Influence of Cross-Linking Degree on Hydrodynamic Behavior and Stimulus-Sensitivity of Derivatives of Branched Polyethyleneimine

Cross-linked derivatives of acylated branched polyethyleneimine containing 2-isopropyl-2-oxazoline units were investigated in chloroform and aqueous solutions using methods of molecular hydrodynamics, static and dynamic light scattering, and turbidity. The studied samples differed by the cross-linker content. The solubility of the polyethyleneimines studied worsened with the increasing mole fraction of the cross-linker. Cross-linked polyethyleneimines were characterized by small dimensions in comparison with linear analogs; the increase in the cross-linker content leads to a growth of intramolecular density. At low temperatures, the aqueous solutions of investigated samples were molecularly dispersed, and the large aggregates were formed due to the dehydration of oxazoline units and the formation of intermolecular hydrogen bonds. For the cross-linked polyethyleneimines, the phase separation temperatures were lower than that for linear and star-shaped poly-2-isopropyl-2-oxazolines. The low critical solution temperature of the solutions of studied polymers decreased with the increasing cross-linker mole fraction. The time of establishment of the constant characteristics of the studied solutions after the jump-like change in temperature reaches 3000 s, which is at least two times longer than for linear polymers.


Introduction
Thermosensitive polymers have become ever more interesting objects for study because of the wide range of their application in various fields, especially in medicine and biotechnology. For example, they are used as components of drug delivery systems and membranes, diagnostics agents, tissue engineering, etc. [1][2][3][4]. During the recent years, the potential of using thermosensitive polymers of complex architecture in medicine has been demonstrated [5,6]. Of particular interest are stimulus-sensitive systems based on branched polyethyleneimines (PEI).
Polyethyleneimines are widely used in biology and medicine due to the high density of amino groups. Moreover, branched representatives of this chemical class, more compact in comparison with linear analogs, are favored as an effective and affordable nonviral gene delivery vector [7][8][9][10][11], as well as an additive to increase the efficiency of the polymerase chain reaction [12] and DNA degradation protection [13]. Besides, branched PEIs are also used as a matrix, stabilizer, and molecular glue in obtaining metal nanoparticles and metal oxides, semiconductor nanoparticles, and carbon nanotubes [14][15][16][17][18][19][20]. the solvent using Specman ACD/Labs (Advanced Chemistry Development, Inc., Toronto, ON, Canada). For dialysis, CellaSep dialysis bags with MWCO 3500 D (Orange Scientific, Braine-l'Alleud, Belgium) were used.
The acetylated derivative of branched polyethyleneimine (PEI-0), namely, poly-N-isobutyroylethyleneimine, was obtained by acylating polyethyleneimine with isobutyroyl chloride under the conditions of the Einhorn reaction (with methylene chloride solvent and triethylamine acceptor). Cross-linking was performed by adding the calculated amount of hexamethylene diisocyanate as previously described [35]. The synthesis scheme for partially cross-linked polyethyleneimines PEI-n is shown in Figure 1. The obtained samples differed in the cross-linking degree, namely, the molar fraction of the added cross-linker w. The w values for prepared samples are listed in Table 1. In Figure 2, 1H NMR spectrum and signal assignment for poly-N-isobutyroylethyleneimine partially cross-linked by 1,6-hexamethylene diisocyanate (sample PEI-3 in Table 1) are shown as an example.
Polymers 2020, 12, x FOR PEER REVIEW 3 of 18 Canada). For dialysis, CellaSep dialysis bags with MWCO 3500 D (Orange Scientific, Braine-l'Alleud, Belgium) were used. The acetylated derivative of branched polyethyleneimine (PEI-0), namely, poly-N-isobutyroylethyleneimine, was obtained by acylating polyethyleneimine with isobutyroyl chloride under the conditions of the Einhorn reaction (with methylene chloride solvent and triethylamine acceptor). Cross-linking was performed by adding the calculated amount of hexamethylene diisocyanate as previously described [35]. The synthesis scheme for partially cross-linked polyethyleneimines PEI-n is shown in Figure 1. The obtained samples differed in the cross-linking degree, namely, the molar fraction of the added cross-linker w. The w values for prepared samples are listed in Table 1. In Figure  2, 1H NMR spectrum and signal assignment for poly-N-isobutyroylethyleneimine partially crosslinked by 1,6-hexamethylene diisocyanate (sample PEI-3 in Table 1) are shown as an example.    The acetylated derivative of branched polyethyleneimine (PEI-0), namely, poly-N-isobutyroylethyleneimine, was obtained by acylating polyethyleneimine with isobutyroyl chloride under the conditions of the Einhorn reaction (with methylene chloride solvent and triethylamine acceptor). Cross-linking was performed by adding the calculated amount of hexamethylene diisocyanate as previously described [35]. The synthesis scheme for partially cross-linked polyethyleneimines PEI-n is shown in Figure 1. The obtained samples differed in the cross-linking degree, namely, the molar fraction of the added cross-linker w. The w values for prepared samples are listed in Table 1. In Figure  2, 1H NMR spectrum and signal assignment for poly-N-isobutyroylethyleneimine partially crosslinked by 1,6-hexamethylene diisocyanate (sample PEI-3 in Table 1) are shown as an example.

