## Abstract

The Pickett effect is the phenomenon of creep enhancement during transient drying. It has been observed for many nanoporous solids, including concrete, wood and Kevlar. While the existing micromechanical models can partially explain this effect, they have yet to consider nanoscale dynamic effects of water in nanopores, which are believed to be of paramount importance. Here, we examine how creep deformations in a slit pore are accelerated by the motion of water due to drying forces using coarse-grained molecular dynamics simulations. We find that the drying that drives water flow in the nanopores lowers both the activation energy of pore walls sliding past one another and the apparent viscosity of confined water molecules. This lowering can be captured with an analytical Arrhenius relationship accounting for the role of water flow in overcoming the energy barriers. Notably, we use this model and simulation results to demonstrate that the drying creep strain is not linearly dependent on the applied creep stress at the nanopore level. Our findings establish the scaling relationships that explain how the creep driving force, drying force and fluid properties are related. Thus, we establish the nanoscale origins of the Pickett effect and provide strategies for minimizing the additional displacements arising from this effect.

## 1. Introduction and background

Nanoporous solids [1,2] contain many internal cavities (pores, channels or both) that are predominantly nanoscale in terms of their average width. These materials are used commonly in applications ranging from chemical separations (zeolites and metal-organic frameworks [3]) to structural materials (concrete [4] and cellulose-based materials [5,6]) in which there exists a tremendous amount of internal surface area through which the solid can interact with fluids confined in the nanoscale cavities. For hydrophilic materials, pore water retained from processing in the form of moisture is always present in these nanopores, even after long periods of drying.

An intriguing aspect of nanoporous solids is the way they respond to changes in their internal moisture content, specifically in terms of their creep behaviour. The so-called ‘Pickett effect’ [7] is one specific phenomenon that is of particular interest for these materials as it has yet to be fully explained [7–9]. This effect is based on the observations of Pickett [7], who noted that the strain of concrete specimens undergoing drying and mechanical loading simultaneously is much greater than the sum of the strains from pure creep (i.e. observed under mechanical loading but in the absence of any drying) and drying shrinkage (i.e. observed upon drying in the absence of any mechanical loading). This concept is illustrated schematically in figure 1.

The Pickett effect, while being central to the creep deformation of concrete [10–12] and thus the life cycle performance of most structural and infrastructural systems, still remains to be fully explained. Since the discovery of the Pickett effect nearly 80 years ago, there has been continued experimentation [13] to quantify this behaviour. In addition to observations of the Pickett effect for cement paste [14] and concrete [15], this drying creep phenomenon has also been observed for a number of other nanoporous solid materials [16] including cellulose-based materials such as wood [17–20] and paper [21,22], keratin fibres [23,24], and synthetic polymers including Kevlar [25] and nylon [26]. Several theories have been proposed to explain the root causes of the Pickett effect.

Initially, the Pickett effect was widely thought to be explained by a difference in microcracking caused by non-uniform moisture distributions. In a companion load-free shrinkage specimen, the microcracks remain open during drying, thus reducing the observed overall shrinkage compared to that in the compressed creep specimen in which, despite drying, there is almost no microcracking. However, later tests under sustained compressive forces of different eccentricities [9] showed that the differences in microcracking explain only a minor part of the Pickett effect in concrete. Currently, the most widely accepted theories [9,27–30] are based on micromechanical models of concrete that are easily extendable to other material systems as they focus on inherent nanostructure as opposed to specific chemistry. These theories attribute the increased strain observed under drying creep to a combination of both microcracking and microprestress developed during concrete formation or drying. In the microstructure of hardened cement gel, the hindered adsorbed water completely fills the nanopores (of width less than 3 nm) and exerts an enormous disjoining pressure [31,32] on the pore walls, which introduces tensile microprestress into the solid skeleton of the material. Especially, the solid bridges across the nanopores must have highly stressed atomic bonds. As these bonds gradually break, the microprestress progressively relaxes, which gradually reduces the rate of creep due to external loads [27,28]. Extended to the transitional thermal creep [33] (an effect similar to the Pickett effect), and combined with solidification theory for the cement hydration process, this micropretress–solidification theory has provided close enough fits of test data.

Nevertheless, the existing theory has some limitations. It is not deep enough. Particularly, the connection of the microprestress increase to the increased creep is not explained in terms of nanomechanics. The water transport along the filled nanopores is not considered. Yet, the viscosity of nanopore water must be expected to play a role. Because of the effect of surface forces at nanopore walls, it is surely not Newtonian, i.e. depends on the shear deformation rate. The non-Newtonian behaviour is supported by a number of recent studies focused on the behaviour of water confined in nanotubes [34–36] or nanochannels [37–39]. In these studies, nanoconfined water exhibits interesting characteristics that vary significantly from its macroscopic behaviour including non-Newtonian fluid behaviour and glassy or ice-like structure and reduced mobility near hydrophilic surfaces. Therefore, it is plausible that the behaviour of nanoconfined water under combined creep and drying forces may lead to physical phenomena that may be an important contributor to the creep enhancement during drying. Some have proposed that the Pickett effect originates at the scale of this nanoconfined water [16].

