Monte Carlo Simulation of the Phase Behavior of Model Dendrimers

6298

Macromolecules 2006, 39, 6298-6305

Monte Carlo Simulation of the Phase Behavior of Model Dendrimers Anastassia N. Rissanou,† Ioannis G. Economou,† and Athanassios Z. Panagiotopoulos*,‡ Molecular Thermodynamics and Modeling of Materials Laboratory, Institute of Physical Chemistry, National Center for Scientific Research “Demokritos”, GR-153 10 Aghia ParaskeVi, Attikis, Greece, and Department of Chemical Engineering, Princeton UniVersity, Princeton, New Jersey 08544 ReceiVed June 15, 2006; ReVised Manuscript ReceiVed July 13, 2006

ABSTRACT: The phase behavior of lattice model dendrimers of generations from 2 to 5 in solution was studied using grand canonical Monte Carlo simulations. The critical properties were calculated using multihistogram reweighting techniques combined with mixed field finite-size scaling. Power law exponents were obtained for the critical volume as well as for the dendrimer radius of gyration. Significant differences in these exponents occur compared to linear molecules. A systematic decrease of the critical volume exponent with the generation and the molecular weight was observed. A decrease of the power law exponent for the radius of gyration with respect to the molecular weight at the critical conditions is also reported. The phase coexistence envelopes of dendrimer-solvent mixtures were calculated. Finally, the local molecular structure and the local density profiles calculations indicate more compact molecules for higher generation dendrimers.

I. Introduction Dendrimers are relatively new polymeric materials that have attracted significant interest from both academic and industrial communities. They are highly branched macromolecules whose structure is based on a multifunctional (usually trifunctional) core with multiple layers of relatively short branches. This treelike structure provides unique properties to the molecules that have been associated with a variety of technological applications such as drug delivery devices,1,2 building blocks for nanostructures,3 separations,4 and many others. At the same time, systematic experimental, theoretical, and computational understanding of their complex microscopic structure and prediction of macroscopic physical properties remain challenging problems. Significant progress has been made in elucidating the structural properties and conformations of dendrimers,5-10 although relatively little attention has been paid to their critical properties.11 Molecular simulations based on atomistic7 and coarsegrained representation5 have been used to calculate the distribution of end groups of dendrimers, and it was shown that end groups are distributed in space throughout the molecule. Recent molecular dynamics investigations have examined the structure and dynamics of dendrimer melts under shear.12-14 The picture of the structure of flexible dendrimers emerging from simulations is in agreement with theoretical models5 and experimental studies using small-angle neutron scattering (SANS)15-17 and small-angle X-ray scattering (SAXS).18-20 However, studies based solely on radius of gyration data21 have been shown to be inconclusive. Thermodynamic properties of dendritic polymers in athermal solutions have been examined on the basis of Monte Carlo (MC) simulations,22 and their PVT behavior has been calculated. Interestingly, it was shown that as dendrimer concentration increases, the equilibrium pressure is much higher than the pressure of a linear polymer of the same molecular weight and concentration. This difference increases for higher dendrimer generation values and lower lengths of spacers between generations. †

National Center for Scientific Research “Demokritos”. Princeton University. * Corresponding author. E-mail: [email protected].

‡

A previous grand canonical Monte Carlo simulation study11 focused on the phase behavior and critical constants of branched polymers, including dendrimers of low generation number (up to g ) 2). It was shown that as polymer branching increases, the critical temperature decreases and the critical volume increases. These trends were attributed to the molecular contraction with the degree of branching and are in qualitative agreement with the lattice cluster theory.23 In addition, the power law behavior of the radius of gyration and of the critical volume fraction with respect to the chain length appeared to be the same as in the case of linear chains. In this work, we studied dendrimers of higher generations, up to the fifth (g ) 5). The phase behavior, critical parameters, and structure of high generation and of high molecular weight dendrimers were investigated. The effects of spacer length (part of the molecule between two consecutive branch points) were also examined. The validity of power scaling laws proposed for linear chains is evaluated at high molecular weight and generation numbers. The rest of the paper is organized as follows: Section II contains details for the systems studied together with the simulation method. Results and associated discussion are presented in section III. Finally, section IV contains the concluding remarks. II. Model and Simulation Method The systems studied consist of regularly branched homopolymers (dendrimers) on a simple cubic lattice with coordination number z ) 26. This coordination number implies that successive beads on a chain can be along coordinate directions (0,0,1), (0,1,1), and (1,1,1) and equivalent vectors on the lattice, thus resulting in possible bond lengths of 1, x2, or x3 lattice spacings. The lattice spacing is taken to be the unit of length. Model dendrimers are built as follows: the innermost part of the molecule consists of a linear subchain with two end beads. There are s spacer beads between the two end beads, where s may also be zero (no spacers). The dendrimer of g ) 0 is then a linear chain of s + 2 beads. In the case of dendrimers with g g 1, each of the end beads of the inner part branches into two additional arms, identical to the inner part, resulting in a first generation dendrimer (g ) 1).