Determination of Hydrodynamic Characteristics
MMs of the synthesized samples were determined previously [35] by the method of static light scattering in dichloromethane (density ρ 0 = 1.33 g cm −3 , dynamic viscosity η 0 = 4.4 × 10 −3 Poise, and refractive index n 0 = 1.424) for the targeted PEI-0 and chloroform (ρ 0 = 1.48 g cm −3 , η 0 = 5.7 × 10 −3 Poise, and n 0 = 1.442) for partially cross-linked PEI-n. Before measurements, the solutions and solvents were filtered through the Millipore syringe filter (Merck KGaA, Darmstadt, Germany) with the pore diameter of 0.20 µ. The intrinsic viscosity [η] was measured with the Ostwald-type Cannon-Manning capillary viscometer (Cannon Instrument Company Inc., State College, PA, USA) at 21 • C. To control the solution temperature, a thermostat with the T-100 temperature control unit (Grant, Cambridge UK) was used. The solvent efflux time was 43.4 s for dichloromethane and 48.7 s for chloroform. Dependencies of reduced viscosity η sp /c on polymer concentration c ( Figure 3) were analyzed using the Huggins equation where k is the Huggins constant.
Polymers 2020, 12, x FOR PEER REVIEW 4 of 18 MMs of the synthesized samples were determined previously [35] by the method of static light scattering in dichloromethane (density 0 = 1.33 g cm -3 , dynamic viscosity 0 = 4.4 ×10 −3 Poise, and refractive index n0 = 1.424) for the targeted PEI-0 and chloroform (0 = 1.48 g cm −3 , 0 = 5.7 ×10 -3 Poise, and n0 = 1.442) for partially cross-linked PEI-n. Before measurements, the solutions and solvents were filtered through the Millipore syringe filter (Merck KGaA, Darmstadt, Germany) with the pore diameter of 0.20 μ. The intrinsic viscosity [η] was measured with the Ostwald-type Cannon-Manning capillary viscometer (Cannon Instrument Company Inc., State College, PA, USA) at 21 °C. To control the solution temperature, a thermostat with the T-100 temperature control unit (Grant, Cambridge UK) was used. The solvent efflux time was 43.4 s for dichloromethane and 48.7 s for chloroform. Dependencies of reduced viscosity ηsp/c on polymer concentration c ( Figure 3) were analyzed using the Huggins equation where k′ is the Huggins constant. Hydrodynamic radii Rh-D were determined by dynamic light scattering using the Photocor Complex E instrument (Photocor Instruments Inc., Moscow, Russia) equipped with the Photocor-DL diode laser (wavelength λ = 635.5 nm), Photoсor-FC correlator with 288 channels, Photoсor-BS device for light backscattering study, and the Photoсor-PD detector for measuring the transmitted light intensity. The autocorrelation function was measured with Photocor Software (Photocor Instruments Inc., Moscow, Russia) and processed with DynaLS soft (ver. 8.2.3, SoftScientific, Tirat Carmel, Israel). Solutions of PEI-0 and PEI-n were unimodal over the entire concentration range studied ( Figure 4). For all samples, the hydrodynamic radii Rh-D(c) of scattering objects that were determined at concentration c decreased with dilution ( Figure 5). The hydrodynamic radii Rh-D of macromolecules (Table  1) were obtained by linear extrapolation to zero concentration. Hydrodynamic radii R h-D were determined by dynamic light scattering using the Photocor Complex E instrument (Photocor Instruments Inc., Moscow, Russia) equipped with the Photocor-DL diode laser (wavelength λ = 635.5 nm), Photocor-FC correlator with 288 channels, Photocor-BS device for light backscattering study, and the Photocor-PD detector for measuring the transmitted light intensity. The autocorrelation function was measured with Photocor Software (Photocor Instruments Inc., Moscow, Russia) and processed with DynaLS soft (ver. 8.2.3, SoftScientific, Tirat Carmel, Israel). Solutions of PEI-0 and PEI-n were unimodal over the entire concentration range studied ( Figure 4). For all samples, the hydrodynamic radii R h-D (c) of scattering objects that were determined at concentration c decreased with dilution ( Figure 5). The hydrodynamic radii R h-D of macromolecules (Table 1) were obtained by linear extrapolation to zero concentration.
