THE JOURNAL OF CHEMICAL PHYSICS 125, 094908 共2006兲
A theoretical study of conformational properties of dendritic block copolymers of first generation M. Kosmasa兲 and C. Vlahos Department of Chemistry, University of Ioannina, 45110 Ioannina, Greece
A. Avgeropoulos Department of Materials Science and Engineering, University of Ioannina, 45110 Ioannina, Greece
共Received 24 May 2006; accepted 25 July 2006; published online 6 September 2006兲 Conformational properties of a dendritic block copolymer of the first generation are studied by means of an analytic calculation and dimensionality techniques. The polymer can have different functionalities and branch lengths in the interior region and the exterior shell. Three parameters are included in order to describe the intensity of the interactions between the same or different monomeric units. Based on the average end to end distances of the branches effective angles are defined in order to study how the microscopic parameters control the position and activity of the end groups, but also the hollowness in the internal region and the tweezing ability of the external shell of the macromolecule. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2337625兴 I. INTRODUCTION
The increasing interest on the conformational behavior of dendritic macromolecules as well as the expanding number of their applications1 are mainly based on three of their properties. First, emerging radialy towards the periphery and possessing a highly branched repeating structure, dendritic macromolecules can have a high surface area and numerous terminal ends. These can yield high density of end groups leading to enhanced catalytic activity or high reactivity generally, depending on the aimed task.2 The second useful property comes from the different number of branches of various kinds which they posses in their interior. The interactions among these branches can create density fluctuations and cavities which are beneficial for the use of these macromolecules as host systems suitable to accept guest molecules.3 This second property is often combined with their third property of high external surface and the capability of an increased permeability to the interior of the polymer. Indeed their controlled loading and release capacity give them an enhanced carrying ability which led to their extensive use and study as drug carriers.4 The reactivity of the end groups depend on their location and early in the development of dendritic homopolymers the scientific interest focused on whether their end groups are mainly found in the periphery5 or in the interior of the macromolecule as well, which is generally believed to be the case.6 Beyond these two aspects about the macroscopic behavior of homodendrimers it is realized that the conformational properties and therefore the position of the end groups for more complicated structures including special interactions depend on the macromolecular characteristics.7 Dendritic polymers can have flexible macromolecular branches and the molecular weights, the number of branches but also the kind of interactions between their monomeric units are some of their mia兲
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croscopic characteristics which can affect their conformational behavior. We study in this work a simple dendritic block copolymer in an effort to determine both the position of the end groups but also the possibility of the formation of cavities in the interior and openings at the surface of the polymer. We employ statistical mechanical properties using our previous experience on studying macromolecules of complicated architectures.8 In order to be able to study the differences coming from the various microscopic parameters on the earlier mentioned properties the simplest dendritic block copolymer of the first generation is chosen. Furthermore, the choice of block copolymer dendrimers came from the fact that recently they are also extensively used for the preparation of nanoscale structure particles with specific properties.9 We will study the effects of different interactions in the same dendritic macromolecule and the influence of different numbers of branches and molecular weights on the macroscopic properties of polymers having dendritic nature. Because the parameters and their combinations are many we choose special values of them in order to see the microscopic origin of the earlier three properties and exhibit the main general trends of the macroscopic behavior of block dendritic copolymers. The macromolecule can have two different kinds of branches made of different monomeric units. The one emerges from the central core of the macromolecules and forms the interior zeroth generation and the other starts from the ends of the first branches and constitute the outside first generation. A picture of such a polymer is given in Fig. 1, where the number of branches emerging from the core is f a = 3 while the number of extra branches from the end of each initial branch is f b = 2. The functionalities f a and f b as well as the molecular weights Na and Nb of the interior and the exterior branches which form the interior A and the exterior B regions of the macromolecule can generally be different. Three different interaction parameters ua, ub, and uab are used to describe the nature and the intensity of he inter-
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© 2006 American Institute of Physics
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094908-2
J. Chem. Phys. 125, 094908 共2006兲
Kosmas, Vlahos, and Avgeropoulos
具R2典 =
再冕
D关r共s兲兴R2 P关R共s兲兴
冎 冒 再冕
冎
D关r共s兲兴P关R共s兲兴 , 共2兲
where R2 is the part’s end to end square distance of each configuration. D关r共s兲兴 is the measure of the path integrals which expresses integrations over all points of the chain in the continuous line limit. FIG. 1. A block copolymer dendritic polymer of first generation. The numbers f a and f b of the branches of the first and second generation is 3 and 2, respectively. In the figure the beginning and the ends of characteristic branches are also shown, ROA , RAB, and RBB⬘, are examples of the end to end distances of some branches. The effective angles between the ends of an A and a B branch, between the ends of two A branches and between two B position vectors are denoted as X, Y, and Z, respectively.
