Zero-parameter theories
for low-temperature the melting lines
of WCA, harmonic repulsive, and Hertzian particles
Ulf R. Pedersen, Roskilde University, "Glass and Time", Denmark
Models for
repulsive particles
Potential energy surface
\[\begin{equation} U({\bf R}) = \sum_{i>j}^N u(r_{ij}) \end{equation}\] where $r_{ij}=|\bf r_i-\bf r_j|$ is a pair distance.
Generalized Hertzian sphere pair-potential
\[\begin{equation} u(r) = (1-r)^\alpha \textrm{ for } r<1 \end{equation} \]
and zero otherwise (reduce units).
Potential energy surface
\[\begin{equation} U({\bf R}) = \sum_{i>j}^N u(r_{ij}) \end{equation}\] where $r_{ij}=|\bf r_i-\bf r_j|$ is a pair distance.
Weeks-Chandler-Andersen (WCA) pair potential
\[\begin{equation} u(r) = 4(r^{-12}-r^{-6})+1 \textrm{ for } r<2^{1/6} \end{equation} \]
and zero otherwise (reduce units).
Pair potentials
\[\begin{equation*} u(r) = (1-r)^\alpha \textrm{ for } r<1 \end{equation*} \]
Pair potentials
\[\begin{equation*} u(r) = (1-r)^\alpha \textrm{ for } r<1 \end{equation*} \]
Pair potentials
\[\begin{equation*} u(r) = (1-r)^\alpha \textrm{ for } r<1 \end{equation*} \]
Pair potentials
Low-temperatures: WCA="harmonic-repulsive"
(in proper reduced units)
Theories for coexistence pressure
Effective Hard-Sphere (HS) diameter
FCC solid-liquid HS coexistence pressure
[Fernandez et al. (2012)]:
\[\begin{equation} p_\sigma = 11.5712(10) T/\sigma^3 \end{equation}\]
\[\begin{equation} \sigma = 1 \textrm{ for } T\to0 \end{equation}\] \[\begin{equation} p_\bullet \equiv 11.5712(10) T \textrm{ for } T \to 0 \end{equation}\] $p = p_\bullet/\sigma^3$. For finite temperatures $\sigma<1$ and $p>p_\bullet$.
Boltzmann's HS criterion (1864)
\[\begin{equation} p_\sigma = 11.5712(10) T/\sigma^3 \end{equation}\]
Hard-sphere (HS) diameter ($\sigma$) for $v(r)=(1-r)^\alpha$:
\[\begin{equation} v(\sigma) = T \end{equation} \]
\[\begin{equation} \sigma = 1-T^\frac{1}{\alpha} \end{equation}\]
Insertion into $p = p_\bullet/\sigma^3$ and $T\to0$ \[\begin{equation} p = p_\bullet(1+3T^\frac{1}{\alpha}) \end{equation}\]
Barker-Henderson theory (1967)
\[\begin{equation} p_\sigma = 11.5712(10) T/\sigma^3 \end{equation}\]
Match Helmholtz free-energy
(identical to AWC theory at $T\to0$):
\[\begin{equation}\sigma\equiv \int_0^\infty[1-\exp(-v(r)/T)]dr \end{equation}\]
\[\begin{equation}\sigma\equiv \int_0^\infty[1-\exp(-v(r)/T)]dr \end{equation}\]
\[\sigma = 1 - \int_0^1 \exp\left(-T^{-1}\left(1-r\right)^\alpha\right)dr\]
Let $t=T^{-1}(1-r)^\alpha$ \[ \sigma = 1 - T^{\frac{1}{\alpha}}\frac{1}{\alpha}\int_0^{T^{-1}} t^{\frac{1}{\alpha}-1}\exp(-t)dt \]
Let $t=T^{-1}(1-r)^\alpha$ \[ \sigma = 1 - T^{\frac{1}{\alpha}}\frac{1}{\alpha}\int_0^{T^{-1}} t^{\frac{1}{\alpha}-1}\exp(-t)dt \]
For $T\to0$ let
\[\Gamma(z)\equiv\int_0^\infty t^{z-1}\exp(-t)dt,\]
\[\Gamma(z)\equiv\int_0^\infty t^{z-1}\exp(-t)dt,\]
\[\begin{equation}\sigma = 1 - T^{\frac{1}{\alpha}}\Gamma\left(\frac{1}{\alpha}+1\right)\end{equation}\]
\[\begin{equation}p = p_\bullet\left[1+3T^{\frac{1}{\alpha}}\Gamma\left(\frac{1}{\alpha}+1\right)\right]\end{equation}\]
- Repulsive harmonic: $\Gamma(\frac{1}{2}+1)=\frac{\sqrt{\pi}}{2}\simeq0.8862$.