In this study, we aim to investigate and quantify these nanoscale effects to further explain the origin of the Pickett effect in nanoporous solids. To examine the notion that nanoscale effects may play a key role in the macroscopic Pickett effect, here we conduct coarse-grained molecular dynamics (CGMD) simulations of the flow of water in a nanochannel. We use a generalized model of water interacting with a hydrophilic solid surface, such that our results could be extended to any nanoporous solid and would not be limited to any specific material chemistry. In our simulations, we focus on quantifying the relative creep displacement or relative creep velocity of the solid walls of the nanopore under an applied creep force and further characterize how these values change when drying occurs (i.e. flow of water within the nanopore).

To explain these nanoscale behaviours, we propose an analytical model, based on probability and nanoscale energy barrier arguments, that helps to explain why the strain due to drying creep is greater than the sum of the contributions due to drying only and to mechanical loading only. Using this analytical model, we can directly compare the results of CGMD simulations and theory to demonstrate how a major part of Pickett effect can be explained by water under nanoconfinement within nanoporous solids.

## 2. Simulation methods

As mentioned previously, we aim to examine the Pickett effect from a nanoscale perspective using CGMD simulations of water flow through a nanopore/nanochannel. Here, we use a slit pore geometry to simulate the flow of water at the nanoscale under drying creep conditions. The nanochannel, illustrated in figure 2*a*, consists of water confined between two solid, hydrophilic, walls that are analogous to the solid phase in nanoporous solids such as cellulose in wood or a calcium silica hydrate (C-S-H) phase in concrete. Generally speaking, the surface chemistry of the walls will vary from one system to another, but for materials exhibiting the Pickett effect, the walls are expected to be strongly hydrophilic, fully wetting surfaces where macroscopically the contact angle of a water droplet on the surfaces should be much less than 90°, in fact virtually 0°. Thus, rather than using chemistry-specific systematic coarse-graining techniques [40–42], here we use a generic coarse-grained (CG) model [43] that captures the essential aspects of the problem such as the bulk properties of the fluid and the contact angle of the fluid with the walls. To ensure that the parameter space studied be relevant for realistic systems, we use a previously developed model [43] that accurately captures the bulk viscosity, surface tension and density of water, and wall–water interactions that represent a fully wetting hydrophilic surface [39]. This (CG) model employs the Morse interatomic potential to represent interactions between CG particles (also referred to as beads). The Morse interatomic potential energy *U* is given by equation (2.1) as a function of the interatomic distance between particles, *r*, where *D*_{0} is the depth of the energy well, *r*_{0} is the equilibrium distance between particles, and *α* governs the decay characteristics and curvature of the interaction potential
*D*_{0} parameter of water–wall interactions to be the same as that for water–water interactions. The specific parameters and Morse interatomic potentials for different pairs of interactions are plotted as a function of interatomic distance *r* in figure 2*b*.

Using this model, CGMD simulations are conducted using the Large-scale Atomic/Molecular Massively Parallel Simulator molecular dynamics package [44]. Periodic boundary conditions (PBCs) are applied in the *x*- and *y*-directions, with the system having a finite thickness in the *z*-direction ( for reference to Cartesian directions see figure 2*a*). The simulation box has dimensions of 15×3×6 nm and the nanochannel width is chosen to be 4 nm—an intermediate nanopore size in many nanoporous solids [45]. In concrete, such width is characteristic of the size of the pores in the C-S-H gel. During simulations, the bottom solid layer is fixed by tethering the atoms to their initial position using virtual springs having a spring constant of 10.0 kcal mol-Å^{−2}. Here, we chose a harmonic constraint over a complete fix/freeze in order to prevent distinct differences between water–wall interactions at the top and bottom surfaces that may arise if the bottom wall is frozen. Additionally, we have verified that changing the value of this spring constant does not significantly change the results of our simulation (e.g. approx. 10–15% change in the measured velocities when the spring constant is varied from 5.0 to 50.0 kcal mol-Å^{−2}, at a given creep stress of 9.5 MPa and drying pressure of 100 MPa). To simulate the effect of basic (or pure) creep, individual forces are added to the atoms in the top layer having a magnitude denoted by *f*_{ic}. To simulate drying, individual forces, of magnitude *f*_{id}, are added to the water beads within the channel.

After constructing the nanopore using a predefined face-centred cubic (FCC) lattice with an edge length of 8.33 Å, the system is equilibrated for 0.5 ns in order to reach a minimum energy configuration. Simulations are run under the microcanonical ensemble (constant number of particles, volume and energy) with a dissipative particle dynamics (DPD) thermostat. A DPD thermostat is employed in simulations to accurately capture the hydrodynamics of the system by preserving momentum of the particles—something that is often neglected by other global thermostats [39]. This is accomplished by applying dissipative forces that depend on the relative velocity between particles/beads and not the absolute, fixed reference frame particle velocity. The DPD potential is overlaid on the interatomic Morse potential and thermostats the system temperature to 300 K and uses a damping coefficient of 2875 kcal-fs mol-Å^{−2}. This damping coefficient has been previously calibrated to match the bulk viscosity of water of 9.0×10^{−4} Pa-s [39]. The calibration conducted in our previous study [39] was performed by inducing Couette flow in a 25 nm width channel and measuring the shear stress as a function of the shear rate. By calculating the slope of the shear stress versus shear rate curve, a viscosity was obtained and the DPD damping coefficient was tuned until the viscosity matched that of bulk water. During equilibration, the top wall is free to move up and down in the *z*-direction, but is constrained to move as a rigid body and remain parallel to the bottom wall. Once the system is equilibrated, forces are applied individually to beads in the top wall and/or the water layer in order to simulate pure creep, pure drying and drying creep. In our simplified geometry, basic creep is simulated by applying forces to the beads only in the top wall, pure drying is simulated by applying forces to only the water beads, and drying creep is simulated by applying forces to both the water beads and those in the top solid wall. Flow simulations are run for 40 ns (2 000 000 time steps of 20 fs) in order to ensure that the system has reached a steady-state condition. The primary output from these simulations is the average, steady-state velocity of the top wall. This velocity, which we will refer to as the creep velocity of the nanochannel from this point onward, is directly related to the total creep displacement and strain that is of interest in the context of the Pickett effect.