10.1021/ma061339u CCC: $33.50 © 2006 American Chemical Society Published on Web 08/11/2006

Macromolecules, Vol. 39, No. 18, 2006

Phase Behavior of Model Dendrimers 6299

Figure 1. (a) Schematic of the topology of a dendrimer of generation g ) 2 with s ) 1 (N ) 27) and (b) actual configuration of a dendrimer of generation g ) 4 with s ) 16 (N ) 1038) at temperature T ) 20 and low density. Colors correspond to beads of successive generations: blue (G ) 0), red (G ) 1), yellow (G ) 2), green (G ) 3), and gray (G ) 4).

All subsequent generations are built up in the same way. The macromolecular architecture of these dendrimers is described by two numbers: the number of spacers, s, and the generation number, g. The total number of generations of a dendrimer is g, while the inner generations are denoted by G (0 e G e g). The total number of beads of a g-generation dendrimer is given by the expression

N ) (bg+2 - b - 1)(s + 1) + 1

(1)

where b ) f - 1 is the branching ratio and f is the functionality of the branch point, here taken to be f ) 3 as is commonly the case. Figure 1a depicts a two-dimensional schematic of the topology of a dendrimer with g ) 2, s ) 1 (N ) 27). Figure 1b shows an actual configuration of a dendrimer with g ) 4, s ) 16 (so that N ) 1038) at temperature T ) 20 and low density. Similar dendrimer models have been used previously in simulation studies.22,24 Mansfield24 considered a model dendrimer on a diamond lattice. The dendrimer had a ternary core, consisting of three individual strands, and binary branches. Each strand was comprised of seven steps on the lattice. Lue22 performed off-lattice Monte Carlo simulations where the dendritic polymers had a central core molecule that branched off into three arms. Furthermore, each arm branched into two

additional branches. The beads were hard spheres that were not allowed to interpenetrate. In this work, we studied dendrimers of generation from g ) 0 (linear polymers) to g ) 5. In addition, the dendrimer spacer s was allowed to vary. In this way, different molecular weights for the same generation were examined. The length of the terminal branches was kept equal to the length of the corresponding spacer. Dendrimers of high molecular weight with polymerization index N up to 3966 in generation 4 and up to 2126 in generation 5 were examined. In our model, solvent molecules occupy lattice sites not excluded by the polymer sites. Interactions between polymerpolymer, polymer-solvent, and solvent-solvent molecules are quantified through the energy parameters pp, ps, and ss, where the polymer-polymer interaction is negative (attractive) while the rest are set to zero. We thus have attraction between “nearestneighbor” polymer segments within x3 lattice sites of each other, consistent with the z ) 26 lattice coordination number. Hence, in the calculations there is a single energy parameter /kB ) (2ps - ss - pp)/kB ) 1, where kB is Boltzmann’s constant. The (reduced) temperature scale is defined by normalizing with the absolute value of the energy parameter. The volume fraction φ is defined as the fraction of lattice sites occupied by polymer beads. MC simulations were performed in the grand canonical (µVT) ensemble. The reference point for the chemical potential was the reversing random walk with no interactions, as in previous work.25 For all systems studied, the box length L was at least 10 times larger than the radius of gyration of the dendrimers. Simulations were performed exclusively on the basis of insertions and deletions of whole dendrimer molecules using configurational bias moves.26 The configurational bias moves were based on inserting new beads in empty lattice positions, without taking into account the energetic interactions that they would experience in these positions. The reason for this choice is that, in order to take into account the energy of interaction in the new positions, z ) 26 neighboring positions would need to be examined, considerably slowing down the calculations and more than compensating for the improved move acceptance that would have resulted. For linear polymers, the acceptance ratio of insertion/deletion moves decreases as molecular weight increases. However, for dendrimers of a given generation, molecular weight increases through the increase of spacers which creates additional free space in the inner part of the molecule and leads to an enhancement of the acceptance ratio. Competition between these two effects results in a nonmonotonic behavior of the acceptance ratio with molecular weight. Typical acceptance ratios at the critical conditions were 7% for N ) 2126, g ) 5 and 65% for N ) 245, g ) 0. Simulations started with preliminary short runs for the approximate estimation of the critical point followed by four long runs at µ and T values in the vicinity of the critical point. The resulting histograms are combined using the procedure of Ferrenberg and Swendsen27-29 and analyzed according to the mixed-field finite-size scaling theory.30,31 The combined histogram (probability distribution) Pµ,β(U,N) is mapped to a distribution in the ordering operator M given by the expression

M ) Nch - smixU

(2)

where β ) 1/kΒT, Nch is the number of polymer chains, U is the internal energy of the system, and smix is a field-mixing parameter which controls the strength of coupling between the energy and density fluctuations near the critical point. At the

6300

Rissanou et al.


