Department of Chemistry, Colorado State University, Fort Collins, Colorado 80523, USA.
J Phys Chem B. 2010 Mar 11;114(9):3185-96. doi: 10.1021/jp1005742.
The persistence length is calculated for polyelectrolyte chains with fixed bond lengths and bond angles (pi-theta), and a potential energy consisting of the screened Coulomb interaction between beads, potential wells alpha phi(i)2 for the dihedral angles phi(i), and coupling terms beta phi(i) phi(i+/-1). This model defines a librating chain that reduces in appropriate limits to the freely rotating or wormlike chains, it can accommodate local crumpling or extreme stiffness, and it is easy to simulate. A planar-quadratic (pq), analytic approximation is based on an expansion of the electrostatic energy in eigenfunctions of the quadratic form that describes the backbone energy, and on the assumption that the quadratic form not only is positive but also adequately confines the chain in an infinite phase space of dihedral angles to the physically unique part with all |phi(i)| < pi. The pq approximation is available under these weak constraints, but the simulations confirm its quantitative accuracy only under the expected condition that alpha is large, that is, for very stiff chains. Stiff chains can also be simulated with small alpha and small theta and compared to an OSF approximation suitably generalized to chains with finite rather than vanishing theta, and increasing agreement with OSF is found the smaller is theta. The two approximations, one becoming exact as alpha --> infinity with fixed theta, the other as theta --> 0 with fixed alpha, are quantitatively similar in behavior, both giving a persistence length P = P0 + aD2 for stiff chains, where D is the Debye length. However, the coefficient apq is about twice the value of aOSF. Under other conditions the simulations show that P may or not be linear in D2 at small or moderate D, depending on the magnitudes of alpha, beta, theta, and the charge density but always becomes linear at large D. Even at a moderately low charge density, corresponding to fewer than 20% of the beads being charged, and with strong crumpling induced by large beta, increasing D dissolves blobs and recovers a linear dependence of P on D2, although a lower power of D gives an adequate fit at moderate D. For the class of models considered, it is concluded that the only universal feature is the asymptotic linearity of P in D2, regardless of flexibility or stiffness.
聚电解质链的持久长度是通过固定键长和键角（π-θ）以及由珠之间的屏蔽库仑相互作用、二面角φ（i）的势阱αφ（i）2以及耦合项βφ（i）φ（i±1）组成的势能来计算的。该模型定义了一个自由摆动的链，在适当的限制下可以简化为自由旋转或线状链，它可以适应局部卷曲或极端刚性，并且易于模拟。基于静电能在描述骨架能量的二次形式本征函数中的展开以及二次形式不仅是正的而且充分限制链在二面角的无限相空间到具有所有|φ（i）| <π的物理唯一部分的假设，提出了平面二次（pq）解析近似。在这些弱约束下，可以使用 pq 近似，但是仅在预期条件下，即α较大时，才可以确认其定量准确性，即对于非常刚性的链。可以用较小的α和θ来模拟刚性链，并将其与适当推广到具有有限而非零θ的链的 OSF 近似进行比较，发现θ越小，与 OSF 的一致性越高。这两个近似值，一个在α趋于无穷大时变为精确，另一个在θ趋于零时变为精确，在行为上是定量相似的，对于刚性链，两者都给出了一个持久长度 P = P0 + aD2，其中 D 是德拜长度。但是，apq 系数约为 aOSF 的两倍。在其他条件下，模拟表明，在较小或中等 D 时，P 可能与 D2 呈线性关系，也可能不呈线性关系，这取决于α、β、θ和电荷密度的大小，但在较大 D 时始终呈线性关系。即使在电荷密度适中的情况下，对应于少于 20%的珠子被充电，并且由于β较大而导致强烈卷曲，增加 D 会溶解斑点并恢复 P 对 D2 的线性依赖性，尽管在中等 D 时，较低的 D 次幂也能提供足够的拟合。对于所考虑的模型类别，可以得出结论，唯一的普遍特征是 P 在 D2 中的渐近线性，无论其灵活性或刚性如何。