## 3. Results and discussion

### (a) Behaviour of water under nanoconfinement

A good starting point for our discussions is the behaviour of water under nanoconfinement. Specifically, we first investigate how the dynamics, specifically the apparent viscosity, of the confined water change as a function of both the creep force *F*_{c} and the drying force *F*_{d}. As shown in figure 1, on a macroscopic specimen, a uniaxial compressive load is applied to cause creep. This macroscopic compressive creep force is translated to shear stress between various molecular components (e.g. C-S-H colloidal particles in concrete or cellulose microfibrils in wood) at interfaces within the material. Recent models have been proposed for hydrated matrix–interface composites that have further quantified the translation of these forces across length scales and support shear displacement of these nanopores due to the applied macroscopic stresses [46–49]. Therefore, the creep force *F*_{c} can be understood in our simulations as the force that causes shear displacement across the pore walls as a result of the uniaxially applied compressive loads. This creep force, or alternatively creep stress *τ*_{c} (here denoted by a shear stress symbol to highlight shear deformation of the nanopores), can be calculated from the forces applied to individual beads in the top layer, *f*_{ic}, by the following equation:
*A*_{s} is the cross-sectional area of the solid wall in the pulling direction (*xy*-plane). The drying forces applied to the water beads in our system are the result of a pressure differential that exists across the nanopore, denoted as *P*. This pressure differential is caused by differences in the relative humidity of the nanopore and the ambient environment. Differences in the relative humidity create a chemical potential and vapour pressure gradient that induces water flow (i.e. drying) along the nanochannel [50]. The pressure differential, Δ*P*, across the nanopore can be calculated from the individual forces applied to the water beads, *f*_{id}, by equation (3.2), where *A*_{w} is the cross-sectional area of the water within the nanochannel. In equation (3.2), we also show a direct relation between this driving pressure and the difference in vapour pressure within the nanopore, *P*_{v,np}, at a relative humidity of *ϕ*_{np} and the vapour pressure of the ambient environment, *P*_{v,env}, at a relative humidity of *ϕ*_{env}. Note that *P*_{sat}|_{T} is the saturation pressure at a given temperature
*η* as given by the following equation, where *F*_{c} is the applied creep force and *v*_{c,mean} is the creep velocity of the top plate:
*v*_{c,mean} since we do not know in which direction the water will flow within the pore relative to the creep force *a priori*. During drying, water will flow out of the pore, but would be equally likely to flow ‘forwards’ or ‘backwards’ relative to the creep force. Therefore, we must calculate the creep velocity both in the case where water flow is in the same direction as creep (*v*_{c+}) and in the case where it is in the opposite direction (*v*_{c−}). Rather than look at these cases individually, we simply can estimate a mean velocity that is given by the following equation and is used in all of our calculations by noting that the two cases are equally likely and there is no net directional preference:

From our simulations, we can then measure the effective viscosity of water as a function of both the applied creep stress and the applied water pressure differential. The results of our simulations are presented in figure 3, with the viscosity calculated from equation (3.3) plotted as a function of the applied creep stress in figure 3*a* and as a function of the applied water pressure in figure 3*b*. In figure 3*a*, the different symbols correspond to a range of the quantity *r*=*f*_{id}/*f*_{ic} which covers a wide range of drying conditions from relatively little drying (low *r* values) to larger amounts of drying (high *r* values). In figure 3*b*, the different line colours and symbols correspond to varying magnitudes of the creep stress applied to the top wall.

When examining these figures, we notice that the viscosity of water is not constant as a function of either the drying force or creep force, indicating that water confined in the nanochannel is exhibiting non-Newtonian behaviour. Under bulk conditions, and since water is a Newtonian fluid at the macroscale, we would expect the viscosity to remain constant and be independent of the shear rate and, indeed, CGMD simulations carried out in large enough channels without any nanoconfinement reproduce the linearity of the shear response. This linearity of the shear response has been verified in a previous study using this model [39] and has also been verified for a larger 25 nm channel in this work. However, from our simulations we see that the effective viscosity decreases with both increasing drying force and increasing creep force. Specifically, we conclude that water exhibits shear thinning, non-Newtonian behaviour with increasing shear rates.

These findings are consistent with previous experimental and computational studies that have shown similar shear thinning behaviour of water [51–54] and other fluids [55] under nanoconfinement. Additionally, this shear thinning behaviour has been observed for a wide range of values of the DPD damping coefficient as discussed in the electronic supplementary material. This illustrates two important facts. First, the shear thinning effect is not an artefact of the DPD thermostat and, second, this shear thinning behaviour is not limited to only water but can be extended to any fluid under nanoconfinement since changing the DPD thermostat damping parameter is equivalent to modelling fluids with a different bulk viscosity.