- 3D Hertzian spheres: $\Gamma(\frac{2}{5}+1)\simeq0.8873$.
- 2D Hertzian spheres: $\Gamma(\frac{2}{7}+1)\simeq0.8997$.
Isomorph theory
(Hidden) Scale-invariance of $U({\bf R})$:
Generalization of the "single parameter ($\eta$)"-idea.
But no effective $\sigma$!
With
Isomorph theory
Isomorph: line of invariant dynamics and structure,
and some thermodynamic quantities.
\[p(T) = p_\star(T) [\alpha_1(T)+\alpha_2(T)+\alpha_3]\]
\[\begin{aligned}\alpha_1(T) &=& [u_s(T)/T - u_s(T_0)/T_0] \\ &-& [u_l(T)/T - u_l(T_0)/T_0], \\ \alpha_2(T) &=& \log(\rho_s(T)/\rho_s(T_0)) \\ &-& \log(\rho_l(T)/\rho_l(T_0)), \\ \alpha_3 &=& \frac{p_0}{T_0} [\rho_l^{-1}(T_0)-\rho_s^{-1}(T_0)], \\ p_\star(T) &=& T/[\rho_l^{-1}(T)-\rho_s^{-1}(T)]\end{aligned}\]
- Need expressions for
$\rho_l(T)$, $\rho_s(T)$, $u_l(T)$ and $u_s(T)$ along isomorphs.
In general, the partition function for configurational degrees of freedom is given by \[Z=\int_{V^N}d{\bf r}_1\ldots d{\bf r}_N\exp(-\sum_{i>j}v(r_{ij})/T)\]
Kissing: $r_{ij}<$"truncation of pair potential"
Low-$T$ on coex.: collisions are uncorrelated
(mean-field approximation valid).
Mean-field approximation
for $T\to0$ and $\rho\to0$,\[\begin{equation} Z=Z_s^N \end{equation}\] $Z_s$: single particle moving in $v_s({\bf r})$ of frozen particles,
\[\begin{equation} Z_s/N = Z_1+Z_0 \end{equation}\]
- Not kissing; $Z_0=V_{free}/N$, or $Z_0=\rho^{-1}$
- Kissing: $Z_1=$ ... (next slide) ...
$Z_1$ for kissing particles
Let $e(r)=\exp(-v(r)/T)$\[\begin{equation} Z_1 = \int_0^1 e(r) d{\bf r} \end{equation}\]
where $d{\bf r} = S_n r^{n-1} dr$ and $
S_n = \frac{2\pi^{\frac{n+1}{2}}}{\Gamma(\frac{n+1}{2})} $ is the surface area of a $n$-dimensional unit sphere.
Specifically $S_2=2\pi$, and $S_3=4\pi$.
Expectation value
Pairwise quantity $A(r)$ \[ \langle A\rangle = \frac{1}{Z_s} \int_0^1 A(r) e(r)d{\bf r} \]
- Pair energy, $\langle u\rangle$: $A(r)=u(r)$
- Square energy, $\langle u^2\rangle$: $A(r)=u(r)u(r)$
- Pair virial, $\langle w\rangle$: $A(r)=w(r)\equiv -\frac{r}{d}\frac{dv(r)}{dr}$
- General, $\langle u^i w^j\rangle$: $A(r)=[u(r)]^i[u(r)]^j$
Example: energy
$\langle v\rangle = \frac{I_1(T)}{I_0(T)+V_\text{free}}$ where
\[\begin{equation}I_i(T) \equiv \int_0^1 [v(r)]^i e(r)d{\bf r}\end{equation} \]
For $T\to0$ \[I_i(T) = \frac{S_nT^{i+\frac{1}{\alpha}}}{(n-1)^{i+1}} \Gamma\left(i+\frac{1}{\alpha}\right)\]similar to what we did for the Barker-Henderson integral.
