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Star-Brush-Shaped Macromolecules: Peculiar Properties in Dilute Solution Georges M. Pavlov,*,†,‡,§ Katrin Knop,†,‡ Olga V. Okatova,§ and Ulrich S. Schubert*,†,‡,∥ †

Laboratory of Organic and Macromolecular Chemistry (IOMC), Friedrich Schiller University Jena, Humboldtstr. 10, 07743 Jena, Germany ‡ Jena Center for Soft Matter (JCSM), Friedrich Schiller University Jena, Philosophenweg 7, 07743 Jena, Germany § Department of Physics, St. Petersburg University, and Institute of Macromolecular Compounds, Russian Academy of Science, 199004 St. Petersburg, Russia ∥ Dutch Polymer Institute (DPI), P.O. Box 902, 5600 AX Eindhoven, The Netherlands S Supporting Information *

ABSTRACT: Star-brush-shaped poly(ε-caprolactone)-blockpoly(oligo(ethylene glycol) methacrylate (PCL-b-POEGMA) macromolecules were synthesized and studied by molecular hydrodynamic methods. The values of the intrinsic viscosity, the velocity sedimentation coefficient, the translational diffusion coefficient, and the frictional ratio were obtained in acetone. Molar masses (M) were determined by the Svedberg relation, and the correlations between the hydrodynamic values and the molar mass were obtained in the range of 19 < M × 10−3 g mol−1 < 124. Comparison of the scaling indexes of the intrinsic viscosity and sedimentation velocity coefficient versus molar mass corresponding to the conventional four-arm stars macromolecules with that of the star-brush-shaped copolymer macromolecules shows that the star-brush-shaped PCL-bPOEGMA macromolecules have the more dense organization in space which is connected with their different topology in contrast to the conventional stars macromolecules. The model of the PCL-b-POEGMA macromolecules based on the ensemble of their hydrodynamic characteristics is discussed.

1. INTRODUCTION The development of the methods for the delivery of biologically active substances into the living body has attracted the attention of researchers for decades. The main objectives are the delivery of biologically active substances to reach their target sites, at the desired concentration, and to secure the withdrawal of the ballast “containers” from the body after completion of the task. The macromolecular systems used in these applications include both natural and synthetic macromolecules. In view of the hydrophobic nature of most biologically active substances, the containers preferentially should possess a hydrophobic core and a hydrophilic periphery. During the development of this approach, appropriate macromolecules and molecular systems as well as nanosystems of different architectures and topologies were synthesized and applied. Based on the original “drugdelivery” concept,1 significant effort has been made to increase the efficiency of pharmaceutically active compounds by attempting to control the temporal as well as spatial drug distribution in vivo,2 evolving into the new interdisciplinary research area of polymer therapeutics.3 Up to now, however, with many or even most container systems studied, knowledge of their structure and the corresponding structure−function relationships is limited or even completely lacking, which significantly hinders progress in the field. This particularly holds © XXXX American Chemical Society

when macromolecules with complex topology are involved. One of the sophisticated macromolecular systems studied as containers for drug delivery is the unimolecular micelle system represented by the star-brush-shaped poly(ε-caprolactone)-bpoly(oligo(ethylene glycol) methacrylate) (PCL-b-POEGMA).4 These macromolecules are different from the standard star-like structure, in which each arm is a conventional linear macromolecule. The peculiarity of the synthesized macromolecules is that the central part is the usual four-arm star, which is then completed by molecular brushes of different lengths, with different chemical structures compared to the core. A series of such macromolecules with different lengths of the peripheral POEGMA part were studied in this work by the methods of macromolecular hydrodynamics. Monitoring of the transport properties of the homologous series of the macromolecules is one of the basic tools to obtain the information about the gross conformation of the solved species.5−8 The main hydrodynamic values are the intrinsic viscosity ([η]), the velocity sedimentation coefficient (s0), the concentration velocity sedimentation coefficient (ks), and the Received: January 23, 2013 Revised: September 30, 2013

A

dx.doi.org/10.1021/ma400160f | Macromolecules XXXX, XXX, XXX−XXX

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N,N,N′,N″,N″-pentamethyldiethylenetriamine (Aldrich, PMDETA) were passed through an aluminum oxide bed containing 5% CaH2 prior to use. Acetone of spectroscopic purity grade for analytical ultracentrifuge purposes was obtained from Sigma-Aldrich showing the following characteristics (at 20 °C): dynamic viscosity η0 = 0.325 × 10−2 mPa s; density ρ0 = 0.7906 g cm−3. All experimental values were determined at 20 °C, except the translational diffusion measurements (25 °C). 2.2. Instrumentation. 1H nuclear magnetic resonance (NMR) spectra were recorded in CDCl3 on a Bruker AC 250 or 300 MHz spectrometer at 298 K. Chemical shifts are given in parts per million (ppm, δ scale) relative to the residual signal of the deuterated solvent. Size exclusion chromatography (SEC) was measured on a Shimadzu system equipped with a SCL-10A system controller, a LC-10AD pump, a RID-10A refractive index detector, and both a PSS Gram30 and a PSS Gram1000 column in series, whereby N,N-dimethylacetamide with 50 mmol of LiCl was used as an eluent at 1 mL min−1 flow rate and the column oven was set to 40 °C. The system was calibrated with PMMA (410−88 000 g mol−1) and PEG (440−44 700 g mol−1) standards. Furthermore, the kinetics of the polymerization was performed on a Shimadzu system equipped with a SCL-10A system controller, a LC-10AD pump, and a RID-10A refractive index detector using a solvent mixture containing chloroform, triethylamine, and isopropyl alcohol (94:4:2) at a flow rate of 1 mL min−1 on a PSS SDV linear M 5 μm column. The system was calibrated with PMMA (410− 88 000 g mol−1) and PEG (440−44 700 g mol−1) standards. Sedimentation velocity experiments were performed using a Beckman XLI analytical ultracentrifuge (ProteomeLab XLI Protein Characterization System) at a rotor speed of 40 000 rpm and at 20 °C, using interference optics at λ = 660 nm and Al double-sector cells of an optical path of 12 mm (Figure SI1). The velocity sedimentation coefficient s0 is one of the key hydrodynamic characteristic of isolated macromolecule or any dispersed species. Its physical meaning is given by Svedberg relation:6,7