Table 1. Critical Parameters for Dendrimers of Generation g with s Spacers and N Total Beads Determined in Systems of Linear Size La µc

103smix

-119.964 -86.703 74.008 288.31 287.793 588.907 1469.21 2073.02 3299.15 59041

-1.286 -0.532 -0.311 -0.221 -0.221 -0.161 -0.0952 -0.0771 -0.0551 -0.0361

g)2 0.17714

-116.201

-1.02

g)3 17.43008 0.14643

-108.093

-0.57302

g)4 0.21535 0.13755 0.13713 0.12721 0.11245 0.0931 0.07974 0.06936 0.06136 0.05116 0.04633

-93.243 -69.11 -68.91 -48.83 -2.11 107.75 291.24 554.58 827.77 13932 1976.19

-0.6201 -0.321 -0.393 -0.2902 -0.2503 -0.2409 -0.1664 -0.08001 -0.0974 -0.0858 -0.0623

-45.36 -22.48 172 65.17 118.54 233.96 487.28 8912

-0.5862 -0.2481 -0.1936 -0.23801 -0.17301 -0.131 -0.12401 -0.07554

s

N

L

Tc

63 243 598 998

65 245 600 1000

1524 2988 3964 5916 9998

1526 2990 3966 5918 10000

50 80 95 120 85 150 200 230 280 350

4

66

30

16.4291

φc

g)0 16.7423 0.15704 18.56668 0.09673 19.35048 0.06822 19.67987 0.05528 19.67682 0.05611 19.90013 0.046331 20.17233 0.034978 20.26394 0.030907 20.37355 0.026064 20.48837 0.02081

4

146

50

0 3

62 245

4 6 10 16 24 32 48 64

306 428 672 1038 1526 2014 2990 3966

35 100 60 80 120 100 120 170 150 180 205

15.8792 17.8152 17.8202 18.0734 18.4221 18.8454 19.1932 19.4603 19.6252 19.8313 19.9621

45 70 80 100 100 100 130 150

g)5 16.782 0.1962 17.591 0.1574 18.032 0.1374 18.3046 0.1233 18.5133 0.1153 18.8023 0.1018 19.1543 0.0832 19.4395 0.0722

0 1 2 3 4 6 10 16

126 251 376 501 626 876 1376 2126

a Statistical uncertainties are listed as subscripts in units of the last decimal point shown.

critical point, the probability distribution should follow a universal distribution corresponding to Ising-type criticality with short-range interactions. The Tsypin and Blo¨te order parameter distribution for the three-dimensional Ising universality class32 was used in the critical point calculations. Even though the mixed-field finite size scaling approach has been used previously for determination of critical points of lattice polymer models,11,25,33,34 values for the field mixing parameters have not been reported. In Table 1, the smix parameter is shown for the models studied, along with the calculated values of the critical parameters Tc (temperature), φc (volume fraction), and µc (chemical potential). All the smix values obtained were negativesat each generation, smix decreases in absolute value with molecular weight. It is interesting to note that for the monomer the field mixing parameter is exactly zero because of symmetry, so there is nonmonotonic behavior at low molecular weights. Furthermore, at high generations some scatter in the smix values is observed because sampling efficiency is lower. The number of MC steps and the corresponding simulation time increase for higher molecular weight and higher generation. Typical CPU times were between 2.5 and 140 h on Pentium 4 3.4 GHz processors for a single state point. Coexistence points at subcritical conditions were obtained through additional simulation runs. Reweighting of the histograms at the critical point provided the initial estimate of coexistence chemical potential at lower temperatures. This chemical potential value has to satisfy the requirement of equal

Figure 2. Scaling of the critical temperature with chain length based on the Flory-Shultz relationship for dendrimers with g ) 0, 4, and 5. Statistical uncertainties are smaller than symbol size; lines are leastsquares fits to the simulation data.

areas under the vapor-phase and liquid-phase peaks. Subsequently, a new simulation run was performed at these conditions, and the resulting histogram was combined with the previous in order to estimate the coexistence chemical potential at even lower temperatures. The procedure was repeated until the desired temperature range was covered. III. Results and Discussion The critical point data for all systems studied are presented in Table 1. Previous literature data on critical parameters are limited to linear homopolymers (g ) 0) or low generation (g e 2) dendrimers.11 For the linear chains, the spacer length s is equal to the total number of beads N between the two end beads, s ) N - 2. Statistical uncertainties were obtained from multiple runs with different pseudorandom sequences and are listed in Table 1 as subscripts in units of the last decimal point shown. Results for the critical temperature (Tc) for dendrimers of generation 0, 4, and 5 are presented in Figure 2. They conform to the Flory-Shultz relationship35

