Solvent induced phenomena in a dendronized linear polymer

Foregoing work [31] has demonstrated that DPs can be described as wormlike structures. The volume of the wormlike chain (Porod–Kratky) [42] of PG4 is defined by V _w = π R _c ² L _c = 26.1 · 10³ nm³ (case a), where R _{c (SAXS)} = 2.8 nm is the cross-section radius as indicated exemplary in Fig. 1 and obtained by SAXS measurements of the dry material (see Fig. 4). Taking the value of the cross-section radius of PG4 determined by MD simulations [22, 43] R _{c (MD)} = 4.0 nm as the maximum radial extension of the corona segments, i.e., where the radial segment density approaches zero, a value of V _w = 53.2 · 10³ nm³ (case b) is obtained. The latter value would also be consistent with data obtained for solutions in methanol by small angle neutron scattering (SANS) experiments (unpublished results of Sigel RL, Schurtenberger P (2011) University of Fribourg, Switzerland) and therefore reflects the maximum extension of the corona segments swollen to maximal extension with methanol (for SANS). The volume occupied by the wormlike chain in its random flight conformation in the solvent dioxane at room temperature is given by V _c = 4/3 π R _G ³ = 8.8 · 10⁶ nm³. Therefore, the volume filling by the wormlike chain can be approximated by 100 · V _w/V _c that is 0.3 % for case a and 0.6 % for case b. Thus, the degree of volume filling is a factor of at least 10³ larger than typically observed for classical polymers like PMMA or poly(styrene). This indicates that excluded volume effects must play a considerably large role in the definition of the persistence length l _p when comparing samples of largely different L _c. Since we investigate the solution behavior of a single polymer sample, we will not attempt to separate excluded volume effects from other effects which control the overall chain conformation. Nevertheless, it is relevant to point out that generalization of the results of our work to all chain lengths needs a detailed consideration of excluded volume effects and its consequences for the random dimension of the macromolecules. For example when establishing the calibration for size exclusion chromatography, this needs to be carefully considered otherwise erroneous conclusions will result.

At this point, a short discussion of the density ρ of the polymer may be of interest. We like to restrict the discussion to the sample PG4 where data obtained by different techniques are available (vide infra). There is general agreement that spherical dendrimers and dendronized linear polymers exhibit a dense packing of the dendron segments in the interior and that the density decreases at the periphery in the manner of an error function [43, 44]. A detailed description of the density profile based on MD simulations can be found in the most recent literature [22]. However, these simulations lack experimental verification so far.

The value R ₁ defines the radius at which the density of the “hard” core starts to deviate toward the lower density of the dendron segments at the periphery of the corona. This could be probably the radius relevant to describe the bending stiffness of the wormlike molecule (vide infra). R ₂ describes the average radius of the wormlike chain in solution and/or in the solid state where the chain–chain interaction is dominated by the equilibrium between attractive and repulsive forces between different chains and/or segments of the same chain in a random flight conformation. Finally R ₃ relates to the radius of maximal extension of structural elements of the corona at which density differences between solvent and polymer attached dendrons vanish. R ₃ is relevant to scattering experiments, e.g., SAXS or SANS. The radius relevant to describe the self-avoiding walk in the random coil conformation would probably be located between R ₂ and R ₃. If one follows the results of MD simulation [22, 43], the difference ΔR = R ₃ − R ₁ is of the order of 1.4 nm and R ₂ = 3.3 nm for PG4. The density of the hard core volume was found to be ρ = 1.18 g cm⁻³ in these simulations, based on a “united atom” type of approach and did not consider solvent incorporation. As shown later, this value is close to that which we have determined for the density of this polymer dissolved in chloroform.