This is distinctly different from water behaviour in the bulk state and is believed to be a result of the glass-like nature of water under nanoconfinement [56]. We hypothesize that this behaviour is related to the Pickett effect since the drying pressure on the fluid introduces this shear thinning behaviour and consequently accelerates the creep of the material. Further, although we are using relatively high shear rates (10^{−2} to 10^{2} s^{−1}) in MD simulations, a recent study has shown that a similar shear thinning behaviour of collagen observed in MD can be extended to the low shear regime (10^{−6} to 10^{2} s^{−1}) using a number of different extrapolation methods [57]. This provides further evidence that our results for the flow of water at unrealistically large flow rates in a system submitted to unrealistically large shear rates can be extended to nanoporous solids submitted to realistic creep and drying rates.

As mentioned earlier, this glass-like nature of water under nanoconfinement has been observed for numerous systems. Typically, this glass-like behaviour is attributed to the molecular ordering of fluid near solid surfaces [58] due to either non-bonded van der Waals interactions or electrostatic interactions caused by the local electrical environment [59]. In our system, we do not explicitly include any electrostatic effects; however, we still observe this glass-like behaviour that arises from reduced mobility induced by strong surface–water interactions. In order to quantify this reduced mobility and increased molecular ordering, we calculate two quantities. First, we calculate the local mean-squared displacement (MSD) of the water beads as defined by the following equation for a single particle:
*n* is the number of time steps, ** r**(

*t*) is the position of the particle at time

*t*, and

*r*_{0}is some reference position of the particle which is taken to be the average position of the particle for the previous

*n*−1 time steps. We calculate the local MSD by binning the water molecules based on their position along the width of the nanochannel (i.e.

*y*-dimension in figure 2

*b*) and averaging over all particles in a given bin. For the local MSD calculation, we use a spatial resolution (i.e. bin size) of 1 Å. Because the water beads are relatively mobile and able to move between bins over the course of the whole simulation (40 ns), we focus on the results of the initial 2 ns when the beads have not yet moved out of their initial spatial bins. Using these relatively short time scales (picoseconds or nanoseconds) to assess the mobility of binned particles has also been employed in Debye–Waller factor calculations for polymer thin films in previous studies [60,61].

Secondly, we calculate the radial distribution function (RDF), *g*(*r*), that describes how the water beads in our system are radially packed around one another. The RDF is calculated using the appropriate plugin in VMD [62], and is given by the following equation:
*p*(*r*) is the average number of atom pairs found within a certain spherical shell extending from *r* to *r*+*dr* over a specified simulation time, *V* is the selection volume and *N*_{pairs} is the total of number of atom pairs in the selection. To determine how the molecular ordering changes throughout the nanochannel, the RDF is calculated for three regions of water along the width of the nanochannel: water beads within 0.75 nm of the lower water–solid interface (*y*=0.00–0.75 nm), water beads within 0.75 nm of the upper water–solid interface (*y*=3.25–4.00 nm), and the remaining water beads in the middle of the channel (*y*=0.75–3.25 nm).

The local MSD as a function of location within the nanochannel is shown in figure 4*a*, and the local MSD as a function of simulation time for each of the different bins is shown in figure 4*b*. The RDFs for each of the three different regions within the nanochannel are shown in figure 4*c*. The spatial profile of the local MSD is averaged from time *t*=0 to time *t*=2 ns and is overlaid over an illustration of the nanochannel. This plot demonstrates that the mobility of the water beads, as measured by the magnitude of the local MSD, is greatly decreased near the walls of the nanochannel. These nanoconfined layers (illustrated as red beads in figure 4*a*) near the walls exhibit vastly decreased mobility compared with the water beads in the middle of the nanochannel (blue beads in figure 4*b*) due to increased molecular ordering.

The increased molecular ordering is confirmed by the RDFs in figure 4*c* which show larger peaks for beads close to the solid–liquid interface and indicate a more densely packed fluid. Further, when examining the MSD versus time plot in figure 4*b*, we see that the MSD is lower for the layers of water near the solid surfaces (red curves). From these plots, we conclude that the confined layers extend approximately 0.50–0.75 nm from both the top and the bottom surfaces. These dimensions correspond to two or three atomic layers of water that are considered to be confined at the liquid–solid interface. Together, these plots help to demonstrate that our relatively simple model is capable of capturing the glass-like behaviour of water due to nanoconfinement. Specifically, our model, using only Morse non-bonded interactions, captures the decreased mobility near solid surfaces due to higher molecular ordering of the fluid.

### (b) Analytical model to describe creep at the nanoscale

After observing the change in water properties as a function of the applied creep force and drying force, a model was developed to try and describe these changes. The central issue we wish to address is: how does the creep velocity change as a function of the applied drying force and the applied creep force? One way to explain creep deformation is to invoke concepts such as energy landscapes and kinetics to describe the breaking and reformation of chemical bonds, in which deformation can be seen to proceed by hopping over energy barriers at the nanoscale. Each hopping event represents the breaking and reformation of weak interactions that resist the relative displacement of the walls. These reorganization events may occur at the wall–fluid interface (slip), and also within the fluid phase.

