Dynamics of (hyper)branched macromolecules
We focus on:

Viscoelastic Relaxation of Cross-Linked and Alternating Copolymers

Authors:  Cristian Satmarel, Andrei Gurtovenko, and Alexander Blumen
Journal-ref: Macromolecules 36, 486 (2003)

We study theoretically the viscoelastic relaxation of linear alternating and cross-linked copolymers in the framework of generalized Gaussian structures (GGS), which are extensions of the Rouse model to arbitrary geometries.

$ \sigma _{A}=\sigma _{C}=1$

Figure 1: The polymer systems under study. Shown are (a) an alternating copolymer chain containing two kinds of beads; (b) an elementary cell (dashed line) of a square lattice built from such (identical) copolymer chains.

We are interested in the dynamic behavior of such polymeric structures and focus mainly on the relation between topology and viscoelastic behavior of such macromolecules. In the GGS picture, the beads are connected to each other by harmonic springs with elastic constant K, and we take the structures to be immersed in a viscous solvent where the inertial forces can be neglected. The corresponding Langevin equation of motion for such systems has the form: 

$\displaystyle \zeta_j\frac{\partial{\bf r}_j(t)}{\partial t}-K\sum_{\alpha =1}^...                                                                                 (1)
Equation (1) expresses the fact that for each bead j at position rj(t), the friction force and the elastic forces acting on it balance the external and fluctuating forces.  In terms of a matrix equation, Eq. (1) may be rewritten as
               
Here, $ \zeta_{i}$ is the friction constant of the $ i$-th bead,  $ {\bf f}_{i}$ the sum of the stochastic forces acting on it, and  $ {\bf A}= (A_{ij}) $ is the connectivity (symmetric) matrix of the system. The nondiagonal $ A_{ij} $ equal $ (-1)$ if the $ i$-th and $ j$-th beads are connected, and 0 otherwise; $ A_{ii} $ equals the number of bonds emanating from the $ i$-th bead. Furthermore the sets  $ \{\zeta_{i}\}$ and  $ \{{\bf f}_{i}\}$are connected via the fluctuation-dissipation theorem. We note here that we are mainly interested in the eigenfrequencies of the system,  which depend solely on $ \bf A$.  It is also well known that in the framework of the Rouse model it is sufficient to focus on the thermally averaged Cartesian components of  $ \{{\bf R}_{i}\}$ in order to get the relaxation spectrum. From now on we will hence view Eq. (1) as relating to thermally averaged variables, under which averaging its right-hand-side vanishes. When the beads in the system differ in their mobility (we consider here different friction coefficients) the above set of Langevin equations may be reformulated as follows: We take the beads of the first kind ($ \zeta _{A}$) as reference and have as their characteristic relaxation time $ \tau _{0}={\zeta}_{A}/K$. For the other kind of beads we set

$\displaystyle \sigma _{z}= \frac{\zeta _{A}} {\zeta _{z}},$ (2)

In this way we have for the thermally averaged  $ {\bf R}_{i}(t)$

$\displaystyle \frac{d{\bf R}_{i}(t)}{dt}+(1/\tau _{0})\sum\limits_{j=1}^{N_{tot}}\tilde{A}_{ij}{\bf R}_{j}(t)=0$ (3)

with

$\displaystyle \tilde{A}_{ij}=\sigma _{z_{i}}{A}_{ij}.$ (4)

Here the matrix  $ {\bf\tilde{A}}=(\tilde{A}_{ij})$ is obtained from $ \bf {A}$ by multiplying its rows with the  $ \sigma_{z_{i}}$ corresponding to the different molecular species $ z_{i}$. Then  $ \bf\tilde{A}$ is, in general, not symmetric anymore. In this way the copolymer system under study differs from a homopolymer system. However, the eigenvalues of  $ \bf\tilde{A}$ are still real and non-negative.

In what the relaxation dynamics of the networks is concerned, we focus on the complex dynamical shear modulus, $ G^{*}({\omega})$. In typical mechanical experiments one measures the complex (shear) modulus  $ G^{*}({\omega})$, as response to a harmonic strain field. In the framework of the Rouse model, it was shown that the following standard expressions for  $ G^{'}({\omega})$, the real, and for  $ G^{''}({\omega})$, the imaginary component of  $ G^{*}({\omega})$ hold:
$\displaystyle [G^{\prime }(\omega )]=\frac{1}{N_{tot}}\sum\limits_{i=2}^{N_{tot}}\frac{({\omega}{\tau}_{i})^{2}}{1+(\omega {\tau}_{i} )^{2}}$ (5)

and
$\displaystyle [G^{\prime \prime }(\omega )]=\frac{1}{N_{tot}}\sum\limits_{i=2}^{N_{tot}}\frac{ \omega {\tau}_{i} }{1+(\omega {\tau}_{i})^{2}}~,$ (6)