actions between two A, two B and one A and one B units, respectively.10 Positive u’s mean average repulsions between the units while negative u’s express attractions. Increasing the temperature or the quality of the solvent ua and ub usually increase while increasing the incompatibility of the two different kinds of units gives rise to the increase of uab. These three interaction parameters as in the initial interaction parameter of the Fixman–Edwards model are defined in terms of the binary cluster integrals u = 共1 / 2兲 兰 dr关1 − Exp共−V共r兲 / kT兲兴, where the average coarse potential V共r兲 is the potential between two A, two B and one A and one B units for the three cases.11 It is straightforward for example to use a realistic potential V共r兲 with a hard core and an attractive well for the analytic determination of the dependence of u’s on the temperature T. The interaction parameters can take both negative and positive values passing from zero values at specific temperatures equivalent to the ⌰ temperature of a homopolymer. Following our previous works we write for the probability distribution of the polymer having its core at an origin the expression
再
P关R共s兲兴 = P0关R共S兲兴Exp − ua − R共sa⬘兲兴 − ub − uab
冕冕
冕冕
冕冕
II. EVALUATION OF THE MEAN END TO END SQUARE DISTANCES
In what follows we will calculate the six different mean 2 2 2 2 end to end square distances, 具ROA 典, 具RAB 典, 具RAA 典, 具ROB 典, and ⬘ 2 具RBB 典, of the segments joining the corresponding points ⬘ shown in Fig. 1. They describe the average sizes of an A and a B branch but also a pair of two A’s, a pair of an A and a B segment as well as the largest segment between two B ends. By means of the squares of these distances considered to form triangles three effective angles are defined as 2 2 2 2 2 X = cos−1兵关具ROA 典 + 具RAB 典 − 具ROB 典兴其/兵2关具ROA 典具RAB 典兴1/2其,
共3a兲
再
冕冕 冕冕
+ uab 共1兲
where R共s兲 is the position vector of each point at the contour length s of the chain considered as a continuous line and P0关R共s兲兴 is the ideal connectivity term of the Gaussian type. ␦ is the d-dimensional Dirac delta function which brings in contact all pairs of the same or different units coming from the same or different branches. As it is seen from Eq. 共1兲 more positive values of u’s mean smaller probabilities of contact of the units. Our description and analysis is based on the mean end to end square distances of all possible branches as well as on specific combinations of them. By means of path integrals the mean end to end distance of a specific part of the macromolecule can be written as
共3c兲
They can be used in order to specify to a first approximation the relative positions of the ends of the branches compared to those of an ideal chain known to have an average angle of / 2 between adjacent chain parts. Writing both numerator and denominator of Eq. 共2兲 to first order in the three interaction parameters u we take to the same order expressions of the form
+ ub
冎
2
2 2 典 − 具RBB⬘典兴其/兵2具ROB 典其. Z = cos−1兵关2具ROB
dsadsa␦关R共Sa兲
dsadsb␦关R共sa兲 − R共sb兲兴 ,
共3b兲
and
具R2典 = 具R2典0 1 + ua
dsbdsb␦关R共sb兲 − R共sb⬘兲兴
2
2 2 典 − 具RAA⬘典兴其/兵2具ROA 典其 Y = cos−1兵关2具ROA
冕冕
didjFa共i, j兲
didjFb共i, j兲
冎
di dj Fab共i, j兲 ,
共4兲
where 具R2典0 is the ideal chain result which considering for simplicity the unit lengths of the branches equal to unities are equal to Na , Nb , 2Na , Na + Nb, and 2共Na + Nb兲 for the cases 2 2 2 2 2 典0 , 具RAB 典0 , 具RAA 典 , 具ROB 典0, and 具RBB 典 , respectively. The 具ROA ⬘0 ⬘0 F functions include contributions from both the numerator and denominator while the two integrations run all over the contour length of the macromolecule. An observation proved and used before in the study of simpler structures, helpful for the evaluation of the nonideal terms of Eq. 共4兲, is that in any chain structure including a loop of length L the mean end to end square distance is given by the ratio 共d / 2ᐉ2兲d/2C2 / Ld/2+1. C is the length of the common part of the corresponding segment whose mean end to end square distance is evaluated and the loop.12 By means of this ex-
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094908-3
Conformational properties of dendritic block polymers
