Siegert relation
1
(1)
1
𝛽 2 𝑔𝑖𝑗 (𝑞, 𝜏) = [𝑔𝑖𝑗 (2) (𝑞, 𝜏) − 1]2
where 𝛽 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
1
1
(1)
ln(𝛽 2 𝑔𝑖𝑗 ) = ln (𝛽 2 ) − Γ̅𝜏 +
𝜇2 2 𝜇3 3
𝜏 − 𝜏 +⋯
2
6
where Γ̅ is the intensity-weighted decay rate of the exponential, and 𝜇2 and 𝜇3 are expansion factors.
̅ is obtained from the relationship
The mean diffusion coefficient 𝐷
̅𝑞2
Γ̅2 = 𝐷
where the subscript refers to the value obtained from the second order cumulant fit.
An LS Instruments AG supplied goniometer in 3D-configuration with the laser wavelength of 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 𝛽 is
approximately 0.95 for autocorrelation and 0.11 for 3D-cross-correlation configuration.
2,0
G (ms-1)
1,5
1,0
0,5
0,0
0
1x10-4
2x10-4
2
3x10-4
4x10-4
-2
q (nm )
Fig S 1 2nd order decay rate with q2 squared for six batches polymerized at 60°C for 15 h, measured at
50°C
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
𝑉𝑀𝐺,𝑡𝑜𝑡 = [𝑉𝑚,𝑁 (1 − 𝑥𝑐 ) + 𝑉𝑚,𝑐 𝑥𝑐 ]𝑛𝑡𝑜𝑡
Here 𝑉𝑀𝐺,𝑡𝑜𝑡 is the total collapsed volume of all the microgels, 𝑉𝑚,𝑁 is the molar volume of collapsed
main monomer, 𝑉𝑚,𝑐 is the collapsed volume of the cross-linker and 𝑥𝑐 the fraction of the cross-linker
of the total monomer amount 𝑛𝑡𝑜𝑡 . 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
𝑉𝑀𝐺,𝑡𝑜𝑡
[𝑉𝑚,𝑁 (1 − 𝑥𝑐 ) + 𝑉𝑚,𝑐 𝑥𝑐 ]
= 𝑁𝑝 =
𝑛𝑡𝑜𝑡
⟨𝑉𝑀𝐺 ⟩
⟨𝑉𝑀𝐺 ⟩
where ⟨𝑉𝑀𝐺 ⟩ is the mean volume of the collapsed particles and 𝑁𝑝 the number of particles. Given that
we deal with dispersions, the concentration of the particles 𝜂𝑝 is then
𝑁𝑝
[𝑉𝑚,𝑁 (1 − 𝑥𝑐 ) + 𝑉𝑚,𝑐 𝑥𝑐 ]
= 𝜂𝑝 =
𝑐𝑡𝑜𝑡
⟨𝑉𝑀𝐺 ⟩
𝑉
⟨𝑉𝑀𝐺 ⟩ =
[𝑉𝑚,𝑁 (1 − 𝑥𝑐 ) + 𝑉𝑚,𝑐 𝑥𝑐 ]
𝑐𝑡𝑜𝑡
𝜂𝑝
We can express the volume of cross-linker units in the polymer by their excess volume Δ𝑉𝑚 = 𝑉𝑚,𝑁 −
𝑉𝑚,𝑐 , which gives the expression in the form of Eq. 1
⟨𝑉𝑀𝐺 ⟩ =
𝑉𝑚,𝑁 + Δ𝑉𝑚 𝑥𝑐
𝑐𝑡𝑜𝑡
𝜂𝑝
(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
102
Intensity (a.u.)
101
100
10-1
10-2
10-3
0,000
0,005
0,010
0,015
0,020
0,025
q (nm-1)
Fig S 2 Effect of the cross-linker fraction on polydispersity. Form factors of batches with total monomer
concentration of 66 mM synthesized at 50 °C. Mole fraction of monomer to initiator is 42.
Characterization in swollen state at 20 °C. From top down: 𝒙𝑩 = 𝟎. 𝟎𝟐𝟓, 𝒙𝑩 = 𝟎. 𝟎𝟓𝟎 and 𝒙𝑩 = 𝟎. 𝟎𝟕𝟓.
Error bars are typically in the size range of the symbols.
102
Intensity (a.u.)