1 1 1 1 ∝ + Tc(r) Tc(∞) xN 2N

(3)

where Tc(∞) is the critical temperature for chains of infinite length. Extrapolation of simulation data to infinite molecular weight values (N f ∞) using eq 3 provides an estimate for the Θ temperature. The following Θ values are obtained: for g ) 0, Θ ) 20.87, for g ) 4, Θ ) 20.79, and for g ) 5, Θ ) 20.62. The combined statistical and extrapolation uncertainty in these values is of the order of 0.01. As the dendrimer generation increases, the Θ temperature decreases, in agreement with previous simulation studies for lower generation number dendrimers and other branched polymers.11 This suggests that the presence of a finite number of topologically constraining branching points results in measurable differences in Θ temperature even for chains with an infinite number of beads. The critical parameters reported in Table 1 and Figure 2 and the extrapolated Θ temperatures are system size dependent. Appropriate corrections may be needed when these values are extrapolated to the thermodynamic limit. However, the size of the simulation box is large, on the order of 10Rg, and so these corrections are expected to be small. Tests of finite-size effects on the critical parameters of two specific systems, g ) 0, s ) 998 and g ) 4, s ) 3 were performed, as shown in Table 1. The effect on the apparent critical temperatures is 0.03% or less.


Figure 3. Scaling of the critical volume fraction with the chain length of dendrimers with g ) 0, 4, and 5. Lines are linear fits to the data and error bars are smaller than symbol size.

Differences in the apparent critical volume fractions of up to 1.6% are observed, but the critical volume fractions are themselves determined with much lower accuracy than the critical temperatures. Only results from the larger system sizes were used in subsequent calculations of the scaling of the critical parameters. The number of dendrimer molecules at the critical conditions can be calculated from the data in Table 1 and ranged from approximately 100 to 300 for the larger system sizes. Our simulation data for the critical volume fraction appear to follow a power law scaling with respect to the molecular weight, φc ∝ N-x2, as depicted in the doubly logarithmic plot of Figure 3. Exponent x2 was determined to be equal to 0.416 ( 0.003, 0.400 ( 0.007, and 0.381 ( 0.010 for generation 0, 4, and 5, respectively, from linear regression of the data. A systematic decrease of critical volume exponent with generation and molecular weight is observed that has not been reported before for low-generation dendrimers.11 In the present work, we studied chain molecules significantly longer than in previous studies,11,25,36 and we noticed that for linear chains there is a slight drift of x2 toward higher values as the molecular weight increases. Previous reported values were in the range 0.370.39 for chain lengths up to 1000,25,36 while the current value is 0.416 ( 0.003 for linear chains of length up to 10 000. Even longer chains could result in further increase of this exponent to its theoretical value of 0.5. All these values should be compared to the experimental measurements37,38 that predict x2 ≈ 0.38-0.40. It is possible that the apparent decrease of the effective volume fraction exponent for higher generation dendrimers simply indicates that they are further away from the limiting value of 0.5 which they will eventually attain at very high molecular weights, inaccessible at the moment to either experiments or simulations. The effect of branching on the critical volume fraction can be quantified by comparing results for three systems with polymer chains of almost the same number of monomers but of different generation. These systems have (a) N ) 65 and g ) 0, (b) N ) 66 and g ) 2, and (c) N ) 62 and g ) 4. The corresponding φc values are 0.1570 ( 0.0004, 0.1771 ( 0.0004, and 0.2154 ( 0.0005, respectively. Obviously, highly branched polymers begin to phase separate at higher volume fractions compared to their linear analogues, which is also in agreement with previous experimental results.39 An alternative way to scale the critical parameters is by keeping the spacer length constant and increasing the dendrimer generation, thus obtaining an increase in dendrimer molecular weight. The study of dendrimers of generation higher than 5 is impossible within our model, due to limitations caused by the


Figure 4. Scaling of the dendrimer critical temperature, Tc, with the chain length using the Flory-Shultz relationship for s ) 4 and g varying from 2 to 5. Error bars are smaller than symbol size, and the line is least-squares fit to the simulation data.