pression it is easily seen that if a common part between the segment and the loop does not exist the corresponding diagram vanishes. All necessary diagrams for the evaluation of the mean end to end square distances are given in Table I after absorbing the constant 共d / 2ᐉ2兲d/2 in u’s. The nonideality of the problem comes through combinations uN共4−d兲 of the three interaction parameters u = ua , ub, or uab and the two lengths N = Na or Nb of the two different branches. This made us understand that working at the critical dimensionality d = 4 is fruitful because it has the less possible difficulties giving more suitable solutions and conclusions. N共4−d兲/2 behaves like the ln N function at d = 4 which remains in the final answers.13 The calculation is done to first order in the three u’s which bring in contact all pairs of chain points coming from the same or different branches and form a loop. In the expressions, Eqs. 共5a兲–共5e兲, the five mean end to end square distances are given in terms of the
J. Chem. Phys. 125, 094908 共2006兲
corresponding diagrams and their multiplicities. The forms and the values of the diagrams of all five expressions where the lengths Na and Nb are replaced for simplicity with a and b, respectively, are given in Table I. In these diagrams the A branch is denoted by a normal line while the heavy line denotes a B branch. Square dots denote the ends of the segment whose mean end to end square distance is evaluated and the circle dots the i and j integration points moving always in a branch. Notice that only the linear parts of these diagrams survive which express the common necessary part of the initial diagrams which may correspond to nonlinear branched structures too. The simplest possible diagrams are used, while larger diagrams proportional to these simple diagrams are substituted. The diagram for example 2 of 具RAA 典, Eq. 共5d兲 is not included ⬘ because it is equal to four times the simpler diagram .
共5a兲
共5b兲
共5c兲
共5d兲
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094908-4
Kosmas, Vlahos, and Avgeropoulos
J. Chem. Phys. 125, 094908 共2006兲
共5e兲
The values of the diagrams, Table I, are used for the evaluation of the five mean end to end square distances and the three interesting angles X , Y, and Z of Fig. 1. Special characteristic graphs are given for the apprehension of the general trends. III. EFFECTS OF MOLECULAR WEIGHTS
The effectiveness and activity of the B ends of the dendritic polymer depend on their positions within the macromolecule and the angle X. When the angle X is large the end groups are mainly in the outside region of the polymer and very active, while for small values of X they turn inside the polymer matrix and their activity is hindered. The extend of diffusion of the B ends into the interior region and their controlled activity can thus be achieved for certain combinations of the values of the microscopic parameters which determine the angle X. The relation between the angle X and the mean end to end square distances which depend on the parameters of the dendritic polymer. Eqs. 共3a兲–共3c兲 and 共5a兲– 共5e兲, are valuable for this purpose. They determine the dependence of X on microscopic parameters which specific examples are presented in the figures. The angle Y, Fig. 1, between the position vectors of two A ends gives us an indication of the average angle between two A branches and the empty spaces which the interior region can have. For larger angles Y larger emptiness in the interior of the dendritic polymer is expected and larger capability to carry quest molecules. Similarly, examples of the dependence of the Y angle on the microscopic parameters taken from Eqs. 共3a兲–共3c兲 and 共5a兲–共5e兲 are given in the figures. A final interesting property of the block dendritic copolymer is the controlled permeability and effective tweezing. Large passages from the outside space through the B matrix permit the easier entrance of cargos to be carried, into the interior generation of the dendritic polymer. Closing these passages after the entrance by changing the solvent or the temperature means better holding of the guests in the polymer matrix and security in the carrying process. A release of the cargos can be done in the end of the process after enlarging the exit passages again. The controlled size of the B pores and the passages can be described by means of the third angle Z formed by the vectors
