A Brownian dynamics computer simulation study of a highly coarse-grained model of telechelic associating polymers has been carried out. In a critical concentration range the model produces the so-called 'loops-to-bridges' transition, thought to exist in the experimental systems, in which the two hydrophobic groups are in different micelles, thereby forming a highly interconnected, ultimately percolating, network. The fraction of bridged polymers produced by the model correlates well with the experimental viscosity at corresponding concentrations. The distribution of micelle sizes compares favorably with the predictions of the Meng-Russell free energy theory. The mean cluster size scales well with volume occupancy according to a simple mean-field theory. The stress relaxation function is a stretched exponential at short times and not too high concentrations but develops a longer time plateau in the percolation region, both in agreement with experiment. New experimental data for the concentration dependence of the self-diffusion coefficient, viscosity, elastic modulus and relaxation time of telechelic associative polymers are presented, which show broad qualitative agreement with the simulation data.

A novel coarse-grained computational model for associating polymers is proposed that is based on a Gaussian "blob" representation of the polymer chains. The model allows a large number of model polymers to be simulated at moderate computational cost over a wide packing fraction range using the Brownian dynamics, BD, technique. The attraction of the hydrophobic part of the polymer to those on other molecules can lead to strong aggregation of the polymer molecules in real systems, and this is included in the model by an attractive potential felt by the Gaussian blobs to a common "nodal" point that represents the center of the micelle. Attention here is confined to model AB diblock copolymers in which the hydrophilic block, A, has a much higher mass than the hydrophobic moiety, B, which leads to relatively small aggregation numbers, Nagg, of approximately 8. The aggregation number at low packing fractions is found to increase with packing fraction, as observed in experiments, with a functional form that closely follows a simple theory derived here that is based on entropy-derived mean-field terms for the free-energy change associated with the incorporation of the polymer molecule into the micelle. The computational model exhibits an extremely low critical micelle concentration (cmc), and micelles with Nagg approximately 5 are observed at the lowest packing fractions, phi, simulated ( approximately 10-4), which is consistent with experiment. The long-time self-diffusion coefficient of the polymers (and hence micelles) decreases logarithmically with packing fraction, and the viscosity increased with concentration according to the Huggins equation. The spherical blob coarse graining results in the simulable time scales being longer than the Rouse time of the chain, and hence for the nonassociating polymers the intrinsic viscosity is an input parameter in the model. The introduction of association leads to the partial inclusion of the intrinsic viscosity in the simulation and has an effect on the computed Huggins coefficient, kH, which is found to be approximately 6 in those cases.

Ruelle’s thermodynamic stability criteria are applied to the separation-shifted Lennard-Jones (SSLJ) fluid, and the domain of its parameters giving “normal” thermodynamic stability in the thermodynamic limit is established. Fluids interacting with the SSLJ both conforming to and breaking these stability criteria were modeled using molecular dynamics computer simulation. For system sizes typical of most simulations, the transition between the two patterns of behavior was found to be smeared out over a range of parameter values. Thermodynamic instability is marked by a collapse of the system into a small “ball” or volume. The collapsed state nevertheless has some statistical mechanical properties typical of systems exhibiting normal thermodynamics (e.g., the kinetic and configurational temperatures were found to be the same in the collapsed state within statistics).

Molecular dynamics simulations have been used to calculate the self-diffusion coefficient, D, of the hard sphere fluid over a wide density range and for different numbers of particles, N, between 32 and 10 976. These data are fitted to the relationship D = D(infinity) - AN(-alpha) where the parameters D(infinity), A, and alpha are all density-dependent (the temperature dependence of D can be trivially scaled out in all cases). The value alpha = 1/3 has been predicted on the basis of hydrodynamic arguments. In the studied system size range, the best value of alpha is approximately 1/3 at intermediate packing fractions of approximately 0.35, but increases in the low- and high-density extremes. At high density, the scaling follows more closely that of the thermodynamic properties, that is, with an exponent of order unity. At low packing fractions (less than approximately 0.1), the exponent increases again, appearing to approach a limiting value of unity in the zero-density limit. The origin of this strong N dependence at low density probably lies in the divergence in the mean path between collisions, as compared with the dimensions of the simulation cell. A new simple analytical fit formula based on fitting to previous simulation data is proposed for the density dependence of the shear viscosity. The Stokes-Einstein relationship and the dependence of D on the excess entropy were also explored. The product Deta(s)p with p = 0.975 was found to be approximately constant, with a value of 0.15 in the packing fraction range between 0.2 and 0.5.