Traditionally, these concepts are employed to describe the failure of materials or interfaces at the nanoscale, as in the works of Bell [63] and Zhurkov [64]. The common feature of the kinetic models is that they employ an Arrhenius relationship that describes the lifetime of nanoscale bonds (or equivalently the probability of a bond breaking) using a single energy barrier and distance to the transition state, regardless of the specific chemistry of the bonds involved. This simplifying assumption works, at least phenomenologically, reasonably well for many types of chemical bonds, including those involved in fracture of glasses, polymers and metals, in cell adhesion [63] and protein unfolding [65,66], and in many other nanoscale phenomena governed by the underlying energy barriers [67]. Where these models fall short and are often incorrect, however, is for processes that involve overcoming multiple energy barriers in succession (i.e. multiple bond-breaking events in the context of Zhurkov’s model). In this case, if the difference between the successive potential troughs is small, both the forward and backward rates must be considered.

In the context of creep of nanoporous solids, we now use this idea to describe how the two solid walls move past one another and overcome nanoscale energy barriers to sliding motion. Overcoming the underlying energy barrier to sliding, *E*_{b}, as is the case in these creep simulations, is analogous to the rupture of atomic bonds as they both require some force or energy input into the system to encourage relative deformation or displacement. For our specific system, we can further relate this probability to the total creep displacement or creep velocity (the quantity measured in simulation) as a function of the applied creep force and drying force. Since these systems are under steady-state conditions, the creep velocity is more or less constant and the creep displacement could be calculated by multiplying this value by the simulation time. The probability of overcoming some underlying energy barrier is given by the following equation, where *E*_{b} is the underlying energy barrier to shear displacement, *k*_{b} is Boltzmann’s constant (1.38×10^{−23} J K^{−1}), *T* is the temperature of the system, *f* is the total force applied to the particle and *x*_{b} is the distance between energy barriers that is a measure of the atomic roughness of the underlying energy landscape,

From a molecular perspective, our system contains a slit pore filled with water molecules between two solid walls interacting with Morse interatomic potentials (figure 2). From a modelling perspective, however, we treat the beads in the top wall as a single object that interacts with the bottom wall with some periodic interatomic potential having an energy barrier *E*_{b} and a period of 2*x*_{b}. Further, although creep forces ( *f*_{ic}) and drying forces ( *f*_{id}) are applied to individual beads in our simulation, we can resolve these forces to net forces acting on the top wall as illustrated in figure 5*a*. The net creep force, *F*_{c}, is simply a sum of the individual creep forces applied directly to each of the beads in the top wall. The net drying force, *F*_{d}, is applied directly to the water beads, but also imparts some force on the top wall through viscous shear transfer. Therefore, the force imparted on the top wall by the water through shear transfer is denoted as *cF*_{d}, where *c* denotes the ratio of the shear force transfer and is a fitting parameter in our current study. Although this is currently a fitting parameter, *c* can be thought of as a measure of the drag coefficient or slip length between the fluid and the top wall. This is a physical quantity that could be measured experimentally.

Although we can directly define the probability of overcoming an energy barrier, the novelty of this model is to relate this probability to the net displacement or velocity of the top wall as a function of the applied creep and drying (i.e. motion of the water within the nanochannel). To calculate the net displacement, we must consider the probabilities of four cases as illustrated in figure 5*b*: (1) the top layer moves forward with positive water flow (*p*_{c+d+}), (2) the top layer moves forward with negative water flow (*p*_{c+d−}), (3) the top layer moves backward with positive water flow (*p*_{c−d+}) and (4) the top layer moves backward with negative water flow (*p*_{c−d−}). Even though the creep force is always in the positive *x*-direction and will assist the top wall in moving forward, the top wall still has some probability (although it is admittedly small) to move backwards due to thermal fluctuations. In this formulation, we improve upon Zhurkov’s classical model by considering both the forward (cases 1 and 2) and backward (cases 3 and 4) motion of the solid walls to overcoming the underlying energy barriers. Further, the cases of both positive and negative water flow must be considered since there is no *a priori* knowledge of the drying directionality. Therefore, the probabilities of overcoming an energy barrier *E*_{b} for these four cases are given by the following equations:

The total displacement, Δ*x*_{c}, or creep velocity, *ν*_{c}, of the top layer can be computed using equation (3.12), which is derived from Bell’s model [63] and also includes the frequency of molecular vibrations *ω*. Although we include the frequency term in our model, we can assume this to be a constant since the frequency of molecular vibrations will not vary greatly between systems. Additionally, in this work all creep displacements are reported in a normalized fashion so the numerical value of *ω* does not matter:

These results are presented in figure 6 and show the predicted displacement for the cases of basic (pure) creep (black, dotted lines) and pure drying shrinkage (green, dashed lines), the summation of these displacement (red, dot-dash lines) and the predicted displacement for the case of drying creep (blue, solid lines) for underlying energy barriers of 4*k*_{b}*T* and 8*k*_{b}*T*. These values are representative of the underlying energy barriers calculated for nanochannels with a width of 4 nm and 3 nm, respectively, as discussed in §3d. In figure 6, we define a ratio of *r*=*f*_{id}/*f*_{ic} and show results for the case of *r*=0.5. We also note that the case of basic creep corresponds to *r* = 0 and the case of pure drying corresponds to *c* is assumed to have a value of 1 for simplicity. Additional plots exploring the effects of changing the values of *r* and *c* are provided in the electronic supplementary material. Finally, we note that both the creep force and the relative creep displacement are presented as dimensionless quantities to evaluate the model and potential importance of underlying energy barriers more easily.