Furthermore, SAXS reveals for bulk (PG4 obtained by freeze-drying solutions in dioxane) the observation of Bragg peaks typical for hexagonally dense packed cylindrical objects (see Fig. 4) that allows the set-up of a unit cell in which the center-to-center distance between next neighbor cylinders is a = 5.5 nm and therefore R _{c (SAXS)} = 2.8 nm. In addition, the density of the material in a unit cell is given by ρ _x  = M _mon/V _el N _A = 1.43 g cm⁻³ with V _el the volume of the elementary cell. Eventually, one may also consider the density of a cylinder with square box dependence of ρ for R _{c (SAXS)} = 2.8 nm which gives ρ′ _x  = M _mon/π R _c ² d N _A = 1.46 g cm⁻³. One may associate ρ _x with the density of a cylinder with effective radius R ₂ in the dry state. The density of PG4 in solution is accessible measuring the density of solutions at known weight fraction w ₂ of solute. With the approximation that the measured density of the mixture ρ _m = w ₂(ρ ₂ − ρ ₁), where the subscripts 1 and 2 refer to the solvent and solute, respectively, and w ₁ + w ₂ = 1, the density of the DP ρ ₂ in the dissolved state can be estimated for dioxane at room temperature. A value of ρ = 1.21 ± 0.03 g cm⁻³ for dioxane and ρ = 1.11 ± 0.03 g cm⁻³ for chloroform is calculated. In other words, the DPs undergo substantial swelling by dissolution. The bond connectivity along the chain trajectory enforces that the swelling is two-dimensional in nature that is stretching of the dendron segments occurs in radial direction. The density ρ _x refers to the density of the bulk material in the form of hexagonally close packed cylinders and contains the contribution of the interstitial volume. However, ρ′ _x refers to the density of individual cylindrical objects void of any solvent (“dry” state). Therefore ρ′ _x is the proper reference to assess the degree of swelling of the object equilibrated with solvent. The degree of swelling is given by Q = 100 (ρ′ _x /ρ _solvent) = ∼120 % for dioxane and ∼130 % for chloroform accordingly. The swelling induces a growth of R _c from 2.8 to 3.1 nm in dioxane and 3.2 nm in chloroform.

The radial extension of PG4 dissolved in methanol was determined by SANS (unpublished results of Sigel RL, Schurtenberger P (2011) University of Fribourg, Switzerland). Two limiting data sets are available. On one hand, a value of R _{c (SANS)} = 2.6 nm has been determined for PG4, which corresponds to our definition of the hard core radius. On the other hand, a second value R _{c (SANS)} = 4.0 nm was obtained by a more refined analysis of the SANS data (unpublished results of Sigel RL, Schurtenberger P (2011) University of Fribourg, Switzerland) and this value refers to the maximum extension of the methanol-swollen sample PG4. The difference ΔR = R ₃ − R ₁ corresponds to the expectations of MD simulations [22, 43]. Since Q can be considered as characteristic for the thermodynamic quality of the used solvent, we conclude dioxane < chloroform < methanol for the respective solvent quality.

We note that previous work [45] of some of the authors has extensively elaborated what the proper value of the density of the same polymer, namely PG4, might be. The present findings corroborate the assumptions made in the context of interpretation of molecularly resolved images of individual chains that had been weakly absorbed to the surface of mica, highly oriented pyrolytic graphite or amorphous carbon. Zhang et al. [45] give an estimate value of 1.35 ≤ ρ ≤ 1.45 g cm⁻³ for the dry sample and ρ = 1.1 g cm⁻³ for the same polymer in solution. Thus, these data are in fair agreement with the present results and also correspond qualitatively with available MD simulations [22].

The swelling of the corona originates from both enthalpic and entropic effects. The main contribution to the latter is the osmotic pressure of the solvent which acts as a diluent for the segments of the dendrons. However, there are also effects due to the heat of mixing to be expected and one will generally formulate Δg _mix = Δh _mix − TΔs _mix where Δg _mix is the free energy change per dendron of generation G upon dissolution, Δh _mix is the respective enthalpy change (heat of mixing), and Δs _mix is the entropy change. The entropy of mixing has been extensively considered in the recent literature [20, 21]. The consequences in terms of chain stretching that are increase in persistence length in thermodynamically good solvents have been pointed out to the extent that scaling laws have been formulated. These scaling laws do not predict the magnitude of the present experimental findings but merely describe the observed trends. For instance, experimentally one finds substantially different values for l _p for different solvents, e.g., for sample PG4 l _p = 12 nm in methanol and 7.0 nm in dioxane. As mentioned earlier, the magnitude of l _p given here needs to be considered as the lower bound. It should be noted, however, that measurements of the heat of mixing solvents and DPs would be most interesting and could help to understand the role of solvents in the molecular mechanics of the swollen wormlike chain much better.