A study of the soft-core to hard-core percolation properties of the hard-sphere fluid as a function of hard core (sphere) packing fraction, ζ, has been made using the Molecular Dynamics computer simulation method. The interparticle separation between the soft shells to achieve percolation, σ p , exhibits a monotonic decrease with ζ from the permeable spheres limit to approximately the glass transition, where σ p tends to the hard-sphere radius, σ. σ p was fitted to a semi-empirical function of ζ, which has the exact low density limiting behaviour. The ζ-dependence of the corresponding packing fraction of the permeable shells, is discussed. The local coordination number at the percolation threshold showed a transition between the soft-core and hard-core limits from ca. 2.74 to 1.51, as found in previous studies. A reasonably accurate simple analytic expression is given for the packing fraction dependence of the coordination number up to the percolation distance. Various key length scales of the hard-sphere system are compared with σ p as a function of ζ. The hard-sphere percolation distance dependence on packing shows a similar behaviour to that of rescaled Weeks–Chandler–Andersen (WCA) fluids, but not the same, which is consistent with the conclusions of that previous study.

Molecular dynamics simulations have been carried out of the radial distribution function of the hard sphere fluid for a range of densities in the equilibrium fluid and just into the metastable region. The first derivative of the hard-sphere radial distribution function at contact was computed and its density dependence fitted to a simple analytic form. Comparisons were made with semi-empirical formulae from the literature, and of these the formula proposed by Tao et al (1992 Phys. Rev. A 46 8007) was found to be in best agreement with the simulation data, although it slightly underestimates the derivative at the higher packing fractions in excess of about 0.45. Close to contact, within a few per cent of the particle diameter, the radial distribution function can be represented well by a second order polynomial. An exponential function, which has some useful analytic features, can also be applied in this region.

We have examined the combined effects of grain morphology and electron trapping on the transient response of photoelectrons moving through the TiO2 grains in a dye-sensitized nanocrystalline solar cell using a multi-time-scale random walk Monte Carlo model. Our use of a multi-time-scale approach enables us to simulate transport for electrons moving through spherical connected grains in a three-dimensional (3D) voided network and look at the effect of the size of interparticle boundaries on carrier dynamics. We can also address similar times to those over which measurements are taken, namely, 0.1 ms. These times are long because of deep traps in the TiO2 grains. The grains have 2-fold connectivity in one dimension (linear chains) or 4-fold or 6-fold connectivity in three dimensions and traps with an exponential distribution of energies. Photoelectrons are generated by a light pulse of short duration. The spatial distribution of the photogenerated electron density from this pulse either has a uniform profile or is peaked on the electrolyte side. We show that the constrictions at the grain necks slow the electrons, making trapping more likely and hence further delaying their passage to the extracting electrode. By comparing our results for 4-fold and 6-fold coordinated particles on a cubic lattice with 2-fold coordinated particles on linear chains, we show that transport is slowed in the former case due to the additional paths available to the electrons in the 3D network. We also find that the charge and current transients cannot be fit to an analytical solution of the continuum equations with an effective diffusion coefficient even at long times. Therefore, caution must be exercised when attempting to fit experimental transient data with an effective diffusion coefficient.

We present a novel multi-timescale approach to simulating transport for electrons moving through a mesoporous film of TiO2 grains in the dye-sensitized solar cell. This method is unique in its ability to integrate microscopic (intra-grain) with mesoscopic (inter-grain) charge transport processes. Comparisons are made with experimental measurements of transient currents.

Microwave reflectance methods have been used to study the kinetics of photogeneration of hydrogen, on p-Si in acidic fluoride solutions. To estimate the rate constant for interfacial electron transfer, the magnitude of the observed steady-state light-induced microwave reflectivity change has been compared quantitatively with theoretical predictions based on numerical calculation of the electron and hole profiles in the silicon sample. In addition, the transient and periodic microwave reflectivity responses to stepped and sinusoidally modulated illumination, respectively, have been analyzed to obtain information about the electrode kinetics. The time constants for relaxation of the light-induced conductivity derived from this analysis confirm that electron transfer during hydrogen evolution is slow. The apparent inconsistency in the values of the phenomenological electron-transfer rate constants derived from the steady-state and transient or periodic responses can be resolved if it is assumed that hydrogen evolution proceeds via electron capture by protons followed by a slow bimolecular step leading to molecular hydrogen. Surface charging as the result of slow kinetics is expected to lead to band edge unpinning even at low light intensities, and the existence of this effect has been confirmed by transient photocapacitance measurements.