translational diffusion coefficient (D0), which are differently related with the molar mass (M) and the size and the shape of the friction species. The most sensitive to the change of the molar mass and the size of the linear macromolecule between the frictional characteristics is the intrinsic viscosity, which is scaled with the molar masse following the Kuhn−Mark− Houwink−Sakurada relationship: [η] = KηMbη, where bη can change in the range from 0 (spheres, globular systems, regularly short-branched macromolecules) to 1.8 (bending rod, extremely rigid macromolecules). At the same time, in the case of the globular systems as well as dendrimers and hyperbranched macromolecules, the most sensitive to the change of the molar mass and the size is the velocity sedimentation coefficient, which scales for these systems with the molar mass as s0 = KsM≈2/3, and simultaneously the intrinsic viscosity for such systems is virtually independent of M ([η] ∼ M≈0). Analysis of the relations of the different hydrodynamic values obtained in the independent experiments with M led to the conclusion about the size and shape of solved species and about the gross conformation of the macromolecules. Over the past decades more complicated macromolecular systems such as dendrimers, hyperbranched macromolecules, and supramolecular systems are under the curious and concerned attention of the researchers.9−13 The hydrodynamic methods show the efficiency in the study of such complicated systems. Nonetheless, during these studies, some of the outstanding problems of molecular hydrodynamics of macromolecular systems resurfaced. Namely, between the nonresolved problems are the coexistence and competition between the intramolecular draining and excluded volume effects, and the depth of the draining in the coils of the linear macromolecule and in the structures of the complex topology.14,15 Thus, hydrodynamic studies, on the one hand, allow to obtain information about dissolved objects (molecular or supramolecular structure) and, on the other hand, stimulate the development of the new theoretical approaches and/or of the computer modeling methods. In this study, based on the hydrodynamic results obtained for PCL-b-POEGMA copolymers, the models of the macromolecules are discussed. The paper is organized as follows. Section 2 describes the materials and instrumentation used during the synthesis and further investigations. The basic equations are cited in this section showing the physical meaning of the measured hydrodynamic characteristics. Section 3 shows the results of the synthesis of the copolymer as well as the initial experimental results of their study. Section 4 is concerned with discussion of the hydrodynamic data: molar masses, scaling relationships, model of the star-brush-shaped PCL-bPOEGMA macromolecules. Finally, in section 5 we present our conclusions. In the Supporting Information, mainly the graphic materials for handling the initial experimental data are summarized.

s0 = M(1 − υρ ̅ 0 )/f0 NA

(1)

where M is molar mass and f 0 is translation friction coefficient:

f0 = P0′η0⟨R2⟩1/2

(2)

where ⟨R ⟩ is the mean-square radius of gyration of a macromolecule, which is adequate size characteristics of any macromolecule, (1 − υρ̅ 0) is a buoyancy factor, P0′ is the Flory hydrodynamic parameter, and NA is Avogadro’s number. For the analysis of the sedimentation velocity data the Sedfit program was used.16,17 This program, which calculates a continuous sedimentation coefficient distribution c(s) obtained at some concentration (where s is the velocity sedimentation coefficient), numerically solves the Lamm equation, the basic differential equation describing the coupled sedimentation and diffusion process.18 In the c(s) method, this is done for a large number of globular species with different sedimentation and diffusion coefficients. The result represents the best combination of species for matching the entire collected set of experimental concentration profiles. The numerical analysis is conducted under appropriate statistical criteria of goodnessof-fit. The Tikhonov−Philips second-derivative regularization method was used with a confidence level of 0.7−0.9 (corresponding F-ratio). The partial specific volume (υ̅), the solvent density (ρ0), and the solvent dynamic viscosity (η0) are additional parameters required to calculate the sedimentation coefficient distributions in the frame of the continuous particle size distribution c(s) of the Sedfit software. Finally, the differential distribution (dc(s)/ds) of the sample is obtained and scaled such that the area under the curve will give the loading concentration of the macromolecules between the minimum and maximum s-value occurring (expressed, in the case of interference optics, as the number of fringes, J) (Figure 1). (Note that the obtained differential distribution of the velocity sedimentation coefficient marked as c(s) instead of dc(s)/ds in original software Sedfit.) For samples with relatively high s-value, in addition the determination of the sedimentation coefficient distribution g*(s) by least-squares (ls) 2

2. EXPERIMENTAL DETAILS 2.1. Materials. Tin(II) 2-ethylhexanoate (Aldrich), α-bromoisobutyryl bromide (Aldrich), anisole (Fluka, 99.0%), copper(I) bromide (Aldrich, 99.999%), and trioxane (Aldrich) were used as received. εCaprolactone (Aldrich) was dried 2 days over CaH2 before distillation and stored under argon. Triethylamine was distilled over CaH2 and stored under argon. Pentaerythritol (Aldrich) was coevaporated with toluene prior to use. Tetrahydrofuran (THF, Aldrich) was dried in a solvent purification system (Pure Solv EN, Innovative Technology) before use. Oligo(ethylene glycol) methyl ether methacrylate with molar mass M00 = 475 g mol−1 (Aldrich, OEGMA 475) and B