Figure 5. Calculated phase envelopes for dendrimers of g ) 4 with s ) 0, 16, and 32 and of g ) 5 with s ) 0 and 16.

steric interactions. Nevertheless, an attempt was made to extrapolate our data to infinite molecular weight in this fashion. In Figure 4, a Flory-Shultz diagram is shown for dendrimers with s ) 4 and g ranging from 2 to 5. By extrapolating the data to infinite molecular weight, one obtains Tc(∞) ) 19.65, which is a value lower than the Θ temperature extrapolated above from the simulations for generations 4 and 5 individually. This lower value can be considered as the limit of Θ temperatures when the number of topological constraints (branching points) becomes extremely large. A similar conclusion can be drawn from the critical volume fraction data as a function of the molecular weight for different generation dendrimers. A fit limited to only the three higher generations studied for fixed spacer length s ) 4 (graph not shown) gives a power law exponent equal to 0.164 ( 0.015, lower than the one calculated above for dendrimers of generation 4 or 5 with varying spacer lengths. Phase coexistence envelopes were calculated for dendrimers of different generations and number of spacers. Results are presented in Figure 5 for three dendrimers of generation 4 with spacer length equal to 0, 16, and 32 and two dendrimers of generation 5 with spacer length equal to 0 and 16. As the dendrimer molecular weight increases, the phase coexistence curve shifts to higher temperatures and lower densities, in agreement also with linear or light branched chains.40 Interestingly, for dendrimers of similar molecular weight (denoted as g5s16 and g4s32), a higher generation results in a suppression of the phase coexistence envelope to lower Tc and higher φc, as also seen in Table 1.

6302

Rissanou et al.


Figure 6. Radius of gyration as a function of chain length for dendrimers of g ) 0, 4, and 5 at the critical point conditions for each molecular weight. Error bars are smaller than symbol size.

We turn now our attention to some structural characteristics for the g ) 4 and the g ) 5 dendrimers and a comparison with linear polymer chains. The radius of gyration at the critical point conditions for each molecular weight, Rg,c, as a function of the molecular weight is presented in Figure 6 for these three polymer systems. A power law fit of the data of the form

Rg,c ∝ N ν

(4)

provides an exponent ν which characterizes the size of the polymer chain and gives an estimation of its degree of packing. We observe that higher generations lead to more compact conformations, as the value of ν decreases with increasing generation. That is, ν ) 0.509 ( 0.003 for g ) 0, ν ) 0.495 ( 0.003 for g ) 4, and ν ) 0.4886 ( 0.0003 for g ) 5. The radius of gyration was computed also at the estimated Θ temperature, as extracted from the extrapolation of the data of Figure 2 to infinite molecular weight. For linear chains, the ν exponents at the Θ temperature are in very good agreement with the theoretically predicted value of 0.5. For dendrimers, the exponent ν is lower than 0.5 at T ) Tc and increases as temperature is increased toward the estimated Θ value, yet it remains below 0.5 at the infinite molecular weight Θ temperature. This implies that the dendrimers are slightly collapsed at their critical point for phase separation and the infinite molecular weight Θ temperature. Simulations were performed also at a high temperature of T ) 200 to calculate Rg of dendrimers in the athermal limit. For the linear chains, an exponent of ν ) 0.574 ( 0.008 was obtained, near the theoretical predicted value of 0.58841 and results from previous simulation studies.11,42 For the highgeneration dendrimers, we obtained ν ) 0.571 ( 0.002 for g ) 4 and ν ) 0.563 ( 0.006 for g ) 5. For the g ) 4 and the g ) 5 dendrimers these values seem to be systematically below the theoretically predicted value for linear chains.41 This difference may be attributed to the same reason for which the asymptotic behavior for the critical volume fraction is not seen in dendrimers, namely that we are not yet in the asymptotic “long chain” limit at accessible molecular weights for dendrimers. The variation of the radius of gyration, Rg, with chain length is plotted in Figure 7 for dendrimers of generation 2-5 and spacer length equal to 4, in all cases. Interestingly again, the power law fit results in a ν exponent value of 0.292 ( 0.001, which is considerably lower than the values reported above when examining dendrimers of generation 4 and 5 with varying spacer lengths. This indicates once more that dendrimer molecules tend to attain more compact conformations (i.e., globules) as the

Figure 7. Radius of gyration as a function of chain length for dendrimers with g ) 2 through 5 and s ) 4 at the critical conditions. Error bars are smaller than symbol size.

generation increases for constant spacer length, while they exhibit a different behavior as the spacer length increases for a given generation. In this case, they tend to form flexible coils that expand with the increase of molecular weight, similarly to the behavior of linear chains under Θ conditions. As suggested earlier, steric interactions in our model do not allow the study of dendrimers with g > 5. This reduces the range of molecular weight values that can be examined, and consequently exponent ν does not attain the exact value of 1/3 for globular conformations.43 Our findings for the Rg scaling law are in agreement with previous simulation studies.8,44,45 By contrast, Sheng et al.46 found that when Rg is considered as a function of N with a fixed number of spacers, s, then a higher exponent, equal to 1/3, is attained, independent of the solvent quality. If one keeps the generation number constant and allows N varying through the change of the spacer length, then a different Rg scaling is attained, of the form Rg ∝ N3/5, which is identical to the behavior of linear polymers in athermal solvents. The fact that in ref 46 the power laws were followed even for small s values (and thus molecular weights) was attributed to the potential model used. The system of ref 46 behaves similar to stiff linear chains that reach the scaling regime at smaller molecular weights. To address the controversy in the literature for the scaling of the radius of gyration with the number of monomers, Chen and Cui47 proposed a single scaling law which describes all dependencies correctly and is given by the following formula:

Rg ∝ (sg)2/5N1/5

(5)

By using this scaling law, we obtained an exponent for N equal to 0.1615 ( 0.0007 for dendrimers of g ) 4 and 0.150 ( 0.002 for dendrimers of g ) 5 with a varying spacer number s. For fixed s and varying g values, we obtained an exponent equal to 0.1612 ( 0.0003. These results are shown in Figure 8 and indicate an almost common law for Rg scaling. The exponent reported here deviates from the 0.2 value proposed by Chen and Cui.47 Having in mind that our findings are in a restricted range of molecular weights, 66 e N e 3966, they should not be interpreted as a universal law, but rather as a local fit to the simulation data. It should be mentioned that Go¨tze and Likos6 reported an even higher exponent of 0.24 for dendrimers in an even narrower molecular weight range 400 e N e 2000. The structure of a single dendrimer molecule is of significant interest for applications such as drug delivery systems. The quantity of interest is the local density of different branch levels, F(r), where r is the radial position. In Figure 9, the local density profile for a dendrimer with g ) 5 and s ) 16 is shown, whereas


Figure 8. A universal scaling law for the radius of gyration which incorporates all the dependencies for dendrimers with variable s and constant g ) 4 and g ) 5 and with variable g and constant s ) 4 at the critical conditions. Error bars are smaller than symbol size.


Figure 10. Intermolecular pair radial distribution function for dendrimers with g ) 5 and different spacer length.

Figure 11. Total density profiles as a function of the distance from the center of mass for dendrimers of g ) 3 through 5 and s ) 4.

Figure 9. Density profile of different generation numbers as a function of the distance from the center of mass for dendrimers with g ) 5 and s ) 16. The total density is scaled by a factor of 1/3 to fit the scale of the graph.

the total density of the molecule is scaled by a factor of 1/3. As the branch level increases, the local density increases, nonmonotonically at short distances and monotonically at larger distances from the molecule center. At short distances, the first generation beads tend to be more localized because they are connected to the molecule core while at larger distances higher generation beads are less localized and spread throughout the entire molecule. On the other hand, the total dendrimer density decreases monotonically with distance, as observed experimentally.16,17 The distribution of terminal groups observed for our model is broadly similar to that observed for flexible dendrimer models in the continuum.12 While we did not perform a systematic study of the variation of dendrimer dimensions with concentration, we observed a slight reduction of overall size when concentration is increased at a constant temperature. The intermolecular pair correlation function, g(r), is presented in Figure 10 for a dendrimer with g ) 5 and different spacer length. The pair correlation function, g(r), is defined on the basis of positions of the centers of mass of dendrimer molecules, which in general does not coincide with any bead of the polymer. The first peak position shifts to higher distances; its intensity decreases and broadens as spacer length increases. This feature indicates that the increase of the molecular weight, for a given generation, facilitates the interpenetration between different molecules. Consequently, a loose structure is observed in high molecular weight systems. The pair correlation functions are again similar to those observed for flexible dendrimer melts in the continuum.14

The feature of more compact molecules for higher generation dendrimers is elucidated by the total density profile, shown in Figure 11 for dendrimers with g ) 3, 4, and 5 and s ) 4. Higher generation dendrimers exhibit higher overall densities at all distances. The density profiles of the individual branches for dendrimers of g ) 3, 4, and 5 at fixed s ) 4 are depicted in Figure 12a-d, corresponding to beads of branching order G ) 0, 1, 2, and 3, respectively. Near the center of the dendrimers, the highest g dendrimers have lower density of segments of low branching order. The physical reason for this is that beads of later generations also occupy space near the center of the dendrimer; there are a lot more such beads in dendrimers of higher molecular weight, thus “pushing out” beads of earlier generations. As the outside surface of the molecule is approached, a crossover of the density occurs so that now higher generation molecules have higher densities, following the overall density profile behavior of Figure 11; this feature is more pronounced at higher branching levels, G. This has been also observed by detailed atomistic molecular dynamics simulations.8 IV. Conclusions In this study we examined the phase behavior of dendrimers of varying generation up to 5 in solution, which provides information for the effect of hyperbranching on the critical properties of polymers. Grand canonical Monte Carlo simulations combined with histogram reweighting techniques were used to obtain the phase coexistence properties of these systems. This is the first time that values for the field mixing parameter, smix, which controls the strength of coupling between the energy and the density fluctuations near the critical point, are reported. The field mixing parameter is an increasing function of the molecular weight in each generation. We studied dendrimers with a number of monomers up to 3966 for generation 4 and 2126 for generation 5 as well as linear chains (i.e., dendrimers

6304

Rissanou et al.