The values for the refractive index increment dn/dc were determined based on the concentration dependence slope of dn = n s -n 0 of the refractive indices n s for the solution of concentration c with n 0 for solvent ( Figure 6). The refractive indices n s and n 0 were measured with an RA-620 refractometer (KEM, Tokyo, Japan). As seen from Table 1, the values of dn/dc are somewhat decreased with the growth of w. Thus, increased cross-linker content results in decreasing the refractive index increment, although this change is not significant.   The values for the refractive index increment dn/dc were determined based on the concentration dependence slope of dn = ns -n0 of the refractive indices ns for the solution of concentration c with n0 for solvent ( Figure 6). The refractive indices ns and n0 were measured with an RA-620 refractometer (KEM, Tokyo, Japan). As seen from Table 1, the values of dn/dc are somewhat decreased with the growth of w. Thus, increased cross-linker content results in decreasing the refractive index increment, although this change is not significant.    The values for the refractive index increment dn/dc were determined based on the concentration dependence slope of dn = ns -n0 of the refractive indices ns for the solution of concentration c with n0 for solvent ( Figure 6). The refractive indices ns and n0 were measured with an RA-620 refractometer (KEM, Tokyo, Japan). As seen from Table 1, the values of dn/dc are somewhat decreased with the growth of w. Thus, increased cross-linker content results in decreasing the refractive index increment, although this change is not significant. The self-organization of PEI-n molecules in aqueous and water-salt solutions on heating was studied by light scattering, light backscattering, and turbidimetry using the Photocor setup described above. The temperature T was varied discretely with a step from 0.5 to 2 °C and maintained with an accuracy of 0.1 °C. The solution concentrations c varied in the range of over 10-fold, from 0.0035 to 0.0380 g cm −3 , and their acidity varied in the range of pH = 2 -8 at c = 0.0280 g cm -3 . The pH of the initial solution was changed by adding 1 N HCl or 1 N NaOH. The solutions were filtered through hydrophilic PTFE Millipore (Merck KGaA, Darmstadt, Germany) membrane filters with the pore di- The self-organization of PEI-n molecules in aqueous and water-salt solutions on heating was studied by light scattering, light backscattering, and turbidimetry using the Photocor setup described Polymers 2020, 12, 1085 6 of 18 above. The temperature T was varied discretely with a step from 0.5 to 2 • C and maintained with an accuracy of 0.1 • C. The solution concentrations c varied in the range of over 10-fold, from 0.0035 to 0.0380 g cm −3 , and their acidity varied in the range of pH = 2 -8 at c = 0.0280 g cm −3 . The pH of the initial solution was changed by adding 1 N HCl or 1 N NaOH. The solutions were filtered through hydrophilic PTFE Millipore (Merck KGaA, Darmstadt, Germany) membrane filters with the pore diameter of 0.20 µ.
The measurement procedure was as follows. After the target temperature was established, the changes in the intensity of the scattered light I and the optical transmission I* in time were analyzed. In this case, I was measured at the scattering angle of 90 • . As a criterion for determining whether the solution has reached the 'equilibrium' state, the constancy of I and I* in time was chosen. The times t eq for establishing the constant characteristics of the solutions were determined accordingly ( Figure 7). In 'equilibrium' conditions, in addition to I and I*, the hydrodynamic radii of the R h particles present in the solutions were determined. The light scattering measurements were carried out in the range of scattering angles θ from 45 • to 135 • to confirm the diffusion nature of the modes and determine the extrapolated values of R h . The relaxation time τ of a correlation function was measured at scattering angles 45 • , 90 • , and 135 • . The values of scattering wave vector q were calculated via equation (2) Polymers 2020, 12, x FOR PEER REVIEW 7 of 18   The magnitudes of translational diffusion coefficient D 0 were obtained from dependencies of inverse relaxation time 1/τ on the squared scattering wave vector ( Figure 8).

Results
The Stokes-Einstein equation was used for the calculation of hydrodynamic radius values where k is the Boltzmann constant and T is the absolute temperature.

Structure and Hydrodynamic Behavior of PEI-n in Dilute Chloroform Solution
To determine the conditions for the preparation of poly-isobutyroylethylenimine with a certain degree of acylation, branched PEI was acylated by isobutyroyl chloride within different synthetic methods as well as with the different ratios of reagents. The best results were obtained using the Einhorn reaction [36], which at a given ratio of reagents allowed obtaining a product with a predictable degree of substitution. Diisocyanate was chosen as a cross-linker, keeping in mind the fact that isocyanates react quantitatively with both primary and secondary amino groups at ambient temperature and are relatively resistant to water at room or lower temperature [37]. Additionally, it should be noted that the complete removal of water from PEI and its acylation products is quite problematic.
The interaction of acylated PEI-0 with hexamethylene diisocyanate leads to the formation of both intramolecular and intermolecular cross-links. At low diisocyanate content in the reaction mixture, intramolecular cross-linking is predominantly observed, which manifests itself in the compaction of a macromolecule without significant MM increase. This is exactly what happens with PEI-n samples ( Table 1). In this case, intramolecular cycles are formed in macromolecules, whose number increases with an increase in the cross-linker content. The prevalence of intramolecular cross-linking can be explained by a smaller change in entropy as compared to intermolecular cross-linking. Given the

Structure and Hydrodynamic Behavior of PEI-n in Dilute Chloroform Solution
To determine the conditions for the preparation of poly-isobutyroylethylenimine with a certain degree of acylation, branched PEI was acylated by isobutyroyl chloride within different synthetic methods as well as with the different ratios of reagents. The best results were obtained using the Einhorn reaction [36], which at a given ratio of reagents allowed obtaining a product with a predictable degree of substitution. Diisocyanate was chosen as a cross-linker, keeping in mind the fact that isocyanates react quantitatively with both primary and secondary amino groups at ambient temperature and are relatively resistant to water at room or lower temperature [37]. Additionally, it should be noted that the complete removal of water from PEI and its acylation products is quite problematic.
