#### 6.2. Scaling of Linear, Branched and Semiflexible Macromolecules

As for bulk condensed matter in general, analysis of the microscopic structure of polymer systems is mostly carried out by scattering experiments. Depending on the system under study and the desired resolution, photons in the X-ray and light scattering range, or neutrons are used. The general set up of a scattering experiment is very simple and indicated schematically in

Figure 33 [

206]: One uses an incident beam of monochromatic radiation with wavelength λ and initial intensity

I_{0}. This beam becomes scattered by a sample and the intensity

I of the scattered waves is registered by a detector (D) at a distance

d, under variation of the direction of observation. The scattering vector

$\overrightarrow{q}$ is defined as

where

${\overrightarrow{k}}_{f}$ and

${\overrightarrow{k}}_{i}$ denote wave vectors of the incident and the scattered plan waves. The result of a scattering experiment is usually expressed by giving the intensity distribution

I(

$\overrightarrow{q}$) in

$\overrightarrow{q}$-space, cf.

Figure 33. In the majority of scattering experiments on polymers the radiation frequency remains practically unchanged, thus one has

where

$\overrightarrow{q}$ is related to the Bragg scattering angle

θ_{B} by

If the scattering can be treated as being due to just one class of materials, the scattering properties can be described by the interference function

S(

$\overrightarrow{q}$), also called

scattering function or

structure function., which is defined as:

Here

N_{m} represents the total number of particles (or monomers in the case of macromolecules) in the sample and

I_{m} is the scattering intensity produced by one particle, if placed in the same incident beam. The scattering function expresses the ration between the actual intensity which would be measured and the intensity which would be measured, if all particles in the sample were to scatter incoherently. Scattering diagrams generally emerge from the superposition and interference of the scattered waves emanating from all particles in the sample. The total scattering amplitude measured at the detector is then given by

Simple geometric considerations [

206] show that the phases

φ_{i} are determined by the particle position

${\overrightarrow{R}}_{i}$ and the scattering vector

$\overrightarrow{q}$ only, being given by

Thus, the scattering amplitude produced by a set of particles at locations

${\overrightarrow{r}}_{i}$ may be formulated as a

$\overrightarrow{q}$-dependent function in the form

It is a well-known fact from electrodynamics that the scattering intensity is proportional to the squared total scattering amplitude,

i.e.,

The brackets indicate an ensemble average which involves, as always in statistical treatments of physical systems, all microscopic states of the sample. For ergodic systems the time average carried out by the detector equals the theoretical ensemble average. As the normalization of the amplitudes of the single scattered waves is already implied in the definition of the structure function,

Equation (56),

Equation (60) can be written as

which is a basic equation of general validity that may serve as starting point for the derivation of other forms of scattering, e.g., for isotropic systems, where

$S(\overrightarrow{q})=S(q)$ with

$q=|\overrightarrow{q}|$, cf.

Equation (68). Inserting

Equation (59) into

(61) one obtains

Equation (62) can be calculated directly in molecular computer simulations as the positions

$\overrightarrow{r}$ of all scattering particles

N_{m} are exactly known at all times (within the boundaries of numerical errors when using floats in double precision as a representation of real numbers).

Polymers usually do not exist in vacuum but in solution. A solvent is referred to as

good when the prevailing form of the effective interaction between polymer segments in this solvent is the repulsive part of the potential energy at shorter distances. In this case the chains tend to swell and the size

R of the polymer (e.g., the end-to-end distance

R_{e} in the case of linear chains or the radius of gyration

R_{g}) scale with an exponent

ν = 3/5. In the opposite case of a

poor solvent, polymers tend to shrink and

R scales with

ν = 1/3. The point were the repulsive and attractive interactions just cancel each other defines the

θ–

point and

θ–

temperature, respectively. Here, the chain configuration is that of a Gaussian random coil with an exponent

ν = 1/2. There are still three-body and higher order interactions present in a

θ–solvent, but their contribution to the free energy is negligibly small [

52]. For the description of the distance of temperature

T from the

θ–temperature, a dimensionless parameter is used, the

reduced temperature ζ which is defined as:

A crossover scaling function

f serves for the description of the scaling behavior of polymer chains in different solvent conditions [

52]. The argument of

f is given by

$\zeta \sqrt{N}$. At

θ–temperature,

At

T <

T_{θ},

At

T >

T_{θ,}In experiments, it is rather difficult to obtain complete and conclusive results in the study of the collapse transition of chains, because one is often restricted to the three distinct limiting cases of polymer solutions, the extreme dilute regime, the

θ-point, and the regime of very high polymer concentrations [

207].

