Network reference state

MCA distinguishes between local and global (systemic) properties of a reaction network. Local properties are characterized by sensitivity coefficients, denoted as elasticities, of a reaction rate v,(S,p) toward a perturbation in substrate concentrations (e-elasticities) or kinetic parameters ( -elasticities). The elasticities measure the local response of a reaction in isolation and are defined as the partial derivatives at a reference state S°... 

Similar to generalized mass-action models, lin-log kinetics provide a concise description of biochemical networks and are amenable to an analytic solution, albeit without sacrificing the interpretability of parameters. Note that lin-log kinetics are already written in term of a reference state v° and S°. To obtain an approximate kinetic model, it is thus sometimes suggested to choose the reference elasticities according to simple heuristic principles [85, 89]. For example, Visser et al. [85] report acceptable result also for the power-law formalism when setting the elasticities (kinetic orders) equal to the stoichiometric coefficients and fitting the values for allosteric effectors to experimental data. 

The challenge is therefore to develop an experiment which allows an experimental separation of the contributions from chain entangling and cross-links. The Two-Network method developed by Ferry and coworkers (17,18) is such a method. Cross-linking of a linear polymer in the strained state creates a composite network in which the original network from chain entangling and the network created by cross-linking in the strained state have different reference states. We have simplified the Two-Network method by using such conditions that no molecular theory is needed (1,21). 

N Nw n n0 = N/V0 total number of units in the network sample total number of molecules of water in the network concentration of monomer units in the network concentration of monomer units in the network in the reference state... 

Q Qo T V V0 Vf Vm x = Bn0m composition of solvent in the network Q = C>a/( a + b) composition of solvent outside the network Q0 = a/C a + < ) temperature expressed in energy units (i.e. kT) volume of the network sample volume of the network sample in the reference state total volume of system volume of micelles inside the network parameter characterizing solvent quality x > 0 corresponds to good solvent x < 0 to poor solvent x = 0 at 0-point... 

In order to connect expressions for FJn, and Fei, it is necessary to define more precisely the reference state with respect to which deformation ratios a, reference state is defined by conditions of network preparation and in many cases it is close to the network state at the preparation conditions. The connection between G>N and otj (i = x, y, z) is given by ... 

Let us suppose that we now apply uniaxial force to the network sample along the axis oZ. Let be the force per cross sectional area of the sample in the reference state (i.e. is the stress normalized in a special way). Applied force leads to some relative deformation along oZ az = p. The network dimensions along axes x and y, ax = cty = a are varied freely. It has been shown [20] that in the case of a homogeneous solvent containing no salt (ns = 0) the equilibrium dimensions of the network are described by the following system of equations ... 

The critical state is evidently an invariant point (terminus of a line) in this case, because it lies at a dimensional boundary between states of / =2 (p = 1) and /= 1 (p = 2). The critical point is therefore a uniquely specified state for a pure substance, and it plays an important role (Section 2.5) as a type of origin or reference state for description of all thermodynamic properties. Note that a limiting critical terminus appears to be a universal feature of liquid-vapor coexistence lines, whereas (as shown in Fig. 7.1) solid-liquid and solid-vapor lines extend indefinitely or form closed networks with other coexistence lines. 

In Eq. (III-9) the deformation ratios are defined with respect to a reference state in which the chain dimensions are such that they do not exert any elastic forces on the crosslinks (state of normal coiling). In general, the chains in a network may not actually be in this state at the beginning of a deformation experiment, because the ciosslinking process may quite well exert a, largely unknown, influence on the chain dimensions. 

Considering next swollen networks, the situation becomes somewhat more complicated. In the first place a free energy of mixing will be needed in addition to the network free energy. Furthermore, the reference state may, in principle, depend upon the nature of the diluent and the amount of it. Also, the effect of the crosslinks on the chain dimensions in the reference state is unknown, and may be a function of the diluent content. Since none of these finer adjustments in the reference state have been given a quantitative molecular basis, we will only formally introduce the... 

In order now to obtain the difference in free energy between a swollen network and a dry network, we need to apply Eq. (111-5) for the elastic free energy twice once with respect to (r2)0s as the swollen reference state and once with respect to [Pg.39]

The reference state dimensions unperturbed dimensions, because of the unknown influence of the presence of crosslinks, possibly specific diluent effects, and perhaps at high swelling even an excluded volume effect. We have pointed out that (r2)0 may depend on the concentration of the diluent. Therefore the reference state is in general not a constant. We have also pointed out that, if (r2)0 contains a molecular expansion term due to an excluded volume effect, the use of the Flory-Huggins free enthalpy of dilution is no longer adequate. A difference between the % parameter in a network and the X parameter of the same polymer material but then in solution, may occur because the presence of crosslinks may modify %. 

At smaller strains this result reduces to the Gaussian Eq. (HI-2), provided we remember that in the derivation of Eq. (IV-9), It was assumed that in the undeformed network the network chains are indeed in the unstrained reference state, so that the front factor (y2 l(r2 is lacking. 

It follows that the nascent b networks are best fitted for defining a reference state, because at the swelling ratio Qe the elastic chains are supposed to have undergone the smallest conformational changes with respect to the free chains they were before crosslinking. 

Another possible choice for the reference state is the equilibrium swelling degree of the network. In that case the thermodynamic characteristics of the actual polymer-swelling solvent system would be taken into account. However this choice does not consider the conditions of network formation. Moreover when the gel is swollen to equilibrium, the elastic chains are extended with respect to the corresponding free chains, and the extension ratio is not easy to evaluate13. ... 

Attention should be drawn to the fact that if the state with the swelling ratio Qc( u"1) of the nascent network is chosen as the reference state, the memory term h2 3 should be independent of the molecular weight of the elastic chains. This point is still somewhat controversial, and though experimental data support this statement in several cases35,36 there are other cases in which the opposite was observed14,22. ... 

The experimentally observed proportionality between Q and M3,s requires that the difference (A h2 3 — B Q 2 3) varies only very little with the molecular weight of the elastic chains or the swelling degree, respectively. If the segment concentration of the nascent network is taken as the swollen reference state, h can be considered as a constant. This implies that either B is zero, or B Q 2 3 is very small compared with A h2 3, provided x is a constant. If h is not a constant — this will be discussed in the next section — one has to assume that the molecular weight dependence of A h2/3 and of B Q 2 3 are compensating, in order for the difference to remain independent of the molecular weight. 

At higher temperatures, a broad transition, present in all epoxy resins independently of the reacting species (amines or anhydrides), is much more complex than the y transition. Its position and shape vary strongly with the chemical structure of the epoxy resin. The dynamic mechanical response and the solid-state 13C NMR of the various model networks referred to in Table 9 focus precisely on the analysis of this /3 transition [62,63]. 

The structure of precursors, the number of functional groups per precursor molecule, and the reaction path leading to the final network all play important roles in the final structure of the polymer network. Some thermosets can be considered homogeneous ideal networks relative to a reference state. It is usually the case when networks are prepared by step copolymerization of two monomers (epoxy-diamine or triol-diisocyanate reactions) at the stoichiometric ratio and at full conversion. 

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