The interaction of acylated PEI-0 with hexamethylene diisocyanate leads to the formation of both intramolecular and intermolecular cross-links. At low diisocyanate content in the reaction mixture, intramolecular cross-linking is predominantly observed, which manifests itself in the compaction of a macromolecule without significant MM increase. This is exactly what happens with PEI-n samples ( Table 1). In this case, intramolecular cycles are formed in macromolecules, whose number increases with an increase in the cross-linker content. The prevalence of intramolecular cross-linking can be explained by a smaller change in entropy as compared to intermolecular cross-linking. Given the equality of changes in enthalpy for intra-and intermolecular reactions, all of the above leads to a still smaller change in the Gibbs free energy, which determines the direction of the process. With an increase in the content of a bifunctional cross-linker in the initial reaction mixture, the probability of intermolecular interactions increases, leading to the formation of large particles with a subsequent loss of polymer solubility.
An adequate interpretation of the results obtained in the study of stimulus sensitivity is impossible without comprehensive information on the characteristics of individual macromolecules. Accordingly, an important research objective was to establish the hydrodynamic characteristics of PEI-n and to analyze the conformation of their molecules.
As seen from Table 1, the refractive index increments dn/dc decrease slightly with increasing w, although this change is not very significant. A decrease in dn/dc is yet another confirmation that the cross-linker fraction becomes larger, and cross-linking occurs at the macromolecule level primarily. Indeed, the refractive indices of hexamethylene diisocyanate and branched PEI are equal to 1.453 and 1.529, respectively. Therefore, if the cross-linking is intramolecular, an increase in the cross-linker fraction should lead to a decrease in the refractive index increment, which we observe experimentally. As known, the Huggins constant characterizes the polymer-solvent hydrodynamic interaction and the hydrodynamic behavior of solutions [38][39][40]. For the PEI-0 and PEI-n under question, k values lie in the range from 0.7 to 1.5 without changing systematically with the variation of w. These values are higher than the usual Higgins constant for linear polymers in good solvents. Elevated k values are often obtained for polymers with complex architecture, for example, for hyperbranched and star-shaped polymers, as well as for molecular brushes in the region of low MM [41][42][43][44][45]. In particular, for a branched PEI, k = 0.7 -0.8 [46]. It can be assumed that the described behavior of k is explained by the compact size and symmetric shape of macromolecules of polymers with complex architecture.
In organic solvents for the studied samples of PEI-0 and PEI-n, low values of intrinsic viscosity [η] were obtained (Table 1). It is typical for polymers with elevated intramolecular density, such as dendrimers, hyperbranched polymers, polymer stars, and low molar mass polymer brushes with high density of grafting of side chains [41,[43][44][45][47][48][49][50][51][52]. Note that at the corresponding MM, the characteristic viscosities of PEI-0 and PEI-n solutions are close to [η] for branched PEI [46,53]. As the cross-linker mole fraction w increases, the intrinsic viscosity of PEI-n decreases, reflecting an increase in intramolecular density. This change is similar to the decrease in [η] at increasing the branching degree and branching functionality in hyperbranched polymers or the arm number in star-shaped polymers [48,[54][55][56]. In the case of the studied polymers, the change in intrinsic viscosity is probably caused by an increase in the number of intramolecular cycles, as well as in hyperbranched polymethylsilsesquioxanes [57]. At passage from PEI-0 to PEI-n, the [η] value decreases by about 20%. Therefore, given that, at first approximation, [η] is inversely proportional to the macromolecule density, we can conclude that the latter characteristic also changes by 20%.
To describe the hydrodynamic behavior of hyperbranched macromolecules, the rotation ellipsoid model with slight shape asymmetry is often used [41,47]. The greater the degree of branching of hyperbranched polymers, the better this model describes their hydrodynamic properties. Similarly, the presence of cycles in a branched molecule and an increase in their number brings the shape of the molecule closer to spherical. The volume of a revolution ellipsoid with a closely similar axis length is well proportional to the cube of the average axis length. Therefore, in terms of the model under discussion, it can be expected that the linear dimensions of PEI-n molecules will change by only 6% with an increase in the cross-linker fraction from 0 to 30 mol %. Therefore, it does not seem surprising that the values of the hydrodynamic radius R h-D determined by the dynamic light scattering method are independent of w ( Figure 9). A possible change in R h-D lies within the experimental measurement error of this characteristic. It was found for the so-called viscosity hydrodynamic radius Rh-η, whose values were calculated from the values of the intrinsic viscosity [η] using the Einstein equation, behaves in much the same way ( Figure 9): which can easily yield 1/3 Figure 9. Dependencies of translation hydrodynamic radius R h-D and viscosity hydrodynamic radius R h-η on cross-linker content w for the studied PEI-0 and PEI-n.
It was found for the so-called viscosity hydrodynamic radius R h-η , whose values were calculated from the values of the intrinsic viscosity [η] using the Einstein equation, behaves in much the same way ( Figure 9): [η] = 2.5v (5) which can easily yield where v is the specific partial volume and N A is the Avagadro number. Note that for the studied polymers, the diffusion radius R h-D is 1.3-1.9 times larger than the viscous size R h-η without a systematic change with increasing w. A similar difference is observed quite often both for linear systems [38] and for polymers of complex architecture [41,56,58]. This is because the principle of dimensional equivalence is not fully satisfied during the translational and rotational motion of the macromolecule. Roughly speaking, the molecule 'flows' differently in terms of translational diffusion and viscosity. For linear polymers, it results in different values of the Kuhn segment length A obtained using the data of viscometry (A η ) and translational friction (A f ). For example, for poly-2-ethyl-2-oxazoline, the difference between A η and A f is 30% [59].