While computer-based molecular dynamics simulations [22] give reasonable prediction of the density profile normal to the backbone trajectory, they will fail in the prediction of the overall topology of the DP, notably the persistence length. The reason is that the computer simulation considers local relaxation processes of the dendron branches on the time scale 10^-12 ≤ t ≤ 10⁻⁹ s while the global structure of the whole chain is the result of the viscoelastic response of the DP to the thermal fluctuations in the solvent bath in which the polymer is observed by, e.g., DLS, that is the time window of 10^-7 ≤ t ≤ 10³ s.

As shown in Fig. 5, both R _G and R _h reveal a strong T dependence. In the following, the data are discussed according to three temperature regimes (as indicated in Fig. 5) which refer to quite different situations: Regime I refers to homogeneous solutions of molecularly dispersed chains of PG4 at T > T _θ . The θ temperature is identified in the context of this work as the temperature where R _G is maximal. Within the small regime II, which extends between 24 ≤ T ≤ 28 °C, both SLS and DLS (see Fig. 5) gave evidence of the presence of individual molecules over extended periods of time, namely hours to days, while in regime III at T < 24 °C strong turbidity and sedimentation appear as a sign of coagulation and the formation of large aggregates.

Since one can take R _G ² as the characteristic magnitude to describe the dimension of a polymer in solution, the ratio of R _G(T)² over R _G(T _θ )² gives a vivid description of the changes in the chain dimension above and below T _θ . A very strong decrease in the dimension of the individual macromolecules both, above and below T _θ, is observed, albeit for very different reasons.

First we investigate the high-temperature regime I (T > T _θ ). Here, we follow the argument of Flory [18, 47] and extract the temperature coefficient

From the graph shown in Fig. 6, we derive a value of C = −3.8 · 10⁻² K⁻¹ that is a very strongly negative coefficient. Remembering a value of C = −0.37 · 10⁻³ K⁻¹ for poly(styrene), 0.09 K⁻¹ < C < 0.28 · 10⁻³ K⁻¹ for poly(isobutylene), and C = −0.23 · 10⁻³ K^-1 for poly(oxyethylene) [18, 47], the most unusual behavior of PG4 comes to the surface. The latter shows a 100-fold stronger temperature dependence of the coil dimension on T in toluene as compared to conventional linear macromolecules. The reason for this surprising behavior becomes obvious looking to the changes in l _p with T obtained from analysis of the SLS data (see ESM). The persistence length decreases from a value of l _p = 7.0 nm at T = T _θ  = 28 °C to 5.8 nm at T = 45 °C, the highest temperature for which data are available. The data indicate an approximately linear decrease of l _p with T in this limited temperature range with a slope of dl _p/dT = −0.07 nm K⁻¹. These results can also be described in the picture of a wormlike chain in which the number of statistical elements which describe the random flight trajectory of the object changes with T (unlike in classical chains). Accordingly, we can calculate the number of statistical elements N per chain by N = L _c/l _k. N changes from a value of 295 statistical elements per chain at T = 28 °C to 356 at T = 45 °C, which is a change by ca. 20 % over a temperature difference of 17 K. This interpretation is corroborated by the extrapolation of the data for the T dependence of l _p in Fig. 6 to 20 °C, which neglecting the occurrence of the θ behavior at T < T _θ gives a value of ∼7.5 nm which coincides very well with our experimental data found for solution in dioxane. The latter is a thermodynamically good solvent for the segments of the dendrons (vide supra). We conclude that the viscoelastic interactions among the branches of the dendrons that form an entanglement network (see Fig. 1) are dominating the elastic behavior of the wormlike chains in the poor solvent. The solvent itself acts as a plasticizer in this case.

The connection of l _p to the molecular mechanics of the wormlike chain is straight forward. The persistence length l _p reflects the Young’s modulus E of the wormlike chain according to [48]where the Boltzmann k _B is constant and where the coefficient C ₀ according to Odjik [48] refers to the topology of the cylindrical object by C ₀ = (π/64)D ⁴ with D = 2R _c being the diameter of its cross section. In other words, the T dependence of l _p and thereby R _G is interpreted as a strong T dependence of the modulus of the chain since there is no reason to assume changes in the topological term, namely D ⁴ in this temperature regime.