This feature is also apparent in the radius of gyration scaling law exponent, which is approximately equal to 0.3 at the critical conditions, indicative of the development of a globular conformation. On the other hand, increasing the molecular weight (i.e., the number of spacers) in the same generation dendrimers leads to the conformation of random coils with behavior similar to that of linear chains under Θ solvent conditions. For dendrimers with generation larger than 5 and larger molecular weights there is a practical difficulty in calculating critical properties because of the extremely low acceptance ratios of our Monte Carlo simulation scheme. More efficient Monte Carlo moves need to be developed for the reliable simulation of such dendrimers. Acknowledgment. The authors thank Dr. Gaurav Arya for help in the initial stages of this work and for a careful reading of the manuscript. A.N.R. gratefully acknowledges financial support for this work by the General Secretariat of Research and Technology, Greece, through the Collaborative Research Projects with Third Countries (Project ΗΠΑ-010). A.Z.P. acknowledges financial support by the U.S. Department of Energy, Office of Basic Energy Sciences (Grant DE-FG0201ER15121), with additional support by ACS-PRF (Grant 38165-AC9). References and Notes

Figure 12. Density profiles for different branch levels as a function of the distance from the center of mass for dendrimers of g ) 3, 4, 5 and s ) 4: (a) G ) 0, (b) G ) 1, (c) G ) 2, (d) G ) 3.

of generation 0) of chain length up to 10 000. The simulations were performed on a cubic lattice with coordination number z ) 26. A scaling analysis of the critical temperature with respect to the Flory-Shultz relationship35 indicated a decrease in the infinite-system critical temperature for higher generation dendrimers. Furthermore, an increase in the critical volume fraction with the generation was observed. The scaling of the critical volume fraction with respect to the molecular weight provided an exponent x2 which decreases as the generation increases, and the specific values are in good agreement with the experiment.19,20 The radius of gyration scales with the molecular weight with an exponent ν equal to 0.5 at the infinite molecular weight Θ temperature only for linear chains. For dendrimers, we find that the exponent ν is lower than T ) Tc at Θ and increases as temperature is increased toward the estimated Θ value, yet it remains below 0.5 at the infinite molecular weight Θ temperature. In the athermal limit, only the linear chains scale with an ν near the theoretically predicted value of 0.58, whereas the dendrimers of generation 4 and 5 attain an exponent ν gradually lower than this value, which can be attributed to the fact that higher generation dendrimers reach their asymptotic “longchain” behavior at higher molecular weights, presently inaccessible by simulation. Interesting information is extracted through the examination of dendrimers of increasing generation with the same number of spacers between consecutive branch points. Scaling analysis of the critical temperature and critical volume fraction for this set of molecules lead to different results for the Θ temperature and the x2 exponent. Specifically, both values were found to be lower than the ones obtained from each generation separately.