The small values of the radii R h-D and R h-η confirm the conclusions about the compact size of PEI-n molecules as compared to linear polymers of the same MM. Such values of hydrodynamic radii at the corresponding MM are characteristic of hyperbranched polymers and even dendrimers [41,[58][59][60][61][62][63]. The small size and symmetrical shape of PEI-n molecules is also evidenced by the values of the hydrodynamic invariant A 0 (Table 1), calculated by Equation (7) [54,64,65]: For linear macromolecules, A 0 is constant over a wide MM range. The average experimental values are A 0 = 3.2 × 10 −10 erg K −1 mol −1/3 for flexible chain polymers and 3.8 × 10 −10 erg K −1 mol −1/3 for rigid chain polymers [38,64] and are in good agreement with the theoretical values of A 0 [38]. On the other hand, for the low molar mass samples of the flexible chain thermosensitive poly-2-ethyl-2-oxazoline, low values of A 0~2 .9 × 10 −10 erg·K −1 ·mol −1/3 were obtained [59].
In Figure 10, the values of A 0 for the PEI-0 and PEI-n are presented as functions of w. The experimental points are rather widely spread, but there is no systematic change in A 0 depending on w. The average value A 0 = (1.9 ± 0.2) × 10 −10 erg K −1 mol −1/3 is noticeably smaller than the theoretical value for the hard sphere (2.88 × 10 −10 erg K −1 mol −1/3 ). Reduced A 0 was previously found for polymers whose molecules have an increased density and shape approaching spherical, namely, for hyperbranched polymers and dendrimers [41,58,66].

Behavior of PEI-n in Aqueous Solution
The solubility of partially cross-linked branched PEI-n in water depended on the cross-linker content, deteriorating sharply with increasing w. It took about three days to dissolve a sample with w = 18.5 mol % with stirring at the temperature of T = 10 °C, and a sample with w = 21.7 mol % Figure 10. Dependence of hydrodynamic invariant A 0 on cross-linker content w for PEI-0 and PEI-n.

Behavior of PEI-n in Aqueous Solution
The solubility of partially cross-linked branched PEI-n in water depended on the cross-linker content, deteriorating sharply with increasing w. It took about three days to dissolve a sample with w = 18.5 mol % with stirring at the temperature of T = 10 • C, and a sample with w = 21.7 mol % completely dissolved under the same conditions in five days. At w ≥ 26.8 mol %, PEI-n did not dissolve within a month at T = 6 -10 • C. Note that a decrease in the solubility of PEI-n was also observed in chloroform, which was manifested in the decrease of the second virial coefficient A 2 from 0.3 × 10 −3 cm 3 mol g −2 for PEI-1 to −0.2 × 10 −3 cm 3 mol g −2 for PEI-4. The decreased solubility is due to the increase in the number of intramolecular cycles with increasing w, which leads, as mentioned above, to the loss of solubility at w > 35 mol %. A similar phenomenon was observed for hyperbranched polymers [57,67,68].
At low temperatures (T = 10 • C) in aqueous solutions of PEI-1 and PEI-2, only one particle type was detected by dynamic light scattering, which had the hydrodynamic radius R f close to the hydrodynamic size R h-D of macromolecules determined in the organic solvent (Figures 9 and 11). Therefore, aqueous solutions of PEI-n were molecularly dispersed, which is typical of stimulus-sensitive polymers that do not contain large hydrophobic fragments [69][70][71][72][73][74]. Moreover, for both samples in aqueous solutions, the radius R f with dilution changed only within the experimental error ( Figure 11) in contrast to solutions in chloroform. On the other hand, a decrease in R f was observed with increasing cross-linker content.
The difference in R f values for PEI-1 and PEI-2 samples was about 20%.

Behavior of PEI-n in Aqueous Solution
The solubility of partially cross-linked branched PEI-n in water depended on the cross-linker content, deteriorating sharply with increasing w. It took about three days to dissolve a sample with w = 18.5 mol % with stirring at the temperature of T = 10 °C, and a sample with w = 21.7 mol % completely dissolved under the same conditions in five days. At w  26.8 mol %, PEI-n did not dissolve within a month at T = 6 -10 °C. Note that a decrease in the solubility of PEI-n was also observed in chloroform, which was manifested in the decrease of the second virial coefficient A2 from 0.3 × 10 -3 cm 3 mol g -2 for PEI-1 to -0.2 × 10 -3 cm 3 mol g -2 for PEI-4. The decreased solubility is due to the increase in the number of intramolecular cycles with increasing w, which leads, as mentioned above, to the loss of solubility at w > 35 mol %. A similar phenomenon was observed for hyperbranched polymers [57,67,68].
