⇦Chinese Physics Letters, 2019, Vol. 36, No. 8, Article code 080501 Critical One-Dimensional Absorption-Desorption with Long-Ranged Interaction * Xiaowei Liu (刘小伟), Jingyuan Guo (郭竞渊), Zhibing Li (李志兵)** Affiliations State Key Laboratory of Optoelectronic Materials and Technologies, and Guangdong Province Key Laboratory of Display Material and Technology, School of Physics, Sun Yat-Sen University, Guangzhou 510275 Received 23 April 2019, online 22 July 2019 *Supported by the National Natural Science Foundation of China under Grant No 11274393, the National Basic Research Program of China under Grant No 2013CB933601, and the National Key Research and Development Program of China under Grant No 2016YFA0202001.
**Corresponding author. Email: stslzb@mail.sysu.edu.cn Citation Text: Liu X W, Guo J Y and Li Z B 2019 Chin. Phys. Lett. 36 080501    Abstract An absorption-desorption model with long-ranged interaction is simulated by the dynamic Monte Carlo method. The dynamic process has an inert phase and an active phase that is controlled by the absorption rate. In the active phase, the number of vacancies increases with time exponentially, while in the inert phase the vacant sites will be occupied by adsorbates rapidly. At the critical absorption rate, both the number of vacancies and the time-depending active probability exhibit power-law behavior. We determine the critical absorption rate and the scaling exponents of the power-laws. The effect of the interaction range of desorption on the critical exponents is investigated. In the short-ranged interaction limit, the critical exponents of Schlögl's first model are recovered. The model may describe the stability of the inner Helmholtz layer, an essential component of the electrochemical double-layer capacitor at a nanowire. DOI:10.1088/0256-307X/36/8/080501 PACS:05.70.Ln, 05.70.Jk, 64.60.Ht © 2019 Chinese Physics Society Article Text Critical processes are classified into universal classes, which depend on the dimensionality and the interaction range but not on the microscopic details of the system.[1–5] The present model is a generalization of the short-ranged adsorption-desorption (SRAD) model that has a critical adsorption rate $r_{\rm c}$. When $r>r_{\rm c}$, the system rapidly reaches the inert phase where all sites are occupied, while for $r < r_{\rm c}$ the system is in the active phase that has a non-vanishing fraction of vacant sites. The active phase can persist for a macroscopic long time. We will show that the corresponding long-ranged adsorption-desorption (LRAD) model has the same phase structure, while the critical exponents for the scaling behavior of the LRAD model are considerably different from those of the SRAD model. A possible realization of the LRAD model is the adsorption/desorption of molecules at a nanowire[6–11] as shown in Fig. 1. The adsorbates at the electrode surface form the inner Helmholtz layer (IHL), which is an essential component of the electric double-layer supercapacitors.[12,13] It has been reported that the addition of 'nano-filler' to simplify polyether-based electrolytes increases the conductivity several fold in the temperature range of 60–80$^{\circ}\!$C, but there is no advantage at room temperature.[14] This observation implies that the IHL would undergo a phase transition driven by temperature or the adsorption rate effectively. The desorption probability is sensitive to the correlation of adsorbates and the interaction between the adsorbates and electrode surface. It has been known that interaction in low-dimensional materials is often long ranged, which depends on the composition of electrolytes, and on the morphology of electrode materials.[15–18] The electric screening length has been found relevant to the performance of the batteries.[19]
cpl-36-8-080501-fig1.png
Fig. 1. A possible realization of the one-dimensional adsorption-desorption model. The grey balls are molecules of the solvent. The molecules can be adsorbed at or desorbed from the nanowire. The adsorption and desorption processes on the nanowire are indicated by the red arrows.