101
100
10-1
10-2
10-3
0,000
0,005
0,010
0,015
0,020
0,025
-1
q (nm )
Fig S 3 Effect of the reaction temperature on polydispersity. Form factors of batches with total
monomer concentration of 66 mM, monomer to initiator mole fraction of 42 and cross-linker fraction of
𝒙𝑩 = 𝟎. 𝟎𝟓𝟎. Characterization in swollen state at 20 °C. From top down: 70°C, 60°C and 50°C. Error
bars are typically in the size range of the symbols.
102
Intensity (a.u.)
101
100
10-1
10-2
0,000
0,005
0,010
0,015
0,020
0,025
-1
q (nm )
Fig S 4 Form factors of uncross-linked collapsed particles prepared at different temperatures. Total
monomer concentration 66 mM, monomer to initiator mole fraction of 42. Characterization in collapsed
state at 50 °C. From top down: 70°C, 60°C and 50°C. Error bars are typically in the size range of the
symbols.
e5
e4
ln(Intensity)
e3
e2
e1
e0
e-1
0,0000
0,0001
2
0,0002
0,0003
-2
q (nm )
Fig S 5 Guinier plots of uncross-linked collapsed particles prepared at different temperatures. Total
monomer concentration 66 mM, monomer to initiator mole fraction of 42. Characterization in collapsed
state at 50 °C. From top down: 70°C, 60°C and 50°C. Error bars are typically in the size range of the
symbols.
Reaction kinetics
50
Conversion (%)
40
30
20
10
0
60
80
100
120
140
160
180
200
220
240
Time (min)
Fig S 6 Conversion with time for DLS-0.0-50°C-1m-1/2i with and without SDS. Reaction temperature
50 °C.
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
𝟏
𝑩′
= 𝑨′[𝑴] +
[𝑴]
𝜼𝒑
(8)
Figure S 7 A 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 7 B 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 6 B. Figure S 8 A 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 8 B.
A' = 0,5; B' = 0,1
A' = 1,0; B' = 0,2
A' = 2,0; B' = 0,4
A' = 4,0; B' = 0,8
8
Volume
Number concentration
2
A' = 0,5; B' = 0,1
A' = 1,0; B' = 0,2
A' = 2,0; B' = 0,4
A' = 4,0; B' = 0,8
10
1
6
4
2
0
0
0,0
0,5
1,0
1,5
2,0
2,5
3,0
0,0
0,5
1,0
[M]
1,5
2,0
2,5
3,0
[M]
Fig S 7 A) Number density functions calculated from Eq. 8 with constant A’-to-B’-ratio. B) Final
particle volumes corresponding to the number density functions.
6
4
5
4
Volume
Number concentration
A' = 0,1; B' = 0,1
A' = 0,2; B' = 0,1
A' = 0,4; B' = 0,1
A' = 0,8; B' = 0,1
A' = 0,1; B' = 0,1
A' = 0,2; B' = 0,1
A' = 0,4; B' = 0,1
A' = 0,8; B' = 0,1
3
2
2
1
0
0
0,0
0,5
1,0
1,5
2,0
2,5
3,0
[M]
0,0
0,5
1,0
1,5
2,0
2,5
3,0
M
Fig S 8 A) Number density functions calculated from Eq. 8 with variable A’-to-B’-ratio. B) Final
particle volumes corresponding to the number density functions.
Relative particle concentration
2,5
2,0
1,5
1,0
0,5
60 ºC
60 ºC constant ionic strength
0,0
0
1
2
3
4
5
6
[M] / [M]min
Fig S 9 Particle concentration relative to the first point in series with constant and non-constant ionic
strength (Figure 22b) calculated from the A’ and B’ parameters according to Eq. 8.
Final particle size data
Table S 1 Synthesis temperature, reagent concentrations and hydrodynamic radii for final particle size
determinations.