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Figure 1. Comparison of the velocity sedimentation distributions obtained with the c(s) model and with the least-squares ls−g*(s) model for sample 7 of the PCL-b-POEGMA macromolecules in acetone; concentration c = 0.059 × 10−2 g cm−3. boundary modeling (ls−g*(s)) was used (Figure 1).19,20 The leastsquares boundary modeling is modeless calculation of the s-values. The two approaches led to virtually identical integral values (the value obtained as sum (integral) of all values under the distribution curve) of the velocity sedimentation coefficient (Figure 1). To obtain the values of the velocity sedimentation coefficient at infinite dilution s0 and the corresponding concentration coefficient (Gralen coefficient) ks, the concentration dependency of s(c) was studied (Figure SI2). Three concentrations of each sample in acetone were studied at an initial concentration (cmax) between 1 × 10−3 and 8 × 10−3 g cm−3 depending on the sample, and the ratio between the initial and lower studied concentration (cmin) in the cells was cmax/cmin ≥ 3.5. All concentrations were in the zone of high dilution, where the Debye parameter c[η] characterizing the degree of dilution is in the range 0.01 ≤ c[η] ≤ 0.1 (c: polymer concentration expressed in g cm−3; [η]: value of the intrinsic viscosity expressed in cm3 g−1), and consequently the concentration dependencies of s followed a linear regression, in accordance with the relationship s−1 = s0−1(1 + ksc + ...) (Table 2). The concentration coefficient ks have also molecular meaning and may be presented as21 2 3/2

ks = B′⟨R ⟩

/M

Figure 2. Translational diffusion interferogramms of the PCL-bPOEGMA block copolymer (sample 5) obtained at different times of the diffusion process: 1−0.25 h, 2−2.25 h, and 3−6.0 h after the formation of the boundary in acetone at the concentration c = 0.0014 g/cm3.

by the method of maximum ordinate and area:6 σ 2 = (a2/8)/ [argerf(aH/Q)]2, where H and Q are the maximum ordinate and the area under the interference curve, respectively; argerf is the argument of the probability integral, and a is the spar twinning (0.11 cm). The dependence of σ 2 on time t (Figure SI3) was approximated by a linear regression and the diffusion coefficient was determined by its slope: D = (1/2)∂σ 2 /∂t. A concentration dependence of the translational diffusion coefficient was absent in the concentration range investigated and the obtained values are considered as5

(3)

where B′ is dimensionless parameter. In the c(s) method of the Sedfit program the frictional ratio value (f/fsph), which is the weight-average frictional ratio of all species, was optimized executing a fit command, where f represents the frictional coefficient of the solute macromolecule at a concentration c and fsph = 1/3 the frictional coefficient of a rigid sphere with the 6πη0(3Mυ/4πN A) ̅ same “anhydrous” volume (free of solvent) as the macromolecule. The frictional ratio (f/fsph) is also extrapolated to zero solute concentration using the linear regression ( f/fsph) = (f/fsph)0(1 + kfc + ...). The combination of the extrapolated values of s0 and ( f/fsph)0 allows to the value of the translation friction coefficient −1/2 f0 = (9π 21/2)η0 3/2(1 − υρ (f /fsph )0 3/2 (s0υ ̅ )1/2 ̅ 0)

D0 = kT /f0

(5)

The refractive index increment of the polymer−solvent system (dn/ dc) was calculated on the basis of Q-values: dn/dc = (λ/abh)(Q/c). Here λ = 546 nm is the light wavelength, and b is the distance between the compensator interference fringes (b = 1.5 mm). Viscosity measurements were conducted using an AMVn viscometer (Anton Paar, Graz, Austria), with the capillary/ball combination of the measuring system. The respective times of the fall of the gilded steel ball in the viscous medium of the solvent and polymer solutions, τ0 and t, were measured at 20 °C, the relative viscosities ηr = t/τ0 being in the range 1.15−1.8. The extrapolation to zero concentration of the reduced (ηsp/c) and inherent (ln ηr/c) viscosities was made by using both the Huggins (6) and the Kraemer (7) equations6

(4)

Thus, the combination of the values s0 and ( f/fsph)0 for some systems allows to determine the molar mass; i.e., the velocity sedimentation method in conjunction with the Sedfit program in some cases becomes self-sufficient from the point of view of molecular characterization of molecularly dispersed macromolecules/substances. The advantage of this method has been shown in the following publications.22−24 The translational friction or the translational diffusion was studied also independently by the classical method of boundary formation between the solution and the solvent in a Tsvetkov polarizing diffusometer;6,25 the boundary was created, at 25 °C, in a cell with a Teflon centerpiece of length h = 2 cm along the beam path. The diffusion process was registered by the digital camera. Diffusion interferogramms (Figure 2) were processed in a Gaussian approach according to the method proposed in ref 26. The dispersion σ 2 of the distribution ∂c/∂x of macromolecular displacements x was calculated

ηsp /c = [η] + k′[η]2 c + ... 2

ln ηr /c = [η] + k″[η] c + ...

(6) (7)

and the average values were considered as the value of the intrinsic viscosity [η] (Table 2, Figure SI4 and Table SI1). The physical meaning of the [η] value is given by the Flory−Fox relation: C


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Figure 3. Chemical structure of the star-brush-shaped poly(ε-caprolactone)-b-poly(oligo(ethylene glycol) methacrylate).