In our model, the state of the one-dimensional lattice is specified by a set of integers $\{s_{i}=0, 1\}$ with $i$ as the site label. Each site could be either occupied ($s_{i}=1$) or unoccupied ($s_{i}=0$, also referred to as damaged). A damaged site has a constant adsorption rate $r$ to be occupied. If the $i$th site is occupied, it would become unoccupied in the next time step with a state-depending desorption probability $w_{i}[s]$. The SRAD model has $w_{i}=w(s_{i-1},s_{i+1})$, which depends only on occupation of the nearest neighbors. It has been known that the SRAD model belongs to the universal class of a number of famous models, including the stochastic cellular automata,[20] the directed percolation,[21] Schlögl's first model,[22,23] and Reggeon field theory.[24,25] Imitating the kinetic spherical model (the large $N$ limit of the dynamic Landau–Ginzburg model) with long-ranged interaction,[26] the present work generalizes the SRAD model to the LRAD model with the desorption rate $$ w_{i}=A\sum_{|j-i|>1}\frac{1-s_{j}}{|j-i|^{1+\sigma}}.~~ \tag {1} $$ In this model, the influence of a damaged site on the desorption of another site is decreasing with the distance between the sites. The interaction range is longer (shorter) for smaller (larger) $\sigma$. The parameter $A$ depends on the timescale, which can be chosen for convenience. Note that the fully occupied state is the attractive point of the inert phase. Obviously the process is non-ergodic and hence has no equilibrium state. To investigate the stability of occupation, we simulate the process initialized in a state that has only one damaged site with the dynamic Monte Carlo method.[27] A review of the method can be found in Refs. [28,29]. The time-dependent order parameter is the fraction of damaged sites $$ n(t)=\frac{1}{N}\sum_{i}(1-s_{i}),~~ \tag {2} $$ where $N$ is the number of total sites. The periodic boundary condition is assumed. The order parameter $n(t)$ should increase with time in a power law at the critical point. Therefore, the critical point can be obtained by searching in a range of parameter $r$ for the power law of damage ratio $$ n(t)\sim t^{\eta}.~~ \tag {3} $$ The scaling behavior emerges at a specific adsorption rate for each given $N$. It becomes more obvious for larger $N$, as shown in Fig. 2, where the double-log plots of $n$ versus $t$ are presented for $\sigma=0.7$ and $N=1024$, 4096 and 8192. For the largest lattice of $N=8192$, the scaling regime is from $10^3$ to $10^5$ (Monte Carlo sweep). In the following, $N=8192$ will be used. Figure 3 presents $n(t)$ for three values of $r$ with $\sigma=0.7$ in double-log coordinates. Each curve is averaged over $6.4\times 10^{5}$ samples of processes with the same initial state. The critical point is estimated by extrapolating a straight line from them. For $\sigma=0.7$ the critical point estimated by such a way is $r_{\rm c}=0.1935(1)$. The critical exponent $\eta$ turns out to be 0.126(1). The numbers in the brackets are the finite-size errors estimated by comparing results of various $N$ that should be of infinity in the thermodynamic limit. The statistical errors presented as error bars in the figure are estimated by separating the samples into five groups. The statistical errors are about one order smaller than the finite-size errors. By the same method we obtain $\eta$ for $\sigma$ ranged from 0.5 to 2.5 as shown in Fig. 4. The statistical error is also indicated by the error bars. One can see that the exponent $\eta$ drops to zero at $\sigma=1/2$ where the Gaussian fixed point plays a role as expected. For $\sigma>1/2$, we assume that $\eta$ varies with $\sigma$ as $$ \sigma_{\rm s}(1-e^{-\frac{2\sigma-1}{a}}).~~ \tag {4} $$ By fitting the numerical data, we obtain $a=0.935$ and $\eta_{\rm s}=0.331$.
cpl-36-8-080501-fig2.png
Fig. 2. Curves of the damage ratio versus time (Monte Carlo sweep) in double-log coordinates, with $\sigma=0.7$ and $N=1024$, 4096 and 8192, respectively. The adsorption rates are chosen so that the fraction of vacant sites exhibits power-law scaling approximately after the initial microscopic slip of the first thousand sweeps.
cpl-36-8-080501-fig3.png
Fig. 3. The critical damage ratio (solid curve) obtained by the method of interpolation from two ratios (dashed and dotted) with $r$ near the critical point. The critical exponent $\eta=0.126$ at $\sigma=0.7$ is estimated for time ranged from $10^3$ to $10^5$ (sweeps). The number of total sites is $N=8192$, as the same as in the following figures.
cpl-36-8-080501-fig4.png
Fig. 4. The critical exponent $\eta$ versus the parameter of interaction range, $\sigma$. The curve is obtained by fitting the data with Eq. (4). The error bars indicate the statistical errors.