where $ N_{tot}$ is the total number of elements (beads) in the system under study and$ {\tau}_{i}$ are the ($ N_{tot}-1$) finite relaxation times of the system. Here we will use the mechanical relaxation functions in the form of the reduced storage and loss moduli, $ [{\it G}^{'}({\omega})]$ and $ [{\it G}^{''}({\omega})]$, given that we are mostly interested in the shapes of $ {\it G}^{'}({\omega})$ and $ {\it G}^{''}({\omega})$, but not in their prefactors. In so doing, a variety of features are observed: alternating copolymers differ from homopolymers in the high-frequency domain, where $ G''(\omega )$ may display two maxima. Cross-linking alternating A-B-copolymer chains into regular networks (lattices) leads to the appearance of a network-dominated, low-frequency relaxation domain. In the case of very large differences in the mobility of the A-, of the B-monomers and of the cross-links, $ G'(\omega )$ is very structured, displaying three relaxation domains, separated by two plateaus, while $ G''(\omega )$ shows three peaks. 

\begin{figure}\centerline{\epsfig{file=fig7.eps,width=0.75\columnwidth}}\end{figure}

Figure 2. Reduced storage modulus [$ G'(\omega )$] plotted in double logarithmic scale vs the reduced frequency. 

\begin{figure}\centerline{\epsfig{file=fig8.eps,width=0.75\columnwidth}}\end{figure}
Figure 3. Reduced storage modulus  $ [G'({\omega})]$ plotted in double logarithmic scale vs the reduced frequency.


Relaxation of copolymeric dendrimers built from alternating monomers

Authors:  Cristian Satmarel, Andrei Gurtovenko, and Alexander Blumen
Journal-ref: Macromol. Theory. Simul. 13, 487 (2004)

In this study we extend our previous work concerning the Rouse dynamics of linear alternating copolymers (Macromolecules 36, 486 (2003)), to tree-like structures and focus on copolymeric dendrimers built from monomers of two kinds $ A$ and $ B$; as before, we let the monomers differ in their interaction with the solvent. In the framework of generalized Gaussian structures (GGS) we consider alternating arrangements of monomers over the dendritic structures. We develop a semi-analytical method to determine for such structures (of arbitrary functionality, $ f$, and number of generations, $ g$), the eigenfrequencies (relaxation times). 
A dendrimer is characterized by its functionality $ f$ and by the number $ g$ of generations. The number $ N_{tot}$ of beads of the dendrimer is then:
$\displaystyle N_{tot}=f\frac{(f-1)^{g}-1}{f-2}+1$ (1)
We should note that $ f=3$ is the lowest $ f$ value which leads to non-trivial dendrimers; in general we take $ f$ to be $ f>2$. Furthermore, here all beads of even generations are assumed to have $ \zeta_{A}$ as friction constant, whereas all beads of odd generations have friction constant $ \zeta_{B}$. Figure 1 shows a particular example for an alternating copolymeric dendrimer, for which $ f=3$ and $ g=3$.
Figure 1:  Dendrimer with alternating generations

The normal modes of the dendrimer are fall into two groups: (i) modes in which the core is mobile and (ii) modes in which the core is immobile.
The advantage here is that our algorithm bypasses the direct diagonalization of the corresponding (asymmetric) connectivity matrix. The method is especially suitable for very large $ g$, where, because of the exponential increase of $ N_{tot}$ with $ g$, limits on computer time and, especially, on computer memory render soon a brute-force diagonalisation (note that we need all the eigenvalues) unfeasible. Even for moderately large $ g$ our method has advantages, because it can be readily implemented on small PCs using existing programs, such as MATHEMATICA or MAPLE.

The storage $ [G'(\omega )]$ and the loss$ [G''(\omega )]$ moduli, show here again a multitude of features, which mainly depend on the difference in the mobilities, or, equivalently, in the friction coefficients $ \zeta_{A}$ and $ \zeta_{B}$ of the $ A$- and $ B$-beads. These features range from the presence of large plateau-type regions in $ [G'(\omega )]$ to the appearance of double-peaks in $ [G''(\omega )]$. In contrast to linear alternating copolymers, the behavior of the dynamic moduli of copolymeric systems with dendritic topology can shed light into their composition, i.e. into the relative numbers of $ A$- and $ B$-beads. 

                   

Figure 2:  The reduced storage modulus $ [G'(\omega )]$ and the reduced loss modulus $ [G''(\omega )]$ plotted in double logarithmic scales versus the reduced frequency $ \omega\tau_{0}$. Shown are results for alternating copolymeric dendrimers. The parameters are $ f=3$ and $ g=6$, while $ \sigma$ ranges from $ 0.01$ to $ 100$.

            
Figure 3: The reduced storage modulus $ [G'(\omega )]$ and the reduced loss modulus $ [G''(\omega )]$ plotted in double logarithmic scales versus the reduced frequency $ \omega\tau_{0}$. Shown are results for alternating copolymeric dendrimers. The parameters are $ f=3$ and $ \sigma =0.01$; $ g$ ranges from 3 to 8.