At the

θ –temperature the chains behave as 〈

R_{g}^{2}〉 ∝ 〈

R_{e}^{2}〉 ∝ (

N – 1)

^{2νθ} with

ν_{θ} = 0.5 besides logarithmic corrections in 3D. Therefore, one expects that a plot of 〈

R^{2}〉 / (

N – 1)

vs. T for different values of

N shows a common intersection point at

T =

T_{θ} where the curvature changes: for

T > T_{θ} the larger

N, the larger the ratio 〈

R^{2}〉 / (

N – 1), while for

T < T_{θ} the larger

N, the

smaller the ratio 〈

R^{2}〉 / (

N – 1). Using the model potential of

Equation (25) – instead of varying temperature (which involves rescaling of the particle velocities), different solvent qualities are obtained by tuning the interaction parameter λ. The corresponding transition curves are displayed in

Figure 34 which show a clear intersection point at roughly λ = λ

_{θ} ≈ 0.65. Moreover it can be seen that the transition becomes sharper with increasing chain length

N. The different curves do not intersect exactly at one single point, but there is an extended region in which the chains behave in a Gaussian manner. The size of this region is ∝

N^{−1/2} [

52]. There is a very slight drift of the intersection point toward a smaller value of λ with increasing chain length

N.

Therefore, to obtain a more precise estimate of the

θ–temperature in the limit of (

N → ∞), one has to chose a different graph that allows an appropriate extrapolation. If one draws straight horizontal lines in

Figure 34 the intersection points of these lines with the curves are points at which the scaling function

$f(\sqrt{N}\zeta )$ of

Equation (65) is constant. Plotting different intersection points over

N^{−1/2} therefore yields different straight lines that intersect each other exactly at

T =

T_{θ} and λ = λ

_{θ} respectively. This extrapolation (

N → ∞) is displayed in

Figure 35. The different lines do not intersect exactly at

N^{−1/2} = 0 which is due to the finite length of the chains. As a result of these plots one yields the value of λ for which the repulsive and attractive interactions in the used model just cancel each other:

An important property of individual chains is the

structure factor S(

q) the spherical average of which is defined as [

3]:

with subscript

$|\overrightarrow{q}|$ denoting the average over all scatter vectors

$\overrightarrow{q}$ of the same magnitude

$|k|\hspace{0.17em}=q,{\overrightarrow{r}}_{i}$ being the position vector to the

ith monomer and

N denoting the total number of monomers (scatter centers). For different ranges of the scatter vectors the following scaling relations hold [

52]:

where

l_{b} is the (constant) bondlength of the monomers (often also called segment length). The importance of

S(

q) lies in the fact that it is

directly measurable in scattering experiments. For

ideal linear chains the function

S(

q) can be explicitly calculated and is given by the monotonously decreasing

Debye function.

where the quantity

x is given by

x =

q^{2} 〈

R_{g}^{2}〉

_{0} with index 0 denoting

θ –conditions. For small values of

x, corresponding to large distances between scattering units, the Debye function

S(

x) also provides a good description of a linear chain in a

good solvent with the scaling variable

x describing the expansion of the chain. For very small scattering vectors

q one obtains the

Guinier approximation [

51] by an expansion of

S(

q), which is used in experiments to calculate the radius of gyration 〈

R_{g}^{2}〉. In the intermediate regime of scattering vectors,

S(

q) obeys a scaling law which, in a double-logarithmic plot, should yield a slope of −1

/ν. For large

q-values finally,

S(

q) is expected to behave as 1

/N. The overall expected behavior of

S(

q) is summarized in

Equation (69).

In the vicinity of the

θ–region, the scaling exponent equals

ν =

ν_{θ} = 0.5. Therefore

q^{2}S(

q), plotted against wave vector

q, which is called a

Kratky plot should approach a constant value.

Figure 36 displays this behavior for different chain lengths with high resolution in terms of λ. The respective dotted horizontal line is a guide to the eye. The larger the chains are, the smaller is the λ-range at which the chains display ideal (Gaussian) random–walk behavior. For large values of λ the chains are collapsed and form compact globules the local structure of which is also reflected in the structure function by several distinct peaks for larger

q-values. These peaks become the more pronounced the longer the chains are, reflecting the fact that the transition curves become ever sharper with increasing chain length. Hence, longer chains are already in the collapsed regime for values of λ at which the smaller chains still exhibit Gaussian behavior. The structure function of the largest system in

Figure 36 for λ = 1.0 already resembles very much the scattering pattern of a sphere.

In

Figure 37 the scaling of 〈

R_{g}^{2}〉 for different star polymers as a function of

N and for different functionalities

f is displayed. Functionality

f = 2 corresponds to linear chains,

f = 3 corresponds three-arm star polymers and so on. The star polymers were generated with the MD simulation package “MD-Cube” developed by Steinhauser [

51,

63] which is capable of handling a very large array of branched polymer topologies, from star polymers to dendrimers, H-polymers, comb-polymers or randomly hyperbranched polymers. Details of the set-up of chains which works the same way for linear and branched polymer topologies can be found in [

208].