(1) Lee, I.; Athey, B. T.; Wetzel, A. W.; Meixner, W.; Baker, J. R. Macromolecules 2002, 35, 4510-4520. (2) Boas, U.; Heegaard, P. M. H. Chem. Soc. ReV. 2004, 33, 43-63. (3) Bosman, A. W.; Janssen, H. M.; Meijer, E. W. Chem. ReV. 1999, 99, 1665-1688. (4) Seiler, M. Fluid Phase Equilib. 2006, 241, 155-174. (5) Ballauff, M.; Likos, C. N. Angew. Chem., Int. Ed. 2004, 43, 29983020. (6) Go¨tze, I. O.; Likos, C. N. Macromolecules 2003, 36, 8189-8197. Go¨tze, I. O.; Archer, A. J.; Likos, C. N. J. Chem. Phys. 2006, 124, 084901. (7) Karatasos, K. Macromolecules 2005, 38, 4472-4483. (8) Zacharopoulos, N.; Economou, I. G. Macromolecules 2002, 35, 18141821. (9) Maiti, P. K.; C¸ agˇın, T.; Wang, G.; Goddard, W. A. Macromolecules 2004, 37, 6236-6254. (10) Mansfield, M. L.; Jeong, M. Macromolecules 2002, 35, 9794-9798. (11) Arya, G.; Panagiotopoulos, A. Z. Macromolecules 2005, 38, 1059610603. (12) Bosko, J. T.; Todd, B. D.; Sadus, R. J. J. Chem. Phys. 2004, 121, 1091-1906. (13) Bosko, J. T.; Todd, B. D.; Sadus, R. J. J. Chem. Phys. 2004, 121, 12050-12059. (14) Bosko, J. T.; Todd, B. D.; Sadus, R. J. J. Chem. Phys. 2006, 124, 044910. (15) Higgins, J. S.; Benoit, H. C. Polymers and Neutron Scattering; Clarendon Press: Oxford, 1994. (16) Po¨tschke, D.; Ballauff, M.; Lindner, P.; Fischer, M.; Vo¨gtle, F. Macromol. Chem. Phys. 2000, 201, 330-339. (17) Po¨tschke, D.; Ballauff, M.; Lindner, P.; Fischer, M.; Vo¨gtle, F. Macromolecules 1999, 32, 4079-4087. (18) Bauer, B. J.; Topp, A.; Prosa, T. J.; Amis, E. J.; Yin, R.; Qin, D.; Tomalia, D. A. Polym. Mater. Sci. Eng. 1997, 77, 87. (19) Bauer, B. J.; Hammouda, B.; Briber, R.; Tomalia, D. A.; Scherrenberg, R. Polym. Mater. Sci. Eng. 1992, 67, 340-341. (20) Prosa, T. J.; Bauer, B. J.; Amis, E. J.; Tomalia, D. A.; Scherrenberg, R. J. Polym. Sci., Part B 1997, 35, 2913-2924. (21) Lyuulin, A. V.; Davies, G. R.; Adolf, D. B. Macromolecules 2000, 33, 6899-6900. (22) Lue, L. Macromolecules 2000, 33, 2266-2272. (23) Nemirovsky, A. M.; Bawendi, M. G.; Freed, K. F. J. Chem. Phys. 1987, 87, 7272-7284. (24) Mansfield, M. L. Macromolecules 2000, 33, 8043-8049. (25) Panagiotopoulos, A. Z.; Wong, V.; Floriano, M. A. Macromolecules 1998, 31, 912-918. (26) Frenkel, D.; Smit, B. Understanding Molecular Simulation, 2nd ed.; Academic Press: San Diego, 2002. (27) Panagiotopoulos, A. Z. J. Phys.: Condens. Matter 2000, 12, R25R52.

Macromolecules, Vol. 39, No. 18, 2006 (28) Ferrenberg, A. M.; Swendsen, R. H. Phys. ReV. Lett. 1998, 61, 26352638. (29) Ferrenberg, A. M.; Swendsen, R. H. Phys. ReV. Lett. 1989, 63, 11951198. (30) Wilding, N. B.; Bruce, A. D. J. Phys.: Condens. Matter 1992, 4, 3087-3108. (31) Wilding, N. B. Phys. ReV. E 1995, 52, 602-611. (32) Tsypin, M. M.; Blo¨te, H. W. J. Phys. ReV. E 2000, 62, 73-76. (33) Frauenkron, H.; Grassberger, P. J. Chem. Phys. 1997, 107, 95999608. (34) Yan, Q. L.; de Pablo, J. J. J. Chem. Phys. 2000, 113, 5954-5957. (35) Shultz, A. R.; Flory, P. J. J. Am. Chem. Soc. 1952, 74, 4760-4767. (36) Wilding, N. B.; Mu¨ller, M.; Binder, K. J. Chem. Phys. 1996, 105, 802-809. (37) Dobashi, T.; Nakata, M.; Kaneko, M. J. Chem. Phys. 1980, 72, 66856691. (38) Shinozaki, K.; van Tan, T.; Saito, Y.; Nose, T. Polymer 1982, 23, 728-734.

Phase Behavior of Model Dendrimers 6305 (39) Yokoyama, H.; Takano, A.; Okado, M.; Nose, T. Polymer 1991, 32, 3218-3224. (40) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (41) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (42) McKenzie, D. S. Phys. Rep. 1976, 27, 35-88. (43) Grosberg, A. Y.; Khokhlov, A. R. Statistical Physics of Macromolecules; AIP Press: New York, 1994. (44) Murat, M.; Grest, G. S. Macromolecules 1996, 29, 1278-1285. (45) Szleifer, I.; O’Toole, E.; Panagiotopoulos, A. Z. J. Chem. Phys. 1992, 97, 6802-6808. (46) Sheng, Y.-J.; Jiang, S.; Tsao, H.-K. Macromolecules 2002, 35, 78657868. (47) Chen, Z. Y.; Cui, S.-M. Macromolecules 1996, 29, 7943-7952.

MA061339U

Monte Carlo Simulation of the Phase Behavior of Model Dendrimers

Recommend Documents