The results presented in figure 6 demonstrate consistent and interesting trends that help to further explain the Pickett effect. First, figure 6 indicates that our model clearly captures the main finding of the Pickett effect—the relative displacement (or equivalently strain) due to drying creep is greater than the sum of the displacement due to basic creep and pure drying. For large underlying energy barriers (figure 6*b*), this relationship holds for the entire range of the applied creep and drying forces. For lower energy barriers (figure 6*a*), we actually see a small region for low applied creep and drying forces where the summation is marginally greater (at most 0.01) than the displacement due to drying creep. This indicates that there may be some range of loading conditions for which the Pickett effect is observed. However, for larger applied creep forces with higher creep displacement and probabilities, we see that the Pickett effect is still observed.

Together these results demonstrate the importance of energy barriers and the nanoscale dimensions of pores within materials that exhibit the Pickett effect. The nanoscale size of these pores leads to higher underlying energy barriers that exhibit excess strain due to drying creep. From our simulations, we see that the systems with a drying force would be expected to exhibit accelerated creep due to a disproportionate tilt of the underlying energy landscape that results from the shear thinning behaviour of the fluid. As the fluid shear thins and effectively tilts the energy landscape, the walls are able to move past one another more easily, because of reduced energy barriers to motion, thereby accelerating the creep of the material beyond an additive effect.

Understanding the importance of energy barriers in this context also lends us theoretical understanding of how the system were to behave if the creep force was perpendicular to the nanochannel, while the drying force would still act along the nanochannel. In that case, the creep force would serve to compress or expand the nanochannel, which would cause an effective increase or decrease, respectively, in the energy barrier to relative plate motion [68]. Therefore, although our model does not explicitly include the cases where creep forces act perpendicular to the nanochannel, our model can adequately capture this scenario by understanding that this will only change the energy barrier to shear motion. The effect of changing the energy barrier, shown in figure 6, qualitatively explains how the results would be different in the presence of a perpendicular component of stress.

The more interesting result of our model is that this increase in the drying creep displacement compared to the creep and drying displacement summation is clearly nonlinear as a function of both creep force and drying force. It is important to note that even though we consider drying forces in both directions and assume them to be equally likely, their net effect does not cancel out because of the exponential terms in our model. This net effect is the reason we observe the distinct nonlinearity. Additionally, the results shown in figure 6 suggest that, for larger energy barriers, these nonlinearity effects are even more pronounced. In this context, these results suggest that characterizing the activation energy of key interfacial sliding mechanisms that govern creep deformation may be an important factor in describing the degree of the Pickett effect. Specifically, greater confinement and also more strongly hydrophilic interfaces should lead to rougher landscapes that accentuate the Pickett effect due to the drying force which induces shear thinning behaviour of the fluid within the nanopores.

### (c) Calculation of underlying energy barriers

The model developed in the previous section clearly demonstrates the importance of nanoscale energy barriers in the context of the Pickett effect. These energy barriers between the two walls dictate the relative sliding motion of the nanochannel and are ultimately related to the energy barrier governing the relative motion of the confined water molecules to move past one another under shear deformation. These barriers are further related to the molecular mobility of the water beads, whose temperature-dependent viscosity follows a simple Arrhenius relationship. The relationship, presented in the following equation relates the measured viscosity *η* to the system temperature *T* and the underlying energy barrier *E*_{b}:

We can algebraically manipulate the above equation to the form shown in equation (3.20), which clearly shows that the natural logarithm of viscosity is linearly related to the inverse of temperature,

At low temperatures, measuring the viscosity, and thus the energy barriers from MD simulations, becomes problematic due to poor sampling. However, by measuring the viscosity of nanoconfined water at different temperatures in simulations, we can estimate the energy barrier by taking a linear fit of the natural log of viscosity versus inverse temperature curve. These data are plotted for a 4 nm width channel in figure 7*a* for a number of different applied creep stresses. The plot shows a clear linear relationship between the natural log of viscosity and inverse temperature as predicted by equation (3.20). By taking a linear fit of these data, we can estimate the underlying energy barrier to shear displacement of the walls to be approximately 2.67 kcal mol^{−1}.

By itself, the estimation of this energy barrier is just a number that describes the ease with which two solid walls can move past another. Greater energy barriers get manifested in a greater resistance by the pore water to creep deformation and to sliding. However, more interesting is how the value of this energy barrier changes when we change the size of our nanochannel, or equivalently, the nanoconfinement effect.