Figure 37a shows a double-logarithmic plot from which one obtains the scaling exponents of

R_{g} for stars with different numbers of arms. The results for linear chains are displayed as well, for which chain lengths of up to

N = 5000 were simulated. Within the errors of the simulation, the exponents do not depend on the number of arms, as expected from theory. The obtained scaling exponents are summarized in

Table 4 and exhibit a reasonable agreement with theory.

In

Figure 37b it is exhibited, that the corrections to scaling due to the finite size of the chains are ∝

N^{−}^{1}. A plot with exponents −2 or −1/2 leads to worse correlation coefficients. This result is consistent with lattice-MC simulations on a fcc-lattice [

209]. More details on finite-size scaling can be found in [

50,

51].

The fundamentals of the dynamics of fully flexible polymers in solution or in the melt were worked out in the pioneering works of Rouse [

210] and Zimm [

211], as well as of Doi and Edwards [

212] and de Gennes [

52]. In contrast to fully flexible polymers, the modeling of

semiflexible and

stiff macromolecules has received recent attention, because such models can be successfully applied to biopolymers such as proteins, DNA, actin filaments or rodlike viruses [

213,

214]. Biopolymers are wormlike chains with persistence lengths

l_{p} (or Kuhn segment lengths

l_{K}) comparable to or larger than their contour length

L and their rigidity and relaxation behavior are essential for their biological functions.

Using the second term of the bonded potential of

Equation (26), a bending rigidity Φ

_{bend}(

θ) can be introduced. Rewriting this term by introducing the unit vector

${\overrightarrow{u}}_{j}=({\overrightarrow{r}}_{j+1}-{\overrightarrow{r}}_{j})/|{\overrightarrow{r}}_{j+1}-{\overrightarrow{r}}_{j}|$ along the macromolecule, cf.

Figure 10, one obtains:

where

θ_{i} is the angle between

${\overrightarrow{u}}_{j}$ and

${\overrightarrow{u}}_{j+1}$. The crossover scaling from coil-like, flexible structures on large length scales to stretched conformations at smaller scales can be seen in the scaling of

S(

q) when performing simulations with different values of

k_{θ} [

208]. Results for linear chains of length

N = 700 are displayed in

Figure 38a. The chains show a scaling according to

q^{ν}. The stiffest chains exhibit a

q^{−}^{1}–scaling which is characteristic for stiff rods. Thus, by varying parameter

k_{θ}, the whole range of bending stiffness of chains from fully flexible chains to stiff rods can be covered. The range of

q–vectors for which the crossover from flexible to semiflexible and stiff occurs shifts to smaller scatter vectors with increasing stiffness

k_{θ} of the chains. The scaling plot in

Figure 38b shows that the transition occurs for

q ≈ 1

/l_{K},

i.e., on a length scale of the order of the statistical Kuhn length. In essence, only the fully flexible chains (red data points) exhibit a deviation from the master curve on large length scales (

i.e., small

q–values), which corresponds to their different global structure compared with semi-flexible macromolecules. Examples for snapshots of stiff and semiflexible chains are finally displayed in

Figure 39.

#### 6.3. Polyelectrolytes

A large variety of synthetic and biological macromolecules are polyelectrolytes [

215]. The most well-known polyelectrolyte biopolymers, proteins, DNA and RNA, are responsible in living systems for functions which are incomparably more complex and diverse than the functions usually discussed for synthetic polymers present in the chemical industry. For example, polyacrylic acid is the main ingredient for diapers and dispersions of copolymers of acrylamide or methacrylamide and methacrylic acid are fundamental for cleaning water [

216]. In retrospect, during the past 30 years, despite the tremendous interest in polyelectrolytes, unlike neutral polymers [

52,

217], the general understanding of the behavior of electrically charged macromolecules is still rather poor. The contrast between our understanding of neutral and charged polymers results form the long range nature of the electrostatic interactions which introduce new length and time scales that render an analytical treatment beyond the Debye-H¨ckel approximation very complicated [

218,

219]. Here, the traditional separation of scales, which allows one to understand properties in terms of simple scaling arguments, does not work in many cases. Experimentally, often a direct test of theoretical concepts is not possible due to idealizing assumptions in the theory, but also because of a lack of detailed control over the experimental system, e.g., in terms of the molecular weight. Quite recently, there has been increased interest in hydrophobic polyelectrolytes which are water soluble, covalently bonded sequences of polar (ionizable) groups and hydrophobic groups which are not [

220]. Many solution properties are known to be due to a complex interplay between short-ranged hydrophobic attraction, long-range Coulomb effects, and the entropic degrees of freedom. Hence, such polymers can be considered as highly simplified models of biologically important molecules, e.g., proteins or lipid bilayers in cell membranes. In this context, computer simulations are a very important tool for the detailed investigation of charged macromolecular systems. A comprehensive review of recent advances which have been achieved in the theoretical description and understanding of polyelectrolyte solutions can be found in [

221].