Dynamics of end-linked star polymers

Authors:  Cristian Satmarel, Christian von Ferber, and Alexander Blumen
Journal-ref: J. Chem. Phys. 123, 034907 (2005)

In this work we focus on the dynamics of macromolecular networks formed by end-linking identical polymer stars. The resulting macromolecular network can then be viewed as consisting of spacers which connect branching points (the cores of the stars).
Figure 1: Network consisting of end-linked polymer stars, of which one is highlighted by a circle. Here, the number of stars is $ N=10$, the functionality of each star is$ f=4$, and the length of each arm is $ k=2$; the total number of beads in the system is $ N_{full}=90$. Note that the structure is very flexible so that in many realizations it will appear considerably less ordered than here.

We succeed in analyzing exactly, in the framework of the generalized Gaussian model, the eigenvalue spectrum of such networks. As applications we focus on several topologies, such as regular networks and dendrimers. The central bead may differ (chemically and functionally) from the other beads, a fact which we take into account by assigning it a different friction coefficient. In Fig. 1 we show such a network formed from end-linked stars. Corresponding to the coloring used in Fig. 1, we call in the following the core bead "black" and the chain beads "white". 
The analytical procedure which we use involves an exact real-space renormalization, which allows to relate the star-network to a (much simpler) network, in which each star is reduced to its core. It turns out that the eigenvalue spectrum of the star-polymer structure consists of two parts: One follows in terms of polynomial equations from the relaxation spectrum of the corresponding renormalized structure, while the second part involves the motion of the spacer chains themselves. We find a set of eigenmodes, belonging to different classes, that fully determine the eigenvalue spectrum of the system. In the first class, the coordinates of some of the black beads are non-vanishing and the black beads do not move in parallel. The second class is that in which all the black beads move in parallel. The next two classes, are such that all the black beads are at rest.
This two-step approach of first finding the eigenvalue spectrum of the reduced black system, and only then calculating the eigenvalue spectrum of the full system, considerably reduces the efforts of solving the problem.

As a specific example of branched polymers, we treat here star-burst dendrimers, which are built from $ f$-functional stars, linked in such a way as to describe a regular Cayley tree, see Figure 2. One may remark that our renormalization procedure reduces the dendrimer with spacers and dangling bonds of Fig. 2 to a simple Cayley tree.
Figure 2. A second generation dendrimer with spacers. Here, $ g=2, f=3$ and each spacer chain contains two beads ($ s=2, k=1$). The dash-dotted line indicates one of the stars from which the structure is built.


The effects of varying the parameters $ s=2k$ and $ \sigma$, which determine the length of the spacers and of the dangling chains, and the ratio of the friction coefficients, respectively are presented in Figs. 3 and 4. The number of white beads (which are more mobile when $ \sigma>1$) is larger than the number of black beads, and their relaxation dominates the spectrum. For $ \sigma<1$ the situation is reversed: The black beads are more mobile and the minor peak corresponding to their relaxation is shifted to higher frequencies, while the major peak induced by the white beads keeps its position. Corresponding effects are seen also in the behavior of the storage moduli$ G'(\omega)$, which for $ \sigma\ne 1$ display a plateau resulting from a gap in the relaxation. The plateau is formed between the domains of high and of low frequency relaxation. For $ \sigma=100$ the white beads dominate the high frequency relaxation, leading to a lower level of the plateau than for $ \sigma=0.01$, where the high frequency relaxation of the black beads is only weakly evident.

                 

Figure 3.  The storage  $ G'(\omega)$ and the loss $ G''(\omega)$ moduli for a dendrimer of generation g=6, functionality f=3, with spacer chains of length s=4 between the branching points and dangling bonds of length k=2. Here, $ \sigma$ varies from $ 0.01$ to $ 100$.


The influence in $ G'(\omega)$ and $ G''(\omega)$ of increasing the length of the spacers and of the dangling chains, can be observed in Fig. 4.  The classical dendrimer ($ s=0$) shows a typical, non-scaling behavior of the relaxation part of the spectrum, both for  $ G'(\omega)$ and for $ G''(\omega)$. Increasing $ s$ the relaxation moduli display a mixture of dendrimer and chain-like behavior. For short chains we observe in  $ G''(\omega)$ a second minor peak at lower frequencies, which, for longer chain lengths becomes less and less pronounced, eventually remaining visible only as a shoulder. For long chains both $ G'(\omega)$ and  $ G''(\omega)$ are dominated by the chain like relaxation behavior of the white beads.

                                   

Figure 4: The storage $ G'(\omega)$ and the loss $ G''(\omega)$ moduli for a dendrimer of generation $ g=6$, when the number of spacers between the branching points grows from $ s=2$ to $ s=50$. In all the situations $ \sigma=100$.



Spacer effects in hyperbranched polymer dynamics

Authors:  Cristian Satmarel, Christian von Ferber, and Alexander Blumen
Journal-ref: in preparation