Grätzel cells are based on a thin film of dye sensitized nanocrystalline TiO2 interpenetrated by a redox electrolyte. Photoexcitation of the dye is followed by fast (less than 100fs) electron injection into the conduction band of the TiO2 nanoparticles. The injected electrons travel through the nanocrystalline TiO2 to the anode. Electron transport is slowed by trapping in surface and/or bulk states. Detrapping from these states competes with the back reaction of electrons with triiodide ions in the electrolyte. The back reaction causes electrons to be removed from the circuit and so reduces efficiency. The requirement to pass through grain necks also slows the electron transport, increasing the likelihood of back reactions occurring. We employ a Monte Carlo simulation to investigate the effects on electron transport of the connectivity. We use a multi-scale simulation method to model the response to a pulse of optical illumination; here the transient current and charge exiting the cell. Our main result is that for small inter-grain necks, transport through the film is slowed considerably. This effect is more pronounced when traps are present, as trapping competes with transfer between grains. Dye sensitized nanocrystalline solar cells (Grätzel cells) have shown great promise because they are comprised of low cost materials and perform well under indirect radiation [1]. They are based on a thin film of porous nanocrystalline TiO2 coated with a dye using a ruthenium complex that harvests the solar energy flux. Photoexcitation of the dye is followed by fast electron injection into the conduction band of the TiO2 nanoparticle electrode. The electrons then travel through the TiO2 to the collecting contact. The oxidized dye reacts with I− ions in the electrolyte, producing I3 which diffuses to the Pt coated cathode where it is reduced to complete the cycle. The efficiency with which the incident photon flux is converted into current is determined in part by the competition between electron transport and back reaction. Optimization of electron collection is therefore clearly important for any practical cell, regardless of whether it employs an electrolyte or another contacting phase such as a polymeric hole conductor. Electron transport in the TiO2 grains occurs by diffusion since the fields across the film are screened by the electrolyte and the small size of the particles does not support a built in field [2]. An electron diffusion coefficient Dbare of 1×10−2cm2s−1 has been deduced from mobility data for single crystalline TiO2 [3]. The effective diffusion coefficient for electrons Deff has been shown to vary with grain morphology from measurements of intensity modulated photocurrent spectra for anatase and rutile TiO2 grains [4]. In addition, Deff has been shown to vary with the background illumination intensity I0 as I 0 [5]. Hence, the transit time for electrons across the films is also sensitive to I0, being several minutes at low light levels and ms at solar illumination intensities [6]. The major reason for such behavior is trapping and detrapping of the electrons. The detrapping time increases exponentially with the trap depth. As I0 increases, so does the generated electron density, thus the traps fill up, so the electrons are only trapped in the shallower traps and they detrap much faster. The results presented here assume transport occurs in the conduction band, punctuated by a series of trapping and detrapping events. It is assumed that traps are located on the surface of the particles and have energies following an exponential distribution. To understand the influence of the morphology on electron transport, we have made predictions of the transient current using a Monte Carlo simulation, in which the electrons execute a random walk through a chain of spherical grains with traps at the surface of the grains. To cope with time scales varying from fs (the time between scattering events in each grain) to ms (the time scale for the transient current), we have adopted a novel approach using a combination of short time-scale and long scale simulations described in detail in [7]. Here, unlike [7], we also look at grains which are connected with 4-fold coordination in 3-dimensions (3D). Fig 1 shows that even without traps, the morphology can significantly affect the transport properties of the film. As the necks get wider their effect on the transport is reduced. Transport is faster for chains than with 3D

Light- and voltage-induced changes in the microwave reflectivity of semiconductors can be used to study the kinetics and mechanisms of electron transfer at semiconductor I electrolyte interfaces. The theory of the method is developed and illustrated by numerical calculations of the steady-state microwave response for low-doped silicon. The results define the range of rate constants that should be experimentally accessible using microwave reflectivity methods. The time and frequency responses of light-induced microwave reflectivity changes are considered, and it is shown that they can be used to derive values of electron transfer and recombination rate constants.

The principles and applications of microwave reflectivity measurements in semiconductor electrochemistry are reviewed and illustrated by theoretical calculations and experimental examples. The microwave response of the illuminated p-Si ∣ NH4F junction has been studied under depletion conditions and related to the calculated concentration profiles of electrons and holes. Time- and frequency-resolved measurements have been used to follow the interfacial transfer of photogenerated electrons to protons in solution. The rate constant for interfacial electron transfer is very small, probably reflecting the absence of low energy sites for stabilisation of the intermediate, in the two-electron reduction of H+ to H2. The time dependent measurements provide evidence for reversible hydrogen absorption into the surface region of the silicon. Potential modulated microwave reflectivity has been used to study the behaviour of p-type silicon in fluoride solutions under depletion and accumulation conditions in the dark. Under depletion conditions, the time-resolved and periodic microwave responses are related to the changes in the width of the space charge region (SCR), and the sensitivity factor that relates the normalised reflectivity changes to changes in carrier concentrations can be obtained by comparison of the microwave response with the potential dependent space charge capacitance. Under accumulation conditions, dissolution of the p-Si occurs, resulting in porous silicon formation or electropolishing depending on the applied potential. In this case, the microwave response gives information about the potential distribution across the Si ∣ (oxide) ∣ solution system.

We present a Monte Carlo model of electron transport in mesoporous films of TiO2 particles in which electrons execute a random walk through a chain of spherical grains with traps at the surface of the grains. This has been used to simulate transient photocurrents in dye sensitized nanocrystalline solar cells. By comparing our results with a model based on solutions of the continuity equation for the free and trapped electron densities in which the film is treated as a homogeneous medium, we find that necks between grains have a noticeable effect on reducing the photocurrent. Values of an effective electron diffusion coefficient have been deduced by comparing an analytical solution to the continuity equation for the free electrons, in which the traps and back-reaction are ignored, with the numerical results from the Monte Carlo and continuum models. To the authors' knowledge, this is the first time that the influence of the grain connectivity on electron transport has been modeled.