The investigation of aggregates between oppositely charged macromolecules plays an important role in technical applications and in in particular biological systems. For example, DNA is associated with histone proteins to form the chromatin. Aggregated of DNA with cationic polymers or dendrimers are discussed in the context of their possible application as DNA vectors in gene therapies [

224,

225]. Here, we present MD simulations of two flexible, oppositely charged polymer chains and illustrate the universal scaling properties of the formed polyelectrolyte complexes that are formed when the chains collapse and build compact, cluster-like structures which are constrained to a small region in space [

223]. The properties are investigated as a function of chain length

N and interaction strength

ξ. Starting with

Equation (10) and using

k = 1 (cgs-system of units) the dimensionless interaction parameter

can be introduced, where the Bjerrum length

ξ_{B} is given by:

where

k_{B} is the Boltzmann constant,

T is temperature,

epsilon is the energy scale from the Lennard-Jones potential of

Equation 21,

σ defines the length scale (size of one monomer) and

e is the electronic charge. The interaction parameter for the presented study is chosen in the range of

ξ = 0

, ..., 100 which covers electrically neutral chains (

ξ = 0) in good solvent as well as highly charged chain systems (

ξ = 100). The monomers in the chains are connected by harmonic bonds using the first term of the bonded potential of

Equation (26). The interaction with the solvent is taken into account by a stochastic force (

${\overrightarrow{\Gamma}}_{i}$) and a friction force with a damping constant

χ, acting on each mass point. The equations of motion of the system are thus given by the Langevin equations

The force

${\overrightarrow{F}}_{i}$ comprises the force due to the sum of potentials of

Equation 21 with cutoff

r_{cut} = 1.5,

Equation 11 with

k = 1 and

z_{i/j} = ±1, and the first term on the right-hand side of the bonded potential in

Equation 26 with

κ = 5000ε

/σ and bond length

l_{0} =

σ_{0} = 1.0. The stochastic force

${\overrightarrow{\Gamma}}_{i}$ is assumed to be stationary, random, and Gaussian (white noise). The electrically neutral system is placed in a cubic simulation box and periodic boundary conditions are applied for the intermolecular Lennard-Jones interaction according to

Equation (21), thereby keeping the density

ρ =

N/V = 2.1 × 10

^{−7}/

σ^{3} constant when changing the chain length

N. The number of monomers

N per chain was chosen as

N = 10

, 20

, 40

, 80 and 160 so as to cover at least one order of magnitude. For the Coulomb interaction a cutoff that is half the box length

r_{cut} = 1/2

L_{B} was chosen. This can be done as the eventually collapsed polyelectolyte complexes which are analyzed are confined to a small region in space which is much smaller than

r_{cut}. In the following we discuss exemplarily some scaling properties of charged linear macromolecules in the collapsed state. The simulations are started with two well separated and equilibrated chains in the simulation box. After turning on the Coulomb interactions with opposite charges

z_{i/j} = ±1 in the monomers of both chains, the chains start to attract each other. In a first step during the aggregation process the chains start to twist around each other and form helical like structures as presented in

Figure 40. In a second step, the chains start to form a compact globular structure because of the attractive interactions between dipoles formed by oppositely charged monomers, see the snapshots in

Figure 41a.

Figure 41a exhibits the universal scaling regime of

R_{g} obtained for intermediate interaction strengths

ξ and scaled by (

N – 1)

^{2/3}. Here, the data of various chain lengths fall nicely on top of each other. This scaling corresponds to the scaling behavior of flexible chains in a bad solvent and is also in accordance with what was reported by Shrivastava and Muthukumar [

226]. The change of

R_{g} is connected with a change of the density

ρ of the polyelectrolyte aggregate. However, in

Figure 41b, which presents an example of

ρ for

ξ = 4, only a slight dependence of the density on the chain length

N can be observed.

ρ measures the radial monomer density with respect to the center of mass of the total system. For longer chains, there is a plateau while for short chains there is a pronounced maximum of the density for small distances from the center of mass. While this maximum vanished with decreasing

ξ it appears also at higher interaction strengths for longer chains. Monomers on the outer part of the polyelectrolyte complex experience a stronger attraction by the inner part of the cluster than the monomers inside of it, and for smaller

ξ, chains of different lengths are deformed to different degrees which leads to a chain length dependence of the density profile.