At low temperatures (T = 10 °C) in aqueous solutions of PEI-1 and PEI-2, only one particle type was detected by dynamic light scattering, which had the hydrodynamic radius Rf close to the hydrodynamic size Rh-D of macromolecules determined in the organic solvent (Figures 9 and 11). Therefore, aqueous solutions of PEI-n were molecularly dispersed, which is typical of stimulus-sensitive polymers that do not contain large hydrophobic fragments [69][70][71][72][73][74]. Moreover, for both samples in aqueous solutions, the radius Rf with dilution changed only within the experimental error ( Figure 11) in contrast to solutions in chloroform. On the other hand, a decrease in Rf was observed with increasing cross-linker content. The difference in Rf values for PEI-1 and PEI-2 samples was about 20%. Figure 11. The concentration dependence of hydrodynamic radius Rf for aqueous solutions of PEI-n samples at 10 °C. Figure 11. The concentration dependence of hydrodynamic radius R f for aqueous solutions of PEI-n samples at 10 • C.
On heating, the phase separation of PEI-n solutions is observed, which manifests itself in a sharp increase in the intensity of scattered light I and drop of optical transmittance I* at temperature T 1 (Figure 12). At temperature T 2 , the optical transmission falls to zero, i.e., T 2 can be considered as the temperature of phase separation completion determined by turbidimetric data. The solutions of PEI-1 and PEI-2 remained molecularly dispersed up to the temperature of the onset of phase separation T 1 when large aggregates with hydrodynamic radius R s were formed in the solution (Figure 13). The phase separation mechanism in PEI-n aqueous solutions involved the dehydration of 2-isopropyl-2-oxazoline units and the formation of intermolecular hydrogen bonds. This led to the formation of aggregates resulting from the association of macromolecules, which, near T 1 , cannot be observed by dynamic light scattering. With further heating, the aggregate size first increases, and then, it begins to decrease near T 2 , which is probably due to the compaction of PEI-n molecules. Note that a decrease in the molecule size in the vicinity of the phase transition can be observed experimentally for very high-molar-mass samples of thermosensitive polymers [75]. oxazoline units and the formation of intermolecular hydrogen bonds. This led to the formation of aggregates resulting from the association of macromolecules, which, near T1, cannot be observed by dynamic light scattering. With further heating, the aggregate size first increases, and then, it begins to decrease near T2, which is probably due to the compaction of PEI-n molecules. Note that a decrease in the molecule size in the vicinity of the phase transition can be observed experimentally for very high-molar-mass samples of thermosensitive polymers [75].  As can be seen from Figure 14, close phase separation temperatures T1 were obtained for the samples studied; however, the nature of their change with concentration is different for PEI-1 and PEI-2. For the sample with lower cross-linker content w, the T1 value is minimal at c = 0.03 g cm -3 , indicating that LCST was found close to 15 °С. For PEI-2, the temperature T1 monotonously decreases with increasing concentration, and in this case, LCST is probably lower than that for PEI-1. Therefore, a rise in the cross-linker content leads to a decrease in LCST. Note that for both samples studied, the phase separation mechanism in PEI-n aqueous solutions involved the dehydration of 2-isopropyl-2oxazoline units and the formation of intermolecular hydrogen bonds. This led to the formation of aggregates resulting from the association of macromolecules, which, near T1, cannot be observed by dynamic light scattering. With further heating, the aggregate size first increases, and then, it begins to decrease near T2, which is probably due to the compaction of PEI-n molecules. Note that a decrease in the molecule size in the vicinity of the phase transition can be observed experimentally for very high-molar-mass samples of thermosensitive polymers [75].  As can be seen from Figure 14, close phase separation temperatures T1 were obtained for the samples studied; however, the nature of their change with concentration is different for PEI-1 and PEI-2. For the sample with lower cross-linker content w, the T1 value is minimal at c = 0.03 g cm -3 , indicating that LCST was found close to 15 °С. For PEI-2, the temperature T1 monotonously decreases with increasing concentration, and in this case, LCST is probably lower than that for PEI-1. Therefore, a rise in the cross-linker content leads to a decrease in LCST. Note that for both samples studied, the Figure 13. Dependencies of radii R f (1, 3) and R s (2, 4) on temperature T for PEI-1 (1, 2) and PEI-2 (3,4) solutions with concentration c = 0.0280 g cm −3 .
As can be seen from Figure 14, close phase separation temperatures T 1 were obtained for the samples studied; however, the nature of their change with concentration is different for PEI-1 and PEI-2. For the sample with lower cross-linker content w, the T 1 value is minimal at c = 0.03 g cm −3 , indicating that LCST was found close to 15 • C. For PEI-2, the temperature T 1 monotonously decreases with increasing concentration, and in this case, LCST is probably lower than that for PEI-1. Therefore, a rise in the cross-linker content leads to a decrease in LCST. Note that for both samples studied, the phase separation temperatures were noticeably lower than LCST for poly-2-isopropyl-2-oxazoline, which is close to 37 • C for linear and maximum 25 • C for star-shaped polymers [76][77][78][79][80]. A decrease in phase separation temperatures for polymers of complex architecture in comparison with linear analogs is observed quite often [3], but there is no reliable explanation for this behavior yet. The effect of the cross-linker content is also observed when analyzing the width of the phase separation interval, namely, the values ∆T = T 2 − T 1 . The ∆T difference varies slightly with dilution for PEI-1, while a strong dependence of ∆T on c was found for PEI-2 ( Figure 15). In the region of low concentrations, the width of the phase separation interval for PEI-2 is accordingly 10 • C higher than ∆T for PEI-1.