ROB, ROB⬘ starting from the core point and ending on two different B ends. The behavior of the Z angle is also exhibited on the figures. We start with the effects which the molecular weights of the two kinds of branches can have on the conformational behavior of the dendritic polymer. For repulsions, u’ s ⬎ 0, the main tendency of all three angles is to decrease on increasing the molecular weight Na of the interior branch, as it is shown in Fig. 2. Though the X, Y, and Z angles are all larger than the angle / 2 of the ideal state they approach in the limit of large A branches the ideal value / 2 = 1.57 shown in Fig. 2 with the horizontal line. The explanation of this lies on the existence of a common origin of the interior branches which forces them to be close in space. Because of the proximity of the branches the monomeric repulsions expand them towards the outer space making the interior branches more extended for larger Na. This extension permits the A ends to come closer in space reducing thus both the Y and Z angles. The expansion of the A branches reduce the density of the interior region and the angle X is also reduced on increasing Nabecause the B ends can easier turn into the less dense A matrix. Notice the smaller effect of the variation of Na on the Z angle which is away of the A branches than those on the other two nearby angles. Opposite trends are observed for attractions between the monomeric units, lower curves of Fig. 2, which make the branches more compact. The behaviors shown in Fig. 2 are also taken place in simple star macromolecules 共limit of small Nb兲 and this star effect is
FIG. 2. The dependence of the angles on the molecular weight of the interior branch for repulsions 共graphs above兲 and attractions 共graphs below兲. X 共dash兲, Y 共dots兲, and Z 共line兲, the horizontal line represents the ideal case of zero interactions with the angles constant and equal to / 2.
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094908-5
Conformational properties of dendritic block polymers
J. Chem. Phys. 125, 094908 共2006兲
TABLE I. Forms and values of the diagrams.
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094908-6
Kosmas, Vlahos, and Avgeropoulos
J. Chem. Phys. 125, 094908 共2006兲
FIG. 3. The dependence of the angles on the molecular weight of the exterior branch for repulsions and attractions, X 共dash兲, Y 共dots兲, and Z 共line兲, while the horizontal line represents the ideal case of zero interactions with the angles constant and equal to / 2.
due to the existence of the core which is the common origin of the A branches. It is not expected to take place when the branches do not have a common origin and indeed this is what is observed in Fig. 3 with the enlargement of the length of the B branches. An increase of the angles is shown to occur on increasing the Nb length of the branches of the outside shell and for repulsions. Opposite behavior is observed also in this case in the region of attractions where the branches are more compact. IV. EFFECTS OF THE NUMBER OF BRANCHES AND THE INTERACTIONS
The behavior seen on the variation of the numbers f a and f b of the branches of the two kinds, Figs. 4 and 5 shares the
FIG. 4. The dependence of the angles on the interaction uab for various numbers f a, of the interior branches.
FIG. 5. The dependence on uab interactions for various numbers f b of the exterior branches.
same origin as that of the variation of the lengths of the branches. The angles Y and Z being of the same nature have similar behavior and decrease on increasing the number f a 关Figs. 4共y兲 and 4共z兲兴. This behavior is also due to the common starting point of the A branches and the smaller space available to each of them when they are more. The X angle gets larger on increasing f a indicating the stretching of the pairs of the A and B branches to the outside space, Fig. 4共x兲. The increase of the number f b of the B branches on the other hand which do not have a common origin but belong to larger volumes the repulsions push them further away and give larger angles Fig. 5. Moving to the study of the effects of the three kinds of interactions we stay on Fig. 4. We see that on increasing the uab repulsions which have both intra- and interbranch origin all angles increase as expected. The similarity of the Y and Z angles and their opposite behavior from X on increasing f a is apparent in these figures. A similarity of the forms of the variation of the Y and Z angles are also seen in Fig. 5 where the dependence on uab for various numbers of f b are seen. Increasing the number f b of the branches of the outside shell though the three angles always increase. The Y and Z angles are less susceptible to the increase of the uab interactions 关Figs. 5共y兲 and 5共z兲兴 and differ from that of the X angle 关Fig. 5共x兲兴. The steeper increase of the X angle indicates the enlarging effects of the uab repulsions between the A and B branches having a common origin. Of interest also is the study of the state of the branches
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094908-7
J. Chem. Phys. 125, 094908 共2006兲
Conformational properties of dendritic block polymers
FIG. 6. The dependence of the three angles on the expansion of the A and B branches, X 共dash兲, Y 共dots兲, and Z 共line兲.