[η] = Φ′0 ⟨R2⟩3/2 /M

delivered corresponding results by comparison of the trioxane signal as internal standard with the methacrylate signals. 3.2. Raw Experimental Results. Viscosity data are shown in Table 2 and Figure S3. The values of the intrinsic viscosity varied slightly for the different samples of the studied set. The average values found are k′ = 1.22 ± 0.07 for the Huggins and k″ = +(0.22 ± 0.03) for the Kraemer parameter (Table SI1). The ks value is usually compared to the intrinsic viscosity values by calculating the dimensionless parameter γ = ks/[η].32,33 For the investigated system, the average γ parameter turned out to be equal to γ = 2.7 ± 0.2 (Table SI1). The values of the translational diffusion coefficient obtained by the classical method of forming a boundary between the solution and the solvent (D0bf) were compared with the translational diffusion coefficients calculated from the s0 and (f/ fsph)0 values by the following expression: D0sf = kT(1 − υρ̅ 0)1/2/ η03/2(9π21/2)( f/fsph)03/2(s0υ)̅ 1/2. The comparison of the intrinsic translational diffusion coefficients [D] ≡ D0η0/T obtained by two independent techniques showed satisfactory agreement between these values (Figure SI8). From the AUC interference scans, the number of fringes J for each concentration c was determined. This value was used to calculate the refractive index increment according to Δn/Δc = (λ/h)(J/c)34 (Figure S8). Independently, the refractive index increment was estimated from the translational diffusion experiments (Table SI1). Both average values are virtually the same and equal to Δn/Δc = 0.127 ± 0.002 cm3 g−1 (Figure SI9). The values of the partial specific volume υ̅ were determined for each sample (Table S1). For the block copolymer samples, the υ̅ values showed fluctuations around a mean value of υ̅ = 0.820 ± 0.006 cm3 g−1. Finally, the set of main hydrodynamic values, [η], s0, ks, (f/ fsph)0, and D0, was measured forming the initial matrix of the experimental data (Table 2).

(8)

where Φ′0 is the Flory hydrodynamic parameter. Density measurements were carried out in a DMA 02 density meter (Anton Paar, Graz, Austria) according to the procedure of Kratky et al.27 (Figure SI5), the density increment Δρ/Δc, which corresponds to the buoyancy factor (1 − υρ̅ 0), was determined (Table S1), and finally the partial specific volume υ̅ was calculated.

3. RESULTS 3.1. Synthesis. The amphiphilic star-brush-shaped macromolecules are composed of a hydrophobic four-armed poly(εcaprolactone) (PCL) core and a hydrophilic shell, which is attached to each arm. The hydrophilic brush-shaped shell is composed of POEGMA (Scheme SI1). Synthesis details are given in refs 4, 28, and 29 (see also the Supporting Information). Briefly, the block copolymers were synthesized by a core-first approach to ensure a regular number of arms.29−31 In this approach, pentaerythritol with four hydroxyl functionalities was used to initiate the ring-opening polymerization of ε-caprolactone to yield a polymer with y = 18 repeating units of ε-caprolactone per arm. To build up the hydrophilic shell, an atom transfer radical polymerization (ATRP) of the oligo(ethylene glycol) methacrylate macromonomer (OEGMA, M00 = 475 g mol−1) was utilized. Therefore, the hydroxyl end functionalities of the PCL core were submitted to an esterification reaction with αbromoisobutyryl bromide. The subsequently performed ATRP yielded star-brush-shaped macromolecules (Figure 3) with varying shell size to study their hydrodynamic behavior (Table 1, Table SI2 and Figure SI6). The conversions were calculated from the decrease of the integral of the OEGMA signal in the SEC (CHCl3:iPrOH:NEt3). Additionally, 1H NMR spectroscopy Table 1. Characteristics of Star-Shaped PCL-b-POEGMA Macromolecules N

polymer

0 1

[PCL18-b-Br]4 [PCL18-bPOEGMA3]4 [PCL18-bPOEGMA12]4 [PCL18-bPOEGMA16]4 [PCL18-bPOEGMA20]4 [PCL18-bPOEGMA23]4 [PCL18-bPOEGMA25]4 [PCL18-bPOEGMA51]4

2 3 4 5 6 7

4. DISCUSSION 4.1. Molar Masses, Hydrodynamic Invariants, and Scaling Relations between Hydrodynamic Characteristics and Molar Masses. The matrix of the hydrodynamic data can be transformed into the matrix of molar masses and hydrodynamic invariants.35 The possibility of these transformations is conditioned by the fact that the experimental hydrodynamic values depend on mass and size of the macromolecule in a nonsimilar way (see eqs 1−5 and 8).5−7,36 In the case of branched macromolecules the appropriate size characteristics is the mean-square radius of gyration of a macromolecule, ⟨R2⟩. The elimination of f 0 from the eqs 1 and 5 results in the Svedberg equation for molar mass determination:

Mnb −1

aimed DP POEGMA

conv in SECa [%]

obtained DPt

[g mol ] calcd

PDIc

6

11

3

8 900 14 600

1.15 1.18

12

25

12

31 700

1.21

15

32

16

39 300

1.25

20

39

20

46 900

1.19

23

47

23

52 600

1.19

25

51

25

56 400

1.19

50

68

51

100 100

1.14

MsD = (RT /(1 − υρ ̅ 0 ))(s0/D0) = R[s]/[D] = NA[s][f ]