Harmonic repulsive, 3D
\[\begin{equation} u(T) \propto T^\frac{3}{2} \quad (T\to0) \end{equation}\]
Along isomorph: \[\begin{equation} u_l(T) = u_l(T_0)[T/T_0]^{3/2} \end{equation}\] \[\begin{equation} u_s(T) = u_s(T_0)[T/T_0]^{3/2} \end{equation}\] where $u_l(T_0)$ and $u_s(T_0)$ is energies at the reference state-point at $T_0$.
For density along isomorph: \[\rho_l(T)=\rho_l(T_0)\left(1+\frac{9\sqrt{\pi}[\sqrt{T}-\sqrt{T_0}]}{2\sqrt{2}}\right)\]
We now have theory for
$u(T)$'s and $\rho(T)$'s along isomorphs.
Let $T_0\to 0$ and assume $\alpha_1(T) =\alpha_2(T) = 0$ \[ p(T) = p_\bullet\left(1+\frac{9\sqrt{\pi T}}{2\sqrt{2}}\right) \]
Summary of theories (harmonic repulsive, 3D)
- Boltzmann: $p(T) = p_\bullet\left(1+3\sqrt{T}\right)$
- Barker-Henderson: $p(T) = p_\bullet\left(1+\frac{3\sqrt{\pi}}{2}\sqrt{T}\right)$
- Isomorph theory: From $T_0$ on coexistence line
Note: Isomorph theory does not refer to a HS reference.
- (simplified HS version): $ p(T) = p_\bullet\left(1+\frac{9\sqrt{\pi}}{2\sqrt{2}}\sqrt{T}\right)$
Numerical determination of coexistence lines
Interface pinning: \[U({\bf R})=U_0({\bf R})+\frac{\kappa}{2}(Q({\bf R})-a)^2\]
$\Delta G$ from average force on bias field.
... then numerical integration
of the Clausius-Clapeyron identity [Kofke (1993)]
\[ \frac{dp}{dT} = \frac{\Delta s}{\Delta v} \]
Results
WCA and harmonic-repulsive
\[p(T) = p_\star(T) [\alpha_1(T)+\alpha_2(T)+\alpha_3]\simeq p_\star(T)\alpha_3 \]
\[\begin{aligned}\alpha_1(T) &=& [u_s(T)/T - u_s(T_0)/T_0] - [u_l(T)/T - u_l(T_0)/T_0], \\
\alpha_2(T) &=& \log(\rho_s(T)/\rho_s(T_0)) - \log(\rho_l(T)/\rho_l(T_0)), \\
\alpha_3 &=& \frac{p_0}{T_0} [\rho_l^{-1}(T_0)-\rho_s^{-1}(T_0)], \\
p_\star(T) &=& T/[\rho_l^{-1}(T)-\rho_s^{-1}(T)]\end{aligned}\]
WCA=harmonic-repulsive at $T\to0$
Results
Hertzian spheres
Hertzian spheres
$u(r)=(1-r)^{5/2} \quad$ for $r<1$
Hertzian spheres
$u(r)=(1-r)^{5/2} \quad$ for $r<1$
References:
- Extreme case of density scaling: The Weeks-Chandler-Andersen system at low temperatures E. Attia, J. C. Dyre and U. R. Pedersen. Phys. Rev. E 103, 62140, (2021).
DOI: 10.1103/PhysRevE.103.062140 - Comparing four hard-sphere approximations for the low-temperature WCA melting line E. Attia, J. C. Dyre and U. R. Pedersen. J. Chem. Phys. 157, 34502, (2022).
DOI: 10.1063/5.0097593 - Comparing zero-parameter theories for the WCA and harmonic-repulsive melting lines J. C. Dyre and U. R. Pedersen
J. Chem. Phys. 158, 164504 (2023)
DOI: 10.1063/5.0147416 - Low-temperature freezing of generalized hertzian spheres*
Ulf R. Pedersen. In preparation ...
*any $\alpha$ in any dimension (2, 3, $\ldots$)