Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Polymers

Field: Soft Matter Physics

Large molecules composed of many repeated subunits.

Sequence of Expressions

Let PP be a polymer chain defined by a sequence of NN structural units (monomers) mim_i, where i∈{1,2,…,N}i \in \{1, 2, \dots, N\}. The polymer structure is represented by the sequence of connected monomers: P=m1−m2−⋯−mNP = m_1 - m_2 - \dots - m_N. The connectivity is governed by covalent bonds, represented by the Hamiltonian Hbond=∑i=1N−1Vcovalent(ri,ri+1)H_{bond} = \sum_{i=1}^{N-1} V_{covalent}(\mathbf{r}_i, \mathbf{r}_{i+1}), where ri\mathbf{r}_i are the spatial coordinates of the ii-th monomer, and VcovalentV_{covalent} is the potential energy associated with the bond linking mim_i and mi+1m_{i+1}. The chain topology is characterized by the repeating unit RR, such that P≈(R)NP \approx (R)^N.
Let F:Body→Body′\mathbf{F}: \text{Body} \to \text{Body}' be the deformation gradient mapping the reference configuration to the current configuration. The Cauchy stress tensor σ\boldsymbol{\sigma} is derived from the strain energy density function W(F)W(\mathbf{F}) via the relationship: \nσ=∂W∂FFT\boldsymbol{\sigma} = \frac{\partial W}{\partial \boldsymbol{\mathbf{F}}} \mathbf{F}^T \nFor an incompressible, neo-Hookean approximation, the strain energy density is defined as W(F)=12μ(tr(FTF)−3)+κ(det(F)−1)W(\mathbf{F}) = \frac{1}{2} \mu (\text{tr}(\mathbf{F}^T \mathbf{F}) - 3) + \kappa (\text{det}(\mathbf{F}) - 1), where μ\mu is the shear modulus and κ\kappa is the bulk modulus.
Define the glass transition temperature TgT_g as a function of the molecular weight MwM_w and the critical entanglement molecular weight MeM_e. The relationship is often modeled by the empirical form: \nTg(Mw)Tg,∞=(1−1Mw/Me)ν\frac{T_g(M_w)}{T_{g, \infty}} = \left(1 - \frac{1}{M_w/M_e}\right)^{\nu} \nwhere Tg,∞T_{g, \infty} is the limiting glass transition temperature for infinite chains, and ν\nu is a critical exponent, typically found to be 1/21/2 or 1/31/3 depending on the specific polymer class and model approximation.
Let r(s)\mathbf{r}(s) be the position of a monomer along the contour length ss of a polymer chain. The chain-entanglement model introduces a characteristic relaxation time τe\tau_e and an effective tube diameter aa. The mean-squared end-to-end distance ⟨R2⟩\langle R^2 \rangle is constrained by the entanglement length aa and the total contour length LL: \n⟨R2⟩≈La\langle R^2 \rangle \approx L a \nFurthermore, the characteristic relaxation time τ\tau scales with the molecular weight MM (or LL) according to the reptation model: \nτ∝M3.0 to M3.6\tau \propto M^{3.0} \text{ to } M^{3.6}
Consider the viscosity η\eta of a polymer solution as a function of the concentration cc and the molecular weight MM. For highly entangled solutions, the zero-shear viscosity η0\eta_0 exhibits a power-law dependence on molecular weight: \nη0∝Ma\eta_0 \propto M^{a} \nwhere the exponent aa is typically found to be 3.43.4 for linear polymers. Additionally, the osmotic pressure Π\Pi scales with the concentration cc and the characteristic volume V0V_0: \nΠ=kBT(c⋅V0)1/2\Pi = k_B T \left(c \cdot V_0 \right)^{1/2}
Consider a polymer solution with concentration cc (mass/volume) and molecular weight MM. The osmotic pressure π\pi is given by the virial expansion: \begin{equation*} \frac{\pi}{c R T} = \frac{1}{M} + A_2 c + A_3 c^2 + \dots \end{equation*} where RR is the ideal gas constant and TT is the absolute temperature. The second virial coefficient A2A_2 quantifies the polymer-solvent interactions and is defined as: \begin{equation*} A_2 = \frac{1}{2 M^2} \left( \frac{V_{2}}{V_{1}} - 2 \right) \end{equation*} Here, V1V_1 and V2V_2 are the interaction volumes between polymer segments and solvent molecules, respectively.
Define the intrinsic viscosity [η][\eta] of a polymer solution, which is the ratio of the solution viscosity η\eta to the concentration cc and the solvent viscosity η0\eta_0: [η]=lim⁡c→0ηc[\eta] = \lim_{c \to 0} \frac{\eta}{c}. The Mark-Houwink equation provides an empirical relationship between [η][\eta] and the molecular weight MM: \begin{equation*} [\eta] = K M^a \end{equation*} where KK is the Mark-Houwink constant, and aa is the exponent, both of which are dependent on the specific solvent and temperature TT. The exponent aa characterizes the polymer chain conformation in solution (e.g., a=0.5a=0.5 for random coil, a=1.0a=1.0 for rigid rod).
Consider a system where the concentration CC of a diffusing species varies spatially over time tt in a domain Ω\Omega. The flux J\mathbf{J} is governed by Fick's First Law: \nJ=−D∇C\mathbf{J} = -D \nabla C \nwhere DD is the diffusion coefficient, assumed constant, and ∇C\nabla C is the concentration gradient. The conservation of mass dictates the time evolution of the concentration field CC: \n∂C∂t=−∇⋅J=D∇2C\frac{\partial C}{\partial t} = -\nabla \cdot \mathbf{J} = D \nabla^2 C
Define the specific enthalpy H(T)H(T) and specific volume V(T)V(T) of an amorphous polymer as functions of temperature TT. The glass transition temperature, TgT_g, is mathematically characterized by the discontinuity in the slope of the enthalpy or volume curve: \begin{equation*} \lim_{T \to T_g^-} \frac{d H}{d T} \neq \lim_{T \to T_g^+} \frac{d H}{d T} \end{equation*} This discontinuity reflects the change in the heat capacity CpC_p: \begin{equation*} C_p(T) = \frac{d H}{d T} \end{equation*} Specifically, the jump in heat capacity is ΔCp=Cp(Tg+)−Cp(Tg−)\Delta C_p = C_p(T_g^+) - C_p(T_g^-). The transition is driven by the onset of large-scale segmental motion, which activates the relaxation time τ\tau such that τ(T)∝exp⁡(EakBT)\tau(T) \propto \exp\left(\frac{E_a}{k_B T}\right).
Let r=(r1,r2,…,rN)\mathbf{r} = (\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_N) be the set of spatial coordinates defining the conformation of a polymer chain of NN monomers. The equilibrium conformation minimizes the total free energy functional F(r)F(\mathbf{r}): \begin{equation*} F(\mathbf{r}) = E_{internal}(\mathbf{r}) + E_{interaction}(\mathbf{r}) - T S(\mathbf{r}) \end{equation*} The internal energy EinternalE_{internal} includes bond stretching and angle bending potentials, while EinteractionE_{interaction} accounts for non-bonded forces (e.g., Lennard-Jones potential VLJV_{LJ}): \begin{equation*} E_{interaction}(\mathbf{r}) = \sum_{ir0\mathbf{r}_0 is the state that minimizes FF: r0=arg⁡min⁡rF(r)\mathbf{r}_0 = \arg\min_{\mathbf{r}} F(\mathbf{r}). The folding tendency is quantified by the free energy difference ΔF=F(r0)−F(rrandom)\Delta F = F(\mathbf{r}_0) - F(\mathbf{r}_{random}).