on the angles shown in Fig. 6. The different nature of the inside A branches from the outside B branches is apparent from the opposite effects of the ua and ub interactions shown in the figure. It is evident again that the reduction of the angles on increasing the ua repulsions 共heavy lines of Fig. 6兲 emanates from the extension of the A branches which share the common origin. They are forced to become extended to larger spaces increasing the freedom to their ends to approach each other. The elongation of the A branches makes them more compatible. The B branches do not start from a common junction and contrary to the ua repulsions the ub repulsions increase the angles 共normal lines of Fig. 6兲. This is mainly due to the larger freedom on the B branches which repelling each other can easier get further apart. The description of the regions of attractions shown in Fig. 7 are more complicated. Though again increasing the uab repulsions the
angles increase in the state of poor solvents of the two branches 共ua , ub ⬍ 0兲, a smaller effect is seen on X with the variation of the number f b of branches, Fig. 7共x兲. The two angles Y and Z behave similarly in this case too, Figs. 7共y兲 and 7共z兲 have an inversion point in their uab dependence. For large uab repulsions increasing f b the Y and Z angles increase but when uab is small which means larger compatibility between different monomers the order is inverted and increasing f b the two angles decrease. This last order is the same as that of the X angle shown in Fig. 7共x兲. The behavior in this state of poor solvent of the branches is a consequence of the smaller volumes which the branches occupy which are disturbed only for large uab repulsions. V. CONCLUSION
We study a block copolymer dendrimer of the first generation which has different number of branches with different molecular weights in the two generations. Three different interaction parameters between the same or different monomeric units are used which affect as well the conformational properties of the polymer. The average sizes of the branches and pairs of branches have been determined and used to find three effective angles between one interior and one exterior branch in the same dendron two branches of the interior region and two exterior branches at different dendrons. The dependence of these angles which describe the penetration of the end groups in the interior and vacancies in the interior and the exterior regions on the molecular weights the number of branches and the interaction parameters is analyzed. For repulsions while the three angles decrease on increasing the length Na of the interior branches because of their enhancing extension the increase of the length of the branches of the outside shell which do not have a common origin has as a result the increase of the three angles. Because of this difference in the interior and exterior branches an opposite behavior is seen with the increase of the ua and ub repulsions also which decrease and increase the three angles, respectively. The forms of the variations of the two angles Y and Z which express angles between similar chain segments starting from the centre of the dendritic copolymer are similar and differs from that of the X angle between the two different kinds of branches. ACKNOWLEDGMENTS
This research was cofunded by the European Union in the framework of the program “Pythagoras I” of the “Operational Program for Education and Initial Vocational Training” of the 3rd Community Support Framework of the Hellenic Ministry of Education, funded by 25% from national sources and by 75% from the European Social Fund 共ESF兲. E. Esfand and D. A. Tomalia, Drug Discov. Tod. 6, 427 共2001兲; J. Luo, M. Haller, H. Ma, S. Liu et al., J. Phys. Chem. B 108, 8523 共2004兲; A. V. Ambade, E. N. Savariar, and S. Thayumanavan, Mol. Pharmacol. 2, 264 共2005兲; X. Shi, I. J. Majoros, and J. R. Baker, Jr., ibid. 2, 278 共2005兲; V. J. Venditto, C. A. S. Regino, and M. W. Brechbiel, ibid. 2, 302 共2005兲; T. D. McCarthy, P. Karellas, S. A. Henderson, M. Giannis, D. F. O’Keefe, G. Heery, J. R. A. Paull, B. R. Matthews, and G. Holan, ibid. 2, 312 共2005兲. 2 G. R. Newkome, C. N. Moorefield, J. M. Keith, G. R. Baker, and G. H. 1
FIG. 7. The dependence on uab when the branches are in poor states with negative ua and ub.
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094908-8
Kosmas, Vlahos, and Avgeropoulos
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