(9)

where [f ] ≡ f 0/η0 is the intrinsic value of the translation friction coefficient. This relation may be converted to one in which [s] and the frictional ratio (f/fsph)0 values are used to determine the molar mass:

a

Obtained from SEC (CHCl3:iPrOH:NEt3) using PEG calibration. b Calculated from the conversion. cObtained from SEC (DMAc:LiCl) using PMMA calibration. D


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Table 2. Hydrodynamic Characteristics and Hydrodynamic Invariants of the Star-Brush-Shaped PCL-b-POEGMA Macromolecules in Acetone at 20 °C

a

N

[η] [cm3 g−1]

s0 × 1013 [s]

ks [cm3 g−1]

( f/fsph)0

A0 × 1010

βs × 10−7

0 1 2 3 4 5 6 7 averagea

10.1 11.0 9.6 10.2 10.4 10.1 9.7 12.7

2.99 5.14 8.80 8.34 9.34 11.3 11.2 17.3

11 23 32 25 25 31 22 41

1.80 1.51 1.60 1.58 1.62 1.61 1.62 1.66 1.60 ± 0.02

2.74 3.37 3.04 3.18 3.11 3.09 2.99 3.23 3.1 ± 0.05

0.95 1.45 1.52 1.44 1.40 1.51 1.32 1.60 1.4 ± 0.03

Averaging performed on all samples, except the core (0).

relatively to molar mass M.25,36 Note that the theoretical values of A0theor and βstheor are related to the hydrodynamic parameters Φ0′, P0′, and B′ by the following relations: A0theor = kΦ0′1/3/P0′ and βstheor = B′1/3/P0′. The theoretical values for A0theor vary in the range from 2.914 × 10−10 g cm2 s−2 K−1 mol−1/3 for rigid spheres to 3.84 × 10−10 g cm2 s−2 K−1 mol−1/3 for Gaussian chains in the absence of volume effects. The theoretical value of βstheor for the model of the rigid spheres is 1.30 × 107 mol−1/3.36 The experimental values of A0 and βs for different samples of the star-brush-shaped PCL-b-POEGMA macromolecules determined in acetone accumulate around the average value of A0 = (3.14 ± 0.05) × 10−10 g cm2 s−2 K−1 mol−1/3 and βs = (1.46 ± 0.03) × 107 mol−1/3. It means that the good correlation is observed between the hydrodynamic characteristics ([η], s0, ks, D0), obtained in independent experiments or inside of the same experiment (s0 and ks). The hydrodynamic characteristics and molar masses evaluated from the experimental data obtained in acetone for all star-brush-shaped PCL-b-POEGMA macromolecules are listed in Tables 2 and 3. Finally, a set of PCL-b-POEGMA macromolecules with molar masses between 19 000 and 124 000 g mol−1 was synthesized and studied. Under the assumption that the functionality of the core in all cases is 4, the degree of polymerization of the terminal POEGMA chains (DPthydr) can be calculated from the values of molar masses determined by the relation

Msf = (RT /(1 − υ ̅ ρ0 ))(s0 /D0) = 9π 21/2NA ([s](f /fsph )0 )3/2 υ ̅ 1/2

(10)

where [s] ≡ s0η0/(1 − υρ̅ 0) is the intrinsic value of the velocity sedimentation coefficient, [D] ≡ D0η0/T is the intrinsic value of the translation diffusion coefficient, and R is the gas constant. Both values of the translation friction coefficient independently determined were used for the molar mass estimation (relations 9 and 10). The obtained values of Msf and MsD correlate fairly well with each other, and in the further discussion the average of these two values was used (Table 3). Table 3. Molar Mass (M × 10−3, g mol−1) of the Star-BrushShaped PCL-b-POEGMA Macromolecules Obtained from the Sedimentation-Diffusion Analysis in Acetone at 20 °C and the Degree of Polymerization of the Terminal POEGMA Chains (DPt) Calculated from MsDav Values N

Msf

MsD

MsDav

DPthydr

0 1 2 3 4 5 6 7

9.2 14.7 38 30 38 52 59 104

11 23 45 39 45 64 66 144

10.1 19 41.5 34.5 41.5 58 62.5 124

4.5 16 12 16 25 27 60

DPt hydr = (Mi − M 0)/4M 00

The elimination of the size value (⟨R ⟩) from the eqs 1 and 8 allows the formation of the equation for the experimental value of the hydrodynamic invariant:25,36 2

where Mi is MsDav molar mass of the ith sample (Table 3), M0 is MsDav molar mass of the core, 4 is the functionality of the core molecule, and M00 is the molar mass of the repeat unit of the terminal chains (M00 = 475 g mol−1). The therewith calculated values for DPthydr (Table 3) can be compared with the DPt (Table 1) values obtained by determination of the conversion of the OEGMA macromonomer by SEC or 1H NMR spectroscopy (Table 1).4 A reasonable correlation is observed between the values of DPt obtained by the different experimental techniques. 4.2. Scaling Relations between Hydrodynamic Characteristics and Molar Masses. Hydrodynamic data obtained for a homologous series of macromolecules are usually related either to each other or to molar mass. This allows to obtain cross (among hydrodynamic characteristics) and canonic (among hydrodynamic characteristics and molar mass) relationships of the Kuhn−Mark−Houwink−Sakurada (KMHS) type or hydrodynamic scaling relationships.25,37

A 0 = R[s][η]1/3 M −2/3 = (R[D]2 [s][η])1/3 = A 0sph ([η]/[η]sph )1/3 (f /fsph )0

(11)