in phase separation temperatures for polymers of complex architecture in comparison with linear analogs is observed quite often [3], but there is no reliable explanation for this behavior yet. The effect of the cross-linker content is also observed when analyzing the width of the phase separation interval, namely, the values T = T2 -T1. The T difference varies slightly with dilution for PEI-1, while a strong dependence of T on c was found for PEI-2 ( Figure 15). In the region of low concentrations, the width of the phase separation interval for PEI-2 is accordingly 10 °С higher than T for PEI-1.  As is known, polyethylene imines are polybases. However, no effect of medium acidity on the studied samples was found. In the pH range from 2.0 to 8.1 at low temperatures (T < 10 °С), PEI-1 solutions are molecularly dispersed, and the hydrodynamic radius Rf is independent of pH within the experimental error. In the same way, no systematic change in the phase separation temperatures was observed with varying pH, although the spread in the T1 values was quite significant, reaching 2.5 °С. Apparently, the number of the remaining unmodified amino groups is too small to lead to conformational rearrangements with varying medium acidity and significantly affects the self-organization of PEI-n molecules.
An important feature of the stimulus-sensitive material is the changing rate of its characteristics after exposure. In the case of thermosensitive polymer solutions, this feature is reflected in the time teq that is necessary for the characteristics of the solution to reach constant values after a jump-like change in temperature (Figure 7). For the studied polymers, teq have the maximum value tmax near the which is close to 37 °С for linear and maximum 25 °С for star-shaped polymers [76][77][78][79][80]. A decrease in phase separation temperatures for polymers of complex architecture in comparison with linear analogs is observed quite often [3], but there is no reliable explanation for this behavior yet. The effect of the cross-linker content is also observed when analyzing the width of the phase separation interval, namely, the values T = T2 -T1. The T difference varies slightly with dilution for PEI-1, while a strong dependence of T on c was found for PEI-2 ( Figure 15). In the region of low concentrations, the width of the phase separation interval for PEI-2 is accordingly 10 °С higher than T for PEI-1.  As is known, polyethylene imines are polybases. However, no effect of medium acidity on the studied samples was found. In the pH range from 2.0 to 8.1 at low temperatures (T < 10 °С), PEI-1 solutions are molecularly dispersed, and the hydrodynamic radius Rf is independent of pH within the experimental error. In the same way, no systematic change in the phase separation temperatures was observed with varying pH, although the spread in the T1 values was quite significant, reaching 2.5 °С. Apparently, the number of the remaining unmodified amino groups is too small to lead to conformational rearrangements with varying medium acidity and significantly affects the self-organization of PEI-n molecules.
An important feature of the stimulus-sensitive material is the changing rate of its characteristics after exposure. In the case of thermosensitive polymer solutions, this feature is reflected in the time teq that is necessary for the characteristics of the solution to reach constant values after a jump-like change in temperature (Figure 7). For the studied polymers, teq have the maximum value tmax near the As is known, polyethylene imines are polybases. However, no effect of medium acidity on the studied samples was found. In the pH range from 2.0 to 8.1 at low temperatures (T < 10 • C), PEI-1 solutions are molecularly dispersed, and the hydrodynamic radius R f is independent of pH within the experimental error. In the same way, no systematic change in the phase separation temperatures was observed with varying pH, although the spread in the T 1 values was quite significant, reaching 2.5 • C. Apparently, the number of the remaining unmodified amino groups is too small to lead to conformational rearrangements with varying medium acidity and significantly affects the self-organization of PEI-n molecules.