To elucidate this issue, here we carried out additional temperature-dependent viscosity simulations for channels having widths of 3, 6, 8 and 10 nm. The calculated energy barriers are plotted as a function of the channel width in figure 7*b*. Here, we see that the underlying energy barrier is a function of the channel width, with the energy barrier increasing with decreasing channel/pore size in a clearly nonlinear manner. This indicates that these nanochannels exhibit nanoconfinement effects; namely, as the water becomes more confined due to decreasing channel size, it becomes more difficult for the channels to slide past one another, owing to increased energy barriers. For example, this nanoconfinement effect is shown in figure 7*b* where we see that the fluid in the 4 nm channel exhibits greater viscosity than the fluid in the larger 10 nm channel, as measured by a doubling of the energy barrier to flow. When examined in the context of our analytical model, these results also suggest that nanoporous solids with smaller nanopores under the same drying conditions would exhibit the Pickett effect to a varying extent as measured by the degree of nonlinearity in the relationship between drying creep strain and the summation of load-induced creep strain and drying shrinkage. These differences are attributed to the relationship between nanopore size and underlying energy barriers between solid walls that are increased due to water under nanoconfinement.

### (d) Comparison of analytical model to coarse-grained molecular dynamics simulations

To verify the results of the analytical model that describes the Pickett effect in nanoporous solids, the results of our CGMD simulations were compared directly to the analytical model derived in the previous section by CGMD simulations conducted under an applied drying creep condition while the total relative displacement velocity of the top solid wall of atoms was recorded during simulation. To compare our model to the measured creep velocity, we must develop an expression for the applied creep velocity as a function of the applied creep force and applied drying force. To determine this function, equations (3.8)–(3.11) are substituted into equation (3.12) (with the accompanying derivation included in the electronic supplementary material). The final result is presented in equation (3.21), where *A* is a fitting constant that is a function of the system temperature, underlying energy barrier and energy landscape roughness given by equation (3.22),
*A*, *x*_{b} and *c*) to predict the creep velocity as a function of the applied creep force. After obtaining the fitting parameters *A*, *c* and *x*_{b}, and using a least-squares regression, we estimate the underlying energy barrier to be *E*_{b}=2.04 kcal mol^{−1}, which is in relatively good agreement with our prediction from the temperature-dependent viscosity simulations (2.67 kcal mol^{−1}). A plot comparing the simulation results and model prediction is presented in figure 8*a*. It demonstrates good agreement between the simulations and our model, with a coefficient of determination of *r*^{2}=0.98 for the case of *f*_{id}/*f*_{ic}=0.75. A more detailed explanation of our calculation of the coefficient of the determination is included in the electronic supplementary material. Although not shown, we have also found good agreement between the simulation and the model for values of *f*_{id}/*f*_{ic} ranging from 0.01 to 0.75. The agreement between simulations and our model suggests that the drying creep behaviour of a nanopore is clearly a nonlinear function of the applied creep force—something that has been commonly neglected when considering drying creep and the Pickett effect.

Although there may be other models that fit the data, ours provides an exceptionally good fit while also having the advantage that our fitting parameters are all physically informed and could be directly extracted from actual experiments or additional simulations. Furthermore, although we only use statistical fitting to obtain values of these parameters in this work, it is promising that their calibrated values are realistic. The parameter *x*_{b} is related to the surface roughness of the solid wall, *A* is related primarily to the underlying energy barrier against relative shear displacement, as well as to the surface roughness of the channel and *c* is related to the shear transfer capability of the fluid and the associated slip length governed by the specific solid–liquid interaction. It should be noted that the fitting parameters for the energy barrier *E*_{b} and energy barrier spacing 2*x*_{b} (2.04 kcal mol^{−1} and 7.88 Å) are in good agreement with the energy barrier measured from the temperature-dependent viscosity of the water beads (2.67 kcal mol^{−1}) and the FCC lattice parameter used in system generation that defines the solid wall’s surface roughness (8.33 Å). Although we have not directly measured the parameter *c* ( fit value of 1.32), this parameter could be obtained from experiments in the future that measure the slip length or drag force/coefficient of a specific solid–liquid interface. The physical basis for these fitting parameters, coupled with the high correlation coefficient (*r*^{2}=0.98) of the analytical model, and the fact that the calibrated values of those parameters are realistic, verifies that our model is capable of explaining the nonlinear relationship between the applied creep force and the magnitude of drying creep.

Traditionally, scientists and engineers working on materials that exhibit a Pickett effect have assumed that there exists an approximately linear relationship between the drying creep magnitude and the applied creep force [70]. Although we have demonstrated clear nonlinearities associated with drying creep, it is of interest to assess the validity of this linearity assumption (i.e. the drying creep strain and velocity being a linear function of the applied creep force). Beyond these conventional assumptions, the functional form of the model given by equation (3.21) itself suggests a range for which this linear relationship between applied creep force and the drying creep strain would be observed. This linear behaviour is also anticipated because of the presence of the hyperbolic sine term and the fact that *x*≪1. In equation (3.21), the argument of the hyperbolic sine function is *F*_{c}*x*_{b}/*k*_{b}*T*. Therefore, we expect that for small values of *F*_{c}, and a constant drying force *F*_{d}, there should be an approximately linear relationship between the drying creep strain (or relative displacement) and the applied creep force.