Eliminating the size value from eqs 1 and 3 allows evaluating the experimental value of the sedimentation parameter:36 βs = R[s]ks1/3M −2/3 = NA(R−2[D]2 [s]ks)1/3 = βssph

(ks/kssph)1/3 (f /fsph )0

(13)

(12)

The analysis of a large number of experimental results obtained mainly for the different linear polymer systems shows that the parameters A0 and βs in the first approximation are invariant E


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For the linear macromolecules their homology is ensured by the same chemical structure of the repeating units and the constancy of the structural parameterthe linear density of the polymer chains ML. Homology among branched macromolecules, in particular branched copolymers, is difficult to define. The studied star-brush-shaped PCL-b-POEGMA macromolecules have the same kernel part, which is a four-arm poly(ε-caprolactone) star, and external layers of different thickness consisting of POEGMA chains with different degrees of polymerization (from DPt = 0 (core) up to DPt = 60). Although the core molecules correspond to the condition DPt = 0 and could be formally included into the full range of data, it became clear throughout the experiments that the data obtained for the core differ from the data received for the samples 1−7. This may be related to different hydrodynamic interactions in the simple star on the one side, and in the starbrush-shaped macromolecules on the other side, as well as to different thermodynamic affinities of PCL (the core of the macromolecule) and POEGMA (the periphery of the macromolecule) toward acetone. The comparison of the hydrodynamic values obtained for the macromolecule samples with different DPt (Table 2) shows that the values of the intrinsic viscosity rather fluctuate around the mean value, whereas those of the velocity sedimentation coefficients differ for a factor of almost 6. At the same time, the frictional ratio also fluctuates around the mean value. This indicates that the friction elements (isolated macromolecules) are similar in the hydrodynamic behavior to the globular-like particles with a constant degree of geometric asymmetry. This view seems to be supported by the values of the scaling indices obtained from the double-logarithmic plots of the hydrodynamic characteristics vs other hydrodynamic characteristics or molar mass. In this situation, the question arises whether the hydrodynamic characteristics of such a series of macromolecules follow the scaling relations (KMHS relationships). In general, canonical scaling relations may be given as follows: Pi = K iM bi

Figure 4. Scaling plots for PCL-b-POEGMA macromolecules in acetone. The double-logarithmic plots of intrinsic viscosity [η] (curve 1), translational diffusion coefficient D0 (curve 2), sedimentation coefficient s0 (curve 3), and concentration coefficient ks (curve 4) vs molar mass are plotted. The values of the slopes yield the scaling indices reported in Table 4. Filled points correspond to the core molecule.

Table 4. Parameters of the Scaling Relationships Pi = KiMbi (KMHS Relationships) for a Series of Star-Brush-Shaped PCL-b-POEGMA in Acetone Pia

bi

Δbi

Ki

rib

[η] s0 ks ( f/fsph)0 [D]

0.07 0.63 0.25 0.046 −0.37

0.07 0.03 0.1 0.006 0.01

5.15 1.07 × 10−15 1.6 0.98 9.6 × 10−10

0.399 0.994 0.691 0.955 −0.998

a

The characteristic Pi of the copolymer samples are related with the molar masse by log Pi = log Ki + bi log M. br is the linear correlation coefficient of these double-logarithmic plots.

model that satisfies these conditions is a set of ellipsoids (spheroids) with a constant asymmetry. The simplest example of such a model is a series of hard spheres. 4.3. Comparison with the Conventional Stars Macromolecules. It is worthy to compare the hydrodynamic characteristics of the studied star-brush-shaped poly(εcaprolactone)-b-poly(oligo(ethylene glycol) methacrylate) macromolecules with whose of the conventional four-arm stars macromolecules.38−41 The comparison of the molar mass dependence of the intrinsic viscosity of polystyrene, polyisoprenes, and polycaprolactone stars shows that for the conventional stars of any chemical structure the values bη ≥ 0.5 are observed depending on the thermodynamic quality of the solvents, whereas for the studied star-brush-shaped copolymer the value of bη ≈ 0 is obtained (Figures 4 and 5). This difference of the intrinsic viscosity scaling indexes bη between the conventional star macromolecules and the PCL-bPOEGMA macromolecules is confirmed by the scaling plots of the velocity sedimentation coefficient and the corresponding bs values (see Figure SI11). The value of bs ≤ 0.5 for the conventional stars macromolecules is observed, whereas for the star-brush-shaped copolymer this value is bs = 0.63. Note that the usual correlation between the scaling indices bη and bs is observed for the series of all type of the star macromolecules: bs ≈ (2 − bη)/3. The difference between scaling indexes of the conventional star macromolecules and the star-brush-shaped copolymer macromolecules reflects their different topology. It is due to the fact that the more part of the macromolecules mass of the studied copolymer is located at its periphery. In

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where Pi is one of the hydrodynamic characteristics [η], s0, ks, (f/fsph)0, or D0 (Figure 5 and see also Table SI3, Figures SI10 and SI11). The following correlations exist in the polymer homologous range among scaling indices: |bD| = (1 + bη)/3, |bD| = bff + 1/3, |bD| + bs = bff + bs + 1/3 = 1, where the underline index refers to correlations of corresponding hydrodynamic characteristics and molar mass. Corresponding plots for the series of PCL-b-POEGMA macromolecules are represented in Figure 4, and Table 4 shows the obtained scaling indices. Apparently, the translational friction data (velocity sedimentation coefficient, frictional ratio, and translational diffusion coefficient) correlate well with the molar mass at the sufficiently high level of the coefficient of linear correlation ri (Table 3). The scaling indices bs = (0.63 ± 0.03), bD = −(0.37 ± 0.01), and bη = (0.07 ± 0.07) are close to the values corresponding to the globular systems. Thus, the macromolecules of the star-brush-shaped PCL-bPOEGMA copolymer follow the scaling relationships of KMHS type and the usual correlations between the scaling indices bi are observed. In this approach the series of the star-brushshaped PCL-b-POEGMA copolymer may be considered as a homologous series. Such values of bi are characteristic to the series of the molecules/particles which keep constant their form and asymmetry going from low M to high M. An example of a F