To bring out the temperature dependence of the modulus more clearly, a plot of l _p · T vs T is shown in Fig. 7. The slope indicates a 0.75 (∼1) % linear decrease in modulus per degree in the small temperature range investigated here and under the assumption that C ₀ = constant. Moreover, these data give evidence that fluctuations in the modulus along the chain trajectory must occur since the energy required for bending is only a few k _B T.

In other words, micro-Brownian motion of the macromolecules and bending fluctuations fall into the same spectral window to be seen by light scattering experiments and this situation is reflected by the KWW distribution of relaxation times seen in the DLS spectra (see Figs. 3 and 8). These results also shed light onto the nature of the corona around the polymer backbone which is composed of the dendrons. We like to describe the material in the bulk of the corona as a gel with strongly T-dependent shear modulus. Therefore, what is observed as T dependence of R _G and of l _p is a reflection of the molecular dynamics of the solvent swollen DPs close to the glass transition temperature T _g. The dry sample PG4 showed T _g = ∼70 °C [20, 35]. However, the precise value needs further discussion since the T _g region is unusually broad. Treating the role of the solvent as that of a plasticizer, one can interpret the data in terms of the viscoelastic behavior of a gel cylinder of given molecular dimensions. Close and around T _θ , the modulus of all polymers show a strong T dependence.

The formula of Odijk [48] describes the mechanics of a cylindrical rod of homogeneous elasticity and homogeneous density. This is, of course, an idealization as already pointed out by Odijk himself but helps to define an equivalent object which shows the observed behavior of the real molecule. Moreover, Eq. 3 gives the scaling with diameter at given elastic constant.

For instance, plugging the value of l _p = 7.2 nm found for PG4 in dioxane at T = 25 °C (vide supra) into Eq. 3 and assuming that E/k _B T = 1 gives R _c = 1.74 nm which is in very reasonable agreement with the experimental result discussed above having in mind that the real molecule exhibits a radial dependence of ρ and thereby of the modulus as well. Assuming a value E/k _B T = 2 would decrease the magnitude of R _c to 1.46 nm, that is further away from the experimentally observed value of R _c. For methanol as solvent, a value of l _p = 12 nm has been reported (unpublished results of Sigel RL, Schurtenberger P (2011) University of Fribourg, Switzerland). Again, setting E/k _B T = 1 gives R _c = 1.98 nm in reasonable agreement to what has been observed by SANS measurements (R _{c (SANS)} = 2.6 nm).

The findings suggest that the diameter of the equivalent cylinder with a stiffness as defined by Eq. 3 is approximately 75 % of the radius estimated from SANS or density measurements in solution. This conclusion holds for PG4 and under the further assumption that E/k _B T = 1. Moreover, E is obviously different for different good solvents as reflected by the dependence of l _p on solvent quality. Furthermore, we cannot assume that E is independent on G. As the number of generations of DPs becomes smaller, the molecular object is less well described by a square box radial distribution of density and modulus. In other words, the boundary conditions which are neglected in Eq. 3 become more and more relevant in a manner which would best be covered by simulations and not by analytical formulations as the one given by Odijk [48]. Of course, one can always define an equivalent object which would behave as the molecule under investigations, meaning that l _p can be correlated to virtual cross sections at given values of E/k _B T. However, the interpretation of the meaning of D and its correlation with chemical structure elements will be a task of further work. So far, we have followed the formulation given by Odijk [48] for the case of a stress-free wormlike chain which is in equilibrium with a thermal bath, i.e., dissolved in a θ solvent. It is worth mentioning that Odijk [48] has considered the case of identical chains under stress as well, that is the eigen-modes of a filament under tension leading to periodic undulation of the chains. The wavelength of the undulation depends on the applied tension and the temperature of the bath in which the chains are embedded. The result is of importance to the measurable end-to-end-distance of the extended wormlike chains which is not an independent parameter but depends on the quality of the thermal excitations of the filaments. This aspect has been subject of a number of theoretical and experimental investigations in the context of developing the biomechanics of naturally occurring polymers, e.g., DNA and collagen [49‐51].

