note that, if the cross-link length is small compared to the particle size, it is unlikely for one monomer in D = 3 (D = 2) dimensions to be connected to more than 12 (6) other monomers. Therefore, average coordination numbers µ2 > 12 (> 6) are inaccessible in these systems. Only if long tethers can form, reaching further than the nearest neighbor, higher average coordination numbersµ2 are possible.
2.10 Conclusions
We have determined the shear modulusGof a randomly cross-linked system. It only depends on the particle density n0 = N/V and the average coordination number µ2. Interestingly, neither the distribution of localization lengths P(ξ2), defined in Eq. (2.43), nor the typical cross-link lengthainfluence the macroscopic behavior of the material. This result is consistent with the observation that the shear modulus determines the response of the system to a shear deformation on the longest scales:
The wavelength of the shear deformation has to be larger than all microscopic length scales. Furthermore, one observes that G∝kBT, i.e. the shear elasticity is purely entropic. This is a consequence of our simple model with entropic springs to account for the cross-links. In more complex systems, e.g. covalently bonded random networks, one expects to find additional energetic contributions toGin the highly cross-linked regime. In any case, the bulk modulus should depend on the strength of the excluded-volume interaction, and hence it is not expected to be as universal as the shear modulus.
It is also noteworthy that the shear modulus does not depend on the spatial dimension D, apart from the fact that the maximal coordination numbers depend on dimensionality and determine the physical range of µ2. Again, this could be expected, because we only consider a Gaussian expansion around the saddle point;
interactions between fluctuations will most likely cause a dependence on dimension-ality.
31
Chapter 3
Cross-Linked Directed Polymers
Contents
3.1 Introduction . . . . 31 3.2 Properties of a Single Chain . . . . 33 3.2.1 Radius of Gyration of an Uncrosslinked Chain . . . . 33 3.2.2 Properties of a Polymer Clamped in Space . . . . 34 3.3 Full Model: Cross-Linked Directed Polymers. . . . 36 3.3.1 Interactions . . . . 36 3.3.2 Disorder Average . . . . 39 3.4 Replica Calculation of the Free Energy . . . . 40 3.4.1 Disorder-Averaged Free Energy: First Steps . . . . 40 3.4.2 Introduction of the Replicated Density Field . . . . 41 3.4.3 Introduction of a Field Theory and Decoupling . . . . 43 3.4.4 Saddle Point Equation . . . . 44 3.5 Ansatz for the Order Parameter . . . . 45 3.6 The Saddle-Point Equation with Ansatz forΩ . . . . 47 3.6.1 Expansion to second order inQ. . . . 48 3.6.2 Sol-Gel Transition . . . . 49 3.7 The Equation for the Localization Length. . . . 49 3.7.1 Normalization of Length Scales . . . . 49 3.7.2 Result for the Distribution of Localization Lengths . . . . 50 3.8 Conclusions . . . . 55
3.1 Introduction
In the previous chapter we investigated a system of cross-linked molecular networks.
Neither the particles nor the cross-linking gave rise to a preferred direction, hence on a macroscopic scale, the system was isotropic. In many cases, however, the experimental setup induces a preferred direction of the polymer chains, like e.g. for so called polymer brushes, where strands are mounted perpendicular to a bottom plate, or DNA strands which are highly stretched along one direction. For polymer chains in a nematic solvent or nematic polymers [Kamien et al., 1992], the alignment and hence the emergence of a preferred direction can also be due
to spontaneous symmetry breaking, whereby the chains align in an arbitrary but uniform direction, when e.g. the temperature is decreased below a critical value.
Hence, the field of directed polymers has become an active area of research during the last twenty years.
This problem of directed chains is also relevant for the behavior of type-II super-conductors [Kamienet al.,1992;Marchetti & Nelson,1993]. A magnetic field, larger than a critical valueHc1, penetrates the superconductor in form offlux lines, which are preferably aligned along the field and each of which is carrying one quantum of magnetic flux. Due to their mutual repulsion, these flux lines can form various lattice or glassy structures in the transverse plane. In this respect, the flux lines are similar to directed polymer chains with excluded volume interaction.
Recently, there has been an increasing interest incross-linkedpolymer brushes with promising applications. This is due to the fact that cross-linking can influence chemical and mechanical stability, permeability and swelling characteristics of the polymer brushes [Huang et al., 2001; Loveless et al., 2006]. For example Ionov et al. [2004] investigated binary polymer brushes, consisting of both hydrophobic and hydrophilic chains; they managed to cross-link the hydrophobic components in the extended or collapsed state, depending on the position of the sample; thereby they could locally, on a µm-scale, tailor the properties of the material. On the other hand, amphiphilic block copolymers (i.e.polymer chains with one hydrophobic and one hydrophilic end) in an aqueous solution are known to form structures like micelles, lipid bilayers, or liposomes, whose interior comprises brush-like structures.
For micelles Iijima et al. [1999] and Xu et al. [2004] also found that cross-linking can be used to achieve stabilization. Stable micelles are of great importance for pharmaceutical applications, since they can be utilized as carriers for targeted drug delivery.
In this chapter we investigate a simple model of flexible cross-linked directed polymers, embedded between two parallel surfaces (see Fig. 3.1). The polymer chains are modeled as functionsr(z), which are preferably aligned along the distin-guished directionz. This preferential alignment may be interpreted as a mechanical stretching of the chains along the z-direction, or as the interaction with a nematic field. We will study the effect of permanent cross-links on this simple structure and observe a gelation transition as the cross-links concentration is increased above a critical value, similar to the RLP model.
In the following section 3.2, we will discuss the behavior of one single polymer chain. The full model, comprising a large ensemble of these chains, is introduced in Sec. 3.3. In Sec. 3.4 we develop a Landau-Wilson free energy, with the replicated density being the order parameter. In Sec.3.5an Ansatz for the order parameter is developed, which is shown to solve the saddle-point equation in Sec.3.6; the sol-gel transition, a consequence of the saddle-point equation, is investigated there (see Sec. 3.6.2). The distribution for the localization lengths is derived in Sec.3.7, and we will conclude in Sec.3.8.