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average intracoil density (∼log(1/ρ)).44 The higher are the curves located along the ordinate, the lower is the density of polymer substance in the volume limited by the polymer molecule. This kind of comparison was made also in other mode and in other plot in (Figure SI12). This comparison shows that the PCL-b-POEGMA macromolecules in solution are denser in comparison with conventional star-like PCL macromolecules. From the point of view of the intracoil density, the star-brush-shaped PCL-b-POEGMA macromolecules are close to the dendrimers (Figure SI12) and to the dendri-star molecules with 64 and 128 arms and situated between the linear brush-like macromolecules and globular systems, dendrimers, and glycogen. 4.4. Experimental Facts. Let summarize the specific features identified during the study and analysis of the hydrodynamic characteristics of the star-brush-shaped PCL-bPOEGMA macromolecules. These peculiar properties are as follows: (1) The values of the intrinsic viscosity of the copolymer samples (in particular samples 1−6) fluctuate around an average value, which is [η] = (10.2 ± 0.2) cm3 g−1. However, this value is about 2 times higher than the values generally observed for globular proteins, dendrimers, and glycogen and 5 times higher than the theoretical values for the rigid spheres.5,7,46−48 (2) The scaling velocity sedimentation coefficient index bs = 0.63 ± 0.03 is close to the value 2/3, while at the same time the scaling intrinsic viscosity index bη = 0.07 ± 0.07 is close to the value ≈0. (3) Similarly, the value of the fictional ratio is virtually constant. The average value for the samples 1−7 is (f/fsph)0 = 1.60 ± 0.02. (4) The obtained Huggins parameter k′ is significantly higher than that for conventional flexible-chain polymers (k′ = 1.22 ± 0.07). Such value is indicative of a compact and/or elongated conformation.49 Note also that for the rigid spheres k′ value is 2.26.50,51 (5) The value of the dimensionless parameter γ = ks/[η] = 2.7 ± 0.2 is close to the theoretical value obtained for a hardsphere model, which is the simplest case of a globular system: kssph/[η]sph = 2.75.6,21,36 Between the experimental parameters characterizing the starbrush-shaped PCL-b-POEGMA macromolecules the more important are the values of the scaling indexes bη and bs (or bD). Such values of bη ≈ 0 and bs ≈ 0.6 are typical for a series of spheroids with a constant asymmetry (p ≠ 1); a particular case of this model is a series of hard spheres (p = 1). This result indicates that studied macromolecules are organized in solution in more dense objects in comparison of the conventional star macromolecules. 4.5. Model of the Star-Brush-Shaped PCL-b-POEGMA Macromolecules. Three zones can be distinguished in the volume occupied by the PCL-b-POEGMA macromolecules which will have different average densities of the polymer material. The first one is the inner region core, which consists of four-armed poly(ε-caprolactone) with a linear density of the polymer chain ML = 1.5 × 109 g/(mol cm). The second one is formed by a comb-like chain of POEGMA with the linear density ML = 21 × 109; the thickness of this zone of the macromolecule varies significantly depending of the degree of polymerization of OEGMA. The third one is the surface layer (corona) of the whole macromolecule which is permeable to the solvent molecules with additional losses due to the friction

Figure 5. Double-logarithmic plot of the intrinsic viscosity values versus molar mass for the conventional four-arm stars: 1: polystyrene in cyclohexane at 35 °C;38,39 2: polyisoprenes in dioxane;40 3: polyisoprenes in toluene;40 4: polycaprolactone in ethyl acetate;41 and 5: star-brush-shaped poly(ε-caprolactone)-b-poly(oligo(ethylene glycol) methacrylate) in acetone (Table 2). The corresponding scaling relationships are the following: 1: [η] = 6.52 × 10−2 M0.497±0.006; 2: [η] = 9.12 × 10−2 M0.509±0.005; 3: [η] = 1.32 × 10−2 M0.737±0.007; 4: [η] = 5.83 × 10−3 M0.87±0.1; 5: [η] = 6.2 M0.05±0.07.

contrast, the architecture of conventional star polymers implies that the monomer density is high in the core region and decreases toward the corona.42 Finally, we compare the results obtained for the star-brushshaped PCL-b-POEGMA macromolecules with other macromolecular systems. This comparison does not require any models for the juxtaposition of the different polymer systems.43,44 It is worthwhile to analyze the evolution of the hydrodynamic volume with respect to the molar masses for the (homologous) series of the studied star-brush-shaped PCL-bPOEGMA macromolecules with those for four arms starshaped PCL,41 64- and 128-arms dendri-star-shaped polystyrene macromolecules,45 and dendrimers.46,47 The comparison can be made in a double-logarithmic coordinate system: [η]M versus the molar mass (Figure 6). In the first approximation the slope of these dependences will be inversely proportional to the