The other interesting quantity is the $t$-depending active probability defined as the ratio of the number of samples possessing damaged sites over the number of total samples at time $t$. At the critical point it should have the scaling behavior $$ P(t)\sim t^{-\delta}.~~ \tag {5} $$ The power-law of Eq. (5) is confirmed by our simulation as shown in Fig. 5. The critical exponent $\delta$ versus $\sigma$ is presented in Fig. 6. For instance, the exponent $\delta$ is 0.63(2) at $\sigma=0.7$ and 0.17(1) at $\sigma=2.5$. The probability that the compact IHL would be recovered at time $t$ is related to $P(t)$ as $$ R(t)=1-P(t)=1-c t^{-\delta},~~ \tag {6} $$ where $c$ is a positive constant. The recover probability is meaningful only after the initial slip of a microscopic timescale such that $R(t) < 1$.
cpl-36-8-080501-fig5.png
Fig. 5. Active probabilities at critical points with $\sigma$ equalling to 0.7, 1.0, 1.5, 2.0 and 2.5.
cpl-36-8-080501-fig6.png
Fig. 6. The exponent $\delta$ for $\sigma$ equalling to 0.7, 1.0, 1.5, 2.0 and 2.5. The error bars indicate the statistical errors.
The LRAD model is obviously distinguished from the SRAD model in the regime $1/2 < \sigma < 1$. The exponent $\eta$ ($\delta$) for $\sigma>2$ coincides with that of the Schlögl's first model which is 0.314 (0.159) within the range of errors. In summary, the one-dimensional adsorption-desorption model with long-ranged interaction as a model for the IHL at nanowires has been simulated with the dynamic Monte Carlo method. An inert phase and an active phase separated by a critical line are found. In the inert phase, the damage of the IHL (i.e., the vacant sites) will be removed automatically in a short time, while in the active phase the number of damaged sites will increase exponentially. Along the critical line, the increase of damaged sites is much slower, increasing only in power law. The critical adsorption rate $r_{\rm c}$ as a function of the interaction range is determined. The inert and active phases are corresponding to $r>r_{\rm c}$ and $r < r_{\rm c}$, respectively. Power-laws of damage ratio and the active probability are confirmed at the critical rate $r_{\rm c}$ and the corresponding critical exponents are calculated as a function of the interaction range. Critical exponents varying along the critical line are common in critical models as the existence of a critical line usually implies that the model has a marginal operator.[30] The critical behavior of the model is universal up to the dimensionality and the interaction range. Therefore, our results would give insight into realistic adsorption-desorption processes involved in various nanowire-based electrochemistry applications such as the electric double-layer supercapacitors.[31] References Theory of dynamic critical phenomenaNew universal short-time scaling behaviour of critical relaxation processesRemanent magnetization decay at the spin-glass critical point: A new dynamic critical exponent for nonequilibrium autocorrelationsThe Universality of Dynamic Exponent θ′ Demonstrated by the Dynamical Two-Dimensional Ising ModelDependence on initial conditions of an adsorption-desorption processSupercapacitor electrodes from multiwalled carbon nanotubesA study of activated carbon nanotubes as electrochemical super capacitors electrode materialsCarbon Coated Fe 3 O 4 Nanospindles as a Superior Anode Material for Lithium-Ion BatteriesCoaxial MnO 2 /Carbon Nanotube Array Electrodes for High-Performance Lithium BatteriesReduced Graphene Oxide-Mediated Growth of Uniform Tin-Core/Carbon-Sheath Coaxial Nanocables with Enhanced Lithium Ion Storage PropertiesOpportunities and challenges for a sustainable energy futureBuilding better batteriesComparison of density matrix renormalization group calculations with electron-hole models of exciton binding in conjugated polymersPotential barrier of graphene edgesRe-orientation transition in molecular thin films: Potts model with dipolar interactionPhase transitions and critical phenomena in the two-dimensional Ising model with dipole interactions: A short-time dynamics studyHierarchical nanowires for high-performance electrochemical energy storagePhase transitions of cellular automataOn the nonequilibrium phase transition in reaction-diffusion systems with an absorbing stationary stateChemical reaction models for non-equilibrium phase transitionsReggeon field theory (Schlögl's first model) on a lattice: Monte Carlo calculations of critical behaviourRecent developments in Reggeon field theoryCritical exponents for the Reggeon quantum spin modelSpherical Model with Long-Range Ferromagnetic InteractionsDynamic Monte Carlo Measurement of Critical ExponentsMonte Carlo Simulations of Short-Time Critical DynamicsStudy of phase transitions from short-time non-equilibrium behaviourBatteries and electrochemical capacitors
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