While most of the work summarized and analyzed in these references point toward the mechanics of biological macromolecules and biogenic filaments as found in animal tissue, the arguments are helpful in the design of synthetic macromolecules as well. These are available in the form of brush-like and dendronized polymers. In fact, phenomena predicted by theory in the above-mentioned literature have been identified experimentally for polymer brushes by Gunari et al. [52]. The predicted thermally activated undulation effects were revealed measuring force–extension curves on individual macromolecules. Analysis of their data in the light of the above mentioned studies allowed them to determine precise values of l _p for the brush-like polymers. We like to note that the interesting case of filaments composed of mechanically heterogeneous elements has also been treated in terms of a theoretical analysis [53]. Translated into the language of dendronized linear polymers, this would mean a copolymer where the co-monomers differ in the chemical structure and/or generation number of the dendrons attached to the repeat units, cases which to our knowledge have only been realized in a few cases [54, 55].

Regime II of the behavior of PG4 dissolved in toluene is characterized by a dramatic decrease in the average chain dimension over a narrow temperature range from T = T _θ  = 28 °C to about 24 °C (see Figs. 5 and 8). However, both SLS and DLS indicate the presence of molecularly disperse macromolecules in stable solutions without signs of aggregation or sedimentation over days. R _G decreases by 43 % and R _h by 53 % (see Fig. 5). Accordingly, <R _G>/<R _h> assumes a value of ca. 1.5 within this temperature regime, that is a value expected for randomly coiled Kuhn chains. Note that in regime I (T > T _θ) we observed fully expanded wormlike chains of the investigated polymer and the determined value for <R _G>/<R _h> is ∼2, which correlate with the theoretical value for rod-like structures very well. Regime II is characterized by a dramatic decrease in the average chain dimension over a temperature range from T = T _θ  = 28 °C to about 24 °C. Here, <R _G>/<R _h > results in a value of ∼1.5. For linear macromolecules with a further decrease of T (T < T _θ ), a very sharp coil-to-globule transition would be expected. In such a case, <R _G>/<R _h> would be around 0.77. In the present case, the situation for dendronized polymers is somewhat different in so far that we observed a transition range between 28 and 24 °C. Within this range, meta-stable structures are observed (vide infra).

Going to T < 24 °C, the solutions become unstable and within a short time turbidity develops rapidly accompanied by sedimentation of the formed aggregates, in other words, regime III refers to a two-phase situation of a polymer–solvent phase diagram, while in regime II, we observe a fractional desolvation of the dendron corona of PG4. This is more precisely documented by the DLS results shown in Fig. 8. In particular, the dramatic change in C(q, t) is documented going from 25 °C where we still determine fluctuating individual macromolecules to 20 °C where the solution becomes unstable (aggregation). Note the reduction of the amplitude of C(q, t) at T < 24 °C and the appearance of aggregates. The decrease in the chain dimensions with decreasing T without precipitation between 28 and 24 °C and keeping β = const. = 0.9 suggests a meta-stable pearl necklace structure of partially desolvated chains. Structures of locally collapsed chain segments and still solvated stretches of segments have been identified as unstable intermediates in the kinetics of the much studied coil-to-globule transition of linear macromolecules [56, 57]. It occurs close to the θ temperature. In fact, many stages of the transition have been identified by simulation and some of them have been verified by time-resolved studies of the precipitation process as kinetic intermediates preceding the final precipitation in the form of multi-molecular aggregates. In the present case, the situation is substantially different in so far as we hypothesize that the gel-like bulk of the corona of the dendrons at first undergoes a desolvation process which leads to fluctuations of the osmotic pressure along the chain trajectory. Presumably this creates regions of fully solvated dendrons in equilibrium with regions that are fully or partially desolvated as indicated in the schematic Fig. 9. This will change the character of the wormlike chain to that of a segmented chain with strongly solvent (and temperature)-dependent segment length. This also means a fluctuating shear modulus along the trajectory of the chain. At this point, it must be left to further studies to find out what the precise segment length distribution looks like.

The foregoing description is also supported by the experimental observation that the transition around T _θ  = 28 °C can be completely suppressed adding 10 vol% of dioxane to the toluene solution (see Fig. SI-5 in ESM). Dioxane acts as a preferential solvent to the segments of the dendrons and the gel-like character of the corona remains intact even if the toluene becomes a non-solvent at T = T _θ .