Relative
concentration
1
0.5
0.33
0.25
0.166
0.125
0.083
1
0.5
0.33
0.25
0.166
0.125
0.083
1
0.5
0.33
0.25
0.166
0.125
0.083
1
0.5
0.33
0.25
0.166
0.125
0.083
1
0.5
0.33
0.25
0.166
0.125
0.083
1
0.5
0.33
0.25
0.166
0.125
0.083
1
0.5
0.33
0.25
0.166
0.125
0.083
T
[NIPAM]
°C
50
50
50
50
50
50
50
60
60
60
60
60
60
60
70
70
70
70
70
70
70
80
80
80
80
80
80
80
60
60
60
60
60
60
60
60
60
60
60
60
60
60
60
60
60
60
60
60
60
mol dm-3
1.30E-01
6.51E-02
4.29E-02
3.25E-02
2.16E-02
1.63E-02
1.08E-02
1.30E-01
6.51E-02
4.29E-02
3.25E-02
2.16E-02
1.63E-02
1.08E-02
1.30E-01
6.51E-02
4.29E-02
3.25E-02
2.16E-02
1.63E-02
1.08E-02
1.30E-01
6.51E-02
4.29E-02
3.25E-02
2.16E-02
1.63E-02
1.08E-02
1.31E-01
6.53E-02
4.31E-02
3.26E-02
2.17E-02
1.63E-02
1.08E-02
1.31E-01
6.55E-02
4.32E-02
3.28E-02
2.17E-02
1.64E-02
1.09E-02
1.30E-01
6.52E-02
4.30E-02
3.26E-02
2.16E-02
1.63E-02
1.08E-02
[NIPAM]/
[KPS]
41
41
41
41
41
41
41
41
41
41
41
41
41
41
41
41
41
41
41
41
41
41
41
41
41
41
41
41
21
21
21
21
21
21
21
85
85
85
85
85
85
85
440
440
440
440
440
440
440
Rh
Std.
Volume
Std.
nm
-203.3
162.3
151.8
132.8
124.4
118.6
-166.3
132.5
114.2
97.6
90.2
85.3
-126.7
100.5
89.0
74.4
70.1
64.8
-103.6
82.8
69.1
62.0
57.3
52.4
-166.7
131.7
105.7
97.8
87.6
80.9
-155.8
125.5
112.8
100.4
111.6
86.1
-162.0
135.3
125.0
107.9
97.3
86.1
nm
-2.7
1.8
1.5
0.9
0.5
1.4
-1.4
1.2
0.6
0.6
0.4
0.4
-1.1
0.6
0.7
0.3
0.5
0.3
-0.7
0.4
0.3
0.3
0.3
0.3
-1.5
0.9
1.3
0.7
0.4
0.4
-1.3
0.9
0.8
1.2
0.6
0.4
-1.0
1.3
0.6
0.7
0.7
0.4
nm3
-3.52E+07
1.79E+07
1.47E+07
9.81E+06
8.06E+06
6.99E+06
-1.93E+07
9.74E+06
6.24E+06
3.89E+06
3.07E+06
2.60E+06
-8.52E+06
4.25E+06
2.95E+06
1.73E+06
1.44E+06
1.14E+06
-4.66E+06
2.38E+06
1.38E+06
9.98E+05
7.88E+05
6.03E+05
-1.94E+07
9.57E+06
4.95E+06
3.92E+06
2.82E+06
2.22E+06
-1.58E+07
8.28E+06
6.01E+06
4.24E+06
5.82E+06
2.67E+06
-1.78E+07
1.04E+07
8.18E+06
5.26E+06
3.86E+06
2.67E+06
nm3
-1.40E+06
5.96E+05
4.34E+05
1.99E+05
9.72E+04
2.47E+05
-4.87E+05
2.65E+05
9.83E+04
7.18E+04
4.09E+04
3.66E+04
-2.22E+05
7.62E+04
6.97E+04
2.09E+04
3.09E+04
1.58E+04
-9.44E+04
3.45E+04
1.80E+04
1.45E+04
1.24E+04
1.04E+04
-5.24E+05
1.96E+05
1.83E+05
8.41E+04
3.86E+04
3.29E+04
-3.97E+05
1.78E+05
1.28E+05
1.52E+05
9.39E+04
3.73E+04
-3.30E+05
2.99E+05
1.18E+05
1.02E+05
8.33E+04
3.73E+04
95% confidence
limits
nm3
-2.75E+06
1.17E+06
8.51E+05
3.91E+05
1.91E+05
4.85E+05
-9.54E+05
5.19E+05
1.93E+05
1.41E+05
8.02E+04
7.17E+04
-4.35E+05
1.49E+05
1.37E+05
4.09E+04
6.05E+04
3.10E+04
-1.85E+05
6.75E+04
3.53E+04
2.84E+04
2.43E+04
2.03E+04
-1.48E-04
9.77E-05
7.40E-05
4.91E-05
3.70E-05
2.46E-05
-7.77E+05
3.49E+05
2.51E+05
2.98E+05
1.84E+05
7.30E+04
-6.46E+05
5.86E+05
2.31E+05
2.01E+05
1.63E+05
7.30E+04
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