An important feature of the stimulus-sensitive material is the changing rate of its characteristics after exposure. In the case of thermosensitive polymer solutions, this feature is reflected in the time t eq that is necessary for the characteristics of the solution to reach constant values after a jump-like change in temperature (Figure 7). For the studied polymers, t eq have the maximum value t max near the temperature of the phase separation onset T 1 , followed by t eq decrease on heating ( Figure 16). Therefore, it can be assumed that the time t eq for PEI-n solutions is determined by the duration of the aggregate formation, which dominates near temperature T 1 . Note that a similar pattern in the behavior of t eq at T > T 1 was previously observed by us for star-shaped poly-2-alkyl-2-oxazolines [81]. temperature of the phase separation onset T1, followed by teq decrease on heating ( Figure 16). Therefore, it can be assumed that the time teq for PEI-n solutions is determined by the duration of the aggregate formation, which dominates near temperature T1. Note that a similar pattern in the behavior of teq at T > T1 was previously observed by us for star-shaped poly-2-alkyl-2-oxazolines [81]. In Figure 17, the tmax values are plotted versus concentration. It can be seen that the maximum value of the time required to establish the 'equilibrium' state of the system for both samples does not change upon dilution. The average values tmax = (2800  300) s for PEI-1 and tmax = (3300  300) s for PEI-2 coincide within the error range, i.e., the duration of the processes does not depend on the fraction of the cross-linker w. As for the absolute values of tmax, they are significantly, sometimes by an order of magnitude, lower than the corresponding characteristic for star-shaped poly-2-alkyl-2-oxazolines and grafted copolymers with side chains of poly-2-alkyl-oxazolines [81,82]. For linear thermosensitive polymers, in most cases, the time teq does not exceed 2000 s [70,[83][84][85][86]. The increase in teq during the passage from linear polymers to polymers with complex architecture can be explained by a growth of intramolecular density [81]. For example, the increased density of the hydrophilic outer layer of star-shaped macromolecules prevents the hydrophobic cores from interacting with each other, slowing down the aggregation process. In branched polymers, hydrophobic fragments are more evenly distributed over the macromolecule volume, which may be the reason for the decrease in teq for the studied PEI-n as compared to star polymers.  In Figure 17, the t max values are plotted versus concentration. It can be seen that the maximum value of the time required to establish the 'equilibrium' state of the system for both samples does not change upon dilution. The average values t max = (2800 ± 300) s for PEI-1 and t max = (3300 ± 300) s for PEI-2 coincide within the error range, i.e., the duration of the processes does not depend on the fraction of the cross-linker w. As for the absolute values of t max , they are significantly, sometimes by an order of magnitude, lower than the corresponding characteristic for star-shaped poly-2-alkyl-2-oxazolines and grafted copolymers with side chains of poly-2-alkyl-oxazolines [81,82]. For linear thermosensitive polymers, in most cases, the time t eq does not exceed 2000 s [70,[83][84][85][86]. The increase in t eq during the passage from linear polymers to polymers with complex architecture can be explained by a growth of intramolecular density [81]. For example, the increased density of the hydrophilic outer layer of star-shaped macromolecules prevents the hydrophobic cores from interacting with each other, slowing down the aggregation process. In branched polymers, hydrophobic fragments are more evenly distributed over the macromolecule volume, which may be the reason for the decrease in t eq for the studied PEI-n as compared to star polymers.

Conclusions
Polymers 2020, 12, x FOR PEER REVIEW 13 of 18 temperature of the phase separation onset T1, followed by teq decrease on heating ( Figure 16). Therefore, it can be assumed that the time teq for PEI-n solutions is determined by the duration of the aggregate formation, which dominates near temperature T1. Note that a similar pattern in the behavior of teq at T > T1 was previously observed by us for star-shaped poly-2-alkyl-2-oxazolines [81]. In Figure 17, the tmax values are plotted versus concentration. It can be seen that the maximum value of the time required to establish the 'equilibrium' state of the system for both samples does not change upon dilution. The average values tmax = (2800  300) s for PEI-1 and tmax = (3300  300) s for PEI-2 coincide within the error range, i.e., the duration of the processes does not depend on the fraction of the cross-linker w. As for the absolute values of tmax, they are significantly, sometimes by an order of magnitude, lower than the corresponding characteristic for star-shaped poly-2-alkyl-2-oxazolines and grafted copolymers with side chains of poly-2-alkyl-oxazolines [81,82]. For linear thermosensitive polymers, in most cases, the time teq does not exceed 2000 s [70,[83][84][85][86]. The increase in teq during the passage from linear polymers to polymers with complex architecture can be explained by a growth of intramolecular density [81]. For example, the increased density of the hydrophilic outer layer of star-shaped macromolecules prevents the hydrophobic cores from interacting with each other, slowing down the aggregation process. In branched polymers, hydrophobic fragments are more evenly distributed over the macromolecule volume, which may be the reason for the decrease in teq for the studied PEI-n as compared to star polymers.

Conclusions
The solution properties of partially cross-linked branched PEI-n in chloroform and water were investigated. In both solvents, the solubility worsening of the studied samples was detected with an increase in the cross-linker content, which is caused by an increase in the number of intramolecular cycles. The hydrodynamic characteristics of PEI-n clearly mirror their elevated intramolecular density in organic solvent. It was shown that the cross-linked PEI-n are more compact and symmetric in shape than linear PEI at similar molar masses. The obtained small values of hydrodynamic radius of macromolecules, intrinsic viscosity, and hydrodynamic invariant are characteristic of hyperbranched polymers. The increase in the cross-linker mole fraction leads to a decrease in magnitudes of the intrinsic viscosity of the PEI-n solutions in chloroform that reflects the growth of intramolecular density.
On heating, the aqueous solutions of PEI-n were molecularly dispersed up to the temperature of the phase separation, and the hydrodynamic radii of macromolecules in water and chloroform coincided essentially. At the phase separation temperature, the large aggregates are formed due to dehydration of 2-isopropyl-2-oxazoline units in the PEI-n molecules and the formation of intermolecular hydrogen bonds. The LCST of the PEI-n solutions decreases with increasing cross-linker mole fraction. Moreover, for the samples studied, the phase separation temperatures are noticeably lower than LCST for the linear and star-shaped poly-2-isopropyl-2-oxazolines. Due to the small number of amino groups in the PEI-n molecules, the influence of medium acidity on the characteristics of aqueous solutions was not found. Unexpectedly, with the same MM, the time to establish the equilibrium characteristics of the solution after a temperature change for cross-linked branched PEI-n is less than that for star-shaped poly-2-alkyl-2-oxazolines, despite that the intramolecular density of polymer stars is lower than for cross-linked and hyperbranched polymers.