Figure 9 describes schematically the situation for the solution of PG4 in pure toluene in terms of a phase diagram. A homogenous solution of wormlike chains corresponding to regime I exists above T _θ , where the dendron corona is homogeneously swollen by the solvent toluene. At T _θ , the intramolecular segregation of dendron segments and solvent starts leading to a locally fluctuating solvent concentration along the trajectory of the macromolecules. Macroscopic phase separation will only occur below 24 °C when the length of the dissolvated regions per macromolecule exceeds a critical value. The process of desolvation is depicted in Fig. 10 and is assumed to occur inside the shaded regime of the phase diagram shown in Fig. 9.

The description given by Fig. 10 bears similarity to what has been extensively discussed for the occurrence of Rayleigh instabilities in thin jets of liquids as well as transition of rod-like filaments or particles to spheres in the solid state [58, 59]. In fact, given the topology and dimensions of DPs, one would expect that Rayleigh instabilities must occur but need to remain incomplete because of the bond connectivity of the polymer main chain. Furthermore, we cannot associate a surface tension vs bulk energy competition as the driving force for the perturbation of the shape. Rather, we invoke a competition between enthalpic and entropic forces acting on the solvent embedded macromolecule. The entropic forces are mainly to be associated to the osmotic pressure of the solvent which likes to swell the bulk of the DPs to the maximum extent while the enthalpic forces refer to the heat of mixing of solvent and branches of the dendrons. The θ temperature would then be understood as T _θ  = ΔH _mix/ΔS _mix. This treatment is common in all theories of the phenomena of the swelling of polymer gels [27]. Below T _θ fluctuations of solvent concentration inside the bulk of the corona of the DP occur, which can also be described as fluctuations of the cross section. This is shown in Fig. 10. The minimum cross-section diameter for PG4 is obviously the one of the dry polymer (2R _{c (SAXS)} = 5.6 nm, vide supra) and the maximum diameter must be in the order of 6 nm for the solvent swollen PG4. Following the literature and without further proof, we assume that the fastest growing wavelength of the shape fluctuation dominates the topology bywith R _max = 3 nm, this yield 6 nm as the order of magnitude for λ which is not surprisingly of the same magnitude as the persistence length observed for this situation in the regime II for PG4.

One needs to emphasize that Fig. 10 depicts merely the local structure of the polymer in toluene at T = 24−28 °C but not the global structure. At present, we have little information on the global structure except values for R _G and R _h (see Fig. 5). However, the foregoing discussion sheds light onto the mechanism of chain collapse below T _θ . The DPs can be seen as anisotropic colloidal objects. In good solvents, the formation of aggregates is prevented by an osmotic barrier which acts against interpenetration of the surface dendron branches of different parts of the same as well as different wormlike chains. In other words, and following DLVO theory [24‐26] and later modifications for steric stabilization of colloids among them most notably latex suspensions [28], the individual DP is subject to osmotic stabilization in good solvents. A schematic diagram that displays the interaction potential W between two segments of the DPs vs their distance H is shown in Fig. SI-6 in ESM. The interaction energy is repulsive for thermodynamically good solvents (W _S) at distances H ≤ 2R _c neglecting terms which describe hydrodynamic interactions. However, attractive forces create a minimum in the energy curve for thermodynamically poor solvents (W _θ ). For a θ solvent, we assume that the magnitude of the attractive force ΔW between two segments can be approximated by k _B T _θ . In consequence at T > T _θ , the thermal energy is sufficient to screen the attractive interactions.

Returning to the case presented here, for PG4, we can now interpret the observation of a temperature regime below T _θ in which still individual chains exist in solution. The still solvated parts of the wormlike chain offer sufficient osmotic stabilization to hold the individual colloidal objects in solution. This situation may be further stabilized by a redistribution of the still solvated regions towards the outside of the volume occupied by the chain in its random walk conformation, in other words by a partial collapse of the whole structure into regions of higher and lesser solvation of the dendrons.

While these considerations are of course rather hypothetical, they may nevertheless serve as a valuable platform or working hypothesis for further investigations.