Figure 6. Comparison of the hydrodynamic volume for the polymer systems of different kinds: star-brush-shaped (1) macromolecules (Table 2), four-arms star-like PCL (2),41 four-arms star-like polysterene (3),39 64-arms dendri-star-like macromolecules with dendrimer core (4),45 128-arms dendri-star-like macromolecules with dendrimer core (5),45 lactosylated polyamidoamine dendrimers (6),46 linear macromolecules (7),44 linear brush-like macromolecules (8),44 and globular systems, dendrimers, and glycogen (9).44 G


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study of the PCL-b-POEGMA copolymers is to establish a scaling relation typical of globular or preglobular organization of macromolecules in solution.

during the macromolecule transport through the solvent. The thickness of this layer is unknown. Such a complex organization of the PCL-b-POEGMA macromolecules cannot be adequately described by a solid body model impermeable for the solvent molecules. Asymmetry defined, for instance, with the model of the solid spheroid will be overestimated because the solid spheroid model does not take into account the density distribution inside the macromolecule, the hydrodynamic losses in the surface layers of macromolecules penetrable for the solvent molecule, and, in general, the “softness” of the macromolecule. Early modeling of the macromolecules in solution by the rigid bodies nowadays is being replaced by the simulation of macromolecules by the soft bodies.52−54 Such mathematical modeling is implemented using a coarse-grained modeling concept.55,56 Coarse-grained model (or blob model) represents a macromolecule as a whole in the form of a soft body, which radius should be equal to the radius of gyration of the macromolecules and is allowed to fluctuate. The models of soft sphere, ellipsoid, and dumbbell were considered for different type of polymers.57−59 The term “soft body” means that the shape of the macromolecule as a whole can be easily changed by external or internal forces. At the end of the force action the macromolecule quickly relaxes. However, the practical implementation of a coarse-grained model and a direct comparison with experimental results is not straightforward. In general, the value of intrinsic viscosity may be represented as [η] = ν(p,h,ε)υ̅, where ν(p,h,ε) is a dimensionless coefficient depending on the asymmetry p of the soft body modeling any macromolecule, on the thickness of corona h accessible for the solvent molecules, and on the thermodynamic quality of the solvent (ε). The significant part of the energy loss due to friction of the macromolecule in solution is contributed by the corona. The different thermodynamic quality of the solvent determines the degree of swelling of the macromolecules, which, in turn, can determine the changes in the degree of draining and in the asymmetry of the macromolecule as a whole. Some attempts to obtain a theoretical expression for the intrinsic viscosity of the soft sphere depending on the thickness of the layer available for the solvent molecules may be found in the literature,60−62 but it still cannot be considered as a theory, allowing the interpretation of experimental data of the intrinsic viscosity of soft macromolecules. Note that the influence of draining effects on the value of intrinsic viscosity of the spheres has been demonstrated by Debye−Bueche on the model of the rigid sphere permeable for the solvent molecules.63 The value of [η] for the uniformly permeable sphere was obtained to be 3.6 times higher than that for the impermeable one. It is a level of the magnitude that is under discussion. The average value of the intrinsic viscosity of the copolymer samples (in particular, samples 1−6) is [η] = 10.2 cm3 g−1, which is 5 times higher than that for the rigid sphere limit. Qualitatively, the [η] values obtained for the PCLb-POEGMA copolymer can be partially attributed to the effects of percolation of the solvent molecules through macromolecules. The solution behavior of such macromolecules can be described by soft spheroids draining by the solvent molecules. Separating the contributions to the small value of [η] from one side due to the draining effect from another side due to the asymmetry is currently not possible. In fact, we discuss in this section the small intrinsic viscosity values and their deviations from the theoretical value, known for hard spheres. However, the main result obtained in this

5. CONCLUSION By combination of the ring-opening polymerization of εcaprolactone with the ATRP of the OEGMA macromonomer, a series of the star-brush-shaped PCL-b-POEGMA macromolecules with four arms were synthesized. The studies of such hybrid macromolecules by the molecular hydrodynamic methods show that isolated macromolecules follow the scaling relationships with scaling indexes, which are characteristics of the particle with constant asymmetry; viz., the intrinsic viscosity virtually does not depend on molar mass ([η] ∼ M≈0), and the sedimentation velocity coefficients strongly depend on molar mass (s0 ∼ M≈0.60). The solution behavior of such macromolecules can be qualitatively described by soft spheroids with tangible degree of draining and unknown asymmetry. From the point of view of average intracoil density of the PCL-bPOEGMA macromolecule in solution, these macromolecules are situated between the linear brush-like macromolecules and the globular systems. We are not aware of any theories or simulations quantitatively describing the behavior of such complex branched systems and hope that our experimental results will be useful for development work in the simulations/ theory of such systems.

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ASSOCIATED CONTENT

* Supporting Information S

Synthesis as well as the characterization of the PCL-bPOEGMA macromolecules; plots showing the hydrodynamics results such as the sedimentation velocity, translational diffusion, and intrinsic viscosity; cross-scaling of the Kuhn− Mark−Houwink−Sakurada type plot and other information. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail [email protected]; Tel +007 812 328 34 01; Fax +007 812 328 68 69 (G.M.P.). *E-mail [email protected]; Tel +49(0) 3641 948200; Fax +49(0) 3641 948202 (U.S.S.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the Dutch Polymer Institute (DPI, technology area HTE), the Fonds der Chemischen Industrie, the Thuringian Ministry for Education, Science and Culture (grant #B51409051, NanoConSens), and the Carl-Zeiss-Stiftung (Strukturantrag JCSM) for financial support. Prof. Dr. Dieter Schubert is gratefully acknowledged for his helpful comments and the careful correction of the manuscript. The authors thank the reviewers and the editor for their helpful comments.

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REFERENCES

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