To mathematically assess this assumption, we linearize our analytical model by taking a Taylor series expansion around *F*_{c}=0 of the analytical function given by equation (3.21). The first derivative of equation (3.21) is given by equation (3.23), in which we define *q*=*x*_{b}/*k*_{b}*T* as a collection of constants, for simplicity. The Taylor series expansion and linearization of our model is then shown in equation (3.24),

Using the values of *A* and *x*_{b} calculated from our previous least-squares regression, we plot this linear fit against the simulation data in figure 8*b*. This linear fit is calculated to have a coefficient of determination of *r*^{2}=0.94 over the entire range of creep forces applied to the top solid wall. Despite the *r*^{2}-value being lower than that for the nonlinear model in equation (3.21), this value is still quite high and indicates that the linear fit still agrees well with the simulation data. However, upon visual inspection, as well as a detailed look at the residuals between the model and simulation data, we see that it neglects the nonlinearity created by shear thinning of nanoconfined water for high values of the creep stress. However, for low values of the creep stress applied to the pore walls (approx. 20–50 MPa), this linearization fits quite well with the simulation data and indicates that a linear relationship is appropriate for a certain range of loading conditions.

Even though the linear model does not fully account for nanoscale behaviours, we can conclude that this commonly used linear approximation is adequate for most practical applications. We believe that the range of the creep stresses applied to the pore walls for which the linearization is applicable (less than or equal to approx. 20 MPa) is on the same order of magnitude of internal shear stresses in real nanoporous solids, and that it is well below the failure strength of pore wall materials (e.g. approx. 0.8–2.0 GPa for both C-S-H [71] and nanocrystalline cellulose [72]). Additionally, the linearization is found to be applicable for drying pressures ranging from approximately 10 to 100 MPa, which are also representative of real nanoporous solid materials [73]. Therefore, our study not only elucidates the clear role of underlying nanoscale energy barriers in drying creep, for the first time, but also is in agreement with the conventionally employed linearity in the small creep stress regime.

## 4. Conclusion

The present CGMD simulations and analytical modelling of water under nanoconfinement lead to the following conclusions.

CGMD simulations corroborate recent studies [51–54] that suggest shear thinning of nanoconfined water, and further help to explain that this behaviour is a result of the applied water pressure across the nanopore. Although interesting by itself, these results are even more impactful in the context of explaining a major part of the Pickett effect for nanoporous solids from a nanoscale perspective.

Shear thinning of the nanoconfined water lowers the effective energy barriers to sliding at the nanoscale, which consequently facilitates the increased displacements/strains observed during drying creep (i.e. the Pickett effect).

This excess strain due to drying creep is clearly a function of the pore size, suggesting a more substantial Pickett effect for smaller pores due to increased shear thinning and increased energy barriers to shear. In fact, our model shows that a large energy barrier to shear is required for a material to exhibit the Pickett effect and the nanoscale size of pores is the origin for these high-energy barriers (greater than approx. 3

*k*_{b}*T*). This behaviour becomes negligible for pores larger than 10 nm, where the energy barrier has been significantly lowered.Owing to limitations of the CG model, we were able to study only pores having a width larger than 3 nm. While there are pores with these dimensions in nanoporous solids, there is in fact a broad distribution of pore sizes. Recent studies of concrete have shown that the mean nanopore width can be as small as 0.75 nm [32]. In these sub-nanometre pores (which would typically hold three water monolayers between the solid walls), we anticipate that there would be an even more extreme degree of nanoconfinement with larger energy barriers to shear, stronger effects of shear thinning, and enhanced glassification of water near the liquid–solid interfaces, with the elimination of the bulk-like water. To confirm these hypotheses and verify our findings for larger pores, fully atomistic MD simulations with chemistry-specific parameters could be conducted.

Beyond the observations from simulations, the present analytical model reveals a quantitative relationship between the creep (relative velocity or displacement), the underlying energy barriers and the applied creep and drying forces. Notably, the model also suggests clear nonlinearities between the amount of drying creep and the applied creep force—something that had not been previously considered.

The present studies illuminate the nonlinearities in the behaviour of drying creep. For the first time, they explain the Pickett effect through measurable nanoscale quantities and thermodynamic principles. These nonlinearities are important to consider when designing nanoporous solid structures for operation under drying creep conditions.

Although clear nonlinearities exist, the linearization of the model around experimentally relevant creep stresses is shown to agree well with the simulations. This suggests that assuming linearity may be adequate for most practical applications.

Using a chemistry-independent CGMD and analytical model made it possible to maintain generality. The results presented here can be extended to any nanoporous solid and have direct implications for the understanding of creep behaviour of not only concrete, but also wood and many synthetic polymers.

Combination of the nanoscale approach with the previous micromechanical models of drying creep has led to a more comprehensive understanding of the Pickett effect. It showed that the movement of nanoconfined water over nanoscale energy barriers is the underlying mechanism of drying creep.

## Authors' contributions

R.S. carried out CGMD simulation work including designing simulations and analysing simulation data, developed the analytical model and drafted the manuscript. S.K. assisted in designing the simulations, conceived the idea for the analytical model and helped draft the manuscript. M.V. assisted in designing the simulations, designed methods for analysing the CGMD simulation data and helped draft the manuscript. Z.B. conceived of the study, designed the study and helped draft the manuscript. All authors gave final approval of the manuscript for publication.

## Competing interests

We declare we have no competing interests.

## Funding

S.K. acknowledges funding from the Army Research Office (award no. W911NF-13-1-0241). R.S. was supported by the DoD through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program. Z.P.B. was partially supported by Army Research Office (award no. W911NF-15-1-0240).

## Acknowledgements

The authors acknowledge support through a supercomputing grant from Northwestern University High Performance Computing Center as well as the Department of Defense (DoD) HPC System.

- Received June 16, 2016.
- Accepted June 30, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.